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Journal of Functional Analysis 260 (2011) 1–29 www.elsevier.com/locate/jfa The range of a class of classifiable separab...

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Journal of Functional Analysis 260 (2011) 1–29 www.elsevier.com/locate/jfa

The range of a class of classifiable separable simple amenable C ∗ -algebras Huaxin Lin a,b,∗ , Zhuang Niu b,c a Department of Mathematics, East China Normal University, Shanghai, China b Department of Mathematics, University of Oregon, Eugene, Oregon 97403, USA c Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada, A1C5S7

Received 9 October 2008; accepted 31 August 2010 Available online 8 October 2010 Communicated by S. Vaes

Abstract We study the range of a classifiable class A of unital separable simple amenable C ∗ -algebras which satisfy the Universal Coefficient Theorem. The class A contains all unital simple AH-algebras. We show that all unital simple inductive limits of dimension drop circle C ∗ -algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable C ∗ -algebras. We also show that there are many other C ∗ -algebras in the class. We prove that, for any partially ordered simple weakly unperforated rationally Riesz group G0 with order unit u, any countable abelian group G1 , any metrizable Choquet simplex S, and any surjective affine continuous map r : S → Su (G0 ) (where Su (G0 ) is the state space of G0 ) which preserves extremal points, there exists one and only one (up to isomorphism) unital separable simple amenable C ∗ -algebra A in the classifiable class A such that

K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A), λA = G0 , (G0 )+ , u , G1 , S, r .

© 2010 Elsevier Inc. All rights reserved. Keywords: Classification of amenable C ∗ -algebras; Range of invariant

* Corresponding author at: Department of Mathematics, University of Oregon, Eugene, Oregon 97403, USA.

E-mail address: [email protected] (H. Lin). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.019

2

H. Lin, Z. Niu / Journal of Functional Analysis 260 (2011) 1–29

1. Introduction Recent years saw some rapid developments in the theory of classification of amenable C ∗ algebras, otherwise known as the Elliott program of classification of amenable C ∗ -algebras. One of the highlights is the Kirchberg–Phillips’s classification of separable purely infinite simple amenable C ∗ -algebras which satisfy the Universal Coefficient Theorem (see [22] and [13]). There are also exciting results for simple C ∗ -algebras of stable rank one. For example, the classification of unital simple AH-algebras with no dimension growth by Elliott, Gong and Li [8]. Limitations of the classification have been also discovered (see [25] and [27], for example). In particular, it is now known that the general class of unital simple AH-algebras cannot be classified by the traditional Elliott invariant. One crucial condition that must be assumed for any general classification (using the Elliott invariant) of separable simple amenable C ∗ -algebras is Z-stability. On the other hand, classification theorems were established for unital separable simple amenable C ∗ -algebras which are not assumed to be AH-algebras, or other inductive limit structures (see [15,18,21]). Winter’s recent result provided a new approach to some more general classification theorems ([31] and [17]). Let A be the class of unital separable simple amenable C ∗ -algebras A which satisfy the UCT and for which A ⊗ Mp has tracial rank no more than one for some supernatural number p of infinite type. A more recent work in [19] shows that C ∗ algebras in A can be classified by the Elliott invariant up to Z-stable isomorphism. All unital simple AH-algebras are in A. One consequence of this is now we know that classifiable class of unital simple AH-algebras is exactly the class of Z-stable ones. But the class A contains more unital simple C ∗ -algebras. Any unital separable simple ASH-algebra A whose state space S(K0 (A)) of its K0 (A) is the same as that tracial state space are in A. It also contains the Jiang– Su algebra Z and many other projectionless simple C ∗ -algebras. We show that the class contains all unital simple so-called dimension drop circle algebras as well as many other C ∗ -algebras whose K0 -groups are not Riesz. It is the purpose of this paper to discuss the range of invariants of C ∗ -algebras in A. 2. Preliminaries Definition 2.1 (Dimension drop interval algebras [12]). A dimension drop interval algebra is a C ∗ -algebra of the form: I(m0 , m, m1 ) = f ∈ C [0, 1], Mm : f (0) ∈ Mm0 ⊗ 1m/m0 and f (1) ∈ 1m/m1 ⊗ Mm1 , where m0 , m1 and m are positive integers with m divisible by m0 and m1 . If m0 and m1 are relatively prime, and m = m0 m1 , then I(m0 , m, m1 ) is called a prime dimension drop algebra. Definition 2.2 (The Jiang–Su algebra [12]). Denote by Z the Jiang–Su algebra, the unital simple inductive limit of prime dimension drop algebras with a unique tracial state, (K0 (Z), K0 (Z)+ , [1Z ]) = (Z, N, 1) and K1 (Z) = 0. Definition 2.3 (Dimension drop circle algebras [20]). Let n be a natural number. Let x1 , . . . , xN be points in the circle T, and let d1 , . . . , dN be natural numbers dividing n. Then a dimension drop circle algebra is a C ∗ -algebra of the form D(n, d1 , . . . , dN ) = f ∈ C(T, MN ): f (xi ) ∈ Mdi ⊗ 1n/di , i = 1, 2, . . . , N .


3

Definition 2.4 (ATD-algebras). By ATD-algebras we mean C ∗ -algebras which are inductive limits of dimension drop circle algebras. Definition 2.5. Denote by I the class of those C ∗ -algebras with the form ni=1 Mri (C(Xi )), where each Xi is a finite CW complex with (covering) dimension no more than one. A unital simple C ∗ -algebra A is said to have tracial rank one if for any finite subset F ⊂ A, > 0, any nonzero positive element a ∈ A, there is a C ∗ -subalgebra C ∈ I such that if denote by p the unit of C, then for any x ∈ F , one has (1) xp − px , (2) there is b ∈ C such that b − pxp , and (3) 1 − p is Murray–von Neumann equivalent to a projection in aAa. Denote by I the class of all unital C ∗ -algebras with the form Xi is a compact metric space with dimension no more than one. Note that, in the above definition, one may replace I by I .

n

i=1 Mri (C(Xi )),

where each

Definition 2.6 (A classifiable class of unital separable simple amenable C ∗ -algebras). Denote by N the class of all unital separable amenable C ∗ -algebras which satisfy the Universal Coefficient Theorem. For a supernatural number p, denote by Mp the UHF algebra associated with p (see [2]). Let A denote the class of all unital separable simple amenable C ∗ -algebras A in N for which TR(A ⊗ Mp ) 1 for all supernatural numbers p of infinite type. Remark 2.7. By Theorem 2.11 below, in order to verify whether a C ∗ -algebra A is in the class A, it is enough to verify TR(A ⊗ Mp ) 1 for one supernatural number p of infinite type. Definition 2.8. Let G be a partially ordered group with an order unit u ∈ G. Denote by Su (G) the state space of G, i.e., Su (G) is the set of all positive homomorphisms h : G → R such that h(u) = 1. The set Su (G) equipped with the weak∗ -topology forms a compact convex set. Denote by Aff(Su (G)) the space of all continuous real affine functions on Su (G). We use ρ for the homomorphism ρ : G → Aff(Su (G)) defined by ρ(g)(s) = s(g)

for all s ∈ Su (G) and for all g ∈ G.

Put Inf(G) = ker ρ. Definition 2.9. Let A be a unital stably finite separable simple amenable C ∗ -algebra. Denote by T (A) the tracial state space of A. We also use τ for τ ⊗ Tr on A ⊗ Mk for any integer k 1, where Tr is the standard trace on Mk . By Ell(A) we mean the following: K0 (A), K0 (A)+ , [1A ], K1 (A), T (A), rA , where rA : T (A) → S[1A ] (K0 (A)) is a surjective continuous affine map such that rA (τ )([p]) = τ (p) for all projections p ∈ A ⊗ Mk , k = 1, 2, . . . .

4


Suppose that B is another stably finite unital separable simple C ∗ -algebra. A map Λ : Ell(A) → Ell(B) is said to be a homomorphism if Λ gives an order homomorphism λ0 : K0 (A) → K0 (B) such that λ0 ([1A ]) = [1B ], a homomorphism λ1 : K1 (A) → K1 (B), and a continuous affine map λρ : T (B) → T (A) such that λρ (τ )(p) = rB (τ ) λ0 [p] for all projection p in A ⊗ Mk , k = 1, 2, . . . , and for all τ ∈ T (B). We say that such Λ is an isomorphism, if λ0 and λ1 are isomorphisms and λρ is an affine homeomorphism. In this case, there is an affine homeomorphism λρ : T (A) → T (B) such that λ−1 ρ = λρ . Theorem 2.10. (See [19, Corollary 11.9].) Let A, B ∈ A. Then A⊗Z ∼ =B ⊗Z if Ell(A ⊗ Z) = Ell(B ⊗ Z). In the next section (Theorem 3.6), we will show the following: Theorem 2.11. Let A be a unital separable amenable simple C ∗ -algebra. Then A ∈ A if only if TR(A ⊗ Mp ) 1 for one supernatural number p of infinite type Definition 2.12. Recall that a C ∗ -algebra A is said to be Z-stable if A ⊗ Z ∼ = A. Denote by AZ the class of Z-stable C ∗ -algebras in A. Denote by A0z the subclass of those C ∗ -algebras A ∈ Az for which TR(A ⊗ Mp ) = 0 for some supernatural number p of infinite type. Corollary 2.13. Let A and B be two unital separable amenable simple C ∗ -algebras in AZ . Then A∼ = B if and only if Ell(A) ∼ = Ell(B). Definition 2.14. Let A be a unital simple C ∗ -algebra and let a, b ∈ A+ . We define a b, if there exists x ∈ A such that x ∗ x = a and xx ∗ ∈ bAb. We write [a] = [b] if there exists x ∈ A such that x ∗ x = a and xx ∗ = b. We write [a] [b] if a b. If e ∈ A is a projection and [e] [a], then there is a projection p ∈ aAa such that e is equivalent to p. If there are n mutually orthogonal elements b1 , b2 , . . . , bn ∈ bAb such that a bi , i = 1, 2, . . . , n, then we write n[a] [b].


5

Let m be another positive integer. We write n[a] m[b], if, in Mm (A), ⎡⎛

⎞⎤ 0 0 ⎟⎥ ⎥, .. ⎟ . ⎠⎦

b ⎢⎜ 0 ⎜ n[a] ⎢ ⎣⎝ ...

0 b .. .

0··· 0··· .. .

0

0

0··· b

where b repeats m times. 3. The classifiable C ∗ -algebras A The purpose of this section is to provide a proof of Theorem 2.11. Lemma 3.1. Let A be a unital simple C ∗ -algebra with an increasing sequence of unital simple C ∗ -algebras {An } such that 1A = 1An and ∞ A n=1 n is dense in A. Suppose that a ∈ A+ \ {0}. Then, there exists b ∈ (An )+ \ {0} for some large n so that b a. Proof. Without loss of generality, we may assume that a = 1. Let 1 > δ > 0. There exists > 0 such that for any c ∈ A+ \ {0} with a − c < one has (see Proposition 2.2 of [24]) that fδ (c) a, where fδ ∈ C0 ((0, ∞)) for which fδ (t) = 1 for all t δ, fδ (t) = 0 for all t ∈ (0, δ/2), and linear between δ/2 and δ. We may assume, for a sufficiently small , fδ (c) = 0. Since ∞ n=1 An is dense in A, it is possible to find such c ∈ A+ \ {0} that c ∈ An for some large n. Put b = fδ (c). Then b ∈ An and b a. 2 Lemma 3.2. Let A be a unital simple C ∗ -algebra and e, a ∈ A+ \ {0}. Then, for any integer n > 0, there exists m(n) > 0 such that n[e] m(n)[a]. Proof. It suffices to show that [1A ] m[a] for some integer m. Since A is simple, there are y1 , y2 , . . . , ym ∈ A for some integer m such that m i=1

yi∗ ayi = 1

6


(see, for example, Lemma 3.3.6 of [14]). Let bi = a 1/2 yi yi∗ a 1/2 , i = 1, 2, . . . , m. Define i−1

bi = diag 0, 0, . . . , 0, bi , 0, . . . , 0 , in Mm (A). Define

⎛ y ∗ a 1/2 ⎜ y=⎜ ⎝

0 .. .

y2∗ a 1/2 0 .. .

··· ··· .. .

0

0

···

1

i = 1, 2, . . . , m ∗ a 1/2 ⎞ ym 0 ⎟ .. ⎟ ⎠. .

0

Then ∗

yy = 1A

∗

and y y =

m

bi .

i=1

Thus

[1A ]

m

bi .

i=1

Clearly

m

bi m[a].

2

i=1

Lemma 3.3. Let A be a unital infinite dimensional simple C ∗ -algebra and let a ∈ A+ \ {0}. Suppose that n 1 is an integer. Then there are nonzero mutually orthogonal elements a1 , a2 , . . . , an ∈ aAa + such that a1 a2 · · · an . Proof. In aAa, there exists a positive element 0 b 1 such that its spectrum has infinitely many points. From this one obtains n nonzero mutually orthogonal positive elements b1 , b2 , . . . , bn . Since A is simple, there is xn−1 ∈ A such that bn−1 xn−1 bn = 0. Set yn−1 = ∗ y ∗ bn−1 xn−1 bn . One then has that yn−1 n−1 ∈ bn Abn and yn−1 yn−1 ∈ bn−1 Abn−1 . In particu∗ ∗ ∗ . One has that bn and yn−1 yn−1 ⊥ bn . Choose an = bn and an−1 = yn−1 yn−1 lar, yn−1 yn−1 an ⊥ an−1 and an−1 an . Then consider b1 , b2 , . . . , bn−2 , an−1 . The lemma follows by applying the argument above n − 1 more times. 2 Lemma 3.4. Let A be a unital separable simple C ∗ -algebra and let {An } be an increasing sequence of unital simple C ∗ -algebras such that 1A = 1An and ∞ n=1 An is dense in A. Then TR(A) 1 if and only if the following holds: For any > 0, any integer n 1, any a ∈ (An )+ \{0} and a finite subset F ⊂ An , there exists an integer N 1 satisfying the following: There is, for each m N, a C ∗ -subalgebra C ⊂ Am with C ∈ I and with 1C = p such that (1) px − xp < for all x ∈ F ;


7

(2) dist(pxp, C) < for all x ∈ F ; and (3) 1 − p is equivalent to a projection q ∈ aAm a. Proof. First we note that “if ” part follows the definition easily. To prove the “only if” part, we assume that TR(A) 1 and there is an increasing sequence of C ∗ -subalgebras An ⊂ A which are unital and the closure of the union ∞ n=1 An is dense in A. In particular, we may assume that 1An = 1A , n = 1, 2, . . . . Let > 0, n 1, a ∈ (An )+ \ {0} and a finite subset F ⊂ An be given. Since TR(A) 1, there is a C ∗ -subalgebra C1 ∈ I with 1C1 = q such that (1) xq − qx < /2 for all x ∈ F ; (2) dist(qxq, C1 ) < /2 for all x ∈ F ; and (3) [1 − q] [a]. From (2) above, there is a finite subset F1 ⊂ C1 such that (2 ) dist(qxq, F1 ) < /2 for all x ∈ F . We may assume that q ∈ F1 . Put d = sup{x; x ∈ F }. Since C1 is generated by stable relations (see, for example [4]), there is δ > 0 and a finite subset G ⊂ C1 satisfying the following: If B ⊂ A is a unital C ∗ -subalgebra and dist(y, B) < δ for all y ∈ G, there is a C ∗ -subalgebra C ∈ I such that C ⊂ B and dist z, C < /2(d + 1) for all z ∈ F1 ∪ G. By the assumption, there is an integer N n such that dist(y, AN ) < δ/2 for all y ∈ G. It follows that there is C ∈ I with 1C = p such that C ∈ AN and dist(z, C) < /2(d + 1) for all z ∈ F1 ∪ G. In particular, p − q < /2(d + 1). One then checks that (i) qx − xq < for all x ∈ F ; (ii) dist(pxp, C) < for all x ∈ F ; and (iii) [1 − p] = [1 − q] [a]. The lemma follows.

2

The following is well known.

8


Lemma 3.5. For any two positive integers p and q with p > q, there exist integers 1 m < q, 0 r < p and s 1 such that p = mq s + r. Proof. Take the largest integer s 1 such that p q s . There are integers 0 r < p and 1 m such that p = mq s + r. Then 1 m < q.

2

Theorem 2.11 follows immediately from the following: Theorem 3.6. Let A be a unital separable simple C ∗ -algebra. Then TR(A ⊗ Mp ) 1 for all supernatural numbers p of infinite type if and only if there exists one supernatural number q of infinite type such that TR(A ⊗ Mq ) 1. Proof. Suppose that there is a supernatural number q ofinfinite type such that TR(A ⊗ Mq ) 1. ∼ Let An ∼ = 1A⊗Mp and ∞ = A ⊗ Mr(n) such that 1An n=1 An is dense in A ⊗ Mp . Let Bn = A ⊗ Mk(n) such that 1Bn = 1A⊗Mq and ∞ B is dense in A ⊗ M . p n=1 n Write q = q1∞ q2∞ · · · qk∞ . . . , where q1 , q2 , . . . , qk , . . . are prime numbers. We may assume that 1 < q1 < q2 < · · · qk < · · · . Fix > 0, n 1, a0 ∈ (An )+ \ {0} and F ⊂ An . Since An is simple, there are mutually orthogonal elements a1 , a2 , . . . , a3(q1 +1) ∈ (An )+ \ {0} such that a1 , a2 , . . . , a3(q1 +1) ∈ a0 An a0 and a1 a2 · · · a3(q1 +1) . By 3.1, there exists an integer m(1) 1 such that [1An ] m(1)[aq1 +1 ]. Note that TR(Mr(n) (A) ⊗ Mq ) 1. Therefore, by 3.4, there exists N 1 satisfying the following: there is, for each m N , a C ∗ -subalgebra Cm ⊂ Mr(n) (A) ⊗ Mk(m) with Cm ∈ I and 1C = e(m) such that (i) e(m)jm (x) − jm (x)e(m) < /2 for all x ∈ F , where jm : Mr(n) (A) → Mr(n) (A) ⊗ Mk(m) is defined by jm (a) = a ⊗ 1Mk(m) for a ∈ Mr(n) (A); (ii) dist(e(m)jm (x)e(m), C) < /2 for all x ∈ F ; and (iii) 1Mr(n) (A)⊗Mr(m) − e(m) is equivalent to a projection in jm (a1 )(An ⊗ Mk(m) )jm (a1 ). s

We may assume that k(N ) = q1s1 q2s2 · · · qkk and k(N ) m(q1 ). To simplify the notation, without loss of generality, we may assume that r(n + 1)/r(n) > k(N). We write r(n + 1) = N1 k(N ) + r0 , r(n)


9

where N1 1 and 0 r0 < k(N ) are integers. Without loss of generality, we may further assume that N1 > m(1)k(N ). We write N1 = n1 q1s + r1 , where q1 > n1 1, s 1, and 0 r1 < q1 are integers. Thus r(n + 1) = n1 q1s K(N) + r1 K(N ) + r0 . r(n) Without loss of generality, to simplify notation, we may assume that k(N 1) = q1s K(N ). + n1 r1 n1 r1 Put C = j =1 CN +1 ⊕ i=1 CN . Put e = j =1 e(N + 1) ⊕ i=1 e(N ). Define j : Mr(n) (A) → Mr(n) (A) ⊗ Mr(n+1)/r(n)−r0 by n1

r1

j (a) = diag jN +1 (a), jN +1 (a), . . . , jN +1 (a) ⊕ diag jN (a), jN (a), . . . , jN (a)

for all a ∈ Mr(n) (A). By what we have proved, we have that ej (x) − j (x)e < /2 for all x ∈ F , dist(ej (x)e, C) < /2 for all x ∈ F and 1Mr(n) (A)⊗Mr(n+1)/r(n) −r0 − e is equivalent to a projection in b1 (Mr(n) (A) ⊗ Mr(n+1)/r(n)−r0 )b1 , where b1 =

n1

j (ai ) +

i=1

p 1 +r1

j (ai ).

i=p1 +1

The last assertion follows from the fact that n1 1Mr(n) (A)⊗Mk(N+1) − e(N + 1) r1 1Mr(n) (A)⊗Mk(N) − e(N )

n1

and

ai

i=1 p 1 +r1

ai .

i=p1 +1

Define j : Mr(n) (A) → Mr(n+1) (A) = Mr(n) (A) ⊗ Mr(n+1)/r(n) by j (a) = a ⊗ 1Mr(n+1)/r(n)

for all a ∈ Mr(n) .

Thus, in Mr(n+1) (A), ej (a) − j (a)e < /2 for all x ∈ F , dist(ej (a)e, C) < /2 for all x ∈ F. Note that r0 [1Mr(n) ] r0 m(1)[a1 ] j (a2q1 +1 ) . Thus

3(q +1) 1 [1An+1 − e] = r0 [1Mr(n) ] + [1Mr(n) (A)⊗Mr(n+1)/r(n) −r0 − e] ai . i=1

It follows that 1An+1 − e is equivalent to a projection in j (a0 )An+1 j (a0 ).

10


This proves that TR(Mr(n) (A) ⊗ Mp ) 1. It follows from Proposition 3.2 of [18] that TR(A ⊗ Mp ) 1. 2 Corollary 3.7. Let A be a unital separable simple C ∗ -algebra. Suppose that there exists a supernatural number q of infinite type for which TR(A ⊗ Mq ) = 0. Then, for all supernatural number p of infinite type, TR(A ⊗ Mp ) = 0. Proof. The proof uses the exactly the same argument used in the proof of 3.6 (and 3.4).

2

TD-algebras 4. Unital simple AT Theorem 4.1. Every unital simple ATD-algebra A for which K0 (A)/ ker ρA Z has tracial rank one or zero. Proof. Since K1 (A) is a countable abelian group, we can write K1 (A) = lim −→ (Gn , ιn ), n→∞

ln with Gn ∼ Z/pn,l Z for some non-negative integers ln and pn,l , where pn,l = 1 (note that = l=1 ln pn,l . if pn,l = 0, one has that Z/pn,l Z ∼ = Z). Denote by pn = l=1 Since K0 (A) is a simple Riesz group (see p. 1304 of [20]) and K0 (A)/ ker ρA ∼ Z, and the = pairing between T (A) and K0 (A) preserves extreme points, by [29], there is a simple inductive limit of interval algebras B such that K0 (A), K0 (A)+ , [1A ] , T (A), λA ∼ = K0 (B), K0 (B)+ , [1B ] , T (B), λB .

(e4.1)

Write B = lim −→

n→∞

kn

Bn,i , ϕn ,

i=1

where Bn,i = Mmn,i (C([0, 1])). For the map [ϕn ]0 induced by ϕn , we write [ϕn ]0 :

kn

Z∼ = K0

kn i=1

Bn,i → K0

kn+1 i=1

Bn+1,i ∼ =

Z

kn+1

in a matrix (rn,i,j ) with rn,i,j ∈ Z+ , where 1 i kn+1 and 1 j kn and where [ϕn ]0 is the map induced by ϕn . Since A is simple, without loss of generality, we may assume that rn,1,j > (n + 1)pn by passing to a subsequence. For each n and each 1 j kn+1 , consider the restriction of the map ϕn to the direct summand Bn,1 and Bn+1,j , that is, consider the map ϕn (1, j ) : Bn,1 → Bn+1,j . It follows from [5] that we may assume that there exist continuous functions s1 , s2 , . . . , srn,1,j such that ϕn (1, j )(f )(t) = W ∗ (t) diag f ◦ s1 (t), . . . , f ◦ srn,1,j (t) W (t)


11

for some W (t) ∈ Mmn,i rn,1,j (C([0, 1])). Factor through the map ϕn (1, j ) by Bn,1

ψ1

ln

⊕ Bn,1 Mpn,l mn,1 C [0, 1]

ψ2

Bn+1,j ,

l=1

where ψ1 (f )(t) = diag {f ◦ s1 , . . . , f ◦ spn,1 }, . . . , {f ◦ spn −pn,ln +1 , . . . , f ◦ spn }, f . and ψ2 (f ⊕ g) = W ∗ diag{f, g ◦ spn +1 , . . . , g ◦ srn,1,j }W is the diagonal embedding. Since rn,1,j npn , one has that the restriction of any tracial state n of B to the unit of ll=1 Mpn,l mn,1 (C([0, 1])) has value less than 1/n. Therefore, by replacing Bn,1 by Mpn,l mn,1 (C([0, 1])) ⊕ ( rn,1,j −pn Bn,1 ), one may assume that there is an inductive limit decomposition k n B = lim Bn,i , ϕn , −→ n→∞

i=1

where Bn,i = Mmn,i (C([0, 1])), such that kn > ln , and mn,j = dn,j pn,j for some natural number dn,j for any 1 j ln . Moreover, one has that τ (e) < 1/n for any τ ∈ T (A), where e is the unit of ljn=1 Bn,j . In other words, ρ(e) < 1/n

(e4.2)

for any ρ ∈ Su (K0 (A)). Fix this inductive limit decomposition. Now, let us replace certain interval algebras at each level n by certain dimension drop interval algebras so that the new inductive limit gives the desired K1 -group, and keep the K0 -group and the pairing unchanged. At level n, for each 1 l ln , if pn,l = 0, denote by Dn,l the dimension drop C ∗ -algebra I[mn,l , mn,l pn,l , mn,l ]; if pn,l = 0, denote by Dn,l the circle algebra Mmn,l (C(T)). Then, one has K0 (Dn,l ), K + (Dn,l ), 1Dn,l = Z, Z+ , mn,l and K1 (Dn,l ) = Z/pn,l Z. Replace each Bn,l by Dn,l , and denote by Dn = Dn,1 ⊕ · · · ⊕ Dn,ln ⊕ Bn,ln +1 ⊕ · · · ⊕ Bn,kn . It is clear that K0 (Dn ) ∼ = K0 (Bn )

and K1 (Dn ) ∼ =

ln l=1

Let us construct maps χn : Dn → Dn+1 as the following.

Z/pn,l Z ∼ = Gn .

12


For the direct summand Dn,i and any direct summand Dn+1,j , if pn,i and pn+1,j are nonzero, by Corollary 3.9 of [12], there is a map χn (i, j ) : Dn+1,i → Dn,j such that χn (i, j ) 0 = ϕn (i, j ) 0 = rn,i,j and χn (i, j ) 1 = ιn (i, j ). If pn,i = 0 and pn+1,j = 0, then define the map χn (i, j ) : Mmn,i (C) ⊗ C(T) ∼ = Dn,i → pDn+1,j p ∼ = Mrn,i,j mn,i (C) ⊗ I[1, pn+1,j , 1], where p is a projection stands for rn,i,j mn,i ∈ K0 (Dn+1,j ), by e ⊗ z → diag{e ⊗ u, e ⊗ z1 , . . . , e ⊗ zrn,i,j −1 }, where z is the standard unitary z → z, zi are certain points in the unit circle, and u is a unitary in pDn+1,j p which represents ιn (i, j )(1). Then, it is clear that χn (i, j ) 0 = ϕn (i, j ) 0 = rn,i,j and χn (i, j ) 1 = ιn (i, j ). A similar argument for pn,i = 0 and pn+1,j = 0 also provides a homomorphism χn (i, j ) which induces the right K-theory map. Moreover, the argument above also applies to the maps between Dn,i and Bn+1,j , and between Bn,i and Dn+1,j , such that there is a map χn (i, j ) with χn (i, j ) 0 = ϕn (i, j ) 0 = rn,i,j and χn (i, j ) 1 = ιn (i, j ). For direct summand Bn,i and Bn+1,j , define χn (i, j ) = ϕn (i, j ). In this way, we have a homomorphism χn : Dn → Dn+1 satisfying [χn ]0 = [ϕn ]

and [χn ]1 = ιn,n+1 .

Let us consider the inductive limit D = lim −→ (Dn , χn ). n→∞


13

It is clear that K0 (D) = K0 (B) and K1 (D) = lim −→(Gn , ιn ) = K0 (A). With a suitable choice of χn (i, j ) between Dn,i and Dn+1,j , Dn,i and Bn+1,j , and Bn,i and Dn+1,j , we may assume that D is a simple C ∗ -algebra. Let us show TR(D) 1. From the construction, it is clear that D has the following property: pn For any finite subset F ⊂ D and any > 0, there exists n such that if denote by In = i=l B and pn = 1In , n +1 n,i then, for any x ∈ F (1) [pn , x] , and (2) there is a ∈ In such that pn xpn − a . By Theorem 3.2 of [18], the C ∗ -algebra has the property (SP), that is, any nonzero hereditary sub-C ∗ -algebra contains a nonzero projection. Thus, in order to show TR(D) 1, one only has to show that for any given projection q ∈ D, one can choose n sufficiently large such that 1 − pn q. Note that the C ∗ -algebra D is an inductive limit of dimension drop interval algebras together with circle algebras, which satisfy the strict comparison on projections, i.e., for any two projections e and f , if τ (e) < τ (f ) for any tracial state τ , then e f . Then D also has the strict comparison on projections. (See, for example, Theorem 4.12 of [9].) Therefore, in order to show 1 − pn q, one only has to show that for any given > 0, there is a sufficiently large n such that τ (1 − pn ) for any t ∈ T (D). However, this condition can be fulfilled by Eq. (e4.2), and thus, the C ∗ -algebra D is tracial rank one. Let us show that B and D have the same tracial simplex, and have the same pairing with the K0 -group. Consider the non-unital C ∗ -algebra C = lim −→

n→∞

kn

Bn,i , ψn ,

i=ln +1

kn kn+1 where the map ψn is the restriction of ϕn to i=ln +1 Bn,i and i=ln+1 +1 Bn+1,i . Then, by # Lemma 10.8 (and its proof) of [18], there are isomorphisms r and r# , and s # and s# such that

T (B)

rB

Su (K0 (B)) r#

r#

T (C)

rC

T (D)

rD

s#

s#

Su (K0 (C)),

T (C)

rC

commutes. Therefore, there are isomorphisms t # and t# such that

T (B)

rB

T (D)

Su (K0 (B)) t#

t# rD

Su (K0 (D))

Su (K0 (D))

Su (K0 (C))

14


commutes. Therefore, the C ∗ -algebra B and D have the same simplex of traces and pairing map. Hence, K0 (D), K0 (D)+ , [1D ] , T (D), λA ∼ = K0 (B), K0 (B)+ , [1B ] , T (B), λB , and therefore by Eq. (e4.1), K0 (D), K0 (D)+ , [1D ] , K1 (D), T (D), λA ∼ = K0 (A), K0 (A)+ , [1A ] , K1 (A)T (A), λA . Since D is also an inductive limit of dimension drop interval algebras together with circle algebras, by (the proof of) Theorem 9.9 of [20], one has that D is a simple inductive limit of dimension drop circle algebras, and hence by Theorem 11.7 of [20], one has that A ∼ = D. Since TR(D) 1, we have that TR(A) 1, as desired. 2 Theorem 4.2. Every unital simple ATD-algebra is in AZ . Proof. If K0 (A)/Inf(K0 (A)) Z, then, 4.1 shows that TR(A) 1. By Theorem 10.4 of [18], A is in fact a unital simple AH-algebra with no dimension growth. It follows from [7] that A is also approximately divisible. It follows from Theorem 2.3 of [28] that A is Z-stable. So A ∈ AZ . Let A be a general unital simple ATD-algebra. Let p be a supernatural number of infinite Z. It follows from 4.1 that TR(A ⊗ Mp ) 1. Thus A ∈ A. type. Then K0 (A ⊗ Mp )/ ker ρA ∼ = It follows from Theorem 4.5 of [28] that A is Z-stable. Therefore A ∈ AZ . 2 4.3. From Theorem 4.2 we see that Theorem 2.13 also unifies the classification theorems of [18] and that of [20]. In the next two sections, we will show that A contains many more C ∗ -algebras. 5. Rationally Riesz groups Theorem 5.1. For any countable abelian groups G00 and G0 , any group extension: π

0 → G00 → G0 → Z → 0, with (G0 )+ = x ∈ G0 : π(x) > 0 or x = 0 , any order unit u ∈ G0 and any metrizable Choquet simple S, there exists a unital simple ASHalgebra A ∈ A such that K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A) = G0 , (G0 )+ , u , G1 , S . Proof. Let S0 be the point corresponding to the unique state on G0 . It follows from a theorem of Elliott [6] that there exists a unital simple ASH-algebra B such that K0 (B), K0 (B)+ , [1B ] , K1 (B), T (B) = G0 , (G0 )+ , u , G1 , S0 . Then B ⊗ Mp is a unital separable simple C ∗ -algebra which is approximately divisible and the projections of B ⊗ Mp separate the tracial state space (in this case it contains a single point).


15

Thus B ⊗ Mp has real rank zero and stable rank one by [23]. Thus, by Proposition 5.4 of [16], TR(B ⊗ Mp ) = 0. Let B0 be the unital simple ATD-algebra (see Theorem 4.5 of [12]) such that K0 (B0 ), K0 (B)+ , [1B0 ] , K1 (B0 ), T (B0 ), λB0 = (Z, N, 1), {0}, S, rS . It follows from 4.2 that B0 ∈ A. By Theorem 11.10(iv) of [19], B0 ⊗ B ∈ A. Define A = B0 ⊗ B. One then calculates that 2 K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A) = G0 , (G0 )+ , u , G1 , S . Theorem 5.2. For any countable weakly unperforated simple Riesz group G0 with order unit u, any countable abelian group G1 and any metrizable Choquet simplex S and any surjective homomorphism rS : S → Su (G0 ) which maps ∂e (S) onto ∂e (Su (G0 )). There exists a unital simple ASH-algebra A ∈ Az such that K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A), λA = G0 , (G0 )+ , u , G1 , S, rS . Proof. First, we assume that G0 /Inf(G0 ) Z. It follows from a theorem of Villadsen [30] that, in this case, there is a unital simple AH-algebra C with no dimension growth (and with tracial rank no more than one – see Theorem 2.5 of [18]) such that K0 (C), K0 (C)+ , [1C ] , K1 (C), T (C), λC = G0 , (G0 )+ , u , G1 , S, rS . The case that G0 /Inf(G0 ) ∼ = Z follows from 5.1.

2

Remark 5.3. Theorem 5.2 includes all unital simple ATD-algebras. Let A be a unital simple C ∗ -algebra in A with weakly unperforated Riesz group K0 (A). If K0 (A)/Inf(K0 (A)) ∼ = Z, then TR(A) 1. By the classification result in [18], A is in fact a unital simple AH-algebra. If K0 (A) = Z, then, by 4.2, A is a unital simple ATD-algebra. However, Theorem 5.2 contains unital simple C ∗ -algebra A for which K1 (A) is an arbitrary countable abelian group. These C ∗ algebras can not be isomorphic to unital simple ATD-algebras (see Theorem 1.4 of [20]). Definition 5.4. Let G be a partially ordered group. We say G has rationally Riesz property if the following holds: For two pairs of elements x1 , x2 , and y1 , y2 ∈ G with xi yj , i, j = 1, 2, there exists z ∈ G such that nz myi

and mxi nz,

i = 1, 2,

where m, n ∈ N \ {0}. Let G and H be two weakly unperforated simple ordered groups with order-unit u and v respectively. Consider the group G ⊗ H . Set the semigroup (G ⊗ H )+ = a ∈ G ⊗ H ; (s1 ⊗ s2 )(a) > 0, ∀s1 ∈ Su (G), ∀s2 ∈ Sv (H ) ∪ {0}. Since G+ ⊗ H+ ⊆ (G ⊗ H )+ and G ⊗ H = G+ ⊗ H+ − G+ ⊗ H+ , one has G ⊗ H = (G ⊗ H )+ − (G ⊗ H )+ .

16


For any a ∈ ((G ⊗ H )+ ) ∩ (−(G ⊗ H )+ ), if a = 0, then (s1 ⊗ s2 )(a) > 0 for any s1 ∈ Su (G) and s2 ∈ Sv (H ), and (s1 ⊗ s2 )(−a) > 0, for any s1 ∈ Su (G) and s2 ∈ Sv (H ), which is a contradiction. Therefore, (G ⊗ H )+ ∩ −(G ⊗ H )+ = {0}. Moreover, since (s1 ⊗ s2 )(u ⊗ v) = 1 for any s1 ∈ Su (G) and s2 ∈ Sv (H ), and Su (G) and Sv (H ) are compact, for any element a ∈ (G ⊗ H ), there is a natural number m such that m(u ⊗ v) − a ∈ (G ⊗ H )+ . Hence (G ⊗ H, (G ⊗ H )+ , u ⊗ v) is a scaled ordered group. Lemma 5.5. Let G and H be simple ordered groups with order units u and v respectively. If H has a unique state τ , then for any state s on the ordered group G ⊗ H , one has s(g ⊗ h) = s(g ⊗ v)τ (h) for any g ∈ G, h ∈ H . Proof. It is enough to show the statement for strictly positive g and h. For each g ∈ G+ \ {0}, since G is simple, one has that u mg for some natural number m. Hence s(g ⊗ v)

1 1 s(u ⊗ v) = , m m

and in particular, s(g ⊗ v) = 0. Consider the map sg : H → R defined by sg (h) =

s(g ⊗ h) . s(g ⊗ v)

Then sg is a state of H . Since τ is the unique state of H . One has that sg = τ , and hence s(g ⊗ h) = τ (h) s(g ⊗ v)

for any h ∈ H.

Therefore, s(g ⊗ h) = s(g ⊗ v)τ (h), as desired.

2

Lemma 5.6. Let G be a countable weakly unperforated simple partially ordered group with an order unit u. Then, for any dense subgroup D of R containing 1, the map λ : Su⊗1 (G ⊗ D) → Su (G) defined by λ(s)(x) = s j (x) for all x ∈ G, where j : G → G ⊗ D defined by j (x) = x ⊗ 1 for all x, is an affine homeomorphism.


17

Proof. Since any dense subgroup of R with the induced order has unique state, the statement follows from Lemma 5.5 directly. 2 Proposition 5.7. Let G be a countable weakly unperforated simple partially ordered group with an order unit u. Then the following are equivalent: (1) (2) (3) (4)

G has the rationally Riesz property; G ⊗ D has the Riesz property for some dense subgroup D of R containing 1; G ⊗ D has the Riesz property for all dense subgroups D of R containing 1; For any two pairs of elements xi and yi ∈ G with xi yj , i, j = 1, 2, there is an element z ∈ G and a real number r > 0 such that s(xi ) rs(z) s(yj ),

for all s ∈ Su (G), i, j = 1, 2.

Proof. It is clear, by the assumption that G is weakly unperforated, that (1) ⇒ (2) for D = Q. It is also clear that (3) ⇒ (2). That (2) ⇒ (4) follows from 5.6. Suppose that (4) holds. Let xi yj be in G, i, j = 1, 2. If one of xi is the same as one of yj , say x1 = y1 , then (2) holds for z = x2 and m = n = 1. Thus, let us assume that xi < yj , i, j = 1, 2. Note that Su (G) is compact. It follows from (4) there is a rational number r ∈ Q such that s(xi ) < rs(z) < s(yj ),

for all s ∈ Su (G), i, j = 1, 2.

Write r = n/m for some m, n ∈ N. Then s(mxi ) < s(nz) < s(myj )

for all s ∈ Su (G), i, j = 1, 2.

It follows from Theorem 6.8.5 of [1] that mxi nz myj

i, j = 1, 2.

Thus (4) ⇒ (1). By applying 5.6, it is even easier to shows that (4) ⇒ (3).

2

Proposition 5.8. Let G be a countable weakly unperforated simple partially ordered group with an order unit u. Then G has the rationally Riesz property if and only if Su (G) is a metrizable Choquet simplex. Proof. Suppose that G has the rationally Riesz property. Then, by 5.7, G ⊗ Q is a weakly unperforated simple Riesz group and (G ⊗ Q)/Inf(G ⊗ Q) Z. Put F = (G ⊗ Q)/Inf(G ⊗ Q). Then, it follows that F is a simple dimension group. It then follows from the Effros–Handelman–Shen Theorem [3] that there exists a unital simple AF-algebra A with ¯ K0 (A), K0 (A), [1A ] = (F, F+ , u), where u¯ is the image of u in F . It follows that T (A) = Su (F ). By Theorem 3.1.18 of [26], T (A) is a metrizable Choquet simplex. It follows from 5.6 that Su (G) is a metrizable Choquet simplex.

18


For the converse, let G be a countable weakly unperforated simple partially ordered group with an order unit u so that Su (G) is a metrizable Choquet simplex. Let F = G ⊗ Q. By 5.7, it suffices to show that F has the Riesz property. By 5.6, Su⊗1 (F ) is a metrizable Choquet simplex. It follows from Theorem 11.4 of [11] that Aff(Su⊗1 (F )) has the Riesz property. Let ρ : F → Aff(Su⊗1 (F )) be the homomorphism defined by ρ(g)(s) = s(g)

for all s ∈ Su⊗1 (G)

and for all g ∈ F.

Define F1 = F + R(u ⊗ 1) and extend ρ from F1 into Aff(Su⊗1 (F )) in an obvious way. Then ρ(F1 ) contains the constant functions. It also separates the points. By Corollary 7.4 of [11], the linear space generated by ρ(F1 ) is dense in Aff(Su⊗1 (G)). Moreover, for any real number r < 1 and any positive element p ∈ F , the positive affine function rρ(p) can be approximated by elements in ρ(F ). It follows that ρ(F ) is dense in Aff(Su⊗1 (G)). It follows that ρ(F ) has the Riesz property. Moreover, since the order of F is given by ρ(F ) (see Theorem 6.8.5 of [1]), this implies that F has the Riesz property. 2 Example 5.9. Let H be any nontrivial group. Then ordered group Z ⊕ H with the order induced by Z is a simple ordered group, which satisfies the rationally Riesz property. However, it is not a Riesz group. LetΓ be a cardinality at most countable and bigger than 1. Then, the ordered group G = Γ Z with the positive cone {0} ∪ Γ Z+ is simple, since any positive element is an order unit. Consider a1 = (1, 0, 0, 0, . . .), a2 = (0, 1, 0, 0, . . .), a3 = (2, 2, 0, 0, . . .), and a4 = (2, 3, 0, 0, . . .). Then a1 , a2 a3 , a4 . However, one can not find an element b such that a1 , a2 b a3 , a4 . Hence, the group G is not Riesz. But this group has rationally Riesz property. Proposition 5.10. Let A ∈ A be a Z-stable C ∗ -algebra. Then (K0 (A), K0 (A)+ , [1A ]) is a countable weakly unperforated simple partially ordered group with order unit [1A ] which has the rationally Riesz property. Moreover S[1A ] (K0 (A)) is a metrizable Choquet simplex. Proof. It follows from [10] that (K0 (A), K0 (A)+ , [1A ]) is a countable weakly unperforated simple partially ordered group with order unit [1A ]. Since TR(A ⊗ Q) 1, (K0 (A ⊗ Q), K0 (A ⊗ Q)+ , [1A⊗Q ]) is a Riesz group. It follows that (K0 (A), K0 (A)+ , [1A ]) has the rationally Riesz property. By 5.6 and 5.8, S[1A ] (K0 (A)) is a metrizable Choquet simplex. 2 Lemma 5.11. Let G0 be a countable weakly unperforated simple partially ordered group with order unit u which also has the rationally Riesz property and G1 any countable abelian group. There exists a unital simple ASH-algebra A ∈ A0z ⊂ Az such that K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A) = G0 , (G0 )+ , u , G1 , Su (G0 ) . Proof. It follows from 5.8 that Su (G0 ) is a Choquet simplex. It follows from [6] that there exists a unital simple ASH-algebra A such that K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A) = G0 , (G0 )+ , u , G1 , Su (G0 ) .


19

We may assume that A ∼ = A ⊗ Z. Note that the set of projections of A ⊗ Mp separates the tracial state space in this case for any supernatural number p. It follows from the argument of 8.2 of [31] that TR(A ⊗ Mp ) = 0. In particular, A ∈ A. 2 Definition 5.12. Let T1 and T2 be two finite simplexes with vertices {e1 , . . . , em } and ˙ 2 the finite simplex spanned by vertices (ei , fj ), 1 i m, {f1 , . . . , fn }. Let us denote by T1 ×T 1 j n. Lemma 5.13. Let A be unital C ∗ -algebra and τ a tracial state of A. Then τ is extremal if and only if πτ (A) in the GNS-representation (πτ , Hτ ) is a II1 factor. Proof. Note that πτ (A) is always of type II1 . Assume that τ is extremal. If πτ (A) were not a factor, then, there is a nontrivial central projection p ∈ πτ (A) . We claim that τ (p) = 0. Suppose that an ∈ A+ such that {πτ (an )} is bounded and πτ (an ) converges to p in the weak operator topology in B(Hτ ). If τ (p) = 0, then τ (an ) → 0. It follows that τ (an2 ) → 0. For any a ∈ A \ {0}, 2 1/2 → 0. τ aan a ∗ = τ an a ∗ a τ an2 τ a ∗ a It follows that ! " lim πτ (an )ξa , ξa H = 0

n→∞

τ

for any a ∈ A, where ξa is the vector given by a in the GNS construction, which implies that πτ (an ) → 0 in the weak operator topology. Therefore p = 0. This proves the claim. By the claim neither τ (p) nor τ (1 − p) is zero. Thus 0 < τ (p) < 1. Define τ1 : a → 1 1 τ (p) τ (pap) and τ2 : a → τ (1−p) τ ((1 − p)a(1 − p)). Note that τ1 (p) = 1. Thus τ1 = τ . Then τ1 and τ2 are traces on πτ (A) , and τ (a) = τ (p)τ1 (a) + τ (1 − p)τ2 (a) for any a ∈ πτ (A) . Therefore, the trace τ on πτ (A) is not extremal. Since both τ1 and τ2 are also normal, we conclude that τ is also not extremal on A, which contradicts to the assumption. Conversely, assume that πτ (A) is a factor. If τ = λτ1 + (1 − λ)τ2 with λ ∈ (0, 1), then it is easy to check that τ1 and τ2 can be extended to normal states on πτ (A) , and hence to traces on πτ (A) . Therefore τ1 = τ2 , and τ is extremal. 2 Lemma 5.14. Let A and B be two unital C ∗ -algebras. Let τ be an extremal tracial state on a C ∗ -algebra tensor product A ⊗ B. Then, the restriction of τ to A or B is an extremal trace. Proof. Denote the restrictions of τ to A and B by τA and τB , respectively. Consider the GNSrepresentation (πτ , Hτ ) of A ⊗ B. Since τ is extremal, by Lemma 5.13, the von Neumann algebra πτ (A ⊗ B) is a II1 factor. Since (A ⊗ 1) commutes with (1 ⊗ B), πτ (A ⊗ 1) and πτ (1 ⊗ B) are also II1 factors. Set p the orthogonal projection to the closure of the subspace spanned by {(a ⊗ 1)(ξ ); ξ ∈ Hτ , a ∈ A}. Then the GNS-representation (πτA , HτA ) of A is unitarily equivalent to the cutdown of πτ to A and pHτ . Hence πτA (A) is a II1 factor in B(HτA ). By Lemma 5.13, τA is an extremal trace of A. The same argument works for B. 2

20


Lemma 5.15. Let A and B be two unital C ∗ -algebras and let τ be a tracial state of a C ∗ algebra tensor product A ⊗ B. Then, if the restriction of τ to B is an extremal trace, one has that τ (a ⊗ b) = τ (a ⊗ 1)τ (1 ⊗ b) for all a ∈ A and b ∈ B. Proof. We may assume that a is a positive element with norm one. If τ (a ⊗ 1) = 0, then τ (a ⊗ b) = 0 by Cauchy–Schwartz inequality, and equation holds. If τ (a ⊗ 1) = 1, then the equation also hold by considering the element (1 − a) ⊗ 1. Therefore, we may assume that τ (a ⊗ 1) = 0, 1. Fix a. Then, we have τ (1 ⊗ b) = τ (a ⊗ 1) Note that both b →

τ (a⊗b) τ (a⊗1)

τ ((1 − a) ⊗ b) τ (a ⊗ b) + 1 − τ (a ⊗ 1) τ (a ⊗ 1) 1 − τ (a ⊗ 1)

and b →

extremal trace, one has that

τ (a⊗b) τ (a⊗1)

τ ((1−a)⊗b) 1−τ (a⊗1)

for any b ∈ B.

are tracial states of B. Since b → τ (1 ⊗ b) is an

= τ (1 ⊗ b) for any b ∈ B. Therefore, the equation

τ (a ⊗ b) = τ (a ⊗ 1)τ (1 ⊗ b) holds for any a ∈ A and b ∈ B.

2

Corollary 5.16. Let A and B be two unital C ∗ -algebras and let τ be an extremal tracial state of a C ∗ -algebra tensor product A ⊗ B. Then τ is the product of its restrictions to A and B. Proof. It follows from Lemma 5.14 and Lemma 5.15.

2

Corollary 5.17. Let A and B be two C ∗ -algebras with simplexes of traces T(A) and T(B). If ˙ T(A) and T(B) have finitely many extreme points, then T (A ⊗ B) = T(A)×T(B). Proof. It follows from Corollary 5.16 directly.

2

Theorem 5.18. Let G0 be a countable weakly unperforated simple partially ordered group with an order unit u which has the rationally Riesz property, let G1 be a countable abelian group, and let T be any finite simplex. Assume that Su (G0 ) has only finitely many extreme points. Then there exists a unital simple ASH-algebra A ∈ A such that ˙ u (G0 ), r , K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A), λA = G0 , (G0 )+ , u , G1 , T ×S ˙ u (G0 ) → Su (G0 ) is defined by (τ, s)(x) = s(x) for all x ∈ G0 and for all extremal where r : T ×S tracial state τ ∈ T and extremal state s ∈ Su (G0 ). Proof. From 5.11, there exists a unital simple ASH-algebra B ∈ A such that K0 (B), K0 (B)+ , [1B ] , K1 (B), T (B) = G0 , (G0 )+ , u , G1 , Su (G0 ) . Then let B0 be a unital simple ATD-algebra with K0 (B0 ), K0 (B0 )+ , [1B0 ] , K1 (B0 ), T (B0 ) = (Z, N, 1), {0}, T .


21

Define A = B0 ⊗ B. Then A ∈ A. One checks that ˙ u (G0 ), r . K0 (A), K0 (A)+ , [1A ] , K1 (A), T (A), λA = G0 , (G0 )+ , u, G1 , T ×S

2

6. The range Suppose that A and B are two stably finite unital Z-stable C ∗ -algebras and suppose that there is a homomorphism Λ : Ell(A) → Ell(B). There is Λp : Ell(A ⊗ Mp ) → Ell(B ⊗ Mp ) and Λq : Ell(A ⊗ Mp ) → Ell(B ⊗ Mq ) induced by Λ so that the following diagram commutes Ell(A ⊗ Mp )

(idA ⊗1)∗

Λp

Ell(A)

(idA ⊗1)∗

Ell(A ⊗ Mq ) Λq

Λ

Ell(B ⊗ Mp )

(idB ⊗1)∗

Ell(B)

(idB ⊗1)∗

Ell(B ⊗ Mq )

Definition 6.1. (See [31, Definition 4.2].) Let A and B be unital C ∗ -algebras and let p and q be supernatural numbers. We say a C([0, 1])-homomorphism ϕ : A ⊗ Zp,q → B ⊗ Zp,q is unitarily suspended, if there are 0 t0 < t1 1, a continuous path (ut )t∈[t0 ,t1 ) of unitaries in B ⊗Mp ⊗Mq and ∗-homomorphisms σ p : A ⊗ Mp → B ⊗ M p and ρ q : A ⊗ Mq → B ⊗ M q such that ut0 = 1B⊗Mp ⊗Mq and ⎧σ p ⎪ ⎪ ⎪ ⎪ σ ⊗ idMq ⎪ ⎨ p (t) ϕ = ad(ut ) ◦ (σp ⊗ idMq ) ⎪ ⎪ ρ [1,3] ⊗ id[2] ⎪ ⎪ q Mp ⎪ ⎩ ρq

for t = 0, for t ∈ (0, t0 ], for t ∈ (t0 , t1 ), for t ∈ [t1 , 1), for t = 1,

where ρq[1,3] ⊗ id[2] Mp : A ⊗ Mp ⊗ Mq → B ⊗ Mp ⊗ Mq is the ∗-homomorphism induced by ρq and idMp in the obvious way. Lemma 6.2. Let A and B be two Z-stable C ∗ -algebras in A and let p and q be two supernatural numbers of infinite type which are relatively prime. Suppose that Λ : Ell(A) → Ell(B)

22


is a homomorphism. Then there is a unitarily suspended C([0, 1])-unital homomorphisms ϕ : A ⊗ Zp,q → B ⊗ Zp,q such that Ell(π0 ◦ ϕ) = Λp and Ell(π1 ◦ ϕ) = Λq so that the following diagram commutes: Ell(A ⊗ Mp )

(idA ⊗1)∗

(π0 ◦ϕ)∗

Ell(B ⊗ Mp )

Ell(A)

(idA ⊗1)∗

(π1 ◦ϕ)∗

Λ

(idB ⊗1)∗

Ell(B)

Ell(A ⊗ Mq )

(idB ⊗1)∗

Ell(B ⊗ Mq )

Proof. Since A, B ∈ A, TR(A ⊗ Mp ) 1, TR(A ⊗ Mq ) 1, TR(B ⊗ Mp ) 1 and TR(B ⊗ Mq ) 1, there is a unital homomorphism ϕp : A ⊗ Mp → B ⊗ Mp and ψq : A ⊗ Mq → B ⊗ Mq such that Ell(ϕp ) = Λp

and Ell(ψq ) = Λq .

Put ϕ = ϕp ⊗ idMq : A ⊗ Q → B ⊗ Q and ψ = ψq ⊗ idMp : A ⊗ Q → B ⊗ Q. Note that (ϕ)∗i = (ψ)∗i

(i = 0, 1) and ϕT = ψT

(they are induced by Λ). Note that ϕT and ψT are affine homeomorphisms. Since K∗i (B ⊗ Q) is divisible, we in fact have [ϕ] = [ψ] (in KK(A ⊗ Q, B ⊗ Q)). It follows from Lemma 11.4 of [19] that there is an automorphism β : B ⊗ Q → B ⊗ Q such that [β] = [idB⊗Q ]

KK(B ⊗ Q, B ⊗ Q)

such that ϕ and β ◦ ψ are asymptotically unitarily equivalent. Since K1 (B ⊗ Q) is divisible, H1 (K0 (A ⊗ Q), K1 (B ⊗ Q)) = K1 (B ⊗ Q). It follows that ϕ and β ◦ ψ are strongly asymptotically unitarily equivalent. Note also in this case βT = (idB⊗Q )T . Let ı : B ⊗ Mq → B ⊗ Q defined by ı(b) = b ⊗ 1 for b ∈ B. We consider the pair β ◦ ı ◦ ϕq and ı ◦ ϕq . By applying 11.5 of [19], there exists an automorphism α : ϕq (A ⊗ Mq ) → ϕq (A ⊗ Mq ) such that ı ◦ α ◦ ψq and β ◦ ı ◦ ψq are asymptotically unitarily equivalent (in M(B ⊗ Q)). So they are strongly asymptotically unitarily equivalent. Moreover, [α] = [idB⊗Mq ]

in KK(B ⊗ Mq , B ⊗ Mq ).

We will show that β ◦ ψ and α ◦ ϕq ⊗ idMp are strongly asymptotically unitarily equivalent. Define β1 = β ◦ ı ◦ ψq ⊗ idMp : B ⊗ Q ⊗ Mp → B ⊗ Q ⊗ Mp . Let j : Q → Q ⊗ Mp defined by j (b) = b ⊗ 1. There is an isomorphism s : Mp → Mp ⊗ Mp with (idMq ⊗ s)∗0 = j∗0 . In this case [idMq ⊗ s] = [j ]. Since K1 (Mp ) = 0. By 7.2 of [19], idMq ⊗ s is strongly asymptotically unitarily equivalent to j . It follows that α ◦ ψq ⊗ idMp and β ◦ ı ◦ ψq ⊗ idMp are strongly asymptotically unitarily equivalent. Consider the C ∗ -subalgebra C = β ◦ ψ(1 ⊗ Mp ) ⊗ Mp ⊂ B ⊗ Q ⊗ Mp . In C, β ◦ ϕ|1⊗Mp and j0 are strongly asymptotically unitarily equivalent, where j0 : Mp → C by


23

j0 (a) = 1 ⊗ a for all a ∈ Mp . There exists a continuous path of unitaries {v(t): t ∈ [0, ∞)} ⊂ C such that lim ad v(t) ◦ β ◦ ϕ(1 ⊗ a) = 1 ⊗ a

t→∞

for all a ∈ Mp .

(3)

It follows that β ◦ ψ and β1 are strongly asymptotically unitarily equivalent. Therefore β ◦ ψ and α ◦ ψq ⊗ idMp are strongly asymptotically unitarily equivalent. Finally, we conclude that α ◦ ψq ⊗ idp and ϕ are strongly asymptotically unitarily equivalent. Note that α ◦ ψq is an isomorphism which induces Λq . Let {u(t): t ∈ [0, 1)} be a continuous path of unitaries in B ⊗ Q with u(0) = 1B⊗Q such that lim ad u(t) ◦ ϕ(a) = α ◦ ψq ⊗ idMq (a)

t→∞

for all a ∈ A ⊗ Q.

One then obtains a unitary suspended C([0, 1])-unital homomorphism which lifts Λ along Zp,q (see [31]). 2 Theorem 6.3. (Cf. [31, Proposition 4.6].) Let A and B be two Z-stable C ∗ -algebras in A. Suppose that there exists a strictly positive unital homomorphism Λ : Ell(A) → Ell(B). Then there exists a unital homomorphism ϕ : A → B such that ϕ induces Λ. Proof. The proof is a simple modification of that of Proposition 4.6 of [31] by applying 6.2. First, it is clear that the proof of Lemma 4.3 of [31] holds if isomorphism is changed to homomorphism without any changes when both A and B are assumed to be simple. Moreover, the one-sided version of Proposition 4.4 also holds. In particular the part (ii) of that proposition holds. It follows that the homomorphism version of Proposition 4.5 of [31] holds since proof requires no changes except that we change the word “isomorphism” to “homomorphism” twice in the proof. To prove this theorem, we apply 6.2 to obtain a unitarily suspended C([0, 1])-homomorphism ϕ : A ⊗ B ⊗ Z which has the properties described in 6.2. The rest of the proof is just a copy of the proof of Proposition 4.6 with only four changes: (1) ϕ˜ : A ⊗ Z → B ⊗ Z ⊗ Z is a homomorphism (instead of an isomorphism); (2) in the diagram (65), λ, λ ⊗ id are homomorphisms (instead of isomorphisms); (3) ϕ˜+ is a homomorphism (instead of an isomorphism); (4) since λ and ϕ˜∗ agree as homomorphisms (instead of isomorphisms) and both are order homomorphisms preserving the order units, they also have to agree as such. 2 Definition 6.4. A C ∗ -algebra A is said to be locally approximated by subhomogeneous C ∗ algebras if for any finite subset F ⊆ A and any > 0, there is a C ∗ -subalgebra H ⊆ A isomorphic to a subhomogeneous algebra such that F ⊆ H . Remark 6.5. It is clear from the definition that any inductive limit of locally approximately subhomogeneous C ∗ -algebras is again a locally approximately subhomogeneous C ∗ -algebra. Lemma 6.6. Let (G0 , (G0 )+ , u) be a countable partially ordered weakly unperforated and rationally Riesz group, let G1 be a countable abelian group, let T be a metrizable Choquet simplex

24


and let λT : T → Su (G0 ) be a surjective affine continuous map sending extremal points to extremal points. Suppose that Su (G0 ) and T have finitely many extremal points. Then there exists one unital Z-stable C ∗ -algebra A ∈ A such that Ell(A) = G0 , (G0 )+ , [1A ] , G1 , T , λT . Moreover, the C ∗ -algebra A can be locally approximated by subhomogeneous C ∗ -algebras. Proof. Denote by e1 , e2 , . . . , en the extreme points of Su (G0 ), and denote by S1 , . . . , Sn the preimage of e1 , . . . , en under λ. Then, each Si is a face of T , and hence a simplex with finitely many extreme points. In each Si , choose an extreme point fi . ˙ → Su (G0 ) by Set an affine map α : Su (G0 )×T α (ei , gj ) = ei , where ei is an extreme point of Su (G0 ) and gj is an extreme point of T . Define an affine map ˙ → T by π((ei , gj )) = gj if gj ∈ Sk , and π(ei , gj ) = fi if gj is not in any of Sk . π : Su (G0 )×T Since there are only finitely many extreme points in both Su (G0 ) and T , π is a continuous affine surjective map. Then λT ◦ π = α. ˙ of π by ı(gj ) = (λT (gj ), gj ) for gj ∈ Sj , j = 1, 2, . . . , n. Choose a lifting ι : T → Su (G0 )×T ˙ → Su (G0 )×T ˙ by β = ι ◦ π . In particular, π ◦ ι = idT . Define an affine map β : Su (G0 )×T By Theorem 5.18, there is a Z-stable ASH-algebra A ∈ A with ˙ ,α . Ell A = G0 , G1 , Su (G0 )×T By Theorem 6.3, there is a unital homomorphism ϕ : A → A such that [ϕ]0 = id,

[ϕ]1 = id,

and (ϕ) = β.

(The compatibility between the map β and [ϕ]0 follows from the commutative diagram below.) Let An = A and let ϕn : An → An+1 be defined by ϕn = ϕ. n = 1, 2, . . . . Put A = limn→∞ (An , ϕn ). Since each An is simple so is A. By Theorem 11.10 of [19], A ∈ A. Since each An is an ASH-algebra, the C ∗ -algebra A can be locally approximated by subhomogeneous C ∗ -algebras. Since the diagram β

˙ Su (G0 )×T

α

α

T λT

Su (G0 )

˙ Su (G0 )×T π

ι

λT

Su (G0 )


25

commutes (the left triangle commutes because α ◦ ι = λT ◦ π ◦ ι = λT ◦ idT = λT ), one has that the inductive limit A = lim −→(A , ϕ) satisfies Ell(A) = (G0 , G1 , T , λT ), as desired.

2

Lemma 6.7. Let G be a countable rationally Riesz group and let T be a metrizable Choquet simplex. Let λ : T → Su (G) be a surjective affine map preserving extreme points. Then, there kn kn are decompositions G = lim −→(Gn , ψn ), Aff T = lim −→(R , ηn ), and maps λn : Gn → R such that each Su (Gn ) is a simplex with finitely many extreme points, and the following diagram commutes: ηn

Rkn

···

Aff T . λ∗

λn+1

λn

Gn

Rkn+1

ψn

Gn+1

···

G

Proof. Consider the ordered group H = G ⊗ Q. It is clear that the ordered group GF := G/Tor(G) (with positive cone the image of the positive cone of G) is a sub-ordered-group of H . Since G is a rationally Riesz group, the group H is a Riesz group. It follows from Effros– mi Handelman–Shen Theorem that there is a decomposition H = lim −→(Hi , ϕi ), where Hi = Z with the usual order for some natural number mi . We may assume that the images of (ϕi,∞ ) is increasing in H . Set −1 Gi = ϕi,∞ GF ∩ ϕi,∞ (Hi ) ⊆ Hi . Then the inductive limit decomposition of H induces an inductive limit decomposition GF = lim −→ Gi , ϕi |Gi . We assert that Su (Gi ) = Su (Hi ). To show the assertion, it is enough to show that any state on Hi is determined by its restriction to Gi , and it is enough to show that two states of Hi are same if their restrictions to Gi are same. Indeed, let τ1 and τ2 be two states of Hi with same restrictions to Gi . For any element h ∈ Hi , consider ϕi,∞ (h) ∈ H . We then can write ϕi,∞ (h) =

p g, q

for some g ∈ GF and relatively prime numbers p and q. In particular, pg = qϕi,∞ (h) ∈ ϕi,∞ (Hi ) ∩ GF , and hence qh ∈ Gi .

26


Therefore, one has τ1 (h) =

1 1 τ1 (qh) = τ2 (qh) = τ2 (h). q q

This proves the assertion. Since Su (Hi ) is a finite dimensional simplex, the convex set Su (Gi ) is also a simplex with finitely many extreme points. Hence we have the inductive decomposition GF = lim −→(Gi , ψi ), where ψi = ϕi |Gi . Consider the extension 0

ι

Tor(G)

G

π

GF

0,

and write Tor(G) = lim −→(Ti , ιi ), where Ti are finite abelian groups. Since the torsion free abelian (g) for any group G1 is finitely generated, there is a lifting γ1 : G1 → G with π ◦ γ1 (g) = ψi,∞ g ∈ G1 . Since an element in G is positive if and only if it is positive in the quotient GF , it is clear that γ1 is positive. Consider the ordered group T1 ⊕ G1 with the order determined by G1 , and set the map ψ1,∞ : T1 ⊕ G1 → G by (a, b) → ι(a) + γ1 (b). It is clear that ψ1,∞ is positive. (G ) ⊆ Using the same argument, one has a positive lifting γ2 : G2 → G. Since ψi,∞ 1 ψ2,∞ (G2 ), one has that γ2 (g) − γ1 (g) ∈ Tor(G) for each g ∈ G1 . By truncating the sequence (Ti ), one may assume that (γ2 − γ1 )(G1 ) ∈ T2 . Define the map ψ2,∞ : T2 ⊕ G2 → G by (a, b) → ι(a) + γ2 (b). Then it is clear that ψ1,∞ (T1 ⊕ G1 ) ⊆ ψ2,∞ (T2 ⊕ G2 ). Define the map ψ1,2 : T1 ⊕ G1 → T2 ⊕ G2 by (a, b) → ι(a) + γ1 (b) − γ2 (b), ψ1,2 (b) . A direct calculation shows that ψ1,∞ = ψ2,∞ ◦ ψ1,2 . Repeating this argument and setting Gi = Ti ⊕ Gi , one has the inductive limit decomposition G = lim −→(Gi , ψi,i+1 ). i

Noting that the order on Gi is determined by the order on Gi , one has that Su (Gi ) = Su (Gi ), and hence the convex set Su (Gi ) is a simplex with finitely many extreme points. Let {an } be a dense sequence in the positive cone Aff+ T . Consider the map λ∗ ◦ ψi,∞ : G → Aff T . Since Su (Gi ) = Su (Hi ) and Hi = Zmi , the image of positive elements of Gi is contained in a finite dimensional cone. Since images of Gi are increasing, we may choose {b1 , . . . , bi , . . .} ⊆ Aff T and natural numbers n1 < · · · < ni < · · · such that {b1 , . . . , bni } is a set of generators for the image of Gi in Aff T . For each i, set ki = i + ni . We identify the affine space Rki ∼ = (Ra1 ⊕ · · · ⊕ Rai ) ⊕ (Rb1 ⊕ · · · ⊕ Rbni ) as the subspace of Aff T spanned by a1 , . . . , ai , b1 , . . . , bni . Define the map λi : Gi → Rki by g → (0 ⊕ · · · ⊕ 0) ⊕ λ∗ ◦ ψi,∞ (g) ,


27

the map ηi : Rki → Rki+1 by (f, g) → ι1 (f ) ⊕ ι2 (g), where ι1 and ι2 are the inclusions of Ra1 ⊕ · · · ⊕ Rai and Rb1 ⊕ · · · ⊕ Rbni to Ra1 ⊕ · · · ⊕ Rai+1 and Rb1 ⊕ · · · ⊕ Rbni+1 in Aff T , respectively. Then, it is a straightforward calculation that Aff T ki has the decomposition lim −→(R , ηi ), and the diagram in the lemma commutes. 2 Finally, we reach the main result of this paper. Theorem 6.8. Let (G0 , (G0 )+ , u) be a countable partially ordered weakly unperforated and rationally Riesz group, let G1 be a countable abelian group, let T be a metrizable Choquet simplex and let λT : T → Su (G0 ) be a surjective affine continuous map sending extremal points to extremal points. Then there exists one (and exactly one, up to isomorphism) unital Z-stable C ∗ -algebra A ∈ A such that Ell(A) =

G0 , (G0 )+ , u , G1 , T , λT .

Moreover, A can be constructed to be locally approximated by subhomogeneous C ∗ -algebras. Proof. Note that the part of the statement about “exactly one, up to isomorphism” follows from [19]. By Lemma 6.7, there exists a decomposition Rkn

ηn

···

ψn

Kn+1

Aff T , λ∗T

λn+1

λn

Kn

Rkn+1

···

G0

where each Kn is a rationally Riesz group with Su (Kn ) having finitely many extreme points. By Lemma 6.6, there is a unital Z-stable algebra An ∈ A such that K0 (An ), K0 (An )+ , [1An ] , K1 (An ), Aff T(An ) , λAn = Kn , (Kn )+ , u , G1 , Rkn , λn , and each An can be locally approximated by subhomogeneous C ∗ -algebras. By Theorem 6.3, there are *-homomorphisms ϕn : An → An+1 such that (ϕn )∗0 = ψn , (ϕn )∗1 = idG1 and (ψn )∗ = ηn , where (ψn )∗ is the induced map from Aff(T(An )) → Aff(T(An+1 )). Then the inductive limit A = lim −→n (An , ψn ) is in the class A and satisfies Ell(A) =

G0 , (G0 )+ , u , G1 , T , λT .

Since each An can be locally approximated by subhomogeneous C ∗ -algebras, so does the C ∗ algebra A. 2

28


Acknowledgments The work of the first named author is partially supported by Shanghai Priority Academic Disciplines and Chang-Jiang Professorship from East China Normal University during the summer 2008 and a grant of NSF. The work of the second named author is supported by an NSERC Postdoctoral Fellowship. References [1] B. Blackadar, K-Theory for Operator Algebras, second ed., Math. Sci. Res. Inst. Publ., vol. 5, Cambridge University Press, 1998. [2] J. Dixmier, On some C ∗ -algebras considered by Glimm, J. Funct. Anal. 1 (1967) 182–203. [3] E.G. Effros, D.E. Handelman, C.L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (2) (1980) 385–407. [4] S. Eilers, T. Loring, G.K. Pedersen, Stability of anticommutation relations: An application of noncommutative CW complexes, J. Reine Angew. Math. 499 (1998) 101–143. [5] G.A. Elliott, A classification of certain simple C*-algebras, in: Quantum and Non-Commutative Analysis, Kyoto, 1992, in: Math. Phys. Stud., vol. 16, Kluwer Acad. Publ., Dordrecht, 1993, pp. 373–385. [6] G.A. Elliott, An invariant for simple C ∗ -algebras, Canad. Math. Soc. 1945–1995 3 (1996) 61–90. [7] G.A. Elliott, G. Gong, L. Li, Approximate divisibility of simple inductive limit C ∗ -algebras, in: Operator Algebras and Operator Theory, Shanghai, 1997, in: Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 87–97. [8] G.A. Elliott, G. Gong, L. Li, On the classification of simple inductive limit C ∗ -algebras, II: The isomorphism theorem, Invent. Math. 168 (2) (2007) 249–320. [9] G.A. Elliott, Z. Niu, On tracial approximation, J. Funct. Anal. 254 (2) (2008) 396–440. [10] G. Gong, X. Jiang, H. Su, Obstructions to Z-stability for unital simple C ∗ -algebras, Canad. Math. Bull. 43 (4) (2000) 418–426. [11] K.R. Goodearl, Partially Ordered Abelian Groups with Interpolation, Math. Surveys Monogr., vol. 20, Amer. Math. Soc., Providence, RI, 1986. [12] X. Jiang, H. Su, On a simple unital projectionless C ∗ -algebra, Amer. J. Math. 121 (2) (1999) 359–413. [13] E. Kirchberg, N.C. Phillips, Embedding of exact C ∗ -algebras in the Cuntz algebra O2 , J. Reine Angew. Math. 525 (2000) 17–53. [14] H. Lin, An Introduction to the Classification of Amenable C ∗ -Algebras, World Scientific Publishing Co., River Edge, NJ, 2001. [15] H. Lin, Classification of simple C ∗ -algebras of tracial topological rank zero, Duke Math. J. 125 (1) (2004) 91–119. [16] H. Lin, Traces and simple C ∗ -algebras with tracial topological rank zero, J. Reine Angew. Math. 568 (2004) 99–137. [17] H. Lin, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, II – An appendix, arXiv:0709.1654v1, 2007. [18] H. Lin, Simple nuclear C ∗ -algebras of tracial topological rank one, J. Funct. Anal. 251 (2) (2007) 601–679. [19] H. Lin, Asymptotically unitary equivalence and classification of simple amenable C ∗ -algebras, arXiv:0806.0636, 2008. [20] J. Mygind, Classification of certain simple C ∗ -algebras with torsion K1 , Canad. J. Math. 53 (6) (2001) 1223–1308. [21] Z. Niu, A classification of certain tracially approximately subhomogeneous C ∗ -algebras, PhD thesis, University of Toronto, 2005. [22] N.C. Phillips, A classification theorem for nuclear purely infinite simple C ∗ -algebras, Doc. Math. 5 (2000) 49–114. [23] M. Rørdam, On the structure of simple C ∗ -algebras tensored with a UHF algebra, J. Funct. Anal. 100 (1991) 1–17. [24] M. Rørdam, On the structure of simple C ∗ -algebras tensored with a UHF algebra. II, J. Funct. Anal. 107 (2) (1992) 255–269. [25] M. Rørdam, A simple C ∗ -algebra with a finite and an infinite projection, Acta Math. 191 (1) (2003) 109–142. [26] S. Sakai, C ∗ -Algebras and W ∗ -Algebras, Ergeb. Math. Grenzgeb., vol. 60, Springer-Verlag, New York–Heidelberg, 1971. [27] A. Toms, On the classification problem for nuclear C ∗ -algebras, Ann. of Math. (2) 167 (2008) 1059–1074. [28] A. Toms, W. Winter, Z-stable ASH algebras, Canad. J. Math. 60 (3) (2008) 703–720. [29] J. Villadsen, The range of the Elliott invariant, J. Reine Angew. Math. 462 (1995) 31–55.


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[30] J. Villadsen, The range of the Elliott invariant of the simple AH-algebras with slow dimension growth, KTheory 15 (1) (1998) 1–12. [31] W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, arXiv:0708.0283v3, 2007.


A Strong Szegö–Widom Limit Theorem for operators with almost periodic diagonal Torsten Ehrhardt a,∗,1 , Steffen Roch b , Bernd Silbermann c a Mathematics Department, University of California, Santa Cruz, CA 95064, USA b Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, Germany c Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany

Received 2 December 2009; accepted 26 July 2010

Communicated by D. Voiculescu

Abstract The classical Strong Szegö–Widom Limit Theorem describes the asymptotic behavior of the determinants of the finite sections Pn T (a)Pn of Toeplitz operators, i.e., of operators which have constant entries along each diagonal. We generalize these results to operators which have almost periodic sequences as their diagonals. © 2010 Elsevier Inc. All rights reserved. Keywords: Szegö–Widom Limit Theorems; Toeplitz operator; Almost Mathieu operator; Determinants of finite sections

1. Introduction The classical results. The n × n Toeplitz matrices are defined as Tn (a) = (aj −k ),

0 j, k n − 1,

* Corresponding author.

E-mail addresses: [email protected] (T. Ehrhardt), [email protected] (S. Roch), [email protected] (B. Silbermann). 1 Research supported in part by NSF Grant DMS-0901434. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.016

T. Ehrhardt et al. / Journal of Functional Analysis 260 (2011) 30–75

31

where a ∈ L∞ (T) is a function defined on the unit circle T = {z ∈ C: |z| = 1} with Fourier coefficients 1 ak = 2π

2π

a eix e−ikx dx,

k ∈ Z.

0

Under certain assumptions on a, one of the several versions of the First Szegö Limit Theorem [23,26] states that det Tn (a) = G[a], n→∞ det Tn−1 (a) lim

while the Strong Szegö–Widom Limit Theorem [24], under additional assumptions, asserts that lim

n→∞

det Tn (a) = E[a]. G[a]n

Therein G[a] and E[a] are well-defined and non-zero constants. At this point we will not discuss further details of the Szegö Limit Theorems, but refer to [8,9] and to Chapter 2 of [22], where also information about the long and rich history can be found. In [18] some milestones in this field are also mentioned. Finite Toeplitz matrices Tn (a) are the finite sections Pn T (a)Pn of Toeplitz operators T (a) = (aj −k ),

j, k ∈ Z+ ,

(1)

which act on 2 (Z+ ), Z+ := {0, 1, 2, . . .}. The projections Pn are defined by Pn : {x0 , x1 , x2 , . . .} ∈ 2 Z+ → {x0 , . . . , xn−1 , 0, 0, . . .} ∈ 2 Z+ . Furthermore, Toeplitz operators T (a) arise as the compressions P L(a)P of the Laurent operators L(a) = (aj −k ),

j, k ∈ Z,

(2)

which act on 2 (Z). Here P is the Riesz projection operator 2 ∞ 2 P : {xn }∞ n=−∞ ∈ (Z) → {yn }n=−∞ ∈ (Z),

yn =

xn

if n 0,

0

if n < 0.

Laurent operators are constant on each diagonal. In other words, they are shift-invariant, i.e., they satisfy U−n L(a)Un = L(a) for all n ∈ Z, where Un denotes the shift operator ∞ Un : {xk }∞ k=−∞ → {xk−n }k=−∞

(3)

acting on 2 (Z). The goal of this paper is to generalize the Strong Szegö–Widom Limit Theorem to finite sections Pn P AP Pn in which A is a bounded linear operator acting on 2 (Z) whose diagonals

32


are almost periodic sequences. More precisely, our generalization will relate to so-called banddominated operators with almost periodic diagonals. The paper [18], where two of the authors dealt with a generalization of the First Szegö Limit Theorem, is thus somewhat related. Let us proceed with explaining the notion of (band-dominated) operators with almost periodic diagonals. Operators with almost periodic diagonals. The set AP(Z) of almost periodic sequences consists of all a ∈ ∞ (Z) for which the set {Un a: n ∈ Z} is relatively compact in the norm topology of ∞ (Z). Here Un : a ∈ ∞ (Z) → b ∈ ∞ (Z),

b(k) := a(k − n),

(4)

is the shift operator acting (isometrically) on ∞ (Z). Clearly, the rules (3) and (4) are the same, only the underlying spaces are different. Moreover, we will prefer the notation a(n) to the usual an for the entries of a sequence a ∈ ∞ (Z) because we will use the second notation for the Fourier coefficients of a sequence a ∈ AP(Z) (see Section 2). There is an equivalent definition of AP(Z) as the closure in ∞ (Z) of the set of all finite linear combinations of sequences eξ ∈ ∞ (Z), where eξ (n) = e2πiξ n ,

n ∈ Z, ξ ∈ R.

(5)

While a Laurent operator can be formally written as L(a) =

a (n) Un ,

n∈Z

where a (n) ∈ C are constants, operators with almost periodic diagonals can be formally written as A=

n∈Z

a (n) Un :=

a (n) I Un ,

(6)

n∈Z

where a (n) ∈ AP(Z) and aI stands for the multiplication operator generated by a ∈ ∞ (Z), ∞ 2 2 aI : {xn }∞ n=−∞ ∈ (Z) → a(n)xn n=−∞ ∈ (Z).

(7)

For sake of brevity, we will usually write (a (n) I )Un as a (n) Un . To make the notion of operators with almost periodic diagonals more precise, let OAP stand for the set of all bounded linear operators A on 2 (Z) such that their n-th diagonal belongs to AP(Z) for each n ∈ Z. In other words, Dn (A) := D(AU−n ) ∈ AP(Z),

(8)


33

where D(A) ∈ ∞ (Z) stands for the main diagonal of a bounded linear operator A acting on 2 (Z). A subclass of OAP are band-dominated operators with almost periodic diagonals. This notion is used rather loosely. It is customary to have it referred to the closure of the set of band operators with almost periodic diagonals with respect to a suitable norm. Band operators are operators of the form (6) with the sum being finite. For some information on band-dominated operators see [17,18]. We are going to establish a generalization of the Strong Szegö–Widom Limit Theorem for certain subclasses of band-dominated operators with almost periodic diagonals. These classes are weighted Wiener type algebras and will be described later on in this introduction. However, in order to get some idea of what to expect as a generalization, let us look first at the case of block Toeplitz determinants. Operators with periodic diagonals and block Toeplitz matrices. The class AP(Z) contains all periodic sequences. An operator A acting on 2 (Z) whose diagonals are periodic sequences with fixed period N can be identified with a block Laurent operator in an obvious way. Block Toeplitz and Laurent operators are defined by formulas (1) and (2), but with a being an N × N matrix-valued function, whose Fourier coefficients an are N × N matrices. Finite block Toeplitz matrices are defined similarly. They arise from the finite sections of the afore-mentioned operators A by Tn (a) = PnN P AP PnN , i.e., when the size of the finite section is a multiple of N . The block case of the Strong Szegö Limit Theorem was established by Widom [26,9]. We are going to state this result for generating functions belonging to the Banach algebra B = W ∩ 2,2 F1/2,1/2 , which by definition consists of all (continuous) functions a ∈ L∞ (T) for which

aB :=

∞

∞

|an | +

n=−∞

1/2 |n| · |an |

< ∞.

2

n=−∞

Theorem 1.1 (Strong Szegö–Widom Limit Theorem). Let a ∈ B N ×N , and assume that det a(t) = 0 for all t ∈ T and that det a(t) has winding number zero. Then lim

n→∞

det Tn (a) = E[a] G[a]n

(9)

where

1 G[a] = exp 2π

2π 0

and E[a] = det T (a)T (a −1 ).

log det a eix dx

34


The winding number condition is equivalent to the existence of a (continuous) logarithm of det a in B. Equivalently, we can require that a belongs to the connected component of the group of all invertible elements of B N ×N containing the unit element. Notice that this last condition is equivalent to requiring that the matrix function a is a finite product of exponentials of functions in B N ×N (see, e.g., [1,20]). The constant E[a] is defined in terms of an operator determinant. In fact, T (a)T (a −1 ) equals identity plus a trace class operator. Only in the case N = 1 (and perhaps in some other very special cases) a more explicit expression is known: E[a] = exp

∞

k(log a)k (log a)−k .

k=1

For more details on the Szegö–Widom Limit Theorem we refer to [8,9,12]. What to expect in the almost periodic case. In view of the results in the block Toeplitz case (i.e., in the case of periodic diagonals), we can now get an idea of what can and, maybe more important, what cannot be expected in the almost periodic case. For instance, for general operators A with almost periodic diagonals one cannot expect an asymptotic formula of the kind det Pn P AP Pn = E. n→∞ Gn lim

(10)

Simple counterexamples can be constructed involving block diagonal Laurent operators. What one can expect is that for certain strictly monotonically increasing sequences h : Z+ → + Z we have lim

n→∞

det Ph(n) P AP Ph(n) = E. Gh(n)

(11)

For instance, in the periodic case, we should take h(n) = nN . Then we have GN = G[a] in comparison with Theorem 1.1. In general such sequences h corresponds to the notion of a distinguished sequence, which plays a crucial role in this paper as it had in [18]. Moreover, we cannot expect an explicit expression for E besides that of an operator determinant. Furthermore, we have to restrict the class of band-dominated operators. This corresponds to take the generating functions in Theorem 1.1 from the Banach algebra B instead of from L∞ (T). Finally, in the almost periodic case a somewhat unexpected difficulty enters the scene. If one wants to obtain the asymptotics of determinants one should be able to analyze the asymptotics of traces in the first place. Indeed, to see that this is a necessary step consider the special case of A = ea I with a ∈ AP(Z), and notice that det(Pn P AP Pn ) = exp(trace(Pn P aI P Pn )). To compute the trace for the finite sections of A ∈ OAP, let a = D(A) ∈ AP(Z). Then trace(Pn P AP Pn ) =

n−1 k=0

a(k).

(12)


35

It turns out that the right-hand side is n · M(a) + o(n), where M(a) stands for the mean of an almost periodic sequence. Desirably we would like to have an error term o(1). However, in the (almost) periodic case this cannot hold true. Yet one could expect that perhaps for distinguished sequences h : Z+ → Z+ we have h(n)−1

a(k) = h(n) · M(a) + o(1),

n → ∞.

(13)

k=0

For instance, in the case of a periodic sequence with period N , we can take h(n) = N n. As will be shown by counterexamples in the last section of this paper, also this is not true in general! In the counterexamples the error term may not even be bounded. Fortunately, for some classes of almost periodic sequences, the asymptotics (13) still holds true. The validity or failure is connected with the Fourier spectrum of the underlying sequence. Illustration of the main results. The proof of the main results is very technical. Unfortunately, even to state the main results in full detail is quite technical, too. One reason is that one has to deal with three different objects: 1. additive subgroups Ξ of R/Z which are related to the Fourier spectrum of almost periodic sequences; 2. Banach algebras A of almost periodic sequences involving a weight; 3. Banach algebras R of operators on 2 (Z) whose diagonals are sequences of the previous Banach algebras. The main result of this paper (Theorem 5.3) will be stated here in the introduction in full generality. It involves the just mentioned additive subgroups Ξ which we will only partially elaborate on here. Section 2 will provide full details on this matter. Sections 3–4 will provide the major steps towards the proof. For the proof it is necessary to introduce further auxiliary Banach algebras. Section 5 will give the main results with their proofs. Section 6 contains the counterexamples related to the asymptotics of the traces. Let us now describe the main results. We start with an additive subgroup Ξ of R/Z, which at this point can be arbitrary. Here R/Z denotes the additive group arising from R by identifying two numbers whose difference is an integer. As for notation, we will write ξ for both a real number and the corresponding equivalence class in R/Z. An admissible weight β on Ξ is a function β : Ξ → R+ for which 1 β(ξ1 + ξ2 ) β(ξ1 )β(ξ2 ). For instance, we can put β(ξ ) = 1. Another example of admissible weights can be constructed for finitely generated groups Ξ = ξ = α1 ξ1 + · · · + αN ξN : α = (α1 , . . . , αN ) ∈ ZN with given generators (ξ1 , . . . , ξN ) ∈ RN by defining ω β(ξ ) = 1 + |α| ,

ω0

(14)

36


with |α| = max |αi |. Here we assume (for simplicity) that 1, ξ1 , . . . , ξN are linearly independent over the field Q to make sure that the coefficients αk are uniquely determined by ξ . For any such additive subgroup Ξ and any admissible weight β we define A = APW(Z, Ξ, β) as the set of all sequences a ∈ ∞ (Z) of the form a=

aξ e ξ

such that aΞ,β :=

ξ ∈Ξ

β(ξ )|aξ | < ∞

ξ ∈Ξ

with aξ ∈ C. We will show that A is a Banach algebra with the above norm which is continuously embedded in AP(Z). All elements in A have Fourier spectrum contained in Ξ . Next, given such a Banach algebra A and α1 , α2 0 define R = Wα1 ,α2 (A) as the set of all operators A acting on 2 (Z) which are of the form A=

a (k) Uk ,

a (k) ∈ A

k∈Z

such that ∞ (1 + k)α1 a (k) A + (1 + k)α2 a (−k) A < ∞. AR := a (0) A + k=1

The last condition ensures that the defining series for A converges. The set R is a unital Banach algebra which is continuously embedded into L(2 (Z)) and contained in OAP. Finally, a strictly increasing sequence h : Z+ → Z+ is called distinguished for Ξ , if lim e2πiξ h(n) = 1

n→∞

for each ξ ∈ Ξ . Clearly, the previous condition is satisfied for all ξ ∈ Ξ if it holds for a set of generators of the group Ξ . Moreover, one can show that all finitely generated groups Ξ possess distinguished sequences. Now we can state an auxiliary result. It corresponds to a special case of Theorem 5.2. Theorem 1.2. Let Ξ , β, A, and R be as above, and assume that α1 + α2 = 1, α1 , α2 > 0. Let h : Z+ → Z+ be a distinguished sequence for Ξ . Then, for A1 , . . . , Ar ∈ R and A = eA1 · · · eAr , we have det(Ph(n) P AP Ph(n) ) = det P AP A−1 P . n→∞ exp(trace(Ph(n) P (A1 + · · · + Ar )P Ph(n) )) lim

Therein the determinant on the right-hand side is a well-defined operator determinant of an operator acting on 2 (Z+ ). If we compare this theorem with the classical Strong Szegö–Widom Theorem (see Theorem 1.1), then the assumption that the operator A is a product of exponentials corresponds to the condition that the symbol of the block Toeplitz operator Tn (a) has a non-vanishing determinant with winding number zero. In fact, we one can recover the classical theorem (although in the setting of a slightly different Banach algebra B) by considering A = L(a), h(n) = nN , Ξ = {k/N: k ∈ Z}, β ≡ 1, and α1 = α2 = 1/2, where N is the block size.


37

The above theorem reduces the asymptotics of a determinant to the asymptotics of a trace. The computation of the trace seems easy. In fact, we have h(n)−1 trace Ph(n) P (A1 + · · · + Ar )P Ph(n) = a(k) k=0

where a = D(A1 + · · · + Ar ) ∈ A = APW(Z, Ξ, β) ⊆ AP(Z) is the sequence arising from the main diagonals of the Ak ’s. However, as mentioned before, the form of the asymptotics that we would expect from the classical (periodic) case does not always hold in the almost periodic case. In order to ensure such an asymptotics, one needs to require a non-trivial extra assumption. This is an assumption on the underlying group Ξ and the weight β. We call the weight β compatible on Ξ if inf

ξ ∈Ξ, ξ =0

β(ξ ) · ξ R/Z > 0

where ξ R/Z = inf{|ξ − n|: n ∈ Z}. Now we have the following result. Theorem 1.3. Let β be an admissible and compatible weight on an additive subgroup Ξ of R/Z, and suppose that h : Z+ → Z+ is a distinguished sequence. Then, for a ∈ APW(Z, Ξ, β), h(n)−1

a(k) = h(n) · M(a) + o(1),

as n → ∞.

k=0

This theorem is stated and proved in Section 2 (see Theorem 2.7). It is now clear that if we can combine the last two results, then we obtain lim

n→∞

det(Ph(n) P AP Ph(n) ) = det P AP A−1 P Gh(n)

with G = exp M(a) ,

which is our main result (Theorem 5.3). The main question which arises in connection with Theorem 1.3 is: Given an additive subgroup Ξ , does there exist an admissible and compatible weight? In addition, we do not want the weights to be growing too fast as this would restrict the classes of almost periodic sequences too much. The counterexamples, which will be presented in Section 6, will imply that such compatible weight do not always exist, not even for every singly-generated group Ξ . On the other hand, there are positive results. They will be presented in Theorem 2.8 and in Section 2.3. First of all, it is shown that almost every finitely generated group Ξ possesses an admissible weight is of the form (14). Here “almost every” is understood in the sense of the Lebesgue’s measure with respect to the generators of the group. To identify concrete groups which possess such weights is more complicated and in fact relies on deep results on diophantine approximation. For instance, positive results are obtained for groups generated by finitely many algebraic numbers.

38


Let us come back to the counterexamples. They concern a singly-generated group Ξ = {kξ : k ∈ Z} with ξ being irrational. In fact, ξ must be a Liouville number. We will construct such numbers ξ (in terms of an infinite series) and corresponding almost periodic sequences a=

∞

ak ekξ

k=1

with exponentially decaying ak ’s such that various properties are fulfilled. In particular, the following examples can be realized: 1. ak = b−k , b > 1, x > 0 arbitrary, and ξ suitably constructed: then there exists a distinguished sequence h such that h(n)−1

a(k) = x + o(1) as n → ∞.

k=0

2. ak = b−k , b > 1, 0 < α < 1 arbitrary, and ξ suitably constructed: then there exists a distinguished sequence h such that h(n)−1

a(k) = h(n)α 1 + o(1)

as n → ∞.

k=0

Notice that therein M(a) = 0, and observe that the asymptotics are different from those asserted in Theorem 1.3. In view of the main result, yet other questions come up. How can one decide whether an operator is a product of exponentials (as is required in the assumptions of the main theorem)? We leave this important question open, although some comments are given at the end of Section 5. Furthermore, is there a way to compute the constant G directly in terms of A? Also this question is left open. Outline of the paper and main idea of the proof. Let us give now a brief outline of the paper. Section 2 is devoted to almost periodic sequences. Besides elaborating on the notion of a distinguished sequence, we are going to define classes of almost sequences for which the asymptotics (13) holds. As mentioned before they are described by a certain condition, and we show, using known results from diophantine approximation, that this condition is in a certain sense generically fulfilled. The proof our main result is based on a Banach algebra approach. This approach was introduced by one of the authors in [12] in the setting of the classical Strong Szegö–Widom Limit Theorem. The potential of this approach seems large and is by no means restricted to the classical results. One could ask if the results of this paper could be proved by other methods. A recent approach to classical Strong Szegö–Widom Theorem is by means of the of Geronimo–Case–Borodin– Okounkov identity. However, it seems that for our operators this approach does not work. The proof of the main result by the Banach algebra approach is done in Sections 3–5. In Section 3 the appropriate classes of operators on 2 (Z) with almost periodic diagonals will be defined. The notions of suitability, rigidity, and shift-invariance are defined, too. The notion of


39

suitability underlies the algebraic approach. In Section 4 certain Banach algebras of operators and of sequences are defined, which are derived from rigid, suitable and shift-invariant Banach algebras. These Banach algebras and their properties are the main ingredients to the Banach algebra approach. The main result is established in Section 5. Finally, in Section 6 we present counterexamples to the asymptotics (13). These counterexamples contrast the positive results of Section 2 and show certain limitations of further generalizing the main results. Our motivation for establishing the Szegö–Widom Limit Theorem for band-dominated operators with almost periodic coefficients comes from its most prominent example: the almost Mathieu operator A = U1 + aI + U−1 Here a(n) = β sin(ξ n + δ), and β, ξ , and δ are certain (real) constants. Our results would apply to the determinants of Pn P (A − λI )P Pn provided that the appropriate assumptions of the main theorem are fulfilled. For more information about almost periodic operators we refer to the monograph [6] and the more recent papers [2,16], where the long standing ten Martini problem is solved. 2. Banach algebras of almost periodic sequences Let us start with some basic facts about AP(Z). A basic reference for almost periodic functions (and thus also for almost periodic sequences) is [11]. For each a ∈ AP(Z) its mean M(a) is well defined by the limit 1 a(k). n→∞ n n−1

M(a) = lim

(15)

k=0

As already mentioned in the introduction, let R/Z denote the additive group arising from R by identifying two numbers whose difference is an integer. We agree to denote a real number which is a representative of the equivalence class ξ ∈ R/Z also by ξ . The Fourier coefficients of a sequence a ∈ AP(Z) are defined by aξ = M(ae−ξ ),

ξ ∈ R/Z,

where eξ ∈ AP(Z) is the sequence defined in (5). The set of all ξ ∈ R/Z for which aξ = 0 is called the Fourier spectrum of a. For each a ∈ AP(Z) this set is at most countable. A subclass of almost periodic sequences is the Wiener class APW(Z), which is the set of all sequences a ∈ AP(Z) with absolutely convergent Fourier series. In other words, it is the set of all sequences which can be represented as a=

aξ e ξ

ξ

where the sum is taken over at most countably many elements ξ ∈ R/Z and where aAPW(Z) := |aξ | < ∞. ξ

(16)

40


The above norm makes APW(Z) a Banach algebra, which is continuously embedded in AP(Z). The goal of this section is to provide information about the asymptotics of n−1

(17)

a(k)

k=0

as n → ∞ for a ∈ AP(Z). By the definition of the mean, this sum equals nM(a) + o(n) as n → ∞. In order to get a better estimate of the error term, let us consider the case where a is a finite sum (16). Using the obvious facts M(eξ ) =

1 if ξ = 0,

(18)

0 if ξ = 0

and n−1

eξ (k) =

k=0

1 − e2πiξ n , 1 − e2πiξ

(19)

it follows that n−1

a(k) = n · M(a) +

aξ

ξ =0

k=0

1 − e2πiξ n . 1 − e2πiξ

The second term behaves oscillatory. Therefore, we are compelled to let run n not through all positive integers, but to restrict ourselves to certain strictly increasing sequence h : Z+ → Z+ . Indeed, if h is such that lim e2πiξ h(n) = 1,

(20)

n→∞

for all ξ in the Fourier spectrum of a, then h(n)−1

a(k) = h(n) · M(a) + o(1),

n → ∞.

k=0

At this point one is tempted to carry over this result to the case where a is an infinite sum (16), e.g., for general a ∈ APW(Z). However, as will be shown in Section 6, this kind of asymptotics does not hold in general. 2.1. Distinguished sequences for subalgebras of AP(Z) Throughout the rest of the paper, let H stand for the set of all strictly increasing sequences h : Z+ → Z+ . All Banach algebras which we will consider have a unit element. When we say that A is a Banach subalgebra of a Banach algebra B we do not imply any relation between the norms of


41

these two Banach algebras. We only require that the algebraic relations are compatible and that the unit elements are the same. A Banach subalgebra A of ∞ (Z) will be called shift-invariant if, for each a ∈ A and n ∈ Z, we have Un a ∈ A and Un aA = aA .

(21)

Let A be a shift-invariant Banach subalgebra of ∞ (Z). A sequence h ∈ H will be called distinguished for A if for each a ∈ A we have lim U−h(n) a − aA = 0.

n→∞

(22)

For certain classes of Banach algebras A, the following proposition provides a simple criterion for a sequence h to be distinguished for A. Proposition 2.1. Let Ξ be an additive subgroup of R/Z, and let A be a shift-invariant Banach subalgebra of AP(Z) such that the linear span of {eξ : ξ ∈ Ξ } is contained and dense in A. Then a sequence h ∈ H is distinguished for A if and only if for each ξ ∈ Ξ we have lim e2πih(n)ξ = 1.

n→∞

(23)

Proof. From U−h(n) eξ = e2πih(n)ξ eξ it follows that U−h(n) eξ − eξ A = e2πih(n)ξ eξ − eξ A = e2πih(n)ξ − 1 · eξ A . This proves the “only if” part as well as the “if” part for a = eξ . Moreover, assuming (23) it follows that (22) holds for finite linear combinations of eξ . Using the density assumption an approximation argument implies the validity of (22) for all a ∈ A. 2 The criterion can be relaxed further. In fact, it is sufficient to require (23) only for a set of generators of the group Ξ . For the classes of Banach algebras to which the previous proposition can be applied, the question of h being distinguished is reduced to the underlying group Ξ . In such a setting we will say that h ∈ H is a distinguished sequence for Ξ . Not for all additive subgroups Ξ there exist distinguished sequences. A necessary condition is that the Lebesgue measure of Ξ is zero, a sufficient condition is that Ξ is countable. (We will not prove these facts here.) Example 2.2 (Trivial example). Let N ∈ N and

k + Z: k ∈ Z . Ξ= N Then h(n) = N n is a distinguished sequence for Ξ .

42


Example 2.3 (Trivial example). Let Ξ = Q/Z. Then h(n) = n! is a distinguished sequence for Ξ . Example 2.4. Let ξ1 , . . . , ξN ∈ R be such that {1, ξ1 , . . . , ξN } are linearly independent over Q, and let Ξ = α1 ξ1 + · · · + αN ξN : (α1 , . . . , αn ) ∈ ZN . Then distinguished sequences exist. This follows from the well-known fact that the set (nξ1 , nξ2 , . . . , nξN ) ∈ (R/Z)N : n ∈ N is dense in (R/Z)N . The next example is a special case of the previous one, where a particular distinguished sequence is described. For details, see [18, Example 2.10]. Example 2.5. Let Ξ = {αξ : α ∈ Z} where ξ ∈ R is irrational. Let pn /qn be the n-th continued fraction of ξ . Then h(n) = qn is a distinguished sequence for Ξ . This follows from the wellknown fact that |qn ξ − pn | < 1/qn . 2.2. Weighted Wiener algebras of almost periodic sequences: the compatibility condition In what follows we are going to define a concrete class of shift-invariant Banach algebras, namely weighted Wiener algebras of almost periodic sequences with their Fourier spectrum contained in a prescribed subgroup of R/Z. Let Ξ be an additive subgroup of R/Z. We call a mapping β : Ξ → R+ an admissible weight on Ξ if 1 β(ξ1 + ξ2 ) β(ξ1 )β(ξ2 )

(24)

for each ξ1 , ξ2 ∈ Ξ . For such Ξ and β, let APW(Z, Ξ, β) stand for the set of all functions a=

aξ e ξ

(25)

ξ ∈Ξ

for which aΞ,β :=

β(ξ )|aξ | < ∞.

ξ ∈Ξ

With the above norm, APW(Z, Ξ, β) becomes Banach space which is continuously embedded in APW(Z). In particular, (25) is an absolutely convergent series. The following theorem guarantees that APW(Z, Ξ, β) is indeed a Banach algebra and that we can apply the results of the previous section. Theorem 2.6. Let β be an admissible weight on an additive subgroup Ξ of R/Z. Then APW(Z, Ξ, β) is a shift-invariant, continuously embedded Banach subalgebra of AP(Z), and the linear span of {eξ : ξ ∈ Ξ } is a dense subset.


43

Proof. Let us first check that APW(Z, Ξ, β) is closed under multiplication. Assume that a=

b=

aξ e ξ ,

ξ ∈Ξ

bξ e ξ

ξ ∈Ξ

belong to APW(Z, Ξ, β). Since eξ1 eξ2 = eξ1 +ξ2 , one has

ab =

aξ1 bξ2 eξ1 +ξ2 =

ξ1 ∈Ξ ξ2 ∈Ξ

ξ ∈Ξ

aξ1 bξ −ξ1 eξ .

ξ1 ∈Ξ

From the following estimate we can conclude that ab ∈ APW(Z, Ξ, β) and also that ab ab: abΞ,β =

ξ ∈Ξ

β(ξ ) aξ1 bξ −ξ1

ξ1 ∈Ξ

β(ξ1 + ξ2 )|aξ1 bξ2 |

ξ1 ∈Ξ ξ2 ∈Ξ

=

β(ξ1 )|aξ1 | β(ξ2 )|bξ2 |

ξ1 ∈Ξ

ξ2 ∈Ξ

= aΞ,β bΞ,β . Notice that we have used that the weight β is admissible. Let us now verify that APW(Z, Ξ, β) is shift-invariant. For n ∈ Z we have U−n eξ = e2πiξ n eξ . Hence a=

aξ e ξ

implies

ξ ∈Ξ

U−n a =

aξ e2πiξ n eξ .

ξ ∈Ξ

Because of |e2πiξ n | = 1 it follows that U−n a belongs to APW(Z, Ξ, β) whenever so does a, and one also has U−n aΞ,β = aΞ,β . The remaining statements are obvious. 2 We are now going to establish the main theorem of this section. Therein we need to require a somewhat restrictive condition on Ξ and β. In order to describe it, let us introduce ξ R/Z := inf |ξ − n|: n ∈ Z ,

(26)

which corresponds to the natural metric on R/Z. Now we say that β is compatible on Ξ if inf

ξ ∈Ξ, ξ =0

β(ξ ) · ξ R/Z > 0.

(27)

We will look in a few moments at the important question whether admissible and compatible weights exist.

44


Theorem 2.7. Let β be an admissible and compatible weight on an additive subgroup Ξ of R/Z. If a ∈ APW(Z, Ξ, β) and h ∈ H is a distinguished sequence for Ξ , then h(n)−1

a(k) = h(n) · M(a) + o(1),

n → ∞.

(28)

k=0

Proof. First observe that (using M(a) = a0 ) h(n)−1

a(k) − h(n)M(a) =

aξ

ξ ∈Ξ, ξ =0

k=0

h(n)−1

eξ (k) .

(29)

k=0

For (fixed) ξ ∈ Ξ , ξ = 0, the expression in the braces can be rewritten as follows, h(n)−1

eξ (k) =

k=0

h(n)−1 k=0

e2πiξ k =

1 − e2πiξ h(n) , 1 − e2πiξ

and thus, by Proposition 2.1, it converges to zero as n → ∞. On the other hand, the same expression can be estimated by h(n)−1 1 Cβ(ξ ), eξ (k) = | sin(πξ )| k=0

where the constant C comes from the infimum (27). Hence, using dominated convergence, the expression (29) represents an absolutely convergent series which converges to zero as n → ∞. 2 Let us now turn to the question of the existence of admissible and compatible weights. We are not able to answer this question completely. In fact, it is connected to some deep question of diophantine approximation. We are first going to show a pure existence result, namely, that for finitely generated subgroups there exists “generically” an admissible and compatible weight. In addition, the weight grows only polynomially. The basis of these observations is the following theorem, which is proved, e.g., in Section 3.5.3 (N = 1) and Section 4.3.2 (N 2) of [5]. Theorem 2.8. Let N 1 and denote by SN ⊆ (R/Z)N the set of all (ξ1 , . . . , ξN ) for which there exist ω > 0 and C > 0 such that −ω α1 ξ1 + · · · + αN ξN R/Z C max |αi | (30) 1iN

for all (α1 , . . . , αN ) ∈ ZN \ {0}. Then the complement of SN in (R/Z)N has Hausdorff dimension N − 1, hence Lebesgue measure zero in (R/Z)N . It is now obvious, that for ξ ∈ SN we can define an admissible and compatible weight on Ξ = ξ = α1 ξ1 + · · · + αN ξN ∈ (R/Z)N : α = (α1 , . . . , αN ) ∈ ZN


45

by ω β(ξ ) = 1 + |α| ,

|α| = max |αi |. 1iN

In the case N = 1, the set S1 is just the set of all numbers which are not Liouville numbers. 2.3. Examples of Banach algebras APW(Z, Ξ, β) The purpose of this section is to provide concrete examples of classes of additive subgroups Ξ and admissible and compatible weights β, thus providing examples of Banach algebras APW(Z, Ξ, β) to which Theorem 2.7 can be applied. Besides two trivial examples, we consider two other examples. In the first one, Ξ is finitely generated by algebraic numbers, in the second one, Ξ is generated by finitely many logarithms of algebraic numbers. Both examples rely on rather deep results from diophantine approximation. Example 2.9 (Trivial example). Let N ∈ N and

k + Z: k ∈ Z . Ξ= N Clearly, Ξ contains precisely N elements and we can define an admissible and compatible weight by β(ξ ) = 1. The Banach algebra APW(Z, Ξ, β) consists of all periodic sequences with period N . Example 2.10 (Trivial example). Let Ξ = Q/Z. We can define an admissible and compatible weight by β(ξ ) = q

where ξ =

p , p ∈ Z, q ∈ N, and p, q are co-prime. q

The Banach algebra APW(Z, Ξ, β) is the closure (with respect to the appropriate norm) of the set of all periodic sequences (with no conditions on the length of the period). In order to establish non-trivial examples we need to resort to the theory of diophantine approximation. We refer to [4,10,25] as some references. Before stating these results, recall the notion of an algebraic number. A number ξ ∈ C is called algebraic if there exists a (not identically vanishing) polynomial with integer coefficients that annihilates ξ . Among all such polynomials there exists one with smallest degree (or, equivalently, which is irreducible). The degree of the algebraic number is by definition the degree of this polynomial. Theorem 2.11 (Roth–Schmidt). Let ξ1 , . . . , ξN be algebraic numbers such that {1, ξ1 , . . . , ξN } is linearly independent over Q. Then for every d > 0 there exists a constant Cd > 0 such that |α0 + α1 ξ1 + · · · + αN ξN | Cd for every (α0 , α1 , . . . , αN ) ∈ ZN +1 .

−N −d max |αi |

1iN

46


This theorem was established by W.E. Schmidt [21]. In the special case N = 1 it is the celebrated theorem of Roth [19]. Roth’s result was the culmination point of a development starting with Liouville’s Theorem, where the exponent d depends on the degree of the algebraic number. The conditions on d were subsequently relaxed by A. Thue, C.L. Siegel, F.J. Dyson, A.O. Gelfond, and finally by K.F. Roth, who obtained, in some sense, the best possible result. Notice that if N = 1 and if the (single) algebraic number ξ1 has degree two, then Liouville’s Theorem gives already a better result, namely the above inequality with d = 0. Example 2.12. Let ξ1 , . . . , ξN ∈ R be algebraic numbers such that {1, ξ1 , . . . , ξN } is linearly independent over Q. Then every Z-linear combination α1 ξ1 + · · · + αN ξN can be considered as an element of R/Z and determines the coefficients α1 , . . . , αN uniquely. Consider Ξ = α1 ξ1 + · · · + αN ξN : (α1 , . . . , αN ) ∈ ZN , and for ε > 0 define the weight N +ε β(ξ ) = 1 + |α| ,

|α| := max |α1 |, . . . , |αN | .

Obviously, β is admissible. The compatibility follows from the Roth–Schmidt Theorem. For the next example we need to state two theorems as preparations. Let Q denote the set of all algebraic numbers (possibly complex), and let Λ = λ ∈ C: exp(λ) ∈ Q .

(31)

In other words, Λ is the set of logarithms of algebraic numbers. Theorem 2.13 (Baker). If λ1 , . . . , λN ∈ Λ are linearly independent over Q, then 1, λ1 , . . . , λN are linearly independent over Q. This theorem was established by Baker in 1966 [3]. Predecessors to this results were established by Gelfond, Kuzmin, and Gelfond–Schneider (see also [4,25]). The next theorem is more general than the previous one. However, the previous theorem usually serves as a prerequisite for the next one. Theorem 2.14 (Feldman). Let λ1 , . . . , λN ∈ Λ be linearly independent over Q, and let d ∈ N. Then there exist (effectively computable) constants C, ω > 0 (depending only on λ1 , . . . , λN and d) such that |α0 + α1 λ1 + · · · + αN λN | Ch−ω for all α0 , . . . , αN ∈ Q of degree d, where h is the maximum of the heights of α0 , . . . , αN .

(32)


47

The height of an algebraic number α = 0 is the maximum of the moduli of the coefficients of the annihilating irreducible polynomial with relatively prime integer coefficients. (The height of α = 0 is zero by definition.) Only the following two cases are of interest to us: if α ∈ Z, then its height equals |α|, if α ∈ iZ, then its height equals |α|2 . The result was established by N.I. Feldman [13] in 1968 (although stated in a slightly different way). The constants, though explicit, are quite complicated. Improvements of those constants have been obtained subsequently. Lower estimates of non-algebraic type have been established earlier by A.O. Gelfond and A. Baker. In the following example we will take generators ξ1 , . . . , ξN for Ξ which belong either to Λ ∩ R or iΛ ∩ R. The first class includes numbers such as log 5 or

√

log(1 +

2 ).

The second class includes numbers such as π = i log(−1) or

√ √ 1−i 2 arctan( 2 ) = i log √ . 1+i 2

Example 2.15. Let ξ1 , . . . , ξM ∈ Λ ∩ R,

ξM+1 , . . . , ξN ∈ (iΛ ∩ R),

and assume that both {ξ1 , . . . , ξM } and {ξM+1 , . . . , ξN } are linearly independent over Q. Then the coefficients α1 , . . . , αN ∈ Z of any linear combination α1 ξ1 + · · · + αN ξN considered as an element in R/Z are uniquely determined. Indeed, we can make an obvious substitution and write α0 + α1 ξ1 + · · · + αN ξN = α0 + α1 λ1 + · · · + αM λM + (iαM+1 )λM+1 + · · · + (iαN )λN

(33)

with λ1 , . . . , λN ∈ Λ. It follows from the assumption that the set {λ1 , . . . , λN } is linearly independent over Q. Hence Theorem 2.13 implies that if the above linear combination is zero, then all their coefficients must be zero, too. Now consider the subgroup Ξ = α1 ξ1 + · · · + αN ξN : (α1 , . . . , αN ) ∈ ZN of R/Z. From Theorem 2.14 it follows that there exist constants C > 0 and ω such that estimate (32) holds, whence by using (33) α1 ξ1 + · · · + αN ξN R/Z C

−2ω max |αi | .

0iN

Note that the exponent 2ω comes from the fact that we have coefficients iαk , which have height |αk |2 . This implies by a simple computation that the weight

48


2ω β(ξ ) = 1 + |α| ,

|α| := max |α1 |, . . . , |αN |

is compatible and (obviously) admissible. Z) 3. Banach algebras of operators on 2 (Z The purpose of this section is to find classes of Banach algebras of operators on 2 (Z) for which a generalization of the Strong Szegö–Widom Limit Theorem can be established. Our proof is based on a “Banach algebra approach”. This approach was established in the case of classical Szegö–Widom Limit Theorem in [12]. The main idea is to reduce the asymptotics of determinants to the asymptotics of traces of certain operators, and it is accomplished by considering Banach algebras of “symbols”, which are characterized by certain properties. The main property that such Banach algebras have to possess is that of suitability. It generalizes the corresponding notion introduced first in [12]. There are also two other notions. They are trivial in the classical setting. The notion of shiftinvariance of such a Banach algebra makes sure that the notion of a distinguished sequence can be considered properly. A third notion, that of rigidity, is of rather technical nature. One could probably work without it, but the changes necessary would make the presentation more tedious. Before we start with this, let us introduce some notation. As usual, L(H ) stands for the Banach algebra of all bounded linear operators on a Hilbert space H . For 1 p < ∞, let Cp (H ) refer to the Schatten–von Neumann class of operators on H , i.e., to the set of all compact operators K ∈ L(H ) for which KCp :=

1/p sn (K)p

0 and α1 + α2 = 1, then Wα1 ,α2 (A) is suitable, (b) if A ⊆ AP(Z), then Wα1 ,α2 (A) is rigid. Proof. It is easy to verify that Wα1 ,α2 (A) is a Banach space. Because of the estimate AL(2 (Z)) = Dk (A)Uk

L(2 (Z))

k∈Z

Dk (A)

∞ (Z)

k∈Z

AWα1 ,α2 (A) , Wα1 ,α2 (A) is continuously embedded in L(2 (Z)). Now let A be given by (43). Then U−n AUn =

U−n a (k) Uk . k∈Z

From this and the shift-invariance of A it is easy to conclude that Wα1 ,α2 (A) is shift-invariant.


53

In order to verify that Wα1 ,α2 (A) is closed under multiplication, let A=

a (k) Uk ,

B=

k∈Z

b(k) Uk

k∈Z

be elements of Wα1 ,α2 (A). Then AB =

k1 ∈Z k2 ∈Z

=

a (k1 ) Uk1 b(k2 ) Uk2 a (k1 ) Uk1 b(k2 ) Uk1 +k2

k1 ∈Z k2 ∈Z

=

c(n) Un

n∈Z

with c(n) =

a (k) Uk b(n−k) .

k∈Z

The fact that c(n) ∈ A can be seen from the estimate n∈Z

α(n)c(n) A α(k)α(n − k)a (k) A Uk b(n−k) A n,k∈Z

= AWα1 ,α2 (A) BWα1 ,α2 (A) , which also implies that AB ∈ Wα1 ,α2 (A) and AB A · B. Here we have used the assumption that A is shift-invariant and that α(n) α(k)α(n − k), which in turn can be verified straightforwardly. (a): We show that if α2 > 0 and A ∈ Wα1 ,α2 (A), then P AQ ∈ Cp with p = 1/α2 . Indeed, write A=

Dk (A)Uk .

k∈Z

Because P commutes with the diagonal operators Dk (A) we then have P AQ =

Dk (A)P Uk Q

k∈Z

=

k>0

Dk (A)P Uk Q.

54


The singular values of P Uk Q are just the square roots of the eigenvalues of P Uk QU−k P = Pk . Thus, 1 is a k-fold singular value of P Uk Q, and it is the only non-zero singular value of that operator. Hence, P Uk QCp = k 1/p = k α2 , which implies the norm estimate P AQCp

Dk (A)

∞ (Z)

P Uk QCp

k>0

k α1 Dk (A)A < ∞.

k>0

In particular, P AQ ∈ Cp . Similarly we obtain QAP ∈ Cq with 1/q = α1 whenever A ∈ Wα1 ,α2 (A) and α1 > 0. Consequently, if α1 + α2 = 1, then Wα1 ,α2 (A) is continuously embedded into the Banach algebra Rp,q with Schatten class corners as defined in Example 3.2 where p = 1/α2 and q = 1/α1 , and one has ARp,q AWα1 ,α2 (A) for all operators A ∈ Wα1 ,α2 (A). The remark made in (iv) above yields finally that Wα1 ,α2 (A) is a suitable Banach algebra. (b): If A ⊆ AP(Z), then Wα1 ,α2 (A) ⊆ OAP. Now Theorem 3.1 applies. 2 Proposition 3.6. Let A be shift-invariant Banach subalgebra of ∞ (Z), and let α1 , α2 0. Then h ∈ H is distinguished for Wα1 ,α2 (A) if and only if h is distinguished for A. Proof. The “only if” part follows by considering the special case of A = aI ∈ Wα1 ,α2 (A) with a ∈ A. As to the “if” part, consider A of the form (43) and assume U−h(n) a (k) → a (k) in the norm of A for each k ∈ Z. Because U−h(n)

k∈Z

U−h(n) a (k) Uk a (k) Uk Uh(n) = k∈Z

we can conclude that U−h(n) AUh(n) → A in the norm of Wα1 ,α2 (A). Indeed, we use the fact that the operators U−h(n) are isometries on A in order to apply dominated converges of the infinite series. 2 The Banach algebra R = Wα1 ,α2 (A) with A = APW(Z, Ξ, β) is the algebra of operators for which we will state later our generalization of the Strong Szegö–Widom Limit Theorem. Of course, we also need to assume that β is admissible and compatible on Ξ . As we have seen, R satisfies the following conditions which one should keep in mind: it is a Banach algebra, it is suitable, rigid, and shift-invariant. It also contains all band operators with diagonals in APW(Z, Ξ, β). There are certainly other algebras with these properties which could take the place of R. Generalizations of the Krein and the Wiener–Krein algebras are good candidates (see Chapter 5 in [7] and [12,15]). We will not pursue this topic further since the main applications involve band operators with almost periodic coefficients which are already covered by this Banach algebra.


55

3.3. An operator determinant identity The goal of this subsection is to generalize an operator determinant identity which was established in [12] in the context of suitable Banach algebras of Laurent operators (or, functions on the unit circle, if we stick to the original setting in [12]). We show that this identity holds for suitable Banach algebras in our context, i.e., it holds also in the non-commutative setting. In what follows we employ the notion of an analytic Banach algebra valued function. We refer to [20] for a definition and basic properties. Proposition 3.7. Let R be a suitable Banach algebra, and let A1 , . . . , Ar ∈ R. Then the functions F0 (λ1 , . . . , λr ) := T eλ1 A1 · · · eλr Ar · e−λr T (Ar ) · · · e−λ1 T (A1 ) − P , F1 (λ1 , . . . , λr ) := T eλ1 A1 · · · eλr Ar · e−λr T (Ar ) · · · e−λ1 T (A1 ) − P are analytic with respect to each variable λk ∈ C and take values in the ideal of the trace class operators on 2 (Z+ ). Proof. The analyticity follows easily since λ → eλB is an analytic function for each bounded linear operator B and since the product of analytic functions is analytic again. To get the trace class property of F0 note that F0 (λ1 , . . . , λr ) − T eλ1 A1 · · · T eλr Ar · e−λr T (Ar ) · · · e−λ1 T (A1 ) + P is trace class by (37). Thus, it is sufficient to prove that T eλA e−λT (A) − P ∈ C1

(44)

for every A ∈ R. Again by (37), one has T Ak = P Ak P = (P AP )k + Rk

for each k ∈ Z+

with Rk ∈ C1 . This implies that λk λk k λk T A = (P AP )k + Rk = eλT (A) + Rk . T eλA = k! k! k! k0

k0

k0

We claim that the latter series converges in C1 . Indeed, by properties (a) and (b) of a suitable Banach algebra, Rk C1 = P Ak P − (P AP )k C 1 k−1 = P AP A P − (P AP )k C + P AQAk−1 P C 1 1 k−1 k−1 k−1 P AP L(2 (Z+ )) P A P − (P AP ) C1 + AR A R k−1 k−1 k AR P A P − (P AP ) C + AR . 1

56


Repeating these arguments r times yields Rk C1 ArR P Ak−r P − (P AP )k−r C + rAkR 1

for each r = 1, . . . , k. In particular, Rk C1 kAkR . Thus, the series |λ|k k1

k!

kAkR =

k1

λAk |λ|k R AkR = λAR (k − 1)! k! k0

k is a convergent majorant for k0 λk! Rk whence the convergence of the latter series in C1 . This settles the proof for F0 . The proof for F1 follows from that for F0 if one takes into account that J RJ is a suitable Banach algebra whenever R is so (see the remark made in (iii) at the beginning of Section 3.2). 2 The preceding proposition implies that det T eλ1 A1 · · · eλr Ar · e−λr T (Ar ) · · · e−λ1 T (A1 ) , det T eλ1 A1 · · · eλr Ar · e−λr T (Ar ) · · · e−λ1 T (A1 ) are well-defined operator determinants which depend analytically on each of the complex parameters λk . The above determinants has to be understood as operator determinants. For basic information about operator determinants and operator traces we refer to [14]. The announced operator determinant identity reads as follows. Theorem 3.8. Let R be a suitable Banach algebra, and let A1 , . . . , Ar ∈ R. Then for each λ1 , . . . , λr ∈ C the operator determinant f (λ1 , . . . , λr ) := det T eλ1 A1 · · · eλr Ar · e−λr T (Ar ) · · · e−λ1 T (A1 ) , is equal to the operator determinant g(λ1 , . . . , λr ) := det eλ1 T (A1 ) · · · eλr T (Ar ) · T e−λr Ar · · · e−λ1 A1 . For the proof of this theorem, we will further need the following auxiliary facts. Fact 1. Let A, B be bounded linear operators on a separable Hilbert space H such that both AB and BA are trace class operators. Then AB and BA have the same trace, and I + AB and I + BA have the same determinant. Fact 2. Let U ⊆ C be a domain, and let K : U → C1 be an analytic function. Assume further that A(λ) := I + K(λ) is invertible (or, equivalently, that det A(λ) = 0) for every λ ∈ U . Then (det A(λ)) = trace A (λ)A−1 (λ) = trace A−1 (λ)A (λ) det A(λ) for every λ ∈ U .


57

Proof of Theorem 3.8. Set A := eλ1 A1 · · · eλr Ar . Due to analyticity, it suffices to prove the equality of f and g for parameters λ1 , . . . , λr which are close to zero. For these parameters, the operators 1 = J AJ = eλ1 J A1 J · · · eλr J Ar J = eλ1 A A · · · eλr Ar

and A−1 = e−λr Ar · · · e−λ1 A1 = P AP and T (A−1 ) = are close to the identity, and the corresponding Toeplitz operators T (A) −1 2 + P A P are invertible on (Z ). Hence, f and g are non-zero. The equality f (λ1 , . . . , λr ) = g(λ1 , . . . , λr ) will be proved by induction on r. For r = 0, the assertion is trivial. Assume the equality is already proved for r − 1, e.g., assume that f (λ1 , . . . , λr−1 , 0) = g(λ1 , . . . , λr−1 , 0) for all parameters sufficiently close to zero. For fixed λ1 , . . . , λr−1 , let B(λr ) := T eλ1 A1 · · · eλr Ar · e−λr T (Ar ) · · · e−λ1 T (A1 ) . By the product rule of differentiation, −λr T (A r ) · · · e−λ1 T (A1 ) B (λr ) = T eλ1 A1 · · · eλr Ar A r ·e 1 ) r )e−λr T (Ar ) · · · e−λ1 T (A − T eλ1 A1 · · · eλr Ar · T (A whence λ1 A 1 −1 · · · eλr Ar B (λr )B −1 (λr ) = T eλ1 A1 · · · eλr Ar A r T e λ1 A 1 −1 − T eλ1 A1 · · · eλr Ar T (A · · · eλr Ar r )T e A r )T (A) −1 − T (A)T (A r )T (A) −1 = T (A (Ar )T (A) −1 = H (A)H due to (34). Thus, by Fact 2, the logarithmic derivative of f with respect to λr is 1 ∂f (Ar )T (A) −1 = trace H (A)H f ∂λr −1 H (A) . = trace H (Ar )T (A)

(45)

−1 1 ∂g = −trace T −1 A−1 H (Ar )H A g ∂λr −1 −1 −1 T A . = −trace H (Ar )H A

(46)

In a similar way, one gets

58


From (35) it further follows that −1 + H (A)T A−1 0 = T (A)H A and, consequently, −1 −1 −1 T (A). 0=H A A + T −1 (A)H In connection with Fact 1, this implies the equality of (45) and (46). Thus, the ratio of the functions f and g does not depend on λr . Since we already know that f (λ1 , . . . , λr ) = g(λ1 , . . . , λr ) for λr = 0, it follows that f (λ1 , . . . , λr ) = g(λ1 , . . . , λr ) for all complex parameters λ1 , . . . , λr . 2 4. Banach algebras associated with suitable Banach algebras 4.1. Banach algebras of operators on 2 (Z+ ) With every rigid and suitable Banach subalgebra R of L(2 (Z)), we associate a Banach algebra O(R) of operators on 2 (Z+ ). Theorem 4.1. Let R be a rigid and suitable Banach subalgebra of L(2 (Z)). The set O(R) := T (A) + K: A ∈ R, K ∈ C1 2 Z+ is a unital Banach algebra with norm T (A) + K

O (R )

:= AR + KC1 .

Proof. The norm is correctly defined since T (A) + K = 0 implies A = 0 and K = 0 due to rigidity. Evidently, O(R) is a linear space. We are going to check that Q(R) is closed under multiplication. Let A, B ∈ R and K, L ∈ C1 . Then using (34), T (A) + K T (B) + L = T (A)T (B) + T (A)L + KT (B) + KL . = T (AB) + T (A)L + KT (B) + KL − H (A)H (B) = P AQBP is This operator is in O(R) because the suitability of R implies that H (A)H (B) trace class. Moreover, the norm can be estimated as follows: T (A) + K T (B) + L O (R ) = ABR + T (A)L + KT (B) + KL − P AQBP C

1

2AR BR + AL(2 (Z)) LC1 + BL(2 (Z)) KC1 + KC1 LC1 2AR BR + AR LC1 + BR KC1 + KC1 LC1 2 AR + KC1 BR + LC1 = 2T (A) + K O(R) T (B) + LO(R) .


59

Finally, the completeness of O(R) follows from that of R and C1 (2 (Z+ )) due to the definition of the norm on O(R). 2 For later purposes, let us make the following observation. Recall that given a Banach alge as the set of all operators A with A ∈ R with the norm A = AR . bra R we have defined R R The definition of rigidity and suitability guarantees that R is suitable and rigid whenever so is R. as well. Clearly, Hence we are able to consider O(R) = T (A) + K: A ∈ R, K ∈ C1 2 Z+ O(R)

(47)

with the norm T (A) + K

) O (R

= AR + KC1 .

will be used later on. Both O(R) and O(R) 4.2. Banach algebras of sequences In this section, we associate with every rigid, suitable, and shift-invariant Banach algebra R, and with each distinguished sequence h, a Banach algebras Sh (R) of sequences that arise from the finite sections of operators in O(R). In the following theorem, we will need the reflection operators Wn : 2 Z+ → 2 Z+ ,

(x0 , x1 , . . .) → (xn−1 , xn−2 , . . . , x0 , 0, 0, . . .).

Theorem 4.2. Let R be a rigid, suitable, and shift-invariant Banach subalgebra of L(2 (Z+ )), and let h ∈ H be distinguished for R. Then the set Sh (R) consisting of all sequences (An )∞ n=1 of operators An : im Ph(n) → im Ph(n) of the form An = Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Wh(n) LWh(n) + Gn

(48)

with A ∈ Rh , K, L ∈ C1 (2 (Z+ )), Gn ∈ C1 (im Ph(n) ) and Gn C1 → 0 forms a unital Banach algebra with respect to the operations (An ) + (Bn ) := (An + Bn ),

(An )(Bn ) := (An Bn ),

λ(An ) := (λAn )

and the norm given by (An )n1

S h (R )

:= AR + KC1 + LC1 + sup Gn C1 .

(49)

n1

Moreover, the set Jh (R) of all sequences (Jn ) of the form Jn = Ph(n) KPh(n) + Wh(n) LWh(n) + Gn with K, L ∈ C1 (2 (Z+ )), Gn ∈ C1 (im Ph(n) ) and Gn C1 → 0 forms a closed two-sided ideal of Sh (R).

60


Proof. First we show that the norm is correctly defined. Assume that Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Wh(n) LWh(n) + Gn = 0 for all n ∈ Z+ . Taking the strong limit n → ∞ yields T (A) + K = 0. Because of rigidity, A = 0 and K = 0. Hence Wh(n) LWh(n) + Gn = 0 2 for all n ∈ Z+ . Now multiply from both sides by Wh(n) , use Wh(n) = Ph(n) , and take the strong limit again in order to obtain L = 0 and finally Gn = 0. It is evident that Sh (R) is a linear space which is complete with respect to the defined norm. In order to check that it is an algebra consider

An = Ph(n) T (A)Ph(n) + Ph(n) K1 Ph(n) + Wh(n) L1 Wh(n) + G(1) n , Bn = Ph(n) T (B)Ph(n) + Ph(n) K2 Ph(n) + Wh(n) L2 Wh(n) + G(2) n . We have to multiply each term in the first sum with each term in the second sum, and to show that the product is in Sh (R) again and moreover that the norm can be estimated appropriately. This is (1) (2) evident if one of the factors is Gn or Gn . The other cases are considered in what follows. (For sake of brevity, we will omit some cases which are completely analogous to the ones considered.) First of all, Ph(n) K1 Ph(n) · Ph(n) K2 Ph(n) = Ph(n) K1 K2 Ph(n) − Ph(n) K1 Qh(n) K2 Ph(n) , Wh(n) L1 Wh(n) · Wh(n) L2 Wh(n) = Wh(n) L1 L2 Wh(n) − Wh(n) L1 Qh(n) L2 Wh(n) , Ph(n) K1 Ph(n) · Wh(n) L2 Wh(n) = Ph(n) K1 Wh(n) L2 Wh(n) . Since Qh(n) := P − Ph(n) converges strongly to zero on 2 (Z+ ) and Wh(n) converges weakly to zero, the last terms in the first two equations as well as the term in the third equation converge to zero in the trace norm. Quite similarly, Ph(n) T (A)Ph(n) · Ph(n) K2 Ph(n) = Ph(n) T (A)K2 Ph(n) − Ph(n) T (A)Qh(n) K2 Ph(n) with the last term converging to zero. The following two cases are slightly more complicated. Therein we are going to use the assumption that h is distinguished and that R is shift-invariant in order to establish that for A ∈ R we have U−h(n) AUh(n) ∈ R and U−h(n) AUh(n) = A + Cn

(50)

where Cn → 0 in the norm of R. In particular, Cn → 0 in the operator norm. We are also going to use the identities Wh(n) P = Ph(n) P J U−h(n) ,

P Wh(n) = Uh(n) J P Ph(n) .

(51)


61

First we consider the product Ph(n) T (A)Ph(n) · (Wh(n) L2 Wh(n) ) = Wh(n) (Wh(n) P AP Wh(n) )L2 Wh(n) = Wh(n) Ph(n) P J (U−h(n) AUh(n) )J P Ph(n) L2 Wh(n) = Wh(n) (P J AJ P L2 )Wh(n) + G n 2 Wh(n) + G n = Wh(n) T (A)L 2 ∈ C1 (2 (Z+ )) and with T (A)L G n = −Wh(n) P J U−h(n) AUh(n) J P Qh(n) L2 Wh(n) + Wh(n) P J (U−h(n) AUh(n) − A)J P L2 Wh(n) . The sequence Gn tends to zero in the trace norm because Qh(n) → 0 strongly, U−h(n) AUh(n) → A in the operator norm, and L2 is trace class. A careful examination, using in particular U−h(n) AUh(n) R = AR , shows that we can estimate G 3AR L2 C . n C 1 1

Let us finally consider Ph(n) T (A)Ph(n) · Ph(n) T (B)Ph(n) h(n) − Ph(n) T (A)Qh(n) T (B)Ph(n) . = Ph(n) T (AB)Ph(n) − Ph(n) H (A)H (B)P is trace class because of the assumption that R is Here we used (34). The term H (A)H (B) suitable. Using (51) and Qh(n) = Uh(n) P U−h(n) , the last term in the sum can be rewritten as Wh(n) Wh(n) P AP Qh(n) P BP Wh(n) Wh(n) = Wh(n) J QU−h(n) AUh(n) P U−h(n) BUh(n) QJ Wh(n) .

(52)

Now we employ the observation made in connection with (50) and substitute U−h(n) AUh(n) = (1) (2) (i) A + Cn and U−h(n) BUh(n) = B + Cn , where all terms belong to R and the Cn converge to zero in the norm of R. Hence (52) equals (1) Wh(n) H A + Cn H B + Cn(2) Wh(n) , (B)Wh(n) plus three terms whose trace norm can be estimated (because which is Wh(n) H (A)H of property (ii) of suitability) by a constant times AR · Cn(2) R + Cn(1) R · BR + Cn(1) R · Cn(2) R . (B) is trace class. In summaThese three terms converge to zero in the trace norm, and H (A)H rizing the previous steps we obtain

62


h(n) Ph(n) T (A)Ph(n) · Ph(n) T (B)Ph(n) = Ph(n) T (AB)Ph(n) − Ph(n) H (A)H (B)P (B)Wh(n) + G

n − Wh(n) H (A)H

(53)

with G

n C1 → 0. In conclusion, the product of the above defined sequences (An ) and (Bn ) can be written as An Bn = Ph(n) T (AB)Ph(n) + Ph(n) KPh(n) + Wh(n) LWh(n) + Gn

(54)

K = K1 K2 + T (A)K2 + K1 T (B) − H (A)H (B),

(55)

2 + L1 T (B) − H (A)H (B) L = L1 L2 + T (A)L

(56)

with

and where Gn is a sequence of trace class operators tending to zero in the trace norm. Elaborating on the precise expression for Gn one can actually show that (An )(Bn )

S h (R )

M · (An )S (R) (Bn )S (R) h h

with some constant M. This concludes the proof that Sh (R) is a Banach algebra. The fact that Jh (R) is closed follows immediately from the definition of the norm and that it is an ideal follows from the formulas (54)–(56). 2 h defined Theorem 4.3. Under the assumptions of the previous theorem, the mappings Wh and W by Wh : (An )n1 ∈ Sh (R) → T (A) + K ∈ O(R),

(57)

h : (An )n1 ∈ Sh (R) → T (A) + L ∈ O(R), W

(58)

where the sequences (An ) of the form (48), are unital Banach algebra homomorphisms. Proof. In view of the definition of the norms by (49) and T (A) + K

O (R )

= AR + KC1 ,

T (B) + L

) O (R

= BR + LC1 ,

the continuity of the mappings is obvious. The linearity is also clear. Their multiplicativity follows from formulas (54)–(56) in connection with (34). 2 5. The Strong Szegö–Widom Limit Theorem Before stating our main result, the generalization of the Strong Szegö–Widom Limit Theorem, we need an auxiliary fact, namely, a generalization of Proposition 9.2 from [12]. It is in some way the most important part of the Banach algebra approach, where its main idea is exhibited.


63

Proposition 5.1. Let R be a rigid, suitable, and shift-invariant Banach algebra, let h ∈ H, and let A1 , . . . , Ar ∈ R. Then the sequence (Bn )n1 defined by Bn := Ph(n) T eA1 · · · eAr Ph(n) · e−Ph(n) T (Ar )Ph(n) · · · e−Ph(n) T (A1 )Ph(n)

(59)

belongs to Sh (R). Moreover, there exist operators K, L ∈ C1 (2 (Z+ )) and Gn ∈ C1 (im Ph(n) ) with Gn C1 → 0 such that Bn = Ph(n) + Ph(n) KPh(n) + Wh(n) LWh(n) + Gn .

(60)

The operators K and L are determined by P + K = T eA1 · · · eAr · e−T (Ar ) · · · e−T (A1 ) , P + L = T eA1 · · · eAr · e−T (Ar ) · · · e−T (A1 ) . Proof. We first observe the general fact that if (An ) ∈ Sh (R), then (eAn ) = e(An ) . From this it follows that the sequence (Bn ) belongs to Sh (R) and that (Bn )n1 = Ph(n) T eA1 · · · eAr Ph(n) n1 · e−(Ph(n) T (Ar )Ph(n) )n1 · · · e−(Ph(n) T (A1 )Ph(n) )n1 . We can write this sequence as (Bn )n1 = Λ eA1 · · · eAr · e−Λ(Ar ) · · · e−Λ(A1 ) ,

(61)

where Λ is the mapping R → Sh (R),

A → Ph(n) T (A)Ph(n) n1 .

Evidently, Λ is linear and bounded. Let Φ denote the canonical homomorphism Φ : Sh (R) → Sh (R)/Jh (R),

(An ) → (An ) + Jh (R).

It follows from (53) that the mapping Φ ◦ Λ : R → Sh (R)/Jh (R) is a continuous homomorphism between Banach algebras. Applying Φ to both sides of (61) yields Φ (Bn )n1 = (Φ ◦ Λ) eA1 · · · eAr · e−(Φ◦Λ)(Ar ) · · · e−(Φ◦Λ)(A1 ) = (Φ ◦ Λ) eA1 · · · eAr e−Ar · · · e−A1 = (Φ ◦ Λ)(I ) = (Ph(n) )n1 + Jh (R).

64


Hence, (Bn − Ph(n) )n1 is in Jh , which proves (60). The representations of I + K and I + L h (which were defined in Theorem 4.3) to follow by applying the homomorphisms Wh and W both sides of (Bn )n1 = (Ph(n) + Ph(n) KPh(n) + Wh(n) LWh(n) + Gn )n1 , respectively.

2

A consequence of the previous proposition is the theorem that we will state next. It is the first part of our main result. Therein the asymptotics of determinants are reduced to the asymptotics of traces. The asymptotics of the traces is left unevaluated, which allows us to stated the result in greater generality. For any Banach algebra R we denote by G1 R the connected component of the group of all invertible elements in R which contains the unit element. It is a well-known result of Lorch – see, e.g., Theorem 3.3.7 in [1], or [20] – that G1 R is the set of all finite products of exponentials of elements in R. Hence the basic assumption on the symbol A is that it belongs to G1 R. This corresponds precisely to the basic assumption in the classical Szegö–Widom Theorem (see Theorem 1.1 and the remarks afterwards). There is, however, a subtlety in the following statement. The operators A1 , . . . , Ar are not uniquely determined by A, and they do appear in the limit expression. Theorem 5.2. Let R be a rigid, suitable, and shift-invariant Banach algebra, and let h ∈ H be a distinguished sequence for R. If A1 , . . . , Ar ∈ R, and A = eA1 · · · eAr , then det(Ph(n) T (A)Ph(n) ) = det T (A)T A−1 . n→∞ exp(trace(Ph(n) T (A1 + · · · + Ar )Ph(n) )) lim

Proof. Consider the sequence (Bn ) defined by (59). Take the determinant and observe that det Bn = det Ph(n) T (A)Ph(n) · e−Ph(n) T (Ar )Ph(n) · · · e−Ph(n) T (A1 )Ph(n) = det Ph(n) T (A)Ph(n) · e−trace(Ph(n) T (Ar )Ph(n) ) · · · e−trace(Ph(n) T (A1 )Ph(n) ) = det Ph(n) T (A)Ph(n) · e−trace(Ph(n) T (Ar +···+A1 )Ph(n) ) . By Proposition 5.1, the sequence Bn is of the form (60). A basic observation (see, e.g., Lemmas 9.1 and 9.3 of [12]) implies lim det Bn = det(P + K) · det(P + L).

n→∞

(62)

Again from Proposition 5.1 we infer that the right-hand side is equal to the product of two operator determinants, det T (A) · e−T (A1 ) · · · e−T (Ar )


65

and 1 ) · e−T (A · · · e−T (Ar ) . det T (A)

The last determinant can be rewritten, using Theorem 3.8, as det T A−1 · eT (Ar ) · · · eT (A1 ) . Thus both determinants multiplied together yield the constant det T (A)T (A−1 ). Remark that −1 ) with the product of the generalized Hankel operators being T (A)T (A−1 ) = I − H (A)H (A trace class. 2 The second part of our main result is the following. Recall that M(a) stands for the mean of an almost periodic function, and D(B) stands for the main diagonal of an operator B ∈ L(2 (Z)). Theorem 5.3 (Generalized Strong Szegö–Widom Limit Theorem). Let β be an admissible and compatible weight on an additive subgroup Ξ of R/Z. Let R = Wα1 ,α2 (APW(Z, Ξ, β)) with α1 , α2 > 0 and α1 + α2 = 1. Suppose that h ∈ H is a distinguished sequence for Ξ . If A1 , . . . , Ar ∈ R, and A = eA1 · · · eAr ,

(63)

then lim

n→∞

det(Ph(n) T (A)Ph(n) ) = det T (A)T A−1 , h(n) G

(64)

with G := exp M(a),

a := D(A1 + · · · + Ar ).

(65)

Proof. Let us first remark that due to Theorem 3.5 the Banach algebra is rigid, suitable, and shiftinvariant. Moreover, h is distinguished for R (see Propositions 2.1 and 3.6, and Theorem 2.6). Hence Theorem 5.2 can be applied. We are left with the asymptotics of h(n)−1 a(k) trace Ph(n) T (A1 + · · · + Ar )Ph(n) = k=0

with a ∈ APW(Z, Ξ, β) being defined as above. Now it remains to apply Theorem 2.7.

2

We conclude this section with a couple of observations and questions. Ambiguity of the constant G. The constant G is defined only implicitly in terms of A. More precisely, different choices for A1 , . . . , Ar could yield the same A, but different constants G. As the following example shows this can happen in the periodic case. We conjecture that this can always happen in the almost periodic case.

66


For the periodic case (with period N = 2) we can give the following simple example (r = 1). Choose A = eA1 = eA2 = I with A1 = 0, A2 = aI , 2πi n even, a(n) = 0 n odd, i.e., a = πie0 + πie1/2 . The corresponding constants G1 = 1 and G2 = −1 are different. On the other hand, since the constant G describes an asymptotics, it cannot be completely arbitrary. The modification which seem to be “admissible” is replacing G by e2πiξ · G for arbitrary ξ ∈ Ξ . In this connection recall that h being distinguished implies that e2πiξ h(n) → 1 so that in fact there is no change in the asymptotic description (64). Problem of inverse closedness. It seems practically very difficult to decide whether an operator belongs to G1 R with R = Wα1 ,α2 (APW(Z, Ξ, β)). The question arises whether (under reasonable assumptions) the Banach algebra R is inverse closed in L(2 (Z)), i.e., GR = R ∩ G L 2 (Z) . If this would be true, then it is conceivable that G1 R = R ∩ G1 L 2 (Z) . One can then replace the assumption by a somewhat more tractable one. For instance, if A ∈ R is self-adjoint and its spectrum sp(A) (in L(2 (Z+ ))) is known, then the theorem applies to A − λI whenever λ ∈ / sp(A). Of course, the almost Mathieu operator, which serves as the main example, is self-adjoint. Let us remark that a necessary condition for the inverse closedness of R in L(2 (Z)) is the inverse closedness of APW(Z, Ξ, β) in AP(Z) (hence in ∞ (Z)). In the case of finitely generated groups Ξ , this seems to be the case if the weight β does not grow exponentially. 6. Counterexamples to the asymptotics of the traces The goal of this section is to show that the asymptotics h(n)−1

a(k) = h(n) · M(a) + o(1),

n → ∞,

(66)

k=0

does not hold in general, i.e., for general a ∈ AP(Z) and h : Z+ → Z+ assumed to be distinguished (say, for the group generated by the Fourier spectrum of a). As has been pointed out in Section 2, the asymptotics (66) does hold if the Fourier spectrum of a is finite, i.e., if the sum (16) is finite. It also holds “generically” if the Fourier spectrum of a belongs, for instance, to a singly-generated group Ξξ := (ξ Z)/Z, assuming that the Fourier coefficients decay sufficiently fast (Theorems 2.7 and 2.8). More precisely, for almost every ξ ∈ R \ Q, there exists ω > 0 such that the asymptotics (66) holds for every a=

∞ k=−∞

ak ekξ

satisfying

∞ k=−∞

ω |ak | 1 + |k| < ∞.

(67)


67

In contrast to that, we are going to show in what follows that there exists ξ ∈ R \ Q and a distinguished sequence for Ξξ such that the asymptotics (66) does not hold for certain a of the form (67). The sequences a we are going to consider have exponentially decaying Fourier coefficients. Moreover, we assume ak = 0 for k 0, i.e., a=

∞

ak ekξ .

k=1

Notice that M(a) = a0 = 0. From the definition of the mean it follows that the error term in (66) is always o(h(n)). Our results suggest that not much improvement is possible in general. We start with constructing irrational numbers ξ which will provide the basis for our counterexamples. They depend on a parameter c > 1, which we can later relate to the exponential decay of the Fourier coefficients ak . In what follows we denote by {x} the fractional part of a real number x. Proposition 6.1. Given c > 1, there exist ξ ∈ R and strictly increasing sequences {qd }∞ d=1 and of natural numbers such that {wd }∞ d=1 ∞ qd+1 1 = 1, ξ = , d→∞ cwd wd d=1 1 1 1+O , d → ∞, {wd ξ } = qd+1 qd+1

wd = q 1 · · · q d ,

lim

(68)

(69)

and kξ R/Z

1 1 , −k·O wd wd+1

(70)

whenever wd k. Proof. Starting with any q1 and p1 = 1 we define recursively wd = q 1 · · · q d ,

qd+1 = cwd + sd ,

pd+1 = pd qd+1 + 1,

where [x] denotes the integer part of x ∈ R and sd ∈ Z+ is chosen such that (i) wd and pd are co-prime, (ii) wd < sd 2wd for each d ∈ N. Clearly, (ii) will guarantee that qd and wd are strictly increasing and that qd+1 ∼ cwd as d → ∞. To show that such a choice of sd is possible we will argue by induction. The case d = 1 is obvious. Assume (i) holds for d. Then, as we will see shortly, there exists sd satisfying (ii) such that (i) holds for d + 1. Indeed, when trying to find qd+1 , let us require pd qd+1 ≡ −1 modulo each prime factor of wd . Because gcd(wd , pd ) = 1 these conditions are equivalent to a

68


system of congruences for qd+1 (or, equivalently, for sd ) modulo each prime factor of wd . By the Chinese Remainder Theorem such a system of congruences has a unique solution modulo the product of the prime factors, and thus we can find the desired sd and qd+1 . It follows that pd+1 = pd qd+1 + 1 ≡ 0 modulo each prime factor of wd , which implies that gcd(pd+1 , wd ) = 1. Clearly, also gcd(pd+1 , qd+1 ) = 1. Thus we obtain condition (i) for d + 1. Let us prove (69) and (70). The recursion for pd can be rewritten as pd+1 pd 1 = + . wd+1 wd wd+1 Hence d 1 pd = , wj wd

ξ=

j =1

∞ 1 pd + . wd wj j =d+1

Multiplying ξ with wd equals pd plus the error term ∞ j =d+1

wd 1 1 1 = + + + ···. wj +1 qd+1 qd+1 qd+2 qd+1 qd+2 qd+3

Since qd is increasing this series can be estimated by a geometric series, and we obtain (69). Now observe that kpd /wd R/Z 1/wd if k is not a multiple of wd (see also (i)). Moreover, using the same estimate as above, ∞ j =d+1

This proves (70).

1 1 . =O wj wd+1

2

The properties stated in (68) and (69) imply that −1 wd ξ R/Z = O qd+1 = O c−wd ,

d → ∞,

which means that ξ is a Liouville number (hence it is irrational). Moreover, if dn is an increasing sequence and kn is some sequence such that h(n) := kn wdn is strictly increasing and kn = o(qdn +1 )

or, equivalently, kn = o cwdn as n → ∞,

(71)

then, using property (69), we have e2πiξ h(n) → 1,

as n → ∞.

Hence h(n) defined in such a way is a distinguished sequence for Ξξ = (ξ Z)/Z.

(72)


69

∞ Proposition 6.2. Let c > 1, ξ ∈ R, {qd }∞ d=1 and {wd }d=1 , be as in the previous proposition. Let 2m b > 1 and m ∈ N be such that c b . Then

k∈N\Sm

b−k < ∞, |1 − e2πiξ k |

(73)

where Sm = {kwd : 1 k 2m, d ∈ N}. Proof. For sufficiently large d, say d d0 , we have (2m + 1)wd−1 < wd and qd+1 2γ (2m + 1)wd , where γ is the constant implied in (70). Using this last estimate and (70) we conclude that kξ R/Z

1 (2m + 1)wd 1 −γ · wd wd+1 2wd

(∗)

whenever d d0 and k ∈ N is such that wd k and k (2m + 1)wd . In regard to the sum (73) we can omit finitely many terms and decompose the set of the remaining indices into ∞ (2m + 1)wd0 −1 , . . . \ Sm = Ad ∪ Bd d=d0

with Ad = (2m + 1)wd−1 , . . . , wd − 1 , Bd = wd , . . . , (2m + 1)wd − 1 \ {kwd : 1 k 2m}. For k ∈ Ad ∪ Bd we have |1 − e2πiξ k |−1 = O(wd ); see (∗). Restricting the sum (73) to all k ∈ Ad and all d d0 yields an upper estimate of a constant times ∞

b−(2m+1)wd−1 (wd )2 C

d=d0

∞

b−(2m+1)wd−1 c2wd−1 (wd−1 )2

d=d0

C

∞

b−wd−1 (wd−1 )2 < ∞.

d=d0

Restricting the sum (73) to all k ∈ Bd and all d d0 yields an upper estimate of a constant times 2m

∞

b−wd (wd )2 < ∞.

d=d0

Thus (73) is finite. This concludes the proof.

2

70


In view of the following theorem, remark that the assumptions therein ensure that h is a distinguished sequence. Furthermore, this theorem has to be understood as follows. For given c > 1 we construct an irrational number ξ as in Proposition 6.1. Then we consider particular distinguished sequences, namely those characterized by (74). The class of such distinguished sequences h is still quite large. Independent of h, we consider almost periodic sequences whose Fourier coefficients behave roughly as b−k where b ∈ (1, c). More precisely, (75) and (76) hold. Then we are able to predict the asymptotics. If this asymptotics is different from o(1), then we have found a counterexample to (66). ∞ Theorem 6.3. Let c > 1, ξ ∈ R, {qd }∞ d=1 and {wd }d=1 be as in Proposition 6.1. Let h(n) = kn wdn be strictly increasing with dn , kn ∈ N such that dn → ∞ and

kn wdn = o(1), qdn +1

n → ∞.

(74)

Moreover, let a=

ak ekξ ,

(75)

k1

and assume 1 < b < c such that b = lim inf |ak |−1/k = lim |awdn |−1/wdn .

(76)

a(j ) = awdn wdn kn 1 + o(1) + o(1),

(77)

n→∞

k→∞

Then h(n)−1

n → ∞.

j =0

Proof. First of all observe that h(n)−1 j =0

a(j ) =

∞ k=1

ak

1 − e2πiξ kh(n) . 1 − e2πiξ k

(78)

Each term in this sum converges to zero because e2πiξ h(n) → 1 as n → ∞. Clearly, (74) implies (71), and hence h(n) is a distinguished sequence. Let 0 < ε < 1/3 be such that b1+ε < c, and choose m ∈ N such that c b(2m+1)(1−ε) . Because of (76) we have b−k(1+ε) |ak | b−k(1−ε)


71

for all sufficiently large k, say k K0 . We split the above sum (78) into a finite sum over k (which is clearly o(1) as n → ∞) and ∞ 2m

akwd

d=d0 k=1

1 − e2πiξ kwd h(n) 1 − e2πiξ kh(n) + a k 1 − e2πiξ kwd 1 − e2πiξ k k∈N\Sm kK0

with d0 sufficiently large and fixed. In fact, we choose d0 such that (a) the error term O(1/qd+1 ) in (69) is bounded by 1/2 for all d d0 , (b) qd+1 > 3m for all d d0 , (c) wd0 K0 . Using Proposition 6.2 with b1−ε instead of b, it follows that the second term converges to zero as n → ∞ because of dominated convergence. Thus we obtain ∞ 2m

akwd

d=d0 k=1

1 − e2πiξ kwd h(n) + o(1), 1 − e2πiξ kwd

n → ∞.

Now assume that n is large enough to ensure that dn d0 . Then one part of the above sum can be estimated as follows, ∞ 2m ∞ 2m 1 − e2πiξ kwd h(n) akwd h(n)|akwd | 1 − e2πiξ kwd d=dn +1 k=1

d=dn +1 k=1

∞

2mh(n)

b−wd (1−ε)

d=dn +1

2mh(n)

b−wdn +1 (1−ε) 1 − b−(1−ε)

= O wdn +1 b−wdn +1 (1−ε) = o(1),

n → ∞.

Here we have used h(n) = o(qdn +1 ) = O(wdn +1 ). Thus we are left with dn 2m d=d0 k=1

akwd

1 − e2πiξ kwd h(n) + o(1), 1 − e2πiξ kwd

n → ∞.

Henceforth we will assume d0 d dn and 1 k 2m. Because of Proposition 6.1, formulas (69) and (70), we have the estimates {ξ kwd } = and

k qd+1

1+O

1 qd+1

72


1 wd kkn 1+O ξ kwd h(n) = {ξ kwd kn wdn } = qdn +1 qdn +1 with the error term bounded by 1/2 because of (a). For the first estimate, we have used that k/qd+1 2m/qd+1 < 2/3, see (b). For the second estimate we have used that wd kkn /qdn +1 2mwdn kn /qdn +1 = o(1), by (74), which is less than 2/3 for n sufficiently large. We can conclude that 1 − e2πiξ kwd h(n) qd+1 wd kn 1 qd+1 wd kn = 1+O =O . qdn +1 qd+1 qdn +1 1 − e2πiξ kwd This bound is uniform in k, d, and dn , as long as n is sufficiently large. In the special case k = 1 and d = dn , we get 1 1 − e2πiξ kwd h(n) , awd = awdn kn wdn 1 + O qdn +1 1 − e2πiξ kwd which yields the leading term. In general, we have to sum over the terms 1 − e2πiξ kwd h(n) akwd qd+1 wd kn akwd wd qd+1 akwd = awdn kn wdn O . =O qdn +1 awdn wdn qdn +1 1 − e2πiξ kwd For d = d0 , . . . , dn − 1 the error term simplifies to

akwd O awdn qdn +1

b(1+ε)wdn −(1−ε)kwd =O , qdn +1

which after summation over d = d0 , . . . , dn − 1 gives O

b(1+ε)wdn qdn +1

=O

b(1+ε)wdn . cwdn

Because we have chosen ε such that b1+ε < c, this error term is o(1). For d = dn the error term is just O

akwdn awdn

= O b(1+ε)wdn −(1−ε)kwdn ,

which is also o(1) for k 2. (Here we use ε < 1/3.)

2

To illustrate the applicability of the previous theorem we present the following corollary. Essentially, we just substitute kn by xn , where xn determines the final asymptotics. ∞ Corollary 6.4. Let c > 1, ξ ∈ R, {qd }∞ d=1 and {wd }d=1 be as in Proposition 6.1. Moreover, let

a=

k1

ak ekξ ,


73

and assume 1 lim xn dn > . b n→∞ b

(79)

Then h(n) = kn wdn with kn =

xn wdn |awdn |

is a distinguished sequence, and h(n)−1 j =0

a(j ) = xn

awdn 1 + o(1) + o(1), |awdn |

n → ∞.

(80)

Proof. Because of |awdn |−1/wdn → b, it is easily seen that kn ∼ 1/wn

and that limn→∞ kn (68) it follows that

xn wdn |awdn |

exists and lies between 1 and c. In particular, kn → ∞ as n → ∞. Using kn wdn xn xn ∼ wd . ∼ qdn +1 qdn +1 |awdn | c n |awdn |

This converges to zero because if we take the wdn -th root, the limit is less than 1. Hence the previous theorem can be applied. In particular, h is a distinguished sequence. It is now easy to conclude (80) from (77). 2 Given 1 < b < c and ξ (and hence qd and wd ), we can now choose xn and dn in different ways in order to realize several kinds of asymptotics. We have also some freedom in choosing ak and dn . We will choose, for simplicity, that ak = b−k , k 1, i.e., a=

∞

b−k ekξ ,

k=1

and dn = n. Example 6.5. Given 1 0. Obviously, lim x 1/wn = 1.

n→∞

74


Hence the corollary can be applied, and we conclude that with the sequence a as above there exists a distinguished sequence h such that h(n)−1

a(j ) = x + o(1),

n → ∞.

j =0 1/w

Example 6.6. Choose xn = 1/wn . Since limn→∞ xn n = 1, the corollary can be applied and it follows that there exists a distinguished sequence h such that h(n)−1

a(j ) = o(1),

n → ∞.

j =0

This is, of course, not a counterexample, but it shows that the “desired” asymptotics holds at least for some distinguished sequences. Example 6.7. Choose xn = bwn α/(1−α) with α such that 0 < α < 1 − log b/ log c < 1. 1/w

Clearly, limn→∞ xn n = bα/(1−α) , which lies between 1/b and c/b because of the previous assumption. A simple computation implies that h(n) ∼ bwn /(1−α) . We can conclude that h(n)−1

a(j ) = bwn α/(1−α) 1 + o(1) = h(n)α 1 + o(1) ,

n → ∞.

j =0

We see that now this term can be even unbounded. Recall the fact that in any case, the asymptotics is o(h(n)) as n → ∞. If we choose b and c properly in the beginning, then each 0 < α < 1 can occur as a possible exponent. References [1] B. Aupetit, A Primer on Spectral Theory, Springer-Verlag, New York, Berlin, Heidelberg, 1990. [2] A. Avila, S. Jitomirskaya, The Ten Martini problem, Ann. of Math. (2) 170 (1) (2009) 303–342. [3] A. Baker, Linear forms in the logarithms of algebraic numbers I, Mathematica 13 (1966) 204–216; A. Baker, Linear forms in the logarithms of algebraic numbers II, Mathematica 14 (1967) 102–107; A. Baker, Linear forms in the logarithms of algebraic numbers III, Mathematica 14 (1967) 220–228. [4] A. Baker, Transcendental Number Theory, Cambridge University Press, London, New York, 1975. [5] V.I. Bernik, M.M. Dodsen, Metric Diophantine Approximation on Manifolds, University Press, Cambridge, 1999. [6] F.P. Boca, Rotation C ∗ -Algebras and Almost Mathieu Operators, Theta Ser. Adv. Math., vol. 1, The Theta Foundation, Bucharest, 2001. [7] A. Böttcher, B. Silbermann, Invertibility and Asymptotics of Toeplitz Matrices, Akademie-Verlag, Berlin, 1983. [8] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1989; Springer-Verlag, Berlin, Heidelberg, New York, 1990. [9] A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, Berlin, Heidelberg, 1999. [10] J.W.S. Cassels, An Introduction to Diophantine Approximation, University Press, Cambridge, 1957. [11] C. Corduneanu, Almost Periodic Functions, Interscience Publishers, John Wiley & Sons, New York, 1961.


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[12] T. Ehrhardt, A new algebraic approach to the Szegö–Widom limit theorem, Acta Math. Hungar. 99 (3) (2003) 233–261. [13] N.I. Feldman, An improvement of the estimate of a linear form in the logarithms of algebraic numbers, Mat. Sb. (N.S.) 77 (119) (1968) 423–436 (in Russian). [14] I. Gohberg, S. Goldberg, N. Krupnik, Traces and Determinants of Linear Operators, Oper. Theory Adv. Appl., vol. 116, Birkhäuser, Basel, 2000. [15] M.G. Krein, On some new Banach algebras and theorems of Wiener–Levy type for Fourier series and integrals, Mat. Issled. 1 (1) (1966) 82–109 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Ser. 2 93 (1970) 177–199. [16] J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys. 244 (2004) 297–309. [17] S. Roch, Finite sections of band-dominated operators, Mem. Amer. Math. Soc. 191 (895) (2008). [18] S. Roch, B. Silbermann, Szegö limit theorems for operators with almost periodic diagonals, Oper. Matrices 1 (1) (2007) 1–29. [19] K.F. Roth, Rational approximations to algebraic numbers, Mathematica 2 (1955) 1–20. [20] W. Rudin, Functional Analysis, McGraw–Hill, Inc., New York, 1991. [21] W.M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math. 125 (1970) 189–201. [22] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory, Amer. Math. Soc. Colloq. Publ., vol. 54, Amer. Math. Soc., Providence, RI, 2005. [23] G. Szegö, Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion, Math. Ann. 76 (1915) 490–503. [24] G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function, in: Festschrift Marcel Riesz, Lund, 1952, pp. 222–238. [25] M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups, Grundlehren Math. Wiss., vol. 326, Springer-Verlag, Berlin, 2000. [26] H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants II, Adv. Math. 21 (1976) 1–29.


From U -bounds to isoperimetry with applications to H-type groups ✩ J. Inglis a,∗ , V. Kontis a , B. Zegarliński b,1 a Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK b CNRS, Toulouse, France

Received 6 January 2010; accepted 5 August 2010 Available online 6 September 2010 Communicated by L. Gross

Abstract In this paper we study U -bounds in relation to L1 -type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space (RN , d). We prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework. The usefulness of our approach is illustrated by an application to the setting of H-type groups, and an extension to infinite dimensions. © 2010 Elsevier Inc. All rights reserved. Keywords: Logarithmic Sobolev inequalities; Isoperimetric inequalities; H-type groups; Infinite dimensional applications

1. Introduction An effective technology to study coercive inequalities involving (sub-)gradients and a variety of probability measures on metric measure spaces was recently introduced in [23]. This approach was based on so-called U -bounds, that is estimates of the following form ✩

Supported by EPSRC grant EP/D05379X/1.


E-mail addresses: [email protected] (J. Inglis), [email protected] (V. Kontis), [email protected] (B. Zegarliński). 1 On leave of absence from Imperial College London. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.003

J. Inglis et al. / Journal of Functional Analysis 260 (2011) 76–116

|f | U (d) dμ Cq q

γq

77

|∇f | dμ + Dq q

|f |q dμ.

Here q ∈ [1, ∞), d is a metric associated to the (sub-)gradient ∇, γq , Cq , Dq ∈ (0, ∞) are constants independent of the function f , and dμ ≡ e−U (d) dλ is a probability measure, where U (d) is a function that is bounded from below and has suitable growth at infinity, and dλ is a natural underlying measure. While the consequences of the bounds corresponding to q > 1 were extensively explored there, the limiting case was left open. In this paper we show that there is a natural direct way from U -bounds with q = 1 to isoperimetric information. In fact we show an essential equivalence of such a bound with an L1 Φ-entropy inequality EntΦ μ (f ) cμ|∇f | where EntΦ μ (f ) ≡ μΦ(f ) − Φ(μf ) is defined with a suitable Orlicz function Φ, as well as the equivalence with an isoperimetric inequality with a suitable profile function. We first recall an interesting result of [26] showing that in case of the Gaussian measures on Euclidean spaces, the functions f such that μ|f | < ∞ 1 belong to the Orlicz space defined by a function Φ(s) = s(log(1 + s)) 2 . Also, on the level of isoperimetry for probability measures, we would like to recall a comprehensive characterisation of isoperimetric profiles for measures on the real line obtained in [10] (see also [5,12,14,29] and references therein) as well as the isoperimetric functional inequalities studied in [8] (see also [2,3,12,33]). These results provided additional motivation to our work. In particular, in [12] the authors conjecture that for super-Gaussian distributions one should expect an analogue of the isoperimetric functional inequality (IFI 2 ) introduced in [8], with a suitable non-Gaussian isoperimetric function and a different than Euclidean length of the gradient. In [2] (an alternative to [27]) the authors gave a proof of the p = 1 (sub-)gradient bound |∇Pt f |p Cp (t)Pt |∇f |p for the heat kernel on the Heisenberg group, and as a consequence obtained an (IFI 2 ) inequality in this case. We mention that, for p > 1, gradient bounds were earlier established in [16], while the logarithmic Sobolev inequality for heat kernels on Heisenberg-type groups was established in [23]. The other interesting question is what are the optimal equivalent conditions, on the one side characterising the properties of the semigroup for which the form associated to the generator is given by the square of a fixed sub-gradient, and on the other side characterising the isoperimetric properties (e.g. in the form of some isoperimetric functional inequality with a given length of the sub-gradient). In the particular situation when p = 1 gradient bounds are known, and an equivalence relation (between (IFI 2 ) and the logarithmic Sobolev inequality) was established in [19]. It seems that we are still away from fully understanding the peculiarity of this situation and in particular answering the question what kind of additional conditions are necessary to establish equivalence between conditions of different orders in the length of the gradient (as well as finding a more direct proof of this equivalence without going through the semigroup route).

78


From the point of view of applications to an infinite dimensional probabilistic set-up involving an infinite product of non-compact Lie groups, it is important that we are dealing with inequalities satisfying the tensorisation property. Then one can attack the interesting question of for which non-product measures one can prove similar properties. This question, when the underlying space is as we wish, appears to have some new challenging features and so far, besides the results of [24] where logarithmic Sobolev inequalities (LSq ), q > 1, are shown for some classes of measures, not much is known. Therefore in the present paper we also contribute to this topic by proving tight L1 Φ-entropy inequalities for suitable infinite dimensional Gibbs measures. The organisation of our paper is as follows. In Section 2, starting from U -bounds, we prove the L1 Φ-entropy inequality via a route involving “dressing up” the classical Sobolev inequality and a tightening procedure using a generalised Rothaus type lemma of [28], extended relative entropy bounds of [20], and the following Cheeger type inequality μ|f − μf | c0 μ|∇f |. In fact, this type of the Cheeger inequality is shown (in Theorem 2.7) to be a simple consequence of a similar inequality in balls together with U -bounds, provided the function U grows to infinity with the size of the ball. In Section 3 we discuss some applications to isoperimetric and functional isoperimetric inequalities. Section 4 contains some consequences of the L1 Φ-entropy inequality. In particular this includes the (LSq ) inequality and U -bounds. In Theorem 4.5 we summarise all interrelations between the properties discussed before. Section 5 is devoted to applications of the theory developed in the previous sections to the important class of H-type groups, where one can check the U -bounds for probability measures with density (essentially) dependent on the Carnot–Carathéodory distance. The interesting outcome, which comes out naturally within the presented approach, includes a proof of the p = 1 sub-gradient bounds for heat kernels on H-type groups which could potentially be extended to more complicated non-compact groups. Finally in Section 6 we prove the L1 Φ-entropy inequalities for non-product probability measures on an infinite product of H-type groups, which allows us in particular to obtain some new isoperimetric information. Additionally we prove here the (IFI 2 ) inequality in such a setup; in fact even in the case of the full gradient setup, this provides an interesting extension of results in [33] allowing us to include the important case of unbounded interactions. 2. L1 Φ-entropy inequalities from U -bounds Throughout this paper we will be working in RN equipped with a metric d : RN × RN → [0, ∞) and Lebesgue measure dλ. For r 0, we will set B(r) := x: d(x) r , where d(x) := d(x, 0). We will also let ∇ be a general sub-gradient in RN i.e. ∇ is a finite collection {X1 , . . . , Xm } of possibly non-commuting fields. Assume that the divergence of each of these fields with respect m 2 2 12 to the Lebesgue measure λ on RN is zero. Set := m i=1 Xi and |∇f | = ( i=1 (Xi f ) ) .


79

Theorem 2.1. Let U be a locally Lipschitz function on RN , which is bounded from below and −U is such that Z = e−U dλ < ∞. Let dμ = e Z dλ, so that μ is a probability measure on RN . Suppose that the following classical Sobolev inequality is satisfied

|f |

1+ε

dλ

1 1+ε

a

|∇f |dλ + b

|f | dλ

(1)

for some constants a, b ∈ [0, ∞), ε > 0 and all locally Lipschitz functions f , and moreover that for some A, B ∈ [0, ∞) we have

μ |f | |U |β + |∇U | Aμ|∇f | + Bμ|f |

(2)

for some β ∈ (0, 1] and all such f . Then there exist constants C, D ∈ [0, ∞) such that |f | β Cμ|∇f | + Dμ|f | μ |f |log μ|f |

(3)

for all locally Lipschitz functions f . Remark 2.2. Inequality (1) should be interpreted as a condition on the gradient ∇. Proof. Without loss of generality, we may suppose that f 0 and U 0. Indeed, otherwise we may apply (3) to the positive and negative parts of f separately. Moreover, if U −K, with K 0, we have that U + K 0 and then we can replace f by f e−K in (3). First note that

f β μf β f β = μ f log + μ f log μ f log 1 {f μf } + μf μf f

f β μ f log+ + e−β β β μ(f ), μf where we have used that supx∈(0,1) x(log x1 )β = e−β β β with x = that

f μf

. Thus it suffices to prove

f β Cμ|∇f | + Dμ(f ), μ f log+ μf

(4)

with some constants C, D ∈ (0, ∞) independent of f . Suppose that μ(f ) = 1. With F ≡ f e−U and ε ∈ (0, 1) sufficiently small, we have

β F log+ (F ) dλ =

{F 1}

F

1 log F ε ε

β dλ.

Now, by Jensen’s inequality (since, for β ∈ (0, 1], the function (log x)β is concave on x 1)

(5)

80


ε β 1 {F 1} F dλ log F F dλ = ε εβ

{F 1}

=

{F 1} F dλ

{F 1}

{F 1} F dλ εβ

(1 + ε)β

(1 + ε)β

F

{F 1} log

F 1+ε dλ

{F 1} F

{F 1} F εβ

{F 1} F εβ

dλ

dλ

β log F ε dλ

β

dλ

1 ( {F 1} F 1+ε dλ) 1+ε β log 1 ( {F 1} F dλ) 1+ε

1

( {F 1} F 1+ε dλ) 1+ε log +1 , 1 ( {F 1} F dλ) 1+ε

using the simple fact that x β x + 1 for all x 0. Thus, since log x x − 1 for all x 0, 1

(1 + ε)β {F 1} F dλ ( {F 1} F 1+ε dλ) 1+ε ε β 1 . log F F dλ 1 ε εβ ( {F 1} F dλ) 1+ε

{F 1}

Since we have assumed that μ(f ) = 1, we have

f e−U dλ/Z ≡

F dλ/Z 1,

{F 1}

{f e−U 1}

so that 1

1 ( {F 1} F dλ) 1+ε

ε

Z 1+ε . {F 1} F dλ

Thus

F {F 1}

ε

1 1+ε β 1 (1 + ε)β Z 1+ε 1+ε log F ε dλ dλ F β ε ε ε (1 + ε)β Z 1+ε a ∇(F ) dλ + bZ , εβ

provided ε > 0 is chosen sufficiently small so that in the last step we can apply the classical Sobolev inequality (1). Dividing both sides by the normalisation factor Z and recalling F ≡ f e−U , this implies

β

f log+ f e−U dμ c1 μ|∇f | + c2 μ f |∇U | + c3 ,

(6)

J. Inglis et al. / Journal of Functional Analysis 260 (2011) 76–116 ε

81 ε

with dμ ≡ Z1 e−U dλ and c1 = c2 = (1 + ε)β aZ 1+ε /ε β , c3 = (1 + ε)β bZ 1+ε /ε β . Now consider the left-hand side of (6). Since β ∈ (0, 1], (x − y)β x β − y β for x y 0. Applying this with x = log f and y = U 0, we have −U β dμ = f (log f − U )β dμ f log+ f e {f eU }

f (log f )β dμ −

{f eU }

= μ f [log+ f ]β − −

f U β dμ

{f eU }

f (log f )β dμ

{1f eU }

f U β dμ

{f eU }

μ f [log+ f ]β −

f U β dμ.

{1f }

Combining this with (6) we see that

μ f [log+ f ]β c1 μ|∇f | + c2 μ f |∇U | + c3 +

f U β dμ

{1f }

c1 μ|∇f | + max{c2 , 1}μ f U β + |∇U | + c3

c1 + max{c2 , 1}A μ|∇f | + c3 + max{c2 , 1}B, where we have used (2) in the last step. Finally, for general f 0, we apply the above inequality to f/μ(f ) to arrive at (4). 2 As a corollary, we can also state the following perturbation result. Corollary 2.3. Let U and μ be as in Theorem 2.1, and suppose conditions (1) and (2) are satisfied. Let W be a locally Lipschitz function such that e−W dμ < ∞ and

|∇W | δ |U |β + |∇U | + C(δ),

|W |β a0 |U |β + |∇U | + a1

(7)

almost everywhere, with some 0 < δ < A1 and C(δ), a0 , a1 ∈ (0, ∞). Then there exist constants C˜ and D˜ such that, for all locally Lipschitz functions f , |f | β C˜ μ|∇f ˜ | + D˜ μ|f ˜ |, μ˜ |f |log μ|f ˜ |

(8)

where μ˜ is the probability measure on RN given by μ(dλ) ˜ := e−W μ(dλ)/Zμ˜ , with Zμ˜ ≡ −W μ(e ).

82


Proof. Take f 0. Since (2) holds by assumption, we can apply it to the function f e−W . This yields

μ˜ f |U |β + |∇U | Aμ|∇f ˜ | + Aμ˜ f |∇W | + B μ(f ˜ )

β ˜ ) Aμ|∇f ˜ | + δAμ˜ f |U | + |∇U | + B + AC(δ) μ(f using (7). Thus, since δA < 1, we have that

μ˜ f |U |β + |∇U | A˜ μ|∇f ˜ | + B˜ μ(f ˜ )

(9)

for A˜ = A/(1 − δA), B˜ = (B + AC(δ))/(1 − δA). Replacing f by f e−W in (3) of Theorem 2.1, we get

f e−W β C μ|∇f ˜ | + C μ˜ f |∇W | + D μ(f ˜ ). μ˜ f log μ(f ˜ )Zμ˜ Using this together with (9) and the elementary inequality (x + y)β x β + y β which holds when x, y 0 and β ∈ (0, 1] yields f β μ˜ f log μ(f ˜ )

˜ ) C μ|∇f ˜ | + μ˜ f |W |β + C|∇W | + D + | log Zμ˜ |β μ(f

β

˜ ) C μ|∇f ˜ | + a0 max{1, C}μ˜ f |U | + |∇U | + a1 + D + | log Zμ˜ |β μ(f C˜ μ|∇f ˜ | + D˜ μ(f ˜ ), ˜ The inequality where C˜ = C + a0 max{1, C}A˜ and D˜ = a1 + D + | log Zμ˜ |β + a0 max{1, C}B. for general f follows in similar way by applying the above inequality to the positive and negative parts of f separately. 2 The resulting inequality in Theorem 2.1 is a defective inequality, in the sense that it contains a term involving μ|f | on the right-hand side. For our purposes this type of inequality is not strong enough, and therefore we now aim to prove a tightened inequality of the following form

EntΦ μ |f | := μ Φ |f | − Φ μ|f | cμ|∇f |,

(10)

where Φ(x) = x(log(1 + x))β , β ∈ (0, 1], and c ∈ (0, ∞) is a constant independent of f . We accomplish this in the situation (see Theorem 2.5 below) when we have the following Cheeger type inequality μ|f − μf | c0 μ|∇f | with a constant c0 ∈ (0, ∞) independent of f . A bound of the form described in (10) will be called in what follows an L1 Φ-entropy inequality. It is an example of a (non-homogeneous) additive Φ-entropy inequality, as studied in [5]


83

and [15]. To arrive at the desired inequality, our strategy will be as follows. We will first use Theorem 2.1 to prove a defective L1 Φ-entropy inequality, that is an inequality of a similar form but containing additionally on its right-hand side a term proportional to μ|f |. Then we will adapt some ideas of Rothaus [30], generalised in [12], to show that such a defective inequality can be tightened. We begin by proving the following lemma. Lemma 2.4. Let Φ(x) = x(log(1 + x))β , β ∈ (0, 1] and let μ be a given probability measure. Then there exists a constant κ ∈ [0, ∞) such that for any functions f and g satisfying 0 g f , μf < ∞, one has β f + κμ(f ). EntΦ (g) μ f log + μ μf Proof. We have that

β

β

EntΦ μ (g) = μ g log(1 + g) − log(1 + μg) β g μ g log 1 + μg β g , μ f log 1 + μg

(11)

since g f . Set F (x) := (log(1 + x))β for x ∈ [0, ∞). Then F is increasing and concave. Moreover, there exists a constant θ ∈ (0, ∞) such that xF (x) θ for all x. Following [20], we now claim that xF (y) xF (x) + θy

(12)

for all x, y 0. Indeed, if y x this is trivial. If x y, we have

F (y) − F (x) x F (y) − F (x) = x (y − x) xF (x)y y −x θy. Setting x = yields

f μf

and y =

g μg

in (12) and integrating both sides with respect to the measure μ

β β f g μ f log 1 + + θ μ(f ). μ f log 1 + μg μf Thus, by (11) β f EntΦ (g) μ f log 1 + + θ μ(f ). μ μf

(13)

84


Now β β f f = μ f log 1 + μ f log 1 + 1{f μf } μf μf β f + μ f log 1 + 1{f μf } μf β 2f β (log 2) μ(f ) + μ f log 1{f μf } μf = (log 2)β μ(f ) β f + μ f log 2 + log 1{f μf } μf β f 2(log 2)β μ(f ) + μ f log+ , μf using once again the inequality (x + y)β x β + y β for x, y 0 and β ∈ (0, 1]. Combining this with (13), we arrive at β

f EntΦ (g) μ f log + 2(log 2)β + θ μ(f ), + μ μf which completes the proof.

2

Theorem 2.5. Suppose U , λ and μ are as in Theorem 2.1. In addition, suppose that the following Cheeger type inequality holds μ|f − μf | c0 μ|∇f |

(14)

for some c0 > 0 and all locally Lipschitz functions f . Then there exists c ∈ (0, ∞) such that (10) holds, i.e. for any locally Lipschitz function f , we have

EntΦ μ |f | cμ|∇f |, where Φ(x) = x(log(1 + x))β . Proof. By Lemma A.1 of the appendix of [28], we have that there exist constants a˜ and b˜ such that 2

Φ 2 2 ˜ aEnt ˜ EntΦ μ f μ (f − μf ) + bμ(f − μf ) . Thus, for any t ∈ R, we have that 1 2 Φ 2 EntΦ μ |f + t| = Entμ |f + t| 1 1 2 1 1

Φ ˜ |f + t| 2 − μ|f + t| 2 2 . 2 2 aEnt ˜ + bμ μ |f + t| − μ|f + t|

(15)

J. Inglis et al. / Journal of Functional Analysis 260 (2011) 76–116 1

85

1

Let G = (|f + t| 2 − μ|f + t| 2 )2 . Note that we can write 1 1 2 G = |f + t| 2 − μ|f + t| 2 =

2 1 1 f (ω) + t 2 − f (ω) ˜ + t 2 dμ(ω) ˜

1 1 2 f (ω) + t 2 − f (ω) ˜ + t 2 dμ(ω) ˜ f (ω) − f (ω) ˜ ˜ dμ(ω)

|f | + μ|f |, 1

1

1

using the elementary inequality ||x + t| 2 − |y + t| 2 | |x − y| 2 in the last but one step. Hence, we have by (15) that Φ ˜ EntΦ ˜ μ |f + t| aEnt μ (G) + 2bμ|f |.

(16)

Since 0 G |f | + μ|f |, by Lemma 2.4 and Theorem 2.1, we have EntΦ μ (G) μ

|f | + μ|f | log+

|f | + μ|f | μ(|f | + μ|f |)

β + 2κμ|f |

Cμ|∇f | + 2(D + κ)μ|f |.

(17)

Combining (16) and (17) yields

sup EntΦ ˜ | + 2 a(D ˜ + κ) + b˜ μ|f |. μ |f + t| aCμ|∇f

(18)

t∈R

This implies the following bound

EntΦ ˜ | + 2 a(D ˜ + κ) + b˜ μ|f − μf |. μ |f | aCμ|∇f

(19)

Finally we can apply the Cheeger type inequality (14) to the last term on the right-hand side of (19) to arrive at

EntΦ μ |f | cμ|∇f |, ˜ with c = aC ˜ + 2c0 (a(D ˜ + κ) + b).

2

In the same spirit as Corollary 2.3, this inequality is stable under perturbations of the following type. Corollary 2.6. Let U , λ and μ be as in Theorem 2.1. Suppose also that the Cheeger type inequality (14) holds. As in Corollary 2.3, let W be a real function which is locally Lipschitz and such that e−W dμ < ∞ and

|∇W | δ |U |β + |∇U | + C(δ),

|W |β a0 |U |β + |∇U | + a1

86


for some δ < such that

1 A , C(δ), a0 , a1

∈ (0, ∞) and β ∈ (0, 1]. Moreover, let V be a measurable function

osc(V ) ≡ sup V − inf V < ∞. Then there exists a constant cˆ such that, for all locally Lipschitz functions f ,

EntΦ ˆ |, μˆ |f | cˆμ|∇f where μˆ is the probability measure on RN given by ˆ μ(dλ) ˆ := e−W −V μ(dλ)/Z, with a normalisation constant Zˆ ∈ (0, ∞) and Φ(x) = x(log(1 + x))β . Proof. In the case V = 0, the result is obtained by following the proof of Theorem 2.5, using Corollary 2.3 where necessary. In the case V = 0, by Lemma 3.4.2 of [1], we may write

EntΦ μˆ |f | =

inf μˆ Φ |f | − Φ (t) |f | − t − Φ(t)

t∈[0,∞)

eosc(V ) Z0 Zˆ

inf

t∈[0,∞)

e−W

dμ Φ |f | − Φ (t) |f | − t − Φ(t) Z0

where Z0 = e−W dμ. Applying the above case when V = 0 to the measure EntΦ μˆ

eosc(V ) Z0 c |f | Zˆ

|∇f |

e−W Z0

dμ yields

e−W dμ Z0

c e2 osc(V ) μ|∇f ˆ |, for some constant c , so that the result holds.

2

In Theorem 2.5 we assume that the Cheeger type inequality (14) holds, together with inequalities (1) and (2). However, we note below that under some conditions it is possible to deduce the Cheeger type inequality directly from a weaker version of the U -bound (2), using the method in [23]. Theorem 2.7. Let dμ = inequality is satisfied

e−U Z

dλ be probability measure on RN , and suppose that the following

μ f |U |β Aμ|∇f | + Bμ|f |,

for some β > 0 and all locally Lipschitz functions f . Suppose also that:

(20)


87

(a) for any L 0 there exists r = r(L) ∈ (0, ∞) such that β |U | L ⊂ B(r)

(21)

for some ball B(r) of radius r; (b) for r = r(L) there exists mr ∈ (0, ∞) such that the following Poincaré inequality in the ball B(r) is satisfied 1 f − dλ 1 f dλ |∇f | dλ λ(B(r)) mr B(r)

B(r)

(22)

B(r)

for all suitable functions f . Then there exists a constant c0 such that μ|f − μf | c0 μ|∇f | for all locally Lipschitz functions f . Proof. We have that μ|f − μf | 2μ|f − m| for all m ∈ R. Now for L 0 we have

μ|f − m| μ |f − m|1{|U |β L} + μ |f − m|1{|U |β L} . We have that {|U |β L} ⊂ B(r) for some r = r(L) ∈ (0, ∞), so that putting m = β B(r) f dλ, and noting that on the set {|U | R} there exists a constant Ar such that

(23) 1 λ(B(r))

×

1 dμ Ar , Ar dλ we can bound the first term using assumption (a). Indeed, 1 f − f dλ dλ λ(B(r))

μ |f − m|1{|U |β L} Ar

B(r)

Ar mr

B(r)

|∇f | dλ B(r)

using (22). On the other hand, using (20), we have

A2r μ|∇f | mr

(24)

88


1

μ |f − m|1{|U |β L} μ |f − m||U |β L B A μ|∇f | + μ|f − m|. L L Using estimates (24) and (25) in (23), and taking L large enough ends the proof.

(25) 2

We can now combine all the results of this section into the following theorem. Theorem 2.8. Let U , λ and μ be as in Theorem 2.1. Suppose also that conditions (a) and (b) of Theorem 2.7 are satisfied. Then there exists c ∈ (0, ∞) such that (10) holds, i.e.

EntΦ μ |f | cμ|∇f | for all locally Lipschitz functions f , where Φ(x) = x(log(1 + x))β . To conclude this section, we finally note that the L1 Φ-entropy inequality (10) can be tensorised in the following sense. Lemma 2.9 (Tensorisation). Let I be a finite index set, and νi , i ∈ I be probability measures. Set νI := i∈I νi . Suppose that for each i ∈ I , νi satisfies the L1 Φ-entropy inequality (10) with a constant c(i) ∈ (0, ∞). Then so does νI with constant maxi∈I {c(i)}. Proof. The proof follows by induction. The key observation is as follows: for J ⊂ I and k ∈ / J, one has

νk ⊗ νJ Φ(f ) − Φ(νk ⊗ νJ f ) = νk νJ Φ(f ) − Φ(νJ f )

+ νk Φ(νJ f ) − Φ νk (νJ f ) νk cJ νJ |∇j f | + ck νk |∇k νJ f | j ∈J

max(cJ , ck )

νk ⊗ νJ |∇j f |.

2

j ∈J ∪k

3. Isoperimetric inequalities In this section our aim is to derive isoperimetric information for the measure μ starting from L1 Φ-entropy inequalities. We assume that μ is non-atomic and that the distance d on RN is related to the modulus of the gradient of a function f : RN → R by |f (x) − f (y)| . d(x, y) d(x,y)↓0

|∇f |(x) = lim sup

As usual, we define the surface measure of a Borel set A ⊂ RN by

(26)


μ+ (A) = lim inf ε↓0

89

μ(Aε \ A) ε

where Aε = {x ∈ Rn : d(x, A) < ε} is the (open) ε-neighbourhood of A (with respect to d). We are concerned with a problem of estimating the isoperimetric profile of the measure μ, that is a function Iμ : [0, 1] → R+ defined by Iμ (t) = inf μ+ (A): A Borel such that μ(A) = t (with Iμ (0) = Iμ (1) = 0). By definition it is the largest function such that the following isoperimetric inequality holds

Iμ μ(A) μ+ (A). For q > 1 and p such that

1 q

+

1 p

(27)

= 1, we define functions Uq = fp ◦ Fp−1 where fp is the

−|x|p

density of the measure dνp (x) = e Zp dx on R and Fp = fp (here, |x| denotes the Euclidean norm of x ∈ R). This is motivated by the fact that Uq is the isoperimetric function of νp in 1

dimension 1. It is known (see [12]) that Uq (t) is symmetric and behaves like G(t) = t (log( 1t )) q near the origin so that for some constant Lq > 0, we have

1 G min(t, 1 − t) Uq (t) Lq G min(t, 1 − t) Lq

(28)

for all t ∈ [0, 1]. Theorem 3.1. Assume that the L1 Φ-entropy inequality

EntΦ μ |f | cμ|∇f | holds for some constant c ∈ (0, ∞) and all locally Lipschitz functions f , where Φ(x) = x(log(1 + x))β and β ∈ (0, 1]. Then Iμ 1c˜ Uq with some constant c˜ > 0, q = β1 and the measure μ satisfies an isoperimetric inequality of the form Uq (t) cμ ˜ + (A)

(29)

for all a Borel sets A of measure t = μ(A). Proof. When applied to a non-negative function f such that μf = 1, the L1 Φ-entropy inequality becomes

β

μ f log(1 + f ) − (log 2)β cμ|∇f |, which implies that for all non-negative f (not identically 0) we have β f μ f log 1 + − (log 2)β cμ|∇f |. μf

(30)

90


Let A be a Borel set with measure t = μ(A). To start with, suppose that t ∈ [0, 12 ]. We can approximate the indicator function of A by a sequence of Lipschitz functions (fn )n∈N satisfying lim sup μ|∇fn | μ+ (A) n→∞

(see [10, Lemma 3.5]). Taking fn in (30) and passing to the limit as n → ∞ yields t

1 β − (log 2)β cμ+ (A). log 1 + t

(31)

We now observe that for t ∈ [0, 12 ] we have β 1 1 β η log log 1 + − (log 2)β t t

(32)

3 β with η = ( log log 2 ) − 1 > 0. This implies

β 1 c t log μ+ (A), t η

(33)

for all t ∈ [0, 12 ]. Thus, by the equivalence relation (28), we have that Uq (t) cμ ˜ + (A)

(34)

for all t ∈ [0, 12 ], with c˜ = ηc Lq . Now suppose that t = μ(A) ∈ ( 12 , 1]. For functions f ∈ [0, 1], we can apply (30) to 1 − f , which yields β 1−f cμ|∇f |. − (log 2)β μ (1 − f ) log 1 + 1 − μf If we now take fn in this inequality (where (fn )n∈N is again the Lipschitz approximation of the characteristic function of A) and pass to the limit as n → ∞, we see that (1 − t) log 1 +

1 1−t

β − (log 2)

β

cμ+ (A).

Writing s = 1 − t ∈ [0, 12 ) and using (32) now gives β 1 c s log μ+ (A). s η

(35)

Thus by (28) again, we have Uq (1 − t) = Uq (s) cμ ˜ + (A) for all t ∈ ( 12 , 1] with c˜ = ηc Lq . By

˜ + (A) for t ∈ ( 12 , 1], which combined with (34) yields the symmetry of Uq therefore Uq (t) cμ result. 2


91

An important corollary of this result is the following: Corollary 3.2. Assume that the L1 Φ-entropy inequality

EntΦ μ |f | cμ|∇f | holds for some constant c ∈ (0, ∞) and all locally Lipschitz functions f , where Φ(x) = x(log(1 + x))β and β ∈ (0, 1]. Then there exists a constant c0 such that μ|f − μf | c0 μ|∇f |.

(36)

Proof. We note that if β = 1/q, Uq (t)

1/q 1 1 (log 2)1/q min(t, 1 − t) log min(t, 1 − t). Lq min(t, 1 − t) Lq

Thus by Theorem 3.1, we have that min(t, 1 − t) c˜

Lq μ+ (A), (log 2)β

for t = μ(A), which is Cheeger’s isoperimetric inequality on sets. This is equivalent (up to a constant) to its functional form μ|f − μf | c0 μ|∇f | (see for example [11]).

2

Following an argument of [26] we can pass from the isoperimetric statement above to inequality (4). We note that in our general setting, the following coarea inequality is available, (for a proof see e.g. [10, Lemma 3.2]), μ|∇f |

μ+ {f > s} ds

(37)

R

for locally Lipschitz functions f . Proposition 3.3. If the measure μ satisfies an isoperimetric inequality of the form (29), then there exist constants K, K > 0 such that

μ f (log+ f )β Kμ|∇f | + K for all positive locally Lipschitz functions f such that μ(f ) = 1, where β = q1 .

(38)

92


Proof. Let f be non-negative, with μ(f ) = 1. The coarea inequality (37) together with our assumption imply μ|∇f | R

1 μ+ {f > s} ds c˜

Uq μ {f > s} ds

R

Now, by Markov’s inequality μ({f > s}) μ({f > 2}) Uq (t)

1 1 Lq t (log( t ))

∞

1 q

for t

1 2

1 2

when s 2. Therefore, since

by (28),

Uq μ {f > s} ds

0

∞

Uq μ {f > s} ds

2

1 Lq

∞

μ {f > s} log

1 μ({f > s})

1 q

ds

2

1 = Lq

∞

μ {f > s} log

1 μ({f > s})

1 q

ds

0

2

1 − Lq

μ {f > s} log

1 μ({f > s})

1 q

ds

0

1 Lq

∞

μ {f > s} log

1 μ({f > s})

1 q

ds −

2 M, Lq

0

where M = supt∈[0,1] t (log 1t )β . Next, again by Markov’s inequality, μ({f > s}) 1s . Therefore, when s 1 we have log

1 log s μ({f > s})

and we always have log μ({f1>s}) 0. Therefore, log μ({f1>s}) log+ s, which implies μ|∇f |

1 cL ˜ q

R

2M 1 (log+ s)β μ {f > s} ds − μ f (log+ f )β − K cL ˜ q cL ˜ q

2M with K = cL ˜ q +1. To see the last inequality, let F (s) =

s

β β 0 (log+ t) dt and H (s) = s(log+ s) −s. F (s) = (log s)β and H (s) = (log s)β +

Then F (s) 0 H (s) on [0, e] and when s e β(log s)β−1 − 1. Therefore, since log s 1 and β ∈ (0, 1], F H for s e from which it follows that F H on [0, ∞). Therefore,


93

∞ ∞

β (log+ s) μ {f > s} ds = F (s)μ {f > s} ds 0

0

= μ F (f )

μ H (f )

= μ f (log+ f )β − μ(f )

= μ f (log+ f )β − 1. 2

Remark 3.4. With the above results, we have thus shown the equivalence of the L1 Φ-entropy inequality with the isoperimetric inequality (29) and with inequality (38) together with the Cheeger inequality (36); see Theorem 4.5 below. Remark 3.5. When β1 = q = 2, the function U2 represents the Gaussian isoperimetric function. In this case, the isoperimetric inequality (29) is known to be equivalent to the following inequalities introduced by Bobkov in [8] and [9]:

˜ | , U2 μ(f ) μ U2 (f ) + c|∇f

U2 μ(f ) μ U2 (f )2 + c˜2 |∇f |2

(39) (40)

for all locally Lipschitz f : R → [0, 1]. The equivalence of these inequalities in this case follows by a transportation argument which uses the fact that the standard Gaussian measure γ on R satisfies (39) and (40) with c˜ = 1 (see [6, Proposition 5]). Remark 3.6. Suppose that the measure μ satisfies an L1 Φ-entropy inequality on a metric space (M, d). Suppose that on the product space (Mn , dn , μ⊗n ) we have |∇f | = ni=1 |∇i f |, where ∇i denotes differentiation with respect to the ith coordinate and where the moduli of the gradients are defined via (26) with the supremum distance. The tensorisation property of the L1 Φ-entropy (Lemma 2.9) then allows us to obtain isoperimetric information on the product space (where the surface measure is now defined with respect to supremum distance). This problem was considered in [4]. 4. Consequences of L1 Φ-entropy inequalities In this section we look at some consequences of the L1 Φ-entropy inequality

EntΦ μ |f | cμ|∇f |,

(41)

with Φ(x) = x(log(1 + x))β , β ∈ (0, 1], for a general probability measure μ. The first result shows that this inequality implies a q-logarithmic Sobolev inequality, as studied in [12] and [23]. Theorem 4.1. Let μ be an arbitrary probability measure which satisfies the L1 Φ-entropy inequality (41) for some β ∈ [ 12 , 1] and set q = β1 ∈ [1, 2]. Then there exists a constant cq such that the following (LSq ) inequality holds

94


|f |q cq μ|∇f |q μ |f |q log μ|f |q

(42)

for all locally Lipschitz functions f . Proof. Without loss of generality we assume that f 0. Applying L1 Φ-entropy inequality (41) to the function f/μf , we obtain the following homogeneous version β f cμ|∇f | + (log 2)β μ(f ). μ f log 1 + μf

(43)

We apply this inequality to the function g = f (1 + log(1 + f ))1−β f 0, where f is such that μ(f ) = 1. Note that μ(g) 1. Then we have β β

1−β g g = μ f 1 + log(1 + f ) log 1 + μ g log 1 + μg μg β

1−β f μ f 1 + log(1 + f ) log 1 + μg 1−β β f f log 1 + μ f 1 + log 1 + μg μg

f = μ f log(μg + f ) − log μ(g) μ f log 1 + μg

μ f log(1 + f ) − μ(g). Thus for all f 0 with μ(f ) = 1,

1−β

+ (log 2)β + 1 μ(g) μ f log(1 + f ) cμ∇ f 1 + log(1 + f )

1−β

cμ 1 + log(1 + f ) |∇f | 1 f |∇f | + c(1 − β)μ (1 + log(1 + f ))β 1 + f

+ (log 2)β + 1 μ(g)

1−β |∇f | + c(1 − β)μ|∇f | cμ 1 + log(1 + f )

+ (log 2)β + 1 μ(g). Since we have assumed β 12 , we have 1 − β β and hence 1−β

1−β

1 + μ f log(1 + f ) μ(g) = μ f 1 + log(1 + f ) β

μ f log(1 + f ) +2 cμ|∇f | + (log 2)β + 2

(44)

(45)


95

by another application of the L1 Φ-entropy inequality (43) in the last step. Using this in (44), we see that for general f 0, 1−β f f cμ 1 + log 1 + |∇f | μ f log 1 + μf μf

2 + c 2 − β + (log 2)β μ|∇f | + (log 2)β + 2 μ(f ). Replacing f by f q with q =

1 β

(46)

in the above yields

1−β fq fq q−1 qcμ 1 + log 1 + f |∇f | μ f q log 1 + μf q μf q

+ cq 2 − β + (log 2)β μ f q−1 |∇f |

2

+ (log 2)β + 2 μ f q fq qcε p−1 μ f q 1 + log 1 + p μf q

c + c 2 − β + (log 2)β μ|∇f |q + ε

2 q

cq β β μ f 2 − β + (log 2) + (log 2) + 2 + p where ε > 0 and we have applied Young’s inequality with indices qcε p−1 /p < 1, we can simplify this bound as follows

1 p

+

1 q

= 1. Choosing

fq μ f q log 1 + C μ|∇f |q + D μ f q q μf where C =

c ε

+ c(2 − β + (log 2)β ) 1−

qcε p−1 p

,

D =

cq p (2 − β

+ (log 2)β ) + ((log 2)β + 2)2 1−

qcε p−1 p

.

From this one obtains the defective (LSq ), which for all f 0 such that μ(f q ) = 1 can be equivalently represented as

μ f q log f q C μ|∇f |q + D .

(47)

Let us now recall that by Corollary 3.2, our assumption implies that there exists a constant c0 such that μ|f − μf | c0 μ|∇f |.

96


From this inequality we can use the arguments of [12, Chapter 2] to deduce that there exists a constant cq such that μ|f − μf |q cq μ|∇f |q . Finally, by Rothaus-type arguments ([12, Chapter 3], see also Appendix B), we can then remove the defective term in (47) to arrive at the result. 2 Theorem 4.1 has a number of corollaries, which follow from known results about the qlogarithmic Sobolev inequality (LSq ) contained in [12] and [23]. We mention here the following one, which is important for our purposes. Corollary 4.2. Let μ be an arbitrary probability measure which satisfies the L1 Φ-entropy inequality (41) with β ∈ [ 12 , 1]. Suppose f is a locally Lipschitz function such that |∇f |q af + b

(48)

with q = β1 , for some constants a, b ∈ [0, ∞). Then for all t > 0 sufficiently small

μ etf < ∞. Proof. Follows from Theorem 4.5 of [23].

2 −U

In Section 2 we proved that, under some conditions, if dμ = e Z dλ is a probability measure which satisfies a Cheeger type inequality of the form (14), and a U -bound of the form

μ |f | |U |β + |∇U | Aμ|∇f | + Bμ|f |,

(49)

then the L1 Φ-entropy inequality (41) holds. We now aim to show the converse i.e. that under some weak conditions, the L1 Φ-entropy inequality (41) implies a bound of the form (49). We first prove the following useful lemma. Lemma 4.3. Let μ be a probability measure. Then −1 μ(f h) s −1 EntΦ μ (f ) + s Θ(sh)

(50)

for all s > 0 and suitable functions f, h 0 such that μ(f ) = 1, where Φ(x) = x(log(1 + x))β , β ≡ q1 ∈ (0, 1] and q β

Θ(h) ≡ θ + (log 2)β + log μeh with θ = supx0 βx(log(1 + x))β−1 /(1 + x). Moreover, suppose that μ satisfies the L1 Φ-entropy inequality (41) for some β ∈ [ 12 , 1] with constant c, and that g 0 is a locally Lipschitz function such that |∇g|q ag + b

(51)


97

for some constants a, b ∈ (0, ∞). Then Θ(s β g β ) < ∞ for sufficiently small s > 0, and

c c μ f g β β μ|∇f | + β Θ s β g β μ(f ), s s

(52)

for all locally Lipschitz functions f 0. Proof. We remark first that for functions f, h 0, μf = 1, with s ∈ (0, ∞) and β ≡ we have

1 q

∈ (0, 1),

q q

β μ(f h) = s −1 μ f log es h

q q β

s q hq es h −1 s q hq s μ f log 1 + s q hq χ e μe μe q q β + s −1 log μes h μ(f ). By the generalised relative entropy inequality of [20], we have β q q β es h f μ f log 1 + s q hq + θ μf μf log 1 + μe μf

β EntΦ μ (f ) + θ + (log 2) μf, since μf = 1. We therefore get the following bound q q β

−1 μ(f h) s −1 EntΦ θ + (log 2)β + log μes h . μ (f ) + s

(53)

This ends the proof of the first part of the lemma. 1

Replacing h by g β ≡ g q and s by s β in (53), we see that the second part is a consequence of Corollary 4.2. 2 −U

Theorem 4.4. Let dμ = e Z dλ be a probability measure on RN , with U a locally Lipschitz function bounded from below. Suppose μ satisfies the L1 Φ-entropy inequality (41) for some β ∈ [ 12 , 1]. Suppose also that |∇U | a|U |β + b

(54)

for some constants a, b ∈ (0, ∞). Then there exist constants A, B ∈ [0, ∞) such that

μ |f | |U |β + |∇U | Aμ|∇f | + Bμ|f |, for all locally Lipschitz f .

(55)

98


Proof. Let f 0. We may also suppose that U 0 (otherwise we can shift it by a constant). Note that from (54), it follows that ˜ + b˜ |∇U |q aU with q = β1 . Hence we may apply Lemma 4.3, to see that

c c μ f U β β μ|∇f | + β Θ s β U β μ(f ) s s with Θ(s β U β ) < ∞ for sufficiently small s.

2

The following theorem summarises the results of the paper so far. Theorem 4.5. Let μ be a non-atomic probability measure on (RN , d), |∇f | be given by (26) and q 1. Then the following statements are equivalent: (i)

EntΦ μ |f | cμ|∇f |, 1

where Φ(x) = x(log(1 + x)) q , for some constant c ∈ (0, ∞) and all locally Lipschitz f ; (ii) f 1/q Kμ|∇f | + K μf, μ f log+ μf for some K > 0 and μ|f − μf | c0 μ|∇f | with some c0 ∈ (0, ∞) and all locally Lipschitz f 0; (iii) Uq (t) cμ ˜ + (A), for some c˜ > 0 and all Borel sets A of measure t = μ(A). Moreover, for q ∈ (1, 2] statements (i)–(iii) imply (iv) |f |q C μ|∇f |q μ |f |q log μ|f |q for some C ∈ (0, ∞) and all locally Lipschitz functions f ,

(LSq )


99

and (v) U2 (μf ) μ U22 (f ) + C |∇f |2

(IFI 2 )

for some C ∈ (0, ∞) and all locally Lipschitz functions 0 f 1. −U

Finally, suppose that the probability measure μ is given by μ(dx) = e Z dλ for some locally Lipschitz function U on RN which is bounded from below. Suppose that the measure dλ satisfies the classical Sobolev inequality (1) together with the Poincaré inequality in balls (22), and that ∀L 0 there exists r = r(L) such that {U L} ⊂ B(r). In this situation the following U -bound

μ |f | |U |β + |∇U | Aμ|∇f | + Bμ|f |

(56)

for all locally Lipschitz functions and constants A, B ∈ [0, ∞), β ∈ (0, 1], implies that statements (i)–(iii) hold with q = β1 . If in addition we have that (54) holds i.e. there exist constants a, b such that |∇U | aU β + b then (56) is actually equivalent to the statements (i)–(iii). Proof. (ii) ⇒ (i) was shown in Section 2. (i) ⇒ (iii) is proved in Theorem 3.1. Finally, Proposition 3.3 together with Corollary 3.2 show that (iii) ⇒ (ii). The rest of the theorem, except (v), is a restatement of the results of Section 2 and the current one. To see (v) we notice that using (28) for small t > 0 (as well as small 1 − t > 0) we have U2 (t) C¯ 0 Uq (t) with some C¯ 0 ∈ (0, ∞), and thus there is a constant C¯ ∈ (0, ∞) such that for all t ∈ (0, 1) ¯ q (t). U2 (t) CU Hence, by (iii), we have the following isoperimetric relation ˜ + (A) U2 (t) Cμ for any set A with μ(A) = t. This isoperimetric inequality was shown in [6, Proposition 5] to be equivalent to (IFI 2 ) in the setting of Riemannian manifolds. The proof remains valid in our setting once we note that the co-area inequality (37) is available. 2 Remark 4.6. We remark that generally perturbation of (IFI 2 ) is a difficult matter if the unbounded log of the density is involved. Our route via U -bounds allows us to achieve that very effectively.

100


Secondly, as conjectured in [12] for q ∈ (1, 2] it would be natural to expect the following functional isoperimetric inequality with optimal isoperimetric function q q Uq (μf ) μ q Uq (f ) + Cq |∇f |q

(IFI q )

with some Cq ∈ (0, ∞) for all locally Lipschitz functions 0 f 1. One of the motivations for such a relation is that (as shown in [12]) it implies (LSq ). Using (IFI 2 ) and the relation of lq norms, in finite dimension one can see that q q U2 (μf ) μ q U2 (f ) + C2 |∇f |q . In the right-hand side, using the asymptotic relation between isoperimetric functions, one could also replace U2 with Uq . The question remains if adjusting the left-hand side in a similar way would still preserve the inequality in the desired sharp form. 5. Application of results In order to see where these results can be applied, suppose we are still working in the general situation described at the start of this paper, and define a probability measure e−αd dλ Z p

dμp :=

(57)

on RN , with α > 0, p ∈ (1, ∞) and normalisation constant Z. Recall that here d : RN × RN → [0, ∞) is a metric on RN . We have the following result which can be found in [23]. Proposition 5.1. Let μp be given by (57). Suppose that we have (i) σ1 |∇d| 1 almost everywhere for some σ ∈ [1, ∞); (ii) d K + αpεd p−1 on {x: d(x) 1}, for some K ∈ [0, ∞), ε ∈ [0, σ12 ). Then there exist constants A, B ∈ [0, ∞) such that

μp |f |d p−1 Aμp |∇f | + Bμp |f | for all locally Lipschitz functions f . This proposition gives conditions under which the bound (56) in Theorem 4.5 holds for a particular choice of U and β. Indeed, we thus have the following corollary: Corollary 5.2. Let μp be given by (57). Suppose that conditions (i) and (ii) of Proposition 5.1 are satisfied. Suppose also that the measure dλ satisfies the classical Sobolev inequality (1) together with the Poincaré inequality in balls (22). Then inequalities (i)–(iii) of Theorem 4.5 are satisfied, with q such that p1 + q1 = 1. Moreover, if p 2 (iv)–(v) are also true.


Proof. For U = αd p and β =

1 q

101

we have

μp |f | U β + |∇U | μp |f | α β d βp + αpd p−1

α β + αp μp |f |d p−1 . Therefore by Proposition 5.1, we have

˜ ˜ | + Bμ|f | μ |f | U β + |∇U | Aμ|∇f where A˜ = (α β + αp)A and B˜ = (α β + αp)B. Thus we can apply Theorem 4.5.

2

We can perturb the measure in this result and all the inequalities will hold for the perturbed measure, as follows. Corollary 5.3. Let d μˆ = e−W −V /Zˆ dμp be the probability measure described in Corollary 2.6 with unbounded locally Lipschitz W and bounded measurable V . Then μˆ enjoys all properties as μp in Corollary 5.2. Remark 5.4. The conditions of Corollary 5.2 are easily seen to be satisfied in the Euclidean case, when we are dealing with the standard gradient and Laplacian in RN , and d(x) = |x|. In this situation, with p = 2, the inequalities we prove are already known (see [26]), though the proof we give here is new. The value of our results is that they can be used in more general situations than the Euclidean one. In particular it can be applied in the following setting. Example 5.5. [H-type groups] Let g be a (finite dimensional real) Lie algebra and let z denote its centre (i.e. [g, z] = 0). We say that g is of H-type if it admits a vector space decomposition g=v⊕z where [v, v] ⊆ z, such that there exists an inner product ·,· on g such that z is an orthogonal complement to v, and the map JZ : v → v given by JZ X, Y = [X, Y ], Z for X, Y ∈ v and Z ∈ z satisfies JZ2 = −|Z|2 I for each Z ∈ z. An H-type group is a simply connected Lie group G whose Lie algebra is of H-type. Such a group is a Carnot group of step 2 (see [13] for details). In particular the Heisenberg group is an H-type group with a one-dimensional centre. However, there also exist H-type groups with centre of any dimension. an orthonormal On an H-type group G we consider vector fields X1 , . . . , Xm which form 2 basis of v. The sub-Laplacian (or Kohn operator) is given by G := m i=1 Xi and sub-gradient by ∇G := (X1 , . . . , Xm ). The associated Carnot–Carathéodory distance is defined by d(x, y) := sup f (x) − f (y): f such that |∇G f | 1 .

102


It is shown in [23] that conditions (i) and (ii) of Proposition 5.1 are satisfied in this setting. Moreover, the Lebesgue measure dλ satisfies the classical Sobolev inequality (1) and Poincaré inequality in balls (22) with the sub-gradient ∇G (see [32]). Thus, by Corollary 5.2 we arrive at the following: Theorem 5.6. Let G be an H-type group, equipped with Carnot–Carathéodory distance d and canonical sub-gradient ∇G as described above. Let e−αd dμp := dλ Z p

with p > 1 and α > 0 be a probability measure on G and d μˆ = e−W −V /Zˆ dμp with W ≡ W (d) satisfying conditions as in Corollary 5.2 with horizontal gradient and V a bounded measurable function. Then inequalities (i)–(iii) of Theorem 4.5 are satisfied with q such that p1 + q1 = 1. Moreover for p 2, the measure μˆ satisfies (LSq ) and (IFI 2 ). See [23, Theorem 2.2] for details of the perturbation technique necessary to achieve the relevant U -bounds. 5.1. U -bounds versus gradient bounds for heat kernel As a conclusion to this section we mention that our setup is naturally inclusive for the following gradient bounds for the heat kernel on H-type groups which has recently attracted considerable attention (see e.g. [2,16–18,27] and references therein). Indeed, in the following let G be an H-type group. Corollary 5.7. The semigroup Pt ≡ etG satisfies the following |∇G Pt f | C1 (t)Pt |∇G f | for all suitable functions, with C1 (t) ∈ (0, ∞) independent of f . Proof. Due to the group covariance, it is sufficient to show the bound at the identity element and thanks to the action of the dilations, one only needs to establish it at t = 1. Denoting the corresponding heat kernel by h, we see that

f ∇G h dλ = f − f ∇G hdλ f − f |∇G log h|h dλ

(58)

with f ≡ f h dλ. To bound the right-hand side of this expression, suppose that we have a function V growing to infinity such that |∇G log h| k1 V (d) + k2

(59)


103

for some k1 , k2 > 0, and for which the following U -bound is satisfied

f − f V (d)h dλ C

|∇G f |h dλ + D

f − f h dλ

(60)

with some C, D ∈ [0, ∞) independent of f . Combining this with (58), we would then be able to conclude that f ∇G h dλ Ck1 |∇G f |h dλ + (Dk1 + k2 ) f − f h dλ. Moreover, under the assumptions of Theorem 2.7, (60) implies that

f − f h dλ α

|∇G f |h dλ,

for some α > 0, allowing us to conclude that

f ∇G h dλ Ck1 + (Dk1 + k2 )α |∇G f |h dλ. Therefore, to complete the proof we must check that, in the setting of H-type groups, there exists a V such that (59) and (60) hold, and that the assumptions of Theorem 2.7 are satisfied, namely that for r > 0, {V < r} is contained in some ball and inequality (22) holds. In an H-type group with dim z = m and dim z⊥ = 2n we have the following heat kernel bounds of [17] (see also [27] and [7]): there exists R > 0 such that for all points (x, z) = (x1 , . . . , x2n , z1 , . . . , zm ) with d(0, (x, z)) > R, 1 d(0, (x, z))2n−m−1 2 e− 4 d(0,(x,z)) , n−1/2 1 + (|x|d(0, (x, z)))

∇G log h(x, z) K 1 + d 0, (x, z) .

h(x, z)

(61) (62)

Here f g means that c1 f g c2 f for some constants c1 , c2 > 0. In this case we may write f = eψ g with a function ψ of bounded oscillation. We can therefore take V (d) = d, so that (59) holds, with k1 = K and k2 = sup{|∇G log h|(x, z): d(0, (x, z)) R} + K, as do the assumptions of Theorem 2.7 (see for example [25,31,32]). Finally, to show that (60) holds we use similar perturbative arguments as in Theorem 7.1 of [23] (where the corresponding U -bound with V (d) = d was established for the first Heisenberg group, where n = m = 1). d(0,(x,z))2n−m−1 More precisely, letting W = log( [1+ε|x|d(0,(x,z))] n−1/2 ) with some ε ∈ (0, 1) to be determined later, we show that we may write 1 2

h(x, z) = e−ψ−W e− 4 d where osc(ψ) < ∞. This follows from the fact that

104


n−1/2

n−1/2 1 + |x|d 0, (x, z) C1 1 + |x|d 0, (x, z)

n−1/2

, C2 1 + |x|d 0, (x, z) for some constants C1 , C2 > 0, together with the estimate

n− 1

n−1/2 2 1 + ε|x|d 0, (x, z) ε n−1/2 1 + |x|d 0, (x, z)

n−1/2 1 + |x|d 0, (x, z) and (61). Moreover, using the triangle inequality we compute |∇G W | (2n − m − 1)

|∇G d(0, (x, z))| d(0, (x, z))

1 |∇G |x||d(0, (x, z)) + |x||∇G d(0, (x, z))| +ε n− 2 1 + ε|x|d(0, (x, z))

2n − m − 1 + ε(2n − 1)d 0, (x, z) R where we have used that |∇G |x|| = |∇G d| = 1 and |x| d(0, (x, z)). Choosing ε small enough, Theorem 2.2 of [23] gives the following U -bound for positive functions f f (x)d(x)h(x) dx C |∇G f |(x)h(x) dx + D f (x)h(x) dx. Applying this to |f − f | we arrive at (60) with V (d) = d.

2

While the gradient bounds still remain a challenge for more complicated groups, it may be useful to keep this observation in mind, as in principle it allows for a heat kernel bound (61) with far less precise description of the slowly varying factor (provided the corresponding control distance d satisfies a sufficiently good Laplacian bound outside some compact set). 5.2. U -bounds versus integrated Gaussian bounds for heat kernel Assuming a bound of the following form μ(f d) Cμ|∇f | + Dμ(f ), 2

for a function f = eλ min(d,L) with L a positive number, we get

2 2 μ eλ min(d,L) min(d, L) 2λCμ eλ min(d,L) min(d, L)∇ min(d, L) 2

+ Dμ eλ min(d,L)

2 2λCμ eλ min(d,L) min(d, L) 2

+ Dμ eλ min(d,L) .

(63)


105

If 2λC < 1, this implies

2 2

μ eλ min(d,L) min(d, L) D μ eλ min(d,L)

(64)

with D ≡ D(1 − 2λC)−1 . Next, choosing f = eλ min(d,L) min(d, L) instead in (63), we obtain 2

2 2 2 μ eλ min(d,L) min(d, L)2 Cμ∇ eλ min(d,L) min(d, L) + Dμ eλ min(d,L) min(d, L)

2 2 2λCμ eλ min(d,L) min(d, L)2 + Dμ eλ min(d,L) min(d, L) 2

+ Cμ eλ min(d,L) . Thus using (64), we obtain

2 2 μ eλ min(d,L) min(d, L)2 2λCμ eλ min(d,L) min(d, L)2

2

+ D + C μ eλ min(d,L) . Rearranging this, for 2λC 2λ0 C < 1,

d λ min(d,L)2

2 2

μ e = μ eλ min(d,L) min(d, L)2 D μ eλ min(d,L) dλ with D ≡ (D + C)(1 − 2λ0 C)−1 . Solving this differential inequality and passing with L → ∞, we arrive at the following: Theorem 5.8 (Integrated Gaussian bound). Suppose the following is true μ(f d) Cμ|∇f | + Dμ(f ) for all locally Lipschitz functions f , with some constants C, D ∈ (0, ∞). Then 2

μ eλd eλD for 2λC 2λ0 C < 1 with some constant D ∈ (0, ∞). See Appendix A for a generalisation of this idea. Remark 5.9. We point out that using this result and idea of [21] one can obtain Gaussian-type upper bounds on the heat kernel. Remark 5.10. From the point of view of the computations of [23] we start with h∇f = ∇(f h) − f ∇h and, with a unitary linear functional α, we get

106


α(∇f )h dλ =

α ∇(f h) dλ +

1 h dλ. f α ∇ log h

Hence, one gets

1 f α ∇ log − div α h dλ |∇f | · |α|h dλ. h

If the expression in the bracket on the left-hand side can be shown to have a treatable bound from below, such a bound can be a useful source of analysis (though the implementation of this idea in case of other than H-type groups remains open). 6. Extension to infinite dimensions In this section we aim to extend the L1 Φ-entropy inequality to the infinite dimensional setting, where we include some bounded interactions. The setup will be as follows. The spin space: Let M = (RN , d) be a metric space equipped with Lebesgue measure dλ, ) consisting of divergence free (possibly non-commuting) general sub-gradient ∇ = (X1 , . . . , Xm 2 vector fields and sub-Laplacian := m i=1 Xi , as above. D The lattice: Let Z be the D-dimensional lattice for some fixed D ∈ N, equipped with the lattice metric dist(·,·) defined by dist(i, j) :=

D

|il − jl |

l=1

for i = (i1 , . . . , iD ), j = (j1 , . . . , jD ) ∈ ZD . For i, j ∈ ZD we will also write i∼j

⇔

dist(i, j) = 1

i.e. i ∼ j when i and j are nearest neighbours in the lattice. For Λ ⊂ ZD , we will write Λc ≡ ZD \ Λ, |Λ| for the cardinality of Λ, and Λ ⊂⊂ ZD when |Λ| < ∞. D The configuration space: Let Ω := (M)Z be the configuration space. Given Λ ⊂ ZD and ω = (ωi )i∈ZD ∈ Ω, let ωΛ := (ωi )i∈Λ ∈ (M)Λ (so that ω → ωΛ is the natural projection of Ω onto MΛ ). Given ω ∈ Ω we introduce the injection: MΛ → Ω, defined by η ∈ MΛ → η •Λ ω where (η •Λ ω)i = ηi when i ∈ Λ and (η •Λ ω)i = ωi when i ∈ Λc . Let f : Ω → R. Then for i ∈ ZD and ω ∈ Ω define fi (·|ω) : M → R by fi (x|ω) := f (x •{i} ω). Let C (n) (Ω), n ∈ N denote the set of all functions f for which we have fi (·|ω) ∈ C (n) (M) for all i ∈ ZD . For i ∈ ZD , k ∈ {1, . . . , m} and f ∈ C (1) (Ω), define Xi,k f (ω) := Xk fi (x|ω)|x=ωi , where X1 , . . . , Xm are the vector fields on M.


107

Define similarly ∇i f (ω) := ∇fi (x|ω)|x=ωi and i f (ω) := fi (x|ω)|x=ωi for suitable f , where ∇ and are the sub-gradient and the sub-Laplacian on M respectively. For Λ ⊂ ZD , set ∇Λ f = (∇i f )i∈Λ and |∇Λ f | :=

|∇i f |.

i∈Λ

Finally, a function f on Ω is said to be localised in a set Λ ⊂ ZD if f is only a function of those coordinates in Λ. Local specification and Gibbs measure: Let Ψ = (ψX )X⊂⊂ZD be a family of C 2 functions such that ψX is localised in X ⊂⊂ ZD . Assume that ψX ≡ 0 whenever the diameter of X is greater than positive constant R. We will also assume that there exists a constant M ∈ (0, ∞) such that ψX ∞ M and ∇i ψX ∞ M for all i ∈ ZD . We say Ψ is a bounded potential of range R. For ω ∈ Ω, define HΛω (xΛ ) =

ψX (xΛ •Λ ω),

Λ∩X =∅

for xΛ = (xi )i∈Λ ∈ MΛ . Let−UU be a locally Lipschitz function on M which is bounded from below and such that dλ < ∞. Suppose also that ∀L 0 there exists r = r(L) such that Me {U L} ⊂ B(r). Let dμ =

e−U Z

dλ, so that μ is a probability measure on M, and let μΛ (dxΛ ) :=

μ(dxi )

i∈Λ

be the product measure on MΛ . Now define ω

EωΛ (dxΛ ) =

eJ HΛ (xΛ )

ω

eJ HΛ (xΛ ) μΛ (dxΛ ) μΛ (dxΛ ) ≡ J H ω (x ) ω ZΛ e Λ Λ μΛ (dxΛ )

(65)

for J ∈ R. We will write μ{i} = μi and Eω{i} = Eωi for i ∈ ZD . We finally define an infinite volume Gibbs measure ν on Ω to be a solution of the (DLR) equation: νE·Λ f = νf

(66)

for all bounded measurable functions f on Ω. ν is a measure on Ω which has EωΛ as its finite volume conditional measures. Following for example [22,24], the extension of Theorem 2.8 to this infinite dimensional setting will take the following form.

108


Theorem 6.1. Suppose that the classical Sobolev inequality (1) and that the Poincaré inequality in balls (22) are both satisfied. Suppose also that inequality (2) is satisfied, i.e. there exist constants A, B ∈ (0, ∞) such that

μ |f | |U |β + |∇U | Aμ|∇f | + Bμ|f | for some β ∈ (0, 1] and locally Lipschitz functions f : M → R. Then there exists J0 > 0 such that for |J | < J0 , the Gibbs measure ν is unique and there exists a constant C such that

EntΦ ν |f | Cν

|∇i f | ,

(67)

i∈ZD

where Φ(x) = x(log(1 + x))β , for all f for which the right-hand side is well defined. For notational simplicity, we will only prove Theorem 6.1 in the case R = 1 and D = 2, but the method can easily be extended to general R and D, (see e.g. [22] for the idea of the general scheme). Define the sets

Γ0 = (0, 0) ∪ j ∈ Z2 : dist j, (0, 0) = 2n for some n ∈ N , Γ1 = Z2 Γ0 . Note that dist(i, j) > 1 for all i, j ∈ Γk , k = 0, 1 and Γ0 ∩ Γ1 = ∅. Moreover Z2 = Γ0 ∪ Γ1 . For the sake of notation, we will write EΓk = EωΓk for k = 0, 1. We will also define P := EΓ1 EΓ0 . The proof will rely on the following few lemmata. Lemma 6.2. Under the conditions of Theorem 6.1, there exist constants cˆ0 and cˆ independent of i ∈ ZD and ω ∈ Ω such that Eωi f − Eωi f cˆ0 Eωi |∇i f |

(68)

ˆ ωi |∇i f | EntΦ Eω |f | cE

(69)

and

i

for all suitable functions f , i ∈ ZD and ω ∈ Ω. Proof. Firstly, by Theorem 2.7, we have that there exists a constant c0 independent of i such that μi |f − μi f | c0 μi |∇i f |. Since


109

osc Hiω 2Hiω ∞ 2 ψX ∞ 8M, {i}∩X =∅

by a standard result about bounded perturbations of Poincaré type inequalities (see [12]), inequality (68) holds. Moreover, by the assumptions and Theorem 2.8, we have

EntΦ μi |f | = μi Φ |f | − Φ μi |f | cμi |∇i f | for all i ∈ ZD . Thus by the bounded perturbation Corollary 2.6, (69) holds.

2

Lemma 6.3. Under the conditions of Theorem 6.1, there exists J0 > 0 such that for |J | < J0 , there exists a constant ε ∈ (0, 1) such that ν ∇Γk (EΓl f ) ν|∇Γk f | + εν|∇Γ1 f | for all suitable f and k, l ∈ {0, 1} such that k = l. Proof. We suppose k = 1 and l = 0. The case k = 0, l = 1 follows similarly. We can write ∇i (EΓ0 f ) ν ∇i (E{∼i} f ) ν ∇Γ1 (EΓ0 f ) = ν i∈Γ1

ν

|∇i f | + |J |ν

i∈Γ1

i∈Γ1

E{∼i} f [∇i H{∼i} − E{∼i} ∇i H{∼i} ]

i∈Γ1

where we have used (66) and denoted {∼ i} = {j: j ∼ i}. Now set Wi = Wi − E{∼i} Wi , where ω . Then since E Wi = ∇i H{∼i} {∼i} Wi = 0, we have that E{∼i} (f − E{∼i} f )Wi . ν ∇Γ1 (EΓ0 f ) ν |∇i f | + |J |ν i∈Γ1

(70)

i∈Γ1

Now, by our assumptions on the potential, we have Wi ∞ 8M for all i ∈ ZD , so that E{∼i} (f − E{∼i} f )Wi 8ME{∼i} |f − E{∼i} f |.

(71)

Note that by construction, E{∼i} is a product measure. Now by Lemma 6.2 together with Lemma 2.9 there exists a constant cˆ0 such that E{∼i} |f − E{∼i} f | cˆ0 E{∼i} |∇{∼i} f |. Using (71) and (72) in (70), we then arrive at

(72)

110


ν ∇Γ1 (EΓ0 f ) ν |∇i f | + 8M cˆ0 |J |ν |∇{∼i} f | i∈Γ1

=ν

i∈Γ1

|∇i f | + 32M cˆ0 |J |ν

i∈Γ1

Thus taking J0 =

1 32M cˆ0

proves the lemma.

|∇i f | .

i∈Γ0

2

Lemma 6.4. Under the conditions of Theorem 6.1, there exists J0 > 0 (given by Lemma 6.3) such that for |J | < J0 , P r f converges almost everywhere to νf , where we recall that P = EΓ1 EΓ0 . In particular ν is unique. Proof. The proof is standard: see for example Lemma 5.6 of [24].

2

Proof of Theorem 6.1. We may suppose f 0. Using (66), write

ν Φ(f ) − Φ(νf ) = νEΓ0 Φ(f ) − ν Φ(EΓ0 f )

+ ν Φ(EΓ0 f ) − Φ(νf )

Φ = ν EntΦ EΓ0 (f ) + ν EntEΓ1 (EΓ0 f )

+ ν Φ(EΓ1 EΓ0 f ) − Φ(νf ). Since probability measures EΓ0 and EΓ1 are product measures by construction, we have by Lemmas 2.9 and 6.2 that they both satisfy L1 Φ-entropy inequalities with constant c. ˆ Therefore, the above yields

ν Φ(f ) − Φ(νf ) cν|∇ ˆ ˆ ∇Γ1 (EΓ0 f ) Γ0 f | + cν

+ ν Φ(Pf ) − Φ(νf ). We can similarly write

2

Φ μ Φ(Pf ) = ν EntΦ EΓ0 (Pf ) + ν EntEΓ1 (EΓ0 Pf ) + ν Φ P f

ˆ ∇Γ1 (EΓ0 f ) + ν Φ P 2 f . cν|∇ ˆ Γ0 Pf | + cν Repeating this process, after r steps we see that r−1 r−1

ν Φ(f ) − Φ(νf ) cˆ ν ∇Γ0 P k f + cˆ ν ∇Γ 1 E Γ 0 P k f k=0

(73)

k=0

+ ν Φ P r f − Φ(νf ). We may control the first and second terms using Lemma 6.3. Indeed

(74)


111

ν ∇Γ0 P k f ε 2 ν ∇Γ0 P k−1 f ε 2k−1 ν|∇Γ1 EΓ0 f | ε 2k−1 ν|∇Γ1 f | + ε 2k ν|∇Γ0 f |.

(75)

Similarly

ν ∇Γ1 EΓ0 P k f ε 2k ν|∇Γ1 f | + ε 2k+1 ν|∇Γ0 f |.

(76)

Using (75) and (76) in (73) yields r−1 r−1

−1 2k 2k+1 ν Φ(f ) − Φ(νf ) cˆ 1 + ε ε ε ν|∇Γ1 f | + ν|∇Γ0 f | k=0

k=0

+ ν Φ P r f − Φ(νf ).

By Lemma 6.4 we have that limr→∞ P r f = νf , ν-almost surely. Therefore taking the limit as r → ∞ in the above (which exists since ε ∈ (0, 1)) yields

ν Φ(f ) − Φ(νf ) Cν|∇ZD f | −1

. where C = cˆ 1+ε 1−ε 2

2

Next, we consider (IFI 2 ) for a family of examples. In particular we restrict ourselves to a situation when M is an H-type group and assume that for i ∈ ZD Ui ≡

p−k

αk di

≡

k=0,...,p−1

αk d p−k (ωi )

(77)

k=0,...,p−1

with d(·) denoting the Carnot–Carathéodory distance from the unit element, p 2, where α0 ∈ (0, ∞) and αk ∈ R. As above we consider an interaction HΛω (xΛ ) =

ψX (xΛ •Λ ω),

(78)

Λ∩X =∅

which is assumed to be bounded with bounded (sub-)gradient and for simplicity is of finite range, as specified at the beginning of the current section. Moreover we are given a family of regular conditional expectations defined by (6.1). Combining the previous results with those of this section the previous we arrive at the following theorem. Theorem 6.5. Suppose p 2. Then there exists J0 > 0 such that for |J | < J0 the unique Gibbs measure ν corresponding to the interaction (77)–(78) satisfies the following inequalities: (i) EntΦ ν

|f | C1 ν

i∈ZD

|∇i f | ,

(79)

112

J. Inglis et al. / Journal of Functional Analysis 260 (2011) 76–116 1

where Φ(x) = x(log(1 + x)) q , q1 + p1 = 1, with some constant C1 ∈ (0, ∞), for any f for which the right-hand side is well defined; (ii) 1 2 U2 (νf ) ν U2 (f )2 + C2 |∇i f |2

(80)

i∈ZD

where U2 is the Gaussian isoperimetric profile function (as defined in Section 3), with some constant C2 ∈ (0, ∞) for any function 0 f 1 for which the right-hand side is well defined. Proof. To begin we notice that the reference measure dμ satisfies a U -bound, and therefore the conditional expectation (as a perturbation of the reference measure by strictly bounded and strictly positive density), also satisfies the following inequality

1

f |U | q dEi A H

|∇i f | dEi + B

H

(81)

f dEi H

with some constants A, B ∈ (0, ∞) independent of i and ωj , where Ei denotes the corresponding conditional expectations. Thus we can apply Theorem 4.5 to conclude that the Ei ’s satisfy Cheeger’s inequality, as well as L1 Φ-entropy and (IFI 2 ) bounds with constants independent of i and ωj ’s. With this bound the proof of (ii) follows via the strategy developed in [33]. 2 Remark 6.6. We remark that once the conditional measures satisfy L1 Φ-entropy or (IFI 2 ) inequalities with constants independent of external conditions, one can show that the Gibbs measure also satisfies (IFI 2 ) even when the interactions Hi contain an unbounded component, provided we have Cheeger’s inequality and appropriate U -bounds. In particular one obtains the following generalisation of the results of [33] where only the bounded interaction case was studied. Theorem 6.7. Suppose M ≡ R and that U is a semibounded polynomial of degree at least 2. Let Hiω (xi ) ≡ ε

ψX (xi •i ω) + ε

{i}∩X =∅

Gij xi ωj

j

with ψX satisfying conditions of Theorem 6.5, j |Gij | < ∞ and ε ∈ (0, ∞). Then, if ε ∈ (0, ∞) is sufficiently small, the corresponding Gibbs measure satisfies (IFI 2 ). Remark 6.8. For cylinder functions dependent on N coordinates, adapting the length of the gradient in part (i) of Theorem 6.5, we get √

EntΦ ν |f | C1 N ν

il

∈ZD ,l=1,...,N

1 |∇il f |2

2

(82)


113

for all functions f for which the right-hand side is well defined. Now, choosing a Lipschitz approximation of a cylinder set AN (specified by conditions on coordinates ωil , l = 1, . . . , N ), by Theorem 3.1 we arrive at √

Uq ν(AN ) c˜ N ν2+ (AN )

(83)

with suitable constant c˜ ∈ (0, ∞) independent of N , and with use of the subscript 2 on the right-hand side to emphasise that we have here the surface measure with respect to the quadratic distance. On the other hand using part (ii) of Theorem 6.5 yields

U2 ν(AN ) C2 ν2+ (AN ).

(84)

Thus we obtain a potentially useful tool for optimisation of isoperimetric relations for finite dimensional marginals of the measure ν. Acknowledgment We are very grateful to the referee for their helpful suggestions and careful reading of this manuscript. Appendix A. Exponential bounds Suppose for dμ ≡ e−U dλ/Z, with U ε, for some ε > 0, and Z a normalisation constant, we have

μ f U β Cμ|∇f | + Dμf for all locally Lipschitz functions f . In particular, for a Lipschitz cut-off function 0 < ε UL U , for f ≡ eλUL ULα , with α, β > 0, α + β = 1, we have

β

μ eλUL UL = μ eλUL ULα · UL Cμ∇ eλUL ULα + Dμ eλUL ULα

λCμ eλUL ULα · |∇UL | + αCμ eλUL ULα−1 · |∇UL |

+ Dμ eλUL ULα . If we assume that β

|∇UL | aUL with a ∈ (0, ∞) independent of L, then we get

β

μ eλUL UL λCμ eλUL ULα · aUL

β

+ αCμ eλUL ULα−1 · aUL + Dμ eλUL ULα

λaCμ eλUL UL + αaCμ eλUL + Dμ eλUL ULα .

114


Using our assumption that UL ε > 0 and a bound ULα λδUL + A(λδ) with some δ, A(λδ) ∈ (0, ∞) independent of L, we get

μ eλUL UL λ(aC + Dδ)μ eλUL UL + αaC + D · A(λδ) μ eλUL . Hence for λ ∈ (0, λ0 ), with λ0 ≡ (aC + Dδ)−1 , we have

d λUL

μ e = μ eλUL UL Bμ eλUL dλ with

−1 B ≡ B(λ0 , δ) ≡ αaC + D · A(λδ) 1 − λ0 (aC + Dδ) . Solving this differential inequality for λ ∈ (0, λ0 ), we obtain

μ eλUL eλB . Since the constant B is independent of L, by the dominated convergence theorem we obtain the following bound

μ eλU eλB , which holds true for λ ∈ (0, λ0 ). Appendix B. Rothaus argument Here we give a brief outline of how to tighten a defective q-logarithmic Sobolev inequality under the additional assumption of a q-Poincaré inequality. To do this we need the following results from [12] which generalise the so-called Rothaus argument. Suppose that q ∈ [1, 2] and define the Orlicz space LNq (μ) generated by the function Nq (x) = x q log(1 + x q ) to be the space of measurable functions f such that f q f Nq := inf λ > 0: dμ 1 < ∞. Nq λ Lemma B.1. (See [12].) If f 0 ∈ LNq (μ) then

q f Nq

fq μ f log +μ fq . μ(f q ) q

Lemma B.2. (See [12].) If f ∈ LNq (μ) and μ(f ) = 0 then sup a∈R

|f + a|q log

|f + a|q q dμ 16f Nq . μ|f + a|q


115

With these results in hand, suppose that we have a defective (LSq ) inequality |f |q μ |f |q log Cμ|∇f |q + Dμ|f |q , μ|f |q

(85)

as well as a q-Poincaré inequality μ|f − μf |q Kμ|∇f |q .

(86)

By Lemma B.1 applied to the function |f − μf | we have |f − μf |q q + μ|f − μf |q f − μf Nq μ |f − μf |q log μ|f − μf |q Cμ|∇f |q + (D + 1)μ|f − μf |q

C + K(D + 1) μ|∇f |q , where we have first used (85) and then (86). By applying Lemma B.2 to f − μf we obtain |f |q |f − μf + μf |q q q = μ |f − μf + μf | log μ |f | log μ|f |q μ|f − μf + μf |q q

16f − μf Nq

16 C + K(D + 1) μ|∇f |q , so that we arrive at the desired tight form of the logarithmic Sobolev inequality. References [1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Panor. Synthèses, vol. 10, Soc. Math. France, Paris, 2000. [2] D. Bakry, F. Baudoin, M. Bonnefont, D. Chafaï, On gradient bounds for the heat kernel on the Heisenberg group, J. Funct. Anal. 255 (2008) 1905–1938. [3] D. Bakry, M. Ledoux, Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator, Invent. Math. 123 (1996) 259–281. [4] F. Barthe, Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance, J. Theoret. Probab. 17 (2) (2004) 293–308. [5] F. Barthe, P. Cattiaux, C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Rev. Mat. Iberoamericana 22 (3) (2006) 993–1067. [6] F. Barthe, B. Maurey, Some remarks on isoperimetry of Gaussian type, Ann. Inst. H. Poincaré Probab. Statist. 36 (4) (2000) 419–434. [7] R. Beals, B. Gaveau, P. Greiner, Hamilton–Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl. 79 (7) (2000) 633–689. [8] S. Bobkov, A functional form of the isoperimetric inequality for the Gaussian measure, J. Funct. Anal. 135 (1) (1996) 39–49. [9] S. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab. 25 (1) (1997) 206–214. [10] S. Bobkov, C. Houdré, Isoperimetric constants for product probability measures, Ann. Probab. 25 (1) (1997) 184– 205. [11] S. Bobkov, C. Houdré, Some Connections Between Isoperimetric and Sobolev-Type Inequalities, Mem. Amer. Math. Soc., vol. 616, 1997.

116


[12] S. Bobkov, B. Zegarliński, Entropy Bounds and Isoperimetry, Mem. Amer. Math. Soc., vol. 829, 2005. [13] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monogr. Math., Springer-Verlag, 2007. [14] P. Cattiaux, N. Gozlan, A. Guillin, C. Roberto, Functional inequalities for heavy tails distributions and application to isoperimetry, preprint. [15] D. Chafaï, Entropies, convexity and functional inequalities: on φ-entropies and φ-Sobolev inequalities, J. Math. Kyoto Univ. 44 (2) (2004) 325–363. [16] B. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (2) (2005) 340–365. [17] N. Eldredge, Precise estimates for the subelliptic heat kernel on H-type groups, J. Math. Pures Appl. 92 (2009) 52–85. [18] N. Eldredge, Gradient estimates for the subelliptic heat kernel on H-type groups, J. Funct. Anal. 258 (2) (2010) 504–533. [19] P. Fougères, Hypercontractivité et isopérimétrie gaussienne. Applications aux systémes de spins, Ann. Inst. H. Poincaré Probab. Statist. 36 (5) (2000) 647–689. [20] P. Fougères, C. Roberto, B. Zegarliński, Sub-Gaussian measures and associated semilinear problems, preprint. [21] A. Grigor’yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1997) 33–52. [22] A. Guionnet, B. Zegarliński, Lectures on logarithmic Sobolev inequalities, in: Séminaire de Probabilités, XXXVI, in: Lecture Notes in Math., vol. 1801, Springer-Verlag, 2003, pp. 1–134. [23] W. Hebisch, B. Zegarliński, Coercive inequalities on metric measure spaces, J. Funct. Anal. 258 (2010) 814–851. [24] J. Inglis, I. Papageorgiou, Logarithmic Sobolev inequalities for infinite dimensional Hörmander type generators on the Heisenberg group, J. Pot. Anal. 31 (1) (2009) 79–102. [25] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (2) (1986) 503–523. [26] M. Ledoux, Isopérimétrie et inégalités de Sobolev logarithmiques Gaussiennes, C. R. Acad. Sci. Paris Sér. I Math. 306 (2) (1988) 79–82. [27] H.-Q. Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Funct. Anal. 236 (2) (2006) 369–394. [28] P. Ługiewicz, B. Zegarliński, Coercive inequalities for Hörmander type generators in infinite dimensions, J. Funct. Anal. 247 (2) (2007) 438–476. [29] E. Milman, On the role of convexity in functional and isoperimetric inequalities, Proc. Lond. Math. Soc. 99 (3) (2009) 32–66. [30] O. Rothaus, Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Funct. Anal. 64 (1985) 296–313. [31] L. Saloff-Coste, Aspects of Sobolev-Type Inequalities, London Math. Soc. Lecture Note Ser., vol. 289, Cambridge University Press, 2002. [32] N.T. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math., vol. 100, Cambridge University Press, 1992. [33] B. Zegarliński, Isoperimetry for Gibbs measures, Ann. Probab. 29 (2001) 802–819.


Spectral triples and manifolds with boundary B. Iochum a,b , C. Levy c,∗ a Centre de Physique Théorique, 1 CNRS–Luminy, Case 907, 13288 Marseille Cedex 9, France b Université de Provence, France c University of Copenhagen, Department of Mathematical Sciences, Universitetsparken 5, DK-2100 Copenhagen,

Denmark Received 22 January 2010; accepted 15 September 2010 Available online 25 September 2010 Communicated by Alain Connes

Abstract We investigate manifolds with boundary in noncommutative geometry. Spectral triples associated to a symmetric differential operator and a local boundary condition are constructed. We show that there is no tadpole for classical Dirac operators with a chiral boundary condition on spin manifolds. © 2010 Elsevier Inc. All rights reserved. Keywords: Boundary pseudodifferential operators; Dirac operators; Elliptic operators

1. Introduction Noncommutative geometry has, in its numerous motivations, the conceptual understanding of different aspects of physics [15,21]. In particular, the spectral approach which is deeply encoded in the notion of a spectral triple, is not only motivated by the algebra of quantum observables A acting on the Hilbert space H of physical states, but also by classical physics like general relativity. For instance, a Riemannian compact spin manifold can be reconstructed only via properties of a commutative spectral triple (A, H, D) [14] where the last piece D is a selfadjoint operator * Corresponding author.

E-mail addresses: [email protected] (B. Iochum), [email protected] (C. Levy). 1 UMR 6207: – Unité Mixte de Recherche du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de

l’Université du Sud Toulon-Var (Aix-Marseille Université); – Laboratoire affilié à la FRUMAM – FR 2291. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.006

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B. Iochum, C. Levy / Journal of Functional Analysis 260 (2011) 117–134

acting on H playing the role of a Dirac operator which can fluctuates: D is then replaced by DA := D + A where A is a selfadjoint one-form. The spectral action S of Chamseddine and Connes [11] associated to a triple (A, H, D) is the 2 /Λ2 ) where Φ is a positive function and Λ plays the role of a cut-off. This can be trace of Φ(DA written (under some conditions on the spectrum) as a series of noncommutative integrals S(DA , Φ, Λ) = (1) Φk Λk − |DA |−k + Φ(0)ζDA (0) + OΛ→∞ Λ−1 k∈Sd+

∞ where Φk = 12 0 Φ(t)t k/2−1 dt, Sd+ is the strictlypositive part of the dimension spectrum of the spectral triple and the noncommutative integral − X for X in the algebra Ψ (A) of pseudod ifferentialoperators, is defined by − X := Ress=0 Tr(X|D|−s ). Since − is a trace on Ψ (A) (non-necessarily positive), it coincides (up to a scalar) with the Wodzicki residue [58,59] in the case of a commutative geometry where A is the algebra of C ∞ functions on a manifold M without boundary: in a chosen coordinate system and local trivialization (x, ξ ) of T ∗ M, this residue is X Wres(X) := Tr σ−d (x, ξ ) b|dξ ||dx|, M Sx∗ M X is the symbol of the classical pseudodifferential operator X which is homogeneous of where σ−d degree −d := −dim(M), dξ is the normalized restriction of the volume form to the unit sphere Sx∗ M Sd−1 . The Dixmier’s trace Trω [23] concerns compact operators X with singular values {μk } satisfying supN →∞ aN < ∞ where aN = log(N )−1 N k=1 μk and Trω (X) is defined after a choice of an averaging procedure ω such that Trω (X) = limN aN when aN converges. As shown in [19], − coincides (still up to a universal scalar) with Trω when X has order −d. When M has a boundary, the choice of an appropriate differential calculus is delicate. In the noncommutative framework, the links between Boutet de Monvel’s algebra, Wodzicki’s residue, Dixmier’s trace or Kontsevich–Vishik’s trace [42] have been clarified [1,25,32,33,48,52] including the case of log-polyhomogeneous symbols [45]. From a physics point of view, applications of noncommutative integrals on manifolds to classical gravity has begun with Connes’ remark that − D−2 coincides in dimension 4 with Einstein–Hilbert action, a fact recovered in [40,41]. Then, a generalization to manifolds with boundaries was proposed in [53,55–57]. From the quantum side, a noncommutative approach of the unit disk is proposed in [10]. Nevertheless, a construction of spectral triples in presence of boundary is not an easy task, although a general approach of boundary spectral triples has been announced in [20]. First examples appear with isolated conical singularity in [46], a work related to some extend to [44,51] when the spectrum dimension is computed. The difficulty is to find an appropriate boundary condition which preserves not only the selfadjointness of the realization of D but also the ellipticity. A special case of boundary also appears in non-compact manifolds when one restricts the operator D to a bounded closed region: for instance, this trick was used in [50,60]. The choice of a chiral boundary condition, already considered in [7,8] for mathematical reasons, is preferred in [13] for physical reasons: firstly, it is consistent with a selfadjoint and elliptic realization, and secondly, it is a local boundary condition contrary to the standard APS’ one


119

which is global [2]. Thirdly, it gives a similar ratio and signs for the second term of the spectral action (the first one being the cosmological constant), namely the scalar curvature of the manifold and the extrinsic curvature of the boundary, as in the Euclidean action used in gravitation [34]. Since there are a lot of possible choices, this last consideration deserves attention. Here, we first show a construction for manifolds with boundary that actually produces a spectral triple, and then give conditions on the algebra of functions on that manifold to get a regular triple (remark that the spectral action has only been computed, until now, for spectral triples which are regular). While in field theory, the one-loop calculation divergences, anomalies and different asymptotics of the effective action are directly obtained from the heat kernel method [26,27,54], we try to avoid this perturbation approach already used in [39] to prove that there are no tadpoles when a reality operator J exists. Tadpoles are the A-linear terms in (1), like for example − AD−1 . In quantum field theory, D−1 is the Feynman propagator and AD−1 is a one-loop graph with fermionic internal line and only one external bosonic line A looking like a tadpole. In Section 2, we derive a technical result on regularity of spectral triples which is sufficient to avoid the use of Sobolev spaces of negative order. Then, we recall few basics on the realization of boundary pseudodifferential operators and their stability by powers using the Grubb’s approach [30]. In Section 4, we define an algebra APT compatible with the realization of an elliptic pseudodifferential boundary system {P , T }. A condition on P is given which guarantees the regularity of the associated spectral triple. The motivating example of a classical Dirac operator is considered in Section 4.2. Moreover, the construction of a spectral triple on the boundary is revisited in Section 4.3. Section 5 is devoted to a reality operator J on a spectral triple with boundary and some consequences on the tadpoles like − AD−1 which can appear in spectral action. 2. Regularity Let N be the non-negative integers and B(H) be the set of bounded operator on a separable Hilbert space H. We shall use the following definition of a spectral triple: Definition 2.1. A spectral triple of dimension d is a triple (A, H, D) such that H is a Hilbert space and – A is an involutive unital algebra faithfully represented in B(H), – D is a selfadjoint operator on H with compact resolvent and its singular values (μn (|D|))n are O(n1/d ), – for any a ∈ A, a Dom D ⊆ Dom D and the commutator [D, a] (with domain Dom D) as an extension in B(H) denoted da. Note that Dom |D| = Dom D. We set δ(T ) := [|D|, T ], where A is the closure of the operator A, with domain

Dom δ := T ∈ B(H): T Dom D ⊆ Dom D and |D|, T has closure in B(H) . Definition 2.2. A spectral triple (A, H, D) is said to be regular if A and dA are included in n n∈N Dom δ .

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As seen in next lemma, it is quite convenient to introduce the following Definition 2.3. Given a selfadjoint operator P on a Hilbert space H, let

HP∞ :=

Dom |P |k =

k1

Dom P k .

k1

A linear map from HP∞ into itself which is continuous for the topologies induced by H is said to be HP∞ -bounded. Note that HP∞ is a core for any power of P or |P |. In particular it is a dense subset of H and |P | is the closure of the essentially selfadjoint operator |P ||HP∞ . Note also that for any k ∈ N, (1 + P 2 )k/2 is a bijection from Dom P k onto H, and thus, (1 + P 2 )−k/2 is a bijection from H onto Dom P k . As a consequence, for any p ∈ Z, the operators (1 + P 2 )p/2 send bijectively HP∞ onto itself, and for any k ∈ N, |P |k send HP∞ into itself. Given a selfadjoint P , let δ , δ1 be defined on operators by

δ (T ) := |P |, T ,

−1/2

δ1 (T ) := P 2 , T 1 + P 2

with domains Dom δ (resp. Dom δ1 ) := T ∈ HP∞ -bounded operators: δ (T ) resp. δ1 (T ) is HP∞ -bounded . We record the following lemma, proven by A. Connes: Lemma 2.4. (See [14, Lemmas 13.1 and 13.2].) (i) If T ∈ Dom δ , then the bounded closure T of T is in Dom δ and δ(T ) = δ (T ). So, by induction, if T ∈

n∈N Dom δ

n

then T ∈

(ii) Let T be an HP∞ -bounded operator. If T ∈

n∈N Dom δ

n∈N Dom δ1

In particular the bounded closure T of T belongs to

n.

n,

then T ∈

n∈N Dom δ

n∈N Dom δ

n.

n.

Definition 2.5. Given a Hilbert space H and a selfadjoint (possibly unbounded) operator P , we call Sobolev scale on N, a family (H k )k∈N of Hilbert spaces such that – H 0 = H, – H k+1 is continuously included in H k for any k ∈ N, – for any k ∈ N, Dom P k is a closed subset of H k . By closed graph theorem, the last point implies that P k is continuous from Dom P k endowed with the H k -topology into H. Similar abstract Sobolev scales, defined as domains of the powers of an abstract differential operator, have been considered in [36]. A corresponding criterion for regularity has been obtained in [36, 4.26 Theorem]. The scale we consider here will correspond in Section 4 to the


121

Sobolev spaces (on the manifold with boundary) and not to the domains of the powers of the realization of a first order pseudodifferential operator. In the case without boundary, these scales coincide. When T is an HP∞ -bounded operator, we shall denote T (k) := [P 2 , ·]k (T ) for any k ∈ N. Lemma 2.6. Let P be a selfadjoint operator and T be an HP∞ -bounded operator. Suppose that there is a Sobolev scale (H k )k∈N such that for any k ∈ N, T (k) is continuous from HP∞ with the H k -topology into HP∞ with the H-topology. Then T ∈ n∈N Dom δ n . Proof. The operator (P − i)k = kj =0 jk (−i)k−j P j is continuous from Dom P k with the H k topology into H. Since P is selfadjoint, P − i is a bijection from Dom P onto H, and by composition, (P − i)k is a bijective map from Dom P k onto H. The inverse mapping theorem now implies that (P − i)−k is continuous from H onto Dom P k with the H k -topology. Moreover, (1 + P 2 )−k/2 = (P − i)−k B where B := (P − i)k (1 + P 2 )−k/2 is, by spectral theory, a bijective operator in B(H). As a consequence, (1 + P 2 )−k/2 is continuous from H onto Dom P k with the H k -topology. In particular,(1 + P 2 )−k/2 is continuous from HP∞ with the H-topology, into HP∞ with the H k -topology. So the hypothesis gives that T (k) (1 + P 2 )−k/2 = ([P 2 , ·](1 + P 2 )−1/2 )k (T ) = δ1k (T ) is an ∞ HP -bounded operator. The result follows from Lemma 2.4. 2 The previous lemma essentially implies that, in order to prove the regularity of a spectral triple (A, H, D), it is sufficient to construct a Sobolev scale (H k )k∈N adapted to H and D, such that the operators Dk and T (k) behave respectively as operators of “order” k with respect to the Sobolev scale, when T is any element of A ∪ dA. Remark 2.7. By Lemma 2.4, it is possible to obtain regularity without using Sobolev spaces of negative order, implicitly used for instance in [29, Theorem 11.1] which follows the original argument of [17]. This shall considerably simplify the proof of the regularity of the spectral triple on manifolds with boundary, since the continuity of realizations of elliptic boundary differential operators is usually established on Sobolev spaces of positive order [5,6,9,30]. Note however that it may be possible to deal with negative orders by using the technique of transposition described in [47]. Another approach of regularity can be found in [35,36,49]. 3. Background on elliptic systems on manifolds with boundary We review in this section a few definitions and basic properties about Sobolev spaces in manifolds with and without boundary and boundary pseudodifferential operators choosing Boutet de Monvel’s calculus. More details and proofs can be found in classical references like [30,38]. be a smooth compact manifold without boundary of dimension d and E be a smooth Let M Hermitian vector bundle on M. Let M be an open submanifold of M of dimension d such that M (topological closure) is a compact manifold with nonempty boundary N := ∂M = M\M. without boundary of dimenAs a consequence, N is a smooth compact submanifold of M sion d − 1.

122


on M (resp. N ) is denoted E (resp. EN ). We denote H s (E), H s (E) the The sub-bundle of E with bundle E and M with bundle E. Recall Sobolev spaces of order s ∈ R respectively on M that by definition H s (E) := H s (M, E) := r + H s (E) where r + is the restriction to M. We refer to [30, p. 496] for the definition of the topology of H s (E). Remark that, for a given manifold M with boundary ∂M, it is always possible to construct with previous properties. Moreover, there exist constructions of invertible double for Dirac M operators and more general first order elliptic operators on closed double of M [3–6]. (resp. Diffk (E)) the space of pseudodifferential (resp. differential) operaWe denote Ψ k (E) E). Any element of Ψ k (E) is a linear continuous operator from H s (E) tors of order k on (M, for any s ∈ R. into H s−k (E), We set E) , C ∞ (M, E) := r + C ∞ (M,

. C ∞ (M) := r + C ∞ (M)

Despite notations, note that the spaces H s (E), C ∞ (M, E) and C ∞ (M) are spaces of functions defined on M, and not on M, as r + is the restriction on M. Remark that C ∞ (M) is identified to C ∞ (M) IdE|M , so that C ∞ (M) can be seen as an algebra of bounded operators on H s (E), and E). in particular on H 0 (E) = L2 (M, E) = r + L2 (M, A differential operator P on M is by definition a differential operator on M with coefficients in C ∞ (M, E). We denote Diffk (M, E) the space of differential operators of order k ∈ N on M. Any element of Diffk (M, E) can be extended uniquely as a linear continuous operator from s H (E) into H s−k (E), for any s ∈ R. E) is dense in any H p (E) and C ∞ (M, E) is also dense in any Finally, note that C ∞ (M, p + p and moreover, H (E) := r H (E),

p∈N

= C ∞ (M, E), H p (E)

H p (E) = C ∞ (M, E).

p∈N

The extension by zero operator e+ is a linear continuous operator from H s (E) into H s (E) 1 1 + + ∞ for any s ∈ ]− 2 , 2 [ such that e (u) = u on M and e (u)(x) = 0 for any u ∈ C (M, E) and x ∈ M\M. we define its truncation to M by For any P ∈ Ψ k (E), P+ := r + P e+ . k ∈ Z, is said to satisfy the transmission condition if P+ maps Recall that P ∈ Ψ k (E), into itself which means that “P+ preserves C ∞ up to the boundary”. In particular, any differential operator satisfies this condition. It turns out that if P satisfies the transmission condition, P+ can be seen as linear continuous operator from H s (E) into H s−k (E) for any s > − 12 [30, 2.5.8 Theorem, 2.5.12 Corollary]. C ∞ (M, E)


123

We refer to [30,31] for all definitions of elliptic boundary system, normal trace operator and singular Green operator but to fix notations, we recall the Green formula [30, 1.3.2 Proposition]: k ∈ N and for any u, v ∈ C ∞ (M, E), for P ∈ Ψ k (E), (P+ u, v)M − u, P ∗ + v M = (AP ρu, ρv)N

(2)

where (u, v)X := X u(x)v(x) dx (if defined) and ρ = {γ0 , . . . , γk−1 } is the Cauchy boundary operator given by γj u = (−i∂d )j u|N (∂d being the interior normal derivative) and AP is the Green matrix associated to P . (x , 0) ∈ N and xd denotes a normal Here, x = (x , xd ) is an element of M = N M with coordinate. By [30, 1.3.1 Lemma], P = A + P with A = kl=0 Sl (−i∂d )l , where Sl is a tangential differential operator of order k − l supported near N = ∂M, and P is a pseudodifferential operator of order k satisfying (P+ u, v)M − (u, (P ∗ )+ v)M = 0. The Green matrix satisfies AP = (Aj,l )j,l=0,...,k−1

with Aj,l x , D := iSj +l+1 x , 0, D + lower-order terms,

and Aj l is zero if j + l + 1 > k. Remark 3.1. When P is a pseudodifferential operator of order 1, AP is an endomorphism on the boundary N . For instance, if P is a classical Dirac operator acting on a Dirac bundle, then AP = −iγd where γ is the (selfadjoint) Clifford action and {ei }, i = 1, . . . , d is a (local) orthonormal frame of TM such that ed is the inward pointing unit normal, and γi := γ (ei ). This AP corresponds to −J0 in [4,6]. n ∈ N, satisfies the transmission condition, G is a singular Green operator When P ∈ Ψ n (E), of order n and class n and T := {T 0 , . . . , T n−1 } is a system of normal trace operators associated with the order n, each T i going from E to EN , then the (H n )-realization of the system {P+ + G, T } is the operator (P + G)T defined as the operator acting like P+ + G with domain Dom(P + G)T := ψ ∈ H n (M, E): T ψ = 0 . These realizations are always densely defined, and since T is continuous from H n (E) into n−1 n−j − 1 2 (EN ), Dom(P + G)T is a closed subset of H n (E). Recall that P+ + G is continj =0 H uous from H s (E) into H s−n (E) for any s > n − 12 . We shall assume that all pseudodifferential operators satisfy the transmission property at the boundary. We record here the following proposition, which is a direct application of [30, 1.4.6 Theorem, 2.5.12 Corollary, 2.7.8 Corollary]: Proposition 3.2. Let {P+ + G, T } be an elliptic system of order n (G being with class n) with T a system of normal trace operators associated with the order n. Then for any k ∈ N, there exist a singular Green operator Gk of class nk and a system of normal trace operators Tk associated with the order nk such that ((P + G)T )k is the realization of the elliptic system {(P k )+ + Gk , Tk } of order nk. Moreover, Dom ((P + G)T )k is a closed subset of H nk (E), and (P k )+ + Gk is continuous from H s (E) into H s−nk (E) for s > nk − 12 .

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the leftover of P and P When P , P are in Ψ ∞ (E), L P , P := P P + − P+ P+ is a singular Green operator of order k + k and class k + k when the order of P and P are k and k . The following result is a consequence of [30, (2.6.27)], but we give a short proof. Lemma 3.3. Let P , P ∈ Ψ ∞ (E). (i) If P is differential, then L(P , P ) = 0. E) (differential operator of order 0), then L(P , P ) = 0. (ii) If P is an endomorphism on (M, Proof. (i) From the locality of differential operators, we see that r + P e+ r + = r + P . It follows that L(P , P ) = 0. (ii) Since e+ r + P e+ = P e+ , the result follows. 2 4. Spectral triples on manifolds with boundary and M 4.1. Spectral triples on M Since it is a first step to the main theorem of this section, we record the following known fact in noncommutative geometry [22]: any elliptic pseudodifferential operator of first order on compact manifolds, whose square has a scalar principal symbol, yields a regular spectral triple with the algebra of smooth functions. is said to be scalar when it is of the Recall that the principal symbol σd (P ) of P ∈ Ψ d (E) ∞ ∗ form σ IdE with σ ∈ C (T M, C). be an elliptic symmetric pseudodifferential operator of order Proposition 4.1. Let P ∈ Ψ 1 (E) such that the principal symbol of P 2 is scalar. Then (C ∞ (M), L2 (E), P ) is a regular one on M spectral triple of dimension d. is an elliptic symmetric operator on L2 (E), it is selfadjoint with doProof. Since P ∈ Ψ 1 (M) Any a ∈ C ∞ (M) is represented by the left multiplication operator on L2 (E) which main H 1 (E). is bounded. Since a is scalar, the commutator [P , a] is a pseudodifferential operator of order 0 and thus can be extended as a bounded operator on L2 (E). for any k ∈ N, where here P k is the composition Ellipticity implies that Dom P k = H k (E) → L2 (E) as unbounded operators in L2 (E). The operator (P − i)d is of operators P : H 1 (E) d thus continuous and bijective from H (E) onto H, and by inverse mapping theorem (P − i)−d By a classical result (see for instance [30, A.4 Lemma]), is continuous from H onto H d (E). −d (P − i) is compact, and the dimension of the given triple is d. E) so that, for any It remains to check the regularity. By Sobolev lemma, HP∞ = C ∞ (M, ∞ ∞ a ∈ C (M), a|HP∞ and da|HP∞ , are HP -bounded operators. Since the principal symbol of P 2 is scalar, (a|HP∞ )(k) and (da|HP∞ )(k) are pseudodifferential operators of order k defined on HP∞ . k∈N yields the result. 2 Applying now Lemma 2.6 with the Sobolev scale (H k (E))


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In the case of manifolds with boundary, the full algebra C ∞ (M) cannot yield, in general, regular spectral triples on M, because there is a conflict between the necessity of selfadjointness for the realization PT which is implemented by a boundary condition given by a trace operator T , and the fact that the elements of the algebra must preserve all the domains Dom PT k . Therefore, we have to consider a subalgebra of C ∞ (M) that will be adapted to a realization PT : Definition 4.2. Let {P+ , T } be an elliptic pseudodifferential boundary system of order one, where T = Sγ0 is a normal trace operator, with γ0 : u → u|N and S an endomorphism P ∈ Ψ 1 (E), of EN . Suppose moreover that PT is selfadjoint (to apply Definition 2.3). We define APT as the ∗-algebra of smooth functions a ∈ C ∞ (M) such that aHP∞T ⊆ HP∞T ,

a ∗ HP∞T ⊆ HP∞T .

Remark 4.3. Note that for any a ∈ C ∞ (M), we have a Dom PT ⊆ Dom PT . As a consequence, when a ∈ APT , [PT , a] is an operator with domain Dom PT , which sends HP∞T into itself. The following lemma provides some lower bounds to APT . Lemma 4.4. (i) APT contains the algebra B := a ∈ C ∞ (M): T d k a, T d k a ∗ ∈ Ψ ∞ (EN )T , for any k ∈ N where d k := [P+ , ·]k . (ii) If P is a differential operator, APT contains the smooth functions that are constant near the boundary. Proof. (i) Suppose that a ∈ B. Since T a = a|N T , we directly check that a Dom PT ⊆ Dom PT . j By induction, for any j ∈ N, [P+ j , a] = i=1 cij d i (a)P+ j −i where cij are scalar coefficients. Choose k ∈ N and ψ ∈ Dom PT k . Thus, ψ ∈ H k (E) and for any 0 j k − 1, T (P+ j ψ) = 0. So, there are Ri ∈ Ψ ∞ (EN ) such that cij T d i (a)P+ j −i ψ = cij Ri T P+ j −i ψ = 0 T P+ j aψ = j

j

i=1

i=1

which proves that aψ ∈ Dom PTk since aψ ∈ H k (E). The same can be obtained for a ∗ . (ii) If a is a smooth function constant near the boundary, there is λ ∈ C and f ∈ Cc∞ (M) with compact support such that a = λ1M + f , where 1M is the function equal to 1 on M. The result follows from inclusions Cc∞ (M) Dom PTk ⊆ Hck (E) ⊆ Dom PTk . 2 Here is the main result of this section: be a symmetric pseudodifferential operator of order one on M Theorem 4.5. Let P ∈ Ψ 1 (E) satisfying the transmission condition.

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Let S ∈ C ∞ (N, End(EN )) be an idempotent selfadjoint endomorphism on the boundary such that the system {P+ , T := Sγ0 } is an elliptic pseudodifferential boundary operator. Then (i) PT is selfadjoint if and only if (1 − S)AP (1 − S) = 0 and SA−1 P S = 0.

(3)

(ii) When PT is selfadjoint, (C ∞ (M), L2 (E), PT ) is a spectral triple of dimension d. (iii) When P is a differential operator such that P 2 has a scalar principal symbol and PT is selfadjoint, the spectral triple (APT , L2 (E), PT ) is regular. (iv) Under the hypothesis of (iii), APT is the largest algebra A in C ∞ (M) such that the triple (A, L2 (E), PT ) is regular. Proof. (i) Since {P+ , T } is elliptic and P ∗ = P (viewed as defined on H 1 (M)), we can apply [30, 1.6.11 Theorem] with the same notations, except that S is here not surjective: it is an en+ − := S(EN ) with kernel EN := (1 − S)(EN ), so EN is the domorphism only surjective on EN + − is S where R direct orthogonal sum of EN and EN and our S is just replaced by the notation the surjective morphism associated to the endomorphism R from its domain to its range R(E) to = T , ρ = γ0 . avoid confusion. In the notation of [30], we take B = PT , 0 = G = K = G = G + − s s ), such that Thus, PT is selfadjoint if there is a homeomorphism Ψ from H (EN ) onto H (EN ∗ ∗ (since P = P yields AP = −AP ) −C ∗ AP γ0 = Ψ Sγ0 ,

(4)

with C satisfying (1 − S) = (1 − S). In other words, C is the injection − S)C = IdE − and C (1 N

− into EN and C ∗ = (1 − S). from EN SC = IdE + By [30, (1.6.52)], when this is the case, Ψ has the form Ψ = C ∗ A∗P C with N × and C S = S (remark that the matrix I is the number 1 here). Note that AP is invertible as a consequence of the ellipticity of P . Now, suppose that (1 − S)AP (1 − S) = 0 and SA−1 P S = 0. We define Ψ = −(1 − S)AP C. + − s s −1 This is a homeomorphism from H (EN ) onto H (EN ). Indeed, if we set Ψ := − SA−1 P C , we get −1 − S)AP C SA−1 Ψ ◦ Ψ −1 = (1 P C = (1 − S)AP SAP C = (1 − S)AP S + (1 − S) A−1 P C = (1 − S)C = IdE − N

and we also have Ψ −1 ◦ Ψ = IdE + using SA−1 P S = 0. Moreover, N

Ψ S = −(1 − S)AP S = −C ∗ AP S + (1 − S) = −C ∗ AP . As a consequence, (4) is satisfied and the if part of the assertion follows. Conversely, suppose that PT is selfadjoint. From Green’s formula (2), we get (AP γ0 u, γ0 v)N = 0 for any u, v ∈ Dom PT . Since γ0 : H 1 (E) → H 1/2 (EN ) is surjective, (AP (1 − S)ψ, (1 − S)φ)N = 0 for any ψ, φ ∈ H 1/2 (EN ) and thus (1 − S)AP (1 − S) = 0. Again,


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+ from [30, 1.6.11 Theorem] we get that Ψ := C ∗ A∗P C is a homeomorphism from H s (EN ) − −1 s −1 := −SAP C is a right-inverse of Ψ , and onto H (EN ) and we check as before that Ψ SA−1 thus, is the inverse of Ψ . The equation Ψ −1 ◦ Ψ = IdE + yields P SAP C = 0, which gives N −1 SA SAP S = 0. Thus, P

−1 −1 SA−1 P SAP = SAP SAP (1 − S) = SAP S + (1 − S) AP (1 − S) = S(1 − S) = 0 so SA−1 P S = 0. (ii) Clearly, C ∞ (M) is represented as bounded operators on L2 (E) by left multiplication. Since PT is a selfadjoint unbounded operator on L2 (E), (PT − i)d is a bijective operator from Dom PTd onto L2 (E). The system {P+ , T } being elliptic, it follows from Proposition 3.2 that Dom PT d is a closed subset of H d (E) and PT d is continuous from Dom PT d , with the topology of H d (E), into L2 (E). Using the inverse mapping theorem as in the proof of Lemma 2.6(ii), we see that (PT − i)−d is a topological isomorphism from L2 (E) onto Dom PT d (with the induced topology of H d (E)). Again, [30, A.4 Lemma] implies that (PT − i)−d is compact, and the singular values μn (|PT |) are O(n1/d ). Let a ∈ C ∞ (M). In particular Dom[PT , a] = Dom PT and if ψ ∈ Dom PT , [PT , a]ψ = is such that ( [P+ , a]ψ . By Lemma 3.3, [P+ , a] = [P , a ]+ where a ∈ C ∞ (M) a )+ = a. Thus, E) satisfying the transmission since [P , a ] is a pseudodifferential operator of order 0 in (M, property, [P+ , a] is continuous from H s (E) into H s (E) for any s > − 12 . In particular, [PT , a] extends uniquely as a bounded operator da on L2 (E). (iii) By (ii) (C ∞ (M), L2 (E), PT ) is a spectral triple. Thus, since APT is a ∗-subalgebra of ∞ C (M), (APT , L2 (E), PT ) is also a spectral triple (of the same dimension). Let A ∈ APT ∪ dAPT . It is clear that A sends HP∞T into itself. We denote B the associated ∞ HPT -bounded operator (so B = A). Clearly, H k (E) is a Sobolev scale associated to L2 (E) and PT . The result will follow by Lemma 2.6 if we check that for any k ∈ N, B (k) := [PT2 , ·]k (B) is continuous from HP∞T with the H k (E)-topology, into HP∞T with the L2 (E)-topology. Since P is differential, Lemma 3.3 yields for any ψ ∈ HP∞T

k

k

k (ψ) B (k) ψ = PT2 , · (B)ψ = (P+ )2 , · (B)(ψ) = P 2 , · (B) + is a differential operator of order 0 on (M, E) satisfying B + = B = A. Since P 2 where B 2 k is a differential operator of order k. The claim folhas scalar principal symbol, [P , ·] (B) lows. (iv) Suppose that A is a subalgebra of C ∞ (M) such that (A, L2 (M, E), PT ) is a regular spectral triple, A acting on L2 (M, E) by left multiplication. By regularity, a direct application of the proof of (3) ⇒ (4) in [14, Lemma 2.1] yields aHP∞T ⊆ HP∞T for any a ∈ A. This shows that A ⊆ APT . 2 Note that if S = 1 or S = 0 then by previous theorem (i), PT is not selfadjoint.

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Remark 4.6. The hypothesis “P is a differential operator” in Theorem 4.5(iii) is crucial in the sense that if P is a non-differential pseudodifferential operator, some non-vanishing leftovers may appear in B (k) and destroy the continuity of order k. This phenomenon does not appear in the boundaryless case since it stems from the singularities generated by the cut-off operator e+ r + at the boundary. We conclude this section with a simple one-dimensional example: = S1 and M = {(x, y) ∈ S1 : x 0} [− π , π ]. Define P on H 1 (M, C2 ) Remark 4.7. Let M 2 2 0 d dθ , where θ is the polar coordinate associated to the map θ → (cos θ, sin θ ), and let by d − dθ 0 T = Sγ0 where S = 10 00 . The system (P+ , T ) is elliptic of order 1 and PT is selfadjoint since 0 1 = −A−1 AP = ε −1 P and (1 − S)AP = AP S where ε is the function on N = ∂M such that 0 ε(0, 1) = 1 and ε(0, −1) = −1. By Theorem 4.5, (APT , L2 (M, C2 ), PT ) is a regular spectral triple of dimension 1. The smooth domain HP∞T is equal to the set of all ψ = (ψ1 , ψ2 ) ∈ C ∞ (M, C2 ) such that for (2k)

(2k+1)

d p )|N = 0, where φ (p) for all p ∈ N denotes ( dθ ) φ. all k ∈ N, (ψ1 )|N = 0 and (ψ2 ∞ Moreover, the algebra APT is equal to the set of all smooth functions a ∈ C (M) such that for all k ∈ N, (a (2k+1) )|N = 0. For example, θ → sin θ ∈ APT while θ → cos θ ∈ / APT . In particular, APT is non-extremal, i.e. it is strictly included in C ∞ (M) and strictly contains the algebra of smooth functions in C ∞ (M) which are constant near the boundary. In the general setting of a compact manifold with boundary M, we conjecture that APT is nonextremal for any differential operator of first order P and any normal trace operator T = Sγ0 such that PT is selfadjoint and {P+ , T } is elliptic.

4.2. Case of a Dirac operator is a Riemannian manifold with metric g, d = dim M is even and E We now assume that M → End(E) such has a Clifford module structure. This means that there is smooth map γ : T M γ (x)γ (y) + γ (y)γ (x) = 2g(x, y). that for any x, y ∈ T M, a Hermitian inner product (·,·) such that γ (x)∗ = γ (x) for any x ∈ T M. We also fix on E Note that for a given γ , such Hermitian inner product always exists. We fix a connection ∇ on E and v, w ∈ E, such that for any x, y ∈ T M x(v, w) = (∇x v, w) + (v, ∇x w), ∇x γ (y)v = γ ∇xLC (y) v + γ (y)∇x v, g). By [27, Lemma 1.1.7], such connecwhere ∇ LC is the Levi-Civita connection of (M, tion always exists. The Dirac operator associated to the connection ∇ is locally defined by D := i j γj ∇j where our conventions are the following: {e1 , . . . , ed } is a local orthonormal frame of the tangent space where ed is the inward pointing unit vector field, γj := γ (ej ) and ∇j := ∇ej . The chiral boundary operator χ investigated in [13,39] is defined the following way: choose χ := (−i)d/2+1 γ1 · · · γd−1 , so χM := iχγd (also sometimes denoted γd+1 ) is the natural chiral Then {χ, γd } = 0 while [χ, γn ] = 0, ∀n ∈ {1, . . . , d − 1}. Let Π± be the projections ity of M.


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associated to eigenvalues ±1 of χ (χ = χ ∗ and χ 2 = 1). This defines a particular case of elliptic boundary condition [27, Lemma 1.4.9, Theorem 1.4.11, Lemma 1.5.3]. The map S := Π− is an idempotent selfadjoint endomorphism on N and 1 − S = Π+ . Since AD = −iγd = −A−1 D by Remark 3.1, we get (1 − S)AD = AD S and DT , where T := S|N γ0 , is selfadjoint by (3). This type of boundary condition is chosen of course to obtain a selfadjoint boundary Dirac operator, which is not the case if we choose a Dirichlet or a Neumann–Robin condition. In this framework, Theorem 4.5 shows the following, a fact not considered in [13]: Theorem 4.8. The triple (ADT , L2 (M, E), DT ) is a regular spectral triple of dimension d. 4.3. A spectral triple on the boundary N := ∂M We intend here to construct a spectral triple on the boundary of the manifold M. The idea is to define a transversal differential elliptic operator on the boundary from a differential elliptic and a Riemannian structure. operator on M Recall that a pair Let D be an elliptic symmetric differential operator of order one on M. (g, h) where g is a metric on M and h is a Hermitian pairing on E is said of the product-type such that (U, g|U ) is near the boundary N if there exists a tubular neighborhood U of N in M 2 isometric to ]−ε, ε[ × N for some ε > 0 with metric dx ⊗ gN (x) where gN (x) is a smooth family of metrics on N , and h is such that h(x) := F∗ h|{x}×N is independent of x ∈ ]−ε, ε[, |U and ]−ε, ε[ × EN . where F is the bundle isomorphism between E is of the As observed in [6, Section 2.1], we can always suppose that the pair (g, h) on M product type near the boundary N . The idea is to let the coefficients of D to absorb any non and if product behavior of (g, h). More precisely, if we start with a general pair (g, h) on M ∞ (g1 , h1 ) is pair of product type near the boundary, we can define s ∈ C (M, L(E)) such that s(p) is the isomorphism between the two equivalent quadratic spaces (E p , h(p)) for any p ∈ M, and (Ep , h1 (p)). We then define the following application E) → ρ −1/2 s −1 u ∈ C ∞ (M, E) Ψ : u ∈ C ∞ (M, where ρ := dω(g1 )/dω(g) and dω(g) (resp. dω(g1 )) is the volume form associated to the met E, g, h) and L2 (M, E, g1 , h1 ). ric g (resp. g1 ). This map extends as an isometry between L2 (M, −1 Thus, we can deal with the differential operator Ψ DΨ which is unitarily equivalent to D. E) is of product-type. From now on, we suppose that (g, h) on (M, 2 Thus, D, as an operator in L (U, E|U ), is unitarily equivalent (see for instance [2,6,9]) to an operator of the form Ix ◦

d + Ax dx

where I ∈ C ∞ (]−ε, ε[, GL(EN )), A ∈ C ∞ (]−ε, ε[, Diff1 (N, EN )), each Ax being elliptic and each Ix being an anti-selfadjoint endomorphism. Note that, when D is a Dirac type operator (in the sense that its square D2 has gx (ξ, ξ ) Id for principal symbol), I can be chosen as constant Ix = I0 with I02 = −1.

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We then define the tangential operator DN := I0 ◦ A0 , which is an elliptic symmetric first-order differential operator on (N, EN ). Note that DN 2 has a scalar principal symbol if D2 has. By Proposition 4.1, we directly obtain a spectral triple on the boundary: such Proposition 4.9. Let D be an elliptic symmetric differential operator of order one on M that D2 has a scalar principal symbol. Then (C ∞ (N ), L2 (EN ), DN ) is a regular spectral triple of dimension d − 1. Remark that if D is a classical Dirac operator associated to a Clifford module, DN corresponds to the hypersurface Witten–Dirac operator and has been intensively studied in [28,37]. 5. Reality and tadpoles 5.1. Conjugation operator and dimension spectrum on a commutative triple Definition 5.1. A commutative spectral triple (A, H, D) provided with a chirality operator χ (a Z/2 grading on H which anticommutes with D) is said to be real if there exists an antilinear isometry J on H such that J aJ −1 = a ∗ for a ∈ A, J D = DJ , J χ = χJ , and J 2 = where , , are signs given by the table quoted in [16,18,29]. As we shall see, it turns out that the existence of an operator J only satisfying J D = ±DJ and J aJ −1 = a ∗ is enough to impose vanishing tadpoles at any order. We thus introduce a weak definition of conjugation operator: Definition 5.2. A conjugation operator on a commutative spectral triple (A, H, D) is an antilinear isometry J on H such that J aJ −1 = a ∗ for any a ∈ A and J D = εDJ with ε ∈ {−1, 1}. In order to be able to compute the spectral action and the corresponding tadpoles on the spectral triples (APT , L2 (E), PT ), we shall need the noncommutative integral and a characterization of the dimension spectrum. We first recall a few definitions of the Chamseddine–Connes pseudodifferential calculus [12]. From now on, (A, H, D) is a regular commutative spectral triple of dimension d endowed with conjugation operator J . We shall use the following convention: X −s when (s) > 0 actually means (X + P0 )−s where P0 is the orthogonal projection on the kernel of X. Note that in our case, J AJ −1 = A, by definition of J . For any α ∈ R, we define OPα := {T : |D|−α T ∈ k Dom δ k }, where δ = [|D|, ·] as defined in Section 2. Lemma 5.3. Suppose that the triple (A, H, D) satisfies the first-order condition [da, b] = 0 for any a, b ∈ A and let A be a one-form (a finite sum of terms of the form adb, for a, b ∈ A). Then A∗ = −εJ AJ −1 . In particular, if A is selfadjoint, A + εJ AJ −1 = 0. Proof. Direct computation.

2


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Definition 5.4. Let D(A) be the polynomial algebra generated by A, D. The operator T is said to be pseudodifferential if there exists d ∈ Z such that for any N ∈ N, there exist p ∈ N, P ∈ D(A) and R ∈ OP−N (p, P and R may depend on N ) such that P D−2p ∈ OPd and T = P D−2p + R. Define Ψ (A) as the set of pseudodifferential operators and Ψ k (A) := Ψ (A) ∩ OPk . Remark 5.5. We use here the algebra of pseudodifferential operators defined by Chamseddine and Connes in [12] and denoted Ψ1 (A) in [24,39]. This algebra does not a priori contain the operators of type |D|k or |D + A|k (A one-form) for k odd, contrarily to the larger algebra considered in [24]. The dimension spectrum Sd(A, H, D) of a spectral triple has initially been defined in [17,22] and is adapted here to the definition of pseudodifferential operator. Definition 5.6. A spectral triple (A, H, D) is said to be simple if all generalized zeta functions P := s → Tr(P |D|−s ), where P is any pseudodifferential operator in OP0 , are meromorphic ζD on the complex plane with only simple poles. The set of these poles is denoted Sd(A, H, D), and called the dimension spectrum of (A, H, D). When (A, H, D) is simple, that for k ∈ N,

:= Ress=0 Tr(T |D|−s ) is a trace on the algebra Ψ (A). Note

−T

−s

Ress=d−k Tr |D + A|

−(d−k) = − K(A)|D| = − |D + A|−(d−k)

where K(A) is a pseudodifferential operator in the sense of Definition 5.4 (see [24, Lemma 4.6, Proposition 4.8]). Suppose now that (A, H, D) is simple and that Sd(A, H, D) ⊆ d − N. Moreover, we suppose, 2 as in [21, p. 197] that for all selfadjoint one-forms A, Tr e−t (D+A) has a complete asymptotic expansion in real powers of t when t ↓ 0. Thus, S(D + A, Φ, Λ) = Tr Φ((D + A)2 /Λ2 ) satisfies (1) q − (AD −1 )q (see [12]). where ζDA (0) = ζD (0) + dq=1 (−1) q We now obtain from [39]: Proposition 5.7. Let A be a selfadjoint one-form. The linear term in A in the Λd−k coefficient of (1), denoted TadD+A (d − k) and called tadpole of order d − k, satisfies TadD+A (d − k) = −(d − k) − AD|D|−(d−k)−2 , TadD+A (0) = − − AD−1 .

∀k = d,

If the triple satisfies the first-order condition [da, b] = 0 for any a, b ∈ A, then (see [39, Corollary 3.7] and Lemma 5.3) TadD+A (d − k) = 0 for any k ∈ N.

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5.2. Tadpoles on (ADT , L2 (M, E), DT ) It is known that real the commutative spectral triple based on the classical Dirac operator on a compact spin manifold [29,43] of even dimension d has no tadpoles [39]. In fact, as we shall see, the same result applies in presence of a boundary. is a spin maniWe now work in the setting of Section 4.2. We suppose moreover that M is the spin bundle and D is the classical Dirac operator in the sense of Atiyah–Singer. fold, E The spin structure brings us an antilinear isometry J (the ordinary conjugation operator) sat and J γi = −γi J for any i ∈ {1, . . . , d} isfying DJ = J D, J bJ −1 = b∗ for all b ∈ C ∞ (M), χ J , and thus J χ = ε χJ , where ε = −1 if d/2 is [29, Theorem 9.20]. Moreover, J χM = ε M odd and ε = 1 if d/2 is even. In particular, if d/2 is even then J S = SJ . If we set J:= J χM and then J is an antilinear isometry satisfying DJ = −JD, JbJ−1 = b∗ for all b ∈ C ∞ (M), J χ = −ε χ J . As a consequence, if d/2 is odd, then J S = S J . Theorem 5.8. The spectral triple (ADT , L2 (M, E), DT ) of Theorem 4.8 has no tadpoles. Proof. By Theorem 4.8, we know that (ADT , L2 (M, E), DT ) is a regular spectral triple of dimension d. In order to prove it is a simple spectral triple with dimension spectrum included B (s) := {d − k: k ∈ N}, it is enough to check that for any B ∈ D(ADT ), the function ζD T Tr(B|DT |−s ) has only simple poles in Z. Since D is differential, by Lemma 3.3, B is a differential operator on (M, E). Since s → Tr(A|DT |−s ) has a meromorphic extension on C with only simple poles in {d + n − k: k ∈ N}, when A is a differential operator of order n on (M, E) (see for instance [26, Theorem 1.12.2]), we can conclude that (ADT , L2 (M, E), DT ) is simple with 2 dimension spectrum included in d − N. Moreover, if A is a selfadjoint one-form, Tr e−t (DT +A) has a complete asymptotic expansion in real powers of t when t → 0 [27, Theorems 1.4.5 and 1.4.11]. Note that the first order condition [da, b] = 0 for any a, b ∈ ADT is clearly satisfied since da is a differential operator of order 0 on (M, E). It only remains to prove that there is conjugation operator (Definition 5.2) on the simple spectral triple (ADT , L2 (M, E), DT ). We define J := J if d/2 is even, and J := J if d/2 is odd. The operator J+ := r + J e+ , is an endomorphism on L2 (M, E). Clearly, J+ is an antilinear isometry satisfying J+ aJ+ −1 = a ∗ for any a ∈ ADT . By Lemma 3.3, L(J , D) = L(D, J ) = 0 and thus, J+ D+ = (J D)+ = (−1)d/2 (DJ )+ = (−1)d/2 D+ J+ . S J −1 Moreover J|N |N |N = S|N and thus T J+ = S|N γ0 J+ = S|N J|N γ0 = J|N S|N γ0 = J|N T.

In particular, J+ preserves Dom DT , and thus J+ DT = (−1)d/2 DT J+ . As a consequence, J+ is a conjugation operator on (ADT , L2 (M, E), DT ). Proposition 5.7 now yields the result. 2


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Acknowledgments We thank Alain Connes, Ali Chamseddine, Gerd Grubb, Ryszard Nest and Uuye Otgonbayar for helpful discussions. References [1] J. Aastrup, R. Nest, E. Schrohe, Index theory for boundary value problems via continuous fields of C∗ -algebras, J. Funct. Anal. 257 (2009) 2645–2692. [2] M. Atiyah, V. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69, II, Math. Proc. Cambridge Philos. Soc. 78 (1975) 405–432, III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99. [3] B. Booß-Bavnbek, The determinant of elliptic boundary problems for Dirac operators, http://mmf.ruc.dk/~booss/ ell/determinant.pdf. [4] B. Booß-Bavnbek, M. Lesch, The invertible double of elliptic operators, Lett. Math. Phys. 87 (2009) 19–46. [5] B. Booß-Bavnbek, K. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, 1993. [6] B. Booß-Bavnbek, M. Lesch, C. Zhu, The Calderón projection: new definitions and applications, J. Geom. Phys. 59 (2009) 784–826. [7] T. Branson, P. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990) 245–272. [8] T. Branson, P. Gilkey, Residues of the eta function for an operator of Dirac type with local boundary conditions, Differential Geom. Appl. 2 (1992) 249–267. [9] J. Brünig, M. Lesch, On boundary value problems for Dirac type operators, J. Funct. Anal. 185 (2001) 1–62. [10] A.L. Carey, S. Klimek, K.P. Wojciechowski, A Dirac type operator on the non-commutative disk, Lett. Math. Phys. 93 (2010) 107–125. [11] A. Chamseddine, A. Connes, The spectral action principle, Comm. Math. Phys. 186 (1997) 731–750. [12] A. Chamseddine, A. Connes, Inner fluctuations of the spectral action, J. Geom. Phys. 57 (2006) 1–21. [13] A. Chamseddine, A. Connes, Quantum gravity boundary terms from the spectral action on noncommutative space, Phys. Rev. Lett. 99 (2007) 071302. [14] A. Connes, On the spectral characterization of manifolds, arXiv:0810.2088v1. [15] A. Connes, Noncommutative Geometry, Academic Press, London/San Diego, 1994. [16] A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995) 6194–6231. [17] A. Connes, Geometry from the spectral point of view, Lett. Math. Phys. 34 (1995) 203–238. [18] A. Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996) 155–177. [19] A. Connes, The action functional in noncommutative geometry, Comm. Math. Phys. 117 (1998) 673–683. [20] A. Connes, Variation sur le thème spectral, Résumé des cours 2006–2007, http://www.college-de-france.fr/media/ ana_geo/UPL53971_2.pdf, 2007. [21] A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, Colloq. Publ., vol. 55, Amer. Math. Soc., 2008. [22] A. Connes, H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995) 174–243. [23] J. Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris, Sér. A 262 (1966) 1107–1108. [24] D. Essouabri, B. Iochum, C. Levy, A. Sitarz, Spectral action on noncommutative torus, J. Noncommut. Geom. 2 (2008) 53–123. [25] B. Fedosov, F. Golse, E. Leichtnam, E. Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996) 1–31. [26] P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theory, CRC Press, Boca Raton, 1995. [27] P.B. Gilkey, Asymptotic Formulae in Spectral Geometry, Chapman & Hall, 2003. [28] N. Ginoux, The Dirac spectrum, Lecture Notes in Math. 1976 (2009). [29] J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser Advanced Texts, Birkhäuser, Boston, 2001. [30] G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, second ed., Progr. Math., vol. 65, Birkhäuser, Boston, 1996.

134


[31] G. Grubb, Distributions and Operators, Grad. Texts in Math., vol. 252, Springer, 2009. [32] G. Grubb, E. Schrohe, Trace expansions and the noncommutative residue for manifolds with boundary, J. Reine Angew. Math. (Crelle’s Journal) 536 (2001) 167–207. [33] G. Grubb, E. Schrohe, Traces and quasi-traces on the Boutet de Monvel algebra, Ann. Inst. Fourier 54 (2004) 1641–1696. [34] S. Hawking, G. Horowitz, The gravitational Hamiltonian, action, entropy and surface terms, Classical Quantum Gravity 13 (1996) 1487–1498. [35] N. Higson, The local index formula in noncommutative geometry, in: Lectures Given at the School and Conference on Algebraic K-Theory and Its Applications, Trieste, 2002. [36] N. Higson, The residue index theorem of Connes and Moscovici, in: Surveys in Noncommutative Geometry, in: Clay Math. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 2006, pp. 71–126. [37] O. Hijazi, S. Montiel, A. Roldán, Eigenvalue boundary problems for the Dirac operator, Comm. Math. Phys. 231 (2002) 375–390. [38] L. Hörmander, The Analysis of Linear Partial Differential Operators I, II, III, Springer-Verlag, Berlin, 1989, 2005, 2007. [39] B. Iochum, C. Levy, Tadpoles and commutative spectral triples, J. Noncommut. Geom., in press, arXiv:0904.0222 [math-ph]. [40] W. Kalau, M. Walze, Gravity, non-commutative geometry, and the Wodzicki residue, J. Geom. Phys. 16 (1995) 327–344. [41] D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166 (1995) 633–643. [42] M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators, in: S. Gindikin, et al. (Eds.), Functional Analysis on the Eve of the 21’st Century, vol. I, in: Progr. Math., vol. 131, Birkhäuser, Boston, 1995, pp. 173–197. [43] H.B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, 1989. [44] M. Lesch, Operators of Fuchs Type, Conical Singularities and Asymptotic Methods, Teubner, Leipzig, 1997. [45] M. Lesch, On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols, Ann. Global Anal. Geom. 17 (1999) 151–187. [46] J.-M. Lescure, Triplets spectraux pour les variétés à singularité conique isolée, Bull. Soc. Math. France 129 (2001) 593–623. [47] J.L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, 1968. [48] R. Nest, E. Schrohe, Dixmier’s trace for boundary value problems, Manuscripta Math. 96 (1998) 203–218. [49] U. Otgonbayar, Pseudo-differential operators and regularity of spectral triples, arXiv:0911.0816 [math.OA]. [50] A. Rennie, Smoothness and locality for nonunital spectral triples, K-Theory 28 (2003) 127–165. [51] E. Schrohe, Noncommutative residues and manifolds with conical singularities, J. Funct. Anal. 150 (1997) 146–174. [52] E. Schrohe, Noncommutative residues, Dixmier’s trace, and heat trace expansions on manifolds with boundary, in: B. Booss-Bavnbek, K. Wojciechowski (Eds.), Geometric Aspects of Partial Differential Equations, in: Contemp. Math., vol. 242, Amer. Math. Soc., Providence, RI, 1999, pp. 161–186. [53] W.J. Ugalde, Some conformal invariants from the noncommutative residue for manifolds with boundary, SIGMA 3 (2007), 104, 18 pages. [54] D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rep. 388 (2003) 279–360. [55] Y. Wang, Differential forms and the Wodzicki residue for manifold with boundary, J. Geom. Phys. 56 (2006) 731– 753. [56] Y. Wang, Differential forms and the noncommutative residue for manifolds with boundary in the non-product case, Lett. Math. Phys. (2006) 41–51. [57] Y. Wang, Gravity and the noncommutative residue for manifolds with boundary, Lett. Math. Phys. 80 (2007) 37–56. [58] M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math. 75 (1984) 143–177. [59] M. Wodzicki, Noncommutative residue. Chapter I: Fundamentals, in: Yu.I. Manin (Ed.), K-Theory, Arithmetic and Geometry, in: Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 320–399. [60] C. Yang, Isospectral deformations of Eguchi–Hanson spaces as non-unital spectral triples, Comm. Math. Phys. 288 (2009) 615–652.


Proper asymptotic unitary equivalence in KK-theory and projection lifting from the corona algebra ✩ Hyun Ho Lee Department of Mathematics, Seoul National University, Seoul, South Korea 151-747 Received 18 February 2010; accepted 10 September 2010 Available online 18 September 2010 Communicated by D. Voiculescu

Abstract In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using KK-theory and prove a result which asserts that there is a unitary of the form ‘identity + compact’ which gives the unitary equivalence of two projections if the ‘essential codimension’ of two projections vanishes for certain C ∗ -algebras employing the proper asymptotic unitary equivalence of KK-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of C(X) ⊗ B where X is [0, 1], (−∞, ∞), [0, ∞), and [0, 1]/{0, 1}. © 2010 Elsevier Inc. All rights reserved. Keywords: KK-theory; Proper asymptotic unitary equivalence; Absorbing representation; Essential codimension

1. Introduction When two projections p and q in B(H ), whose difference is compact, are given, an integer [p : q] is defined as the Fredholm index of v ∗ w where v, w are isometries on H with vv ∗ = p and ww ∗ = q. This number is called the essential codimension because it gives the codimension of p in q if p q [2]. A modern interpretation of this essential codimension is provided using the Kasparov group KK(C, C). Indeed, a ∗-homomorphism from C to B(H ) is determined by the image of 1 which is a projection. Thus we can associate to the essential codimension a Cuntz pair. An important result of the essential codimension is the following: [p : q] = 0 if and only if there is a unitary u of the form ‘identity + compact’ such that upu∗ = q. Motivated by this ✩

Research partially supported by NRF-2009-0068619. E-mail address: [email protected].

0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.003

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H.H. Lee / Journal of Functional Analysis 260 (2011) 135–145

result, Dadarlat and Eilers defined a new equivalence relation on KK-group [4]. When π, σ : A → L(E) are two representations, with E being a Hilbert B-module, we say π and σ are properly asymptotically unitarily equivalent and write π σ if there is a continuous path of unitaries u : [0, ∞) → U(K(E) + C1E ), u = (ut )t∈[0,∞) , such that • limt→∞ σ (a) − ut π(a)u∗t = 0 for all a ∈ A, • σ (a) − ut π(a)u∗t ∈ K(E) for all t ∈ [0, ∞), and a ∈ A. Note that the word ‘proper’ reflects the fact that implementing unitaries are of the form ‘identity + compact’. The main result of them is [4, Theorem 3.8] which asserts that if φ, ψ : A → M(B ⊗ K(H )) is a Cuntz pair of representations, then the class [φ, ψ] vanishes in KK(A, B) if and only if there is another representation γ : A → M(B ⊗ K(H )) such that φ ⊕ γ ψ ⊕ γ . When B = C, which corresponds to K-homology, the result is improved as a non-stable version. In fact, if (φ, ψ) is a Cuntz pair of faithful, nondegenerate representations from A to B(H ) such that both images do not contain any nontrivial compact operator, then the cycle [φ, ψ] = 0 in KK(A, C) if and only if φ ψ [4, Theorem 3.12]. This fits nicely with the above aspect of the essential codimension. An abstract version of this is proved by given a Cuntz pair of absorbing representations (see Theorem 2.11). Thus the proper asymptotic unitary equivalence must be the right notion and tool for further developments of the non-stable K-theory. Our intrinsic interest lies in when this non-stable version of proper asymptotic unitary equivalence happens as shown in Khomology case. We show a similar result for K-theory. In fact, we prove that if (φ, ψ) is a Cuntz pair of faithful representations from C → M(B ⊗ K) whose images are not in B ⊗ K, then [φ, ψ] = 0 in K(B) if and only if φ ψ provided that B is non-unital, separable, purely infinite simple C ∗ -algebra such that M(B) has real rank zero (see Theorem 2.14). Besides our intrinsic interest, Theorem 2.14 was motivated by the projection lifting problem from the corona algebra to the multiplier algebra of a C ∗ -algebra of the form C(X) ⊗ B. To lift a projection from a quotient algebra to a projection has been a fundamental question related to K-theory (see [5]). We show that a projection in the corona algebra is ‘locally’ liftable to a projection in the multiplier algebra but not ‘globally’ in general. In other words, it can be represented by finitely many projection valued functions so that their discontinuities are described in terms of Cuntz pairs. They give rise to K-theoretical obstructions. We show that these discontinuities can be resolved if corresponding K-theoretical terms are vanishing. In this process, the crucial point of proper asymptotic unitary equivalence is exploited as a key step (see Theorem 3.3). 2. Proper asymptotic unitary equivalence Let E be a (right) Hilbert B-module. We denote by L(E, F ) the C ∗ -algebra of adjointable, bounded operators from E to F . The ideal of ‘compact’ operators from E to F is denoted by K(E, F ). When E = F , we write L(E) and K(E) instead of L(E, E) and K(E, E). Throughout the paper, A is a separable C ∗ -algebra, and all Hilbert modules are assumed to be countably generated over a separable C ∗ -algebra. We use the term representation for a ∗-homomorphism from A to L(E). We let HB be the standard Hilbert module over B which is H ⊗ B where H is a separable infinite dimensional Hilbert space. We denote by M(B) the multiplier algebra of B.


137

It is well known that L(HB ) = M(B ⊗ K) and K(HB ) = B ⊗ K where K is the C ∗ -algebra of the compact operators on H [8]. Definition 2.1. (See [4, Definition 2.1].) Let π, σ be two representations from A to E and F respectively. We say π and σ are approximately unitarily equivalent and write π ∼ σ , if there exists a sequence of unitaries un ∈ L(E, F ) such that for any a ∈ A (i) limn→∞ σ (a) − un π(a)u∗n = 0, (ii) σ (a) − un π(a)u∗n ∈ K(F ) for all n. Definition 2.2. (See [4, Definition 2.5].) A representation π : A → L(E) is called absorbing if π ⊕ σ ∼ π for any representation σ : A → L(F ). We say that π and σ are asymptotically unitarily equivalent, and write π asym ∼ σ if there is a unitary valued norm continuous map u : [0, ∞) → L(E, F ) such that t → σ (a) − ut π(a)u∗t lies in C0 ([0, ∞)) ⊗ K(E) for any a ∈ A, or if (i) limt→∞ σ (a) − ut π(a)u∗t = 0, (ii) σ (a) − ut π(a)u∗t ∈ K(F ) for all t ∈ [0, ∞). If π : A → L(E) is a representation, we define π (∞) : A → L(E (∞) ) by π (∞) = π ⊕ π ⊕ · · · where E (∞) = E ⊕ E ⊕ · · · . Lemma 2.3. Let ψ be an absorbing representation, and φ be a representation of a separable C ∗ -algebra A on the standard Hilbert B-module HB . Then there exists a sequence of isometries (∞) {vn } ⊂ L(HB , HB ) such that for each a ∈ A (∞) vn φ (∞) (a) − ψ(a)vn ∈ K HB , HB , vn φ (∞) (a) − ψ(a)vn → 0 as n → ∞, vj∗ vi = 0 for i = j . ∗ Proof. Let ∗Si , i = 1, 2, 3, . . . , be a sequence of isometries of L(H∗B ) such that Si Sj = 0, i = j , and i Si Si = 1 in the strict topology. Let φ∞ (a) = i Si φ(a)Si . Since ψ is absorbing, there is a unitary U ∈ L(HB , HB ) such that

U ∗ ψ(a)U − φ∞ (a) ∈ K(HB ), (∞)

Define T : HB

a ∈ A.

(1)

→ HB by T = (S1 , S2 , . . .). Then φ∞ (a) = T φ (∞) (a)T ∗ .

Thus Eq. (1) is rewritten as (∞) T ∗ U ∗ ψ(a)U T − φ (∞) (a) ∈ K HB ,

a ∈ A.

(2)

138


If we identify φ (∞) as (φ (∞) )(∞) , there is a partition Ni , i = 1, 2, 3, . . . , of N so that we generate (∞) a sequence of isometries vi ∈ L(HB , HB ) from U T = (U S1 , U S2 , . . .). More concretely, if we let νi : Ni → N be bijections, we can define vi = (U Sν −1 (1) , U Sν −1 (2) , . . .). It is easily checked i i that vi vj∗ = 0 for i = j . Eq. (2) implies that (∞) vi∗ ψ(a)vi − φ (∞) (a) ∈ K HB , ∗ v ψ(a)vi − φ (∞) (a) → 0 as i → ∞. i Finally, our claim follows from ∗ vn φ (∞) (a) − ψ(a)vn vn φ (∞) (a) − ψ(a)vn = φ (∞) a ∗ φ (∞) (a) − vn∗ ψ(a)vn + φ (∞) a ∗ − vn∗ ψ(a)vn φ (∞) (a) − φ (∞) a ∗ a − vn∗ ψ a ∗ a vn .

2

Lemma 2.4. (See [4, Lemma 2.6].) Let π : A → L(E) and σ : A → L(F ) be two representations. Suppose that there is a sequence of isometries vi : F (∞) → E such that for a ∈ A vi σ (∞) (a) − π(a)vi ∈ K F (∞) , E ,

lim vi σ (∞) (a) − π(a)vi → 0,

i→∞

and vj∗ vi = 0 for i = j . Then π ⊕ σ asym ∼ π. We say φ : A → B(H ) is admissible if φ is faithful, non-degenerate, and φ(A) ∩ K = {0}. The main result in [14] states that any pair of admissible representations φ and ψ satisfies that φ ∼ ψ . Dadarlat and Eilers proved a much stronger version which states that any pair of admissible representations φ and ψ satisfies φ asym ∼ ψ [4, Theorem 3.11]. Since the admissible representation is absorbing, the following result is the appropriate generalization of Voiculescu’s result. Theorem 2.5. If two representations ψ , φ of a separable C ∗ -algebra A on the standard Hilbert ∼ ψ. B-module HB are absorbing, then we have φ asym Proof. By Lemma 2.3 and Lemma 2.4, we have ψ ⊕ φ asym ∼ ψ, and the proof is complete by symmetry. 2 Definition 2.6. Let φ be a representation from A to M(B ⊗ K). Then we define a C ∗ -algebra by Dφ (A, B) = x ∈ M(B ⊗ K) xφ(a) − φ(a)x ∈ B ⊗ K, a ∈ A . Lemma 2.7. If M(B ⊗ K) has real rank zero, then Dφ (C, B) has real rank zero for any representation φ : C → M(B ⊗ K). Proof. The proof of the lemma is essentially based on the argument due to Brown and Pedersen [1]. Note that any representation φ : C → M(B ⊗ K) is determined by φ(1), which is a projection in M(B ⊗ K). Say φ(1) = p. Then we see that Dφ (C, B) = {x ∈ M(B ⊗ K) | xp − px ∈ B ⊗ K}.


139

To show Dφ (C, B) has real rank zero, it is enough to show any self-adjoint element in Dφ (C, B) is approximated by a self-adjoint, invertible element. Let x be a self-adjoint element. Using the obvious matrix notation x=

a

c

c∗

b

,

xp − px ∈ B ⊗ K implies that c is ‘compact’, i.e., it is in B ⊗ K. Since M(B ⊗ K) has real rank zero, pM(B ⊗ K)p and (1 − p)M(B ⊗ K)(1 − p) have real rank zero. Given > 0 we can find b0 invertible in (1 − p)M(B ⊗ K)(1 − p) with b0 = b0∗ and b − b0 < . Then considering a − cb0−1 c∗ , we can find a0 in pM(B ⊗ K)p with a0 = a0∗ and a − a0 < , such that a0 − −1 p 0 cb0−1 c∗ is invertible in pM(B ⊗ K)p. Then p cb0 , b−1 c∗ 1−p are in Dφ (C, B) since cb0−1 is 0 1−p 0 ‘compact’. Thus x0 =

a0 c∗

c b0

=

p 0

cb0−1 1−p

a0 − cb0−1 c∗ 0

0 b0

p b0−1 c∗

is invertible in Dφ (C, B). Evidently x − x0 < , so we are done.

0 1−p

2

Let us recall the definition of Kasparov group KK(A, B). We refer the reader to [9] for the general introduction of the subject. A KK-cycle is a triple (φ0 , φ1 , u), where φi : A → L(Ei ) are representations and u ∈ L(E0 , E1 ) satisfies that (i) uφ0 (a) − φ1 (a)u ∈ K(E0 , E1 ), (ii) φ0 (a)(u∗ u − 1) ∈ K(E0 ), φ1 (a)(uu∗ − 1) ∈ K(E1 ). The set of all KK-cycles will be denoted by E(A, B). A cycle is degenerate if uφ0 (a) − φ1 (a)u = 0,

φ0 (a) u∗ u − 1 = 0,

φ1 (a) uu∗ − 1 = 0.

An operator homotopy through KK-cycles is a homotopy (φ0 , φ1 , ut ), where the map t → ut is norm continuous. The equivalence relation ∼ is generated by operator homotopy and addition oh of degenerate cycles up to unitary equivalence. Then KK(A, B) is defined as the quotient of E(A, B) by ∼. When we consider non-trivially graded C ∗ -algebras, we define a triple (E, φ, F ), oh where φ : A → L(E) is a graded representation, and F ∈ L(E) is of odd degree such that F φ(a) − φ(a)F , (F 2 − 1)φ(a), and (F − F ∗ )φ(a) are all in K(E) and call it a Kasparov (A, B)module. Other definitions like degenerate cycle and operator homotopy are defined in similar ways. Let v be a unitary in Mn (Dφ (A, B)). Define φ n : A → LB (B n ) by φ n (a)(b1 , b2 , . . . , bn ) = (φ(a)b1 , φ(a)b2 , . . . , φ(a)bn ). Let B n ⊕ B n be graded by (x, y) → (x, −y). Then

φn B ⊕B , 0 n

n

0 φn

0 , ∗ v

v 0

is a Kasparov (A, B)-module. The class of this module depends only on the class of v in K1 (Dφ (A,B)) so that the construction gives rise to a group homomorphism Ω : K1 (Dφ (A,B))→ KK(A, B).

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Lemma 2.8. Let φ be an absorbing representation from A to L(HB ) = M(B) where B is a stable C ∗ -algebra. Then Ω : K1 (Dφ (A, B)) → KK(A, B) is an isomorphism. D (A,B)

Proof. See [13, Theorem 3.2]. In fact, Thomsen proved K1 ( (Dφ φ(A,A,B)) ) is isomorphic to KK(A, B) via a map Θ where Dφ (A, A, B) = {x ∈ Dφ (A, B) | xφ(A) ⊂ B} is the ideal of Dφ (A, B). However, the same proof shows Ω is an isomorphism. Alternatively we can show that Ki (Dφ (A, A, B)) = 0 for i = 0, 1 by the argument of [7, Lemma 1.6] with the fact that K∗ (M(B)) = 0. Thus, using the six term exact sequence, K∗ (Dφ (A, B)) is isomorphic to D (A,B) K∗ ( (Dφ φ(A,A,B)) ). This implies the map Ω which is the composition with Θ and q1 is an isomorphism. Here q1 is the induced map between K-groups from the quotient map from Dφ (A, B) D (A,B) onto (Dφ φ(A,A,B)) . 2 Definition 2.9. (See [4, Definition 3.2].) If π, σ : A → L(E) are representations, we say that π and σ are properly asymptotically unitarily equivalent and write π σ if there is a continuous path of unitaries u : [0, ∞) → U(K(E) + CIE ), u = (ut )t∈[0,∞) such that for all a ∈ A (i) limt→∞ σ (a) − ut π(a)u∗t = 0, (ii) σ (a) − ut π(a)u∗t ∈ K(E) for all t ∈ [0, ∞). In the above, we introduced the Fredholm picture of KK-group. There is an alternative way to describe the element of KK-group. The Cuntz picture is described by a pair of representations φ, ψ : A → L(HB ) = M(B ⊗ K) such that φ(a) − ψ(a) ∈ K(HB ) = B ⊗ K. Such a pair is called a Cuntz pair. They form a set denoted by Eh (A, B). A homotopy of Cuntz pairs consists of a Cuntz pair (Φ, Ψ ) : A → M(C([0, 1]) ⊗ (B ⊗ K)). The quotient of Eh (A, B) by homotopy equivalence is a group KKh (A, B) which is isomorphic to KK(A, B) via the mapping sending [φ, ψ] to [φ, ψ, 1] [3]. Dadarlat and Eilers proved that [φ, ψ] = 0 in KKh (A, B) if and only if there is a representation γ : A → M(B ⊗ K) = L(HB ) such that φ ⊕ γ ψ ⊕ γ [4, Proposition 3.6]. The point is that the equivalence is implemented by unitaries of the form compact + identity. Sometimes, we can have a non-stable equivalence keeping this useful point. its unitization. We say that A has K1 Definition 2.10. Let A be a C ∗ -algebra. Denote by A to K1 (A) is injective where U(A) is the unitary group 0 (A) injectivity if the map from U(A)/U is the connected component of the identity. We note that H. Lin proved in [11, and U0 (A) Lemma 2.2] that real rank zero implies K1 -injectivity. Theorem 2.11. Let A be a separable C ∗ -algebra and let ψ, φ : A → HB be a Cuntz pair of absorbing representations. Suppose that the composition of φ with the natural quotient map ˙ is faithful. Further, we suppose π : M(B ⊗ K) → M(B ⊗ K)/B ⊗ K, which will be denoted by φ, that Dφ (A, B) satisfies K1 -injectivity. If [φ, ψ] = 0 in KK(A, B), then φ ψ . Proof. The proof of this theorem is almost identical to the one given in [4, Theorem 3.12]. We just give the proof to illustrate how our assumptions play the roles. By Theorem 2.5, we get a continuous family of unitaries (ut )t∈[0,∞) in M(B ⊗ K) such that ut φ(a)u∗t − ψ(a) ∈ C0 [0, ∞) ⊗ (B ⊗ K).

(3)


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Note that (3) implies [φ, ψ] = [φ, u1 φu∗1 ] (see [4, Lemma 3.1]). We assume that [φ, ψ] = 0 and we conclude that [φ, u1 φu∗1 ] = 0. Since (φ, φ, u∗1 ) is unitarily equivalent to (φ, u1 φu∗1 , 1),

(φ, φ, u1 ) = φ, φ, u∗1 = 0. Since the isomorphism Ω : K1 (Dφ (A, B)) → KK(A, B) sends [u1 ] to [φ, φ, u1 ] by Lemma 2.8, K1 -injectivity implies that u1 is homotopic to 1 in Dφ (A, B). Thus we may assume that u0 = 1 in (3). Let Eφ be a C ∗ -algebra φ(A) + B ⊗ K. We define (αt )t∈[0,∞) in Aut0 (Eφ ) by Ad(ut ). Note that α0 = id and (αt ) is a uniform continuous family of automorphisms. Thus we apply Proposition 2.15 in [4] and get a continuous family (vt )[0,∞) of unitaries in Eφ such that lim αt (x) − Ad vt (x) = 0

t→∞

(4)

for any x ∈ Eφ . Combining (4) with (3), we obtain (vt )[0,∞) of unitaries in Eφ such that lim vt φ(a)vt∗ − ψ(a) = 0

t→∞

for any a ∈ A. Since φ˙ is faithful, we can replace (vt )[0,∞) by a family of unitaries in B ⊗ K + C1 by the argument shown in Step 1 of the proof of Proposition 3.6 in [4]. 2 Recall the definition of the essential codimension of Brown, Douglas, and Fillmore defined by two projections p, q in B(H ) whose difference is compact as we have defined in Introduction. Using KK-theory, or K-theory, we generalize this notion as follows, keeping the same notation. Definition 2.12. Given two projections p, q ∈ M(B ⊗ K) such that p − q ∈ B ⊗ K, we consider representations φ, ψ from C to M(B ⊗ K) such that φ(1) = p, ψ(1) = q. Then (φ, ψ) is a Cuntz pair so that we define [p : q] as the class [φ, ψ] ∈ KK(C, B) K(B). Lemma 2.13. (See [12].) Let B be a non-unital (σ -unital) purely infinite simple C ∗ -algebra. Let φ, ψ be two monomorphisms from C(X) to M(B ⊗ K) where X is a compact metrizable space. ˙ ψ˙ are still injective, then they are approximately unitarily equivalent. If φ, The following theorem is a sort of generalization of BDF’s result about the essential codimension. Theorem 2.14. Let B be a non-unital (σ -unital) purely infinite simple C ∗ -algebra such that M(B ⊗ K) has real rank zero. Suppose two projections p and q in M(B ⊗ K) = L(HB ) such that p − q ∈ B ⊗ K and neither of them are in B ⊗ K. If [p, q] ∈ K0 (B) vanishes, then there is a unitary u in id + B ⊗ K such that upu∗ = q. Proof. Step 1: Let φ, ψ : C → M(B ⊗ K) be representations from p and q respectively. Evidently φ is injective. Moreover, it does not contain any “compacts” since p does not belong to B ⊗ K. Thus φ˙ is faithful. Recall ψ∞ is defined by ψ∞ (a) = Si ψ(a)Si∗ where {Si } is a sequence of isometries in M(B ⊗ K) such that Si Sj∗ = 0 for i = j . Suppose that ψ∞ (λ) = 0 for

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λ ∈ C. Then Si∗ ψ∞ (λ)Si = ψ(λ) = 0 or λq = 0. Thus λ = 0. Similarly, ψ˙ ∞ is injective. Then they are approximately unitarily equivalent by applying Lemma 2.13 to X = {x0 }. Thus we have a unitary U in L(HB ) such that U ∗ φ(a)U − ψ∞ (a)

(5)

for a ∈ C. Note that to get a sequence of isometries {vi } ∈ L(HB(∞) , HB ) satisfying the conditions of Lemma 2.3, what we needed was Eq. (5). Following the same argument in the proof of Theorem 2.5, we get φ asym ∼ ψ . In other words, we have a continuous family of unitaries (ut )t∈[0,∞) in M(B ⊗ K) such that ut φ(a)u∗t − ψ(a) ∈ C0 [0, ∞) ⊗ (B ⊗ K)

for any a in A.

Since Dφ (C, B) has real rank zero, it satisfies K1 -injectivity. Thus it follows that φ ψ as in the proof of Theorem 2.11. Step 2: For large enough t, we can take ut = u of the form ‘identity + compact’ such that upu∗ − q < 1. For the moment we write upu∗ as p. Thus p − q < 1. Note that p − q ∈ B ⊗ K. Then z = pq + (1 − p)(1 − q) ∈ 1 + B ⊗ K is invertible and pz = zq. If we consider the polar decomposition of z as z = v|z|. It is easy to check that v ∈ 1 + B ⊗ K and vpv ∗ = q. Now w = vu is also a unitary of the form ‘identity + compact’ such that wpw ∗ = q.

2

3. Application: projection lifting In this section, we show an application of proper asymptotic unitary equivalence of two projections. In this application, with an additional real rank zero property, the unitary of the form ‘identity + compact’ plays a crucial role as we shall see. Let B be a stable C ∗ -algebra such that the multiplier algebra M(B) has real rank zero. Let X be [0, 1], [0, ∞), (−∞, ∞) or T = [0, 1]/{0, 1}. When X is compact, let I = C(X) ⊗ B which is the C ∗ -algebra of (norm continuous) functions from X to B. When X is not compact, let I = C0 (X) ⊗ B which is the C ∗ -algebra of continuous functions from X to B vanishing at infinity. Then M(I ) is given by Cb (X, M(B)s ), which is the set of bounded functions from X to B(H ), where M(B) is given the strict topology. Let C(I ) = M(I )/I be the corona algebra of I and also let π : M(I ) → C(I ) be the natural quotient map. Then an element f of the corona algebra can be represented as follows: Consider a finite partition of X, or X {0, 1} when X = T, which is given by partition points x1 < x2 < · · · < xn all of which are in the interior of X and divide X into n + 1 (closed) subintervals X0 , X1 , . . . , Xn . We can take fi ∈ Cb (Xi , M(B)s ) such that fi (xi ) − fi−1 (xi ) ∈ B for i = 1, 2, . . . , n and f0 (x0 ) − fn (x0 ) ∈ B where x0 = 0 = 1 if X is T. Lemma 3.1. The coset in C(I ) represented by (f0 , . . . , fn ) consists of functions f in M(I ) such that f − fi ∈ C(Xi ) ⊗ B for every i and f − fi vanishes (in norm) at any infinite end point of Xi .


143

Proof. If X is compact, then we set x0 = 0, xn+1 = 1. Otherwise, we set x0 = x1 − 1 when X contains −∞, and xn+1 = xn + 1 when X contains +∞. Then we define a function in C(X) ⊗ B by ⎧ x−xi−1 ⎪ ⎨ xi −xi−1 (fi (xi ) − fi−1 (xi )), mi (x) = x−xi+1 (fi (xi ) − fi−1 (xi )), ⎪ ⎩ xi −xi+1 0,

if xi−1 x xi , if xi x xi+1 , otherwise

for each i = 1, . . . , n. In addition, we set m0 = mn+1 = 0. Then we define a function ffrom fi ’s by f(x) = fi (x) − mi (x)/2 + mi+1 (x)/2 on each Xi . It follows that fi (xi ) − mi (xi )/2 + mi+1 (xi )/2 = fi−1 (xi ) − mi−1 (xi )/2 + mi (xi )/2. Thus fis well defined. The conditions f − fi ∈ C(Xi ) ⊗ B for each i imply that f − fis norm continuous function from X to B since f |Xi (xi ) − f|Xi (xi ) = f |Xi−1 (xi ) − f|Xi−1 (xi ). 2 Similarly (f0 , . . . , fn ) and (g0 , . . . , gn ) define the same element of C(I ) if and only if fi − gi ∈ C(Xi ) ⊗ B for i = 0, . . . , n if X is compact. (f0 , . . . , fn ) and (g0 , . . . , gn ) define the same element of C(I ) if and only if fi − gi ∈ C(Xi ) ⊗ B for i = 0, . . . , n − 1, fn − gn ∈ C0 ([xn , ∞)) ⊗ B if X is [0.∞). (f0 , . . . , fn ) and (g0 , . . . , gn ) define the same element of C(I ) if and only if fi − gi ∈ C(Xi ) ⊗ B for i = 1, . . . , n − 1, fn − gn ∈ C0 ([xn , ∞)) ⊗ B, f0 − g0 ∈ C0 ((−∞, x1 ]) ⊗ B if X = (−∞, ∞). The following theorem says that any projection in the corona algebra of C(X) ⊗ B for some C ∗ -algebras B is described by a “locally trivial fiber bundle” with the fiber HB in the sense of Dixmier and Duady [6]. Theorem 3.2. Let I be C(X) ⊗ B or C0 (X) ⊗ B where B is a stable C ∗ -algebra such that M(B) has real rank zero. Then a projection f in M(I )/I can be represented by (f0 , f1 , . . . , fn ) as above where fi is a projection valued function in C(Xi ) ⊗ M(B)s for each i. Proof. Let f be the element of M(I ) such that π(f ) = f. Without loss of generality, we can assume f is self-adjoint and 0 f 1. (i) Suppose X does not contain any infinite point. Choose a point t0 ∈ X. Then there is a selfadjoint element T ∈ M(B) such that T − f (t0 ) ∈ B and the spectrum of T has a gap around 1/2 by [1, Theorem 3.14]. So we consider f (t) + T − f (t0 ) which is still self-adjoint whose image is f. Thus we may assume f (t0 ) is a self-adjoint element whose spectrum has a gap around 1/2. Since r(f (t)) : t → f (t) − f (t)2 is norm continuous where r(x) = x − x 2 , if we pick a point z / σ (f (t0 ) − f (t0 )2 ), then σ (f (s)) omits r −1 (J ) for s sufficiently close to in (0, 14 ) such that z ∈ t where J is an interval containing z. In other words, there is δ > 0 and b > a > 0 such that if |t0 − s| < δ, then σ (f (s)) ⊂ [0, a) ∪ (b, 1]. If we let ft0 (s) = χ(b,1] (f (s)) for s in (t0 − δ, t0 + δ) where χ(b,1] is the characteristic function on (b, 1], then it is a continuous projection valued function such that ft0 − f ∈ C(t0 − δ, t0 + δ) ⊗ B. By repeating the above procedure, since X is compact, we can find n + 1 points t0 , . . . , tn , n + 1 functions ft0 , . . . , ftn , and an open covering {Oi } such that ti ∈ Oi , Oi ∩ Oi−1 = ∅, and

144


fti is projection valued function on Oi . Now let fi = fti as above. Take the point xi ∈ Oi−1 ∩ Oi for i = 1, . . . , n. Then fi (xi ) − fi−1 (xi ) = fi (xi ) − f (xi ) + f (xi ) − fi−1 (xi ) ∈ B and f0 (x0 ) − fn (x0 ) ∈ B if applicable. Let Xi = [xi , xi+1 ] for i = 1, . . . , n − 1, X0 = [0, x1 ], and Xn = [xn , 1]. Since each fi is also defined on Xi , (f0 , . . . , fn ) is what we want. (ii) Let X be [0, ∞). Since f 2 (t) − f (t) → 0 as t goes to ∞, for given δ in (0, 1/2), there is M > 0 such that whenever t M then f 2 (t) − f (t) < δ − δ 2 . It follows that σ (f (t)) ⊂ [0, δ) ∪ (1 − δ, 1] for t M. Then again χ(1−δ,1] (f (t)) is a continuous projection valued function for t M such that f (t) − χ(1−δ,1] (f (t)) vanishes in norm as t goes to ∞. By applying the argument in (i) to [0, M], we get a closed sub-intervals Xi for i = 0, . . . , n − 1 of [0, M] and fi ∈ Cb (Xi , B(H )). Now if we let Xn = [M, ∞) and fn (t) = χ(1−δ,1] (f (t)), we are done. (iii) The case X = (−∞, ∞) is similar to (ii). 2 When a projection f ∈ C(I ) is represented by (f0 , f1 , . . . , fn ) by Theorem 3.2, we note that fi (x) is a projection in M(B ⊗ K) for each x ∈ Xi and fi (xi ) − fi−1 (xi ) ∈ B. Applying Definition 2.12 we have K-theoretical terms ki = [fi (xi ) : fi−1 (xi )] ∈ KK(C, B) for i = 1, 2, . . . , n. The following theorem shows that if all ki ’s are vanishing, then a projection f in C(I ) lifts to a projection in M(I ). Theorem 3.3. Let I be C(X) ⊗ B where B is a σ -unital, non-unital, purely infinite simple C ∗ algebra such that M(B) has real rank zero or K1 (B) = 0 (see [15]). Let a projection f in M(I )/I be represented by (f1 , f2 , . . . , fn ), where fi is a projection valued function in C(Xi ) ⊗ M(B)s for each i, as in Theorem 3.2. If ki = [fi (xi ) : fi−1 (xi )] = 0 for all i, then the projection f in M(I )/I lifts. Proof. Note that, by Zhang’s dichotomy, B is stable [15, Theorem 1.2]. By induction, assume that fj (xj ) = fj −1 (xj ) for j = 1, 2, . . . , i − 1. Let fi (xi ) = pi , fi−1 (xi ) = pi−1 . Since [pi : pi−1 ] = 0, we have a unitary u of the form ‘identity + compact’ such that pi − u∗ pi−1 u < 1/2 by Theorem 2.14. Since B has real rank zero, given 0 < < 1/4 there is a unitary v ∈ C1 + B with finite spectrum such that u − v < [10,11]. Then pi − vpi−1 v ∗ pi − upi−1 u∗ + upi−1 u∗ − vpi−1 v ∗ < 1. Note that pi − vpi−1 v ∗ ∈ B. Thus we have wpi w ∗ = vpi−1 v ∗ for some unitary w ∈ id + B. (Recall that Step 2 of the proof of Theorem 2.14.) Let gi = wfi w ∗ , then fi − gi ∈ C(Xi ) ⊗ B since w is of the form ‘identity + compact’. On the other hand, we can write v as eih where h is a self-adjoint element in B since v has the finite spectrum. A homotopy of unitaries t → eith , which are of the form “identity + compact”, connects 1 to v. Now we define gi−1 (t) as

t − xi−1 t − xi−1 h fi−1 (t)exp i h exp i xi − xi−1 xi − xi−1 for t ∈ [xi−1 , xi ]. Then we see that gi−1 (xi ) = gi (xi ), gi−1 − fi−1 ∈ C(Xi−1 ) ⊗ K, and gi−1 (xi−1 ) = fi−1 (xi−1 ). Moreover, if we let gi+1 = wfi+1 w ∗ , then fi+1 −gi+1 ∈ C(Xi+1 )⊗B, and


145

gi+1 (xi+1 ) : gi (xi+1 ) = wfi+1 (xi+1 )w ∗ : wfi (xi + 1)w ∗

= fi+1 (xi+1 ) : fi (xi+1 ) = 0. Then (f0 , f1 , . . . , fn ) and (f0 , f1 , . . . , gi−1 , gi , gi+1 , fi+2 , . . . , fn ) define the same element f while the ki ’s are unchanged and i-th discontinuity is resolved. So we take the latter as (f0 , . . . , fn ) such that fj (xj ) = fj −1 (xj ) for j = 1, . . . , i. We can repeat the same procedure until we have fi (xi ) = fi−1 (xi ) for all i. It follows that (f0 , . . . , fn ) is a projection in M(C(X)⊗B) which lifts f. 2 Remark 3.4. When I = C0 (X) ⊗ B where X is [0, ∞) or (−∞, ∞), the similar result holds replacing C(Xi ) ⊗ B with C0 (−∞, x1 ] ⊗ B or C0 [xn , ∞) ⊗ B for i = 0 or i = n respectively. Acknowledgments Although this work was not carried out at Purdue, a significant influence on the author was made by Larry Brown and Marius Dadarlat who have acquainted him with geometric ideas in operator algebras. He also would like to thank Huaxin Lin for answering the question related to Lemma 2.14. References [1] L.G. Brown, G.K. Pedersen, C ∗ -algebras of real rank zero, J. Funct. Anal. 99 (1991) 131–149. [2] L.G. Brown, R.G. Douglas, P.A. Fillmore, Unitary equivalence modulo the compact operators and extensions of C ∗ -algebras, in: Proc. Conf. Operator Theory, in: Lecture Notes in Math., vol. 345, Springer, New York, 1973, pp. 58–128. [3] J. Cuntz, Generalized homomorphisms between C ∗ -algebras and KK-theory, in: Dynamics and Processes, in: Lecture Notes in Math., vol. 1031, Springer, New York, 1983, pp. 31–45. [4] M. Dadarlat, S. Eilers, Asymptotic unitary equivalence in KK-theory, K-theory 23 (2001) 305–322. [5] K.R. Davidson, C ∗ -algebras by Example, Fields Inst. Monogr., vol. 6, Amer. Math. Soc., Providence, RI, 1996. [6] J. Dixmier, A. Duady, Champs continus d’space hilbertiens et de C ∗ -algebres, Bull. Soc. Math. France 91 (1963) 227–284. [7] N. Higson, C ∗ -algebra extension theory and duality, J. Funct. Anal. 129 (1995) 349–363. [8] G. Kasparov, Hilbert C ∗ -modules: Theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980) 133–150. [9] G. Kasparov, The operator K-functor and extensions of C ∗ -algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (3) (1981) 571–636. [10] H. Lin, Exponential rank of C ∗ -algebras of real rank zero and the Brown–Pedersen conjectures, J. Funct. Anal. 114 (1993) 1–11. [11] H. Lin, Approximation by normal elements with finite spectra in C ∗ -algebra of real rank zero, Pacific J. Math. 173 (1996) 397–411. [12] H. Lin, private communication. [13] K. Thomsen, On absorbing extensions, Proc. Amer. Math. Soc. 129 (2001) 1409–1417. [14] D. Voiculescu, A non-commutative Weyl–von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1) (1976) 97–113. [15] S. Zhang, Certain C ∗ -algebras with real rank zero and their corona and multiplier algebras. Part I, Pacific J. Math. 155 (1) (1992) 169–197.


Rieffel deformation of homogeneous spaces P. Kasprzak 1,2,3 Department of Mathematical Sciences, University of Copenhagen, Denmark Received 2 March 2010; accepted 12 August 2010 Available online 6 September 2010 Communicated by S. Vaes

Abstract Let G1 ⊂ G be a closed subgroup of a locally compact group G and let X = G/G1 be the quotient space of left cosets. Let X = (C0 (X), X ) be the corresponding G-C∗ -algebra where G = (C0 (G), ). Suppose that Γ is a closed abelian subgroup of G1 and let Ψ be a 2-cocycle on the dual group Γˆ . Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ -C∗ algebra XΨ – the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of Ψ the subgroup G1 obtaining the closed quantum subgroup GΨ 1 ⊂ G , which in turn, by the results of S. Vaes, Ψ Ψ ∗ Ψ ∼ Ψ leads to the G -C -algebra G /G1 . In this paper we show that GΨ /GΨ 1 = X . We also consider the case where Γ ⊂ G is not a subgroup of G1 , for which we cannot construct the subgroup GΨ 1 . Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ -simple object in the category of GΨ -C∗ -algebras. © 2010 Elsevier Inc. All rights reserved. Keywords: Operator algebras; Quantum groups; Quantum homogeneous spaces

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . 2.1. G-C∗ -category . . . . . . . . . . . 2.2. Quantum homogeneous spaces

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E-mail address: [email protected]. 1 Supported by the Marie Curie Research Training Network Non-Commutative Geometry MRTN-CT-2006-031962. 2 Supported by Geometry and Symmetry of Quantum Spaces, PIRSES-GA-2008-230836. 3 On leave from Department of Mathematical Methods in Physics, Faculty of Physics, Warsaw University, Poland.


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3. Group algebra twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Rieffel deformation of group coactions . . . . . . . . . . . . . . . . . 5. Induction of the regular corepresentation . . . . . . . . . . . . . . . . 6. Rieffel deformation of homogeneous spaces – the quotient case 7. Rieffel deformation of G -simple C∗ -algebras . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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150 152 152 160 162 163

1. Introduction The theory of locally compact quantum groups (LCQG) has already reached its maturity. Almost ten years have passed since the appearance of the seminal paper of J. Kustermans and S. Vaes [11], where the axiomatic theory was formulated. A locally compact quantum group is a C∗ -bialgebra (A, ) equipped with a left and a right Haar weight φ and ψ. Imposing some natural conditions on (weak cancellation) and on the Haar weights φ and ψ (KMS-type conditions) the authors were able to develop the theory, proving among other things the existence of the coinverse which admits the polar decomposition κ = R ◦ τi/2 and showing that the theory is self dual. The assumption of the existence of the Haar weights, which is a theorem for the classical groups and for the compact quantum groups, may be perceived as a drawback. It seems that a Haar weights free axiomatization is out of reach. There exists second formulation of the LCQG theory, due to T. Masuda, Y. Nakagami and S.L. Woronowicz [12], in which the authors include the existence of the coinverse κ in the axioms. But to develop the theory they still need to assume the existence of a left Haar weight. In fact, it can be shown that both theories are equivalent. One way of constructing examples of LCQGs is to start with a classical group G and search for its deformations Gq . In general, the link between G and Gq is not rigorously described. There is at least one mathematical procedure – the Rieffel deformation – where this correspondence is clear. For the original approach of Rieffel we refer to [15]. In this paper we shall use our recent approach to the Rieffel deformation which describes it in terms of crossed product construction (see [9]). The Rieffel deformation of G will be denoted by GΨ and its dual will be denoted Ψ . The deformation procedure of a locally compact group in terms of the transition from G by G Ψ to G is described in Section 3. Let G be a locally compact group and D a G-C∗ -algebra. Using the results of [8] one may apply the Rieffel deformation to D, obtaining the deformed GΨ -C∗ -algebra D Ψ . The concise account of the deformation procedure of G-C∗ -algebras is the subject of Section 4. In Section 2 beside giving some preliminaries on G-C∗ -algebras we also discuss the notion of a quantum homogeneous space and the C∗ -algebraic quantum quotient G/G1 , a construction due to S. Vaes [16]. His construction may be performed for any closed quantum subgroup G1 of a regular LCQG G – for regularity we refer to [1]. The relation of G/G1 with the induction procedure of the regular corepresentation is explained. The awareness of this relation is crucial in the understanding of the proof that XΨ ∼ = GΨ /GΨ 1 . In Section 5 we perform the induction procedure of Ψ the regular corepresentation W1 of the deformed group GΨ 1 and compare the resulting objects with their undeformed counterparts. This enables us to prove that XΨ ∼ = GΨ /GΨ 1 which is the subject of Section 6. Finally, in Section 7 we comment on the case where XΨ is not necessarily of the quotient type. In connection with it we show that the Rieffel deformation of a G-simple C∗ -algebra is GΨ -simple.

148


The particular case of the Rieffel deformation of a homogeneous space has been discussed in [17]. In this paper J. Varilly treats the situation where there is given a pair Γ ⊂ G1 ⊂ G of closed subgroups with Γ being abelian and G compact. He shows that it is possible to perform a covariant deformation of X = G/G1 obtaining a quantum homogeneous GΨ -space XΨ . In this specific situation the difficulties that one encounters in general do not manifest themselves. Throughout the paper we will freely use the language of C∗ -algebras and the theory of locally compact quantum groups. For the notion of multipliers and morphism of C∗ -algebras we refer reader to [18]. For the theory of locally compact quantum groups we refer to [11] and [12]. I would like to express my gratitude to W. Szymański for many stimulating discussions, which greatly influenced the final form of this paper. 2. Preliminaries 2.1. G-C∗ -category Let us fix a notation related with a locally compact quantum group G. The C∗ -algebra and the comultiplication of G will be denoted by C0 (G) and G respectively. The von Neumann algebra associated with G and the Hilbert space obtained by the GNS-representation related with we shall denote the left Haar weight will be denoted respectively by L∞ (G) and L2 (G). By G the reduced version of the dual of G. The modular conjugations related with the left Haar weight will be denoted by J (by Jˆ). on G (on G) The main subject of this paper is related with the quantum groups coactions. The following definition may be traced back to [13]. To formulate it we adopt the following notation: a closed linear span of a subspace W ⊂ V of a Banach space V will be denoted by [W]. Definition 2.1. Let G be a LCQG. A G-C∗ -algebra is a pair D = (D, D ) consisting of a C∗ algebra D and a coaction D ∈ Mor(D, C0 (G) ⊗ D): (ι ⊗ D ) ◦ D = (G ⊗ ι) ◦ D , such that [D (D)(C0 (G) ⊗ 1)] = C0 (G) ⊗ D. The C∗ -algebra D will also be denoted by C0 (D) and the coaction D will be denoted by D . Remark 2.2. Suppose that G corresponds to an ordinary locally compact group G. There is a 1-1 correspondence between G-C∗ -algebras and G-C∗ -algebras, i.e. C∗ -algebras equipped with a continuous action of G. In order to describe it we introduce the characters χg : C0 (G) → C that are associated with the points of G: χg (f ) = f (g), for any g ∈ G and f ∈ C0 (G). For any G-C∗ algebra D we define the corresponding continuous action α : G → Aut(C0 (D)) by the formula: αg (a) = (χg −1 ⊗ ι)D (a) where g ∈ G and a ∈ C0 (D). The category of G-C∗ -algebras was consider either explicitly or implicitly in many different contexts – see for instance [6] where it appears explicitly in the context of the Landstad duality. In order to specify the category of G-C∗ -algebras we adopt the following notion of G-morphisms. Definition 2.3. Let G be a LCQG and suppose that B and D are G-C∗ -algebras. We say that a morphism π : C0 (B) → C0 (D) is a G-morphism from B to D if D ◦ π = (ι ⊗ π) ◦ B . The set of G-morphism from B to D will be denoted by MorG (B, D).


149

2.2. Quantum homogeneous spaces Let G be a locally compact quantum group and let X be a G-C∗ -algebra. The concept of a quantum homogeneous G-space exists so far only in the case of G being compact and C0 (X) being unital – see Definition 1.8 of [14]. We then say that X is a quantum homogeneous space if X is ergodic. Let G be a compact quantum group corresponding to a compact group G. It may be checked that there is a 1-1 correspondence between quantum homogeneous G-spaces with the underlying commutative C∗ -algebra C0 (X) and the classical homogeneous G-spaces. Unfortunately the ergodicity assumption in the non-compact case cannot serve as a replacement for homogeneity. The next approximation to the homogeneity is the notion of G-simplicity: Definition 2.4. Let D be a G-C∗ -algebra. We say that D is G-simple if for any G-C∗ -algebra B and any G-morphism π ∈ MorG (D, B) we have ker π = {0}. In order to motivate the introduction of the G-simplicity let us prove that in the case of the compact quantum groups the homogeneity of X implies its G-simplicity. Lemma 2.5. Let G = (A, ) be a compact quantum group with a faithful Haar measure h. Let X be a quantum homogeneous space with X being injective. Then X is G-simple. Proof. Using the ergodicity of X we may introduce the state ρ : C0 (X) → C such that (h ⊗ ι)X (a) = ρ(a)1 for any a ∈ C0 (X).

(1)

The faithfulness of h and the injectivity of X imply the faithfulness of ρ. Let π ∈ MorG (X, B) and suppose that a ∈ ker π . Note that (h ⊗ ι)X (a ∗ a) ∈ ker π : π (h ⊗ ι)X a ∗ a = (h ⊗ ι)B π a ∗ a = 0. On the other hand, using (1) we may see that π((h ⊗ ι)X (a ∗ a)) = ρ(a ∗ a)1 which together with the above computation shows that ρ(a ∗ a) = 0. The faithfulness of ρ implies that a = 0 hence ker π = {0}. 2 Let G be a locally compact quantum group and let G be the corresponding LCQG. It may be checked that there is a 1-1 correspondence between G-simple C∗ -algebras with the underlying C∗ -algebra being commutative and minimal G-spaces – the G-spaces with all orbits being dense (see [2], p. 49). As was already mentioned, a definition of a quantum homogeneous space appropriate for locally compact quantum groups is not yet known. What has been generalized so far is the notion of the quantum quotient space, due to Vaes [16]. But since the work of Podle´s [13] on quantum spheres we know that a generic quantum homogeneous space is not of the quotient type. The Rieffel deformation of the homogeneous spaces provides a new class of examples of non-compact quantum spaces which in our opinion should also be considered as homogeneous and which generically seems not to be of the quotient type. Let us move on to the discussion of the quantum quotient spaces. Let G be a regular LCQG (for the notion of regularity we refer to [1]). Let G1 be a closed quantum subgroup in the sense of Definition 2.5 of [16]. As a part of the structure we have the injective normal ∗-homomorphism

150


1 ) → L∞ (G). Its commutant counterpart is denoted by πˆ : L∞ (G 1 ) → L∞ (G) . The πˆ : L∞ (G definition of the measurable quotient space L∞ (G/G1 ) goes as follows: 1 ) . L∞ (G/G1 ) = a ∈ L∞ (G): a πˆ (x) = πˆ (x)a for any x ∈ L∞ (G A remarkable Theorem 6.1 of [16] provides the existence and the uniqueness of the C∗ -algebraic version C0 (G/G1 ) of L∞ (G/G1 ). For the needs of this paper we shall formulate it as a definition: Definition 2.6. Let G be a LCQG which is regular and let G1 be a closed quantum subgroup. The C∗ -algebraic quotient G/G1 is the unique G-C∗ -algebra characterized by the following conditions: • C0 (G/G1 ) ⊂ L∞ (G/G1 ) is a strongly dense C∗ -subalgebra; • G/G1 is given by the restriction of G to C0 (G/G1 ); • G (L∞ (G/G1 )) ⊂ M(K(L2 (G)) ⊗ C0 (G/G1 )) and the ∗-homomorphism G/G1 : L∞ (G/G1 ) → L L2 (G) ⊗ C0 (G/G1 )

is strict.

For the notion of strictness we refer to Definition 3.1 of [16]. Note that the symbol G/G1 denotes the coaction on C0 (G/G1 ) as well as the strict map defined on L∞ (G/G1 ). Remark 2.7. The difficulty of the proof of Theorem 6.1 of [16] lies in the existence part. To explain the idea of the proof (which will be also important to understand this paper) let 1 ) treated as a Hilbert C∗ -module over itself. Performing the us consider the C∗ -algebra C0 (G 1 )) we get the ininduction procedure of the regular corepresentation W1 ∈ M(C0 (G1 ) ⊗ C0 (G 1 )-module Ind(C0 (G 1 )) which is equipped with the induced corepresentation duced C∗ -C0 (G 1 )) → M(Ind(C0 (G 1 )) ⊗ C0 (G)) of the dual quanInd(W1 ) and the coaction γ : Ind(C0 (G (note that we use Notation 12.1 of [16]: for any C∗ -B-module E we denote tum group G M(E) = L(B, E)). The coaction γ is obtained by the induction procedure applied to the right 1 ) → M(C0 (G 1 ) ⊗ C0 (G)), β(a) = (ι ⊗ πˆ ) (a) for any a ∈ C0 (G 1 ). This coaction β : C0 (G G1 additional structure consisting of γ and Ind(W1 ) enables one to prove that the C∗ -algebra of 1 ))) is canonically isomorphic with the C∗ -algebra of a crossed compact operators K(Ind(C0 (G 1 ))) (which will also be product by G. The coaction γ gives rise to the coaction on K(Ind(C0 (G denoted by γ ) and then it may be identified with the dual coaction on the crossed product, while Ind(W1 ) is identified with the corepresentation implementing the coaction on the γ -invariants. The C∗ -algebra C0 (G/G1 ) is defined as the Landstad–Vaes C∗ -algebra of γ -invariants, by which we mean the C∗ -algebra satisfying the conditions of [16], Theorem 6.7 and we have 1 ))) ∼ K(Ind(C0 (G = G C0 (G/G1 ). 3. Group algebra twist Let G be a locally compact group and let δ : G → R+ be the modular function. Suppose that Γ is a closed abelian subgroup of G1 , which in turn is a closed subgroup of G. For reasons which will become clear later we shall assume that the modular function δ : G → R+ when restricted to Γ is identically 1: δ(γ ) = 1 for any γ ∈ Γ . In order to perform the twisting procedure one has to fix a 2-cocycle Ψ on Γˆ , i.e. a continuous function Ψ : Γˆ × Γˆ → T1 satisfying:


151

(i) Ψ (e, γˆ ) = Ψ (γˆ , e) = 1 for all γˆ ∈ Γˆ ; (ii) Ψ (γˆ1 , γˆ2 + γˆ3 )Ψ (γˆ2 , γˆ3 ) = Ψ (γˆ1 + γˆ2 , γˆ3 )Ψ (γˆ1 , γˆ2 ) for all γˆ1 , γˆ2 , γˆ3 ∈ Γˆ . For the theory of 2-cocycles we refer to [10]. In order to simplify the notation and make further computations less cumbersome we shall assume that Ψ is a skew symmetric bicharacter on Γˆ , Ψ (γˆ1 + γˆ2 , γˆ3 ) = Ψ (γˆ1 , γˆ3 )Ψ (γˆ2 , γˆ3 ), Ψ (γˆ1 , γˆ2 ) = Ψ (γˆ2 , γˆ1 ),

(2) (3)

satisfying Ψ (γˆ , γˆ ) = 1. We will only prove our results for such 2-cocycles although we have reasons to believe that they hold for arbitrary 2-cocycles. The role of the bicharacter Ψ ∈ M(C0 (Γˆ ) ⊗ C0 (Γˆ )) will vary in the course of this paper. The simplest variation is connected with the identification C0 (Γˆ ) ∼ = C∗ (Γ ), which enables us to treat ∗ ∗ Ψ as an element of M(C (Γ ) ⊗ C (Γ )). Furthermore, the morphism ι ∈ Mor(C∗ (Γ ), C∗l (G1 )) which corresponds to the representation Γ γ → Lγ ∈ M(C∗l (G1 )) enables us to treat Ψ as an element of M(C∗l (G1 ) ⊗ C∗l (G1 )). Finally, in some cases Ψ will be treated as an operator acting on L2 (G1 ) ⊗ L2 (G1 ) or L2 (G) ⊗ L2 (G). ˆ In what follows we shall use the notation Let us describe the twisting procedure of (C∗l (G), ). ˆ by of Section 2.1, denoting the locally compact quantum group related to the pair (C∗l (G), ) ∗ ˆ G. In particular, C0 (G) = Cl (G) and G = . The comultiplication G may be twisted by ⊗ C0 (G)): Ψ (a) = Ψ ∗ (a)Ψ . Using Corollary 5.3 of [4] we see means of Ψ ∈ M(C0 (G) G G Ψ such that C0 (G Ψ ) = C0 (G) with the that there exists a locally compact quantum group G . (For more general results concerning the twist of a quantum comultiplication given by G Ψ Ψ is of the group by a 2-cocycle we refer to [3].) Furthermore, with our modular assumption G Kac type – the coinverse κG Ψ is an involutive anti-automorphism. Ψ will be denoted by GΨ . Its description in The dual locally compact quantum group of G 2 = id – which terms of the Rieffel deformation was given in [9]. It is also of the Kac type – κG Ψ by Example 3.4 of [1] implies that GΨ is regular. The only reason for the assumption δ|Γ = 1 is to ensure the regularity of GΨ which is important in the construction of the quotient of a locally compact quantum group by its closed quantum subgroup. In what follows we shall give the formula for the multiplicative operators W Ψ ∈ L∞ (GΨ ) ⊗ ∞ Ψ ) and Vˆ Ψ ∈ L∞ (GΨ ) ⊗ L∞ (G Ψ ) where we adopted the notation of Section 2.1 of [16]. L (G Using Theorem 1 of [5] and the properties of Ψ we may see that the operator W Ψ acts on B(L2 (G) ⊗ L2 (G)) and it is of the form W Ψ = Ψ W Ψ where Ψ = (Jˆ ⊗ J )Ψ ∗ (Jˆ ⊗ J ).

(4)

The assumption of the co-stability introduced in Section 2.7 of [5] is satisfied trivially in the discussed case. The required group homomorphism t → γˆt is the trivial homomorphism: γˆt = e ∈ Γˆ (we replaced the γt introduced in [5] by γˆt which is more appropriate in our context). To give the formula for the multiplicative unitary Vˆ Ψ we only have to invoke the fact that it Ψ is the fundamental multiplicative unitary corresponding to the co-opposite quantum group G cop . Ψ The comultiplication of G is the flip of the comultiplication . By the skew-symmetry Ψ cop G of Ψ we get Vˆ Ψ = Ψ ∗ Vˆ Ψ ∗ .

(5)

152


can be also applied to G 1 leading to G Ψ and the The twist construction that we applied to G 1 Ψ Ψ Ψ ˆ dual G1 . Again using Theorem 1 of [5] one may see that W1 and V1 are related to W1 and V1 in the way analogous to the case of G described above (see Eq. (5)). The aforementioned assumption of co-stability is also satisfied but the homomorphism t → γˆt may be non-trivial. In order to see that, we shall use the modular function δ1 : G1 → R+ . For any t ∈ R we may define the character γˆt ∈ Γˆ given by γˆt , γ = δ1it (γ ). The homomorphism t → γˆt is the one that ensures the co-stability. It may be checked that the quantum group GΨ 1 is a closed quantum subgroup of GΨ in the sense of Definition 2.5 of [16]. In particular, there exists the embedΨ ) → L∞ (G Ψ ). Its counterpart acting between commutants will be denoted by ding πˆ : L∞ (G 1 Ψ ) → L∞ (G Ψ ) . πˆ : L∞ (G 1 4. Rieffel deformation of group coactions Suppose that G is a locally compact group containing an abelian closed subgroup Γ and let Ψ be a skew-symmetric bicharacter on Γˆ (see Section 3). Let X = (C0 (X), X ) be a G-C∗ -algebra (see Definition 2.1). In paper [8] we defined a GΨ -C∗ -algebra XΨ – the Rieffel deformation of X. The deformation procedure may be viewed as a two steps procedure: first extend X to the morphism of the appropriate crossed products and then twist the extension by a unitary obtained from Ψ . The reader may notice some differences between the description of the deformation procedure of X that we give below and the one given in [8]. They are due to the adopted definition Ψ and the fact X is a left G-C∗ -algebra whereas in [8] we consider the of GΨ as the dual of G right case. Let β : G → Aut(C0 (X)) be the continuous action corresponding to the coaction X (see Remark 2.2). Let α : Γ → Aut(C0 (X)) be the restriction of β to the subgroup Γ . The C∗ algebra C0 (XΨ ) is defined as the Landstad algebra of the Γ -product (Γ α C0 (X), λ, ρˆ Ψ ) where ρˆ Ψ : Γˆ → Aut(Γ α C0 (X)) is the Ψ -deformed dual action (see [9], Section 3). Let us note that X ◦ αγ = (ργ ,e ⊗ ι) ◦ X , where ρ : Γ 2 → Aut(C0 (G)) is the action defined by ργ1 ,γ2 (f )(g) = f (γ1−1 gγ2 ) for any γ1 , γ2 ∈ Γ and g ∈ G. The universal property of the crossed product Γ α C0 (X) enables us to define the extension ΓX ∈ Mor(Γ α C0 (X), Γ 2 ρ C0 (G) ⊗ Γ α C0 (X)) of X to the level of crossed product. In order to describe the aforementioned twisting step we define Υ ∈ M(Γ 2 ρ C0 (G) ⊗ Γ α C0 (X)) as follows. Let Φ ∈ Mor(C∗ (Γ 2 ), Γ 2 ρ C0 (G) ⊗ Γ α C0 (X)) be the morphism that corresponds to the representation Γ (γ1 , γ2 ) → λe,γ1 ⊗ λγ2 ∈ M(Γ 2 ρ C0 (G) ⊗ Γ α C0 (X)). Applying Φ to Ψ¯ we introduce Υ = Φ(Ψ¯ ). We define XΨ by the formula XΨ (b) = Υ ΓX (b)Υ ∗ for any b ∈ Γ α C0 (X). By Theorems 4.3 and 4.4 of [8] XΨ restricts to a morphism XΨ : C0 (XΨ ) → M(C0 (GΨ ) ⊗ C0 (XΨ )) which is a continuous coaction of GΨ on C0 (XΨ ). This defines GΨ -C∗ -algebra XΨ . 5. Induction of the regular corepresentation In this section we shall apply the induction procedure to the regular corepresentation W1Ψ ∈ Ψ ∗ Ψ Ψ Ψ M(C0 (GΨ 1 ) ⊗ C0 (G1 )) of G1 on C0 (G1 ), where C0 (G1 ) is treated as a C -Hilbert module over itself. For the induction procedure in the framework of LCQGs we refer to [16] (we Ψ )-Hilbert shall also adopt the notation of this paper). As a result, we obtain the induced C0 (G 1 Ψ Ψ module Ind(C0 (G1 )) together with the induced corepresentation Ind(W1 ) ∈ L(C0 (GΨ ) ⊗


153

Ψ ))). We shall also apply the induction procedure to the right coaction β Ψ : C0 (G Ψ ) → Ind(C0 (G 1 1 Ψ Ψ Ψ Ψ M(C0 (G1 ) ⊗ C0 (G )) of G on C0 (G1 ), defined by β Ψ (a) = (ι ⊗ π) ˆ G Ψ (a), 1

(6)

Ψ ), C0 (G Ψ )) is the standard embedding. The result is the coaction where πˆ ∈ Mor(C0 (G 1 Ψ Ψ Ψ Ind β Ψ : Ind C0 G → M Ind C0 G ⊗ C0 G , 1 1 Ψ )) ⊗ C0 (G Ψ )) we refer to Remark 2.7. Finally, the where for the definition of M(Ind(C0 (G 1 Ψ )), Ind(W Ψ ) and Ind(β Ψ ) will be compared with their untwisted induced objects Ind(C0 (G 1 1 counterparts Ind(C0 (G1 )), Ind(W1 ) and Ind(β). Ψ remains unLet us first recall that the von Neumann algebra of the twisted quantum group G ∞ Ψ ∞ Ψ changed L (G ) = L (G). This implies that the imprimitivity bimodule I , which is defined by Ψ I Ψ = v ∈ B L2 (G1 ), L2 (G) vm = πˆ (m)v for any m ∈ L∞ G 1 Ψ ): αI Ψ (v) = stays undeformed: I Ψ = I. Nevertheless, the coaction αI Ψ : I Ψ → I Ψ ⊗ L∞ (G Ψ ∗ Ψ ˆ Vˆ1 gets twisted. The relation with its untwisted counterpart is established by Vˆ (v ⊗ 1)(ι ⊗ π) the following formula: αI Ψ (v) = Ψ ∗ αI (v)Ψ.

(7)

Ψ ∞ Ψ ) → L(L2 (G) ⊗ C (G Let us analyze the strict ∗-homomorphism πG Ψ : L (G 0 1 )) (in [16], 1 1 Lemma 4.5 it was denoted by πl ), given by:

ˆ ⊗ ι) W1Ψ (m ⊗ 1)W1Ψ ∗ πG Ψ (m) = (π 1

Ψ ). It may be noted that the map πΨ coincides with (πˆ ⊗ id)Ψ when for any m ∈ L∞ (G G1 G1,cop 1 Ψ ∞ 2 appropriately interpreted. Its relation with πG 1 : L (G1 ) → L(L (G) ⊗ C0 (G1 )) is expressed by the twisting formula: ∗ πG 1 (m)Ψ Ψ (m) = Ψ πG 1

Ψ ) = L∞ (G 1 ). Using πΨ we may introduce the C0 (G Ψ ) – module for any m ∈ L∞ (G G 1 1 1

Ψ . F Ψ = I Ψ ⊗ L2 (G) ⊗ C0 G 1 πG Ψ 1

Ψ ) → L(F Ψ ) and the strict ∗-antiIt is equipped with the strict ∗-homomorphism πlΨ : L∞ (G Ψ ∞ Ψ Ψ homomorphism πr : L (G ) → L(F ), which are defined as follows:

154


πlΨ (m) i ⊗ h ⊗ a1 = (mi) ⊗ h ⊗ a1 , πG Ψ

πG Ψ

1

1

πrΨ (m) i ⊗ h ⊗ a1 = i ⊗ Jˆm∗ Jˆh ⊗ a1 πG Ψ

πG Ψ

1

1

Ψ ), i ∈ I Ψ , h ∈ L2 (G) and a1 ∈ C0 (G Ψ ). In order to compare F Ψ and F we for any m ∈ L∞ (G 1 prove: Proposition 5.1. There exists a unitary transformation U ∈ L(F , F Ψ ) such that U i ⊗ h ⊗ a1 = i ⊗ Ψ (h ⊗ a1 ) πG

πG Ψ

1

1

Ψ ). U intertwines πl with π Ψ and πr with πrΨ . for any i ∈ I, h ∈ L2 (G) and a1 ∈ C0 (G l 1 Proof. For the existence of U it is enough to note that: U ia ⊗ h ⊗ a1 = ia ⊗ Ψ (h ⊗ a1 ) πG

πG Ψ

1

1

= i ⊗ Ψ πG 1 (a)(h ⊗ a1 ) πG Ψ

1

= U i ⊗ πG 1 (a)(h ⊗ a1 ) . πG

1

The fact that U is unitary and that it possesses the required intertwining properties can be verified by a straightforward computation. 2 Ψ )). It is defined as a prodLet us now introduce the coaction αF Ψ : F Ψ → M(F Ψ ⊗ C0 (G 2 Ψ ) → uct of the coaction αI Ψ given by (7) and the coaction αL2 (G)⊗C0 (G Ψ ) : L (G) ⊗ C0 (G 1 1 Ψ ) ⊗ C0 (G Ψ )) given by: M(L2 (G) ⊗ C0 (G 1

Ψ αL2 (G)⊗C0 (G Ψ ) (ξ ) = Vˆ13 (ξ ⊗ 1), 1

Ψ ). The formula defining αF Ψ goes as follows: for any ξ ∈ L2 (G) ⊗ C0 (G 1 αF Ψ i ⊗ ξ = αI Ψ (i) ⊗ αL2 (G)⊗C0 (G Ψ ) (ξ ). πG Ψ 1

πG Ψ ⊗ι

1

1

Ψ )) of G Ψ on F Ψ : The coaction αF Ψ corresponds to the corepresentation Y Ψ ∈ L(F Ψ ⊗ C0 (G Y Ψ (f ⊗ a) = αF Ψ (f )(1 ⊗ a).


155

we compute: In order to compare Y Ψ with its untwisted counterpart Y ∈ L(F ⊗ C0 (G)) ∗ U ⊗ 1 Y Ψ (U ⊗ 1) i ⊗ ξ ⊗ a πG

1

= U∗ ⊗ 1 YΨ i ⊗ Ψ ξ ⊗ a πG Ψ 1

= U ∗ ⊗ 1 αF Ψ i ⊗ Ψ ξ (1 ⊗ a) πG Ψ 1

= U ∗ ⊗ 1 Ψ ∗ αI (i)Ψ

∗ ˆ

∗ ⊗ Ψ13 Ψ12 (ξ ⊗ 1) (1 ⊗ a) V13 Ψ13

πG Ψ ⊗ι 1

∗ ∗ ˆ

∗ = Ψ ∗ αI (i)Ψ ⊗ Ψ12 Ψ13 V13 Ψ13 Ψ12 (ξ ⊗ 1) (1 ⊗ a) πG ⊗ι 1

∗ ∗ ∗ ˆ

∗ Ψ13 Ψ12 Ψ23 (ξ ⊗ 1) (1 ⊗ a) = Ψ ∗ αI (i)Ψ ⊗ Ψ12 V13 Ψ13

πG ⊗ι 1

∗ ∗ ∗ ˆ

∗ Ψ13 Ψ12 Ψ23 (ξ ⊗ 1) (1 ⊗ a) = Ψ ∗ αI (i) ⊗ Ψ13 Ψ23 Ψ12 V13 Ψ13

πG ⊗ι 1

∗ (ξ ⊗ 1) (1 ⊗ a) = Ψ ∗ αI (i) ⊗ Vˆ13 Ψ13

πG ⊗ι 1

= (πl ⊗ id) Ψ ∗ Y (πr ⊗ id) Ψ ∗ i ⊗ ξ ⊗ a . πG

1

∗ = Ψ Ψ ∗ and in the sixth In the fifth equality we used the easy to verify formula Vˆ13 Ψ12 Vˆ13 12 23

equality we used: (πG 1 ⊗ ι)Ψ = Ψ13 Ψ23 . For the formula for Ψ we refer to (4). This computation and Y Ψ ∈ L(F Ψ ⊗ shows that the relation between the corepresentations Y ∈ L(F ⊗ C0 (G)) Ψ C0 (G )) is given by

∗ U ⊗ id Y Ψ (U ⊗ id) = (πl ⊗ id) Ψ ∗ Y (πr ⊗ id) Ψ ∗ , → L(F ) as the homomorphism of the comwhere we treat the anti-homomorphism πr : L∞ (G) (note that Ψ ∈ L∞ (G) ⊗ L∞ (G)). mutant L∞ (G) Ψ )) and the induced corepresenWe are now ready to compare the induced module Ind(C0 (G 1 1 )) and Ind(W1 ). Let us first recall tation Ind(W1Ψ ) with their untwisted counterparts Ind(C0 (G the construction of the untwisted objects. By the results of [16] there exists a (unique) strict ∗-homomorphism Θ : B(L2 (G)) → L(F ) such that (Θ ⊗ ι)Vˆ = Y, Θ Jˆx ∗ Jˆ = πr (x), The induced C∗ -module is defined as for any x ∈ L∞ (G). 1 ) = v : L2 (G) → F : vx = Θ(x)v for all x ∈ B L2 (G) , Ind C0 (G

156


where by assumption v : L2 (G) → F is a continuous linear map. It follows from the above 1 )) of C0 (G 1 )definition that there exists a unitary transformation Φ : F → L2 (G) ⊗ Ind(C0 (G modules, such that Φ ∗ (h ⊗ v) = v(h)

(8)

1 )). The induced corepresentation Ind(W1 ) ∈ L(C0 (G) ⊗ for any h ⊗ v ∈ L2 (G) ⊗ Ind(C0 (G Ind(C0 (G1 ))) of G is defined by the equation (ι ⊗ πl )W = W12 Ind(W1 )13 , where the leg numbering notation on the right-hand side of this equation is to be understood in the 1 )). It turns out that the isomorphism Φ sense of the above identification F ∼ = L2 (G) ⊗ Ind(C0 (G defined in (8) intertwines the strict ∗-homomorphism πl with the AdInd(W1 ) (see Proposition 3.7 and Section 4 of [16]): Φπl (x)Φ ∗ = Ind(W1 )(x ⊗ 1) Ind(W1 )∗ .

(9)

Let us write the corepresentation equation (G ⊗ ι) Ind(W1 ) = Ind(W1 )13 Ind(W1 )23 in terms of W and Ind(W1 ): Ind(W1 )23 W12 Ind(W1 )∗23 = W12 Ind(W1 )13 .

(10)

Applying the character χg : C0 (G) → C, g ∈ G to the first leg of the above equation we get the following formula Ind(W1 )(Lg ⊗ 1) Ind(W1 )∗ = Lg ⊗ Ind(W1 )(g),

(11)

1 ))) with the where we identified the induced corepresentation Ind(W1 ) ∈ L(C0 (G) ⊗ Ind(C0 (G corresponding representation of the group G on K(Ind(C0 (G1 ))): Ind(W1 )(g) = (χg ⊗ ι) Ind(W1 ) . Ψ )) and Ind(W Ψ ) are defined similarly. Starting with the The twisted objects Ind(C0 (G 1 1 Ψ strict ∗-homomorphism Θ : B(L2 (G)) → L(F Ψ ) we may define the induced C∗ -module Ψ )) and the isomorphism Φ Ψ : F Ψ → L2 (G) ⊗ Ind(C0 (G Ψ )). Defining Ind(W Ψ ) by Ind(C0 (G 1 1 1 the equation Ψ ι ⊗ πlΨ W Ψ = W12 Ind W1Ψ 13 we may relate πlΨ , Φ Ψ and Ind(W1Ψ ) by the formula: ∗ Φ Ψ πlΨ (x)Φ Ψ ∗ = AdInd(W Ψ ) (x) = Ind W1Ψ (x ⊗ 1) Ind W1Ψ . 1

1 )) ⊗ L2 (G)) by: In order to compare Θ and Θ Ψ let us define Ψ˜ ∈ L(Ind(C0 (G Ind(W1 )12 Ψ13 Ind(W1 )∗12 = Ψ13 Ψ˜ 23 .

(12)


157

The existence of such Ψ˜ follows from Eq. (11). Let us also introduce Ψ˜ = Φ ∗ Σ Ψ˜ ΣΦ ∈ L(F )

(13)

where Σ is the flip operation on the tensor product. Theorem 5.2. Let Θ : B(L2 (G)) → L(F ) and Θ Ψ : B(L2 (G)) → L(F Ψ ) be the strict ∗homomorphisms introduced above and let U ∈ L(F , F Ψ ) be the unitary transformation introduced in Proposition 5.1. Let Ψ˜ ∈ L(F ) be the element introduced in Eq. (13). Then Θ Ψ (x) = U (AdΨ˜ ◦Θ)(x)U ∗

(14)

1 )) and Ind(C0 (G Ψ )) are isomorphic, via the for any x ∈ B(L2 (G)). In particular, Ind(C0 (G 1 Ψ 1 )) → Ind(C0 (G )) given by isomorphism K : Ind(C0 (G 1 Ψ 1 ) v → K(v) = U Ψ˜ v ∈ Ind C0 G Ind C0 (G 1 .

(15)

Ψ ) and L∞ (GΨ ) generate B(L2 (G)), it is enough to check Proof. Since the algebras L∞ (G Ψ ) , Eq. (14) on them separately. It is easy to see that for any x ∈ L∞ (G (AdΨ˜ ◦Θ) Jˆx ∗ Jˆ = U ∗ πr (x)U,

(16)

Ψ ) . In order to prove it on L∞ (GΨ ) it is enough to check which proves equality (14) on L∞ (G Ψ ∗ ˆ that (AdΨ˜ ◦Θ ⊗ id)V = (U ⊗ id)Y Ψ (U ⊗ id). In the following computation we shall identify 1 )) by means of the isomorphism Φ (see (8)). Using (9) and (12) it F with L2 (G) ⊗ Ind(C0 (G may be shown that under the aforementioned identification we have: (πl ⊗ id)Ψ = Ψ13 Ψ˜ 23 .

(17)

We compute Ψ

∗ ˆ

∗ ˜ ∗ Ψ12 = Ψ˜ 12 Ψ13 V13 Ψ13 (AdΨ˜ ◦Θ ⊗ id) Vˆ13

∗ ˆ ˜ ∗ ∗ = Ψ˜ 12 Ψ13 V13 Ψ12 Ψ13

˜ ∗ ˜ ∗ ∗ ˆ

∗ = Ψ˜ 12 Ψ12 Ψ23 Ψ13 V13 Ψ13 ∗ = (πl ⊗ id) Ψ Vˆ13 (πr ⊗ id) Ψ ∗ = Y Ψ ,

(18)

where in the third equality we used the formula:

∗ ˆ ∗

∗ ˜ ∗ Ψ23 V13 = Ψ˜ 12 Vˆ13 Ψ˜ 12

and in the fourth equality we used (17).

2

Let us now move on to the comparison of the induced corepresentations Ind(W1Ψ ) and 1 ))) such Ind(W1 ). In order to do that we introduce two elements Ψˇ , Ψˇ ∈ L(L2 (G) ⊗ Ind(C0 (G that

158


(ι ⊗ πl )(Ψ ) = Ψ12 Ψˇ 13 ,

ˇ

Ψ13 . (ι ⊗ πl ) Ψ = Ψ12

(19) (20)

Their existence follows from the bicharacter equation for Ψ and Eq. (11). Let us note that Ψˇ and the element Ψ˜ satisfying (17) are related by the formula Ψˇ = Σ Ψ˜ ∗ Σ.

(21)

We compute: ∗ ι ⊗ πlΨ W Ψ = Ψˇ 23 (ι ⊗ πl ) Ψ W Ψ Ψˇ 23

ˇ ˇ∗ Ψ13 Ψ23 = Ψˇ 23 Ψ12 Ψˇ 13 W12 Ind(W1 )13 Ψ12

= Ψ12 W12 Ψˇ 13 Ψ12 Ind(W1 )13 Ψˇ 13

ˇ

Ψ13 Ind(W1 )13 Ψˇ 13 = Ψ12 W12 Ψ12 Ψ ˇ

Ψ13 Ind(W1 )13 Ψˇ 13 = W12 ,

where in the third equality we used ∗ W12 Ψˇ 13 W12 = Ψˇ 13 Ψˇ 23 ,

ˇ∗ Ind(W1 )∗13 Ψ12 Ind(W1 )13 = Ψ12 Ψ23 .

The first of these equalities follows from the bicharacter equation for Ψ whereas the second one follows from the equality below: Ind(W1 )∗ (Rg ⊗ 1) Ind(W1 ) = Rg ⊗ Ind(W1 )(g). 1 )) ∼ Ψ )) of Theorem 5.2 we have This shows that with the identification Ind(C0 (G = Ind(C0 (G 1 Ind W1Ψ = Ψˇ Ind(W1 )Ψˇ , 1 ))). where the product on the right-hand side is taken in the C∗ -algebra L(L2 (G) ⊗ Ind(C0 (G Finally, we shall compare the induced right coactions Ψ Ψ Ψ Ind β Ψ : Ind C0 G → M Ind C0 G ⊗ C0 G , 1 1 1 ) → M Ind C0 (G 1 ) ⊗ C0 (G) . Ind(β) : Ind C0 (G Eq. (6.2) of [16] defines the induced coaction Ind(β) as ∗ (ι ⊗ β)(x) . ι ⊗ Ind(β) Φ i ⊗ x = W13 (Φ ⊗ ι) (i ⊗ 1) ⊗ W13 πG

1

πG ⊗ι 1


159

Similarly, Ind(β Ψ ) is defined by: Ψ Ψ Ψ∗ ι ⊗ Ind β Ψ Φ Ψ i ⊗ x = W13 Φ ⊗ ι (i ⊗ 1) ⊗ W13 ι ⊗ β Ψ (x) . πG Ψ ⊗ι

πG Ψ 1

1

Combining (8), (15) and (21) one can easily check that (ι ⊗ K)Ψˇ Φ = Φ Ψ U.

(22)

Using this formula we compute: ι ⊗ Ind β Ψ (ι ⊗ K)Ψˇ Φ i ⊗ x πG

1

= ι ⊗ Ind β Ψ Φ Ψ U i ⊗ x πG

1

Ψ Ψ Ψ∗ Φ ⊗ ι (i ⊗ 1) ⊗ W13 ι ⊗ β Ψ (Ψ x) = W13 πG Ψ ⊗ι 1

Ψ Ψ Ψ∗ Φ ⊗ ι (i ⊗ 1) ⊗ W13 Ψ13 Ψ12 ι ⊗ β Ψ (x) = W13 πG Ψ ⊗ι 1

Ψ = W13

Ψ

∗ ∗ Φ ⊗ ι (i ⊗ 1) ⊗ Ψ13 W13 Ψ12 ι ⊗ β Ψ (x)

Ψ = W13

Ψ

∗ ∗ ∗ Φ ⊗ ι (i ⊗ 1) ⊗ Ψ13 W13 Ψ12 Ψ23 (ι ⊗ β)(x)Ψ23

Ψ = W13

Ψ

∗ ∗ Φ ⊗ ι (i ⊗ 1) ⊗ Ψ13 Ψ12 W13 (ι ⊗ β)(x)Ψ23

Ψ = W13

Ψ

∗ ∗ Φ ⊗ ι (U ⊗ id) (i ⊗ 1) ⊗ Ψ13 W13 (ι ⊗ β)(x)Ψ23

πG Ψ ⊗ι 1

πG Ψ ⊗ι 1

πG Ψ ⊗ι 1

πG ⊗ι 1

∗ = Ψ13 W13 Φ Ψ ⊗ ι (U ⊗ id) (i ⊗ 1) ⊗ W13 (ι ⊗ β)(x)Ψ23 πG ⊗ι 1

∗ = Ψ13 W13 (ι ⊗ K ⊗ ι)(Ψˇ ⊗ 1)(Φ ⊗ id) (i ⊗ 1) ⊗ W13 (ι ⊗ β)(x)Ψ23 πG ⊗ι 1

∗ ∗ = (ι ⊗ K ⊗ ι)Ψ13 Ψˇ 12 Ψ˜ 23 W13 (Φ ⊗ id) (i ⊗ 1) ⊗ W13 (ι ⊗ β)(x) Ψ23 πG ⊗ι 1

∗ ι ⊗ Ind(β) Φ ι ⊗ x Ψ23 . = (ι ⊗ K ⊗ ι)Ψ13 Ψˇ 12 Ψ˜ 23 πG

1

Using the equality ι ⊗ Ind(β) (Ψˇ ) = Ψ13 Ψˇ 12 we get

160


ι ⊗ Ind β Ψ (ι ⊗ K)Ψˇ Φ i ⊗ x ∗

= (ι ⊗ K ⊗ ι)Ψ˜ 23

πG

1

ι ⊗ Ind(β) Ψˇ Φ ι ⊗ x Ψ23 . πG

1

Hence we see that ∗ K ⊗ ι Ind β Ψ K(v) = Ψ˜ ∗ Ind(β)(v)Ψ 1 )). The coaction Ind(β Ψ ) gives rise to the coaction on K(Ind(C0 (G Ψ ))) for any v ∈ Ind(C0 (G 1 (which we denote by the same symbol) and using the above equation we have ∗ K ⊗ ι Ind β Ψ KxK ∗ (K ⊗ ι) = Ψ˜ ∗ Ind(β)(x)Ψ˜

(23)

1 ))). In what follows we summarize the above considerations: for any x ∈ K(Ind(C0 (G Ψ ) be the C∗ -algebra of the quantum group G Ψ treated as the C∗ -Hilbert Theorem 5.3. Let C0 (G Ψ Ψ Ψ module over itself. Let W1 ∈ M(C0 (G ) ⊗ C0 (G )) be the left regular corepresentation of GΨ Ψ defined in (6). Let K : Ind(C0 (G 1 )) → Ind(C0 (G Ψ )) be and β Ψ be the right coaction of G 1 Ψ )) with the isomorphism introduced in Eq. (15) (in what follows we shall identify Ind(C0 (G 1 1 )) by means of K). Ind(C0 (G 1 ))) is given by The induced coaction Ind(W1Ψ ) ∈ L(L2 (G) ⊗ Ind(C0 (G Ind W1Ψ = Ψˇ Ind(W1 )Ψˇ

(24)

1 ))) are the unitary elements defined by (19) and (20). The where Ψˇ , Ψˇ ∈ L(L2 (G) ⊗ Ind(C0 (G twisted induced coaction Ψ Ψ 1 ) ⊗ C0 G → M Ind C0 (G Ind β Ψ : Ind C0 G 1 is related with its untwisted counterpart Ind(β) by the following formula Ind β Ψ (v) = Ψ˜ ∗ Ind(β)(v)Ψ

(25)

1 )), where Ψ˜ ∈ L(Ind(C0 (G 1 )) ⊗ C0 (G Ψ )) is the unitary element defined for any v ∈ Ind(C0 (G by (17). 6. Rieffel deformation of homogeneous spaces – the quotient case Let G be a locally compact group, G1 its closed subgroup and let X = G/G1 be the homogeneous space of left G1 -cosets. The standard action of G on X gives rise to the G-C∗ -algebra X = (C0 (X), X ). Suppose that Γ ⊂ G1 is a closed abelian subgroup and let Ψ be a 2-cocycle on the dual group Γˆ . Using the results described in Section 4 we know that X may be deformed to GΨ -C∗ -algebra XΨ .


161

On the other hand, applying the Rieffel deformation to the subgroup G1 ⊂ G we get a closed Ψ Ψ Ψ ∗ quantum subgroup GΨ 1 ⊂ G in the sense of Definition 2.5 of [16]. Let G /G1 be the C algebraic quotient space in the sense of Definition 2.6. The aim of this section is to show that ∼ Ψ GΨ /GΨ 1 =X . Let ι ∈ Mor(C0 (X), C0 (G)) be the standard embedding – it maps a function f ∈ C0 (X) to the same function on G which is constant on the right G1 -cosets. Let α : Γ → Aut(C0 (X)) and ρ : Γ 2 → Aut(C0 (G)) be the actions introduced in Section 4. The action ρ of Γ 2 on C0 (G) restricts to the action α on C0 (X). To be more precise, the second copy of Γ in Γ 2 acts trivially on ι(C0 (X)) and the action of the first copy coincides with α. By the results of Section 3.2 of [9] the embedding ι may be twisted to the embedding ιΨ ∈ Mor(C0 (XΨ ), C0 (GΨ )). Note that the comultiplication GΨ restricts to the coaction XΨ on ιΨ (C0 (XΨ )). The last statement follows from Theorem 4.11 of [9] and the description of XΨ given in Section 4. In particular, XΨ is implemented by the multiplicative unitary: XΨ (a) = W Ψ ∗ (1 ⊗ a)W Ψ ,

(26)

for any a ∈ C0 (XΨ ). Ψ As was explained in Remark 2.7, the crossed product of C0 (GΨ /GΨ 1 ) by the coaction of G Ψ Ψ ))). Using Theorem 5.3 we may identify K(Ind(C0 (G ))) with coincides with K(Ind(C0 (G 1 1 ∗ K(Ind(C0 (G1 ))), which in turn may be identified with the C -algebra G C0 (G/G1 ). The later 0 (X)] ⊂ B(L2 (G)) – the C∗ -algebra generated C∗ -algebra may be identified with B = [C0 (G)C 2 by C0 (G) and C0 (X) inside B(L (G)). Ψ ))) = B the crossed product structure of Under the identification K(Ind(C0 (G 1 Ψ K(Ind(C0 (G1 ))) may be described as follows. Using Eq. (25) one can see that the dual coaction Ψ )) is implemented by the unitary Vˆ Ψ introduced in (5). Eq. (24) Ind(β Ψ ) : B → M(B ⊗ C0 (G shows in turn that the induced corepresentation Ind(W1Ψ ) ∈ M(C0 (GΨ ) ⊗ B) can be identified Ψ )) ⊂ M(C0 (GΨ ) ⊗ B). We are now with the regular corepresentation W Ψ ∈ M(C0 (GΨ ) ⊗ C0 (G well prepared to prove: Theorem 6.1. Let XΨ be the Rieffel deformation of the homogeneous space X and GΨ /GΨ 1 the Ψ /GΨ . G Vaes’ quotient considered above. Then XΨ ∼ = 1 Proof. By the universal properties of the crossed product C∗ -algebra Γ ρ C0 (X) there exists a unique morphism ιΓ ∈ Mor(Γ ρ C0 (X), B) which is identity on C0 (X) ⊂ M(Γ ρ C0 (X)) and such that it sends the unitary generator uγ ∈ M(C∗ (Γ )) to the left shift Lγ ∈ M(B). Using Theorem 3.6 of [9] we may conclude that the restriction of ιΓ to C0 (XΨ ) gives rise to the injective morphism ι|C0 (XΨ ) which we shall denote by ιΨ ∈ Mor(C0 (XΨ ), B). Let ρˆ Ψ be the twisted dual action on Γ ρ C0 (X). In what follows we shall interpret it as a coaction ρˆ Ψ ∈ Mor(Γ ρ C0 (X), Γ ρ C0 (X) ⊗ C∗ (Γ )) (see Remark 2.2). Under this interpretation, the relation of ρˆ Ψ with its untwisted counterpart ρˆ is established by the following formula: ρˆ Ψ (x) = Ψ ∗ ρ(x)Ψ for any x ∈ Γ ρ C0 (X).

(27)

162


Let us note that ιΓ intertwines the twisted dual coactions: Ind β Ψ ◦ ιΓ = ιΓ ⊗ id ◦ ρˆ Ψ .

(28)

In order to see it we have to observe that: • ιΓ intertwines ρˆ and Ind(β): Ind(β) ◦ ιΓ = ιΓ ⊗ id ◦ ρ. ˆ

(29)

Indeed, the equality Ind(β)(ιΓ (f )) = (ιΓ ⊗ id)(ρ(f ˆ )) for any f ∈ C0 (X) is the consequence of the simultaneous invariance of f under ρˆ and Ind(β). Moreover Ind(β)(ιΓ (uγ )) = Lγ ⊗ Lγ = (ιΓ ⊗id)(ρ(u ˆ γ )) for any γ ∈ Γ . Using the fact C0 (X) and C∗ (Γ ) generate Γ ρ C0 (X) we get (29). • Let Ψ˜ be the unitary element introduced in (12). Note that (ιΓ ⊗ id)Ψ = Ψ˜ . Using the above observations and Eqs. (25), (27) one easily gets (28). This in turn implies that Ψ ιΨ (C0 (XΨ )) ⊂ M(B)Ind(β ) . Now we may prove the equality ιΨ (C0 (XΨ )) = C0 (GΨ /GΨ 1 ) using Theorem 6.7 of [16]. Its application requires to check the following two conditions: • The map x → W Ψ ∗ (1 ⊗ x)W Ψ defines a continuous coaction on ιΨ (C0 (XΨ )). Indeed, this follows from (26). = B. In order to see that we compute • We have [ιΨ (C0 (XΨ ))C0 (G)]

Ψ Ψ = ιΨ C0 XΨ C∗ (Γ )C0 (G) = ιΓ Γ ρ C0 XΨ C0 (G) ι C0 X C0 (G)

= ιΓ C0 (X) C∗ (Γ )C0 (G) = ιΓ Γ ρ C0 (X) C0 (G)

=B = C0 (X)C0 (G) = C0 (G) and where in the first and the fourth equality we used the fact that [C∗ (Γ )C0 (G)] Ψ in the third equality we used the fact that Γ ρ C0 (X ) = Γ ρ C0 (X) (note that the action of Γ on C0 (X) and on C0 (XΨ ) is denoted by the same ρ). This ends the proof of the isomorphism XΨ ∼ = GΨ /GΨ 1 .

2

7. Rieffel deformation of G -simple C∗ -algebras Let G be a locally compact quantum group corresponding to a locally compact group G. The aim of this section is to prove that the Rieffel deformation XΨ of a G-simple C∗ -algebra X is GΨ -simple (see Definition 2.4). In particular, the Rieffel deformation of a quotient space X = G/G1 , also in the case when Γ is a subgroup of G but not of G1 is GΨ -simple (note that in this case we cannot construct the closed quantum subgroup GΨ 1 ). The idea of the following proof is based on the functorial properties of the Rieffel deformation for which we refer to Section 3.2 of [9].


163

Theorem 7.1. Let X be a G-simple C∗ -algebra in the sense of Definition 2.4. Let Γ be a closed abelian subgroup of G and Ψ a 2-cocycle on the dual group Γˆ . The Rieffel deformation XΨ of X is GΨ -simple. Proof. Let B be a GΨ -C∗ -algebra and let π ∈ MorGΨ (XΨ , B) be a GΨ -morphism in the sense of Definition 2.3. The Rieffel deformation Γ Ψ of Γ is a quantum closed subgroup of GΨ . Actually, Γ Ψ = Γ which easily follows from the abelianity of Γ (see Appendix B of [7]). Let α : Γ → Aut(C0 (XΨ )) be the action that corresponds to the Γ -restriction of the coaction XΨ . Similarly, we may introduce β : Γ → Aut(C0 (B)). Obviously, the morphism π is covariant: π ◦ αγ = βγ ◦ π for any γ ∈ Γ . Using deformation data (C0 (B), β, Ψ¯ ) we may construct the deformed ¯ C∗ -algebra that we shall denote by C0 (BΨ ) and by the covariance of π we get the deformed ¯ ¯ Ψ Ψ morphism π ∈ Mor(C0 (X), C0 (B )) (see Section 3.2 of [9]). Note that we used the fact that ¯ ¯ C0 (XΨ )Ψ = C0 (X). Similarly we have (GΨ )Ψ = G. Employing the ideas presented in Section 4 ¯ ¯ we may construct the coaction BΨ¯ of G on BΨ and check that π Ψ is a G-morphism. The G¯

simplicity of X implies that ker π Ψ = {0}. Using Proposition 3.8 of [9] we see that ker π = {0}, which ends the proof of GΨ -simplicity of XΨ . 2 References [1] S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de C ∗ -algebres, Ann. Sci. Ecole Norm. Sup. (4) 26 (1993) 425–488. [2] M. Brin, G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, ISBN 0-52180841-3, 2002, xii+240 pp. [3] K. de Commer, Galois objects and cocycle twisting for locally compact quantum groups, J. Operator Theory, in press. [4] M. Enock, L. Vainerman, Deformation of a Kac algebra by an abelian subgroup, Comm. Math. Phys. 178 (3) (1996) 571–596. [5] P. Fima, L. Vainerman, Twisting and Rieffel’s deformation of locally compact quantum groups. Deformation of the Haar measure, Comm. Math. Phys. 286 (3) (2009) 1011–1050. [6] S. Kaliszewski, J. Quigg, Categorical Landstad duality for actions, Indiana Univ. Math. J. 58 (1) (2009) 415–441. [7] P. Kasprzak, The Heisenberg–Lorentz quantum group, J. Noncommut. Geom. 4 (2010) 577–611. [8] P. Kasprzak, Rieffel deformation of groups coactions, Comm. Math. Phys., doi:10.1007/s00220-010-1093-9, in press. [9] P. Kasprzak, Rieffel deformation via crossed products, J. Funct. Anal. 257 (5) (2009) 1288–1332. [10] A. Kleppner, Multipliers on abelian groups, Math. Ann. 158 (1965) 11–34. [11] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup. 33 (4) (2000) 837–934. [12] T. Masuda, Y. Nakagami, S.L. Woronowicz, A C∗ -algebraic framework for quantum groups, Internat. J. Math. 14 (9) (2003) 903–1001. [13] P. Podle´s, Quantum spheres, Lett. Math. Phys. 14 (3) (1987) 193–202. [14] P. Podle´s, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170 (1) (1995) 1–20. [15] M.A. Rieffel, Deformation quantization for action of Rd , Mem. Amer. Math. Soc. 106 (506) (1993). [16] S. Vaes, A new approach to induction and imprimitivity results, J. Funct. Anal. 229 (2) (2005) 317–374. [17] J.C. Varilly, Quantum symmetry groups of noncommutative spheres, Comm. Math. Phys. 221 (3) (2001) 511–523. [18] S.L. Woronowicz, C∗ -algebras generated by unbounded elements, Rev. Math. Phys. 7 (3) (1995) 481–521.


Improved bounds in the metric cotype inequality for Banach spaces Ohad Giladi a , Manor Mendel b , Assaf Naor a,∗ a Courant Institute, New York University, United States b Computer Science Division, The Open University of Israel, Israel

Received 5 March 2010; accepted 26 August 2010 Available online 6 September 2010 Communicated by N. Kalton

Abstract It is shown that if (X, · X ) is a Banach space with Rademacher cotype q then for every integer n there 1+ q1

exists an even integer m n n j =1

such that for every f : Znm → X we have

q q m q Ex f x + ej − f (x) m Eε,x f (x + ε) − f (x) X , 2 X

(1)

where the expectations are with respect to uniformly chosen x ∈ Znm and ε ∈ {−1, 0, 1}n , and all the implied constants may depend only on q and the Rademacher cotype q constant of X. This improves the 2+ 1

bound of m n q from Mendel and Naor (2008) [13]. The proof of (1) is based on a “smoothing and approximation” procedure which simplifies the proof of the metric characterization of Rademacher cotype of Mendel and Naor (2008) [13]. We also show that any such “smoothing and approximation” approach to 1

metric cotype inequalities must require m n 2 © 2010 Elsevier Inc. All rights reserved.

+ q1

.

Keywords: Metric cotype; bi-Lipschitz embeddings; Coarse embeddings


E-mail addresses: [email protected] (O. Giladi), [email protected] (M. Mendel), [email protected] (A. Naor). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.015

O. Giladi et al. / Journal of Functional Analysis 260 (2011) 164–194

165

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The smoothing and approximation scheme . . . . . . . . . . . . . . . . . . . . . 1.2. A lower bound on smoothing and approximation with general kernels . . 1.3. The relation to nonembeddability results and some open problems . . . . . 2. Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Estimates for the bivariate Bernoulli numbers . . . . . . . . . . . . . . . . . . . 3.2. Some combinatorial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Putting things together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. A lower bound for general convolution kernels: Proof of Proposition 4.1 . 4.2. A sharp lower bound for Ej averages: Proof of Proposition 4.2 . . . . . . . 4.3. Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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165 167 170 171 173 178 178 179 183 185 186 190 191 193 193

1. Introduction A metric space (M , dM ) is said [13] to have metric cotype q > 0 with constant Γ > 0 if for every integer n there exists an even integer m such that for every f : Znm → X we have n j =1

Ex

q

q m dM f x + ej , f (x) Γ q mq Eε,x dM f (x + ε), f (x) . 2

(2)

In (2) the expectations are taken with respect to x chosen uniformly at random from the discrete torus Znm , and ε chosen uniformly at random from {−1, 0, 1}n (the ∞ generators of Znm ). Also, in (2) and in what follows, {ej }nj=1 denotes the standard basis of Znm . A Banach space (X, · X ) is said to have Rademacher cotype q > 0 if there exists a constant C < ∞ such that for every n ∈ N and for every x1 , x2 , . . . ., xn ∈ X, n

q xj X

j =1

q n C Eε εj xj . q

j =1

(3)

X

X is said to have Rademacher type p > 0 if there exists a constant T < ∞ such that for every n ∈ N and for every x1 , x2 , . . . , xn ∈ X, p n n p Eε εj xj T p xj X . j =1

X

(4)

j =1

The smallest possible constants C, T in (3), (4) are denoted Cp (X), Tp (X), respectively. We refer to [16,9] for more information on the notions of type and cotype, though the present paper

166


requires minimal background of this theory. We shall use throughout standard Banach space notation and terminology, as appearing in, say, [21]. The following theorem was proved in [13]: Theorem 1.1. (See [13].) A Banach space (X, · X ) has Rademacher cotype q if and only if it has metric cotype q. Thus, for Banach spaces the linear notion of Rademacher cotype q is equivalent to the notion of metric cotype q, which ignores all the structure of the Banach space except for its metric properties. Theorem 1.1 belongs to a comprehensive program, first formulated by Bourgain in [2], which is known as the Ribe program, whose goal is to recast the local theory of Banach spaces as a purely metric theory. A byproduct of this program is that linear properties such as Rademacher cotype can be made to make sense in general metric spaces, with applications to metric geometry in situations which lack any linear structure. We refer to [13] and the references therein for more information on the Ribe program and its applications. Definition (2) and Theorem 1.1 suppress the value of m, since it is irrelevant for the purpose of a metric characterization of Rademacher cotype. Nevertheless, good bounds on m are important for applications of metric cotype to embedding theory, some of which will be recalled in Section 1.3. It was observed in [13] that if the metric space M contains at least two points then the value of m in (2) must satisfy m n1/q (where the implied constant depends only on Γ ). If X is a Banach space with Rademacher type p > 1 and Rademacher cotype q, then it was shown in [13] that X satisfies the metric cotype q inequality (2) for every m n1/q (in which case Γ depends only on p, q, Tp (X), Cq (X)). Such a sharp bound on m is crucial for certain applications [13,14] of metric cotype, and perhaps the most important open problem in [13] is whether this sharp bound on m holds true even when the condition that X has type p > 1 is dropped. The bound on m from [13] in Theorem 1.1 is m n to m n

1+ q1

2+ q1

. Our main result improves this bound

:

Theorem 1.2. Let X be a Banach space with Rademacher cotype q 2. Then for every n ∈ N, every integer m 6n n j =1

1+ q1

which is divisible by 4, and every f : Znm → X, we have

q q m X mq Eε,x f (x + ε) − f (x)X . Ex f x + ej − f (x) 2 X

(5)

In (5), and in what follows, X , X indicate the corresponding inequalities up to constants which may depend only on q and Cq (X). Similarly, we will use the notation q , q to indicate the corresponding inequalities up to constants which may depend only on q. Though a seemingly modest improvement over the result of [13], the strengthened metric cotype inequality (5) does yield some new results in embedding theory, as well as a new proof of a result of Bourgain [3]; these issues are discussed in Section 1.3. More importantly, our proof of Theorem 1.2 is based on a better understanding and sharpening of the underlying principles behind the proof of Theorem 1.1 in [13]. As a result, we isolate here the key approach to the metric characterization of Rademacher cotype in [13], yielding a simpler and clearer proof of Theorem 1.1, in addition to the improved bound on m. This is explained in detail in Section 1.1. 1+ 1 While the bound m n q is far from the conjectured optimal bound m n1/q , our second


167

main result is that (a significant generalization of) the scheme for proving Theorem 1.1 and Theorem 1.2 (which is implicit in [13] and formulated explicitly here) cannot yield a bound 1

+1

better than m n 2 q . Our method for proving this lower bound is presented in Section 4, and might be of independent interest. We remark in passing that in [13] a one parameter family of variants of the notion of metric cotype is studied, corresponding to raising the distances to powers other than q, and modifying the right-hand side of (2), (5) accordingly (we refer to [13] for more details). The argument presented here can be modified to yield simplifications and improvements of all the corresponding variants of Theorem 1.1. While these variants are crucial for certain applications of metric cotype [13,11], we chose to present Theorem 1.2 only for the simplest “vanilla” version of metric cotype (2), for the sake of simplicity of exposition. Notation for measures. Since our argument uses a variety of averaging procedures over several spaces, it will be convenient to depart from the expectation notation that we used thus far. In particular, throughout this paper μ will denote the uniform probability measure on Znm (m, n will always be clear from the context), σ will denote the uniform probability measure on {−1, 0, 1}n , and τ will denote the uniform probability measure on {−1, 1}n . 1.1. The smoothing and approximation scheme We start with a description of an abstraction of the approach of [13] to proving the metric characterization of Rademacher cotype of Theorem 1.1. For a Banach space X, a function f : Znm → X and a probability measure ν on Znm , we use the standard notation for the convolution f ∗ ν : Znm → X: f ∗ ν(x) =

f (x − y) dν(y).

Znm

Assume that we are given n probability measures ν1 , . . . , νn on Znm , and two additional probability measures β1 , β2 on the pairs in Znm × Znm of ∞ distance 1, i.e., on the set

def E∞ Znm = (x, y) ∈ Znm × Znm : x − y ∈ {−1, 0, 1}n .

(6)

For A, S, q 1, we shall say that the measures ν1 , . . . , νn , β1 , β2 are a (q, A, S)-smoothing and approximation scheme on Znm if for every Banach space (X, · X ) and every f : Znm → X we have the following two inequalities: (A) Approximation property: 1 n n

j =1 Zn

m

f ∗ νj (x) − f (x)q dμ(x) Aq X

E∞ (Znm )

f (x) − f (y)q dβ1 (x, y). X

(7)

168


(S) Smoothing property:

Znm {−1,1}n

n q

εj f ∗ νj (x + ej ) − f ∗ νj (x − ej ) dτ (ε) dμ(x) j =1

Sq

X

f (x) − f (y)q dβ2 (x, y). X

(8)

E∞ (Znm )

Often, when the underlying space Znm is obvious from the context, we will not mention it explicitly, and simply call ν1 , . . . , νn , β1 , β2 a (q, A, S)-smoothing and approximation scheme. In some cases, however, it will be convenient to mention the underlying space Znm so as to indicate certain restrictions on m. We introduce these properties for the following simple reason. We wish to deduce the metric cotype inequality (5) from the Rademacher cotype inequality (3). In essence, the Rademacher cotype condition (3) is the same as the metric cotype inequality (5) when restricted to linear mappings. This statement is not quite accurate, but it suffices for the purpose of understanding the intuition behind the ensuing argument; we refer to Section 5.1 in [13] for the precise argument. In any case, it stands to reason that in order to prove (5) from (3), we should first smooth out f , so that it will be locally well approximated (on average) by a linear function. As we shall see momentarily, it turns out that the appropriate way to measure the quality of such a smoothing procedure is our smoothing property (8). Of course, while the averaging operators corresponding to convolution with the measures ν1 , . . . , νn yield a better behaved function, we still need the resulting averaged function to be close enough to the original function f , so as to deduce a meaningful inequality such as (5) for f itself. Our approximation property (7) is what’s needed for carrying out such an approach. The above general scheme is implicit in [13]. Once we have isolated the crucial approximation and smoothing properties, it is simple to see how they relate to metric cotype. For this purpose, assume that the Banach space X has Rademacher cotype q, and for each x ∈ Znm apply the Rademacher cotype q inequality to the vectors {f ∗ νj (x + ej ) − f ∗ νj (x − ej )}nj=1 (where the averaging in (3) is with respect to ε ∈ {−1, 1}n , rather than ε ∈ {−1, 0, 1}n ; it is an easy standard fact that these two variants of Rademacher cotype q coincide): q n

εj f ∗ νj (x + ej ) − f ∗ νj (x − ej ) dτ (ε) {−1,1}n

X

j =1

n f ∗ νj (x + ej ) − f ∗ νj (x − ej )q . X

X

(9)

j =1

The triangle inequality, combined with the convexity of the function t → t q , implies that for every x ∈ Znm and j ∈ {1, . . . , n} we have q q f x + m ej − f (x) 3q−1 f ∗ νj x + m ej − f ∗ νj (x) 2 2 X X q m m f ∗ ν x + − f x + e e + 3q−1 j j j 2 2 X q + 3q−1 f ∗ νj (x) − f (x)X . (10)


169

At the same time (recalling that m is divisible by 4), a combination of the triangle inequality and Hölder’s inequality bounds the first term in the right-hand side of (10) as follows: q f ∗ νj x + m ej − f ∗ νj (x) 2 X m/4 q

f ∗ νj (x + 2tej ) − f ∗ νj x + 2(t − 1)ej X

t=1

m 4

q−1 m/4

f ∗ νj (x + 2tej ) − f ∗ νj x + 2(t − 1)ej q . X

(11)

t=1

Substituting (11) into (10), summing up over j ∈ {1, . . . , n}, and integrating with respect to x ∈ Znm while using the translation invariance of the measure μ, we deduce the inequality q n f x + m ej − f (x) dμ(x) 2 X j =1 Zn

m

3q

n f ∗ νj (x) − f (x)q dμ(x) X j =1 Zn

m

n q f ∗ νj (x + ej ) − f ∗ νj (x − ej )q dμ(x). +m X

(12)

j =1Zn

m

We can now bound the first term in the right-hand side of (12) using the approximation property (7), and the second term in the right-hand side of (12) using (9) and the smoothing property (8). The inequality thus obtained is q n f x + m ej − f (x) dμ(x) 2 X j =1 Zn

m

X nAq + mq S q

f (x) − f (y)q dβ3 (x, y), X

(13)

E∞ (Znm )

where β3 = (β1 + β2 )/2. Note in passing that when m A, an inequality such as (13), with perhaps a different measure β3 on E∞ (Znm ), is a consequence of the triangle inequality, and therefore holds trivially on any Banach space X. Thus, for our purposes, we may assume throughout that a (q, A, S)-smoothing and approximation scheme on Znm satisfies m A. Assuming that m inequality (13) becomes

A 1/q ·n , S

(14)

170


q n f x + m ej − f (x) dμ(x) 2 X j =1 Zn

m

X S m q

q

f (x) − f (y)q dβ3 (x, y). X

(15)

E∞ (Znm )

If we could come up with a smoothing and approximation scheme for which S 1, and m satisfied (14), then inequality (15) would not quite be the desired metric cotype inequality (5), but it would be rather close to it. The difference is that the probability measure β3 is not uniformly distributed on all ∞ edges E∞ (Znm ), as required in (5). Nevertheless, for many measures β3 , elementary triangle inequality and symmetry arguments can be used to “massage” inequality (15) into the desired inequality (5). This last point is a technical issue, but it is not the heart of our argument: we wish to design a smoothing and approximation scheme satisfying S 1 with A as small as possible. In [13] such a scheme was designed with A n2 . Here we carefully optimize the approach of [13] to yield a smoothing and approximation scheme with A n, in which 1+ 1

case (14) becomes the desired bound m n q . The bounds that we need in order to establish this improved estimate on m are based on the analysis of some quite delicate cancellations; indeed the bounds that we obtain are sharp for our smoothing and approximation scheme, as discussed in Section 1.2. In proving such sharp bounds, a certain bivariate extension of the Bernoulli numbers arises naturally; these numbers, together with some basic asymptotic estimates for them, are presented in Section 3.1. The cancellations in the Rademacher sums corresponding to our convolution kernels are analyzed via certain combinatorial identities in Section 3.2. 1.2. A lower bound on smoothing and approximation with general kernels One might wonder whether our failure to prove the bound m n1/q without the non-trivial Rademacher type assumption is due to the fact we chose the wrong smoothing and approximation scheme. This is not the case. In Section 4 we show that any approach based on smoothing and approximation is doomed to yield a sub-optimal dependence of m on n (assuming that the conjectured n1/q bound is indeed true). Specifically, we show that for any√(q, A, S)-smoothing and approximation scheme on Znm , with m A, we must have AS q n. Thus the bound 1 1 √ + S 1 forces the bound A q n, and correspondingly (14) becomes m q n 2 q . Additionally, we show in Section 4 that for the specific smoothing and approximation scheme used here, the 1+ 1

bound m n q is sharp. It remains open what is the best bound on m that is achievable via a smoothing and approximation scheme. While this question is interesting from an analytic perspective, our current lower bound shows that we need to use more than averaging with respect to positive measures in order to prove the desired bound m n1/q . 1

+1

Note that the lower bound m q n 2 q for smoothing and approximation schemes rules out the applicability of this method to some of the most striking potential applications of metric cotype to embedding theory in the coarse, uniform, or quasisymmetric categories, as explained in Section 1.3; these applications rely crucially on the use of a metric cotype inequality with m n1/q .


171

The cancellation that was exploited in [13] in order to prove the sharp bound on m in the presence of non-trivial Rademacher type was also related to smoothing properties of convolution kernels, but with respect to signed measures: the smoothed Rademacher sums in the left-hand side of (8) are controlled in [13] via the Rademacher projection, and the corresponding smoothing inequality (for signed measures) is proved via an appeal to Pisier’s K-convexity theorem [15]. It would be of great interest to understand combinatorially/geometrically the cancellations that underly the estimate m n1/q from [13], though there seems to be a lack of methods to handle smoothing properties of signed convolution kernels in spaces with trivial Rademacher type and finite Rademacher cotype. 1.3. The relation to nonembeddability results and some open problems We recall some standard terminology. Let (X, dX ) and (Y, dY ) be metric spaces. X is said to embed with distortion D into Y if there exists a mapping f : X → Y and (scaling factor) λ > 0, such that for all x, y ∈ X we have λdX (x, y) dY (f (x), f (y)) DλdX (x, y). X is said to embed uniformly into Y if there exists an into homeomorphism f : X → Y such that both f and f −1 are uniformly continuous. X is said to embed coarsely into Y if there exists a mapping f : X → Y and two non-decreasing functions α, β : [0, ∞) → [0, ∞) such that limt→∞ α(t) = ∞, and for all x, y ∈ X we have α(dX (x, y)) dY (f (x), f (y)) β(dX (x, y)). X is said to admit a quasisymmetric embedding into Y if there exists a mapping f : X → Y and an increasing (x),f (y)) (modulus) η : (0, ∞) → (0, ∞) such that for all distinct x, y, z ∈ X we have ddYY (f (f (x),f (z))

η( ddXX (x,y) (x,z) ). For a Banach space X, let qX denote the infimum over those q 2 such that X has Rademacher (equiv. metric) cotype q. It was shown in [13,14] that if X, Y are Banach spaces, Y has Rademacher type p > 1, and X embeds uniformly, coarsely, or quasisymmetrically into Y , then qX qY . Thus, under the Rademacher type > 1 assumption on the target space, Rademacher cotype q is an invariant that is stable under embeddings of Banach spaces, provided that the embedding preserves distances in a variety of (seemingly quite weak) senses. The role of the assumption that Y has non-trivial Rademacher type is via the metric cotype inequality with optimal m: the proofs of these results only use that Y satisfies the metric cotype q inequality (2) for some m n1/q (under this assumption, Y can be a general metric space and not necessarily a Banach space). This fact motivates our conjecture that for any Banach space Y with Rademacher cotype q, the metric cotype inequality (5) holds for every m Y n1/q . The same assertion for general metric spaces of metric cotype q is too much to hope for; see [19]. Perhaps the simplest Banach spaces for which we do not know how to prove a sharp metric cotype inequality are L1 and the Schatten–von Neumann trace class S1 (see, e.g., [21]). Both of these spaces have Rademacher cotype 2 (for S1 see [18]), yet the currently best known bound on m in the metric cotype inequality (5) (with q = 2) for both of these spaces is the bound m n3/2 obtained here. The above embedding results in the uniform, coarse or quasisymmetric categories do hold true for embeddings into L1 (i.e., a Banach space X that embeds in one of these senses into L1 satisfies qX = 2). This fact is due to an ad-hoc argument, which fails for S1 (see Section 8 in [13] for an explanation). We can thus ask the following natural questions (many of which were already raised in [13]): Question 1. Can Lr admit a uniform, coarse, or quasisymmetric embedding into S1 when r > 2? More ambitiously, can a Banach space X with qX > 2 embed in one of these senses into S1 ? In

172


greatest generality: can a Banach space X embed in one of these senses into a Banach space Y with qY < qX ? If a Banach space X admits a uniform or coarse embedding into S1 , then X must have finite cotype. This fact, which could be viewed as a (non-quantitative) step towards Question 1, was communicated to us by Nigel Kalton. To prove it, note that it follows from [17, Lem. 3.2] that for any ultrapower (S1 )U of S1 , the unit ball of (S1 )U is uniformly homeomorphic to a subset of Hilbert space. Thus (S1 )U has Kalton’s property Q (see [8] for a detailed discussion of this property). If the unit ball of X is uniformly homeomorphic to a subset of S1 (resp. X admits a coarse embedding into S1 ), then the unit ball in any ultrapower of X is uniformly homeomorphic to a subset of (S1 )U (resp. any ultrapower of X admits a coarse embedding into (S1 )U ). By the proof of [8, Thm. 4.2], it follows that any ultrapower of X has property Q, and hence it cannot contain c0 . Thus X cannot have infinite cotype by the Maurey–Pisier Theorem [10] and standard Banach space ultrapower theory (see [4, Thm. 8.12]). Question 2. Does S1 admit a uniform, coarse, or quasisymmetric embedding into a Banach space Y with Rademacher type p > 1? More ambitiously, does S1 embed in one of these senses into Banach space Y with Rademacher type p > 1 and qY = 2? In greatest generality: does every Banach space X embed in one of these senses into a Banach space Y with Rademacher type p > 1? Perhaps we can even ensure in addition that qY = qX ? Question 2 relates to Question 1 since embeddings into spaces with type > 1 would allow us to use the nonembeddability results of [13]. While the improved bound on m in Theorem 1.2 does not solve any of these fundamental questions, it does yield new restrictions on the possible moduli of embeddings in the uniform, coarse, or quasisymmetric categories. Instead of stating our nonembedding corollaries in greatest generality, let us illustrate our (modest) improved nonembeddability results for snowflake embeddings of L4 into S1 (this is just an illustrative example; the method of [13] yields similar results for embeddings of any Banach space X with qX > 2 into S1 , and S1 itself can be replaced by general Banach spaces of finite cotype). Take θ ∈ (0, 1) and assume the metric space (L4 , x − yθ4 ) admits a bi-Lipschitz embedding into S1 . Our strong conjectures imply that this cannot happen, but at present the best we can do is give bounds on θ . An application of Theorem 1.2 shows that θ 4/5, i.e., we have a definite quantitative estimate asserting that a uniform embedding of L4 into S1 must be far from bi-Lipschitz. The previous bound from [13] for S1 was m = n5/2 , yielding θ 8/9. Our lower bound shows that by using a smoothing and approximation scheme we cannot hope to get a bound of θ < 2/3. Turning to bi-Lipschitz embeddings, consider the grid {0, 1, . . . , m}n ⊆ Rn , equipped with the n ∞ metric. We denote this metric space by [m]n∞ . Bourgain [3] proved that if Y is a Banach space 1+ 1

with Rademacher cotype q, then any embedding of [n q ]n∞ into Y incurs distortion Y n1/q . The same result follows from Theorem 1.2, while the previous estimate on m from [13] only q+1

1+ 1

yields the weaker distortion lower bound of Y n q(2q+1) for embeddings of [n q ]n∞ into Y . The sharp bound on m from [13] when Y has Rademacher type > 1 implies that in this case, any embedding of [n1/q ]n∞ into Y incurs distortion n1/q (where the implied constant is now allowed to depend also on the Rademacher type parameters of Y ). Our main conjecture implies the same improvement of Bourgain’s result without the assumption that Y has non-trivial Rademacher type.


173

Bourgain’s theorem [3] is part of his more general investigation of embeddings of ε-nets in unit balls of finite dimensional normed spaces. Bourgain’s approach in [3] is based on ideas similar to ours, that are carried out in the continuous domain. Specifically, given a mapping f : [m]n∞ → Y , he finds a mapping g : Rn → Y which is L-Lipschitz and close in an appropriate sense (depending on L, m, n) to f on points of the grid [m]n∞ . Once this is achieved, it is possible to differentiate g to obtain the desired distortion lower bound. Bourgain’s approximate Lipschitz extension theorem (an alternative proof of which was found in [1]) is a continuous version of a smoothing and approximation scheme; it seems plausible that our method in Section 4 for proving impossibility results for such schemes can be used to prove similar restrictions on Bourgain’s approach to approximate Lipschitz extension. When Y has non-trivial Rademacher type, the improvement in [13] over Bourgain’s nonembeddability result for grids is thus based on a more delicate cancellation than was used in [3,1]. Question 3. Is it true that for any Banach space Y of Rademacher cotype q, any embedding of [n1/q ]n∞ into Y incurs distortion Y n1/q (if true, this is a sharp bound). Specializing√to the n Schatten–von √ Neumann trace class S1 , we do not even know whether1/6the distortion of [ n ]∞ in S1 is n. Theorem 1.2 implies a distortion lower bound of n , while the bound on m from [13] only yields a distortion lower bound of n1/10 . Our results in Section 4 show that one cannot get a distortion lower bound asymptotically better than n1/4 by using smoothing and approximation schemes. We did not discuss here metric characterizations of Rademacher type. We refer to [12] for more information on this topic. It turns out that our approach to Theorem 1.2 yields improved bounds in [12] as well; see [5]. 2. Proof of Theorem 1.2 For n ∈ N denote [n] = {1, . . . , n}. When B ⊆ [n], and x ∈ ZB m , we will sometimes slightly abuse notation by treating x as an element of Znm , with the understanding that for i ∈ [n] \ B we have xi = 0. For y ∈ Znm , we denote by yB the restriction of y to the coordinates in B. As in [13], for j ∈ [n] and an odd integer k < m/2, we define S(j, k) ⊆ Znm by def S(j, k) = y ∈ [−k, k]n ⊆ Znm : yj is even ∧ ∀ ∈ [n] \ {j } y is odd .

(16)

The parameter k will be fixed throughout the ensuing argument, and will be specified later. For every j ∈ [n] let νj be the uniform probability measure on S(j, k). Following the notation of [13], for a Banach space (X, · X ) and f : Znm → X, we write f ∗ νj = Ej f , that is, 1 Ej f (x) = μ(S(j, k)) def

f (x + y) dμ(y).

(17)

S(j,k)

Recall that E∞ (Znm ), defined in (6), is the set of all ∞ edges of Znm . Similarly, we denote the 1 edges of Znm by E1 (Znm ), i.e.,

def E1 Znm = (x, y) ∈ Znm × Znm : x − y ∈ {±e1 , . . . , ±en } . Clearly E1 (Znm ) ⊆ E∞ (Znm ).

(18)

174


Let β1◦ denote the uniform probability distribution on the pairs (x, y) ∈ E∞ (Znm ) with x − y ∈ {−1, 1}n , and let β1◦◦ denote the uniform probability distribution on E1 (Znm ). We shall consider the probability measure on E∞ (Znm ) given by β1 = (β1◦ + β1◦◦ )/2. Lemma 5.1 in [13] implies that for all q 1 and f : Znm → X we have: 1 n n

q

f (x) − f (y)q dβ1 (x, y). X

Ej f − f X dμ (2k)q

j =1 Zn

(19)

E∞ (Znm )

m

Inequality (19) corresponds to the approximation property (7), with A k. The relevant smoothing inequality is the main new ingredient in our proof of Theorem 1.2, and it requires a more delicate choice of probability measure β2 on E∞ (Znm ). If (x, y) ∈ E∞ (Znm ) then x − y ∈ {−1, 0, 1}n . Let S = {i ∈ [n]: xi = yi }, and define def

β2 (x, y) =

1 (n/k)q|S| · n−|S| n n , Z 2 m |S|

(20)

where Z is a normalization factor ensuring that β2 is a probability measure, i.e., Z=

n q n =0

k

1,

(21)

provided that, say, k 2n.

(22)

Our final choice of k will satisfy (22), so we may assume throughout that Z satisfies (21). The key smoothing property of the averaging operators {Ej }nj=1 is contained in the following lemma: Lemma 2.1. Let X be a Banach space, q 1, n, m ∈ N, where m > 4n is divisible by 4, and f : Znm → X. Suppose that k is an odd integer satisfying 2n k < m2 . Then,

Znm {−1,1}n

n q εj Ej f (x + ej ) − Ej f (x − ej ) dτ (ε) dμ(x) j =1

Sq

X

f (x) − f (y)q dβ2 (x, y), X

(23)

E∞ (Znm )

where S q 1. We shall postpone the proof of Lemma 2.1 to Section 3, and proceed now to deduce Theorem 1.2 assuming its validity. Before doing so, we recall for future use the following simple lemma from [13]:


175

Lemma 2.2. (See Lemma 2.6 from [13].) For every q 1 and for every f : Znm → X, 1 n n

j =1Zn

f (x + ej ) − f (x)q dμ(x) X

m

2

{−1,0,1}n

Znm

q

f (x + δ) − f (x)q dμ(x) dσ (δ). X

(24)

Proof of Theorem 1.2. The argument in the introduction leading to (15), when specialized to our smoothing and approximation scheme using (19) and (23), shows that if k 2n and m 2kn1/q , then q n f x + m ej − f (x) dμ(x) X mq 2 X j =1 Zn

m

f (x) − f (y)q dβ3 (x, y), X

(25)

E∞ (Znm )

where β3 =

β1 + β2 β1◦ + β1◦◦ + β2 . 2

Note that β1◦ β2 due to the contribution of S = ∅ in (20). Thus, (25) implies the following bound: q n f x + m ej − f (x) dμ(x) 2 X j =1 Zn

m

n mq f (x + ej ) − f (x)q dμ(x) X X n j =1Zn

m

+ mq

(n/k)q|S|

n S⊆[n]

|S|

f (x + ε) − f (x)q dμ(x) dτ (ε), X

(26)

{−1,1}[n]\S Znm

where the first term in the right-hand side of (26) corresponds to β1◦◦ . In order to deduce the desired metric cotype inequality (5) from (26), we shall apply (26) to lower dimensional sub-tori of Znm . Note that we are allowed to do so since our requirements on k, namely k 2n and m 2kn1/q , remain valid for smaller n. [n]\B Fix ∅ = B ⊆ [n] and x[n]\B ∈ Zm . We can then consider the mapping g : ZB m → X given by g(xB ) = f (x[n]\B , xB ). Applying (26) to g, and averaging the resulting inequality over x[n]\B ∈ [n]\B Zm , we obtain q f x + m ej − f (x) dμ(x) 2 X

j ∈B Zn

m

176


X

mq |B|

f (x + ej ) − f (x)q dμ(x) X

j ∈B Zn

m

{−1,1}B\S

Znm

(|B|/k)q|S| + mq

|B| |S|

S⊆B

f (x + ε) − f (x)q dμ(x) dτ (ε). X

def 2|B|−1 . 3n−1

For B ⊆ [n] define the weight W|B| = ∅ = B ⊆ [n], we obtain the bound

(27)

Multiplying (27) by W|B| and summing over

q n 1 f x + m ej − f (x) dμ(x) q m 2 X j =1 Zn

m

W|B| f (x + ej ) − f (x)q dμ(x) X |B|

X

j ∈B Zn

B⊆[n] B=∅

+

m

W|B|

B⊆[n] B=∅

(|B|/k)q|S|

|B| |S|

S⊆B

f (x + ε) − f (x)q dμ(x) dτ (ε), X

(28)

{−1,1}B\S Znm

where we used the identity B⊆[n]

W|B|

q n f x + m ej − f (x) dμ(x) = 2 X

j ∈B Zn

j =1 Zn

m

q f x + m ej − f (x) dμ(x). 2 X

m

The first term in the right-hand side of (28) is easy to bound, using Lemma 2.2, as follows: W|B| f (x + ej ) − f (x)q dμ(x) X |B| j ∈B Zn

B⊆[n] B=∅

m

1 n

n

f (x + ej ) − f (x)q dμ(x) X

j =1 Zn

m

(24)

2

q

f (x + δ) − f (x)q dμ(x) dσ (δ), X

(29)

{−1,0,1}n Znm

2−1 1 where in the first inequality of (29) we used the fact that n=1 n−1 −1 3n−1 n . To bound the second term in the right-hand side of (28), note that it equals def

C=

S⊆[n] S⊆B⊆[n] ε∈{−1,1}B\S B=∅

1 3n

T ⊆[n] ε∈{−1,1}T

aT Znm

2|B|−1 (|B|/k)q|S| ·

3n−1 2|B|−|S| |B| |S|

f (x + ε) − f (x)q dμ(x) X

Znm

f (x + ε) − f (x)q dμ(x), X

(30)


177

where we used the change of variable T = B \ S, and for every T ⊆ [n] we write, def

aT =

n 2|B|−|T | (|B|/k)q(|B|−|T |) n − |T | 2−|T | (/k)q(−|T |) · = .

|B|

− |T | |B|−|T |

B⊇T

Fix T ⊆ [n]. Using the standard bounds ( uv )v v u, we can bound aT as follows: aT

u

v ( eu v ) , which hold for all integers 0

v

n e(n − |T |) −|T | − |T | −|T | q(−|T |) =|T |

=

−|T |

=|T |

− |T |

n 2e(n − |T |)q−1 −|T | kq

=|T |

k

2−|T |

.

Thus, assuming that k 3n, and recalling that q 2, we get the bound

aT

n 2e(n − |T |)nq−1 −|T | (3n)q

=|T |

n 2e (−|T |) =|T |

9

1.

(31)

Combining (31) with (30), we see that the second term in the right-hand side of (28) is 1 C n 3

T ⊆[n] ε∈{−1,1}T Zn

f (x + ε) − f (x)q dμ(x) X

m

f (x + δ) − f (x)q dμ(x) dσ (δ). X

= {−1,0,1}n Znm

In combination with (29), inequality (28) implies that q n 1 f x + m ej − f (x) dμ(x) q m 2 X j =1 Zn

m

X

f (x + δ) − f (x)q dμ(x) dσ (δ), X

{−1,0,1}n Znm

which is precisely the desired inequality (5). Recall that in the above argument, our requirement on k was k 3n, and our requirement on m was m 2kn1/q (and that it is divisible by 4). This implies the requirement m 6n

1+ q1

of Theorem 1.2.

2

178


3. Proof of Lemma 2.1 Lemma 2.1 is the main new ingredient of the proof of Theorem 1.2. Its proof is based on combinatorial identities which relate the “smoothed out Rademacher sum” n

εj Ej f (x + ej ) − Ej f (x − ej )

(32)

j =1

to a certain bivariate extension of the Bernoulli numbers. We shall therefore first, in Section 3.1, do some preparatory work which introduces these numbers and establishes estimates that we will need in the ensuing argument. We shall then derive, in Section 3.2, certain combinatorial identities that relate (32) to the bivariate Bernoulli numbers. In Section 3.3 we shall combine the results of Section 3.1 and Section 3.2 to complete the proof of Lemma 2.1. 3.1. Estimates for the bivariate Bernoulli numbers There are two commonly used definitions of the Bernoulli numbers {Br }∞ r=0 . For more information on these two conventions, we refer to http://en.wikipedia.org/wiki/Bernoulli_number. Here we shall refer to the variant of the Bernoulli numbers that was originally defined by J. Bernoulli, for which B1 = 12 , and which is defined via the recursion r=

r−1 a=0

Ba

r . a

(33)

Observe that (33) contains the base case B0 = 1 when substituting r = 1. The recursion (33) extends naturally to a bivariate sequence {Br,s }nr,s=0 , given by r −s =

r−1 a=0

s−1 r s − . Ba,s Br,b a b

(34)

b=0

It is well known (cf. [20, Sec. 2.5]) that the exponential generating function for {Br }∞ r=0 is def

F (x) =

∞

xr xex = Br . ex − 1 r! r=0

We shall require the following analogous computation of the bivariate exponential generating function of {Br,s }nr,s=0 : Lemma 3.1. For all x, y ∈ C with |x|, |y| < π we have def

F (x, y) =

∞

∞

xr ys (x − y)ex+y , = B r,s ex − ey r! · s!

(35)

r=0 s=0

where the series in (35) is absolutely convergent on {(x, y) ∈ C × C: |x|, |y| r} for all r < π .


179

Proof. The function F (x, y) is analytic on Dπ = {(x, y) ∈ C × C: |x|, |y| < π}, since its only non-removable are when x − y ∈ 2πi(Z \ {0}). It follows that we can write ∞singularities r y s , for some {z }∞ F (x, y) = ∞ z x r,s r,s=0 ⊆ C, where the series converges absolutely r=0 s=0 r,s on any compact subset of Dπ (see, e.g., [6, Thm. 2.2.6]). Note that

x e − ey F (x, y) =

∞ n x − yn n=1

=

n!

r−1 ∞ ∞ r=0 s=0

a=0

∞ ∞

r s

zr,s x y

r=0 s=0

s−1 za,s zr,b − xr ys . (r − a)! (s − b)!

(36)

b=0

At the same time, ∞ ∞ ∞ r s ∞

x x y xr ys y x y e − e F (x, y) = (x − y)e e = (x − y) (r − s) = . r!s! r!s! r=0 s=0

(37)

r=0 s=0

By equating coefficients in (36) and (37), we see that for all r, s ∈ N ∪ {0}, r−1 r − s = r!s! a=0

zr,b za,s − (r − a)! (s − b)! s−1

b=0

=

r−1 r a=0

a

a!s!za,s −

Since z0,0 = 1, the recursive definition (34) implies that zr,s =

s−1 s b=0

Br,s r!s! ,

b

as required.

r!b!zr,b .

2

An immediate corollary of Lemma 3.1 is that since F (x, y) = F (y, x), ∀r, s ∈ N ∪ {0},

Br,s = Bs,r .

(38)

Another (crude) corollary of Lemma 3.1 is that since the power series in (35) converges absolutely on {(x, y) ∈ C × C: |x|, |y| 2}, for all but at most finitely many r, s ∈ N ∪ {0} we have |Br,s /(r!s!)|1/(r+s) 1/2. Thus, ∀r, s ∈ N ∪ {0}, Remark 3.1. Since B2m =

(−1)m−1 2ζ (2m)(2m)! , (2π)2m

|Br,s |

r!s! . 2r+s

(39)

where ζ (s) is the Riemann zeta function (and

2(2m)! B2m+1 = 0 for m 1), one has the sharp asymptotics |B2m | ∼ (2π) 2m for the classical Bernoulli numbers. We did not investigate the question whether similar sharp asymptotics can be obtained for the bivariate Bernoulli numbers.

3.2. Some combinatorial identities We start by introducing some notation. For y ∈ Znm write:

180


def ↑y↑ = l: yl = k (mod m) , def ↓y↓ = l: yl = −k (mod m) , def

def

y = ↑y↑ + ↓y↓ and ↑y↓ = ↑y↑ − ↓y↓. We also define def S = y ∈ [−k, k]n ⊆ Znm : yt is odd ∀t ∈ [n] . the coordinate wise multiplication, i.e., For x ∈ Znm and ε ∈ {−1, 1}n , let x ε ∈ Znm be (x ε)j = xj εj . Also for ε, ε ∈ {−1, 1}n let ε, ε = nj=1 εj εj . We need to define additional auxiliary averaging operators. Definition 3.2. For f : Znm → X, k
k for some t0 ∈ [n], then a(z, ε) = bi,j (z, ε) = 0. Proof. We may assume that z ∈ [−m/2, m/2]n . If there is t0 ∈ [n] for which |zt0 | > k then all the terms in the right-hand side of (40) and (41) are 0. If |zt | < k for all t ∈ [n], then all the terms in the right-hand side of (40) and (41) cancel out. 2 It follows that for z ∈ / S we have a(z, ε) = bi,j (z, ε) = 0 for every ε ∈ {−1, 1}n and every 0 j i n. Thus, in particular, identity (42) can be rewritten as:

bi,j (y − x, ε)f (y)

y∈x+S

= k n−i

[n]\S f (x + δk + ε[n]\S ) − [n]\S f (x + δk − ε[n]\S ) .

(44)

S⊆[n] δ∈{−1,1}S |S|=i δ,εS =i−2j

Note that the definition (41) shows that for z ∈ S we have a(z, ε) =

εt −

t∈[n] zt =k

εt = ↑z ε↓.

(45)

t∈[n] zt =−k

Using Claims 3.3 and 3.4, in conjunction with (43) and (45), we conclude that: Lemma 3.5. The following identity holds for all x ∈ Znm and ε ∈ {−1, 1}n : n

εj Ej f (x + ej ) − Ej f (x − ej ) =

j =1

⏐ 1 ⏐(y − x) ε f (y). k(k + 1)n−1

(46)

y∈x+S

Lemma 3.6. If z ∈ S and i z then ∀j ∈ {0, . . . , i} and ∀ε ∈ {−1, 1}n , we have bi,j (z, ε) = 0. Proof. If i > z then z∈ / (δk + ε[n]\S + L[n]\S ) ∪ (δk − ε[n]\S + L[n]\S ), for every S ⊆ [n] with |S| = i, and δ ∈ {−1, 1}S . If i = z then there exists exactly one subset S ⊆ [n] in (40) where z can appear, namely S = { ∈ [n]: z ∈ {−k, k}}. If

182


z ∈ (δk + ε[n]\S + L[n]\S ) ∪ (δk − ε[n]\S + L[n]\S ), for some δ ∈ {−1, 1}S , then z ∈ (δk + ε[n]\S + L[n]\S ) ∩ (δk − ε[n]\S + L[n]\S ), since for all coordinates i ∈ [n] \ S we have |zi | < k. Hence in this case the terms in the sum in the right-hand side of (40) cancel out. 2 Lemma 3.7. For every z ∈ S, ε ∈ {−1, 1}n , and 0 j i < z, ⎧ z−j ⎪ ⎨ i−j , bi,j (z, ε) = − z−(i−j ), ⎪ j ⎩ 0

↓z ε↓ = j, ↑z ε↑ = i − j, otherwise.

Proof. By looking at the elements of (δk + ε[n]\S + L[n]\S ) ∪ (δk − ε[n]\S + L[n]\S ), it is clear that we must have S ⊆ {h: zh ∈ {−k, k}} in order to get a non-zero contribution to the right-hand side of (40). For such an S there is at most one δ ∈ {−1, 1}S which can contribute to the sum, namely δh = sgn(zh ) for every h ∈ S. But since this δ should also satisfy δ, εS = i − 2j , we conclude that a non-zero contribution can occur only when ↓zS εS ↓ = j . In those cases, there is an actual contribution only if either sgn(zh εh ) = 1 for every h ∈ {: z ∈ {−k, k}} \ S, or sgn(zh εh ) = −1 for every h ∈ {: z ∈ {−k, k}} \ S, and those contributions have different signs. The claim now follows. 2 The following lemma relates, via Lemma 3.7, what we have done so far to the bivariate Bernoulli numbers. Lemma 3.8. There exists a sequence {hα,β } 0αn ⊆ R such that for all y ∈ Znm and all 0βα

ε ∈ {−1, 1}n , ↑y ε↓ =

n α

hα,β bα,β (y, ε),

(47)

α=0 β=0

(α − β)!β! , 2α = hα,α−β .

|hα,β | hα,β

for all 0 β α,

(48) (49)

Proof. Write r = ↑z ε↑ and s = ↓z ε↓. Thus r + s = z and r − s = ↑z ε↓. With this notation, if we substitute the values of bα,β (y, ε) from Lemma 3.7, the desired identity (47) becomes:


r −s =

n

hα,s

α=s r−1 (♣)

=

183

n n−s n r s (♠) r s − − hβ+r,β = ha+s,s hb+r,b α−s β a b β=0

ha+s,s

a=0

a=0

b=0

s−1 r s − , hb+r,b a b

(50)

b=0

where in (♠) we used the change of variable β = b, α = a + s, and in (♣) we noted that r + s = z n and that the terms corresponding to a > r or b > s vanish, while the terms corresponding to a = r and b = s cancel out. Thus, the desired identity (50) shows that we must take ha+b,b = Ba,b , or hα,β = Bα−β,β . The bound (48) is now the same as (39), and the identity (49) is the same as (38). 2 3.3. Putting things together We are now ready to complete the proof of Lemma 2.1 using the tools developed in the previous two sections. Lemma 3.9. Let {hα,β } 0αn be the sequence from Lemma 3.8. Then for all f : Znm → X and 0βα

all ε ∈ {−1, 1}n we have

q n i 1 h b (y − x, ε)f (y) dμ(x) i,j i,j k(k + 1)n−1 i=0 j =0

Znm

y∈x+S

X

n (n/k)q f (x + ε[n]\S ) − f (x)q dμ(x).

n q X =0

(51)

S⊆[n] Zn |S|= m

Proof. For every x ∈ Znm and 0 j i n write

def

1 Di,j (x) = k(k + 1)n−1

bi,j (y − x, ε)f (y) .

X

y∈x+S

Note that,

n i

q |hi,j |Di,j (x)

=

i=0 j =0

n

2

−(i+1)

i=0 n (∗)

2−(i+1)

j =0 i

q 2

i+1

|hi,j |Di,j (x) q

2i+1 |hi,j |Di,j (x)

j =0

i=0 n (∗∗)

i

i

i=0 j =0

2(i+1)(q−1) (i + 1)q−1 |hi,j |q Di,j (x)q ,

(52)

184


−(i+1) = 1, and in where in (∗) we used the convexity of the function t → t q and that ∞ i=0 2 (∗∗) we used Hölder’s inequality. It follows from (52), combined with the bound (48) on hi,j , that, q n i 1 h b (y − x, ε)f (y) i,j i,j k(k + 1)n−1 n

i=0 j =0

n

i

y∈Zm

X

q

|hi,j |Di,j (x)

i=0 j =0

n i

2

(i+1)(q−1)

(i + 1)

q−1

i=0 j =0

(i − j )!j ! 2i

q Di,j (x)q .

(53)

Now, Di,j (x) can be estimated using the identity (44) as follows: k i Di,j (x)

(k + 1)n−1 D (x) i,j k n−i−1

[n]\S f (x + δk + ε[n]\S )

S⊆[n] δ∈{−1,1}S |S|=i δ,εS =i−2j

− [n]\S f (x + δk − ε[n]\S )X .

Note that the number of terms in the sum in the right-hand side of (54) is Di,j (x)q

(54)

n i i

j

. Thus

1 n q−1 i q−1 j k iq i [n]\S f (x + δk + ε[n]\S ) − [n]\S f (x + δk − ε[n]\S )q . (55) · X S⊆[n] δ∈{−1,1}S |S|=i δ,εS =i−2j

If we integrate inequality (55) with respect to x, use the translation invariance of μ to eliminate the additive term δk in the argument of the integrands, and use the fact that B is an averaging operator for all B ⊆ [n], we obtain the bound

1 n q−1 i q f (x + ε[n]\S ) − f (x − ε[n]\S )q dμ(x) Di,j (x) dμ(x) iq X i j k q

Znm

S⊆[n] Zn |S|=i m

2q n q−1 i q f (x + ε[n]\S ) − f (x)q dμ(x), X iq i j k S⊆[n] Zn |S|=i m

where in the last step of (56) we used the triangle inequality as follows: f (x + ε[n]\S ) − f (x − ε[n]\S )q 2q−1 f (x + ε[n]\S ) − f (x)q X X q q−1 f (x) − f (x − ε[n]\S )X , +2

(56)


185

while noticing that upon integration with respect to x, by translation invariance, both terms become equal. Integrating (53) with respect to x, and using (56), we see that the left-hand side of (51) is at most n i 2(i+1)(q−1)+q (i + 1)q−1

n i=0 j =0

(i−j )!j ! n i q 2i k i

i

i

= 22q−1

f (x + ε[n]\S ) − f (x)q dμ(x) X

j

S⊆[n] Zn |S|=i m

q n n! (i + 1)q f (x + ε[n]\S ) − f (x)q dμ(x).

n X i i k (n − i)! 2 i S⊆[n] i=0 |S|=i

(57)

Znm

Inequality (57) implies the desired bound (51), since (i + 1)q 2−i q 1 and n!/(n − i)! ni .

2

Proof of Lemma 2.1. It follows from (43) and (46) that q n εj Ej f (x + ej ) − Ej f (x + ej ) dμ(x) dτ (ε)

{−1,1}n Znm

= {−1,1}n

l=1

X

q ⏐ 1 ⏐(y − x) ε f (y) dμ(x) dτ (ε). k(k + 1)n−1

(58)

X

y∈x+S

Znm

An application of identity (47) now shows that q ⏐ 1 ⏐(y − x) ε f (y) dμ(x) dτ (ε) k(k + 1)n−1

{−1,1}n Znm

=

X

y∈x+S

q n i 1 h b (y − x, ε)f (y) dμ(x) dτ (ε). (59) i,j i,j k(k + 1)n−1

{−1,1}n Znm

i=0 j =0

Lemma 2.1 now follows from Lemma 3.9.

y∈x+S

X

2

4. Lower bounds In this section we establish lower bounds for the best possible value of m in Theorem 1.2 that is achievable via a smoothing and approximation scheme. Our first result deals with general convolution kernels: Proposition 4.1. Assume that the probability measures ν1 , . . . , νn , β1 , β2 are a (q, A, S)smoothing and approximation scheme on Znm , i.e., conditions (7) and (8) are satisfied for every Banach space X and every f : Znm → X. Assume also that m > cA for a large enough universal

186


constant c > 0. Then √ n . S q A

(60)

Recall, as explained in Section 1.1, that in order for a smoothing and approximation scheme to yield the metric cotype inequality (5), we require S 1, in which case √ the bound on m becomes m An1/q . Proposition 4.1 shows that S 1 forces the bound A q n, and correspondingly 1

+1

m q n 2 q . For the particular smoothing and approximation scheme used in our proof of Theorem 1.2, the following proposition establishes asymptotically sharp bounds. Proposition 4.2. Fix an odd integer k m/2 and consider the averaging operators {Ej }nj=1 used in our proof of Theorem 1.2, i.e., they are defined as in (17). If there exist probability measures β1 , β2 on E∞ (Znm ) for which the associated approximation and smoothing inequalities (7) and (8) are satisfied for every Banach space X and every f : Znm → X, then Ak

and S min

n n , . k k

(61)

Proposition 4.2 shows that in order to have S 1 we need to require k n, in which case A n, and correspondingly m n

1+ q1

, matching the bound obtained in Theorem 1.2.

4.1. A lower bound for general convolution kernels: Proof of Proposition 4.1 Assume that the probability measures ν1 , . . . , νn , β1 , β2 are a (q, A, S)-smoothing and approximation scheme, i.e., they satisfy (7) and (8). It will be convenient to think of these measures as functions defined on the appropriate (finite) spaces, i.e., ν1 , . . . , νn : Znm → [0, 1] and β1 , β2 : E∞ (Znm ) → [0, 1]. For a probability measure ν on Znm , let Pj (ν) be the probability measure on Zm which is the marginal of ν on the j th coordinate, i.e., def

Pj (ν)(r) =

ν(x).

x∈Znm xj =r

Define the absolute value of x ∈ Zm to be |x| = min{x, m − x}. Lemma 4.3. Assume that ν1 , . . . , νn , β1 satisfy (7). Then for every s ∈ N we have: n 1 A νj (x) . n s n

(62)

j =1 x∈Zm |xj |>s

Proof. We shall apply (7) with X = n∞ . Let gs : R → R be the truncated jigsaw function with period 12s, depicted in Fig. 1.


187

Fig. 1. gs is truncated jigsaw function.

Define f : Znm → X by def

f (x) = gs (x1 ), gs (x2 ), . . . , gs (xn ) . The Lipschitz constant of f with respect to the ∞ metric on Znm is 1, and therefore it follows from (7) that

1 n n

q

f ∗ νj − f

j =1 Zn

n∞

dμ

1 n n

j =1 Zn

m

q

f ∗ νj − f n dμ Aq . ∞

(63)

m

For every x ∈ Znm and j ∈ [n], (f ∗ νj − f )(x) =

y∈Znm

=

νj (y) f (x − y) − f (x)

νj (y) gs (x1 − y1 ) − gs (x1 ), . . . , gs (xn − yn ) − gs (xn ) .

y∈Znm

Assume that (xj mod 12s) ∈ [0, s] ∪ [12s − s, 12s − 1].

(64)

When 3s |yj | 4s, we have gs (xj − yj ) − gs (xj ) s, and for every yj ∈ Zm , we have gs (xj − yj ) − gs (xj ) 0. Hence,

(f ∗ νj − f )(x) n νj (y) gs (xj − yj ) − gs (xj ) ∞

y∈Znm

sPj (νj ) z ∈ Zm : 3s |z| 4s .

(65)

Note that (64) holds for a constant fraction of x ∈ Znm , and hence by integrating (65) over Znm we obtain:

188


y∈Znm 3s|yj |4s

1 νj (y) s

(f ∗ νj − f )(x) n dμ(x).

(66)

∞

Znm

Therefore

νj (y) =

y∈Znm |yj |3s

∞

νj (y)

=0 3( 4 ) s|yj |4( 4 ) s 3 3 (66)

∞ =0

1 s

1 s · (4/3)

(f ∗ νj − f )(x) n dμ(x) ∞

Znm

(f ∗ νj − f )(x) n dμ(x). ∞

Znm

Averaging the above inequality over j ∈ [n], and using (63), we obtain (62).

2

Corollary 4.4. Assume that m > cA for a large enough universal constant c ∈ N. Then: n 1 1 Pj (νj )(z + 1) − Pj (νj )(z − 1) . n A j =1 z∈Zm

Proof. We may assume that cA is an integer. By Lemma 4.3, for c large enough we have n 1 3 Pj (νj )(z) n 4 j =1 |z|cA

and

n 3cA+2 1 1 Pj (νj )(z) . n 4 j =1 z=cA+2

Therefore, n n 1 1 1 Pj (νj )(z) − 2 n n j =1 |z|cA

Pj (νj )(z)

j =1 |z−2cA−2|cA

n 1 = Pj (νj )(z) − Pj (νj )(z + 2cA + 2) n j =1 |z|cA

=

n cA+1

1 Pj (νj ) z + 2(t − 1) − Pj (νj )(z + 2t) n j =1 |z|cA t=1

n A Pj (νj )(z + 1) − Pj (νj )(z − 1), n j =1 z∈Zm

as required.

2

(67)


189

Proof of Proposition 4.1. We shall apply the smoothing inequality (8) when X = L1 (Znm , μ) and f : Znm → X is defined as f (x) = mn · δ{x} , i.e., for x ∈ Znm the function f (x) : Znm → R is def

f (x)(y) =

mn , 0

x = y, otherwise.

(68)

For every ε ∈ {−1, 1}n and x ∈ Znm we have: n

εj f ∗ νj (x + ej ) − f ∗ νj (x − ej )

j =1

=

n

νj (y + ej ) − νj (y − ej ) f (x − y) . εj

j =1

(69)

y∈Znm

By Kahane’s inequality [7,21] and the fact that L1 (Znm , μ) has cotype 2 (see [21]), {−1,1}n

q

n

q νj (y − ej ) − νj (y + ej ) f (x − y) εj n j =1

dτ (ε)

L1 (Znm ,μ)

y∈Zm

q/2

2 n

νj (y − ej ) − νj (y + ej ) f (x − y)

.

(70)

L1 (Znm ,μ)

j =1 y∈Znm

Note that by the definition of f , for every x ∈ Znm and j ∈ [n] we have,

νj (y − ej ) − νj (y + ej ) f (x − y) y∈Znm

L1 (Znm ,μ)

νj (z − ej ) − νj (z + ej ) = z∈Znm

νj (z − ej ) − νj (z + ej ) w∈Zm z∈Znm zj =w

=

Pj (νj )(w − 1) − Pj (νj )(w + 1).

(71)

w∈Zm

Hence, 2 n

1 ν (y − e ) − ν (y + e ) f (x − y) j j j j n L1 (Znm ,μ) n j =1 y∈Zm (71)

n 1 Pj (νj )(w − 1) − Pj (νj )(w + 1) n j =1 w∈Zm

2

(67)

1 . A2

(72)

190


Finally, since f (x) − f (y)L1 (Znm ,μ) f (x)L1 (Znm ,μ) + f (y)L1 (Znm ,μ) 2 for all x, y ∈ Znm , we can use the smoothing inequality (8) to deduce that S S q

q

L1 (Znm ,μ)

E∞ (Znm ) (8)

f (x) − f (y)q

Znm {−1,1}n

dβ2 (x, y)

n q

εj f ∗ νj (x + ej ) − f ∗ νj (x − ej ) j =1

dτ (ε) dμ(x)q

L1 (Znm ,μ)

nq/2 , Aq

where in the last step we used the identity (69), combined with the inequalities (70) and (72). The proof of Proposition 4.1 is complete. 2 4.2. A sharp lower bound for Ej averages: Proof of Proposition 4.2 Recall that S(j, k) is defined in (16), and in the setting of Proposition 4.2 we have: νj (x) =

1S(j,k) (x) . k(k + 1)n−1

Let s ∈ {(k + 1)/2, (k + 3)/2} be an odd integer. By the definition of S(j, k) we have x∈Znm |xj |>s

νj (x) =

(k − s)(k + 1)n−1 1. k(k + 1)n−1

Plugging this estimate into (62) we see that A/k 1, proving the first assertion in (61). To prove the second assertion of Proposition 4.2, we shall apply the smoothing inequality (8), as in Section 4.1, to the Banach space X = L1 (Znm , μ) and the function f from (68), i.e., f (x) = mn δ{x} ∈ L1 (Znm , μ). We shall use here notation from Section 3.2. In our setting, the value of n εj Ej f (x + ej ) − Ej f (x − ej ) j =1

L1 (Znm ,μ)

does not depend on x ∈ Znm and ε ∈ {−1, 1}n . Thus the left-hand side of (8) equals (by Lemma 3.5),

q E f (e ) − E f (−e ) j j j j

=

L1 (Znm ,μ)

j

q 1 ↑y↓ . k(k + 1)n−1 y∈S

At the same time, as noted in Section 4.1, the right-hand side of (8) is S q . It follows that S

1 ↑y↓ = E[Z], k(k + 1)n−1 y∈S

(73)


191

where Z = | ni=1 ξi |, and {ξi }ni=1 are i.i.d. random variables taking the 0 with probability k−1 k+1 , 1 and each of the values {−1, 1} with probability k+1 . The last equality in (73) is an immediate 2 , we have E[Z 2 ] = np and consequence of the definitions of S and |↑y↓|. Writing p = k+1 4 2 E[Z ] = np + n(n − 1)p . By Hölder’s inequality it then follows that we have S E[Z] √ Z32 /Z24 min{ np, np}, completing the proof of Proposition 4.2. 2 4.3. Symmetrization We do not know what is the smallest m for which the metric cotype inequality (5) can be shown to hold true via a smoothing and approximation scheme: all we know is that it is be1+ 1

1

+1

tween n q and n 2 q . In this short section, we note that the special symmetric structure of the smoothing and approximation scheme that we used in the proof of Theorem 1.2 can be always assumed to hold true without loss of generality. This explains why our choice of convolution kernels is natural. Additionally, this fact might be useful in improving the lower bound on m of Proposition 4.1, though we do not know how to use it in our current proof of Proposition 4.1. For π ∈ Sn , i.e., a permutation of [n], and x ∈ Znm , write def

x π = (xπ(1) , xπ(2) , . . . , xπ(n) ). For f : Znm → X we define f π : Znm → X by f π (x) = f (x π ). Note that if ν is a probability measure on Znm then for all x ∈ Znm we have

−1 π f ∗ νπ = f π ∗ ν .

(74)

Indeed, f ∗ ν (x) =

f (x − y)ν y

π

π

dμ(y)=

Znm

Znm

=

−1 f x − zπ ν(z) dμ(z)

fπ

−1

−1 π −1 x π − z ν(z) dμ(z) = f π ∗ ν x π = f π ∗ ν (x).

Znm

It follows from (74) that f ∗ ν π − f L

n q (Zm ,X)

−1 −1 = f π ∗ ν − f π L

n q (Zm ,X)

.

(75)

Lemma 4.5. Assume that the probability measures ν1 , . . . , νn , β1 , β2 are a (q, A, S)-smoothing and approximation scheme. Then there exist probability measures ν¯ 1 , . . . , ν¯ n on Znm and two probability measures β¯1 , β¯2 on E∞ (Znm ), such that: 1. The sequence ν¯ 1 , . . . , ν¯ n , β¯1 , β¯2 is also a (q, A, S)-smoothing and approximation scheme. (j,h) 2. For any j, h ∈ [n] we have ν¯ j = ν¯ h , where (j, h) ∈ Sn is the transposition of j and h. 3. For every j, h ∈ [n] \ {i} we have Pj (¯νi ) = Ph (¯νi ).

192


Proof. Define for j ∈ [n], def

ν¯ j =

1 π −1 νπ(j ) . n!

(76)

π∈Sn

We also define for (x, y) ∈ E∞ (Znm ), def β¯1 (x, y) =

1 π π def 1 and β¯2 (x, y) = β1 x , y β2 x π , y π . n! n! π∈Sn

(77)

π∈Sn

Fix f : Znm → X and assume the validity of the approximation and smoothing inequaliq ties (7), (8). Then, by the convexity of · X , 1 n n

q

f ∗ ν¯ j − f X dμ

(75)∧(76)

j =1 Zn

n 1 1 f π ∗ νπ(j ) − f π q n Lq (Zm ,X) n! n π∈Sn

m

(7)∧(77)

Aq

j =1

f (x) − f (y)q d β¯1 (x, y). X

(78)

E∞ (Znm )

This is precisely the approximation property for ν¯ 1 , . . . , ν¯ n , β¯1 , β¯2 . Similarly,

Znm {−1,1}n

n q

εj f ∗ ν¯ j (x + ej ) − f ∗ ν¯ j (x − ej ) dτ (ε) dμ(x) j =1

1 n!

(77)

X

π∈Sn Zn {−1,1}n m

−f

π −1 ∗ νπ(j ) (x

n

π −1 εj f ∗ νπ(j ) (x + ej ) j =1

q − ej ) dτ (ε) dμ(x).

(79)

X

Note that n

π −1 π −1 εj f ∗ νπ(j ) (x + ej ) − f ∗ νπ(j ) (x − ej )

j =1 n (74)

=

−1

−1 επ −1 (i) f π ∗ νi x π + ei − f π ∗ νi x π − ei ,

(80)

i=1

where we made the change of variable j = π −1 (i) and used the fact that erπ r ∈ [n] and π ∈ Sn . Hence,

−1

= eπ(r) for all


Znm {−1,1}n

n q

π −1 π −1 εj f ∗ νπ(j ) (x + ej ) − f ∗ νπ(j ) (x − ej ) dτ (ε) dμ(x) j =1

Znm

{−1,1}n

(80)

193

=

X

n q

εr f π ∗ νr (x + er ) − f π ∗ νr (x − er ) dτ (ε) dμ(x). r=1

(81)

X

The smoothing inequality for ν¯ 1 , . . . , ν¯ n , β¯1 , β¯2 now follows:

Znm {−1,1}n

n q

εj f ∗ ν¯ j (x + ej ) − f ∗ ν¯ j (x − ej ) dτ (ε) dμ(x) j =1

(79)∧(81)∧(8)

X

Sq

f (x) − f (y)q d β¯2 (x, y). X

E∞ (Znm )

Assertions 2 and 3 of Lemma 4.5 follow directly from the definition (76).

2

Acknowledgments O.G. was partially supported by NSF grant CCF-0635078. M.M. was partially supported by ISF grant no. 221/07, BSF grant no. 2006009, and a gift from Cisco research center. A.N. was supported in part by NSF grants CCF-0635078 and CCF-0832795, BSF grant 2006009, and the Packard Foundation. References [1] B. Begun, A remark on almost extensions of Lipschitz functions, Israel J. Math. 109 (1999) 151–155. [2] J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math. 56 (2) (1986) 222–230. [3] J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, in: Geometrical Aspects of Functional Analysis (1985/86), in: Lecture Notes in Math., vol. 1267, Springer, Berlin, 1987, pp. 157–167. [4] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43, Cambridge University Press, Cambridge, 1995. [5] O. Giladi, A. Naor, Improved bounds in the scaled Enflo type inequality for Banach spaces, available at http://arxiv. org/abs/1004.4221, 2010. [6] L. Hörmander, An Introduction to Complex Analysis in Several Variables, third edition, North-Holland Math. Library, vol. 7, North-Holland, Amsterdam, 1990. [7] J.-P. Kahane, Sur les sommes vectorielles ±un , C. R. Acad. Sci. Paris 259 (1964) 2577–2580. [8] N.J. Kalton, Coarse and uniform embeddings into reflexive spaces, Q. J. Math. 58 (3) (2007) 393–414. [9] B. Maurey, Type, cotype and K-convexity, in: Handbook of the Geometry of Banach Spaces, vol. 2, North-Holland, Amsterdam, 2003, pp. 1299–1332. [10] B. Maurey, G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1) (1976) 45–90. [11] M. Mendel, Metric dichotomies, in: Limits of Graphs in Group Theory and Computer Science, EPFL Press, Lausanne, 2009, pp. 59–76. [12] M. Mendel, A. Naor, Scaled Enflo type is equivalent to Rademacher type, Bull. Lond. Math. Soc. 39 (3) (2007) 493–498. [13] M. Mendel, A. Naor, Metric cotype, Ann. of Math. (2) 168 (1) (2008) 247–298.

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[14] A. Naor, An application of metric cotype to quasisymmetric embeddings, preprint, available at http://arxiv.org/abs/ math/0607644, 2006. [15] G. Pisier, Holomorphic semigroups and the geometry of Banach spaces, Ann. of Math. (2) 115 (2) (1982) 375–392. [16] G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis, Varenna, 1985, in: Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986, pp. 167–241. [17] Y. Raynaud, On ultrapowers of non commutative Lp spaces, J. Operator Theory 48 (1) (2002) 41–68. [18] N. Tomczak-Jaegermann, The moduli of smoothness and convexity and the Rademacher averages of trace classes Sp (1 p < ∞), Studia Math. 50 (1974) 163–182. [19] A. Veomett, K. Wildrick, Spaces of small metric cotype, preprint, available at http://arxiv.org/abs/1001.3326, 2010. [20] H.S. Wilf, Generatingfunctionology, third edition, A K Peters Ltd., Wellesley, MA, 2006. [21] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Stud. Adv. Math., vol. 25, Cambridge University Press, Cambridge, 1991.


Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis Sergey V. Astashkin Department of Mathematics and Mechanics, Samara State University, Akad. Pavlov St. 1, 443011 Samara, Russia Received 10 March 2010; accepted 20 August 2010

Communicated by N. Kalton

Abstract Let X be a rearrangement invariant function space on [0, 1]. We consider the subspace Radi X of X which consists of all functions of the form f = ∞ k=1 xk rk , where xk are arbitrary independent functions from X and rk are usual Rademacher functions independent of {xk }. We prove that Radi X is complemented in X if and only if both X and its Köthe dual space X possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [0, ∞). This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)]. © 2010 Elsevier Inc. All rights reserved. Keywords: Rademacher functions; Rearrangement invariant spaces; Isomorphism of Banach spaces; Kruglov property

0. Introduction Let rk (t) = sign sin 2k πt (k = 1, 2, . . .) be the Rademacher functions on the interval [0, 1]. By classical Khintchine inequality [13], for each p > 0 there exist positive constants Ap and Bp

E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.013

196

S.V. Astashkin / Journal of Functional Analysis 260 (2011) 195–207

such that for arbitrary real ak (k = 1, 2, . . .) we have Ap

∞

1/2 ak2

k=1

∞ ak rk k=1

Bp

Lp [0,1]

∞

1/2 ak2

.

k=1

This result caused a great number of investigations and generalizations and found many applications in various areas of analysis (see, for instance, [19,3]). In particular, in 1975, Rodin and Semenov [20] proved that the inequality ∞ ∞ 1/2 ak rk C ak2 k=1

X

k=1

holds in a rearrangement invariant (r.i.) space X on [0, 1] if and only if X contains the space G, the closure of L∞ in the Orlicz space exp L2 generated by the function exp(u2 ) − 1 (u > 0). It was proved also that the closed linear span [rn ] is complemented in X if and only if G ⊂ X ⊂ G , where G is the Köthe dual space to G [21], [17, Theorem 2.b.4(ii)]. Moreover, Lindenstrauss and Tzafriri [17, Proposition 2.d.1] showed that the analogous relation for series with arbitrary coefficients xk from an r.i. space X ∞ xk (s)rk (t) k=1

X([0,1]×[0,1])

∞ 1/2 C xk2 k=1

(0.1)

X

holds provided that the lower Boyd index αX of X is positive. They proved as well that in the case when the Boyd indices of X are non-trivial, i.e., 0 < αX βX < 1, the subspace Rad X of all functions defined on the square [0, 1] × [0, 1] and representable in the form f (s, t) =

∞

xk (s)rk (t),

xk ∈ X the series converges a.e. on [0, 1] × [0, 1]

(0.2)

k=1

is complemented in X([0, 1] × [0, 1]) [17, Proposition 2.d.2]. Later, Astashkin and Braverman [4] have proved the opposite statements: if (0.1) is fulfilled for arbitrary xk ∈ X, with some C > 0, then αX > 0, and if Rad X is a complemented subspace of X([0, 1] × [0, 1]), then 0 < αX βX < 1. In [1] (see also [2]), the analogous problem was considered in the case of Rademacher series with independent coefficients. More precisely, there it was proved that inequality (0.1) holds for a some constant C > 0 and for every sequence of independent functions {fk }∞ k=1 ⊂ X if and only if the r.i. space X has the so-called Kruglov property (which is less restrictive than the condition αX > 0; see the next section). In this paper we solve the problem of complementability of the subspace Radi X consisting of all functions f (s, t) which can be represented in the form (0.2), where xk are arbitrary independent functions from an r.i. space X. We prove that Radi X is a complemented subspace in X if and only if both X and its Köthe dual X possess the Kruglov property (Theorem 2.1). The last result can be applied to construct isomorphisms between r.i. spaces on the finite interval and on the semi-axis. The general problem of existence of such isomorphisms in the case


197

of r.i. spaces other than Lp -spaces was first posed by Mityagin in [18]. This and other related problems were extensively studied in [11] (see also [17]) via the approach using a stochastic integral with respect to a symmetrized Poisson process. We will follow here an other approach established on applying the Kruglov property of an r.i. space [7]. It is technically rather simpler; a somewhat similar approach was appeared earlier in a special case of r.i. (Lorentz) spaces Lp,q [10]. We show that an r.i. space X on [0, 1] is isomorphic to some r.i. space on the semi-axis provided that both X and its Köthe dual space X possess the Kruglov property (Theorem 2.4). This strengthens the analogous results of [11, §8] (see also [17, p. 203]) proved under a stronger condition when an r.i. space X has non-trivial Boyd indices. 1. Preliminaries 1.1. Rearrangement invariant spaces Detailed exposition of theory of rearrangement invariant spaces see in [15,17,8]. A Banach space X of real-valued Lebesgue-measurable functions on the interval [0, α), where 0 < α ∞, is called rearrangement invariant (r.i.) if from the conditions y ∈ X and x ∗ (t) y ∗ (t) a.e. on [0, α) it follows that x ∈ X and xX yX . Here and throughout, λ is the Lebesgue measure and x ∗ (t) is the right-continuous non-increasing rearrangement of |x(s)|, i.e.,

(t > 0). x ∗ (t) := inf τ 0: λ s ∈ [0, α): x(s) > τ < t If X is an r.i. space on the interval [0, α), then the Köthe dual space X consists of all measurable functions y such that α yX = sup

x(t)y(t) dt: xX 1 < ∞.

0

The space X is r.i. as well; it is embedded into the dual space X ∗ of X isometrically, and X = X ∗ if and only if X is separable. An r.i. space X is said to have the Fatou property if the conditions xn ∈ X (n = 1, 2, . . .), supn=1,2,... xn X < ∞, and xn → x a.e. imply that x ∈ X and xX lim infn→∞ xn X . X has the Fatou property if and only if the natural embedding of X into its second Köthe dual X is an isometric surjection. In what follows we suppose that every r.i. space X has the Fatou property or separable. For each r.i. space X on [0, 1] we have the continuous embeddings L∞ ⊂ X ⊂ L1 . By X◦ we will denote the separable part of X, i.e., the closure of L∞ in X; X◦ is an r.i. space which is separable provided that X = L∞ . If τ > 0 then the dilation operator στ x(t) := x(t/τ ) · χ[0,min(1,τ )] (t) is bounded in every r.i. space X and στ X→X max(1, τ ). The numbers αX = lim

τ →0+

lnστ X→X ln τ

and βX = lim

τ →+∞

lnστ X→X ln τ

are called the Boyd indices of X; we always have 0 αX βX 1.

198


Important examples of r.i. spaces are Lp -spaces (1 p ∞) and their generalization, the Orlicz spaces [14]. Let M(u) be an Orlicz function on [0, ∞), that is, a continuous convex increasing function on [0, ∞) such that M(0) = 0. Then the Orlicz space LM = LM [0, 1] consists 1 of all measurable functions x(t) on [0, 1] such that 0 M(|x(t)|/λ) dt 1, for some λ > 0. The Luxemburg norm xLM := inf λ, where the infimum is taken over all λ satisfying the last inequality. An Orlicz space LM always has the Fatou property and it is separable if and only if the function M satisfies the 2 -condition at infinity (i.e., there exist u0 > 0 and C > 0 such that M(2u) CM(u) for all u > u0 ). A special interest for us will be classical exponential Orlicz spaces. The space exp Lq , q > 0, is generated by an Orlicz function equivalent to the function q et − 1 for sufficiently small t > 0. 2 as the set of Following [11] (see also [17, 2.f]), for each r.i. space X on [0, 1] we define ZX all measurable on (0, ∞) functions f such that f Z 2 := f ∗ χ[0,1] X + f ∗ χ[1,∞)

L2 [1,∞)

X

< ∞.

2 is an It can easily be shown that the quasinorm · Z 2 is equivalent to an r.i. norm, so that ZX X r.i. space on [0, ∞).

1.2. The Kruglov property Let f be a measurable function on [0, 1] (equivalently, a random variable). Denote by π(f ) the random variable N i=1 fi , where fi are independent copies of f (that is, independent random variables equidistributed with f ) and N is a random variable independent of the sequence {fi } and having the Poisson distribution with parameter 1. In other words, π(f ) is equidistributed with the sum ∞ k

fi (s)χEk (t)

(0 s, t 1),

k=1 i=1

where Ek are disjoint subsets of [0, 1], λ(Ek ) = 1/(ek!) (k = 1, 2, . . .). It is not hard to check that the characteristic function θπ(f ) (t) of π(f ) is equal to the function exp(θf (t) − 1) for all t ∈ R, where θf is the characteristic function of the random variable f . The following property has its origin in Kruglov’s paper [16] and was actively studied and used by Braverman [9]. Definition 1.1. We say that an r.i. space X on [0, 1] has the Kruglov property (X ∈ K) if the relation f ∈ X implies that π(f ) ∈ X. Roughly speaking, an r.i. space X has the Kruglov property if it is situated sufficiently “far” from the space L∞ . In particular, if X contains some Lp with p < ∞ or, all the more, if its lower Boyd index αX > 0, then X ∈ K. However, this is not necessary; for instance, the exponential Orlicz spaces exp Lq do not contain Lp with any p < ∞ but exp Lq ∈ K if and only if 0 < q 1 [9, §2.4], [5]. The Kruglov property is closely related to the well-known Rosenthal inequality [22] and to the problem of the comparison of sums of independent functions and their disjoint copies in r.i.


199

spaces [9,12]. Using an operator approach developed in [5,6], Astashkin and Sukochev proved that in every r.i. space X with the Kruglov property the inequality ∞ ∞ fk C fk k=1

X

k=1

(1.1)

2 ZX

holds for some constant C > 0 and for each sequence of independent functions {fk }∞ k=1 ⊂ X such 1 that 0 fk (t) dt = 0 (k = 1, 2, . . .) [7]. Here, fk are disjoint copies of fk defined on the semiaxis [0, ∞) (for instance, we may take fk (t) = fk (t − k + 1)χ[k−1,k) (t), k = 1, 2, . . .). Earlier inequality (1.1) was proved in [12] under a stronger assumption that an r.i. space X contains some Lp with p < ∞. 2. Results and proofs Let X be an r.i. space on [0, 1]. Equivalently, the set Radi X (see Introduction) can be defined ∞ ¯ in the following ∞ way. Denote Ω=∞[0, 1] (respectively, Ω = Ω × [0, 1]) with the probability measure k=1 λk (respectively, k=0 λk ), where λk is the usual Lebesgue measure on [0, 1]. If ¯ consists of all functions f (t0 , t1 , . . .) measurable X is an r.i. space on [0, 1] then the space X(Ω) ∗ on Ω¯ such that the norm f X(Ω) ¯ := f X < ∞. Then Radi X is the set of all functions from ¯ which are representable in the form X(Ω) f (t0 , t1 , . . .) =

∞

xk (tk )rk (t0 ),

¯ xk ∈ X (the series converges a.e. on Ω),

(2.1)

k=1

where, as above, rk are usual Rademacher functions. It is easily to check that the functions ¯ an unconditional basic sequence with xk (tk )rk (t0 ), where xk ∈ X (k = 1, 2, . . .), form in X(Ω) constant 1 [9, Prop. 1.14]. This implies immediately that Radi X is a closed linear subspace ¯ Since there exists a measure-preserving mapping from ([0, 1], λ) onto of the r.i. space X(Ω). ∞ ¯ ¯ are isometric and we may (and will) identify them. (Ω, k=0 λk ), the spaces X and X(Ω) The following theorem is the main result of this paper. Theorem 2.1. Let X be an r.i. space on [0, 1]. Then Radi X is a complemented subspace of X if and only if X ∈ K and X ∈ K. This theorem is an immediate consequence of the following two propositions. Denoting Ωk := (t1 , . . . , tk−1 , tk+1 , . . .): 0 ti 1, i = 1, . . . , k − 1, k + 1, . . .

(k ∈ N),

¯ the sequence of following conditional expectations: we define on L1 (Ω)

1 E(f |tk )(tk ) := Ωk 0

f (t0 , t1 , . . . , tk−1 , tk , tk+1 , . . .) dt0 dt1 . . . dtk−1 dtk+1 . . .

(k 1).

200


Moreover, let f rk := f (t0 , t1 , . . .)rk (t0 ) (k = 1, 2, . . .) and

Pf (t0 , t1 , t2 , . . .) :=

∞ E(f rk |tk )rk (t0 ). k=1

Proposition 2.2. The operator P is bounded in an r.i. space X if and only if X ∈ K and X ∈ K. ¯ If f ∈ Proof. Firstly, suppose that X ∈ K and X ∈ K and prove that P is bounded in X(Ω). ¯ L∞ (Ω), then the function x(t) :=

∞ E(f rk |tk )(t − k + 1)χ[k−1,k) (t)

belongs to L∞ ∩ L2 (0, ∞).

(2.2)

k=1

In fact, it is clear that |E(f rk |tk )| f ∞ (k = 1, 2, . . .), whence x ∈ L∞ . Moreover, from Minkowski and Bessel inequalities it follows

x2L2 (0,∞)

=

∞

1

f (u, t1 , . . . , tk−1 , t, tk+1 , . . .)rk (u) du dt1 . . . dtk−1 dtk+1 . . .

k=1 0

dt

Ωk 0

1

∞ Ω k=1

2

1

2 f (u, t1 , t2 , . . . , tk , . . .)rk (u) du

dt1 dt2 . . . dtk . . .

0

1

f (u, t1 , t2 , . . .)2 du dt1 dt2 . . . = f 2L

¯ 2 (Ω)

f 2L

¯ ∞ (Ω)

.

Ω 0 2 . Furthermore, Thus, (2.2) is proved. Since L∞ [0, 1] ⊂ X, then from (2.2) it follows that x ∈ ZX 2 it is not hard to check that ZX has the Fatou property or separable just as X itself and that 2 ) = Z 2 . Note that Pf = Qx, where the linear operator Q is defined as follows (ZX X

Qx(t0 , t1 , t2 , . . .) :=

∞ xk (tk )rk (t0 ), k=1

where xk (tk ) = x(k − 1 + tk )

(0 tk 1).

(2.3)

2 Since X ∈ K, then, by [7], inequality (1.1) holds and therefore Q is bounded from the space ZX ¯ into X(Ω). Thus,


201

Pf X(Ω) ¯ = QxX(Ω) ¯ CxZ 2

X

∞

= C sup x(t)y(t) dt : yZ 2 1 .

X

(2.4)

0

¯ and g ∈ X (Ω) ¯ denote For any f ∈ X(Ω)

1 f, g :=

f (t0 , t1 , . . .)g(t0 , t1 , . . .) dt0 dt1 . . . . Ω 0

2 and y ∈ Z 2 we have It can be easily checked that for any x ∈ ZX X

∞ Qx, Qy =

x(t)y(t) dt

(2.5)

0

and that the projection P is self-conjugate, i.e., if f, P g is well defined then Pf, g = f, P g .

(2.6)

Thus, using (2.4), (2.5) and (2.6), the condition X ∈ K and again (1.1), we obtain that

Pf X(Ω) ¯ C sup Qx, Qy : yZ 2 1 X

= C sup Pf, Qy : yZ 2 1 X

= C sup f, Qy : yZ 2 1 X C sup QyX : yZ 2 1 · f X(Ω) ¯ X

CC f X(Ω) ¯ . Since X is separable or has the Fatou property, then applying standard arguments (see also the proof of Proposition 2.3 below for this argument in the Fatou case) we are able to extend the last ¯ ≈ X. inequality to the whole X. Thus, P is bounded in X(Ω) Conversely, suppose that P is bounded in an r.i. space X. Taking into account (2.6), we see that it is bounded in X as well, with the same norm. Denote A := P X→X = P X →X . Assume that a function f (s0 , s1 , . . .) = ∞ x (s )r (s ) ∈ Radi X. Since Pf = f, then k k k 0 k=1 again by (2.6) f X(Ω) ¯ = sup f, g : gX 1 = sup f, P g : gX 1 sup f, g : g ∈ Radi X, gX A .

(2.7)

202


If g(s0 , s1 , . . .) =

∞

k=1 yk (sk )rk (s0 )

f, g =

(yk ∈ X ), then it is easy to see that

∞

yk (sk )xk (sk ) ds1 ds2 . . . .

Ω k=1

Therefore,

f, g

∞ Ω

1/2 2

xk (sk )

k=1

∞

1/2 2

yk (sk )

ds1 ds2 . . .

k=1

∞ 1/2 xk (sk )2 k=1

X(Ω)

∞ 1/2 yk (sk )2

.

(2.8)

X (Ω)

k=1

It is an easy consequence of Khintchine inequality (see also [17, Proposition 2.d.1]) that for an arbitrary r.i. space Y there is a constant M(Y ) such that ∞ ∞ 1/2 2 yk yk (t)rk (s) M(Y ) k=1

k=1

Y

.

(2.9)

Y ([0,1]×[0,1])

Therefore, from (2.7) and (2.8) it follows that ∞ xk (sk )rk (s0 ) k=1

∞ 1/2 sup f, g : yk (sk )2

M X A

¯ X(Ω)

k=1

∞ 1/2 M X A xk (sk )2 k=1

X (Ω)

.

X(Ω)

Thus, inequality (0.1) holds in X for arbitrary independent xk (k = 1, 2, . . .). Since X has the Fatou property or separable, then from [1, Theorem 2] we infer that X ∈ K. The second condition X ∈ K may be proved in the same way only inequality (2.9) should be applied to the space X. 2 Proposition 2.3. If Radi X is a complemented subspace of an r.i. space X, then the projection P is bounded in X. ∞ −i and v = −i (α , β = 0, 1) be dyadic expansions of numProof. Let u = ∞ i i i=1 αi 2 i=1 βi 2 bers u, v ∈ [0, 1]. For a dyadic rational number we choose the nonterminating binary expansion (note that the set of all dyadic rational numbers from [0, 1] has the Lebesgue measure zero and, therefore, it is not essential in the question). Following [21] (see also [4]), we set u ⊕ v :=

∞ i=1

2−i (αi + βi ) mod 2 ,


203

where by (α + β) mod 2 (α, β = 0, 1) we define, as usual, the addition modulo 2, i.e., (α + β) mod 2 =

0, if α + β = 0 or α + β = 2, 1, if α + β = 1.

Moreover, the set Ω¯ is a compact abelian group with respect to the operation t ⊕ s := {ti ⊕ si }∞ i=0 , ¯ and s = {si }∞ ∈ Ω. ¯ It is clear that wp (t) := t ⊕ p, where p = ∈ Ω where t = {ti }∞ i=0 i=0 ∞ ¯ ¯ {pi }∞ i=0 ∈ Ω, is a measure-preserving mapping from the probability space (Ω, k=0 λk ) onto itself and therefore the operators Tp f (t0 , t1 , . . .) := f ◦ wp (t0 , t1 , . . .) = f (t ⊕ p)

¯ (p ∈ Ω)

¯ Note that the subspace Radi X is invariant related act isometrically in arbitrary r.i. space X(Ω). to them. In fact, if γi = u ∈ [0, 1]: u =

∞

αj 2−j , αi = 0

and γi = [0, 1]\γi

(i = 1, 2, . . .),

j =1

then ri (t ⊕ u) =

ri (t), −ri (t),

u ∈ γi , u ∈ γi .

(2.10)

Furthermore, the mapping f → f (· ⊕ u) is an isometry in an r.i. space X for every u ∈ [0, 1]. Therefore, by Rudin’s theorem [23] (or [24, Theorem 5.18]), there is a bounded projection ¯ → Radi X commuting with all Tp (p ∈ Ω). ¯ We will show that the operators S and S : X(Ω) P coincide on the separable part X◦ of X. First of all, since the Rademacher functions form a basic sequence in every r.i. space then the operator S is representable in the form: Sf (t0 , t1 , . . .) =

∞

Si f (ti )ri (t0 ),

(2.11)

i=1

¯ into X. Since the operators Tp and S commute, where Si are bounded linear operators from X(Ω) then S(f ◦ wp ) = S(Tp f ) = Tp Sf. Hence, using (2.11) and (2.10), we obtain that Si (f ◦ wp ) =

Si f,

p0 ∈ γi , pi = 0,

−Si f, p0 ∈ γi , pi = 0.

Moreover, taking into account that λ(γi ) = λ(γi ) = 1/2, we have that

Ωi γi

and

1 Si (f ◦ wp ) dp0 . . . dpi−1 dpi+1 . . . = Si f 2

(2.12)

204


1 Si (f ◦ wp ) dp0 . . . dpi−1 dpi+1 . . . = − Si f, 2

(2.13)

Ωi γi

where it is assumed that pi = 0. It is easy to see that the closed linear span [rn ] is a complemented subspace of the space Radi X. Since, by hypothesis, Radi X, in turn, is a complemented subspace of X, then [rn ] is complemented in X. Then, by [21], X = L∞ , and X◦ is separable. ¯ ◦ and the fact that the operators ¯ ◦ . Using the separability of (X(Ω)) Now, suppose f ∈ (X(Ω)) Tp act isometrically in this space, it is not hard to check that the mapping p → f ◦ wp is an ¯ Thus, since Si is a linear bounded operator ¯ ◦ -valued Bochner-integrable function on Ω. (X(Ω)) ¯ to X, it can be taken out of the integral in equalities (2.12) and (2.13). Therefore, we from X(Ω) obtain that

Si f = Si (f ◦ wp ) dp0 − (f ◦ wp ) dp0 dp1 . . . dpi−1 dpi+1 . . . . (2.14) γi

Ωi

γi

On the other hand, for any t ∈ [0, 1]

and

Thus,

γi , v ∈ [0, 1]: v = t ⊕ u, u ∈ γi = γi ,

t ∈ γi , t ∈ γi

γi , v ∈ [0, 1]: v = t ⊕ u, u ∈ γi = γi ,

t ∈ γi , t ∈ γi .

(f ◦ wp ) dp0 dp1 . . . dpi−1 dpi+1 . . . Ωi γi

=

f (v, t1 , t2 , . . .) dv dt1 . . . dti−1 dti+1 . . . · χγi (t0 ) Ωi γi

+


and

(f ◦ wp ) dp0 dp1 . . . dpi−1 dpi+1 . . . Ωi γi

=


+

f (v, t1 , t2 , . . .) dv dt1 . . . dti−1 dti+1 . . . · χγi (t0 ). Ωi γi


205

Since ri = χγi − χγi (i = 1, 2, . . .), then we infer that

(f ◦ wp ) dp0 − γi

Ωi

γi

=

(f ◦ wp ) dp0 dp1 . . . dpi−1 dpi+1 . . .

f (v, t1 , t2 , . . .) dv − Ωi

γi

f (v, t1 , t2 , . . .) dv dt1 . . . dti−1 dti+1 . . . · ri (t0 )

γi

1 =

f (v, t1 , . . . , ti−1 , ti , ti+1 , . . .)ri (v) dv dt1 . . . dti−1 dti+1 . . . · ri (t0 ) Ωi 0

= E(f ri |ti ) · ri (t0 ). Moreover, we have that x(tk ), Si x(tk )rk (t0 ) = 0,

i = k, i = k,

and from (2.14) it follows that Si f = E(f ri |ti ) (i = 1, 2, . . .). Taking into account the definition of the projection P and (2.11), we obtain that Sf = Pf, and the coincidence of the operators S and P on X0 is proved. If X is separable, then S = P on the whole X, and the proof is completed. ¯ ≈ X be arbitrary. There exists a sequence Now, let X have the Fatou property. Let f ∈ X(Ω) {fn } ⊂ L∞ such that |fn | |f | and fn → f a.e. It is obvious that E(fn rk |tk ) → E(f rk |tk ) a.e. as n → ∞ (k ∈ N). At the same time, as it is proved, Pfn X = Sfn X Sfn X Sf X

(n ∈ N).

Since {yk (tk )rk (t0 )}∞ k=1 is a basic sequence with constant 1 in X for any yk ∈ X, then the previous inequality yields m E(fn rk |tk )rk (t0 ) Sf X k=1

X

for all m = 1, 2, . . . and n = 1, 2, . . . . Since X has the Fatou property, then passing to the limit firstly as n → ∞ and then as m → ∞ we obtain that ∞ Pf X = E(f rk |tk )rk (t0 ) Sf X , k=1

which implies that P is bounded in X.

X

2

As a consequence of Theorem 2.1 we will show that an r.i. space X on [0, 1] such that X ∈ K and X ∈ K is isomorphic to some r.i. space on the semi-axis. This strengthens the analogous results of [11, §8] (see also [17, p. 203]) proved using a different approach under a stronger condition when an r.i. space X has non-trivial Boyd indices.

206


Theorem 2.4. If X is an r.i. space on [0, 1] such that X ∈ K and X ∈ K, then the spaces X and 2 are isomorphic. ZX Proof. By Theorem 3.1 from [7], for every r.i. space X satisfying the Kruglov property there exists a constant C > 0 such that for each sequence of mean zero independent functions {fk }∞ k=1 ⊂ X inequality (1.1) holds. In particular, this implies that the operator Q (see (2.3)) 2 into X(Ω). ¯ Moreover, by [12, Theorem 1] (see inequality (3)), is bounded from the space ZX 2 there is a constant c > 0 such that Qf X(Ω) ¯ cf Z 2 (f ∈ ZX ). Hence, the image of the X

2 . Therefore, by Theooperator Q, which coincides with Radi X, is isomorphic to the space ZX 2 ¯ rem 2.1, ZX is a complemented subspace of X(Ω) ≈ X. On the other hand, it is obvious that X is 2 . Since X ⊕ X (Z 2 ⊕ Z 2 , respectively) isomorphic to a complemented subspace of the space ZX X X 2 is isomorphic to X (to ZX , respectively), then by the decomposition method due to Pelczynski 2 are isomorphic. 2 [17, p. 172], we conclude that the spaces X and ZX

References [1] S.V. Astashkin, Independent functions in symmetric spaces and Kruglov property, Mat. Sb. 199 (7) (2008) 3–20 (in Russian); English transl. in: Sb. Math. 199 (7) (2008) 945–963. [2] S.V. Astashkin, A generalized Khintchine inequality in rearrangement invariant spaces, Funktsional. Anal. i Prilozhen. 42 (2) (2008) 78–81 (in Russian); English transl. in: Funct. Anal. Appl. 42 (2) (2008) 144–147. [3] S.V. Astashkin, Rademacher functions in symmetric spaces, Sovrem. Mat. Fundam. Napravl. 32 (2009) 3–161 (in Russian). [4] S.V. Astashkin, M.Sh. Braverman, On a subspace of a symmetric space generated by the Rademacher system with vector coefficients, in: Operator Equations in Function Spaces, Voronezh, 1986, pp. 3–10 (in Russian). [5] S.V. Astashkin, F.A. Sukochev, Sums of independent random variables in rearrangement invariant spaces: an operator approach, Israel J. Math. 145 (2005) 125–156. [6] S.V. Astashkin, F.A. Sukochev, Comparison of the sums of independent and disjoint functions in symmetric spaces, Mat. Zametki 76 (4) (2004) 483–489 (in Russian); English transl. in: Math. Notes 76 (4) (2004) 449–454. [7] S.V. Astashkin, F.A. Sukochev, Series of independent mean zero random variables in rearrangement-invariant spaces having the Kruglov property, Zap. Nauchn. Sem. POMI Issled. Lin. Oper. Teor. Funkts. 345 (2007) 25–50 (in Russian); English transl. in: J. Math. Sci. 148 (2008) 795–809. [8] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, London, 1988. [9] M.Sh. Braverman, Independent Random Variables and Rearrangement Invariant Spaces, Cambridge University Press, 1994. [10] N.L. Carothers, S.J. Dilworth, Inequalities for sums of independent random variables, Proc. Amer. Math. Soc. 194 (1988) 221–226. [11] W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979). [12] W.B. Johnson, G. Schechtman, Sums of independent random variables in rearrangement invariant function spaces, Ann. Probab. 17 (1989) 789–808. [13] A. Khintchine, Über dyadische Brüche, Math. Z. 18 (1923) 109–116. [14] M.A. Krasnosel’skii, Ya.B. Rutickii, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958 (in Russian); English transl. in: Noordhoff, Gröningen, 1961. [15] S.G. Krein, Yu.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English transl. in: Amer. Math. Soc., Providence, 1982. [16] V.M. Kruglov, A remark on the theory of infinitely divisible laws, Teor. Veroyatn. Primen. 15 (1970) 331–336 (in Russian). [17] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces. II. Function Spaces, Ergeb. Math. Grenzgeb. (Results in Mathematics and Related Areas), vol. 97, Springer-Verlag, Berlin/New York, 1979. [18] B.S. Mityagin, The homotopy structure of the linear group of a Banach space, Uspekhi Mat. Nauk 25 (5 (155)) (1970) 63–106 (in Russian); English transl. in: Russian Math. Surveys 25 (5) (1970) 59–103. [19] G. Peshkir, A.N. Shiryaev, The Khintchine inequality and martingale expanding sphere of their action, Uspekhi Mat. Nauk 50 (5 (305)) (1995) 3–62 (in Russian); English transl. in: Russian Math. Surveys 50 (5) (1995) 849–904.


207

[20] V.A. Rodin, E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (3) (1975) 207–222. [21] V.A. Rodin, E.M. Semenov, The complementability of a subspace that is generated by the Rademacher system in a symmetric space, Funktsional. Anal. i Prilozhen. 13 (2) (1979) 91–92 (in Russian); English transl. in: Funct. Anal. Appl. 13 (2) (1979) 150–151. [22] H.P. Rosenthal, On the subspaces of Lp (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970) 273–303. [23] W. Rudin, Projections on invariant subspaces, Proc. Amer. Math. Soc. 13 (3) (1962) 429–432. [24] W. Rudin, Funktsional’nyi Analiz, Mir, Moscow, 1975 (in Russian).


On the lack of compactness in the 2D critical Sobolev embedding Hajer Bahouri a , Mohamed Majdoub b,∗,1 , Nader Masmoudi c,2 a Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris 12, 61 avenue du Général de Gaulle,

94010 Créteil cedex, France b University of Tunis ElManar, Faculty of Sciences of Tunis, Department of Mathematics, ElManar 2092, Tunisia c New York University, The Courant Institute for Mathematical Sciences, 251 Mercer Street, New York,

NY 10012-1185, USA Received 17 March 2010; accepted 25 August 2010 Available online 15 September 2010 Communicated by H. Brezis

Abstract 1 (R2 ) in the Orlicz space. This paper is devoted to the description of the lack of compactness of Hrad Our result is expressed in terms of the concentration-type examples derived by P.-L. Lions (1985) in [24]. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and takes into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of nonlinear wave equation with exponential growth, where the Orlicz norm plays a decisive role. © 2010 Elsevier Inc. All rights reserved.

Keywords: Sobolev critical exponent; Trudinger–Moser inequality; Orlicz space; Lack of compactness; Nonlinear wave equation; Strichartz estimates


E-mail addresses: [email protected] (H. Bahouri), [email protected] (M. Majdoub), [email protected] (N. Masmoudi). 1 M.M. is grateful to the Laboratory of PDE and Applications at the Faculty of Sciences of Tunis. 2 N.M. is partially supported by an NSF Grant DMS-0703145. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.016

H. Bahouri et al. / Journal of Functional Analysis 260 (2011) 208–252

209

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Lack of compactness in the Sobolev embedding in Lebesgue spaces . . . 1.2. Critical 2D Sobolev embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Lack of compactness in 2D critical Sobolev embedding in Orlicz space 1.4. Fundamental remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Extraction of scales and profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Extraction of the first scale and the first profile . . . . . . . . . . . . . . . . . 2.2. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Qualitative study of nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . 3.1. Technical tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Strichartz estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Logarithmic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Convergence in measure . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Subcritical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Critical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Some known results on Sobolev embedding . . . . . . . . . . . . . . . . . . . A.2. Some additional properties on Orlicz spaces . . . . . . . . . . . . . . . . . . . A.3. BMO and L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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209 209 210 213 216 221 223 223 224 228 233 237 237 238 238 239 243 246 246 246 248 249 250

1. Introduction 1.1. Lack of compactness in the Sobolev embedding in Lebesgue spaces Due to the scaling invariance of the critical Sobolev embedding H˙ s Rd −→ Lp Rd , in the case where 0 s < d/2 and p = 2d/(d − 2s), no compactness properties may be expected. Indeed if u ∈ H˙ s \ {0}, then for any sequence (yn ) of points of Rd tending to the infinity and for any sequence (hn ) of positive real numbers tending to 0 or to infinity, the sequences (τyn u) and 1 (δhn u), where we denote δhn u(·) = d/p u( h·n ), converge weakly to 0 in H˙ s but are not relatively hn

compact in Lp since τyn uLp = uLp and δhn uLp = uLp . After the pioneering works of P.-L. Lions [24] and [25], several works have been devoted to the study of the lack of compactness in critical Sobolev embeddings, for the sake of geometric problems and the understanding of features of solutions of nonlinear partial differential equations. This question was investigated through several angles: for instance, in [11] the lack of compactness is describe in terms of microlocal defect measures, in [12] by means of profiles and in [19] by the use of nonlinear wavelet approximation theory. Nevertheless, it has been shown in all these results that translational and scaling invariance are the sole responsible for the defect of compactness of the embedding of H˙ s into Lp and more generally in Sobolev spaces in the Lq frame.

210


As it is pointed above, the study of the lack of compactness in critical Sobolev embedding supply us numerous information on solutions of nonlinear partial differential equations whether in elliptic frame or evolution frame. For example, one can mention the description of bounded energy sequences of solutions to the defocusing semi-linear wave equation u + u5 = 0 in R × R3 , up to remainder terms small in energy norm in [3] or the sharp estimate of the span time life of the focusing critical semi-linear wave equation by means of the size of energy of the Cauchy data in the remarkable work of [21]. Roughly speaking, the lack of compactness in the critical Sobolev embedding H˙ s Rd → Lp Rd in the case where d 3 with 0 s < d/2 and p = 2d/(d − 2s), is characterized in the following terms: a sequence (un )n∈N bounded in H˙ s (Rd ) can be decomposed up to a subsequence extraction on a finite sum of orthogonal profiles such that the remainder converges to zero in Lp (Rd ) as the number of the sum and n tend to ∞. This description still holds in the more general case of Sobolev spaces in the Lq frame (see [19]). 1.2. Critical 2D Sobolev embedding It is well known that H 1 (R2 ) is continuously embedded in all Lebesgue spaces Lp (R2 ) for 2 p < ∞, but not in L∞ (R2 ). A short proof of this fact is given in Appendix A for the convenience of the reader. On the other hand, it is also known (see for instance [21]) that H 1 (R2 ) embed in BMO(R2 ) ∩ L2 (R2 ), where BMO(Rd ) denotes the space of bounded mean oscillations which is the space of locally integrable functions f such that def

f BMO = sup B

1 |B|

|f − fB | dx < ∞

def

with fB =

B

1 |B|

f dx. B

The above supremum being taken over the set of Euclidean balls B, | · | denoting the Lebesgue measure. In this paper, we rather investigate the lack of compactness in Orlicz space L (see Definition 1.1 below) which arises naturally in the study of nonlinear wave equation with exponential growth. As, it will be shown in Appendix A.2, the spaces L and BMO are not comparable. Let us now introduce the so-called Orlicz spaces on Rd and some related basic facts. (For the sake of completeness, we postpone to Appendix A.2 some additional properties on Orlicz spaces.) Definition 1.1. Let φ : R+ → R+ be a convex increasing function such that φ(0) = 0 = lim φ(s), s→0+

lim φ(s) = ∞.

s→∞


211

We say that a measurable function u : Rd → C belongs to Lφ if there exists λ > 0 such that

|u(x)| dx < ∞. φ λ

Rd

We denote then uLφ

|u(x)| dx 1 . = inf λ > 0, φ λ

(1)

Rd

It is easy to check that Lφ is a C-vector space and · Lφ is a norm. Moreover, we have the following properties. • For φ(s) = s p , 1 p < ∞, Lφ is nothing else than the Lebesgue space Lp . 2 • For φα (s) = eαs − 1, with α > 0, we claim that Lφα = Lφ1 . It is actually a direct consequence of Definition 1.1. • We may replace in (1) the number 1 by any positive constant. This change the norm · Lφ to an equivalent norm. • For u ∈ Lφ with A := uLφ > 0, we have the following property |u(x)| λ > 0, dx 1 = [A, ∞[. φ λ

(2)

Rd 2

In what follows we shall fix d = 2, φ(s) = es − 1 and denote the Orlicz space Lφ by L endowed with the norm · L where the number 1 is replaced by the constant κ that will be fixed in identity (6) below. As it is already mentioned, this change does not have any impact on the definition of Orlicz space. It is easy to see that L → Lp for every 2 p < ∞. The 2D critical Sobolev embedding in Orlicz space L states as follows: Proposition 1.2. 1 uL √ uH 1 . 4π

(3)

Remarks 1.3. a) Inequality (3) is insensitive to space translation but not invariant under scaling nor oscillations. b) The embedding of H 1 (R2 ) in L is sharp within the context of Orlicz spaces. In other words, the target space L cannot be replaced by an essentially smaller Orlicz space. However, this target space can be improved if we allow different function spaces than Orlicz spaces. More precisely H 1 R2 → BW R2 ,

(4)

212


where the Brezis–Wainger space BW(R2 ) is defined via 1 uBW :=

u∗ (t) log(e/t)

0

2

dt t

∞

1/2 +

1/2 ∗

2

u (t) dt

,

1

where u∗ denotes the rearrangement function of u given by

u∗ (t) = inf λ > 0; x; u(x) > λ t . The embedding (4) is sharper than (3) as BW(R2 ) L(R2 ). It is also optimal with respect to all rearrangement invariant Banach function spaces. For more details on this subject, we refer the reader to [5,8,10,13,14,27]. c) In higher dimensions (d = 3 for example), the equivalent of embedding (4) is H 1 R3 → L6,2 R3 , where L6,2 is the classical Lorentz space. Notice that L6,2 is a rearrangement invariant Banach space but not an Orlicz space. To end this short introduction to Orlicz spaces, let us point out that the embedding (3) derives immediately from the following Trudinger–Moser type inequalities: Proposition 1.4. Let α ∈ [0, 4π[. A constant cα exists such that α|u|2 e − 1 dx cα u2L2 (R2 )

(5)

R2

for all u in H 1 (R2 ) such that ∇uL2 (R2 ) 1. Moreover, if α 4π , then (5) is false. A first proof of these inequalities using rearrangement can be found in [1] (see also [28,39]). In other respects, it is well known (see for instance [33]) that the value α = 4π becomes admissible in (5) if we require uH 1 (R2 ) 1 rather than ∇uL2 (R2 ) 1. In other words, we have Proposition 1.5. sup

uH 1 1 R2

4π|u|2 e − 1 dx := κ < ∞,

(6)

and this is false for α > 4π . Now, it is obvious that estimate (6) allows to prove Proposition 1.2. Indeed, without loss of generality, we may assume that uH 1 = 1 which leads under Proposition 1.5 to the inequality uL √1 , which is the desired result. 4π

Remark 1.6. Let us mention that a sharp form of Trudinger–Moser inequality in bounded domain was obtained in [2].


213

1.3. Lack of compactness in 2D critical Sobolev embedding in Orlicz space The embedding H 1 → L is noncompact at least for two reasons. The first reason is the lack of compactness at infinity. A typical example is uk (x) = ϕ(x + xk ) where 0 = ϕ ∈ D and |xk | → ∞. The second reason is of concentration-type derived by P.-L. Lions [24,25] and illustrated by the following fundamental example fα defined by: ⎧ 0 if |x| 1, ⎪ ⎪ ⎨ √ log |x| − if e−α |x| 1, fα (x) = 2απ ⎪ ⎪ ⎩ α if |x| e−α , 2π

where α > 0. Straightforward computations show that fα 2L2 (R2 ) =

1 −2α ) − 1 e−2α 4α (1 − e 2 in H 1 (R2 ) as α → ∞ or α

and

∇fα L2 (R2 ) = 1. Moreover, it can be seen easily that fα 0 → 0. However, the lack of compactness of this sequence in the Orlicz space L occurs only when α goes to infinity. More precisely, we have Proposition 1.7. For fα denoting the sequence defined above, we have the following convergence results: a) fα L → √1 as α → ∞. 4π b) fα L → 0 as α → 0. Proof. Let us first go to the proof of the first assertion. If

|fα (x)|2 e λ2 − 1 dx κ,

then e−α α 2π e 2πλ2 − 1 r dr κ, 0

which implies that λ2

α 2π log(1 +

κe2α π )

.

It follows that 1 lim inf fα L √ . α→∞ 4π To conclude, it suffices to show that 1 lim sup fα L √ . α→∞ 4π

214


Let us fix ε > 0. Taking advantage of Trudinger–Moser inequality and the fact that fα L2 → 0, we infer (4π−ε)|f (x)|2 α − 1 dx Cε fα 2L2 , e κ,

for α αε .

Hence, for any ε > 0, lim sup fα L √ α→∞

1 4π − ε

,

which ends the proof of the first assertion. To prove the second one, let us write

(x)|2 |fα1/2 − 1 dx = 2π e α

e−α √ 1 log2 r α 2π e − 1 r dr + 2π e 2πα3/2 − 1 r dr e−α

0 √

α α1/2 π e 2π − 1 e−2α + 2π 1 − e−α e 2π . This implies that, for α small enough, fα L α 1/4 , which leads to the result.

2

The difference between the behavior of these families in Orlicz space when α → 0 or α → ∞ comes from the fact that the concentration effect is only displayed by this family when α → ∞. Indeed, in the case where α → ∞ we have the following result which does not occur when α → 0. Proposition 1.8. For fα being the family of functions defined above, we have |∇fα |2 −→ δ(x = 0)

and e4π|fα | − 1 −→ 2πδ(x = 0) 2

as α −→ ∞ in D R2 .

Proof. Straightforward computations give for any smooth compactly supported function ϕ

∇fα (x) 2 ϕ(x) dx = 1 2πα

1 2π e−α 0

ϕ(r cos θ, r sin θ ) dr dθ r

1 = ϕ(0) + 2πα

1 2π e−α 0

ϕ(r cos θ, r sin θ ) − ϕ(0) dr dθ. r

Since | ϕ(r cos θ,r rsin θ)−ϕ(0) | ∇ϕL∞ , we deduce that which ensures the result. Similarly, we have

4π|f (x)|2 α e − 1 ϕ(x) dx =

|∇fα (x)|2 ϕ(x) dx → ϕ(0) as α → ∞,

e−α2π 2α e − 1 ϕ(r cos θ, r sin θ )r dr dθ 0

0


215

1 2π 2 log2 r eα + − 1 ϕ(r cos θ, r sin θ )r dr dθ e−α 0

= πϕ(0) 1 − e−2α + 2πϕ(0)

1

2 log2 r eα − 1 r dr

e−α

e−α2π 2α + e − 1 ϕ(r cos θ, r sin θ ) − ϕ(0) r dr dθ 0

0

1 2π 2 log2 r + eα − 1 ϕ(r cos θ, r sin θ ) − ϕ(0) r dr dθ. e−α 0

2

We conclude by using the following lemma. Lemma 1.9. When α goes to infinity 1 Iα :=

re α log

2

2r

dr −→ 1

(7)

2

2r

1 dr −→ . 3

(8)

e−α

and 1 Jα :=

r 2 e α log

e−α

Proof. The change of variable y :=

2 α (− log r

Iα = 2

− α2 ) yields

α −α e 2 2

√α 2

2

ey dy. 0

Taking advantage of the following obvious equivalence at infinity which can be derived by integration by parts A

2

2

ey dy ∼ 0

we deduce (7).

eA , 2A

(9)

216


Similarly for the second term, the change of variable y := Jα =

α −9α e 8 2

√α 2

=

α −9α e 8 2

− 34 α) implies that

2

ey dy

√α 2

√α 1 2 2

2

ey dy +

α −9α e 8 2

0

According to (9), we get (8).

2 α (− log r

1 2

− 32

√α 2

3 2

2

ey dy. 0

2

Remark 1.10. When α goes to zero, we get a spreading rather than a concentration. Notice also that for small values of the function, our Orlicz space behaves like L2 (see Proposition A.9) and simple computations show that fα L2 (R2 ) goes to zero when α goes to zero. In fact, the conclusion of Proposition 1.8 is available for more general radial sequences. More precisely, we have the following result due to P.-L. Lions (in a slightly different form): 1 (R2 ) such that Proposition 1.11. Let (un ) be a sequence in Hrad

lim inf un L > 0 and

un 0 in H 1 ,

n→∞

lim lim sup

R→∞ n→∞

un (x) 2 dx = 0.

|x|>R

Then, there exists a constant c > 0 such that ∇un (x) 2 dx μ cδ(x = 0)

(n −→ ∞)

(10)

weakly in the sense of measures. Remark 1.12. The hypothesis of compactness at infinity lim lim sup

R→∞ n→∞

un (x) 2 dx = 0

|x|>R x 2

is necessary to get (10). For instance, un (x) = n1 e−| n | satisfies un L2 = C > 0, ∇un L2 → 0 and lim infn→∞ un L > 0. 1.4. Fundamental remark 1 (R2 ) in Orlicz In order to describe the lack of compactness of the Sobolev embedding of Hrad space, we will make the change of variable s := − log r, with r = |x|. We associate then to any radial function u on R2 a one space variable function v defined by v(s) = u(e−s ). It follows that


u2L2

= 2π

217

v(s) 2 e−2s ds,

(11)

2 v (s) ds,

(12)

R

∇u2L2 = 2π

R

| u(x) |2 e λ − 1 dx = 2π

and

| v(s) |2 e λ − 1 e−2s ds.

(13)

R

R2

The starting point in our analysis is the following observation related to the Lions’ example f˜α (s) := fα e−s =

s α L , 2π α

where L(t) =

0 if t 0, t if 0 t 1, 1 if t 1.

The sequence α → ∞ is called the scale and the function L the profile. In fact, the Lions’ example generates more elaborate situations which help us to understand the defect of compactness of Sobolev embedding in Orlicz space. For example, it can be seen that for the sequence gk := fk + f2k we have gk (s) =

k s 2π ψ( k ),

ψ(t) =

where

⎧0 ⎪ ⎪ ⎨t +

√t

2

if t 0, if 0 t 1,

if 1 t 2, 1 + √t ⎪ ⎪ ⎩ √2 1 + 2 if t 2.

This is due to the fact that the scales (k)k∈N and (2k)k∈N are not orthogonal (see Definition 1.13 below) and thus they give a unique profile. However, for the sequence hk := fk + fk 2 , the situation is completely different and a decomposition under the form hk (x)

− log |x| αk ψ 2π αk

is not possible, where the symbol means that the difference is compact in the Orlicz space L. The reason behind is that the scales (k)k∈N and (k 2 )k∈N are orthogonal. It is worth noticing that in the above examples the support is a fixed compact, and thus at first glance the construction cannot be adapted in the general case. But as shown by the following example, no assumption on the support is needed to display lack of compactness in the Orlicz space. Indeed, let Rα in (0, ∞) be such that Rα √ −→ 0, α

α −→ ∞,

(14)

218


and log Rα > −∞. a := lim inf α→∞ α

(15)

We can take for instance Rα = α θ with θ < 1/2 and then a = 0, or Rα = e−γ α with γ 0 and then a = −γ . Remark that assumption (14) implies that a is always negative. Now, let us define the sequence gα (x) := fα ( Rxα ). It is obvious that the family gα is not uniformly supported in a fixed compact subset of R2 , in the case when Rα = α θ with 0 < θ < 1/2. Now, arguing exactly as for Lions’ example, we can easily show that Rα gα L2 ∼ √ , 2 α

∇gα L2 = 1 and gα 2L

α . α 2π log(1 + πκ ( Re α )2 )

Hence, gα 0 in H 1 and lim infα→∞ gα L > 0. Up to a subsequence extraction, straightforward computation yields the strong convergence to zero in H 1 for the difference fα − gα , in the case when a = 0, which implies that gα (x) − log |x| α ). However, in the case when a < 0, the sequence (gα ) converges 2π L( α strongly to |x| · α fα ( eαa ) in H 1 and then the profile is slightly different in the sense that gα (x) 2π La ( − log ) α where La (s) = L(s + a). To be more complete and in order to state our main result in a clear way, let us introduce some definitions as in [12] for instance.

Definition 1.13. A scale is a sequence α := (αn ) of positive real numbers going to infinity. We shall say that two scales α and β are orthogonal (in short α ⊥ β) if log(βn /αn ) −→ ∞. According to (11) and (12), we introduce the profiles as follows. Definition 1.14. The set of profiles is

P := ψ ∈ L2 R, e−2s ds ; ψ ∈ L2 (R), ψ|]−∞,0] = 0 . Some remarks are in order: a) The limitation for scales tending to infinity is justified by the behavior of fα L stated in Proposition 1.7. b) The set P is invariant under negative translations. More precisely, if ψ ∈ P and a 0 then ψa (s) := ψ(s + a) belongs to P. c) It will be useful to observe that a profile (in the sense of Definition 1.14) is a continuous 1 (R). function since it belongs to Hloc d) For a scale α and a profile ψ , define gα,ψ (x) :=

− log |x| αn ψ . 2π αn


219

It is clear that, for any λ > 0, gα,ψ = gλα,ψλ , where ψλ (t) =

√1 ψ(λt). λ

The next proposition illustrates the above definitions of scales and profiles. Proposition 1.15. Let ψ ∈ P be a profile, (αn ) be any scale and set gn (x) :=

− log |x| αn . ψ 2π αn

Then |ψ(s)| 1 lim gn L = √ max √ . n→∞ s 4π s>0

(16)

Proof. Let us first prove that |ψ(s)| 1 lim inf gn L √ max √ . n→∞ s>0 s 4π Setting L = lim infn→∞ gn L , we have according to (2) for fixed ε > 0 and n large enough (up to a subsequence extraction)

| gn (x) |2 e L+ε − 1 dx κ.

R2

A straightforward computation yields ∞ e

αn

2αn s(

1 √ )2 −1) ( ψ(s) s 4π(L+ε)2

ds C,

0

for some absolute constant C and for n large enough. Since ψ is continuous, we deduce that necessarily, for any s > 0, 1 |ψ(s)| √ √ L + ε, s 4π which ensures the result. Now, to obtain formula (16), it is enough to prove that for any δ > 0, |ψ(s)| 1+δ lim sup gn L √ max √ . s n→∞ 4π s>0

(17)

220


To prove it, let us for fixed δ > 0 prove that, if λ =

e

|gn (x)|2 λ2

1+δ √ 4π

maxs>0

|ψ(s)| √ , then, as s

n tends to infinity,

− 1 dx −→ 0.

R2

In fact the left hand side reads

∞ e

2π αn

4πλ s

∞

2

−2αn s(1− |ψ(s)|2 )

ds − αn

0

e

−2αn s

ds .

0

According to the choice of λ, the main contribution of both integrals lies in a neighborhood of s = 0. It suffices then to prove that, for a suitable η > 0, we have η e

αn

η

2

−2αn s(1− |ψ(s)|2 ) 4πλ s

ds − αn

0

e−2αn s ds −→ 0.

0

To do so, let us first observe that ψ(s) √ −→ 0 as s −→ 0. s

(18)

Indeed s

1/2 √ s 2 ψ (τ ) dτ ψ(s) = ψ (τ ) dτ s , 0

0

which ensures the result since ψ ∈ L2 (R). Taking advantage of (18), we infer that for any ε > 0 there exists η > 0 such that |ψ(s)|2 0, n→∞

and

(20)

|un |2 dx = 0.

(21)

lim lim sup

R→∞ n→∞

|x|>R

Then, there exist a sequence (α (j ) ) of pairwise orthogonal scales and a sequence of profiles (ψ (j ) ) in P such that, up to a subsequence extraction, we have for all 1,

un (x) =

j =1

(j ) − log |x| αn + r() ψ (j ) n (x), (j ) 2π αn

→∞ lim supr() n L −−−→ 0.

(22)

n→∞

Moreover, we have the following stability estimates

∇un 2L2 =

(j ) 2 ψ 2 + ∇r() 2 2 + o(1), j =1

L

n

L

n −→ ∞.

(23)

Remarks 1.17. a) As in higher dimensions, the decomposition (22) is not unique (see [12]). b) The assumption (21) means that there is no lack of compactness at infinity. It is in particular satisfied when the sequence (un )n∈N is supported in a fixed compact of R2 and also by gα . c) Also, this assumption implies the condition ψ|]−∞,0] = 0 included in the definition of the set of profiles.Indeed, first let us observe that under condition (21), necessarily each element (j )

|x| ψ (j ) ( − log (j ) ) of decomposition (22) does not display lack of compactness αn |x| αn at infinity. The problem is then reduced to prove that if a sequence gn = 2π ψ( − log αn ), (j )

gn (x) :=

αn 2π

where (αn ) is any scale and ψ ∈ L2 (R, e−2s ds) with ψ ∈ L2 (R), satisfies hypothesis (21)

222


then consequently ψ|]−∞,0] = 0. Let us then consider a sequence gn satisfying the above assumptions. This yields lim lim sup

R→∞ n→∞

ψ(t) 2 e−2αn t dt = 0.

R − log αn

αn2 −∞

Now, if ψ(t0 ) = 0 for some t0 < 0 then by continuity, we get |ψ(t)| 1 for t0 − η t t0 + η < 0. Hence, for n large enough, R − log αn

αn2 −∞

ψ(t) 2 e−2αn t dt αn e−2αn (t0 −η) − e−2αn (t0 +η) , 2

which leads easily to the desired result. d) Compared with the decomposition in [12], it can be seen that there’s no core in (22). This is justified by the radial setting. e) The description of the lack of compactness of the embedding of H 1 (R2 ) into Orlicz space in the general frame is much harder than the radial setting. This will be dealt with in a forthcoming paper. f) Let us mention that M. Struwe in [36] studied the loss of compactness for the functional 1 2 E(u) = e4π|u| dx, |Ω| Ω

where Ω is a bounded domain in R2 . It should be emphasized that, contrary to the case of Sobolev embedding in Lebesgue spaces, where the asymptotic decomposition derived by P. Gérard in [12] leads to p

un Lp −→

ψ (j ) p p , j 1

L

Theorem 1.16 induces to un L −→ sup

(j ) lim gn L ,

j 1 n→∞

(24)

thanks to the following proposition. Proposition 1.18. Let (α (j ) )1j be a family of pairwise orthogonal scales and (ψ (j ) )1j be a family of profiles, and set gn (x) =

j =1

(j ) αn (j ) (j ) − log |x| := ψ gn (x). (j ) 2π αn j =1


223

Then gn L −→ sup

(j ) lim gn L .

1j n→∞

(25)

A consequence of this proposition is that the first profile in the decomposition (22) can be chosen such that up to extraction αn(1) (1) − log |x| . lim sup un L = A0 = lim ψ (1) n→∞ 2π n→∞ αn L 1.6. Structure of the paper Our paper is organized as follows: we first describe in Section 2 the algorithmic construc1 (R2 ), up to a subsequence tion of the decomposition of a bounded sequence (un )n∈N in Hrad extraction, in terms of asymptotically orthogonal profiles in the spirit of the Lions’ examples − log |x| α ψ( ), and then prove Proposition 1.18. Section 3 is devoted to the study of nonlinear 2π α wave equations with exponential growth, both in the subcritical and critical cases. The purpose is then to investigate the influence of the nonlinear term on the main features of solutions of nonlinear wave equations by comparing their evolution with the evolution of the solutions of the Klein–Gordon equation. Finally, we deal in Appendix A with several complements for the sake of completeness. Finally, we mention that, C will be used to denote a constant which may vary from line to line. We also use A B to denote an estimate of the form A CB for some absolute constant C and A ≈ B if A B and B A. For simplicity, we shall also still denote by (un ) any subsequence of (un ). 2. Extraction of scales and profiles This section is devoted to the proofs of Theorem 1.16 and Proposition 1.18. Our approach to extract scales and profiles relies on a diagonal subsequence extraction and uses in a crucial way the radial setting and particularly the fact that we deal with bounded functions far away from the origin. The heart of the matter is reduced to the proof of the following lemma. 1 (R2 ) satisfying the assumptions of Theorem 1.16. Lemma 2.1. Let (un ) be a sequence in Hrad Then there exist a scale (αn ) and a profile ψ such that

ψ

L2

CA0 ,

(26)

where C is a universal constant. Roughly speaking, the proof is done in three steps. In the first step, according to Lemma 2.1, we extract the first scale and the first profile satisfying the condition (26). This reduces the problem to the study of the remainder term. If the limit of its Orlicz norm is null we stop the process. If not, we prove that this remainder term satisfies the same properties as the sequence start which allow us to apply the lines of reasoning of the first step and extract the second scale and the

224


second profile which verify the above key property (26). By contradiction arguments, we get the property of orthogonality between the two first scales. Finally, we prove that this process converges. 2.1. Extraction of the first scale and the first profile 1 (R2 ) satisfying hypothesis (19)–(21) and let Let us consider a bounded sequence (un ) in Hrad −s us set vn (s) = un (e ). The following lemma summarizes some properties of the sequence (un ) that will be useful to implement the proof strategy.

Lemma 2.2. Under the above assumptions, the sequence (un ) converges strongly to 0 in L2 and we have, for any M ∈ R, vn L∞ (]−∞,M[) −→ 0,

n −→ ∞.

(27)

Proof. Let us first observe that for any R > 0, we have un L2 = un L2 (|x|R) + un L2 (|x|>R) . Now, by virtue of Rellich’s theorem, the Sobolev space H 1 (|x| R) is compactly embedded in L2 (|x| R). Therefore, lim sup un L2 lim sup un L2 (|x|R) . n→∞

n→∞

Taking advantage of the compactness at infinity of the sequence, we deduce the strong convergence of the sequence (un ) to zero in L2 . On the other hand, property (27) derives immediately from the boundedness of (un ) in H 1 , the strong convergence to zero of (un ) in L2 and the following well-known radial estimate recalled in Lemma A.2 1 1 C u(r) √ uL2 2 ∇uL2 2 . r

2

The first step is devoted to the determination of the first scale and the first profile. Proposition 2.3. For any δ > 0, we have vn (s) 2 − s −→ ∞, sup A −δ

s0

0

n −→ ∞.

(28)

Proof. We proceed by contradiction. If not, there exists δ > 0 such that, up to a subsequence extraction sup s0, n∈N

vn (s) 2 − s C < ∞. A − δ 0

(29)


225

On the one hand, thanks to (27) and (29), we get by virtue of Lebesgue theorem

| Aun (x) |2 e 0 −δ − 1 dx = 2π

|x| 0 such that 1 Wn αn(1) an − . n (1)

(1)

In other respects under (28), an → ∞ and then Wn (αn ) → ∞. It remains to prove that αn → ∞. If not, up to a subsequence extraction, the sequence (αn(1) ) is bounded and so is (Wn (αn(1) )) by (27). This completes the proof. 2

226


An immediate consequence of the previous corollary is the following result. Corollary 2.5. Under the above hypothesis, we have for n large enough A0 2

(1) (1) αn vn αn(1) C αn + o(1),

√ with C = (lim supn→∞ ∇un L2 )/ 2π and where, as in all that follows, o(1) denotes a sequence which tends to 0 as n goes to infinity. Proof. The left hand side inequality follows immediately from Corollary 2.4. In other respects, noticing that by virtue of (27), the sequence vn (0) → 0, one can write for any positive real s s n 2 vn (s) = vn (0) + v (τ ) dτ vn (0) + s 1/2 v 2 vn (0) + s 1/2 ∇u √ L , n n L 2π 0

from which the right hand side of the desired inequality follows.

2

Now we are able to extract the first profile. To do so, let us set ψn (y) =

2π

vn αn(1)

(1) αn y .

The following lemma summarizes the principle properties of (ψn ). Lemma 2.6. Under notations of Corollary 2.5, there exists a constant C such that A0 √ 2π ψn (1) C + o(1). 2 Moreover, there exists a profile ψ (1) ∈ P such that, up to a subsequence extraction ψn

ψ (1)

2

in L (R)

and ψ (1) L2

√ 2π A0 . 2

Proof. The first assertion is contained in Corollary 2.5. To prove the second one, let us first remark that since ψn L2 = ∇un L2 then the sequence (ψn ) is bounded in L2 . Thus, up to a subsequence extraction, (ψn ) converges weakly in L2 to some function g ∈ L2 . In addition, (ψn (0)) converges in R to 0 and (still up to a subsequence extraction) (ψn (1)) converges in R to √ 2π some constant a satisfying |a| 2 A0 . Let us then introduce the function s ψ

(1)

(s) :=

g(τ ) dτ. 0


227

Our task now is to show that ψ (1) belongs to the set P. Clearly ψ (1) ∈ C(R) and (ψ (1) ) = g ∈ L2 (R). Moreover, since s (1) ψ (s) = g(τ ) dτ s 1/2 g 2 , L (R) 0

we get ψ (1) ∈ L2 (R+ , e−2s ds). It remains to prove that ψ (1) (s) = 0 for all s 0. Using the boundedness of the sequence (un ) in L2 (R2 ) and the fact that un 2L2

2 = αn(1)

(1) ψn (s) 2 e−2αn s ds,

R

we deduce that 0

ψn (s) 2 ds

C (1) (αn )2

−∞

.

Hence, (ψn ) converges strongly to zero in L2 (] − ∞, 0[), and then almost everywhere (still up to a subsequence extraction). In other respects, since (ψn ) converges weakly to g in L2 (R) and 1 (R), we infer that ψn ∈ Hloc s ψn (s) − ψn (0) =

ψn (τ ) dτ

s −→

0

g(τ ) dτ = ψ (1) (s), 0

from which it follows that ψn (s) −→ ψ (1) (s),

for all s ∈ R.

that ψ (1) (s) = 0 for all s 0. Finally, we have As ψn goes to zero for all s 0, we deduce √ 2π (1) (1) proved that ψ ∈ P and |ψ (1)| = |a| 2 A0 . The fact that 1 (1) (1) ψ (1) = ψ (τ ) dτ ψ (1) L2 , 0

yields (ψ (1) ) L2

√

2π 2 A0 .

2

Set r(1) n (x) =

(1)

αn 2π

ψn

− log |x| (1)

αn

−ψ

(1)

− log |x| (1)

αn

.

(31)

228


It can be easily seen that (1) 2 ∇r 2 2 = ψ − ψ (1) 2 2 . n L (R ) n L (R) Taking advantage of the fact that (ψn ) converges weakly in L2 (R) to (ψ (1) ) , we get the following result. 1 (R2 ) satisfying the assumptions of Theorem 1.16. Lemma 2.7. Let (un ) be a sequence in Hrad (1) Then there exist a scale (αn ) and a profile ψ (1) such that

(1) ψ

L2

√ 2π A0 , 2

and (1) 2 2 2 2 lim ∇r(1) n L2 = lim ∇un L2 − ψ L

n→∞

n→∞

(32)

(1)

where rn is given by (31). 2.2. Conclusion Our concern now is to iterate the previous process and to prove that the algorithmic construc(1) tion converges. Observing that, for R 1, and thanks to the fact that ψ|]−∞,0] = 0, − log(1)R

(1) 2 r 2 n

L

2 = αn(1) (|x|R)

αn

(1) ψn (t) − ψ (1) (t) 2 e−2αn t dt

−∞ − log(1)R

2 = αn(1)

αn

(1) ψn (t) 2 e−2αn t dt

−∞

= un 2L2 (|x|R) , (1)

we deduce that (rn ) satisfies the hypothesis of compactness at infinity (21). This leads, accord1 ing to (32), that (r(1) n ) is bounded in Hrad and satisfies (19). (1)

Let us define A1 = lim supn→∞ rn L . If A1 = 0, we stop the process. If not, we apply the (1) (2) above argument to rn and then there exists a scale (αn ) satisfying the statement of Corollary 2.4 with A1 instead of A0 . In particular, there exists a constant C such that (1) (2) A1 (2) (2) C αn + o(1), αn ˜rn αn 2

(33)


229 (2)

where ˜rn (s) = rn (e−s ). Moreover, we claim that αn ⊥ αn , or equivalently that log | αn(1) | → (1)

(1)

(2)

(1)

αn

∞. Otherwise, there exists a constant C such that (2) 1 αn (1) C. C αn Now, according to (31), we have (2) ˜r(1) = n αn

(1)

αn 2π

ψn

(2)

αn

−ψ

(1)

αn

(1)

(2)

αn

(1)

.

αn

This yields a contradiction in view of (33) and the following convergence result (up to a subsequence extraction) ψn

(2)

αn

(1)

αn

− ψ (1)

(2)

αn

(1)

αn

−→ 0.

Moreover, there exists a profile ψ (2) in P such that r(1) n (x) = with (ψ (2) ) L2

(2) − log |x| αn + r(2) ψ (2) n (x), (2) 2π αn

√

2π 2 A1

and

(2) 2 2 (1) 2 2. lim ∇r(2) n L2 = lim ∇rn L2 − ψ L

n→∞

n→∞

This leads to the following crucial estimate 2 lim supr(2) n H1 C − n→∞

√ √ 2π 2 2π 2 A0 − A1 , 2 2

with C = lim supn→∞ ∇un 2L2 . At iteration , we get un (x) =

j =1

(j ) − log |x| αn + r() ψ (j ) n (x), (j ) 2π αn

with 2 2 2 2 lim supr() n H 1 1 − A0 − A1 − · · · − A−1 . n→∞

Therefore A → 0 as → ∞ and the proof of the decomposition (22) is achieved. This ends the proof of the theorem.

230


Let us now go to the proof of Proposition 1.18. Proof of Proposition 1.18. We restrict ourselves to the example hα := afα + bfα 2 where a, b are two real numbers. The general case is similar except for more technical complications. Set M := sup(|a|, |b|). We want to show that M hα L −→ √ 4π

as α −→ ∞.

We start by proving that M lim inf hα L √ . α→∞ 4π

(34)

Let λ > 0 such that

|hα (x)|2 e λ2 − 1 dx κ.

R2

This implies e−α

2

|hα (r)|2 κ e λ2 − 1 r dr , 2π

(35)

0

and e−α e−α

|hα (r)|2 κ . e λ2 − 1 r dr 2π

(36)

2

Since ⎧ α ⎨ a 2π + b √α 2π hα (r) = ⎩ a α − √b log r 2π α 2π

if r e−α , 2

if e−α r e−α , 2

we get from (35) and (36) λ2

√ α(a + b α )2 2π log(1 + Ce

2α 2

)

=

b2 + o(1), 4π

and λ2 This leads to (34) as desired.

√ a2 a 2 α + 2ab α + b2 + o(1). = 4π 2π log(1 + Ce2α )


231

In the general case, we have to replace (35) and (36) by estimates of that type. Indeed, (j )

assuming that

αn (j +1) αn

→ 0 when n goes to infinity for j = 1, 2, . . . , l − 1, we replace (35) and

(36) by the fact that

(j )

−αn e

(j +1) e−αn

|hα (r)|2 κ , e λ2 − 1 r dr 2π

j = 1, . . . , l − 1

(37)

and (l)

−αn e

|hα (r)|2 κ . e λ2 − 1 r dr 2π

(38)

0

Our concern now is to prove the second (and more difficult) part, that is M lim sup hα L √ . α→∞ 4π

(39)

To do so, it is sufficient to show that for any η > 0 small enough and α large enough

4π−η |hα (x)|2 e M2 − 1 dx κ.

(40)

R2

Actually, we will prove that the left hand side of (40) goes to zero when α goes to infinity. We shall make use of the following lemma. Lemma 2.8. Let p, q be two real numbers such that 0 < p, q < 2. Set e−α Iα = epα

2

e

e

q log2 r α

r dr.

−α 2

Then Iα → 0 as α → ∞. Proof. The change of variable y =

√

q α (− log r

−

yields

√ α q−1 q

2

− αq

e Iα = αepα √ q

α2 q )

2

ey dy. √

q− √αq

(41)

232


A

Since

0

2

ey dy

2

eA A

for every nonnegative real A, we get (for q > 1 for example) 2

Iα epα e(q−2)α + e(p−2)α , and the conclusion follows. We argue similarly if q 1.

2

We return now to the proof of (39). To this end, write 4π − η 2 2 4π − η 2 2 4π − η (4π − η) |hα |2 = a fα + b fα 2 + 2 abfα fα 2 M2 M2 M2 M2 := Aα + Bα + Cα .

(42) (43)

The simple observation ex+y+z − 1 = ex − 1 ey − 1 ez − 1 + ex − 1 ey − 1 + ex − 1 e z − 1 + e y − 1 e z − 1 + ex − 1 + e y − 1 + e z − 1 , yields

4π−η |hα (x)|2 e M2 − 1 dx =

R2

A e α − 1 eBα − 1 eCα − 1 +

A e α − 1 eBα − 1

A B e α − 1 eCα − 1 + e α − 1 eCα − 1 A B C + e α −1 + e α −1 + e α −1 . +

(44)

By Trudinger–Moser estimate (5), we have for ε 0 small enough, A e α − 1 1+ε + eBα − 1 1+ε −→ 0 as α −→ ∞. L L

(45)

To check that the last term in (44) tends to zero, we use Lebesgue theorem in the region e−α r 1. Observe that one can replace Cα with γ Cα for any γ > 0. The two terms containing both Aα and Cα or Bα and Cα can be handled in a similar way. Indeed, by Hölder inequality and (45), we infer (for ε > 0 small enough)

A e α − 1 eCα − 1 +

B e α − 1 eCα − 1 eAα − 1L1+ε eCα − 1

1

L1+ ε

+ eBα − 1L1+ε eCα − 1

1

L1+ ε

−→ 0.

Now, we claim that

A e α − 1 eBα − 1 −→ 0,

α −→ ∞.

(46)


233

The main difficulty in the proof of (46) comes from the term e−α e−α

4π−η 2 α A (r) a e α − 1 eBα (r) − 1 r dr Iα := e M 2 2π

2

e−α e e−α

2

4π−η 2 log2 r b M2 2πα 2

r dr.

2

2

4π−η b a Setting p := 4π−η 2π M 2 and q := 2π M 2 , we conclude thanks to Lemma 2.8 since 0 < p, q < 2. It easy to see that (46) still holds if Aα and Bα are replaced by (1 + ε)Aα and (1 + ε)Bα respectively, where ε 0 is small. Finally, for the first term in (44), we use Hölder inequality and (46). Consequently, we obtain

M lim sup hα L √ . α→∞ 4π In the general case we replace (43), by + (−1) terms and the rest of the proof is very similar. 2 This completes the proof of Proposition 1.18. 2 3. Qualitative study of nonlinear wave equation This section is devoted to the qualitative study of the solutions of the two-dimensional nonlinear Klein–Gordon equation u + u + f (u) = 0,

u : Rt × R2x −→ R,

(47)

where 2 f (u) = u e4πu − 1 . Exponential type nonlinearities have been considered in several physical models (see e.g. [23] on a model of self-trapped beams in plasma). For decreasing exponential nonlinearities, T. Cazenave in [6] proved global well-posedness together with scattering in the case of NLS. It is known (see [29,31]) that the Cauchy problem associated to Eq. (47) with Cauchy data small enough in H 1 × L2 is globally well posed. Moreover, subcritical, critical and supercritical regimes in the energy space are identified (see [18]). Global well-posedness is established in both subcritical and critical regimes while well-posedness fails to hold in the supercritical one (we refer to [16,18] for more details). Very recently, M. Struwe [37] has constructed global smooth solutions for the 2D energy critical wave equation with radially symmetric data. Although the techniques are different, this result might be seen as an analogue of Tao’s result [38] for the 3D energy supercritical wave equation. Let us emphasize that the solutions of the two-dimensional nonlinear Klein–Gordon equation formally satisfy the conservation of energy 2 2 1 e4πu(t)2 − 1 1 E(u, t) = ∂t u(t)L2 + ∇u(t)L2 + L 4π = E(u, 0) := E0 .

(48)

234


The notion of criticality here depends on the size of the initial energy3 E0 with respect to 1. This relies on the so-called Trudinger–Moser type inequalities stated in Proposition 1.4 (see [17] and references therein for more details). Let us now precise the notions of these regimes: Definition 3.1. The Cauchy problem associated to Eq. (47) with initial data (u0 , u1 ) ∈ H 1 (R2 ) × L2 (R2 ) is said to be subcritical if E0 < 1. It is said critical if E0 = 1 and supercritical if E0 > 1. It is then natural to investigate the feature of solutions of the two-dimensional nonlinear Klein– Gordon equation taking into account the different regimes, as in earlier works of P. Gérard [11] and H. Bahouri and P. Gérard [3]. The approach that we adopt here is the one introduced by P. Gérard in [11] which consists in comparing the evolution of oscillations and concentration effects displayed by sequences of solutions of the nonlinear Klein–Gordon equation (47) and solutions of the linear Klein–Gordon equation v + v = 0.

(49)

More precisely, let (ϕn , ψn ) be a sequence of data in H 1 × L2 supported in some fixed ball and satisfying ϕn 0

in H 1 ,

ψn 0 in L2 ,

(50)

n∈N

(51)

such that E n 1,

where E n stands for the energy of (ϕn , ψn ) given by E n = ψn 2L2 + ∇ϕn 2L2 +

1 e4πϕn2 − 1 1 L 4π

and let us consider (un ) and (vn ) the sequences of finite energy solutions of (47) and (49) such that (un , ∂t un )(0) = (vn , ∂t vn )(0) = (ϕn , ψn ). Arguing as in [11], we introduce the following definition. Definition 3.2. Let T be a positive time. We shall say that the sequence (un ) is linearizable on [0, T ], if sup Ec (un − vn , t) −→ 0 as n −→ ∞

t∈[0,T ]

3 This is in contrast with higher dimensions where the criticality depends on the nonlinearity.


235

where Ec (w, t) denotes the kinetic energy defined by: |∂t w|2 + |∇x w|2 + |w|2 (t, x) dx. Ec (w, t) = R2

The following results illustrate the critical feature of the condition E0 = 1. Theorem 3.3. Under the above notations, let us assume that lim supn→∞ E n < 1. Then, for every positive time T , the sequence (un ) is linearizable on [0, T ]. Remark 3.4. Let us recall that in the case of dimension d 3, the same kind of result holds. More precisely, P. Gérard proved in [11] that in the subcritical case, the nonlinearity does not induce any effect on the behavior of the solutions. In the critical case i.e. lim supn→∞ E n = 1, it turns out that a nonlinear effect can be produced and we have the following result: Theorem 3.5. Assume that lim supn→∞ E n = 1 and let T > 0. Then the sequence (un ) is linearizable on [0, T ] provided that the sequence (vn ) satisfies 1 lim sup vn L∞ ([0,T ];L) < √ . n→∞ 4π

(52)

Remark 3.6. In Theorem 3.5, we give a sufficient condition on the sequence (vn ) which ensures the linearizability of the sequence (un ). Similarly to higher dimensions, this condition concerns the solutions of linear Klein–Gordon equation. However, unlike in higher dimensions, we are not able to prove the converse, that is if the sequence (un ) is linearizable on [0, T ] then lim supn→∞ vn L∞ ([0,T ];L) < √1 . The main difficulty in our approach is that we do not know 4π whether 1 lim sup vn L∞ ([0,T ];L) = √ n→∞ 4π

and f (vn )L1 ([0,T ];L2 (R2 )) −→ 0.

Nevertheless, combining Theorem 1.16 and Proposition 1.18, we get the following complement to Theorem 3.5. Proposition 3.7. Under the above notations, let us assume that the sequence (un ) is radial and linearizable on [0, T ] with 1 E n −→ 1 and vn L∞ ([0,T ];L) −→ √ . 4π Then there exist a sequence (tn ) of [0, T ] and s0 > 0 such that |x| αn ψ( − log i) un (tn , x) = 2π αn ) + rn (x), rn H 1 → 0, √ ii) ψ(s) = √ss for 0 s s0 ; ψ(s) = s0 for s s0 , 0

(53)

236


iii) ∂t un (tn )L2 (R2 ) → 0, 2

iv) e4πun (tn ) − 1L1 (R2 ) → 0. Remark 3.8. It is not clear whether sequences (un ) satisfying hypothesis of Proposition 3.7 exist. Proof. The fact that vn belongs to C([0, T ]; L) ensures the existence of a sequence (tn ) of [0, T ] such that vn (tn )L → √1 . By linearization, we also have un (tn )L → √1 . Then 4π 4π properties iii) and iv) result from E n → 1 and Sobolev embedding (3). Now, the application of Theorem 1.16 and Proposition 1.18 to un (tn ) shows that un (tn ) has only one profile in its decomposition and the remainder term tends to 0 in H 1 (R2 ). In other words, un (tn , x) =

− log |x| αn ψ + rn (x), 2π αn

rn H 1 −→ 0.

(54)

|x| αn ψ( − log On the one hand, it is obvious that ψn L → √1 , where ψn (x) = 2π αn ). On the 4π other hand, thanks to estimate (23) we necessarily have ψ L2 = 1. Taking advantage of Proposition 1.15, we deduce that ψ 2 = max |ψ(s)| √ = 1. L s>0 s By continuity, there exists s0 > 0 such that ψ

L2

|ψ(s0 )| = √ = 1. s0

Therefore √

√ s0 = ψ(s0 ) s0

1/2 s0 2 ψ (t) dt . 0

Hence s0 +∞ 2 2 ψ (t) dt = 1. 1 ψ (t) dt 0

0

This implies that ψ = 0 on [s0 , +∞[ and then by continuity ψ(s) = the equality case of the Cauchy–Schwarz inequality

√ s0 for any s s0 . Finally,

s0

1/2 s0 √ ψ 2 (τ ) dτ , ψ (τ ) dτ = s0 0

leads to ψ(s) =

√s s0

0

for s s0 which ends the proof of Proposition 3.7.

2


237

Before going into the proofs of Theorems 3.3 and 3.5, let us recall some well-known and useful tools. The main basic tool that we shall deal with is Strichartz estimate. 3.1. Technical tools 3.1.1. Strichartz estimate Let us first begin by introducing the definition of admissible pairs. Definition 3.9. Let ρ ∈ R. We say that (q, r) ∈ [4, ∞] × [2, ∞] is a ρ-admissible pair if 1 2 + = ρ. q r

(55)

When ρ = 1, we shall say admissible instead of 1-admissible. For example, (4, ∞) is a 1/4-admissible pair, and for every 0 < ε 1/3, the couple (1 + 1/ε, 2(1 + ε)) is an admissible pair. The following Strichartz inequalities that can be for instance found in [30] will be of constant use in what follows. Proposition 3.10 (Strichartz estimate). Let ρ ∈ R, (q, r) be a ρ-admissible pair and T > 0. Then vLq ([0,T ];Bρ

r,2 (R

2 ))

∂t v(0, ·)L2 (R2 ) + v(0, ·)H 1 (R2 ) + v + vL1 ([0,T ];L2 (R2 )) ,

(56)

ρ

where Br,2 (R2 ) stands for the usual inhomegenous Besov space (see for example [7] or [34] for a detailed exposition on Besov spaces). Now, for any time slab I ⊂ R, we shall denote vST(I ) :=

sup (q,r) admissible

vLq (I ;B1

r,2 (R

2 ))

.

By interpolation argument, this Strichartz norm is equivalent to vL∞ (I ;H 1 (R2 )) + vL4 (I ;B1

8/3,2 (R

2 ))

.

As B1r,2 (R2 ) → Lp (R2 ) for all r p < ∞ (and r p ∞ if r > 2), it follows that vLq (I ;Lp ) vST(I ) ,

2 1 + 1. q p

(57)

Proposition 3.10 will often be combined with the following elementary bootstrap lemma. Lemma 3.11. Let X(t) be a nonnegative continuous function on [0, T ] such that, for every 0 t T, X(t) a + bX(t)θ ,

(58)

238


where a, b > 0 and θ > 1 are constants such that 1 1 a< 1− , θ (θ b)1/(θ−1)

X(0)

1 . (θ b)1/(θ−1)

(59)

Then, for every 0 t T , we have X(t)

θ a. θ −1

(60)

Proof. We sketch the proof for the convenience of the reader. The function f : x → bx θ − x + a is decreasing on [0, (θ b)1/(1−θ) ] and increasing on [(θ b)1/(1−θ) , ∞[. The first assumption in (59) implies that f ((θ b)1/(1−θ) ) < 0. As f (X(t)) 0, f (0) > 0 and X(0) (θ b)1/(1−θ) , we deduce by continuity (60). 2 3.1.2. Logarithmic inequalities It is well known that the space H 1 (R2 ) is not included in L∞ (R2 ). However, resorting to an interpolation argument, we can estimate the L∞ norm of functions in H 1 (R2 ), using a stronger norm but with a weaker growth (namely logarithmic). More precisely, we have the following logarithmic estimate which also holds in any bounded domain. Lemma 3.12 (Logarithmic inequality). (See [17, Theorem 1.3].) Let 0 < α < 1. For any real 1 , a constant Cλ exists such that for any function ϕ belonging to H01 (|x| < 1) ∩ number λ > 2πα C˙α (|x| < 1), we have ϕ2L∞

λ∇ϕ2L2

ϕC˙α , log Cλ + ∇ϕL2

(61)

where C˙α denotes the homogeneous Hölder space of regularity index α. We shall also need the following version of the above inequality which is available in the whole space. 1 and any 0 < μ 1, a Lemma 3.13. (See [17, Theorem 1.3].) Let 0 < α < 1. For any λ > 2πα 1 2 α 2 constant Cλ > 0 exists such that for any function u ∈ H (R ) ∩ C (R ), we have

8α μ−α uC α , u2L∞ λu2Hμ log Cλ + uHμ

(62)

where C α denotes the inhomogeneous Hölder space of regularity index α and Hμ the Sobolev space endowed with the norm u2Hμ := ∇u2L2 + μ2 u2L2 . 3.1.3. Convergence in measure Similarly to higher dimensions (see [11]), the concept of convergence in measure occurs in the process of the proof of Theorems 3.3 and 3.5. For the convenience of the reader, let us give an outline of this notion. In many cases, the convergence in Lebesgue space L1 is reduced to the convergence in measure.


239

Definition 3.14. Let Ω be a measurable subset of Rm and (un ) be a sequence of measurable functions on Ω. We say that the sequence (un ) converges in measure to u if, for every ε > 0,

y ∈ Ω; un (y) − u(y) ε −→ 0 as n −→ ∞, where |B| stands for the Lebesgue measure of a measurable set B ⊂ Rm . It is clear that the convergence in L1 implies the convergence in measure. The contrary is also true if we require the boundedness in some Lebesgue space Lq with q > 1. More precisely, we have the following well-known result. Proposition 3.15. Let Ω be a measurable subset of Rm with finite measure and let (un ) be a bounded sequence in Lq (Ω) for some q > 1. Then, the sequence (un ) converges to u in L1 (Ω) if, and only if, it converges to u in measure in Ω. Proof. The fact that the convergence in L1 implies the convergence in measure follows immediately from the following Tchebychev’s inequality

ε y ∈ Ω; un (y) − u(y) ε un − uL1 . To prove the converse, let us show first that u belongs to Lq (Ω). Since the sequence (un ) converges to u in measure, we get thanks to Egorov’s lemma, up to subsequence extraction un −→ u

a.e. in Ω.

The Fatou’s lemma and the boundedness of (un ) in Lq imply then

u(y) q dy lim inf

n→∞

Ω

un (y) q dy C.

Ω

According to Hölder inequality, we have for any fixed ε > 0 un − uL1 = {|un −u| 0 and E0 < 1, there exists a constant C(T , E0 ), such that every solution u of the nonlinear Klein–Gordon equation (47) of energy E(u) E0 , satisfies uL4 ([0,T ];C 1/4 ) C(T , E0 ).

(63)

Proof. By virtue of Strichartz estimate (56), we have 1/2

uL4 ([0,t];C 1/4 ) E0

+ f (u)L1 ([0,t];L2 (R2 )) .

To estimate f (u) in L1 ([0, t]; L2 (R2 )), let us apply Hölder inequality f (u)

L2

2 uL2+2/ε e4πu − 1L2(1+ε) ,

where ε > 0 is chosen small enough. This leads in view of Lemma A.1 to f (u)

L2

1 2 2 uH 1 e2πuL∞ e4π(1+ε)u − 1L2(1+ε) . 1

The logarithmic inequality (62) yields for any fixed λ > π2 , 2 uC 1/4 2πλE0 e2πuL∞ C + 1/2 E0 and Trudinger–Moser inequality implies that for ε > 0 small enough 4π(1+ε)u2 e − 1

L1

κ.

Plugging these estimates together, we obtain

t uC 1/4 θ C+ dτ 1+ 1/2 E0

1/2 uL4 ([0,t];C 1/4 ) E0

0

where θ := 2πλE0 . Since E0 < 1, we can choose λ > in time, we deduce that

2 π

such that θ < 4. Using Hölder inequality

1/2

uL4 ([0,t];C 1/4 ) E0

1/2

E0

θ −1/2 1 + t 1−θ/4 t 1/4 + E0 uL4 ([0,t];C 1/4 ) 1−θ

+ T + E0 2 t 1−θ/4 uθL4 ([0,t];C 1/4 ) .

In the case where θ > 1, we set tmax :=

1

CE02 1/2 E0

+T

4(θ−1) 4−θ

,


241

where C is some constant. Then we obtain the desired result on the interval [0, tmax ] by absorption argument (see Lemma 3.11). Finally, to get the general case we decompose [0, T ] = i=n−1 i=0 [ti , ti+1 ] such that ti+1 − ti tmax . Applying the Strichartz estimate on [ti , t] with t ti+1 and using the conservation of the energy, we deduce uL4 ([ti ,ti+1 ],C 1/4 ) C(T , E0 ), which yields the desired inequality. In the case where θ 1 we use a convexity argument and proceed exactly as above. Notice that similar argument was used in higher dimension (see [15,26]). 2 Remark 3.17. Let us emphasize that in the critical case (E0 = 1) with the additional assumption δ uL∞ ([0,T ];L) √ , 4π

(64)

for some δ < 1, the conclusion of Lemma 3.16 holds with a constant which depends also on δ. 2 The key point consists in estimating differently the term e4πu − 1L2(1+ε) . More precisely, taking advantage of (64) we write 4πu2 e − 1

L2(1+ε)

1 2π 2 2 e 1+ε uL∞ e4π(1+2ε)u − 1L2(1+ε) 1 1

2π

κ 2(1+ε) e 1+ε uL∞ , 2

which leads to the result along the same lines as above. Let us now go to the proof of Theorem 3.3. Denoting by wn = un − vn , we can easily verify that wn is the solution of the nonlinear wave equation wn + wn = −f (un ) with null Cauchy data. Under energy estimate, we obtain wn T f (un )L1 ([0,T ],L2 (R2 )) , def

where wn 2T = supt∈[0,T ] Ec (wn , t). Therefore, to prove that the sequence (un ) is linearizable on [0, T ], it suffices to establish that f (un )

L1 ([0,T ],L2 (R2 ))

−→ 0

as n −→ ∞.

Thanks to finite propagation speed, for any time t ∈ [0, T ], the sequence f (un (t, ·)) is uniformly supported in a compact subset K of R2 . So, to prove that the sequence (f (un )) converges strongly to 0 in L1 ([0, T ], L2 (R2 )), we shall follow the strategy of P. Gérard in [11] which is firstly to demonstrate that this sequence is bounded in L1+ ([0, T ], L2+ (R2 )), for some nonnegative , and secondly to prove that it converges to 0 in measure in [0, T ] × R2 .

242


Let us then begin by estimate f (un )L1+ ([0,T ],L2+ (R2 )) , for small enough. Since, by defi2

nition we have f (un ) = −un (e4πun − 1), straightforward computations imply that f (un )2+ 2+ L

Ce4π(1+)un L∞ (R2 ) 2

2 |un |2+ e4πun − 1 dx.

R2

In other respects, using the obvious estimate 2 sup x m e−γ x =

x0

m 2γ

m 2

m

e− 2 ,

we get, for any positive real η

2 |un |2+ e4πun − 1 dx Cη

R2

(4π+η)u2 n − 1 dx. e

R2

In conclusion f (un )2+ 2+ L

Cη e4π(1+)un L∞ (R2 ) 2

(4π+η)(1−ρ)2 ( un )2 1−ρ e − 1 dx.

R2

Thanks to Trudinger–Moser estimate (5), we obtain for η small enough f (un )2+ 2+ L

Cη e4π(1+)un L∞ un 2L2 2

(R2 )

Cη e4π(1+)un L∞ E n 2

Cη e4π(1+)un L∞ , 2

by energy estimate, using the fact that lim supn→∞ E n < 1 − ρ. Now, taking advantage of the logarithmic estimate (62), we get for any λ > μ1 e

4π(1+)un 2L∞

Cλ +

un

2 π

and any 0
0, we are then reduced as it is mentioned above to prove that the sequence (f (un )) converges to 0 in measure in [0, T ] × R2 . Thus, by definition we have to prove that for every > 0,

(t, x) ∈ [0, T ] × R2 , f (un ) −→ 0 as n −→ ∞. The function f being continuous at the origin with f (0) = 0, it suffices then to show that the sequence (un ) converges to 0 in measure. Using the fact that (un ) is supported in a fixed compact subset of [0, T ]×R2 , we are led thanks to Rellich’s theorem and Tchebychev’s inequality to prove that the sequence (un ) converges weakly to 0 in H 1 ([0, T ] × R2 ). Indeed, assume that the sequence (un ) converges weakly to 0 in H 1 ([0, T ] × R2 ), then by Rellich’s theorem (un ) converges strongly to 0 in L2 ([0, T ] × R2 ). The Tchebychev’s inequality

2 (t, x) ∈ [0, T ] × R2 , un (t, x) un 2L2

(66)

implies the desired result. Let u be a weak limit of a subsequence (un ). By virtue of Rellich’s theorem and Tchebychev’s inequality (66), the sequence (un ) converges to u in measure. This leads to the convergence in measure of the sequence f (un ) to f (u) under the continuity of the function f . Combining this information with the fact that (f (un )) is bounded in some Lq with q > 1 and is uniformly compactly supported, we infer by Proposition 3.14 that the convergence is also distributional and u is a solution of the nonlinear Klein–Gordon equation (47). Taking advantage of Lemma 3.16, the compactness of the support and estimate (65), we deduce that f (u) ∈ L1 ([0, T ], L2 (R2 )). This allows to apply energy method, and shows that the energy of u at time t equals the energy of the Cauchy data at t = 0, which is 0. Hence u ≡ 0 and the proof is complete. 3.3. Critical case Our purpose here is to prove Theorem 3.5. Let T > 0 and assume that 1 L := lim sup vn L∞ ([0,T ];L) < √ . n→∞ 4π As it is mentioned above, wn = un − vn , is the solution of the nonlinear wave equation wn + wn = −f (un ) with null Cauchy data. Under energy estimate, we have wn T C f (un )L1 ([0,T ],L2 (R2 )) ,

(67)

244

H. Bahouri et al. / Journal of Functional Analysis 260 (2011) 208–252 def

where wn 2T = supt∈[0,T ] Ec (wn , t). It suffices then to prove that f (un )

L1 ([0,T ],L2 (R2 ))

−→ 0,

as n −→ ∞.

The idea here is to split f (un ) as follows applying Taylor’s formula 1 f (un ) = f (vn + wn ) = f (vn ) + f (vn )wn + f (vn + θn wn )wn2 , 2 for some 0 θn 1. The Strichartz inequality (56) yields (with I = [0, T ]) wn ST(I ) f (vn )L1 ([0,T ];L2 (R2 )) + f (vn )wn L1 ([0,T ];L2 (R2 )) + f (vn + θn wn )w 2 1 2 2 n L ([0,T ];L (R ))

In + Jn + Kn .

(68)

The term In is the easiest term to treat. Indeed, by assumption (67) we have 1 , vn L∞ ([0,T ];L) √ 4π(1 + ε)

(69)

for some ε > 0 and n large enough. This leads by similar arguments to the ones used in the proof of the subcritical case 4π(1+3η)v 2 4π(1−η)vn 2L∞ f (vn )2+η n − 1 dx. e Ce L2+η (R2 ) R2

In view of (69) and the Logarithmic inequality, we obtain for 0 < η < f (vn )1+η 1+η L

([0,T ],L2+η (R2 ))

ε 4

and n large enough

θ 1 C(η, T ) T 4 + vn L4 ([0,T ],C 1/4 ) ,

2

) with θ = 4πλ(1−η and 0 < λ − π2 1. It follows by Strichartz estimate that (f (vn )) is bounded 2+η in L1+η ([0, T ]; L2+η (R2 )). Since vn solves the linear Klein–Gordon equation with Cauchy data weakly convergent to 0 in H 1 × L2 , we deduce that (vn ) converges weakly to 0 in H 1 ([0, T ] × R2 ). This implies that f (vn ) converges to 0 in measure. This finally leads, using Proposition 3.15, the fixed support property and interpolation argument, to the convergence of the sequence (f (vn )) to 0 in L1 ([0, T ]; L2 (R2 )). Concerning the second term Jn , we will show that

Jn εn wn ST(I ) , where εn → 0. Using Hölder inequality, we infer that Jn = f (vn )wn L1 ([0,T ];L2 (R2 )) wn

1+ η1

L

2+ η2

([0,T ];L

(70)

(R2 ))

f (vn )

L1+η ([0,T ];L2+2η (R2 ))

.


245

Arguing exactly in the same manner as for In , we prove that for η η0 small enough the sequence (f (vn )) is bounded in L1+η ([0, T ]; L2(1+η) (R2 )) and converges to 0 in measure which ensures its convergence to 0 in L1 ([0, T ]; L2 (R2 )). Hence the sequence (f (vn )) converges to 0 in L1+η ([0, T ]; L2+2η (R2 )), for η < η0 , by interpolation argument. This completes the proof of (70) under the Strichartz estimate (57). For the last (more difficult) term we will establish that Kn εn wn 2ST(I ) ,

εn −→ 0,

(71)

provided that √ 1 − L 4π lim sup wn L∞ ([0,T ];H 1 ) . 2 n→∞

(72)

By Hölder inequality, Strichartz estimate and convexity argument, we infer that Kn wn2

1+ η1

L

2+ η2

([0,T ];L

(R2 ))

f (vn + θn wn )

L1+η ([0,T ];L2+2η (R2 ))

wn 2ST(I ) f (vn )L1+η ([0,T ];L2+2η (R2 )) + f (un )L1+η ([0,T ];L2+2η (R2 )) . According to the previous step, we are then led to prove that for η small enough f (un )

L1+η ([0,T ];L2+2η (R2 ))

−→ 0.

(73)

Arguing exactly as in the subcritical case, it suffices to establish that the sequence (f (un )) is bounded in L1+η0 ([0, T ]; L2+2η0 (R2 )) for some η0 > 0. Let us first point out that the assumption (72) implies that lim sup un L∞ ([0,T ];L) lim sup vn L∞ ([0,T ];L) + lim sup wn L∞ ([0,T ];L) n→∞

n→∞

n→∞

1 L + √ wn L∞ ([0,T ];H 1 ) 4π 1 1 1 0. Assume that T ∗ < T and apply the same arguments as above, we deduce that Xn (T ∗ ) → 0. By continuity this implies that lim supn→∞ wn L∞ ([0,T ∗ +];H 1 ) ν for some small enough. Obviously, this contradicts the definition of T ∗ and hence T ∗ = T . Acknowledgment We are grateful to the anonymous referee for a careful reading of the manuscript and fruitful remarks and suggestions. Appendix A A.1. Some known results on Sobolev embedding Lemma A.1. H 1 (R2 ) is embedded into Lp (R2 ) for all 2 p < ∞ but not in L∞ (R2 ). Proof. Using Littlewood–Paley decomposition and Bernstein inequalities (see for instance [9]), we infer that vLp

j vLp ,

j −1

C

2

− 2j p j

2 j vL2 .

j −1

Taking advantage of Schwartz inequality, we deduce that vLp C

2

− 4j p

1 2

vH 1 Cp vH 1 ,

j −1

which achieves the proof of the embedding for 2 p < ∞. However, H 1 (R2 ) is not included in L∞ (R2 ). For the convenience, it suffices to consider the function u defined by u(x) = ϕ(x) log − log |x| for some smooth function ϕ supported in B(0, 1) with value 1 near 0.

2

It will be useful to notice, that in the radial case, we have the following estimate which implies the control of the L∞ -norm far away from the origin. 1 (R2 ) and 1 p < ∞. Then Lemma A.2. Let u ∈ Hrad p 2 p+2 u(x) Cp u p+2 2 Lp ∇uL2 , r 2+p


247

with r = |x|. In particular 1 1 u(x) C2 u 2 2 ∇u 2 2 C2 u 1 . H 1 1 L L r2 r2

(76)

Proof. By density, it suffices to consider smooth compactly supported functions. Let us then consider u(x) = ϕ(r), with ϕ ∈ D([0, ∞[). Obviously, we have ϕ(r)

p+2 =− 2

p 2 +1

∞

p

ϕ (s)ϕ 2 (s) ds.

r

Hence p ϕ(r) 2 +1 p + 2 2r

∞ p ϕ (s) ϕ(s) 2 s ds, r

This achieves the proof of the lemma.

p p+2 ∇uL2 uL2 p . 2r

2

Remark A.3. In the general case, the embedding of H 1 (R2 ) into Lp (R2 ) is not compact. This observation can be illustrated by the following example: un (x) = ϕ(x + xn ) with 0 = ϕ ∈ D and |xn | → ∞. However, by virtue of Rellich–Kondrachov’s theorem, this embedding is compact in the case of HK1 (R2 ) the subset of functions of H 1 (R2 ) supported in the compact K. Moreover, in the radial case, the following compactness result holds. 1 (R2 ) in Lp (R2 ) is compact. Lemma A.4. Let 2 < p < ∞. The embedding Hrad

Proof. The proof is quite standard and can be found in many references (see for example [4, 1 (R2 ) 20,35]). We sketch it here for the sake of completeness. For (un ) being a sequence in Hrad 1 2 which converges weakly to u ∈ Hrad (R ), let us set vn := un − u. The problem is then reduced to the proof of the fact that vn Lp tends to zero. On the one hand, using the above lemma, we get for any R > 0, p−2 vn (x) p dx = vn (x) p−2 vn (x) 2 dx CR − 2 . |x|>R

|x|>R

On the other hand, we know by Rellich–Kondrachov’s theorem that the injection H 1 (|x| R) into Lp (|x| R) is compact. This ends the proof. 2 1 (R2 ) is not compactly embedded in L2 (R2 ). To see this, it suffices to consider Remark A.5. Hrad x 2

the family un (x) = α1n e−| αn | where (αn ) is a sequence of nonnegative real numbers tending to infinity. One can easily show that (un ) is bounded in H 1 but cannot have a subsequence converging strongly in L2 .

248


A.2. Some additional properties on Orlicz spaces Here we recall some well-known properties of Orlicz spaces. For a complete presentation and more details, we refer the reader to [32]. The first result that we state here deals with the connection between Orlicz spaces and Lebesgue spaces L1 and L∞ . Proposition A.6. We have a) (Lφ , · Lφ ) is a Banach space. b) L1 ∩ L∞ ⊂ Lφ ⊂ L1 + L∞ . c) If T : L1 → L1 with norm M1 and T : L∞ → L∞ with norm M∞ , then T : Lφ → Lφ with norm C(φ) sup(M1 , M∞ ). The following result concerns the behavior of Orlicz norm against convergence of sequences. Lemma A.7. We have the following properties a) Lower semi-continuity: un −→ u a.e.

⇒

uL lim inf un L .

b) Monotonicity: |u1 | |u2 |

a.e.

⇒

u1 L u2 L .

⇒

un L uL .

c) Strong Fatou property: 0 un u

a.e.

Let us now stress that besides the topology induced by its norm, the Orlicz space L is equipped with one other topology, namely the mean topology. More precisely, Definition A.8. A sequence (un ) in L is said to be mean (or modular) convergent to u ∈ L, if φ(un − u) dx −→ 0. It is said strongly (or norm) convergent to u ∈ L, if un − uL −→ 0. Clearly there is no equivalence between these convergence notions. Precisely, the strong convergence implies the modular convergence but the converse is false as shown by taking the Lions’ functions fα .


249

To end this subsection, let us mention that our Orlicz space L behaves like L2 for functions in H 1 ∩ L∞ . Proposition A.9. For every μ > 0 and every function u in H 1 ∩ L∞ , we have u2 ∞ L 2μ2

1 e √ uL2 uL μ + √ uL2 . κ κ

(77)

Proof. The left hand side of (77) is obvious. The second inequality follows immediately from the following simple observation u2

L∞ |u(x)|2 e 2μ2 λ μ + √ uL2 ⊂ λ > 0; e λ2 − 1 dx κ . κ u2 ∞ L 2μ2

Indeed, assuming λ μ +

e √

κ

, we get

|u(x)|2 e λ2 − 1 dx

e

2 |u(x)|2 |u(x)| e λ2 dx 2 λ u2 ∞ L μ2

λ2

κ.

u2L2 2

A.3. BMO and L Now, we shall discuss the connection between the Orlicz space L and BMO. At first, let us recall the following well-known embeddings H 1 → BMO ∩ L2 ,

L∞ → BMO → B0∞,∞ ,

!

H 1 → L →

2p 0. Then there exists a non-trivial solution ψ ∈ C 1 (M, S(M)) to the equation 2

Dψ = λψ + |ψ| m−1 ψ

on M.

For the case λ = 0, in [4], Ammann obtained an existence criterion which is similar to the condition given by Aubin [8] in the resolution of the Yamabe problem. However, such a condition is only verified for some special cases and general existence result is still lacking (cf. [7,23]). Ammann’s result is recovered from our result: Ammann’s solution corresponds to the mountain pass critical point of the dual action and such a critical point exists if the condition of Ammann is satisfied. See also [28], where Raulot proved the existence of a solution to the equation Dψ = 2 H (x)|ψ| m−1 ψ when D is invertible and a certain condition is satisfied for H . Both of the proofs of Ammann [4] and Raulot [28] rely on a subcritical approximation argument which is similar to Yamabe [33] and Aubin [8]. This paper is organized as follows: In Section 2, we give some preliminary materials: a basis of spin geometry and Dirac operators on spin manifolds. We also introduce H 1/2 -spinors in this section. In Section 3, we introduce the dual action L∗λ for the functional Lλ and show that there exists a one-to-one correspondence between the critical points of Lλ on H 1/2 (M, S(M)) + and the critical points of L∗λ on L2 (M, S(M)). In Section 4, we prove an energy estimate ∗ for non-trivial solutions to Dψ = |ψ|2 −2 ψ in Rm . More precisely, for any non-trivial solu1 m n 1/2 m m ( 2 ) ωm , where L0 (ψ) = tion ψ ∈ D (R , S(R )) to that equation, we prove L0 (ψ) 2m ∗ 1 m−1 2 m 2 Rm ψ, Dψ dx − 2m Rm |ψ| dx and ωm is the volume of S . In Section 5, we study compactness properties of the functionals Lλ and L∗λ . In particular, we prove that Lλ and L∗λ satisfy 1 m m ( 2 ) ωm . We also prove a global the Palais–Smale compactness condition below the level 2m compactness theorem for the Palais–Smale sequences which will be useful for the future development in this direction of research. In Section 6, we study the mountain pass critical level 1 m m ( 2 ) ωm if λ > 0 (λ ∈ / Spec(D)) and for L∗λ in detail. It is shown that it is strictly smaller than 2m m 4. This implies the existence of a non-trivial critical point and proves our main theorem. In ∗ Appendix A, we prove that any L2 -weak solution to Eq. (1.1) is C 1,α for some 0 < α < 1. This extends the previously known regularity results for weak solutions to nonlinear Dirac equations (cf. [15,32]) and it seems independent of interest.

256

T. Isobe / Journal of Functional Analysis 260 (2011) 253–307

2. Preliminaries 2.1. Spin structures on manifolds and Dirac operators We collect here some basic definitions and facts about spin structures on manifolds and Dirac operators. Since our purpose is only to introduce notations which are used throughout this paper, we do not enter this subject in detail. For more detailed exposition, please consult [18,25]. Let M be an m-dimensional oriented Riemannian manifold. We henceforth assume that m 2. Since M is orientable, the tangent bundle T M admits an SO(m)-structure: it can be defined by an open covering {Uα } of M and transition maps gαβ : Uαβ := Uα ∩ Uβ → SO(m) satisfying the cocycle condition: gαβ · gβγ · gγ α = 1 in Uαβγ := Uα ∩ Uβ ∩ Uγ , where 1 ∈ SO(m) is the identity. Recall that SO(m) is not simply connected (indeed, π1 (SO(2)) = Z and π1 (SO(m)) = Z2 for m 3.) Thus there exists the universal covering ρ : Spin(m) → SO(m) for the case m 3 and the double covering ρ : Spin(2) ∼ = S 1 z → z2 ∈ S 1 ∼ = SO(2) for the case m = 2. The manifold M is said to possess a spin structure if there exist smooth maps g˜ αβ : Uαβ → Spin(m) satisfying the cocycle condition g˜ αβ · g˜ βγ · g˜ γ α = 1˜ in Uαβγ , where 1˜ is the identity element of Spin(m), and ρ(g˜ αβ ) = gαβ for all α, β. A pair of manifold and its spin structure is called a spin manifold. There is a topological obstruction for the existence of a spin structure, namely, the vanishing of the second Stiefel–Whitney class w2 (M) ∈ H 2 (M; Z2 ). Moreover, there may be many different spin structures on the same manifold. For these and more, please consult [18,25]. For a spin manifold M, {g˜ αβ } defines a principal Spin(m)-bundle which we denote by PSpin (M). It is a double covering of the oriented frame bundle PSO (M) of M whose restriction to each fiber is ρ : Spin(m) → SO(m). We can regard PSO (M) as the bundle associated to PSpin (M) via ρ : Spin(m) → SO(m). In order to introduce the spinor bundle, we first assume that m is even. Recall that the Clifford algebra Clm is the associative R-algebra with unit, generated by Rm subject to the relations uv + vu = −2(u, v) for u, v ∈ Rm ((u, v) is the Euclidean inner product of u and v in Rm ). Spin(m) is a group generated by even number of unit vectors in Rm . There exists a complex Clm -module Sm such that Clm := Clm ⊗ C ∼ = EndC (Sm ) as C-algebras. Sm is the unique (up to isomorphism) irreducible complex Clm -module, usually called the spinor module. The isomorphism Clm ∼ = EndC (Sm ) induces the representation (unique up to isomorphism) σ : Spin(m) → End(Sm ), the spinor representation. On the other hand, the orientation on Rm even − induces a Z2 -grading on Sm ; Sm = S+ m ⊕ Sm , see [18,25] for the details. Since Spin(m) ⊂ Clm m (the subalgebra of Clm generated by the multiples of even number of vectors in R ), each of the spinor spaces S± m is a representation space for Spin(m). They are in fact irreducible, non-isomorphic complex Spin(m)-modules. They are called positive/negative complex spin representations and we denote them as σ ± : Spin(m) → End(S± m ). Associated to these, we obtain Hermitian vector bundles: S(M) := PSpin (M) ×σ Sm , S± (M) := PSpin (M) ×σ ± S± m. These are called complex (positive/negative) spinor bundles. These are Dirac bundles in the sense that (i) X ·Y ·ψ +Y ·X ·ψ = −2g(X, Y )ψ for any X, Y ∈ C ∞ (M, T M) and ψ ∈ C ∞ (M, S(M)), (ii) (X · ψ1 , ψ2 ) = −(ψ1 , X · ψ2 ) for X ∈ C ∞ (M, T M) and ψ1 , ψ2 ∈ C ∞ (M, S(M)) ((·,·) is the Hermitian metric on S(M)), (iii) (·,·) is metric, i.e., X(ψ1 , ψ2 ) = (∇X ψ1 , ψ2 ) + (ψ1 , ∇X ψ2 )


257

for any X ∈ C ∞ (M, T M), ψ1 , ψ2 ∈ C ∞ (M, S(M)), where ∇ is a connection on S(M) induced naturally from the Levi-Civita connection on T M (cf. [18,25]) which we also call the Levi-Civita connection on S(M) and (iv) ∇X (Y · ψ) = (∇X Y ) · ψ + Y · ∇X ψ. The associated metric on these bundles (i.e., the real part of the Hermitian metric) is denoted by ·,·. 0 1 For odd m, we first observe that there is an isomorphism Clm ∼ = Cleven m+1 defined by x + x → odd 1 x 0 + e0 · x 1 , where x 0 ∈ Cleven m , x ∈ Clm and {e0 , e1 , . . . , em } and {e1 , . . . , em } are orthonormal m+1 m and R , respectively. (Thus we shall regard Rm as a subspace of Rm+1 .) We then bases of R have the isomorphism − even ∼ (Sm+1 ) ∼ Clm ∼ = End S+ = Cleven m+1 = End m+1 ⊕ End Sm+1 . Thus both of S± m+1 are representation spaces of Clm . In fact, they are both irreducible and also become representation spaces of Spin(m). It is known that they are irreducible but isomorphic as ∼ − Spin(m)-modules. We denote Sm ∼ = S+ m+1 = Sm+1 and call it as the complex spinor representation. Denoting by σ : Spin(m) → Sm the representation so obtained, as in the even case, we form the spinor bundle S(M) as S(M) := PSpin (M) ×σ Sm . As in the even case, it is also a Dirac bundle. The sections of the spinor bundle S(M) are simply called spinors on M. The Dirac operator D acts on spinors on M, D : C ∞ (M, S(M)) → C ∞ (M, S(M)), and is defined by ∇ D = c ◦ ∇ : C ∞ M, S(M) − → C ∞ M, T ∗ M ⊗ S(M) c ∞ ∼ → C M, S(M) , = C ∞ M, T M ⊗ S(M) − where c denotes the Clifford multiplication c : T M ⊗ S(M) X ⊗ ψ → X · ψ ∈ S(M) and we have used the identification T ∗ M ∼ = T M by the metric g. ∞ S∓ (M)), where For the even dimensional case, we have D± : C ∞ (M, S± (M)) O→D−C (M, ± ∞ ± ∞ : C (M, S+ (M)) ⊕ D is the restriction of D to C (M, S (M)) and D = D+ O C ∞ (M, S− (M)) → C ∞ (M, S+ (M)) ⊕ C ∞ (M, S− (M)). 2.2. H 1/2 -spinors To treat Eq. (1.1) from a variational point of view, it is necessary to give a suitable functional analytic framework. A suitable function space to work with the functional Lλ is the Sobolev space H 1/2 (M, S(M)) of H 1/2 -spinors which we now define. Recall that the Dirac operator D on a compact spin manifold M is essentially self-adjoint in L2 (M, S(M)) and has compact resolvents (see [18,25]). In particular, there exists a complete orthonormal basis ψ1 , ψ2 , . . . of the Hilbert space L2 (M, S(M)) consisting of the eigenspinors of the Dirac operator D: Dψk = λk ψk . Moreover, |λk | → ∞ as k → ∞. For s 0, we define the (unbounded) operator |D|s : L2 (M, S(M)) → L2 (M, S(M)) by |D|s ψ =

∞ k=1

where ψ =

∞

k=1 ak ψk

∈ L2 (M, S(M)).

|λk |s ak ψk ,

258


s s The domain by H s (M, S(M)). Thus ψ = ∞ k=1 ak ψk ∈ H (M, S(M)) if ∞of |D| 2sis denoted and only if k=1 |λk | |ak |2 < ∞. H s (M, S(M)) coincides with the usual L2 -Sobolev space of order s, W s,2 (M, S(M)) (cf. [1,4]). For s < 0, H s (M, S(M)) is defined as the dual space of H −s (M, S(M)). For s > 0, the inner product on H s (M, S(M)) is defined by (ψ, ϕ)s,2 := |D|s ψ, |D|s ϕ 2 + (ψ, ϕ)2 , where (ψ, ϕ)2 = M ψ, ϕ dvolg is the L2 -inner product on spinors. 1/2 We denote by ψs,2 = (ψ, ψ)s,2 and ψp = ( M |ψ|p dvolg )1/p (for p 1) the H s -norm p of ψ ∈ H s (M, S(M)) and of ψ ∈ Lp (M, S(M)), respectively. We sometimes use thepL -norm 1/p the notation ϕp,B = ( B |ϕ| dvolg ) . In this paper, we are mainly concerned with the Sobolev space H 1/2 (M, S(M)) because it is the largest Sobolev space on which the integral M ψ, Dψ dvolg is well defined. By the Sobolev embedding theorem (cf. [1]), we have a continuous embedding H 1/2 (M, 2m S(M)) ⊂ Lp (M, S(M)) for 1 p m−1 =: 2∗ . Moreover, it is compact for 1 p < 2∗ . From this, the functional Lλ defined in (1.2) is well defined on H 1/2 (M, S(M)). We also introduce a similar function space which will be used in this paper. We denote by D1/2 (M, S(M)) the set of all spinors ψ on M such that ψ1/2,2 := |D|1/2 ψ2 + ψ 2m < ∞. m−1 In this definition, it is not necessary that M is compact. We use that space for both compact manifolds and M = Rm . For the latter case, |D|1/2 is defined by the Fourier transformation (ξ ) and |D|1/2 ψ2 = |ξ |1/2 ψ (ξ )2 . For a compact manifold M, as |D |1/2 ψ(ξ ) = |ξ |1/2 ψ 1/2 D (M, S(M)) coincides with H 1/2 (M, S(M)). The dual space of D1/2 (M, S(M)) is denoted by D−1/2 (M, S(M)). It is easy to check that Lλ is C 1 on H 1/2 (M, S(M)) and if ψ ∈ H 1/2 (M, S(M)) is a critical point of Lλ , i.e., dLλ (ψ) = 0, then ψ is a weak solution to (1.1). In fact, we have

1 dLλ (ψ), ϕ = 2

M

1 2

−λ =

ϕ, Dψ dvolg M

ϕ, ψ dvolg − M

∗ −2

|ψ|2

ϕ, ψ dvolg

M

ϕ, Dψ dvolg − λ M

ψ, Dϕ dvolg +

ϕ, ψ dvolg − M

∗ −2

|ψ|2

ϕ, ψ dvolg

M

for ϕ ∈ H 1/2 (M, S(M)), where the self-adjointness of D is used. ∗ As we shall show in Appendix A, any H 1/2 -spinor (more generally, L2 -spinor) which weakly satisfies Eq. (1.1) is C 1,α for some 0 < α < 1. Thus to prove the existence of solutions to (1.1), it suffices to prove the existence of critical points of Lλ in H 1/2 (M, S(M)). Before ending this section, we introduce some more notations which will be frequently used throughout this paper. The complete orthonormal basis {ψk } of L2 (M, S(M)) consisting of the + ∞ 0 κ eigenspinors of D is decomposed into three parts: {ψk } = {ψk− }∞ k=1 ∪ {ψk }k=1 ∪ {ψk }k=1 , where − − − − − 0 the eigenvalue of ψk is negative; Dψk = λk ψk with λk < 0, the eigenvalue of ψk is 0; Dψk0 = 0


259

+ + and the eigenvalue of ψk+ is positive; Dψk+ = λ+ k ψk with λk > 0 and κ = dim ker D < ∞. By the elliptic regularity, we have ψk ∈ C ∞ (M) for any eigenspinor ψk . We also set

∞

κ

∞

H− := span ψk− k=1 , H0 := span ψk0 k=1

and H+ := span ψk+ k=1 ,

where the closure is taken in the H 1/2 -topology. We have the orthogonal decomposition of the Hilbert space H 1/2 (M, S(M)): H 1/2 M, S(M) = H− ⊕ H0 ⊕ H+ .

(2.1)

3. The dual action We assume λ ∈ / Spec(D) throughout this section. For such a case, we have isomorphisms

D − λ : H 1/2 M, S(M) → H −1/2 M, S(M)

(3.1)

and +

D − λ : W 1,2

+ M, S(M) → L2 M, S(M) ,

(3.2)

2m 2m where 2+ = m+1 is the dual exponent of the critical exponent 2∗ = m−1 . We denote by Aλ and Bλ the inverses of these operators, respectively. Thus we have

Aλ := (D − λ)−1 : H −1/2 M, S(M) → H 1/2 M, S(M)

(3.3)

+ + Bλ := (D − λ)−1 : L2 M, S(M) → W 1,2 M, S(M) .

(3.4)

∗ i : H 1/2 M, S(M) → L2 M, S(M)

(3.5)

+ j : W 1,2 M, S(M) → H 1/2 M, S(M)

(3.6)

and

We denote by

and

the Sobolev embeddings.

260


Consider the following sequences of maps: i∗ Aλ i 2∗ + Kλ : L2 M, S(M) − → H −1/2 M, S(M) −−→ H 1/2 M, S(M) − → L M, S(M)

(3.7)

and Bλ j + + L2 M, S(M) −→ W 1,2 M, S(M) − → H 1/2 M, S(M) ,

(3.8)

+

where i ∗ : L2 (M, S(M)) → H −1/2 (M, S(M)) is the dual of i. We then have a relation Aλ ◦ i ∗ = j ◦ Bλ .

(3.9)

By the self-adjointness of D, we also have Kλ∗ = Kλ .

(3.10)

The functional Lλ is defined precisely as Lλ (ψ) =

1 (D − λ)ψ, ψ H −1/2 ×H 1/2 − H i(ψ) 2

for ψ ∈ H 1/2 (M, S(M)), where ·,·H −1/2 ×H 1/2 is the duality paring between H −1/2 (M, S(M)) ∗ and H 1/2 (M, S(M)) and H is a functional on L2 (M, S(M)) defined by H(ψ) =

1 2∗

∗

|ψ|2 dvolg M

∗

for ψ ∈ L2 (M, S(M)). By the Sobolev embedding, it is easy to see that Lλ is C 1 on H 1/2 (M, S(M)) and dLλ (ψ) = (D − λ)ψ − i ∗ dH i(ψ) ∈ H −1/2 M, S(M)

(3.11)

for ψ ∈ H 1/2 (M, S(M)). In order to define the dual functional for Lλ , we first consider the Legendre transformation + of H (see [30]). It is defined as a functional H∗ on L2 (M, S(M)) by ∗ H∗ (ϕ) = max ψ, ϕL2∗ ×L2+ − H(ψ): ψ ∈ L2 M, S(M) 1 + = + |ϕ|2 dvolg , 2 M ∗

+

where ·,·L2∗ ×L2+ is the duality paring between L2 and L2 . We see that dH∗ is the inverse of dH: dH ◦ dH∗ = 1L2+ ,

dH∗ ◦ dH = 1L2∗ .

(3.12)


261

The dual functional L∗λ is defined as 1 L∗λ (ϕ) = H∗ (ϕ) − Kλ ϕ, ϕL2∗ ×L2+ 2 1 1 2+ = + |ϕ| dvolg − Kλ ϕ, ϕ dvolg 2 2 M

M

+

for ϕ ∈ L2 (M, S(M)). + It is also easy to see that L∗λ is C 1 on L2 (M, S(M)). We have the following: Lemma 3.1. There is a one-to-one correspondence between the critical points of Lλ in + H 1/2 (M, S(M)) and the critical points of L∗λ in L2 (M, S(M)). Proof. Suppose ψ ∈ H 1/2 (M, S(M)) is a critical point of Lλ in H 1/2 (M, S(M)). By (3.11), we + have (D − λ)ψ = i ∗ dH(i(ψ)). Define ϕ := dH(i(ψ)) ∈ L2 (M, S(M)). We then have (D − λ)ψ = i ∗ ϕ. From this, we have ψ = Aλ ◦ i ∗ (ϕ) and i(ψ) = i ◦ Aλ ◦ i ∗ (ϕ) = Kλ ϕ.

(3.13)

On the other hand, by (3.12), we have i(ψ) = dH∗ (ϕ).

(3.14)

Combining (3.13) and (3.14), we have dH∗ (ϕ) − Kλ ϕ = 0. This is equivalent to dL∗λ (ϕ) = 0, i.e., ϕ is a critical point of L∗λ . + Conversely, suppose ϕ ∈ L2 (M, S(M)) is a critical point of L∗λ . Define ψ = Aλ ◦ i ∗ (ϕ) ∈ H 1/2 (M, S(M)). By the criticality of ϕ, we have dH∗ (ϕ) − Kλ ϕ = 0. Combining this with (3.12), we obtain ϕ = dH ◦ Kλ (ϕ) = dH ◦ i ◦ Aλ ◦ i ∗ (ϕ) = dH i(ψ) .

(3.15)

From (3.15), we have i ∗ (ϕ) = i ∗ ◦ dH(i(ψ)) and (D − λ)ψ = i ∗ ◦ dH(i(ψ)). But the last condition is equivalent to dLλ (ψ) = 0, i.e., ψ is a critical point of Lλ . This completes the proof. 2 More generally, there is a one-to-one correspondence between Palais–Smale sequences for + Lλ on H 1/2 (M, S(M)) and Palais–Smale sequences for L∗λ on L2 (M, S(M)). We first recall the definition of the Palais–Smale sequence: Definition 3.1. Let L be a C 1 -functional defined on a Banach space B. For c ∈ R, a sequence {xn } ⊂ B is a (PS)c -sequence if it satisfies L(xn ) → c,

dL(xn )

B∗

→ 0.

262


We say that L satisfies the (PS)c -condition if any (PS)c -sequence {xn } ⊂ B has a convergent subsequence. In order to establish a correspondence between the Palais–Smale sequences for Lλ and L∗λ , we need the following two lemmas: Lemma 3.2. For any c ∈ R, any (PS)c -sequence {ψn } ⊂ H 1/2 (M, S(M)) for Lλ is bounded. Proof. With respect to the spectral decomposition H 1/2 (M, S(M)) = H− ⊕ H0 ⊕ H+ (see (2.1)), for any ψ ∈ H 1/2 (M, S(M)), we write ψ = ψ − + ψ 0 + ψ + . Let {ψn } ⊂ H 1/2 (M, S(M)) be a (PS)c -sequence, i.e., it satisfies

dLλ (ψn )

Lλ (ψn ) → c,

−1/2,2

→ 0.

(3.16)

Thus, for large n, we have

C + ψn 1/2,2 2Lλ (ψn ) − dLλ (ψn ), ψn 1 2 ∗ ∗ |ψn |2 dx = ψn 22∗ . = 1− ∗ 2 m

(3.17)

M

We also have, for large n,

dLλ (ψn ), ψ + = ψ + , (D − λ)ψn dx − |ψn |2∗ −2 ψ + , ψn dx ψ + n n n n 1/2,2 . M

M

From this and the Hölder and Sobolev inequalities, we have

+

ψ , Dψn dx |λ| ψ + 2 dx + |ψn |2∗ −1 ψ + dx + ψ + n n n n 1/2,2 M

M

M

2

∗ |λ| ψn+ 2 + ψn 22∗ −1 ψn+ 2∗ + ψn+ 1/2,2

∗ Cψn 2∗ + Cψn 2∗ −1 ψ + + ψ + 2

2

n

1/2,2

n

1/2,2

.

(3.18)

−1 −1 > 0, we have On the other hand, for C+ = (1 + (λ+ 1) )

2

ψn+ , Dψn dx C+ ψn+ 1/2,2 .

(3.19)

M

From (3.17), (3.18) and (3.19), we obtain

+ 2

ψ n

1/2,2

2/2∗ 1−1/2∗ C 1 + ψn 1/2,2 + C 1 + ψn 1/2,2 ψn+ 1/2,2 .

(3.20)


263

Similarly, we have

− 2 2/2∗ 1−1/2∗

ψ −

ψ n 1/2,2 C 1 + ψn 1/2,2 + C 1 + ψn 1/2,2 n 1/2,2 .

(3.21)

On the other hand, on H0 , any two different norms are equivalent since dim H0 < ∞. Therefore, by (3.17), we have

0 2

ψ

n 1/2,2

2 2/2∗ C ψn0 2 Cψn 22 Cψn 22∗ C 1 + ψn 1/2,2 .

(3.22)

From (3.20), (3.21) and (3.22), we obtain 2/2∗ 1−1/2∗ ψn 21/2,2 C 1 + ψn 1/2,2 + C 1 + ψn 1/2,2 ψn 1/2,2 . Since 2∗ =

2m m−1

> 2, it follows from (3.23) that {ψn } is bounded in H 1/2 (M, S(M)).

(3.23) 2

Similarly, we have the boundedness of the Palais–Smale sequences for L∗λ : +

Lemma 3.3. For any c ∈ R, any (PS)c -sequence {ϕn } ⊂ L2 (M, S(M)) for L∗λ is bounded. +

Proof. Let {ϕn } ⊂ L2 (M, S(M)) be a (PS)c -sequence, i.e., it satisfies

∗

dL (ϕn )

L∗λ (ϕn ) → c,

λ

∗

L2

→ 0.

(3.24)

Since L∗λ (ϕn ) −

1 ∗ dLλ (ϕn ), ϕn = 2

1 1 − + 2 2

+

|ϕn |2 dx =

1 + ϕn 22+ , 2m

(3.25)

M

(3.24) implies 1 + ϕn 22+ C + ϕn 2+ . 2m Since 2+ =

2m m+1

+

> 1 for m 2, it follows from this that {ϕn } is bounded in L2 (M, S(M)).

2

By the above two lemmas, we have a one-to-one correspondence between Palais–Smale sequences for Lλ and L∗λ . Lemma 3.4. There exists a one-to-one correspondence between Palais–Smale sequences for Lλ + on H 1/2 (M, S(M)) and Palais–Smale sequences for L∗λ on L2 (M, S(M)). Moreover, for any c ∈ R, (PS)c -condition is satisfied for Lλ on H 1/2 (M, S(M)) if and only if (PS)c -condition is + satisfied for L∗λ on L2 (M, S(M)).

264


Proof. Let {ψn } ⊂ H 1/2 (M, S(M)) be a Palais–Smale sequence for Lλ . Thus we have

dLλ (ψn )

Lλ (ψn ) → c,

−1/2,2

→ 0.

(3.26)

The second condition of (3.26) is written as (D − λ)ψn − i ∗ ◦ dH i(ψn ) → 0 in H −1/2 M, S(M) .

(3.27)

+

We set ϕn = dH(i(ψn )) ∈ L2 (M, S(M)). By (3.27), we have ψn − Aλ i ∗ ϕn → 0 in H 1/2 M, S(M) and ∗ iψn − Kλ ϕn → 0 in L2 M, S(M) .

(3.28)

Since iψn = (dH)−1 (ϕn ) = dH∗ (ϕn ), (3.28) implies ∗ dH∗ (ϕn ) − Kλ ϕn → 0 in L2 M, S(M) .

(3.29)

But this is equivalent to dL∗λ (ϕ)2∗ → 0. One the other hand, we have L∗λ (ϕn ) = H∗ (ϕn ) −

1 2

Kλ ϕn , ϕn dvolg M

1 1 = + dH∗ (ϕn ), ϕn L2∗ ×L2+ − 2 2


1 ∗ 1 dH (ϕn ), ϕn L2∗ ×L2+ − dH∗ (ϕn ), ϕn + o(1) 2+ 2

1 ∗ dH (ϕn ), ϕn L2∗ ×L2+ + o(1) = 2m

1 iψn , dH(iψn ) L2∗ ×L2+ + o(1) = 2m 1 2∗ H(iψn ) + o(1) = H(iψn ) + o(1), = 2m m−1 =

(3.30)

where we have used the boundedness of {ϕn } (since {ψn } ⊂ H 1/2 (M, S(M)) is bounded by Lemma 3.2) and (3.29) in the third line and the homogeneity of H∗ and H in the second and the sixth lines.


265

Similarly, by the boundedness of {ψn }, (3.27) and the homogeneity of H, we have Lλ (ψn ) =

1 2

(D − λ)ψn , ψn dvolg − H(iψn )

M

1 = 2

i ∗ dH(iψn ), ψn dvolg − H(iψn ) + o(1)

M

=

1

+ ∗ − H(iψn ) + o(1) L2 ×L2 2 2∗ 1 = H(iψn ) − H(iψn ) + o(1) = H(iψn ) + o(1). 2 m−1

dH(iψn ), iψn

(3.31)

Combining (3.30) and (3.31), we have L∗λ (ϕn ) → c and {ϕn } is a (PS)c -sequence for L∗λ . + Assume conversely that {ϕn } ⊂ L2 (M, S(M)) is a (PS)c -sequence for L∗λ . Then we have L∗λ (ϕn ) → c,

dLλ 2∗ → 0.

(3.32)

As before, the second condition is equivalent to ∗ dH∗ (ϕn ) − i ◦ Aλ i ∗ ϕn → 0 in L2 M, S(M) .

(3.33)

∗

We set ψn = Aλ (i ∗ ϕn ). By (3.33), we have dH∗ (ϕn ) − iψn → 0 in L2 (M, S(M)). Operating + dH = (dH∗ )−1 on both sides, we have ϕn − dH(iψn ) → 0 in L2 (M, S(M)). This implies i ∗ ϕn − i ∗ ◦ dH(iψn ) → 0 in H −1/2 M, S(M) . But this is equivalent to (D − λ)ψn − i ∗ dH(iψn ) → 0 in H −1/2 (M, S(M)), i.e., dLλ (ψn )−1/2,2 → 0. We also have

1 Lλ (ψn ) = ψn , (D − λ)ψn dvolg − H(iψn ) 2 M

1 = 2

1 ψn , i ∗ ϕn dvolg − ∗ dH(iψn ), iψn L2+ ×L2∗ 2

M

1 = 2

iψn , ϕn dvolg −

1 ∗ i dH(iψn ), ψn H −1/2 ×H 1/2 ∗ 2

M

1 = 2

1 dH∗ (ϕn ), ϕn dvolg − ∗ (D − λ)ψn , ψn H −1/2 ×H 1/2 + o(1) 2

1 dH∗ (ϕn ), ϕn dvolg − ∗ i ∗ ϕn , ψn H −1/2 ×H 1/2 + o(1) 2

M

=

1 2

M

266


1 = 2

1 dH∗ (ϕn ), ϕn dvolg − ∗ ϕn , iψn L2+ ×L2∗ + o(1) 2

M

1 = 2

1 dH∗ (ϕn ), ϕn dvolg − ∗ ϕn , dH∗ (ϕn ) L2+ ×L2∗ + o(1) 2

M

=

1 H∗ (ϕn ) + o(1), m+1

(3.34)

where we have used the boundedness of {ψn } and {ϕn }, and the homogeneity of H and H∗ . Similarly, we have L∗λ (ϕn ) = H∗ (ϕn ) −

1 2


= H∗ (ϕn ) −

1 2

dH∗ (ϕn ), ϕn dvolg + o(1)

M

=

1 H∗ (ϕn ) + o(1). m+1

(3.35)

From (3.34) and (3.35), we have Lλ (ψn ) → c as n → ∞ and {ψn } is a (PS)c -sequence for Lλ . + Finally, assume Lλ satisfies the (PS)c -condition. Let {ϕn } ⊂ L2 (M, S(M)) be a (PS)c sequence for L∗λ . Then ψn := Aλ (i ∗ ϕn ) is a (PS)c -sequence by the above argument and has a + convergent subsequence by the assumption. Then, since ϕn = dH(iψn ) + o(1) in L2 (M, S(M)) by the above argument, {ϕn } also has a convergent subsequence, i.e., (PS)c -condition is satisfied for L∗λ . Conversely, assume L∗λ satisfies the (PS)c -condition. Let {ψn } be a (PS)c -sequence for Lλ . Then ϕn = dH(i(ψn )) is a (PS)c -sequence by the above argument and has a convergent sub+ sequence in L2 (M, S(M)) by the assumption. Since ψn = Aλ i ∗ ϕn + o(1) in H 1/2 (M, S(M)) by the above argument, {ψn } also has a convergent subsequence in H 1/2 (M, S(M)). Thus Lλ satisfies the (PS)c -condition. This completes the proof. 2 ∗ −2

4. Energy gap for solutions to Dψ = |ψ|2

ψ on R m

We consider weak solutions to the equation ∗ Dψ = |ψ|2 −2 ψ

on Rm

(4.1)

belonging to the class D1/2 (Rm , S(Rm )). These correspond “bubbles” for our variational problem. By the regularity result proved in Appendix A, these solutions are in fact C 1,α for some 0 < α < 1 and are classical solutions to (4.1). In this section, we shall prove the following proposition:


267

Proposition 4.1. Let ψ ∈ D1/2 (Rm , S(Rm )) be a non-trivial solution to Eq. (4.1). Then we have 1 m m LRm (ψ) ωm , 2m 2 where LRm (ψ) =

1 2 Rm ψ, Dψ dx

−

1 2∗

∗

Rm

|ψ|2 dx.

Before proving the proposition, we begin with some preliminary materials. First of all, we need to observe that Eq. (4.1) is interpreted as an equation on S m by a conformal change of the Euclidean metric gRm on Rm . To see this, we recall that Rm and S m \ {N } (N ∈ S m is the North pole) are conformally equivalent, i.e., denoting by π : S m \ {N } → Rm the stereographic projection from N , we have (π −1 )∗ gS m = f 2 gRm , where gS m is the round metric 2 on S m with constant sectional curvature 1 and f (x) = 1+r 2 (r = |x|). On the other hand, the ∗

equation Dψ = |ψ|2 −2 ψ on the manifold M is invariant under conformal change of the metric on M. To see this, let h = f 2 g for some positive function f on M. There is an isomorphism of vector bundles F : S(M, g) → S(M, h) (S(M, g) and S(M, h) are spinor bundles on M with respect to the metrics g and h, respectively) which is a fiberwise isometry such that (cf. [21])

Dh F (ϕ) = F f −

m+1 2 Dg

m−1 f 2 ϕ ,

(4.2)

where Dg and Dh are the Dirac operators on M with respect to the metrics g and h, respectively. ∗

m−1

Thus when ψ is a solution to the equation Dg ψ = |ψ|2 −2 ψ on (M, g), then ϕ := F (f − 2 ψ) ∗ also satisfies the same equation on (M, h): Dh ϕ = |ϕ|2 −2 ϕ on (M, h). Moreover, since the volume form dvolh on (M, h) is related to the one on (M, g) as dvolh = f m dvolg , we have

ψ, Dg ψ dvolg = M

ϕ, Dh ϕ dvolh ,

M

∗

|ψ|2 dvolg = M

∗

|ϕ|2 dvolh . M

In particular, the Lagrangian L0 is invariant under conformal change of the metric. m−1 Returning to our case M = Rm , if ψ satisfies (4.1), ϕ = F (f − 2 ψ) (f (x) = solution to ∗ DgS m ϕ = |ϕ|2 −2 ϕ

on S m \ {N }

(4.3)

2 ) 1+r 2

is a

(4.4)

∗ and S m |ϕ|2 dvolgS m < ∞. ∗ Since ϕ ∈ L2 (S m ), by [5, Theorem 5.1] (see also [4]), ϕ extends as a weak solution on S m . Then, by the regularity result proved in Appendix A (Theorem A.1), ϕ is in fact a C 1,α -solution on S m for some 0 < α < 1. Summing up the argument, we have proved that there exists a oneto-one correspondence between weak solutions to (4.1) in the class D1/2 (Rm , S(Rm )) and weak ∗ solutions to DgS m ϕ = |ϕ|2 −2 ϕ on S m in the class H 1/2 (S m , S(S m )). Now we give the proof of the proposition: Proof of Proposition we shall give the estimate of the ac 4.1. By the above observation, ∗ tion LS m (ϕ 0 ) = 12 S m ϕ 0 , DgS m ϕ 0 dvolgS m − 21∗ S m |ϕ 0 |2 dvolgS m for 0 = ϕ 0 ∈ H 1/2 (S m , ∗ S(S m )) which satisfies the equation DgS m ϕ 0 = |ϕ 0 |2 −2 ϕ 0 on S m .

268

T. Isobe / Journal of Functional Analysis 260 (2011) 253–307 ∗ −2

By DgS m ϕ 0 = |ϕ 0 |2

ϕ 0 , we have

LS m ϕ 0 =

1 1 − ∗ 2 2

0 2∗ ϕ dvolg

Sm

Sm

=

1 2m

0 2∗ ϕ dvolg m . S

(4.5)

Sm

Recall that the Bär–Hijazi–Lott invariant of (S m , [gS m ], σ0 ) (see [19, §8.5]) is defined by λmin S m , [gS m ], σ0 :=

1 inf λ1 (g) vol S m , g m ,

g∈[gS m ]

where [gS m ] is the conformal class of metrics on S m containing gS m , σ0 is the canonical spin structure on S m and λ1 (g) is the smallest positive eigenvalue of Dg . We have λmin (S m , [gS m ], σ0 ) = lows:

1

m m 2 ωm

(see [19, §8.5]). By [4,5], it is characterized variationally as fol 2m m+1 ( S m |ψ| m+1 dvolgS m ) m inf ψ∈imC ∞ DgS m \{0} | m ψ, D−1 gS m ψ dvolgS m | S 2m m+1 ( S m |DgS m ϕ| m+1 dvolgS m ) m , = inf | S m ϕ, DgS m ϕ dvolgS m | ϕ∈W 1,2m/m+1 , ϕ=0

λmin S m , [gS m ], σ0 =

(4.6) 2m

where imC ∞ DgS m is the image of DgS m : C ∞ (S m , S(S m )) → C ∞ (S m , S(S m )) and W 1, m+1 = 2m

2m

2m

W 1, m+1 (S m , S(S m )) is the Sobolev space of W 1, m+1 -spinors, i.e., spinors ϕ ∈ L m+1 (S m ) with 2m ∇ϕ ∈ L m+1 (S m ). The first characterization is used in [3,4], while the second one is used, for example, in [5,6,28]. Writting ψ = DgS m ϕ for ϕ ∈ C ∞ (S m , S(S m )) and noting that ψ = DgS m ϕ = 0 if and only if ϕ = 0 (this is because (S m , gS m ) has positive scalar curvature and ker DgS m = {0} 2m

via the Lichnerowicz formula), and the density of C ∞ (S m , S(S m )) in W 1, m+1 (S m , S(S m )), we have the equality in (4.6). ∗ Since DgS m ϕ 0 = |ϕ 0 |2 −2 ϕ 0 and ϕ 0 = 0, we thus have

0 2∗ ϕ dvolg

1

m

Sm

Sm

By (4.5)–(4.7), we have LS m (ϕ 0 )

2m m+1 ( S m DgS m ϕ 0 m+1 dvolgS m ) m m m1 ωm . = 2 | S m ϕ 0 , DgS m ϕ 0 dvolgS m | 1 m m 2m ( 2 ) ωm .

This completes the proof.

(4.7)

2

5. Local and global compactness properties of the Palais–Smale sequences for Lλ ∗

Due to the non-compactness of the Sobolev embeddings i : H 1/2 (M, S(M)) ⊂ L2 (M, S(M)) + and i ∗ : L2 (M, S(M)) ⊂ H −1/2 (M, S(M)), it is not difficult to see that Lλ and L∗λ do not satisfy + the Palais–Smale condition on H 1/2 (M, S(M)) and L2 (M, S(M)), respectively. However, they satisfy that condition for certain energy levels. In this section, we shall first prove the following:


269

1 m m Theorem 5.1. For any c < 2m ( 2 ) ωm , Lλ and L∗λ satisfy the (PS)c -condition on H 1/2 (M,S(M)) + 2 and L (M, S(M)), respectively.

In fact, our argument provides a more general global compactness result for general Palais– Smale sequences, see Theorem 5.2 below. By Lemma 3.4, it suffices to prove the assertion of Theorem 5.1 for Lλ . The proof of Theorem 5.1 is divided into several steps. Let {ψn } ⊂ H 1/2 (M, S(M)) be a (PS)c -sequence for Lλ on H 1/2 (M, S(M)), i.e., it satisfies

dLλ (ψn )

H −1/2

→ 0,

Lλ (ψn ) → c.

(5.1)

By Lemma 3.2, {ψn } ⊂ H 1/2 (M, S(M)) is bounded and, after taking a subsequence if necessary, we may assume that there exists ψ∞ ∈ H 1/2 (M, S(M)) such that weakly in H 1/2 (M),

(5.2)

strongly in Lp (M) for any 1 p < 2∗

(5.3)

ψn ψ∞ ψn → ψ∞ and

ψn → ψ∞

a.e. in M.

(5.4)

The limit spinor ψ∞ satisfies Eq. (1.1), that is, we have ∗ −2

Lemma 5.1. ψ∞ weakly satisfies Dψ∞ = λψ∞ + |ψ∞ |2

ψ∞ on M.

Proof. We need to show

ψ∞ , Dϕ dvolg − λ

M

ψ∞ , ϕ dvolg −

M

∗ −2

|ψ∞ |2

ψ∞ , ϕ dvolg = 0

(5.5)

M

for any ϕ ∈ C ∞ (M, S(M)). Let ϕ ∈ C ∞ (M, S(M)) be arbitrary. By (5.1), we have dLλ (ψn )(ϕ) =

ψn , Dϕ dvolg − λ

M

ψn , ϕ dvolg −

M

∗ −2

|ψn |2

ψn , ϕ dvolg

M

= o(1)

(5.6)

as n → ∞. By (5.2) and (5.3), it is easy to see that the first, the second ∗and the third integrals of (5.6) converge to M ψ∞ , Dϕ dvolg , M ψ∞ , ϕ dvolg and M |ψ∞ |2 −2 ψ∞ , ϕ dvolg , respectively. Therefore, (5.5) holds. 2 To investigate the detailed behavior of ψn as n → ∞, we set ϕn = ψn − ψ∞ . We have:

270


Lemma 5.2. {ϕn } satisfies the following: L0 (ϕn ) = Lλ (ψn ) − Lλ (ψ∞ ) + o(1)

(5.7)

and

dL0 (ϕn )

→0

H −1/2

(5.8)

as n → ∞, i.e., {ϕn } is a Palais–Smale sequence for L0 . Proof. We first prove (5.7). We have Lλ (ψn ) =

1 2

λ ϕn + ψ∞ , D(ϕn + ψ∞ ) dvolg − 2

M

1 − ∗ 2

|ϕn + ψ∞ |2 dvolg M

2∗

|ϕn + ψ∞ | dvolg . M

By the weak convergence ϕn 0 in H 1/2 (M, S(M)) and the convergence ϕn → 0 in L2 (M, S(M)) (see (5.2), (5.3)), we have Lλ (ψn ) =

1 2

ϕn , Dϕn dvolg + M

λ − 2

1 2

ψ∞ , Dψ∞ dvolg M

1 |ψ∞ |2 dvolg − ∗ 2

M

∗

|ϕn + ψ∞ |2 dvolg + o(1)

(5.9)

M

as n → ∞. To investigate the behavior of the last integral in (5.9) as n → ∞, we set ∗

∗

∗

Φn := |ϕn + ψ∞ |2 − |ϕn |2 − |ψ∞ |2 . It is easy to see that there exists C > 0 (independent of n) such that ∗ −1

|Φn | C|ϕn |2

∗ −1

|ψ∞ | + C|ϕn ||ψ∞ |2

(5.10)

.

On the other hand, by the Egorov theorem, for any > 0, there exists E ⊂ M such that meas(M \ E ) < and ϕn → 0 uniformly on E as n → ∞. By (5.10) and the Hölder inequality, we have:

|Φn | dvolg =

M

|Φn | dvolg +

E

M\E

|Φn | dvolg

|Φn | dvolg + C E

2∗

|ϕn | dvolg M\E

1 2+

2∗

|ψ∞ | dvolg M\E

1 2∗


+C

2∗

|ψ∞ | dvolg

1 2+

2∗

|ϕn | dvolg

M\E

271 1 2∗

(5.11)

.

M\E

The first integral in (5.11) converges to 0 as n → 0 and the remaining integrals go to 0 uniformly in n as → 0. Therefore, we have |Φn | dvolg → 0 M

as n → ∞. We thus have: 1 Lλ (ψn ) = 2

1 ϕn , Dϕn dvolg + 2

M

λ − 2

ψ∞ , Dψ∞ dvolg M

1 |ψ∞ |2 dvolg − ∗ 2

∗

|ϕn |2 dvolg −

M

1 2∗

M

∗

|ψ∞ |2 dvolg + o(1) M

= L0 (ϕn ) + Lλ (ψ∞ ) + o(1). This proves (5.7). To prove (5.8), let ϕ ∈ H 1/2 (M, S(M)) with ϕ1/2,2 1 be arbitrary. We have

dLλ (ψn ), ϕ =

ψn , Dϕ dvolg − λ

M

M

=

ψn , ϕ dvolg −

− =

∗ −2

|ϕn + ψ∞ |2

ϕn + ψ∞ , ϕ dvolg

M

−

ϕn + ψ∞ , ϕ dvolg M

ψ∞ , Dϕ dvolg + M

ψn , ϕ dvolg

M

ϕn + ψ∞ , Dϕ dvolg − λ M

∗ −2

|ψn |2

ϕn , Dϕ dvolg − λ

M ∗ −2

|ϕn + ψ∞ |2

ψ∞ , ϕ dvolg M

ϕn + ψ∞ , ϕ dvolg + O ϕn 2 ϕ2 .

(5.12)

M ∗

∗

To estimate the last integral in (5.12), we set Ψn := |ϕn + ψ∞ |2 −2 (ϕn + ψ∞ ) − |ϕn |2 −2 ϕn − ∗ |ψ∞ |2 −2 ψ∞ . It is easy to see that there exists a constant C > 0 (independent of n) such that ∗ −2

|Ψn | C|ϕn |2

∗ −2

|ψ∞ | + C|ϕn ||ψ∞ |2

.

Thus we have, by the Hölder inequality and the argument as given before

272


Ψn , ϕ dvolg C |ϕn |2∗ −2 |ψ∞ ||ϕ| dvolg + C |ψ∞ |2∗ −2 |ϕn ||ϕ| dvolg M

M

M

∗ ∗ C |ϕn |2 −2 |ψ∞ | 2+ ϕ2∗ + C |ψ∞ |2 −2 |ϕn | 2+ ϕ2∗

∗ ∗ C |ϕn |2 −2 |ψ∞ | 2+ + C |ψ∞ |2 −2 |ϕn | 2+

∗ ∗ C |ϕn |2 −2 |ψ∞ | L2+ (E ) + C |ψ∞ |2 −2 |ϕn | L2+ (E )

2∗ −2 2∗ −2

+ C |ϕn | |ψ∞ | L2+ (M\E ) + C |ψ∞ | |ϕn | L2+ (M\E

→0

)

as n → ∞ and → 0.

(5.13)

Therefore, we have

dLλ (ψn ), ϕ =

ϕn , Dϕ dvolg +

M

−

ψ∞ , Dϕ dvolg − λ

M 2∗ −2

|ϕn | M

M

ϕn , ϕ dvolg −

ψ∞ , ϕ dvolg

2∗ −2

|ψ∞ |

ψ∞ , ϕ dvolg + o(1)

M

= dL0 (ϕn ), ϕ + dLλ (ψ∞ ), ϕ + o(1)

= dL0 (ϕn ), ϕ + o(1),

(5.14)

where we have used Lemma 5.1, i.e., dLλ (ψ∞ ), ϕ = 0 in the last equality. (5.14) implies

dL0 (ϕn )

H −1/2

= dLλ (ψn ) H −1/2 + o(1) → 0

as n → ∞. This proves (5.8) and the proof is complete.

2

If {ϕn } has a subsequence which converges to 0 in H 1/2 (M, S(M)), the same subsequence of {ψn } converges to ψ∞ in H 1/2 (M, S(M)) and the Palais–Smale condition is satisfied. Thus we are interested in the case where any subsequence of {ϕn } does not converge to 0 in H 1/2 (M, S(M)). Henceforth, we assume that {ϕn } has that property. Let 0 > 0 be a positive number (which will be specified soon), and we define the singular set of {ϕn } as Σ = a ∈ M: lim lim

r↓0 n→∞ Br (a)

|ϕn | dvolg 0 . 2∗

Lemma 5.3. Let {ϕn } be as above. There exists 0 > 0 depending only on the geometry of M such that Σ = ∅. Proof. Assume contrary that Σ = ∅ for any choice of 0 > 0. Let 0 > 0 be small to be specified below. By our assumption, for any a ∈ M, there exist r0 > 0 and a subsequence of {ϕn } (for simplicity, it is also denoted by {ϕn }) such that


273

∗

|ϕn |2 dvolg < 0

(5.15)

B2r0 (a)

for n 1. By the Sobolev embedding, taking a further subsequence if necessary, we may assume that ϕn → 0 in Lp (M) for any 1 p < 2∗ . Take a cut off function ρ ∈ C ∞ (M) such that ρ = 1 on Br0 (a) and supp ρ ⊂ B2r0 (a). By (5.8), we have ∗ Dϕn = |ϕn |2 −2 ϕn + fn ,

fn → 0 in H −1/2 (M).

(5.16)

We thus have D(ρϕn ) = ρ Dϕn + ∇ρ · ϕn ∗ −2

= ρ|ϕn |2

ϕn + ρfn + ∇ρ · ϕn ,

(5.17)

where · denotes the Clifford multiplication. By (5.17) and the elliptic estimate, there exists C > 0 depending only on M such that there holds

ρϕn 1/2,2 C D(ρϕn ) −1/2,2 + Cρϕn 2

∗ C ρ|ϕn |2 −2 ϕn + ρfn + ∇ρ · ϕn −1/2,2 + Cρϕn 2

∗ C ρ|ϕn |2 −2 ϕn −1/2,2 + Cρfn + ∇ρ · ϕn −1/2,2 + Cρϕn 2 . +

(5.18)

∗

Here, by the Sobolev embeddings L2 (M) ⊂ H −1/2 (M), H 1/2 (M) ⊂ L2 (M) and the Hölder inequality, we have

ρ|ϕn |2∗ −2 ϕn

−1/2,2

∗ ∗ C ρ|ϕn |2 −2 ϕn 2+ C |ϕn |2 −2 m,B

2r0 (a)

2

C ϕn 2m−1 ∗ ,B 2r

(a) ρϕn 1/2,2 0

ρϕn 2∗

1

C 0m ρϕn 1/2,2 ,

(5.19)

where C , C > 0 depend only on M, and in the last inequality, we have used (5.15). 1

We choose 0 > 0 such that CC 0m < 12 . Notice that 0 > 0 can be chosen so that it depends only on the geometry of M. Then by (5.18) and (5.19), we have ρϕn 1/2,2 Cρfn + ∇ρ · ϕn −1/2,2 + Cρϕn 2 → 0

(5.20)

as n → ∞. Since a ∈ M is arbitrary and M is compact, we have ϕn → 0 in H 1/2 (M, S(M)). But this contradicts our assumption. The proof is complete. 2 In order to investigate the behavior of ϕn near points in Σ , we introduce the concentration function (cf. [26,27,11]) for t 0 as

274


∗

Qn (t) = sup

|ϕn |2 dvolg .

a∈M Bt (a)

Choose > 0 such that 3 < 0 , where 0 > 0 is as in Lemma 5.3. By Lemma 5.3, there exist Rn > 0, Rn ↓ 0 and an ∈ M such that

∗

Qn (Rn ) =

|ϕn |2 dvolg = .

BRn (an )

By taking a subsequence if necessary, we may assume that an → a ∈ M as n → ∞. We define ρn (x) = expan (Rn x) for x ∈ Rn such that Rn |x| ι(M), where ι(M) > 0 is the injectivity radius of M. Henceforth, we may assume without loss of generality ι(M) 3. Denoting BR0 = {x ∈ Rm : |x| R} ⊂ Rm , where | · | is the Euclidean norm in Rm , we have a conformal equivalence Rm ⊃ (BR0 , Rn−2 ρn∗ g) ∼ = (BRn R (an ), g) ⊂ M for large n. Define a metric gn on BR0 by gn = Rn−2 ρn∗ g. For any R > 0 we have gn → gRm in C ∞ (BR0 ) as n → ∞, where gRm is the Euclidean metric on Rm . Let (ρn )∗ : Sp (BR0 , gn ) → Sρn (p) (M, g) be the spinor identification map as constructed in [21, m−1

10] (see also Section 4 and Section 6.1). We define spinors φn on BR0 by φn = Rn 2 ρn∗ ϕn , where ρn∗ ϕn := (ρn )−1 ∗ ◦ ϕn ◦ ρn . By the transformation property of the Dirac operator under conformal change of the metric (see Section 4, (4.2)) and (4.3), we have m+1

Dgn φn = Rn 2 ρn∗ (Dg ϕn ),

φn , Dgn φn dvolgn = BR0

ϕn , Dg ϕn dvolg ,

(5.21) (5.22)

BRn R (an ) ∗

∗

|φn |2 dvolgn = BR0

|ϕn |2 dvolg .

(5.23)

BRn R (an )

We have, in particular, lim

n→∞ BR0

2∗

|φn | dvolgn sup n1

∗

|ϕn |2 dvolg < +∞

(5.24)

M

for any R > 0. Moreover, we have ∗ Lemma 5.4. Let {φn } be as above. Define fñ := Dgn φn − |φn |2 −2 φn . Then fñ satisfies fñ → 0 −1/2 in Hloc (Rm ) in the sense that for any R > 0, there holds

sup fñ , ϕ: ϕ ∈ H 1/2 Rm , S Rm , supp ϕ ⊂ BR0 , ϕ1/2,2 1 → 0 as n → ∞.


275

Proof. By the definition of φn , we have ∗ ∗ Dgn φn − |φn |2 −2 φn = Rn 2 ρn∗ Dg ϕn − |ϕn |2 −2 ϕn = Rn 2 ρn∗ fn , m+1

m+1

(5.25)

m+1

∗ where fn := Dg ϕn − |ϕn |2 −2 ϕn . Thus fñ = Rn 2 ρn∗ fn . Let ϕ ∈ H 1/2 (Rm , S(Rm )) be such that supp ϕ ⊂ BR0 and ϕ1/2,2 1. For large n, we have

fñ , ϕ =

fñ , ϕ dvolgn

B 0 −1 Rn

=

m+1

ρn∗ fn , Rn 2 ϕ dvolgn

B 0 −1 Rn

=

− m−1 2

ρn∗ fn , Rn

ϕ dvolρn∗ g

B 0 −1 Rn

=

−1 ∗

− m−1 2 ρn ϕ dvolg ,

fn , Rn

(5.26)

B1 (an )

where we have used gn = Rn−2 ρn∗ g and therefore dvolgn = Rn−m dvolρn∗ g in the third equality and −1 (ρn )−1 ∗ ϕ := (ρn )∗ ◦ ϕ ◦ ρn in the last equation. 1/2 m Since ϕ ∈ H (R , S(Rm )) satisfies supp ϕ ⊂ BR0 and ϕ1/2,2 1, there exists a constant − m−1 2

C > 0 independent of n and ϕ such that Rn implies the assertion of the lemma. 2

(ρn−1 )∗ ϕ1/2,2 C. By (5.8) and (5.26), this

We next prove: Lemma 5.5. If we choose > 0 as above suitably small, there exists φ∞ ∈ D1/2 (Rm , S(Rm )) 1/2 such that, after taking a subsequence if necessary, we have φn → φ∞ in Hloc (Rm , S(Rm )). Proof. Since {φn } is Hloc (Rm , S(Rm ))-bounded, i.e., for any β ∈ C0∞ (Rm ), {βφn } ⊂ H 1/2 (Rm , S(Rm )) is bounded, by the Sobolev embedding, we may assume, after taking a subsequence 1/2 p if necessary, φn φ∞ weakly in Hloc (Rm , S(Rm )) and φn → φ∞ in Lloc (Rm ) for any 1 ∗ p < 2∗ . It is also easy to see that φ∞ ∈ L2 (Rm ) (cf. (5.24)) and satisfies 1/2

∗ DgRm φ∞ = |φ∞ |2 −2 φ∞

in Rm .

(5.27)

This implies, by the elliptic regularity (or directly by taking the Fourier transform of both sides of 2m (5.27) and applying the Hausdorff–Young inequality), ∇φ∞ ∈ L m+1 (Rm ). This combined with 2m the Sobolev embedding L m+1 (Rm ) ⊂ H −1/2 (Rm ) imply φ∞ ∈ D1/2 (Rm , S(Rm )).

276


By the similar argument for the proof of Lemma 5.2, {φn − φ∞ } also satisfies the same conclusion of Lemma 5.4 and considering φn − φ∞ instead of φn if necessary, we may assume that φ∞ = 0. Let a ∈ Rm be arbitrary. By our choice of Rn and an , we have

∗

|φn |2 dvolgn

(5.28)

B10 (a)

for large n. By the Fourier transformation, it is easy to see that the following elliptic estimate holds for DgRm : For any φ ∈ H 1/2 (Rm , S(Rm )), there holds φ1/2,2 CDgRm φ−1/2,2 + Cφ2 ,

(5.29)

where C > 0 depends only on m. We apply (5.29) for φ = β 2 φn , where β ∈ C0∞ (Rm ) is an arbitrary function such that supp β ⊂ 0 B1 (a). We have

2

β φn

1/2,2

C DgRm β 2 φn −1/2,2 + C β 2 φn 2

C Dgn β 2 φn −1/2,2 + C (DgRm − Dgn ) β 2 φn −1/2,2 + C β 2 φn 2 .

(5.30)

The last term of (5.30) obviously converges to 0 as n → ∞. To show that the second term also converges to 0, we argue as follows: First, we observe that

(DgRm − Dgn ) β 2 φn , ϕ = βφn , β DgRm − D∗gn ϕ

for any ϕ ∈ H 1/2 (Rm , S(Rm )), where D∗gn is the adjoint of Dgn with respect to the metric gRm . Since gn → gRm in C ∞ (Rm ) as n → ∞, it is easy to see that β(DgRm − D∗gn ) : H 1 (Rm , S(Rm )) → L2 (Rm , S(Rm )) satisfies

β Dg

Rm

− D∗gn H 1 →L2 → 0

(5.31)

as n → ∞. Similarly, (DgRm − Dgn )(β·) : H 1 (Rm , S(Rm )) → L2 (Rm , S(Rm )) satisfies

(Dg

Rm

− Dgn )(β·) H 1 →L2 → 0

(5.32)

as n → ∞. Taking the dual of (5.32), we obtain β(DgRm − D∗gn ) : L2 (Rm , S(Rm )) → H −1 (Rm , S(Rm )) satisfies

β Dg as n → ∞.

Rm

− D∗gn L2 →H −1 → 0

(5.33)


277

Interpolating (5.31) and (5.33), we see that β(DgRm − Dgn ) : H 1/2 (Rm , S(Rm )) → H −1/2 (Rm , S(Rm )) satisfies

β D g

Rm

− D∗gn H 1/2 →H −1/2 → 0

(5.34)

as n → ∞. Thus we have (Dg

Rm

− Dgn ) β 2 φn , ϕ = βφn , β DgRm − D∗gn ϕ

βφn 1/2,2 β(DgRm − Dgn )ϕ −1/2,2 o(1)ϕ1/2,2

(5.35)

for any ϕ ∈ H 1/2 (Rm , S(Rm )) and therefore,

(Dg

Rm

− Dgn ) β 2 φn −1/2,2 → 0

(5.36)

as n → ∞. By (5.30) and (5.36), we have

2

β φn

1/2,2

C Dgn β 2 φn −1/2,2 + o(1)

∗ C β 2 |φn |2 −2 φn + β 2 fñ + ∇ β 2 ·gn φn −1/2,2 + o(1)

∗ C β 2 |φn |2 −2 φn −1/2,2 + C β 2 fñ −1/2,2

+ C ∇ β 2 ·gn φn −1/2,2 + o(1),

(5.37)

where ·gn denote the Clifford multiplication with respect to the metric gn . 2m

By the Sobolev embedding H −1/2 (Rm ) ⊂ L m+1 (Rm ), we have

2

∇ β ·g φn C ∇ β 2 ·gn φn n −1/2,2

2m m+1

→0

(5.38)

as n → ∞. By Lemma 5.4, we also have

2

β fñ

−1/2,2

→0

(5.39)

as n → ∞. Combining (5.37), (5.38) and (5.39), we have, by the Hölder inequality and (5.28),

2

β φn

1/2,2

∗ C β 2 |φn |2 −2 φn −1/2,2 + o(1)

∗ C β 2 |φn |2 −2 φn 2m ,B 0 (a) + o(1) m+1 1

2 2∗ −2

β φn 1/2,2 + o(1) Cφn ∗ 0 2 ,B1 (a)

C β 2 φn 1/2,2 + o(1). 1 m

(5.40)

278

T. Isobe / Journal of Functional Analysis 260 (2011) 253–307 1

If we choose > 0 such that C m < 12 , (5.40) implies that β 2 φn → 0 in H 1/2 (Rm , S(Rm )). Since a ∈ Rm and β ∈ C ∞ (Rm ) with supp β ⊂ B10 (a) are arbitrary, the assertion follows. 2 By Lemma 5.5 and

∗

|φn |2 dvolgn = B10

we have

∗

|ϕn |2 dvolg = Q(Rn ) = ,

BRn (an )

∗

B10

|φ∞ |2 dvolgRm = . In particular, φ∞ = 0 and φ∞ ∈ D1/2 (Rm , S(Rm )) satisfies

∗ DgRm φ∞ = |φ∞ |2 −2 φ∞ on Rm . By the regularity result proved in Appendix A, we have φ∞ ∈ ∗

C 1,α (Rm , S(Rm )) for some 0 < α < 1. Also, since Rm |φ∞ |2 dvolgRm < ∞ and the conformal equivalence Rm ∼ = S m \ {N }, it also follows from the regularity result in Appendix A that φ∞ ∗ is extended as a C 1,α -solution to the equation DgS m φ = |φ|2 −2 φ on S m . We denote the latter solution on S m as φ 0 and call it the first bubble for the sequence {ϕn }. In the following, we shall not distinguish between φ∞ and φ 0 . To proceed, let us assume that (taking a subsequence if necessary) an → a as n → ∞. Let β ∈ C ∞ (M) be such that β = 1 on B1 (a) and supp β ⊂ B2 (a). We define a spinor ωn ∈ C ∞ (M, S(M)) by − m−1 2

ωn (x) := Rn

∗ − m−1 β(x) ρn−1 φ 0 (x) = Rn 2 β(x)(ρn )∗ ◦ φ 0 Rn−1 exp−1 an x .

We set un := ϕn − ωn . We have Lemma 5.6. There exists a subsequence of {un } (which we still denote by {un }) such that un 0 weakly in H 1/2 (M, S(M)). Proof. Since ϕn 0 weakly in H 1/2 (M, S(M)), it suffices to prove that (taking a subsequence if necessary) ωn 0 weakly in H 1/2 (M, S(M)). Since {ωn } is bounded in H 1/2 (M, S(M)), it suffices to prove ωn , ϕ dvolg → 0

(5.41)

M

as n → ∞ for any ϕ ∈ C ∞ (M, S(M)). (5.41) is proved as follows: Fix R > 0 arbitrary. For ϕ ∈ C ∞ (M, S(M)), we first consider the integral BRn R (an )

− m−1 ωn , ϕ dvolg = Rn 2

BRn R (an )

∗

β ρn−1 φ 0 , ϕ dvolg


− m−1 = Rn 2

279

ρn∗ β φ 0 , ρn∗ ϕ dvolρn∗ g

BR0

m+1

ρn∗ β φ 0 , ρn∗ ϕ dvolgn .

= Rn 2

BR0

From this, we have

m+1 2 ωn , ϕ dvolg CRn ϕ∞ φ 0 dvolgRm .

(5.42)

BR0

BRn R (an )

Similarly, we have for large n

ωn , ϕ dvolg =

M\BRn R (an )

ωn , ϕ dvolg

B3 (an )\BRn R (an )

m+1

ρn∗ β φ 0 , ρn∗ ϕ dvolgn

= Rn 2

B0

−1 3Rn

\BR0

and

m+1 ωn , ϕ dvolg CRn 2 ϕ∞

0 φ dvolg

Rm

B 0 −1 \BR0 3Rn

M\BRn R (an )

0 2∗ φ dvolg

Cϕ∞ B0

−1 3Rn

1 2∗

(5.43)

,

Rm

\BR0

where we have used the Hölder inequality. Combining (5.42) and (5.43), we have m+1 ωn , ϕ dvolg CRn 2 ϕ∞ ϕ 0 dvolg m R M

BR0

+ Cϕ∞

0 2∗ φ dvolg

1 2∗

Rm

(5.44)

.

B 0 −1 \BR0 3Rn

In (5.44), we first let n → ∞ and then R → ∞. We then have (5.41) as desired. The following lemma shows that {un } is a Palais–Smale sequence for L0 :

2

280


Lemma 5.7. We have dL0 (ωn )−1/2,2 → 0 and dL0 (un )−1/2,2 → 0 as n → ∞. Proof. We first prove dL0 (ωn )−1/2,2 → 0 as n → ∞. ∗ Set fn := Dg ωn − |ωn |2 −2 ωn . We shall prove fn → 0 in H −1/2 (M, S(M)). Let ϕ ∈ H 1/2 (M, S(M)) be arbitrary. We have − m−1 2

fn , ϕ = Rn

M − m+1 2

∗

− m+1 ∇β · ρn−1 φ 0 , ϕ dvolg + Rn 2

= Rn

M − m+1 2

∗

β ρn−1 Dgn φ 0 , ϕ dvolg

M

∗ 2∗ −2 0

m+1 β m−1 ρn−1 φ 0 φ , ϕ dvolg

− Rn

− m−1 2

M

∗

∇β · ρn−1 φ 0 , ϕ dvolg

∗

β ρn−1 Dgn φ 0 − DgRm φ 0 , ϕ dvolg

+ Rn

M − m+1 2

+ Rn

∗ 2∗ −2 0

m+1 β − β m−1 ρn−1 φ 0 φ , ϕ dvolg

M

= I1 + I2 + I3 ,

(5.45)

where we have used (4.2) and the fact that φ 0 satisfies Eq. (5.27). In order to estimate I1 , let γ ∈ C ∞ (M) be such that γ = 1 on supp β. We have

− m−1 I 1 = Rn 2

∗

∇β · ρn−1 φ 0 , γ ϕ dvolg

M

− m−1 2

ρn∗ (∇β) · φ 0 , ρn∗ (γ ϕ) dvolρn∗ g

= Rn

Rm

m+1 2

= Rn

ρn∗ (∇β) · φ 0 , ρn∗ (γ ϕ) dvolgn .

Rm

Therefore, by the Hölder inequality and the Sobolev embedding, we have

m−1

|I1 | Rn ρn∗ (∇β) · φ 0 2+ ;Rm Rn 2 ρn∗ (γ ϕ) 2∗ ;Rm = Rn ρn∗ (∇β) · φ 0 2+ ;Rm γ ϕ2∗ ;M

CRn ρn∗ (∇β) · φ 0 2+ ;Rm ϕ1/2,2;M (5.46) (norms on Rm are with respect to the metric gn on Rm ), where we have, by the Hölder inequality, Rm

∗ + ρ (∇β) · φ 0 2 dvolg C n n

−1 −1 Rn−1 exp−1 an (B2 (a))\Rn expan (B1 (a))

0 2+ φ dvolg

Rm


+ CRn−2

B0

−1 3Rn

0 2∗ φ dvolg

281

m−1 m+1

(5.47)

Rm

\B 01

−1 2 Rn

for n large. Since φ 0 ∈ D1/2 (Rm , S(Rm )), combining (5.46) and (5.47), we obtain |I1 | o(1)ϕ1/2,2

(5.48)

as n → ∞. I2 is estimated similarly. As before, we have I2 =

m−1

ρn∗ β Dgn φ 0 − DgRm φ 0 , Rn 2 ρn∗ (γ ϕ) dvolgn

Rm

and

|I2 | C ρn∗ β Dgn φ 0 − DgRm φ 0 2+ ;Rm ϕ1/2,2;M ,

(5.49)

where

∗

ρ β D g φ 0 − D g m φ 0 + m n n R 2 ;R

∗

0

ρn β Dgn φ − DgRm φ 0 2+ ;B 0 + ρn∗ β Dgn φ 0 − DgRm φ 0 2+ ;Rm \B 0 . R

(5.50)

R

+

Since ∇φ 0 ∈ L2 (Rm ) (see the proof of Lemma 5.5) and gn → gRm in C ∞ (BR0 ) as n → ∞, we see that (5.50) tend to 0 as n → ∞ and R → ∞. Thus by (5.49), we have |I2 | o(1)ϕ1/2,2

(5.51)

as n → ∞. Similarly, we have I3 =

2∗ −2 0 m−1

m+1 ρn∗ β − β m−1 φ 0 φ , Rn 2 ρn∗ (γ ϕ) dvolgn

Rm

and

2∗ −2 0 m+1 |I3 | C ρn∗ β − β m−1 φ 0 φ 2+ ;Rm ϕ1/2,2;M , where Rm

∗ 2∗ −2 0 2+ ρ β − β m+1 m−1 φ 0 φ dvolgn C n B0

−1 3Rn

0 2∗ φ dvolg m → 0 R

\B 01

−1 2 Rn

282


as n → ∞, and we have |I3 | o(1)ϕ1/2,2

(5.52)

as n → ∞. Combining (5.48), (5.51) and (5.52), we have fn −1/2,2 → 0 as n → ∞. We next prove the second assertion dL0 (un )−1/2,2 → 0 as n → ∞. Let ϕ ∈ H 1/2 (M, S(M)) be arbitrary. We have

dL0 (un ), ϕ =

ϕ, Dun dvolg −

M

∗ −2

|un |2 M

= dL0 (ϕn ), ϕ − dL0 (ωn ), ϕ −

un , ϕ dvolg Φn , ϕ dvolg ,

(5.53)

M ∗

∗

∗

where Φn = |un |2 −2 un − |ϕn |2 −2 ϕn + |ωn |2 −2 ωn . By Lemma 5.2 (see (5.8)) and the first assertion of the lemma, it only necessary to prove + Φn −1/2,2 → 0 as n → ∞. By the Sobolev embedding L2 (M) ⊂ H −1/2 (M), it suffices to estimate Φn 2+ . First, observe that there exists C > 0 such that ∗ −2

|Φn | C|ϕn |2

∗ −2

|ωn | + C|ωn |2

|ϕn |.

Therefore, we have

∗ Φn 2+ ;M\BRn R (an ) C |ϕn |2 −2 |ωn | 2+ ;M\B (a ) Rn R n

∗ + C |ωn |2 −2 |ϕn | 2+ ;M\B (a ) .

(5.54)

n

Rn R

Here, we have, by the Hölder inequality,

+

|ϕn |2∗ −2 |ωn | 2+

2 ;M\BRn R (an )

2∗

|ϕn | dvolg

2 m+1

M\BRn R (an )

2∗

|ωn | dvolg

m−1 m+1

M\BRn R (an )

4m (m−1)(m+1)

Cϕn 1/2,2;M

0 2∗ φ dvolg

m−1 m+1

n

→0

(5.55)

B 0 −1 \BR0 3Rn

as R → ∞ (uniformly for n 1 since dvolgn CdvolgRm ). Similarly, we have

|ωn |2∗ −2 |ϕn |

2+ ;M\BRn R (an )

as R → ∞ (uniformly for n 1).

→0

(5.56)


On the other hand, we have

+

|Φn |2 dvolg =

283

+

|Φ˜ n |2 dvolgn ,

BR0

BRn R (an )

where 2∗ −2 2∗ −2 ∗ 0 ∗ φn − ρn∗ βφ 0 − |φn |2 −2 φn + ρn∗ βφ 0 ρn βφ Φ˜ n = φn − ρn∗ βφ 0 ∗ −2 ∗ −2 ∗ 2 2 φn − φ 0 − |φn |2 −2 φn + φ 0 = φn − φ 0 φ 0 in BR0 for n large enough. 1/2 Since φn → φ 0 in Hloc (Rm , S(Rm )) (see Lemma 5.5), we thus have

+

|Φ˜ n |2 dvolgn → 0

(5.57)

BR0

as n → ∞. Combining (5.54)–(5.57), we have Φn 2+ → 0 as n → ∞. This completes the proof.

2

We also have: Lemma 5.8. We have L0 (un ) = L0 (ϕn ) − LRm φ 0 + o(1) as n → ∞, where 1 LRn φ 0 = 2

Rm

1 φ 0 , DgRm φ 0 dvolgRm − ∗ 2

0 2∗ φ dvolg

Rm

.

Rm

Proof. We have L0 (un ) =

1 2

un , Dg un dvolg − M

1 2∗

∗

|un |2 dvolg . M

By un = ϕn − ωn , we have

un , Dg un dvolg =

M

ϕn , Dg ϕn dvolg − 2

M

M

+

ωn , Dg ωn dvolg . M

As for the second term of (5.58), we have

ϕn , Dg ωn dvolg

(5.58)

284


φn , Dgn ρn∗ βφ 0 dvolgn

ϕn , Dg ωn dvolg = Rm

M

φn , ∇ ρn∗ β ·gn φ 0 dvolgn +

= Rm

φn , ρn∗ β Dgn φ 0 dvolgn .

(5.59)

Rm

Here, the first term of (5.59) is estimated as, by the Hölder inequality,

φn , ∇ ρ ∗ β ·g φ 0 dvolg n n n Rm

Cφn 2∗ ;B 0

0 −1 \B 1

−1 2 Rn

3Rn

0

φ

2∗ ;B 0

−1 3Rn

\B 01

−1 2 Rn

as n → ∞. In order to estimate the second term of (5.59), we write

φn , ρn∗ β Dgn φ 0 dvolgn = φn , ρn∗ β Dgn φ 0 dvolgn + Rm

= o(1)

(5.60)

φn , ρn∗ β Dgn φ 0 dvolgn . (5.61)

Rm \BR0

BR0

Here, by Lemma 5.5, we have

φn , ρn∗ β Dgn φ 0

dvolgn =

BR0

φ 0 , DgRm φ 0 dvolgRm + o(1)

(5.62)

BR0

as n → ∞ and

φn , ρn∗ β Dgn φ 0 dvolgn φn 2∗ ;B 0

−1 3Rn

Rm \BR0

Dg φ 0 + m 0 n 2 ;R \B

R

C ∇φ 0 2+ ;Rm \B 0 = o(1)

(5.63)

R

as R → ∞ (uniformly for n 1). By (5.61), (5.62) and (5.63), we obtain

φn , ρn∗ β Dgn φ 0 dvolgn =

Rm


(5.64)

Rm

as n → ∞. Therefore, by (5.59), (5.60) and (5.64), we have

ϕn , Dg ωn dvolg =

M

as n → ∞.

Rm


(5.65)


285

We next estimate the third term of (5.58). We have

ρn∗ βφ 0 , Dgn ρn∗ βφ 0 dvolgn

ωn , Dg ωn dvolg = Rm

M

ρn∗ βφ 0 , ∇ ρn∗ β ·gn φ 0 dvolgn

= Rm

ρn∗ βφ 0 , ρn∗ β Dgn φ 0 dvolgn .

+

(5.66)

Rm

(5.66) is estimated as before and we have

∗ 0 ∗ 0

ρ βφ , ∇ ρ β · φ dvolg C φ 0 2∗ 0 n n n 2 ;B

−1 3Rn

Rm

\B 01

−1 2 Rn

= o(1)

(5.67)

and

ρn∗ βφ 0 , ρn∗ β Dgn φ 0 dvolgn =

Rm


(5.68)

Rm

as n → ∞. Therefore, we have


ωn , Dg ωn dvolg =

(5.69)

Rm

M

as n → ∞. Combining (5.65) and (5.69), we obtain

un , Dg un dvolg = M

ϕn , Dg ϕn dvolg −


(5.70)

Rm

M

as n → ∞. ∗ We next estimate M |un |2 dvolg . We write

∗

|un |2 dvolg = M

∗

|un |2 dvolg +

BRn R (an )

∗

|un |2 dvolg .

M\BRn R (an )

The first integral of (5.71) is estimated, by Lemma 5.5,

∗

|un |2 dvolg = BRn R (an )

BR0

∗ φn − ρ ∗ βφ 0 2 dvolg n

n

(5.71)

286


∗ φn − φ 0 2 dvolg = o(1) n

=

(5.72)

BR0

as n → ∞. In order to estimate the second integral of (5.71), since ϕn = un + ωn , we remark, as before, that there exists C > 0 such that |ϕn |2∗ − |un |2∗ − |ωn |2∗ C|un |2∗ −1 |ωn | + C|ωn |2∗ −1 |un |.

(5.73)

We have

∗ −1

|un |2

∗ φn − ρ ∗ βφ 0 2 −1 ρ ∗ βφ 0 dvolg m n n R

|ωn | dvolg C B0

M\BRn R (an )

−1 3Rn

\BR0

2∗ −1 C φn − ρn∗ βφ 0 2∗ ;B 0

0 −1 \BR

0

φ

2∗ ;B 0

3Rn

C φ 0 2∗ ;Rm \B 0 = o(1)

−1 3Rn

\BR0

(5.74)

R

as R → ∞, uniformly for n 1. Similarly, we have

∗ −1

|ωn |2

|un | dvolg = o(1)

(5.75)

M\BRn R (an )

as R → ∞, uniformly for n 1. From (5.73)–(5.75), we obtain

2∗

|un | dvolg = M\BRn R (an )

2∗

|ϕn | dvolg −

M\BRn R (an )

∗

|ωn |2 dvolg + o(1)

(5.76)

M\BRn R (an )

as R → ∞, uniformly for n 1. Moreover, we have

2∗

|ωn | dvolg = M\BRn R (an )

n

n

B 0 −1 \BR0 3Rn

C Rm \BR0

as R → ∞, uniformly for n 1.

∗ 0 2∗ ρ βφ dvolg

0 2∗ φ dvolg

Rm

= o(1)

(5.77)


287

Combining (5.76) and (5.77), we have

∗

∗

|un |2 dvolg = M\BRn R (an )

|ϕn |2 dvolg + o(1)

M\BRn R (an )

2∗

=

|ϕn | dvolg − M

∗

|φn |2 dvolgn + o(1)

(5.78)

BR0

as R → ∞, uniformly for n 1. By (5.71), (5.72) and (5.78), we obtain

∗

|un |2 dvolg = M

∗

|ϕn |2 dvolg − M

∗

|φn |2 dvolgn + o(1)

(5.79)

BR0

as R → ∞, uniformly for n 1. Since, by Lemma 5.5,

∗

|φn |2 dvolgn → BR0

0 2∗ φ dvolg

Rm

BR0

as n → ∞ for any R > 0, we finally obtain

2∗

|un | dvolg = M

2∗

|ϕn | dvolg −

0 2∗ φ dvolg

Rm

+ o(1)

(5.80)

Rm

M

as n → ∞. Combining (5.70) and (5.80), we obtain L0 (un ) = L0 (ϕn ) − LRm φ 0 + o(1) as n → ∞. This completes the proof.

(5.81)

2

We are now in a position to complete the proof of Theorem 5.1. Proof of Theorem 5.1. It suffices to prove ϕn → 0 in H 1/2 (M, S(M)) under the assumption of the theorem. Assume contrary that ϕn 0 in H 1/2 (M, S(M)). Then by Lemma 5.8, we have (5.81). Since {un } is a Palais–Smale sequence for L0 (see Lemma 5.7), by Lemma 3.2, {un } ⊂ H 1/2 (M, S(M)) is bounded and

o(1) = dL0 (un ), un =

M

as n → ∞.

un , Dg un dvolg −

∗

|un |2 dvolg M

288


Thus we have L0 (un ) =

1 2

un , Dg un dvolg − M

=

1 2m

1 2∗

∗

|un |2 dvolg M

∗

|un |2 dvolg + o(1)

(5.82)

M

as n → ∞. Combining (5.81), (5.82) and Proposition 4.1, we have 0 1 m m lim L0 (ϕn ) LRm φ ωm . 2m 2 n→∞ ∗ −2

On the other hand, since Dg ψ∞ = λψ∞ + |ψ∞ |2 Lλ (ψ∞ ) =

1 2

ψ∞ , Dg ψ∞ dvolg − M

=

1 2m

ψ∞ , we have

λ 2

(5.83)

|ψ∞ |2 dvolg − M

1 2∗

∗

|ψ∞ |2 dvolg M

∗

|ψ∞ |2 dvolg 0.

(5.84)

M

By (5.7), (5.83) and (5.84), we finally obtain c = lim Lλ (ψn ) n→∞

This contradicts to the assumption c
0 such that + inf L∗λ (ϕ): ϕ ∈ L2 M, S(M) , ϕ2+ = ρ > 0. +

Moreover, for ϕ ∈ L2 (M, S(M)) with

M ϕ, Aλ ϕ dvolg

> 0, there holds

lim L∗λ (tϕ) = −∞.

t→∞

Proof. Since 2+ < 2, the first assertion is satisfied if ρ > 0 is small. The second assertion is also trivial. 2 Recall that the mountain pass level for L∗λ is defined as + cλ = inf max L∗λ (tϕ): ϕ ∈ L2 M, S(M) , ϕ, Aλ ϕ dvolg > 0 . t0

M

Easy computation shows that + 1 ( M |ϕ|2 dvolg )m+1 2+ cλ = inf : ϕ ∈ L M, S(M) , ϕ, Aλ ϕ dvolg > 0 . 2m ( M ϕ, Aλ ϕ dvolg )m

(6.1)

M

We shall prove the following: Lemma 6.2. Assume λ ∈ / Spec(D), λ > 0 and m 4. We then have cλ
0 such that 2δ < ι(M). Let p0 ∈ M 0 and η = 1 on B 0 . be fixed and η ∈ C ∞ (Rm ) be such that η = 0 on Rm \ B2δ δ x For > 0 small enough, define ψ (x) = η(x)ψ( ) and ϕ = D ψ , where D is the Dirac operator acting on S(V , g). We shall use ϕ as a test spinor to estimate cλ .

292


6.3. Estimate of cλ We shall estimate the following quantity (cf. (6.1)) + 1 ( M |ϕ |2 dvolg )m+1 . 2m ( M ϕ , Aλ ϕ dvolg )m By [6, Proposition 3.2], we have ϕ = Dψ + W · ψ + X · ψ +

j j bi − δi ∂i · ∇∂j ψ ,

(6.7)

i,j

where W ∈ C ∞ (V , Cl(T V )) and X ∈ C ∞ (V , T V ) are given as W=

X=

k bir ∂r bjl b−1 l ei · ej · ek ,

(6.8)

1 i 1 i Γ ik − Γ kii ek = Γ ik ek , 4 2

(6.9)

1 4

i,j,k i=j =k=i

i,k

i,k

where Γ kij = g(∇ ei ej , ek ). Since B = (G−1 )1/2 and G = I + O(r 2 ) as r → 0, we have bir = δir + O(r 2 ), ∂r bjs = O(r) and Γrsl = O(r) as r → 0. Thus Γ kij = g(∇ ei ej , ek ) = bir bkt g ∇ ∂r bjs ∂s , ∂t = bir bkt g ∂r bjs ∂s + bjs Γrsl ∂l , ∂t = O(r) and X = O(r)

as r → 0.

(6.10)

Similarly, we have W=

δir ∂r bjl δlk ei · ej · ek + O r 3

i,j,k i=j =k=i

=

i=j =k=i j

since bjk = bk .

∂i bjk ei · ej · ek + O r 3 = O r 3 ,

(6.11)


293

In fact, as remarked in [6], we have

1 1 (Ric)αk x α + (Ric)αk;β x α x β + O r 3 ek , 4 6 Rlβγ k (Rj iαl + Rj lαi )x α x β x γ ei · ej · ek + O r 4 .

X=− W =−

1 144

i,j,k i=j =k=i

On the other hand, by (6.4), we have x 1 x η(x)Dψ + ∇η(x) · ψ m x x x = η(x)f ψ + ∇η(x) · ψ . 2

Dψ (x) =

(6.12)

Plugging (6.12) into (6.7) and squaring both sides, we have: |ϕ |2 = f1 + f2 + · · · + f21 , where (identifying x ∈ Rm with expp0 x ∈ M for simplicity of the notation) 2 2 m2 x x 2 f1 = 2 η(x) f ψ , 4 2 x f2 = ∇η(x) · ψ , 2 x 2 f3 = η(x) W (x) · ψ , 2 x f4 = η(x)2 X(x) · ψ , 2 j 1 x j f5 = 2 η(x)2 bi (x) − δi ∂i · ∇∂j ψ , i,j

2 j x j bi (x) − δi ∂j η(x)∂i · ψ f6 = , i,j

x m x x , ∇η(x) · ψ , ψ f7 = η(x)f m x x x f8 = η(x)2 f , W (x) · ψ , ψ x x x m f9 = η(x)2 f ψ , X(x) · ψ ,

(6.13)

294


f10 =

m x x j x j 2 , b , η(x) f ψ (x) − δ ∂ · ∇ ψ i ∂ i i i 2 i,j

f11 =

m

i,j

η(x)∂j η(x)f

x x j x j , bi (x) − δi ∂i · ψ , ψ

x x f12 = 2η(x) ∇η(x) · ψ , W (x) · ψ , x x , X(x) · ψ , f13 = 2η(x) ∇η(x) · ψ η(x) x j x j , bi (x) − δi ∂i · ∇∂j ψ , f14 = 2 ∇η(x) · ψ ij

f15 = 2

x j x j , bi (x) − δi ∂j η(x)∂i · ψ , ∇η(x) · ψ i,j

x x f16 = 2η(x)2 W (x) · ψ , X(x) · ψ , η(x)2 x j x j W (x) · ψ , bi (x) − δi ∂i · ∇∂j ψ , f17 = 2 i,j

f18 = 2

i,j

f19 = 2

η(x)2 i,j

f20 = 2

i,j

f21 = 2

x j x j , bi (x) − δi ∂j η(x)∂i · ψ , η(x) W (x) · ψ

X(x) · ψ

x j x j , bi (x) − δi ∂i · ∇∂j ψ ,

x j x j η(x) X(x) · ψ , bi (x) − δi ∂j η(x)∂i · ψ ,

η(x) j x q x j q bi (x) − δi ∂i · ∇∂j ψ , bp (x) − δp ∂q η(x)∂p · ψ .

i,j,p,q

For simplicity of the notation, in the following, for two functions f and g on M, we denote f g when there exists a constant C > 0 depending only on the geometry of M such that f (x) Cg(x) for any x ∈ M. 0 , it is sufficient to give estimates on B (p ). By Since the support of η is contained in B2δ 2δ 0 (6.10) and (6.11), we have the following estimates on B2δ (p0 ): m+1 m2 x f1 2 f , 4 2 m−1 2 x x 6 |f3 | W (x) ψ r f ,

(6.14) (6.15)


2 m−1 2 x x 2 |f4 | X(x) ψ r f , m r4 x |f5 | 2 f ,

295

(6.16) (6.17)

where we have used an easily checked inequality ∇ψ(x) f (x) m2 , m x r3 , |f8 | f

(6.18) (6.19)

f9 = 0,

(6.20)

since X is a vector field, m+ 1 2 x r2 |f10 | 2 f , m−1 x 4 , |f16 | r f m− 1 2 r5 x |f17 | f , m− 1 2 x r3 |f19 | f .

(6.21) (6.22)

(6.23)

(6.24)

By using rf (x) 1 and 1 2

rf (x) =

2r 2 1 + r2

1 2

1

22 ,

we have, from (6.14)–(6.24), the following estimate on B2δ (p0 ): f1 + f3 + f4 + f5 + f8 + f9 + f10 + f16 + f17 + f19 m+1 m−1 m+ 1 2 m2 x x r2 x 2 2f + Cr f +C 2f . 4

(6.25)

The remaining terms are easier to handle, since we have |∇ k η(x)| Ck,p r p for any k, p. Using this inequality, on B2δ (p0 ), we have m−1 x . f2 + f6 + f7 + f11 + f12 + f13 + f14 + f15 + f18 + f20 + f21 r f 2

Combining (6.25) and (6.26), on B2δ (p0 ), we obtain

(6.26)

296


m+1 m−1 m+ 1 2 m2 x x r2 x 2 |ϕ | 2 f +C r f + 2f 4 − 1 2 m2 x m+1 x −2 x 2 2 2 1 + C r f . 2f + Cr f 4 2

m

By the elementary inequality (1 + a) m+1 1 + 2+

|ϕ |

m m+1 a

for a 0, on B2δ (p0 ), we have

m −2 − 1 2 x x x 2 2 2 1 + C r f f + Cr f 2m m m−2 m− 1 2 2 2m m m+1 x x x f + C m+1 r 2 f + C − m+1 r 2 f . 2

m 2

(6.27)

2m m+1

(6.28)

Taking into account dvolg = dvolgRm + O(r 2 ) in normal coordinate at p0 , we have I1 :=

m 2

2m m+1

f

m x dvolg

B2δ (p0 )

m 2

m m 2m x x − m+1 2 f dvolgRm + C r f dvolgRm

2m m+1

0 B2δ

2

m

m 2

0 B2δ

2m m+1

ωm−1

m(m−1) m+1

∞ 0

m(m−1) s m−1 ds + O m+1 +2 , (1 + s 2 )m

(6.29)

provided m 3, I2 :=

m−2 x r f dvolg

2 m+1

2

B2δ (p0 )

2

m+1

r 2f

m−2 x dvolgRm

0 B2δ 2δ

2 m+1 +m+2

0

s m+1 (1 + s 2 )m−2

⎧ 2 +m+2 ⎪ ⎪ ⎨ m+1 58 ds 7 |log | ⎪ ⎪ ⎩ 2 +2m−4 m+1

and I3 :=

2m − m+1

m− 1 2 x r f dvolg 2

B2δ (p0 )

if m 7, if m = 6, if 2 m 5

(6.30)


m− 1 2 x r f dvolgRm

2m − m+1

297

2

0 B2δ

2δ

2 m+ m+1

s m+1 1 (1 + s 2 )m− 2

0

ds

2

m+ m+1

if m 4,

7 2

|log |

if m = 3.

(6.31)

Combining (6.29)–(6.31), we obtain

2+

|ϕ |

dvolg 2

m

m 2

2m m+1

ωm−1

m(m−1) m+1

I+

2

O( m+ m+1 ) 5 2

O( )

M

if m 4, if m = 3,

(6.32)

∞ s m−1 where I = 0 (1+s 2 )m ds. We next give the estimate of M ϕ , Aλ ϕ dvolg . We define η by Aλ ϕ = ψ + η . Since Aλ = (D − λ)−1 , we have ϕ = (D − λ)(ψ + η ) = (D − λ)η + ϕ − λψ and η = λAλ ψ . We have

ϕ , Aλ ϕ dvolg =

M

ϕ , ψ dvolg +

M

ϕ , η dvolg . M

Here, we have

ϕ , η dvolg = λ

M

ϕ , Aλ ψ dvolg M

=λ

Aλ ϕ , ψ dvolg M

=λ

ψ + η , ψ dvolg M

(6.33)

298


=λ

|ψ | dvolg + λ

η , ψ dvolg .

2

M

(6.34)

M

By the elliptic regularity, we have η 1,p Cψ p 2m

for any 1 < p < ∞. This combined with the Sobolev embedding W 1, m+2 (M) ⊂ L2 (M) implies η 2 Cψ

2m m+2

(6.35)

.

Therefore, the second term of (6.34) is estimated as η , ψ dvolg Cψ 2m ψ 2 . m+2

(6.36)

M

Thus to estimate (6.33), we need to estimate three integrals and M ϕ , ψ dvolg . Estimate of M ϕ , ψ dvolg : By (6.7) and (6.12), we have

M

|ψ |2 dvolg ,

2m

M

|ψ | m+2 dvolg

ϕ , ψ dvolg = J1 + J2 + · · · + J6 , M

where m J1 = 2

M

x x ,ψ dvolg , η(x)2 W (x) · ψ

M

x x ,ψ dvolg , η(x) X(x) · ψ

J4 = J5 =

i,j

M

x x η(x) ∇η(x) · ψ ,ψ dvolg ,

J3 =

1

2

M 2

η(x)

M

2

J2 =

2 x x η(x) f ψ dvolg ,

x x ,ψ dvolg , ∂i · ∇∂j ψ

j j bi (x) − δi

j x x j ,ψ dvolg . J6 = bi (x) − δi η(x)∂j η(x) ∂i · ψ i,j M

(6.37)


We have J1

m 2

f

299

m x dvolg

Bδ (p0 )

m 2

f

m m x 1 x dvolgRm + O r 2f dvolgRm

Bδ0

Bδ0

δ

2

m−1

2

m−1

mωm−1

m−1

mωm−1

m−1

0

s m−1 (1 + s 2 )m

I +O

m+1

ds + O

δ

m+1 0

s m+1 ds (1 + s 2 )m

(6.38)

if m 3, J2 = 0, 2 x r 3 ψ dvolg

|J3 |

(6.39)

B2δ (p0 )

m−1 x f r m+2 dvolgRm

0 B2δ

2δ

m+3 0

|J5 |

s m+2 (1 + s 2 )m−1

if m 6, if m = 5, if 2 m 4,

J4 = 0, m− 1 2 x r 2f dvolg

1

⎧ m+3 ⎪ ⎨ ds 8 |log | ⎪ ⎩ 2m−2

(6.40) (6.41)

B2δ (p0 )

1

r 2f

m− 1 2 x dvolgRm

0 B2δ 2δ

m+1 0

|J6 | B2δ (p0 )

s m+1 m− 12

(1 + s 2 )

ds

m+1

if m 4,

4 |log |

if m = 3,

⎧ m+3 2 ⎪ ⎨ x 3 r ψ dvolg 8 |log | ⎪ ⎩ 2m−2

where we have used |∇η(x)| r and the same estimate for J3 .

if m 6, if m = 5, if 2 m 4,

(6.42)

(6.43)

300


From (6.38)–(6.43), we obtain

ϕ , ψ dvolg 2m−1 mωm−1 m−1 I + M

Estimate of We have

M

O( m+1 )

if m 4,

O( 4 |log |)

if m = 3.

(6.44)

|ψ |2 dvolg :

|ψ | dvolg = 2

M

M

=

2 m−1 x x 2 η(x) ψ dvolg = η(x) f dvolg 2

M

m−1 x f dvolgRm + O

Bδ0

m−1 x f dvolgRm

0 \B 0 B2δ δ

m−1 x r f dvolgRm

+O

2

0 B2δ

δ = ωm−1

2m−1 r m−1 (1 +

0

2δ +O

r 2 m−1 ) 2

dr + O

r m+1 (1 +

0

2δ

r 2 m−1 ) 2

∞ = 2m−1 ωm−1 m 0

δ

(1 +

r 2 m−1 ) 2

dr

s m−1 ds (1 + s 2 )m−1

∞ s m−1 where A = 2m−1 ωm−1 0 (1+s 2 )m−1 ds > 0. 2m Estimate of M |ψ | m+2 dvolg : We have |ψ | M

2m m+2

dvolg B2δ (p0 )

dr

⎧ m+2 ) ⎪ ⎨ O( 2m−2 (if m 3) + O( 6 |log |) +O ⎪ ⎩ O( 4 ) ⎧ m+2 ) ⎪ if m 5, ⎨ O( m = A + O( 6 |log |) if m = 4, ⎪ ⎩ O( 4 ) if m = 3,

r m−1

m(m−1) x m+2 f dvolg

if m 5, if m = 4, if m = 3 (6.45)


2δ 0

r m−1 (1 +

r 2 m(m−1) ) m+2 2

⎧ m ⎨ dr 4 |log | ⎩ 12 5

if m 5, if m = 4,

301

(6.46)

if m = 3.

Combining all these estimates, we finally obtain

ϕ , Aλ ϕ dvolg 2m−1 mωm−1 m−1 I 1 + λB + C() ,

(6.47)

M

where ∞

B=

s m−1 (1+s 2 )m−1 ∞ s m−1 m 0 (1+s 2 )m 0

ds >0 ds

and ⎧ 2 ⎪ ⎨ 3 C() 2 |log | 4 ⎪ ⎩ 3 2

if m 5, if m = 4, if m = 3.

Combining this with (6.32), we have + 1 ( M |ϕ |2 dvolg )m+1 1 m m m 2 ωm−1 I 1 − mλB + D() , m 2m ( M ϕ , Aλ ϕ dvolg ) 2m 2 where ⎧ 2 ⎪ ⎨ D() 2 |log | 34 ⎪ ⎩

if m 5, if m = 4, if m = 3.

On the other hand, the integral I is calculated as 2 ωm−1 I = m

f (x)m dx = ωm ,

Rm

since gS m = f 2 gRm . Therefore, we finally obtain cλ

1 m m 1 m m ωm 1 − mλB + D() < ωm 2m 2 2m 2

if m 4 and > 0 is small enough. This completes the proof of Lemma 6.2.

(6.48)

302


Proof of Theorem 1.1. By Theorem 5.1, Lemma 6.1 and Lemma 6.2, by the standard argument (cf. [30]), cλ > 0 is a critical value of L∗λ . Therefore, by Lemma 3.1, cλ is also a critical value of Lλ . This completes the proof. 2 Acknowledgment The author wishes to express his gratitude to the referee for his/her helpful suggestions. ∗ −2

Appendix A. Regularity of weak solutions to Dψ = λψ + |ψ|2

ψ ∗

In this appendix, we prove a regularity result for weak solutions ψ ∈ L2 (M) to Eq. (1.1). Eq. (1.1) is critical in the sense that the standard bootstrap argument does not yield regularity of finite action weak solutions (cf. [4,5]). Such critical equations are common in geometric analysis, see [13,20,31], etc. Concerning critical nonlinear Dirac equations, regularity of weak solutions is studied in [15,32] for m = 2. We henceforth assume m 3, but similar argument is applied for m = 2 with slight modifications. ∗ Let ψ ∈ L2 (M) be a weak solution to (1.1), i.e., it satisfies

Dϕ, ψ dvolg = λ M

ϕ, ψ dvolg +

M

∗ −2

|ψ|2

ϕ, ψ dvolg

M

for all ϕ ∈ C ∞ (M). We shall prove ∗

Theorem A.1. Assume m 3. Let ψ ∈ L2 (M) be a weak solution to (1.1). Then ψ ∈ C 1,α (M) for some 0 < α < 1. Proof. Let ρ ∈ C ∞ (M) be arbitrary. Let η ∈ C ∞ (M) be such that η = 1 on supp ρ. We have D(ρψ) = ρ Dψ + ∇ρ · ψ

∗ = ρ λψ + |ψ|2 −2 ψ + ∇ρ · ψ ∗ −2

= λ(ρψ) + η|ψ|2

(ρψ) + ∇ρ · ψ.

(A.1)

Let μ ∈ / Spec(D) be arbitrary and fixed in the following argument. We write (A.1) in the form ∗ −2

(D − μ)(ρψ) − η|ψ|2

(ρψ) = (λ − μ)(ρψ) + ∇ρ · ψ.

(A.2)

We begin the proof for m = 3. Since ψ ∈ L3 (M), we have (λ − μ)(ρψ) + ∇ρ · ψ ∈ L3 (M).

(A.3)

Let 1 < p < 3 be arbitrary and consider the map W 1,p M, S(M) ϕ → η|ψ|ϕ ∈ Lp (M).

(A.4)


303

3p

Notice that, by the Sobolev embedding W 1,p (M) ⊂ L 3−p (M) and the Hölder inequality, we have η|ψ|ϕ ∈ Lp (M) for ϕ ∈ W 1,p (M, S(M)) and the above map is well defined. The operator norm is estimated as

η|ψ|(·)

W 1,p →Lp

Cp ψ3,B

(A.5)

for some constant Cp depending only on p, where B = supp η. Thus, since μ ∈ / Spec(D), (D − μ) − η|ψ|(·) : W 1,p M, S(M) → Lp M, S(M)

(A.6)

is invertible if ψ3,B is small. Therefore, by (A.3), there exists a unique solution ϕ ∈ W 1,p (M, S(M)) to the equation (D − μ)ϕ − η|ψ|ϕ = (λ − μ)(ρψ) + ∇ρ · ψ

in M.

(A.7)

On the other hand, we have a well-defined map L3 M, S(M) ϕ → η|ψ|ϕ ∈ W −1,3 M, S(M) .

(A.8)

Indeed, by the Hölder inequality, we have η|ψ|ϕ ∈ L3/2 (M). Since L3/2 (M) ⊂ W −1,3 (M) by the Sobolev embedding, the map (A.8) is well defined and the operator norm is estimated as before:

η|ψ|(·)

L3 →W −1,3

Cψ3,B .

(A.9)

Therefore, (D − μ) − η|ψ|(·) : L3 M, S(M) → W −1,3 M, S(M)

(A.10)

is invertible if ψ3,B is small and there exists a unique solution ϕ˜ ∈ L3 (M, S(M)) to the equation (D − μ)ϕ˜ − η|ψ|ϕ˜ = (λ − μ)(ρψ) + ∇ρ · ψ.

(A.11)

Notice that ϕ˜ = ρψ by (A.2). On the other hand, since W 1,p (M) ⊂ L3 (M) if 3/2 p < 3, ϕ ∈ W 1,p (M, S(M)) is also an L3 -solution to (A.11) provided 3/2 p < 3. Thus by the uniqueness, we have ϕ = ρψ and ρψ ∈ W 1,p (M, S(M)) for any 3/2 p < 3 provided that supp η = B is small. Since ρ and η arbitrary (under the assumption that supp ρ is small and η = 1 on supp ρ), this implies that ψ ∈ W 1,p (M, S(M)) for any 3/2 p < 3. Then by the Sobolev embedding, we have ψ ∈ Lp (M, S(M)) for any 1 < p < ∞. Therefore, Dψ = λψ + |ψ|ψ ∈ Lp (M)

for any 1 < p < ∞ and this implies that ψ ∈ W 1,p (M, S(M)) for any 1 < p < ∞ by the elliptic estimate. Thus, by the Sobolev embedding, we have ψ ∈ C 0,α (M) for any 0 < α < 1 and

304


Dψ = λψ + |ψ|ψ ∈ C 0,α (M)

implies that ψ ∈ C 1,α (M, S(M)) by the elliptic estimate. This completes the proof for the case m = 3. 2m ∗ We assume m 4. By the Sobolev embedding W 1,2 (M) ⊂ L m−3 (M) and the Hölder inequality, the map ∗ ∗ ∗ W 1,2 M, S(M) ϕ → η|ψ|2 −2 ϕ ∈ L2 M, S(M)

(A.12)

is well defined and its operator norm is estimated as

η|ψ|2∗ −2 (·)

∗

∗

W 1,2 →L2

∗

Cψ22∗ −2 ,B .

(A.13) ∗

Therefore, if supp η = B is small, there exists a unique solution ϕ ∈ W 1,2 (M, S(M)) to the equation ∗ −2

(D − μ)ϕ − η|ψ|2

ϕ = (λ − μ)(ρψ) + ∇ρ · ψ

in M. +

(A.14) ∗

On the other hand, the Hölder inequality and the Sobolev embedding L2 (M) ⊂ W −1,2 (M) implies that the map ∗ ∗ ∗ L2 M, S(M) ϕ → η|ψ|2 −2 ϕ ∈ W −1,2 M, S(M)

(A.15)


η|ψ|2∗ −2 (·)

∗

∗

L2 →W −1,2

∗

Cψ22∗ −2 ,B .

(A.16)

∗

Therefore, if B is small, there exists a unique solution ϕ˜ ∈ L2 (M) to the equation ∗ −2

(D − μ)ϕ˜ − η|ψ|2

ϕ˜ = (λ − μ)(ρψ) + ∇ρ · ψ ∗

in M.

(A.17)

∗

Notice that ϕ˜ = ρψ by (A.2). Since ϕ ∈ W 1,2 (M) ⊂ L2 (M) also satisfies (A.17), we have ∗ ∗ ρψ = ϕ ∈ W 1,2 (M, S(M)) by the uniqueness of solutions in L2 (M). Therefore, we conclude ∗ ψ ∈ W 1,2 (M, S(M)). Once this is proved, we argue inductively as follows: 2m Let us assume m > 2k + 1 and ψ ∈ W 1, m−2k−1 (M, S(M)) for some Z k 0. As we have proved, the assumption is satisfied for k = 0. By the Sobolev embedding, we have ⎧ 2m ⎪ ⎨ L m−2k−3 (M) 2m W 1, m−2k−1 (M) ⊂ Lp (M) (∀p < ∞) ⎪ ⎩ α C (M) (0 < ∃α < 1)

if m > 2k + 3, if m = 2k + 3, if m < 2k + 3.

When m = 2k + 2, we have ψ ∈ C 0,α (M) for some 0 < α < 1 and Eq. (1.1) implies that ψ ∈ C 1,β (M) for some 0 < β < 1. When m = 2k + 3, we have ∗ Dψ = λψ + |ψ|2 −2 ψ ∈ Lp (M)


305

for any 1 < p < ∞. By elliptic regularity, we have ψ ∈ W 1,p (M) for any 1 2k + 3. As observed, we have ψ ∈ L m−2k−3 (M). By the Sobolev embedding ⎧ 2m ⎪ if m > 2k + 5, ⎨ L m−2k−5 (M) 2m W 1, m−2k−3 (M) ⊂ Lp (M) (∀p < ∞) if m = 2k + 5, ⎪ ⎩ α C (M) (0 < ∃α < 1) if m < 2k + 5. When m > 2k + 5, the above and the Hölder inequality imply that the map 2m ∗ ∗ W 1, m−2k−3 M, S(M) ϕ → η|ψ|2 −2 ϕ ∈ L2 M, S(M)

(A.18)


η|ψ|2∗ −2 (·)

W

2m 1, m−2k−3

2m

→L m−2k−3

∗ −2

Cψ2

2m m−2k−3 ,B

(A.19)

. 2m

Therefore, if B = supp η is small, there exists a unique solution ϕ ∈ W 1, m−2k−3 (M, S(M)) to 2m Eq. (A.14). By arguing similarly as before, we have ρψ = ϕ ∈ W 1, m−2k−3 (M, S(M)) and ψ ∈ 2m W 1, m−2k−3 (M, S(M)). 2m When m = 2k + 5, we also have (A.18) and (A.19) and obtain ψ ∈ W 1, m−2k−3 (M, S(M)). Similarly, when m < 2k + 5, i.e., m = 2k + 4, we have again (A.18) and (A.19) and obtain ψ ∈ 2m 1, m−2k−3 (M, S(M)). W 2m Summing up all the argument, when m > 2k + 1 and ψ ∈ W 1, m−2k−1 (M, S(M)), we have

ψ ∈ C 1,α M, S(M) (0 < ∃α < 1) if m = 2k + 2, 2k + 3, 2m ψ ∈ W 1, m−2k−3 M, S(M) if m > 2k + 3.

(A.20)

Now the induction works as follows. Let us assume m > 3 > 2 × 0 + 1. We have proved ψ ∈ 2m W 1, m−1 (M, S(M)). Then (A.20) for k = 0 implies that 2m ψ ∈ W 1, m−3 M, S(M) .

(A.21)

Since m > 3 = 2 × 1 + 1, (A.21) together with (A.20) for k = 1 imply

ψ ∈ C 1,α M, S(M) (0 < ∃α < 1) if m = 4, 5, 2m if m > 5. ψ ∈ W 1, m−5 M, S(M)

(A.22)

Therefore, the proof is completed for m = 4, 5. For the case m > 5 = 2 × 2 + 1, (A.22) together with (A.20) for k = 2 imply

ψ ∈ C 1,α M, S(M) (0 < ∃α < 1) if m = 6, 7, 2m if m > 7. ψ ∈ W 1, m−7 M, S(M)

(A.23)

306


Thus the proof is completed for m = 6, 7. Continuing similarly, we complete the proof for all the case m 3. 2 References [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. ∗ [2] A. Ambrosetti, M. Struwe, A note on the problem −u = λu + u|u|2 −2 , Manuscripta Math. 54 (1986) 373–379. [3] B. Ammann, A spin-conformal lower bound of the first positive Dirac eigenvalue, Differential Geom. Appl. 18 (2003) 21–32. [4] B. Ammann, A variational problem in conformal spin geometry, Habilitationsschift, Universität Hamburg, 2003. [5] B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Comm. Anal. Geom. 17 (2009) 429–479. [6] B. Ammann, J.-F. Grosjean, E. Humbert, B. Morel, A spinorial analogue of Aubin’s inequality, Math. Z. 260 (2008) 127–151. [7] B. Ammann, E. Humbert, J.-F. Grosjean, Mass endomorphism and spinorial Yamabe type problem on conformally flat manifolds, Comm. Anal. Geom. 14 (2006) 163–182. [8] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976) 269–296. [9] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monogr. Math., Springer, Berlin– Heidelberg, 1998. [10] J.-P. Bourguignon, P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992) 581–599. [11] H. Brezis, J.-M. Coron, Convergence of solutions of H -systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89 (1985) 21–56. [12] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 31 (1983) 437–477. [13] S.Y. Chang, M. Gursky, P. Yang, Regularity of a fourth order nonlinear PDE with critical exponent, Amer. J. Math. 121 (1999) 215–257. [14] Q. Chen, J. Jost, J. Li, G. Wang, Dirac-harmonic maps, Math. Z. 254 (2006) 409–432. [15] Q. Chen, J. Jost, G. Wang, Nonlinear Dirac equations on Riemann surfaces, Ann. Global Anal. Geom. 33 (2008) 253–270. [16] O. Druet, E. Hebey, F. Robert, Blow-Up Theory for Elliptic PDEs in Riemannian Geometry, Math. Notes, vol. 45, Princeton University Press, Princeton and Oxford, 2004. [17] T. Friedrich, On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phys. 28 (1998) 143–157. [18] T. Friedrich, Dirac Operators in Riemannian Geometry, Grad. Stud. Math., vol. 25, Amer. Math. Soc., Providence, RI, 2000. [19] N. Ginoux, The Dirac Spectrum, Lecture Notes in Math., vol. 1976, Springer, Dordrecht–Heidelberg–London–New York, 2009. [20] F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris 312 (1991) 591–596. [21] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974) 1–55. [22] J. Hulshof, E. Mitidieri, R. van der Vorst, Strongly indefinite systems with critical Sobolev exponents, Trans. Amer. Math. Soc. 350 (1998) 2349–2365. [23] E. Humbert, Generic metrics and the mass endmorphism on spin three manifolds, Ann. Global Anal. Geom. 37 (2010) 163–171. [24] T. Isobe, Existence results for solutions to nonlinear Dirac equations on compact spin manifolds, preprint. [25] H.B. Lawson, M.L. Michelson, Spin Geometry, Princeton University Press, 1989. [26] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part 1, Rev. Mat. Iberoamericana 1 (1) (1985) 145–201. [27] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part 2, Rev. Mat. Iberoamericana 1 (2) (1985) 45–121. [28] S. Raulot, A Sobolev-like inequality for the Dirac operator, J. Funct. Anal. 26 (2009) 1588–1617. [29] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984) 511–517. [30] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th edition, Springer, Berlin–Heidelberg–New York, 2008.


307

[31] K. Uhlenbeck, J. Viaclovsky, Regularity of weak solutions to critical exponent variational equations, Math. Res. Lett. 7 (2000) 651–656. [32] C.Y. Wang, A remark on nonlinear Dirac equations, arXiv:0810.2051. [33] H. Yamabe, On the deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960) 21–37.


Spectrum is periodic for n-intervals Debashish Bose ∗ , Shobha Madan Department of Mathematics and Statistics, I.I.T. Kanpur, India Received 12 April 2010; accepted 17 September 2010

Communicated by L. Gross

Abstract In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set Ω is periodic, with the period an integral multiple of the measure of Ω. As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case. © 2010 Elsevier Inc. All rights reserved. Keywords: Spectral sets; Spectrum; Tiling; Fuglede’s conjecture; Zeros of exponential polynomials; Sets of sampling and interpolation; Landau’s density theorem

1. Introduction In this paper we study the structure of the spectrum associated to a spectral set Ω ⊂ R, which is a finite union of intervals. In order to describe our result and its context, we begin with a brief account of some of the relevant history of the problem. Let Ω be a Lebesgue measurable subset of Rd with finite positive measure. For λ ∈ Rd , let eλ (x) := |Ω|−1/2 e2πiλ.x χΩ (x),

x ∈ Rd .

Ω is said to be a spectral set if there exists a subset Λ ⊂ Rd such that the set of exponential functions EΛ := {eλ : λ ∈ Λ} is an orthonormal basis for the Hilbert space L2 (Ω). The set Λ is said to be a spectrum for Ω and the pair (Ω, Λ) is called a spectral pair. * Corresponding author.

E-mail addresses: [email protected] (D. Bose), [email protected] (S. Madan). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.011

D. Bose, S. Madan / Journal of Functional Analysis 260 (2011) 308–325

309

The study of spectral properties of sets has its origin in some questions of functional-analysis. It began with the work of B. Fuglede [15], who while investigating a question suggested to him by I. Segal concerning sets Ω which have the ‘extension property’ (namely, the existence of commuting self-adjoint extensions of the operators −i ∂x∂ 1 , . . . , −i ∂x∂ n defined on C0∞ (Ω) to a dense subspace of L2 (Ω)) observed that spectral sets have this property. Further it was shown that if Ω is assumed to be connected then having the extension property is equivalent to Ω being a spectral set [15,26,47]. For a detailed and very interesting account of the early history and motivation behind the origin of these problems we refer the interested reader to [11]. In his study of spectral sets Fuglede observed that the spectral pair problem has interesting connections to tiling problems. A measurable set T ⊂ Rd , having positive measure is said to be a prototile if T tiles Rd by translations. In other words, we say a set T as above is a prototile if there exists a subset T ⊂ Rd such that {T + t: t ∈ T } forms a partition a.e. of Rd . The set T is said to be a tiling set for T and the pair (T , T ) is called a tiling pair. Fuglede proved the following theorem: Theorem 1.1. (See Fuglede [15].) Let L be a full rank lattice in Rd and L∗ be the dual lattice. Then (Ω, L) is a tiling pair if and only if (Ω, L∗ ) is a spectral pair. He went on to make the following conjecture, which is also known as the spectral set conjecture: Conjecture 1.2 (Fuglede’s conjecture). A set Ω ⊂ Rd is a spectral set if and only if Ω tiles Rd by translations. This led to an intense study of spectral and tiling properties of sets. In recent years, this conjecture, in its full generality, has been shown to be false in both directions if the dimension d 3 [50,34,35,44,14,13]. However, interest in the conjecture is alive and the conjecture has been shown to be true in many cases under additional assumptions. For example, the case where Ω is assumed to be convex received a lot of attention recently. It is known that if a convex body K tiles Rd by translations then it is necessarily a symmetric polytope and there is a lattice L such that (K, L) is a tiling pair [53,45]. Thus the “tiling implies spectral” part of the Fuglede conjecture follows easily from Fuglede’s result. In the converse direction, it has been shown that a convex set which is spectral has to be symmetric [31], and such sets do not have a point of curvature [20,32,25] (i.e., they are symmetric polytopes). However it is only in dimension 2 that the “spectral implies tiling” part of the Fuglede conjecture has been proved [21]. In its full generality Fuglede’s conjecture remains open in dimensions 1 and 2. In one dimension the conjecture is known to be related to some interesting number theoretic questions and conjectures [6,37,40,51]. It is generally believed that the conjecture is true in dimension 1. An interesting recent development is the discovery of spectral measures [28,49], these are probability measures μ which have a spectrum Λ (i.e., the set of exponentials EΛ is an orthonormal basis for L2 (μ)). Research on these problems has led to the spectral theory for fractal measures and has received a lot of attention in recent years [10,7,8,12]. Not surprisingly, spectral sets have found application in various fields, most notably in the study of wavelets. Gabardo and Nashed introduced a generalization of Mallat’s classical mul-

310


tiresolution analysis using spectral pairs [16–19,3]. Later in a very influential work Wang [54] studied wavelets with irregular translation and dilation sets and established a surprising connection of this question to that of spectral pair and tiling sets. Starting with the work of Dutkay and Jorgensen [9] the subject of wavelets on spectral measures has gained considerable attention recently [11,4,1,2]. Attempts to answer the question about sets (measures) which admit such Fourier expansions, have revealed a plethora of connections between functional analysis, number theory, representation theory, combinatorics, commutative algebra, dynamical systems, operator theory and Fourier analysis. In both of these instances there are intriguing duality questions related to tiling problems about which we talk in more detail in the next section. 1.1. Spectral-tiling duality Starting with Fuglede’s original work, many results demonstrate that there exists a deep relationship between spectra and tiling sets. For example, when I is the unit cube in Rd , then (I, Γ ) is a tiling pair if and only if (I, Γ ) is a spectral pair. This was first conjectured by Jorgensen and Pedersen [29] who proved it for d 3. Subsequently several authors gave proofs of this result using different techniques [38,23,30,43]. It is worth mentioning here that tiling by cubes can be very complicated [39]. In fact there is a dual conjecture due to Jorgensen and Pedersen. Conjecture 1.3 (The dual spectral set conjecture [29]). A subset Γ of R is a spectrum for some spectral set Ω if and only if it is a tiling set for some prototile T . Approaching the spectral set conjecture by studying the associated spectra or tiling sets has been very fruitful, specially when these have some additional structure like periodicity. A set Γ ⊂ Rd is said to be periodic if there exists a full-rank lattice L of Rd such that Γ = L + {γ1 , . . . , γm }, and if, in addition, all coset differences γi − γj are commensurate with the lattice L, then Γ is said to be rational periodic. Pedersen [48] gave a classification of spectral sets which have a periodic spectrum expressed in terms of complex Hadamard matrices. On the other hand, Lagarias and Wang [41] gave a characterization of prototiles which tile Rd by a rational periodic tiling set in terms of factorization of abelian groups. Further, they introduced the concept of universal spectrum [41]. A tiling set T is said to have a universal spectrum ΛT , if every set Ω that tiles R d by T is a spectral set with spectrum ΛT . Lagarias and Wang [41] proved that a large class of tiling sets T have a universal spectrum and then conjectured that all rational periodic tiling sets have a universal spectrum which is also rational periodic. This is known as the Universal Spectrum conjecture. Given a rational periodic tiling set T they gave necessary and sufficient conditions for a rational periodic spectrum ΛT to be a universal spectrum for T . These developments were instrumental in disproving the “tiling implies spectral” part of Fuglede’s conjecture. Later Farkas, Matolcsi and Móra [13] proved that the “tiling implies spectral” part of Fuglede’s conjecture is equivalent to the Universal Spectrum conjecture in any dimension. Unlike in higher dimensions where the tiling sets can be very irregular (e.g. consider the case of tiling by a cube [39]) the tiling sets in one dimension are quite rigid in that they exhibit a lot of structure. This gives further credence to the belief that Fuglede conjecture may be true in this case.


311

1.2. Structure of one-dimensional tiling sets and spectrum Many results are known concerning the structure of tiling sets associated with onedimensional prototiles. The fundamental work in this setting is due to Lagarias and Wang [40], who gave a complete characterization of the structure of a tiling set T associated with a compactly supported prototile T whose boundary has measure zero. They proved that in this case T is always rational periodic and the period is an integral multiple of the measure of T . Equipped with this knowledge they manage to give a characterization of T itself. Further they show that for every tiling pair (T , T ) there exists a tiling pair (T1 , T ) where T1 is a cluster i.e., a union of equal intervals, and the problem of finding all possible tiling pairs (T , T ) is then related to finding all possible factorizations of finite cyclic groups. Thus, in essence, the entire complexity is contained in the equal interval case itself. Later Kolountzakis and Lagarias extended the periodicity result to all compactly supported prototiles [33]. Given that the tiling sets in one dimension are such highly regular object one would expect the spectra to possess a similar property in this case. Indeed, to date all known spectra associated with one-dimensional spectral sets are rational periodic. But rather surprisingly, comparatively little progress has been made in classifying the structure of spectra associated with one-dimensional spectral sets. The main result in this direction which the authors could find in the literature is by Jorgensen and Pedersen in [27], where the following assertion is proved: that a spectral set Ω ⊂ R which is a finite union of equal intervals has finitely many distinct spectra, which are all periodic. Further, under an additional hypothesis that such a set Ω is contained in a “small” interval, Laba has proved that the associated spectra for such spectral sets Ω are rational periodic [37]. For the case when Ω is a finite union of intervals (intervals can have unequal length), even less is known. Only the 2-interval case has been completely resolved by Laba [36], where she proved that Fuglede’s conjecture holds true. In [5] the 3-interval case was investigated, where it was shown that for such sets “tiling implies spectral” holds; whereas the “spectral implies tiling” part of the conjecture was proved for this case under some additional hypothesis. The general case of spectral sets Ω which are unions of finitely many intervals (not necessarily equal) was studied in [5]. It was shown there that a spectrum Λ associated with a spectral set Ω, which is a union of n-intervals has a highly “arithmetical structure”, namely, if the spectrum Λ contains an arithmetic progression of length 2n, then the complete arithmetic progression is contained in it. 1.3. Results Our objective in this paper is to study the structure of a spectrum Λ associated with a spectral set Ω ⊂ R, when Ω is a union of n-intervals. We prove that all associated spectra for such spectral sets are periodic. The essential idea behind our proof is to show that similar to the case of a tiling set a finite section of a spectrum essentially determines the complete spectrum. Theorem 2.2 and Theorem 2.8 are manifestations of this phenomenon and will be central to our proof. The other key ingredient of the proof is a density result of Landau for sets of sampling and interpolation (see Theorem 2.10). In Section 2, we state this theorem, explore the geometry of the zero set of the Fourier transform of a spectral set and prove Theorem 2.2 and Theorem 2.8. In Section 3 we prove our main theorem

312


Theorem 1.4. Let Ω be a union of n intervals, Ω =

n

such that |Ω| = 1. If (Ω, Λ) is a spectral pair, then Λ is a d-periodic set with d ∈ N. Thus Λ has the form Λ = dj =1 {λj + dZ}. j =1 Ij ,

The structure of spectral sets which have a periodic spectrum has been studied in [48] and [41]. As a consequence of Theorem 1.4 we get a structure theorem for such spectral sets and observe that the equal interval case is the generic case. Theorem 1.5. Let (Ω, Λ) be a spectral pair such that Ω is a bounded region in R and Λ is d-periodic. Then there exists a disjoint partition of [0, 1/d) into finite number of sets E1 , E2 , . . . , Ek such that Ω = kj =1 (Ej + Aj ); Aj ⊆ Z/d. Further, each set Ωj := [0, 1/d) + Aj is a spectral set with Λ as a spectrum. Our results are based on the study of the geometry of the spectrum, more specifically the study of zero sets of exponential polynomials and Landau’s density theorem about sets of sampling and interpolation which we describe in the following section. 2. The geometry of the spectrum Let (Ω, Λ) be a spectral pair. Since spectral properties of sets are invariant under affine transformations, we will henceforth assume that Ω has measure 1 and that 0 ∈ Λ ⊂ Λ − Λ. In this paper we will always assume that Ω is bounded. Then χ Ω , the Fourier transform of the characteristic function of Ω, is an entire function. Let Z(χ Ω ) be the zero set of χ Ω union {0} i.e., Z(χ Ω ) := ξ ∈ R: χ Ω (ξ ) = 0 ∪ {0}. If λ, λ ∈ Λ, then by orthogonality of eλ and eλ we have λ − λ ∈ Z(χ Ω ). Hence 0 ∈ Λ ⊂ Λ − Λ ⊂ Z(χ ). Thus the geometry of the zero set of χ plays a crucial role in determining the Ω Ω structure of Λ. Observe that, as χ Ω (0) = 1, there exists a neighborhood around 0, which does not intersect Z(χ ) except at 0. Hence, Λ is uniformly discrete. Let Λs be the set of spectral gaps for a Ω spectrum Λ i.e., Λs := {λn+1 − λn | λn ∈ Λ}. Clearly Λs ⊆ Λ − Λ ⊆ Z(χ Ω ) and Λs is bounded below. On the other hand, as a consequence of Landau’s density results (see Theorem 2.10 below), we see easily that Λs is also bounded above. So, by the analyticity of χ Ω we can conclude that Λs is finite. Thus the spectrum can be seen as a bi-infinite word made up of a finite alphabet, in terms of the spectral gaps. When Ω is a union of finite number of intervals, a much more precise estimate is known for spectral gaps [42,22,24]. nwill assume that Ω is a union of a finite number of intervals. Let Ω = nFrom now on we [a , a + r ), i i i i=1 i=1 ri = 1. Then, n χ Ω (ξ ) =

i=1 [e

2πi(ai +ri )ξ

2πiξ

− e2πi(ai )ξ ]

,


313

and Z(χ Ω ) is precisely the zero set of the exponential polynomial given by

PΩ (ξ ) :=

n 2πi(a +r )ξ i i e − e2πi(ai )ξ , i=1

which is the numerator in the expression of χ Ω . Thus we are naturally led to the study of exponential polynomials and their zeros. There is a beautiful result by Turan [52,46] which gives size estimates of exponential polynomials along arithmetic progressions. This result has the interesting consequence that if an arithmetic progression a, a + d, . . . , a + (2n − 1)d of length 2n occurs in Z(χ Ω ) then the complete arithmetic progression a + dZ ⊂ Z(χ ). This suggests that the zero sets of exponential Ω polynomials are highly structured and we are naturally led to ask the question whether Λ inherits this kind of structure? In the next section we will prove an analog of Turan’s lemma for the spectrum. 2.1. Arithmetic progressions in Λ As we have mentioned before, it was shown in [5] that the existence of an arithmetic progression of length 2n in Λ implies that the complete arithmetic progression is in Λ. Here, we improve on that result and using Newton’s Identities about symmetric polynomials, give a proof that the occurrence of an arithmetic progression of length n + 1 in the spectrum ensures that the complete arithmetic progression is in the spectrum. Let

P (z) :=

n

(z − αn ) = zn + S1 zn−1 + S2 zn−2 + · · · + Sn . i=1

Let Wk be the sum of kth power of the roots of P (z), namely Wk := α1k + α2k + · · · + αnk ;

k = 1, . . . , n.

Then the coefficients Si and Wi are related by Newton’s Identities: Wk + S1 Wk−1 + S2 Wk−2 + · · · + Sk−1 W1 + kS1 = 0;

k = 1, . . . , n.

(1)

Thus W1 , W2 , . . . , Wn uniquely determine the polynomial P (z). Proposition 2.1. If Z(χ Ω ) contains an arithmetic progression of length n + 1 with its first term 0, say 0, d, . . . , nd ∈ Z(χ Ω ) then (a) the whole arithmetic progression dZ ⊂ Z(χ Ω ), (b) d ∈ Z, and (c) Ω d-tiles R.

314


Proof. Note that if t ∈ Z(χ Ω ), then n 2πit (a +r ) j j − e 2πitaj = 0. e j =1

The hypothesis says that χ Ω (ld) = 0; l = 1, . . . , n, hence n 2πild(a +r ) j j − e 2πildaj = 0, e

l = 1, . . . , n.

j =1

We write ζ2j = e2πidaj ; ζ2j −1 = e2πid(aj +rj ) ; j = 1, . . . , n. Then the above system of equations can be rewritten as ζ1 + ζ3 + · · · + ζ2n−1 = ζ2 + ζ4 + · · · + ζ2n = W1 , 2 2 ζ12 + ζ32 + · · · + ζ2n−1 = ζ22 + ζ42 + · · · + ζ2n = W2 ,

.. . n n = ζ2n + ζ4n + · · · + ζ2n = Wn . ζ1n + ζ3n + · · · + ζ2n−1

(2)

Let P1 (z) :=

n

(z − ζ2j −1 )

and P2 (z) :=

j =1

n

(z − ζ2j ).

j =1

Then by (1) and (2) we get P1 (z) = P2 (z). Thus we get a partition of ζi ’s into n distinct pairs (ζi , ζj ) such that ζi = ζj ; i ∈ 1, 3, . . . , 2n − 1 and j ∈ 2, 4, . . . , 2n. We can relabel the ζ2j ’s, j = 1, . . . , n so that ζ2j −1 = ζ2j . But then k k ζ2j −1 = ζ2j , ∀k ∈ Z and we get χ Ω (kd) =

n 1 k k ζ2j −1 − ζ2j = 0; 2πikd

∀k ∈ Z \ {0}.

(3)

j =1

Thus dZ ⊂ Z(χ Ω ). Now consider, F (x) =

χΩ (x + k/d),

x ∈ [0, 1/d).

(4)

k∈Z

Thus F is

1 d

periodic and integer valued and

(ld) = d F

1 d

χΩ (x + k/d)e−2πildx dx = d χ Ω (ld) = dδl,0 .

k∈Z 0

Thus F (t) = d a.e. so d ∈ Z and Ω d-tiles the real line.

2

(5)


315

Using Proposition 2.1, we now prove the corresponding result for the spectrum. Theorem 2.2. Let (Ω, Λ) be a spectral pair. If for some a, d ∈ R, an arithmetic progression of length n+1, say a, a +d, . . . , a +nd ∈ Λ, then the complete arithmetic progression a +dZ ⊆ Λ. Further d ∈ Z and Ω d-tiles R. Proof. Since a, a + d, . . . , a + nd ∈ Λ, shifting Λ by a we get that Λ1 = Λ − a is a spectrum for Ω and 0, d, . . . , nd ∈ Λ1 ⊂ Λ1 − Λ1 ⊂ Z(χ Ω ). Thus surely dZ ⊂ Z(χ Ω ) by Proposition 2.1. Now, let λ ∈ Λ1 . Then by orthogonality, −λ, d − λ, 2d − λ, . . . , nd − λ ∈ Z(χ Ω ). Put ξ2j = e−2πiλaj , ζ2j = e

2πidaj

,

ξ2j −1 = e−2πiλ(aj +rj ) ; ζ2j −1 = e

2πid(aj +rj )

;

j = 1, . . . , n, j = 1, . . . , n.

Since χ Ω (kd − λ) = 0, for k = 0, . . . , n we have k k ξ1 ζ1k − ξ2 ζ2k + · · · + ξ2n−1 ζ2n−1 − ξ2n ζ2n = 0 for k = 0, . . . , n.

(6)

But the ζi ’s can be partitioned into n disjoint pairs (ζi , ζj ) such that ζi = ζj where i ∈ 1, 3, . . . , 2n − 1 and j ∈ 2, 4, . . . , 2n. Without loss of generality, we relabel the ζ2j ’s and simultaneously, the corresponding ξ2j ’s so that ζ2j −1 = ζ2j , j = 1, . . . , n. Thus from (6) we get ⎛ ⎜ ⎜ ⎜ ⎝

⎞⎛

1 ζ1 .. .

1 ζ3 .. .

··· ··· .. .

⎜ ζ2n−1 ⎟ ⎟⎜ .. ⎟ ⎜ . ⎠⎝

ζ1n−1

ζ3n−1

···

n−1 ζ2n−1

1

ξ1 − ξ2 ξ3 − ξ4 .. . ξ2n−1 − ξ2n

⎛ ⎞ 0 ⎟ ⎜0⎟ ⎟ ⎜ ⎟ ⎟ = ⎜ .. ⎟ . ⎠ ⎝.⎠ ⎞

(7)

0

Now, if [ξ1 − ξ2 , ξ3 − ξ4 , . . . , ξ2n−1 − ξ2n ]t is the trivial solution, i.e., ξ2j −1 − ξ2j = 0, ∀j = 1, . . . , n then ∀k ∈ Z, we have k 1 k k ξ1 ζ1 − ξ2 ζ2k + · · · + ξ2n−1 ζ2n−1 − ξ2n ζ2n 2πi(kd − λ) k 1 k ζ1 (ξ1 − ξ2 ) + · · · + ζ2n−1 = (ξ2n−1 − ξ2n ) = 0. 2πi(kd − λ)

χ Ω (kd − λ) =

t Thus dZ − λ ∈ Z(χ Ω ). If, however, [ξ1 − ξ2 , ξ3 − ξ4 , . . . , ξ2n−1 − ξ2n ] is not the trivial solution, then ζ2l−1 = ζ2k−1 for some l, k ∈ 1, . . . , n; l = k. l Removing all the redundant variables and writing the remaining variables as η2j +1 , j, l = 0, 1, . . . , k − 1, we get a non-singular Vandermonde matrix satisfying

316


⎛

1 η1 .. .

1 η3 .. .

··· ··· .. .

η1k−1

η3k−1

···

⎜ ⎜ ⎜ ⎝

⎞⎛ ⎞ ⎛ ⎞ 0 1 ⎟ ⎜ ⎟ ⎜ η2k−1 ⎟ 3 ⎟ ⎟⎜ ⎜0⎟ = . ⎟ ⎟ ⎜ ⎟ ⎜ .. . . ⎠ ⎝ .. ⎠ ⎝ .. ⎠ k−1 0 η2k−1 2k−1 1

(8)

where k

Then each of the

i

=

ξ2j −1 − ξ2j .

j : ζ2j −1 =ηk

= 0; i = 1, . . . , k. But, then once again ∀p ∈ Z,

χ Ω (pd − λ) =

1 p p p η1 = 0. + η3 + · · · + η2k−1 1 3 2k−1 2πi(pd − λ)

Thus dZ − λ ⊆ Z(χ Ω ). We already have dZ ⊆ Z(χ Ω ) and now we have seen if λ ∈ Λ1 then dZ − λ ∈ Z(χ Ω ). Thus dZ ⊆ Λ1 , hence a + dZ ⊂ Λ. That d ∈ Z and Ω d-tiles R follow from Proposition 2.1. 2 Remark 2.3. Theorem 2.2 is the best possible result in this direction, as existence of an arithmetic progression of shorter length in a spectrum does not ensure the complete arithmetic progression is in the spectrum. For example, consider Ω = [0, 1/3] ∪ [1, 4/3] ∪ [2, 7/3] then Λ = {0, 1/3, 2/3} + 3Z is a spectrum for Ω which contains the 3 term arithmetic progression 0, 1/3, 2/3 but clearly the complete arithmetic progression Z/3 Λ. 2.2. Embedding Λ in a vector space In this section we will investigate the spectrum in a geometric manner. The setting is again that of a set Ω, which is a union of finitely many intervals, namely, Ω = n1 [aj , aj + rj ]. We assume that Ω is spectral with a spectrum Λ. We will embed Λ in a vector space and incorporate the orthogonality of the corresponding set EΛ = {eλ : λ ∈ Λ}, via a conjugate linear form. Consider the 2n-dimensional vector space Cn × Cn . We write its elements as v = (v1 , v2 ) with v1 , v2 ∈ Cn . For v, w ∈ Cn × Cn define v w := v1 , w1 − v2 , w2 , where ·,· denotes the usual inner product on Cn . Note that this conjugate linear form is degenerate, i.e., there exists v ∈ Cn × Cn , v = 0 such that v v = 0. We call such a vector a null-vector. For example, every element of Tn × Tn is a null-vector. A subset S ⊆ Cn × Cn is called a set of mutually null-vectors if ∀v, w ∈ S we have v w = 0. It is clear from the definition that elements of a set of mutually null-vectors are themselves nullvectors. Lemma 2.4. Let S = {v1 , v2 , . . . , vm } be a set of mutually null-vectors in Cn × Cn . Let V be the linear subspace spanned by S. Then, V is a set of mutually null-vectors and dim(V ) n.


317

i Proof. v= m i=1 ai v and w = m Letj v, w ∈ V . Since the subspace V is spanned by S, we have i vj ) = 0; ∀i, j = 1, . . . , m b v . Now, as the set S is a set of mutually null-vectors (v j =1 j i j and so, we have v w = m i,j =1 ai bj (v v ) = 0. Hence, V is a set of mutually null-vectors. Let wj := (ej , 0), j = 1, . . . , n where ej ’s are the standard basis vectors of Cn . Consider the subspace W of Cn × Cn spanned by the vectors wj , j = 1, . . . , n. Since, these vectors are linearly independent in Cn × Cn , dim(W ) = n. Further, note that for w ∈ W , w = 0 we have w w > 0. Thus W ∩ V = {0} and hence dim(V ) n. 2 Suppose Ω = nj=1 [aj , aj + rj ) is a union of n disjoint intervals with a1 = 0 < a1 + r1 < a2 < a2 + r2 < · · · < an < an + rn and n1 rj = 1. We define a map ϕΩ from R to Tn × Tn ⊆ Cn × Cn by x → ϕΩ (x) = ϕ1 (x), ϕ2 (x) , where ϕ1 (x) = e2πi(a1 +r1 )x , e2πi(a2 +r2 )x , . . . , e2πi(an +rn )x and ϕ2 (x) = 1, e2πia2 x , . . . , e2πian x . The following lemma, which is immediate from the definitions, makes clear the connection between a spectral pair (Ω, Λ) and the image of Λ under the map ϕΩ . Lemma 2.5. Let Ω be a union of n intervals, as above, and suppose Γ ⊆ R. Then the set of exponentials EΓ = {eγ : γ ∈ Γ } is an orthogonal set in L2 (Ω) if and only if ϕΩ (Γ ) := {ϕΩ (γ ): γ ∈ Γ } is a set of mutually null-vectors. Thus, if (Ω, Λ) is a spectral pair, ϕΩ (Λ) is a set of mutually null-vectors. What about the converse? We will now try to find some criterion to decide whether a given pair (Ω, Λ) is a spectral pair. First, observe that from Lemma 2.4, we already know that if (Ω, Λ) is a spectral pair then the vector space VΩ (Λ) := span{ϕΩ (λ): λ ∈ Λ} has dimension at most n. We will now show that Λ has a “local finiteness property”, in the sense that there exists a finite subset B of Λ, #B n, such that Λ gets uniquely determined by B. Lemma 2.6. Let (Ω, Λ) be a spectral pair and B = {y1 , . . . , ym } ⊆ Λ be such that ϕΩ (B) := {ϕΩ (y1 ), . . . , ϕΩ (ym )} forms a basis of VΩ (Λ). Then x ∈ Λ iff ϕΩ (x) ϕΩ (yi ) = 0, ∀i = 1, . . . , m. Proof. Let x ∈ Λ. Since B ⊆ Λ, by orthogonality we have ex , eyi = 0, ∀yi ∈ B and the result follows from Lemma 2.5. For the converse, let dim(VΩ (Λ)) = m and B = {y1 , . . . , ym } ⊆ Λ be such that ϕΩ (B) is a / Λ such that ϕΩ (x) ϕΩ (yj ) = 0, ∀yj ∈ B. basis for VΩ (Λ). Suppose there exists some x ∈ Since ϕΩ (B) is a basis for VΩ (Λ), we have for any λ ∈ Λ, ϕΩ (λ) = m j =1 aj ϕΩ (yj ). Now by

318


linearity we get ϕΩ (x) ϕΩ (λ) = m j =1 aj (ϕΩ (x) ϕΩ (yj )) = 0. Hence by Lemma 2.5 we get ex , eλ = 0, ∀λ ∈ Λ. But EΛ = {eλ : λ ∈ Λ} is total in L2 (Ω), and ex ≡ 0, a contradiction. Thus x must be in Λ. 2 The following lemma, gives a rather nice criterion for a spectrum Λ to be periodic. Lemma 2.7. Let dim(VΩ (Λ)) = m n and B = {y1 , . . . , ym } ⊆ Λ be such that ϕΩ (B) is a basis for VΩ (Λ). If for some d ∈ R, we have B + d = {y1 + d, . . . , ym + d} ⊆ Λ then Λ is d-periodic, i.e., Λ = {λ1 , . . . , λd } + dZ. Proof. By Lemma 2.6 x ∈ Λ iff ϕΩ (x) ϕΩ (yj ) = 0, j = 1, . . . , m. Let λ ∈ Λ, since B + d ⊆ Λ we get ϕΩ (λ) ϕΩ (yj + d) = 0, j = 1, . . . , m ⇔ ϕΩ (λ − d) ϕΩ (yj ) = 0, j = 1, . . . , m ⇔ λ − d ∈ Λ and hence Λ is d-periodic. By Theorem 2.2 we get d ∈ N and since Λ has density 1 by Theorem 2.10, we conclude that Λ = {λ1 , . . . , λd } + dZ. 2 Recall, that if Γ is periodic, has density 1 and ϕΩ (Γ ) is a set of mutually null-vectors, then by [48,41] (Ω, Γ ) is a spectral pair. Let (Ω, Γ ) be such that ϕΩ (Γ ) is a set of mutually null-vectors. The natural queston is: Can we extend Γ to a spectrum of Ω? The following theorem gives a criterion for periodic orthogonal extension of a set Γ and will be central to our proof of periodicity of a spectrum in the next section. Theorem 2.8. Let Γ ⊂ R be such that the set of exponentials EΓ is orthogonal in L2 (Ω). Let dim(VΩ (Γ )) = r and B0 = {μ1 , . . . , μr } be such that ϕΩ (B0 ) forms a basis of VΩ (Γ ). Further suppose a translate of B0 is contained in Γ , i.e., B1 = B0 + d ⊆ Γ . Then Γ can be extended periodically to obtain a d-periodic subset Γd ⊆ R such that the set of exponentials EΓd are orthogonal in L2 (Ω). Proof. Let Γd := Γ + dZ. As in Lemma 2.7, we will prove that ϕΩ (Γd ) is a mutually null set. We will first show by induction that ϕΩ (μk ) ϕΩ (μj + ld) = 0 for all l ∈ Z, and j, k = 1, . . . , r. Observe that both ϕΩ (B0 ) and ϕΩ (B1 ) span the same vector space VΩ (Γ ). Let us assume that the orthogonality relations hold for all s = 1, . . . , l − 1 i.e., ϕΩ (μk ) ϕΩ (μj + sd) = 0 for all j, k = 1, . . . , r. We have to show ϕΩ (μk ) ϕΩ (μj + ld) = 0 for all j, k = 1, . . . , r. But by the induction hypothesis, we have ϕΩ (μk + d) ϕΩ (μj + ld) = ϕΩ (μk ) ϕΩ μj + (l − 1)d = 0,

∀j, k = 1, . . . , r.


319

But, we know that ϕΩ (B0 ) ⊆ span{ϕΩ (B1 )}. Hence, ϕΩ (μk ) ϕΩ (μj + ld) = 0,

∀j, k = 1, . . . , r.

Now if γ , γ ∈ Γd , then γ = γp + ld, γ = γp + l d for some γp , γp ∈ Γ and l, l ∈ Z. Since ϕΩ (γp ), ϕΩ (γp ) ∈ VΩ (Γ ) = Span{ϕΩ (B0 )}, we have ϕΩ (γp ) =

r

αk ϕΩ (μk )

r and ϕΩ γp = αj ϕΩ (μj ). j =1

k=1

Now, ϕΩ (γ ) ϕΩ γ = ϕΩ (γp + ld) ϕΩ γp + l d = ϕΩ γp + l − l d ϕΩ γp r αj ϕΩ (μj ) = ϕΩ γp + l − l d 1

=

r

αj ϕΩ (γp ) ϕΩ μj + l − l d

j =1

=

r r

αj αk ϕΩ (μk ) ϕΩ μj + l − l d = 0.

2

j =1 k=1

Remark 2.9. Under the assumption of Lemma 2.7 the Λd obtained in Theorem 2.8 is Λ itself. 2.3. Density of the spectrum Let Γ ⊂ R be a uniformly discrete set. Then we define n+ (R), n− (R) respectively, as the largest and smallest number of elements of Γ contained in any interval of length R, i.e., n+ (R) = max # Γ ∩ [x − R, x + R] , x∈R − n (R) = min # Γ ∩ [x − R, x + R] . x∈R

A uniformly discrete set Γ is called a set of sampling for L2 (Ω), if there exists a con2 stant K such that f 2 K λ∈Λ |fˆ(λ)|2 , ∀f ∈ L2 (Ω), and Γ is called a set of interpolation for L2 (Ω), if for every square summable sequence {aγ }γ ∈Γ , there exists an f ∈ L2 (Ω) with fˆ(γ ) = aγ , γ ∈ Γ . Clearly if (Ω, Λ) is a spectral pair, then Λ is both a set of sampling and a set of interpolation for L2 (Ω). The following result of Landau, regarding sets of sampling and interpolation gives an estimate on the numbers n+ (R) and n− (R) for a spectrum Λ, when Ω is a union of a finite number of intervals. Theorem 2.10. (See Landau [42].) Let Ω be a union of a finite number of intervals with total measure 1, and Λ a uniformly discrete set. Then:

320


(1) If Λ is a set of sampling for L2 (Ω), n− (R) R − A log+ R − B. (2) If Λ is a set of interpolation for L2 (Ω), n+ (R) R − A log+ R − B where A and B are constants independent of R. It follows from Theorem 2.10 that Λ has asymptotic density 1, that is ρ(Λ) := lim

R→∞

#(Λ ∩ [−R + x, R + x]) = 1, 2R

uniformly in x ∈ R.

3. Proof of periodicity of the spectrum n nOnce again in this section Ω ⊂ R is a union of finitely many intervals, Ω = 1 [aj , aj + rj ], j =1 rj = 1. We assume that Ω is spectral with a spectrum Λ. We will continue to use the notations introduced in Section 2. We begin with some definitions. Let Λ = {λj }j ∈Z where λj < λj +1 and λ0 = 0. Recall that the consecutive distance set of Λ, namely Λs = {λj +1 − λj : j ∈ Z} is finite. So we can view Λ as an infinite word with a finite alphabet Λs = {d1 , d2 , . . . , dl }. For a finite word W = [dj1 , dj2 , . . . , djn ], dji ∈ Λs we write length(W ) = ni=1 dji . Suppose dim(VΩ (Λ)) = m n and let {μ1 , μ2 , . . . , μm } be such that {ϕΩ (μj ), j = 1, 2, . . . , m} is a basis for VΩ (Λ). Choose L0 such that {μ1 , μ2 , . . . , μm } ⊆ [0, L0 ] and then for any L L0 , partition R as R=

kL, (k + 1)L . k∈Z

Let ΛL k = Λ ∩ kL, (k + 1)L . Now, for each k ∈ Z, ΛL k , corresponds to a finite word of length at most L, and there are only finitely many, say NL , words of length at most L. Let VkL = Span ϕΩ (λ): λ ∈ ΛL k . Let us first consider the special case that for some large enough L we have dim VkL = m

for every k ∈ Z.

(9)


321

k k k In this case, each ΛL k has a set of m elements Bk := {μ1 , μ2 , . . . , μm } such that ϕΩ (Bk ) := k k {ϕΩ (μ1 ), . . . , ϕΩ (μm )} forms a basis of VΩ (Λ). Also by the remarks above, at least two of the L L L words ΛL k1 and Λk2 must be the same. Hence for some d ∈ R, Λk2 = Λk1 + d. In particular, k

k

0 0 there exists k0 , such that ΛL k0 contains a set of elements {μ1 , . . . , μm } which form a basis of

k

k

VΩ (Λ) and also {μ10 , . . . , μm0 } + d ⊆ Λ. Thus the hypothesis of Lemma 2.7 holds, and so Λ is d-periodic. Observe that in the above argument, we do not require as much as (9). It would be enough if {k: dim(VkL ) = m} is an infinite set, or for that matter, has at least NL + 1 elements. But once we conclude that Λ is periodic, it will follow that for some, possibly larger L (if d is the period L = 3d will do) that dim(VkL ) = m, ∀k ∈ Z. For the general case, let 1 s m, and L > 0 and write EsL = k: dim VkL s . L = Z, then Λ is periodic. Suppose this is not the We have just seen that if for some L > 0, Em case. Then we need the following lemma:

L = Z. Lemma 3.1. Let m m be the largest integer such that there exists an L > 0 so that Em L Then m itself will occur infinitely often in the set {dim(Vk )}k∈Z .

Proof. First note that for s = 1, we can choose L > max{dj }, and then E1L = Z so clearly m 1. If dim(VkL ) = m only for finitely many k’s then we can take L˜ large enough so that ˜ ˜ (k + 1)L]}. ˜ Let L = 2L, ˜ then dim(V L ) = m for precisely one interval of the partition {[k L, k

L observe that Em +1 = Z, and this contradicts maximality of m . (Without loss of generality we may choose L ∈ N.) 2

We will now prove Theorem 1.4. Proof of Theorem 1.4. Step 1. We will first prove that the spectrum Λ can be modified to a set Λd which is d-periodic and is such that (Ω, Λd ) is a spectral pair. For this we use Landau’s density result to extract a “patch” from Λ which has some periodic structure and has a large enough density. Then we use Theorem 2.8 to show that a suitable periodization of this patch is a spectrum. With L as above, let L = Then choose L∗ >

1 2L

1 . 2L (NL + 1)

= L (NL + 1) such that n− (L∗ )/L∗ > 1 − L . ∗

L = Z and also that the cardinality of the In the case under consideration, we know that Em ∗ ∗ L set {p: dim(Vp ) = m } is infinite. We choose and fix one such p such that dim(VpL ) = m . By the choice of L∗ , the interval [pL∗ , (p + 1)L∗ ) contains at least (NL + 1) disjoint intervals ∗ ∗ of length L . Now for j = 1, . . . , NL + 1 each of the ΛL j ⊂ [pL , (p + 1)L ) has a word Wj of length at most L associated with it. Further, observe that by the choice of L , each of these

322


ΛL j contains at least m elements whose image under ϕΩ is a linearly independent set, and that,

∗

by the choice of p, there can be at most m such elements. Notice this implies VpL = VjL , j = 1, . . . , NL + 1. L Hence by the pigeon hole principle, there exists k1 and k2 such that the words ΛL k1 and Λk2

L are the same, and therefore ΛL k2 = Λk1 + d for some d ∈ R, where d (NL + 1)L = To complete the proof, we will need the following lemma: ∗

1 2L

.

∗

L Lemma 3.2. Let Λd be the d-periodization of ΛL p , i.e. Λd = {Λp + dZ}. Then Λd is orthogonal.

∗

L Proof. Let B0 = {μ1 , . . . , μm } ⊆ ΛL k1 ⊆ Λp be such that ϕΩ (B0 ) := {ϕΩ (μ1 ), . . . , ϕΩ (μm )} ∗

L L is a basis of VpL and also of VkL1 . Now since ΛL k1 + d = Λk2 , B1 = B0 + d ⊆ Λk2 , this subset ∗ again gives a basis for VpL . By Theorem 2.8 we see that the set of exponentials EΛd are mutually orthogonal in L2 (Ω). 2

Now since Λd is orthogonal it is a set of interpolation and by Landau’s density theorem we get ρ(Λd ) 1. But by our choice of L∗ we get ρ(Λd ) > n− (L∗ )/L∗ > 1 − L . On the other hand, since Λd is d-periodic, if ρ(Λd ) < 1, we have ρ(Λd ) 1 − d1 < 1 − 2L as d1 2L . This is a contradiction. It follows that Λd is a periodic set whose density is 1 and EΛd is orthogonal in L2 (Ω). Thus for Ω [48,41]. Since Λd has density of 1 and is d-periodic it can be we get Λd is a spectrum written in the form Λd = dj =1 (μj + dZ). Step 2. We now prove that Λ itself is periodic. Once again we will be using Landau’s density theorem and Theorem 2.8 along with Theorem 2.2 which will be crucial. ∗ Choose L∗ as above, so that {p: dim(VpL ) = m } is infinite. Then let L∗∗ be such that n− L∗∗ /L∗∗ > 1 −

1 2(n + 1)L∗

and L∗∗ (n + 1)L∗ .

(Recall that n is the number of intervals in Ω.) Here by we mean that many blocks of intervals, each of length (n + 1)L∗ are contained in any interval of the L∗∗ -grid. ∗∗ Then we can find a p such that dim(VpL ) = m (since there are infinitely many such). Now d ∗∗ ∗ ∗ ∗ extend ΛL p d-periodically to a spectrum Λd of Ω, where d < L . Write Λd = j =1 (μj + dZ), ∗∗ ∗∗ with μ1 , μ2 , . . . , μd ∈ [pL , (p + 1)L ). We end the proof by showing that in fact Λ∗d = Λ. For this it will be enough to show that for each μj , there are (n + 1) consecutive terms from the arithmetic progression μj + dZ in Λ. Suppose this is not the case, then for each a ∈ Z such that [μj + ad, μj + (a + n)d] ⊂ [pL∗∗ , (p + 1)L∗∗ ] we have at least one element from the n + 1 length AP μj + ad, μj + ∗∗ (a + 1)d, . . . , μj + (a + n)d is missing from ΛL p . But that will affect the density, so that 1 1 1 − (n+1)L n− (L∗∗ )/L∗∗ 1 − (n+1)d ∗ , which is a contradiction. Now By Theorem 2.2 we get that Λ is indeed periodic. 2 The structure of spectral sets Ω which have a periodic spectrum is well known (see [48, 41]). Here for the sake of completeness we give a structure theorem for Ω using a result of Kolountzakis.


323

Theorem. [30].) Let Ω be a bounded open set, Λ a discrete set in Rd , and (See Kolountzakis 2 + Λ is a tiling if and only if Λ has uniformly bounded density and | δΛ = λ∈Λ δλ . Then |χ Ω (Ω − Ω) ∩ supp(δ Λ ) = {0}. We will now prove Theorem 1.5. 2 Proof of Theorem 1.5. Recall that (Ω, Λ) is a spectral pair if and only if |χ Ω | + Λ is a tiling. Further if Λ is d-periodic, then Ω d-tiles R, i.e. n χΩ (x + n/d) = d. In particular,

dχ[0, 1 ) (x) = χ[0, 1 ) d

d

χΩ (x + k/d).

k∈Z

So for each x ∈ [0, dk ), the set Ax = {k ∈ Z: x + k/d ∈ Ω} has cardinality d. Define an equivalence relation ≈ on [0, 1/d) by x ≈ y if and only if Ax = Ay . Since Ω is bounded, the above equivalence relation gives a partition of [0, 1/d) into finitely many equivalence classes E1 , E2 , . . . , Ek . For each Ej we write Aj for the common set defined above. Then Ω = kj =1 (Ej + Aj ) and [0, 1/d) = kj =1 Ej and we may assume |Ej | > 0, ∀j = 1, 2, . . . , k. Now let Ωj := [0, 1/d) + Aj . Our claim is (Ωj , Λ) is a spectral pair. We will need the above mentioned theorem due to Kolountzakis [30]. Now as (Aj ) = d we have |Ωj | = 1. If Λ = Γ + dZ with Γ = {λ1 , λ2 , . . . , λd }, then supp(δ Λ ) = {k/d: δ Λ (k/d) = 0} ⊆ Z/d and supp(Ωj − Ωj ) ⊆ (−1/d, 1/d) + Aj − Aj . But Aj − Aj is 1/d-separated, so supp(Ωj − Ωj ) ∩ supp(δ Λ ) = {0} for otherwise as Ej + Aj ⊆ Ω and |Ej | > 0 we get supp(Ω −Ω)∩supp(δ Λ ) = {0} and thus (Ω, Λ) cannot be a spectral set. 2 Acknowledgments The authors would like to thank Krishnan Rajkumar and C.P. Anil Kumar for the many insightful comments and suggestions they made at several stages of this work and for providing us with much needed encouragement. References [1] L.W. Baggett, V. Furst, K.D. Merrill, J.A. Packer, Classification of generalized multiresolution analyses, J. Funct. Anal. 258 (12) (2010) 4210–4228. [2] L.W. Baggett, N.S. Larsen, J.A. Packer, I. Raeburn, A. Ramsay, Direct limits, multiresolution analyses, and wavelets, J. Funct. Anal. 258 (8) (2010) 2714–2738. [3] B. Behera, Wavelet packets associated with nonuniform multiresolution analyses, J. Math. Anal. Appl. 328 (2) (2007) 1237–1246. [4] J. Bohnstengel, M. Kesseböhmer, Wavelets for iterated function systems, J. Funct. Anal. 259 (3) (2010) 583–601. [5] D. Bose, C.P. Anil Kumar, R. Krishnan, S. Madan, On Fuglede’s conjecture for three intervals, http://arxiv.org/ abs/0803.0049; Online J. Anal. Comb., in press. [6] E.M. Coven, A. Meyerowitz, Tiling the integers with translates of one finite set, J. Algebra 212 (1) (1999) 161–174. [7] D.E. Dutkay, D. Han, P.E.T. Jorgensen, Orthogonal exponentials, translations, and Bohr completions, J. Funct. Anal. 257 (9) (2009) 2999–3019. [8] D.E. Dutkay, D. Han, Q. Sun, On the spectra of a Cantor measure, Adv. Math. 221 (1) (2009) 251–276. [9] D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoam. 22 (1) (2006) 131–180.

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[10] D.E. Dutkay, P.E.T. Jorgensen, Fourier frequencies in affine iterated function systems, J. Funct. Anal. 247 (1) (2007) 110–137. [11] D.E. Dutkay, P.E.T. Jorgensen, Fourier series on fractals: a parallel with wavelet theory, in: Radon Transforms, Geometry, and Wavelets, in: Contemp. Math., vol. 464, Amer. Math. Soc., Providence, RI, 2008, pp. 75–101. [12] D.E. Dutkay, P.E.T. Jorgensen, Quasiperiodic spectra and orthogonality for iterated function system measures, Math. Z. 261 (2) (2009) 373–397. [13] B. Farkas, M. Matolcsi, P. Móra, On Fuglede’s conjecture and the existence of universal spectra, J. Fourier Anal. Appl. 12 (2006) 483–494. [14] B. Farkas, Sz.Gy. Revesz, Tiles with no spectra in dimension 4, Math. Scand. 98 (2006) 44–52. [15] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974) 101–121. [16] J.P. Gabardo, M.Z. Nashed, Nonuniform multiresolution analyses and spectral pairs, J. Funct. Anal. 158 (1) (1998) 209–241. [17] J.P. Gabardo, M.Z. Nashed, An analogue of Cohen’s condition for nonuniform multiresolution analyses, in: Wavelets, Multiwavelets, and Their Applications, San Diego, CA, 1997, in: Contemp. Math., vol. 216, Amer. Math. Soc., Providence, RI, 1998, pp. 41–61. [18] J.P. Gabardo, X. Yu, Wavelets associated with nonuniform multiresolution analyses and one-dimensional spectral pairs, J. Math. Anal. Appl. 323 (2) (2006) 798–817. [19] J.P. Gabardo, X. Yu, Nonuniform wavelets and wavelet sets related to one-dimensional spectral pairs, J. Approx. Theory 145 (1) (2007) 133–139. [20] A. Iosevich, N.H. Katz, T. Tao, Convex bodies with a point of curvature do not have Fourier bases, Amer. J. Math. 123 (1) (2001) 115–120. [21] A. Iosevich, N.H. Katz, T. Tao, The Fuglede spectral conjecture holds for convex planar domains, Math. Res. Lett. 10 (5–6) (2003) 559–569. [22] A. Iosevich, M.N. Kolountzakis, A Weyl type formula for Fourier spectra and frames, Proc. Amer. Math. Soc. 134 (11) (2006) 3267–3274. [23] A. Iosevich, S. Pedersen, Spectral and tiling properties of the unit cube, Int. Math. Res. Not. 1998 (16) (1998) 819–828. [24] A. Iosevich, S. Pedersen, How large are the spectral gaps?, Pacific J. Math. 192 (2) (2000) 307–314. [25] A. Iosevich, M. Rudnev, A combinatorial approach to orthogonal exponentials, Int. Math. Res. Not. 2003 (50) (2003) 2671–2685. [26] P.E.T. Jorgensen, Spectral theory of finite volume domains in Rn , Adv. Math. 44 (1982) 105–120. [27] P.E.T. Jorgensen, S. Pedersen, Estimates on the spectrum of fractals arising from affine iterations, in: Fractal Geometry and Stochastics, Finsterbergen, 1994, in: Progr. Probab., vol. 37, Birkhäuser, Basel, 1995, pp. 191–219. [28] P.E.T. Jorgensen, S. Pedersen, Dense analytic subspaces in fractal L2 -spaces, J. Anal. Math. 75 (1998) 185–228. [29] P.E.T. Jorgensen, S. Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (4) (1999) 285–302. [30] M. Kolountzakis, Packing, tiling, orthogonality and completeness, Bull. Lond. Math. Soc. 32 (5) (2000) 589–599. [31] M. Kolountzakis, Non-symmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (3) (2000) 542–550. [32] M. Kolountzakis, Distance sets corresponding to convex bodies, Geom. Funct. Anal. 14 (4) (2004) 734–744. [33] M.N. Kolountzakis, J.C. Lagarias, Structure of tilings of the line by a function, Duke Math. J. 82 (3) (1996) 653–678. [34] M. Kolountzakis, M. Matolcsi, Tiles with no spectra, Forum Math. 18 (3) (2006) 519–528. [35] M. Kolountzakis, M. Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collect. Math. Extra (2006) 281–291. [36] I. Laba, Fuglede’s conjecture for a union of two intervals, Proc. Amer. Math. Soc. 129 (10) (2001) 2965–2972. [37] I. Laba, The spectral set conjecture and multiplicative properties of roots of polynomials, J. Lond. Math. Soc. (2) 65 (3) (2002) 661–671. [38] J.C. Lagarias, J.A. Reeds, Y. Wang, Orthonormal bases of exponentials for the n-cube, Duke Math. J. 103 (1) (2000) 25–37. [39] J.C. Lagarias, P. Shor, Keller’s conjecture on cube tilings is false in high dimensions, Bull. Amer. Math. Soc. (N.S.) 27 (2) (1992) 279–283. [40] J.C. Lagarias, Y. Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1–3) (1996) 341–365. [41] J.C. Lagarias, Y. Wang, Spectral sets and factorizations of finite abelian groups, J. Funct. Anal. 145 (1) (1997) 73–98. [42] H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52.


[43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

325

Jian-Lin Li, On characterizations of spectra and tilings, J. Funct. Anal. 213 (1) (2004) 31–44. M. Matolcsi, Fuglede’s conjecture fails in dimension 4, Proc. Amer. Math. Soc. 133 (10) (2005) 3021–3026. P. McMullen, Convex bodies which tile space by translation, Mathematika 27 (1) (1980) 113–121. F.L. Nazarov, Local estimates of exponential polynomials and their applications to inequalities of uncertainty principle type, Algebra i Analiz 5 (4) (1993) 3–66, translation in: St. Petersburg Math. J. 5 (4) (1994) 663–717. S. Pedersen, Spectral theory of commuting self-adjoint partial differential operators, J. Funct. Anal. 73 (1987) 122– 134. S. Pedersen, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (2) (1996) 496–509. R.S. Strichartz, Remarks on: “Dense analytic subspaces in fractal L2 -spaces”, J. Anal. Math. 75 (1998) 229–231. T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2–3) (2004) 251–258. R. Tijdeman, Decomposition of the integers as a direct sum of two subsets, in: Number Theory, Paris, 1992–1993, in: London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 261–276. P. Turan, Eine neue Methode in der Analyses und deren Anwendungen, Acad. Kiado, Budapest, 1953. B.A. Venkov, On a class of Euclidean polyhedra, Vestn. Leningrad. Univ. Ser. Mat. Fiz. Him. 9 (2) (1954) 11–31. Y. Wang, Wavelets, tiling, and spectral sets, Duke Math. J. 114 (1) (2002) 43–57.


Orthogonality-preserving, C ∗ -conformal and conformal module mappings on Hilbert C ∗ -modules ✩ Michael Frank a,∗ , Alexander S. Mishchenko b , Alexander A. Pavlov c,d a Hochschule für Technik, Wirtschaft und Kultur (HTWK) Leipzig, Fakultät Informatik,

Mathematik und Naturwissenschaften, PF 301166, 04251 Leipzig, Germany b Moscow State University, Faculty of Mechanics and Mathematics, Main Building, Leninskije Gory,

119899 Moscow, Russia c Moscow State University, 119 922 Moscow, Russia d Università degli Studi di Trieste, I-34127 Trieste, Italy

Received 12 August 2009; accepted 6 October 2010


Abstract We investigate orthonormality-preserving, C ∗ -conformal and conformal module mappings on full Hilbert C ∗ -modules to obtain their general structure. Orthogonality-preserving bounded module maps T act as a multiplication by an element λ of the center of the multiplier algebra of the C ∗ -algebra of coefficients combined with an isometric module operator as long as some polar decomposition conditions for the specific element λ are fulfilled inside that multiplier algebra. Generally, T always fulfills the equality T (x), T (y) = |λ|2 x, y for any elements x, y of the Hilbert C ∗ -module. At the contrary, C ∗ -conformal and conformal bounded module maps are shown to be only the positive real multiples of isometric module operators. © 2010 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebra; Hilbert C ∗ -modules; Orthogonality-preserving mappings; Conformal mappings; Isometries

✩ The research has been supported by a grant of Deutsche Forschungsgemeinschaft (DFG) and by the RFBR-grant 07-01-91555. * Corresponding author. E-mail addresses: [email protected] (M. Frank), [email protected], [email protected] (A.S. Mishchenko), [email protected] (A.A. Pavlov).


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The set of all orthogonality-preserving bounded linear mappings on Hilbert spaces is fairly easy to describe, and it coincides with the set of all conformal linear mappings there: a linear map T between two Hilbert spaces H1 and H2 is orthogonality-preserving if and only if T is the scalar multiple of an isometry V with V ∗ V = idH1 . Furthermore, the set of all orthogonalitypreserving mappings {λ · V : λ ∈ C, V ∗ V = idH1 } corresponds to the set of all those maps which transfer tight frames of H1 into tight frames of (norm-closed) subspaces V (H1 ) of H2 , cf. [12]. The latter fact transfers to the more general situation of standard tight frames of Hilbert C ∗ modules in case the image submodule is an orthogonal summand of the target Hilbert C ∗ -module, cf. [9, Prop. 5.10]. Also, module isometries of Hilbert C ∗ -modules are always induced by module unitary operators between them [15], [13, Prop. 2.3]. However, in case of a non-trivial center of the multiplier algebra of the C ∗ -algebra of coefficients the property of a bounded module map to be merely orthogonality-preserving might not infer the property of that map to be (C ∗ -)conformal or even isometric. So the goal of the present note is to derive the structure of arbitrary orthogonality-preserving, C ∗ -conformal or conformal bounded module mappings on Hilbert C ∗ -modules over (non-)unital C ∗ -algebras without any further assumption. Partial solutions can be found in a publication by D. Ilišević and A. Turnšek for C ∗ -algebras A of coefficients which admit a faithful ∗-representation π on some Hilbert space H such that K(H ) ⊆ π(A) ⊆ B(H ), cf. [13, Thm. 3.1]. Orthogonality-preserving mappings have been mentioned also in a paper by J. Chmieliński, D. Ilišević, M.S. Moslehian, Gh. Sadeghi, [7, Thm. 2.2]. For C ∗ -algebras results can be found in [4]. In two working drafts [16,17] by Chi-Wai Leung, Chi-Keung Ng and Ngai-Ching Wong found by a Google search in May 2009 we obtained further partial results on orthogonality-preserving linear mappings on Hilbert C ∗ -modules. Orthogonality-preserving bounded linear mappings between C ∗ -algebras have been considered by J. Schweizer in his Habilitation thesis in 1996 [21, Props. 4.5–4.8]. His results are of interest in application to the linking C ∗ -algebras of Hilbert C ∗ -modules. A bounded module map T on a Hilbert C ∗ -module M is said to be orthogonalitypreserving if T (x), T (y) = 0 in case x, y = 0 for certain x, y ∈ M. In particular, for two Hilbert C ∗ -modules M, N over some C ∗ -algebra A a bounded module map T : M → N is orthogonality-preserving if and only if the validity of the inequality x, x x + ay, x + ay for some x, y ∈ M and any a ∈ A forces the validity of the inequality T (x), T (x) T (x) + aT (y), T (x) + aT (y) for any a ∈ A, cf. [13, Cor. 2.2]. So the property of a bounded module map to be orthogonality-preserving has a geometrical meaning considering pairwise orthogonal one-dimensional C ∗ -submodules and their orthogonality in a geometric sense. Orthogonality of elements of Hilbert C ∗ -modules with respect to their C ∗ -valued inner products is different from the classical James–Birkhoff orthogonality defined with respect to the norm derived from the C ∗ -valued inner products, in general. Nevertheless, the results are similar in both situations, and the roots of both these problem fields coincide for the particular situation of Hilbert spaces. For results in this parallel direction the reader might consult publications by A. Koldobsky [14], by A. Turnšek [24,25], by J. Chmieliński [5,6], and by A. Blanco and A. Turnšek [3], among others. Further resorting to C ∗ -conformal or conformal mappings on Hilbert C ∗ -modules, i.e. bounded module maps preserving either a generalized C ∗ -valued angle x, y/xy for any x, y of the Hilbert C ∗ -module or its normed value, we consider a particular situation of orthogonality-preserving mappings. Surprisingly, both these sets of orthogonality-preserving and of (C ∗ -)conformal mappings are found to be different in case of a non-trivial center of the multiplier algebra of the underlying C ∗ -algebra of coefficients.


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The content of the present paper is organized as follows. In the following section we investigate the general structure of orthogonality-preserving bounded module mappings on Hilbert C ∗ -modules. The results are formulated in Theorems 3 and 4. In the last section we characterize C ∗ -conformal and conformal bounded module mappings on Hilbert C ∗ -modules, see Theorems 6 and 8. Since we rely only on the very basics of ∗-representation and duality theory of C ∗ -algebras and of Hilbert C ∗ -module theory, respectively, we refer the reader to the monographs by M. Takesaki [23] and by V.M. Manuilov and E.V. Troitsky [18], or to other relevant monographical publications for basic facts and methods of both these theories. 1. Orthogonality-preserving mappings The set of all orthogonality-preserving bounded linear mappings on Hilbert spaces is fairly easy to describe. For a given Hilbert space H it consists of all scalar multiples of isometries V , where an isometry is a map V : H → H such that V ∗ V = idH . Any bounded linear orthogonalitypreserving map T induces a bounded linear map T ∗ T : H → H . For a non-zero element x ∈ H set T ∗ T (x) = λx x + z with z ∈ {x}⊥ and λx ∈ C. Then the given relation x, z = 0 induces the equality 0 = T (x), T (z) = T ∗ T (x), z = λx x + z, z = z, z. Therefore, z = 0 by the non-degenerateness of the inner product, and λx 0 by the positivity of T ∗ T . Furthermore, for two orthogonal elements x, y ∈ H one has the equality λx+y (x + y) = T ∗ T (x + y) = λx x + λy y which induces the equality λx+y x, x = λx x, x after scalar multiplication by x ∈ H . Since the element x, x is invertible in C we can conclude that the orthogonality-preserving operator T induces an operator T ∗ T which acts as a positive scalar multiple λ · idH of the identity operator on any orthonormal basis of the Hilbert space H . So T ∗ T = λ · idH on the Hilbert space H by linear continuation. The polar decomposition of T inside the √ von Neumann algebra B(H ) of all bounded linear operators on H gives us √ the equality T = λV for an isometry V : H → H , i.e. with V ∗ V = idH . The positive number λ can be replaced by an arbitrary complex number of the same modulus multiplying by a unitary u ∈ C. In this case the isometry V has to be replaced by the isometry u∗ V to yield another decomposition of T in a more general form. As a natural generalization of the described situation one may change the algebra of coefficients to arbitrary C ∗ -algebras A and the Hilbert spaces to C ∗ -valued inner product A-modules, the (pre-)Hilbert C ∗ -modules. Hilbert C ∗ -modules are an often used tool in the study of locally compact quantum groups and their representations, in noncommutative geometry, in KK-theory, and in the study of completely positive maps between C ∗ -algebras, among other research fields. To be more precise, a (left) pre-Hilbert C ∗ -module over a (not necessarily unital) C ∗ -algebra A is a left A-module M equipped with an A-valued inner product ·,· : M × M → A, which is A-linear in the first variable and has the properties x, y = y, x∗ , x, x 0 with equality if and only if x = 0. We always suppose that the linear structures of A and M are compatible. A pre-Hilbert A-module M is called a Hilbert A-module if M is a Banach space with respect to the norm x = x, x1/2 .

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Consider bounded module orthogonality-preserving maps T on Hilbert C ∗ -modules M. For several reasons we cannot repeat the simple arguments given for Hilbert spaces in the situation of an arbitrary Hilbert C ∗ -module, in general. First of all, the bounded module operator T might not admit a bounded module operator T ∗ as its adjoint operator, i.e. satisfying the equality T (x), y = x, T ∗ (y) for any x, y ∈ M. Secondly, orthogonal complements of subsets of a Hilbert C ∗ -module might not be orthogonal direct summands of it. Last but not least, Hilbert C ∗ -modules might not admit analogs (in a wide sense) of orthogonal bases. So the understanding of the nature of bounded module orthogonality-preserving operators on Hilbert C ∗ -modules involves both more global and other kinds of localization arguments. Example 1. Let A be the C ∗ -algebra of continuous functions on the unit interval [0, 1] equipped with the usual Borel topology. Let I = C0 ((0, 1]) be the C ∗ -subalgebra of all continuous functions on [0, 1] vanishing at zero. I is a norm-closed two-sided ideal of A. Let M1 = A ⊕ A be the Hilbert A-module that consists of two copies of A, equipped with the standard A-valued inner product on it. Consider the multiplication T1 of both parts of M1 by the function a(t) ∈ A, a(t) := t for any t ∈ [0, 1]. Obviously, the map T1 is bounded, A-linear, injective and orthogonality-preserving. However, its range is even not norm-closed in M1 . Let M2 = I ⊕ l2 (A) be the orthogonal direct sum of a proper ideal I of A and of the standard countably generated Hilbert A-module l2 (A). Consider the shift operator T2 : M2 → M2 defined by the formula T2 ((i, a1 , a2 , . . .)) = (0, i, a1 , a2 , . . .) for ak ∈ A, i ∈ I . It is an isometric A-linear embedding of M2 into itself and, hence, orthogonality-preserving, however T2 is not adjointable. To formulate the result on orthogonality-preserving mappings we need a construction by W.L. Paschke [19]: for any Hilbert A-module M over any C ∗ -algebra A one can extend M canonically to a Hilbert A∗∗ -module M# over the bidual Banach space and von Neumann algebra A∗∗ of A [19, Thm. 3.2, Prop. 3.8, §4]. For this aim the A∗∗ -valued pre-inner product can be defined by the formula [a ⊗ x, b ⊗ y] = ax, yb∗ , for elementary tensors of A∗∗ ⊗M, where a, b ∈ A∗∗ , x, y ∈ M. The quotient module of A∗∗ ⊗M by the set of all isotropic vectors is denoted by M# . It can be canonically completed to a selfdual Hilbert A∗∗ -module N which is isometrically algebraically isomorphic to the A∗∗ -dual A∗∗ -module of M# . N is a dual Banach space itself (cf. [19, Thm. 3.2, Prop. 3.8, §4]). Every A-linear bounded map T : M → M can be continued to a unique A∗∗ -linear map T : M# → M# preserving the operator norm and obeying the canonical embedding π (M) of M into M# . Similarly, T can be further extended to the self-dual Hilbert A∗∗ -module N . The extension is such that the isometrically algebraically embedded copy π (M) of M in N is a w ∗ -dense A-submodule of N , and that A-valued inner product values of elements of M embedded in N are preserved with respect to the A∗∗ -valued inner product on N and to the canonical isometric embedding π of A into its bidual Banach space A∗∗ . Any bounded A-linear operator T on M extends to a unique bounded A∗∗ -linear operator on N preserving the operator norm, cf. [19, Prop. 3.6, Cor. 3.7, §4]. The extension of bounded A-linear operators from M to N is continuous with respect to the w ∗ -topology on N . For topological characterizations of self-duality of Hilbert C ∗ -modules over W ∗ -algebras we refer to [19], [8, Thm. 3.2] and to [21,22]: a Hilbert C ∗ -module K over a W ∗ -algebra B is self-dual, if and only if its unit ball is complete with respect


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to the topology induced by the semi-norms {|f (., x)|: x ∈ K, f ∈ B ∗ , x 1, f 1}, if and only if its unit ball is complete with respect to the topology induced by the semi-norms {f (·,·)1/2 : f ∈ B ∗ , x 1, f 1}. The first topology coincides with the w ∗ -topology on K in that case. Note, that in the construction above M is always w ∗ -dense in N , as well as for any subset of M the respective construction is w ∗ -dense in its biorthogonal complement with respect to N . However, starting with a subset of N its biorthogonal complement with respect to N might not have a w ∗ -dense intersection with the embedding of M into N , cf. [20, Prop. 3.11.9]. Example 2. Let A be the C ∗ -algebra of all continuous functions on the unit interval, i.e. A = C([0, 1]). In case we consider A as a Hilbert C ∗ -module over itself and an orthogonalitypreserving map T0 defined by the multiplication by the function a(t) = t · (sin(1/t) + i cos(1/t)) we obtain that the operator T0 cannot be written as the combination of a multiplication by a positive element of A and of an isometric module operator U0 on M = A. The reason for this phenomenon is the lack of a polar decomposition of a(t) inside A. Only a lift to the bidual von Neumann algebra A∗∗ of A restores the simple description of the continued operator T0 as the combination of a multiplication by a positive element (of the center) of A and an isometric module operator on M# = N = A∗∗ . The unitary part of a(t) is a so-called local multiplier of C([0, 1]), i.e. a multiplier of C0 ((0, 1]). But it is not a multiplier of C([0, 1]) itself. We shall show that this example is a very canonical one. We are going to demonstrate the following fact on the nature of orthogonality-preserving bounded module mappings on Hilbert C ∗ -modules. Without loss of generality, one may assume that the range of the A-valued inner product on M in A is norm-dense in A. Such Hilbert C ∗ modules are called full Hilbert C ∗ -modules. Otherwise A has to be replaced by the range of the A-valued inner product which is always a two-sided norm-closed ∗-ideal of A. The sets of all adjointable bounded module operators and of all bounded module operators on M, respectively, are invariant with respect to such changes of sets of coefficients of Hilbert C ∗ -modules, cf. [19]. Theorem 3. Let A be a C ∗ -algebra, M be a full Hilbert A-module and M# be its canonical A∗∗ -extension. Any orthogonality-preserving bounded A-linear operator T on M is of the form T = λV , where V : M# → M# is an isometric A-linear embedding and λ is a positive element of the center Z(M(A)) of the multiplier algebra M(A) of A. If any element λ ∈ Z(M(A)) with |λ | = λ admits a polar decomposition inside Z(M(A)) then the operator V preserves π (M) ⊂ M# . So T = λ · V on M. In [13, Thm. 3.1] D. Ilišević and A. Turnšek proved Theorem 3 for the particular case if for some Hilbert space H the C ∗ -algebra A admits an isometric representation π on H with the property K(H ) ⊂ π(A) ⊂ B(H ). In this situation Z(M(A)) = C. Proof. We want to make use of the canonical non-degenerate isometric ∗-representation π of a C ∗ -algebra A in its bidual Banach space and von Neumann algebra A∗∗ of A, as well as of its extension π : M → M# → N and of its operator extension. That is, we switch from the triple {A, M, T } to the triple {A∗∗ , M# ⊆ N , T }. We have to demonstrate that for orthogonality-preserving bounded A-linear mappings T on M the respective extended bounded A∗∗ -linear operator on N is still orthogonalitypreserving for N . Let x be an element of N and denote by K its biorthogonal complement

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with respect to N . Then K is a direct orthogonal summand of N because N and K are self-dual Hilbert A∗∗ -modules. Consider any positive normal state f on A∗∗ with f (x, x) = 0. Since the A-valued inner product ·,· on M continues to an A∗∗ -valued inner product ·,· on N in a unique way by [19, Thm. 3.2], the possibly degenerated complex-valued inner product f (·,·) on M continues to a possibly degenerated complex-valued inner product f (·,·) on N in a unique way. Consider x ∈ N and its module-biorthogonal complement K with respect to N . The intersection of K with the isometrically embedded copy of M in N has to be a weakly-dense subset of K after factorization by the kernel of f (·,·)1/2 , otherwise the continuation of f (·,·) from M∩K to K would be non-unique. So x can be represented as a weak limit of a Hilbert space sequence of the subset (K ∩ M)/kernel(f (·,·)1/2 ) in N /kernel(f (·,·)1/2 ). Now, take another non-trivial element y ∈ N with x, y = 0. Then the module-biorthogonal complement L of y with respect to N is orthogonal to K. Repeat the construction for y fixing f . Since f (z, t) = 0 for any z ∈ (K ∩ M)/kernel(f (·,·)1/2 ) and any t ∈ (L ∩ M)/kernel(f (·,·)1/2 ), and since these sets are weakly dense in K/kernel(f (·,·)1/2 ) and L/kernel(f (·,·)1/2 ), respectively, the weak continuity of the map T and the jointly weak continuity of inner products forces f (T (z), T (t)) = 0. Since f has been selected arbitrarily, x, y = 0 for some x, y ∈ N forces T (x), T (y) = 0. Note, that the arguments are so complicated because K or L might have nonw ◦ ∗-dense intersections with M ⊆ N by [20, Prop. 3.11.9]. Next, we want to consider only discrete W ∗ -algebras, i.e. W ∗ -algebras for which the supremum of all minimal projections contained in them equals their identity. (We prefer to use the word discrete instead of atomic.) To connect to the general C ∗ -case we make use of a theorem by Ch.A. Akemann stating that the ∗-homomorphism of a C ∗ -algebra A into the discrete part of its bidual von Neumann algebra A∗∗ which arises as the composition of the canonical embedding π of A into A∗∗ followed by the projection ρ to the discrete part of A∗∗ is an injective ∗-homomorphism, [1, p. 278] and [2, p. I]. The injective ∗-homomorphism ρ is partially implemented by a central projection p ∈ Z(A∗∗ ) in such a way that A∗∗ multiplied by p gives the discrete part of A∗∗ . Applying this approach to our situation we reduce the problem further by investigating the triple {pA∗∗ , pN , pT } instead of the triple {A∗∗ , N , T }, where we rely on the injectivity of the algebraic embeddings ρ ◦ π : A → pA∗∗ and ρ ◦ π : M → pN . The latter map is injective since x, x = 0 forces px, px = px, x = ρ ◦ π(x, x) = 0. Obviously, the bounded pA∗∗ -linear operator pT is orthogonality-preserving for the self-dual Hilbert pA∗∗ module pN because the orthogonal projection of N onto pN and the operator T commute, and both they are orthogonality-preserving. In the sequel we have to consider the multiplier algebra M(A) and the left multiplier algebra LM(A) of the C ∗ -algebra A. By [20] every non-degenerate injective ∗-representation of A in a von Neumann algebra B extends to an injective ∗-representation of the multiplier algebra M(A) in B and to an isometric algebraic representation of the left multiplier algebra LM(A) of A preserving the strict and the left strict topologies on M(A) and on LM(A), respectively. In particular, the injective ∗-representation ρ ◦ φ extends to M(A) and to LM(A) in such a way that ρ ◦ φ M(A) = b ∈ pA∗∗ : bρ ◦ φ(a) ∈ A, ρ ◦ φ(a)b ∈ A for every a ∈ A , ρ ◦ φ LM(A) = b ∈ pA∗∗ : bρ ◦ φ(a) ∈ A for every a ∈ A . Since Z(LM(A)) = Z(M(A)) for the multiplier algebra of A of every C ∗ -algebra A, we have the description ρ ◦ φ Z M(A) = b ∈ pA∗∗ : bρ ◦ φ(a) = ρ ◦ φ(a)b ∈ A for every a ∈ A .


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Since the von Neumann algebra pA∗∗ is discrete the identity p can be represented as the orthogonal atomic projections {qα : α ∈ I } of the center w ∗ -sum of a maximal set of pairwise Z(pA∗∗ ) of pA∗∗ . Note, that α∈I qα = p. Select a single atomic projection qα ∈ Z(pA∗∗ ) of this collection and consider the part {qα pA∗∗ , qα pN , qα pT } of the problem for every single α ∈ I. By [13, Thm. 3.1] the operator qα pT can be described as a non-negative constant λqα multiplied by an isometry Vqα on the Hilbert qα pA∗∗ -module qα pN , where the isometry Vqα preserves the qα pA-submodule qα pM inside qα pN since the operator qα pT preserves it, and multiplication by a positive number does not change this fact. In case λqα = 0 we set simply Vqα = 0. We have to show the existence of global operators on the Hilbert pA∗∗ -module pN build as ∗ w -limits of nets of finite sums with pairwise distinct summands of the sets {λqα qα : α ∈ I } and {qα Vqα : α ∈ I }, respectively. Additionally, we have to establish key properties of them. First, note that the collection of all finite sums with pairwise distinct summands of {λqα qα : α ∈ I } form an increasingly directed net of positive elements of the center of the operator algebra EndpA∗∗ (pN ), which is ∗-isomorphic to the von Neumann algebra Z(pA∗∗ ). This net is bounded by pT · idpN since the operator pT admits an adjoint operator on the self-dual Hilbert pA∗∗ -module pN by [19, Prop. 3.4] and since for any finite subset I0 of I the inequality 0

λ2qα · idqα pN =

α∈I0

qα pT ∗ T pT ∗ T pT 2 · idpN

α∈I0

holds in the operator algebra EndpA∗∗ (pN ), the center of which is ∗-isomorphic to Z(pA∗∗ ). Therefore, the supremum of this increasingly directed bounded net of positive elements exists as an element of the center of the operator algebra EndpA∗∗ (pN ), which is ∗-isomorphic to the von Neumann algebra Z(pA∗∗ ). We denote the supremum of this net by λp . By construction and by the w ∗ -continuity of transfers to suprema of increasingly directed bounded nets of self-adjoint elements of von Neumann algebras we have the equality λp = w ∗ - lim

I0 ⊆I

λqα · qα ∈ Z pA∗∗ ≡ Z EndpA∗∗ (pN )

α∈I0

where I0 runs over the partially ordered net of all finite subsets of I . Since qα pT ∗ T (z), z = λ2qα qα z, z for any z ∈ qα N and for any α ∈ I , we arrive at the equality ∗ pT T (z), z = λ2p · pz, z for any z ∈ pN and for the constructed positive element λp ∈ Z(pA∗∗ ) ≡ Z(EndpA∗∗ (pN )). Consequently, the operator pT can be written as pT = λp Vp for some isometric pA∗∗ -linear map Vp ∈ EndpA∗∗ (pN ), cf. [13, Prop. 2.3]. Consider the operator pT on pN . Since the formula pT (x), pT (x) = λ2p x, x ∈ ρ ◦ π(A)

(1)

holds for any x ∈ ρ ◦ π (M) ⊆ pN and since the range of the A-valued inner product on M is supposed to be the entire C ∗ -algebra A, the right side of this equality and the multiplier theory of C ∗ -algebras forces λ2p ∈ LM(pA) ∩ Z(pA∗∗ ) = Z(M(ρ ◦ π(A))) = ρ ◦ π(Z(M(A))) [20].

334


Taking the square root of λ2p in a C ∗ -algebraical sense is an operation which results in a (unique) positive element of the C ∗ -algebra itself. So we arrive at λp ∈ ρ ◦ π(Z(M(A))) as the square root of λ2p 0. In particular, the operator λp · idpN preserves the ρ ◦ π(A)-submodule ρ ◦ π (M). As a consequence, we can lift the bounded pA∗∗ -linear orthogonality-preserving operator pT on pN back to M# since A∗∗ allows polar decomposition for any element, the embedding ρ ◦ π : A → pA∗∗ and the module and operator mappings, induced by ρ ◦ π and by Paschke’s embedding were isometrically and algebraically, just by multiplying with or, respectively, acting by p in the second step. So we obtain a decomposition T = λV of T ∈ EndA (M) with a positive function λ ∈ Z(M(A)) ≡ Z(EndA (M)) derived from λp , and with an isometric A-linear embedding V ∈ EndA (M# ), V derived from Vp . In case any element λ ∈ Z(M(A)) with |λ | = λ admits a polar decomposition inside Z(M(A)) then the operator V preserves π (M) ⊂ M# . So T = λ · V on M. For completeness just note, that the adjointability of V goes lost on this last step of the proof in case T has not been adjointable on M in the very beginning. 2 Theorem 4. Let A be a C ∗ -algebra and M be a Hilbert A-module. Any orthogonality-preserving bounded A-linear operator T on M fulfills the equality

T (x), T (y) = κx, y

for a certain T -specific positive element κ ∈ Z(M(A)) and for any x, y ∈ M. Proof. We have only to remark that the values of the A-valued inner product on M do not change if M is canonically embedded into M# or N . Then the obtained formula works in the bidual situation, cf. (1). 2 Problem 1. We conjecture that any orthogonality-preserving map T on Hilbert A-modules M over C ∗ -algebras A are of the form T = λV for some element λ ∈ Z(M(A)) and some A-linear isometry V : M → M. To solve this problem one has possibly to solve the problem of general polar decomposition of arbitrary elements of (commutative) C ∗ -algebras inside corresponding local multiplier algebras or in similarly derived algebras. Corollary 5. Let A be a C ∗ -algebra and M be a Hilbert A-module. Let T be an orthogonalitypreserving bounded A-linear operator on M of the form T = λV , where V : M → M is an isometric bounded A-linear embedding and λ is an element of the center Z(M(A)) of the multiplier algebra M(A) of A. Then the following conditions are equivalent: (i) T is adjointable. (ii) V is adjointable. (iii) The graph of the isometric embedding V is a direct orthogonal summand of the Hilbert A-module M ⊕ M. (iv) The range Im(V ) of V is a direct orthogonal summand of M. Proof. Note, that a multiplication operator by an element λ ∈ Z(M(A)) is always adjointable. So, if T is supposed to be adjointable, then the operator V has to be adjointable, and vice versa. By [10, Cor. 3.2] the bounded operator V is adjointable if and only if its graph is a direct orthogonal summand of the Hilbert A-module M ⊕ M. Moreover, since the range of the isometric


335

A-linear embedding V is always closed, adjointability of V forces V to admit a bounded A-linear generalized inverse operator on M, cf. [11, Prop. 3.5]. The kernel of this inverse to V mapping serves as the orthogonal complement of Im(V ), and M = Im(V ) ⊕ Im(V )⊥ as an orthogonal direct sum by [11, Thm. 3.1]. Conversely, if the range Im(V ) of V is a direct orthogonal summand of M, then there exists an orthogonal projection of M onto this range and, therefore, V is adjointable. 2 2. C ∗ -conformal and conformal mappings We want to describe generalized C ∗ -conformal mappings on Hilbert C ∗ -modules. A full characterization of such maps involves isometries as for the orthogonality-preserving case since we resort to a particular case of the latter. Let M be a Hilbert module over a C ∗ -algebra A. An injective bounded module map T on M is said to be C ∗ -conformal if the identity x, y T x, T y = T xT y xy

(2)

holds for all non-zero vectors x, y ∈ M. It is said to be conformal if the identity T x, T y x, y = T xT y xy

(3)

holds for all non-zero vectors x, y ∈ M. Theorem 6. Let M be a Hilbert A-module over a C ∗ -algebra A and T be an injective bounded module map. The following conditions are equivalent: (i) T is C ∗ -conformal; (ii) T = λU for some non-zero positive λ ∈ R and for some isometrical module operator U on M. Proof. The condition (ii) implies condition (i) because the condition U x = x for all x ∈ M implies the condition U x, Uy = x, y for all x, y ∈ M by [13, Prop. 2.3]. So we have only to verify the implication (i) → (ii). Assume an injective bounded module map T on M to be C ∗ -conformal. We can rewrite (2) in the following equivalent form: T x, T y = x, y

T xT y , xy

x, y = 0.

(4)

Consider the left part of this equality as a new A-valued inner product on M. Consequently, the right part of (4) has to satisfy all the conditions of a C ∗ -valued inner product, too. In particular, the right part of (4) has to be additive in the second variable, what exactly means x, y1 + y2

T xT (y1 + y2 ) T xT y1 T xT y2 = x, y1 + x, y2 xy1 + y2 xy1 xy2

336


for all non-zero x, y1 , y2 ∈ M. Therefore, (y1 + y2 )

T y1 T y2 T (y1 + y2 ) = y1 + y2 , y1 + y2 y1 y2

by the arbitrarity of x ∈ M, which can be rewritten as

y1

T (y1 + y2 ) T y1 T (y1 + y2 ) T y2 − + y2 − =0 y1 + y2 y1 y1 + y2 y2

(5)

for all non-zero y1 , y2 ∈ M. In case the elements y1 and y2 are not complex multiples of each other both the complex numbers inside the brackets have to equal to zero. So we arrive at T (y1 ) T (y2 ) = y1 y2

(6)

for any y1 , y2 ∈ M which are not complex multiples of one another. Now, if the elements would be non-trivial complex multiples of each other both the coefficients would have to be equal, what again forces equality (6). x Let us denote the positive real number T x by t. Then the equality (6) provides 1 T (z) = z, t which means U = 1t T is an isometrical operator. The proof is complete.

2

Example 7. Let A = C0 ((0, 1]) = M and T be a C ∗ -conformal mapping on M. Our aim is to demonstrate that T = tU for some non-zero positive t ∈ R and for some isometrical module operator U on M. To begin with, let us recall that the Banach algebra EndA (M) of all bounded module maps on M is isomorphic to the algebra LM(A) of left multipliers of A under the given circumstances. Moreover, LM(A) = Cb ((0, 1]), the C ∗ -algebra of all bounded continuous functions on (0, 1]. So any A-linear bounded operator on M is just a multiplication by a certain function of Cb ((0, 1]). In particular, T (g) = fT · g,

g ∈ A,

for some fT ∈ Cb ((0, 1]). Let us denote by x0 the point of (0, 1], where the function |fT | achieves its supremum, i.e. |fT (x0 )| = fT , and set t := fT . We claim that the operator 1t T is an isometry, what exactly means |fT (x)| =1 fT

(7)

for all x ∈ (0, 1]. Indeed, consider any point x = x0 . Let θx ∈ C0 ((0, 1]) be an Urysohn function for x, i.e. 0 θx 1, θx (x) = 1 and θx = 0 outside of some neighborhood of x, and let θx0 be an Urysohn function for x0 . Moreover, we can assume that the supports of θx and θx0 do not intersect


337

each other. Now the condition (2) written for T and for coinciding vectors x = y = θx +θx0 yields the equality |fT |2 (θx + θx0 )2 (θx + θx0 )2 = , fT (θx + θx0 )2 θx + θx0 2 which implies |fT |2 (θx + θx0 )2 = (θx + θx0 )2 . fT 2 This equality at point x takes the form (7) for any x ∈ (0, 1]. Theorem 8. Let M be a Hilbert A-module over a C ∗ -algebra A and T be an injective bounded module map. The following conditions are equivalent: (i) T is conformal; (ii) T = λU for some non-zero positive λ ∈ R and for some isometrical module operator U on M. Proof. As in the proof of the theorem on orthogonality-preserving mappings we switch from the setting {A, M, T } to its faithful isometric representation in {pA∗∗ , pM# ⊆ pN , T }, where p ∈ A∗∗ is the central projection of A∗∗ mapping A∗∗ to its discrete part. First, consider a minimal projections e ∈ pA∗∗ . Then the equality (3) gives ex, ey T (ex), T (ey) = exey T (ex)T (ey) for any x, y ∈ pM# . Since {eM# , ·,·} becomes a Hilbert space after factorization by the set {x ∈ pM# : ex, xe = 0}, the map T acts as a positive scalar multiple of a linear isometry on eM# , i.e. eT = λe Ue . Secondly, every two minimal projections e, f ∈ pA∗∗ with the same minimal central support projection q ∈ pZ(A∗∗ ) are connected by a (unique) partial isometry u ∈ pA∗∗ such that u∗ u = f and uu∗ = e. Arguments analogous to those given at [13, p. 303] show λ2e · ex, xe = uf u∗ T (x), T (x) uf u∗ = uf T u∗ x t u∗ x f u∗ = uλ2f f u∗ x, u∗ x f = λ2f · ex, xe. Therefore, qT = λq U for some positive λq ∈ R, for a qA-linear isometric mapping U : qM# → qM# and for any minimal central projection q ∈ pA∗∗ .

338


Thirdly, suppose e, f are two minimal central projections of pA∗∗ that are orthogonal. For any x, y ∈ pM# consider the supposed equality (e + f )x, (e + f )y T ((e + f )x), T ((e + f )y) = . (e + f )x(e + f )y T ((e + f )x)T ((e + f )y) Since T is a bounded module mapping which acts on epM# like λe · id and on fpM# like λf · id we arrive at the equality (e + f )x, (e + f )y (λe e + λf f )x, (λe e + λf f )y = . (e + f )x(e + f )y (λe e + λf f )x(λe e + λf f )y Involving the properties of e, f to be central and orthogonal to each other and exploiting modular linear properties of the pA∗∗ -valued inner product we transform the equality further to ex, ey + f x, fy ex, ex + f x, f x1/2 · ey, ey + fy, fy1/2 =

λ2e ex, ey + λ2f f x, fy λ2e ex, ex + λ2f f x, f x1/2 · λ2e ey, ey + λ2f fy, fy1/2

.

Since e, f are pairwise orthogonal central projections we can transform the equality further to sup{ex, ey, f x, fy} sup{ex, ex, f x, f x}1/2 · sup{ey, ey, fy, fy}1/2 =

sup{λ2e ex, ey, λ2f f x, fy} sup{λ2e ex, ex, λ2f f x, f x}1/2 · sup{λ2e ey, ey, λ2f fy, fy}1/2

.

By the w ∗ -density of pM# in pN we can distinguish a finite number of cases at which of the central parts the respective six suprema may be admitted, at epA∗∗ or at fpA∗∗ . For this aim we may assume, in particular, that x, y belong to pN to have a larger set for these elements to be selected specifically. Most interesting are the cases when (i) both (e + f )x and (λe e + λf f )x admit their norm at the e-part, (ii) both (e + f )y and (λe e + λf f )y admit their norm at the f -part, and (iii) both (e + f )x, (e + f )y and (λe e + λf f )x, (λe e + λf f )y admit their norm (either) at the e-part (or at the f -part). In these cases the equality above gives λe = λf . (All the other cases either give the same result or do not give any new information on the interrelation of λe and λf .) Finally, if for any central minimal projection f ∈ pA∗∗ the operator T acts on fpN as λU for a certain (fixed) positive constant λ and a certain module-linear isometry U then T acts on pN in the same way. Consequently, T acts on M in the same manner since U preserves pM# inside pN . 2 Remark 1. Obviously, the C ∗ -conformity of a bounded module map follows from the conformity of it, but the converse is not obvious, even it is true for Hilbert C ∗ -modules.


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Acknowledgment We are grateful to Chi-Keung Ng who pointed us to the results by G.K. Pedersen in September 2009. So we had to correct a crucial argument in the second paragraph of Theorem 3 giving other and much more detailled arguments. References [1] Ch.A. Akemann, The general Stone–Weierstrass problem for C ∗ -algebras, J. Funct. Anal. 4 (1969) 277–294. [2] Ch.A. Akemann, A Gelfand representation theory for C ∗ -algebras, Pacific J. Math. 39 (1971) 1–11. [3] A. Blanco, A. Turnšek, On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 709–716. [4] M. Burgos, F.J. Fernández-Polo, J.J. Garcés, J. Martínez Moreno, A.M. Peralta, Orthogonality preservers in C ∗ algebras, JB∗ -algebras and JB∗ -triples, J. Math. Anal. Appl. 348 (2008) 220–233. [5] J. Chmieliński, On an ε-Birkhoff orthogonality, JIPAM. J. Inequal. Pure Appl. Math. 6 (3) (2005), article 79. [6] J. Chmieliński, Remarks on orthogonality-preserving in normed spaces and some stability problems, Banach J. Math. Anal. 1 (2007) 117–124. [7] J. Chmieliński, D. Ilišević, M.S. Moslehian, Gh. Sadeghi, Perturbation of the Wigner equation in inner product C ∗ -modules, J. Math. Phys. 40 (3) (2008) 033519, 8 pp. [8] M. Frank, Self-duality and C ∗ -reflexivity of Hilbert C ∗ -modules, Z. Anal. Anwend. 9 (1990) 165–176. [9] M. Frank, D.R. Larson, Frames in Hilbert C ∗ -modules and C ∗ -algebras, J. Operator Theory 48 (2002) 273–314. [10] M. Frank, K. Sharifi, Adjointability of densely defined closed operators and the Magajna–Schweizer theorem, J. Operator Theory 63 (2010) 271–282. [11] M. Frank, K. Sharifi, Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C ∗ -modules, J. Operator Theory 64 (2010) 377–386. [12] Deguang Han, D.R. Larson, Frames, bases and, group representations, Mem. Amer. Math. Soc. 147 (697) (2000). [13] D. Ilišević, A. Turnšek, Approximately orthogonality preserving mappings on C ∗ -modules, J. Math. Anal. Appl. 341 (2008) 298–308. [14] A. Koldobsky, Operators preserving orthogonality are isometries, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 835–837. [15] E.C. Lance, Unitary operators on Hilbert C ∗ -modules, Bull. Lond. Math. Soc. 26 (1994) 363–366. [16] Chi-Wai Leung, Chi-Keung Ng, Ngai-Ching Wong, Automatic continuity and C0 (Ω)-linearity of linear maps between C0 (Ω)-modules, preprint, arXiv:1005.4561 [math.OA] at http://arxiv.org, May 2010. [17] Chi-Wai Leung, Chi-Keung Ng, Ngai-Ching Wong, Linear orthogonality preservers of Hilbert bundles, preprint, arXiv:1005.4502 [math.OA] at http://arxiv.org, May 2010. [18] V.M. Manuilov, E.V. Troitsky, Hilbert C ∗ -Modules, Faktorial Press, Moscow, 2001; Transl. Math. Monogr., vol. 226, American Mathematical Society, Providence, RI, 2005. [19] W.L. Paschke, Inner product modules over B ∗ -algebras, Trans. Amer. Math. Soc. 182 (1973) 443–468. [20] G.K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, Academic Press, London, New York, San Francisco, 1979. [21] J. Schweizer, Interplay between noncommutative topology and operators on C ∗ -algebras, Habilitation thesis, Mathematische Fakultät der Universität Tübingen, Tübingen, Germany, 1996. [22] J. Schweizer, Hilbert C ∗ -modules with a predual, J. Operator Theory 48 (2002) 621–632. [23] M. Takesaki, Theory of Operator Algebras I–III, Encyclopaedia Math. Sci., vols. 124, 125, 127, Springer-Verlag, Berlin, 2002–2003. [24] A. Turnšek, On operators preserving James’ orthogonality, Linear Algebra Appl. 407 (2005) 189–195. [25] A. Turnšek, On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (2007) 625–631.


Perturbations of embedded eigenvalues for the planar bilaplacian Gianne Derks a , Sara Maad Sasane b,∗,1 , Björn Sandstede c,2 a Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK b Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden c Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Received 11 September 2009; accepted 3 October 2010

Communicated by I. Rodnianski

Abstract Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials. © 2010 Elsevier Inc. All rights reserved. Keywords: Embedded eigenvalues; Persistence; Perturbation; Bilaplacian

1. Introduction Determining the dependence of the spectrum of operators on perturbations is an important issue that is of relevance in many applications. Of course, much is known in this direction: the persistence of point eigenvalues and the behaviour of the essential spectrum under small bounded * Corresponding author.

E-mail address: [email protected] (S. Maad Sasane). 1 Partially supported by the European Union under the Marie Curie Fellowship MEIF-CT-2005-024191 and by the

Swedish Research Council. 2 Partially supported by a Royal Society Wolfson Research Merit Award and by the NSF under grant DMS-0907904. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.001

G. Derks et al. / Journal of Functional Analysis 260 (2011) 340–398

341

perturbations, for instance, have been analysed comprehensively, and we refer to [11] for many results along these lines. Here, we consider differential operators that are posed on unbounded domains and are interested in the interaction between eigenvalues, with proper eigenfunctions in the underlying domain of the operator, and the essential spectrum. More precisely, we study the fate of eigenvalues that are embedded in the essential spectrum under small perturbations of the operator. Typically, such eigenvalues will disappear under generic perturbations of the potential, and it is therefore of interest to determine the class of perturbations for which an embedded eigenvalue persists. For the bilaplacian on cylindrical domains, we showed in our previous work [8] that the set of perturbations for which an embedded eigenvalue persists is an infinite-dimensional manifold of finite codimension. Furthermore, we showed that the codimension of this set is given by the multiplicity of the essential spectrum, defined as the number of independent continuum eigenfunctions or, more rigorously, via the spectral resolution of the Fourier transform of the bilaplacian (see e.g. [3, Definition 2 in §85]). In this paper, we continue the investigation that we began in [8] and consider the bilaplacian posed on the plane: the challenge is that the essential spectrum of the planar bilaplacian has infinite multiplicity. Thus, we may expect that the set of potentials for which an embedded eigenvalue persists is an infinite-dimensional manifold of infinite codimension, and this is indeed what we shall prove for an appropriate class of potentials. For a different approach on persistence of embedded eigenvalues, see [2]. Before stating our results, we briefly outline why embedded eigenvalues are of interest. Our first motivation comes from quantum mechanics: the eigenfunctions associated with eigenvalues of an energy operator correspond to bound states that can be attained by the physical system modelled by the energy operator. If such an eigenvalue is embedded in the essential spectrum, then its fate under perturbations of the potential determines whether the associated bound states persist or not (see [10,16] for examples). The second example comes from inverse scattering theory, where eigenvalues correspond to coherent soliton structures of the underlying integrable system, while the essential spectrum describes radiative scattering behaviour. Thus, bifurcations of solitons are reflected by the disappearance or persistence of embedded eigenvalues [13,14]. Finally, embedded eigenvalues provide a common mechanism for the destabilisation of travelling waves in near-integrable Hamiltonian partial differential equations, and we refer to [17] for further background information and pointers to the literature. As mentioned above, we focus in this paper on the persistence of embedded eigenvalues for the planar bilaplacian. Our primary reason for considering the bilaplacian is that this operator is complex enough to exhibit the underlying difficulties, while not adding technical complications that have nothing to do with the issue we are interested in. In other words, the planar bilaplacian provides a useful paradigm for the issues that we expect to encounter for other more complicated differential operators. Note also that the applications we mentioned above all involve self-adjoint operators, a feature shared by the bilaplacian. We now describe the precise setting that we consider. Let r0 > 0, and assume that θ ∈ C0∞ (Br0 (0); R) is a radially symmetric potential. Hence, we use polar coordinates (r, ϕ), write θ = θ (r), and consider the multiplication operator on L2 (R2 ) (also denoted by θ ) defined by [θ u](r, ϕ) := θ (r)u(r, ϕ). We define L := 2 + θ on L2 (R2 ), where 2 is the bilaplace operator which is densely defined on L2 (R2 ) with domain H 4 (R2 ). It is known that the spectrum of 2 is σ (2 ) = [0, ∞). Since θ has compact support, the essential spectra of L and 2 coincide, and so σc (L) = [0, ∞). We assume that θ is chosen so that L has a simple positive eigenvalue λ0 :

342


(A1) L has an eigenvalue λ0 > 0 of multiplicity 1. We are mainly interested in the case where λ0 is an embedded eigenvalue, i.e. when λ0 0, since when λ0 is isolated from the rest of the spectrum, the persistence of eigenvalues is well known, [11, pp. 213–215]. We also exclude the case λ0 = 0 which lies on the boundary between spectrum and resolvent set. We denote by u∗ (r, ϕ) the eigenfunction associated with the embedded eigenvalue λ0 . Since θ is radially symmetric and the laplacian is invariant under rotations of the underlying cartesian coordinates, we see that the functions u∗ (r, ϕ + ϕ0 ) are, for each fixed ϕ0 , also eigenfunctions of L belonging to the eigenvalue λ0 . The simplicity of λ0 required in assumption (A1) therefore implies that u∗ is a radial function, and we henceforth write u∗ = u∗ (r). It is clear by existence and uniqueness of solutions of ODEs that u∗ (r) cannot vanish for all r r1 , and so we assume that (A2) r1 > r0 is such that u∗ (r1 ) = 0. Lemma 1 below shows that our hypotheses can be satisfied. We now perturb the potential θ by potentials ρ in the weighted L2 -space R := L2 ([0, r1 ], H 1/2 (S 1 ), r dr) of functions that map the interval [0, r1 ] into H 1/2 (S 1 ), where the interval [0, r1 ] is the domain of the radial variable r, while H 1/2 (S 1 ) describes the dependence on the angular variable ϕ. Our main result is as follows. Theorem 1. Let 0 < r0 r1 , θ ∈ C0∞ (Br0 (0); R) be radially symmetric, and assume that (A1) and (A2) hold. Then there exist δ > 0 and a neighbourhood O of 0 in R = L2 ([0, r1 ], H 1/2 (S 1 ), r dr) such that the set Remb := ρ ∈ O; L + ρ has an embedded eigenvalue in (λ0 − δ, λ0 + δ) is a smooth manifold in R of infinite dimension and codimension. Before commenting on the ideas behind the proof of Theorem 1, we illustrate that our hypotheses can be met. Lemma 1. There exists a smooth radial potential θ (r) with compact support such that L = 2 + θ satisfies hypothesis (A1). Proof. Let K0 (r) denote the modified Bessel function of the second kind and define a smooth, strictly positive function u0 (r) via u0 (r) =

1, K0 (r),

0 r 1, 2r

together with a smooth interpolation in the intermediate region r ∈ [1, 2]. Note that u0 decays to exponentially as r → ∞ and can be chosen so that u0 (r) > 0 for all r. Thus, the radial potential θ :=

1 1 1, 0 r 1, −2 + 1 u0 = ( + 1)(− + 1)u0 = 0, 2 r u0 u0


343

is well defined, smooth, and has support in [0, 2] since K0 (r) satisfies (− + 1)K0 = 0. Furthermore, we have 1 Lu0 = 2 + θ u0 = 2 u0 + −2 + 1 u0 u0 = u0 , u0 and u0 is a positive radial eigenfunction belonging to the embedded eigenvalue λ0 = 1 of L. It remains to show that λ0 = 1 is simple. Using the radial symmetry of θ , the results presented in the rest of this paper imply that it suffices to show that the equation ∂r2

1 k2 + ∂r − 2 r r

2 u = (1 − θ )u for r ∈ (0, 2),

ur (0) = urrr (0) = 0,

u(r) = Kk (r)

for r 2

(1)

does not have a solution u(r) for each integer k 1. We will now outline why (1) will not have solutions for k 1 provided θ is modified appropriately but omit the straightforward details. Using variations of parameters, it can be shown that (1) cannot have solutions for k 1. If it does have solutions for some or all of the remaining finitely many integers k 1, then we can modify the potential θ to remove these solutions while retaining the eigenfunction for k = 0. Indeed, any solution of (1) for k 1 is of the explicit form u(r) = r for some integer = (k) 1 since θ (r) − 1 = 0 for 0 r 1. Replacing u0 in the above construction of θ by u0 + v0 for bounded functions v0 with support in ( 12 , 1) and using the necessary expressions (64) derived in Section 7 for the persistence of eigenvalues, it is then not difficult to see that any nonzero choice of v0 0 removes the solutions of (1) for k 1 for 0 1. 2 The idea for proving Theorem 1 is to characterise embedded eigenvalues as roots of a regular function, since such a characterisation would allow us to use the implicit function theorem. As it appears difficult to find a functional-analytic characterisation of embedded eigenvalues, we pursue here a dynamical-systems formulation similar to that used in our precursor work [8] for the bilaplacian on cylinders. The eigenvalue problem can be written as a system of differential equations in the radial evolution variable r posed on an appropriate function space X of functions that are defined in the angular variable ϕ. The issue is that this system is ill-posed in the sense that, for given initial data, solutions may not exist. Using a similar approach as in Scheel [19], we will show, however, that this dynamical system has an exponential dichotomy: there are two infinite-dimensional subspaces X cu and X s of X at r = r1 so that solutions with initial data in X cu exist and stay bounded for r r1 , while solutions with data in X s exist and decay as r → ∞. The intersection of these spaces corresponds to eigenfunctions of the underlying operator, and our goal is therefore to characterise those perturbations for which this intersection is nontrivial. We show that there are infinitely many conditions that characterise such intersections and prove that we can solve them using an implicit function theorem. A key issue is the space for the perturbation ρ. For the conditions of the implicit function theorem to be satisfied, the space for ρ needs to be L2 ([0, r1 ]; H 1/2 (S 1 ), r dr), a space with very low regularity. This low regularity forces us to work with different function spaces for r r1 (where ρ has its support) and for r r1 (where we have an explicit formulation of the solutions of the system in terms of Bessel functions), and so we need to take extra care when matching the solutions at r = r1 .

344


The rest of this paper is organised as follows. In Section 2, we introduce the spatial-dynamics formulations of the eigenvalue problem. In Sections 3 and 4, we prove the existence of exponential dichotomies for the bilaplacian and for the operator L, respectively, near the core r = 0. We then construct dichotomies for L in the far field for r 1 in Section 5 and discuss similar properties for the adjoint spatial dynamical system in Section 6. These results are then used in Section 7 where we match the solutions from the core and the far field by using Lyapunov– Schmidt reduction and prove Theorem 1. The paper is concluded with suggestions for extensions and some open problems. 2. Spatial-dynamics formulation If λ is an eigenvalue of L + ρ, then there exists u ∈ H 4 (R2 ) such that 2 u + (θ + ρ)u = λu.

(2)

Let r3 > r2 > max(1, r1 ). We introduce a new radial variable s(r) =

log r r

if r r2 , if r r3 ,

(3)

and for r ∈ (r2 , r3 ), we define s such that s ∈ C ∞ (R+ ; R) is strictly increasing. Note that this implies that there exist constants c and C such that 0 < c < C and c s (r) C for every r2 r r3 . We define θ˜ and ρ˜ by θ˜ (s(r)) = θ (r), etc. Since s is an increasing function, it is invertible, and we denote the inverse function by r(s). Let sj := s(rj ), j = 1, . . . , 3. Under the given by coordinate transformation (3), the space R transforms into the space R := L2 (−∞, s1 ]; H 1/2 S 1 , e2s ds , R that is, the weighted L2 space with values in H 1/2 (S 1 ) and weight e2s . Setting v = u, Eq. (2) is equivalent to the system u = v, v = (λ − θ˜ − ρ)u, ˜

(4)

where in the variables s and ϕ, the laplacian is given by 2

2 2

∂ r (s) ∂ r (s) r (s) ∂ 1 . − + + = 2 2 r(s) r (s) ∂s r(s) ∂ϕ 2 r (s) ∂s Rewriting this intermediate system as a first order system, with u1 = u, u2 = u , u3 = v and u4 = v , where denotes differentiation with respect to s, we obtain a system of the form ˜ U (s) = A(s; λ, ρ)U, where A(s; λ, ρ) ˜ is given by

(5)


⎛ ⎜ A(s; λ, ρ) ˜ := ⎜ ⎝

0

(s)2 2 − rr(s) 2 ∂

r (s) r (s)

0 (λ − θ˜ − ρ)r ˜ (s)2

1 − 0 0

r (s) r(s)

0

r (s)2 0

r (s)2

− r(s)2 ∂ 2

r (s) r (s)

345

0 0 1 −

⎞

r (s)

⎟ ⎟ ⎠

(6)

r(s)

∂ where ∂ denotes differentiation with respect to ϕ, i.e., ∂ = ∂ϕ . The expression for A(s; λ, ρ) ˜ simplifies significantly for s < s2 or s > s3 ; see Sections 3 and 5. is, in general, not continuous or even bounded, so we need to study The perturbation ρ˜ ∈ R weak solutions of (5). Let

X = H 2 S 1 × H 1 S 1 × H 1 S 1 × L2 S 1 , Y = H 3 S1 × H 2 S1 × H 2 S1 × H 1 S1 . Definition 1. Let J be an interval of R. A function U : J → X is a weak solution of (5) in J if 1 (J ; X), 1. U ∈ L2loc (J ; Y ) ∩ Hloc 2. for every V ∈ C0∞ (J ; X) we have

− J

U (s)V (s) ds =

A(s; λ, ρ)U ˜ (s)V (s) ds. J

4 (R2 ) if and only if (5) Lemma 2. Let λ ∈ R. The eigenvalue equation (2) has a solution u ∈ Hloc 1 2 ∞ has a weak solution U ∈ Hloc (R; X) ∩ Lloc (R; Y ) ∩ L (R− ; X). 4 (R2 ) is a solution of (2), and let U := (u, u , u, (u) )T , where Proof. Suppose that u ∈ Hloc denotes differentiation with respect to s. We first consider u as a function of r, and let BR (0) be a ball centered at 0, with R any positive radius. Then

u ∈ H 1 (0, R); H 3 S 1 , r dr ∩ L2 (0, R); H 4 S 1 , r dr ⊂ H 1 (0, R); H 2 S 1 , r dr ∩ L2 (0, R); H 3 S 1 , r dr , u = r

(u) = r

du ∈ H 1 (0, R); H 2 S 1 , r dr ∩ L2 (0, R); H 3 S 1 , r dr dr ⊂ H 1 (0, R); H 1 S 1 , r dr ∩ L2 (0, R); H 2 S 1 , r dr , u ∈ H 1 (0, R); H 1 S 1 , r dr ∩ L2 (0, R); H 2 S 1 , r dr ,

d(u) ∈ H 1 (0, R); L2 S 1 , r dr ∩ L2 (0, R); H 1 S 1 , r dr , dr

where r = dr/ds = r for r < r2 . By the Sobolev embedding theorem, U ∈ C([0, R]; X), and so U (s(r)) has a limit as r → 0+, or equivalently, as s → −∞. Hence, viewing U as a function of s, 1 (R; X) ∩ L2 (R; Y ). It is clear from the construction U ∈ L∞ (R− ; X). We also see that U ∈ Hloc loc that U is a weak solution of (5).

346


1 (R; X) ∩ L2 (R; Y ) ∩ L∞ (R ; X) be a weak solution of (5), and let Conversely, let U ∈ Hloc − loc 1 (R ; H 2 (S 1 )) ⊂ u = U1 . Viewing u as a function of r rather than of s, it is clear that u ∈ Hloc + 1 C((0, ∞); C(S )), and so by (4),

2 u = (λ − θ − ρ)u ∈ L2loc R+ ; H 1/2 S 1 , r dr ⊂ L2loc R+ ; L2 S 1 , r dr = L2loc R2 \ {0} . Since we also have u ∈ L∞ [0, r2 ]; H 2 S 1 ⊂ L∞ [0, r2 ]; C S 1 , λ − θ − ρ ∈ L2 [0, r2 ]; H 1/2 S 1 , r dr ⊂ L2 [0, r2 ]; L2 S 1 , r dr , we see that 2 u = (λ − θ − ρ)u ∈ L2 ([0, r2 ]; L2 (S 1 ), r dr) = L2 (Br2 (0)). We have proved that 4 (R2 \ {0}) = H 4 (R2 ). u ∈ H 4 (Br2 (0)) ∩ Hloc loc Since it is also clear that u solves (2), the proof is complete. 2 Note that a weak solution satisfies U ∈ C(R; X) (see e.g. [9, p. 286]), and so the following definition for an exponential dichotomy makes sense (see also [7] for the standard definition for ODEs and [15] for an extension to PDEs): Definition 2. Let J be an unbounded subinterval of R. We say that Eq. (5) has an exponential dichotomy in X on J if there exists a family of projections P (s) for s ∈ J such that for any s ∈ J , P (s) ∈ L(X), P (s)2 = P (s) and P (·)U ∈ C(J ; X) for every U ∈ X, and there exist constants K > 0 and κ s < κ u with the following properties: (i) For each t ∈ J and U ∈ X there exists a unique weak solution Φ s (s, t)U of (5) defined for s t, s, t ∈ J such that Φ s (t, t)U = P (t)U and s Φ (s, t)U Keκ s (s−t) U X X for all s t, s, t ∈ J . (ii) For each t ∈ J and U ∈ X there exists a unique weak solution Φ u (s, t)U of (5) defined for s t, s, t ∈ J such that Φ u (t, t)U = (I − P (t))U and u Φ (s, t)U Keκ u (s−t) U X X for all s t, s, t ∈ J . (iii) The solutions Φ s (s, t)U and Φ u (s, t)U satisfy Φ s (s, t)U ∈ Ran P (s) Φ u (s, t)U ∈ ker P (s)

for every s t, s, t ∈ J, for every s t, s, t ∈ J.

We also need the definition of time-dependent exponential dichotomy, which will be used for J = [s1 , ∞) and with X s := H 1 × L2 × H 1 × L2 with the s-dependent norm


U 2X s :=

347

1 u1 2H 1 (S 1 ) + u1 2L2 (S 1 ) + u2 L2 (S 1 ) s2 1 + 2 u3 2H 1 (S 1 ) + u3 2L2 (S 1 ) + u4 L2 (S 1 ) , s

where uj are the components of U , j = 1, . . . , 4. Definition 3. Let J be an unbounded subinterval of R. We say that Eq. (5) has a time-dependent exponential dichotomy in X s on J if there exists a family of projections P (s) for s ∈ J such that for any s ∈ J , P (s) ∈ L(X s ), P (s)2 = P (s) and P (·)U ∈ C(J ; X ) for every U ∈ X , and there exist constants K > 0 and κ s < κ u with the following properties: (i) For each t ∈ J and U ∈ X t there exists a unique solution Φ s (s, t)U of (5) defined for s t, s, t ∈ J such that Φ s (t, t)U = P (t)U and s Φ (s, t)U

Xs

Keκ

s (s−t)

U X t

for all s t, s, t ∈ J . (ii) For each t ∈ J and U ∈ X t there exists a unique solution Φ u (s, t)U of (5) defined for s t, s, t ∈ J such that Φ u (t, t)U = (I − P (t))U and u Φ (s, t)U

Xs

Keκ

u (s−t)

U X t

for all s t, s, t ∈ J . (iii) The solutions Φ s (s, t)U and Φ u (s, t)U satisfy Φ s (s, t)U ∈ Ran P (s) Φ u (s, t)U ∈ ker P (s)

for every s t, s, t ∈ J, for every s t, s, t ∈ J.

In the following sections, we will consider the intervals J− := (−∞, s1 ]

and J+ := [s1 , ∞),

and show that the system (5) has an exponential dichotomy on J− and a time-dependent exponential dichotomy on J+ . 3. Dichotomies for the system at −∞ For s s1 , r(s) = es , and hence the system (5) is given by ⎧ u1 = u2 , ⎪ ⎪ ⎨ u = −∂ 2 u + e2s u , 1 3 2 =u , u ⎪ 4 ⎪ ⎩ 3 ˜ − ρ(s, ˜ ·) e2s u1 − ∂ 2 u3 . u4 = λ − θ(s)

(7)

348


In the limit as s → −∞ we have the system ⎞ ⎛ u 1 0 ⎜ u ⎟ ⎜ −∂ 2 ⎜ 2⎟=⎝ ⎝ u ⎠ 0 3 0 u4 ⎛

⎞⎛ ⎞ 0 u1 0 ⎟ ⎜ u2 ⎟ ⎠⎝ ⎠, u3 1 0 u4

1 0 0 0 0 0 0 −∂ 2

or U = A− U.

(8)

We expand U = (u1 , u2 , u3 , u4 )T as a Fourier series in the ϕ variable, and denote the k-th k (s). For j ∈ R, we define the weighted l 2 spaces l 2 with norm defined Fourier coefficient by U j by j {ak }k∈Z 22 := 1 + k 2 |ak |2 . l j

k∈Z

The function space induced by X is := l22 × l12 × l12 × l 2 . X

(9)

The system (8) decouples in the Fourier space and for k ∈ Z we have k (s) = A − (k)U k (s), U

(10)

where ⎛

0 2 k ⎜ − (k) := ⎝ A 0 0

1 0 0 0 0 0 0 k2

⎞ 0 0⎟ ⎠. 1 0

− (k) are ±|k|, and for k = 0 both eigenvalues have geometric multiplicThe eigenvalues of A ity 2. The eigenvectors for k = 0 are (±1/k 2 , 1/|k|, 0, 0)T and (0, 0, ±1/|k|, 1) (we normalise norm is approximately constant and bounded away from 0 as the eigenvectors so that their X k → ∞). Let ⎛

−1/k 2 ⎜ 1/|k| Mk := ⎝ 0 0

0 0 −1/|k| 1

⎞ 1/k 2 0 1/|k| 0 ⎟ ⎠ 0 1/|k| 0 1

and ⎛

−|k| ⎜ 0 Dk := ⎝ 0 0

0 −|k| 0 0

0 0 |k| 0

⎞ 0 0 ⎟ ⎠, 0 |k|


349

− (k) = Mk Dk M −1 for k = 0. Note that so that A k ⎛

−k 2 1 ⎜ 0 Mk−1 = ⎝ 2 k 2 0

|k| 0 |k| 0

0 −|k| 0 |k|

⎞ 0 1⎟ ⎠. 0 1

− (0) For k = 0, the eigenvalue 0 has algebraic multiplicity 4 and geometric multiplicity 2, so A − (0) is already in Jordan normal form. We define is not diagonalisable. Note however that A − (0). M0 = I and D0 = A Lemma 3. The operator A− : X → X is a closed densely defined operator with spectrum σ (A− ) = Z. Proof. Recall that X = H 2 (S 1 ) × H 1 (S 1 ) × H 1 (S 1 ) × L2 (S 1 ), and Y = H 3 (S 1 ) × H 2 (S 1 ) × H 2 (S 1 ) × H 1 (S 1 ). It is easy to check that the domain of A− is Y , which is dense in X. To see that A− is closed, let Uj ∈ Y be such that Uj → U in X and A− Uj → f in X. We write Uj = (u1,j , u2,j , u3,j , u4,j )T , etc. By the definition of A− we have u1,j → u1

in H 2 ,

u2,j → u2

in H 1 ,

u3,j → u3

in H 1 ,

u4,j → u4

in L2 ,

while u2,j → f1

in H 2 ,

−∂ 2 u1,j → f2

in H 1 ,

u4,j → f3

in H 1 ,

−∂ 2 u3,j → f4

in L2 .

It follows that u2 = f1 ∈ H 2 , and that u1,j converges in H 3 . Since u1,j → u1 in H 2 ⊃ H 3 , and since limits (in H 2 ) are unique if they exist, we also have u1,j → u1 in H 3 , and so −∂ 2 u1 = f2 . It follows in exactly the same way that u4 = f3 ∈ H 1 , that u3 ∈ H 2 and that −∂ 2 u3 = f4 . This shows that U ∈ Y and A− U = F , and so A− : X → X is closed. − : X →X defined by The operator A− : X → X induces an operator A − U − (k)U k . )k := A (A with domain Y := l 2 × l 2 × l 2 × l 2 . − is a densely defined operator on X Then A 3 2 2 1 − − μI ) : X →X It is clear that (A− − μI ) : X → X has a bounded inverse if and only if (A has a bounded inverse. It is also clear that k ∈ σ (A− ) for k ∈ Z. To prove that there are no other points in the spectrum of A− , let μ ∈ C \ Z.

350


: l 2 × l 2 × l 2 × l 2 → l 2 × l 2 × l 2 × l 2 by Define M 2 1 1 U )k = Mk U k , (M is a linear homeomorphism between these spaces. Define also the unbounded and note that M operator D on l 2 × l 2 × l 2 × l 2 by U )k = Dk U k . (D is a closed densely defined operator with domain l 2 ×l 2 ×l 2 ×l 2 , and that σ (D) = Z. Note that D 1 1 1 1 If μ ∈ C \ Z, then − − μI )−1 = M( D − μI )−1 M −1 . (A − − μI )−1 : X →X is bounded, and consequently also It is now easy to see that (A −1 (A− − μI ) : X → X. 2 Having established that the spectrum of A− consists exactly of its eigenvalues, we define the (generalised) spectral projections P s , P c , P u in X, corresponding to the negative, the zero and the positive eigenvalues of A− , respectively. Let X s = P s X, etc. so that X = X s ⊕ X c ⊕ X u , where X s and X u are infinite-dimensional whereas X c is four-dimensional. We also define cor− (k), in the spaces Xk , k ∈ Z \ {0} and note that if responding spectral projections Pks , Pku of A ik· U = k∈Z Uk e ∈ X, then P sU =

k eik· , Pks U

k∈Z\{0}

P uU =

k eik· , Pku U

k∈Z\{0}

0 . P cU = U Lemma 4. The operator A− possesses an exponential dichotomy in X on J− = (−∞, s1 ] with constant K and rates κ s = 0 and κ u = 1, and another exponential dichotomy in X on J− with constant K and rates κ s = −1 and κ u = 0. Proof. Let η ∈ (0, 1) be arbitrary. We apply Lemma 2.1 of [15] for the operators A− − ηI and A− + ηI , and obtain exponential dichotomies with constant K and rates κ s = −η and κ u = 1 − η, and κ s = −1 + η and κ u = η, respectively. The existence of exponential dichotomies for A− with rates κ s = −1 and κ u = 0, and κ s = 0 and κ u = 1, respectively then follows by using the transformation V = e±η· U . We only consider the operator A− − ηI , since the proof for A− + ηI is similar. The result follows from Lemma 2.1 of [15] if we can verify condition (H1) of [15] for the operator A− − ηI , namely:


351

(H1) Suppose that there exists a constant C > 0 such that (A− − ηI − iμI )−1 L(X)

C 1 + |μ|

for every μ ∈ R. such that As in the proof of Lemma 3, it suffices to prove that there exists a constant C (D − ηI − iμI )−1 U 2 2 2 2 l ×l ×l ×l

C l 2 ×l 2 ×l 2 ×l 2 U 1 + |μ|

∈ l 2 × l 2 × l 2 × l 2 . Note that for k = 0 we have for every U (Dk − ηI − iμI )−1 U k 2 =

k |2 |U 2 k |2 , |U 2 2 2 (k − η) + μ min(η , (1 + η)2 )(1 + |μ|)2

0 |2 . Hence and it is not difficult to see that a similar estimate holds for |(D0 − ηI − iμI )U (Dk − ηI − iμI )−1 U (D − ηI − iμI )−1 U 22 2 2 2 k 2 l ×l ×l ×l k∈Z

2 C k |2 . |U 2 (1 + |μ|)

2

k∈Z

4. Dichotomies near the core The system (7) can be abbreviated and written as U = A− + B(s; λ, ρ) ˜ U,

(11)

where ⎛

0 0 ⎜ B(s; λ, ρ) ˜ := e2s ⎝ 0 λ − θ˜ (s) − ρ(s, ˜ ·)

0 0 0 1 0 0 0 0

⎞ 0 0⎟ ⎠. 0 0

(12)

We will show that the system (11) has an exponential dichotomy on the interval J− = (−∞, s1 ]. To show this, we would like to apply Theorem 1 of [15]. This is not possible, however, since is not smooth enough in s. We are interested in ρ˜ small and consider therefore first ρ˜ = 0, ρ˜ ∈ R and show that the λ-perturbed system U = A− + B(s; λ, 0) U

(13)

possesses an exponential dichotomy in X on J− . Then we will use the implicit function theorem to show that also the system (11) possesses an exponential dichotomy. Note that from its definition, it follows immediately that B(s; λ, 0) ∈ L(X).

352


As θ˜ does not depend on ϕ, the system (13) decouples in Fourier space, just as the limiting system (8). Using the same notation in the Fourier spaces as before, we get for k ∈ Z − (k) + B(s; λ, 0) U k (s). k (s) = A U

(14)

k and s to get estimates which are uniform in k. For − (k) = Mk Dk M −1 , we rescale both U As A k k (τ/|k| + s1 ). Then (14) becomes k = 0, define τ = |k|(s − s1 ) and Vk (τ ) = Mk−1 U

d 1 −1 Vk = D1 + M B τ/|k|; λ, 0 Mk Vk . dτ |k| k

(15)

A short calculation shows that 2τ/|k| −1 M B τ/|k|; λ, 0 Mk e ˜ /k 2 . sup 1, λ − θ(s) k 2 ss1

Hence there exists a constant C such that for all k = 0, |Mk−1 B(τ/|k|; λ, 0)Mk | 2Ce2τ/|k| and 0 −∞

1 −1 Mk B τ/|k|; λ, 0 Mk dτ |k|

0

−∞

2Ce2τ/|k| dτ = C. |k|

(16)

By the proof of the roughness theorem for ordinary dichotomies (see [7] for details), the u/s system (15) has an exponential dichotomy which we denote by Ψk (τ, σ ), with constants K, κ u = 1, κ s = −1. We choose the dichotomy in such a way that Ran Ψ−s (s1 , s1 ; λ, 0) ⊂ span{e1 , e2 }, where ej , j = 1, . . . , 4, are the standard basis vectors of C4 (again see [7]). This implies that the stable and unstable solutions satisfy s Ψ (σ, τ )Vk Ke−(σ −τ ) |Vk |, τ σ 0, k u Ψ (σ, τ )Vk Ke(σ −τ ) |Vk |, σ τ 0. k The norm in X induces a norm on the Fourier space Xk with 2 ik· 2 k ]1 2 + k 2 + 1 [U k ]2 2 + k 2 + 1 [U k ]3 2 + [U k ]4 2 . k 2X := U e = k 2 + 1 [U U k X k As seen in the proof of Lemma 3, Mk is a linear homeomorphism between C4 and Xk . Thus if u/s u/s we denote the exponential dichotomy of the unscaled system (14) by Φk , then Φk (s, t) = u/s k ∈ Xk Mk Ψk (|k|(s − s1 ), |k|(t − s1 ))Mk−1 , and they satisfy for U s Φ (s, t)U k Ke−|k|(s−t) U k Xk Ke−(s−t) U k Xk , t s s1 , k Xk u Φ (s, t)U k Ke|k|(s−t) U k Xk Ke(s−t) U k Xk , s t s1 , k X k

for some constant K, which is independent of k.

(17)


353

0 (s) and the integraFor the central space, corresponding to k = 0, the scaling V0 (s) = e±s U bility of B(s; λ, 0) shows that, for any > 0, Φ0 (s, t)U 0 Ke(s−t) |U 0 |, t s s1 , Φ0 (s, t)U 0 Ke−(s−t) |U 0 |, s t s1 .

(18)

Thus for the full solutions, we can define the stable and center-unstable solutions s Φ− (s, t; λ, 0)U =

k eik· , Φks (s, t)U

k∈Z\{0} cu 0 + Φ− (s, t; λ, 0)U = Φ0 (s, t)U

s t s1 ,

k eik· , Φku (s, t)U

t s s1 ,


s t s1 ,

k∈Z\{0}

and the unstable and center-stable solutions cs 0 + Φ− (s, t; λ, 0)U = Φ0 (s, t)U u Φ− (s, t; λ, 0)U =

k∈Z\{0}

k eik· , Φku (s, t)U

t s s1 .

k∈Z\{0}

These solutions are related to dichotomies for (13) in X on J− . Lemma 5. Let −1 = κ s < κ cu < 0 < κ cs < κ u = 1 and λ ∈ R. Then the system (13) has an exponential dichotomy in X on J− with constant K and rates κ cu and κ s , and another with constant s (s , s ; λ, 0) = P s K and rates κ u and κ cs . The dichotomies can be chosen such that Ran Φ− 1 1 cs cs and Ran Φ− (s1 , s1 ; λ, 0) = P . Moreover, for any t ∈ (−∞, s1 ] and U0 ∈ X, the solutions cu (·, t; λ, 0)U and Φ s (·, t; λ, 0)U belong to C ∞ ((−∞, t); X) and C ∞ ((t, s ); X), respecΦ− 0 0 1 − u (·, t; λ, 0)U and Φ cs (·, t; λ, 0)U belong to C ∞ ((−∞, t); X) tively. Similarly, the solutions Φ− 0 0 − and C ∞ ((t, s1 ); X), respectively. All solutions also depend smoothly on the parameter λ. Proof. The scaling e±ηs U for 0 < η < 1 and the dichotomy estimates in (17) and (18) immes (s , s ; λ, 0) = P s and diately prove the first part of the lemma. The dichotomies satisfy Ran Φ− 1 1 u/s cs (s , s ; λ, 0) = P c + P s since we have chosen the Ψ above to satisfy Ran Ψ−s (s1 , s1 ; Ran Φ− 1 1 k λ, 0) ⊂ span{e1 , e2 } (cf. the definition of Dk ). The smoothness with respect to s follows since θ is smooth in s and smoothness in λ can be proved using an implicit function theorem argument. First observe that for any λ, λ˜ close to each other, the solutions Φ cu and Φ s satisfy the integral equations s cu cu s cu ˜ 0) + Φ− ˜ 0) dτ 0 = −Φ− (s, t; λ, (s, t; λ, 0) + (λ˜ − λ) Φ− (s, τ ; λ, 0)e2τ B0 Φ− (τ, t; λ, −∞

t − s

cu cu ˜ 0) dτ Φ− (s, τ ; λ, 0)e2τ B0 Φ− (τ, t; λ,

s1 + t

cu s ˜ 0) dτ Φ− (s, τ ; λ, 0)e2τ B0 Φ− (τ, t; λ,

,

354


t s s s cu 0 = −Φ− (s, t; λ˜ , 0) + Φ− (s, t; λ, 0) − (λ˜ − λ) Φ− (s, τ ; λ, 0)e2τ B0 Φ− (τ, t; λ˜ , 0) dτ −∞

s −

s s ˜ 0) dτ Φ− (s, τ ; λ, 0)e2τ B0 Φ− (τ, t; λ,

s1 +

cu s ˜ 0) dτ Φ− (s, τ ; λ, 0)e2τ B0 Φ− (τ, t; λ,

,

s

t

for s t s1 and t s s1 , respectively, where B0 is the matrix ⎛

0 ⎜0 B0 := ⎝ 0 1

0 0 0 0

0 0 0 0

⎞ 0 0⎟ ⎠. 0 0

(19)

Define the function spaces X s := Φ s ; Φ s (s, t) ∈ L(X) is defined and continuous for t s s1 ! s with Φ s s := sup e−κ (s−t) Φ s (s, t)L(X) , tss1

X cu := Φ cu ; Φ cu (s, t) ∈ L(X) is defined and continuous for s t s1 ! cu with Φ cu cu := sup e−κ (s−t) Φ cu (s, t)L(X) . sts1

For λ fixed, the integral equations can be written as F (Φ cu , Φ s ; λ˜ ) = 0, where F : X cu × X s × R → X cu × X s . The estimates of the exponential dichotomies immediately give that F is indeed a mapping between those spaces, for example, s s 2τ cu ˜ 0) dτ Φ− (s, τ ; λ, 0)e B0 Φ− (τ, t; λ, −∞

L(X)

s

K 2 eκ

s (s−τ )

e2τ eκ

cu (τ −t)

dτ

−∞ cu

=

K 2 e2s1 eκ (s−t) . 2 + κ cu − κ s

The other integrals can be estimated in a similar way. Since D(Φ cu ,Φ s ) F (Φ cu (s, t; λ, 0), Φ s (s, t; λ, 0); λ) = I , the implicit function theorem can be applied and the smoothness with respect to λ follows immediately. 2 Remark 1. The ϕ-independence of θ˜ is not essential in Lemma 5. The lemma can be proved for smooth ϕ-dependent functions θ by using Theorem 1 of [15] and verifying the conditions (H1), (H2), (H3) and (H5) of that paper.


355

Next, we prove four technical lemmas needed in the proof of the existence of exponential We work in exponentially weighted spaces, and dichotomies for the full system (11) with ρ˜ ∈ R. for an unbounded interval J ⊂ J− and η ∈ R, we let Cη (J ; X) be the space defined by ! Cη (J ; X) := U ∈ C(J ; X); U Cη := sup eηs U (s)X < ∞ . s∈J

Hence C0 (J, X) is the space of continuous functions with an X-norm that is uniformly bounded in J . Lemma 6. Let J ⊂ J− and pick u ∈ C0 (J ; H 2 (S 1 )) and ρ ∈ L2 (J ; H 1/2 (S 1 )), then ρu ∈ L2 (J ; H 1/2 (S 1 )). Proof. We need to prove that for s fixed, ρ(s)u(s)

H 1/2 (S 1 )

C u(s)H 2 (S 1 ) ρ(s)H 1/2 (S 1 ) .

(20)

Indeed, if this is proved, the claim follows, since ρu 2L2 (J ;H 1/2 (S 1 )) =

ρ(s)u(s)2

H 1/2 (S 1 )

J

C2

ρ(s)2

H 1/2 (S 1 )

J

2 C 2 supu(s)H 2 (S 1 ) s∈J

=C

2

ds

u 2H 2 (S 1 ) ds

ρ(s)2

H 1/2 (S 1 )

ds

J

u 2C (J ;H 2 (S 1 )) ρ 2L2 (J ;H 1/2 (S 1 )) . 0

To prove (20), let u ∈ H 2 (S 1 ) and ρ ∈ H 1/2 (S 1 ) (we suppress the variable s for simplicity of notation). Let ρˆk and uˆ k be the Fourier coefficients of ρ and u, respectively. We have u 2H 2 =

k∈Z

ρ 2H 1/2 =

2 uˆ 2k 1 + k 2 , 1/2 ρˆk2 1 + k 2 .

k∈Z

Then (" uρ)k = j ∈Z uˆ j ρˆk−j , and so uρ 2H 1/2 = k∈Z ( j ∈Z uˆ j ρˆk−j )2 (1 + k 2 )1/2 . Let v and σ be the functions with Fourier coefficients uˆ k (1 + k 2 )1/4 and ρˆk (1 + k 2 )1/4 , respectively. Note that v ∈ H 3/2 (S 1 ) and σ ∈ L2 (S 1 ). Now observe that 2 1 + k 2 = 1 + (k − j ) + j 2 1 + j 2 + 2 1 + (k − j )2 , and hence

356


1/4 1/4 1/4 1 + k2 21/4 1 + j 2 + 1 + (k − j )2 for any j ∈ Z. Thus uρ 2H 1/2

√ 1/4 1/4 2 2 uˆ j ρˆk−j 1 + j 2 + uˆ j ρˆk−j 1 + (k − j )2 k∈Z

√ 2 2

j ∈Z

j ∈Z

1/4 uˆ j ρˆk−j 1 + j 2

2

+

j ∈Z

k∈Z

1/4 uˆ j ρˆk−j 1 + (k − j )2

2

j ∈Z

√ = 2 2 vρ 2L2 + uσ 2L2 $ √ # 2 2 2 2 sup v(ϕ) ρ 2L2 + sup u(ϕ) σ 2L2 ϕ∈S 1

ϕ∈S 1

C 2 v 2H 3/2 ρ 2L2 + u 2H 2 σ 2L2 C 2 u 2H 2 ρ 2H 1/2 for some constant C > 0. This completes the proof.

2

and s ∈ J− , let For each ρ˜ ∈ R δB(s; ρ) ˜ := B(s; λ, ρ) ˜ − B(s; λ, 0) = −e2s ρ(s)B ˜ 0, where B0 has been defined in (19). Note that for any s ∈ J− , δB(s; ρ) ˜ L(X)

2s e ρ(s)u ˜ 1

sup

L2 (S 1 )

u1 ∈H 2 (S 1 ) u1 H 2 =1

˜ L2 (S 1 ) , Ce2s ρ(s)

C

sup

sup u1 (ϕ)e2s ρ(s) ˜ L2 (S 1 )

u1 ∈H 2 (S 1 ) ϕ∈S 1 u1 H 2 =1

(21)

where we use the notation C for the different constants occurring. It follows that s1

2 δB(s; ρ) ˜

s1

L(X) ds

C

2

−∞

e

2 ρ(s) ˜ L2 (S 1 ) e2s ds C 2 e2s1

2s

−∞

s1

2 ρ(s) ˜ 2

L (S 1 )

e2s ds

−∞

C 2 e2s1 ρ ˜ 2R .

(22)

Lemma 7. For η ∈ (−1, κ cu ), where κ cu is as in Lemma 5, pick U−cu ∈ Cη (J− ; X), and ρ˜ ∈ R. Let s ∈ J− . Then the integral s I := −∞

belongs to X.

A− eA− P

s (s−τ )

P s δB(τ ; ρ)U ˜ −cu (τ ) dt


357

Proof. Let H (τ ) := e(η−1)τ δB(τ ; ρ)U ˜ −cu (τ ). By the definition of δB(τ, ρ), ˜ T T H (τ ) = 0, 0, 0, −e(η+1)τ ρ(τ ˜ )u(τ ) =: 0, 0, 0, h(τ ) , where u(τ ) is the first component of U−cu (τ ). Then eη· u ∈ C0 (J− ; H 2 (S 1 )) and e· ρ˜ ∈ L2 (J− ; H 1/2 (S 1 )), and so by Lemma 6, h ∈ L2 (J− ; H 1/2 (S 1 )). For k ∈ Z, let Hk (τ ) and hk (τ ) s := M −1 P s Mk . To show that be the Fourier coefficients of H (τ ) and h(τ ), respectively. Let P k k k (see (9)), where I exists in X, it suffices to show that {Ik }k∈Z ∈ X s

s

e(1−η)τ Mk Dk eDk Pk (s−τ ) Mk−1 P s Hk (τ ) dτ

Ik := −∞

=

1 2

s

T e(1−η)τ e−|k|(s−τ ) hk (τ ) dτ 0, 0, 1, −|k| .

−∞

We therefore need to prove that %

&

s |k|

e

(1−η)τ −|k|(s−τ )

e

∈ l2.

hk (τ ) dτ

−∞

k∈Z

Using that η < 0 and hk ∈ L2 (J− ) (as h ∈ L2 (J− ; H 1/2 (S 1 ))), we note that s (1/2 ' s (1/2 ' s (1−η)τ −|k|(s−τ ) −|k|s 2(1−η+|k|)τ 2 |k| e e hk (τ ) dτ |k|e e dτ hk (τ ) dτ −∞

−∞

−∞

1 e(1−η)s hk L2 ((−∞,s]) 2(1 − η + |k|) 1/4 e(1−η)s1 1 + |k|2 hk L2 (J− ) .

= |k| √

Since {(1 + |k|2 )1/4 hk L2 (J− ) }k∈Z ∈ l 2 , the proof is complete.

2

Lemma 8. For −1 < η < κ cu , where κ cu < 0 is as in Lemma 5, pick U−cu ∈ Cη (J− ; X) and Then the integrals ρ˜ ∈ R. s

s s Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −cu (τ ) dτ

−∞

exist in X for each s ∈ J− .

s B(s; λ, 0)Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −cu (τ ) dτ

and −∞

358


Proof. We use (21) and compute s s cu Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ − (τ ) dτ −∞

X

s

s Φ (s, τ ; λ, 0)

C

−

L(X) e

ρ(τ ˜ )L2 (S 1 ) U−cu (τ )X dτ

2τ

−∞

s C

e2τ eκ

s (s−τ )

ρ(τ ˜ )

L2 (S 1 )

cu U (τ ) dτ − X

−∞

C U cu −

Cη (J− ;X)

e

κs s

' s e

2(1−κ s −η)τ

(1/2 ' s dτ

e

−∞

C U−cu C

η (J− ;X)

√

2 ρ(τ ˜ ) 2

2τ

L (S 1 )

(1/2 dτ

−∞

1 ˜ R e(1−η)s ρ . 2(1 − κ s − η)

Using that B(s; λ, 0) ∈ L(X), it follows that both integrals converge in X.

2

Lemma 9. Let −1 < η < κ cu , where κ cu < 0 is as in Lemma 5. Let U−cu ∈ Cη (J− ; X) and ρ˜ ∈ R. For every s ∈ J− , the integral s s A− Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −cu (τ ) dτ

(23)

−∞

exists in X. Proof. By (3.1) of [15], for τ s s1 ,

s s Φ− (s, τ ; λ, 0) = eA− P (s−τ ) P s

τ −

eA− P

s (s−ξ )

cu P s B(ξ ; λ, 0)Φ− (ξ, τ ; λ, 0) dξ

−∞

s +

eA− P

s (s−ξ )

eA− P

cu (s−ξ )

s P s B(ξ ; λ, 0)Φ− (ξ, τ ; λ, 0) dξ

τ

s1 −

s P cu B(ξ ; λ, 0)Φ− (ξ, τ ; λ, 0) dξ.

s s (s , s ; λ, 0) has been chosen so that it coincides with Ran P s . Note that we used here that Ran Φ− 1 1


359

Substituting this into (23), we have four integrals to estimate, the first of which was dealt with in Lemma 7. The other three integrals are s I1 :=

τ eA− P

A−

−∞

−∞

s

s

I2 :=

A−

−∞

eA− P

s (s−ξ )

eA− P

cu (s−ξ )

cu P s B(ξ ; λ, 0)Φ− (ξ, τ ; λ, 0) dξ δB(τ ; ρ)U ˜ −cu (τ ) dτ,

s P s B(ξ ; λ, 0)Φ− (ξ, τ ; λ, 0) dξ δB(τ ; ρ)U ˜ −cu (τ ) dτ,

τ

s I3 :=

s (s−ξ )

s1 A−

−∞

s P cu B(ξ ; λ, 0)Φ− (ξ, τ ; λ, 0) dξ δB(τ ; ρ)U ˜ −cu (τ ) dτ.

s

We carry out the calculations for I1 , since the others are similar. Let (φj l (ξ, τ )), j, l = 1, . . . , 4, cu (ξ, τ ; λ, 0), and as in the proof of Lemma 7, let be the entries of the matrix corresponding to Φ− ˜ )u(τ ). Recall that h ∈ L2 (J− ; H 1/2 (S 1 )). A short calculation shows that h(τ ) = −e(η+1)τ ρ(τ ⎞ 0 φ34 (ξ, τ )h(τ ) ⎟ ⎜ cu B(ξ ; λ, 0)Φ− (ξ, τ ; λ, 0)δB(τ ; ρ)U ˜ −cu (τ )e2ξ +(1−η)τ ⎝ ⎠. 0 (λ − θ˜ (τ ))φ14 (ξ, τ )h(τ ) ⎛

Note that φ34 (ξ, τ ) and φ14 (ξ, τ ) map L2 (S 1 ) boundedly into H 1 (S 1 ) and H 2 (S 1 ), respectively, and that by Lemma 5 for ξ τ s1 φ34 (ξ, τ ) φ14 (ξ, τ )

L(L2 ;H 1 ) L(L2 ;H 2 )

∗Keκ ∗Ke

cu (ξ −τ )

,

κ cu (ξ −τ )

.

Introducing the notation f (ξ, τ ) := e−κ (ξ −τ ) φ34 (ξ, τ )h(τ ), and g(ξ, τ ) := e−κ (ξ −τ ) (λ − θ˜ (τ ))φ14 (ξ, τ )h(τ ), we note that max( f (ξ, τ ) H 1 , g(ξ, τ ) H 2 ) K h(τ ) L2 , for ξ < τ < s1 . The Fourier coefficients of f (ξ, τ ) and g(ξ, τ ) are denoted by fˆk (ξ, τ ) and gˆ k (ξ, τ ), respectively. where To prove that I1 ∈ X, it suffices to prove that {Jk }k∈Z ∈ X, cu

cu

⎞ 0 ˆ cu s ⎜ f (ξ, τ ) ⎟ Mk D k e2ξ +(1−η)τ eκ (ξ −τ ) eDk Pk (s−ξ ) Mk−1 P s ⎝ k Jk : = ⎠ dξ dτ 0 −∞ −∞ gˆ k (ξ, τ ) ⎞ ⎛ ˆ fk (ξ, τ ) s τ ˆ 1 2ξ +(1−η)τ κ cu (ξ −τ ) −|k|(s−ξ ) ⎜ −|k|fk (ξ, τ ) ⎟ = e e e ⎠ dξ dτ, ⎝ gˆ k (ξ, τ ) 2 −∞ −∞ −|k|gˆ k (ξ, τ ) s

τ

⎛

s = M −1 P s Mk as before. The first component of Jk can be written where P k k k

(24)

360


1 −|k|s e 2

s e(1−κ

cu −η)τ

−∞

1 e−|k|s 2

τ e(1+κ

cu +|k|)ξ

eξ fˆk (ξ, τ ) dξ dτ

−∞

s −∞

1 e(2+|k|−η)τ √ 2(1 + κ cu + |k|)

1 e(2−η)s √ 4 (1 + κ cu + |k|)(2 + |k| − η)

' τ

' s τ

(1/2 e fˆk (ξ, τ ) dξ 2ξ

2

dτ

−∞

(1/2 e fˆk (ξ, τ ) dξ dτ 2ξ

2

.

−∞ −∞

The square of the l22 norm of the first component of {Jk }k∈Z can then be estimated by s τ 1 + k2 e2(2−η)s 1 + k 2 e2ξ fˆk (ξ, τ )2 dξ dτ cu 16 (1 + κ + |k|)(2 + |k| − η) k∈Z

−∞ −∞

e2(2−η)s 16(1 + κ cu )

s

e2(2−η)s = 32(1 + κ cu )

s

τ

e2ξ hˆ k (τ )2 dξ dτ

k∈Z−∞ −∞

e2τ hˆ k (τ )2 dτ

k∈Z−∞

e2(3−η)s1 h 2L2 (J ;L2 (S 1 )) . − 32(1 + κ cu )

The other three components of {Jk }k∈Z are estimated in a completely similar way, and by adding these estimates we see that I1 ∈ X. 2 We are now ready to prove the existence of exponential dichotomies for the full system. Theorem 2. Let −1 < κ s < κ cu < 0 and 0 < κ cs < κ u < 1. Then there exists a neighbourhood such that for any ρ˜ ∈ U and any λ ∈ R, the system (5) has an exponential dichotomy U of 0 in R on J− with constants K and rates κ cu , κ s , and another with constants K and rates κ u , κ cs . Moreover, the projections and evolution operators depend smoothly on λ ∈ R and ρ˜ ∈ U . The s (s, t; λ, ρ), cu (s, t; λ, ρ), cs (s, t; λ, ρ), u (s, t; λ, ρ), dichotomies are denoted by Φ− ˜ Φ− ˜ and Φ− ˜ Φ− ˜ s s respectively. The associated projections will be denoted by P− (s; λ, ρ)(:= ˜ Φ− (s, s; λ, ρ)), ˜ P−cu (s; λ, ρ), ˜ P−cs (s; λ, ρ), ˜ and P−u (s; λ, ρ), ˜ respectively. such that if ρ˜ belongs to this Proof. We will show that there exists a neighbourhood of 0 in R neighbourhood then there exist exponential dichotomies for the system (11) with this ρ. ˜ Let U0 ∈ X and t ∈ J− be fixed but arbitrary. We will use the implicit function theorem to solve the system of integral equations for the pair of functions (U−cu , U−s ) as functions of the parameters near 0 λ ∈ R and ρ˜ ∈ R


361

s cu 0 = Φ− (s, t; λ, 0)U0 − U−cu (s) +

s Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −cu (τ ) dτ

−∞

t −

cu Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −cu (τ ) dτ s

s1 cu Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −s (τ ) dτ,

+

for s t s1 ,

t

t s 0 = Φ− (s, t; λ, 0)U0

− U−s (s) −

s Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −cu (τ ) dτ

−∞

s +

s Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ −s (τ ) dτ t

s1 cu Φ− (s, τ ; λ, 0)δB(τ ; ρ, ˜ 0)U−s (τ ) dτ,

−

for t s s1 .

(25)

s cu (s, t; λ, 0)U and Φ s (s, t; λ, 0)U exist and have constants K, By Lemma 5, the dichotomies Φ− 0 0 − s cu κ˜ = −1 and κ˜ ∈ (−1, 0). Let η ∈ (κ s , κ cu ) and rewrite Eq. (25) as F (U−cu , U−s ; λ, ρ) ˜ = 0, where F : Cη ((−∞, t]; X) × → Cη ((−∞, t]; X) × Cη ([t, s1 ]; X) is the right-hand side of (25). Cη ([t, s1 ]; X) × R × R We first verify that F is indeed a map between the above spaces. We do the estimates for the first integral in the first equation of (25). The other estimates are similar. Lemma 5 gives that for any s ∈ (−∞, t] and U cu ∈ Cη ((−∞, t]; X):

s s cu e Φ− (s, τ ; λ, 0)δB(τ ; ρ)U ˜ − (τ ) dτ ηs

−∞

K

X

sup

τ ∈(−∞,s]

ητ cu e U− (τ )X

s e(κ

s −η)(s−τ )

δB(τ ; ρ) ˜

L(X) dτ

−∞

s

K U−cu C

η ((−∞,t],X)

e2(κ

s −η)(s−τ )

2 ˜ L(X) dτ + δB(τ ; ρ)

−∞

K U cu −

Cη ((−∞,t],X)

1 2s1 2 + e ρ ˜ , R 2(η − κ s )

(26)

where we have used (22). After taking the supremum over all s ∈ (−∞, s1 ] we see that the function defined by the first integral in (25) belongs to Cη (J− , X). Using similar estimates for the other integrals, we can conclude that F is indeed a map between the spaces as stated.

362


That F is smooth with respect to λ and ρ˜ follows since the evolution operators cu (·, t; λ, 0)U and Φ s (·, t; λ, 0)U are smooth in λ by Lemma 5 (using that the H 1 norm Φ− 0 0 − is weaker than the C 1 norm on bounded intervals), and since δB depends smoothly on ρ˜ (indeed, δB is a bounded linear mapping with respect to ρ). ˜ Note that cu s (·, t; λ, 0)U0 , Φ− (·, t; λ, 0)U0 ; λ, 0 = 0. F Φ− cu (·, t; λ, 0)U , The Fréchet derivative of F with respect to its two first variables evaluated at (Φ− 0 s Φ− (·, t; λ, 0)U0 ; λ, 0) is −I on Cη ((−∞, t]; X) × Cη ([t, s1 ]; X). In particular, this derivative is a linear homeomorphism on this space, and so the implicit function theorem is applicable, cu (·, t; λ, ρ)U s (·, t; λ, ρ)U and we obtain solutions Φ− ˜ 0 := U−cu and Φ− ˜ 0 := U−s of the integral Smoothness of these solutions equation (25), which exist in a neighbourhood of (λ0 , 0) in R × R. with respect to parameters also follows from a corollary of the implicit function theorem (see e.g. [5, p. 115]). cu (·, t; λ , ρ)U s Next, we need to verify that Φ− ˜ 0 are weak solutions 0 ˜ 0 and Φ− (·, t; λ0 , ρ)U cu (·, t; λ, ρ)U of (11), and that they satisfy the conditions of Definition 2. We first check that Φ− ˜ 0 cu (·, t; λ, 0)U is a C ∞ solution of is a weak solution on the interval (−∞, t]. By Lemma 5, Φ− 0

U = A− + B(s; λ, 0) U on (−∞, t], and hence it is also a weak solution of this equation. Next we deal with the integral terms. For the first integral we use the abbreviation s g(s) :=

f (s, τ ) dτ,

s cu with f (s, τ ) = Φ− (s, τ ; λ, 0)δB(τ ; ρ)Φ ˜ − (τ, t; λ0 , ρ)U ˜ 0.

−∞

Thus f is C ∞ in the first variable and L1 in the second. From its definition, it follows immediately that g is continuous. We will see that g is weakly differentiable and that

s

g (s) = f (s, s) + −∞

∂f (s, τ ) dτ. ∂s

(27)

In order to prove this, we need to check that the integral on the right-hand side of (27) exists, and that the equality (27) holds. The integral in the right-hand side of (27) is s −∞

∂f (s, τ ) dτ = ∂s

s

s cu A− + B(s; λ, 0) Φ− (s, τ ; λ, 0)δB(τ ; ρ)Φ ˜ − (τ, t; λ, ρ)U ˜ 0 dτ,

−∞

and it exists in X by Lemmas 8 and 9. Next, we calculate the distributional derivative of g and let V ∈ C0∞ ((−∞, t]; X) be a test function. Then by Fubini’s Theorem and integration by parts


t

t

g (s)V (s) ds = − −∞

363

g(s)V (s) ds

−∞

t s

f (s, τ ) dτ V (s) ds

=− −∞ −∞

t t =−

f (s, τ )V (s) ds dτ

−∞ τ

t ' =

t f (τ, τ )V (τ ) +

−∞

t =

( ∂f (s, τ )V (s) ds dτ ∂s

τ

'

s f (s, s) +

−∞

−∞

( ∂f (s, τ ) dτ V (s) ds, ∂s

and we see that the weak derivative of g is indeed given by (27). Hence d ds

s s cu Φ− (s, τ ; λ, 0)δB(τ ; ρ)Φ ˜ − (τ, t; λ, ρ)U ˜ 0 dτ −∞

cu = I − P−cu (s; λ, 0) δB(s; ρ)Φ ˜ − (s, t; λ, ρ)U ˜ 0 s +

s cu A− + B(s; λ, 0) Φ− (s, τ ; λ, 0)δB(τ ; ρ)Φ ˜ − (τ, t; λ, ρ)U ˜ 0 dτ.

(28)

−∞

)s s (s, τ ; λ, 0)δB(τ ; ρ)Φ cu (τ, t; λ, ρ)U We have already noticed that g(s) = −∞ Φ− ˜ 0 dτ is con− 2 tinuous, and so it belongs to Lloc ((−∞, t]; X). The right-hand side of (28) also belongs to L2loc ((−∞, t]; X) since the first term belongs to L2loc ((−∞, t]; X) and the second term is continuous on (−∞, t]. This shows that s s cu Φ− (s, τ ; λ, 0)δB(τ ; ρ)Φ ˜ − (τ, t; λ, ρ)U ˜ 0 dτ −∞ 1 ((−∞, t]; X). belongs to Hloc Similar calculations for the other integral terms of the first equation of (25) show that these are 1 ((−∞, t], X). After adding the terms also weakly differentiable on (−∞, t] and belong to Hloc up, we conclude that

cu d cu Φ− (s, t; λ, ρ)U ˜ 0 = A− + B(s; λ, ρ) ˜ Φ− (s, t; λ, ρ)U ˜ 0, ds cu (·, t; λ, ρ)U i.e. Φ− ˜ 0 is a weak solution of (11).

364


s (·, t; λ, ρ)U Similar calculations for the terms of the second equation of (25) show that Φ− ˜ 0 is a weak solution of (11) on the interval [t, s1 ]. Finally we check that the conditions of Definition 2 are satisfied. A similar computation as cu (s, t; λ, ρ) in (26) also shows that the estimates in (i) and (ii) of Definition 2 are satisfied for Φ− s cu s s s cu cu ˜ for any κ and κ such that −1 = κ˜ < κ < κ < κ˜ < 0. Since κ˜ cu can be and Φ− (s, t; λ, ρ) taken arbitrarily close to 1, the same is true also for κ cu . Note that (iii) of Definition 2 is satisfied s (s, s; λ, ρ). ˜ := Φ cu (s, s; λ, ρ) ˜ and P−s (s; λ, ρ) ˜ := Φ− ˜ 2 with P−cu (s; λ, ρ)

To finish this section, we derive some more details about the solutions of (5) in the case when λ = λ0 and ρ˜ = 0. We are particularly interested in the solutions on J− , and we study the exact growth/decay rate of solutions as s → −∞. As we have seen before, the space X decouples into a direct sum of four-dimensional pairwise orthogonal Fourier subspaces Xk , and that since θ is radially symmetric, the subspaces Xk are invariant both under the flow of (5) with ρ˜ = 0 and under the flow of the asymptotic system (8). Lemma 10. Let ej , j = 1, . . . , 4, be the standard basis of C4 and consider the four-dimensional invariant central space corresponding to k = 0 of the unperturbed equation obtained when ρ˜ = 0 and λ = λ0 in (5). Then there exist two unique solutions U0,j (s) with j = 1, 3 such that lim U0,j (s) = ej ,

s→−∞

j = 1, 3.

We may also pick two solutions U0,j with j = 2, 4 which grow algebraically as s → −∞ and satisfy 1 U0,j (s) = ej −1 , s→−∞ s lim

j = 2, 4.

The solutions U0,j , j = 1, . . . , 4, are linearly independent. Proof. It is straightforward to check the assertions of the lemma, using [6, Chapter 3.8].

2

In Section 6 we will specify the solutions U0,2 and U0,4 using the adjoint system. Lemma 11. For every k ∈ Z \ {0}, there exist solutions Uk,j of (5) with ρ˜ = 0 and λ = λ0 such that (together with the solutions specified in Lemma 10 for k = 0) we have span Uk,j (s1 ); k ∈ Z, j = 1, . . . , 4 = X, and for s → −∞, T e|k|(s−s1 ) Uk,1 (s) → −1/k 2 , 1/|k|, 0, 0 eik· , T e|k|(s−s1 ) Uk,2 (s) → 0, 0, −1/|k|, 1 eik· , T e−|k|(s−s1 ) Uk,3 (s) → 1/k 2 , 1/|k|, 0, 0 eik· , T e−|k|(s−s1 ) Uk,4 (s) → 0, 0, 1/|k|, 1 eik· .


365

Proof. As seen in the beginning of this section, the system (5) with ρ˜ = 0 and λ = λ0 leaves the subspaces Xk invariant. The estimates on the matrix B(s; λ0 , 0) in (16) now show that there are solutions of (14) (and hence of (5)) which converge to the solutions of the system at infinity, see e.g. [6, Chapter 3.8]. In Section 3 we have seen that (8) has two solutions in Xk with decay rate e|k|s and two with growth rate e−|k|s for s → −∞. A comparison with the eigenvectors of − (k) in Section 3, we obtain solutions Uk,j , with k ∈ Z \ {0} and j = 1, . . . , 4 with the desired A properties. 2 Next, we perturb the solutions U0,j (s) with j = 1, 3 described in Lemma 10 to solutions of (5) for all sufficiently small potentials ρ. ˜ First, we will show that the four-dimensional central subcs (s , s ; λ, ρ) space corresponding to k = 0 persists in (5) as the intersection of the ranges of Φ− ˜ 1 1 cu and Φ− (s1 , s1 ; λ, ρ). ˜ Note that the difference between the operators A(s; λ, ρ) ˜ and A(s; λ0 , 0) in (5) is A(s; λ, ρ) ˜ − A(s; λ0 , 0) = r (s)2 (λ − λ0 − ρ)B ˜ 0 e2s (λ − λ0 − ρ)B ˜ 0, as r (s)2 = e2s for s s1 (see (19) for the definition of B0 ). The function eτ ρ(τ ) belongs to L2 (J− , H 1/2 ). By Lemma 5, eτ ρ(τ )u1 (τ ) also belongs to this space. Thus eτ ρ(τ )B0 U X ∈ L2 (J− ) and e2τ ρ(τ )B0 U X is the product of an L2 function and the exponentially decaying function eτ . This allows us to use the Gap Lemma as in [17, §4.3 and (4.12)] and [4, Proof of Lemma 4.1] cb (s; λ, ρ) to show that (5) has two linearly independent solutions U0,j ˜ for j = 1, 3 that converge to ej as s → −∞, and two other solutions which grow algebraically. In fact, the results in these cb (s; λ, ρ) works show that any linear combination of the bounded solutions U0,j ˜ with j = 1, 3 can be found as a fixed point of the equation cu U (s) = Φ− (s; λ0 , 0)U0cb

s

cs Φ− (s, τ ; λ0 , 0)e2τ λ − λ0 − ρ(τ ˜ ) B0 U (τ ) dτ

+ −∞

s1 −

u Φ− (s, τ ; λ0 , 0)e2τ λ − λ0 − ρ(τ ˜ ) B0 U (τ ) dτ,

(29)

s

where U0cb belongs to the unperturbed bounded central subspace spanned by U0,j (s1 ) for j = ˜ U0cb ) and write 1, 3. We denote the fixed point by U−cb (s; λ, ρ, P−cb (s1 ; λ, ρ)U ˜ 0cb := U−cb s1 ; λ, ρ, ˜ U0cb s1 = U0cb

+ −∞

cs ˜ U0cb dτ. Φ− (s1 , τ ; λ0 , 0)e2τ λ − λ0 − ρ(τ ˜ ) B0 U−cb τ ; λ, ρ,

(30)

366


Similarly, we can use (25) to describe the solutions of (5) with exponential decay as s → −∞ by ˜ = P−u (s1 ; λ0 , 0) P−u (s1 ; λ, ρ) s1 +

cs u Φ− (s1 , τ ; λ0 , 0)e2τ λ − λ0 − ρ(τ ˜ ) B0 Φ− (τ, s1 ; λ, ρ) ˜ dτ.

(31)

−∞

These results will be used later to characterise eigenfunctions of the perturbed operator. 5. Dichotomies for the far field The method of Section 4 is not available for determining the dichotomies for s large. Going back to (2), we observe that θ and ρ have support in a ball with radius r1 , and thus for r r1 the eigenvalue problem (2) reduces to (2 − λ)u = 0, which can be factorised: ( −

√ √ √ √ λ)( + λ)u = ( + λ)( − λ)u = 0.

Expanding u(r, ϕ) as a Fourier series in the angular variable ϕ, we see that the Fourier coefficients uˆ k satisfy the differential equations

k2 √ ∂2 1 ∂ − + − λ ∂r 2 r ∂r r2

k2 √ ∂2 1 ∂ − 2 + λ uˆ k = 0. + ∂r 2 r ∂r r

(32)

For k fixed, this is a fourth order linear ODE, so it has a four-dimensional space of solutions. The general solution can then be obtained as a linear combination of the solutions of

k2 √ 1 ∂ ∂2 − + − λ uˆ k = 0 ∂r 2 r ∂r r2

(33)

∂2 k2 √ 1 ∂ − + + λ uˆ k = 0, ∂r 2 r ∂r r2

(34)

and the solutions of

so that the general solution of (32) is given by uˆ k (r) = C1 Ik λ1/4 r + C2 Kk λ1/4 r + C3 Jk λ1/4 r + C4 Yk λ1/4 r , where Jk and Yk are Bessel functions of the first and second kind, respectively, which satisfy Eq. (34), and Ik and Kk are modified Bessel functions of the first and second kind, respectively, which satisfy Eq. (33). For r r3 = s3 , we have s = r. Thus, we can define the systems corresponding to Eqs. (33) and (34) with the variable s for s s3 as u 1 = u2 , 2

√ k 1 u 2 = 2 + λ u1 − u2 , s s

(35)


367

and u 1 = u2 , 2

√ k 1 u 2 = 2 − λ u1 − u2 , s s

(36)

respectively. We will consider these systems for s r1 and define φk (s, t) and ψk (s, t) to be the evolution operators corresponding to the systems (35) and (36), respectively. After deriving dichotomies for those systems, we will derive dichotomies for the original system (5). s = To derive dichotomies for (35) and (36), we introduce for s r1 the function spaces X 1 1 2 1 s 2 1 1 1 s 3 1 2 1 = H (S ) × H (S ) and Z = H (S ) × H (S ) with norms H (S ) × L (S ), Y 1 u1 2H 1 (S 1 ) + u1 2L2 (S 1 ) + u2 2L2 (S 1 ) , s2 1 := 2 u1 2H 2 (S 1 ) + u1 2H 1 (S 1 ) + u2 2H 1 (S 1 ) , s 1 := 2 u1 2H 3 (S 1 ) + u1 2H 2 (S 1 ) + u2 2H 2 (S 1 ) . s

u 2X s := u 2Ys u 2Zs

s , Y s , and Z s into their Fourier subspaces We decompose the spaces X s = X

*

s , X k

s = Y

k∈Z

* k∈Z

s Y k

s = and Z

*

s , Z k

(37)

k∈Z

where ks = Y ks = Z ks = X

T ik· ae , beik· ; a, b ∈ C ,

s , Y s and Z s . The norms on X s , Y s and the completion in (37) is in the respective norms of X k k s s s s are given by the restriction of the norms of X ,Y , and Z , respectively, and so and Z k 2 ik· ae , beik· T 2s = 1 + k |a|2 + |b|2 , Xk s2 ik· ae , beik· T 2s = 1 + k 2 aeik· , beik· T 2s , Y X k

k

ik· ae , beik· T 2s = 1 + k 2 2 aeik· , beik· T 2s . Z X k

(38)

k

For each ∈ (0, λ1/4 ), we now prove the existence of a time-dependent exponential dichotomy for (35) with constant K > 0 and rates κ s = −(λ1/4 − ) and κ u = (λ1/4 − ) that are independent of k. For (36), we will show that the evolution operator always acts in the center-unstable manifold and derive that its growth can be bounded by any exponential.

368


Lemma 12. There exists an 0 > 0 such that for any ∈ (0, 0 ) there exists a K > 0 such that for any k ∈ Z and λ ∈ (λ0 /2, 2λ0 ) there exists a time-dependent exponential dichotomy of (35) on J+ so that φks (s, t; λ) and φku (s, t; λ) satisfy s φ (s, t; λ) t s = φ s (s, t; λ) t s Ke−(λ1/4 −)(s−t) , k k L(X ,X ) L(Y ,Y )

s t r1 ,

u φ (s, t; λ)

t s r1 .

k

k

k

t ,X s ) L (X k k

k

k

k

k

1/4 = φku (s, t; λ)L(Yt ,Ys ) Ke−(λ −)(t−s) ,

Proof. Let (u1 , u2 )T satisfy Eq. (35). To get estimates which are uniform in k, we follow [19], and let

√ k 2 1/2 u˜ 1 (s) := λ+ 2 u1 (s). s Note that + + ˜ 1 , u2 )T 2 max(1, 2λ0 )(u1 , u2 )T s , min(1, λ0 /2)(u1 , u2 )T X s (u X C k

k

and that the constants above are independent of k and λ. This shows that, when using the new variables u˜ 1 and u2 , we can use the standard norm in C2 . Next, we rewrite the system (35) in the new variables u˜ 1 , u2 :

√ k 2 1/2 k2 √ k 2 −1 λ+ 2 u2 − 3 λ+ 2 u˜ 1 , s s s

√ 1 k 2 1/2 u2 = − u2 + λ+ 2 u˜ 1 . s s

u˜ 1 =

√ Now, we change the independent variable by making the substitution dτ/ds = ( λ + k 2 /s 2 )1/2 . We write s(τ ) to describe the dependence of s on τ . We then obtain (where now denotes differentiation with respect to τ )

k2 √ k 2 −3/2 λ + u˜ 1 + u2 , s(τ )3 s(τ )2

1 √ k 2 −1/2 u2 = u˜ 1 − λ+ u2 . s(τ ) s(τ )2

u˜ 1 = −

(39)

Noting that s(τ ) → ∞ as τ → ∞ we find that the limiting system at +∞ is u˜ 1 = u2 , u 2 = u˜ 1 ,

(40)


369

which is independent of k. The matrix associated with this system has eigenvalues ±1. Hence Eq. (40) possesses exponential dichotomies with κ u = −κ s = 1. To get estimates for the perturbated system (39), we will use the estimates 2

√ k 1 1 k 2 /s(τ )2 k 2 −3/2 1 , = , λ + 1/4 √ s(τ )3 2 √ 2 2 s(τ ) s(τ ) λ s(τ ) λ + k 2 /s(τ )2 λ + k /s(τ )

1 √ 1 k 2 −1/2 1/4 . λ+ s(τ ) 2 s(τ ) λ s(τ ) This estimate is uniform in λ in a neighbourhood of λ0 . The roughness theorem for exponential dichotomies [7, Chapter 4] now guarantees the existence of an exponential dichotomy also for the system (39), and we denote the corresponding evolution operators by φ˜ ks (σ, τ ) and φ˜ ku (σ, τ ). For each positive ˜ sufficiently small, there exists a K 0 such that s φ˜ (σ, τ ) k

L(C2 )

Ke−(1−˜ )(σ −τ ) ,

σ τ,

k

L(C2 )

Ke−(1−˜ )(τ −σ ) ,

σ τ.

u φ˜ (σ, τ )

Moreover, K does not depend on λ in a neighbourhood of λ0 or on k ∈ Z. It remains to translate this result back to the s variable. We write s = s(σ ) and t = s(τ ). Note that ds/dτ λ1/4 , and so by the chain rule we have for s > t s φ (s, t) k

t ,X s ) L (X k k

C φ˜ ks (σ, τ )L(C2 ) Ke−(1−˜ )(σ −τ ) Ke−(1−˜ )λ

1/4 (s−t)

Ke−(λ

1/4 −)(s−t)

,

where we have put = ˜ λ1/4 . A similar calculation proves that for t > s u φ (s, t) k

t ,X s ) L (X k k

Ke−(λ

1/4 −)(t−s)

.

s also follow from these estimates, since it is only a matter of multiplying The estimates for Y k both sides of the inequalities by a factor (1 + k 2 ). 2 Lemma 13. Let > 0 be given. Then there exists a K > 0 such that for any k ∈ Z and λ ∈ (λ0 /2, 2λ0 ) we have ψk (s, t; λ)

t ,X s ) L (X k k

= ψk (s, t; λ)L(Yt ,Ys ) Ke(t−s) k

k

for t s r1 .

Proof. First we note a scaling invariance in (36). If (u1 (s, t), u2 (s, t)) is a solution of (36) with 1/4 1/4 1/4 1/4 1/4 λ = 1, then (u1 (s, t), u2 (s, t)) = (u1 (λ1 s, λ1 t), λ1 u2 (λ1 s, λ1 t)) is a solution of (36) with λ = λ1 . So it is sufficient to prove the estimate in the lemma in case λ = 1. Using the explicit expressions for the solution in terms of Bessel function, it follows that, for λ = 1, ψk (s, t) is given by

370


−1 Jk (t) Yk (t) Jk (s) Yk (s) ψk (s, t) = Jk (s) Yk (s) Jk (t) Yk (t)

πt Jk (s) Yk (s) Yk (t) −Yk (t) = −Jk (t) Jk (t) 2 Jk (s) Yk (s)

πt ak (s, t) bk (s, t) = 2 ck (s, t) dk (s, t)

where ak (s, t) := Jk (s)Yk (t) − Yk (s)Jk (t) , bk (s, t) := −Jk (s)Yk (t) + Yk (s)Jk (t), ck (s, t) := Jk (s)Yk (t) − Yk (s)Jk (t) , dk (s, t) := −Jk (s)Yk (t) + Yk (s)Jk (t) , and we have used that the Wronskian of Jk (t) and Yk (t) is t , we have (u1 , u2 )T ∈ X k 2 2 ψk (s, t)u2s = π t Xk 4

2 πt

[1, (9.1.16)]. Writing u =

2 2 k2 , 1 + 2 ak (s, t)u1 + bk (s, t)u2 + ck (s, t)u1 + dk (s, t)u2 s

and so we need to show that there exists a K > 0 such that for every u1 , u2 ∈ R, k ∈ Z and t s r1 , t2

2 2 k2 a (s, t)u + b (s, t)u + c (s, t)u + d (s, t)u k 1 k 2 k 1 k 2 s2

k2 2 2 2(t−s) 2 1 + 2 u1 + u2 . K e t 1+

By choosing u1 and u2 appropriately, we note that this inequality holds if and only if the following two inequalities hold for some K > 0 and all k ∈ Z and t > s > r1 :

k2 k2 2 2 2 2 2(t−s) a t 1 + , (s, t) + c (s, t) K e k k s2 t2

k2 2 2 2 1 + 2 bk (s, t) + dk (s, t) t K 2 e2(t−s) . s 1+

To simplify further, we note that the above two inequalities hold if there exists a constant K > 0 such that for t s r1 ,


-

371

k 2 k2 (t−s) 1 + 2 ak (s, t)t Ke 1+ 2, s t 1+

k 2 t Ke(t−s) , b (s, t) k s2 2 ck (s, t)t Ke(t−s) 1 + k , t2 dk (s, t)t Ke(t−s) .

(41)

First we will prove the fourth inequality of (41). Let fk (s, t) := e−(t−s) tdk (s, t). We need to show that fk (s, t) is uniformly bounded for k ∈ Z and t s r1 . Since J−k = (−1)k Jk and Y−k = (−1)k Yk , it is sufficient to consider k ∈ N. We start with showing that fk is bounded for k ∈ N fixed. In the slightly smaller sector t (1 + δ)s (1 + δ)r1 (where δ > 0 is arbitrary), we have |fk (s, t)| → 0 as s 2 + t 2 → ∞, or equivalently, as t → ∞. Indeed, Yk (s), Jk (s), Yk (s) and Jk (s) are bounded by a constant Ck for s r1 [1, (9.2.1)], and so fk (s, t) C 2 e−(t−s) t C 2 te−δt/(1+δ) → 0 k

(42)

k

√ √ √ √ as t → ∞. Furthermore, sYk (s), sJk (s), sYk (s) and sJk (s) are bounded by a constant Dk for s r1 [1, (9.2.1)], and so for r1 s t (1 + δ)s we have + √ fk (s, t) e−(t−s) (1 + δ)st dk (s, t) 1 + δD 2 . k

Altogether this implies that fk (s, t) is bounded in the whole sector t s r1 by a constant, possibly depending on k. To show that in fact fk (s, t) is bounded by a k-independent constant, we consider s r1 as being fixed for the moment. First we note that in (42), we proved that for fixed s and k, fk (s, t) → 0 as t → ∞. Thus for s r1 fixed, the function fk (s, t) attains its maximum in an interior point t > s or at the boundary t = s. We use a method by L. Landau [12] to analyse the behaviour of fk (s, ·) at its critical points. At a critical point, we have ∂fk /∂t (s, t) = 0. By Eq. (11) of [12], at the points where ∂fk /∂t = 0 we have ∂ fk (s, t)2 ∂ −2(t−s) 2 fk (s, t)2 = 2t −2 e t Ak (t) , 2 ∂k e (t − s)t ∂t where Ak (t) = kind, satisfying

)∞ 0

K0 (2t cosh τ )e−2kτ dτ , and K0 is a modified Bessel function of the second ∞ K0 (x) = 0

e−x cosh τ dτ.

372


In particular (since 2tfk (s, t)2 /e2(t−s) t 2 > 0), fk (s, t)2 is decreasing in k at a point where ∂fk /∂t = 0 if and only if e−2(t−s) t 2 Ak (t) is decreasing in t. Note that Ak (t) is monotonically decreasing for t > 0, and since e−2(t−s) t 2 is monotonically decreasing for t > 1/, we conclude that |fk (s, ·)| is monotonically decreasing in k at its critical points for t > 1/. Note that in the case when r1 > 1/, we have proved that if the maximum of fk (s, ·) occurs for t > s r1 , then the maximum is decreasing in k, and hence stays bounded as k increases. At the boundary t = s r1 , we have fk (s, s) = 2/π , which is independent of k. As each function fk (s, t) is bounded, in particular f1 (s, t) is bounded, it follows that fk (s, t) is bounded in the whole sector t s r1 by a k-independent constant for all k ∈ Z. This shows that fk is uniformly bounded in the case when r1 > 1/. When r1 < 1/, we also need to estimate fk (s, t) in the triangle 1/ > t > s r1 . Here we use the estimate e−(t−s) t 1/. It follows that |fk (s, t)| |gk (s, t)|, where gk (s, t) :=

1 Yk (s)Jk (t) − Jk (s)Yk (t) .

Applying Section 3 of [12] we conclude that gk (s, t)2 is decreasing in k at the points where 2 2 ∂gk (s, t)/∂t = 0. Furthermore, gk (s, t) → 0, for t → ∞ and gk (s, s) = πs πr for s r1 . 1 The proof of the fourth inequality of (41) is complete. Next, we prove the second inequality of (41). By [1, (9.1.27)] we have

k 1 1+ Yk (s) = Yk (s) + Yk−1 (s) + Yk+1 (s) , s 2

1 k 1+ Jk (s) = Jk (s) + Jk−1 (s) + Jk+1 (s) . s 2 Note that 1+

k2 s2

1/2 1+

k √ k 2 1/2 2 1+ 2 , s s

and so the second inequality of (41) is equivalent to

t Yk (s) + 1 Yk−1 (s) + 1 Yk+1 (s) Jk (t) − Jk (s) + 1 Jk−1 (s) + 1 Jk+1 (s) Yk (t) 2 2 2 2 Ke(t−s) . To prove the inequality, we use the same method as above, with

1 1 Yk (s) + Yk−1 (s) + Yk+1 (s) Jk (t) 2 2

1 1 − Jk (s) + Jk−1 (s) + Jk+1 (s) Yk (t) 2 2

fk (s, t) := te−(t−s)


373

and

1 1 Yk (s) + Yk−1 (s) + Yk+1 (s) Jk (t) 2 2

1 1 − Jk (s) + Jk−1 (s) + Jk+1 (s) Yk (t) . 2 2

gk (s, t) := te−s

e−1

Then |fk (s, t)| → 0 as s 2 + t 2 → ∞ in the sector t s r1 and as above, fk (s, t)2 is decreasing in k at the points where ∂fk /∂t = 0 if t > 1/. In the triangle 1/ t s r1 , |fk (s, t)| |gk (s, t)| and gk is decreasing in k at the points where ∂gk /∂t = 0. Finally, at the boundary points where s = t r1 , we have fk (s, s) = gk (s, s) = 0. We conclude that the second inequality is valid in the whole sector t s r1 . For the third inequality of (41), we use the last identity of [1, (9.1.27)] which shows that the inequality is equivalent to the two inequalities

k 2 1/2 k (t−s) 1+ 2 J (s)Yk (t) − Yk (s)Jk (t) t Ke , t k t 2 1/2 J (s)Yk+1 (t) − Y (s)Jk+1 (t)t Ke(t−s) 1 + k . k k t2 The first of these inequalities follows from the fourth inequality of (41), and the second can be handled as in the proof of the second and fourth inequalities after noting that at the boundary where t = s r1 we have J (s)Yk+1 (s) − Y (s)Jk+1 (s) = k J (s)Yk (s) − Y (s)Jk (s) = 2k , k k k k s πs 2 thus fk (s, s) = √ π

2 1+s 2 /k 2

π2 . We omit the details.

It remains to prove the first inequality of (41). This can be handled as the second inequality by using Jk (s)Yk (t) − Yk (s)Jk (t) =

k Jk (s)Yk (t) − Yk (s)Jk (t) + Yk (s)Jk+1 (t) − Jk (s)Yk+1 (t) t

and splitting the inequality up into the two inequalities

k 2 1/2 k 1+ 2 Jk (s)Yk (t) − Yk (s)Jk (t)t Ke(t−s) 1 + t s 2 1/2 (t−s) Yk (s)Jk+1 (t) − Jk (s)Yk+1 (t) 1 + k 1+ t Ke s2

k2 t2 k2 t2

1/2 ,

1/2 .

The first inequality follows directly from the second inequality of (41), and the second inequality is proved in the same way as the second inequality of (41), except that at the boundary where

374


t = s we have Yk (s)Jk+1 (s) − Jk (s)Yk+1 (s) = 2/πs. The details are omitted. It is also clear that the estimates above are uniform in λ for λ ∈ (λ0 /2, 2λ0 ). s follow by the same estimates, since it is just a matter of multiplying each The estimates for Y k side of the inequalities by the factor (1 + k 2 ). 2 We are ready to prove that there exist time-dependent exponential dichotomies on J+ for the s × X s and Y s := Y s × Y s . Note that Y s ⊂ full system (5). First we define the spaces X s := X . . s s s X ⊂ X . As before, we can decompose those spaces into X = k∈Z Xk and Y s = k∈Z Yks s × X s and Y s = Y s × Y s . with Xks = X k k k k k For s > s3 = r3 , we have that r = s and hence, for s ∈ [s3 , ∞), the system (5) reduces to ⎛ U = A(s; λ)U,

0

⎜ − 1 ∂2 with A(s; λ) = ⎝ s 2 0 λ

1 − 1s 0 0

0 1 0 − s12 ∂ 2

⎞ 0 0 ⎟ ⎠. 1 − 1s

(43)

We consider the system (43) for s ∈ [r1 , ∞) and record that (43) and (5) coincide on the smaller interval [s3 , ∞). The exponential dichotomy for (5) on the whole interval J+ will follow from the fact that the systems (5) and (43) are linked by the smooth transformation r(s) of the independent variable on the compact interval [s1 , s3 ]. Thus, since we are using r = s in (43), we see that if (s) is a solution of (43) for s ∈ [r1 , ∞), then U (s) = diag(1, r (s), 1, r (s))U (r(s)) is a solution U of (5) for all s ∈ [s1 , ∞). Recall that there are constants 0 < c < C such that c r (s) C for will immediately give similar dichotomy results for U . all s ∈ R, thus dichotomy results for U It is easy to check (similarly to Lemma 3) that A(s; λ) : X s → X s is closed and densely defined with domain Y s and that A(s; λ) : Y s → Y s is closed and densely defined with domain s × Z s . Z The Fourier coefficients of U = (u1 , u2 , u3 , u4 )T satisfy the system u 1 = u2 , u 2 =

k2 1 u1 − u2 + u3 , 2 s s

u 3 = u4 , u 4 = λu1 +

k2 1 u3 − u4 , s s2

(44)

of ODEs, where we omit the subscript k. We denote by Φk (s, t) the evolution operator corresponding to the system (44) and consider this evolution operator in either X s or Y s . Now we can use the earlier dichotomy results to show the existence of a uniform (s-dependent) exponential dichotomy for the system (44) and hence for (43).


375

Lemma 14. There exists an 0 > 0 such that for any ∈ (0, 0 ) there exists a K > 0 such that for any λ ∈ (λ0 /2, 2λ0 ) there exists an s-dependent exponential dichotomy of (43) on J+ such that s (s, t; λ) and Φ cu (s, t; λ) solve (43) and the evolution operators Φ+ + s Φ (s, t; λ)

s 1/4 = Φ+ (s, t; λ)L(Y t ,Y s ) Ke−(λ −)(s−t) , s t r1 , cu cu (t−s) Φ (s, t; λ) , t s r1 . + L(X t ,X s ) = Φ+ (s, t; λ) L(Y t ,Y s ) Ke

+

L(X t ,X s )

(45)

The dichotomy is smooth in λ for λ near λ0 . Proof. Since X s = U ∈ Xt

.

Xks and each Xkt is mapped into Xks under the flow of (43), we write for s Φ+ (s, t)U :=


s t r1 ,

k∈Z

cu Φ+ (s, t)U :=

k eik· , Φkcu (s, t)U

t s r1 .

(46)

k∈Z

Moreover, for each k ∈ Z the evolution operator Φk (s, t) associated with (44) can be expressed in terms of φk (s, t) and ψk (s, t). Indeed, it can be seen that Φk (s, t) =

1 2

φk (s, t) + ψk (s, t) √ λ(φk (s, t) − ψk (s, t))

√1 (φk (s, t) − ψk (s, t)) λ . φk (s, t) + ψk (s, t)

Similarly, Φks (s, t) =

1 2

φ s (s, t) √ ks λφk (s, t)

√1 φ s (s, t) λ k φks (s, t)

and Φkcu (s, t) =

1 2

φku (s, t) + ψk (s, t)

√ λ(φku (s, t) − ψk (s, t))

√1 (φ u (s, t) − ψk (s, t)) λ k . φku (s, t) + ψk (s, t)

k = (u, v)T ∈ X t , where u(u1 , u2 )T and v = (u3 , u4 )T , Introducing the temporary notation U k we have 2 s s 2 1 1 s Φ (s, t)U k X s = (1 + λ)φk (s, t)u + √ φk (s, t)v . k k 4 s λ X k

(47)

k ∈ X t such that U k X t = 1, we may without loss Since we will take the supremum over all U k k √ of generality assume that v = λu, since all other choices will result in a smaller value of the t , the condition that U k 2 t = 1 implies that right-hand side of (47). For any such u and v ∈ X k

u 2X t (1 + λ) = 1. k

We therefore have

Xk

376


s Φ (s, t)U k 2 s = k X

sup

k 2 t =1 U

1 + λ φ s (s, t)u2s k Xk =1/(1+λ) 2

sup

k

u 2t

Xk

Xk

2 1 sup φks (s, t)uX s , k 2 u 2 =1

=

t X k

which shows that s Φ (s, t) k

L(Xkt ,Xks )

1 = √ φks (s, t)L(X t ,X s ) . k k 2

Likewise, cu Φ (s, t) k

L(Xkt ,Xks )

Φkc (s, t)L(X t ,X s ) + Φku (s, t)L(X t ,X s ) , k

k

k

k

where Φkc (s, t) =

− ψk√(s,t)

ψk (s, t) √ − λψk (s, t)

and Φku (s, t) = Φkcu (s, t) − Φkc (s, t).

λ

ψk (s, t)

As above, c Φ (s, t)

2 1 = √ ψk (s, t)L(X t ,X s ) , k k 2 2 u 1 u t s . Φ (s, t) k L(Xkt ,Xks ) = √ φk (s, t) L(X k ,Xk ) 2 k

L(Xkt ,Xks )

From Lemmas 12 and 13 it follows that s Φ (s, t) k

cu Φ (s, t) k

L(Xkt ,Xks ) L(Xkt ,Xks )

Ke−(λ

1/4 −)(s−t)

Ke(t−s) ,

s t r1 ,

,

t s r1 .

(48)

By (46) and (48) we see that for s t r1 s Φ (s, t)2 +

L(X t ,X s )

= =

2 sup Φ s (s, t)U Xs =

U X t =1

sup

U X t =1 k∈Z

Φ s (s, t)U k 2 s k X

U X t =1 k∈Z

sup

2 s ik· k e sup Φk (s, t)U

U X t =1 k∈Z

k

K 2 e−2(λ

1/4 −)(s−t)

sup

Φ s (s, t)2

U X t =1 k∈Z

Xs

2 L(X t ,X s ) Uk Xkt

k

k 2 t = K 2 e−2(λ U X

1/4 −)(s−t)

.

k

cu satisfies the second equation of (45). The estimates for Y s A similar calculation shows that Φ+ k follow by the same estimates since it is just a matter of multiplying each side of the inequalities by a factor (1 + k 2 )2 .


377

Finally, the smoothness in λ follows from the implicit function theorem [5, Corollary 3.1.11]. s/cu As A(s; λ) depends linearly on λ, we get that Φ+ (s, t; λ) satisfies s/cu d s/cu Φ+ (s, t; λ) = A(s; λ0 ) + (λ − λ0 )B0 Φ+ (s, t; λ), ds where B0 is defined by (19). s (s, t; λ), Φ cu (s, t; λ)) satisfies For λ ∈ (λ0 /2, 2λ0 ) and sufficiently close to λ0 , the pair (Φ+ + the fixed point equation s s s cu Φ+ (s, t; λ) = Φ+ (s, t; λ0 )+Φ+ (s, t; λ0 )Φ+ (s1 , t; λ) t

+ (λ − λ0 ) −

s cu Φ+ (s, τ ; λ0 )B0 Φ+ (τ, t; λ) dτ r1

s +

s s Φ+ (s, τ ; λ0 )B0 Φ+ (τ, t; λ) dτ t

∞ −

cu s Φ+ (s, τ ; λ0 )B0 Φ+ (τ, t; λ) dτ

,

s t r1 ,

s cu cu s cu (s, t; λ) = Φ+ (s, t; λ0 )−Φ+ (s, t; λ0 )Φ+ (s1 , t; λ) Φ+

s

+ (λ − λ0 )

s cu Φ+ (s, τ ; λ0 )B0 Φ+ (τ, t; λ) dτ r1

t −

cu cu Φ+ (s, τ ; λ0 )B0 Φ+ (τ, t; λ) dτ s

∞ +

cu s Φ+ (s, τ ; λ0 )B0 Φ+ (τ, t; λ) dτ

,

t s r1 .

t

s × X cu , where X s/cu are defined as This fixed point equation is considered as a mapping on X s = Φ s (s, t) ∈ L X t , X s ; X ! s Φ (s, t) s = sup e((λ0 /2)1/4 −)(s−t) Φ s (s, t) < ∞ , t s X L(X ,X ) str1

cu = Φ cu (s, t) ∈ L X t , X s ; X ! cu Φ (s, t) cu = sup e−2(t−s) Φ cu (s, t) X L(X t ,X s ) < ∞ . tsr1

378


s (s, t; λ) and Φ cu (s, t; λ), it is easy to check that the With the estimates derived before on Φ+ + right-hand side is well defined in this space. With the implicit function theorem, it follows immediately that the mapping is smooth for λ near λ0 . 2

6. The unperturbed adjoint system The dual space of X is X = H −2 × H −1 × H −1 × L2 (using the L2 pairing). For the space X , we make the decomposition X =

*

Xk ,

k∈Z

where Xk are four-dimensional pairwise orthogonal subspaces span{((a, b, c, d)eik· ); a, b, c, d ∈ C} ⊂ X . For W ∈ Xk and U ∈ Xk , we have the pairing W, U := w1 u1 + w 2 u2 + w3 u3 + w 4 u4 ,

(49)

where the bar denotes the complex conjugate. This means that we may use the standard inner product on C4 when computing the adjoint equation. Similarly, the dual space of X := H 1 × L2 × H 1 × L2 is X = H −1 × L2 × H −1 × L2 (using the L2 pairing) and we can make the same decomposition as above, i.e., X =

*

Xk ,

k∈Z

where Xk are the same four-dimensional subspaces as above but are now regarded as subspaces of X . For W ∈ Xk and U ∈ Xk , the pairing is again as in (49). At the end of Section 4, we have investigated the solutions of the unperturbed linear system U = A(s; λ0 , 0)U on J− . The adjoint unperturbed system is W = −A(s; λ0 , 0)∗ W = − A∗− + B(s; λ0 , 0)∗ W.

(50)

Just as in the case of the unperturbed linear system itself, expanding W in a Fourier series shows that the Fourier spaces Xk are invariant under the flow of the adjoint system (50), and the Fourier coefficients satisfy the adjoint equation of (14), i.e., k (s). − (k)∗ + B(s; λ, 0)∗ W k (s) = − A W

(51)

It is well known and straightforward to check that the pairing of a solution of a linear system with a solution of its adjoint is constant. For our systems, this means that any two solutions k (s) of (14) and W k (s) of (51) satisfy U 0 d / k (s) = 0, Wk (s), U ds

and thus

/

0 / 0 k (s), U k (s) = W k (s1 ), U k (s1 ) W

for any s ∈ R. (52)


379

From [18, p. 56] it follows that if a finite-dimensional linear system has an exponential dichotomy on an interval J with constants K, κ s and κ u , then the adjoint system has an exponential dichotomy on J with the dichotomy constants K, −κ u and −κ s . Furthermore, if we denote evolution operators corresponding to the exponential dichotomy of the adjoint system (51) u (s, t), respectively, then Φ s (s, t) = Φ u (t, s)∗ for t s with s, t ∈ J and s (s, t) and Φ by Φ k± k k k u (s, t) = Φ s (t, s)∗ for s t with s, t ∈ J . Φ k k On J− , the dichotomy constant K in (17) is independent of k, and so we immediately get the following estimates about the solutions of the adjoint system in the Fourier spaces Xk with norm k 2 := W k eik· 2 = W X X k

|w3 |2 |w1 |2 |w2 |2 + + |w4 |2 . + (k 2 + 1)2 k 2 + 1 k 2 + 1

k ∈ X Lemma 15. There exists a K > 0 such that for every k ∈ Z \ {0} and W k s Φ k (s, t)W k

Xk

k X , Ke−|k|(s−t) W k

Xk

k X , Ke|k|(s−t) W k

u Φ k k (s, t)W

t s s1 , s t s1 .

k (s) with W k (s1 ) ∈ Ran(Φ u (s1 , s1 )), we have Furthermore, for any solution W k− wk (s) Ke|k|(s−s1 ) W k (s1 )

X

k (s). for all s s1 , where wk (s) denotes the fourth component of W Similar arguments give the dichotomy of the adjoint system on J+ . From (48), it follows that the solutions of the linearised system in the Fourier spaces Φks (s, t) and Φkcu (s, t) have an exponential dichotomy with a uniform constant K. The dual space of X s is denoted by (X s ) , and for s fixed, this space is equivalent to H −1 × L2 × H −1 × L2 . The dual Fourier space is denoted by (Xks ) , and its norm is k 2 s := W k eik· 2 s = W (X ) (X ) k

s2 s2 2 2 |w | + |w | + |w3 |2 + |w4 |2 . 1 2 k2 + s 2 k2 + s 2

On J+ , we have the following estimates: k ∈ X Lemma 16. For every > 0, there exists a K > 0 such that for every k ∈ Z \ {0} and W k cs Φ k (s, t)W

(X s )

k (X t ) , Ke(s−t) W

u Φ k (s, t)W

(X s )

Ke(λ0

k

k

1/4

−)(s−t)

s1 t s,

k (X t ) , W

s1 s t.

k (s1 ) ∈ Ran(Φ k (s) with W cs (s1 , s1 )), we have Moreover, for any solution W k wk (s) Ke(s−s1 ) W k (s1 )

(X s1 )

k (s). for all s s1 , where wk (s) denotes the fourth component of W

380


Next we look at the adjoint system associated with k = 0. In Lemma 10, we have seen that the solutions space of the linear system at k = 0 is spanned by U0,j , j = 1, . . . , 4. Now let Z0,j (s) be solutions of the adjoint system (51) with k = 0 such that {Z0,l (s1 )}l=1,...,4 is a dual basis of {U0,j (s1 )}j =1,...,4 (i.e., Z0,l (s1 ), U0,j (s1 ) = δlj ). With (52), this implies δlj = Z0,l (s), U0,j (s) for any s s1 . The unbounded solutions U0,2 and U0,4 are not unique, but they can be chosen such that Z0,2 and Z0,4 are bounded on J− , whereas Z0,1 and Z0,3 grow algebraically as s → −∞. A convenient choice for Z0,2 and Z0,4 is Z0,j (z) :=

r(s) ⊥ U (s), r (s) 0,j −1

j = 2, 4,

(53)

⊥ (s) are where U ⊥ = (−u4 , u3 , −u2 , u1 ) if U = (u1 , u2 , u3 , u4 )T . It is easy to check that Z0,j solutions of the adjoint system (50) using that U0,j −1 (s) are solutions of the original system (5). As will be shown below, a similar exponential dichotomy on J− as in Lemma 15 also holds with norms in X .

k ∈ (Xk ) , we have Lemma 17. There exists a K > 0 such that, for every k ∈ Z \ {0} and W s Φ k Ke−|k|(s−t) W k X , t s s1 , k (s, t)W X u Φ k Ke|k|(s−t) W k X , s t s1 . k (s, t)W X k (s) with W k (s1 ) ∈ Ran(Φ u (s1 , s1 )) and with wk (s) denoting the fourth comFor any solution W k |k|(s−s 1 ) W k (s1 ) X for all s s1 . ponent of Wk (s), we have |wk (s)| Ke Proof. First we will show that the linear system (14) has an exponential dichotomy in Xk . The proof is very similar to the one in Section 4 with a slightly modified matrix Mk . Define the matrix k whose columns consist of eigenvectors of A− (k) that are scaled different to those in Mk : M ⎛

−1/|k| ⎜ k = ⎝ 1 M 0 0

0 1/|k| 0 1 −1/|k| 0 1 0

⎞ 0 0 ⎟ ⎠. 1/|k| 1

k Dk M k consists of eigenvectors of A− (k), it follows immediately that A− (k) = M −1 . As M k 4 k is a homeomorphism between C and Xk with It is also straightforward to verify that M √ k L(C4 ,X ) → 2 as |k| → ∞. M k k and exploiting the Using the same ideas as in the proof of (17), with Mk replaced by M observation that −1 e2τ/|k| M k B τ/|k|; λ0 , 0 M sup 1, λ − θ˜ (s) , k 2|k| ss1 we find that s Φ (s, t)U k Ke−|k|(s−t) U k X , t s s1 , k X u Φ (s, t)U k Ke|k|(s−t) U k X , s t s1 k X

(54)


381

k ∈ Xk for some constant K that is independent of k. As Xk is finite-dimensional, this for any U immediately implies the estimates of the lemma. 2 Finally we will show that, for large values of k, the solutions of (51) are close to the solutions (s) = −A− (k)∗ W k (s). Recall that we denote the spectral projection of the asymptotic system W k onto the eigenspace of A− (k) associated with the positive eigenvalue |k| by Pku and the complementary projection onto the eigenspace of A− (k) associated with the negative eigenvalue −|k| by Pks . Lemma 18. For every > 0, there exists an N ∈ N and a δ > 0 such that, for every |k| > N , we have u Φ (s, s1 ) − e|k|(s−s1 ) P s ∗

e|k|(s−s1 ) ,

for all s1 − δ s s1 ,

u Φ (s, s1 ) − e|k|(s−s1 ) P s ∗

e|k|(s−s1 ) ,

for all s s1 .

k

k

L(Xk )

k

k

L(Xk )

(55)

k (s1 ) ∈ Ran(Φ u (s1 , s1 )), we have Thus, for |k| > N and W k W k (s1 ) − P s ∗ W k (s1 ) k

Xk

k

Xk

W k (s1 ) − P s ∗ W k (s1 )

k (s1 ) , W Xk k (s1 ) . W Xk

u (s, s1 ) satisfies Proof. The evolution operator Φ k ku (s, s1 ) = e|k|(s−s1 ) Φ

s ∗ Pk −

s

∗ ku (t, s1 ) dt e−|k|(s−t) Pku B(t; λ0 , 0)∗ Φ

−∞

s1 +

∗ ku (t, s1 ) dt. e|k|(s−t) Pks B(t; λ0 , 0)∗ Φ

s

From its definition (12), we immediately see that B(t; λ0 , 0)∗ L(Xk ) Ce2t and B(t; λ0 , 0)∗ L(Xk ) √ C2 e2t for some constant C, independent of k. Hence, with the dik +1

chotomy estimates from Lemma 15, we get that for s s1 and k ∈ Z \ {0} u Φ (s, s1 ) − e|k|(s−s1 ) P s ∗ k

k

L(Xk )

s CK

e−|k|(s−t) e2t e|k|(t−s1 ) dt

−∞

s1 + CK s

e|k|(s−t) e2t e|k|(t−s1 ) dt

CK |k|(s−s1 ) e2s1 e + e2s1 − e2s . 2 1 + |k|

382


It is now easy to see that we can choose N > 0 large enough and δ > 0 small enough such that the first inequality of (55) is satisfied. With the dichotomy estimates from Lemma 17, we get u C Φ k (s, s1 ) − e|k|(s−s1 ) Pks ∗ K L(Xk ) √ 2 k +1

s

e−|k|(s−t) e2t e|k|(t−s1 ) dt

−∞

C

+√ K k2 + 1

s1

e|k|(s−t) e2t e|k|(t−s1 ) dt

s

e2s1 CK + e2s1 e|k|(s−s1 ) √ (1 + |k|) 2 k2 + 1 CKe2s1 |k|(s−s1 ) e . √ k2 + 1 It follows that also the second inequality of (55) is valid when N is sufficiently large.

2

7. Matching the core and far field solutions In the next lemma we show that u is an eigenfunction that belongs to an embedded eigen˜ ∩ value λ of L + ρ if and only if U is a solution of (5) such that U (s1 ) ∈ Ran P+s (s1 ; λ, ρ) ˜ ⊕ P−cb (s1 ; λ, ρ)). ˜ (Ran P−u (s1 ; λ, ρ) Lemma 19. Let u be an L2 solution of (2). Then the corresponding solution U (s) of the system (5) is bounded in X as s → −∞ and decays exponentially with rate λ1/4 − as s → +∞ in the sense that for any ∈ (0, λ1/4 ) there exists a constant K > 0 such that U (s)

Xs

Ke−(λ

1/4 −)s

(56)

for every s s1 . Conversely, if U is a weak solution of (5) such that U (s) X is bounded as s ∈ J− and such that U (s) X s decays exponentially as s → +∞ (with any decay rate), then it corresponds to an H 4 solution u of (2). Proof. If u is an eigenfunction of (5), it belongs to H 4 (R2 ). By Lemma 2, the associated solution U (s) of (5) is bounded in X as s ∈ J− . In J+ , for s s3 , the system (5) reduces to (43) and the decaying solutions of this system √ are series formed by Bessel functions Kk , Jk and Yk . The decay of Jk and Yk is asymptotic to 1/ s as s → +∞, and so these solutions do not give rise to L2 solutions of (2). It now follows from Lemma 14 that U (s) decays exponentially as s → +∞, in the sense that (56) holds. Assume that U is a bounded weak solution of (5) which decays exponentially as s ∈ J+ . We need to show that the first component of U which we denote by u belongs to H 4 (R2 ) when regarded as a function of the two variables (r, ϕ) in radial coordinates. As U is a weak solution 1 (J ; X). Also, U is bounded on J , and hence of (5), by Definition 1, U ∈ L2loc (J ; Y ) ∩ Hloc X − ∞ 4 (R2 ), so we only need to worry about U ∈ Lloc (R− ; X). From Lemma 2 we know that u ∈ Hloc


383

the decay properties of u as r → ∞ (i.e. as s → ∞). From Lemma 14 and the definition of X s it follows that for any 0 < < 0 and s s1 u(s)

L2 (S 1 )

Ke−(λ

1/4 −)(s−s

1)

U (s1 )

X s1

and so it is clear that u ∈ L2 (R2 ). From (2) it then follows that u ∈ H 4 (R2 ).

2

Recall that u∗ is the radially symmetric eigenfunction associated with the embedded eigenvalue λ0 when ρ˜ = 0. Let U∗ be the associated solution of (5) with ρ˜ = 0 and λ = λ0 , defined for s ∈ R, i.e. U∗ = (u∗ , u ∗ , u∗ , (u∗ ) )T . Define X := H 1 × L2 × H 1 × L2 and recall that X = H 2 × H 1 × H 1 × L2 . Let s E+ := U ∈ X ; P+s (s1 ; λ0 , 0)U = U , u E− := U ∈ X; P−u (s1 ; λ0 , 0)U = U , cb := span U0,1 (s1 ), U0,3 (s1 ) ⊂ X, E− s and E u consist of the initial values where U0,j are defined in Lemma 10. Roughly speaking, E+ − at s1 of solutions of (5) with ρ˜ = 0 and λ = λ0 which decay exponentially as s → ∞ and as u ⊕ E cb consists of the bounded solutions on J . Note that the s → −∞, respectively, and E− − − s u and E cb . We have E s ∩ (E u ⊕ norm of X is used for E+ , while the norm of X is used for E− − + − cb ) = span{U (s )} since λ is an eigenvalue of L with multiplicity 1. E− ∗ 1 0 Next, we introduce a new Hilbert space X such that X ⊂ X ⊂ X , and special solutions Vk,j , k ∈ Z, j = 1, . . . , 4, such that {Vk,j (s1 )} is a basis for X. We have seen that the unperturbed system (5) decouples when ρ˜ = 0 so that the subspaces Xk and Xk are invariant under the flow of (5) with ρ˜ = 0. For k = 0, we pick V0,1 (s1 ) := cb ∩ E s . Note that there are no other solutions in X which decay exponentially as U∗ (s1 ) ∈ E− 0 + cb such that V (s ) ∈ s and E cb = span{V (s ), V (s )} / E s → ∞. We pick V0,2 (s1 ) ∈ E− 0,2 1 0,1 1 0,2 1 + − (thus span{V0,1 (s1 ), V0,2 (s1 )} = span{U0,1 (s1 ), U0,3 (s1 )}). Next, we will choose V0,3 (s1 ) and V0,4 (s1 ) ∈ X0 such that they belong to the span of U0,2 (s1 ) and U0,4 (s2 ). In order to do this, we introduce a dual basis W0,j (s1 ) of V0,j (s1 ) and choose W0,3 (s1 ) := U∗⊥ (s1 ), where U∗⊥ (s1 ) = (−u4 (s1 ), u3 (s1 ), −u2 (s1 ), u1 (s1 )) and uj (s1 ) are the components of U∗ (s1 ), while W0,4 (s1 ) is any other vector so that span{W0,3 (s1 ), W0,4 (s1 )} = span{Z0,2 (s1 ), Z0,4 (s1 )}. The remaining vectors W0,1 (s1 ), W0,2 (s1 ), V0,3 (s1 ) and V0,4 (s1 ) are determined by the conditions that {W0,j (s1 ); j = 1, . . . , 4} and {V0,j (s1 ); j = 1, . . . , 4} are dual bases:

/

0 W0,j (s1 ), V0,l (s1 ) = δj l ,

l = 1, . . . , 4.

ca := span{V (s ), V (s )}. We normalise the vectors such that We use the notation E− 0,3 1 0,4 1 V0,j (s1 ) X = 1 for j = 1, . . . , 4 and note that, for V ∈ X0 , we have V X = V X . Define W0,j (s) so that it satisfies the adjoint system (50) (and hence (51) with k = 0) and passes through W0,j (s1 ) for s = s1 . From (52) and the relation above, it follows immediately that W0,j (s), V0,l (s) = δj l for all s s1 . Furthermore, from (53), we conclude that W0,3 (s) and W0,4 (s) are bounded solutions of the adjoint system on J− .

384


s ∩ X and Next we consider k = 0. The spaces Xk and Xk are four-dimensional, and E+ k s u u u s ∩X = E+ ∩ Xk are one-dimensional, E− ∩ Xk and E− ∩ Xk are two-dimensional, and E− ∩ E+ k u ∩ E s ∩ X , since the multiplicity of the eigenfunction U is 1. Using this, we de{0} = E− k ∗ + s (and hence fine base vectors in Xk and Xk as follows: For k = 0 we pick Vk,1 (s1 ) ∈ E+ u u Vk,1 (s1 ) ∈ / E− ). We also pick Vk,2 (s1 ) and Vk,3 (s1 ) so that they belong to E− (and hence do s ). Thus {V (s ), V (s ), V (s )} span a three-dimensional subspace in the not belong to E+ k,1 1 k,2 1 k,3 1

four-dimensional spaces Xk and Xk . We normalise the solutions Vk,j such that for k ∈ Z \ {0}: Vk,1 (s1 ) X = 1 and Vk,j (s1 ) X = 1 for j = 2, 3. Hence there exists a unique (up to multiplication by a unimodular constant) vector Wk,4 ∈ Xk such that Wk,4 , Vk,j = 0 for j = 1, 2, 3 and Wk,4 (s1 )X = 1. k

s (s , s )) + Ran(Φ u (s , s )) and hence W (s ) ∈ Then Wk,4 (s1 ), V = 0 for V ∈ Ran(Φk,+ 1 1 k,4 1 k,− 1 1 cu s ∗ ∗ cs (s1 , s1 )) ∩ Ran(Φ u (s1 , s1 )). We take the Ran(Φk,+ (s1 , s1 ) ) ∩ Ran(Φk,− (s1 , s1 ) ) = Ran(Φ k,+ k,− one remaining solution in Xk and Xk such that Wk,4 (s1 ), Vk,4 (s1 ) = 1 and Vk,4 (s1 ) X = 1. u ∪ E s as W (s ), V = 0 for V ∈ E u + E s . Then Vk,4 (s1 ) ∈ / E− k,4 1 + − + Let X be defined by

X := U =

ak,j Vk,j (s1 ) ∈ X ;

k∈Z j =1,...,4

U 2X

2 2 1 := ak,j Vk,j (s1 ) + ak,j Vk,j (s1 ) < ∞ . k∈Z j =1,4

X

k∈Z j =2,3

X

Note that X is the direct sum of two Hilbert spaces, which are closed subspaces of X and X, respectively. It follows that X is a Hilbert space. s and E u are both closed subspaces of X: Indeed, Note that E+ − s E+ = clX span Vk,1 (s1 ) k∈Z = clX span Vk,1 (s1 ) k∈Z , u = clX span Vk,2 (s1 ), Vk,3 (s1 ) k∈Z\{0} = clX span Vk,2 (s1 ), Vk,3 (s1 ) k∈Z\{0} . E− cb is a closed subspace of X since it is finite-dimensional. It is clear that E− s × (E u ⊕ E cb ) × R × R → X by Define ι : E+ − −

˜ 0u + P−cb (s1 ; λ, ρ)U ˜ 0cb − P+s (s1 ; λ, ρ)U ˜ 0s ι U0s , U0u + U0cb ; λ, ρ˜ := P−u (s1 ; λ, ρ)U = P−u (s1 ; λ, ρ)U ˜ 0u + P−cb (s1 ; λ, ρ)U ˜ 0cb − P+s (s1 ; λ, 0)U0s ,

(57)

where we recall the definition (30) and note that the last equality holds since ρ(s) ˜ = 0 for s s1 . We will see that the range of ι is a subspace of X, and that ι is smooth into this space. For this we need a more explicit formula for ι. By (30), (31) and (49) evaluated at s = t = s1 , we have (see (19) for the definition of B0 )


385

P−cb (s1 ; λ, ρ)U ˜ 0cb s1 = U0cb

+

cs ˜ U0cb dτ, Φ− (s1 , τ ; λ0 , 0)r (τ )2 λ − λ0 − ρ(τ ˜ ) B0 U−cb τ ; λ, ρ,

(58)

−∞

P−u (s1 ; λ, ρ) ˜ s1 = P−u (s1 ; λ0 , 0) +

cs u Φ− (s1 , τ ; λ0 , 0)r (τ )2 λ − λ0 − ρ(τ ˜ ) B0 Φ− (τ, s1 ; λ, ρ) ˜ dτ,

(59)

−∞

P+s (s1 ; λ, 0) ∞ = P+s (s1 ; λ0 , 0) −

cu s Φ+ (s1 , τ ; λ0 , 0)r (τ )2 (λ − λ0 )B0 Φ+ (τ, s1 ; λ, 0) dτ,

(60)

s1

where we recall that r(τ ) = eτ for τ < s2 , so that r (τ )2 = e2τ in this interval. Hence we may write ι U0s , U0u + U0cb ; λ, ρ˜ ∞ = U0u

+ U0cb

− U0s

+

cu s Φ+ (s1 , τ ; λ0 , 0)r (τ )2 (λ − λ0 )B0 Φ+ (τ, s1 ; λ, 0)U0s dτ

s1

s1 +

cs ˜ U0cb Φ− (s1 , τ ; λ0 , 0)e2τ λ − λ0 − ρ(τ ˜ ) B0 U−cb τ ; λ, ρ,

−∞

u (τ, s1 ; λ, ρ)U ˜ 0u dτ. + Φ−

(61)

s × (E u ⊕ E cb ) × R × R → X is smooth. Lemma 20. The map ι : E+ − −

˜ → P−u (s1 ; λ, ρ)U ˜ 0u and (U0cb , λ, ρ) ˜ → Proof. We have seen in Theorem 2 that (U0u , λ, ρ) cu cb u cb × R × R to X ⊂ X and from E− P− (s1 ; λ, ρ)U ˜ 0 are smooth as functions from E− × R × R to X ⊂ X, respectively. s × R to X. We do this by Hence it suffices to prove that P+s (s1 ; λ, 0) is smooth from E+ s studying the terms of (60) separately. It is clear that U0 ∈ X. s (τ, s ; λ, 0) : Next, we study the integral term, and note that by Lemma 14, for τ s1 , Φ+ 1 1/4 cu s τ (λ −)(τ −s ) τ s 1 1 1 and Φ+ (s1 , τ ; λ0 , 0) : Y → Y with norm X → X with norm bounded by Ke bounded by Ke(τ −s1 ) . Recall that X s = H 1 × L2 × H 1 × L2 and Y s = H 2 × H 1 × H 2 × H 1 with s-dependent norms. Thus B0 : X τ → Y τ is bounded with norm τ . Using the exponential estimates of Lemma 14 and that Y s1 ⊂ X ⊂ X ⊂ X s1 , there exists a constant C > 0 such that for any > 0 sufficiently small

386


∞ cu 2 s s Φ+ (s1 , τ ; λ0 , 0)r (τ ) B0 Φ+ (τ, s1 ; λ, 0)U0 dτ

X

s1

∞ cu s C Φ+ (s1 , τ ; λ0 , 0)B0 Φ+ (τ, s1 ; λ, 0)U0s dτ s1

2 (λ1/4 −2)s1

∞

CK e

τ e−(λ

1/4 −2)τ

Y s1

dτ U0s X s1

s1

2 s CK U , (λ1/4 − 2)2 0 X

To show that the integral term is smooth in λ into X, it suffices to prove for some constant C. that for n 1 ∞

cu Φ+ (s1 , τ ; λ0 , 0)r (τ )2 B0

dn s Φ (τ, s1 ; λ, 0)U0s dτ dλn +

s1 n

d s belongs to X. This follows since, by Lemma 14, dλ n Φ+ (τ, s1 ; λ, 0) satisfies a similar exponential s decay estimate as Φ+ (τ, s1 ; λ, 0) does. It follows as above that the integral in question converges in Y s1 ⊂ X. Smoothness in U0s is immediate, since ι is bounded and linear in U0s into X. 2

Lemma 21. The operator 2 + θ + ρ has an embedded eigenvalue λ > 0 if and only if there s , U u ∈ E u and U cb ∈ E cb such that exist U0s ∈ E+ − − 0 0 ι U0s , U0u + U0cb ; λ, ρ˜ = 0.

(62)

Proof. If λ is an eigenvalue of L + ρ, then by Lemma 19, the corresponding solution of the system (5) is bounded as s → −∞ and decays exponentially as s → +∞. Hence there exists a solution of (5) with initial condition P+s (s1 ; λ, ρ)U ˜ 0s = P−u (s1 ; λ, ρ)U ˜ 0u + P−cb (s1 ; λ, ρ)U ˜ 0cb at s = s1 , i.e. (62) holds. s ×(E u ⊕E cb )× Conversely, suppose that (62) is satisfied for some (U0s , U0u +U0cb , λ, ρ) ˜ ∈ E+ − − Then there exists a solution of (5) with initial condition U s = U u + U cb . By Lemma 19, R × R. 0 0 0 this implies that λ is an eigenvalue of L + ρ. 2 cb ⊕ E u and E s have complements in X denoted by E ca ⊕ E s and Lemma 22. The subspaces E− − + − − cu ca s cu is infinite-dimensional and has a basis with elements V (s ), E+ . Moreover, (E− ⊕ E− ) ∩ E+ 0,3 1 V0,4 (s1 ) ∈ X0 , Vk,4 (s1 ) ∈ Xk , k ∈ Z \ {0}.


387

ca = span{V (s ), V (s )}. Let Proof. Recall that E− 0,3 1 0,4 1

s := span Vk,1 (s1 ), Vk,4 (s1 ); k ∈ Z \ {0} , E− cu := span Vk,2 (s1 ), Vk,3 (s1 ), Vk,4 (s1 ); k ∈ Z , E+ where the closures are taken in X. It is easy to see that these spaces have the desired properties. 2 s + (E u ⊕ E cb ) such that Let Q be the projection in X onto Ran ι(·, ·; λ0 , 0) = E+ − −

ca cu s ker Q = E+ . ∩ E− ⊕ E− Note that Ran Q and ker Q are closed subspaces of X, and Q is therefore continuous. Eq. (62) is equivalent to the pair of equations Qι U0s , U0u + U0cb ; λ, ρ˜ = 0, (I − Q)ι U0s , U0u + U0cb ; λ, ρ˜ = 0.

(63)

the first equation of (63) has Lemma 23. For (λ, ρ) ˜ in a neighbourhood of (λ0 , 0) ∈ R × R, a unique (up to constant multiples) nonzero solution (U0s , U0u + U0cb ) which depends smoothly ˜ U0u (λ, ρ) ˜ and U0cb (λ, ρ). ˜ Furthermore, on λ and ρ˜ in this neighbourhood. We write U0s (λ, ρ), s cb u U0 (λ0 , 0) = U∗ (s1 ) = U0 (λ0 , 0) and U0 (λ0 , 0) = 0. s × (E u ⊕ E cb ) to Ran Q. It is clear that Proof. For (λ, ρ) ˜ fixed, Qι is a linear mapping from E+ − −

ker Qι(·, ·; λ0 , 0) = span U∗ (s1 ), U∗ (s1 ) . By Lemma 20 and since Ran Q is closed, it follows that Qι is a smooth mapping in its arguments. s × (E u ⊕ E cb ) such that D ∩ span{(U (s ), U (s ))} = Let D be an affine hyperplane of E+ ∗ 1 ∗ 1 − − {(U∗ (s1 ), U∗ (s1 ))}. The implicit function theorem then implies that for (λ, ρ) ˜ close to (λ0 , 0) ˜ U0u (λ, ρ) ˜ + the first equation of (63) has a unique solution (U0s , U0u + U0cb ) = (U0s (λ, ρ), cb s u cb ˜ ∈ D in a neighbourhood of (U∗ (s1 ), U∗ (s1 )). Moreover, U0 , U0 and U0 are smooth U0 (λ, ρ)) in their arguments. 2 For (λ, ρ) ˜ in the neighbourhood obtained in Lemma 23, we let F (λ, ρ) ˜ := (I − Q)ι U0s (λ, ρ), ˜ U0u (λ, ρ) ˜ + U0cb (λ, ρ) ˜ s1 = (I − Q)

cs Φ− (s1 , τ ; λ0 , 0)e2τ λ − λ0 − ρ(τ ˜ ) B0 U0cb (τ ; λ, ρ) ˜

−∞

u + Φ− (τ, s1 ; λ, ρ)U ˜ 0u (λ, ρ) ˜

∞ + (I − Q) s1

dτ

cu s Φ+ (s1 , τ ; λ0 , 0)r (τ )2 (λ − λ0 )B0 Φ+ (τ, s1 ; λ, 0)U0s (λ, ρ) ˜ dτ

388


where U0cb (s; λ, ρ) ˜ corresponds to U−cb (s, λ, ρ, ˜ U0cb (λ, ρ)) ˜ so that U0cb (s; λ0 , 0) = U∗ (s). We see that solving (63) is equivalent to solving F (λ, ρ) ˜ = 0. ˜ =0 On Ran(I − Q) ⊂ X, the X-norm is the same as the X -norm, and so we solve F (λ, ρ) in X . For k ∈ Z \ {0} let Fk (λ, ρ) ˜ := Wk,4 (s1 ), F (λ, ρ), ˜ and for k = 0 and j = 3, 4 we let ˜ := W0,j (s1 ), F (λ, ρ). ˜ Define Wk,4 (s) so that it satisfies the adjoint system (50) F0,j (λ, ρ) cu (s , s )∗ ) ∩ (and hence (51)) and passes through Wk,4 (s1 ) for s = s1 . As Wk,4 (s1 ) ∈ Ran(Φk,+ 1 1 s (s , s )∗ ) = Ran(Φ cs (s1 , s1 )) ∩ Ran(Φ u (s1 , s1 )), we get for k = 0 Ran(Φk,− 1 1 k,+ k,− s1 Fk (λ, ρ) ˜ =

/

cs Wk,4 (s1 ), Φ− (s1 , τ ; λ0 , 0)e2τ λ − λ0 − ρ(τ ˜ ) B0 U0cb (τ ; λ, ρ) ˜

−∞

0 u + Φ− (τ, s1 ; λ, ρ)U ˜ 0u (λ, ρ) ˜ dτ ∞ /

0 cu s Wk,4 (s1 ), Φ+ (s1 , τ ; λ0 , 0)r (τ )2 (λ − λ0 )B0 Φ+ (τ, s1 ; λ, 0)U0s (λ, ρ) ˜ dτ

+

s1

s1 =

/

0 u Wk,4 (τ ), e2τ λ − λ0 − ρ(τ ˜ ) B0 U0cb (τ ; λ, ρ) ˜ + Φ− (τ, s1 ; λ, ρ)U ˜ 0u (λ, ρ) ˜ dτ

−∞

+

∞ /

0 s Wk,4 (τ ), r (τ )2 (λ − λ0 )B0 Φ+ (τ, s1 ; λ, 0)U0s (λ, ρ) ˜ dτ

s1

and for k = 0 and j = 3, 4 we have similarly s1 ˜ = F0,j (λ, ρ)

/

0 u W0,j (τ ), e2τ λ − λ0 − ρ(τ ˜ ) B0 U0cb (τ ; λ, ρ) ˜ + Φ− (τ, s1 ; λ, ρ)U ˜ 0u (λ, ρ) ˜ dτ

−∞

+

∞ /

0 s W0,j (τ ), r (τ )2 (λ − λ0 )B0 Φ+ (τ, s1 ; λ, 0)U0s (λ, ρ) ˜ dτ

(64)

s1

Eq. (63) has a nontrivial solution Lemma 24. For (λ, ρ) ˜ in a neighbourhood of (λ0 , 0) ∈ R × R, s u cb s u cb if and only if Fk (λ, ρ) ˜ U0 (λ, ρ) ˜ + U0 (λ, ρ), ˜ λ, ρ) ˜ ∈ E+ × (E− ⊕ E− ) × R × R ˜ =0 (U0 (λ, ρ), for k ∈ Z \ {0} and F0,j (λ, ρ) ˜ = 0 for j = 3, 4. s × (E u ⊕ E cb ) × R × R solves ˜ U0u (λ, ρ) ˜ + U0cb (λ, ρ), ˜ λ, ρ) ˜ ∈ E+ Proof. Suppose that (U0s (λ, ρ), − − ˜ = 0 for k ∈ Z \ {0} and F0,j (λ, ρ) ˜ =0 (63). It is then clear from the definition of Fk that Fk (λ, ρ) ˜ = 0 for k ∈ Z \ {0} and F0,j (λ, ρ) ˜ = 0 for j = 3, 4. By for j = 3, 4. Conversely, let Fk (λ, ρ) Lemma 23, the first equation of (63) is satisfied, so it remains to check the second equation of (63). Recall that the basis vectors in Ran(I − Q) are Vk,j (s1 ), where j = 4 for k = 0 and j = 3, 4 for k = 0. The coefficients of (I − Q)ι(U0s , U0u + U0cb ; λ, ρ) ˜ with respect to this basis are then F0,j (j = 3, 4) and Fk , k ∈ Z \ {0}. Since all these coefficients vanish, the conclusion follows. 2


389

Lemma 25. The equation F0,3 (λ, ρ) ˜ = 0 defines λ as a smooth function of ρ˜ in a neighbourhood of ρ˜ = 0 such that λ(0) = λ0 . Furthermore, ) s1

λ (0)ρ˜ = − ) −∞ ∞

u∗ (τ )2 ρˆ˜ 0 (τ )e2τ dτ

2 −∞ u∗ (τ ) r(τ )r (τ ) dτ

where ρˆ˜ 0 is the Fourier coefficient of ρ˜ corresponding to k = 0. Proof. By Lemma 20 it follows that F0,3 is a smooth function of λ and ρ˜ in a neighbourhood of (λ0 , 0). Note that ∂F0,3 (λ0 , 0) = ∂λ

∞

/

0 W0,3 (τ ), r (τ )2 B0 U∗ (τ ) dτ

−∞

∞ =

w0,3 (τ )u∗ (s)r (τ )2 ds,

−∞

where we have used that U0s (λ0 , 0) = U∗ (s) = U0cb (λ0 , 0) and U0u (λ0 , 0) = 0. We recall that W0,3 (s1 ) = U∗⊥ (s1 ) and that W0,3 (s) satisfies the adjoint system (50) for s ∈ R. It can be verified that W0,3 (s) =

r(s) ⊥ U (s) r (s) ∗

for s ∈ R, where U ⊥ = (−u4 , u3 , −u2 , u1 ), and uj are the components of U , j = 1, . . . , 4. Hence ∂F0,3 (λ0 , 0) = ∂λ

∞

u∗ (s)2 r(τ )r (τ ) ds > 0.

(65)

−∞

The last inequality follows since the integral is positive (using that u∗ is an eigenfunction). ˜ = 0 by the implicit funcSince ∂F0,3 /∂λ(λ0 , 0) = 0, we can solve the equation F0,3 (λ, ρ) tion theorem for λ as a function of ρ, ˜ and this solution is a smooth function λ(ρ), ˜ defined in a neighbourhood of ρ˜ = 0, such that λ(0) = λ0 , and λ (0) is given by ∂F0,3 ∂F0,3 (λ0 , 0)ρ/ (λ0 , 0) ˜ ∂ ρ˜ ∂λ ) s1 u∗ (τ )2 ρˆ˜ 0 (τ )e2τ dτ = − ) −∞ ∞ 2 −∞ u∗ (τ ) r(τ )r (τ ) dτ

λ (0)ρ˜ = −

as claimed.

2

Since we have solved F0,3 = 0 for λ in terms of ρ, ˜ the remaining equation corresponding to ˜ ρ) ˜ = 0. We define k = 0 is F0,4 (λ(ρ),

390


s1

/

W0,4 (τ ), e2τ λ(ρ) ˜ − λ0 − ρ(τ ˜ ρ˜ ˜ ) B0 U0cb τ ; λ(ρ),

G0 (ρ) ˜ := −∞

0 u τ, s1 ; λ(ρ), ˜ ρ˜ dτ ˜ ρ˜ U0u λ(ρ), + Φ− ∞ /

0 s W0,4 (τ ), r (τ )2 λ(ρ) ˜ − λ0 B0 Φ+ τ, s1 ; λ(ρ), ˜ ρ˜ dτ, ˜ 0 U0s λ(ρ),

+

s1

and for k = 0, ˜ ρ˜ ˜ := Fk λ(ρ), Gk (ρ) s1 =

/

Wk,4 (τ ), e2τ λ(ρ) ˜ − λ0 − ρ(τ ˜ ρ˜ ˜ ) B0 U0cb τ ; λ(ρ),

−∞

0 u τ, s1 ; λ(ρ), ˜ ρ˜ dτ. ˜ ρ˜ U0u λ(ρ), + Φ− +

∞ /

0 s Wk,4 (τ ), r (τ )2 λ(ρ) ˜ − λ0 B0 Φ+ τ, s1 ; λ(ρ), ˜ ρ˜ dτ. ˜ 0 U0s λ(ρ),

s1

→ l 2 defined by Lemma 26. The mapping G : R 1 ˜ k∈Z G(ρ) ˜ = Gk (ρ) is smooth. Proof. We first verify that the range of G belongs to l12 . To do this, we split the expression for ˜ (k = 0) into three terms, which we deal with separately: Gk (ρ) Gk (ρ) ˜ = λ(ρ) ˜ − λ0

s1

/

0 Wk,4 (τ ), e U (τ ) dτ − 2τ

−∞

+ λ(ρ) ˜ − λ0

s1

cu

/

0 Wk,4 (τ ), e2τ ρ(τ ˜ )U cu (τ ) dτ

−∞

∞

/

0 Wk,4 (τ ), r (τ )2 U s (τ ) dτ,

(66)

s1

where we used the notation u ˜ ρ˜ + Φ− τ, s1 ; λ(ρ), ˜ ρ˜ , ˜ ρ˜ U0u λ(ρ), U cu (τ ) := U0cb τ ; λ(ρ), s τ, s1 ; λ(ρ), ˜ ρ˜ . ˜ ρ˜ U0s λ(ρ), U s (τ ) := Φ+ Then B0 U cu (τ ) ∈ {0} × {0} × {0} × H 2 (S 1 ) and B0 U s (τ ) ∈ {0} × {0} × {0} × H 1 (S 1 ). Furtheru (s1 , s1 )) ∩ Ran(Φ cs (s1 , s1 )) for all more, by its construction, we have that Wk,4 (s1 ) ∈ Ran(Φ k,− k,+


391

k ∈ Z \ {0}. Thus Lemma 16 implies that for any > 0 there exists a constant K such that for every k ∈ Z \ {0} and s s1 , Wk,4 (s)

(X s )

Ke(s−s1 )

as the norms on X and (X s1 ) are equivalent and Wk,4 (s1 ) X = 1. Now observe that B0 U s (τ ) vanishes for all components except the last one, so we only need an estimate on the last component of Wk,4 (s), which we denote by wk,4 . Then the estimate above gives that there exists a constant K independent of k such that wk,4 (s) Ke(s−s1 ) ,

for all k ∈ Z \ {0} and s s1 ,

(67)

u (s1 , s1 )), Lemma 15, as X ≡ H −1 × L2 × H −1 × L2 . Similarly, from Wk,4 (s1 ) ∈ Ran(Φ k,− −2 −1 −1 2 X = H × H × H × L and Wk,4 (s1 ) X Wk,4 (s1 ) X = 1, it follows that there is some constant K such that wk,4 (s) Ke|k|(s−s1 ) ,

for all k ∈ Z \ {0} and s s1 .

(68)

First we look at the last integral in (66). Let usk (τ ) be the first component of the k-th Fourier s (τ, s ; λ(ρ), ˜ ρ)U ˜ 0s (λ(ρ), ˜ ρ), ˜ then the definition of B0 in (19) gives coefficient of Φ+ 1 ∞ /

0 Wk,4 (τ ), r (τ ) B0 U (τ ) dτ =

2

∞

s

s1

r (τ )2 wk,4 (τ )usk (τ ) dτ.

s1

From Lemma 14, it follows that, for any > 0 and τ s1 , s Φ τ, s1 ; λ(ρ), ˜ ρ˜ ˜ ρ˜ U s λ(ρ), +

0

Xτ

˜ Ke−(λ(ρ)

1/4 −)(τ −s

1)

s U λ(ρ), ˜ ρ˜ 0

This implies for the Fourier coefficients usk that

2 2 k2 ˜ 1/4 −)(τ −s1 ) s U0 λ(ρ), 1 + 2 usk (τ ) K 2 e−2(λ(ρ) ˜ ρ˜ X s1 . τ

k∈Z\{0}

Combining this with (67), we see that for any ρ, ˜ we have ' ∞ (2 1 + k2 r (τ )2 wk,4 (τ )usk (τ ) dτ k∈Z\{0}

s1

(2 ' ∞ 2 (τ −s1 ) s uk (τ ) dτ 1+k C Ke k∈Z\{0}

C

k∈Z\{0}

s1

' ∞ e s1

−(τ −s1 )

(' ∞ dτ

e s1

3(τ −s1 )

2 1 + k 2 usk (τ ) dτ

(

X s1 .

392


∞

C

e ∞

˜ τ 2 e−2(λ(ρ)

k 2 s 2 2τ 1 + 2 uk (τ ) dτ τ 2

k∈Z\{0}

s1

C

3(τ −s1 )

1/4 −4)(τ −s

1)

s 2 U λ(ρ), ˜ ρ˜

X s1

0

dτ C,

s1

where C = C() denotes the different constants occurring. cu be the first component of the k-th Fourier Next, we look at the first integral in (66). Let u" k cb u ˜ ρ) ˜ + Φ− (τ, s1 ; λ(ρ), ˜ ρ)U ˜ 0u (λ(ρ), ˜ ρ). ˜ The definition of B0 gives that coefficient of U0 (τ ; λ(ρ), the first integral can be written as s1

/

0 Wk,4 (τ ), e B0 U (τ ) dτ = 2τ

s1

cu

−∞

cu (τ ) dτ. e2τ wk,4 (τ )u" k

−∞

u leads to solutions with an X-norm that is exponentially decaying at −∞ and U cb is As Φ− 0 cu (τ ) satisfy bounded in the X-norm, there exists a constant K such that the Fourier coefficients u" k

cu 2 2 cu 2 2 1 + k 2 u" 1 + k 2 u" k (τ ) k (τ ) K k∈Z\{0}

k∈Z\{0}

for all τ s1 . Together with the fact that (68) implies that |wk,4 (τ )| K for all τ s1 , this gives ' s1 (2 cu (τ ) dτ 1 + k2 e2τ wk,4 (τ )u" k k∈Z\{0}

K

−∞

(2 ' s1 cu 2 2s1 2(τ −s1 ) " 1+k e e uk (τ ) dτ

2

k∈Z\{0}

K e

2 2s1

' s1

−∞

e

2(τ −s1 )

k∈Z\{0} −∞

K 2 e2s1 2

s1

e2(τ −s1 )

(' s1 dτ

cu 2 1 + k 2 e2(τ −s1 ) u" k (τ ) dτ

(

−∞ 4 2s1 cu 2 dτ K e . 1 + k 2 u" (τ ) k 4

k∈Z\{0}

−∞

Finally, let ν(τ ) be the first component of ρ(τ ˜ )U cu (τ ), so that ν(τ ) = ρ(τ ˜ )ucu (τ ). As ucu ∈ 2 2 its H norm is uniformly bounded on (−∞, s1 ] and ρ˜ ∈ L (J− ; H 1/2 (S 1 ), e2s ds), Lemma 6 implies that ν ∈ L2 (J− ; H 1/2 (S 1 ), e2s ds). Denote the Fourier coefficients of ν by νˆ k . Then the second integral of (66) can be written as H 2 (S 1 ),

s1 −∞

/

0 Wk,4 (τ ), e ρ(τ ˜ )U (τ ) dτ = 2τ

s1

cu

−∞

e2τ wk,4 (τ )νˆ k (τ ) dτ


393

and the estimate (68) on the decay of wk,4 implies that there exists a constant C such that (2

' s1

' s1

e wk,4 (τ )νˆ k (τ ) dτ

2τ

−∞

2 2(|k|+1)(τ −s1 )

(

(' s1 e νˆ k (τ ) dτ 2τ

K e −∞

2

−∞

K2 2(|k| + 1)

s1 e νˆ k (τ ) dτ √ 2τ

2

−∞

s1

C 1 + k2

e2τ νˆ k (τ )2 dτ, −∞

and so ' s1 (2 s1 2 2τ 2 1/2 1+k 1+k e wk,4 (τ )νˆ k (τ ) dτ C e2τ νˆ k (τ )2 dτ k∈Z\{0}

k∈Z\{0}

−∞

−∞

= C ν 2L2 (J ;H 1/2 (S 1 ),e2s ds) −

< ∞.

Hence the second term also belongs to l12 , and so the proof of the claim that the range of G is contained in l12 is complete. Smoothness follows since the integrands are smooth in ρ˜ and since the derivatives of arbis (τ, s ; λ(ρ), u (τ, s ; λ(ρ), trary order of the evolution operators Φ+ ˜ ρ) ˜ and Φ− ˜ ρ) ˜ belong to the 1 1 same exponentially weighted space as the evolution operators themselves (see Theorem 2 and Lemma 14). 2 Finally we consider G (0). Since U∗ is radially symmetric (and hence belongs to X0 ) we have for k = 0 that G k (0)ρ˜

∞ =

/

0 Wk,4 (τ ), r (τ )2 λ (0)ρ˜ − ρ(τ ˜ ) B0 U∗ (τ ) dτ

−∞

s1 =−

wk,4 (τ )ρˆ˜ k (τ )u∗ (τ )e2τ dτ,

−∞

˜ For k = 0 we have where ρˆ˜ k is the k-th Fourier coefficient of ρ. G 0 (0)ρ˜

s1 =− −∞

w0,4 (s)ρˆ˜ 0 (s)u∗ (s)e ds − 2s

)∞

2 −∞ w0,4 (τ )u∗ (τ )r (τ ) dτ )∞ 2 −∞ u∗ (τ ) r(τ )r (τ ) dτ

s1

ρˆ˜ 0 (s)u∗ (s)2 e2s ds.

−∞

To rewrite the preceding expressions, we define ηk (s) := wk,4 (s)u∗ (s)χ(−∞,s1 ] (s)

(69)

394


for k ∈ Z \ {0}, and set )∞

w0,4 (τ )u∗ (τ )r (τ )2 dτ )∞ u∗ (s)χ(−∞,s1 ] (s). η0 (s) := w0,4 (s) + u∗ (s) −∞ 2 −∞ u∗ (τ ) r(τ )r (τ ) dτ

Then we may write % G (0)ρ˜ = −

s1

& e2τ ηk (τ )ρˆ˜ k (τ ) dτ

−∞

. k∈Z

Indeed, (68) shows that |wk,4 (τ )| Ke|k|(τ −s1 ) for any For any k ∈ Z, we have ηk eik· ∈ R: τ s1 and k ∈ Z\{0}, while |w0,3 | and |w0,4 | are bounded on J− , so that there exists a constant C such that s1

ηk (τ )2 e2τ dτ

−∞

sup

τ ∈(−∞,s1 )

u∗ (τ )2 C

sup

τ ∈(−∞,s1 )

s1

e(2|k|+2)(τ −s1 ) dτ

−∞

u∗ (τ )2

C . 2|k| + 2

(70)

From the definition of G (0)ρ, ˜ it follows immediately that G (0)ρ˜ = 0 if and only if s1

e2τ ηk (τ )ρˆ˜ k (τ ) dτ = 0

(71)

−∞

for all k ∈ Z. Thus, if we define M := span ηk eikϕ ; k ∈ Z , then it can be seen that M is the orthogonal complement in R where the closure is taken in R, of ker G (0), and so R = ker G (0) ⊕ M. Lemma 27. G (0) : M → l12 is a linear homeomorphism. Furthermore, the spaces ker G (0) and M are both infinite-dimensional. → l 2 is bounded since by Lemma 26, G is smooth in a neighProof. It is clear that G (0) : R 1 bourhood of 0. We need to investigate the subspace M. Let η ∈ M be arbitrary, then η(s, ϕ) =

k∈Z

ak ηk (s)eikϕ .

(72)


395

The upper bound estimate (70) implies that

η 2R

s1 2 1/2 2 ηk (τ )2 e2τ dτ C 1+k = |ak | |ak |2 . k∈Z

(73)

k∈Z

−∞

Next we derive a lower bound for η 2 . Since u∗ (s1 ) = 0, there exist ˆ and δ > 0 such R that u∗ (s)2 > ˆ 2 for every s ∈ (s1 − δ, s1 ). Lemma 18 shows that, for k large, Wk,4 is close to solutions of the system at infinity, both in the X and X norms. This allows us to get a lower bound u (s1 , s1 )), it on |wk,4 (s)| for k large. Let > 0 and K be as in Lemma 18. As Wk,4 (s1 ) ∈ Ran(Φ k u (s, s1 )Wk,4 (s1 ) = Wk,4 (s), and hence follows that Φ k Wk,4 (s) − e|k|(s−s1 ) P s ∗ Wk,4 (s1 ) e|k|(s−s1 ) Wk,4 (s1 ) = e|k|(s−s1 ) k X X for |k| > K. Thus we get for the fourth component wk,4 (s) wk,4 (s) e|k|(s−s1 ) P s ∗ Wk,4 (s1 ) − wk (s) − e|k|(s−s1 ) P s ∗ Wk,4 (s1 ) k k 4 4 s ∗ s ∗ |k|(s−s1 ) |k|(s−s1 ) Pk Wk,4 (s1 ) 4 − Wk,4 (s) − e e Pk Wk,4 (s1 )X ∗ e|k|(s−s1 ) Pks Wk,4 (s1 ) 4 − Wk,4 (s1 )X e|k|(s−s1 ) wk,4 (s1 ) − 2 Wk,4 (s1 )X e|k|(s−s1 ) wk,4 (s1 ) − 2 . To get a lower bound on wk,4 (s1 ), first note that Lemma 18 implies that Wk,4 (s1 ) − (Pks )∗ Wk,4 (s1 ) X and that Ran(Pks )∗ = span{(−|k|, 1, 0, 0)T , (0, 0, −|k|, 1)T }. Thus there exist αk , βk ∈ C and Wk ∈ Xk with Wk X 1 such that T T Wk,4 (s1 ) = αk −|k|, 1, 0, 0 eik· + βk 0, 0, −|k|, 1 eik· + Wk . 1/4

s , its first component has to be a multiple of K (λ As Vk,1 (s1 ) ∈ E+ |k| 0 r(s1 )) and hence

⎛

⎞ 1/4 K|k| (λ0 r1 )

⎜ r (s )λ1/4 K (λ1/4 r ) ⎟ vk 1 0 1 ⎟ ⎜ |k| 0 ik· , C Vk,1 (s1 ) = Ck eik· ⎜ e ⎟ k 1/2 λ0 v k ⎝ λ1/2 K|k| (λ1/4 r1 ) ⎠ 0

0

with

(λ r (s1 )λ0 K|k| 0 r1 )

1/4 K|k| (λ0 r1 ) vk = 1/4 1/4 r (s1 )λ0 K|k| (λ0 r1 ) 3/4

1/4

where r1 = r(s1 ) and Ck is such that Vk,1 (s1 ) X = 1, i.e., Ck2 =

1 1/4 (1 + λ0 )[(k 2 + 1)(K|k| (λ0 r1 ))2

+

. √ (λ1/4 r ))2 ] λ0 (r (s1 )K|k| 1 0

396


Since Wk,4 (s1 ), Vk,1 (s1 ) = 0, we have + 1/4 1/4 / 0 1/4 λ0 r1 (αk + βk λ0 ) + Wk , Vk,1 (s1 ) . (74) 0 = Ck −|k|K|k| λ0 r1 + λ0 r (s1 )K|k| (z) = From (9.6.23) in [1], we see that K|k| (z) > 0 for any z > 0 and (9.6.26) implies K|k|

−K|k|−1 (z) −

|k| z K|k| (z)

(z) < 0 for any z > 0. So we see that for any z > 0, hence K|k|

(λ −|k|K|k| (λ0 r1 ) < 0 and λ0 r (s1 )K|k| 0 r1 ) < 0. A short analysis gives that 1/4

1/4

1/4

+ + 1/4 1/4 1/4 λ0 r1 −1/ 2(1 + λ0 ). −2/ 1 + λ0 Ck −|k|K|k| λ0 r1 + λ0 r (s1 )K|k| Furthermore, |Wk , Vk,1 (s1 )| Wk X Vk,1 (s1 ) X 1, and we can conclude from (74) that √ 2 αk = −βk λ0 + O(). Finally, Wk,4 (s1 ) X = 1 gives 1+2k (αk2 + βk2 ) = 1 − O( 2 ), and hence 1+k 2 2

1+k (1 + λ0 )βk2 = 1+2k 2 − O(). Thus there exists a C > 0 such that |wk,4 (s1 )| > C for all |k| > K. |k|(s−s1 ) for every s s1 and > 0 such that |wk,4 (s)| > Ce This implies that there exists a C |k| > K. Combining the lower bounds on u∗ (s) and wk,4 (s), we find that there exists a δ > 0 such that for |k| > K

s1 2 2τ

ηk (τ ) e

2 2

s1

dτ ˆ C

−∞

e(2|k|+2)(τ −s1 ) dτ =

s1 −δ

2 ˆ 2 C 2|k| + 2

1 − e−2δ

2 ˆ 2 C 1 − e−(2|k|+2)δ 2|k| + 2

C , (1 + k 2 )1/2

for some positive k-independent constant C. Since for all k ∈ Z, constant C above can be modified so that s1 ηk (τ )2 e2τ dτ −∞

(75) ) s1

2 2τ −∞ ηk (τ ) e

dτ > 0, the

C (1 + k 2 )1/2

also for |k| K. Hence it follows that

η 2R

s1 2 1/2 2 1+k = |ak | ηk (τ )2 e2τ dτ C |ak |2 . k∈Z

−∞

k∈Z

The upper and lower bounds on η 2 show that η ∈ M if and only if η is given by (72) and R {ak }k∈Z ∈ l 2 . As we have seen that the mapping G (0) is bounded above, it is sufficient to show that it is bounded below to conclude that G (0) is a linear homeomorphism from M to l12 . From its definition, it follows that


% G (0)η = ak

s1

ηk (τ )2 e2τ dτ

397

&

−∞

, k∈Z

and so by (75) and (73), we see that G (0)η22 = 1 + k 2 |ak |2 l 1

k∈Z

' s1

ηk (s)2 e2s ds

−∞

(2 C

k∈Z

|ak |2 C η 2R .

It remains to show that the spaces ker G (0) and M have infinite dimension. For M, this follows directly from its definition. Next, consider the characterisation of ker G (0) given in (71). We proved above that the functions wk,4 (s) that appear in the definition (69) of ηk satisfy |wk,4 (s1 )| C uniformly in |k| K, which implies that the space ker G (0) has infinite dimension as claimed. 2 We are now ready to complete the proof of Theorem 1. Proof of Theorem 1. By Lemma 21, if (λ, ρ) ˜ is sufficiently close to (λ0 , 0) then λ is an embedded eigenvalue for 2 + θ˜ + ρ˜ if and only if (62) holds. We have also seen that (62) is equivalent to F (λ, ρ) ˜ = 0, which allowed us to solve for λ as a function of ρ˜ and finally obtain the equation = ker G (0) ⊕ M, and G (0) : M → l 2 is a linear → l 2 . By Lemma 27, R G(ρ) ˜ = 0, where G : R 1 1 homeomorphism. Hence for ρ˜ ∈ R we may write ρ˜ = ξ + η, where ξ ∈ ker G (0) and η ∈ M. By the implicit function theorem, we can solve for η in terms of ξ , and this equation defines a smooth manifold in a neighbourhood of 0 with infinite dimension and codimension, since the spaces ker G (0) and M are infinite-dimensional by Lemma 27. 2 8. Conclusions and open problems In this paper, we considered the planar bilaplacian with a smooth, radially symmetric and compactly supported potential θ and described the set of perturbations of the potential in the space R = L2 ([0, r1 ]; H 1/2 (S 1 ), r dr) for which an embedded eigenvalue persists. We expect that the space R can be replaced by the Sobolev space H 1/2 (Br1 (0)) of H 1/2 -functions of two variables that have support in the ball Br1 (0). One restriction of our work is that we consider only potentials with compact support: The reason is that we were forced to work with different function spaces of solutions for r small and for r large. For r small, we have some freedom in choosing the space, as any space of the form X = H 1+α × H α × H 1 × L2 , with 0 < α 1, ensures that an exponential dichotomy exists. For r large, due to the structure of the equations, there is no such freedom, the regularity on the first two components has to be same as the regularity of the last two. So it is unclear whether there exists an exponential dichotomy when the support of ρ is not compact. It would be interesting to see whether our hypothesis that ρ has compact support could be replaced by an appropriate decay condition on ρ. For the original potential θ , we see no obstacles in removing the condition that θ has compact support. It should be possible to replace this condition by the long range condition |θ (r)| C(1 + r)−1−β for some β > 0. It should also be possible to remove the condition that θ is radially symmetric, although considerably more work will be needed without this condition.

398


We believe that the methods put forward in this paper can be used to study other operators. In particular, the exponential-dichotomy results established in [19] are for systems of reaction– diffusion equations, so we believe that the only obstacle for extending our results to self-adjoint systems is the presence of nonsmooth potentials. For other operators, it might not be possible to modify the function spaces involved to prove the existence of exponential dichotomies. These are difficult problems that have to be studied in future work. References [1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, 1972. [2] S. Agmon, I. Herbst, S. Maad Sasane, Persistence of embedded eigenvalues, J. Funct. Anal. (2010), doi:10.1016/ j.jfa.2010.09.005, in press. [3] N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space I, Pitman, 1981. [4] M. Beck, B. Sandstede, K. Zumbrun, Nonlinear stability of time-periodic viscous shocks, Arch. Ration. Mech. Anal. 196 (2010) 1011–1076, doi:10.1007/s00205-009-0274-1. [5] M. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977. [6] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw–Hill, 1955. [7] W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer-Verlag, 1978. [8] G.A. Derks, S. Maad, B. Sandstede, Perturbations of embedded eigenvalues for the bilaplacian on a cylinder, Discrete Contin. Dyn. Syst. Ser. A 21 (2008) 801–821. [9] L.C. Evans, Partial Differential Equations, Amer. Math. Soc., 1998. [10] P.D. Hislop, I.M. Sigal, Introduction to Spectral Theory with Application to Schrödinger Operators, Springer-Verlag, New York, 1996. [11] T. Kato, Perturbation Theory for Linear Operators, Springer, 1976. [12] L. Landau, Bessel functions: Monotonicity and bounds, J. Lond. Math. Soc. 61 (2000) 197–215. [13] D. Pelinovsky, C. Sulem, Bifurcations of new eigenvalues for the Benjamin–Ono equation, J. Math. Phys. 39 (1998) 6552–6572. [14] D. Pelinovsky, C. Sulem, Eigenfunctions and eigenvalues for a scalar Riemann–Hilbert problem associated to inverse scattering, Comm. Math. Phys. 39 (2000) 713–760. [15] D. Peterhof, B. Sandstede, A. Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Differential Equations 140 (1997) 266–308. [16] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV, Academic Press, New York, 1978. [17] B. Sandstede, Stability of travelling waves, in: B. Fiedler (Ed.), Handbook of Dynamical Systems II, North-Holland, 2002, pp. 983–1055. [18] B. Sandstede, A. Scheel, On the structure of spectra of modulated travelling waves, Math. Nachr. 232 (2001) 39–93. [19] A. Scheel, Bifurcation to spiral waves in reaction–diffusion systems, SIAM J. Math. Anal. 29 (1998) 1399–1418.


Functional inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion ✩ Shui Feng b , Wei Sun c , Feng-Yu Wang a,d,∗ , Fang Xu a,b a School of Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing 100875, China b Department of Mathematics and Statistics, McMaster University, Hamilton, L8S 4K1, Canada c Department of Mathematics and Statistics, Concordia University, Montreal, H3G 1M8, Canada d Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK

Received 2 October 2009; accepted 8 October 2010 Available online 20 October 2010 Communicated by L. Gross

Abstract By explicitly identifying the transition density function, we derived the super-Poincaré and super-logSobolev inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion, which in particular implies the Gross log-Sobolev inequality. © 2010 Elsevier Inc. All rights reserved. Keywords: Poisson–Dirichlet distribution; Two-parameter Poisson–Dirichlet distribution; Infinitely-many-neutral-alleles diffusion; Transition function; Log-Sobolev inequality

1. Introduction The log-Sobolev inequality introduced by Gross [11] has become a powerful tool in the study of symmetric Markov semigroups. In general, let (E , D(E )) be a symmetric Dirichlet form

✩

Supported in part by WIMICS, NNSFC (10721091), the 973-Project, the Natural Science and Engineering Research Council of Canada. * Corresponding author at: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. E-mail address: [email protected] (F.-Y. Wang). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.005

400

S. Feng et al. / Journal of Functional Analysis 260 (2011) 399–413

on L2 (μ) for μ a probability measure on a Polish space. Let μ(f ) denote the integration of a function f w.r.t. μ. Then the log-Sobolev inequality μ f 2 log f 2 CE (f, f ),

f ∈ D(E ), μ f 2 = 1

(1.1)

holds for some constant C > 0 if and only if the associated semigroup Pt is hypercontractive, i.e. Pt 2→4 1 holds for some t > 0, where · p→q (p, q 1) stands for the operator norm from Lp (μ) to Lq (μ). Moreover, (1.1) implies that Pt decays exponentially in entropy: μ (Pt f ) log Pt f e−4t/C μ(f log f ),

f 0, μ f 2 = 1, t > 0.

(1.2)

Indeed, (1.1) and (1.2) are equivalent for diffusion processes. The log-Sobolev inequality can also be used to derive concentration, deviation and transportation-cost inequalities, see e.g. [1,3, 4,12,14,15,20] and references within for accounts on criteria and applications of the log-Sobolev inequality. Since the log-Sobolev inequality has sub-additivity property, it is available in many infinitedimensional cases. The purpose of this paper is to establish the log-Sobolev inequality for diffusion processes arisen from genetic models. It was shown by Stannat [18] that the log-Sobolev inequality holds for the Fleming–Viot process with parent independent mutation if and only if the number of alleles or type space is finite. The invalidity of the log-Sobolev inequality indicates the extreme singularity in the Fleming–Viot process when there is infinite number of alleles. Our main result shows that not only a log-Sobolev inequality but a super-log-Sobolev inequality holds for the two-parameter extension of the infinite-many-neutral-alleles diffusion (see Theorem 4.1 below). Let ∞ ∇∞ := x = (x1 , x2 , . . .): x1 x2 · · · 0, xi = 1 i=1

denote the infinite-dimensional ordered simplex and ∇ := x = (x1 , x2 , . . .): x1 x2 · · · 0,

∞

xi 1

i=1

be the closure of ∇∞ in the product space [0, 1]∞ . For any 0 α < 1 and θ > −α, the two-parameter extension of the infinitely-many-neutral-alleles diffusion (henceforth, the twoparameter diffusion) discussed in this paper is an infinite-dimensional symmetric diffusion process taking values in ∇ with generator

Aα,θ

1 = 2

∞ i,j =1

∞ ∂2 ∂ xi (δij − xj ) − (θ xi + α) , ∂xi ∂xj ∂xi

(1.3)

i=1

defined on an appropriate domain specified in the next section. It was introduced in [16] and further studied in [10]. The unique reversible measure is the two-parameter Poisson–Dirichlet


401

distribution, simply denoted by μα,θ , and is defined as follows. Let Uk , k = 1, 2, . . . , be a sequence of independent random variables such that Uk has Beta(1 − α, θ + kα) distribution. Set V1α,θ = U1 ,

Vnα,θ = (1 − U1 ) · · · (1 − Un−1 )Un ,

n 2,

and let P(α, θ ) = (ρ1 , ρ2 , . . .) denote (V1α,θ , V2α,θ , . . .) in descending order. The distribution of (V1α,θ , V2α,θ , . . .) is called the two-parameter GEM distribution. The reversible measure μα,θ is the law of P(α, θ ). The case α = 0 corresponds to the infinitely-many-neutral-alleles diffusion constructed in [8]. The corresponding reversible measure is the one-parameter Poisson–Dirichlet distribution first introduced by Kingman in [13]. The labeled version of the one-parameter Poisson–Dirichlet distribution is the law of the Dirichlet process ∞

ρi δξi ,

i=1

where ξ1 , ξ2 , . . . are i.i.d. with common diffuse distribution ν and are independent of P(0, θ ) = (ρ1 , . . .). Here ξi is the label or type of the allele and ρi is the corresponding relative frequency of the allele in the population. The symmetric diffusion process associated with the Dirichlet process is the Fleming–Viot process with parent independent mutation (cf. [5]). Several known results in the case of α = 0 provide evidence for the dramatic change caused by the ordering of atoms. It was shown in [6] that the complete set of eigenvalues of generator A0,θ is {0, −λ2 , −λ3 , . . .}, with λn = n(n − 1 + θ )/2,

n 0.

(1.4)

But the set of eigenvalues of the Fleming–Viot process has one more element −λ1 than infinitelymany-neutral-alleles diffusion. In other words, the ordering and un-labeling increase the spectral gap by an amount of 1 + θ/2. Also the infinitely-many-neutral-alleles diffusion has a transition density with respect to the reversible measure while the Fleming–Viot process does not (cf. [7]). Our result shows that the ordering also reduces the singularity through the uniform regularization of small frequencies. As we are not able to verify the Bakry–Emery [2] condition for the present model, the establishment of functional inequalities is achieved by estimating the transition density of the associated semigroup. In Section 2, we first collect several results that are either known or could be easily derived from known results. A uniform upper bound for the transition density function is derived in Section 3 following the strategy in [6], which leads to explicit super-Poincaré/logSobolev inequalities in Section 4 for the two-parameter diffusion. These inequalities are stronger than the Gross log-Sobolev inequality, so that the associated process converges exponentially in entropy to μα,θ . 2. Preliminaries In this section, we collect several needed results that are known or easily derived. Let N denote the set of positive integers. For each n 1, set φ1 (x) = 1,

φn (x) =

∞ i=1

xin ,

n 2, x ∈ ∇∞ .

402


For n > 1, these functions are continuous, while for n = 1 the extension is unity. Let C(∇) be the space of continuous functions on ∇ and D0 be the subalgebra of C(∇) generated by {φn : n 1}. For any real number a and n in N, set a(0) = a[0] = 1,

a(n) = a(a + 1) · · · (a + n − 1),

a[n] = a(a − 1) · · · (a − n + 1),

n 1.

etc. For each For each l 1, Nl is the l-fold product of N with elements denoted by n , m, n = (n1 , . . . , nl ) in Nl , define | n| =

l

nj ,

j =1

n|, ai = #{nj = i: j = 1, . . . , l}, i = 1, . . . , | 1 | n| n , C( n) = n 1 , . . . , nl i=1 ai ! ψn (x) = xin11 · · · xinl l , distinct i1 ,...,il ∈Nl

n)ψn (x). φn (x) = C( According to [10], the operator (Aα,θ , D0 ) is symmetric on L2 (μα,θ ) with

∞ 1 −μα,θ (f Aα,θ g) = E (f, g) := xi (δij − xj )(∂i f ∂j g)(x) μα,θ (dx), 2 ∇

f, g ∈ D0 ,

i,j =1

where grad(f ) is the gradient of f and D(x) is the infinite matrix whose (i, j )-th entry is xi (δij − xj ). Thus, (E , D0 ) is closable in L2 (μα,θ ) and the closure (E , D(E )) is a symmetric Dirichlet form with generator (Aα,θ , D(Aα,θ )) being the Friedrich extension of (Aα,θ , D0 ). Theorem 2.1. (1) (Aα,θ , D(Aα,θ )) generates a ∇-valued diffusion process Xα,θ (t), the two-parameter diffusion; (2) The process Xα,θ (t) is reversible with respect to μα,θ ; (3) For each l 1 and n in Nl , we have | n| l−1 ((1 − α)(j −1) )aj | n|! , PSF( n) := μα,θ (φn ) = (θ + iα) θ(|n|) (j !)aj aj !

(2.1)

j =1

i=0

which implies that l−1 μα,θ (ψn ) =

i=0 (θ

| n| + iα)

θ(|n|)

j =1

(1 − α)(j −1)

aj

.

(2.2)


403

Proof. (1) and (2) are obtained in [16]. Alternate proof of (1) using Dirichlet forms can be found in [10]. The formula (3) is the well-known Pitman sampling formula (cf. [17]). 2 The next result characterizes the spectrum and eigenspaces of the self-adjoint operator (Aα,θ , D(Aα,θ )). Theorem 2.2. (1) For any m 2, let λm be defined as in (1.4). Then for any α 0 the spectrum of Aα,θ is {0, −λ2 , −λ3 , . . .} and 0 is a simple eigenvalue and for m 2, the multiplicity of −λm is π(m) − π(m − 1), where π(m) is the total number of partitions of integer m; (2) For m = 0, 2, 3, . . . , let Wm be the eigenspace corresponding to eigenvalue −λm and ⊕ denote the direct sum. Then

L (μα,θ ) = 2

∞

Wm :=

m=0

∞

fm : fm ∈ W m ,

m=0

∞

2 μα,θ fm < ∞ ;

m=0

= (m1 , . . . , mk ) ∈ Nk : m1 m2 · · · mk 2}, and N∞ = (3) For

∞ k 1, let Nk = {m 2 ∈ N∞ } such that : m i=1 Ni . The space L (μα,θ ) has an orthonormal basis {1} ∪ {gm {gm : m ∈ N∞ , |m| = m} is an orthonormal basis for Wm . Proof. The spectrum in (1) is identified in [16]. The multiplicity of −λm is identified in [16] as the number of partitions of m without singletons. Noting that each partition of m with at least a singleton can be obtained by adding a singleton to a partition of m − 1, it follows that the multiplicity of −λm can be written as π(m) − π(m − 1). Results in (2) and (3) can be derived following an argument similar to that used in the proof of Theorem 2.3 in [6]. 2 3. Transition density function In this section, we derive the density function of the transition function of the two-parameter diffusion. Since the main idea of proof is similar to that in [6], we focus here only on the derivations of results that require additional efforts due to the introduction of the additional parameter α. Lemma 3.1. For any x, y ∈ ∇, and n 1, define

pn (x, y) =

n∈N∞ : | n|=n

φn (x)φn (y) . PSF( n)

Then sup pn (x, y) x,y∈∇

θ(n) , (1 − α)n (θ n ∧ 1)

n 1.

(3.1)

404


Proof. Since for 1 l n, θ l θ n ∧ 1, it follows from a direct calculation that l−1 n! PSF( n) = (θ + pα) n1 ! · · · nl !a1 ! · · · an !

n

aj j =1 ((1 − α)(j −1) )

θ(n)

p=0

(1 − α)n (θ n ∧ 1) , θ(n)

which, combined with the fact that

φn (x)φn (y) 1,

n∈N∞ : | n|=n

implies the result.

2

The next result plays a key role in deriving the explicit form of the density function. = m 2, and any n m, Lemma 3.2. For any m = (m1 , . . . , mk ) in N∞ satisfying |m|

n[m] φm (·) ∈ Wn . (n + θ )(m) m−1

pn (x, ·)φm (x) μα,θ (dx) −

n=0

∇

Proof. Fix m = (m1 , . . . , mk ), |m| = m 2. For each n = (n1 , . . . , n ) ∈ N∞ satisfying n = n m, set J (m, n) =

μα,θ (ψm ψn ) . μα,θ (ψn )

For any 0 j l ∧ k, let Π(l + k − j ) denote the set of permutations of {1, . . . , l + k − j }. For each σ in Π(l + k − j ), set nσ m = (nσ (1) , . . . , nσ (l+k−j ) ) + (m1 , . . . , ml+k−j ). Then we have by a direct calculation that J (m, n) =

l∧k

j =0 σ ∈Π(l+k−j ) σ (i)l if i>k

μα,θ (ψnσ m ) 1 . (l − j )!(k − j )! μα,θ (ψn )

It now follows from (2.2) that J (m, n) =

∧k +k−j −1 (θ + pα) j =0

·

i ∈Λ /

p=

(1 − α)(mi −1)

(nσ (i) − α)(mi )

Λ⊂{1,...,k} σ : Λ→{1,..., } i∈Λ one to one |Λ|=j

1 . (n + θ )(m)


405

For Γ ⊂ {1, 2 . . . , k}, set Ω(m, Γ ) = r = (r1 , . . . , rk ) ∈ Nk : 1 ri mi , i ∈ Γ ; ri = 0, i ∈ /Γ . Noting that (nσ (i) − α)(mi ) = (−α)(1 − α)(mi −1) +

mi

cmi ri (α)(nσ (i) )[ri ] ,

ri =1

it follows that

(nσ (i) − α)(mi ) =

j γ =0 Γ ⊂Λ |Γ |=γ

i∈Λ

(−α)j −γ

(1 − α)(mi −1)

cmi ri (α)(nσ (i) )[ri ] ,

r∈Ω(m,Γ ) i∈Γ

i∈Λ\Γ

and J (m, n) =

∧k +k−j −1 j =0

cmi ri (α)(nσ (i) )[ri ]

∧k +k−j −1 j =0

(1 − α)(mi −1)

j

(θ + tα)

(−α)j −γ

Λ⊂{1,...,k} γ =0 Γ ⊂Λ |Γ |=γ |Λ|=j

t=

·

i ∈Γ /


=

j

(−α)j −γ

Λ⊂{1,...,k} σ : Λ→{1,..., } γ =0 Γ ⊂Λ |Γ |=γ one to one |Λ|=j

t=

·

(θ + tα)

σ : Γ →{1,..., } one to one

( − γ )! (j − γ )!

1 (n + θ )(m)

(1 − α)(mi −1)

i ∈Γ /

cmi ri (α)(nσ (i) )[ri ] ·


1 . (n + θ )(m)

Reorganizing the terms, yields

J (m, n) =

∧k +k−j −1 j =0

(θ + tα)

σ : Γ →{1,..., } r∈Ω(m,Γ ) i∈Γ one to one

=

∧k +k−j −1

∧k γ =0

j =γ

γ =0 Γ ⊂{1,...,k} |Γ |=γ

t=

·

j

t=

k−γ ( − γ )! (−α)j −γ j −γ (j − γ )!


1 (1 − α)(mi −1) (n + θ )(m)

k−γ j −γ ( − γ )! (θ + tα) (−α) (j − γ )! j −γ

i ∈Γ /

/ Γ ⊂{1,...,k} i ∈Γ |Γ |=γ

(1 − α)(mi −1)

406


·



1 . (n + θ )(m)

(3.2)

The term inside the braces can be expressed as

∧k +k−j −1 ( − γ )! k−γ (θ + tα) (−α)j −γ (j − γ )! j −γ

j =γ

=

t=

(θ + α) · · · (θ + ( + k − j − 1)α) − γ (−α)j −γ · (k − γ )! (k − j )! j −γ

∧k j =γ

=

∧k θ ( α + ) · · · ( αθ + + k − j − 1) − γ (−1)j −γ α k−γ (k − γ )! (k − j )! j −γ

j =γ

=

( −γ )∧(k−γ ) j =0

( αθ + ) · · · ( αθ + + k − j − 1) − γ (−1)j −γ α k−γ (k − γ )!. (k − j )! j −γ

(3.3)

Next we show that ( −γ )∧(k−γ ) θ α j =0

θ + +k−j −γ −1 −γ +k−1 (−1)j = α . k−j −γ j k−γ

(3.4)

θ

Let b = − γ . Considering the Taylor expansion of (1 − x)−( α +γ ) , we have −( αθ +γ )

(1 − x)

∞ θ +γ +i −1 i α x. = i i=0

The coefficient of x k−γ in the expansion is writing we get

θ +γ +k−γ −1 α

k−γ −( αθ +γ ) −( αθ +b+γ ) = (1 − x) · (1 − x)b , and (1 − x)

−( αθ +b+γ )

(1 − x)

=

θ +k−1 α

k−γ

. On the other hand, by

doing the Taylor expansion separately,

∞ θ +b+γ +i −1 i α x, = i i=0

(1 − x) =

b b i=0

i

(−1)i x i ,

and −( αθ +b+γ )

(1 − x)

∞ b θ + b + γ + i1 − 1 b α (−1)i2 x i1 +i2 . · (1 − x) = i2 i1 b

i1 =0 i2 =0


407

The coefficient of x k−γ now has the form θ

α

i1 +i2 =k−γ

+ b + γ + i1 − 1 i1

b∧(k−γ ) θ + k + b − i2 − 1 b b i2 α (−1) = (−1)i2 , i2 i2 k − γ − i2 i2 =0

which leads to (3.4). Now putting together (3.3) and (3.4) together, we obtain

∧k +k−j −1 j =γ

t=

θ +k−1 k−γ j −γ ( − γ )! k−γ α (θ + tα) (−α) (k − γ )! =α (j − γ )! k−γ j −γ = (θ + γ α) · · · θ + (k − 1)α

(3.5)

which combined with (3.2) implies that J (m, n) =

∧k

Γ ⊂{1,...,k} |Γ |=γ

i ∈Γ /

(θ + γ α) · · · θ + (k − 1)α

γ =0

·

(1 − α)(mi −1)



1 . (n + θ )(m)

(3.6)

Denoting the multiplicities of m by βi , 1 i m, it follows from (3.6) and the definition of the function pn (x, y) in (3.1) that

pn (x, y)φm (x) μα,θ (dx) =

n

=1 n∈N | n|=n

=

m 1 J (m, n)φn (y) m1 , . . . , mk β1 ! · · · βm !

n

=1 n∈N | n|=n

·

∧k m 1 (θ + γ α) · · · θ + (k − 1)α m1 , . . . , mk β1 ! · · · βm ! γ =0

(1 − α)(mi −1) ·

/ Γ ⊂{1,...,k} i ∈Γ |Γ |=γ



k m 1 n[m] = (θ + γ α) · · · θ + (k − 1)α (n + θ )(m) m1 , . . . , mk β1 ! · · · βm ! ·

/ Γ ⊂{1,...,k} i ∈Γ |Γ |=γ

γ =0

(1 − α)(mi −1)

φn (y) (n + θ )(m)

408


·

r∈Ω(m,Γ )

n n[r] cmi ri (α) n[m]

i∈Γ (nσ (i) )[ri ]

n[r]

=γ ∨1 n∈I σ : Γ →{1,..., } one to one | n|=n

i∈Γ

φn (y) .

(3.7)

Noting that for r = (r1 , . . . , rγ ),

ψr (y) =

r

distinct i1 ,...,iγ ∈Nγ

yir11 · · · yiγγ

is the probability of the event that a random sample of size r = |r | from a population with allele frequencies y1 , y2 , . . . containing γ families and each family has ri alleles for 1 i γ . To calculate this probability, we can first select a random sample of size n with types ( γ ), one with n1 individuals, another with n2 individuals and so on. The probability of such an event is φn (y). We then choose γ types denoted by σ out of types from the random sample. Since for each type one need to select ri individuals, the probability would be these procedures we can see that ψr (y) =

n

i∈Γ (nσ (i) )[ri ] n[r]

. Following

i∈Γ (nσ (i) )[ri ]

n[r]

=γ ∨1 n∈I σ : Γ →{1,..., } one to one | n|=n

φn (y),

2

which combined with (3.7) implies the lemma.

Let Pt denote the semigroup defined on C(∇) generated by Aα,θ , and {1, gm : m ∈ N∞ } be the orthonormal basis of L2 (α, θ ) in Theorem 2.2. Then for f in C(∇), the following holds Pt f (x) = (f, 1) +

∞

e−λm t

(f, gm )gm (x),

m∈N ∞ : |m|=m

m=2

where (·,·) denotes the scalar product in L2 (α, θ ). Thus, if ∞

e−λm t

m=2

m∈N ∞ : |m|=m

2 1 gm ∈ L (α, θ ),

then Pt has a transition density p(t, x, y) = 1 +

∞

e−λm t qm (x, y)

m=2

with qm (x, y) =

m∈N ∞ : |m|=m

gm (x)gm (y),

x, y ∈ ∇.

(3.8)


409

As in [6], we can show that for any m 1, m

Wn = span φn : n ∈ {0} ∪ N∞ , | n| m ,

n=0

and Lemma 3.2 can be applied to the orthonormal basis {gm : m ∈ N∞ }, and for any m, m ∈ 0 ∪ N∞ with m = |m| |m | n,

pn (x, y)gm (x)gm (y) μα,θ (dx) μα,θ (dy) = ∇ ∇

n[m] δm m . (n + θ )m

(3.9)

Write pn (x, y) as a linear combination of {gm (x)gm (y): |m| ∨ |m | n}. It follows from (3.9) that the coefficient of gm (x)gm (y) is zero if m = m . Therefore, we get a linear expression of pn (x, y) in terms of {qm (x, y): m n}. Solving for qm (x, y) in terms of pn (x, y), yields that for any m 2, qm (x, y) =

m m 2m − 1 + θ (n + θ )(m−1) pn (x, y) (−1)m−n n m! n=0

where p0 (x, y) = 1. Taking the result in Lemma 3.1 into account, we obtain that for m 2, there exist C, d > 1 such that sup qm (x, y) C m mdm ,

(3.10)

x,y∈∇

which leads to (3.8) and the following upper bound for the transition density function. Theorem 3.3. The transition function of the process Xα,θ (t) has a density function p(t, x, y) with respect to μα,θ given by p(t, x, y) = 1 +

∞

e−λm t qm (x, y),

(3.11)

m=2

and there exists a constant c > 1 such that p(t, x, y) ct c(log t)/t ,

t > 0, x, y ∈ ∇.

(3.12)

Proof. It is not difficult to see that (3.11) follows from (3.10). Hence to prove the theorem it suffices to prove (3.12). Since supx,y p(t, x, y) is decreasing in t, we only need to consider the case that t ∈ (0, 1/2]. By (3.10), (3.11) and the fact that λm = 12 m(m − 1 + θ ), we have p(t, x, y) 1 +

∞ m=2

2−m exp Ψt (m) ,

(3.13)

410


where 1 Ψt (m) = − m(m − 1 + θ )t + m log(2C) + dm log m, 2

m 2.

Noting that d log m

mt 4d + d log − d, 4 t

we conclude that Ψt (m) −

m2 t c2 (log t −1 )2 + c1 m log t −1 , 4 t

t ∈ (0, 1/2]

holds for some constants c1 , c2 > 0 and all m > 0. The proof is then completed by combining this with (3.13). 2 4. Functional inequalities It is well known that the uniform upper bound of the transition density (i.e. the ultracontractivity of the semigroup) is corresponding to the super-log-Sobolev inequality (see e.g. [4]). In this section we aim to derive various functional inequalities using the heat kernel bound given in Theorem 3.3. Following the line of [19], explicit super-Poincaré inequality, super-log-Sobolev inequality and the F -Sobolev inequality will be obtained. Let (E , D(E )) be the Dirichlet form associated with the process Xα,θ introduced in Section 2. By Theorem 3.3, we have the following result on functional inequalities. Theorem 4.1. Let (E , D(E )) be defined above on L2 (μα,θ ). Then: (1) There exists a constant c > 0 such that the super-Poincaré inequality 2 μα,θ f 2 rE (f, f ) + c exp cr −1/2 log 1 + r −1 μα,θ |f | ,

r > 0, f ∈ D(E)

holds. (2) There exists a constant c > 0 such that the super-log-Sobolev inequality μα,θ f 2 log f 2 rE (f, f ) + cr −1 log2 1 + r −1 ,

r > 0, f ∈ D(E ), μα,θ (f 2 ) = 1

holds. In particular, there exists a constant C > 0 such that the log-Sobolev inequality μα,θ f 2 log f 2 CE (f, f ) + μα,θ f 2 log μα,θ f 2 ,

f ∈ D(E )

holds. (3) There exist two constants c1 , c2 > 0 such that the F -Sobolev inequality μα,θ f 2 F f 2 c1 E (f, f ) + c2 ,

f ∈ D(E ), μα,θ f 2 = 1


411

holds for F (r) :=

log(r + e) log log(4 + r)

2 ,

r > 0.

Proof. (1) Since the generator has a spectral gap λ2 > 0, the Poincaré inequality 1 μα,θ f 2 E (f, f ) + μα,θ (f )2 , λ2

f ∈ D(E )

(4.1)

holds. So, it suffices to prove the desired super-Poincaré inequality for r ∈ (0, 1/2). To this end, we make use of [19, Theorem 4.5], which implies that 2 μα,θ f 2 rE (f, f ) + μα,θ |f |

inf

sr, t>0

s Pt 1→∞ et/s−1 , t

r > 0, f ∈ D(E ),

(4.2)

where and in what follows · p→q stands for the operator norm from Lp (μα,θ ) to Lq (μα,θ ) for q, p 1. By Theorem 3.3 we have Pt 1→∞ ct c(log t)/t . So, for r ∈ (0, 1/2], s 1 Pt 1→∞ et/r−1 inf Pt 1→∞ et/r t>0 t sr, t>0 t c r c(log t)2 c exp c r −1/2 log r −1 inf exp + t>0 t t t inf

√ for some constant c > 0 by taking e.g. t = r log r −1 1 in the last step. Therefore, the desired inequality follows from (4.2). (2) Noting that the log-Sobolev inequality follows from (4.1) and the defective log-Sobolev inequality μα,θ f 2 log f 2 C1 E (f, f ) + C2 , f ∈ D(E ), μα,θ f 2 = 1 for some constants C1 , C2 > 0, it suffices to prove the desired super-log-Sobolev inequality. According to [4, Theorem 2.2.3], we have μα,θ f 2 log f 2 rE (f, f ) + 2 log Pr/2 2→∞ , μα,θ f 2 = 1. Since by the symmetry and the interpolation theorem one has Pr/2 22→∞ = Pr 1→∞ , the proof is completed by Theorem 3.3. (3) By (1) and [19, Theorem 1.2] with ε = 12 , the F -Sobolev inequality holds for 1 c − exp 2cr −1 log 2r −1 F (t) = sup . rt r>0 r It remains to note that F (t) c holds for some constant c > 0.

2

log(t + e) log log(4 + t)

2 ,

t >0

412


Finally, we remark that since functional inequalities included in Theorem 4.1 are stable under bounded perturbations of log-density, this result remains true if use eV (x) μα,θ (dx) V (x) μ α,θ (dx) ∇e

μVα,θ (dx) := and 1 EV (f, g) = 2

∞ ∇

xi (δij − xj )(∂i f ∂j g)(x) μVα,θ (dx)

i,j =1

to replace μα,θ and E respectively, where V is a bounded measurable function on ∇. Since V is bounded, (EV , D(E )) is again a symmetric Dirichlet form. In particular, taking V = σ φ2 for a constant σ , the associated generator restricted on D0 reads Aα,θ,σ = Aα,θ

∞ ∂ +σ xi xi − φ2 (x) . ∂xi i=1

This corresponds to the two-parameter diffusion process with a special selection studied in [9]. Acknowledgment We would like to thank the referee for careful reading and valuable comments on the first version of the paper. References [1] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in: K.D. Elworthy, S. Kusuoka, I. Shigekawa (Eds.), New Trends in Stochastic Analysis, World Scientific, Singapore, 1997. [2] D. Bakry, M. Emery, Hypercontractivité de semi-groupes de diffusion, C. R. Math. Acad. Sci. Paris 299 (1984) 775–778. [3] S.G. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton–Jacobi equations, J. Math. Pures Appl. 80 (2001) 669–696. [4] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. [5] S.N. Ethier, The infinitely-many-neutral-alleles diffusion model with ages, Adv. in Appl. Probab. 22 (1990) 1–24. [6] S.N. Ethier, Eigenstructure of the infinitely-many-neutral-alleles diffusion model, J. Appl. Probab. 29 (1992) 487– 498. [7] S.N. Ethier, R.C. Griffiths, The transition function of a Fleming–Viot process, Ann. Probab. 21 (1993) 1571–1590. [8] S.N. Ethier, T.G. Kurtz, The infinitely-many-neutral-alleles diffusion model, Adv. in Appl. Probab. 13 (1981) 429– 452. [9] S.N. Ethier, T.G. Kurtz, Coupling and ergodic theorems for Fleming–Viot processes, Ann. Probab. 26 (1998) 533– 561. [10] S. Feng, W. Sun, Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process, Probab. Theory Related Fields 148 (2010) 501–525. [11] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061–1083. [12] L. Gross, Logarithmic Sobolev inequalities and contractivity properties of semigroups, in: Dirichlet Forms, Varenna, 1992, in: Lecture Notes in Math., vol. 1563, Springer-Verlag, Berlin, 1993, pp. 54–88. [13] J.C.F. Kingman, Random discrete distributions, J. Roy. Statist. Soc. Ser. B 37 (1975) 1–22. [14] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, in: Séminaire de Probabilités, XXXIII, in: Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 120–216.


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[15] F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000) 361–400. [16] L. Petrov, A two-parameter family of infinite-dimensional diffusions in the Kingman simplex, Funktsional. Anal. i Prilozhen. 43 (2009) 45–66. [17] J. Pitman, Exchangeable and partially exchangeable random partitions, Probab. Theory Related Fields 102 (1995) 145–158. [18] W. Stannat, On the validity of the log-Sobolev inequality for symmetric Fleming–Viot operators, Ann. Probab. 28 (2000) 667–684. [19] F.-Y. Wang, Functional inequalities, semigroup properties and spectrum estimates, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000) 263–295. [20] F.-Y. Wang, Probability distance inequalities on Riemannian manifolds and path spaces, J. Funct. Anal. 206 (2004) 167–190.


Free product actions with relative property (T) and trivial outer automorphism groups Damien Gaboriau CNRS – Université de Lyon, ENS-Lyon, UMPA UMR 5669, 69364 Lyon cedex 7, France Received 5 February 2010; accepted 22 September 2010 Available online 20 October 2010 Communicated by S. Vaes

Abstract We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S. Popa (2006) [Pop06, Def. 4.1]. There are continuum many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives, in particular, the first examples of such actions for the free group on 2 generators. © 2010 Elsevier Inc. All rights reserved. Keywords: Free products; Relative property (T); Measured equivalence relations; Group measure space construction; Outer automorphism group

1. Introduction Several breakthroughs in von Neumann Algebras Theory and Orbit Equivalence have been made possible, during the last years, by the introduction by S. Popa of the notion of rigidity for pairs B ⊂ N of von Neumann algebras [Pop06]. This property is inspired by the relative property (T) of Kazhdan and is satisfied by the inclusion L∞ (T2 ) ⊂ L∞ (T2 ) SL(2, Z) associated with the standard action of SL(2, Z) on the 2-torus. It has proved to be extremely useful, more generally, for inclusions A ⊂ M(R) arising from standard countable probabilE-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.013

D. Gaboriau / Journal of Functional Analysis 260 (2011) 414–427

415

ity measure preserving (p.m.p.) equivalence relations R, where A is a Cartan subalgebra in the generalized crossed product or group-measure-space construction von Neumann algebra [MvN36,FM77b]. In case the pair A ⊂ M(R) is rigid, the relation is said to have the relative property (T). When combined with antagonistic properties like Haagerup property [Pop06] or (amalgamated) free product decomposition [IPP08], the rigid Cartan subalgebra were shown to be essentially unique, thus allowing orbit equivalence invariants (like the cost [Gab00], L2 -Betti numbers βn (R) [Gab02] or the fundamental group F (R) [Gab02, Cor. 5.7], . . .) to translate to von Neumann algebras invariants. This led to the solution of the long standing problem of finding von Neumann II1 factors with trivial fundamental group [Pop06]. While the class of groups that admit a free p.m.p. ergodic relative property (T) action is closed under certain algebraic constructions (like direct products, commensurability [Pop06], or free product with an arbitrary group [IPP08, Cor. 7.15]), the building blocks were very few and relying on some arithmetic actions [Pop06,Val05,Fer06] leaving open the general problem [Pop06, Prob. 5.10.2.]: “Characterize the countable discrete groups Γ0 that can act with relative property (T) on the probability space (X, μ), i.e., for which there exist free ergodic measure preserving actions σ on (X, μ) such that L∞ (X, μ) ⊂ L∞ (X, μ) σ Γ0 is a rigid embedding”. The purpose of this paper is first to prove that the class of groups that admit such a free p.m.p. ergodic relative property (T) action contains all non-amenable free products of groups. Moreover, we show that they have continuously many different such actions (Theorem 1.3) and we remove any arithmetic assumption on the individual actions of the building pieces. In fact, given the state of art, an arithmetic flavor remains hidden in the way the individual actions are mutually arranged. Outer automorphisms groups of standard equivalence relations are usually hard to calculate and there are only few special families of group actions for which one knows that Out(R) = {1}. The first examples are due to S. Gefter [Gef93,Gef96] and more examples were produced by A. Furman [Fur05]. They all take advantage of the setting of higher rank lattices. N. Monod and Y. Shalom [MS06] produced an uncountable family of non-orbit-equivalent actions with trivial outer automorphisms group, by left–right multiplication on the orthogonal groups of certain direct products of subgroups. Using rigidity results from [MS06], A. Ioana, J. Peterson and S. Popa [IPP08] gave the first shift-type examples. However, all these examples are concerned with very special kind of actions. And the free groups keep out of reach by these techniques. The first general result appeared recently in a paper by S. Popa and S. Vaes [PV10] and concerns free products Λ ∗ F∞ of any countable group Λ with the free group on infinitely many generators F∞ , with no condition at all on the Λ-action. This relies heavily on the existence of a relative property (T) action for F∞ . The second purpose of our paper is twofold: extend the result from F∞ to free groups on finitely many generators, and using the first part extend it to any non-elementary free products Γ1 ∗ Γ2 of groups (i.e. (|Γ1 | − 1)(|Γ2 | − 1) 2). This leads, in particular, to the first examples of actions of the free groups Fn , ∞ > n > 1, with trivial outer automorphisms group. ————oooOOOOOooo———— Before stating more precisely our results, let’s recall some definitions. We only consider measure preserving actions on the standard Borel space. First, from [Pop06, Def. 4.1] the definition of a rigid inclusion (or of relative rigidity of a subalgebra).

416


Definition 1.1. Let M be a factor of type II1 with normalized trace τ and let A ⊂ M be a von Neumann subalgebra. The inclusion A ⊂ M is called rigid if the following property holds: for every > 0, there exist a finite subset J ⊂ M and a δ > 0 such that whenever M HM is a Hilbert M–M-bimodule admitting a unit vector ξ with the properties • a · ξ − ξ · a < δ for all a ∈ J , • |a · ξ, ξ − τ (a)| < δ and |ξ · a, ξ − τ (a)| < δ for all a in the unit ball of M, then, there exists a vector ξ0 ∈ H satisfying ξ − ξ0 < and a · ξ0 = ξ0 · a for all a ∈ A. σ

Definition 1.2. (See [Pop06, Def. 5.10.1].) A free p.m.p. ergodic action Γ (X, μ) (respectively a countable standard p.m.p. equivalence relation R on (X, μ)) is said to have the relative property (T) if the inclusion of the Cartan subalgebra L∞ (X, μ) is rigid in the (generalized) crossed-product L∞ (X, μ) ⊂ L∞ (X, μ) σ Γ (resp. L∞ (X, μ) ⊂ M(R)). One could say that the equivalence relation has “the property (T) relative to the space (X, μ)”. Observe that A. Ioana [Ioa07, Th. 4.3] exhibited for every non-amenable group, some actions σ satisfying a weak form of relative rigidity, namely for which there exists a diffuse Q ⊂ L∞ (X, μ) ⊂ M(Rσ ) such that Q is relatively rigid in M(Rσ ) and has relative commutant contained in L∞ (X, μ). Recall the following weaker and weaker notions of equivalence for p.m.p. actions or standard equivalence relations R, S on (X, μ): α

β

Two actions Γ (X, μ) and Λ (X, μ) are Conjugate α

Conj

β

Γ (X, μ) ∼ Λ (X, μ)

(1)

if there is a group isomorphism h : Γ → Λ and a p.m.p. isomorphism of the space f : X → X that conjugate the actions ∀γ ∈ Γ (almost every) x ∈ X: f (α(γ )(x)) = β(h(γ ))(f (x)). Two p.m.p. standard equivalence relations R, S are Orbit Equivalent OE

R∼S

(2)

if there is a p.m.p. isomorphism of the space that sends classes to classes, or equivalently [FM77b] if the associated pairs are isomorphic ∞ L (X, μ) ⊂ M(R) L∞ (X, μ) ⊂ M(S) .

(3)

This makes the relative property (T) an orbit equivalence invariant. The standard equivalence relations are von Neumann Equivalent if solely the generalized crossed products are isomorphic M(R) M(S).

(4)


417

They are von Neumann Stably Equivalent if (they are ergodic and) the generalized crossed product factors are stably isomorphic for some amplification r ∈ R∗+ M(R) M(S)r .

(5)

We obtained in [GP05] that the non-cyclic free groups admit continuously many relative property (T) orbit inequivalent, and even von Neumann stably inequivalent, free ergodic actions. We show that free products admit relative property (T) free actions and that we have a full freedom for the conjugacy classes of the restrictions of the action to the free factors. In what follows, a free product decomposition Γ = ∗i∈I Γi is called non-elementary if (|I | − 1) i∈I (|Γi | − 1) 2, i.e. there are at least 2 free factors, none of them is ∼ {1} and if |I | = 2, then one of the Γi has at least 3 elements. Theorem 1.3. Every non-elementary free product Γ = ∗i∈I Γi admits continuum many von Neumann stably inequivalent relative property (T) free ergodic p.m.p. actions, whose restriction to each factor is conjugate with any (non-necessarily ergodic) prescribed free action. σi

More precisely, let Γi (X, μ) be an at most countable collection of p.m.p. (non-necessarily ergodic) free actions of countable groups Γi on the standard probability space. There exist continuum many von Neumann inequivalent free ergodic actions (αt )t∈T of the free product Γ = ∗i∈I Γi that have relative property (T), and such that for every t ∈ T and i ∈ I , the restriction αt |Γi of αt to Γi is conjugate with σi αt |Γi Conj σi Γi (X, μ) ∼ Γi (X, μ) .

(6)

We introduced in [Gab00] in connection with cost, the notion of freely independent equivalence relations and of free decomposition of an equivalence relation (see [Alv08] for a geometric approach): S = ∗ Si . i∈I

(7)

The following, essentially due to A. Törnquist [Tör06], see also [IPP08, Prop. 7.3], states that countable standard equivalence relations may be put in general position: Theorem 1.4 (Törnquist). Let (Si )i∈I be a countable collection of standard p.m.p. equivalence relations on (X, μ), then there exists an equivalence relation S on (X, μ) generated by a family OE

of freely independent subrelations Si such that Si ∼ Si , ∀i ∈ I . We obtain in fact the following more general form of Theorem 1.3 involving equivalence relations instead of free actions. Recall that a p.m.p. standard equivalence relation is called aperiodic if almost all its classes are infinite. Theorem 1.5. Let (Si )i∈I be a countable collection (with |I | 2) of p.m.p. standard countable aperiodic equivalence relations on the standard non-atomic probability space (X, μ). Then there exist continuum many von Neumann stably inequivalent relative property (T) ergodic p.m.p.

418


equivalence relations on (X, μ) generated by a freely independent family of subrelations Si OE

such that for every i ∈ I , Si ∼ Si . More precisely, there exists a strictly increasing continuum of ergodic equivalence relations St , t ∈ (0, 1] such that OE

1. for every t ∈ (0, 1], we have a free decomposition St = ∗i∈I Si,t such that Si,t ∼ Si for all i ∈ I, 2. for every t ∈ (0, 1], lims→t, s 1 is called the fundamental unit of O by O+ Dirichlet’s unit theorem. We refer the reader to [15] for details. The proof of Theorem 3.14 of [14] implies the following proposition. Proposition 3.11. Let λ be a fundamental unit of an order of a real quadratic field or a cubic field with one real embedding. Then there exists a simple AF-algebra A with unique trace such that F (A) = Gλ . × generated by Note that if p is a prime number and n 2, then the subgroup Gλ of R+ n λ = p cannot be the positive inner multiplier group IM + (E) for any additive subgroup E of R containing 1. In fact, on the contrary, suppose that Gλ = IM + (E) for some E. Then there exists

434

N. Nawata, Y. Watatani / Journal of Functional Analysis 260 (2011) 428–435

× a unital subring R of R such that Gλ = R+ by Lemma 3.6 of [14]. Then

This contradicts that

1 p

1 p

=

1 λ

+ ··· +

1 λ

× ∈ R+ .

∈ / Gλ . However, we have another construction.

Example 3.12. For λ = 32 = 9, Matui shows us the following example: Let A be an AF-algebra such that

b a, b, c ∈ Z, b ≡ c mod 8 , , c ∈ R × Z K0 (A) = 9a

b b K0 (A)+ = , c ∈ K0 (A): a > 0 ∪ (0, 0) and [1A ]0 = (1, 1). a 9 9 Then F (A) = G9 := 9n ∈ R× + n∈Z . Moreover τ∗ : K0 (A) → τ∗ (K0 (A)) is not an order isomorphism and F (A) = IM + (τ∗ (K0 (A))). Furthermore Katsura suggests us the following examples: Let λ = p n for a prime number p and a natural number n 2. Then there exists a simple AF-algebra A with unique trace such that F (A) = Gλ . First consider the case that λ 8. Define

n b E= , c ∈ R × Z a, b, c ∈ Z, b ≡ c mod p − 1 , p na

b b E+ = , c ∈ E: > 0 ∪ (0, 0) and [u]0 = (1, 1). p na p na Then there exists a simple AF-algebra A such that (K0 (A), K0 (A)+ , [1A ]0 ) = (E, E+ , u) by [4]. The classification theorem of [5] and some computation yield that F (A) = Gλ . Next consider the case that λ = 22 = 4. Let

b , c ∈ R × Z a, b, c ∈ Z, b ≡ c mod 5 , E= 16a

b b E+ = , c ∈ E: > 0 ∪ (0, 0) and [u]0 = (1, 1). a a 16 16 Consider a simple AF-algebra A such that (K0 (A), K0 (A)+ , [1A ]0 ) = (E, E+ , u). Then F (A) = G4 . References [1] B. Blackadar, Weak expectations and nuclear C ∗ -algebras, Indiana Univ. Math. J. 27 (1978) 1021–1026. [2] L.G. Brown, P. Green, M.A. Rieffel, Stable isomorphism and strong Morita equivalence of C ∗ -algebras, Pacific J. Math. 71 (1977) 349–363. [3] A. Connes, A factor of type II 1 with countable fundamental group, J. Operator Theory 4 (1980) 151–153. [4] E. Effros, D. Handelman, C.L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (2) (1980) 385–407.

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435

[5] G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976) 29–44. [6] M. Frank, D. Larson, A module frame concept for Hilbert C ∗ -modules, in: Contemp. Math., vol. 247, 1999, pp. 207– 233. [7] T. Kajiwara, Y. Watatani, Jones index theory by Hilbert C ∗ -bimodules and K-theory, Trans. Amer. Math. Soc. 352 (2000) 3429–3472. [8] K. Kodaka, Full projections, equivalence bimodules and automorphisms of stable algebras of unital C ∗ -algebras, J. Operator Theory 37 (1997) 357–369. [9] K. Kodaka, Picard groups of irrational rotation C ∗ -algebras, J. London Math. Soc. (2) 56 (1997) 179–188. [10] K. Kodaka, Projections inducing automorphisms of stable UHF-algebras, Glasg. Math. J. 41 (3) (1999) 345–354. [11] E.C. Lance, Hilbert C ∗ -modules, London Math. Soc. Lecture Note Ser., vol. 210, Cambridge University Press, Cambridge, 1995. [12] V.M. Manuilov, E.V. Troitsky, Hilbert C ∗ -Modules, Transl. Math. Monogr., vol. 226, Amer. Math. Soc., Providence, RI, 2005. [13] F. Murray, J. von Neumann, On rings of operators IV, Ann. of Math. 44 (1943) 716–808. [14] N. Nawata, Y. Watatani, Fundamental group of simple C ∗ -algebras with unique trace, Adv. Math. 225 (2010) 307– 318. [15] J. Neukirch, Algebraic Number Theory, Grundlehren Math. Wiss., vol. 322, Springer-Verlag, New York, 1999. [16] N.C. Phillips, A simple separable C ∗ -algebra not isomorphic to its opposite algebra, Proc. Amer. Math. Soc. 132 (10) (2004) 2997–3005. [17] S. Popa, Strong rigidity of II 1 factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006) 369–408. [18] S. Popa, S. Vaes, Actions of F∞ whose II 1 factors and orbit equivalence relations have prescribed fundamental group, J. Amer. Math. Soc. 23 (2010) 383–403. [19] F. Radulescu, The fundamental group of the von Neumann algebra of a free group with infinitely many generators is R∗+ , J. Amer. Math. Soc. 5 (1992) 517–532. [20] I. Raeburn, D.P. Williams, Morita Equivalence and Continuous-Trace C ∗ -Algebras, Math. Surveys Monogr., vol. 60, Amer. Math. Soc., Providence, RI, 1998. [21] M.A. Rieffel, Morita equivalence for operator algebras, in: Operator Algebras and Applications, Part I, Kingston, Ont., 1980, in: Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, RI, 1982, pp. 285–298. [22] D. Voiculescu, Circular and semicircular systems and free product factors, in: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, in: Progr. Math., vol. 92, Birkhäuser, Boston, 1990, pp. 45–60. [23] Y. Watatani, Index for C ∗ -subalgebras, Mem. Amer. Math. Soc. 424 (1990).


The IVP for the Benjamin–Ono equation in weighted Sobolev spaces Germán Fonseca a , Gustavo Ponce b,∗ a Departamento de Matemáticas, Universidad Nacional de Colombia, Bogota, Colombia b Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

Received 29 April 2010; accepted 19 September 2010 Available online 29 September 2010 Communicated by I. Rodnianski

Abstract We study the initial value problem associated to the Benjamin–Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces Zs,r = H s (R) ∩ L2 (|x|2r dx), s ∈ R, s 1 and s r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp. © 2010 Elsevier Inc. All rights reserved. Keywords: Benjamin–Ono equation; Weighted Sobolev spaces

1. Introduction This work is concerned with the initial value problem (IVP) for the Benjamin–Ono (BO) equation

∂t u + H∂x2 u + u∂x u = 0, u(x, 0) = u0 (x),

t, x ∈ R,

where H denotes the Hilbert transform * Corresponding author.

E-mail addresses: [email protected] (G. Fonseca), [email protected] (G. Ponce). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.010

(1.1)

G. Fonseca, G. Ponce / Journal of Functional Analysis 260 (2011) 436–459

1 1 p.v. ∗ f (x) π x ∨ 1 f (x − y) = lim dy = −i sgn(ξ )fˆ(ξ ) (x). π ↓0 y

437

Hf (x) =

(1.2)

|y|

The BO equation was deduced by Benjamin [3] and Ono [28] as a model for long internal gravity waves in deep stratified fluids. It was also shown that it is a completely integrable system (see [2,6] and references therein). Several works have been devoted to the problem of finding the minimal regularity, measured in the Sobolev scale H s (R) = (1 − ∂x2 )−s/2 L2 (R), which guarantees that the IVP (1.1) is locally or globally wellposed (LWP and GWP, resp.), i.e. existence and uniqueness hold in a space embedded in C([0, T ] : H s (R)), with the map data-solution from H s (R) to C([0, T ] : H s (R)) being locally continuous. Let us recall them: in [31] s > 3 was proven, in [1] and [16] s > 3/2, in [30] s 3/2, in [23] s > 5/4, in [20] s > 9/8, in [34] s 1, in [4] s > 1/4, and finally in [15] s 0 was established. Real valued solutions of the IVP (1.1) satisfy infinitely many conservation laws (time invariant quantities), the first three are the following: ∞ I1 (u) =

∞ u(x, t) dx,

I2 (u) =

−∞

u2 (x, t) dx,

−∞

∞ I3 (u) = −∞

1/2 2 u3 Dx u − (x, t) dx, 3

(1.3)

where Dx = H∂x . Roughly, for k 2 the k-conservation law Ik provides an a priori estimate of the L2 -norm of (k−2)/2 u(t)2 . This allows one to deduce the derivatives of order (k − 2)/2 of the solution, i.e. Dx GWP from LWP results. For existence of solutions with non-decaying at infinity initial data we refer to [18] and [11]. In the BO equation the dispersive effect is described by a non-local operator and is significantly weaker than that exhibited by the Korteweg–de Vries (KdV) equation ∂t u + ∂x3 u + u∂x u = 0. Indeed, it was proven in [25] that for any s ∈ R the map data-solution from H s (R) to C([0, T ] : H s (R)) is not locally C 2 , and in [24] that it is not locally uniformly continuous. This implies that no LWP results can be obtained by an argument based only on a contraction method. This is certainly not the case of the KdV (see [22]). Our interest here is to study real valued solutions of the IVP (1.1) in weighted Sobolev spaces Zs,r = H s (R) ∩ L2 |x|2r dx ,

s, r ∈ R,

(1.4)

and decay properties of solutions of Eq. (1.1). In this direction R. Iorio [16] proved the following results:

438


Theorem A. (See [16].) (i) The IVP (1.1) is GWP in Z2,2 . (ii) If uˆ 0 (0) = 0, then the IVP (1.1) is GWP in Z˙ 3,3 . (iii) If u(x, t) is a solution of the IVP (1.1) such that u ∈ C([0, T ] : Z4,4 ) for arbitrary T > 0, then u(x, t) ≡ 0. Above we have introduced the notation

Z˙ s,r = f ∈ H s (R) ∩ L2 |x|2r dx : fˆ(0) = 0 ,

s, r ∈ R.

(1.5)

Notice that the conservation law I1 in (1.3) tells us that the property uˆ 0 (0) = 0 is preserved by the solution flow. We observe that the linear part of the equation in (1.1) L = ∂t + H∂x2 commutes with the operator Γ = x − 2tH∂x , i.e. [L; Γ ] = LΓ − Γ L = 0. In fact, one can deduce (see [16]) that for a solution v(x, t) of the associated linear problem ∨ 2 v(x, t) = U (t)v0 (x) = e−it H∂x v0 (x) = e−itξ |ξ | vˆ0 (x),

(1.6)

to satisfy that v(·, t) ∈ L2 (|x|2k dx), t ∈ [0, T ], one needs v0 ∈ Zk,k , k ∈ Z+ for k = 1, 2 and ∞ x j v0 (x) dx = 0,

j = 0, 1, . . . , k − 3, if k 3.

−∞

Also one notices that the traveling wave φc (x + t), c > 0 for the BO equation φ(x) =

−4 , 1 + x2

φc (x + t) = cφ c(x + ct) ,

has very mild decay at infinity. In this case, the traveling wave is negative and travels to the left. To get a positive traveling wave moving to the right one needs to consider the equation ∂t v − H∂x2 v + v∂x v = 0,

t, x ∈ R,

(1.7)

and observes that if u(x, t) is a solution of (1.1) then v(x, t) = −u(x, −t), satisfies Eq. (1.7). In particular, (1.7) has the traveling wave solution v(x, t) = ψc (x − t) = cψ c(x − ct) , c > 0 with ψ(x) = −φ(x). In [17] R. Iorio strengthened his unique continuation result in Z4,4 found in [16] (Theorem A, part (iii)) by proving:


439

Theorem B. (See [17].) Let u ∈ C([0, T ] : H 2 (R)) be a solution of the IVP (1.1). If there exist three different times t1 , t2 , t3 ∈ [0, T ] such that u(·, tj ) ∈ Z4,4 ,

j = 1, 2, 3, then u(x, t) ≡ 0.

(1.8)

Our goal in this work is to extend the results in Theorem A and Theorem B from integer values to the continuum optimal range of indices (s, r). Our main results are the following: Theorem 1. (i) Let s 1, r ∈ [0, s], and r < 5/2. If u0 ∈ Zs,r , then the solution u(x, t) of the IVP (1.1) satisfies that u ∈ C([0, ∞) : Zs,r ). (ii) For s > 9/8 (s 3/2), r ∈ [0, s], and r < 5/2 the IVP (1.1) is LWP (GWP resp.) in Zs,r . (iii) If r ∈ [5/2, 7/2) and r s, then the IVP (1.1) is GWP in Z˙ s,r . Theorem 2. Let u ∈ C([0, T ] : Z2,2 ) be a solution of the IVP (1.1). If there exist two different times t1 , t2 ∈ [0, T ] such that u(·, tj ) ∈ Z5/2,5/2 ,

j = 1, 2, then uˆ 0 (0) = 0,

so u(·, t) ∈ Z˙ 5/2,5/2 .

(1.9)

Theorem 3. Let u ∈ C([0, T ] : Z˙ 3,3 ) be a solution of the IVP (1.1). If there exist three different times t1 , t2 , t3 ∈ [0, T ] such that u(·, tj ) ∈ Z7/2,7/2 ,

j = 1, 2, 3, then u(x, t) ≡ 0.

(1.10)

Remarks. (a) Theorem 2 shows that the condition uˆ 0 (0) = 0 is necessary to have persistence property of the solution in Zs,5/2 , with s 5/2, so in that regard Theorem 1 parts (i)–(ii) are sharp. Theorem 3 affirms that there is an upper limit of the spacial L2 -decay rate of the solution (i.e. |x|7/2 u(·, t) ∈ / L∞ ([0, T ] : L2 (R)), for any T > 0) regardless of the decay and regularity of the non-zero initial data u0 . In particular, Theorem 3 shows that Theorem 1 part (iii) is sharp. (b) In part (ii) of Theorem 1 we shall use that in that case the solution u(x, t) satisfies ∂x u ∈ L1 [0, T ] : L∞ (R) (see [20,23], and [30]) to establish that the map data-solution is locally continuous from Zs,r into C([0, T ] : Zs,r ). (c) The condition in Theorem 3 involving three times seems to be technical and may be reduced to two different times as that in Theorem 2. We recall that unique continuation principles for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation have been established in [9] and [10] resp. under assumptions on the solutions at two different times. Following the idea in [17] one finds from Eq. (1.1) that d dt

∞ −∞

2 1 1 xu(x, t) dx = u(t)2 = u0 22 , 2 2

(1.11)

440


so the first momentum of a non-null solution of the BO equation is strictly increasing. On the other hand, using the integral equation version of the BO equation from the hypotheses one can deduce that the first momentum must vanish somewhere in the time intervals (t1 , t2 ) and (t2 , t3 ). This implies that u(x, t) ≡ 0. (d) We recall that if for a solution u ∈ C([0, T ] : H s (R)) of (1.1) one has that ∃t0 ∈ [0, T ] such that u(x, t0 ) ∈ H s (R), s > s, then u ∈ C([0, T ] : H s (R)). So we shall mainly consider the most interesting case s = r in (1.4). (e) Consider the IVP for generalized Benjamin–Ono (gBO) equation

∂t u + H∂x2 u ± uk ∂x u = 0, u(x, 0) = u0 (x),

t, x ∈ R, k ∈ Z+ ,

(1.12)

with u0 a real valued function. In this case the best LWP available results are: for k = 2, s 1/2 (see [21]), for k = 3, s > 1/3 (see [35]), and for k 4, s 1/2 − 1/k (see [35]). So for any power k = 1, 2, . . . with focussing (+) or defocussing (−) non-linearity the IVP (1.12) is LWP in H 1 (R). So the local results in Theorems 1 and 2 and their proofs extend to the IVP (1.12) with possible different values s = s(k) for the minimal regularity required. This is also the case for Theorem 3 when the power k in (1.12) is odd in the focusing and defocusing regime. (f) In [19] the number 7/2 was mentioned as a possible threshold in the spaces (1.4). The proof of Theorem 1 is based on weighted energy estimates and involves several inequalities for the Hilbert transform H. Among them we shall use the Ap condition introduced in [26] (see Definition 1). It was proven in [14] that this is a necessary and sufficient condition for the Hilbert transform H to be bounded in Lp (w(x) dx) (see [14]), i.e. w ∈ Ap , 1 < p < ∞ if and only if ∞

1/p |Hf | w(x) dx p

−∞

c

∗

1/p

∞ |f | w(x) dx p

(1.13)

−∞

(see Theorem 4). In order to justify some of our arguments in the proofs we need some further continuity properties of the Hilbert transform. More precisely, our proof requires the constant c∗ in (1.13) to depend only on c(w) the constant describing the Ap condition (see (2.2)) and on p. In [29] precise bounds for the constant c∗ in (2.3) were given which are sharp in the case p = 2 and sufficient for our purpose (see Theorem 5). It will be essential in our arguments that some commutator operators involving the Hilbert transform H are of “order zero”. More precisely, we shall use the following estimate: ∀p ∈ (1, ∞), l, m ∈ Z+ ∪ {0}, l + m 1 ∃c = c(p; l; m) > 0 such that l ∂ [H; a]∂ m f c∂ l+m a f p . x x x p ∞

(1.14)

In the case l + m = 1, (1.14) is Calderón’s first commutator estimate [5]. In the case l + m 2, (1.14) was proved in [7]. The rest of this paper is organized as follows: Section 2 contains some preliminary estimates to be utilized in the coming sections. Theorem 1 will be proven in Section 3. Finally, the proofs of Theorem 2 and Theorem 3 will be given in Sections 4 and 5, respectively.


441

2. Preliminary estimates We shall use the following notations: ∞

1/p p f (x) dx f p = , ∞

f ∞ = sup f (x),

1 p < ∞,

x∈R

s/2 f s,2 = 1 − ∂x2 f 2 ,

s ∈ R.

(2.1)

Let us first recall the definition of the Ap condition. We shall restrict here to the cases p ∈ (1, ∞) and the 1-dimensional case R (see [26]). Definition 1. A non-negative function w ∈ L1loc (R) satisfies the Ap inequality with 1 < p < ∞ if p−1 1 1 w w 1−p = c(w) < ∞, (2.2) sup |Q| Q interval |Q| Q

Q

where 1/p + 1/p = 1. Theorem 4. (See [14].) The condition (2.2) is necessary and sufficient for the boundedness of the Hilbert transform H in Lp (w(x) dx), i.e.

1/p

∞ |Hf | w(x) dx p

c

−∞

∗

1/p

∞ |f | w(x) dx p

.

(2.3)

−∞

In the case p = 2, a previous characterization of w in (2.3) was found in [13] (for further references and comments we refer to [8,12], and [33]). However, even though we will be mainly concerned with the case p = 2, the characterization (2.3) will be the one used in our proof. In particular, one has that in R |x|α ∈ Ap

⇐⇒

α ∈ (−1, p − 1).

(2.4)

In order to justify some of the arguments in the proof of Theorem 1 we need some further continuity properties of the Hilbert transform. More precisely, our proof requires the constant c∗ in (2.3) to depend only on c(w) in (2.2) and on p (in fact, this is only needed for the case p = 2). Theorem 5. (See [29].) For p ∈ [2, ∞) the inequality (2.3) holds with c∗ c(p)c(w), with c(p) depending only on p and c(w) as in (2.2). Moreover, for p = 2 this estimate is sharp. Next, we define the truncated weights wN (x) using the notation x = (1 + x 2 )1/2 as x if |x| N, wN (x) = 2N if |x| 3N, (x) 1 for all x 0. wN (x) are smooth and non-decreasing in |x| with wN

(2.5)

442


θ (x) satisfies the A inequality (2.2). Proposition 1. For any θ ∈ (−1, 1) and any N ∈ Z+ , wN 2 θ (x) dx) with a constant depending on θ Moreover, the Hilbert transform H is bounded in L2 (wN but independent of N ∈ Z+ .

The proof of Proposition 1 follows by combining the fact that for a fixed θ ∈ (−1, 1) the family θ (x), N ∈ Z+ satisfies the A inequality in (2.2) with a constant c independent of N , of weights wN 2 and Theorem 5. Next, we have the following generalization of Calderón commutator estimates [5] founded in [7] and already commented in the Introduction: Theorem 6. For any p ∈ (1, ∞) and l, m ∈ Z+ ∪ {0}, l + m 1 there exists c = c(p; l; m) > 0 such that l ∂ [H; a]∂ m f c∂ l+m a f p . x x x p ∞

(2.6)

We shall also use the pointwise identities [H; x]∂x f = H; x 2 ∂x2 f = 0, and more generally [H; x]f = 0 if and only if

f dx = 0.

We recall the following characterization of the Ls (Rn ) = (1 − )−s/2 Lp (Rn ) spaces given in [32]. p

p

Theorem 7. Let b ∈ (0, 1) and 2n/(n + 2b) < p < ∞. Then f ∈ Lb (Rn ) if and only if f ∈ Lp R n , 1/2 |f (x) − f (y)|2 b dy ∈ Lp R n , (b) D f (x) = n+2b |x − y| (a)

(2.7)

Rn

with f b,p ≡ (1 − )b/2 f p = J b f p f p + D b f p f p + Db f p .

(2.8)

Above we have used the notation: for s ∈ R D s = (−)s/2

with D s = (H∂x )s ,

if n = 1.

For the proof of this theorem we refer the reader to [32]. One sees that from (2.7) for p = 2 and b ∈ (0, 1) one has b D (f g) f Db g + gDb f . 2 2 2

(2.9)


443

We shall use this estimate in the proof of Theorem 3. As applications of Theorem 7 we have the following estimate: Proposition 2. Let b ∈ (0, 1). For any t > 0 Db e−itx|x| c |t|b/2 + |t|b |x|b .

(2.10)

For the proof of Proposition 2 we refer to [27]. As a further direct consequence of Theorem 7 we deduce the following result to be used in the proof of Theorem 3. Proposition 3. Let p ∈ (1, ∞). If f ∈ Lp (R) such that there exists x0 ∈ R for which f (x0+ ), p f (x0− ) are defined and f (x0+ ) = f (x0− ), then for any δ > 0, D1/p f ∈ / Lloc (B(x0 , δ)) and consep quently f ∈ / L1/p (R). Also as consequence of the estimate (2.9) one has the following interpolation inequality. Lemma 1. Let a, b > 0. Assume that J a f = (1 − )a/2 f ∈ L2 (R) and xb f = (1 + |x|2 )b/2 f ∈ L2 (R). Then for any θ ∈ (0, 1) 1−θ θ θa (1−θ)b J x f 2 cxb f 2 J a f 2 .

(2.11)

Moreover, the inequality (2.11) is still valid with wN (x) in (2.5) instead of x with a constant c independent of N . Proof. It will suffice to consider the case: a = 1 + α, α ∈ (0, 1). We denote by ρ(x) a function equal to x or equal to wN (x) as in (2.5) and consider the function F (z) = e

(z2 −1)

∞

J az ρ b(1−z) f (x) g(x) dx

−∞

with g ∈ L2 (Rn ) with g2 = 1, which is continuous in {z = η + iy: 0 η 1} and analytic in its interior. Moreover, F (0 + iy) e−(y 2 +1) ρ b f , 2 and since |ρ /ρ| + |ρ /ρ| c (independent of N ) combining (2.7) and (2.9) one has F (1 + iy) e−y 2 J a ρ iby f e−y 2 ρ iby f + D α ∂x ρ iby f 2 2 2 α iby α iby−1 −y 2 f 2 + D ρ ∂x f 2 + |by| D ρ ρf 2 e 2 e−y f 2 + Dα ρ iby ∂x f 2 + |by|Dα ρ iby−1 ρ f 2 2 e−y f 2 + Dα ρ iby ∂x f 2 + ρ iby Dα ∂x f 2 + |by|Dα ρ iby−1 ρ f 2 + |by| ρ iby−1 ρ Dα f 2

444


2 cα e−y 1 + |yb|2 f 2 + Dα f 2 + ∂x f 2 + Dα ∂x f 2 2 2 cα e−y 1 + |yb|2 J 1+α f 2 = cα e−y 1 + |yb|2 J a f 2 , using that for α ∈ (0, 1) α D h

∞

cα h∞ + ∂x h∞ . 2

Therefore, the three lines theorem yields the desired result. We shall also employ the following simple estimate. Proposition 4. If f ∈ L2 (R) and φ ∈ H 1 (R), then

1/2 D ; φ f cφ1,2 f 2 . 2

(2.12)

Finally, to complete this section we recall the result obtained in [30] concerning regularity properties of the solutions of the IVP (1.1) with data u0 ∈ H s (R), s 3/2. This will be used in the proof of Theorem 3. Theorem 8. For any u0 ∈ H s (R) with s 3/2 the IVP (1.1) has a unique global solution u ∈ C([0, T ] : H s (R)) such that for any T > 0 J s+1/2 u ∈ lk∞ L2 [k, k + 1] × [0, T ] ,

J u ∈ lk2 L∞ [k, k + 1] × [0, T ]

and J s−3/2 ∂x u ∈ L4 [0, T ] : L∞ (R) . 3. Proof of Theorem 1 We consider several cases: Case 1: s = 1 and r = θ ∈ (0, 1]. Part (i) in Theorem 1. 2θ u (see (2.5)) with 0 < θ 1 and integrate on R We multiply the differential equation by wN to obtain 1 d 2 dt

θ 2 wN u dx +

θ θ wN H∂x2 uwN u dx +

2θ 2 wN u ∂x u dx = 0.

To handle the second term on the left hand side (l.h.s.) of (3.1) we write θ θ 2 θ wN H∂x2 u = wN ; H ∂x2 u + H wN ∂x u θ θ θ = A1 + H∂x2 wN u − 2H ∂x wN ∂x u − H ∂x2 wN u = A1 + A2 + A3 + A4 .

(3.1)


445

We observe that by Theorem 6 and our assumption on θ ∈ (0, 1] the terms A1 , A4 are bounded by the L2 -norm of the solution u and A3 is bounded by the H 1 -norm of the solution with constants independent of N , thus they are bounded uniformly on N ∈ Z+ by M1 = sup u(t)1,2 . t∈[0,T ]

We insert the term A2 in (3.1) and use integration by parts, to get that θ θ H∂x2 wN u wN u dx = 0. Finally, using integration by parts, we bound the nonlinear term (the third term on the l.h.s.) in (3.1) as w 2θ u2 ∂x u dx cu∞ u2 w θ u cu2 w θ u . (3.2) 1,2 N N N 2 2 Inserting this information in (3.1) we get d w θ u(t) cM, 2 dt N

with c independent of N,

which tells us that θ sup wN u(t)2 cxθ u0 2 eT M ,

t∈[0,T ]

with c independent of N,

which yields the result u ∈ L∞ ([0, T ] : L2 (|x|2θ )) for any T > 0. To see that u ∈ C([0, T ] : L2 (|x|2θ )) one considers the sequence θ wN u N ∈Z+ ⊆ C [0, T ] : L2 (R) , and reapply the above argument to find that it is a Cauchy sequence. Finally, we point out that the use of the differential equation in (1.1) can be justified by the locally continuous dependence of the solution upon the data from H s (R) to C([0, T ] : H s (R)). Case 2: s ∈ (1, 2] and r = s. Part (i) in Theorem 1. 2+2θ We multiply the differential equation by wN u (see (2.5)) with 0 θ 1 and integrate on R to obtain 1+θ 2 1 d 1+θ 1+θ 2+2θ 2 2 u ∂x u dx = 0. (3.3) wN u dx + wN H∂x uwN u dx + wN 2 dt To control the second term on the l.h.s. of (3.3) we write 1+θ 2 1+θ 2 1+θ wN H∂x2 u = wN ; H ∂x u + H wN ∂x u 1+θ 1+θ 1+θ 2 ∂x u − H ∂x2 wN u = B1 + H∂x wN u − 2H ∂x wN = B1 + B2 + B3 + B4 .

446


We observe that by Theorem 6 and our assumption θ ∈ (0, 1] the terms B1 , B4 are bounded by the L2 -norm of the solution. Inserting the term B2 in (3.3) and using integration by parts one 1+θ ∂x u). Since finds that its contribution is null. So it remains to control B3 = −2H(∂x wN ∂x w 1+θ = (1 + θ )w θ ∂x wN cw θ , N N N

c independent of N,

one has θ θ θ θ B3 2 cwN ∂x u2 c∂x wN u 2 + c∂x wN u 2 c∂x wN u 2 + cu2 .

(3.4)

Then by the interpolation inequality in (2.11) it follows that θ ∂x w u J w θ u cw 1+θ uθ/(1+θ) J 1+θ u1/(1+θ) , N N N 2 2 2 2

(3.5)

with a constant c independent of N . So by Young’s inequality in (3.5) and (3.4) the term B3 is appropriately bounded. Finally, for the last term on the l.h.s. of (3.3) we write w 2+2θ u2 ∂x u dx cu∞ w 1+θ u2 cu1,2 w 1+θ u2 , N N N 2 2

(3.6)

with c independent of N . So inserting the above information in (3.3) we obtain the result. Case 3: s ∈ (9/8, 2] and r = s. Part (ii) in Theorem 1. In this case it remains to establish the continuous dependence of the solution C([0, T ] : Zs,r ) upon the data in Zs,r . We are considering the most interesting case s = r ∈ (9/8, 2]. Suppose that u, v ∈ C([0, T ] : Zs,s ) are two solutions of the BO equation in (1.1) corresponding to data u0 , v0 respectively. Hence, ∂t (u − v) + H∂x (u − v) + ∂x u(u − v) + v∂x (u − v) = 0.

(3.7)

We will reapply the argument used in the previous case. However, we notice that the nonlinear term in (3.7) is different than that in (3.3). So we recall the result in [20] which affirms that for s > 9/8 ∂x u, ∂x v ∈ L1 [0, T ] : L∞ x (R) ,

(3.8)

and use integration by parts to obtain that w 2+2θ ∂x u(u − v)2 + v(u − v)∂x (u − v) dx N 1+θ 2 c ∂x u(t)∞ + ∂x v(t)∞ + v(t)∞ wN (u − v)2 .

(3.9)

Hence, combining the argument in the previous section, the estimates (3.9) and (3.8), and the continuous dependence of the solution in C([0, T ] : H s (R)) upon the data in H s (R) the desired result follows.


447

Case 4: s = r ∈ (2, 5/2). Part (ii) in Theorem 1. We recall that from the previous cases we know the result for s r ∈ (0, 2]. Also we shall 2+2θ write r = 2 + θ , θ ∈ (0, 1/2), and we multiply the differential equation by x 2 wN u (see (2.5)) and integrate on R to obtain 1 d 2 dt

1+θ 2 wN xu dx +

1+θ 1+θ wN xH∂x2 uwN xu dx +

2+2θ 2 x 2 wN u ∂x u dx = 0. (3.10)

From our previous proofs it is clear that we just need to handle the second term on the l.h.s. of (3.10). First we write the identity xH∂x2 u = H x∂x2 u = H ∂x2 (xu) − 2H∂x u = E1 + E2 .

(3.11)

θ with θ ∈ To bound the contribution of the term E2 inserted in (3.10) we shall use that wN (0, 1/2) satisfies the A2 inequality uniformly in N (see Proposition 1) so

1+θ w E2 = 2w 1+θ H∂x u cw θ H∂x u + cw θ xH∂x u N N N N 2 2 2 2 θ θ c wN ∂x u 2 + c wN H(x∂x u) 2 θ θ c w N ∂x u2 + cwN x∂x u2 = F1 + F2 .

(3.12)

Now using complex interpolation one gets (see Lemma 1) θ w ∂x u ∂x w θ u + ∂x w θ u N N N 2 2 2 θ θ ∂x wN u 2 + cu2 cJ wN u 2 + cu2 1/2 1/2 cJ 2 u2 x2θ u2 + cu2 ,

(3.13)

which has been bounded in the previous cases. So it remains to bound the term θ F 2 = w N x∂x u2 ,

(3.14)

which will be considered later. Inserting the term E1 in (3.11) into (3.10) one obtains the term G1 =

1+θ 1+θ wN H∂x2 (xu)wN xu dx.

(3.15)

As before we write 1+θ 2 1+θ 1+θ 2 wN ∂x (xu) + H wN H∂x2 (xu) = − H; wN ∂x (xu) 1+θ 1+θ 1+θ xu − 2H ∂x wN ∂x (xu) − H ∂x2 wN (xu) = K1 + H ∂x2 wN = K1 + K2 + K3 + K4 .

(3.16)

448


Thus, by Theorem 6 and the results in the previous cases the contribution of K1 , K4 in (3.15) is bounded. Also inserting the term K2 in (3.15) one has by integration by parts that its contribution is null. So in (3.16) it only remains to consider the contribution from K3 in (3.15). But using that 1+θ 1+θ ∂x (xu) 2 = ∂x wN ∂x (xu)2 K3 2 = H ∂x wN 1+θ 1+θ ∂x wN u 2 + ∂x wN x∂x u2 θ θ c wN u2 + wN x∂x u2 = R1 + R2 ,

(3.17)

since R1 was previously bounded, it remains to estimate R2 which is equal to the term F2 in (3.14). To estimate this term we use the BO equation in (1.1) to obtain the new equation ∂t (x∂x u) + H∂x2 (x∂x u) − 2H∂x2 u + x∂x (u∂x u) = 0.

(3.18)

2θ x∂ u leads to the identity The differential equation (3.18) multiplied by wN x

2 θ θ θ H∂x2 (x∂x u)wN (x∂x u) dx wN x∂x u dx + wN θ 2 θ θ θ − 2 wN H∂x uwN x∂x u dx + wN x∂x (u∂x u)wN x∂x u dx = 0.

1 d 2 dt

(3.19)

Sobolev inequality and integration by parts lead to w θ x∂x (u∂x u)w θ x∂x u dx cu2,2 w θ x∂x u w θ x∂x u + w θ u , (3.20) N N N N N 2 2 2 and since 2 θ 2 θ θ ∂x (x∂x u) + H wN H∂x2 (x∂x u) = − H; wN ∂x (x∂x u) wN θ θ θ = V1 + H∂x2 wN x∂x u x∂x u − 2H ∂x wN ∂x (x∂x u) − H ∂x2 wN = V1 + V2 + V3 + V4 ,

(3.21)

Theorem 6, the previous results, and interpolation allow to bound the L2 -norm of the terms V1 and V4 . As before by integration by parts the contribution of the term V2 in (3.19) is null. So it just remains to consider the term V3 in (3.21). In fact, 2 θ θ V3 = −2H ∂x wN x∂x u = V3,1 + V3,2 , ∂x u − 2H ∂x wN so one just needs to handle the term V3,2 . Using that ∂x w θ x cw θ , N N

c independent of N,

it suffices to consider θ 2 w ∂ u cJ 2 w θ u + cu1,2 + cw θ u , N x N N 2 2 2

(3.22)


449

with c independent of N . So we just need to consider the first term on the r.h.s. of the inequality (3.22). Using interpolation it follows that 2 θ J w u cJ 2+θ u2/(2+θ) w 2+θ uθ/(2+θ) . N N 2 s 2

(3.23)

We notice that the first term on the r.h.s. of (3.23) is bounded and the second one is bounded by the one we were estimating in (3.10). Therefore, (3.10) and (3.19) yield closed differential 1+θ θ x∂ u , and consequently the desired result. u2 and wN inequalities for xwN x 2 Case 5: s = r ∈ [5/2, 7/2). Part (iii) in Theorem 1. First, by differentiating the BO equation in (1.1) one gets ∂t (∂x u) + H∂x2 (∂x u) + u∂x (∂x u) + ∂x u∂x u = 0, so by reapplying the argument in the previous cases it follows that sup xs−1 ∂x u(t)2 M,

(3.24)

t∈[0,T ]

with M depending on u0 s,2 , xs u0 2 , and T . θ˜ with θ˜ ∈ [1/2, 3/2) to get Next, we multiply the BO equation in (1.1) by x 2 wN ˜

˜

˜

θ θ θ ∂t x 2 wN u + x 2 wN H∂x2 u + x 2 wN u∂x u = 0,

(3.25)

so a familiar argument leads to 1 d 2 dt

2 θ˜ 2 x wN u dx +

˜

˜

θ θ x 2 wN H∂x2 ux 2 wN u dx +

˜

˜

θ θ x 2 wN u∂x ux 2 wN u dx = 0.

(3.26)

Using the identity x 2 H∂x2 u = H∂x2 x 2 u + 4H∂x (xu) + Hu, the linear dispersive part of (3.25) (the second term on the l.h.s. of (3.25)) can be written θ˜ 2 θ˜ θ˜ θ˜ x H∂x2 u = wN H∂x2 x 2 u + 4wN H∂x (xu) + wN Hu wN = Q1 + Q2 + Q3 .

(3.27)

Since ∞

∞ u0 (x) dx =

−∞

u(x, t) dx = 0,

then H (xu) = xH u,

−∞

for θ˜ ∈ [1/2, 1] one has θ˜ Q3 2 = wN Hu2 1 + |x| Hu2 u + xu2 ,

450


and for θ˜ ∈ (1, 3/2) using Proposition 1 θ˜ θ−1 ˜ Q3 2 = wN Hu2 1 + |x| wN Hu2 ˜ ˜ w θ−1 u + w θ−1 xu , N

N

2

so in both cases by the previous results Q3 in (3.27) is bounded in L2 . To control Q2 we first consider the case θ˜ ∈ [1/2, 1] and use Calderón commutator theorem to get θ˜ H∂x (xu)2 Q2 2 = 4wN θ˜ θ˜ ∂x (xu)2 + H wN c H; wN ∂x (xu) 2 θ˜ θ˜ x∂x u2 + wN u2 . c xu2 + wN Thus, in the case θ˜ ∈ [1/2, 1], (3.24) provides the appropriate bound on the L2 -norm of Q2 . For the case θ˜ = 1 + θ , θ ∈ (0, 1/2) we combine Proposition 1 and the hypothesis on the mean value of u0 to deduce that θ˜ θ θ H∂x (xu)2 c wN xH∂x (xu)2 + wN H∂x (xu)2 Q2 2 = 4wN θ θ c w N H x∂x (xu) 2 + wN ∂x (xu)2 θ θ c wN x∂x (xu)2 + wN ∂x (xu)2 . Hence, (3.24) yields the appropriate bound on the L2 -norm of Q2 . Finally, we turn to the contribution of the term Q1 when inserted in (3.27). Thus, we write 2 2 θ˜ 2 2 θ˜ θ˜ ∂x x u + H wN H∂x2 x 2 u = − H; wN ∂x x u wN 2 θ˜ 2 θ˜ θ˜ = V1 + H ∂x2 wN x u x u − 2H ∂x wN ∂x x 2 u − H ∂x2 wN = V1 + V2 + V3 + V4 . From the previous cases it follows that the L2 -norm of the terms V1 , V4 are bounded. By integration by parts, the contribution of the term V2 is null. So it just remains to consider V3 = θ˜ ∂ (x 2 u)) in L2 , but −2H(∂x wN x 2 θ˜ θ˜ ∂x wN x ∂x u + 2xu = V2,1 + V2,2 . ∂x x 2 u = ∂x wN Since θ˜ ∈ (1/2, 3/2] V2,2 2 cx2 u2 which has been found to be bounded in the previous cases. Now since ∂x w θ˜ x 2 x1+θ˜ , N


451

it follows that ˜ V2,1 2 x1+θ ∂x u2 , so (3.24) gives the bound. Gathering the above information one completes the proof of Theorem 1. 4. Proof of Theorem 2 Without loss of generality we assume that t1 = 0 < t2 . − Since u(t1 ) ∈ Z 5 , 5 , we have that u ∈ C([0, T ] : H 2+1/2 ∩ L2 (|x|5 dx)). 2 2

Let us denote by U (t)u0 = (e−it|ξ |ξ uˆ 0 )∨ the solution of the IVP for the linear equation associated to the BO equation with datum u0 . Therefore, the solution to the IVP (1.1) can be represented by Duhamel’s formula t u(t) = U (t)u0 −

U t − t u t ∂x u t dt .

(4.1)

0

From Plancherel’s equality we have that for any t, |x|2+1/2 U (t)u0 ∈ L2 (R) if and only if and since

1/2 Dξ ∂ξ2 (e−it|ξ |ξ uˆ 0 ) ∈ L2 (R)

∂ξ2 e−it|ξ |ξ uˆ 0 = −e−it|ξ |ξ 4t 2 ξ 2 uˆ 0 + 2it sgn(ξ )uˆ 0 + 4it|ξ |∂ξ uˆ 0 − ∂ξ2 uˆ 0 ,

(4.2)

we show that with the hypothesis on the initial data, all terms in Duhamel’s formula for our solution u except the one involving sgn(ξ ), arising from the linear part in (4.2), have the appropriate decay at a later time. The argument in our proof requires localizing near the origin in Fourier frequencies by a function χ ∈ C0∞ , supp χ ⊆ (−, ) and χ = 1 on (−/2, /2). Let us start with the computation for the linear part in (4.1) by introducing a commutator as follows 1/2 1/2 1/2 χDξ ∂ξ2 e−it|ξ |ξ uˆ 0 = χ; Dξ ∂ξ2 e−it|ξ |ξ uˆ 0 + Dξ χ∂ξ2 e−it|ξ |ξ uˆ 0 = A + B.

(4.3)

From Proposition 4 and identity (4.2) we have that 1/2 A2 = χ; Dξ ∂ξ2 e−it|ξ |ξ uˆ 0 2 c∂ξ2 e−it|ξ |ξ uˆ 0 2 c t 2 ξ 2 uˆ 0 2 + t sgn(ξ )uˆ 0 2 + t |ξ |∂ξ uˆ 0 2 + ∂ξ2 uˆ 0 2 c t 2 ∂x2 u0 2 + tu0 2 + t ∂x (xu0 )2 + x 2 u0 2 , which are all finite since u0 ∈ Z2,2 .

(4.4)

452


On the other hand, 1/2

χ∂ξ2 e−it|ξ |ξ uˆ 0 1/2 1/2 = 4Dξ χe−it|ξ |ξ t 2 ξ 2 uˆ 0 + 2iDξ χe−it|ξ |ξ t sgn(ξ )uˆ 0 1/2 1/2 + 4iDξ χe−it|ξ |ξ t|ξ |uˆ 0 − Dξ χe−it|ξ |ξ ∂ξ2 uˆ 0

B = Dξ

= B1 + B2 + B3 + B4 .

(4.5)

Next, we shall estimate B4 in L2 (R). From Theorem 7, Proposition 2, and the fractional product rule type inequality (2.10) we get that 1/2 B4 2 c χe−it|ξ |ξ ∂ξ2 uˆ 0 2 + Dξ χe−it|ξ |ξ ∂ξ2 uˆ 0 2 1/2 −it|ξ |ξ 2 1/2 c ∂ 2 uˆ 0 + D e χ∂ uˆ 0 + e−it|ξ |ξ D χ∂ 2 uˆ 0 ξ

2

ξ

ξ

2

ξ

ξ

2

1/2 c x 2 u0 2 + t 1/4 + t 1/2 |ξ |1/2 χ∂ξ2 uˆ 0 2 + Dξ χ∂ξ2 uˆ 0 2 1/2 1/2 c(T ) x 2 u0 2 + Dξ (χ)2 ∂ξ2 uˆ 0 ∞ + χ∞ Dξ ∂ξ2 uˆ 0 2 c(T )x2+1/2 u0 2 .

(4.6)

Estimates for B1 and B3 in L2 (R) are easily obtained in a similar manner involving lower decay and regularity of the initial data. On the other hand for the analysis of B2 we introduce χ˜ ∈ C0∞ (R) such that χ˜ ≡ 1 on supp(χ). Then we can express this term as 1/2

Dξ

1/2 χe−it|ξ |ξ t sgn(ξ )uˆ 0 = tDξ e−it|ξ |ξ χ˜ χ sgn(ξ )uˆ 0 1/2 1/2 = t e−it|ξ |ξ χ˜ , Dξ χ sgn(ξ )uˆ 0 + e−it|ξ |ξ χ˜ Dξ χ sgn(ξ )uˆ 0 = t (S1 + S2 ).

(4.7)

Proposition 4 can be applied to bound S1 in L2 (R) as 1/2 S1 2 e−it|ξ |ξ χ, ˜ Dξ χ sgn(ξ )uˆ 0 2 cχ sgn(ξ )uˆ 0 2 cu0 2 .

(4.8)

Therefore, once we show that the integral part in Duhamel’s formula (4.1) lies in L2 (|x|5 dx), we will be able to conclude that 1/2

1/2

S2 , χ˜ Dξ χ sgn(ξ )uˆ 0 , Dξ

χ˜ χ sgn(ξ )uˆ 0 ∈ L2 (R),

then from Proposition 3 it will follow that uˆ 0 (0) = 0, and from the conservation law I1 in (1.3), this would necessarily imply that uˆ 0 (0) = u(x, t) dx = 0.


453

As we just mentioned above, in order to complete the proof, we consider the integral part in Duhamel’s formula. We localize again with the help of χ ∈ C0∞ (R) so that the integral in Eq. (4.1) after weights and a commutator reads now in Fourier space as t 0

2 1/2 −i(t−t )|ξ |ξ χ; Dξ e 4 t − t ξ 2 zˆ + 2i t − t sgn(ξ )ˆz + 4i t − t |ξ |∂ξ zˆ − ∂ξ2 zˆ 2 1/2 + Dξ χ e−i(t−t )|ξ |ξ 4 t − t ξ 2 zˆ dt + 2i t − t sgn(ξ )ˆz + 4i t − t |ξ |∂ξ zˆ − ∂ξ2 zˆ = A1 + A2 + A3 + A4 + B 1 + B 2 + B 3 + B 4

(4.9)

ξ 2 ˆ ∗ u. ˆ where zˆ = 12 ∂ xu = i 2 u We limit our attention to the terms in (4.9) involving the highest order derivatives of u, i.e. A1 and B1 , and remark that the others can be treated in a similar way by using that the function zˆ vanishes at ξ = 0. Combining Proposition 4, Holder’s inequality and Theorem 8 one has that

2 A1 L∞ L2x c t − t ξ 2 ξ uˆ ∗ uˆ L1 L2 T T x 3 c(T ) ∂x (uu) L2 L2 T x 3 c(T ) u∂x uL2 L2 + ∂x u∂x2 uL2 L2 T x T x 3 2 ∂ u ∞ 2 c(T ) ul 2 L∞ L∞ (QT ) ∂x u l ∞ L2 L2 (QT ) + ∂x uL∞ L∞ x L L x T x k T k k T x k T x 3 2 c(T ) ul 2 L∞ L∞ (QT ) ∂x u l ∞ L2 L2 (QT ) + uL∞ H 2 , (4.10) k

x

T

k

k

T

x

k

T

where QTk = [k, k + 1] × [0, T ]. For B1 we obtain from Theorem 7 T B1 L∞ L2x c T

1/2 −i(t−t )|ξ |ξ 2 D e χξ ξ uˆ ∗ uˆ 2 dt ξ

0

c e−i(t−t )|ξ |ξ χξ 2 ξ uˆ ∗ uˆ

L1T L2x

1/2 + Dξ e−i(t−t )|ξ |ξ χξ 2 ξ uˆ ∗ uˆ L1 L2 T

= Y1 + Y2 .

x

(4.11)

These terms can be handled as follows 2 ˆ L1 L2 cu∞ u2 L1 cT sup u(t)1,2 , Y1 cuˆ ∗ u T

x

and using Proposition 2, (2.9), (2.10), and (4.12)

T

[0,T ]

(4.12)

454


1/2 Y2 = Dξ e−i(t−t )|ξ |ξ χξ 2 ξ uˆ ∗ uˆ L1 L2 T x 1/2 −i(t−t )|ξ |ξ 2 1/2 χξ ξ uˆ ∗ uˆ L1 L2 + cDξ χξ 2 ξ uˆ ∗ uˆ L1 L2 c Dξ e T x T x 1/2 1/2 3 2 1/2 1/2 χξ ξ uˆ ∗ uˆ L1 L2 + c Dξ χξ ∞ uˆ ∗ u c t + t |ξ | ˆ 2 L1 T x T 3 1/2 + cχξ ∞ Dξ (uˆ ∗ u) ˆ 2 L1 T 1/2 c(T )uˆ ∗ u ˆ 1 2 + cD (uˆ ∗ u) ˆ 1 2 LT Lx

ξ

LT Lx

2 c(T ) sup u(t)1,2 + c(T )|x|1/2 uL∞ L2 sup u(t)1,2 . [0,T ]

T

x

(4.13)

[0,T ]

Hence the terms in (4.9) are all bounded, so by applying the argument after inequality (4.8) we complete the proof. 5. Proof of Theorem 3 From the previous results and the hypothesis we have that for any > 0 u ∈ C [0, T ] : Z˙ 7/2,7/2− and u(·, tj ) ∈ L2 |x|7 dx , j = 1, 2, 3. Hence, uˆ ∈ C [0, T ] : H 7/2− (R) ∩ L2 |ξ |7 dξ and u(·, ˆ tj ) ∈ H 7/2 (R),

j = 1, 2, 3

for any > 0. Thus, in particular it follows that uˆ ∗ uˆ ∈ C [0, T ] : H 6 (R) ∩ L2 |ξ |7 dξ .

(5.1)

Let us assume that t1 = 0 < t2 < t3 . An explicit computation shows that F (t, ξ, uˆ 0 ) = ∂ξ3 e−it|ξ |ξ uˆ 0 = e−it|ξ |ξ 8it 3 ξ 3 uˆ 0 − 12t 2 ξ uˆ 0 − 12t 2 ξ 2 ∂ξ uˆ 0

− 6it sgn(ξ )∂ξ uˆ 0 − 6it|ξ |∂ξ2 uˆ 0 − 2itδ uˆ 0 + ∂ξ3 uˆ 0 ,

(5.2)

where we observe that since the initial data u0 have zero mean value the term involving the Dirac function in (5.2) vanishes. Hence in order to prove our theorem, via Plancherel’s theorem and Duhamel’s formula (4.1), it is enough to show that the assumption that 1/2 Dξ F (t, ξ, uˆ 0 ) −

t

1/2 Dξ F t − t , ξ, zˆ t dt ,

(5.3)

0 ξ 2 lies in L2 (R) for times t1 = 0 < t2 < t3 , where zˆ = 12 ∂ ˆ ∗ u, ˆ leads to a contradiction. Let xu = i 2 u us show that the first term in Eq. (5.3) which arises from the linear part in Duhamel’s formula persists in L2 .


455

We proceed as in the proof of Theorem 2 and localize one more time by introducing χ ∈ C0∞ , supp χ ⊆ (−, ) and χ = 1 on (−/2, /2) so that 1/2 1/2 1/2 χDξ ∂ξ3 e−it|ξ |ξ uˆ 0 = χ; Dξ ∂ξ3 e−it|ξ |ξ uˆ 0 + Dξ χ∂ξ3 e−it|ξ |ξ uˆ 0 + B. =A

(5.4)

from Proposition 4, this is bounded in L2 (R) by ∂ 3 (e−it|ξ |ξ uˆ 0 )2 , As for the first term, A, ξ which is finite as can easily be observed from its explicit representation in (5.2), the assumption on the initial data u0 , and the quite similar computation already performed in (4.4), therefore we omit the details. we notice that On the other hand, for B, = D 1/2 χ∂ξ3 e−it|ξ |ξ uˆ 0 B ξ 1/2 1/2 = 8iDξ χe−it|ξ |ξ t 3 |ξ |3 uˆ 0 − 12Dξ χe−it|ξ |ξ t 2 ξ uˆ 0 1/2 1/2 − 12Dξ χe−it|ξ |ξ t 2 ξ 2 ∂ξ uˆ 0 − 6iDξ χe−it|ξ |ξ t sgn(ξ )∂ξ uˆ 0 1/2 1/2 − 6iDξ χe−it|ξ |ξ t|ξ |∂ξ2 uˆ 0 + Dξ χe−it|ξ |ξ ∂ξ3 uˆ 0 2 + B 3 + B 4 + B 5 + B 7 . 1 + B =B

(5.5)

6 does not appear, and that B 1 and Notice that from the remark made after the identity (5.2) B 7 are the terms involving the highest regularity and decay of the initial data. Therefore we show B 4 , in detail their L2 estimates along with the argument to exploit a nice cancellation property of B and a term arising in the integral part in Duhamel’s formula (4.1). 1 we obtain from Theorem 7, fractional product rule type estimate (2.9), (2.10), and For B Holder’s inequality that 1 2 c χe−it|ξ |ξ t 3 ξ 3 uˆ 0 + D1/2 χe−it|ξ |ξ t 3 |ξ |3 uˆ 0 B ξ 2 2 1/2 −it|ξ |ξ −it|ξ |ξ 1/2 3 3 ct u0 2 + Dξ e χ|ξ | uˆ 0 2 + e Dξ χ|ξ |3 uˆ 0 2 1/2 ct 3 u0 2 + t 1/4 + t 1/2 |ξ |1/2 χ|ξ |3 uˆ 0 2 + Dξ χ|ξ |3 uˆ 0 2 1/2 1/2 c(T ) u0 2 + Dξ χξ 3 ∞ uˆ 0 2 + χξ 3 ∞ Dξ uˆ 0 2 c(T ) u0 2 + |x|1/2 u0 2 , and similarly 7 2 c χe−it|ξ |ξ ∂ξ3 uˆ 0 + D1/2 χe−it|ξ |ξ ∂ξ3 uˆ 0 B ξ 2 2 1/2 −it|ξ |ξ 3 −it|ξ |ξ 1/2 3 3 e χ∂ uˆ 0 + e χ∂ uˆ 0 c ∂ uˆ 0 + D D ξ

2

ξ

ξ

2

ξ

ξ

2

1/2 c x 3 u0 2 + t 1/4 + t 1/2 |ξ |1/2 χ∂ξ3 uˆ 0 2 + Dξ χ∂ξ3 uˆ 0 2 1/2 1/2 c(t) x 3 u0 2 + Dξ (χ)∞ ∂ξ3 uˆ 0 2 + χL∞ Dξ ∂ξ3 uˆ 0 2

(5.6)

456


1/2 c(T ) x 3 u0 2 + Dξ ∂ξ3 uˆ 0 2 c(T )x3+1/2 u0 2 .

(5.7)

Now, let us go over the integral part that can be written in Fourier space and with the help of a commutator as t

3 2 1/2 −i(t−t )|ξ |ξ χ; Dξ e 8i t − t ξ 3 zˆ − 12 t − t ξ zˆ

0

2 − 12 t − t ξ 2 ∂ξ zˆ − 6i t − t |ξ |∂ξ2 zˆ − 2i t − t δ zˆ + ∂ξ3 zˆ 3 2 1/2 + Dξ χ 8i t − t ξ 3 zˆ − 12 t − t ξ zˆ 2 − 12 t − t ξ 2 ∂ξ zˆ − 6i t − t |ξ |∂ξ2 zˆ − 2i t − t δ zˆ + ∂ξ3 zˆ dt 2 + A 3 + A 5 + A 6 + A 7 + B 1 + B 2 + B 3 + B 5 + B 6 + B 7 + C, 1 + A =A

(5.8)

where C= −6i

t

1/2 −i(t−t )|ξ |ξ

Dξ

e

χ t − t sgn(ξ )∂ξ zˆ dt ,

(5.9)

0 ξ 2 ˆ ∗ u. ˆ and zˆ = 12 ∂ xu = i 2 u 6 vanish since u∂x u has zero mean value and for A 1 , A 2 , A 3 , A 5 , 6 , B Notice that A 1 , B 2 , B 3 , B 5 and B 7 the estimates in L2 (R) are essentially the same for their counterparts 7 , B A in Eq. (4.9), in the proof of Theorem 2, so we omit the details of their estimates. Therefore from the assumption that u0 , u(t2 ) ∈ Z˙ 7 , 7 , Eq. (5.8), and the estimates above, we 2 2 conclude that 1/2 −it|ξ |ξ

e χt sgn(ξ )∂ξ uˆ 0 − C 1/2 = −6iDξ e−it|ξ |ξ χt sgn(ξ )∂ξ uˆ 0

R = −6iDξ

t + 6i

1/2

Dξ

iξ e−i(t−t )|ξ |ξ χ t − t sgn(ξ )∂ξ uˆ ∗ uˆ dt , 2

(5.10)

0

is a function in L2 (R) at time t = t2 . But t iξ iξ 1/2 Dξ e−i(t−t )|ξ |ξ χ t − t sgn(ξ ) ∂ξ R = 6i uˆ ∗ uˆ − ∂ξ uˆ ∗ uˆ (0) dt 2 2 0

1/2 −it|ξ |ξ

e χt sgn(ξ ) ∂ξ uˆ 0 − ∂ξ uˆ 0 (0) 1/2 − 6iDξ e−it|ξ |ξ χt sgn(ξ )∂ξ uˆ 0 (0)

− 6iDξ


t + 6i

1/2 Dξ

e

−i(t−t )|ξ |ξ

457

iξ χ t − t sgn(ξ ) ∂ξ uˆ ∗ u(0) ˆ dt 2

0

= R1 + R2 + R3 + R4 .

(5.11)

We shall show that R1 and R2 are L2 (R) functions. This will imply that (R3 + R4 )(t2 ) is also an L2 (R) function. For R1 we observe that from (5.1) iξ iξ −i(t−t )|ξ |ξ χ(ξ ) sgn(ξ ) ∂ξ e uˆ ∗ uˆ ξ, t − ∂ξ uˆ ∗ uˆ 0, t 2 2 is a Lipschitz function with compact support in the ξ variable. Therefore, using Theorem 7 one sees that R1 (t) ∈ L2 (R). A similar argument shows that R2 (t) ∈ L6 (R). Therefore, we have that (R3 + R4 )(t2 ) ∈ L2 (R). On the other hand iξ x u(0) = −i xu∂x u dx = i u22 , uˆ ∗ uˆ (0) = −ixu∂ ∂ξ 2 2 and from the Benjamin–Ono equation we have d dt

x∂x2 Hu dx

xu dx +

+

xu∂x u dx = 0,

(5.12)

which implies that d dt

xu dx = −

1 xu∂x u dx = u0 22 , 2

and hence ∂ξ

iξ d uˆ ∗ uˆ (0) = i xu dx. 2 dt

Substituting this into R4 gives us after integration by parts

t R4 = −6

1/2

Dξ 0

1/2 = −6Dξ

t +6 0 1/2

= 6Dξ

e

−i(t−t )|ξ |ξ

1/2 Dξ

d e−i(t−t )|ξ |ξ χ t − t sgn(ξ ) xu dx dt dt

χ t − t sgn(ξ )

t =t xu dx t =0

−i(t−t )|ξ |ξ e χ i|ξ |ξ t − t − 1 sgn(ξ ) xu dx dt

e−it|ξ |ξ χt sgn(ξ )

xu0 (x) dx

(5.13)

458


t + 6i

1/2 Dξ

−i(t−t )|ξ |ξ xu dx dt χ t − t |ξ |ξ sgn(ξ ) e

0

t −6

1/2 Dξ

e

−i(t−t )|ξ |ξ

χ sgn(ξ ) xu dx dt .

(5.14)

0

We observe that the second term after the last equality in (5.14) belongs to L2 (R) and the first cancels out with R3 since ∂ξ uˆ 0 (0) = −i x u0 (0) = −i xu0 (x) dx, (5.15) and therefore 1/2

R3 = −6Dξ

e−it|ξ |ξ χt sgn(ξ ) xu0 (x) dx .

(5.16)

So the argument above implies that t −6

1/2

Dξ

e−i(t−t )|ξ |ξ χ sgn(ξ ) dt xu x, t dx

(5.17)

0

is in L2 (R) at time t = t2 , and from Theorem 7 this is equivalent to have that 1/2 Dξ

t2

χ(ξ ) sgn(ξ )

e−i(t1

−t )|ξ |ξ

xu x, t dx dt ∈ L2 (R),

(5.18)

0

which 3 (choosing the support (−, ) of χ sufficiently small) implies that t2 from Proposition ) dx) dt = 0 and consequently xu(x, t) dx must be zero at some time in (0, t ). ( xu(x, t 2 0 We reapply the same argument to conclude that xu(x, t) dx is again zero at some other time in (t2 , t3 ). Finally, the identity (1.11) completes the proof of the theorem. Acknowledgments The authors would like to thank J. Duoandikoetxea for fruitful conversations concerning this paper. This work was done while G.F. was visiting the Department of Mathematics at the University of California-Santa Barbara whose hospitality he gratefully acknowledges. G.P. was supported by NSF grant DMS-0800967. References [1] L. Abdelouhab, J.L. Bona, M. Felland, J.-C. Saut, Nonlocal models for nonlinear dispersive waves, Phys. D 40 (1989) 360–392. [2] M.J. Ablowitz, A.S. Fokas, The inverse scattering transform for the Benjamin–Ono equation, a pivot for multidimensional problems, Stud. Appl. Math. 68 (1983) 1–10.


[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

459

T.B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967) 559–592. N. Burq, F. Planchon, On the well-posedness of the Benjamin–Ono equation, Math. Ann. 340 (2008) 497–542. A.P. Calderón, Commutators of singular integral operators, Proc. Natl. Acad. Sci. USA 53 (1965) 1092–1099. R. Coifman, M. Wickerhauser, The scattering transform for the Benjamin–Ono equation, Inverse Problems 6 (1990) 825–860. L. Dawson, H. McGahagan, G. Ponce, On the decay properties of solutions to a class of Schrödinger equations, Proc. Amer. Math. Soc. 136 (2008) 2081–2090. J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., 2000. L. Escauriaza, C.E. Kenig, G. Ponce, L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations, J. Funct. Anal. 244 (2007) 504–535. L. Escauriaza, C.E. Kenig, G. Ponce, L. Vega, The sharp hardy uncertainty principle for Schrödinger evolutions, Duke Math. J., in press. G. Fonseca, F. Linares, Benjamin–Ono equation with unbounded data, J. Math. Anal. Appl. 247 (2000) 426–447. J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, 1985. H. Helson, G. Szegö, A problem in prediction theory, Ann. Mat. Pura Appl. (4) 51 (1960) 107–138. R. Hunt, B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973) 227–251. A.D. Ionescu, C.E. Kenig, Global well-posedness of the Benjamin–Ono equation on low-regularity spaces, J. Amer. Math. Soc. 20 (3) (2007) 753–798. R.J. Iorio, On the Cauchy problem for the Benjamin–Ono equation, Comm. Partial Differential Equations 11 (1986) 1031–1081. R.J. Iorio, Unique continuation principle for the Benjamin–Ono equation, Differential Integral Equations 16 (2003) 1281–1291. R.J. Iorio, F. Linares, M. Scialom, KdV and BO equations with bore-like data, Differential Integral Equations 11 (1998) 895–915. E. Kaikina, K. Kato, P.I. Naumkin, T. Ogawa, Wellposedness and analytic smoothing effect for the Benjamin–Ono equation, Publ. Res. Inst. Math. Sci. 38 (2002) 651–691. C.E. Kenig, K.D. Koenig, On the local well-posedness of the Benjamin–Ono and modified Benjamin–Ono equations, Math. Res. Lett. 10 (2003) 879–895. C.E. Kenig, H. Takaoka, Global well-posedness of the Benjamin–Ono equation with initial data in H 1/2 , Int. Math. Res. Not. (2006) 1–44, Art. ID 95702. C.E. Kenig, G. Ponce, L. Vega, On the unique continuation of solutions to the generalized KdV equation, Math. Res. Lett. 10 (2003) 833–846. H. Koch, N. Tzvetkov, On the local well-posedness of the Benjamin–Ono equation on H s (R), Int. Math. Res. Not. 26 (2003) 1449–1464. H. Koch, N. Tzvetkov, Nonlinear wave interactions for the Benjamin–Ono equation, Int. Math. Res. Not. 30 (2005) 1833–1847. L. Molinet, J.C. Saut, N. Tzvetkov, Ill-posedness issues for the Benjamin–Ono and related equations, SIAM J. Math. Anal. 33 (2001) 982–988. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. J. Nahas, G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Comm. Partial Differential Equations 34 (2009) 1–20. H. Ono, Algebraic solitary waves on stratified fluids, J. Phys. Soc. Japan 39 (1975) 1082–1091. S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical Ap characteristic, Amer. J. Math. 129 (2007) 1355–1375. G. Ponce, On the global well-posedness of the Benjamin–Ono equation, Differential Integral Equations 4 (1991) 527–542. J.-C. Saut, Sur quelques généralisations de l’ équations de Korteweg–de Vries, J. Math. Pures Appl. 58 (1979) 21–61. E.M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc. 67 (1961) 102–104. E.M. Stein, Harmonic Analysis, Princeton University Press, 1993. T. Tao, Global well-posedness of the Benjamin–Ono equation on H 1 , J. Hyperbolic Differ. Equ. 1 (2004) 27–49, Int. Math. Res. Not. (2006) 1–44, Art. ID 95702. S. Vento, Well posedness for the generalized Benjamin–Ono equations with arbitrary large data in the critical space, Int. Math. Res. Not. IMRN 2 (2010) 297–319.


Commuting Toeplitz operators on the Segal–Bargmann space Wolfram Bauer a,∗,1 , Young Joo Lee b a Mathematisches Institut, Georg-August-Universität, Bunsen-str. 3-5, 37073 Göttingen, Germany b Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea

Received 31 May 2010; accepted 15 September 2010 Available online 22 September 2010 Communicated by L. Gross

Abstract Consider two Toeplitz operators Tg , Tf on the Segal–Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg , Tf ] = 0 does not imply the radial dependence of f . Finally, we consider Toeplitz operators on the Segal–Bargmann space over Cn and n > 1, where the commuting property of Toeplitz operators can be realized more easily. © 2010 Elsevier Inc. All rights reserved. Keywords: Toeplitz operator; Mellin transform; Reproducing kernel Hilbert space; Radial symbol

1. Introduction The study of commuting Toeplitz operators on the Bergman and Hardy spaces over various domains and related operator algebras has a long lasting history; cf. [1,6,8,10] and recently [7,11,13, 14,16,17]. The problem of characterizing commuting Toeplitz operators with arbitrary bounded * Corresponding author.

E-mail addresses: [email protected] (W. Bauer), [email protected] (Y.J. Lee). 1 The first named author has been supported by an “Emmy-Noether scholarship” of DFG (Deutsche Forschungs-

gemeinschaft). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.007

W. Bauer, Y.J. Lee / Journal of Functional Analysis 260 (2011) 460–489

461

symbols seems quite challenging and is not fully understood until now. However, methods for an analysis are available if one restricts attention to certain sub-classes of symbols. In the present paper we study classes of commuting Toeplitz operators acting on the Segal– Bargmann space H 2 (Cn , dμ) of all Gaussian square integrable entire functions on Cn . Here the case n = 1 is of particular interest. Moreover, we admit the (not so frequently studied) situation of unbounded operator symbols of a certain type since their growth behavior near infinity essentially influences our results; cf. Theorem 4.17 and Example 5.8. As a consequence we have to deal with a space of i.g. unbounded, densely defined Toeplitz operators. We use the construction in [3] in order to check that they can be embedded into an operator algebra and therefore commutators are well defined. An easy calculation shows that a Toeplitz operator with a radial symbol f is diagonal with respect to the standard orthonormal basis of H 2 (Cn , dμ) and therefore such type of operators commute. Here a function f is called radial if f (z) = f (|z|). Conversely, one can ask whether for a non-trivial radial symbol f and an arbitrary symbol g (both in our symbol class) the commutator condition [Tf , Tg ] = 0 implies that g is radial. The analog question in the case of Toeplitz operators with bounded symbols acting on the unweighted Bergman space over the unit disc D ⊂ C has been answered before in [10]. Theorem A below gives the result of Theorem 6 in [10]: Theorem A. (See [10].) Let ψ, ϕ ∈ L∞ (D, dA) where dA is the usual area measure on D. Let ϕ be a non-trivial radial function. If the Toeplitz operators Tψ and Tϕ commute on the Bergman space, then ψ is a radial function. In order to prove Theorem A, the authors use an expansion of ψ into an L2 -convergent series. Then the commutator condition [Tψ , Tϕ ] = 0 can be converted into a functional equation for the Mellin transform of ϕ and coefficient functions of that expansion. From this equation the result follows, but as an essential ingredient in the argument the Blaschke condition for the possible distribution of zeros of non-vanishing bounded holomorphic functions on D (or a right halfplane) is used. Since here we consider Toeplitz operators with i.g. unbounded symbols acting on a function space over the complex plane, we cannot use such type of arguments and this fact causes the main complications. However, we can prove a result similar to Theorem A above. Let S be a space of measurable complex valued functions on the complex plane and of at most polynomial growth at infinity (see Definition 4.1), then we show: Theorem B. Let u, v ∈ S and assume that u is radial and non-constant. If the Toeplitz operators Tu and Tv commute on the Segal–Bargmann space H 2 (C, dμ), then v is a radial function. The restriction to symbols in the space S in Theorem B is necessary. We provide an example of a unitary Toeplitz operator Tf with radial symbol f ∈ / S acting on H 2 (C, dμ) such that [Tf , Tg ] = 0 where g is a bounded non-radial function on the complex plane; cf. Example 5.8. We do not know whether a similar effect is possible in case of the Toeplitz operators with certain unbounded symbols acting on the Bergman space over the unit disc D. In the last part of this paper we discuss commuting Toeplitz operators with polynomial symbols acting on H 2 (Cn , dμ). It has been shown in [3,9] that the corresponding Toeplitz operators form an algebra under composition and the symbol of the product of two operators can be calculated as a Moyal-type product. In case of dimension n > 1 and for polynomials p and q where p is non-constant and radial it is shown that the condition [Tp , Tq ] = 0 is equivalent with q belong-

462


ing to a certain subspace P1 of all polynomials (see (5.3) for the definition of P1 ). In particular, the space P1 strictly includes all polynomials that are radial in each component. In Section 2 we define a generalized Segal–Bargmann space by replacing the usual Gaussian density by a suitable radial function, cf. [2,5,15]. In this setting we explain the notion of Toeplitz operators and state our main question on commuting Toeplitz operators. In the case of the dimension n = 1 we convert the commutator condition on Toeplitz operators into a functional equation for Mellin transforms of the symbols and their components, respectively; cf. Proposition 2.4. In Section 3 we specialize to the classical Segal–Bargmann space of Gaussian square integrable entire functions on the complex plane. The construction in [3] of an algebra of operators containing Toeplitz operators with a certain type of unbounded symbols is recalled. As an important feature, the Berezin transform is one-to-one on this algebra. Section 4 contains the proof of Theorem B which is deduced from a detailed discussion of the functional equation we have obtained in Proposition 2.4. In particular, we need to analyze the Mellin convolutions of certain functions in Propositions 4.8 and 4.9. In Section 5 we discuss commuting Toeplitz operators with polynomial symbols acting on H 2 (Cn , dμ) where n > 1. Using the product structure of Cn it is easy to produce commuting Toeplitz operators. Finally, we give a counterexample to Theorem B in case of u ∈ / S in the end of Section 5. 2. Preliminaries Throughout this section let ϕ be a nonnegative integrable radial function on Cn . With z = (z1 , . . . , zn ) ∈ Cn we write z¯ := (¯z1 , . . . , z¯ n ) for its complex conjugate. By dv we denote the usual Lebesgue measure on Cn ∼ = R2n and additionally we suppose that ϕ satisfies the following two conditions: ϕ(k) ˆ :=

|z|2k ϕ(z) dv(z) < ∞,

lim sup k ϕ(k) ˆ =∞ k

Cn

for every k = 0, 1, . . . and with the Euclidean norm | · | on Cn . Let Fϕ be the set of all entire functions in L2 (Cn , ϕ dv). Then it is known that Fϕ is a closed linear subspace of L2 (Cn , ϕ dv) with the inner product f, gϕ =

f gϕ ¯ dv Cn

and the usual L2 -norm f ϕ = f, f ϕ where f, g ∈ L2 (Cn , ϕ dv). In fact, Fϕ is a reproducing kernel Hilbert space and the corresponding reproducing kernel Kϕ (z, w) can be given by

Kϕ (z, w) =

∞ (n − 1 + |α|)! zα w¯ α (n)k = (z · w)k (n − 1)!α! ϕ(|α|) ˆ k! ϕ(k) ˆ n

α∈N0

k=0

(2.1)


463

where z · w := z1 w1 + · · · + zn wn and (n)k := n(n + 1) · · · (n + k − 1) denotes the usual Pochhammer symbol. Also, the notation Nn0 denotes the set of all n-tuples of nonnegative integers. Note that by Stirling’s formula we have the asymptotics: (n)k ∼ k n−1 (n − 1)! k!

(2.2)

as k → ∞. Therefore, it follows from the above assumptions on ϕ(k) ˆ that (2.1) converges uniformly on every compact subsets of Cn × Cn . See [15] for details and related facts. Let Pϕ be the orthogonal projection from L2 (Cn , ϕ dv) onto Fϕ . Let u : Cn → C be a measurable function. Then, for all f ∈ Du := g ∈ Fϕ : u · g ∈ L2 Cn , ϕ dv ϕ

the Toeplitz operator Tu with symbol u is defined by Tuϕ f := Pϕ (uf ). ϕ

Note that in general Tu is an unbounded linear operator on Du ⊂ Fϕ . Clearly, in case of ϕ ϕ u ∈ L∞ (Cn ) the Toeplitz operator Tu is bounded with Tu u ∞ . In the following we are considering products of two Toeplitz operators. Hence, we restrict ourself to spaces of measurable complex valued symbols S which have the following property: Assumption. There is a dense subspace HS ⊂ ϕ Toeplitz operators Tu with symbol u ∈ S.

u∈S

Du ⊂ Fϕ which is invariant under all

Remark 2.1. In case of S = L∞ (Cn ) we can choose HS = Fϕ . If ϕ is a Gaussian density we construct a space S containing unbounded symbols and a corresponding invariant subspace HS in the next section. For each multi-index α = (α1 , . . . , αn ) ∈ Nn0 , we put eα (z) := zα zα −1 ϕ and |α| = α1 + · · · + αn . Note, that {eα }α∈Nn0 forms a dense subset of Fϕ . We have: Lemma 2.2. Let u ∈ S be radial and assume that eβ ∈ HS for all β ∈ Nn0 . Moreover, assume that for all m ∈ N u(z)

∞ n−1 k k=0

ϕ(k) ˆ

|z|k+m ∈ L1 Cn , ϕ dv . ϕ

(2.3)

Then, we have Pϕ (uzβ ) = 0 for β = 0 and Tu is diagonal with respect to the orthonormal ϕ basis {eα }α∈N0 . More precisely, Tu eβ =

u(β)eβ where 1

u(β) := ϕ(|β|) ˆ Note that

u(β) only depends on |β|.

u(z)|z|2|β| ϕ(z) dv(z). Cn

464


Proof. Let ρ ∈ {¯zβ , zβ } with β ∈ Nn0 . Using the expression (2.1) of the reproducing kernel function, we see Pϕ (uρ)(w) = u(z)ρ(z)Kϕ (w, z)ϕ(z) dv(z) Cn

=

u(z)ρ(z)

k=0

Cn

=

∞ (n)k (w · z)k ϕ(z) dv(z) k!ϕ(k) ˆ

(n)|α| w α u(z)ρ(z)zα ϕ(z) dv(z). α! ϕ(|α|) ˆ n

α∈N0

(2.4)

Cn

Here we have used (2.2) and (2.3) together with Lebesgue’s dominated convergence theorem in order to interchange the integration and summation. In case of ρ(z) = z¯ β where β = 0 the first assertion follows from Cn u(z)zβ+α ϕ(z) dv(z) = 0 for all α ∈ Nn0 . In order to prove the second assertion choose ρ(z) = zβ and let S2n−1 ⊂ Cn denote the − 1)-dimensional unit sphere with the usual measure dσ2n−1 . Using the relation (real) (2n β |2 dσ n |ζ 2n−1 2n−1 (ζ ) = 2β!π /(n + |β| − 1)!, we have S

uz z¯ ϕ dv = δα,β β α

Cn

β 2 ζ dσ2n−1 (ζ )

S2n−1

= δα,β

β!(n − 1)! (n + |β| − 1)!

∞ r 2(n+|β|)−1 u(r)ϕ(r) dr 0

|z|2|β| u(z)ϕ(z) dv(z), Cn

and the second assertion follows from (2.4). The proof is complete.

2

Let u ∈ S be separately radial, i.e. u only depends on |z1 |, |z2 |, . . . , |zn | and assume that (2.3) holds. Then for all α ∈ Nn0 we have: (a) Cn u(z)zβ+α ϕ(z) dv(z) = 0, in case of β = 0, (b) Cn u(z)zα zβ ϕ(z) dv(z) = cα δα,β where cα ∈ C is a suitable number. In fact, the relations (a) and (b) follow from the invariance of both integrals under the linear ϕ transformation U (z) := iz. By the same argument as before Pϕ (uzβ ) = 0 for β = 0 and Tu is diagonal with respect to the orthonormal basis {eα }α∈Nn0 . Due to this observation it holds: ϕ

ϕ

ϕ

ϕ

Proposition 2.3. Let u, v ∈ S be separately radial, then we have Tu Tv = Tv Tu on span{eα | α ∈ Nn0 }. In the following we assume that all functions u ∈ S fulfill condition (2.3). Let us specialize now to the complex one-dimensional case. With our notations before recall that 1

u(k) = u(w)|w|2k ϕ(w) dv. ϕ(k) ˆ C


465

Assume that u ∈ S is non-constant radial and (2.3) is fulfilled. Let v ∈ S be of the form ∞ v(z) = v reiθ = vj (r)eij θ ,

z := reiθ

(2.5)

j =−∞

where each vj can be interpreted as a radial function on C with (2.3). Moreover, we assume that ϕ the sum in (2.5) converges in the topology of L2 (C, ϕ dv). Suppose that the Toeplitz operators Tu ϕ and Tv commute

zk k ∈ N0 ⊂ HS → HS . Tuϕ Tvϕ = Tvϕ Tuϕ : span ek (z) = ϕ(k) ˆ

(2.6)

Since u is radial, we have by Lemma 2.2 Tuϕ ek =

u(k)ek for all k ∈ N0 . By Lemma 2.2 and our assumptions on vj it follows that u(k)Pϕ [vek ](z) Tvϕ Tuϕ ek (z) =

=

u(k)

∞

Pϕ vj eij θ ek (z)

j =−∞ ∞

u(k) = Pϕ vj r k ei(j +k)θ (z) ϕ(k) ˆ j =−∞

u(k) zj +k = vj (w)|w|j +2k ϕ(w) dv(w), ϕ(j ˆ + k) ϕ(k) ˆ j −k

(2.7)

C

for every z ∈ C. Moreover, we have Tvϕ ek (z) = =

1

∞

ϕ(k) ˆ j =−∞ 1

ϕ(k) ˆ j −k

Pϕ vj r k ei(j +k)θ (z)

zj +k ϕ(j ˆ + k)

vj (w)|w|j +2k ϕ(w) dv(w).

C

It follows from Lemma 2.2 again and the assumption on u ∈ S that Tuϕ Tvϕ ek (z) =

Pϕ [uw j +k ](z) vj (w)|w|j +2k ϕ(w) dv(w) ϕ(j ˆ + k) ϕ(k) ˆ j −k 1

C

=

zj +k

u(j + k) ϕ(j ˆ + k) ϕ(k) ˆ j −k 1

C

vj (w)|w|j +2k ϕ(w) dv(w).

(2.8)

466

W. Bauer, Y.J. Lee / Journal of Functional Analysis 260 (2011) 460–489 ϕ

ϕ

Since the operators Tu and Tv are commuting by assumption, it follows from (2.7) and (2.8) that for all integers k 0, j with j + k 0

u(k) −

u(j + k)

vj (w)|w|j +2k ϕ(w) dv(w) = 0,

C

or equivalently

u(k) −

u(j + k)

∞

vj (r)r j +2k+1 ϕ(r) dr = 0.

(2.9)

0

Note that 1

u(k) = ϕ(k) ˆ

2π u(w)|w| ϕ(w) dv = ϕ(k) ˆ

∞

2k

C

u(r)r 2k+1 ϕ(r) dr. 0

Thus,

u(k) can be expressed as values of the Mellin transform of uϕ. Given a (suitable) function ψ on the half line (0, ∞), the Mellin transform M[ψ](z) of the complex parameter z is defined by ∞ M[ψ](z) =

ψ(t)t z−1 dt. 0

Recall that each M[ψ] for suitable ψ is complex analytic on a strip in the complex plane parallel to the imaginary axis. Moreover, the Mellin transform M is injective. For all k ∈ N0 one can write ϕ(k) ˆ = 2πM[ϕ](2k + 2) and hence

u(k) =

1 ϕ(k) ˆ

u(w)|w|2k ϕ(w) dv = C

M[uϕ](2k + 2) . M[ϕ](2k + 2)

We can rewrite (2.9) by using the Mellin transform and summarizing the above calculations we have shown: Proposition 2.4. Let u, v ∈ S such that u is non-constant and radial. Assume that v can be written in the form (2.5) which converges in L2 (C, ϕ dv). Under the assumption (2.6) it follows

M[uϕ](2k + 2) M[uϕ](2k + 2j + 2) − M[vj ϕ](j + 2k + 2) = 0 M[ϕ](2k + 2) M[ϕ](2k + 2j + 2)

for all integers k 0 and j with j + k 0.

(2.10)


467

3. An enveloping algebra for Toeplitz operators on the Segal–Bargmann space In the remaining part of this paper we specialize our analysis to the case of a standard Gaussian weight-function ϕ in order to study Eq. (2.10) in a precise way. On Cn we consider the 2 normalized Gaussian measure dμ given by dμ = ϕ dv where ϕ is defined by ϕ(z) := π −n e−|z| , 2 n cf. [2,4]. Then Fϕ = H (C , dμ) is called the Segal–Bargmann space and it has the reproducing kernel: Kϕ = K : Cn × Cn → C : (z, w) → K(z, w) = ez·w . ϕ

To keep the notations short we will write P := Pϕ for the projection and Tu := Tu for the Toeplitz operator with symbol u. Since in the following we are considering products of i.g. unbounded Toeplitz operators on Fϕ , we must carefully choose the space of symbols to obtain densely defined operators on an invariant domain of definition. We follow the construction in [3] and for completeness we give a short summary here. We write M(Cn ) for the space of measurable complex valued functions on Cn . For c ∈ R we set Dc := f ∈ M Cn : ∃d > 0 such that f (z) d exp c|z|2 a.e. and we define a space of symbols by ∞ D1 , Sym>0 Cn := j =1

j

which is a ∗-algebra under the complex conjugation and pointwise multiplication, cf. [3]. Clearly, this space contains all essentially bounded functions, but also functions having a polynomial or even linear exponential growth at infinity. Consider the following sequence (cj )j ∈N0 of positive real numbers: cj :=

1 1 − 2 2j + 2

and put Hj := Dcj ∩ Fϕ .2 Then we obtain a scale of Banach spaces C∼ = H0 ⊂ H1 ⊂ · · · ⊂ Hj ⊂ Hj +1 · · · ⊂ H :=

Hj ⊂ Fϕ ,

(3.1)

j ∈N

where the norm · j of Hj is given by f j := exp{−cj | · |2 }f L∞ (Cn ) . It is well known that the last inclusion H ⊂ Fϕ is dense in the topology of Fϕ . Given two linear spaces X and Y we write L(X, Y ) for all linear operators from X to Y . If in addition X and Y are normed spaces we denote by L(X, Y ) the subspace of all bounded linear 2 The specific choice of the sequence (c ) is needed in Proposition 3.2. j j

468


operators. As usual we shortly write L(X) := L(X, X) and L(X) := L(X, X). With k ∈ N0 and the notations in (3.1) we define Lk (H) := A ∈ L(H): A|Hj ∈ L(Hj , Hj +k ) for all j ∈ N0 . We say that operators in Lk (H) act on the scale (3.1) by an order shift k. Definition 3.1. The space of operators acting on (3.1) by a finite order shift is given by Lfos (H) :=

Lk (H).

k∈N0

Since for Ak ∈ Lk (H) where ∈ {1, 2}, Ak1 ◦ Ak2 ∈ Lk1 +k2 (H) we see that Lfos (H) in fact defines an algebra of linear operators on H. The normalized reproducing kernel of H 2 (Cn , dμ) is

|z|2 kz (u) = exp u · z − , 2

z, u ∈ Cn .

By a straightforward calculation we have kz ∈ H for all z ∈ Cn and for all operators A ∈ Lfos (H) we can define the Berezin transform of A as usual by

= Akz , kz , ∼ : Lfos (H) → C ω Cn : A → A(z) where C ω (Cn ) denotes the space of all real analytic functions on Cn and ·,· is the inner product of L2 (Cn , dμ). It has been shown in [3]: Proposition 3.2. Let f, g ∈ Sym>0 (Cn ) and Tf the Toeplitz operator on Fϕ . Then we have: (a) The restriction of Tf to H defines an element in the algebra Lfos (H). In particular, the product Tg Tf exists as a densely defined operator on Fϕ . (b) The Berezin transform ∼ : Lfos (H) → C ω (Cn ) is one-to-one. From this we obtain a simple result which in the case of bounded symbols f, g ∈ L∞ (Cn ) directly follows from Tf¯ = Tf∗ . Lemma 3.3. Let f, g ∈ Sym>0 (Cn ) such that [Tf , Tg ] = 0, then [Tf¯ , Tg¯ ] = 0. Proof. Note that the Berezin transform of Tf Tg is given by T f Tg (z) = f · Tg kz , kz 2 = e−|z| f (u)g(w)ew·z+w·u+u·z dμ(u, w), Cn ×Cn


469

where the existence of the integrals for all z ∈ Cn follows from Tg kz ∈ H. Hence, Proposition 3.2 shows that the relation [Tf , Tg ] = 0 is equivalent to 0=

f (u)g(w) − f (w)g(u) ew·z+w·u+u·z dμ(u, w),

Cn ×Cn

for all z ∈ Cn . After applying the complex conjugation to this equation and using the transform (u, w) → (w, u) we obtain 0=

w·z+w·u+u·z f¯(u)g(w) ¯ − f¯(w)g(u) ¯ e dμ(u, w),

Cn ×Cn

which (again, by Proposition 3.2) implies that [Tf¯ , Tg¯ ] = 0.

2

4. Commuting Toeplitz operators on the complex plane We consider the case of dimension n = 1 and define a symbol space for the Toeplitz operators in consideration: Definition 4.1. Let S be the subspace of measurable functions with at most polynomial growth at infinity: m S := f : C → C: ∃C, m > 0 such that f (z) C 1 + |z| for all z ∈ C . From S ⊂ Sym>0 (C) and Proposition 3.2 it is clear that {Tf |H | f ∈ S} ⊂ Lfos (H) and therefore all finite products of Toeplitz operators with symbols in S are well defined at least as densely defined operators with domain H. 2 The Gaussian density ϕ is given by ϕ(z) := π −1 e−|z| and for Re(z) > 0 its Mellin transform can be expressed by the usual Gamma function as 1 M[ϕ](z) = π

∞ e

−t 2 z−1

t

1 dt = 2π

0

∞

z

e−t t 2 −1 dt =

z 1 Γ . 2π 2

0

From this one has ϕ(k) ˆ = 2πM[ϕ](2k + 2) = Γ (k + 1) = k!, and therefore it is easy to check that the condition (2.3) is fulfilled for all u ∈ S. We need the following simple observation. Lemma 4.2. Each v ∈ S has an L2 (C, dμ)-convergent expansion of the form ∞ vj (r)eij θ , v reiθ =

z = reiθ .

j =−∞

By interpreting vj as radial functions on the complex plane we have vj ∈ S for all j ∈ Z.

(4.1)

470


Proof. Let v ∈ S, then v 2L2 (C,dμ)

1 = π

∞2π iθ 2 v re dθ re−r 2 dr < ∞. 0 0

By Fubini’s theorem, it follows Φv,r (λ) := v(rλ) ∈ L2 (S1 ) for a.e. r > 0. For such r > 0 we can expand Φv,r into an L2 (S1 )-convergent Fourier-series: Φv,r eiθ = vj (r)eij θ

(4.2)

j ∈Z

with (measurable) coefficients vj (r) = such that vj (r) 1 2π

1 2π

2π 0

v(reiθ )e−ij θ dθ . In particular, there are C, m > 0

2π iθ v re dθ C 1 + r 2 m , 0

which shows that, interpreted as a radial function on the complex plane, vj defines an element in S for all j ∈ Z. Finally, for all ε > 0 we can choose a finite set J ⊂ Z such that ∞ 2 ij θ vj (r) 2 re−r 2 dr < ε. vj (r)e dμ(z) = 2 C

j ∈J /

j ∈J / 0

This shows the L2 (C, dμ)-convergence of the series in (4.1).

2

Now, we analyze Eq. (2.10) in Proposition 2.4. Let u, v ∈ S and j ∈ Z, then we put Fj (z) = Φj (z)Θj (z)

(4.3)

where Θj (z) := M[vj ϕ](j + 2z + 2) and M[uϕ](2z + 2) M[uϕ](2z + 2j + 2) − . Φj (z) := 2π Γ (z + 1) Γ (z + 1 + j )

Since the Gamma function does not have zeros in the complex plane, it follows that: Lemma 4.3. Fj (z) is holomorphic on the half-plane Re(z) > max{−1, −j − 1}. Note that with these notations the condition (2.10) can be written as Fj (k) = 0 for all integers k 0 and j with j + k 0. The following example shows that Φj (k) = 0 for all k ∈ N is possible in case of a non-vanishing symbol u of exponential growth. Hence, with such u the relation (2.10) is fulfilled independently of the choice of vj where j ∈ N.


471

2

Example 4.4. Consider u(t) := eλt where λ ∈ C and Re(λ) < 1. Then we have 1 (λ−1)t 2 Me (z) π ∞ z 1 1 z 2 = . e−(1−λ)t t z−1 dt = (1 − λ)− 2 Γ π 2π 2

M[uϕ](z) =

0

Therefore Φj (z) = (1 − λ)−z−1 1 − (1 − λ)−j . Fix λ := λn ∈ C where n ∈ N such that (1 − λn )−1 = ei be sufficiently large with Re(λn ) < 1. It follows: Φj (k) = e2πi

k+1 n

2π n

i.e. λn = 1 − e−i

2π n

. Moreover, let n

j 1 − e2πi n .

Let j = n, then Φj (k) = 0 for all k 0 with j + k 0. Note that in case of λ = 0 we have Re(λn ) = 1 − cos( 2π n ) > 0 and u(t) is of exponential growth at infinity. Now, we define the function:

j (z) := Φ

1 Fj (z)Γ (z + 1) 2π

= M[uϕ](2z + 2) −

j

−1

(z + )

M[uϕ](2z + 2j + 2) M[vj ϕ](2z + 2 + j ).

=1

=:Hj (z)

(4.4)

j (z) as a Mellin transform. Recall that for a We wish to express the function Hj (z) and finally Φ

(z) on a right half-plane Re(z) > δ the inverse Mellin transform suitable holomorphic function Ψ is given by −1 1

(x) = M Ψ 2πi

c+i∞

(s) ds x −s Ψ

c−i∞

j (z) = Ψ

(z) := whenever the integral exists. Here we put Ψ Then we have

j

=1 (z

+ )−1 with Re(z) > −1.

j }(x) has support in [0, 1]. MoreLemma 4.5. The inverse Mellin transform mj (x) := {M−1 Ψ α over mj (x) = O(x ) for all α < 1 as x → 0.

472


Proof. In case of j = 1 it is known that:

m1 (x) =

x, 0,

if 0 < x < 1, if x > 1

and the assertion directly follows. Now, we assume that j ∈ {2, 3, . . .}. With c > −1 we have 1 mj (x) = 2πi

c+i∞

x −s

c−i∞

j =1

1 1 ds = (s + ) 2πix c

j

x −it

R

=1

1 dt. (c + + it)

j (z) is holomorphic on Re(z) > −1 the above integral is independent of c > −1. Now: Since Ψ mj (x)

1 2πx c

j

R =1

1 (c + )2 + t 2

dt.

(4.5)

As c → ∞ the integral on the right-hand side tends to zero. In particular, it is bounded as a function of c by some η > 0. Hence, for all c > 0 0 mj (x)

η , 2πx c

and this shows that mj (x) = 0 for x > 1. Now, we study the behavior of mj (x), j 2 as x → 0. Since the integral (4.5) converges for c > −1, the assertion follows. 2 Recall that for suitable functions f, g : R+ → C and x > 0 the Mellin convolution is defined by ∞ (f ∗ g)(x) :=

x dy . f (y)g y y

0

Definition 4.6. We define a space A of complex valued measurable functions on R+ by

c A := u(x): R+ → C: ∃C, c > 0 and ∃ρ, η 0 such that u(x) ρ x for all x ∈ (0, 1] and u(x) Cx η for all x ∈ [1, ∞) . In the following we will often identify radial functions on C and functions on R+ in the obvious way. In this sense we have S ⊂ A (see Definition 4.1). We need a simple technical lemma: Lemma 4.7. Let ρ 0. Then g(x) := ex

2

∞ x

e−y y ρ−1 dy is of order O(x ρ ) as x → ∞. 2

Proof. Let 2n + 1 with n ∈ N0 be the smallest odd number ρ − 1. With a real parameter a > 0 we see that there is a polynomial P (x) of degree 2n such that


∞ e

−y 2 2n+1

y

dn dy = (−1) da n

∞

n

x

473

e−ay y dy 2

|a=1

x −ax 2

= (−1)n

dn e da n 2a

= P (x)e

−x 2

|a=1

.

Therefore, we obtain in the case x 1: 0 g(x) ex

2

∞

2 e−y y 2n+1 dy P (x) Dx 2n .

x

Finally, note that 2n ρ due to our choice of n. The proof is complete.

2

Proposition 4.8. With u, v ∈ A we write fu (x) := u(x)e−x and fv (x) := v(x)e−x . Then the Mellin convolution (fu ∗ fv )(x) exists for all x 0 and there is h ∈ A such that 2

(fu ∗ fv )(x) = h(x)e−x ,

2

x ∈ R+ .

Proof. We choose Cw , cw > 0 and ρw , ηw 0 such that w(x) cw x ρw

w(x) Cw x ηw

for all x ∈ (0, 1],

for all x ∈ [1, ∞),

where w ∈ {u, v}. Let x > 1, then (fu ∗ fv )(x)

1 0

fu (y) ycρuu

fv x dy y y x2 ηv − y 2

Cv xy ηv e

x

fu (y)

+ 1

Cu y ηu e−y

2

∞ x dy + f v fu (y) fv x dy . y y y x 2 η −y ηv Cv xy ηv

Cu y

2 − x2 e y

ue

ρv

cv yx ρv

Therefore, we have (fu ∗ fv )(x) Cv cu x ηv

1

x

2

y

x −ρu −ηv −1 − y 2

e

dy + Cu Cv x

0

+ Cu cv x −ρv

ηv

y ηu −ηv e

2

−y 2 − x 2 y

dy

1

∞ x

y ηu +ρv −1 e−y dy. 2

(4.6)

474


We estimate the three integrals on the right as functions of x as x → ∞. Using the transformation rule in the first equality below and Lemma 4.7, we find 1 x

ηv

2

y

x −ρu −ηv −1 − y 2

e

dy = x

−ρu

∞

r ρu +ηv −1 e−r dr D1 x ηv e−x , 2

2

x

0

where D1 > 0 is a suitable constant independent of x > 1. Now, we look at the last term on the right of (4.6). Applying Lemma 4.7 again, it follows

x

−ρv

∞

y ηu +ρv −1 e−y dy D2 x ηu e−x . 2

2

x

In order to estimate the middle term on the right of (4.6) note that for y ∈ [1, x] y2 +

x2 2x x + y. y2

This implies that x x

ηv

2

y

2 x ηu −ηv −y − y 2

e

dy x

ηu +ηv

1

x

2

e

−y 2 − x 2 y

dy

1

x

x

ηu +ηv −x

x

ηu +ηv −x−1

e

e−y dy

1

e

.

Summarizing these estimates, we have with a suitable constant C > 0 and x > 1: (fu ∗ fv )(x) C x ηv e−x 2 + x ηu e−x 2 + x ηv +ηu e−x . In case of x ∈ [1, ∞) we define h(x) := ex (fu ∗ fv )(x). Then we have h(x) Ch x ηv +ηu with a suitable constant Ch > 0. Next, we estimate the Mellin convolution (fu ∗ fv )(x) with x ∈ (0, 1]. We can assume that ρu ρ v :


(fu ∗ fv )(x)

x 0

475

2 x2 u(y) v x e−y − y 2 dy + y y cu y ρu

1 + x

ηv

Cv xy ηv

∞ 2 x2 x −y 2 − x 22 dy e u(y) v u(y) v x e−y − y 2 dy . y + y y y y cu 1 C y ηu y ρu

u

ρv

cv yx ρv

ρv

cv yx ρv

Therefore, we have (fu ∗ fv )(x) cu Cv x ηv

x

y −ρu −ηv −1 e

2

− x2 y

dy

0

cu cv + ρ x v

1

2

y

x ρv −ρu −1 − y 2

e

Cu c v dy + ρ x v

x

∞

y ηu +ρv −1 e−y dy. 2

(4.7)

1

We estimate the three terms on the right separately. Applying the transformation rule we have as x → 0: x x

ηv

2

y

x −ρu −ηv −1 − y 2

e

dy = x

−ρu

0

∞

2 r ρu +ηv −1 e−r dr = O x −ρu .

1

As for the second integral, we have

x −ρv

1

y ρv −ρu −1 e

2

− x2 y

dy = x −ρu

x

=

1

r ρu −ρv −1 e−r dr 2

x

O(x −ρu ), O(x −ρu log x1 ),

if ρu > ρv , if ρu = ρv .

Since the last term on the right of (4.7) is of order O(x −ρv ) as x → 0 we see that there is a constant C > 0 such that (fu ∗ fv )(x) C x −ρu log 1 + x −ρv . x Again, we set h(x) = ex (fu ∗ fv )(x) with x ∈ (0, 1]. Then there is ch > 0 such that |h(x)| ch and hence h ∈ A as desired. The proof is complete. 2 x ρu +ρv +1 Let u, v ∈ A and in addition assume that the support supp v of v is contained in [0, 1]. As 2 before we write fu (x) := u(x)e−x . Then we have:

476


Proposition 4.9. Under the above assumptions there is h ∈ A such that (fu ∗ v)(x) = h(x)e−x ,

x ∈ R+ .

2

Proof. We use the same notations as in the proposition before. Since supp v ⊂ [0, 1], we have for x 1 and together with Lemma 4.7 (fu ∗ v)(x)

∞ x

fu (y) v x dy y y η −y 2

Cu y

Cu c v ρ x v

ue

∞

ρv

cv yx ρv

y ηu +ρv −1 e−y dy Cx ηu e−x , 2

2

x 2

where C > 0 is a suitable constant. We set h(x) := ex (fu ∗ v)(x) in case of x 1. In case of x ∈ (0, 1] we can apply the same calculation as in the proof of Proposition 4.8. 2

j (z) = j (z + )−1 . According Now, we return to (4.4). Let j ∈ N and as before we write Ψ =1

j ⊂ [0, 1] such that to Lemma 4.5 there is a function m

j : R+ → C with supp m

j (z). M[

mj ](z) = Ψ Now, the transformation y :=

j (z) = Ψ

√ x shows

1

1 m

j (x)x

0

z−1

dx = 2

m

j y 2 y 2z−1 dy = M[mj ](2z),

0

mj (y 2 ). Clearly, it holds supp mj ⊂ [0, 1]. Hence (4.4) can be written where we write mj (y) := 2! in the form:

j (z) = M ux 2 ϕ (2z) − M[mj ](2z)M ux 2j +2 ϕ (2z) M vj x 2+j ϕ (2z). Φ Assume u, vj ∈ A, then we have ux 2 , ux 2j +2 , vj x 2+j ∈ A. Due to Proposition 4.9, there is a function h ∈ A such that 2 mj ∗ ux 2j +2 ϕ = he−x . According to the Mellin convolution theorem we have for Re(z) sufficiently large 2 M[mj ](2z) · M ux 2j +2 ϕ (2z) = M mj ∗ ux 2j +2 ϕ (2z) = M he−x (2z) and with this notations

j (z) = M ux 2 ϕ (2z) − M he−x 2 (2z) M vj x 2+j ϕ (2z). Φ


477

Next, we can apply Proposition 4.8 to see that there are functions g1 , g2 ∈ A such that (i) [(ux 2 ϕ) ∗ (vj x 2+j ϕ)](x) = g1 (x)e−x , 2 (ii) [(he−x ) ∗ (vj x 2+j ϕ)](x) = g2 (x)e−x . Again, we can use the Mellin convolution theorem in order to see that

j (z) = M ux 2 ϕ ∗ vj x 2+j ϕ (2z) − M he−x 2 ∗ vj x 2+j ϕ (2z) Φ = M (g1 − g2 )e−x (2z). We have shown with g := g1 − g2 :

j (z) = M[ge−x ](2z). Proposition 4.10. With the notations in (4.4) there is g ∈ A such that Φ The next proposition is essential in our proof. It is a replacement for the Blaschke condition which is used by the authors of [10] to prove Theorem A of the introduction. Proposition 4.11. Let u ∈ A and a ∈ (0, 2]. For fixed number m0 ∈ N the condition: ∞

u(t)e−t t ak dt = 0,

(4.8)

0

for all k m0 implies that u = 0 a.e. on R+ . Proof. Since with m0 ∈ N0 we have ∞

−t a(m0 +k)

u(t)e t

∞ dt = t

0

m0 a

u(t)e−t t ak dt = 0

0

for all k ∈ N. Hence, we can assume that u is integrable over [0, 1] and that m0 = 0. First, we consider the case a ∈ (0, 2). The transformation r = t a in the integral (4.8) implies for all k ∈ N0 : 1 a

∞ 0

1 1 a 1 u r a e−r r a −1 r k dr = 0.

=:h(r)

With x ∈ R consider the sum: √ ∞ (x r )2k (−1)k h(r) dr (2k)! k=0 0 ∞

0=

=0

478


∞ =

√ h(r) cos(x r ) dr

0

∞ =2

rh r 2 cos(xr) dr.

(4.9)

0 √ √ " (x r )2k x r for all m ∈ N and 1 > 1 together In the second equality we have used m k=0 (2k)! e a 2 with Lebesgue dominated convergence theorem in order to interchange the sum and integral. We extend s(r) := rh(r 2 ) ∈ L1 (R+ ) by zero to an integrable function on the real line. Then we have ixr −ixr 0 = s(r) e + e dr = s(r) + s(−r) eixr dr. R

R

Since the Fourier transform is one-to-one on L1 (R) it follows s(r) + s(−r) = 0 a.e. on R. Since s(−r) = 0 in case of r > 0 we have s(r) = 0 a.e. on R+ and this implies that u = 0, a.e. on R+ . Next, we consider the case a = 2. We can exchange the integration and summation in (4.9) at least in the case |x| < 1. Hence, we have for all |x| < 1 and with our former notations ∞ s(r) + s(−r) eixr dr. 0= 0

Let us replace x by the complex variable z = σ + it. From |eizr | = e−tr and 0 s(r) = rh r 2 = u(r)e−r Cu r ηu e−r with some positive Cu > 0 and ηu > 0 it follows that F (z) = s(r) + s(−r) eizr dr R

defines a holomorphic function on N := {z ∈ C | −1 < Im(z) = t < 1}. Since F (x) = 0 whenever |x| < 1 we have F ≡ 0 on N . In particular, the Fourier transform of s(r) + s(−r) vanishes identically. As before we obtain u(r) = 0 a.e. on R+ . 2 The next example shows that Proposition 4.11 fails in the case of a > 2. Example 4.12. Consider ga (r) := sin(ar)e−r where a > 0. Then: M[ga ](z) =

1 2i

∞ 0

=

e−(1−ia)r r z−1 dr −

1 2i

∞

e−(1+ia)r r z−1 dr

0

1 1 (1 − ia)−z Γ (z) − (1 + ia)−z Γ (z) 2i 2i


479

− z 1 1 + a 2 2 eiz arctan a − e−iz arctan a Γ (z) 2i − z = 1 + a 2 2 sin(z arctan a)Γ (z).

=

Let β > 2, then M[ga ](βz) vanishes exactly for βz arctan a = kπ with k ∈ Z. Choose a with arctan a = πβ < π2 , then M[ga ](βk) = 0 for all k ∈ N. In our arguments below we need some well-known estimates on the Gamma function: Lemma 4.13. There are constants A > 0 and C > 0 independent of z, such that for Re z = σ > A σ −1

2 4 Γ (σ ) σ2 |t| C 2 . exp |t| arctan |Γ (z)| σ σ + t2

(4.10)

In particular, we have for Re z = σ > A > 1

π Γ (σ ) C exp |t| . |Γ (z)| 2

(4.11)

Proof. We write z = σ + it ∈ C and we use a well-known formula on the asymptotic expansion of log Γ (z + 1) (cf. [18, p. 279]): 1 1 log Γ (z) = z − log z − z − log(2π) + ψ(z), 2 2 where lim|z|→∞ ψ(z) = 0. Consider the line Lσ := {z ∈ C | Re(z) = σ }, which for σ > 0 is parametrized by

t 2 2 Lσ = z := σ + t exp i arctan t ∈R . σ If we insert such z ∈ Lσ into the above formula, we obtain

1 t 1 log σ 2 + t 2 + i arctan − z − log 2π + ψ(z) . Γ (z) = exp z − 2 σ 2 Therefore, there is ρ : C → R such that lim|z|→∞ ρ(z) = 1 and

1 1 t log σ 2 + t 2 − t arctan − σ − log 2π 2 σ 2

σ −1 e−σ 2 |t| = ρ(z) √ , σ + t 2 2 4 exp −|t| arctan σ 2π

Γ (z) = ρ(z) exp

σ−

which gives the desired result. The proof is complete.

2

(4.12)

480


Let u ∈ A and assume that Ψj (z) :=

M[uϕ](2z + 2) Γ (z + 1)

is periodic on Re(z) A with period j ∈ N. Therefore, it can be considered as an entire function on the complex plane. With the notations in the proof of Proposition 4.8 we have for σ = Re(z) ρ2u M[uϕ](2z + 2) 1 π

∞ u(t) e−t 2 t 2σ +1 dt 0

cu π

1 e

−t 2 2σ −ρu +1

t

Cu dt + π

0

∞

e−t t 2σ +ηu +1 dt 2

1

cu ρu Cu ηu Γ σ− +1 + Γ σ+ +1 2π 2 2π 2 ηu +1 , CΓ σ + 2 where C > 0 is a suitable constant. Without lost of generality we can assume that ηu is an even integer. According to (4.11) in Lemma 4.13, it follows Γ (σ + η2u + 1) Ψj (z) C |Γ (z + 1)| ηu Γ (σ ) ηu σ+ − 1 · · · (σ + 1) C σ + 2 2 |Γ (z)|

m π |t| C1 |σ | + 1 exp 2 where C1 > 0 and m ∈ N are sufficiently large. Let z = σ + it ∈ C and choose k ∈ Z with ρu ρu + j σ + jk . 2 2 By the periodicity of Ψj (z), we have

m Ψj (z) = Ψj (z + j k) C1 ρu + j + 1 exp π |t| C2 exp π |z| . 2 2 2 We have proved: Lemma 4.14. If the entire function Ψj is periodic of period j , then Ψj has linear exponential growth as |z| → ∞.


481

The next proposition characterizes the class of all periodic entire functions of at most linear exponential growth at infinity. Proposition 4.15. Let f be an entire function of period j ∈ N. Assume that there are A, B > 0 such that f (z) AeB|z| . Then f is a trigonometric polynomial of the following type: f (z) =

n

a e

2πi z j

a ∈ C.

,

(4.13)

=−n

Proof. Let w = re2πiϕ ∈ C∗ with r > 0 and ϕ ∈ [0, 1) and define ji ji log w = f j ϕ − log r . g(w) := f − 2π 2π Note that g is an entire function on the complex plane since f has the period j ∈ N. If n is an integer greater than B we have in case of r > 1 nj w g(w) Ar nj exp{nj ϕ + nj log r} = Ar

2nj = A|w|

2nj .

where A

> 0 is a In case of 0 < r 1, we have |w nj g(w)| Ar nj exp{nj ϕ + nj log r −1 } = A, suitable constant. Therefore

O(|w|2nj ), as |w| → ∞, nj w g(w) = O(1), as |w| → 0. The second estimate shows that w nj g(w) has a removable singularity at 0. We remove it. The first inequality shows that w nj g(w) is a polynomial of degree 2nj . Therefore, 2πiz f (z) = g e j is a trigonometric polynomial of the type (4.13).

2

So, we see from Lemma 4.14 and Proposition 4.15 that Ψj must be a trigonometric polynomial of the form Ψj (z) =

n

a e

2πi z j

=−n

=

n

a e

2πi z j

+

=−n |4 |>j

=: Ψj+ (z) + Ψj− (z).

n

a e

2πi z j

=−n |4 |j

(4.14)

482


First, we observe that the trigonometric polynomial Ψj− (z) can be written in the form: Ψj− (z) =

M[wϕ](2z + 2) Γ (z + 1)

with a suitable function w (which is of exponential growth as |z| → ∞). This follows from our calculations in Example 4.4. More precisely, for λ ∈ C and Re(λ) < 1 2 z 1 z (1 − λ)− 2 Γ . M eλr ϕ (z) = 2π 2 Therefore: 2

M[2π(1 − λ)eλr ϕ](2z + 2) = (1 − λ)−z . Γ (z + 1) If we choose 1 − λ = e

− 2πi j

where |4 | < j , then we have Re(λ) = 1 − cos 2π j < 1 and with the 2

definition w(r) := 2π(1 − λ)eλr it follows that 2πi z M[wϕ](2z + 2) =e j . Γ (z + 1)

We still need to consider the case |4 | = j , which means that e we have in case of −1 < Re(z) < 0:

− 2πi j

= ±i. Let 4 = −j , then

2

πiz M[−2ieit ](2z + 2) =e 2 . Γ (z + 1)

The case 4 = j can be treated in a similar way. Note that in these calculations and for λ = 0 we have Re(λ) > 0, which implies that w ∈ / A. With our notations in (4.14) we prove: Proposition 4.16. Let u ∈ A such that Ψj (z) =

M[uϕ](2z + 2) Γ (z + 1)

is an entire function of period j ∈ N. Then u must be a constant function. Proof. By what was said before Ψj (z) is a trigonometric polynomial of the form (4.14) and using the assumption and the notations above we can write M[uϕ](2z + 2) = Γ (z + 1) Ψj+ (z) + Ψj− (z) . First we show that Ψj+ must vanish identically. Otherwise there is 0 ∈ {−n, . . . , 0, . . . , n} such that | 0 | = max{| |: a = 0} > j4 . To fix a particular case let us assume that 0 < 0. It is easy to check that Ψj+ (ir) ∼ a 0 e

−

2π 0 r j

,

as r → ∞.


483

√ From (4.12), we known that Γ (ir + 1) ∼ const · 1 + r 2 exp{−r arctan r} as r → ∞. Therefore, + π 0 we conclude from − 2π j > 2 = limr→∞ arctan r that r → Γ (ir + 1)Ψj (ir) has growth of exponential order as r → ∞. A similar argument shows that r → Γ (ir + 1)Ψj− (ir) can have at most (linear) polynomial growth as r → ∞. Finally, note that r → M[uϕ](2ir + 2) is bounded: M[uϕ](2ir + 2)

∞ |uϕ|(t)t 2 dt < ∞. 0

Hence, we must find u with M[uϕ](2z + 2) = Γ (z + 1)Ψj− (z).

(4.15)

Due to our calculations above we can find λ , a ∈ C with 1 Re(λ ) 0 such that for all z ∈ C with −1 < Re(z) < 0: n 2 Γ (z + 1)Ψj− (z) = M ϕ a eλ r (2z + 2). =−n

The Mellin transform is one-to-one and therefore we conclude that (4.15) is uniquely solved by u(r) =

n

2

a eλ r .

=−n

Since u ∈ A is of at most polynomial growth at infinity, we have a = 0 if λ = 0 and u must be constant. 2 Now, we can formulate and prove our main theorem in this section: Theorem 4.17. Let u, v ∈ S and assume that u is radial and non-constant. If Tu Tv = Tv Tu on H, cf. (3.1), then v is a radial function. We remark that the condition u, v ∈ S is essential. A counter example to the above statement in case of u ∈ / S and even bounded v is given in Example 5.8. Proof of Theorem 4.17. According to Lemma 4.2 there is an expansion of v: ∞ v reiθ = vj (r)eij θ ,

z = reiθ ,

(4.16)

j =−∞

which is convergent in L2 (C, dμ) and it holds vj ∈ A for all j ∈ Z. Assume, that v is not radial, then there is j ∈ Z \ {0} such that vj (r) = 0. Using Lemma 3.3 we can assume that j ∈ N. Using the notation in (4.3) we see that the holomorphic function Θj does not vanish on a right half-plane. Proposition 2.4 shows that Fj (k) = Φj (k)Θj (k) = 0

484


j (k) = 0 for k ∈ N where for all k ∈ N. From this we have Φ

j (z) := Φ

1 Fj (z)Γ (z + 1) 2π

is the holomorphic function on Re(z) > −1 in (4.4). According to Proposition 4.10 there is g ∈ A such that

j (z) = M ge−x (2z) Φ

j (k) = ∞ g(x)e−x x 2k−1 dx = 0 for all k ∈ N. Using Proposition 4.11 with and therefore Φ 0 a = 2 it follows that g = 0 a.e. on R+ and therefore

j (z) = Φ

1 Φj (z)Θj (z)Γ (z + 1) = 0 2π

on Re(z) > −1. Consequently, we have M[uϕ](2z + 2) M[uϕ](2z + 2j + 2) ≡ 0, Φj (z) = 2π − Γ (z + 1) Γ (z + 1 + j ) which shows that Ψj (z) :=

M[uϕ](2z + 2) Γ (z + 1)

is an entire function on the complex plane of period j ∈ N. Finally, using Proposition 4.16 it follows that u is constant, which gives a contradiction and the proof is finished. 2 As a direct consequence of Theorem 4.17 we remark: Corollary 4.18. Let v ∈ S and assume that the Toeplitz operator Tv is diagonal with respect to 1 the standard orthonormal basis {(j !)− 2 zj : j ∈ N0 }. Then v has to be a radial function. Proof. Fix a non-constant radial function u ∈ S. Due to our assumption on Tv we have [Tu , Tv ] = 0 and the assertion follows from Theorem 4.17. 2 Note that Corollary 4.18 can also be proved directly. Let Rv denote the radialization of the symbol v. In case the Toeplitz operator Tv is diagonal, it can be checked that TRv = Tv , which implies that TRv−v = 0. It follows that v = Rv is a radial function. 5. Discussion on C n We use the notations of Section 3, and we write P[z, z] for the space of all polynomials in the complex variables z = (z1 , . . . , zn ) and z = (z1 , . . . , zn ). For simplicity we only consider Toeplitz operators Tf with symbols f ∈ P[z, z]. In general, such operators are unbounded but as a common dense domain of definition we can choose: D := P[z, z] ∩ H 2 Cn , dμ


485

which is the space of all holomorphic polynomials on Cn . Moreover, one can easily check that D is invariant under all Toeplitz operators Tf with f ∈ P[z, z]. The linear space: O := Tf : D → D f ∈ P[z, z] is also invariant under the operator product. More precisely, it holds (cf. [3,9]): Lemma 5.1. Let p, q ∈ P[z, z], then the product Tp Tq is a Toeplitz operator Tpq , where the symbol p q can be calculated as pq =

(−1)|γ | ∂γ p · ∂ γ q. γ! n

(5.1)

γ ∈N0

Here we write ∂γ :=

∂ |γ | ∂zγ

∂ |γ | ∂zγ

and ∂ γ :=

where γ ∈ Nn0 .

By using this result, we can easily construct non-radial commuting Toeplitz operators in case of dimension n 2. Consider the linear subspace P0 of P[z, z] defined by m 2 a α z α . P0 := p ∈ P[z, z]: ∃m ∈ N, ∃aα ∈ C such that p(z) = j =0 |α|=j

Lemma 5.2. The space (P0 , ) is a commutative -sub-algebra of (P[z, z], ). Proof. First assume that p, q ∈ P0 have the form p(z) = |zα |2 and q(z) = |zβ |2 where α, β ∈ Nn0 . Note that

∂ |γ | p ∂ |γ | q β! α! α−γ α β β−γ · z z · = z z ¯ ∂zγ ∂ z¯ γ (α − γ )! (β − γ )! α!β! zα+β−γ 2 = (α − γ )!(β − γ )! =

∂ |γ | q ∂ |γ | p · . ∂zγ ∂ z¯ γ

By linearity it follows that P0 is closed under the -product and the -multiplication is commutative. 2 As an immediate consequence of Lemmas 5.1 and 5.2 we find that O0 := {Tp : p ∈ P0 } is a commutative sub-algebra in O. Let p, q ∈ P[z, z] be of degree k and m with principal parts pk and qm , respectively. We see from Lemma 5.1 that the relation [Tp , Tq ] = Tp Tq − Tq Tp = 0 is equivalent to 0=pq −q p=

∞ 1 (−1) {∂γ p · ∂ γ q − ∂γ q · ∂ γ p}. γ! =0

|γ |=

486


In general, the right-hand side is a polynomial of degree m + k − 2 having the principal part: PP(p q − q p) =

n ∂pk ∂qm ∂qm ∂pk . − ∂zj ∂zj ∂zj ∂zj j =1

Corollary 5.3. A necessary condition for [Tp , Tq ] = Tp Tq − Tq Tp = 0 is 0=

n ∂pk ∂qm ∂qm ∂pk . − ∂zj ∂zj ∂zj ∂zj

(5.2)

j =0

Consider the space Pa [z] := {p ∈ P[z, z] | p is complex analytic}, which we also have denoted by D before. As a simple consequence of Corollary 5.3 we have: Proposition 5.4. Let n = 1 and assume that p ∈ Pa [z] is non-constant. If q ∈ P[z, z] such that [Tp , Tq ] = 0, then q ∈ Pa [z]. k Proof. Using our former notations we conclude from Corollary 5.3 and ∂p ∂z = 0 that ∂pk ∂qm ∂qm ∂z · ∂z = 0. If pk is non-constant we see that ∂z = 0, which implies that qm ∈ Pa [z]. Now, we can apply the same argument to p and q − qm , using [Tp , Tq−qm ] = 0 to show that qm−1 (the principal part of q − qm ) is in Pa [z]. The assertion follows by induction. 2

Next, we consider the space of radial symmetric polynomials: Prad [z, z] := p |z|2 ∈ P[z, z] p = p(r) is a polynomial in one variable r and the space P1 of all polynomials p ∈ P[z, z] which fulfill the invariance p(eit z) = p(z) for all t ∈ R and z ∈ Cn :

α β P1 := p(z) = aα,β z z ∈ P[z, z] aα,β = 0 if |α| = |β| . (5.3) α,β∈Nn0

We clearly have Prad [z, z] ⊂ P0 ⊂ P1 and all the inclusions are strict. Generalizing Lemma 5.2 we can prove the following: Lemma 5.5. Let p ∈ Prad [z, z] and q ∈ P1 , then the Toeplitz operators Tp and Tq commute. Proof. Let ∈ N and α, β, γ ∈ Nn0 . Then it follows by a direct calculation: ∂ |γ | 2 ∂ |γ | α β β |γ |!γ !zα zβ |z|2( −|γ |) , |z| · γ z z = γ ∂z ∂z |γ | γ ∂ |γ | α β ∂ |γ | 2 α z z · γ |z| = |γ |!γ !zα zβ |z|2( −|γ |) . ∂zγ ∂z |γ | γ Hence, we have


|z|2 zα zβ = zα zβ

487

β |γ |!(−1)|γ | |z|2( −|γ |) , |γ | γ n

γ ∈N0

zα zβ |z|2 = zα zβ

α |γ |!(−1)|γ | |z|2( −|γ |) . |γ | γ

γ ∈N0

" In order to see that |z|2 zα zβ = zα zβ |z|2 in case of |α| = |β| note that |γ |=m γβ = |β| m for m ∈ N0 . The assertion now follows by linearity of the f g-product in f and g. 2 Finally, we state the converse result. A similar statement in case of Toeplitz operators acting on Bergman spaces over the unit ball has been given in [12]. Proposition 5.6. Let p ∈ Prad [z, z] be non-constant and q ∈ P[z, z] such that [Tq , Tp ] = 0, then q ∈ P1 . " Proof. Let p(z) = kj =0 aj |z|2j be non-constant with principal part pk (z) = ak |z|2k . The gen" "m " α β eral form of q is q(z) = m j =0 j =0 qj (z). We define the differential |α|+|β|=j aα,β z z = operators Lj := zj

∂ ∂ − zj , ∂zj ∂zj

where j = 1, . . . , n. Then condition (5.2) can be written in the form: 0 = |z|2k−2

n

Lj q m ,

j =1

which implies that 0=

n

aα,β Lj zα zβ =

j =1 |α|+|β|=m

=

n

aα,β (αj − βj )zα zβ

j =1 |α|+|β|=m

aα,β |α| − |β| zα zβ .

|α|+|β|=m

It follows that aα,β = 0 for |α| = |β| and therefore qm ∈ P1 . According to our assumptions and due to Lemma 5.5 the Toeplitz operator Tq−qm also commutes with Tp and we can use induction to finish the proof. 2 We give a generalization of Corollary 4.18 to Toeplitz operators on H 2 (Cn , dμ), n > 1 with polynomial symbols: Corollary 5.7. Let q ∈ P[z, z] and assume that Tq is diagonal with respect to the standard or1

thonormal basis B := [eα := (α!)− 2 zα : α ∈ Nn0 ]. Then q ∈ P0 , i.e. q is radial in all components z1 , . . . , zn .

488


Proof. Let pj (z) := |zj |2 for j = 1, . . . , n such that pj ∈ P0 . Then the operators Tpj are diagonal with respect to the basis B. Since by assumption Tq is diagonal with respect to B as well, it follows that [Tpj , Tq ] = 0 and (5.2) shows that Lj qm = 0 for all j = 1, . . . , n. In the same way as in the proof of Proposition 5.6 we obtain that q ∈ P0 . 2 There are bounded Toeplitz operators with radial symbols of exponential growth at infinity that are commuting with Toeplitz operators having non-radial bounded symbol. The construction is closely related to our observations in Example 4.4: 2

Example 5.8. On Cn we consider the radial function gλ (z) := eλ|z| where λ ∈ C such that Re(λ) < 12 . The latter condition ensures that the Toeplitz operator Tgλ is densely defined on H 2 (Cn , dμ). Next, we consider the composition operators Uγ : Dγ ⊂ H 2 Cn , dμ → H 2 Cn , dμ : f → f (γ z) where γ ∈ C. By using the reproducing kernel K(z, w) = ez·w of H 2 (Cn , dμ) we see that the

γ (z) of Uγ is given by Berezin transform U # $

γ (z) = e−|z|2 Uγ K(·, z), K(·, z) = e(γ −1)|z|2 . U On the other hand, the Berezin transform of Tgλ is given by −|z|2 T! gλ (z) = e

gλ (u)ez·u+z·u dμ(u) Cn

= π −n e−|z|

2

2

ez·u+z·u−(1−λ)|u| dv(u) Cn

= (1 − λ)−1 eλ(1−λ)

−1 |z|2

.

Since the Berezin transform is one-to-one on (suitable) operators we find that (1 − λ)−1 U(1−λ)−1 = Tgλ . For all m ∈ N we define λm := 1 − e− Then Tgλm is well defined and

2iπ m

e

. By choosing m sufficiently large, it follows Re λm < 12 . 2iπ m

U

e

2iπ m

= Tg λ m .

(5.4)

Note that by this equation the Toeplitz operator Tgλm is unitary with unbounded symbol. We shortly write Vk := U 2iπ where k ∈ Z and remark that for any bounded symbol f e

k

Vk Tf V−k = TVk f .

(5.5)

It immediately follows from (5.5) and (5.4) that Tgλm commutes with all Toeplitz operators with symbols that are invariant under Vm . To give an explicit example, choose n = 1 and m = 8 such


that Re λm = 1 − cos( π4 ) = 1 − non-radial function f (z) =

z8 . |z|8

√

2 2

489

< 12 . Then Tgλ8 commutes with Tf where f is the bounded

Problem. Is there an extension of the results in Section 5 to Toeplitz operators with arbitrary measurable symbols that have at most polynomial growth at infinity? Acknowledgments We thank Trieu Le for his very helpful remarks on the proof of Proposition 5.6. We are also grateful to the referee of this paper who gave valuable comments, especially regarding a generalization of results in Section 2. References ˇ cković, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator The[1] S. Axler, Ž. Cuˇ ory 14 (1991) 1–11. [2] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961) 187–214. [3] W. Bauer, Berezin Toeplitz quantization and composition formulas, J. Funct. Anal. 256 (2009) 3107–3142. [4] C.A. Berger, L.A. Coburn, Toeplitz operators on the Segal–Bargmann space, Trans. Amer. Math. Soc. 301 (1987) 813–829. [5] H. Bommier-Hato, E.H. Youssfi, Hankel operators on weighted Fock spaces, Integral Equations Operator Theory 59 (2007) 1–17. [6] A. Brown, P. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1964) 89–102. [7] B.R. Choe, H. Koo, Y.J. Lee, Commuting Toeplitz operators on the polydisk, Trans. Amer. Math. Soc. 356 (2004) 1727–1749. [8] B.R. Choe, Y.J. Lee, Pluriharmonic symbols of commuting Toeplitz operators, Illinois J. Math. 37 (1993) 424–436. [9] L.A. Coburn, On the Berezin Toeplitz calculus, Proc. Amer. Math. Soc. 129 (11) (2001) 3331–3338. ˇ cković, N.V. Rao, Mellin transform, monomial symbols and commuting Toeplitz operators, J. Funct. Anal. 154 [10] Ž. Cuˇ (1998) 195–214. [11] S. Grudsky, R. Quiroga-Barranco, N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators and quantization on the unit disc, J. Funct. Anal. 234 (2006) 1–44. [12] T. Le, The commutants of certain Toeplitz operators on weighted Bergman spaces, J. Math. Anal. Appl. 348 (1) (2008) 1–11. [13] Y.J. Lee, Commuting Toeplitz operators on the Hardy space of the polydisc, Proc. Amer. Math. Soc. 138 (1) (2010) 189–197. [14] R. Quiroga-Barranco, N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball, I. Bargmanntype transforms and spectral representations of Toeplitz operators, Integral Equations Operator Theory 59 (3) (2007) 379–419. [15] Sangadji, K. Stroethoff, Compact Toeplitz operators on generalized Fock space, Acta Sci. Math. (Szeged) 47 (1998) 387–400. [16] N. Vasilevski, Bergman space structure, commutative algebras of Toeplitz operators and hyperbolic geometry, Integral Equations Operator Theory 46 (2003) 235–251. [17] N. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space, Oper. Theory Adv. Appl., Birkhäuser, 2008. [18] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, fourth ed., Cambridge Univ. Press, 1927.


Monotonicity properties of the Neumann heat kernel in the ball ✩ Mihai N. Pascu ∗ , Maria E. Gageonea Faculty of Mathematics and Computer Science, Transilvania University of Bra¸sov, Bra¸sov 500091, Romania Received 1 June 2010; accepted 24 August 2010 Available online 1 September 2010 Communicated by Daniel W. Stroock

Abstract A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball U ⊂ Rn is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball. © 2010 Elsevier Inc. All rights reserved. Keywords: Neumann heat kernel; Reflecting Brownian motion; Hot Spots conjecture

1. Introduction We learned from Rodrigo Bañuelos the following conjecture of Richard Laugesen and Carlo Morpurgo which arose in connection with their work on conformal extremals of zeta functions of eigenvalues under Neumann boundary conditions in [9]: Conjecture 1 (Laugesen–Morpurgo Conjecture). Let pU (t, x, y) denote the heat kernel for the Laplacian with Neumann boundary conditions on the unit ball U = {z ∈ Rn : z < 1} in Rn (n 1). ✩

The authors kindly acknowledge the support from CNCSIS - UEFISCSU research grant PNII - IDEI 209/2007.


E-mail addresses: [email protected] (M.N. Pascu), [email protected] (M.E. Gageonea). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.014

M.N. Pascu, M.E. Gageonea / Journal of Functional Analysis 260 (2011) 490–500

491

For any t > 0 we have pU (t, x, x) < pU (t, y, y),

(1)

for all x, y ∈ U with x < y. Remark 2. The fact that the Laugesen–Morpurgo conjecture is true in the 1-dimensional case is known (see for example [4], Remark 5.4 for an analytic proof, or [12,14] for two different probabilistic proofs). Our main result in Theorem 9 (in the case n = 1) gives another probabilistic proof of the Laugesen–Morpurgo conjecture in the 1-dimensional case. Surprisingly, despite the seemingly simple nature of this conjecture and the fact that it seems to have been well known since 1994, we do not know of any progress on it, aside from some partial related results (see [12–14]). A more recent result related to this conjecture is due to Bañuelos et al. [4], in which the authors show that if we replace pU (t, x, y) by the transition density of the n-dimensional Bessel process on (0, 1] reflected at 1, then the monotonicity (1) in the Laugesen–Morpurgo conjecture holds in the case n > 2, and that this is false for n = 2. Since the absolute value of an n-dimensional Brownian motion is a Bessel process of order n, this is equivalent to the monotonicity with respect to r ∈ [0, 1] of the integral mean r n−1

pU (t, re1 , ru) dσ (u),

∂U

where e1 = (1, 0, . . . , 0) ∈ Rn and σ is the normalized surface measure on ∂U. In this paper, we use probabilistic arguments (couplings of reflecting Brownian motions) to settle the Laugesen–Morpurgo conjecture. The paper is organized as follows: in Section 2 we present the mirror coupling introduced by Kendall [8] and developed by Burdzy et al. ([6], and more recently [1,2]), and we establish the notation. In Section 3, we begin with a detailed analysis of mirror coupling of reflecting Brownian motions in the unit ball, which shows that the hyperplane of symmetry between the two reflecting Brownian motions (the mirror of the coupling) moves towards the origin (Lemma 5). Using Lemma 5, in Theorem 6 we obtain a comparison result for the transition probabilities of reflecting Brownian motion in the unit ball, which is the key for our proof of the Laugesen– Morpurgo conjecture. Using this result, we obtain a double inequality for Neumann heat kernel of the unit ball (the double inequality in Theorem 7), and as a corollary we conclude with a short proof of Laugesen–Morpurgo conjecture (Theorem 9). 2. Preliminaries Our proof of Laugesen–Morpurgo conjecture relies on a certain property of the mirror coupling of reflecting Brownian motions in the unit disk and a representation of the Neumann heat kernel as an occupation time density of reflecting Brownian motion. We begin with a presentation of these results. We denote by U = {z ∈ Rn : z < 1} the open unit ball in Rn (n 1) and by ν(z) = −z, z ∈ ∂U, the inward unit vector field on the boundary of U.

492


Given a hyperplane H ⊂ Rn , we say that the points x, y ∈ Rn are separated by H if x and y lie in different components of Rn − H, and we say that they are not separated by H otherwise. We define the reflecting Brownian motion in U as a solution of the stochastic differential equation:

1 Xt = X0 + Bt + 2

t ν(Xs ) dLs ,

t 0.

(2)

0

Formally we have: Definition 3. Xt is a reflecting Brownian motion in U starting at x0 ∈ U if it satisfies (2), where: (a) Bt is an n-dimensional Brownian motion started at 0, (b) Lt is a continuous non-decreasing process which increases only when Xt ∈ ∂U, (c) Xt is (FtB )-adapted, and almost surely X0 = x0 and Xt ∈ U for all t 0. Remark 4. For pathwise existence and uniqueness of reflecting Brownian motion in the sense of the above definition see for example [5]. The notion of mirror coupling was introduced by W.S. Kendall in [8] in the case of Brownian motions on a complete Riemannian manifold with nonnegative Ricci curvature, and was considered in [15] in the case of reflected processes. In [6], and more recently in [1] and [2], K. Burdzy et al. gave a detailed construction of the mirror coupling of reflecting Brownian motions Xt and Yt in a smooth planar domain D ⊂ Rn , and used it in order to derive various properties related to Neumann eigenvalues and eigenfunctions of the Laplacian on D. The construction of mirror couplings was extended recently by the first author in [10] to the case when the two reflecting Brownian motions live in different smooth domains D1 , D2 ⊂ Rn satisfying an additional assumption (this condition is satisfied in particular if D1 , D2 have non-tangential boundaries and D1 ∩ D2 is a convex domain). We will briefly present the construction of the mirror coupling in a smooth domain D ⊂ Rn , and then we give the formal construction in the case of the unit ball U in Rn , n 1. The idea of the mirror coupling is that the two processes behave like ordinary Brownian motions (symmetric with respect to a hyperplane, called the mirror of the coupling) when both of them are inside the domain D. When one of the processes hits the boundary, the mirror Mt gets a minimal push towards the inward unit normal at the corresponding point at the boundary, needed in order to keep both processes in D. Considering the coupling time τ = inf{t > 0: Xt = Yt }, the mirror coupling evolves as described above for t τ , and we let Xt = Yt for t τ (the two processes move together after the coupling time). For definiteness, for t τ we define the mirror Mt as the hyperplane parallel to Mτ passing through Xt = Yt . Formally, in the case D = U, given two arbitrarily fixed points x, y ∈ U, we define the mirror coupling of reflecting Brownian motions in the unit ball U ⊂ Rn as a pair (Xt , Yt )t0 of stochastic processes given by


493

⎧ t ⎪ ⎪ ⎪ ⎪ Xt = x + Wt + ν(Xs ) dLX ⎪ s , ⎪ ⎪ ⎨ 0

t ⎪ ⎪ ⎪ ⎪ ⎪ Yt = y + Zt + ν(Ys ) dLYs , ⎪ ⎪ ⎩

(3)

0

where Wt is an n-dimensional Brownian motion starting at W0 = 0, Zt is the mirror image of the Brownian motion Wt with respect to the hyperplane Mt of symmetry between Xt and Yt , that is t Zt = Wt − 2 0

Xs − Ys (Xs − Ys ) · dWs , Xs − Ys 2

(4)

Y and LX t and Lt denote the boundary local times of the reflecting Brownian motions Xt and respectively Yt . The processes Xt and Yt evolve according to (3) above for t τ , where τ is the coupling time

τ = inf{t > 0: Xt = Yt } ∈ R ∪ {∞}, and they evolve together after the coupling time (i.e. Xt = Yt for t τ ). 3. Main results The key for proving the Laugesen–Morpurgo conjecture (Conjecture 1) is the double inequality (14) in Theorem 7, which in turn relies on proving the following inequality: pU (t, y, z) pU (t, x, z),

t > 0,

(5)

for all x, y, z ∈ U satisfying x − z y − z and y x. Consider a mirror coupling Xt , Yt of reflecting Brownian motions in U given by (3)–(4), with starting points X0 = x, Y0 = y ∈ U. For t < τ = inf{t > 0: Xt = Yt }, the mirror Mt of the coupling (the hyperplane of symmetry between Xt and Yt ) is given by

Xt + Yt · (Xt − Yt ) = 0 . M t = z ∈ Rn : z − (6) 2 The idea for proving the inequality (5) is that the mirror Mt moves towards the origin, in the sense of Lemma 5 below. This property is a rigorous version of Example 4.5 in [6], used by the authors to prove the efficiency of the mirror coupling in the case of the unit disk. Lemma 5. Let Xt , Yt be a mirror coupling of reflecting Brownian motions in U with starting points X0 = x, Y0 = y ∈ U, and let τ = inf{t > 0: Xt = Yt } be the coupling time and τ1 = inf{t > 0: 0 ∈ Mt }. For all times t < τ ∧ τ1 , the mirror Mt moves towards the origin, in such a way that if a point P ∈ U and the origin are separated by Mt1 for some t1 ∈ [0, τ ∧ τ1 ), then the point P and the origin are separated by Mt2 for all t2 ∈ [t1 , τ ∧ τ1 ) (see Fig. 1).

494


Fig. 1. Mirror coupling of reflecting Brownian motions in the unit disk (the case n = 2).

Proof. If x = y, then τ1 = 0 and there is nothing to prove in this case (the mirror M0 passes through the origin). Without loss of generality we may therefore assume that x > y. Setting

Ut = Xt − Yt , Vt = Xt + Yt ,

t 0,

(7)

from the definition (3)–(4) of the mirror coupling we obtain ⎧ t t ⎪ ⎪ ⎪ i i i i i i X ⎪ U = x − y + Wt − Zt − Xs dLs + Ysi dLYs , ⎪ ⎪ ⎪ t ⎨ 0

0

t t ⎪ ⎪ ⎪ ⎪ i i i i i i X i Y ⎪ ⎪ Vt = x + y + Wt + Zt − Xs dLs − Ys dLs , ⎪ ⎩ 0

i = 1, . . . , n,

0

for all t τ , where the superscript i indicates the ith Cartesian coordinate of the given point. Using the definition (4) of Zt , we have ⎧ t t t ⎪ ⎪ Usi ⎪ i i i i X ⎪ Ut = x − y + 2 U · dWs − Xs dLs + Ysi dLYs , ⎪ ⎪ 2 s ⎪ U s ⎨ 0

0

0

t t t ⎪ ⎪ ⎪ Usi ⎪ i i i i i X ⎪ V = x + y + 2Wt − 2 Us · dWs − Xs dLs − Ysi dLYs , ⎪ ⎪ Us 2 ⎩ t 0

0

(8)

0

for all i = 1, . . . , n and t < τ , and therefore we obtain the following formulae for the quadratic variation of the processes U and V :


⎧ t i j ⎪ i j Us Us ⎪ ⎪ ⎪ U ,U t = 4 ds, ⎪ 2 ⎪ U ⎪ s ⎪ ⎪ 0 ⎨ t j i j Usi Us ⎪ ⎪ V , V = 4 δ − ds, ⎪ ij ⎪ t ⎪ Us 2 ⎪ ⎪ 0 ⎪ ⎪ ⎩ i j U , V t = 0,

i, j, = 1, . . . , n.

495

(9)

Note that since X0 = x > y = Y0 , it follows that for all t < τ ∧ τ1 we have Ut · Vt = (Xt − Yt ) · (Xt + Yt ) = Xt 2 − Yt 2 > 0,

(10)

and therefore for t < τ ∧ τ1 we may define the process At by At =

2 Ut . Ut · Vt

(11)

We will first show that for t < τ ∧ τ1 the components of the process At are processes of bounded variation, satisfying dAit =

2 U i + Vti Ait − t dLX t , Ut · Vt 2

Applying the Itô formula to the C 2 function f (u, v) = have

ui u·v

i = 1, . . . , n.

(12)

and to the processes Ut and Vt , we

1 Uti dAit = d 2 Ut · Vt =

n

1 j j j j δij Ut · Vt − Uti Vt dUt − Uti Ut dVt 2 (Ut · Vt ) j =1

+

n

i j k 1 j 2Ut Vt Vt − δij Vtk Ut · Vt − δik Vt Ut · Vt d U j , U k t 3 2(Ut · Vt ) j,k=1

+

n

i j k j k 1 2Ut Ut Ut d V , V t 2(Ut · Vt )3 j,k=1

n

i k j 1 2Ut Ut Vt − δij Utk Ut · Vt − δj k Uti Ut · Vt d U j , V k t . + (Ut · Vt )3 j,k=1

Using the relations in (8) it can be seen that the martingale part in the above expression reduces to zero, and combining with (9) we obtain

496

M.N. Pascu, M.E. Gageonea / Journal of Functional Analysis 260 (2011) 490–500 n

1 1 j j j Y dAit = δij Ut · Vt − Uti Vt −Xt dLX t + Yt dLt 2 (Ut · Vt )2 j =1

−

n

i j 1 j j Y Ut Ut −Xt dLX t − Yt dLt 2 (Ut · Vt ) j =1

+

n

i j k Utj Utk 1 j k 2U 4 V V − δ V U · V − δ V U · V dt ij t t ik t t t t t t t 2(Ut · Vt )3 Ut 2 j,k=1

n j

i j k Ut Utk 1 dt 2Ut Ut Ut 4 δj k − + 2(Ut · Vt )3 Ut 2 j,k=1

=

1 (Ut · Vt )2 +

n

i j j j Ut Ut + Uti Vt − δij Ut · Vt Xt dLX t

j =1

n

i j j 1 j Ut Ut − Uti Vt + δij Ut · Vt Yt dLYt . 2 (Ut · Vt ) j =1

Using the fact that LYt ≡ 0 on the time interval [0, τ ∧ τ1 ) (the process Yt cannot reach the boundary ∂U before either coupling first with Xt or before the first time when Xt = Yt = 1, that is before 0 ∈ Mt ), and that LX t increases only when Xt ∈ ∂U, that is only when Xt = t = 1, we obtain Ut +V 2 n

i j j 1 1 j dAit = Ut Ut + Uti Vt − δij Ut · Vt Xt dLX t 2 2 (Ut · Vt ) j =1

=

n

j Uti Uti + Vti j 2 X U + V dL − dLX t t t t 2Ut · Vt 2(Ut · Vt )2

=

Uti + Vti 2 X U + V dL − dLX t t t t 2Ut · Vt 2(Ut · Vt )2

j =1

Uti

2Uti Uti + Vti dLX dLX t − t 2 2Ut · Vt (Ut · Vt ) 1 U i + Vti Ait − t dLX = t , Ut · Vt 2 =

thus proving the claim (12). To prove the claim of the lemma, assume by contradiction that there exists a point P ∈ U and times 0 < t1 < t2 < τ ∧ τ1 such that the point P and the origin are separated by Mt1 , but are not separated by Mt2 . By eventually changing the point P , without loss of generality we may assume that P ∈ / Mt2 , and using (6) and (7) we obtain 1 1 P · Ut2 − Ut2 · Vt2 < 0 0 for t ∈ [0, τ ∧ τ1 )) P · At2 < 1 t1 : P · At < 1} ∈ (t1 , t2 ) and using (12), we obtain t0 P · At0 = P · At1 +

P · dAt t1

t0 = P · At1 + t1

2 1 P · At − P · (Ut + Vt ) dLX t Ut · Vt 2

P · At1 > 1, since P · At 1 for t ∈ [t1 , t0 ] and 1 P · (Ut + Vt ) = |P · Xt | P Xt 1. 2 By the continuity of the process At and the choice of t0 we must also have P · At0 = 1, contradiction which concludes the proof of the lemma. 2 From the previous lemma we obtain the following: Theorem 6. For any points x, y ∈ U with y x and any z ∈ U such that x − z y − z, we have (13) P y Yt − z < ε P x Xt − z < ε , for any t 0 and ε ∈ (0, min{z, 1 − z}), where Xt and Yt are reflecting Brownian motions in U starting at x, respectively y, and P x , P y denote the corresponding probability measures. Proof. Without loss of generality we may assume that x and y are distinct points. Let Xt , Yt be a mirror coupling of reflecting Brownian motions in U with starting points X0 = x and Y0 = y, and let τ be the coupling time and τ1 = inf{t > 0: 0 ∈ Mt }. If Mt separates Xt and z for some t < τ ∧ τ1 , there exists a point P ∈ U such that the origin and the point P are separated by M0 , but are not separated by Mt , contradicting Lemma 5. It follows that the mirror Mt does not separate the points Xt and z for all t < τ ∧ τ1 , and therefore the distance from Xt to z is not greater than the distance from Yt to z in this case. Since for t τ ∧ τ1 , either the processes Xt and Yt are symmetric with respect to the (fixed) hyperplane Mτ ∧τ1 passing through the origin (for t ∈ (τ ∧ τ1 , τ )), or they have coupled (for t ∈ (τ, ∞)), it follows that the distance from Xt to z is also not greater than the distance from Yt to z. In all cases we obtained that the distance from Xt to z is not greater than the distance from Yt to z, and the claim follows. 2

498


Denoting by pU (t, x, y) the heat kernel for the Laplacian with Neumann boundary conditions on the unit ball U ⊂ Rn (or equivalently, the transition density of reflecting Brownian motion in U), we can now prove the following double inequality: Theorem 7. For any x ∈ U − {0}, r ∈ (0, min{x, 1 − x}) and t > 0 we have

pU (t, x + ru, x) dσ (u) pU ∂U

x x x t, x + r , x pU t, x + r ,x + r , x x x

(14)

where σ is the normalized surface measure on ∂U. Proof. Using the continuity of the transition density pU (t, x, y) of reflecting Brownian motion in the space variable, it follows pU (t, x, y) can be written as 1 ε0 cn ε n

pU (t, x, y) = lim

1 P x Wt − y < ε , n ε0 cn ε

pU (t, x, z) dz = lim y−z 0 we have (17)

pU (t, x, x) < pU (t, y, y), for all x, y ∈ U with x < y.

Proof. First note that for t > 0 fixed, by the radial symmetry of the problem it follows that pU (t, x, x) is a function of x ∈ [0, 1]. For an arbitrarily fixed x ∈ U − {0}, from Theorem 7 we obtain x x pU t, x + r ,x + r − pU (t, x, x) pU (t, x + ru, x) dσ (u) − pU (t, x, x) x x ∂U

=

pU (t, x + ru, x) − pU (t, x, x) dσ (u),

∂U

for any r ∈ (0, min{x, 1 − x}). Dividing by r and passing to the limit with r 0, we obtain pU (t, x + r x , x + r x ) − pU (t, x, x) d pU (t, x, x) = lim r0 dx r pU (t, x + ru, x) − pU (t, x, x) dσ (u). lim r0 r x

x

∂U

By bounded convergence theorem (pU (t, ·, x) is a C 2 function in the second variable, hence ∇pU (t, ·, x) is bounded in a neighborhood of x), we obtain d pU (t, x, x) ∇pU (t, x, x) · u dσ (u) = 0, dx ∂U

where we denoted by ∇pU the gradient of ∇pU (t, ·, x) in the second variable. Since x ∈ U − {0} was arbitrarily fixed, we have d pU (t, x, x) 0, dx

x ∈ (0, 1),

which shows that pU (t, x, x) is a non-decreasing function of x ∈ (0, 1), and by continuity this also holds for x ∈ [0, 1]. Since pU (t, x, x) is the diagonal of a heat kernel of an operator with real analytic coefficients, pU (t, x, x) is a real analytic function. If pU (t, x, x) were constant on a non-empty open subset of U, then it would be identically constant in U, which is impossible (it can be shown that pU (t, 0, 0) < pU (t, 1, 1) for any t > 0). This, together with the fact that pU (t, x, x) is a nondecreasing radial function shows that pU (t, x, x) is in fact a strictly increasing radial function for any t > 0, concluding the proof. 2

500


We conclude with the remark that the Laugesen–Morpurgo conjecture implies the famous Hot Spots conjecture of J. Rauch (see for example [3,7,11]) in the case of the unit ball U ⊂ Rn , and that extending the Laugesen–Morpurgo conjecture to more general domains would also give a resolution of the Hot Spots conjecture for the corresponding domains (the Hot Spots conjecture is only partially solved at the present moment). References [1] R. Atar, K. Burdzy, On Neumann eigenfunctions in lip domains, J. Amer. Math. Soc. 17 (2004) 243–265. [2] R. Atar, K. Burdzy, Mirror couplings and Neumann eigenfunctions, Indiana Univ. Math. J. 57 (3) (2008) 1317– 1351. [3] R. Bañuelos, K. Burdzy, On the “Hot Spots” conjecture of J. Rauch, J. Funct. Anal. 164 (1999) 1–33. [4] R. Bañuelos, T. Kulczycki, B. Siudeja, Neumann heat kernel monotonicity, Potential Anal. 3 (1) (2009) 65–83. [5] R.F. Bass, E.P. Hsu, Pathwise uniqueness for reflecting Brownian motion in Euclidean domains, Probab. Theory Related Fields 117 (2000) 183–200. [6] K. Burdzy, W. Kendall, Efficient Markovian couplings: Examples and counterexamples, Ann. Appl. Probab. 10 (2) (2000) 362–409. [7] D. Jerison, N. Nadirashvili, The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc. 13 (4) (2000) 741–772. [8] W.S. Kendall, Nonnegative Ricci curvature and the Brownian coupling property, Stochastics 19 (1–2) (1986) 111– 129. [9] R. Laugesen, C. Morpurgo, Extremals for eigenvalues of Laplacians under conformal mapping, J. Funct. Anal. 155 (1998) 64–108. [10] M.N. Pascu, Mirror coupling of reflecting Brownian motion and an application to Chavel’s conjecture, preprint, available online at http://arxiv.org/abs/1004.2398. [11] M.N. Pascu, Scaling coupling and the Hot spots conjecture, Trans. Amer. Math. Soc. 354 (2002) 4681–4702. [12] M.N. Pascu, A. Nicolaie, On a discrete version of the Laugesen–Morpurgo conjecture, Statist. Probab. Lett. 79 (6) (2009) 797–806. [13] M.N. Pascu, N.R. Pascu, Monotonicity properties of reflecting Brownian motion, in: 21st Scientific Session on Mathematics and its Applications, Transilvania University Press, Bra¸sov, June 2007, pp. 109–112. [14] M.N. Pascu, N.R. Pascu, Brownian motion on the circle and applications, Bull. Transilv. Univ. of Bra¸sov Ser. III 1 (50) (2008) 469–478. [15] F.Y. Wang, Application of coupling methods to the Neumann eigenvalue problem, Probab. Theory Related Fields 98 (3) (1994) 299–306.


Asymptotic Euler–Maclaurin formula over lattice polytopes ✩ Tatsuya Tate Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan Received 7 June 2010; accepted 12 August 2010 Available online 1 September 2010 Communicated by Daniel W. Stroock

Abstract Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler–Maclaurin formulas. An asymptotic Euler–Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin and Sternberg (2007) [11]. Then, the problem is to find a concrete formula for each term of the expansion. In this paper, an asymptotic Euler–Maclaurin formula of the Riemann sums over general lattice polytopes is given. The formula given here is an asymptotic form of the so-called local Euler–Maclaurin formula of Berline and Vergne (2007) [3]. For Delzant polytopes, our proof given here is independent of the local Euler–Maclaurin formula. Furthermore, a concrete description of differential operators which appear in each term of the asymptotic expansion for Delzant lattice polytopes is given. By using this description, when the polytopes are Delzant lattice, a concrete formula for each term of the expansion in two dimension and a formula for the third term of the expansion in arbitrary dimension are given. © 2010 Elsevier Inc. All rights reserved. Keywords: Euler–Maclaurin formula; Lattice polytopes; Asymptotic expansion; Toric varieties

✩

Research partially supported by JSPS Grant-in-Aid for Scientific Research (No. 21740117). E-mail address: [email protected].


502

T. Tate / Journal of Functional Analysis 260 (2011) 501–540

0. Introduction In this paper, we consider asymptotic behavior of the Riemann sums over lattice polytopes, RN (P ; ϕ) :=

1 N dim(P )

(0.1)

ϕ(γ /N ),

γ ∈(N P )∩Zm

where P is a lattice polytope in Rm , which means that each vertex has integer coordinates, and ϕ is a smooth function on P . Formulas for RN (P ; ϕ), which are often called Euler–Maclaurin formulas, are extensively investigated in combinatorics and geometry of toric varieties. If we take ϕ = 1, the Riemann sum RN (P ; 1) is reduced to the so-called Ehrhart polynomial EP (N ) := (N P ) ∩ Zm = N dim(P ) RN (P ; 1), which is closely related to the Todd class of a toric variety corresponding to the polytope P . In this context, geometry of toric varieties is a suitable and powerful tool to analyze the function EP (N). Indeed, as in [8], one can show that EP (N ) is a polynomial in N by using the Hirzebruch–Riemann–Roch theorem. The problems concerning (exact) Euler–Maclaurin formulas and Ehrhart polynomials are investigated by various authors, for example [6,3,4,14]. See [16] and references therein for various results on these topics. Before explaining some of the results closely related to the present paper, we state one of our theorems. Theorem 1. Let P be a lattice polytope in Rm . For each face f of P and non-negative integer n with dim(f ) dim(P ) − n, there exists a homogeneous differential operator Dn (P ; f ) of order n − dim(P ) + dim(f ) with rational constant coefficients which involves derivatives only in directions orthogonal to the face f such that for each smooth function ϕ on P , we have the following asymptotic Euler–Maclaurin formula: RN (P ; ϕ) ∼

n0

N

−n

Dn (P ; f )ϕ

(N → ∞),

(0.2)

f ∈F (P ), dim(f )dim(P )−n f

where F (P ) denotes the set of faces of P . The integration in the right-hand side is performed with respect to the measure on the affine hull f of f which is the parallel translation of the Lebesgue measure on the subspace L(f ) parallel to f defined by the lattice L(f ) ∩ Λ. In this section, we explain some of the previous works on the Euler–Maclaurin formula closely related to Theorem 1 and mention other results obtained in the present paper. An exact Euler–Maclaurin formula for Delzant polytopes was originally obtained by Khovanskii and Pukhlikov [14], and Brion and Vergne [4] generalized it to simple polytopes without using the theory of toric varieties. One of their results can be stated as (assuming that P is a Delzant polytope) RN (P ; ϕ) = Todd(P ; ∂/N ∂h) Ph

ϕ(x) dx

h=0

,

(0.3)


503

where ϕ is a polynomial, h = (h1 , . . . , hd ) ∈ Rd is a small parameter with d the number of faces of P of codimension one, Todd(z) = 1−ez −z is an analytic function around the origin, called the Todd function,

Todd(P ; ∂/N ∂h) =

d

Todd(∂/N ∂hi )

i=1

is a differential operator (of infinite order), and when the polytope P is given by P = {x; ui , x ci , i = 1, . . . , d}, then Ph = {x; ui , x ci − hi , i = 1, . . . , d}. Note that Brion and Vergne [4] obtained the same formula for simple polytopes with a modification of the differential operator Todd(P ; ∂/N ∂h). In [3], Berline–Vergne obtained an effective formula for RN (P ; ϕ) (still ϕ being assumed to be polynomial), which they call a local Euler–Maclaurin formula. This formula is of the form (setting N = 1 for simplicity) R1 (P ; ϕ) =

D(P , f )ϕ,

(0.4)

f f

where the sum runs over all faces f of P , D(P , f ) is a differential operator (of infinite order) with rational constant coefficients on Rm which involves derivatives only in directions perpendicular to the face f . One of remarkable points is that the formula (0.4) of Berline–Vergne holds for any rational polytopes, which means that each vertex of the polytope has rational coordinates. They constructed a meromorphic function μ(a) for any affine rational polyhedral cone a and use a sort of inclusion-exclusion property (which is called a valuation property) of μ to show that it is analytic near the origin, and they define the symbol of the operator D(P , f ) by using μ. The operators Dn (P ; f ) in our formula (0.2) is, by definition, the homogeneous parts of the operator D(P , f ) in (0.4). Thus, one can think the formula (0.2) as an asymptotic form of the local Euler–Maclaurin formula (0.4) due to Berline–Vergne. As we point out in Section 1.3, one can deduce (0.2) by using one of results in [3] directly and formally. However, the method mentioned in Section 1.3 is formal, and we use a different method to prove Theorem 1. Moreover, any transparent formula for the homogeneous parts of D(P , f ) is, in general, not known. We will see that, when P is a Delzant lattice polytope, the operators Dn (P ; f ) can be, to some extent, expressible concretely (Definition 3.6, Theorem 3.9). Note that our formula (0.2) is valid for any smooth function ϕ on P . Our construction of the operator Dn (P ; f ) makes us to obtain concrete formula for Delzant lattice polytopes in two dimension (Corollary 5.4). A part of our construction of these operators Dn (P ; f ) uses an induction procedure, and they are still complicated. This complication comes from the “angles” at each face of the polytopes, and hence it would be rather natural. The complication involving the “angles” is embodied in an integration by parts procedure. In this paper, by the name asymptotic Euler–Maclaurin formula, we mean formulas of asymptotic expansion of the Riemann sum RN (P ; ϕ). In one dimension (m = 1 and P = [0, 1]), the following asymptotic Euler–Maclaurin formula is well know.

504


N ϕ(0) 1 1 ϕ(1) − ϕ(0) ϕ(k/N ) = RN [0, 1]; ϕ − ∼ ϕ(x) dx + N N 2N 1

k=1

0

(−1)n−1 Bn + ϕ (2n−1) (1) − ϕ (2n−1) (0) N −2n , (2n)!

(0.5)

n1

where ϕ is any smooth function on [0, 1], and bn are the coefficients of the Taylor expansion of the Todd function: Todd(−z) =

∞ bn n=0

n!

zn ,

and Bn = (−1)n−1 b2n (n 1) are the Bernoulli numbers. A higher dimensional analogue of (0.5) was given by Guillemin and Sternberg [11]. Namely, Guillemin–Sternberg obtained the asymptotic Euler–Maclaurin formula of the form (assuming that P is Delzant) RN (P ; ϕ) ∼ Todd(P ; ∂/∂N h) ϕ(x) dx . (0.6) h=0

Ph

This formula also holds true for simple lattice polytopes under a modification. Note that this formula is, at least its appearance, similar to the Brion–Vergne formula (0.3). The proof of (0.6) given in [11] is different from the proof of (0.3) given in [4], and it does not use geometry of toric varieties. There are some applications of the above formula for spectral analysis on toric Kähler manifolds. In fact, in [12], the asymptotic Euler–Maclaurin formula obtained in [11], combined with an asymptotic expansion of ‘twisted Mellin transform’ studied in [19], is applied to analyze a spectral measure on Cm which is, in a GIT setting, related to the pair (X, L) where X is a toric manifold corresponding to a Delzant polytope and L is a Hermitian line bundle on X. (See also [5] where the same spectral measure as in [12] is discussed.) One more asymptotic Euler–Maclaurin formula was brought to us by Zelditch [20]. The formula obtained in [20] is stated as 1 ϕ(x) dσ + N −n En (P )ϕ(x) dx, (0.7) RN (P ; ϕ) ∼ ϕ dx + 2N P

∂P

n2

P

where P is a Delzant lattice polytope, En (P ) is a differential operator (of finite order), and dσ is the Leray measure on the boundary ∂P . In [20], Zelditch introduced the notion of Bergman– Bernstein measures (this name is taken from [18]) and obtained its asymptotic expansion. Then, integration (over the toric Kähler manifold corresponding to the Delzant polytope P ) of the asymptotic expansion yields the formula (0.7). In [20], the formula (0.7) is called a ‘metric expansion’ to distinguish it from the Euler–Maclaurin formula of the form (0.6), since the differential operators En (P ) depend on the choice of a Hermitian metric on a line bundle over the toric manifold. But, the Riemann sum itself does not depend on such a metric. A point is that such a metric dependence would be disappeared after an integration by parts. Indeed, in [20], the second term is computed by using an integration by parts identity due to Donaldson [7].


505

As is mentioned in [20], comparison of asymptotic Euler–Maclaurin formula and the metric expansion of the form (0.7) will give some further identities in the lower order terms. One of our motivation is to give another asymptotic Euler–Maclaurin formula which is computable to some extent. Indeed, we have a concrete formula for the third term of the expansion when the polytope is Delzant. See Corollary 5.6 in Section 5.3. Thus, if one can compute the differential operator E2 (P ) in (0.7) in terms of curvatures, then one will obtain an integration by parts identity in the third term in (0.7), which might be useful to geometry of toric manifolds. An idea of proof of Theorem 1 is to reduce the problem to that for unimodular cones, which are cones generated by a part of an integral basis, by using a subdivision of a rational cones into unimodular cones (see [8, Section 2.6]) and a canonical decomposition of the characteristic functions of polytopes (see Eqs. (5.3), (5.6) in Section 5). The asymptotic Euler–Maclaurin formula of Riemann sums over unimodular cones can be deduced by a method in [11] (see also [1,15,16]). However, we deduce it here by a quite different method. This method is rather similar to the Bergman–Bernstein approach in [20]. But, we work on unimodular cones instead of polytopes themselves. Thus, we use the Szasz measures introduced in Section 2 instead of Bernstein or Bergman–Bernstein measures discussed in [20] or [18]. More concretely, an asymptotic property of the Szasz functions is used to show Proposition 3.1 in Section 3, which is an asymptotic Euler–Maclaurin formula for unimodular cones. Proposition 3.1 can be deduced directly from Theorem 3.2 in [11], and one can consider that the Proposition 3.1 is a starting point for the subsequent sections. Thus, one might be able to perform similar computations in sections after Section 3 at least for simple polytopes, by using Theorem 3.3 in [11] instead of Proposition 3.1. However, the asymptotic behavior of Szasz functions would be a general interest in its own right. Furthermore, there would be a possibility of using a version of Szasz functions to get asymptotics of the Riemann sum over general rational cones without using a subdivision of cones into unimodular cones, if one could resolve a problem on ‘rare events’ along with the lines in [18]. (See also Remark after the proof of Theorem 5.1 on this point.) In one dimension, we compute explicitly each term of the expansion for twisted Riemann sum by using this approach. This computation uses the twisted version of the Szasz function, and it shows that coefficients in the Taylor expansion of the ‘twisted’ Todd function can be represented by the Stirling numbers of the second kind (in particular, Eq. (2.19)), which is a generalization of a well-known formula among Bernoulli numbers, Catalan numbers and the Stirling numbers of the second kind (see (2.17) or [10]). Thus, this approach might have some advantages also in higher dimension. These are the reasons why we use the approach with the Szasz functions in this paper. We here mention that an asymptotic expansion of the Szasz function was first obtained in [9]. In [9], Feng also obtained an asymptotic formula of the Riemann sum over the positive orthant Rm + in the same strategy as ours. However, concrete formulas for each term of the asymptotic expansion are not discussed fully in [9]. We give an explicit formula for each term of the expansion of the Szasz function in Section 2. (The main purpose in [9] was to give a non-compact analogue of Bergman–Bernstein approximation in [20]. Indeed the Szasz function, defined in Section 2 in the present paper, is closely related to the Bergman kernel for the Bargmann–Fock space as explained in [9].) We close Introduction with some comments on the organization of this paper. We collect some of the notation used in this paper in Section 1.1, and then, we review and define the Berline– Vergne operators Dn (P ; f ) in Section 1.2. As we mentioned above, a heuristic argument to find a formula (0.2) is given in Section 1.3. In Section 1.4, we prove a uniqueness theorem on the expression of each term of the asymptotic expansion of the form (0.2) (Theorem 1.2). In Section 2, we study asymptotic behavior of Szasz functions. Some computations for the twisted Riemann

506


sum in one dimension is given in Section 2.1. In Sections 2.2, 2.3, we define the Szasz functions and prove their asymptotic expansion formula by using an idea coming from [13]. Section 3 is devoted to the study of asymptotic behavior of the Riemann sums over unimodular cones. First, we prove an asymptotic expansion formula (Proposition 3.1) by using the asymptotic property of the Szasz functions studied in Section 2. Asymptotic formula obtained in Proposition 3.1 uses differential operators in direction transversal to each face of the unimodular cone. Then, one can perform further integration by parts. This is done in Section 3.2. In Section 3.3, we define differential operators obtained by the integration by parts procedure discussed in Section 3.2 which is used to renormalize each term of the expansion in Proposition 3.1. The fact that the operators so defined coincide with the Berline–Vergne operators is proved also in this subsection (Theorem 3.9). In Section 4, we prove the asymptotic Euler–Maclaurin formula for general pointed rational cones by using the Berline–Vergne operators and the subdivision of pointed rational cones into a finite number of unimodular cones. Finally, in Section 5, we prove our main Theorem 1, which is reformulated in Theorem 5.1, and a uniqueness result (Theorem 5.3), and give some explicit computation. 1. Berline–Vergne operators and heuristic argument In this section, we review the symbol of differential operators defined in [3]. Then, we give a heuristic argument to obtain an asymptotic Euler–Maclaurin formula of the form (0.2). Furthermore, we deduce a uniqueness theorem on expression of coefficients in asymptotic Euler– Maclaurin formula of the form (0.2). 1.1. Notation Let X be a finite dimensional vector space over R, and let Λ be a lattice in X. Such a pair (X, Λ) is called a rational vector space. The dual space X ∗ of a rational space (X, Λ) is a rational space with the dual lattice Λ∗ of Λ. A point x ∈ X is said to be rational if qx ∈ Λ for some q ∈ Z \ {0}. The set of rational points in X is denoted by XQ . A basis of Λ over Z is called an integral basis of Λ. For each rational vector space (X, Λ), we fix a Lebesgue measure on X normalized so that the measure of the fundamental domain of the action of Λ on X has measure 1. A subspace L in X is said to be rational if L ∩ Λ is a lattice in L. We fix a Lebesgue measure on a rational subspace (L, L ∩ Λ) as above. An affine subspace A is said to be rational if A is a parallel translation of a rational subspace. (Note that a rational affine subspace A is allowed to be a translation of a rational subspace by a point which is not rational.) For a rational affine subspace A, we fix a Lebesgue measure on A which is a translation of the fixed Lebesgue measure on the rational subspace parallel to A. Any integration on a subset in a rational affine subspace is performed by using the Lebesgue measure normalized in this way. For each vector u ∈ X, let ∇u denote the derivative in the direction u. For each non-empty subset S in X, let L(S) be the subspace spanned by the vectors y − x with x, y ∈ S, which is parallel to the affine hull, denoted by S, of S. If S ⊂ XQ , then L(S) is a rational subspace in X. Let L be a rational subspace in a rational space (X, Λ). The natural projection from X onto X/L is denoted by πL : X → X/L. If L is a subspace in X, let L⊥ ⊂ X ∗ denote the annihilator of L. The quotient space X/L of X by a rational subspace L is again a rational space with the lattice πL (Λ). An inner product Q on a rational space (X, Λ) is said to be rational if Q(x, y) ∈ Q for each x, y ∈ XQ . Let Q be a rational inner product on (X, Λ). The rational inner product on (X ∗ , Λ∗ )


507

induced by the inner product Q on X is also denoted by Q. Let L be a subspace in X. The orthogonal complement of L in X is denoted by L⊥Q . Note that we have a natural identification (X/L)∗ ∼ = L⊥ . The orthogonal projection from X ∗ onto (X/L)∗ ∼ = L⊥ is denoted by pL : X ∗ → (X/L)∗ . When L is rational, the rational space X/L is equipped with the rational inner product obtained by identifying X/L with L⊥Q . Note that, with this identification, the lattice πL (Λ) of X/L is identified with the orthogonal projection pL (Λ) of Λ, where the orthogonal projection from X onto L⊥Q is also denoted by pL : X → L⊥Q , which is different from the lattice L⊥Q ∩ Λ in L⊥Q . A subset P in a rational space (X, Λ) is called a rational polyhedron if P is an intersection of a finite number of half spaces each of which is bounded by a rational affine hyperplane. Let P be a rational polyhedron. Then the set of faces of P is denoted by F (P ), and, for non-negative integer k, the set of faces of P of dimension k is denoted by F (P )k . We set V(P ) = F (P )0 , the set of vertices of P . A face of codimension one is called a facet. For each f ∈ F (P ), we set πf = πL(f ) , the natural projection from X onto X/L(f ). When, a rational inner product on X is fixed, we set pf = pL(f ) , the orthogonal projection from X ∗ onto (X/L(f ))∗ . A rational polyhedron C in X is called a rational cone if C is a cone generated by a finite number of elements in Λ. Note that a rational cone C might contain straight lines. The largest subspace contained in the rational cone C is C ∩ (−C), which is a rational subspace in X. If C ∩ (−C) = {0}, then the rational cone C is said to be pointed. If a rational cone C is generated by a subset of an integral basis of Λ, then C is said to be unimodular. A subset a of X is called a rational affine cone if a is of the form a = s + C where s ∈ XQ and C is a rational cone. If C is pointed, then a is also said to be pointed. 1.2. The Berline–Vergne operators In this subsection, we recall the construction of operators given in [3]. Let (X, Λ) be a rational space with a rational inner product Q. For each rational polyhedron P in X, we set S(P )(ξ ) =

γ ∈P ∩Λ

eξ,γ ,

I (P ) =

eξ,x

(1.1)

P

if the sum and the integral converge absolutely, where ξ ∈ X ∗ . These functions are defined as meromorphic functions on X ∗ . Let f be a face of a rational polyhedron P in X. Let CP (f ) be the cone generated by the vectors of the form y − x with y ∈ P , x ∈ f . This is actually a rational cone in X with CP (f ) ∩ (−CP (f )) = L(f ). Then, the pointed affine cone t(P , f ) := πf (f + CP (f )) in X/L(f ) is called the transverse cone of P along f . For any rational quotient W = X/L of X by a rational subspace L, let C(W ) denote the set of all rational affine cones in W . Let H(W ∗ ) denote the ring of analytic functions with rational Taylor coefficients defined in a neighborhood of 0 in W ∗ with respect to an (and hence all) integral basis of the dual lattice of the lattice πL (Λ) in W = X/L. Then, it is shown in Theorem 20 in [3] that there is a unique family of maps μW , indexed by rational quotient spaces W of X, from C(W ) to H(W ∗ ) such that the following conditions hold: (1) If W = {0}, then μW ({0}) = 1. (2) If the affine cone a ∈ C(W ) contains a straight line, then μW (a) = 0.

508


(3) For any a ∈ C(W ), one has

S(a)(ξ ) =

μW/L(F ) t(a, F ) (ξ )I (F )(ξ ),

ξ ∈ W ∗.

(1.2)

F ∈F (a)

Moreover, one of main theorems in [3] is that, for each rational polyhedron P in W = X/L, one has S(P )(ξ ) =

μX/L(f ) t(P , f ) (ξ )I (f )(ξ ),

ξ ∈ W ∗.

(1.3)

f ∈F (P )

(See Theorem 21 in [3].) Note that the functions μX/L(f ) in (1.3) (and also in (1.2)) is the lift to W ∗ of functions defined on (W/L(f ))∗ through the orthogonal projection pf : W ∗ → (W/L(f ))∗ . Let a be a pointed rational affine cone in the rational quotient X/L of X. For any non-negative integer k, let μkX/L (a) denote the homogeneous polynomial of degree k on (X/L)∗ which is the homogeneous part of the Taylor expansion of the analytic function μX/L (a) near 0 ∈ (X/L)∗ . We set μkX (a) = pL∗ μkX/L (a), which is a homogeneous polynomial of degree k on X ∗ . Definition 1.1. Let (X, Λ) be a rational space with a rational inner product Q. For any rational polyhedron P in X, any face f of P and any non-negative integer n such that n − dim(P ) + dim(f ) 0, we define the homogeneous differential operator DnX (P ; f ) on X with rational constant coefficients of order n − dim(P ) + dim(f ), which involves derivatives only in directions perpendicular to the subspace L(f ), as the differential operator whose symbol n−dim(P )+dim(f ) n−dim(P )+dim(f ) is given by μX (t(P , f )) = pf∗ μX/L(f ) (t(P , f )). We call the operators DnX (P ; f ) the Berline–Vergne operators. We note that, when C is a pointed rational cone in X and F is a face of C, then t(C, F ) = πF (C), and hence we have DnX (C; F ) = DnX (πF (C); 0). Let P be a lattice polytope in X, which means that each vertex is an element in Λ, and let f ∈ F (P ). Then, we have t(P , f ) = πf (v) + πf (CP (f )) where v ∈ f ∩ Λ. Since the function μX/L(f ) is invariant under translation by elements in the lattice (Theorem 21 in [3]), we have DnX (P ; f ) = DnX (πf (CP (f )); 0). 1.3. Heuristic arguments In this subsection, we give a heuristic argument to find the formula (0.2) by using the result (1.3) in [3]. Let (X, Λ) be a rational space. Let P be a lattice polytope in X. For simplicity, assume that m := dim(P ) = dim(X). For each f ∈ F (P ), we set μ(P , f ) := pf∗ μX/L(f ) (t(P , f )) which is a meromorphic function on X ∗ analytic in a neighborhood of the origin. Now let us compute the Riemann sum RN (P ; ϕ) by using (1.3). Let ϕ be a smooth function on P . Since P is compact, one may assume that ϕ ∈ C0∞ (X). Normalize the Lebesgue measure dξ on X ∗ so that it satisfy the Fourier inversion formula ϕ(x) = (2π)−m

X∗

eiξ,x ϕ(ξ ˆ ) dξ,

ϕ(ξ ˆ )= X

e−iξ,x ϕ(x).


509

Inserting the above for x = γ /N with γ ∈ N P ∩ Λ into the definition of RN (P ; ϕ) and using the formula (1.3), we have 1 RN (P ; ϕ) = μ(N P , Nf )(iξ/N )I (Nf )(iξ/N ) ϕ (ξ ) dξ, (2πN)m f X∗

But, since P is a lattice polytope, we have μ(N P , Nf ) = μ(P , f ) (see [3, Remark 29]). Changing the variable x → x/N , we have I (Nf )(iξ/N ) = N dim(f ) I (f )(iξ ). Thus we have 1 dim(f ) N μ(P , f )(iξ/N )I (f )(iξ ) ϕ (ξ ) dξ. RN (P ; ϕ) = (2πN)m f

X∗

Formally, substituting the Taylor expansion μ(P , f )(iξ/N ) =

μk t(P , f ) (iξ )N −k

k0

into the above formula, we could have RN (P ; ϕ) “∼”

n0

N

−n

DnX (P ; f )ϕ,

(1.4)

f ∈F (P ); dim(f )m−n f

where DnX (P ; f ) is defined in Definition 1.1. However, the above computation is formal because we do not know much about global properties of the functions μ(P , f ). Even if we could prove the formula (1.4) along with the method explained above, we do not know much about homogeneous parts of its Taylor expansion. One of our purposes in this paper is to give an effective formula for the operator DnX (P ; f ) given in Definition 1.1, at least for Delzant lattice polytopes, by a method different from the above strategy. 1.4. A uniqueness property In this subsection, we discuss a uniqueness property of an expression of each term of the asymptotic expansion of RN (C; ϕ) for unimodular cones C. Let C be a unimodular cone in a rational space (X, Λ) with a rational inner product Q. Then, note that, for each face F of C, we have t(C, F ) = πF (C). Note also that, we give a rational inner product in each rational quotient space X/L by identifying X/L with L⊥Q . Theorem 1.2. Suppose that, for any rational space (X, Λ) with a rational inner product Q, any rational subspace L of X, any unimodular cone C in X/L and any non-negative integer n such that n dim(C), there exists a homogeneous differential operator DnX (C) on X of order n − dim(C) with symbol νnX (C) such that X/L (1) If C ⊂ X/L, then νnX (C) = pL∗ νn (C) where pL : X → L⊥Q ∼ = (X/L)∗ denote the orthogonal projection. L(C) (2) If C ⊂ X with dim(C) < dim(X), then νnX (C) = t ι∗C νn (C), where t ιC : X ∗ → L(C)∗ is the transpose of the inclusion ιC : L(C) → X.

510


(3) When dim(X) = 0, we have D0X ({0}) = 1, DnX ({0}) = 0 (n 1). When dim(X) = 1 and C = R+ u with a generator u of Λ, we have DnX (C) = − bn!n ∇un−1 (n 1). (4) For any unimodular cone C ⊂ X, any F ∈ F (C), any n ∈ Z+ with dim(F ) dim(C) − n and any Schwartz function ϕ ∈ S(X) on X, the following holds: RN (C; ϕ) ∼

N

−n

DnX πF (C) ϕ

(N → ∞).

(1.5)

F ∈F (C);dim(F )dim(C)−n F

n0

Then, we have n−dim(C)

νnX (C) = μX

(1.6)

(C)

for any such X, C and n satisfying n − dim(C) 0. Proof. First, we note that, the symbols of the Berline–Vergne operators satisfy the assumption (2) in the statement (Proposition 13 in [3]). We prove the assertion by induction on the dimension of X. Consider the case where dim X = 1. Take a generator u of the lattice Λ and identify u with 1 in Z. Let C = R+ u. Then, as is computed in [3], we have ∞

μX (C)(ξ ) =

bn 1 1 =− + ξ, un−1 , ξ,u ξ, u 1 − e n!

ξ ∈ X∗ .

n=1

bn n−1 (n 1), We also have μ{0} ({0}) = 1. From this, we have μn−1 X (C)(ξ ) = − n! ξ, u 0 n μX ({0}) = 1, μX ({0}) = 0 (n 1). By the assumption (3), this shows the assertion when dim(X) = 1. Next, assume that, for any rational space (X, Λ) with dim(X) m − 1, any unimodular cone C in a rational quotient X/L and any non-negative integer n such that n dim(C), Eq. (1.6) holds. Let X be an m-dimensional rational space, and let C ⊂ X be a unimodular cone. If dim(C) < m, then by the assumption (2) and the induction hypothesis, we have (1.6). Thus, we assume that dim(C) = m. Let F ∈ F (C). If dim(F ) > 0, then, by the assumption (1), we X/L(F ) have νnX (πF (C)) = pF∗ νn (πF (C)). Since dim(X/L(F )) m − 1 and πF (C) is a unimodular cone in X/L(F ), we can use the induction hypothesis, and hence the latter function coincides n−m+dim(F ) n−m+dim(F ) with pF∗ μX/L(F ) (πF (C)) = μX (πF (C)). To prove νnX (C) = μn−m X (C) for n m, ∗ take ξ ∈ X such that ξ, x < 0 for each x ∈ C. Then, for any N > 0, we have

S(C)(ξ/N) = N m RN (C; eξ ),

eξ (x) = eξ,x .

Note that there is a ϕ ∈ S(X) such that ϕ(x) = eξ (x) for x ∈ C. Thus, by the assumption (4), we have S(C)(ξ/N ) ∼

n0

N m−n

F ∈F (C); dim(F )m−n

νnX πF (C) (ξ )I (F )(ξ )

(1.7)


511

as N → ∞. By (1.2) and the identity I (F )(ξ/N ) = N dim(F ) I (F )(ξ ), we have S(C)(ξ/N ) =

N m−n

n−m+dim(F )

πF (C) (ξ )I (F )(ξ )

μX

F ∈F (C); dim(F )m−n

n0

for every sufficiently large N . Let n m. By using the induction hypothesis, the coefficient of N m−n in the above can be written as μn−m νnX πF (C) (ξ )I (F )(ξ ). (1.8) X (C)(ξ ) + F ∈F (C); 0=dim(F )m−n X Equating (1.8) with the coefficient of N m−n in (1.7) shows μn−m X (C) = νn (C).

2

2. Szasz functions and their asymptotic behavior In this section, we define Szasz functions over unimodular cones and investigate their asymptotic behavior. First of all, let us compute in one dimension, which illustrate the general case. 2.1. Computation in one dimension The Szasz function associated with a function ϕ on R, originally introduced and discussed in [17], is defined by SN (ϕ)(x) =

∞

k (N x)ϕ(k/N ),

k (x) =

k=0

x k −x e , k!

x ∈ R.

Szasz introduced the function SN (ϕ) as an analogue of the Bernstein polynomial BN (ϕ)(x) =

N

mkN (x)ϕ(k/N ),

mkN (x) =

k=0

N k x (1 − x)N −k . k

Indeed, these two functions are related through Poisson’s law of rare events lim mkN (x/N) = k (x).

N →∞

For us, an important property of the Szasz function SN (ϕ) is the following: ∞ SN (ϕ)(x) dx = 0

∞ 1 ϕ(k/N ) =: RN [0, +∞); ϕ N k=0

for any ϕ ∈ S(R). We put ∞ 1 ϕ(−k/N ). RN (−∞, 0]; ϕ := N k=0

(2.1)

512


Then, once we obtain the asymptotic expansion of SN (ϕ) as N → ∞ with a suitable reminder estimate, then integrating it on [0, ∞) will give the asymptotic expansion of RN ([0, +∞); ϕ). But then we have the formula RN [0, 1]; ϕ = RN [0, +∞); ϕ + RN (−∞, 0]; T1 ϕ − RN (R; ϕ),

(2.2)

where we set T1 ϕ(x) = ϕ(1 + x). In this formula, note that we have RN (R; ϕ) = R ϕ(x) dx + O(N −∞ ) (see [11] or see Lemma 3.2). We also have RN ((−∞, 0]; T1 ϕ) = RN ([0, +∞); ψ), where we set ψ(x) = ϕ(1 − x), and hence the asymptotics of RN ([0, +∞); ϕ) will give the classical asymptotic Euler–Maclaurin formula (0.5). Thus, to obtain (0.5), it is enough to consider RN ([0, +∞); ϕ). In one dimension, we can consider a bit more general situation. We choose a positive integer q 1 and a qth root of unity ω. We consider the twisted Riemann sum ∞ 1 k ω ϕ(k/N ), N

ω RN (ϕ) :=

(2.3)

k=0

ω (ϕ) is discussed in [11] and the asymptotic where ϕ ∈ C0∞ (R). The twisted Riemann sum RN formula ω RN (ϕ) ∼

(−1)n−1 bnω

n1

ϕ (n−1) (0) Nn

(2.4)

was obtained, where the coefficients bnω is defined by the Taylor expansion of the function τω (s) :=

s = bnω s n , −s 1 − ωe

b1ω =

n1

1 . 1−ω

(2.5)

The formula (2.4) is used in [11] to obtain asymptotic Euler–Maclaurin formula for simple polyω (ϕ) along with topes. Now, to obtain the asymptotic expansion of the twisted Riemann sum RN our strategy, we use the twisted version of the Szasz function, which is defined by ω SN (ϕ)(x) =

∞

ωk k (N x)ϕ(k/N ).

(2.6)

k=0

From the definition, we have ∞ ω ω SN (ϕ)(x) dx = RN (ϕ).

(2.7)

0 ω (ϕ), we need to prepare To state a result on asymptotic expansion of the twisted Szasz function SN some properties of the Stirling numbers of the second kind and related polynomials. The Stirling numbers of the second kind, denoted by S(n, k) where n, k are integers satisfying 0 k n, are defined by the following recursion formula:


S(0, 0) = 1,

S(n, 0) = 0,

S(n, n) = 1

S(n + 1, k) = kS(n, k) + S(n, k − 1)

513

(n 1),

(1 k n).

(2.8)

For example, we have S(n, 1) = 1 (n 1) and S(n, n − 1) = n2 (n 2). For convenience, we set S(n, k) = 0 for 0 n < k. For any integer n, k with 0 k n, we define the polynomial p(n, k; z) in z ∈ C of degree k by p(n, k; z) :=

k

n t=0

t

(−1)t S(n − t, k − t)zk−t .

(2.9)

Some of p(n, k; z) are computed as follows. p(0, 0; z) = 1,

p(n, 0; z) = 0,

p(n, n; z) = (z − 1)n (n 1),

n p(n, n − 1; z) = z(z − 1)n−2 (n 2). 2

p(n, 1; z) = z,

(2.10)

Lemma 2.1. (1) For any non-negative integer n, we have ez

n

S(n, k)zk =

k=0

∞ n k k=0

k!

zk .

(2) The polynomials p(n, k; z) satisfy the following recursion formula: p(n + 1, k; z) = (z − 1)p(n, k − 1; z) + kp(n, k; z) + np(n − 1, k − 1; z),

1 k n.

(3) For [n/2] + 1 k n, the polynomial p(n, k; z) is divisible by (z − 1)2k−n . In particular, we have p(n, k; 1) = 0 for [n/2] + 1 k n. Proof. (1) is proved easily by using induction on n and the recurrence formula for the Stirling numbers S(n, k) of the second kind. To prove (2), let 1 k n. By using the relation n+1 = t n n + for 1 t n, we have t t−1 p(n + 1, k; z) =

k

n (−1)t S(n + 1 − t, k − t)zk−t − p(n, k − 1; z). t t=0

Denote S the sum above. Then, by the recursion formula (2.8), we have S=

k

n t=0

t

(−1)t (k − t)S(n − t, k − t)zk−t +

= kp(n, k; z) − n

k−1

n t=0

k t=1

t

(−1)t S(n − t, k − 1 − t)zk−t

n−1 (−1)t S(n − t, k − t)zk−t + zp(n, k − 1; z). t −1

Minus the sum in the middle of the above equals np(n − 1, k − 1; z), and hence (2) is proved.

514


Let us prove (3). Since the statement is obvious from (2.10) for n = 1, 2, we assume that, for some n 2, p(m, k; z) is divisible by (z − 1)2k−m for each 1 m n and [m/2] + 1 k m, and use the induction on n. So, we take l with [(n + 1)/2] + 1 l n + 1. If l = n + 1, p(n + 1, n + 1; z) = (z − 1)n+1 and hence (3) is clear. Thus, we assume that [(n + 1)/2] + 1 l n. By the induction hypothesis, p(n, l; z) is divisible by (z − 1)2l−n . We have [(n − 1)/2] + 1 = [(n + 1)/2] and hence, by induction hypothesis, p(n − 1, l − 1; z) is divisible by (z − 1)2l−n−1 . If [n/2] = l − 1, then n is even and 2l − n − 1 = 1, and hence, by the recurrence relation (2), p(n+1, l; z) is divisible by (z −1). Otherwise, we have [n/2]+1 l −1, and hence p(n, l −1; z) is divisible by (z −1)2l−n−2 . Then, again by (2), p(n +1, l; z) is divisible by (z − 1)2l−n−1 . 2 ω (ϕ) by using Now, we can state the asymptotic expansion of the twisted Szasz functions SN the polynomials p(n, k; z) as follows.

Proposition 2.2. Let ϕ ∈ S(R). Let ω be a qth root of unity. Then, for any positive integer n and positive number K such that n < K < 2n, there exists a constant CK,n > 0 such that we have ω SN (ϕ)(x) =

2n−1 μ=0

ϕ (μ) (x) −μ ω ω N Jμ (N x) + S2n,N (x), μ!

x > 0,

(2.11)

ω where the function S2n,N (x) satisfies the following estimate:

ω S

CK,n N −n (1 + x)n−K ,

2n,N (x)

x > 0, N > 0.

(2.12)

The function Jμω (x) is given by Jμω (x) = e−(1−ω)x

μ

p(μ, k; ω)x k .

k=0

When ω = 1, the function Jμ1 (x) is a polynomial in x of degree at most [μ/2]. Proof. Let x > 0. Substituting the Taylor expansion ϕ(k/N ) =

0μ2n−1

ϕ (μ) (x) (k/N − x)2n (k/N − x)μ + R2n (k/N, x), μ! (2n − 1)! 1

R2n (k/N, x) =

(1 − t)2n−1 ϕ (2n) x + t (k/N − x) dt,

0

we have ω SN (ϕ)(x) =

2n−1 μ=0

ϕ (μ) (x) ω ω J (N x) + S2n,N (x), μ!N μ μ

(2.13)


515

where Jμω (x) and S2n,N (x) are given by

Jμω (x) =

∞

ωk k (x)(k − x)μ ,

k=0 ω (x) = S2n,N

∞ 1 ωk k (N x)(k − N x)2n R2n (k/N, x). (2n − 1)!N 2n k=0

By using Lemma 2.1, (1) and the definition (2.1) of the function k (x), it is easy to show 1 ω the formula (2.13) for Jμω (x). We set S2n,N (x) = (2n−1)!N 2n S2n,N (x). Take K as in the state-

ment, and choose C > 0 so that |ϕ (2n) (y)| C(1 + |y|)−K for any y ∈ R. Then, we have |x + t (k/N − x)| (1 − t)x for any t ∈ [0, 1], x 0, k 0, and hence |R2n (k/N, x)| CK,n x −K , x > 1, k 0. When 0 x 1, |R2n (k/N, x)| is bounded uniformly in N . Thus, 1 (N x) for x > 1. When 0 x 1, we have |S we have |S2n,N (x)| Cx −K J2n 2n,N (x)| 1 1 (x) is a polynomial in x −2n CN J2n (N x). But, by Lemma 2.1, (3) and the formula (2.13), J2n of degree at most n. Therefore, we obtain (2.12). 2 In general, for any τ ∈ C with Re(τ ) > 0 and any n > 0, we have ∞ e

−τ N x

ϕ(x) dx =

n−1 (j −1) ϕ (0) j =1

0

(τ N )j

+ O N −n .

Taking K > 0 in Proposition 2.2 so that n + 1 < K < 2n and integrating (2.11), we conclude the following. Proposition 2.3. When ω = 1 is the qth root of unity, we have ω RN (ϕ) ∼

n1

cnω =

cnω

ϕ (n−1) (0) , Nn

n−1 α (n − k − 1)! p(α, α − k; ω) . α!(n − α − 1)! (1 − ω)n−k

(2.14)

α=0 k=0

When ω = 1, we have RN [0, ∞ ; ϕ) ∼

∞ ϕ(x) dx + 0

cn =

2n (α − n)! α=n

α!

n1

cn

ϕ (n−1) (0) , Nn

(−1)α−n+1 p(α, α − n),

(2.15)

516


where we set p(n, k) := p(n, k; 1) =

k

n

t

t=0

(−1)t S(n − t, k − t),

0 k n.

(2.16)

Note that a direct computation and the well-known formula for the relation among the 1 2n Bernoulli numbers, Catalan numbers n+1 n , and the Stirling numbers [10] shows

n bn (n + 1) 2n −1 (−1)l 2n S(n + l, l) = − , cn = − n! n l+1 n+l n!

(2.17)

l=0

which shows that, for ω = 1, we have RN [0, ∞); ϕ ∼

∞ ϕ(x) dx −

bn n1

0

n!

ϕ (n−1) (0)N −n ,

(2.18)

from which we have (0.5). For ω = 1, we compare each term of the asymptotics (2.4), (2.14) to get bnω = (−1)n−1 cnω =

n−1 α (n − k − 1)! p(α, α − k; ω) (−1)k+1 . α!(n − α − 1)! (ω − 1)n−k

(2.19)

α=0 k=0

2.2. Definition of Szasz functions Let C be a unimodular cone in X. Since the Riemann sum RN (C; ϕ) depends only on the restriction of ϕ to L(C), replacing (X, Λ) by (L(C), L(C) ∩ Λ) if necessary, we assume, for a moment, that dim(C) = dim(X). Then, C is written in the form C=

R+ e,

e∈E

where E is an integral basis of Λ, and R+ denotes the set of non-negative real numbers. For abstract two sets S and T , let S T be the set of all functions from T to S. The whole space X is identified with RE . Since E is an integral basis, Λ is identified with ZE . Note that, C and C ∩ Λ E are identified with RE + and Z+ , respectively, where Z+ denotes the set of non-negative integers. E For any α ∈ Z+ and x ∈ X, we set α! =

α(e)!,

e∈E

xα =

x(e)α(e) ,

e∈E

where x(e) is the value of x at e ∈ E when we identify X = RE . For each γ ∈ ZE + , we define the function γ on X by

γ (x) =

x γ − x(e) e∈E e . γ!

(2.20)


517

Then, the function γ is non-negative, integrable on C and satisfies

γ ∈ ZE + ,

γ (x) dx = 1

γ (x) = 1 (x ∈ X).

(2.21)

γ ∈ZE +

C

Definition 2.4. We define the Szasz measure S(x) = dSx on X = RE , parametrized by x ∈ X, by

S(x) = dSx :=

γ (x)δγ .

γ ∈ZE +

By the second property of (2.21), the measure dSx is a probability measure on C. For each N ∈ N, the N th dilated convolution powers, denoted by dSxN , of dSx is given by

γ (N x)δγ /N , dSxN := (D1/N )∗ S(x) ∗ · · · ∗ S(x) = γ ∈ZE +

where D1/N : X → X is the dilation D1/N (x) = x/N , x ∈ X. Definition 2.5. We define the Szasz function SN (ϕ) associated to a function ϕ on X, by SN (ϕ)(x) :=

ϕ(z) dSxN (z) =

γ (N x)ϕ(γ /N )

(2.22)

γ ∈ZE +

C

if the sum in the right-hand side converges absolutely. By (2.21), the Szasz function SN (ϕ) satisfies that RN (C; ϕ) :=

1 N dim(C)

γ ∈C∩Λ

ϕ(γ /N ) =

SN (ϕ)(x) dx

(2.23)

C

if the sum converges absolutely. 2.3. Asymptotics of Szasz functions For each μ, ν ∈ ZE + with μ ν, we define pE (μ, ν) =

p μ(e), ν(e) ,

(2.24)

e∈E

μ(e) μ and where p(n, k) is an integer defined by (2.16). For each μ ∈ ZE + , we set ∇ = e∈E ∇e |μ| = e∈E μ(e). Then, a relevant asymptotic formula for the Szasz function SN (ϕ) is given as follows.

518


Proposition 2.6. For each positive integer r and positive number K with r < K < 2r, there exists a positive constant Cr,K such that we have

SN (ϕ)(x) =

μ∈ZE + ; |μ|2r−1

∇ μ ϕ(x) Jμ (N x) + S2r,N (x), μ!N |μ|

(2.25)

where the function S2r,N (x) satisfies the following estimate; S2r,N (x) Cr,K N −r 1 + |x| r−K , where the norm |x| of x ∈ X is defined by |x|2 = nomial in x of degree at most [|μ|/2] given by

2 e∈E x(e) ,

Jμ (x) =

x ∈ C,

and the function Jμ (x) is a poly-

pE (μ, ν)x ν ,

ν∈ZE + ; ν[μ/2]

where [μ/2] ∈ ZE + is defined by [μ/2](e) = [μ(e)/2]. Proof. The proof is the same as that for Proposition 2.2. Inserting the Taylor expansion ϕ(z) =

μ∈ZE + ; |μ|2r−1

1 ∇ μ ϕ(x) (z − x)μ + R2r,μ (z, x)(z − x)μ , μ! μ! |μ|=2r

1 R2r,μ (z, x) = 2r

(1 − t)2r−1 ∇ μ ϕ x + t (z − x) dt

0

with z = γ /N into the definition (2.22) of the Szasz function SN (ϕ), we have SN (ϕ)(x) =

μ∈ZE + ; |μ|2r−1

∇ μ ϕ(x) Jμ (N x) + S2r,N (x), μ!N |μ|

where the functions Jμ (x), S2r,μ (x) are given by Jμ (x) =

γ (N x)(γ − x)μ ,

γ ∈ZE +

S2r,N (x) =

|μ|=2r

1

γ (N x)R2r,μ (γ /N, x)(γ − N x)μ . μ!N |μ| E γ ∈Z+

The formula (2.27) is easily obtained by the relation γν x γ = e e∈E x(e) SE (ν, α)x α , γ ! E

γ ∈Z+

αν

(2.26)

(2.27)


519

which follows from Lemma 2.1(1), where SE (ν, α) is given by SE (ν, α) =

S ν(e), α(e) .

(2.28)

e∈E

Next, we estimate the term S2r,N (x). Note that x and γ /N are in C. Thus, we have |x + t (γ / N −x)| (1−t)|x| for each 0 t 1. We choose a positive constant Cr,K such that |∇ μ ϕ(x)| Cr,K (1 + |x|)−K for each μ ∈ ZE + with |μ| = 2r. Hence, if |x| 1 and |μ| = 2r, we have μ ∇ ϕ x + t (γ /N − x) Cr,K 1 + x + t (γ /N − x) −K Cr,K (1 − t)−K |x|−K . (2.29) Thus, for |x| 1, we have |R2r,μ (γ /N, x)| Cr,K |x|−K Cr,K (1 + |x|)−K , where Cr,K is a constant. Therefore, we obtain 1 S2r,N (x) Cr,K 1 + |x| −K (γ − N x)μ

γ (N x) 2r μ! N E |μ|=2r

γ ∈Z+

−K Cr,K 1 + |x| J2μ (N x). N 2r |μ|=r

As is mentioned above, the function J2μ (x) with |μ| = r is a polynomial in x of degree at most r. Thus, we have |Jμ (x)| Cμ |x|r where Cμ does not depend on x. Therefore, we obtain the estimate (2.26). When |x| 1, we estimate R2r,μ (γ /N, x) as |R2r,μ (γ /N, x)| Cr,K , and hence S2r,N (x) is bounded by Cr,K N −r Cr,K N −r (1 + |x|)r−K , which completes the proof. 2 3. Asymptotic Euler–Maclaurin formula over unimodular cones In this section, we deduce asymptotic Euler–Maclaurin formula of the Riemann sum over unimodular cones in a rational space (X, Λ). At first, we deduce it by using Proposition 2.6. The result coincide a well-known result due to Guillemin and Sternberg [11]. We don’t need to use a rational inner product on X so far. Then, we renormalize each term of the expansion using an integration by parts procedure to find explicit form of Berline–Vergne operators. This step involves a rational inner product. 3.1. An Euler–Maclaurin formula for unimodular cones As before, let C be a unimodular cone in X with dim(C) = dim(X) and let E be the integral basis of Λ generating C. For each I ⊂ E, let |I | be the number of elements in I . For such I , E we regard ZI+ as a subset of ZE + consisting of α ∈ Z+ with the property that α(e) = 0 for each J I e ∈ E \ I . Clearly we have Z+ ⊂ Z+ if I ⊂ J . For any e ∈ E, we define λe ∈ ZE + by λe (e) = 1, λe (v) = 0, v ∈ E \ {e}. Then, we obviously have α = e∈E α(e)λe for each α ∈ ZE + . For ∅ = I ⊂ E, we set ZI>0 = {α ∈ ZI+ ; α(e) = 0, e ∈ I }. For I = ∅, we set Z∅>0 = {0}. Each I ⊂ E corresponds to a face C(I ) of C defined by C(I ) :=

e∈E\I

R+ e,

C(E) := {0},

(3.1)

520


and for each face F of C, there is a unique IF ⊂ E such that F = C(IF ). Thus, we identify subsets in E and faces of C. Note that F ⊂ G if and only if IG ⊂ IF . For each μ, ν ∈ ZE + with ν μ and ∅ = I ⊂ E, we set pI (μ, ν) :=

p μ(e), ν(e) .

(3.2)

e∈I

If μ, ν ∈ ZI+ , we clearly have pJ (μ, ν) = pI (μ, ν) for each J with I ⊂ J because p(0, 0) = 1. For each ν ∈ ZI+ , we set

pI (ν) =

(−1)|μ|

μ∈ZI+ ; νμ2ν

(μ − ν)! pI (μ, μ − ν). μ!

(3.3)

Then, we have pJ (ν) = pI (ν) if ν ∈ ZI+ and I ⊂ J . Note that pI (ν) = e∈I p(ν(e)), where we have p(n) = (−1)n−1 cn = (−1)n bn /n! as in (2.15), (2.17). For each non-negative integer n and a subset I of E with |I | n, we define a homogeneous differential operator Ln (C; I ) of order n − |I | on X with constant coefficients by

Ln (C; I ) = (−1)n

e(I ) =

pI (ν)∇ ν−e(I ) ,

ν∈ZI>0 ; |ν|=n

λe

(n 1),

(3.4)

e∈I

and L0 (C; ∅) = 1. When n 1 we set Ln (C; I ) = 0 for |I | > n or I = ∅. Proposition 3.1. For each ϕ ∈ S(X), we have RN (C; ϕ) ∼

N −n

(−1)|I |

I ⊂E; |I |n

n0

Ln (C; I )ϕ.

(3.5)

C(I )

Proof. We use Proposition 2.6. We take r ∈ N and K > 0 so that r + dim(X) < K < 2r. By the estimate (2.26), one can integrate the asymptotic expansion (2.25) over C. Then, by (2.23) and (2.25), we have RN (C; ϕ) =

μ,ν; |μ|2r−1, ν[μ/2]

1 −|μ−ν| N pE (μ, ν) μ!

x ν ∇ μ ϕ + O N −r .

C

Integrating by parts, we have C x ν ∇ μ ϕ = (−1)|ν| ν! C ∇ μ−ν ϕ, and hence, substituting this into the formula for RN (C; ϕ) above, we obtain

RN (C; ϕ) =

r−1 k=0

N −k

C

Lk (C)ϕ + O N −r ,


Lk (C) = (−1)k

521

pE (ν)∇ ν

ν∈ZE + , |ν|=k

=

(−1)|μ|+k

ν,μ, |ν|=k, νμ2ν

(μ − ν)! pE (μ, μ − ν)∇ ν . μ!

To integrate by parts further in the right-hand side, note that we have ZE += a disjoint union. For ν ∈ ZI>0 , we have ν |I | ∇ ϕ = (−1) ∇ ν−e(I ) ϕ. C

I I ⊂E Z>0 ,

(3.6)

which is

C(I )

From this, we obtain Lk (C)ϕ =

(−1)|I |

I ⊂E; |I |k

C

which shows the assertion.

Lk (C; I )ϕ,

C(I )

2

Next, we consider cones containing straight lines. Let E be an integral basis of Λ, and let I ⊂ E. Consider the cone C in X of the form C= R+ e + L, (3.7) e∈I

where L is a subspace in X spanned by vectors e ∈ E \ I . If I = E, then L = {0} and in this case C is a unimodular cone discussed above. When I = ∅, we set C = X. Lemma 3.2. Let E be an integral basis of Λ, and let C be a cone of the form (3.7) with I ⊂ E. Then, for each ϕ ∈ C0∞ (X), we have RN (C; ϕ) = RN πL (C); (πL )∗ ϕ + O N −∞ , (3.8) where (πL )∗ ϕ is a compactly supported smooth function on X/L defined by (πL )∗ ϕ(x) = ϕ, x ∈ X/L. πL−1 (x)

Proof. For simplicity, we write π = πL : X → X/L for the natural projection. Take ϕ ∈ C0∞ (X). For any v ∈ X, we set Tv ϕ(x) := ϕ(x + v). Let M be the subspace spanned by I so that X = M ⊕ L. We identify L with RE\I and M with RI in a natural way. Then, we can choose v ∈ ZE\I E\I so that supp(Tv ϕ) ⊂ M + R>0 . Clearly we have RN (C; ϕ) = RN (RE + ; Tv ϕ), where we note that E R+ is a unimodular cone in X. Therefore, by (3.6), we have N −n Ln RE RN (C; ϕ) ∼ + Tv ϕ, n0

RE +

522


where the differential operator Ln (RE + ) is given in (3.6). Note that π(C) is a unimodular cone in X/L with respect to the lattice π(Λ) generated by the integral basis π(I ) of π(Λ). Since v ∈ ZE\I , we have π∗ Tv ϕ = π∗ ϕ. Therefore, according to Proposition 3.1, we only need to show that E Ln R+ ψ dx = Ln π(C) π∗ ψ dx (3.9) RE +

π(C)

E for any ψ ∈ C0∞ (X) with supp(ψ) ⊂ M + R >0 . If ν ∈ Z+ has some e ∈ E \ I such that ν(e) 1, E\I

E\{e}

then, since supp(ψ) ∩ R+

= ∅, we have

∇ ν ψ = 0 and hence RE +

Ln R E + ψ dx =

RE +

If we denote ∇πν =

pI (ν)∇ ν .

ν∈ZI+ ; |ν|=n

RE +

L˜ n = (−1)n

L˜ n ψ,

ν(e)

e∈I

∇π(e) for each ν ∈ I , then, by the definition of the function π∗ ψ on E\I

X/L, we have ∇πν π∗ ψ = π∗ ∇ ν ψ for each ν ∈ ZI+ . Since supp(ψ) ⊂ M + R+ , we obtain, for ν ∈ ZI+ , ∇πν π∗ ψ = π∗ ∇ ν ψ = ∇ ν ψ. π(C)

π(C)

RE +

From this and the definition of Ln (π(C)), we obtain (3.9).

2

Remark. As is mentioned in Introduction, Proposition 3.1 is deduced directly from Theorem 3.2 in [11]. Lemma 3.2 is also obtained in [11]. 3.2. Integration by parts In Proposition 3.1 and Lemma 3.2 in the previous subsection, we have derived an asymptotic formula for the Riemann sums over unimodular cones and their variants. In each term in these asymptotic formulas, integration over faces of homogeneous differential operators Ln (C; I ) defined in (3.6) appears. The differential operators Ln (C; I ) involve derivatives only in directions transversal to the face C(I ). However, these derivatives are not ‘perpendicular’ to the face C(I ), and hence we can perform further integration by parts. If one performs integration by parts in (3.5), then one will find the differential operators which involves derivatives only in directions perpendicular to faces. However, we need to perform this procedure systematically to define the operators all at once. This step is one of the main points which makes the final formula complicated. In the rest of this paper, we fix a rational inner product Q on the rational space (X, Λ). Let E be an integral basis of Λ. For each I ⊂ E, we set Re ∼ X(E) = {0}. (3.10) X(I ) = = RE\I , e∈E\I


523

Note that X = X(∅), and if I ⊂ J , then X(J ) ⊂ X(I ) and hence X(I )⊥Q ⊂ X(J )⊥Q . As before, for each I ⊂ E, we define the unimodular cone C(I ) in X by (3.1). For each α ∈ ZE + , we set

α(e) α ∇ = e∈E ∇e . Proposition 3.3. There exists a family L(E; I, J ; α); ∅ = I ⊂ J ⊂ E, α ∈ ZI+ , |J | |α| + |I | of homogeneous differential operators L(E; I, J ; α) of order |α| − |J | + |I | on X with rational constant coefficients which involves derivatives only in directions perpendicular to the rational subspace X(J ) such that for each (I, α) with ∅ = I ⊂ E, α ∈ ZI+ , we have

∇αϕ = C(I )

(−1)|J |−|I |

J ;I ⊂J, |J ||α|+|I |

L(E; I, J ; α)ϕ

(3.11)

C(J )

for any ϕ ∈ S(X). Furthermore, fix α and ∅ = I ⊂ E with α ∈ ZI+ . Suppose that a family {L(J ); I ⊂ J ⊂ E, |J | |α| + |I |} of homogeneous differential operators with constant coefficients of order |α| − |J | + |I | which involves derivatives only in directions perpendicular to X(J ) satisfy Eq. (3.11) for any ϕ ∈ S(X). Then, we have L(J ) = L(E; I, J ; α). Note that a differential operator on X is said to have rational coefficients if it has rational coefficients with respect to an (and hence all) integral basis of Λ. We first prove the existence of such family of differential operators. Proof of the existence in Proposition 3.3. For a given ∅ = I ⊂ E and e ∈ I , we decompose e along with the orthogonal decomposition X = X(I )⊥Q ⊕ X(I ), which is denoted by e = u(I ; e) +

c(I ; e, v)v,

u(I ; e) ∈ X(I )⊥Q ∩ XQ ,

c(I ; e, v) ∈ Q,

e ∈ I.

v∈E\I

(3.12)

We set u(E; e) = e for each e ∈ E. We construct the operators L(E; I, J ; α) inductively as follows. (0) When |α| = 0, then |J | = |I | and I ⊂ J implies I = J . In this case, we set L(E; I, I ; 0) = 1.

(3.13)

(1) We take ∅ = I ⊂ J ⊂ E and α ∈ ZI+ with |α| = 1 and |J | |α| + |I |. In this case, J = I or J = I ∪ {v} with v ∈ E \ I and α = λe with e ∈ I . We then define L(E; I, J ; λe ) by

L(E; I, I ; λe ) = ∇u(I ;e) L E; I, I ∪ {v}; λe = c(I ; e, v)

(when I = J ), when J = I ∪ {v} with v ∈ E \ I .

(3.14)

524


Then, by the identity

∇v ϕ = −

ϕ,

v ∈ E \ I, ϕ ∈ C0∞ (X),

C(I ∪{v})

C(I )

it is easy to show that the operators L(E; I, I ∪ {v}; λe ) satisfy ∇e ϕ = L(E; I, I ; λe )ϕ + (−1) L E; I, I ∪ {v}; λe ϕ. C(I )

v∈E\I

C(I )

(3.15)

C(I ∪{v})

(2) Suppose that, for a positive integer n 2, we have defined differential operators L(E; I, J ; β) satisfying the formula (3.11) for any ϕ ∈ C0∞ (X) for each I , J , β with ∅ = I ⊂ J ⊂ E, β ∈ ZI+ satisfying |β| n − 1, |J | |β| + |I |. (3) For ∅ = I ⊂ J ⊂ E and α ∈ ZI+ satisfying |α| = n and |J | n + |I |, we take e ∈ I such that α(e) 1. Then, we can decompose α as α = λe + β,

β ∈ ZI+ ,

|β| = n − 1.

(3.16)

We define L(E; I, J ; α) by the formula ⎧ L(E; I, I ; β)∇u(I ;e) (when J = I ), ⎪ ⎪ ⎨ L(E; I, J ; β)∇ u(I ;e) + v∈J \I c(I ; e, v)L(E; I ∪ {v}, J ; β) L(E; I, J ; α) = (3.17) (when |I | + 1 |J | |I | + |α| − 1), ⎪ ⎪ ⎩ v∈J \I c(I ; e, v)L(E; I ∪ {v}, J ; β) (when |J | = |I | + |α|). A direct computation shows that the differential operators L(E; I, J ; α) satisfy (3.11).

2

Next, we proceed to prove the uniqueness of such family {L(E; I, J ; α)}. Fix α and ∅ = I ⊂ E such that α ∈ ZI+ . If α = 0, then clearly the operator L(E; I, I ; 0) satisfying (3.11) is just the constant 1. If I = E, then for any α ∈ ZE + , the operator L(E; E, E; α) is uniquely determined as L(E; E, E; α) = ∇ α . Thus, in the following, we assume α = 0 and ∅ = I E. If the homogeneous differential operators {L(J )}I ⊂J ⊂E;|J ||α|+|I | with constant coefficients of order |α| − |J | + |I | which involves derivatives only in directions perpendicular to X(J ) satisfy Eq. (3.11) for any ϕ ∈ S(X), then their symbols σ (L(J )) must satisfy the equation ξα =

J ;I ⊂J,|J ||α|+|I |

σ L(J ) (ξ ) ξ, e,

ξα =

e∈J \I

σ L(J ) (ξ ) = σ L(J ) pJ (ξ ) ,

ξ, eα(e) ,

e∈E ∗

ξ ∈X ,

(3.18)

where the symbol σ (D) of a differential operator D on X (with constant coefficients) is a polynomial function on X ∗ characterized by σ (D)(ξ ) = e−ξ Deξ , eξ (x) = eξ,x , x ∈ X, ξ ∈ X ∗ . In (3.18), pJ denotes the orthogonal projection from X ∗ onto the annihilator X(J )⊥ of X(J ). Therefore, to prove the uniqueness in the statement of Proposition 3.3, it is enough to show the uniqueness of the family of homogeneous polynomials {σ (L(J ))} satisfying (3.18). First of all, consider the following expression.


σ (I, I ; α)(ξ ) = pI (ξ )α , σi (I ; α)(pJ (ξ )) pJ (ξ )α − k−1

i=0 , I ⊂ J, |J | = k + |I |, σ (I, J ; α)(ξ ) = e∈J \I pJ (ξ ), e σi (I ; α)(ξ ) = σ (I, J ; α)(ξ ) ξ, e,

525

(3.19)

(3.20)

e∈J \I

J ;I ⊂J,|J |=|I |+i

where k is an integer satisfying 1 k |α|. Note that σ0 (I ; α) = σ (I, I ; α) = pI (ξ )α is a welldefined homogeneous polynomial of degree |α| on X ∗ . Thus, the above Eqs. (3.19), (3.20) define rational functions σ (I, J ; α), σi (I ; α) for ∅ = I ⊂ J ⊂ E, α ∈ ZI+ , |J | |α| + |I |, 0 i |α|, which are homogeneous of degree |α| − |J | + |I |, |α|, respectively. Note also that the functions σ (I, J ; α) satisfy the second line of (3.18). Lemma 3.4. The functions defined by (3.19), (3.20) are homogeneous polynomials. Proof. First of all, let us examine the function σ (I, J ; α) with I ⊂ J , |J | = 1 + |I |. In this case, we can write J = I ∪ {u} with some u ∈ E \ I . By (3.20), we have pI ∪{u} (ξ )α − pI (ξ )α σ I, I ∪ {u}; α (ξ ) = . pI ∪{u} (ξ ), u Take ξ ∈ X(I ∪ {u})⊥ , which means that ξ, e = 0 for each e ∈ E \ I , e = u. Thus, there exists an η ∈ X(I ∪ {u})⊥ perpendicular to X(I )⊥ with respect to the inner product Q such that Q(η, η) = 1. Note that η, u = 0. Let q(ξ ) = Q(ξ, η)η denote the orthogonal projection onto the onedimensional subspace Rη. Then, we have pI ∪{u} = pI + q, and hence (pI (ξ ) + q(ξ ))α − pI (ξ )α σ I, I ∪ {u}; α (ξ ) = , q(ξ ), u which is a homogeneous polynomial of degree |α| − 1. Next, to use the induction, suppose that the functions σ (I, J ; α) with I ⊂ J , |J | k + |I | (1 k |α| − 1) are homogeneous polynomials of degree |α| − |J | + |I |. By (3.20), the functions σi (I ; α) (i = 0, . . . , k) are homogeneous polynomials of degree |α|. Take J0 ⊂ E such that I ⊂ J0 , |J0 | = k + 1 + |I |. Set f (ξ ) = pJ0 (ξ )α − ki=1 σi (I ; α)(pJ0 (ξ )). Note that the polynomial f is determined on X(J0 )⊥ . So, let ξ ∈ X(J0 )⊥ . Assume that ξ, e = 0 for some e ∈ J0 \ I , which means that ξ ∈ X(K)⊥ with K = J0 \ {u}. Note that I ⊂ K J , |K| = k + |I |. Since ξ ∈ X(K)⊥ ⊂ X(J0 )⊥ , we have pJ0 (ξ ) = ξ , and hence by (3.20), σk (I ; α)(ξ ) = σ (I, K; α)(ξ )

e∈K\I

ξ, e = ξ − α

k−1

σi (I ; α)(ξ ).

i=0

This shows that f (ξ ) = 0 for ξ ∈ X(K)⊥ . Thus, the homogeneous polynomial f (ξ ) is divisible by the linear function pJ0 (ξ ), e for each e ∈ J0 \ I . Note that the elements in

J0 \ I are linearly independent. Therefore, the homogeneous polynomial f (ξ ) is divisible by e∈J0 \I pJ0 (ξ ), e, and hence σ (I, J0 ; α) is a homogeneous polynomial. This completes the proof. 2

526


Proof of the uniqueness in Proposition 3.3. Fix α, I such that 0 = α ∈ ZI+ , ∅ = I E. Suppose that the set of functions {s(J )}I ⊂J ⊂E,|J ||α|+|I | , where each s(J ) is a homogeneous function on X ∗ of degree |α| − |J | + |I |, satisfy Eq. (3.18). Let σ (I, J ; α) denote the homogeneous polynomials defined by (3.19), (3.20). We need to prove s(J ) = σ (I, J ; α). Let ξ ∈ X(I )⊥ . Then, we have ξ, e = 0 for each e ∈ E \ I . Thus, by (3.18), we have s(I )(ξ ) = σ (I, I ; α)(ξ ) for each ξ ∈ X(I )⊥ . Since s(I ) satisfies the second equation of (3.18), we have s(I ) = σ (I, I ; α) on X ∗ . Next, take J0 ⊂ E such that I ⊂ J0 , |J0 | = 1 + |I | |α| + |I |. We write J0 = I ∪ {u}. Take ξ ∈ X(J0 )⊥ . Then ξ, e = 0 for each e ∈ E \ J0 , and hence the function s(J0 ) must satisfy ξ α = σ0 (I ; α)(ξ ) + s(J0 )(ξ )ξ, u,

ξ ∈ X(J0 )⊥ .

Since the function s(J0 ) satisfies the second line of (3.18), we have s(J0 )(ξ ) =

(pI (ξ ) + q(ξ ))α − pI (ξ )α , q(ξ ), u

J0 = I ∪ {u},

where q is the orthogonal projection onto the one-dimensional subspace of X(J0 )⊥ perpendicular to X(I )⊥ . This equation shows s(J0 ) = σ (I, J0 ; α) for J0 = I ∪ {u}. Now, suppose that for any J ⊂ E with I J , |J | k + |I |, k + 1 |α|, we have s(J ) = σ (I, J ; α). Take J0 ⊂ E with I ⊂ J0 , |J0 | = k + 1 + |I | |α| + |I |. Take ξ ∈ X(J0 )⊥ . Since ξ, e = 0 for each e ∈ E \ J0 , Eq. (3.18) shows ξ α = sk (ξ ) + s(J0 )(ξ ) ξ, e,

sk (ξ ) =

e∈J0 \I

σ (I, J ; α)(ξ )

I ⊂J J0 , |J |k+|I |

ξ, e,

(3.21)

e∈J \I

for each ξ ∈ X(J0 )⊥ . If J ⊂ E with I ⊂ J satisfy J \ J0 = ∅, then we have e∈J \I ξ, e = 0 and hence sk (ξ ) = ki=0 σi (I ; α)(ξ ). Since s(J0 ) satisfies the second line of (3.18), we have s(J0 ) = σ (I, J0 ; α). 2 Remark. One can prove directly that the polynomials σ (I, J ; α) defined by (3.19), (3.20) actually a solution to (3.18). However, its proof is similar to the proof of the existence in Proposition 3.3, and hence we omit it. In the above, we have defined the differential operators L(E; I, J ; α) for each ∅ = I ⊂ J ⊂ E and α ∈ ZI+ satisfying |J | |α| + |I |. But we need to work on the quotient space X/L and the unimodular cone π(C) which is the image of a unimodular cone C under the natural projection π : X → X/L where L is a subspace spanned by a subset of E. To state the next lemma, we need to fix some notation. Let E be an integral basis of Λ. For ∅ = K ⊂ E, we set, as before, X(K) = v∈E\K Rv. Let πK : X → X/X(K) be the natural projection. For each e ∈ E, we set e = πK (e). Then, we have πK (E) = πK (K) = {e; e ∈ K}, and the set πK (K) is an integral basis of the lattice πK (Λ) in X/X(K). Note that πK is a bijective map from K onto π(E). For each π(I ) ∅ = I ⊂ K and α ∈ ZI+ , denote πK (α) ∈ Z+ the Z+ -valued function on πK (I ) defined by πK (α)(e) := α(e), π(I )

e ∈ I.

(3.22)

We note that, for each α ∈ Z+ , there is a unique α ∈ ZI+ with the property that πK (α) = α.


527

Lemma 3.5. In the notation as above, we identify X/X(K) with X(K)⊥Q to give X/X(K) the inner product induced by the inner product Q on X. Then, for each ∅ = I ⊂ J ⊂ K and α ∈ ZI+ with |J | |α| + |I |, we have L(E; I, J ; α) = L πK (K); πK (I ), πK (J ); πK (α)

(3.23)

where the operator L(πK (K); πK (I ), πK (J ); πK (α)) is regarded as an operator on X by the identification X/X(K) ∼ = X(K)⊥Q . Proof. For simplicity, we set π = πK . Let σK (I, J ; α) denote the symbol of the differential operator L(π(K); π(I ), π(J ); π(α)) which is a homogeneous polynomial on (X/X(K))∗ , and note that we have the identification (X/X(K))∗ ∼ = X(K)⊥ under the transpose t π : (X/X(K))∗ → ∗ X of π . Then, the symbol of the lift of the differential operator L(π(K); π(I ), π(J ); π(α)) is given by σK (I, J ; α)(pK (ξ )), ξ ∈ X ∗ . The symbols of the operators L(E; I, J ; α) for J ⊂ K, which are as above denoted by σ (I, J ; α), are determined on X(K)⊥ . By (3.18), we have ξα =

σ (I, J ; α)(ξ )

ξ, e,

ξ ∈ X(K)⊥ .

(3.24)

e∈J \I

J ;I ⊂J ⊂K |J ||α|+|I |

In Proposition 3.3, we can replace X by X(E \ K) which is identified, as a rational space, with X/X(K). With this identification, the symbols σK (I, J ; α) also satisfy Eq. (3.24). Noting that Eq. (3.24) is nothing but Eq. (3.18) on X(E \ K), and using the uniqueness in Proposition 3.3, we conclude the assertion. 2 3.3. Berline–Vergne operators over unimodular cones We use the results obtained in the previous subsections and Theorem 1.2 to find an explicit expression of Berline–Vergne operators for unimodular cones. Definition 3.6. (1) Let C be a unimodular cone in a rational space (X, Λ) with a rational inner product Q. Assume that dim(C) = dim(X). Let E be the integral basis of Λ generating C. For each F ∈ F (C), we take, as before, a unique IF ⊂ E such that F = C(IF ). Then, for each F ∈ F (C) and n ∈ Z+ with dim(F ) dim(C) − n, we define a homogeneous differential operator DnX (C; F ) of order n − dim(C) + dim(F ) with rational constant coefficients which involves derivatives only in directions perpendicular to the face F by DnX (C; F ) := (−1)n−dim(C)+dim(F )

pI (ν)L E; I, IF ; ν − e(I ) , (3.25)

I ⊂IF ν∈ZI , |ν|=n >0

and D0X (C; C) := 1, DnX (C; C) := 0 (n 1). (2) Let C ⊂ X be a unimodular cone. For any F ∈ F (C) and n ∈ Z+ with n − dim(C) + dim(F ), let DnX (C; F ) be the differential operator DnL(C) (C; F ) regarded as an operator on X L(C) through the inclusion ιC : L(C) → X, where the operator Dn (C; F ) is defined as in (1) replacing (X, Λ) by (L(C), L(C) ∩ Λ).

528


For unimodular cones C in X with dim(C) < dim(X), the differential operator DnX (C; F ) is L(C) characterized by the identity ι∗C DnX (C; F )ϕ = Dn (C; F )ι∗C ϕ for ϕ ∈ C ∞ (X). Thus, a direct computation using Definition 3.6, (3.11), (3.4) combined with Proposition 3.1 (replacing (X, Λ) by (L(C), L(C) ∩ Λ) if necessary) shows the following. Theorem 3.7. Let C be a unimodular cone in the rational space (X, Λ) with a rational inner product. Then, for any ϕ ∈ S(X), we have RN (C; ϕ) ∼

n0

N −n

DnX (C; F )ϕ.

F ∈F (C), dim(F )dim(C)−n F

Let C be a unimodular cone in the rational space (X, Λ), and let F ∈ F (C). The order of the differential operator DnX (C; F ) is n − (dim(C) − dim(F )), and which is equal to the order of the X/L(F ) differential operator Dn (πF (C); 0). Moreover, we have the following. Lemma 3.8. Let C be a unimodular cone in (X, Λ), and let F ∈ F (C). Then, the operator X/L(F ) DnX (C; F ) coincides with the lift of the operator Dn (πF (C); 0) on X through the identifi⊥ ∼ Q cation X/L(F ) = L(F ) . Proof. Let K be the integral basis of L(C) ∩ Λ generating C, and let F = C(IF ) with a subset IF of K. Then, the cone πF (C) in the rational space (X/L(F ), πF (Λ)) is a unimodular cone with the generator IF = {e; e = πF (e), e ∈ IF }. Thus, by Definition 3.6, we have πF (C); 0 = (−1)n−dim(πF (C))

X/L(F )

Dn

pI (ν)L IF ; I , IF ; ν − e(I ) .

I ⊂IF ν∈ZI ;|ν|=n >0

The subsets I of K correspond to the subsets I of IF by the projection πF , and the elements ν in ZI+ corresponds to the elements ν in ZI+ . Therefore, Lemma 3.5 shows L IF ; I , IF ; ν − e(I ) = L πF (IF ); πF (I ), πF (IF ); πF ν − e(I ) = L K; I, IF ; ν − e(I ) as an operator on X. From this, the assertion follows.

2

Example. In one dimension, it is easy to compute the differential operators DnX (C; F ). Let X be a 1-dimensional vector space with the lattice Λ. Let u ∈ Λ be a generator and set C = R+ u. The faces of C are 0 and C itself. Then, E = {u}. By definition, we have D0X (C; C) = 1, DnX (C; C) = 0 (n 1). By (3.17), we have L(E; {u}, {u}; k) = ∇uk for k ∈ Z+ . Thus, by Definition 3.6, we have DnX (C; 0) = (−1)n−1 p(n)∇un−1 = −

bn n−1 ∇ , n! u

and its symbol is given by − bn!n ξ, un−1 , and we have RN (C; ϕ) ∼

n 1, C

ϕ−

(3.26) bn n−1 n1 n! ∇u ϕ(0).


529

Theorem 3.9. For each unimodular cone C in a rational space (X, Λ), each face F ∈ F (C) and each non-negative integer n such that dim(F ) dim(C) − n, we have DnX (C; F ) = DnX (C; F ), where DnX (C; F ) is the differential operator defined in Definition 3.6 and DnX (C; F ) is the Berline–Vergne operator defined in Definition 1.1. Proof. For any rational space (X, Λ), any rational subspace L in X, any unimodular cone C in X/L and any non-negative integer n satisfying n dim(C), define the operator DnX (C) on X by X/L the lift of Dn (C; 0) to X under the identification X/L ∼ = L⊥Q . We need to check that these operators satisfy the conditions in Theorem 1.2. The condition (1) in Theorem 1.2 follows from this definition. The condition (4) in Theorem 1.2 follows from Theorem 3.7 and Lemma 3.8. The condition (3) follows from Example above. The condition (2) follows from Definition 3.6. Therefore, the assertion follows from Theorem 1.2. 2 4. Asymptotic Euler–Maclaurin formula over rational cones In this section, we derive an asymptotic Euler–Maclaurin formula of RN (C; ϕ) for general rational cone C. To discuss asymptotic expansion of RN (C; ϕ) for pointed rational cones C, we define, for such a cone C and non-negative integer n, the distribution An (C; ·) ∈ S (X) by DnX (C; F )ϕ, (4.1) An (C; ϕ) := F ∈F (C), dim(F )dim(C)−n F

where DnX (C; F ) is the Berline–Vergne operator defined in Definition 1.1. d Lemma 4.1. Let {Ci }i=1 be a family of pointed rational cones in a rational space (X, Λ) satisfying i ri χ(Ci ) = 0, where, for each subset S ⊂ X, χ(S) denotes the characteristic function of S. We set m = maxi dim(Ci ). Suppose further that there exists a vector η ∈ X ∗ such that η, x < 0 for each 0 = x ∈ ∪i Ci . Then, for each ϕ ∈ S(X), we have ri An−m+dim(Ci ) (Ci ; ϕ) = 0. (4.2) i,dim(Ci )m−n

Proof. Since Ak (Ci ; ·) are distributions and C0∞ (X) is dense in S(X), it is enough to prove (4.2) for each ϕ ∈ C0∞ (X). Note that the function S(C) defined in (1.1) have a valuation property (see [2]). By this and Eq. (1.2), we have ri μ πG (Ci ) I (G) = 0, i G∈F (Ci )

where the subscript X/L(G) in μX/L(G) is dropped since these functions are lift to X ∗ . Substituting tξ (t ∈ R, ξ ∈ X ∗ ) in these functions and taking the Taylor expansion of each function, we have t k μdim(G)+k πG (Ci ) (ξ )I (G)(ξ ) = 0. k−m i,G∈F (Ci ),dim(G)+k0

530


Thus, each coefficient of t k in the above vanishes, and hence we have ri μn−m+dim(G) πG (Ci ) (iξ + η)I (G)(iξ + η) = 0

(4.3)

i,G∈F (Ci ), dim(G)m−n

for each n 0, where, η ∈ X ∗ is as in the statement of the lemma and ξ ∈ X ∗ is arbitrary. Let ϕ ∈ C0∞ (X). We have Dn−m+dim(Ci ) (Ci ; G)ϕ(x) 1 eiξ +η (x)μn−m+dim(G) πG (Ci ) (iξ + η)ϕ(ξ ˆ − iη) dξ, = m (2π)

(4.4)

X∗

where the Lebesgue measure dξ on X ∗ is normalized as in Section 1.3. Taking the integral over G, we have Dn−m+dim(Ci ) (Ci ; G)ϕ G −m

= (2π)

I (G)(iξ + η)μn−m+dim(G) πG (Ci ) (iξ + η)ϕ(ξ ˆ − iη) dξ,

(4.5)

X∗

where we have used the fact that eiξ +η is integrable on G for each i and G ∈ F (Ci ). Thus, multiplying (4.5) by ri , taking the sum over all i and G ∈ F (Ci ) with dim(G) m − n and using Eq. (4.3), we have (4.2). 2 Theorem 4.2. Let C be a pointed rational cone in a rational space (X, Λ) with a rational inner product Q. Then, for any ϕ ∈ S(X), we have N −n An (C; ϕ), RN (C; ϕ) ∼ n0

where An (C; ϕ) is defined in (4.1). Furthermore, the uniqueness statement of Theorem 1.2 still true if we replace the unimodular cones in the statement of Theorem 1.2 with the pointed rational cones. Proof. By replacing X with L(C), we may assume that m := dim(C) = dim(X). It is well known that, for any pointed rational cone C, one can find a finite set of unimodular cones C = {σi }di=1 such that C is a subdivision of the pointed cone C, namely, the collection C satisfies the following. (1) C = F (σ ) (2) σ, τ ∈ C ⇒ σ ∩ τ ∈ F (σ ) ∩ F (τ ) (3) C = σ. σ ∈C

σ ∈C

(For a proof of this fact, see [8, Section 2.6].) By the inclusion-exclusion principle, there is a relation χ(C) = σ ∈C rσ χ(σ ) with some rσ ∈ Z. Then, we have RN (C; ϕ) =

1 r χ(σ )(γ )ϕ(γ /N ) = N −m+dim(σ ) rσ RN (σ ; ϕ). σ Nm σ ∈C γ ∈Λ

σ ∈C


531

Each cone σ ∈ C is a unimodular cone in X, and hence we can apply Theorem 3.7 to RN (σ ; ϕ) for each σ ∈ C. By a direct computation, we have N −n rσ An−m+dim(σ ) (σ ; ϕ), RN (C; ϕ) ∼ n0

σ ∈C ,dim(σ )m−n

and hence Lemma 4.1 shows the first part of the assertion. The last assertion on the uniqueness follows from the same discussion as in Theorem 1.2, and hence we omit the proof. 2 In the next section, we need the following lemma, which generalizes Lemma 3.2. Lemma 4.3. Let C be a rational cone in a rational space (X, Λ). Let L = C ∩ (−C). Then, for any ϕ ∈ C0∞ (X), we have RN (C; ϕ) = RN πL (C); (πL )∗ ϕ + O N −∞ . Proof. If L = {0}, we have the conclusion without the term O(N −∞ ). So, we assume that L = {0}. For simplicity, we write π = πL : X → X/L, the natural projection. Take ϕ ∈ C0∞ (X). Since L is rational, one can take a complementary rational subspace W to L such that X = L⊕W and Λ = (L ∩ Λ) ⊕ (W ∩ Λ). Set G = C ∩ W , which is a pointed rational cone in W . We have C = L + G. Take a subdivision C of G into unimodular cone in W . The set {C σ = L + σ ; σ ∈ C} is a subdivision of C into rational cones. Then, there is a relation χ(C) = σ ∈C rσ χ(Cσ ), and hence RN (C; ϕ) = N −m+dim(Cσ ) rσ RN (Cσ ; ϕ). σ ∈C

Note that the cones Cσ is of the form discussed in Lemma 3.2. By Lemma 3.2, we have RN (Cσ ; ϕ) = RN (π(σ ); π∗ ϕ) + O(N −∞ ), and hence RN (C; ϕ) = N −m+dim(σ )+dim(L) rσ RN π(σ ); π∗ ϕ + O N −∞ . σ ∈C

The set {π(σ ); σ ∈ C} is a subdivision of the pointed rational cone π(C) in X/L into unimodular cones. Furthermore, since χ(C) = σ ∈C rσ χ(Cσ ) we have χ(π(C)) = σ ∈C rσ χ(π(Cσ )). (π defines a valuation. See [2].) Thus, the sum in the right-hand side of the last equation is RN (π(C); π∗ ϕ), which proves the assertion. 2 5. Results and their proofs In this section, we restate Theorem 1 on the asymptotic Euler–Maclaurin formula of the Riemann sum RN (P ; ϕ) :=

1 N dim(P )

ϕ(γ /N ),

γ ∈(N P )∩Λ

for a lattice polytope P in a rational space (X, Λ) and a smooth function ϕ on P in the abstract notation we used as before and give its proof. We also state and give proofs of its corollaries.

532


5.1. Main theorems and their proofs Theorem 5.1. Let P be a lattice polytope in a rational space (X, Λ) with a rational inner product. For each f ∈ F (P ) and n ∈ Z+ satisfying dim(f ) dim(P ) − n, let DnX (P ; f ) be the differential operator defined in Definition 1.1. Then, for each ϕ ∈ C ∞ (P ), we have the following asymptotic expansion: RN (P ; ϕ) ∼

An (P ; ϕ)N −n ,

n0

An (P ; ϕ) =

DnX (P ; f )ϕ.

(5.1)

f ∈F (P );dim(f )dim(P )−n f

To prove Theorem 5.1, we need the following lemma. Lemma 5.2. Let f ∈ F (P ) and let n ∈ Z+ satisfy dim(f ) dim(P )−n. Then, for any g ∈ F (P ) such that g ⊂ f , we have (5.2) DnX (P ; f ) = DnX πg CP (g) ; πg Cf (g) . Proof. First of all, as in Section 1.2, note that we have DnX (P ; f ) = DnX (πf (CP (f )); 0). We set C = πg (CP (g)) and G = πg (Cf (g)). Then, C is a pointed rational cone in X/L(g) and G ∈ F (C). Furthermore, we have DnX (πg (CP (g)); πg (Cf (g))) = DnX (πG (C); 0), where πG : X/L(g) → (X/L(g))/L(G) = X/L(f ) is the natural projection. Since πG ◦πg = πf : X → X/L(f ) and CP (f ) = L(f ) + CP (g), we have πG (C) = πf (CP (g)) = πf (CP (f )), and hence Eq. (5.2) follows. 2 Proof of Theorem 5.1. For any g ∈ F (P ) and v ∈ g, we set CP+ (g) = CP (g) + v which does not depend on the choice of v ∈ g. Then, we use the following version of Euler’s formula [4, Proposition 3.2(1)]: + (−1)dim(g) δ CN (5.3) δ (N P ) ∩ Λ = P (Ng) ∩ Λ , g∈F (P )

where, N is a positive integer and, for any subset S of Λ, δ(S) is a distribution defined by δ(S), ϕ = ϕ(s), ϕ ∈ C0∞ (X). s∈S ∗ ϕ)(x) = ϕ(x/N ). For each g ∈ F (P ), we fix For each N ∈ Z>0 and ϕ ∈ C ∞ (X), we set (D1/N vg ∈ g ∩ Λ. Clearly we have

∗ δ(N P ∩ Λ), D1/N ϕ = N dim(P ) RN (P ; ϕ), + ∗ ϕ = N dim(P ) RN CP (g); Tvg ϕ , δ CN P (Nf ) ∩ Λ , D1/N where, for v ∈ X, we set Tv ϕ(x) = ϕ(v + x). Take ϕ ∈ C ∞ (P ) and extend ϕ as a compactly supported smooth function on X. Then, by (5.3) and Lemma 4.3, we have


RN (P ; ϕ) =

533

(−1)dim(g) RN CP (g); Tvg ϕ

g∈F (P )

∼

(−1)dim(g) RN πg CP (g) ; (πg )∗ Tvg ϕ .

(5.4)

g∈F (P )

Since πg (CP (g)) is a pointed rational cone in X/L(g) with respect to the lattice πg (Λ), we can use Theorem 4.2 for RN (πg (CP (g)); (πg )∗ Tvg ϕ)) and hence RN (P ; ϕ) ∼

An (P ; ϕ)N −n ,

n0

g∈F (P )

G∈F (πg (CP (g))), dim(G)dim(P )−n−dim(g)

An (P ; ϕ) =

×

(−1)dim(g)

DnX πg CP (g) ; G (πg )∗ Tvg ϕ.

(5.5)

G

Each faces G ∈ F (πg (CP (g))) with dim(G) dim(P ) − n − dim(g) can be written as G = πg (Cf (g)) with a face f ∈ F (P ) such that g ⊂ f and dim(f ) dim(P ) − n. Furthermore, the correspondence

f ∈ F (P ); g ⊂ f f → πg Cf (g) ∈ F πg CP (g)

defines a bijective correspondence between the above two sets. Thus, by Lemma 5.2 and the definition of the function (πg )∗ Tvf ϕ, we can write An (P ; ϕ) =

=

=

g∈F (P )

f ∈F (P ), g⊂f dim(f )dim(P )−n

g∈F (P )

f ∈F (P ), g⊂f dim(f )dim(P )−n

πg (Cf (g))

(−1)dim(g)

DnX (P ; f )ϕ

Cf+ (g)

DnX (P ; f )(πg )∗ Tvg ϕ

(−1)dim(g)

f ∈F (P ), dim(f )dim(P )−n g∈F (f )

(−1)dim(g)

χ f Cf+ (g) DnX (P ; f )ϕ,

f

where f is the affine hull of f , and for each S ⊂ f , we denote χ f (S) the characteristic function of S on f . In the first line above, we used an obvious identity DnX (P ; f )(πg )∗ ψ = (πg )∗ DnX (P , f )ψ for ψ ∈ C0∞ (X). To simplify the above, we use the formula (Proposition 3.1(1) in [4])

(−1)dim(g) χ CP+ (g) = χ(P ).

g∈F (P )

(5.6)

534


Note that in [4], the above formula is proved for P with non-empty interior. Replacing P by f ∈ F (P ), which is regarded as a polytope in the affine subspace f with non-empty relative interior, we have (−1)dim(g) χ f Cf+ (g) = χ f (f ). g∈F (f )

Therefore, we obtain the formula (5.1) for An (P ; ϕ), which complete the proof of Theorem 5.1. 2 Remark. In the proof above, we used Theorem 4.2. However, if the lattice polytope P is Delzant, then the cone πf (CP (f )) for each f ∈ F (P ) is a unimodular cone in X/L(f ). Therefore, we only need to use Theorem 3.7. Hence, for Delzant lattice polytopes, it turns out that our proof of Theorem 5.1 is independent of [3]. However, for general lattice polytopes, it does not seem to be easy to construct the operator DnX (P ; f ) in such a way given in Definition 3.6. Indeed, Definition 3.6 is based on Proposition 3.1. This means that if we could obtain a result like Proposition 3.1 for general rational cones, then one might be able to find such an expression as in Definition 3.6. Hence, it might be better to prove a result like Proposition 3.1 for rational cones without using a subdivision of rational cones into unimodular cones. However, to do this, it seems that one need to find a different method. Next, we show that, under some assumptions, the asymptotic expansion of RN (P ; ϕ) of the form (5.1) is unique. Theorem 5.3. Suppose that, for any rational space (X, Λ) with a rational inner product, rational subspace L of X, pointed rational cone C in X/L and non-negative integer n such that n dim(C), there exists a homogeneous differential operator DnX (C) of order n − dim(C) with symbol νnX (C) such that they satisfy the conditions (1), (2) and (3) in Theorem 1.2. Furthermore, suppose that, for any lattice polytope P in X and ϕ ∈ C ∞ (P ), the following holds: RN (P ; ϕ) ∼

n0

N

−n

DnX πf CP (f ) ϕ.

(5.7)

f ∈F (P );dim(f )dim(P )−n f

Then, we have DnX (C) = DnX (C; 0) for any pointed rational cone C in X and non-negative integer n with n dim(C), where the operator DnX (C; 0) is defined in Definition 1.1. Proof. Let us prove the assertion by the induction on dim(X). For dim(X) = 0, 1, the assertion is true by the condition (3) in Theorem 1.2. Suppose that for each (X, Λ) with dim(X) m − 1, the assertion holds. Let dim(X) = m. Take a pointed rational cone C in X. We may assume that dim(C) = m. Take a vector ξ ∈ Λ∗ such that ξ, x > 0 for any x ∈ C. Set P1 = C ∩ {x; ξ, x 1}, which is a rational polytope in X. Hence, each vertex of P1 is a rational point in X. We take a positive integer q such that P = qP1 is a lattice polytope. Let U be a small open ball around the origin such that U ∩ V(P ) = {0} and U ⊂ {x; ξ, x < q}. Then, by the assumption, for each ϕ ∈ C0∞ (U ), the Riemann sum RN (P ; ϕ) admits the asymptotic expansion (5.7). In (5.7), if dim(f ) > 0, then since dim(πf (CP (f ))) = m − dim(f ) < m, the differential operators DnX (πf (CP (f ))) coincide with DnX (P , f ) = DnX (πf (CP (f )), 0) by the induction hypothesis.


535

Take a vertex v of P . Suppose v = 0. Since ϕ is zero near v, the contribution from the vertex v to the expansion (5.7) vanishes. Thus, by Theorem 5.1, we have X (5.8) Dn π0 CP (0) ϕ (0) = DnX (P ; 0)ϕ (0) (n m). Take ρ ∈ C0∞ (U ) such that ρ = 1 near 0. For any ψ ∈ C ∞ (X), we have (5.8) for ϕ = ρψ. But since ρ = 1 near 0, Eq. (5.8) holds for any ϕ ∈ C ∞ (X). Take ϕ ∈ C ∞ (X) and x ∈ X. Applying (5.8) for the function Tx ϕ, we have X Dn π0 CP (0) ϕ (x) = DnX (P ; 0)ϕ (x) (n m) for any ϕ ∈ C ∞ (X). Since π0 (CP (0)) = C, we conclude the assertion.

2

5.2. Computation in one and two dimensions A polytope P in a rational space (X, Λ) is said to be Delzant if for each vertex v of P , the number of edges incident to v is dim(X) and there exists an integral basis E of Λ such that each edge incident to v is of the form {v + te; t 0} with an e ∈ E. In this and the next subsection, we give explicit computations for Delzant lattice polytopes. To compute each coefficient An (P ; ϕ) in the asymptotic expansion of the Riemann sum RN (P ; ϕ), it is important to compute in low dimensions. In this subsection, we perform these computation. In this and the next subsections, we drop the superscript X in DnX (P , g). 5.2.1. In one dimension Let X be a 1-dimensional vector space with the lattice Λ. Let u ∈ Λ be a generator and set C = R+ u. We have computed the differential operator Dn (C; 0) in Example at the end of Section 3.3. Let P be an interval given by P = {tu ∈ X; a t b} with a, b ∈ Z, a < b. Since Dn (P ; P ) = 0 for n 1, we have n−1 An (P ; ϕ) = Dn P ; {a} ϕ(a) + Dn P ; {b} ϕ(b) = (−1)n−1 p(n) ∇un−1 ϕ(a) + ∇−u ϕ(b) . Identifying X = R and u = 1 so that Λ = Z, we have An (P ; ϕ) = −

bn (n−1) ϕ (a) − (−1)n ϕ (n−1) (b) . n!

Substituting b2m+1 = 0 (m 1) and b2m = (−1)m−1 Bm with the Bernoulli number Bm , we have A2m+1 (P ; ϕ) = 0,

A2m (P ; ϕ) = (−1)m−1

Bm (2m−1) ϕ (b) − ϕ (2m−1) (a) , (2m)!

which shows the classical asymptotic Euler–Maclaurin formula. 5.2.2. In two dimension Next, we compute in two dimension. Let (X, Λ) be a two-dimensional rational vector space with a rational inner product Q. Let E = {e1 , e2 } be an integral basis of the lattice Λ, and set C = R+ e1 + R+ e2 . Set e1 = u1 + c1 e2 ,

e2 = u2 + c2 e1 ,

536


where the non-zero vectors u1 , u2 ∈ X satisfy Q(u1 , e2 ) = Q(u2 , e1 ) = 0, and c1 , c2 ∈ Q are given by c1 =

Q(e1 , e2 ) , Q(e2 , e2 )

c2 =

Q(e1 , e2 ) . Q(e1 , e1 )

(5.9)

Define λ1 , λ2 ∈ ZE by λi (ej ) = δi,j . A straightforward computation shows L(C; E, E; kλ1 + lλ2 ) = ∇ek1 ∇el 2 , L C; {e2 }, {e2 }; lλ2 = ∇ul 2 , L C; {e1 }, {e1 }; kλ1 = ∇uk1 , k−1 ∇us 1 ∇ek−1−s , L C; {e1 }, E; kλ1 = c1 1

l−1 L C; {e2 }, E; lλ2 = c2 ∇us 2 ∇el−1−s . 2

s=0

s=0

Set F1 = R+ e2 , F2 = R+ e1 . Then, we have Dn (C; F1 ) = (−1)n−1 p(n)∇un−1 , 1 Dn (C; 0) = (−1)n

n−1

Dn (C; F2 ) = (−1)n−1 p(n)∇un−1 2

(n 1),

p(k)p(n − k)∇ek−1 ∇en−1−k 1 2

k=1

+ (−1) p(n) c1 n

n−2

∇us 1 ∇en−2−s 1

+ c2

s=0

n−2

∇us 2 ∇en−2−s 2

(n 2).

s=0

Let P be a Delzant lattice polytope in (X, Λ). For each facet f of P , Dn (P ; f ) is the lift of Dn (πf (CP (f )); 0). Let αf ∈ Λ be the inward primitive normal of f . (Such a vector αf exists because the dual basis of an integral basis of Λ with respect to Q is rational.) We identify πf (CP (f )) with R+ αf by the map ϕf : X/L(f ) x + L(f ) →

Q(x, αf ) αf ∈ Rαf . Q(αf , αf )

Let e1 ∈ Λ be a generator of L(f ) ∩ Λ. Since P is Delzant, we can find e2 ∈ CP (f ) ∩ Λ such that {e1 , e2 } forms an integral basis of Λ. Then, the vector uf :=

Q(e2 , αf ) αf Q(αf , αf )

(5.10)

is a generator of ϕf (πf (Λ)) such that ϕf (πf (CP (f ))) = R+ uf . Note that the definition of uf does not depend on the choice of e2 ∈ CP (f ) ∩ Λ whenever e1 , e2 forms an integral basis of Λ. Hence, by (3.26), the differential operator Dn (P ; f ) is given by =− Dn (P ; f ) = (−1)n−1 p(n)∇un−1 f Therefore, we have the following.

bn n−1 ∇ n! uf

(n 1).


537

Corollary 5.4. Let (X, Λ) be a two-dimensional rational vector space with a rational inner product Q. Let P be a Delzant lattice polytope in (X, Λ). Then, the coefficients An (P ; ϕ) (n 2) in the asymptotic expansion (5.1) of the Riemann sum RN (P ; ϕ) is given by Dn (P ; f )ϕ + Dn (P ; v)ϕ(v). An (P ; ϕ) = f ∈F (P )1 f

v∈V (P )

In the above, the differential operators Dn (P ; f ) and Dn (P ; v) are given by Dn (P ; f ) = − Dn (P ; v) =

bn n−1 ∇ , n! uf

n−1 bk bn−k ∇ k−1 ∇ n−1−k k!(n − k)! e1 (v) e2 (v) k=1 n−2 n−2 bn n−2−s n−2−s s s + ∇u1 (v) ∇e1 (v) + c2 (v) ∇u2 (v) ∇e2 (v) , c1 (v) n! s=0

s=0

where, for a face f ∈ F (P )1 , uf ∈ XQ denotes the inward normal defined in (5.10), and for a vertex v ∈ V(P ), the vectors e1 (v), e2 (v) ∈ Λ denote the integral basis of Λ such that two facets meeting at v lie on the half lines v + tei (v), t 0, i = 1, 2, and u1 (v), u2 (v) ∈ X satisfy e1 (v) = u1 (v) + c1 (v)e2 ,

Q u1 (v), e2 (v) = 0,

e2 (v) = u2 (v) + c2 (v)e1 ,

Q u2 (v), e1 (v) = 0,

Q(e1 (v), e2 (v)) , Q(e2 (v), e2 (v)) Q(e1 (v), e2 (v)) c2 (v) = . Q(e1 (v), e1 (v))

c1 (v) =

Note that, in the following, we use D2 (C; 0) for two-dimensional unimodular cone C. The explicit formula for D2 (C; 0) is given by D2 (C; 0) = p(1)2 + (c1 + c2 )p(2) =

1 1 + (c1 + c2 ) , 4 12

(5.11)

where c1 , c2 are given in (5.9). 5.3. Computation of the coefficient in the third term Our main Theorem 5.1, or rather the construction of the operators Dn (P ; f ), allows us to compute the coefficient A2 (P ; ϕ) in the third term of the asymptotic expansion (5.1). Before computing the third term, let us compute the first and second terms. Corollary 5.5. For any Delzant lattice polytope P in a rational space (X, Λ) with a rational inner product Q, we have 1 A0 (P ; ϕ) = ϕ dx, A1 (P ; ϕ) = ϕ, 2 P

g∈F (P )m−1 g

538


where the integration on facets g ∈ F (P )m−1 is performed with respect to the measure on g induced by the lattice Λ. Proof. The first term is obvious. For the second term A1 (P ; ϕ), note that the dimension of faces which contribute to A1 (P ; ϕ) is m − 1 and m. But the operator D1 (P ; P ) is the lift of D1 (0; 0) (see Definition 3.6) which is zero. Thus, the contribution to A1 (P ; ϕ) comes from facets. Let g ∈ F (P )m−1 . Then the operator D1 (P ; g) is the lift of D1 (πg (CP (g)); 0), which is a rational constant. Let αg ∈ Λ be inward primitive normal of the facet g. As in the computation in two dimension, let ϕg : X/L(g) → L(g)⊥Q be the isomorphism defined by Q(x, αg ) αg . ϕg x + L(g) = Q(αg , αg ) We take an integral basis e1 , . . . , em−1 of L(g) ∩ Λ. Since P is Delzant, one can take em ∈ CP (g) such that e1 , . . . , em form an integral basis of Λ. We set ug =

Q(em , αg ) αg ∈ L(g)⊥Q . Q(αg , αg )

(5.12)

As before, the definition of ug above does not depend on the choice of em above. By (3.26), we have D1 (πg (CP (g)); 0) = − b1!1 = 12 . Hence, we have A1 (P ; ϕ) =

1 2

ϕ,

g∈F (P )m−1 g

2

which completes the proof.

Note that the above formula for the second term A1 (P ; ϕ) coincides with that in (0.7). Indeed, if X = Rm , Λ = Zm and Q is the standard Euclidean inner product, the primitive inward primitive normal αg for each facet g of a Delzant polytope P is a part of an integral basis of Zm . Next, we compute the third term, which does not seem to have been obtained before. For simplicity, we work in the Euclidean space X = Rm with the standard lattice Zm and the standard inner product. Corollary 5.6. Let P be a Delzant lattice polytope in the Euclidean space (Rm , Zm ) with the standard inner product Q. Then, we have the following: A2 (P ; ϕ) = −

1 12

+

g∈F (P )m−1

g∈F (P )m−2

1 Q(αg , αg )

∇αg ϕ g

1 1 Q(α1 (g), α2 (g)) Q(α1 (g), α2 (g)) − + 4 12 Q(α1 (g), α1 (g)) Q(α2 (g), α2 (g))

ϕ, g

where, for g ∈ F (P )m−1 , the vector αg is the inward primitive normal to g, and for g ∈ F (P )m−2 , the vectors α1 (g), α2 (g) are the inward primitive normal to the facets g1 , g2 ∈ F (P )m−1 such that g = g1 ∩ g2 .


539

Proof. By (5.1), the faces which contribute to A2 (P ; ϕ) is of m − 1 or m − 2 dimension. Let g be a facet of P . Then, Dn (P ; g) is the lift of Dn (πg (CP (g)); 0). Hence, as before, we have =− Dn (P ; g) = (−1)n−1 p(n)∇un−1 g

bn n−1 ∇ n! ug

(n 1),

where the rational vector ug ∈ L(g)⊥Q is given in (5.12). But, we are working in the standard Euclidean space with the integral lattice Zm and the standard inner product. Since P is Delzant, we can take an integral basis e1 , . . . , em of Zm such that e1 , . . . , em−1 is an integral basis of L(g) ∩ Zm and if we denote the dual basis of e1 , . . . , em by α1 , . . . , αm , then αm = αg . Thus, we have ug = αg /Q(αg , αg ) and hence D2 (P ; g) = −

b2 1 ∇αg /Q(αg ,αg ) = − ∇α . 2! 12Q(αg , αg ) g

Next, suppose that g is a face of dimension m − 2. Take two facets g1 , g2 such that g = g1 ∩ g2 . Denote αi (g) ∈ Λ the primitive inward normal to gi (i = 1, 2). Let v be a vertex in g, and take g3 , . . . , gm ∈ F (P )m−1 such that {v} = g1 ∩ · · · ∩ gm . Let E = {e1 , . . . , em } be an integral basis of Zm such that each vector v + ej defines an edge incident to v and v + ej ∈ / gj . We have CP (g) = R+ e1 + R+ e2 + L(g), and e3 , . . . , em is an integral basis of L(g) ∩ Zm . Let α1 , . . . , αm be the dual basis of e1 , . . . , em . Then αi = αi (g) for i = 1, 2, and α1 , α2 form a basis of L(g)⊥ . We write e1 = u1 + v1 ,

u1 , u2 ∈ L(g)⊥ ,

e2 = u2 + v2 ,

v1 , v2 ∈ L(g).

Under the identification X/L(g) x + L(g) → Q(x, α1 )u1 + Q(x, α2 )u2 ∈ L(g)⊥ , the cone πg (CP (g)) is identified with R+ u1 + R+ u2 and the generator of πg (Zm ) is identified with u1 , u2 . Thus, by (5.11), we have

1 1 Q(u1 , u2 ) Q(u1 , u2 ) + . D2 (P ; g) = D2 πg CP (g) ; 0 = + 4 12 Q(u1 , u1 ) Q(u2 , u2 )

(5.13)

But then it is straight forward to show that Q(u1 , u1 ) =

Q(α2 , α2 ) , D

Q(u2 , u2 ) =

Q(α1 , α1 ) , D

Q(u1 , u2 ) = −

D = Q(u1 , u1 )Q(u2 , u2 ) − Q(u1 , u2 )2 . From this and (5.13), we conclude the assertion.

2

Q(α1 , α2 ) , D

540


Acknowledgments The author would like to thank to Dr. Micheal Stolz who informed him about the work of O. Szasz [17]. He would also like to thank to Prof. Steve Zelditch for his helpful comments on the earlier version of the paper. References [1] J. Agapito, L. Godinho, New polytope decompositions and Euler–Maclaurin formulas for simple integral polytopes, Adv. Math. 214 (2007) 379–416. [2] A.I. Barvinok, J.E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in: New Perspectives in Algebraic Combinatorics, in: Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 91– 147. [3] N. Berline, M. Vergne, Local Euler–Maclaurin formula for polytopes, Mosc. Math. J. 7 (3) (2007) 355–386. [4] M. Brion, M. Vergne, Lattice points in simple polytopes, J. Amer. Math. Soc. 10 (2) (1997) 371–392. [5] L. Charles, Toeplitz operators and Hamiltonian torus actions, J. Funct. Anal. 236 (1) (2006) 299–350. [6] S.E. Cappel, J.L. Shaneson, Euler–Maclaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Sér. I Math. 321 (7) (1995) 885–890. [7] S.K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2) (2002) 289–349. [8] W. Fulton, Introduction to Toric Varieties, Ann. Math. Study, vol. 131, Princeton University Press, Princeton, 1993. [9] R. Feng, Szasz analytic functions and noncompact Kähler toric manifolds, arXiv:0809.2436v4 [math.DG]. [10] R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics, second ed., Addison–Wesley Publ., Reading, MA, 1994. [11] V. Guillemin, S. Sternberg, Riemann sums over polytopes, in: Festival Yves Colin de Verdière, Ann. Inst. Fourier (Grenoble) 57 (7) (2007) 2183–2195. [12] V. Guillemin, Z. Wang, The Mellin transform and spectral properties of toric varieties, Transform. Groups 13 (3–4) (2008) 575–584. [13] L. Hörmander, The multinomial distribution and some Bergman kernels, in: Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 249–265. [14] A.G. Khovanskii, A.V. Pukhlikov, A Riemann–Roch theorem for integrals and sums of quasipolynomials over virtual polytopes, St. Petersburg Math. J. 4 (1993) 789–812. [15] Y. Karshon, S. Sternberg, J. Weitsman, Euler Maclaurin with reminder for a simple integral polytope, Duke Math. J. 130 (3) (2005) 401–434. [16] Y. Karshon, S. Sternberg, J. Weitsman, Exact Euler Maclaurin formulas for simple lattice polytopes, Adv. in Appl. Math. 39 (1) (2007) 1–50. [17] O. Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. National Bureau of Standards 45 (3) (1950) 239–244. [18] T. Tate, Bernstein measures on convex polytopes, in: Spectral Analysis in Geometry and Number Theory, in: Contemp. Math., vol. 484, Amer. Math. Soc., Providence, RI, 2009, pp. 295–319. [19] Z. Wang, The twisted Mellin transform, arXiv:0706.2642v2 [math.CO]. [20] S. Zelditch, Bernstein polynomials, Bergman kernels and toric Kähler varieties, J. Symplectic Geom. 7 (2) (2009) 51–76.


Two-state free Brownian motions Michael Anshelevich 1 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States Received 15 June 2010; accepted 10 September 2010 Available online 28 September 2010 Communicated by D. Voiculescu

Abstract In a two-state free probability space (A, ϕ, ψ), we define an algebraic two-state free Brownian motion to be a process with two-state freely independent increments whose two-state free cumulant generating function R ϕ,ψ (z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states ϕ, ψ are normal, and ϕ is faithful, then there is only a one-parameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to ϕ = ψ), these processes only exist for finite time. © 2010 Elsevier Inc. All rights reserved. Keywords: Free probability; Free Brownian motion; Two-state non-commutative probability space; Free stochastic integral

1. Introduction The study of free probability was initiated by Voiculescu in the early 1980s [24]. While free probability has crucial applications to the study of operator algebras and random matrices, it has also developed into a deep and sophisticated theory in its own right. As one illustration, consider the free Central Limit Theorem. Its formulation is the same as for the usual CLT, with two changes. First, the objects involved are non-commutative random variables, that is, elements of a non-commutative ∗-algebra (or C ∗ -algebra, or von Neumann algebra A, or the algebra of operators affiliated to it), with a state ϕ which replaces the expectation functional. Second, E-mail address: [email protected]. 1 This work was supported in part by NSF grant DMS-0900935.


542

M. Anshelevich / Journal of Functional Analysis 260 (2011) 541–565

independence is replaced by Voiculescu’s free independence, which is more appropriate for noncommuting objects. The algebraic version of the theorem was proved in [24], followed by the full analytic version for identically distributed triangular arrays in [21] and general triangular arrays in [18]. Note that in the analytic theorems, the hypothesis on the distributions are identical to those in the usual CLT. On the other hand, in [9] and [26] the authors showed that the mode of convergence in the free CLT is actually much stronger than the classical convergence in distribution. In all these results, the limiting distribution is the semicircle law. It is characterized by having zero free cumulants of order greater than 2 (of course, in most of these results, no a priori assumption on the existence of free cumulants is made). An important point about free probability is that, as mentioned above, there are different settings in which the theory can be studied. Consider the notion of (reduced) free product, related to the discussion above by the property that different components in a free product are freely independent. One can take a reduced free product of ∗-algebras with states, or C ∗ -algebras with representations, or of Hilbert spaces, or of von Neumann algebras with states, and all of these constructions are consistent. One can frequently extend purely algebraic results to the more analytic context of normed algebras (although, as illustrated in [8], such extensions are often non-trivial). This paper is about a related theory where that is no longer the case. In [12,13], Bo˙zejko, Leinert, and Speicher constructed what they called a conditionally free probability theory, which we will refer to as two-state free probability theory. The setting is now a ∗-algebra (von Neumann algebra, etc.) A with two states, say ϕ and ψ . Initially the authors had a single example of such a structure, but the theory has since been quite successful, at least in two settings. For results concerning single distributions, including the study of limit theorems, see [19,6,25]; on the other hand, for results in the purely algebraic setting, see [22,11,2]. However, very little work on this theory has been done in the analytic setting; in fact, we are only aware of one article [23]. We show here that this is not a coincidence, by the following example. We define what is natural to call (algebraic) two-state free Brownian motions. This is a very large class of processes, since the “Brownian motion” property only determines the relative position of the expectations ϕ and ψ, but the choice of ψ is arbitrary, at least in the algebraic setting. We then show that if A is a von Neumann algebra, and the expectation ϕ is faithful, then out of this infinite-dimensional family only a one-parameter family of processes can actually be realized. Moreover, with the exception of the actual free Brownian motion (corresponding to the case ϕ = ψ ), these processes only exist on a finite time interval. The paper is organized as follows. After the introduction and a background section, in Section 3 we define the two-state free Brownian motions, and show that if ϕ is faithful, only a one-parameter family of these processes may exist. The method of proof involves stochastic integration. In Section 4 we show that this one-parameter family actually does exist, by using a Fock space construction. We show that these processes are not Markov, even though they have classical versions, the time-reversed free Poisson processes of [16]. We also compute the generators of these processes. Finally, Section 5 contains some comments on the case when A is a C ∗ -rather than a von Neumann algebra. In particular, in this section we give another characterization of the one-parameter family mentioned above: in a large class, these are the only processes whose higher variation processes converge to the appropriate limits in L∞ (ϕ) rather than just in L2 (ϕ).


543

2. Preliminaries 2.1. Partitions P(n) will denote the lattice of all partitions of a set of n elements (into non-empty, pairwise disjoint subsets called blocks). The number of blocks of π is denoted |π|. The partitions are ordered by reverse refinement, so that 0ˆ = {(1), (2), . . . , (n)} is the smallest and 1ˆ = {(1, 2, . . . , n)} is the largest partition. In the lattice, σ ∨ π is the smallest partition which is larger than both σ and π , and σ ∧ π is the largest partition which is smaller than both σ and π . NC(n) is the sub-lattice of non-crossing partitions, which have the property that whenever x1 < y1 < x2 < y2 with x1 , x2 ∈ U and y1 , y2 ∈ V , where U, V are blocks of the partition π , then U = V . In a non-crossing partition π , a block V is inner if for some y1 , y2 ∈ / V and all x ∈ V , y1 < x < y2 , otherwise it is called outer. Denote by Inn(π) all the inner blocks of the non-crossing partition π , and by Out(π) the outer blocks. Also, denote by NC1,2 (n) all the noncrossing partitions into pairs and singletons, in other words partitions with all |V | 2. Sing(π) are all the singleton blocks of a partition. If f is a function of k < n arguments and V ⊂ {1, . . . , n}, V = {i(1) < i(2) < · · · < i(k)}, then we denote f (x1 , x2 , . . . , xn : V ) = f (xi(1) , xi(2) , . . . , xi(k) ). 2.2. Jacobi parameters If ν is a probability measure on R all of whose moments are finite, it has associated to it two sequences of Jacobi parameters, J (ν) =

β0 , γ1 ,

β1 , γ2 ,

β2 , γ3 ,

β3 , γ4 ,

... . ...

There are numerous ways of defining these parameters, using orthogonal polynomials, tridiagonal matrices, or Viennot–Flajolet theory. For our purposes, the most convenient definition is the following. The Cauchy transform of ν can be expanded into a formal power series Gν (z) = R

where mn (ν) = expansion

Rx

n dν(x)

Gν (z) =

∞

1 1 dν(x) = mn (ν) n+1 , z−x z n=0

is the n-th moment of ν. Then we also have a continued fraction

1 z − β0 −

.

γ1 z − β1 −

γ2 z − β2 −

γ3 z − β3 −

γ4 z − ···

544


If some γ = 0, the continued fraction terminates, in which case the subsequent β and γ coefficients can be defined arbitrarily. See [17] for more details. The monic orthogonal polynomials {Pn } for ν satisfy a recursion relation xPn (x) = Pn+1 + βn Pn (x) + γn Pn−1 (x), with P−1 (x) = 0. Finally, we define the map Φt (Jacobi shift, an inverse of coefficient stripping) on probability measures with finite moments by

J Φt [ν] =

0, β0 , t, γ1 ,

β1 , γ2 ,

β2 , γ3 ,

β3 , γ4 ,

... ...

for ν as above. Equivalently, GΦt [ν] (z) =

1 , z − tGν (z)

and this last definition makes sense for general probability measures, see Remark 4.3 of [7]. 2.3. Two-state free probability theory In this section, (A, ϕ, ψ) is an algebraic two-state non-commutative probability space, that is, A is a star-algebra and ϕ, ψ are positive unital functionals on it. Asa will denote the self-adjoint part of A. For X1 , X2 , . . . , Xn ∈ A, define the free cumulant functionals R ψ (X1 , X2 , . . . , Xn ) via ψ[X1 X2 . . . Xn ] =

R ψ (X1 , X2 , . . . , Xn : V )

π∈NC(n) V ∈π

and the two-state free cumulant functionals R ϕ,ψ (X1 , X2 , . . . , Xn ) via ϕ[X1 X2 . . . Xn ] =

R ϕ,ψ (X1 , X2 , . . . , Xn : V )

π∈NC(n) V ∈Out(π)

R ψ (X1 , X2 , . . . , Xn : V ).

V ∈Inn(π)

Then both R ψ and R ϕ,ψ are multilinear functionals. ϕ,ψ Denote Rn (X) = R ϕ,ψ (X, X, . . . , X), where X is repeated n times. If X has distribution μ with respect to ϕ and ν with respect to ψ, its two-state free cumulant generating function is ϕ,ψ

R μ,ν (z) = RX (z) =

∞

Rnϕ,ψ (X)zn .

n=1 ψ

The definition of the free cumulant generating function R ν (z) = RX (z) is similar.


545

Definition 1. Let A1 , A2 , . . . , Ak ⊂ A be a family of subalgebras. (a) This family is ψ -freely independent if for any a1 , a2 , . . . , an ∈ A, aj ∈ Ai(j ) ,

i(j ) = i(j + 1)

with all ψ[aj ] = 0, we have ψ[a1 a2 . . . an ] = 0. (b) This family is two-state freely independent if it is ψ-freely independent and, under the same assumptions on a1 , . . . , an , also ϕ[a1 a2 . . . an ] = ϕ[a1 ]ϕ[a2 ] . . . ϕ[an ]. Theorem. (See Theorem 3.1 of [13].) Let A1 , A2 , . . . , Ak ⊂ A be a family of subalgebras. (a) This family is ψ -freely independent if and only if for any a 1 , a 2 , . . . , an ∈

k

Ai ,

i=1

we have R ψ (a1 , a2 , . . . , an ) = 0 unless all a1 , a2 , . . . , an ∈ Aj for the same j . (b) This family is two-state freely independent if and only if it is ψ -freely independent and also R ϕ,ψ (a1 , a2 , . . . , an ) = 0 unless all a1 , a2 , . . . , an ∈ Aj for the same j . 3. Almost uniqueness of the two-state free Brownian motion 3.1. Algebraic framework Definition 2. A family {X(t), 0 t T } ⊂ Asa is a process with two-state freely independent increments if the increments X [s, t) = X(t) − X(s) of this process corresponding to disjoint intervals are two-state freely independent. For convenience, we will also assume that X(0) = 0, ϕ[X(t)] = 0 for all t, and the distributions of the increments of the process with respect to both ϕ and ψ are stationary. These assumptions can be dropped.

546


From now on, whenever we are considering such a process, we will assume that {X(t): 0 t T } generate A. ϕ,ψ

Definition 3. X has a two-state normal distribution if Rn (X) = 0 for n > 2. Remark 1. The justification for this definition is that such random variables appear in the twostate free central limit theorem, see Theorem 4.3 in [13], Proposition 3.1 and Theorem 4.1 in [11], and Lemma 7 and Remark 3 in [2]. Theorem 1 shows that alternatively, we could define a two-state free Brownian motion below by requiring it to have zero higher variations, computed as limits in L2 (ϕ). Definition 4. A family {X(t), 0 t T } is an algebraic two-state free Brownian motion if it is a process with two-state freely independent increments which have two-state normal distributions, ϕ-mean zero, and ϕ-variance 2 ϕ X(t) − X(s) = (t − s). 3.2. Analytic framework Throughout most of the paper, (A, ϕ, ψ) will be a W ∗ -non-commutative probability space, that is, A a von Neumann algebra, ϕ a faithful normal state on A, and ψ a normal (typically not faithful) state on A. We will call algebraic two-state free Brownian motions that exist in this setting simply twostate free Brownian motions. We say that An → A in L2 (ϕ) if ϕ[|An − A|2 ] → 0. Notation 5. In a number of proofs, we will fix a time T > 0. In that case, we denote X = X(T ) and i−1 i T −X T . Xi,N = X N N Note that the two-state free independence of increments implies ψ

Rk (Xi,N ) =

1 ψ R (X), N k

ϕ,ψ

Rk

(Xi,N ) =

1 ϕ,ψ R (X). N k

Theorem 1. For a process with two-state freely independent increments,

lim ϕ

N →∞

N

2 k Xi,N

ϕ,ψ

= Rk

ϕ,ψ

(X)2 + R2k (X)

i=1

and

lim ϕ

N →∞

N i=1

2 2 Xi,N

ϕ,ψ − R2 (X)

ϕ,ψ

= R4

(X).


547

In particular, for a two-state free Brownian motion, in L2 (ϕ) N

lim

N →∞

ϕ,ψ

2 Xi,N = R2

(X)

i=1

and for k > 2, lim

N

N →∞

k Xi,N = 0.

i=1

We record these results symbolically as T

2 dX(t) = T

0

and

T 0

(dX(t))k = 0 for k > 2.

Proof.

ϕ

N

2 =

k Xi,N

N N k 2k k + ϕ Xi,N Xj,N ϕ Xi,N i=j

i=1

i=1

2k k 2 . + N ϕ X1,N = N (N − 1)ϕ X1,N Now using free cumulant expansions, this expression equals 1 1 ϕ,ψ ϕ,ψ Rk (X) + R|V | (X) = N(N − 1) |π| N N +N

π∈NC(k) π=1ˆ

1 ϕ,ψ R (X) + N 2k

N →∞ − −−−→ Rk

ϕ,ψ

Also,

ϕ

N

π∈NC(2k) π=1ˆ

1 N |π|

V ∈Out(π)

V ∈Out(π)

ϕ,ψ

R|V | (X)

2

V ∈Inn(π)

V ∈Inn(π)

ψ R|V | (X)

ψ R|V | (X)

ϕ,ψ

(X)2 + R2k (X). 2

2 Xi,N

ϕ,ψ − R2 (X)

i=1 N N N 2 4 2 ϕ,ψ ϕ,ψ 2 = + R2 (X)2 ϕ Xi,N Xj,N + ϕ Xi,N − 2R2 (X) ϕ Xi,N i=j

i=1

N →∞

ϕ,ψ −−−−→ R2 (X)2

The result follows.

2

ϕ,ψ ϕ,ψ + R4 (X) − 2R2 (X)2

i=1 ϕ,ψ

+ R2

ϕ,ψ

(X)2 = R4

(X).

548


Proposition 2.

lim ψ

N →∞

N

k Xi,N

ψ

= Rk (X).

i=1

We record this result symbolically as T

k ψ ψ dX(t) = Rk X(T ) .

0

The proof is similar to the preceding theorem, using the ψ-free independence of the increments of the process {X(t)}. ψ

ψ

Corollary 3. For a two-state free Brownian motion, Rk (X) = 0 for k > 2 and R2 (X(T )) = ϕ,ψ ψ R2 (X(T )). If R1 (X(T )) = αT , we call the corresponding process a two-state free Brownian motion with parameter α. Corollary 4. Denote by μt the distribution of X(t) with respect to ϕ, and by νt the corresponding distribution with respect to ψ. Then 1 4t − (x − αt)2 + dx 2πt

dνt (x) =

(1)

and for α = 0, 1 dμt (x) = 2πt

1 (4t − (x − αt)2 )+ dx + max 1 − 2 , 0 δ−1/α . 1 + αx α t

(2)

Also denote CT = 1 + αX(T ). Then the distribution of CT with respect to ϕ is 1 1 2π α 2 T

√ √ (((1 + α T )2 − y)(y − (1 − α T )2 ))+ y

1 dy + max 1 − 2 , 0 δ0 . α T

(3)

Proof. Since {X(t)} is a two-state free Brownian motion, its two-state free cumulant generating function is R μt ,νt (z) = tz2 . Moreover, the preceding corollary implies that the free cumulant generating function of νt is R νt (z) = αtz + tz2 for some α. Therefore νt is a semicircular distribution with Jacobi parameters

(4)


J (νt ) =

αt, t,

αt, t,

αt, t,

αt, t,

... ...

549

and formula (1) holds. If α = 0, then μt = νt and the process is the free Brownian motion, so throughout the rest of the paper we will assume that α = 0. By Lemma 7 of [2], Eq. (4) implies that μt = Φt [νt ], so μt is a free Poisson distribution with Jacobi parameters J (μt ) =

0, t,

αt, t,

αt, t,

... . ...

αt, t,

(5)

This implies the density formula (2). Note that even though these distributions are free Poisson, t is not the free convolution parameter. The last formula follows via the substitution y = 1 + αx. 2 Remark 2. The classical version of the process {X(t)} is a particular case of the free bi-Poisson process from [16], corresponding to η = α and the rest of the parameters equal zero. In fact, this process is a time-reversed free Poisson process: if Y (t) = tX(1/t), then the ψ-distribution of Y (t) has Jacobi parameters

α, t,

α, α, t, t,

α, t,

... ...

and so is semicircular with mean α and variance t, and its ϕ-distribution τt has Jacobi parameters J (τt ) =

0, α, α, α, . . . . t, t, t, t, . . .

So τt is the centered free Poisson distribution with parameter α, and {τt } form a free (rather than a two-state free) convolution semigroup. Proposition 5. The process {Y (t)} itself is not the free Poisson process. Proof. We compute, for s < t, Y (t) − Y (s) = tX(1/t) − sX(1/s) = (t − s)X(1/t) − sX [1/t, 1/s) . The free cumulant generating function of the ψ -distribution of (t − s)X(1/t) is ψ

R(t−s)X(1/t) (z) = α

(t − s)2 2 s2 2 t −s s z+ z =α 1− z + t − 2s + z . t t t t

Similarly, the free cumulant generating function of the ψ -distribution of −sX([1/t, 1/s)) is the same as that of −sX(1/s − 1/t), in other words 1 1 1 1 2 s s2 2 ψ R−sX([1/t,1/s)) (z) = α(−s) − − z + s2 z = α −1 + z+ s− z . s t s t t t

550


Since (t − s)X(1/t) and −sX([1/t, 1/s)) are freely independent with respect to ψ , it follows that the free cumulant generating function of Y (t) − Y (s) is the sum ψ

ψ

R(t−s)X(1/t) (z) + R−sX([1/t,1/s)) (z) = (t − s)z2 . Similarly, (t − s)2 2 s2 2 z = t − 2s + z , t t 1 2 s2 2 ϕ,ψ 2 1 − z = s− z , R−sX([1/t,1/s)) (z) = s s t t ϕ,ψ R(t−s)X(1/t) (z) =

and ϕ,ψ

ϕ,ψ

R(t−s)X(1/t) (z) + R−sX([1/t,1/s)) (z) = (t − s)z2 . We conclude that Y (t) − Y (s) has, with respect to ϕ, the centered semicircular distribution with variance t − s, which is clearly different from the distribution of Y (t − s). 2 Proposition 6. Let {X(t): 0 t S}, S > T be a process with two-state freely independent increments and A ∈ Asa two-state free from it. Assume that ψ[A] = 0. Then in L2 (ϕ), lim

N →∞

Symbolically,

T 0

N

Xi,N AXi,N = 0.

i=1

dX(t) A dX(t) = 0.

Proof.

ϕ

N

2 Xi,N AXi,N

=

N

ϕ[Xi,N AXi,N Xj,N AXj,N ] +

i=j

i=1

N 2 ϕ Xi,N AXi,N AXi,N . i=1

For the first term, since Xi,N are ϕ-centered, A is ψ -centered, and they are two-state freely independent among themselves, a cumulant expansion shows that each term of the sum is zero. For the second term, 2 2 2 1 ϕ,ψ 2 AXi,N = ϕ Xi,N ψ A = 2 R2 (X)2 ψ A2 . ϕ Xi,N AXi,N N So as N → ∞, both terms above converge to zero.

2

Corollary 7. If A is two-state free from {X(t): S1 t S2 }, then S2 S1

S2 2 dX(t) . dX(t) A dX(t) = ψ[A] S1


551

In particular, for a two-state free Brownian motion, 1 ψ[A] = S2 − S1

S2 dX(t) A dX(t). S1

Corollary 8. Let {X(t): 0 t T } be a process with two-state freely independent increments in (A, ϕ, ψ), which generates A. Then ψ is uniquely determined by ϕ and the process. Proof. For each S, for A ∈ W ∗ ({X(t): t < S}) we have T

T dX(t) A dX(t) = ψ[A]

S

2 dX(t) ,

S

the integrals being defined in L2 (ϕ). But

W ∗ X(t): t < S

0S 1/α 2 , ϕ is not a faithful state on At . Proof. For t > 1/α 2 , by Corollary 4 the distribution of CT has an atom at zero, and so CT has a non-trivial kernel. Indeed, the vector η=

∞ 1 n ⊗n − 1[0,t) αt n=0

is in this kernel; note that the norm of this vector is ∞ 1 n < ∞. α2t n=0

∞

⊗n Suppose ξ ∈ ker Ct . We can write ξ = n=0 ξn with ξn ∈ H⊗n . Note that on ∞ n=1 H , Ct acts in the same way as 1 + αS(t), where S(t) has the semicircular distribution, which has no atoms. So ξ is not in this subspace, and ξ0 = 0, so without loss of generality, ξ0 = Ω. But then (η − ξ ) is also in the kernel, and (η − ξ )0 = 0. It follows that ξ = η and so ker Ct = Cη. We conclude that At , and in fact C ∗ (X(t)), contains a rank-one operator Pη : ζ → η, ζ η. The non-zero positive operator s s X(s) + Pη X(s) + αt αt is in At . We compute, for s < t, s X(s) + η = 1[0,s) ⊗ η + αs(η − Ω) ⊥ Ω. αt Denote this vector by η(s). Then ϕ

2 s s Pη X(s) + = Ω, η(s) = 0. X(s) + αt αt

Therefore ϕ is not faithful on At .

2

Corollary 14. A two-state free Brownian motion {X(t)} with parameter α can be realized in a two-state non-commutative probability space (A, ϕ, ψ) with faithful normal ϕ and normal ψ for t ∈ [0, 1/α 2 ] but not for larger values of t.

556


Proof. Note first that the Fock space representation of At on F (L2 ([0, t], dx)) considered in this section is exactly the GNS representation of (At , ϕ). The assumption that {X(t)} is an (algebraic) two-state free Brownian motion determines the values of ϕ on the (non-closed) algebra generated by {X(t)}, and therefore, via the GNS representation, on the von Neumann algebra At . For the same reason, the values of ψ on the non-closed algebra are determined, and since ψ is normal, it extends uniquely to At . We conclude that the Fock realization is the unique realization of the two-state free Brownian motion {X(t)} with parameter α. But then the result follows from the preceding proposition. 2 Remark 4. The two-state free central limit theorems were proved in [13,11], where the (ϕ-)distributions of the limit objects were computed. These distributions do not belong to our class, but rather to the family considered in [23] and [27]. Therefore the processes corresponding to those distributions do not exist in the analytic sense. Note that in the first of the papers just cited, the time convolution parameter is n, while t is a fixed parameter. A related question concerns the 8-parameter family of two-state free convolution semigroups constructed in [5]: which of these semigroups are distributions of a process with two-state freely independent increments, which can be realized in a von Neumann algebra with a faithful state? Proposition 15. For α = 0, there is no ϕ-preserving conditional expectation from AT to At . Proof. Suppose the desired conditional expectation E : AT → At exists. It then satisfies E[B1 AB2 ] = B1 E[A]B2 and ϕ E[A] = ϕ[A] for A ∈ AT , B1 , B2 ∈ At . We compute, for t < s < T and B ∈ At , ϕ B ∗ E X(s)X(t) = ϕ B ∗ X(s)X(t) = BΩ, X(s)X(t)Ω = BΩ, 1[0,s) ⊗ 1[0,t) + tΩ + αs1[0,t) = BΩ, 1[0,t) ⊗ 1[0,t) + tΩ + αs1[0,t) = BΩ, X(t)2 + α(s − t)X(t) Ω = ϕ B ∗ X(t)2 + α(s − t)X(t) . Since ϕ is faithful on At , this implies that E X(s)X(t) = X(t)2 + α(s − t)X(t). On the other hand, by a similar argument E[X(s)] = X(t) and so E[X(s)X(t)] = X(t)2 . We arrive at a contradiction. 2


557

Remark 5 (Generator). Even though the two-state free Brownian motion with parameter α is not (for α = 0) a Markov process, as noted in Remark 2, it has a classical version which is a Markov process. Denote by Ks,t the transition functions of the classical version. The operator At is the generator of the process at time t if for some dense domain D ⊂ L2 (R, dμt ) and any f ∈ D, ∂ ∂h

Kt,t+h (f ) = At f. h=0

See [3] for related ideas. Proposition 16. On the dense domain of polynomial functions, the generator of the two-state free Brownian motion with parameter α is α(∂x − Lμt ) + ∂x Lνt , where we use the notation [4] Lν [f ](x) = R

f (x) − f (y) dν(y) = (1 ⊗ ν)(∂f ), x −y

and ∂ is the difference quotient. Proof. For B ∈ At and s > t, ⊗n ∗ ϕ B ∗ Qn X(s), s = 1⊗n [0,s) , BΩ = 1[0,t) , BΩ = ϕ B Qn X(t), t . So the orthogonal polynomials {Qn (x, t)} are polynomial martingales for the classical version of the process. This also follows from Proposition 3.3 of [15] and Lemma 2.4 of [16]. It is easy to see that, to show that At is the generator of the process with the domain consisting of all polynomial functions, it suffices to show that ∂t Qn (x, t) = −At Qn (x, t) for all n. Note that since μt = Φt [νt ], by Lemma 7 of [2] the polynomials Qn (x, t) are precisely the c-free Appell polynomials for this pair. By Proposition 4 from the same paper, the generating function for these polynomials is H (x, t, z) =

∞

Qn (x, t)zn =

n=0

1 + tαz 1 − xz + t (αz + z2 )

since R νt (z) = t (αz + z2 ) and R νt (z) − R μt ,νt (z) = tαz; this result is also easy to obtain directly. On the other hand, ∞ n=0

Pn (x, t)zn =

1 . 1 − xz + t (αz + z2 )

558


We conclude that ∂H (x, t, z) = zH (x, t, z)

1 1 − yz + t (αz + z2 )

and so Lνt H (x, t, z) = zH (x, t, z) and Lμt H (x, t, z) =

z . 1 − xz + t (αz + z2 )

Now we compute α(∂x − Lμt ) + ∂x Lνt H (x, t, z) = αz

1 + tαz 1 + tαz 1 + z2 − αz 2 2 2 (1 − xz + t (αz + z )) 1 − xz + t (αz + z ) (1 − xz + t (αz + z2 ))2

= −∂t

1 + tαz = −∂t H (x, t, z). 1 − xz + t (αz + z2 )

The result follows.

2

Remark 6 (Itô formula). By the same methods as in [10] and [1], for sufficiently nice f , f X(t) = f X(0) +

t

∂f X(s) dX(s) +

0

t

(∂x ⊗ ψ)∂f X(s) ds,

(7)

0

where we use the notation t

A(x) ⊗ B(s) dX(s) =

0

t A(x) dX(s)B(s). 0

Using Lemma 2.1 of [13] and the observation that the process {X(t)} is ϕ-centered, we see that

t ϕ 0

t ∂f X(s) dX(s) = ϕ ∂x f X(s) − (ϕ ⊗ ϕ) (∂f ) X(s) dψ X(s) . 0

Therefore taking ϕ-expectations in the Itô formula (7) gives


ϕ f X(t) = ϕ f X(0) +

t

559

ϕ ∂x f X(s) − (ϕ ⊗ ϕ) (∂f ) X(s) dψ X(s)

0

t +

ϕ (∂x ⊗ ψ)∂f X(s) ds

0

= ϕ f X(0) +

t

ϕ α∂x − α(1 ⊗ ϕ)∂ + (∂x ⊗ ψ)∂ f X(s) ds.

0

This result is consistent with the generator formula in the preceding proposition. 5. C ∗ -algebra setting T We saw in Corollary 3 that for any algebraic two-state free Brownian motion, 0 (dX(t))k = 0 T for k > 2 and 0 (dX(t))2 = T as limits in L2 (ϕ). If ϕ is a faithful state, these limits can be identified with elements in A. We now investigate the same limits in L∞ (ϕ). Here A∞ = lim A2n n→∞

and for A ∈ Asa , 1/2n . A2n = ϕ A2n Note that if ϕ is faithful, then A∞ = A, the operator norm on A. Lemma 17. Recall that the Stirling number of the second kind S(n, k) is the number of set partitions of a set of n elements into k non-empty blocks. Then lim

n→∞

N

1/n = N.

S(n, k)

k=1

Proof. It is easy to see that n! S(n, N ) S(n, k) N n . N N!((n/N)!) N

k=1

The result now follows by Stirling’s formula.

2

Theorem 18. Suppose {X(t)} is an algebraic two-state free Brownian motion. T (a) Assume further that all the ψ -free cumulants of X are non-negative. Then 0 (dX(t))p = 0 ψ as a limit in L∞ (ϕ) for some p > 2 if and only if Rk (X) = 0 for all k > 2. In this case, in T fact 0 (dX(t))p = 0 for all p > 2.

560


T ψ (b) Assume now that Rk (X) = 0 for k > 2. Then 0 (dX(t))2 = T as a limit in L∞ (ϕ) if and ψ only if R2 (X) = T , so that {X(t)} is a two-state free Brownian motion with parameter α. Proof. For part (a), using both assumptions on the process,

N n

N p pn 1 ψ 1 1 ϕ,ψ ψ R (X) R Xi,N Xi,N N (X) = T Rpn−2 (X). ϕ ϕ N 2 N pn−2 N i=1

i=1

Therefore

! !N ! ! ! p ! Xi,N ! ! ! ! i=1

and

2n

! ! N ! ! ! p ! Xi,N ! ! ! !

ψ

lim sup R2pn−2 (X)1/2n . n→∞

∞

i=1

1/2n 1 ψ T R2pn−2 (X) N

p ψ 1/2n = 0. Denote by So to have limN →∞ N i=1 Xi,N ∞ = 0, we need lim supn→∞ R2pn−2 (X) ν t R (z) the generating function for the ψ -free cumulants of X(t). Since {νt } form a free convolution semigroup, and all their moments are finite, we have the free canonical representation (Theorem 6.2 of [20]) R νt (z) = t αz + R

z2 dλ(x) 1 − xz

for a finite positive measure λ (our R is z times the usual R-transform). In particular, for n 2, ψ Rn+2 (X(t)) = t R x n dλ(x). Since by Hölder’s inequality,

|x|n−1 dλ(x) R

(n−1)/n |x|n dλ(x) λ(R)1/n ,

R

ψ

in fact lim supn→∞ |Rn (X)|1/n = 0. This says that R νt (z) is analytic in the complex plane. It follows that the Cauchy transform Gλ (z) = R

z 1 dλ(x) = R νt (1/z) − α z−x t

is also analytic, except possibly at z = 0. But then by the Stieltjes inversion formula dλ(x) = −

1 lim Im Gλ (x + iy), π y↓0

λ is a multiple of δ0 . So R νt (z) = t (αz + βz2 ), and νt is a semicircular distribution.

M. Anshelevich / Journal of Functional Analysis 260 (2011) 541–565 ψ

ψ

561

ψ

On the other hand, if R1 (X) = αT , R2 (X) = βT , and Rk (X) = 0 for k > 2, then

ϕ

N

n k Xi,N

Nn

i=1

! N ! ! ! ! k ! Xi,N ! ! ! !

∞

i=1

kn NC(kn) max 1, T , |α|T , βT ,

1 N kn/2

k 1 4k max 1, T , |α|T , βT , N k/2−1

and ! N ! ! ! ! k ! lim ! Xi,N ! ! N →∞ ! i=1

= 0.

∞

For part (b), we first assume that β = 1 and use Lemma 19.

ϕ

N

n 2 Xi,N

−T

=

σ ∈NC1,2 (2n) Out(σ )∩Sing(σ )=∅ Out(σ )∩τn =∅

i=1

N|π|

π∈P (2n) π(σ ∨τn )

T N

|σ |

× α |Sing(σ )| (β − 1)| Inn(σ )∩τn | , where Nn = N (N − 1) . . . (N − n + 1) and τn = (1, 2), (3, 4), . . . , (2n − 1, 2n) . Note also that N|π| = 0 for |π| > N . Now take σ = (1, 2n), (2, 2n − 1), (3, 4), (5, 6), . . . , (2n − 3, 2n − 2) . Then σ ∨ τn = (1, 2, 2n − 1, 2n), (3, 4), (5, 6), . . . , (2n − 3, 2n − 2) and |σ ∨ τn | = n − 1. Therefore N π ∈ P(2n): π (σ ∨ τn ), |π| N = π ∈ P(n − 1), π| N = S(n − 1, k). k=1

So using only the term corresponding to σ ,

562


ϕ

N

n 2 Xi,N

−T

i=1

N 1 S(n − 1, k)T n (β − 1)n−2 Nn k=1

and by Lemma 17, ! N ! ! ! ! ! 2 Xi,N − T ! ! ! !

T |β − 1|

∞

i=1

for all N . On the other hand, if β = 1, then

ϕ

N

n 2 Xi,N

−T

=

σ ∈NC1,2 (2n) Out(σ )∩Sing(σ )=∅ σ ∧τn =0ˆ

i=1

N|π|

π∈P (2n) π(σ ∨τn )

T N

|σ |

α |Sing(σ )| .

In this case, the conditions σ ∈ NC1,2 (2n), σ ∧ τn = 0ˆ guarantee that |σ ∨ τn | |σ | − n2 , and so for each such σ ,

π ∈ P(2n): π (σ ∨ τn ), |π| N

=

π ∈ P |σ ∨ τn | , |π| N

N |σ ∨τn | N |σ |−n/2 . This time we also note that N|π| N N . Then

ϕ

N

n 2 Xi,N

−T

2n N |σ ∨τn |−|σ | N N max 1, T , |α|T

σ ∈NC1,2 (2n) σ ∧τn =0ˆ

i=1

2n 42n N −n/2 N N max 1, T , |α|T and ! N ! ! ! ! ! 2 Xi,N − T ! ! ! ! i=1

2 1 √ 42 max 1, T , |α|T , N ∞

so ! N ! ! ! ! ! 2 lim ! Xi,N − T ! ! N →∞ ! i=1

The result follows.

2

= 0. ∞


563

Proof of Lemma 19. We first note that n−k N n 2 ϕ Xi,N Tk = k i=1

N

2 2 2 , ϕ Yi(1),1,N Yi(2),2,N . . . Yi(n),n,N

i(1),i(2),...,i(n)=1 S⊂{1,2,...,n} |S|=k

where " Yi,j,N =

j∈ / S,

Xi,N , T N,

j ∈ S.

This expression equals N

R ϕ,ψ (Yi(1),1,N , Yi(1),1,N , . . . , Yi(n),n,N : V )

i(1),i(2),...,i(n)=1 S⊂{1,2,...,n} σ ∈NC(2n) V ∈Out(σ ) |S|=k

×

R ψ (Yi(1),1,N , Yi(1),1,N , . . . , Yi(n),n,N : V ).

V ∈Inn(σ ) ϕ,ψ

ψ

Since Rm (Xi,N ) = 0 for m = 2 and Rm (Xi,N ) = 0 for m > 2, this simplifies to N

i(1),i(2),...,i(n)=1 U ⊂τn |U |=k

×

σ ∈NC1,2 (2n) (u,v)∈U U ⊂σ Out(σ )∩Sing(σ )=∅

T N

R ϕ,ψ (Xi([(u+1)/2]),N , Xi([(v+1)/2]),N )

(u,v)∈Out(σ )\U

×

R ψ (Xi([(u+1)/2]),N , Xi([(v+1)/2]),N )

(u,v)∈Inn(σ )\U

R ψ (Xi([(u+1)/2]),N ),

(u)∈Inn(σ )

where [a] denotes the integer part. π For each choice of (i(1), i(2), . . . , i(n)), we define the partition π ∈ P(2n) by (2j −1) ∼ (2j ) and π

2j1 ∼ 2j2

⇔

i(j1 ) = i(j2 ).

Then the sum is transformed into U ⊂τn |U |=k

σ ∈NC1,2 (2n) π∈P (2n) πσ U ⊂σ Out(σ )∩Sing(σ )=∅ πτn

Therefore

N|π|

(u,v)∈U

T N

(u,v)∈Out(σ )\U

T T α N N (u)∈σ

(u,v)∈Inn(σ )\U

β

T . N

564


ϕ

N

n 2 Xi,N

−T

i=1

k=0

n = (−1)k k=0

×

(u,v)∈U

(u,v)∈Out(σ )\U

σ ∈NC1,2 (2n) Out(σ )∩Sing(σ )=∅ Out(σ )∩τn =∅

The result follows.

1

π∈P (2n) π(σ ∨τn )

T

N|π|

π∈P (2n) π(σ ∨τn )

k

T N

|σ |

β

(u,v)∈Inn(σ )\U

N|π|

π∈P (2n) π(σ ∨τn )

n−k 2 Xi,N

α

(u)∈σ

N i=1

1

σ ∈NC1,2 (2n) Out(σ )∩Sing(σ )=∅

=

σ ∈NC1,2 (2n) U ⊂(σ ∩τn ) |U |=k Out(σ )∩Sing(σ )=∅

=

n n ϕ (−1)k = k

T N

N|π|

|σ |

T N

α |Sing(σ )| (1 − 1)| Out(σ )∩τn | (β − 1)| Inn(σ )∩τn |

|σ |

α |Sing(σ )| (β − 1)| Inn(σ )∩τn | .

2

Corollary 20. Let (A, ϕ, ψ) be a C ∗ -non-commutative probability space, so that A is a C ∗ algebra, ϕ and ψ states on it, and ϕ is faithful. Suppose that A is generated by an algebraic two-state free Brownian motion {X(t)} all of whose ψ-free cumulants are non-negative. Suppose T T also that 0 (dX(t))k = 0 for k > 2 and 0 (dX(t))2 = T , where the limits are taken in the operator norm. Then {X(t)} is a two-state free Brownian motion with parameter α. Remark 7. In the setting of the preceding corollary, our results do not imply directly that if A is generated by an algebraic two-state free Brownian motion {X(t)} without any extra assumptions, then {X(t)} has to be a two-state free Brownian motion with parameter α. So it is possible that new examples may arise if we only assume that the state ϕ is faithful on the C ∗ -algebra and not on the von Neumann algebra generated by the process. On the other hand, note that the argument in Proposition 13 shows that for a two-state free Brownian motion with parameter α, for t > 1/α 2 the state ϕ is not faithful even on the C ∗ algebra. Acknowledgments I am grateful to Ken Dykema and Laura Matusevich for important comments. Thanks also to Włodek Bryc for giving me an early version of the article [14], which led (in a very indirect way) to the present work. References [1] Michael Anshelevich, Itô formula for free stochastic integrals, J. Funct. Anal. 188 (1) (2002) 292–315, MR1878639 (2002m:46095). [2] Michael Anshelevich, Appell polynomials and their relatives. III. Conditionally free theory, Illinois J. Math. 53 (1) (2009) 39–66, MR2584934.


565

[3] Michael Anshelevich, Generators of some non-commutative stochastic processes, 2010. [4] Michael Anshelevich, Bochner–Pearson-type characterization of the free Meixner class, Adv. Appl. Math., in press, arXiv:0909.1097 [math.CO]. [5] Michael Anshelevich, Wojciech Młotkowski, Semigroups of distributions with linear Jacobi parameters, arXiv:1001.1540 [math.CO], 2010. [6] Serban Teodor Belinschi, C-free convolution for measures with unbounded support, in: Von Neumann Algebras in Sibiu, in: Theta Ser. Adv. Math., vol. 10, Theta, Bucharest, 2008, pp. 1–7, MR2512322. [7] Serban T. Belinschi, Alexandru Nica, On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution, Indiana Univ. Math. J. 57 (4) (2008) 1679–1713, MR2440877 (2009f:46087). [8] Hari Bercovici, Dan Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (3) (1993) 733–773, MR1254116 (95c:46109). [9] H. Bercovici, D. Voiculescu, Superconvergence to the central limit and failure of the Cramér theorem for free random variables, Probab. Theory Related Fields 103 (2) (1995) 215–222, MR1355057 (96k:46115). [10] Philippe Biane, Roland Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Related Fields 112 (3) (1998) 373–409, MR99i:60108. [11] Marek Bo˙zejko, Włodzimierz Bryc, A quadratic regression problem for two-state algebras with an application to the central limit theorem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2) (2009) 231–249, MR2541395 (2010j:46119). [12] Marek Bo˙zejko, Roland Speicher, ψ -Independent and symmetrized white noises, in: QP–PQ: Quantum Prob. Relat. Top., vol. VI, World Sci. Publ., River Edge, NJ, 1991, pp. 219–236, MR1149828. [13] Marek Bo˙zejko, Michael Leinert, Roland Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (2) (1996) 357–388, MR1432836 (98j:46069). [14] Włodek Bryc, Markov processes with free-Meixner laws, Stochastic Process. Appl. 120 (8) (2010) 1393–1403. [15] Włodzimierz Bryc, Jacek Wesołowski, Conditional moments of q-Meixner processes, Probab. Theory Related Fields 131 (3) (2005) 415–441, MR2123251 (2005k:60233). [16] Włodzimierz Bryc, Jacek Wesołowski, Bi-Poisson process, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2) (2007) 277–291, MR2337523 (2008d:60097). [17] T.S. Chihara, An Introduction to Orthogonal Polynomials, Math. Appl., vol. 13, Gordon and Breach Science Publ., New York, 1978, MR0481884 (58#1979). [18] G.P. Chistyakov, F. Götze, Limit theorems in free probability theory. I, Ann. Probab. 36 (1) (2008) 54–90, MR2370598 (2009d:46116). [19] Anna Dorota Krystek, Infinite divisibility for the conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (4) (2007) 499–522, MR2376439 (2009d:46118). [20] Hans Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (2) (1992) 409–438, MR1165862 (94g:46069). [21] Vittorino Pata, The central limit theorem for free additive convolution, J. Funct. Anal. 140 (2) (1996) 359–380, MR1409042 (97e:46090). [22] Mihai Popa, Multilinear function series in conditionally free probability with amalgamation, Commun. Stoch. Anal. 2 (2) (2008) 307–322, MR2446696 (2010i:46102). [23] Éric Ricard, The von Neumann algebras generated by t -Gaussians, Ann. Inst. Fourier (Grenoble) 56 (2) (2006) 475–498, MR2226024 (2007g:46101). [24] Dan Voiculescu, Symmetries of some reduced free product C ∗ -algebras, in: Operator Algebras and Their Connections with Topology and Ergodic Theory, Bu¸steni, 1983, in: Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588, MR799593 (87d:46075). [25] Jiun-Chau Wang, Limit theorems for additive c-free convolution, Canad. J. Math., in press, arXiv:0805.0607v2 [math.OA], 2010. [26] Jiun-Chau Wang, Local limit theorems in free probability theory, Ann. Probab. 38 (4) (2010) 1492–1506. [27] Janusz Wysoczański, The von Neumann algebra associated with an infinite number of t -free non-commutative Gaussian random variables, in: Quantum Probability, in: Banach Center Publ., vol. 73, Polish Acad. Sci. Inst. Math., Warsaw, 2006, pp. 435–438, MR2423148 (2009h:46130).


The spatial product of Arveson systems is intrinsic B.V. Rajarama Bhat a , Volkmar Liebscher b,∗ , Mithun Mukherjee a , Michael Skeide c a Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, R.V. College Post, Bangalore 560059, India b Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald, 17487 Greifswald, Germany c Dipartimento S.E.G.e S., Università degli Studi del Molise, Via de Sanctis, 86100 Campobasso, Italy

Received 17 June 2010; accepted 2 September 2010 Available online 15 September 2010 Communicated by D. Voiculescu

Abstract We prove that the spatial product of two spatial Arveson systems is independent of the choice of the reference units. This also answers the same question for the minimal dilation of the Powers sum of two spatial CP-semigroups: It is independent up to cocycle conjugacy. © 2010 Elsevier Inc. All rights reserved. Keywords: Continuous product systems of Hilbert spaces; Spatial E0 -semigroups

1. Introduction Arveson [1] associated with every E 0 -semigroup (a semigroup of unital endomorphisms) on B(H ) its Arveson system (a family of Hilbert spaces E = (Et )t0 with an associative identification Es ⊗ Et = Es+t ). He showed that E0 -semigroups are classified by their Arveson system up to cocycle conjugacy. By a spatial Arveson system we understand a pair (E , u) of an Arveson system E and a unital unit u (that is a section u = (ut )t0 of unit vectors ut ∈ Et that factor * Corresponding author.

E-mail addresses: [email protected] (B.V.R. Bhat), [email protected] (V. Liebscher), [email protected] (M. Mukherjee), [email protected] (M. Skeide). URLs: http://www.isibang.ac.in/Smubang/BHAT/ (B.V.R. Bhat), http://www.math-inf.uni-greifswald.de/mathe/index.php?id=97:volkmar-liebscher (V. Liebscher), http://www.math.tu-cottbus.de/INSTITUT/lswas/_skeide.html (M. Skeide). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.001

B.V.R. Bhat et al. / Journal of Functional Analysis 260 (2011) 566–573

567

as us ⊗ ut = us+t ). Spatial Arveson systems have an index, and this index is additive under the tensor product of Arveson systems. Much of this can be carried through also for product systems of Hilbert modules and E0 semigroups on B a (E), the algebra of all adjointable operators on a Hilbert module; see the conclusive paper Skeide [19] and its list of references. However, there is no such thing as the tensor product of product systems of Hilbert modules. To overcome this, Skeide [18] (preprint, 2001) introduced the product of spatial product systems (henceforth, the spatial product), under which the index of spatial product systems of Hilbert modules is additive. It is known that the spatial structure of a spatial Arveson system (Et )t0 depends on the choice of the reference unit (ut )t0 . In fact, Tsirelson [22] showed that if (vt )t0 is another unital unit, then there need not exist an automorphism of (Et )t0 that sends (ut )t0 to (vt )t0 . Also the spatial product depends a priori on the choice of the reference units of its factors. This immediately raises the question if different choices of references units give isomorphic products or not. In these notes we answer this question in the affirmative sense for the spatial product of Arveson systems. For two Arveson systems (Et )t0 and (Ft )t0 with reference units (ut )t0 and (vt )t0 , respectively, their spatial product can be identified with the subsystem of the tensor product generated by the subsets ut ⊗ Ft and Et ⊗ vt . This raises another question, namely, if that subsystem is all of the tensor product or not. This has been answered in the negative sense by Powers [13], resolving the same question for a related problem. Let us describe this problem very briefly. Suppose we have two E0 -semigroups ϑ i = (ϑti )t0 on B(H i ) with intertwining semigroups i (Ut )t0 of isometries in B(H i ) (that is, ϑti (a i )Ut = Ut a i ). Intertwining semigroups correspond one-to-one with unital units of the associated Arveson systems (Eti )t0 , so that these are spatial. Then by T

a11 a21

a12 a22

:=

ϑt1 (a11 ) ∗ Ut2 a21 Ut1

∗

Ut1 a12 Ut2 ϑt2 (a22 )

we define a Markov semigroup on B(H 1 ⊕ H 2 ). Its unique minimal dilation (see Bhat [5]) is an E0 -semigroup (fulfilling some properties). At the 2002 Workshop Advances in Quantum Dynamics in Mount Holyoke, Powers asked for the cocycle conjugacy class (that is, for the Arveson system) of that E0 -semigroup. More precisely, he asked if it is the cocycle conjugacy class of the tensor product of ϑ 1 and ϑ 2 , or not. Still during the workshop Skeide (see the proceedings [17]) identified the Arveson system of that Powers sum as the spatial product of the Arveson systems of ϑ 1 and ϑ 2 . So, Powers’ question is equivalent to the question if the spatial product is the tensor product, or not. In [13] Powers answered the former question in the negative sense and, henceforth, also the latter. He left open the question if the cocycle conjugacy class of the minimal dilation of the Powers sum depends on the choice of the intertwining isometries. Our result of the present notes tells, no, it doesn’t depend. We should say that Powers in [13] to some extent considered the Powers sum not only for E0 -semigroups but also for those CP-semigroups he called as spatial. We think that his definition of spatial CP-semigroup is too restrictive, and prefer to use Arveson’s definition [2], which is much wider; see Bhat, Liebscher, and Skeide [6]. The definition of Powers sum easily extends to those CP-semigroups and the relation of the associated Arveson system of the minimal dilations is stills the same: The Arveson system of the sum is the spatial product of the Arveson systems of the addends; see Skeide [20]. Therefore, our result here also applies to the more general situation.

568


Remark 1.1. It should be noted that the result is visible almost at a glance when the intuition of random sets to describe spatial Arveson systems is available; see [10,21]. However, in order to make this clear a lot of random set techniques had to be explained, so we opted to give a plain Hilbert space proof. Although this is, maybe, not too visible, the proof here is nevertheless very much inspired by the intuition coming from random sets. We will explain that intuition elsewhere [7]. 2. Arveson systems Definition 2.1. An Arveson system is a measurable family E = (Et )t0 of separable Hilbert spaces endowed with a measurable family of unitaries Vs,t : Es ⊗ Et → Es+t for all s, t 0 such that for all r, s, t 0 Vr,s+t ◦ (1Er ⊗ Vs,t ) = Vr+s,t ◦ (Vr,s ⊗ 1Et ). Remark 2.2. In the sequel, we shall omit the Vs,t and simply identify Es ⊗ Et with Es+t . This lightens the formulae, but requires a certain flexibility (see Proposition 2.7 or the proof of Lemma 3.2) when interpreting correctly operators on tensor products of Arveson systems. Remark 2.3. Note that Definition 2.1 is equivalent to Arveson’s in [1]; see [10, Lemma 7.39]. The only difference is that Definition 2.1 allows for one-dimensional and zero-dimensional Arveson systems. The latter is necessary in view of the following property. By [10, Theorem 5.7], for every Arveson system E the set S (E ) := {F : product subsystem of E } forms a (complete) lattice with the lattice operations E ∧ F = (Et ∩ Ft )t0 and E ∨ F defined as the smallest Arveson subsystem containing both E and F . Remark 2.4. By [10, Theorem 7.7], the algebraic structure of an Arveson system determines the measurable structure completely. Definition 2.5. A unit u of an Arveson system is a measurable non-zero section (ut )t0 through (Et )t0 , which satisfies for all s, t 0 us+t = us ⊗ ut . If u is unital ( ut = 1 for all t 0), the pair (E , u) is also called a spatial Arveson system. For Hilbert spaces, the spatial product from Skeide [18] can be defined as a subsystem of the tensor product in the following way. Definition 2.6. Let (E , u) and (F , v) be two spatial Arveson systems. We define their spatial product as E u ⊗v F := (u ⊗ F ) ∨ (E ⊗ v) ⊂ E ⊗ F .


569

That this coincides with the product in [18] follows either from the universal property [18, Theorem 5.1] that characterizes it, or after Proposition 2.7 below, that identifies directly the pieces from the inductive limit by which the product is constructed in [18]. Let, with N = {1, 2, . . .}, Π t := (t1 , . . . , tn ): n ∈ N, t1 > 0, t2 > 0, . . . , tn > 0, t1 + · · · + tn = t denote the set of interval partitions of [0, t] (parametrized suitably for our purposes). For t = (t1 , . . . , tn ) ∈ Π t and s = (s1 , . . . , sm ) ∈ Π s , denote by t s := (t1 , . . . , tn , s1 , . . . , sm ) ∈ Π t+s their join. We order Π t by saying that (t1 , . . . , tn ) ≺ (s1 , . . . , sm ) if there exist ti ∈ Π ti such that t1 · · · tn = s. For any Hilbert subspace H denote by H ⊥ the orthogonal complement of H in the space containing it. Proposition 2.7. Let (E , u) and (F , v) be two spatial Arveson systems, and define ⊥ ⊥ Gu,v t := ut ⊗ vt ⊕ Cut ⊗ vt ⊕ ut ⊗ vt .

Then for all t > 0 (E u ⊗v F )t =

lim

(t1 ,...,tn )∈Π t

u,v u,v u,v Gu,v tn ⊗ Gtn−1 ⊗ · · · ⊗ Gt2 ⊗ Gt1 .

(∗)

u,v u,v Proof. (See Remark 2.2 about notation.) Since Gu,v ⊂ Et ⊗ Ft and Gu,v t s+t ⊂ Gs ⊗ Gt , the limit exists due to monotonicity and (E u ⊗v F )t ⊂ Et ⊗ Ft for all t 0. From the properties of the interval partitions it is easy to see that in fact the RHS of (∗) is a product system in its own right. u,v Clearly, Gu,v t ⊃ Et ⊗ vt and Gt ⊃ ut ⊗ Ft . Therefore, the RHS of (∗) contains both E ⊗ v and u ⊗ F . On the other hand, let H ⊂ E ⊗ F contain both E ⊗ v and u ⊗ F . Then, obviously, Gu,v t ⊂ Ht . Consequently, E u ⊗v F contains the RHS of (∗) and the assertion is proved. 2

Remark 2.8. The structure Gs ⊗ Gt ⊃ Gs+t is a recurrent theme in the analysis of quantum dynamics, in particular, of CP-semigroup; see [14,9,4,18,12,11,17,8]. Recently, it has been formalized by Shalit and Solel [16] under the name of subproduct systems (of Hilbert modules), and by Bhat and Mukherjee under the name of inclusion systems (only the Hilbert case). Once for all, [8] prove by the same inductive limit construction that every subproduct or inclusion system of Hilbert spaces embeds into an Arveson system. In Shalit and Skeide [15], the same will be shown for modules by reducing it to the case of CP-semigroups considered by Bhat and Skeide [9]. While the spatial product may be viewed as amalgamation of two spatial product systems over their reference units, [8] generalize this to an amalgamation over a contraction morphism between two (not necessarily spatial) Arveson systems. This applies, in particular, to the amalgamation of two spatial Arveson systems of not necessarily unital units, and answers Powers’ question for the Markov semigroup obtained from not necessarily isometric intertwining semigroups.

570


3. Universality of the spatial product Our aim is to prove the following theorem. Theorem 3.1. Let (E , u), (E , u ), (F , v) and (F , v ) be spatial product systems. Then E u ⊗v F ∼ =E

u ⊗v

F.

Actually, we will prove even more, namely, E u ⊗v F = E u ⊗v F as subsystems of E ⊗ F . The key of the proof is the following lemma (whose proof we postpone to the very end, after having illustrated the immediate consequences). Lemma 3.2. E ⊗ v ⊂ E u ⊗v F . Corollary 3.3. E u ⊗v F ⊂ E u ⊗v F and, by symmetry, E u ⊗v F ⊃ E u ⊗v F , so E u ⊗v F = E u ⊗v F . Once more, by symmetry E u ⊗v F = E u ⊗v F . This proves E u ⊗v F = E

u ⊗v

F and, therefore, Theorem 3.1.

Corollary 3.4. Denote by E 0 , F 0 the product subsystems of E and F generated by all units of E and F respectively. Then for the product with amalgamation over all units E ⊗0 F := E ⊗ F 0 ∨ E 0 ⊗ F we find E ⊗0 F = E u ⊗v F . Proof. For every pair of unital units u and v we have E⊗ F= 0

E ⊗v

∨

v

=

u ⊗F

u

E ⊗ v ∨ u ⊗ F = (E v ,u

because E

u ⊗v

F = E u ⊗v F .

u ⊗v

F ) = E u ⊗v F ,

v ,u

2

Corollary 3.5. Suppose F is type I, that is, F = F 0 . Then E u ⊗v F = E ⊗ F . Proof. E u ⊗v F 0 = E ⊗0 F 0 = E ⊗ F 0 ∨ E 0 ⊗ F 0 = E ⊗ F 0 , because E ⊗ F 0 ⊃ E 0 ⊗ F 0. 2 Proof of Lemma 3.2. By Proposition 2.7, it is enough to show that for ψ ∈ E1 we have (E u ⊗v F )1 lim PrG⊗2n ψ ⊗ v1 = ψ ⊗ v1 n→∞

2−n


571

which proves that E1 ⊗ v1 ⊂ (E u ⊗v F )1 . (The proof of Et ⊗ vt ⊂ (E u ⊗v F )t is analogous.) Since PrG⊗2n increases strongly to a projection (the projection onto (E u ⊗v F )1 ), it is sufficient 2−n

to show that

lim PrG⊗2n ψ ⊗ v1 = ψ v1 = ψ . 2−n

n→∞

For 0 s < t 1, we define the projections Ps,t := 1Es ⊗ Prut−s ⊗ 1E1−t ∈ B(E1 ) in the factorization E1 = Es ⊗ Et−s ⊗ E1−t . We put Pt,t := 1E1 . Similarly, we define Qs,t := 1Fs ⊗ Prvt−s ⊗ 1F1−t ∈ B(F1 ). Then Pr(Es ⊗Fs )⊗Gt−s ⊗(E1−t ⊗F1−t ) = Ps,t ⊗ (1 − Qs,t ) + (1 − Ps,t ) ⊗ Qs,t + Ps,t ⊗ Qs,t = (1 − Ps,t ) ⊗ Qs,t + Ps,t ⊗ 1. (See Remark 2.2 about notation!) This gives n

PrG⊗2n 2−n

2

(1 − P i−1 = i ) ⊗ Q i−1 n , n n , 2

i=1

=

S⊂{1,...,2n }

=

2

S⊂{1,...,2n }

2

+ P i−1 n , 2

(1 − P i−1 i ) ⊗ Q i−1 n , n n , 2

i∈S

i 2n

2

2

(1 − P i−1 i ) n , n 2

i∈S

2

i ∈S /

i 2n

⊗1

i 2n

P i−1 n , 2

i 2n

P i−1 n , 2

i ∈S /

⊗

i∈S

i 2n

⊗1

Q i−1 n , 2

i 2n

.

(∗∗)

Since the vt , vt form a (measurable) contractive semigroup, there is a complex number γ (with Re γ 0) such that vt , vt = eγ t . If we put wiS :=

v 1n ,

i ∈ S,

,

i∈ / S,

2

v

1 2n

then

i∈S

Q i−1 n , 2

#S v1 = eγ 2n w1S ⊗ · · · ⊗ w2Sn . i n

2

S Note that w1S ⊗ · · · ⊗

w2n are unit vectors. Note, too, that in the last line of (∗∗) the projections

i∈S (1 − P i−1 / P i−1 , i ) i ∈S , i in the first factor are orthogonal for different choices of S. We 2n 2n 2n 2n conclude that

572


Pr

n G⊗2 2−n

2 ψ ⊗ v1 =

e

S⊂{1,...,2n }

=

γ

#S 2n

(1 − P i−1 i ) n , n 2

i∈S

2

i ∈S /

2 P i−1 i ψ , 2n 2n

2

−n

eγ 2 (1 − P i−1 P i−1 i ψ . i ) , , n n n n 2 2 2 2 i ∈S /

S⊂{1,...,2n } i∈S

Next recall that f (p) = f (0)1 + (f (1) − f (0))p for every entire function f and every projection p. We find for the commuting projections P i−1 i n , n 2

2

−n

eγ 2 (1 − P i−1 P i−1 i ) n , n n , 2

S⊂{1,...,2n } i∈S

2

n

=

2

2

n

=

2

i=1 n

=

2

2

2

i 2n

−n

eγ 2 (1 − P i−1 i ) + P i−1 n , n n ,

i=1

2

i ∈S /

i 2n

−n 1 + eγ 2 − 1 (1 − P i−1 i ) n , n 2

2

2n −n −n exp γ 2 (1 − P i−1 (1 − P i−1 i ) = exp γ 2 i ) . n , n n , n 2

i=1

2

i=1

2

2

From [10, Proposition 3.18] (see also [3, Proposition 8.9.9]), we know that (s, t) → Ps,t is strongly continuous. The simplex {(s, t): 0 s t 1} is compact, so the function is even uniformly strongly continuous. This implies that 1 − Ps,t → 0 strongly uniformly as (t − s) → 0. Thus we obtain that 2 n→∞ (1 − P i−1 − − − → (1 − Pt,t ) dt = 0 i )− n , n 1

n

2

−n

i=1

2

2

0

strongly. Since entire functions are strongly continuous, this shows

−n

eγ 2 (1 − P i−1 P i−1 i ) n , n n ,

S⊂{1,...,2n } i∈S

2

2

2

i ∈S /

in the strong topology, which completes the proof.

i 2n

n→∞ − −−− → exp(0) = 1

2

Acknowledgments This work began with an RiP stay at Mathematisches Forschungsinstitut Oberwolfach in May 2007 and was almost completed during a stay of V.L. at Università del Molise, Campobasso, 2009. V.L. thanks M.S. for warm hospitality during the latter stay. M.S. is supported by funds from the Italian MIUR (PRIN 2007) and the Dipartimento S.E.G.e S.


573

References [1] W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 409 (1989). [2] W. Arveson, Minimal E0 -semigroups, in: P. Fillmore, J. Mingo (Eds.), Operator Algebras and Their Applications, in: Fields Inst. Commun., vol. 13, American Mathematical Society, 1997, pp. 1–12. [3] W. Arveson, Noncommutative Dynamics and E-Semigroups, Monogr. Math., Springer, 2003. [4] S.D. Barreto, B.V.R. Bhat, V. Liebscher, M. Skeide, Type I product systems of Hilbert modules, J. Funct. Anal. 212 (2004) 121–181, preprint, Cottbus, 2001. [5] B.V.R. Bhat, An index theory for quantum dynamical semigroups, Trans. Amer. Math. Soc. 348 (1996) 561–583. [6] B.V.R. Bhat, V. Liebscher, M. Skeide, Subsystems of Fock need not be Fock: Spatial CP-semigroups, Proc. Amer. Math. Soc. 138 (2010) 2443–2456; electronically: arXiv:0804.2169v2, Feb. 2010. [7] B.V.R. Bhat, V. Liebscher, M. Skeide, The relation of spatial product and tensor product of Arveson systems, preprint, 2010, in preparation. [8] B.V.R. Bhat, M. Mukherjee, Inclusion systems and amalgamated products of product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010) 1–26, arXiv:0907.0095v1. [9] B.V.R. Bhat, M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000) 519–575, Rome, Volterra-preprint 1999/0370. [10] V. Liebscher, Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces, Mem. Amer. Math. Soc. 930 (2009), arXiv:math.PR/0306365. [11] D. Markiewicz, On the product system of a completely positive semigroup, J. Funct. Anal. 200 (2003) 237–280. [12] P.S. Muhly, B. Solel, Quantum Markov processes (correspondences and dilations), Int. J. Math. 51 (2002) 863–906, arXiv:math.OA/0203193. [13] R.T. Powers, Addition of spatial E0 -semigroups, in: Operator Algebras, Quantization, and Noncommutative Geometry, in: Contemp. Math., vol. 365, American Mathematical Society, 2004, pp. 281–298. [14] M. Schürmann, White Noise on Bialgebras, Lecture Notes in Math., vol. 1544, Springer, 1993. [15] O.M. Shalit, M. Skeide, CP-Semigroups, dilations, and subproduct systems: The multi-parameter case and beyond, preprint, 2010, in preparation. [16] O.M. Shalit, B. Solel, Subproduct systems, Doc. Math. 14 (2009) 801–868, preprint, arXiv:0901.1422v2. [17] M. Skeide, Commutants of von Neumann modules, representations of Ba (E) and other topics related to product systems of Hilbert modules, in: G.L. Price, B.M. Baker, P.E.T. Jorgensen, P.S. Muhly (Eds.), Advances in Quantum Dynamics, in: Contemp. Math., vol. 335, American Mathematical Society, 2003, pp. 253–262, preprint, Cottbus, 2002, arXiv:math.OA/0308231. [18] M. Skeide, The index of (white) noises and their product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 617–655, Rome, Volterra-preprint 2001/0458, arXiv:math.OA/0601228. [19] M. Skeide, Classification of E0 -semigroups by product systems, preprint, arXiv:0901.1798v2, 2009. [20] M. Skeide, The Powers sum of spatial CPD-semigroups and CP-semigroups, Banach Center Publ. 89 (2010) 247– 263, arXiv:0812.0077. [21] B. Tsirelson, From random sets to continuous tensor products: answers to three questions of W. Arveson, preprint, arXiv:math.FA/0001070, 2000. [22] B. Tsirelson, On automorphisms of type II Arveson systems (probabilistic approach), New York J. Math. 14 (2008) 539–576; available at http://nyjm.albany.edu/j/2008/14-25.html, arXiv:math.OA/0411062v3.


Finitely correlated representations of product systems of C ∗-correspondences over Nk Adam Hanley Fuller Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L-3G1, Canada Received 23 June 2010; accepted 7 October 2010 Available online 20 October 2010 Communicated by D. Voiculescu

Abstract We study isometric representations of product systems of correspondences over the semigroup Nk which are minimal dilations of finite-dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be a complete unitary invariant. For a certain class of graph algebras the nonself-adjoint WOT-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex. © 2010 Elsevier Inc. All rights reserved. Keywords: Correspondences; Product systems; Graph algebras; Higher-rank graph algebras

1. Introduction A C ∗ -correspondence over a C ∗ -algebra A is a Hilbert bimodule with an A-valued inner product. The C ∗ -algebras of representations of C ∗ -correspondences were first studied by Pimsner [31]. In a series of papers beginning with [27], Muhly and Solel studied representations of C ∗ -correspondences and their algebras. Remarkably they managed to achieve many results from single operator theory in this very general setting. In [27] they include a dilation theorem which supersedes the classical Sz.-Nagy [44] dilation theorem for contractions and the Frazho, Bunce and Popescu [17,6,32] dilation theorem for row-contractions. In [28] a Wold decomposition is presented as well as a Beurling-type theorem. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.004

A.H. Fuller / Journal of Functional Analysis 260 (2011) 574–611

575

Product systems of C ∗ -correspondences over the semigroup R+ were introduced by Arveson in [2]. The study of product systems over discrete semigroups began with Fowler’s work in [16], where the generalised Cuntz–Pimsner algebra associated to a product system was introduced. In recent years there have been several papers considering product systems of C ∗ -correspondences over discrete semigroups, e.g. [37,36,38,39,42,43]. There has been work on dilation results for representations of product systems generalizing dilation results for commuting contractions. For example, Solel [42] shows the existence of a dilation for contractive representations of product systems over N2 . This result is analogous to the well-known Ando’s theorem for two commuting contractions [1]. Solel [43] gives necessary and sufficient conditions for a contractive representation of a product system over Nk to have what is known as a regular dilation. This result is analogous to a theorem of Brehmer [5]. Skalski and Zacharias [39] have presented a Wold decomposition for representations of product systems over Nk . The generalised Cuntz–Pimsner C ∗ -algebras associated to product systems over the semigroup Nk are not in general GCR, i.e. they can be NGCR. A theorem due to Glimm [18, Theorem 2] tells us that NGCR C ∗ -algebras do not have smooth duals, i.e. there is no countable family of Borel functions on the space of unitary equivalence classes of irreducible representations which separates points. It follows that trying to classify all irreducible representations up to unitary equivalence of a generalised Cuntz–Pimsner algebra would be a fruitless task. However, in this paper we find a complete unitary invariant for a certain class of representations: finitely correlated representations. An isometric representation of a product system of C ∗ -correspondences is finitely correlated if it is the minimal isometric dilation of a finite-dimensional representation. We show the existence of a unique minimal cyclic coinvariant subspace for finitely correlated, isometric, fully coisometric representations of product systems over the semigroup Nk . The compression of the representation to this minimal subspace will be the complete unitary invariant. This result generalises the work of Davidson, Kribs and Shpigel [10] for the minimal isometric dilation [S1 , . . . , Sn ] of a finite-dimensional row-contraction. Indeed, studying row-contractions is equivalent to studying representations of the C ∗ -correspondence Cn over the C ∗ -algebra C. In [10], it is shown that the projection onto the minimal coinvariant subspace is contained in the WOT-closed algebra generated by the Si ’s. This is an important invariant for free semigroup algebras [9]. We are able to establish this in a number of interesting special cases. Finitely correlated representations were first introduced by Bratteli and Jorgensen [3] via finitely correlated states on On . When ω is a finitely correlated state on On , the GNS construction on ω will give a representation π of On with the property that [π(s1 ), . . . , π(sn )] is a finitely correlated row isometry, where s1 , . . . , sn are pairwise orthogonal isometries generating On . This relates [10] with [3]. Similarly, following the work of Skalski and Zacharias [40], we will define what it means for a state on the Cuntz–Pimsner algebra OΛ for finite k-graph Λ to be finitely correlated. Finitely correlated states will give rise to finitely correlated representations of the product system associated to Λ. In [11] Davidson and Pitts classified atomic representations of On , which include as a special case the permutation representations studied by Bratteli and Jorgensen [4]. If s1 , . . . , sn are pairwise orthogonal isometries which generate On then a representation π of On on a Hilbert space H is atomic if there is an orthonormal basis for H which is permuted by each π(si ) up to multiplication by scalars in T ∪ {0}. There exist finitely correlated atomic representations of On [11]. Atomic representations have been a used in the study of other objects. In [8] Davidson and Katsoulis show that the C ∗ -envelope of An ×ϕ Z+ is On ×ϕ Z, where An is the noncommutative disc algebra. Finitely correlated atomic representations of On are used as a tool to

576


get to this result, see [8, Theorem 4.4]. For a general C ∗ -correspondence or product system of C ∗ -correspondences it is not clear what it could mean for a representation to be atomic. Thus the finitely correlated representations presented in this paper are possibly the nearest analogy to finitely correlated atomic representations. In [13,15] atomic representations of single vertex k-graphs have been classified. In Section 2 we study finitely correlated representations of C ∗ -correspondences. To this end we follow the same program of attack as [10]. Many of the proofs follow the same line of argument as the corresponding proofs in [10]. When this is the case it is remarked upon. Lemma 2.12 corresponds to [10, Lemma 4.1], and is the key technical tool to our analysis in this section. It should be noted that Lemma 2.12 not just generalises [10, Lemma 4.1], but the proof presented here greatly simplifies the argument in [10]. The main results of this section are summarised in Theorem 2.27 and Corollary 2.28. Every graph can be associated with a C ∗ -correspondence. Thus results on representations of ∗ C -correspondences also apply to graph algebras. In Section 4.1 we apply our results to nonselfadjoint graph algebras. The study of nonself-adjoint graph algebras has received attention in several papers in recent years, e.g. [7,19–23,41]. We strengthen our results from Section 2 for the case of an algebra of a finite graph with the strong double-cycle property, i.e. for finite graphs where every vertex has a path to a vertex which lies on two distinct minimal cycles. We show that the nonself-adjoint WOT-closed algebra generated by a finitely correlated, isometric, fully coisometric representation of such a graph contains the projection onto its unique minimal cyclic coinvariant subspace. Aided by the work of Kribs and Power [22] and Muhly and Solel [28] on the algebras of directed graphs we use the same method of proof as in [10] to prove this result. This includes the case studied in [10]. In Section 3 we prove the prove the main results of the paper (Theorem 3.19 and Corollary 3.21) by generalising the results of Section 2 to product systems of C ∗ -correspondences over Nk . Our main tool in this section is Theorem 3.12. A representation of a product system of C ∗ -correspondences provides a representation for each C ∗ -correspondence in the product system. An isometric dilation of a contractive representation of a product system of C ∗ correspondences gives an isometric dilation of each of the representations of the individual C ∗ -correspondences. Theorem 3.12 tells us that if we have a minimal isometric dilation of a fully coisometric representation of a product system over Nk , then the dilations of the corresponding representations of certain individual C ∗ -correspondences in the product system will also be minimal. This allows us to deduce the existence of a unique minimal cyclic coinvariant subspace for finitely correlated, isometric, fully coisometric representations of product systems from the C ∗ -correspondence case. In fact, we will show in Theorem 3.19 that the unique minimal cyclic coinvariant subspace for a representation of a product system will be the same unique minimal cyclic coinvariant subspace for a certain C ∗ -correspondence. Higher-rank graph algebras were introduced by Kumjian and Pask in [25]. A k-graph is, roughly speaking, a set of vertices with k sets of directed edges (k colours), together with a commutation rule between paths of different colours. In the last decade there has been a lot of study on the C ∗ -algebras generated by representations of higher-rank graphs. In more recent years there has been some study on their nonself-adjoint counterparts, see e.g. [24,33]. The case of algebras of higher-rank graphs with a single vertex has proved to be rather interesting. Their study was begun by Kribs and Power [24]. Further study has been carried out by Davidson, Power and Yang [33,12–15,45]. A k-graph can be associated with a product system of C ∗ -correspondences over the discrete semigroup Nk . Thus results on product systems of C ∗ -correspondences over Nk apply to higher-


577

rank graph algebras. In Section 4.2 we remark that since certain 1-graphs contained in a kgraph Λ share the same unique minimal cyclic coinvariant subspace for a finitely correlated representation, if Λ contains a 1-graph with the strong double-cycle property, then the WOTclosed algebra generated by a finitely correlated, isometric, fully coisometric representation will contain the projection onto its minimal cyclic coinvariant subspace. A k-graph with only one vertex satisfies this condition. In [10] the case of non-fully coisometric, finitely correlated row isometries are also studied. The case of finitely correlated representations of product systems of C ∗ -correspondences which are not fully coisometric are not studied in this paper. The reason for this is because, unlike the Frazho–Bunce–Popescu dilation used in [10], dilations of representations of product systems need not be unique if they are not fully coisometric. See Section 3.2 for a discussion of dilation theorems for representations of product systems of C ∗ -correspondences over Nk . 2. C ∗ -correspondences 2.1. Preliminaries and notation We will assume throughout that all C ∗ -algebras are unital and that representations of C ∗ algebras are unital. The theory will also work for the non-unital case. The details are left to the reader. Most of the background on C ∗ -correspondences needed in this paper can be found in the works of Muhly and Solel [27,28]. Provided here is a brief summary of the necessary definitions. Let E be a right module over a C ∗ -algebra A. An A-valued inner product on E is a map ·,· : E × E → A which is conjugate linear in the first variable, linear in the second variable and satisfies (i) ξ, ηa = ξ, ηa, (ii) ξ, η∗ = η, ξ , and (iii) ξ, ξ 0 where ξ, ξ = 0 if and only if ξ = 0, 1

for ξ, η ∈ E and a ∈ A. We can define a norm on E by setting ξ = ξ, ξ 2 . If E is complete with respect to this norm then it is called a Hilbert C ∗ -module. We denote by L(E) the space of all adjointable bounded linear functions from E to E, i.e. the bounded operators on E with a (necessarily unique) adjoint with respect to the inner product on E. The adjointable operators on a Hilbert C ∗ -module form a C ∗ -algebra. For ξ, η ∈ E define ξ η∗ ∈ L(E) by ξ η∗ (ζ ) = ξ ζ, η for each ζ ∈ E. Denote by K(E) the closed linear span of {ξ η∗ : ξ, η ∈ E}. The space K(E) forms a C ∗ -subalgebra of L(E) referred to as the compact operators on E. More on Hilbert C ∗ -modules can be found in [26]. If there is a homomorphism ϕ from A to L(E), then the Hilbert C ∗ -module E, together with the left action on E defined by ϕ, is a C ∗ -correspondence over A. If E and F are two C ∗ correspondences over A we will write ϕE and ϕF to describe the left action of A on E and F respectively. With that said, when there is little chance of confusion we will write aξ in place of ϕ(a)ξ .

578


Suppose E and F are two C ∗ -correspondences over a C ∗ -algebra A. We define the following A-valued inner product on the algebraic tensor product E ⊗A F , of E and F : for ξ1 , ξ2 in E and η1 , η2 in F we let ξ1 ⊗ η1 , ξ2 ⊗ η2 = η1 , ϕF ξ1 , ξ2 η2 . Taking the Hausdorff completion of E ⊗A F with respect to this inner product gives us the interior tensor product of E and F denoted E ⊗ F . This is the only tensor product of C ∗ correspondences that we will use in this paper so we will omit the word “interior” and merely say we are taking the tensor product of C ∗ -correspondences. When taking the tensor product of a C ∗ -correspondence E with itself we will write E 2 in place of E ⊗ E, and similarly we will write E n in place of the n-fold tensor product of E with itself. We will also set E 0 = A. The Fock space F (E) is defined to be the C ∗ -correspondence F (E) =

⊕

En.

n0

The left action of A on F (E) is denote by ϕ∞ and defined by ϕ∞ (a)ξ1 ⊗ · · · ⊗ ξn = (aξ1 ) ⊗ · · · ⊗ ξn . We define creation operators Tξ in L(F (E)) for ξ ∈ E by Tξ (ξ1 ⊗ · · · ⊗ ξn ) = ξ ⊗ ξ1 ⊗ · · · ⊗ ξn ∈ E n+1 for ξ1 ⊗ · · · ⊗ ξn ∈ E n . The norm closed algebra in L(F (E)) generated by

Tξ , ϕ∞ (a): ξ ∈ E, a ∈ A

is denoted by T+ (E) and called the tensor algebra over E. The C ∗ -algebra generated by T+ (E) is denoted T (E) and called the Toeplitz algebra over E. A completely contractive covariant representation (A, σ ) of a C ∗ -correspondence E over A on a Hilbert space H is a completely contractive linear map A from E to B(H) and a unital, non-degenerate representation σ of A on H which satisfy the following covariant property: A(aξ b) = σ (a)A(ξ )σ (b) for a, b ∈ A and ξ ∈ E. We will abbreviate completely contractive covariant representation to merely representation, as these will be the only representations of C ∗ -correspondences we will consider. A representation (A, σ ) is called isometric if it satisfies A(ξ )∗ A(η) = σ ξ, η . Why this is called isometric will become clear presently. If σ is a representation of A on a Hilbert space H and E is a C ∗ -correspondence over A, then we can form a Hilbert space E ⊗σ H by taking the algebraic tensor product of E and H and taking the Hausdorff completion with respect to the inner product defined by


579

ξ1 ⊗ h1 , ξ2 ⊗ h2 = h1 , σ ξ1 , ξ2 h2 for ξ1 , ξ2 ∈ E and h1 , h2 ∈ H. We will write E ⊗ H in place of E ⊗σ H when it is understood which representation we are talking about. We can induce σ to a representation σ E of L(E) on E ⊗ H. This is defined by σ E (T )(ξ ⊗ h) = (T ξ ) ⊗ h for T ∈ L(E), ξ ∈ E and h ∈ H. In particular we can induce σ to σ F (E) . We define an isometric representation (V , ρ) of E on F (E) ⊗ H by ρ(a) = σ F (E) ◦ ϕ∞ (a) for each a ∈ A and V (ξ ) = σ F (E) (Tξ ) for each ξ ∈ E. We call (V , ρ) the representation of E induced by σ . If (A, σ ) is a representation of E on H, then we define the operator A˜ from E ⊗σ H to H by ˜ ⊗ h) = A(ξ )h. A(ξ This operator was introduced by Muhly and Solel in [27], where they show that A˜ is a contraction. Furthermore, they show that A˜ is an isometry if and only if (A, σ ) is an isometric representation. A representation is called fully coisometric when A˜ is a coisometry. We write A˜ n for the operator from E n ⊗σ H to H defined by A˜ n (ξ1 ⊗ · · · ⊗ ξn ⊗ h) = A(ξ1 ) . . . A(ξn )h. ˜ E (ϕ(a)). Note also that σ (a)A˜ = Aσ If σ is a representation of A on H and X is in the commutant of σ (A), then we can define a bounded operator I ⊗ X on E ⊗ H by (I ⊗ X)(ξ ⊗ h) = ξ ⊗ Xh. It is readily verifiable that I ⊗ X is a bounded operator and that I ⊗ X X. In particular if M is a subspace of H with PM ∈ σ (A) then I ⊗ PM is a projection in B(E ⊗ H). Thus E ⊗ H decomposes into a direct sum E ⊗ H = (E ⊗ M) ⊕ (E ⊗ M⊥ ). Let (S, ρ) be a representation of a C ∗ -correspondence E on a Hilbert space H. We denote by I be the identity in B(H ). We call the weak-operator topology closed algebra S = Alg I, S(ξ ), ρ(a): ξ ∈ E, a ∈ A WOT the unital WOT-closed algebra generated by the representation (S, ρ).

580


2.2. Minimal isometric dilations Definition 2.1. Let E be a C ∗ -correspondence over a C ∗ -algebra A and let (A, σ ) be a representation of E on a Hilbert space V. A representation (S, ρ) of E on H is a dilation of (A, σ ) if V ⊆ H and (i) V reduces ρ and ρ(a)|V = σ (a) for all a ∈ A. (ii) V ⊥ is invariant under S(ξ ) for all ξ ∈ E. (iii) PV S(ξ )|V = A(ξ ) for all ξ ∈ E. A dilation (S, ρ) of (A, σ ) is an isometric dilation if (S, ρ) is an isometric representation. A dilation (S, ρ) of (A, σ ) on H is called minimal if H=

Sñ E n ⊗ V .

n0

Theorem 2.2. (See Muhly and Solel [27].) If (A, σ ) is a contractive representation of a C ∗ correspondence E on a Hilbert space V, then (A, σ ) has an isometric dilation (S, ρ). Further, we can choose (S, ρ) to be minimal; and the minimal isometric dilation of (A, σ ) is unique up to a unitary equivalence which fixes V. The following lemma uses a standard argument in dilation theory. Lemma 2.3. If (A, σ ) is a representation of a C ∗ -correspondence E on a Hilbert space V and (S, ρ) is its minimal isometric dilation on H, then (S, ρ) is fully coisometric if and only if (A, σ ) is fully coisometric. Proof. Clearly, if S˜ S˜ ∗ = IH then for v ∈ V, A˜ A˜ ∗ v = PV S˜ S˜ ∗ v = PV v = v, and so A˜ is a coisometry. Conversely, suppose that A˜ is a coisometry. Let M = (I − S˜ S˜ ∗ )H. It is not hard to see that M is an S∗ -invariant subspace, where S is the unital WOT-closed algebra generated by the representation (S, ρ). Also since A˜ A˜ ∗ = IV we have that PV S˜ S˜ ∗ |V = IV , hence M is an S∗ invariant space orthogonal to V. But, since our dilation is minimal the only S∗ -invariant subspace orthogonal to V is the zero space. Therefore M = {0}. 2 The following two results have been proved in [10] for the case when E = Cn (where 2 n ∞) and A = C. We follow much the same line of proof as found there. Lemma 2.4. Let (A, σ ) be a representation of a C ∗ -correspondence E on a Hilbert space V, and let (S, ρ) be the unique minimal isometric dilation of (A, σ ) on a Hilbert space H. Let ˜ ⊗ V)) V. Then W is a ρ-reducing subspace and V ⊥ is isometrically isomorphic W = (V + S(E to F (E) ⊗ W. Furthermore, the representation of E obtained by restricting (S, ρ) to V ⊥ is the representation induced by ρ(·)|W . Proof. First note that W is ρ-reducing. This follows since V is ρ-reducing and hence so is V ⊥ and ρ(a)S(ξ )V = S(aξ )V for each a ∈ A and ξ ∈ E.


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The subspace V ⊥ is invariant under S(ξ ) for each ξ ∈ E. So for any n and ξ1 , . . . , ξn ∈ E, the space S(ξ1 ) . . . S(ξn )W is orthogonal to V. It follows that if n 1, then S(ξ1 ) . . . S(ξn )W is ˜ ⊗ V), orthogonal to S(ξ )V for all ξ ∈ E. Therefore S(ξ1 ) . . . S(ξn )W is orthogonal to V + S(E which contains W. Also note that if η1 , . . . , ηm are in E, with m < n and w1 and w2 in W then S(ξ1 ) . . . S(ξn )w1 , S(η1 ) . . . S(ηm )w2 = ρ η1 ⊗ · · · ⊗ ηm , ξ1 ⊗ · · · ⊗ ξm S(ξm+1 ) . . . S(ξn )w1 , w2 = 0.

By minimality we have that V⊥ =

n0

=

⊕ ⊕

S(ξ1 ) . . . S(ξn )W

ξ1 ,...,ξn ∈E

Sñ E n ⊗ W

n0

F (E) ⊗ W.

2

Remark 2.5. When (A, σ ) is a fully coisometric representation of a C ∗ -correspondence E on ˜ ⊗ V). Hence, when (A, σ ) is fully a Hilbert space V, we have that V = S˜ S˜ ∗ V = S˜ A˜ ∗ V ⊆ S(E ˜ coisometric the space W in Lemma 2.4 is simply W = S(E ⊗ V) V. Lemma 2.6. Let (A, σ ) be a representation of a C ∗ -correspondence E on a Hilbert space V, and let (S, ρ) be the unique minimal isometric dilation of (A, σ ) on a Hilbert space H. Let A be the WOT-closed unital algebra generated by (A, σ ) and let S be the WOT-closed unital algebra generated by (S, ρ). Suppose V1 is an A∗ -invariant subspace of V. Then H1 = S[V1 ] reduces S. If V1 and V2 are orthogonal A∗ -invariant subspaces, then Hj = S[Vj ] for j = 1, 2 are mutually orthogonal. If V = V1 ⊕ V2 , then H = H1 ⊕ H2 and Hj ∩ V = Vj for j = 1, 2. Proof. Note that for any a ∈ A, ρ(a)V1 = σ (a)V1 ⊆ V1 . Also for any ξ ∈ E, S(ξ )∗ V1 = A(ξ )∗ V1 ⊆ V1 . Hence V1 is S∗ -invariant. Now H1 is spanned by vectors of the form S(ξ1 ) . . . S(ξn )v, where ξ1 , . . . , ξn ∈ E and v ∈ V1 . If n 2 then for any ξ ∈ E, S(ξ )∗ S(ξ1 ) . . . S(ξn )v = S ξ, ξ1 ξ2 . . . S(ξn )v ∈ H1 . If n = 1 we have S(ξ )∗ S(ξ1 )v = ρ(ξ, ξ1 )v = σ (ξ, ξ1 )v ∈ H1 . Hence H1 reduces S. Now suppose V1 and V2 are orthogonal A∗ -invariant subspaces. Take v1 ∈ V1 , v2 ∈ V2 and ξ1 , . . . , ξn , η1 , . . . , ηm be in E. Suppose n m then S(ξ1 ) . . . S(ξn )v1 , S(η1 ) . . . S(ηm )v2 = v1 , S(ξn )∗ . . . S(ξm+1 )∗ ρ ξm , ηm . . . ξ1 , η1 v2 = 0.

It follows that H1 and H2 are orthogonal.

582


If V = V1 ⊕ V2 then, since H1 contains V1 and is orthogonal to V2 , H1 ∩ V = V1 . Finally, H1 ⊕ H2 is an S-reducing subspace containing V, so it is all of H by the minimality of the dilation. 2 Given an isometric representation (S, ρ) of a C ∗ -correspondence E on H with corresponding unital WOT-closed algebra S which is the minimal isometric dilation of a representation (A, σ ) on V ⊆ H, Lemma 2.6 shows that S∗ -invariant subspaces of V give rise to S-reducing subspaces of H. In Corollary 2.8 we give a weak converse of this: that S-reducing subspaces in H are uniquely determined by their projections onto V. This follows from the following more general result. Lemma 2.7. Let (A, σ ) be a representation of a C ∗ -correspondence E on a Hilbert space V, and let (S, ρ) be the unique minimal isometric dilation of (A, σ ) on a Hilbert space H. Let S be the unital WOT-closed algebra generated by (S, ρ). Suppose B is a normal operator in B(H) such that the range of B is contained in V ⊥ and B is in C ∗ (S(E), ρ(A)) , the commutant of the C ∗ -algebra generated by S(E) and ρ(A). Then B = 0. Proof. Suppose that B is non-zero. Take any δ such that 0 < δ < B and let Dδ be the open disc of radius δ about 0. Let Q be the spectral projection Q = EB (spec(B)\Dδ ), where spec(B) denotes the spectrum of B. Then Q ∈ W ∗ (B) ⊆ C ∗ (S(E), ρ(A)) and QH is orthogonal to V. In particular QH is a non-zero S∗ -invariant space orthogonal to V. But no such space can exist since our dilation is minimal. 2 Corollary 2.8. Suppose M and N are two S-reducing subspaces of H and the compressions of PM and PN to V are equal, i.e. PV PM PV = PV PN PV . Then M = N . Proof. Let M and N be two S-reducing subspaces with PV PM PV = PV PN PV . Elements of the form S(ξ1 ) . . . S(ξn )v, with v ∈ V and ξ1 , . . . , ξn ∈ E, span a dense subset of H and (PM − PN )S(ξ1 ) . . . S(ξn )v = S(ξ1 ) . . . S(ξn )(PM − PN )v = S(ξ1 ) . . . S(ξn )PV ⊥ (PM − PN )v ∈ V ⊥ . It follows that the range of PM − PN lies in V ⊥ . Hence, by Lemma 2.7 PM − PN = 0 and M = N. 2 2.3. Finitely correlated representations Definition 2.9. An isometric representation (S, ρ) of a C ∗ -correspondence E on a Hilbert space H is called finitely correlated if (S, ρ) is the minimal isometric dilation of a representation (A, σ ) on a non-zero finite-dimensional Hilbert space V ⊆ H. In particular, if S is the unital WOT-closed algebra generated by (S, ρ), then (S, ρ) is finitely correlated if there is a finite-dimensional S∗ -invariant subspace V of H such that (S, ρ) is the minimal isometric dilation of the representation (PV S(·)|V , ρ(·)|V ). Remark 2.10. It should be noted that not all C ∗ -algebras can be represented non-trivially on a finite-dimensional Hilbert spaces, e.g. if A is a properly infinite C ∗ -algebra then there are no


583

non-zero finite-dimensional representations of A since A contains isometries with pairwise orthogonal ranges. Likewise, any simple infinite-dimensional C ∗ -algebra has no finite-dimensional representations. In this section we are concerned with finitely correlated fully coisometric representations. If we assume that a C ∗ -correspondence E over a C ∗ -algebra A has a fully coisometric representation then we are assuming that there are non-zero representations of A on finite-dimensional Hilbert spaces. Under this assumption there are still a wide range of C ∗ -correspondences which can be studied, e.g. the following example and the C ∗ -correspondences associated to graphs in Section 4. Example 2.11. The case when A = C and E = Cn has been studied previously in [10]. A representation of E on a finite-dimensional space V is simply a row-contraction A = [A1 , . . . , An ] from V (n) to V. The representation is fully-coisometric when A is defect free, i.e. n

Ai A∗i = IV .

i=1

The dilation of A will be the Frazho–Bunce–Popescu dilation of A to a row-isometry S = [S1 , . . . , Sn ]. The dilation S will be defect free as A is. These representations can alternatively be viewed as representations of a graph with 1 vertex and n edges, see Section 4.1. Let (S, ρ) be a fully coisometric, finitely correlated representation on H of the C ∗ correspondence E over the C ∗ -algebra A, and let S be the unital WOT-closed algebra generated by (S, ρ). A key tool in the analysis in [10] is that every non-zero S∗ -invariant subspace of H has non-trivial intersection with V [10, Lemma 4.1], for the case A = C and E = Cn . The main idea of the proof is that, because the representation is fully coisometric and the unit ball in V is compact, one can “pull back” any non-zero element of H with elements in S∗ to V, without the norm going to zero. However, the proof in [10] that the norm does not go to zero is quite complicated. We prove the analogous result for more general C ∗ -correspondences than those studied in [10] below. The proof presented below simplifies the approach in [10] by “pulling back” not in H but in F (E) ⊗ H, making use of Muhly and Solel’s ˜ operators. Lemma 2.12. Let (S, ρ) be a finitely correlated, fully coisometric representation of a C ∗ correspondence E on H. Let S be the unital WOT-closed algebra generated by (S, ρ) and let V be a finite-dimensional S∗ -invariant subspace of H such that (S, ρ) is the minimal isometric dilation of the representation (PV S(·)|V , ρ(·)|V ). Then if M is a non-zero, S∗ -invariant subspace of H, the subspace M ∩ V is non-trivial. Proof. Let μ = PV PM . If μ = 1 then for each n there is a unit vector hn ∈ M such that PV ⊥ hn < n1 . Let vn = PV hn . We have that (vn )n is a sequence in the unit ball of V therefore it has a convergent subsequence (vni )ni . Let v0 be the limit of (vni )ni . We have then that hni − v0 hni − vni + vni − v0 → 0, as ni → ∞ and so the subsequence (hni )ni converges to v0 . Therefore v0 is a non-zero vector in M ∩ V. Thus showing that μ = 1 will prove the lemma.

584


Let h be a unit vector in M. Since our dilation is minimal there is a sequence (kn )n converging to h where each kn is of the form

kn =

Nn

S(ξn,i,1 ) . . . S(ξn,i,wn,i )vn,i

i=1

with ξn,i,j ∈ E and vn,i ∈ V. Without loss of generality we can assume that kn = 1 for each n. If we let Mn = max{wn,i : 1 i Nn } for each n, then for any ξ1 , . . . , ξMn ∈ E we have S(ξ1 )∗ . . . S(ξMn )∗ kn ∈ V. ∗ k ∈ E Mn ⊗ V. Note that S˜ ∗ It follows that S˜M Mn is a coisometry so S˜Mn kn = 1. We also have n n ∗ ∗ M that S˜Mn h ∈ E n ⊗ M and S˜Mn h = 1. ∗ k and h = S˜ ∗ h. We have that Let un = S˜M n Mn n n

un − hn → 0 as n → 0. If μ < 1 we can choose ε > 0 such that 1 − ε μ and take n large enough so that hn − un 2 = hn 2 + un 2 − 2 Rehn , un < 2ε. It follows that 1 − ε < Rehn , un

(IEMn ⊗ PV )hn un , with last inequality being the Cauchy–Schwarz inequality. So our choice of ε tells us that μ < (IEMn ⊗ PV PM ) PV PM . This is a contradiction. Thus μ = 1. 2 Proposition 2.13. Let (A, σ ) be a representation of a C ∗ -correspondence E on a finitedimensional Hilbert space V, and let (S, ρ) be the unique minimal isometric dilation of (A, σ ) on a Hilbert space H. Let A be the unital algebra generated by the representation (A, σ ) and let S be the unital WOT-closed algebra generated by (S, ρ). If V1 is an A∗ -invariant subspace of V and H1 = S[V1 ]. Then H1 ∩ V = A[V1 ]. Proof. If w ∈ V A[V1 ] then A∗ w is an A∗ -invariant space orthogonal to V1 , hence by Lemma 2.6 S[A∗ w] ⊆ H1⊥ . Therefore H1 ∩ V ⊆ A[V1 ]. If w ∈ H1⊥ ∩ V then for any A ∈ A and v ∈ V1 then we have that 0 = A∗ w, v = w, Av. Hence A[V1 ] ⊆ H1 ∩ V. 2 Corollary 2.14. If M is an S-reducing subspace then M = S[M ∩ V].


585

Proof. M is an S-reducing subspace and, by Lemma 2.6, S[M∩V] is an S-reducing subspace. Hence M S[M ∩ V] is S-reducing. If M S[M ∩ V] is non-zero then by Lemma 2.12, M S[M ∩ V] has non-zero intersection with V. This yields a contradiction as the intersection will be orthogonal to M ∩ V. 2 Corollary 2.15. If A = B(V) then every S∗ -invariant subspace of H contains V. Proof. Suppose M is a non-zero S-reducing subspace. Then M ∩ V is a non-zero S∗ -invariant, and hence A∗ -invariant, subspace of V. Hence M ∩ V = V. 2 Corollary 2.16. If V1 and V2 are minimal A∗ -invariant subspaces of V such that S[V1 ] = S[V2 ], then V1 = V2 . Proof. Let H = S[V1 ] = S[V2 ]. Define representations (B, σ1 ) and (C, σ2 ) of E on V1 and V2 respectively by B(ξ ) = PV1 A(ξ )|V1

and C(ξ ) = PV2 A(ξ )|V2

for all ξ ∈ E, and σi (a) = σ (a)|Vi for all a ∈ A, i = 1, 2. The representations (B, σ1 ) and (C, σ2 ) share a unique minimal isometric dilation (S(·)|H , σ (·)|H ). By Corollary 2.15, any S∗ -invariant subspace of H contains both V1 and V2 . In particular V1 ⊆ V2 and V2 ⊆ V1 . Hence V1 = V2 . 2 Definition 2.17. Let E be a C ∗ -correspondence over a C ∗ -algebra A. When (A, σ ) is a representation of E on H, we denote by ΦA the completely positive map from σ (A) to σ (A) defined by ˜ ⊗ X)A˜ ∗ ΦA (X) = A(I for every X in σA (A) . Remark 2.18. For any a ∈ A and X ∈ σ (A) we have ˜ ⊗ X)A˜ ∗ = Aσ ˜ E (a)(I ⊗ X)A˜ ∗ σ (a)ΦA (X) = σ (a)A(I ˜ ⊗ X)σ E (a)A˜ ∗ = A(I ˜ ⊗ X)A˜ ∗ σ (a). = A(I So ΦA maps from σ (A) to σ (A) as claimed. In [28] isometric representations that are not necessarily fully coisometric are studied. It is shown there that the corresponding ΦA function for an isometric representation (A, σ ) will be an endomorphism of σ (A) . It is also shown that the fixed point set of ΦA is the commutant of A, where A is the algebra generated by the representation. In our setting, when (A, σ ) is a finitedimensional, fully coisometric representation on a Hilbert space V, we get that the commutant of

586


A is fixed by ΦA (Lemma 2.19). Later, in Lemma 2.26, when we compress (A, σ ) to a certain A∗ invariant subspace Vˆ ⊆ V, we will get that the fixed point set of the corresponding ΦAˆ map for the ˆ σˆ ) := (P ˆ A(·)| ˆ , σ (·)| ˆ ), is the commutant of the compression compressed representation (A, V V V ˆ of A to V. The map ΦA is a generalisation of the map Φ introduced in Section 4 of [10]. Indeed Lemma 2.20 and Lemma 2.21 are direct analogues of [10, Lemma 5.10] and [10, Lemma 5.11] respectively. We follow the same line of proof as in [10] when proving these results. Lemma 2.19. Let (A, σ ) be a fully coisometric representation of a C ∗ -correspondence E over a C ∗ -algebra A on a finite-dimensional Hilbert space V. Let A be the unital algebra generated by the representation (A, σ ) and let ΦA be the map from σ (A) to σ (A) defined in Definition 2.17. Then if X is in the commutant of A, X is a fixed point of ΦA . Proof. Suppose that X ∈ A . Then for any ξ ∈ E and v ∈ V we have ˜ ⊗ v) = XA(ξ )v = A(ξ )Xv X A(ξ ˜ ⊗ Xv) = A(I ˜ ⊗ X)(ξ ⊗ v). = A(ξ ˜ ⊗ X). Multiplying on the right by A˜ ∗ gives X = ΦA (X). Hence X A˜ = A(I

2

Lemma 2.20. Let (A, σ ) be a fully coisometric representation of a C ∗ -correspondence E over a C ∗ -algebra A on a finite-dimensional Hilbert space V. Let A be the unital algebra generated by the representation (A, σ ) and let ΦA be the map from σ (A) to σ (A) defined in Definition 2.17. Suppose there is an X ∈ σ (A) which is non-scalar and ΦA (X) = X. Then V has two pairwise orthogonal minimal A∗ -invariant subspaces. Proof. Since ΦA is unital and self-adjoint there is a positive, non-scalar X ∈ σ (A) such that ΦA (X) = X. Assume X = 1. Note that, as X ∈ σ (A) , the eigenspaces of X are invariant under σ (A). Let μ be the smallest eigenvalue of X and let M = ker(X − I ) and N = ker(X − μI ). Take any non-zero x ∈ M. x2 = ΦA (X)x, x = (I ⊗ X)A˜ ∗ x, A˜ ∗ x A˜ ∗ x, A˜ ∗ x = x2 . From this we must have (I ⊗ X)A˜ ∗ x = A˜ ∗ x and hence A˜ ∗ x ∈ E ⊗ M. Note that if x, y are eigenvectors for X for different eigenvalues then ξ ⊗ x, η ⊗ y = x, σ ξ, η y = 0, for any ξ, η ∈ E. Hence if we take any non-zero x ∈ M and let y be any eigenvector of X orthogonal to M we get A(ξ )∗ x, y = A˜ ∗ x, ξ ⊗ y = 0


587

for any ξ ∈ E. Hence M is A∗ -invariant. The same argument works for N , as both M and N are eigenspaces for extremal values in the spectrum of X. As M and N are distinct eigenspaces for a self-adjoint operator, they are orthogonal. Since V is a finite-dimensional, there exists a space {0} = M ⊆ M of minimal dimension which is A∗ -invariant and a space {0} = N ⊆ N of minimal dimension which is A∗ -invariant. 2 Lemma 2.21. Let (A, σ ) be a fully coisometric representation of a C ∗ -correspondence E over a C ∗ -algebra A on a finite-dimensional Hilbert space V. Let A be the unital algebra generated by the representation (A, σ ) and let ΦA be the map from σ (A) to σ (A) defined in Definition 2.17. Suppose V = V1 ⊕ V2 where both V1 and V2 are minimal A∗ -invariant subspaces. Further suppose the representation (A, σ ) decomposes into (B, σ1 ) ⊕ (C, σ2 ) with respect to V1 ⊕ V2 with B(V1 ) = Alg{B(ξ ), σ1 (a): ξ ∈ E, a ∈ A} and B(V2 ) = Alg{C(ξ ), σ2 (a): ξ ∈ E, a ∈ A}. If there exists X ∈ σ (A) such that (i) ΦA (X) = X, and (ii) X21 := PV2 XPV1 = 0 then there is a unitary W such that C(ξ ) = W ∗ B(ξ )W and σ2 (a) = W ∗ σ1 (a)W all ξ ∈ E and a ∈ A. Moreover the fixed point set of ΦA consists of all matrices of the form for a11 IV1 a12 W ∗ . a W a I 21

22 V2

Proof. We can assume that X = X ∗ and X21 = 1 as ΦA is self-adjoint. We denote by B˜ and C˜ the usual maps from E ⊗ V1 and E ⊗ V2 respectively. Let M = {x ∈ V1 : X21 v = v}. ∗ X v = v. It As V is finite-dimensional, M is non-empty. Note that for any v ∈ M we have X21 21 follows that M is a subspace of V1 . Thus if v ∈ M and a ∈ A we have

X21 σ (a)v 2 = X21 σ (a)v, X21 σ (a)v = X ∗ X21 v, σ a ∗ a v = σ (a)v 2 . 21 So M reduces σ (A). This tells us that E ⊗ M and E ⊗ (V1 M) are orthogonal spaces. Now take any v in M. We have that ˜ ⊗ X21 )B˜ ∗ v. X21 v = C(I This implies that (I ⊗ X21 )B˜ ∗ v = B˜ ∗ v = v. Thus B˜ ∗ v ∈ E ⊗ M for all v ∈ M. Take any ξ ∈ E, v ∈ M and w ∈ V1 M.

B(ξ )∗ v, w = B˜ ∗ v, ξ ⊗ w = 0.

588


Thus B(ξ )∗ v ∈ M. We conclude that M is A∗ -invariant. Hence, by the minimality of V1 , M is all of V1 . Therefore X21 is a unitary. Let W = X21 . For v ∈ V1

˜ ⊗ W )B˜ ∗ v (I ⊗ W )B˜ ∗ v v. v = W v = C(I Hence C˜ is an isometry from Ran(I ⊗ W )B˜ ∗ to the Ran W = V2 . C˜ is a contraction and so must be zero on the orthogonal complement of Ran(I ⊗ W )B˜ ∗ . It follows that C˜ ∗ is an isometry from V2 to Ran(I ⊗ W )B˜ ∗ . Hence C˜ ∗ W = (I ⊗ W )B˜ ∗ . From this it follows that C(ξ )∗ = W B(ξ )∗ W ∗ for all ξ ∈ E. Since W is also in the commutant of σ (A) it is the desired unitary. Suppose Y ∈ B(V1 , V2 ) and Y0 00 is fixed by ΦA , then ˜ ⊗ Y )B˜ ∗ = W B˜ I ⊗ W ∗ (I ⊗ Y )B˜ ∗ = W B˜ I ⊗ W ∗ Y B˜ ∗ . Y = C(I It follows from Lemma 2.20 that W ∗ Y is a scalar and so Y is a scalar multiple of W . A similar argument works for the other coordinates. 2 By Proposition 2.13, if V is an A∗ -invariant subspace of V such that A[V ] = V (i.e. V is cyclic for A) then S[V ] = H. Hence the minimal isometric dilation of the completely contractive representation (PV A(·)|V , σ (·)|V ) is (S, ρ). Definition 2.22. Suppose A is an algebra acting on a Hilbert space V, and that V is an A∗ invariant subspace of V which is cyclic for A. If V has no proper A∗ -invariant subspaces which are cyclic for A then we say that V is a minimal cyclic coinvariant subspace (for A) of V. When (A, σ ) is representation of a C ∗ -correspondence on a Hilbert space V and A is the unital WOT-closed algebra generated by (A, σ ), we call a minimal cyclic coinvariant subspace for A a minimal cyclic coinvariant subspace for (A, σ ). The following proof is due to Ken Davidson. Lemma 2.23. Let V be a finite-dimensional Hilbert space. Suppose A ⊆ B(V) is an algebra and that V is a minimal cyclic coinvariant space for A. Then A is a C ∗ -algebra. Proof. Suppose L is an A∗ -invariant subspace such that V L is not A∗ -invariant. Let M = A[L]. Then L M and M V. So V M is a non-zero A∗ -invariant subspace such that V M V L. We have that L ⊕ (V M) is an A∗ -invariant subspace and A[L ⊕ (V M)] = V. Hence, by our assumption that V is a minimal cyclic coinvariant space, V = L ⊕ (V M). This is a contradiction. Hence if L is an A∗ -invariant subspace then V L must also be A∗ -invariant. Since V is finite-dimensional, it follows that A is a C ∗ -algebra. 2 Lemma 2.24. Let (A, σ ) be a fully coisometric representation of a C ∗ -correspondence E on a finite-dimensional Hilbert space V, and let (S, ρ) be the unique minimal isometric dilation of (A, σ ) on a Hilbert space H. Let A be the unital algebra generated by the representation (A, σ ) and let S be the unital WOT-closed algebra generated by (S, ρ). If V1 , V2 , . . . , Vk is a maximal set of pairwise orthogonal minimal A∗ -invariant spaces of V then Vˆ = V1 ⊕ · · · ⊕ Vk is the unique minimal cyclic coinvariant subspace of V.


589

ˆ is an S-reducing subspace by Lemma 2.6. If S[V] ˆ is not all of H then its Proof. Firstly, S[V] orthogonal complement in H, M, is also an S-reducing space. By Lemma 2.12 M ∩ V is a nonzero A∗ -invariant space orthogonal to each Vj for 1 j k. This contradicts the maximality of our choice of V1 , . . . , Vk . Hence, by Proposition 2.13, Vˆ is A-cyclic. Since each Vj is a minimal A∗ -invariant space and since the S-reducing spaces S[Vj ] are orthogonal by Lemma 2.6 it follows that Vˆ is indeed a minimal cyclic coinvariant subspace of V. Now suppose that W is an A∗ -invariant subspace of V such that S[W] = H, i.e. A[W] = V. Let Hj = S[Vj ] for each j . We have that Hj ⊆ S[W] for each j and hence Hj ∩ W is non-zero. But each Hj is irreducible by Corollary 2.15 and hence Vj is the unique minimal S∗ -invariant subspace of Hj by Corollary 2.16. It follows that Vj is contained in Hj ∩W for each j . Therefore Vˆ ⊆ W. 2 Remark 2.25. In [10], Lemma 2.23 is proved for the case when A is the unital algebra generated by a finite-dimensional, fully coisometric representation (A, σ ) of the C ∗ -correspondence Cn over C [10, Part of Theorem 5.13]. The proof uses analysis of ΦA and the fact that Vˆ is a direct sum of minimal A∗ -invariant subspaces. We note that the proof presented here shows that the result is in fact just a general result about cyclic, coinvariant subspaces in finite-dimensions, independent of any deeper analysis. However, that the minimal cyclic coinvariant space is unique is not a general result in finitedimensional linear algebra. For example, the algebra C=

λ 0 : λ, γ ∈ C γ −λ γ

in B(C2 ) has both {(x, 0): x ∈ C} and {(x, x): x ∈ C} as minimal cyclic coinvariant spaces. While it is shown that the minimal cyclic coinvariant space Vˆ in Lemma 2.24 is unique, the decomposition of Vˆ into a direct sum of minimal coinvariant subspaces is not necessarily unique. For example, suppose A∗ has two 1-dimensional invariant, orthogonal subspaces V1 and V2 and that the representation (PV1 A(·)|V , σ (·)|V1 ) is unitarily equivalent to (PV2 A(·)|V , σ (·)|V2 ). Let U be the unitary defining the equivalence. Take a unit vector v1 ∈ V1 and let v2 = U v1 . Then V1 = span{v1 + v2 } and V2 = span{v1 − v2 } are orthogonal, A∗ -invariant subspaces and V1 ⊕ V2 = V1 ⊕ V2 . We follow the argument given in [10, Theorem 5.13] for the following result. This serves as a converse to Lemma 2.19. Lemma 2.26. Let (A, σ ) be a fully coisometric representation of a C ∗ -correspondence E over a C ∗ -algebra A. Let A be the unital algebra generated by the representation (A, σ ) and let ΦA be the map from σ (A) to σ (A) defined in Definition 2.17. ˆ where Vˆ = V1 ⊕ · · · ⊕ Vk is as in Lemma 2.24. Then the fixed point set of ΦA Suppose V = V, is equal to the commutant of A. Proof. We have already shown in Lemma 2.19 that if X ∈ A then ΦA (X) = X. Take X ∈ σ (A) such that ΦA (X) = X. Suppose that X is non-scalar. If there is no unitary between Vk and Vl

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intertwining A then, by Lemma 2.21, PVk XPVl = 0. On the other hand, if Wk,l is an intertwining unitary from Vk to Vl then Lemma 2.21 tells us that PVk XPVl = xkl Wk,l for some scalar xkl , and hence PVk XPVl is in A . It follows that X ∈ A . 2 The following theorem summarises our main results. Theorem 2.27. Suppose E is a C ∗ -correspondence over a C ∗ -algebra A. Let (A, σ ) be a fully coisometric, finite-dimensional representation of E on a Hilbert space V, and let (S, ρ) be the minimal isometric dilation of (A, σ ) on H. Let A be the unital algebra generated by (A, σ ) and S be the unital WOT-closed algebra generated by (S, ρ). If

Vˆ =

n ⊕

Vj

j =1

is a maximal direct sum of minimal, orthogonal A∗ -invariant subspaces of V, then Vˆ is the unique ˆ = H. Further minimal A∗ -invariant subspace such that S[V]

H=

n ⊕

Hj

j =1

where Hj = S[Vj ]. The representation (PVˆ ⊥ S(·)|Vˆ ⊥ , ρ(·)|Vˆ ⊥ ) is an induced representation and S∗ |Vˆ is a C ∗ algebra. We now show that the compression to the minimal cyclic coinvariant space for a finitely correlated, fully coisometric representation is a complete unitary invariant. Corollary 2.28. Suppose (S, σ ) and (T , τ ) are finitely correlated, isometric, fully coisometric representations of a C ∗ -correspondence E on HS and HT respectively. Let VS be the unique minimal cyclic coinvariant subspace for (S, σ ) and let VT be the unique minimal cyclic subspace for (T , τ ). Then (S, σ ) and (T , τ ) are unitarily equivalent if and only if the finite-dimensional representations (PVS S(·)|VS , σ (·)|VS ) and (PVT T (·)|VT , τ (·)|VT ) are unitarily equivalent. Proof. Suppose (S, σ ) and (T , τ ) are unitarily equivalent. Let U be the unitary from HS to HT intertwining (S, σ ) and (T , τ ). It follows that U VS is invariant under T (·)∗ and is cyclic, hence VT ⊆ U VS . Similarly VS ⊆ U ∗ VT . It follows that U VS = VT and (PVS S(·)|VS , σ (·)|VS ) and (PVT T (·)|VT , τ (·)|VT ) are unitarily equivalent. Conversely, suppose that (PVS S(·)|VS , σ (·)|VS ) and (PVT T (·)|VT , τ (·)|VT ) are unitarily equivalent. Then, by the uniqueness of the minimal isometric dilation, (S, σ ) and (T , τ ) are unitarily equivalent. 2


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3. Product systems of C ∗ -correspondences over N k We will now extend our results to product systems of C ∗ -correspondences. This is the analogue of multivariate operator theory, and so relies on a more sophisticated dilation theory. The key to our analysis will be a trick to reduce to the consideration of a certain C ∗ correspondence contained inside our product system (Theorem 3.12). 3.1. Preliminaries and notation Recall that we are restricting our attention to unital C ∗ -algebras, and we are only considering unital representations of C ∗ -algebras. The following description of product systems of C ∗ -correspondences over Nk follows that of [16,43]. Let A be a unital C ∗ -algebra. A semigroup E is a product system of C ∗ -correspondences over Nk if there is a semigroup homomorphism p : E → Nk such that E(n) := p −1 (n) is a C ∗ -correspondence over A and the map (ξ, η) ∈ E(n) × E(m) → ξ η ∈ E(n + m) extends to an isomorphism tn,m from E(n) ⊗ E(m) onto E(n + m). By E(0) we mean the C ∗ algebra A. Letting e1 , e2 , . . . , ek be the standard generating set of Nk , we write Ei for the C ∗ -correspondence p −1 (ei ). We identify E(n) with E1n1 ⊗ · · · ⊗ Eknk when n = (n1 , . . . , nk ). −1 It follows that ti,j := tei ,ej is an isomorphism from Ei ⊗ Ej to Ej ⊗ Ei , for i j and tj,i = ti,j for i j . We write ti,i for the identity on Ei2 . We will often suppress the isomorphism and write E(n) ⊗ E(m) = E(n + m). If, for each i, (A(i) , σ ) is a representation of Ei on a Hilbert space H and we have the following commutation relation A˜ (i) IEi ⊗ A˜ (j ) = A˜ (j ) IEj ⊗ A˜ (i) (ti,j ⊗ IH ) then (A(1) , . . . , A(k) , σ ) is a (completely contractive covariant) representation of E on H. A representation (A(1) , . . . , A(k) , σ ) is said to be isometric (resp. fully coisometric) if each representation (A(i) , σ ) is isometric (resp. fully coisometric). For n = (n1 , . . . , nk ) ∈ Nk we define a map A˜ n from E(n) ⊗ H to H by ˜ (2) ˜ (k) A˜ n = A˜ (1) n1 IE n1 ⊗ An2 . . . IE n1 ⊗ · · · ⊗ IE nk−1 ⊗ Ank . 1

1

k−1

We define a representation (An , σ ) of the C ∗ -correspondence E(n) by letting An (ξ )h = A˜ n (ξ ⊗ h) for each ξ ∈ E(n) and h ∈ H. A representation (A(1) , . . . , A(k) , σ ) of E is said to be doubly commuting if it satisfies A˜ (j )∗ A˜ (i) = IEj ⊗ A˜ (i) (ti,j ⊗ IH ) IEi ⊗ A˜ (j )∗ . It has been shown in [16,43] that the doubly commuting condition is equivalent to what is known as Nica covariance [29]. It is easy to check that an isometric, fully coisometric representation is doubly commuting.

592


Note that if (A(1) , . . . , A(k) , σ ) is an isometric representation, then for n = (n1 , . . . , nk ), m = (m1 , . . . , mk ) ∈ Nk we have A˜ ∗m A˜ n = IE(n−(n−m)+ ) ⊗ A˜ ∗(n−m)− A˜ (n−m)+ where (n − m)+ is equal to ni − mi in the ith coordinate if ni mi and zero in the ith coordinate otherwise, and (n − m)− ∈ Nk satisfies n − m = (n − m)+ − (n − m)− . We define the Fock space F (E) of a product space of C ∗ -correspondences by F (E) =

⊕

E(n).

n∈Nk

For more details on the construction see [16]. For each n and ξ ∈ E(n) define the creation operator Tξ : F (E) → F (E) by Tξ (η) = ξ ⊗ η for each η ∈ F (E). The C ∗ -algebra in L(F (E)) generated by the creation operators is called the Toeplitz algebra associated to E and denoted T (E). A product system (E, A) is said to have the normal ordering property if ∗ E(n) . T (E) = span L(ξ )L(η) : ξ, η ∈ n∈Nk

Let (S (1) , . . . , S (k) , ρ) be a representation of a product system (E, A) on a Hilbert space H. We denote by I be the identity in B(H ). We call the weak-operator topology closed algebra S = Alg I, S (i) (ξi ), ρ(a): a ∈ A, ξi ∈ Ei for 1 i k WOT the unital WOT-closed algebra generated by the representation (S (1) , . . . , S (k) , ρ). 3.2. Minimal isometric dilations Definition 3.1. Let (E, A) be a product system over Nk and let (A(1) , . . . , A(k) , σ ) be a representation of E on V. A representation (S (1) , . . . , S (k) , ρ) on a Hilbert space H is a dilation of (A(1) , . . . , A(k) , σ ) if H contains V and, for each i, (S (i) , ρ) dilates (A(i) , σ ). A dilation (S (1) , . . . , S (k) , ρ) of (A(1) , . . . , A(k) , σ ) is an isometric dilation if (S (1) , . . . , S (k) , ρ) is an isometric representation. A dilation (S (1) , . . . , S (k) , ρ) of (A(1) , . . . , A(k) , σ ) on H is minimal if H=

Sñ E(n) ⊗ V .

n∈Nk

Given an arbitrary representation of a product system (E, A) over Nk it is not always possible to find an isometric dilation. Indeed, if k 3 and A = E = C, then a representation of E is simply k commuting contractions A1 , . . . , Ak . It is known that there are examples of commuting contractions which cannot be dilated to commuting isometries, see e.g. [30]. With that said, there


593

are a number of dilation theorems for product systems of C ∗ -correspondences. We will now review a number of these dilations results that will be useful. The subsequent remarks may help clarify some of the distinctions. Theorem 3.2. (See Solel [42].) Let (E, A) be a product system of C ∗ -correspondences over N2 . Then any representation of E has an isometric dilation. Definition 3.3. Let (A(1) , . . . , A(k) , σ ) be a representation of a product system (E, A) on H. For each n ∈ Zk we define A(n) to be A(n) = A˜ ∗n− A˜ n+ . Let (S (1) , . . . , S (k) , ρ) be an isometric dilation of (A(1) , . . . , A(k) , σ ). If for each n ∈ Zk , (S (1) , . . . , S (k) , ρ) satisfies (IE(n+ ) ⊗ PH )S(n)|E(n+ )⊗H = A(n) then (S (1) , . . . , S (k) , ρ) is a regular isometric dilation of (A(1) , . . . , A(k) , σ ). Theorem 3.4. (See Solel [43].) Let (E, A) be a product system of C ∗ -correspondences over Nk and let (A(1) , . . . , A(k) , σ ) be a representation of E. If (A(1) , . . . , A(k) , σ ) satisfies the additional condition that, for every v ⊆ {1, . . . , k} (−1)|u| Ie(v)−e(u) ⊗ A˜ ∗e(u) A˜ e(u) 0,

(3.1)

u⊆v

where e(u) ∈ Nk is 1 in the ith coordinate if i ∈ u and zero in the ith coordinate otherwise, then it has a unique minimal regular isometric dilation. Theorem 3.5. (See Solel [43].) Let (E, A) be a product system of C ∗ -correspondences over Nk and let (A(1) , . . . , A(k) , σ ) be a doubly commuting representation of E. Then (A(1) , . . . , A(k) , σ ) will satisfy (3.1). Further, the minimal regular isometric dilation of (A(1) , . . . , A(k) , σ ) will be doubly commuting. Theorem 3.6. (See Shalit [36].) Let (E, A) be a product system of C ∗ -correspondences over Nk and let (A(1) , . . . , A(k) , σ ) be a fully coisometric representation of E. Then (A(1) , . . . , A(k) , σ ) has a minimal isometric dilation which is fully coisometric. Definition 3.7. Let (E, A) be product system of C ∗ -correspondences over Nk . For a representation (A(1) , . . . , A(k) , σ ) of E on a Hilbert space H define the defect operator for s ∈ (0, 1) s =

2 (|n|) ˜ A(n) ˜ ∗, −s A(n)

n∈Nk n(1,1,...,1)

where |n| = n1 + · · · + nk when n = (n1 , n2 , . . . , nk ). The representation (A(1) , . . . , A(k) , σ ) is said to satisfy the Popescu condition if there is a t ∈ (0, 1) such that s is positive for all s ∈ (t, 1).

594


Theorem 3.8. (See Skalski [38].) Let (E, A) be a product system of C ∗ -correspondences over Nk having the normal ordering property. Let (A(1) , . . . , A(k) , σ ) be a representation of E. If (A(1) , . . . , A(k) , σ ) satisfies the Popescu condition then it has an isometric dilation. Remark 3.9. (See remarks on Theorems 3.2, 3.4, 3.6 and 3.8.) The dilation given in Theorem 3.2 is not necessarily unique. Examples of representations which do not dilate uniquely are given by Davidson, Power and Yang in [12]. They also provide an alternative proof of Theorem 3.2 for the case that A = C and Ei = Cni for i = 1, 2. Further it is proved that in this setting a minimal isometric dilation of a fully coisometric representation is fully coisometric and unique. A fully coisometric representation does not necessarily satisfy (3.1). For example if T1 = T2 = S ∗ , where S is a unilateral shift on a separable Hilbert space, then the commuting coisometries T1 and T2 do no satisfy (3.1). The atomic representations of single vertex k-graphs studied in [13,15] do satisfy (3.1) since they are doubly commuting. For another example of a non-doubly commuting, fully coisometric representation see Example 4.17. An alternative proof of Theorem 3.4 was given by Shalit in [37]. The method of proof in [37,36] is to construct a semigroup of commuting contractions from a contractive representation. The result is then deduced from dilation results for semigroups of commuting contractions. Skalski and Zacharias [39] show that if (A(1) , . . . , A(k) , σ ) is a doubly commuting representation of E then its minimal isometric dilation is fully coisometric if and only if (A(1) , . . . , A(k) , σ ) is fully coisometric. We will show in Lemma 3.10 that a minimal, isometric dilation of a representation (A(1) , . . . , A(k) , σ ) is fully coisometric if and only if (A(1) , . . . , A(k) , σ ) is fully coisometric, without the assumption that (A(1) , . . . , A(k) , σ ) is doubly commuting. It is noted in [38] that if a representation (A(1) , . . . , A(k) , σ ) is doubly commuting or coisometric then it will satisfy the Popescu condition. Theorem 3.8 is a more general version of a dilation theorem for k-graphs proved by Skalski and Zacharias in [40]. We will look more closely at k-graphs in Section 4. The following result is just a higher-rank version of Lemma 2.3 and follows much the same argument. Lemma 3.10. Let (A(1) , . . . , A(k) , σ ) be a representation of a product system E on a Hilbert space V with a minimal isometric dilation (S (1) , . . . , S (k) , ρ) on a Hilbert space H. Then (S (1) , . . . , S (k) , ρ) is fully coisometric if and only if (A(1) , . . . , A(k) , σ ) is fully coisometric. Proof. That (A(1) , . . . , A(k) , σ ) is fully coisometric when (S (1) , . . . , S (k) , ρ) is follows the same argument as in Lemma 2.3. Conversely, assume that (A(1) , . . . , A(k) , σ ) is fully coisometric. We will show that S˜ := S˜1 is a coisometry. That S˜i is a coisometry, for 2 i k, follows similarly. Note that S˜ is an isometry and so S˜ S˜ ∗ is a projection on H. Let M = (I − S˜ S˜ ∗ )H. Take any x ∈ M and y ∈ H. We have

S(ξ1 )∗ x, S(ξ2 )y = x, S˜ ξ1 ⊗ S(ξ2 )y = 0

for all ξ1 , ξ2 ∈ E1 . For 2 i k we have

S (i) (η)∗ x, S(ξ )y = x, S (i) (η)S(ξ )y


595

˜ = x, S˜ (i) (IEi ⊗ S)(η ⊗ ξ ⊗ y) = x, S˜ IE ⊗ S˜ (i) ◦ (t ⊗ IH )(η ⊗ ξ ⊗ y) =0 for all ξ ∈ E and η ∈ Ei (where t = t1,i ). It follows that M is S∗ -invariant, where S is the unital WOT-closed algebra generated by (S (1) , . . . , S (k) , ρ). The rest of the proof follows the same argument as Lemma 2.3. 2 Lemma 3.11. Let (A(1) , . . . , A(k) , σ ) be a fully coisometric representation of a product system E. Then the minimal isometric dilation of (A(1) , . . . , A(k) , σ ) is unique up to unitary equivalence. Proof. Since (A(1) , . . . , A(k) , σ ) is fully coisometric it can be dilated by Theorem 3.6. It follows from [38, Theorem 2.7] that all doubly commuting, minimal, isometric dilations of a representation (A(1) , . . . , A(k) , σ ) are unitarily equivalent. By Lemma 3.10, if (A(1) , . . . , A(k) , σ ) is a fully coisometric representation then all minimal, isometric dilations are also fully coisometric, and hence they are doubly commuting. It follows that the minimal isometric dilation is unique up to unitary equivalence. 2 We now prove a key technical tool. We show that taking the minimal isometric dilation of a fully coisometric representation (A(1) , . . . , A(k) , σ ) gives rise to the minimal isometric representation of the representation (An , σ ) when n (1, . . . , 1). This allows us, in Lemma 3.18, to prove the analogous result of Lemma 2.12 for product systems. In fact, Theorem 3.12 allows us to deduce Lemma 3.18 from Lemma 2.12. Lemma 3.18 will play an important role in our analysis, just as Lemma 2.12 did in the study of the C ∗ -correspondence case. Theorem 3.12. Let (A(1) , . . . , A(k) , σ ) be a fully coisometric representation of a system of C ∗ -correspondences E on a Hilbert space V with minimal isometric (S (1) , . . . , S (k) , ρ) on a Hilbert space H. If n = (n1 , n2 , . . . , nk ) ∈ Nk satisfies ni 1 i k then the C ∗ -correspondence representation (Sn , ρ) of E(n) is the (unique) isometric dilation of (An , σ ).

product dilation = 0 for minimal

Proof. It is clear that (Sn , ρ) is an isometric dilation of (An , σ ) for any n ∈ Nk . It remains to show that the dilation is minimal when ni = 0 for each i. For any n ∈ Nk we define Hn to be the space mapped out by (Sn , σ ), i.e. Hn =

Sñm E(n)m ⊗ V .

m∈Z m0

Claim 1. If m, n ∈ Nk and m n, then Hm ⊆ Hn . Let p = n − m. Take any v ∈ V and ξ ∈ E(m) then S˜m (ξ ⊗ v) = S˜m (IE(m) ⊗ S˜p ) IE(m) ⊗ S˜p∗ (ξ ⊗ v) ∈ Sñ E(n) ⊗ V .

596


l is contained in the range of S˜ l for positive integers l follows by a similar That the range of S˜m n argument.

Claim 2. If m, n ∈ Nk and n = lm for some positive integer l, then Hm = Hn . We know from the first claim that Hm ⊆ Hn . The reverse inclusion follows from the fact that Sñ is isomorphic to S˜m (IE(m) ⊗ S˜m ) . . . (IE(m)p−1 ⊗ S˜m ). Claim 3. If m, n ∈ Nk such that ni , mi = 0 for 1 i k, then Hm = Hn . Choose an integer l such that lm n. Then, by the previous two claims, Hn ⊆ Hlm = Hm . The reverse inclusion follows similarly. Now, since (S (1) , . . . , S (k) , ρ) is a minimal dilation, we have that H = n∈Nk Hn . However, if we fix n ∈ Nk such that ni = 0 for each i, then the previous three claims tell us that Hm ⊆ Hn for every m ∈ Nk . Hence H = Hn and so (Sn , ρ) is the minimal isometric dilation of (An , σ ). 2 Remark 3.13. The condition in Theorem 3.12 that n (1, 1, . . . , 1) is necessary to guarantee that (Sn , ρ) is the minimal isometric dilation of (An , σ ). For example, let H be a separable Hilbert space with orthonormal basis {en : n 0}. Define commuting isometries T1 and T2 on H by T1 en = e2n and T2 en = e3n . Then T1∗ and T2∗ are commuting coisometries. Let U1 and U2 be the minimal commuting unitaries dilating T1∗ and T2∗ . Note that commuting unitaries are necessarily doubly commuting. We have that for any n, k 0

U1 e3 , U2k en = e3 , U2k e2n = e3 , T2∗k e2n = 0,

and so U2 is not the minimal isometric dilation of T2∗ . In the case of fully coisometric, atomic representations of single vertex k-graphs, however, it is not necessary for n (1, . . . , 1) for Theorem 3.12 to be satisfied. See Example 4.15, or [13,15]. We now prove a higher rank version of Lemma 2.6. Lemma 3.14. Let (A(1) , . . . , A(k) , σ ) be a representation of a product system E on a Hilbert space V with a minimal isometric dilation (S (1) , . . . , S (k) , ρ) on a Hilbert space H. Let A and S be the unital WOT-closed algebra generated by (A(1) , . . . , A(k) , σ ) and (S (1) , . . . , S (k) , ρ) respectively. Further, suppose that the representation (S (1) , . . . , S (k) , ρ) is doubly commuting. Then if V1 is an A∗ -invariant subspace of V, H1 = S[V1 ] reduces S.


597

If V1 and V2 are orthogonal A∗ -invariant subspaces the Hj = S[Vj ] for j = 1, 2 are mutually orthogonal. If V = V1 ⊕ V2 , then H = H1 ⊕ H2 and Hj ∩ V = Vj for j = 1, 2. Proof. We will prove the first part of the theorem. The remaining parts follow in a similar manner as in Lemma 2.6. First, V1 is A∗ -invariant, and so V1 is S∗ -invariant. Elements of the form Sñ (η ⊗ v), with n ∈ Nk , η ∈ E(n) and v ∈ V1 , span a dense subset of H1 . Take n = (n1 , . . . , nk ) ∈ Nk and i ∈ {1, . . . , k}. Then for any ξ ∈ Ei , η ∈ E(n), v ∈ V1 , w ∈ S[V1 ]⊥ , if ni = 0 then ∗ S (i) (ξ )∗ Sñ (η ⊗ v), w = S ˜(i) Sñ (η ⊗ v), ξ ⊗ w = IE i ⊗ Sñ−ei (η ⊗ v), ξ ⊗ w

= 0, and so S (i) (ξ )∗ Sñ (η ⊗ v) ∈ H1 . If ni = 0 then, since our dilation is doubly commuting, ∗ S (i) (ξ )∗ Sñ (η ⊗ v), w = S ˜(i) Sñ (η ⊗ v), ξ ⊗ w ∗ = (IEi ⊗ Sñ )(t ⊗ IH ) IE(n) ⊗ S ˜(i) (η ⊗ v), ξ ⊗ w ∗ = (IEi ⊗ Sñ )(t ⊗ IH ) IE(n) ⊗ A˜(i) (η ⊗ v), ξ ⊗ w

=0 and so again S (i) (ξ )∗ Sñ (η ⊗ v) ∈ H1 . Thus H1 is S-reducing.

2

Remark 3.15. It is natural to ask if there is a higher rank analogy of Lemma 2.4. If (A(1) , . . . , A(k) , σ ) is a representation of E on V with a minimal isometric dilation (S (1) , . . . , S (k) , ρ) on H, is the restriction of (S (1) , . . . , S (k) , ρ) to V ⊥ an induced representation? The answer is no. From [16] it is known that induced representations are doubly commuting. Looking at the atomic representations studied in [13,15], or looking at Example 4.15, we see that the restriction to V ⊥ is not, in general, doubly commuting. 3.3. Finitely correlated representations Definition 3.16. An isometric representation (S (1) , . . . , S (k) , ρ) of a product system E on a Hilbert space H is called finitely correlated if (S (1) , . . . , S (k) , ρ) is the minimal isometric dilation of a representation (A(1) , . . . , A(k) , σ ) on a non-zero finite-dimensional Hilbert space V ⊆ H. In particular, if S is the unital WOT-closed algebra generated by (S (1) , . . . , S (k) , ρ), then (1) (S , . . . , S (k) , ρ) is finitely correlated if there is a finite-dimensional S∗ -invariant subspace V of H such that (S (1) , . . . , S (k) , ρ) is the minimal isometric dilation of the representation (PV S (1) (·)|V , . . . , PV S (k) (·)|V , ρ(·)|V ). Remark 3.17. In this section we are concerned with finitely correlated fully coisometric representations of product systems. Let (E, A) be a product system of C ∗ -correspondences over Nk .

598


As in the C ∗ -correspondence case, assuming existence of a finitely correlated fully coisometric representation of E puts restrictions on the C ∗ -algebra A. See Remark 2.10. The class of product systems of C ∗ -algebras which exhibit finitely correlated representations includes the k-graphs studied in Section 4. A class of finitely correlated representations of k-graphs have been studied in [13] (2-graphs) and [15] (k-graphs). These papers consider finitely correlated atomic representations of k-graphs. These representations are both isometric and fully coisometric. Atomic representations are an example of partially isometric representations, i.e. they are representations defined by rowcontractions of partial isometries. Atomic representations of k-graphs are looked at more closely in Section 4.2.1. As in the rank 1 case above, the existence of a unique minimal generating space is shown. We will now prove the existence of such a space for a general finitely correlated, isometric, fully coisometric representation of a product system of C ∗ -correspondences over Nk . We begin with a higher rank version of Lemma 2.12. Lemma 3.18. Let (S (1) , . . . , S (k) , ρ) be a finitely correlated, fully coisometric representation of a product system E on a Hilbert space H. Let S be the unital WOT-closed algebra generated by (S (1) , . . . , S (k) , ρ) and let V be a finite-dimensional S∗ -invariant subspace of H such that (S (1) , . . . , S (k) , ρ) is the minimal isometric dilation of the representation (PV S (1) (·)|V , . . . , PV S (k) (·)|V , ρ(·)|V ). Then if M is a non-zero, S∗ -invariant subspace of H, the subspace M ∩ V is non-trivial. Proof. Take any n = (n1 , . . . , nk ) ∈ Nk with ni = 0 for 1 i k. By Theorem 3.12, (Sn , ρ) is the unique minimal isometric dilation of (An , σ ). The subspace M is S∗ -invariant and so for any ξ ∈ E(n), Sn (ξ )∗ M ⊆ M. Let Sn be the unital WOT-closed algebra generated by Sn (E(n)) and ρ(A). It follows that M is invariant under S∗n . Hence, by Lemma 2.12, M ∩ V is nontrivial. 2 Theorem 3.19. Let (A(1) , . . . , A(k) , σ ) be a fully coisometric representation of a product system E on a finite-dimensional Hilbert space V, and let (S (1) , . . . , S (k) , ρ) be the unique minimal isometric dilation of (A(1) , . . . , A(k) , σ ) on a Hilbert space H. Let A be the unital algebra generated by the representation (A(1) , . . . , A(k) , σ ) and let S be the unital WOT-closed algebra generated by (S (1) , . . . , S (k) , ρ). If V1 , V2 , . . . , Vk is a maximal set of pairwise orthogonal minimal A∗ -invariant spaces of V then Vˆ = V1 ⊕ · · · ⊕ Vk is the unique minimal cyclic coinvariant subspace of V and S∗ |Vˆ is a C ∗ -algebra. Further, if Wm is the unique, minimal cyclic space for the C ∗ -correspondence representation ˆ (Sm , ρ), where m = (m1 , m2 , . . . , mk ) (with mi = 0 for 1 i k), then Wm = V. ˆ = H follows the same argument as in Proof. Using Lemma 3.14 and Lemma 3.18, that S[V] the C ∗ -correspondence case. That S∗ |Vˆ is a C ∗ -algebra follows by Lemma 2.23. Let m = (m1 , m2 , . . . , mk ) ∈ Nk where mi = 0 for 1 i k. Let Am be the unital algebra generated by the representation (Am , σ ) and let W be the unique minimal cyclic coinvariant space for Am . By Theorem 3.12 and since W is unique, W is contained in any minimal cyclic ˆ Note also that A[W] = V since Am [W] = V and coinvariant space for A. In particular W ⊆ V. ∗ Am ⊆ A. We will show W is A -invariant.


599

Define the subspace U ⊆ V by

U =

Sm−ek (ξ )∗ W.

ξ ∈E(m−ek )

Note that, by the commutation relations S˜m−ek (IE(m−ek ) ⊗ S˜m ) = S˜m (IE(m) ⊗ S˜m−ek )(t ⊗ IH ) where t is the isomorphism t : E(m − ek ) ⊗ E(m) → E(m) ⊗ E(m − ek ). So, if we take vectors w ∈ W, v ∈ V U and η ∈ E(m) and ξ ∈ E(m − ek ) then ∗ ˜∗ Sm−ek w, ξ ⊗ η ⊗ v Sm (η)∗ Sm−ek (ξ )∗ w, v = IE(m−ek ) ⊗ S˜m ∗ ∗ S˜m w, (t ⊗ IH )(ξ ⊗ η ⊗ v = IE(m) ⊗ S˜m−e k ∗ = S˜m w, (IE(m) ⊗ S˜m−ek )(t ⊗ IH )(ξ ⊗ η ⊗ v) .

∗ w ∈ E(m) ⊗ W, (I Note that S˜m E(m) ⊗ S˜ m−ek )(t ⊗ IH )(ξ ⊗ η ⊗ v) is in the space

E(m) ⊗ S˜m−ek E(m − ek ) ⊗ V U , and W and U are both σ reducing subspaces. It follows that Sm (η)∗ Sm−ek (ξ )∗ w, v = 0,

and so U is A∗m -invariant. By Lemma 2.12, U has non-trivial intersection with W. Let U = W ∩ U . Suppose U = W. A similar argument to above will show that

S (k) (ξ )∗ (W U)

ξ ∈Ek

has intersection with W. Choose w1 , . . . , wn ∈ W U and ζ1 , . . . , ζn ∈ Ek such that n non-trivial (k) ∗ (1) (k) i=1 S (ζi ) wi is a non-zero vector in W. Since (A , . . . , A , σ ) is fully coisometric we can choose η ∈ E(m − ek ) such that w := Sm−ek (η)∗

n

S (k) (ζi )∗ wi

i=1

is non-zero. Now w is in W and hence w is in U ∩ (W U). This contradiction shows that we must have U = W. By construction of U , we have that for any u ∈ U and ξ ∈ Ek , S (k) (ξ )∗ u is in W. Hence W is invariant under A∗k , where Aj is the unital algebra generated by (A(j ) , σ ). We can similarly show that W is A∗j -invariant for 1 j k − 1, and so W is A∗ -invariant. Therefore ˆ and thus Vˆ is unique. 2 W = V,

600


Remark 3.20. Take a non-zero m ∈ Nk with m (1, . . . , 1). Let U be the minimal cyclic coinvariant subspace for the representation (Am , σ ) of the C ∗ -correspondence E(m) and Vˆ be the minimal cyclic coinvariant subspace for the representation of the product system, as in Theˆ However given an arbitrary finitely correlated orem 3.19. We necessarily have that U ⊆ V. ˆ ˆ For the case when k = 2 and m = (0, 1), representation we cannot say whether U = V or U V. ˆ ˆ Example 4.15 satisfies U = V and Example 4.16 satisfies U V. We again conclude that the compression to the unique minimal cyclic subspace for a finitely correlated, fully coisometric representation is a complete unitary invariant. Corollary 3.21. Suppose (S (1) , . . . , S (k) , σ ) and (T (1) , . . . , T (k) , τ ) are finitely correlated, fully coisometric representations of a product system (E, A) on HS and HT respectively. Let VS be the unique minimal cyclic coinvariant subspace for the representation (S (1) , . . . , S (k) , σ ) and let VT be the unique minimal cyclic subspace for the representation (T (1) , . . . , T (k) , τ ). Then (S (1) , . . . , S (k) , σ ) and (T (1) , . . . , T (k) , τ ) are unitarily equivalent if and only if the finite-dimensional representations (PVS S (1) (·)|VS ,. . . ,PVS S (k)(·)|VS,σ(·)|VS ) and (PVT T (1)(·)|VT , . . . , PVT T (k) (·)|VT , τ (·)|VT ) are unitarily equivalent. 4. Higher rank graph algebras 4.1. Graph algebras Let G be a directed graph with a countable number of vertices V(G) and a countable number of edges E(G). If e ∈ E(G) is an edge from a vertex v to a vertex w then we say that v is the source of e, denoted s(e), and that w is the range of e, denoted r(e). A vertex x is called a source if there is no edge e with r(e) = x. A path of length k in G is a finite collection of edges ek ek−1 . . . e1 such that r(ei ) = s(ei+1 ) for 1 i k − 1. A cycle is a path ek ek−1 . . . e1 with s(e1 ) = r(ek ). If x = s(e1 ) and y = r(ek ) then we say that ek ek−1 . . . e1 is a path from x to y. A graph G is transitive if, for any vertices x, y ∈ V(G), there is a path from x to y. A graph is strongly transitive if it is transitive and it is neither a single cycle nor a graph with one vertex and no edges. As described in [35,38,28] a graph can be described by a C ∗ -correspondence. We follow the construction of [35] as presented in [38]. Note that in the case of a finite graph this construction is the same as that given in [28]. Let A = C0 (V(G)) be the C ∗ -algebra of all functions on V(G) vanishing at infinity. Let E(G) be the set of functions ξ : E(G) → C which satisfy for each v ∈ V(G) ξv :=

ξ(e)2 < ∞ e∈E (G) s(e)=v

and the function v → ξv vanishes at infinity. Define an A-valued inner product on E(G) by ξ, η(v) =

ξ(e)η(e),

e∈E (G) s(e)=v

for ξ, η ∈ E(G). Define a left action of A on E(G) by


601

(aξ )(e) = a r(e) ξ(e) and a right action by (ξ a)(e) = ξ(e)a s(e) for ξ ∈ E(G), a ∈ A and e ∈ E(G). These make E(G) into a C ∗ -correspondence over A. We identify the vertex v ∈ V(G) with function δv ∈ A which sends v to 1 and all other vertices to 0. Similarly, we identify an edge e ∈ E(G) with the function δe ∈ E(G) which sends e to 1 and all other edges to 0. For a good introduction to graph algebras see [34]. We remark that representations of E(G) coincide with completely contractive representations of G and that the dilation theorem for contractive representations of graphs in [21,7] is implied by Theorem 2.2. Denote by LG the WOT-closed algebra generated by Tξ , ϕ∞ (a): ξ ∈ E, a ∈ A acting on the space HG := F (E(G)). The algebra LG is known as a free semigroupoid algebra, see [22]. When G has a single vertex and n edges then LG is a free semigroup algebra, more commonly denoted Ln . Finite-dimensional representations of graphs are plentiful. Indeed Davidson and Katsoulis show that the finite-dimensional representations of a graph G separate points in LG [7]. Thus finitely correlated, isometric representations are also plentiful. Provided in [7] is an algorithm for creating finite-dimensional representations. Below is a method for creating finite-dimensional, fully coisometric representations. A similar example can be found in [21]. Example 4.1. Let G be a finite graph with no sources. Let V(G) = {v1 , . . . , vn }. Let E(G)i = {e ∈ E(G): r(e) = vi } = {ei1 , ei2 , . . . , eiCi }, where Ci is the number of elements in E(G)i . Let A and E(G) be as described above. Let H be a finite-dimensional Hilbert space and let K = H1 ⊕ · · · ⊕ Hn , where Hi = H for each i. We will define a representation (A, σ ) of E(G) on K. For each vertex vi let Ti = [Ti1 , . . . , TiCi ] be a defect free row-contraction on Hi , i.e. Ci

Tij Tij∗ = IHi .

j =1

Suppose eij ∈ E(G)i with s(eij ) = vl . Define A(eij ) ∈ B(K) = Mn (B(H)) by (A(eij ))i,l = Ti,j and (A(eij ))k,m = 0 when (k, m) = (i, l). We define a representation σ of A on K by σ (vi ) = PHi =: Pvi for 1 i n. Thus

A(e)A(e)∗ = IK

e∈E (G)

and Pr(e) A(e)Ps(e) = A(e).

602


It follows that (A, σ ) is a finite-dimensional, fully coisometric representation of E(G), see [7,21]. This method readily extends to any graph containing a finite subgraph with no sources. 4.1.1. Strong double-cycle property We now strengthen our results from Section 2 for the special case of C ∗ -correspondences defined by finite graphs with the strong double-cycle property. Definition 4.2. A vertex in G is said to lie on a double-cycle if it lies on two, distinct, minimal cycles. We say that G has the strong double-cycle property if for every vertex x in G there is a path from x to a vertex lying on a double-cycle. Example 4.3. When n 2, a single vertex graph with n edges has the strong double-cycle property. This is the case studied in [10]. Example 4.4. If each connected component of G is strongly transitive, then G has the strong double-cycle property. The following result is proved in [28,22] for finite graphs with the strong double-cycle property, and in [11] for when LG = Ln is a free semigroup algebra. Theorem 4.5. Suppose G is a finite graph with the strong double-cycle property and ϕ is a weak-∗ continuous linear functional on LG with f < 1. Then there are vectors ξ and ζ in HG , with ξ , ζ < 1, such that ϕ(A) = Aξ, ζ for all A in LG . We fix such a graph G with a finitely correlated fully coisometric representation (S, ρ) of E(G) on a Hilbert space H. Let U be the unique minimal cyclic coinvariant subspace for (S, ρ) and let (A, σ ) be the compression of (S, ρ) to U , so that (S, ρ) is the unique minimal dilation of (A, σ ). Let A be the unital algebra generated by (A, σ ) and let S be the unital WOT-closed algebra generated by (S, ρ). By Theorem 2.27, U = U1 ⊕ · · · ⊕ Un is a direct sum of minimal A∗ -invariant subspaces and A is a C ∗ -algebra. For each j let Hj = S[Uj ]. Let d = dim U and let {f1 , . . . , fd } form an orthonormal basis of U . We now follow the methods in [10] in order to give a full description of S. In particular, we will show that S contains the projection onto U . For 1 i n, let qi be the compression of A to Ui , i.e. qi (A) = PUi APUi = B(Ui ). Choose a ⊕ minimal set H ⊆ {1, . . . , n} such that ⊕ h∈H qh is faithful. The minimal ideal ker h∈H \{h0 } qh is isomorphic to B(Uh0 ). This kernel can be supported on more than one of the U i ’s. We let q . For each h ∈ H let Hh ⊆ H be the set of indices i where Ui is supported on ker ⊕ g g∈H \{h} ⊕ ⊕ mh be the number of elements in Hh . If we let Wh = i∈Hh Ui , then U = h∈H Wh . For each j ∈ Hh there is a spatial, algebra isomorphism σj of B(Uh ) onto B(Uj ) such that A|Wh =

⊕

σj (X): X ∈ B(Uh ) .

j ∈Hh

For each h ∈ H let Ph be the projection onto Wh . For each h ∈ H the projection Ph lies in the centre of A.


603

A closer look at Lemma 2.4 tells us that for each ξ ∈ E and a ∈ A

A(ξ ) S(ξ ) = Xξ

0 (α)

Tξ

σ (a) ρ(a) = 0

,

0 ρ(a)|V ⊥

˜ ⊗ U)) U as in Lemma 2.4. Hence where α = dim W, with W = (U + S(E S=

A ∗

0 (α) . LG (α)

We denote by B the WOT-closed operator algebra on H spanned by B(H)PU and 0U ⊕ LG . The following three proofs follow the arguments of [10, Lemma 4.4], [10, Lemma 5.14] and [10, Corollary 5.3] respectively. Lemma 4.6. Every weak-∗ continuous functional on B is given by a trace class operator of rank at most d + 1, where d = dim U . Hence the WOT and weak-∗ topologies coincide on B and S. Proof. Let ϕ be a weak-∗ continuous functional on B. If B ∈ B then ϕ(B) is determined by ϕ(BPU ) and ϕ(BPU ⊥ ). By the Riesz Representation Theorem there are vectors y1 , . . . , yd ∈ U such that

ϕ(BPU ) =

d Bfi , yi . i=1

By Theorem 4.5 there are vectors ξ, ζ ∈ U ⊥ such that ϕ(A) = Aξ, ζ for all A ∈ LG . Hence (α)

ϕ(B) =

d Bfi , yi + Bξ, ζ , i=1

and ϕ is trace-class of rank at most d + 1.

2

Lemma 4.7. For h ∈ H , let Ph denote the minimal central projections of A as above. Then Ph lies in S. Hence PU is in S. Proof. Fix a minimal central projection P of A. Let ϕ be a non-zero weak-∗ continuous functional on B which is zero on S. We will show that ϕ(P ) = 0. It follows immediately that P ∈ S. By Lemma 4.6 there are vectors x, y ∈ H(d+1) such that ϕ(A) = A(d+1) x, y for all A ∈ B. Let M = S∗(d+1) [y]. Since ϕ is zero on S it follows that x is orthogonal to M. Let M0 = M ∩ U (d+1) . By Lemma 2.12, M0 is non-zero. The subspace M0 is invariant under the C ∗ algebra A(d+1) and hence M0 is the range of a projection Q in the commutant of A(d+1) .

604


We decompose U (d+1) into the following spaces P (d+1) QU (d+1) ⊕ P ⊥(d+1) QU (d+1) ⊕ P (d+1) Q⊥ U (d+1) ⊕ P ⊥(d+1) Q⊥ U (d+1) =: Mpq ⊕ Mp⊥ q ⊕ Mpq ⊥ ⊕ Mp⊥ q ⊥ . Note that, as Q and P (d+1) are projections in the commutant of A(d+1) , the four spaces Mij are A(d+1) -reducing. Also M0 = Mpq ⊕ Mp⊥ q . Letting Hij = S[Mij ] we see that H decomposes into H = Hpq ⊕ Hp⊥ q ⊕ Hpq ⊥ ⊕ Hp⊥ q ⊥ . (d+1)

It follows that y ∈ Hpq ⊕ Hp⊥ q = S[M0 ] and so PU P and so P (d+1) y ∈ M0 . Hence

y ∈ M0 . The projection PU dominates

ϕ(P ) = P (d+1) x, y = x, P (d+1) y = 0. Lemma 4.8. The algebra SPU

⊕

(mh ) , h∈H (B(Hh )Ph )

2

where mh = |Hh |.

Proof. First suppose that A = B(U), i.e. U is a minimal A∗ -invariant subspace. By Lemma 4.7, the projection PU is in S. Hence SPU = B(U) is in S. In particular, for any v ∈ U the rank 1 operator vv ∗ is in S. Note also that S[v] = H for any non-zero v ∈ U . Hence for any x ∈ H there are operators Tk in S such that Tk v converges to x. Hence Tk vv ∗ is in S. Hence xv ∗ is in S for all x ∈ H and v ∈ U . Therefore B(H)PU is in S. Returning to the general case, note that there is a unitary equivalence between ⊕ j ∈Hh Hj and Hh ⊗ C(mh ) . Lemma 4.7 tells us that PWh Ph(mh ) lies in S for each h ∈ H . From the first (mh ) . 2 paragraph it now follows that SPU decomposes as ⊕ h∈H (B(Hh )Ph )

Combining Lemma 4.7 and Lemma 4.8 with Theorem 2.27 we get the following theorem. When G is a single vertex graph with 2 or more edges, Theorem 4.9 is the same as [10, Theorem 5.15]. Theorem 4.9. Let G be a finite graph with the strong double cycle property. Let (A, σ ) be fully coisometric, finite-dimensional representation of G on a Hilbert space U . Let (S, ρ) be the unique minimal isometric dilation of (A, σ ) to a Hilbert space K. Let A = Alg{A(ξ ), σ (a): ξ ∈ E, a ∈ A} and S = Alg{S(ξ ), ρ(a): ξ ∈ E, a ∈ A}WOT . If Uˆ =

n ⊕

Uj

j =1

is a maximal direct sum of minimal A∗ -invariant subspaces of U then Uˆ is the unique minimal ˆ of A to Uˆ is a C ∗ -algebra. ˆ = H. The compression A A∗ -invariant subspace such that S[U]


Writing Uˆ as

⊕

h∈H

(mh )

Uh

605

, where Uh has dimension dh and multiplicity mh then ˆ = A

⊕

Mdh ⊗ Cmh .

h∈h

Let Ph be the projection onto Uh . Then the dilation acts on the space K=

⊕

(mh )

Kh

(α) = Uˆ ⊕ HG

h∈H (αh )

where Kh = Uh ⊕ Hh

˜ , αh = dim(S(E(G) ⊗ Uh ) Uh ) and α=

αh mh .

h∈H

The algebra S decomposes as S

⊕ (m ) (α) B(Hh )Ph h + 0Uˆ ⊕ LG . h∈H

4.2. Higher rank graph algebras Definition 4.10. A k-graph (Λ, d) consists of a countable small category Λ, together with a degree functor d from Λ to Nk , satisfying the factorization property: for every λ ∈ Λ and m, n ∈ Nk with d(λ) = m + n, there are unique elements μ, ν ∈ Λ such that λ = μν and d(μ) = m and d(ν) = n. For each n ∈ Nk let Λn = d −1 (n). Each k-graph (Λ, d) has a source map s : Λ → Λ0 and a range map r : Λ → Λ0 . A k-graph Λ is said to be finitely aligned if for each λ, μ ∈ Λ the set {ν ∈ Λ: ∃α,β∈Λ ν = λα = μβ, d(ν) = d(λ) ∨ d(μ)}, is finite. A 1-graph (Λ, d) is simply a graph with vertices Λ0 and edges Λ1 . A k-graph can be visualized as a multi-coloured graph with vertices Λ0 and Λei representing a different coloured set of edges for each i. As in the 1-graph case, a k-graph can be associated with a product system of C ∗ correspondences over Nk . Briefly, define a C ∗ -algebra A by Λ0 , in the same manner that we used the vertices of a 1-graph to define a C ∗ -algebra. For 1 i k define a C ∗ -correspondence Ei over A by Λei in the same manner that we defined a C ∗ -correspondence using the edges of a 1-graph. The factorisation rule of (Λ, d) will define the isomorphisms ti,j : Ei ⊗ Ej → Ej ⊗ Ei , and this in turn will define a product system of C ∗ -correspondences (E(Λ), A) over Nk , see [35] or [38] for the details. In [35] it is shown that Toeplitz Λ-families of contractions coincide with isometric representations of E(Λ). In [38] it is shown that Λ-contractions coincide with representations of E(Λ). Thus there is a 1−1 correspondence between representations of the k-graph (Λ, d) and representations of (E(Λ), A). When Λ is finitely aligned then E(Λ) will satisfy the normal ordering condition. Hence Theorem 3.8 can be applied to finitely aligned k-graphs. This is the dilation theorem originally proved in [40].

606


Let Λ be a k-graph with no sources and with Λ0 finite. In [40, Theorem 4.7] it is shown that there is a 1−1 correspondence between states ω on the Cuntz–Pimsner algebra OΛ and (the unitary equivalence classes of) triples (V, Ω, (S (1) , . . . , S (k) , ρ)) where V is a vector space, Ω ∈ V is norm 1 vector, (S (1) , . . . , S (k) , ρ) is an isometric representation of E(Λ), and V = S∗ Ω (where S is the algebra generated by (S (1) , . . . , S (k) , ρ)). It is noted in [40] that (S (1) , . . . , S (k) , ρ) is the minimal isometric dilation of the compression of (S (1) , . . . , S (k) , ρ) to V. Given this result, it is natural to define what it means for a state on OΛ to be finitely correlated as follows: Definition 4.11. A state ω on OΛ is finitely correlated if its corresponding triple (Vω , Ωω , (1) (k) (Sω , . . . , Sω , ρω )) has the property that Vω is finite-dimensional. When ω is a finitely correlated state on the Cuntz–Pimsner algebra OΛ with corresponding (1) (k) (1) (k) triple (Vω , Ωω , (Sω , . . . , Sω , ρω )), the representation (Sω , . . . , Sω , ρω ) will be finitely correlated. When Λ is a 1-graph with a single vertex and n edges, OΛ is the Cuntz algebra On and the above definition coincides with the definition of finitely correlated states in [3]. Theorem 3.19 and Theorem 4.9 together give us the following result. Proposition 4.12. Let (Λ, d) be a k-graph. Suppose there is an n = (n1 , . . . , nk ) ∈ Nk with ni = 0 for 1 i k, such that the 1-graph with vertices Λ0 and edges defined by Λn has the strong double-cycle property. Let (S (1) , . . . , S (k) , ρ) be a finitely correlated, isometric, fully coisometric representation of E(Λ) generating a WOT-closed algebra S. Then S contains the projection onto its minimal cyclic coinvariant subspace. 4.2.1. Graphs with a single vertex Suppose (Λ, d) is a k-graph where Λ0 is a singleton and Λei is finite for 1 i k. Let (i) Λei = {el : 1 l mi }, where mi is the number of elements in Λei . Let Smi ×mj be the set of permutations on the set of tuples {(a, b): 1 a mi , 1 b mj }. By the factorisation property, for each pair i, j with 1 i < j k there is a permutation θij ∈ Smi ×mj such that (i) (j )

(j ) (i)

el em = em el

when θij (l, m) = (l , m ). Let θ = {θij : 1 i < j k}. The k-graph Λ can be described as being + a unital semigroup F+ θ , where Fθ is the semigroup (j ) (i) (i) (i) (j ) el : el em = em el when θij (l, m) = l , m .

That is, for each i, ei(1) , . . . , ei(mi ) form a copy of the free semigroup F+ mi and, when i = j and i < j , a commutation relation between the ei ’s and the ej ’s is defined by the permutation θij . Note that if we are given arbitrary permutations θij ∈ Smi ×mj for 1 i < j k we cannot necessarily form a cancellative semigroup F+ θ . However, if k = 2 and θ ∈ Sm1 ×m2 is any permutation, + Fθ will form a cancellative semigroup, and hence a 2-graph on a single vertex. ∗ Let (E(F+ θ ), A) be the product system of C -correspondences defined by a k-graph on + a single vertex Fθ . It is not hard to see that A = C and that Ei = Cmi for 1 i m. Let (A(1) , . . . , A(k) , σ ) be a representation of (E(F+ θ ), A) on a Hilbert space H and define (i) (i) (i) Al = A (el ). For each i we have that


607

(i) A(i) = A1 , . . . , A(i) mi is a row-contraction. A representation (A(1) , . . . , A(k) , σ ) is fully coisometric when mi

(i)

(i)∗

Aj Aj

= IH ,

(4.1)

j =1

for 1 i k i.e. when each row-contraction is defect free. A representation is isometric when (i) (i) [A1 , . . . , Ami ] is a row-isometry for 1 i k. (i) (i) Conversely, if [A1 , . . . , Ami ] are k row-contractions which satisfy for 1 i < j k (i)

(j )

(j )

(i)

Al Am = Am Al

when θij (l, m) = (l , m ), then they define a representation of the k-graph F+ θ . + The k-graph Fθ is finite and so it is finitely aligned. Thus, by either Theorem 3.6 or Theorem 3.8 together with Lemma 3.11, all fully coisometric representations of F+ θ have a unique minimal isometric, coisometric dilation. Let F+ be the unital free semigroup with m1 m2 . . . mk generators

el(2) . . . el(k) : 1 lj mj . el(1) k 1 2

This corresponds to the graph with 1-vertex and C ∗ -correspondence E(1, 1, . . . , 1). If m1 . . . mk = 1, i.e. if F+ Z0 , then it is clear that F+ has the strong double-cycle property. (i) Thus by Proposition 4.12, if [S1(i) , . . . , Sm i ] are defect free row-isometries defining a finitely cor+ related representation of Fθ , then the WOT-closed algebra they generate contains the projection onto the minimal cyclic coinvariant subspace. Definition 4.13. Let [A1 , . . . , Ami ], for 1 i k, define a representation of F+ θ on a Hilbert (i) space H. The representation is atomic if each Al is a partial isometry and there is an orthonormal basis {ξn : n 1} of H which is permuted, up to scalars, by each partial isometry, (i) i.e. Al ξn = αξm for some m and some α ∈ T ∪ {0}. (i)

(i)

Atomic representations of k-graphs on a single vertex were studied by Davidson, Power and Yang for 2-graphs [13] and by Davidson and Yang for k-graphs [15]. There the existence of the minimal cyclic coinvariant subspace is shown. The minimal cyclic coinvariant subspace for a finitely correlated, isometric, fully coisometric atomic representation is exhibited by a group construction. That is, a finitely correlated, isometric, fully coisometric atomic representation is shown to be a dilation of a certain representation on B(2 (G)) where G is a group with k generators. The following theorem shows that finitely correlated atomic representations are plentiful. Theorem 4.14. (See Davidson, Power and Yang [13,15].) There are irreducible finite-dimensional defect free atomic representations of F+ θ of arbitrarily large dimension.

608


Example 4.15. Let F+ θ be the two graph where θ ∈ S2×2 is the permutation defined by the cycle ((1, 1), (2, 2)). Let V be a 4-dimensional vector space with orthonormal basis {ζ1 , ζ2 , ζ3 , ζ4 }. We define a fully coisometric, atomic representation of F+ θ on V by row-contractions [A1 , A2 ] and [B1 , B2 ], where A1 ζ1 = ζ2 ,

A1 ζ3 = ζ4 ,

A1 ζi = 0 for i = 2, 4,

A2 ζ2 = ζ1 ,

A2 ζ4 = ζ3 ,

A1 ζi = 0 for i = 1, 3,

B1 ζ2 = ζ3 ,

B1 ζ4 = ζ1 ,

B1 ζi = 0

for i = 1, 3,

B2 ζ1 = ζ4 ,

B2 ζ3 = ζ2 ,

B1 ζi = 0

for i = 2, 4.

Let [S1 , S2 ] and [T1 , T2 ] define the unique minimal isometric dilation of this representation. The representation defined by [S1 , S2 ] and [T1 , T2 ] will also be atomic [12]. Clearly V is the minimal cyclic coinvariant subspace for this representation. For u, w ∈ F+ 2 , where u = i1 . . . il and w = ji . . . jm , we write Su Tw for S i 1 . . . S i l T j 1 . . . Tj m . The set {Su Tw ζi : u, w ∈ F+ 2 , i = 1, 2, 3, 4} will form an orthonormal basis of H. Since the representation is atomic and fully coisometric each of these basis vectors will be in the range of exactly one Si and exactly one Tj . It follows that [S1 , . . . , Sn ] is the minimal isometric Frazho–Bunce– Popescu dilation of the row-contraction [A1 , . . . , An ]. That is, in this case, it is not necessary to have m (1, 1) in order for the conclusion of Theorem 3.12 to be satisfied. This is true of all finitely correlated atomic representations. Recall, by Remark 3.13, that in general we do need the condition that m (1, 1) for Theorem 3.12 to hold. We also have that the minimal cyclic coinvariant subspace for [S1 , . . . , Sn ] is all of V. Thus, again, it is not necessary to have m (1, 1) for the conclusion of Theorem 3.19 to be satisfied. This is also a general fact about atomic representations. Again, recall that we do require that m (1, 1) in the general case. See Remark 3.20 and the following example. There are examples of finite-dimensional, fully coisometric representations which are not partially isometric. Example 4.16. Let θ ∈ S2×2 be the permutation defined by θ (1, 1) = (1, 2), θ (1, 2) = (1, 1), θ (2, 1) = (2, 2) and θ (2, 2) = (2, 1), and let F+ θ be the single vertex 2-graph defined by θ . Let [a1 , a2 ] be a defect free row-contraction on a finite-dimensional Hilbert space V and [b1 , b2 ] be a defect free row-contraction on a finite-dimensional Hilbert space W. We will define a repre(2) sentation of F+ θ on V ⊗ W . Define IW , A1 = a1 ⊗ IW 0 0 b , B1 = IV ⊗ 1 0 b2

0

IW A2 = a2 ⊗ , IW 0 0 b B2 = IV ⊗ 2 . 0 b1

0

Then [A1 , A2 ] and [B1 , B2 ] define a finite-dimensional, fully coisometric representation of F+ θ .


609

Let V = W = C2 and let 1 0 a1 = , 0 √1 2 0 1 b1 = , 1 0

a2 = b2 =

0

0

1 2

1 2

,

0 0 . 0 0

Construct [A1 , A2 ] and [B1 , B2 ] as above. Let U be the minimal cyclic coinvariant subspace for the row-contraction [A1 , A2 ]. A calculation shows that U = span{e1 , e3 , e5 , e7 }, where {e1 , . . . , e8 } is the standard orthonormal basis for C8 . However, we have that B1∗ e1 = / U , and so U is not the minimal cyclic coinvariant subspace for the representation of F+ e2 ∈ θ defined by [A1 , A2 ] and [B1 , B2 ]. In fact, the minimal cyclic coinvariant subspace for this representation is all of C8 . This example shows that atomic representations are special in not needing m (1, 1) in order to satisfy Theorem 3.19. It is not true of all representations single vertex 2-graphs. The construction above works because the permutation θ is very simple. Precisely, if we fix i, θ satisfies θ (i, j ) = θ (i, j ), i.e. i is not changed. Similar constructions of fully coisometric representations of 2-graphs will work for any 2-graph defined by a permutation satisfying this condition. These representations will be doubly commuting. A general method of constructing finite-dimensional, fully coisometric representations of 2graphs which are not partially isometric has proved hard to find. We give below an example of finite-dimensional, fully coisometric representation of a 2-graph which is not doubly commuting. Example 4.17. Let [A1 , A2 ] and [B1 , B2 , B3 ] be row-contractions on C3 with A1 =

0 0

0 0

1 2

1 2

0 0 , 0

⎡

1 0 A2 = ⎣ 0 1 0 0

⎤ 0 0 ⎦ √1 2

and ⎡1 B1 = ⎣

2 1 2

0

1 2 1 2

0

⎡

⎤ 0 0⎦, 0

1 2

B2 = ⎣ − 12 0

1 2

− 12 0

⎤ 0 0 ⎦, √1 2

B3 =

0 0

0 0

1 2

1 2

0 0 . 0

3 Then [A1 , A2 ] and [B1 , B2 , B3 ] define a fully coisometric representation of F+ θ on C where θ ∈ S2×3 is the cycle

(1, 1), (2, 3), (1, 2), (1, 3) .

This fully coisometric representation is not doubly commuting.

610


It is not hard to see that the minimal cyclic coinvariant space for this representation is C2 = span{e1 , e2 }, where {e1 , e2 , e3 } is the standard orthonormal basis for C3 . Acknowledgments The author would like to thank his advisor, Ken Davidson, for his invaluable advice and support. The author would also like to thank the anonymous reviewer for their careful reading of the manuscript and for providing the author with many helpful notes. References [1] T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963) 88–90. [2] W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (409) (1989). [3] O. Bratteli, P.E.T. Jorgensen, Endomorphisms of B(H). II. Finitely correlated states on On , J. Funct. Anal. 145 (2) (1997) 323–373. [4] O. Bratteli, P.E.T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (663) (1999). [5] S. Brehmer, Über vetauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged) 22 (1961) 106– 111. [6] J.W. Bunce, Models for n-tuples of noncommuting operators, J. Funct. Anal. 57 (1) (1984) 21–30. [7] K.R. Davidson, E. Katsoulis, Nest representations of directed graph algebras, Proc. London Math. Soc. (3) 92 (3) (2006) 762–790. [8] K.R. Davidson, E. Katsoulis, Dilating covariant representations of the non-commutative disc algebras, J. Funct. Anal. 259 (4) (2010) 817–831. [9] K.R. Davidson, E. Katsoulis, D.R. Pitts, The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001) 99–125. [10] K.R. Davidson, D.W. Kribs, M.E. Shpigel, Isometric dilations of non-commuting finite rank n-tuples, Canad. J. Math. 53 (3) (2001) 506–545. [11] K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. (3) 78 (2) (1999) 401–430. [12] K.R. Davidson, S.C. Power, D. Yang, Dilation theory for rank 2 graph algebras, J. Operator Theory 63 (2) (2010) 245–270. [13] K.R. Davidson, S.C. Power, D. Yang, Atomic representations of rank 2 graph algebras, J. Funct. Anal. 255 (4) (2008) 819–853. [14] K.R. Davidson, D. Yang, Periodicity in rank 2 graph algebras, Canad. J. Math. 61 (6) (2009) 1239–1261. [15] K.R. Davidson, D. Yang, Representations of higher rank graph algebras, New York J. Math. 15 (2009) 169–198. [16] N.J. Fowler, Discrete product systems of Hilbert bimodules, Pacific J. Math. 204 (2) (2002) 335–375. [17] A. Frazho, Models for noncommuting operators, J. Funct. Anal. 48 (1) (1982) 1–11. [18] J. Glimm, Type I C ∗ -algebras, Ann. of Math. (2) 73 (1961) 572–612. [19] F. Jaëck, S.C. Power, Hyper-reflexivity of free semigroupoid algebras, Proc. Amer. Math. Soc. 134 (7) (2006) 2027– 2035. [20] M.T. Jury, D.W. Kribs, Ideal structure in free semigroupoid algebras from directed graphs, J. Operator Theory 53 (2) (2005) 273–302. [21] M.T. Jury, D.W. Kribs, Partially isometric dilations of noncommuting N -tuples of operators, Proc. Amer. Math. Soc. 133 (1) (2005) 213–222. [22] D.W. Kribs, S.C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc. 19 (2) (2004) 117–159. [23] D.W. Kribs, S.C. Power, Partly free algebras from directed graphs, in: Current Trends in Operator Theory and Its Applications, in: Oper. Theory Adv. Appl., vol. 149, Birkhäuser, Basel, 2004, pp. 373–385. [24] D.W. Kribs, S.C. Power, The analytic algebras of higher rank graphs, Math. Proc. R. Ir. Acad. 106A (2) (2006) 199–218. [25] A. Kumjian, P. Pask, Higher rank graph C ∗ -algebras, New York J. Math. 6 (2000) 1–20. [26] E. Christopher Lance, Hilbert C ∗ -Modules. A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser., vol. 210, Cambridge University Press, Cambridge, 1995. [27] P.S. Muhly, B. Solel, Tensor algebras over C ∗ -correspondences: representations, dilations, and C ∗ -envelopes, J. Funct. Anal. 158 (2) (1998) 389–457.


611

[28] P.S. Muhly, B. Solel, Tensor algebras, induced representations, and the Wold decomposition, Canad. J. Math. 51 (4) (1999) 850–880. [29] A. Nica, C ∗ -algebras generated by isometries and Wiener–Hopf operators, J. Operator Theory 27 (1) (1992) 17–52. [30] S. Parrott, Unitary dilations for commuting contractions, Pacific J. Math. 34 (1970) 481–490. [31] M.V. Pimsner, A class of C ∗ -algebras generalizing both Cuntz–Krieger algebras and crossed products by Z, Free Probability Theory, Waterloo, ON, 1995, pp. 189–212. [32] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (2) (1989) 523–536. [33] S.C. Power, Classifying higher rank analytic Toeplitz algebras, New York J. Math. 13 (2007) 271–298. [34] I. Raeburn, Graph Algebras, CBMS Reg. Conf. Ser. Math., vol. 103, American Mathematical Society, Providence, RI, 2005. [35] I. Raeburn, A. Sims, Product systems of graphs and the Toeplitz algebras of higher-rank graphs, J. Operator Theory 53 (2) (2005) 399–429. [36] O.M. Shalit, E0 -dilation of strongly commuting CP0 -semigroups, J. Funct. Anal. 255 (1) (2008) 46–89. [37] O.M. Shalit, Representing a product system representation as a contractive semigroup and applications to regular isometric dilations, Canad. Math. Bull. 153 (3) (2010) 550–563. [38] A. Skalski, On isometric dilations of product systems of C ∗ -correspondences and applications to families of contractions associated to higher-rank graphs, Indiana Univ. Math. J. 58 (5) (2009) 2227–2252. [39] A. Skalski, J. Zacharias, Wold decomposition for representations of product systems of C ∗ -correspondences, Internat. J. Math. 19 (4) (2008) 455–479. [40] A. Skalski, J. Zacharias, Poisson transform for higher-rank graph algebras and its applications, J. Operator Theory 63 (2) (2010) 425–454. [41] B. Solel, You can see the arrows in a quiver operator algebra, J. Aust. Math. Soc. 77 (1) (2004) 111–122. [42] B. Solel, Representations of product systems over semigroups and dilations of commuting CP maps, J. Funct. Anal. 235 (2) (2006) 593–618. [43] B. Solel, Regular dilations of representations of product systems, Math. Proc. R. Ir. Acad. 108 (1) (2008) 89–110. [44] B. Sz.-Nagy, Sur les contractions de l’espace de Hilbert, Acta Sci. Math. (Szeged) 15 (1953) 87–92. [45] D. Yang, Endomorphisms and modular theory of 2-graph C ∗ -algebras, Indiana Univ. Math. J. 59 (2) (2010) 495– 520.


On the unitary equivalence of absolutely continuous parts of self-adjoint extensions Mark M. Malamud a , Hagen Neidhardt b,∗ a Institute of Applied Mathematics and Mechanics, Universitetskaya str. 74, 83114 Donetsk, Ukraine b Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany

Received 13 July 2009; accepted 25 October 2010 Available online 5 November 2010 Communicated by Alain Connes Dedicated to the memory of M.S. Birman

Abstract The classical Weyl–von Neumann theorem states that for any self-adjoint operator A0 in a separable Hilbert space H there exists a (non-unique) Hilbert–Schmidt operator C = C ∗ such that the perturbed operator A0 + C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering the set ExtA of self-adjoint extensions of a given densely deac and fined symmetric operator A in H and some fixed A0 = A∗0 ∈ ExtA . We show that the ac-parts A = A ∗ ∈ ExtA and A0 are unitarily equivalent provided that the resolvent difference K := of A Aac A 0 − i)−1 − (A0 − i)−1 is compact and the Weyl function M(·) of the pair {A, A0 } admits weak boundary (A limits M(t) := w-limy→+0 M(t + iy) for a.e. t ∈ R. This result generalizes the classical Kato–Rosenblum theorem. Moreover, it demonstrates that for such pairs {A, A0 } the Weyl–von Neumann theorem is in general not true in the class ExtA . © 2010 Elsevier Inc. All rights reserved. Keywords: Symmetric operators; Self-adjoint extensions; Boundary triplets; Weyl functions; Unitary equivalence


E-mail addresses: [email protected] (M.M. Malamud), [email protected] (H. Neidhardt). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.021

614

M.M. Malamud, H. Neidhardt / Journal of Functional Analysis 260 (2011) 613–638

1. Introduction Let A0 be a self-adjoint operator in a separable Hilbert space H and let C = C ∗ be a trace class operator in H, C ∈ S1 (H). Recall, that according to the Kato–Rosenblum theorem, cf. [19,30] ac the absolutely continuous parts Aac 0 and A , in short the ac-parts, of the operators A0 and = A0 + C are unitarily equivalent. In other words, the absolutely continuous spectrum, in short A (·) of Aac ac-spectrum, of A0 and the spectral multiplicity function NAac 0 are stable under additive 0 trace class perturbations. At the same time, the Weyl–von Neumann–Kuroda theorem [1, Theorem 94.2], [31,23,24] shows that the condition C ∈ S1 (H) cannot be replaced by C ∈ Sp (H) with p ∈ (1, ∞] (where Sp (H) denotes the Neumann–Schatten operator ideals). Theorem 1.1. (See [20, Theorems 10.2.1 and 10.2.3].) For any operator A0 = A∗0 in H and any p ∈ (1, ∞] there exists an operator C = C ∗ ∈ Sp (H) such that the perturbed operator = A0 + C has purely point spectrum. In particular, σac (A0 + C) = ∅. A The Kato–Rosenblum theorem was generalized by Birman [4] and Birman and Krein [6] to ac the case of non-additive perturbations. Namely, it was shown that Aac 0 and A still remain unitary equivalent whenever − i)−1 − (A0 − i)−1 ∈ S1 (H). (A are self-adjoint extensions of a symmetric operator A (in In particular, this is true if A0 and A short A0 , A ∈ ExtA ). This rises the following Weyl–von Neumann problem for extensions: Given p ∈ (1, ∞] and a self-adjoint extension A0 of A. Does there exist a self-adjoint extension A −1 −1 of A such that A has purely point spectrum and the difference (A − i) − (A0 − i) belongs to Sp (H)? To the best of our knowledge this problem was not investigated. In the present paper we show that the Weyl–von Neumann theorem for extensions becomes false in general. We show that under an additional assumption on the symmetric operator A the ac-part of a certain extension A0 = A∗0 is unitarily equivalent to the ac-part of any extension = A ∗ of A provided that their resolvent difference is compact, that is, A − i)−1 − (A0 − i)−1 ∈ S∞ (H). KA := (A

(1.1)

The additional assumption on the pair {A, A0 } is formulated in terms of the Weyl function of the pair {A, A0 }. The latter is the main object in the boundary triplet approach to the extension theory extensively developed in the last three decades, see [11,12,17] and references therein. The core of this approach is the following abstract version of Green’s formula ∗ A f, g − f, A∗ g = (Γ1 f, Γ0 g)H − (Γ0 f, Γ1 g)H ,

f, g ∈ dom A∗ ,

(1.2)

where H is an auxiliary Hilbert space and Γ0 , Γ1 : dom(A∗ ) → H are linear mappings. A triplet Π = {H, Γ0 , Γ1 } is called a boundary triplet for the operator A∗ if (1.2) holds and the mapping Γ := {Γ0 , Γ1 } : dom(A∗ ) → H ⊕ H is surjective. With a boundary triplet Π for A∗ one associates in a natural way the Weyl function M(·) = MΠ (·) (see Definition 2.11), which is the key object of this approach. It is an operator-valued Nevanlinna function with values in [H] and its role in the extension theory is similar to that of the classical Weyl–Titchmarsh function in the spectral theory of Sturm–Liouville operators.


615

In particular, if A is simple, then M(·) determines the pair {A, A0 }, where A0 := A∗ ker Γ0 , uniquely, up to unitary equivalence. Moreover, M(·) is regular (holomorphic) precisely on the resolvent set (A0 ) of A0 and the spectral properties of A0 are described in terms of the limits M(t + i0) at the real line (see [8]). Our main result (Theorem 4.3) reads now as follows. Theorem 1.2. Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ such that the corresponding Weyl function M(·) admits weak limits M(t + i0) := w-lim M(t + iy) y↓0

for a.e. t ∈ R.

(1.3)

of A satisfies condition (1.1), then the ac-parts A ac and Aac of A If a self-adjoint extension A 0 ∗ and A0 = A ker(Γ0 ), respectively, are unitarily equivalent. We also present a certain local version of this result (cf. Corollary 4.6). Namely, we show that if the condition (1.3) holds for a.e. t of a measurable subset D of R, then the corresponding parts ac EA(D) and Aac EA0 (D) are unitarily equivalent provided that condition (1.1) is satisfied. A and A0 , respectively. Here EA(·) and EA0 (·) stand for the spectral measures of A The condition (1.3) is independent from a choice of a boundary triplet. Moreover, it is rather strong. For instance, there exist operators for which the only condition (1.3) (without the comac pactness assumption (1.1)) yields ac-minimality of Aac 0 . The latter means that A0 is contained in of A. In particular, this effect (i.e. is unitarily equivalent to a part of ) any self-adjoint extension A takes place for some Schrödinger operators in the half-spaces (see [28] and Section 5 below). We plan to discuss this problem for elliptic operators in general unbounded domains in a separate publication. The paper is organized as follows. In Section 2 we give a short introduction into the theory of ordinary and generalized boundary triplets and the corresponding Weyl functions. In Secac of A = A ∗ (∈ ExtA ) by tion 3 we express the spectral multiplicity function of the ac-part A means of the corresponding Weyl function. Here we substantially use the multiplicity theory for non-orthogonal operator-valued measures on R developed in [27]. In Section 4 we apply this technique for proving Theorem 1.2. Moreover, we present a simple independent proof of the Kato–Rosenblum theorem without using a concept of the wave operators. Finally, Section 5 contains a short description of applications of Theorem 1.2 as well as condition (1.3) itself to Schrödinger operators which will be discussed in a forthcoming paper. The main results of the paper have been announced (without proofs) in [29], a preliminary version with applications to elliptic operators in half-space has been published as a preprint [28]. Notations. We consider only separable Hilbert spaces which are denoted by H, H etc. The symbols C(H1 , H2 ) and [H1 , H2 ] stand for the set of closed densely defined linear operators and the set of bounded linear operators from H1 to H2 , respectively. We set C(H) := C(H, H) and [H] := [H, H]. The symbols dom(·), ran(·), (T ) and σ (T ) denote the domain, the range, the resolvent set and the spectrum of an operator T ∈ C(H), respectively; T ac and σac (T ) stand for the ac-part and the ac-spectrum of an operator T = T ∗ ∈ C(H); ET and NT (·) denote the resolution of the identity and the multiplicity function of T = T ∗ ∈ C(H), respectively. For operator-valued measures Σ the multiplicity function is denoted by NΣ (t). If ET (·) is the orthogonal spectral measure associated with a self-adjoint operator T , then we usually write NT (t) instead of NET (t).

616


Sp (H), p ∈ [1, ∞], stand for the Schatten–von Neumann ideals in H. Denote by B(R) the Borel σ -algebra of the line R and by Bb (R) the algebra of bounded subsets in Bb (R). The Lebesgue measure of a set δ ∈ B(R) is denoted by |δ|. 2. Preliminaries 2.1. Operator measures Definition 2.1. Let H be a separable Hilbert space. A mapping Σ(·) : Bb (R) → [H] is called an operator (operator-valued) measure if (i) Σ(·) is σ -additive in the strong sense and (ii) Σ(δ) = Σ(δ)∗ 0 for δ ∈ Bb (R). The operator measure Σ(·) is called bounded if it extends to the Borel algebra B(R) of R, i.e. Σ(R) ∈ [H]. Otherwise, it is called unbounded. A bounded operator measure Σ(·) = E(·) is called orthogonal if, in addition the conditions (iii) E(δ1 )E(δ2 ) = E(δ1 ∩ δ2 ) for δ1 , δ2 ∈ B(R) and E(R) = IH are satisfied. Setting in (iii) δ1 = δ2 , one gets that an orthogonal measure E(·) takes its values in the set of orthogonal projections on H. Every orthogonal measure E(·) defines an operator T = T ∗ = λ dE(λ) in H with E(·) being its spectral measure. Conversely, by the spectral theorem, every R operator T = T ∗ in H admits the above representation with the orthogonal spectral measure E =: ET . By Σ ac , Σ s , Σ sc and Σ pp we denote absolutely continuous, singular, singular continuous and pure point parts of the measure Σ, respectively. The Lebesgue decomposition of Σ is given by Σ = Σ ac + Σ s = Σ ac + Σ sc + Σ pp . The operator measure Σ1 is called subordinated to the operator measure Σ2 , in short Σ1 ≺ Σ2 , if Σ2 (δ) = 0 yields Σ1 (δ) = 0 for δ ∈ Bb (R). If the measures Σ1 and Σ2 are mutually subordinated, then they are called equivalent, in short Σ1 ∼ Σ2 . Note, that there always exists a scalar measure ρ defined on Bb (R) such that Σ ∼ ρ, see [27, Remark 2.2]. In particular, there always exists a scalar measure such that Σ ≺ ρ. Usually, with the operator-valued measure Σ(·) one associates a distribution operator-valued function Σ(·) defined by Σ(t) =

Σ([0, t)), 0, −Σ([t, 0)),

t > 0, t = 0, t < 0,

(2.1)

which is called the spectral function of Σ. Clearly, Σ(·) is strongly left continuous, Σ(t − 0) = Σ(t), and satisfies Σ(t) = Σ(t)∗ , Σ(s) Σ(t), s t. Definition 2.2. (See [27, Definition 4.5].) Let Σ be an operator measure in H and let ρ be a scalar measure on B(R) such that Σ ≺ ρ. Further, let e = {ej }∞ j =1 be an orthonormal basis in H. Let


Σij (t) := Σ(t)ei , ej , n Ψne (t) := Ψij (t) i,j =1 ,

617

Ψij (t) := dΣij (t)/dρ, ∞ Ψ e (t) := Ψij (t) i,j =1 .

We call e (t) := rank Ψ e (t) := sup rank Ψne (t) mod(ρ) NΣ

(2.2)

n1

the multiplicity function. e (·) does not depend on the orthogonal basis e. Therefore one By [27, Proposition 4.6] NΣ e always has NΣ (t) := NΣ (t) and one can omit the index e in (2.2). When applying this definition to the absolutely continuous part Σ ac of Σ the scalar measure ρ ac can be chosen to be the Lebesgue measure | · | on B(R). The concept of the multiplicity function allows one to introduce the following definitions.

Definition 2.3. Let Σ1 and Σ2 be two operator measures. (i) The operator measure Σ1 is called spectrally subordinate to the operator measure Σ2 , in short Σ1 ≺≺ Σ2 , if Σ1 ≺ Σ2 and NΣ1 (t) NΣ2 (t) (mod(Σ2 )). (ii) The operator measures Σ1 and Σ2 are called spectrally equivalent, in short Σ1 ≈ Σ2 , if Σ1 ∼ Σ2 and NΣ1 (t) = NΣ2 (t) (mod(Σ2 )). In application to self-adjoint operators it makes sense to introduce the following definition. Definition 2.4. Let Tj = Tj∗ ∈ C(Hj ), j = 1, 2. We say that T1 is a part of T2 if there is an isometry V from H1 into H2 such that V T1 V ∗ ⊆ T2 . Crucial for us in the sequel is the following theorem. Theorem 2.5. Let Tj be self-adjoint operators acting in Hj with corresponding spectral measures ETj (·), j = 1, 2. Let D ∈ B(R). (i) T1 ET1 (D) is a part of T2 ET2 (D) if and only if ET1 ,D ≺≺ ET2 ,D , where ETj ,D (δ) := ETj (δ ∩ D), j = 1, 2. (ii) The parts T1 ET1 (D) and T2 ET2 (D) are unitarily equivalent if and only if ET1 ,D ≈ ET2 ,D . The proof follows immediately from [7, Theorem 7.5.1]. For D = R Theorem 2.5 gives conditions for T1 to be unitarily equivalent either to a part of T2 or to T2 itself. If T1 is a part of T2 , then σ (T1 ) ⊆ σ (T2 ) and NT1 (t) NT2 (t) for a.e. t ∈ R (mod(ET2 )). Obviously, if T1 is a part of T2 and T2 is a part of T1 , then T1 and T2 are unitarily equivalent. Using Definition 2.4 Theorem 1.2 can be reformulated as follows: If the conditions (1.1) ac and (1.3) are satisfied, then Aac 0 and A are parts of each other. 2.2. R-functions Let H be a separable Hilbert space. We recall that an operator-valued function F (·) with values in [H] is called to be a Herglotz, Nevanlinna or R-function [1,3,17,22], if it is holomor-

618


phic in C+ and its imaginary part is non-negative, i.e. Im(F (z)) := (2i)−1 (F (z) − F (z)∗ ) 0, z ∈ C+ . In what follows we prefer the notion of R-function. The class of R-functions with values in [H] will be denoted by (RH ). Any (RH )-function F (·) admits an integral representation ∞ F (z) = C0 + C1 z + −∞

1 t dΣF , − t − z 1 + t2

z ∈ C+

(2.3)

(see, for instance, [1,3,22]), where C0 = C0∗ , C1 0 and ΣF is an operator-valued Borel measure on R satisfying R (1 + t 2 )−1 dΣF ∈ [H]. The integral is understood in the strong sense. In contrast to spectral measures of self-adjoint operators the measure ΣF is not necessarily orthogonal. However, the operator-valued measure ΣF is uniquely determined by the R-function F (·). It is called the spectral measure of F (·). The associated spectral function is denoted by ΣF (t), t ∈ R, cf. (2.1). Let us calculate NΣFac (t), t ∈ R. For any Hilbert–Schmidt operator D ∈ S2 (H) satisfying ker(D) = ker(D ∗ ) = {0} let us consider the modified or sandwiched RH -function D F (z) := D ∗ F (z)D,

z ∈ C+ .

For F D (·) the strong limit F D (t) := F D (t + i0) := s-limy→+∞ F D (t + iy) exists for a.e. t ∈ R. We set dF D (t) := dim ran Im F D (t) ,

for a.e. t ∈ R.

(2.4)

Proposition 2.6. Let F (·) ∈ (RH ), D ∈ S2 (H) and ker(D) = ker(D ∗ ) = {0}. Then NΣFac (t) = dF D (t) for a.e. t ∈ R. Proof. It follows from (2.3) that Im F (λ + iy) = yC1 +

∞

−∞

y dΣF , (t − λ)2 + y 2

λ ∈ R.

(2.5)

By Berezanski˘ı–Gel’fand–Kostyuchenko theorem [3,7] the derivative ΨD ∗ ΣF D (t) := d ∗ dt D ΣF (t)D exists for a.e. t ∈ R and the representation D

∗

ΣFac (δ)D

=

ΨD ∗ ΣF D (t) dt,

δ ∈ Bb (R)

δ

holds. Applying the Fatou theorem (see [22]) to (2.5) and using (2.4) we obtain Im F D (λ) = πΨD ∗ ΣF D (λ)

for a.e. λ ∈ R.

(2.6)

By [27, Corollary 4.7] NΣFac (λ) = rank(ΨD ∗ ΣF D (λ)) = dim(ran(ΨD ∗ ΣF D (λ))) for a.e. λ ∈ R. Finally, using (2.6) we get NΣFac (λ) = dF D (λ) for a.e. λ ∈ R. 2


619

Notice that Proposition 2.6 implies that dF D (t) does not dependent on D. Assuming the existence of the limit F (t) := s-limy→+0 F (t + iy) for a.e. t ∈ R, we set dF (t) := dim ran Im F (t) for a.e. t ∈ R. In this case Proposition 2.6 can be modified as follows. Corollary 2.7. Let F (·) ∈ (RH ). If the limit F (t) := s-limy→+0 F (t + iy) exists for a.e. t ∈ R, then NΣFac (t) = dF (t) for a.e. t ∈ R. 2.3. Boundary triplets and self-adjoint extensions In this section we briefly recall the basic facts on boundary triplets and the corresponding Weyl functions, cf. [10–12,17]. Let A be a densely defined closed symmetric operator in the separable Hilbert space H with equal deficiency indices n± (A) = dim(ker(A∗ ∓ i)) ∞. Definition 2.8. (See [17].) A triplet Π = {H, Γ0 , Γ1 }, where H is an auxiliary Hilbert space and Γ0 , Γ1 : dom(A∗ ) → H are linear mappings, is called an (ordinary) boundary triplet for A∗ if the “abstract Green’s identity” ∗ A f, g − f, A∗ g = (Γ1 f, Γ0 g)H − (Γ0 f, Γ1 g)H ,

f, g ∈ dom A∗ ,

(2.7)

holds and the mapping Γ := (Γ0 , Γ1 ) : dom(A∗ ) → H ⊕ H is surjective. Definition 2.9. (See [17].) A closed extension A of A is called a proper extension, in short A ∈ ExtA , if A ⊂ A ⊂ A∗ . Two proper extensions A , A are called disjoint if dom(A ) ∩ dom(A ) = dom(A) and transversal if in addition dom(A ) + dom(A ) = dom(A∗ ). = A ∗ is proper, A ∈ ExtA . A boundary triplet Π = Clearly, any self-adjoint extension A ∗ {H, Γ0 , Γ1 } for A exists whenever n+ (A) = n− (A). Moreover, the relations n± (A) = dim(H) and ker(Γ0 ) ∩ ker(Γ1 ) = dom(A) are valid. Besides, Γ0 , Γ1 ∈ [H+ , H], where H+ denotes the Hilbert space obtained by equipping dom(A∗ ) with the graph norm of A∗ . With any boundary triplet Π one associates two extensions Aj := A∗ ker(Γj ), j ∈ {0, 1}, which are self-adjoint in view of Proposition 2.10 below. Conversely, for any extension A0 = A∗0 ∈ ExtA there exists a (non-unique) boundary triplet Π = {H, Γ0 , Γ1 } for A∗ such that A0 := A∗ ker(Γ0 ). Using the concept of boundary triplets one can parameterize all proper, in particular, self adjoint extensions of A. For this purpose denote by C(H) the set of closed linear relations in H, that is, the set of (closed) linear subspaces of H ⊕ H. The adjoint relation Θ ∗ ∈ C(H) of a linear relation Θ in H is defined by

h k ∗ : h , k = h, k for all ∈Θ . Θ = h k A linear relation Θ is called symmetric if Θ ⊂ Θ ∗ and self-adjoint if Θ = Θ ∗ .

620


The multivalued part mul(Θ) of Θ ∈ C(H) is mul(Θ) = {h ∈ H: {0, h} ∈ Θ}. Setting ⊥ we get H = H ⊕ H . This decomposition yields an orthogH∞ := mul(Θ) and Hop := H∞ op ∞ onal decomposition Θ = Θop ⊕ Θ∞ where Θ∞ := {0} ⊕ mul(Θ) and Θop := {{f, g} ∈ Θ: f ∈ dom(Θ), g ⊥ mul(Θ)}. For the definition of the inverse and the resolvent set of a linear relation Θ we refer to [13]. Proposition 2.10. Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ . Then the mapping

→ Γ dom(A) = {Γ0 f, Γ1 f }: f ∈ dom(A) =: Θ ∈ C(H) ExtA A

(2.8)

where establishes a bijective correspondence between the sets ExtA and C(H). We put AΘ := A Θ is defined by (2.8). Moreover, the following hold: (i) AΘ = A∗Θ if and only if Θ = Θ ∗ ; (ii) The extensions AΘ and A0 are disjoint if and only if Θ ∈ C(H). In this case (2.8) becomes AΘ = A∗ ker(Γ1 − ΘΓ0 ); (iii) The extensions AΘ and A0 are transversal if and only if Θ = Θ ∗ ∈ [H]. In particular, Aj := A∗ ker(Γj ) = AΘj , j ∈ {0, 1} where Θ0 := {0} × H and Θ1 := H × {0}. Hence Aj = A∗j since Θj = Θj∗ . In the sequel the extension A0 is usually regarded as a reference self-adjoint extension. 2.4. Weyl functions and γ -fields It is well known that Weyl functions give an important tool in the direct and inverse spectral theory of singular Sturm–Liouville operators. In [10–12] the concept of Weyl function was generalized to the case of an arbitrary symmetric operator A with n+ (A) = n− (A). Following [10–12] we recall basic facts on Weyl functions and γ -fields associated with a boundary triplet Π . Definition 2.11. (See [10,11].) Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ . The functions γ (·) : (A0 ) → [H, H] and M(·) : (A0 ) → [H] defined by γ (z) := (Γ0 Nz )−1

and M(z) := Γ1 γ (z),

z ∈ (A0 ),

(2.9)

Nz := ker(A∗ − z), are called the γ -field and the Weyl function, respectively, corresponding to Π . ˙ Nz , z ∈ (A0 ), where A0 = A∗ ker(Γ0 ), It follows from the identity dom(A∗ ) = ker(Γ0 ) + ∗ and Nz := ker(A − z), that the γ -field γ (·) is well defined and takes values in [H, H]. Since Γ1 ∈ [H+ , H], it follows from (2.9) that M(·) is well defined too and takes values in [H]. Moreover, both γ (·) and M(·) are holomorphic on (A0 ) and satisfy the following relations (see [11]) γ (z) = I + (z − ζ )(A0 − z)−1 γ (ζ ), z, ζ ∈ (A0 ), (2.10) and M(z) − M(ζ )∗ = (z − ζ )γ (ζ )∗ γ (z),

z, ζ ∈ (A0 ).

(2.11)


621

The last identity yields that M(·) is an RH -function, that is, M(·) is an [H]-valued holomorphic function on C \ R satisfying M(z) = M(z)∗

and

Im(M(z)) 0, Im(z)

z ∈ C \ R.

Moreover, it follows from (2.11) that M(·) satisfies 0 ∈ (Im(M(z))), z ∈ C \ R. If A is a simple symmetric operator, then the Weyl function M(·) determines the pair {A, A0 } uniquely up to unitary equivalence (see [12,21]). Therefore M(·) contains (implicitly) full information on spectral properties of A0 . We recall that a symmetric operator is said to be simple if there is no non-trivial subspace which reduces it to a self-adjoint operator. For a fixed extension A0 = A∗0 the boundary triplet Π = {H, Γ0 , Γ1 } satisfying dom(A0 ) = j j ker(Γ0 ) is not unique. Let Πj = {Hj , Γ0 , Γ1 }, j ∈ {1, 2}, be two such triplets. Then the corresponding Weyl functions M1 (·) and M2 (·) are related by M2 (z) = R ∗ M1 (z)R + R0 ,

(2.12)

where R0 = R0∗ ∈ [H2 ] and R ∈ [H2 , H1 ] is boundedly invertible. According to Proposition 2.10 the extensions AΘ and A0 are not disjoint whenever mul(Θ) = {0}. Considering AΘ and A0 as extensions of an intermediate extension S := A0 (dom(A0 ) ∩ dom(AΘ )) we can avoid this inconvenience. Lemma 2.12. Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ , M(·) the corresponding Weyl and Θ = Θop ⊕ Θ∞ its orthogonal decomposition. Further let S := function, Θ = Θ ∗ ∈ C(H) Γ0 , Γ1 }, defined by = {H, A0 (dom(A0 ) ∩ dom(AΘ )). Then the triplet Π := Hop = dom(Θ), H

Γ0 := Γ0 dom S ∗ ,

Γ1 := πop Γ1 dom S ∗ ,

is a boundary triplet for S ∗ , where πop is the orthogonal projection from H onto Hop , A0 = S ∗ ker(Γ0 ) and AΘ = SΘop . The corresponding Weyl function is := πop M(z) Hop , M(z)

z ∈ C± .

(2.13)

The proof can be found in [9]. Hence without loss of generality we can very often assume that of an extension A = AΘ = A∗ ∈ ExtA corresponds to the graph of the “coordinate” Θ := Γ A Θ a self-adjoint operator. In what follows, without loss of generality, we always assume that the closed symmetric A is simple and, due to Lemma 2.12, the “coordinate” Θ of the extension AΘ = A∗Θ ∈ ExtA is the graph of a self-adjoint operator. 2.5. Krein-type formula for resolvents and comparability With any boundary triplet Π = {H, Γ0 , Γ1 } for A∗ and any proper (not necessarily selfadjoint) extension AΘ ∈ ExtA it is naturally associated the following (unique) Krein-type formula (cf. [10–12]) −1 (AΘ − z)−1 − (A0 − z)−1 = γ (z) Θ − M(z) γ (z)∗ ,

z ∈ (A0 ) ∩ (AΘ ).

(2.14)

622


Formula (2.14) is a generalization of the well-known Krein formula for resolvents. We note also, that all objects in (2.14) are expressed in terms of the boundary triplet Π (cf. [10–12]). In other words, (2.14) gives a relation between Krein-type formula for canonical resolvents and the theory of abstract boundary value problems (framework of boundary triplets). The following result is deduced from formula (2.14) (cf. [11, Theorem 2]). Proposition 2.13. Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ , Θi = Θi∗ ∈ C(H), i∈ {1, 2}. Then for any Schatten–von Neumann ideal Sp , p ∈ (0, ∞], and any z ∈ C \ R the following equivalence holds (AΘ1 − z)−1 − (AΘ2 − z)−1 ∈ Sp (H)

⇐⇒

(Θ1 − z)−1 − (Θ2 − z)−1 ∈ Sp (H).

In particular, (AΘ1 − z)−1 − (A0 − z)−1 ∈ Sp (H) ⇐⇒ (Θ1 − i)−1 ∈ Sp (H). If in addition Θ1 , Θ2 ∈ [H], then for any p ∈ (0, ∞] the equivalence holds (AΘ1 − z)−1 − (AΘ2 − z)−1 ∈ Sp (H)

⇐⇒

Θ1 − Θ2 ∈ Sp (H).

2.6. Generalized boundary triplets and proper extensions In applications the concept of boundary triplets is too restrictive. Here we recall some facts on generalized boundary triplets following [12]. Definition 2.14. (See [12, Definition 6.1].) A triplet Π = {H, Γ0 , Γ1 } is called a generalized boundary triplet for A∗ if H is an auxiliary Hilbert space and Γj : dom(Γj ) → H, j = 0, 1, are linear mappings such that dom(Γ ) := dom(Γ0 ) ∩ dom(Γ1 ) is a core for A∗ , Γ0 is surjective, A0 := A∗ ker(Γ0 ) is self-adjoint and the following Green’s formula holds (A∗ f, g) − (f, A∗ g) = (Γ1 f, Γ0 g)H − (Γ0 f, Γ1 g)H ,

f, g ∈ dom(A∗ ),

(2.15)

where A∗ := A∗ dom(Γ ). By definition, A∗ := A∗ dom(Γ ) and A∗ ⊆ A∗ = A∗ and (A∗ )∗ = A. Clearly, every ordinary boundary triplet is a generalized boundary triplet. Lemma 2.15. (See [12, Proposition 6.1].) Let A be a densely defined closed symmetric operator and let Π = {H, Γ0 , Γ1 } be a generalized boundary triplet for A∗ . Then the following assertions are true: (i) N∗z := dom(A∗ ) ∩ Nz is dense in Nz and dom(A∗ ) = dom(A0 ) + N∗z ; (ii) Γ1 dom(A0 ) = H; (iii) ker(Γ ) = dom(A) and ran(Γ ) = H ⊕ H. Lemma 2.16. Let A be a densely defined closed symmetric operator and let Π = {H, Γ0 , Γ1 } be a generalized boundary triplet for A∗ . Then the mapping Γ = {Γ0 , Γ1 } is closable and Γ ∈ C(H+ , H).


623

Proof. The Green’s formula can be rewritten as (A∗ f, g) − (f, A∗ g) = (J Γf, Γ g) where Γ := 0 I . Let fn ∈ dom(Γ0 ) ∩ dom(Γ1 ) = dom(A∗ ), fn H+ → 0 and Γfn = (Γ0 , Γ1 ) and J := −I 0 {Γ0 fn , Γ1 fn } → {ϕ, ψ} as n → ∞. Hence 0 = lim (A∗ fn , g) − (fn , A∗ g) = (Jf∞ , Γ g), n→∞

where f∞ := {ϕ, ψ} .

Since ran(Γ ) is dense in H ⊕ H one has Jf∞ = 0. Thus, ϕ = ψ = 0 and Γ is closable.

2

For any generalized boundary triplet Π = {H, Γ0 , Γ1 } we set Aj := A∗ ker(Γj ), j ∈ {0, 1}. The extensions A0 and A1 are disjoint but not necessarily transversal. The latter holds if and only if Π is an ordinary boundary triplet. In general, the extension A1 is only essentially self-adjoint. Starting with Definition 2.14, one easily extends the definitions of γ -field and Weyl function to the case of a generalized boundary triplet Π by analogy with Definition 2.11 (cf. [12, Definition 6.2]). Definition 2.17. Let Π = {H, Γ0 , Γ1 } be a generalized boundary triplet for A∗ . Then the operator-valued functions γ (·) and M(·) defined by −1 γ (z) := Γ0 N∗z : H → Nz

and M(z) := Γ1 γ (z),

z ∈ (A0 ),

(2.16)

are called the (generalized) γ -field and the Weyl function associated with the generalized boundary triplet Π , respectively. It follows from Lemma 2.15(i) that γ (·) takes values in [H, H], ran(γ (z)) = N∗z := dom(A∗ ) ∩ Nz and it satisfies the identity similar to that of (2.10) which shows that γ (z) is a holomorphic operator-valued function on (A0 ). Further, one has dom(M(z)) = H since ran γ (z) ⊂ dom(Γ1 ), z ∈ (A0 ). By (2.16) M(z) is closable since γ (z) is bounded and Γ1 is closable, by Lemma 2.16. Hence, by the closed graph theorem M(·) takes values in [H]. Moreover, it is holomorphic on (A0 ), because so is γ (·), and satisfies the relation (2.11). It follows that ker(Im M(z)) = {0}, z ∈ C+ , though the stronger condition 0 ∈ (Im M(i))(⇐⇒ ran(γ (i)) = Ni ) is satisfied if and only if Π is an ordinary boundary triplet (in the sense of Definition 2.8). In the sequel we need the following simple but useful statement. Proposition 2.18. Let Π = {H, Γ0 , Γ1 } be an ordinary boundary triplet for A∗ , M(·) the corresponding Weyl function, B = B ∗ ∈ C(H) and AB = A∗ ker(Γ1 − BΓ0 ). Let Γ1B := Γ0 and Γ0B := BΓ0 − Γ1 . Then (i) ΠB = {H, Γ0B , Γ1B } is a generalized boundary triplet for A∗ such that it holds dom(A∗ ) := dom(Γ ) := dom(A0 ) + dom(AB ) ⊆ dom(A∗ ), A∗∗ = A; (ii) the corresponding (generalized) Weyl function MB (·) is −1 MB (z) = B − M(z) ,

z ∈ C± ;

(iii) ΠB is an (ordinary) boundary triplet if and only if B = B ∗ ∈ [H]. In this case MB (·) is an ordinary Weyl function in the sense of Definition 2.8.

624


Note, an analogon of Proposition 2.10 does not hold for generalized boundary triplets. Nevertheless, since the corresponding Weyl function determines the pair {A, A0 } uniquely, up to unitary equivalence, it is possible to describe the spectral properties of A0 in terms of the (generalized) Weyl function M(·). 3. Weyl function and spectral multiplicity Throughout of this section A is a densely defined simple closed symmetric operator in H with n+ (A) = n− (A). Let Π = {H, Γ0 , Γ1 } be a generalized boundary triplet for A∗ , and let M(·) be the corresponding generalized Weyl function. Since M(·) ∈ (RH ) it admits representation (2.3). Since A is densely defined (see [12,26]), one gets C1 = 0, i.e. ∞ M(z) = C0 + −∞

1 t dΣM . − t − z 1 + t2

Proposition 3.1. Let A be a densely defined, simple closed symmetric operator and let Π = {H, Γ0 , Γ1 } be a generalized boundary triplet for A∗ (⊆ A∗ ), A∗∗ = A, and let M(·) be the corresponding Weyl function. If EA0 is the spectral measure of A0 := A∗ ker(Γ0 ), then ΣM ≈ EA0 ac ≈ E ac . and ΣM A0 0 (·), Proof. Alongside ΣM (·) we introduce the bounded operator measure ΣM

0 ΣM (δ) = δ

1 dΣM , 1 + t2

δ ∈ Bb (R).

0 (·) ≈ Σ (·). According to [2, formula (2.16)] one has Clearly, ΣM M 0 (δ) = γ (i)∗ EA0 (δ)γ (i), ΣM

δ ∈ B(R),

(3.1)

where γ (·) is the generalized γ -field of Π . Note, that though formula (3.1) is proved in [2] for ordinary boundary triplets, the proof remains valid for generalized boundary triplets. Due to the simplicity of A one has

span (A0 − z)−1 ran γ (i) : z ∈ C+ ∪ C− = H. Hence the subspace Ni := N∗i , where N∗i := ran(γ (i)) is cyclic for A0 . Next, let Pi be the 0 (·) := Pi EA0 (·) Ni . orthogonal projection from H onto Ni . We set Σ M 0 Clearly, ΣM (·) is an operator measure. Since the linear manifold N∗i is cyclic for A0 , one gets 0 and EA0 are spectrally equivalent. from [27, Theorem 4.15] that the measures Σ M 0 (·) = γ (i)∗ Σ 0 (·)γ (i). Since ran(γ (i)) is dense in Ni , the latter yields Note that ΣM M 0 ∼Σ 0 . Let D ∈ S2 (H) and ker(D) = ker(D ∗ ) = {0}. We set ΣM M ΨD ∗ Σ 0 D (t) := M

0 (t)D dD ∗ ΣM dρ(t)

and ΨD ∗ Σ (t) := 0 D M

∗ Σ 0 (t)D dD M dρ(t)


625

:= γ (i)D : H → Ni . We note that ker(D) = 0 ∼ ρ and D where ρ is a scalar measure such that Σ M ∗ ) = {0}. By [27, Corollary 4.7] we have ker(D NΣ 0 (t) = rank ΨD ∗ Σ 0 D (t) M

M

and NΣ0 (t) = rank ΨD ∗ Σ (t) 0 D M

M

for a.e. t ∈ R (mod(ρ)). Since ΨD ∗ Σ 0 D (t) = ΨD ∗ Σ (t) for a.e. t ∈ R (mod(ρ)) we get 0 D M M 0 N 0 (t) = N 0 (t) for a.e. t ∈ R (mod(ρ)). Hence Σ and Σ 0 are spectrally equivalent. Since ΣM

M

ΣM

M

0 and EA0 are spectrally equivalent, the measures Σ 0 and EA0 are spectrally equivalent. This Σ M M proves the first statement. 0,ac ac (δ)γ (i), δ ∈ B(R) (δ) = γ (i)∗ EA The second statement follows from the equality ΣM 0

0,ac 0 . where ΣM is the ac-part of ΣM

2

The proof of Proposition 3.1 leads to the following computing procedure for NΣMac (t): choosing D ∈ S2 (H) such that ker(D) = ker(D ∗ ) = {0} we introduce the sandwiched Weyl function M D (·), D M (z) := D ∗ M(z)D,

z ∈ C+ .

It turns out that the limit (M D )(t) := s-limy→+0 M D (t + iy) exists for a.e. t ∈ R. We define in accordance with (2.13) the function dM D (·) : R → N ∪ {∞}, dM D (t) := rank Im M D (t) = dim ran Im M D (t) , which is well defined for a.e. t ∈ R. For a measurable non-negative function ξ : R → R+ defined for a.e. t ∈ R we introduce its support supp(ξ ) := {t ∈ R: ξ(t) > 0}. By clac (·) we denote the absolutely continuous closure of a Borel set of R, cf. Appendix A. Proposition 3.2. Let A be as in Proposition 3.1, let Π = {H, Γ0 , Γ1 } be a generalized boundary triplet for A∗ (⊆ A∗ ), A∗∗ = A, and let M(·) be the corresponding Weyl function. Further, let EA0 (·) be the spectral measure of A0 = A∗ ker(Γ0 ) = A∗0 . If D ∈ S2 (H) and satisfies ker(D) = (t) = dM D (t) for a.e. t ∈ R and σac (A0 ) = clac (supp(dM D )). ker(D ∗ ) = {0}, then NAac 0 (t) = If, in addition, the limit M(t) := s-limy→+0 M(t + iy) exists for a.e. t ∈ R, then NAac 0 dM (t) for a.e. t ∈ R and σac (A0 ) = clac (supp(dM )). Proof. The relation NAac (t) = dM D (t) follows from Propositions 2.6 and 3.1. To prove 0 σac (A0 ) = clac (supp(dM D )) we choose a total set {gk }N k=1 , 1 N ∞, in H. We set hk := Dgk . is also a total set. We set Mhn (z) := (M(z)hn , hn ), z ∈ C+ . One easily verifies that {hn }N n=1 Clearly, Mhn (z) is R-function for every n ∈ {1, 2, . . . , N} and Mhn (t) := lim Mhn (t + iy) = M(t)hn , hn y→+0

exists for a.e. t ∈ R. Set

Ωac (Mhn ) := t ∈ R: 0 < Im Mhn (t) < ∞ .

626


Combining [8, Proposition 4.1] with Lemma A.3 we obtain

σac (A0 ) =

N

N clac Ωac (Mhn ) = clac Ωac (Mhn ) .

k=1

(3.2)

k=1

If t ∈ supp(dM D ), then Im((M D )(t)) = 0. Hence t ∈ Ωac (Mhn ) for some n ∈ {1, 2, . . . , N}. Therefore supp(dM D ) ⊆ N k=1 Ωac (Mhn ) which yields N clac supp(dM D ) ⊆ clac Ωac (Mhn ) .

(3.3)

k=1

Conversely, if t ∈ Ωac (Mhn ) ∩ EM D , where EM D := {t ∈ R: ∃(M D )(t)}, for some n, then 0 < dM D (t). Hence Ωac (Mhn ) ∩ EM D ⊆ supp(dM D ) which yields N k=1 Ωac (Mhn ) ∩ EM D ⊆ supp(dM D ). Hence clac

N

Ωac (Mhn ) ∩ EM

k=1

= clac

N

Ωac (Mhn ) ⊆ clac supp(dM D ) .

k=1

Combining this equality with (3.2) and (3.3) we obtain σac (A0 ) = clac (supp(dM D )).

2

Corollary 3.3. Let A be as in Proposition 3.2, let Π = {H, Γ0 , Γ1 } be an ordinary boundary triplet for A∗ and let M(·) be the corresponding Weyl function. Further, let B = B ∗ ∈ C(H), AB = A∗ ker(Γ1 − BΓ0 ) and EAB (·) the spectral measure of AB . If D ∈ S2 (H) and satisfies ker(D) = ker(D ∗ ) = {0}, then NAac (t) = dM D (t) for a.e. t ∈ R and σac (AB ) = clac (supp(dM D )). B B B If, in addition, the limit MB (t) := s-limy→+0 MB (t + iy) exists for a.e. t ∈ R, then NAac (t) = B dMB (t) for a.e. t ∈ R and σac (AB ) = clac (supp(dMB )). Proof. By Proposition 2.18 ΠB = {H, Γ0B , Γ1B } is a generalized boundary triplet for A∗ := A∗ dom(A∗ ), dom(A∗ ) = dom(A0 ) + dom(AB ), and MB (z) = (B − M(z))−1 , z ∈ C+ , is the corresponding generalized Weyl function. Clearly, AB = A∗ ker(Γ0B ). It remains to apply Proposition 3.2. 2 This leads to the following theorem. Theorem 3.4. Let A be a densely defined closed symmetric operator, let Π = {H, Γ0 , Γ1 } be an ordinary boundary triplet for A∗ and let M(·) be the corresponding Weyl function. Further, let AB := A∗ ker(Γ1 − BΓ0 ), B = B ∗ ∈ C(H), and EAB (·) the spectral measure of AB . Let D ∈ S2 (H) and ker(D) = ker(D ∗ ) = {0}. Then: ac (D) is a part of A E ac (D) if and only if d (i) A0 EA B AB M D (t) dMBD (t) for a.e. t ∈ D. 0 ac (D) and A E ac (D) are unitarily equivalent if and only if d (ii) A0 EA B AB M D (t) = dMBD (t) for a.e. 0 t ∈ D.


627

Proof. Without loss of generality we assume that A is simple since the self-adjoint part of A ac (δ) = 0 is contained as a direct summand in any self-adjoint extension of A. We show that ΣM for some δ ∈ Bb (R) if and only if dM D (t) = 0 for a.e. t ∈ δ. By the Berezanski˘ı–Gel’fand– d D ∗ Σ(t)D exists and the relation Kostyuchenko theorem [3,7] the derivative ΨD ∗ ΣM D (t) := dt ac D ∗ ΣM (δ ∩ D)D =

δ ∈ Bb ,

ΨD ∗ ΣM D (t) dt,

δ∩D ac (δ) = 0 if and only if Ψ ∗ holds. One has ΣM D ΣM D (t) = 0 for a.e. t ∈ δ. Since dM D (t) = ac (δ ∩ D) = 0 if and only if d dim(ran(ΨD ∗ ΣM D (t))) for a.e. t ∈ R we find that ΣM M D (t) = 0 ac for a.e. t ∈ δ ∩ D. Similarly we prove that ΣMB (δ ∩ D) = 0 if and only if dD ∗ MB D (t) = 0 for a.e. t ∈ δ ∩ D. (i) Since by assumption dM D (t) dM D (t) for a.e. t ∈ D, one gets by the considerations B ac (δ ∩ D) ≺ Σ ac (δ ∩ D). By Proposition 2.6 we have N ac (t) = d above that ΣM ΣM M D (t) and MB ac ac ac NΣM (t) = dM D (t) for a.e. t ∈ R. Hence NΣM (t) NΣM (t) for a.e. t ∈ D which proves that the B B B ac (· ∩ D) are spectrally subordinated to Σ ac (· ∩ D), cf. Definition 2.3(i). restricted measures ΣM M ac ≈ E ac and Σ ac ≈ E ac , by Proposition 3.1, we get Bthat E ac (· ∩ D) is spectrally Since ΣM A0 MB AB A0 ac (· ∩ D). Applying Theorem 2.5(i) we complete the proof. subordinated to EA B ac (· ∩ D) ∼ Σ ac (· ∩ D). By Proposi(ii) If dM D (t) = dD ∗ MB D (t) for a.e. t ∈ D, then ΣM MB tion 2.6, NΣMac (t) = dM D (t) and NΣMac (t) = dM D (t) for a.e. t ∈ R which implies that the operator B B ac (· ∩ D) and Σ ac (· ∩ D) are spectrally equivalent, cf. Definition 2.3(ii). By Proposimeasures ΣM MB ac (·∩D) and E ac (·∩D) are spectrally equivalent. Applying Theorem 2.5(ii) we prove tion 3.1, EA AB 0 ac (D) and A E ac (D) are unitarily equivalent. 2 that the absolutely continuous parts A0 EA B AB 0

Theorem 3.4 reduces the problem of unitary equivalence of ac-parts of certain self-adjoint extensions of A to investigation of the functions dM D (·) and dM D (·). B

Definition 3.5. Let A be a symmetric operator in H and A0 = A∗0 ∈ ExtA . We say that A0 is ∗ ac-minimal if Aac 0 is a part of any A = A ∈ ExtA . ⊇ σac (A0 ) and NAac (t) NAac (t) for a.e. In particular, if A0 is ac-minimal, then σac (A) 0 ∈ ExtA . Notice that an ac-minimal extension t ∈ R (mod(EAac )) for any self-adjoint extension A of A is not unique. However, the following corollary holds. and A of A are Corollary 3.6. Let A be as in Theorem 3.4. If the self-adjoint extensions A ac-minimal, then their ac-parts are unitarily equivalent. 4. Unitary equivalence 4.1. Preliminaries In what follows we assume that A is a densely defined simple closed symmetric operator in H. = By A0 we denote a self-adjoint extension of A which is fixed. Alongside A0 we consider A ∗ A ∈ ExtA . It is known (see [11]) that there exists a boundary triplet Π := {H, Γ0 , Γ1 } for A∗ such that A0 := A∗ ker(Γ0 ). Of course, the boundary triplet Π is not uniquely determined by

628


the assumption A0 := A∗ ker(Γ0 ). If Π1 and Π2 are two such boundary triplets for A∗ , then their Weyl functions M1 (·) and M2 (·) are related by (2.12) (cf. [11]). Fix a boundary triplet Π := {H, Γ0 , Γ1 } for A∗ such that A0 = A∗ ker(Γ0 ). By Proposi = AΘ . In general, Θ is not such that A tion 2.10 there is a linear relation Θ = Θ ∗ ∈ C(H) the graph of an operator, Θ ∈ / C(H). However, let us assume that Θ is the graph an operator B. By condition (1.1) and Proposition 2.13 we get that (B − i)−1 ∈ S∞ (H), that means, that B is a self-adjoint operator with discrete spectrum. Hence, (B) ∩ R = ∅. In what follows we assume without loss of generality that 0 ∈ (B). According to the polar decomposition we have B −1 = DJ D where D := |B|−1/2 = D ∗ ∈ S∞ (H)

and J := sign(B) = J ∗ = J −1 .

(4.1)

Clearly, D ∈ S∞ (H), ker(D) = {0}, and D commutes with J . We set G(z) := J − M D (z),

z ∈ C+ ,

(4.2)

M D (z) := DM(z)D, z ∈ C+ , as usually. Obviously, M D (z) and −G(z) are R-functions. Moreover, ker(G(z)) = {0} for any z ∈ C+ . Indeed, if G(z)f = 0, then Jf = DM(z)Df . Hence, Im(M(z)Df, Df ) = Im(Jf, f ) = 0 which yields Df = 0 or f = 0. Since J is a Fredholm operator satisfying ker(J ) = ker(J ∗ ) = {0} we find by [20, Theorem 5.26] that G(z) is boundedly invertible for z ∈ C+ . We set T (z) := G(z)−1 , z ∈ C+ and note that T (·) is a Nevanlinna function because so is M D (·). Moreover, T (z) − J = T (z)M D (z)J ∈ S∞ (H) for z ∈ C+ . 4.2. Trace class perturbations: Rosenblum–Kato theorem Here we apply the Weyl function technique in order to obtain a simple and quite different proof of the classical Rosenblum–Kato theorem. In fact, we prove a generalization of the Rosenblum–Kato theorem due to Birman and Krein [6] which includes non-additive (trace class) perturbations. Our proof demonstrates the main idea of the proof of more general results contained in the next subsection. be self-adjoint operators in H satisfying Theorem 4.1. Let A0 and A − i)−1 − (A0 − i)−1 ∈ S1 (H). (A

(4.3)

ac and Aac of A and A0 , respectively, are unitarily equivThen the absolutely continuous parts A 0 alent. ac and Aac in the framework of extension theory we set Proof. To include the operators A 0 A := A0 dom(A),

∩ dom(A0 ): A0 f = Af . dom(A) = f ∈ dom(A)

dom(A). Clearly, A is a closed symmetric operator in H with equal Obviously, we have A := A ∈ ExtA . deficiency indices and A0 , A First we assume that A is densely defined. Let Π = {H, Γ0 , Γ1 } be an (ordinary) boundary triplet for A∗ , such that A0 := A∗ ker(Γ0 ), and M(·) the corresponding Weyl function. By and A0 are disjoint, that is, dom(A) = dom(A0 ) ∩ dom(A). = A ∗ ∈ ExtA and A definition, A ∗ Hence, by Proposition 2.10(ii), there exists an operator B = B ∈ C(H) such that A = AB .


629

It follows from (2.14) and (4.3) that MB (z) := (B − M(z))−1 ∈ S1 (H) for z ∈ C+ . In accordance with [5, Lemma 2.4], see also [32], the limits MB (t) := limy→+0 MB (t + iy) exist in S2 (H), for a.e. t ∈ R. By Theorem 3.4 it suffices to calculate the multiplicity function dMB (t) := rank(MB (t)) = dim(ran(Im(MB (t)))). It follows from (4.1) and (4.2) that −1 −1 T (z) = G(z)−1 = J − M D (z) = J − DM(z)D −1 −1 = D −1 D −1 J D −1 − M(z) D −1 = |B|1/2 B − M(z) |B|1/2 ,

(4.4)

z ∈ C+ . Combining this relation with (4.1) yields −1 = DT (z)D, MB (z) := B − M(z) In turn, this equality implies Im MB (z) = DT (z)∗ Im M D (z) T (z)D,

z ∈ C+ .

z ∈ C+ .

(4.5)

Moreover, since M D (z) ∈ S1 (H) and T (z) − J ∈ S1 (H) for z ∈ C+ , by [5, Lemma 2.4] (see also [32]), for a.e. t ∈ R and y → 0 there exist the limits M D (t) and T (t) in S2 (H)-norm of the RH -functions M D ((t + iy)) and T (t + iy), respectively. Therefore passing to the limit in (4.5) as y → 0 we get (4.6) Im MB (t) = DT (t)∗ Im M D (t) T (t)D for a.e. t ∈ R. Therefore we find dMB (t) = dim ran Im MB (t) = dim ran Im M D (t) T (t)D . = dim ran Im MB (t)

(4.7)

Since (J − M D (t))T (t) = T (t)(J − M D (t)) = I for a.e. t ∈ R, we find ran(T (t)) = H for a.e. t ∈ R. Combining this relation with ran(D) = H and (4.7) we obtain dMB (t) = dim ran Im M D (t) = dim ran Im M D (t) = dM D (t) (4.8) for a.e. t ∈ R. Applying Theorem 3.4(ii) we complete this part of the proof. If A is not densely defined one can repeat the above reasonings applying only the boundary triplet technique for non-densely defined symmetric operators developed in [12,26]. It turns out that the proof above can easily be carried over to this case. 2 ∈ ExtA In the following corollary we show that in proving of unitary equivalence of A0 and A it suffices to restrict the consideration to the case of disjoint extensions. Corollary 4.2. Let A be a densely defined closed symmetric operator in H, let Π = {H, Γ0 , Γ1 } be an ordinary boundary triplet for A∗ , and let M(·) be the corresponding Weyl function. Let also A0 := A∗ ker(Γ0 ) and D ∈ B(R).

630


ac (D) for any extension A = A ∗ ∈ ExtA disjoint with A0 , (i) If Aac 0 EA0 (D) is a part of A EA ac ac ∗ ∈ ExtA . then A0 EA0 (D) is a part of A EA(D) for any extension A = A ac ac = A ∗ ∈ ExtA dis(ii) If A0 EA0 (D) is unitarily equivalent to A EA(D) for any extension A ac joint with A0 , then A0 EA0 (D) is unitarily equivalent to the absolutely continuous part = A ∗ ∈ ExtA . ac EA(D) of any extension A A ∈ ExtA which is not disjoint with A0 admits a Proof. By Proposition 2.10 an extension A Θ with Θ = Θ ∗ ∈ C(H) representation A \ C(H). However, Θ admits a decomposition H = ∗ ∈ C(H ) Hop ⊕ H∞ , Θ = Θop ⊕ Θ∞ where Θop is the graph of the operator Bop = Bop op (cf. Section 2). Denoting by πop the orthogonal projection from H onto Hop and Mop (z) := πop M(z) Hop , we get (Θ − M(z))−1 = (Bop − Mop (z))−1 πop . Therefore formula (2.14) takes the form −1 (AΘ − z)−1 − (A0 − z)−1 = γ (z) Bop − Mop (z) πop γ (z)∗ ,

z ∈ C± .

∗ ∈ C(H ) such that (B −1 ∈ S (H ) and put B = Choose an operator B∞ = B∞ ∞ ∞ − i) 1 ∞ Bop ⊕ B∞ . It follows from Proposition 2.13 that

(AΘ − z)−1 − (AB − z)−1 ∈ S1 (H), ac since (B∞ − i)−1 ∈ S1 (H∞ ). By Theorem 4.1 the absolutely continuous parts Aac Θ and AB of AΘ and AB , respectively, are unitarily equivalent. ac ac (i) Since by assumption Aac 0 EA0 (D) is a part of AB EAB (D) and AB is unitarily equivalent ac ac ac to AΘ we get that A0 EA0 (D) is a part of AΘ EAΘ (D). ac ac (ii) Since, by assumption, Aac 0 EA0 (D) is unitarily equivalent to AB EAB (D) and AB is uniac ac tarily equivalent to AΘ , we get that A0 EA0 (D) is unitarily equivalent to AΘ EAΘ (D). 2

4.3. Compact non-additive perturbations Here we generalize the Rosenblum–Kato theorem for the case of compact perturbations. To this end we assume that the maximal normal function m+ (t) := sup M(t + iy) 0 0, the limit Ξ (t) := o-limy→+0 Ξ (t + iy) exists in the operator norm and the following representation holds −1 D (t) . Ξ (t) = J1 − M11

(4.11)

D (z) = J (I − J M D (z)). Using (4.10) we get J M D (t) < 1 for First we note that J1 − M11 1 1 1 11 1 11 D (t))−1 exists for t ∈ δ . Using (J − M D (t))−1 = t ∈ δa . Hence the inverse operator (I1 − J1 M11 a 1 11 D (t))−1 J we find that the inverse operator (J − M D (t))−1 exists for t ∈ δ . Since (I1 − J1 M11 1 1 a 11 D (z) has limits M D (t) for a.e. t ∈ R one gets that J M D (t) = o-lim D (t + iy) for J M M11 1 y→+0 1 11 11 11 a.e. t ∈ R. Fix any such t0 ∈ δa . Then due to estimate (4.10) there exists η = η(t0 ) such that D (t + iy) 1/2. Therefore, the family {(I − J M D (t + iy))−1 } supy∈(0,η) J1 M11 0 1 1 11 0 y∈(0,η) is uniformly bounded for any fixed t0 ∈ δa . Using this fact and (4.9) we can pass to the limit as y → 0 in the identity

−1 −1 D D I1 − J1 M11 (t0 + iy) − I1 − J1 M11 (t0 ) −1 −1 D D D D J1 M11 = I1 − J1 M11 (t0 + iy) (t0 + iy) − J1 M11 (t0 ) I1 − J1 M11 (t0 ) . D (t + iy))−1 = (I − J M D (t))−1 for a.e. t ∈ δ which yields We obtain o-limy→+0 (I1 − J1 M11 1 1 11 a the existence of Ξ (t) := o-limy→+0 Ξ (t + iy) and proves representation (4.11). (ii)3 Next we set

−1 D D D D (z) := M22 (z) + M21 (z) J1 − M11 (z) M12 (z),

z ∈ C+

and show that the function T2 (·) := (J2 − (·))−1 is RH2 -function. Clearly, (·) is holomorphic in C+ and it acts in a finite dimensional Hilbert space H2 . Since det(J2 − (·)) is also holomorphic in C+ , the determinant det(J2 − (·)) has only a discrete set of zeros in C+ . Hence the inverse operator T2 (·) := (J2 − (·))−1 exists for z ∈ Ω ⊂ C+ where C+ \ Ω is at most countable discrete set, that is, T2 (·) is meromorphic in C+ . As we just mentioned the inverse operator (J2 − (z))−1 exists for z ∈ Ω ⊂ C+ . Choose any z ∈ Ω. Then, by the Frobenius formula, −1 T (z) := J − M D (z) =

T1 (z)

D (z)T (z) Ξ (z)M12 2

D (z)Ξ (z) T2 (z)M21

T2 (z)

(4.12)


633

where D D T1 (z) := Ξ (z) + Ξ (z)M12 (z)T2 (z)M21 (z)Ξ (z).

(4.13)

Hence T2 (z) = P2 T (z) H2 ,

z ∈ Ω.

Since T (·) is an RH -function, we get that Im(T2 (z)) > 0 for z ∈ Ω. Since in addition T2 (·) is meromorphic in C+ , we conclude that it is holomorphic. Thus, T2 (·) = (J2 − (·))−1 is RH2 function, too. (ii)4 In this step we show that for any a > 0 the limit T (t) := o-limy→+0 T (t + iy) exists in the operator norm for a.e. t ∈ δa . Since T2 (·) is the matrix RH2 -function, the limit T2 (t) = o-limy→+0 T2 (t + iy) exists for a.e. t ∈ R. Besides, (4.9) yields D D lim M12 (t + iy) − M12 (t) = 0

y→+0

and

D D lim M21 (t + iy) − M21 (t) = 0

y→+0

for a.e. t ∈ R. Combining these relations with (4.11) and (4.13) yields the existence of the limit T1 (t) := o-limy→+0 T1 (t + iy) for a.e. t ∈ δa . Finally, combining all these relations with the block-matrix representation (4.12) we complete the proof of (ii). (iii) Using the results of (ii) we are now going to complete the proof of the theorem. We set δn := {t ∈ R: m+ (t) n} and note that n∈N δn differs from R by a setof Lebesgue measure zero. By step (ii) the limit T (t) := o-limy→+0 T (t + iy) exists for a.e. t ∈ n∈N δn in the operator norm. Hence the limit T (t) := o-limy→+0 T (t + iy) exists for a.e. t ∈ R. Combining this fact with (4.9) we can pass to the limit in the identity (J − M D (t + iy))T (t + iy) = I as y → 0. We get J − M D (t) T (t) = T (t) J − M D (t) = I

for a.e. t ∈ R.

(4.14)

is disjoint The rest of the proof is similar to that of Theorem 4.1. First we assume that A with A0 , hence, it admits a representation A = AB with B ∈ C(H). Therefore, setting MB (·) := (B − M(·))−1 and assuming without loss of generality that 0 ∈ (B) we arrive at the representation (4.6) with D = |B|−1/2 for a.e. t ∈ R. Moreover, (4.14) yields ran(T (t)) = H for a.e. t ∈ R. Therefore arguing as in (4.7) and (4.8) we obtain = dim ran Im M(t) D dMB (t) = dim ran Im M D (t) = dim ran Im M(t) = dim ran Im M(t) = dM (t) for a.e. t ∈ R. Applying Theorem 3.4(ii) we complete the proof. not disjoint with A0 . Finally, we apply Corollary 4.2 to extend the proof for extensions A

2

Remark 4.4. Note that in passing we proved the following “individual” version of Theorem 4.3. = A ∗ = A B (∈ ExtA ) satisfies conditions (1.1) and (4.9) with D = |B|−1/2 , If the extension A and A0 , respectively, are unitarily equivac and Aac of A then the absolutely continuous parts A 0 alent.

634


This observation shows that the classical Kato–Rosenblum theorem, as well as its generalization, Theorem 4.1, is implied by Theorem 4.3. Indeed, the condition (4.3) is equivalent to D ∈ S2 , hence the limit (4.9) exists even in S2 -norm (cf. [5,15]). However, we presented the direct proof of Theorem 4.1 because of its simplicity. Remark 4.5. Theorem 4.3 as well as its proof remains valid if A is non-densely defined. In this case it suffices to use the boundary triplet technique for non-densely defined operators developed in [12,26], cf. proof of Theorem 4.1. However, the assumptions on the Weyl function are indispensable. The following local version of Theorem 4.3 is implied by combining Theorem 3.4(ii) with the proof of Theorem 4.3. Corollary 4.6. Let the assumptions of Theorem 4.3 be satisfied and let

F := t ∈ R: m+ (t) < ∞ .

(4.15)

and A0 , respectively, ac EAac (F ) and Aac EAac (F ) of A If condition (1.1) holds, then the parts A 0 0 are unitarily equivalent. Remark 4.7. Let us define the invariant maximal normal function −1/2 −1/2 , M(t + iy) − Re M(i) Im M(i) m+ (t) := sup Im M(i)

(4.16)

y∈(0,1]

for t ∈ R. For Weyl functions one easily proves that m+ (t) is finite if and only if m+ (t) is finite. (i) The quantity m+ (t) has the advantage that it is invariant: Let A be a densely defined closed symmetric operator, Π = {H, Γ0 , Γ1 } a boundary triplet for A∗ , and M(·) the corresponding Γ0 , Γ1 } be another boundary triplet for A∗ with the Weyl = {H, Weyl function. Further, let Π are related and let A0 := A∗ ker(Γ0 ) = A∗ ker(Γ0 ). In this case M(·) and M(·) function M(·) + (t) = m+ (t) for t ∈ R, where m+ (t) is obtained by replacing in (4.16) by (2.12) However, m M(·) by M(·). (ii) Further, if the Weyl function M(·) satisfies M(i) = i, then m+ (t) = m+ (t) for t ∈ R. of H. If m+ (t) is finite, then (iii) Let π be an orthogonal projection onto a subspace H + := (t), obtained from (4.16) replacing M(·) by M(·) the invariant maximal normal function m is also finite and satisfies m + (t) m+ (t) for t ∈ R. πM(·) H, 5. Concluding remarks Here we demonstrate that condition (1.3) might have much stronger conclusions than Theorem 1.2. For this purpose we complete Definition 3.5. Definition 5.1. Let A be a symmetric operator in H, A0 = A∗0 ∈ ExtA and σ0 := σac (A0 ). We say = A ∗ ∈ ExtA the ac-part A ac EA(σ0 ) of AE A(σ0 ) is that A0 is strictly ac-minimal if for any A ac unitarily equivalent to A0 . In [28] we applied Theorem 1.2 as well as technique elaborated in this paper to direct sums A := ∞ n=1 Sn of closed symmetric operators Sn with finite deficiency indices. It turns out that in


635

this case for a suitable boundary triplet Π for A∗ the corresponding extension A0 is ac-minimal provided that condition (1.3) is satisfied, cf. [28, Theorem 5.12]. Moreover, if the symmetric operators Sn are mutually unitary equivalent, then for a suitable boundary triplet Π for A∗ the extension A0 is actually strictly ac-minimal. Moreover, in [28] the Sturm–Liouville operator (Af )(x) = −f (x) + Tf (x) with non-negative unbounded operator potential T was considered. It is shown in [28] that condition (1.3) is satisfied for the Weyl function of the pair {A, AF } and, by [28, Theorem 6.11(ii)], for the Friedrichs extension AF =: A0 is ac-minimal. In particular, this yields σac (AF ) ⊆ σac (A) F any A ∈ ExtA , i.e. σac (A ) is stable under non-additive perturbations preserving the class ExtA . (t) holds for the spectral multiplicity functions. MoreIn this case the inequality NAac (t) NAac 0 over, if inf σess (T ) = inf σ (T ), then both AF and the Krein extension AK are strictly ac-minimal, cf. [28, Corollary 6.12]. Finally, in [28] we apply the above mentioned results for the investigation of self-adjoint realizations of partial differential expressions of the form

∂2 ∂2 + L=− ∂t 2 ∂xj2 n

(t, x) ∈ R+ × Rn , 0 q = q ∈ L∞ Rn ,

+ q(x),

j =1

in the half-space R+ × Rn . Let L := Lmin be the minimal symmetric operator associated with the differential expression L in H := L2 (R+ × Rn ). Denote also by LD , LN and LK the Dirichlet, Neumann and Krein realizations of L (extensions of L), respectively. Note that the realizations LD and LN are always self-adjoint (cf. [25, Theorem 2.8.1], [18]). Theorem 5.2. Let q(·) ∈ L∞ (R), q(·) 0, and lim

|x|→∞ |x−y|1

q(y) dy = 0.

= L ∗ ∈ ExtL . Then: Let also L (i) The realizations LD and LN are absolutely continuous, LD = (LD )ac and LN = (LN )ac . − i)−1 − ac and LD are unitarily equivalent provided that either the condition (L (ii) L D −1 −1 K −1 (L − i) ∈ S∞ or (L − i) − (L − i) ∈ S∞ is satisfied. (iii) The realizations LD , LN and LK are strictly ac-minimal, σ LD = σac LD = σac LK = σ LN = σac LN = [0, ∞), and NLD (t) = NLN (t) = N(LK )ac (t) = ∞ for a.e. t ∈ [0, ∞). The proof is contained in our preprint [28]. Note only that, since condition (1.3) is now satisfied, the statement (ii) follows from Theorem 1.2.

636


Acknowledgment The first author thanks Weierstrass Institute of Applied Analysis and Stochastics in Berlin for financial support and hospitality. Appendix A. Absolutely continuous closure The concept of the ac-closure has been introduced in [8] (see also [14]). Its properties can also be found in [8,14]. Here we recall some basic facts on the ac-closure of a Borel subset of R that were used in Section 3. Definition A.1. (See [8].) Let δ ∈ B(R). The set clac (δ) defined by

clac (δ) := x ∈ R: (x − ε, x + ε) ∩ δ > 0 ∀ε > 0 is called the absolutely continuous closure of the Borel set δ ∈ B(R). Obviously, two Borel sets δ1 , δ2 ∈ B(R) have the same ac-closure if their symmetric difference δ1 δ2 has Lebesgue measure zero. Moreover, the set clac (δ) is always closed and clac (δ) ⊆ δ. In particular, if we have two measurable non-negative functions ξ1 and ξ2 which differ only on a set of Lebesgue measure zero, then clac (supp(ξ1 )) = clac (supp(ξ2 )). Lemma A.2. If δ ∈ B(R), then |δ \ clac (δ)| = 0. Proof. Since clac (δ) is closed the set := R \ clac (δ) is open. The open set is decomposed as = L l=1 l , 1 L ∞, where l = (al , bl ) are disjoint open intervals. We set l = δ ∩ l , l = 1, 2, . . . , L. Obviously, δ \ clac (δ) = δ ∩ =

L

l .

l=1

We note that l ∩ clac (δ) = ∅, l = 1, 2, . . . , L. Hence for each t ∈ l there is a sufficiently small neighborhood Ot such that |Ot ∩ δ| = 0. If η is sufficiently small, then [al + η, al − η] ⊆ (al , bl ) and {Ot }t∈l forms a covering of [al + η, al − η]. Since [al + η, al − η] is compact we can M chosen a finite covering {Otm }M m=1 Otm we find m=1 of [al + η, al − η]. By [al + η, al − η] ⊆ |[al + η, al − η] ∩ δ| = 0 for each sufficiently small η > 0. Using that we get (al , bl ) ∩ δ = (al , al + η) ∩ δ + (bl − η, bl ) ∩ δ = (al , al + η) ∩ δ + (bl − η, bl ) ∩ δ 2η for sufficiently small η > 0. Hence |l | = |(al , bl )∩δ| = 0 which yields that |δ \clac (δ)| = 0.

2

Lemma A.3. If {δk }k∈N , δk ⊆ R, is a sequence of Borel subsets, then clac (δ) =

k∈N

clac (δk ),

δ=

k∈N

δk .

(A.1)


637

δk Proof. Weset δk = δk ∩ clac (δk ) and k := δk \ clac (δk ). We have δ = δ ∪ , where δ := k∈N and := k∈N k . By Lemma A.2, |k | = 0, k ∈ N, which yields || = 0. Hence clac (δ) = δ ). Similarly one gets clac (δk ) = clac ( δk ), k ∈ N. Notice that δk ⊆ clac ( δk ), k ∈ N. We have clac ( δ)⊇ clac (

clac ( δk ) ⊇

k∈N

δk = δ.

k∈N

Hence δ ) = clac ( δ)⊇ clac (

δ ⊇ clac ( clac ( δk ) ⊇ δ)

k∈N

which yields clac ( δ) = we prove (A.1). 2

k∈N clac ( δk ).

Since clac ( δ ) = clac (δ) and clac ( δk ) = clac (δk ), k ∈ N,

References [1] N.I. Achieser, I.M. Glasmann, Theorie der linearen Operatoren im Hilbert–Raum, eighth ed., Verlag Harri Deutsch, Thun, 1981. [2] S. Albeverio, J.F. Brasche, M.M. Malamud, H. Neidhardt, Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions, J. Funct. Anal. 228 (1) (2005) 144–188. [3] Yu.M. Berezanski˘ı, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr., vol. 17, American Mathematical Society, Providence, RI, 1968, translated from the Russian by R. Bolstein, J.M. Danskin, J. Rovnyak and L. Shulman. [4] M.Š. Birman, Existence conditions for wave operators, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963) 883–906. [5] M.Š. Birman, S.B. Èntina, Stationary approach in abstract scattering theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967) 401–430. [6] M.Š. Birman, M.G. Kre˘ın, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962) 475–478. [7] M.Š. Birman, M.Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Math. Appl. (Soviet Ser.), D. Reidel Publishing Co., Dordrecht, 1987. [8] J.F. Brasche, M.M. Malamud, H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions, Integral Equations Operator Theory 43 (3) (2002) 264–289. [9] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo, Generalized resolvents of symmetric operators and admissibility, Methods Funct. Anal. Topology 6 (3) (2000) 24–55. [10] V.A. Derkach, M.M. Malamud, On the Weyl function and Hermite operators with lacunae, Dokl. Akad. Nauk SSSR 293 (5) (1987) 1041–1046. [11] V.A. Derkach, M.M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1) (1991) 1–95. [12] V.A. Derkach, M.M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (2) (1995) 141–242, Analysis 3. [13] A. Dijksma, H.S.V. de Snoo, Symmetric and selfadjoint relations in Kre˘ın spaces. I, in: Oper. Theory Adv. Appl., vol. 24, Birkhäuser, Basel, 1987, pp. 145–166. [14] F. Gesztesy, K.A. Makarov, M. Zinchenko, Essential closures and AC spectra for reflectionless CMV, Jacobi, and Schrödinger operators revisited, Acta Appl. Math. 103 (3) (2008) 315–339. [15] Ju.P. Ginzburg, Multiplicative representations of bounded analytic operator-functions, Dokl. Akad. Nauk SSSR 170 (1966) 23–26. [16] I.C. Gohberg, M.G. Kre˘ın, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, Moscow, 1965. [17] V.I. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Math. Appl. (Soviet Ser.), vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1991. [18] G. Grubb, Distributions and Operators, Grad. Texts in Math., vol. 252, Springer, New York, 2009. [19] T. Kato, Perturbation of continuous spectra by trace class operators, Proc. Japan Acad. 33 (1957) 260–264.

638


[20] T. Kato, Perturbation Theory for Linear Operators, second ed., Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin, 1976. [21] M.G. Kre˘ın, G.K. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space Πκ , Funktsional. Anal. i Prilozhen. 5 (3) (1971) 54–69. [22] M.G. Kre˘ın, A.A. Nudel’man, The Markov Moment Problem and Extremal Problems, American Mathematical Society, Providence, RI, 1977. [23] S.T. Kuroda, On a theorem of Weyl–von Neumann, Proc. Japan Acad. 34 (1958) 11–15. [24] S.T. Kuroda, Diagonalization modulo norm ideals; spectral method and modulus of continuity, RIMS Kôyûroku Bessatsu B16 (2010) 101–126. [25] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss., vol. 181, Springer-Verlag, New York, 1972. [26] M.M. Malamud, On a formula for the generalized resolvents of a non-densely defined Hermitian operator, Ukraïn. Mat. Zh. 44 (12) (1992) 1658–1688. [27] M.M. Malamud, S.M. Malamud, Spectral theory of operator measures in a Hilbert space, Algebra i Analiz 15 (3) (2003) 1–77. [28] M.M. Malamud, H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, Preprint, arXiv:0907.0650v1 [math-ph], 2009. [29] M.M. Malamud, H. Neidhardt, On the Kato–Rosenblum and the Weyl–von Neumann theorems, Dokl. Akad. Nauk 432 (2) (2010) 162–166. [30] M. Rosenblum, Perturbation of the continuous spectrum and unitary equivalence, Pacific J. Math. 7 (1957) 997– 1010. [31] J. von Neumann, Charakterisierung des Spektrums eines Integraloperators, Actualités Sci. Indust. 229 (1935) 33–55. [32] D.R. Yafaev, Mathematical Scattering Theory, Transl. Math. Monogr., vol. 105, American Mathematical Society, Providence, RI, 1992.


Commutators and localization on the Drury–Arveson space Quanlei Fang, Jingbo Xia ∗ Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 23 October 2009; accepted 19 October 2010 Available online 27 October 2010 Communicated by D. Voiculescu

Abstract Let f be a multiplier for the Drury–Arveson space Hn2 of the unit ball, and let ζ1 , . . . , ζn denote the coordinate functions. We show that for each 1 i n, the commutator [Mf∗ , Mζi ] belongs to the Schatten class Cp , p > 2n. This leads to a localization result for multipliers. © 2010 Elsevier Inc. All rights reserved. Keywords: Multiplier; Drury–Arveson space

1. Introduction Let B denote the open unit ball {z: |z| < 1} in Cn . Throughout the paper, the complex dimension n is assumed to be greater than or equal to 2. A multivariable analogue of the classical Hardy space of the unit circle is the Drury–Arveson space Hn2 on B [3,9]. Because of its close relation to a number of topics in operator theory, among which we mention the von Neumann inequality for commuting row contractions, Hn2 has been the subject of intense study of late [2–7,10,12,13]. The space Hn2 is a reproducing kernel Hilbert space with the kernel K(z, w) =

1 , 1 − z, w

z, w ∈ B,


E-mail addresses: [email protected] (Q. Fang), [email protected] (J. Xia). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.013

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Q. Fang, J. Xia / Journal of Functional Analysis 260 (2011) 639–673

which is a multivariable generalization of the one-variable Szegö kernel. An orthonormal basis of Hn2 is given by {eα : α ∈ Zn+ }, where eα (ζ ) =

|α|! α ζ . α!

In this paper we use the standard multi-index notation: For α = (α1 , . . . , αn ) ∈ Zn+ , α! = α1 !α2 ! · · · αn !,

ζ α = ζ1α1 · · · ζnαn .

|α| = α1 + · · · + αn ,

For functions f, g ∈ Hn2 with Taylor expansions f (ζ ) =

cα ζ α

and g(ζ ) =

α∈Zn+

dα ζ α ,

α∈Zn+

the inner product is given by f, g =

α! cα dα . n |α|!

α∈Z+

Throughout the paper, we let Mζ1 , . . . , Mζn denote the operators of multiplication by the coordinate functions ζ1 , . . . , ζn on Hn2 . With the identification of each ζi with each Mζi , Hn2 is often called the Drury–Arveson module over the polynomial ring C[ζ1 , . . . , ζn ]. A holomorphic function f on B is called a multiplier for the space Hn2 if f Hn2 ⊂ Hn2 . If f is a multiplier, then the multiplication operator Mf defined by Mf (g) = f g is necessarily bounded on Hn2 [3], and the multiplier norm of f is defined to be the operator norm of Mf . In [3], Arveson showed that, when n 2, the collection of multipliers of Hn2 is strictly smaller than H ∞ . On Hn2 , multipliers can be used to express orthogonal projections. Suppose that E is a submodule of the Drury–Arveson module, i.e., E is a closed linear subspace of Hn2 which is invariant under Mζ1 , . . . , Mζn . Then there exist multipliers {f1 , . . . , fk , . . .} of Hn2 such that the operator Mf1 Mf∗1 + · · · + Mfk Mf∗k + · · · is the orthogonal projection from Hn2 onto E (see p. 191 in [4]). Among the recent results related to multipliers, we would like to mention the following developments. Interpolation problems for multipliers and model theory related to the Drury–Arveson space also have been intensely studied over the past decade or so [5,6,10,12,13]. Recently, Arcozzi, Rochberg and Sawyer gave a characterization of the multipliers in terms of Carleson measures for Hn2 [2]. In another study, Costea, Sawyer and Wick [7] proved a corona theorem for the Drury–Arveson space multipliers. Since Hn2 is a natural analogue of the Hardy space, it is natural to take a list of Hardy-space results and try to determine which ones have analogues on Hn2 and which ones do not. Commutators are certainly very high on any such list. One prominent part of the theory of the Hardy space is the Toeplitz operators on it. Since there is no L2 associated with Hn2 , the only analogue of Toeplitz operators on Hn2 are the multipliers. In this paper we are interested in the commutators of the form [Mf∗ , Mζi ], where f is a multiplier for the Drury–Arveson space. Since the story


641

about the commutators of the form [Mf∗ , Mζi ] is well known on the Hardy space, one would certainly like to know the analogous story on Hn2 . Recall that for each 1 p < ∞, the Schatten class Cp consists of operators A satisfying the condition Ap < ∞, where the p-norm is given by the formula p/2 1/p . Ap = tr A∗ A Arveson showed in his seminal paper [3] that commutators of the form [Mζ∗j , Mζi ] all belong to Cp , p > n. As the logical next step, one certainly expects a Schatten class result for commutators on Hn2 involving multipliers other than the simplest coordinate functions. The following is the main result of the paper: Theorem 1.1. Let f be a multiplier for the Drury–Arveson space Hn2 . For each 1 i n, the commutator [Mf∗ , Mζi ] belongs to the Schatten class Cp , p > 2n. Moreover, for each 2n < p < ∞, there is a constant C which depends only on p and n such that ∗

M , Mζ CMf i f p for every multiplier f of Hn2 and every 1 i n. This Schatten-class result has C ∗ -algebraic implications. Throughout the paper, we denote the unit sphere {z ∈ Cn : |z| = 1} in Cn by S. Let Tn be the C ∗ -algebra generated by Mζ1 , . . . , Mζn on Hn2 . Recall that Tn was introduced by Arveson in [3]. In more ways than one, Tn is the analogue of the C ∗ -algebra generated by Toeplitz operators with continuous symbols. Indeed Arveson showed that there is an exact sequence τ

{0} → K → Tn −→ C(S) → {0},

(1.1)

where K is the collection of compact operators on Hn2 . But there is another natural C ∗ -algebra on Hn2 which is also related to “Toeplitz operators”, where the symbols are not necessarily continuous. We define T Mn = the C ∗ -algebra generated by Mf : f Hn2 ⊂ Hn2 . Theorem 1.1 tells us that Tn is contained in the essential center of T Mn , in analogy with the classic situation on the Hardy space of the unit sphere S. This opens the door for us to use the classic localization technique [8] to analyze multipliers. Recall that the essential norm of a bounded operator A on a Hilbert space H is AQ = inf A + K: K is compact on H . Alternately, AQ = π(A), where π denotes the quotient map from B(H) to the Calkin algebra Q = B(H)/K(H). To state our localization result, we need to introduce a class of Schur multipliers. For each z ∈ B, let sz (ζ ) =

1 − |z| . 1 − ζ, z

(1.2)

642


The reason we call sz a Schur multiplier is that the norm of the operator Msz on Hn2 is 1, as we will see in Section 2. Using Theorem 1.1, we will prove Theorem 1.2. Let A ∈ T Mn . Then for each ξ ∈ S, the limit lim AMsrξ r↑1

(1.3)

exists. Moreover, we have AQ = sup lim AMsrξ . ξ ∈S r↑1

The C ∗ -algebraic meaning of the “localized limit” (1.3) will be explained in Section 6. Alternately, we can state Theorem 1.2 in a version which may be better suited for applications: Theorem 1.3. For each A ∈ T Mn , we have AQ = lim sup AMsz . r↑1 r|z| 0. By Proposition 5.1.4 in [14], there is a constant A0 ∈ (2−n , ∞) such that 2−n r 2n σ B(x, r) A0 r 2n

(4.1)

√ for all x ∈ S and 0 < r 2. Note that the upper bound actually holds for all r > 0. Before getting to the main estimates of the section, let us recall: Lemma 4.1. (See Lemma 4.1 in [15].) Let X be a set and let E be a subset of X × X. Suppose that m is a natural number such that card y ∈ X: (x, y) ∈ E m and

card y ∈ X: (y, x) ∈ E m

for every x ∈ X. Then there exist pairwise disjoint subsets E1 , E2 , . . . , E2m of E such that E = E1 ∪ E2 ∪ · · · ∪ E2m and such that for each 1 j 2m, the conditions (x, y), (x , y ) ∈ Ej and (x, y) = (x , y ) imply both x = x and y = y . For each z ∈ B, define the functions

uz (ζ ) = mn+3 z (ζ ) =

1 − |z|2 1 − ζ, z

n+3 and vz (ζ ) = mn+4 z (ζ ) =

1 − |z|2 1 − ζ, z

n+4 .

(4.2)

656


The proofs of our next three lemmas have much in common. More specifically, they all use a counting argument based on Lemma 4.1. However, because the estimates involved vary in details, it is difficult to reduce them to one. Therefore we present all three proofs. It should be pretty clear from Lemma 2.1 that Msz = 1 for each z ∈ B. Therefore Mmz = 1 + |z|. This fact will be used several times in this section. Lemma 4.2. Let 2n < p < ∞. Suppose that 0 < t < 1 and that {ξj : j ∈ J } is a subset of S satisfying the condition B(ξi , t) ∩ B(ξj , t) = ∅ for all i = j.

(4.3)

Define zj = (1 − t 2 )1/2 ξj , j ∈ J . Let {fj : j ∈ J } be a set of vectors in Hn2 with norm at most 1, and let {ej : j ∈ J } be an orthonormal set. For each ν ∈ {1, . . . , n}, define the operator Eν =

Mζ∗ν −(zj )ν vzj fj ⊗ ej , j ∈J

where (zj )ν denotes the ν-th component of zj . Then there exists a constant C4.2 (p) depending only on p and n such that Eν p C4.2 (p)t 1−(2n/p) . Proof. Let ν ∈ {1, . . . , n} be given. By Lemma 2.1, Mζν has a normal extension. More precisely, there is a Hilbert space Lν containing Hn2 and a normal operator Mν on Lν such that Mν h = Mζν h,

for each h ∈ Hn2 .

(4.4)

Let Pν : Lν → Hn2 be the orthogonal projection. Define the operator E˜ ν =

Mν∗ − (zj )ν vzj fj ⊗ ej . j ∈J

Since Mζ∗ν = Pν Mν∗ |Hn2 , we have Eν = Pν E˜ ν . Thus it suffices to estimate E˜ ν p . For the convenience of the reader, we will denote the inner product and the norm on Lν by ·,·Lν and · Lν respectively, whereas those on the subspace Hn2 will still be denoted by ·,· and · . We have E˜ ν∗ E˜ ν =

∞ Mν∗ − (zj )ν vzj fj , Mν∗ − (zi )ν vzi fi L ei ⊗ ej = B + Yk , ν

i,j ∈J

k=0

where B=

M ∗ − (zj )ν vz fj 2 ej ⊗ ej ν j L j ∈J

ν

(4.5)


657

and

Yk =

∗ Mν − (zj )ν vzj fj , Mν∗ − (zi )ν vzi fi L ei ⊗ ej , ν

2k td(ξi ,ξj ) 0 be given. By (6.1), there exist T1 , . . . , Tn ∈ Tn and a K ∈ K such that B − (T1 Mζ

1 −ξ1

+ · · · + Tn Mζn −ξn + K) δ.

The conclusion of the lemma follows from (6.6) and (6.7).

(6.7)

2

Proposition 6.2. For every A ∈ T Mn and every ξ ∈ S, we have lim AMsrξ = A + Iξ . r↑1

Proof. Let ξ ∈ S be given. We first show that lim W Msrξ = 0 r↑1

(6.8)

for every W ∈ Iξ . Applying Lemma 2.5, we have lim M(ζj −ξj ) Msrξ = lim M(ζj −ξj )srξ r↑1

r↑1

lim M(ζj −rξj )srξ + |rξj − ξj |Msrξ = 0 r↑1

(6.9)

for each j ∈ {1, . . . , n}, where ξj is the j -th component of ξ . Let K be a compact operator. Then Corollary 2.2 gives us KMsrξ = Ms∗rξ K ∗ Msrξ K ∗ . Since K ∗ is also compact, it follows from Lemma 2.6 that lim KMsrξ limMsrξ K ∗ = 0. r↑1

r↑1

Combining (6.9), (6.10) and Lemma 6.1, (6.8) is proved. Let A ∈ T Mn be given. Then by (6.8), for every W ∈ Iξ we have lim sup AMsrξ = lim sup(A + W )Msrξ A + W . r↑1

r↑1

(6.10)

672


Since this holds for every W ∈ Iξ , it follows that lim sup AMsrξ A + Iξ .

(6.11)

AMsrξ A + Iξ

(6.12)

r↑1

Next we show that

for every 0 < r < 1. Note that, since |ξ | = 1, 1 − srξ (ζ ) = 1 −

rξ − ζ, ξ 1−r = . 1 − rζ, ξ 1 − rζ, ξ

(6.13)

This and (6.1) together imply 1 − Msrξ ∈ Iξ . Thus A − AMsrξ ∈ Iξ , which clearly implies (6.12). The proposition follows from (6.11) and (6.12). 2 Proof of Theorem 1.2. It follows immediately from Proposition 6.2 and (6.5).

2

Proof of Theorem 1.3. Let A ∈ T Mn be given. Then we obviously have sup lim AMsrξ lim sup AMsz . ξ ∈S r↑1

r↑1 r|z| 0. For a Hilbert space Z we will be using the following function spaces. Cb (Z) = {u : Z → R: u is continuous and bounded}, Lipb (Z) = u ∈ Cb (Z): u is Lipschitz continuous , C 2 (Z) = u : Z → R: Du, D 2 u are continuous , C 1,2 (0, T ) × Z = u : (0, T ) × Z → R: ut , Du, D 2 u are continuous , 2 (Z) = u : Z → R: u, Du, D 2 u are uniformly continuous , Cuc where Du, D 2 u denote the Fréchet derivatives of u with respect to the spatial variable. We will denote by S(·) the C0 -semigroup generated by −A. For λ > 0 we denote by Aλ the Yosida approximation of A, Aλ = λARλ , where Rλ = (λI + A)−1 . The C0 -semigroup generated by −Aλ will be denoted by Sλ (·). Both S(·) and Sλ (·) are semigroups of contractions. It is well known (see for instance [23]) that 1 Rλ , λ

and

lim λRλ x = x

λ→+∞

for x ∈ H.

For C ∈ L(H ) we will denote by CHS its Hilbert–Schmidt norm. We will need the following simple fact which we record for future use.

(2.9)

´ ech, J. Zabczyk / Journal of Functional Analysis 260 (2011) 674–723 A. Swi˛

679

Lemma 2.2. If f ∈ C 2 (H ) then for every x, y ∈ H f (x + y) = f (x) + Df (x), y +

1 1

D 2 f (x + sσy)y, y σ ds dσ.

0 0

2.2. Viscosity solutions To minimize the technicalities we will be using a slightly simplified definition of viscosity solution. This simplified definition will be enough since in this paper we only deal with bounded solutions. We also point out that Definition 2.4 applies to terminal value problems. Definition 2.3. A function ψ is a test function if ψ = ϕ + h(x), where: (i) ϕ ∈ C 1,2 ((0, T ) × H ) is B-lower semicontinuous, ϕ, ϕt , Dϕ, D 2 ϕ, A∗ Dϕ are uniformly continuous on [, T − ] × H for every > 0, and ϕ is bounded on every set [, T − ] × {x−1 r}. (ii) h ∈ C 2 ([0, +∞)) is such that h (0) = 0, h (r) 0 for r ∈ (0, +∞), and h, h , h

are uniformly continuous on [0, +∞). We will be concerned with terminal value problems for integro-PDE of the form vt − Ax, Dv + F t, x, Dv, v(t, ·) = 0 in (0, T ) × H,

(2.10)

2 (H ) → R. where F : (0, T ) × H × H × Cuc

Definition 2.4. A locally bounded B-upper semicontinuous function u : (0, T ) × H → R is a viscosity subsolution of (2.10) if whenever u − ϕ − h( · ) has a maximum over (0, T ) × H at a point (t, x) for some test functions ϕ, h(y) then ψt (t, x) − x, A∗ Dϕ(t, x) + F t, x, Dψ(t, x), ψ(t, ·) 0, where ψ(s, y) = ϕ(s, y) + h(y). A locally bounded B-lower semicontinuous function u : (0, T ) × H → R is a viscosity supersolution of (2.10) if whenever u + ϕ + h( · ) has a minimum over (0, T ) × H at a point (t, x) for some test functions ϕ, h(y) then ψt (t, x) + x, A∗ Dϕ(t, x) + F t, x, Dψ(t, x), ψ(t, ·) 0, where ψ(s, y) = −ϕ(s, y) − h(y). A viscosity solution of (2.10) is a function which is both a viscosity subsolution and a viscosity supersolution.

680


3. Estimates for solutions of stochastic PDE with Lévy noise In this section we recall basic facts and show various estimates about mild solutions of the equations, dXn (s) = −AXn (s) + F Xn (s) ds + G Xn (s−) dLn (s),

Xn (t) = x ∈ H,

(3.1)

on a fixed time interval [0, T ], where Ln are the processes defined in (1.3). Let us recall that if (1.3) holds then p

Eep,Ln (t) = entH0 ( n ) = ent

H [e

1 n p,z −1− 1 p,z ] ν(dz) n

,

p ∈ H.

(3.2)

The covariance operator of the process L will be denoted by Q and then the covariance operator of Ln is n1 Q. We refer the readers to Chapter 9 of [25] for the definition of a mild solution. We will also need solutions Xnm of the equations dXnm (s) = −Am Xnm (s) + F Xnm (s) ds + G Xnm (s−) dLn (s),

Xnm (t) = x ∈ H,

(3.3)

where the operators Am are Yosida approximations of A for λ = m = 1, 2, . . . . Proposition 3.1. Let 0 t T . Let (2.5) be satisfied and let G(x) − G(y), F (x) − F (y) Cx − y for all x, y ∈ H,

(3.4)

for some C 0. Then: (i) There exists a unique mild solution Xn of (3.1). The solution Xn has a càdlàg modification. (ii) If Xnm is the solution of (3.3) then lim E

m→+∞

2 sup Xnm (s) − Xn (s) = 0.

(3.5)

tsT

(iii) If in addition (2.4) holds then there exist constants c1 > 0, c2 > 0 (depending only on T , M, with c2 depending also on x) such that E

sup enc1 Xn (s) enc2 .

(3.6)

tsT

Remark 3.2. It follows from the proof that (3.6) is also satisfied for the processes Xnm with the same constants c1 , c2 . In particular this implies that there exists a constant C(x, T ) such that for every n, m E

m sup ec1 Xn (s) C x, T tsT

with the same estimate being also true for the processes Xn .

(3.7)


681

Proof of Proposition 3.1. (i) This is a standard result, see Theorem 9.29 in [25]. (ii) We will need two general results on convergence of stochastic and deterministic convolutions, Propositions 3.3 and 3.4. The proof of Proposition 3.3 will be postponed to Appendix A and the classical proof of Proposition 3.4 will be omitted. Denote by L the space of all predictable processes ψ(·) whose values are linear operators from the space Q1/2 (H ) into H , equipped with the scalar product ψ1 , ψ2 L =

+∞

ψ1 Q1/2 en , ψ2 Q1/2 en

H

ψ1 , ψ2 ∈ L.

,

n=1

Here (en ) is any orthonormal basis in H . Moreover two operators on H , even unbounded, identical on Q1/2 (H ), are identified. The norm on L is given by the formula. T 1/2 2 < +∞. |ψ|1 = E ψ(s)Q1/2 HS ds 0

Proposition 3.3. Let L(t) be a square integrable Lévy martingale in H with the covariance operator Q, and ψ ∈ L. Then the processes t

t S(t − s)ψ(s) dL(s),

Sλ (t − s)ψ(s) dL(s),

0

t ∈ [0, T ], λ > 0,

(3.8)

0

have càdlàg modifications and t 2 t lim E sup S(t − s)ψ(s) dL(s) − Sλ (t − s)ψ(s) dL(s) = 0. λ→+∞ 0tT 0

0

Proposition 3.4. Assume that ψ is an H -valued predictable process such that T E

ψ(s)2 ds < +∞.

0

Then the processes t

t S(t − s)ψ(s) ds,

0

Sλ (t − s)ψ(s) ds,

t ∈ [0, T ], λ > 0,

0

have continuous modifications and t 2 t lim E sup S(t − s)ψ(s) ds − Sλ (t − s)ψ(s) ds = 0. λ→+∞ 0tT 0

0

(3.9)

682


We can now proceed with the proof of (ii). Let X denote the space of all càdlàg, adapted to the filtration Ft , H -valued processes X, equipped with the norm | · |0 :

2 1/2 . |X|0 = E sup X(t) tT

Define transformations Kn , Knm , n, m = 1, 2, . . . by the formulae t Kn (X)(t) = S(t)X0 +

S(t − s)F X(s) ds +

0

t Knm (X)(t) = Sm (t)X0 +

t

S(t − s)G X(s−) dLn (s),

0

Sm (t − s)F X(s) ds +

0

t

Sm (t − s)G X(s−) dLn (s).

0

It will follow from the first part of the proof of Proposition 3.3 that the processes Kn (X), Knm (X) have càdlàg modifications. Moreover, as in the proof of existence of mild solutions, see e.g. [25] and using arguments similar to the proof of (A.1) one can show that for arbitrary α ∈ (0, 1) there exists Tα such that all transformations Kn , Kn satisfy Lipschitz conditions on X with a constant smaller than α. Moreover processes Xn , Xnm are unique solutions in X of the following fixed point problems X = Kn (X),

X = Kmn (X).

Therefore, it is easy to see, that to prove the results it is enough to show that for each X ∈ X , lim Kmn (X) = Kn (X), m

and this follows from Propositions 3.3, 3.4. The case of arbitrary T > 0 follows by repeating the same argument on intervals [0, Tα ], [Tα , 2Tα ], . . . , [(k − 1)Tα , kTα ], where kTα > T . (iii) Without loss of generality we will assume that t = 0. We will denote by πn (dt, dz), respectively πnk (dt, dz), k 1, the Poisson random measure for the process L(nt), respectively Lk (nt), where Lk (nt) is the process L(nt) with jumps restricted to size k. It is easy to see that the intensity measure of L(nt) is equal to n ν(dz) and the intensity measure of Lk (nt) is equal to nν k (dz), where ν k (dz) = χ{zk} ν(dz). Denote by Xnmk , m, k = 1, 2, . . . the solution of (3.1) with A replaced by Am and Ln replaced by Lkn , where Lkn = n1 Lk (nt). We will show (3.6) for the processes Xnmk and then pass to the limit as k → +∞ and m → +∞. Let h : R → R be a smooth even function such that h(0) = 1, h is increasing on √ (0, +∞), h (0) = 0, |h (r)| 1, h(r) (1 + r)/2 for r > 0. (We can take for instance h(r) = 1 + r 2 .) For l > 0 denote by τl the exit time of Xnmk from {y l}. Let α > 0 be a number which will be specified later. By Ito’s formula, see [22, Theorem 27.2, page 190], we have


ene

683

−α(s∧τl ) h(X mk (s∧τ )) l n

s∧τ l

=e

nh(x)

−αr mk αne−αr h Xnmk (r) ene h(Xn (r)) dr

− 0

s∧τ l

ne−αr ene

+

−αr h(X mk (r)) n

0

s +

ne−αr ene

Xnmk (r) dr h Xnmk (r) −Am Xnmk (r) + F Xnmk (r) , Xnmk (r)

−αr h(X mk (r−)) n

h Xnmk (r−) 1[0,τl ]

0

mk k Xnmk (r−) , G X (r−) dL (r) n n Xnmk (r−)

1 −αr mk mk −αr mk 1[0,τl ] ene h(Xn (r−)+ n G(Xn (r−))z) − ene h(Xn (r−))

s + 0 H \{0}

− e−αr ene

−αr h(X mk (r−)) n

h Xnmk (r−)

Xnmk (r−) , G Xnmk (r−) z mk Xn (r−)

πnk (dr, dz). (3.10)

To proceed further we compensate the measure πnk and recall that stochastic integrals with respect to the compensated random measures form martingales. Thus taking expectation in (3.10), using (2.4), (3.4), martingale property, the fact that −Am y, y 0 for y ∈ H and 1 + r 2h(r), we therefore obtain Eene

−α(s∧τl ) h(X mk (s∧τ )) l n

s∧τ l

e

nh(x)

ne−αr ene

+E


C 1 + Xnmk (r) − αh Xnmk (r) dr

0 s∧τ l

+E 0

−αr 1 mk mk −αr mk nene h(Xn (r)+ n G(Xn (r))z) − ene h(Xn (r))

H

− e−αr ene


s∧τ l

e

nh(x)

h Xnmk (r)

ne−αr ene

+E

Xnmk (r) , G Xnmk (r) z mk Xn (r)


ν(dz) dr

(2C − α)h Xnmk (r) dr

0 s∧τ l

+E

I (r) dr,

(3.11)

0

where I (r) is the integrand of the last term in the middle line of (3.11). Applying Lemma 2.2 to −αr the function f (x) = ene h(x) we have


684

1 1 tσ mk 1 mk 2 mk I (r) = n D f Xn (r) + G Xn (r) z G Xn (r) z, n n H

0 0

1 mk G Xn (r) z σ dt dσ ν(dz). n

(3.12)

Elementary calculation gives us D 2 f (x) = ne−αr ene

−αr h(x)

nψ1 (x) + ψ2 (x) ,

where 2 x x ψ1 (x) = e−αr h x ⊗ , x x h (x) x x h (x)

ψ2 (x) = h x − ⊗ + I. x x x x We observe that both ψ1 , ψ2 are bounded as functions from H to L(H ). Therefore

I (r) e

ne−αr h(Xnmk (r))

1 1

M 2 e−αr

H 0 0

×e e

ne−αr |h(Xnmk (r)+ tσ n


G(Xnmk (r))z)−h(Xnmk (r))|

M 2 e−αr eMz nψ1 ∞ + ψ2 ∞ z2 ν(dz)

H

nM1 e−αr e

nψ1 ∞ + ψ2 ∞ z2 dt dσ ν(dz)


z2 eMz ν(dz) nM2 e−αr ene


(3.13)

H

for some M1 , M2 > 0. Plugging (3.13) into (3.11), choosing α = 2C + M2 + 1 and recalling that h(r) 1 we thus obtain Ee

ne−α(s∧τl ) h(Xnmk (s∧τl ))

s∧τ l

ne−αr ene

+E


dr enh(x)

0

which in particular implies that Eene

−αs h(X mk (s∧τ )) l n

enh(x) .

Since liml→+∞ (T ∧ τl ) = T a.s., letting l → +∞ and using Fatou’s lemma we obtain Eene

−αs h(X mk (s)) n

enh(x) .


685

We can now send k → +∞, employ once again Fatou’s lemma and the fact that Xnmk (s) → Xnm (s) a.s. (at least along a subsequence). This can be shown using the arguments from the proof of (ii). This way we arrive at Eene

−αs h(X m (s)) n

enh(x) .

(3.14) n −αr h(x)

We can now go back to Ito’s formula (3.10) but apply it to the function e 2 e process Xnm and without stopping time. It yields

, the

n −αs h(Xnm (s))

e2e

=e

n 2 h(x)

s −

n −αr n m α e−αr h Xnm (r) e 2 e h(Xn (r)) dr 2

0

s + 0

s + 0

Xnm (r) n −αr n e−αr h(Xnm (r)) m h Xn (r) −Am Xnm (r) + F Xnm (r) , e e2 dr 2 Xnm (r) m Xnm (r−) n −αr n e−αr h(Xnm (r−)) m e e2 , G X h Xn (r−) (r−) dL (r) n n 2 Xnm (r−)

s +

n −αr h(Xnm (r−)+ n1 G(Xnm (r−))z)

e2e

n −αr h(Xnm (r−))

− e2e

0 H \{0}

m Xnm (r−) 1 −αr n e−αr h(Xnm (r−)) m 2 , G Xn (r−) z πn (dr, dz). − e e h Xn (r−) 2 Xnm (r−) Arguing like in (3.11) and (3.13), applying sup0sT to both sides and taking expectation give us n −αs h(Xnm (s))

E sup e 2 e 0sT

e

n 2 h(x)

s + E sup 0sT

n −αr n e−αr h(Xnm (r)) e e2 (2C + M2 − α)h Xnm (r) dr 2

0

s n m m Xnm (r−) −αr n2 e−αr h(Xnm (r−))

+ E sup e e , G Xn (r−) dLn (r) h Xn (r−) 2 Xnm (r−) 0sT 0

s n −αr 1 n −αr m m m + E sup e 2 e h(Xn (r−)+ n G(Xn (r−))z) − e 2 e h(Xn (r−)) 0sT 0 H \{0}

m Xnm (r−) 1 −αr n e−αr h(Xnm (r−)) m , G X − e e2 h Xn (r−) (r−) z π ˆ (dr, dz) . (3.15) n n 2 Xnm (r−)


686

Denote s N(s) = 0

Xnm (r−) m n −αr n e−αr h(Xnm (r−)) m h Xn (r−) e e2 , G Xn (r−) dLn (r) . 2 Xnm (r−)

Then N is a square integrable martingale. From the definition of the quadratic variation process, see [26], E[N, N ]T = EN 2 (T ). Therefore, from the Burkholder–Davis–Gundy inequality [26,25], 1 1 1 E sup N (s) C1 E[N, N ]T2 C1 E[N, N ]T 2 = C1 EN 2 (T ) 2

0sT

1 T 2 2 1 n M −αr m C2 E n2 ene h(Xn (r)) 2 n dr M3 n 2 e 2 h(x) n

(3.16)

0

for some constant M3 > 0, where we used (3.14) to get the last inequality. As regards the last term of (3.15), by a straightforward generalization of Lemma 8.22 of [25] to predictable p-integrable fields, with p = 1, s n −αr 1 n −αr m m m e 2 e h(Xn (r−)+ n G(Xn (r−))z) − e 2 e h(Xn (r−)) E sup 0sT 0 H \{0}

m Xnm (r−) 1 −αr n e−αr h(Xnm (r−)) m , G X − e e2 h Xn (r−) (r−) z π ˆ (dr, dz) n n 2 Xnm (r−) T n e−αr h(Xm (r)+ 1 G(Xm (r))z) n −αr m e 2 n n n D1 nE − e 2 e h(Xn (r)) 0 H

m 1 −αr n e−αr h(Xnm (r)) m Xnm (r) 2 , G Xn (r) z ν(dz) dr − e e h Xn (r) 2 Xnm (r) n

M4 ne 2 h(x)

(3.17)

if we once again argue like in (3.13) and then use (3.14). Therefore, plugging (3.16) and (3.17) in (3.15) we finally obtain n −αs h(Xnm (s))

E sup e 2 e

n

M5 ne 2 h(x) eM6 nh(x)

(3.18)

0sT

for some M6 > 0. We can now pass to the limit as m → +∞ using (3.5) and use that (1 + r)/2 h(r) to complete the proof. 2


687

Proposition 3.5. Let 0 t T and let (2.2)–(2.5) be satisfied. Let Xn (s) and Yn (s) are solutions of (3.1) with initial conditions x and y respectively. Then 2 EXn (s) − Yn (s)−1 C1 (T )x − y2−1 , 2 EXn (s) − x −1 C2 x, T (s − t),

(3.19) (3.20)

and 2 EXn (s) − x ωx (s − t)

(3.21)

for some modulus ωx . Proof. The proofs are rather typical for these kinds of estimates. We first show (3.19). By Ito’s formula we have 2 EXnm (s) − Ynm (s)−1 s = x

− y2−1

+ 2E

m Xn (τ ) − Ynm (τ ), A∗m B Xnm (τ ) − Ynm (τ )

t

+ F Xnm (τ ) − F Ynm (τ ) , B Xnm (τ ) − Ynm (τ ) dτ 1 + E n

s

m G X (τ ) − G Y m (τ ) z2 ν(dz) dτ. n n −1

(3.22)

t H

Using (3.5) and moment estimates (3.7) for Xnm and Ynm we can pass to the limit above to obtain that (3.22) is still true if Xnm and Ynm are replaced by Xn and Yn respectively and Am is replaced by A. We then use (2.1), (2.2) and (2.3) to get 2 EXn (s) − Yn (s)−1 x

− y2−1

1 + 2c0 + M B 2 E

s

Xn (τ ) − Yn (τ )2 dτ −1

t 1

+

MB 2 E n

s

Xn (τ ) − Yn (τ )2 z2 ν(dz) dτ −1

t H

s x

− y2−1

+C

2 EXn (τ ) − Yn (τ )−1 dτ

t

and the claim follows from Gronwall’s inequality. To show (3.20) we again employ Ito’s formula and (2.2), (2.4) to find that


688

2 EXnm (s) − x −1 s = 2E

m − Xn (τ ), A∗ B Xnm (τ ) − x

t

1 + F Xnm (τ ) , B Xnm (τ ) − x dτ + E n

s

m 2 G X (τ ) z ν(dz) dτ n −1

t H

C x E

s

2 1 + Xnm (τ ) dτ C2 x, T (s − t).

(3.23)

t

As regards (3.21) it follows from the definition of mild solution that s Xn (s) = S(s − t)x +

S(s − τ )F Xn (τ ) dτ +

t

s

S(s − τ )G Xn (τ ) dLn (τ ).

t

Therefore 2 s 2 2 EXn (s) − x 4 S(s − t)x − x + E M 1 + Xn (τ ) dτ t

s 2 + E S(s − τ )G Xn (τ ) dLn (τ ) t

s 2 1 2 dτ , C S(s − t)x − x + (s − t) + E n

(3.24)

t

where we have used the isometric formula to obtain the last inequality.

2

Finally we state for future use the following lemma which can be shown rather easily using again Ito’s formula applied first to the process Xnm and then letting m → +∞. Its proof will thus be omitted. Lemma 3.6. Let the assumptions of Proposition 3.1 be satisfied. Let t s T . Let ψ = ϕ + h( · ) be a bounded test function. Then s Ee

ψ(s,Xn (s))

e

ψ(t,x)

+E t

eψ(τ,Xn (τ )) ψt τ, Xn (τ )

+ F Xn (τ ) , Dψ τ, Xn (τ ) + Xn (τ ), A∗ Dϕ τ, Xn (τ ) dτ


+ nE

s 1 eψ(τ,Xn (τ )+ n G(Xn (τ ))z) − eψ(τ,X(τ )) t H

−e

689

ψ(τ,X(τ ))

1 Dψ τ, Xn (τ ) , G Xn (τ ) z n

ν(dz) dτ.

4. Associated nonlinear integro-PDE For g ∈ Cb (H ) we define the function vn (t, x) =

1 log E eng(Xn (T )) , n

(4.1)

where Xn solves (3.1). As we have stated earlier one of our main aims is to establish convergence of the sequence (vn ) and to identify its limit as a solution of a Hamilton–Jacobi–Bellman equation. In the present section we investigate the approximating and the limiting equations. 4.1. Approximating equations We first show that for each n the function vn is a viscosity solution of an integro-PDE. Theorem 4.1. Let (2.2)–(2.5) be satisfied and let g ∈ Lipb (H−1 ). Then there exist a constant C1 and, for every R > 0, a constant C2 = C2 (R) (both possibly depending on n) such that vn (t, x) − vn (s, y) C1 x − y−1 + C2 max x, y |t − s| 12 for x, y ∈ H, t, s ∈ [0, T ],

(4.2)

and vn is a viscosity solution of an integro-PDE ⎧ (vn )t + −Ax + F (x), Dvn ⎪ ⎪ ⎪ ⎪ ⎨

n(v (t,x+ 1 G(x)z)−v (t,x)) n n e n − 1 − Dvn , G(x)z ν(dz) = 0, + ⎪ ⎪ ⎪ H ⎪ ⎩ vn (T , x) = g(x) in (0, T ) × H.

(4.3)

Proof. Estimate (4.2) is a direct consequence of (3.19), (3.20), and the Markov property of the process Xn . The proof that vn is a viscosity solution of (4.3) is similar to the proof of Theorem 7.1 in [32]. We will only show that vn is a viscosity subsolution since the supersolution part is similar. Suppose that vn − h( · ) − ϕ has a global maximum at (t, x). Since vn is bounded by Remark 4.3 of [32] without loss of generality we can also assume that h, h , h

and ϕ are bounded. Denote ψ(s, y) = h(y) + ϕ(s, y). Then for small > 0 vn t + , Xn (t + ) − ψ t + , Xn (t + ) vn (t, x) − ψ(t, x). Therefore, setting un = envn we have


690

un (t + , Xn (t + )) enψ(t+,Xn (t+)) e−nψ(t,x) , un (t, x) which, upon taking the expectation of both sides of the above inequality and using the Markov property of Xn (s), produces enψ(t,x) Eenψ(t+,Xn (t+)) . Therefore, applying Lemma 3.6, we obtain 1 nψ(t+,Xn (t+)) e − enψ(t,x) t+

1 E nenψ(τ,Xn (τ )) ψt τ, Xn (τ )

0E

t

+ F Xn (τ ) , Dψ τ, Xn (τ ) dτ − Xn (τ ), A∗ Dϕ τ, Xn (τ ) dτ n +E

t+ t

−e

nψ(τ,X (τ )+ 1 G(X (τ ))z) n n n e − enψ(τ,Xn (τ ))

H

nψ(τ,Xn (τ ))

Dψ τ, Xn (τ ) , G Xn (τ ) z ν(dz) dτ.

(4.4)

Using (3.21), (2.2), boundedness of ψ , uniform continuity of ψ, ψt , Dψ, A∗ ϕ, and moment estimates (in particular (3.6)) it is easy to see that 1 E

t+

nenψ(τ,Xn (τ )) ψt τ, Xn (τ ) t

+ F Xn (τ ) , Dψ τ, Xn (τ ) dτ − Xn (τ ), A∗ Dϕ τ, Xn (τ ) dτ t+

1 = nenψ(t,x) ψt (t, x) t

∗ + F (x), Dψ(t, x) dτ − x, A Dϕ(t, x) dτ + o() .

(4.5)

As regards the other term, by Lemma 2.2, (2.3), (2.4), (2.5), (3.6), (3.21), boundedness of ψ and uniform continuity of ψ, Dψ, D 2 ψ , we have n E

t+ t

H

nψ(τ,X (τ )+ 1 G(X (τ ))z) n n n e − enψ(τ,Xn (τ ))

− enψ(τ,Xn (τ )) Dψ τ, Xn (τ ) , G Xn (τ ) z ν(dz) dτ


n =E

691

t+ 1 1 1 1 D 2 enψ(τ,Xn (τ )+sσ n G(Xn (τ ))z) G Xn (τ ) z, n t

H 0 0

t

H

1 G Xn (τ ) z σ ds dσ ν(dz) dτ n t+ 1 1 1 n 2 nψ(t,x+sσ n1 G(x)z) 1 D e G(x)z, G(x)z σ ds dσ E n n 0 0

2 ν(dz) dτ + C1 1 + Xn (τ ) + z2 z2 ω Xn (τ ) − x 1 + z n =

t+ t

−e

nψ(t,x+ 1 G(x)z) n e − enψ(t,x)

H nψ(t,x)

Dψ(t, x), G(x)z ν(dz) + ω1 () dτ.

(4.6)

(Above ω, ω1 are some moduli and C1 , C2 are constants, all depending on ψ.) Therefore plugging (4.5) and (4.6) into (4.4) and sending → 0 we obtain nψ(t,x) ψt (t, x) − x, A∗ Dϕ(t, x) + F (x), Dψ(t, x) 0 ne +

n(ψ(t,x+ 1 G(x)z)−ψ(t,x)) n e − 1 − Dψ(t, x), G(x)z ν(dz)

H

which completes the proof after we divide both sides by nenψ(t,x) .

2

4.2. Limiting Hamilton–Jacobi–Bellman equation The limiting equation (obtained by letting n → +∞ in (4.3)) can be formally identified as vt + −Ax + F (x), Dv + H0 G∗ (x)Dv = 0, (4.7) v(T , x) = g(x) in (0, T ) × H, where

H0 (p) =

p,z e − 1 − p, z ν(dz).

H

It is the Bellman equation corresponding to a deterministic control problem. For 0 t T , x ∈ H , and u(·) ∈ Mt = {u : [t, T ] → H : u is strongly measurable} we consider the state equation X(t) = x, (4.8) X (s) = −AX(s) + F X(s) + G X(s) u(s),


692

and we want to maximize the cost functional J t, x; u(·) =

T

−L0 u(s) ds + g X(T )

t

over all controls u(·) ∈ Mt , where L0 is the Legendre transform of H0 , i.e. L0 (z) = sup z, y − H0 (y) .

(4.9)

y∈H

The value function for the problem is v(t, x) = sup J t, x; u(·) .

(4.10)

u(·)∈Mt

The Hamiltonian H0 and Lagrangian L0 are both convex. By (2.5) and the definition of H0 we see that 0 H0 (y) < +∞ for every y ∈ H , H0 (0) = 0, and H0 is locally Lipschitz continuous on H . Therefore L0 (0) = 0, L0 (z) 0 for every z ∈ H , and moreover L0 (z) z − H0

z z

→ +∞ as z → +∞

(4.11)

(but L0 can possibly take infinite values). Since g is bounded it is then obvious that v(t, x) = sup J t, x; u(·) , u(·)∈M˜ t

where M˜ t = u(·) ∈ Mt :

T

! L0 u(s) ds K = 2g∞ .

(4.12)

t

We will need the following simple lemma. Lemma 4.2. For every > 0 there exists a constant N = N (ν) such that for every z ∈ H z L0 (z) + N . Proof. It follows from (4.9), (2.5), and L0 (0) = 0 that z = z,

z z L0 (z) + H0 L0 (z) + N . z z

Lemma 4.3. Let (2.2)–(2.4) be satisfied. Let 0 t T and u(·) ∈ M˜ t . Then:

2


693

(i) There exists a unique mild solution X ∈ C([t, T ]; H ) of (4.8). Moreover there exists a constant C1 = C1 (T , K, M) such that sup X(s) C1 1 + x .

(4.13)

tsT 1

(ii) There exists a constant C2 = C2 (T , K, M, c0 , B 2 ), such that if X, and Y are solutions of (4.8) with initial conditions x and y respectively then X(s) − Y (s) C2 x − y−1 −1

for t s T ,

(4.14)

(iii) For every R > 0 there exists a modulus ωR , depending on R, K, T , A∗ B, such that if x R then X(s) − x

−1

ωR (s − t) for t s T ,

(4.15)

and for every x ∈ H there exists a modulus ωx , independent of u(·), such that X(s) − x ωx (s − t)

for t s T .

(4.16)

Proof. We first notice that by Lemma 4.2 (applied with = 1) T

u(τ ) dτ K + N1

(4.17)

t

for every u(·) ∈ M˜ t . Therefore the existence and uniqueness of a mild solution of (4.8) and estimate (4.13) are well known. We refer for instance to [21, Chapter 2, Proposition 5.3]. To show (4.14) we notice that X(s) − Y (s)2

−1

s = x

− y2−1

−2

A∗ B X(τ ) − Y (τ ) , X(τ ) − Y (τ ) dτ

t

s +2

B X(τ ) − Y (τ ) , F X(τ ) − F Y (τ ) + G X(τ ) − G Y (τ ) u(τ ) dτ

t

and therefore using (2.1), (2.2) and (2.3) we have X(s) − Y (s)2 x − y2 + C −1 −1

s

X(τ ) − Y (τ )2 1 + u(τ ) dτ. −1

t

Therefore (4.14) follows from (4.17) and Gronwall’s inequality.

694


To prove (4.15) we write X(s) − x 2 = −2 −1

s

A∗ B X(τ ) − x , X(τ ) dτ

t

s +2

B X(τ ) − x , F X(τ ) + G X(τ ) u(τ ) dτ

t

and thus using (2.2)–(2.4), (4.13) and Lemma 4.2 we obtain X(s) − x 2 −1

s

CR 1 + u(τ ) dτ

t

s CR

L u(τ ) dτ + CR N (s − t) CR K + CR N (s − t).

t

Therefore we obtain (4.15) with 1

ωR (τ ) = inf (CR K + CR N τ ) 2 . >0

Estimate (4.16) is proved similarly noticing that X(s) − x S(s − t)x − x +

s

CR 1 + u(τ ) dτ.

2

t

The definition of viscosity solution of (4.7) is the same as Definition 2.4 after we disregard the nonlocal part and of course it is enough to have test functions which are only once continuously differentiable. For more on viscosity solutions of first-order PDE in Hilbert spaces we refer to [9, 10,21]. Theorem 4.4. Let (2.2)–(2.4) be satisfied and let g ∈ Lipb (H−1 ). There exist a constant D1 and, for every R > 0, a modulus ωR such that the value function v satisfies v(t, x) − v(s, y) D1 x − y−1 + ωR |t − s| for x, y ∈ H, x, y R, t, s ∈ [0, T ].

(4.18)

Moreover v is a viscosity solution of the HJB equation (4.7). Proof. The proof is very similar to the proof of Theorem 7.3 in [32]. We include it here for completeness. The Lipschitz continuity in x follows from (4.14) and the fact that g ∈ Lipb (H−1 ). To show the continuity in time let x ∈ H and s < t and let > 0. Let u (·) ∈ Mt be such that


695

v(t, x) J t, x; u (·) + . Extending u (·) by 0 to [s, T ] we can assume that u (·) ∈ Ms . Therefore v(s, x) − v(t, x) J s, x; u (·) − J t, x; u (·) − g X(T ; s, x) − g X(T ; t, x) + −C2 D2 ωR |s − t| − , where we have used (4.14), (4.15), and D2 is the Lipschitz constant of g. For the opposite inequality if u (·) ∈ Ms is such that v(s, x) J s, x; u (·) + then u (·) ∈ Mt and by (4.14), (4.15) we again have v(s, x) − v(t, x) J s, x; u (·) + − J t, x; u (·) g X(T ; s, x) − g X(T ; t, x) − C2 D2 ωR |s − t| + .

t

L0 u (τ ) dτ +

s

Therefore since was arbitrary we have obtained v(s, x) − v(t, x) C2 D2 ωR |s − t| . We will only show that v is a viscosity subsolution as the proof of the supersolution property is similar but easier. We will use the dynamic programming principle. It asserts that if 0 t < t + T , x ∈ H then

v(t, x) = sup u(·)∈Mt

! t+ −L0 u(s) ds + v t + , X(t + ) . t

Let now v − ϕ − h( · ) have a local maximum at (t, x). By the dynamic programming principle for every 0 < < T − t there exists a control u (·) such that. t+ −L0 u (s) ds + v t + , X (t + ) + 2 . v(t, x) t

We recall that in particular this implies that u (·) is integrable. Denote ψ(s, y) = −ϕ(s, y) − h(y). For simplicity we will write h(y) := h(y).


696

We have ϕ t + , X (t + ) = ϕ t, X (t) +

t+

− X (s), A∗ Dϕ X (s) t

+ F X (s) + G X (s) u (s), Dϕ X (s) ds and h X (t + ) h(x) +

t+ F X (s) + G X (s) u (s), Dh X (s) ds. t

The first equality above is proved for instance in [21, Chapter 2, Proposition 5.5] and the inequality is also standard and can be shown using Yosida approximations similarly to what we have done in the stochastic case. Using this we therefore have 1 − v t + , X (t + ) − v(t, x) −

t+ L0 u (s) ds t

1 ϕ t + , X (t + ) − ϕ(t, x) + h X (t + ) − h(x) −

t+ L u (s) ds t

1

t+

ϕt s, X (s) − X (s), A∗ Dϕ s, X (s) t

+ F X (s) + G X (s) u (s), Dψ s, X (s) − L0 u (s) ds 1

!

t+

ϕt s, X (s) − X (s), A∗ Dϕ s, X (s) t

! ∗ + F X (s) , Dψ s, X (s) + H0 G X (s) Dψ s, X (s) ds .

(4.19)

Therefore, using (4.16), we can pass to the limit as → 0 in (4.19) to obtain 0 ψt (t, x) − x, A∗ Dϕ(t, x) + F (x), Dψ(t, x) + H0 G∗ (x)Dψ(t, x) .

2


697

5. Existence of Laplace limit Define H (x, p) = H0 G∗ (x)p . By (2.3), (2.4) and local Lipschitz continuity of H0 we have that for every R > 0 there exists a constant KR such that H (x, p) − H (y, q) KR x − y−1 + p − q for all x, y, p, q ∈ H, p, q R.

(5.1)

The theorems below are our key results on the existence of the Laplace limit. Theorem 5.1. Let (2.2)–(2.5) hold. Let g ∈ Lipb (H−1 ). Let vn be bounded viscosity solutions of (4.3), and v be a bounded viscosity solution of (4.7) such that lim vn (t, x) − g(x) + v(t, x) − g(x) = 0,

t→T

uniformly on bounded sets,

(5.2)

for every n and v(t, x) − v(t, y) D1 x − y−1

(5.3)

for some D1 0 and all t ∈ (0, T ], x, y ∈ H . Let K := v∞ + supn vn ∞ < +∞. Then vn − v∞ → 0 as n → +∞.

(5.4)

In particular, the value function (4.10) of the control problem of Section 4.2 is the unique bounded viscosity solution of (4.7) satisfying (5.2) and (5.3). The proof of this theorem is postponed until the end of the section. For a function g and t > 0 we denote Vn (t)f (x) =

1 log E eng(Xn (t)) : Xn (0) = x , n

where Xn (t) is the solution of (1.2). Theorems 4.1, 4.4, and 5.1 yield the following corollary. Corollary 5.2. Let (2.2)–(2.5) hold and let g ∈ Lipb (H−1 ). Then Λ(g) := lim Vn (t)f (x) = v(0, x), n→∞

where v is the value function defined by (4.10). We will need a more general convergence result which is now rather standard.

(5.5)

698


Proposition 5.3. Let (2.2)–(2.5) hold and let g be bounded and weakly sequentially continuous on H . Then: (i) For every n the function Vn (t)g is weakly sequentially continuous on H . (ii) For every x ∈ H V (t)g(x) = lim Vn (t)g(x) = v(0, x) n→+∞

exists and is uniform on bounded subsets of H . In particular V (t)g(x) is weakly sequentially continuous on H . Above, v is the value function defined by (4.10). (iii) If gn are weakly sequentially continuous on H , such that gn ∞ M for n 1 and gn → g uniformly on bounded subsets of H then lim Vn (t)gn (x) = V (t)g(x)

n→+∞

(5.6)

uniformly on bounded subsets of H . Proof. We use Corollary 5.2, exponential moment estimate (3.6) and the fact that g can be approximated uniformly on balls in H by functions in Lipb (H−1 ). Since (5.5) is true for every g ∈ Lipb (H−1 ), it will be preserved in the limit. The proof repeats directly the argument of the proofs of Lemma 7.6 and Proposition 7.7 of [32]. We refer the reader to this paper. 2 We now pass to the proof of Theorem 5.1. Proof of Theorem 5.1. If (5.4) is not satisfied then without loss of generality we can assume that there exist > 0 and a subsequence nk such that sup(vnk − v) 4.

(5.7)

Let a > 0 be such that aT and let m > 0 be such that mK +

D12 ,

and

1 D1 2D12 a c0 + M B 2 + K . 1 2 m m 2D1 B +1 2

Let ψ : [0, +∞) → [0, +∞) be a smooth and nondecreasing function such that ψ(r) = r 2 for 0 r 1 and ψ(r) = 2 for r 2. For each k we choose μk > 0 such that μk μk − 3. sup vnk − v − t s For δ, β > 0 we now consider the function Φ(t, s, x, y) = vnk (t, x) − v(s, y) − a(T − t) − " " − δ 1 + x2 − δ 1 + y2 .

(t − s)2 μk μk − − mψ x − y2−1 − t s 2β (5.8)


699

Since Φ is B-upper semicontinuous, by a perturbed optimization technique of [10] (see [10, page 424] or [21, Chapter 6.4]), which is a version of the Ekeland–Lebourg lemma [13], we find for every sufficiently big i > 0 elements pi , qi ∈ H and ai , bi ∈ R such that pi + qi + |ai | + |bi | 1/i and such that Φ(t, s, x, y) + ai t + bi s + Bpi , x + Bqi , r has a global maximum over [0, T ] × H at some points t¯, s¯ , x, ¯ y, ¯ where 0 < t¯, s¯ . Following standard arguments (see for instance [17]) it is easy to see that lim sup lim sup lim sup δ δ→0

β→0

" " 1 + x ¯ 2 + 1 + y ¯ 2 =0

for fixed k,

(5.9)

i→+∞

lim sup lim sup β→0

i→+∞

(t¯ − s¯ )2 = 0 for fixed k, δ. 2β

(5.10)

Moreover it is clear that ψ(x¯ − y ¯ 2−1 ) = x¯ − y ¯ 2−1 and, since Φ(t¯, s¯ , x, ¯ x) ¯ Φ(t¯, s¯ , x, ¯ y), ¯ we obtain " ¯ −1 + δ 1 + x ¯ 2 + qi , y¯ − x ¯ mx¯ − y ¯ 2−1 D1 x¯ − y which, in light of (5.9) and the fact that x, ¯ y ¯ cδ for every i for some constant cδ , implies lim sup lim sup lim sup mx¯ − y ¯ −1 D1 . δ→0

β→0

(5.11)

i→+∞

Therefore, by (5.7), (5.9), (5.10), (5.11) and the definition of m, for small δ, β, and big i we have 0 < t¯, s¯ < T . We now use (5.8) and the definition of viscosity solution to obtain μk t¯ − s¯ − x, ¯ A∗ B 2m(x¯ − y) + ¯ − pi β t¯2 δ x¯ − Bpi + F (x), ¯ 2mB(x¯ − y) ¯ +# 1 + x ¯ 2 " √ nk m(ψ(x+ ¯ n1 G(x)z− ¯ y ¯ 2−1 )−ψ(x− ¯ y ¯ 2−1 ))+δnk ( 1+x+ ¯ n1 G(x)z ¯ 2 − 1+x ¯ 2 )−Bpi ,G(x)z ¯ k k e +

−a − ai −

H

δ x¯ − 1 − 2mB(x¯ − y) ¯ +# − Bpi , G(x)z ¯ ν(dz) 0 1 + x ¯ 2

(5.12)

and μk t¯ − s¯ δ y¯ ∗ bi + 2 + ¯ 2mB(x¯ − y) ¯ −# + Bqi − y, ¯ A 2mB(x¯ − y¯ + qi ) + F (y), β s¯ 1 + y ¯ 2 δ y¯ (5.13) + Bqi 0. + H y, ¯ 2mB(x¯ − y) ¯ −# 1 + y ¯ 2


700

But e

" √ nk (ψ(x+ ¯ n1 G(x)z− ¯ y ¯ 2−1 )−ψ(x− ¯ y ¯ 2−1 ))+δnk ( 1+x+ ¯ n1 G(x)z ¯ 2 − 1+x ¯ 2 )−Bpi ,G(x)z ¯ k

=e

k

2mB(x− ¯ y)+ ¯ √ δ x¯ 2 −Bpi ,G(x)z +σ ¯ k (z) 1+x ¯

,

(5.14)

where for small δ, β and big i 2 σk (z) Cm min z, z nk for some constant Cm independent of k. Using this in (5.12) we therefore obtain that for small δ, β and big i μk t¯ − s¯ − x, ¯ A∗ B 2m(x¯ − y) + ¯ − pi β t¯2 δ x¯ − Bpi + F (x), ¯ 2mB(x¯ − y) ¯ +# 1 + x ¯ 2 δ x¯ + H x, ¯ 2mB(x¯ − y) ¯ +# − Bpi 1 + x ¯ 2 C˜ m z2 ν(dz) − nk

−a − ai −

{z1}

+

e(2D1 B

1 2 +1)Mz

σ (z) e k − 1 ν(dz) −ω(k, δ, β, i),

(5.15)

{z>1}

where limk→+∞ lim supδ→0 lim supβ→0 lim supi→+∞ ω(k, δ, β, i) = 0 by (2.5) and the Lebesgue dominated convergence theorem. Combining (5.13) and (5.15) and using (5.9), (5.11), (2.2), (2.3), (2.4) we thus obtain 1 μk ¯ 2−1 + K + 2m c0 + M B 2 x¯ − y x¯ − y ¯ −1 + ω1 (k, δ, β, i) 1 2 2D1 B 2 +1 T 1 D1 2D12 c 0 + M B 2 + K + ω2 (k, δ, β, i) 1 m m 2D1 B 2 +1 a + ω2 (k, β, δ, i), (5.16) 2

a −2

where lim supk→+∞ lim supδ→0 lim supβ→0 lim supi→+∞ ωj (k, β, δ, i) = 0 for fixed j = 1, 2. This yields a contradiction after we send i → +∞, β → 0, δ → 0 and then k → +∞. Similar argument gives us that limn→+∞ sup(v − vn ) = 0 and therefore (5.4) follows. Finally we notice that, by Theorem 4.4, the value function (4.10) of the control problem of Section 4.2 is a bounded viscosity solution of (4.7) satisfying (5.2) and (5.3). Since the functions vn converge to a single solution of (4.7), the value function must be the unique bounded viscosity solution of (4.7) satisfying (5.2) and (5.3). 2


701

6. Large deviation principle at a single time Let V be a Hilbert space such that H ⊂ V and H → V is compact. We remark that on every closed ball in H , the convergence in V is equivalent to the weak sequential convergence in H . We have the following large deviation result. Theorem 6.1. Let (2.2)–(2.5) hold. Let T > 0, x ∈ H, and let Xn be the solutions of (1.2). Then the random variables Xn (T ) satisfy large deviation principle in V with the rate function T I (y) = lim inf inf

z→y u(·)∈M0

L0 u(s) ds: X satisfies (4.8), X(0) = x, X(T ) = z

! (6.1)

0

(where the liminf above is taken in the topology of V ). Proof. By Bryc’s theorem (see for instance [12, Theorem 1.3.8]) to show that Xn (T ) satisfy large deviation principle in V it is enough to prove that Xn (T ) are exponentially tight in V and that for every g ∈ Cb (V ) the Laplace limit Λ(g) exists. Since closed balls in H are compact in V , exponential tightness of Xn (T ) follows from the exponential moment estimates (3.6). Since every g ∈ Cb (V ) is weakly sequentially continuous on H , the Laplace limit Λ(g) exists by Theorem 5.3. It remains to prove the representation formula for the rate function. We recall that T Λ(g) = sup u(·)∈M0

! −L0 u(s) ds + g X(T ) ,

0

where X(0) = x. We have (see [12, page 27] or [15, page 47]) I (y) =

−Λ(g)

sup g∈Cb (V ),g(y)=0

T =

sup

inf

g∈Cb (V ),g(y)=0,g0 u(·)∈M0

! L0 u(s) ds + g X(T ) .

0

Denote the right-hand side of (6.1) by I1 (y) and for m > 0 define the function gm (z) = mz − yV , where · V is the norm in V . Then for m, n 1 T I (y)

inf

u(·)∈M0 0

L0 u(s) ds + g X(T )

!


702

m , inf min n u(·)∈M0

T

1 L0 u(s) ds: X(T ) − y V n

!! .

0

Therefore, letting m → +∞ we obtain T I (y)

inf

u(·)∈M0

! 1 L0 u(s) ds: X(T ) − y V , n

0 y

which implies I (y) I1 (y). To show the reverse inequality, for g ∈ Cb (V ) let ωg be a modulus of continuity of g at y. Then for n 1 we have T inf

u(·)∈M0

L0 u(s) ds + g X(T )

!

0

T

inf

u(·)∈M0

1 L0 u(s) ds: X(T ) − y V n

! y + ωg

1 . n

0

Taking the lim infn→+∞ in the above inequality and then supremum over g gives us I (y) I1 (y). 2 T Remark 6.2. Since if 0 L0 (u(s)) ds n the solution of (4.8) with X(0) = x satisfies X(T ) Cn for some absolute constant Cn it is clear that I (y) = +∞ if y ∈ V \ H . In some cases lim infz→y can be removed from (6.1). We present below one such case. Proposition 6.3. Suppose that, in addition to the assumptions of Theorem 6.1, there exists p > 1 such that zp C 1 + L0 (z)

for all z ∈ H,

(6.2)

and that for every x ∈ H and K > 0 there exists a modulus ωx,K such that if X satisfies (4.8), T X(0) = x, 0 u(s)p ds K, then X(s1 ) − X(s2 ) ωx,K |s1 − s2 | V

for all s1 , s2 ∈ [0, T ].

(6.3)

Then T I (y) =

inf

u(·)∈M0 0

! L0 u(s) ds: X satisfies (4.8), X(0) = x, X(T ) = y .

(6.4)


703

Proof. To show (6.4), suppose that Xm satisfies (4.8) with um (·) ∈ M0 , Xm (0) = x, Xm (T ) = zm , where zm → y in V , and T

L0 u(s) ds → α ∈ R as m → +∞.

0

Then by (4.13), (6.2) and (6.3) the family {Xm } is equibounded in H and equicontinuous in V and since balls in H are compact in V , by the Arzela–Ascoli theorem, a subsequence, still denoted by Xm , converges uniformly in C([0, T ]; V ) to Y : [0, T ] → H which also satisfies (6.3). Moreover we can assume that um u in Lp (0, T ; H ) for some u. By the definition of mild solution for 0 s T s

S(s − τ ) F Xm (τ ) + G Xm (τ ) um (τ ) dτ.

Xm (s) = S(s)x + 0

Since the topology of V on closed balls of H is equivalent to the weak topology in H , we have that sup0τ T Xm (τ ) − Y (τ )−1 → 0 as m → +∞, and thus sup F Xm (τ ) − F Y (τ ) + G Xm (τ ) − G Y (τ ) → 0 as m → +∞. (6.5) 0τ T

Therefore (6.5), combined with um u in Lp (0, T ; H ), yields that for every p ∈ H Y (s), p = lim Xm (s), p m→+∞

$

s

= S(s)x +

% S(s − τ ) F Y (τ ) + G Y (τ ) u(τ ) dτ, p .

0

This means that Y is the mild solution of (4.8) with u(·) ∈ M0 , Y (0) = x, Y (T ) = y. Since um u in Lp (0, T ; H ) k

λki umk → u i

in Lp (0, T ; H )

(6.6)

i=1

& where for every k 1, ki=1 λki = 1 and inf1ik mki k. Moreover, upon taking another subsequence, we can assume that we have pointwise convergence in (6.6) a.e. on [0, T ]. It now follows from Fatou’s lemma that k T T k L0 u(s) ds = lim L0 λi umk (s) ds k→+∞

0

i

i=1

0

T lim inf

L0

k→+∞

i=1

0

which completes the proof.

k

2

λki umk (s) i

ds lim inf k→+∞

k i=1

T λki

L0 umk (s) ds = α i

0

704


Remark 6.4. Condition (6.3) is satisfied for instance if S(·) is a compact semigroup. We also remark that in the above proof, (2.2) cannot be replaced by (2.8) even if (2.7) is satisfied. 7. Large deviation principle in path space In this section we will show that the sequence {Xn (·)} satisfies the large deviation principle in DH−α [0, T ] for every α > 0, where DH−α [0, T ] is the space of H−α -valued right continuous with left limit functions on [0, T ] equipped with the Skorohod topology. To do it we will follow the method used in [32] which in turn was based on the general program described in [15]. To make the presentation self-contained we had to repeat some arguments of Section 8 in [32]. The main result of this section is the following theorem. Theorem 7.1. Let T > 0. Let (2.1)–(2.5) hold and let in addition the operator B be compact. Then the sequence {Xn (·)} of the solutions of (1.2) satisfies the large deviation principle in DH−α [0, T ] for every α > 0. The proof of the above theorem is based on a result which we state below in the form adapted to our case. Proposition 7.2. (See [15, Corollary 4.29].) Let Z be a complete, separable metric space. Suppose that: (i) The sequence {Xn (·)} is exponentially tight in DZ [0, T ], and (ii) For each 0 t1 · · · tm T and g1 , . . . , gm ∈ Cb (Z) the limit lim

n→+∞

1 log E en(g1 (Xn (t1 ))+···+gm (Xn (tm ))) n

(7.1)

exists. Then the sequence {Xn (·)} satisfies the large deviation principle in DZ [0, T ]. We remark that Corollary 4.29 of [15] also contains a general formula for the rate function based on the existence of limits in (7.1). We refer the readers to [15] for the details. Proof of Theorem 7.1. We need to show that (i) and (ii) of Proposition 7.2 are satisfied for Z = H−α . Condition (i) will follow from Theorems 7.3 and 7.6. The existence of the limit (7.1) is a consequence of Proposition 5.3 and the Markov property of the processes Xn (s). We repeat the arguments from [32]. Since B is compact, the functions in Cb (H−α ) are weakly sequentially continuous on H . Thus, using the Markov property of Xn (·) we have

E en(f1 (Xn (t1 ))+···+fm (Xn (tm )))

= E E en(f1 (Xn (t1 ))+···+fm (Xn (tm ))) Ftm−1

= E en(f1 (Xn (t1 ))+···+Vn (tm −tm−1 )fm (Xn (tm−1 )))


705

= · · · = E en(f1 +Vn (t2 −t1 )(f2 +···+Vn (tm −tm−1 )fm )···)(Xn (t1 )) = enVn (t1 )(f1 +Vn (t2 −t1 )(f2 +···+Vn (tm −tm−1 )fm )···)(x) . By Proposition 5.3, the functions Vn (tm − tm−1 )gm are uniformly bounded, weakly sequentially continuous and they converge uniformly on bounded subsets of H to V (tm − tm−1 )gm , and then Vn (tm−1 − tm−2 )(gm−1 + Vn (tm − tm−1 )gm ) are also uniformly bounded, weakly sequentially continuous and converge uniformly on bounded subsets of H to V (tm−1 − tm−2 )(gm−1 + V (tm − tm−1 )gm ). Continuing this process we therefore obtain

1 log E en(g1 (Xn (t1 ))+···+gm (Xn (tm ))) n = V (t1 ) g1 + V (t2 − t1 ) g2 + · · · + V (tm − tm−1 )gm · · · (x).

lim

n→+∞

2

(7.2)

It remains to show that the sequence {Xn (·)} is exponentially tight in DH−α [0, T ]. As in [32] this will be done with the help of the following theorem. Theorem 7.3. (See [15, Theorem 4.4].) Let Z be a complete, separable metric space. The sequence of Z-valued processes {Xn (·)} is exponentially tight in DZ [0, T ] if and only if: (i) (exponential compact containment) for every M > 0 there exists a compact set KM ⊂ Z such that lim sup n→+∞

1 log P there exists 0 t T such that Xn (t) ∈ / KM −M; n

(7.3)

(ii) there exists a family of functions A ⊂ C(Z) that is closed under additions and separates points in Z such that for each f ∈ A, {f (Xn )} is exponentially tight in DR [0, T ]. For α > 0 let k = [α] + 1. We now define Aα =

m

fi x − xi −k : m ∈ N, xi ∈ H, fi ∈ C 2 [0, +∞) ,

i=1

!

fi (0) = 0, fi , fi , fi

are bounded .

It is clear that the family Aα is closed under addition and, since H is dense in H−α , it separates points in H−α for every α > 0. Moreover, for every f ∈ Aα and r > 0 m

∗ ∗ k A B (x − xi ) fi (x − xi −k ) C(f, r) sup A Df (x) sup x − xi −k xr xr

(7.4)

i=1

for some constant C(f, r) 0. We will need two lemmas. The first one is a version of Ito’s formula for integrals with unbounded integrands.


706

Lemma 7.4. Let 0 t t + s T . Let (2.2)–(2.5) be satisfied and let f ∈ Aα . Then a.s. f Xn (t + s) − f Xn (t) =

t+s

−A∗ Df Xn (τ ) , Xn (τ ) + Df Xn (τ ) , F Xn (τ ) dτ

t

+

t+s

t+s

t

t H \{0}

Df Xn (τ −) , G Xn (τ −) dLn (τ ) +

αn (τ, y) πn (dτ, dy),

(7.5)

where αn (τ, y) = f Xn (τ −) + G Xn (τ −) y − f Xn (τ −) − Df Xn (τ −) , G Xn (τ −) y . Proof. Let Xnm be the solutions of (3.3) with initial conditions Xnm (0) = x. By (3.5), and passing to a subsequence if necessary, we can assume that lim

sup Xnm (s) − Xn (s) = 0

m→+∞ 0sT

for a.e. ω.

(7.6)

By Ito’s formula we have that for a.e. ω f Xnm (t + s) − f Xnm (t) =

t+s

−A∗m Df Xnm (τ ) , Xnm (τ ) + Df Xnm (τ ) , F Xnm (τ ) dτ

t

+

t+s

Df Xnm (τ −) , G Xnm (τ −) dLn (τ )

t

t+s +

m f Xn (τ −) + G Xnm (τ −) y − f Xnm (τ −)

t H \{0}

− Df Xnm (τ −) , G Xnm (τ −) y πn (dτ, dy).

(7.7)

Using (7.6), (3.6), (2.2)–(2.4) and the properties of the function f it is easy to see that for a.e. ω the absolute value of the integrand of the first term in (7.7) is bounded by a constant independent of m, and the absolute value of the integrand of the third term in (7.7) is bounded by C|y|2 for some constant C independent of m. Moreover for a.e. ω these integrands converge for every τ ∈ [t, t + s], y ∈ H to the respective integrands of (7.5). We can thus use the Lebesgue dominated convergence theorem to obtain the convergence of the respective integrals. As regards the second term in (7.7), using the isometric formula, (2.2)–(2.4), the properties of the function f , and finally (7.6), we obtain


707

t+s 2 t+s m m Df Xn (τ −) , G Xn (τ −) dLn (τ ) − Df Xn (τ −) , G Xn (τ −) dLn (τ ) E t

t

CE

t+s m X (s) − Xn (s)2 ds → 0 as m → +∞. n t

Therefore, passing to a subsequence if necessary, we can assume that for a.e. ω t+s

Df

Xnm (τ −)

, G Xnm (τ −) dLn (τ ) →

t

t+s

Df Xn (τ −) , G Xn (τ −) dLn (τ )

t

as m → +∞. Therefore, letting m → +∞ along a subsequence in (7.7) gives us (7.5).

2

The second lemma is a simplified restatement of Corollary 5.2.2 from [4] suitable for our purposes. We remark that even though Corollary 5.2.2 in [4] was formulated for the case when ν is a measure on R, its conclusion is true when ν and πˆ (called N˜ in [4]) are defined on any measure space. Lemma 7.5. Let (2.5) be satisfied and let h(s, y) be a predictable field (see [4, page 192]) such that t E

h(s, y)2 ν(dy) ds < +∞,

t E

eh(s,y) ν(dy) ds < +∞. 0 y1

0 H

Let t Y (t) =

t h(s, y) πˆ (ds, dy) −

0 H \{0}

h(s,y) e − 1 − h(s, y) ν(dy) ds.

0 H

Then the process eY (t) is a local martingale. Theorem 7.6. Let (2.1)–(2.5) be satisfied and let the operator B be compact. Then the sequence {Xn (·)} is exponentially tight in DH−α [0, +∞) for every α > 0. Proof. We need to verify conditions (i) and (ii) of Theorem 7.3 with Z = H−α and Aα defined above. The exponential compact containment condition (7.3) follows from (3.6). Since B is compact, if KM = {x r} then KM is compact in H−α , and moreover, by (3.6), 1 / KM log P there exists 0 t T such that Xn (t) ∈ n 1 log en(c2 −c1 r) = c2 − c1 r −M n if r = (M + c2 )/c1 .


708

It remains to show that for every f ∈ Aα , {f (Xn )} is exponentially tight in DR [0, T ]. To do this, according to [15, Theorem 4.1] and the fact that e|f | ef + e−f , it is enough to show that for s > 0, λ ∈ R there exist random variables γn (s, λ), nondecreasing in s, such that for 0 t t + s T

E enλ(f (Xn (t+s))−f (Xn (t))) Ft E eγn (s,λ) Ft ,

(7.8)

and lim lim sup

s→0 n→+∞

1 log E eγn (s,λ) = 0. n

(7.9)

Let r > 0 be big enough so that if Ω1n = ω: there exists 0 t T such that Xn (t) > r then P Ω1n e−2n|λ|f ∞ .

(7.10)

Denote Ω2n = Ω \ Ω1n . We have

E enλ(f (Xn (t+s))−f (Xn (t))) Ft E enλ(f (Xn (t+s))−f (Xn (t))) 1Ω1n Ft

+ E enλ(f (Xn (t+s))−f (Xn (t))) 1Ω2n Ft = J1 + J2 .

(7.11)

Obviously

J1 E e2n|λ|f ∞ 1Ω1n Ft . Now, since Ln (τ ) =

(7.12)

τ 0

ˆ n (du, dy), H \{0} y π

t+s

t+s

t

t H \{0}

Df Xn (τ −) , G Xn (τ −) dLn (τ ) =

Df Xn (τ −) , G Xn (τ −) y πˆ n (dτ, dy).

It does follow directly from the definition of the transport measure that for (nonnegative) Borel measurable functions ψ

ψ(y) νn (dy) = n

H

ψ

y ν(dy). n

(7.13)

H

Thus (7.13) is also true for Borel measurable functions ψ such that |ψ(y)| Cy2 eKy for some C, K. Using Lemma 7.4, properties of f , (7.4), and (2.2), we have


709

t+s ∗ J2 = E enλ t (−A Df (Xn (τ )),Xn (τ ) +Df (Xn (τ )),F (Xn (τ )) ) dτ t+s

t+s ˆ n (dτ,dy)+ t H \{0} Df (Xn (τ −)),G(Xn (τ −))y π H \{0} αn (τ,y) πn (dτ,dy)]

1Ω2n Ft t+s t+s

en|λ|C(1+r)s E enλ[ t H \{0} α˜ n (τ,y) πˆ n (dτ,dy)+ t H αn (τ,y) νn (dy) dτ ] 1Ω2n Ft , · enλ[

t

where α˜ n (τ, y) = Df Xn (τ −) , G Xn (τ −) y + αn (τ, y). Now, by Lemma 2.2, 1 1 αn (τ, y) =

D 2 f Xn (τ −) + ησ G Xn (τ −) y G Xn (τ −) y, G Xn (τ −) y σ dη dσ.

0 0

(7.14) Thus, (7.13), (7.14) and (2.4), (2.5) yield t+s

t+s αn (τ, y) νn (dy) dτ 1Ω2n = n λ 2

nλ t

t

H

y ν(dy) dτ 1Ω2n αn τ, n

H

t+s n2 |λ|

C t

|y|2 ν(dy) dτ C|λ|s. n2

H

Therefore

t+s J2 en|λ|C(1+r)s+C|λ|s E enλ t H \{0} α˜ n (τ,y) πˆ n (dτ,dy) Ft

en|λ|C(1+r)s+C|λ|s E Mn (t, t + s)eΛn (s) Ft ,

(7.15)

where Mn (t, t + s) := enλ

t+s t

˜ n (τ,y) πˆ n (dτ,dy)−Λn (s) H \{0} α

,

and t+s Λn (s) = t

nλα˜ (τ,y) e n − 1 − nλα˜ n (τ, y) νn (dy) dτ.

H

The purpose of the above decomposition in (7.15) is to us the fact that, according to Lemma 7.5, Mn (t, t + s) is a local martingale with respect to the filtration Ft+s . We notice that, since Df and G are bounded, α˜ n (τ, y) C|y|.

(7.16)


710

By (7.13) we have t+s y nλα˜ n (τ, yn ) e n ν(dy) dτ. − 1 − nλα˜ n τ, Λn (s) = n t

H

Applying Lemma 2.2 with f (x) = ex , and using (7.16) and (2.5), we thus have t+s 1 1 Λn (s) =

e t

H

ησ nλα˜ n (τ, yn )

2 y nλα˜ n τ, σ dη dσ n ν(dy) dτ n

0 0

t+s

neC|λ||y| C|λ|2 |y|2 ν(dy) dτ nCλ s. t

(7.17)

H

Using (7.17) in (7.15) we thus obtain

J2 enC(f,λ,r)s E Mn (t, t + s)Ft

(7.18)

for some constant C(f, λ, r). Let τk be a nondecreasing sequence of stopping times such that limk→+∞ τk = +∞ a.s. and Mn (t, (t + s) ∧ τk ) is a martingale respect to Ft+s for every k 1. By Conditional Fatou’s lemma (see for instance [34, page 88]), ' (

E Mn (t, t + s)Ft = E lim Mn t, (t + s) ∧ τk Ft k→+∞

lim inf E Mn t, (t + s) ∧ τk Ft = 1. k→+∞

Therefore J2 enC(f,λ,r)s . It thus follows from (7.11), (7.12) and (7.19) that

E enλ(f (Xn (t+s))−f (Xn (t))) Ft E e2n|λ|f ∞ 1Ω1n + enC(f,λ,r)s Ft . We can now take γn (s, λ) = log enC(f,λ,r)s + e2n|λ|f ∞ χΩ1n . Then, by (7.10) and the inequality log(1 + x) x for x > 0,

1 1 log E eγn (s,λ) = log E e2n|λ|f ∞ 1Ω1n + enC(f,λ,r)s n n 1 1 log 1 + enC(f,λ,r)s C(f, λ, r)s + log 1 + e−nC(f,λ,r)s n n 1 −nC(f,λ,r)s C(f, λ, r)s + e , n which obviously implies (7.9).

2

(7.19)


711

8. Examples of noise processes We will consider two specific cases of small perturbations: compound Poisson processes and subordinated Wiener processes. We will try to calculate the functions

p,z e − 1 − p, z ν(dz),

H0 (p) =

(8.1)

H

L0 (z) = sup z, y − H0 (y) .

(8.2)

y∈H

8.1. Compound Poisson noise Let L be a compound Poisson process with the Gaussian jump measure ν = N (0, Q) with the trace class covariance operator Q 0, Tr Q < +∞. It is easy to see, compare also Proposition 4.18 in [25], that the operator Q is identical with the covariance of L. It is well known, see e.g. [11], that in this specific case for each k > 0 ekz ν(dz) < +∞.

(8.3)

H

To calculate the function H0 (·) remark that for a random variable ξ such that L(ξ ) = ν,

2 p, z 2 ν(dz) = Ep, ξ 2 = Qp, p = Q1/2 p .

Moreover, for a real-valued random variable η such that L(η) = N (0, 1), 1 2

Eeλη = e 2 λ ,

λ ∈ R.

Consequently

ep,z ν(dz) = EeηQ

1/2 p

1

= e 2 Qp,p .

(8.4)

H

Thus, in the present situation 1

1

1

H0 (p) = e 2 Qp,p − 1 = e 2 Q 2 p − 1. 2

(8.5)

We denote by Q−1/2 the pseudo inverse of Q1/2 . Since Q1/2 is self-adjoint we have an orthogonal decomposition H = Im Q1/2 × Ker Q1/2 and we notice that Q−1/2 z is the unique element p0 ∈ Im Q1/2 such that Q−1/2 p0 = z. For x ∈ H we will write x = x0 + x ⊥ to indicate the orthogonal decomposition of x. We have the following general result.


712

Proposition 8.1. Assume that 1 H0 (p) = h Q 2 p ,

p ∈ H,

where Q is a trace class nonnegative operator and h is a convex,even function with the Legendre transform l. Then the Legendre transform L0 of H0 is of the form: L0 (z) =

l(Q−1/2 z),

if z ∈ Im Q1/2 ,

+∞,

if z ∈ / Im Q1/2 .

Proof. Let z = z0 + z⊥ . If z⊥ = 0 then

L0 (z) = sup z, p − h Q1/2 p p

sup p ⊥ ∈Ker Q1/2

z⊥ , p ⊥ − h(0) = +∞.

¯ p¯ ∈ Im Q1/2 = H1 , then If z = Q1/2 p,

L0 (z) = sup z, p − h Q1/2 p = sup p, ¯ Q1/2 p − h Q1/2 p p

p

'

( = sup p, ¯ v − h v = sup sup p, ¯ v − h(t) t0 v=t

v∈H1

v ¯ t − h(t) = sup pt ¯ − h(t) = l p ¯ = l Q−1/2 z , = sup sup p, v t0 v=t t0 as required. Let now z ∈ Im Q1/2 \ Im Q1/2 . When restricted to Im Q1/2 , Q1/2 is a positive, self-adjoint, compact operator and Q−1/2 exists in the usual sense. Let {e1 , e2 , . . .} be an orthonormal basis & of Im Q1/2 composed of eigenvectors of Q1/2 . Then zn = ni=1 z, ei ei ∈ Im Q1/2 . Let Hn be the linear subspace of H spanned by the vectors {e1 , . . . , en } and p = pn + pn⊥ , z = zn + zn⊥ be the orthogonal decompositions of p and z with respect to Hn and Hn⊥ . Thus L0 (z) = sup

pn +pn⊥

z, pn + pn⊥ − h Q1/2 pn + pn⊥ sup z, pn − h Q1/2 pn

pn

sup zn + zn⊥ , pn − h Q1/2 pn sup zn , pn − h Q1/2 pn pn

pn

1 = sup zn , p − h Q1/2 p = l Q− 2 zn . p

1

But the sequence (Q− 2 zn ) tends to +∞ and since l(+∞) = +∞, L(z) = +∞, as required. 2 As a corollary we get the following proposition.


713

Proposition 8.2. Assume that H0 is given by (8.5). Let f : R1+ → R1+ be the inverse function to 1

2

g(σ ) = σ e 2 σ , σ 0. Then 1

L0 (z) =

([f (Q−1/2 z)]2 − 1)e 2 [f (Q +∞,

−1/2 z)]2

+ 1,

if z ∈ Im Q1/2 , if z ∈ / Im Q1/2 .

Remark 8.3. It is immediate that f is a concave function and for every 0 < a < 2 we have √ √ a ln x f (x) 2 ln x,

for large x.

8.2. Subordinated Wiener process Take L(t) = W (Z(t)), t 0, where W is a Wiener process on H , say L(W (1)) = N (0, QW ) and Z is a subordinator with the jump measure ρ on [0, +∞). Thus Z is an increasing process starting from 0 and such that Ee−λZ(t) = e−tψ(λ) ,

λ 0,

+∞

1 − e−λσ ρ(dσ ),

ψ(λ) = γ λ +

λ 0,

(8.6)

0

1 +∞ where γ 0 and 0 σ ρ(dσ ) < +∞, 1 ρ(dσ ) < +∞. If γ = 1, ρ ≡ 0, then Z(t) = t, t 0 and we have L identical with the Wiener process W . We will assume that γ = 0, find the function H0 and check under what assumptions on ρ the crucial condition (2.5) is satisfied. It is well known, see e.g. [29,25], that for the Lévy process L, the measure ν is of the form +∞ ν= N (0, tQW ) ρ(dt).

(8.7)

0

By direct calculations we get that the covariance operator Q of L is equal to +∞

Q= t ρ(dt) QW = EZ(1) QW .

(8.8)

0

To simplify notation we will assume that EZ(1) = 1, Therefore,

and then QW = Q.

(8.9)

714


H0 (p) =

ρ,z e − 1 ν(dz) =

H

+∞ 0

ρ,z e − 1 N (0, tQ)(dz) ρ(dt)

H

+∞ 1 tQp,p e2 − 1 ρ(dt). = 0

Thus H0 (p) = h Q1/2 p ,

+∞ 1 tu2 where h(u) = e 2 − 1 ρ(dt),

u 0,

0

and Proposition 8.1 applies. An explicit formula for L0 can be easily derived. Note that z e

I=

2 κz

+∞ 2 κz ν(dz) = ρ(dt) z e N (0, tQ)(dz)

H

0

H

+∞ 2

ρ(dt) E W (t) eκW (t) . = 0

√ But L(W (t)) = L( tW (1)). Therefore +∞ 2 √

I= t ρ(dt) EW (1) eκ tW (1) . 0

We will need the following lemma. Lemma 8.4. There exists a > 0 such that for all s 0, 2

EesW (1) eas . Proof. By [20, page 55], there exists δ > 0 such that 2 P W (1) > u e−δu ,

u > 0.

Therefore E s sW (1) =

+∞ +∞ ln u sW (1) du. P e u du = 1 + P W (1) > s 0

Note that

1

(8.10)


715

+∞ +∞ ln u 2 du P W (1) > e−δ(ln u/s) du. s 1

Substituting v =

ln u s ,

1

du = us dv = sevs dv,

+∞ +∞ +∞ 2 −δ(ln u/s)2 −δv 2 vs −δ(v−s/(2δ))2 e du = s e e dv = s e dv es /(4δ) 1

0

0

+∞ 2 −δv 2 s e dv es /(4δ) . −∞

The required result now follows.

2

Proposition 8.5. If +∞ t ρ(dt) = 1

+∞ eλt ρ(dt) < +∞,

and

0

λ 0,

1

then the measure ν given by (8.7) satisfies (2.5) and H0 is given by (8.10). Proof. It is enough to remark that 2 √ 4 1/2 2κ √tW (1) 1/2 a 2 EW (1) eκ tW (1) EW (1) Ee ce 2 κ t .

2

Example 8.6. The assumptions of the above proposition are satisfied if, for instance, ρ(dt) =

1 t 1+α

e−t dt 2

for α < 1.

In some cases asymptotic behavior of the function ψ can be determined. Example 8.7. ρ(dt) = 1[0,1] (t) 1 −ψ(−λ) = 0

After substitution, λσ = u, for λ > 1,

1 dt, t 1+α

α < 1,

λσ 1 e − 1 1+α dσ. σ


716

1

e

λσ

1 1 − 1 1+α dσ = λ σ

λ

0

u e −1

0

1 λ

α

1 α u 1+α du = λ (λ)

eu − 1 1 · α du + u u

0

λ

u 1 e − 1 1+α du u

0

λ u

e du . 1

Thus, for large λ, 1

λσ 1 e − 1 1+α dσ ∼ cλα eλ . σ

0

Remark 8.8. In the considered examples, the Legendre transforms L0 of H0 were of the form 1 l(Q− 2 z), z ∈ H . Thus the control system, which defines the rate function, can be written in a more convenient way, X (s) = −AX(s) + F X(s) + G X(s) Q1/2 u(s),

X(t) = x,

(8.11)

and to find the rate function one has to look for the infimum of the cost functional J x; u(·) =

T

l u(s) ds + g X(T )

0

over all controls u(·) ∈ M0 . 9. Stochastic PDE of hyperbolic type We present an example of a class of stochastic PDE which can be handled by the developed theory. To begin consider a nonlinear stochastic wave equation which can be formally written as ⎧ 2 ∂ ∂ u ⎪ ⎪ (t, ξ ) = u(t, ξ ) + f u(t, ξ ) + L˜ n (t, ξ ), ⎪ ⎪ 2 ∂t ⎪ ∂t ⎪ ⎨ u(t, ξ ) = 0, ⎪ u(0, ξ ) = u0 (ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂u (0, ξ ) = v (ξ ), 0 ∂t

t > 0, ξ ∈ O, t > 0, ξ ∈ ∂O, ξ ∈ O,

(9.1)

ξ ∈ O,

with L˜ n , L2 (O)-valued Lévy process (properly normalized), O a bounded regular domain in Rd , f : R → R is a Lipschitz function and u0 ∈ H01 (O), v0 ∈ L2 (O). Setting X(t) =

u(t) , v(t)

t 0,


717

we can rewrite (9.1) in an abstract way: dX(t) =

0 I X(t) + F X(t) dt + dLn (t), −A 0

(9.2)

where u 0 , F = v F1 (u)

0 Ln (t) = L˜ n (t)

(9.3)

and A = − in H = L2 (O) with D(A) = H 2 (O) ∩ H01 (O). Moreover the same setup applies to other equations of hyperbolic type. Therefore let us assume that A in (9.2) is a strictly positive, self-adjoint operator in a Hilbert space H with a bounded inverse. It is then well known that the operator A=

0 A

−I 0

,

D(A) =

is maximal monotone in the Hilbert space H = type inner product

D(A1/2 ) × H

D(A) × D(A1/2 )

, equipped with the following “energy”

u u¯ , = A1/2 u, A1/2 u¯ H + v, v ¯ H, v v¯ H

u u¯ , ∈ H. v v¯

Moreover, A∗ = −A. It is easy to check that the operator B=

A−1/2 0

0

A−1/2

is bounded, positive, self-adjoint on H, and such that A∗ B is bounded. Moreover (2.1) holds with constant c0 = 1. In fact 2 2 ∗ u u u u A +I B , = B , = A1/4 u + A−1/4 v . v v H v v H In particular we see that u 1/4 2 −1/4 2 1/2 v . v = A u + A −1 Thus F =

0 F1

is Lipschitz from H−1 into H (condition (2.2)) if and only if

−1/4 A F1 (u) − F1 (u) ¯ H cA1/2 (u − u) ¯ , It is easy to see that if

u, u¯ ∈ D A1/2 .

(9.4)

718


F1 (u)(ξ ) = f u(ξ ) ,

ξ ∈ O,

and f is a Lipschitz function, then (9.4) is satisfied. Acknowledgment The authors would like to thank the referee for comments which helped to improve the final version of the paper. Appendix A. Proof of Proposition 3.3 Let us recall that the spaces X , L were introduced in Section 4. Define, for each ψ ∈ L, processes t K(ψ)(t) =

S(t − s)ψ(s) dL(s),

t ∈ [0, T ],

0

t Kλ (ψ)(t) =

Sλ (t − s)ψ(s) dL(s),

λ > 0, t ∈ [0, T ].

0

We can treat K and Kλ as linear transformations from the space L into X . We prove this now and establish that there exists a constant C1 > 0 such that Kλ C1

for λ > 1.

(A.1)

ˆ be the extensions, In the proof we omit the subscript λ. Let Hˆ , and the unitary semigroup S, respectively of H and of the semigroup S, given by the dilation theorem, see e.g. [25, Theorem 9.24]. Thus H → Hˆ is an isometry and the semigroup S is the restriction of P Sˆ to H , where P is the orthogonal projection of Hˆ onto H . Therefore we have t

t S(t − s)ψ(s) dL(s) =

0

ˆ − s)ψ(s) dL(s) = P S(t) ˆ P S(t

t

0

ˆ S(−s)ψ(s) dL(s),

t ∈ [0, T ].

0

Moreover the process Yˆ (t) =

t

ˆ S(−s)ψ(s) dL(s),

t 0,

0

is an Hˆ martingale and therefore has càdlàg modification. This implies that the stochastic convolution has H -valued, càdlàg modifications and t S(t − s)ψ(s) dL(s) Yˆ (t)Hˆ , 0

t ∈ [0, T ].


719

However, Yˆ (t)Hˆ , t ∈ [0, T ], is a submartingale and by the classical Doob inequality for all p>1 E

p sup Yˆ (t) ˆ

H

0tT

p p−1

p

p EYˆ (T ) ˆ . H

In particular t 2 E sup S(t − s)ψ(s) dL(s) 0tT

H

0

2 T ˆ 4E S(−s)ψ(s) dL(s) ˆ

H

0

T 4E

1/2 2 S(−s)ψ(s)Q ˆ

LHS

T ds 4E (H,Hˆ )

0

ψ(s)Q1/2 2 ds. HS

0

Thus the existence of the constant C1 follows, and by the Banach–Steinhaus theorem it is enough to establish (3.9) for a dense set of ψ . Lemma A.1. For each k = 1, 2, . . . the set T Lk = ψ ∈ L: E

! k A ψ(u)Q1/2 2 du < +∞ HS

0

is dense in L. Proof. Let ψ ∈ L. Since for μ > 0 the operator μARμ is bounded we have T E

k A (μRμ )k ψ(u)Q1/2 2 du = E HS

0

T

(μARμ )k ψ(u)Q1/2 2 du < +∞, HS

0

and thus (μRμ )k ψ ∈ Lk . Moreover it follows from (2.9) that (μRμ )k − I ψ(u)Q1/2 2 C ψ(u)Q1/2 2 , HS HS and limμ→+∞ (μRμ )k x = x for every x ∈ H . Therefore the dominated convergence theorem yields T lim E

μ→+∞

0

(μRμ )k − I ψ(u)Q1/2 2 du = 0. HS

2


720

Lemma A.2. Assume that M(t), t 0, is a D(A)-valued process with locally bounded trajectories, H -square integrable martingale, and M(0) = 0. Then t

t S(t − s) dM(s) = M(t) −

0

S(t − s)AM(s) ds.

(A.2)

0

Proof. Let e ∈ D((A∗ )2 ) and ϕ(s, x) = S(t − s)x, e = x, S ∗ (t − s)e . Then ϕ ∈ C 2 ((−∞, t) × H ) and has uniformly continuous derivatives. In fact it can be extended to a function in C 2 (R × H ) in an obvious way. Therefore, applying Ito’s formula for Hilbert space valued semimartingales (see [22, Theorem 27.2] or [25, Theorem D2]) we obtain

t

M(t), e =

t

S(t − s)AM(s), e ds +

0

Applying Lemma A.2 to the martingale M(t) = following lemma. T 0

S(t − s) dM(s), e ds

0

which proves the claim since D((A∗ )2 ) is dense in H .

Lemma A.3. If E

t 0

2 ψ(u) dL(u), t ∈ [0, T ] we arrive at the

Aψ(u)Q1/2 2HS du < +∞ then for all t ∈ [0, T ], λ > 0,

t

t S(t − s)ψ(s) dL(s) =

0

ψ(s) dL(s) − 0

t

t Sλ (t − s)ψ(s) dL(s) =

0

s

t

S(t − s)

Aψ(u) dL(u) ds,

0

0

t

s

ψ(s) dL(s) − 0

Sλ (t − s)

Aλ ψ(u) dL(u) ds.

0

0

We can now continue the proof of the theorem. We will show that (3.9) holds for every ψ ∈ L2 . Note that s

t Kψ(t) − Kλ ψ(t) =

S(t − s) 0

−Aψ(u) dL(u) + 0

t +

s

S(t − s) − Sλ (t − s)

0

= Iλ1 ψ(t) + Iλ2 ψ(t).

Aλ ψ(u) dL(u) ds 0

s

−Aλ ψ(u) dL(u) ds

0


721

Now t Iλ1 ψ(t) =

s S(t − s)(Aλ − A)

0

ψ(u) dL(u) ds, 0

and sup Iλ1 ψ(t)

0tT

T s (A − Aλ ) ψ(u) dL(u) ds. 0

0

But (A − Aλ )x = Rλ A2 x λ1 A2 x, x ∈ D(A2 ). Therefore, since T E

2 A ψ(u)Q1/2 2 du < +∞, HS

0

we have, by isometric identity, 2 E sup Iλ1 ψ(t) E 0tT

2 T s (A − Aλ ) ψ(u) dL(u) ds 0

0

T T

s E

0

1 2T λ

0

T

s E

0

(A − Aλ )ψ(u)Q1/2 2 du ds HS

1 2 T λ2

2 A ψ(u)Q1/2 2 du ds HS

0

T

2 EA2 ψ(u)Q1/2 HS du.

0

Therefore, if (A.3) holds, 2 lim EIλ1 ψ(t) = 0.

λ→+∞

Since for every x ∈ D(A), λ > 0, Sλ (t)x − S(t)x tAλ x − Ax (see for instance [23, page 10]), we have

(A.3)


722

2 t s

S(t − s) − Sλ (t − s) Aλ ψ(u) dL(u) ds 0tT

2 sup Iλ2 ψ(t) sup

0tT

0

t sup 0tT

0

2 s (t − s)(A − Aλ )Aλ ψ(u) dL(u) ds

0

0

2 t s 2 T sup (A − Aλ )Aλ ψ(u) dL(u) ds . 0tT 0

0

Moreover, (A − Aλ )Aλ = (A − λRλ A)λRλ A = λRλ (I − λRλ )A2 . Therefore 2 E sup Iλ2 ψ(t) T 2 E 0tT

2 T s 2 ψ(u) dL(u) ds (I − λRλ )A 0

T

0

s

T 2E

(I − λRλ )A2 ψ(u)Q1/2 2 ds du HS

0 0

T T E 3

(I − λRλ )A2 ψ(u)Q1/2 2 du. HS

0

Thus, if (A.3) holds, we can conclude by the dominated convergence theorem that 2 lim E sup Iλ2 ψ(t) = 0.

λ→+∞

0tT

This finishes the proof of the proposition. References [1] A. de Acosta, Exponential moments for vector valued random series and triangular arrays, Ann. Probab. 8 (1980) 381–389. [2] A. de Acosta, Large deviations for vector-valued Lévy processes, Stochastic Process. Appl. 51 (1994) 75–115. [3] A. de Acosta, A general non-convex large deviation result with applications to stochastic equations, Probab. Theory Related Fields 118 (2000) 483–521. [4] D. Applebaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge Stud. Adv. Math., vol. 116, Cambridge University Press, 2009. [5] J. Bertoin, Lévy Processes, Cambridge University Press, 1996. [6] S. Cerrai, M. Rockner, Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab. 32 (2004) 1–40. [7] F. Chenal, A. Millet, Uniform large deviations for parabolic SPDE’s and applications, Stochastic Process. Appl. 72 (1997) 161–187.


723

[8] P.L. Chow, Large deviation problem for some parabolic Ito equations, Comm. Pure Appl. Math. 45 (1) (1992) 97–120. [9] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton–Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms, J. Funct. Anal. 90 (1990) 237–283. [10] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton–Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions, J. Funct. Anal. 97 (1991) 417–465. [11] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, MA, 1992. [12] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley Ser. Probab. Stat., John Wiley & Sons, Inc., New York, 1997. [13] I. Ekeland, G. Lebourg, Generic Frechet-differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1977) 193–216. [14] J. Feng, Large deviations for diffusions and Hamilton–Jacobi equations in Hilbert spaces, Ann. Probab. 34 (1) (2006) 321–385. [15] J. Feng, T. Kurtz, Large Deviations for Stochastic Processes, Math. Surveys Monogr., vol. 131, American Mathematical Society, Providence, RI, 2006. [16] M.I. Freidlin, Random perturbations of reaction–diffusion equations: The quasi deterministic approximation, Trans. Amer. Math. Soc. 305 (1988) 665–697. [17] H. Ishii, Viscosity solutions for a class of Hamilton–Jacobi equations in Hilbert spaces, J. Funct. Anal. 105 (1992) 301–341. [18] G. Kallianpur, J. Xiong, Large deviations for a class of stochastic partial differential equations, Ann. Probab. 24 (1996) 320–345. [19] V.M. Kruglov, Integrals with respect to infinitely divisible distributions in Hilbert spaces, Mat. Zametki 11 (1972) 669–676. [20] S. Kwapień, W.A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Probab. Appl., Birkhäuser Boston, Inc., Boston, MA, 1992. [21] X. Li, J.M. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Birkhäuser Boston, Cambridge, MA, 1995. [22] M. Métivier, Semimartigales, de Gruyter, 1982. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, 1983. [24] S. Peszat, Large deviation principle for stochastic evolution equations, Probab. Theory Related Fields 98 (1994) 113–136. [25] S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia Math. Appl., vol. 113, Cambridge University Press, Cambridge, MA, 2007. [26] P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, 1995. [27] M. Renardy, Polar decomposition of positive operators and a problem of Crandall and Lions, Appl. Anal. 57 (3–4) (1995) 383–385. [28] M. Röckner, T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal. 26 (2007) 255–279. [29] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. [30] R. Sowers, Large deviations for a reaction–diffusion equation with non-Gaussian perturbation, Ann. Probab. 20 (1992) 504–537. [31] S.S. Sritharan, P. Sundar, Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (11) (2006) 1636–1659. ´ ech, A PDE approach to large deviations in Hilbert spaces, Stochastic Process. Appl. 119 (4) (2009) 1081– [32] A. Swi˛ 1123. [33] A.D. Wentzell, Theorems on Large Deviations for Markov Stochastic Processes, Kluwer, 1990. [34] D. Williams, Probability with Martingales, Cambridge University Press, 1991. [35] T. Xu, T. Zhang, Large deviation principles for 2-D stochastic Navier–Stokes equations driven by Lévy processes, J. Funct. Anal. 257 (2009) 1519–1545.


Moment asymptotics for the parabolic Anderson problem with a perturbed lattice potential Ryoki Fukushima a,∗,1,2 , Naomasa Ueki b,3 a Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan b Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan

Received 15 March 2010; accepted 23 October 2010 Available online 3 November 2010 Communicated by Daniel W. Stroock

Abstract The parabolic Anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied. The moment asymptotics of the total mass of the solution is derived. The results show that the total mass of the solution concentrates on a small set in the space of configuration. © 2010 Elsevier Inc. All rights reserved. Keywords: Brownian motion; Random media; Perturbed lattice; Parabolic Anderson problem

1. Introduction This paper is a continuation of [4]. We consider the initial value problem of the heat equation with a random potential ∂ 1 v(t, x) = v(t, x) − Vξ (x)v(t, x), ∂t 2

(t, x) ∈ (0, ∞) × Rd ,


E-mail addresses: [email protected] (R. Fukushima), [email protected] (N. Ueki). 1 Current address: Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan. 2 Partially supported by JSPS Fellowships for Young Scientists. 3 Partially supported by KAKENHI (21540175).


(1.1)

R. Fukushima, N. Ueki / Journal of Functional Analysis 260 (2011) 724–744

v(0, x) = δx0 (x),

725

x ∈ Rd ,

where is the Laplacian, x0 ∈ Rd , and Vξ (x) :=

u(x − q − ξq )

(1.2)

q∈Zd

with ξ = (ξq )q∈Zd a collection of independent and identically distributed random vectors. Under appropriate assumptions, (1.2) has a solution vξ (t, x; x0 ) represented by the Feynman–Kac formula vξ (t, x; x0 ) = Ex0

t exp − Vξ (Bs ) ds Bt = x

1 |x − x0 |2 , exp − 2t (2πt)d/2

(1.3)

0

where (Bs )s0 is the Brownian motion on Rd and Ex0 is the expectation of the Brownian motion starting at x0 . In this paper, we investigate the long time asymptotics of the moment of the total mass

vξ (t; x0 ) :=

vξ (t, x; x0 ) dx0 = Ex0

t exp − Vξ (Bs ) ds .

Rd

(1.4)

0

Our main result is Theorem 1.2, which deals with the first moment. We also obtain results on the higher moments in Section 3 below. The operator Hξ = −/2 + Vξ is the Hamiltonian of the so-called random displacement model in the theory of random Schrödinger operators and there has recently been an increase in research, see e.g. [1–4,9]. Also, the initial value problem (1.2) itself is called the “parabolic Anderson problem” in literature (see e.g. a survey article by Gärtner and König [7]). The solution of the parabolic Anderson problem is believed to concentrate on a relatively small region and there are many results support this concentration. We shall discuss this aspect in more detail in Subsection 3.2 below. 1.1. Basic assumptions We are mainly interested in the case where the single site potential and the displacement variables satisfy the following: (i) u is a nonnegative function belonging to the Kato class Kd (cf. [11]) and u(x) = C0 |x|−α 1 + o(1)

(1.5)

as |x| → ∞ for some α > d and C0 > 0; (ii) each ξq has the explicit distribution Pθ (ξq ∈ dx) =

1 exp −|p|θ δp (dx) Z(d, θ ) d p∈Z

for some θ > 0 and the normalizing constant Z(d, θ ).

(1.6)

726


We also consider the case that u is a nonpositive function. For this case, we assume inf u = u(0) > −∞, (1.5) for some C0 < 0, and that for any ε > 0, there exists Rε > 0 such that u(x) u(0) + ε for |x| < Rε . Nevertheless, our main interest is the nonnegative case and we assume u 0 unless otherwise specified. 1.2. Motivation In Theorem 6.3 of the preceding paper [4], we have shown the following: Theorem 1.1. Let us define

c(d, α, θ, C0 ) :=

dq inf

y∈Rd

Rd

C0 θ . + |y| |q + y|α

(1.7)

(i) Assume that d = 1 and that ess infB(R) u > 0 for any R 1 if α 3. Then we have ⎧ ⎪ ∼ −t (1+θ)/(α+θ) c(1, α, θ, C0 )

⎨ log Eθ vξ (t; x0 ) −t (1+θ)/(3+θ) ⎪ ⎩ π 2 (1+θ)/(3+θ) ∼ −t (1+θ)/(3+θ) 3+θ 1+θ ( 8 )

(1 < α < 3), (α = 3),

(1.8)

(α > 3)

as t → ∞, where f (t) ∼ g(t) means limt→∞ f (t)/g(t) = 1 and f (t) g(t) means 0 < lim t→∞ f (t)/g(t) limt→∞ f (t)/g(t) < ∞. (ii) Assume that d = 2 and that ess infB(R) u > 0 for any R 1 if α 4. Then we have ⎧ ∼ −t (2+θ)/(α+θ) c(2, α, θ, C0 )

⎨ log Eθ vξ (t; x0 ) −t (2+θ)/(4+θ) ⎩ −t (2+θ)/(4+θ) (log t)−θ/(4+θ)

(2 < α < 4), (α = 4), (α > 4)

(1.9)

as t → ∞. (iii) Assume that d 3 and that ess infB(R) u > 0 for any R 1 if α d + 2. Then we have

∼ −t (d+θ)/(α+θ) c(d, α, θ, C0 ) log Eθ vξ (t; x0 ) −t (d+θμ)/(d+2+θμ)

(d < α < d + 2), (α d + 2)

(1.10)

as t → ∞, where μ=

2(α − 2) . d(α − d)

(1.11)

(iv) Assume u 0, sup u = u(0) > −∞, and the existence of Rε > 0 for any ε > 0 such that ess supB(Rε ) u u(0) + ε. Then we have

log Eθ vξ (t; x0 ) ∼ t 1+d/θ c− d, θ, u(0) as t → ∞, where

(1.12)


c− (d, θ, K) :=

2π d/2 θ |K|1+d/θ d(d + θ )Γ (d/2)

727

(1.13)

for K ∈ R. We have precise forms of the leading terms for the one-dimensional case with α = 3, the general dimensional case with d < α < d + 2, and the case of u 0. Furthermore, if one goes into the proof of these results, it will be observed that only a very small set in ξ -space contributes the leading terms of the asymptotics. More precisely, when u 0 and d < α < d + 2 for instance, the y-variable in the definition of c(d, α, θ, C0 ) corresponds to the displacement ξq from q. Therefore taking the infimum in the definition of c(d, α, θ, C0 ) with respect to y means minimizing the sum of the contribution of u(−q − ξq ) to Vξ (0) and the cost for displacement for each q. With these interpretation, the above theorem says that only the optimal configuration contributes the leading term. This kind of concentration in ξ -space is sometimes regarded as a collateral evidence of the aforementioned spatial irregularity of vξ (t, x; x0 ), see Subsection 1.3 of [7]. The aim of this paper is to find a variational expression for the leading part in the remaining cases to see a concentration phenomenon similar to above. 1.3. Main result We need to introduce some notations to state the results. We write Λr for [−r/2, r/2]d and introduce scaling factors ⎧ 1/(3+θ) ⎨t r = t 1/(4+θ) (log t)θ/(8+2θ) ⎩ 1/(d+2+μθ) t

(d = 1 and α = 3), (d = 2 and α > 4), (d 3 and α d + 2 or (d, α) = (2, 4)).

(1.14)

For any open set U and ξ = (ξq )q∈Zd ∈ (Zd )Z , we denote by λrξ (U ) the bottom of the spectrum of d

1 − + Vξr 2 in U with the Dirichlet boundary condition, where Vξr (x) :=

r 2 u(rx − q − ξq ).

q∈Zd

Finally, let Ωt = (Zd )Λt ∩Z , which is the set of possible configurations of (ξq )q∈Λt ∩Zd , and we write λrξ (U ) for the same object as above also for ξ ∈ Ωt with the potential replaced by d

Vξr (x) :=

r 2 u(rx − q − ξq ).

q∈Zd ∩Λt

Theorem 1.2. Assume that α = 3 for d = 1 and α d + 2 for d 2. Under the above setting, we have

728


−2 log Eθ vξ (t; x0 ) = −tr inf λrζ (Λt/r ) + γ (r)θ

ζ ∈Ωt

r

θ r 1 + o(1) (1.15)

−d ζq

q∈Λt ∩Zd

as t goes to ∞, where ⎧ ⎨ 1 γ (r) = (4 + θ ) log r ⎩ 1−μ r

(d = 1 and α = 3), (d = 2 and α > 4), (d 3 or (d, α) = (2, 4)),

(1.16)

and μ is the number defined in (1.11). The interpretation of this result is as follows. For a given configuration ξ = ζ , the eigenfunction expansion indicates that vζ (t, x) = exp −λ1ζ (Λt )t 1 + o(1)

(1.17)

since the contribution from outside Λt is negligible. On the other hand, the probability to have such a configuration is formally given by |ζq |θ 1 + o(1) . Pθ (ξ = ζ ) = exp −

(1.18)

q∈Zd

Therefore, the variational problem to minimize the sum of the decay rate for fixed configuration and the cost to realize it has the form |ζq |θ , inf λ1ζ (Λt )t + ζ

(1.19)

q∈Zd

which becomes almost the same as the right hand side of (1.15) after the scaling. Hence, the above theorem says that only the optimal configuration contributes the leading part of the asymptotics, just as in the heavy tailed case. 2. Proof of Theorem 1.2 In Theorem 2.9 of [3], the leading term for log Eθ [vξ (t; x0 )] with compactly supported u was investigated by using Sznitman’s “method of enlargement of obstacles”. We shall apply the same method here. 2.1. Method of enlargement of obstacles for the multidimensional case Let us first recall the elements of the methods developed in [3]. It is basically a coarse graining method to establish a certain variational principle by reducing the number of configurations contributing the asymptotics. In this subsection, we define a set of reduced configurations and show that its cardinality is indeed negligible compared with the decay of Eθ [vξ (t; x0 )] (see (2.7) and (2.8) below).


729

We take χ ∈ ((μ − 2/d)θ, μθ ) and η ∈ (0, 1) so small that

2 2θ θ + 2η2 + d − 2 + η χ > μ− d d and define γ :=

d − 2 2η + < 1. d d

We further introduce a notation concerning a diadic decomposition of Rd . For each k ∈ Z+ , let Ik be the collection of indices iı = (i0 , i1 , . . . , ik ) with i0 ∈ Zd and i1 , . . . , ik ∈ {0, 1}d . For each iı ∈ Ik , we associate the box Ciı = qiı + 2−k [0, 1]d , where qiı = i0 + 2−1 i1 + · · · + 2−k ik . For iı ∈ Ik and iı ∈ Ik with k k, iı iı means that the first k coordinates coincide. Finally, we introduce log r nβ = β log 2 for β > 0 so that 2−nβ −1 < r −β 2−nβ . We can now define the density set, which we can discard from the consideration. Definition 2.1. We call a unit cube Cq with q ∈ Zd a density box if all q iı ∈ Inηγ satisfy the following: for at least half of iı iı ∈ Inγ , qiı + 2−nγ −1 [0, 1]d ∩ (q + ξq )/r: q ∈ Zd = ∅.

(2.1)

The union of all density boxes is denoted by D r (ξ ). The following theorem tells us that we can replace D r (ξ ) by a hard trap without causing a substantial increase in the principal eigenvalue. Spectral control. There exists ρ > 0 such that for all M > 0 and sufficiently large r, sup d

ξ ∈(Rd )Z

r λξ R r (ξ ) ∧ M − λrξ (Λt/r ) ∧ M r −ρ ,

where R r (ξ ) = Λt/r \ D r (ξ ).

(2.2)

730


By Proposition 2.7 in [3], the proof of this theorem is reduced to the extension of Theorem 4.2.3 in [11] from the compactly supported single site potentials to the Kato class single site potentials, which is straightforward. For R r (ξ ), we can give the following quantitative estimate on its volume: Lemma 2.2. (i) There exists a positive constant c1 independent of r such that |R r (ξ )| r χ implies

|ξq |θ c1 r d(1−ηγ )+(1−γ )θ+χ .

(2.3)

q∈Λt ∩Zd

(ii) There exists a positive constant c2 independent of r such that Pθ R r (ξ ) r χ exp −c2 r d(1−ηγ )+(1−γ )θ+χ .

(2.4)

In particular, Pθ (|R r (ξ )| r χ ) = o(Eθ [vξ (t; x0 )]). Proof. Throughout the proof, c1 and c2 are positive constants whose values may change line by line. We consider the following necessary condition of Cq ⊂ D r (ξ ): there exists an iı q in Inηγ such that for a half of iı iı in Inγ , −1

r q + r −1 ξq : q ∈ (rCiı ) ∩ Zd ⊂ qiı + 2−nγ −1 [0, 1]d .

(2.5)

Note first that

|ξq |θ

q ∈(rCiı )∩Zd

d q , ∂(rCiı ) θ c1 r (1−γ )(d+θ)

q ∈(rCiı )∩Zd

for any configurations satisfying the second line in (2.5). Thus Cq ⊂ D r (ξ ) implies

|ξq |θ c1 r (1−γ )(d+θ) 2d(nγ −nηγ )−1

q ∈(rCq )∩Zd

c2 r (1−γ )(d+θ)+dγ (1−η)

(2.6)

and the first assertion follows from this. For the second assertion, we use (2.6) and take the sum over the possibilities of the indices iı and iı ’s in (2.5) to obtain

2d(nγ −nηγ ) (1−γ )(d+θ)+dγ (1−η) r exp −c 1 d(n −n )−1 2 γ ηγ exp −c2 r d(1−ηγ )+(1−γ )θ

Pθ (2.5) is satisfied 2dnηγ


731

for large r. In the second line, the first factor represents the choice of the index iı and the second factor the choice of the indices iı ’s. Since the variables {ξq : q ∈ Cq ∩ Zd } are independent in q ∈ Zd , we have r χ χ Pθ Λt/r \ D r (ξ ) r χ t dr exp −c2 r d(1−ηγ )+(1−γ )θ exp −c2 r d(1−ηγ )+(1−γ )θ+χ , which is the desired estimate. Finally the third assertion follows from Theorem 1.1 and our choice of χ .

2

With the help of this lemma, we may restrict ourselves on some special configurations. To see this, we introduce some more notations. A domain R is called a lattice animal if it is represented as

◦ Λ1 (q) , R= q∈S(R)

where S(R) ⊂ Zd consists of adjacent sites. This means that R is a combination of unit cubes connected via faces. We set Sr =

Rr , ζ = (ζq )q∈(r[Rr : l])∩Zd : Rr is a lattice animal included in Λt/r ,

|Rr | < r χ , q + ζq ∈ T : t 1/(μθ) ∩ Zd for all q ∈ r[Rr : l] ∩ Zd ,

(2.7)

where l is a positive number specified later, and [A : l] = {x ∈ Rd : d(x, A) < l} for any A ⊂ Rd . For any (Rr , ζ ) ∈ Sr , we write Vζr (x) =

r 2 u(rx − q − ζq )

q∈(r[Rr : l])∩Zd

with a slight abuse of the notation and define λrζ (Rr ) accordingly. We now see that the relevant configurations of (R r (ξ ), ξ ) are only the pairs in Sr . In fact removing the points {q + ξq : q ∈ Zd \ (r[Rr : l])}, which should be cared in proving the lower bound, is permitted as we will show in Lemma 2.5 below. We also have λrξ R r (ξ ) = λrξ (Rr ) for some lattice animal Rr included in R r (ξ ) and

/ T : t 1/(μθ) for some q ∈ r[Rr : l] ∩ Zd Pθ q + ξq ∈ decays exponentially in t. The latter easily follows by observing that c

d r[Rr : l], T : t 1/(μθ) > t 1/θ , which is due to lr + t 1/θ < t 1/(μθ) , for large t.

732


The key point in our coarse graining method is that the number of relevant configurations is estimated as dr d+χ c(1+l) −1 χ = o Eθ vξ (t; x0 ) #Sr t dr t + 2t 1/(μθ)

(2.8)

by an elementary counting argument, where c is a finite constant depending only on d. The second relation comes from our choice of χ . 2.2. Proof of a modified statement for the multidimensional case We state and prove slightly modified versions of Theorem 1.2 in this section. They are shown to be equivalent to Theorem 1.2 in Subsection 2.4 below. Let us start with the multidimensional case. Theorem 2.3. Let d 2 and assume the setting of Theorem 1.2. Then we have the following: (i) For any ε > 0 and l > 0, there exists tε,l > 0 such that

t −1 r 2 log Eθ vξ (t; x0 ) λrζ (Rr ) + γ (r)θ −(1 − ε) inf (Rr ,ζ )∈Sr

r


θ r

−d ζq

(2.9)

for any t tε, l , where γ (r) is the function defined in (1.16). (ii) If α > d + 2, then for any ε > 0 and l > 0, there exists tε,l > 0 such that

t −1 r 2 log Eθ vξ (t; x0 ) λrζ (Rr ) + γ (r)θ −(1 + ε) inf (Rr ,ζ )∈Sr

r


θ r

−d ζq

(2.10)

for any t tε,l . (iii) If α = d + 2, then for any ε > 0, there exist tε > 0 and lε > 0 such that

t −1 r 2 log Eθ vξ (t; x0 ) λrζ (Rr ) + γ (r)θ −(1 + ε) inf (Rr ,ζ )∈Sr

θ ζq r −d r d

(2.11)

q∈(r[Rr : l])∩Z

for any t tε and l lε . Proof. We first prove the upper bound in (i). By a standard Brownian estimate and scaling, we have t

t Eθ vξ (t; x0 ) Eθ ⊗ Ex0 exp − Vξ (Bs ) ds : sup |Bs |∞ < + e−ct 2 0st 0


Eθ ⊗ Ex0 /r

tr−2 r Vξ (Bs ) ds : exp −

0

t sup |Bs |∞ < 2r −2 0str

733

+ e−ct . (2.12)

For any ε ∈ (0, 1), there exists a finite constant cε depending only on d and ε such that the first term of the right hand side is less than

cε Eθ exp −(1 − ε)λrξ (Λt/r )tr−2 by (3.1.9) of [11]. By the spectral control (2.2), Lemma 2.2, and (2.8), this quantity is less than −1 o Eθ vξ (t; x0 )

sup

(Rr ,ζ )∈Sr

Pθ ξq = ζq for all q ∈ r[Rr : l] ∩ Zd

× exp −(1 − ε) λrζ (Rr ) ∧ M − r −ρ tr−2 . Thus, we have

t −1 r 2 log Eθ vξ (t; x0 ) −(1 − 2ε) +t

inf

(Rr ,ζ )∈Sr

λrζ (Rr ) ∧ M − r −ρ

−1 2

r

θ |ζq | + log Z(d, θ )

(2.13)


for sufficiently large t. We can drop M and r −ρ from the right hand side since Theorem 1.1 tells us that the left hand side is bounded from below. Moreover, we can also neglect log Z(d, θ ) since # r[Rr : l] ∩ Zd cr d+χ = o tr−2 .

(2.14)

After removing the above three terms, (2.13) gives us the upper bound. We next proceed to the lower bound. We pick a pair (Rr∗ , ζ ∗ ) which attains the infimum in the right hand side of (2.10). Then we have the following estimate for the L2 -normalized nonnegative eigenfunction φ ∗ corresponding to λrζ ∗ (Rr∗ ). Lemma 2.4. There exist p ∗ ∈ (rRr∗ ) ∩ Zd and c0 > 0 such that sup

x∈Λ2/r (p ∗ /r)

Vζr∗ (x) c0 r d+χ+2

and Λ1/r (p ∗ /r)

φ ∗ (x) dx

1 r −d−χ . 2φ ∗ ∞

Proof. We fix 1 < r0 < ∞ so that 2C0 C0 u(x) α α 2|x| |x|

(2.15)

734


for all |x| > r0 and take k ∈ N satisfying 2−k−3 r0 /r < 2−k−2 . We divide Rr∗ into subboxes of sidelength 2−k as

Rr∗ =

for some I ∗ ⊂ Ik .

Ciı

iı∈I ∗

Let C be the union of all boxes Ciı in Rr∗ whose enlarged boxes qiı + 2−k [−1, 2]d intersect with {r −1 (q + ζq∗ ): q ∈ (r[Rr∗ : l]) ∩ Zd }. Then it is easy to see that if Ciı ⊂ C, there exist a ∈ Ciı and c1 > 0 for which Vζr∗ c1 r 2 1B(a,1/r) . Thus, by using Lemma 3.5 in [4], which states −d inf λ1 (− + 1B(b,1) )N R : b ∈ ΛR cR ,

(2.16)

and the scaling with the factor r, we have inf ∞

φ∈C (Ciı )

1 φ22

2 1 ∇φ(x) + Vζr∗ (x)φ(x)2 dx c2 r 2 2

Ciı

for all Ciı ⊂ C and consequently c2 r

∗

φ (x) dx

2

2

C

C

1 ∗ 2 r ∗ 2 ∇φ (x) + Vζ ∗ (x)φ (x) dx. 2

Since the right hand side is bounded from above by λrζ ∗ (Rr∗ ), it follows that

φ ∗ (x)2 dx c3 r −2 .

C

This implies

φ ∗ (x)2 dx 1/2

Rr∗ \C

for large r and hence we can find a Λ1/r (p ∗ /r) in Rr∗ \ C such that ∗ φ

∞ Λ1/r (p ∗ /r)

∗

φ (x) dx Λ1/r (p ∗ /r)

1 φ ∗ (x)2 dx r −d−χ . 2

Finally, we show the bound supx∈Λ2/r (p∗ /r) Vζr∗ (x) c0 r d+χ+2 . Note first that we have supx∈Λ2/r (p∗ /r) r 2 u(rx − q − ζq∗ ) c4 r 2 for each q since Rr∗ \ C keeps the distance larger than (r0 + 1)/r from {r −1 (q + ζq∗ ): q ∈ (r[Rr∗ : l]) ∩ Zd }. Multiplying the total number of points #{r −1 (q + ζq∗ ): q ∈ (r[Rr∗ : l]) ∩ Zd } (2l + 1)d r d+χ , we obtain the result. 2


735

We bound Eθ [vξ (t; x0 )] from below by Pθ ξq = ζp∗∗ +q for q ∈ r Rr∗ : l ∩ Zd − p ∗ sup × Pθ u(x − q − ξq ) < x∈(rRr∗ −p ∗ )∪Λ2

× Ex0

q∈Zd \{(r[Rr∗ : l])∩Zd −p ∗ }

t exp −

c1 (rl)α−d

∗ u Bs − q − ζp∗ +q ds :

∗ d ∗ 0 q∈(r[Rr : l])∩Z −p

Bs ∈ Λ2 for 0 s 1, B1 ∈ Λ1 , Bs ∈ rRr∗ − p ∗ for 1 s t × exp −

c1 t . (rl)α−d

(2.17)

The first factor is greater than or equal to exp −

|ζq | − cr θ

d+χ


by the same argument using (2.14) as for the upper bound. The last factor is greater than exp(−εtr−2 ) for sufficiently large r if α > d + 2, and for sufficiently large r and l if α = d + 2. To bound the second factor we use the following: Lemma 2.5. Let {Rr : r 1} be a family of lattice animals satisfying Rr ⊂ Λt/r and |Rr | < r χ . Let k, l > 0. Then there exist c1 , c2 , c3 > 0 independent of Rr such that Pθ

sup

x∈[rRr :k]

−α+d

u(x − q − ξq ) < c1 (rl)

c2

(2.18)

q∈Zd \(r[Rr : l])

for any r c3 . Proof. We consider the event 1 d q + ξq , [rRr : k] d q, [rRr : k] for all q ∈ Zd \ r[Rr : l] . 2 On this event, we have q∈Zd \(r[Rr : l])

|x − q − ξq |−α

q∈Zd \(r[Rr : l])

c4

2 d(q, [rRr : k])

q∈Zd :d(q,rRr )rl

α

−α d q, [rRr : k] c5 (rl)−α+d

(2.19)

736


for any x ∈ [rRr : k] and large r. By this estimate and the assumption u(x) = C0 |x|−α (1 + o(1)), we see that the event in (2.19) implies the event in (2.18). Since the inequality in (2.19) is satisfied if |ξq | d q, [rRr : k] /2

for all q ∈ Zd \ r[Rr : l] ,

the probability of the event (2.19) is greater than or equal to

q∈Zd \(r[R

1− r : l])

1 Z(d, θ )

exp −|y|θ .

y∈Zd :

(2.20)

|y|d(q,[rRr :k])/2

It is easy to see that 1 Z(d, θ )

θ exp −|y|θ exp −c6 d q, [rRr : k]

y∈Zd : |y|d(q,[rRr :k])/2

and # q ∈ Zd : n d q, [rRr : k] < n + 1 c7 r χ+d nd−1 . By using also an elementary inequality (1 − x)p 1 − px for any p 1 and 0 < x < 1, the quantity in (2.20) is greater than or equal to

c r χ+d nd−1 1 − exp −c6 nθ 7

rl−kn∈N

1 − c8 r χ+d exp −c9 nθ .

rl−kn∈N

Since the right hand side is a convergent infinite product, we conclude (2.18).

2

It remains to bound the third factor in (2.17). We use the bound sup

x∈Λ2/r (p ∗ /r)

Vζr∗ (x) c0 r d+χ+2

in Lemma 2.4 for 0 s 1 and the positivity of inf exp D 2 /2 (x, y),

x,y∈Λ1

where exp(tD 2 /2)(x, y), (t, x, y) ∈ (0, ∞) × Λ2 × Λ2 is the integral kernel of the heat semigroup generated by the Dirichlet Laplacian on Λ2 multiplied by −1/2. Then, we can show that the third factor is greater than r exp −c0 r d+χ

d

Λ1/r

dy Rr∗ −p ∗ /r

dz exp −(t − 1)r −2 H ∗ (y, z)

(2.21)


737

for large r by using a scaling, where exp(−tH ∗ )(x, y), (t, x, y) ∈ (0, ∞) × (Rr∗ − p ∗ /r) × (Rr∗ − p ∗ /r) is the integral kernel of the heat semigroup generated by the Schrödinger operator

H ∗ = −/2 +

r 2 u rx − q − ζp∗∗ +q

q∈(r[Rr∗ : l])∩Zd −p ∗

in Rr∗ − p ∗ /r with the Dirichlet boundary condition. By (2.15), the integral in (2.21) is greater than or equal to

dy Λ1/r

Rr∗ −p ∗ /r

φ ∗ (z + p ∗ /r) dz exp −(t − 1)r −2 H ∗ (y, z) φ ∗ ∞

2 exp −(t − 1)r −2 λrζ ∗ Rr∗ / 2φ ∗ ∞ r d+χ .

Finally φ ∗ ∞ is bounded since φ ∗ (y) = exp λrζ ∗ Rr∗

exp −H 0 (y, z)φ ∗ (z) dz,

exp(−H 0 )(y, ·)2 1, and λrζ ∗ (Rr∗ ) is bounded by Theorem 1.1 and the upper bound in (i), where exp(−tH 0 )(x, y), (t, x, y) ∈ (0, ∞) × Rr∗ × Rr∗ , is the integral kernel of the heat semigroup generated by the Schrödinger operator H 0 = −/2 + Vζr∗ in Rr∗ with the Dirichlet boundary condition. By all these the lower bounds (ii) and (iii) are proven. 2 2.3. Proof of a modified statement for the one-dimensional case We first fix a constant M > 0 such that Pθ {q + ξq : q ∈ Z} ∩ 0, Mt 1/(3+θ) = ∅ exp −cM 1+θ t (1+θ)/(3+θ) = o Eθ vξ (t; x0 ) , which is possible in view of Theorem 1.1. We define the set Sr of relevant configurations by Sr =

(m, n), ζ = (ζq )q∈(m−lr,n+lr)∩Z : m, n ∈ Z, −t m < n t, n − m Mr, |ζq | t 1/θ , q + ζq : q ∈ (m − lr, n + lr) ∩ Z ∩ (m, n) = ∅

in this case. Now we can state the result. Theorem 2.6. Let d = 1 and assume the setting of Theorem 1.2. Then, for any ε > 0, there exist tε > 0 and lε > 0 such that −(1 + ε)

inf

((m,n),ζ )∈Sr

λrζ (m/r, n/r) +

t −(1+θ)/(3+θ) log Eθ vξ (t; x0 )

q∈(m−lr,n+lr)∩Z

θ ζq r −1 r

738


−(1 − ε)

inf

((m,n),ζ )∈Sr

λrζ (m/r, n/r) +

r


θ r ,

−1 ζq

for all t > tε and l > lε . Proof. We only prove the upper bound. After having it, the lower bound follows exactly in the same way as for Theorem 2.3. We use a simple version of the method of enlargement of obstacles where γ = 1 and any 2−n1 -box containing a point of {r −1 (q + ξq ): q ∈ Z} is a density box. Such a box indeed satisfies the quantitative Wiener criterion (2.12) in p. 152 of [11] since even a point has positive capacity when d = 1 (cf. p. 153 of [11]). Then, the spectral control (2.2) implies that we can impose the Dirichlet boundary condition on each point in {r −1 (q + ξq )}q∈Z . Combining this observation with a standard Brownian estimate and (3.1.9) in [11], we find

1/2 exp −λ1ξ (−t, t) t + e−ct Eθ vξ (t; x0 ) Eθ c 1 + λ1ξ (−t, t) t cε Eθ sup exp −(1 − ε)λrξ r −1 Ik tr−2 + e−ct , k

where ε is an arbitrary positive constant and {Ik }k are the random open intervals such that k Ik = (−t, t) \ {q + ξq : q ∈ Z}. By considering all possibilities of Ik , we can bound the Eθ -expectation in the right hand side by

Eθ exp −(1 − ε)λrξ (m/r, n/r) tr−2 : {q + ξq : q ∈ Z} ∩ (m, n) = ∅ .

m,n∈Z: −tm Mr thanks to our choice of M. Hence, we can restrict our consideration on Sr and we can also show #Sr = exp{o(t (1+θ)/(3+θ) )} by an elementary counting argument. Now, we have

Eθ vξ (t; x0 )

exp −(1 − ε)λrζ (m/r, n/r) tr−2 Pθ (ξq = ζq for all q)

((m,n),ζ )∈Sr

+ o Eθ vξ (t; x0 ) exp −(1 − 2ε)t (1+θ)/(3+θ) ×

inf

((m,n),ζ )∈Sr

which is the desired estimate.

λrζ (m/r, n/r) +


r

θ , r

−1 ζq

2

2.4. Proof of Theorem 1.2 In this section, we complete the proof of Theorem 1.2 by simplifying the variational expression in Theorem 2.3. We treat only the multidimensional case since the modification for the


739

one-dimensional case is straightforward. Recall that Ωt = (Zd )Λt ∩Z is the set of possible configurations of {ξq }q∈Λt ∩Zd . We first show d

inf

(Rr ,ζ )∈Sr

λrζ (Rr ) + γ (r)θ

r


λrζ (Λt/r ) + γ (r)θ

(1 − ε) inf

ζ ∈Ωt

θ r

−d ζq

r

q∈Λt ∩Zd

θ r

−d ζq

(2.22)

for sufficiently large t (and l) if α ∈ (d, d + 2) (resp. α = d + 2). Let (Rr∗ , ζ ∗ ) be a minimizer of the variational problem in the first line. We extend ζ ∗ to ζ ∗∗ ∈ Ωt by setting ζq∗∗ = 0 for q ∈ (Λt \ r[Rr∗ : l]) ∩ Zd . Then, it is obvious that ∗ θ ζq r −d r d

q∈(r[Rr∗ : l])∩Z

∗∗ θ ζq . r −d r d

(2.23)

q∈Λt ∩Z

Moreover, we can prove

sup

x∈rRr∗

x − q − ζ ∗∗ −α c1 (rl)−α+d q

(2.24)

q∈Zd \(r[Rr∗ : l])

for this ζ ∗∗ . Therefore, we have λrζ ∗ Rr∗ + c2 r −α+d+2 l −α+d λrζ ∗∗ (Λt/r )

(2.25)

and this yields (2.22). We next show inf

(Rr ,ζ )∈Sr


ζ ∈Ωt

θ ζq r −d r d

q∈(r[Rr : l])∩Z

(1 + ε) inf

λrζ (Λt/r ) + γ (r)θ

q∈Λt ∩Zd

r

θ r

−d ζq

(2.26)

for sufficiently large t. It follows from Lemma 2.2 that if a sequence {ζ t }t of configurations satisfies ζ t ∈ Ωt and |R r (ζ t )| r χ for any t, then we have γ (r)θ

t θ ζq r −d −→ ∞, r d

(2.27)

q∈Λt ∩Z

as t → ∞. Thus if each ζ t is a minimizer of the right hand side of (2.26), then we have |R r (ζ t )| < r χ for large t. We may also assume that q + ζqt ∈ [T : t 1/(μθ) ] for all q ∈ (r[R r (ζ t ) : l]) ∩ Zd since otherwise (2.27) holds. We here extend ζ t to (r[R r (ζ t ) : l]) ∩ Zd by ζqt = 0 for

740


q ∈ (r[R r (ζ t ) : l]) ∩ Zd \ Λt . There exists a lattice animal Rrt in R r (ζ t ) such that λrζ t (R r (ζ t )) = λrζ t (Rrt ). Then it follows that (Rrt , (ζqt )q∈(r[Rrt :l])∩Zd ) ∈ Sr for sufficiently large t. Combining with Spectral control (2.2), we obtain (2.26). 3. Asymptotics of higher moments In [3], a result on the asymptotics for higher moments of the survival probability is shown as an application of the precise form of the leading term. We shall extend the result to our cases in this section. Our objects are the p-th moments Eθ [vξ (t; x0 )p ] for p 1. We consider their asymptotics in Subsection 3.1. In Subsection 3.2, we discuss a related quantitative estimate on intermittency for the parabolic Anderson problem. 3.1. Asymptotics for each case Proposition 3.1. Under the settings in Theorem 1.2, there exist c1 , c2 ∈ (0, ∞) depending on d, θ and u such that for any p 1,

−c1 p (d+μθ)/(d+2+μθ) t −1 r 2 log Eθ vξ (t; x0 )p −c2 p (d+μθ)/(d+2+μθ) holds for sufficiently large t, uniformly in x0 ∈ Λ1 , where we take μ = 1 in the case d = 1. Proof. We first assume d 3 and α > d + 2. The same argument as in Section 1.3, using the scaling with factor s = (pt)1/(d+2+μθ) instead of r = t 1/(d+2+μθ) in (2.12) and (2.21), yields

log Eθ vξ (t; x0 )p ∼ −(pt)(d+μθ)/(d+2+μθ) × inf λsζ (Rs ) + s (1−μ)θ (Rs ,ζ )∈Ss

s

q∈(s[Rs : l])∩Zd

θ −d ζq s

(3.1)

as t → ∞ for any l. Since we know 0 < lim

inf

s→∞ (Rs ,ζ )∈Ss

lim

inf


λsζ (Rs ) + s (1−μ)θ λsζ (Rs ) + s (1−μ)θ

s

s

−d ζq



θ s

−d ζq

θ s d + 2, the existence of the above limit implies lim

inf

t→∞ (Rr ,ζ )∈Sr


θ ζq r −d = L r d

q∈(r[Rr : l])∩Z

by Theorem 2.3 and then (3.3) is obvious from the proof of the last proposition. When α = d + 2, we know only that the superior limit and the inferior limit in (3.2) tend to L as l → ∞. This is still enough to show (3.3). The above remark actually applies for the case d = 1 and α > 3: Proposition 3.3. Under the conditions of Theorem 1.1(i) with α > 3, we have

3+θ lim t −(1+θ)/(3+θ) log Eθ vξ (t; x0 )p = − t↑∞ 1+θ

pπ 2 8

(1+θ)/(3+θ) (3.4)

for any p 1, uniformly in x0 ∈ Λ1 . Proof. As in the proof of the last proposition we have

log Eθ vξ (t; x0 )p ∼ −(pt)(1+θ)/(3+θ) λsζ (m/s, n/s) + × inf (Rs ,ζ )∈Ss

q∈(m−ls,n+ls)∩Z

θ ζq s −1 s

(3.5)

as t → ∞ for any l in the notations of Subsection 2.3, where s = (pt)1/(3+θ) . When α > 3, we know the limit lim

inf


λsζ (m/s, n/s) +

s

θ

3 + θ π 2 (1+θ)/(3+θ) . s = 1+θ 8

−1 ζq

q∈(m−ls,n+ls)∩Z

2

Proposition 3.4. Under the conditions of Theorem 1.1 with α < d + 2, we have

lim t −(d+θ)/(α+θ) log Eθ vξ (t; x0 )p = −p (d+θ)/(α+θ) c(d, α, θ, C0 )

t↑∞

for any p 1, uniformly in x0 ∈ Λ1 . Proof. We have only to show

lim t −(d+θ)/(α+θ) log Eθ vξ (t; x0 )p = −

t↑∞

Rd

The upper estimate is easy since we have

dq inf

y∈Rd

pC0 θ . + |y| |q + y|α

(3.6)

742


p t

Eθ vξ (t; x0 )p Eθ Ex0 exp − Vξ (Bs ) ds 0

t

Eθ ⊗ Ex0 exp −p

Vξ (Bs ) ds 0

by removing the Dirichlet condition and using the Hölder inequality. For the lower estimate, we take R, R1 and β as in the proof of Proposition 2.2 in [4] and restrict the integral as p t

p Eθ vξ (t; x0 ) Eθ Ex0 exp − Vξ (Bs ) ds : Bs ∈ ΛR for 0 s t : Ξt 0

√ for t β 2(R1 + R d ), where Ξt is the set of configurations defined by √ |ξq | |q|/2 for |q| t β , and |q + ξq | R1 + R d for |q| < t β . The right hand side is bounded from below by ! Eθ exp −pt sup Vξ (y) : Ξt exp −cptR −2 . y∈ΛR

This is estimated by the same method as in our proof of Proposition 2.2 in [4].

2

Proposition 3.5. Under the conditions of Theorem 1.1 with u 0, we have

lim t −(1+d/θ) log Eθ vξ (t; x0 )p = c− d, θ, pu(0) t↑∞

for any p 1, uniformly in x0 ∈ Λ1 . Proof. The upper and lower estimates are obtained by similar ways to the proof of Proposition 3.4 and that of (1.12) respectively. 2 3.2. Intermittency The initial value problem of the form (1.2) is called the “parabolic Anderson problem” in literature, see e.g. a survey article by Gärtner and König [7]. For a wide class of random potentials, it is believed that the solution of parabolic Anderson problem consists of high peaks which are far from each other. A manifestation of this phenomenon formulated by Gärtner and Molchanov [8] is so-called “intermittency” defined by Eθ [vξ (t; x0 )p2 ]1/p2 t→∞ −−−→ ∞ Eθ [vξ (t; x0 )p1 ]1/p1

for p1 < p2 .

(3.7)

Although (3.7) implies the concentration of vξ (t; x0 ) in the ξ -space, there is a way to relate this to the spatial concentration of the solution through the ergodic theorem. See Subsection 1.3 of [7]


743

for this point. Gärtner and Molchanov also proved in [8] that the intermittency holds for a quite general class of potentials. In particular, if we consider a slightly different moment Eθ

p

(3.8)

vξ (t; x0 ) dx0 Λ1

in our model, then the intermittency follows by the same argument as for Theorem 3.2 of [8]. Our main result Theorem 1.2 gives a more detailed description of the concentration in the configuration space. Indeed, it says that the main contribution to Eθ [vξ (t; x0 )] comes only from minimizers of the right hand side of (1.15). Furthermore, we can derive the rates of the divergence in (3.7) from the results in the previous subsection as follows: (i) Under the settings in Theorem 1.2, we have Eθ [vξ (t; x0 )p2 ]1/p2 Eθ [vξ (t; x0 )p1 ]1/p1

−2/(d+2+μθ)

exp{tr−2 (c2 p1

−2/(d+2+μθ) exp{tr−2 (c1 p1

−2/(d+2+μθ)

− c1 p2

)},

−2/(d+2+μθ) − c2 p2 )},

for sufficiently large t, where ∞ > c1 c2 > 0 are the constants in Proposition 3.1. (ii) Under the settings in Theorem 1.1 with d = 1 and α > 3, it holds that

Eθ [vξ (t; x0 )p2 ]1/p2 3 + θ π 2 t (1+θ)/(3+θ) −2/(3+θ) −2/(3+θ) p = exp − p + o(1) 1 2 1+θ 8 Eθ [vξ (t; x0 )p1 ]1/p1 as t goes to ∞. (iii) Under the settings in Theorem 1.1 with α < d + 2, it holds that Eθ [vξ (t; x0 )p2 ]1/p2 Eθ [vξ (t; x0 )p1 ]1/p1 (d−α)/(α+θ) (d−α)/(α+θ) = exp c(d, α, θ, C0 )t (d+θ)/(α+θ) p1 − p2 + o(1) as t goes to ∞. (iv) Under the settings in Theorem 1.1 with u 0, it holds that d/θ Eθ [vξ (t; x0 )p2 ]1/p2 d/θ = exp c− d, θ, u(0) t 1+d/θ p2 − p1 + o(1) p 1/p Eθ [vξ (t; x0 ) 1 ] 1 as t goes to ∞. Note that in the first case, the left hand side goes to infinity only when p2 /p1 is sufficiently large. On the other hand, the left hand sides go to infinity for any p2 /p1 > 1 in other cases. This is slightly better than Theorem 3.2 of [8] where p2 2 is required. Note also that all these estimates hold uniformly in x0 ∈ Λ1 and therefore, the same estimates hold for (3.8) as well.

744


4. Note added in proof The problem similar to ours has been studied for Poisson type potentials. The interested reader could consult [5,6,10]. References [1] J. Baker, M. Loss, G. Stolz, Minimizing the ground state energy of an electron in a randomly deformed lattice, Comm. Math. Phys. 283 (2) (2008) 397–415. [2] J. Baker, M. Loss, G. Stolz, Low energy properties of the random displacement model, J. Funct. Anal. 256 (8) (2009) 2725–2740. [3] R. Fukushima, Brownian survival and Lifshitz tail in perturbed lattice disorder, J. Funct. Anal. 256 (9) (2009) 2867– 2893. [4] R. Fukushima, N. Ueki, Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice, Ann. Henri Poincaré, in press. [5] J. Gärtner, W. König, Moment asymptotics for the continuous parabolic Anderson model, Ann. Appl. Probab. 10 (1) (2000) 192–217. [6] J. Gärtner, W. König, S.A. Molchanov, Almost sure asymptotics for the continuous parabolic Anderson model, Probab. Theory Related Fields 118 (4) (2000) 547–573. [7] J. Gärtner, W. König, The parabolic Anderson model, in: Interacting Stochastic Systems, Springer, Berlin, 2005, pp. 153–179. [8] J. Gärtner, S.A. Molchanov, Parabolic problems for the Anderson model. I. Intermittency and related topics, Comm. Math. Phys. 132 (3) (1990) 613–655. [9] F. Ghribi, F. Klopp, Localization for the random displacement model at weak disorder, Ann. Henri Poincaré 11 (2010) 127–149. [10] L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, Grundlehren Math. Wiss., vol. 297, Springer-Verlag, Berlin, 1992. [11] A.-S. Sznitman, Brownian Motion, Obstacles and Random Media, Springer Monogr. Math., Springer-Verlag, Berlin, 1998.


Global well-posedness for the Euler–Boussinesq system with axisymmetric data Taoufik Hmidi ∗ , Frédéric Rousset IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France Received 21 April 2010; accepted 19 October 2010

Communicated by Cédric Villani

Abstract In this paper we prove the global well-posedness for the three-dimensional Euler–Boussinesq system with axisymmetric initial data without swirl. This system couples the Euler equation with a transport-diffusion equation governing the temperature. © 2010 Elsevier Inc. All rights reserved. Keywords: Axisymmetric flows; Global well-posedness; Harmonic analysis

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Dyadic decomposition and functional spaces . . . . . . . . . . . 2.2. Lorentz spaces and interpolation . . . . . . . . . . . . . . . . . . . . 2.3. Some useful commutator estimates . . . . . . . . . . . . . . . . . . 2.4. Some algebraic identities . . . . . . . . . . . . . . . . . . . . . . . . . Commutator estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The commutator between the advection operator and ∂rr −1 3.2. Commutation between the advection operator and q . . . . . A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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E-mail addresses: [email protected] (T. Hmidi), [email protected] (F. Rousset). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.012

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746 751 751 753 756 756 759 759 768 770 771

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T. Hmidi, F. Rousset / Journal of Functional Analysis 260 (2011) 745–796

4.2. Estimates of the moments of ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Strong estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. De Giorgi–Nash–Moser estimates for convection–diffusion equations Appendix B. Proof of Lemma 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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772 776 786 788 793 796

1. Introduction Boussinesq systems are widely used to model the dynamics of the ocean or the atmosphere. They arise from the density dependent fluid equations by using the so-called Boussinesq approximation which consists in neglecting the density dependence in all the terms but the one involving the gravity. This approximation can be justified from compressible fluid equations by a simultaneous low Mach number/Froude number limit, we refer to [15] for a rigorous justification. In this paper we shall assume that the fluid is inviscid but heat-conducting and hence the system reads ⎧ ∂t v + v · ∇v + ∇p = ρez , (t, x) ∈ R+ × R3 , ⎪ ⎪ ⎪ ⎨ ∂ ρ + v · ∇ρ − ρ = 0, t (1) ⎪ div v = 0, ⎪ ⎪ ⎩ v|t=0 = v0 , ρ|t=0 = ρ0 . Here, the velocity v = (v 1 , v 2 , v 3 ) is a three-component vector field with zero divergence, the scalar function ρ denotes the density or the temperature and p the pressure of the fluid. Note that we have assumed that the heat conductivity coefficient is one, one can always reduce the problem to this situation by a change of scale (as soon as the fluid is assumed to be heat conducting) which is not important for global well-posedness issues with data of arbitrary size that we shall consider here. The term ρez where ez = (0, 0, 1)t takes into account the influence of the gravity and the stratification on the motion of the fluid. Note that when the initial density ρ0 is identically zero (or constant) then the above system reduces to the classical incompressible Euler equation: ⎧ ⎨ ∂t v + v · ∇v + ∇p = 0, div v = 0, (2) ⎩ v|t=0 = v0 . From this observation, one cannot expect to have a better theory for the Boussinesq system than for the Euler equation. For the Euler equation, a well-known criterion for the existence of global smooth solution is the Beale–Kato–Majda criterion [3]. It states that the control of the vorticity of the fluid ω = curl v in L1loc (R+ , L∞ ) is sufficient to get global well-posedness. In space dimension two, the vorticity ω can be identified to a scalar function which solves the transport equation ∂t ω + v · ∇ω = 0. From this transport equation, one immediately gets that ω(t) p ω0 Lp L


747

for every p 1 and hence the global well-posedness follows from the Beale–Kato–Majda criterion. In a similar way, the global well-posedness for two-dimensional Boussinesq systems which has recently drawn a lot of attention seems to be in a satisfactory state. More precisely global well-posedness has been shown in various function spaces and for different viscosities, we refer for example to [1,5,7,12,13,17–20,22]. In particular, for the model (1) in 2D, the main idea is that by studying carefully the coupling between the two equations and by using the smoothing 0 ) effect of the second equation, it is still possible to get an a priori estimate in L∞ (or in B∞,1 for ω and hence the global well-posedness. In the three-dimensional case, very few is known: even for the Euler equation, the vorticity ω solves the equation ∂t ω + v · ∇ω = ω · ∇v

(3)

and the way to control the vortex stretching term ω · ∇v in the right-hand side is a widely open problem. Nevertheless, a classical situation where one can get global existence is the case that v is axisymmetric without swirl [30,23]. Our aim here is to study how this classical global existence result for axisymmetric data for the Euler equation can be extended to the Boussinesq system (1). Before stating our main result, let us recall the main ingredient in the global existence proof for the Euler equation with axisymmetric data. The assumption that the vector field v is axisymmetric without swirl means that it has the form: v(t, x) = v r (t, r, z)er + v z (t, r, z)ez ,

x = (x1 , x2 , z),

1 r = x12 + x22 2 ,

(4)

where (er , eθ , ez ) is the local basis of R3 corresponding to cylindrical coordinates. Note that we assume that the velocity is invariant by rotation around the vertical axis (axisymmetric flow) and that the angular component v θ of v is identically zero (without swirl). For these flows, the vorticity is under the form ω = ∂3 v r − ∂r v z eθ := ωθ eθ and the vortex stretching term reads ω · ∇v =

vr ω. r

In particular ωθ satisfies the equation ∂t ωθ + v · ∇ωθ = The crucial fact is then that the quantity ζ :=

ωθ r

vr ωθ . r

(5)

solves the transport equation

∂t ζ + v · ∇ζ = 0 from which we get that for every p ∈ [1, ∞] ζ (t) p ζ0 Lp . L

(6)

748


It was shown by Ukhoviskii and Yudovich [30] and independently by Ladyzhenskaya [23] that these new a priori estimates are strong enough to prevent the formation of singularities in finite time for axisymmetric flows without swirl. More precisely global existence and uniqueness was established for axisymmetric initial data with finite energy and satisfying in addition ω0 ∈ L2 ∩ L∞ and ωr0 ∈ L2 ∩ L∞ . In terms of Sobolev regularity these assumptions are satisfied if the velocity v0 belongs to H s with s > 72 . This condition was improved more recently. In [29], it was proven that global well-posedness still holds if v0 is in H s with s > 52 (note that this is the natural regularity requirement for the initial velocity in the Sobolev scale in view of the standard local existence result) and in the recent work [11], Danchin has obtained global existence and uniqueness for initial data such that ω0 ∈ L3,1 ∩L∞ and ζ0 ∈ L3,1 (here, L3,1 denotes the Lorentz space, the definition of the Lorentz spaces Lp,q as interpolation spaces is recalled below). Their proof is based on the observation that one can deduce from the Biot–Savart law, the pointwise estimate r v 1 ∗ |ζ |. (7) r | · |2 By convolution laws in Lorentz spaces (again recalled below) and (6), this yields the estimate r v (t) ζ (t) 3,1 ζ0 3,1 . L ∞ r L L Since one gets from a crude estimate on (5) that ωθ (t)

L∞

t r ω0 (t)L∞ e 0 v /rL∞ ,

the global well-posedness in H s s > 5/2 (the assumption ζ0 ∈ L3,1 is automatically satisfied) then follows from the Beale–Kato–Majda criterion. It is actually possible, as shown in [2], to get 3

+1

p , ∀p ∈ [1, ∞], in the global well-posedness in the critical Besov regularity, that is, v0 ∈ Bp,1 sense that it is possible to propagate globally the critical Besov regularity if ζ0 ∈ L3,1 . Our aim here is to extend these global well-posedness results to the Boussinesq system (1). Our main result reads:

Theorem 1.1. Consider the Boussinesq system (1). Let s > 52 , v0 ∈ H s be an axisymmetric divergence free vector field without swirl and let ρ0 be an axisymmetric function belonging to H s−2 ∩ Lm with m > 6 and such that r 2 ρ0 ∈ L2 . Then there is a unique global solution (v, ρ) such that (v, ρ) ∈ C R+ ; H s × C R+ ; H s−2 ∩ Lm ∩ L1loc R+ ; W 1,∞

and r 2 ρ ∈ C R+ ; L2 .

Let us give a few comments about our result. Remark 1.2. By axisymmetric scalar function we mean again a function that depends only on the variables (r, z) but not on the angle θ in cylindrical coordinates. One can easily check that for smooth local solutions, if (ρ0 , v0 ) is axisymmetric (and v0 without swirl), this property is preserved by the evolution.


749

Remark 1.3. The assumption on the moment of ρ is probably technical. The control of the moments of ρ are needed in our proof in some commutator estimates (see (9) for example). Note that in view of the proof for the Euler equation, the crucial part is to get an a priori estimate for ζ in L3,1 . The equation for ζ = ωθ /r becomes ∂t ζ + v · ∇ζ = −

∂r ρ r

(8)

and consequently, the main difficulty is to find some strong a priori estimates on ρ to control the term in the right-hand side of (8). The rough idea is that on the axis r = 0 the singularity 1 r scales as a derivative and hence that the forcing term ∂r ρ/r can be thought as a Laplacian of ρ and thus one may try to use smoothing effects to control it. We observe that if we neglect for the moment the advection term v · ∇ρ in the equation of the density then using the maximal smoothing effects of the heat semigroup we can gain two derivatives by integrating in time which is exactly what we need. From this point of view we see that our model is in some sense critical for the global well-posedness analysis. The main difficulty if one wants to use this argument is to deal with the advection term. Indeed, the only control on v that we have at our disposal is a 2 L∞ loc L estimate (which comes from the basic energy estimate) and this is not sufficient to obtain an estimate for D 2 ρ in L1loc (Lp ) by considering the convection term as a source term and by using the maximal smoothing effect of the heat equation. Even more refined maximal regularity estimates on convection–diffusion equations ([9,16] for example) do not seem to provide useful information when the control of the velocity field is so poor. Consequently, our strategy for the proof will be to use more carefully the structure of the coupling between the two equations of (1) in order to find suitable a priori estimates for (ζ, ρ). Since the coupling between the two equations does not make the original Boussinesq system well suited for a priori estimates, our main idea is to use an approach that was successfully used for the study of two-dimensional systems with a critical dissipation, see [19,20] and the Navier–Stokes–Boussinesq system with axisymmetric data [21]. It consists in diagonalizing the linear part of the system satisfied by ζ and ρ. We introduce a new unknown Γ which here formally reads Γ =ζ +

∂r −1 ρ r

and we study the system satisfied by (Γ, ρ) which is given by:

∂r −1 ∂t Γ + v · ∇Γ = − , v · ∇ ρ, r

∂t ρ + v · ∇ρ = ρ

where [ ∂rr −1 , v · ∇] is the commutator defined by

∂r −1 ∂r ∂r −1 , v · ∇ ρ = −1 (v · ∇ρ) − v · ∇ ρ . r r r

Note that if we forget the commutator for a while, we immediately get an a priori Lp estimate for Γ for every p from which we can hope to get an Lp estimate for ζ , if the operator ∂rr −1 behaves well.

750


To make this argument rigorous, we need first to study the action of the operator ∂rr −1 over axisymmetric functions. This is done in Proposition 2.9 where we prove that this operator takes the form ∂r −1 = aij (x)Rij r i,j

where Rij = ∂i ∂j −1 are Riesz operators and the functions aij are bounded. This yields that ∂r −1 acts continuously on L3,1 and hence that r ∂r −1 ρ(t) r 3,1 ρ(t) L3,1 ρ0 L3,1 . L It follows that the control of Γ is equivalent to the control of ζ in L3,1 . Now it remains to estimate in a suitable way the commutator term [ ∂rr −1 , v · ∇]ρ which is the main technical part. It seems that there is no hope to bound the commutator without using unknown quantities because there is no other known a priori estimates of the velocity except that given by energy estimate which is not strong enough. We shall prove (Theorem 3.1) that (∂r /r)−1 , v · ∇ ρ 3,1 ωθ /r 3,1 ρxh 0 (9) L B ∩L2 + ρ 1 . L ∞,1

2 B2,1

This estimate is the heart of our argument, its proof combines the use of paradifferential calculus and some harmonic analysis results and also requires a careful use of the property that velocity v is axisymmetric without swirl in the Biot–Savart law. The main reason for which we need some moments of ρ in the right-hand side of (9) is that we want an estimate of the commutator involving ωθ /r and not ω. In the right-hand side of (9), ρ 1 and ρxh L2 can be controlled in terms of the initial 2 B2,1

data only by using the smoothing effect of the convection–diffusion equation for ρ and standard energy estimates. Consequently, from this commutator estimate, we obtain that ζ (t)

L3,1

C(t)e

Cρxh L1 B 0

t ∞,1

(10)

and the next difficult step is to control ρxh L1 B 0 . This is done in two steps. The first step is t ∞,1 to get a global L∞ estimate of ρxh in terms of the initial data only and then in a second step, we 0 norm of xh ρ in terms of the L3,1 norm of ζ . shall prove a logarithmic estimate for the B∞,1 For the first step, let us observe that f = ρxh solves the equation ∂t f + v · ∇f − f = v h ρ − 2∇h ρ.

(11)

Note that for the moment, we only have at our disposal the standard energy estimate for v (thus ∞ estimate for f we need to use an we control vL∞ 2 only), consequently to obtain an L t L L2 → L∞ estimate for the convection–diffusion equation since the source term in the righthand side can be estimated only in L2 . Note that the convection term cannot be neglected (again because of the weak control on v that we have at this stage) and hence this estimate cannot be obtained from heat kernel estimates. We shall obtain this estimate by using the Nash–Moser– De Giorgi iterations [14,26,25]. Indeed, the main interest of this approach is that since it is based


751

on energy type estimates, the convection term does not contribute. A general result is recalled in Appendix A. For technical reasons, some higher order moment estimates which are easier to obtain are also needed, they are stated in Proposition 4.2. Once the estimate of ρxh L∞ is known in terms of the initial data, one can establish logarithmic Besov space estimates for the convection–diffusion equation (11) by using a special time dependent frequency cut-off of xh ρ where we combine the L∞ estimate with some smoothing effects for xh ρ. This yields (see (56)) ρxh L1 B 0 t

∞,1

t

C0 (t) 1 +

h(τ ) log 2 + ζ L∞ 3,1 dτ τ L

(12)

0

where C0 (t) is a given continuous function and h is some L1loc (R+ ) function. We point out that the use of the moment of order two |xh |2 ρ is due to the treatment of the commutator [q , v · ∇](xh ρ) which appears when we deal with the smoothing effects. The combination of the estimates (12) and (10) with Gronwall inequality allows to control ζ (t)L3,1 globally in time. The final step is to deduce, as for the incompressible Euler equation, from the control of ζ L∞ 3,1 an estimate of ωL∞ L∞ and of ∇vL∞ L∞ . This is the aim of Propositions 4.5 t t t L and 4.6. Estimates in Sobolev spaces then follow in a rather classical way. Once a priori estimates for sufficiently smooth functions are known, the result of Theorem 1.1 follows from an approximation argument. The paper is organized as follows. In Section 2 we fix the notations, give the definitions of the functional spaces, in particular Besov and Lorentz spaces, that we shall use and state some of their useful properties. We also study the operator ∂rr −1 in Proposition 2.9. Next, in Section 3, we study the commutator [ ∂rr −1 , v · ∇]. In Section 4, we turn to the proof of a priori estimates for sufficiently smooth solutions of (1). We first prove in Proposition 4.1 some basic energy estimates, next, we study the moments of ρ in Proposition 4.2 and then we control ζ L3,1 in Proposition 4.4. Lipschitz and Sobolev estimates are finally obtained in Proposition 4.5 and Proposition 4.6. In Section 5, we give the proof of Theorem 1.1: we obtain the existence part by using the a priori estimates and an approximation argument and then we prove the uniqueness part. Finally, Appendix A is devoted to the proof of a priori estimates for convection–diffusion equations by the Nash–De Giorgi iterations which are needed in the estimate of the moments of ρ. In Appendix B we give the proof of Lemma 2.7 which is a technical commutator lemma used in several places. 2. Preliminaries 2.1. Dyadic decomposition and functional spaces Throughout this paper, C stands for some real positive constant which may be different in each occurrence and C0 denotes a positive number depending on the initial data only. We shall sometimes alternatively use the notation X Y for the inequality X CY . p When B is a Banach space, we shall use the shorthand LT (B) for Lp (0, T , B).

752


Now to introduce Besov spaces which are a generalization of Sobolev spaces we need to recall the dyadic decomposition of the unity in the whole space (see [8]). Proposition 2.1. There exist two positive radial functions χ ∈ D(R3 ) and ϕ ∈ D(R3 \{0}) such that (1) χ(ξ ) +

1 ϕ 2 (2−q ξ ) 1 ∀ξ ∈ R3 , χ 2 (ξ ) + 3

ϕ(2−q ξ ) = 1,

q∈N

q∈N

supp ϕ(2−p ·) ∩ supp ϕ(2−q ·) = ∅,

(2) if |p − q| 2, (3) q 1 ⇒ supp χ ∩ supp ϕ(2−q ) = ∅. For every u ∈ S (R3 ) we define the nonhomogeneous Littlewood–Paley operators by, −1 u = χ(D)u;

∀q ∈ N,

q u = ϕ 2−q D u

and Sq u =

j u.

−1j q−1

One can easily prove that for every tempered distribution u,

u=

(13)

q u.

q−1

In the sequel we will frequently use Bernstein inequalities (see for example [8]). Lemma 2.2. There exists a constant C such that for k ∈ N, 1 a b and u ∈ La , we have 1 1 sup ∂ α Sq uLb C k 2q(k+3( a − b )) Sq uLa ,

|α|=k

and for q ∈ N C −k 2qk q uLa sup ∂ α q uLa C k 2qk q uLa . |α|=k

The basic tool of the paradifferential calculus is Bony’s decomposition [4]. It distinguishes in a product uv three parts as follows: uv = Tu v + Tv u + R(u, v), where Tu v =

Sq−1 uq v,

and R(u, v) =

q

q

with q =

1 i=−1

q+i .

q v, q u


753

The term Tu v is called the paraproduct of v by u and R(u, v) the remainder term. The main interest of the paraproduct term is that each term Sq−1 uq v has the support of its Fourier transform still localized in an annulus of size 2q and thus Tu v is a sum of almost orthogonal functions. s is the set of Let (p, r) ∈ [1, +∞]2 and s ∈ R, then the nonhomogeneous Besov space Bp,r tempered distributions u such that qs s := 2 q uLp r < +∞. uBp,r s . Also, by using the We remark that the Sobolev space H s coincides with the Besov space B2,2 Bernstein inequalities we get easily the embeddings s+3( p1 − p1 )

Bps 1 ,r1 → Bp2 ,r2

2

1

p1 p2 and r1 r2 .

,

Finally, let us notice that we can also characterize Lp spaces in terms of the dyadic decomposition, see [28]. For p ∈ ]1, +∞[, there exists C > 0 such that: f belongs to Lp if and only if (q f )q−1 ∈ Lp l 2 and C

1 2 2 |q f |

−1

f Lp

Lp

q−1

1 2 2 C |q f |

(14)

.

Lp

q−1

2.2. Lorentz spaces and interpolation For p ∈ ]1, ∞[, q ∈ [1, +∞], the Lorentz space Lp,q can be defined by real interpolation from Lebesgue spaces: p L 0 , Lp1 (θ,q) = Lp,q , where 1 p0 < p < p1 ∞, θ satisfies From this definition, we get:

1 p

=

1−θ p0

+

θ p1

and 1 q ∞.

Lp,p = Lp

Lp,q → Lp,q ,

(15)

for every 1 < p < ∞, 1 q q ∞. Lorentz spaces will arise in a natural way in our problem because of the following classical convolution results, for the proof see for instance [24,27]. Theorem 2.3. For every α, 0 < α < d, pi ∈ ]1, +∞[, qi ∈ [1, +∞], such that 1 + and

1 q1

=

1 q2

+

1 q3 ,

1 p1

=

1 p2

+

1 p3

there exists C > 0 such that f ∗ gLp1 ,q1 Cf Lp2 ,q2 gLp3 ,q3 .

(16)

Moveover, in the case that p1 = ∞, we have f ∗ gL∞ (Rd ) Cf

d

L α ,∞ (Rd )

g

d ,1

L d−α (Rd )

.

(17)

754

T. Hmidi, F. Rousset / Journal of Functional Analysis 260 (2011) 745–796 3

In particular, by using this result and the fact that 1/|x|2 belongs to L 2 ,∞ (R3 ), we have that ∇−1 f

L∞ (R3 )

f L3,1 (R3 )

and thanks to the pointwise estimate (7) that vr ζ 3,1 . L r ∞ L

(18)

(19)

To establish some functional inequalities involving Lorentz spaces the following classical interpolation result (see [24] for example) will be very useful. Theorem 2.4. Let 1 p1 < p2 ∞, 1 r1 < r2 ∞, q ∈ [1, ∞] and T be a linear bounded 1 θ 1−θ operator from Lpi to Lri . Let θ ∈ ]0, 1[ and p, r such that p1 = pθ1 + 1−θ p2 and r = r1 + r2 . p,q r,q Then T is also bounded from L to L with T L(Lp,q ;Lr,q ) CT θL(Lp1 ;Lr1 ) T 1−θ L(Lp2 ;Lr2 ) . As a consequence, we obtain the following results. Proposition 2.5. For 1 0 such that the following estimates hold true (1) uvLp,q CuL∞ vLp,q . (2) Tu vLp,q CuL∞ vLp,q . (3) Let us define the Riesz transform Rij = ∂i ∂j −1 , i, j ∈ {1, 2}, then Rij uLp,q CuLp,q . (4) For s >

1 2

3

−1

p we have H s → L3,1 . For 1 p < 3 we have Bp,1 → L3,1 .

Proof. (1) For a fixed function u ∈ L∞ , the linear operator T : v → uv belongs to L(Lp , Lp ) with norm smaller that uL∞ and hence the result follows by interpolation from Theorem 2.4. (3) In a similar way, for every p ∈ ]1, +∞[, Rij ∈ L(Lp , Lp ) thanks to the Calderón– Zygmund theorem and hence (3) follows again by using Theorem 2.4. (2) To establish the inequality, it is again sufficient thanks to Theorem 2.4 to prove that for u ∈ L∞ , v ∈ Lp we have Tu vLp CuL∞ vLp . For this last purpose we will make use of the maximal functions tool. We will start with some classical results in this subject. For a locally integrable function f : R3 → R, we shall define its maximal function Mf by 1 f (y) dy. Mf (x) = sup 3 r>0 r B(x,r)

From the definition we get 0 M(f g)(x) gL∞ Mf (x).

(20)


755

It is well known that M maps continuously Lp to itself for p ∈ ]1, ∞]. Moreover, we have the following lemma. We refer to [28] for a proof. Lemma 2.6. (1) Let ψ ∈ S(R3 ) and define ψε (x) = ε −3 ψ(ε −1 x) for ε > 0. Then there exists C > 0 such that for every p ∈ [1, ∞], ε ∈ (0, 1], we have sup ψε f (x) CMf (x). ε>0

In particular, we have sup q f (x) CMf (x). q−1

(2) Let p ∈ ]1, ∞] and {fq , q −1} be a sequence belonging to Lp 2 . Then we have 1 2 Mfq (x) 2

Lp

q

1 2 2 fq (x) C

.

Lp

q

Let us now come back to the proof of (2). By using (14), we have

Tu v

Lp

1 2 2 j (Tu v)

Lp

j −1

2 1 2 = S u v j q−1 q j −1

.

Lp

|j −q|4

This yields according to Lemma 2.6 and (20),

2 1 2 M(S u v) Tu vLp q−1 q j −1

Lp

|j −q|4

2 1 2 uL∞ M v q j −1

Lp

|j −q|4

1 2 2 uL∞ (Mq v) q−1

Lp

1 2 2 uL∞ (q v) q−1

Lp

uL∞ vLp where the last estimate follows from a new use of (14). This ends the proof of (2). (4) The first embedding follows from Sobolev embeddings combined with Theorem 2.4. This is left to the reader. For the second one we refer for example to the proof of Proposition 2.2 of [2]. 2

756


2.3. Some useful commutator estimates This section is devoted to the study of some basic commutators which will be needed in our main commutator estimates, especially in Theorem 3.1 and Proposition 3.2 . Our first result reads as follows. The proof is postponed to Appendix B. Lemma 2.7. Given (p, r, ρ, m) ∈ [1, +∞]4 such that 1 1 1 1 1+ = + + , p m ρ r

1 p r and ρ > 3 1 − . r

Let f, g and h be three functions such that ∇f ∈ Lρ , g ∈ Lm and xF −1 h ∈ Lr . Then h(D), f g

Lp

C xF −1 hLr ∇f Lρ gLm ,

where C is a constant. As an application of Lemma 2.7 we get the following commutator estimates. Lemma 2.8. Let p, m, ρ ∈ [1, +∞] such that p1 = ∇f ∈ Lρ , g ∈ Lm and for every q ∈ N ∪ {−1}

1 m

+ ρ1 . Then, there exists C > 0 such that for

[q , f ]g ˙ 1,p C∇f Lρ gLm , W with the following definition ϕW˙ 1,p = ∇ϕLp . Proof. We write for i = 1, 2, 3, ∂i [q , f ]g = [∂i q , f ]g − ∂i f q g = hq (D), f g − ∂i f q g, with hq (ξ ) = 2q φ(2−q ξ ), and φ ∈ S(R3 ). Using Lemma 2.7 we get hq (D), f g

Lp

C xF −1 hq L1 ∇f Lρ gLm C∇f Lρ gLm .

For the other term, the Hölder inequality yields ∂i f q gLp C∇f Lρ q gLm C∇f Lρ gLm .

2

2.4. Some algebraic identities We intend in this paragraph to describe first the action of the operator ∂rr −1 u over axisymmetric functions. We will show that it behaves like Riesz transforms. The second part is


757

concerned with the study of some algebraic identities involving some multipliers which will appear in a natural way when try to study our main commutator [∂r /r)−1 , v · ∇]ρ. Proposition 2.9. We have for every axisymmetric smooth scalar function u (∂r /r)−1 u(x) =

x22 x2 x1 x2 R11 u(x) + 12 R22 u(x) − 2 2 R12 u(x), 2 r r r

(21)

with Rij = ∂ij −1 . Moreover, for p ∈ ]1, ∞[, q ∈ [1, ∞] there exists C > 0 such that (∂r /r)−1 u

Lp,q

CuLp,q .

(22)

Proof. We set f = −1 u, then we can show from Biot–Savart law that f is also axisymmetric. Hence we get by using polar coordinates that ∂11 f + ∂22 f = (∂r /r)f + ∂rr f

(23)

where ∂r =

x1 x2 ∂1 + ∂2 . r r

By using this expression of ∂r , we obtain

∂rr = =

x1 x2 ∂1 + ∂2 r r

2

= ∂r

x2 x2 x1 x2 2x1 x2 ∂1 + ∂r ∂2 + 12 ∂11 + 22 ∂22 + ∂12 . r r r r r2

x12 x22 2x1 x2 ∂ + ∂22 + ∂12 11 r2 r2 r2

since

∂r

xi r

= 0,

∀i ∈ {1, 2}.

This yields by using (23) that

x12 x22 ∂r 2x1 x2 f = 1 − 2 ∂11 f + 1 − 2 ∂22 f − ∂12 f r r r r2 =

x22 x12 2x1 x2 ∂ f + ∂22 f − ∂12 f. 11 r2 r2 r2

To get (21), it suffices replace f by −1 u. The estimate (22) is a consequence of (21) and the estimates (1) and (3) of Proposition 2.5 xx since for every i, j ∈ {1, 2}, ri 2j ∈ L∞ . 2 We shall also need the following identities and estimates.

758


Lemma 2.10. For every f ∈ S(R3 , R), we have (1) For i, j ∈ {1, 2, 3} −1 (xi ∂j f ) = xi ∂j −1 f + Lij f where Lij f = −2Rij −1 f. Moreover, we have the estimates: 2 ∇ Lij f

∇Lij f L∞ Cf L3,1 , Lp,q

Cf Lp,q ,

(24)

p ∈ ]1, +∞[, q ∈ [1, +∞].

(25)

(2) For i, j, k ∈ {1, 2, 3} Rij (xk f ) = xk Rij f + Lkij f, with Lkij := −2∂k −1 Rij + δik ∂j −1 + δj k ∂i −1 where δij denotes the Kronecker symbol. Moreover we have the estimates k L f ∞ Cf 3,1 , L ij L

k ∇L f p,q Cf Lp,q , ij L

(26)

p ∈ ]1, +∞[, q ∈ [1, +∞].

(27)

Proof. (1) We first expand xi ∂j −1 f − 2Rij −1 f = 2∂ij −1 f + xi ∂j f − 2Rij f = xi ∂j f. This yields −1 (xi ∂j f ) = xi ∂j −1 f − 2Rij −1 f + P (x), with P a harmonic polynomial. We can easily see that the r.h.s of this identity and Rij −1 f are decreasing at infinity. Thus to prove that P is zero it suffices to prove that xi ∂j −1 f goes to zero at infinity. Since |yj f (y)| |f (y)| |f (y)| xi ∂j −1 f |xj | dy + dy dy. |x − y| |x − y|2 |x − y|2 R3

R3

R3

Using Theorem 2.3, we get that xi ∂j −1 f ∈ Lp , for every p > 3. Hence we get P = 0. The estimates (24), (25) are a direct consequence of the above expression and (18) and the estimate (3) of Proposition 2.5. (2) We use the same idea as previously. We first get the identity Lkij f = Rij (xk f ) − xk Rij f = ∂ij (xk f ) − 2∂k Rij f − xk ∂ij f = δik ∂j f + δj k ∂i f − 2∂k Rij f


759

and by the same argument as above, we finally obtain that Rij (xk f ) − xk Rij f = δik ∂j −1 f + δj k ∂i −1 f − 2∂k Rij −1 f. The estimates (26), (27) are a direct consequence of the above expression and (18) and the estimate (3) of Proposition 2.5. 2 3. Commutator estimates 3.1. The commutator between the advection operator and

∂r −1 r

In this part we discuss the commutation between the operators ∂rr −1 and v · ∇. This is a crucial estimate in order to get better a priori estimates for the solution of (1) by using our transformation. Our result reads as follows. Theorem 3.1. Let v be an axisymmetric smooth and divergence free without swirl vector field and ρ an axisymmetric smooth scalar function. Then we have, with the notation xh = (x1 , x2 ), that (∂r /r)−1 , v · ∇ ρ

L3,1

ωθ /rL3,1 ρxh B 0

2 ∞,1 ∩L

+ ρ

1 2 B2,1

.

Proof. Since the functions ρ and v · ∇ρ are axisymmetric then using the identity of Proposition 2.9 we have (∂r /r)−1 ρ(x) =

2 x22 x12 x1 x2 R ρ(x) + R ρ(x) − 2 R ρ(x) := aij (x)Ri,j ρ(x) 11 22 12 r2 r2 r2 i,j =1

and also (∂r /r)−1 (v · ∇ρ)(x) =

2

aij (x)Ri,j (v · ∇ρ)(x).

i,j =1

Since v has no swirl and the functions ai,j do not depend on r and z, we have for every 1 i, j 2 v · ∇ai,j (x) = v r ∂r ai,j + v z ∂3 ai,j = 0. Consequently our commutator can be rewritten as 2 2 (∂r /r)−1 , v · ∇ ρ(x) = ai,j (x)[Rij , v · ∇]ρ = ai,j (x) div [Rij , v]ρ i,j =1

i,j =1

760


where we have used the fact that v is divergence free to get the last equality. By using that aij ∈ L∞ and the estimate (1) of Proposition 2.5, we first obtain that (∂r /r)−1 , v · ∇ ρ

L3,1

2 div [Rij , v]ρ i,j =1

L3,1

.

(28)

The terms ∂1 ([Rij , v 1 ]ρ) and ∂2 ([Rij , v 2 ]ρ) can be treated in same way and hence, we shall prove the estimate of the first one only. The estimate of ∂3 ([Rij , v 3 ]ρ) which is easier will be done in a second step. • Estimate of ∂1 ([Rij , v 1 ]ρ). Since v is divergence free, we have that v = −∇ ∧ ω. Hence for axisymmetric flows (where in particular ω = ωθ eθ ), we obtain that v 1 (x) = −1 ∂3 ω2 = −1 ∂3 x1 (ωθ /r) . Applying Lemma 2.10(1) we get v 1 (x) = x1 −1 ∂3 (ωθ /r) + L(ωθ /r),

with L = −2∂13 −2

(29)

(we omit the subscript ij for notational convenience). Consequently the commutator can be rewritten under the form ∂1 Rij , v 1 ρ = ∂1 Ri,j , L(ωθ /r) ρ + ∂1 Ri,j , −1 ∂3 (ωθ /r) x1 ρ = ∂1 Ri,j , L(ωθ /r) ρ + ∂1 Ri,j , −1 ∂3 (ωθ /r) x1 ρ + ∂1 −1 ∂3 (ωθ /r) [Rij , x1 ]ρ = ∂1 −1 ∂3 (ωθ /r) L1ij ρ + ∂1 Ri,j , L(ωθ /r) ρ + ∂1 Ri,j , −1 ∂3 (ωθ /r) x1 ρ = I + II + III

(30)

where we have used the identity (2) of Lemma 2.10. Estimate of I. We write ∂1 ∂3 −1 (ωθ /r)L1ij ρ = R13 (ωθ /r)L1ij ρ + ∂3 −1 (ωθ /r)∂1 L1ij ρ.

(31)

By using (1) and (3) of Proposition 2.5 and (26) we have R13 (ωθ /r)L1 ρ ij

L3,1

R13 (ωθ /r)L3,1 L1ij ρ L∞ Cωθ /rL3,1 ρL3,1

and by using Proposition 2.5(1) and (18), (27), we also obtain ∂3 −1 (ωθ /r)∂1 L1 ρ 3,1 ∂3 −1 (ωθ /r) ∞ ∂1 L1 ρ 3,1 Cωθ /r 3,1 ρ 3,1 . L L ij ij L L L Combining these estimates we find IL3,1 Cωθ /rL3,1 ρL3,1 .

(32)


761

Estimate of II. We will use Bony decomposition II = II1 + II2 + II3 , with II1 = ∂1

Rij , Sq−1 L(ωθ /r) q ρ, q0

II2 = ∂1

Rij , q L(ωθ /r) Sq−1 ρ,

q0

II3 = ∂1

˜ q ρ. Rij , q L(ωθ /r)

q−1

For the first term we easily get that there exists a function ψ ∈ S(R3 ) such that II1 = ∂1 ψq (D), Sq−1 L(ωθ /r) q ρ , q0

with ψq = 23q ψ(2q ·). By using the Bernstein inequality, this yields ∂1 ψq (D), Sq−1 L(ωθ /r) q ρ

L2

C2q ψq (D), Sq−1 L(ωθ /r) q ρ L2 .

Thanks to Lemma 2.7 and (24), we find ∂1 ψq (D), Sq−1 L(ωθ /r) q ρ

L2

C2q xψq L1 ∇L(ωθ /r)L∞ q ρL2 CxψL1 ωθ /rL3,1 q ρL2 .

It follows that II1

1 2 B2,1

C

1 2q 2 ∂1 ψq (D), Sq−1 L(ωθ /r) q ρ L2 Cωθ /rL3,1 ρ

q∈N

1

2 B2,1

1

2 and hence by using the embedding B2,1 → L3,1 (see Proposition 2.5(4)), we obtain

II1 L3,1 Cωθ /rL3,1 ρ

1

2 B2,1

.

(33)

To estimate the term II2 we do not need to detect cancellation in the structure of the commutator, we just write II2 =

q0

∂1 Rij q L(ωθ /r) Sq−1 ρ − ∂1 q L(ωθ /r) Rij Sq−1 ρ . q0

A useful remark is that thanks to the Bernstein inequalities and (25), we have q Lf Lp 2−2q ∇ 2 Lq f Lp 2−2q f Lp , ∀q 0, p ∈ ]1, +∞[.

(34)

762


This yields by using the Hölder inequality and Proposition 2.5(3) that II2

1 2 B2,1

3 3 2 2 q q L(ωθ /r) Sq−1 ρ L2 + 2q 2 q L(ωθ /r) Rij Sq−1 ρ L2

q0

q0

2

q 32

q L(ωθ /r) 3 Sq−1 ρ 6 + Rij Sq−1 ρ 6 L L L

q0

ωθ /rL3

1

2−q 2 Sq−1 ρL6

q0

ωθ /rL3

k 1 2 2 (k−q) 2 2 k ρL2

q0 kq−2

ωθ /rL3 ρ

1

2 B2,1

.

Hence we get from Proposition 2.5(4) that II2 L3,1 Cωθ /rL3,1 ρ

1

2 B2,1

(35)

.

For the term II3 we write II3 = ∂1

˜ q ρ + ∂1 Rij , q L(ωθ /r)

˜ qρ Rij , q L(ωθ /r) −1q0

q1

:= II31 + II32 . To estimate the first term we first use the Bernstein inequality to get k II31 L2 2k

Rij , q L(ωθ /r) ˜ q ρ 2 . L qk−4

Next, to estimate the terms inside the sum we do not need to use the structure of the commutator. By using again the Hölder inequality, (34) and the Bernstein inequality, we obtain Rij , q L(ωθ /r) ˜ q ρ 2 L ˜ q ρL6 ˜ q L(ωθ /r) L3 q ρL6 + q L(ωθ /r) L3 Rij ˜ q ρL2 . 2−q ωθ /rL3 It follows by using again Proposition 2.5(4) that II31 L3,1 II31

1 2 B2,1

ωθ /rL3

k−1 qk−4

˜ q ρL3 ωθ /rL3 ρ 2 2 (k−q) 2q 2 3

1

1

2 B2,1

.


763

For the estimate of the low frequencies term II32 we need to use more deeply the structure of the commutator. We first write ˜ q ρ. ˜ qρ − ∂1 Rij , q L(ωθ /r) ∂1 Lq (ωθ /r)Rij II32 = −1q0

−1q0

The last term of the above identity is estimated as follows by using again Proposition 2.5 (1) and (3) and (24)

−1q0

˜ q ρ ∂1 Lq (ωθ /r)Rij

1 2 B2,1

∂1 Lq (ωθ /r)Rij ˜ q ρ

L2

−1q0

∂1 L(ωθ /r)L∞ ρL2 ωθ /rL3,1 ρ

1

2 B2,1

.

To estimate the first term of II32 we write for every −1 q 0 thanks to Lemma 2.7 that ∂1 Rij , q L(ωθ /r) ˜ q ρL2 , ˜ q ρ 5 xh 10 ∇L(ωθ /r) ∞ L L2

L

9

where h(ξ ) = ξ1 |ξi |2j χ˜ (ξ ) and χ˜ ∈ D(R3 ). Using Mikhlin–Hörmander Theorem we have ξξ

h(x) C 1 + |x| −4 ,

∀x ∈ R3 .

10

This gives in particular xh ∈ L 9 . Therefore we get by using again (24) that

˜ ∂1 Rij , q L(ωθ /r) q ρ

1 B 55 2 ,1

−1q0

∇L(ωθ /r)L∞ ρL2 ωθ /rL3,1 ρL2 .

1

By using the embedding B 55 → L3,1 , (see Proposition 2.5(4)), we find that 2 ,1

II32 L3,1 ωθ /rL3,1 ρL2 . We have thus obtained that the term II3 enjoys the estimate II3 L3,1 ωθ /rL3,1 ρL2 . Consequently, by gathering this last estimate and the estimates (33), (35), we finally get that IIL3,1 Cωθ /rL3,1 ρ

1

2 B2,1

.

Estimate of III. We also decompose the term III by using Bony’s formula as follows: III = III1 + III2 + III3 ,

(36)

764


with III1 = ∂1

Rij , Sq−1 ∂3 −1 (ωθ /r) q (x1 ρ), q0

III2 = ∂1

Rij , q ∂3 −1 (ωθ /r) Sq−1 (x1 ρ),

q0

III3 = ∂1

˜ q (x1 ρ). Rij , q ∂3 −1 (ωθ /r)

q−1

As we have done to handle the term II1 , we can use that there exists a function ψ ∈ S(R3 ) such that ∂1 ψq (D), Sq−1 ∂3 −1 (ωθ /r) q (x1 ρ) , III1 = q0

with ψq = 23q ψ(2q ·). For every p ∈ ]1, ∞[, we first write thanks to the Bernstein inequality that ∂1 ψq (D), Sq−1 ∂3 −1 (ωθ /r) q (x1 ρ) p L q −1 C2 ψq (D), Sq−1 ∂3 (ωθ /r) q (x1 ρ)Lp . Then by using successively Lemma 2.7 and the continuity of the Riesz transform (i.e. Proposition 2.5(3)), we get ψq (D), Sq−1 ∂3 −1 (ωθ /r) q (x1 ρ) p L −1 xψq L1 Sq−1 ∇∂3 (ωθ /r) Lp q (x1 ρ)L∞ 2−q xψL1 ∇∂3 −1 (ωθ /r)Lp q (x1 ρ)L∞ 2−q ωθ /rLp q (x1 ρ)L∞ . It follows that III1 Lp

ωθ /rLp q (x1 ρ)L∞ ωθ /rLp x1 ρB 0 . ∞,1

q0

This proves that the linear operator T f → ∂1 ψq (D), Sq−1 ∂3 −1 f q (x1 ρ) q0

is continuous from Lp into itself for every p ∈ ]1, ∞[ and that T L(Lp ) Cp x1 ρB 0 . ∞,1

Consequently, by using the interpolation result of Theorem 2.4, we get that T is continuous on Lp,q for every 1 < p < ∞ and q ∈ [1, ∞]. In particular, this yields III1 L3,1 ωθ /rL3,1 x1 ρB 0 . ∞,1

(37)


765

For the term III3 , we use split it into ˜ q (x1 ρ) Rij , q ∂3 −1 (ωθ /r) III3 = ∂1 q1

+ ∂1

˜ q (x1 ρ) Rij , q ∂3 −1 (ωθ /r)

−1q0

:= III31 + III32 . Let p ∈ ]1, ∞[ from the Bernstein inequality, we have that Rij , q ∂3 −1 (ωθ /r) ˜ q (x1 ρ) p k III31 Lp 2k L qk−4

and the terms inside the sum can be controlled without using the structure of the commutator. We just write Rij , q ∂3 −1 (ωθ /r) ˜ q (x1 ρ) ∞ ˜ q (x1 ρ) p q ∂3 −1 (ωθ /r) p L L L −1 ˜ q (x1 ρ) ∞ + q ∂3 (ωθ /r) Lp Rij L −q ˜ q (x1 ρ) ∞ . 2 ωθ /rLp L Note that we have used the Bernstein inequality, the continuity of the Riesz transform on Lp and ˜ q does not contain zero which gives that the the fact that the support of the Fourier transform of ∞ operator Rij q also acts continuously on L (since it can be written as the convolution with an L1 function). It follows that for every p ∈ ]1, +∞[, we have ˜ q (x1 ρ) ∞ ωθ /rLp x1 ρ 0 . 2k−q III31 Lp ωθ /rLp B L ∞,1

k−1 qk−4

By using again the interpolation result of Theorem 2.4, this yields III31 L3,1 ωθ /rL3,1 x1 ρB 0 . ∞,1

We can also estimate the term III32 without using the structure of the commutator. By using the continuity of the Riesz transform on L2 and (18), we obtain Rij , q ∂3 −1 (ωθ /r) ˜ q (x1 ρ) 2 ˜ q (x1 ρ) 2 q ∂3 −1 (ωθ /r) ∞ L L L −1 ˜ q (x1 ρ) 2 + q ∂3 (ωθ /r) L∞ Rij L −1 ˜ q (x1 ρ) 2 ∂3 (ωθ /r) ∞ L

ωθ /rL3,1 x1 ρL2 . Therefore we get III32

1

2 B2,1

ωθ /rL3,1 x1 ρL2 .

L

766


Consequently we obtain III3 B 0 ωθ /rL3,1 x1 ρB 0

2 ∞,1 ∩L

3,1

.

(38)

Let us now turn to the estimate of the term III2 . We write III2 =

Rij , q (∂13 −1 (ωθ /r) Sq−1 (x1 ρ) + Rij , q (∂3 −1 (ωθ /r) ∂1 Sq−1 (x1 ρ) q0

= III21 + III22 . We have by definition of the paraproducts that III21 = Rij Tx1 ρ R13 (ωθ /r) − TRij (x1 ρ) R13 (ωθ /r). Thanks to Proposition 2.5, we get that III21 L3,1 R13 (ωθ /r)L3,1 x1 ρL∞ + R13 (x1 ρ)L∞ ωθ /rL3,1 x1 ρL∞ + R13 (x1 ρ)L∞ ωθ /rL3,1 x1 ρB 0

2 ∞,1 ∩L

.

Note that the L2 norm in the right-hand side comes from the low frequency term in the Littlewood–Paley decomposition: we have R13 −1 (x1 ρ) ∞ R13 −1 (x1 ρ) 2 x1 ρ 2 (39) L L L thanks to the Bernstein inequality and the L2 continuity of the Riesz transform. For the estimate of III22 , we shall use that thanks to the Bernstein inequality, we have for every f that, q ∂3 −1 f p 2−q f Lp , ∀q 0, p ∈ ]1, +∞[. L This yields III22 Lp ωθ /rLp

q0

ωθ /rLp

2−q ∂1 Sq−1 (x1 ρ)L∞ + ∂1 Sq−1 Rij (x1 ρ)L∞

2p−q p (x1 ρ)L∞ + p Rij (x1 ρ)L∞

q0 q−2p−1

ωθ /rLp x1 ρB 0

∞,1

ωθ /r

Lp

+ Rij (x1 ρ)B 0 ∞,1

x1 ρB 0

2 ∞,1 ∩L

by using again (39). Consequently, by interpolation, we also find III22 L3,1 ωθ /rL3,1 x1 ρB 0

2 ∞,1 ∩L

.


767

We have thus shown that III2 L3,1 ωθ /rL3,1 x1 ρB 0

.

(40)

.

(41)

2 ∞,1 ∩L

Gathering (37), (38) and (40), we obtain IIIL3,1 ωθ /rL3,1 x1 ρB 0

2 ∞,1 ∩L

Finally we obtain ∂1 Rij , v 1 ρ

L3,1

ωθ /rL3,1 x1 ρB 0

2 ∞,1 ∩L

+ ρ

1 2 B2,1

thanks to (41), (36), (32) and (30). In the same way, we also obtain the estimate ∂2 Rij , v 2 ρ 3,1 ωθ /r 3,1 x2 ρ 0 L B ∩L2 + ρ 1 . L ∞,1

2 B2,1

In view of (28), it remains to estimate the term ∂3 ([Rij , v 3 ]ρ) which has a different structure. • Estimate of ∂3 ([Rij , v 3 ]ρ). Since we can write that

ωθ ωθ ωθ ωθ ωθ = − r∂r +2 = −xh · ∇h −2 , v = −(curl ω)3 = − ∂r ωθ + r r r r r 3

we obtain that −1

v (x) = − 3

xh · ∇h

ωθ r

−1

− 2

ωθ r

and hence by using Lemma 2.10 that −v (x) = xh · ∇h

−1

= xh · ∇h

−1

3

(ωθ /r) − 2

2

−1 Rii (ωθ /r) + 2−1 (ωθ /r)

i=1

(ωθ /r) + 2−1 R33 (ωθ /r).

Thus, we have a decomposition of the commutator under the form 2 ∂3 ∂k −1 (ωθ /r)[Rij , xk ]ρ + 2∂3 Rij , −1 R33 (ωθ /r) ρ −∂3 Rij , v 3 ρ = k=1

+

2

∂3 Rij , ∂k −1 (ωθ /r) (xk ρ)

k=1

= I + II + III. To estimate the first term I, we use Lemma 2.10(2) to obtain that

(42)

768


ωθ ωθ [Rij , xk ]ρ = ∂3 ∂k −1 Lkij ρ ∂3 ∂k −1 r r

ωθ k −1 ωθ Lij ρ + ∂k ∂3 Lkij ρ. = R3k r r It follows that IL3,1

2 k L ρ ij

L∞

R3k (ωθ /r)

L3,1

+ ∂k −1 (ωθ /r)L∞ ∂3 Lkij ρ L3,1

k=1

ρL3,1 ωθ /rL3,1 thanks to (26), (27). The estimates of the terms II and III are similar to the ones of II and III in (30) (indeed, the operator −1 R33 = ∂33 −2 has the same properties as L = −2∂13 −2 which arises in (30)) consequently, we also get as in (36) and (41) that IL3,1 ωθ /rL3 ρ

1

2 B2,1

,

IIL3,1 ωθ /rL3,1 ρxh B 0

2 ∞,1 ∩L

.

Consequently, we also find that ∂3 Rij , v 3 ρ

L3,1

This ends the proof of Theorem 3.1.

ωθ /rL3,1 ρxh B 0

2 ∞,1 ∩L

+ ρ

1 2 B2,1

.

2

3.2. Commutation between the advection operator and q The last commutator estimate which is needed in the proof of our main result is the following. Proposition 3.2. Let v be an axisymmetric divergence free vector field without swirl and ρ a smooth scalar function. Then there exists C > 0 such that for every q ∈ N ∪ {−1} we have [q , v · ∇]ρ

L2

Cωθ /rL3,1 ρxh L6 + ρL2 .

Proof. From the incompressibility of the velocity we have [q , v · ∇]ρ =

3 ∂i q , v i ρ = I + II + III.

(43)

i=1

The first and the second terms can be handled in the same way, so we shall only detail the proof of the estimate of the first one. Thanks to (29), we have that v 1 (x) = x1 −1 ∂3 (ωθ /r) + L(ωθ /r), and hence we get

with L = −2R13 −1


769

I = ∂1 q , x1 −1 ∂3 (ωθ /r) ρ + ∂1 q , L(ωθ /r) ρ = I1 + I2 . The estimate of the second term in the right-hand side is again a direct consequence of Lemma 2.8 and (24). Indeed, we write I2 L2 ∇L(ωθ /r)L∞ ρL2 ωθ /rL3,1 ρL2 . The first term I1 in the right-hand side can be expanded under the form I1 = ∂1 q , −1 ∂3 (ωθ /r) (x1 ρ) + ∂1 −1 ∂3 (ωθ /r)[q , x1 ]ρ = I11 + I12 . We start with the estimate of I12 . By definition of q , we have x1 q ρ = x1 23q =2

3q

ϕ 2q (x − y) ρ(y) dy

R3

ϕ 2q (x − y) y1 ρ(y) dy + 23q

R3

= q (x1 ρ) + 2−q 23q ϕ1 2q · ρ,

ϕ 2q (x − y) (x1 − y1 )ρ(y) dy

R3

where ϕ1 (x) = x1 ϕ(x) ∈ S(R3 ). Consequently we get the expression of the commutator: [q , x1 ]ρ = −2−q 23q ϕ1 2q · ρ.

(44)

This yields I12 = − R13 (ωθ /r) 22q ϕ1 2q · ρ − −1 ∂3 (ωθ /r) 23q (∂1 ϕ1 ) 2q · ρ. Therefore we get by using again the Hölder inequality, the continuity of the Riesz transform, (18) and the Young inequality for convolutions that: I12 L2 R13 (ωθ /r)L3 22q ϕ1 2q · ρ L6 + −1 ∂3 (ωθ /r)L∞ 23q (∂1 ϕ1 ) 2q · ρ L2 ωθ /rL3 ϕ1

3

L2

ρL2 + ωθ /rL3,1 ∂1 ϕ1 L1 ρL2

ωθ /rL3,1 ρL2 . Note that we have also used the embedding (15). To estimate I11 we use again Lemma 2.8: I11 L2 ∇−1 ∂3 (ωθ /r)L3 x1 ρL6 ωθ /rL3 x1 ρL6 ωθ /rL3,1 x1 ρL6 . We have thus shown that IL2 ωθ /rL3,1 ρL2 + x1 ρL6 .

(45)

770


In the same way, we obtain that IIL2 ωθ /rL3,1 ρL2 + x2 ρL6 .

(46)

It remains to estimate the last term III. By using (42), we get −III = ∂3 q , ∇h −1 (ωθ /r) (xh ρ) + ∂3 ∇h −1 (ωθ /r)[q , xh ]ρ + 2∂3 q , −1 R33 (ωθ /r) ρ = III1 + III2 + III3 . The estimates of the first and last terms follow again from Lemma 2.8: we write that III1 L2 C ∇ 2 −1 (ωθ /r)L3 xh ρL6 ωθ /rL3 xh ρL6 ωθ /rL3,1 xh ρL6 and that III3 L2 C ∇−1 R33 (ωθ /r)L∞ ρL2 R33 (ωθ /r)L3,1 ρL2 Cωθ /rL3,1 ρL2 . Note that we have used again the estimate (18). Finally, to estimate the second term III2 we can use the expression of the commutator [q , xh ] given by (44): III2 = 2−q ∂3 ∇h −1 (ωθ /r) 23q ϕh 2q · ρ = 2−q ∂3 ∇h −1 (ωθ /r) 23q ϕh 2q · ρ + ∇h −1 (ωθ /r) 23q (∂3 ϕh ) 2q · ρ , with ϕh (x) = −xh ϕ(x). It follows as before that III2 L2 2−q ωθ /rL3 23q ϕh 2q · ρ L6 + ∇h −1 (ωθ /r)L∞ 23q (∂3 ϕh ) 2q · ρ L2 ωθ /rL3 ϕh

3

L2

ρL2 + ωθ /rL3,1 ∂3 ϕh L1 ρL2

ωθ /rL3,1 ρL2 . Gathering these estimates we also find that IIIL3 ωθ /rL3,1 ρLp + xh ρL6 . In view of (43), (45), (46) and the last estimate, this ends the proof of Proposition 3.2.

2

4. A priori estimates In this section we intend to establish the global a priori estimates needed for the proof of Theorem 1.1. We shall first prove some basic weak estimates that can be obtained easily through energy type estimates. In a second step, we shall prove the control of some stronger norms such as ω(t)L∞ and ∇v(t)L∞ . This part requires more refined analysis: we use the special structure of the Boussinesq model combined with the previous commutator estimates.


771

4.1. Energy estimates We start with some elementary energy estimates. Proposition 4.1. Let (v, ρ) be a smooth solution of (1) then (1) For p ∈ ]1, ∞[, q ∈ [1, ∞] and t ∈ R+ , we have ρ2L∞ L2 + 2∇ρ2L2 L2 ρ0 2L2 , t

t

p,q Cρ0 Lp,q . ρL∞ t L

(2) For v0 ∈ L2 , ρ0 ∈ L2 and t ∈ R+ we have v(t)

L2

v0 L2 + tρ0 L2 .

(3) For ρ0 ∈ L2 we have the dispersive estimate ρ(t)

L∞

1 C 1 + 3 ρ0 L2 . t4

The constant C is absolute. Note that the axisymmetric assumption is not needed in this proposition. Proof. (1) By taking the L2 -scalar product of the second equation of (1) with ρ and integrating by parts, we get since v is divergence free that 1 d ρ(t)2 2 + L 2 dt

∇ρ(t, x) 2 dx = 0.

R3

Integrating in time this differential inequality gives the desired result. Let us now move to the estimate of the density in Lorentz spaces. First, the same argument yields that for every p ∈ [1, ∞], we have ρ(t)

Lp

ρ0 Lp .

It suffices now to use the interpolation result of Theorem 2.4. (2) We take the L2 -scalar product of the velocity equation with v and we integrate by parts 1 d v(t)2 2 v(t) 2 ρ(t) 2 L L L 2 dt and this implies that d v(t) 2 ρ(t) 2 . L L dt

772


Thus, integrating in time gives v(t)

t L2

v0 L2 +

ρ(τ )

L2

dτ.

0

Since ρ(t)L2 ρ0 L2 , we infer v(t)

L2

v0 L2 + tρ0 L2 .

(3) The estimate is a direct consequence of Lemma A.1 The proof of the proposition is now achieved. 2 4.2. Estimates of the moments of ρ. We have seen in Section 3.1 and Section 3.2 that the estimates of the commutators involve some moments of the density. Thus we aim in this paragraph at giving suitable estimates for the moments that will be needed later when we shall perform our diagonalization of the Boussinesq system. Two types of estimates are discussed: the energy estimates of the horizontal moments |xh |k ρ, with k = 1, 2 and some dispersive estimates. More precisely we prove the following. Proposition 4.2. Let v be a vector field with zero divergence and satisfying the energy estimate of Proposition 4.1. Let ρ be a solution of the transport-diffusion equation ∂t ρ + v · ∇ρ − ρ = 0,

ρ(0, x) = ρ0 .

Then we have the following estimates. (1) For ρ0 ∈ L2 and xh ρ0 ∈ L2 , there exists C0 > 0 such that for every t 0 5 xh ρL∞ 2 + xh ρL2 H˙ 1 C0 1 + t 4 . t L t

(2) For ρ0 ∈ L2 ∩ Lm , m > 6 and xh ρ0 ∈ L2 , there exists C0 > 0 such that for every t > 0 xh ρ(t) ∞ C0 t 14 + t − 34 . L (3) For ρ0 ∈ L2 and |xh |2 ρ0 ∈ L2 , there exists C0 > 0 such that for every t 0 |xh |2 ρ

2 L∞ t L

5 + |xh |2 ρ L2 H˙ 1 C0 1 + t 2 . t

(4) For ρ0 ∈ L2 ∩ L6 and |xh |2 ρ0 ∈ L2 , there exists C0 > 0 such that for every t > 0 |xh |2 ρ(t)

L6

13 1 C0 t 6 + t − 2 .


773

Remark 4.3. Note that when ρ0 ∈ L2 and |xh |2 ρ0 ∈ L2 then automatically the moment of order one belongs to L2 , that is xh ρ0 ∈ L2 . This is an easy consequence of the Hölder inequality 1 1 xh ρL2 ρL2 2 |xh |2 ρ L2 2 .

Proof. (1) Setting f = xh ρ, we can easily check that f solves the equation ∂t f + v · ∇f − f = v h ρ − 2∇h ρ

(47)

with the notations v h = (v 1 , v 2 ) and ∇h = (∂1 , ∂2 ). Now, taking the L2 -scalar product with f , integrating by parts and using the Hölder inequality 1 d f (t)2 2 + ∇f (t)2 2 = L L 2 dt

v ρf dx − 2 h

R3

∇h ρf dx

R3

vL2 ρL3 f L6 + 2ρL2 ∇f L2 . By using the Sobolev embedding H˙ 1 → L6 combined with the Young inequality, we obtain d f (t)2 2 + ∇f (t)2 2 v2 2 ρ2 3 + ρ2 2 . L L L L L dt Since the Gagliardo–Nirenberg inequality gives that ρ2L3 ρL2 ∇ρL2 , we infer d f (t)2 2 + ∇f (t)2 2 v2 2 ρ 2 ∇ρ 2 + ρ2 2 . L L L L L L dt

(48)

Integrating in time and using the energy estimate of Proposition 4.1(1), we thus obtain f (t)2 2 + ∇f (t)2 2 L

Lt L2

f0 2L2 + v2L∞ L2 ρ0 L2 ∇ρL1 L2 + ρ0 2L2 t t

t

1 f0 2L2 + C0 1 + t 2 t 2 + ρ0 2L2 t 5 C0 1 + t 2 .

(2) We shall apply Lemma A.1 to (47) with F = ρ ei and G = vi ρ. First, we observe that we have obviously from the Hölder inequality combined with Proposition 4.1(1) that for m 2 G

2m

m+2 L∞ t L

vL∞ 2 ρL∞ Lm t t L C0 (1 + t)

and F L∞ 6 ρ0 L6 . t L

774


Consequently, we get from Lemma A.1 and Proposition 4.1 that for m > 6 and for t > 0, 3 1 f (t) ∞ C 1 + t − 34 f0 2 + C0 1 + t 14 − 2m + 1 + t 4 ρ0 L6 L L 3 1 C0 t − 4 + t 4 . (3) The second moment g = |xh |2 ρ solves the following equation ∂t g + v · ∇g − g = 2v h (xh ρ) − 2∇h ρ − 4 divh (xh ρ) = vh f − 2∇h ρ − 4 divh f. By using again an L2 energy estimate, we find that d g(t)2 2 + ∇g(t)2 2 v(t)2 2 f (t) 2 ∇f (t) 2 + ρ(t)2 2 + f (t)2 2 . L L L L L L L dt Thus, by integrating in time and using the energy estimates for ρ, v and f we get g(t)

1

L2

1

1

1

1

2 2 + ∇gL2 L2 g0 L2 + vL∞ 2 f ∞ 2 ∇f 2 2 t 4 + ρ0 L2 t 2 + f L∞ L2 t 2 L L t L t t

Lt L

t

5 C0 1 + t 2 ,

where C0 is a constant depending on the quantities |xh |k ρ0 L2 for k = 0, 1, 2. (4) By setting g1 (t, x) = tg(t, x), we have that ∂t g1 + v · ∇g1 − g1 = g + 2v h (txh ρ) − 2t∇h ρ − 4 divh (txh ρ). Multiplying this equation by |g1 |4 g1 , integrating by parts and the obvious inequality |xh ρ| 1 1 |ρ| 2 |g| 2 , we thus get 1 d g1 6L6 + 5 6 dt

|∇g1 | |g1 | dx 2

4

gL6 g1 5L6

R3

+t

1 2

1

R3

11

|v||ρ| 2 |g1 | 2 dx 1

1

|ρ||∇g1 ||g1 |4 dx + t 2

+t R3

R3

It follows from Hölder inequality that 1

R3

Consequently

1

9

1

11

1

|ρ| 2 |g1 | 2 |∇g1 | dx t 2 vL2 g1 L218 ρ 2 18 .

t2

9

|ρ| 2 |g1 | 2 |∇g1 | dx.

L

7


1 d g1 6L6 + 5 6 dt

775

|∇g1 |2 |g1 |4 dx

R3 11

1

1

gL6 g1 5L6 + t 2 vL2 g1 L218 ρ 2 18 L

1 5 1 + tρL6 g1 2L6 + t 2 ρL2 6 g1 L2 6

7

1 |∇g1 | |g1 | dx 2

4

2

.

R3

Now we can use the Young inequality combined with the following Sobolev inequality 2 ∇ g13 L2 = 9 g1 18 L18

|∇g1 |2 |g1 |4 dx

R3

to obtain that d g1 6L6 + cg1 6L18 + c dt

|∇g1 |2 |g1 |4 dx gL6 g1 5L6 + t 6 v12 ρ6 18 L2 L

R3

7

+ t 2 ρ2L6 g1 4L6 + tρL6 g1 5L6 . By using Proposition 4.1, we deduce that d g1 6L6 + cg1 6L18 + c dt

|∇g1 |2 |g1 |4 dx gL6 g1 5L6 + C0 t 6 1 + t 12

R3

+ t 2 ρ0 2L6 g1 4L6 + tρ0 L6 g1 5L6 . Next, by using again the Young inequality, we infer d g1 (t)6 6 C0 t 6 + t 18 + C0 t + g(t) 6 g1 (t)5 6 . L L L dt By integrating in time this differential inequality, we obtain that g1 (t)6 6 C0 t 7 + t 19 + C0 L

t

5 τ + g(τ )L6 g1 (τ )L6 dτ.

0

Therefore we get from Proposition 4.2(3) combined with the Sobolev embedding H˙ 1 ⊂ L6 that g1 (t)

L6

7 19 C0 t 6 + t 6 + C0 gL1 L6 t 7 19 1 C0 t 6 + t 6 + t 2 ∇gL2 L2 t 7 19 5 1 C0 t 6 + t 6 + C0 1 + t 2 t 2 .

776


Therefore, we obtain that |xh |2 ρ(t) 6 C0 t 136 + t − 12 . L This ends the proof of Proposition 4.2.

2

4.3. Strong estimates As in the study of the axisymmetric Euler equation, the main important quantity that one should estimate in order to get the global existence of smooth solutions is ωr (t)L3,1 . Indeed, this will enable us to bound stronger norms such as ω(t)L∞ and ∇v(t)L∞ which are the significant quantities to propagate higher regularities. 4.3.1. Estimate of ωr (t)L∞ First, we will introduce the following notation: we denote by Φk any function of the form 19 Φk (t) = C0 exp . . . exp C0 t 6 . . . , k times

where C0 depends on the involved norms of the initial data and its value may vary from line to line up to some absolute constants. We will make an intensive use (without mentioning it) of the following trivial facts t

t Φk (τ ) dτ Φk (t)

and

exp

0

Φk (τ ) dτ

Φk+1 (t).

0

We first establish the following result. Proposition 4.4. Let v0 ∈ L2 be an axisymmetric vector field such that ωr0 ∈ L3,1 and ρ0 ∈ L2 ∩ Lm , for m > 6, axisymmetric and such that |xh |2 ρ0 ∈ L2 . Then, we have for every t ∈ R+ ω (t) r

r v (t) + ∞ Φ2 (t), r L3,1 L

where C0 is a constant depending on the norms of the initial data. Proof. Recall that the equation of the scalar component of the vorticity ω = ωθ eθ is given by ∂t ωθ + v · ∇ωθ = It follows that the evolution of the quantity

ωθ r

vr ωθ − ∂r ρ. r

(49)

is governed by the equation

(∂t + v · ∇)

∂r ρ ωθ =− · r r

(50)

T. Hmidi, F. Rousset / Journal of Functional Analysis 260 (2011) 745–796 ∂r −1 r

By applying the operator

777

to the equation of the density in (1), we obtain that

1 1 ∂r ρ = − ∂r −1 , v · ∇ ρ. (∂t + v · ∇) ∂r −1 ρ − r r r By setting Γ :=

ωθ r

+

∂r −1 r ρ,

we infer

(∂t + v · ∇)Γ = −

1 ∂r −1 , v · ∇ ρ. r

Observe that the incompressibility of the velocity field allows us to get that for every p ∈ [1, ∞] Γ (t)

Γ0 Lp

Lp

t 1 −1 + ∂r , v · ∇ ρ p dτ. r L 0

Therefore we get by the interpolation result of Theorem 2.4 that for 1 < p < ∞ and q ∈ [1, ∞] Γ (t)

Lp,q

Γ0 Lp,q

t 1 −1 ∂ + , v · ∇ ρ(τ ) r r p,q dτ. L 0

In particular, we have Γ (t)

L3,1

t 1 −1 Γ0 L3,1 + ∂r , v · ∇ ρ(τ ) 3,1 dτ. r L 0

Applying Theorem 3.1 we find Γ (t) 3,1 Γ0 3,1 + L L

t

(ωθ /r)(τ )

L3,1

xh ρ(τ )

0 ∩L2 B∞,1

+ ρ(τ )

0

1 2 B2,1

dτ.

Moreover, thanks to Proposition 2.9 and Proposition 4.1 we have (ωθ /r)(t)

L3,1

1 −1 ∂ Γ (t)L3,1 + ρ(t) r r 3,1 Γ (t) L3,1 + Cρ0 L3,1 . L

The combination of these last estimates yield (ωθ /r)(t)

L3,1

C ω0 /rL3,1 + ρ0 L3,1 t + 0

(ωθ /r)(τ )

L3,1

xh ρ(τ )

0 ∩L2 B∞,1

+ ρ(τ )

1 2 B2,1

dτ.

778


Thus we get by the Gronwall inequality that (ωθ /r)(t) The term ρ mate

L3,1

C ω0 /rL3,1 + ρ0 L3,1 exp Cxh ρL1 (B 0

1

2 L1t B2,1

2 ∞,1 ∩L )

t

+ Cρ

1 2 L1t B2,1

.

(51)

will be controlled only by energy estimates. Indeed, the interpolation esti-

ρ

1

1 2 B2,1

1

ρL2 2 ∇ρL2 2

combined with Proposition 4.1 and the Hölder inequality give ρ

1

1 2 L1t B2,1

3

1

ρL2 ∞ L2 t 4 ∇ρ 2 2

3

Lt L2

t

t 4 ρ0 L2 .

To control the term xh ρL1 L2 in the right-hand side of (51), we can use Proposition 4.2: t

9 xh ρL1 L2 txh ρL∞ 2 C0 1 + t 4 . t L t

Consequently we obtain in view of (51) 9 Cρx ω h L1 B 0 C0 t 4 t ∞,1 . (t) C e e 0 r 3,1 L

(52)

Now it remains to estimate the right term of (52) inside the exponential. Let us first sketch the strategy of our approach. We will introduce an integer N (t) ∈ N that will be chosen in an optimal way in the end and we will split in frequency the involved quantity into two parts: low frequencies corresponding to q N (t) and high frequencies associated to q > N (t). To estimate the low frequencies we use the dispersive result of Proposition 4.2(2). The estimate of high frequencies is based on a smoothing effect. By using Proposition 4.2(2) and the Bernstein inequality, we find that

xh ρL1 B 0 t

∞,1

t q (xh ρ)(τ ) =

t q (xh ρ)(τ ) ∞ dτ dτ + L∞ L

0 qN (τ )

t C0 0

1 3 τ 4 + τ − 4 N (τ ) dτ + C

0 q>N (τ )

t

3 2q 2 q (xh ρ)(τ )L2 dτ.

(53)

0 q>N (τ )

Now we intend to estimate the last sum in the above inequality. For this purpose we localize in frequency the equation for f = xh ρ which is ∂t f + v · ∇f − f = v h ρ − 2∇h ρ := F.


779

By setting fq := q f , we infer ∂t fq + v · ∇fq − fq = −[q , v · ∇]f + Fq . From an L2 energy estimate, we obtain that

1 d fq (t)2 2 − L 2 dt

(fq )fq dx fq L2 [q , v · ∇]f L2 + Fq L2 .

R3

Since the Bessel identity yields c22q fq 2L2 −

(fq )fq dx,

R3

it follows that d fq (t) 2 + c22q fq (t) 2 C [q , v · ∇]f 2 + Fq 2 . L L L L dt Therefore we obtain by integration in time that fq (t)

L2

e

fq (0)

t

−ct22q

L2

+

e−c(t−τ )2

2q

[q , v · ∇]f

L2

+ Fq L2 dτ.

0

To estimate the commutator in the right-hand side, we can use Proposition 3.2 and Proposition 4.2, [q , v · ∇]f (τ ) 2 C (ωθ /r)(τ ) 3,1 |xh |2 ρ(τ ) 6 + xh ρ(τ ) 2 L L L L 13 1 C0 (ωθ /r)(τ )L3,1 τ 6 + τ − 2 . Hence we get fq (t)

L2

e

fq (0)L2 +

−ct22q

t

2q e−c(t−τ )2 Fq (τ )L2 dτ

0

t + C0

13 1 2q e−c(t−τ )2 (ωθ /r)(τ )L3,1 τ 6 + τ − 2 dτ.

0

Let us set K(τ ) = τ

13 6

1

+ τ − 2 , then (54) and convolution inequalities yield

(54)

780


t

3 2q 2 q (xh ρ)(τ )L2 dτ

0 q>N (τ )

1 2− 2 q fq (0)L2 + Fq L1 L2 t

q−1

+ C0

t

2

q 32

τ

0 q>N (τ )

e−c(τ −τ

)22q

K τ (ωθ /r) τ L3,1 dτ

0

t f0 L2 + F L1 L2 + C0

ωθ /rL∞ 3,1 τ L

t

2

τ

q 32

q>N (τ )

0

e

−c(τ −τ )22q

τ

− 34

K τ dτ dτ.

0

Moreover, from Proposition 4.1 and Lemma A.1, we also have 1

F L1 L2 t 2 ∇ρL2 L2 + vL∞ 2 ρL1 L∞ t L t

t

t

t

1 2 L2

Cρ0 t + C0 (1 + t)ρ0 L2

dτ + t

0

C0 1 + t 2 . Inserting these estimates into (53) yields xh ρL1 B 0 t

∞,1

C0 1 + t 2 + C0

t

1 3 τ 4 + τ − 4 N (τ ) dτ

0

t + C0

ωθ /rL∞ 3,1 τ L

2

q 32

τ

q>N (τ )

0

C0 1 + t 2 + C 0

t

e

−c(τ −τ )22q

13 − 1 6 2 τ dτ dτ + τ

0

1 3 τ 4 + τ − 4 N (τ ) dτ

0

t + C0

ωθ /rL∞ 3,1 τ τ L

0

By a change of variables we get

13 6

2

− 12 N (τ )

+

q>N (τ )

2

q 32

τ e 0

−c(τ −τ )22q

− 1 2 dτ dτ. (55) τ


2

q 32

q>N (τ )

τ e

−c(τ −τ )22q

781

22q τ − 1 1 2q − 1 q −cτ 2 2 dτ = τ 2 2e ecτ τ 2 dτ q>N (τ )

0

=

0

2

q 12

e

22q τ

−cτ 22q

q∈Λ1 (τ )

0

+

− 1 ecτ τ 2 dτ

2

q 12

e

−cτ 22q

q∈Λ2 (τ )

22q τ

− 1 ecτ τ 2 dτ

0

:= I(τ ) + II(τ ). with Λ1 (τ ) = q > N(τ ) and τ 22q 1 and Λ2 (τ ) = q > N (τ ) and τ 22q 1 . To estimate the first term we use the following inequality which can, be proven by integration by parts: there exists C > 0 such that for every x 1 x

1

1

y − 2 ecy dy Cx − 2 ecx .

0

It follows that

1

I(τ ) τ − 2

1

1

1

2− 2 q 2− 2 N (τ ) τ − 2 .

q>N (τ )

To estimate the second term, we observe that the integral is bounded by a fixed number and hence, we find that

II(τ )

1

1

2 2 q 2− 2 N (τ )

1 1 2q 2− 2 N (τ ) 1 + τ − 2 .

22q τ −1

q∈Λ2 (τ )

Gathering these estimates, we obtain q>N (τ )

2

q 32

τ

e−c(τ −τ

)22q

− 1 1 1 2 dτ 2− 2 N (τ ) 1 + τ − 2 . τ

0

By plugging this estimate into (55), we get

xh ρL1 B 0 t

∞,1

C0 1 + t 2 + C 0

t 0

13 3 − 1 N (τ ) τ 6 + τ − 4 N (τ ) + ωθ /rL∞ dτ. 3,1 2 2 τ L

782


We choose N such that N (τ ) = 2 log2 2 + ωθ /rL∞ 3,1 τ L and then we find xh ρL1 B 0 t

∞,1

19 C0 1 + t 6 + C 0

t

13 3 τ 6 + τ − 4 log 2 + ωθ /rL∞ 3,1 dτ. τ L

(56)

0

Putting together (52), (56) and Proposition 4.2(2), we find that 19 log 2 + ωθ /rL∞ + C0 3,1 C0 1 + t 6 t L

t

13 3 τ 6 + τ − 4 log 2 + ωθ /rL∞ 3,1 dτ . τ L

0

From the Gronwall inequality, we infer 19 1 19 6 +t 4 ) 6 e C0 (t log 2 + ωθ /rL∞ C 1 + t Φ1 (t). 3,1 0 t L

Therefore we get by using again (56) that ωθ (t) r 3,1 Φ2 (t). L Since

ω r

=

ωθ r eθ ,

(22) implies that ω (t) r 3,1 Φ2 (t). L

(57)

Finally, thanks to (19), we obtain r v (t) C ω (t) ∞ r r 3,1 Φ2 (t). L L This ends the proof of Proposition 4.4.

2

4.3.2. Estimate of ω(t)L∞ Our purpose now is to bound the vorticity. Proposition 4.5. Let v0 ∈ L2 be an axisymmetric divergence free vector field without swirl such that ω0 ∈ L∞ , ωr0 ∈ L3,1 . Let ρ0 be axisymmetric scalar function, belonging to L2 ∩ Lm , m > 6 and such that |xh |2 ρ0 ∈ L2 . Then we have for every t ∈ R+ ω(t)

L∞

+ ∇ρL1 L∞ Φ4 (t). t


783

Proof. From the maximum principle for Eq. (49), we obtain that t

ω(t) ∞ ω0 L∞ + L

r v /r(τ )

∞ ω(τ )

L

t L∞

dτ +

0

∇ρ(τ )

L∞

dτ.

0

By combining Proposition 4.4 and the Gronwall inequality, this yields ω(t)

L∞

t

Φ3 (t) 1 +

∇ρL∞ dτ . 0

Now we claim that,

t

∇ρL1 L∞ C0 1 + t + 2

t

ω(τ )

L∞

dτ .

0

Let us first finish the proof by using this estimate. We deduce that ω(t)

L∞

t

Φ3 (t) 1 +

ω(τ )

L∞

dτ

0

and thanks to the Gronwall inequality that ω(t) ∞ Φ4 (t). L This gives in turn ∇ρL1 L∞ Φ4 (t). t

Let us now come back to the proof of (58). For q ∈ N we set ρq := q ρ, then ∂t ρq + v · ∇ρq − ρq = −[q , v · ∇]ρ. Let p 2 then multiplying this equation by |ρq |p−2 ρq and using Hölder inequality 1 d ρ(t)p p − L p dt

p−1 (ρq )|ρq |p−2 ρq dx ρq Lp [q , v · ∇]ρ Lp .

R3

Now we use the generalized Bernstein inequality, see [24], 1 2q p 2 ρq Lp − p

R3

(ρq )|ρq |p−2 ρq dx.

(58)

784


Hence we get d ρ(t) p + cp 22q ρq Lp [q , v · ∇]ρ p . L L dt This gives ρq (t)

Lp

e

−cp t22q

t q ρ0

Lp

e−cp 2

2q (t−τ )

+

[q , v · ∇]ρ

Lp

dτ.

(59)

0

Integrating in time implies that ρq L1 Lp 2−2q ρ0 Lp + 2−2q [q , v · ∇]ρ L1 Lp . t

t

According to Proposition 2.3 [18] and Proposition 4.1 we have [q , v · ∇]ρ

Lp

CρLp (q + 1)ωL∞ + ∇−1 vL∞ Cρ0 Lp (q + 1)ωL∞ + vL2 Cρ0 Lp (q + 1)ωL∞ + C0 (1 + t) .

It follows that ρq L1 Lp C0 1 + t 2 2−2q + Cρ0 Lp (q + 1)2−2q ωL1 L∞ . t

t

By using the Bernstein inequality, we find for p > 3 that ∇ρL1 L∞ Ctρ0 L2 + C

t

2

q(1+ p3 )

ρq L1 Lp t

q∈N

q(−1+ 3 ) q(−1+ 3 ) p + C ω 1 p (q + 1) C0 1 + t 2 2 2 0 L L∞ t

q∈N

C0 1 + t 2 + C0 ωL1 L∞ .

q∈N

t

This ends the proof of the desired inequality.

2

4.3.3. Lipschitz bound of the velocity We shall now deal with the global propagation of the sub-critical Sobolev regularities. This is basically related to the control of the Lipschitz norm of the velocity. Proposition 4.6. Let 52 < s < 3 and (v0 , ρ0 ) ∈ H s × H s−2 and (v, ρ) be a solution of the Boussinesq system (1). Then we have for every t 0 vL s + ρL s−2 + ρL ∞ ∞ 1 H s C0 (1 + t)e t H t H t

C∇vL1 L∞ t

.


785

If in addition ρ0 ∈ Lm with m > 6 and |xh |2 ρ0 ∈ L2 , then we get for every t 0 ∇v(t) ∞ Φ5 (t); L

vL s + ρL s−2 + ρL ∞ ∞ 1 H s Φ6 (t). t H t H t

Remark 4.7. We point out that we can extend the results of Proposition 4.6 to higher regularities s 3 but for the sake of simplicity we restrict ourselves here to the case of s < 3. Proof. We localize in frequency the equation of the velocity. For q ∈ N ∪ {−1} we set vq := q v and ρq := q ρ. ∂t vq + v · ∇vq + ∇πq = ρq ez − [q , v · ∇]v. Thus taking the L2 -scalar product with vq and using the incompressibility of v and vq we get d vq (t) 2 ρq 2 + [q , v · ∇]v 2 . L L L dt Integrating in time we obtain vq (t)

L2

vq (0)L2 + ρq L1 L2 + [q , v · ∇]v L1 L2 . t

t

Thus we get qs [q , v · ∇]v 1 2 2 . vL s v0 H s + ρL ∞ 1 H s + 2 L L q t H t

t

We will use the commutator estimate, see for instance Lemma B.5 of [10], qs 2 [q , v · ∇]v

C L1 L2 q 2

t

t

∇v(τ )

L∞

v(τ )

Hs

dτ.

0

Putting together these estimates and using Gronwall inequality yield C∇vL1 L∞ t vL . s v0 H s + ρL ∞ 1 H s e t H

(60)

t

Using the estimate (59) we get for q ∈ N 2q ρq L∞ 2 + 2 ρq L1 L2 C ρq (0) 2 + [q , v · ∇]ρ 1 2 . L L L L t t

t

Therefore we find q(s−2) [q , v · ∇]ρ 1 2 2 ρL 1 H s −1 ρL1 L2 + ρ0 H s−2 + 2 s−2 + ρL ∞ H L L q t t t t q(s−2) [q , v · ∇]ρ 1 2 2 . Ctρ0 L2 + ρ0 H s−2 + 2 L L q t

786


Since −1 < s − 2 < 1 then we have the estimate, see [10], q(s−2) [q , v · ∇]ρ 2

C L1 L2 q 2

t

∇v(τ )

L∞

t

ρ(τ )

H s−2

dτ.

0

Consequently, t ρL s−2 + ρL ∞ 1 H s C0 (1 + t) + C t H

∇v(τ )

L∞

t

ρ(τ )

H s−2

dτ.

0

By Gronwall inequality ρL s−2 + ρL ∞ 1 H s C0 (1 + t)e t H

C∇vL1 L∞ t

t

.

(61)

Combining this estimate with (60) gives the desired estimates. Now to get a global bound for Lipschitz norm of the velocity we use the classical logarithmic estimate: for s > 52 ∇vL∞ vL2 + ωL∞ log e + vH s . Combining this estimate with the first result of Proposition 4.6 and Proposition 4.5

t

∇vL∞ Φ4 (t) 1 +

∇v(τ )

L∞

dτ .

0

It follows from Gronwall inequality that ∇v(t)

L∞

Φ5 (t).

Plugging this estimate into (60) and (61) gives vL s + ρL s−2 + ρL ∞ ∞ 1 H s Φ6 (t). t H t H t

This ends the proof of the proposition.

2

5. Proof of the main result The proof of the existence part of Theorem 1.1 can be done in a classical way by smoothing out the initial data as follows v0,n = Sn v0 = 23n χ 2n · v0 ,

ρ0,n = Sn ρ0 = 23n χ 2n · ρ0

where Sn is the cut-off in frequency defined in the preliminaries. Since χ is radial then the functions v0,n and ρ0,n remain axisymmetric. Moreover this family is uniformly bounded in the


787

space of initial data: this is obvious in Sobolev and Lebesgue spaces but it remains to check the uniform boundedness of the horizontal moment of the density. We will show that sup|xh |2 ρn,0 L2 C ρ0 L2 + |xh |2 ρ0 L2 .

(62)

n∈N

For this purpose we write n 2 |xh | ρn,0 (x) = |xh | χ 2 (x − y) ρ(y) dy 2

R3

n n 2 |xh − yh | |χ| 2 (x − y) |ρ|(y) dy + 2 |χ| 2 (x − y) |yh | ρ(y) dy

2

2

R3

R3

2 2−2n 23n χ1 2n · ρ (x) + 2 |χ| 2n · |yh |2 ρ (x) with χ1 (x) = |xh |2 |χ(x). From convolution laws we get |xh |2 |ρn,0

L2

C2−2n ρL2 + C |xh |2 ρ0 L2 .

This achieves the proof of (62). Now, by using standard arguments based on the a priori estimates described in Proposition 4.6, Proposition 4.5 and Proposition 4.2 we can construct a unique global solution (vn , ρn ) in the following space vn ∈ C R+ ; H s ∩ L1 R+ ; W 1,∞

and ρn ∈ C R+ ; H s−2 ∩ L1 R+ ; W 1,∞ .

The control is uniform with respect to the parameter n. Therefore we can prove the strong convergence of a subsequence of (vn , ρn )n∈N to some (v, ρ) belonging to the same space and satisfying the initial value problem. It remains to prove the uniqueness problem. This gives the existence of a solution. The uniqueness will be proven in the following space 2 (v, ρ) ∈ X := C R+ ; L2 ∩ L1 R+ ; W 1,∞ . Let (v i , ρ i ) ∈ X , 1 i 2 be two solutions of the system (1) with the same initial data (v0 , θ0 ) and denote δv = v 2 − v 1 , δρ = ρ 2 − ρ 1 . Then ⎧ ⎨ ∂t δv + v 2 · ∇δv + ∇Π = −δv · ∇v 1 + δρez , 2 1 ⎩ ∂t δρ + v · ∇δρ − δρ = −δv · ∇ρ , i div v = 0. Taking the L2 -scalar product of the first equation with δv and integrating by parts gives 1 d δv(t)2 2 ∇v 1 ∞ δv2 2 + δρ 2 δv 2 . L L L L L 2 dt

(63)

788


Consequently, d δv(t) 2 ∇v 1 ∞ δv 2 + δρ 2 . L L L L dt Using the Gronwall inequality yields

e

δv(t)

−∇v 1 L1 L∞ t

L2

t

δv0 L2 +

e

δρ(τ )

−∇v 1 Lτ L∞ t

L2

dτ .

0

By the same computations we get δρ(t)

L2

t δρ0 L2 +

1 ∇ρ (τ )

L∞

δv(τ )

L2

dτ.

0

It suffices now to put together these estimates and to use the Gronwall inequality. Appendix A. De Giorgi–Nash–Moser estimates for convection–diffusion equations We intend to prove some dispersive estimates for a parabolic equation. We use De Giorgi– Nash–Moser method similarly to [6]. Lemma A.1. Consider the equation ∂t f + u · ∇f − f = ∇ · F + G,

t > 0,

x ∈ R3 ,

f (0, x) = f0 (x).

(64)

Consider p, q, p1 , q1 , ∈ [1, +∞], r ∈ [2, +∞] with 3 2 + < 1, p q

2 3 + < 2. p1 q 1

There exists C > 0 such that for every smooth divergence free vector field u and every T > 0, if p F ∈ LT Lq and f0 ∈ Lr , the solution of (64) satisfies the estimate: f (T )

L∞

√ 1−( p2 + q3 ) 1 F Lp Lq C 1 + 3 f0 Lr + C 1 + T T T 2r √ 2−( p2 + q3 ) 1 1 GLp1 Lq1 . +C 1+ T T

(65)

Proof. Since the equation is linear, we can study separately the three problems

Pf = ∇ · F, f (0, x) = 0,

Pf = G, f (0, x) = 0,

where we have set Pf = ∂t f + u · ∇f − f .

Pf = 0, f (0, x) = f0 (x),

(66)


789

Let us start with the first problem in (66). We shall prove that there exists C > 0 such that for every F with F Lp Lq 1, we have the estimate 1

∞ C. f L∞ 1 L

(67)

Once this estimate, is proven, the estimate involving F in (65) will just follow by a scaling argument. The first step is to use the standard Lq a priori estimate (obtained by multiplying by Pf by |f |q−1 sign f ). Since u is divergence free, we have that d dt

1 f (t)q q L q

+ (q − 1)

|∇f |2 |f |q−2 dx (q − 1)

R2

|F ||∇f ||f |q−2 dx.

R3

From the Young inequality and the Holder inequality (note that since 2/p + 3/q < 1, we necessarily have that q > 3 and p > 2) this yields d dt

1 f (t)q q L q

q−2

(q − 1)F 2Lq f Lq

and hence by integration in time, we obtain since q > 3 that 1 q Cq f L∞ 1 L

F (t)2 q dt

1 2

Cq F Lp Lq Cq

L

1

(68)

0

where Cq depends only on q. To improve this estimate that is to go from the above Lq estimate to an L∞ estimate, we shall follow the De Giorgi, Nash iteration argument. For M > 0 to be chosen, let us take a positive increasing sequence (Mk )k0 such that Mk M and Mk converges towards M. A good choice is for example

Mk = M 1 −

1 , k+1

k 0.

(69)

We shall use the standard notation x+ = max(x, 0). Since u is divergence free, we obtain the level set energy estimate d dt

1 (f − Mk )+ (t)2 2 + ∇(f − Mk )+ 2 2 L L 2

|F ||∇f | dx

f Mk

1

|F |

2

f Mk

2

∇(f − Mk )+ 2 L

790


where the last inequality comes from Cauchy–Schwarz. By using the Young inequality, we thus obtain 1 Uk

|F |2 dx dt

(70)

0 f Mk

where 2 2 Uk = (f − Mk )+ L∞ L2 + ∇(f − Mk )+ L2 L2 . 1

1

The main idea is to prove that the right-hand side of (70) can estimated by a power of Uk−1 strictly larger than 1. By using the Hölder inequality, we first get that 1 Uk

2 F (t)2 q mk (t)1− q dt F 2 p q L L L

1− 2

1

mk (t)

1

0

p (1− q2 )( p−2 )

p

dt

(71)

,

0

where mk (t) = |{x, f (t) Mk }|. To estimate mk (t), we note that if f (t, x) Mk then f (t, x) − Mk−1 Mk − Mk−1 0 and thus we have 1f (t,x)Mk

(k + 1)2 f (t, x) − Mk−1 + . M

(72)

This yields mk (t)

(k + 1)2m f (t) − Mk−1 mm + L m M

(73)

for every m 1. We shall choose m carefully below. By plugging this last estimate in (71), we get

Uk F 2Lp Lq 1

(k + 1)2 M

m(1− 2 ) 1 q

m(1− 2 )( p ) f (t) − Mk−1 m q p−2

1− 2

+ L

p

.

(74)

0

Now let us notice that if α 1 and β ∈ [2, 6] are such that f (t) − Mk−1 2 α

+ L1 Lβ

2 α

+

3 β

3 2

then we have

Uk−1 .

2 1 1 Indeed the control of Uk−1 gives a control of the L∞ 1 L and the L1 H norm. By Sobolev em2 6 bedding this gives a control of the L1 L norm and then the inequality follows by standard


791

interpolation in Lebesgue spaces. Consequently, to achieve our program, we need to choose m ∈ [2, 6] such that

2 m 1− > 2, q

1 − p2 3 3 + . 2 m 2 m 1 − q2

(75)

The first constraint can be satisfied as soon as 2/(1 − 2/q) < 6 which is equivalent to q > 3 while the second constraint can be satisfied as soon as

2 2 3 2 +3 1− m 1− > ·2=3 2 1− p q q 2 which is equivalent to 2 3 + < 1. p q Consequently, since we have q > 3 and 2/p + 3/q < 1 by assumption we can choose m such that the constraint (75) are matched. This yields that there exists γ > 1 such that

Uk F 2Lp Lq 1

(k + 1)2 M

m(1− 2 ) p

γ Uk−1

(k + 1)2 M

m(1− 2 ) p

γ

Uk−1 ,

∀k 2.

If U1 is sufficiently small, this yields that limk→+∞ Uk = 0. Since, we have from (71), the Tchebychev inequality and the energy inequality (68) that 1 U1

1− 2 p (1− q2 )( p−2 )

m1 (t)

p

dt

q 4f L∞ 1 L

q−2

M

4Cq M

q−2 ,

0

we can indeed make U0 arbitrarily small by taking M sufficiently large and thus limk→+∞ Uk = 0. From Fatou’s Lemma, we obtain that for every t ∈ [0, 1] f (t, x) − M + dx 0 R3

and therefore that almost everywhere f (t, x) M. By changing f into −f , we obtain in a similar way that f −M almost everywhere and thus (67) is proven. To obtain the part of estimate (65) √ involving F if T 1 we can use a change of scale argument. Let us set K f˜(τ, X) = f (T τ, T X) for K > 0 to be chosen. Then we have ∞ = f L∞ L∞ and f˜(τ, X) solves the equation Kf˜L∞ T 1 L ∂τ f˜ + u˜ · ∇X f˜ − X f˜ = ∇X · F˜

792


where u˜ is still divergence free and √ √ T F (τ T , T X). F˜ (τ, X) = K In particular, with the choice K=

√

T

1−( p2 + q3 )

F Lp Lq , T

we get that F˜ Lp Lq = 1 and thus that 1

√ 1−( p2 + q3 ) ∞ = Kf˜L∞ L∞ MK = M T F Lp Lq . f L∞ T L 1 T

This gives the part of the estimate (65) involving F . Let us turn to the study of the second problem in (66) in order to get the part of the estimate (65) involving G. The estimate can be deduced from the previous one when q1 < +∞ which is the interesting case (when q1 = ∞, the estimate is a direct consequence of the Maxi∗ p p p mum principle). Indeed, if G ∈ LT 1 Lq1 , we can write G = ∇ · F with F ∈ LT 1 W 1,q1 ⊂ LT 1 Lq1 3q1 by Sobolev embedding (where q1∗ = 3−q ) and moreover, we have 1 F

p

q1∗

LT 1 L

GLp1 Lq1 . T

Consequently, by using the estimate that we have already proven, we get that √ 1−( p2 + q3∗ ) 1 1 F ∞ M f L∞ T L T

p

q1∗

LT 1 L

if 2/p1 + 3/q1∗ < 1. This gives the claimed estimate. It remains to study the third problem in (66) that is the problem with no source term but a nontrivial initial data. Again, we shall first prove that there exists M > 0 such that for every f0 ∈ Lr with f0 L2 1, we have the estimate supf (t)L∞ M.

(76)

t1

The standard energy estimate gives that f 2L∞ L2 + f 2L2 L2 f0 L2 .

(77)

To improve this estimate, we shall also use the De Giorgi–Nash iteration method. We take a 1 which tends sequence Mk as previously, and we also choose a sequence of times Tk = 1 − k+1 to 1. The energy estimate for (f − Mk )+ yields that for every t, s with t Tk s, we have 2 sup (f − Mk )+ (t)L2 +

tTk

+∞ ∇(f − Mk )+ )(τ )2 2 dτ 2(f − Mk )+ (s)2 2 L L s


793

and hence, by integrating in s for Tk−1 s Tk , we obtain that +∞ (f − Mk )+ (s)2 2 ds Uk 2(k + 1) L 2

(78)

Tk−1

with 2 Uk = sup (f − Mk )+ (t)L2 + tTk

+∞ ∇(f − Mk )+ )(τ )2 2 dτ. L Tk

The aim is again to estimate the right-hand side of (78) by a power of Uk−1 strictly greater than 1. By using the same notations as previously, we get from (72) that

Uk 2(k + 1)

2

(k + 1)2 M

4 +∞ 3

10

(f − Mk−1 )+3 dt dx

Tk−1 R3

2(k + 1)

2

(k + 1)2 M

4 3

5 3 Uk−1 .

(79)

Since we have from (77) that U0 (f0 )+ 2L2 f0 2L2 , U1 can be made arbitrarily small by taking M sufficiently large and hence we get from (79) that limk→+∞ Uk = 0. This proves that f (t, x) M for t 1 and then we get (76) by changing f into −f . We have thus proven that for every t 1 the linear operator f → f (t, ·) is bounded from L2 into L∞ with norm smaller than M. Since by the standard maximum principle, it is also bounded from L∞ to L∞ with norm 1, we get by interpolation that it also maps Lr to L∞ for every r 2. To get the claimed estimate in (65) for t 1, it suffices to use again a scaling argument. This ends the proof. 2 Appendix B. Proof of Lemma 2.7 Proof. Set φ = F −1 h, then we have by definition and from Taylor formula h(D), f g =

φ(x − y)g(y) f (y) − f (x) dy

Rd

1 =

g(y)Φ(x − y) · ∇f x + t (y − x) dy dt,

0 Rd

with Φ(x) = xφ(x). Let α, β ∈ ]0, 1[ with α + β = 1. Using Hölder inequality and a change of variables we get with Φt = t −3 Φ( xt )

794


h(D), f g(x)

1

g(y) Φ(x − y) α ∇f x + t (y − x) Φ(x − y) β dy dt

0 Rd

1 Φ(x − y) g(y) α dy

α 1

R3

0

1 α |Φ| |g| α (x)

1

1 Φt (y) ∇f (x − y) β dy

β

R3

1 β |Φt | |∇f | β (x) dt.

0

Let p1 , p2 ∈ [1, ∞] such 1 1 1 = + . p p1 p2

αp1 , βp2 1 and

(80)

Then by Hölder inequality h(D), f g

1 α |Φ| |g| α αp

Lp

L

1 1

1 |Φt | |∇f | β β βp dt. L 2

0

We choose p1 , p2 , α and β such that αm 1,

βρ 1,

1+

1 1 1 = + αp1 r αm

and 1 +

1 1 1 = + βp2 r βρ

(81)

then the classical convolution laws give 1

h(D), f g

ΦαLr gLm ∇f Lρ Lp

β

Φt Lr dt 0

1 ΦLr gLm ∇f Lρ

1

t 3β(−1+ r ) dt. 0

This last integral is finite provided that β
1. From (81)

1 1 r − . α= r − 1 m p1


795

To get α ∈ [ m1 , 1[ we must choose p1 such that 1 1 1 1 − < . p ρ p1 rm The condition βρ 1 is equivalent, by the use of 1 +

1 p

=

(83) 1 m

+

1 ρ

+ 1r , to

1 1 1 − . p rρ p1

(84)

The condition αp1 1 is automatically satisfied from (83) since αp1 rαm r 1. The condition βp2 1 is also a consequence of (83) and (84). Indeed, from the value of α and 1 1 1 p = p1 + p2 , this condition is equivalent to 1 r 1 1 − . p 2r − 1 ρ p1 This condition is weaker than (84). We can easily check that (83) and (84) are equivalent to 1 1 1 1 − . p rρ p1 rm The set of p1 described by the above condition is nonempty if 1 1 1 − + 0. rm p rρ Using the identity 1 + p1 = (82) is equivalent to

1 m

+ ρ1 + 1r , this is satisfied under the condition p r. The condition

1 1 1 1 < + − . p1 3 p ρ Now there is a compatibility between this condition and (84) if

1 < ρ. 3 1− r This ends the proof of Lemma 2.7.

2

796


References [1] H. Abidi, T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations 233 (1) (2007) 199–220. [2] H. Abidi, T. Hmidi, K. Sahbi, On the global well-posedness for the axisymmetric Euler equations, Math. Ann. 347 (1) (2010) 15–41. [3] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984) 61–66. [4] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Ec. Norm. Super. 14 (1981) 209–246. [5] Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, J. Nonlinear Sci. 19 (5) (2009) 547–570. [6] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (3) (2010). [7] D. Chae, Global regularity for the 2-D Boussinesq equations with partial viscous terms, Adv. Math. 203 (2) (2006) 497–513. [8] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, 1998. [9] R. Danchin, Poches de tourbillon visqueuses, J. Math. Pures Appl. (9) 76 (7) (1997) 609–647. [10] R. Danchin, The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math. (6) 15 (4) (2006) 637–688. [11] R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys 62 (3) (2007) 73– 94. [12] R. Danchin, M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Comm. Math. Phys. 290 (1) (2009) 1–14. [13] R. Danchin, M. Paicu, R. Danchin, M. Paicu, Le théorème de Leary et le théorème de Fujita–Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France 136 (2008) 261–309. [14] E. De Giorgi, Sulla differenziabilit‘a e l’analiticit‘a delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur. (3) 3 (1957) 25–43. [15] E. Feireisl, A. Novotny, The Oberbeck–Boussinesq approximation as a singular limit of the full Navier–Stokes– Fourier system, J. Math. Fluid. Mech. 11 (2009) 274–302. [16] T. Hmidi, Régularité höldérienne des poches de tourbillon visqueuses, J. Math. Pures Appl. (9) 84 (11) (2005) 1455–1495. [17] T. Hmidi, S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations 12 (4) (2007) 461–480. [18] T. Hmidi, S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J. 58 (4) (2009) 1591–1618. [19] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for an Euler–Boussinesq system with critical dissipation, Comm. Partial Differential Equations, in press, arXiv:0903.3747. [20] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for a Navier–Stokes–Boussinesq system with critical dissipation, J. Differential Equations 249 (2010) 2147–2174. [21] T. Hmidi, F. Rousset, Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010) 1227–1246. [22] T. Hmidi, M. Zerguine, On the global well-posedness of the Euler–Boussinesq system with fractional dissipation, Phys. D 239 (2010) 1387–1401. [23] O.A. Ladyzhenskaya, Unique solvability in large of a three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry, Zap. Nauchn. Sem. LOMI 7 (1968) 155–177. [24] P.-G. Lemarié, Recent Developments in the Navier–Stokes Problem, CRC Press, 2002. [25] J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960) 457–468. [26] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958) 931–954. [27] R. O’Neil, Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963) 129–142. [28] E.M. Stein, Harmonic Analysis, Real–Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. [29] T. Shirota, T. Yanagisawa, Note on global existence for axially symmetric solutions of the Euler system, Proc. Japan Acad. Ser. A Math. Sci. 70 (10) (1994) 299–304. [30] M.R. Ukhovskii, V.I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Mekh. 32 (1) (1968) 59–69.


Classification of homomorphisms into simple Z -stable C ∗ -algebras Hiroki Matui Graduate School of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan Received 19 May 2010; accepted 2 November 2010


Abstract We classify unital monomorphisms into certain simple Z-stable C ∗ -algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C ∗ -algebra, or any unital simple separable nuclear Z-stable C ∗ -algebra satisfying the UCT such that C ⊗ B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C ∗ algebra such that A ⊗ B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C ∗ -algebra whose projections separate traces and whose extremal traces are finitely many. © 2010 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebras; K-theory; Classification; The Jiang–Su algebra

1. Introduction Consider unital monomorphisms ϕ, ψ : C → A from a C ∗ -algebra C to a simple C ∗ algebra A. In this paper we study the problem to determine when ϕ and ψ are approximately unitarily equivalent, i.e. when there exists a sequence of unitaries (un )n in A such that ϕ(x) = lim un ψ(x)u∗n holds for any x ∈ C. This problem is known to be closely related to the classification problem for the simple C ∗ -algebra A. In the recent progress of Elliott’s program to classify nuclear C ∗ -algebras via K-theoretic invariants (see [30] for an introduction to this subject), the Jiang–Su algebra plays a central role. The Jiang–Su algebra Z, which was E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.001

798

H. Matui / Journal of Functional Analysis 260 (2011) 797–831

introduced by X. Jiang and H. Su in [12], is a unital, simple, separable, infinite-dimensional, stably finite and nuclear C ∗ -algebra KK-equivalent to C. A C ∗ -algebra A is said to be Z-stable if A ⊗ Z is isomorphic to A. Z-stability implies many nice properties from the point of view of classification theory. Among other things, if A is a unital separable simple Z-stable C ∗ -algebra, then A is either purely infinite or stably finite. If, in addition, A is stably finite, then A must have stable rank one and weakly unperforated K0 (A) (see [12,9,31]). All classes of unital simple infinite-dimensional C ∗ -algebras for which Elliott’s classification conjecture is confirmed consist of Z-stable algebras. It is then natural to consider classification of unital monomorphisms from certain C ∗ -algebras into simple Z-stable C ∗ -algebras which are not necessarily of real rank zero. In the present paper we give a positive solution for large classes of unital stably finite C ∗ -algebras (Theorem 6.6, Corollary 6.8, Theorem 7.1). Classification of homomorphisms from C(X) into a unital simple algebra has a long history. The earliest result for this subject is the classical Brown–Douglas–Fillmore theory [3]. They showed that two unital monomorphisms ϕ and ψ from C(X) to the Calkin algebra B(H )/K(H ) are unitarily equivalent if and only if KK(ϕ) = KK(ψ). M. Dadarlat [4] showed that two monomorphisms from C(X) to a unital simple purely infinite C ∗ -algebra are approximately unitarily equivalent if and only if they give the same element in KL(C(X), A). In the case that the target algebra A is stably finite, G. Gong and H. Lin [7] showed that for a unital simple separable C ∗ -algebra A with real rank zero, stable rank one, weakly unperforated K0 (A) and a unique quasitrace τ , two unital monomorphisms ϕ, ψ : C(X) → A are approximately unitarily equivalent if and only if KL(ϕ) = KL(ψ) and τ ◦ ϕ = τ ◦ ψ . H. Lin [17] obtained the same result for the case that the target algebra A is of tracial rank zero. P.W. Ng and W. Winter [26] also obtained the same result for the case that X is a path connected space and A is a Z-stable C ∗ -algebra of real rank zero. Similar classification up to approximate unitary equivalence is also known for more general domain algebras. G.A. Elliott [6] showed that two homomorphisms ϕ and ψ between AT algebras of real rank zero are approximately unitarily equivalent if and only if Ki (ϕ) = Ki (ψ) for each i = 0, 1. K.E. Nielsen and K. Thomsen [25] obtained the analogous result for general AT algebras. H. Lin [17,20] classified unital homomorphisms from AH algebras into simple separable C ∗ -algebras of tracial rank no more than one. Classification up to asymptotic unitary equivalence is also studied in [27,13,18]. It should be noted that all the target algebras in these results are assumed to have many nontrivial projections (and most of them are of real rank zero). Indeed almost nothing is known so far when the target algebra does not contain non-trivial projections. The present paper gives a first non-trivial general result for this subject. Our target algebras consist of two classes C and C . The class C is the family of all unital simple separable Z-stable C ∗ -algebras A such that A ⊗ Q has tracial rank zero, where Q denotes the universal UHF algebra. The classification theorems in [35,21] assert that any nuclear C ∗ -algebras A, B ∈ C satisfying the UCT are isomorphic if and only if their K-groups are isomorphic as graded ordered groups. The other class C is the family of all unital simple separable stably finite Z-stable exact C ∗ -algebras whose extremal traces are finitely many and whose projections separate traces. The Jiang–Su algebra Z itself is in C ∩ C and any unital simple separable Z-stable exact C ∗ -algebra with a unique trace is in C . In order to extend the target to C ∗ -algebras not necessarily of real rank zero, we need a new invariant Θϕ,ψ , which is a homomorphism from K1 (C) to Aff(T (A))/ Im DA (Lemma 3.1). Roughly speaking, if A is of real rank zero, then the range of the dimension map DA is uniformly dense in Aff(T (A)). Therefore this invariant trivially vanishes. When A is not of real rank zero, it is not the case that Im DA is dense in Aff(T (A)), and so the homomorphism Θϕ,ψ must be taken into account.


799

The paper is organized as follows. In Section 2 we collect preliminary material. The most important one is the notion of Bott(ϕ, u). In Section 3 we introduce the homomorphism Θϕ,ψ for a pair of unital monomorphisms ϕ, ψ : C → A. In Section 4 we give a classification theorem of unital monomorphisms from commutative C ∗ -algebras to certain unital simple C ∗ -algebras of real rank zero. The results in Section 4 (especially Theorem 4.5) partly overlap with those obtained in [17]. But the proof given in [17] is quite lengthy, and so we provide a simpler and self-contained proof for the reader’s convenience. Section 5 is devoted to the proof of a version of the so-called basic homotopy lemma (see [19]). In Section 6 we prove the classification theorem for the case that the domain algebra is commutative (Theorem 6.6) by combining the results obtained in Sections 4 and 5. We also extend the classification theorem to the case that the domain is a unital AH algebra (Corollary 6.8). In Section 7 we prove the classification theorem for the case that the domain is a nuclear C ∗ -algebra in C satisfying the UCT (Theorem 7.1). 2. Preliminaries 2.1. Notations We let log be the standard branch defined on the complement of the negative real axis. For a Lipschitz continuous function f , we denote its Lipschitz constant by Lip(f ). We denote by K the C ∗ -algebra of all compact operators on 2 (Z). The normalized trace on Mn is written by tr and the unnormalized trace on Mn or K is written by Tr. The finite cyclic group of order n is written by Zn = Z/nZ. Let A be a C ∗ -algebra. For a, b ∈ A, we mean by [a, b] the commutator ab − ba. We write a ≈ε b when a − b < ε. The set of tracial states on A is denoted by T (A) and the collection of all continuous bounded affine maps from T (A) to R is denoted by Aff(T (A)). We regard Aff(T (A)) as a real Banach space with the sup norm. The dimension map DA : K0 (A) → Aff(T (A)) is defined by DA ([p])(τ ) = τ (p). For a unital positive linear map ϕ : A → B between unital C ∗ -algebras, T (ϕ) : T (B) → T (A) denotes the affine continuous map induced by ϕ. We say that a C ∗ -algebra A has strict comparison of projections if for projections p, q ∈ A ⊗ K, (τ ⊗ Tr)(p) < (τ ⊗ Tr)(q) for any τ ∈ T (A) implies that p is Murray–von Neumann equivalent to a subprojection of q. When ϕ is a homomorphism between C ∗ -algebras, K0 (ϕ) and K1 (ϕ) mean the induced homomorphisms on K-groups. A unital completely positive linear map is called a ucp map for short. A ucp map ϕ : A → B is said to be (G, δ)-multiplicative if ϕ(ab) − ϕ(a)ϕ(b) < δ holds for any a, b ∈ G, where G is a subset of A. For two ucp maps ϕ, ψ : A → B, we write ϕ ∼G,δ ψ , when there exists a unitary u ∈ B such that ϕ(a) − uψ(a)u∗ < δ holds for any a ∈ G. 2.2. The entire K-group We recall the mod p K-theory introduced by C. Schochet [33]. The Ki -group of a C ∗ -algebra A with the coefficient module Zn for i = 0, 1, n ∈ N is defined by

800


Ki (A; Zn ) = Ki (A ⊗ On+1 ), where On+1 is the Cuntz algebra. For notational convenience, we set Ki (A; Z0 ) = Ki (A). Although our definition looks different from the original one in [33], it gives an equivalent theory to the conventional one [33, Theorem 6.4]. The entire K-group K(A) of A is defined by K(A) =

∞ K0 (A; Zn ) ⊕ K1 (A; Zn ) . n=0

For each i = 0, 1 and n ∈ N, we have the Künneth exact sequence 0 → Ki (A) ⊗ Zn → Ki (A; Zn ) → Tor Ki (A), Zn → 0. It is known that this exact sequence splits unnaturally. For C ∗ -algebras A, B, we denote by HomΛ (K(A), K(B)) the set of all homomorphisms from K(A) to K(B) preserving the direct sum decomposition and commuting with natural coefficient transformations and the Bockstein operations (see [5,14] for details). M. Dadarlat and T.A. Loring [5] proved the following universal multicoefficient theorem. Theorem 2.1. Let A be a C ∗ -algebra satisfying the UCT and let B be a σ -unital C ∗ -algebra. Then there exists a short exact sequence 0→

Pext Ki (A), K1−i (B) → KK(A, B) → HomΛ K(A), K(B) → 0,

i=0,1

where Pext(Ki (A), K1−i (B)) is the subgroup of Ext(Ki (A), K1−i (B)) consisting of the pure extensions. The sequence is natural in each variable. Let A and B be C ∗ -algebras. Suppose that A satisfies the UCT and B is σ -unital. In [30, Section 2.4], the KL-group KL(A, B) is defined as the quotient of KK(A, B) by the image of Pext(K∗ (A), K1−∗ (B)). Thus, by the theorem above, KL(A, B) is identified with HomΛ (K(A), K(B)). Throughout this paper we keep this identification. For a homomorphism ϕ : A → B, we denote by Ki (ϕ; Zn ) the homomorphism from Ki (A; Zn ) to Ki (B; Zn ) induced by ϕ. We set KL(ϕ) = Ki (ϕ; Zn ) i,n ∈ HomΛ K(A), K(B) . If ϕ : A → B and ψ : A → B are approximately unitarily equivalent, then KL(ϕ) = KL(ψ) holds (see [30]). For κ ∈ KL(A, B) = HomΛ (K(A), K(B)) and i = 0, 1, we denote its Ki -component by Ki (κ) ∈ Hom(Ki (A), Ki (B)). 2.3. Almost multiplicative ucp maps For a C ∗ -algebra A, we mean by P (A) the set of all projections of A. When A is unital, we mean by U (A) the set of all unitaries of A. The connected component of the identity in U (A) is


801

denoted by U (A)0 . Let U∞ (A) be the union of U (A ⊗ Mn )’s via the embedding U (A ⊗ Mn ) u →

u 0 0 1

∈ U (A ⊗ Mn+1 ).

Likewise, we let U∞ (A)0 denote the union of U (A ⊗ Mn )0 ’s. For a unital C ∗ -algebra A, we set K0 (A) = P (A ⊗ K) ∪

∞

P (A ⊗ On+1 ),

n=1

K1 (A) = U∞ (A) ∪

∞

U (A ⊗ On+1 )

n=1

and K(A) = K0 (A) ∪ K1 (A). Let ϕ : A → B be a (G, δ)-multiplicative ucp map. For p ∈ K0 (A), if G is sufficiently large and δ is sufficiently small, then (ϕ ⊗ id)(p) is close to a projection and one can consider its equivalence class in K0 (B; Zn ). We denote this class by ϕ# (p). In a similar fashion, for u ∈ K1 (A), if G is sufficiently large and δ is sufficiently small, then (ϕ ⊗ id)(u) is close to a unitary and one can consider its equivalence class in K1 (B; Zn ). We denote this class by ϕ# (u). Thus, for any finite subset L ⊂ K(A), if ϕ is a sufficiently multiplicative ucp map, then ϕ# |L : L → K(B) is well defined. In this paper, whenever we write ϕ# (x) or ϕ# |L, the ucp map ϕ is always assumed to be sufficiently multiplicative so that they are well defined. When ϕ is sufficiently multiplicative, we can verify the following easily: ϕ# (p) = ϕ# (q) for Murray–von Neumann equivalent projections p, q ∈ K0 (A), ϕ# (p + q) = ϕ# (p) + ϕ# (q) for orthogonal projections p, q ∈ K0 (A), ϕ# (u) = 0 for any u ∈ U∞ (A)0 ∪ U (A ⊗ On+1 )0 and ϕ# (uv) = ϕ# (u) + ϕ# (v) for any u, v ∈ K1 (A). Therefore ϕ# gives rise to a ‘partial homomorphism’ from K(A) to K(B). Next, we would like to recall the notion of Bott(ϕ, w) introduced in [19]. Let ϕ : A → B be a unital homomorphism between unital C ∗ -algebras and let w ∈ B be a unitary satisfying

ϕ(a), w < δ for every a ∈ G, where G is a large finite subset of A and δ > 0 is a small positive real number. For a projection p ∈ A ⊗ C, (w ⊗ 1)(ϕ ⊗ id)(p) + (ϕ ⊗ id)(1 − p) in B ⊗ C is close to a unitary, where C is Mn or On+1 . We denote the equivalence class of this unitary by Bott(ϕ, w)(p) ∈ K1 (A ⊗ C). Next, we would like to introduce Bott(ϕ, w)(u) ∈ K0 (A ⊗ C) for a unitary u ∈ A ⊗ C. To this end we need to recall the notion of Bott elements associated with almost commuting unitaries [19, 2.11]. There exists a universal constant δ0 > 0 such that for any unitaries v1 , v2 in a C ∗ -algebra D satisfying [v1 , v2 ] < δ0 , the self-adjoint element e(v1 , v2 ) =

f (v1 ) g(v1 ) + v2∗ h(v1 )

g(v1 ) + h(v1 )v2 1 − f (v1 )

∈ M2 (D)

has a spectral gap at 1/2, where f, g, h are certain universal real-valued continuous functions on T [22, Section 3]. Then one can consider the K0 -class

802


1[1/2,∞) e(v1 , v2 ) −

1 0 0 0

∈ K0 (D)

and call it the Bott element associated with v1 , v2 . In our setting, for a unitary u ∈ A ⊗ C, we can consider the Bott element in K0 (A ⊗ C) corresponding to the almost commuting unitaries (ϕ ⊗ id)(u) and w ⊗ 1. We denote it by Bott(ϕ, w)(u) ∈ K0 (A ⊗ C). Thus, for a finite subset L ⊂ K(A), when G is large enough and δ is small enough, then Bott(ϕ, w)|L : L → K(B) is well defined. In this paper, whenever we write Bott(ϕ, w)|L, G and δ are always assumed to be chosen so that Bott(ϕ, w)|L is well defined. In the same way as above, we can see that Bott(ϕ, w) gives rise to a ‘partial homomorphism’ from Ki (A ⊗ C) to K1−i (B ⊗ C). 2.4. The target algebras We denote the Jiang–Su algebra by Z [12]. When a C ∗ -algebra A satisfies A ∼ = A ⊗ Z, we say that A is Z-stable. We let Q denote the universal UHF algebra, that is, Q is the UHF algebra satisfying K0 (Q) = Q. We introduce four classes T , T , C and C of unital simple separable stably finite C ∗ -algebras as follows. Definition 2.2. We define T to be the class of all infinite-dimensional unital simple separable C ∗ -algebras with tracial rank zero. Let T be the class of infinite-dimensional unital simple separable exact C ∗ -algebras A with real rank zero, stable rank one, weakly unperforated K0 (A) and finitely many extremal tracial states. We let C be the class of unital simple separable Z-stable C ∗ -algebras A such that A ⊗ Q is in T . Let C be the class of unital simple separable stably finite Z-stable exact C ∗ -algebras A whose projections separate traces and whose extremal traces are finitely many. Remark 2.3. (1) Any A ∈ T has real rank zero, stable rank one, weakly unperforated K0 (A) and strict comparison of projections (see [14, Chapter 3]). (2) Exactness of A ∈ T is assumed only for the purpose of using the fact that any quasitrace on an exact C ∗ -algebra is a trace [10]. By [1, Corollary 6.9.2], any A ∈ T has strict comparison of projections. (3) If A ∈ C, then A ⊗ B has tracial rank zero for any UHF algebra B by [24, Lemma 2.4], that is, A ⊗ B belongs to T . (4) Let A ∈ C and let B be a UHF algebra. Then A ⊗ B has real rank zero by [2, Theorem 1.4(f)] and has stable rank one by [28, Corollary 6.6] (or [31]). By [29, Theorem 5.2] (or [9]), K0 (A ⊗ B) is weakly unperforated. It follows that A ⊗ B is in T . (5) Of course, Z itself is in C ∩ C . To continue, we fix a notation. Let A and B be unital stably finite C ∗ -algebras and let ξ ∈ Hom(K0 (A), K0 (B)). We say that ξ is unital when ξ([1]) = [1]. We say that ξ is positive (resp. strictly positive) when ξ(K0 (A)+ ) ⊂ K0 (B)+ (resp. ξ(K0 (A)+ \ {0}) ⊂ K0 (B)+ \ {0}). Assume further that A satisfies the UCT. We denote by KL(A, B)+,1 the set of all κ ∈ KL(A, B) such that K0 (κ) is unital and strictly positive.


803

Lemma 2.4. Let X be a connected compact metrizable space and let B be a unital stably finite C ∗ -algebra. For ξ ∈ Hom(K0 (C(X)), K0 (B)) the following are equivalent. (1) ξ is unital and strictly positive. (2) ξ is unital and positive. If K0 (B) is simple and weakly unperforated, then the two conditions above are equivalent to the following condition. (3) ξ is unital and ξ(Ker DC(X) ) ⊂ Ker DB . Proof. (1) ⇒ (2) is clear. To show (2) ⇒ (1), assume that ξ is unital and positive. By [1, Corollary 6.3.6], K0 (C(X)) is a simple ordered group. Hence for any x ∈ K0 (C(X))+ \ {0} there exists n ∈ N such that [1] nx. Then [1] nξ(x) in K0 (A), and so ξ(x) = 0. Assume that K0 (B) is simple and weakly unperforated. Let us show (2) ⇒ (3). Take x ∈ Ker DC(X) and τ ∈ T (B). The composition of τ and ξ is a state on K0 (C(X)). By [1, Corollary 6.10.3(e)], any state on K0 (C(X)) comes from a trace. Therefore τ (ξ(x)) = 0. It remains to show that (3) implies (1). Take x ∈ K0 (C(X))+ \ {0}. Let ρ be a state on K0 (B). The composition of ρ and ξ is a state on K0 (C(X)). By [1, Corollary 6.10.3(e)] it comes from a trace on C(X), and so ρ(ξ(x)) > 0. By [1, Theorem 6.8.5], K0 (B) has the strict ordering from its states. It follows that ξ(x) is in K0 (B) \ {0}. 2 We recall the following three theorems from [15,26]. Theorem 2.5. (See [15, Corollary 4.6].) Let A be a unital simple separable C ∗ -algebra with real rank zero, stable rank one and weakly unperforated K0 (A). Then there exist a unital simple separable AH algebra B with real rank zero and slow dimension growth and a unital homomorphism ϕ : B → A which induces a graded ordered isomorphism from K∗ (B) to K∗ (A). Theorem 2.6. (See [26, Theorem 0.1].) Let X be a path connected compact metrizable space and let A be a unital simple separable exact C ∗ -algebra with real rank zero, stable rank one and weakly unperforated K0 (A). Let κ ∈ KL(C(X), A)+,1 and let λ : T (A) → T (C(X)) be an affine continuous map such that λ(τ ) gives a strictly positive measure on X for any τ ∈ T (A). Then there exists a unital monomorphism ϕ : C(X) → A such that KL(ϕ) = κ and T (ϕ) = λ. Theorem 2.7. (See [26, Theorem 0.2].) Let X be a path connected compact metrizable space and let A be a unital simple separable exact Z-stable C ∗ -algebra with real rank zero. Let ϕ, ψ : C(X) → A be unital monomorphisms. Then ϕ and ψ are approximately unitarily equivalent if and only if KL(ϕ) = KL(ψ) and τ ◦ ϕ = τ ◦ ψ for all τ ∈ T (A). Remark 2.8. (1) The proof of [26, Theorem 0.1] uses [15, Corollary 4.6], and in the statement of [15, Corollary 4.6] A is assumed to be nuclear. But the proof given there does not use nuclearity, and so we omit it. In the statement of [26, Theorem 0.2], A is also assumed to be nuclear. But its proof needs only exactness of A.

804


(2) The condition (b) of [26, Theorem 0.1] automatically follows from other assumptions, because any traces on C(X) induce the same state on K0 (C(X)) [1, Corollary 6.10.3(a)] and K0 (C(X)) has no other states [1, Corollary 6.10.3(e)]. We give a generalization of Theorem 2.6 for later use.

Corollary 2.9. Let C = ni=1 pi (C(Xi ) ⊗ Mki )pi , where Xi is a path connected compact metrizable space and pi ∈ C(Xi ) ⊗ Mki is a projection. Let A be a unital simple separable exact C ∗ -algebra with real rank zero, stable rank one and weakly unperforated K0 (A). Let κ ∈ KL(C, A)+,1 and let λ : T (A) → T (C) be an affine continuous map such that λ(τ ) is a faithful trace for any τ ∈ T (A). Suppose that λ(τ )(pi ) = τ (K0 (κ)([pi ])) holds for any τ ∈ T (A) and i = 1, 2, . . . , n. Then there exists a unital monomorphism ϕ : C → A such that KL(ϕ) = κ and T (ϕ) = λ. Proof. It is clear that the case C = C(X) ⊗ Mk follows immediately from Theorem 2.6. Let us consider the case C = p(C(X) ⊗ Mk )p, where X is a path connected compact metrizable space. Let m ∈ N be the rank of p. There exist l ∈ N and a projection q ∈ C ⊗ Ml ⊂ C(X) ⊗ Mkl such that p ⊗ e is a subprojection of q and q is Murray–von Neumann equivalent to 1C(X) ⊗ r, where e ∈ Ml is a minimal projection of Ml and r ∈ Mkl is a projection of rank k. We can find a projection q˜ ∈ A ⊗ Ml such that K0 (κ)([q]) = [q]. ˜ Set C0 = q(C ⊗ Ml )q ∼ = C(X) ⊗ Mk and A0 = q(A ˜ ⊗ Ml )q. ˜ For any tracial state τ ∈ T (A), mk −1 (τ ⊗ Tr) gives a tracial state on A0 , and this correspondence induces a homeomorphism between T (A) and T (A0 ). Likewise there exists a natural homeomorphism between T (C) and T (C0 ). The identifications KL(C, A)+,1 ∼ = KL(C0 , A0 )+,1 ,

T (A) ∼ = T (A0 )

and T (C) ∼ = T (C0 )

allow us to regard κ as an element of KL(C0 , A0 )+,1 and λ as an affine continuous map from T (A0 ) to T (C0 ). Therefore the previous case shows that there exists ϕ : C0 → A0 realizing κ and λ. From [ϕ(p ⊗ e)] = K0 (κ)([p ⊗ e]) = [1A ⊗ e], there exists a unitary u ∈ A ⊗ Ml such that uϕ(p ⊗ e)u∗ = 1A ⊗ e. The restriction of Ad u ◦ ϕ to (p ⊗ e)C0 (p ⊗ e) gives a desired unital monomorphism from C to A.

We now turn to the general case. Let C = ni=1 pi (C(Xi ) ⊗ Mki )pi , κ and λ be as in the statement. Let γi : pi (C(Xi ) ⊗ Mki )pi → C be the canonical embedding. Choose projections p˜ 1 , p˜ 2 , . . . , p˜ n ∈ A so that K0 (κ)([pi ]) = [p˜ i ] and p˜ 1 + p˜ 2 + · · · + p˜ n = 1. For each i = 1, 2, . . . , n, κ ◦ KL(γi ) is regarded as an element of KL(pi C, p˜ i Ap˜ i )+,1 . For τ ∈ T (A), the restriction of τ/τ (p˜ i ) to p˜ i Ap˜ i is a tracial state. Similarly the restriction of λ(τ )/τ (p˜ i ) to pi C is also a tracial state, because λ(τ )(pi ) = τ (K0 (κ)([pi ])) = τ (p˜ i ). It is not so hard to check that λi : τ/τ (p˜ i ) → λ(τ )/τ (p˜ i ) gives rise to an affine continuous map from T (p˜ i Ap˜ i ) to T (pi C). We have already shown that κ ◦ KL(γi ) and λi are realized by a unital monomorphism ϕi : pi C → p˜ i Ap˜ i . Then ϕ = ϕ1 + ϕ2 + · · · + ϕn is a unital monomorphism from C to A satisfying KL(ϕ) = κ and T (ϕ) = λ. 2


805

3. Determinants of unitaries In this section, we would like to introduce a homomorphism Θϕ,ψ : K1 (C) → Aff T (A) / Im DA , which plays an important role in the main theorems of this paper (Theorem 6.6, Corollary 6.8, Theorem 7.1). Let A be a unital C ∗ -algebra. For τ ∈ T (A), the de la Harpe–Skandalis determinant [11] associated with τ is written by τ : U∞ (A)0 → R/DA K0 (A) (τ ). It is well known that A (u)(τ ) = τ (u) gives a homomorphism A : U∞ (A)0 → Aff T (A) / Im DA . Let C, A be unital C ∗ -algebras and let ϕ, ψ : C → A be unital homomorphisms satisfying K1 (ϕ) = K1 (ψ) and T (ϕ) = T (ψ). In what follows, we use the same notation ϕ, ψ for the homomorphisms from C ⊗ Mn to A ⊗ Mn induced by ϕ, ψ. For u ∈ U∞ (C), we can consider A (ϕ(u∗ )ψ(u)), as ϕ(u∗ )ψ(u) belongs to U∞ (A)0 . Lemma 3.1. In the setting above, we have the following. (1) There exists a homomorphism Θϕ,ψ : K1 (C) → Aff T (A) / Im DA such that Θϕ,ψ ([u]) = A (ϕ(u∗ )ψ(u)) for any u ∈ U∞ (C). (2) For any w ∈ U (A), Θϕ,Ad w◦ψ = Θϕ,ψ . (3) If ϕ and ψ are approximately unitarily equivalent, then Im Θϕ,ψ ⊂ Im DA . (4) If C satisfies the UCT and KL(ϕ) equals KL(ψ), then the homomorphism Θϕ,ψ factors through K1 (C)/ Tor(K1 (C)). Proof. (1) We first show that A (ϕ(u∗ )ψ(u)) equals A (ϕ(v ∗ )ψ(v)) when u, v ∈ U∞ (C) satisfy uv ∗ ∈ U∞ (C)0 . We can find n ∈ N and piecewise smooth paths of unitaries x : [0, 1] → U (A ⊗ Mn ), y : [0, 1] → U (A ⊗ Mn ) and z : [0, 1] → U (C ⊗ Mn ) such that x(0) = ϕ(u), x(1) = ψ(u), y(0) = ϕ(v), y(1) = ψ(v) and z(0) = u, z(1) = v. Define h : [0, 1] → U (A ⊗ Mn ) by ⎧ x(4t), ⎪ ⎪ ⎪ ⎨ ψ(z(4t − 1)), h(t) = ⎪ y(3 − 4t), ⎪ ⎪ ⎩ ϕ(z(4 − 4t)), Since h is a closed path of unitaries, one has

1 t 1/4, 1/4 t 1/2, 1/2 t 3/4, 3/4 t 1.

806


1 √ 2π −1

1

∗ ˙ dt ∈ DA K0 (A) (τ ) (τ ⊗ Tr) h(t)h(t)

0

for any τ ∈ T (A). It is easy to see that the contribution from t → ψ(z(4t − 1)) and t → ϕ(z(4 − 4t)) cancels out, because of T (ϕ) = T (ψ). It follows that 1 √ 2π −1

1 0

∗ dt − (τ ⊗ Tr) x(t)x(t) ˙

1

∗ dt ∈ DA K0 (A) (τ ) (τ ⊗ Tr) y(t)y(t) ˙

0

for any τ ∈ T (A), which implies A (ϕ(u∗ )ψ(u)) = A (ϕ(v ∗ )ψ(v)). It follows that Θϕ,ψ : K1 (C) → Aff(T (A))/ Im DA is well defined as a map by Θϕ,ψ ([u]) = A (ϕ(u∗ )ψ(u)) for any u ∈ U∞ (C). For any u, v ∈ U∞ (C), diag(uv, 1) is homotopic to diag(u, v), and so A diag ϕ(uv)∗ ψ(uv), 1 = A diag ϕ(u)∗ ψ(u), ϕ(v)∗ ψ(v) = A ϕ u∗ ψ(u) + A ϕ v ∗ ψ(v) . Hence we can conclude that Θϕ,ψ is a homomorphism. (2) can be shown in a similar fashion to the proof of (ii) ⇒ (i) of [13, Theorem 3.1]. We leave the details to the reader. (3) follows from (1) and (2). (4) Let Mϕ,ψ = f ∈ C [0, 1], A f (0) = ϕ(c), f (1) = ψ(c) for some c ∈ C be the mapping torus of ϕ, ψ : C → A. Since Ki (ϕ) = Ki (ψ) for i = 0, 1, from the short exact sequence π 0 → SA → Mϕ,ψ − → C→0

of C ∗ -algebras, we obtain the following short exact sequence of abelian groups: K (π)

0 → K0 (A) → K1 (Mϕ,ψ ) −−1−−→ K1 (C) → 0. By KL(ϕ) = KL(ψ), this exact sequence is pure (see Theorem 2.1). Thus, the quotient map K1 (π) has a right inverse on any finitely generated subgroup of K1 (C). Let Rϕ,ψ : K1 (Mϕ,ψ ) → Aff(T (A)) be the rotation map introduced in [19, Section 2] (see also [13, Section 1]). It is easy to verify that Rϕ,ψ (x) + Im DA = Θϕ,ψ K1 (π)(x) holds for every x ∈ K1 (Mϕ,ψ ). Therefore Θϕ,ψ kills torsion of K1 (A), because Aff(T (A)) is torsion free. In other words, Θϕ,ψ factors through K1 (C)/ Tor(K1 (C)). 2


807

4. C ∗ -algebras of real rank zero In this section we give a classification result of unital monomorphisms from C(X) to a C ∗ algebra in T ∪ T (Theorem 4.8). We begin with the following lemma, which is a variant of [7, Lemma 2.2]. A similar argument is also found in [14, Lemma 6.2.7]. Lemma 4.1. Let X be a compact metrizable space. For any finite subset F ⊂ C(X) and ε > 0, there exist a finite subset G ⊂ C(X) and δ > 0 such that the following hold. Let ϕ and ψ be (G, δ)-multiplicative ucp maps from C(X) to Mn such that tr ϕ(f ) − tr ψ(f ) < δ,

∀f ∈ G.

Then there exist a projection p ∈ Mn , (F, ε)-multiplicative ucp maps ϕ , ψ : C(X) → pMn p and a unital homomorphism σ : C(X) → (1−p)Mn (1−p) such that ϕ ∼F,ε ϕ ⊕ σ , ψ ∼F,ε ψ ⊕ σ and tr(p) < ε. Proof. Suppose that we are given a finite subset F ⊂ C(X) and ε > 0. We may assume that elements of F are of norm one. The proof is by contradiction. If the lemma was false, then we would have a sequence of pairs of ucp maps ϕn and ψn from C(X) to Mmn such that ϕn (f g) − ϕn (f )ϕn (g) → 0,

ψn (f g) − ψn (f )ψn (g) → 0

and tr ϕn (f ) − tr ψn (f ) → 0 as n → ∞ for any f, g ∈ C(X), and the conclusion of the lemma does not hold for any ϕn , ψn . Let ω ∈ βN \ N be a free ultrafilter on N. Define

Mmn = (an )n ∈ Mmn lim an = 0 . n→ω

ω

We set A = Mmn / ω Mmn and let π : Mmn → A be the quotient map. Define ucp maps ϕ˜ ˜ ) = (ψn (f ))n for f ∈ C(X). Clearly and ψ˜ from C(X) to Mmn by ϕ(f ˜ ) = (ϕn (f ))n and ψ(f π ◦ ϕ˜ and π ◦ ψ˜ are unital homomorphisms from C(X) to A. One can define a tracial state τ ∈ T (A) by τ π (an )n = lim tr(an ) n→ω

for (an )n ∈ Mmn . Then we have τ ◦ π ◦ ϕ˜ = τ ◦ π ◦ ψ˜ . Let μ be the probability measure on X corresponding to τ ◦ π ◦ ϕ˜ = τ ◦ π ◦ ψ˜ . Any x ∈ X has an open neighborhood Ux such that μ(Ux \ Ux ) = 0 and |f (y) − f (y )| < ε/3 for any y, y ∈ Ux and f ∈ F . (Such Ux exists by the following reason. Let d(·,·) be a metric compatible with the topology of X and let Cr = {y ∈ X | d(x, y)=r} for r > 0. There exist only countably many r such that μ(Cr ) > 0, because μ is a probability measure. Hence it is easy to find r > 0 so that μ(Cr ) = 0 and |f (x) − f (y)| < ε/6 for any y ∈ X with d(x, y) < r and

808


f ∈ F . Then Ux = {y ∈ X | d(x, y) < r} meets the requirement.) Since X is compact, we can find x1 , x2 , . . . , xk ∈ X such that Ux1 ∪ · · · ∪ Uxk = X. Consider open subsets of the form W = V1 ∩ V2 ∩ · · · ∩ Vk satisfying μ(W ) > 0, where Vi is either Uxi or X \ Uxi . Let W1 , W2 , . . . , Wl be these open subsets. Evidently Wi ’s are pairwise disjoint. Then we have μ X \ (W1 ∪ W2 ∪ · · · ∪ Wl ) = 0 )| < ε/3 for any y, y ∈ W and f ∈ F . Choose z ∈ W for each i = 1, 2, . . . , l. and |f (y) − f (y i i i ∗ The C -algebra Mmn has real rank zero and so does A. Accordingly, the hereditary subalgebra of A generated by π(ϕ(C ˜ 0 (Wi ))) contains an approximate unit consisting of projections. It follows that there exists a projection

pi ∈ π ϕ˜ C0 (Wi ) Aπ ϕ˜ C0 (Wi ) satisfying τ (pi ) > μ(Wi )−ε/ l. It is easy to see that π(ϕ(f ˜ ))pi −f (zi )pi < ε/3 holds for any f ∈ F . Extend π ◦ ϕ˜ to a unital homomorphism from C(X)∗∗ to A∗∗ and define p¯ i = π(ϕ(1 ˜ Wi )). Then p¯ i commutes with π(ϕ(C(X))) ˜ and pi p¯ i . Similarly one can find projections qi in the satisfying anal˜ 0 (Wi ))) and q¯i in A∗∗ ∩ π(ψ(C(X))) ˜ hereditary subalgebra generated by π(ψ(C ogous properties. It is not so hard to find projections pi pi , qi qi and a unitary u ∈ A such that pi = uqi u∗ and τ pi = τ qi = min τ (pi ), τ (qi ) for any i = 1, 2, . . . , l. Set p = 1 − (p1 + p2 + · · · + pl ) and q = 1 − (q1 + q2 + · · · + ql ). We have l l τ (p) = 1 − μ(Wi ) − ε/ l = ε. τ pi < 1 − i=1

i=1

Moreover, l l l (1 − p)π ϕ(f ˜ ) = pi π ϕ(f ˜ ) = pi p¯ i π ϕ(f ˜ ) = pi π ϕ(f ˜ ) p¯ i i=1

≈ε/3

l i=1

i=1

f (zi )pi p¯ i =

l

i=1

f (zi )pi

i=1

˜ )) − li=1 f (zi )q < ε/3 for any holds for any f ∈ F . Similarly one has (1 − q)π(ψ(f i ) , (p ) in f ∈ F . It is well known that the projections pi , p in A lift to projections (pi,n n n n + · · · + p = 1. Similarly the unitary u ∈ A lifts to a unitary Mmn satisfying pn + p1,n l,n (un )n ∈ Mmn . Define ucp maps ϕn , ψn : C(X) → pn Mmn pn and a unital homomorphism


809

σn : C(X) → (1−pn )Mmn (1−pn ) by ϕn (f ) = pn ϕn (f )pn ,

ψn (f ) = pn un ψn (f )u∗n pn

and σn (f ) =

l

f (zi )pi,n .

i=1

It follows that there exists n ∈ N such that ϕn and ψn are (F, 2ε/3)-multiplicative, ϕn ∼F,ε ϕn ⊕ σn and ψn ∼F,ε ψn ⊕ σn . This contradicts the assumption, and so the proof is completed. 2 We can prove the following lemma in the same way as above. Lemma 4.2. Let X be a compact metrizable space. For any finite subset F ⊂ C(X), ε > 0 and m ∈ N, there exist a finite subset G ⊂ C(X) and δ > 0 such that the following hold. Let A ∈ T be a C ∗ -algebra with at most m extremal tracial states. Let ϕ and ψ be (G, δ)-multiplicative ucp maps from C(X) to A such that τ ϕ(f ) − τ ψ(f ) < δ,

∀f ∈ G, τ ∈ T (A).

Then there exist a projection p ∈ A, (F, ε)-multiplicative ucp maps ϕ , ψ : C(X) → pAp and a unital homomorphism σ : C(X) → (1−p)A(1−p) with finite-dimensional range such that ϕ ∼F,ε ϕ ⊕ σ , ψ ∼F,ε ψ ⊕ σ and τ (p) < ε for any τ ∈ T (A). Proof. Suppose that we are given a finite subset F ⊂ C(X), ε > 0 and m ∈ N. We may assume that elements of F are of norm one. The proof is by contradiction. If the lemma was false, then we would have a sequence of C ∗ -algebras (An )n in T and a sequence of pairs of ucp ˜ ψ˜ , maps ϕn and ψn from C(X) to An as in the proof of Lemma 4.1. Define ϕ, B = An / ω An and π : An → B in the same way. For each n, choose extremal tracial states τ1,n , τ2,n , . . . , τm,n ∈ T (An ) so that {τ1,n , τ2,n , . . . , τm,n } exhausts all the extremal traces on An . For each j = 1, 2, . . . , m, one can define τj,ω ∈ T (B) by τj,ω π (an )n = lim τj,n (an ). n→ω

˜ In the We obtain a probability measure μj on X corresponding to τj,ω ◦ π ◦ ϕ˜ = τj,ω ◦ π ◦ ψ. same way as in Lemma 4.1, we can find pairwise disjoint open subsets W1 , W2 , . . . , Wl of X such that max μj (Wi ) > 0, j

∀i = 1, 2, . . . , l,

max μj X \ (W1 ∪ W2 ∪ · · · ∪ Wl ) = 0 j

and |f (y) − f (y )| < ε/3 for any y, y ∈ Wi and f ∈ F . Choose zi ∈ Wi for each i = 1, 2, . . . , l. In the same way as in Lemma 4.1, we also get a family of mutually orthogonal non-zero projections p1 , p2 , . . . , pl in B such that τj,ω (pi ) > μj (Wi ) − ε/2l and π(ϕ(f ˜ ))pi − f (zi )pi < ε/3 for all f ∈ F . Similarly one can find mutually orthogonal non-zero projections q1 , q2 , . . . , ql in B for ψ˜ . It is well known that the projections pi (resp. qi ) lift to mutually orthogonal projections (pi,n )n (resp. (qi,n )n ) in An . Then there exists N ∈ ω such that

810


pi,n = 0, qi,n = 0,

τj,n (pi,n ) > μj (Wi ) − ε/2l,

τj,n (qi,n ) > μj (Wi ) − ε/2l

holds for every i = 1, 2, . . . , l, j = 1, 2, . . . , m and n ∈ N . For each n ∈ N, the image of DAn is dense in Aff(T (An )) by [1, Theorem 6.9.3]. It follows that for each n ∈ N and i = 1, 2, . . . , l there exist projections ri,n ∈ An such that μj (Wi ) − ε/2l < τj,n (ri,n ) < min τj,n (pi,n ), τj,n (qi,n ) ,

∀j = 1, 2, . . . , m.

Besides An satisfies strict comparison of projections (see Remark 2.3(2)). Therefore, for n ∈ N , p , q q we can find projections pi,n i,n i,n and a unitary un ∈ An such that τj,n (pi,n ) > i,n ∗ / N , set pi,n = qi,n = 0 and un = 1. Let pi , qi , u ∈ B μj (Wi ) − ε/2l and pi,n = un qi,n un . For n ∈ ) , (q ) and (u ) by π . Then we have p p , q q , p = uq u∗ and be the image of (pi,n n n n i i i i,n n i i i τj,ω (pi ) > μj (Wi ) − ε/ l. The rest of the proof is exactly the same as Lemma 4.1. 2 We can show the same statement for the case that the target algebra is of tracial rank zero. Lemma 4.3. Let X be a compact metrizable space. For any finite subset F ⊂ C(X) and ε > 0, there exist a finite subset G ⊂ C(X) and δ > 0 such that the following hold. Let A ∈ T and let ϕ and ψ be (G, δ)-multiplicative ucp maps from C(X) to A such that τ ϕ(f ) − τ ψ(f ) < δ,

∀f ∈ G, τ ∈ T (A).

Then there exist a projection p ∈ A, (F, ε)-multiplicative ucp maps ϕ , ψ : C(X) → pAp and a unital homomorphism σ : C(X) → (1−p)A(1−p) with finite-dimensional range such that ϕ ∼F,ε ϕ ⊕ σ , ψ ∼F,ε ψ ⊕ σ and τ (p) < ε for any τ ∈ T (A). Proof. Suppose that we are given a finite subset F ⊂ C(X) and ε > 0. Applying Lemma 4.1 for F and ε, we obtain G ⊂ C(X) and δ > 0. Let A be a unital simple separable C ∗ -algebra with tracial rank zero and let ϕ and ψ be (G, δ)-multiplicative ucp maps from C(X) to A such that τ ϕ(f ) − τ ψ(f ) < δ for any f ∈ G and τ ∈ T (A). Since A has tracial rank zero, there exist a sequence of projections en ∈ A, a sequence of finite-dimensional subalgebras Bn of A with 1Bn = en and a sequence of ucp maps πn : A → Bn such that the following hold. • [a, en ] → 0 as n → ∞ for any a ∈ A. • πn (a) − en aen → 0 as n → ∞ for any a ∈ A. • τ (1 − en ) → 0 as n → ∞ uniformly on T (A). It is easy to see that πn ◦ ϕ and πn ◦ ψ are (G, δ)-multiplicative for sufficiently large n ∈ N. We would like to show that τ πn ϕ(f ) − τ πn ψ(f ) < δ holds for every f ∈ G, τ ∈ T (Bn ) and sufficiently large n ∈ N. To this end, we assume that there


811

exist τn ∈ T (Bn ) such that maxτn πn ϕ(f ) − τn πn ψ(f ) δ. f ∈G

Let τ ∈ A∗ be an accumulation point of τn ◦ πn . Clearly τ is a tracial state of A and |τ (ϕ(f )) − τ (ψ(f ))| δ for some f ∈ G, which is a contradiction. Hence, Lemma 4.1 implies that, for sufficiently large n ∈ N, there exist a projection pn ∈ Bn , (F, ε)-multiplicative ucp maps ϕn , ψn : C(X) → pn Bn pn and a unital homomorphism σn : C(X) → (en −pn )Bn (en −pn ) such that πn ◦ ϕ ∼F,ε ϕn ⊕ σn ,

πn ◦ ψ ∼F,ε ψn ⊕ σn

and τ (pn ) < ε for any τ ∈ T (Bn ). Therefore the proof is completed.

2

The following is taken from [8, Theorem 3.1]. We remark that its origin is found in [4]. Theorem 4.4. (See [8, Theorem 3.1].) Let X be a compact metrizable space. For any finite subset F ⊂ C(X) and ε > 0, there exist a finite subset G ⊂ C(X), δ > 0, l ∈ N and a finite subset L ⊂ K(C(X)) satisfying the following: For any unital C ∗ -algebra A with real rank zero, stable rank one and weakly unperforated K0 (A) and any (G, δ)-multiplicative ucp maps ϕ, ψ : C(X) → A satisfying ϕ# |L = ψ# |L, there exist a unitary u ∈ Ml+1 (A) and {x1 , x2 , . . . , xl } ⊂ X such that u diag ϕ(f ), f (x1 ), f (x2 ), . . . , f (xl ) u∗ − diag ψ(f ), f (x1 ), f (x2 ), . . . , f (xl ) < ε for any f ∈ F . The following theorem is a variant of [17, Theorem 4.6]. Theorem 4.5. Let X be a compact metrizable space, let F ⊂ C(X) be a finite subset and let ε > 0. Then there exist a finite subset L ⊂ K(C(X)) and a family of mutually orthogonal positive elements h1 , h2 , . . . , hk ∈ C(X) of norm one such that the following holds. For any ν > 0, one can find a finite subset G ⊂ C(X) and δ > 0 satisfying the following. For any A ∈ T and any (G, δ)-multiplicative ucp maps ϕ, ψ : C(X) → A such that ϕ# |L = ψ# |L, τ ϕ(hi ) ν,

∀τ ∈ T (A), i=1, 2, . . . , k

and τ ϕ(f ) − τ ψ(f ) < δ,

∀τ ∈ T (A), f ∈ G,

there exists a unitary u ∈ A such that uϕ(f )u∗ − ψ(f ) < ε holds for any f ∈ F .

812


Proof. We say that a subset Y ⊂ X is (F, ε)-dense if for any x ∈ X there exists y ∈ Y such that |f (x) − f (y)| < ε for every f ∈ F . Choose an (F, ε/7)-dense finite subset {y1 , y2 , . . . , yk } ⊂ X. For each i = 1, 2, . . . , k, choose an open neighborhood Ui of yi so that x ∈ Ui implies |f (x) − f (yi )| < ε/7 for any f ∈ F and that U1 , U2 , . . . , Uk are mutually disjoint. Choose a positive function hi ∈ C0 (Ui ) of norm one. By applying Theorem 4.4 to F and ε/7, we obtain a finite subset G1 ⊂ C(X), δ1 > 0, l ∈ N and a finite subset L ⊂ K(C(X)). There exist a finite subset G2 ⊂ C(X) and δ2 > 0 such that the following holds: For any unital C ∗ -algebra A and any (G2 , δ2 )-multiplicative ucp maps ϕ, ψ : C(X) → A, if ϕ(f ) − ψ(f ) < δ2 for every f ∈ G2 , then ϕ# |L = ψ# |L. Suppose that ν > 0 is given. Let G3 = F ∪ G1 ∪ G2 ∪ {h1 , h2 , . . . , hk } and δ3 = min ε/7, δ1 , δ2 , ν/(l+2) . By applying Lemma 4.3 to G3 and δ3 , we obtain a finite subset G ⊂ C(X) and δ > 0. Suppose that A is a unital simple separable C ∗ -algebra A with tracial rank zero and that ϕ, ψ : C(X) → A are (G, δ)-multiplicative ucp maps satisfying ϕ# |L = ψ# |L, τ ϕ(hi ) ν,

∀τ ∈ T (A), i=1, 2, . . . , k

and τ ϕ(f ) − τ ψ(f ) < δ,

∀τ ∈ T (A), f ∈ G.

By Lemma 4.3, there exist a projection p ∈ A, (G3 , δ3 )-multiplicative ucp maps ϕ , ψ : C(X) → pAp, a unital homomorphism σ : C(X) → (1−p)A(1−p) with finite-dimensional range such that ϕ ∼G3 ,δ3 ϕ ⊕ σ , ψ ∼G3 ,δ3 ψ ⊕ σ and τ (p) < δ3 for any τ ∈ T (A). Since G2 is contained in G3 and δ2 is not greater than δ3 , by the choice of G2 and δ2 , we obtain (ϕ ⊕ σ )# |L = (ψ ⊕ σ )# |L, and hence ϕ# |L = ψ# |L. Besides, ϕ and ψ are (G1 , δ1 )-multiplicative, because G1 is contained in G3 and δ1 is not greater than δ3 . By Theorem 4.4, there exist a unitary u ∈ Ml+1 (pAp) and {x1 , x2 , . . . , xl } ⊂ X such that u diag ϕ (f ), f (x1 ), f (x2 ), . . . , f (xl ) u∗ − diag ψ (f ), f (x1 ), f (x2 ), . . . , f (xl ) < ε/7 for any f ∈ F . In what follows, for a positive linear functional ρ on C(X), we let μρ denote the corresponding measure on X. For any τ ∈ T (A) and i = 1, 2, . . . , k, one has μτ ◦σ (Ui ) τ σ (hi ) > τ ϕ(hi ) − τ ϕ (hi ) − δ3 > ν − 2δ3 lδ3 . It follows that there exists a unital homomorphism σ : C(X) → (1−p)A(1−p) with finitedimensional range such that σ (f ) − σ (f ) < ε/7


813

for any f ∈ F and μτ ◦σ ({yi }) > lδ3 for any τ ∈ T (A) and i = 1, 2, . . . , k. Since {y1 , y2 , . . . , yk } is (F, ε/7)-dense, we can find a unital homomorphism σ : C(X) → (1−p)A(1−p) with finitedimensional range such that σ (f ) − σ (f ) < ε/7 for any f ∈ F and μτ ◦σ ({xj }) > δ3 for any τ ∈ T (A) and j = 1, 2, . . . , l. Then it is not so hard to see ϕ ⊕ σ ∼F,ε/7 ψ ⊕ σ . Consequently we have ϕ ∼F,ε/7 ϕ ⊕ σ ∼F,ε/7 ϕ ⊕ σ ∼F,ε/7 ϕ ⊕ σ ∼F,ε/7 ψ ⊕ σ ∼F,ε/7 ψ ⊕ σ ∼F,ε/7 ψ ⊕ σ ∼F,ε/7 ψ.

2

In the same fashion as above, one can prove the following by using Lemma 4.2 instead of Lemma 4.3 (see also [7, Corollary 2.17]). Theorem 4.6. Let X be a compact metrizable space, let F ⊂ C(X) be a finite subset and let ε > 0, m ∈ N. Then there exist a finite subset L ⊂ K(C(X)) and a family of mutually orthogonal positive elements h1 , h2 , . . . , hk ∈ C(X) of norm one such that the following holds. For any ν > 0, one can find a finite subset G ⊂ C(X) and δ > 0 satisfying the following. Let A ∈ T be a C ∗ -algebra with at most m extremal tracial states. For any (G, δ)-multiplicative ucp maps ϕ, ψ : C(X) → A such that ϕ# |L = ψ# |L, τ ϕ(hi ) ν, ∀τ ∈ T (A), i=1, 2, . . . , k and τ ϕ(f ) − τ ψ(f ) < δ,

∀τ ∈ T (A), f ∈ G,

there exists a unitary u ∈ A such that uϕ(f )u∗ − ψ(f ) < ε holds for any f ∈ F . By using the theorems above, we obtain the following generalization of [17, Theorem 3.3]. Theorem 4.7. Let X be a compact metrizable space and let A ∈ T ∪ T . Let ϕ : C(X) → A be a unital monomorphism. Then for any finite subset F ⊂ C(X) and ε > 0, there exist a finite subset L ⊂ K(C(X)), a finite subset G ⊂ C(X) and δ > 0 such that the following hold. If ψ : C(X) → A is a (G, δ)-multiplicative ucp map satisfying ϕ# |L = ψ# |L and τ ϕ(f ) − τ ψ(f ) < δ for any τ ∈ T (A) and f ∈ G, then there exists a unitary u ∈ A such that uϕ(f )u∗ − ψ(f ) < ε holds for any f ∈ F .

814


Proof. Applying Theorem 4.5 or Theorem 4.6, we obtain a finite subset L ⊂ K(C(X)) and positive elements h1 , h2 , . . . , hk ∈ C(X) of norm one. Since A is simple and ϕ is injective, ν = min τ ϕ(hi ) τ ∈ T (A), i = 1, 2, . . . , k is positive. Using Theorem 4.5 or Theorem 4.6 for ν, we find a finite subset G ⊂ C(X) and δ > 0. It is clear that G and δ meet the requirement. 2 The following is an immediate consequence of the theorem above. Theorem 4.8. Let X be a compact metrizable space and let A ∈ T ∪ T . Let ϕ, ψ : C(X) → A be unital monomorphisms. Then ϕ and ψ are approximately unitarily equivalent if and only if KL(ϕ) = KL(ψ) and τ ◦ ϕ = τ ◦ ψ for all τ ∈ T (A). Corollary 4.9. Let C be a unital AH algebra and let A ∈ T ∪ T . Let ϕ, ψ : C → A be unital monomorphisms. Then ϕ and ψ are approximately unitarily equivalent if and only if KL(ϕ) = KL(ψ) and τ ◦ ϕ = τ ◦ ψ for all τ ∈ T (A). Proof. Although the proof is essentially the same as [17, Corollary 4.8], we present it for completeness. Without loss of generality, we may assume C = p(C(X) ⊗ Mk )p, where X is a compact metrizable space and p ∈ C(X) ⊗ Mk is a non-zero projection. We may further assume that the rank of p(x) ∈ Mk is strictly positive for every x ∈ X. We first consider the case p = 1 ∈ C(X) ⊗ Mk . It is easy to see that there exists a unitary u ∈ A such that ϕ(1 ⊗ a) = uψ(1 ⊗ a)u∗ holds for any a ∈ Mk . Let e be a minimal projection of Mk . Then ϕ (f ) = ϕ(f ⊗ e) and ψ (f ) = uψ(f ⊗ e)u∗ are unital monomorphisms from C(X) to ϕ(e)Aϕ(e). By Theorem 4.8, they are approximately unitarily equivalent. Hence ϕ and ψ are approximately unitarily equivalent. Let us consider the general case. There exist l ∈ N and a projection q ∈ C ⊗ Ml ⊂ C(X) ⊗ Mkl such that p ⊗ e is a subprojection of q and q is Murray–von Neumann equivalent to 1C(X) ⊗ r, where e ∈ Ml is a minimal projection of Ml and r ∈ Mkl is a projection of rank k. By the argument above, the restrictions of ϕ ⊗ idMl and ψ ⊗ idMl to q(C ⊗ Ml )q are approximately unitarily equivalent. It follows that their restrictions to (p ⊗ e)(C ⊗ Ml )(p ⊗ e) = C are also approximately unitarily equivalent, which completes the proof. 2 In Section 6 we will generalize the results above to the case that the target algebra A belongs to C ∪ C . 5. Homotopy of unitaries In this section, we prove the so-called basic homotopy lemma for A in T ∪ T (Theorem 5.3 and Theorem 5.4). The basic idea of the proof is similar to that of [19, Theorem 8.1], but there are two main differences. One is the use of Theorem 2.6, which claims the existence of a unital monomorphism ϕ : C(X) → A realizing the given κ ∈ KL(C(X), A)+,1 and λ : T (A) → T (C(X)). The other point is that we allow G ⊂ C(X) and δ > 0 in Theorem 5.3 to depend on the given homomorphism ϕ : C(X) → A. Although, as shown in [19], it is possible


815

to state the theorem in a more general form, we do not pursue this here because the actual application discussed in Section 6 does not need that general form. These two points enable us to simplify the proof given in [19]. We let√T denote the unit √ circle in the complex plane and let z ∈ C(T) be the identity function z(exp(π −1t)) = exp(π −1t). The following is a variant of [19, Lemma 6.4]. Lemma 5.1. Let X be a compact metrizable space and let A ∈ T ∪ T . For any finite subsets ⊂ C(X × T) and ε > 0, there exist a finite subset G ⊂ C(X) and δ > 0 such that F ⊂ C(X), F the following hold. For any k ∈ N, any unital monomorphism ϕ : C(X) → A and a unitary u ∈ A satisfying [ϕ(f ), u] < δ for any f ∈ G, there exist a path of unitaries w : [0, 1] → A and an , ε)-multiplicative ucp map ψ : C(X × T) → A such that (F w(1) − ψ(1 ⊗ z) < ε, w(0) − u < ε, Lip(w) π,

ϕ(f ), w(t) < ε, ψ(f ⊗ 1) − ϕ(f ) < ε hold for any f ∈ F and t ∈ [0, 1], and τ ψ f ⊗ zj < ε f

holds for any τ ∈ T (A), f ∈ C(X) and j ∈ Z with 1 |j | < k. Proof. Without loss of generality, we may assume that all the elements of F are of norm one. Applying Lemma 4.2 or Lemma 4.3 to ∪ {f ⊗ 1 | f ∈ F } ∪ {1 ⊗ z} ⊂ C(X × T) G1 = F and δ1 = min{ε/8, ε 2 }, we obtain a finite subset G2 ⊂ C(X × T) and δ2 > 0. We may assume that G2 contains G1 and that δ2 is less than δ1 . Clearly there exist a finite subset G ⊂ C(X) and δ > 0 such that the following holds: If ϕ : C(X) → A is a unital monomorphism and u ∈ A is a unitary satisfying [ϕ(f ), u] < δ for any f ∈ G, then one can find a (G2 , δ2 )-multiplicative ucp map ϕ0 : C(X × T) → A such that ϕ0 (1 ⊗ z) − u < δ2 ,

ϕ0 (f ⊗ 1) − ϕ(f ) < δ2

for every f ∈ G2 . Suppose that we are given k ∈ N, a unital monomorphism ϕ : C(X) → A and a unitary u ∈ A satisfying [ϕ(f ), u] < δ for every f ∈ G. We find ϕ0 : C(X × T) → A as above. By using Lemma 4.2 or Lemma 4.3, there exist a projection p ∈ A, a (G1 , δ1 )-multiplicative ucp map ϕ0 : C(X × T) → pAp and a unital homomorphism σ : C(X × T) → (1−p)A(1−p) with finitedimensional range such that ϕ0 (f ) − ϕ ⊕ σ (f ) < δ1 , 0

∀f ∈ G1

and τ (p) < δ1 for any τ ∈ T (A). We may further assume that there exists a unitary u ∈ pAp such that u − ϕ0 (1 ⊗ z) < δ1 . Since σ has finite-dimensional range, one can find x1 , x2 , . . . , xl ∈ X, y1 , y2 , . . . , yl ∈ T and projections p1 , p2 , . . . , pl ∈ A such that

816

H. Matui / Journal of Functional Analysis 260 (2011) 797–831 l

pi = 1−p,

σ (f ⊗ g) =

i=1

l

f (xi )g(yi )pi

i=1

holds for any f ∈ C(X) and g ∈ C(T). By replacing each pi with its subprojection if necessary, we may assume that DA ([pi ]) belongs to k Im(DA ), because Im DA is dense in Aff(T (A)). Choose projections qi,j for i = 1, 2, . . . , l and j = 1, 2, . . . , k so that k

kDA [qi,j ] = DA [pi ] ,

qi,j = pi ,

∀j = 1, 2, . . . , k.

j =1

Define a homomorphism σ : C(X × T) → (1−p)A(1−p) with finite-dimensional range by σ (f ⊗ g) =

l k

f (xi )g ζ j qi,j ,

i=1 j =1

√ where ζ = exp(2π −1/k). Define a ucp map ψ : C(X × T) → A by ψ = ϕ0 ⊕ σ . It is clear , ε)-multiplicative. Moreover, one has that ψ is (G1 , δ1 )-multiplicative, and hence is (F ψ(f ⊗ 1) = ϕ0 (f ⊗ 1) ⊕ σ (f ⊗ 1) = ϕ0 (f ⊗ 1) ⊕ σ (f ⊗ 1) ≈δ1 ϕ0 (f ⊗ 1) ≈δ2 ϕ(f ) for any f ∈ F . For any τ ∈ T (A), f ∈ C(X) and j ∈ Z with 1 |j | < k, it is easy to see τ ψ f ⊗ z j τ ϕ f ⊗ z j + τ σ f ⊗ z j 0 = τ ϕ0 f ⊗ zj f τ (p)1/2 < ε f . We construct a path of unitaries w : [0, 1] → A. By the definition of σ , we can find a path of unitaries v : [0, 1] → (1−p)A(1−p) such that v(0) = σ (1 ⊗ z),

v(1) = σ (1 ⊗ z),

Lip(v) π

and [qi,j , v(t)] = 0 for any i, j and t ∈ [0, 1]. Define w : [0, 1] → U (A) by w(t) = u ⊕ v(t). Evidently we have w(0) = u ⊕ σ (1 ⊗ z) ≈δ1 ϕ0 (1 ⊗ z) ⊕ σ (1 ⊗ z) ≈δ1 ϕ0 (1 ⊗ z) ≈δ2 u, w(1) = u ⊕ σ (1 ⊗ z) ≈δ1 ϕ0 (1 ⊗ z) ⊕ σ (1 ⊗ z) = ψ(1 ⊗ z), and Lip(w) π . Besides, for any f ∈ F and t ∈ [0, 1], one can verify

ϕ(f ), w(t) ≈2δ2 ϕ0 (f ⊗ 1), w(t) ≈2δ1 ϕ0 ⊕ σ (f ⊗ 1), u ⊕ v(t)

= ϕ0 (f ⊗ 1), u ≈2δ1 ϕ0 (f ⊗ 1), ϕ0 (1 ⊗ z) ≈2δ1 0, thereby completing the proof.

2


817

Remark 5.2. In the lemma above, for A in T , one can see that G ⊂ C(X) and δ depend only on F ⊂ C(X) and ε. For A in T , G ⊂ C(X) and δ depend only on F ⊂ C(X), ε and the cardinality of extremal tracial states on A. The following is a generalization of [19, Corollary 8.4]. Theorem 5.3. Let X be a path connected compact metrizable space and let A ∈ T ∪ T . Let ϕ : C(X) → A be a unital monomorphism. For any finite subset F ⊂ C(X) and ε > 0, there exist a finite subset L ⊂ K(C(X)), a finite subset G ⊂ C(X) and δ > 0 such that the following hold. If u ∈ A is a unitary satisfying

ϕ(f ), u < δ,

∀f ∈ G and

Bott(ϕ, u)(x) = 0,

∀x ∈ L,

then there exists a path of unitaries w : [0, 1] → A such that w(0) = u,

w(1) = 1,

Lip(w) < 2π + ε

and

ϕ(f ), w(t) < ε,

∀f ∈ F, t ∈ [0, 1].

Proof. Let p : C([−1, 1]) → C be the point evaluation at 1 ∈ [−1, 1] √ and let q : C(T) → C([−1, 1]) be the unital monomorphism defined by q(f )(t) = f (exp(π −1t)). By Lemma 2.4, KL(ϕ) ◦ KL(id ⊗p) belongs to KL(C(X × [−1, 1]), A)+,1 . Define τ0 ∈ T (C([−1, 1])) by 1 τ0 (f ) = 2

1 f (t) dμ(t), −1

where μ is the Lebesgue measure on R. Let λ : T (C(X)) → T (C(X × [−1, 1])) be the affine continuous map defined by λ(τ ) = τ ⊗ τ0 . By applying Theorem 2.6 to KL(ϕ) ◦ KL(id ⊗ p) and λ ◦ T (ϕ), we get a unital monomorphism σ : C(X × [−1, 1]) → A such that KL(σ ) = KL(ϕ) ◦ KL(id ⊗p) and T (σ ) = λ ◦ T (ϕ). Then σ = σ ◦ (id ⊗ q) is a unital monomorphism from C(X × T) to A such that KL σ = KL σ ◦ (id ⊗ q) = KL(ϕ) ◦ KL id ⊗(p ◦ q) and T (σ ) = T (id ⊗ q) ◦ λ ◦ T (ϕ). Under the canonical isomorphism K C(X × T) ∼ = K C(X) ⊕ K C0 X × T \ {−1} , KL(σ ) ∈ HomΛ (K(C(X × T)), K(A)) corresponds to KL(ϕ) ⊕ 0. It is also easy to see that T (σ )(τ ) = (τ ◦ ϕ) ⊗ τ0 for any τ ∈ T (A), where τ0 ∈ T (C(T)) is the tracial state corresponding to the Haar measure on T. From the construction, there exists a path of unitaries w1 : [0, 1] → A such that

818


w1 (0) = 1,

w1 (1) = σ (1 ⊗ z),

Lip(w1 ) = π

and [σ (f ⊗ 1), w1 (t)] = 0 for any f ∈ C(X) and t ∈ [0, 1]. By applying Theorem 4.7 to σ : C(X × T) → A, {f ⊗ 1 | f ∈ F } ∪ {1 ⊗ z} and ε/4, we obtain a finite subset L ⊂ K(C(X × T)), a finite subset G1 ⊂ C(X × T) and δ1 > 0. Choose a sufficiently large finite subset L0 ⊂ K(C(X)), a sufficiently large finite subset G2 ⊂ C(X) and a sufficiently small real number δ2 > 0. By applying Lemma 5.1 to G2 ⊂ C(X), G1 ⊂ C(X × T) and δ2 > 0, we obtain a finite subset G ⊂ C(X) and δ > 0. Suppose that we are given a unitary u ∈ A satisfying

ϕ(f ), u < δ,

∀f ∈ G and

Bott(ϕ, u)(x) = 0,

∀x ∈ L0 .

Let k ∈ N be a sufficiently large natural number. By Lemma 5.1, one can find a path of unitaries w0 : [0, 1] → A and a (G1 , δ2 )-multiplicative ucp map ψ : C(X × T) → A such that w0 (0) − u < δ2 , w0 (1) − ψ(1 ⊗ z) < δ2 , Lip(w0 ) π,

ψ(f ⊗ 1) − ϕ(f ) < δ2 ϕ(f ), w0 (t) < δ2 , hold for any f ∈ G2 and t ∈ [0, 1], and τ ψ f ⊗ zj < δ2 f

holds for any τ ∈ T (A), f ∈ C(X) and j ∈ Z with 1 |j | < k. Hence, if L0 ⊂ K(C(X)) is large enough, G2 ⊂ C(X) is large enough and δ2 > 0 is small enough, then one can conclude ψ# |L = σ# |L. In addition, if k ∈ N is chosen to be large enough, then we may assume τ ψ(f ) − τ σ (f ) < δ1 ,

∀τ ∈ T (A), f ∈ G1 .

It follows from Theorem 4.7 that there exists a unitary v ∈ A such that vσ (1 ⊗ z)v ∗ − ψ(1 ⊗ z) < ε/4,

vσ (f ⊗ 1)v ∗ − ψ(f ⊗ 1) < ε/4,

∀f ∈ F.

We define w : [0, 1] → U (A) by w(t) = w0 (t)vw1 (t)∗ v ∗ . Clearly one has Lip(w) 2π , w(0) − u < δ2 ,

w(1) − 1 < δ2 + ε/4

and

ϕ(f ), w(t) < 3δ2 + ε/2,

∀f ∈ F, t ∈ [0, 1].

It is easy to perturb the path w : [0, 1] → A a little bit so that w(0) = u and w(1) = 1. The following is an easy generalization of the theorem above.

2


819

Theorem 5.4. Let C be a unital C ∗ -algebra of the form ni=1 pi Mki (C(Xi ))pi , where Xi is a path connected compact metrizable space and pi is a non-zero projection of Mki (C(Xi )). Let A ∈ T ∪ T . Let ϕ : C → A be a unital monomorphism. For any finite subset F ⊂ C and ε > 0, there exist a finite subset L ⊂ K(C), a finite subset G ⊂ C and δ > 0 such that the following hold. If u ∈ A is a unitary satisfying

ϕ(f ), u < δ,

∀f ∈ G and

Bott(ϕ, u)(x) = 0,

∀x ∈ L,

then there exists a path of unitaries w : [0, 1] → A such that w(0) = u,

w(1) = 1,

Lip(w) < 2π + ε

and

ϕ(f ), w(t) < ε,

∀f ∈ F, t ∈ [0, 1].

Proof. We can prove this in a similar fashion to [19, Lemma 17.5] by using the theorem above. We omit the detail. It is worth noting that if A is in T ∪ T and e ∈ A is a non-zero projection, then eAe is also in T ∪ T . See also the proof of Corollary 4.9. 2 Remark 5.5. In the theorems above, if the target algebra A satisfies A ∼ = A ⊗ Q (i.e. A is Q-stable), then Ki (A; Zn ) = 0 for any i = 0, 1 and n 1 because Ki (A) is torsion free and divisible. Therefore the entire K-group K(A) is canonically isomorphic to K0 (A) ⊕ K1 (A). Consequently we may assume that the finite subset L ⊂ K(C(X)) in the statement is actually a finite subset of P (C(X) ⊗ K) ∪ U∞ (C(X)). 6. Z-stable C ∗ -algebras In this section we prove Theorem 6.6 and Corollary 6.8. When X is a finite CW complex, it is well known that K∗ (C(X)) is finitely generated. Lemma 6.1. Let C be a C ∗ -algebra of the form p(C(X) ⊗ Mk )p, where X is a finite CW complex and p ∈ C(X) ⊗ Mk is a projection. Let A ∈ T ∪ T . Let L ⊂ U∞ (C) be a finite subset which generates K1 (C) and let ϕ : C → A be a unital monomorphism. For any finite subset F ⊂ C and ε > 0, there exists δ > 0 such that the following holds. If ξ : K1 (C) → K0 (A) satisfies

DA (ξ([w])) < δ for any w ∈ L, then there exists a unitary u ∈ A such that

ϕ(f ), u < ε for every f ∈ F and Bott(ϕ, u)(w) = ξ [w] for every w ∈ L.

820


Proof. When A is in T , this lemma is contained in [18, Lemma 6.11]. Assume A ∈ T . By Theorem 2.5, there exist a unital simple AH algebra B with real rank zero and slow dimension growth and a unital homomorphism ψ : B → A such that K∗ (ψ) gives a graded ordered isomorphism. The tracial simplexes T (A) and T (B) are naturally isomorphic to the state spaces of K0 (A) and K0 (B), respectively. Hence T (ψ) induces an affine isomorphism from T (A) to T (B). It follows from Corollary 2.9 that there exists a unital homomorphism ϕ0 : C → B such that KL(ϕ0 ) = KL(ψ)−1 ◦ KL(ϕ) and T (ϕ0 ) = T (ϕ) ◦ T (ψ)−1 . By Corollary 4.9, ψ ◦ ϕ0 and ϕ are approximately unitarily equivalent. As B is in T , we have already known that the lemma holds for ϕ0 : C → B. Therefore the lemma holds for ψ ◦ ϕ0 : C → A, and hence for ϕ : C → A. 2 Remark 6.2. In the lemma above the finite subset L ⊂ U∞ (C) is allowed to be any finite subset which generates K1 (C), though this point is not clearly mentioned in [18, Lemma 6.11]. This readily follows from the fact that (if F is large enough and ε is small enough, then) Bott(ϕ, u) gives rise to a ‘partial homomorphism’ from K1 (C) to K0 (A), as mentioned in Section 2.3. In what follows, we frequently omit ‘⊗id’, ‘⊗1’ and ‘⊗Tr’ to simplify notation. For example, u ⊗ 1 ∈ A ⊗ Mn is denoted by u. Lemma 6.3. Let C be a C ∗ -algebra of the form p(C(X) ⊗ Mk )p, where X is a finite CW complex and p ∈ C(X) ⊗ Mk is a projection. Let A ∈ T ∪ T . Suppose that unital monomorphisms ϕ, ψ : C → A satisfy KL(ϕ) = KL(ψ), T (ϕ) = T (ψ). Let L ⊂ U∞ (C) be a finite subset which generates K1 (C). For any finite subset F ⊂ C and ε > 0, there exists δ > 0 such that the following holds. If η : K1 (C) → Aff(T (A)) is a homomorphism satisfying η(x) + Im DA = Θϕ,ψ (x),

∀x ∈ K1 (C)

and η [w] < δ,

∀w ∈ L,

then there exists a unitary u ∈ A such that ϕ(f ) − uψ(f )u∗ < ε,

∀f ∈ F

and 1 √ τ log ϕ(w)∗ uψ(w)u∗ = η [w] (τ ), 2π −1

∀τ ∈ T (A), w ∈ L.

Proof. Applying Lemma 6.1 to ψ , F and ε/2, we obtain δ > 0. Suppose that η ∈ Hom(K1 (C), Aff(T (A))) satisfies η(x) + Im DA = Θϕ,ψ (x),

∀x ∈ K1 (C)

and η [w] < δ/2,

∀w ∈ L.


821

Choose a large finite subset F0 ⊂ C and a small real number ε0 > 0. By virtue of Corollary 4.9, there exists a unitary u1 ∈ A such that ϕ(f ) − u1 ψ(f )u∗ < min{ε0 , ε/2}, 1

∀f ∈ F0 ∪ F.

Put ψ = Ad u1 ◦ ψ. For each w ∈ U∞ (C) satisfying ϕ(w) − ψ (w) < 2, the function zw : τ →

1 √

2π −1

τ log ϕ(w)∗ ψ (w)

gives an element of Aff(T (A)). By [11, Lemma 1], we can see the following (see also the proof of Lemma 3.1). • If w1 , w2 ∈ U∞ (C) satisfy ϕ(w1 ) − ψ (w1 ) + ϕ(w2 ) − ψ (w2 ) < 2, then zw1 w2 = zw1 + zw2 . • If w : [0, 1] → U (C ⊗ Mn ) is a path of unitaries satisfying ϕ(w(t)) − ψ (w(t)) < 2, then zw(0) = zw(1) . Therefore, if F0 is large enough and ε0 is small enough, then there exists a homomorphism ζ : K1 (C) → Aff(T (A)) such that ζ [w] (τ ) = zw (τ ) =

1 √ τ log ϕ(w)∗ ψ (w) , 2π −1

∀τ ∈ T (A), w ∈ L.

Clearly we may further assume that ϕ(w) and u1 ψ(w)u∗1 are close enough to imply ζ ([w]) < δ/2 for every w ∈ L. We also have η([w]) − ζ ([w]) ∈ Im DA by Lemma 3.1(2). Hence there exists ξ ∈ Hom(K1 (C), K0 (A)) such that DA (ξ(x)) = η(x)−ζ (x) for any x ∈ K1 (C) and ξ(x) = 0 for any x ∈ Tor(K1 (C)). Moreover one has DA (ξ([w])) < δ/2 + δ/2 = δ. It follows from Lemma 6.1 that there exists a unitary u2 ∈ A such that

ψ(f ), u2 < ε/2,

∀f ∈ F

and Bott(ψ, u2 )(w) = ξ [w] ,

∀w ∈ L.

Set u = u1 u2 . It is straightforward to check that ϕ(f ) − uψ(f )u∗ < ε holds for any f ∈ F . Besides, for any τ ∈ T (A) and w ∈ L, τ log ϕ(w)∗ uψ(w)u∗ = τ log ϕ(w)∗ u1 u2 ψ(w)u∗2 u∗1 = τ log ϕ(w)∗ u1 ψ(w)u∗1 u1 ψ(w)∗ u2 ψ(w)u∗2 u∗1 = τ log ϕ(w)∗ u1 ψ(w)u∗1 + τ log ψ(w)∗ u2 ψ(w)u∗2 √ = 2π −1 ζ [w] (τ ) + DA Bott(ψ, u2 )(w) (τ )

822


√ = 2π −1 ζ [w] (τ ) + DA ξ [w] (τ ) √ = 2π −1η [w] (τ ), where we have used [18, Theorem 3.6].

2

The following lemma is an easy exercise and we leave it to the reader. Lemma 6.4. Let L be a finitely generated abelian group and let M be an abelian group. Let N0 and N1 be subgroups of Q and let N ⊂ Q be the subgroup generated by N0 , N1 . Then for any ξ ∈ Hom(L, M ⊗ N ), there exist ξj ∈ Hom(L, M ⊗ Nj ) such that ξ = ξ1 − ξ0 . For each infinite supernatural number p we let Mp denote the UHF algebra of type p. Let p, q be relatively prime infinite supernatural numbers such that Mp ⊗ Mq ∼ = Q. As in [32], define a C ∗ -algebra Z by Z = f ∈ C [0, 1], Mp ⊗ Mq f (0) ∈ Mp ⊗ C, f (1) ∈ C ⊗ Mq . The following proposition is the main part of the proof of Theorem 6.6. Proposition 6.5. Let X be a connected finite CW complex and let A ∈ C ∪ C . Suppose that two unital monomorphisms ϕ, ψ : C(X) → A satisfy KL(ϕ) = KL(ψ), T (ϕ) = T (ψ) and Im Θϕ,ψ ⊂ Im DA . Then for any finite subset F ⊂ C(X) and ε > 0, there exists a unitary u ∈ A ⊗ Z such that ϕ(f ) ⊗ 1 − u ψ(f ) ⊗ 1 u∗ < ε holds for any f ∈ F . Proof. We write Q = Mp ⊗ Mq , B0 = Mp ⊗ C and B1 = C ⊗ Mq . By Remark 2.3, A ⊗ Q, ¯ ) = ψ(f ) ⊗ 1. We regard ϕ¯ and A ⊗ B0 and A ⊗ B1 are in T ∪ T . Set ϕ(f ¯ ) = ϕ(f ) ⊗ 1 and ψ(f ¯ ψ as homomorphisms from C(X) into A ⊗ Q or A ⊗ Bj . We identify T (A ⊗ Q), T (A ⊗ Bj ) with T (A). In the same way as Lemma 6.3, to simplify notation, for u ∈ A we denote u ⊗ 1 ∈ A ⊗ Mn by u. Similarly, for τ ∈ T (A), τ ⊗ Tr on A ⊗ Mn is written by τ for short. Applying Theorem 5.3 to ψ¯ : C(X) → A ⊗ Q, F and ε/2, we obtain a finite subset L ⊂ K(C(X)), a finite subset G1 ⊂ C(X) and δ1 > 0. By Remark 5.5, we may and do assume that L is written as L = L0 ∪ L1 , where L0 is a finite subset of P (C(X) ⊗ K) and L1 is a finite subset of U∞ (C(X)). We may further assume that L1 generates K1 (C(X)). Since Ki (C(X)) is finitely generated, one can find a finite subset G2 ⊂ C(X) and δ2 > 0 such that the following ¯ ), w] < δ2 for any f ∈ G2 , there exist ξi ∈ holds: For any unitary w ∈ A ⊗ Q satisfying [ψ(f ¯ w)(s) for any s ∈ Li and i = 0, 1 Hom(Ki (C(X)), K1−i (A ⊗ Q)) such that ξi ([s]) = Bott(ψ, [19, Section 2]. We may assume that G2 contains F ∪ G1 and that δ2 is less than min{ε/2, δ1 /2}. By applying Lemma 6.3 to ϕ, ¯ ψ¯ : C(X) → A ⊗ Bj , G2 ⊂ C(X) and δ2 /2, we get δ3,j > 0 for each j = 0, 1. Since K1 (C(X)) is finitely generated and the homomorphism Θϕ,ψ factors through K1 (C(X))/ Tor(K1 (C(X))) by Lemma 3.1(4), there exists η ∈ Hom(K1 (C(X)), Aff(T (A)))


823

such that η(x) + Im DA = Θϕ,ψ (x),

∀x ∈ K1 C(X) .

Moreover, we may assume η([w]) < min{δ3,0 , δ3,1 } for all w ∈ L1 because Im Θϕ,ψ is contained in the closure of Im DA . It follows from Lemma 6.3 that there exists a unitary uj ∈ A ⊗ Bj such that ϕ(f ¯ )u∗j < δ2 /2, ¯ ) − uj ψ(f

∀f ∈ G2

and 1 ∗ ∗ ¯ = η [w] (τ ), uj ψ(w)u ¯ √ τ log ϕ(w) j 2π −1

∀τ ∈ T (A ⊗ Bj ), w ∈ L1 .

¯ ), u∗ u0 ] < δ2 for f ∈ G2 . From the choice of G2 and δ2 , we can find In particular one has [ψ(f 1 ¯ u∗ u0 )(s) holds for any s ∈ Li ξi ∈ Hom(Ki (C(X)), K1−i (A ⊗ Q)) such that ξi ([s]) = Bott(ψ, 1 and i = 0, 1. By [18, Theorem 3.6], ¯ u∗1 u0 (w) (τ ) DA⊗Q ξ1 [w] (τ ) = DA⊗Q Bott ψ, 1 ∗ ∗ ¯ ¯ √ τ log u∗1 u0 ψ(w)u 0 u1 ψ(w) 2π −1 1 ∗ ∗ ∗ ∗ ¯ ¯ = ¯ u0 ψ(w)u u1 ψ(w)u τ log ϕ(w) ¯ √ 0 − τ log ϕ(w) 1 2π −1 = η [w] (τ ) − η [w] (τ ) = 0

=

for any τ ∈ T (A ⊗ Q). Thus Im ξ1 is contained in Ker DA⊗Q . By Lemma 6.4, we can find ξ1,j : K1 (C(X)) → Ker DA⊗Bj such that ξ1 = ξ1,1 − ξ1,0 , where Ker DA⊗C is naturally identified with (Ker DA ) ⊗ K0 (C) for C = Q, B0 , B1 . In the same way, one obtains ξ0,j : K0 (C(X)) → K1 (A ⊗ Bj ) such that ξ0 = ξ0,1 − ξ0,0 . We consider the following exact sequence of C ∗ -algebras: ι π 0 → C0 X × T \ {−1} − → C(X × T) − → C(X) → 0, where π is the evaluation at −1 ∈ T. We write S = C0 (T \ {−1}) for short. Let ρ : C(X) → C(X × T) be the homomorphism defined by ρ(f ) = f ⊗ 1. Then π ◦ ρ is the identity on C(X). This split exact sequence induces the isomorphism (a, b) → KL(ρ)(a) + KL(ι)(b) from K(C(X)) ⊕ K(C(X) ⊗ S) to K(C(X × T)). Let ωi : Ki (C(X) ⊗ S) → K1−i (C(X)) be the canonical isomorphism for each i = 0, 1. For each j = 0, 1, choose κj ∈ KL(C(X) ⊗ S, A ⊗ Bj ) such that Ki (κj ) = ξ1−i,j ◦ ωi . Define κ˜ j ∈ KL(C(X × T), A ⊗ Bj ) by ¯ κ˜ j ◦ KL(ρ) = KL(ψ)

and κ˜ j ◦ KL(ι) = κj .

824


¯ is (strictly) positive and the image of Clearly K0 (κ˜ j ) is unital. Also, K0 (κ˜ j ) ◦ K0 (ρ) = K0 (ψ) K0 (κ˜ j ) ◦ K0 (ι) = K0 (κj ) = ξ1,j ◦ ω0 is contained in Ker DA⊗Bj . It follows from Lemma 2.4 that K0 (κ˜ j ) is unital and (strictly) positive. Thus, κ˜ j is in KL(C(X × T), A ⊗ Bj )+,1 . Let τ0 ∈ T (C(T)) be the tracial state corresponding to the Haar measure on T and define the affine continuous map λ : T (A ⊗ Bj ) → ¯ ) ⊗ τ0 . Thanks to Theorem 2.6, there exists a unital monomorT (C(X × T)) by λ(τ ) = T (ψ)(τ phism σj : C(X × T) → A ⊗ Bj such that KL(σj ) = κ˜ j and T (σj ) = λ. Since KL(σj ◦ ρ) = ¯ ψ¯ and σj ◦ ρ are approximately unitarily equivalent by Theo¯ and T (σj ◦ ρ) = T (ψ), KL(ψ) rem 4.8. Hence there exists a unitary vj ∈ A ⊗ Bj such that

ψ(f ¯ ), vj < δ2 /2,

∀f ∈ G2

and ¯ vj )(s) = K1−i (σj ) ◦ K1−i (ι) ◦ ω−1 [s] Bott(ψ, 1−i −1 [s] = K1−i (κj ) ◦ ω1−i −1 = ξi,j ◦ ω1−i ◦ ω1−i [s] = ξi,j [s] for any s ∈ Li and i = 0, 1. It is easy to see that ϕ(f ¯ )v ∗ u∗ < δ2 /2 + δ2 /2 = δ2 ¯ ) − uj vj ψ(f j j

holds for any f ∈ G2 . In particular one has

ψ(f ¯ ), v ∗ u∗ u0 v0 < 2δ2 < δ1 , 1 1

∀f ∈ G2 .

Besides, when G2 is sufficiently large and δ2 is sufficiently small, we get ¯ v0 ) [s] ¯ v1∗ u∗1 u0 v0 [s] = Bott ψ, ¯ v1∗ [s] + Bott ψ, ¯ u∗1 u0 [s] + Bott(ψ, Bott ψ, = −ξi,1 [s] + ξi [s] + ξi,0 [s] = 0 for any s ∈ Li and i = 0, 1, where we have used [18, (e2.6)]. Therefore, by Theorem 5.3, we can find a path of unitaries w : [0, 1] → A ⊗ Q such that w(0) = v1∗ u∗1 u0 v0 , w(1) = 1 and

ψ(f ¯ ), w(t) < ε/2,

∀f ∈ F, t ∈ [0, 1].

Define a unitary U ∈ Z by U (t) = u1 v1 w(t). It is easy to see that ϕ(f ) ⊗ 1 − U ψ(f ) ⊗ 1 U ∗ < ε/2 + δ2 < ε holds for any f ∈ F .

2


825

Theorem 6.6. Let X be a compact metrizable space and let A ∈ C ∪ C . For unital monomorphisms ϕ, ψ : C(X) → A, the following two conditions are equivalent. (1) ϕ and ψ are approximately unitarily equivalent. (2) KL(ϕ) = KL(ψ), τ ◦ ϕ = τ ◦ ψ for any τ ∈ T (A) and Im Θϕ,ψ ⊂ Im DA . Proof. It is straightforward to check that (1) implies (2). Indeed, KL(ϕ) = KL(ψ) and T (ϕ) = T (ψ) are clear. By Lemma 3.1, Im Θϕ,ψ ⊂ Im DA also follows. We would like to show the other implication. We first consider the case that X is a connected finite CW complex. Let ϕ, ψ : C(X) → A be unital monomorphisms satisfying KL(ϕ) = KL(ψ), T (ϕ) = T (ψ) and Im Θϕ,ψ ⊂ Im DA . We may replace the target algebra A with A ⊗ Z, because A is Z-absorbing. Since Z is strongly self-absorbing [34], there exists an approximately inner endomorphism π : A ⊗ Z → A ⊗ Z such that π(A ⊗ Z) = A ⊗ C. Hence ϕ and π ◦ ϕ (resp. ψ and π ◦ ψ ) are approximately unitarily equivalent. By [32, Proposition 3.3], the C ∗ -algebra Z embeds unitally into Z. It follows from Proposition 6.5 that π ◦ ϕ and π ◦ ψ are approximately unitarily equivalent. Therefore ϕ and ψ are approximately unitarily equivalent. A general finite CW complex is a finite union of pairwise disjoint connected finite CW complexes. Since A has cancellation by [31, Theorem 6.7] and eAe is in C ∪ C for any non-zero projection e ∈ A, the conclusion follows from the previous case. Let X be a compact metrizable space. Let {f1 , f2 , . . . , fn } be a finite subset of C(X) and let ε > 0. By [23, Lemma 1], there exist a finite CW complex (actually a finite simplicial complex) Y , a finite subset {g1 , g2 , . . . , gn } of C(Y ) and a unital monomorphism σ : C(Y ) → C(X) such that fi − σ (gi ) < ε/3 for any i = 1, 2, . . . , n. Clearly KL(ϕ ◦ σ ) = KL(ψ ◦ σ ), T (ϕ ◦ σ ) = T (ψ ◦ σ ) and Im Θϕ◦σ,ψ◦σ is contained in the closure of Im DA . It follows from the argument above that ϕ ◦ σ and ψ ◦ σ are approximately unitarily equivalent. Hence there exists a unitary u ∈ A such that ϕ(σ (gi )) − uψ(σ (gi ))u∗ < ε/3, which implies ϕ(fi ) − uψ(fi )u∗ < ε. Thus, ϕ and ψ are approximately unitarily equivalent. 2 Remark 6.7. In the theorem above, if A has real rank zero, then the image of DA is dense in Aff(T (A)). Hence the condition Im Θϕ,ψ ⊂ Im DA is trivially satisfied. Corollary 6.8. Let C be a unital AH algebra and let A ∈ C ∪ C . For unital monomorphisms ϕ, ψ : C → A, the following two conditions are equivalent. (1) ϕ and ψ are approximately unitarily equivalent. (2) KL(ϕ) = KL(ψ), τ ◦ ϕ = τ ◦ ψ for any τ ∈ T (A) and Im Θϕ,ψ ⊂ Im DA . Proof. We can prove this in the same way as Corollary 4.9.

2

7. Homomorphisms between simple Z-stable C ∗ -algebras In this section we prove Theorem 7.1. The main idea is almost the same as Proposition 6.5 and Theorem 6.6. The proof is, however, somewhat lengthy because we must work with finitely generated subgroups of K∗ (C) so as to use Lemma 6.4. Theorem 7.1. Let C be a nuclear C ∗ -algebra in C satisfying the UCT and let A ∈ C ∪ C . For any unital homomorphisms ϕ, ψ : C → A, the following are equivalent.

826


(1) ϕ and ψ are approximately unitarily equivalent. (2) KL(ϕ) = KL(ψ) and Im Θϕ,ψ ⊂ Im DA . Proof. The implication (1) ⇒ (2) is trivial. We would like to show the other implication. Note that KL(ϕ) = KL(ψ) implies T (ϕ) = T (ψ), because projections of C separate traces. Since Z is strongly self-absorbing [34] and C (resp. A) is Z-stable by assumption, there exists an isomorphisms πC : C → C ⊗ Z (resp. πA : A → A ⊗ Z) which is approximately unitarily equivalent to the unital monomorphism c → c ⊗ 1. It is not so hard to see that any unital homomorphism ϕ : C → A is approximately unitarily equivalent to πA−1 ◦ (ϕ ⊗ id) ◦ πC . Hence, for given homomorphisms ϕ, ψ satisfying KL(ϕ) = KL(ψ) and Im Θϕ,ψ ⊂ Im DA , it suffices to show that ϕ ⊗ id : C ⊗ Z → A ⊗ Z is approximately unitarily equivalent to ψ ⊗ id : C ⊗ Z → A ⊗ Z. Suppose that we are given a finite subset F ⊂ C ⊗ Z and ε > 0. We would like to show ϕ ⊗ id ∼F,ε ψ ⊗ id. Without loss of generality, we may assume that F is contained in C ⊗ C. Let Z, B0 , B1 , Q be as in Proposition 6.5. We identify T (A ⊗ Q), T (A ⊗ Bj ) with T (A). Put ϕ¯ = ϕ ⊗ id, ψ¯ = ψ ⊗ id. As in the proof of Proposition 6.5, we omit ‘⊗C’, ‘⊗1’ and ‘⊗ Tr’ to simplify notation. By the classification theorem in [16], C ⊗ Q is a unital simple AT algebra with real rank zero.

Thus, C ⊗Q can be written as an inductive limit of C ∗ -algebras of the form ni=1 Mki (C(T)). By using Theorem 5.4 to ψ¯ : C ⊗ Q → A ⊗ Q, F and ε/2, we obtain a finite subset L ⊂ K(C ⊗ Q), a finite subset G1 ⊂ C ⊗ Q and δ1 > 0. By Remark 5.5, we may and do assume that L is written as L = L0 ∪ L1 , where L0 is a finite subset of P (C ⊗ Q ⊗ K) and L1 is a finite subset of U∞ (C ⊗ Q). We may further assume that L0 and L1 are finite subsets of P (C ⊗ K) and U∞ (C) respectively, because Bott gives rise to a ‘partial homomorphism’ (see Section 2.3). Let Hi ⊂ Ki (C) be the subgroup generated by Li . Since Hi is finitely generated, one can find a finite subset G2 ⊂ C ⊗ Q and δ2 > 0 such that the following holds: For any unitary w ∈ A ⊗ Q ¯ satisfying [ψ(c), w] < δ2 for any c ∈ G2 , there exist ξi ∈ Hom(Hi , K1−i (A ⊗ Q)) such that ¯ ξi ([s]) = Bott(ψ, w)(s) for any s ∈ Li and i = 0, 1 [19, Section 2]. We may assume that G2 contains G1 and that δ2 is less than δ1 . As C ⊗ Q is generated by C, B0 and B1 , one may choose finite subsets G3 ⊂ C, G3,0 ⊂ B0 , G3,1 ⊂ B1 and δ3 > 0 so that if a unitary w ∈ A ⊗ Q satisfies ¯ ¯ w] < δ2 holds for all c ∈ G2 .

[ψ(c), w] < δ3 for every c ∈ G3 ∪ G3,0 ∪ G3,1 , then [ψ(c), We assume that G3 contains F and that δ3 is less than ε. For each j = 0, 1, by the classification theorem in [16], C ⊗ Bj is a unital simple AH algebra with real rank zero and slow dimension growth, and so one can find a unital subalgebra Cj ⊂ C ⊗ Bj such that the following hold. • Cj is a finite direct sum of C ∗ -algebras of the form p(C(X) ⊗ Mk )p, where X is a connected finite CW complex (with dimension at most three) and p ∈ C(X) ⊗ Mk is a projection. • There exists a finite subset G3,j ⊂ Cj such that any elements of G3 ∪ G3,j are within distance δ3 /12 of G3,j . • There exist finite subsets L0,j ∈ P (Cj ⊗ K) and L1,j ⊂ U∞ (Cj ) such that any elements of Li are within distance 1/2 of Li,j for each i = 0, 1. Let Li s → sj ∈ Li,j be a map such that s − sj < 1/2. We further require that L1,j generates K1 (Cj ).


827

Let γj : Cj → C ⊗ Bj denote the embedding map. For each i = 0, 1, we choose a finitely generated subgroup Hi ⊂ Ki (C) so that Hi is contained in Hi and Im Ki (γj ) is contained in Hi ⊗ Ki (Bj ) for each j = 0, 1. Applying Lemma 6.3 to ϕ¯ ◦ γj : Cj → A ⊗ Bj , ψ¯ ◦ γj : Cj → A ⊗ Bj , G3,j and δ3 /12, we get δ4,j > 0 for each j = 0, 1. As in the proof of Proposition 6.5, we can find a homomorphism η : H1 → Aff(T (A)) such that ∀x ∈ H1

η(x) + Im DA = Θϕ,ψ (x), and

η˜ j γj (w) < δ4,j ,

∀w ∈ L1,j , j = 0, 1,

where η˜ j ∈ Hom(H1 ⊗ K0 (Bj ), Aff(T (A))) denotes the homomorphism induced from η. It follows from Lemma 6.3 that there exists a unitary uj ∈ A ⊗ Bj such that ∗ ϕ(c) ¯ ¯ − uj ψ(c)u j < δ3 /12,

∀c ∈ G3,j

and 1 √

2π −1

∗ ∗ ¯ = η˜ j γj (w) (τ ), uj ψ(w)u τ log ϕ(w) ¯ j

∀τ ∈ T (A ⊗ Bj ), w ∈ L1,j .

∗ is ¯ By choosing G3,j large enough in advance, we may also assume that ϕ(w) ¯ − uj ψ(w)u j ∗ < δ /4 ¯ less than 1/2 for every w ∈ L1,j . From the choice of G3,j , we obtain ϕ(c) ¯ − uj ψ(c)u 3 j ¯ for all c ∈ G3 ∪ G3,j . If c is in G3,1−j , then uj ∈ A ⊗ Bj commutes with ψ(c) ∈ B1−j , and so ∗ . Therefore ¯ ¯ ϕ(c) ¯ = ψ(c) = uj ψ(c)u j

ψ(c), ¯ u∗ u0 < δ3 /2, 1

∀c ∈ G3 ∪ G3,0 ∪ G3,1 .

From the choice of G3 , G3,0 , G3,1 and δ3 , one has

ψ(c), ¯ u∗ u0 < δ2 , 1

∀c ∈ G2 .

Then, from the choice of G2 and δ2 , we can find ξi ∈ Hom(Hi , K1−i (A ⊗ Q)) such that ξi ([s]) = ¯ u∗ u0 )(s) holds for any s ∈ Li and i = 0, 1. By [18, Theorem 3.6], Bott(ψ, 1 ¯ u∗1 u0 (w) (τ ) DA⊗Q ξ1 [w] (τ ) = DA⊗Q Bott ψ, 1 ∗ ∗ ¯ ¯ √ τ log u∗1 u0 ψ(w)u 0 u1 ψ(w) 2π −1 1 ∗ ∗ ∗ ∗ ¯ ¯ u0 ψ(w)u u1 ψ(w)u ¯ = τ log ϕ(w) ¯ √ 0 − τ log ϕ(w) 1 2π −1 ∗ ∗ 1 = τ log ϕ¯ w0 u0 ψ¯ w0 u∗0 − τ log ϕ¯ w1 u1 ψ¯ w1 u∗1 √ 2π −1

=

828


= η˜ 0 w0 (τ ) − η˜ 1 w1 (τ ) = η˜ 0 [w] (τ ) − η˜ 1 [w] (τ ) = η [w] (τ ) − η [w] (τ ) = 0 for any w ∈ L1 and τ ∈ T (A ⊗ Q). Thus Im ξ1 is contained in Ker DA⊗Q . Since Ker DA⊗Q = (Ker DA ) ⊗ Q is divisible, ξ1 extends to a homomorphism ξ1 : H1 → Ker DA⊗Q . Likewise ξ0 extends to a homomorphism ξ0 : H0 → K1 (A ⊗ Q) = K1 (A) ⊗ Q. By Lemma 6.4, we can find ξ1,j : H1 → Ker DA⊗Bj such that ξ1 = ξ1,1 − ξ1,0 . We let ξ˜1,j : H1 ⊗ K0 (Bj ) → Ker DA⊗Bj denote the homomorphism induced from ξ1,j . In the same way, one obtains ξ0,j : H0 → K1 (A ⊗ Bj ) such that ξ0 = ξ0,1 − ξ0,0 . We let ξ˜0,j : H0 ⊗ K0 (Bj ) → K1 (A ⊗ Bj ) denote the homomorphism induced from ξ0,j . In the same way as in the proof of Proposition 6.5, for each j = 0, 1, we consider the following exact sequence of C ∗ -algebras: ιj πj 0 → Cj ⊗ C0 T \ {−1} − → Cj ⊗ C(T) −→ Cj → 0, where πj is the evaluation at −1 ∈ T. We write S = C0 (T \ {−1}) for short. Let ρj : Cj → Cj ⊗ C(T) be the homomorphism defined by ρj (c) = c ⊗ 1. Then πj ◦ ρj is the identity on Cj . This split exact sequence induces the isomorphism (a, b) → KL(ρj )(a) + KL(ιj )(b) from K(Cj )⊕K(Cj ⊗S) to K(Cj ⊗C(T)). Let ωi,j : Ki (Cj ⊗S) → K1−i (Cj ) be the canonical isomorphism for each i, j = 0, 1. For each j = 0, 1, choose κj ∈ KL(Cj ⊗ S, A ⊗ Bj ) so that Ki (κj ) = ξ˜1−i,j ◦ K1−i (γj ) ◦ ωi,j ,

∀i = 0, 1.

Notice that the composition of ξ˜1−i,j and K1−i (γj ) is well defined, because Im K1−i (γj ) is ⊗ K0 (Bj ). Define κ˜ j ∈ KL(Cj ⊗ C(T), A ⊗ Bj ) by contained in H1−i κ˜ j ◦ KL(ρj ) = KL(ψ¯ ◦ γj )

and κ˜ j ◦ KL(ιj ) = κj .

Clearly K0 (κ˜ j ) is unital. Also for any x ∈ K0 (Cj ⊗ C(T))+ \ {0}, one has K0 (πj )(x) ∈ K0 (Cj )+ \ {0}, and so τ (K0 (ψ¯ ◦ γj ◦ πj )(x)) > 0 for every τ ∈ T (A ⊗ Bj ). Since the image of ξ˜1,j is contained in the kernel of DA⊗Bj , we obtain τ K0 (κ˜ j )(x) = τ K0 (ψ¯ ◦ γj ◦ πj )(x) > 0, which entails K0 (κ˜ j )(x) ∈ K0 (A ⊗ Bj )+ \ {0}. It thus follows that K0 (κ˜ j ) is unital and strictly positive, and hence κ˜ j is in KL(Cj ⊗ C(T), A ⊗ Bj )+,1 . Let τ0 ∈ T (C(T)) be the tracial state corresponding to the Haar measure on T and define the affine continuous map λj : T (A ⊗ Bj ) → T (Cj ⊗ C(T)) by λj (τ ) = T (ψ¯ ◦ γj )(τ ) ⊗ τ0 . For each minimal central projection p ⊗ 1 ∈ Cj ⊗ C(T) and τ ∈ T (A ⊗ Bj ), it is easy to verify

τ K0 (κ˜ j ) [p ⊗ 1] = τ K0 (ψ¯ ◦ γj ) πj (p ⊗ 1) = τ ψ¯ γj (p) = λj (τ )(p ⊗ 1).


829

Hence the hypotheses of Corollary 2.9 are satisfied. Thanks to Corollary 2.9, there exists a unital monomorphism σj : Cj ⊗ C(T) → A ⊗ Bj such that KL(σj ) = κ˜ j and T (σj ) = λj . Since KL(σj ◦ ρj ) = KL(ψ¯ ◦ γj ) and T (σj ◦ ρj ) = T (ψ¯ ◦ γj ), Corollary 4.9 implies that ψ¯ ◦ γj and σj ◦ ρj are approximately unitarily equivalent. Hence there exists a unitary vj ∈ A ⊗ Bj such that ψ¯ γj (c) , vj < δ3 /12,

∀c ∈ G3,j

and −1 [s] Bott(ψ¯ ◦ γj , vj )(s) = K1−i (σj ) ◦ K1−i (ιj ) ◦ ω1−i,j −1 = K1−i (κj ) ◦ ω1−i,j [s] −1 = ξ˜i,j ◦ Ki (γj ) ◦ ω1−i,j ◦ ω1−i,j [s]

= ξ˜i,j γj (s) for any s ∈ Li,j and i = 0, 1. As before,

ψ(c), ¯ vj < δ3 /4 holds for any c ∈ G3 ∪ G3,0 ∪ G3,1 . By choosing G3,j large enough and δ3 small enough in ¯ vj )(s) = Bott(ψ¯ ◦ γj , vj )(s ) for any s ∈ Li and i = 0, 1. advance, we have Bott(ψ, j It is easy to see that ∗ ∗ ϕ(c) ¯ ¯ − uj vj ψ(c)v j uj < δ3 /4 + δ3 /4 = δ3 /2 holds for any c ∈ G3 ∪ G3,0 ∪ G3,1 . In particular one has

ψ(c), ¯ v ∗ u∗ u0 v0 < δ3 , 1 1

∀c ∈ G3 ∪ G3,0 ∪ G3,1 ,

and hence

ψ(c), ¯ v ∗ u∗ u0 v0 < δ2 < δ1 , 1 1

∀c ∈ G1 ,

because G1 is contained in G2 . Besides, when G3 is sufficiently large and δ3 is sufficiently small, we get ¯ v1∗ [s] + Bott ψ, ¯ u∗1 u0 [s] + Bott(ψ, ¯ v0 ) [s] ¯ v1∗ u∗1 u0 v0 [s] = Bott ψ, Bott ψ, = −ξ˜i,1 s1 + ξi [s] + ξ˜i,0 s0 = −ξi,1 [s] + ξi [s] + ξi,0 [s] = 0 for any s ∈ Li and i = 0, 1, where we have used [18, (e2.6)]. Therefore, by Theorem 5.4, we can

830


find a path of unitaries w : [0, 1] → A ⊗ Q such that w(0) = v1∗ u∗1 u0 v0 , w(1) = 1 and

ψ(c), ¯ w(t) < ε/2,

∀c ∈ F, t ∈ [0, 1].

Define a unitary U ∈ Z by U (t) = u1 v1 w(t). It is easy to see that ϕ(c) ⊗ 1 − U ψ(c) ⊗ 1 U ∗ < ε/2 + δ3 /2 < ε holds for any c ∈ F .

2

Remark 7.2. In the theorem above, if A has real rank zero, then the image of DA is dense in Aff(T (A)). Hence the condition Im Θϕ,ψ ⊂ Im DA is trivially satisfied. Acknowledgments I would like to thank Huaxin Lin for valuable comments. I also like to thank the referee for a number of helpful comments. References [1] B. Blackadar, K-Theory for Operator Algebras, Math. Sci. Res. Inst. Publ., vol. 5, Cambridge University Press, Cambridge, 1998. [2] B. Blackadar, A. Kumjian, M. Rørdam, Approximately central matrix units and the structure of non-commutative tori, K-Theory 6 (1992) 267–284. [3] L.G. Brown, R.G. Douglas, P.A. Fillmore, Extensions of C ∗ -algebras and K-homology, Ann. of Math. 105 (1977) 265–324. [4] M. Dadarlat, Approximately unitarily equivalent morphisms and inductive limit C ∗ -algebras, K-Theory 9 (1995) 117–137. [5] M. Dadarlat, T.A. Loring, A universal multicoefficient theorem for the Kasparov groups, Duke Math. J. 84 (1996) 355–377. [6] G.A. Elliott, On the classification of C ∗ -algebras of real rank zero, J. Reine Angew. Math. 443 (1993) 179–219. [7] G. Gong, H. Lin, Classification of homomorphisms from C(X) to simple C ∗ -algebras of real rank zero, Acta Math. Sin. (Engl. Ser.) 16 (2000) 181–206. [8] G. Gong, H. Lin, Almost multiplicative morphisms and K-theory, Internat. J. Math. 11 (2000) 983–1000. [9] G. Gong, X. Jiang, H. Su, Obstructions to Z-stability for unital simple C ∗ -algebras, Canad. Math. Bull. 43 (2000) 418–426. [10] U. Haagerup, Every quasitrace on an exact C ∗ -algebra is a trace, preprint, 1991. [11] P. de la Harpe, G. Skandalis, Déterminant associé à une trace sur une algébre de Banach, Ann. Inst. Fourier (Grenoble) 34 (1984) 241–260. [12] X. Jiang, H. Su, On a simple unital projectionless C ∗ -algebra, Amer. J. Math. 121 (1999) 359–413. [13] A. Kishimoto, A. Kumjian, The Ext class of an approximately inner automorphism, II, J. Operator Theory 46 (2001) 99–122. [14] H. Lin, An Introduction to the Classification of Amenable C ∗ -Algebras, World Scientific Publishing Co., River Edge, NJ, 2001. [15] H. Lin, Embedding an AH-algebra into a simple C ∗ -algebra with prescribed KK-data, K-Theory 24 (2001) 135– 156. [16] H. Lin, Classification of simple C ∗ -algebras with tracial topological rank zero, Duke Math. J. 125 (2004) 91–119. [17] H. Lin, Classification of homomorphisms and dynamical systems, Trans. Amer. Math. Soc. 359 (2007) 859–895, arXiv:math/0404018. [18] H. Lin, Asymptotically unitary equivalence and asymptotically inner automorphisms, Amer. J. Math. 131 (2009) 1589–1677, arXiv:math/0703610. [19] H. Lin, Approximate homotopy of homomorphisms from C(X) into a simple C ∗ -algebra, Mem. Amer. Math. Soc. 205 (963) (2010), arXiv:math/0612125.


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[20] H. Lin, Approximate unitary equivalence in simple C ∗ -algebras of tracial rank one, preprint, arXiv:0801.2929. [21] H. Lin, Z. Niu, Lifting KK-elements, asymptotical unitary equivalence and classification of simple C ∗ -algebras, Adv. Math. 219 (2008) 1729–1769, arXiv:0802.1484. [22] T.A. Loring, K-theory and asymptotically commuting matrices, Canad. J. Math. 40 (1988) 197–216. [23] S. Mardešić, On covering dimension and inverse limits of compact spaces, Illinois J. Math. 4 (1960) 278–291. [24] H. Matui, Y. Sato, Z-stability of crossed products by strongly outer actions, preprint, arXiv:0912.4804. [25] K.E. Nielsen, K. Thomsen, Limits of circle algebras, Expo. Math. 14 (1996) 17–56. [26] P.W. Ng, W. Winter, Commutative C ∗ -subalgebras of simple stably finite C ∗ -algebras with real rank zero, Indiana Univ. Math. J. 57 (2008) 3209–3239, arXiv:math/0701035. [27] N.C. Phillips, A classification theorem for nuclear purely infinite simple C ∗ -algebras, Doc. Math. 5 (2000) 49–114. [28] M. Rørdam, On the structure of simple C ∗ -algebras tensored with a UHF-algebra, J. Funct. Anal. 100 (1991) 1–17. [29] M. Rørdam, On the structure of simple C ∗ -algebras tensored with a UHF-algebra, II, J. Funct. Anal. 107 (1992) 255–269. [30] M. Rørdam, Classification of nuclear, simple C ∗ -algebras, in: Classification of Nuclear C ∗ -Algebras. Entropy in Operator Algebras, in: Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. [31] M. Rørdam, The stable and the real rank of Z-absorbing C ∗ -algebras, Internat. J. Math. 15 (2004) 1065–1084, arXiv:math/0408020. [32] M. Rørdam, W. Winter, The Jiang–Su algebra revisited, J. Reine Angew. Math. 642 (2010) 129–155, arXiv:0801.2259. [33] C. Schochet, Topological methods for C ∗ -algebras, IV, Mod p homology, Pacific J. Math. 114 (1984) 447–468. [34] A.S. Toms, W. Winter, Strongly self-absorbing C ∗ -algebras, Trans. Amer. Math. Soc. 359 (2007) 3999–4029, arXiv:math/0502211. [35] W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, preprint, arXiv:0708.0283.


Some results on the controllability of forward stochastic heat equations with control on the drift ✩ Qi Lü a,b,∗ a School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China b School of Mathematics, Sichuan University, Chengdu 610064, China

Received 2 June 2010; accepted 24 October 2010 Available online 4 November 2010 Communicated by J. Coron

Abstract In this paper, we establish the null/approximate controllability for forward stochastic heat equations with control on the drift. The null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators. As a consequence of the null controllability, we obtain the observability estimate for backward stochastic heat equations, which leads to a unique continuation property for backward stochastic heat equations, and hence the desired approximate controllability for forward stochastic heat equations. It deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability. This is a new phenomenon arising in the study of the controllability problem for stochastic heat equations. © 2010 Elsevier Inc. All rights reserved. Keywords: Stochastic heat equation; Null controllability; Approximate controllability

✩ This work is partially supported by National Basic Research Program of China (973 Program) under grant 2011CB808000 and the NSF of China under grants 10831007 and 60974035. This paper is an improved version of one chapter of the author’s PhD thesis [10] accomplished at Sichuan University under the guidance of Professor Xu Zhang. The author would like to take this opportunity to thank him deeply for his help. * Address for correspondence: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China. E-mail address: [email protected].


Q. Lü / Journal of Functional Analysis 260 (2011) 832–851

833

1. Introduction Let T > 0, G ⊂ Rn be a given bounded domain with a C 2 boundary ∂G, and G0 a given nonempty open subset of G. Denote by χG0 the characteristic function of G0 . Put Q (0, T )×G and Σ (0, T ) × ∂G. Let (Ω, F , {Ft }t0 , P ) be a complete filtered probability space on which a one-dimensional standard Brownian motion {w(t)}t0 is defined so that {Ft }t0 is its natural filtration augmented by all the P -null sets. Let H be a Banach space. We denote by L2F (0, T ; H ) the Banach space consisting of all H -valued {Ft }t0 -adapted processes X(·) such that E(|X(·)|2L2 (0,T ;H ) ) < ∞,

with the canonical norm; by LrF (0, T ; L2 (Ω; H )) the Banach space consisting of all H -valued {Ft }t0 -adapted processes X such that |E|X|2H |Lr (0,T ) < ∞ (1 r ∞), with the canonical norm. Put L∞ F (0, T ; H ) the Banach space consisting of all H -valued {Ft }t0 -adapted bounded processes. Denote by L2F (Ω; C([0, T ]; H )) the Banach space consisting of all H -valued {Ft }t0 -adapted processes X(·) such that E(|X(·)|2C(0,T ;H ) ) < ∞, with the canonical norm. Let a ij ∈ C 1 (G) (i, j = 1, 2, . . . , n) satisfy a ij = a j i and for some constant μ > 0, n

a ij ξi ξj μ|ξ |2 ,

∀(x, ξ ) ∈ G × Rn .

i,j =1

This paper is devoted to a study of the null/approximate controllability for the following stochastic heat equation ⎧ n ⎪ ij ⎪ ⎪ dy − a yxi x dt = a(t)y dw + χE χG0 f dt ⎪ ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij yxi ν j + ly = 0 ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ y(0) = y0

in Q, (1.1) on Σ, in G,

where a(t) ∈ L∞ F (0, T ; R), E is a measurable subset in (0, T ) with a positive Lebesgue measure (i.e., m(E) > 0), χE is the characteristic function of E, ν = (ν 1 , ν 2 , . . . , ν n ) = ν(x) is the unit outward normal vector of G at x ∈ ∂G, both l and l˜ belong to L∞ (∂G) and satisfy either l˜ = 1 and l 0 or l˜ = 0 and l > 0, y0 ∈ L2 (Ω, F0 , P ; L2 (G)), the control f belongs to 2 2 L∞ F (0, T ; L (Ω; L (G))). We refer to [2, Chapter 6] for the well-posedness of system (1.1) in 1

the class y ∈ L2F (Ω; C([0, T ]; L2 (G))) ∩ L2 (0, T ; D(A 2 )). The equation we study here is a particular case of stochastic heat equations since a(·) is independent of x. The way to manage the general potential that depends both on t and x is unknown. Put τ = |a|2L∞ (0,T ;R) . Throughout this paper, we will use C to denote a generic positive conF stant depending only on G, G0 , T , (a ij )n×n , l, l˜ and τ , which may change from one place to another. In this paper, we will prove the following theorem on the null controllability of system (1.1).

834


Theorem 1.1. System (1.1) is null controllable at time T , i.e., for each initial datum y0 ∈ 2 2 L2 (Ω, F0 , P ; L2 (G)), there is a control f ∈ L∞ F (0, T ; L (Ω; L (G))) such that the solution y of system (1.1) satisfies y(T ) = 0 in G, P-a.s. Moreover, the control f satisfies the following estimate: |f |2L∞ (0,T ;L2 (Ω;L2 (G))) CE|y0 |2L2 (G) . F

(1.2)

System (1.1) is said to be approximately controllable at time T if for any initial datum y0 ∈ L2 (Ω, F0 , P ; L2 (G)), any final state y1 ∈ L2 (Ω, FT , P ; L2 (G)) and any ε > 0, there exists a control f ∈ L2F (0, T ; L2 (G)) such that the solution of system (1.1) with initial datum y0 and control f satisfies |y(T ) − y1 |L2 (Ω,FT ,P ;L2 (G)) ε. Starting from Theorem 1.1, we will show the following approximate controllability result for system (1.1) under a little stronger assumption on the controller than that for the null controllability: Theorem 1.2. System (1.1) is approximately controllable at time T if and only if m((s, T ) ∩ E) > 0 for any s ∈ [0, T ). Remark 1.1. It seems that Theorem 1.2 is unreasonable at the first glance. If a ≡ 0, then system (1.1) is like a deterministic heat equation with a random parameter. The readers may guess that one can obtain the approximate controllability by only assuming m((0, T ) ∩ E) > 0. However, this is untrue. The reason for this comes from our definition of the approximate controllability for system (1.1). We want any element belongs to L2 (Ω, FT , P ; L2 (G)) other than L2 (Ω, Fs , P ; L2 (G)) (s < T ) can be gained on as close as one wants. Hence we need to put control act until the time T . The more details can be found in the proof of Theorem 1.2. There are many studies on the controllability of deterministic parabolic equations (e.g. [4,5,7,16,17]). However, very little is known for the stochastic counterpart. To the best of our knowledge, one can find only a very few papers concerned with the controllability problems for stochastic parabolic equations. In [3], the authors announced an approximate controllability result for linear forward stochastic parabolic equations with time-invariant coefficients (it seems that the detailed proof of the result in [3] has never been published). In [1] and [14], the null controllability of both linear forward and backward stochastic parabolic equations was studied. Note however that, in [1], only a reachable set was presented for some linear forward stochastic parabolic equations; while in [14], the authors needed to introduce two controls (one is put on the drift term and the other on the diffusion term) to establish the null controllability result for general linear forward stochastic parabolic equations. In [13], the authors proved that the null controllability of a general class of stochastic parabolic equations can be reduced to suitable deterministic partial differential equations by simple computations on the related Riccati equations. Based on their result, the null controllability of system (1.1) can be established when a is deterministic. Note that the dual system of system (1.1) is a backward stochastic heat equation. As remarked in [1], it is very hard to establish the observability estimate for this system with only one observer. In [14], the authors introduced two observers to overcome this difficulty, and therefore, they needed to use two controls to achieve the desired null controllability result. The system considered in this paper is simpler than that in [14], but the advantage is that we need to introduce only one control into the system. Moreover, our control is chosen to belong to a small space, i.e., 2 2 L∞ F (0, T ; L (Ω; L (G))), and also we only put control in the measurable subset E. To do this,


835

we need to borrow some ideas developed in [8] and [16]. As far as we know, Theorem 1.1 is the first null controllability result for forward stochastic parabolic equations with only one control. It would be quite interesting to derive a similar controllability result for linear forward stochastic parabolic equations with general (both time- and space-variant) coefficients or for some nonlinear stochastic parabolic equations but these seem to be challenging open problems. In some sense, it is surprising that we need a little more assumption in Theorem 1.2 for the approximate controllability of system (1.1) than that in Theorem 1.1 for the null controllability. Indeed, it is well known that in the deterministic setting, the null controllability is usually stronger than the approximate controllability. But this does not remain to be true in the stochastic case. Indeed, from Theorem 1.2, we see that the additional condition (compared to the null controllability) that m((s, T ) ∩ E) > 0 for any s ∈ [0, T ) is not only sufficient but also necessary for the approximate controllability of system (1.1). Therefore, for stochastic heat equations, the null controllability does NOT imply the approximate controllability. This indicates that there exists some essential difference between the controllability theory between deterministic heat equations and stochastic heat equations. The rest of the paper is organized as follows. In Section 2, we show some preliminary results. In Section 3, we will prove Theorem 1.1. In Section 4, we will prove Theorem 1.2. 2. Preliminaries In this section, we collect some preliminary results that will be used subsequently. Firstly, we recall the following known and useful property about Lebesgue measurable sets. Lemma 2.1. (See [9, pp. 256–257].) For almost all t˜ ∈ E, there exists a sequence of numbers {ti }∞ i=1 ⊂ (0, T ) such that t1 < t2 < · · · < ti < ti+1 < · · · < t˜, ti → t˜ as i → ∞, m E ∩ [ti , ti+1 ] ρ(ti+1 − ti ), i = 1, 2, . . . , ti+1 − ti C0 , ti+2 − ti+1

i = 1, 2, . . . ,

(2.1) (2.2) (2.3)

where ρ and C0 are two positive constants which are independent of i. Nextly, let A be an unbounded operator on L2 (G) as follows ⎧ n

⎪

⎪ ⎪ D(A) = u ∈ H 2 (G) l˜ a ij uxi ν j + lu = 0 on ∂G , ⎪ ⎪ ⎨ i,j =1 n ⎪ ⎪ ij ⎪ ⎪ a uxi x , Au = − ⎪ ⎩ j

(2.4)

∀u ∈ D(A).

i,j =1

∞ Let {λi }∞ i=1 be the eigenvalues of A, and {ei }i=1 be the corresponding eigenfunctions satisfying |ei |L2 (G) = 1, i = 1, 2, 3, . . . . We recall the following explicit observability estimate (for partial sums of the eigenfunctions of A), established in [10] (we refer to [7,8] for a special case of this result).

836


Lemma 2.2. There exist two positive constants C1 and C2 such that

|ai |2 C1 eC2

√

λi r

r

2

dx a e (x) i i

G0

(2.5)

λi r

for every finite r > 0 and every choice of the coefficients {ai }λi r with ai ∈ C. Further, we need to introduce the following backward stochastic heat equation: ⎧ n ⎪ ij ⎪ ⎪ dz + a zxi x dt = −a(t)Z dt + Z dw ⎪ ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij zxi ν j + lz = 0 ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎪ ⎩ z(T ) = η

in Q, (2.6) on Σ, in G.

For any terminal datum η ∈ L2 (Ω, FT , P ; L2 (G)), according to the well-posedness result for backward stochastic parabolic equations (e.g., [6,12,15]), Eq. (2.6) admits one and only one solution (z, Z) ∈ (L2F (Ω; C([0, T ]; L2 (G))) ∩ L2F (0, T ; H01 (G))) × L2F (0, T ; L2 (Ω)). For each r > 0, we set Xr = span{ei (x)}λi r and denote by Pr the orthogonal projection from L2 (G) to Xr . We need to derive some observation results for system (2.6) with the final state belonging to Xr . The desired observation results, with an explicit estimate on the cost of the observation, can be stated as follows. Proposition 2.1. For each r λ1 , the solution of Eq. (2.6) with η ∈ L2 (Ω, FT , P ; Xr ) satisfies: i) If 2λ1 > τ , then

2

E z(0) 2 L

√

2 C1 eC2 r

χE (t)χG0 z L1 (0,T ;L2 (Ω;L2 (G))) ; (G) 2 F (m(E))

(2.7)

ii) For the general case, it holds that

2

E z(0) 2 L

√

2 C1 eC2 r+τ T

χE (t)χG0 z L1 (0,T ;L2 (Ω;L2 (G))) . (G) 2 F (m(E))

(2.8)

Proof. Each element η in L2 (Ω, FT , P ; Xr ) can be written as η = λi r ηi ei (x) for a sequence of FT -measurable random variable {ηi }λi r . In this case, system (2.6) can be reduced to a backward stochastic ordinary differential systems. Indeed, the solution (z, Z) of Eq. (2.6) can be expressed as z=

λi r

zi (t)ei ,

Z=

λi r

Zi (t)ei ,


837

where zi (·) ∈ L2F (Ω; C[0, T ]) and Zi (·) ∈ L2F (0, T ) (λi r), and satisfies the following equation

dzi − λi zi dt = −a(t)Zi dt + Zi dw zi (T ) = ηi .

in [0, T ],

By Lemma 2.2, we have

2 √

2 C2 r

zi (t) C1 e E E

zi (t)ei

dx λi r

G0 λi r

= C1 e

√ C2 r

E

|z|2 dx,

∀t ∈ [0, T ].

(2.9)

G0

Let us prove first conclusion i). By Itô’s formula, we see that d|z|2 = 2z dz + (dz)2 . Hence we obtain that

z(t) 2 dx − E

E G

z(0) 2 dx

G

= 2E

t

2

λi zi (t) dt + E

0 λi r

2E

t

E

−2a(t)zZ + Z 2 dx dt

0 G

2

λi zi (t) dt − E

0 λi r

t

t

t

a(t)z 2 dx dt

0 G

2 (2λi − τ ) zi (t) dt 0.

(2.10)

0 λi r

From (2.9) and (2.10), we obtain that E

z2 (x, 0) dx C1 eC2 G

√

r

E

z(x, t) 2 dx,

∀t ∈ [0, T ].

G0

Therefore, 1 1 √ 1 2 2

2

2 C2 r 2

E z (x, 0) dx dt C1 e E z(x, t) dx dt. E

G

E

Hence we obtained that for each η ∈ L2 (G, FT , P ; Xr ),

G0

838


E

√

C1 eC2 r z (x, 0) dx (m(E))2 2

T 1 2 2

2

E χE (t)χG0 (x)z(x, t) dx dt

G

0

=

G

√ eC2 r

C1 |χE χG0 z|2L1 (0,T ;L2 (Ω;L2 (G))) . (m(E))2 F

(2.11)

This gives the desired estimate (2.7) in conclusion i). Next, let us prove conclusion ii). By Itô’s formula, we find d(eτ t |z|2 ) = 2eτ t z dz + eτ t (dz)2 + τ τ e t |z|2 . Hence we see that

z(t) 2 dx − E

Eeτ t G

z(0) 2 dx

G

t

=E

2

2e λi zi (s) ds + E

0 λi r

t

+E

t

τs

eτ s −2a(s)zZ + Z 2 dx ds

0 G

2

τ eτ s z(s) dx ds

0 G

E

t

2

2eτ s λi zi (s) ds 0.

(2.12)

0 λi r

From (2.9) and (2.12), we obtain that E

z (x, 0) dx C1 e 2

√ C2 r+τ T

G

E

z(x, t) 2 dx,

∀t ∈ [0, T ].

G0

Now, proceeding exactly as in the case considered above, we end up with the desired estimate (2.8). This completes the proof. 2 By means of the usual duality argument (e.g., [14,16]), Proposition 2.1 yields the following partial controllability results for system (1.1), with explicit estimates on the control cost. 2 Proposition 2.2. For each r λ1 , there exists a control fr ∈ L∞ F (0, T ; L (Ω; Xr )) such that the solution y of system (1.1) with f = fr satisfies Pr (y(·, T )) = 0 in G, P -a.s. Moreover, fr verifies:

i) If 2λ1 > τ , then √

|fr |2L∞ (0,T ;L2 (Ω;X )) r F

C1 eC2 r E|y0 |2L2 (G) ; (m(E))2

(2.13)


839

ii) For the general case, it holds that √

|fr |2L∞ (0,T ;L2 (Ω;X )) r F

C1 eC2 r+τ T E|y0 |2L2 (G) . (m(E))2

(2.14)

Proof. We consider only case i) (case ii) can be analyzed similarly). Let us introduce a linear subspace of L1F (0, T ; L2 (Ω; Xr )) as follows:

Λ ≡ χE χG0 z z solves Eq. (2.6) with some η ∈ L2 (Ω, FT , P ; Xr ) , and define a linear functional L on Λ as follows: L(χE χG0 z) = −E

y0 z(0) dx, G

where y0 is the initial datum of system (1.1). By Proposition 2.1, we see that L is a bounded linear √ C2 r

1

2 1e 2 functional (on Λ) whose norm is not larger than ( C(m(E)) 2 E|y0 |L2 (G) ) . By Hahn–Banach Theo-

rem, L can be extended to a bounded linear functional on L1F (0, T ; L2 (Ω; Xr )) whose norm is √ C2 r

1

2 1e 2 not larger than ( C(m(E)) 2 E|y0 |L2 (G) ) . For simplicity, we use the same notation for the extension. Now, by means of a Riesz-type Representation Theorem for general stochastic processes [11], 2 we conclude that there is an fr ∈ L∞ F (0, T ; L (Ω; Xr )) such that

E

χE χG0 fr z dx dt = −E

Q

y0 z(0) dx,

(2.15)

G

and √

|fr |2L∞ (0,T ;L2 (Ω;X )) r F

C1 eC2 r E|y0 |2L2 (G) . (m(E))2

We claim that fr is the desired control. In fact, a direct computation shows that

y(T )η dx − E

E G

y0 z(0) dx = E

G

d(yz) dx Q

=E

(z dy + y dz + dy dz) dx Q

n n ij ij =E a yxi zxj + a zxi yxj + χE χG0 f z dx dt − Q

=E

i,j =1

χE χG0 f z dx dt. Q

i,j =1

(2.16)

840


From (2.15) and (2.16), we know that E

y(T )η dx = 0.

(2.17)

G

Since η is an arbitrary element in L2 (Ω, FT , P ; Xr ), equality (2.17) allows us to conclude that Pr (y(T )) = 0, P -a.s. 2 Finally, we show a decay result for system (1.1) without control. Proposition 2.3. If f ≡ 0 in system (1.1), then for any y0 ∈ L2 (Ω, F0 , P ; L2 (G)) with Pλk−1 (y0 ) = 0 for some k = 2, 3, . . . , the corresponding solution y of system (1.1) satisfies

2

E y(t) L2 (G) e−(2λk −τ )t E|y0 |2L2 (G) .

(2.18)

i Proof. By y0 ∈ L2 (Ω, F0 , P ; L2 (G)) satisfying Pλk−1 (y0 ) = 0, we see that y0 = ∞ i=k y0 ei for suitable y0i ∈ L2 (Ω, F0 , P ). Clearly, the solution y of system (1.1) can be expressed as y=

∞

y i (t)ei ,

i=k

where y i (·) ∈ L2F (Ω; C[0, T ]) solves the following equation

dy i + λi y i dt = a(t)y i dw y i (0) = y0i .

in [0, T ],

By Itô’s formula, we have that d e(2λk −τ )t |y|2 = e(2λk −τ )t 2y dy + e(2λk −τ )t (dy)2 + (2λk − τ )e(2λk −τ )t |y|2 . Hence we know

2

2

E e(2λk −τ )t y(t) dx − E y(0) dx G

G

T =E

e 0

(2λk −τ )s

T ∞

i 2

(−2λi ) y ds + E e(2λk −τ )s a 2 (s)|y|2 dx ds i=k

T + (2λk − τ )E

0 G

e(2λk −τ )s |y|2 dx ds

0

0, which gives the desired estimate (2.18) immediately.

2


841

3. Proof of Theorem 1.1 This section is devoted to giving a proof of Theorem 1.1. Firstly, we explain the main ideas of our proof, some of which are borrowed from [8,16]. We distinguish two cases. The first case is that 2λ1 > τ . In this case, by means of Proposition 2.2, one can show that the projection of solutions of system (1.1) over Xr can be controlled to zero √ C r 2 /(m(E))2 . On the other hand, by Proposition 2.3, solutions of and the control cost is C1 e system (1.1) without control (f ≡ 0) but with a vanishing projection of the initial data over Xr , decay in L2 (Ω, Ft , P ; L2 (G)) at a rate of the order of exp(−(2r − τ )t). Therefore, if we divide the set E into two parts E1 = (0, T1 ) ∩ E and E2 = (T1 , T ) ∩ E where T1 is a chosen positive number such that m(E1 ) > 0, we control the projection of the solution over Xr to zero in the first subset and then allow the equation to evolve without control in (T1 , T ). It follows that, at time t = T , the projection of the solution y over Xr vanishes and the norm of the high frequencies does not exceed the norm of the initial datum y0 . This argument allows us to control to zero the projection of the solutions of (1.1) over Xr for any r > 0 but not the whole solution. For the later an iterative argument is needed in which the set E is decomposed into a suitable chosen sequence of subsets [ti , ti+1 ] ∩ E given by Lemma 2.1 and the argument above is applied in each subset to control an increasing range of frequencies with λj ri and ri going to infinity at suitable rate. The difficulty here is reduced to estimate the cost of the control and prove that it is finite. The latter is guaranteed by the energy decay of system (1.1). This is a key point in the proof of Theorem 1.1 in the first case. The second case is that 2λ1 τ . In this case, noting that λi → ∞ as i → ∞, we see that there exists a k ∈ N such that 2λk > τ . Therefore, by choosing first a control f0 to make Pr (y(T1 )) = 0 (this follows from Proposition 2.2), the problem can be reduced to the first case considered before. We now prove Theorem 1.1. Without loss of generality, in what follows we assume that 2λ1 > τ and C1 1. By Lemma 2.1, we can take a number t˜ ∈ E with t˜ < T and a sequence {tN }∞ N =1 in the open interval (0, T ) such that (2.1)–(2.3) hold for some positive numbers ρ and C0 , and t˜ − t1 min{λ1 , 1}. Let us consider the following equation ⎧ n ⎪ ij ⎪ ⎪ d y ˜ − a y˜xi x dt = a(t)y˜ dw + χE χG0 f˜ dt ⎪ ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij y˜xi ν j + l y˜ = 0 ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ y(t ˜ 1 ) = y˜0

in G × (t1 , t˜ ), on ∂G × (t1 , t˜),

(3.1)

in G.

We will show that for any given initial datum y˜0 ∈ L2 (Ω, Ft1 , P ; L2 (G)), there exists a control 2 ˜ (t ,t˜;L2 (Ω;L2 (G))) CE|y˜0 |L2 (Ω) , such ˜ 2 function f˜ ∈ L∞ F (t1 , t ; L (Ω; L (G))) satisfying |f |L∞ F 1 that the solution y˜ of system (3.1) vanishes at time t˜, i.e. y(t˜) = 0 in G, P -a.s.

842


Set IN = [t2N −1 , t2N ], JN = [t2N , t2N +1 ] for N = 1, 2, . . . . Then ∞

[t1 , t˜ ) =

(IN ∪ JN ).

N =1

Notice that for each N 1, it holds that m(E ∩ IN ) > 0 and m(E ∩ JN ) > 0. We will put control on IN and allow the equation to evolve freely on JN . Also, we fix a strictly monotone increasing sequence (λ1 )r1 < r2 < · · · < rm → ∞ as m → ∞. Firstly, let us consider the following controlled equation on the interval I1 = [t1 , t2 ], ⎧ n ⎪ ij ⎪ ⎪ dy a (y1 )xi x dt = a(t)y1 dw + χE χG0 f1 dt − ⎪ 1 ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij (y1 )xi ν j + ly1 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎩ y1 (t1 ) = y˜0

in G × (t1 , t2 ), (3.2)

on ∂G × (t1 , t2 ), in G.

2 2 By Proposition 2.2, there exists a control f1 ∈ L∞ F (t1 , t2 ; L (Ω; L (G))) with the estimate: √

|f1 |2L∞ (t ,t ;L2 (Ω;L2 (G))) F 1 2

C1 eC2 r1 E|y˜0 |2L2 (G) , (m(E ∩ [t1 , t2 ]))2

such that Pr1 (y(·, t2 )) = 0 in G, P -a.s. Then, by (2.2) and (2.3), we see that √

|f1 |2L∞ (t ,t ;L2 (Ω;L2 (G))) F 1 2

C1 eC2 r1 2 E|y˜0 |2L2 (G) . ρ (t2 − t1 )2

Moreover, using Itô’s formula, we obtain that

2 E y1 (·, t2 ) 2

L (G)

t2

2 E y1 (·, t1 ) 2

L (G)

+ 2E t1

n ij a (y1 )xi x y1

+E

a

(s)y12 dx ds

1

+ 2E

f1 y1 dx ds t1 G

2 E y1 (·, t1 ) 2

t2

L (G)

− 2λ1 E

t2 |y| dx ds + τ E

t2

t1 G

t2 |f1 |2 dx ds + λ1 E

t1 G

E|y˜0 |2L2 (Ω) +

|y|2 dx ds

2

t1 G

1 E λ1

1

D(A 2 ),D(A 2 )∗

t2 2

t1 G

+

ds

j

i,j =1

t2

|y|2 dx ds t1 G

t2 − t1 |f1 |2L∞ (t ,t ;L2 (Ω;L2 (G))) . F 1 2 λ1


843

Hence √

2

C1 eC2 r1 E|y˜0 |2L2 (G) . E y1 (·, t2 ) L2 (G) 2 2 ρ (t2 − t1 )2 Here we have used the facts that (t2 − t1 ) min(λ1 , 1), ρ 1 and C1 > 1. On the interval J1 ≡ [t2 , t3 ], we consider the following equation without control: ⎧ n ⎪ ij ⎪ ⎪ dz1 − a (z1 )xi x dt = a(t)z1 dw in G × (t2 , t3 ), ⎪ ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij (z1 )xi ν j + lz1 = 0 on ∂G × (t2 , t3 ), ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ in G. z1 (t2 ) = y1 (t2 ) Since Pr (y1 (·, t2 )) = 0 in G, P -a.s., we have

2

2

E z1 (·, t3 ) L2 (G) exp (−2r1 + τ )(t3 − t2 ) E y1 (·, t2 ) L2 (G) √

C1 eC2 r1 2 2 exp (−2r1 + τ )(t3 − t2 ) E|y˜0 |2L2 (G) . ρ (t2 − t1 )2 Next, we consider the following equation ⎧ n ⎪ ij ⎪ ⎪ a (y2 )xi x dt = a(t)y2 dw + χE χG0 f2 dt ⎪ dy2 − ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ⎪ l˜ a ij (y2 )xi ν j + ly2 = 0 ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ y2 (t3 ) = z1 (t3 )

(3.3)

in G × (t3 , t4 ), on ∂G × (t3 , t4 ), in G.

With a similar argument to system (3.2), one can show that for any r2 > r1 > 0, there exists a 2 2 control f2 ∈ L∞ F (t3 , t4 ; L (Ω; L (G))) satisfying √

|f2 |2L∞ (t ,t ;L2 (Ω;L2 (G))) 3 4 F

2 C1 eC2 r2 E z1 (t3 ) L2 (G) 2 (m(E ∩ [t3 , t4 ])) √

2 C1 eC2 r2

2 z 2 E (t ) 1 3 L (G) ρ (t4 − t3 )2 such that Pr2 (y(·, t4 )) = 0 in G, P -a.s. From (2.3), (3.3) and (3.4), we can get |f2 |2L∞ (t F

2 2 3 ,t4 ;L (Ω;L (G)))

ρ 2 (t

√

2 C1 C04 eC2 r2 E z1 (t3 ) L2 (G) 2 2 − t1 )

(3.4)

844


C1 2 2 ρ (t2 − t1 )2

2 C04 eC2

√

√ r1 C2 r2

e

exp (−2r1 + τ )(t3 − t2 ) E|y˜0 |2L2 (G) .

(3.5)

On the interval IN , we consider the controlled equation: ⎧ n ⎪ ij ⎪ ⎪ dy a (yN )xi x dt = a(t)yN dw + χE χG0 fN dt − ⎪ N ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij (yN )xi ν j + lyN = 0 ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ yN (t2N −1 ) = zN −1 (t2N −1 )

in G × (t2N −1 , t2N ), on ∂G × (t2N −1 , t2N ), in G.

On the interval JN , we consider the following equation without control: ⎧ n ⎪ ij ⎪ ⎪ dz a (zN )xi x dt = a(t)zN dw − ⎪ N ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij (zN )xi ν j + lzN = 0 ⎪ ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎩ zN (t2N ) = yN (t2N )

in G × (t2N , t2N +1 ), on ∂G × (t2N , t2N +1 ), in G.

By induction, utilizing (2.2) and (2.3), we can conclude that, for any given rN > 0, there exists a 2 2 control function fN ∈ L∞ F (t2N −1 , t2N ; L (Ω; L (G))) satisfying: |fN |2L∞ (t F

2 2 2N−1 ,t2N ;L (Ω;L (G)))

2N −1

C1 2 ρ (t2 − t1 )2

N

4(N −1)

C04 C04×2 · · · C0

α1 α2 · · · αN E|y˜0 |2L2 (G) ,

where αN =

√ exp(C2 r1 ), √ −2(N −2) exp(C2 rN ) exp((−2rN −1 + τ )(t3 − t2 )C0 ),

N = 1, N 2,

(3.6)

and C0 is defined in (2.3) such that PrN (yN (·, t2N )) = 0 in G, P -a.s. Let = C

2C1 C2, 2 0 2 − t1 )

ρ 2 (t

(3.7)

then we have |fN |2L∞ (t F

where N > 1.

2 2 2N−1 ,t2N ;L (Ω;L (G)))

N (N−1) α1 α2 · · · αN E|y0 |2 2 , C L (G)

(3.8)


845

Now we choose rN as: N −1 4 rN = C C + τ 4,

N 1,

(3.9)

where C=

2 . (t3 − t2 )

> C 2 > 1 and t3 − t2 < 1, it follows Since C 0 24 < r1 < r2 < · · · < rN < rN +1 < · · · ,

and rN → ∞ as N → ∞.

Moreover, we have −2(N −2)

1

(rN −1 ) 4 (t3 − t2 )C0

−2(N −2)

− τ (t3 − t2 )C0

2,

∀N 2.

Therefore 3 exp −(2rN −1 − τ )(t3 − t2 )C0−2(N −2) exp −4rN4 −1 ,

∀N 2.

(3.10)

Note that 3 N (N+1) exp −r 4 C N −1 =

N (N+1) C 1

1 2 rN−1

N (N +1) C

1 2

N −1 ))rN−1 (exp(2C

(exp(rN4 −1 ))

N (N −1) C

1 2

(N −1)·2rN−1 C for each N 2, we derive from (3.9) that there exists a natural number N1 with N1 2 such that 1 N −1 )2 > N . Hence we have that for any N > N1 , it holds for each N N1 , it holds r 2 (C N −1

3 N (N−1) exp −r 4 C N −1 1.

(3.11)

By using (3.9) again, we obtain that for each N 2, 1 3 3 √ 4(N −1) + τ 4 2 exp − C 4 C 4(N −2) + τ 4 4 exp(C2 rN ) exp −rN4 −1 = exp C2 C 4 C 2(N −1) exp −C 3 C 3(N −2) exp C2 C 2 C 2(N −1) − C 3 C 3(N −2) . = exp C2 C 2 C (3.12) Thus, there exists a natural number N2 2 such that for each N N2 , 3 √ exp(C2 rN ) exp −rN4 −1 1.

(3.13)

846


Now, put N0 = max{N1 , N2 }.

(3.14)

Combining (3.10), (3.11) and (3.13), we see that for all N N0 , N (N−1) αN = C N (N−1) exp(C2 √rN ) exp −(2rN −1 − τ )(t3 − t2 )C −2(N −2) C 0 3 N (N−1) exp(C2 √rN ) exp −4r 4 C N −1 3 exp −2rN4 −1 .

(3.15)

Moreover, it is obviously that αN 1,

∀N N0 .

(3.16)

We set N (N−1) C = max (C) α1 α2 · · · αN , 1 N N0 < ∞.

(3.17)

It follows from (3.8), (3.15), (3.16), (3.17) that for all N 1, |fN |2L∞ (t F

2 2 2N−1 ,t2N ;L (Ω;L (G)))

CE|y0 |2L2 (G) .

(3.18)

We now construct a control f˜ by setting fN (x, t), ˜ f (x, t) = 0,

x ∈ G, t ∈ IN , N 1, x ∈ G, t ∈ JN , N 1,

(3.19)

2 ˜ 2 from which and by (3.18), we see that the control f˜ ∈ L∞ F (t1 , t ; L (Ω; L (G))) and satisfies the estimate

|f˜|2L∞ (t F

2 2 1 ,t˜;L (Ω;L (G)))

CE|y˜0 |2L2 (G) .

Let y˜ be the solution of system (3.1) corresponding to the control constructed in (3.19). Then on the interval IN , y(·, ˜ t) = yN (·, t). Since PrN (yN (·, t2N )) = 0 for all N 1, we see that PrN yN (·, t2M ) = 0 for all M N, P -a.s.

(3.20)

On the other hand, since t2M → t˜ as M → ∞, we obtain that ˜ t˜) y(·, ˜ t2M ) → y(·,

strongly in L2 (G), as M → ∞, P -a.s.,

which, combining with (3.20), imply that PrN (y(·, ˜ t˜ )) = 0 for all N 1, P -a.s. Since rN → ∞ as N → ∞, it holds that y(·, ˜ t˜ ) = 0, P -a.s. Thus, we have proved that for each y˜0 ∈ 2 ˜ 2 L2 (Ω, Ft1 , P ; L2 (G)), there exists a control f ∈ L∞ F (t1 , t ; L (Ω; L (G))) with the estimate


|f |2L∞ (t F

2 2 1 ,t˜;L (Ω;L (G)))

847

CE|y˜0 |2L2 (Ω) , where the constant C is given by (3.17), such that the

solution y˜ to system (3.1) vanishes at time t˜, namely, y( ˜ t˜ ) = 0 in Ω, P -a.s. Next, we take y˜0 to be ψ(x, t1 ), where ψ(x, t) is the solution to the following equation ⎧ n ⎪ ij ⎪ ⎪ dψ − a ψxi x dt = a(t)ψ dw ⎪ ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨ n ⎪ ˜ ⎪ l a ij ψxi ν j + lψ = 0 ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ ψ(0) = y0

in G × (0, t1 ), on ∂G × (0, t1 ), in G,

and construct a control f by setting ⎧ in G × (0, t1 ), ⎨0 f (x, t) = f˜(x, t) in G × (t1 , t˜ ), ⎩ 0 in G × (t˜, T ).

(3.21)

2 2 It is clear that the control f belongs to L∞ F (0, T ; L (Ω; L (G))) and that the corresponding solution y of system (1.1) verifies y(T ) = 0 in Ω, P -a.s. Moreover, the control f constructed in (3.21) satisfies the following estimate:

|f |2L∞ (0,T ;L2 (Ω;L2 (G))) CE|y0 |2L2 (Ω) , F

where C is given by (3.17). This completes the proof of Theorem 1.1. 4. Proof of Theorem 1.2 In this section, we shall give a proof of Theorem 1.2. In the sequel, C is a generic positive constant depending also on s ∈ [0, T ) (as before, it may change from line to line). As a preliminary, we first show the two following propositions which have their independent interests in the theory of stochastic partial differential equations. Proposition 4.1. If m((s, T ) ∩ E) > 0 for any s ∈ [0, T ), then for arbitrary given η ∈ L2 (Ω, FT , P ; L2 (G)), the corresponding solution of Eq. (2.6) satisfies

z(s) 2 2

L (Ω,Fs

,P ;L2 (G))

CE

z(t) 2 dx dt.

(4.1)

(s,T )∩E G0

Remark 4.1. Proposition 4.1 is an observability inequality for Eq. (2.6) with only one observer. It seems that it is very difficult (if is not impossible) to establish it directly (as remarked at pp. 99 and 108–110 in [1]). Here we use a duality argument to derive this inequality from the null controllability result.

848


Proof of Proposition 4.1. Consider the following controlled system ⎧ n ⎪ ij ⎪ ⎪ dy − a yxi x dt = a(t)y dw + χ(s,T )∩E χG0 f dt ⎪ ⎪ j ⎪ ⎪ i,j =1 ⎪ ⎨

in (s, T ) × G,

n ⎪ ˜ ⎪ l a ij yxi ν j + l(x)y = 0 ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ y(s) = ys

on (s, T ) × ∂G,

(4.2)

in G,

where the state variable ys ∈ L2 (Ω, Fs , P ; L2 (G)) and the control variable f ∈ L2F (s, T ; L2 (G)). By Theorem 1.1, system (4.2) is null controllable, i.e., for any ys ∈ L2 (Ω, Fs , P ; L2 (G)), we can find a control f ∈ L2F (s, T ; L2 (G)) such that y(T ) = 0 in G, P -a.s. Moreover, by (1.2), it holds |f |2L2

2 F (s,T ;L (G))

C|f |2L∞ (s,T ;L2 (Ω;L2 (G))) C|ys |2L2 (Ω,F F

2 s ,P ;L (G))

(4.3)

.

Applying Itô’s formula to d(yz), where y and z solve respectively systems (4.2) and (2.6), we end up with E

y(T )z(T ) dx − E

G

ys z(s) dx = E G

f z dx dt.

(s,T )∩E G0

Since y(T ) = 0 in G, P -a.s., we arrive at

−E

ys z(s) dx = E G

f z dx dt.

(s,T )∩E G0

Choosing ys = −z(s), it follows that E G

z(s) 2 dx = E

f z dx dt

(s,T )∩E G0

1

C E

|f | dx dt 2

2

(s,T )∩E G0

which gives immediately the desired estimate (4.1).

1

E

|z| dx dt 2

2

(s,T )∩E G0

1 2

2

E C E z(s) dx G

1

|z|2 dx dt

2

,

(s,T )∩E G0

2

As an easy corollary of Proposition 4.1, we have the following unique continuation property of solutions to Eq. (2.6).


849

Proposition 4.2. If m((s, T ) ∩ E) > 0 for any s ∈ [0, T ), then any solution (z, Z) of Eq. (2.6) vanishes identically in G provided that z = 0 in G0 × E, P -a.s. Remark 4.2. If the condition m((s, T )∩E) > 0 for any s ∈ [0, T ) is not assumed, Proposition 4.2 may fail to be true. This can be shown by the following counterexample. Let E satisfy that m(E) > 0 and m((s0 , T ) ∩ E) = 0 for some s0 ∈ [0, T ). Let (z1 , Z1 ) = 0 in G × (0, s0 ), P -a.s. and let ξ2 be a nonzero process belonging to L2F (s0 , T ) (then Z2 ≡ ξ2 e1 is a nonzero process in L2F (s0 , T ; L2 (G))). Solving the following forward stochastic differential equation

dζ1 − λ1 ζ1 dt = −a(t)ξ2 dt + ξ2 dw

in [s0 , T ],

ζ1 (s0 ) = 0, we find a nonzero ζ1 ∈ L2F (Ω; C[s0 , T ]). In this way, we find a nonzero solution (z2 , Z2 ) ≡ (ζ1 e1 , ξ2 e1 ) ∈ L2F (Ω; C([s0 , T ]; L2 (G))) × L2F (s0 , T ; L2 (G)) to the following forward stochastic partial differential equation ⎧ n ⎪ ij ⎪ ⎪ dz a (z2 )xi x dt = −a(t)Z2 dt + Z2 dw + ⎪ 2 ⎪ j ⎪ ⎪ ⎪ i,j =1 ⎨ n ⎪ ˜ ⎪ l a ij (z2 )xi ν j + l(x)z2 = 0 ⎪ ⎪ ⎪ ⎪ i,j =1 ⎪ ⎪ ⎩ z2 (s0 ) = 0

in (s0 , T ) × G, on (s0 , T ) × ∂G,

(4.4)

in G.

(Note however that one cannot solve system (4.4) directly because this system is non-wellposed.) Put (z, Z) =

(z1 , Z1 ) (z2 , Z2 )

in G × (0, s0 ), in G × (s0 , T ).

Then, (z, Z) is a nonzero solution of system, for which z = 0 in G0 × E, P -a.s. Note also that, the nonzero solution constructed for system (4.4) indicates that forward uniqueness does NOT hold for backward stochastic differential equations. Proof of Proposition 4.2. Since z = 0 in G0 × E, P -a.s., we have E (s,T )∩E G0 |z|2 dx dt = 0 for any s ∈ [0, T ). By Proposition 4.1, we know that for any s ∈ [0, T ), it holds z(s) = 0 in G, P -a.s. Therefore, Z = 0 in G, P -a.s. and for a.e. t ∈ [0, T ]. 2 We are now in a position to prove Theorem 1.2. Proof of Theorem 1.2. The “if” part. Since system (1.1) is linear, it suffices to show that its attainable set AT at time T with initial datum y(0) = 0 is dense in L2 (Ω, FT , P ; L2 (G)). Let us prove this by the contradiction argument. Assume that there exists an η ∈ L2 (Ω, FT , P ; L2 (G)) such that η = 0 and E G y(T )η dx = 0 for any y(T ) ∈ AT . Using d(yz) = y dz + z dy + dy dz

850


again (where y solves system (1.1) with y0 = 0 and arbitrarily given f ∈ L2F (0, T ; L2 (G)); while z solves Eq. (2.6) with the above given final datum η), we obtain E y(T )η dx = E f z dx dt. (4.5) G

E G0

Hence E E G0 f z dx dt = 0 for any f ∈ L2F (0, T ; L2 (G)). Therefore we get z = 0 in G0 × E, P -a.s. By Proposition 4.2, we arrive at η = 0, a contradiction. The “only if” part. We use the contradiction argument again. Assume that m((s0 , T ) ∩ E) = 0 for some s0 ∈ [0, T ) and system (1.1) is approximately controllable at time T . If z = 0 in G0 × E, P -a.s.,from (4.5) (since (4.5) is obtained by integration by parts, it holds for any E), we know that E G y(T )η dx = 0 for any y(T ) ∈ AT . Since AT is dense in L2 (Ω, FT , P ; L2 (G)), for any ε > 0, we can find a yTε ∈ AT such that |η − yTε |L2 (Ω,FT ,P ;L2 (G)) < ε. Therefore we have

yTε η dx = E

0=E G

η2 dx − E

G

η − yTε η dx.

G

1 1 Hence it holds that E G η2 dx ε(E G η2 dx) 2 , which implies that (E G η2 dx) 2 ε. Since ε 2 is an arbitrarily positive number, we have E G η dx = 0, which, in turn, contradicts the counterexample in Remark 4.2. This completes the proof of Theorem 1.2. 2 References [1] V. Barbu, A. R˘ascanu, G. Tessitore, Carleman estimate and cotrollability of linear stochastic heat equations, Appl. Math. Optim. 47 (2003) 97–120. [2] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [3] E. Fernández-Cara, M.J. Garrido-Atienza, J. Real, On the approximate controllability of a stochastic parabolic equation with multiplicative noise, C. R. Acad. Sci. Paris Sér. I 328 (1999) 675–680. [4] X. Fu, Null controllability for parabolic equation with a complex principle part, J. Funct. Anal. 257 (2009) 1333– 1354. [5] A.V. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser., vol. 34, Seoul National University, Seoul, 1996. [6] Y. Hu, S. Peng, Adapted solution of backward stochastic evolution equations, Stoch. Anal. Appl. 9 (1991) 445–459. [7] G. Lebeau, L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations 20 (1995) 335–356. [8] G. Lebeau, E. Zuazua, Null controllability of a system of linear thermoelasticity, Arch. Ration. Mech. Anal. 141 (1998) 297–329. [9] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1971. [10] Q. Lü, Control and Observation of Stochastic Partial Differential Equations, PhD thesis, Sichuan University, 2010. [11] Q. Lü, J. Yong, X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, preprint. [12] J. Ma, J. Yong, Forward-Backward Differential Equations and Their Applications, Lecture Notes in Math., vol. 1702, Springer, Berlin, 1999. [13] M. Sîrbu, G. Tessitore, Null controllability of an infinite dimensional SDE with state- and control-dependent noise, Systems Control Lett. 44 (2001) 385–394. [14] S. Tang, X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim. 48 (2009) 2191–2216. [15] G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stoch. Anal. Appl. 14 (1996) 461–486.


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[16] G. Wang, L∞ -null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim. 47 (2008) 1701–1720. [17] E. Zuazua, Controllability and observability of partial differential equations: some results and open problems, in: Evolutionary Differential Equations, in: Handb. Differ. Equ., vol. 3, Elsevier Science, 2006, pp. 527–621.


Regularity of eigenstates in regular Mourre theory Jacob S. Møller, Matthias Westrich ∗ Department of Mathematics, Aarhus Universitet, Denmark Received 2 July 2010; accepted 8 October 2010 Available online 5 November 2010 Communicated by L. Gross

Abstract The present paper gives an abstract method to prove that possibly embedded eigenstates of a self-adjoint operator H lie in the domain of the kth power of a conjugate operator A. Conjugate means here that H and A have a positive commutator locally near the relevant eigenvalue in the sense of Mourre. The only requirement is C k+1 (A) regularity of H . Regarding integer k, our result is optimal. Under a natural boundedness assumption of the multiple commutators we prove that the eigenstate ‘dilated’ by exp(iθA) is analytic in a strip around the real axis. In particular, the eigenstate is an analytic vector with respect to A. Natural applications are ‘dilation analytic’ systems satisfying a Mourre estimate, where our result can be viewed as an abstract version of a theorem due to Balslev and Combes (1971) [3]. As a new application we consider the massive Spin-Boson Model. © 2010 Elsevier Inc. All rights reserved. Keywords: Analytic vectors; Positive commutators; Dilation analyticity; Massive spin-boson model

1. Introduction and main results In this paper we study regularity of eigenstates ψ of a self-adjoint operator H , with respect to an auxiliary operator A for which i[H, A] satisfies a so-called Mourre estimate near the associated eigenvalue λ. Our results are partly an extract of a recent work of Faupin, Skibsted and one of us [8], and partly an improvement of a result of Cattaneo, Graf and Hunziker [4]. We consider in the present work the case of regular Mourre theory, where the derivation of the bounds on Ak ψ is simpler compared to [8]. In fact we derive explicit bounds which are independent of * Corresponding author.

E-mail addresses: [email protected] (J.S. Møller), [email protected] (M. Westrich). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.006

J.S. Møller, M. Westrich / Journal of Functional Analysis 260 (2011) 852–878

853

proof technical constructions. The bounds are good enough to formulate a natural condition on the growth of norms of multiple commutators which ensures that eigenstates are analytic vectors with respect to A. We discuss how these growth conditions may be checked in concrete examples and illustrate this for dilation analytic N -body Hamiltonians and the massive Spin-Boson Model. The general strategy in this paper, as well as in [4] and [8], is to implement a Froese–Herbst type argument, [10], in an abstract setting. In a formal computation the Mourre estimate suffices to extract results of the type presented here but to make the argument rigorous one has to impose enough conditions on the pair of operators H and A to enable a calculus of operators. This is usually done by requiring a number of iterated commutators between H and A to exist and be controlled by operators already present in the calculus. The type of conditions imposed is typically guided by a set of applications that the authors have in mind. Most examples, like many-body quantum systems with or without external classical fields, have been possible to treat using natural extensions of conditions originally introduced by Mourre in [20]. The same goes for a number of models in non-relativistic QED like confined massive Pauli–Fierz models and massless models, with A being the generator of dilations. These are the type of conditions used in [4]. Over the last 10 years a number of models that fall outside the scope of Mourre’s original conditions, and hence not covered by [4], have appeared. We split them into two types. The first type are models that, while not covered by Mourre type conditions on iterated commutators, still satisfy weaker conditions developed over some years by Amrein, Boutet de Monvel, Georgescu and Sahbani [2,24]. These conditions play the same role as Mourre’s original conditions in that they enable the same type of calculus of the operators H and A. We call this setting for regular Mourre theory. Examples of models that fall in this category but are not covered by Mourre type conditions as in [4], are: P (φ)2 -models [6] (with P (ϕ) = ϕ 4 ), the renormalised massive Nelson model [1], Pauli–Fierz type models without confining potential [11], the standard model of nonrelativistic QED near the ground state energy, where only local C k conditions are available [12], and the translation invariant massive Nelson model [18]. The second type of models we wish to highlight are those for which the commutator H = i[H, A] is not comparable to H (or A). Here one views the commutator as a new operator in the calculus and impose assumptions of mixed iterated commutators between the three possibly unbounded operators H, A and H . This type of analysis goes back to [25] and was further developed in [19] and [13]. This situation we call singular Mourre theory and is the topic considered in [8]. There are two examples where this type of analysis is natural. The first is massless Pauli–Fierz models with A being the generator of radial translations [7,14,8,9,25,15] and the second is many-body systems with time-periodic pair-potentials, in particular AC-Stark Hamiltonians [19,8]. The technical complications arising from having to deal with a calculus of three unbounded operators are significant. Part of the motivation of this work is to extract the essence of [8] in the context of regular Mourre theory, where the technical overhead is more manageable. A second motivating factor is drawn from the paper [9], which is in fact intimately connected to [8]. We remind the reader of the Fermi Golden Rule (FGR) which we now formulate. Let P denote the orthogonal projection onto the span of the eigenvector ψ, and abbreviate P¯ = I − P . The FGR states that a, for simplicity isolated and simple, embedded eigenvalue is unstable under a perturbation W provided Im lim W ψ, P¯ (H¯ − λ − i)−1 P¯ W ψ = 0. →0+

(1.1)

854


Here H¯ = P¯ H P¯ is an operator on the range of P¯ . In the above statement the existence of the limit is of course implicitly assumed. Due to the presence of the projection P¯ , the operator H¯ has purely continuous spectrum near the eigenvalue λ, and the existence of the limit can thus be inferred from the limiting absorption principle (LAP). The LAP can be deduced using positive commutator estimates, see e.g. [2], provided there exists an auxiliary operator A such that H and A satisfy a Mourre estimate near λ and (H¯ − i)−1 admits two bounded commutators with A, or more precisely H is of class C 2 (P¯ AP¯ ) (see the next subsection). This implies in particular that ran(P ) ⊆ D(A2 ), i.e. ψ ∈ D(A2 ). Even by the improvement of [8], and in turn this paper, we would still need H to be of class C 3 (A) in order to verify this property. This would for example preclude application to the model considered in [18]. In [9] the authors study the limit in (1.1) directly, bypassing the general limiting absorption theorems, albeit applying the same differential inequality technique, and prove existence of the limit assuming only ψ ∈ D(A). Combined with [8] (or this paper) this establishes the existence of the limit in the Fermi Golden Rule [9] abstractly under a C 2 (A) condition. The price to pay is that one needs a priori control of the norm Aψ locally uniformly in possibly existing perturbed eigenstates. While it is clear that such a locally uniform bound does hold, provided all the input in [8] is controlled locally uniformly in the perturbation, it is however impractical due to the complexity of the setup to extract such bounds in closed form. In this paper we do just that in the simpler context of regular Mourre theory. As a last motivation, we had in mind a consequence of having good explicit bounds on the norms Ak ψ. Namely, provided one imposes natural conditions on the norms of all iterated commutators, we show as a consequence of our explicit bounds on Ak ψ that the power series (iθA)k k=1 k! ψ has a positive radius of convergence, thus establishing that ψ is an analytic vector for A. Here however, we have to work with conditions of the type considered in [4]. Having established analyticity of the map θ → exp(iθ A)ψ in a ball around 0 one may observe that this map is actually analytic in a strip around the real axis, and thus this result reproduces a result of Balslev and Combes [3, Thm. 1] on analyticity of dilated non-threshold eigenstates. As an example of a new result, we prove for the massive Spin-Boson Model that non-threshold eigenstates are analytic vectors with respect to the second quantised generator of dilations. 1.1. Commutator calculus We pause to introduce the commutator calculus of [2] before formulating our main results. Let A be a self-adjoint operator with domain D(A) in a Hilbert space H. We denote with B(X, Y ) the set of bounded operators on the normed space X with images in the normed space Y and B(X) := B(X, X). Definition 1.1. A bounded operator B ∈ B(H) is said to be of class C k (A), in short B ∈ C k (A), if R t → eitA Be−itA

(1.2)

is strongly in C k (R). A possibly unbounded self-adjoint operator S is said to be of class C k (A) if (i − S)−1 ∈ C k (A).


855

The property, that B ∈ B(H) is of class C 1 (A) is equivalent to the statement that φ, [B, A]χ := B ∗ φ, Aχ − (Aφ, Bχ), ∀φ, χ ∈ D(A) extends to a bounded form on H × H, which in turn is implemented by a bounded operator, adA (B), see e.g. [14]. If B ∈ C 2 (A), then an argument using Duhamel’s formula shows adA (B) ∈ C 1 (A) and thus there exists a bounded extension of the form [adA (B), A]. This allows (k−1) to construct iteratively the bounded operator adkA (B) := adA (adA (B)), for B ∈ C k (A). We 0 set adA (B) := B. Commutators involving two possibly unbounded self-adjoint operators H and A will in general not extend to bounded operators on H and the definition of the quadratic form [H, A] requires further restrictions on its domain. Thus we denote by [H, A] the form φ, [H, A]χ := (H φ, Aχ) − (Aφ, H χ),

∀φ, χ ∈ D(A) ∩ D(H ).

If H ∈ C 1 (A), then D(A) ∩ D(H ) is dense in D(H ) in the graph norm of H and [H, A] extends to an H -form bounded quadratic form, which in turn defines a unique element of B(D(H ), D(H )∗ ) denoted by adA (H ) : D(H ) → D(H )∗ , see [13]. The space D(H )∗ is the dual of D(H ) in the sense of rigged Hilbert spaces. Our result on the analyticity of eigenvectors of H with respect to A requires a construction of multiple commutators of H and A which are bounded as maps from D(H ) to H in the graph norm of H . The construction is as follows: Let H ∈ C 1 (A). We assume that adA (H ) ∈ B(D(H ), H). Then, [adA (H ), A] is defined as

ψ, adA (H ), A φ := −adA (H )ψ, Aφ − Aψ, adA (H )φ ,

(1.3)

for all ψ, φ ∈ D(A) ∩ D(H ). Here we used, that adA (H ) is skew-symmetric on the domain D(A) ∩ D(H ). Assume that this form extends in graph norm of H to a form which is implemented by an element ad2A (H ) ∈ B(D(H ), H). Proceeding iteratively, we construct adkA (H ) ∈ B(D(H ), H). Lemma 1.2. Let H, A be self-adjoint operators on the Hilbert space H and assume H ∈ C 1 (A). j If adA (H ) ∈ B(D(H ), H) for 0 j k, then H ∈ C k (A). The proof of this lemma may be found in Section 5. In several places we need an appropriate class of functions to regularise the self-adjoint operators H, A, defined on D(H ), D(A) respectively, and enable a calculus for them. Definition 1.3. Define B := {r ∈ Cb∞ (R, R) | r (0) = 1, r(0) = 0, ∀k ∈ N: supt∈R |r k (t)tk | < ∞, r is real analytic in some ball around 0}. Let h ∈ B. For λ = 0 redefine hλ (x) := h(x − λ). In the following we will drop the index λ as well as the argument of hλ (H ) and other regularisations of H and A, if the context is clear. The following condition is a local C 1 (A) condition, as in [24], plus a Mourre estimate.

856


Condition 1.4. Let H, A be self-adjoint operators on H and λ ∈ R. There exist an h ∈ B, hλ (s) := h(s − λ), with hλ (H ) ∈ C 1 (A) and an floc ∈ C0∞ (R, [0, 1]), such that floc (λ) = 1 and hλ (x) > 0 for all x ∈ supp(floc ). Assume there is a smooth Mourre estimate, i.e. ∃C0 , C1 > 0 and a compact operator K, such that 2 iadA hλ (H ) C0 − C1 floc,⊥ (H ) − K,

(1.4)

floc,⊥ is defined as floc,⊥ := 1 − floc . Remark 1.5. (1) The requirement hλ (x) > 0, ∀x ∈ supp(floc ), implies floc ∈ C k (A) if hλ ∈ C k (A) for k ∈ N, since hλ is smoothly invertible (on each connected component of supp(floc )) and floc may be written as a smooth function of hλ . (2) The assumption of K being compact is not necessary. In fact we could replace this by the requirement that 1|A|Λ K, where 1|A|Λ denotes the spectral projection on [Λ, ∞), can be made arbitrarily small. (3) For a comparison of the ‘local’ Mourre estimate (1.4) with the standard form of the Mourre estimate see Section 6. Theorem 1.6 (Finite regularity). Let H, A be self-adjoint operators on the Hilbert space H and ψ be an eigenvector of H with eigenvalue λ. Assume Condition 1.4 to be satisfied with respect to λ and hλ (H ) ∈ C k+1 (A) for some k ∈ N. There exists ck > 0, only depending on supp(floc ), j C0 , C1 , K, ad A (floc (H )), adA (hλ (H )), 1 k, 1 j k + 1, such that k A ψ ck ψ.

(1.5)

Remark 1.7. In [8, Ex. 1.4] it is shown, that the statement of Theorem 1.6 is false in general if one requires hλ ∈ C k (A) only. Therefore, the result is optimal concerning integer values of k. Condition 1.8. The self-adjoint operator H is of class C 1 (A) and there exists a v > 0, such that for all k ∈ N k ad (H )(i − H )−1 k!v −k . A

(1.6)

Theorem 1.9 (Analyticity). Let H, A be self-adjoint operators on the Hilbert space H and ψ be an eigenvector of H with eigenvalue λ. Assume Condition 1.4 to be satisfied with respect to λ and that Condition 1.8 holds. Then, the map R θ → eiθA ψ ∈ H extends to an analytic function in a strip around the real axis.

(1.7)


857

2. Applications The applications of our result on ‘finite regularity of eigenstates’ are well known and discussed in the literature [23,4,16,19,9]. In contrast results on the analyticity of eigenvalues in regular Mourre theory are to our knowledge unknown. Even though the condition under which our result holds appears difficult to verify in concrete situations, we will illustrate for some deformation analytic models that it is strikingly simple to check the assumptions of Theorem 1.9. Let H be a self-adjoint operator on the Hilbert space H and U (t) := exp(itA) a strongly continuous one parameter group of unitary operators U (t). The self-adjoint operator A is the generator of this group. Assume that U (t) b-preserves D(H ), i.e. U (t)D(H ) ⊆ D(H ),

sup U (t)φ D(H ) < ∞,

∀t ∈ R and

∀φ ∈ D(H ),

t∈[−1,1]

where ψD(H ) denotes the graph norm of H . Remark 2.1. Observe that the following are equivalent: • U (t) b-preserves D(H ). • There exist μ0 > 0 and C > 0 such that for all μ ∈ R with |μ| μ0 , we have (A − iμ)−1 : D(H ) → D(H ) and (A − iμ)−1

B(D (H ),H)

C|μ|−1 .

By [13, Lemma 2.33] one observes that U ◦ (·) := U (·) D(H ) is a C0 -group in the topology of D(H ). Proposition 2.2. Let H, A be self-adjoint operators and U (t) := exp(itA). Assume that U (·) b-preserves D(H ). Then for any k ∈ N the following statements are equivalent. j

(1) H admits k H -bounded commutators with A, denoted by adA (H ), j = 1, . . . , k. (2) The map t → I (t) = (ϕ, U (t)H U (t)∗ ψ) ∈ C k ([−1, 1]), for all ψ, ϕ ∈ D(H ) ∩ D(A). There dj (j ) (0)ψ), exist H -bounded operators H (j ) (0), j = 1, . . . , k, such that dt j I (t)|t=0 = (ϕ, H for j = 1, . . . , k and all ψ, ϕ ∈ D(H ) ∩ D(A). (3) t → ψ(t) := U (t)H U (t)∗ ψ ∈ C k ([−1, 1]; H) for all ψ ∈ D(H ), and there exist H -bounded dj (j ) (0)ψ, for all operators H (j ) (0), j = 1, . . . , k, with the property that dt j ψ(t)|t=0 = H j = 1, . . . , k and ψ ∈ D(H ). If one of the three statements holds, then the pertaining H -bounded operators are uniquely determined and we have j

i j adA (H ) = (−1)j H (j ) (0),

j = 1, . . . , k.

(2.1)

Proof. Assume the commutator form [H, A] has an extension from D(H ) ∩ D(A) to an H -bounded operator. Then an argument of Mourre [20, Prop. II.2], keeping Remark 2.1 in mind, implies that (H + i)−1 : D(A) → D(A). Hence, it follows that (H + i)−1 is of class C 1 (A).

858


A consequence of this is that D(A) ∩ D(H ) is dense in D(H ) (as well as in D(A)). (Alternatively use Remark 2.1 backwards in conjunction with Nelson’s theorem [22, Thm. X.49].) This remark implies that any extension of the commutator form [H, A] to an H -bounded operator is necessarily unique. j (1) ⇒ (2): A consequence of the above observation is that adA (H ), for j = 1, . . . , k, is symmetric for j even and anti-symmetric for j odd. Compute first for ϕ, ψ ∈ D(H ) ∩ D(A) d I (t) = − ϕ, U (t)i[H, A]U (t)∗ ψ = − ϕ, U (t)iadA (H )U (t)∗ ψ . dt If we evaluate at t = 0 we observe that H (1) (0) = −iadA (H ) can be used as a weak derivative on D(H ) ∩ D(A). Iteratively we now conclude that

dk ∗ k k k ∗ I (t) = (−1)k ϕ, U (t)i k adk−1 A (H ), A U (t) ψ = (−1) ϕ, U (t)i adA (H )U (t) ψ . k dt j

Taking t = 0 implies (2). The computation here also establishes the formula connecting adA (H ) and H (j ) (0). (2) ⇒ (3): From the computation of I ’s first derivative above, evaluated at 0, we observe that [H, A] extends from the intersection domain to an H -bounded operator. Hence this extension is unique, and indeed all the derivatives H (j ) (0), j = 1, . . . , k are unique extensions by continuity. In particular H (j ) (0) are symmetric operators on D(H ) and, for j = 1, . . . , k and ϕ, ψ ∈ D(H ) ∩ D(A),

dj I (t) = ϕ, U (t)i A, H (j −1) (0) U (t)∗ ψ = ϕ, U (t)H (j ) (0)U (t)∗ ψ . j dt That ψ(t) := U (t)H U (t)∗ ψ is itself continuous is a consequence of U ◦ being a C0 -group on D(H ). We assume inductively that ψ(t) is C k−1 ([−1, 1]; H) and d k−1 ψ(t) = U (t)H (k−1) (0)U (t)∗ ψ. dt k−1 Assume now ψ, ϕ ∈ D(A) ∩ D(H ) and compute 1 t −s

d k−1 d k−1 ϕ, k−1 ψ(t) − ϕ, k−1 ψ(s) − ϕ, U (t)H (k) (0)U (t)∗ ψ dt dt

1 = t −s

t

ϕ, U (r)H (k) (0)U (r)∗ − U (t)H (k) (0)U (t)∗ ψ dr.

s

This identity now extends by continuity to ϕ ∈ H and ψ ∈ D(H ). We can furthermore estimate (for s < t)


859

k−1

1 d d k−1 (k) ∗ ψ(t) − ψ(s) − U (t)H (0)U (t) ψ t − s dt k−1 k−1 dt

1 t −s

t

U (r)H (k) (0)U (r)∗ − U (t)H (k) (0)U (t)∗ ψ dr.

s

That the right-hand side converges to zero when s → t (from the left) now follows from the strong continuity of U ◦ on D(H ). A similar argument works for s > t. (3) ⇒ (1): Compute for ϕ, ψ ∈ D(H ) ∩ D(A) dj ϕ, ψ(t) t=0 = ϕ, H (j ) (0)ψ . j dt Conversely one can compute the j th derivative in terms of iterated commutators, and hence (1) follows. Note again, that the very first step in particular ensures that extensions are unique. 2 2.1. Examples 2.1.1. N -body Schrödinger operators Consider the operator 1,...,N 1 H =− + Vij (xi − xj ), 2 i<j

with Coulomb pair potentials Vij (x) := cik /(|xi − xj |), cik ∈ R, on L2 (X), where X := x = (x1 , . . . , xN ) ∈ R3N

N xj = 0 xj ∈ R3 , 1 j N,

j =1

[16]. As a shorthand we write x = (x1 , . . . , xN ). The unitary group of dilations, U (·) is defined by 3(N−1) U (t)ψ (x) := et 2 ψ et x , and U (t) = exp(itA) for the generator of dilations A. From Proposition 2.2 infer for some C > 0 k ad (H ) A

B(D (p 2 ),H)

C2k .

It is well known, that there is a Mourre estimate for a much more general class than the Coulomb N -body Hamiltonian, including the following example [16]. This enables Theorem 1.9. Another example for N -body Schrödinger operators to which Theorem 1.9 is applicable is defined with Yukawa pair potentials. The pair potentials Vik are now given by Vij (x) :=

cik e−μ|xi −xj | , |xi − xj |

cik ∈ R, μ > 0.

860


Observe the estimate k −t d e μret e k!a k , dt k r |t=0

r := |xi − xj |,

for some a > 0. The r-dependent functions on the right-hand side of this inequality are infinitesimally p 2 -bounded, which again shows the applicability of Theorem 1.9. Hence non-threshold eigenvectors are analytic vectors with respect to A. This reproduces known results of [3]. 2.1.2. The Spin-Boson Model The ‘matter’ Hamiltonian is defined as Hat := σ3 ,

> 0,

with the 2 × 2 Pauli-matrices σ1 , σ2 , σ3 . The corresponding Hilbert space is Hat := C2 . We briefly list the definition of the quantised bosonic field, but for the details of second quantisation we refer to [5]. The Hilbert space of the bosonic field is the bosonic Fock space, F+ :=

∞

h := L2 R3 , d 3 k ,

Sn h⊗n ,

n=0

where Sn denotes the orthogonal projection onto the totally symmetric n-particle wave functions. We denote for k ∈ R with a(k) and a † (k) the annihilation and creation operator, respectively. The energy of the free field, Hf , is defined as Hf = a † (k)ω(k)a(k) d 3 k, ω(k) := k 2 + m2 , m > 0. R3

The Hilbert space of the compound system is H := Hat ⊗ F . We define the coupling between atom and field by 1 Φ(v) := √ v(k) G ⊗ a † (k) + G∗ ⊗ a(k) d 3 k, 2 R3

with a complex 2 × 2 matrix G. The function v is given by 2

v(k) :=

e

− k2 Λ

1

ω(k) 2

,

∀k ∈ R3 .

The constant Λ > 0 plays the role of an ultraviolet cutoff. We define the Hamiltonian of the compound system, H , as H := Hat ⊗ 1 + 1 ⊗ Hf + Φ(v).


861

Define, i α := (∇k · k + k · ∇k ). 2 This operator is symmetric and densely defined on L2 (R3 ) as it is the well-known generator of the strongly continuous unitary group 3 u(t)ψ (k) := e− 2 t ψ e−t k . We will denote the second-quantised operators of α and u(t) by A := dΓ (α) and U (t) := Γ (u(t)), respectively. A is the generator of the strongly continuous unitary group U (t). Observe that i ad A (H ) = dΓ i ad α (ω) + (−1) +1 Φ (iα) v and Φ (iα) v (Hf + 1)− 12 ω− 12 (iα) v 2 . L Since (iα) v =

d

(eiαt v)|t=0 , dt

(2.2)

we have to estimate the multiple derivatives. Consider the map

1 π

z → k 2 e−2z + m2 2 = ω e−z k , B 0, 4

k ∈ R3 ,

where B(0, π4 ) denotes the closed ball of radius π/4, centred at 0. Observe, that π m ω e−z k e 4 ω(k)

(2.3)

1

where the lower bound implies that z → ω(e−z k)− 2 is holomorphic in B(0, π4 ), for all k ∈ R3 . The upper bound ensures that D(1 ⊗ Hf ) is b-stable with respect to U (·). Below, we will also show that adA (H ) ∈ B(D(H ), H), which implies by Proposition 2.2 that H ∈ C 1 (A). Analogously we define the holomorphic map 2

−e−z k 2 Λ e π

z → B 0, = v e−z k , 1 4 ω(e−z k) 2

k ∈ R3 .

We may compute by Cauchy’s formula,

!( π4 )−

d −z − 3 z 2 v e k e = z=0 dz

2π

2π 0

3 e− 2 γ (ϕ) v e−γ (ϕ) k e−i ϕ dϕ,

862


γ (ϕ) := (π/4)eiϕ , ϕ ∈ [0, 2π). Using the estimate

−π d −z − 3 z m 12 e 3π8 e−e 2 2 v e k e dz

z=0

k2 Λ2

−

π

! , 4

∀k ∈ R3 ,

one finds together with (2.2) Φ (iα) v (Hf + 1)− 12 !R − , for some R > 0. Analogously, we get from (2.3)

−

d −z π π e 4 ω(k), dz ω e k z=0 ! 4 so that

dΓ i ad (ω) (Hf + 1)−1 i ad (ω)ω−1 !c− , α α ∞ for some c > 0. From [5] we may infer a Mourre estimate for our model. Dereziński and Gérard use a different generator of dilations, namely αω :=

i (∇k ω)(k) · ∇k + ∇k · (∇k ω)(k) . 2

It is also possible to prove a Mourre estimate using their techniques if ω(k) is radially increasing, ω(k) > 0, ∀k ∈ R3 and 0 is the only critical point of ω. Thus, we conclude by Theorem 1.9 and Proposition 2.2 that any eigenstate pertaining to an embedded non-threshold eigenvalue is an analytic vector with respect to A. 3. Preliminaries In what follows, we need some regularisation techniques from operator theory. It is convenient to perform calculations involving multiple commutators by using the so-called Helffer–Sjöstrand functional calculus. Part and parcel of this calculus are certain extensions of a subclass of the smooth functions on R, the almost analytic extensions. The following proposition allows us to define such extensions. Proposition 3.1. Consider a family of continuous functions (fn )n∈N ⊂ C ∞ (R), for which there (k) is an m ∈ R, such that xk−m fn is uniformly bounded for all n 0. There exists a family of ˜ ˜ functions (fn )n∈N , with fn R = fn R for any n ∈ N, such that (1) supp(fñ ) ⊂ {z ∈ C | Re z ∈ supp(fn ) and | Im z| Re z}. (2) |∂¯ fñ (z)| CN zm−N −1 | Im z|N for all N 0. The constant CN does not depend on n. For a proof of this statement see [17].


863

Remark 3.2. We will call these extensions for almost analytic extensions, because ∂¯ fñ vanishes approaching the real axis. Let ε > 0. For any self-adjoint operator L and any f ∈ C ∞ (R) with supf (k) (t)tk+ε

(3.1)

t∈R

we may define a bounded operator f (L), by 1 2πi

f (L) :=

∂¯ f˜(z)(z − L)−1 dz ∧ d z¯ .

(3.2)

C

The integral on the right-hand side converges in operator norm. It is well known that this definition coincides with the operator defined by functional calculus. Concerning the class B however, we cannot directly apply this definition. Inspired by a construction in [19] we consider the following instead. Lemma 3.3. Let r ∈ B. There is an almost analytic extension of t → r(t)/t =: ρ(t), which satisfies due to Proposition 3.1 the bounds N ∂¯ ρ(z) ˜ CN z−N −2 Im(z) .

(3.3)

Proof. Since r is real analytic around 0 we observe sup ρ (k) (t)tk+1 < ∞.

|t|1

On the other hand, the Leibniz rule yields r (k) (t) = ρ (k) (t)t + kρ (k−1) (t) and thus by induction sup ρ (k) (t)tk+1 < ∞.

2

|t|1

For any r ∈ B, set rn (t) := nr(t/n), ρ(t) := r(t)/t, ∀t ∈ R and define rn (A) by functional ˜ = ρ(¯ ˜ z) the well-known formula calculus. If we require ρ(z) rn (t) =

1 2πi

C

∂¯ ρ(z) ˜

t z−

t n

dz ∧ d z¯

(3.4)

may be recovered. Observe, that t z−

t n

= −n 1 −

z z−

t n

.

(3.5)

The first term on the right-hand side is constant and vanishes when computing commutators. Although we cannot use the formula (3.4) directly as a representation of rn (A) on H, it is possible to use it on the domain of A; a fact which is useful in the next lemma.

864


Lemma 3.4. Let B ∈ C 1 (A), where B ∈ B(H). For any r ∈ B we have

B, rn (A) = rn (A)adA (B) + R(rn , B),

(3.6)

with R(rn , B) :=

1 n2πi

2 ∂¯ ρ(z)zJ ˜ n (z) adA (B), A Jn (z) dz ∧ d z¯ ,

(3.7)

C

where Jn (z) := n(nz − A)−1 and the integral being norm convergent. Moreover, there is a c > 0 s-lim R(rn , B) = 0, n→∞

and R(rn , B) cadA (B).

(3.8)

If B ∈ C 2 (A), we have for any n ∈ N and some α, β > 0 AR(rn , B) α ad2 (B), A

R(rn , B) β ad2 (B). A n

(3.9)

In addition, s-lim AR(rn , B) = 0. n→∞

(3.10)

Proof. Let first B ∈ C 1 (A). If we consider [rn (A), B] as a form on D(A) × D(A), the commutator may be represented using (3.4) with t replaced by A, more precisely for all ψ, φ ∈ D(A)

1 φ, B, rn (A) ψ = 2πi

∂¯ ρ(z) ˜ Aφ, Jn (z)Bψ − φ, BJn (z)Aψ dz ∧ d z¯ .

C

Observe, that the sum in the integrand is by definition

Aφ, Jn (z)Bψ − φ, BJn (z)Aψ = φ, AJn (z), B ψ . But since B ∈ C 1 (A), we obtain using (3.5)

φ, AJn (z), B ψ = φ, nzJn (z), B ψ = φ, zJn (z)adA (B)Jn (z)ψ

z 2 2 = φ, zJn (z)adA (B)ψ + φ, Jn (z) adA (B), A Jn (z)ψ . n There is an almost analytic extension ρ(z) ˜ such that |z| ∂¯ ρ(z) ˜ 2 CN |y|N −2 z−N −1 , |y|

(3.11)


865

with z = x + iy, x, y ∈ R. Choose N = 2 and observe that the integral

1 2πi

2 ∂¯ ρ(z)zJ ˜ n (z) dz ∧ d z¯

(∗)

C

converges in norm. Moreover, |z|2 ∂¯ ρ(z) ˜ 3 C3 z−3 . |y| Thus from r (t) = ρ(t) + ρ (t)t we may infer that this integral in (∗) equals rn (A). Estimate (3.11) shows that the integral (3.7) converges in norm. Since s-lim n→∞

A Jn (z) = 0, n

(3.12)

the Theorem of Dominated Convergence implies (3.8). Let now B ∈ C 2 (A). Choose in (3.3) N = 3, replace in (3.7) [adA (B), A] with ad2A (B) and observe that the integrand of AR(gn , h)(B) is point-wise bounded by a constant times z−3 . The term R(gn , h)(B) is point-wise bounded by a constant times z−4 . Both functions are in L1 (R2 ) and hence the bounds follow. Eq. (3.10) is a consequence of (3.7), (3.12) and an application of the Theorem of Dominated Convergence. 2 Lemma 3.5. Let r ∈ B and k ∈ N. If B ∈ C k (A), then s-lim adkrn (B) = adkA (B). n→∞

Proof. For k = 1 the statement follows from Lemma 3.4. Let k ∈ N and assume k−1 s-lim adk−1 rn (B) = adA (B). n→∞

The first term on the right-hand side of k−1 adrn (B) = rn adk−1 adA (B) + adk−1 R(rn , B) adrn adk−1 rn (B) = adrn rn rn converges strongly by the induction hypothesis and Lemma 3.4 since adA (B) ∈ C k−1 (A). R(rn , adk−1 rn (B)) is a sum of two integrals: 1 R(rn , B) = adk−1 rn 2πi

C

1 − 2πi

A 2 Jn (z)adk−1 adA (B) Jn (z) dz ∧ d z¯ ∂¯ ρ(z)z ˜ rn n C

A 2 k−1 adA (B) Jn (z) dz ∧ d z¯ . ∂¯ ρ(z)zJ ˜ n (z)adrn n

866


Observe, that A Jn (z) = s-lim A(nz − A)−1 = 0. n→∞ n n→∞

s-lim

The integrands are strongly convergent by the uniform boundedness principle and converge to the product of the strong limits. Lemma 3.4 and the Theorem of Dominated Convergence imply that we may exchange integration with the strong limit n → ∞. 2 We use of the following expansion formula for commutators. Lemma 3.6. Let K, L ∈ B(H). Then, for any k ∈ N, k

k k−j j k K, L = L adL (K). j

(3.13)

j =1

It is convenient to regularise the operator A such that we may use the Helffer–Sjöstrand calculus and have sufficient flexibility in the proof. Let g ∈ Cc∞ (R, R) be such that g(t) = t

∀t ∈ [−1, 1],

g(t) = 2 ∀t 3,

g(t) = −2 ∀t −3,

g 0, (3.14)

and that tg (t)/g(t) has a smooth square root; clearly g ∈ B. We set gn (t) := ng(t/n) and define gn (A) by functional calculus. Observe, that n → gn2 (t)

(3.15)

is monotonously increasing for all t ∈ R. Set γ (t) := g(t)/t, for the function g defined in (3.14). We may pick an almost analytic extension of γ , denoted by γ˜ , such that γ˜ satisfies, up to a possibly different constant CN , the same bounds as ρ˜ in (3.3). 4. Finite regularity of eigenstates Proof of Theorem 1.6. Using the convention A0 = 1, the statement is correct for k = 0. Let now be k ∈ N and assume ψ ∈ D(Ak−1 ). The starting point for the proof is

0 = ψ, i h, gnk gm gnk ψ ,

(4.1)

which may be rewritten as

0 = ψn(k) , iadgm (h)ψn(k) + 2 Re ψ, gm i h, gnk ψn(k) + 2 Re ψ, i h, gnk , gm ψn(k) , (4.2) (k)

where we introduced the notation ψn := gnk ψ . We abbreviate I0 (n, m) := ψn(k) , iadgm (h)ψn(k) ,

I1 (n, m) := 2 Re ψ, gm i h, gnk ψn(k)

(4.3) (4.4)


867

and

I2 (n, m) := 2 Re ψ, i h, gnk , gm ψn(k) = 2 Re ψ, i [h, gm ], gnk ψn(k) .

(4.5)

We organise the proof in three steps. In the first step we extract from I1 a term I0 which is of a similar type as I0 . Then, starting with (4.2) upper bounds to I0 , I0 are established. Finally, using Mourre’s estimate we find lower bounds to I0 , I0 , from which we conclude ψ ∈ D(Ak ). Step 1. By an application of Lemma 3.6 we rewrite I1 (n, m) as

k k I1 (n, m) = 2 Re i E1 (j, k, n, m) + 2k Re i ψn(k−1) , gm R(gn , h)ψn(k) j j =2

+ 2k Re i ψn(k−1) , gm gn adA (h)ψn(k) , (k−j )

j

(k)

(4.6) (k−1)

(k)

, gm adgn (h)ψn ) and 2k Re(i(ψn , gm R(gn , h)ψn )) are where E1 (j, k, n, m) := (ψn present if k 2 only, in which case ψ ∈ D(A) by induction hypothesis. We discuss the term in the last line of (4.6) first. One computes 2k Re i ψn(k−1) , gm gn adA (h)ψn(k) = 2k Re i ψn(k) , γm pn2 adA (h)ψn(k) = 2k Re i ψn(k) , γm pn adA (h)pn ψn(k)

+ 2k Re i ψn(k) , γm pn pn , adA (h) ψn(k) , with γm being the operator γm (A) and p(t) :=

tg (t) , g(t)

pn (t) := p(t/n).

Hence, with (k−j ) j E1 (j, k, n) := lim E1 (j, k, n, m) = Aψn , adgn (h)ψn(k) , m→∞

k j 2,

we obtain I1 (n) := lim I1 (n, m) m→∞

k k E1 (j, k, n) + 2k Re i ψn(k−1) , AR(gn , h)ψn(k) = 2 Re i j j =2

+ 2k Re i ψn(k) , pn pn , adA (h) ψn(k) + 2k ψn(k) , pn iadA (h)pn ψn(k) .

(4.7)

Set I0 (n) := 2k ψn(k) , pn iadA (h)pn ψn(k) ,

I1 (n) := I1 (n) − I0 (n).

(4.8)

868


Step 2. First note that by an application of Lemma 3.5

I2 (n) := lim I2 (n, m) = 2 Re ψ, i adA (h), gnk ψn(k) m→∞

= 2 Re i

k k j =1

j

E2 (j, k, n) ,

with (k−j ) j E2 (j, k, n) := ψn , adgn adA (h) ψn(k) ,

k j 1.

Eq. (4.2) may be rewritten as I0 (n) + I0 (n) = −I1 (n) − I2 (n).

(4.9)

In order to find an upper bound for the right-hand side, we first estimate E1 (j, k, n), E2 (j, k, n) by 2 j k−j + j k ψ (k) 2 , 2E1 (j, k, n) j−1 n k adgn (h)gn Aψ j (k−j ) 2 + μj k ψ (k) 2 , 2E2 (j, k, n) μ−1 j k adgn adA (h) ψn for all μj k , j k > 0. The terms j adg (h)gnk−j Aψ , n

j adg adA (h) ψn(k−j ) n

are uniformly bounded in n by Lemma 3.5, h ∈ C k+1 (A) and the induction hypothesis. For the remaining terms in (4.7) we have 2 2 2k ψn(k−1) , AR(gn , h)ψn(k) k δ −1 R(gn , h)Aψ (k−1) + δ ψ (k) , 2 2

2k ψn(k) , pn pn , adA (h) ψn(k) k ν −1 pn , iadA (h) gn ψn(k−1) + ν ψn(k) . R(gn , h)A is uniformly bounded in virtue of Lemma 3.4. The function t → p(t) is by assumption smooth. Note that

pn , iadA (h) gn = pn , iadA (h) Aγn . Further, since p ∈ Cc∞ (R), an application of Proposition 3.1 together with

−1 pn , adA (h) A = 2πi

C

A 2 ∂¯ p(z)J ˜ n (z)adA (h) Jn (z) dz ∧ d z¯ n


869

(j )

shows the uniform boundedness of [pn , adA (h)]gn . For 1 j k − 1 is (ψn )n∈N convergent −1 j in norm to Aj ψ and hence (ψn )n∈N is bounded. Choose now μj k := jk k −1 C0 /12, j k := k −1 (k − 1)−1 C0 /12, ν := C0 /(12k) =: δ and observe j I0 (n) + I0 (n) −

C0 I3 (n), 3

(4.10)

where (I3 (n))n∈N is a bounded sequence. Step 3. Note, that we may assume floc (x) = χ(h(x)), ∀x ∈ R, for some compactly supported smooth function χ because h is chosen to be invertible on the support of floc . This implies floc (H ) ∈ C k+1 (A), since h ∈ C k+1 (A), see [13, Prop. 2.23]. Inserting the Mourre estimate from Condition 1.4 yields 2 2 (k) ψn , i[h, A]ψn(k) C0 ψn(k) − C1 floc,⊥ ψn(k) − ψn(k) , Kψn(k) . The second term is evaluated by floc,⊥ gnk ψ

=−

k k l=1

l

(−1)l adlgn (floc )gnk−l ψ,

where we used, that ψ is an eigenstate and an adjoint version of (3.13). Thus, the contributions from this term are uniformly bounded in n by Lemma 3.5 and the induction hypothesis. The spectral projection 1|A|Λ (A) defines a partition of unity, 1 = 1|A|Λ (A) + 1|A|>Λ (A). Hence we may write (k) ψn , Kψn(k) = ψn(k) , 1|A|Λ (A)Kψn(k) + ψn(k) , 1|A|>Λ (A)Kψn(k) . Furthermore, we may estimate

(k) 2 (k) ψ , 1|A|Λ (A)Kψ (k) 1 K1|A|Λ (A)ψn + ν ψ (k) 2 n n n 2 ν and

2 (k) ψ , 1|A|>Λ (A)Kψ (k) 1 1|A|>Λ (A)K + δ ψ (k) 2 . n n n 2 δ Observe that since K is compact and s-limΛ→∞ χ|A|>Λ = 0 we have ∀ > 0 ∃Λ > 0: χ|A|>Λ K < , but this implies ∀Λ Λ 1|A|>Λ (A)K = 1|A|>Λ (A)1|A|>Λ (A)K .

870


Thus, we may choose ν = C0 /9, δ = C0 /9 and pick then a Λ > 0 big enough, such that 2 21|A|>Λ (A)K C02 /(9)2 ,

(4.11)

i.e. C0 − ν − δ − = C0 /3. Thus we arrive at k 2 (k) k 9K1|A|Λ (A)ψn 2 2C0 l k−l ψ k 2 . adgn (floc )gn ψ + C1 I0 (n) + n l 2C0 3 l=1

The left-hand side is bounded in n by Step 2 and the induction hypothesis. Analogously, one finds for I0 (n) I0 (n) + bn

C0 pn ψ (k) 2 , n 3

for some bn 0, n ∈ N and supn∈N bn < ∞. Let k 2 (k) k 9K1|A|Λ (A)ψn 2 l k−l I4 (n) := bn + adgn (floc )gn ψ . + C1 l 2C0 l=1

Finally, this gives with (4.10) C0 pn ψ (k) 2 + ψ (k) 2 I3 (n) + I4 (n), n n 3 where the right-hand side is bounded in n. By definition of g the result is now a consequence of the Theorem of Monotone Convergence applied to the left-hand side. 2 5. Eigenstates as analytic vectors To obtain explicit bounds, independent of the regularisations of A, we apply Lemma 3.5 and use (4.9) as a starting point. Proposition 5.1. Let k ∈ N, hλ (H ) ∈ C k+1 (A) and Condition 1.4 be satisfied. Then, for any eigenstate ψ of H with eigenvalue λ ∈ supp(floc ) and Λ 0 being chosen as in (4.11) we have k 2 k k 2 27K1|A|Λ (A)Ak ψ2 6C1 l k−l A ψ ad + (f )A ψ loc A l C0 C02 l=1 k+1 96 ad (h)ψ 2 + k 2 ad2 (h)Ak−1 ψ 2 A A 2 ((1 + 2k)C0 )

k−1 k + 1 k+1−j 12 j +1 A + ψ, adA (h)Ak−1 ψ j +1 (1 + 2k)C0

+

j =2

j +2 + Ak−j ψ, ad (h)Ak−1 ψ . A

(5.1)


871

Remark 5.2. The bounds derived in this proposition make the locally uniform boundedness of Ak ψ in the sense of Condition 1.10 of [9] apparent. Proof. Note that ψ ∈ D(Ak ) by Theorem 1.6. We observe

−1 2 ∂¯ p(z)J ˜ lim pn , adA (h) = lim n (z)adA (h)Jn (z) dz ∧ d z¯ = 0, n→∞ n→∞ n2πi C

since ∂¯ p˜ has compact support and h ∈ C k+1 (A). Further with ψ (l) := Al ψ , for 0 l k, j lim E1 (j, k, n) = ψ (k+1−j ) , adA (h)ψ (k) =: E1 (j, k),

k j 2,

j +1 lim E2 (j, k, n) = ψ (k−j ) , adA (h)ψ (k) =: E2 (j, k),

k j 1.

n→∞

n→∞

Note that E1 (j + 1, k) = E2 (j, k) for k − 1 j 1. Thus, Eq. (4.9) reads after taking the limit n→∞ k−1 k + 1 (k) (k) E2 (j, k) + 2 Re iE2 (k, k). = 2 Re i (1 + 2k) ψ , iadA (h)ψ j +1 j =1

The term E2 (k, k) is singular in the sense that one cannot commute one power of A to the lefthand side and the estimate for E2 (1, k) does not improve under such a manipulation. To estimate E2 (1, k) we note 2 2 1 −2 Re ψ (k−1) , iad2A (h)ψ (k) ad2A (h)ψ (k−1) + ψ (k) . We pick up a combinatorial factor (k + 1)k/2 and thus choose =

(1 + 2k)C0 −3 2 . (k + 1)k

For E2 (k, k), the combinatorial factor is 1 and we estimate 2 2 1 k+1 (k) −2 Re ψ, iadk+1 adA (h)ψ + μψ (k) . A (h)ψ μ Choose now μ = (1 + 2k)C0 2−4 . This gives with (k + 1)k/2 k 2 the inequality 2 (k) ψ , iadA (h)ψ (k) − C0 2−3 ψ (k)

k−1 k+1 16 2 k + 1 ad (h)ψ 2 + k 2 ad2 (h)ψ (k−1) 2 . E2 (j, k) + A A j +1 1 + 2k (1 + 2k)2 C0 j =2

872


Note, that the upper bounds are modified as compared to the bounds in Step 2 of the proof of Theorem 1.6. Namely we use for 2 j k − 1, j +1 j +2 E2 (j, k) = ψ (k+1−j ) , adA (h)ψ (k−1) + ψ (k−j ) , adA (h)ψ (k−1) . Next, lower bounds are established using an analogous argument as in Step 3 of the proof of Theorem 1.6. Observe that 9K1|A|Λ (A)ψ (k) 2 (k) ψ , iadA (h)ψ (k) + 2C0 k 2 k 2 C0 (k) 2 ψ . adlA (floc )ψ (k−l) − C0 2−3 ψ (k) + C1 l 6 l=1

Finally, we arrive at k 2 k k 2 C0 l k−l ψ (k) 2 9K1|A|Λ (A)A ψ + C1 adA (floc )A ψ l 6 2C0 l=1

k+1 ad (h)ψ 2 + k 2 ad2 (h)Ak−1 ψ 2

16 A (1 + 2k)2 C0

k−1 2 k + 1 + E2 (j, k), j +1 1 + 2k

+

A

j =2

which implies (5.1).

2

Lemma 5.3. Let K, L ∈ B(H) and J (z) := (z − K)−1 for z ∈ ρ(K). Then, adkL J (z) = a∈C(k)

a k! J (z) adaLi (K)J (z), a1 ! · · · · · ana !

n

(5.2)

i=1

where C(k) denotes the set of all possible decompositions of k = a1 + · · · + ana in sums of natural numbers and further a := (a1 , . . . , ana ). The formula may easily be observed to be correct. For a proof of similar statement see [21]. Proof of Lemma 1.2. We proof the statement by establishing the formula (5.2) inductively for K replaced by H and L replaced by A. For k = 1 we observe adA (J (z)) = J (z)adA (H )J (z), since H ∈ C 1 (A). Assume now for k − 1 ∈ N, ρ(H ), J (z) = adk−1 A

a∈C(k−1)

a (k − 1)! a J (z) adAj (H )J (z). a1 ! · · · · · ana !

n

j =1

(5.3)


873

a

Observe, that adAj (H )J (z) ∈ B(H), for all 1 j na . It is well known that the bounded elements in C 1 (A) form an algebra. This means that it suffices to check that each of the a operators adAj (H )J (z) is in C 1 (A). For 0 m k − 1 we consider [adm A (H )J (z), A]. Let ψ, φ ∈ D(A) ∩ D(H ), then

m m ψ, adA (H )J (z), A φ = (−1)m J (¯z)adm A (H )ψ, Aφ + Aψ, adA (H )J (z)φ

= ψ, adm A (H ), A J (z)φ + (−1)m adm A (A)ψ, J (z)adA (H )J (z)φ , where in the last line we used AJ (z)ψ = J (z)Aψ + J (z)adA (H )J (z)ψ,

∀ψ ∈ D(H ).

m+1 By assumption, [adm A (H ), A] extends to an element adA (H ) ∈ B(D(H ), H), which implies m that [adA (H )J (z), A] extends to a bounded operator for 0 m k − 1, i.e. adm A (H )J (z) ∈ C 1 (A). Hence H ∈ C k (A). 2

We devote the rest of this section to prove Theorem 1.9. Proof of Theorem 1.9. We organise the proof for analyticity in two steps and, for simplicity, we suppose the eigenvalue λ with respect to H, ψ is 0. We consider h(x) := x(1 + νx 2 )−1 , for sufficiently small ν > 0, see Section 6 and replace floc by fana , defined in (6.6). By assumption and Section 6, this h satisfies Condition 1.4. The first step consists of proving that ψ is an analytic vector for A under the condition k ad (h), adk (fana ) k!w −k , A

A

∀k ∈ N,

(5.4)

for some w ∈ R+ to be fixed later in the proof. In the second step we prove (5.4) using Condition 1.8. Note, that it is sufficient to prove analyticity of the map θ → exp(iθ A)ψ =: ψ(θ ) ˜ + θ ) := exp(itA)ψ(θ ), in some ball around 0. Namely, if ψ(·) is analytic in a ball then ψ(t t ∈ R defines an analytic extension of this map to a strip. Alternatively, one observes the bounds in (5.1) to be invariant under conjugation of H with exp(itA), t ∈ R and hence ψ(·) extends to an analytic function in a strip around the real axis. Step 1. Assume Condition (5.4) to be satisfied and abbreviate

k + 1 (k+1−j ) j +1 12 ψ , adA (h)ψ (k−1) , (1 + 2k)C0 j + 1

12 k + 1 (k−j ) j +2 β(j, k) := ψ , adA (h)ψ (k−1) . (1 + 2k)C0 j + 1 α(j, k) :=

Motivated by Condition (5.4), we use the ansatz (l) ψ l!q −l , for 1 l k − 1, for some q ∈ R+ , q < w, independent of l. Employing the assumptions gives

874


2 −2k

α(j, k) k! q

j q 12 (k + 1 − j ) , C0 wk w

thus

2 −2k −1

k! q

k−1 j =2

k−3 1 12 q 2 q j 12 q 2 α(j, k) . C0 w w w C0 w w 1 − ( wq ) j =0

Analogously, β(j, k) k!2 q −2k

j +1 q 12 (j + 2) C0 wk w

and consequently k−3 k−1 2 −2k −1 1 24 q 3 q j 24 q 3 β(j, k) . k! q C0 w w w C0 w w 1 − ( wq ) j =2

j =0

We continue by estimating (3.13),

6C1 C0

1 2

fana,⊥ ψ (k)

6C1 C0

k!q

−k

1 j k 2 k q j !(k − j )! q −k j w j =1

6C1 C0

1 2

q 1 . w 1 − ( wq )

Further, 2 q 24 k!2 q −2k , 2 2 2 2 2 C0 (1 + 2k) C0 k w w 2k 2 96adk+1 q 96 A (h)ψ 2 k!2 q −2k 2 2 2 C0 (1 + 2k) C0 w w

96k 2 ad2A (h)ψ (k−1) 2

and finally 2 2k 27 K1|A|Λ (A)ψ (k) 2 27K (Λq) k!2 q −2k . C02 C02 k!2

Pick now q sufficiently small, such that all pre-factors of k!2 q −2k are less than 1/6 and observe that this can be done uniformly in k. Then, we obtain for our specified q (k−1) ψ (k − 1)!q −(k−1)

⇒

(k) ψ k!q −k .

This proves that ψ is an analytic vector for A, given condition (5.4).


875

Step 2. We first compute the multiple commutators of h. For some n0 ∈ N, see Section 6, the function h(x) = −

1 (i − x/n0 )−1 + (−i − x/n0 )−1 2

and (6.6) satisfy Condition 1.4. It follows from Condition 1.8 and (5.3) in the proof of Lemma 1.2 that the multiple commutators of h may be expressed in terms of the multiple commutators of J (z) := (z − H /n0 )−1 , adkA J (±i) = n−k 0 a∈C(k)

a k! J (±i) adaAi (H )J (±i), a1 ! · · · · · ana !

n

(5.5)

i=1

for any z in the resolvent set of H . The number of elements in C(k) is given by 2k − 1, which may be verified by induction. Thus, we may estimate (5.5) further in virtue of (1.6).

k k ad J (±i) k!v −k 2k − 1 k!w −k 2w . A v Choose now R w > 0 such that 4w v and conclude as in Step 1 by induction that for h, Condition 1.8 implies (5.4) and in particular, h ∈ C ∞ (A). It is obvious that fana gives the same bounds, which completes the proof. 2 Remark 5.4. (1) If we had used arctan(x) instead of h(x) = x(1 + x 2 )−1 , we would have encountered the problem that the bounds (5.4) are easily obtained from (1.6) in graph norm w.r.t. H , only. In contrast, the decay at infinity of our choice of h allows naturally for bounds in operator norm. (2) Note, that the first step in the proof uses the relations (5.4) only and is, abstractly, independent of the stronger assumption (1.6). 6. The Mourre estimate in localised form The Mourre estimate is usually cast in a different form than it is used here. Let H, A be selfadjoint operators, H ∈ C 1 (A). Let now C˜ 0 > 0 and K˜ be a compact operator. We denote by 1I (H ) spectral projections of H for an interval I ⊂ R. Suppose, that in the sense of quadratic forms on H × H ˜ 1I (H )i[H, A]1I (H ) C˜ 0 1I (H ) − K.

(6.1)

This inequality is usually referred to as a Mourre estimate. Choose floc ∈ Cc∞ (R) such that supp(floc (H )) ⊆ I and floc (λ) = 1. Set floc,⊥ := 1 − floc . Then, multiplying (6.1) from the left and the right with floc (H ) yields 2 floc i[H, A]floc C˜ 0 + C˜ 0 floc,⊥ − 2C˜ 0 floc,⊥ − K,

876


˜ loc is compact. As forms we observe ∀ > 0 where K := floc Kf 1 2 2floc,⊥ + floc,⊥ . Pick = 1/4. Therefore, we may rewrite (6.3) as 3 2 floc i[H, A]floc C˜ 0 − 3C˜ 0 floc,⊥ − K. 4

(6.2)

Let h ∈ B. Set h(t) := h(t − λ). By possibly shrinking the support of floc we may assume supp(floc ) ⊆ supp(hλ ). To avoid obscuring the computations notationally, we refrain from writing hλ and use h instead. Set hn (t) := nh(t/n), ∀t ∈ R and abbreviate Kn (z) := (z − H /n)−1 . Then, by similar arguments as in Lemma 3.4, floc iadA (hn )floc = floc hn iadA (H )floc + R, where 1 R := 2πn

C

¯∂ h (z)zKn (z)2 floc adA (H ), H floc Kn (z) dz ∧ d z¯ . t

Note that floc iadA (H )floc = floc 1I (H )iadA (H )1I (H )floc is a bounded operator on H. Analogue estimates as in the proof of Lemma 3.4 yield R

C , n

for a C 0. This gives floc iadA (H − hn )floc 1 − h floc iadA (H )floc + C n n

1 , C 1 − hn 1supp(floc ) (H ) + n for some C > 0. Taylor’s theorem implies for positive t ∈ supp(floc ) t

1 − h (t)

n

n

0

supt∈supp(floc ) |t| h (s) ds sup h (s) n s∈supp(floc )

and analogously for negative t ∈ supp(floc ). Thus, there is a C > 0 such that floc iadA (H − hn )floc

C . n


877

Choose n0 ∈ N large enough such that floc iadA (H − hn0 )floc

C˜ 0 . 4

(6.3)

Using floc,⊥ = 1 − floc we obtain from (6.3), (6.2) i[hn0 , A]

C˜ 0 2 − 3C˜ 0 floc,⊥ − K − floc,⊥ i[hn0 , A]floc,⊥ − 2 Re floc,⊥ i[hn0 , A] . 2

(6.4)

Note, that all operators appearing in (6.4) are self-adjoint. With 2 floc,⊥ iadA (hn0 )floc,⊥ adA (hn0 )floc,⊥ , ∀δ > 0:

2 1 2 ±2 Re floc,⊥ iadA (hn0 ) δ adA (hn0 ) + floc,⊥ , δ

and a choice of δ such that δadA (hn0 )2 C˜ 0 /4 we find 2 i[hn0 , A] C0 − C1 floc,⊥ − K,

(6.5)

where 0 < C0 := C˜ 0 /4. The other constant is C1 := 3C˜ 0 + δ −1 + adA (hn0 ). We may choose an h which is real analytic and extends to an analytic function in a strip around the real axis. Thus it is possible to reformulate inequality (6.5) using analytic functions only; a fact we rely on in the proof of our analyticity result. Consider the real analytic function

1 1 1 1 + , fana (x) := = 1 + (x − λ)2 2 1 + i(x − λ) 1 − i(x − λ)

∀x ∈ R.

(6.6)

Replacing the constant C1 with

floc,⊥, (x) , C1 sup x∈R fana,⊥ (x) where fana,⊥ := 1 − fana , we may rewrite the Mourre estimate (6.5) as 2 − K. i[h, A] C0 − C1 fana,⊥

(6.7)

2 in a slight abuse of notation again with C1 . We denote the constant in front of fana,⊥

Acknowledgments M. Westrich thanks Johannes-Gutenberg Universität Mainz, and V. Bach in particular, for support. Moreover, both authors thank the Erwin Schrödinger Institut (ESI) for hospitality, and in the case of M. Westrich for financial support in the form of a ‘Junior Research Fellowship’.

878


References [1] Z. Ammari, Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: The Nelson model, Math. Phys. Anal. Geom. 3 (2000) 217–285. [2] W. Amrein, A. Boutet de Monvel, V. Georgescu, C0 -Groups, Commutator Methods, and Spectral Theory of N -Body Hamiltonians, Birkhäuser, 1996. [3] E. Balslev, J.M. Combes, Spectral properties of many-body Schrödinger operators with Dilatation-analytic interactions, Comm. Math. Phys. 22 (1971) 280–294. [4] L. Cattaneo, G.M. Graf, W. Hunziker, A general resonance theory based on Mourre’s inequality, Ann. Henri Poincaré 7 (2006) 583–601. [5] J. Dereziński, C. Gérard, Asymptotic completeness in quantum field theory. Massive Pauli–Fierz Hamiltonians, Rev. Math. Phys. 11 (1999) 383–450. [6] J. Dereziński, C. Gérard, Spectral and scattering theory of spatially cut-off P (φ)2 Hamiltonians, Comm. Math. Phys. 213 (2000) 39–125. [7] J. Dereziński, V. Jaksic, Spectral theory of Pauli–Fierz operators, J. Funct. Anal. 180 (2001) 243–327. [8] J. Faupin, J.S. Møller, E. Skibsted, Regularity of bound states, arXiv:1006.5871, 2010. [9] J. Faupin, J.S. Møller, E. Skibsted, Second order perturbation theory for embedded eigenvalues, arXiv:1006.5869, 2010. [10] R. Froese, I. Herbst, A new proof of the Mourre estimate, Duke Math. J. 49 (1982) 1075–1085. [11] J. Fröhlich, M. Griesemer, B. Schlein, Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field, Adv. Math. 164 (2001) 349–398. [12] J. Fröhlich, M. Griesemer, I.M. Sigal, Spectral theory for the standard model of non-relativistic QED, Comm. Math. Phys. 283 (2008) 613–646. [13] V. Georgescu, C. Gérard, J.S. Møller, Commutators, C0 -semigroups and resolvent estimates, J. Funct. Anal. 216 (2004) 303–361. [14] V. Georgescu, C. Gérard, J.S. Møller, Spectral theory of massless Pauli–Fierz models, Comm. Math. Phys. 249 (2004) 29–78. [15] S. Golénia, Positive commutators, Fermi golden rule and the spectrum of zero temperature Pauli–Fierz Hamiltonians, J. Funct. Anal. 256 (2009) 2587–2620. [16] W. Hunziker, I.M. Sigal, The quantum N -body problem, J. Math. Phys. 41 (2000) 3448–3510. [17] J.S. Møller, An abstract radiation condition and applications to N -body systems, Rev. Math. Phys. 12 (2000) 767– 803. [18] J.S. Møller, M.G. Rasmussen, The massive translation invariant Nelson model II, 2011, in preparation. [19] J.S. Møller, E. Skibsted, Spectral theory for time-periodic many-body systems, Adv. Math. 188 (2004) 137–221. [20] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys. 78 (1981) 391–408. [21] M.G. Rasmussen, PhD thesis, Aarhus University, 2010. [22] M. Reed, B. Simon, Methods of Modern Mathematical Physics: II. Fourier Analysis and Self-Adjointness, 1 edition, Academic Press, San Diego, 1975. [23] I. Herbst, S. Agmon, E. Skibsted, Perturbation of embedded eigenvalues in the generalized N -body problem, Comm. Math. Phys. 122 (1989) 411–438. [24] J. Sahbani, The conjugate operator method for locally regular Hamiltonians, J. Operator Theory 38 (1997) 297–322. [25] E. Skibsted, Spectral analysis of N-body systems coupled to a bosonic field, Rev. Math. Phys. 10 (1998) 989–1026.


Minimal initial data for potential Navier–Stokes singularities W. Rusin 1 , V. Šverák ∗,2 University of Minnesota, United States Received 5 July 2010; accepted 19 September 2010 Available online 8 October 2010 Communicated by I. Rodnianski

Abstract Assuming some initial data u0 ∈ H˙ 1/2 (R 3 ) lead to a singularity for the 3d Navier–Stokes equations, we show that there are also initial data with the minimal H˙ 1/2 -norm which will produce a singularity. © 2010 Elsevier Inc. All rights reserved. Keywords: Navier–Stokes; Singularities; Minimal data

1. Introduction We consider the Cauchy problem for the Navier–Stokes equations in R 3 × (0, ∞) ut + u∇u + ∇p − u = 0 div u = 0 u(·, 0) = u0

in R 3 × (0, ∞), in R 3 .

(1.1) (1.2)

In this paper we will be interested in the case when the initial condition u0 belongs to the space H˙ 1/2 (R 3 ). The H˙ 1/2 -norm is invariant under the natural scaling of the initial data * Corresponding author.

E-mail address: [email protected] (V. Šverák). 1 Supported in part by the Graduate School Dissertation Fellowship. 2 Supported in part by NSF Grant DMS-0800908.


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W. Rusin, V. Šverák / Journal of Functional Analysis 260 (2011) 879–891

u0 (x) → λu0 (λx), and the Cauchy problem is known to be globally well-posed for sufficiently small u0 ∈ H˙ 1/2 , and locally well-posed for any u0 ∈ H˙ 1/2 , as proved by Fujita and Kato [10]. These statements have to be made more precise by specifying the exact notion of the solution. The solutions constructed by Kato are usually called the mild solutions. See Section 3 for details. For u0 ∈ H˙ 1/2 we denote by Tmax (u0 ) the maximal time of existence of the mild solution starting at u0 . Let Bρ = {u0 ∈ H˙ 1/2 , u0 H˙ 1/2 < ρ}, and let ρmax be the supremum of all ρ > 0 such that the Cauchy problem (1.1)–(1.2) is globally well-posed for u0 ∈ Bρ . It is not known if ρmax is finite or infinite. Here we will be interested in the hypothetical situation when ρmax is finite. In principle ρmax could be finite for various reasons, which depend on the exact notion of the solution. However, one can show that with the natural definition of the mild solution, the only reason ρmax could be finite is the appearance of finite-time singularities in the solution u for some initial data u0 .3 We will consider the following question, motivated by a discussion of one of the authors with Isabelle Gallagher: (Q) If ρmax is finite, does there exist an initial datum u0 ∈ H˙ 1/2 with u0 H˙ 1/2 = ρmax , such that the solution u of the Cauchy problem (1.1)–(1.2) develops a singularity in finite time? We will show that the answer to the question is affirmative, see Corollary 4.3. The initial data u0 with u0 H˙ 1/2 = ρmax leading to a singularity will be called H˙ 1/2 -minimal singularity-generating data. We will show that, if singularities exist, the set of the H˙ 1/2 -minimal singularity-generating data is in fact a (non-empty) subset of H˙ 1/2 which is compact, modulo the action of the scalings u0 (x) → λu0 (λx) and translations u0 (x) → u0 (x − x0 ). The main idea of the proof is straightforward: take a suitable minimizing sequence and pass to the limit. More precisely, let us consider initial data uk0 , k = 1, 2, . . . for which the corresponding solutions uk of the initial value problem develop a finite-time singularity and uk0 H˙ 1/2 → ρmax . By suitable rescalings and shifts uk0 (x) → λk uk0 (λk x − xk ) we can assume that the first singularity of uk appears at t = 1 and x = 0. We can also assume that uk0 converge weakly in H˙ 1/2 to some u0 . We suspect that the solution u with initial datum u0 will also have a singularity at t = 1 and x = 0.4 There are a few difficulties which have to be overcome. First, we have to obtain sufficiently good estimates of uk which enable us to pass to some weak limit, say, u˜ for these solutions. As the sequence of the values of the global initial energy uk0 L2 , k = 1, 2, . . . may in principle be unbounded, the usual energy inequality is insufficient for obtaining the necessary estimates. This problem can be overcome by using a local energy estimate due to LemariéRiuesset, see Lemma 4.1. Next, one has to show that the singularity at t = 1, x = 0 will “survive” in the limit, and that u˜ will also be singular at t = 1, x = 0. This is proved in Lemma 2.1, using techniques of the partial regularity theory. Finally, one has to show that u = u. ˜ (Note that u is defined in terms of the initial data u0 , and not as a limit of the sequence uk .) This is essentially a uniqueness problem for weak solutions, and in our situation we can apply a “weak-strong uniqueness” theorem, see Theorem 4.1, to obtain the conclusion u = u. ˜ Once the above technical issues are clarified, the proofs are quite straightforward.

3 The proof of the statement uses some special properties of the system (1.1)–(1.2), and can fail for other equations with similar non-linearities covered by the same perturbation theory, such as the complex viscous Burgers equation. In particular, the energy inequality plays an important role in the proof. 4 In principle u could also become singular at an earlier time, in which case the proof is of course finished.


881

At the technical level, the main new results of the paper are Theorem 4.2, and the above mentioned Lemma 2.1. Theorem 4.2 can be thought of as a strengthening of the weak-strong uniqueness theorem Theorem 4.1 and says, roughly speaking, that the solutions of the Cauchy problem are stable with respect to the weak convergence of the initial data in H˙ 1/2 .5 This question was studied by I. Gallagher in [7] and Theorem 4.2 can be thought of as a continuation of those studies. Our results can also be used to show that the absence of singularities in all (reasonable) solutions is equivalent to certain a priori estimates. Such statements were already proved in [7,22], and we give another illustration of this principle. Throughout this paper our main space for the initial data is the space H˙ 1/2 , which is the unique H˙ s space invariant under the natural scaling of the equation. It is natural to ask if our results are true for other scale-invariant spaces, such as L3 , the Morrey space M with the norm 2 −1 2 uM = supx,r r Bx,r |u| studied in [23], or some other suitable spaces covered by [14]. We plan to address these questions in the future. In the case of critical dispersive equations, the notion of minimal blow-up solutions (with a definition quite different from ours) and related profile decomposition has played an important role in the recent remarkable advances, see for example [2,6,12,1]. These techniques have been recently also applied to the Navier–Stokes regularity in critical spaces, see [11]. The situation considered here is different, in that we focus only on the initial data, since we do not have bounds in critical norms for general solutions. 2. Suitable weak solutions We first define suitable weak solutions of the Navier–Stokes equations, as introduced by [4]. See also [18,20]. This is a local notion. Let O be an open subset of the space–time R 3 × R and let u = u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)), p = p(x, t) be functions in O such that 2 2 ˙1 • u belongs locally to the energy space L∞ t Lx ∩ Lt Hx , 3/2 3/2 • p belongs locally to the space Lt Lx , • the equations div u = 0 and ut + div(u ⊗ u) + ∇p − u = 0 are satisfied on O in the sense of distributions, and • the local energy inequality

2

|∇u|2 φ dx dt

2 |u| (φt + u) + |u|2 + 2p u∇φ dx dt

(2.1)

is satisfied for every non-negative smooth test function φ = φ(x, t) compactly supported in O. In what follows we will use standard notation for euclidean balls centered at x0 ∈ R n and parabolic balls Qz0 ,r centered at z0 = (x0 , t0 ) ∈ R n × R: 5 There are several definitions of solutions and therefore one has to formulate the result with some care – see Section 4 for details.

882


Bx0 ,r = x ∈ R n ; |x − x0 | < r , Qz0 ,r = Bx0 ,r × t0 − r 2 , t0 . Given a suitable weak solution (u, p), a point z0 = (x0 , t0 ) ∈ O is called a regular point of (u, p) if u is Hölder continuous in a neighborhood of z0 . A singular point z0 ∈ O of (u, p) is any point which is not regular. We will use the following two propositions, the various versions of which can be found in [4,18,20,15]. The version below contains some quantitative estimates which are often not explicitly stated in the literature, although they are implicit in the proofs. A sketch of the proof of the spatial derivatives estimates can be found for example in [19]. Proposition 2.1 (ε-Regularity criterion). There exists ε0 > 0 such that the following statement is true: If (u, p) is a suitable weak solution in O, such that 1 r2

3 |u| + |p|3/2 dx dt < ε0 ,

(2.2)

Qz0 ,r

for some Qz0 ,r compactly contained in O, then all points in Qz0 ,r/2 are regular points of (u, p). Moreover, in Qz0 ,r/2 one has k ∇ u Ck r −1−k ,

k = 0, 1, . . .

(2.3)

u(x, t) − u x, t C t − t 1/3 .

(2.4)

and

Remark. The regularity in t cannot be improved, due to solutions of the form u(x, t) = ∇h(x, t) with h harmonic in x and having arbitrary dependence on t. The Hölder exponent in t for these solutions is dictated by the assumptions on the integrability of the pressure p = −|∇h|2 /2 − ht , and under the assumptions of the lemma the Hölder exponent 1/3 is optimal. Proposition 2.2 (Compactness). Let (uk , p k ), k = 1, 2, . . . be a sequence of suitable weak solu2 2 ˙1 tions such that uk are uniformly bounded in the energy space L∞ t Lx ∩ Lt Hx on compact subsets 3/2 3/2 k of O and p are uniformly bounded in Lt Lx on compact subsets of O. Then the sequence uk is compact in L3t L3x on compact subsets of O. Moreover, if uk → u in L3t L3x on compact subsets 3/2 3/2 of O and p k p in Lt Lx on compact subsets of O, then (u, p) is again a suitable weak solution. The two previous propositions imply the following lemma, which will be important for the proof of our main results. Lemma 2.1 (Stability of singularities). In the situation of Proposition 2.2, assume that zk ∈ O are singular points of (uk , p k ), k = 1, 2, . . . , and that zk → z0 ∈ O. Then z0 is a singular point of (u, p).


883

Proof. If the regularity criterion in Proposition 2.1 did not contain the pressure p, the statement of the lemma would be immediate: indeed, if z0 is a regular point of u, then r −2 Qz ,r |u|3 dx dt = O(r 3 ) as r → 0+ . Choosing a sufficiently small r, one sees that 0 r −2 Qz ,r |uk |3 dx dt is small for large k by the strong convergence of uk in L3t,x . However, 0

such argument cannot be applied to the pressure term, since the sequence p k may not have a 3/2 subsequence which is compact in Lt,x . It is well known how to deal with this difficulty: the trick can be found in one form or another in the proofs of partial regularity [4,18,20,15]. The pressure p k solves the equation −p k = ∂i ∂j uki ukj .

(2.5)

3/2

Recall that the term uki ukj is compact in Lt,x on compact subsets of O. Therefore we can invert the Laplacian in (2.5) using a suitable boundary condition (or just taking the Riesz transforms p˜ k = Ri Rj (uki ukj χBx0 ,r )) and decompose p k as p k = p˜ k + hk

(2.6)

3/2

with p˜ k compact in Lt,x (Qz0 ,r ) (by Calderón–Zygmund estimates) and hk bounded in 3/2 Lt,x (Qz0 ,r ) and harmonic in x in Qz0 ,r . The term with p˜ k can be dealt with in the same way as the term with uk . The term hk is handled by using classical estimates for harmonic functions: γ let γ 1 and let h ∈ Lx (Bx0 ,r ) be harmonic in Bx0 ,r . We denote (h)r = |Bx0 ,r |−1 B h. For x0 ,r

r r/2 and x ∈ Bx0 ,r we have

γ h x − (h)r γ Cγ r r −3 |h|γ dx. r

(2.7)

Bx0 ,r

We recall that we can change the pressure by any function depending on t only. Therefore we can use(2.7) with h = hk , and integrating over Qz0 ,r , we get the required smallness of the term (r )−2 Q |hk − (hk (·, t))r |3/2 dx dt. 2 z0 ,r

In fact, the above proof together with the estimates in Proposition 2.1 give the following version of Lemma 2.1. Lemma 2.2. Under the assumptions of Proposition 2.2, let K be a compact subset of O. If each point of K is a regular point of u, then, for sufficiently large k, each point of K is also a regular point of uk , and on the set K the functions uk converge to u, together with all spatial derivatives. 3. Mild solutions In this section we review the results we need about the so-called “mild solutions” of the problem (1.1)–(1.2). This approach was introduced by Fujita and Kato [10], see also [9], although the terminology was introduced later. Let us first recall basic facts about the linear Stokes problem

884


ut + ∇p − u =

∂ ∂xk fk

div u = 0 u(·, 0) = u0

in R n × (0, ∞),

(3.1)

in R n .

(3.2)

Here fk = (f1k , . . . , fnk ) for k = 1, . . . , n. Let S(t) be the solution operator of the heat equation and let P be the Helmholtz projection of vector fields onto the divergence-free vector fields. By definition, a mild solution of the linear problem above is given by the representation formula t u(t) = S(t)u0 +

S(t − s)P ∇ · f (s) ds.

(3.3)

0

A mild solution of the Cauchy problem (1.1)–(1.2) is the mild solution of the linear problem above with fij = −ui uj . We will denote the “heat extension” S(t)u0 of the initial datum u0 by t U = U (x, t). The term 0 S(t − s)P ∇ · f (s) ds with fij = −ui vj will be denoted by B(u, v). In this notation, a mild solution of the Cauchy problem (1.1)–(1.2) in R 3 × (0, T ) is defined as a solution of the integral equation u = U + B(u, u)

(3.4)

in a suitably defined space of functions X on R 3 × (0, T ). In this approach, a key property of X is the continuity of the bilinear form (u, v) → B(u, v) as a map from X × X to X. This is equivalent to B(u, v) cuX vX . (3.5) X For initial datum u0 ∈ H˙ 1/2 there are many possible choices of X. A good choice is for example X = L4t H˙ x1 . In this case the proof of (3.5) is particularly simple: using the inequality 1/2 fgH˙ 1/2 (R 3 ) cf H˙ 1 (R 3 ) gH˙ 1 (R 3 ) we see that for u, v ∈ X we have uv ∈ L2t H˙ x . Recalling the energy inequality for the linear system (3.1), s 2 |∇| u ∞

Lt L2x

2 2 2 + |∇|s uL2 H˙ 1 |∇|s u0 L2 + |∇|s f L2 L2 t

x

x

t

x

(3.6)

one easily gets (3.5). Also, the energy inequality implies that u0 ∈ H˙ 1/2 gives U ∈ X, with U X u0 H˙ 1/2 . 1/2 3/2 In fact, the above proof gives that B maps L4t H˙ x1 × L4t H˙ x1 into Ct H˙ x ∩ L2t H˙ x (where the first space denotes the space of continuous functions of t with values in H˙ 1/2 ), which shows that Eq. (3.4) can be treated as an ODE in t. In particular, one always has local-in-time existence of the solution u, and one can define the maximal time of existence Tmax (u0 ) on which the solution of (3.4) exists. If Tmax (u0 ) is finite, one has lim

T →Tmax (u0 )

uL4 H˙ 1 (R 3 ×(0,T )) = ∞. t

x

(3.7)

We note that for sufficiently small u0 H˙ 1/2 we have Tmax (u0 ) = +∞. This justifies the definition of ρmax from the Introduction.


885

The mild solutions have the same regularity as U since, roughly speaking, for short times they are just a perturbation of U , and this can be iterated forward to the time interval where the solution exists.6 In particular, the mild solutions are smooth in R 3 × (0, Tmax (u0 )). One obvious reason for Tmax (u0 ) to be finite would be the development of a singularity in the solution u at time Tmax (u0 ). A priori it is not clear that this is the only reason. One could also imagine a scenario where the L4t H˙ x1 norm of the solution would blow up even though the solution would remain smooth on each compact subset. However, this scenario can be ruled out. The only reason for the blow-up of the L4t H˙ x1 norm of the mild solutions u with u0 ∈ H˙ 1/2 are the possible finite-time singularities. This will be justified in the next section. 4. Leray’s solutions In his pioneering work [17] Leray proved the existence of weak solutions to the Cauchy problem (1.1)–(1.2). A key ingredient in his approach is the energy inequality

u(x, t)2 dx + 2

t 0

R3

∇u(x, s)2 dx ds

R3

u0 (x)2 dx.

(4.1)

R3

This inequality is the only known a priori bound for general solutions. At the first glance it would seem that for its application it is crucial that u0 ∈ L2 (R 3 ), which would rule out using Leray’s techniques in the situation of the preceding section, where the basic assumption is u0 ∈ H˙ 1/2 , which is not a subset of L2 . However, Lemarié-Rieusset [16] found a generalization of (4.1) to the situation when the energy is only (uniformly) locally finite, and this makes it possible to extend the theory of Leray’s weak solutions to a much more general setting. See also [13]. In this paper we will not need the full version of Lemarié-Rieusset’s local theory, but we will need a version of his inequality for local energy, see Lemma 4.1. In our setting with u0 ∈ H˙ 1/2 one can use the following trick by C. Calderon [5] to construct the weak solutions in a simple way. We can write u0 = a0 + v0 with a0 smooth and small in H˙ 1/2 , and v0 in L2 . (For example, a0 can be defined in terms of the Fourier transform as aˆ 0 (ξ ) = uˆ 0 (ξ )ϕ(ξ ), where ϕ is a suitable smooth function equal to 1 in a small neighborhood of 0.) Since a0 is small, the Cauchy problem (1.1)–(1.2) has a global solution a which is small in L4t H˙ x1 . We now seek solutions u in the form u = a + v, where v is a new unknown function satisfying the equation vt + a∇v + v∇a + v∇v + ∇q − v = 0.

(4.2)

The energy identity for this equation is R3

v(x, t)2 dx + 2

t 0 R3

∇v(x, s)2 dx ds =

R3

v0 (x)2 + 2

t (a∇v)v.

(4.3)

0 R3

6 One has to be cautious with this statement if “regularity” also means decay properties of u as x → ∞. Due to the non-local effect of the constraint div u = 0, the solutions u can have slower decay at infinity than U . See for example [3].

886


The last integral on the right-hand side can be estimated by 1/2

caL4 H˙ 1 vL∞ L2 ∇v t

x

t

x

3/4 L2t L2x

and we see that we have a good energy estimate for v when aL4 H˙ 1 is sufficiently small. t x Leray’s construction of weak solutions can therefore be applied to Eq. (4.2) for v. This way we can construct a global weak solution u = a + v to the Cauchy problem with u0 ∈ H˙ 1/2 . The pressure can be recovered from the equation −p = ∂i ∂j ui uj . Moreover, one can do the construction in such a way that (u, p) is a suitable weak solution in R 3 × (0, ∞) and u(t) → u0 in L2 on every compact subset of R 3 . The weak solution u with these properties will be called the Leray solution. The relation of the Leray solution and the mild solution introduced in the previous section is clarified by the following “weak-strong uniqueness” theorem. In the case u0 ∈ L2 ∩ H˙ 1 the theorem was proved by Leray. Leray’s result was generalized in various directions, see for example [21,24,16]. We will use the following version which is a special case of Theorem 33.2, p. 354 from Lemarié-Rieusset’s book [16]. Theorem 4.1. Let u be a Leray solution of the initial value problem (1.1)–(1.2) with u0 ∈ H˙ 1/2 . Let Tmax (u0 ) be the maximal time of existence of the mild solution with of (1.1)–(1.2) with the same initial value u0 . Then the mild solution coincides with u in R 3 × [0, Tmax (u0 )). The problem of uniqueness of u after Tmax (u0 ) is open. At the time of this writing we cannot rule out that Tmax (u0 ) is finite and that the Leray solution is not unique after Tmax (u0 ). We will denote the set of all Leray solutions with initial data u0 ∈ H˙ 1/2 by NS(u0 ). Proposition 2.1 can be used to show that the only reason for Tmax (u0 ) < ∞ can be a finite time singularity. We√will now sketch a proof of this statement. Let us assume that T = Tmax (u0 ) is finite. Set r = T /2. We consider the decomposition u = a + v as above, where a is a solution generated by a0 with small a0 H˙ 1/2 (and hence a satisfies global estimates) and v satisfying the energy estimates. The key point is that these estimates do not deteriorate as we approach T . Using these estimates, together with corresponding estimates for the pressure, it is not hard to see that for sufficiently large R > 0, the assumptions of Proposition 2.1 are satisfied for our solution (u, p) and Qz0 ,r with z0 = (x0 , T ) and |x0 | > R. If u does not develop a singularity at time T in the ball BR , it means that u and ∇u will be bounded in (t1 , T ) for any t1 > 0. We can now write the Navier–Stokes equation for u as ut − u + ∇p = − div(u ⊗ u).

(4.4)

Using the standard estimates for the small solution a, the energy estimates for v, together with the pointwise bound for u and ∇u, one can easily show that the term u ⊗ u = (a + v) ⊗ (a + v) is bounded both in L2t L2x (R 3 × (T /2, T )) and L2t H˙ x1 (R 3 × (T /2, T )), and therefore 1/2 also in L2t H˙ x (R 3 × (T /2, T )). Viewing (4.4) as a linear equation with the right-hand side − div(u ⊗ u), we see by the energy estimate that u ∈ L4t H˙ x1 (R 3 × (0, T )), which means that T was not the maximal time of existence of the mild solution, a contradiction. 2 2 ˙1 The weak solution v of Eq. (4.2) always belongs to the energy space L∞ t Lx ∩ Lt Hx . As noticed already by Leray in [17], this implies that v is smooth and small for large times. In our set-up we can see this from the fact that v(t)2H˙ 1/2 v(t)L2 ∇v(t)L2 and the expression on the right-hand side of this inequality clearly has to be small on a large set of times if v is in the


887

energy space. Following the reasoning of Leray [17, p. 246], we notice that at a time t0 when v(t0 )H˙ 1/2 is small we can apply the theory of mild solutions and the weak-strong uniqueness results to see that after time t0 the solution v coincides with a global mild solutions. Similar considerations have been used for example in [8]. We can summarize the above facts in the following statement: Proposition 4.1. Let u be a Leray solution of the Cauchy problem (1.1)–(1.2) with u0 ∈ H˙ 1/2 . Then for some compact set K ⊂ R 3 × (0, ∞) we have ∇u ∈ L4t L2x (R 3 × (0, ∞) \ K). In particular, u is regular at every point of R 3 × (0, ∞) \ K. Proof. The proof of the first statement follows from the comments above. The second statement follows from the first and the standard regularity criteria, such as the Ladyzhenskaya–Prodi– Serrin criterion, or from Proposition 2.1. 2 The energy estimate for v which can be obtained from Eq. (4.2) depends on the decomposition of the initial data u0 = a0 + v0 . For our purposes in this paper we need an energy estimate which is “more uniform” (although more local). Fortuitously, an estimate found by Lemarié-Rieusset in his work on weak solutions with locally finite energy provides exactly what we need. We will use the following notation: for x0 ∈ R 3 and r > 0 we will denote by Q˜ x0 ,r the space–time cylinder Bx0 ,r × (0, r 2 ). We will also use the notation uE (Q˜ x ,r ) to denote the energy norm defined by 0

u2E (Q˜

x0 ,r )

= u2L∞ L2 (Q˜ t

x

x0 ,r )

+ ∇u2L2 L2 (Q˜ t

x

x0 ,r )

.

(4.5)

Lemma 4.1. Let u0 ∈ H˙ 1/2 and let u be a Leray solution of the Cauchy problem (1.1)–(1.2) with initial condition u0 . Then for each r > 0 and x0 ∈ R 3 u2E (Q˜

x0 ,r )

C u0 H˙ 1/2 r

(4.6)

and, for a suitable function px0 ,r (t) of t,

p − px

0 ,r

3/2 (t) dx dt C u0 H˙ 1/2 r 2 .

(4.7)

Q˜ x0 ,r

Proof. The first estimate can be easily derived from Proposition 32.1, p. 342 and its proof in [16], and the second estimate follows from Lemma 32.2, p. 343 in the same book. There are two crucial points in the proof of these estimates. One is that the energy flux in the localized energy estimate (2.1) is bounded by ∼ |u|3 , if we count the pressure as p ∼ |u|2 . The energy itself controls ∼ |u|10/3 , and it is important for the proof that this is a higher power than the one in the energy flux. This is no longer the case in higher dimensions and therefore a possible generalization to higher dimensions would not be straightforward. Similar issue arises in the proof of partial regularity. The second point is that the non-local effects of the pressure are under control, so that the heuristics p ∼ |u|2 is valid, at least as far as the estimates are concerned. To see this, we notice that the kernel of the pressure equation −p = ∂i ∂j (Fij ),

with Fij = ui uj

(4.8)

888


is Gij = ∂i ∂j G,

with G(x) =

1 . 4π|x|

(4.9)

Therefore the kernel for expressing the gradient ∇p in terms of Fij decays as |x|−4 as x → ∞, and is integrable near ∞. This fast decay makes it possible to estimate the contributions to ∇p from far away. This would not be the case for p, and hence we have to work with ∇p, which is the reason for the appearance of the function px0 ,r in the estimate (4.7). This part of the argument would work in the higher dimensions as well. 2 We can now formulate the main new result of this section. Theorem 4.2. Let uk0 be a bounded sequence of initial conditions in H˙ 1/2 converging weakly in H˙ 1/2 to u0 . Let uk ∈ NS(uk0 ) be Leray solutions of the Cauchy problem with initial conditions uk0 . Assume that uk converge weakly to u in distributions. Then u ∈ NS(u0 ), i.e. u is a Leray solution of the Cauchy problem with initial condition u0 . Proof. Using Lemma 4.1, Proposition 2.2 and Theorem 4.1, we see that it is enough to show that u(t) → u0 in L2 as t → 0 on every compact subset of R 3 . This can be seen as follows. We take a non-negative smooth test function φ(x, t) compactly supported in R 3 × [0, ∞). Note that we are taking the interval [0, ∞), which is closed at zero, and φ(x, 0) does not have to vanish everywhere. We write a version of the local energy inequality with such test functions in the following form. ∞

2 k 2 −u φt + 2∇uk φ dx dt

0 R3

k 2 u φ(x, 0) dx +

∞

0

k 2 u φ + uk 2 + 2p k uk ∇φ dx dt.

(4.10)

0 R3

R3

Since for every compactly supported smooth test function ψ the sequence uk0 ψ is compact in L2 , we see that in the limit k → ∞ the inequality (4.10) will be preserved. Hence ∞

−|u|2 φt + 2|∇u|2 φ dx dt

0 R3

∞ |u0 | φ(x, 0) dx + 2

R3

0

2 |u| φ + |u|2 + 2p u∇φ dx dt,

(4.11)

R3

where p is a suitable pressure corresponding to the solution u. The last inequality implies the required local L2 -continuity property at time t = 0 for the solution u. The key points, well known in the theory of weak solutions and going back to Leray’s paper [17] are that


889

(a) the equation implies that u(t) → u0 weakly in L2 on compact subsets as t → 0, and (b) inequality (4.11) implies that for every compactly supported smooth test function ψ one has lim supt→0+ u(t)ψ u0 ψ. 2 Corollary 4.1. Let uk0 , u0 , uk be as in Theorem 4.2. Let (0, Tmax (u0 )) be the maximal interval of existence of the mild solution u starting at u0 . Then for any compact set K ⊂ R 3 × (0, Tmax (u0 )) and any k k0 = k0 (K) the solutions uk are regular at all points of K and converge uniformly to u in K, together with all spatial derivatives. Proof. Apply the theorem, together with Lemma 4.1, Proposition 2.2 and Lemma 2.2.

2

Corollary 4.2. Let uk0 , u0 , uk be as in the theorem. Assume that Tmax (uk0 ) = T < +∞ for each k and that the singular points zk of u at t = T (which exist by Proposition 4.1) stay in a compact subset of R 3 × {T }. Then Tmax (u0 ) T . Proof. Apply the theorem, together with Lemma 4.1, Proposition 2.2 and Lemma 2.1.

2

Let us recall that

ρmax = sup ρ; Tmax (u0 ) = +∞ for every u0 ∈ H˙ 1/2 with u0 H˙ 1/2 < ρ .

(4.12)

Let us also define

M = u0 ∈ H˙ 1/2 ; Tmax (u0 ) < ∞, u0 H˙ 1/2 = ρmax .

(4.13)

Corollary 4.3. The set M is non-empty. Moreover, M is compact modulo scalings and translations, i.e. if uk0 ∈ M is a sequence in M, then there exist λk > 0 and x0k ∈ R 3 such the sequence v k ∈ H˙ 1/2 defined by v k (x) = λk uk (λk x − x k ) is compact in H˙ 1/2 . 0

0

Proof. Let uk0 ∈ H˙ 1/2 be a sequence of initial data with Tmax (uk0 ) finite and uk0 H˙ 1/2 → ρmax . Find λk > 0 and x0k so that the functions given by v k (x) = λk uk0 (λk x − x0k ) develop their first singularity at time t = 1 and that (x, t) = (0, 1) is a singular point of v k . We can assume that the functions v0k (x) = v k (x, 0) converge weakly in H˙ 1/2 to v0 ∈ H˙ 1/2 . By Corollary 4.2 we know that Tmax (v0 ) 1, and by definition of ρmax this means that v0 H˙ 1/2 = ρmax . This shows that M is non-empty. We also see that v0k → v0 and hence v0k → v0 strongly. 2 The following corollary can be thought of as a variant of results in [7,22]. See also Theorem 33.3, p. 359 in [16] for a related statement about “individual solutions”. Corollary 4.4. Assume that every solution of the Cauchy problem (1.1)–(1.2) with u0 ∈ H˙ 1/2 is regular, i.e. Tmax (u0 ) = +∞ for each u0 ∈ H˙ 1/2 . Then, for l = 0, 1, 2, . . . there exist functions Fl : [0, ∞) → [0, ∞) such that t (l+1)/2 sup∇ l u(x, t) Fl u0 H˙ 1/2 . x

(4.14)

890


Proof. Let us prove the case l = 0, the proof for the derivatives being the same. By scaling invariance, it is enough to prove the statement for t = 1. If the statement fails we can assume by the translational invariance that there exists a sequence of initial data uk0 bounded in H˙ 1/2 such that for the corresponding solutions uk one has |uk (0, 1)| k. Let u0 be a weak limit of uk0 . By our assumption, the solution u corresponding to u0 is regular at (x, t) = (0, 1) and by Theorem 4.2 and Lemma 2.2 we get a contradiction. 2 Acknowledgment The authors thank the referee for useful comments, which lead to an improvement of the first version of the paper. References [1] H. Bahouri, P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131–175. [2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1) (1999) 145–171. [3] L. Brandolese, Localisation, oscillation et comportement asymptotique pour les equations de Navier–Stokes, These de doctorat, ENS Cachan, 2001. [4] L. Caffarelli, R.-V. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. XXXV (1982) 771–831. [5] C.P. Calderon, Existence of weak solutions for the Navier–Stokes equations with initial data in Lp , Trans. Amer. Math. Soc. 318 (1) (1990) 179–200. [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3 , Ann. of Math. (2) 167 (3) (2008) 767–865. [7] I. Gallagher, Profile decomposition for solutions of the Navier–Stokes equations, Bull. Soc. Math. France 129 (2) (2001) 285–316. [8] I. Gallagher, D. Iftimie, F. Planchon, Asymptotics and stability for global solutions to the Navier–Stokes equations, Ann. Inst. Fourier (Grenoble) 53 (5) (2003) 1387–1424. [9] T. Strong Kato, Lp -solutions of the Navier–Stokes equation in Rm , with applications to weak solutions, Math. Z. 187 (4) (1984) 471–480. [10] T. Kato, H. Fujita, On the nonstationary Navier–Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962) 243–260. [11] C. Kenig, G. Koch, An alternative approach to regularity for the Navier–Stokes equations in a critical space, preprint, arXiv:0908.3349. [12] C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schroedinger equation in the radial case, Invent. Math. 166 (3) (2006) 645–675. [13] N. Kikuchi, G. Seregin, Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality, in: Nonlinear Equations and Spectral Theory, in: Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 141–164. [14] H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (1) (2001) 22–35. [15] O.A. Ladyzhenskaya, G.A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations, J. Math. Fluid Mech. 1 (4) (1999) 356–387. [16] P.G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Res. Notes Math., vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. [17] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934) 193–248. [18] F.-H. Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Comm. Pure Appl. Math. 51 (3) (1998) 241–257. [19] J. Neˇcas, M. R˚užiˇcka, V. Šverák, On Leray’s self-similar solutions of the Navier–Stokes equations, Acta Math. 176 (2) (1996) 283–294. [20] V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pacific J. Math. 66 (2) (1976) 535– 552. [21] J. Serrin, The initial value problem for the Navier–Stokes equations, in: Nonlinear Problems, Proc. Sympos., Madison, Wis., Univ. of Wisconsin Press, Madison, WI, 1963, pp. 69–98.


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[22] T. Tao, A quantitative formulation of the global regularity problem for the periodic Navier–Stokes equation, Dyn. Partial Differ. Equ. 4 (4) (2007) 293–302. [23] M.E. Taylor, Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations, Comm. Partial Differential Equations 17 (9–10) (1992) 1407–1456. [24] W. von Wahl, The Equations of Navier–Stokes and Abstract Parabolic Equations, Aspects Math., vol. E8, Friedr. Vieweg & Sohn, Braunschweig, 1985.


Two results on the equivariant Ginzburg–Landau vortex in arbitrary dimension Adriano Pisante Department of Mathematics, Sapienza, University of Rome, P.le Aldo Moro 5, 00185 Roma, Italy Received 8 July 2010; accepted 3 September 2010 Available online 18 September 2010 Communicated by J. Coron

Abstract We characterize the O(N )-equivariant vortex solution for Ginzburg–Landau type equations in the N dimensional Euclidean space and we prove its local energy minimality for the corresponding energy functional. © 2010 Elsevier Inc. All rights reserved. Keywords: Ginzburg–Landau equation; Harmonic maps; Local minimizers

1. Introduction In this paper we continue the study of energy minimality property for maps u : RN → RN which are entire (smooth) solutions of the system u + u 1 − |u|2 = 0

(1.1)

in dimension N 3. The case N = 3 has been extensively treated in [18] in the spirit of the important work [19] concerning the case N = 2 which is the truly relevant one in the study of vortices in Ginzburg–Landau theory of superconductivity (see e.g. [3,20] and references therein). E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.09.002

A. Pisante / Journal of Functional Analysis 260 (2011) 892–905

893

The system (1.1) is naturally associated to the energy functional E(v, Ω) :=

1 1 2 2 2 dx |∇v| + 1 − |v| 2 4

(1.2)

Ω 1 (RN ; RN ) ∩ L4 (RN ; RN ) and a bounded open set Ω ⊂ RN . Indeed, defined for v ∈ X := Hloc loc if u ∈ X is a critical point of E(·, Ω) for every Ω then u is a weak solution of (1.1) and thus a classical solution according to the standard regularity theory for elliptic equations. In addition, any weak solution u ∈ X of (1.1) satisfies the natural bound |u| 1 in the entire space, see [10, Proposition 1.9]. A natural “boundary condition” at infinity, namely

u(x) → 1 as |x| → +∞,

(1.3)

is usually added to rule out solutions with values in a lower dimensional Euclidean space and to single out genuinely N -dimensional solutions of (1.1) with nontrivial topology at infinity. More precisely, under the assumption (1.3) the map u has a well-defined topological degree at infinity given by u deg∞ u := deg , ∂BR |u| whenever R is large enough, and we are interested in solutions satisfying deg∞ u = 0. A special symmetric solution U to (1.1) has been constructed in [1] and [13] in the form U (x) =

x f |x| , |x|

(1.4)

for a unique function f vanishing at zero and increasing to one at infinity. Actually, the map U given by (1.4) is the unique O(N ) equivariant solution of (1.1), i.e. U (T x) ≡ T U (x) for any T ∈ O(N ) (see [13]). Taking into account the obvious invariance properties of (1.1) and (1.2), infinitely many solutions can be obtained from (1.4) by translations on the domain and orthogonal transformations on the image. In addition, these solutions satisfy R 2−N E(U, BR ) → 1 N −1 N −1 | as R → +∞, so that U has infinite energy in RN . It is also easy to check that U 2 N −2 |S as in (1.4) satisfies |U (x)| = 1 + O(|x|−2 ) as |x| → +∞ and deg∞ U = 1. In [4], H. Brezis has formulated the following very natural problem: Is any solution u to (1.1) satisfying (1.3) (possibly with a “good” rate of convergence) and deg∞ u = ±1, of the form (1.4) (up to a translation on the domain and an orthogonal transformation on the image)? The answer to the previous problem is affirmative when N = 3, see [18], at least under the assumption |u(x)| = 1 + O(|x|−2 ) as |x| → +∞. In higher dimension the answer turns out to be negative in general. Indeed, following [1] it is possible to look for solutions of (1.1) in the form u(x) = ω

x f |x| , |x|

(1.5)

894


for suitable harmonic maps ω ∈ C ∞ (S N −1 ; S N −1 ) with constant energy density on S N −1 (this constant being an eigenvalue of the Laplace–Beltrami operator on the sphere and the components of the maps being in turn corresponding eigenfunctions) and for suitable profile functions f ∈ C 2 (R+ ) increasing from zero to one (depending only on this constant density). At least for N = 8 a solution of (1.1) in the form (1.5) has been constructed in [11] with degree one at infinity for a harmonic map ω different from the identity. However, if we add a further assumption on the energy growth at infinity then the previous problem has a positive answer. Indeed we have the following characterization of the equivariant vortex solution (1.4). Theorem 1.1. Let N 3 and let u ∈ X be an entire solution of (1.1). The following are equivalent: −1 N −1 N −2 |R + (i) u satisfies |u(x)| → 1 as |x| → +∞, deg∞ u = ±1 and E(u, BR ) = 12 N N −2 |S N −2 o(R ) as R → ∞; (ii) up to a translation on the domain and an orthogonal transformation on the image, u is O(N )-equivariant, i.e., u = U as given by (1.4).

The previous characterization of the equivariant solution relies on the division trick introduced in [19] and a suitable improvement of the integral identity used in [18] in the case N = 3. As a consequence, the result in [18] extends to every dimension but no precise behaviour of the solution at infinity is needed in the proof except its energy growth at infinity. Note that, the assumptions on the modulus and the degree are only used to infer that u vanish at some point, which readily gives the translation parameter in the final formula. In the three dimensional situation a more precise characterization of (1.4) was given in [18] in terms of local energy minimality according to the following general definition. 1 (RN ; RN ) ∩ L4 (RN ; RN ) is a local minimizer of E(·) if Definition 1.2. A map u ∈ X := Hloc loc

E(u, Ω) E(v, Ω)

(1.6)

for any bounded open set Ω ⊂ RN and for every v ∈ X satisfying v − u ∈ H01 (Ω; RN ). Obviously local minimizers are smooth entire solutions of (1.1) but it is not clear that for each N 3 nonconstant local minimizers do exist or if the solutions obtained from (1.4) are locally minimizing. The main goal of this paper is to discuss local minimality in the sense of the definition above for the solutions given by (1.4) in any dimension N 3. Following ideas introduced in [18] in the three dimensional case, first we show existence of a nonconstant local minimizer u vanishing at the origin and satisfying the correct energy growth at infinity (see Theorem 3.4 for the precise statement) and then, arguing as in the proof of Theorem 1.1 we show its symmetry, i.e. we show that u is given by (1.4). The existence of a nonconstant local minimizer of E(·) is ultimately related to the minimality x for the Dirichlet integral on the unit ball among maps and uniqueness property of u∞ (x) = |x| 1 N N −1 in HId (R ; S ), which makes a strong connection of our problem with the theory of minimizing harmonic maps. These two properties of u∞ are well known for N = 3 (see [5]) and for N 7 (see [14] and [2] respectively). Some years later a striking simple proof of the minimality


895

property of u∞ was given in [15] for any N 3 and then uniqueness follows readily from the unique continuation argument in [2]. The construction of a nonconstant local minimizer relies indeed on the analysis of the vorticity set for solutions uλ to (Pλ )

u + λ2 u 1 − |u|2 = 0

in B1 ,

u = Id

on ∂B1 ,

λ > 0,

(1.7)

1 (B ; RN ) which are absolute minimizers of the Ginzburg–Landau functional Eλ (u, B1 ) on HId 1 where

Eλ (u, Ω) :=

eλ (u) dx

2 1 λ2 1 − |u|2 . with eλ (u) := |∇u|2 + 2 4

Ω

We will show that uλ → u∞ in H 1 (B1 ; RN ) as λ → ∞, so that the zeros of uλ will tend to the origin. Thus, up to translations, we will obtain a locally minimizing solution to (1.1) as a limit of uλn (x/λn ) for some sequence λn → +∞. In addition, the correct energy bound, namely N −1 N −1 N −2 E(u, BR ) 12 N |R for all R > 0, will follow from the explicit boundary condition −2 |S −1 N −1 |, and the following celebrated monoin (1.7) which gives the bound Eλ (uλ , B1 ) 12 N N −2 |S tonicity formula proved in [17]. Lemma 1.3 (Monotonicity formula). Assume that u : Ω → RN is a smooth solution of the system u + λ2 u(1 − |u|2 ) = 0 in some open set Ω ⊂ RN and λ > 0. Then, E u, BR (x0 ) = N −2 λ 1

R

E u, Br (x0 ) + N −2 λ 1

r

BR (x0 )\Br (x0 )

λ2 + 2

R

1

t N −1 r

1 ∂u 2 dx |x − x0 | ∂|x − x0 |

2 1 − |u|2 dx dt,

(1.8)

Bt (x0 )

for any x0 ∈ Ω and any 0 < r R dist(x0 , ∂Ω). As already outlined above, once we have a local energy minimizer vanishing at the origin and with the correct bound on the energy at infinity, we can argue as in the proof of Theorem 1.1 and we obtain the main result of the paper. Theorem 1.4. Let N 3 and let U be the solution of (1.1) given by (1.4). Then U is a local minimizer of the energy E according to Definition 1.2. In particular, U is stable and the following inequality holds for any bounded open set Ω ⊂ RN and for any ϕ ∈ C0∞ (Ω; RN ),

|∇ϕ|2 + |U |2 − 1 |ϕ|2 + 2|U · ϕ|2 dx 0.

(1.9)

896


The stability inequality was already known. Indeed, in [13] a direct stability analysis for the linearized operator at U was performed in any dimension N 3, in the same spirit of the two dimensional result in [9], using block diagonalization and Perron–Frobenius type arguments. Here, instead, inequality (1.9) is obtained as a straightforward consequence of a much deeper property of U , namely the local energy minimality property given in Definition 1.2, with respect to arbitrarily large (but compactly supported) perturbation. Finally, note that both Theorem 1.1 and Theorem 1.4 also apply to the case N = 3, which was essentially covered in [18]. Here, however, the proofs are much simpler and do not rely neither on the deep concentration-compactness and quantization results in [17,16], nor on a precise asymptotic analysis at infinity inspired to the one for harmonic maps at isolated singularities given in [21], which was an important ingredient in [18]. The plan of the paper is the following. In Section 2 we review the properties of the equivariant solution (1.4) and we prove Theorem 1.1. In Section 3 we study minimizing solutions to (Pλ ), we prove Theorem 3.4 and the main result of the paper and finally we suggest two open problems. 2. A characterization of the equivariant solution In this section first we collect some preliminary results about the equivariant entire solution (1.4) and then we prove its characterization in terms of topological degree and asymptotic growth rate of the energy at infinity. The existence and uniqueness statement and the qualitative study of the profile function f in (1.4) are essentially contained in [1,12,13]. In the following lemma we stress the asymptotic behaviour at infinity. The proof is exactly the same as in [18] and will be omitted. Lemma 2.1. There is a unique solution f ∈ C 2 ([0, +∞)) of ⎧ ⎨

N − 1 N − 1 f − f + f + f 1 − f 2 = 0, 2 r r ⎩ f (0) = 0 and f (+∞) = 1.

(2.1)

In addition, 0 < f (r) < 1 for each r > 0, f (0) > 0, f is strictly increasing, R 2 f

(R) + Rf (R) + N − 1 − R 2 1 − f (R)2 = o(1)

as R → +∞,

(2.2)

and 1

R

R N −2 0

2 2 r 2 2 N − 1 2 1N −1 2 (1 − f ) f + f +r r N −3 dr → 2 2 4 2N −2

as R → +∞.

(2.3)

A straightforward consequence of the previous lemma is the following result. Proposition 2.2. Let x0 ∈ RN and T ∈ O(N ). Consider the function f : [0, +∞) → [0, 1) given by Lemma 2.1 and define w(x) :=

T (x − x0 ) f |x − x0 | . |x − x0 |


897

Then w is a smooth solution of (1.1). In addition, 0 < |w(x)| < 1 for each x = x0 , w satisfies |w(x)| = 1 + O(|x|−2 ) as |x| → ∞, deg∞ w = det T = ±1 and lim

R→+∞

1

R N −2 BR (x0 )

2 (1 − |w(x)|2 )2 1 1 N − 1 N −1 . ∇w(x) + dx = S 2 4 2N −2

(2.4)

Proof. As in [1] and [13], w is smooth and it is a classical solution of (1.1) and clearly |w(x)| → 1 as |x| → ∞, deg∞ w = det T . Finally, a simple calculation yields 2 (1 − |w(x)|2 )2 1 2 (N − 1) (f (|x − x0 |))2 1 ∇w(x) + = f |x − x0 | + 2 4 2 2 |x − x0 |2 + whence (2.4) follows easily from (2.3).

(1 − |f (|x − x0 |)|2 )2 , 4

(2.5)

2

Remark 2.3. Note that, in view of (2.2) and (2.5), the function w(x) above also satisfies the 2 )2 condition 12 |∇w(x)|2 + (1−|w(x)| = N 2−1 |x|1 2 + o(|x|−2 ) as |x| → ∞ for any x0 ∈ RN , whence 4 E(w, BR ) =

1 N −1 N −1 N −2 |R 2 N −2 |S

+ o(R N −2 ) as R → ∞.

The main ingredient in the proof of Theorem 1.1 is given by the following auxiliary result which is of independent interest and will be used also in the next section. Proposition 2.4. Let u ∈ C 2 (RN ; RN ) an entire solution of (1.1) and suppose that u(0) = 0 −1 N −1 N −2 and E(u, BR ) 12 N |R for each R > 0. Then, there exists T ∈ O(N ) such that N −2 |S u(x) = T U (x), where U is given by (1.4). Proof. First we apply the division trick of [19] to prove that u has the form (1.5) with the function f as in Lemma 2.1. Then a simple argument calculating the energy at infinity will give the conclusion. Let f ∈ C 2 ([0, ∞)) given by Lemma 2.1 and define v(x) :=

u(x) . f (|x|)

(2.6)

The following lemma gives the basic properties of the function v that we need in the sequel. Lemma 2.5. Let v as defined in (2.6). Then v ∈ C 2 (RN \ {0}; RN ), v(x) = B

x x + o |x|−1 , + o(1) and ∇v(x) = ∇ B |x| |x|

as |x| → 0 and finally

where B :=

∇u(0) , f (0)

(2.7)

898


1

lim

R→∞ R N −2

1

lim

E(v, BR )

R→∞ R N −2 BR

1 N − 1 N −1 , S 2N −2

N − 1 (1 − |v|2 ) dx = 0. 2 |x|2

(2.8)

Proof. Since u is smooth the same holds for v outside the origin and (2.7) follows easily from Taylor expansion of u near the origin. In order to prove (2.8) is suffices to show that

(1 − |v|2 )2 dx = o R N −2 = 4

BR

(1 − |u|2 )2 dx 4

BR

and lim

1

R→∞ R N −2

1 1 |∇v|2 dx = lim N −2 R→∞ R 2

BR

1 |∇u|2 dx 2

BR

as R → ∞, where the last limit exists because of the monotonicity formula (1.8). Indeed, (2.8) follows easily from the two equalities above combining the definition of E, the energy growth of u at infinity and Young’s inequality. To prove the first statement above, it is enough to note that by definition |1−|v|2 | f −2 (1−f 2 +|1−|u|2 |) when |x| 1. Thus, the claim on the potential part of the energy follows easily from Young’s inequality and the corresponding property for u (the latter being a simple consequence of the monotonicity formula exactly as in [18], Lemma 4.1). To prove the second claim above, note that f (|x|) = 1 + O(|x|−2 ) and f (|x|) = o(|x|−1 ) at infinity. Since (2.2) yields |∇v|2 =

2 1 f ∂ 2 2 f |∇u| + |u| − 3 |u|2 = 1 + o(1) |∇u|2 + o |x|−2 2 f f f ∂r

as |x| → ∞, the conclusion follows by integration and straightforward manipulations.

2

As u solves (1.1) and f solves (2.1), simple computations lead to N −1 f x · ∇v − v + f 2 v 1 − |v|2 = −2 v. f |x| |x|2 On the other hand, as r 3−N ∂v ∂r = v · r

3−N

x ∇v |x|N−2

straightforward calculations give

N − 2 ∂v 2 ∂v x 1 1 ∂v 2 = N −2 + div − |∇v| , + ∇v · ∂r ∂r 2 ∂r |x| |x|N −2 |x|N −3

∂v f 2 v 1 − |v|2 · r 3−N = ∂r

(1 − |v|2 )2 f f 2f 2 2 N−3 + N −2 4 |x| |x| 2 2 x 2 (1 − |v| ) − div f 4 |x|N −2

(2.9)


899

and N −1 x N −1 3−N ∂v 2 = div 1 − |v| . v·r − ∂r 2 |x|N |x|2 Thus, multiplying Eq. (2.9) by r 3−N ∂v ∂r and taking the previous identities into account yields 2 ∂v N − 2 (1 − |v|2 )2 f f f 1 2f 2 + 2 0 G(x) := + 2 + ∂r f |x|N −3 4 |x|N −2 |x|N −3 |x|N −2 = div Φ(x),

(2.10)

where 1 x (1 − |v|2 )2 N − 1 x 1 ∂v x Φ(x) := |∇v|2 N −2 − N −3 ∇v · 1 − |v|2 . + N −2 f 2 + N 2 ∂r 4 2 |x| |x| |x| |x| When integrating (2.10) over an annulus, the inner boundary integral is controlled by the following lemma. Lemma 2.6. For each N 3 we have

x |x|=δ Φ(x) · |x|

dHN −1 →

N −1 N −1 | 2 |S

as δ → 0.

Proof. By definition of Φ we have Φ(x) · |x|=δ

x dHN −1 |x|

=

1 |x|N −3

|x|=δ

2 ∂v 1 f 2 (1 − |v|2 )2 (N − 1) (1 − |v|2 ) dHN −1 . |∇v|2 − + N −3 + 2 ∂r 4 2 |x| |x|N −1 (2.11)

Taking (2.7) into account, as |x| → 0 we have x 2 |∇v| = ∇ B + o |x|−2 , |x| 2

∂v = o |x|−1 , ∂r

1 − |v|2 =

|x|2 − |Bx|2 + o(1). |x|2

Consequently, Φ(x) · {|x|=δ}

= {|x|=δ}

x dHN −1 |x|

1−N x 2 N − 1 |x|2 − |Bx|2 1 ∇ B dHN −1 + + o |x| |x| 2 |x|N −3 2 |x|N +1 1

900


= {|x|=1}

x 2 N − 1 |Bx|2 1 N − 1 N −1 N −1 ∇ B + o(1) dH S − + |x| 2 |x|N +1 2 |x|N −3 2 1

as δ → 0.

(2.12)

Since a direct computation gives

{|x|=1}

x 2 N − 1 |Ax|2 1 dHN −1 = 0 ∇ A − |x| 2 |x|N +1 |x|N −3 2 1

for any constant matrix A ∈ RN ×N , the conclusion of the lemma follows.

2

Integrating (2.10) on {δ < |x| < R} and taking Lemma 2.6 into account, as δ → 0 we obtain N − 1 N −1 x + g(R) = S dHN −1 , Φ(x) · (2.13) 2 |x| |x|=R

where g(R) = |x|=R

BR

G(x) dx and

x dHN −1 = Φ(x) · |x|

1 |x|N −3

|x|=R

+ |x|=R

2 ∂v 1 2 dHN −1 |∇v| − 2 ∂r

1 |x|N −3

f

2 2 2 (1 − |v| )

4

1 N − 1 2 dHN −1 . + 1 − |v| 2 |x|N −1 (2.14)

Multiplying (2.13) by R N −3 , integrating from 0 to R¯ and dividing by R¯ N −2 we have 1 1 N − 1 N −1 + N −2 S ¯ 2N −2 R

R¯ g(R)R

N −3

dR +

1 R¯ N −2

BR¯

0

1 1 E(v, BR¯ ) + N −2 N −2 ¯ ¯ R R

2 ∂v dx ∂r

N −1 2

(1 − |v|2 )

BR¯

|x|2

dx.

(2.15)

Letting R¯ → ∞ and taking Lemma 2.5 into account we infer lim

¯ R→∞

1

R¯ g(R)R

R¯ N −2 0

N −3

dR +

1 R¯ N −2

2 ∂v dx = 0, ∂r BR¯

whence |v| ≡ 1 and ∂v ∂r ≡ 0, because g(R) is an increasing function. As a consequence of (2.6) we x ) for some smooth see that |u(x)| = f (|x|) and it is a radial function. In addition, v(x) = ω( |x|


901

harmonic map ω ∈ C ∞ (S N −1 ; S N −1 ) (harmonic being the limit of u at infinity, see [17]), i.e. (1.5) holds with the profile function f given by Lemma 2.1. Clearly u(x) · u(x) = −|u(x)|2 (1 − |u(x)|2 ) = −f 2 (|x|)(1 − f 2 (|x|)), so it is a radial function. On the other hand (1.5) implies N −1

f N −1

f2 f ω + 2 0 ω · ωf = ff

+ ff + 2 0 ω · ω, u · u = f ω + |x| |x| |x| |x|

where 0 is the Laplace–Beltrami operator on S N −1 . Since ω has values on the sphere and 0 ω and ω are parallel, from the previous formula we conclude that 0 ω · ω is a radial function in RN , therefore −0 ω = λω on S N −1 for some λ 0, i.e. ω is an eigenharmonic map and hence |∇0 ω|2 ≡ λ on S N −1 (here ∇0 is the tangential gradient on the sphere). Finally, since x 2 )| = |x|λ 2 and (1.5) holds, the assumption on the asymptotic energy bound of u together |∇ω( |x| with (2.2) easily implies λ = N −1. Thus, the components of ω are spherical harmonics of degree one, i.e. they are restrictions to the unit sphere of entire affine functions in RN and this fact in x x ) = T |x| for some constant matrix T . Since v takes values on the sphere turn yields v(x) = ω( |x| we infer T ∈ O(N ) and in view of (2.6) the proof is complete. 2 As a direct consequence of the previous results we have a straightforward proof of Theorem 1.1. Proof of Theorem 1.1. (i) ⇒ (ii) Since u satisfies (1.3) and deg∞ u = 0 we deduce that u(x0 ) = 0 for some x0 ∈ RN . Thus, without loss of generality we may assume u(0) = 0 up to translations. Then, the monotonicity formula (1.8) and the asymptotic energy growth yield −1 N −1 N −2 |R for any R > 0, and the conclusion follows from Proposition 2.4. E(u, BR ) 12 N N −2 |S (ii) ⇒ (i) Since u is given by (1.4) the claim follows from Proposition 2.2. 2 3. Local minimality of the equivariant solution A basic ingredient in the construction of a nonconstant local minimizer is the following small energy regularity result taken from [17] (see also [8]). Lemma 3.1. There exist two positive constants η0 > 0 and C0 > 0 such that for any λ 1 and any u ∈ C 2 (B2R (x0 ); RN ) satisfying u + λ2 u 1 − |u|2 = 0 with

1 E (u, B2R (x0 )) η0 , (2R)N−2 λ

then

R 2 sup eλ (u) C0 BR (x0 )

in B2R (x0 ),

1 Eλ u, B2R (x0 ) . N −2 (2R)

(3.1)

We will also make use of the following boundary version of Lemma 3.1 (see [6,7]). Lemma 3.2. Let g : ∂B1 → SN −1 be a smooth map. There exist two positive constants η1 > 0 and C1 > 0 such that for any λ 1, 0 < R < η1 /2, x0 ∈ ∂B1 and any u ∈ C 2 (B¯ 1 ∩ B2R (x0 ); RN ) satisfying u = g on ∂B1 ∩ B2R (x0 ) and

902


u + λ2 u 1 − |u|2 = 0 in B1 ∩ B2R (x0 ), with

1 E (u, B1 (2R)N−2 λ

R2

∩ B2R (x0 )) η1 , then sup

B1 ∩BR (x0 )

eλ (u) C1

1 u, B E ∩ B (x ) . λ 1 2R 0 (2R)N −2

(3.2)

The key result of this section is the following proposition on the behaviour of minimizers in the minimization problems (Pλ ) defined in (1.7). This fact is a weaker extension to higher dimension of the corresponding one in [18]. Proposition 3.3. Let N 3 and B1 = {x ∈ RN s.t. |x| < 1}. For each λ 1 let uλ ∈ 1 (B ; RN ). Then u (x) → u (x) := x H 1 (B1 ; RN ) be a global minimizer of Eλ (·, B1 ) over HId 1 λ ∞ |x| 0 (B 1 N ¯ 1 \ {0}) and for any δ ∈ (0, 1), in H (B1 ; R ) as λ → ∞. In addition, uλ (x) → u∞ (x) in Cloc distH ({|uλ | δ}, {0}) = o(1) as λ → +∞ where distH denotes the Hausdorff distance. Proof. Let us consider an arbitrary sequence λn → +∞, and for every n ∈ N let un ∈ H 1 (B1 ; RN ) be a global minimizer of Eλn (·, B1 ) under the boundary condition un |∂B1 = x (which clearly exists by standard direct method). It is well known that un satisfies |un | 1 and un ∈ C 2 (B¯ 1 ) for every n ∈ N by a simple truncation argument and elliptic regularity respectively. Step 1. We claim that un (x) → u∞ (x) := x/|x| strongly in H 1 (B; RN ). Since the map u∞ is admissible, one has 1 2

|∇un |2 Eλn (un , B1 ) Eλn (u∞ , B1 ) = B1

1 2

|∇u∞ |2 = B1

1 N − 1 N −1 S 2N −2

for every n ∈ N.

(3.3)

As a consequence, {un } is bounded in H 1 (B1 ; RN ) and up to a subsequence, un → u weakly in H 1 (B; RN ) for some SN −1 -valued map u satisfying u|∂B1 = x. By [15,14] and [2] the map u∞ is the unique minimizer of u ∈ H 1 (B1 ; SN −1 ) → B1 |∇u|2 under the boundary condition u|∂B1 = x. In particular, B1 |∇u |2 B1 |∇u∞ |2 which, combined with (3.3), yields 1 2

1 |∇un | → 2

2

B1

1 |∇u | = 2

|∇u∞ |2

2

B1

as n → +∞.

B1

Therefore u ≡ u∞ and un → u∞ strongly in H 1 (B; RN ). Step 2. Let δ ∈ (0, 1) be fixed. We now prove that the family of compact sets Vn := {|un | δ} → {0} in the Hausdorff sense. It suffices to prove for any given 0 < ρ < 1, Vn ⊂ Bρ for every n large enough. Since u∞ is smooth outside the origin, we can find 0 < σ min(ρ/8, η1 /4) such that 1 σ N −2

B1 ∩B4σ (x)

|∇u∞ |2 < min(η0 , η1 ) := for every x ∈ B¯ 1 \ Bρ ,


903

where η0 and η1 are given by Lemma 3.1 and Lemma 3.2 respectively. From the strong convergence of un to u∞ in H 1 , we infer that u E , B (x) < for every x ∈ B¯ 1 \ Bρ λ n 4σ n N −2 1

σ

(3.4)

whenever n N1 for some integer N1 independent of x. Next consider a finite family of points {xj }j ∈J ⊂ B¯ 1 \ Bρ satisfying B2σ (xj ) ⊂ B1 if xj ∈ B1 and B¯ 1 \ Bρ ⊂

Bσ (xj ) ∪ B2σ (xj ) .

xj ∈B1

xj ∈∂B1

In view of (3.4), for each j ∈ J we can apply Lemma 3.1 in B2σ (xj ) if xj ∈ B1 and Lemma 3.2 in B1 ∩ B4σ (xj ) if xj ∈ ∂B1 to deduce sup eλn (un ) Cσ −2

B¯ 1 \Bρ

for every n N1 ,

for some constant C = max{C0 , C1 } independent of n. By Ascoli Theorem the sequence {un } is compact in C 0 (B¯ 1 \ Bρ ), thus un → u∞ and |un | → 1 uniformly in B¯ 1 \ Bρ . In particular |un | > δ in B¯ 1 \ Bρ whenever n is large enough, i.e. Vn ⊂ Bρ for every n sufficiently large. 2 The main step in our study of local minimality of (1.4) consists in the following result giving the existence of nonconstant local minimizers. Theorem 3.4. For each N 3 there exists a smooth nonconstant solution u : RN → RN of (1.1) −1 N −1 | for which is a local minimizer of E(·). In addition, u(0) = 0 and R 2−N E(u, BR ) 12 N N −2 |S any R > 0. 1 (B ; RN ). Proof. Consider a sequence λn → +∞ and let un be a minimizer of Eλn (·, B1 ) on HId 1 2 N Since un ∈ C (B¯ 1 ; R ) and Proposition 3.3 holds, by elementary degree theory we may find an ∈ B1/2 such that un (an ) = 0 for every n sufficiently large and an → 0 as n → ∞. Setting Rn := λn (1 − |an |), Rn → +∞ as n → +∞, and we define for x ∈ BRn , u¯ n (x) := ¯ n clearly satisfies un (λ−1 n x + an ) so that u

u¯ n + u¯ n 1 − |u¯ n |2 in BRn , u¯ n (0) = 0 and |u¯ n | 1 for every n. Moreover taking (3.3) and the strong convergence of un in H 1 into account, it is easy to see that 2−N lim sup Rn2−N E(u¯ n , BRn ) = lim sup λ−1 Eλn un , Bλ−1 Rn (an ) n Rn n→+∞

n→+∞

1 N − 1 N −1 . S 2N −2

n

(3.5)

Then we infer from standard elliptic regularity that, up to a subsequence, u¯ n → u in 2 (RN ; RN ) for some map u : RN → RN solving u + u(1 − |u|2 ) = 0 in RN and satisCloc fying u(0) = 0. Next we deduce from (3.5), the monotonicity formula (1.8) and the smooth

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N −1 N −1 convergence of u¯ n to u, that supR>0 R 2−N E(u, BR ) 12 N |. Finally, the local minimal−2 |S ity of u easily follows from the minimality of u¯ n (i.e. of un ) and the convergence of u¯ n to u in 2 (RN ; RN ). 2 Cloc

Proof of Theorem 1.4. Let u be the local minimizer given by Theorem 3.4. Since u(0) = 0 and u has the correct energy bound at infinity we can apply Proposition 2.4 to conclude that up to isometries u = U as given by (1.4), i.e. the equivariant solution U is locally energy minimizing. Finally, the stability inequality (1.9) is a straightforward consequence of the energy minimality by computing the second variation. 2 As a concluding remark we would like to mention two open problems connected to Theorem 1.4. When N = 3, in the main result of [18] a complete characterization of nonconstant local minimizers of E(·) was given assuming an energy bound at infinity. It is very natural to ask the same question in all dimensions. Open problem 1. Is any entire nonconstant local energy minimizer u of E(·) (possibly under the assumption supR>0 R 2−N E(u, BR ) < ∞) of the form (1.4) (up to translations on the domain and orthogonal transformation on the image)? The answer is affirmative for N = 3 essentially because of the classification of the blowdown maps from infinity, i.e. the locally energy minimizing homogeneous harmonic maps u∞ ∈ 1 (RN ; S N −1 ) obtained scaling u from infinity. In that case, using Theorems 7.3 and 7.4 in [5], Hloc x it has been proved that, up to an orthogonal transformation, uR (x) := u(Rx) → u∞ (x) = |x| 1 along subsequences R → ∞. The analogous result for N 4 seems unknown and it in Hloc n would be the main ingredient in proving the converse of Theorem 1.4, giving a positive answer to the open problem stated above. Thus, we ask the same question for harmonic maps in higher dimension. Open problem 2. If N > 3, is any homogeneous nonconstant local energy minimizer of the x 1 (RN ; S N −1 ) of the form Dirichlet integral among sphere-valued maps u(x) = ω( |x| ) ∈ Hloc x u(x) = |x| (possibly up to rotations)? It would be interesting to address the previous question even under the additional assumption that ω ∈ C ∞ (S N −1 ; S N −1 ) is an eigenharmonic map (i.e. the components are spherical harmonics on S N −1 with the same eigenvalue). In that case the quantization property of the energy for eigenharmonic maps (see [11]) seems to suggest that the answer could be affirmative, the value for the identity map being the least possible. Acknowledgments The work was initiated after the workshop “Differential and topological problems in modern theoretical physics” held at SISSA in Trieste, on April 2010. The author would like to thank Bob Hardt for a useful discussion which renewed his interest on the subject. References [1] V. Akopian, A. Farina, Sur les solutions radiales de l’équation −u = u(1 − |u|2 ) dans RN (N 3), C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 601–604.


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[2] A. Baldes, Stability and uniqueness properties of the equator map from a ball into an ellipsoid, Math. Z. 185 (1984) 505–516. [3] F. Bethuel, H. Brezis, F. Hélein, Ginzburg–Landau Vortices, Progr. Nonlinear Differential Equations Appl., vol. 13, Birkhäuser, Boston, 1994. [4] H. Brezis, Symmetry in nonlinear PDE’s, in: Differential Equations: La Pietra 1996, Florence, in: Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, 1999, pp. 1–12. [5] H. Brezis, J.M. Coron, E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986) 649–705. [6] Y. Chen, Dirichlet problems for heat flows of harmonic maps in higher dimensions, Math. Z. 208 (1991) 557–565. [7] Y. Chen, F.H. Lin, Evolution of harmonic maps with Dirichlet boundary conditions, Comm. Anal. Geom. 1 (1993) 327–346. [8] Y. Chen, M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989) 83–103. [9] M. del Pino, P. Felmer, M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg–Landau vortex as a Hardy’s type inequality, Int. Math. Res. Not. 30 (2004) 1511–1527. [10] A. Farina, Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg– Landau system in R k , Differential Integral Equations 11 (1998) 875–893. [11] A. Farina, Two results on entire solutions of Ginzburg–Landau system in higher dimensions, J. Funct. Anal. 214 (2) (2004) 386–395. [12] A. Farina, M. Guedda, Qualitative study of radial solutions of the Ginzburg–Landau system in RN (N 3), Appl. Math. Lett. 13 (2000) 59–64. [13] S. Gustafson, Symmetric solutions of the Ginzburg–Landau equation in all dimensions, Int. Math. Res. Not. 16 (1997) 807–816. [14] W. Jäger, H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew. Math. 343 (1983) 146–161. [15] F.H. Lin, A remark on the map x/|x|, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 529–531. [16] F.H. Lin, T. Rivière, Energy quantization for harmonic maps, Duke Math. J. 111 (2002) 177–193. [17] F.H. Lin, C.Y. Wang, Harmonic and quasi-harmonic spheres. II, Comm. Anal. Geom. 10 (2002) 341–375. [18] V. Millot, A. Pisante, Symmetry of local minimizers for the three dimensional Ginzburg–Landau functional, J. Eur. Math. Soc. (JEMS) 12 (2010) 1069–1096. [19] P. Mironescu, Les minimiseurs locaux pour l’équation de Ginzburg–Landau sont à symétrie radiale, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 593–598. [20] F. Pacard, T. Rivière, Linear and Non-linear Aspects of Vortices, Birkhäuser, 2000. [21] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983) 525–571.


Composition operators on noncommutative Hardy spaces ✩ Gelu Popescu Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA Received 19 July 2010; accepted 21 September 2010


Abstract 2 , In this paper we initiate the study of composition operators on the noncommutative Hardy space Hball which is the Hilbert space of all free holomorphic functions of the form

f (X1 , . . . , Xn ) =

∞

aα Xα ,

|aα |2 < 1,

α∈F+ n

k=0 |α|=k

where the convergence is in the operator norm topology for all (X1 , . . . , Xn ) in the noncommutative operatorial ball [B(H)n ]1 and B(H) is the algebra of all bounded linear operators on a Hilbert space H. When the symbol ϕ is a free holomorphic self-map of [B(H)n ]1 , we show that the composition operator Cϕ f := f ◦ ϕ,

2 , f ∈ Hball

2 . Several classical results about composition operators (boundedness, norm estimates, is bounded on Hball spectral properties, compactness, similarity) have free analogues in our noncommutative multivariable setting. The most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of B(H)n , the operator algebras generated by the left creation operators on the full Fock space with n generators, and the classical complex function theory in the unit ball of Cn . In a more general setting, we establish basic properties concerning the composition operators acting on Fock spaces associated with noncommutative varieties VP0 (H) ⊆ [B(H)n ]1 generated by sets P0 of noncommutative polynomials in n indeterminates such that p(0) = 0, p ∈ P0 . In particular, when P0 consists of the ✩

Research supported in part by an NSF grant. E-mail address: [email protected].


G. Popescu / Journal of Functional Analysis 260 (2011) 906–958

907

commutators Xi Xj − Xj Xi for i, j = 1, . . . , n, we show that many of our results have commutative counterparts for composition operators on the symmetric Fock space and, consequently, on spaces of analytic functions in the unit ball of Cn . © 2010 Elsevier Inc. All rights reserved. Keywords: Composition operator; Noncommutative Hardy space; Fock space; Creation operator; Free holomorphic function; Free pluriharmonic function; Compact operator; Spectrum; Similarity; Noncommutative variety

Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Noncommutative Littlewood subordination principle . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ................. 2. Composition operators on the noncommutative Hardy space Hball 3. Noncommutative Wolff theorem for free holomorphic self-maps of [B(H)n ]1 . . . . . . . . . 4. Composition operators and their adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 .................................. 5. Compact composition operators on Hball 6. Schröder equation for noncommutative power series and spectra of composition operators 7. Composition operators on Fock spaces associated to noncommutative varieties . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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907 911 917 929 932 937 945 951 957

0. Introduction An important consequence of Littlewood’s subordination principle [12,6] is the boundedness of the composition operator Cϕ on the Hardy space H 2 (D), when ϕ : D → D is an analytic self-map of the open unit disc D := {z ∈ C: |z| < 1} and Cϕ f := f ◦ ϕ. This result was the starting point of the modern theory of composition operators on spaces of analytic functions, which has been developed since the 1960’s through the fundamental work of Ryff [42], Nordgren [18,19], Schwartz [46], Shapiro [44], Cowen [2] and many others (see [45,3,1], and the references therein). They answered basic questions about composition operators such as boundedness, compactness, spectra, cyclicity, revealing a beautiful interaction between operator theory and complex function theory. In the multivariable setting, when ϕ is a holomorphic self-map of the open unit ball Bn := z = (z1 , . . . , zn ) ∈ Cn : z2 < 1 , the composition operator Cϕ is no longer a bounded operator on the Hardy space H 2 (Bn ). However, significant work was done concerning the spectra of automorphism-induced composition operators and compact composition operators on H 2 (Bn ) by MacCluer [13–15] and others (see [3] and its references). The study of composition operators on the Hardy space H 2 (Bn ) is close connected to the several variable function theory in the unit ball of Cn [41]. There is an extensive literature on composition operators on other spaces of analytic functions in several variables (see [3]). For the interested reader, we mention two very nice books on composition operators: Shapiro’s monograph [45], which is an excellent account of composition operators on H 2 (D) and the

908


monograph [3] by Cowen and MacCluer, which is a comprehensive treatment of composition operators on spaces of analytic functions in one or several variables. It is our hope that the present paper will open a new chapter in the theory of composition operators. The goal is to initiate the study of composition operators on the noncommutative 2 (which will be introduced shortly) and, more generally, on subspaces of the Hardy space Hball full Fock space with n generators associated to noncommutative varieties. The most prominent feature of this paper is the interplay between the noncommutative analytic function theory in the unit ball of B(H)n , the operator algebras generated by the left creation operators S1 , . . . , Sn on the full Fock space with n generators: the Cuntz–Toeplitz algebra C ∗ (S1 , . . . , Sn ) [4], the noncommutative disk algebra An and the analytic Toeplitz algebra Fn∞ [26–29], as well as the classical function theory in the unit ball of Cn [41]. To present our results we need some notation and preliminaries on free holomorphic functions. Initiated in [33], the theory of free holomorphic (resp. pluriharmonic) functions on the unit ball of B(H)n , where B(H) is the algebra of all bounded linear operators on a Hilbert space H, has been developed very recently (see [34–39]) in the attempt to provide a framework for the study of arbitrary n-tuples of operators on a Hilbert space. Several classical results from complex analysis and hyperbolic geometry have free analogues in this noncommutative multivariable setting. Related to our work, we mention the papers [8,16,17], and [48], where several aspects of the theory of noncommutative analytic functions are considered in various settings. We recall that the algebra Hball of free holomorphic functions on the open operatorial n-ball of radius one is defined as the set of all power series α∈F+n aα Zα with radius of convergence 1, i.e., {aα }α∈F+n are complex numbers with lim supk→∞ ( |α|=k |aα |2 )1/2k 1, where F+ n is the free semigroup with n generators g1 , . . . , gn and the identity g0 . The length of α ∈ F+ is defined by |α| := 0 if n α = g0 and |α| := k if α = gi1 · · · gik , where i1 , . . . , ik ∈ {1, . . . , n}. If (X1 , . . . , Xn ) ∈ B(H)n , we denote Xα := Xi1 · · · Xik and Xg0 := IH . A free holomorphic function on the open ball 1/2 B(H)n 1 := (X1 , . . . , Xn ) ∈ B(H)n : X1 Xn∗ + · · · + Xn Xn∗ < 1 , is the representation of an element f ∈ Hball on the Hilbert space H, that is, the mapping ∞ B(H)n 1 (X1 , . . . , Xn ) → f (X1 , . . . , Xn ) := aα Xα ∈ B(H), k=0 |α|=k

where the convergence is in the operator norm topology. Due to the fact that a free holomorphic function is uniquely determined by its representation on an infinite dimensional Hilbert space, throughout this paper, we identify a free holomorphic function with its representation on a separable infinite dimensional Hilbert space. A free holomorphic function f on [B(H)n ]1 is bounded if f ∞ := sup f (X) < ∞, where the supremum is taken over all X ∈ [B(H)n ]1 and H is an infinite dimensional Hilbert space. Let ∞ be the set of all bounded free holomorphic functions and let A Hball ball be the set of all elements ∞ such that the mapping f ∈ Hball B(H)n 1 (X1 , . . . , Xn ) → f (X1 , . . . , Xn ) ∈ B(H) ∞ has a continuous extension to the closed unit ball [B(H)n ]− 1 . We showed in [33] that Hball and Aball are Banach algebras under pointwise multiplication and the norm · ∞ , which can be


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identified, via the noncommutative Poisson transform [30], with the noncommutative analytic Toeplitz algebra Fn∞ and the noncommutative disc algebra An , respectively. If f : [B(H)n ]1 → B(H) and ϕ : [B(H)n ]1 → [B(H)n ]1 are free holomorphic functions then f ◦ ϕ is a free holomorphic function on [B(H)n ]1 (see [38]), defined by (f ◦ ϕ)(X1 , . . . , Xn ) =

∞

aα ϕα (X1 , . . . , Xn ),

k=0 |α|=k

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

2 where the convergence is in the operator norm topology. The noncommutative Hardy space Hball n is the Hilbert space of all free holomorphic functions on [B(H) ]1 of the form

f (X1 , . . . , Xn ) =

∞

aα X α ,

|aα |2 < 1,

α∈F+ n

k=0 |α|=k

∞ with the inner product f, g := ∞ k=0 k=0 |α|=k aα bα , where g = |α|=k bα Xα is another 2 free holomorphic function in Hball . The main question that we answer in this paper is whether 2 for any f ∈ H 2 and whether the corresponding composition operator is bounded. f ◦ ϕ ∈ Hball ball This will be the starting point in our attempt to develop a theory of compositions operators on noncommutative Hardy spaces. We are interested in extracting properties of the composition operator Cϕ (boundedness, spectral properties, compactness) from the operatorial or dynamical properties of the model boundary function ϕ := SOT- limr→1 ϕ(rS1 , . . . , rSn ) ∈ Fn∞ ⊗ Cn or the scalar representation of ϕ, i.e., the holomorphic function Bn λ → ϕ(λ) ∈ Bn . In Section 1, we characterize the free holomorphic self-maps of [B(H)n ]1 in terms of the model boundary functions with respect to the left creation operators on the full Fock space 2 with F 2 (Hn ), to F 2 (Hn ). This will be used, together with the natural identification of Hball 2 . More provide a noncommutative Littlewood subordination theorem for the Hardy space Hball n precisely, we show that if ϕ is a free holomorphic self-map of the ball [B(H) ]1 such that 2 , then f ◦ ϕ ∈ H 2 and f ◦ ϕ f . ϕ(0) = 0 and f ∈ Hball 2 2 ball Section 2 contains the core material on boundedness of compositions operators on the non2 and estimates for their norms. An important role in our investigacommutative Hardy space Hball 2 in terms of pluriharmonic majorants [34] and tion will be played by the characterization of Hball the Herglotz–Riesz type representation for positive free pluriharmonic functions [37]. The key result of this section asserts that if ϕ is a free holomorphic automorphism of the noncommutative ball [B(H)n ]1 (see [38]), then

1 − ϕ(0) 1 + ϕ(0)

1/2

f Cϕ f

1 + ϕ(0) 1 − ϕ(0)

1/2 f

2 . Moreover, these inequalities are best possible and we have a formula for the for any f ∈ Hball norm of Cϕ . Combining this result with the noncommutative Littlewood subordination theorem from the previous section, we obtain the main result which asserts that, for any free holomorphic 2 and self-map ϕ of [B(H)n ]1 , the composition Cϕ f := f ◦ ϕ is a bounded operator on Hball

1 Cϕ (1 − ϕ(0)2 )1/2

1 + ϕ(0) 1 − ϕ(0)

1/2 .

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This leads to an extension of Cowen’s [2] one-variable spectral radius formula for composition operators to our noncommutative multivariable setting. More precisely, we obtain −1/2k r(Cϕ ) = lim 1 − ϕ [k] (0) , k→∞

where ϕ [k] is the k-iterate of ϕ. Another consequence of the above-mentioned result is that Cϕ is similar to a contraction if and only if there is ξ ∈ Bn such that ϕ(ξ ) = ξ . This will also show 2 to contractions is equivalent to power (resp. that similarity of composition operators on Hball polynomial) boundedness. This is interesting in light of Pisier’s [22] famous example of a polynomially bounded operator which is not similar to a contraction, and Paulsen’s [20] result that every completely polynomially bounded operator is similar to a contraction. For more information on similarity problems we refer the reader to [21] and [23]. In Section 3, extending the classical result obtained by Wolff [50,51] and MacCluer’s version for Bn (see [13]), we provide a noncommutative analogue of Wolff’s theorem for free holomorphic self-maps of [B(H)n ]1 . We show that if ϕ : [B(H)n ]1 → [B(H)n ]1 is a free holomorphic function such that its scalar representation has no fixed points in Bn , then there is a unique point ζ ∈ ∂Bn (the Denjoy–Wolff point of ϕ) such that each noncommutative ellipsoid Ec (ζ ) (see Section 3 for the definition) is mapped into itself by every iterate of the symbol ϕ. We also show 2 is 1 when the symbol is elliptic or that the spectral radius of a composition operator on Hball parabolic, which extends some of Cowen’s results [2] from the single variable case. 2 . It is shown In Section 4, we obtain a formula for the adjoint of a composition operator on Hball that if ϕ = (ϕ1 , . . . , ϕn ) is a free holomorphic self-map of the noncommutative ball [B(H)n ]1 , then Cϕ∗ f = f, ϕα eα , α∈F+ n

where f and ϕ1 , . . . , ϕn are seen as elements of the Fock space F 2 (Hn ). As a consequence we prove that Cϕ is normal if and only if ϕ(X1 , . . . , Xn ) = [X1 . . . Xn ]A for some normal scalar matrix A ∈ Mn×n with A 1. This leads to characterizations of 2 . A nice connection between Fredholm self-adjoint or unitary composition operators on Hball 2 composition operators on Hball and the automorphisms of the open unit ball Bn is also presented. In Section 5, we study compact composition operators on the noncommutative Hardy space 2 . Using some of Shapiro’s arguments from the single variable case (see [44]) in our setting Hball as well as some results from Section 4, we obtain a formula for the essential norm of the compo2 . In particular, this implies that C is a compact operator if and only sition operator Cϕ on Hball ϕ if f, ϕα 2 = 0. lim sup k→∞

2 ,f 1 f ∈Hball 2 |α|k

2 , then the scalar representation of ϕ Moreover, we show that if Cϕ is a compact operator on Hball is a holomorphic self-map of Bn which


911

(i) cannot have finite angular derivative at any point of ∂Bn , and (ii) has exactly one fixed point in the open ball Bn . 2 is similar to As a consequence, we deduce that every compact composition operator on Hball a contraction. In the end of this section, we prove that the set of compact composition operators 2 is arcwise connected in the set of all composition operators with respect to the operator on Hball norm topology. In Section 6, we consider a noncommutative multivariable extension of Schröder equation 2 [43] which is used to obtain results concerning the spectrum of composition operators on Hball (see Theorem 6.4). Combining these results with those from Section 5, we determine the spectra 2 . More precisely, if ϕ is a free holomorphic self-map of compact composition operators on Hball 2 , then n of the noncommutative ball [B(H) ]1 and Cϕ is a compact composition operator on Hball the scalar representation of ϕ has a unique fix point ξ ∈ Bn and the spectrum σ (Cϕ ) consists of 0, 1, and all possible products of the eigenvalues of the matrix

ψi , ej n×n , where ψ = (ψ1 , . . . , ψn ) := Φξ ◦ ϕ ◦ Φξ and Φξ is the involutive free holomorphic automorphism of [B(H)n ]1 associated with ξ , the functions ψ1 , . . . , ψn are seen as elements of the Fock space F 2 (Hn ), and the Hilbert space Hn has e1 , e2 , . . . , en as orthonormal basis. In Section 7, we consider composition operators on Fock spaces associated to noncommutative varieties in unit ball [B(H)n ]1 . Given a set P0 of noncommutative polynomials in n indeterminates such that p(0) = 0, p ∈ P0 , we define a noncommutative variety VP0 (H) ⊆ [B(H)n ]1 by setting VP0 (H) := (X1 , . . . , Xn ) ∈ B(H)n 1 : p(X1 , . . . , Xn ) = 0 for all p ∈ P0 . According to [32], there is a universal model (B1 , . . . , Bn ) associated with the noncommutative variety VP0 (H), where Bi = PNP0 Si |NP0 and NP0 is a subspace of the full Fock space F 2 (Hn ). Let Fn∞ (VP0 ) be the w ∗ -closed algebra generated by B1 , . . . , Bn and the identity. Using the results from Section 2 and the noncommutative commutant lifting theorem [24] (see [47] for the ∈ Fn∞ (VP ) ⊗ Cn with ψ 1, one can define classical case n = 1), we show that given any ψ 0 a composition operator Cψ : NP0 → NP0 , which turns out to be bounded. Many results from the previous sections have analogues in this more general setting. In particular, if Pc := {Xi Xj − Xj Xi : i, j = 1, . . . , n}, then NPc coincides with the symmetric Fock space. As a consequence, many of our results have commutative counterparts for composition operators on the symmetric Fock space and on spaces of analytic functions in the unit ball of Cn . 1. Noncommutative Littlewood subordination principle In this section, we characterize the free holomorphic self-maps of the unit ball [B(H)n ]1 in terms of the model boundary functions with respect to the left creation operators on the full Fock space F 2 (Hn ). This will be used to provide a noncommutative Littlewood subordination theorem 2 . for the Hardy space Hball

912


Let Hn be an n-dimensional complex Hilbert space with orthonormal basis e1 , e2 , . . . , en , where n ∈ {1, 2, . . .}. We consider the full Fock space of Hn defined by F 2 (Hn ) := C1 ⊕

Hn⊗k ,

k1

where Hn⊗k is the (Hilbert) tensor product of k copies of Hn . We denote eα := ei1 ⊗ · · · ⊗ eik if α = gi1 · · · gik , where i1 , . . . , ik ∈ {1, . . . , n}, and eg0 := 1. Note that {eα }α∈F+n is an orthonormal basis for F 2 (Hn ). Define the left (resp. right) creation operators Si (resp. Ri ), i = 1, . . . , n, acting on F 2 (Hn ) by setting Si ϕ := ei ⊗ ϕ,

ϕ ∈ F 2 (Hn ),

(resp. Ri ϕ := ϕ ⊗ ei ). Note that Si Rj = Rj Si for i, j ∈ {1, . . . , n}. The noncommutative disc algebra An (resp. Rn ) is the norm closed algebra generated by the left (resp. right) creation operators and the identity. The noncommutative analytic Toeplitz algebra Fn∞ (resp. Rn∞ ) is the weakly closed version of An (resp. Rn ). These algebras were introduced in [26] in connection with a noncommutative version of the classical von Neumann inequality [49]. Let C ∗ (S1 , . . . , Sn ) be the Cuntz–Toeplitz C ∗ -algebra generated by the left creation operators (see [4]). The noncommutative Poisson transform at X := (X1 , . . . , Xn ) ∈ [B(H)n ]− 1 is the unital completely contractive linear map PX : C ∗ (S1 , . . . , Sn ) → B(H) defined by ∗ (f ⊗ IH )KrX , PX [f ] := lim KrX r→1

f ∈ C ∗ (S1 , . . . , Sn ),

where the limit exists in the operator norm topology of B(H). Here, KrX : H → F 2 (Hn ) ⊗ H, 0 < r 1, is the noncommutative Poisson kernel defined by KrX h :=

∞

eα ⊗ r |α| rX Xα∗ h,

h ∈ H,

k=0 |α|=k

where rX := (IH − r 2 X1 X1∗ − · · · − r 2 Xn Xn∗ )1/2 . We recall that PX Sα Sβ∗ = Xα Xβ∗ ,

α, β ∈ F+ n.

When X := (X1 , . . . , Xn ) is a pure row contraction, i.e. SOT- limk→∞ we have PX [f ] = KX∗ (f ⊗ IH )KX ,

∗ |α|=k Xα Xα

= 0, then

f ∈ C ∗ (S1 , . . . , Sn ) or f ∈ Fn∞ .

α = gik · · · gik is Under an appropriate modification of the Poisson kernel (eα becomes e α where ∗ (R , . . . , R ) of R ∞ . For simplicthe reverse of α = gi1 · · · gik ∈ F+ ), similar results hold for C 1 n n n ity, we use the same notation for the noncommutative Poisson transform. We refer to [30,31,35] for more on noncommutative Poisson transforms on C ∗ -algebras generated by isometries. ∞ (see the introduction) can According to [33] and [37], the noncommutative Hardy space Hball ∞ be identified with the noncommutative analytic Toeplitz algebra Fn . More precisely, a bounded


913

free holomorphic function ψ on [B(H)n ]1 is uniquely determined by its (model) boundary func (S1 , . . . , Sn ) ∈ Fn∞ defined by tion ψ =ψ (S1 , . . . , Sn ) := SOT- lim ψ(rS1 , . . . , rSn ). ψ r→1

(S1 , . . . , Sn ) at X := (X1 , . . . , Xn ) ∈ Moreover, ψ is the noncommutative Poisson transform of ψ [B(H)n ]1 , i.e., (S1 , . . . , Sn ) . ψ(X1 , . . . , Xn ) = PX ψ Similar results hold for bounded free holomorphic functions on the noncommutative ball [B(H)n ]1 with operator-valued coefficients. There are also versions of these results when the boundary function is taken with respect to the right creation operators R1 , . . . , Rn . Throughout this paper, we deal with free holomorphic self-maps of the unit ball [B(H)n ]1 . The following results gives us, in particular, a characterization of these maps in terms of the model boundary functions with respect to the left creation operators on the full Fock space F 2 (Hn ). For simplicity, [X1 , . . . , Xn ] denotes either the n-tuple (X1 , . . . , Xn ) ∈ B(H)n or the operator row matrix [X1 . . . Xn ] acting from H(n) , the direct sum of n copies of a Hilbert space H, to H. Theorem 1.1. Let ϕ : [B(H)n ]1 → [B(H)m ]− 1 be a free holomorphic function. Then the following statements hold. (i) Either ϕ([B(H)n ]1 ) ⊆ [B(H)m ]1 or there exists ζ ∈ ∂Bm such that ϕ(X) = ζ for all X ∈ [B(H)n ]1 . (ii) ϕ is constant if and only if ϕ(0) = ϕ∞ . (iii) If ϕ is non-constant and ϕr (X) := ϕ(rX), X ∈ [B(H)n ]1 , then the map [0, 1) r → ϕr ∞ is strictly increasing. (iv) If ϕ is the boundary function of ϕ with respect to S1 , . . . , Sn , then ϕ([B(H)n ]1 ) ⊆ [B(H)m ]1 if and only if either ϕ = ζ I for some ζ ∈ Bn or ϕ is non-constant with ϕ 1. Proof. If ϕ∞ < 1, then (i) holds. Assume that ϕ∞ = 1. In this case, if ϕ(0) < 1 then, according to the maximum principle for free holomorphic functions (see Proposition 5.2 from [38]), we have ϕ(X) < 1 for all X ∈ [B(H)n ]1 . It remains to consider the case when ϕ(0) = 1. Set ζ = [ζ1 , . . . , ζm ] := ϕ(0) ∈ ∂Bm and let U ∈ Mm×m be a unitary matrix such that [ζ1 , . . . , ζm ]U = ξ1 := [1, 0, . . . , 0] ∈ ∂Bm . Let ϕU (X) := [X1 , . . . , Xm ]U and note that g := ϕU ◦ ϕ : [B(H)n ]1 → [B(H)m ]− 1 is a free holomorphic function with g(0) = ξ1 . Setting g = (g1 , . . . , gm ), we deduce that gi are free holomorphic functions with g1 (0) = 1 and gi (0) = 0 if i = 2, . . . , m. Applying Theorem 5.1 from [38] to g1 , we deduce that g1 (X) = 1 for all X ∈ [B(H)n ]1 . Hence g2 = · · · = gm = 0. This implies that ϕ(X) = ζ for all X ∈ [B(H)n ]1 , and completes the proof of item (i). Since the direct implication in item (ii) is obvious, we assume that ϕ(0) = ϕ∞ and ϕ∞ = 1. The rest of the proof of (ii) is contained in the proof of item (i). To prove item (iii), assume that ϕ is non-constant. Due to part (ii), we must have ϕ(0) < ϕ∞ . Using again Proposition 5.2 from [38], we have ϕ(X) < ϕ∞ for all X ∈ [B(H)n ]1 . ϕr is in An ⊗ M1×m , Let 0 r1 < r2 < 1. We recall that, if r ∈ [0, 1), then the boundary function ϕr = ϕr (S1 , . . . , rSn ). Using where An is the noncommutative disc algebra and ϕr ∞ =

914


the noncommutative von Neumann inequality (see [26]) and applying the above-mentioned result to ϕr2 and (X1 , . . . , Xn ) := ( rr12 S1 , . . . , rr12 Sn ), we obtain

r1 r1 ϕ ϕr1 ∞ = ϕr1 (S1 , . . . , Sn ) = S , . . . , S n < ϕr2 (S1 , . . . , Sn ) = ϕr2 ∞ , r2 r 1 r2 2 which shows that (iii) holds. ϕ = ϕ∞ 1 and the result folNow we prove (iv). If ϕ([B(H)n ]1 ) ⊆ [B(H)m ]1 , then lows. Conversely, assume that ϕ 1 and ϕ is not of the form ζ I for some ζ ∈ Bn . Then ϕ is not a constant and due to (ii) we have ϕ(0) < ϕ∞ . Using now item (iii), we deduce that the map [0, 1) r → ϕr ∞ is strictly increasing. If X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 , then there is r ∈ [0, 1) such that X < r. Consequently, due to the noncommutative von Neumann inequality, we have ϕ(X1 , . . . , Xn ) ϕ(rS1 , . . . , rSn ) = ϕr ∞ < 1. The proof is complete.

2

2 if and only sup Note that if f ∈ Hball , then f ∈ Hball r∈[0,1) f (rS1 , . . . , rSn )1 < ∞. Moreover, in this case, we have

f 2 = lim f (rS1 , . . . , rSn )1 = sup f (rS1 , . . . , rSn )1. r→1

If f =

∞ k=0

|α|=k aα Xα

r∈[0,1)

and g =

∞ k=0

|α|=k bα Xα

2 , then are in Hball

f, g = lim f (rS1 , . . . , rSn )1, g(rS1 , . . . , rSn )1 F 2 (H ) n r→1 = aα eα , bα eα . α∈F+ n

α∈F+ n

F 2 (Hn )

2 can be identified with the full Fock space Consequently, the noncommutative Hardy space Hball 2 → F 2 (H ) defined by the mapping F 2 (Hn ), via the unitary operator U : Hball n 2 Hball

∞

aα Xα →

k=0 |α|=k

∞

aα eα ∈ F 2 (Hn ).

k=0 |α|=k

This identification will be used throughout the paper whenever necessary. We recall from [38] that if f : [B(H)n ]1 → B(H) and ϕ : [B(H)n ]1 → [B(H)n ]1 are free holomorphic functions then f ◦ ϕ is a free holomorphic function on [B(H)n ]1 defined by (f ◦ ϕ)(X1 , . . . , Xn ) =

∞

aα ϕα (X1 , . . . , Xn ),

k=0 |α|=k

where the convergence is in the operator norm topology.

(X1 , . . . , Xn ) ∈ B(H)n 1 ,


915

We can prove now the following noncommutative Littlewood subordination theorem for the 2 , which will play an important role in this paper. Hardy space Hball Theorem 1.2. Let ϕ be a free holomorphic self-map of the ball [B(H)n ]1 such that ϕ(0) = 0, 2 . Then f ◦ ϕ ∈ H 2 and f ◦ ϕ f . and let f ∈ Hball 2 2 ball Proof. Let ϕ := (ϕ1 , . . . , ϕn ) be a free holomorphic self-map of the ball [B(H)n ]1 such that ϕn ) ∈ Fn∞ ⊗ Cn be the model boundary function with respect to ϕ(0) = 0, and let ϕ = ( ϕ1 , . . . , the left creation operators S1 , . . . , Sn . Thus ϕi := SOT- limr→1 ϕi (rS1 , . . . , rSn ) for i = 1, . . . , n. 2 Let Pn be the set of all polynomials in F 2 (Hn ) and define C ϕ : Pn → F (Hn ) by setting

C aα eα := aα ϕα (1). ϕ |α|m

|α|m

2 , then p := Uq = If q := |α|m aα Xα is a polynomial in Hball |α|m aα eα is a polynomial in F 2 (Hn ). Note that p = p(0) + ni=1 Si (Si∗ p), where p(0) = PC p = a0 := ag0 . Hence, we deduce that C ϕ p = a0 +

n

∗ ϕi C ϕ Si p .

i=1

∗ 2 Since ϕ(0) = 0, the vector ni=1 ϕi C ϕ (Si p) is orthogonal to the constants in F (Hn ). Consequently, using the fact that [ ϕ1 , . . . , ϕn ] is a row contraction, we have 2 C ϕ p2

n 2 ∗ = |a0 | + ϕi C ϕ Si p 2

i=1

n 2

2 ∗ |a0 | + C ϕ Si p . i=1

Note that, for each i = 1, . . . , n, we have n ∗ ∗ ∗ ∗ S p = S p (0) + ϕj C C ϕ i ϕ Sj Si p . i j =1

Hence, using again that ϕ(0) = 0 and that [ ϕ1 , . . . , ϕn ] is a row contraction, we deduce that n 2 n 2 n n 2 ∗ ∗

∗ ∗ Si p (0) + C ϕj C ϕ Si p = ϕ Sj Si p i=1

i=1

i=1

j =1

n 2 n ∗ ∗ 2 |aα | + ϕj C ϕ Sj Si p |α|=1

i=1 j =1

∗ 2 2 |aα | + C ϕ Sβ p . |α|=1

|β|=2

916


Similarly, for any k ∈ {1, . . . , m + 1}, we obtain ∗ 2 ∗ 2 2 . S S C p |a | + C p ϕ β α ϕ β |β|=k−1

|β|=k

|α|=k−1

Using these relations and the fact that Sγ∗ p = 0 for |γ | m + 1, we obtain 2 C ϕ p2

|aα |2 = p22 .

|α|m

Since UCϕ U −1 p = C ϕ p, we deduce that Cϕ q2 q2

2 for any polynomial q ∈ Hball .

(1.1)

2 for any f ∈ H 2 and f ◦ϕ f . Let f (X , . . . , X ) = Now, we prove that f ◦ϕ is in Hball 2 2 1 n ball ∞ 2 . Then f ◦ ϕ is a free holomorphic c X be a free holomorphic function in H α α k=0 |α|=k ball function on [B(H)n ]1 , defined by

(f ◦ ϕ)(X1 , . . . , Xn ) =

∞

cα ϕα (X1 , . . . , Xn ),

k=0 |α|=k

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

where the convergence is in the operator norm topology. In particular, we have (f ◦ ϕ)(rS1 , . . . , rSn )1 =

∞

cα ϕα (rS1 , . . . , rSn )1,

(1.2)

k=0 |α|=k

where the convergence is in F 2 (Hn ). On the other hand, setting pm (X1 , . . . , Xn ) := m 2 k=0 |α|=k cα Xα , we have pm → f in Hball as m → ∞. Therefore, {pm } is a Cauchy se2 quence in Hball . Due to relation (1.1), we have pm ◦ ϕ − pk ◦ ϕ2 pm − pk 2 ,

m, k ∈ N.

2 and, consequently, there is g ∈ H 2 such that Hence, {pm ◦ ϕ} is a Cauchy sequence in Hball ball 2 pm ◦ ϕ → g in Hball . Hence, for each r ∈ [0, 1), we have

lim (pm ◦ ϕ)(rS1 , . . . , rSn )1 = g(rS1 , . . . , rSn )1.

m→∞

Combining this relation with (1.2), we get g(rS1 , . . . , rSn )1 = (f ◦ ϕ)(rS1 , . . . , rSn )1,

r ∈ [0, 1).

2 . Now, since Since f ◦ ϕ and g are free holomorphic functions, we deduce that f ◦ ϕ = g ∈ Hball 2 2 pm ◦ ϕ → f ◦ ϕ in Hball , relation (1.1) implies f ◦ ϕ2 f 2 for any f ∈ Hball . The proof is complete. 2


917

If in addition to the hypothesis of Theorem 1.2, we assume that ϕ is inner, i.e. the boundary function ϕ is an isometry, then we can prove the following result. Theorem 1.3. Let ϕ be an inner free holomorphic self-map of the ball [B(H)n ]1 such that 2 . ϕ(0) = 0. Then the composition operator Cϕ is an isometry on Hball Proof. Let ϕ := [ ϕ1 , . . . , ϕn ] be the boundary function of ϕ with respect to the left creation opeartors. Note that due to the fact that ϕ(0) = 0, we have 1, ϕα 1 = 0 for any α ∈ F+ n with |α| 1. On the other hand, since [ ϕ1 , . . . , ϕn ] is an isometry, we have ϕi∗ ϕj = δij IF 2 (Hn ) for i, j ∈ {1, . . . , n}. Consequently, ϕα , ϕβ H 2 = ϕα 1, ϕβ 1 ball ⎧ ϕγ 1, 1 if α = βγ , ⎨ if α = β, = 1 ⎩ 1, ϕγ 1 if β = αγ 1 if α = β, = 0 if α = β. ∞ 2 . If f = 2 This shows that {ϕα }α∈F+n is an orthonormal set in Hball k=0 |α|=k cα Xα is in Hball , m 2 , as m → ∞. then setting pm (X1 , . . . , Xn ) := k=0 |α|=k cα Xα , we have pm → f in Hball Note that pm ◦ ϕ22

=

m k=0 |α|=k

cα ϕα ,

m k=0 |β|=k

cβ ϕβ =

m

|cα |2 = pm 22 .

(1.3)

k=0 |α|=k

2 and there is g ∈ H 2 such that p ◦ ϕ → g Consequently, {pm ◦ ϕ} is a Cauchy sequence in Hball m ball 2 in Hball . Hence, we deduce that

g(rS1 , . . . , rSn )1 = lim (pm ◦ ϕ)(rS1 , . . . , rSn )1 = (f ◦ ϕ)(rS1 , . . . , rSn )1, m→∞

r ∈ [0, 1).

Since f ◦ ϕ and g are free holomorphic functions, the identity theorem for free holomorphic functions implies f ◦ ϕ = g. Therefore, relation (1.3) implies that Cϕ is an isometry and the proof is complete. 2 2 2. Composition operators on the noncommutative Hardy space Hball

This section contains the core material on the boundedness of compositions operators on the 2 and the estimates of their norms. We also characterize the noncommutative Hardy space Hball 2 to contractions. similarity of composition operators on Hball Let θ be an analytic function on the open disc D. It is well known that the map ϕ : D → R+ defined by ϕ(λ) := |θ (λ)|2 is subharmonic. A classical result on harmonic majorants (see Sec-

918


tion 2.6 in [6]) states that θ is in the Hardy space H 2 (D) if and only if ϕ has a harmonic majorant. Moreover, the least harmonic majorant of ϕ is given by the Herglotz–Riesz [9,40] formula 1 h(λ) = 2π

2π 0

eit + λ it 2 θ e dt, eit − λ

λ ∈ D.

In [34], we obtained free analogues of these results. Since these results play an important role in our investigation we shall recall them. We say that a map h : [B(H)n ]1 → B(H) is a self-adjoint free pluriharmonic function on [B(H)n ]1 if h = f := 12 (f ∗ + f ) for some free holomorphic function f on [B(H)n ]1 . An arbitrary free pluriharmonic function is a linear combination of self-adjoint free pluriharmonic functions. A pluriharmonic curve in C ∗ (S1 , . . . , Sn ) is a map ϕ : [0, 1) → An + An · satisfying the Poisson mean value property, i.e., ϕ(r) = P rt S ϕ(t)

for 0 r < t < 1,

where S := (S1 , . . . , Sn ) and PX [u] is the noncommutative Poisson transform of u at X. According to [37], there exists a one-to-one correspondence u → ϕ between the set of all free pluriharmonic functions on the noncommutative ball [B(H)n ]1 , and the set of all pluriharmonic curves ϕ : [0, 1) → A∗n + An · . Moreover, we have u(X) = P 1 X ϕ(r) r

for X ∈ B(H)n r and r ∈ (0, 1),

and ϕ(r) = u(rS1 , . . . , rSn ) if r ∈ [0, 1). We say that a map ψ : [0, 1) → An + An · is selfadjoint if ψ(r) = ψ(r)∗ for r ∈ [0, 1). We call ψ a sub-pluriharmonic curve provided that for each γ ∈ (0, 1) and each self-adjoint pluriharmonic curve ϕ : [0, γ ] → An + An · , if ψ(γ ) ϕ(γ ), then ψ(r) ϕ(r) for any r ∈ [0, γ ]. We proved that a self-adjoint map g : [0, 1) → A∗n + An · is a sub-pluriharmonic curve in C ∗ (S1 , . . . , Sn ) if and only if g(r) P γr S g(γ ) for 0 r < γ < 1. We obtained a characterization for the class of all sub-pluriharmonic curves that admit free pluriharmonic majorants, and proved the existence of the least pluriharmonic majorant. We mention that all these results can be written for sub-pluriharmonic curves in C ∗ (R1 , . . . , Rn ), where R1 , . . . , Rn are the right creation operators on the full Fock space. In [34], we showed that, for any free holomorphic function Θ on the noncommutative ball [B(H)n ]1 , the mapping ϕ : [0, 1) → C ∗ (R1 , . . . , Rn ),

ϕ(r) = Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ),

is a sub-pluriharmonic curve in the Cuntz–Toeplitz algebra generated by the right creation operators R1 , . . . , Rn . We proved that a free holomorphic function Θ is in the noncommutative 2 if and only if ϕ has a pluriharmonic majorant. In this case, the least pluriharHardy space Hball monic majorant ψ for ϕ is given by ψ(r) := W (rR1 , . . . rRn ), r ∈ [0, 1), where W is the free holomorphic function having the Herglotz–Riesz type representation


W (X1 , . . . , Xn ) = (μθ ⊗ id)

I+

n

Ri∗

⊗ Xi

I−

i=1

n

919

−1 Ri∗

⊗ Xi

i=1

for (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where μθ : R∗n + Rn → C is a positive linear map uniquely determined by the function Θ. Now, we need to recall from [38] some basic facts concerning the free holomorphic automorphisms of the noncommutative ball [B(H)n ]1 . A map ϕ : [B(H)n ]1 → [B(H)n ]1 is called free biholomorphic if ϕ is free homolorphic, one-to-one and onto, and has free holomorphic inverse. The automorphism group of [B(H)n ]1 , denoted by Aut([B(H)n ]1 ), consists of all free biholomorphic functions of [B(H)n ]1 . It is clear that Aut([B(H)n ]1 ) is a group with respect to the composition of free holomorphic functions. We used the theory of noncommutative characteristic functions for row contractions [25] to find all the involutive free holomorphic automorphisms of [B(H)n ]1 , which turned out to be of the form Φλ (X1 , . . . , Xn ) = −Θλ (X1 , . . . , Xn ),

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

for some λ = [λ1 , . . . , λn ] ∈ Bn , where Θλ is the characteristic function of the row contraction λ, acting as an operator from Cn to C. We recall that the characteristic function of the row contraction λ is the boundary function (with respect to R1 , . . . , Rn ) λ := SOT- lim Θλ (rR1 , . . . , rRn ) Θ r→1

of the free holomorphic function Θλ : [B(H)n ]1 → [B(H)n ]1 given by Θλ (X1 , . . . , Xn ) := −λ + λ IH −

n

−1 λi X i

[X1 , . . . , Xn ]λ∗

i=1

for (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where λ = (1 − λ22 )1/2 IC and λ∗ = (IK − λ∗ λ)1/2 . For simplicity, we used the notation λ := [λ1 IG , . . . , λn IG ] for the row contraction acting from G (n) to G, where G is a Hilbert space. 1 , then Φλ := −Θλ is a free In [38], we proved that if λ := (λ1 , . . . , λn ) ∈ Bn \{0} and γ := λ 2 n holomorphic function on [B(H) ]γ which has the following properties: (i) Φλ (0) = λ and Φλ (λ) = 0; (ii) the identities

−1

−1 IH − Φλ (X)Φλ (Y )∗ = λ I − Xλ∗ I − XY ∗ I − λY ∗ λ ,

−1

−1 λ∗ , I − X ∗ Y I − λ∗ Y IH⊗Cn − Φλ (X)∗ Φλ (Y ) = λ∗ I − X ∗ λ (iii) (iv) (v) (vi)

hold for all X and Y in [B(H)n ]γ ; Φλ is an involution, i.e., Φλ (Φλ (X)) = X for any X ∈ [B(H)n ]γ ; Φλ is a free holomorphic automorphism of the noncommutative unit ball [B(H)n ]1 ; n − Φλ is a homeomorphism of [B(H)n ]− 1 onto [B(H) ]1 ; Φλ is inner, i.e., the boundary function Φλ is an isometry.

(2.1)

920


Moreover, we determined all the free holomorphic automorphisms of the noncommutative ball [B(H)n ]1 by showing that if Φ ∈ Aut([B(H)n ]1 ) and λ := Φ(0), then there is a unitary operator U on Cn such that Φ = Φ λ ◦ ΦU , where ΦU (X1 , . . . Xn ) := [X1 , . . . , Xn ]U,

(X1 , . . . , Xn ) ∈ B(H)n 1 .

We have now all the ingredients to prove the key result of this section. Theorem 2.1. If ϕ is a free holomorphic automorphism of the noncommutative ball [B(H)n ]1 , 2 for all f ∈ H 2 , and then Cϕ f ∈ Hball ball

1 − ϕ(0) 1 + ϕ(0)

1/2

f Cϕ f

1 + ϕ(0) 1 − ϕ(0)

1/2 f

2 . Moreover, these inequalities are best possible and for all f ∈ Hball

Cϕ =

1 + ϕ(0) 1 − ϕ(0)

1/2 .

Proof. Let ϕ := (ϕ1 , . . . , ϕn ) be an inner free holomorphic self-map of the noncommutative ball [B(H)n ]1 . Then the boundary function with respect to the right creation operators R1 , . . . , Rn , i.e., ϕn ), ϕ := ( ϕ1 , . . . ,

where ϕi := SOT- lim ϕi (rR1 , . . . , rRn ), r→1

ϕj = δij IF 2 (Hn ) for i, j ∈ {1, . . . , n}. Recall that R1 , . . . , Rn are is an isometry. Consequently, ϕi∗ isometries with orthogonal ranges, so Ri∗ Rj = δij IF 2 (Hn ) for i, j ∈ {1, . . . , n}. Consequently, we have ⎧ ⎧ ϕγ if β = αγ , ⎨ Rγ if β = αγ , ⎨ if α = β, if α = β, Rα∗ Rβ = I ϕβ = I and ϕα∗ ⎩ R ∗ if α = βγ , ⎩ ϕγ∗ if α = βγ . γ Fix a noncommutative polynomial p(X1 , . . . , Xn ) := |α|m aα r |α| Xα . Note that, using the above-mentioned relations and applying the noncommutative Poisson transform (with respect ϕ1 , . . . , ϕn ], we obtain to R1 , . . . , Rn ) at [ ∗ P[ ϕ1 , . . . , r ϕn )∗ p(r ϕ1 , . . . , r ϕn ) ϕ1 ,..., ϕn ] p(rR1 , . . . , rRn ) p(rR1 , . . . , rRn ) = p(r 2 , Theorem 2.3 from [34] shows that the map for any r ∈ [0, 1). Since p ∈ Hball

0, 1) r → p(rR1 , . . . , rRn )∗ p(rR1 , . . . , rRn ) ∈ C ∗ (R1 , . . . , Rn )

(2.2)


921

has a pluriharmonic majorant. In this case, the least pluriharmonic majorant is given by [0, 1) r → W (rR1 , . . . rRn ) ∈ C ∗ (R1 , . . . , Rn ), where W is the free holomorphic function on [B(H)n ]1 having the Herglotz–Riesz type representation W (X1 , . . . , Xn ) = (μp ⊗ id)

I+

n i=1

Ri∗

⊗ Xi

I−

n

−1 Ri∗

⊗ Xi

(2.3)

i=1

for (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where μp : R∗n + Rn → C is the completely positive linear map uniquely determined by the equation ∗ ∗ ∗ μp R α := lim p(rR1 , . . . , rRn ) S α p(rR1 , . . . , rRn )1, 1 r→1

(2.4)

α is the reverse of α ∈ F+ α = gik · · · gik if α = gi1 · · · gik ∈ F+ for α ∈ F+ n , where n , i.e., n . Therefore, we have p(rR1 , . . . , rRn )∗ p(rR1 , . . . , rRn ) W (rR1 , . . . , rRn ) for any r ∈ [0, 1). Hence, using relation (2.2) and the fact that the noncommutative Poisson transform is a completely positive map, we deduce that p(r ϕ1 , . . . , r ϕn )∗ p(r ϕ1 , . . . , r ϕn ) W (r ϕ1 , . . . , r ϕn ) for any r ∈ [0, 1). The latter relation implies 2

p(r ϕ1 , . . . , r ϕn )1 Re W (r ϕ1 , . . . , r ϕn )1, 1 = W rϕ1 (0), . . . , rϕn (0) . On the other hand, according to the Harnak type theorem for positive free pluriharmonic functions (see [36]), we have

1 + rϕ(0) . Re W rϕ1 (0), . . . , ϕn (0) W (0) 1 − rϕ(0) Combining the latter two inequalities and taking r → 1, we deduce that 2 1 + ϕ(0) ϕ1 , . . . , ϕn )1 W (0) p ◦ ϕ22 = p( . 1 − ϕ(0) Using the Herglotz–Riesz representation (2.3) and relation (2.4), we obtain 2 W (0) = μp (I ) = lim p(rR1 , . . . , rRn )1 = p22 . r→1

(2.5)

922


Hence, and using relation (2.5), we have

p ◦ ϕ2 p2

1 + ϕ(0) 1 − ϕ(0)

1/2 (2.6)

∞ 2 . Let f (X , . . . , X ) = for any noncommutative polynomial p ∈ Hball 1 n k=0 |α|=k cα Xα be a 2 . Then f ◦ ϕ is a free holomorphic function on [B(H)n ] and free holomorphic function in Hball 1 (f ◦ ϕ)(rS1 , . . . , rSn )1 =

∞

cα ϕα (rS1 , . . . , rSn )1,

(2.7)

k=0 |α|=k

where the convergence is in F 2 (Hn ). Setting pm (X1 , . . . , Xn ) := m k=0 |α|=k cα Xα , we have 2 as m → ∞. Therefore, {p } is a Cauchy sequence in H 2 . Due to relation pm → f in Hball m ball (2.6), we have

pm ◦ ϕ − pk ◦ ϕ2

1 + ϕ(0) 1 − ϕ(0)

1/2 pm − pk 2 ,

m, k ∈ N.

2 and there is g ∈ H 2 such that p ◦ ϕ → g Consequently, {pm ◦ ϕ} is a Cauchy sequence in Hball m ball 2 as m → ∞. Hence, and using relation (2.7), we deduce that in Hball

g(rS1 , . . . , rSn )1 = lim (pm ◦ ϕ)(rS1 , . . . , rSn )1 = (f ◦ ϕ)(rS1 , . . . , rSn )1, m→∞

r ∈ [0, 1).

Since f ◦ ϕ and g are free holomorphic functions, the identity theorem for free holomorphic 2 and relation (2.6), we functions implies f ◦ ϕ = g. Using that fact that pm ◦ ϕ → f ◦ ϕ in Hball obtain

f ◦ ϕ2

1 + ϕ(0) 1 − ϕ(0)

1/2 f 2 ,

2 f ∈ Hball .

(2.8)

Since any free holomorphic automorphism of [B(H)n ]1 is inner, i.e., its boundary function with respect to R1 , . . . , Rn is an isometry, the result above implies the right-hand inequality of the theorem. Now, we the left-hand inequality. For each μ := (μ1 , . . . , μn ) ∈ Bn , we define the vecprove tor zμ := k=0 |α|=k μα eα , where μα := μi1 · · · μip if α = gi1 · · · gip ∈ F+ n and i1 , . . . , ip ∈ 2 . 2 {1, . . . , n}, and μg0 = 1. Note that zμ ∈ F (Hn ) and Zμ (X) := k=0 |α|=k μα Xα is in Hball 2 Since Cϕ is a bounded operator on Hball , we have ∗ Cϕ Zμ (X) = bα X α , k=0 |α|=k

X ∈ B(H)n 1 ,

for some coefficients bα ∈ C with α∈F+n |bα |2 < ∞. Since the monomials {Xα }α∈F+n form an 2 , for each α ∈ F+ , we have orthonormal basis for Hball n


923

bα = Cϕ∗ Zμ , Xα = Zμ , Cϕ (Xα ) = zμ , ϕα (S1 , . . . , Sn )1 = ϕα (S1 , . . . , Sn )∗ zμ , 1 . Since Si∗ zμ = μi zμ , one can see that ϕα (S1 , . . . , Sn )∗ zμ = ϕα (μ)zμ . Consequently, we deduce that bα = ϕα (μ), α ∈ F+ n , and Cϕ∗ Zμ =

ϕα (μ)Xα = Zϕ(μ) ,

μ := (μ1 , . . . , μn ) ∈ Bn .

(2.9)

k=0 |α|=k

A straightforward computation shows that ∗ C Zμ = zϕ(μ) = ϕ

1 1 − ϕ(μ)2

1/2 .

Now, we assume that ϕ = Φλ ∈ Aut([B(H)n ]1 ). Then, using relation (2.1), we deduce that

∗ Z ∗ CΦ λ μ = CΦλ = CΦ λ Zμ

1 − μ2 1 − Φλ (μ)2

1/2

=

|1 − μ, λ |2 1 − λ2

1/2

λ for any μ ∈ Bn . Taking μ → − λ and using the fact that Φλ (0) = λ, we obtain

CΦλ

1 + Φλ (0) 1 − Φλ (0)

1/2 .

Combining this inequality with relation (2.8), we obtain

CΦλ =

1 + Φλ (0) 1 − Φλ (0)

1/2 (2.10)

,

which also shows that the right-hand inequality in the theorem is sharp. Now, we assume that ϕ ∈ Aut([B(H)n ]1 ) with ϕ(0) = λ. Then, due to [38], we have ϕ = Φλ ◦ ΦU , where U ∈ B(Cn ) is a unitary operator. Since ΦU is inner and ΦU (0) = 0, Theorem 1.3 shows that CΦU is an isometry. Consequently, using relation (2.10) and the fact that Cϕ = CΦU CΦλ , we deduce that

Cϕ =

1 + ϕ(0) 1 − ϕ(0)

1/2 .

Taking into account that Φλ ◦ Φλ = id, we deduce that

f CΦλ CΦλ f

1 + Φλ (0) 1 − Φλ (0)

1/2 CΦλ f

924


2 . Now, we assume that ϕ ∈ Aut([B(H)n ] ) with ϕ(0) = λ. As above, ϕ = for any f ∈ Hball 1 Φλ ◦ ΦU and Cϕ = CΦU CΦλ . Since CΦU is an isometry, the latter inequality implies

Cϕ f = CΦλ CΦU f

1 − ϕ(0) 1 + ϕ(0)

1/2 f ,

which shows that the left-hand inequality of the theorem holds. To prove that this inequality is 2 with g = 1 and C = lim sharp, let gk ∈ Hball k 2 Φλ k→∞ CΦλ gk . Set fk := CΦλ gk and note 1−Φλ (0) 1/2 λ (0) 1/2 , that the inequality ( 1+Φλ (0) ) fk CΦλ fk is equivalent to CΦλ gk ( 1+Φ 1−Φλ (0) ) which is sharp due to (2.10), and proves our assertion. The proof is complete. 2 Theorem 2.2. If ϕ is an inner free holomorphic self-map of the noncommutative ball [B(H)n ]1 , 2 for all f ∈ H 2 , and then Cϕ f ∈ Hball ball

1 − ϕ(0) 1 + ϕ(0)

1/2 f Cϕ f

1 + ϕ(0) 1 − ϕ(0)

1/2 f

2 . Moreover, these inequalities are best possible and for any f ∈ Hball

Cϕ =

1 + ϕ(0) 1 − ϕ(0)

1/2 .

Proof. First, we consider the case when ϕ is an inner free holomorphic self-map of the noncommutative ball [B(H)n ]1 with ϕ(0) = 0. Then Theorem 1.3 shows that the composition operator 2 and, therefore, the theorem holds. Cϕ is an isometry on Hball Now, we consider the case when λ := ϕ(0) = 0. Since ϕ is a free holomorphic self-map of the noncommutative ball [B(H)n ]1 , we must have λ2 < 1. Let Φλ be the corresponding involutive free holomorphic automorphism of [B(H)n ]1 and let Ψ := Φλ ◦ ϕ. Since Φλ is inner and the composition of inner free holomorphic functions is inner (see Theorem 1.2 from [39]), we deduce that Ψ is also inner. Since Ψ (0) = 0, the first part of the proof implies CΨ f = f ,

2 f ∈ Hball .

Consequently, using Theorem 2.1 and the fact that Φλ ◦ Φλ = id, we get

Cϕ f = CΨ CΦλ f = CΦλ f

=

1 + ϕ(0) 1 − ϕ(0)

1/2

1 + Φλ (0) 1 − Φλ (0)

1/2 f

f

(2.11)

2 . Similarly, one can show that for any f ∈ Hball

Cϕ f = CΦλ f

1 − Φλ (0) 1 + Φλ (0)

1/2 f =

1 − ϕ(0) 1 + ϕ(0)

1/2 f


925

2 . Therefore, the inequalities in the theorem hold. Now, we show that they are for any f ∈ Hball 2 with f = 1 such that sharp. According to Theorem 2.1, we can find fk ∈ Hball k 2

lim CΦλ fk =

k→∞

1 + Φλ (0) 1 − Φλ (0)

1/2 .

Hence, using relation (2.11) and the fact that Φλ (0) = ϕ(0), we obtain

lim Cϕ fk = lim CΦλ fk =

k→∞

k→∞

1 − ϕ(0) 1 + ϕ(0)

1/2 ,

which shows that the right-hand inequality in the theorem is sharp. Similarly, one can show that the left-hand inequality is also sharp. The proof is complete. 2 Now, we can prove the main result of this section. Theorem 2.3. If ϕ is a free holomorphic self-map of the ball [B(H)n ]1 , then the composition 2 . Moreover, operator Cϕ f := f ◦ ϕ is bounded on Hball

1/2

1 − λ2 1 1 + ϕ(0) 1/2 sup Cϕ . 1 − ϕ(0) (1 − ϕ(0)2 )1/2 λ∈Bn 1 − ϕ(λ)2 Proof. If ϕ(0) = 0, then the right-hand inequality follows from the noncommutative Littlewood subordination principle of Theorem 1.2. Now, we consider the case when λ := ϕ(0) = 0. Since λ2 < 1, let Φλ be the corresponding involutive free holomorphic automorphism of [B(H)n ]1 and let Ψ := Φλ ◦ ϕ. Since Ψ is a free holomorphic self-map of the ball [B(H)n ]1 with Ψ (0) = 0, Theorem 1.2 implies CΨ 1. Using Theorem 2.1 and the fact that Φλ ◦ Φλ = id, we deduce that

Cϕ = CΨ CΦλ CΨ CΦλ

1 + ϕ(0) 1 − ϕ(0)

1/2 .

On the other hand, as in the proof of Theorem 2.1, we have Cϕ∗ Zμ = Cϕ = Cϕ∗ Zμ

1 − μ2 1 − ϕ(μ)2

1/2

for any μ ∈ Bn . Hence, we deduce the left-hand inequality. The proof is complete.

2

2 with the full Fock space Under the identification of the noncommutative Hardy space Hball 2 2 via the unitary operator U : Hball → F (Hn ) defined by

F 2 (Hn ),

2 F → f := lim F (rS1 , . . . , rSn )1 ∈ F 2 (Hn ), Hball r→1

2 → H 2 associated with ϕ, a free holomorphic self-map of the composition operator Cϕ : Hball ball n 2 2 [B(H) ]1 , can be identified with the composition operator C ϕ : F (Hn ) → F (Hn ) defined by

926


C ϕ

∞

aα eα := lim

r→1

k=0 |α|=k

∞

aα ϕα (rS1 , . . . , rSn )1

(2.12)

k=0 |α|=k

2 −1 for any ∞ ϕ = UCϕ U . |α|=k aα eα ∈ F (Hn ). Indeed, note that C k=0 A consequence of Theorem 2.3 is the following result. Corollary 2.4. If ϕ is a free holomorphic self-map of the ball [B(H)n ]1 , then the composition 2 2 operator C ϕ : F (Hn ) → F (Hn ) satisfies the equation C ϕ

∞ k=0 |α|=k

aα eα =

∞

aα ( ϕα 1),

k=0 |α|=k

ϕ := SOT- limr→1 ϕ(rS1 , . . . , rSn ) is the where the convergence of the series is in F 2 (Hn ) and boundary function of ϕ with respect to the left creation operators S1 , . . . , Sn . 2 Proof. Let ϕ := ( ϕ1 , . . . , ϕn ) be the boundary of ϕ and let f = ∞ k=0 |α|=k aα Xα be in Hball . 2 with F 2 (H ), we have Due to Theorem 2.3 and the identification of Hball n p|α|m

1 + ϕ(0) 1/2 aα ϕα 1 1 − ϕ(0)

1/2

|aα |2

(2.13)

p|α|m

2 , the sequence { m ϕα 1}∞ for any p, m ∈ N, p m. Consequently, since f ∈ Hball k=0 |α|=k aα m=1 is Cauchy in F 2 (Hn ) and therefore convergent to an element in F 2 (Hn ). Hence, and using relation (2.13), we deduce that ∞

1 + ϕ(0) 1/2 aα ϕα 1 f . 1 − ϕ(0) k=0 |α|=k

Similarly, one can show that

∞ k=0

|α|=k aα ϕα (rS1 , . . . , rSn )1

is in F 2 (Hn ) and

∞

1 + ϕ(0) 1/2 aα ϕα (rS1 , . . . , rSn )1 f 1 − ϕ(0) k=0 |α|=k

for each r ∈ [0, 1). Consequently, taking into account that ϕ := SOT- limr→1 ϕ(rS1 , . . . , rSn ), a simple approximation argument shows that lim

r→1

∞ k=0 |α|=k

aα ϕα (rS1 , . . . , rSn )1 =

∞

aα ϕα 1

k=0 |α|=k

in F 2 (Hn ), which together with relation (2.12) completes the proof.

2

In this paper, we will use either one of the representations Cϕ or C ϕ for the composition operator with symbol ϕ.


927

Corollary 2.5. Let ϕ = (ϕ1 , . . . , ϕn ) be a free holomorphic self-map of the noncommutative ball 2 . Then the following statements hold. [B(H)n ]1 and let Cϕ be the composition operator on Hball (i) Cϕ 1. (ii) Cϕ is a contraction if and only if ϕ(0) = 0. 2 . (iii) Cϕ is an isometry if and only if {ϕα }α∈Fn is an orthonormal set in Hball Proof. Since Cϕ 1 = 1, we have Cϕ 1. To prove part (ii), note that if Cϕ = 1, then according to Theorem 2.3, we have 1 Cϕ = 1, (1 − ϕ(0)2 )1/2 which implies ϕ(0) = 0. Conversely, if ϕ(0) = 0, the same theorem implies Cϕ = 1. Now, assume that Cϕ is an isometry. Then δα,β = Cϕ (Xα ), Cϕ (Xβ ) = ϕα , ϕβ , α, β ∈ F+ n. 2 . Then, for any Conversely, assume that {ϕα }α∈Fn is an orthonormal set in Hball

f=

∞

aα X α

k=0 |α|=k 2 , we have in the Hardy space Hball

2 ∞ ∞ Cϕ f 2 = aα ϕα = |aα |2 = f 2 . k=0 |α|=k

The proof is complete.

k=0 |α|=k

2

Halmos’ famous similarity problem [7] asked whether any polynomially bounded operator is similar to a contraction. This long standing problem was answered by Pisier [22] in a remarkable paper where he shows that there are polynomially bounded operator which are not similar to 2 , similarity to contractions. In what follows we show that, for compositions operators on Hball contractions is equivalent polynomial boundedness. Theorem 2.6. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 and 2 . Then the following statements are equivalent: let Cϕ be the composition operator on Hball (i) (ii) (iii) (iv)

Cϕ is similar to a contraction; Cϕ is polynomially bounded; Cϕ is power bounded; there is ξ ∈ Bn such that ϕ(ξ ) = ξ .

Proof. The fact that an operator similar to a contraction is power bounded and polynomially bounded is a consequence of the well-known von-Neumann inequality [49]. We prove that

928


(iii) ⇒ (iv). Assume that Cϕ is power bounded, i.e., there is a constant M > 0 such that Cϕk M for any k ∈ N. Note that the scalar representation of ϕ, i.e. Bn λ → ϕ(λ) ∈ Bn , is a holomorphic self-map of Bn . Suppose there is no ξ ∈ Bn such that ϕ(ξ ) = ξ . Then, due to MacCluer’s result [13], there is γ ∈ ∂Bn , called the Denjoy–Wolff point of the map Bn λ → ϕ(λ) ∈ Bn , such that the sequence of iterates ϕ [k] := ϕ ◦ · · · ◦ ϕ converges to γ uniformly on any compact subset of Bn . In particular, we have ϕ [k] (0) → 1 as k → ∞. On the other hand, Theorem 2.3 implies k C = C ϕ

ϕ [k]

1 (1 − ϕ [k] (0)2 )1/2

.

Consequently, Cϕk → ∞ as k → ∞, which contradicts the fact that Cϕ is a power bounded operator. Therefore, item (iv) holds. Finally, to prove that (iv) ⇒ (i), assume that there is ξ ∈ Bn such that ϕ(ξ ) = ξ . Set Ψ := Φξ ◦ ϕ ◦ Φξ , where Φξ is the involutive free holomorphic automorphism of [B(H)n ]1 associated with ξ . Note that Ψ is a bounded free holomorphic function on [B(H)n ]1 and Ψ (0) = 0. Due to Theorem 1.2, we have CΨ 1. On the other hand, since −1 Φξ ◦ Φξ = id and Cϕ = CΦ CΨ CΦξ , the result follows. The proof is complete. 2 ξ Corollary 2.7. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 and 2 . If there is ξ ∈ B such that ϕ(ξ ) = ξ , then the let Cϕ be the composition operator on Hball n spectral radius of Cϕ is 1. Proof. According to the proof of Theorem 2.6, Cϕ is similar to a composition operator CΨ with Ψ (0) = 0. Since Ψ [k] (0) = 0, Theorem 1.2 implies CΨ [k] = 1 for any k ∈ N. Consequently, we have r(Cϕ ) = r(CΨ ) = lim CΨ [k] 1/k = 1. k→∞

The proof is complete.

2

Corollary 2.8. Let ϕ be an inner free holomorphic self-map of the noncommutative ball 2 . Then the following statements hold. [B(H)n ]1 and let Cϕ be the composition operator on Hball (i) Cϕ is an isometry if and only if ϕ(0) = 0. (ii) Cϕ is similar to an isometry if and only if there is ξ ∈ Bn such that ϕ(ξ ) = ξ . Proof. Assume that Cϕ is an isometry. Due to Theorem 2.2, we have

1 = Cϕ =

1 + ϕ(0) 1 − ϕ(0)

1/2 .

Consequently, ϕ(0) = 0. The converse follows also from Theorem 2.2. Therefore, item (i) holds. The direct implication in item (ii) follows from Theorem 2.6. To prove the converse, assume that there is ξ ∈ Bn such that ϕ(ξ ) = ξ and set Ψ := Φξ ◦ ϕ ◦ Φξ , where Φξ is the involutive free holomorphic automorphism of [B(H)n ]1 associated with ξ .


929

According to [39], the composition of inner free holomorphic functions on [B(H)n ]1 is inner. Consequently, Ψ is an inner free holomorphic function and Ψ (0) = 0. Due to part (i), the −1 composition operator CΨ is an isometry. Since Cϕ = CΦ CΨ CΦξ , the result follows. 2 ξ The following result is an extension to our noncommutative multivariable setting of Cowen’s [2] one-variable spectral radius formula for composition operators. Theorem 2.9. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 and 2 . Then the spectral radius of C satisfies the relation let Cϕ be the composition operator on Hball ϕ −1/2k r(Cϕ ) = lim 1 − ϕ [k] (0) . k→∞

Moreover,

r(Cϕ ) = lim

k→∞

1 − ϕ [k] (0) 1 − ϕ [k+1] (0)

1/2

if the latter limit exists. Proof. Note that Theorem 2.3 implies

1 1 − ϕ [k] (0)2

1/2k

1/k Cϕk

1 + ϕ [k] (0) 1 − ϕ [k] (0)

1/2k

2 1 − ϕ [k] (0)

1/2k .

Taking k → ∞, we obtain the first formula for the spectral radius of Cϕ . To prove the second formula, note that −1/2k r(Cϕ ) = lim 1 − ϕ [k] (0) k→∞

k−1 1/2k 1 − ϕ [p] (0) = lim k→∞ 1 − ϕ [p+1] (0) p=0

= lim

k→∞

1 − ϕ [k] (0) 1 − ϕ [k+1] (0)

if the latter limit exists. The proof is complete.

1/2

2

3. Noncommutative Wolff theorem for free holomorphic self-maps of [B(H)n ]1 In this section, we use Julia type lemma for free holomorphic functions [39] and the ideas from the classical result obtained by Wolff [50,51] and MacCluer’s extension to Bn (see [13]), to provide a noncommutative analogue of Wolff’s theorem for free holomorphic self-maps of 2 is 1 when [B(H)n ]1 . We also show that the spectral radius of a composition operator on Hball the symbol is elliptic or parabolic, which extends some of Cowen’s results [2] from the single variable case.

930


Julia’s lemma [10] says that if f : D → D is an analytic function and there is a sequence (zk )| {zk } ⊂ D with zk → 1, f (zk ) → 1, and such that 1−|f 1−|zk | is bounded, then f maps each disc in D tangent to ∂D at 1 into a disc of the same kind. Wolff [50,51] used this result to show that if f has no fixed points in D, then there is a unique point ξ ∈ ∂D such that any closed disc in D which is tangent to ∂D at ξ is mapped into itself by every iterate of f , i.e., f [1] = f , f [k+1] := f [k] ◦ f , k ∈ N. The Denjoy–Wolff theorem [50,5] asserts that, under the above-mentioned conditions, the sequence of iterates of f converges uniformly on compact subsets of D to the constant map g(z) = ξ , z ∈ D. The point ξ is called the Denjoy–Wolff point of f . This result was extended to the unit ball of Cn by MacCluer [13]. If A, B ∈ B(K) are selfadjoint operators, we say that A 0 such that (B − A)h, h γ h2 for any h ∈ K. Note that T ∈ B(K) is a strict contraction (T < 1) if and only if T T ∗ < I . For 0 < c < 1 and ξ1 = (1, 0, . . . , 0), we define the noncommutative ellipsoid Ec (ξ1 ) := (X1 , . . . , Xn ) ∈ B(H)n : [X1 − (1 − c)I ][X1∗ − (1 − c)I ] X2 X2∗ Xn Xn∗ + + ··· + 0 and

−1

−1

I − F1 (X)∗ I − F (X)F (X)∗ I − F1 (X) L I − X1∗ I − XX ∗ (I − X1 ) for any X = (X1 , . . . , Xn ) ∈ [B(H)n ]1 . Moreover, if 0 < c < 1, then

F Ec (ξ1 ) ⊂ Eγ (ξ1 ),

where γ :=

Lc . 1 + Lc − c

In what follows we provide a noncommutative analogue of Wolff’s theorem for free holomorphic self-maps of [B(H)n ]1 . Theorem 3.1. Let ϕ : [B(H)n ]1 → [B(H)n ]1 be a free holomorphic function such that its scalar representation has no fixed points in Bn . Then there is a unique point ζ ∈ ∂Bn such that each noncommutative ellipsoid Ec (ζ ), c ∈ (0, 1), is mapped into itself by every iterate of ϕ. Proof. Let rk ∈ (0, 1) be a convergent sequence to 1. Define the map ψk : [B(H)n ]− rk → n − n − [B(H) ]rk by ψk := rk ϕ(X), X ∈ [B(H) ]rk , and note that ψk is a free holomorphic funcn − n − tion in [B(H)n ]− rk . Consequently, its scalar representation χk : [C ]rk → [C ]rk , defined by


931

n − χk (λ) := ψk (λ), λ ∈ [Cn ]− rk , is holomorphic in [C ]rk . According to Brouwer fixed point theorem n − there exists λk ∈ [C ]rk such that χ(λk ) = λk . Hence, ϕ(λk ) = λrkk . Passing to a subsequence and taking into account that the scalar representation of ϕ has no fixed point in Bn , we may assume that λk → ζ ∈ ∂Bn . This implies that ϕ(λk ) → ζ and 1 2 1 − ϕ(λk )2 1 − rk2 λk = < 1. 1 − λk 2 1 − λk 2

Consequently, we may assume that 1 − ϕ(λk )2 = L 1. k→∞ 1 − λk 2 lim

Without loss of generality, we may also assume that ζ = ξ1 := (1, 0, . . . , 0) ∈ ∂Bn . Using the above-mentioned Julia type lemma for free holomorphic functions, we deduce that L > 0 and

ϕ Ec (ξ1 ) ⊂ Eγ (ξ1 ),

where γ :=

Lc . 1 + Lc − c

(3.1)

Note that X ∈ Ec (ξ1 ) if and only if

(I − X1 ) I − X1∗ <

c I − XX ∗ . 1−c

Since L 1, it is easy to see that γ c, which implies Eγ (ξ1 ) ⊆ Ec (ξ1 ). Combining this with relation (3.1), we obtain ϕ(Ec (ξ1 )) ⊆ Ec (ξ1 ) for any c ∈ (0, 1), which proves the first part of the theorem. To prove the uniqueness, assume that there two distinct points ζ, ζ ∈ ∂Bn such that ϕ(Ec (ζ )) ⊆ Ec (ζ ) and ϕ(Ec (ζ )) ⊆ Ec (ζ ) for any c ∈ (0, 1). Let EC c (ζ ) be the scalar representation of the noncommutative ellipsoid Ec (ζ ) and let ϕ C be the scalar representation of ϕ. C Choose c, c ∈ (0, 1) such that EC c (ζ ) and Ec (ζ ) are tangent to each other at some point ξ ∈ Bn . C Note that ϕ C (ξ ) ∈ EC c (ζ ) ∩ Ec (ζ ) = {ξ }, which contradicts the hypothesis. The proof is complete. 2

The point ζ of Theorem 3.1 is called the Denjoy–Wolff point of ϕ. We remark that Theorem 3.1 shows that 0 < lim inf z→ζ

1 − ϕ(z)2 = α 1. 1 − z2

The number α is called the dilatation coefficient of ϕ. When n = 1, α is the angular derivative of ϕ at ζ . Combining Theorem 3.1 with Julia type lemma for free holomorphic functions [39], we obtain the following result.

932


Theorem 3.2. Let ϕ : [B(H)n ]1 → [B(H)n ]1 be a free holomorphic function with Denjoy–Wolff point ζ ∈ ∂Bn and dilatation coefficient α. Then, for any X ∈ [B(H)n ]1 , −1

−1

I − ζ ϕ(X)∗ I − ϕ(X)ϕ(X)∗ I − ϕ(X)ζ ∗ α I − ζ X ∗ I − XX ∗ I − Xζ ∗ . Let ϕ : [B(H)n ]1 → [B(H)n ]1 be a free holomorphic self-map. Following the classical case, ϕ will be called: (i) elliptic if ϕ fixes a point in Bn ; (ii) parabolic if ϕ has no fixed points in Bn and dilatation coefficient α = 1; (iii) hyperbolic if ϕ has no fixed points in Bn and dilatation coefficient α < 1. In the single variable case, when ϕ : D → D, Cowen [2] proved that the spectral radius of the composition operator Cϕ on H 2 (D) is 1 if ϕ is elliptic or parabolic, and √1α if ϕ is hyperbolic. 2 when the symbol ϕ is elliptic or We can extend his result to composition operators on Hball parabolic.

Theorem 3.3. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 . If ϕ 2 is 1. is elliptic or parabolic, then the spectral radius of the composition operator Cϕ on Hball Proof. The case when ϕ is elliptic was considered in Corollary 2.7. Now, we assume that ϕ is parabolic and let ζ ∈ ∂Bn be the corresponding Denjoy–Wolff point. According to MacCluer version [13] of Denjoy–Wolff theorem, the iterates of the scalar representation of ϕ converge uniformly to ζ on compact subsets of Bn . In particular, we have ϕ [k] (0) → ζ as k → ∞. Since [k+1]

(0) 1/2 the dilatation coefficient of ϕ is 1, we must have lim infk→∞ ( 1−ϕ ) 1. Consequently, 1−ϕ [k] (0) as in the proof of Theorem 2.9, we deduce that

1 − ϕ [k] (0) 1/2 r(Cϕ ) lim sup 1. 1 − ϕ [k+1] (0) k→∞ Taking into account that Cϕ 1 = 1, the result follows.

2

2 when the symbol is hyperTo calculate the spectral radius of a composition operator on Hball bolic remains an open problem. Another open problem is to find a Denjoy–Wolff type theorem (see [5,50]) for free holomorphic self-maps of [B(H)n ]1 .

4. Composition operators and their adjoints 2 . As In this section, we obtain a formula for the adjoint of a composition operator on Hball 2 . We also present a a consequence we characterize the normal composition operators on Hball 2 nice connection between Fredholm composition operators on Hball and the automorphisms of the open unit ball Bn .


933

Proposition 4.1. Let ϕ = (ϕ1 , . . . , ϕn ) be a free holomorphic self-map of the noncommutative 2 satisfies the relation ball [B(H)n ]1 . Then the adjoint of the composition Cϕ on Hball ∗ Cϕ f (X1 , . . . , Xn ) = f, ϕα Xα ,

2 f ∈ Hball .

α∈F+ n

Proof. According operator Cϕ is bounded on the Hardy space ∞ toTheorem 2.3, then composition 2 . If f = 2 , then, Hball c X is in H |α|=k α α k=0 ball Cϕ∗ f =

bα X α ,

k=0 |α|=k

X ∈ B(H)n 1 ,

for some coefficients bα ∈ C with α∈F+n |bα |2 < ∞. Since the monomials {Xα }α∈F+n form an 2 , we have orthonormal basis for Hball bα = Cϕ∗ f, Xα = f, Cϕ (Xα ) = f, ϕα , The proof is complete.

α ∈ F+ n.

2

2 with the Fock space F 2 (H ), the operator C We remark that under the identification of Hball n ϕ is unitarily equivalent to C ϕ (see Corollary 2.4) and

C ϕg =

g, ϕα (1) eα ,

g ∈ F 2 (Hn ).

α∈F+ n

By abuse of notation, we also write Cϕ∗ f = elements in the Fock space F 2 (Hn ).

f, ϕα eα , α∈F+ n

where f, ϕ1 , . . . , ϕn are seen as

Theorem 4.2. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 . Then 2 is normal if and only if the composition operator Cϕ on Hball ϕ(X1 , . . . , Xn ) = [X1 , . . . , Xn ]A for some normal scalar matrix A ∈ Mn×n with A 1. Proof. Assume that A = [aij ]n×n is a scalar matrix and A 1. Then it is clear that the relation ϕ(X1 , . . . , Xn ) = [X1 , . . . , Xn ]A,

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

defines a bounded free holomorphic function ϕ : [B(H)n ]1 → [B(H)n ]1 . According to Theo2 rem 2.3, the composition operator n Cϕ is bounded on Hball . Setting ϕ = (ϕ1 , . . . , ϕn ), +we have the Fock representation ϕj = p=1 apj ep for each j = 1, . . . , n. Fix β = gi1 · · · gik ∈ Fn and let α = ej1 · · · ejk . Note that eβ , ϕγ = 0 if |α| = |γ |, γ ∈ F+ n , and eβ , ϕα = a i1 j1 · · · a ik jk .

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Consequently, using Proposition 4.1, we deduce that Cϕ∗ eβ =

eβ , ϕα eα =

|α|=k

a i 1 j1 · · · a i k jk e α .

α=ej1 ···ejk , i1 ,...ik ∈{1,...,n}

Now, define ψ(X1 , . . . , Xn ) = [X1 , . . . , Xn ]A∗ ,

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

and note that ψ : [B(H)n ]1 → [B(H)n ]1 is a bounded free holomorphic function. Once again. 2 . Setting ψ = Theorem 2.3 shows that the composition operatorCψ is bounded on Hball n (ψ1 , . . . , ψn ), we have the Fock representation ψi = j =1 a ij ej for each i = 1, . . . , n. Hence, if β = gi1 · · · gik ∈ F+ n , we have

Cψ (eβ ) = ψi1 · · · ψik =

a i 1 j1 · · · a i k jk e α .

α=ej1 ···ejk ,i1 ,...ik ∈{1,...,n}

This shows that Cϕ∗ = Cψ . If we assume that A is a normal matrix, then ϕ ◦ ψ = ψ ◦ ϕ. Indeed, for any (X1 , . . . , Xn ) ∈ [B(H)n ]1 , we have (ϕ ◦ ψ)(X1 , . . . , Xn ) = [X1 , . . . , Xn ]A∗ A = [X1 , . . . , Xn ]AA∗ = (ψ ◦ ϕ)(X1 , . . . , Xn ). Consequently, we deduce that Cϕ Cϕ∗ = Cϕ Cψ = Cψ◦ϕ = Cϕ◦ψ = Cψ Cϕ = Cϕ∗ Cϕ . Now we prove the direct implication. Assume that ϕ is a free holomorphic self-map of the noncommutative ball [B(H)n ]1 and the composition operator Cϕ is normal. Since Cϕ 1 = 1, the vector 1 ∈ F 2 (Hn ) is also an eigenvector for Cϕ∗ . Since, due to Theorem 4.1, Cϕ∗ 1 = 1, ϕα eα , we deduce that 1, ϕα = 0 for all α ∈ F+ n with |α| 1. In particular, we α∈F+ n have 1, ϕi = 0 which implies ϕi (0) = 0 for i = 1, . . . , n. Therefore ϕ(0) = 0 and Cϕ∗ 1 = 1. Consequently, we have ϕ(X1 , . . . , Xn ) = [X1 , . . . , Xn ]A + (ψ1 , . . . , ψn ) (i) for some matrix A ∈ Mn×n and bounded free holomorphic functions ψi = |α|2 cα eα , i = 1, . . . , n. Consequently, using again the Fock space representation formula for the adjoint of Cϕ , we obtain egi , ϕα eα , Cϕ∗ (egi ) = α∈F+ n

which implies that the subspace M := span{egi : i = 1, . . . , n} is invariant under Cϕ∗ . Since M is finite dimensional, it is also invariant under Cϕ and Cϕ |M is a normal operator. This implies that, for each j = 1, . . . , n, Cϕ (ej ) is a linear combination of e1 , . . . , en and, consequently, ϕ(X1 , . . . , Xn ) = [X1 , . . . , Xn ]A for (X1 , . . . , Xn ) ∈ [B(H)n ]1 . Since ϕ : [B(H)n ]1 → [B(H)n ]1 , we must have A 1. Setting ψ(X1 , . . . , Xn ) = [X1 , . . . , Xn ]A∗ for (X1 , . . . , Xn ) ∈


935

2 and C ∗ = C . [B(H)n ]1 , the first part of the proof shows that Cψ is a bounded operator on Hball ψ ϕ Since Cϕ is normal, we have

Cψ◦ϕ = Cϕ Cψ = Cϕ Cϕ∗ = Cϕ∗ Cϕ = Cψ Cϕ = Cϕ◦ψ , which implies ψ ◦ ϕ(X) = ϕ ◦ ψ(X), X ∈ [B(H)n ]1 . Hence, we deduce that [X1 , . . . , Xn ]A∗ A = [X1 , . . . , Xn ]AA∗ for any (X1 , . . . , Xn ) ∈ [B(H)n ]1 , which implies A∗ A = AA∗ . The proof is complete. 2 Due to Theorem 4.2, characterizations of self-adjoint or unitary composition operators on 2 are now obvious. Hball Lemma 4.3. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 and let 2 . If the kernel of C ∗ is finite dimensional, then the scalar Cϕ be the composition operator on Hball ϕ representation of ϕ is one-to-one. (j )

(j )

Proof. Let λ(j ) = (λ1 , . . . , λn ), j = 1, . . . , k, be k distinct points in Bn and fix p ∈ {1, . . . , k}. (p) (j ) For each j ∈ {1, . . . , k} with j = p, there exists qj ∈ {1, . . . , n} such that λqj = λqj . Define the n free holomorphic function ϕp : [B(H) ]1 → B(H) by setting ϕp (X1 , . . . , Xn ) =

(p) j ∈{1,...,k}, j =p λqj

1 (j ) − λqj

(j ) Xqj − λqj I .

Note that ϕp (λ(p) ) = 1 and ϕp (λ(j ) ) = 0 for any j ∈ {1, . . . , k} with j= p. For each μ := (μ1 , . . . , μn ) ∈ Bn , we define the vector zμ := k=0 |α|=k μα eα , where μα := μi1 · · · μip if α = gi1 · · · gip ∈ F+ n and i1 , . . . , ip ∈ {1, . . . , n}, and μg0 = 1. Since zμ ∈ ∗ 2 F (Hn ) and Si zμ = μi zμ , one can see that q(S1 , . . . , Sn )∗ zμ = q(μ)zμ for any noncommutative polynomial q. Now we prove that the vectors zλ(1) , . . . , zλ(k) are linearly independent. Let a1 , . . . , ak ∈ C be such that a1 zλ(1) + · · · + ak zλ(k) = 0. Due to the properties of the free holomorphic function ϕp , p ∈ {1, . . . , k}, we deduce that

ϕp (S1 , . . . , Sn )∗ (a1 zλ(1) + · · · + ak zλ(k) ) = a1 ϕp λ(1) zλ(1) + · · · + ak ϕp λ(k) zλ(k)

= ap ϕp λ(p) zλ(p) = ap zλ(p) = 0. Hence, we deduce that a1 = · · · = ak = 0, which proves our assertion. Let ψ : Bn → Bn be the scalar representation of ϕ, i.e., ψ(λ) = ϕ(λ), λ ∈ Bn . Assume that there is ξ ∈ Bn such that ψ −1 (ξ ) is an infinite set. Let {λ(j ) }k∈N ⊂ ψ −1 (ξ ) be a sequence of distinct points. Due to relation (2.9), we have Cϕ∗ (zλ(j ) ) = Cϕ∗ (zλ(k) ) = zξ , which implies zλ(j ) − zλ(k) ∈ ker Cϕ∗ . As shown above, {zλ(j ) }j ∈N is a set of linearly independent vectors. Consequently, ker Cϕ∗ is infinite dimensional, which contradicts the hypothesis. Therefore, for each ξ ∈ Bn , the inverse image ψ −1 (ξ ) is a finite set. According to Rudin’s result (Theorem 15.1.6 from [41]), ψ : Bn → Bn is an open map. Suppose that ψ is not one-to-one. Let u, v ∈ Bn , u = v, be such that ψ(u) = ψ(v), and let U, V be open sets in Bn with the property that u ∈ U , v ∈ V , and U ∩ V = ∅. Since ψ is an open map, we deduce that ψ(U ) ∩ ψ(V ) is a nonempty open set. Consequently, we can find sequences {λ(j ) }j ∈N ⊂ U and {μ(j ) }j ∈N ⊂ V of distinct points such

936


that ψ(λ(j ) ) = ψ(μ(j ) ) for all j ∈ N. As above, we deduce that zλ(j ) − zμ(j ) ∈ ker Cϕ∗ for j ∈ N. Using the linear independence of the set {zλ(j ) }j ∈N ∪ {zμ(j ) }j ∈N , we deduce that ker Cϕ∗ is infinite dimensional, which contradicts the hypothesis. Therefore, ψ is a one-to-one map. The proof is complete. 2 Note that, unlike the single variable case, if n 2, then the composition operator Cϕ is not 2 . For example, one can take ϕ = (ϕ , ϕ ) : [B(H)2 ] → [B(H)2 ] and f = one-to-one on Hball 1 1 1 1 e1 e2 − e2 e1 , and note that Cϕ f = 0. 2 We remark that if ϕ ∈ Aut([B(H)n ]1 ), then the composition operator Cϕ is invertible on Hball and therefore Fredholm. It will be interesting to see if the converse is true. At the moment, we can prove the following result. Theorem 4.4. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 . If Cϕ 2 , then the scalar representation of ϕ is a holomorphic automoris a Fredholm operator on Hball phism of Bn . Proof. Let ψ : Bn → Bn be the scalar representation of ϕ, i.e., ψ(λ) := ϕ(λ), λ ∈ Bn . Due to Lemma 4.3, ψ is a one-to-one holomorphic map. We need to prove that ψ is surjective. To this end, assume that ψ is not surjective. Then there is a sequence {λ(k) } ⊂ Bn and ζ ∈ ∂Bn such that λ(k) → ζ as k → ∞ and ψ(λ(k) ) → w for some w ∈ Bn . z As we will see in the proof of Theorem 5.4 (see relation (5.2)), zλ(k) → 0 weakly as k → ∞. λ(k) On the other hand taking into account relation (2.9), we have Cϕ∗ zλ(k) =

ϕα λ(k) eα = zϕ(λ(k) ) ,

k ∈ N.

k=0 |α|=k

Hence, we get

∗ zλ(k) zϕ(λ(k) ) C ϕ z (k) = z (k) . λ λ z

Since zϕ(λ(k) ) → zw < ∞ and zλ(k) → ∞ as k → ∞, we deduce that Cϕ∗ ( zλ(k) ) → 0 as (k) B(F 2 (Hn )) such that we have

λ

zλ(k) 2 zλ(k) . Since Cϕ is a Fredholm operator on Hball , there is an operator Λ ∈ ΛCϕ∗ − I = K for some compact operator K ∈ B(F 2 (Hn )). Consequently,

k → ∞. Denote fk :=

ΛC ∗ fk 2 = fk + Kfk 2 = fk 2 + Kfk 2 + 2fk , Kfk . ϕ

(4.1)

Since K is a compact operator, fk = 1 and fk → 0 weakly as k → ∞, we must have Kfk → 0. Consequently, we have |fk , Kfk | fk Kfk → 0 as k → ∞. On the other hand, we have Cϕ∗ fk → 0. Now it is easy to see that relation (4.1) leads to a contradiction. Therefore, ψ is surjective. In conclusion ψ is an automorphism of Bn . 2


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2 5. Compact composition operators on Hball

In this section we obtain a formula for the essential norm of the composition operators Cϕ on 2 . In particular, this implies a characterization of compact composition operators. We show Hball 2 , then the scalar representation of ϕ is a holomorphic selfthat if Cϕ is a compact operator on Hball map of Bn which cannot have finite angular derivative at any point of ∂Bn and has exactly one fixed point in the open ball Bn . As a consequence, we deduce that every compact composition 2 is similar to a contraction. In the end of this section, we prove that the set operator on Hball 2 is arcwise connected in the set of all composition of compact composition operators on Hball operators. We recall that the essential norm of a bounded operator T ∈ B(H) is defined by T e := inf T − K: K ∈ B(H) is compact . Theorem 5.1. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 . Then 2 satisfies the equality the essential norm of the composition operator Cϕ on Hball Cϕ e = lim

k→∞

1/2 f, ϕα 2 .

sup

2 , f 1 f ∈Hball 2

|α|k

Consequently, Cϕ is a compact operator if and only if lim

k→∞

sup

f, ϕα 2 = 0.

2 , f 1 f ∈Hball 2 |α|k

Proof. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 . Since Cϕ 2 (see Theorem 2.3), one can use standard arguments is a bounded composition operator on Hball (see Proposition 5.1 from [44]) to show that the essential norm of the composition operator Cϕ 2 satisfies the equality on Hball Cϕ e = lim Cϕ Pk , k→∞

(5.1)

where Pk is the orthogonal projection of F 2 (Hn ) onto the closed linear span of all eα with α ∈ F+ n and |α| k. Indeed, note that the sequence {Cϕ Pk }∞ k=1 is decreasing and, due to the fact that I − Pk is a finite rank projection, we have Cϕ e Cϕ Pk for any k ∈ N. Hence Cϕ e limk→∞ Cϕ Pk . On the other hand, let K be a compact operator and a := limk→∞ KPk . Assume that a > 0 and let > 0 with 0 < a − . Then there is a sequence hk ∈ F 2 (Hn ) with hk 1, such that Pk K ∗ hk a − for any k N and some N ∈ N. Since K ∗ is a compact operator, there is a subsequence km ∈ N such that K ∗ hkm → v for some v ∈ F 2 (Hn ). Consequently, taking into account that Pkm v → 0, Pk 1, and Pk K ∗ hk Pk v + Pk v − K ∗ hk , m m m m m

938


we deduce that Pkm K ∗ hkm → 0, which is a contradiction. Therefore, limk→∞ KPk = 0. Note also that Cϕ − K (Cϕ − K)Pk Cϕ Pk − Pk K ∗ . Now, taking k → ∞, we obtain Cϕ − K limk→∞ Cϕ Pk , which proves relation (5.1). According to Proposition 4.1 and the remarks that follow, we have Pk Cϕ∗ f = f, ϕα eα , f ∈ F 2 (Hn ), |α|k

where Pk is the orthogonal projection of the full Fock space F 2 (Hn ) onto the closed span of the 2 vectors {eα : α ∈ F+ n , |α| k}, and f , ϕ1 , . . . , ϕn are seen as elements of the Fock space F (Hn ). Hence, we deduce that Pk C ∗ = ϕ

sup

1/2 f, ϕα 2 .

2 , f 1 f ∈Hball

|α|k

Combining this result with relation (5.1), we obtain the formula for the essential norm of Cϕ . The last part of the theorem is now obvious. 2 Proposition 5.2. Let ϕ := (ϕ1 , . . . , ϕn ) be a free holomorphic self-map of the noncommutative 2 . Then the following statements ball [B(H)n ]1 and let Cϕ be the composition operator on Hball hold. (i) (ii) (iii) (iv)

If ϕ is inner then Cϕ is not compact. If ϕ∞ < 1 then Cϕ is compact. If ϕ1 ∞ + · · · + ϕn ∞ < 1, then Cϕ is a trace class operator. If ϕ1 2∞ + · · · + ϕn 2∞ < 1, then Cϕ is a Hilbert–Schmidt operator.

Proof. To prove item (i), assume first that ϕ is an inner free holomorphic self-map of the noncommutative ball [B(H)n ]1 with ϕ(0) = 0. As in the proof of Theorem 2.2, {ϕα }α∈F+n is an 2 . Consequently, if {a } 2 orthonormal set in Hball α |α|k ⊂ C is such that |α|k |aα | = 1, then g := |β|k aβ ϕβ is in F 2 (Hn ) and g2 = 1. Note also that g, ϕα 2 = |aα |2 = 1. |α|k

|α|k

2 , we have Since {ϕα }α∈F+n is an orthonormal set in Hball 2 . Now, one can deduce that Hball

sup 2 , f 1 f ∈Hball

2 |α|k |f, ϕα |

f 2 for any f ∈

1/2 f, ϕα 2 = 1. |α|k

Due to Theorem 5.1, we deduce that Cϕ e = 1. Now, we consider the case when ξ := ϕ(0) = 0. Since the involutive free holomorphic automorphism Φξ is inner and the composition of inner


939

free holomorphic functions is inner (see [39]), we deduce that Ψ := Φξ ◦ ϕ is an inner free holomorphic self-map of [B(H)n ]1 . Since Ψ (0) = 0, the first part of the proof shows that CΨ is not compact. Taking into account that CΨ = Cϕ CΦξ , we deduce that Cϕ is not compact. To prove item (ii), let ϕ := ( ϕ1 , . . . , ϕn ) be the boundary function with respect to the left creation operators S1 , . . . , Sn , and set ϕ = s < 1. It is easy to see that [ ϕα : |α| = k] [ ϕ1 , . . . , ϕn ]k = s k , k ∈ N. For any g ∈ F 2 (Hn ) and m ∈ N, we have m C = g − g, e ϕ (1) g, e ϕ (1) ϕ α α α α k=0 |α|=k

k=m+1 |α|=k

⎡ ⎤ g, eα [ ⎣ .. ⎦ . ϕα : |α| = k] k=m+1 |α| = k

1/2 g, eα 2 sk |α|=k

k=m+1

1/2 2 1/2 g, eα s

2k

k=m+1 |α|=k

k=m+1

sm

g2 √ . 1 − s2 Consequently, the operator Gm : F 2 (Hn ) → F 2 (Hn ) defined by Gm (g) :=

m

g, eα ϕα (1)

k=0 |α|=k

has finite rank and converges to the composition operator C ϕ in the operator norm topology. Therefore, Cϕ is a compact operator. To prove item (iii), note that α∈F+ n

C ϕ eα =

∞ ∞

k ϕα (1) ϕ1 + · · · + ϕα ϕn < ∞. α∈F+ n

k=0 |α|=k

k=0

Consequently, Cϕ is a trace class operator. Finally, we prove item (iv). First, note that Cϕ is a Hilbert–Schmidt operator if and only if α∈F+n ϕα 22 < ∞. On the other hand, as above, one ca show that α∈F+ n

2 C ϕ eα

∞

k ϕ1 2 + · · · + ϕn 2 < ∞, k=0

which shows that Cϕ is a Hilbert–Schmidt operator. The proof is complete.

2

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Corollary 5.3. If ϕ is an inner free holomorphic self-map of the noncommutative ball [B(H)n ]1 2 is 1. such that ϕ(0) = 0, then the essential norm of the composition operator Cϕ on Hball Theorem 5.4. Let ϕ be a free holomorphic self-map of the noncommutative ball [B(H)n ]1 and 2 . Then the following statements hold. let Cϕ be the composition operator on Hball 2 satisfies the inequality (i) The essential norm of Cϕ on Hball

1/2 1 − λ2 Cϕ e lim sup . 2 λ→1 1 − ϕ(λ) 2 , then the scalar representation of ϕ cannot have finite (ii) If Cϕ is a compact operator on Hball angular derivative at any point of ∂Bn .

Proof. For each μ := (μ1 , . . . , μn ) ∈ Bn , we define the vector zμ := ∞ |α|=k μα eα , where k=0 + μα := μi1 · · · μip if α = gi1 · · · gip ∈ Fn and i1 , . . . , ip ∈ {1, . . . , n}, and μg0 = 1. Since zμ ∈ F 2 (Hn ) and Si∗ zμ = μi zμ , one can see that q(S1 , . . . , Sn )∗ zμ = q(μ)zμ for any noncommuta(j ) (j ) tive polynomial q. Let λ(j ) := (λ1 , . . . , λn ) ∈ Bn be such that λ(j ) → 1 as j → ∞. Since 1 zμ = √ , we deduce that 2 1−μ

# $ q(λ(j ) ) z (j ) lim q, λ = 0, = lim j →∞ j →∞ zλ(j ) zλ(j ) where q is seen as a noncommutative polynomial in F 2 (Hn ). Consequently, since the unit ball of F 2 (Hn ) is weakly compact and the polynomials are dense in F 2 (Hn ), there is a subsequence z (jk ) λ z (j ) which converges weakly to 0 as jk → ∞. Since this is true for any subsequence, we λ k

deduce that zλ(j ) → 0 weakly as λ(j ) 2 → 1. zλ(j )

(5.2) z

If K ∈ B(F 2 (Hn )) is an arbitrary compact operator, then limλ(j ) →1 K ∗ ( zλ(j ) ) = 0. On the λ other hand, due to relation (2.9), we have ∗ C z ϕ

λ(j )

=

1 1 − ϕ(λ(j ) )2

1/2 .

Using all these facts, we deduce that Cϕ e = inf T − K: K ∈ B(H) is compact

zλ(j ) ∗ (C lim sup − K) ϕ zλ(j ) λ(j ) →1

(j )


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∗ zλ(j ) C = lim sup ϕ z (j ) (j ) λ

→1

λ

= lim sup

λ(j ) →1

1 − λ(j ) 2 1 − ϕ(λ(j ) )2

1/2 ,

which proves item (i). To prove part (ii), we recall that the Julia–Carathéodory theorem in Bn asserts that if ψ : Bn → Bn is analytic and ξ ∈ ∂Bn , then ψ has finite angular derivative at ξ if and only if lim inf λ→ξ

1 − ψ(λ) < ∞, 1 − λ

2 , where the limit is taking as λ → ξ unrestrictedly in Bn . If Cϕ is a compact operator on Hball then according to part (i), we have

1/2 1 − λ2 lim sup = 0. 1 − ϕ(λ)2 λ→ξ Now, combining these results when ψ : Bn → Bn is defined by ψ(λ) := ϕ(λ), λ ∈ Bn , the result in part (ii) follows. The proof is complete. 2 We need the following lemma which can be extracted from [14]. We include a proof for completeness. Lemma 5.5. Let ψ = (ψ1 , . . . , ψn ) be a holomorphic self-map of the open unit ball Bn with the property that ψ(E(L, ζ1 )) ⊆ E(L, ζ1 ) for each ellipsoid 2

E(L, ζ1 ) := λ ∈ Bn : 1 − λ, ζ1 L 1 − λ2 ,

L > 0,

where ζ1 := (1, 0, . . . , n) ∈ Bn . Then the slice function φζ1 : D → D defined by φζ1 (z) := ψ1 (z, 0 . . . , 0), z ∈ D, has the property that lim inf z→1

1 − |φζ1 (z)| 1. 1 − |z|

Proof. Note that when w = (r, 0, . . . , 0) ∈ Bn with r ∈ (0, 1) and L := ψ(E(L, ζ1 )) ⊆ E(L, ζ1 ) implies |1 − ψ1 (w)|2 L. 1 − ψ(w)2 Hence, and using the inequality 1 − |ψ1 (w)| |1 − ψ1 (w)|, we obtain 1 − |ψ1 (w)| 1 − r , 1 + |ψ1 (w)| 1 + r

1−r 1+r ,

the inclusion

942


which implies |ψ1 (w)| r = w and, therefore, 1 − |ψ1 (w)| 1 1 − w for w = (r, 0, . . . , 0) ∈ Bn . The latter inequality can be used to complete the proof.

2

In what follows we also need the following lemma. Since the proof is straightforward, we space of all free holomorphic functions shall omit it. We denote by H 2 ([B(H)] 1 ) thekHilbert ∞ 2 c X with on [B(H)]1 of the form f (X) = ∞ k=0 k k=0 |ak | < ∞. It is easy to see that 2 2 H ([B(H)]1 ) can be identified with the classical Hardy space H (D). Lemma 5.6. Let F : [B(H)n ]1 → B(H) be a free holomorphic function and let ζ1 := (1, 0, . . . , 0) ∈ ∂Bn . The slice function Fζ1 : [B(H)]1 → B(H) defined by Fζ1 (Y ) := F (ζ1 Y ), Y ∈ B(H) 1 , has the following properties. (i) (ii) (iii) (iv)

Fζ1 is a free holomorphic function on [B(H)]1 . 2 then F ∈ H 2 ([B(H)] ) and F F . If F ∈ Hball ζ1 1 ζ1 2 2 2 is an isometry. The inclusion H 2 ([B(H)]1 ) ⊂ Hball 2 with the full Fock space F 2 (H ), Under the identification of Hball n Fζ1 = PF 2 (H1 ) F,

where PF 2 (H1 ) is the orthogonal projection of F 2 (Hn ) onto F 2 (H1 ) ⊂ F 2 (Hn ). (v) If F is bounded on [B(H)n ]1 , then Fζ1 is bounded on [B(H)]1 and Fζ1 ∞ F ∞ . Now, we have all the ingredients to prove the following result. Theorem 5.7. Let ϕ = (ϕ1 , . . . , ϕn ) be a free holomorphic self-map of the noncommutative ball 2 , then the scalar representation of ϕ [B(H)n ]1 . If Cϕ is a compact composition operator on Hball is a holomorphic self-map of Bn which has exactly one fixed point in the open ball Bn . Proof. Let ψ = (ψ1 , . . . , ψn ) be the scalar representation of ϕ, i.e. the map ψ : Bn → Bn defined by ψ(λ) := φ(λ), λ ∈ Bn . It is clear that ψ is a holomorphic self-map of the open unit ball Bn . Assume that ψ has no fixed points in Bn . According to [13] (see also Theorem 3.1), there exists a unique Denjoy–Wolff point ζ ∈ ∂Bn such that ψ(E(L, ζ )) ⊆ E(L, ζ ) for each ellipsoid E(L, ζ ), L > 0. Without loss of generality we can assume that ζ = ζ1 := (1, 0, . . . , 0) ∈ Bn . Then, due to Lemma 5.5, the slice function φζ1 : D → D defined by φζ1 (z) := ψ1 (z, 0 . . . , 0) has the property that lim inf z→1

1 − |φζ1 (z)| 1. 1 − |z|

According to Julia–Carathéodory theorem (see [41]), φζ1 has finite angular derivative at 1 which is less than or equal to 1. On the other hand, it is well known (see also Theorem 5.4 when n = 1)


943

that if a composition operator is compact on H 2 (D), then its symbol cannot have a finite angular derivative at any point. Consequently, Cφζ1 is not a compact operator on H 2 (D). 2 with the full Fock space F 2 (H ), set Under the identification of Hball n Γ = PF 2 (H1 ) ϕ1 ,

(5.3)

where PF 2 (H1 ) is the orthogonal projection of F 2 (Hn ) onto F 2 (H1 ) ⊂ F 2 (Hn ). According to Lemma 5.5, Γ : [B(H)]1 → [B(H)]1 is a bounded free holomorphic function. Now we show that CΓ is a compact composition operator on F 2 (H1 ). Let {f (m) }∞ m=1 be a bounded sequence in F 2 (H1 ) such that f (m) → 0 weakly in F 2 (H1 ). Since F 2 (H1 ) ⊂ F 2 (Hn ) and F 2 (Hn ) = F 2 (H1 ) ⊕ F 2 (H1 )⊥ , it is easy to see that f (m) → 0 weakly in F 2 (Hn ). Due to the compactness of Cϕ on F 2 (Hn ), we must have Cϕ f (m)

F 2 (Hn )

→0

as m → ∞.

(5.4)

(m) k Since f (m) ∈ F 2 (H1 ), it has the representation f (m) = ∞ k=0 ak e1 for some coefficients ∞ (m) (m) k 2 ak ∈ C with k=0 |ak |2 < ∞. Hence Cϕ f (m) = ∞ k=0 ak ϕ1 , where ϕ1 is seen in F (Hn ), k k 2 i.e., ϕ1 := ϕ1 (1), and the convergence of the series is in F (Hn ). Note also that, due to (5.3), for each k ∈ N, ϕ1k = Γ k + χk for some χk ∈ F 2 (Hn ) F 2 (H1 ). Consequently, we have Cϕ f

(m)

=

∞ k=0

(m) ak ϕ1k

=

∞

(m)

ak Γ k + g = f (m) ◦ Γ + g

k=0

for some g ∈ F 2 (Hn ) F 2 (H1 ). Hence, we deduce that CΓ f (m) F 2 (H1 ) Cϕ f (m) F 2 (Hn ) . Using relation (5.4), we have CΓ f (m) F 2 (H1 ) → 0 as m → ∞. This proves that the composition operator CΓ is compact on F 2 (H1 ). Note alsothat, under the natural identification of ∞ k 2 2 k F (H1 ) with H (D), i.e., f = ∞ k=0 ck e1 → g(z) = k=0 ck z , the composition operator CΓ 2 on F (H1 ) is unitarily equivalent to the composition operator Cφζ on H 2 (D). Consequently, Cφζ is compact, which is a contradiction. Therefore the map ψ has fixed points in Bn . Now we prove that ψ has only one fixed point in Bn . Assume that there are two distinct points ξ (1) , ξ (2) ∈ Bn such that ψ(ξ (1) ) = ξ (1) and ψ(ξ (2) ) = ξ (2) . It is well known [41] that the fixed point set of the map ψ is affine. As in the proof of Theorem 2.1, we have Cϕ∗ zμ =

ϕα (μ)eα = zϕ(μ) ,

μ := (μ1 , . . . , μn ) ∈ Bn ,

k=0 |α|=k

where the vector zμ ∈ F 2 (Hn ) is defined by zμ := ∞ k=0 |α|=k μα eα . As a consequence, we deduce that Cϕ∗ zξ = zξ for any ξ in the fixed point set Λ of ψ . Since Λ is infinite and according to the proof of Lemma 4.3 the vectors {zξ }ξ ∈Λ are linearly independent, we deduce that ker(I − Cϕ∗ ) is infinite dimensional. This contradicts the fact that Cϕ is a compact operator on 2 . In conclusion, ψ has exactly on fixed point in B . This completes the proof. 2 Hball n Combining now Theorem 5.7 and Theorem 2.6, we can deduce the following similarity result. 2 is similar to a contraction. Corollary 5.8. Every compact composition operator on Hball

944


2 is arcwise connected, with Theorem 5.9. The set of compact composition operators on Hball respect to the operator norm topology, in the set of all composition operators.

Proof. Let ϕ = (ϕ1 , . . . , ϕn ) be a non-constant free holomorphic self-map of the noncommuta2 . For each r ∈ [0, 1], tive ball [B(H)n ]1 such that Cϕ is a compact composition operator on Hball consider the free holomorphic map ϕr : [B(H)n ]1 → [B(H)n ]1 defined by ϕr (X) = ϕ(rX), X ∈ [B(H)n ]1 . If ϕ∞ < 1, then ϕr ∞ < 1 and due to Proposition 5.2, the operator Cϕr is 2 . Now assume that ϕ = 1. Since ϕ is non-constant, Theorem 1.1 implies compact on Hball ∞ ϕ(0) < 1 and the map [0, 1) r → ϕr ∞ is strictly increasing. Therefore ϕr ∞ < 1 for 2 all r ∈ [0, 1). Using again Proposition 5.2, we deduce that the operator Cϕr is compact on Hball 2 ) denote the algebra of all compact operators on H 2 and define for any r ∈ [0, 1). Let K(Hball ball 2 ) by setting γ (r) := C . Now we show that γ is a continuous the function γ : [0, 1] → K(Hball ϕr 2 set map in the operator norm topology. Fix r0 ∈ [0, 1]. For any g(X) := α∈F+n aα Xα ∈ Hball 2 and note that gr (X) := α∈F+n aα r |α| Xα ∈ Hball gr − gr0 2 → 0

as r → r0 .

(5.5)

2 and f 1, we have In particular, taking g = Cϕ f where f ∈ Hball 2 (f ◦ ϕ)r − (f ◦ ϕ)r → 0 as r → r0 . 0 2 2 with f 1. We need to show that the latter convergence is uniform with respect to f ∈ Hball 2 Indeed, if we assume the contrary, then there is 0 > 0 such that for any n ∈ N there is rn ∈ [0, 1] 2 with f 1 such that with |rn − r0 | < n1 and there exists fn ∈ Hball n 2

(fn ◦ ϕ)r − (fn ◦ ϕ)r > 0 . n 0 2

(5.6)

2 under C is relatively compact. Since Cϕ is a compact operator the image of the unit ball of Hball ϕ Therefore there is a subsequence {fnk } such that 2 fnk ◦ ϕ → ψ ∈ Hball .

(5.7)

Now, note that (fn ◦ ϕ)r − (fn ◦ ϕ)r nk k k 0 2 (fnk ◦ ϕ)rnk − ψrnk 2 + ψrnk − ψr0 2 + ψr0 − (fnk ◦ ϕ)r0 2 2fnk ◦ ϕ − ψ2 + ψrnk − ψr0 2 . Due to relations (5.5) and (5.7), we deduce that (fn ◦ ϕ)r − (fn ◦ ϕ)r → 0 as r → r0 , nk k k 0 2 which contradicts relation (5.6). Therefore Cϕr − Cϕr0 → 0 as r → r0 , which proves the continuity of the map γ . Let χ = (χ1 , . . . , χn ) be another non-constant free holomorphic self-map 2 . of the noncommutative ball [B(H)n ]1 such that Cχ is a compact composition operator on Hball


945

2 ) given by (r) := C is continuous in the operator As above, the function : [0, 1] → K(Hball χr 2 ) such norm topology. It remains to show that there is a continuous mapping ω : [0, 1] → K(Hball that ω(0) = Cϕ0 and ω(1) = Cχ0 . To this end, since ϕ(0) < 1 and χ(0) < 1, we can define the map σ : [0, 1] → Bn by setting σ (t) := (1 − t)ϕ(0) + tχ(0) for t ∈ [0, 1]. Using again Propo2 for any t ∈ [0, 1]. sition 5.2, we deduce that Cσ (t)I is a compact composition operator on Hball 2 Now we define ω : [0, 1] → K(Hball ) by setting ω(t) := Cσ (t)I . To prove continuity of this map in the operator norm topology, note that

Cσ (t)I f − Cσ (t )I f = f, zσ (t) − zσ (t ) f 2 zσ (t) − zσ (t ) 2 ,

(5.8)

where zλ = α∈F+n λα eα for λ ∈ Bn . On the other hand, consider the noncommutative Cauchy kernel Cλ := (I −λ1 S1 −· · ·−λn Sn )−1 , λ := (λ1 , . . . , λn ) ∈ Bn . Note that λ1 S1 +· · ·+λn Sn = λ2 < 1 and Cλ ∈ Fn∞ for any λ ∈ Bn . We have zσ (t) − zσ (t ) 2 = (Cσ (t) − Cσ (t ) )1 Cσ (t) − Cσ (t ) Cσ (t) Cσ (t ) σ (t) − σ t 2 . Consequently, since Bn λ → Cλ ∈ Fn∞ is continuous, we deduce that [0, 1] t → zσ (t) ∈ F 2 (Hn ) is continuous as well. Combining this result with relation (5.8), we deduce the continuity of ω, which completes the proof. 2 6. Schröder equation for noncommutative power series and spectra of composition operators In this section, we consider a noncommutative multivariable Schröder type equation and use it 2 . As a consequence, to obtain results concerning the spectrum of composition operators on Hball using the results from the previous section, we determine the spectra of compact composition 2 . operators on Hball First, we provide the following noncommutative Schröder [43] type result. Theorem 6.1. Let A ∈ Mn×n be a scalar matrix and let Λ = (Λ1 , . . . Λn ) be an n-tuple of power series in noncommuting indeterminates Z1 , . . . , Zn , of the form Λ = [Z1 , . . . , Zn ]A + [Γ1 , . . . , Γn ], where Γ1 , . . . , Γn are noncommutative power series containing only monomials of degree greater than or equal to 2. If there is a noncommutative power series F which is not identically zero and satisfies the Schröder type equation F ◦ Λ = cF for some c ∈ C, then either c = 1 or c is a product of eigenvalues of the matrix A.

946


Proof. Since A ∈ Mn×n there is a unitary matrix U ∈ Mn×n such that U −1 AU is an upper triangular matrix. Setting ΦU = [Z1 , . . . , Zn ]U , the equation F ◦ Λ = cF is equivalent to F ◦ Λ = cF , where F := ΦU ◦ F ◦ ΦU −1 and Λ := ΦU ◦ Λ ◦ ΦU −1 = [Z1 , . . . , Zn ]U −1 AU + U −1 [Γ1 , . . . , Γn ]U. Therefore, we can assume that A = [aij ] ∈ Mn×n is an upper triangular matrix. We introduce + a total order on the free semigroup F+ n as follows. If α, β ∈ Fn with |α| |β| we say that + α < β. If α, β ∈ Fn are such that |α| = |β|, then α = gi1 · · · gik and β = gj1 · · · gjk for some i1 , . . . , ik , j1 , . . . , jk ∈ {1, . . . , k}. We say that α < β if either i1 < j1 or there exists p ∈ {2, . . . , k} such that i1 = j1 , . . . , ip−1 = jp−1 and ip < jp . It is easy to see that relation is a total order on F+ n. According to the hypothesis and due to the fact that A is an upper triangular matrix, we have

Λj =

j

aij Xi + Γj ,

j = 1, . . . , n.

(6.1)

i=1

Consequently, if α = gi1 · · · gik ∈ F+ n , i1 , . . . ik ∈ {1, . . . , n}, then Λα := Λi1 · · · Λik = Ψ 0 is a real parameter. Let ψ (k) −itH N ψN , and let γN,t be the corresponding k-particle marginal. Then, there exists a constant C e independent of k and there exists an integer N0 (k) for every k 1, such that for all N > N0 (k), (k)

γN,t C k . Tr(1 − x1 ) · · · (1 − xk )

(2.39)

3. Compactness and convergence to the infinite hierarchy In this section, we summarize the main steps of the proof in [7,15] of the compactness of the sequence of k-particle marginals and the convergence to the infinite hierarchy as N → ∞. The arguments developed in these works can be adopted almost verbatim for the current context. We outline them here for the convenience of the reader, essentially quoting the exposition in [15]. The following topology on the space of density matrices is picked in [7]. We let Kk = K(L2 ((Rd )k )) denote the space of compact operators on L2 (Rd ) equipped with the operator norm topology, and L1k := L1 (L2 ((Rd )k )) denotes the space of trace-class operators on L2 ((Rd )k ) equipped with the trace-class norm. It is a standard fact that L1k = Kk∗ . Kk is a separable Banach space, hence the closed unit ball in L1k , which is weak-* compact by the (k) Banach–Alaoglu theorem, is metrizable in the weak-* topology. Let {Jj } be a countable dense (k)

subset of the unit ball of Kk , that is, Jj 1. Then, −j (k) γ (k) = 2 Tr Jj γ (k) − γ (k) ηk γ (k) ,

(3.1)

j ∈N

is a metric, and the associated metric topology is equivalent to the weak-* topology. A uniformly bounded sequence γN(k) ∈ L1k converges to γ (k) ∈ L1k with respect to the weak-* topology if and (k) only if ηk (γN , γ (k) ) → 0 as N → ∞. Let C([0, T ], L1k ) be the space of L1k -valued functions of t ∈ [0, T ] that are continuous with respect to the metric ηk . One can endow C([0, T ], L1k ) with the metric ηk (γ (k) (·), γ (k) (·)) = (k) (k) supt∈[0,T ] ηk (γ (t), γ (t)) [7]. This induces the product topology τprod on k∈N C([0, T ], 1 ηk on C([0, T ], L1k ), for k ∈ N. For more details, see [7]. Lk ), generated by the metrics (k) Proposition 3.1. The sequence of marginal densities ΓN,t = { γN,t }N k=1 is compact with respect ηk [7]. Any subsequential limit point to the product topology τprod generated by the metrics (k) (k) Γ∞,t = {γ∞,t }k1 has the property that the components γ∞,t are symmetric under permutations, (k) is positive, and Tr γ∞,t 1 for every k 1.

Proof. The proof is completely analogous to the one given for a related result in [7], and for Theorem 4.1 in [15]. We summarize the main steps.

T. Chen, N. Pavlović / Journal of Functional Analysis 260 (2011) 959–997

971 (k)

Using a Cantor diagonal argument, it is sufficient to prove the compactness of γN,t for a fixed k. By the Arzela–Ascoli theorem, this is achieved by proving equicontinuity of ΓN,t = (k) ηk . It is sufficient to prove that for every observable J (k) { γN,t }N k=1 with respect to the metric from a dense subset of Kk and for every > 0, there exists δ = δ(J (k) , ) such that (k) (k) 0 that is introduced in (2.38) can be removed by the same limiting procedure as in [7], see also [15]. We quote the main steps for the convenience of the reader, from [7,15]. For the limiting hierarchy ΓN,t → Γ∞,t as N → ∞, it is proven below that for every κ > 0, (k) η( γN,t , |φt φt |⊗k ) → 0 as N → ∞, for every fixed k. This also implies the convergence γN,t → |φt φt |⊗k (k)

(3.19)

in the weak-* topology of L1k . (k) It remains to be proven that also γN,t → |φt φt |⊗k . To this end, one may assume κ > 0 to be sufficiently small such that

(k) Tr J (k) γ (k) − N < Cκ , γN,t J (k)

ΨN − Ψ N,t 2

(3.20)

974


N < Cκ, uniformly in N , which can be easily uniformly in N . This follows from ΨN − Ψ verified. On the other hand, for all N > N0 with N0 sufficiently large, we have (k) Tr J (k) γN,t − |φt φt |⊗k , 2

(3.21)

(k)

due to the convergence of γN,t described above. This implies that for arbitrary > 0, Tr J (k) γ (k) − |φt φt |⊗k , N,t

(3.22)

for all N > N0 . Thus, for every t ∈ [0, T ] and every fixed k, γN,t → |φt φt |⊗k in the weak-* topology of L1k . Because the limiting density is an orthogonal projection, this is equivalent to the convergence in trace norm topology. For details, we refer to [7,15]. (k)

4. A priori energy bounds on the limiting hierarchy (k)

In this section we prove some spatial bounds for the limit points {γ∞,t }k1 that shall be used in order to prove uniqueness of the hierarchy. More precisely, first we state the a priori bound which follows from the estimates (2.39) (k) for γN,t . (k) (k) N Proposition 4.1. If Γ∞,t = {γ∞,t }k1 is a limit point of the sequence ΓN,t = { γN,t }k=1 with respect to the product topology τprod , then there exists C > 0 such that (k)

Tr(1 − 1 ) · · · (1 − k )γ∞,t C k ,

(4.1)

for all k 1. (k)

Proof. The proof follows from the fact that the a priori estimates (2.39) for γN,t hold uniformly in N . 2 As in [15], we prove uniqueness of the infinite hierarchy following the approach introduced by Klainerman and Machedon [16]. In order to apply the approach of [16] we establish another a priori bound on the limiting density. Such a bound is formulated in Theorem 4.2 below. In what follows S (k,α) denotes S (k,α) =

k j =1

α

α

(1 − xj ) 2 (1 − xj ) 2 .

(k) Theorem 4.2. Suppose that d ∈ {1, 2}. If Γ∞,t = {γ∞,t }k1 is a limit point of the sequence (k) N ΓN,t = { γN,t }k=1 with respect to the product topology τprod , then, for every α < 1 if d = 2, and every α 1 if d = 1, there exists C > 0 such that

(k,α) (k+2)

S Bj ;k+1,k+2 γ∞,t

L2 (Rdk ×Rdk )

for all k 1 and all t ∈ [0, T ].

C k+2 ,

(4.2)


975

Proof. We modify the proof of an analogous result presented in Theorem 5.2 of [15]. We note () that for the argument employed here, the fact is used that γ∞,t is positive, and thus, especially, hermitean. We note that Theorem 5.1 below states a similar result, but for a different quantity () than γ∞,t which may be neither positive nor hermitean. Thus, the proof of Theorem 5.1 is based on a different approach that necessitates a lower bound on α, instead of an upper bound as required here. By (4.1) it suffices to prove

(k,α) (k+2)

S Bj ;k+1,k+2 γ∞,t

L2 (Rdk ×Rdk )

(k+2)

Tr(1 − 1 ) · · · (1 − k+2 )γ∞,t .

(4.3)

We will consider the case k = 1, j = 1 (the argument for k 2 can be carried out in a sim(3) ilar way). We start by calculating the Fourier transform of B1;2,3 γ∞,t . It suffices to do that for (3) (3) + − γ∞,t (the calculations for B1;2,3 γ∞,t can be carried out in an analogous way): B1;2,3 (3) + γ∞,t p; p B1;2,3 −ix1 ·p ix1 ·p = dx1 dx1 e e dx2 dx2 dx3 dx3 (3) × δ(x1 − x2 )δ x1 − x2 δ(x1 − x3 )δ x1 − x3 γ∞,t x1 , x2 , x3 ; x1 , x2 , x3 = dq dκ dr ds dx1 dx1 dx2 dx2 dx3 dx3 (3) × e−ix1 ·p eix1 ·p eiq(x1 −x2 ) e−iκ(x1 −x2 ) eir(x1 −x3 ) e−is(x1 −x3 ) γ∞,t x1 , x2 , x3 ; x1 , x2 , x3 = dq dκ dr ds dx1 dx1 dx2 dx2 dx3 dx3 (3) × e−ix1 ·(p−q+κ−r+s) e−ix2 ·q e−ix3 ·r eix1 ·p eix2 ·κ eix3 ·s γ∞,t x1 , x2 , x3 ; x1 , x2 , x3 (3) = dq dκ dr ds γ∞,t p − q + κ − r + s, q, r; p , κ, s .

(4.4)

Hence (3) + γ∞,t p; p S (1,α) B1;2,3

(3) α α = p p p − q + κ − r + s, q, r; p , κ, s , dq dκ dr ds γ∞,t

(4.5)

which in turn implies

(1,α) + 2 (3)

S p; p L2 (Rd ×Rd ) B1;2,3 γ∞,t

2α = dp dp dq1 dq2 dκ1 dκ2 dr1 dr2 ds1 ds2 p 2α p (3) × γ∞,t p − q1 + κ1 − r1 + s1 , q1 , r1 ; p1 , κ1 , s1 (3) × γ∞,t p − q2 + κ2 − r2 + s2 , q2 , r2 ; p2 , κ2 , s2 .

(4.6)

976


Substituting (3) γ∞,t p1 , p2 , p3 ; p1 , p2 , p3 = λj ψj (p1 , p2 , p3 )ψ j p1 , p2 , p3

(4.7)

j

into (4.6) and keeping in mind that λj 0 for all j and j λj 1 thanks to γ (k+2) being a non-negative trace-class operator with trace at most one, we obtain

(1,α) +

S B =

(3) 2 1;2,3 γ∞,t p; p L2 (Rd ×Rd )

λi λj s

2α dp dp dq1 dq2 dκ1 dκ2 dr1 dr2 ds1 ds2 p 2α p

i,j

× ψj (p − q1 + κ1 − r1 + s1 , q1 , r1 )ψ j p , κ1 , s1 × ψj (p − q2 + κ2 − r2 + s2 , q2 , r2 )ψ j p , κ2 , s2 .

(4.8)

We observe that for l = 1, 2 we have

p α C p − ql + κl − rl + sl α + ql α + rl α + κl α + sl α which implies that

p 2α C p − q1 + κ1 − r1 + s1 α + q1 α + r1 α + κ1 α + s1 α × p − q2 + κ2 − r2 + s2 α + q2 α + r2 α + κ2 α + s2 α .

(4.9)

Substituting (4.9) into (4.8), we obtain 16 terms. We will illustrate how to control one of them, the remaining cases are similar. Using a weighted Schwarz inequality, we find

dp dp dq1 dq2 dκ1 dκ2 dr1 dr2 ds1 ds2

2α × p p − q1 + κ1 − r1 + s1 α p − q2 + κ2 − r2 + s2 α × ψj (p − q1 + κ1 − r1 + s1 , q1 , r1 )ψ j p , κ1 , s1 × ψj (p − q2 + κ2 − r2 + s2 , q2 , r2 )ψ j p , κ2 , s2 I + II,

where I=


p 2α p − q1 + κ1 − r1 + s1 2 q1 2 r1 2 κ2 2 s2 2

p − q2 + κ2 − r2 + s2 2−2α q2 2 r2 2 κ1 2 s1 2 2 2 × ψj (p − q1 + κ1 − r1 + s1 , q1 , r1 ) ψj p , κ2 , s2 , ×


977

and


II =

p 2α p − q2 + κ2 − r2 + s2 2 q2 2 r2 2 κ1 2 s1 2

p − q1 + κ1 − r1 + s1 2−2α q1 2 r1 2 κ2 2 s2 2 2 2 × ψj (p − q2 + κ2 − r2 + s2 , q2 , r2 ) ψj p , κ1 , s1 . ×

Below we illustrate how to estimate I . The expression II can be estimated in a similar manner. We will use the bound dy C , (4.10)

P − y 2−2α y 2 P 2−2α Rd

which is valid for d = 1 if α 1, and for d = 2 if α < 1; it is easily obtained by rescaling y → P y. To estimate I , we integrate over q2 , using (4.10), followed by integrating over r2 , using (4.10) again, to obtain

dp dp dq1 dκ1 dκ2 dr1 ds1 ds2

I

p 2α p − q1 + κ1 − r1 + s1 2 q1 2 r1 2 κ2 2 s2 2

p + κ2 + s2 2−2α κ1 2 s1 2 2 2 × ψj (p − q1 + κ1 − r1 + s1 , q1 , r1 ) ψj p , κ2 , s2 . ×

The change of variable p˜ = p − q1 + κ1 − r1 + s1 gives I

d p˜ dp dq1 dκ1 dκ2 dr1 ds1 ds2

p ˜ 2 q1 2 r1 2 p 2 κ2 2 s2 2

p˜ + q1 − κ1 + r1 − s1 + κ2 + s2 2−2α κ1 2 s1 2 2 2 × ψj (p, ˜ q1 , r1 ) ψj p , κ2 , s2 2 Cα d p˜ dq1 dr1 p ˜ 2 q1 2 r1 2 ψj (p, ˜ q1 , r1 ) ×

×

2 2 dp dκ2 ds2 p κ2 2 s2 2 ψj p , κ2 , s2 .

(4.11)

To obtain (4.11) we have used that, as a consequence of (4.10), Cα = sup

P ∈Rd

dy dz < ∞,

P − y − z 2−2α y 2 z 2

for all α 1 if d = 1, and all α < 1 if d = 2.

(4.12)

978


The other 15 contributions to (4.8) can be obtained in a similar way. Therefore, using the above analysis and (4.8), we conclude that

(1,α) +

S B

(3) 2 1;2,3 γ∞,t p; p L2 (Rd ×Rd )

C

λi λj

2 ˜ 2 q1 2 r1 2 ψj (p, ˜ q1 , r1 ) d p˜ dq1 dr1 p

i,j

×

2 2 dp dκ2 ds2 p κ2 2 s2 2 ψj p , κ2 , s2

˜ q1 d p˜ dq1 dr1 p 2

2

2 (3)

r1 2 γ∞,t (p, ˜ q1 , r1 ; p, ˜ q1 , r1 )

2

(3) 2 = C Tr(1 − 1 )(1 − 2 )(1 − 3 )γ∞,t , which gives (4.3) in the case k = 1, j = 1.

(4.13)

2

In addition to the above results, we derive a third type of spatial bounds, which is more restrictive in terms of the condition on α (it requires α > d2 ). Note that for d = 1 we can afford this range of α. In particular, we shall use this new bound iteratively in the proof of uniqueness of the limiting hierarchy when d = 1. The proof of the bound is inspired by the proof of a spacetime bound for the freely evolving limiting hierarchy given in Theorem 1.3 of [16]. However, the (k) bound that we derive here is obtained for any γ∞,t . Theorem 4.3. Suppose that d 1. If Γ∞,t = {γ∞,t }k1 is a limit point of the sequence ΓN,t = (k) d { γN,t }N k=1 with respect to the product topology τprod , then, for every α > 2 there exists a constant C = C(α) such that the estimate (k)

(k,α) (k+2)

S Bj ;k+1,k+2 γ∞,t

L2 (Rdk ×Rdk )

(k+2) C S (k+2,α) γ∞,t L2 (Rd(k+2) ×Rd(k+2) )

(4.14)

holds. Proof. Let (uk , uk ), q := (q1 , q2 ), and q := (q1 , q2 ) denote the Fourier conjugate variables cor , xk+2 ), respectively. responding to (x k , x k ), (xk+1 , xk+2 ), and (xk+1 Without any loss of generality, we may assume that j = 1 in Bj ;k+1,k+2 . Then, we have

(k,α) (k+2) 2

S B1;k+1,k+2 γ∞,t 2

L (Rdk ×Rdk )

=

duk duk

×

k

2α

uj 2α uj

j =1

dq dq

(k+2) γ∞,t u1

+ q1 + q2 − q1

− q2 , u2 , . . . , uk , q; uk , q

2

,

(4.15)

where now, the Fourier transform in only performed in the spatial coordinates. Applying the Schwarz inequality, we find the upper bound


duk duk Iα

τ, uk , uk

979

dq dq

k k 2α

α 2α

2α

uj × u1 + q1 + q2 − q1 − q2 q1 α q2 2α q1 q2

uj 2α j =2

(k+2) 2 γ∞,t u1 + q1 + q2 − q1 − q2 , u2 , . . . , uk , q; uk , q ×

j =1

(4.16)

where Iα uk , uk :=

dq dq

u1 2α .

u1 + q1 + q2 − q1 − q2 2α q1 2α q2 2α q1 2α q2 2α

(4.17)

Using 2α

2α 2α

u1 2α C u1 + q1 + q2 − q1 − q2 , + q1 2α + q2 2α + q1 + q2

(4.18)

and shifting some of the momentum variables, one immediately obtains that Iα uk , uk < C

dq dq

1

q1

2α q

, 2α 2α 2α 2 q1 q2

(4.19)

which is finite for all α> This proves the claim.

d . 2

(4.20)

2

5. Bounds on the freely evolving infinite hierarchy In this section, we prove bounds on the infinite hierarchy for b0 = 0, i.e., in the absence of particle interactions; see (1.9) for the definition of b0 . These will be used for the recursive estimation of terms appearing in the Duhamel expansions studied in Section 7. Our approach is similar to the one of Klainerman and Machedon in [16]. In dimension d = 2, we prove space-time bounds in complete analogy to [16,15] which are global in time.2 From here on and for the rest of this paper, we will write (r) γ (r) t, x k ; x k ≡ γ∞,t t, x k ; x k

(5.1)

which is notationally more convenient for the discussion of space-time norms. Theorem 5.1. Assume that d = 2 and

5 6

< α < 1. Let γ (k+2) denote the solution of

i∂t γ (k+2) t, x k+2 ; x k+2 + (x k+2 − x k+2 )γ (k+2) t, x k+2 ; x k+2 = 0

(5.2)

2 In dimension d = 1, the argument used for d = 2 would produce a divergent bound; accordingly, when d = 1 we shall use the a priori bounds obtained in Theorem 4.3.

980


with initial condition (k+2)

γ (k+2) (0, ·) = γ0

∈ Hα .

(5.3)

Then, there exists a constant C = C(α) such that

(k,α)

S Bj ;k+1,k+2 γ (k+2)

L2

(k+2)

2 C S (k+2,α) γ0 L

(R×R t,x k ,x k

(R x k+2 ,x k+2

2k ×R2k )

(5.4)

2(k+2) ×R2(k+2) )

holds. Proof. We give a proof using the arguments of [16,15]. We note that the arguments presented in the proof of Theorem 4.2 cannot be straightforwardly employed here because here, Bj ;k+1,k+2 γ (k+2) are not hermitean so that (4.7) is not available. Let (τ, uk , uk ), q := (q1 , q2 ), and q := (q1 , q2 ) denote the Fourier conjugate variables corre , xk+2 ), respectively. sponding to (t, x k , x k ), (xk+1 , xk+2 ), and (xk+1 Without any loss of generality, we may assume that j = 1 in Bj ;k+1,k+2 . Then, abbreviating

k 2 2 2 δ(· · ·) := δ τ + u1 + q1 + q2 − q1 − q2 + u2j + |q|2 − uk − q

(5.5)

j =2

we find

2

(k,α)

S B1;k+1,k+2 γ (k+2) 2 L

t,x k ,x k

=

duk duk

dτ

k

(R×R2(k+2) ×R2(k+2) )

2α

uj 2α uj

j =1

R

×

2 γ (k+2) τ, u1 + q1 + q2 − q1 − q2 , u2 , . . . , uk , q; uk , q . dq dq δ(· · ·)

(5.6)

Using the Schwarz estimate, this is bounded by

dτ

duk duk Iα τ, uk , uk

dq dq δ(· · ·)

R

×

u1 + q1 + q2 − q1

2α

2α 2α q2 − q2 q1 2α q2 2α q1

k j =2

uj

(k+2) 2 γ τ, u1 + q1 + q2 − q1 − q2 , u2 , . . . , uk , q; uk , q ×

2α

k

uj

2α

j =1

(5.7)


981

where Iα τ, uk , uk :=

dq dq

u1 + q1 + q2 − q1

δ(· · ·) u1 2α . − q2 2α q1 2α q2 2α q1 2α q2 2α

(5.8)

Similarly as in [16,15], we observe that 2α

2α 2α , + q1 2α + q2 2α + q1 + q2

u1 2α C u1 + q1 + q2 − q1 − q2

(5.9)

so that 5 Iα τ, uk , uk J

(5.10)

=1

where J is obtained from bounding the numerator of (5.8) using (5.9), and from canceling the -th term on the rhs of (5.9) with the corresponding term in the denominator of (5.8). Thus, for instance, δ(· · ·) J1 < dq dq , (5.11) 2α

q1 q2 2α q1 2α q2 2α and each of the terms J with = 2, . . . , 5 can be brought into a similar form by appropriately translating one of the momenta qi , qj . Further following [16,15], we observe that the argument of the delta distribution equals k 2 2 2 τ + u1 + q1 + q2 − q1 + u2j + |q|2 − uk − q1 − 2 u1 + q1 + q2 − q1 · q2 , j =2

and we integrate out the delta distribution using the component of q2 parallel to (u1 + q1 + q2 − q1 ). This leads to the bound 1 J1 < Cα C dq dq1 (5.12) |u1 + q1 + q2 − q1 | q1 2α q2 2α q1 2α where

Cα := R

dζ .

ζ 2α

(5.13)

Clearly, Cα is finite for any α > 12 . To bound J1 , we pick a spherically symmetric function h 0 with rapid decay away from the unit ball in R2 , such that h∨ (x) 0 decays rapidly outside of the unit ball in R2 , and 1 1 (q) (5.14) < h ∗

q 2α | · |2α (for example, h(u) = c1 e−c2 u , for suitable constants c1 , c2 ); since α < 1, the right-hand side is in L∞ (R2 ). Then, 2

982


1 1 1 1 ∗ h ∗ , h ∗ ∗ h∗ |·| | · |2α | · |2α | · |2α L2 (R2 ) ∨ 3 ∨ 1 1 = Cα C dx (x) h ∗ (x) |·| | · |2α 3 1 1 ∨ 3 h (x) = Cα C . dx |x| |x|2−2α

J 1 < Cα C

(5.15)

The integral on the last line is finite if the singularity at x = 0 is integrable. In dimension d = 2, this is the case if 5 α> . 6

(5.16)

Finiteness of the integral for the region |x| 1 is obtained from the decay of h∨ . We remark that if 0 < 1 − α 1, the upper bound (5.14) may overestimate the left-hand side by as much as 1 1 1 pointwise in q, for small |q|, due to the singularity of |·|2(1−α) at zero. But the a factor 1−α integral in (5.15) is uniformly bounded in the limit α 1, implying that the argument is robust. The terms J2 , . . . , J5 can be bounded in a similar manner. For more details, we refer to [16,15]. This proves the statement of the theorem. 2 6. Uniqueness of solutions of the infinite hierarchy Collecting our results derived in the previous sections, we now prove the uniqueness of solutions of the infinite hierarchy. (k) We recall the notation ± = x k − x k and ±,xj = xj − xj . Let us fix a positive integer r. Using Duhamel’s formula we can express γ (r) in terms of the iterates γ (r+2) , γ (r+4) , . . . , γ (r+2n) as follows: tr γ

(r)

(tr , ·) =

(r) ei(tr −tr+2 )± Br+2 γ (r+2) (tr+2 ) dtr+2

0

tr tr+2 (r) (r+2) = ei(tr −tr+2 )± Br+2 ei(tr+2 −tr+4 )± Br+4 γ (r+4) (tr+4 ) dtr+2 dtr+4 0

0

= ··· tr+2n−2 tr = ... J r (t r+2n ) dtr+2 . . . dtr+2n , 0

(6.1)

0

where t r+2n = (tr , tr+2 , . . . , tr+2n ), (r)

(r+2(n−1))

J r (t r+2n ) = ei(tr −tr+2 )± Br+2 · · · ei(tr+2(n−1) −tr+2n )±

Br+2n γ (r+2n) (tr+2n ) .


983

Our main results are given in the following two theorems. Theorem 6.1. Assume that d = 1 and tr ∈ [0, T ]. The estimate

tr+2n−2 tr

(r,α) r ... J (t r+2n ) dtr+2 . . . dtr+2n

S

0

holds for

1 2

< C r (C0 T )n

(6.2)

< C r (C0 T )n

(6.3)

→0

(6.4)

L2 (Rdr ×Rdr )

0

< α 1, and for constants C, C0 independent of r and T .

Theorem 6.2. Assume that d = 2 and tr ∈ [0, T ]. The estimate

tr+2n−2 tr

(r,α) ... J r (t r+2n ) dtr+2 . . . dtr+2n

S

0

holds for

5 6

L2 (Rdr ×Rdr )

0

< α < 1, and for constants C, C0 independent of r and T .

Theorems 6.1 and 6.2 imply that for sufficiently small T ,

tr+2n−2 tr

(r,α)

r ... J (t r+2n ) dtr+2 . . . dtr+2n

S

0

L2 (Rdr ×Rdr )

0

as n → ∞. Since n is arbitrary, we conclude that γ (r) (tr , ·) = 0, given the initial condition γ (r) (0, ·) = 0. This establishes the uniqueness of γ (r) (tr , ·), and since r is arbitrary, we conclude that the solution of the infinite hierarchy is unique. First, we shall present a proof of Theorem 6.1, which is done via iterative applications of the spatial bound obtained in Theorem 4.3 followed by the use of the a priori spatial bound given in Theorem 4.2. We thank the referee and Aynur Bulut who independently observed that Theorem 6.1 can be proved without using the combinatorial argument of Section 7 (which was used in an earlier version of the manuscript). Proof of Theorem 6.1 (joint with Aynur Bulut). Let us fix α such that e

(r) i(tr −tr+2 )±

commutes with the operator

S (r,α)

and e

(r) i(tr −tr+2 )±

1 2

< α 1. Since

is unitary we have

tr+2n−2 tr

(r,α)

... J r (t r+2n ) dtr+2 . . . dtr+2n

S

0

tr

tr+2n−2

(r,α) i(t −t )(r) (r+2)

S e r r+2 ± Br+2 ei(tr+2 −tr+4 )± Br+4 · · ·

... 0

L2 (Rr ×Rr )

0

0 (r+2(n−1))

× ei(tr+2(n−1) −tr+2n )±

Br+2n γ (r+2n) (tr+2n ) L2 (Rr ×Rr ) dtr+2 . . . dtr+2n

984


tr

tr+2n−2

(r,α) (r+2) (r+2(n−1))

S Br+2 ei(tr+2 −tr+4 )± Br+4 · · · ei(tr+2(n−1) −tr+2n )±

... 0

0

× Br+2n γ (r+2n) (tr+2n ) L2 (Rr ×Rr ) dtr+2 . . . dtr+2n tr

r

tr+2n−2

(r+2,α) i(t −t )(r+2) (r+2(n−1))

S e r+2 r+4 ± Br+4 · · · ei(tr+2(n−1) −tr+2n )±

... 0

0

× Br+2n γ (r+2n) (tr+2n ) L2 (Rr+2 ×Rr+2 ) dtr+2 . . . dtr+2n

(6.5)

··· tr r(r + 2) · · · (r + 2n − 4)

tr+2n−2

(r+2n−2,α) i(t (r+2(n−1))

S e r+2(n−1) −tr+2n )±

... 0

0

× Br+2n γ (r+2n) (tr+2n ) L2 (Rr+2n−2 ×Rr+2n−2 ) dtr+2 . . . dtr+2n tr r(r + 2) · · · (r + 2n − 4)(r + 2n − 2)

tr+2n−2

C r+2n dtr+2 . . . dtr+2n

... 0

(6.6) (6.7)

0

( r − 1 + n)! trn C r+2n 2n 2 r ( 2 − 1)! n! r r+2n n 2 − 1 + n n =C 2 tr

(6.8) (6.9)

n

r

C r+2n 2n 2 2 −1+n trn C r (C0 T )n ,

(6.10)

where to obtain (6.5) we applied Theorem 4.3 and consequently continued iterative applications of this bound in order to obtain (6.6). Then to obtain (6.7) we use the fact that (r+2(n−1)) (r+2(n−1)) ei(tr+2(n−1) −tr+2n )± commutes with S (r+2n−2,α) , the unitarity of ei(tr+2(n−1) −tr+2n )± and an application of the spatial a priori bound stated in Theorem 4.2. Finally, (6.8) was obtained by integrating and using the following simple observation: r r r + 1 ··· +n−1 2 2 2 r r n r + 1 ··· +n−1 0 such that Tr J (k) ha (xj − xk+1 )ha (xj − xk+2 ) − δ(xj − xk+1 )δ(xj − xk+2 ) γ (k+2) Ca κ J (k) TrSj Sk+1 Sk+2 γ (k+2) Sk+2 Sk+1 Sj ,

(A.1)

for all non-negative γ (k+2) ∈ L1k+2 . Proof. We shall prove the lemma by modifying the proof of Lemma A.2 from [15]. As in [15], we demonstrate the argument for k = 1 (the proof is analogous for k > 1). We start by using the representation γ (3) = j λj |φj φj |, where φj ∈ L2 (R3d ), for eigenvalues λj 0. Therefore, Tr J (1) ha (x1 − x2 )ha (x1 − x3 ) − δ(x1 − x2 )δ(x1 − x3 ) γ (3)

= λj φj , J (1) ha (x1 − x2 )ha (x1 − x3 ) − δ(x1 − x2 )δ(x1 − x3 ) φj j

=

j

λj ψj , ha (x1 − x2 )ha (x1 − x3 ) − δ(x1 − x2 )δ(x1 − x3 ) φj ,

(A.2)


993

where ψj = (J (1) ⊗ 1)φj . From Parseval’s identity, we find

ψj , ha (x1 − x2 )ha (x1 − x3 ) − δ(x1 − x2 )δ(x1 − x3 ) φj j (p1 , p2 , p2 )φ j (q1 , q2 , q3 ) = dp1 dp2 dp3 dq1 dq2 dq3 ψ × dx dy h(x) h(y) eiax·(p2 −q2 ) eiay·(p3 −q3 ) − 1 × δ(p1 + p2 + p3 − q1 − q2 − q3 ).

(A.3)

For arbitrary 0 < κ < 1, we have iax·(p −q ) iay·(p −q ) 2 2 e 3 3 − 1 a κ |x||p − q | + |y||p − q | κ e 2 2 3 3 a κ |x|κ |p2 − q2 |κ + |y|κ |p3 − q3 |κ . 1

1

(A.4)

1

Here we use (a + b)κ = ((a κ ) κ + (bκ ) κ )κ ((a κ + bκ ) κ )κ = a κ + bκ for a, b 0, which holds whenever κ1 > 1. Thus, taking absolute values in (A.3), and recalling that h = 1, we find

ψj , ha (x1 − x2 )ha (x1 − x3 ) − δ(x1 − x2 )δ(x1 − x3 ) φj κ κ a dx h(x)|x| dp1 dp2 dp3 dq1 dq2 dq3 |p2 − q2 |κ + |p3 − q3 |κ j (p1 , p2 , p2 )φ j (q1 , q2 , q3 )δ(p1 + p2 + p2 − q1 − q2 − q3 ). × ψ

(A.5)

Clearly, |pj − qj |κ |pj |κ + |qj |κ , for j = 2, 3, and it suffices to show how to control one of the four terms thereby obtained, for instance the one containing the factor |p2 |κ . By applying Cauchy–Schwarz, we obtain dp1 dp2 dp3 dq1 dq2 dq3 δ(p1 + p2 + p3 − q1 − q2 − q3 ) j (p1 , p2 , p2 )φ j (q1 , q2 , q3 )|p2 |κ × ψ = dp1 dp2 dp3 dq1 dq2 dq3 δ(p1 + p2 + p3 − q1 − q2 − q3 ) ×

q1 q2 q3

p1 p2 p3 φ j (q1 , q2 , q2 ) j (p1 , p2 , p2 ) ψ

q1 q2 q3

p1 p2 1−κ p3 dp1 dp2 dp3 dq1 dq2 dq3 δ(p1 + p2 + p3 − q1 − q2 − q3 )

p1 2 p2 2 p3 2 j (p1 , p2 , p2 )2 ψ 2 2 2

q1 q2 q3 + −1 dp1 dp2 dp3 dq1 dq2 dq3 δ(p1 + p2 + p3 − q1 − q2 − q3 ) ×

×

q1 2 q2 2 q3 2 j (q1 , q2 , q2 )2 φ 2 2(1−κ) 2

p1 p2

p3

994


ψj , S12 S22 S32 ψj

sup

dq1 dq2

P

+ −1 φj , S12 S22 S32 φj sup

1

q1

2 q

dp1 dp3

Q

2

2 P

(A.6)

− q1 − q2 2

1 ,

p1 2 Q − p1 − p3 2(1−κ) p3 2

(A.7)

for arbitrary > 0. Now we apply (4.12) to bound the integral appearing in (A.6) and (A.7) for all κ 1 when d = 1 and for all κ < 1 if d = 2. Hence (A.2), (A.5), (A.6) and (A.7) imply Tr J (1) ha (x1 − x2 )ha (x1 − x3 ) − δ(x1 − x2 )δ(x1 − x3 ) γ (3) Ca κ Tr J (1) S12 S22 S32 J (1) γ (3) + −1 Tr S12 S22 S32 γ (3) Ca κ Tr S1−1 J (1) S12 J (1) S1−1 S1 S2 S3 γ (3) S3 S2 S1 + −1 Tr S12 S22 S32 γ (3)

Ca κ S1−1 J (1) S1

S1−1 J (1) S1 + −1 Tr S12 S22 S32 γ (3) Ca κ J (1) Tr S12 S22 S32 γ (3) , where to obtain (A.8) we choose = |||J (k) |||−1 . Hence, the lemma is proved.

(A.8)

2

Appendix B. Regularization of the initial data In this section we give a statement and a sketch of the proof of the result on regularization of the initial data, which was used in the proof of Theorem 3.2. Proposition B.1. Suppose that ψN ∈ L2 (RdN ) with ψN = 1 is a family of N -particle wave functions with the associated marginal densities denoted by γN(k) , for k = 1, . . . , N . N is defined according to (2.38) i.e. For κ > 0, suppose that ψ N := ψ

χ( Nκ HN )ψN ,

χ( Nκ HN )ψN

(B.1) (k)

where χ is a bump function supported on [0, 1]. Also suppose that γN , for k = 1, . . . , N denote N . the marginal densities associated with ψ If

ψN , HN ψN CN,

(B.2)

and (1)

γN → |φ φ|

as N → ∞,

(B.3)

for a φ ∈ H 1 (Rd ), then for κ > 0 small enough and for every fixed k 1 we have (k) γN − |φ φ|⊗k = 0. lim Tr

N →∞

(B.4)


995

In the context of the two-body potentials, the above proposition was proved by Erdös, Schlein and Yau in [8] (see Proposition 9.1) and under a supplementary assumption in the earlier work [9] (see Proposition 8.1). We shall give a sketch of the proof of Proposition B.1 by following [9] and [8]. Proof. Since the proof of this proposition has only minor modifications compared to (iii) of Proposition 8.1 [9] and Proposition 9.1 [8], we shall give just a sketch of the proof here. As in [9] and [8], we note that it suffices3 to prove (1) γN − |φ φ| = 0. lim Tr

(B.5)

N →∞

However, the limiting density is a rank one (orthogonal) projection, so the trace-norm convergence is equivalent to weak-* convergence. Hence it is enough to prove that for every compact operator J (1) ∈ K(1) and for every > 0, there exists N0 such that (1) Tr J (1) γ − |φ φ| , N

for N > N0 .

(B.6)

Following [9] and [8], we note that (B.6) can be proved as follows: (N −1)

Step 1 First, we note that there exists a sequence ξN ing

(N −1)

∈ L2 (Rd(N −1) ), ξN

ψN − φ ⊗ ξ (N −1) → 0, N

as N → 0.

= 1 satisfy-

(B.7)

This was assumed in Proposition 8.1 in [9] and was consequently proved in Proposition 9.1 in [8] following the proof by Michelangeli [20]. The proof in our case is identical to the proof presented in [8]. Step 2 Let us choose φ∗ ∈ H 2 (Rd ) with φ = 1 and such that

φ − φ∗

. 32|||J (1) |||

(B.8)

Step 3 As in [9] and [8], let Ξ denote χ( Nκ HN ). In the same way as in [9] and [8], by combining (B.7) and (B.8), one can find κ > 0 small enough such that

(N −1)

Ξ φN Ξ (φ∗ ⊗ ξN )

− .

Ξ φ (N −1) (1) ||| 6|||J N

Ξ (φ∗ ⊗ ξN )

(B.9)

Step 4 Inspired by the Hamiltonian introduced in [9] and [8], we introduce a slightly different Hamiltonian, to take into the account the three-body interactions studied in this paper. More precisely, we introduce the Hamiltonian: H˘ N :=

N 1 (−xj ) + 2 N j =2

VN (xi − xj , xi − xk ).

10

δ0 (e, vi )bi = 0

i: δ0 (e,vi )0

δ0 (e, vi )bi .

(3.1)

i: δ0 (e,vi ) 0 such that δ B (x, y), δ ∩ Gk = (x + v, y + v): v ∈ B 0, √ 2 for all δ δk . Proposition 5.4. Lebesgue measure is Gk -invariant. Proof. Let g ∈ Cc (Gk ). Notice that we can cover the support of g by a finite number of open sets as in Lemma 5.3. By a partition of unity argument, see [20], we can write g as a finite sum of functions in Cc (Gk ), where each summand has support contained in an open set as in Lemma 5.3. So it is enough to show that for any U as in Lemma 5.3, if f ∈ Cc (U) then f (x, z) dμ(x) = f (z, x) dμ(x). Rd

z∈[x]

Rd

z∈[x]

So suppose U = {(x0 + v, y0 + v): v ∈ B(0, )} and f ∈ Cc (U). Observe that we only need to show that f (x, z) dμ(x) = f (z, x) dμ(x), (5.1) B(x0 , ) z∈[x]

B(y0 , ) z∈[x]

since if x ∈ / B(x0 , ) then r −1 {x} ∩ U = ∅ and if x ∈ / B(y0 , ) then s −1 {x} ∩ U = ∅ (recall that r and s are the range and source maps). Now if x ∈ B(x0 , ), say x = x0 + v for some v ∈ Rd , then there exists one and only one point (x, y) in r −1 {x} ∩ U. Notice that y = y0 − x0 + x, so the left-hand side in (5.1) is equal to d B(x0 , ) f (x, y0 − x0 + x) dμ(x). Analogously if y ∈ B(y0 , ), say y = y0 + u for some u ∈ R , then there exists one and only one point (t, y) in s −1 {y} ∩ U and t = x0 − y0 + y, so the right-hand side in (5.1) is equal to B(y0 , ) f (x0 − y0 + y, y) dμ(y). Finally since μ(B(x0 , )) = μ(B(y0 , )) we have that f (x, y0 − x0 + x) dμ(x) = f (x0 − y0 + y, y) dμ(y) B(x0 , )

as desired.

B(y0 , )

2

The Gk -invariance of Lebesgue measure, denoted by μ, allows us to define a trace functional in Cc (Gk ) via: Definition 5.5. τ : Cc (Gk ) → C, f → f (x, x) dμ(x). Rd

1010

D. Gonçalves / Journal of Functional Analysis 260 (2011) 998–1019

Observation 5.6. τ cannot be extended to a bounded trace on C∗r (Gk ). Observation 5.7. It follows from the fact that Lebesgue measure is Gk -invariant that τ (f ∗ f ) = τ (ff ∗ ), for all f ∈ Cc (Gk ), and hence τ has the tracial property.

Observation 5.8. Lebesgue measure is defined in Cc (∪Gk ).

Gk -invariant and hence the trace functional is well

Later, in the examples, we will need a more computable formula for the trace. We do this for d = 1, 2 in the next two lemmas. Lemma 5.9. For x ∈ Rd , d = 1, 2, and f ∈ C(Gk ), let πx be the representation on l 2 ([x]) as in the definition of the reduced norm. Then τ (f ) = Tr πx (f ) dx. [p]∈Px∈p

If f is a projection then

τ (f ) =

Tr πx (f ) μ(p),

[p]∈P

where for each [p] ∈ P, x is a point in the interior of p and μ is the Lebesgue measure. Proof. Recall that if {δy }y∈[x] is the orthonormal basis of l 2 ([x]) given by δy (ξ ) = 1 if ξ = y and δy (ξ ) = 0 if ξ = y then " ! Tr πx (f ) = πx (f )δy , δy = f (y, y). y∈[x]

y∈[x]

d Also we already know that for each [p] ∈ P we can write [p] = {p + wi }∞ i=1 , where wi ∈ R . p ∞ Notice that if x is a point in the interior of p then [x] = {x + wi }i=1 . Denote the interior of p by int(p). We now have that f (x, x) dμ(x) = f (x, x) dx = f (x, x) dx p

[p]∈P t∈[p] t

t∈{tiles} t

Rd

=

∞ [p]∈P i=1

=

f (x, x) dx

p p+wi

∞

p p f x + wi , x + wi dx

[p]∈P i=1 p

=

∞ p p f x + wi , x + wi dx

[p]∈P int(p) i=1

p


=

1011

f (y, y) dx

[p]∈P int(p) y∈[x]

=

Tr πx (f ) dx = Tr πx (f ) dx

[p]∈P int(p)

[p]∈P p

and the first part of the proposition is proved. Now suppose f is a projection. Then Tr(πx (f )) ∈ Z and for any tile t the map x → Tr(πx (f )) is a continuous function from the interior of t into the integers, and hence a constant function. This implies the second part of the proposition. 2 At first sight, for two-dimensional examples, the formula for the trace above does not appear to be helpful, since we usually do not know the values of a projection on the interior of a tile. But we usually know the values at the vertices and we may use this to get an even better description of the trace: Lemma 5.10. For each [p] ∈ P we choose one and only one vertex on the tile p, which we p p d call vp . Using the language of Lemma 5.9 we have that [p] = {p + wi }∞ i=1 , where wi ∈ R , d = 1, 2. Let p p Vvp = [v] ∈ V: ∃ wi such that vp + wi ∈ int T(v) . Then for any projection f ∈ Cc (Gk ) we have that τ (f ) =

Tr πv (f ) μ(p).

[p]∈P [v]∈Vvp

Proof. Observe that for x ∈ int(p) we have Tr(πx (f )) = vp we have ∞ p p f v p + w i , v p + wi =

p p i=1 f (x + wi , x + wi )

f (z, z) =

[v]∈Vvp z∈[v]

i=1

#∞

and for each

Tr πv (f ) .

[v]∈Vvp

Let [p] ∈ P. Notice that x → Tr(πx (f )) is not necessarily continuous on the border of p. Actually we have that lim Tr πx (f ) = Tr πv (f ) .

x→vp x∈int(p)

[v]∈Vvp

Now since x # → Tr(πx (f )) is constant in the interior of p we have that for any x ∈ int(p), Tr(πx (f )) = [v]∈Vvp Tr(πv (f )) and the lemma follows. 2 We finish the paper with K-theory and trace computations for a number of examples.

1012


6. The Fibonacci tiling The Fibonacci tiling is associated with the substitution matrix stant γ , the golden mean. We can tile the line as shown below

1 1 10

and has inflation con-

Remember that in the one-dimensional case C∗r (G) = C∗r (G|X1 ) and we need to give an orientation to all edges in the tiling. We do so by giving all edges the same orientation, to the right. v2 = 1 • 0 and We can see that there are three equivalence classes of vertices, namely v1 = 0 • 1, So δ0 is v3 = 0 • 0. Also there are two equivalence classes of edges, namely e1 = 0 and e2 = 1. the matrix given by [δ0 ] =

−1 1 0 , 1 −1 0

as described in Proposition 2.1. The vectors (0, 0, 1) and (1, 1, 0) generate the kernel of δ0 and (1, −1) generates the image of δ0 . So we conclude that K0 C∗r (G0 ) ∼ = ker(δ0 ) ∼ = Z2

Z⊕Z ∼ and K1 C∗r (G0 ) ∼ = = Z. Im(δ0 )

• K0 (C∗r ( Gk )). We now proceed to compute K0 (C∗r ( Gk )). From what is done above and Proposition 4.2 we know that K0 (C∗r (Gk )) is isomorphic to Z2 for all k ∈ Z+ and the connecting maps from Z2 to Z2 are all the same. In order to compute the connecting map we need to specify the isomorphism between ker(δ0 ) and Z2 . As expected we will use the isomorphism that maps (0, 0, 1) to (0, 1) and (1, 1, 0) to (1, 0). Since the connecting map is a group homomorphism, it is enough to find where the basis elements (0, 1) and (1, 0) are mapped. We start with (0, 1). By our choice of isomorphism, (0, 1) is mapped to the vector (0, 0, 1) in the kernel of δ0 , which by its turn is mapped to [f ]0 ∈ K0 (C∗r (G0 |X0 )), where the function f ∈ Cc (G0 |X0 ) is defined to be 1 at one representative of the vertex v3 and 0 otherwise. Letting (a + 1, a + 1), for some a ∈ R, be the representative of v3 , we can write f as f (x, y) =

$

1 if (x, y) = (a + 1, a + 1), 0 otherwise.

Unfortunately we cannot include this function in K0 (C∗r (G1 |X0 )) directly as we do not have this map. But we know that [f ]0 lifts to an element in K0 (C∗r (G0 )). Moreover [f ]0 lifts to [f˜]0 where f˜ ∈ Cc (G0 ) is the projection defined below. ⎧ t ⎪ ⎪ ⎪ ⎪ ⎨ 1√− t ˜ f (x, y) = t − t2 √ ⎪ ⎪ 2 ⎪ ⎪ ⎩ t −t 0

if (x, y) = (a + t, a + t) for 0 t 1, if (x, y) = (a + 1 + t, a + 1 + t) for 0 t 1, if (x, y) = (a + t, a + 1 + t) for 0 < t < 1, if (x, y) = (a + 1 + t, a + t) for 0 < t < 1, otherwise.


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So we have that (0, 1) is “lifted” to [f˜]0 ∈ K0 (C∗r (G0 )), which we can include in K0 (C∗r (G1 )) via K0 (ι), where ι is the inclusion map. Next we follow the isomorphism from K0 (C∗r (G1 )) to Z2 to find out that K0 (ι)([f˜]0 ) is mapped to (1, 0). Proceeding analogously as above we get that (1, 0) is mapped to the vector (1, 1) and the connecting map in Z2 is given by the matrix c = 11 10 , which is an isomorphism. We conclude that K0 (C∗r (∪Gk )) ∼ = Z2 . Before proceeding to K1 we compute the trace of K0 (C∗r (∪Gk )) using the formula of Lemma 5.9. Proposition 6.1. K0 (τ )(K0 (C∗r (∪Gk ))) = Z[ γ1 ] ⊕ γ Z where K0 (τ ) is the map in K-theory induced by τ , and γ is the golden mean. ∼ (1, 1, 0) and Remember that for any k ∈ N, K0 (C∗r (Gk )) = [f1k ]0 , [f2k ]0 , where [f1k ]0 = k 0 0 ∼ [f2 ]0 = (0, 0, 1). We have explicitly defined f1 and f2 and the definitions of f1k , f2k are analogous. Once we have this we can just use the trace formula to compute τ (f1k ) and τ (f2k ) (even the definition works). Just observe that edges in Gk are scaled down by γ −1 . For example if (a + γ1k , a + γ1k ) is a representative of v3 in Gk then ⎧ t ⎪ ⎪ ⎨ f2k (x, y) = 1 − t ⎪ ⎪ ⎩0 ∗

if (x, y) = (a +

t , a + γtk ), 0 t 1, γk 1+t , a + 1+t ), 0 t 1, γk γk

if (x, y) = (a + if x = y not as above, if x = y.

Notice that the off diagonal values of f2k are irrelevant for the trace. With this description we get K0 (τ )([f2k ]0 ) = τ (f2k ) = γ1k . 1 1 Analogously we get that K0 (τ )([f1k ]0 ) = γ1k + γ k+1 = γ k−1 . Observe that K0 (τ )([f10 ]0 ) = γ . Finally for all k ∈ N, let K0 (τk ) = K0 (τ ). Then K0 (τk ) = K0 (τk+1 ) ◦ K0 (ι) and hence the ∗ collection of maps {K0 (τk )}∞ i=0 gives a well-defined map τu , from the direct limit K0 (Cr (∪Gk )) 1 onto Z[ γ ] ⊕ γ Z as desired.

Observation 6.2. As we did for the Fibonacci tiling, we can compute the trace for the Thue– Morse example and we get that K0 (τ )(K0 (C∗r (∪Gk ))) = Z[ 12 ] where K0 (τ ) is the map in Ktheory induced by τ . • K1 (C∗r ( Gk )). We proceed in the same fashion as we did for K0 (C∗r ( Gk )). We know that K1 (C∗r (G0 )) ∼ = ∼ = Z and we only need to compute the connecting map. For that it is enough to find where Z⊕Z we have that 1 is mapped to (0, 1) 1 is mapped by c. Following the isomorphism from Z to Im(δ 0) ∼ which in turn is mapped to (0, 1) in E Z = Z ⊕ Z. (We could also choose to take 1 to (1, 0).) say (a + λ, a + λ) for some a ∈ R and 0 λ 1 . We now choose a representative of the edge 1, γ Proceeding analogously to the second part of the proof of Proposition 2.1 we have that (1, 0) is mapped to [g]1 ∈ K1 (C∗r (G0 |X1 −X0 )), where the function g in C∗r (G 0 |X1 −X0 ) is defined as follows Z⊕Z Im(δ0 )

1014


⎧ ⎨ exp(2πiλ) g(x, y) = 1 ⎩ 0

if (x, y) = (a + λ γ1 , a + λ γ1 ) for λ ∈ [0, 1], if x = y, otherwise

and we can include [g]1 in K1 (C∗r (G0 )) and then in K1 (C∗r (G1 )) via K1 (ι). Our final step is to follow back the isomorphism from K1 (C∗r (G1 )) to Z. Observe that our edge representative is still an edge in G1 . Actually (a + λ, a + λ), with 0 λ γ1 , is a representative of the edge 0 in G1 , and we have that g winds once in this edge (g here means the inclusion of g in C∗r (G1 )). So proceeding again analogously to the second part of the proof of Proposition 2.1, we have that Z⊕Z [g]1 is mapped to (0, 1) in Im(δ , which is mapped to 1 in Z. 0) So the connecting map c in K1 is just the identity map and hence K1 C∗r (∪Gk ) ∼ = Z. 7. The octagonal tiling We will use the triangular version of the octagonal tiling, as described in [16]. We can obtain the tiling by means of the substitutions below:

So the tiles in the octagonal tiling are triangles and rhombi. The prototiles consist of the two tiles on the left-hand side above and all its rotates around nπ 4 and reflections along the boundaries of the tiles. The substitution is extended to all prototiles by symmetry. Thus the octagonal tiling has twenty prototiles; four of them congruent to the rhombus and the remaining sixteen congruent to the triangle. We refer the reader to [16] to check that the substitution is primitive and recognizable. Local finite complexity follows readily. The detailed computation of the K-theory groups can be found in [9]. Our aim here is to give a general idea. We assume all tiles are oriented counterclockwise and edges are oriented as in [9]. To obtain all vertex and edge patterns, we apply the substitution enough times to the rhombus prototile (p), so that the vertex and edge patterns in w n (p) are the same as in w n+1 (p) (in this case n = 5). To compute the K-theory groups of C∗r (G0 ), we will need both exact sequences (2.1) and (2.3). We start by restricting our equivalence relation to the edges, GX1 . Using Proposition 2.1 we get a 56 × 49 matrix for δ0 , which has kernel generated by 17 vectors. So K0 (C∗r (G0 |X1 ) is isomorphic to Z17 . For K1 we use the software GAP, see [8], to compute the Smith Normal Form of δ0 , (see [19] for Smith Normal Form), which is the diagonal matrix δ = diag(1, . . . , 1, 0, . . . , 0)


with 32 1’s and 24 0’s. This allow us to conclude that ∼ Z56 ∼ K1 (C∗ (G0 |X ) = = Z24 . r

1

Z56 Im δ

1015

is isomorphic to Z24 and hence

Im δ0

We have computed the K-groups for C∗r (G|X1 ) and can now use the six-term exact sequence (2.3) to compute the K-groups of C∗r (G0 ). Corollary 2.6 implies that K0 (C∗r (G0 )) is isomorphic to ker δ0 ⊕ Z ∼ = Z18 . In order to compute K1 , we first need to compute the index map and this is done using Proposition 2.4. We have five vectors as a basis for the kernel of δ1 and hence K1 (C∗r (G0 )) ∼ = Z5 . 7.1. K-theory of the inductive limit C∗ -algebra As sketched above we have that K0 C∗r (G0 ) ∼ = Z18

and K1 C∗r (G0 ) ∼ = Z5 .

To compute the K-theory of C∗r (∪Gk ), we proceed similarly to what was done for the onedimensional tilings. But in the two-dimensional case it is not so easy to follow all isomorphisms. For example, to compute K0 , we would need to extend a projection in the vertices of G0 to a projection on the whole tiling and then restrict this projection to the vertices of G1 . The problem is to find this extension. So instead we find an equivalent projection in K0 , but such that we know its values at the vertices of G1 . The same idea can be used for K1 , only with functions winding on edges instead of projections. Diagram (7.1) illustrates the argument above. The choice of the equivalent projection above will depend mostly on the map F defined below. Roughly F is a map that collapses the new vertices and edges obtained by applying the inflation map to a tile, into the “old” vertices and edges of this tile. We start by defining F on a rectangle prototile and a triangle prototile. We then extend F to R2 by symmetry (using the fact that the octagonal tiling covers the plane). To make it easier to define F , we name the vertices of this prototiles and their inflation as show in the pictures below:

1016


Now we define F as a map that takes the rhombus vertices A, E, H, I to the vertex A, the vertices J, F, G, C to the vertex C and leaves the vertices B and D fixed. Moreover we require F to be a homeomorphism from the interior of the rhombus defined by BIDJ into the interior of the rhombus ABCD. Also F should take the triangle vertices A, D, G to the vertex A, the vertices F, C to C and the vertices H, E, B to the vertex B. Moreover F should be a homeomorphism from the interior of the triangle GHF into the interior of the triangle ABC. Finally we require F to be a continuous map. Since F agrees on the borders of the matching tiles of the octagonal tiling we can extend it to R2 by symmetry. Observe that vertices in ω(T) are mapped to vertices in T under F and edges in ω(T) are mapped to either vertices or edges in T. Furthermore notice that the deformation of the tiles induced by F can be done in a continuous way, so that there exists a path of continuous maps Ft such that F0 is the identity and F1 = F . Now for f ∈ Cc (G0 ) we define αt : Cc (G0 ) → Cc (G0 ) by $ f (Ft (x), Ft (y)) if (Ft (x), Ft (y)) ∈ G0 , αt (f )(x, y) = 0 otherwise and this implies that α(f ) := α1 (f ) is homotopic to f . The proof that each αt is well defined may be found in [9]. We are now able to compute some inductive limits. • K0 (Cr∗ (∪Gk )). From what was done previously we have that K0 (Cr∗ (Gk )) ∼ = Z18 . ∗ We need to make a choice of isomorphism. Since K0 (Cr (G0 )) ∼ = ker δ0 ⊕ Z we will say that, for i = 1, . . . , 17, the canonical basis vectors ei in Z18 is isomorphic to the column vector ci in the kernel of δ0 and e18 is isomorphic to an element [g]0 ∈ K0 (IX1 ), included in K0 (C∗r (G0 )), that is not on ker(K0 (i)) = Im δ1 . We proceed describing the isomorphism chase for the vectors ci , i = 1, . . . , 17. Notice that ci is isomorphic to [f ]0 , where f ∈ ker δ0 . We lift [f ]0 to K0 (Cr∗ (G0 )) and call this lift [f˜]0 . From the definition of the F map we have that [f˜]0 = [α(f˜)]0 . We then include [α(f˜)]0 in K0 (Cr∗ (G1 )) and restrict it to the vertices of G1 , to arrive at ker δ0 . Finally, we follow the isomorphism from ker δ0 to Z17 . The diagram below illustrates the isomorphism chase: Z17

ker δ0 ⊆ K0 Cr∗ (G0 |X0 )

K0 Cr∗ (G0 )

K0 Cr∗ (G0 )

ei

ci → [f ]0

[f˜]0

[α(f˜)]0 (7.1)

Z17

α(f˜)|X0 0

α(f˜) 0

ker δ0 ⊆ K0 Cr∗ (G1 |X0 )

K0 Cr∗ (G1 ) .

So proceeding as described above we can find where e1 to e17 are mapped under the connecting map. We still need to find where e18 is mapped. If we take [g ]0 to be the Bott element in the


1017

tile p17 and 1 everywhere else then [g ]0 do not belong to the ker(K0 (i)) = Im δ1 and hence e18 is isomorphic to [g ]0 . But this is not a great choice of representative. Instead we take [g]0 to be equal to the Bott element on one of the copies of p17 obtained by applying the inflation map to p17 and 1 everywhere else. One can see that g and g are homotopic and hence e18 is isomorphic to [g]0 . Now once we include [g]0 in K0 (C∗r (G1 )) we have that [g]0 is a Bott element in exactly one tile of G1 (the copy of p17 ) and 1 otherwise and hence [g]0 in K0 (Cr∗ (G1 )) is isomorphic to e18 in Z18 . With all this in mind we get the following connecting map in K0 : ⎡

1 0 −1 1 1 ⎢ 3 0 −3 2 2 ⎢ ⎢ 2 −1 −2 1 2 ⎢ ⎢ 1 −1 −2 1 2 ⎢ ⎢ 2 0 −2 3 2 ⎢ ⎢ 3 0 −3 3 2 ⎢ ⎢ 1 0 −1 2 1 ⎢ ⎢ 3 −1 −3 2 2 ⎢ ⎢ 4 0 −4 3 3 C=⎢ 0 1 0 ⎢0 0 ⎢ 0 0 1 ⎢0 0 ⎢ 0 0 0 ⎢0 1 ⎢ 0 0 0 ⎢0 0 ⎢ ⎢ 1 1 −1 1 1 ⎢ 1 0 0 ⎢0 0 ⎢ 0 0 0 ⎢0 0 ⎣ 1 0 0 0 0 0 0 0 0 0

−1 −3 −2 −1 −3 −3 −2 −3 −4 0 0 0 1 −1 0 0 0 0

0 0 1 1 −1 −1 0 1 −1 0 0 0 0 −1 0 1 0 0

2 2 2 2 2 2 2 2 3 0 0 0 0 0 0 0 0 0

2 2 2 2 2 2 2 3 2 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎦ 0 1

and hence K0 (Cr∗ (∪Gk )) is isomorphic to the direct limit in Z18 induced by C. Unfortunately we can not describe this limit any further. We do have a better description for K1 though: • K1 (Cr∗ (∪Gk )). ∼ Z5 . To find the connecting map From what was done previously we have that K1 (Cr∗ (Gk )) = c we use the fact proved above that α(f ) is homotopic to f , for any f ∈ Cc (G0 ), and then read the values of α(f ) on the edges of G1 from the values of f on the edges in G0 . Proceeding like this we get that the connecting map in K1 is given by the matrix ⎡

⎤ −1 0 0 0 0 1 −2 0 2 ⎥ ⎢ 0 ⎢ ⎥ c = ⎢ 2 −1 0 1 1 ⎥ , ⎣ ⎦ 3 −1 0 1 2 2 0 1 1 2 which is an isomorphism and hence K1 (Cr∗ (∪Gk )) is isomorphic to Z5 .

1018


8. The table tiling The table tiling is given by the substitution below (with inflation constant λ = 2):

This tiling has the interesting property that tiles do not necessarily match edge to edge. This will induce two natural choices for a cell complex on the tiling. One being just the vertices and edges on the tiling and the other being the previous one with added vertices on the middle of some edges so that the tiles meet edge to edge. Notice that the choice of cell complex do not change the equivalence relation in R2 × R2 and so do not change the C∗ -algebra of the tiling at all. This implies we can use either of the two choices to compute the K-theory groups. Actually it has proven easier to use the cell complex with added vertices to compute K0 and the cell complex induced by the tiling to compute K1 . The K-theory computations for the table tiling are very much alike what was done for the octagonal tiling. The only problem is that the F map we used do not map edges in G1 into edges of G0 . For K0 this does not have any further implications but for K1 this means we have to use other method. So we actually extended the unitaries in the edges to the whole tiling. This was done to some very special generators of K1 and in some cases, by choosing linear combinations of the generators. Full details can be found in [9]. The K-groups are: K0 C∗r (G0 ) ∼ = Z10

and K1 C∗r (G0 ) ∼ = Z2 ⊕ Z2 , + , + , 1 1 ⊕Z ⊕ Z2 , K1 Cr∗ (∪Gk ) ∼ =Z 2 2

and we can write the following exact sequence for K0 of the inductive limit C∗ -algebra: + ,4 + , ∗ 1 1 2 0 → Z → K0 Cr (∪Gk ) → Z ⊕ Z → 0. ⊕Z 2 4 2

Finally we notice that using Lemma 5.9 we get that Proposition 8.1. K0 (τ )(K0 (C∗r (∪Gk ))) maps onto Z[ 12 ], where K0 (τ ) is the map in K-theory induced by τ .


1019

Observation 8.2. The detailed proofs of all statements above about the table tiling require a few pages of pictures (labeling all vertices and edge patterns of the tiling) and big matrices (obtained via software GAP) which we omit here, since they can be found in [9]. Acknowledgments This is the second part of my PhD thesis. I thank Dr. Ian Putnam for his superb orientation. References [1] J.E. Anderson, I.F. Putnam, Topological invariants for substitution tilings and their associated C∗ -algebras, Ergodic Theory Dynam. Systems 18 (1998) 509–537. [2] J. Bellissard, K-Theory of C∗ -algebras in solid state physics, in: T.C. Dorlas, M.N. Hugenholtz, M. Winnink (Eds.), Statistical Mechanics and Field Theory, Mathematical Aspects, in: Lecture Notes in Phys., vol. 257, 1986, pp. 99– 156. [3] J. Bellissard, Gap labelling theorems for Schrödinger’s operators, in: J.M. Luck, P. Moussa, M. Waldschmidt (Eds.), From Number Theory to Physics, Les Houches, March 1989, Springer, 1993, pp. 538–630. [4] Blackadar, K-Theory for Operator Algebras, Math. Sci. Res. Inst. Publ., second ed., Cambridge University Press, 1998. [5] N.P. Brown, Invariant measures and finite representation theory of C∗ -algebras, Mem. Amer. Math. Soc. 184 (865) (2006). [6] A.H. Forrest, K-groups associated with substitution minimal systems, Israel J. Math. 98 (1997) 101–139. [7] A.H. Forrest, J.R. Hunton, J. Kellendonk, Topological invariants for projection method patterns, Mem. Amer. Math. Soc. 159 (758) (2002). [8] GAP, Groups algebras and permutations, http://www-groups.dcs.st-and.ac.uk/~gap. [9] D. Gonçalves, C∗ -algebras from substitution tilings: A new approach. PhD Thesis, Univ. of Victoria, Victoria, 2005. [10] D. Gonçalves, New C∗ -algebras from substitution tilings, J. Operator Theory 57 (2) (2007) 391–407. [11] J. Kaminker, I.F. Putnam, K-theoretic duality for subshifts of finite type, Comm. Math. Phys. 187 (1997) 509–522. [12] J. Kaminker, I.F. Putnam, J. Spielberg, Operator algebras and hyperbolic dynamics, in: S. Doplicher, R. Longo, J.E. Roberts, L. Zsido (Eds.), Operator Algebras and Quantum Field Theory, International Press, 1997. [13] J. Kaminker, I.F. Putnam, M. Whittaker, K-Theoretic duality for hyperbolic dynamical systems, Eprint arXiv: 1009.4999v1. [14] J. Kellendonk, Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys. 7 (1995) 1133–1180. [15] J. Kellendonk, The local structure of tilings and their integer group of coinvariants, Comm. Math. Phys. 187 (1997) 115–157. [16] J. Kellendonk, I.F. Putnam, Tilings, C∗ -algebras, and K-Theory, CRM Monogr. Ser., vol. 13, Centre de Recherches Mathematiques, 2000. [17] I.F. Putnam, A homology theory for Smale spaces, Eprint arXiv:0801.3294 [math.DS]. [18] M. Rordan, F. Larsen, N.J. Laustsen, An Introduction to K-theory for C∗ -algebras, London Math. Soc. Stud. Texts, vol. 49, Cambridge University Press, 2000. [19] Joseph J. Rotman, Advanced Modern Algebra, Prentice Hall, 2002. [20] W. Rudin, Real and Complex Analysis, third ed., McGraw–Hill, 1986. [21] L.A. Sadun, Tilings, tilings spaces and topology, Philos. Mag. 86 (2006) 875–881. [22] J. Savinien, J. Bellissard, A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam. Systems 29 (2009) 997–1031. [23] N.E. Wegge-Olsen, K-Theory and C∗ -Algebras, Oxford Sci. Publ., Oxford University Press, 1993.


Solution to a conjecture on the norm of the Hardy operator minus the identity Santiago Boza a,∗ , Javier Soria b,1 a Department of Applied Mathematics IV, EPSEVG, Polytechnical University of Catalonia, E-08880 Vilanova i Geltrú,

Spain b Department of Applied Mathematics and Analysis, University of Barcelona, E-08007 Barcelona, Spain

Received 9 September 2009; accepted 15 November 2010 Available online 7 December 2010 Communicated by G. Godefroy

Abstract We prove that for a decreasing weight w, the following inequality is sharp: ∞ ∞ p ∗∗ ∗ p ∗ f (t) − f (t) w(t) dt wBp f (t) w(t) dt, 0

0

where Bp is the Ariño and Muckenhoupt class of weights, and p 2. The case w ≡ 1 gives a positive answer to a conjecture formulated in Kruglyak and Setterqvist (2008) [8], where this estimate is proved only when p 2 is an integer. Simple examples show that, for 1 < p < 2, or if w is not decreasing, the result is false. Finally, using a different argument, we also prove that in the case p = 1, and for arbitrary weights w ∈ B1 , wB1 is the best constant in the corresponding inequality. © 2010 Elsevier Inc. All rights reserved. Keywords: Hardy operator; Bp weights; Lorentz spaces; Best constants


E-mail addresses: [email protected] (S. Boza), [email protected] (J. Soria). 1 Both authors have been partially supported by Grant MTM2007-60500.


S. Boza, J. Soria / Journal of Functional Analysis 260 (2011) 1020–1028

1021

1. Introduction Let C p be the cone of nonnegative decreasing functions on Lp = Lp (0, ∞) and let S be the Hardy operator 1 Sf (x) = x

x f (t) dt,

x > 0.

(1)

0

It was proved in [8] that, if S − I C p =

sup

(S − I )f

f ∈C p , f p 1

Lp

,

then, S − I C p =

1 , (p − 1)1/p

p ∈ {2, 3, 4, . . .},

(2)

and it was also conjectured that the same sharp estimate would hold true for all p 2. We will show in Theorem 2.2 that this conjecture is true and, moreover, the result can be extended to weights in the Bp class of Ariño and Muckenhoupt [1], satisfying some monotonicity property (see (4)). The main technique used in [8] to prove (2) is based on the fact that, for a simple function p p f in C p and p a whole number, the expressions Sf − f C p and f C p are homogeneous polynomials of degree p. Instead, in our proof, the result is obtained by making use of some cancellation properties, after collecting terms in an appropriate way (see Lemma 2.1), together with the monotonicity property assumed on the weight. We observe that, if f is in the cone of nonincreasing functions we have, for almost every x > 0, (S − I )f (x) = f ∗∗ (x) − f ∗ (x), where f ∗ is the classical nonincreasing rearrangement of f with respect to the Lebesgue measure and f ∗∗ (t) = Sf ∗ (t) is its maximal function [3]. Estimates concerning functional spaces involving the expression f ∗∗ − f ∗ have been obtained in the last years (see [2,4,5]). More recently, in [10], the same kind of results for the norm of S − I have been studied, but for spaces of restricted type. For a measurable function f and w a weight (i.e., a nonnegative locally integrable function on (0, ∞)), the weighted space Lp (w) is defined as

∞ 1/p

p f (t) w(t) dt Lp (w) = f : f p,w = 0, p 2 and the difference (x + c)α − x α defines an increasing function if α 1 and c > 0. Then (3) will be proved if we check that the following function, defined on 0 < x 1 g(x) =

(γ1 x +

n

p i=2 γi )

− (γ1 x +

n−1 i=2

p γi )p − ( ni=2 γi )p + ( n−1 i=2 γi )

x

is also increasing and attains its maximum at x = 1. Defining A = n−1 i=2 γi > 0 and B = γn > 0, we can rewrite the function g as g(x) =

(γ1 x + A + B)p − (γ1 x + A)p − (A + B)p + Ap . x

,


1023

We compute its derivative

g (x) = x −2 pγ1 x (γ1 x + A + B)p−1 − (γ1 x + A)p−1 − (γ1 x + A + B)p + (γ1 x + A)p + (A + B)p − Ap . The sign of g will be determined by the sign of the following function h defined, for x 0 h(x) = pγ1 x (γ1 x + A + B)p−1 − (γ1 x + A)p−1 − (γ1 x + A + B)p + (γ1 x + A)p + (A + B)p − Ap . Since h(0) = 0, p 2, and its derivative is h (x) = p(p − 1)γ12 x (γ1 x + A + B)p−2 − (γ1 x + A)p−2 0, we deduce that h(x) h(0) = 0 and hence g (x) 0, for x > 0, as we wanted to see.

2

For p > 0, we recall that a weight w is in the Bp -class (see [1]), if there exists a positive constant C > 0 such that, for every r > 0, ∞ r

p

w(x) dx C xp

r

r w(x) dx. 0

If w ∈ Bp we denote by wBp the best constant in the above inequality: wBp

∞ r p r w(x) x p dx . = sup r w(x) dx r>0 0

Bp weights were introduced in [1] (see also [9]) and characterize the boundedness of the Hardy operator (1) on monotone functions. We are going to prove our main theorem for weights w ∈ Bp , p 2, satisfying that ∞ r

p−1

w(x) dx is a decreasing function of r > 0. xp

(4)

r

It is easy to see that an equivalent condition for this to hold is ∞ r

p−1 r

w(x) w(r) dx , p x p−1

a.e. r > 0.

Observe that this inequality holds for any nonincreasing weight w (in particular if w ≡ 1), although there are also weights satisfying (4) which are not decreasing (consider, for instance, w(x) = χ(0,1) (x) + (1 + δ)χ(1,2) (x), with δ > 0 small enough).

1024


Theorem 2.2. Let p 2 and w be a weight in the Bp -class satisfying condition (4). Then, for any decreasing function f , we have p

p

Sf − f p,w wBp f p,w , and wBp is the best constant. Proof. Without loss of generality, using Fatou’s lemma, we can restrict ourselves to consider the subset consisting of all simple functions of the form (see [3]) fN (x) =

N

ci > 0, i = 1, 2, . . . , N, 0 < a1 < a2 < · · · < aN .

ci χ(0,ai ) (x),

i=1

Easy calculations show that

(SfN − fN )(x) =

⎧ 0, ⎪ ⎪ ⎪ c1 a1 , ⎪ ⎪ ⎪ ⎨ c xa +c 1 1

0 < x < a1 , a1 < x < a2 , 2 a2

x

⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩

,

c1 a1 +c2 a2 +···+cN aN x

a2 < x < a3 , .. . x > aN .

,

Assuming aN +1 = ∞, then the weighted norm of (S − I )fN is p SfN − fN p,w

∞ = (SfN − fN )p (x)w(x) dx 0

=

k N k=1

p ∞ cj aj

j =1

w(x) dx − xp

ak

∞

w(x) dx . xp

ak+1

We use the standard convention that, if the upper index in the sum is smaller than the lower one, then the sum is equal to zero. Now, the last expression above can be rewritten as k N

p −

cj aj

j =1

k=1

k−1

p ∞ cj aj

j =1

w(x) dx. xp

(5)

ak

Also, introducing a telescopic sum, we can show that (5) is equal to k k N k=1

j =1

i=j

p ci ai

−

k−1 i=j

p ci ai

−

k i=j +1

p ci ai

+

k−1 i=j +1

p ∞ ci ai

w(x) dx. xp

ak

(6)


1025

Similarly, assuming a0 = 0, the weighted norm of fN is p fN p,w

∞ = (fN )p (x)w(x) dx 0

N N

=

w(x) dx

0

N

N

k=1

j =k

0

p

N

−

cj

ak−1

w(x) dx −

cj

j =k

k=1

=

p ak

p ak cj

(7)

w(x) dx.

j =k+1

0

We claim that this last expression can be written as k p k−1 p k p k−1 p aj N k ci − ci − ci + ci w(x) dx. k=1 j =1

i=j

i=j +1

i=j

i=j +1

(8)

0

To check this equality, just observe that, for 1 n N , the corresponding coefficient any fixed n, a p −( N p . And also, looking at expression (8), we c ) c ) of 0 n w(x) dx in (7) is ( N j =n j j =n+1 j observe that if we put j = n, the sum in k is extended to n k N and, hence, the coefficient a that multiplies 0 n w(x) dx is equal to N

k=n

=

k

−

ci

i=n N

p

p −

ci

i=n

k−1

p −

ci

i=n

ci

p +

ci

i=n+1

p

N

k

k−1

p ci

i=n+1

.

i=n+1

Therefore, we have shown that in order to prove the inequality p

p

SfN − fN p,w wBp fN p,w

(9)

it suffices to show the following k k N k=1

j =1

wBp

p ci ai

−

i=j N k k=1 j =1

k−1

p −

ci ai

k i=j

p ci

p ci ai

−

k−1 i=j

p ci

−

p ∞

k−1

+

i=j +1

i=j

k

ci ai

i=j +1 k i=j +1

p ci

+

k−1

w(x) dx xp

ak

p aj ci

i=j +1

The inequality above will be guaranteed if, for any 1 j k N , we show that

w(x) dx. 0

1026


k

p −

ci ai

i=j

k−1

p ci ai

−

wBp

k

p ci

ci ai

−

i=j

k−1

−

ci

k

p +

ci

i=j +1

i=j

p ∞ ci ai

i=j +1

p

k−1

+

i=j +1

i=j

p

k

w(x) dx xp

ak k−1

p aj ci

w(x) dx,

i=j +1

0

which is equivalent to showing that, for any 1 j k N , k k−1 p p p ∞ ( ki=j ci ai )p − ( k−1 w(x) i=j ci ai ) − ( i=j +1 ci ai ) + ( i=j +1 ci ai ) dx k k−1 k k−1 xp ( i=j ci )p − ( i=j ci )p − ( i=j +1 ci )p + ( i=j +1 ci )p ak

aj wBp

(10)

w(x) dx. 0

For any j and k with 1 j k N , we apply Lemma 2.1, with α = {ai }j ik and γ = {ci }j ik , and condition (4) (recall that aj ak ) to obtain k k−1 p p p ∞ ( ki=j ci ai )p − ( k−1 w(x) i=j ci ai ) − ( i=j +1 ci ai ) + ( i=j +1 ci ai ) dx k k−1 p k k−1 p p p xp ( i=j ci ) − ( i=j ci ) − ( i=j +1 ci ) + ( i=j +1 ci ) ak

p−1

∞

aj ak

w(x) p dx aj xp

ak

∞

w(x) dx wBp xp

aj

aj w(x) dx, 0

which is (10). In order to see that wBp is the sharp constant Cp,w , just observe that for N = 1 we have p

Cp,w

sup c1 >0,a1 >0

Sf1 − f1 p,w p

f1 p,w

p∞ a1 a1 w(x) x p dx = wBp . = sup a1 a1 >0 0 w(x) dx

2

Remark 2.3. Theorem 2.2 is not true, in general, without the hypothesis (4). In fact, fix p 2 and consider the power weights w(t) = t α . Then, t α ∈ Bp , for −1 < α < p − 1, and α t

Bp

=

1+α . p−α−1

(11)

It is easy to check that, for any 0 < α < p − 1, (4) fails. To show that (11) is not the best constant in this case, take f (x) = 2χ(0,1) (x) + χ(1,2) (x), and for any p 2, let us consider, for example, α = (p − 1)/2. Easy computations show that

S. Boza, J. Soria / Journal of Functional Analysis 260 (2011) 1020–1028 p

Sf − f p,w p f p,w

=

1027

(2(−p+1)/2 (3p − 1) + 1)/(p − 1) p+1 > t (p−1)/2 B = , (p+1)/2 p p p−1 (2 + 2 − 1)/(p + 1)

which implies that t (p−1)/2 Bp is not the optimal constant. The following corollary gives a positive answer to the conjecture formulated in [8]. Corollary 2.4. S − I C p =

1 , (p−1)1/p

for all p 2.

Proof. Just observe that the constant weight w(t) ≡ 1 is in Bp , for all p > 1. Moreover, wBp =

1 . p−1

It obviously verifies condition (4) and, hence, Theorem 2.2 applies.

2

The following proposition shows, with the use of very different techniques than those used in Theorem 2.2, that also in the case p = 1, and for any weight w in B1 , the constant wB1 is sharp in the corresponding inequality. Proposition 2.5. Let w ∈ B1 . Then, for any f ∈ C 1 , we have the following sharp inequality Sf − f 1,w wB1 f 1,w . Proof. First we use Fubini’s theorem to write ∞ t 1 f (s) ds − f (t) w(t) dt Sf − f 1,w = t 0

0

∞ ∞ = 0

∞ w(t) dt f (s) ds − f (t)w(t) dt. t

s

(12)

0

We observe that first integral in the last equality is the norm of f in the Lorentz space Λ1 (v) ∞the w(t) where v(s) = s t dt. It is proved in [7] (see also [6]) that the best constant for the embedding Λ1 (w) → Λ1 (v) is given by t sup 0t t>0

0

v(s) ds w(s) ds

.

Hence, taking into account the expression in (12), the best constant Cw in the inequality Sf − f 1,w Cw f 1,w is given by

1028


t ∞ w(x) dx ds 0 v(s) ds − 1 = sup 0 s t x −1 Cw = sup t t>0 0 w(s) ds t>0 0 w(s) ds t ∞ w(s) 0 w(s) ds + t t s ds − 1 = wB1 . = sup t t>0 0 w(s) ds t

2

Remark 2.6. Looking at Theorem 2.2 and Proposition 2.5, it is natural to ask whether Theorem 2.2 is true if 1 < p < 2. However, we can show that the constant wBp is not sharp, in general, in that range. To see this, we take the constant weight w(t) ≡ 1 which, as we have previously remarked, satisfies condition (4) with wBp = 1/(p − 1). Straightforward calculations for the simple function f (x) = 2χ(0,1) (x) + χ(1,2) (x) show that, for every 1 . (p − 1)(2p + 1) p−1

Acknowledgment We want to thank professor María J. Carro for her helpful suggestions that have contributed to improve the final version of this paper. References [1] M.A. Ariño, B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990) 727–735. [2] C. Bennett, R.A. DeVore, R. Sharpley, Weak-L∞ and BMO, Ann. of Math. 113 (1981) 601–611. [3] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988. [4] S. Boza, J. Martín, Equivalent expressions for norms on classical Lorentz spaces, Forum Math. 17 (2005) 375–404. [5] M.J. Carro, A. Gogatishvili, J. Martín, L. Pick, Functional properties of rearrangement invariant spaces defined in terms of oscillations, J. Funct. Anal. 229 (2) (2005) 375–404. [6] M.J. Carro, L. Pick, J. Soria, V. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl. 4 (2001) 397–428. [7] M.J. Carro, J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112 (1993) 480–494. [8] N. Kruglyak, E. Setterqvist, Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions, Proc. Amer. Math. Soc. 136 (7) (2008) 2505–2513. [9] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990) 145–158. [10] J. Soria, Optimal bounds of restricted type for the Hardy operator minus the Identity on the cone of radially decreasing functions, Studia Math. 197 (2010) 69–79.


Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation Uri Kaluzhny, Yoram Last ∗ Institute of Mathematics, The Hebrew University, 91904 Jerusalem, Israel Received 30 October 2009; accepted 25 May 2010 Available online 3 December 2010 Communicated by J. Bourgain

Abstract We consider random self-adjoint Jacobi matrices of the form (Jω u)(n) = an (ω)u(n + 1) + bn (ω)u(n) + an−1 (ω)u(n − 1) on 2 (N), where {an (ω) > 0} and {bn (ω) ∈ R} are sequences of random variables on a probability space (Ω, dP (ω)) such that there exists q ∈ N, such that for any l ∈ N, β2l (ω) = al (ω) − al+q (ω)

and

β2l+1 (ω) = bl (ω) − bl+q (ω)

are independent random variables of zero mean satisfying ∞

βn2 (ω) dP (ω) < ∞.

n=1Ω

Let Jp be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in Jω . We prove that for a.e. ω, the a.c. spectrum of the operator Jω equals to and fills the spectrum of Jp . If, moreover, ∞

βn4 (ω) dP (ω) < ∞,

n=1Ω


E-mail addresses: [email protected] (U. Kaluzhny), [email protected] (Y. Last). 0022-1236/$ – see front matter © 2010 Published by Elsevier Inc. doi:10.1016/j.jfa.2010.05.014

1030

U. Kaluzhny, Y. Last / Journal of Functional Analysis 260 (2011) 1029–1044

then for a.e. ω, the spectrum of Jω is purely absolutely continuous on the interior of the bands that make up the spectrum of Jp . © 2010 Published by Elsevier Inc. Keywords: Random Jacobi matrices; Absolutely continuous spectrum

1. Introduction In this paper we study sufficient conditions for preservation of a.c. spectrum of periodic Jacobi matrices under a natural class of random slowly decaying perturbations. ∞ For any two bounded sequences of real numbers a = {an }∞ n=1 and b = {bn }n=1 , where an > 0, we define a Jacobi matrix J = J(a, b) by (Ju)(n) = an u(n + 1) + bn u(n) + an−1 u(n − 1).

(1)

We consider only such matrices whose elements are bounded, so they define bounded selfadjoint operators on 2 (N). The special case of a = 1 (namely, an = 1 for all n) is also called a discrete Schrödinger operator and its diagonal is then referred to as a potential. The discrete Schrödinger operator with zero potential = J(1, 0) is called the free discrete Laplacian. We say that the absolutely continuous spectrum of an operator of the form (1) fills a set S if μac (Q) > 0 for any Q ⊂ S of positive Lebesgue measure. We say that its spectrum is purely absolutely continuous on S if, in addition, μsing (S) = 0. Here μ = μac + μsing is the decomposition of the spectral measure of the operator into absolutely continuous and singular parts. The essential support of the absolutely continuous spectrum of such an operator, denoted Σac (·), is the equivalence class (up to sets of zero Lebesgue measure) of the largest set filled by its absolutely continuous spectrum. Preservation of the absolutely continuous spectrum of an operator under decaying perturbations has been the subject of extensive research over the last two decades. One can start, e.g, from the free Laplacian that has purely a.c. spectrum filling the interval [−2, 2] and add to it a decaying potential. Two basic facts have been known for a long time: Theorem 1. If ∞ n=1 |an − 1| + |bn | < ∞, then the a.c. spectrum of J(a, b) is the same as that of in the sense that it is equal to and fills [−2, 2] and that J(a, b) has purely a.c. spectrum on the interior of [−2, 2]. Theorem 2. If an → 1, bn → 0 as n → ∞, and ∞

|an+1 − an | + |bn+1 − bn | < ∞,

n=1

then the a.c. spectrum of J(a, b) is the same as that of in the same sense as in Theorem 1. The first fact has been known at least from the 1950’s and follows, in particular, from Kato– Birman theory of trace class perturbations (see [25, vol. III]). The second has been essentially proven by Weidemann [34] in 1967. (Weidemann actually proved a variant of this for continuous Schrödinger operators. For a proof of the discrete case, see Dombrowski and Nevai [10] or Simon [28].)


1031

On the other hand, the works of Delyon, Simon and Souillard [7,8,27] and Kotani and Ushiroya [20,21] on decaying random potentials showed in the 1980’s that perturbations that are not square-summable can result in purely singular spectrum. In the 1990’s much work (see, e.g., [3,5,6,17,18,26]) has been done towards showing that square-summable perturbations of the free Laplacian do not change its a.c. spectrum. Eventually, Killip and Simon [15] proved, in particular, the following. Theorem 3. If a perturbation is square-summable, that is ∞

|an − 1|2 + |bn |2 < ∞,

n=1

then Σac (J(a, b)), the essential support of the a.c. spectrum of J(a, b), is equal to [−2, 2]. In [19], Kiselev, Last and Simon conjectured the following. Conjecture 1. If a potential bˆ is square-summable, then, for any Jacobi matrix J(a, b), ˆ is equal to Σac (J(a, b)). Σac (J(a, b + b)) Results towards the full Conjecture 1 seem to be scarce so far. Killip [14] has proven it for the case of discrete Schrödinger operators with periodic potentials. Breuer and Last [1] have recently shown that, for any Jacobi matrix, the a.c. spectrum which is associated with bounded generalized eigenfunctions (like the spectrum of a periodic Jacobi matrix) is stable under squaresummable random perturbations. In [12] we have shown that if both a square-summable random perturbation and a decaying perturbation of bounded variation are added to the free Laplacian, the a.c. spectrum is still preserved. But, to the best of our knowledge, there has been no significant progress made so far with the general deterministic case. Another natural direction of research in this area is to extend and generalize Theorem 2 in the direction in which Theorem 1 has been extended. A notable result in this direction has been obtained by Kupin [22] who showed that the essential support of the a.c. spectrum of is still preserved if a decaying potential of a square-summable variation is added to it under an additional restriction that this perturbation lies in m for some m ∈ N. But for a perturbation of a bounded variation of a general Jacobi matrix, a guess even weaker than an analog of Conjecture 1 would be wrong. Indeed, one of us have recently constructed [24] an example of a Jacobi matrix ˆ with a = 1, limn→∞ bñ = limn→∞ bˆn = 0, so that {bñ }∞ is of bounded variation, J(a, b˜ + b) n=1 ˆ and J(a, b) ˜ have purely a.c. spectrum on (−2, 2) with essential support (−2, 2), but both J(a, b) ˆ has empty absolutely continuous spectrum. In particular, adding a decaying perturJ(a, b˜ + b) bation of bounded variation to a Jacobi matrix can fully “destroy” its absolutely continuous spectrum. Thus, it would be natural to confine the consideration first to the simple case of perturbations of a summable variation added to a periodic Jacobi matrix. In particular, we note the following result of Golinskii and Nevai [11] concerning variations of order q (before [11], some related results were obtained by Stolz [30,31]):

1032


ˆ ∞ Theorem 4. Let J(a, b) be a periodic Jacobi matrix of period q and let {aˆ n }∞ n=1 and {bn }n=1 be decaying sequences obeying, ∞

|aˆ n+q − aˆ n | + |bˆn+q − bˆn | < ∞.

(2)

n=1

ˆ is equal to the spectrum of Then the essential support of the a.c. spectrum of J(a + aˆ , b + b) ˆ J(a, b) and, moreover, the spectrum of J(a + aˆ , b + b) is purely absolutely continuous on the interior of the bands that make up the spectrum of J(a, b). Remark. Note that, in particular, this theorem extends Theorem 2 to the case where the free Laplacian is replaced by a periodic Jacobi matrix. We believe that the following statement (which generalizes [24, Conjecture 1.6]) should be true: ˆ ∞ Conjecture 2. Let J(a, b) be a periodic Jacobi matrix of period q and let {aˆ n }∞ n=1 and {bn }n=1 be decaying sequences obeying, ∞

|aˆ n+q − aˆ n |2 + |bˆn+q − bˆn |2 < ∞.

(3)

n=1

ˆ is equal to the spectrum of Then the essential support of the a.c. spectrum of J(a + aˆ , b + b) J(a, b). We note that a variant of this conjecture for the special case q = 1 has also been made by Simon [29, Chapter 12]. Kim and Kiselev [16] made some progress towards Conjecture 2 by extending to the discrete case some of the results and techniques previously used by Christ and Kiselev [4] to treat continuous (namely, differential) Schrödinger operators. They studied the discrete Schrödinger case where a = 1, b = 0 (so J(a, b) is just the discrete Laplacian), aˆ =p 0 and ˆ ˆ where bˆ is a bounded (but not necessarily decaying) sequence obeying ∞ n=1 |bn+1 − bn | < ∞ for some p < 2. They show that in this case the essential support of the a.c. spectrum coincides with [−2 + lim sup bˆn , 2 + lim sup bˆn ] ∩ [−2 + lim inf bˆn , 2 + lim inf bˆn ]. We note, however, that the case p = 2 appears to be outside the scope of their techniques. Some very significant progress towards Conjecture 2 has been recently made by Denisov [9], who proved the full [24, Conjecture 1.6], namely, Conjecture 2 for the discrete Schrödinger case where a = 1, b = 0 and aˆ = 0. We believe that his ideas are likely to be extensible to the more general setting of Conjecture 2 and we hope that it will thus be soon possible to prove it in full [13]. Our present work explores perturbations whose variations of order q are square-summable random variables. As mentioned above, the study of random perturbations have indicated the exact rate of the decay that still preserves the a.c. spectrum in the extension of Theorem 1. From this point of view, part of the interest in our present result is that it can be considered as “evidence” in support of Conjecture 2. We note, however, that for appropriate cases (namely, when the variations are also fourth-order-summable), our result also yields stable purity of the a.c. spectrum which is not expected to hold in the general deterministic setting of Conjecture 2. Indeed, we obtain for such cases the same kind of a.c. spectrum preservation as in Theorem 4.


1033

Thus, our result here goes beyond being a random version of Conjecture 2 and we believe that it cannot be deduced from any deterministic statement of its type. Our main result in this paper is the following: Theorem 5. Let Jω be a self-adjoint random Jacobi matrix (Jω u)(n) = an (ω)u(n + 1) + bn (ω)u(n) + an−1 (ω)u(n − 1) on 2 (N), where {an (ω) > 0} and {bn (ω) ∈ R} are sequences of random variables on a probability space (Ω, dP (ω)) such that there exists q ∈ N, so that for any l ∈ N, β2l (ω) = al (ω) − al+q (ω)

and β2l+1 (ω) = bl (ω) − bl+q (ω)

(4)

are independent random variables of zero mean satisfying ∞

βn2 (ω) dP (ω) < ∞.

(5)

n=1 Ω

˜ be the deterministic periodic (of the period q) Jacobi matrix whose coefficients Let Jp = J(ã, b) are a˜ l =

al (ω) dP (ω) = a˜ l+q , Ω

b˜l =

bl (ω) dP (ω) = b˜l+q .

Ω

Then, for a.e. ω, the a.c. spectrum of the operator Jω equals to and fills the spectrum of Jp . If, moreover, ∞

βn4 (ω) dP (ω) < ∞,

n=1 Ω

then for a.e. ω, the spectrum of Jω is purely absolutely continuous on the interior of the bands that make up the spectrum of Jp . Remark. For cases where Theorem 5 yields purity of the a.c. spectrum, our proof actually shows something a bit stronger, namely, that for a.e. fixed ω, the purity of the absolutely continuous spectrum will be stable under changing any finite number of entries in the Jacobi matrix. Theorem 5 is proven in Section 2.

1034


2. Proof of Theorem 5 Let us start from building explicitly a random Jacobi operator Jω on 2 (N), satisfying the condition (4) and introducing some notations. Let Jp be a periodic (of period q) Jacobi matrix (Jp u)(n) = a˜ n u(n + 1) + bñ u(n) + a˜ n−1 u(n − 1), such that a˜ n+q = a˜ n , bñ+q = bñ and min a˜ n = ε0 > 0. Consider a sequence {βn }∞ n=1 of independent random variables on a probability space (Ω, dP (ω)) that satisfies Def ∀n E(βn ) = βn (ω) dP (ω) = 0 (6) Ω

and ∞ E βn2 < ∞. n=1

In such a case, for n, m ∈ N and 0 m < q, αn (ω) =

∞

β2qi+n (ω),

i=0

anq+m (ω) = a˜ m + α2(nq+m) (ω),

bnq+m (ω) = b˜m + α2(nq+m)+1 (ω)

(7)

are well defined for a.e. ω (see, e.g., [33]). To get an (ω) > 0 for all n ∈ N, we can take, e.g., βi such that |βi (ω)| < C/i, for a.e. ω, for an appropriate constant C. The operator ∞ Jω = J({an (ω)}∞ n=1 , {bn (ω)}n=m ) will be then a well-defined self-adjoint random Jacobi matrix. Jω satisfying the conditions of Theorem 5 is given. Denote a˜ n = So, suppose an operator ˜ a (ω) dP (ω), b = b n Ω n Ω n (ω) dP (ω) and αn (ω) as in (7). Our analysis of the a.c. spectrum of the operator Jω will be built upon establishing the nearboundedness of its generalized eigenfunctions, which are the solutions of the difference equation (here and in what follows we denote by the same letter an operator on 2 (N) and the corresponding difference operator on the vector space of all real-valued sequences) Jω u = Eu. Definition. The sequence {Tn (E, ω)} of transfer matrices for the operator Jω at the energy E is defined by Tn (E, ω) =

E−bn (ω) an (ω)

−an−1 (ω) an (ω)

1

0

u(n) u(n + 1) : −→ , u(n − 1) u(n)

where u is a solution of the difference equation Jω u = Eu.


1035

For n m, we define Def

Tn,m (E, ω) = Tn−1 (E, ω)Tn−2 (E, ω) . . . Tm (E, ω);

Def

Tn,n = I.

The transfer matrices Tñ (E) and Tñ,m (E) for the operator Jp are defined similarly. Of course, Tñ (E) = Tñ+q (E), so we will be writing, e.g., T˜0 (E) understanding it as T˜q (E). The growth of generalized eigenfunctions for Jω will be controlled by the growth of

Tn,m (E, ω) as n → ∞. In particular, we shall use the following theorems from [23]: Theorem 6. Let J be a Jacobi matrix and Tn,m (E) its transfer matrices. Suppose S is a set such that for a.e. E ∈ S,

lim Tn,m (E) < ∞ n→∞

(a fact that does not depend on m). Then the absolutely continuous spectrum of the operator J fills S. Theorem 7. Suppose there is some m ∈ N so that d lim n→∞

Tn,m (E, ω) p dE < ∞

c

for some p > 2. Then (c, d) is in the essential support of the absolutely continuous spectrum of Jω and the spectrum of Jω is purely absolutely continuous on (c, d). Remarks. 1. Theorem 7 is essentially Theorem 1.3 of [23]. While [23] only discusses Jacobi matrices with an = 1, the result easily extends to our more general context. 2. As noted in [23], Theorem 7 is an extension of an idea of Carmona [2]. 3. While the fact that (c, d) is in the essential support of the absolutely continuous spectrum isn’t explicitly stated in [23, Theorem 1.3], this easily follows from spectral averaging and the fact that the d lim n→∞

Tn,m (E, ω) p dE < ∞

c

condition is invariant to changing any finite number of entries in the Jacobi matrix. To single out the independent random variables we will rather consider for n m the matrices 1 0 −1 . Pn,m (E, ω) = Am (ω)Tn,m (E, ω)A−1 (ω), A (ω) = n n 0 an−1 (ω) In particular, Pn,m (E, ω) = Pm (E, ω) . . . Pn−1 (E, ω), where 0 1/an (ω) . Pn (E, ω) = Pn+1,n (E, ω) = n (ω) −an (ω) E−b an (ω) Note that, as a random variable, Pn,m (E, ω) depends only on {βj }∞ j =2m .

1036


Let ε1 = ε0 /2 = min1nq a˜ n /2. Under the assumption that sup αn (ω) ε1 ,

(8)

nm

the norms of the matrices Tl (E, ω) are uniformly bounded and to get a uniform bound on the norms of the matrices Tl,mq (E, ω) as l → ∞ it is sufficient to prove that limn→∞ T(m+n)q,mq (E, ω) < ∞, since between (m + n)q and (m + n + 1)q the norm of a transfer matrix cannot be more than supl Tl (E, ω) q−1 times larger. But (8) also implies that det Tn,m (E, ω) =

−am−1 (ω) ∼ 1, an−1 (ω)

Am (ω) ∼ 1 and A−1 n (ω) ∼ 1, so

T(m+n)q,mq (E, ω) ∼ T −1

(m+n)q,mq (E, ω) ∼ P(m+n)q,mq (E, ω) . Define ˜ M(E) =

1 0 0 a˜ 0

1 0 −1 −1 T˜q,0 (E) . 0 a˜ q

Note that because of periodicity of a˜ n and bñ , we have, for every E ˜ det M(E) = det T˜q,0 (E) = 1.

(9)

˜ The tr M(E) is a polynomial function of E and, because of (9), −1 ˜ (E) = tr T˜q,0 (E). tr M(E) = tr T˜q,0

(10)

So, from the general Floquet theory (see, e.g., [32, Chapter 7]) we know that the spectrum of the ˜ 2. In other words, operator Jp is the union of the intervals of R that are defined by | tr M(E)| 1 ˜ the matrix M(E) is quasi-unitary inside the spectrum of Jp . ˜ Let a compact set I ⊂ {E|| tr M(E)| < 2} and some m ∈ N be given. Following Golinskii and Nevai [11], we will use a theorem due to Kooman to prove our result. For a proof of Theorem 8, see [29, Chapter 12]. Theorem 8. Let A(E) be a quasi-unitary matrix which depends continuously on some parameter E varying in a compact Hausdorff space. Then there exists γ > 0 and functions CE (Q) and BE (Q), jointly continuous in E and Q, such that 1 A (2 × 2 in our case) matrix A is called quasi-unitary if it has two different eigenvalues both of absolute value 1. Obviously, a real matrix is quasi-unitary iff it is similar to a unitary matrix and iff det A = 1 and |tr A| < 2. A quasiunitary matrix is indeed unitary in a norm associated with its eigenvectors as will be defined in what follows.


1037

(i) for each fixed E, CE (Q) and BE (Q) are analytic on

Nγ A(E) = Q A(E) − Q < γ , (ii) (iii) (i) (iv)

CE (A(E)) = I, BE (A(E)) = A(E), ∀Q ∈ Nγ (A(E)), CE (Q) is invertible, ∀Q ∈ Nγ (A(E)), Q = CE (Q)−1 BE (Q)CE (Q), ∀Q ∈ Nγ (A(E)), BE (Q) commutes with A(E).

˜ We shall apply Kooman’s theorem to the quasi-unitary matrix M(E) for E ∈ I . ˜ Let γ be the provided for M(E) by Theorem 8. Since, because of (ii), ˜ ˜ tr BE M(E) = tr M(E) ∈ (−2, 2), ˜ we can choose γ small enough to assure |tr B˜E (Q)| < 2 for all Q ∈ N¯ γ /2 (M(E)), where ¯ Nγ /2 (M) = {Q | M − Q γ /2}. Note that for n ∈ N, the matrix Mn+m (E, ω) = P(m+n+1)q,(m+n)q (E, ω), ˜ corresponding to a period corresponding to the (m + n)-th period for Jω is like the matrix M(E) ˜ perturbed by of Jp , with a’s ˜ and b’s α2(m+n)q (ω), . . . , α2(m+n+1)q−1 (ω). We think of a˜ 1 , . . . , a˜ q and b˜1 , . . . , b˜q as of fixed values and the dependence of Mn (E, ω) on α’s and E is continuous and even analytic. Hence, there exists ε2 > 0 (we take ε2 ε1 to satisfy also the assumption (8)), such that sup αl (ω) ε2 ε1

(11)

l>2qm

implies that for all E ∈ I and for all n ∈ N, ˜ Mn (E, ω) ∈ N¯ γ /2 M(E) . Then, as mentioned above, |tr BE (Mn (E, ω))| < 2 and, because of (iv), det BE Mn (E, ω) = det Mn (E, ω) = 1. This means that the matrix BE (Mn (E, ω)) is then quasi-unitary. Let, under the assumption (11), for an E ∈ I , {v1 , v2 } be the basis, depending on E, in which ˜ the matrix M(E) and also, because of (v), each BE (Mn (E, ω)), for n ∈ N, are diagonal. Let

· E be the norm in this basis, that is, for a vector v = xv1 + yv2 its norm will be v 2E = |x|2 + |y|2 , and for a matrix M, M E = sup v E =1 M(v) E . In this norm BE (Mn (E, ω)) is unitary for every n ∈ N and for every E ∈ I .

1038


Let us denote, for 0 l n, Mn+m,l+m (E, ω) as Ml+m (E, ω) . . . Mn+m−1 (E, ω) = P(n+m)q,(l+m)q (E, ω). In what follows, we will show that the assumption (11) is eventually fulfilled for a.e. ω and then we can bound the norm of Mn,0 (E, ω) uniformly as n → ∞. Let, for E ∈ I , v ∈ R2 be an arbitrary vector, such that v E = 1. Define wl (E, ω) = BE Ml (E, ω) · CE Ml (E, ω) · Mn,l (E, ω)v. Now, denoting Cl = CE (Ml (E, ω)) and Bl = BE (Ml (E, ω)), we have wl = Bl Cl Mn,l v = Bl Cl Ml+1 Mn,l+1 v −1 −1 Bl+1 Cl+1 Mn,l+1 v = Bl Cl Cl+1 wl+1 . = Bl Cl Cl+1

Hence,

−1

2

2 −1 wl+1 E wl+1 E = I + (Cl − Cl+1 )Cl+1

wl 2E = Cl Cl+1

2 −1 −1 = wl+1 2E + 2 wl+1 , (Cl − Cl+1 )Cl+1 wl+1 E + (Cl − Cl+1 )Cl+1 wl+1 E . Notice that Cl = CE (Ml (E, ω)) is analytic as a function of α2(m+l)q (ω), . . . , α2(m+l+1)q−1 (ω)

and E,

and, hence, since βi = αi − αi+2q , we can represent, for some matrix functions Di (E, ω), Fi (E, ω) and Gi (E, ω) that depend only on {βj }∞ j =i , −1 (Cl − Cl+1 )Cl+1

=

2(m+l+1)q−1

βi (ω)Gi (E, ω),

(12)

βi (ω)Di+1 (E, ω) + βi2 (ω)Fi (E, ω) .

(13)

i=2(m+l)q

and, also, −1 (Cl − Cl+1 )Cl+1 =

2(m+l+1)q−1 i=2(m+l)q

The norms of Di (E, ω), Fi (E, ω) and Gi (E, ω) are (uniformly in n) bounded, provided ˜ Mn (E, ω) ∈ N¯ γ /2 (M(E)), E ∈ I. −1 −1 When we plug (13) into wl+1 , (Cl − Cl+1 )Cl+1 wl+1 E and (12) into (Cl − Cl+1 )Cl+1 wl+1 2E , we see that

wl 2E < Bl (E, ω) wl+1 2E ,

(14)


1039

where Bl (E, ω) = 1 +

2(m+l+1)q−1

βi (ω)Di+1 (E, ω) + βi2 (ω)Fi (E, ω)

i=2(m+l)q

and Di (E, ω), Fi (E, ω) are some scalar (uniformly in n) bounded functions, provided ˜ E ∈ I. Mn (E, ω) ∈ N¯ γ /2 (M(E)), Going from w0 to wn−1 , we get from (14) that Mn,0 (E, ω)v 2E is bounded by 2(m+l+1)q−1 n−1

−1

2

C Cn E βi (ω)Di+1 (E, ω) + βi (ω) Fi (E, ω) . 1+ 0 E l=0

i=2(m+l)q

˜ Let B = max{ CE (M) E | E ∈ I, M ∈ N¯ γ /2 (M(E))}. Since, for ω’s satisfying the assump˜ tion (11), Ml (E, ω) lays in N¯ γ /2 (M(E)), we have

CE Ml (E, ω) B E for any E ∈ I and l ∈ N. In the same way we can assure2 that

CE Mn (E, ω) −1 < B, E ∀i, Di (E, ω) < B, ∀i, Fi (E, ω) < B. For {v1 , v2 } an orthonormal basis in R2 , for any 2 × 2 matrix A,

A 2 max Av1 , Av2 . This, in particular, implies that if for any unit vector v, Av < B, then A < 2B. So, using the fact that 1 + x ex for x 0,

2

Mn,0 (E, ω) < B exp E We note that

∞

2 i=1 βi (ω)

∞ Ω i=1

2(m+n)q i=2mq+1

∞ 2 βi (ω)Di+1 (E, ω) + B βi (ω) . i=1

is finite for a.e. ω, from (5), since βi (ω) dP (ω) = 2

∞

βi (ω)2 dP (ω) < ∞.

i=1 Ω

To control the rest of the factors in the above estimate and to get along with the assumptions (8) and (11), we will prove the following lemma. 2 We adopt the convention of denoting different constants by the same letter.

1040


Let Dl (ω) be an integrable measurable function depending only on {βi (ω)}∞ i=l . Remember β (ω) and define that αn (ω) = ∞ i=0 2qi+n l−1 l = βi Di+1 . Sm i=m

Lemma 1. For every ε > 0 and δ > 0, there exist Ω ⊆ Ω and m ∈ N, such that P{Ω } > 1 − δ and for every ω ∈ Ω sup αn (ω) ε,

(15)

n sup Sm (ω) 2.

(16)

n>m

n>m

Proof. Fix n > m and define Ω(m, n) = ω ∈ Ω

max

l=m,...,(n−1)

|αl | > ε or

Ωm =

max

l=(m+1),...,n

l Sm >2 ,

Ω(m, n).

n>m

Note that for n < n , Ω(m, n) ⊆ Ω(m, n ), so

P{Ωm } = lim P Ω(m, n) . n→∞

The lemma will be proved if we show that limm→∞ P{Ωm } = 0. l S n + S n , so For m l n, Sm m l max

l=m,...,(n−1)

Sln 1

⇒

max

l=(m+1),...,n

l Sm 2.

Hence, for Ω (m, n) = ω ∈ Ω

max

l=m,...,(n−1)

|αl | > ε or

max

l=m,...,(n−1)

l Sm >1 ,

Ω(m, n) ⊆ Ω (m, n). Now we can proceed as in a standard martingale inequality. Define ∞

Al =

1 βi , ε i=l

Bl =

n−1

βi Di+1 ,

i=l

Cj = ω ∈ Ω ∀i > j : |Ai | 1, |Bi | 1 and |Aj | > 1 or |Bj | > 1 . Note that for i < j , since βi is independent from βj , we have, from (6), βi Aj χCj dP (ω) = βi dP (ω) Aj χCj dP (ω) = 0. Ω

Ω

Ω


1041

Also, since Dl depends only on {βi }∞ i=l , for i < j , βi Di+1 Bj χCj dP (ω) = 0. Ω

Hence,

A2m χCj

dP (ω)

Ω

A2j χCj dP (ω), Ω

j −1 since, as we expand the square A2m = [ 1ε ( i=m βi ) + Aj ]2 , the expectation of the cross terms vanishes. In the same way, 2 Bm χCj dP (ω) Bj2 χCj dP (ω). Ω

Hence

Ω

2 2 Am + Bm χCj dP (ω)

Ω

Since Ω (m, n) ⊆

2 Aj + Bj2 χCj dP (ω)

Ω

n−1

j =m Cj ,

χCj dP (ω). Ω

we have

n−1

P Ω(m, n) P Ω (m, n)

χCj dP (ω)

j =m Ω

n−1

2 2 Am + Bm χCj dP (ω)

j =m Ω

2 2 Am + Bm dP (ω).

Ω

If we expand the squares, using the mutual independence of the β’s, (6) and (5), we see that ∞ 2 m→∞ 2 Am + Bm dP (ω) B E βi2 −→ 0. Ω

This proves the lemma.

i=m

2

The lemma says that for any given E ∈ I , the set of ω’s such that • there exists m such that the assumptions (8) and (11) hold for n m, and then • limn→∞ Mm+n,m (E, ω) < ∞, is of full measure. Since I is an arbitrary compact set in the spectrum of Jp , this, along with Theorem 6, proves the first part of our theorem. To prove the second part of the theorem, first note that, when E varies in the compact set I , ˜ the matrix M(E) always has two separate eigenvalues, and hence, we can pick the eigenvectors

1042


for it in a continuous manner. This, in particular, will imply that there will be a constant B, such that for any vector v and a matrix M, 1/B v < v E < B v ,

1/B M < M E < B M ,

(17)

where · is the canonical norm of R2 . Now the inequality (14) implies that, for l− = 2(m + l)q, l + = 2(m + l + 1)q − 1,

wl 2E

+

< 1+B

l

+

wl+1 2E

βi2

i=l−

+

l

βi wl+1 , Di+1 (E, ω)wl+1 E .

i=l−

Squaring this and grouping the similar powers of β’s, we get, after some tedious but straightforward calculations, that (remember that B is just a generic name for some constant) l+ 2 4 βi + βi wl+1 4E < 1+B

wl 4E

i=l− +

+

l

βi wl+1 , Di+1 (E, ω)wl+1 E wl+1 2E .

i=l−

Integrating the last inequality over I , we get l+ 2 4 βi D˜ i+1 (ω) + Bβi + Bβi < 1+

wl+1 4E dE,

wl 4E dE

i=l−

I

I

where D˜ i (ω) =

2 I wl+1 (E, ω), Di (E, ω)wl+1 (E, ω) E wl+1 (E, ω) E dE 4 I wl+1 (E, ω) E dE

is an integrable measurable function depending only on {βj (ω)}∞ j =i .

˜ Now we can proceed as in the first part, use the same lemma, this time for D(ω), and the fact that we now have ∞ Ω i=1

so

∞

4 i=1 βi (ω)

βi (ω) dP (ω) = 4

∞

βi (ω)4 dP (ω) < ∞,

i=1 Ω

is finite for a.e. ω, to establish that the set of ω’s for which

• there exists m such that the assumptions (8) and (11) hold for n m, and then • limn→∞ I Mm+n,m (E, ω) 4 dE < ∞, is of full measure. By the fact that I is an arbitrary compact set in the spectrum of Jp and Theorem 7, this proves the second part of Theorem 5.


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Acknowledgments We would like to thank J. Breuer, M. Shamis, and B. Simon for useful discussions. This research was supported in part by The Israel Science Foundation (Grant No. 1169/06) and by Grant 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. References [1] J. Breuer, Y. Last, Stability of spectral types for Jacobi matrices under decaying random perturbations, J. Funct. Anal. 245 (2007) 249–283. [2] R. Carmona, Exponential localization in one-dimensional disordered systems, Duke Math. J. 49 (1982) 191–213. [3] M. Christ, A. Kiselev, Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results, J. Amer. Math. Soc. 11 (1998) 771–797. [4] M. Christ, A. Kiselev, WKB asymptotics of generalized eigenfunctions of one-dimensional Schrödinger operators, J. Funct. Anal. 179 (2001) 426–471. [5] M. Christ, A. Kiselev, C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials: Some optimal results, Math. Res. Lett. 4 (1997) 719–723. [6] P. Deift, R. Killip, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys. 203 (1999) 341–347. [7] F. Delyon, Appearance of a purely singular continuous spectrum in a class of random Schrödinger operators, J. Stat. Phys. 40 (1985) 621–630. [8] F. Delyon, B. Simon, B. Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré 42 (1985) 283–309. [9] S.A. Denisov, On a conjecture by Y. Last, J. Approx. Theory 158 (2009) 194–213. [10] J. Dombrowski, P. Nevai, Orthogonal polynomials, measures and recurrence relations, SIAM J. Math. Anal. 17 (1986) 752–759. [11] L. Golinskii, P. Nevai, Szegö difference equations, transfer matrices and orthogonal polynomials on the unit circle, Comm. Math. Phys. 223 (2001) 223–259. [12] U. Kaluzhny, Y. Last, Purely absolutely continuous spectrum for some random Jacobi matrices, in: D.A. Dawson, V. Jaksic, B. Vainberg (Eds.), Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov, in: CRM Proc. Lecture Notes, vol. 42, Amer. Math. Soc., Providence, RI, 2007, pp. 273–282. [13] U. Kaluzhny, M. Shamis, Preservation of absolutely continuous spectrum of periodic Jacobi operators under perturbations of square-summable variation, Constr. Approx., in press. [14] R. Killip, Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum, Int. Math. Res. Not. 38 (2002) 2029–2061. [15] R. Killip, B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. (2) 158 (2003) 253–321. [16] A. Kim, A. Kiselev, Absolutely continuous spectrum of discrete Schrödinger operators with slowly oscillating potentials, Math. Nachr. 282 (2009) 552–568. [17] A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. 179 (1996) 377–400. [18] A. Kiselev, Y. Last, B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Comm. Math. Phys. 194 (1998) 1–45. [19] A. Kiselev, Y. Last, B. Simon, Stability of singular spectral types under decaying perturbations, J. Funct. Anal. 198 (2003) 1–27. [20] S. Kotani, Lyapunov exponents and spectra for one-dimensional random Schrödinger operators, in: Random Matrices and Their Applications, Brunswick, Maine, 1984, in: Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 276–286. [21] S. Kotani, N. Ushiroya, One-dimensional Schrödinger operators with random decaying potentials, Comm. Math. Phys. 115 (1988) 247–266. [22] S. Kupin, Spectral properties of Jacobi matrices and sum rules of special form, J. Funct. Anal. 227 (2005) 1–29. [23] Y. Last, B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999) 329–367.

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[24] Y. Last, Destruction of absolutely continuous spectrum by perturbation potentials of bounded variation, Comm. Math. Phys. 274 (2007) 243–252. [25] M. Reed, B. Simon, Methods of Modern Mathematical Physics, I–IV, Academic Press, New York, 1979. [26] C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193 (1998) 151–170. [27] B. Simon, Some Jacobi matrices with decaying potential and dense point spectrum, Comm. Math. Phys. 87 (1982) 253–258. [28] B. Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 124 (1996) 3361–3369. [29] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2, Spectral Theory, Amer. Math. Soc., Providence, RI, 2005. [30] G. Stolz, Spectral theory for slowly oscillating potentials. I. Jacobi matrices, Manuscripta Math. 84 (1994) 245–260. [31] G. Stolz, Spectral theory for slowly oscillating potentials. II. Schrödinger operators, Math. Nachr. 183 (1997) 275– 294. [32] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Amer. Math. Soc., Providence, RI, 2000. [33] H. Tucker, A Graduate Course in Probability, Academic Press, New York, 1967. [34] J. Weidmann, Zur Spektraltheorie von Sturm–Liouville-Operatoren, Math. Z. 98 (1967) 268–302.


Selfadjoint operators in S-spaces Friedrich Philipp a , Franciszek Hugon Szafraniec b , Carsten Trunk a,∗ a Technische Universität Ilmenau, Institut für Mathematik, Postfach 100565, D-98684 Ilmenau, Germany b Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland

Received 10 December 2009; accepted 28 October 2010 Available online 13 November 2010 Communicated by D. Voiculescu

Abstract We study S-spaces and operators therein. An S-space is a Hilbert space (S, ( · , −)) with an additional inner product given by [ · , −] := (U · , −), where U is a unitary operator in (S, ( · , −)). We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator A in an S-space we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well-know and in general unsolved problem. © 2010 Elsevier Inc. All rights reserved. Keywords: S-space; Krein space; Indefinite inner products; Selfadjoint operators; Invariant subspaces

1. Introduction A complex linear space H with a Hermitian sesquilinear form [ · , −] is called a Krein space if there exists a fundamental decomposition H = H+ ⊕ H−

(1)

with subspaces H± being orthogonal to each other with respect to [ · , −] such that (H± , ±[ · , −]) are Hilbert spaces. If H− or H+ is finite dimensional, then (H, [ · , −]) is called a Pontryagin * Corresponding author.

E-mail addresses: [email protected] (F. Philipp), [email protected] (F. Hugon Szafraniec), [email protected] (C. Trunk). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.10.023

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F. Philipp et al. / Journal of Functional Analysis 260 (2011) 1045–1059

space. To each decomposition (1) there correspond a Hilbert space inner product ( · , −) and a selfadjoint operator J (the fundamental symmetry) with J J ∗ = I, J = J ∗ such that [x, y] = (J x, y)

for x, y ∈ H,

(2)

see, e.g., [2,5,14]. Conversely, every bounded and boundedly invertible selfadjoint operator G in a Hilbert space (H, ( · , −)) defines an inner product via [ · , −] := (G · , −)

(3)

and (H, [ · , −]) becomes a Krein space. In particular, if the spectrum of G consists on the positive (or negative) semiaxis only of finitely many isolated eigenvalues of finite multiplicity, then (H, [ · , −]) is a Pontryagin space. Eq. (3) is the starting point for various generalizations. E.g., if G is a bounded selfadjoint operator (but no more boundedly invertible) in H such that σ (G) ∩ (−∞, ε) consists of finitely many eigenvalues of G with finite multiplicities for some ε > 0, then (H, [ · , −]), where [ · , −] is defined by (3), is called an Almost Pontryagin space, see [9]. Observe that in this case zero is allowed to be an eigenvalue of G with finite multiplicity. Almost Pontryagin spaces and operators therein were considered in various situations, we mention only [1,9–12,17,21,26,27]. The more general case that G is a bounded selfadjoint operator in H such that zero is an isolated eigenvalue of G with finite multiplicity gives rise to Almost Krein spaces, see [3]. Spaces with an inner product given by an arbitrary bounded selfadjoint operator were studied, e.g., in [16,22]. For applications we refer to [4,6,8–12,15,17–20,26,27]. In all the above-mentioned generalizations of (3) the selfadjointness of the operator G in H is maintained and the bounded invertibility is dropped. Obviously, this is the same as generalizing (2) by dropping J J ∗ = I and preserving J = J ∗ . From this point of view, it seems natural to generalize (2) the other way: dropping selfadjointness and preserving unitarity of J . The inner product space (H, [ · , −]), where [ · , −] is defined by (2) with a unitary operator J is called an S-space, cf. [23] and also Definition 2.1 below. Moreover, the pair (( · , −), J ) is called a Hilbert space realization of the S-space (H, [ · , −]). Evidently, by definition every Krein space is a special case of an S-space. In this paper we continue the study of S-spaces and operators therein started in [23,24]. It is known from [24] that the inner products of two Hilbert space realizations (( · , −)1 , U1 ) and (( · , −)2 , U2 ) define the same topology. Here, we show in particular that U1 and U2 are similar operators with respect to this topology, cf. Proposition 2.4. In Section 3 we introduce the notion of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that to each selfadjoint operator A in an S-space (S, [ · , −]) we find an inner product · , − on S such that (S, · , −) is a Krein space with the same topology as (S, [ · , −]) and A is a selfadjoint operator in the Krein space (S, · , −), cf. Theorem 3.13. Moreover, if (( · , −), U ) is a Hilbert space realization, we show in Theorem 3.13 below that each spectral subspace of U related to a Borel subset of the unit circle which is symmetric with respect to the origin (i.e. x ∈ implies −x ∈ ) is invariant under A. Hence, in this paper we obtain the rather unexpected result: Each selfadjoint operator in an S-space is a selfadjoint operator in a Krein space with many invariant subspaces, provided the spectrum of the operator U from some Hilbert space realization (( · , −), U ) of (S, [ · , −]) is sufficiently rich, i.e., if it consists of more than two points.


1047

2. Definition and basic properties The following definition is taken from [23]. Definition 2.1. A complex linear space S with an inner product [ · , −], that is a mapping from S × S into C which is linear in the first variable and conjugate linear in the other, is said to be an S-space if there is a Hilbert space structure in S given by a positive definite inner product ( · , −) and if there is a unitary operator U in the Hilbert space (S, ( · , −)) such that [f, g] = (Uf, g)

for all f, g ∈ S.

We refer to [ · , −] as the inner product of S. The pair (( · , −), U ) is called a Hilbert space realization of (S, [ · , −]). Note, that the inner product [ · , −] is not Hermitian, in general. An S-space is a Krein space if and only if the operator U in Definition 2.1 is in addition selfadjoint in the Hilbert space (S, ( · , −)). For the theory of operators in Krein spaces we refer to [2,5]. Proposition 2.2. Let S be a complex linear space with an inner product [ · , −]. Then the pair (S, [ · , −]) is an S-space if and only if there exist a Hilbert space inner product ( · , −) on S and a bounded and boundedly invertible normal operator T in (S, ( · , −)) such that [f, g] = (Tf, g)

for all f, g ∈ S.

Proof. We define the operator U := T (T ∗ T )−1/2 and the inner product x, y :=

∗ 1/2 T T x, y ,

x, y ∈ S.

Since T is bijective, this is a Hilbert space inner product on S. From the relation (T (T ∗ T )−1/2 T ∗ )2 = T T ∗ it follows that ∗ 1/2 ∗ 1/2 −1/2 ∗ TT = T T = T T ∗T T .

(4)

Hence, for x, y ∈ S we obtain U x, y =

∗ 1/2 ∗ −1/2 −1/2 ∗ ∗ −1/2 T T T T T x, y = T T ∗ T T T T T x, y = [x, y]

and ∗ 1/2 ∗ −1/2 −1/2 T T T T T x, T T ∗ T y ∗ −1/2 ∗ −1/2 ∗ ∗ −1/2 T T T T x, T T T y = T T T ∗ −1/2 ∗ −1/2 ∗ y = T T T T x, y = T x, T T T ∗ 1/2 x, y = x, y, = T T

U x, Uy =

which shows that U is unitary in (S, ·,·) and (S, [ · , −]) is an S-space.

2

1048


Lemma 2.3. Let (S, [ · , −]) be an S-space. Then there exists a uniquely defined linear operator D : S → S such that [x, y] = [y, Dx]

for all x, y ∈ S.

(5)

If (( · , −), U ) is a Hilbert space realization of (S, [ · , −]), then D = U 2 . Proof. Let (( · , −), U ) be a Hilbert space realization of (S, [ · , −]). Then it is easily seen that U 2 satisfies the relation (5) (with D replaced by U 2 ). Let D : S → S be a linear operator satisfying (5). Then from [y, Dx] = [y, U 2 x] for all x, y ∈ S we conclude (Uy, Dx − U 2 x) = 0 for all x, y ∈ S. And since U is bijective, it follows that D = U 2 . 2 The topology of an S-space (S, [ · , −]) is given by the topology induced by the Hilbert space inner product ( · , −) of some Hilbert space realization of (S, [ · , −]). The following proposition states in particular that it does not depend on the choice of the Hilbert space realization, see also [24]. Proposition 2.4. Let (S, [ · , −]) be an S-space and assume that there are two Hilbert space realizations (( · , −)1 , U1 ) and (( · , −)2 , U2 ) with [f, g] = (U1 f, g)1 = (U2 f, g)2

for all f, g ∈ S.

Then ( · , −)1 and ( · , −)2 are equivalent and the Gram operator S, defined by (f, g)2 = (Sf, g)1

for f, g ∈ S,

is bounded, boundedly invertible and selfadjoint with respect to ( · , −)1 and with respect to ( · , −)2 . Moreover, the following statements hold: (i) U12 = U22 . (ii) The spectral measures of S in (S, ( · , −)1 ) and (S, ( · , −)2 ) coincide and we have S = U1 U2−1 = U1−1 U2 ,

and U1−1 SU1 = S −1 = U2−1 SU2 .

(6)

Hence, the operator S is unitarily equivalent to its inverse. (iii) The operators U1 and U2 are similar. We have U1 = S 1/2 U2 S −1/2 . Hence σ (U1 ) = σ (U2 ). Proof. Denote by · 1 and · 2 the norms induced by ( · , −)1 and ( · , −)2 , respectively, and set B1 := {y ∈ S: y 1 = 1}. Then, for y ∈ B1 the linear functional Fy := [· , y] = (U1 · , y)1 = (U2 · , y)2


1049

is continuous on both (S, ( · , −)1 ) and (S, ( · , −)2 ). For its corresponding operator norms

Fy L((S,( · ,−)1 ),C) and Fy L((S,( · ,−)2 ),C) , respectively, we obtain Fy L((S,( · ,−)1 ),C) = 1 and

Fy L((S,( · ,−)2 ),C) = y 2 . For all x ∈ S we have supy∈B1 |Fy (x)| x 1 < ∞. Due to the principle of uniform boundedness there exists some c ∈ (0, ∞) with sup Fy L((S,( · ,−)2 ),C) c.

y∈B1

This yields y 2 c y 1 for all y ∈ S. By interchanging the roles of · 1 and · 2 we obtain that these two norms are equivalent. Hence, by the well-known Lax–Milgram Theorem there exists a unique bounded linear operator S, selfadjoint in (S, ( · , −)1 ), such that (f, g)2 = (Sf, g)1

for f, g ∈ S.

It is boundedly invertible since Sfn 1 → 0 and fn 1 = 1 would imply fn 22 = (Sfn , fn )1 → 0 which contradicts the above proven fact that · 1 and · 2 are equivalent. For f, g ∈ S we have (Sf, g)2 = S 2 f, g 1 = (Sf, Sg)1 = (f, Sg)2 . Thus, S is also selfadjoint with respect to ( · , −)2 . Moreover, as ( · , −)1 and ( · , −)2 are positive definite, the operator S is uniformly positive. Now we will show (i)–(iii). Statement (i) follows directly from Lemma 2.3. The equality of the spectral measures E1 and E2 of S in (S, ( · , −)1 ) and (S, ( · , −)2 ) follows from the equivalence of the norms · 1 and · 2 and Stone’s formula (see, e.g., [7, XII.2]), E1 (a, b) = lim lim

1 δ→0+ →0+ 2πi

= E2 (a, b) ,

b−δ −1 −1 S − (λ + i) − S − (λ − i) dλ a+δ

(7)

where the limit is taken in the strong operator topology. As (Sf, g)1 = (f, g)2 = U2 U2−1 f, g 2 = U2−1 f, g = U1 U2−1 f, g 1 , we have S = U1 U2−1 and, with (i), we conclude S = U1−1 U12 U2−1 = U1−1 U2 . We will denote the adjoint with respect to ( · , −)1 by the symbol ∗1 and the adjoint with respect to ( · , −)2 by ∗2 . For f, g ∈ S we have (U2 f, g)2 = (SU2 f, g)1 = (U2 f, Sg)1 = f, U2∗1 Sg 1 = S −1 f, U2∗1 Sg 2 = f, S −1 U2∗1 Sg 2 , thus U2∗2 = S −1 U2∗1 S.

(8)

1050


This implies ∗ ∗ −1 S = S ∗1 = U1 U2−1 1 = U2−1 1 U1−1 = SU2∗2 S −1 U1−1 = SU2 S −1 U1−1 , hence, with S = U1 U2−1 we get S −1 = U1−1 SU1 . Replacing U1 by U2 and U2 by U1 also S −1 = U2−1 SU2 holds and formula (6) and (ii) are proved. By (ii) the square root of S in (S, ( · , −)1 ) and in (S, ( · , −)2 ) coincide. We denote the unique positive square root of the operator S by S 1/2 . Since, by (6), (U1−1 S −1/2 U1 )2 = U1−1 S −1 U1 = S, we have the relation S 1/2 = U1−1 S −1/2 U1 , which yields S 1/2 U2 S −1/2 = S 1/2 S −1 U1 S −1/2 = S −1/2 U1 S −1/2 = U1 and (iii) is proved.

2

3. Linear operators in S-spaces For the rest of this paper let (S, [ · , −]) be an S-space and let (( · , −), U ) be a fixed Hilbert space realization of (S, [ · , −]). In the following all topological notions are related to the Hilbert space topology given by ( · , −). Its topology is independent of the particular choice of a Hilbert space realization (see Proposition 2.4). Let T be a densely defined operator in a Hilbert space with a Hilbert space inner product ( · , −). As usual, we denote by T ∗ the adjoint of T with respect to ( · , −). As T is densely defined, T ∗ is unique. If T is, in addition, a closed operator, then T ∗ is densely defined, see, e.g., [13, Theorem III, §5.5]. Definition 3.1. Let A be a closed, densely defined operator in an S-space. An adjoint A with respect to [ · , −] is defined via the following relations: dom A := g ∈ S: ∃h ∈ S with [Af, g] = [f, h] for all f ∈ dom A , [Af, g] = f, A g for all f ∈ dom A and g ∈ dom A . Analogously, we define A via dom A := f ∈ S: ∃h ∈ S with [f, Ag] = [h, g] for all g ∈ dom A , [f, Ag] = Af, g for all g ∈ dom A and f ∈ dom A. In the following proposition (see [24]) we collect some of the properties of A and A. We provide here a short proof in order to make this exposition self-contained. Proposition 3.2. The operators A and A are closed, densely defined and satisfy dom A = U dom A∗ = dom A∗ U ∗

and A = U A∗ U ∗

(9)


1051

and dom A = U ∗ dom A∗ = dom A∗ U

and

A = U ∗ A∗ U.

(10)

Proof. Obviously, we have f ∈ dom(A∗ U ∗ ) if and only if U ∗ f ∈ dom A∗ which in turn holds if and only if f ∈ U dom A∗ . Hence U dom A∗ = dom(A∗ U ∗ ). Let g ∈ dom A . By Definition 3.1 we have for all f ∈ dom A f, U ∗ A g = f, A g = [Af, g] = Af, U ∗ g . Thus U ∗ g ∈ dom A∗ and U ∗ A ⊂ A∗ U ∗ . If g ∈ dom(A∗ U ∗ ), then we have for all f ∈ dom A f, U A∗ U ∗ g = f, A∗ U ∗ g = Af, U ∗ g = [Af, g]. Hence g ∈ dom A and A ⊂ U A∗ U ∗ . This gives U ∗ A = A∗ U ∗ and (9) is proved. The proof of (10) is similar and we omit it here. 2 Recall that for a densely defined operator T and a bounded operator X in a Hilbert space we have (see [25, Section 4.4]) (XT )∗ = T ∗ X ∗

and, if X is boundedly invertible,

(T X)∗ = X ∗ T ∗ .

(11)

Proposition 3.3. If A = A then AD = DA where D = U 2 . Proof. If A = A , then from Proposition 3.2 and (11) we conclude

∗ A = U A∗ U ∗ = U ∗ U A∗ U ∗ U = A,

and hence, with A = A , ∗ ∗ 2 A = A = U ∗ A U = U ∗ U ∗ A∗ U U = U ∗ AU 2 = D ∗ AD. And since D is unitary, the assertion follows.

2

Corollary 3.4. If A = A and U has no eigenvalues, then A does not have eigenvalues with finite geometric multiplicity. Proof. By Proposition 3.3 we have AD = DA. Assume that λ is an eigenvalue of A with finite geometric multiplicity. From AD = DA it follows that ker(A − λ) is invariant under D. Therefore, D (and hence U ) has eigenvalues. 2 Definition 3.5. A densely defined operator A in the S-space (S, [ · , −]) is called selfadjoint if A = A . We have the following characterization for selfadjointness of operators in S-spaces.

1052


Proposition 3.6. For a densely defined operator A in S the following assertions are equivalent: (i) (ii) (iii) (iv)

A = A , i.e., A is selfadjoint in (S, [ · , −]). U ∗ A = A∗ U ∗ . U A = A∗ U . A = A.

If one of these equivalent statements holds true we have f ∈ dom A

⇐⇒

U ∗ f ∈ dom A∗

⇐⇒

Uf ∈ dom A∗ .

(12)

Proof. The equivalence of (i) and (ii) follows from (9), the equivalence of (iii) and (iv) follows from (10). Assume that (ii) holds. For f ∈ dom A we conclude U ∗ f ∈ dom A∗ . This implies for f, g ∈ dom A: (f, U Ag) = A∗ U ∗ f, g = U ∗ Af, g = (Af, Ug) and we have Ug ∈ dom A∗ , hence U A ⊂ A∗ U . For the other inclusion, we observe by (ii) that dom A∗ = U ∗ dom A. For Ug ∈ dom A∗ and f ∈ dom A we have U ∗ f ∈ dom A∗ and ∗ U f, U ∗ A∗ Ug = f, A∗ Ug = (Af, Ug) = U ∗ Af, g = A∗ U ∗ f, g , thus g ∈ dom(A∗ )∗ = dom A. This gives U ∗ A∗ Ug = Ag and A∗ U ⊂ U A. This proves (iii). Assume that (iii) holds. For f ∈ dom A we conclude Uf ∈ dom A∗ . This gives for f, g ∈ dom A ∗ U Ag, f = (Ag, Uf ) = g, A∗ Uf = (g, U Af ) = U ∗ g, Af and we have U ∗ g ∈ dom A∗ , hence U ∗ A ⊂ A∗ U ∗ . For the other inclusion, we observe by (iii) that dom A∗ = U dom A. For U ∗ g ∈ dom A∗ and f ∈ dom A we have Uf ∈ dom A∗ and Uf, U A∗ U ∗ g = f, A∗ U ∗ g = Af, U ∗ g = (U Af, g) = A∗ Uf, g , thus g ∈ dom(A∗ )∗ = dom A. This gives A∗ U ∗ g = U ∗ Ag and A∗ U ∗ ⊂ U ∗ A. This proves (ii). Moreover, we have shown that (12) holds. 2 Proposition 3.7. Let A be a selfadjoint operator in the S-space (S, [ · , −]). Then the spectrum of A is symmetric with respect to the real axis. Proof. Since A = A = U A∗ U ∗ , cf. Proposition 3.2, the operator A is unitarily equivalent to its adjoint. Hence, σ (A) = σ (A∗ ) = {λ: λ ∈ σ (A)}. 2 Let A be a selfadjoint operator in the S-space (S, [ · , −]). If (S, [ · , −]) is a Krein space, then U is selfadjoint and thus σ (U ) = σp (U ) ⊂ {−1, 1}. It is well known that the spectrum of A may be rather arbitrary. For example, it can happen that σ (A) = C.


1053

Example 3.8. Assume that – in contrast to the Krein space case – σ (U ) consists of two eigenvalues λ1 , λ2 with λ1 = −λ2 , e.g., σ (U ) = {1, i}. Then σ (U 2 ) = {1, −1}, and since A commutes with D = U 2 by Proposition 3.3 the spectral subspaces of D are A-invariant. Since these coincide with the eigenspaces of U corresponding to 1 and i, respectively, we have A = A1 ⊕ Ai and U = I ⊕ iI with respect to the decomposition S = ker(U − 1) ⊕ ker(U − i). From the selfadjointness of A in (S, [ · , −]) we conclude that both A1 and Ai are selfadjoint with respect to the Hilbert space scalar product ( · , −) in ker(U − 1) and ker(U − i), respectively. Hence, A is selfadjoint in (S, ( · , −)). In particular its spectrum is real. This simple example shows that it is not necessarily “better” to know that an operator is selfadjoint in a Krein space than in an S-space. In fact, we will show in the following that every selfadjoint operator in an S-space is also selfadjoint in some Krein space. However, in general (if σ (U ) = {eit , −eit } for some t ∈ [0, π)) the selfadjointness in the S-space gives us more information about the operator. E.g., we automatically know a whole bunch of invariant subspaces of the operator – namely the spectral subspaces of D. Definition 3.9. Let G be a bounded selfadjoint operator in the Hilbert space (S, ( · , −)). A closed and densely defined linear operator T in S will be called G-symmetric if GT ⊂ (GT)∗ . The operator T is called G-selfadjoint if GT = (GT)∗ . In the following we will deal with the operators G(t) :=

1 it e U − e−it U ∗ , 2i

t ∈ [0, π).

It is easily seen that all these operators are bounded selfadjoint operators in the Hilbert space (S, ( · , −)). We have G(0) = Im U and G(π/2) = Re U . Moreover, the operator G(t) can be factorized in the following way G(t) =

eit eit ∗ 2 U U − e−2it = U ∗ U − e−it U + e−it . 2i 2i

Therefore, G(t) is boundedly invertible if and only if e−it , −e−it ∈ ρ(U ). In this case (S, (G(t) · , −)) is a Krein space. Proposition 3.10. Let A be a selfadjoint operator in the S-space (S, [ · , −]). Then A is G(t)symmetric for all t ∈ [0, π). If for some t ∈ [0, π) we have e−it , −e−it ∈ ρ(U ), then the operator A is G(t)-selfadjoint. Proof. Let t ∈ [0, π). Then by Proposition 3.6 we have 1 it 1 it e U − e−it U ∗ A = e U A − e−it U ∗ A 2i 2i 1 it ∗ e A U − e−it A∗ U ∗ = 2i ∗ ⊂ A∗ G(t) = G(t)A .

G(t)A =

This shows that A is G(t)-symmetric.

1054


We have by Proposition 3.3 AD = DA, therefore for each complex number λ (D − λ)A ⊂ A(D − λ).

(13)

We will show that for λ ∈ ρ(D) equality holds, (D − λ)A = A(D − λ).

(14)

Let λ ∈ ρ(D). We have to show dom(A(D − λ)) ⊂ dom A. Consider the Hilbert space SA := (dom A, ( · , −)A ), where the inner product ( · , −)A is defined by (f, g)A := (f, g) + (Af, Ag),

f, g ∈ dom A.

Due to AD = DA the linear manifold dom A is D-invariant. Hence, define DA : S A → S A ,

DA f := Df,

f ∈ dom A.

For f, g ∈ SA we have (DA f, DA g)A = (Df, Dg) + (ADf, ADg) = (f, g) + (DAf, DAg) = (f, g)A and DA is an isometric operator in SA . Assume that there exists z ∈ SA with (DA f, z)A = 0 for all f ∈ SA . That gives − f, D ∗ z = (DAf, Az) = Af, D ∗ Az for all f ∈ SA and, hence, D ∗ Az ∈ dom A∗ with A∗ D ∗ Az = −D ∗ z. By (11) and AD = DA we obtain −D ∗ z = (DA)∗ Az = (AD)∗ Az = D ∗ A∗ Az. It follows A∗ Az = −z and 0 (A∗ Az, z) = −(z, z) 0. Therefore z = 0 and DA has a dense range in SA . The operator DA is a unitary operator in SA . For λ ∈ ρ(D) \ {0}, we have −1 −1 − λ = ker DA λ λ−1 − DA = {0}, ran(DA − λ)⊥A = ker DA where ⊥A denotes the orthogonal complement in SA with respect to ( · , −)A . Hence, for λ ∈ ρ(D), the operator DA − λ has a dense range in SA . In order to show (14) let f ∈ dom(A(D − λ)). Then (D − λ)f ∈ dom A. As ran(DA − λ) is dense in SA , there exists a sequence (fn ) in dom A such that (DA − λ)fn − (D − λ)f → 0 as n → ∞. A From this we conclude fn → f

and A(D − λ)fn → A(D − λ)f

as n → ∞


1055

(in S). But fn ∈ dom A and from (13) it follows that fn → f

and Afn → (D − λ)−1 A(D − λ)f

as n → ∞.

Now, it is a consequence of the closedness of A that f ∈ dom A and (D − λ)Af = A(D − λ)f . This shows (14). The selfadjointness of A in (S, [ · , −]) is equivalent to A∗ U ∗ = U ∗ A, cf. Proposition 3.6. With ±e−it ∈ ρ(U ) we have e−2it ∈ ρ(D). This and (14) yield eit eit ∗ ∗ A U D − e−2it = U ∗ A D − e−2it 2i 2i it e = U ∗ D − e−2it A = G(t)A, 2i

A∗ G(t) =

which is the G(t)-selfadjointness of A.

2

Note that in general the operator A in Proposition 3.11 is not G(t)-selfadjoint. For example let U := iI and suppose that A is unbounded. Then G(π/2) = 0 and G(π/2)A is the restriction of the zero operator to dom A, whereas (G(π/2)A)∗ equals the zero operator on S. Hence, in this case, A is not G(π/2)-selfadjoint. If G(t) is boundedly invertible, then the space S equipped with the inner product (G(t) · , −) is a Krein space. The following theorem follows immediately from Proposition 3.10. Theorem 3.11. Let A be a selfadjoint operator in the S-space (S, [ · , −]). If for some t ∈ [0, π) we have e−it , −e−it ∈ ρ(U ), then the operator A is selfadjoint in the Krein space (S, (G(t) · , −)). If in the situation of Theorem 3.11 the operator U satisfies some additional assumptions, more can be said about the spectrum of A. Theorem 3.12. Let A be a selfadjoint operator in the S-space (S, [ · , −]) and assume that there ˙ T2 be a decomposition of the unit is some t ∈ [0, π) such that e−it , −e−it ∈ ρ(U ). Let T = T1 ∪ circle, where and T2 := eis : −t + π s < −t + 2π . T1 := eis : −t s < −t + π If T1 ∩ σ (U ) = ∅ or T2 ∩ σ (U ) = ∅ then A is selfadjoint in the Hilbert space (S, (G(t) · , −)). In particular, σ (A) ⊂ R. If T1 ∩ σ (U ) or T2 ∩ σ (U ) consists of finitely many κ isolated eigenvalues (counted with multiplicity) of U , then the non-real spectrum of A in the open upper half-plane consists of at most κ isolated eigenvalues with finite algebraic multiplicities (counted with multiplicity), σ (A) \ R = {λ1 , λ1 , λ2 , λ2 , . . . , λκ0 , λκ0 } ⊂ σp (A), for some κ0 with 0 κ0 κ.

1056


Proof. We define

:= eit U. U

). The operator A is selfadjoint in the S-space (S, [ · , −]∼ ), where [ · , −]∼ is Then ±1 ∈ ρ(U given by

f, g) [f, g]∼ := (U

for all f, g ∈ S.

· , ·)). If T1 ∩ σ (U ) = ∅ then By Theorem 3.11, A is selfadjoint in the Krein space (S, (Im U

is a uniformly negative operator in the Hilbert space (S, ( · , −)), and hence A is a selfIm U

· , ·)). A similar argument holds for the case adjoint operator in the Hilbert space (S, −(Im U T2 ∩ σ (U ) = ∅ and the first assertion of the theorem is proved. If T1 ∩ σ (U ) consists of finitely many isolated eigenvalues of U with finite multiplicity then

is a bounded and boundedly invertible selfadjoint operator in the Hilbert space (S, ( · , −)). Im U

corresponding to the positive real numbers is finite Moreover, the spectral subspace of Im U

· , ·)) and dimensional. Therefore A is a selfadjoint operator in the Pontryagin space (S, (Im U the second assertion of the theorem follows from well-known properties of selfadjoint operators in Pontryagin spaces, see, e.g., [2,5]. Similar arguments apply if T2 ∩ σ (U ) consists of finitely many isolated eigenvalues of U . 2 The following theorem is the main result of this paper. It shows that the notions of S-space selfadjointness and Krein space selfadjointness coincide. Theorem 3.13. Let A be a selfadjoint operator in the S-space (S, [ · , −]). Then there exists a Krein space inner product · , − such that A is selfadjoint in the Krein space (S, · , −). Moreover, if EU denotes the spectral measure of U and if is a Borel subset of the unit circle T with the property that λ ∈ implies −λ ∈ , then the spectral subspace EU ()S is an invariant subspace for A. Proof. We choose some ε ∈ (0, π/2) and define 1 := eit : t ∈ (−ε, ε) ∪ −eit : t ∈ (−ε, ε) ,

2 := T \ 1 .

Let S1 and S2 be the spectral subspaces of U corresponding to 1 and 2 , respectively, i.e. S1 = EU (1 )S

and S2 = EU (2 )S.

Then we have S = S1 ⊕ S2 . We define the sets 21 := eit : t ∈ (−2ε, 2ε)

and 22 := T \ 21 = z2 : z ∈ 2 .


1057

If EU 2 denotes the spectral measure of U 2 and h : C → C denotes the function given by h(z) = z2 , then we deduce from the properties of the functional calculus for unitary operators for j = 1, 2 EU 2 2j = 12 U 2 = (12 ◦ h)(U ) = 1h−1 (2 ) (U ) = EU (j ), j

j

j

where 1 is the indicator function corresponding to a Borel set and h−1 (2j ) denotes the preimage of 2j under h. Therefore, the spectral subspace of D = U 2 corresponding to 2j coincides with Sj , j = 1, 2. For λ ∈ ρ(D) the operator (D − λ)−1 commutes with A, cf. (14). With some obvious modifications due to the fact that U is a unitary operator, the projector EU (j ), j = 1, 2, can be written in a similar form as in (7). From this, we conclude EU (j )A ⊂ AEU (j ).

(15)

Hence, for x ∈ dom A we have EU (j )x ∈ dom A and dom A = (S1 ∩ dom A) ⊕ (S2 ∩ dom A). Moreover, if x ∈ Sj ∩ dom A then with (15) Ax = EU (j )Ax, which implies that the subspaces S1 and S2 are A-invariant. Thus, with respect to the decomposition S = S1 ⊕ S2 the operators A and U decompose as A = A1 ⊕ A2 and U = U1 ⊕ U2 , where Aj = A|Sj and Uj = U |Sj , j = 1, 2. It is easy to see that A1 is selfadjoint in the S-space (S1 , (U1 ·, −)) and that A2 is selfadjoint in the S-space (S2 , (U2 ·, −)). Since i, −i ∈ ρ(U1 ) and 1, −1 ∈ ρ(U2 ), it follows from Theorem 3.11 that there are Krein space inner products · , −1 and · , −2 in S1 and S2 , respectively, such that Aj is selfadjoint in the Krein space (Sj , · , −j ), j = 1, 2. Hence, A is obviously selfadjoint in the Krein space (S, · , −), where · , − is given by x, v := x1 , y1 1 + x2 , y2 2 , x = x1 + x 2 , y = y 1 + y 2 , x 1 , x 2 ∈ S 1 , x 2 , y 2 ∈ S 2 .

2

Remark 3.14. Each Krein space is also an S-space, hence, obviously, every selfadjoint operator in a Krein space is simultaneously selfadjoint in an S-space. Theorem 3.13 shows that in a sense the contrary is true as well. Hence, for each selfadjoint operator A in an S-space S we find an inner product which turns S into a Krein space and A into a selfadjoint operator with respect to this inner product. In addition, as revealed in the proof of Theorem 3.13, all spectral subspaces of U which correspond to Borel sets symmetric with respect to z → −z are invariant subspaces of A.

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Example 3.15. As an illustration of Theorem 3.13 we consider a simple example with 2 × 2 matrices. Let U be unitary in C2 and choose an orthonormal basis of C2 such that the corresponding matrix is diagonal with entries z1 , z2 ∈ T. A matrix with entries a, b, c, d ∈ C which is selfadjoint in the S-space given by U has to satisfy a b z1 0 a c z1 0 = , c d b d 0 z2 0 z2 cf. Proposition 3.6, part (iii). We assume cb = 0. From this we see that a and d are real, z1 = ±z2 and b = ±c. Hence, either the matrix is selfadjoint (in the case z1 = z2 ) or, if z1 = −z2 , we have b = −c and the matrix is selfadjoint in the (finite dimensional) Krein space with fundamental symmetry 1 0 J= . 0 −1 4. Concluding remarks S-spaces are Hilbert spaces with an additional inner product given by a unitary Gramian U . Krein spaces are special cases of S-spaces as their Gramian can be chosen to be selfadjoint and simultaneously unitary. From this point of view, the class of S-spaces is larger than the class of Krein spaces. It is the main result of this paper that the class of selfadjoint operators in S-spaces is not larger than the corresponding class in Krein spaces. Moreover, Theorem 3.13 reveals an interesting fact: A selfadjoint operator in an S-space having a Gramian U with spectrum larger than the set {−1, 1} has invariant subspaces – a fact which is not known a priori for selfadjoint operators in Krein spaces. An interesting, and so far unanswered, question is: Which class of selfadjoint operators in Krein spaces are selfadjoint in an S-space with a Gramian U which has a spectrum larger than {−1, 1} (and, hence, gives rise to many invariant subspaces)? Acknowledgments The first and the second author would like to acknowledge an assistance of the EU Sixth Framework Programme for the Transfer of Knowledge “Operator theory methods for differential equations” (TODEQ) # MTKDCT-2005-030042. The first author is grateful to the Deutsche Forschungsgesellschaft (DFG) for their support under grant BE 3765/5-1 TR 903/4-1. Moreover, the work of the second author was partially supported by the MNiSzW grant N201 026 32/1350. References [1] T.Ya. Azizov, About compact operators which are selfadjoint with respect to a degenerated indefinite inner product, in: Mat. Issled., vol. 26, 1972, pp. 237–240, Kishinyov, UII:4 (in Russian). [2] T.Ya. Azizov, I.S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, Ltd., Chichester, 1989. [3] T.Ya. Azizov, L.I. Soukhotcheva, Linear operators in almost Krein spaces, in: Oper. Theory Adv. Appl., vol. 175, 2007, pp. 1–11. [4] P. Binding, R. Hryniv, Full and partial range completeness, in: Oper. Theory Adv. Appl., vol. 130, 2002, pp. 121– 133.


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[5] J. Bognár, Indefinite Inner Product Spaces, Springer-Verlag, New York, Heidelberg, 1974. [6] V. Bolotnikov, C.-K. Li, P. Meade, C. Mehl, L. Rodman, Shells of matrices in indefinite inner product spaces, Electron. J. Linear Algebra 9 (2002) 67–92. [7] N. Dunford, J.T. Schwartz, Linear Operators. Part II: Spectral Theory, John Wiley & Sons, 1963. [8] M.-A. Henn, C. Mehl, C. Trunk, Hyponormal and strongly hyponormal matrices in inner product spaces, Linear Algebra Appl. 433 (2010) 1055–1076. [9] M. Kaltenbäck, H. Winkler, H. Woracek, Almost Pontryagin spaces, in: Oper. Theory Adv. Appl., vol. 160, 2005, pp. 253–371. [10] M. Kaltenbäck, H. Woracek, Selfadjoint extensions of symmetric operators in degenerated inner product spaces, Integral Equations Operator Theory 28 (1997) 289–320. [11] M. Kaltenbäck, H. Woracek, On extensions of Hermitian functions with a finite number of negative squares, J. Operator Theory 40 (1998) 147–183. [12] M. Kaltenbäck, H. Woracek, The Krein formula for generalized resolvents in degenerated inner product spaces, Monatsh. Math. 127 (1999) 119–140. [13] T. Kato, Perturbation Theory for Linear Operators, second edition, Springer, 1976. [14] M.G. Krein, Introduction to the theory of indefinite J -spaces and to the theory of operators in those spaces, in: Amer. Math. Soc. Transl., vol. 93, 1970, pp. 103–176. [15] P. Lancaster, A.S. Markus, P. Zizler, The order of neutrality for linear operators on inner product spaces, Linear Algebra Appl. 259 (1997) 25–29. [16] H. Langer, A.S. Markus, V.I. Matsaev, Locally definite operators in indefinite inner product spaces, Math. Ann. 308 (1997) 405–424. [17] H. Langer, R. Mennicken, C. Tretter, A self-adjoint linear pencil Q − λP of ordinary differential operators, Methods Funct. Anal. Topology 2 (1996) 38–54. [18] C. Mehl, A.C.M. Ran, L. Rodman, Semidefinite invariant subspaces: degenerate inner products, in: Oper. Theory Adv. Appl., vol. 149, 2004, pp. 467–486. [19] C. Mehl, L. Rodman, Symmetric matrices with respect to sesquilinear forms, Linear Algebra Appl. 349 (2002) 55–75. [20] C. Mehl, C. Trunk, Normal matrices in inner product spaces, in: Oper. Theory Adv. Appl., vol. 175, 2007, pp. 193– 209. [21] F. Philipp, C. Trunk, G-selfadjoint operators in almost Pontryagin spaces, in: Oper. Theory Adv. Appl., vol. 188, 2008, pp. 207–235. [22] B.C. Ritsner, The Theory of Linear Relations, No. 846-82, Dep. VINITI, Voronezh, 1982 (in Russian). [23] F.H. Szafraniec, Two-sided weighted shifts are almost Krein normal, in: Oper. Theory Adv. Appl., vol. 188, 2008, pp. 245–250. [24] F.H. Szafraniec, Dissymmetrising inner product spaces, manuscript, in preparation. [25] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, New York, Heidelberg, Berlin, 1980. [26] H. Woracek, An operator theoretic approach to degenerated Nevanlinna–Pick interpolation, Math. Nachr. 176 (1995) 335–350. [27] H. Woracek, Resolvent matrices in degenerate inner product spaces, Math. Nachr. 213 (2000) 155–175.


Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation Felipe Linares a,∗ , Ademir Pastor b a Instituto Nacional de Matemática Pura e Aplicada – IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro,

RJ, Brazil b IMECC-UNICAMP, Departamento de Matemática, Rua Sérgio Buarque de Holanda 651, 13083-859, Campinas,

SP, Brazil Received 29 December 2009; accepted 7 November 2010 Available online 16 November 2010 Communicated by I. Rodnianski

Abstract This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov–Kuznetsov equation, namely,

ut + ∂x u + uk ux = 0, u(x, y, 0) = u0 (x, y).

(x, y) ∈ R2 , t > 0,

For 2 k 7, the IVP above is shown to be locally well posed for data in H s (R2 ), s > 3/4. For k 8, local well-posedness is shown to hold for data in H s (R2 ), s > sk , where sk = 1 − 3/(2k − 4). Furthermore, for k 3, if u0 ∈ H 1 (R2 ) and satisfies u0 H 1 1, then the solution is shown to be global in H 1 (R2 ). √ For k = 2, if u0 ∈ H s (R2 ), s > 53/63, and satisfies u0 L2 < 3 ϕL2 , where ϕ is the corresponding ground state solution, then the solution is shown to be global in H s (R2 ). © 2010 Elsevier Inc. All rights reserved. Keywords: Zakharov–Kuznetsov equation; Local well-posedness; Global well-posedness


E-mail addresses: [email protected] (F. Linares), [email protected] (A. Pastor). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.005

F. Linares, A. Pastor / Journal of Functional Analysis 260 (2011) 1060–1085

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1. Introduction This paper is concerned with the initial value problem (IVP) associated with the generalized Zakharov–Kuznetsov (gZK) equation in two space dimensions, namely,

ut + ∂x u + uk ux = 0, u(x, y, 0) = u0 (x, y),

(x, y) ∈ R2 , t > 0,

(1.1)

where u is a real-valued function, and k 2 is an integer number. When k = 1, Eq. (1.1) (termed simply as ZK equation) was formally deduced by Zakharov and Kuznetsov [17] (see also [14] and references therein) as an asymptotic model to describe the propagation of nonlinear ion-acoustic waves in a magnetized plasma. Eq. (1.1) may also be seen as a natural, two-dimensional extension of the one-dimensional generalized Korteweg–de Vries (KdV) equation ut + uxxx + uk ux = 0. The aim of this paper is to establish local and global well-posedness to the IVP (1.1). The notion of well-posedness will be the usual one in the context of nonlinear dispersive equations, that is, it includes existence, uniqueness, persistence property, and continuous dependence upon the data. Before describing our results, let us recall what has been done so far regarding the gZK equation. In [8], Faminskii considered the IVP associated with the ZK equation. He showed local and global well-posedness for initial data in H m (R2 ), m 1 integer. In [2], Biagioni and Linares dealt with the IVP associated with the modified ZK equation (i.e. that one in (1.1) with k = 2). They proved local and global well-posedness for data in H 1 (R2 ). Linares and Pastor [13] studied the IVP associated with both the ZK and modified ZK equations. They improved the results in [2,8] by showing that both IVP’s are locally well posed for initial data in H s (R2 ), s > 3/4. Moreover, by using the techniques introduced in Birnir et al. [3,4], they proved that the IVP associated with the modified ZK equation is ill posed, in the sense that the flow-map data-solution is not uniformly continuous, for data in H s (R2 ), s 0. It should be noted that the method employed in [2,8,13] to show local well-posedness, was the one developed by Kenig, Ponce, and Vega [11] (when dealing with the generalized KdV equation), which combines smoothing effects, Strichartz-type estimates, and a maximal function estimate together with the Banach contraction principle. 1 2 It is worth √ mentioning that in [13], the authors proved that if u0 ∈ H (R ) and satisfies u0 L2 < 3 ϕL2 , where ϕ is the unique positive radial solution (hereafter refereed to as the ground state solution) of the elliptic equation −ϕ + ϕ − ϕ 3 = 0,

(1.2)

1 2 then the solution u(t) of (1.1), √ with k = 2, may be globally extended in H (R ). It1should be pointed out that if u0 L2 3 ϕL2 , the question of showing whether or not the H -solution, with u(0) = u0 , blows up in finite time is currently open. It should also be observed that questions of existence and orbital stability of solitary-wave solutions, and unique continuation were addressed, respectively, by de Bouard [7], and Panthee [15]. In [7], the author proved that the positive radially symmetric solitary waves are orbitally

1062


stable if k = 1, and orbitally unstable otherwise. In [15], the author established that if the solution of the ZK equation is sufficiently regular and is compactly supported in a nontrivial time interval, then it vanishes identically. Now, let us describe our results. We first recall that the quantities (1.3) I1 u(t) = u2 (t) dx dy R2

and I2 u(t) =

u2x (t) + u2y (t) − R2

2 uk+2 (t) dx dy (k + 1)(k + 2)

(1.4)

are conserved by the flow of the gZK equation, that is, I1 (u(t)) = I1 (u(0)) and I2 (u(t)) = I2 (u(0)), as long as the solution exists. Thus, these quantities could lead local rough solutions to global ones. So, it is natural to ask what would be the largest Sobolev space where local wellposedness holds. To answer this question, we perform a scaling argument, by noting that if u solves (1.1), with initial data u0 , then uλ (x, y, t) = λ2/k u λx, λy, λ3 t also solves (1.1), with initial data uλ (x, y, 0) = λ2/k u0 (λx, λy), for any λ > 0. Hence, u(·,·, 0)

H˙ s

= λ2/k+s−1 u0 H˙ s ,

(1.5)

where H˙ s = H˙ s (R2 ) denotes the homogeneous Sobolev space of order s. As a consequence of (1.5), the scale-invariant Sobolev space for the gZK equation is H sc (k) (R2 ), where sc (k) = 1 − 2/k. Therefore, one expects that the Sobolev spaces H s (R2 ) for studying the well-posedness of (1.1) are those with indices s > sc (k). We divide the paper into two parts. The first one deals with local and global well-posedness in the case k 3, whereas the second part is devoted to establishing the global well-posedness in the case k = 2 (the critical case). Our first result regards local well-posedness of (1.1) for 3 k 7. More precisely, we prove the following. Theorem 1.1. Assume 3 k 7. For any u0 ∈ H s (R2 ), s > 3/4, there exist T = T (u0 H s ) > 0 and a unique solution of the IVP (1.1), defined in the interval [0, T ], such that u ∈ C [0, T ]; H s R2 , s D ux ∞ 2 + D s ux ∞ 2 < ∞, x y L L L L x

x

yT

(1.6) (1.7)

yT

uLpk L∞ + ux L12/5 L∞ < ∞,

(1.8)

uL4x L∞ < ∞,

(1.9)

T

xy

T

xy

and yT


1063

where pk = 12(k−1) 7−12γ and γ ∈ (0, 1/12). Moreover, for any T ∈ (0, T ) there exists a neighs 2 ˜ from W into the class defined by borhood W of u0 in H (R ) such that the map u˜ 0 → u(t) (1.6)–(1.9) is smooth.

To prove Theorem 1.1 we use the technique introduced by Kenig, Ponce, and Vega [11] to study the IVP associated with the KdV equation. We point out that the proof of Theorem 1.1 in [13] does not apply to the case k 3. Here, instead of using an L2x maximal function estimate, we use an L4x one (see Proposition 2.6 below). However, the main new ingredient is the embedding given in Lemma 2.4. Observe that for 3 k 7, we obtain sc (k) < 3/4, so that, our result does not reach the indices conjectured by the scaling argument. Next, we deal with the case k 8. Our main result in this case reads as follows. 3 . For any u0 ∈ H s (R2 ), s > sk , there exist T = Theorem 1.2. Let k 8 and sk = 1 − 2(k−2) T (u0 H s ) > 0 and a unique solution of the IVP (1.1), defined in the interval [0, T ], such that

u ∈ C [0, T ]; H s R2 , s D ux ∞ 2 + D s ux ∞ 2 < ∞, x y L L L L x

u

p˜

x

yT

LT k L∞ xy

(1.11)

yT

+ ux L12/5 L∞ < ∞, T

(1.10)

(1.12)

xy

and uL4x L∞ < ∞, yT

(1.13)

where p˜ k = 2(k−2) 1−2γ and γ > 0 is sufficiently small. Moreover, for any T ∈ (0, T ) there exists a ˜ from U into the class defined by neighborhood U of u0 in H s (R2 ) such that the map u˜ 0 → u(t) (1.10)–(1.13) is smooth.

The proof of Theorem 1.2 is very close to that of Theorem 1.1. In this case, because of the scaling argument, we do not expect to prove local well-posedness for all s > 3/4. Indeed, note that, for k 8, we always have, sk sc (k) 3/4. Moreover, sk = sc (k) if and only if k = 8. Also observe that in the case k = 8, we get s8 = sc (8) = 3/4. This implies that our result, for k = 8, is “almost” sharp, but for k > 8 there is still a gap between the scaling and our results, which is evidenced by the theorem below. Theorem 1.3. Let k 3. Then, the IVP (1.1) is ill posed for data in H sc (k) (R2 ), sc (k) = 1 − 2/k, in the sense that the map data-solution is not uniformly continuous. Note that the well-posedness sense in Theorem 1.3 requires additional smoothness of the map data-solution, and not only that of mere continuity. However, this is not too strong because as affirmed in Theorems 1.1 and 1.2, the map data-solution, in those cases, is sufficiently smooth. The argument to establish Theorem 1.3 is similar to that of Theorem 1.2 in [13], and goes back to the techniques introduced in [3] and [4]. One of the main difficulties to obtain possible sharp results is the lack of some needed estimates in mixed spaces. There is not an available Leibniz rule for fractional derivatives in mixed

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F. Linares, A. Pastor / Journal of Functional Analysis 260 (2011) 1060–1085 q

p

spaces Lx Ly LrT for instance. This makes a difference with the analysis for the generalized KdV for k 4. Another point we should remark is the gain of derivatives we have for the Strichartz estimate for the linear group. We only get 1/4 − derivatives, 0 < 1 (see Lemma 2.2 below) in contrast to the gain of exactly 1/4 derivatives of the KdV linear group. Because of this we also loose some regularity. We now turn our attention to the global well-posedness issue. Our main result is proved under a smallness condition on the initial data. Theorem 1.4. Let k 3. Let u0 ∈ H 1 (R2 ) and assume u0 H 1 1, then the local solutions given in Theorems 1.1 and 1.2 can be extended to any time interval [0, T ]. Theorem 1.4 is proved in a standard fashion, and relies on a combination of the conserved quantities (1.3) and (1.4) with the Gagliardo–Nirenberg interpolation inequality. Next, we will focus on the second part of the paper. As we already mentioned, the local well-posedness of (1.1) with k = 2 for initial data in H s (R2 ), s > 3/4, was obtained in [13]. Furthermore, we announced that√(1.1) was globally well posed for initial data u0 in H s (R2 ), s > 19/21 satisfying u0 L2 < 3ϕL2 , where ϕ is the ground state solution of Eq. (1.2). In the present paper, we reaffirm that this result holds, however, we slightly modify the proof of the local well-posedness in [13], to improve that announced result. More precisely, we prove the following. Theorem 1.5. Let k = 2. Let u0 ∈ H s (R2 ), s > 53/63, and assume that u0 L2 < where ϕ is the ground state solution of Eq. (1.2), then (1.1) is globally well posed.

√

3ϕL2 ,

The method we use to prove Theorem 1.5 is that one developed in [9] and [10], which combines the smoothing effects for the solution of the linear problem with the iteration process introduced by Bourgain [5]. Since we are in the critical case, as in [10], controlling the L2 -norm of the initial data could bring some difficulty. Nevertheless, with a suitable decomposition of the initial data into low and high frequencies, we are able to handle this. Let us highlight what enables us to improve the global result announced in [13]. The reason is quite simple. In [13], to apply the contraction principle, we get a factor of T 2/3 in front of the estimates for the nonlinear terms. Here, modifying a little bit the function spaces, we get a factor of T 5/12− (see proof of Theorem 4.1), this in turn, is relevant to the method described in [9,10]. As we have pointed out in [13], the Fourier restriction method does not seem to work to proving a local well-posedness result for the generalized ZK equation. So, it is not clear that the I-method, introduced by Colliander et al. [6], works either to establish a better global wellposedness result. The paper is organized as follows. In Section 2, we state the results concerned with the linear problem associated with (1.1). In Section 3, we deal with the case k 3. We show our local (and global) well-posedness result as well as the ill-posedness one. Finally, in Section 4 we establish the global well-posedness for k = 2 announced in Theorem 1.5. Notation. The symbol a± means that there exists an ε > 0, small enough, such that a± = α a ± ε. For α ∈ C, the operators Dxα and Dyα are defined via Fourier transform by D x f (ξ, η) =


1065

α ˆ α |ξ |α fˆ(ξ, η) and D y f (ξ, η) = |η| f (ξ, η). The mixed space–time norm is defined as (for 1 p, q, r < ∞)

+∞ +∞ T f Lpx Lqy Lr = T

−∞

−∞

f (x, y, t) r dt

q/r

p/q dy

1/p dx

,

0

with obvious modifications if either p = ∞, q = ∞ or r = ∞. 2. Preliminary results In this section, we recall some results concerning the linear IVP associated to the gZK equation, which will be useful throughout the paper. Consider the linear IVP ut + ∂x u = 0, (x, y) ∈ R2 , t ∈ R, (2.1) u(x, y, 0) = u0 (x, y). The solution of (2.1) is given by the unitary group {U (t)}∞ t=−∞ such that u(t) = U (t)u0 (x, y) =

ei(t (ξ

3 +ξ η2 )+xξ +yη)

uˆ 0 (ξ, η) dξ dη.

(2.2)

R2

We begin by remembering the smoothing effect of Kato type, and the Strichartz-type estimates. Lemma 2.1 (Smoothing effect). Let u0 ∈ L2 (R2 ). Then, ∂x U (t)u0

2 L∞ x LyT

cu0 L2xy

(2.3)

and t ∂x U −t f ·,·, t dt

cf L1 L2 .

L2xy

0

x

yT

(2.4)

Moreover, the same still holds if we replace ∂x with ∂y . Proof. See Faminskii [8, Theorem 2.2] for the proof of (2.3). The inequality (2.4) is just the dual version of (2.3). 2 Proposition 2.2 (Strichartz-type estimates). Let 0 ε < 1/2 and 0 θ 1. Then, the group {U (t)}∞ t=−∞ satisfies θε/2 Dx U (t)f

q

p

Lt Lxy

where p =

2 1−θ

and

2 q

=

θ(2+ε) 3 .

cf L2xy ,

(2.5)

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Proof. See Linares and Pastor [13, Proposition 2.4].

2

The next lemmas are useful to recover the “loss of derivative” present in the nonlinear term of the gZK equation. Lemma 2.3. Let 0 ε < 1/2. Then, the group {U (t)}∞ t=−∞ satisfies U (t)f p ∞ cT γ1 Dx−ε/2 f 2 , L L L xy

T

where 1 p

6 2+ε

and γ1 =

1 p

−

2+ε 6 .

(2.6)

xy

In particular, if 0 < T 1, then

U (t)f

12/5 ∞ Lxy

LT

−ε/2 cDx f L2 .

(2.7)

xy

Proof. By using Hölder’s inequality (in t), we get U (t)f

cT γ1 U (t)f Lq L∞ ,

p

LT L∞ xy

T

xy

where p1 = γ1 + q1 . Thus, taking θ = 1 and q = 6/(2 + ε) in Proposition 2.2, the estimate (2.6) then follows. 2 Lemma 2.4. Let δ > 0. Then, c f Lpxyδ + Dxδ f Lpδ + Dyδ f Lpδ , f L∞ xy xy

xy

where pδ > 2/δ. In particular, pδ → ∞ as δ → 0. Proof. See Kenig and Ziesler [12, Lemma 3.4].

2

Lemma 2.5. Let 0 < δ < 1. Assume 1 − δ < θ < 1 and 2 r 3/θ . Then, U (t)f for some γ2 =

1 r

−

θ 3

LrT L∞ xy

cT γ2 f L2xy + Dxδ f L2 + Dyδ f L2 , xy

xy

0.

Proof. We first note that taking ε = 0 in Proposition 2.2, we obtain U (t)f

3/θ

2/(1−θ)

LT Lxy

cf L2xy .

(2.8)

Now, applying Hölder’s inequality followed by Lemma 2.4, we deduce U (t)f

LrT L∞ xy

cT γ2 U (t)f Lr L∞ T xy

γ2 U (t)f Lr Lpδ + Dxδ U (t)f Lr Lpδ + Dyδ U (t)f Lr Lpδ . cT T

xy

T

xy

T

xy


1067

If we now choose r = 3/θ and pδ = 2/(1 − θ ), then an application of (2.8) yields the affirmation. Note that pδ > 2/δ implies 1 − δ < θ , and γ2 0 implies r 3/θ . This completes the proof of the lemma. 2 As we commented before, Kenig, Ponce, and Vega’s technique combines the smoothing effect and Strichartz estimate with a maximal function estimate. Here, we present the L2x and L4x maximal function estimates that we will use in our arguments. Proposition 2.6 (Maximal function). (i) For any s1 > 1/4, r1 > 1/2 and 0 < T 1, we have U (t)f

L4x L∞ yT

c(1 + Dx )s1 (1 + Dy )r1 f L2 . xy

(ii) For any s > 3/4, we have U (t)f

L2x L∞ yT

s , c(s, T )f Hxy

where c(s, T ) is a positive constant depending only on T and s. Proof. See Linares and Pastor [13, Proposition 1.5] for part (i), and Faminskii [8, Theorem 2.4] for part (ii). 2 Corollary 2.7. For any s > 3/4 and 0 < T 1, we have U (t)f

L4x L∞ yT

s . cf Hxy

Proof. Let s1 and r1 be as in Proposition 2.6(i). In view of Plancherel’s theorem, we obtain (1 + Dx )s1 (1 + Dy )r1 f 2 2 = L

2 s r

1 + |ξ |2 1 1 + |η|2 1 fˆ(ξ, η) dξ dη

xy

R2

c

2

1 + |ξ |2s1 + |η|2r1 + |ξ |2s1 |η|2r1 fˆ(ξ, η) dξ dη

R2

c

2

1 + |ξ |2s1 + |η|2r1 + |ξ |6s1 + |η|3r1 fˆ(ξ, η) dξ dη,

R2

where in the last inequality we applied the Young inequality. Now, splitting the integral into B1 (0) and R2 \ B1 (0), where B1 (0) denotes the ball of radius 1 centered at the origin, it is easy to see that

2

1 + |ξ |2s1 + |η|2r1 + |ξ |6s1 + |η|3r1 fˆ(ξ, η) dξ dη R2

1068


c

2

1 + |ξ |6s1 + |η|3r1 fˆ(ξ, η) dξ dη.

(2.9)

R2

Write s1 = 1/4 + ρ/3 and r1 = 1/2 + 2ρ/3, where ρ > 0. Thus, 3/4+ρ 1 + |ξ |6s1 + |η|3r1 c 1 + |ξ |2 + |η|2 .

(2.10)

2

Using (2.10) in (2.9) one easily shows the desired conclusion.

Finally, we also recall the Leibniz rule for fractional derivatives. Lemma 2.8. Let 0 < α < 1 and 1 < p < ∞. Then, α D (f g) − f D α g − gD α f

Lp (R)

cgL∞ (R) D α f Lp (R) ,

where D α denotes either Dxα or Dyα . Proof. See Kenig, Ponce, and Vega [11, Theorem A.12].

2

3. Proofs of Theorems 1.1–1.4 We begin this section by showing Theorem 1.1. Since the proof of Theorem 1.2 is similar, we only sketch it. We finish the section by proving Theorem 1.4. Proof of Theorem 1.1. As usual, we consider the integral operator t Ψ (u)(t) = Ψu0 (u)(t) := U (t)u0 +

U t − t uk ux t dt ,

(3.1)

0

and define the metric spaces

YT = u ∈ C [0, T ]; H s R2 ; |||u||| < ∞ and

YTa = u ∈ YT ; |||u||| a , with s s s + u pk ∞ + ux 12/5 ∞ + Dx ux ∞ 2 + Dy ux ∞ 2 + uL4 L∞ , |||u||| := uL∞ L L L L L L L L T Hxy x T

xy

T

xy

x

yT

x

yT

yT

where a, T > 0 will be chosen later. We assume that 3/4 < s < 1 and 0 < T 1. First we estimate the H s -norm of Ψ (u). Let u ∈ YT . By using Minkowski’s inequality, group properties and then Hölder’s inequality, we have


Ψ (u)(t)

T L2xy

1069

uL2xy uk−1 ux L∞ dt

cu0 H s + c

xy

0

T cu0 H s + cuL∞ L2xy T

∞ uk−1 L∞ ux Lxy dt xy

0

cu0 H s + cT γ uL∞ L2xy ux L12/5 L∞ uk−1 pk T

LT L∞ xy

xy

T

(3.2)

.

Using group properties, Minkowski and Hölder’s inequalities and twice Lemma 2.8, we have s D Ψ (u)(t) x

L2xy

Dxs u0 L2 +

T

s k D u ux x

xy

L2xy

dt

0

T cu0 H s + c

s k D u 2 dt + c ux L∞ x xy L

T

0

D s u 2 ux L∞ uk−1 x L∞ xy Lxy xy

T

dt + c

k s u D ux

dt

k s u D ux

dt .

L2xy

0

T cu0

dt

x

0

Hs

L2xy

0

T cu0 H s + c

k s u D ux x

xy

+ cu

ux

s L∞ T Hxy

L∞ xy

uk−1 dt L∞ xy

T +c

0

x

L2xy

0

(3.3) As in (3.2), from Hölder’s inequality, we get T

k−1 γ ux L∞ uk−1 L∞ dt cT ux L12/5 L∞ u pk xy xy

LT L∞ xy

xy

T

(3.4)

.

0

Moreover, T

k s u D ux x

0

L2xy

T

dt

2 s uk−2 L∞ u Dx ux L2 dt xy

0

T γ uk−2 p˜ k

LT L∞ xy

xy

u2L4 L∞ Dxs ux L∞ L2 , x

yT

x

(3.5)

yT

where p˜ k = 2(k−2) 1−2γ . Note that for 3 k 7 we have p˜ k < pk . Thus, combining (3.4)–(3.5) with (3.3), we then deduce

1070


s D Ψ (u)(t) x

L2xy

k−1 s ux 12/5 ∞ u pk cu0 H s + cT γ uL∞ L L T Hxy T

+ cT γ uk−2 pk

s ∞ D ux

u2L4 L

LT L∞ xy

x

yT

LT L∞ xy

xy

2 L∞ x LyT

x

(3.6)

.

A similar analysis can be carried out to see that s D Ψ (u)(t) y

L2xy

k−1 s ux 12/5 ∞ u pk cu0 H s + cT γ uL∞ L L T Hxy T

+ cT γ uk−2 pk

s ∞ D ux

u2L4 L

LT L∞ xy

x

yT

LT L∞ xy

xy

2 L∞ x LyT

y

(3.7)

.

Therefore, from (3.2), (3.6), and (3.7), we deduce Ψ (u)

s L∞ T H

cu0 H s + cT γ |||u|||k+1 .

(3.8)

Next, we estimate the remaining norms. By taking δ = 3/4 and θ = 1/4 + σ , 0 < σ 1−12γ 24 , in Lemma 2.5, we see that pk 3/θ . Thus, Lemma 2.5, group properties and the arguments used to obtain (3.8) yield Ψ (u)

p

LT k L∞ xy

U (t)u0

p

LT k L∞ xy

t k + U (t) U −t u ux t dt

T cu0 H 3/4 + c

k u ux

H 3/4

p

LT k L∞ xy

0

dt

0

T cu0 H s + c

k u ux

Hs

dt

0

cu0 H s + cT γ |||u|||k+1 .

(3.9)

By choosing ε ∼ 1/2 such that 1 − ε/2 s, an application of Lemma 2.3 together with arguments similar to those ones used to derive (3.8) imply ∂x Ψ (u)

12/5 ∞ Lxy

LT

U (t)∂x u0

t k + U (t) U −t ∂x u ux t dt

12/5 ∞ Lxy

LT

−ε/2 cDx ∂x u0

T

L2xy

+c

12/5 ∞ Lxy

LT

0

−ε/2 k Dx ∂x u ux

L2xy

dt

0

T cu0 H s + c

k u ux

L2xy

0

T

dt + c

s k D u ux x

L2xy

dt

0

cu0 H s + cT |||u||| γ

k+1

.

(3.10)


1071

Applying Lemma 2.1, group properties, and the Minkowski and Hölder inequalities, we obtain s D ∂x Ψ (u) x

2 L∞ x LyT

∂x U (t)D s u0 x

2 L∞ x LyT

t s k + ∂x U (t) U −t Dx u ux t dt

2 L∞ x LyT

0

T

cD s u0 x

L2xy

+c

s k D u ux x

L2xy

dt

0

cu0 H s + cT |||u|||k+1 γ

(3.11)

and s D ∂x Ψ (u) y

2 L∞ x LyT

∂x U (t)D s u0 y

2 L∞ x LyT

t s k + ∂x U (t) U −t Dy u ux t dt

2 L∞ x LyT

0

cD s u0 y

L2xy

T +c

s k D u ux y

L2xy

dt

0

cu0 H s + cT |||u|||k+1 . γ

(3.12)

Finally, an application of Corollary 2.7, Minkowski’s inequality, group properties, and arguments previously used yield t k Ψ (u) 4 ∞ U (t)u0 4 ∞ + U −t u ux t dt U (t) Lx LyT Lx LyT 0

T cu0 H s + c

k u ux

Hs

L4x L∞ yT

dt

0

cu0 H s + cT γ |||u|||k+1 .

(3.13)

Therefore, from (3.8)–(3.13), we infer

Ψ (u)

cu0 H s + cT γ |||u|||k+1 . Choose a = 2cu0 H s and T > 0 such that 1 ca k T γ . 4 Then, it is easy to see that Ψ : YTa → YTa is well defined. Moreover, similar arguments show that Ψ is a contraction. To finish the proof we use standard arguments, thus, we omit the details. This completes the proof of Theorem 1.1. 2

1072


Proof of Theorem 1.2. The proof is very similar to that of Theorem 1.1. So, we give only the main steps. Assume sk < s < 1 and 0 < T 1. Define the metric space

XT = u ∈ C [0, T ]; H s R2 ; |||u|||s,k < ∞ with s + u |||u|||s,k := uL∞ T Hxy

+ D s ux

2 L∞ x LyT

x

p˜

LT k L∞ xy

+ ux L12/5 L∞

+ D s ux

2 L∞ x LyT

y

We first note that since k 8 we have p˜ k > pk , where pk = Hence, similarly to estimates (3.2)–(3.7), we establish that Ψ (u)

s L∞ T H

xy

T

+ uL4x L∞ . yT

12(k−1) 7−12γ

is given in Theorem 1.1.

cu0 H s + cT γ |||u|||k+1 s,k ,

(3.14)

where Ψ is the integral operator given in (3.1). The estimates (3.10)–(3.13) also hold here without any change. What is left, is to show a similar estimate as (3.9). Here, to use Lemma 2.5 we take δ = s and θ = 1 − s + σ , where σ and γ are chosen such that s > sk +

6γ + σ. 2(k − 2)

(3.15)

The inequality (3.15) promptly implies that p˜ k 3/θ . Thus, in view of Lemma 2.5, we obtain Ψ (u)

T p˜

LT k L∞ xy

cu0 H s + c

k u ux

Hs

dt

0

cu0 H s + cT γ |||u|||k+1 s,k . Collecting all of our estimates, we then deduce

Ψ (u)

The rest of the proof runs as before.

s,k

cu0 H s + cT γ |||u|||k+1 s,k .

2

Proof of Theorem 1.3. We start by recalling some facts about solitary wave for the generalized ZK equation. In fact, solitary wave are special solutions of Eq. (1.1) having the form u(x, y, t) = ϕc (x − ct, y), for some c ∈ R. Thus, substituting this form of u in (1.1) and integrating once, we see that ϕc must satisfy −cϕc + ϕc +

1 ϕ k+1 = 0. k+1 c

The following lemma is well known and will be sufficient to establish our result.

(3.16)


1073

Lemma 3.1. Let c > 0. Then Eq. (3.16) admits a positive, radially symmetric solution ϕc ∈ H 1 (R2 ). Moreover, ϕc ∈ C ∞ (R2 ), and there exists ρ > 0 such that for all multi-index α ∈ N2 with |α| 2, one has |D α ϕc (x)| Cα e−ρ|x| , where Cα depends only on α. Proof. See Berestycki and Lions [1].

2

It is easy to see that √ √ ϕc (x, y) = c1/k ϕ1 ( cx, cy),

for all c > 0,

where ϕ1 is the solution of (3.16) with c = 1. Thus, since ϕˆ c (ξ, η) = c

1/k−1

ϕˆ1

ξ η √ ,√ , c c

(3.17)

one easily checks that ϕc H˙ sc (k) = c1/k−1/2+sc (k)/2 ϕ1 H˙ sc (k) = ϕ1 H˙ sc (k) =: a0 .

(3.18)

Note that the constant a0 does not depend on c. Next, for any c > 0 fixed, we consider uc (x, y, t) = ϕc (x − ct, y). Hence, at t = 0, we have uc (0) = ϕc . Moreover, for any c1 , c2 > 0, we obtain ϕc1 − ϕc2 2H˙ sc (k) = ϕc1 2H˙ sc (k) + ϕc2 2H˙ sc (k) − 2 ϕc1 , ϕc2 H˙ sc (k) .

(3.19)

But, using (3.17) again, we obtain ϕc1 , ϕc2 H˙ sc (k) =

D sc (k) ϕc1 (x, y)D sc (k) ϕc2 (x, y) dx dy R2

=

(ξ, η) 2sc (k) ϕˆc (ξ, η)ϕˆ c (ξ, η) dξ dη 1 2

R2 1

= (c1 c2 ) k −1 =

R2

c2 c1

1 −1 k

(ξ, η) 2sc (k) ϕˆ 1 √ξ , √η ϕˆ 1 √ξ , √η dξ dη c1 c1 c2 c2

(ξ, η) 2sc (k) ϕˆ1 (ξ, η)ϕˆ 1

R2

c1 ξ, c2

c1 η dξ dη. c2

Therefore, as θ := c1 /c2 → 1, we get lim ϕc1 , ϕc2 sc (k) = a02 .

θ→1

(3.20)

1074


As a consequence of (3.18)–(3.20), we then have lim ϕc1 − ϕc2 H˙ sc (k) = 0.

θ→1

On the other hand, for any t > 0, uc (t) − uc (t)2˙ s (k) = uc (t)2˙ s (k) + uc (t)2˙ s (k) − 2 uc (t), uc (t) ˙ s (k) . 1 2 1 2 1 2 H c H c H c H c But, since ξ η 1/k−1 −icξ t uc (t)(ξ, η) = c e ϕˆ 1 √ , √ , c c we deduce uc1 (t), uc2 (t) H˙ sc (k)

2s (k)

1 ξ ξ η η dξ dη = (c1 c2 ) k −1 e−itξ(c1 −c2 ) (ξ, η) c ϕˆ 1 √ , √ ϕˆ 1 √ , √ c1 c1 c2 c2

R2

=

c2 c1

1 k −1

e−itξ

√

c1 (c1 −c2 )

2s (k) (ξ, η) c ϕˆ 1 (ξ, η)ϕˆ 1

R2

c1 ξ, c2

c1 η dξ dη. c2

By choosing c1 = m + 1 and c2 = m ∈ N and letting m → ∞, an application of the Riemann– Lebesgue lemma, yields lim uc1 (t), uc2 (t) H˙ sc (k) = 0.

m→∞

Therefore, for any t > 0, √ lim uc1 (t) − uc2 (t)H˙ sc (k) = 2 a0 .

θ→1

This completes the proof of the theorem.

2

Proof of Theorem 1.4. By using the Gagliardo–Nirenberg interpolation theorem it follows that 2 k u(t)k+2 cu(t)L2 ∂x u(t)L2 . Lk+2 Combining (1.3), (1.4) and (3.21), we obtain that u(t)2 1 = I1 u(t) + I2 u(t) + cu(t)k+2 H Lk+2 k I1 (u0 ) + I2 (u0 ) + cu0 2L2 ∂x u(t)L2 .

(3.21)


1075

Denote X(t) = u(t)2H 1 . Since k 3, we then have k−2 X(t) C u0 H 1 + cu0 2L2 X(t)1+ 2 . Thus, if u0 H 1 is small enough, a standard argument leads to u(t)H 1 C(u0 H 1 ) for t ∈ [0, T ]. Therefore, we can apply the local theory to extend the solution. 2 4. Global well-posedness for the modified ZK In this section, we consider the Cauchy problem associated with the modified ZK. The main goal is to prove the global well-posedness result stated in Theorem 1.5. 4.1. Auxiliary results We start with the following local well-posedness result. The proof is slightly different from that of Theorem 1.1 in [13]. Theorem 4.1. Let k = 2. For any u0 ∈ H s (R2 ), s > 3/4, there exist T = T (u0 H s ) > 0 and a unique solution of the IVP (1.1), defined in the interval [0, T ], such that u ∈ C [0, T ]; H s R2 , s D ux ∞ 2 + D s ux ∞ 2 < ∞, x y L L L L x

x

yT

(4.1) (4.2)

yT

uLp L∞ + ux L12/5 L∞ < ∞,

(4.3)

uL2x L∞ < ∞,

(4.4)

T

xy

xy

T

and yT

where p =

2 1−2γ

and γ ∈ (0, 5/12). In addition, the following statements hold:

(i) For any T ∈ (0, T ) there exists a neighborhood V of u0 in H s (R2 ) such that the map ˜ from V into the class defined by (4.1)–(4.4) is smooth. u˜ 0 → u(t) (ii) The existence time T is given by −2/γ

T ∼ u0 H s .

(4.5)

To simplify the exposition and for further references, we prove first the following lemma. Lemma 4.2. Assume u, v, w are sufficiently smooth. Let p be as in Theorem 4.1. (i) For any T > 0, T

vwux L2xy dt cT γ ux L12/5 L∞ vL∞ L2xy wLp L∞ . T

0

xy

T

T

xy

1076


(ii) For any T > 0 and s ∈ (0, 1), T

s D (vwux ) x

L2xy

s w p ∞ + wL∞ H s v p ∞ dt cT γ ux L12/5 L∞ vL∞ L L L L xy T Hxy T T

xy

T

0

xy

T

xy

+ cT γ wLp L∞ vL2x L∞ Dxs ux L∞ L2 . T

xy

yT

x

yT

The same still holds if we replace Dxs by Dys . Proof. The estimate (i) follows after applying Hölder’s inequality. The proof of (ii) is roughly an application of Lemma 2.8 combined with the Hölder inequality. Indeed, applying twice Lemma 2.8, we deduce s D (vwux ) x

L2xy

cDxs w L2 vL∞ ux L∞ + cDxs v L2 wL∞ ux L∞ xy xy xy xy xy xy s + cwvDx ux L2 . xy

(4.6)

For the first two terms, from Hölder’s inequality, we obtain T

s D w x

0

L2xy

dt vL∞ ux L∞ + Dxs v L2 wL∞ ux L∞ xy xy xy xy xy

s w p ∞ + wL∞ H s v p ∞ . cT γ ux L12/5 L∞ vL∞ L L L L xy T Hxy T T

xy

T

xy

T

xy

For the last term in (4.6), we obtain T

wvD s ux x

L2xy

T

dt

0

s vD ux 2 dt wL∞ x xy L xy

0

T

1/2 w2L∞ xy

dt

s vD ux x

L2xyT

0

cT γ wLp L∞ vL2x L∞ Dxs ux L∞ L2 . T

This completes the proof of the lemma.

xy

yT

x

yT

2

Sketch of proof of Theorem 4.1. The proof is similarly carried out as the proof of Theorem 1.1 (see also [13]). The main difference is that instead of using the maximal function in Proposition 2.6(i), we use the one in (ii). Thus, we consider the integral operator t Φ(u)(t) = Φu0 (u)(t) := U (t)u0 + 0

U t − t u2 ux t dt ,


1077

and define the metric spaces

ZT = u ∈ C [0, T ]; H s R2 ; |||u|||s,2 < ∞ and

ZTa = u ∈ ZT ; |||u|||s,2 a , with s + u p ∞ + ux 12/5 ∞ |||u|||s,2 := uL∞ LT Lxy LT Lxy T Hxy s s + Dx ux L∞ L2 + Dy ux L∞ L2 + uL2x L∞ , x

x

yT

yT

yT

where a, T > 0 will be chosen later. We assume that 3/4 < s < 1 and 0 < T 1. s Here, we only estimate the L∞ T Hxy -norm, because the others ones are obtained as in Theorem 1.1. From group properties, Minkowski’s inequality, and Lemma 4.2 it follows that Φ(u)(t) s u0 H s + xy H

T

2 u ux

s Hxy

xy

dt

0

γ s + cT ux 12/5 ∞ uL∞ H s u p ∞ + uL∞ H s u p ∞ u0 Hxy L L L L xy xy LT Lxy T T T xy T xy s γ + cT uLp L∞ uL2x L∞ Dx ux L∞ L2 yT T xy x yT s γ + cT u p ∞ u 2 ∞ D ux ∞ 2 . (4.7) LT Lxy

Lx LyT

y

Lx LyT

Thus, Φ(u)

s L∞ T Hxy

γ 3 s + cT |||u||| u0 Hxy s,2 .

Finally, gathering together all estimates we see that

Φ(u)

cu0 H s + cT γ |||u|||3 . s,2 xy s,2 Choosing a = 2cu0 H s , and then T such that cT γ a 2
0, v(x, y, 0) = v0 (x, y) ∈ H 1 R2 .

(4.9)

1078

F. Linares, A. Pastor / Journal of Functional Analysis 260 (2011) 1060–1085 −2/γ

Let T ∼ v0 H 1

be the existence time given by Theorem 4.1. If v0 satisfies v0 L2
3/4, and let v be the solution given in Proposition 4.3. Then there exists a unique solution w of the IVP

wt + ∂x w + w 2 wx + 2wvvx + 2wvwx + v 2 wx + w 2 vx = 0, w(x, y, 0) = w0 (x, y),

(x, y) ∈ R2 , t > 0, (4.13)

defined in the same interval of existence of v, [0, T ], such that w ∈ C [0, T ]; H ρ R2 , ρ D wx ∞ 2 + D ρ wx ∞ 2 < ∞, x y L L L L x

x

yT

(4.14) (4.15)

yT

wLp L∞ + wx L12/5 L∞ < ∞,

(4.16)

wL2x L∞ < ∞,

(4.17)

T

xy

T

xy

and

yT

where p =

2 1−2γ

and γ ∈ (0, 5/12).


1079

Sketch of the proof. Define the integral operator w0 (w)(t) := U (t)w0 + Φ(w)(t) =Φ

t

U t − t F t dt ,

0

where F = w 2 wx + 2wvvx + 2wvwx + v 2 wx + w 2 vx .

(4.18)

Consider the metric spaces

WT = w ∈ C [0, T ]; H ρ R2 ; |||w|||ρ,2 < ∞ and

WTa = w ∈ WT ; |||w|||ρ,2 a , with ρ + w p ∞ + wx 12/5 |||w|||ρ,2 := wL∞ Hxy LT Lxy LT L∞ T xy ρ ρ + D wx ∞ 2 + D wx ∞ 2 + w

x

y

Lx LyT

Lx LyT

. L2x L∞ yT

As before, we only estimate the L∞ T Hxy -norm, because from our linear estimates, all the others estimates reduce to this one. First, we note that ρ

ρ D Φ(w) x

T L2

w0 H ρ +

ρ D F x

L2

dt .

0

But, successive applications of Lemma 4.2(ii) lead to T

ρ 2 D w wx x

0

L2xy

ρ w p ∞ dt cT γ wx L12/5 L∞ wL∞ Hxy L L xy

T

ρ D (wvvx ) x

0

T

xy

+ cT γ wLp L∞ wL2x L∞ Dxρ wx L∞ L2 , T

T

T

L2xy

xy

yT

x

yT

ρ w p ∞ + w ∞ ρ v p ∞ dt cT γ vx L12/5 L∞ vL∞ Hxy L Hxy L L L L xy

T

T

T

xy

T

+ cT γ wLp L∞ vL2x L∞ Dxρ vx L∞ L2 , T

xy

yT

x

yT

T

xy

1080

T


ρ D (wvwx ) x

L2xy

ρ w p ∞ + w ∞ ρ v p ∞ dt cT γ wx L12/5 L∞ vL∞ Hxy L Hxy L L L L T

xy

T

0

T

xy

T

T

xy

+ cT γ wLp L∞ vL2x L∞ Dxρ wx L∞ L2 , T

T

ρ 2 D v wx x

L2xy

xy

yT

yT

ρ v p ∞ dt cT γ wx L12/5 L∞ vL∞ Hxy L L xy

T

0

T

T

xy

+ cT γ vLp L∞ vL2x L∞ Dxρ wx L∞ L2 , T

T

x

ρ 2 D w vx x

L2xy

xy

yT

x

yT

ρ w p ∞ dt cT γ vx L12/5 L∞ wL∞ Hxy L L

0

xy

T

T

T

xy

+ cT γ wLp L∞ wL2x L∞ Dxρ vx L∞ L2 . T

xy

yT

x

yT

Thus, we see that ρ D Φ(w) x

L2

w0 H ρ + cT γ |||w|||2ρ,2 + |||v|||21,2 + |||w|||ρ,2 |||v|||1,2 |||w|||ρ,2 .

Analogously, we deduce ρ D Φ(w) y

L2

w0 H ρ + cT γ |||w|||2ρ,2 + |||v|||21,2 + |||w|||ρ,2 |||v|||1,2 |||w|||ρ,2

and Φ(w)

L2

w0 L2 + cT γ |||w|||2ρ,2 + |||v|||21,2 + |||w|||ρ,2 |||v|||1,2 |||w|||ρ,2 .

Therefore, we have established that

Φ(w)

ρ,2

w0 H ρ + cT γ |||w|||2ρ,2 + |||v|||21,2 + |||w|||ρ,2 |||v|||1,2 |||w|||ρ,2 .

(4.19)

Now, by choosing a = 2c max{v0 H 1 , w0 H ρ }, we see that cT γ a 2
1. For the self-affine measure μM,D corresponding to

p1 M= 0

0 p2

0 1 0 1 and D = , , , , 0 0 1 1

(4.7)

we have shown in [18] that if p1 , p2 ∈ 2Z + 1, then there are at most 4 mutually orthogonal exponential functions in L2 (μM,D ), and the number 4 is the best. In the case when p1 = ±2 and p2 = ±2, we know that D is a complete residue system (mod M), hence μM,D is a spectral measure. Now, in the case when p1 , p2 ∈ 2Z \ {0, 2}, we can show that μM,D is a spectral measure with one of its spectra Λ ⊆ Z2 . In fact, by letting S be the following set p1 /2 0 0 p1 /2 , , , S= , 0 p2 /2 p2 /2 0

(4.8)

we get a compatible pair (M −1 D, S). Furthermore, we have

∗ 0 p1 /2 T (M, D) ⊆ [2/3, 2/3]2 , T M , S = T (M, S) = 0 p2 /2

(4.9)

and

p1 Z mM −1 D (t) = Z mD M ∗−1 t = 0

0 Z mD (t) ⊆ R2 \ [−2/3, 2/3]2 . p2

(4.10)

It follows from (4.9) and (4.10) that T (M ∗ , S) ∩ Z(mM −1 D (t)) = ∅. Hence, we obtain, from Lemma 3.2, that (μM,D , Λ(M, S)) is a spectral pair. That is, Λ(M, S) is an integer spectrum for μM,D .

J.-L. Li / Journal of Functional Analysis 260 (2011) 1086–1095

1095

Note that by applying the invariance of spectrality under the similarity [17, Proposition 4.2], we also have the spectral measure μM,D corresponding to the pair (M, D) given by

2k M= 0

0 2k

k ∈ Z \ {0} and D =

0 a c a+c , , , , 0 b d b+d

where a, b, c, d ∈ R satisfy ad − bc = 0. In particular, let k = 2, a = 0, b = 2, c = 1, d = 4, we get the spectral measure μM,D in [4, Theorem 5.2]. Acknowledgments The present research is partially supported by the Key Project of Chinese Ministry of Education (No. 108117) and the National Natural Science Foundation of China (No. 10871123). References [1] D.E. Dutkay, P.E.T. Jorgensen, Fourier frequencies in affine iterated function systems, J. Funct. Anal. 247 (2007) 110–137. [2] D.E. Dutkay, P.E.T. Jorgensen, Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. 256 (2007) 801–823. [3] D.E. Dutkay, P.E.T. Jorgensen, Fourier series on fractals: a parallel with wavelet theory, in: Radon Transform, Geometry, and Wavelets, in: Contemp. Math., vol. 464, 2008, pp. 75–101. [4] D.E. Dutkay, P.E.T. Jorgensen, Probability and Fourier duality for affine iterated function systems, Acta Appl. Math. 107 (2009) 293–311. [5] D.E. Dutkay, P.E.T. Jorgensen, Quasiperiodic spectra and orthogonality for iterated function system measures, Math. Z. 261 (2009) 373–397. [6] D.E. Dutkay, D. Han, P.E.T. Jorgensen, Orthogonal exponentials, translations, and Bohr completions, J. Funct. Anal. 257 (2009) 2999–3019. [7] D.E. Dutkay, D. Han, Q. Sun, On the spectra of a Cantor measure, Adv. Math. 221 (2009) 251–276. [8] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974) 101–121. [9] T.-Y. Hu, K.-S. Lau, Spectral property of the Bernoulli convolutions, Adv. Math. 219 (2008) 554–567. [10] J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981) 713–747. [11] P.E.T. Jorgensen, S. Pedersen, Dense analytic subspaces in fractal L2 -spaces, J. Anal. Math. 75 (1998) 185–228. [12] P.E.T. Jorgensen, K. Kornelson, K. Shuman, Orthogonal exponentials for Bernoulli iterated function systems, in: Representations, Wavelets, and Frames, in: Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2008, pp. 217– 237. [13] P.E.T. Jorgensen, K. Kornelson, K. Shuman, Families of spectral sets for Bernoulli convolutions, available on http://arxiv.org/abs/0911.2435v1, 2009. [14] I. Łaba, Y. Wang, On spectral Cantor measures, J. Funct. Anal. 193 (2002) 409–420. [15] J.-L. Li, Spectral sets and spectral self-affine measures, PhD thesis, The Chinese University of Hong Kong, November 2004. [16] J.-L. Li, μM,D -Orthogonality and compatible pair, J. Funct. Anal. 244 (2007) 628–638. [17] J.-L. Li, Spectral self-affine measures in Rn , Proc. Edinb. Math. Soc. 50 (2007) 197–215. [18] J.-L. Li, The cardinality of certain μM,D -orthogonal exponentials, J. Math. Anal. Appl. 362 (2010) 514–522. [19] R. Strichartz, Remarks on “Dense analytic subspaces in fractal L2 -spaces”, J. Anal. Math. 75 (1998) 229–231. [20] R. Strichartz, Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math. 81 (2000) 209–238. [21] R. Strichartz, Convergence of mock Fourier series, J. Anal. Math. 99 (2006) 333–353.


A variant of the Johnson–Lindenstrauss lemma for circulant matrices Jan Vybíral Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria Received 15 February 2010; accepted 26 November 2010 Available online 4 December 2010 Communicated by K. Ball

Abstract We continue our study of the Johnson–Lindenstrauss lemma and its connection to circulant matrices started in Hinrichs and Vybíral (in press) [7]. We reduce the bound on k from k = Ω(ε−2 log3 n) proven there to k = Ω(ε−2 log2 n). Our technique differs essentially from the one used in Hinrichs and Vybíral (in press) [7]. We employ the discrete Fourier transform and singular value decomposition to deal with the dependency caused by the circulant structure. © 2010 Elsevier Inc. All rights reserved. Keywords: Johnson–Lindenstrauss lemma; Circulant matrix; Discrete Fourier transform; Singular value decomposition

1. Introduction Let x 1 , . . . , x n ∈ Rd be n points in the d-dimensional Euclidean space Rd . The classical Johnson–Lindenstrauss lemma tells that, for a given ε ∈ (0, 12 ) and a natural number k = Ω(ε −2 log n), there exists a linear map f : Rd → Rk , such that 2 2 2 (1 − ε)x j 2 f x j 2 (1 + ε)x j 2 for all j ∈ {1, . . . , n}. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.014

J. Vybíral / Journal of Functional Analysis 260 (2011) 1096–1105

1097

Here · 2 stands for the Euclidean norm in Rd or Rk , respectively. Furthermore, here and any time later, the condition k = Ω(ε −2 log n) means, that there is an absolute constant C > 0, such that the statement holds for all natural numbers k with k Cε −2 log n. We shall also always assume, that k d. Otherwise, the statement becomes trivial. The original proof of this fact was given by Johnson and Lindenstrauss in [9]. We refer to [6] for a beautiful and self-contained proof. Since then, it has found many applications for example in algorithm design. These applications inspired numerous variants and improvements of the Johnson–Lindenstrauss lemma, which try to minimize the computational costs of f (x), the memory used, the number of random bits used and to simplify the algorithm to allow an easy implementation. We refer to [8,1–3,12] for details and to [12] for a nice description of the history and the actual “state of the art”. All the known proofs of the Johnson–Lindenstrauss lemma work with random matrices and proceed more or less in the following way. One considers a probability measure P on a some subset P of all k × d matrices (i.e. all linear mappings Rd → Rk ). The proof of the Johnson– Lindenstrauss lemma then emerges by some variant of the following two estimates 2 1 P f ∈ P: f (x)2 1 + ε < 1 − 2n and 2 1 P f ∈ P: f (x)2 1 − ε < 1 − , 2n which have to be proven for all unit vectors x ∈ Rd , and a simple union bound over all points x j /x j 2 , j = 1, . . . , n. Here and later on we assume, without loss of generality, that x j = 0 for all j = 1, . . . , n. The biggest breakthrough in the attempts to minimize the running time of f was achieved by Ailon and Chazelle in [2] (with improvements by Matoušek [12] and Ailon and Liberty [4]). The mapping f is given in [2] as the composition of a sparse matrix, a certain random Fourier matrix and a random diagonal matrix. The value f (x) can be computed with high probability very efficiently, i.e. using O(d log d + min{dε −2 log n, ε −2 log3 n}) operations. This was later further improved by Ailon and Liberty to O(d log k) for k = O(d 1/2−δ ), for any arbitrary small fixed δ > 0. In [7], we studied a different construction of f , namely the possibility of a composition of a random circulant matrix with a random diagonal matrix. As a multiple of a circulant matrix may be implemented with the help of a discrete Fourier transform, it provides the running time of O(d log d), requires very few random bits (only 2d random bits in the case of Bernoulli variables) and allows a very simple implementation, as the Fast Fourier Transform is a part of every standard mathematical software package. The main difference between this approach and the usual constructions available in the literature is that the components of f (x) are now no longer independent random variables. Decoupling this dependence, we were able to prove in [7] the Johnson–Lindenstrauss lemma for composition of a random circulant matrix and a random diagonal matrix, but only for k = Ω(ε −2 log3 n). It is the main aim of this note to improve this bound to k = Ω(ε −2 log2 n). This comes essentially closer to the standard bound k = Ω(ε −2 log n). Reaching this optimal bound (and keeping the control of the constants involved) remains an open problem and a subject of a challenging research.

1098


We use a completely different technique here. We use the discrete Fourier transform and the singular value decomposition of circulant matrices. That is the reason, why we found it more instructive to state and prove our variant of Johnson–Lindenstrauss lemma for complex vectors and Gaussian random variables. As a corollary, we obtain of course a corresponding real version. Before we state our main result, we give the necessary definitions. Definition 1.1. Let α and β be independent real Gaussian random variables with Eα = Eβ = 0

and E|α|2 = E|β|2 = 1.

Then we call a = α + iβ a complex Gaussian variable. Let us note, that if a is a complex Gaussian variable, then Ea = Eα + iEβ = 0 and E|a|2 = Eα 2 + Eβ 2 = 2. Definition 1.2. (i) Let k d be natural numbers. Let a = (a0 , . . . , ad−1 ) ∈ Cd be a fixed complex vector. We denote by Ma,k the partial circulant matrix ⎛

a0

⎜ ad−1 ⎜ ad−2 Ma,k = ⎜ ⎜ . ⎝ . . ad−k+1

ad−1 .. .

a2 a1 a0 .. .

··· ··· ··· .. .

⎞ ad−1 ad−2 ⎟ ⎟ ad−3 ⎟ ∈ Ck×d . .. ⎟ ⎠ .

ad−k+2

ad−k+3

···

ad−k

a1 a0

If k = d, we denote by Ma = Ma,d the full circulant matrix. This notation extends naturally to the case, when a = (a0 , . . . , ad−1 ) are independent complex Gaussian variables. (ii) If = (0 , . . . , d−1 ) are independent Bernoulli variables, we put ⎛

0 ⎜0 D = diag() := ⎜ ⎝ ...

0 1 .. .

··· ··· .. .

0

0

···

⎞ 0 0 ⎟ ∈ Rd×d . .. ⎟ . ⎠ d−1

Of course, D : Cd → Cd is an isomorphism. Theorem 1.3. Let ε ∈ (0, 12 ), n d be natural numbers, and let x 1 , . . . , x n ∈ Cd be n arbitrary points in Cd . Let a = (a0 , . . . , ad−1 ) be d independent complex Gaussian variables and let = (0 , . . . , d−1 ) be independent Bernoulli variables. If k = Ω(ε −2 log2 n) is a natural number, then the mapping f : Cd → Cd given by f (x) = 1 √ Ma,k D x satisfies 2k


1099

2 2 2 (1 − ε)x j 2 f x j 2 (1 + ε)x j 2 for all j ∈ {1, . . . , n} with probability at least 2/3. Here · 2 stands for the 2 -norm in Cd or Ck , respectively. For reader’s convenience, we formulate also a variant of Theorem 1.3, which deals with real Euclidean spaces. Corollary 1.4. Let ε ∈ (0, 12 ), n d be natural numbers, and let x 1 , . . . , x n ∈ R2d be n arbitrary points in R2d . Let α0 , . . . , αd−1 , β0 , . . . , βd−1 be 2d independent real Gaussian variables and let = (0 , . . . , d−1 ) be independent Bernoulli variables. If k = Ω(ε −2 log2 n) is a natural number, then the mapping f : R2d → R2k given by 1 f (x) = √ 2k

Mα,k

−Mβ,k

Mβ,k

Mα,k

D 0

0 D

x

satisfies 2 2 2 (1 − ε)x j 2 f x j 2 (1 + ε)x j 2 for all j ∈ {1, . . . , n} with probability at least 2/3. Here · 2 stands for the 2 -norm in R2d or R2k , respectively. The proof follows trivially from Theorem 1.3 by considering complex Gaussian variables j j j j a = (α0 +iβ0 , . . . , αd−1 +iβd−1 ) and complex vectors y j = (x0 +ixd , . . . , xd−1 +ix2d−1 ) ∈ Cd , j = 1, . . . , n. 2. Used techniques We give an overview of the techniques used in the proof of Theorem 1.3. 2.1. Discrete Fourier transform Our main tool in this note is the discrete Fourier transform. If d is a natural number, then the discrete Fourier transform Fd : Cd → Cd is defined by d−1 1 2πiuξ . (Fd x)(ξ ) = √ xu exp − d d u=0 With this normalization, Fd is an isomorphism of Cd onto itself. The inverse discrete Fourier transform is given by d−1 −1 1 2πiuξ . Fd x (ξ ) = √ xu exp d d u=0

1100


Observe, that the matrix representation of Fd−1 is the conjugate transpose of the matrix representation of Fd , i.e. Fd−1 = Fd∗ . The fundamental connection between discrete Fourier transform and circulant matrices is given by √ Ma = Fd diag( dFd a)Fd−1 ,

(2.1)

which may be verified by direct calculation. Hence every circulant matrix may be diagonalized with the use of a discrete Fourier transform, its inverse and a multiple of the discrete Fourier transform of its first row. 2.2. Singular value decomposition The last tool needed in the proof is the singular value decomposition. Let M : Cd → Ck be a k × d complex matrix with k d. Then there exists a decomposition M = U ΣV ∗ , where U is a k × k unitary complex matrix, Σ is a k × k diagonal matrix with nonnegative entries on the diagonal, V is a d × k complex matrix with k orthonormal columns and V ∗ denotes the conjugate transpose of V . Hence V ∗ has k orthonormal rows. The entries of Σ are the singular values of M, namely the square roots of the eigenvalues of MM ∗ . If a = (a0 , . . . , ad−1 ) ∈ Cd is a complex vector and Ma is the corresponding circulant matrix, then its singular values may be calculated using (2.1). We obtain √ √ ∗ Ma Ma∗ = Fd diag( dFd a)Fd−1 Fd diag( dFd a)Fd−1 √ √ = Fd diag( dFd a) diag( dFd a)Fd−1 = Fd diag d|Fd a|2 Fd−1 . √ Hence, the singular values of Ma are { d|(Fd a)(ξ )|}d−1 ξ =0 . The action of an arbitrary projection onto a vector of independent real Gaussian variables is very well known. It may be described as follows. Lemma 2.1. Let a = (a0 , . . . , ad−1 ) be independent real Gaussian variables. Let k d be a natural number and let x 1 , . . . , x k be mutually orthogonal unit vectors in Rd . Then k a, x j j =1 is equidistributed with a k-dimensional vector of independent real Gaussian variables. A direct calculation shows, that Lemma 2.1 holds also for complex vectors a and x 1 , . . . , x k . We present the following formulation of this fact. Lemma 2.2. Let a = (a0 , . . . , ad−1 ) be independent complex Gaussian variables. Let W be a k × d matrix with k orthonormal rows. Then W a is equidistributed with a k-dimensional vector of independent complex Gaussian variables.


1101

3. Proof of Theorem 1.3 We shall need the following statement, which describes the preconditioning role of the diagonal matrix D . A similar fact has been used also in [2]. Nevertheless, using discrete Fourier transform instead of a Hadamard matrix does not pose any restrictions on the underlying dimension d. Without repeating the details, we point out, that we discussed briefly in [7, Remark 2.5], why this preconditioning may not be omitted. Lemma 3.1. Let n d be natural numbers and let x 1 , . . . , x n ∈ Cd be complex vectors. Let = (0 , . . . , d−1 ) be independent Bernoulli variables. Then there is an absolute constant C > 0, such that with probability at least 5/6, Fd D x j

∞

C

√ log n · x j 2 √ d

(3.1)

holds for all j ∈ {1, . . . , n}. Proof. Let x = α + iβ be a unit complex vector in Cd . We put y = (y0 , . . . , yd−1 ) = Fd D (x). Combining the inclusion s 2 2 2 z ∈ C: |z| > s = z ∈ C: ( z) + ( z) > s ⊂ z ∈ C: | z| > √ 2 s ∪ z ∈ C: | z| > √ 2 with s s P | yl | > √ = 2P yl > √ , 2 2 we may estimate s s P |yl | > s 2P yl > √ + 2P yl > √ , 2 2

l = 0, . . . , d − 1,

where d−1 1 yl = √ u αu cos(2πlu/d) + βu sin(2πlu/d) d u=0

and d−1 1 u βu cos(2πlu/d) − αu sin(2πlu/d)

yl = √ d u=0

are the real and the imaginary part of yl , respectively.

(3.2)

1102


Let t > 0 be a real parameter to be chosen later. Using Markov’s inequality we may proceed in a standard way: s st P yl > √ = P exp t yl − √ >1 2 2 st exp − √ E exp(t yl ) 2 d−1 t st cosh √ αu cos(2πlu/d) + βu sin(2πlu/d) = exp − √ d 2 u=0 d−1 2 2 st t αu cos(2πlu/d) + βu sin(2πlu/d) exp − √ exp 2d 2 u=0 2 d−1 t 2 st t2 st 2 exp − √ α + βu = exp − √ + exp . 2d u 2 u=0 2 2d We have used the inequality cosh(v) exp(v 2 /2), which holds for all v ∈ R, and the inequality 2 sd between geometric and quadratic means. For the optimal t = √ , this is equal to exp(− s 4d ). 2 As the second summand in (3.2) may be estimated in the same way, we obtain 2 s d , P |yl | > s 4 exp − 4

l = 0, . . . , d − 1.

(3.3)

√ Choosing s = Ω(d −1/2 log n ) and applying the union bound over all nd n2 components of {Fd D (x j /x j 2 )}nj=1 , we obtain the result. 2 Proof of Theorem 1.3. Let us choose a vector = (0 , . . . , d−1 ) ∈ {−1, +1}d , such that (3.1) holds. According to Lemma 3.1 this happens with probability at least 5/6. j Let us take x˜ = xxj for any fixed j = 1, . . . , n. We show, that there is an absolute constant 2 c > 0, such that ckε 2 2 Pa Ma,k D x ˜ 2 2(1 + ε)k exp − log n

(3.4)

ckε 2 2 Pa Ma,k D x ˜ 2 2(1 − ε)k exp − log n

(3.5)

and

hold. From (3.4) and (3.5), Theorem 1.3 follows again by a union bound over all j = 1, . . . , n. ˜ ∈ Cd , j = 0, . . . , k − 1, where S is the shift operator defined by Let y j = S j (D x) S : Cd → Cd ,

S(z0 , . . . , zd−1 ) = (z1 , . . . , zd−1 , z0 ).

We denote by Y the k × d matrix with rows y 0 , . . . , y k−1 .


1103

Then it holds Ma,k D x ˜ 22

d−1 2 k−1 d−1 2 k−1

j = a(u−j ) mod d u x˜u = yu au = Y a22 . j =0 u=0

j =0 u=0

Let Y = U ΣV ∗ be the singular value decomposition of Y . As mentioned above, b := V ∗ a is a k-dimensional vector of independent complex Gaussian variables. Hence, 2 Pa Y a22 > τ = Pa U ΣV ∗ a 2 > τ = Pb U Σb22 > τ k−1

2 2 2 λj |bj | > τ , = Pb Σb2 > τ = Pb j =0

holds for every τ > 0. Here, λj , j = 0, . . . , k − 1, are the singular values of Y . Let us denote μj = λ2j . Then

μ1 =

k−1

λ2j = Y 2F = k,

j =0

where Y F is the Frobenius norm of Y . Moreover, μ∞ = λ2∞ =

sup z∈Cd ,z2 1

sup z∈Cd ,z2 1

Y z22

2 MD x˜ z22 = d Fd D (x) ˜ ∞ C 2 log n,

(3.6)

˜ where MD x˜ stands for the d × d complex circulant matrix with the first row equal to D x. This leads finally also to μ2

μ1 · μ∞ C k log n.

Then k−1

2 2 Pa Y a2 > 2(1 + ε)k = Pb μj |bj | − 2 > 2εk . j =0

We denote Z :=

k−1

j =0

μj |bj |2 − 2 .

(3.7)

1104


The complex version of Lemma 1 from Section 4.1 of [11] (cf. also Lemma 2.2 of [12]) states that √ √ Pb Z 2 2μ2 t + 2μ∞ t exp(−t).

(3.8)

Using (3.6) and (3.7), we arrive at √ Pb Z 2 2C tk log n + 2C 2 t log n exp(−t). Choosing t =

c kε 2 C 2 log n

for c > 0 small enough, we get ckε 2 . Pb (Z 2εk) exp − log n

This finishes the proof of (3.4). Let us note, that (3.5) follows in the same manner with (3.8) replaced by √ √ Pb Z −2 2μ2 t exp(−t), which may be again found in Lemma 1, Section 4.1 of [11].

2

Remark 3.2. The statement and the proof of Theorem 1.3 do not change, if we replace the partial circulant matrix Ma,k with any k × d submatrix of Ma . Note added in proof Interesting new work of Ailon and Liberty [5] appeared during the review process of this paper. Their transformation is the composition of a random sign matrix with a random selection of a suitable number k of rows from a Fourier matrix. Their bound on k, namely k = Ω(ε−4 log n· polylog d), is optimal up to the polylog d factor. Depending on d and n, this may be better than our bound. In another very recent preprint [10], Krahmer and Ward applied the RIP bounds of [13] to prove that partial circulant matrices satisfy the Johnson–Lindenstrauss lemma if k = Ω max ε −1 log3/2 n · log3/2 d, ε −2 log n · log4 d . Acknowledgments The author would like to thank Aicke Hinrichs for valuable comments and the anonymous referee for the careful reading of the manuscript. The author also acknowledges the financial support provided by the FWF project Y 432-N15 START-Preis “Sparse Approximation and Optimization in High Dimensions”.


1105

References [1] D. Achlioptas, Database-friendly random projections: Johnson–Lindenstrauss with binary coins, J. Comput. System Sci. 66 (4) (2003) 671–687. [2] N. Ailon, B. Chazelle, Approximate nearest neighbors and the fast Johnson–Lindenstrauss transform, in: Proc. 38th Annual ACM Symposium on Theory of Computing, 2006. [3] N. Ailon, B. Chazelle, The fast Johnson–Lindenstrauss transform and approximate nearest neighbors, SIAM J. Comput. 39 (1) (2009) 302–322. [4] N. Ailon, E. Liberty, Fast dimension reduction using Rademacher series on dual BCH codes, Discrete Comput. Geom. 42 (4) (2009) 615–630. [5] N. Ailon, E. Liberty, Almost optimal unrestricted fast Johnson–Lindenstrauss transform, http://arxiv.org/ abs/1005.5513. [6] S. Dasgupta, A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures Algorithms 22 (2003) 60–65. [7] A. Hinrichs, J. Vybíral, Johnson–Lindenstrauss lemma for circulant matrices, Random Structures Algorithms, in press. [8] P. Indyk, R. Motwani, Approximate nearest neighbors: Towards removing the curse of dimensionality, in: Proc. 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 604–613. [9] W.B. Johnson, J. Lindenstrauss, Extensions of Lipschitz Mappings into a Hilbert Space, Contemp. Math., vol. 26, 1984, pp. 189–206. [10] F. Krahmer, R. Ward, New and improved Johnson–Lindenstrauss embeddings via the Restricted Isometry Property, http://arxiv.org/abs/1009.0744. [11] B. Laurent, P. Massart, Adaptive estimation of a quadratic functional by model selection, Ann. Statist. 28 (5) (2000) 1302–1338. [12] J. Matoušek, On variants of the Johnson–Lindenstrauss lemma, Random Structures Algorithms 33 (2) (2008) 142– 156. [13] H. Rauhut, J. Romberg, J. Tropp, Restricted isometries for partial random circulant matrices, http://arxiv.org/abs/ 1010.1847.


Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers Xuan Thinh Duong a , Adam Sikora a,∗ , Lixin Yan b a Department of Mathematics, Macquarie University, NSW 2109, Australia b Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou 510275, PR China

Received 16 February 2010; accepted 9 November 2010

Communicated by Alain Connes

Abstract Let L be a non-negative self-adjoint operator acting on L2 (X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt (x, y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp -norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L2 estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials. © 2010 Elsevier Inc. All rights reserved. Keywords: Hörmander-type spectral multiplier theorems; Non-negative self-adjoint operator; Weights; Heat semigroup; Plancherel-type estimate; Space of homogeneous type

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107 Singular integrals and weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109


E-mail addresses: [email protected] (X.T. Duong), [email protected] (A. Sikora), [email protected] (L. Yan). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.006

X.T. Duong et al. / Journal of Functional Analysis 260 (2011) 1106–1131

3. 4. 5. 6.

General spectral multiplier theorems on weighted spaces . . . . . . . . . . . . . . . . . . Proofs of Theorems 3.1 and 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two interpolation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Compact manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Laplace operators on irregular domains with Dirichlet boundary conditions . 6.4. Schrödinger operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Estimates on operator norms of holomorphic functional calculi . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1111 1115 1123 1126 1126 1127 1128 1129 1129 1130 1130

1. Introduction Suppose that L is a non-negative self-adjoint operator acting on L2 (X). Let E(λ) be the spectral resolution of L. By the spectral theorem, for any bounded Borel function F : [0, ∞) → C, one can define the operator ∞ F (L) =

F (λ) dE(λ),

(1.1)

0

which is bounded on L2 (X). A natural problem considered in the spectral multipliers theory is to give sufficient conditions on F and L which imply the boundedness of F (L) on various functional spaces defined on X. This topic has attracted a lot of attention and has been studied extensively by many authors: for example, for sub-Laplacian on nilpotent groups in [5,12], for sub-Laplacian on Lie groups of polynomial growth in [1], for Schrödinger operator on Euclidean space Rn in [17], for sub-Laplacian on Heisenberg groups in [26] and many others. For more information about the background of this topic, the reader is referred to [1,2,4,5,10,12,14,15,23] and the references therein. We also refer the reader to [37] and the references therein for examples of potential applications of the spectral multiplier results. We wish to point out [14], which is closely related to this paper. In [14], a sharp spectral multiplier for a non-negative self-adjoint operator L was obtained under the assumption of the kernel pt (x, y) of the analytic semigroup e−tL having a Gaussian upper bound. As there was no assumption on smoothness of the space variables of pt (x, y), the singular integral F (L) does not satisfy the standard kernel regularity condition of a so-called Calderón–Zygmund operator, thus standard techniques of Calderón–Zygmund theory are not applicable. The lacking of smoothness of the kernel was indeed the main obstacle in [14] and it was overcome by shrewd exploitation of the analyticity of the kernel pt (x, y) in variable t, together with a so-called Plancherel estimate, see Remark 2 after Corollary 3.4. We will now recall some of main features of the spectral multipliers theory. An interesting example of a spectral multiplier result comes from the paper [1] where Alexopoulos considers the operators acting on Lie groups of polynomial growth. He proved that if L is a group invariant Laplacian and n is the maximum of the local and global dimension of the group then F (L)

1108


is bounded on Lp (Rn ) for all 1 < p < ∞ if the function F is differentiable s times where s = [ n2 ] + 1 and satisfies k (k) λ F (λ) C for some constant C and k = 0, 1, . . . , s, see also Section 6.1 and Proposition 6.2 below. The philosophy is that we need function F to possess just more than n/2 derivatives (with suitable bounds) for F (L) to be bounded on all Lp spaces, 1 0

where δt F (λ) = F (tλ), F Wsp = (I − d 2 /dx 2 )s/2 F Lp and η is an auxiliary non-zero cutoff function such that η ∈ Cc∞ (R+ ). We note that condition (1.2) is actually independent of the choice of η. It is well known that condition (1.2) can be generalized with positive numbers s > 0 and it is sufficient to take real value s > n/2, see [2,14]. It is an interesting question when condition (1.2) can be replaced by the following weaker condition sup ηδt F Ws2 < ∞

(1.3)

t>0

for some s > n/2. Already for the standard Laplace operator on the Euclidean space Rn , the classical Fourier multiplier result of Hörmander [18] applied to radial functions says that the weaker Ws2 condition for any s > n/2 is enough to guarantee Lp boundedness of F () for all 1 < p < ∞, see also [5] for further discussion. Actually, replacing the Ws∞ norm in condition (1.2) by the Ws2 norm in condition (1.3) is essentially the same problem which one encounters in sharp Bochner–Riesz summability analysis, see [6,14,30,33,34]. Discussion of possibility of replacing condition (1.2) by (1.3) is one of the main themes of [14]. The aim of this paper is to extend the study of sharp spectral multipliers in [14] to the setting of weighted Lp spaces. It turns out that for a function F having more than n/2 suitable derivatives, the range of p that we can obtain for F (L) to be bounded depends also on the weight w. Most of the results of [14] follow from Theorems 3.1, 3.2 and 3.3 which are the main results of this paper; see Remark 1 after Corollary 3.4. We use the techniques developed in [14] to estimate the kernels of spectral multipliers. The new contribution of this paper is a development of an original technique to deal with singular integral nature of the considered spectral multipliers to obtain generalization of unweighted results described in [14] to weighted Lp spaces. This paper is organized as follows. In Section 2, we recall basic properties of spaces of homogeneous type, the class of Muckenhoupt weights and a sufficient condition for boundedness of weighted singular integrals from [3]. We state the main results on weighted spectral multipliers, Theorems 3.1 and 3.3 in Section 3. Section 4 is devoted to the proofs of these theorems. In Section 5, we use complex interpolation to obtain boundedness for spectral multipliers on weighed Lp spaces. In Section 6, we give applications of our results to various operators in different settings, including Laplace operators on homogeneous groups and on irregular domains of Euclidean spaces, elliptic pseudo-differential operators on compact manifolds, Schrödinger operators with positive potentials and holomorphic functional calculi of non-negative self-adjoint operators.


1109

2. Singular integrals and weights Let (X, d, μ) be a space endowed with a distance d and a non-negative Borel measure μ on X. Set B(x, r) = {y ∈ X: d(x, y) < r} and V (x, r) = μ(B(x, r)). We shall often just use B instead of B(x, r). Recall that (X, d, μ) satisfies the doubling volume property provided that there exists a constant C > 0 such that V (x, 2r) CV (x, r)

∀r > 0, x ∈ X,

(2.1)

∀r s > 0, x ∈ X.

(2.2)

more precisely if there exist n, Cn > 0 such that n V (x, r) r , Cn V (x, s) s

The parameter n is a measure of the doubling dimension of the space. It also follows from the doubling condition that there exist C and D, 0 D n so that d(x, y) D V (y, r) C 1 + V (x, r) r

∀r > 0, x, y ∈ X

(2.3)

uniformly for all x, y ∈ X and r > 0. Indeed, property (2.3) with D = n is a direct consequence of the triangle inequality for the metric d and (2.2). In many cases like the Euclidean space Rn or Lie groups of polynomial growth, D can be chosen to be 0. Muckenhoupt weights. Next we review the definitions of Muckenhoupt classes of weights. We use the notation h=

1 V (E)

E

h(x) dμ(x) E

and we often forget the measure and variable of the integrand in writing integrals. In what follows for any number or symbol s with value in [1, ∞] by s we denote it’s conjugate, that is 1s + s1 = 1. A weight w is a non-negative locally integrable function. We say that w ∈ Ap , 1 < p < ∞, if there exists a constant C such that for every ball B ⊂ X,

w 1−p

w B

p−1 C.

B

For p = 1, we say that w ∈ A1 if there is a constant C such Mw Cw a.e. where M denotes the uncentered maximal operator over balls in X, that is Mw(x) = sup B x

w. B

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The reverse Hölder classes are defined in the following way: w ∈ RH q , 1 < q < ∞, if there is a constant C such that for every ball B ⊂ X, 1/q

w

C

q

B

w .

B

The endpoint q = ∞ is given by the condition: w ∈ RH ∞ whenever, for any ball B, w(x) C

for a.e. x ∈ B.

w, B

Note that we have excluded the case q = 1 since the class RH 1 consists of all weights, and that is the way RH 1 is understood in what follows. We sum up some properties of the Ap and RH q classes in the following lemmas. Lemma 2.1. Suppose that (X, d, μ) is a metric, measure space, which satisfies doubling condition (2.1). Then the following properties hold for the weights classes Ap and RH q defined on (X, d, μ): (i) (ii) (iii) (iv) (v)

A1 ⊂ Ap ⊂ Aq for 1 < p q < ∞. RH ∞ ⊂ RH q ⊂ RH p for 1 p q < ∞. If w ∈ Ap , 1 0 and all Borel functions F such that supp F ⊆ [0, R],

K

√ 2 F ( m L ) (x, y) dμ(x)

C δR F 2Lq V (y, R −1 )

(3.1)

X

for some q ∈ [2, ∞]. Then for any bounded Borel function F such that supt>0 ηδt F Wsq < ∞, the operator F (L) is bounded on Lp (X, w) for all p and w satisfying r0 0

Note that Gaussian bound (GE) implies estimates (3.1) for q = ∞. This means that one can omit condition (3.1) if the case q = ∞ is consider. We describe the details in Theorem 3.2 below. Theorem 3.2. Let L be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy Gaussian bound (GE). Let s > n2 and let r0 = max(1, 2(n+D) 2s+D ). Then for any bounded Borel function F such that supt>0 ηδt F Ws∞ < ∞, the operator F (L) is bounded on Lp (X, w) for all p and w satisfying r0 0

Proof. Note that it was proved in Lemma 2.2 of [14], that for any Borel function F such that supp F ⊂ [0, R],

K

2 √

F ( m L ) (·, y) L2 (X)

2 = KF ( m√L ) (y, ·) L2 (X)

where F denotes the complex conjugate of F .

C F 2L∞ V (y, R −1 )

(3.2)


1113

This shows that estimate (3.1) always holds for q = ∞, and Theorem 3.2 follows from Theorem 3.1. 2 From the point of view of some applications of spectral multipliers the sharp results and the required number of derivatives are not essential for the final outcome, see for example [37]. For this kind of applications Theorem 3.2 is the best solution because using it one does not have to consider or prove condition (3.1). Nevertheless, Theorem 3.1 and condition (3.1) are of significant interest independent of their applications. In the case of standard Laplace operator condition (3.1) is equivalent with (1, 2) restriction theorem and both Theorem 3.1 and condition (3.1) are a new part of Bochner–Riesz analysis. Estimates (3.1) are also closely related to Strichartz and other dissipative type estimates. For further discussion of condition (3.1), see also [14]. It is not difficult to see that condition (3.1) with some q < ∞ implies that the set of point spectrum of the considered operator is empty because the Lq norm √ of characteristic function of any singleton subset of R is zero. Hence if q < ∞ then F ( m L ) does not depend on the value of F (0) because then the point spectrum is empty and the spectral projection on zero eigenvalue E({0}) = 0. Therefore if q < ∞ then one can skip |F (0)| in the concluding estimates of Theorem 3.1. See [14, (3.3)] for more detailed explanation. The fact that the point spectrum of the considered operator is empty implies also that for elliptic operators on compact manifolds condition (3.1) cannot hold for any q < ∞. To be able to study these operators as well, similarly as in [7,14] we introduce some variation of condition (3.1). For a Borel function F such that supp F ⊆ [−1, 2] we define the norm F N,q by the formula

F N,q =

2N 1 3N

sup

=1−N λ∈[

F (λ)q

1/q ,

−1 N ,N )

where q ∈ [1, ∞) and N ∈ Z+ . For q = ∞, we put F N,∞ = F L∞ . It is obvious that F N,q increases monotonically in q. The next theorem is a variation of Theorem 3.1. This variation can be used in case of operators with nonempty point spectrum, see also [7, Theorem 3.6] and [14, Theorem 3.2]. Theorem 3.3. Assume that μ(X) < ∞. Let L be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy Gaussian bound (GE). Let s > n2 and let r0 = max(1, 2(n+D) 2s+D ). Suppose that for any N ∈ Z+ and for all Borel functions F such that supp F ⊆ [−1, N + 1],

K

√ 2 F ( m L ) (x, y) dμ(x)

C δN F 2N,q V (y, N −1 )

(3.3)

X

for some q 2. Then for any bounded Borel function F such that supt>1 ηδt F Wsq < ∞, the operator F (L) is bounded on Lp (X, w) for all p and w satisfying r0 1

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We will discuss the proofs of Theorems 3.1 and 3.3 in Section 4. These results have the following corollary. Corollary 3.4. Let s > 10

(b) Alternatively assume in addition that the operator L satisfies the assumptions of Theorem 3.3 for some 2 q ∞, then

F (L) p q + F L∞ . sup C ηδ F s t p Ws L (X,w)→L (X,w) t>1

Proof. Suppose 1 < p < r0 and w ∈ Ap ∩ RH

r

( p0 )

. We have that w

1 − p−1

∈ A p . Then for f ∈ r0

L∞ c (X) (i.e. bounded with compact support),

,

F (L)f p = F (L)f (x)g(x) dμ(x) L (X,w) X

where the supremum is taken over all functions g ∈ L∞ c (X) such that g

Lp (X,w

1 − p−1

)

= 1. Let

(L) be the operator with multiplier F , the complex conjugate of F . Then F satisfies the same F estimates as F , and we have

F (L)f p f (x)F (L)g(x) dμ(x) = sup L (X,w) X

(L)g

sup f Lp (X,w) F

Lp (X,w

1 − p−1

)

Cf Lp (X,w) since p > r0 , and we can apply Theorems 3.1 or 3.3 to the weight w

1 − p−1

∈ A p . r0

2

Remarks. 1) Note that Theorems 3.1 and 3.3 imply the main results obtained in [14]. Indeed the trivial weight w = 1 is in all Ap classes, so under the assumptions of Theorems 3.1 and 3.3 the operator F (L) is bounded on all Lp spaces 1 < p < ∞. Note that for p < 2, Lp boundedness (L). Similarly to the results of F (L) follows by considering the adjoint operator F (L)∗ = F in [14] the important point of this paper is that if one can obtain (3.1) or (3.3) for some q < ∞ then one can prove stronger multiplier results than in case q = ∞. The estimates (3.1) for q = ∞ are not necessary because estimates (3.1) with q = ∞ follow from Gaussian bound assumption (GE), see Theorem 3.2. If one has (3.1) or (3.3) for q = 2, then this implies the


1115

sharp weighted Hörmander-type multiplier result. Actually, we believe that to obtain any sharp weighted Hörmander-type multiplier theorem one has to investigate conditions of the same type as (3.1) or (3.3), i.e. conditions which allow us to estimate the norm KF m√L (·, y)2L2 (X,μ) in terms of some kind of Lq norm of the function F . 2) We call hypothesis (3.1) or (3.3) the Plancherel estimates or the Plancherel conditions. For the standard Laplace operator on Euclidean spaces Rn , this is equivalent to (1, 2) Stein–Tomas restriction theorem (which is also the Plancherel estimate of the Fourier transform). Assumption that q 2 is not necessary in the proofs of Theorems 3.1 and 3.3. However we do not expect that there are any examples where estimates (3.1) or (3.3) hold with q < 2 because this would imply the Riesz summability for the index α < (n − 1)/2 which is false for the standard Laplace operator. 3) If we take s > n/2 in Theorems 3.1 and 3.3, then for every w ∈ A1 ∩ RH 2 , the operator F (L) maps L1 (X, w) into L1,∞ (X, w), that is, there is a constant C > 0, independent of f and λ, such that C w x ∈ X: F (L)f (x) > λ f L1 (X,w) , λ

λ > 0.

The proof follows from the line of Theorem 5.8 in [24], together with the proofs of Theorems 3.1 and 3.2 in [14], respectively. The details are left to the reader. 4. Proofs of Theorems 3.1 and 3.3 Recall that B = B(xB , rB ) is the ball of radius rB centered at xB . Given λ > 0, we will write λB for the ball with the same center as B and with radius rλB = λrB . We set U0 (B) = B,

and Uj (B) = 2j B\2j −1 B

for j = 1, 2, . . . .

(4.1)

As a preamble to the proof of Theorem 3.1, we record a useful auxiliary result. For a proof, see pp. 453–454, Lemma 4.3 of [14]. Lemma 4.1. (a) Suppose that L satisfies (3.1) for some q ∈ [2, ∞] and that R > 0, s > 0. Then for any > 0, there exists a constant C = C(s, ) such that C K m√ (x, y)2 1 + Rd(x, y) s dμ(x) δR F 2W q (4.2) F( L) s V (y, R −1 ) 2 + X

for all Borel functions F such that supp F ⊆ [R/4, R]. (b) Suppose that L satisfies (3.3) for some q ∈ [2, ∞] and that N > 8 is a natural number. Then for any s > 0, > 0 and function ξ ∈ Cc∞ ([−1, 1]) there exists a constant C = C(s, , ξ ) such that 2 C √ 1 + N d(x, y) s dμ(x) K δN F 2W q (4.3) F ∗ξ( m L ) (x, y) s V (y, N −1 ) 2 + X

for all Borel functions F such that supp F ⊆ [N/4, N].

1116


Proof of Theorem 3.1. We fix s such that s > n2 , and thus 2(n+D) 2s+D < 2. In this case, we take one parameter p0 in the sequel such that p0 belongs to the interval (max{ 2(n+D) 2s+D , 1}, 2). Let M ∈ N such that M > s/m, where m is the constant in (GE). We will show that for all balls B x,

F (L) I − e−rBm L M f p0 dμ

1/p0

1 CM |f |p0 p0 (x)

(4.4)

B

for all f ∈ L∞ c (X). Let us prove (4.4). Observe that supt>0 ηδt F Wsp ∼ supt>0 η δt GWsp where G(λ) = √ √ F ( m λ ). For this reason, we can replace F (L) by F ( m L ) in the proof. Notice that F (λ) = F (λ) − F (0) + F (0) and hence F

√ √ m m L = F (·) − F (0) L + F (0)I.

Replacing F by F − F (0), we may assume in the sequel = 0. Let ϕ ∈ Cc∞ (0, ∞) be ∞that F (0)− 1 a non-negative function satisfying supp ϕ ⊆ [ 4 , 1] and =−∞ ϕ(2 λ) = 1 for any λ > 0, and let ϕ denote the function ϕ(2− ·). Then F (λ) =

∞

∞ ϕ 2− λ F (λ) = F (λ),

=−∞

∀λ 0.

(4.5)

=−∞

√ This decomposition implies that the sequence N F ( m L ) converges strongly in L2 (X) to =−N √ F ( m L ) (see for instance, Reed and Simon [29, Theorem VIII.5]). For every ∈ Z, r > 0 and λ > 0, we set m M Fr,M (λ) = F (λ) 1 − e−(rλ) , m M Fr,M (λ) = F (λ) 1 − e−(rλ) . Given a ball B ⊂ X, we use the decomposition f = were defined in (4.1). We may write F

∞

j =0 fj

(4.6) (4.7)

in which fj = f χUj (B) , and Uj (B)

√ √ m M m m L 1 − e−rB L f = FrB ,M L f =

2 j =1

FrB ,M

∞ N √ √ m m L fj + lim FrB ,M L fj , (4.8) N →∞

=−N j =3

where the sequence converges strongly in L2 (X). From Gaussian condition (GE), we have that for any t > 0, √ e−tL f Lp (X) Cf Lp (X) . m p This, in combination with L -boundedness of the operator F ( L ) (see Theorem 3.1 [14]), gives that for all balls B x,


F r

√ p m L fj 0 dμ B ,M

1/p0

1117

√

m V (B)−1/p0 FrB ,M L fj Lp0 (X)

B

CV (B)−1/p0 fj Lp0 (X) 1 CM |f |p0 p0 (x) for j = 1, 2. Fix j 3. Let p1 2 and B x,

1 p0

−

1 p1

= 12 . By Hölder’s inequality, we have that for all balls

√ p m L fj 0 dμ rB ,M

F

(4.9)

1/p0

B − p1

F

√

m L fj Lp1 (B)

− p1

F

√

m L Lp0 (U

V (B) V (B)

1

1

rB ,M rB ,M

p j (B))→L 1 (B)

jn √

1

m C2 p0 V (B) 2 FrB ,M L Lp0 (U

Let

1 p0

=

θ 1

+

1−θ 2

and

1 p1

fj Lp0 (X)

p j (B))→L 1 (B)

1 M |f |p0 p0 (x).

(4.10)

= θ2 , that is θ = 2( p10 − 12 ). By interpolation,

√

m

F L Lp0 (U rB ,M

p j (B))→L 1 (B)

√ 1−θ m FrB ,M L L2 (U

j

(B))→L∞ (B)

F

rB ,M

√ θ m L L2 (B)→L∞ (U

j (B))

.

(4.11)

√ Next we estimate FrB ,M ( m L )L2 (Uj (B))→L∞ (B) . For every ∈ Z, let KF ( m√L ) (y, z) be the rB ,M √ Schwartz kernel of operator FrB ,M ( m L ). Then we have

√ 2 m

F L L2 (U (B))→L∞ (B) rB ,M j 2 √ K = sup F ( m L ) (y, z) dμ(z) y∈B Uj (B)

C2

−2sj

rB ,M

−2s 2 rB sup y∈B

K

Fr

B

2s 1 + 2 d(y, z) dμ(z).

√ (y, z)2 m ,M ( L )

(4.12)

X

We then apply Lemma 4.1 with F = FrB ,M and R = 2 to obtain

K

Fr

B

X

2s 1 + 2 d(y, z) dμ(z)

√ (y, z)2 m ,M ( L )

2 Cs

q.

δ F V (y, 2− ) 2 rB ,M Ws

(4.13)

1118

X.T. Duong et al. / Journal of Functional Analysis 260 (2011) 1106–1131 q

Now for any Sobolev space Ws (R), if k is an integer greater than s, then

−(2 rB t)m M

δ F q 2 rB ,M Wsq = ϕ(t)F 2 t 1 − e Ws

m M

k 1 δ [ϕ F ] q C 1 − e−(2 rB t) C ([ 4 ,1]) 2 Ws

mM

δ [ϕ F ] q . C min 1, 2 rB 2 W

(4.14)

s

Note that for all y ∈ B, B ⊂ B(y, 2rB ) so by (2.2) n 1 C V (y, 2rB ) C sup max 1, 2 rB . − − V (y, 2 ) V (B) y∈B V (y, 2 )V (B)

(4.15)

Hence by (4.14) and (4.15),

F

rB ,M

√

m L L2 (U

C 2

−2sj

∞ j (B))→L (B)

−2s 2mM n 2 rB max 1, 2 rB min 1, 2 rB

1 V (B)

1/2

δ [ϕ F ] q . 2 W s

(4.16) √ We now turn to estimate the term F rB ,M ( m L )L2 (B)→L∞ (Uj (B)) . The calculations symmetric to (4.12), (4.13) and (4.14) with supy∈B replaced by supz∈Uj (B) yield,

F

rB ,M

√

m L L2 (B)→L∞ (U

j (B))

−2s 2mM sup C 2−2sj 2 rB min 1, 2 rB

1 − z∈Uj (B) V (z, 2 )

1/2

δ [ϕ F ] q . 2 W s

Next by (2.2) and (2.3) V (z, rB ) 1 d(z, xB ) D 1 C sup × 1+ sup − − rB V (xB , rB ) z∈Uj (B) V (z, 2 ) z∈Uj (B) V (z, 2 ) C

n 2j D max 1, 2 rB . V (B)

Hence

F

rB ,M

√

m L L2 (B)→L∞ (U

j (B))

−2s j D 2mM n max 1, 2 rB C 2−2sj 2 rB 2 min 1, 2 rB

1 V (B)

1/2

δ [ϕ F ] q . 2 W s

(4.17) It then follows from estimates (4.16) and (4.17), in combination with (4.11) and (4.10) that


√ p m F L fj 0 dμ

1119

1/p0

r,M

B

C2

−j s+ pj n + j Dθ 2 0

−s mM n 2 rB max 1, 2 rB 2 min 1, 2 rB

1 × M |f |p0 p0 (x) sup δ2 [ϕ F ] W q .

(4.18)

s

∈Z

Therefore, 1/p0 ∞ ∞ √ p0 m F L fj dμ r,M

j =3 =−∞

C

∞

B

2

−j s+ pj n + j Dθ 2

∞ −s mM n 2 rB max 1, 2 rB 2 min 1, 2 rB

0

j =3

=−∞

1 × M |f |p0 p0 (x) sup δ2 [ϕ F ] W q s

∈Z

C

∞

2

D ( n+D p −(s+ 2 ))j

0

j =3

−s+ n 2 + 2 rB

mM−s 2 rB : 2 rB 1

: 2 rB >1

1 × M |f |p0 p0 (x) sup δ2 [ϕ F ] W q s

∈Z

C

∞ j =3

2

D ( n+D p −(s+ 2 ))j 0

1 M |f |p0 p0 (x) sup δ2 [ϕ F ] W q ∈Z

1 CM |f |p0 p0 (x) sup δ2 [ϕ F ] W q . ∈Z

s

s

(4.19)

Here, the second inequality is obtained by using condition θ = 2( p10 − 12 ), and the third inequality follows from the convergence of power series with common ratio 1/2. In the last inequality we have used the fact that p0 > 2(n+D) 2s+D . Combining estimates (4.9) and (4.19), we have therefore proved (4.4), and then estimate (2.4) m holds for T = F (L) and ArB = I − (I − e−rB L )M . Note also that estimate (2.5) always holds for m m ArB = I − (I − e−rB L )M . Indeed note that T = F (L) and ArB = I − (I − e−rB L )M commutes so it is enough to show that 1 ArB f L∞ (B) CM |f |p0 p0 (x). It is not difficult to see that it is enough to prove the above inequality for p0 = 1. However m m ArB = I − (I − e−rB L )M is a finite linear combination of the terms e−j rB L , j = 1, . . . , M, which all satisfy Gaussian bounds and the above inequality and in turn (2.5) follows from that observation.

1120


It then follows from Theorem 2.3 that for all p > p0 > r0 = 2(n+D) 2s+D , the operator F (L) is bounded on Lp (X, w) provided that w ∈ A p . On the other hand, we note that p0

Ap = r0

p0 >r0

Ap. p0

This implies for all p > r0 and all w ∈ A p , that the operator F (L) is bounded on Lp (X, w). r0

2

Proof of Theorem 3.3. Note that the condition μ(X) < ∞ implies that X is bounded. Hence X = B(x0 , r0 ) for some x0 ∈ X and 0 < r0 < ∞ [24]. It follows from condition (2.3) that for any x ∈ X, V (x0 , 1) C(1 + d(x, x0 ))D V (x, 1) CV (x, 1). This shows that for any x, y ∈ X, |Ke−L (x, y)| CV (x0 , 1)−1 . As a consequence,

(4.20) max e−L L1 (X)→L2 (X) , e−L L2 (X)→L∞ (X) C. On √ the other hand, for any bounded Borel function F such that supp F ⊆ [0, 16], the operator F ( m L )e2L is bounded on L2 (X). This, together with (4.20), yields

√ √

m

F m L 1 = e−L F L e2L e−L L1 (X)→L∞ (X) L (X)→L∞ (X)

√

m e−L L1 (X)→L2 (X) F L e2L L2 (X)→L2 (X) e−L L2 (X)→L∞ (X) CF L∞ < ∞. √ This implies that the kernel KF ( m√L ) (x, y) of the operator F ( m L ) satisfies sup KF ( m√L ) (x, y) C < ∞. y∈X

Hence, for any x ∈ X,

√ F m L f (x) = K m√ (x, y)f (y) dμ(y) F( L) X

C

f (y) dμ(y)

X

CM(f )(x), and for any 1 < p < ∞ and w ∈ Ap ,

√

F m L f p

L (X,w)

C M(f ) Lp (X,w) Cf Lp (X,w) .

Therefore, in order to prove Theorem 3.3, we can assume that supp F ⊂ [8, ∞]. Following the proof of Theorem 3.1, we set F (λ) = ϕ(2− λ)F (λ), and F˜ =

∞

F ∗ ξ,

=3

where ξ is a function defined in (b) of Lemma 4.1.


1121

By repeating √ the proof of Theorem 3.1 and using (4.3) in place of (4.2) we can prove that the operator F˜ ( m L ) is bounded on Lp (X, w) for all p and w satisfying (i) and (ii) in Theorem 3.3. To prove Theorem 3.3, it follows by Theorem 2.3 again that it suffices to show that for all balls B x,

√ √ F m L − F˜ m L I − e−rBm L M f p0 dμ

1/p0

1 CM |f |p0 p0 (x)

(4.21)

B

for all f ∈ L∞ c (X). (λ) = (F (λ) − F ∗ ξ(λ))(1 − Let us prove (4.21). For every 3 and r > 0, we set Hr,M m M −(rλ) ) , λ > 0. (Note that supp HrB ,M ⊆ [0, 2 + 1].) Now for a given ball B ⊂ X, we put e ∞ f = j =0 fj , where fj = f χUj (B) , and Uj (B) were defined in (4.1). We may write 2 √ √ √ √ m M m m m m −rBm L M ˜ F L −F L I −e f= L − F˜ L I − e−rB L fj F j =1

+ lim

N →∞

N ∞ =3 j =3

HrB ,M

√ m L fj .

(4.22)

A similar argument as in the proof of Theorem 3.1 gives the desired estimates for j = 1, 2. Next, fix j 3. For every 3, let KH ( m√L ) (y, z) be the Schwartz kernel of operator rB ,M √ 1 θ 1 1 HrB ,M ( m L ). Let p10 − p11 = 12 , and denote by p10 = θ1 + 1−θ 2 and p1 = 2 , that is θ = 2( p0 − 2 ). Following (4.10) and (4.11), we use Hölder’s inequality and interpolation again to obtain that for all balls B x,

√ p m L fj 0 dμ rB ,M

H

1/p0

B jn

1 √ 1−θ 1 m C2 p0 V (B) 2 M |f |p0 p0 (x) HrB ,M L L2 (U (B))→L∞ (B) j

√ θ m × H rB ,M L L2 (B)→L∞ (U (B)) . j

The Hölder inequality, together with condition that X is bounded give

√ 2 m

H L L2 (U (B))→L∞ (B) rB ,M j 2 √ K = sup H ( m L ) (y, z) dμ(z) y∈B Uj (B)

rB ,M

−2s C 2j rB sup y∈B

K

Hr

B

X

√ (y, z)2 d(y, z)2s m ,M ( L )

dμ(z)

(4.23)

1122


−2s CX 2j rB sup y∈B

K

Hr

( B ,M

√ m

2 dμ(z)

L ) (y, z)

X

j −2s 2 CX

δ H 2 rB sup 2 rB ,M 2 ,q , − y∈B V (y, 2 )

(4.24)

where the last inequality follows from the fact that supp HrB ,M ⊆ [0, 2 + 1], and then from (3.3) with N = 2 , we have that 2

2 C √

K

Hr ,M ( m L ) (y, z) dμ(z) V (y, 2− ) δ2 HrB ,M 2 ,q . B X

From the expression HrB ,M (λ) = (F (λ) − F ∗ ξ(λ))(1 − e−(rB λ) )M , one obtains m

−(2 rB λ)m M

δ H 2 rB ,M 2 ,q = δ2 F (λ) − F ∗ ξ(λ) 1 − e 2 ,q mM

δ F (λ) − F ∗ ξ(λ) . C min 1, 2 rB 2 2 ,q

(4.25)

Everything then boils down to estimating · 2 ,q norm of δ2 [F (λ) − F ∗ ξ(λ) ]. We make the following claim. For its proof, we refer to p. 26, claim (3.29) of [7] or p. 459, Proposition 4.6 of [14]. Proposition 4.2. Suppose that ξ ∈ Cc∞ is a function such that supp ξ ⊂ [−1, 1], ξ 0, ξˆ (0) = 1, ξˆ (κ) (0) = 0 for all 1 κ [s] + 2 and set ξN (t) = N ξ(N t). Assume also that supp G ⊂ [0, 1]. Then G − G ∗ ξN N,q CN −s GWsq for all s > 1/q. By Proposition 4.2

δ F (λ) − F ∗ ξ(λ) = δ [ϕ F ] − ξ ∗ δ [ϕ F ] C2−s δ [ϕ F ] q , 2 2 2 2 2 2 ,q 2 ,q W s

and thus

mM

−s

δ H

δ [ϕ F ] q . min 1, 2 rB 2 2 rB ,M 2 ,q C2 W s

Substituting (4.26) back into (4.24), we then use the doubling property (2.2) to obtain

H

rB ,M

√ 2 m L L2 (U

∞ j (B))→L (B)

2 2mM n

C −2sj −2s 2 2 rB max 1, 2 rB δ2 [ϕ F ] W q , min 1, 2 rB s V (B)

which is exactly the same estimate as in (4.16).

(4.26)


1123

Following the proof√of Theorem 3.1, an argument as above shows the same estimate (4.17) for the term H rB ,M ( m L )L2 (B)→L∞ (Uj (B)) . The rest of the proof of (4.21) is just a repetition of the proof of Theorem 3.1, so we skip it. Hence, we complete the proof of Theorem 3.3 when X has a finite measure, i.e., μ(X) < ∞. 2 5. Two interpolation results In this section we continue to assume that L is a non-negative self-adjoint operator on L2 (X), which has a kernel pt (x, y) satisfying a Gaussian upper bound (GE). Using interpolation, other conditions on the weight can be found which guarantee that F (L) is a bounded operator. We first prove the following result. Theorem 5.1. Let s > n2 and let r0 = max{ 2(n+D) 2s+D , 1}. Suppose that the operator L satisfies condition (3.1) with some q ∈ [2, ∞]. If 1 0 ηδt F Wsq < ∞, the operator F (L) is bounded on Lp (X, w). Moreover,

F (L) p q + F (0) . sup C ηδ F s t p Ws L (X,w)→L (X,w) t>0

Proof. We will derive Theorem 5.1 from Theorem 3.1 by using Proposition 2.4 and the characterization of Ap functions that if w ∈ Ap , then there are A1 weights u and v such that w = uv 1−p [22]. Following the proof of Theorem 2 of [23], we fix p, 1 < p < ∞, and w so that w r0 ∈ Ap where r0 =

2(n+D) 2s+D .

−1

We have that w r0 = uv 1−p , u, v ∈ A1 , or w = ur0 v −1

w = ur0 v

1−p r0

1−p r0

. Next, write this as

t 1−t = uα v β uγ v δ = w0t w11−t ,

in which αt + γ (1 − t) = r0−1 ,

(5.1)

βt + δ(1 − t) = r0−1 (1 − p).

(5.2)

Then in order to use Proposition 2.4 for weights which satisfy Theorem 3.1, we require 1 − r−1

w0

∈ A r , r0

w1 ∈ A q , r0

t=

1 < r < min r0 , p ,

(5.3)

q > max{r0 , p},

(5.4)

q −p . q −r

(5.5)

Recall that u ∈ A1 (similarly v ∈ A1 ) implies u Cu(x) for almost all x ∈ B. B

1124


Therefore, if α > 0 and β < 0, letting s =

1 − r−1

s−1

1

B α

β

u− r−1 v − r−1

B

we have

w0(r−1)(s−1)

w0 B

r r0 ,

β

α

s−1

u (r−1)(s−1) v (r−1)(s−1) B

β s−1 − α r−1 r−1 α β − r−1 (r−1)(s−1) u v v u

B

B

B

B

C, if r r − 1 = − r + 1 and β = −(r − 1); α = (r − 1) r0 r0 1 − r−1

this is w0

∈ A r for these values of α and β. Similarly, we can show w1 ∈ A q if γ = 1 r0

r0

and δ = −( rq0 − 1). Using these values of α and γ , we have (5.1) if t = r.1 Next, solving (5.2) for q, we get q = r (p − 1). This value of q also satisfies (5.5). Therefore, if we choose r < min{r0 , p} close enough to 1 so that q = r (p − 1) > max{r0 , p}, then (5.1)–(5.5) hold. This proves Theorem 5.1. 2 If X = Rn then Theorem 5.1 can be strengthen for the following polynomial weights. When w(x) = |x|β , we have w ∈ Ap if −n < β < n(p − 1). Applying Theorem 3.1 and Theorem 3 of [23] to such w and using interpolation with change of measures, we have the following theorem. Theorem 5.2. Let s > n2 . Suppose that the operator L satisfies condition (3.1) with some q ∈ [2, ∞]. If 1 0 ηδt F Wsq < ∞, the operator F (L) is bounded on Lp (Rn , |x|β ). In addition,

F (L) p n β Cs sup ηδt F Wsq + F (0) . L (R ,|x| )→Lp (Rn ,|x|β ) t>0

In particular, if s < n and and p = ( ns ) .

n s

< p < ( ns ) , we get −n < β < n(p − 1); we may also take p =

n s

Proof. The proof of Theorem 5.2 can be obtained by making minor modifications with the proof of Theorem 3 in [23] and using Theorem 3.1. We give a brief argument of this proof for completeness and convenience of the reader. Notice that −n −sp if n/s p, and n(p − 1) sp if p (n/s) . Therefore, for s < n the conclusion of Theorem 5.2 can be divided into three cases:


n and − sp < β < n(p − 1), s n n p and − n < β < n(p − 1), s s n < p < ∞ and − n < β < sp. s 1 0

√ 2

F ( L )

2→∞

∞ 2 = C F (t) t n−1 dt. 0

See for example Eq. (7.1) of [14] or [5, Proposition 10]. It follows from the above equality that the operator L satisfies estimate (3.1) with q = 2. Hence Theorem 3.1 holds for spectral multipliers F (L) with q = 2, D = 0 and n = n0 = n∞ . 2


1127

This result can be extended to “quasi-homogeneous” operators acting on homogeneous groups, see [31] and [14]. In the setting of general Lie groups of polynomial growth, spectral multipliers were investigated by Alexopoulos. The following weighted spectral multiplier result extends Alexopoulos’s unweighted result in [1]. Proposition 6.2. Let L be a group invariant operator acting on a Lie group G of polynomial growth defined by (6.1). Then Theorem 3.2 holds for spectral multipliers F (L) with the doubling dimension n = max{n0 , n∞ } and D = 0. Proof. It is well known that the heat kernel corresponding to the operator L satisfies Gaussian bound (GE) so the operator L satisfies estimate (3.1) for q = ∞, see the proof of Theorem 3.2 above and Lemma 2.2 of [14]. Hence Theorem 3.2 holds for spectral multipliers F (L). 2 6.2. Compact manifolds For a general non-negative self-adjoint elliptic operator on a compact manifold, Gaussian bound (GE) holds by general elliptic regularity theory. Further, one has the Avakumovi˘c– Agmon–Hörmander theorem. Theorem 6.3. Let L be a non-negative elliptic pseudo-differential operator of order m on a compact manifold X of dimension n. Then

χ[R,R+1] L1/m 2 1

L (X)→L2 (X)

CR n−1 ,

∀R ∈ R+ .

(6.3)

Theorem 6.3 was proved by Hörmander [19]. This theorem has the following useful consequence. Corollary 6.4. Condition (3.3) with q = 2 holds for non-negative elliptic pseudo-differential operators on compact manifolds. Proof. By spectral theorem

2 √

F ( m L ) (·, y) L2 (X)

sup K y∈X

N

√

χ[−1, ] F m L 2 1

1/2

L (X)→L2 (X)

=1

CN n/2 δN F N,2 as required.

2

The importance of estimate (6.3) for multiplier theorems was noted by Sogge [33], who used it to establish the convergence of the Riesz means up to the critical exponent (n − 1)/2 (see also [6] and [30]). Proposition 6.5. Suppose that L is a non-negative self-adjoint elliptic differential operator of order m 2 acting on a compact Riemannian manifold X of dimension n. Then the operator L

1128


satisfies estimate (3.3) for q = 2, and hence Theorem 3.3 holds for spectral multipliers F (L) under the same conditions with q = 2, D = 0 and with the doubling dimension n. That is the exponent n in (2.2) is equal to the topological dimension of the manifold X. Proof. This result is a direct consequence of Theorem 3.3 and Corollary 6.4.

2

Proposition 6.5 applied to an elliptic operator on a compact Lie group gives a stronger result than Proposition 6.2. One can say that for elliptic operators on a compact Lie group Proposition 6.1 holds. However, we do not know if the Avakumovi˘c–Agmon–Hörmander condition holds for sub-elliptic operators on a compact Lie group (see also [7]). Hence, Proposition 6.2 gives the strongest known result for sub-elliptic operators on a compact Lie group. 6.3. Laplace operators on irregular domains with Dirichlet boundary conditions Let Ω be a connected open subset of Rn . Note that if the boundary of Ω is not smooth enough, then Ω is not necessarily a homogeneous space because the doubling condition might not hold. In this section we are interested in dealing with weighted norm estimates in those contexts. As it is pointed out in [13], one can extend the singular operators defined in Ω to the space Rn . Since there is no assumption on the regularity of the kernels in space variables, the extension of the kernel still satisfies similar conditions. Given T , a bounded linear operator on Lp (Ω), 1 < p < ∞, the extension of T to Rn is defined as Tf (x) = T (f χΩ )(x)χΩ (x) for f ∈ Lp (Rn ). Then, T is bounded on Lp (Ω) if and only if T is bounded on Lp (Rn ). If K is the kernel of T , then y) = K(x, y) for (x, y) ∈ Ω × Ω and K(x, y) = 0 the associated kernel of T is given by K(x, otherwise. As it is observed in [13], the assumptions on the kernels do not involve their regularity so they imply similar properties on the kernels of the extended operators. We are going to use the notation Ap (Rn ) in order to make clear that the Muckenhoupt weights are considered in the whole space Rn . The following result gives examples of singular integral multipliers on spaces without the doubling condition. Proposition 6.6. Suppose that Ω is the Laplace operator with Dirichlet boundary condition Ω ⊂ Rn . Let s > n/2 and r0 = max(1, n/s). Then for any bounded Borel function F such that supt>0 ηδt F Ws∞ < ∞, the operator F (Ω ) is bounded on Lp (Ω, w) for all p and w satisfying r0 0

Proof. Note that 0 Kexp(−tΩ ) (x, y)

1 |x − y|2 exp − 4t (4πt)n/2

(see e.g., Example 2.18 [9]). That is the heat kernels corresponding to Ω satisfy Gaussian bound (GE), and the operator Ω satisfies estimate (3.1) for q = ∞. Then, Proposition 6.6 follows from estimate (3.2) and Theorem 3.2 applied to the extended operator F (Ω ). Hence the same weighted norm estimates hold for the original operator F (Ω ). 2


1129

6.4. Schrödinger operators In this section we discuss applications of our main results to spectral multipliers of Schrödinger operators. Let be the standard Laplace operator acting on Rn . We consider the Schrödinger operator L = − + V where V : Rn → R, V ∈ L1loc (Rn ) and V 0. The operator L is defined by the quadratic form. If pt (x, y) denotes the heat kernel corresponding to L then as a consequence of the Trotter product formula 0 pt (x, y) p˜ t (x, y),

(6.4)

where p˜ t (x, y) denotes the standard Gauss heat kernel corresponding to , see also [27, Section 2.3]. The estimate (6.4) holds also for heat kernel pt (x, y) of Schrödinger operator with electromagnetic potentials, see [32, Theorem 2.3] and [14, (7.9)]. For the Schrödinger operator in this setting, estimate (3.1) holds for q = ∞ as in the next result. Proposition 6.7. Assume that L = − + V where is the standard Laplace operator acting on Rn and V ∈ L1loc (Rn ) is a non-negative function. Then the operator L satisfies estimate (3.1) for q = ∞, and hence Theorem 3.2 holds for spectral multipliers F (L) under the same conditions with q = ∞, D = 0 and the doubling constant n. We note that under suitable additional assumptions this result can be extended by a similar proof to situation of magnetic Schrödinger operators acting on a complete Riemannian manifold with non-negative potentials. Proof. This result is a consequence of (6.4) and Theorem 3.2.

2

6.5. Estimates on operator norms of holomorphic functional calculi For θ > 0, we put θ = {z ∈ C − {0}: | arg z| < θ }. Let Fbe a bounded holomorphic function on θ . By F θ,∞ we denote the supremum of F on θ . We are interesting in finding sharp bounds, in terms of θ , of the norm of F (L) as the operator acting on Lp (X, w). The following proposition, which is a weighted version of [14, Proposition 8.1], is a consequence of Theorem 3.2. Proposition 6.8. Let L be an operator satisfying assumptions of Theorem 3.2. Let s > n2 and let p r0 = max{1, 2(n+D) 2s+D }. Then the operator F (L) is bounded on L (X, w) for all p and w satisfying r0 0, r0 0

∀k ∈ Z+ .

k (k) ∞ C sup For any > 0, supt>0 ηδt F Wk− λ>0 |λ F (λ)| so by interpolation

sup ηδt F Ws∞ t>0

C F θ,∞ . θ s+

Applying the above inequality and Theorem 3.2 we obtain Proposition 6.8 (see also, Theorem 4.10 [8]). 2 Acknowledgments The authors are grateful to the referee who gave detailed comments and suggestions for improving the original manuscript. This work was started during the third named author’s stay at Macquarie University. L.X. Yan would like to thank the Department of Mathematics of Macquarie University for its hospitality. The research of X.T. Duong was supported by Australia Research Council (ARC). The research of L.X. Yan was supported by Australia Research Council (ARC) and NNSF of China (Grant Nos. 10771221 and 10925106). References [1] G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. 46 (1994) 457– 468. [2] G. Alexopoulos, Spectral multipliers for Markov chains, J. Math. Soc. Japan 56 (3) (2004) 833–852. [3] P. Auscher, J.M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights, Adv. Math. 212 (2007) 225–276. [4] S. Blunck, A Hörmander-type spectral multiplier theorem for operators without heat kernel, Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 449–459. [5] M. Christ, Lp bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991) 73–81. [6] M. Christ, C.D. Sogge, The weak type L1 convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988) 421–453. [7] M. Cowling, A. Sikora, A spectral multiplier theorem on SU(2), Math. Z. 238 (2001) 1–36. [8] M. Cowling, I. Doust, A. McIntosh, A. Yagi, Banach spaces operators with a bounded H ∞ functional calculus, J. Aust. Math. Soc. 60 (1996) 51–89. [9] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1989. [10] L. De Michele, G. Mauceri, H p multipliers on stratified groups, Ann. Mat. Pura Appl. 148 (1987) 353–366. [11] J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., Providence, 2000. [12] X.T. Duong, From the L1 norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups, Pacific J. Math. 173 (1996) 413–424. [13] X.T. Duong, A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999) 233–265. [14] X.T. Duong, E.M. Ouhabaz, A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002) 443–485. [15] G. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, 1982. [16] J. García-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud., vol. 116, North-Holland, Amsterdam, 1985. [17] W. Hebisch, A multiplier theorem for Schrödinger operators, Colloq. Math. 60/61 (1990) 659–664. [18] L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960) 93–140.


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[19] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968) 193–218. [20] A. Hulanicki, E.M. Stein, Marcinkiewicz multiplier theorem for stratified groups, unpublished manuscript. [21] R. Johnson, C.J. Neugebauer, Change of variable results for Ap and reverse Hölder RHr -classes, Trans. Amer. Math. Soc. 328 (1991) 639–666. [22] P. Jones, Factorization of Ap weights, Ann. of Math. 111 (1980) 511–530. [23] D.S. Kurtz, R.L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979) 343–362. [24] J.M. Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2004) 113–145. [25] G. Mauceri, S. Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (1990) 141–154. [26] D. Müller, E.M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. 73 (1994) 413–440. [27] E.M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monogr. Ser., vol. 31, Princeton Univ. Press, 2005. [28] D.W. Robinson, Elliptic Operators and Lie Groups, The Clarendon Press, Oxford University Press, New York, 1991. [29] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. I, Academic Press, 1980. [30] A. Seeger, C.D. Sogge, On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J. 59 (1989) 709–736. [31] A. Sikora, On the L2 → L∞ norms of spectral multipliers of “quasi-homogeneous” operators on homogeneous groups, Trans. Amer. Math. Soc. 351 (9) (1999) 3743–3755. [32] B. Simon, Maximal and minimal Schrödinger forms, J. Operator Theory 1 (1979) 37–47. [33] C.D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math. 126 (1987) 439–447. [34] C.D. Sogge, Fourier Integral in Classical Analysis, Cambridge University Press, Cambridge, 1993. [35] E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956) 482–492. [36] E.M. Stein, G. Weiss, Interpolation of operators with change of measure, Trans. Amer. Math. Soc. 87 (1958) 159– 172. [37] R.S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series, Math. Res. Lett. 12 (2005) 269–274. [38] J. Strömberg, A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., vol. 1381, Springer-Verlag, Berlin, 1989. [39] N. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups, Cambridge Univ. Press, London, 1993.


Global weak solutions for a two-component Camassa–Holm shallow water system Chunxia Guan, Zhaoyang Yin ∗ Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China Received 24 February 2010; accepted 30 November 2010

Communicated by J. Bourgain

Abstract In this paper, we prove the existence of global weak solution for an integrable two-component Camassa– Holm shallow water system provided the initial data satisfying some certain conditions. © 2010 Elsevier Inc. All rights reserved. Keywords: An integrable two-component Camassa–Holm shallow water system; Global existence; Weak solution

1. Introduction In this paper we consider the following integrable two-component Camassa–Holm shallow water system:

mt + 2ux m + umx + σρρx = 0, ρt + (uρ)x = 0,

t > 0, x ∈ R, t > 0, x ∈ R,

(1.1)

where m = u − uxx , σ = ±1. Eq. (1.1) was recently derived by Constantin and Ivanov [18] in the context of shallow water theory. The variable u(t, x) describes the horizontal velocity of the fluid and the variable ρ(t, x) is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units [18]. The case σ = 1 (σ = −1) corresponds to the situation in which the gravity acceleration points downwards (upwards) [18]. Eq. (1.1) * Corresponding author.

E-mail addresses: [email protected] (C. Guan), [email protected] (Z. Yin). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.015

C. Guan, Z. Yin / Journal of Functional Analysis 260 (2011) 1132–1154

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with σ = −1 was originally proposed by Chen et al. in [6] and Falqui in [32]. Eq. (1.1) with σ = −1 is identified with the first negative flow of the AKNS hierarchy and has peakon and multi-kink solutions [6]. The extended N = 2 supersymmetric Camassa–Holm equation was presented recently by Popowicz in [43]. The mathematical properties of Eq. (1.1) have been studied in many works, cf. [6,18,29,36]. For ρ ≡ 0, Eq. (1.1) becomes the Camassa–Holm equation, modeling the unidirectional propagation of shallow water waves over a flat bottom. Here u(t, x) stands for the fluid velocity at time t in the spatial x direction [4,21,28,35,37,38]. The Camassa–Holm equation is also a model for the propagation of axially symmetric waves in hyperelastic rods [24,25]. It has a bi-Hamiltonian structure [8,33] and is completely integrable [4,10]. Also there is a geometric interpretation of Eq. (1.1) in terms of geodesic flow on the diffeomorphism group of the circle [20,39]. Its solitary waves are peaked [5]. They are orbitally stable and interact like solitons [1,23]. The peaked traveling waves replicate a characteristic for the waves of great height – waves of largest amplitude that are exact solutions of the governing equations for water waves, cf. [11,17,45]. Recently, it was claimed in the paper [40] that the equation might be relevant to the modeling of tsunami, see also the discussion in [19]. The Cauchy problem and initial–boundary value problem for the Camassa–Holm equation have been studied extensively [13,14,26,30,31,41,44,48]. It has been shown that this equation is locally well-posed [13,14,26,41,44] for initial data u0 ∈ H s (R), s > 32 . More interestingly, it has global strong solutions [9,13,14] and also finite time blow-up solutions [9,12–14,16,26,41,44]. On the other hand, it has global weak solutions in H 1 (R) [2,3,15,22,47]. The advantage of the Camassa–Holm equation in comparison with the KdV equation lies in the fact that the Camassa–Holm equation has peaked solitons and models wave breaking [5,12] (by wave breaking we understand that the wave remains bounded while its slope becomes unbounded in finite time [46]). For ρ ≡ 0, the Cauchy problems of Eq. (1.1) with σ = −1 and with σ = 1 have been discussed in [29] and [18], respectively. Recently, a new global existence result and several new blow-up results of strong solutions for the Cauchy problem of Eq. (1.1) with σ = 1 were obtained in [34]. However, the existence and uniqueness of global weak solutions to Eq. (1.1) have not been discussed yet. Note that Eq. (1.1) with σ = −1 or with σ = 1 has peakon solutions which are not strong solutions. Our aim of this paper is to prove the existence of global week solutions to Eq. (1.1) with σ = 1 provided the initial data satisfying some certain conditions. We hope that our result sheds some light on important physical phenomena of Eq. (1.1) with σ = 1 such as wave breaking. Up to now, we have no uniqueness result for the obtained global weak solutions to Eq. (1.1) with σ = 1. This problem will be discussed later on. Since Eq. (1.1) with σ = 1 is a system, there are more difficulties in analyzing it than a single equation. The main difficulties are the mutual effect between two components ρ and u and the estimates of ux and ρ. One cannot follow directly the same argument as in [22] or in [7,47] to deal with this problem. To solve the problem, we combine the method in [22] with the method in [7,47] to show the existence of global weak solution to Eq. (1.1) with σ = 1 provided the initial data satisfying some certain conditions. The paper is organized as follows. In Section 2, we recall and present some useful lemmas which will be used in the sequel. In Section 3, we prove the existence of global weak solution to Eq. (1.1) with σ = 1. Notation. In the following, we denote by ∗ the spatial convolution. Given a Banach space Z, we denote its norm by · Z .

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2. Preliminaries In this section, we will recall and present some useful lemmas which will be used in the sequel. Notice that in the hydrodynamical derivation of Eq. (1.1) [18] it is required that u(t, x) → 0 and ρ(t, x) → 1 as |x| → ∞, at any instant t. Then, letting ρ¯ = ρ − 1, we have ρ¯ → 0 as |x| → ∞. With m = u − uxx and ρ¯ = ρ − 1, we can rewrite Eq. (1.1) with σ = 1 as follows: ⎧ ut − utxx + 3uux = 2ux uxx + uuxxx − ρ¯ ρ¯x − ρ¯x , ⎪ ⎪ ⎨ ρ¯ + (uρ) ¯ x + ux = 0, t ⎪ u(0, x) = u0 (x), ⎪ ⎩ ρ(0, ¯ x) = ρ¯0 (x),

t > 0, x ∈ R, t > 0, x ∈ R, x ∈ R, x ∈ R.

(2.1)

Note that if p(x) := 12 e−|x| , x ∈ R, then (1 − ∂x2 )−1 f = p ∗ f for all f ∈ L2 (R) and p ∗ m = u. Using this identity, Eq. (2.1) takes the form of a quasi-linear evolution equation of hyperbolic type: ⎧ 1 2 1 2 2 ⎪ ⎪ ut + uux = −∂x p ∗ u + ux + ρ¯ + ρ¯ , ⎪ ⎪ 2 2 ⎨ ρ¯t + uρ¯x = −ux ρ¯ − ux , ⎪ ⎪ ⎪ ⎪ u(0, x) = u0 (x), ⎩ ρ(0, ¯ x) = ρ¯0 (x),

t > 0, x ∈ R, t > 0, x ∈ R, x ∈ R, x ∈ R.

(2.2)

Consider now the following initial value problem

qt = u(t, q), q(0, x) = x,

t ∈ [0, T ), x ∈ R,

(2.3)

where u denotes the first component of the solution to Eq. (2.1). Applying classical results in the theory of ordinary differential equations, one can obtain the two following results on q which are crucial in studying global existence of strong solutions to Eq. (2.1). Lemma 2.1. (See [18,29].) Let u ∈ C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)), s 2. Then Eq. (2.3) has a unique solution q ∈ C 1 ([0, T ) × R; R). Moreover, the map q(t, ·) is an increasing diffeomorphism of R with t qx (t, x) = exp

ux s, q(s, x) ds > 0,

∀(t, x) ∈ [0, T ) × R.

0

u Lemma 2.2. (See [34].) Let z0 = ρ¯00 ∈ H s (R) × H s−1 (R), s 2 and let T > 0 be the maximal existence time of the corresponding solution z = (u, ρ) ¯ to Eq. (2.1). Then we have ρ¯ t, q(t, x) + 1 qx (t, x) = ρ¯0 (x) + 1 , ∀(t, x) ∈ [0, T ) × R. Then, we give a useful conservation law for strong solutions to Eq. (2.1).

(2.4)


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u Lemma 2.3. (See [34].) Let z0 = ρ¯00 ∈ H s (R) × H s−1 (R), s 2 and let T > 0 be the maximal existence time of the corresponding solution z = (u, ρ) ¯ to Eq. (2.1). Then we have

E(t) =

2

u

+ u2x

+ ρ¯

2

R

dx =

2 u0 + u20,x + ρ¯02 dx,

∀t ∈ [0, T ).

R

Moreover, we have u(t, ·)2 ∞

L (R)

1 u0 2H 1 (R) + ρ¯0 2L2 (R) , 2

∀t ∈ [0, T ).

Next, we present a very useful lemma on global strong solutions to Eq. (2.1). u Lemma 2.4. Let z0 = ρ¯00 ∈ H 2 (R) × H 1 (R). If ρ¯0 (x) = −1 for all x ∈ R, then the correu sponding strong solution z = ρ¯ to Eq. (2.1) exists globally in time, i.e. z ∈ C(R+ ; H 2 (R) × H 1 (R)) ∩ C 1 (R+ ; H 1 (R) × L2 (R)). Moreover, there exists β > 0 such that for all t ∈ R, 32 (u0 2 1 +ρ¯0 2 2 +1)t ux (t, ·) ∞ 1 3 + 2ρ¯0 2 ∞ + u0,x 2 ∞ H (R) L (R) e , L L (R) (R) L (R) 2β where β = infx∈R |ρ¯0 (x) + 1|. Proof. Differentiating the first equation in (2.2) with respect to x and using the identity ∂x2 p ∗ f = p ∗ f − f , we have 1 1 1 1 utx + uuxx + u2x = u2 + ρ¯ 2 + ρ¯ − p ∗ u2 + u2x + ρ¯ 2 + ρ¯ . 2 2 2 2

(2.5)

Set M(t, x) := ux (t, q(t, x)), γ (t, x) =: ρ(t, ¯ q(t, x)) + 1 and

1 2 1 2 f (t, x) := u (t, q) − p ∗ u + ux + ρ¯ + ρ¯ (t, q). 2 2 2

2

Using Lemma 2.3, we get f (t, x) 3 u0 2 1 + ρ¯0 2 2 + 1, H L (R) (R) 2

∀(t, x) ∈ [0, T ) × R.

(2.6)

∂γ = −γ M. ∂t

(2.7)

By Lemma 2.1 and Eq. (2.2), we have ∂M = (utx + uuxx ) t, q(t, x) ∂t

and

Then by (2.5), we have 1 1 Mt = − M 2 + γ 2 + f (t, x), 2 2

(t, x) ∈ [0, T ) × R.

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By Lemmas 2.1–2.2, we know that γ (t, x) has the same sign with γ (0, x) = ρ0 (x) for all x ∈ R. Since ρ¯0 (x) ∈ H 1 (R), it follows that ρ¯0 (x) ∈ C(R) and there exists R0 such that |ρ¯0 (x)| < 12 for all |x| R0 . Note that ρ¯0 (x) ∈ C(R) and ρ¯0 (x) = −1 for all x ∈ R. Then we have inf γ (0, x) = inf ρ¯0 (x) + 1 > 0.

|x|R0

|x|R0

Set β := min{ 12 , inf|x|∈R |γ (0, x)|}. Then |γ (0, x)| β > 0 for all x ∈ R. Therefore γ (t, x)γ (0, x) > 0 for all x ∈ R. Next, we consider the following Lyapunov positive function first introduced in [18], w(t, x) = γ (0, x)γ (t, x) +

γ (0, x) 1 + M 2 (t, x) , γ (t, x)

(t, x) ∈ [0, T ) × R.

(2.8)

By Sobolev imbedding theorem, we have 0 < w(0, x) = γ 2 (0, x) + 1 + M 2 (0, x) 2ρ¯02 (x) + 3 + u20,x (x) 3 + 2ρ¯0 2L∞ (R) + u0,x 2L∞ (R) .

(2.9)

Differentiating Eq. (2.8) with respect to t, in view of (2.7), we obtain ∂w γ (0, x) 1 (t, x) = 2 M(t, x) f (t, x) + ∂t γ (t, x) 2 γ (0, x) 3 u0 2H 1 (R) + ρ0 2L2 (R) + 1 1 + M2 2 γ (t, x) 3 u0 2H 1 (R) + ρ0 2L2 (R) + 1 w(t, x) 2 for all (t, x) ∈ [0, T ) × R. Thus by Gronwall’s inequality and (2.9), we get 3

(u0 2

+ρ0 2

+1)t

H 1 (R) L2 (R) w(t, x) w(0, x)e 2 3 (u0 2 1 +ρ¯0 2 2 +1)t H (R) L (R) 3 + 2ρ¯0 2L∞ (R) + u0,x 2L∞ (R) e 2

for all (t, x) ∈ [0, T ) × R. On the other hand, w(t, x) 2 γ 2 (0, x) 1 + M 2 2β M(t, x),

∀(t, x) ∈ [0, T ) × R.

Therefore 32 (u0 2 1 +ρ¯0 2 2 +1)t M(t, x) 1 w(t, x) 1 3 + 2ρ¯0 2 ∞ + u0,x 2 ∞ H (R) L (R) L (R) L (R) e 2β 2β for all (t, x) ∈ [0, T ) × R. By Lemma 2.1 and the above inequality, we have


ux (t, ·)

L∞ (R)

1137

= ux t, q(t, ·) L∞ (R) 3 (u0 2 1 +ρ¯0 2 2 +1)t 1 H (R) L (R) 3 + 2ρ¯0 2L∞ (R) + u0,x 2L∞ (R) e 2 2β

for all (t, x) ∈ [0, T ) × R. By Lemma 2.3 in [34], we have T = ∞. This completes the proof of the lemma. 2 We finally recall the two following useful lemmas. Lemma 2.5. (See Appendix C of [42].) Let X be a separable reflexive Banach space and let f n be bounded in L∞ (0, T ; X) for some T ∈ (0, ∞). We assume that f n ∈ C([0, T ]; Y ) where Y is a Banach space such that X → Y , Y is separable and dense in X . Furthermore, (φ, f n (t))Y ×Y is uniformly continuous in t ∈ [0, T ] and uniformly in n 1. Then f n is relatively compact in C w ([0, T ]; X), the space of continuous functions from [0, T ] with values in X when the latter space is equipped with its weak topology. Remark 2.1. If the conditions which f n satisfies in Lemma 2.5 are replaced by the following conditions: f n ∈ L∞ (0, T ; X),

∂t f n ∈ Lp (0, T ; Y )

for some p ∈ (1, ∞),

and n f

L∞ (0,T ;X)

,

n ∂t f

Lp (0,T ;Y )

C,

∀n 1,

then the conclusion of Lemma 2.5 holds true. Throughout this paper, we will denote by {φn }n1 the mollifiers

φn (x) :=

−1 φ(ξ ) dξ

nφ(nx),

x ∈ R, n 1,

R

where φ ∈ Cc∞ (R) is defined by

1/(x φ(x) = e 0,

2 −1)

, |x| < 1, |x| 1.

Lemma 2.6. (See [42].) Let f ∈ W 1,α (R) and g ∈ Lp (R) with 1 p ∞. Then f g ∗ φn − f (g ∗ φn ) → 0, in Lβ (R), where

1 β

=

1 α

+ p1 .

Remark 2.2. Note that if the domain R is replaced by a bounded domain, then Lemma 2.6 has already been shown in [42]. We observe that Lemma 2.6 holds also true for an unbounded domain R. In the following, we would like to write down the whole proof although it might be similar to the one in [42].

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Proof of Lemma 2.6. We first observe that

f g ∗ φn − f (g ∗ φn ) = − g(y) f (y)φn (x − y) y + f (x)φn (x − y) dy

=

R

g(y) f (y) − f (x) φn (x − y) dy − gf ∗ φn .

R

By standard results on convolutions, the second term converges in Lβ (R) as n → ∞ to gf . Next, we estimate the first term as follows for n large enough

g(y) f (y) − f (x) φ (x − y) dy n R

CgLp (R) R

Lβ (R)

α nf (x) − f (y) dy dx

1/α ,

|x−y| Cn

where C denotes various constants independent of n, f and g. Then we deduce that

α nf (x) − f (y) dy dx

1/α

R |x−y| C n

=

1/α α

1 f x + tz dt dz dx C f Lα (R) . n

R |z|C

0

To this end, we just need to observe that it is now enough to show that

g(y) f (y) − f (x) φn (x − y) dy → gf in Lβ (R), R

when f and g are smooth. Indeed, the general case follows by density using the above bounds. But, this convergence is clear if f and g are smooth since

g(y) f (y) − f (x) φn (x − y) dy → −g(x) −f (x) = g(x)f (x). 2 R

3. The existence and regularity of global weak solution In this section, we first establish the global existence of approximate solutions. By acquiring the precompactness of approximate solutions, we then prove the existence of the global weak solutions to Eq. (2.1). We finally show the regularity of the obtained global weak solutions. Before giving the precise statements of the main result, we first introduce the definition of an admissible weak solution to the Cauchy problem (2.2).


Definition 3.1. Give z0 =

u0 ρ¯0

1139

∈ H 1 (R) × L2 (R). If

z(t, x) =

u ∈ L∞ (0, ∞); H 1 (R) × L2 (R) ρ¯

satisfies Eq. (2.2), z(t, ·) → z0 as t → 0+ in the sense of distribution and zH 1 (R)×L2 (R) z0 H 1 (R)×L2 (R) , then z =

u ρ¯

is called an admissible weak solution to Eq. (2.2).

The main result of this paper can be stated as follows: u Theorem 3.1. Let z0 = ρ¯00 ∈ (H 1 (R) ∩ W 1,∞ (R)) × (L2 (R) ∩ L∞ (R)). If there exists α > −1 such that ρ¯0 (x) α for all x ∈ R, then Eq. (2.2) has an admissible weak solution z=

u ∈ C R+ ; H 1 (R) × L2 (R) ∩ L∞ R+ ; H 1 (R) × L2 (R) . ρ¯

Moreover, E(z(t, ·)) = z(t, ·)2H 1 (R)×L2 (R) is a conservation law. Furthermore, 1,∞ (R) u ∈ L∞ loc R+ ; W

∞ and ρ¯ ∈ L∞ loc R+ ; L (R) .

Proof. The proof of the theorem is divided into the four following steps. Step 1. The global existence of approximate solutions. u Let z0 = ρ¯00 ∈ (H 1 (R) ∩ W 1,∞ (R)) × (L2 (R) ∩ L∞ (R)) and let there exist α > −1 such that ρ¯0 (x) α > −1 for all x ∈ R. φ ∗u un Define z0n := φnn ∗ρ¯00 = ρ¯0n ∈ H 2 (R) × H 1 (R) for n 1. Note that ρ¯0 (x) α > −1 and 0 φn (x) 0 for all x ∈ R. Then we get

ρ¯0n (x) = φn

∗ ρ¯0 (x) α

φn (y) dy = α > −1,

∀x ∈ R.

R

Obviously, we have z0n → z0

in H 1 (R) × L2 (R) as n → ∞,

(3.1)

and n z

0 H 1 (R)×L2 (R)

z0 H 1 (R)×L2 (R) .

(3.2)

By Lemma 2.4, we obtain that the corresponding solution zn ∈ C(R+ ; H 2 (R) × H 1 (R)) to Eq. (2.1) with initial data z0n exists globally in time.

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Remark 3.1. By Lemma 2.3 and (3.2), we have n u (t, ·)2

H 1 (R)

2 2 + ρ¯ n (t, ·)L2 (R) = z0n H 1 (R)×L2 (R) z0 2H 1 (R)×L2 (R) ,

∀t ∈ R+ .

(3.3)

Then, we obtain n u (t, ·)

√

√ 2 2 n u (t, ·) H 1 (R) z0 H 1 (R)×L2 (R) , L∞ (R) 2 2

∀t ∈ R+ .

(3.4)

Step 2. The precompactness of approximate solutions. Let us denote P n (t, x) =: p ∗ ((un )2 + 12 (unx )2 + 12 (ρ¯ n )2 + ρ¯ n )(t, x) in the following text. Lemma 3.1. There exist a pair of subsequences {znk (t, x)} ⊂ {zn (t, x)} and {P nk (t, x)} ⊂ {P n (t, x)} and a pair of functions z(t, x) ∈ L∞ (R+ ; H 1 (R) × L2 (R)) and P (t, x) ∈ L∞ (R+ ; H 1 (R)) such that znk z nk

u

weakly in H 1 (0, T ) × R × L2 (0, T ) × R as nk → ∞, ∀T > 0, → u uniformly on each compact subset of R+ × R as nk → ∞

(3.5) (3.6)

and P nk → P

uniformly on each compact subset of R+ × R as nk → ∞.

(3.7)

Proof. By Lemma 2.3, we can easily obtain that {zn (t, x)} is uniformly bounded in L∞ (R+ ; H 1 (R) × L2 (R)). We claim that the sequence {un } is uniformly bounded in H 1 ((0, T ) × R) for fixed T > 0. Indeed, unt ∈ L2 ((0, T ) × R), in view of (3.3)–(3.4), we have n n u u

x L2 ((0,T )×R)

√ un L∞ ((0,T )×R) unx L2 ((0,T )×R) z0 2H 1 (R)×L2 (R) T

and 2 px ∗ un 2 + 1 un 2 + 1 ρ¯ n 2 + ρ¯ n 2 2 x 2 L ((0,T )×R) px 2L2 (R)

2

T n 2 1 n 2 1 n 2 u + u ρ¯ (t, ·) dt x + 1 2 2 L ((0,T )×R) 0

T + px 2L1 (R)

n ρ¯ (t, ·)2 2

L ((0,T )×R)

dt

0

T z0 4H 1 (R)×L2 (R)

+ T z0 2H 1 (R)×L2 (R) ,


1141

where we used the inequalities px 2L2 (R) 1 and px 2L1 (R) 1. Then by the first equation of (2.2), we obtain that {unt } is uniformly bounded in L2 ((0, T ) × R). From Lemma 2.3, we u can easily get that {ρ¯ n } is uniformly bounded in L2 ((0, T ) × R). Therefore, there exist z = ρ¯ ∈ L∞ (R+ ; H 1 (R) × L2 (R)) and a subsequence {znk (t, x)} such that weakly in H 1 (0, T ) × R × L2 (0, T ) × R as nk → ∞.

znk z

By the classical Lions–Aubin lemma, {unk (t, x)} is weakly compact in L∞ (R+ ; H 1 (R)). Furthermore, {unk } converges to u(t, x) uniformly on each compact subset of R+ × R as nk → ∞. Next, we turn to the compactness of {P n }. For fixed t ∈ (0, T ), in view of p2L2 (R) p2L1 (R) = 1, we have n P (t, ·)2 2 L

2 2 p ∗ un 2 + 1 un 2 + 1 ρ¯ n 2 + p ∗ ρ¯ n L2 (R) x (R) 2 2 L2 (R) 2 n 2 1 n 2 1 n 2 2 (t, ·) pL2 (R) u + ux + ρ¯ 1 2 2 L (R) 2 + p2L1 (R) ρ¯ n (t, ·)L2 (R) 4 2 zn (t, ·) 1 + zn (t, ·) 1 2 2 H (R)×L (R)

z0 4H 1 (R)×L2 (R)

H (R)×L (R)

+ z0 2H 1 (R)×L2 (R) .

In a similar way, in view of ∂x p2L2 (R) ∂x p2L1 (R) 1, we can get ∂x P n (t, ·)2 2

L (R)

z0 4H 1 (R)×L2 (R) + z0 2H 1 (R)×L2 (R) .

(3.8)

Therefore, {P n } is uniformly bounded in L∞ (R+ ; H 1 (R)). Since u0 ∈ W 1,∞ (R) and un0 = φn ∗ u0 , we have n u ∞ φn 1 u0,x L∞ (R) = u0,x L∞ (R) . L (R) 0,x L (R) By Lemma 2.4, for fixed T > 0, the sequence unx is uniformly bounded in L∞ ((0, T ) × R). Since ρ¯0 ∈ L∞ (R), then we have n ρ¯ ∞ φn 1 ρ¯0 L∞ (R) = ρ¯0 L∞ (R) . L (R) 0 L (R) Note that unx is uniformly bounded in L∞ ((0, T ) × R). Then by Lemmas 2.1–2.2, we obtain that {ρ¯ n } is uniformly bounded in L∞ ((0, T ) × R). Thus, there exists M(T ) > 0 such that n u (t, x), ux (t, x), ρ¯ n (t, x), ρ(t, ¯ x) M(T ), x

∀(t, x) ∈ (0, T ) × R.

Moreover, by Eq. (2.2), we can obtain untx + un unxx +

1 n 2 n 2 1 n 2 u = u + ρ¯ + ρ¯ n − P n 2 x 2

(3.9)

1142


and ρ¯tn + un ρ¯ n x + unx = 0. Therefore, we have ∂P n = p ∗ 2un unt + unx untx + ρ¯ n ρ¯tn + ρ¯tn ∂t = p ∗ 2un −un unx − ∂x P n + ρ¯ n −unx ρ¯ n − un ρ¯xn − unx 1 2 2 1 2 + p ∗ unx −un unxx − unx + un + ρ¯ n + ρ¯ n − P n 2 2 n n n + p ∗ − u ρ¯ x − ux = I1 + I2 + I3 .

(3.10)

Next, we estimate I1 , I2 and I3 , respectively. I1 = p ∗ 2un −un unx − ∂x P n + ρ¯ n −unx ρ¯ n − un ρ¯xn − unx

2 1 = p ∗ 2un −un unx − ∂x P n − unx ρ¯ n − ρ¯ n unx − e−|x−y| un ρ¯ n ρ¯yn dy 2 R

2 1 = p ∗ 2un −un unx − ∂x P n − unx ρ¯ n − ρ¯ n unx + 4

1 2 = p ∗ 2un −un unx − ∂x P n − unx ρ¯ n − ρ¯ n unx 2

2 1 + e−|x−y| un sign(x − y) ρ¯ n dy. 4

−|x−y| n n 2 e u y ρ¯ dy

R

R

Using (3.3)–(3.4), (3.8)–(3.9) and Hölder’s inequality, we get

I1 2L2 ((0,T )×R)

p2L2 (R)

2

T n n n 2u −u u − ∂x P n − 1 un ρ¯ n 2 − ρ¯ n un dt x x x 2 L1 (R) 0

T

+ un ρ¯ n

p2L1 (R) L∞ (0,T )×R

n ρ¯ (t, ·)2 2

L (R)

dt

0

T 0

n 2 n n u + u u + ∂x P n 2 + 1 M(T ) + 1 ρ¯ n 2 + un 2 dt x x 2 L1 (R)

+ un ρ¯ n

T

L∞ (0,T )×R

n ρ¯ (t, ·)2 2

L (R)

0

dt


1143

C T , z0 H 1 (R)×L2 (R) , M(T ) , here we used pL2 (R) pL1 (R) 1. Since −un unx unxx − 12 (unx )3 = − 12 (un (unx )2 )x , it follows that 1 2 2 1 2 I2 = p ∗ unx −un unxx − unx + un + ρ¯ n + ρ¯ n − P n 2 2 2 1 2 2 1 = p ∗ unx un + ρ¯ n + ρ¯ n − P n − p ∗ un unx x 2 2 2 1 2 2 1 − ∂x p ∗ un unx . = p ∗ unx un + ρ¯ n + ρ¯ n − P n 2 2 Using (3.3)–(3.4), (3.8)–(3.9) and Hölder’s inequality, we obtain I2 2L2 ((0,T )×R)

T

2 n n 2 1 n 2 n n ux u + ρ¯ (t, ·) + ρ¯ − P 1 dt 2 L (R)

p2L2 (R)

0

1 + un unx L∞ (0,T )×R 2

T

2 p2L1 (R) unx (t, ·)L2 (R) dt

0

2 M(T ) + 1 + z0 H 1 (R)×L2 (R) C T , z0 H 1 (R)×L2 (R) , M(T ) ,

T

n 2 n 2 n 2 n 2 u + u + ρ¯ + P x

L1 (R)

dt

0

here we used pL2 (R) , pL1 (R) 1. Next, we estimate the last term. Using Hölder’s inequality, (3.3)–(3.4) and (3.9), in view of ∂x pL2 (R) , pL1 (R) 1, we get

T I3 2L2 ((0,T )×R)

2 2 2 2 p2L1 (R) unx (t, ·)L2 (R) + ∂x p2L2 (R) un + ρ¯ n L1 (R) dt

0

C T , z0 H 1 ×L2 , M(T ) . n

2 By (3.10) and the above estimates, we deduce that { ∂P ∂t } is uniformly bounded in L ((0, T )×R). Thus, by the classical Lions–Aubin lemma again, it has a subsequence (we denote it again by {P nk }) and P ∈ L∞ (R+ ; H 1 (R)) such that {P nk (t, x)} is weakly compact in L∞ (R+ ; H 1 (R)). Moreover, {P nk (t, x)} converges to P (t, x) uniformly on each compact subset of R+ × R as nk → ∞. This completes the proof of the lemma. 2

1144


By (3.4)–(3.6) and (3.9), we have that for any T > 0, unk ρ¯ nk uρ¯

weakly in L2 (0, T ) × R

(3.11)

unk unx k uux

weakly in L2 (0, T ) × R .

(3.12)

and

Remark 3.2. By the above argument, we see that for any fixed T > 0, there exists a pair of subsequences {(unx k )2 } ⊂ {(unx )2 } and {(ρ¯ n )2 } ⊂ {(ρ¯ nk )2 } converging weakly in Lr ((0, T ) × R) for all 1 < r < ∞, i.e. there exists a pair of functions u2x ∈ Lr ((0, T ) × R) and ρ¯ 2 ∈ Lr ((0, T ) × R) such that n 2 ux k u2x

and

n 2 ρ¯ k ρ¯ 2

weakly in Lr (0, T ) × R .

Moreover, we have weakly in Lp (0, T ) × R and unx k ux weakly* in L∞ R+ ; L2 (R) , ρ¯ weakly in Lp (0, T ) × R and ρ¯ nk ρ¯ weakly* in L∞ R+ ; L2 (R) ,

unx k ux ρ¯ nk

where p 2. Furthermore, we have u2x (t, x) u2x (t, x)

and ρ¯ 2 (t, x) ρ¯ 2 (t, x)

a.e. on (R+ × R).

(3.13)

Note that unx k ux weakly in Lp ((0, T ) × R), unx k ux weakly* in L∞ (R+ ; L2 (R)), ρ¯ weakly in Lp ((0, T ) × R) and ρ¯ nk ρ¯ weakly* in L∞ (R+ ; L2 (R)). Then, we conclude that for any η ∈ C 1 (R) with η bounded, Lipschitz continuous on R and η(0) = 0, we have ρ¯ nk

η unx k η(ux ) weakly in Lp (0, T ) × R , η unx k η(ux ) weakly* in L∞ R+ ; L2 (R) and ¯ weakly in Lp (0, T ) × R , η ρ¯ nk η(ρ) η ρ¯ nk η(ρ) ¯ weakly* in L∞ R+ ; L2 (R) . Lemma 3.2. There hold ∂ ∂ux 1 1 + (uux ) = u2x + ρ¯ 2 + ρ¯ + u2 − P ∂t ∂x 2 2

(3.14)

∂ ρ¯ ∂ + (uρ) ¯ + ux = 0 ∂t ∂x

(3.15)

and

in the sense of distributions on R+ × R.


1145

Proof. Note that znk is the solution of (2.2). Differentiating the first equation in (2.2) with respect to x and using the relation ∂x2 p ∗ f = p ∗ f − f , we have 1 2 2 1 2 untxk + unk unx k x = unx k + unk + ρ¯ nk + ρ¯ nk − P nk . 2 2

(3.16)

Then by the second equation of (2.2), we get n ρ¯t k + unk ρ¯ nk x + unx k = 0.

(3.17)

By (3.5)–(3.7), (3.11)–(3.12) and Remark 3.2, we obtain (3.14) and (3.15) immediately from (3.16) and (3.17) as nk → ∞, respectively. 2 The next lemma contains renormalized formulations of (3.14) and (3.15). Lemma 3.3. For any η ∈ C 1 (R) with η bounded, Lipschitz continuous on R and η(0) = 0, we have ∂ 1 2 ∂η(ux ) + uη(ux ) = ux η(ux ) + ux − u2x η (ux ) + u2 η (ux ) ∂t ∂x 2 1 + ρ¯ 2 η (ux ) + ρη ¯ (ux ) − P η (ux ) 2

(3.18)

and ∂η(ρ) ¯ ∂ + uη(ρ) ¯ = ux η(ρ) ¯ − ux ρη ¯ (ρ) ¯ − ux η (ρ) ¯ ∂t ∂x

(3.19)

in the sense of distributions on R+ × R. Proof. Denote un,x (t, x) := (ux (t, ·) ∗ φn )(x) and ρ¯n (t, x) := (ρ(t, ¯ ·) ∗ φn )(x). According to Lemma II.1 of [27], we may deduce from (3.14)–(3.15) that un,x and ρ¯n solve ∂un,x ∂un,x 1 2 2 2 +u = −ux + ux + ρ¯ + ρ¯ ∗ φn + u2 − P ∗ φn + τn ∂t ∂x 2

(3.20)

∂ ρ¯n ∂ ρ¯n +u + (ux ρ) ¯ ∗ φn + ux ∗ φn = σn , ∂t ∂x

(3.21)

and

where the errors τn and σn tend to zero in L1loc (R+ × R). Multiplying (3.20) and (3.21) by η (un,x ) and η (ρ¯n ) respectively, we get ∂ ∂η(un,x ) + uη(un,x ) = ∂t ∂x

1 2 1 2 2 u + ρ¯ − ux + ρ¯ ∗ φn η (un,x ) 2 x 2 + ux η(un,x ) + u2 − P ∗ φn + τn η (un,x )

(3.22)

1146


and ∂ ∂η(ρ¯n ) + uη(ρ¯n ) = − (ux ρ¯ + ux ) ∗ φn η (ρ¯n ) + ux η(ρ¯n ) + τn η (ρ¯n ). ∂t ∂x

(3.23)

Using the boundedness of η and η , we can send n → ∞ in (3.22) and (3.23) to obtain (3.18) and (3.19). 2 Multiplying (3.16) and (3.17) by η (unx k ) and η (ρ¯ nk ) respectively, we get 1 2 ∂ nk ∂ nk nk η ux + u η ux = unx k η unx k − unx k η unx k ∂t ∂x 2 n 2 1 n 2 nk nk k k η unx k + u + ρ¯ + ρ¯ − P 2 and ∂ nk ∂ nk nk η ρ¯ + u η ρ¯ = unx k η ρ¯ nk − unx k ρ¯ nk (η) ρ¯ nk − unx k (η) ρ¯ nk . ∂t ∂x Then we have ∂η(ux ) 1 ∂ uη(ux ) = ux η(ux ) − u2x η (ux ) + ∂t ∂x 2 1 2 ¯ η (ux ) + ρη ¯ (ux ) − P η (ux ) + u2 η (ux ) + (ρ) 2

(3.24)

and ∂ ∂ uη(ρ) η(ρ) ¯ + ¯ = ux η(ρ) ¯ − ux ρη ¯ (ρ) ¯ − ux η (ρ) ¯ ∂t ∂x

(3.25)

in the sense of distributions on R+ × R. Here f is the limit of f nk in the sense of distributions on R+ × R. Lemma 3.4. There hold

lim

t→0+

u2x (t, x) dx

R

= lim

t→0+

u2x (t, x) = R

u20,x (x) dx

(3.26)

R

and

ρ¯ (t, x) dx = lim

lim

t→0+

R

ρ¯ 2 (t, x) dx

2

t→0+

R

=

ρ¯0 2 (x) dx. R

(3.27)


1147

Proof. By Lemma 3.1 and Lemma 2.4, for any T > 0, we have that un ∈ L∞ ((0, T ); H 1 (R)), {unt } is uniformly bounded in L∞ ((0, T ); L2 (R)), and un ∈ C([0, T ]; H 1 (R)). Then in view of Lemma 2.5, Remark 2.1 and the proof of Lemma 3.1, we have that {un } contains a subsequence denoted again by {unk }, which converges to u weakly in H 1 (R) uniformly in t. This implies that u is weakly continuous from (0, T ) into H 1 (R), i.e. u ∈ C w [0, T ]; H 1 (R) .

(3.28)

Similarly, since ρ¯ n ∈ L∞ ((0, T ); L2 (R)) and for all t ∈ (0, T ), n ρ¯ (t, ·) t

= H −1 (R)

sup

φH 1 (R)

n n u ρ¯ + un φx dx un ρ n + un L2 (R)

R

2z0 2H 1 (R)×L2 (R) + 2z0 H 1 (R)×L2 (R) in view of (3.3) and (3.4), it follows that {ρ¯tn } is uniformly bounded in L∞ ((0, T ); H −1 (R)). Then by Remark 2.1, we have that {ρ¯ n } contains a subsequence denoted again by {ρ¯ nk }, which converges to ρ¯ weakly in L2 (R) uniformly in t. This implies that ρ¯ is weakly continuous from (0, T ) into L2 (R), i.e. ρ¯ ∈ C w [0, T ]; L2 (R) .

(3.29)

Then by (3.28) and (3.29), we get ρ(t, ¯ ·) ρ¯0

weakly in L2 (R) as t → 0+ .

and ux (t, ·) u0,x

Thus, we have

ρ¯ (t, x) dx

lim inf t→0+

ρ¯0 2 (x) dx

2

R

R

and

u2x (t, x) dx

lim inf t→0+

R

u20,x (x) dx. R

Therefore, we deduce that

lim inf t→0+

u2x (t, x) dx

+ ρ¯ (t, x) dx lim inf 2

t→0+

R

+ lim inf

R

u20,x (x) + ρ¯0 2 (x) dx.

R

Moreover, from Lemma 2.3 we have

u2x (t, x) dx

t→0+

ρ¯ 2 (t, x) dx R

(3.30)

1148


2 u (t, x) + u2x (t, x) + ρ¯ 2 (t, x) dx

R

lim inf nk →∞

R

= lim inf nk →∞

=

n 2 2 2 u k (t, x) + unx k (t, x) + ρ¯ nk (t, x) dx nk 2 nk 2 n 2 u0 (x) + u0,x (x) + ρ¯0 k (x) dx

R

2 u0 (x) + u20,x (x) + ρ¯02 (x) dx.

R

Using the continuity of u and limt→0+

lim sup t→0+

2 R u (t, x) dx

u2x (t, x) + ρ¯ 2 (t, x) dx

R

=

2 R u0 (x) dx,

we obtain

2 u0,x (x) + ρ¯02 (x) dx.

(3.31)

R

In view of (3.13), (3.30) and (3.31), we get (3.26) and (3.27).

2

Step 3. The existence of global weak solution. Define 1 2 ξ , |ξ | R, ηR (ξ ) := 2 1 2 R|ξ | − 2 R , |ξ | > R. is bounded. Then ηR (0) = 0 and ηR

Remark 3.3. Let R > 0. Then for each ξ ∈ R 1 1 ηR (ξ ) = ξ 2 − R − |ξ |2 χ(−∞,−R)∪(R,∞) (ξ ), 2 2 ηR (ξ ) = ξ + R − |ξ | sign(ξ )χ(−∞,−R)∪(R,∞) (ξ ). Lemma 3.5. For almost all t ∈ (0, T ) with T > 0, we have

R

u2x

− u2x

(t, x) dx

t

ρ¯ 2 ux

− ρ¯ 2 ux (s, x)

0 R

t

dx ds + 2

ρu ¯ x (s, x) − ρu ¯ x (s, x) dx ds

0 R

t

+ M(T )

u2x (s, x) − u2x (s, x) dx ds.

(3.32)

0 R

Proof. Let R > M(T ) (see (3.9)). Subtracting (3.18) from (3.24) and using the entropy ηR , we get


1149

∂ ∂ u ηR (ux ) − ηR (ux ) ηR (ux ) − ηR (ux ) + ∂t ∂x 1 ux ηR (ux ) − ux ηR (ux ) − u2x (ηR ) (ux ) − u2x (ηR ) (ux ) 2 1 2 − ux − u2x (ηR ) (ux ) + u2 − P (ηR ) (ux ) − (ηR ) (ux ) 2 1 ¯ R ) (ux ) − ρ(η ¯ R ) (ux ) . + ρ¯ 2 (ηR ) (ux ) − ρ¯ 2 (ηR ) (ux ) + ρ(η 2 From Remark 3.3, we know that 1 ux ηR (ux ) − u2x (ηR ) (ux ) = 0, ∀(t, x) ∈ (0, T ) × R, 2 1 ux ηR (ux ) − u2x (ηR ) (ux ) = 0, ∀(t, x) ∈ (0, T ) × R 2 and (ηR ) (ux ) = (ηR ) (ux ) = ux , ηR (ux ) = 12 u2x , ηR (ux ) = 12 u2x . Then the following inequality holds in (0, T ) × R, i.e. ∂ 2 ∂ 2 u ux − u2x ρ¯ 2 ux − ρ¯ 2 ux + 2(ρu ux − u2x + ¯ x − ρu ¯ x ) − u2x − u2x ux . ∂t ∂x Integrating the above inequality over (ε, t) × R, we obtain

u2x − u2x (t, x) dx

R

t

ρ¯ 2 ux

− ρ¯ 2 ux (s, x)

t

dx ds + 2

ε R

+

ρu ¯ x (s, x) − ρu ¯ x (s, x) dx ds

ε R

u2x (ε, x) − u2x (ε, x)

R

t

dx −

u2x − u2x ux

ε R

for almost all t ∈ (0, T ). Sending ε → 0 and using Lemma 3.4 and (3.9), we get (3.32).

2

We now estimate the second component of z. Lemma 3.6. For any T > 0, we have

R

ρ¯ 2 − ρ¯ 2 (t, x) dx =

t

ux ρ¯ 2 − ux ρ¯ 2 (s, x) dx ds + 2

0 R

t

(ux ρ¯ − ux ρ)(s, ¯ x) dx ds 0 R

(3.33) in the sense of distributions on (0, T ) × R.

1150


Proof. Let R > M(T ) (see (3.9)). Subtracting (3.19) from (3.25) and using the entropy ηR , in ¯ = ρ¯ 2 , 2ηR (ρ) ¯ = ρ¯ 2 , (ηR ) (ρ) ¯ = ρ, ¯ and (ηR ) (ρ) ¯ = ρ. ¯ Thus, view of Remark 3.3, we have 2ηR (ρ) we deduce that ∂ 2 ∂ 2 u ρ¯ − ρ¯ 2 = ux ρ¯ 2 − ux ρ¯ 2 + 2ux ρ¯ 2 − 2ux ρ¯ 2 + 2ux ρ¯ − 2ux ρ¯ ρ¯ − ρ¯ 2 + ∂t ∂x ¯ = ux ρ¯ 2 − ux ρ¯ 2 + 2ux ρ¯ − 2ux ρ. Integrating the above equality over (ε, t) × R, we obtain

ρ¯ 2 − ρ¯ (t, x) dx =

2

R

ρ¯ 2 − ρ¯ (ε, x) dx +

t

2

R

ux ρ¯ 2 − ux ρ¯ 2 (s, x) dx ds

ε R

t

+2

(ux ρ¯ − ux ρ)(s, ¯ x) dx ds. ε R

Letting ε → 0 and using Lemma 3.4, we get (3.33). This completes the proof of Lemma 3.6.

2

Lemma 3.7. There hold u2x (t, x) = u2x (t, x)

and ρ¯ 2 (t, x) = ρ¯ 2 (t, x)

a.e. on R+ × R.

Proof. Adding (3.32) and (3.33), we have that for fixed T > 0 and all t ∈ (0, T )

u2x − u2x + ρ¯ 2 − ρ¯ 2 (t, x) dx

R

t

ux ρ¯ 2 − ρ¯ 2 (s, x) ds dx + M(T )

t

0 R

u2x (s, x) − u2x (s, x) dx ds.

0 R

Using (3.9) again, we get

u2x − u2x

+ ρ¯ 2 − ρ¯ (t, x) dx M(T )

t

2

R

u2x − u2x + ρ¯ 2 − ρ¯ 2 (s, x) dx ds.

0 R

Using Gronwall’s inequality and Lemma 3.4, we conclude that

R

u2x − u2x + ρ¯ 2 − ρ¯ 2 (t, x) dx 0.

(3.34)


1151

By (3.13), it yields

0

u2x − u2x + (ρ) ¯ 2 − ρ¯ 2 (t, x) dx 0,

R

that is

u2x − u2x (t, x) dx =

R

(ρ) ¯ 2 − ρ¯ 2 (t, x) dx = 0.

R

This implies that (3.34) holds true.

2

By Lemma 3.7, we have p∗

n 2 1 n 2 1 n 2 u k + ux k + ρ¯ k + ρ¯ nk 2 2

1 1 → p ∗ u2 + u2x + ρ¯ 2 + ρ¯ . 2 2

(3.35)

From the relations (3.3)–(3.5), (3.11)–(3.12) and (3.35), we infer that z satisfies Eq. (2.2) in D ((0, T ) × R) and z ∈ C w ([0, T ]; H 1 (R) × L2 (R)) for any T > 0. Step 4. The regularity of the obtained global weak solution. w (R ; H 1 (R) × L2 (R)). In order to conclude that z ∈ C(R ; H 1 (R) × Note that z ∈ Cloc + + 2 L (R)), it suffices to show that the functional 2 E z(t) = z(t, ·)H 1 (R)×L2 (R) is conserved in time. Indeed, if this holds, then z(t) − z(s)2

H 1 (R)×L2 (R)

2 2 = z(t)H 1 (R)×L2 (R) − 2 z(t), z(s) H 1 (R)×L2 (R) + z(s)H 1 (R)×L2 (R) = 2z0 2H 1 (R)×L2 (R) − 2 z(t), z(s) H 1 (R)×L2 (R) , ∀t, s ∈ R+ .

Thus, the scalar product in the last line converges to z(t)2

H 1 (R)×L2 (R)

= z0 2H 1 (R)×L2 (R)

as s → t.

The conservation of E(z(t)) in time can be proved by a regularization technique. Denote fn := f ∗ φn . Since z solves Eq. (2.2) in the sense of distribution, we see that for a.e. t ∈ R+ , ∂un 1 1 + (uux ) ∗ φn + px ∗ u2 + u2x + ρ¯ 2 + ρ¯ ∗ φn = 0, ∂t 2 2

n 1.

(3.36)

Multiplying (3.36) by un and integrating by parts, we have 1 d 2 dt

(un )2 dx + R

R

un (uux ) ∗ φn dx +

R

1 1 un px ∗ u2 + u2x + ρ¯ 2 + ρ¯ ∗ φn dx = 0. 2 2 (3.37)

1152


By (3.20), we get ∂un,x ∂un,x 1 1 +u − u2 − u2x + ρ¯ 2 + ρ¯ ∗ φn ∂t ∂x 2 2 1 1 + p ∗ u2 + u2x + ρ¯ 2 + ρ¯ ∗ φn + τn = 0, 2 2

(3.38)

here the error τn tends to zero in L1 (R) by applying Lemma 2.6 with α = 2, p = 2 and β = 1. Multiplying (3.38) by un,x and integrating by parts, we obtain 1 d 2 dt

(un,x )2 dx + R

−

+

un,x R

u u2n,x x dx

R

1 2 1 2 u − ux + ρ¯ + ρ¯ ∗ φn dx 2 2 2

un,x R

1 2

1 2 1 2 p ∗ u + ux + ρ¯ + ρ¯ ∗ φn dx + un,x τn dx = 0. 2 2 2

(3.39)

R

By the second equation in (2.2), we have ∂ ρ¯n ∂ ρ¯n ¯ ∗ φn + ux ∗ φn + σn = 0, +u + (ux ρ) ∂t ∂x

(3.40)

here again the error σn tends to zero in L1 (R) by applying Lemma 2.6 with α = 2, p = 2 and β = 1. Multiplying (3.40) by ρ¯n and integrating by parts, we deduct that 1 d 2 dt

(ρ¯n )2 dx + R

1 2

u ρ¯n2 x dx +

R

ρ¯n (ux ρ) ¯ ∗ φn + ux ∗ φn dx +

R

ρ¯n σn dx = 0. R

(3.41) Adding (3.37), (3.39) and (3.41), and then integrating by parts, we infer that 1 d 2 dt

R

2 un + u2n,x + ρ¯n2 dx

=− R

1 + 2 1 + 2

un (uux ) ∗ φn dx +

ux u2n,x dx −

R

R

ux ρ¯n2 dx −

un,x R

1 2 1 2 u − ux + ρ¯ + ρ¯ ∗ φn dx 2 2 2

ρ¯n (ux ρ) ¯ ∗ φn + ux ∗ φn dx R

un,x τn dx − R

ρ¯n σn dx. R


1153

Note that for fixed T > 0, u, ux and ρ¯ are bounded in (0, T ) × R. An application of Lebesgue’s dominated convergence theorem yields 1 d 2 dt

2 u + u2x + ρ¯ 2 dx = 0.

R

This completes the proof of Theorem 3.1.

2

Acknowledgments This work was partially supported by NNSFC (No. 10971235), RFDP (No. 200805580014), NCET (No. 08-0579) and the key project of Sun Yat-sen University. The authors thank the referee for valuable comments and suggestions. References [1] R. Beals, D. Sattinger, J. Szmigielski, Multipeakons and a theorem of Stieltjes, Inverse Problems 15 (1999) 1–4. [2] A. Bressan, A. Constantin, Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal. 183 (2007) 215–239. [3] A. Bressan, A. Constantin, Global dissipative solutions of the Camassa–Holm equation, Anal. Appl. 5 (2007) 1–27. [4] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661– 1664. [5] R. Camassa, D. Holm, J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 1–33. [6] M. Chen, S.-Q. Liu, Y. Zhang, A 2-component generalization of the Camassa–Holm equation and its solutions, Lett. Math. Phys. 75 (2006) 1–15. [7] G.M. Coclite, H. Holden, K.H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal. 37 (2006) 1044–1069. [8] A. Constantin, The Hamiltonian structure of the Camassa–Holm equation, Expo. Math. 15 (1997) 53–85. [9] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000) 321–362. [10] A. Constantin, On the scattering problem for the Camassa–Holm equation, Proc. R. Soc. Lond. Ser. A 457 (2001) 953–970. [11] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006) 523–535. [12] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998) 229–243. [13] A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa 26 (1998) 303–328. [14] A. Constantin, J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998) 475–504. [15] A. Constantin, J. Escher, Global weak solutions for a shallow water equation, Indiana Uni. Math. J. 47 (1998) 1527–1545. [16] A. Constantin, J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation, Math. Z. 233 (2000) 75–91. [17] A. Constantin, J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. 44 (2007) 423–431. [18] A. Constantin, R. Ivanov, On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A 372 (2008) 7129–7132. [19] A. Constantin, R.S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res. 40 (2008) 175–211. [20] A. Constantin, B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003) 787–804. [21] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186.

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[22] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys. 211 (2000) 45–61. [23] A. Constantin, W.A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610. [24] A. Constantin, W.A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A 270 (2000) 140–148. [25] H.H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod, Acta Mech. 127 (1998) 193–207. [26] R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (2001) 953–988. [27] R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev space, Invent. Math. 98 (1989) 511–547. [28] H.R. Dullin, G.A. Gottwald, D.D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett. 87 (2001) 4501–4504. [29] J. Escher, O. Lechtenfeld, Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation, Discrete Contin. Dyn. Syst. Ser. A 19 (2007) 493–513. [30] J. Escher, Z. Yin, Initial boundary value problems of the Camassa–Holm equation, Comm. Partial Differential Equations 33 (2008) 377–395. [31] J. Escher, Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal. 256 (2009) 479–508. [32] G. Falqui, On a Camassa–Holm type equation with two dependent variables, J. Phys. A 39 (2006) 327–342. [33] A. Fokas, B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D 4 (1981) 47–66. [34] C. Guan, Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system, J. Differential Equations 248 (2010) 2003–2014. [35] D. Ionescu-Krus, Variational derivation of the Camassa–Holm shallow water equation, J. Nonlinear Math. Phys. 14 (2007) 303–312. [36] R.I. Ivanov, Extended Camassa–Holm hierarchy and conserved quantities, Z. Naturforsch. 61a (2006) 133–138. [37] R.I. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A 365 (2007) 2267–2280. [38] R.S. Johnson, Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid Mech. 457 (2002) 63–82. [39] B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philos. Trans. R. Soc. Lond. Ser. A 365 (2007) 2333–2357. [40] M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in: Tsunami and Nonlinear Waves, Springer, Berlin, 2007, pp. 31–49. [41] Y. Li, P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 (2000) 27–63. [42] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. I. Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3, Clarendon, Oxford University Press, New York, 1996. [43] Z. Popowicz, A 2-component or N = 2 supersymmetric Camassa–Holm equation, Phys. Lett. A 354 (2006) 110– 114. [44] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa–Holm equation, Nonlinear Anal. 46 (2001) 309– 327. [45] J.F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996) 1–48. [46] G.B. Whitham, Linear and Nonlinear Waves, J. Wiley & Sons, New York, 1980. [47] Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (2000) 1411– 1433. [48] Z. Yin, Well-posedness, global existence and blowup phenomena for an integrable shallow water equation, Discrete Contin. Dyn. Syst. Ser. A 10 (2004) 393–411.


Outer actions of measured quantum groupoids Michel Enock Institut de Mathématiques de Jussieu, Unité Mixte Paris 6 / Paris 7 / CNRS de Recherche 7586, 175, rue du Chevaleret, Plateau 7E, F-75013 Paris, France Received 3 March 2010; accepted 7 November 2010 Available online 17 November 2010 Communicated by S. Vaes

Abstract Mimicking a recent article of Stefaan Vaes, in which was proved that every locally compact quantum group can act outerly, we prove that we get the same result for measured quantum groupoids, with an appropriate definition of outer actions of measured quantum groupoids. This result is used to show that every measured quantum groupoid can be found from some depth 2 inclusion of von Neumann algebras. © 2010 Elsevier Inc. All rights reserved. Keywords: Quantum groupoids

1. Introduction 1.1. In two articles [44,45], J.-M. Vallin has introduced two notions (pseudo-multiplicative unitary, Hopf-bimodule), in order to generalize, up to the groupoid case, the classical notions of multiplicative unitary [1] and of Hopf–von Neumann algebras [21] which were introduced to describe and explain duality of groups, and leaded to appropriate notions of quantum groups [21,49,51,1,28,50,25,26,29]. In another article [22], J.-M. Vallin and the author have constructed, from a depth 2 inclusion of von Neumann algebras M0 ⊂ M1 , with an operator-valued weight T1 verifying a regularity condition, a pseudo-multiplicative unitary, which leaded to two structures of Hopf bimodules, dual to each other. Moreover, we have then constructed an action of one of these structures on the algebra M1 such that M0 is the fixed point subalgebra, the algebra M2 given by the basic

E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.004

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construction being then isomorphic to the crossed-product. We construct on M2 an action of the other structure, which can be considered as the dual action. If the inclusion M0 ⊂ M1 is irreducible, we recovered quantum groups, as proved and studied in former papers [19,13]. Therefore, this construction leads to a notion of “quantum groupoid”, and a construction of a duality within “quantum groupoids”. 1.2. In a finite-dimensional setting, this construction can be mostly simplified, and is studied in [30,7,8,37,46–48], and examples are described. In [31], the link between these “finite quantum groupoids” and depth 2 inclusions of II 1 factors is given, and in [10] had been proved that any finite-dimensional connected C∗ -quantum groupoid can act outerly on the hyperfinite II 1 factor. 1.3. In [14], the author studied, in whole generality, the notion of pseudo-multiplicative unitary introduced par J.-M. Vallin in [45]; following the strategy given by [1], with the help of suitable fixed vectors, he introduced a notion of “measured quantum groupoid of compact type”. Then F. Lesieur in [27], using the notion of Hopf-bimodule introduced in [44], then there exist a left-invariant operator-valued weight and a right-invariant operator-valued weight, mimicking in this wider setting the axioms and the technics of Kustermans and Vaes [25,26], obtained a pseudo-multiplicative unitary, which, as in the quantum group case, “contains” all the informations about the object (the von Neuman algebra, the coproduct, the antipod, the co-inverse). Lesieur gave the name of “measured quantum groupoids” to these objects. A new set of axioms for these had been given in an appendix of [16]. In [15] had been shown that, with suitable conditions, the objects constructed in [22] from depth 2 inclusions, are “measured quantum groupoids” in the sense of Lesieur. 1.4. In [16] have been developed the notions of action (already introduced in [22]), crossedproduct, etc., following what had been done for locally compact quantum groups in [12,20,41]; a biduality theorem for actions had been obtained in [16, 11.6]. Moreover, we proved in [16, 13.9] that, for any action of a measured quantum groupoid, the inclusion of the initial algebra (on which the measured quantum groupoid is acting) into the crossed-product is depth 2, which leads, thanks to [15], to the construction of another measured quantum groupoid [16, 14.2]. In [18] was proved a generalization of Vaes’ theorem [41, 4.4] on the standard implementation of an action of a locally compact quantum group; namely, we had obtained such a result when there exists a normal semi-finite faithful operator-valued weight from the von Neumann algebra on which the measured quantum groupoid is acting, onto the copy of the basis of this measured quantum groupoid which is put inside this algebra by the action. 1.5. One question remained open: can any measured quantum groupoid be constructed from a depth 2 inclusion? For locally compact quantum groups, the answer is positive [16, 14.9], but the most important step in that proof was Vaes’ [42], who proved that any locally compact quantum group has an outer action. 1.6. In that article, we answer positively to that question, and we follow the same strategy than for locally compact groups: thanks to the construction of the measured quantum groupoid associated to an action [16, 14.2], we show that this question is equivalent to the existence of an outer action; to prove that last result, we mimic what was done in [42], by proving that any measured quantum groupoid has a faithful action. In [42], it was constructed on some free product


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of factors; here we clearly need to construct this action on an amalgamated free product of von Neumann algebras, as described, for instance, by Ueda [40]. 1.7. This article is organized as follows: In Section 2, we recall very quickly all the notations and results needed in that article; we have tried to make these preliminaries as short as possible, and we emphazise that this article should be understood as the continuation of [16] and [18]. In Section 3, we define outer actions of a measured quantum groupoid, and, in Section 4, faithful actions and minimal actions, and obtain links between faithful and outer actions. In Section 5, we construct an outer action of any measured quantum groupoid on some amalgamated free product of von Neumann algebras. Finally, in Section 6, we study if and when it is possible for a measured quantum groupoid to act outerly on a semi-finite (or finite) von Neumann algebra, or a finite factor. 2. Preliminaries This article is the continuation of [16] and [18]; preliminaries are to be found in [16], and we just recall herafter the following definitions and notations: 2.1. Spatial theory; relative tensor products of Hilbert spaces and fiber products of von Neumann algebras [9,35,38,22] Let N be a von Neumann algebra, ψ a normal semi-finite faithful weight on N ; we shall denote by Hψ , Nψ , . . . the canonical objects of the Tomita–Takesaki theory associated to the weight ψ ; let α be a non-degenerate faithful representation of N on a Hilbert space H; the set of ψ-bounded elements of the left-module α H is: D(α H, ψ) = ξ ∈ H; ∃C < ∞, α(y)ξ C Λψ (y), ∀y ∈ Nψ Then, for any ξ in D(α H, ψ), there exists a bounded operator R α,ψ (ξ ) from Hψ to H, defined, for all y in Nψ by: R α,ψ (ξ )Λψ (y) = α(y)ξ which intertwines the actions of N . If ξ , η are bounded vectors, we define the operator product ξ, η α,ψ = R α,ψ (η)∗ R α,ψ (ξ ) belongs to πψ (N) , which, thanks to Tomita–Takesaki theory, will be identified to the opposite von Neumann algebra N o , which will be equiped with a canonical normal semi-finite faithful weight ψ o . If y in N is analytical with respect to ψ , and if ξ ∈ D(α H, ψ), then we get that α(y)ξ belongs to D(α H, ψ) and that: ψ R α,ψ α(y)ξ = R α,ψ (ξ )Jψ σ−i/2 y ∗ Jψ

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If now β is a non-degenerate faithful antirepresentation of N on a Hilbert space K, the relative tensor product K β ⊗α H is the completion of the algebraic tensor product K D(α H, ψ) by the ψ

scalar product defined, if ξ1 , ξ2 are in K, η1 , η2 are in D(α H, ψ), by the following formula: (ξ1 η1 |ξ2 η2 ) = β η1 , η2 α,ψ ξ1 |ξ2 If ξ ∈ K, η ∈ D(α H, ψ), we shall denote ξ β ⊗α η the image of ξ η into K β ⊗α H, and, writing ψ

β,α ρη (ξ )

ψ

= ξ β ⊗α η, we get a bounded linear operator from K into K β ⊗α H, which is equal to ν

ψ

1K ⊗ψ R α,ψ (η). Changing the weight ψ will give a canonical isomorphic Hilbert space, but the isomorphism will not exchange elementary tensors! We shall denote σψ the relative flip, which is a unitary sending K β ⊗α H onto H α ⊗β K, ψo

ψ

defined, for any ξ in D(Kβ , ψ o ), η in D(α H, ψ), by: σψ (ξ β ⊗α η) = η α ⊗β ξ ψo

ψ

In x ∈ β(N ) , y ∈ α(N ) , it is possible to define an operator x β ⊗α y on K β ⊗α H, with natψ

ψ

ural values on the elementary tensors. As this operator does not depend upon the weight ψ , it will be denoted x β ⊗α y. We can define a relative flip ςN at the level of operators such N

that ςN (x β ⊗α y) = y α ⊗β x. If P is a von Neumann algebra on H, with α(N ) ⊂ P , and Q N

No

a von Neumann algebra on K, with β(N) ⊂ Q, then we define the fiber product Q β ∗α P as N

{x β ⊗α y, x ∈ Q , y ∈ P } . N

It is straightforward to verify that, if Q1 and P1 are two other von Neumann algebras satisfying the same relations with N , we have: Qβ

∗N α P ∩ Q1 β∗N α P1 = (Q ∩ Q1) β∗N α (P ∩ P1)

In particular, we have: Qβ

∗N α α(N ) =

Q ∩ β(N ) β ⊗γ 1 N

Moreover, this von Neumann algebra can be defined independently of the Hilbert spaces on which P and Q are represented; if (i = 1, 2), αi is a faithful non-degenerate homomorphism from N into Pi , βi is a faithful non-degenerate anti-homomorphism from N into Qi , and Φ (resp. Ψ ) a homomorphism from P1 to P2 (resp. from Q1 to Q2 ) such that Φ ◦ α1 = α2 (resp. Ψ ◦ β1 = β2 ), then, it is possible to define a homomorphism Ψ β1 ∗α1 Φ from Q1 β1 ∗α1 P1 into Q2 β2 ∗α2 P2 . N

N

N

Slice maps with vector states, normal faithful semi-finite weights and operator-valued weights had been defined in [15] and recalled in [16, 2.5].


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The operators θ α,ψ (ξ, η) = R α,ψ (ξ )R α,ψ (η)∗ , for all ξ , η in D(α H, ψ), generates a weakly dense ideal in α(N ) . Moreover, there exists a family (ei )i∈I of vectors in D(α H, ψ) such that the operators θ α,ψ (ei , ei ) are 2 by 2 orthogonal projections (θ α,ψ (ei , ei ) being then the projection on the closure of α(N )ei ). Such a family is called an orthogonal (α, ψ)-basis of H. 2.2. Measured quantum groupoids [27,16] A quintuplet (N, M, α, β, Γ ) will be called a Hopf-bimodule, following [45], [22, 6.5], if N , M are von Neumann algebras, α a faithful non-degenerate representation of N into M, β a faithful non-degenerate anti-representation of N into M, with commuting ranges, and Γ an injective involutive homomorphism from M into M β ∗α M such that, for all X in N : N

(i) Γ (β(X)) = 1 β ⊗α β(X); N

(ii) Γ (α(X)) = α(X) β ⊗α 1; N

(iii) Γ satisfies the co-associativity relation:

(Γ

β

∗N α id)Γ = (id β∗N α Γ )Γ

This last formula makes sense, thanks to the two preceding ones and 2.1. The von Neumann algebra N will be called the basis of (N, M, α, β, Γ ). If (N, M, α, β, Γ ) is a Hopf-bimodule, it is clear that (N o , M, β, α, ςN ◦ Γ ) is another Hopfbimodule, we shall call the symmetrized of the first one. (Recall that ςN ◦ Γ is a homomorphism from M to M r ∗s M.) No

If N is abelian, α = β, Γ = ςN ◦ Γ , then the quadruplet (N, M, α, α, Γ ) is equal to its symmetrized Hopf-bimodule, and we shall say that it is a symmetric Hopf-bimodule. A measured quantum groupoid is an octuplet G = (N, M, α, β, Γ, T , T , ν) such that [16, 3.8]: (i) (N, M, α, β, Γ ) is a Hopf-bimodule. (ii) T is a left-invariant normal, semi-finite, faithful operator valued weight T from M to α(N ), which means that, for any x ∈ M+ T , we have (id β ∗α T )Γ (x) = T (x) β ⊗α 1. ν

N

(iii) T is a right-invariant normal, semi-finite, faithful operator-valued weight T from M to

β(N ), which means that, for any x ∈ M+ T , we have (T β ∗α id)Γ (x) = 1 β ⊗α T (x). ν

N

(iv) ν is normal semi-finite faitfull weight on N , which is relatively invariant with respect to T and T , which means that the modular automorphisms groups of the weights Φ = ν ◦α −1 ◦T and Ψ = ν o ◦ β −1 ◦ T commute. ˆ We shall write H = HΦ , J = JΦ , and, for all n ∈ N , β(n) = J α(n∗ )J , α(n) ˆ = Jβ(n∗ )J . The weight Φ will be called the left-invariant weight on M. Examples are explained in 2.5 and 2.6.

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Then, G can be equipped with a pseudo-multiplicative unitary W which is a unitary from ˆ β in the following way: for all X ∈ N , H β ⊗α H onto H α ⊗βˆ H [16, 3.6], which intertwines α, β, νo

ν

we have: W α(X) β ⊗α 1 = 1 α ⊗βˆ α(X) W No

N

W 1 β ⊗α β(X) = 1 α ⊗βˆ β(X) W No

N

ˆ ˆ W β(X) β ⊗α 1 = β(X) α ⊗βˆ 1 W No

N

ˆ = β(X) α ⊗βˆ 1 W W 1 β ⊗α β(X) No

N

and the operator W satisfies: 2,3 (W βˆ ⊗α 1)(1H β ⊗α σν o )(1H β ⊗α W ) 1H α ⊗βˆ W (W β ⊗α 1H ) = W α ⊗βˆ 1H σα,β No

No

N

N

N

N

2,3 Here, σα,β goes from (H α ⊗βˆ H ) β ⊗α H to (H β ⊗α H ) α ⊗βˆ H , and 1H β ⊗α σν o goes from νo

ν

ν

H β ⊗α (H α ⊗βˆ H ) to H β ⊗α H βˆ ⊗α H . νo

ν

ν

νo

N

ν

All the intertwining properties allow us to write such a formula, which will be called the “pentagonal relation”. Moreover, W , M and Γ are related by the following results: (i) M is the weakly closed linear space generated by all operators of the form (id ∗ ωξ,η )(W ), where ξ ∈ D(α H, ν), and η ∈ D(Hβˆ , ν o ). (ii) for any x ∈ M, we have Γ (x) = W ∗ (1 α ⊗βˆ x)W . No

Moreover, it is also possible to construct many other data, namely a co-inverse R, a scaling group τt , an antipod S, a modulus δ, a scaling operator λ, a managing operator P , and a canonical one-parameter group γt of automorphisms on the basis N [16, 3.8]. Instead of G, we shall mostly use (N, M, α, β, Γ, T , RTR, ν) which is another measured quantum groupoid, denoted G, which is equipped with the same data (W , R, . . . ) as G. which is denoted (N, M, α, β, TR, ν), can be ˆ Γ, T, R A dual measured quantum group G, constructed, and we have G = G. Canonically associated to G, can be defined also the opposite measured quantum groupoid Go = (N o , M, β, α, ςN Γ, RTR, T , ν o ) and the commutant measured quantum groupoid Gc = o = (G) c, G c = (G) o , and ˆ α, ˆ Γ c , T c , R c T cR c , ν o ); we have (Go )o = (Gc )c = G, G (N o , M , β, oc co G = G is canonically isomorphic to G [16, 3.12]. (resp. Go , Gc ) will be denoted W (resp. W o , W c ). The pseudo-multiplicative unitary of G o c o The left-invariant weight on G (resp. G , G ) will be denoted Φ (resp. Φ , Φ c ). Let a Hb be a N − N -bimodule, i.e. a Hilbert space H equipped with a normal faithful nondegenerate representation a of N on H and a normal faithful non-degenerate anti-representation b on H, such that b(N ) ⊂ a(N ) . A corepresentation of G on a Hb is a unitary V from H a ⊗β H onto H b ⊗α H , satisfying, for all n ∈ N : ν

νo


1161

V b(n) a ⊗β 1 = 1 b ⊗α β(n) V No

N

V 1 a ⊗β α(x) = a(n) b ⊗α 1 V No

N

such that, for any ξ ∈ D(a H, ν) and η ∈ D(Hb , ν o ), the operator (ωξ,η ∗ id)(V ) belongs to M α,β (then, it is possible to define (id ∗ θ )(V ), for any θ in M∗ which is the linear set generated by o the ωξ , with ξ ∈ D(α H, ν) ∩ D(Hβ , ν )), and such that the application θ → (id ∗ θ )(V ) from α,β M∗ into L(H) is multiplicative [16, 5.1, 5.5]. 2.3. Action of a measured quantum groupoid [16] An action [16, 6.1] of G on a von Neumann algebra A is a couple (b, a), where: (i) b is an injective ∗-anti-homomorphism from N into A; (ii) a is an injective ∗-homomorphism from A into A b ∗α M; N

(iii) b and a are such that, for all n in N : a b(n) = 1 b ⊗α β(n) N

(which allow us to define a b ∗α id from A b ∗α M into A b ∗α M β ∗α M) and such that: N

N

(a b

N

N

∗N α id)a = (id b∗N α Γ )a

The invariant subalgebra Aa is defined by: Aa = x ∈ A ∩ b(N) ; a(x) = x b ⊗α 1 N

Let us write, for any x ∈ A+ , Ta (x) = (id b ∗α Φ)a(x); this formula defines a normal faithful ν

operator-valued weight from A onto Aa ; the action a will be said integrable if Ta is semi-finite ([16], 6.11, 12, 13 and 14). If the von Neumann algebra acts on a Hilbert space H, and if there exists a representation a of N on H such that b(N ) ⊂ A ⊂ a(N ) , a corepresentation V of G on the bimodule a Hb will be called an implementation of a if we have a(x) = V (x a ⊗b 1)V ∗ , for all x ∈ A [16, 6.6]. No

2.4. Crossed-product [16] The crossed-product of A by G via the action a is the von Neumann algebra generated by [16, 9.1] and is denoted A a G; then there exists [16, 9.3] an integrable a(A) and 1 b ⊗α M N

c on A a G, called the dual action. ˆ a˜ ) of (G) action (1 b ⊗α α, N

o is canoniThe biduality theorem [16, 11.6] says that the bicrossed-product (A a G) a˜ G cally isomorphic to A b ∗α L(H ); more precisely, this isomorphism is given by: N

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Θ(a b

∗N α id) A b∗N α L(H )

o = (A a G) a˜ G

where Θ is the spatial isomorphism between L(H b ⊗α H β ⊗α H ) and L(H b ⊗α H αˆ ⊗β H ) imν

ν

νo

ν

plemented by 1H b ⊗α σν W o σν ; the biduality theorem says also that this isomorphism sends the ν

ˆ a) of G on A b ∗α L(H ), defined, for any X ∈ A b ∗α L(H ), by: action (1 b ⊗α β, N

N

N

a(X) = (1 b ⊗α σν o W σν o )(id b N

∗N α ςN )(a b∗N α id)(X)(1 b ⊗N α σν W σν )∗ o

o

o . on the bidual action (of Gco ) on (A a G) a˜ G ˜ a We have (A a G) = a(A) [16, 11.5], and, therefore, the normal faithful semi-finite operatorvalued weight Ta˜ sends A a G onto a(A); therefore, starting with a normal semi-finite weight ψ on A, we can construct a dual weight ψ˜ on A a G by the formula ψ˜ = ψ ◦ a−1 ◦ Ta˜ [16, 13.2]. Moreover [16, 13.3], the linear set generated by all the elements (1 b ⊗α a)a(x), for all x ∈ Nψ , N

a ∈ NΦc ∩ NTc , is a core for Λψ˜ , and it is possible to identify the GNS representation of A a G associated to the weight ψ˜ with the natural representation on Hψ b ⊗α HΦ by writing: ν

Λψ (x) b ⊗α ΛΦc (a) = Λψ˜ (1 b ⊗α a)a(x) ν

N

which leads to the identification of Hψ˜ with Hψ b ⊗α H . Let us write, for all n ∈ N , a(n) = ν

Jψ b(n∗ )Jψ . Then, the unitary Uψa = Jψ˜ (Jψ a ⊗β JΦ) from Hψ a ⊗β HΦ onto Hψ b ⊗α HΦ satisfies: νo

No

ν

∗ Uψa (Jψ b ⊗α JΦ) = (Jψ b ⊗α JΦ) Uψa N

N

and we have [16, 13.4]: (i) for all y ∈ A: ∗ a(y) = Uψa (y a ⊗β 1) Uψa No

(ii) for all b ∈ M: (1 b ⊗α JΦ bJΦ )Uψa = Uψa (1 a ⊗β JΦ bJΦ ) No

N

(iii) for all n ∈ N : Uψa b(n) a ⊗β 1 = 1 b ⊗α β(n) Uψa No

Uψa

N

1 a ⊗β α(n) = a(n) b ⊗α 1 Uψa No

N


1163

Finally, if there exists a normal faithful semi-finite operator-valued weight T from A on b(N ) ψ such that ψ = ν o ◦ b−1 ◦ T, then, we can prove [18, 5.7 and 5.8] that Ua is a corepresentation, which, by (i), implements a, that we shall call a standard implementation of a. In [18] was introduced the notion of invariant weight with respect to an action; namely, be given an action (b, a) of a measured quantum groupoid G on a von Neumann algebra A, a normal faithful semi-finite weight ψ on A is said invariant by a, if, for any η ∈ D(α H, ν) ∩ D(Hβ , ν o ) and x ∈ Nψ , we have: ψ (id b

∗N α ωη )a x ∗x

2 = Λψ (x) a ⊗β η νo

and if ψ bear a density property [16, 8.1], namely that D((Hψ )b , ν o ) ∩ D(a Hψ , ν) is dense in Hψ . Then, it was proved [18, 7.7(vi)] that there exists a normal semi-finite faithful operator-valued ψ weight T from A on b(N), such that ψ = ν o ◦ b−1 ◦ T (and, therefore, Ua is a standard implementation of a); moreover, T satisfies, for all x ∈ Nψ ∩ NT : (T b

∗N α id)a x ∗x

= 1 b ⊗α β ◦ b−1 T x ∗ x = a T x ∗ x N

and (ψ b ∗α id)a(x ∗ x) = β ◦ b−1 T(x ∗ x). Such an operator-valued weight T will be called invariν

ant under a. A normal faithful conditional expectation E from A to b(N ) will be called invariant if (E b ∗α id)a = a ◦ E. N

2.5. Depth 2 inclusions Let M0 ⊂ M1 be an inclusion of σ -finite von Neumann algebras, equipped with a normal faithful semi-finite operator-valued weight T1 from M1 to M0 (to be more precise, from M1+ to the extended positive elements of M0 (cf. [38], IX.4.12)). Let ψ0 be a normal faithful semi-finite weight on M0 , and ψ1 = ψ0 ◦ T1 ; for i = 0, 1, let Hi = Hψi , Ji = Jψi , i = ψi be the usual objects constructed by the Tomita–Takesaki theory associated to these weights. We shall write ji for the mirroring on L(Hi ) defined by ji (x) = Ji x ∗ Ji . We shall write also j1 for the restriction of the mirroring to M0 ∩ M2 (which is an anti-automorphism of M0 ∩ M2 ), or for the restriction of the mirroring to M0 ∩ M1 (which is an injective anti-homomorphism from M0 ∩ M1 into M0 ∩ M2 ). Following [24, 3.1.5(i)], the von Neumann algebra M2 = J1 M0 J1 defined on the Hilbert space H1 will be called the basic construction made from the inclusion M0 ⊂ M1 . We have M1 ⊂ M2 , and we shall say that the inclusion M0 ⊂ M1 ⊂ M2 is standard. Using Haagerup’s theorem [38, 4.24], we can construct from T1 another normal faithful semifinite operator-valued weight T1 from M0 onto M1 , and, by defintion of M2 , a normal faithful semi-finite operator-valued weight T2 from M2 onto M1 ; T2 will be called the basic construction made from T1 ; we can go on and construct M2 ⊂ M3 and T3 by the basic construction made from M1 ⊂ M2 and T2 . Following now [23, 4.6.4], we shall say that the inclusion M0 ⊂ M1 is depth 2 if the inclusion (called the derived tower): M0 ∩ M1 ⊂ M0 ∩ M2 ⊂ M0 ∩ M3

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is also standard, and, following [19, 11.12], we shall say that the operator-valued weight T1 is regular if both restrictions T2 = T2|M ∩M2 and T3 = T3|M ∩M3 are semi-finite. 0 1 In [22] was proved that, with such a hypothesis, there exists a coproduct Γ from M0 ∩ M2 into (M0 ∩ M2 ) j1 ∗id (M0 ∩ M2 ) (where here id means the injection of M0 ∩ M1 into M0 ∩ M2 )

M0 ∩M1

such that (M0 ∩ M1 , M0 ∩ M2 , id, j1 , Γ ) is a Hopf-bimodule; moreover, T2 is then a left-invariant weight, and j1 ◦ T2 ◦ j1 a right-invariant weight; if there exists a normal faithful semi-finite weight χ on M0 ∩ M1 invariant under the modular automorphism group σtT1 [15, 8.2 and 8.3], we get

that: G1 = M0 ∩ M1 , M0 ∩ M2 , id, j1 , Γ, T2 , j1 ◦ T2 ◦ j1 , χ is a measured quantum groupoid. We shall write G1 = G(M0 ⊂ M1 ). Moreover, the inclusion M1 ⊂ M2 satisfies the same hypothesis, and leads to another mea1 o , and there exists a canonical sured quantum groupoid G2 , which can be identified with G action a of G2 on M1 [22, 7.3], which can be described as follows: the anti-representation of the basis M1 ∩ M2 (which, using j1 , is anti-isomorphic to M0 ∩ M1 ), is given by the natural inclusion of M0 ∩ M1 into M1 , and the homomorphism from M1 is given by the natural inclusion of M1 into M3 (which is, thanks to [22, 4.6], isomorphic to M1 j1 ∗id L(Hχ2 ), where χ2 = χ ◦ T2 ). We M0 = M1a

M0 ∩M1

then get that and that M2 is isomorphic to M1 a G2 [22, 7.5 and 7.6]. So, from a depth 2 inclusion M0 ⊂ M1 equipped with a regular operator-valued weight, and an invariant weight on the first relative commutant, one can construct a measured quantum groupoid G2 given, in fact, by a specific action a of G2 on M1 , with M0 being the invariant elements under this action. If G is any measured quantum groupoid, and (b, a) an action of G on a von Neumann algebra A, then the inclusion a(A) ⊂ A a G is depth 2 [16, 13.9], and the operator-valued weight Ta˜ is regular [16, 13.10]; so, we can construct a Hopf-bimodule from this depth 2 inclusion, equipped with a left-invariant operator-valued weight and a right-invariant operator-valued T weight; moreover, if there exists a weight χ on A a G ∩ a(A) , invariant by σt a˜ , we get another measured quantum groupoid G(a) = G(a(A) ⊂ A a G) [16, 14.2], which contains, in a sense, Goc [16, 14.7]. More precisely, the inclusion a(A) ⊂ A a G ⊂ A b ∗α L(H ) is standard, and, if we write N

B = A a G ∩ a(A) and b = a˜ |B , the derived inclusion B ⊂ A b ∗α L(H ) ∩ a(A) is isomorphic N

c [16, 13.9], and there exist a ∗-anti-automorphism j1 of B b G c and a to b(B) ⊂ B b G coproduct Γ1 such that [16, 14.2]: c , b, j1 ◦ b, Γ1 , T ˜ , j1 ◦ T ˜ ◦ j1 , χ G(a) = B, B b G b b 2.6. Examples of measured quantum groupoids Examples of measured quantum groupoids are the following: (i) Locally compact quantum groups, as defined and studied by J. Kustermans and S. Vaes [25,26,41]; these are, trivially, the measured quantum groupoids with the basis N = C.


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(ii) Measured groupoids, equipped with a left Haar system and a quasi-invariant measure on the set of units, as studied mostly by T. Yamanouchi [52–55]; it was proved in [17] that these measured quantum groupoids are exactly those whose underlying von Neumann algebra is abelian. (iii) the finite-dimensional case had been studied by D. Nikshych and L. Vainermann [30– 32], J.-M. Vallin [46,47] and M.-C. David [10]; in that case, non-trivial examples are given, for instance Temperley–Lieb algebras [32,10], which had appeared in subfactor theory [24]. (iv) Continuous fields of (C∗ -version of) locally compact quantum groups, as studied by E. Blanchard in [4,5]; it was proved in [17] that these measured quantum groupoids are exactly those whose basis is central in the underlying von Neumann algebras of both the measured quantum groupoid and its dual. As a particular case, we find in [27, 17.1] that, be given a family Gi = (Ni , Mi , αi , βi , Γi , Ti , Ti , νi ) a measured quantum groupoids, Lesieur showed that it is possible to construct another measured quantum groupoid G = i∈I Gi = ( i∈I Ni , i∈I Mi , i∈I αi , i∈I βi , i∈I Γi , i∈I Ti , i∈I Ti , i∈I νi ). (v) In [11], K. De Commer proved that, in the case of a monoidal equivalence between two locally compact quantum groups (which means that these two locally compact quantum group have commuting ergodic and integrable actions on the same von Neumann algebra), it is possible to construct a measurable quantum groupoid of basis C2 which contains all the data. Moreover, this construction was usefull to prove new results on locally compact quantum groups, namely on the deformation of a locally compact quantum group by a unitary 2-cocycle; he proved that these measured quantum groupoids are exactly those whose basis C2 is central in the underlying von Neumann algebra of the measured quatum groupoid, but not in the underlying von Neumann algebra of the dual measured quantum groupoid. (vi) In [43] and [2] was given a specific procedure for constructing locally compact quantum groups, starting from a locally compact group G, whose almost all elements belong to the product G1 G2 (where G1 and G2 are closed subgroups of G whose intersection is reduced to the unit element of G); such (G1 , G2 ) is called a “matched pair” of locally compact groups (more precisely, in [43], the set G1 G2 is required to be open, and it is not the case in [2]). Then, G1 acts naturally on L∞ (G2 ) (and vice versa), and the two crossed-products obtained bear the structure of two locally compact quantum groups in duality. In [48], J.-M. Vallin generalizes this constructions up to groupoids, and, then, obtains examples of measured quantum groupoids; more specific examples are then given by the action of a matched pair of groups on a locally compact space, and also more exotic examples. 3. Outer actions of a measured quantum groupoid In this section, we define (3.2) outer actions of a measured quantum groupoid, and prove (3.9) that a measured quantum groupoid can be constructed by a geometric construction from a depth 2 inclusion if and only if it has an outer action on some von Neumann algebra. 3.1. Theorem. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) an action of G on a von Neumann algebra A; then, are equivalent: ˆ ); (i) A a G ∩ a(A) = 1 b ⊗α α(N N

(ii) A b ∗α L(H ) ∩ a(A) = 1 b ⊗α M ; N

N

(iii) it is possible to define the measured quantum groupoid G(a), and G(a) = Goc .

1166


Proof. Let us suppose (i); using then [16, 14.1(iii)], we get (ii). Let us suppose (ii); using [16, 14.7], we see the application x → 1 b ⊗α x from M onto N

A b ∗α L(H ) ∩ a(A) is an isomorphism of Hopf-bimodules, from Goc onto the Hopf-bimodule N

constructed from the depth 2 inclusion a(A) ⊂ A a G, and sends the left- (resp. right-) invariant operator-valued weights of Goc on the left- (resp. right-) invariant operator-valued weights of Hopf-bimodule constructed from the depth 2 inclusion a(A) ⊂ A a G; therefore, we get that the T weight ν is invariant under σt a˜ , which means that we can define the measured quantum groupoid oc G(a), and that G(a) = G , which is (iii). Let us suppose (iii); the application x → 1 b ⊗α x from M onto A b ∗α L(H ) ∩ a(A) is an N

isomorphism between Goc and G(a); in particular, we get (i).

N

2

3.2. Definition. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) an action of G on a von Neumann algebra A; we shall say that the action (b, a) is outer if it satisfies one of the equivalent conditions of 3.1. 3.3. Theorem. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) an outer action of G on a von Neumann algebra A; then, the dual action of the measured quantum c on the crossed product A a G is outer. groupoid G Proof. Let us put the von Neumann algebra A on its standard Hilbert space L2 (A); we have, using 3.1 and [16, 3.11]: Ab

∗N α L(H ) ∩ (A a G) = A b∗N α L(H ) ∩ a(A) ∩ L L2(A) b∗N α M = 1 b ⊗α M ∩ L L2 (A) b∗α M N N = 1 b ⊗α M ∩ M N

ˆ ) = 1 b ⊗α β(N N

from which we get the result, using 3.1 again.

2

3.4. Proposition. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) an outer action of G on a von Neumann algebra A; then, we have: Z(A) = b(n), n ∈ Z(N ), β(n) ∈ Z(M) Moreover, we have: Z(A a G) = 1 b ⊗α α(n), ˆ α(n) ∈ Z(M) N

Proof. As we have a(Z(A)) ⊂ A a G ∩ a(A) , we get that, for any z ∈ Z(A), there exists n ∈ N ; therefore, ˆ But, we then infer that α(n) ˆ belongs to M ∩ M ∩ M such that a(z) = 1 b ⊗α α(n). N


1167

we have α(n) ˆ = β(n) ∈ Z(M), n ∈ Z(N ), and a(z) = 1 b ⊗α β(n) = a(b(n)), from which we get N

that Z(A) ⊂ b(n), n ∈ Z(N ), β(n) ∈ Z(M) Conversely, if n ∈ Z(N ), such that β(n) ∈ Z(M), we get that a(b(n)) = 1 b ⊗α β(n) commutes N

with all elements a(x) ∈ A b ∗α M, for any x ∈ A; therefore, we get that b(n) ∈ Z(A). Applying N

this result to the outer action a˜ , we get that: Z(A a G) = 1 b ⊗α α(n), ˆ α(n) ˆ ∈Z M N

and, as α(n) ˆ = J Jˆα(n)JˆJ , where Jˆ stands for JΦ, [16, 3.11], we get the result.

2

3.5. Corollary and Definition. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) an outer action of G on a von Neumann algebra A; (i) the algebra A is a factor if and only if we have: ˆ )=C n ∈ N, α(n) ∈ Z(M) = n ∈ N, β(n) ∈ Z(M) = α(N ) ∩ β(N Such a measured quantum groupoid is called connected. Then, the scaling operator of G is a scalar. is connected. (ii) A a G is a factor if and only if G are connected; then, we shall (iii) Both A and A a G are factors if and only if both G and G say that G is biconnected. We have then: ˆ )=C α(N ) ∩ β(N) = α(N ) ∩ β(N ˆ that: Proof. We clearly have, by definition of β, ˆ ˆ ˆ ) α(N ) ∩ β(N) ⊂ n ∈ N, α(n) ∈ Z(M) = n ∈ N, α(n) = β(n) ⊂ α(N ) ∩ β(N moreover, using the co-inverse R, it is clear that {n ∈ N, α(n) ∈ Z(M)} = C is equivalent to {n ∈ N, β(n) ∈ Z(M)} = C; then, the first part of (i) is given by 3.4. In that situation, we get immediately that the scaling operator of G, which belongs to α(N ) ∩ Z(M), must be a scalar, which finishes the proof of (i). c (which is outer by 3.3), we get (ii), and (i) and (ii) By applying (i) to the action a˜ of G give (iii). 2 3.6. Corollary. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, such that α(N ) ⊂ Z(M), and (b, a) an outer action of G on a von Neumann algebra A; then, we have Z(A) = b(N ); let us N = L∞ (X, ν); the von Neumann algebra A is decomposable and

⊕write x can be written A = X A dν(x), and, for ν almost all x ∈ X, the algebras Ax are factors.

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Proof. If α(N) ⊂ Z(M), we have, using the co-inverse R, that β(N ) ⊂ Z(M), and, therefore, using 3.4, Z(A) ⊂ b(N ), and, as a(b(n)) = 1 b ⊗α β(n) commutes with A b ∗α M, and, therefore, N

with a(A), we get that b(N) ⊂ Z(A), which finishes the proof.

N

2

3.7. Example. Let M0 ⊂ M1 a depth 2 inclusion, equipped with a regular operator-valued weight T1 from M1 onto M0 , and a normal semi-finite faithful weight χ on M0 ∩ M1 , invariant under σtT1 ; let us use all notations of 2.5. There exists a measured quantum groupoid G2 and an action a of G2 on M1 , with M0 = M1a . Then, this action a is outer: in fact, the crossed-product M1 a G2 is isomorphic with M2 , and this isomorphism, described in [22, 7.6], sends a(M1 ) on M1 , and ˆ on n, for any n ∈ M1 ∩ M2 . Here b is the restriction of the mirroring j1 to M1 ∩ M2 , 1 b ⊗α α(n) M1 ∩M2

which sends the basis M1 ∩ M2 on M0 ∩ M1 , α is the injection of M1 ∩ M2 into M1 ∩ M3 , and αˆ is the restriction to M1 ∩ M2 of the standard representation of M0 ∩ M2 . 3.8. Example. (i) Let G be a locally compact quantum group; then an action of G is outer (in the sense of 3.2) if and only if it is strictly outer in the sense of Vaes [42, 2.5]. and (bi , ai ) an action of Gi on (ii) let Gi be a family of measured quantum groupoids, then, let us define b = a von Neumann algebra Ai . Let us construct G = i∈I Gi (2.6(v)); b , which will be an injective ∗-anti-homomorphism from N into i i i∈I i∈I Ai , and a = i∈I a , which will be an injective ∗-homomorphism from A into (A i i i bi ∗αi Mi ) = i∈I i∈I i∈I Ni ( i∈I Ai ) b ∗α M, where α = i∈I αi and M = i∈I Mi ; then (b, a) is an action of G on N i∈I Ai , and this action is outer if and only if all the actions ai are outer. 3.9. Theorem. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid; then, are equivalent: (i) there exists a depth 2 inclusion M0 ⊂ M1 , equipped with a regular operator-valued weight T1 from M1 onto M0 , and a normal semi-finite faithful weight χ on M0 ∩ M1 , invariant under σtT1 , such that G = G(M0 ⊂ M1 ); (ii) there exists a von Neumann algebra A, and (b, a) a outer action of G on A. Proof. Let us suppose (i); let M0 ⊂ M1 ⊂ M2 ⊂ · · · be Jones’ tower associated to the ino = G(M1 ⊂ M2 ), and, therefore, Goc = G(M2 ⊂ M3 ), and clusion M0 ⊂ M1 ; then (2.5), G G = G(M4 ⊂ M5 ). Applying 3.7 to the inclusion M3 ⊂ M4 , we get (ii). Let us suppose (ii); then, using 3.1, we have Goc = G(a) = G(a(A) ⊂ A a G). Using [16, 13.9], we get that Jones’tower associated to the inclusion a(A) ⊂ A a G is: c ⊂ (A a G) αˆ a(A) αˆ ⊗β 1 ⊂ a˜ (A a G) ⊂ (A a G) a˜ G

∗

β No

No

L(HΦ )

c ⊂ (Aa G) αˆ ∗β L(HΦ )], which gives (i). and, therefore, we get that G = G[(Aa G)a˜ G No

2


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4. Faithful actions of a measured quantum groupoid In this section, we define faithful actions of a measured quantum groupoid (4.1), and we prove some links between faithful and outer actions (4.4, 4.8). 4.1. Definition. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) an action of G on a von Neumann algebra A; we shall say that the action (b, a) is faithful if

(ωη b

∗N α id)a(x), η ∈ D L2(A)b , ν o , x ∈ A

= M

We shall say that (b, a) is minimal if it is faithful and if A ∩ (Aa ) = b(N ). 4.2. Example. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, (β, Γ ) the action of G on M defined in [16, 6.10]. Using [16, 3.6(ii) and 3.8(vii)], we get that the von Neumann algebra generated by the set {(ωη β ∗α id)Γ (x), η ∈ D(Hβ , ν o ), x ∈ M} is equal to M, N

which says that (β, Γ ) is faithful. 4.3. Proposition. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) an action of G on a von Neumann algebra A; let A1 be a von Neumann subalgebra of A such that b(N ) ⊂ A1 ⊂ A, and such that a(A1 ) ⊂ A1 b ∗α M; therefore (b, a|A1 ) is an action of G on A1 ; N

moreover, if (b, a|A1 ) is faithful, then (b, a) is faithful. Proof. Trivial.

2

4.4. Proposition. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid, and (b, a) a minimal action of G on a von Neumann algebra A; then (b, a) is an outer action. Proof. Let z ∈ A b ∗α L(H ) ∩ a(A) ; then, using 4.1 and 2.1, z belongs to: N

Ab

∗N α L(H ) ∩

a A b ⊗α 1H = A b N

∗N α L(H ) ∩

= A ∩ Aa b = b(N) b

a

A b

∗N α L(H )

∗N α L(H )

∗N α L(H )

= 1 b ⊗α α(N )

N

So, there exists y ∈ α(N ) such that z = 1 b ⊗α y. But, as z commutes with a(A), we get that y N

commutes with all elements of the form (ωη b ∗α id)a(x), for all η ∈ D(L2 (A)b , ν o ) and x ∈ A. N

Therefore, by 4.1, we get that y ∈ M , which finishes the proof, by 3.1.

2

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4.5. Definition. Let G be a measured quantum groupoid, A be a von Neumann algebra (with separable predual), and θ be a normal faithful state on A; let us denote idN the canonical antihomomorphism from N into N o ⊗ B, and idA the identity of A; then, as the fiber product (A ⊗ N o )∗α M can be identified with A ⊗ (M ∩ α(N ) ), we get that (idN , idA ⊗ β) is an action of G on A ⊗ N o , we shall call the trivial action of G on A ⊗ N o ; this generalizes the example [16, 6.2]. If θ denotes a faithful state on A, we shall denote Eθ the normal faithful conditional expectation from A ⊗ N o onto N o given by the slice map θ ⊗ idN ; this conditional expectation satisfies (Eθ idN ∗α id)(idA ⊗ β) = (idA ⊗ β) ◦ Eθ and is therefore invariant by the trivial action N

in the sense of 2.4. Moreover, we have, trivially (A ⊗ N o )id⊗β = A ⊗ C. 4.6. Proposition. Let G be a measured quantum groupoid; for i = (1, 2), let (bi , ai ) be an action of G on a von Neumann algebra Ai ; let us suppose that there exists a normal faithful conditional expectation Ei from Ai onto bi (N ), invariant under ai , i.e. ([18], 7.6 and 7.7, recalled in 2.4) such that (Ei bi ∗α id)ai = ai ◦ Ei . Then, there exists an action (b, a) of G on the amalgamated N

free product A1 A2 , as defined in [40, 2], where b is the anti-isomorphism from N into the No

canonical subalgebra of A1 A2 isomorphic to N o , and a is given by the composition of the No

isomorphism a1 a2 from A1 A2 onto a1 (A1 ) a2 (A2 ) constructed, as [40, p. 366], using No

No

No

[40, 2.5], and the inclusion: a1 (A1 ) a2 (A2 ) ⊂ A1 A2 b No

No

∗N α M

which is given by the formulae (Ei bi ∗α id)ai = ai ◦ Ei . For any xi ∈ Ai , considered as a subalN

gebra of A1 A2 , we have a(xi ) = ai (xi ). No

Proof. The construction of the application a is an application of [40, 2.5]. Then, it is straightforward to get it is an action by verifying it on each Ai . 2 4.7. Definition. Let G be a measured quantum groupoid, and let (b, a) be a faithful action of G on a von Neumann algebra A; let us suppose that there exists a normal faithful conditional expectation E from A onto b(N ), invariant under a. Moreover, let A be a von Neumann algebra with separable predual, and θ a normal faithful state on A, and let us consider the trivial action on G on A ⊗ N o , as defined in 4.5. Let us construct now the action a1 of G on the amalgamated free product A (A ⊗ N o ) of A No

over its subalgebra b(N ) with (A ⊗ N o ) over its subalgebra N o , constructed as in 4.6 using the normal faithful conditional expectation E from A onto b(N ) and the normal faithful conditional expectation Eθ from A ⊗ N o onto N o defined in 4.5, which are invariant, respectively, towards the action a and the trivial action. As the action a is a restriction of a1 and is faithful, we get that the action a1 is faithful also. Moreover, we have trivially A ⊗ C ⊂ [A (A ⊗ N o )]a1 . No


1171

4.8. Theorem. Let G be a measured quantum groupoid; let us suppose that G has a faithful action on a von Neumann algebra A, such that there exists a normal faithful conditional expectation E from A onto b(N ), invariant under a; then G has an outer action. Proof. Let’s use Barnett’s result [3, Theorem 2]; using the notations of 4.6, let us take A = (A1 , θ1 ) (A2 , θ2 ), each θi being a faithful state on the von Neumann algebra Ai , and let us suppose that there exists a in the centralizer Aθ11 such that θ1 (a) = 0, and b, c in the centralizer Aθ22 such that θ2 (b) = θ2 (c) = θ2 (b∗ c) = 0; let’s use the normal faithful conditional expectations (θ1 ⊗ id) from A1 ⊗ N o onto N o and (θ2 ⊗ id) from A2 ⊗ N o onto N o ; it is straightforward to get that the amalgamated free product (A1 ⊗ N o ) (A2 ⊗ N o ) is equal to A ⊗ N o , which, by N

the associativity of the amalgamated free product, leads to: A A ⊗ N o = A A1 ⊗ N o A2 ⊗ N o N

N

N

Then, we get that the elements a ⊗ 1, b ⊗ 1, c ⊗ 1 satisfy the conditions of [40], condition I-A of Appendix I, which leads to [40], Prop. I-C of Appendix I: A A ⊗ N o ∩ {a ⊗ 1, b ⊗ 1, c ⊗ 1} = N o N

from which we get: a A A ⊗ N o ∩ A A ⊗ N o 1 ⊂ A A ⊗ N o ∩ (A ⊗ C)

No

No

No

⊂ A A ⊗ N o ∩ {a ⊗ 1, b ⊗ 1, c ⊗ 1}

No

=N

o

which proves that the action a1 is minimal, in the sense of 4.1, and, therefore, outer, using 4.4. 2 5. Any measured quantum groupoid has an outer action In this section, following the strategy of [42], we construct a faithful action of a measured quantum groupoid G on a von Neumann algebra acting on a “relative Fock space”. We prove that this action leaves invariant some conditional expectation; then, using 4.8, inspired again by [42], we construct a strictly outer action on some amalgamated free product. 5.1. Definition and notations. For any n ∈ N, let us write H (n) for L2 (N ) (that we shall identify with the Hilbert space Hν given by the G.N.S. construction made from the weight ν) if n = 0, for H if n = 1, for H α ⊗βˆ H for n = 2, and, if n 3, for the relative tensor product (n-times) H α ⊗βˆ H α . . .βˆ H . νo

νo

1172


Each of these Hilbert spaces is equipped with a surjective involutive anti-linear isometry, Jν on H (0) , J on H (1) , σν (J α ⊗βˆ J ) on H (2) , and Σn (J α ⊗βˆ J α . . .βˆ J ) on H (n) where Σn means νo

νo

Σn (ξ1 βˆ ⊗α ξ2 βˆ . . .α ξn ) = ξn α ⊗βˆ ξn−1 α . . .βˆ ξ1 . ν ν Let us write F (H ) = n H (n) , and let J be the surjective involutive anti-linear isometry constructed by taking the direct sum of all these isometries on H (n) . Let us consider the canonical representation of N on H (0) , the representation α on H (1) , the representation 1 α ⊗βˆ α on H (2) , and the representations αn = 1 α ⊗βˆ 1 α . . .βˆ α on H (n) , and let No

No

us write a for the direct sum of all these representations, which is a normal faithful representation of N on F (H ). Writing b(n) = J a(n∗ )J , we construct a normal faithful antirepresentation of N on F (H ); we easily get that b is, on H (0) , equal to the canonical antirepresentation of N , that b is, on H (1) , ˆ that b is, on H (2) , equal to βˆ α ⊗ ˆ 1, and, on H (n) , equal to βˆn = βˆ α ⊗ ˆ 1 α . . . ˆ 1. equal to β, β β β No

No

For any ξ ∈ D(α H, ν), let us define on F (H ) bounded operator l(ξ ) by: – for any n ∈ N , l(ξ )Λν (n) = α(n)ξ ; – for any η ∈ H , l(ξ )η = ξ α ⊗βˆ η; νo

– for any ξ1 α ⊗βˆ ξ2 α . . .βˆ ξn ∈ H (n) , l(ξ )(ξ1 α ⊗βˆ ξ2 α . . .βˆ ξn ) = ξ α ⊗βˆ ξ1 α ⊗βˆ ξ2 α . . .βˆ ξn . νo

νo

νo

νo

Then, we get that l(ξ ) belongs to a(N ) , and, for ξ , ξ in D(α H, ν), we have l(ξ )∗ l(ξ ) = b(ξ, ξ α,ν ). We can easily check that l(ξ )l(ξ )∗ is equal to 0 on H (0) , is equal to θ α,ν (ξ, ξ ) on H (1) , and to θ α,ν (ξ, ξ ) α ⊗βˆ 1 on H (n) . Therefore, if (ξi )i∈I is an orthogonal (α, ν) basis of H , in the sense No recalled in 2.1, we get that i l(ξi )l(ξi )∗ = 1 − PH (0) . Let us write A for the von Neumann algebra generated by all the operators l(ξ ), for ξ ∈ D(α H, ν). From these remarks, we infer that b(N ) ⊂ A ⊂ a(N ) , and that PH (0) ∈ A. Taking the final support of l(ξ )PH (0) , we get that θ α,ν (ξ, ξ )PH (1) belongs to A, and taking again an (α, ν)orthogonal basis of H , we get that PH (1) belongs to A. By recurrence, we get that, for all n ∈ N, PH (n) belongs to A. For any η ∈ D(Hβˆ , ν o ), we define on F (H ) a bounded operator r(η) by: ˆ ∗ )η; – for any n ∈ N , r(η)Λν (n) = β(n – for any ξ ∈ H , r(η)ξ = ξ α ⊗βˆ η; νo

– for any ξ1 α ⊗βˆ ξ2 α . . .βˆ ξn ∈ H (n) , r(η)(ξ1 α ⊗βˆ ξ2 α . . .βˆ ξn ) = ξ1 α ⊗βˆ ξ2 α . . .βˆ ξn α ⊗βˆ η. νo

ωo

ωo

ωo

Then, we easily get that r(η) = J l(J η)J , and that r(η) ∈ A , from which we get that J AJ ⊂ A . Let us now consider a faithful normal state ω on N and the G.N.S. construction (Hω , πω , Λω (1)) made from ω. There exists a unique unitary u from Hω onto Hν = L2 (N ), such that u∗ nu = πω (n), for all n ∈ N , and uJω = Jν u; then, for any p ∈ N , analytic with respect to ν, using 2.1 and these properties of u, we have: ∗ ν ∗ ν p Jν uΛω (1) = l(ξ )uJω πω σ−i/2 p Jω Λω (1) l α(p)ξ uΛω (1) = l(ξ )Jν σ−i/2


1173

Using the weak density of the analytic elements in N , we get that the closure of l(A)uΛω (1) contains, for any ξ ∈ D(α H, ν), the subspace l(ξ )uπω (N ) Λω (1); therefore, it contains l(ξ )L2 (N ), and, by definition, it contains ξ . On the other hand, we have Jν uΛω (1) = uJω Λω (1) = uΛω (1); in the sequel, we shall skip the unitary u, and consider the vector Λω (1) as an element of L2 (N ), invariant by Jν . 5.2. Proposition. Let’s take the notations of 5.1; we have: (i) the state Ω(X) = (XΛω (1)|Λω (1)) on A is faithful; (ii) let E(X) = b(XΛω (1), Λω (1) a,ω ); then, E is a normal faithful conditional expectation from A onto b(N). Proof. We had seen in 5.1 that any ξ ∈ D(α H, ν) belongs to AΛω (1); therefore, AΛω (1) contains H ; the same way, we get that, for all n ∈ N, AH (n) contains H (n+1) , and, therefore, the vector Λω (1) is cycling for A; as J Λω (1) = Λω (1), by 5.1 again, we see that this vector is cycling also for J AJ , and, therefore, also for A ; so, Λω (1) is separating for A, from which we get (i). Let us write, for X ∈ A, E(X) = b(XΛω (1), Λω (1) a,ω ); E is a positive bounded application from A on b(N); moreover, for any n ∈ N , we have b(n)Λω (1) = Jω n∗ Jω Λω (1) = ω (n)Λ (1), and R a,ω (b(n)Λ (1)) = R a,ω (Λ (1))J n∗ J ; so, we get that E(b(n)) = b(n), σ−i/2 ω ω ω ω ω 2 and, therefore E = E, and E is a conditional expectation. As Ω(X) = Ω ◦ E(X), we get, using (i), that E is faithful, which finishes the proof. 2 5.3. Proposition. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid; let us use the notations of 5.1. Then: (i) σν o W σν o is a corepresentation of G on the N − N bimodule α Hβˆ . (ii) There exists a unitary

(σν o W σν o )1,n (σν o W σν o )2,n . . . (σν o W σν o )n−2,n (σν o W σν o )n−1,n from H (n−1) αn ⊗β H to H (n−1) βˆn ⊗α H , which is a corepresentation of G on the N − N ν (n) H α ˆ n . βn

νo

bimodule (iii) By taking the sum of all these, we can define a corepresentation F (σν o W σν o ) of G on the N − N bimodule b F (H )a . Proof. Result (i) is nothing but [16, 5.6]. It is then easy to get, at least formally, that (σν o W σν o )1,3 (σν o W σν o )2,3 is, using [16, 5.1], a corepresentation of G on the N − N bimodule (2) βˆ2 Hα2 , and we can get by recurrence a proof of (ii). The proof of (iii) is then straightforward. 2 5.4. Theorem. Let G = (N, M, α, β, Γ, T , T , ν) be a measured quantum groupoid; let us use the notations of 5.1, 5.2 and 5.3. Then the corepresentation F (σν o W σν o ) of G on the N − N

1174


bimodule b F (H )a implements, in the sense of [16, 6.6] an action (b, a) of G on A defined, for all X ∈ A by: a(X) = F (σν o W σν o )(X a ⊗β 1)F (σν o Wω σν o )∗ No

Moreover, this action is faithful, and the faithful conditional expectation E is invariant by a. Proof. Using [16, 6.6], we get that F (σν o W σν o ) implements an action on a(N ) . Moreover, we get, for ξ and η in D(α H, ν), and η ∈ D(α H, ν) ∩ D(β H, ν o ), that: (id b

∗N α ωη,η ) F (σν W σν ) l(ξ ) aN⊗β 1 F (σν W σν )∗

o

o

o

o

o

= l (i

∗ ωη,η )(σν W σν )ξ

o

o

From which we get that F (σν o W σν o )(A a ⊗β 1)F (σν o W σν o )∗ ⊂ A b No

∗N α M

which gives that (b, a) is an action of G on A. Moreover, using the formula: (id b

∗N α ωη,η )a l(ξ )

= l (id

∗ ωη,η )(σν W σν )ξ

o

o

we get that, for any ζ ∈ D(Hβˆ , ν o ) and n ∈ Nν , we get: (ωΛν (n),ζ b

∗N α id)a l(ξ ) η|η

= l (id

∗ ωη,η )(σν W σν )ξ Λν (n)|ζ = α(n)(id ∗ ωη,η )(σν W σν )ξ |ζ = (i ∗ ωα(n)η,η )(σν W σν )ξ |ζ = (ωξ,ζ ∗ id)(σν W σν )α(n)η|η

o

o

o

o

o

o

o

o

from which we get that: (ωΛν (n),ζ b

∗N α id)a l(ξ )

= (ωξ,ζ

∗ id)(σν W σν )α(n) o

o

So, using 2.2, we get that the weak closure of all the elements of the form (ωη b ∗α id)a(x), for N

η ∈ D(L2 (A)b , ν o ) and x ∈ A, contains all elements in M, and, therefore, this action a is faithful. Finally, we have, for any X ∈ A: (Ω b and, therefore:

∗N α id)a(X) = β

XΛω (1), Λω (1) α,ω


(E b

1175

∗N α id)a(X) = 1 b ⊗N α (Ω b∗N α id)a(X)

= 1 b ⊗α β XΛω (1), Λω (1) α,ω N

= a b XΛω (1), Λω (1) α,ω = a E(X) which finishes the proof.

2

5.5. Theorem. Let G be a measured quantum groupoid; then: (i) there exists an outer action of G; (ii) there exists a depth 2 inclusion M0 ⊂ M1 , equipped with a regular operator-valued weight T1 from M1 onto M0 , and a normal semi-finite faithful weight χ on M0 ∩ M1 , invariant under σtT1 , such that G = G(M0 ⊂ M1 ). 2

Proof. Result (i) is clear by 4.8 and (ii) by 3.9.

6. Outer actions on semi-finite and finite von Neumann algebras In this section, we study the case when a measured quantum groupoid is acting outerly on a semi-finite von Neumann algebra (6.3, 6.5, 6.8), or a finite von Neumann algebra (6.9, 6.10). S. Vaes had proved [42, 3.5] that, if a locally compact quantum group acts outerly on an II 1 factor, then its scaling group τt is trivial. Here the situation is much more complicated, as it is known, since M.-C. David’s result [10], that any connected finite-dimensional measured quantum groupoid (with an antipode which is involutive on the two copies of the basis) acts outerly on the hyperfinite II 1 factor (and, instead of the locally compact quantum case), there are finitedimensional quantum groupoids with a non-trivial scaling group. 6.1. Definition. Let G be a measured quantum groupoid, and let (b, a) be an action on a von Neumann algebra A. We shall say [18, 4.1] that this action is weighted if there exists a normal, semi-finite faithful operator-valued weight T from A onto b(N ). Then, the weight ψ = ν o ◦ b−1 ◦ T will be called lifted from ν o (or lifted). Then, for any lifted weight ψ on A, it is possible it ) in A

to define a 2-cocycle (Dψ ◦a : Dψ)t = it˜ (−it b ∗α (M ∩β(N ) ) which satisfies, ψ b ⊗ α Φ ψ

N

for all s, t in R [18, 7.2 and 7.3]:

(Dψ ◦ a : Dψ)s+t = (Dψ ◦ a : Dψ)s σsψ b (id b

∗N α Γ ) (Dψ ◦ a : Dψ)t

= (a b

N

∗N α τs

∗N α id) (Dψ ◦ a : Dψ)t

(Dψ ◦ a : Dψ)t

(Dψ ◦ a : Dψ)t β ⊗α 1 N

This 2-cocycle is, by definition, Connes’ cocycle derivative (Dψ : Dψ)t , where ψ is the bidual weight of ψ, defined on A b ∗α L(H ) which is canonically isomorphic to the bicrossed product N

[16, 11.6], and the weight ψ is equal to ν o ◦ (ψ b ∗α id), where ν o is a normal semi-finite faithful N

1176


weight on α(N ) , such that

dν o dν o

= −1 ([18], 4.6), and ψ b ∗α id is a normal semi-finite faithful Φ N

operator-valued weight from A b ∗α L(H ) onto α(N ) [18, 4.4]; moreover, we have then

dψ dψ o

=

ψ˜ , where ψ˜ is the dual weight of ψ , defined on the crossed-product A a G, and

dψ dψ o

=

N

ψ b ⊗α −1 , which leads to the result. Φ N

6.2. Proposition. Let G be a measured quantum groupoid, and let (b, a) be a weighted outer action of G on a von Neumann algebra A. Let t ∈ R be in Connes’ invariant T (A) [36, 27.1]; then, there exists v ∈ M ∩ α(N ) ∩ β(N) such that Γ (v) = v b ⊗α v, and τt (x) = vxv ∗ , for all N

x ∈ M; moreover, we have σtν = id. ψ

ψ

Proof. Let ψ be a lifted weight; as σt is interior, there exists a unitary w ∈ A such that σt (x) = wxw ∗ for all x ∈ A; therefore, we get that itψ = wJψ wJψ , and, using [18, 7.1 and 7.2], we get that: (Dψ ◦ a : Dψ)t = itψ˜ w ∗ Jψ w ∗ Jψ b ⊗α itΦ N

One should note that it is possible to define the unitary w ∗ b ⊗α itΦ on elementary tensors (and ν

then extends it to the Hilbert space Hψ b ⊗α H ) because we have, for all n ∈ N , w ∗ b(n)w = ν

ν σ−t (b(n)) = b(σtν (n)) and itΦα(n)−it = τt (α(n)) = α(σt (n)). Moreover, we have: Φ ψ

it (Dψ ◦ a : Dψ)t wJψ wJψ b ⊗α −it a(x) = ψ˜ a(x) Φ N

ψˆ = σt a(x) itψ˜

ψ = a σt (x) itψ˜ = a wxw ∗ itψ˜ which is equal to a(w)a(x)a(w ∗ )(Dψ ◦ a : Dψ)t (wJψ wJψ b ⊗α −it ). Φ N

From which we get that a(w ∗ )(Dψ ◦ a : Dψ)t (w b ⊗α −it ) (which belongs to A b ∗α L(H )) Φ N

N

commutes with a(x), for all x ∈ M. Using then 3.2 and 3.1, we get that there exists u ∈ M such that: a w ∗ (Dψ ◦ a : Dψ)t w b ⊗α −it = 1 b ⊗α u Φ N

N

or: (Dψ ◦ a : Dψ)t = a(w) w ∗ b ⊗α uitΦ N


1177

from which we deduce that uitΦ = v belongs to M ∩ β(N ) ; so v−it belongs to M , Φ

and vxv ∗ = itΦx−it = τt (x). So, the automorphism τt is interior; as v ∈ β(N ) , we get Φ ν ν that β(σt (n)) = τt (β(n)) = β(n), which implies that σt = id. So, we get that wb(n)w ∗ = ψ ν (n)) = b(n), and, therefore, w ∈ A ∩ b(N ) . Moreover, as: σt (b(n)) = b(σ−t

(Dψ ◦ a : Dψ)t = a(w) w ∗ b ⊗α v N

and w commutes with β(N), we get that v ∈ α(N ) ; moreover, the cocycle property with respect to a gives that: (id b

∗N α Γ )a(w) w∗ b ⊗N α Γ (v)

∗N α id)a(w) a w∗ b ⊗N α v a(w) w∗ b ⊗N α v = (a b∗α id)a(w) w ∗ b ⊗α v β ⊗α v N N N

= (a b

from which we deduce that Γ (v) = v β ⊗α v. N

β ⊗α 1 N

2

6.3. Proposition. Let G be a measured quantum groupoid, and let (b, a) be a weighted outer action of G on a semi-finite von Neumann algebra A; then: (i) there exists a positive non-singular operator ρ affiliated to M ∩ α(N ) ∩ β(N ) , such that, for any t ∈ R, x ∈ M, we have: Γ (ρ) = ρ β ⊗α ρ N

τt (x) = ρ xρ −it it

from which we deduce that ρ commutes with Φ. (ii) The weight ν is a trace, and, for any normal semi-finite faithful trace θ on A, θ is lifted, and (Dθ : Dθ )t = 1 b ⊗α ρ it , with the notations of 6.1. N

(iii) We have Z(A) ⊂ Aa , Z(A) b ⊗α C ⊂ Z(A a G), and α(N ) ∩ Z(M) ⊂ α(N ) ∩ Z(M). N

Proof. Let us apply 6.2 to the hypothesis; we get that ν is a trace, and, therefore, that any normal semi-finite faithful trace θ on A is lifted from ν o ; we obtain then, for any t ∈ R, the existence of a unitary vt in M ∩ α(N ) ∩ β(N) such that Γ (vt ) = vt β ⊗α vt , τt (x) = vt xvt∗ , for all x ∈ M, N

and (Dθ ◦ a : Dθ )t = 1 b ⊗α vt ; it is therefore clear that the application t → vt is continuous; N

moreover, the cocycle relation (with respect to (id b ∗α τt )) of (Dθ ◦ a : Dθ )t leads to vs+t = N

vs τs (vt ). But we get also that, for all t ∈ R, we have it˜ = 1 b ⊗α vt −it . From which we deduce that Φ

θ N −it −it t → vt Φ is a one-parameter group of unitaries. So, for any s, t in R, we have vs −is v t Φ = Φ −i(s+t) −is −is vs+t Φ . From the cocycle relation of vt , we then get that Φ vt = τs (vt )Φ , and, therefore, that τs (vt ) = τ−s (vt ); from which we deduce that the unitaries vt are invariant under τs ,

1178


and, therefore, that t → vt is a one-parameter group of unitaries in M ∩ α(N ) ∩ β(N ) . From which, with the help of 6.2, we finish the proof of (i) and (ii). Moreover, let now k ∈ Z(N)+ , such that α(k) belongs to Z(M); then β(k) = R(α(k)) be k longs also to Z(M), using 3.4, we get that b(k) ∈ Z(A); Let us write k = 0 λdeλ , and

k kn = 1/n λdeλ ; then kn is invertible, and, for any normal semi-finite faithful trace θ on A, there exists a normal semi-finite faithful trace θn on A such that (Dθn : Dθ )t = b(kn )it . We then obtain that: (Dθn : Dθ)t = a b(kn )it = 1 b ⊗α β(kn )it N

and, on the other hand: (Dθn : Dθ )t = b(kn )it b ⊗α 1 = 1 b ⊗α α(kn )it N

N

Applying then (ii) to the traces θ and θn , as ρ commutes with α(N ) and β(N ), we get that α(kn ) = β(kn ), and, when n goes to ∞, α(k) = β(k), from which we get that α(k) belongs to Z(M). Moreover, let now x ∈ Z(A); using again 3.4, we get that there exists k ∈ Z(N ) such that α(k) belongs to Z(M), and x = b(k); but, now, we have, as we proved that α(k) = β(k): a(x) = 1 b ⊗α β(k) = 1 b ⊗α α(k) = b(k) b ⊗α 1 N

N

N

which proves that Z(A) ⊂ Aa . On the other hand, as β(k) belongs to Z(M), we have also: ˆ x b ⊗α 1 = a(x) = 1 b ⊗α β(k) = 1 b ⊗α α(n) N

N

N

and, using again 3.4, we get that x b ⊗α 1 belongs to Z(A a G), which finishes the proof. N

2

6.4. Corollary. Let G be a measured quantum groupoid, and let (b, a) be a weighted outer action of G on a semi-finite von Neumann algebra A; then, if A a G is a factor, then A is a factor; is connected, then G is connected also. equivalently, if G Proof. This is clear, using 6.3(iii).

2

6.5. Theorem. Let G be a measured quantum groupoid, and let (b, a) be a weighted outer action of G on a semi-finite von Neumann algebra A; let θ be a normal semi-finite faithful trace on A; then, ν is a trace, there exists a normal semi-finite faithful operator-valued weight T from A onto b(N ) such that θ = ν ◦ b−1 ◦ T , and we have, for all x ∈ Nθ ∩ NT : (θ b

∗N α id)a x ∗x

= β ◦ b−1 ◦ T x ∗ x ρ −1

where ρ is a non-singular positive operator affiliated to M ∩ α(N ) ∩ β(N ) had been defined in 6.3 and satisfies, for any t ∈ R, x ∈ M:


1179

Γ (ρ) = ρ β ⊗α ρ N

τt (x) = ρ xρ −it it

Proof. We had got in 6.3 that ν is a trace,

∞ and the existence of the operator

n ρ; as ρ is affiliated to M ∩ α(N) ∩ β(N ) , if we write ρ = o λ deλ and, for all n ∈ N, fn = 1/n deλ , we get, for any ξ ∈ D(α H, ν) ∩ D(Hβ , ν o ), that fn ξ belongs to D(α H, ν) ∩ D(Hβ , ν o ) ∩ D(ρ −1/2 ), and that ρ −1/2 fn ξ belongs to D(α H, ν) ∩ D(Hβ , ν o ). As (Dθ : Dθ )t = 1 b ⊗α ρ it , by 6.3, we have, for all ζ , ζ in D(α H, ν) and t ∈ R: N

θ σtθ 1 b ⊗α θ α,ν ζ, ζ = 1 b ⊗α ρ it σt 1 b ⊗α θ α,ν ζ, ζ 1 b ⊗α ρ −it N

N

N

N

= 1 b ⊗α ρ it σt θ α,ν ζ, ζ ρ −it νo

N

α,ν ζ, ζ itΦρ −it = 1 b ⊗α ρ it −it θ Φ N

it −it

= 1 b ⊗α θ α,ν ρ it −it ζ, ρ Φ ζ Φ N

−1/2

Let x ∈ Nθ , ξ ∈ D(α H, ν) ∩ D(Hβ , ν o ) ∩ D(ρ −1/2 ), η in D(Φ

−1/2 Φ η

) ∩ D(ρ −1/2 ), such that

and ρ −1/2 η belong to D(α H, ν); we have then:

Λθ (x) a ⊗β ρ −1/2 ξ α ⊗β J −1/2 η2 = Λθ x ∗ b ⊗α J ρ −1/2 ξ β ⊗α −1/2 η2 νo

Φ

νo

Φ

Φ

ν

ν

Φ

which, using [18, 4.11], is equal to: Λ 1 b ⊗α θ α,ν −1/2 η, 1/2 ρ −1/2 ξ a x ∗ 2 θ Φ Φ N

−1/2

The hypothesis about ξ and η give that (1 b ⊗α θ α,ν (Φ N

η, Φ ρ −1/2 ξ ))∗ belongs to 1/2

θ D(σ−i/2 ), and, therefore, we get that:

Λθ (x) a ⊗β ρ −1/2 ξ α ⊗β J −1/2 η2 = Λ a(x) 1 b ⊗α θ α,ν ξ, ρ −1/2 η 2 Φ Φ θ νo

νo

N

which, using again the Radon–Nikodym derivative between θ and θ , is equal to: Λθ a(x) 1 b ⊗α θ α,ν (ξ, η) 2 = ν o θ α,ν (ξ, η)∗ (θ b N

−1/2

We had got in [18, 4.11] that Λν o (θ α,ν (ξ, η)) = ξ α ⊗β JΦΦ νo

∗N α id)a x ∗x θ α,ν (ξ, η)

η, and, therefore, we get:

1180


Λθ (x) a ⊗β ρ −1/2 ξ α ⊗β J −1/2 η2 νo

= (θ b

Φ

νo

Φ

−1/2 Φ ∗N α id)a x ∗x ξ α ⊗ν β JΦ−1/2 η|ξ α ⊗β JΦ η Φ ν o

o

from which we infer that: −1/2 −1/2 Λθ (x) a ⊗β α JΦΦ η, JΦΦ η β,ν o ρ −1/2 ξ |Λθ (x) a ⊗β ρ −1/2 ξ = νo

(θ b

νo

−1/2 −1/2 ∗ Φ Φ α id)a x x α JΦ η, JΦ η β,ν o ξ |ξ

∗N

which, by density gives, for all n ∈ N : Λθ (x) a ⊗β α(n)ρ −1/2 ξ |Λθ (x) a ⊗β ρ −1/2 ξ = (θ b νo

νo

∗N α id)a x ∗x α(n)ξ |ξ

and, therefore Λθ (x) a ⊗β ρ −1/2 ξ 2 = ((θ b ∗α id)a(x ∗ x)ξ |ξ ). If now x belongs to Nθ ∩ NT , νo

N

Λθ (x) belongs to D(a Hθ , ν), and Λθ (x), Λθ (x) a,ν = b−1 (T (x ∗ x)). We get then that: β ◦ b−1 T x ∗ x ρ −1 ξ |ξ = (θ b

∗N α id)a x ∗x ξ |ξ

and, as we deal on both sides with positive closed operators, by density, we get the result.

2

6.6. Notations. On the constructive point of view, we shall now prove that any locally compact groupoid acts outerly on a semi-finite von Neumann algebra (6.8); in the case of finite groupoids, this result had been obtained by J.-M. Vallin in [48, 3.3.11]; let’s fix the notations: let G be a locally compact groupoid, in the sense of [34], equipped with a left Haar system (λu )u∈G (0)

and a quasi-invariant measure ν on the set of units G (0) . Let us denote μ = G (0) λx dν(x). Let us consider the left regular representation λ(g) of G; for any g ∈ G, λ(g) is a unitary from L2 (G r(g) , λr(g) ) onto L2 (G s(g) , λs(g) ), where, as usual, for any x ∈ G (0) , G x = r −1 (x). We can consider as well λg as an orthogonal operator from the real Hilbert space L2R (G r(g) , λr(g) ) onto the real Hilbert space L2R (G s(g) , λs(g) ); this operator extends to an isomorphism from the Clifford algebra Cl(L2R (G r(g) , λr(g) )) (see [6] for details) onto the Clifford algebra Cl(L2R (G s(g) , λs(g) )); on each of these algebras, there exists a finite trace, and each GNS representation of these algebras generates a factor, which is the hyperfinite II 1 factor if G r(g) is infinite (and a finitedimensional factor if it is finite). This construction is just a generalization of [6] up to groupoids. So, for any x ∈ G (0) , we get a copy Ax of the hyperfinite II 1 factor R (or a finite-dimensional factor), and, for any g ∈ G, an isomorphism ag from Ar(g) onto As(g) ; we obtain then an action

⊕ a of the groupoid G on the von Neumann algebra A = G (0) Ax dν(x) (which is hyperfinite II 1 ); let us write τ x for the canonical finite trace on Ax . By the unicity of the normalized trace τ x on Ax , we have, for any g ∈ G, and y positive in Ar(g) , τ s(g) (ag (y)) = τ r(g) (x). If we consider now G(G) the canonical abelian measured associated to G, we get that the

⊕ quantum groupoid

⊕ application a given by the formula a[ G (0) a x dν(x)] = G ag (a s(g) ) dμ(g) is an action of G(G) on A, together with the isomorphism b of L∞ (G (0) , ν) with Z(A) ([16] 6.3); it is clear that the


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⊕ formula E( G (0) a x dν(x)) = b(x → τ x (a x )) is a normal faithful conditional expectation from A onto Z(A); moreover, we have then (E b ∗α id)a = a ◦ E. L∞ (G (0) ,ν)

6.7. Proposition. Let G be a locally compact separable groupoid, and let a be a faithful action of ⊕ G on a von Neumann algebra A = G (0) Ax dν(x), where the algebras Ax are factors, equipped with a faithful state ωx , invariant by a, i.e. such that ωs(g) ◦ ag = ωr(g) , for all g ∈ G. Let us x x x ωx ). It is clear consider the infinite tensor product (B , ω∞ ) = N (Ax , ωx ) of copies of (A ,r(g) x that (B )x∈G (0) is a continuous field of factors. Then the action a˜ g = N (ag , ω ) defines an

⊕ outer action of G on B = G (0) B x dν(x). Proof. The proof is completely taken from [42, 5.1], and we shall give the arguments only when it differs. Let’s take a ∈ B a˜ G ∩ a˜ (B) ; as in [42, 5.1], we can prove that a commutes with all elements of the form 1 b ⊗r (ωξ b ∗r id)a(x), for all x ∈ B, and ξ ∈ D(⊗N (L2 (Ax ), ωx )b , ν), L∞ (G (0) )

L∞ (G (0) )

where b means the isomorphism from L∞ (G (0) , ν) onto Z(B), and r the injection of L∞ (G (0) , ν) into L∞ (G, μ) given the range function. As a is faithful, these elements are functions on G which separate the points of G, and we get that the commutant of these elements is equal to L(L2 (B)) b ∗r L∞ (G, μ). Using then [16, 9.4 and 11.5], we get that a belongs to a˜ (B), L∞ (G (0) ,ν)

and, therefore, to a˜ (Z(B)) = 1

b ⊗r

L∞ (G (0) ,ν)

s(L∞ (G (0) , ν)), where s is the injection of L∞ (G (0) , ν)

into L∞ (G, μ) given the source function, which is here equal to rˆ (because s(L∞ (G (0) , ν)) is central in L∞ (G, μ)). So, we get that (b, a˜ ) is outer. 2 6.8. Theorem. Let G be locally compact separable infinite groupoid; then G has an outer action on a hyperfinite semi-finite von Neumann algebra. Proof. Let us apply 6.7 to the action constructed in 6.6.

2

6.9. Theorem. Let G be a measured quantum groupoid, and let (b, a) be an outer action of G on a finite von Neumann algebra A; let θ be a faithful tracial state on A; then: (i) there exists a positive non-singular operator h affiliated to the center of N , such that (Dθ ◦b : Dν)t = hit ; moreover, for all x ∈ M, we have τt (x) = α(h−it )β(hit )xα(hit )β(h−it ); (ii) there exists a normal faithful conditional expectation from A onto b(N ), which is invariant under a. Proof. As θ ◦ b is a trace on N , we get that there exists a normal faithful conditional expectation E from A onto b(N ); therefore, the action (b, a) is weighted, and we can apply 6.3, from which we get that ν is a trace, and, therefore, the existence of the operator h defining the Radon– Nikodym derivative between θ ◦ b and ν. Moreover, as, for any positive n in N , we have θ ◦ b(n) = ν(hn), we get that there exists also a normal semi-finite faithful operator-valued weight T from A onto b(N ) such that θ = ν ◦ b−1 ◦ T , which verify, for all positive x in A, T (x) = b(h)E(x); moreover, using then 6.5, we get, for any x ∈ M+ T:

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(E b

∗N α id)a(x) 1 b ⊗N α α(h)

∗N α id)a(x) b(h) b ⊗N α 1 = (T b∗α id)a(x) N = 1 b ⊗α (θ b∗α id)a(x) N N

= (E b

= 1 b ⊗α β ◦ b−1 T (x) ρ −1 N

= 1 b ⊗α β ◦ b−1 E(x) β(h)ρ −1 N

And, making now x increase towards 1, we get that ρ = β(h)α(h−1 ), which is (i). Using this result in the calculation above, we get, for any x ∈ M+ T: (E b

∗N α id)a(x) = 1 b ⊗N α β ◦ b−1 E(x)

= a E(x)

which, by increasing limits, remains true for any positive x in A; which is (ii).

2

6.10. Theorem. Let G be a measured quantum groupoid, and let (b, a) be an outer action on a von Neumann algebra A; let us suppose that the crossed-product A a G is a finite von Neumann algebra. Then: is a measured quantum groupoid of compact type, in the sense of [27, 13.2] and [14, (i) G 5.11], i.e. there exists a left-invariant normal conditional expectation on the Hopf-bimodule N, α, β, ˆ Γ), which implies that δˆ = λ = 1. Moreover, we get also that M is semi-finite. (M, (ii) We have: Z(A) b ⊗α C = Z(A a G) N

α(N ) ∩ Z(M) = α(N ) ∩ Z(M) Therefore, A is a factor if and only if A a G is a factor, and G is connected if and only if is connected. G (iii) Let us suppose A is a factor; then, G is finite-dimensional if and only if the depth 2 inclusion a(A) ⊂ A a G is of finite index. Proof. Let θ be a faithful tracial normal state on A a G; then, the restriction of θ to a(A) is also a faithful normal state, and, there exists a normal faithful conditional expectation E from A a G onto a(A). On the other hand, using [16, 9.8], we get that there exists a normal semifinite faithful operator-valued weight Ta˜ from A a G onto a(A), and we get that (DTa˜ : DE)t belongs to A a G ∩ a(A) , which is, as a is outer, equal to 1 b ⊗α α(N ˆ ). N

Moreover, let us consider θ ◦ a which is a faithful tracial normal state on A (and allows us to apply 6.9); we have: (DTa˜ : DE)t = (D θ ◦ a : Dθ )t


1183

and, as θ is a trace, we get that there exists k positive invertible affiliated to N such that: (DTa˜ : DE)t = 1 b ⊗α αˆ k it N

which gives that, for any positive X in A a G, we have: Ta˜ (X) = E 1 b ⊗α αˆ k 1/2 X 1 b ⊗α αˆ k 1/2 N

N

, we have, using [16, 9.8]: and, taking now a positive y in M 1 b ⊗α Tc (y) = E 1 b ⊗α αˆ k 1/2 y αˆ k 1/2 N

N

, we get a normal faithful conditional From which, by taking the restriction of E to 1 b ⊗α M N

onto β(N) which satisfies, for all positive y in M : expectation F from M Tc (y) = F αˆ k 1/2 y αˆ k 1/2 Or, equivalently, we get the existence of a normal faithful conditional expectation G from M we have: onto α(N ) such that, for all positive z in M, T(z) = G βˆ k 1/2 zβˆ k 1/2 It is then straightforward to verify that G is left-invariant, which gives the beginning of (i). As G which implies δˆ = λˆ = 1; as λˆ = λ−1 , [16, 3.10(vii)], =Φ ◦ R, is of compact type, we have Φ we get that λ = 1. Moreover, we get also that Φ = P , as P it is the standard implementation of τt , which is, thanks to 6.9, equal to the interior automorphism implemented by α(h−it )β(hit ), where h is defined as θ (1 b ⊗α β(n)) = ν(hn), for all positive n in N . So, σtΦ = τt is interior, N

which finishes the proof of (i). By applying again 6.9, we get that the action a is weighted, so that we may apply 6.3 to the action a; but, we may also apply 6.9 and 6.3 to the action a˜ ; so, we get that: Z(A) b ⊗α C ⊂ Z(A a G) ⊂ Z A b N

∗N α L(H )

= Z(A) b ⊗α C N

and that: ⊂ α(N ) ∩ Z(M) α(N ) ∩ Z(M) ⊂ α(N ) ∩ Z(M) which finishes the proof of (ii). When A is a factor, it is well known [23, 4.6.2] that, if the inclusion a(A) ⊂ A a G is of finite index, then all relative commutants in the tower are finite-dimensional; in particular A b ∗α L(H ) ∩ a(A) is finite-dimensional, and we get that G is finite-dimensional by 3.1(ii). N

Conversely, if G is finite-dimensional, so is H , and the factor A b ∗α L(H ) is finite, so is a(A) , and the index of a(A) ⊂ A a G is finite [24, 2.1.7].

2

N

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6.11. Remarks. (i) In [33, 4.2.5] was proved that any connected finite-dimensional quantum groupoid outerly acting on a factor is biconnected. (ii) In [31] was proved that any depth 2 finite index subfactor of the hyperfinite II 1 factor R leads to a biconnected finite-dimensional quantum groupoid outerly acting on R (such that the subfactor is the algebra of invariants elements by this action). (iii) In [10] was proved that any finite-dimensional biconnected quantum groupoid, whose antipode is involutive on the two copies of the basis (called target and source Cartan subalgebras, or target and source co-unital subalgebras) is outerly acting on R; this hypothesis on the antipode is equivalent to the fact of having a finite normal quasi-invariant trace on the basis. 6.12. Examples. (i) As recalled in 6.11(iii), any connected measured quantum groupoid G = (N, M, α, β, Γ, T , T , ν) such that dimM < ∞, and ν is a trace has an outer action a on R such that the crossed-product R a G is isomorphic to R, and the index of the inclusion a(R) ⊂ Ra G is finite. Moreover, by 6.11(ii), these are the only outer actions of finite index of measured quantum groupoids on R. (ii) Let τ be the tracial normal faithful state on R, and let us write H for Hτ , J for Jτ , T r the canonical semi-finite faithful trace on L(H ). Let us denote by Tτ the normal faithful semi-finite operator-valued weight from L(H ) onto R, such that τ ◦ Tτ = Tr. Let us recall the construction of the “R-quantum groupoid”, as made in [27, 14]. Let us consider the von Neumann algebra Ro ⊗ R, equipped with its canonical structure of R-bimodule, i.e. we define, for any x ∈ R, α(x) = 1 ⊗ x, and β(x) = x o ⊗ 1; then, the fiber product (Ro ⊗ R) β ∗α (Ro ⊗ R) is R

canonically isomorphic to Ro ⊗ R; moreover G(R) = (R, Ro ⊗ R, α, β, id, id ⊗ τ, τ o ⊗ id, τ ) is a measured quantum groupoid, which is of compact type, because id ⊗ τ is a conditional expectation. Constructing its dual, we obtain the von Neumann algebra L(H ), with its canonical structure of R-bimodule; let us write id|R for the inclusion of R into L(H ), and ido|R for the canonical anti-homomorphism x → J x ∗ J from R into L(H ); then, we get that the fiber prod is equal uct L(H ) ido|R ∗id|R L(H ) is canonically isomorphic to L(H ), and we get that G(R) R

to (R, L(H ), id|R , ido|R , id, Tτ , (Tτ )o , τ ), where id means the identity of L(H ), and (Tτ )o is defined, for any y ∈ L(H ), by (Tτ )o (y) = J Tτ (JyJ )J . on R [16, 6.2]: it’s crossedLet us consider now the trivial action (ido|R , id|R ) of G(R) c product [16, 9.5] is the von Neumann algebra of G(R) , i.e. R ⊗ Ro , which is isomorphic to R; moreover, the relative commutant of R into the crossed-product is Ro , and, so, this action is outer. The inclusion of R into the crossed-product is isomorphic to R ⊗ C ⊂ R ⊗ Ro , which is of infinite index [24, 2.1.19]. (iii) Let G be a finite-dimensional measured quantum groupoid, with a relatively invariant be the dual R-quantum trace ν on the basis N , and (b, a) an outer action of G on R; let G(R) groupoid, and (id, id) its trivial action on R, which is outer by (ii). We can construct now the (2.6(v)) and construct the action (b ⊕ id, a ⊕ id) of measured quantum groupoid G ⊕ G(R) on R ⊕ R (or, G ⊕ G(R) on R ⊕ R. We obtain this way (3.8(ii)) an outer action of G ⊕ G(R) equivalently, on R), whose crossed-product is also isomorphic to R. This action is clearly of infinite index.


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(iv) Let G be a locally compact quantum group having a strictly outer action (in the sense of [42]) on R; for instance, any locally compact group G [42, 5.2], or any amenable Kac algebra of discrete type [42, 8.1] (or, equivalently, using [39, 3.17], any Kac algebra of discrete type, such that the underlying von Neumann algebra of the dual Kac algebra of compact type is injective), or, by duality, any Kac algebra of compact type whose underlying von Neumann algebra is injective. Using again 3.8(ii) and the example given in (iii), we ob on R, for any such G, any finitetain the existence of an outer action of G ⊕ G ⊕ G(R) dimensional measured quantum groupoid G, with a relative invariant trace on the basis. Using 6.10(i), we get that its crossed-product is finite if and only if G is of discrete type. This crossed-product is then a finite factor, by 6.10(ii), and, in the case when this Kac algebra of discrete type has an invariant mean (which, by [39, 3.17], implies that the underlying von Neumann algebra of the dual Kac algebra of compact type is injective), we get that the crossedproduct is injective and is therefore isomorphic to R, and this inclusion will be of infinite index. (v) Let’s now give examples of outer actions of measured quantum groupoids on semi-finite von Neumann algebras. We had got in 6.8 that any locally compact separable infinite groupoid has an outer action on a hyperfinite semi-finite von Neumann algebra; therefore, using (iv), we has on outer action on a hyperfinite semi-finite von Neueasily get that G(G) ⊕ G ⊕ G ⊕ G(R) mann algebra, where G(G) is the measured quantum groupoid constructed from G, G is any locally compact quantum group having a strictly outer action on R, G is any finite-dimensional has been measured quantum groupoid, with a relatively invariant trace on the basis, and G(R) defined in (ii). References [1] S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de C ∗ -algèbres, Ann. Sci. Ec. Norm. Super. 26 (1993) 425–488. [2] S. Baaj, G. Skandalis, S. Vaes, Non-semi-regular quantum groups coming from number theory, Comm. Math. Phys. 235 (2003) 139–167. [3] L. Barnett, Free product von Neumann algebras of type III, Proc. Amer. Math. Soc. 123 (1995) 543–553. [4] E. Blanchard, Tensor products of C(X)-algebras over C(X), Astérisque 232 (1995) 81–92. [5] E. Blanchard, Déformations de C ∗ -algèbres de Hopf, Bull. Soc. Math. France 24 (1996) 141–215. [6] R.J. Blattner, Automorphic group representations, Pacific J. Math. 8 (1958) 665–677. [7] G. Böhm, K. Szlachányi, A coassociative C ∗ -quantum group with non integral dimensions, Lett. Math. Phys. 38 (1996) 437–456. [8] G. Böhm, K. Szlachányi, Weak C ∗ -Hopf algebras: the coassociative symmetry of non-integral dimensions, in: Quantum Groups and Quantum Spaces, in: Banach Center Publ., vol. 40, 1997, pp. 9–19. [9] A. Connes, On the spatial theory of von Neumann algebras, J. Funct. Anal. 35 (1980) 153–164. [10] M.-C. David, C ∗ -groupoïdes quantiques et inclusions de facteurs, structure symétrique et autodualité, action sur le facteur hyperfini de type II 1 , J. Operator Theory 54 (2005) 27–68. [11] K. De Commer, Galois objects and cocycle twisting for locally compact quantum groups, J. Operator Theory, in press, arXiv:0804.2405v3 [math.OA]. [12] M. Enock, Produit croisé d’une algèbre de von Neumann par une algèbre de Kac, J. Funct. Anal. 26 (1977) 16–46. [13] M. Enock, Inclusions irréductibles de facteurs et unitaires multiplicatifs II, J. Funct. Anal. 137 (1996) 466–543. [14] M. Enock, Quantum groupoids of compact type, J. Inst. Math. Jussieu 4 (2005) 29–133. [15] M. Enock, Inclusions of von Neumann algebras and quantum groupoids III, J. Funct. Anal. 223 (2005) 311–364. [16] M. Enock, Measured quantum groupoids in action, Mem. Soc. Math. Fr. 114 (2008) 1–150. [17] M. Enock, Measured quantum groupoids with a central basis, J. Operator Theory, in press, arXiv:0808.4052v2 [math.OA]. [18] M. Enock, The unitary implementation of a measured quantum groupoid action, Ann. Math. Blaise 17 (2) (2010) 247–316, arXiv:0808.4049v2 [math.OA].

1186


[19] M. Enock, R. Nest, Inclusions of factors, multiplicative unitaries and Kac algebras, J. Funct. Anal. 137 (1996) 466–543. [20] M. Enock, J.-M. Schwartz, Produit croisé d’une algèbre de von Neumann par une algèbre de Kac II, Publ. Res. Inst. Math. Sci. 16 (1980) 189–232. [21] M. Enock, J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag, Berlin, 1989. [22] M. Enock, J.-M. Vallin, Inclusions of von Neumann algebras and quantum groupoids, J. Funct. Anal. 172 (2000) 249–300. [23] F.M. Goodman, P. de la Harpe, V.R. Jones, Coxeter Graphs and Towers of Algebras, Math. Sci. Res. Inst. Publ., vol. 14, Springer-Verlag, Berlin, 1989. [24] V. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25. [25] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. Ec. Norm. Super. 33 (2000) 837–934. [26] J. Kustermans, S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003) 68–92. [27] F. Lesieur, Measured quantum groupoids, Mem. Soc. Math. Fr. 109 (2007) 1–122. [28] T. Masuda, Y. Nakagami, A von Neumann Algebra framework for the duality of the quantum groups, Publ. Res. Inst. Math. Sci. 30 (1994) 799–850. [29] T. Masuda, Y. Nakagami, S.L. Woronowicz, A C ∗ -algebraic framework for quantum groups, Internat. J. Math. 14 (2003) 903–1001. [30] D. Nikshych, L. Va˘ınerman, Algebraic Versions of a Finite Dimensional Quantum Groupoid, Lect. Notes Pure Appl. Math., Marcel Dekker, 2000. [31] D. Nikshych, L. Va˘ınerman, A characterization of depth 2 subfactors of II 1 factors, J. Funct. Anal. 171 (2000) 278–307. [32] D. Nikshych, L. Va˘ınerman, Finite quantum groupoids and their applications, in: S. Montgomery, H.-J. Schneider (Eds.), New Directions in Hopf Algebras, in: Math. Sci. Res. Inst. Publ., vol. 43, Cambridge University Press, 2002, pp. 211–262. [33] F. Nill, K. Szlachányi, H.-W. Wiesbrock, Weak Hopf algebras and reducible Jones inclusions of depth 2, I: From crossed products to Jones Tower, math/9806130v1 [math.QA]. [34] J. Renault, A Groupoid Approach to C ∗ -Algebras, Lecture Notes in Math., vol. 793, Springer-Verlag, 1980. [35] J.-L. Sauvageot, Sur le produit tensoriel relatif d’espaces de Hilbert, J. Operator Theory 9 (1983) 237–352. [36] S. ¸ Str˘atil˘a, Modular Theory in Operator Algebras, Abacus Press, Turnbridge Wells, England, 1981. [37] K. Szlachányi, Weak Hopf algebras, in: S. Doplicher, R. Longo, J.E. Roberts, L. Zsido (Eds.), Operators Algebras and Quantum Field Theory, International Press, 1996. [38] M. Takesaki, Theory of Operator Algebras II, Springer-Verlag, Berlin, 2003. [39] R. Tomatsu, Amenable discrete quantum groups, J. Math. Soc. Japan 58 (2006) 949–964. [40] Y. Ueda, Amalgamated free product over Cartan subalgebra, Pacific J. Math. 191 (1999) 359–391. [41] S. Vaes, The unitary implementation of a locally compact quantum group action, J. Funct. Anal. 180 (2001) 426– 480. [42] S. Vaes, Strictly outer actions of groups and quantum groups, J. Reine Angew. Math. (2005) 147–184. [43] S. Vaes, L. Va˘ınerman, Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math. 175 (2003) 1–101. [44] J.-M. Vallin, Bimodules de Hopf et Poids opératoriels de Haar, J. Operator Theory 35 (1996) 39–65. [45] J.-M. Vallin, Unitaire pseudo-multiplicatif associé à un groupoïde, applications à la moyennabilité, J. Operator Theory 44 (2000) 347–368. [46] J.-M. Vallin, Groupoïdes quantiques finis, J. Algebra 239 (2001) 215–261. [47] J.-M. Vallin, Multiplicative partial isometries and finite quantum groupoids, in: V. Turaev, L. Vainerman (Eds.), Locally Compact Quantum Groups and Groupoids, in: IRMA Lect. Math. Theor. Phys., vol. 2, de Gruyter, 2002. [48] J.-M. Vallin, Actions and coactions of finite quantum groupoids on von Neumann algebras, extensions of the matched pair procedure, J. Algebra 314 (2007) 789–816. [49] S.L. Woronowicz, Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N ) group, Invent. Math. 93 (1988) 35–76. [50] S.L. Woronowicz, From multiplicative unitaries to quantum groups, Int. J. Math. 7 (1996) 127–149. [51] S.L. Woronowicz, Compact quantum group, in: Symétries quantiques, Les Houches, 1995, North-Holland, Amsterdam, 1998, pp. 845–884. [52] T. Yamanouchi, Crossed product by groupoid actions and their smooth flows of weights, Publ. Res. Inst. Math. Sci. 28 (1992) 535–578.


1187

[53] T. Yamanouchi, Dual weights on crossed products by groupoid actions, Publ. Res. Inst. Math. Sci. 28 (1992) 653– 678. [54] T. Yamanouchi, Duality for actions and coactions of measured Groupoids on von Neumann Algebras, Mem. Amer. Math. Soc. 101 (1993) 1–109. [55] T. Yamanouchi, Takesaki duality for weights on locally compact quantum group covariant systems, J. Operator Theory 50 (2003) 53–66.


Module maps on duals of Banach algebras and topological centre problems ✩ Zhiguo Hu a , Matthias Neufang b,c,∗ , Zhong-Jin Ruan d a Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada N9B 3P4 b School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6 c Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, Canada M5T 3J1 d Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Received 26 March 2010; accepted 23 October 2010 Available online 4 December 2010 Communicated by S. Vaes

Abstract We study various spaces of module maps on the dual of a Banach algebra A, and relate them to topological centres. We introduce an auxiliary topological centre Zt (A∗ A∗ )♦ for the left quotient Banach algebra A∗ A∗ of A∗∗ . Our results indicate that Zt (A∗ A∗ )♦ is indispensable for investigating properties of module maps over A∗ and for understanding some asymmetry phenomena in topological centre problems as well as the interrelationships between different Arens irregularity properties. For the class of Banach algebras of type (M) introduced recently by the authors, we show that strong Arens irregularity can be expressed both in terms of automatic normality of A∗∗ -module maps on A∗ and through certain commutation relations. This in particular generalizes the earlier work on group algebras by Ghahramani and McClure (1992) [13] and by Ghahramani and Lau (1997) [12]. We link a module map property over A∗ to the space WAP(A) of weakly almost periodic functionals on A, generalizing a result by Lau and Ülger (1996) [34] for Banach algebras with a bounded approximate identity. We also show that for a locally compact quantum group G, the quotient strong Arens irregularity of L1 (G) can be obtained from that of M(G) and can be characterized via the canonical C0 (G)-module structure on LUC(G)∗ . © 2010 Published by Elsevier Inc.

✩ The first and the second authors were partially supported by NSERC. The third author was partially supported by the National Science Foundation DMS-0901395. * Corresponding author. E-mail addresses: [email protected] (Z. Hu), [email protected], [email protected] (M. Neufang), [email protected] (Z.-J. Ruan).

0022-1236/$ – see front matter © 2010 Published by Elsevier Inc. doi:10.1016/j.jfa.2010.10.017

Z. Hu et al. / Journal of Functional Analysis 260 (2011) 1188–1218

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Keywords: Banach algebras; Module maps; Topological centres; Locally compact groups and quantum groups

1. Introduction Let A be a Banach algebra. We use and ♦, respectively, to denote the left and the right Arens products on A∗∗ , and Zt (A∗∗ , ) and Zt (A∗∗ , ♦), respectively, to denote the left and the right topological centres of A∗∗ (see Section 2 for the definitions). The dual space A∗ is canonically a Banach A-bimodule and a Banach left (A∗∗ , )-module. The present paper has been motivated by the study of the following two properties of A. (i) A is left strongly Arens irregular; that is, Zt (A∗∗ , ) = A. (ii) Every bounded left (A∗∗ , )-module map on A∗ is w ∗ -w ∗ continuous. For a locally compact group G, it was shown by Lau and Losert [31] that the convolution group algebra L1 (G) is (left) strongly Arens irregular. Properties (i) and (ii) were considered together in [13, Theorem 1.8], where Ghahramani and McClure showed implicitly that (i) implies (ii) for A = L1 (G). In [38, Satz 3.7.7], Neufang proved that (i) and (ii) are equivalent if A = L1 (G) and G is metrizable. He further proved in [39, Theorem 2.3] that, for Banach algebras A of type (MF ) (i.e., A has the Mazur property of level κ and A∗ has the left A∗∗ factorization property of level κ for some cardinal κ ℵ0 ) such as L1 (G) with G non-compact, property (i) holds, and property (ii) can even be strengthened to “every left (A∗∗ , )-module map on A∗ is automatically bounded and w ∗ -w ∗ continuous”. For general Banach algebras, (i) always implies (ii) (cf. [39, Proposition 2.6]). Among other results, we shall show that for the class of Banach algebras of type (M) introduced and studied recently by the authors [21], properties (i) and (ii) are equivalent. We note that A∗ is also a Banach right (A∗∗ , ♦)-module via the action (f, m) −→ f ♦ m, and all the results mentioned above have their right-hand side versions. In this paper, we shall focus on the Banach algebra BA (X) of bounded right A-module maps on a Banach right A-module X, and hence most results are stated in their right-hand side versions even if X is an A-bimodule. When a right A-module X is the dual space of a given Banach space, we let BAσ (X) denote the subalgebra of BA (X) consisting of w ∗ -w ∗ continuous maps in BA (X). We use BA∗∗ (A∗ ) to denote the Banach algebra of bounded right (A∗∗ , ♦)-module maps on A∗ . Then we have BAσ A∗ = BAσ ∗∗ A∗ ⊆ BA∗∗ A∗ ⊆ BA A∗ .

(1.1)

Combining the right-hand side versions of the above results by Lau and Losert [31] and by Ghahramani and McClure [13], we can conclude that BLσ 1 (G) (L∞ (G)) = BL1 (G)∗∗ (L∞ (G)) for all locally compact groups G. In [26, Theorem 2], Lau showed that BLσ 1 (G) (L∞ (G)) = BL1 (G) (L∞ (G)) precisely when G is compact. For more general Banach algebras, we shall study under what circumstances the equalities in (1.1) hold, and how these equalities are related to topological centre problems. The paper is organized as follows. We start Section 2 with notation conventions followed by introducing an auxiliary topological centre Zt (A∗ A∗ )♦ for (A∗ A∗ , ), the canonical quotient Banach algebra of (A∗∗ , ). We show that Zt (A∗ A∗ )♦ happens to be the hidden piece that is responsible for some asymmetry phenomena occurring in topological centre problems as observed in [20,34]. Results in the paper indicate that Zt (A∗ A∗ )♦ is indispensable for the

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comparison between the algebras BAσ (A∗ ), BA∗∗ (A∗ ), and BA (A∗ ), and for the study of the interrelationships between different Arens irregularity (respectively, Arens regularity) properties. We close this section by summarizing some general results on module maps. In Section 3, we study various spaces of module maps on A∗ , and relate them to topological centres. We show that some of the automatic normality properties of module maps define new concepts between strong Arens irregularity and quotient strong Arens irregularity, the two notions introduced, respectively, by Dales and Lau [6] and by the authors [20]. More characterizations of module map properties over A∗ are obtained for Banach algebras A of type (M). We also characterize the left faithfulness of the algebras (A∗∗ , ) and (A∗ A∗ , ) through topological centres. We note that there is another notion of “maximal” Arens irregularity in the literature, the notion of extreme non-Arens regularity introduced by Granirer [14]. The reader is referred to [19] and references therein for results on this type of Arens irregularity and its relationship to strong Arens irregularity. In Section 4, we discuss commutation relations for module maps on A∗ . We obtain several bicommutant theorems, which in particular improve and extend the commutant theorem [12, Theorem 5.1] on L1 (G) by Ghahramani and Lau to all Banach algebras of type (M). For this class of Banach algebras, we show that left and right strong Arens irregularities are in fact equivalent to certain commutation and double commutation relations between module maps. In particular, we show that a unital weakly sequentially complete Banach algebra A (e.g., A = A(G) with G compact) is right strongly Arens irregular if and only if we have the bicommutant theorem Acc = A for the canonical embedding A → BA (A∗ ). We introduced in [20] the Banach algebra A∗ A∗R , which is a subspace of (A∗ A∗ , ) but equipped with a distinct multiplication. In Section 5, we consider the corresponding Banach algebra of module maps on A∗ . This new Banach algebra structure is used in particular to characterize the introversion of A∗ A in A∗ and the equality A∗ A = AA∗ A. We further relate a module map property over A∗ to the algebra A∗ A∗R and the space WAP(A) of weakly almost periodic functionals on A, in particular generalizing [34, Theorem 3.6] by Lau and Ülger for Banach algebras with a bounded approximate identity. For a locally compact quantum group G, let L1 (G) and M(G) be the quantum group algebra and quantum measure algebra of G, respectively. Let LUC(G) = L1 (G)∗ L1 (G) be the space of left uniformly continuous functionals on L1 (G). In Section 6, we obtain a natural completely isometric M(G)-module isomorphism LUC(G) ∼ = M(G)∗ L1 (G), and characterize the quotient strong Arens irregularity of L1 (G) in terms of the canonical C0 (G)-module structure on LUC(G)∗ . We prove that if M(G) is quotient strongly Arens irregular, then L1 (G) is also quotient strongly Arens irregular. This in particular shows that for all amenable locally compact groups G, the strong Arens irregularity of the Fourier–Stieltjes algebra B(G) implies that of the Fourier algebra A(G). In the subsequent work [22], we will investigate module maps over locally compact quantum groups through a general Banach algebra approach as used in the present paper. The authors are grateful to the referee for valuable suggestions. 2. Definitions and preliminary results Let B be an algebra equipped with a topology such that B is a topological linear space. Suppose that B is also a right topological semigroup under the multiplication. That is, for any fixed y ∈ B, the map x −→ xy is continuous on B (cf. [2]). In this case, the topological centre Zt (B) of B is defined to be the set of all y ∈ B such that the map x −→ yx is continuous on B. When


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B is a left topological semigroup under the multiplication, the topological centre Zt (B) of B is defined analogously. Throughout this paper, A denotes a Banach algebra with a faithful multiplication. As is well known, on the bidual A∗∗ of A, there are two Banach algebra multiplications called, respectively, the left and the right Arens products, each extending the multiplication on A. By definition, the left Arens product is induced by the left A-module structure on A. That is, for m, n ∈ A∗∗ , f ∈ A∗ , and a, b ∈ A, we have f · a, b = f, ab,

n f, a = n, f · a,

and m n, f = m, n f .

The right Arens product ♦ is defined by considering A as a right A-module. More precisely, we have b, a · f = ba, f , f, m ♦ n = f ♦ m, n

a, f ♦ n = a · f, n, and a, b ∈ A, f ∈ A∗ , m, n ∈ A∗∗ .

It is known that m n = w ∗ -limα w ∗ -limβ aα bβ and m ♦ n = w ∗ -limβ w ∗ -limα aα bβ whenever (aα ) and (bβ ) are nets in A converging, respectively, to m and n in the weak∗ -topology on A∗∗ . The Banach algebra A is said to be Arens regular if and ♦ coincide on A∗∗ . Since the multiplication on a von Neumann algebra is separately w ∗ -w ∗ continuous, every C ∗ -algebra is Arens regular. Hence, by [4, Corollaries 6.3 and 6.4], every operator algebra and every quotient algebra thereof are Arens regular (see [8] for more information on the structure of the bidual of an operator algebra). Under the weak∗ -topology, (A∗∗ , ) is a right topological semigroup and (A∗∗ , ♦) is a left topological semigroup. By definition, we have Zt A∗∗ , = m ∈ A∗∗ : the map n −→ m n is w ∗ -w ∗ continuous on A∗∗ and Zt A∗∗ , ♦ = m ∈ A∗∗ : the map n −→ n ♦ m is w ∗ -w ∗ continuous on A∗∗ . The algebras Zt (A∗∗ , ) and Zt (A∗∗ , ♦) are called, respectively, the left and the right topological centres of A∗∗ . It is easy to see that Zt A∗∗ , = m ∈ A∗∗ : m n = m ♦ n for all n ∈ A∗∗ and Zt A∗∗ , ♦ = m ∈ A∗∗ : n ♦ m = n m for all n ∈ A∗∗ . Therefore, A is Arens regular if and only if Zt (A∗∗ , ) = Zt (A∗∗ , ♦) = A∗∗ . For a set X in a Banach space, we use X to denote the closed linear span of X in the space. By Cohen’s factorization theorem, A∗ A = A∗ A if A has a bounded right approximate identity (BRAI), and AA∗ = AA∗ if A has a bounded left approximate identity (BLAI). The

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canonical quotient map A∗∗ −→ A∗ A∗ , m −→ m|A∗ A yields a Banach algebra multiplication on A∗ A∗ (also denoted by ) such that we have the isometric Banach algebra identification

∗ ∗∗ ∗ ⊥ A∗ A , ∼ = A , A A .

Under the weak∗ -topology, (A∗ A∗ , ) is also a right topological semigroup. In the rest of the paper, (A∗ A∗ , ) and its topological centre are simply denoted by A∗ A∗ and Zt (A∗ A∗ ), respectively. The Banach algebra (AA∗ ∗ , ♦) (or AA∗ ∗ for short) is defined analogously as a quotient algebra of (A∗∗ , ♦). Recall that the weak∗ operator topology (w ∗ot) on B(A∗ ) is the locally convex topology determined by the seminorms T ∈ B(A∗ ) −→ |T (f ), a| (a ∈ A, f ∈ A∗ ), and B(A∗ ) is a dual Banach space via the isometric Banach space identification B(A∗ ) ∼ = (A∗ ⊗γ A)∗ , where ⊗γ ∗ ∗ is the projective tensor product. Clearly, w -convergence in B(A ) implies w ∗ot-convergence, and the converse holds on bounded subsets of B(A∗ ). Then BA (A∗ ) is w ∗ot-closed and hence w ∗ -closed in B(A∗ ). For each T ∈ BA (A∗ ), the map BA (A∗ ) −→ BA (A∗ ), S −→ S ◦ T is w ∗otw ∗ot continuous and also w ∗ -w ∗ continuous. Thus (BA (A∗ ), w ∗ot) and (BA (A∗ ), w ∗ ) are both right topological semigroups. Let ∗ BAw ot A∗ = T ∈ BA A∗ : S −→ T ◦ S is w ∗ot-w ∗ot continuous on BA A∗

(2.1)

∗ BAw A∗ = T ∈ BA A∗ : S −→ T ◦ S is w ∗ -w ∗ continuous on BA A∗ .

(2.2)

and

∗

∗

Then BAw ot (A∗ ) and BAw (A∗ ) are, respectively, the topological centres of (BA (A∗ ), w ∗ot) and ∗ ∗ (BA (A∗ ), w ∗ ), and we have BAw ot (A∗ ) ⊆ BAw (A∗ ) by the Krein–Šmulian theorem. For m in A∗∗ or A∗ A∗ and f in A∗ , let mL (f ) ∈ A∗ be defined by mL (f )(a) = m f, a = m, f · a (a ∈ A).

(2.3)

Then mL (f ) ∈ A∗ A if f ∈ A∗ A. Let Φ : A∗ A∗ −→ BA (A∗ ) be the map m −→ mL . Note that Φ is just the adjoint of the right A-module map A∗ ⊗γ A −→ A∗ A, f ⊗ a −→ f · a. Then Φ is a w ∗ -w ∗ continuous, contractive, and injective algebra homomorphism, and we have ∗ ∗ Φ A∗ A = mL : m ∈ A∗ A = mL : m ∈ A∗∗ .

(2.4)

We call Φ the canonical representation of A∗ A∗ on A∗ . If A has a BRAI, then A∗ A∗ has an identity. In this case, Φ is surjective (since T = T ∗ (E)L for all T ∈ BA (A∗ ), where E is a right identity of (A∗∗ , )), and is in fact a w ∗ -w ∗ot (and hence w ∗ -w ∗ ) homeomorphism, and the w ∗ ot and the weak∗ -topology coincide on BA (A∗ ). Furthermore, Φ is isometric if A has a contractive BRAI. Conversely, if Φ is surjective, then A∗ A∗ has an identity, which implies that A has a BRAI if A2 = A (cf. [16, Theorem 4(ii)]). Let RM(A) be the right multiplier algebra of A (with opposite composition as the multiplication). It is clear that RM(A) −→ BAσ (A∗ ), μ −→ μ∗ is an isometric algebra isomorphism. Then we have RM(A) ∼ = BAσ A∗ = T ∈ BA A∗ : T ∗ (A) ⊆ A .

(2.5)


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If A has a BRAI, then BAσ (A∗ ) ⊆ Φ(A∗ A∗ ). Conversely, if BAσ (A∗ ) ⊆ Φ(A∗ A∗ ), then A∗ A∗ is unital, since BAσ (A∗ ) contains the identity of BA (A∗ ) and Φ is an injective algebra homomorphism. By the definition of Zt (A∗∗ , ♦), we have ∗ BA∗∗ A∗ = BAr,w A∗ := T ∈ BA A∗ : T ∗ (A) ⊆ Zt A∗∗ , ♦ .

(2.6)

Corresponding to (2.3), for m in A∗∗ or AA∗ ∗ and f in A∗ , we let mR (f ) ∈ A∗ be defined by mR (f )(a) = a, f ♦ m = a · f, m (a ∈ A).

(2.7)

Then we obtain ∗ BAr,w A∗ = T ∈ BA A∗ : A∗∗ −→ B A∗ , m −→ T ◦ mR is w ∗ -w ∗ continuous .

(2.8)

Replacing Zt (A∗∗ , ♦) in (2.6) by Zt (A∗∗ , ), we can define the space ∗ BAl,w A∗ = T ∈ BA A∗ : T ∗ (A) ⊆ Zt A∗∗ , .

(2.9)

In fact, comparing with (2.8), we can show that ∗ BAl,w A∗ = T ∈ BA A∗ : A∗∗ −→ B A∗ , m −→ T ◦ mL is w ∗ -w ∗ continuous .

(2.10)

Therefore, we have ∗ ∗ ∗ BAσ A∗ ⊆ BAw ot A∗ ⊆ BAw A∗ ⊆ BAl,w A∗ ∗ . ⊆ T ∈ BA A∗ : T ∗ (A)|A∗ A ⊆ Zt A∗ A

(2.11)

We note that w ∗ -w ∗ continuity in (2.8) and (2.10) can be replaced by w ∗ -w ∗ ot continuity. It is known from [20, Corollary 3(i)] that ∗ ∗ Zt A∗ A = m ∈ A∗ A : A · m ⊆ Zt A∗∗ , .

(2.12)

∗

Hence, we also have Φ(Zt (A∗ A∗ )) ⊆ BAl,w (A∗ ). As mentioned above, if A has a BRAI, then the map Φ : A∗ A∗ −→ BA (A∗ ) is a w ∗ -w ∗ homeomorphism; in this case, we obtain ∗ ∗ ∗ ∗ Φ Zt A∗ A = BAw ot A∗ = BAw A∗ = BAl,w A∗ ∗ , = T ∈ BA A∗ : T ∗ (A)|A∗ A ⊆ Zt A∗ A ∗

∗

and, in particular, BAl,w (A∗ ) is a Banach algebra. Clearly, the module map spaces BAl,w (A∗ ) and ∗ BAr,w (A∗ ) (= BA∗∗ (A∗ )) are closely related to topological centres. They will be used to study several Arens irregularity properties in the paper. It is seen that Zt (A∗∗ , ) and Zt (A∗∗ , ♦) both appear in the study of (the one-sided) right A-module maps on A∗ . We observe that one natural A-submodule of A∗ A∗ has not been considered in the discussions so far. This A-bimodule relates A∗ A∗ to Zt (A∗∗ , ♦), playing

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a similar role as Zt (A∗ A∗ ) does in (2.12) where A∗ A∗ is linked to Zt (A∗∗ , ). The results obtained in the paper indicate that this Banach A-bimodule is exactly the missing piece in the study of Arens irregularity, without which some asymmetries occur in topological centre problems (cf. [20, Remark 28] and [34, Remark 5.2]). This A-bimodule shall be used to describe the pre-image of BA∗∗ (A∗ ) under Φ, to compare the algebras BAσ (A∗ ), BA∗∗ (A∗ ), and BA (A∗ ), and to study further interrelationships between various topological centre problems and properties of module maps on A∗ (see the results presented in this and the next sections). Before giving the definition, let us first note that A∗ ♦ A∗∗ A ⊆ A∗ A holds, since we have f ♦ (n · a) = f ♦ (n ♦ a) = (f ♦ n) ♦ a = (f ♦ n) · a

a ∈ A, f ∈ A∗ , n ∈ A∗∗ .

For m ∈ A∗ A∗ , let m ∈ A∗∗ be any extension of m. Then, for n ∈ A∗∗ A and p ∈ A∗∗ , we see that n♦m=n♦m and p m = p m

(2.13)

are well-defined elements of A∗∗ . Therefore, for m ∈ A∗ A∗ , we obtain that A · m ⊆ Zt (A∗∗ , ♦) if and only if n ♦ m = n m in A∗∗ for all n ∈ A∗∗ A. Definition 2.1. Let A be a Banach algebra. The auxiliary topological centre of A∗ A∗ is defined by ∗ ∗ Zt A∗ A ♦ = m ∈ A∗ A : n ♦ m = n m in A∗∗ for all n ∈ A∗∗ A . The auxiliary topological centre Zt (AA∗ ∗ ) of AA∗ ∗ is defined similarly. Obviously, Zt (A∗ A∗ )♦ is just the right topological centre Zt (A∗∗ , ♦) of A∗∗ if A is unital. In general, Zt (A∗ A∗ )♦ is a Banach A-submodule of A∗ A∗ . Comparing with (2.11) and (2.12), we have ∗ ∗ Zt A∗ A ♦ = m ∈ A∗ A : A · m ⊆ Zt A∗∗ , ♦

(2.14)

and ∗ ∗ BAσ A∗ ⊆ BA∗∗ A∗ = BAr,w A∗ ⊆ T ∈ BA A∗ : T ∗ (A)|A∗ A ⊆ Zt A∗ A ♦ .

(2.15)

Therefore, Zt (A∗ A∗ )♦ = Zt (A∗ A∗ ) if Zt (A∗∗ , ) = Zt (A∗∗ , ♦). In general, Φ(Zt (A∗ A∗ )♦ ) ⊆ BA∗∗ (A∗ ), and Φ(Zt (A∗ A∗ )♦ ) = BA∗∗ (A∗ ) = {T ∈ BA (A∗ ): T ∗ (A)|A∗ A ⊆ Zt (A∗ A∗ )♦ } if A has a BRAI. ∗ In the sequel, we use the symbol BAr,w (A∗ ) for the algebra BA∗∗ (A∗ ) when we want to ∗ ∗ show the parallel between BAl,w (A∗ ) and BAr,w (A∗ ). The proposition below on Zt (A∗ A∗ ) and Zt (A∗ A∗ )♦ is clearly true.


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Proposition 2.2. Let A be a Banach algebra. Then the following assertions hold. (i) The canonical quotient map ϕ : A∗∗ −→ A∗ A∗ satisfies ∗ ∗ and ϕ Zt A∗∗ , ♦ ⊆ Zt A∗ A ♦ . ϕ Zt A∗∗ , ⊆ Zt A∗ A (ii) The canonical representation Φ : A∗ A∗ −→ BA (A∗ ) satisfies ∗ ∗ ∗ ∗ ⊆ BAl,w A∗ and Φ Zt A∗ A ♦ ⊆ BAr,w A∗ . Φ Zt A∗ A ∗

∗

(iii) Zt (A∗∗ , ) = Zt (A∗∗ , ♦) ⇒ BAl,w (A∗ ) = BAr,w (A∗ ) ⇒ Zt (A∗ A∗ ) = Zt (A∗ A∗ )♦ . (iv) Zt (A∗ A∗ )♦ is a subalgebra of (A∗ A∗ , ) in the following cases: (a) Zt (A∗∗ , ♦) = A; (b) A has a BRAI; (c) A2 = A and A · Zt (A∗∗ , ♦) ⊆ A; (d) Zt (A∗∗ , ♦)A∗∗ ⊆ Zt (A∗∗ , ♦). For an easy comparison, we record the following results on topological centres from [20]. Proposition 2.3. (See [20].) Let A be a Banach algebra. Then the following assertions hold. (i) Φ(Zt (A∗ A∗ )) ⊆ BAσ (A∗ ) ⇐⇒ A · Zt (A∗ A∗ ) ⊆ A. (ii) If A2 = A, then we have ∗ A · Zt A∗∗ , ⊆ A ⇐⇒ A · Zt A∗ A ⊆ A and Zt A∗∗ , ♦ · A ⊆ A

⇐⇒

∗ Zt AA∗ · A ⊆ A.

(iii) If A2 = A, in particular, if A has a BRAI or a BLAI, then we have ∗ ∗ and Zt A∗∗ , ♦ · A = Zt AA∗ · A. A · Zt A∗∗ , = A · Zt A∗ A In Proposition 2.3, Zt (A∗∗ , ) and Zt (A∗∗ , ♦) are on the “wrong” sides, i.e., the left (respectively, right) topological centre stays on the right-hand (respectively, left-hand) side. The proposition below shows that they can be pulled to the “correct” sides by using the auxiliary topological centres. Proposition 2.4. Let A be a Banach algebra. Then the following assertions hold. (i) Φ(Zt (A∗ A∗ )♦ ) ⊆ BAσ (A∗ ) ⇐⇒ A · Zt (A∗ A∗ )♦ ⊆ A. (ii) If A2 = A, then we have ∗ Zt A∗∗ , · A ⊆ A ⇐⇒ Zt AA∗ · A ⊆ A and A · Zt A∗∗ , ♦ ⊆ A

⇐⇒

∗ A · Zt A∗ A ♦ ⊆ A.

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(iii) If A2 = A, in particular, if A has a BRAI or a BLAI, then we have ∗ Zt A∗∗ , · A = Zt AA∗ · A

∗ and A · Zt A∗∗ , ♦ = A · Zt A∗ A ♦ .

Proof. (i) is obvious. Clearly, a · m = a · m|A∗ A for all a ∈ A and m ∈ A∗∗ , and m|A∗ A ∈ Zt (A∗ A∗ )♦ if m ∈ Zt (A∗∗ , ♦). Then we obtain A · Zt (A∗∗ , ♦) ⊆ A · Zt (A∗ A∗ )♦ ⊆ Zt (A∗∗ , ♦). Similarly, we have Zt (A∗∗ , ) · A ⊆ Zt (AA∗ ∗ ) · A ⊆ Zt (A∗∗ , ). Therefore, (ii) and (iii) hold. 2 Proposition 2.5. Let A be a Banach algebra. Then the following assertions hold. (i) Zt (A∗ A∗ ) = A∗ A∗ ⇐⇒ A · A∗∗ ⊆ Zt (A∗∗ , ), which is equivalent to A · A∗ A∗ ⊆ Zt (A∗ A∗ ) if A satisfies A2 = A. (ii) Zt (A∗ A∗ )♦ = A∗ A∗ ⇐⇒ A · A∗∗ ⊆ Zt (A∗∗ , ♦), which is equivalent to A · A∗ A∗ ⊆ Zt (A∗ A∗ )♦ if A satisfies A2 = A. Proof. We prove (i); assertion (ii) can be shown similarly. The first equivalence follows from (2.12). Clearly, if Zt (A∗ A∗ ) = A∗ A∗ , then A · ∗ A A∗ ⊆ Zt (A∗ A∗ ). Conversely, suppose that A2 = A and A · A∗ A∗ ⊆ Zt (A∗ A∗ ). Let a, b ∈ A and m ∈ A∗∗ , and let p = m|A∗ A ∈ A∗ A∗ . Then b · p ∈ Zt (A∗ A∗ ). Thus, by (2.12) again, we have ∗ (ab) · m = a · (b · m) = a · (b · p) ∈ A · Zt A∗ A ⊆ Zt A∗∗ , . That is, A2 · A∗∗ ⊆ Zt (A∗∗ , ). Therefore, A · A∗∗ ⊆ Zt (A∗∗ , ) since A2 = A.

2

For convenience, we summarize the facts discussed and implied above in the following two propositions. Most of these facts can be found in some form in the existing literature, but some of them were stated only for certain classes of Banach algebras. See, for example, [1,3,5,6,16, 20,21,26,27,30,34]. The reader is also referred to [15] for a systematic study of left (respectively, right) Banach modules of the form V ∗ A (respectively, AV ∗ ), where V is a left (respectively, right) Banach A-module. Proposition 2.6. Let A be a Banach algebra satisfying A2 = A. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v) (vi)

A has a BRAI; Φ : A∗ A∗ −→ BA (A∗ ) is surjective; A∗ A∗ is unital; BAσ (A∗ ) ⊆ Φ(A∗ A∗ ); BAσ (A∗ ) ⊆ Φ(Zt (A∗ A∗ )) (respectively, BAσ (A∗ ) ⊆ Φ(Zt (A∗ A∗ )♦ )); ∗ ∗ Φ(Zt (A∗ A∗ )) = BAl,w (A∗ ) (respectively, Φ(Zt (A∗ A∗ )♦ ) = BAr,w (A∗ )).

Proposition 2.7. Let A be a Banach algebra with a BRAI. Then Φ : A∗ A∗ −→ BA (A∗ ) is a w ∗ w ∗ homeomorphism which is isometric if A has a contractive BRAI, and the following assertions hold.


(i) (ii) (iii) (iv)

∗

∗

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∗

BAw ot (A∗ ) = BAw (A∗ ) = BAl,w (A∗ ). ∗ BAr,w (A∗ ) = Φ(Zt (A∗ A∗ )♦ ) = {mL : m ∈ A∗∗ and A · m ⊆ Zt (A∗∗ , ♦)}. ∗ BAl,w (A∗ ) = Φ(Zt (A∗ A∗ )) = {mL : m ∈ A∗∗ and A · m ⊆ Zt (A∗∗ , )}. r,w ∗ ∼ A, BA (A∗ ) ∼ If A is unital, then we have BAσ (A∗ ) = = (A∗∗ , ), BA (A∗ ) ∼ = Zt (A∗∗ , ♦), ∗ and BAl,w (A∗ ) ∼ = Zt (A∗∗ , ); in this case, A is Arens regular ⇐⇒ BA (A∗ ) = ∗ l,w ∗ BA (A∗ ) ⇐⇒ BA (A∗ ) = BAr,w (A∗ ).

Therefore, for every unital C ∗ -algebra A, we have r,w ∗ l,w ∗ BAσ A∗ ∼ = A and BA A∗ = BA A∗ = BA A∗ ∼ = A∗∗ ∼ = πu (A) , where πu (A) is the enveloping von Neumann algebra of A. It is known that for a general left introverted subspace X of A∗ (that is, X is a closed right Asubmodule of A∗ satisfying A∗∗ X ⊆ X), the Banach algebra (X ∗ , ) and its topological centre Zt (X ∗ ) are also defined. In this case, the map ΨX : X ∗ −→ BA (X), m −→ mL is a contractive algebra homomorphism, where mL is given as before by mL (f ), a = m, f · a (m ∈ X ∗ , f ∈ X, a ∈ A). The proposition below can be shown easily. We omit the proof. Proposition 2.8. Let A be a Banach algebra and let X be a left introverted subspace of A∗ . Then the following assertions hold. (i) ΨX is injective ⇐⇒ [X = X · A] ⇐⇒ the product on X ∗ is right faithful, which is equivalent to X ⊆ A∗ A if A has a BRAI. (ii) ΨX is surjective ⇐⇒ X ∗ is right unital. (iii) Suppose that A is w ∗ -dense in X ∗ . Then ΨX is bijective ⇐⇒ X ∗ is left unital ⇐⇒ X ∗ is unital. Remark 2.9. Note that T (A∗ A) ⊆ A∗ A for all T ∈ BA (A∗ ). Let Π : BA (A∗ ) −→ BA (A∗ A) be the map T −→ T |A∗ A . Then Π is a contractive algebra homomorphism, and is injective if A2 = A. By Propositions 2.6 and 2.8 (with X = A∗ A), the map Π is bijective if A has a BRAI. However, the converse is not true. For example, if A = A(F2 ) is the Fourier algebra of the free group F2 of two generators, then A has no BRAI, but Π : BA (A∗ ) −→ BA (A∗ A) is bijective, since A∗ A is the reduced group C ∗ -algebra Cλ∗ (F2 ) of F2 and, for every S ∈ BA (A∗ A), we have S = Π(T ) with T = (S ∗ |A )∗ ∈ BA (A∗ ). In fact, the above arguments can be applied to a general locally compact quantum group G, which show that if G is co-amenable or compact, then Π yields an isometric algebra isomorphism BL1 (G) (L∞ (G)) ∼ = BL1 (G) (LUC(G)). Recall that a locally compact quantum group G is co-amenable if the quantum group algebra L1 (G) has a BAI, and is compact if the C ∗ -algebra C0 (G) is unital. See Section 6 and references therein for more information on locally compact quantum groups. Analogously, right introverted subspaces of A∗ and their topological centres can be considered. In the present paper, however, we shall focus on the case where X = A∗ A. The corresponding results with A∗ A replaced by AA∗ also hold. Some of the results given in the paper do not require the faithfulness of the multiplication on A, and many of them have their cb-versions. In particular, if A is a completely contractive Banach algebra, then the canonical representation Φ induces a completely contractive injection A∗ A∗ −→ CBA (A∗ ), where

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CBA (A∗ ) is the algebra of completely bounded maps in BA (A∗ ). Therefore, we have the following immediate consequence of Proposition 2.6. Corollary 2.10. Let A be a completely contractive Banach algebra with a BRAI. Then we have BA A∗ = CBA A∗ and RM A∗ = RMcb A∗ , where RMcb (A∗ ) denotes the completely bounded right multiplier algebra of A. We note that recently Viktor Losert showed that RM(A(SL(2, R))) = RM cb (A(SL(2, R))) though the Fourier algebra A(SL(2, R)) does not have a BRAI. 3. Module maps and topological centres The results in Section 2 show in particular that the comparison between BAσ (A∗ ), BA∗∗ (A∗ ), ∗ and BAl,w (A∗ ) does describe Arens irregularity of a Banach algebra A. In other words, the automatic normality of certain module maps is intrinsically related to topological centre problems. We start this section with a corollary of Propositions 2.2–2.5 (cf. (2.4) and (2.5)). As in Sec∗ tion 2, we shall use the symbol BAr,w (A∗ ) for the algebra BA∗∗ (A∗ ) when a result contains both ∗ BAl,w (A∗ ) and BA∗∗ (A∗ ). Corollary 3.1. Let A be a Banach algebra. Then the following assertions hold. (i) (ii) (iii) (iv) (v)

∗

If BAr,w (A∗ ) = BAσ (A∗ ), then A · Zt (A∗ A∗ )♦ ⊆ A. ∗ If BA (A∗ ) = BAr,w (A∗ ), then A · A∗∗ ⊆ Zt (A∗∗ , ♦). ∗ If BAl,w (A∗ ) = BAσ (A∗ ), then A · Zt (A∗ A∗ ) ⊆ A. ∗ If BA (A∗ ) = BAl,w (A∗ ), then A · A∗∗ ⊆ Zt (A∗∗ , ). If BA (A∗ ) = BAσ (A∗ ), then A · A∗∗ ⊆ A.

We shall show that under the mild condition that “A2 = A”, the converse of each of (i)–(v) holds. Note that the predual A of any Hopf–von Neumann algebra M satisfies this condition, where the multiplication on A is the pre-adjoint of the co-multiplication on M and hence is a (complete) quotient map. In the sequel, the right multiplier algebra RM(A) and the algebra A∗ A∗ are identified with their canonical images in BA (A∗ ). Theorem 3.2. Let A be a Banach algebra satisfying A2 = A (e.g., A is the predual of a Hopf– von Neumann algebra). Then the statements (i)–(iv) in each of (I)–(V) are equivalent. (I)

(i) (ii) (iii) (iv) (II) (i) (ii) (iii) (iv) (III) (i) (ii)

∗

BAr,w (A∗ ) = BAσ (A∗ ). Zt (A∗ A∗ )♦ ⊆ RM(A). A · Zt (A∗ A∗ )♦ ⊆ A. A · Zt (A∗∗ , ♦) ⊆ A. ∗ BA (A∗ ) = BAr,w (A∗ ). Zt (A∗ A∗ )♦ = A∗ A∗ . A · A∗ A∗ ⊆ Zt (A∗ A∗ )♦ . A · A∗∗ ⊆ Zt (A∗∗ , ♦). ∗ BAl,w (A∗ ) = BAσ (A∗ ). Zt (A∗ A∗ ) ⊆ RM(A).


(iii) (iv) (IV) (i) (ii) (iii) (iv) (V) (i) (ii) (iii) (iv)

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A · Zt (A∗ A∗ ) ⊆ A. A · Zt (A∗∗ , ) ⊆ A. ∗ BA (A∗ ) = BAl,w (A∗ ). Zt (A∗ A∗ ) = A∗ A∗ . A · A∗ A∗ ⊆ Zt (A∗ A∗ ). A · A∗∗ ⊆ Zt (A∗∗ , ). BA (A∗ ) = BAσ (A∗ ). A∗ A∗ ⊆ RM(A). A · A∗ A∗ ⊆ A. A · A∗∗ ⊆ A.

In addition, the inclusions in (I), (III), and (V) can be replaced by the equalities if A has a BRAI. Proof. (I) Due to Proposition 2.4 and Corollary 3.1(i), we need only show that (iv) ⇒ (i). ∗ Suppose that A · Zt (A∗∗ , ♦) ⊆ A and T ∈ BAr,w (A∗ ). Let a, b ∈ A. By (2.6), we have T ∗ (b) ∈ Zt (A∗∗ , ♦), and thus T ∗ (ab) = a · T ∗ (b) ∈ A · Zt (A∗∗ , ♦) ⊆ A. Hence, T ∗ (A) = T ∗ (A2 ) ⊆ A, ∗ and T ∈ BAσ (A∗ ) (cf. (2.5)). Therefore, BAr,w (A∗ ) = BAσ (A∗ ). (II) Due to Proposition 2.5(ii) and Corollary 3.1(ii), we need only prove that (iv) ⇒ (i). Suppose that A · A∗∗ ⊆ Zt (A∗∗ , ♦) and T ∈ BA (A∗ ). Let a, b ∈ A. Then T ∗ (ab) = a · T ∗ (b) ∈ ∗ A · A∗∗ ⊆ Zt (A∗∗ , ♦). Hence, T ∗ (A) = T ∗ (A2 ) ⊆ Zt (A∗∗ , ♦), and T ∈ BAr,w (A∗ ) (cf. (2.6)). ∗ Therefore, we have BA (A∗ ) = BAr,w (A∗ ). (III) The proof is similar to that of (I), where Proposition 2.4, Corollary 3.1(i), and (2.6) are replaced by Proposition 2.3, Corollary 3.1(iii), and (2.9), respectively. (IV) The proof is similar to that of (II), where Proposition 2.5(ii), Corollary 3.1(ii), and (2.6) are replaced by Proposition 2.5(i), Corollary 3.1(iv), and (2.9), respectively. (V) This is true by (I), (II) and Corollary 3.1(v). The final assertion holds by Proposition 2.6. 2 Remark 3.3. (a) The equivalence between (ii)–(iv) in (III) has been proved in [20, Corollary 4 and Theorem 24]. (IV) generalizes the equivalence between b) and d) in [34, Theorem 3.6], which shows that if A has a bounded approximate identity (BAI), then A · A∗∗ ⊆ Zt (A∗∗ , ) if and only if Zt (A∗ A∗ ) = A∗ A∗ (cf. Proposition 2.6). Correspondingly, (V) generalizes [34, Corollary 3.7], which is equivalent to saying that for a Banach algebra A with a BAI, we have A · A∗∗ ⊆ A if and only if A∗ A∗ = RM(A). (b) Recall that a Banach algebra A is left strongly Arens irregular (LSAI) if Zt (A∗∗ , ) = A, a concept introduced and studied by Dales and Lau [6]. Right strong Arens irregularity (RSAI) is defined similarly. The algebra A is strongly Arens irregular (SAI) if it is both LSAI and RSAI. In [20], we say that A is left quotient strongly Arens irregular (LQ-SAI) if Zt (A∗ A∗ ) ⊆ RM(A), and left quotient Arens regular (LQ-AR) if Zt (A∗ A∗ ) = A∗ A∗ . Similarly, RQ-SAI and RQ-AR are defined via Zt (AA∗ ∗ ) and LM(A) (the left multiplier algebra of A). The algebra A is called quotient strongly Arens irregular (Q-SAI) if it is both LQ-SAI and RQ-SAI. Proposition 2.6 shows that, in a sense, A being LQ-SAI is opposite to A having a BRAI. In the spirit of this terminology, we say that A is right-left quotient strongly Arens irregular (RLQ-SAI) if Zt (A∗ A∗ )♦ ⊆ RM(A), and right-left quotient Arens regular (RLQ-AR) if Zt (A∗ A∗ )♦ = A∗ A∗ . Analogously, LRQ-SAI and LRQ-AR are defined through LM(A) and

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Zt (AA∗ ∗ ) . Therefore, for Banach algebras A satisfying A2 = A, the statements in Theorem 3.2(I)–(IV) indeed characterize RLQ-SAI, RLQ-AR, LQ-SAI, and LQ-AR, respectively. (c) Let G be a locally compact group. It is known that L1 (G) is Q-SAI and SAI (cf. [29,31]). The situation for the Fourier algebra A(G) is very different. Firstly, since A(G) is commutative, both topological centres of A(G)∗∗ are equal to the algebraic centre Z(A(G)∗∗ ) of A(G)∗∗ (with either Arens product). Secondly, on the one hand, for many amenable groups G, the Fourier algebra A(G) is SAI (cf. [10,17,18,32,33]). On the other hand, as shown by Losert [35,36], both A(F2 ) and A(SU(3)) are non-SAI, though A(F2 ) is Q-SAI (by Theorem 3.2 and [28, Theorem 3.7]) and SU(3) is compact. Recently, Matthias Neufang and his Ph.D student Denis Poulin introduced and studied the strong left and right topological centres of A∗∗ , which are defined by SZt A∗∗ , := m ∈ A∗∗ : λm = T ∗∗ for some T ∈ B(A) and SZt A∗∗ , ♦ r := m ∈ A∗∗ : ρ m = T ∗∗ for some T ∈ B(A) , where λm and ρ m are the maps on A∗∗ given by n −→ m n and n −→ n ♦ m, respectively (cf. [42]). By [21, Theorem 17], we have (3.1) SZt A∗∗ , = Zt A∗∗ , ∩ m ∈ A∗∗ : m · A ⊆ A ; ∗∗ ∗∗ SZt A , ♦ r = Zt A , ♦ ∩ m ∈ A∗∗ : A · m ⊆ A . (3.2) We can define the other two strong topological centres of A∗∗ and the strong topological centres of the two canonical quotient Banach algebras of A∗∗ as follows: (3.3) SZt A∗∗ , := Zt A∗∗ , ∩ m ∈ A∗∗ : A · m ⊆ A ; ∗∗ ∗∗ ∗∗ SZt A , ♦ := Zt A , ♦ ∩ m ∈ A : m · A ⊆ A ; (3.4) ∗ ∗ ∗ ∗ SZt A A := m ∈ A A : A · m ⊆ A and ∗ ∗ (3.5) SZt AA∗ := m ∈ AA∗ : m · A ⊆ A . Due to (2.12), (2.14), and the corresponding AA∗ -versions, we have ∗ ∗ ∗ SZt A∗ A ⊆ Zt A∗ A ∩ Zt A∗ A ♦ and ∗ ∗ ∗ SZt AA∗ ⊆ Zt AA∗ ∩ Zt AA∗ .

(3.6)

The algebra SZt (A∗ A∗ ) in A∗ A∗ plays a role as M(G) does in LUC(G)∗ (cf (6.7)). It is clear that ∗ ∗ A is LQ-SAI ⇐⇒ Zt A∗ A = SZt A∗ A and A is RQL-SAI

⇐⇒

∗ ∗ Zt A∗ A ♦ = SZt A∗ A .


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Therefore, for Banach algebras A satisfying A2 = A, Theorem 3.2 together with its AA∗ version shows in particular the following relations between these twelve topological centres: Zt A∗∗ , = SZt A∗∗ , Zt A∗∗ , = SZt A∗∗ , Zt A∗∗ , ♦ = SZt A∗∗ , ♦ Zt A∗∗ , ♦ = SZt A∗∗ , ♦ r

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

∗ ∗ Zt A∗ A = SZt A∗ A ; ∗ ∗ Zt AA∗ = SZt AA∗ ; ∗ ∗ Zt AA∗ = SZt AA∗ ; ∗ ∗ Zt A∗ A ♦ = SZt A∗ A .

(3.7) (3.8) (3.9) (3.10)

The following result is evident by Propositions 2.2–2.4, and Theorem 3.2(I) and (III) (cf. (2.5), (2.6), and (2.9)). It shows that certain form of automatic normality yields properties between strong Arens irregularity and quotient strong Arens irregularity. More precisely, for a general ∗ Banach algebra A, the equality BAl,w (A∗ ) = BAσ (A∗ ) defines a property between LSAI and LQ∗ SAI, and the equality BAr,w (A∗ ) = BAσ (A∗ ) defines a property between RSAI and RLQ-SAI. The question of when LSAI and LQ-SAI are equivalent was investigated by the authors in [20]. Proposition 3.4. Let A be a Banach algebra. Then, in each of (I) and (II), we have (i) ⇒ (ii) ⇒ (iii), and (ii) ⇐⇒ (iii) if A2 = A. (I)

(i) (ii) (iii) (II) (i) (ii) (iii)

A is left strongly Arens irregular; ∗ BAl,w (A∗ ) = BAσ (A∗ ); A is left quotient strongly Arens irregular. A is right strongly Arens irregular; ∗ BAr,w (A∗ ) = BAσ (A∗ ); A is right-left quotient strongly Arens irregular.

The corollary below follows from Proposition 2.2 and Theorem 3.2, noticing that BA∗∗ (A∗ ) =

∗ BAr,w (A∗ ).

Corollary 3.5. Let A be a Banach algebra with a BRAI. If Zt (A∗∗ , ) = Zt (A∗∗ , ♦), then we have (i) BA∗∗ (A∗ ) = BAσ (A∗ ) ⇐⇒ Zt (A∗ A∗ ) = RM(A); (ii) BA (A∗ ) = BA∗∗ (A∗ ) ⇐⇒ A∗ A∗ = Zt (A∗ A∗ ). Example 3.6. It is known that there exists a unital weakly sequentially complete Banach algebra A such that Zt (A∗∗ , ) = A Zt (A∗∗ , ♦) (cf. [20, page 636]). In this case, due to Proposition 2.7(iv), we have ∗ Zt A∗ A = Zt A∗∗ , = A = RM(A), ∗ but BA∗∗ A∗ = Zt A∗∗ , ♦ = Zt A∗ A ♦ = A = BAσ A∗ . This shows that Corollary 3.5(i) does not hold in general if the topological centres Zt (A∗∗ , ) and Zt (A∗∗ , ♦) are different.

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It is seen that various module map properties over A∗ are closely related to the Banach modules Zt (A∗∗ , ), Zt (A∗∗ , ♦), Zt (A∗ A∗ ), Zt (A∗ A∗ )♦ , and RM(A), and the relationship between Zt (A∗∗ , ) and Zt (A∗∗ , ♦). We shall further explore such connections for the classes of Banach algebras introduced recently by the authors [21]. Let us first recall the following definition. Definition 3.7. (See [21].) Let A be a Banach algebra with a BAI. Then A is said to be of type (RM) if for every μ ∈ RM(A), there is a closed subalgebra B of A with a BAI such that (I) μ|B ∈ RM(B); (II) f |B ∈ BB ∗ for all f ∈ AA∗ ; (III) there is a family {Bi } of closed right ideals in B satisfying (i) each Bi is weakly sequentially complete with a sequential BAI, (ii) for every i, there exists a left Bi -module projection from B onto Bi , and (iii) μ ∈ A if μ|Bi ∈ Bi for all i. Furthermore, the algebra A is said to be of type (RM + ) if the words “left” and “right” can be removed from condition (III). Similarly, Banach algebras of type (LM) (respectively, (LM + )) are defined. The algebra A is said to be of type (M) if it is both of type (LM) and of type (RM), and of type (M + ) if it is both of type (LM + ) and of type (RM + ). Obviously, a unital weakly sequentially complete Banach algebra is of type (M + ). For every locally compact group G, any convolution Beurling algebra L(G, ω) with ω 1 is of type (M + ), and so is the Fourier algebra A(G) if G is amenable. Also, every separable quantum group algebra of a co-amenable locally compact quantum group is of type (M + ). It is known from [21, Theorem 18] that SZt A∗∗ , = A if A is of type (LM) and SZt A∗∗ , ♦ r = A if A is of type (RM).

(3.11)

The reader is referred to [21] for results on Banach algebras from these classes. We have the following generalization of [38, Satz 3.7.7], where the equivalence between (i) and (v) below was shown for the group algebra L1 (G) with G metrizable. Corollary 3.8. Let A be a Banach algebra of type (RM). Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

BA∗∗ (A∗ ) = BAσ (A∗ ); Zt (A∗ A∗ )♦ = RM(A); A · Zt (A∗ A∗ )♦ = A; A · Zt (A∗∗ , ♦) = A; Zt (A∗∗ , ♦) = A.

Proof. The equivalence between (i)–(iv) holds due to Theorem 3.2, and (ii) and (v) are equivalent by (3.10) and (3.11). 2


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Analogous to Corollary 3.8 on right (A∗∗ , ♦)-module maps, we can obtain the following characterizations of automatic normality of right A-module maps. In this situation, a natural subspace of A∗∗ as discussed below plays a similar role as Zt (A∗∗ , ♦) does in Corollary 3.8. Let us assume that A has a BAI. Then A∗∗ has a mixed identity E (that is, E is a right identity of (A∗∗ , ) and a left identity of (A∗∗ , ♦)). Let E(A∗∗ ) be the set of mixed identities of A∗∗ and let Λr A∗∗ = m ∈ A∗∗ : E m = m for all E ∈ E A∗∗ .

(3.12)

The space Λr (A∗∗ ) was denoted as F1 in [34], and as Λ(G) in [33] where A = A(G). Clearly, Λr (A∗∗ ) is a closed right ideal in (A∗∗ , ), and we have A ⊆ Zt A∗∗ , ♦ ⊆ Λr A∗∗ ⊆ A∗∗ .

(3.13)

The equivalence between (iv) and (v) below was shown in [33, Proposition 5.4] for A(G) with G amenable. Corollary 3.9. Let A be a Banach algebra of type (RM + ). Then the following statements are equivalent: (i) BA (A∗ ) = BAσ (A∗ ); (ii) A∗ A∗ = RM(A); (iii) A · A∗ A∗ = A; (iv) A · A∗∗ = A; (v) A · Λr (A∗∗ ) = A; (vi) Λr (A∗∗ ) = A. Proof. The equivalence between (i)–(iv) follows from Theorem 3.2. Note that we have A · A∗∗ ⊆ Λr (A∗∗ ). Thus (vi) ⇒ (iv). It is obvious that (iv) ⇒ (v). Finally, (v) ⇒ (vi) holds by the definition of Λr (A∗∗ ) and [21, Proposition 27]. 2 With the space Λr (A∗∗ ) involved in the discussion, we characterize below Banach algebras A with a BAI such that every bounded right A-module map on A∗ is automatically a right (A∗∗ , ♦)module map. Corollary 3.10. Let A be a Banach algebra with a BAI. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

BA (A∗ ) = BA∗∗ (A∗ ); Zt (A∗ A∗ )♦ = A∗ A∗ ; A · A∗ A∗ ⊆ Zt (A∗ A∗ )♦ ; A · A∗∗ ⊆ Zt (A∗∗ , ♦); A · Λr (A∗∗ ) ⊆ Zt (A∗∗ , ♦).

Proof. Due to Theorem 3.2(II), we only have to prove that (v) ⇒ (iv). To show this, we suppose that A · Λr (A∗∗ ) ⊆ Zt (A∗∗ , ♦). Let m ∈ A∗∗ and a ∈ A. Since A has a BAI, we have a = bc

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for some b, c ∈ A. Then c · m ∈ Λr (A∗∗ ) and a · m = b · (c · m) ∈ A · Λr (A∗∗ ) ⊆ Zt (A∗∗ , ♦). Therefore, we have A · A∗∗ ⊆ Zt (A∗∗ , ♦). 2 Remark 3.11. Note that A · Zt (A∗∗ , ♦) ⊆ Zt (A∗∗ , ♦) ⊆ Λr (A∗∗ ). Hence, we have Λr A∗∗ = Zt A∗∗ , ♦

A · Λr A∗∗ ⊆ Zt A∗∗ , ♦ A · Λr A∗∗ = A · Zt A∗∗ , ♦ .

⇒ ⇐⇒

It is not clear for us when the reverse implication holds. ∗

We consider below under what circumstances the equality BAw (A∗ ) = BAσ (A∗ ) is equivalent to the left strong Arens irregularity of A. Corollary 3.12. Let A be a Banach algebra of type (RM). If Zt (A∗∗ , ) ⊆ Zt (A∗∗ , ♦), then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

∗

BAw (A∗ ) = BAσ (A∗ ); Zt (A∗ A∗ ) = RM(A); A · Zt (A∗ A∗ ) = A; A · Zt (A∗∗ , ) = A; Zt (A∗∗ , ) = A. ∗

∗

Proof. In this case, we have BAw (A∗ ) = BAl,w (A∗ ) (cf. Proposition 2.7(i)). The equivalence between (i)–(iv) then follows from Theorem 3.2, and (ii) and (v) are equivalent by [21, Theorem 25]. 2 Comparing with Corollary 3.8, we are unable to show in Corollary 3.12 that (v) is equivalent to the rest (i)–(iv) without the extra assumption that “Zt (A∗∗ , ) ⊆ Zt (A∗∗ , ♦)”; notice that, unlike the situation in Corollary 3.8(iv), the left topological centre Zt (A∗∗ , ) in Corollary 3.12(iv) is on the “wrong” side. The AA∗ -version of Corollary 3.8 together with Corollary 3.12 shows that for a Banach algebra A of type (M), the LSAI (i.e., Zt (A∗∗ , ) = A) is in fact intrinsically related to the LRQ-SAI (i.e., Zt (AA∗ ∗ ) = LM(A)) rather than the LQ-SAI (i.e., Zt (A∗ A∗ ) = RM(A)). We have the following partial converse to the first half of Proposition 2.2(iii) (see also Example 3.6). ∗

∗

Corollary 3.13. Let A be a Banach algebra of type (RM) that is LSAI. If BAl,w (A∗ ) = BAr,w (A∗ ), then Zt (A∗∗ , ♦) = Zt (A∗∗ , ) = A (i.e., A is SAI). ∗

Proof. Since A has a BAI and Zt (A∗∗ , ) = A, we have BAl,w (A∗ ) = BAσ (A∗ ) by Theorem 3.2. ∗ ∗ Suppose that BAl,w (A∗ ) = BAr,w (A∗ ). Then BA∗∗ (A∗ ) = BAσ (A∗ ), and hence Zt (A∗∗ , ♦) = A by Corollary 3.8. 2 We remark that we are unable to derive the conclusion of Corollary 3.13 when (RM) and LSAI are replaced by (LM) and RSAI, respectively (see the paragraph after Corollary 3.12).


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It is easy to see that if A is Arens regular, then A is (left and right) quotient Arens regular (cf. Remark 3.3(b)). The converse is not true. For instance, if G is an infinite compact group, then L1 (G) is quotient Arens regular but non-Arens regular. As left and right-left quotient Arens regularity properties are characterized in Theorem 3.2, we can also characterize Arens regularity of A in terms of module maps on A∗ (see Proposition 2.7(iv) for the case where A is unital). To see this, we recall that the weak operator topology (wot) on B(A∗ ) is the locally convex topology determined by the seminorms T ∈ B(A∗ ) −→ |T x, n| (x ∈ A∗ , n ∈ A∗∗ ). Let BAl,w A∗ = T ∈ BA A∗ : the map A∗∗ −→ B A∗ , m −→ T ◦ mL is w ∗ -wot continuous

(3.14)

BAr,w A∗ = T ∈ BA A∗ : the map A∗∗ −→ B A∗ , m −→ T ◦ mR is w ∗ -wot continuous .

(3.15)

and

Then we have ∗ BAl,w A∗ = T ∈ BA A∗ : T ∗ A∗∗ ⊆ Zt A∗∗ , ⊆ BAl,w A∗ ⊆ BA A∗ and ∗ BAr,w A∗ = T ∈ BA A∗ : T ∗ A∗∗ ⊆ Zt A∗∗ , ♦ ⊆ BAr,w A∗ ⊆ BA A∗ . However, we may not have BAσ (A∗ ) ⊆ BAl,w (A∗ ) or BAσ (A∗ ) ⊆ BAr,w (A∗ ). In fact, these inclusions hold if and only if A is Arens regular. Comparing with Theorem 3.2(II) and (IV) on quotient Arens regularity, the proposition below on Arens regularity is clear. Proposition 3.14. Let A be a Banach algebra. Then the following statements are equivalent: (i) (ii) (iii) (iv)

A is Arens regular; BA (A∗ ) = BAl,w (A∗ ) (respectively, BA (A∗ ) = BAr,w (A∗ )); BAσ (A∗ ) ⊆ BAl,w (A∗ ) (respectively, BAσ (A∗ ) ⊆ BAr,w (A∗ )); id ∈ BAl,w (A∗ ) (respectively, id ∈ BAr,w (A∗ )).

Therefore, for every C ∗ -algebra A, we have BA (A∗ ) = BAl,w (A∗ ) = BAr,w (A∗ ). Since A is faithful, we have A ∩ AA∗ ⊥ = {0}. It is known that (A∗∗ , ) is right faithful if and only if A∗ = A∗ A, and A∗ A∗ is right faithful if and only if A∗ A = A∗ A2 (cf. ⊥ ∗ ∗ Proposition 2.8(i)). For a subset X of A∗ A, let XA ∗ A∗ denote the annihilator of X in A A . ∗ ∗ ∗ ∗∗ ∗ Then we have RM(A)∩AA∗ A⊥ A∗ A∗ = {0}. We also note that A A A A = A A A, ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ and [A A = A ] ⇒ [A A A A = A A]. Therefore, we can derive from a brief calculation the following proposition on the algebras (A∗∗ , ) and A∗ A∗ . Part of this proposition has been considered in [17] and [23] for commutative Banach algebras and quantum group algebras, respectively.

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Proposition 3.15. Let A be a Banach algebra. Then we have (i) Zt (A∗∗ , ) ∩ AA∗ ⊥ = A∗∗ A∗ ⊥ ; ⊥ ∗ ∗ ∗ (ii) Zt (A∗ A∗ ) ∩ AA∗ A⊥ A∗ A∗ = A A A AA∗ A∗ . Therefore, the following assertions hold. (a) (b) (c) (d) (e)

(A∗∗ , ) is left faithful ⇐⇒ Zt (A∗∗ , ) ∩ AA∗ ⊥ = {0}. A∗ A∗ is left faithful ⇐⇒ Zt (A∗ A∗ ) ∩ AA∗ A⊥ A∗ A∗ = {0}. A∗ A∗ is left faithful if (A∗∗ , ) is left faithful. (A∗∗ , ) is left faithful if A has a BRAI or is LSAI, or A∗ = AA∗ . A∗ A∗ is left faithful if A has a BRAI or is LQ-SAI, or A∗ A = AA∗ A.

Remark 3.16. (i) The converse of (c) is not true, since A(F2 ) is Q-SAI, but A(F2 )∗∗ A(F2 )∗ = A(F2 )∗ as shown by Losert [35]. Furthermore, the Banach algebra B2 constructed in [19, Theorem 5] together with B2 ⊕ L1 (G) of any non-SIN group G shows that none of the converses of (d) and (e) holds. (ii) It follows from Proposition 3.15(i) that AA∗ = A∗∗ A∗ if A is Arens regular. The converses is not true in general, since this equality always holds for every unital Banach algebra A. However, Ülger [44, Theorem 3.3] proved that if A is commutative, semisimple, weakly sequentially complete, and completely continuous with A∗ a von Neumann algebra, then A is Arens regular if and only if AA∗ = A∗∗ A∗ . Therefore, we have A(F2 )A(F2 )∗ A(F2 )∗∗ A(F2 )∗ A(F2 )∗ . (iii) It is easy to see that A∗ A + A∗∗ (AA∗ )⊥ is always a two-sided ideal in A∗∗ . On the other hand, A∗ A + AA∗ ⊥ is a two-sided ideal in A∗∗ in the following two cases: (1) AA∗ ⊆ A∗ A; (2) A∗∗ AA∗ ⊆ AA∗ , which is true if AA∗ = A∗∗ A∗ (e.g., A is Arens regular; see (ii) above). For Banach algebras A satisfying A2 = A, we can show that

A∗ A + AA∗

⊥

is a two-sided ideal in A∗∗

⇐⇒

A∗∗ AA∗ ⊆ A∗ A + AA∗ .

It is still open whether A∗ A + AA∗ ⊥ is a two-sided ideal in A∗∗ even for the case when A has a contractive BAI (see [15, Remark 4.38]). Remark 3.17. For a Banach algebra A, let A = (A∗∗ , ). Then BA (A∗ ) ⊆ BA (A∗ ). For A = L1 (G), when studying whether the involution on A can be extended to an involution on A, Farhadi and Ghahramani [9] raised the following question: Do we have T ∈ BA A∗ if T ∈ BAσ A∗ is surjective?

(3.16)

In [40], Neufang proved that the answer to question (3.16) is negative for all A = L1 (G) with G discrete, abelian, and countably infinite. We observe that the argument given in [40] shows that the answer to (3.16) is negative whenever A is a commutative Banach algebra with a BAI such that the map mL : A∗ −→ A∗ is bounded from below for some m in A∗∗ \ Zt (A∗∗ , ).


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4. Commutation relations, bicommutant theorems, and strong Arens irregularity Let A be a Banach algebra. For a subset X of B(A∗ ), let X c be the commutant of X in B(A∗ ). We should point out that X c is not w ∗ -closed in B(A∗ ) in general. Let A B(A∗ ) (respectively, A∗∗ B(A∗ )) be the Banach algebra of bounded left A-module (respectively, (A∗∗ , )module) maps on A∗ . Note that the embedding A −→ BA (A∗ ) via the canonical embedding A −→ RM(A) is the same as the one obtained via the canonical embedding A −→ A∗ A∗ , a −→ a|A∗ A . It is easy to see that aL (f ) = a · f for all a ∈ A and f ∈ A∗ . Therefore, we have BAσ (A∗ )c ⊆ A B(A∗ ). Recall from (2.7) that for m in A∗∗ or AA∗ ∗ and f in A∗ , mR (f ) ∈ A∗ is given by mR (f ) = f ♦ m. Then mR (f ) ∈ AA∗ if f ∈ AA∗ (that is, AA∗ is right introverted in A∗ ). We note that the canonical representation AA∗ ∗ −→ A B(A∗ ), m −→ mR is an anti-algebra homomorphism. Clearly,

∗ mR : m ∈ A∗∗ = mR : m ∈ AA∗ ⊆ A B A∗ .

Thus we have BA∗∗ (A∗ )c ⊆ BAσ (A∗ )c ⊆ A B(A∗ ), and A B(A∗ )c ⊆ {mR : m ∈ A∗∗ }c = BA∗∗ (A∗ ). Noticing that a · T (f ) = T ∗ (a) f (a ∈ A, f ∈ A∗ , T ∈ BA (A∗ )), we can derive the following proposition. Proposition 4.1. Let A be a Banach algebra satisfying A2 = A. Then we have c c c BAσ A∗ ⊆ A B A∗ ⊆ BA∗∗ A∗ ⊆ A∗∗ B A∗ = A B σ A∗ = BA A∗ . We note that all the results established for BA (A∗ ) have their A B(A∗ )-versions. In particular, if A has a BLAI, then A B(A∗ ) = {mR : m ∈ A∗∗ } (cf. Proposition 2.6). In this case, A B(A∗ )c = BA∗∗ (A∗ ). Therefore, by Theorem 3.2(I), Proposition 4.1, and their A B(A∗ )-versions (see also Remark 3.3(b)), we can obtain the theorem below regarding commutants and bicommutants of module map algebras. Theorem 4.2. Let A be a Banach algebra. Then the following assertions hold. (i) Suppose that A has a BRAI. Then we have BA (A∗ )c = A∗∗ B(A∗ ) = A B σ (A∗ )cc . Moreover, σ ∗ cc σ ∗ A B (A ) = A B (A ) ⇐⇒ A is left-right quotient strongly Arens irregular. (ii) Suppose that A has a BLAI. Then we have A B(A∗ )c = BA∗∗ (A∗ ) = BAσ (A∗ )cc . Moreover, BAσ (A∗ )cc = BAσ (A∗ ) ⇐⇒ A is right-left quotient strongly Arens irregular. In particular, if A is a commutative Banach algebra with a BAI, then A is Q-SAI

⇐⇒

M(A)cc = M(A),

where M(A) is the multiplier algebra of A and M(A) → B(A∗ ). The corollary below is immediate by Theorem 4.2, Corollary 3.8, and the A B(A∗ )-version of Corollary 3.8. It shows that for Banach algebras of type (M), left and right strong Arens irregularities are in fact equivalent to certain commutation relations.

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Corollary 4.3. Let A be a Banach algebra of type (M). Then we have (i) A is LSAI ⇐⇒ BA (A∗ )c = A B σ (A∗ ) ⇐⇒ A B σ (A∗ )cc = A B σ (A∗ ); (ii) A is RSAI ⇐⇒ A B(A∗ )c = BAσ (A∗ ) ⇐⇒ BAσ (A∗ )cc = BAσ (A∗ ). Recall that L1 (G) is of type (M) and is SAI. Therefore, Corollary 4.3(i) improves and extends [12, Theorem 5.1], which shows that BL1 (G) (L∞ (G))c = {λ∗μ : μ ∈ M(G)}, where G is a locally compact group and λμ (f ) = μ ∗ f (f ∈ L1 (G)). For the canonical embedding A ⊆ A B σ (A∗ ), we have BA (A∗ ) = Ac . Similarly, A B(A∗ ) = Ac if A is embedded into BAσ (A∗ ). Therefore, by Theorem 3.2(V), Proposition 4.1, and Corollary 4.3, we obtain the following two corollaries, where LM(A) and RM(A) are identified with their canonical images in B(A∗ ). Corollary 4.4. Let A be a Banach algebra satisfying A2 = A. Then we have (i) Ac = LM(A)c via A → LM(A); (ii) Ac = RM(A)c via A → RM(A); (iii) RM(A) = LM(A)c ⇐⇒ A · A∗∗ ⊆ A. Corollary 4.5. Let A be a Banach algebra of type (M). Then we have (i) A is LSAI ⇐⇒ LM(A)cc = LM(A); (ii) A is RSAI ⇐⇒ RM(A)cc = RM(A). Therefore, for every unital weakly sequentially complete Banach algebra A, we have (a) A is LSAI ⇐⇒ Acc = A, where A ∼ = LM(A); (b) A is RSAI ⇐⇒ Acc = A, where A ∼ = RM(A). We have the following immediate corollary of Theorem 4.2 and Corollary 4.5. For Fourier algebras of amenable locally compact groups, the implication (ii) ⇒ (i) below was shown in [32, Theorem 6.4]. Corollary 4.6. Let A be a commutative Banach algebra of type (M). Then the following statements are equivalent: (i) A is SAI; (ii) A is Q-SAI; (iii) M(A)cc = M(A). Let M be a von Neumann algebra standardly represented on a Hilbert space H with M∗ a Banach algebra of type (M) (e.g., M∗ is a unital Banach algebra or a separable Banach algebra with a BAI). It is interesting to compare certain commutation theorems over CB(B(H )), B(M), B(H ), and B(M∗ ). Firstly, on the one hand, we have in CB(B(H )) that c σ B(H ) = CBM B(H ) CBM

c σ and CBM B(H ) = CBM B(H ) ,


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where CBM (B(H )) is the algebra of completely bounded M-bimodule maps on B(H ), and M is the commutant of M in B(H ) (cf. (2.7) and (2.8) in [41] and references therein). On the other hand, by Proposition 4.1 and Corollary 4.3, we have M∗ B

σ

(M)c = BM∗ (M),

σ and M∗ B(M)c = BM (M) ∗

⇐⇒

M∗ is RSAI.

Secondly, the von Neumann double commutation theorem says that M = M. However, when M∗ is unital, for the canonical representation M∗ −→ BM∗ (M), the double commutation relation (M∗ )cc = M∗ holds precisely when M∗ is RSAI, which is not true, for example, if M = VN(SU(3)) (cf. [36]). Finally, in contrast to the bicommutant theorem M = M in B(H ), we have [Mcc = M in B(M∗ )] ⇐⇒ [dim(M) < ∞] due to Theorem 4.2(ii) and the fact that M is Arens regular. We end this section with the following remark on measure algebras of locally compact groups. Remark 4.7. For a locally compact group G, by Proposition 2.7(iv), we have w∗ BM(G) M(G)∗ = Zt M(G)∗∗ ,

and BM(G)∗∗ M(G)∗ = Zt M(G)∗∗ , ♦ .

It is known from [39, Theorem 3.5] that M(G) is SAI if either G is non-compact with |G| a nonmeasurable cardinal or 2χ(G) κ(G), where κ(G) and χ(G) denote, respectively, the compact covering number and the local weight of G. Recently, Losert, Neufang, Pachl and Stepr¯ans [37] showed that M(G) is SAI for all locally compact groups G. Therefore, by Corollary 4.5, we always have w∗ M(G)∗ = M(G) = M(G)cc , BM(G)∗∗ M(G)∗ = BM(G) σ (M(G)∗ ) and the commutant is taken in B(M(G)∗ ). Note that M(G)cc = where M(G) ∼ = BM(G) M(G) also holds in B(L∞ (G)) by Corollary 4.5. The reader is referred to [7] for other recent results on M(G).

5. A new product for module maps and some applications Let A be a Banach algebra. In [20], we introduced the closed subspace A∗ A∗R of A∗ A∗ given by

∗ ∗ A∗ A R = m ∈ A∗ A : A∗ A ♦ m ⊆ A∗ A .

(5.1)

For m ∈ A∗ A∗R and n ∈ A∗ A∗ , the functional m ♦ n ∈ A∗ A∗ is naturally defined by f, m ♦ n = f ♦ m, n

f ∈ A∗ A .

Then (A∗ A∗R , ♦) is a Banach algebra, and we showed [20, Theorem 2(i)] that ∗ ∗ ∗ Zt A∗ A = m ∈ A∗ A R : m n = m ♦ n for all n ∈ A∗ A .

(5.2)

On the other hand, (A∗ A∗R , ♦) is also a left topological semigroup under the σ (A∗ A∗R , A∗ A)-topology. By definition, the topological centre Zt (A∗ A∗R ) of (A∗ A∗R , ♦) is the set

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of all m ∈ A∗ A∗R such that the map n −→ n ♦ m is σ (A∗ A∗R , A∗ A)-continuous. Since A is w ∗ -dense in A∗ A∗ , we have ∗ ∗ ∗ Zt A∗ A R = m ∈ A∗ A R : n ♦ m = n m for all n ∈ A∗ A R .

(5.3)

We note that, in general, the topological centre Zt (A∗ A∗R ) of A∗ A∗R and the auxiliary topological centre Zt (A∗ A∗ )♦ of A∗ A∗ are not related. For example, if A is the group algebra L1 (G) of a locally compact group G, then Zt (A∗ A∗ )♦ = Zt (A∗ A∗ ) = M(G); however, δeG ∈ / Zt (A∗ A∗R ) unless G is an SIN group (cf. [20, Theorem 19]). We consider below the new Banach algebra structure on BA (A∗ ) associated with (A∗ A∗R ,♦). Note that, for each T ∈ BA (A∗ ), we have T (A∗ A) ⊆ A∗ A. Let BA A∗ R = T ∈ BA A∗ : A∗ A ♦ T ∗ (A) ⊆ A∗ A .

(5.4)

Then BA (A∗ )R is a norm closed subspace of BA (A∗ ). Notice that we have A∗ ♦ Zt (A∗∗ , ) ⊆ A∗ A (cf. [6, Proposition 2.20]). Therefore, due to (2.11) and (2.12), we have ∗ ∗ BAl,w A∗ ⊆ T ∈ BA A∗ : T ∗ (A)|A∗ A∗ ⊆ Zt A∗ A ⊆ BA A∗ R ⊆ BA A∗ .

(5.5)

Proposition 5.1. Let A be a Banach algebra and let Φ : A∗ A∗ −→ BA (A∗ ) be the canonical representation of A∗ A∗ . Then the following assertions hold. (i) Φ(A∗ A∗R ) ⊆ BA (A∗ )R , and the equality holds if A has a BRAI. (ii) If A∗ A is two-sided introverted in A∗ , then BA (A∗ ) = BA (A∗ )R ; the converse holds if A2 = A. Proof. (i) This is true, since (A∗ A) ♦ (mL )∗ (A) = (A∗ A2 ) ♦ m. (ii) Suppose that A∗ A is two-sided introverted in A∗ . Then we have A∗ A ♦ A∗∗ ⊆ A∗ A. Therefore, BA (A∗ ) = BA (A∗ )R . Conversely, suppose that BA (A∗ ) = BA (A∗ )R and A2 = A. If m ∈ A∗ A∗ , then mL ∈ BA (A∗ )R , and thus (A∗ A2 ) ♦ m = (A∗ A) ♦ (mL )∗ (A) ⊆ A∗ A. Therefore, A∗ A ♦ m ⊆ A∗ A for all m ∈ A∗ A∗ , since A2 = A; that is, A∗ A is right and hence two-sided introverted in A∗ . 2 In the following, we assume that A has a contractive BRAI. Then (A∗∗ , ) has a right identity E of norm 1, and (A∗ A∗ , ) ∼ = (BA (A∗ ), ◦) via the isometric and w ∗ -w ∗ homeomorphic algebra isomorphism Φ (cf. Proposition 2.6). For T ∈ BA (A∗ )R and S ∈ BA (A∗ ), we define

(T ♦ S)(f ), a = f, T ∗ (a) ♦ S ∗ (E) = f ♦ T ∗ (a), S ∗ (E)

f ∈ A∗ , a ∈ A .

(5.6)

It is easily seen that T ♦ S ∈ BA (A∗ ) with T ♦ S T S, and T ♦ S is independent of the choice of E, since A∗ ♦ T ∗ (A) = A∗ ♦ T ∗ A2 ⊆ A∗ A ♦ T ∗ (A) ⊆ A∗ A .


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Let T1 , T2 ∈ BA (A∗ )R , f ∈ A∗ , and a ∈ A. Since f ♦ T1∗ (a) ∈ A∗ A = A∗ A, we have f ♦ T1∗ (a) = g · b for some g ∈ A∗ and b ∈ A. Then we obtain f ♦ (T1 ♦ T2 )∗ (a) = f ♦ T1∗ (a) ♦ T2∗ (E) = (g · b) ♦ T2∗ (E) = g ♦ T2∗ (b) ∈ A∗ A . It follows that T1 ♦ T2 ∈ BA (A∗ )R . Also, we have (T1 ♦ T2 ) ♦ S = T1 ♦ (T2 ♦ S) (S ∈ BA (A∗ )). Therefore, (BA (A∗ )R , ♦) is an associative Banach algebra, and (A∗ A∗R , ♦) ∼ = (BA (A∗ )R , ♦) via the isometric algebra isomorphism Φ|A∗ A∗R (cf. Proposition 5.1(i)). Since (BA (A∗ )R , ♦) is a left topological semigroup under the relative w ∗ot-topology, its topological centre Zt (BA (A∗ )R ) is defined. In fact, we have Zt (BA (A∗ )R ) = Φ(Zt (A∗ A∗R )). Therefore, Zt (BA (A∗ )R ) is a norm closed subalgebra of (BA (A∗ ), ◦) and of (BA (A∗ )R , ♦), and Zt BA A∗ R = T ∈ BA A∗ R : S ♦ T = S ◦ T for all S ∈ BA A∗ R .

(5.7)

Summarizing the above, we have the following theorem on the Banach algebra (BA (A∗ )R , ♦). Theorem 5.2. Let A be a Banach algebra with a contractive BRAI. Then the canonical representation Φ : A∗ A∗ −→ BA (A∗ ) induces the isometric algebra isomorphisms ∗ ∗ ∗ ∗ BA A R , ♦ ∼ = Zt A∗ A R . = A A R , ♦ and Zt BA A∗ R ∼ Clearly, id is the identity of (BA (A∗ ), ◦), id ∈ BA (A∗ )R , and id ♦ S = S for all S ∈ BA (A∗ ). It is also easy to show that S ♦ id is the w ∗ -limit of (S ◦ (eα )L ) in BA (A∗ ) if S ∈ BA (A∗ )R , where (eα ) is a contractive BRAI of A. Corollary 5.3. Let A be a Banach algebra with a contractive BRAI. Then the following statements are equivalent: (i) (ii) (iii) (iv)

BAσ (A∗ ) ⊆ Zt (BA (A∗ )R ); id ∈ Zt (BA (A∗ )R ); id is the identity of (BA (A∗ )R , ♦); A∗ A = AA∗ A.

Proof. Obviously, we have (i) ⇒ (ii) ⇒ (iii). The equivalence (iii) ⇐⇒ (iv) holds by Theorem 5.2 and [20, Theorem 10]. The implication (iii) ⇒ (i) follows from Theorem 5.2 and [20, Lemma 9(ii)], which shows that RM(A) ⊆ Zt (A∗ A∗R ) if (A∗ A∗R , ♦) is unital. 2 Let WAP(A) be the space of weakly almost periodic functionals on A. That is, WAP(A) is the subspace of A∗ consisting of all f ∈ A∗ such that the map A −→ A∗ , a −→ f · a is weakly compact. Notice that the composition of the canonical embedding A −→ A∗∗ and the adjoint of the above map is the map A −→ A∗ , a −→ a · f . Therefore, if f ∈ A∗ , then f ∈ WAP(A) if and only if A −→ A∗ , a −→ a · f is weakly compact. It is easy to show that WAP(A) ⊆ A∗ A if A has a BLAI, and WAP(A) ⊆ AA∗ if A has a BRAI (cf. the proof of [6, Proposition 3.12]). The following theorem links the module map property described in Theorem 3.2(IV) to the spaces WAP(A) and A∗ A∗R . This in particular generalizes [34, Theorem 3.6], where the equivalence between (i)–(iii) below was shown for A with a BAI.

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Theorem 5.4. Let A be a Banach algebra satisfying A2 = A. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

A∗ A ⊆ WAP(A); A∗ A∗ = Zt (A∗ A∗ ); A · A∗∗ ⊆ Zt (A∗∗ , ); A · A∗ A∗ ⊆ Zt (A∗ A∗ ); ∗ BA (A∗ ) = BAl,w (A∗ ).

In addition, if A is an involutive Banach algebra such that A∗ A∗ is left faithful (e.g., A has a BRAI or is LQ-SAI, or A∗ A = AA∗ A), then (i)–(v) are equivalent to (vi) A∗ A∗R = Zt (A∗ A∗ ). Furthermore, if A is an involutive Banach algebra with a contractive BRAI, then the above ∗ (i)–(vi) are also equivalent to BA (A∗ )R = BAw (A∗ ). Proof. It is known that A∗ A ⊆ WAP(A) if and only if A∗ A is two-sided introverted in A∗ and x, m n = x, m ♦ n for all x ∈ A∗ A and m, n ∈ A∗∗ (cf. [6, Propositions 3.11 and 5.7]). Due to (5.2), we have (i) ⇐⇒ (ii). The equivalence between (ii)–(v) has been proved in Theorem 3.2(IV). Obviously, (ii) ⇒ (vi) holds. Suppose that A is an involutive Banach algebra such that A∗ A∗ is left faithful, which is the case, by Proposition 3.15(e), if A has a BRAI or is LQ-SAI, or A∗ A = AA∗ A. By Proposition 3.15(b), we have Zt (A∗ A∗ ) ∩ AA∗ A⊥ A∗ A = {0}. To show ∗ ∗ ∗ ∗ ∗ ∗ (vi) ⇒ (ii), we suppose that A AR = Zt (A A ). It is evident that AA∗ A⊥ A∗ A ⊆ A AR , ⊥ ∗ ∗ ∗ and thus AA AA∗ A = {0}, or equivalently, A A = AA A. According to [20, Proposition 11(v)], the space A∗ A is two-sided introverted in A∗ . It follows that A∗ A∗ = A∗ A∗R = Zt (A∗ A∗ ). Therefore, we have (vi) ⇒ (ii). The final assertion follows from Theorem 5.2 and Proposition 2.7. 2 6. Quotient strong Arens irregularity over locally compact quantum groups Let us start this section with a brief recalling of some notation related to locally compact quantum groups. The reader is referred to [24,25] by Kustermans and Vaes for more information. See also [20,21,23]. Let G = (L∞ (G), Γ, ϕ, ψ) be a von Neumann algebraic locally compact quantum group. Then the pre-adjoint of the co-multiplication Γ induces on L1 (G) = L∞ (G)∗ an associative multiplication such that L1 (G) is a faithful completely contractive involutive Banach algebra satisfying L1 (G) L1 (G) = L1 (G). In the case where L∞ (G) is L∞ (G) or VN(G) for a locally compact group G, the algebra L1 (G) is the usual convolution group algebra L1 (G), respectively, the Fourier algebra A(G). Let C0 (G) be the reduced C ∗ -algebra of G, let M(C0 (G)) be the multiplier algebra of C0 (G), and let LUC(G) be the subspace L1 (G)∗ L1 (G) of L∞ (G). It is known from [43] that C0 (G) ⊆ LUC(G) ⊆ M C0 (G) ⊆ L∞ (G).

(6.1)

The space M(G) = C0 (G)∗ is also a faithful completely contractive involutive Banach algebra with a multiplication induced by Γ , and L1 (G) can be identified with a closed two-sided ideal in


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M(G) via the restriction map f −→ f |C0 (G) . When L∞ (G) = L∞ (G) (respectively, VN(G)), M(G) is the measure algebra M(G) of G (respectively, the reduced Fourier–Stieltjes algebra Bλ (G) of G). Now we let X be a left introverted subspace of L∞ (G) such that C0 (G) ⊆ X ⊆ M(C0 (G)). ∗∗ Let M(C 0 (G)) be the idealizer of C0 (G) in C0 (G) . That is, M C0 (G) = x ∈ C0 (G)∗∗ : a x, x a ∈ C0 (G) for all a ∈ C0 (G) ,

(6.2)

where a is the canonical image of a in C0 (G)∗∗ . It is well known that we have a C ∗ -algebra ∼ isomorphism M(C0 (G)) ∼ = M(C 0 (G)), extending the canonical identification C0 (G) = C 0 (G). This yields a complete isometry C0 (G) ⊆ C0 (G)∗∗ τ : X ⊆ M C0 (G) ∼ = M

(6.3)

= τ (X) ⊆ M(G)∗ . Then M(G) is satisfying τ (C0 (G)) = C 0 (G). Correspondingly, we let X ∗ It was shown via μ −→ μ|X isometrically identified with a subspace of X since C 0 (G) ⊆ X. in [22, Proposition 2.1] that there exists a completely isometric algebra homomorphism π : M(G) −→ X ∗

(6.4)

which is an L1 (G)-module and C0 (G)-module map such that π ∗ |X = τ . Clearly, X is always a left M(G)-submodule of L∞ (G). Therefore, τ : X −→ M(G)∗ is an L1 (G)-module map, and is an M(G)-module map if X is also a right M(G)-submodule of L∞ (G) (cf. [22, Corollary 2.4]). Proposition 6.1. Let G be a locally compact quantum group and let X be a left introverted subspace of L∞ (G) such that C0 (G) ⊆ X ⊆ M(C0 (G)). Then the following assertions hold. is a completely isometric L1 (G) is an L1 (G)-submodule of M(G)∗ and τ : X −→ X (i) X module isomorphism. is an M(G)-submodule of M(G)∗ and (ii) If X is a right M(G)-submodule of L∞ (G), then X τ : X −→ X is an M(G)-module map. is left introverted in M(G)∗ and τ ∗ : X ∗ −→ X ∗ is a completely (iii) If X = X L1 (G), then X isometric algebra isomorphism and an M(G)-module map. In this case, we have ∗ = M(G) ⊕ C X 0 (G)

⊥

∗ , and τ ∗ M(G) = π M(G) ⊆ Zt X ∗ = τ ∗ Zt X

⊥

∗ : m m∈X |C = 0}. where C 0 (G) = { (G) 0

Proof. (i) and (ii). This is clear by the above discussions. (iii) Suppose that X = X L1 (G). Then X is an M(G)-submodule of L∞ (G) and hence, is an M(G)-submodule of M(G)∗ and both τ and τ ∗ are completely isoby (i) and (ii), X ∗ , with x ∈ X, and μ ∈ M(G). Then metric M(G)-module maps. Let m ∈X x = τ (x) ∈ X ∗ ∗ m) ∈ X , and hence m x ∈ X. By [22, Proposition 2.1(i)] and noticing that X = m = τ ( X L1 (G), we have π(μ) f, x = f, x μ for all f ∈ L1 (G). Since L1 (G) is w ∗ -dense

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in X ∗ , there exists a net (fi ) in L1 (G) such that fi −→ m in the weak∗ -topology on X ∗ . We obtain that π(μ) fi −→ π(μ) m in the weak∗ -topology on X ∗ , since π(μ) ∈ Zt (X ∗ ) (cf. [22, Proposition 2.1(iii)]). Then we have

π(μ) m, x = lim π(μ) fi , x = limfi , x μ = m, x μ. i

i

On the other hand, π(μ) m, x = π(μ), m x = τ (m x), μ. Therefore, we have m x , μ = m , τ (x) μ = m , τ (x μ) = m, x μ = π(μ) m, x = τ (m x), μ . That is, Thus m x = τ (m x) ∈ τ (X) = X. m ∗ , x ∈ X . m τ (x) = τ τ ∗ ( m) x ∈ X ∈X

(6.5)

is left introverted in M(G)∗ and τ ∗ : X ∗ −→ X ∗ is an algebra isoIt follows from (6.5) that X morphism. It is obvious that the final assertion holds, noticing that π(M(G)) ⊆ Zt (X ∗ ) (cf. [22, Proposition 2.1]). 2 By definition, the space LUC(G) is a left introverted M(G)-submodule of L∞ (G) satisfying is left introverted in M(G)∗ . As we shall LUC(G) = LUC(G) L1 (G). Therefore, LUC(G) can be obtained naturally as a space of certain left uniformly continuous show, the space LUC(G) functionals on M(G). The equality given in (6.6) below is a quantum group version of [11, Lemma 2.18] on the classical space LUC(G) of left uniformly continuous functions on a locally compact group G. Theorem 6.2. Let G be a locally compact quantum group and let τ be the map given in (6.3) for X = LUC(G). Then τ : LUC(G) −→ C0 (G)∗∗ is a completely isometric M(G)-module map and we have = M(G)∗ L1 (G) . LUC(G)

(6.6)

Therefore, M(G)∗ L1 (G) is a unital C ∗ -subalgebra of C0 (G)∗∗ if G is semi-regular. Let y ∈ M(G)∗ , f , g ∈ L1 (G), Proof. We only need show that M(G)∗ L1 (G) ⊆ LUC(G). and μ ∈ M(G). Let x = y|L1 (G) ∈ L∞ (G). Then we have

y (f g), μ = y, f (g μ) = x, f (g μ) = x f, g μ.

Since x f ∈ LUC(G) and g μ ∈ L1 (G), by [22, Proposition 2.1], we have x f, g μ = x f, π(g μ) = τ (x f ), g μ = τ (x f ) g, μ = τ x (f g) , μ . It follows that y (f g) = τ (x (f g)) ∈ τ (LUC(G)). Therefore, we have M(G)∗ L1 (G) ⊆ since L1 (G) L1 (G) = L1 (G). The final assertion holds by [23, Theτ (LUC(G)) = LUC(G), orem 5.6]. 2


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Remark 6.3. Note that the adjoint of the inclusion map L1 (G) −→ M(G) is the normal and surjective ∗-homomorphism C0 (G)∗∗ −→ L∞ (G), x −→ x|L1 (G) , which extends the inclusion C0 (G) ⊆ L∞ (G). The kernel of this ∗-homomorphism is the w ∗ -closed ideal L1 (G)⊥ in C0 (G)∗∗ . Then there exists a central projection p in C0 (G)∗∗ such that L1 (G)⊥ = (1 − p)C0 (G)∗∗ , and thus we have C0 (G)∗∗ = pC0 (G)∗∗ ⊕∞ L1 (G)⊥ ∼ = L∞ (G) ⊕∞ L1 (G)⊥

via x ⊕ y −→ x|L1 (G) ⊕ y.

Let κ : L∞ (G) −→ C0 (G)∗∗ be the induced normal and injective ∗-homomorphism. By [22, Proposition 3.6], L1 (G) = M(G) if and only if κ(C0 (G)) = C 0 (G) (respectively, (G))). It follows that the following statements are equivalent: κ(M(C0 (G))) = M(C 0 (i) L1 (G) = M(G); (ii) κ(LUC(G)) = LUC(G); (iii) κ(L∞ (G)) = C0 (G)∗∗ . Therefore, we do not have κ(LUC(G)) = M(G)∗ L1 (G) in general. It is known from [23, Proposition 6.2] that L1 (G) is Q-SAI if and only if the embedding π : M(G) −→ LUC(G)∗ maps M(G) onto Zt (LUC(G)∗ ). Note that, by the definition of the strong topological centre SZt (LUC(G)∗ ) of LUC(G)∗ (cf. (3.5)) and the decomposition LUC(G)∗ = π(M(G)) ⊕ C0 (G)⊥ , we have SZt LUC(G)∗ = π M(G) .

(6.7)

Let • denote the canonical C0 (G)-module actions on M(G) and L∞ (G). Since M(G) = M(G) • C0 (G) = C0 (G) • M(G) and π : M(G) −→ LUC(G)∗ is a C0 (G)-module map, we obtain that C0 (G) • LUC(G)∗ • C0 (G) = LUC(G)∗ • C0 (G) = C0 (G) • LUC(G)∗ = π M(G) ⊆ Zt LUC(G)∗ .

(6.8)

Therefore, we can characterize the quotient strong Arens irregularity of L1 (G) in terms of the strong topological centre SZt (LUC(G)∗ ) of LUC(G)∗ and the canonical C0 (G)-module structure on LUC(G)∗ . Proposition 6.4. For any locally compact quantum group G, the following statements are equivalent: (i) L1 (G) is Q-SAI; (ii) Zt (LUC(G)∗ ) = SZt (LUC(G)∗ ); (iii) Zt (LUC(G)∗ ) = C0 (G) • LUC(G)∗ • C0 (G). For a general Banach algebra A, it is known from (2.12) that A is LSAI ⇒ A is LQ-SAI. Similarly, we have A is RSAI ⇒ A is RQ-SAI. Therefore, for every locally compact quantum group G, we have

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M(G) is SAI

⇒

M(G) is Q-SAI,

and L1 (G) is SAI

⇒

L1 (G) is Q-SAI.

Note that in general an SAI Banach algebra A (e.g., A = L1 (T)) can even have an infinite dimensional closed ideal which is Arens regular (cf. [19] and references therein). For the quantum measure algebra M(G) and its closed ideal L1 (G), however, we have the following theorem. Theorem 6.5. Let G be a locally compact quantum group. Then we have M(G) is Q-SAI

⇒

L1 (G) is Q-SAI. ∗

) ⊆ M(G) due to Proof. Suppose that M(G) is Q-SAI. We only need prove that Zt (LUC(G) ∗ ). We show below that m Proposition 6.1(iii). Let m ∈ Zt (LUC(G) ∈ M(G). is a well-defined element of M(G)∗∗ if we let Let f ∈ L1 (G). By Theorem 6.2, f m , y = m, y f (y ∈ M(G)∗ ). For all p ∈ M(G)∗∗ and y ∈ M(G)∗ , we have f m p, y = f m , p y = m , (p y) f = m , p (y f ) f m = m (p|LUC(G) ), y f . Thus we obtain that f m ∈ Zt (M(G)∗∗ , ). Let q = (f m )|M(G)∗ M(G) . Then q ∈ ∗ ∗ Zt (M(G) M(G) ) (cf. Proposition 2.2(i)). Let μ ∈ M(G). By the assumption, we have we have x ∈ LUC(G), μq = μ q ∈ M(G). In particular, for all x = μ q, x = q, x μ = f m , x μ = m , x (μ f ) = (μ f ) m , x. μq , ∗

. On the other hand, by ProposiThat is, we also have (μ f ) m = μq ∈ M(G) in LUC(G) ⊥

tion 6.1(iii), m = ν + n for some ν ∈ M(G) and n∈C 0 (G) . Then we have (μ f ) m = (μ f ) ν + (μ f ) n = μq ∈ M(G). ⊥

⊥

Notice that L1 (G) is an ideal in M(G) and L1 (G) C 0 (G) ⊆ C 0 (G) . Therefore, (μ f ) = n = 0 for all f ∈ L1 (G) and μ ∈ M(G). It follows that we have n = 0, since LUC(G) LUC(G) (M(G) L1 (G)). Consequently, we have m = ν ∈ M(G). 2 The corollary below is immediate by Theorem 6.5, [20, Corollary 32], and Corollary 4.5. Corollary 6.6. Let G be a locally compact quantum group such that L1 (G) is of type (M) with a central BAI. Then we have M(G) is SAI

⇒

L1 (G) is SAI.

Equivalently, we have [M(G)cc = M(G) in B(M(G)∗ )] ⇒ [M(G)cc = M(G) in B(L1 (G)∗ )]. Therefore, for every amenable locally compact group G, we have [B(G) is SAI] ⇒ [A(G) is SAI], or equivalently, [B(G)cc = B(G) in B(W ∗ (G))] ⇒ [B(G)cc = B(G) in B(VN(G))].


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References [1] J. Baker, A.T.-M. Lau, J. Pym, Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras, J. Funct. Anal. 158 (1998) 186–208. [2] J.F. Berglund, H.D. Junghenn, P. Milnes, Analysis on Semigroups. Function Spaces, Compactifications, Representations, Canad. Math. Soc. Ser. Monogr. Adv. Texts, A Wiley–Interscience Publication, John Wiley & Sons, Inc., New York, 1989. [3] J. Cigler, V. Losert, P. Michor, Banach Modules and Functors on Categories of Banach Spaces, Lect. Notes Pure Appl. Math., vol. 46, Marcel Dekker, Inc., New York, 1979. [4] P. Civin, B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961) 847– 870. [5] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. New Ser., vol. 24, Oxford University Press, New York, 2000. [6] H.G. Dales, A.T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (836) (2005). [7] H.G. Dales, A.T.-M. Lau, D. Strauss, Second duals of measure algebras, preprint, 2009. [8] E.G. Effros, Z.-J. Ruan, On non-self-adjoint operator algebras, Proc. Amer. Math. Soc. 110 (1990) 915–922. [9] H. Farhadi, F. Ghahramani, Involutions on the second duals of group algebras and a multiplier problem, Proc. Edinb. Math. Soc. (2) 50 (2007) 153–161. [10] M. Filali, M. Neufang, M. Sangani Monfared, Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications, Math. Ann., in press. [11] F. Ghahramani, A.T.-M. Lau, Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal. 132 (1995) 170–191. [12] F. Ghahramani, A.T.-M. Lau, Multipliers and modulus on Banach algebras related to locally compact groups, J. Funct. Anal. 150 (1997) 478–497. [13] F. Ghahramani, J.P. McClure, Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull. 35 (1992) 180–185. [14] E.E. Granirer, Day points for quotients of the Fourier algebra A(G), extreme nonergodicity of their duals and extreme non-Arens regularity, Illinois J. Math. 40 (1996) 402–419. [15] M. Grosser, Bidualräume und Vervollständigungen von Banachmoduln, Lecture Notes in Math., vol. 717, Springer, Berlin, 1979. [16] M. Grosser, V. Losert, The norm-strict bidual of a Banach algebra and the dual of Cu (G), Manuscripta Math. 45 (1984) 127–146. [17] Z. Hu, Open subgroups and centre problem for the Fourier algebra, Proc. Amer. Math. Soc. 134 (2006) 3085–3095. [18] Z. Hu, M. Neufang, Decomposability of von Neumann algebras and the Mazur property of higher level, Canad. J. Math. 59 (2006) 768–795. [19] Z. Hu, M. Neufang, Distinguishing properties of Arens irregularity, Proc. Amer. Math. Soc. 137 (2009) 1753–1761. [20] Z. Hu, M. Neufang, Z.-J. Ruan, On topological centre problems and SIN quantum groups, J. Funct. Anal. 257 (2009) 610–640. [21] Z. Hu, M. Neufang, Z.-J. Ruan, Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres, Proc. Lond. Math. Soc. 100 (2010) 429–458. [22] Z. Hu, M. Neufang, Z.-J. Ruan, Module maps over locally compact quantum groups and regularity properties, preprint, 2010. [23] Z. Hu, M. Neufang, Z.-J. Ruan, Completely bounded multipliers over locally compact quantum groups, Proc. Lond. Math. Soc., doi:10.1112/plms/pdq041. [24] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. Éole Norm. Sup. 33 (2000) 837–934. [25] J. Kustermans, S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003) 68–92. [26] A.T.-M. Lau, Operators which commute with convolutions on subspaces of L∞ (G), Colloq. Math. 39 (1978) 351– 359. [27] A.T.-M. Lau, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 251 (1979) 39–59. [28] A.T.-M. Lau, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 267 (1981) 53–63. [29] A.T.-M. Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986) 273–283. [30] A.T.-M. Lau, Uniformly continuous functionals on Banach algebras, Colloq. Math. 51 (1987) 195–205.

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[31] A.T.-M. Lau, V. Losert, On the second conjugate algebra of L1 (G) of a locally compact group, J. Lond. Math. Soc. (2) 37 (1988) 464–470. [32] A.T.-M. Lau, V. Losert, The C ∗ -algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993) 1–30. [33] A.T.-M. Lau, V. Losert, The centre of the second conjugate algebra of the Fourier algebra for infinite products of groups, Math. Proc. Cambridge Philos. Soc. 138 (2005) 27–39. [34] A.T.-M. Lau, A. Ülger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996) 1191– 1212. [35] V. Losert, The centre of the bidual of Fourier algebras (discrete groups), preprint, 2002. [36] V. Losert, On the centre of the bidual of Fourier algebras (the compact case), presentation at the 2004 Istanbul International Conference on Abstract Harmonic Analysis, 2004. [37] V. Losert, M. Neufang, J. Pachl, J. Stepr¯ans, Proof of the Ghahramani–Lau conjecture, preprint, 2010. [38] M. Neufang, Abstrakte Harmonische Analyse und Modulhomomorphismen über von Neumann-Algebren, PhD thesis at University of Saarland, Saarbrücken, Germany, 2000. [39] M. Neufang, On a conjecture by Ghahramani–Lau and related problems concerning topological centres, J. Funct. Anal. 224 (2005) 217–229. [40] M. Neufang, Solution to Farhadi–Ghahramani’s multiplier problem, Proc. Amer. Math. Soc. 138 (2010) 553–555. ˆ ∗ , Trans. Amer. [41] M. Neufang, Z.-J. Ruan, N. Spronk, Completely isometric representations of Mcb A(G) and UCB(G) Math. Soc. 360 (2008) 1133–1161. [42] D. Poulin, The strong topological centre and factorization properties, PhD thesis, Carleton University, in preparation. [43] V. Runde, Uniform continuity over locally compact quantum groups, J. Lond. Math. Soc. 80 (2009) 55–71. [44] A. Ülger, Central elements of A∗∗ for certain Banach algebras A without bounded approximate identities, Glasg. Math. J. 41 (1999) 369–377.


Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras Kunyu Guo a , Hansong Huang b,∗ a School of Mathematical Sciences, Fudan University, Shanghai 200433, China b Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 2 April 2010; accepted 3 November 2010

Communicated by S. Vaes

Abstract In this paper, we combine methods of complex analysis, operator theory and conformal geometry to construct a class of Type II factors in the theory of von Neumann algebras, which arise essentially from holomorphic coverings of bounded planar domains. One will see how types of such von Neumann algebras are related to algebraic topology of planar domains. As a result, the paper establishes a fascinating connections to one of the long-standing problems in free group factors. An interplay of analytical, geometrical, operator and group theoretical techniques is intrinsic to the paper. © 2010 Elsevier Inc. All rights reserved. Keywords: Holomorphic covering map; The fundamental group; Von Neumann algebra; Type II factor

1. Introduction Let D be the open unit disk in the complex plane C, and dA be the normalized area measure on D. Denote by L2a (D) the Bergman space consisting of all holomorphic functions over D, which are square integrable with respect to dA. For any bounded holomorphic function φ over D, let Mφ be the multiplication operator defined on L2a (D) by the symbol φ. In this paper, we set W ∗ (φ) * Corresponding author.

E-mail addresses: [email protected] (K. Guo), [email protected] (H. Huang). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.11.002

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K. Guo, H. Huang / Journal of Functional Analysis 260 (2011) 1219–1255

as the von Neumann algebra generated by Mφ and denote by V ∗ (φ) W ∗ (φ) the commutant algebra of W ∗ (φ). We emphasize that both W ∗ (φ) and V ∗ (φ) are von Neumann algebras. It is interesting and important to study von Neumann algebras W ∗ (φ) and V ∗ (φ). One of motivations is to understand connections of complex analysis and von Neumann algebras; secondly, when φ is a holomorphic covering map from D onto a bounded planar domain Ω, we will see how both W ∗ (φ) and V ∗ (φ) are related to geometric properties of domains, and their fascinating connections to one of the long-standing problems in free group factors ∗

L(Fn ) ∼ = L(Fm )? for n = m and n, m 2; here Fn denotes the free group on n generators, and L(Fn ) is the von Neumann algebra generated by left regular representation of Fn on l 2 (Fn ). Thirdly, it is known that the famous Invariant Subspace Problem is equivalent to the problem whether there exists some single operator in the Aℵ0 -class that is saturated [6]. For “most bounded planar domains”, we find that reducing subspace lattice of Mφ is saturated; and Mφ has rich invariant subspaces. There also exist many other motivations from the theory of groups that have the infinite conjugacy class property and free groups. In [3], M. Abrahamse and R. Douglas constructed a class of subnormal operators related to multiply-connected domains, and considered the von Neumann algebras generated by such operators for which they act on vector-valued Hardy space. Our paper is related to [3] for which we will give a comment at the end of Section 5. However, the techniques and ideas developed in this paper are quite different from [3]. We first begin with notations from von Neumann algebras. A von Neumann algebra A is a unital C ∗ -algebra on a Hilbert space H , which is closed in the weak operator topology. Two projections p, q ∈ A are called equivalent, p ∼ q if there is a partial isometry v in A such that v ∗ v = p, vv ∗ = q. A projection p ∈ A is called finite if there exists no projection q ∈ A such that q < p and q ∼ p. Otherwise p is called infinite. The von Neumann algebra A is called finite if its identity is finite, otherwise, infinite. Recall that the center Z(A) of A is defined by Z(A) = {A ∈ A: AB = BA, ∀B ∈ A}. When Z(A) = CI , A is called a factor. (a) Type I factor—if there is a minimal projection E = 0, i.e., a projection E such that there is no other projection F with 0 < F < E; (b) Type II factor—if there are no minimal projections but there are nonzero finite projections. By a II1 factor we mean that it is a Type II factor and its identity is finite, otherwise, II∞ . (c) Type III factor—if it does not contain any nonzero finite projections at all. It is well known that the study of von Neumann algebras is essentially reduced to the study of factors [14,11,27]. Let Ω be a bounded planar domain. Recall that φ : D → Ω is called a holomorphic covering map if every point of Ω has a connected open neighborhood U in Ω such that φ maps each component of φ −1 (U ) biholomorphically onto U . It is well known that such a φ always exists, and is unique up to a conformal automorphism of the unit disk [18,30,39].


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Now we consider the von Neumann algebra V ∗ (φ) on L2a (D), where φ is a holomorphic covering map from D onto a bounded planar domain Ω. By the punctured disk we mean D − {0}. Theorem 1.1. Suppose φ : D → Ω is a holomorphic covering map. Then the following are equivalent. (1) V ∗ (φ) is abelian; (2) the fundamental group π1 (Ω) of Ω is abelian; (3) Ω is conformally isomorphic to one of the disk, annuli or the punctured disk. In other cases, V ∗ (φ) is a Type II1 factor, and W ∗ (φ) is a Type II∞ factor. Remark 1.2. We mention that the condition (3) of the above theorem is equivalent to the fact that C − Ω has no more than one bounded component. Otherwise, π1 (Ω) is always a nonabelian free group [4]. Furthermore, we find that V ∗ (φ) is closely related to the free group factor L(Fn ). Precisely, we have the following result. Theorem 1.3. Suppose φ : D → Ω is a holomorphic covering map. Then V ∗ (φ) is ∗-isomorphic to L(π1 (Ω)). By Remark 1.2, if Ω is not simply connected, then π1 (Ω) is a free group, i.e. π1 (Ω) ∼ = Fn , where n = the cardinality of the bounded components of C − Ω. Later we will see that, for two holomorphic covering maps φi : D → Ω (i = 1, 2), W ∗ (φ1 ) and W ∗ (φ2 ) (respectively, V ∗ (φ1 ) and V ∗ (φ2 )) are unitarily isomorphic. In this sense, we rewrite W ∗ (Ω) for W ∗ (φ1 ), and V ∗ (Ω) for V ∗ (φ1 ). Under this setting, we have the following consequence of Theorem 1.3. Corollary 1.4. Suppose both Ω1 and Ω2 are not conformally isomorphic to one of the disk, annuli and the punctured disk. Then the following are equivalent. ∗

∼ V ∗ (Ω2 ); 1. V ∗ (Ω1 ) = ∗ 2. V (Ω1 ) is unitarily isomorphic to V ∗ (Ω2 ); 3. W ∗ (Ω1 ) is unitarily isomorphic to W ∗ (Ω2 ). ∼ π1 (Ω2 ), then V ∗ (Ω1 ) is unitarily isomorphic Consequently, by Theorem 1.3, if π1 (Ω1 ) = ∗ ∗ to V (Ω2 ), and W (Ω1 ) is unitarily isomorphic to W ∗ (Ω2 ). Theorem 1.3 and its corollary are closely related to one of the long-standing problems in free group factors: ∗

Is L(Fn ) ∼ = L(Fm ) for n = m and n, m 2? This problem is thus equivalent to the following:

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Problem. If π1 (Ω1 ) π1 (Ω2 ), then is V ∗ (Ω1 ) unitarily isomorphic to V ∗ (Ω2 ); or equivalently, is W ∗ (Ω1 ) unitarily isomorphic to W ∗ (Ω2 )? To generalize the above study to a general situation, we need some notations. For a holomorphic map φ : D → Ω, write G(φ) = {γ ; γ is a conformal automorphism of D satisfying φ ◦ γ = φ}, and call G(φ) the deck transformation group of φ. Definition. (See [30, Appendix E].) A holomorphic map φ : D → Ω is called a regular branched covering map if it satisfies the following: (a) every point of Ω has a connected open neighborhood U such that each connected component V of φ −1 (U ) maps onto U by a proper map φ|V ; (b) for any z1 , z2 ∈ D, φ(z1 ) = φ(z2 ) implies that there is a γ ∈ G(φ) such that γ (z1 ) = z2 . A holomorphic map φ : D → Ω is called a branched covering map if it satisfies the condition (a); and it is called regular if (b) is satisfied. For each w ∈ Ω, φ −1 (w) is called a fiber. Notice that the condition (b) means that the deck transformation group G(φ) acts transitively on each fiber. In particular, for a holomorphic covering map φ : D → Ω, φ is a universal covering map since D is simply connected. It is well known that all universal covering maps are regular. Regular branched covering maps have several good properties. For example, let φ : D → Ω be such a map, and it is not difficult to check that for each w ∈ Ω, the multiplicity of the zero point of φ − w at z(z ∈ φ −1 (w)) only depends on w. In this way, we can define a function ν from Ω to {1, 2, . . .}, such that ν(w) equals the above multiplicity. One can also check that ν takes the value 1 except on a discrete subset of Ω. Given a planar bounded domain Ω, and a discrete closed subset Σ of Ω, and a function ν : Ω → {1, 2, . . .}, the triple (Ω, Σ, ν) is called an orbifold domain if ν takes the value 2 on Σ, and 1 on other points. For such a triple (Ω, Σ, ν), there exists a regular branched covering map φ : D → Ω with the property that for each w ∈ Ω, ν(w) equals the multiplicity of the zero point of φ − w at any z in φ −1 (w) (see [30, Theorem E1], [36, Theorem 2.3] and [30, Theorem 1.1]). Usually, we write (Ω, ν) for (Ω, Σ, ν). Such a map φ : D → (Ω, ν) is unique up to an automorphism of D [9, Theorem 2.6], and φ is called the universal covering of (Ω, ν) (for details, see [30]). The function ν is called a ramified function and Σ is called the singular locus of (Ω, ν). If Σ is empty and ν ≡ 1, then (Ω, ν) is identified with the domain Ω. In this situation, φ is the usual universal covering map. Now we state our result as follows, whose special case is Theorem 1.1. Theorem 1.5. Let (Ω, ν) be an orbifold domain with the singular locus Σ , and suppose φ : D → (Ω, ν) is a holomorphic universal covering. Then V ∗ (φ) is a Type II1 factor with the following exceptions: 1. Σ is empty, and Ω is conformally isomorphic to one of the disk, annuli or the punctured disk. In this case, φ is a covering map. 2. Σ = {w0 } is a singleton and Ω is conformally isomorphic to the unit disk D. In this case, φ = ψ ◦ ρ n , where ψ is a conformal map from D onto Ω, and ρ ∈ Aut(D), n = ν(w0 ). 3. Σ = {w0 , w1 } with ν(w0 ) = ν(w1 ) = 2 and Ω is conformally isomorphic to the unit disk D. In this case, V ∗ (φ) is a finite von Neumann algebra whose center is infinite dimensional. Whenever V ∗ (φ) is a Type II1 factor, W ∗ (φ) always is a Type II∞ factor.


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The paper is organized as follows. In Section 2, we give the representation of operators in V ∗ (φ) and show that V ∗ (φ) is the SOT-closure of unitary operators Uρ : h → h ◦ ρρ ,

h ∈ L2a (D),

where ρ run over the deck transformation group G(φ). In Section 3, we completely characterize when V ∗ (φ) is abelian in the case of φ being a holomorphic covering map. It is shown that there exist a lot of Blaschke products B such that V ∗ (B) are not abelian (see Example 3.7 and Proposition 3.8). In Section 4, we complete the proof of Theorem 1.1 by establishing a technical lemma (Lemma 4.8). In Section 5, we show that V ∗ (φ) is ∗-isomorphic to the free group factors L(Fn ), and the proof of Corollary 1.4 is done. In Section 6, we discuss the relation between holomorphic regular branched covering maps and orbifold domains, and Theorem 1.5 is established. 2. Representations of operators in V ∗ (φ) In this section, we give the representations of those operators in V ∗ (φ), where φ is a holomorphic regular branched covering map from the unit disk D onto a bounded planar domain Ω. Also, we give some lemmas which will be needed later. Now let G(φ) be the deck transformation group of φ; that is, G(φ) consists of those automorphisms ρ of D which satisfy φ ◦ ρ = φ. In this case, for any points z1 , z2 in D, φ(z1 ) = φ(z2 ) if and only if there is a ρ ∈ G(φ) such that z2 = ρ(z1 ). Since G(φ) is at most countable, then we write G(φ) = {ρk }. Let Eφ denote the critical value set of φ; that is, Eφ = φ(z): there is a z ∈ D such that φ (z) = 0 . We need the following lemma, which is implied by the proof of [12, Theorem 6]. Lemma 2.1. (See [12].) Suppose φ : D → Ω is a holomorphic regular branched covering map. Then for each w0 ∈ D − φ −1 (Eφ ), {ρk (w0 )} is an interpolating sequence for H ∞ (D). In general, for each ρk ∈ G(φ), define Uρk on L2a (D) by Uρk h = h ◦ ρk ρk ,

h ∈ L2a (D).

Clearly, Uρk is a unitary operator, and it is easy to verify that Uρ∗k Mφ Uρk = Mφ◦ρ −1 = Mφ , k

and hence all Uρk are in V ∗ (φ). For each w ∈ D, set πw : h → Uρk h(w) ,

h ∈ L2a (D),

and we will see that each πw is a bounded operator from the Bergman space to l 2 or Cn . The following theorem gives the representation of unitary operators in V ∗ (φ).

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Theorem 2.2. Suppose φ : D → Ω is a holomorphic regular branched covering map, and S is a unitary operator on the Bergman space which commutes with Mφ . If G(φ) is infinite (or finite), then there is a unique operator W : l 2 → l 2 (or W : Cn → Cn ) such that W πw (h) = πw (Sh),

h ∈ L2a (D), w ∈ D.

(2.1)

This W is necessarily a unitary operator. Moreover, there is a unique square-summable sequence {ck } satisfying Sh(z) =

ck Uρk h(z),

h ∈ L2a (D), z ∈ D,

(2.2)

k

where ρk run over G(φ). Remark. If φ is just a branched covering map and its critical value set Eφ is discrete, then for any w0 ∈ D − φ −1 (Eφ ) there is a neighborhood of w0 on which (2.1) holds. But all ρk , coming from branches of φ −1 ◦ φ, are only locally holomorphic. In this case, we do not know whether W is unique. In particular, for the situation when φ is a finite Blaschke product, a similar version of (2.2) is obtained by [38] and [15], with a different proof. Proof of Theorem 2.2. Suppose φ : D → Ω is a holomorphic regular branched covering map. Without loss of generality, we assume that G(φ) is infinite. When G(φ) is finite, the discussion is similar and easier. For the reader’s convenience, the proof is divided into several steps because of its length, and the difficulty lies in (2.1). Step 1. First we have the following claim. Claim. For each z0 ∈ D, there exists a disk ⊆ D containing z0 such that {Uρk h} is a squaresummable sequence of L2a ( ) for any h ∈ L2a (D). It follows that for any {ci } ∈ l 2 , ∞

ci Uρi h(z),

z ∈ D,

i=0

converges uniformly on each compact set in D, and hence is holomorphic in D. To see this, for each z0 ∈ D, there exists a connected neighborhood U0 of φ(z0 ) such that each connected component Vi (i 0) of φ −1 (U0 ) maps onto U0 by a proper map, namely φ|Vi . Without loss of generality, we assume z0 ∈ V0 . We first give the following statement: For each Vj , there are m distinct ρk ∈ G(φ) satisfying ρk (V0 ) = Vj ; moreover, this m does not depend on j , and m is finite. The reasoning is as follows. Assume that for each Vj , there are exactly mj distinct ρk ∈ G(φ) satisfying ρk (V0 ) = Vj . Since the identity is in G(φ), m0 1. Since φ|V0 : V0 → U0 is a proper map, for a given w ∈ U0 − Eφ , (φ|V0 )−1 (w) is a compact set, and hence is finite [30, Appendix E]. Observe that for a z ∈ (φ|V0 )−1 (w), (φ|V0 )−1 (w) ⊇ ρk (z); ρk ∈ G(φ) and ρk (V0 ) = V0 ,


1225

which implies that m0 is no more than the cardinality of the set (φ|V0 )−1 (w), and then m0 is finite. It remains to show that mj = m0 for each j . To see this, notice that φ|Vj : Vj → U0 is surjective, and then there exists at least one point zj in Vj satisfying φ(zj ) = φ(z0 ). Since φ is regular, this implies that there is a ρ ∈ G(φ) such that ρ(zj ) = z0 . Since all Vi are pairwisely disjoint, then the above identity implies that ρ(Vj ) = V0 . Therefore, for each ρk ∈ G(φ) with ρk (V0 ) = Vj , ρ ◦ ρk (V0 ) = V0 . This implies that mj m0 , and similar discussion shows that m0 mj , forcing mj = m0 . Therefore, mj does not depend on j . For such m distinct ρk , we have

Uρ h(z)2 dA(z) = k

V0

h ◦ ρk (z)ρ (z)2 dA(z) =

h(z)2 dA(z),

k

V0

Vj

which implies that ∞ Uρ h(z)2 dA(z) = m k k=0 V

0

h(z)2 dA(z) m

h(z)2 dA(z) < ∞.

D

j 0 Vj

Therefore, Uρk h is a square-summable sequence of L2a (V0 ). Just take an open disk ⊆ V0 such that z0 ∈ . Clearly, Uρk h is a square-summable sequence of L2a ( ). 2 2 Notice that for each {ci } ∈ l and h ∈ La (D), ∞ i=0 ci Uρi h converges to some function in the norm of L2a ( ), and hence converges uniformly locally in . By the arbitrariness of z0 , it is easy to see that for each {ci } ∈ l 2 , ∞

ci Uρi h(z)

i=0

converges uniformly locally in D; equivalently, it converges uniformly on each compact set in D. So it is holomorphic in D. Step 2. We will show that for each z0 ∈ D − φ −1 (Eφ ), there is a small disk containing z0 , on which (2.1) holds and W is unique. The idea of this part comes from [23]. It is not difficult to see that Eφ is discrete in Ω, and thus φ −1 (Eφ ) is discrete in D [30]; moreover, the map φ|D−φ −1 (Eφ ) : D − φ −1 (Eφ ) → Ω − Eφ is a covering map [7]. This immediately shows that there is an enough small disk containing z0 such that ∩ φ −1 (Eφ ) is empty and

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φ −1 (V 0 ) =

ρk ( ),

k

where V 0 φ( ). Now let H = L2a ( ) and Λ = L2a (D). For each h ∈ Λ, set ehk (z) = Uρk h(z),

fhk (z) = Uρk (Sh)(z),

z ∈ , k = 0, 1, . . . .

Clearly ehk and fhk are in H . For f1 , f2 in H , a rank-one operator f1 ⊗ f2 on H is defined by (f1 ⊗ f2 )f = f, f2 f1 . By Step 1, for any h ∈ L2a (D), {ehk } is square-summable sequence of H , and hence for any g ∞ k k k k and h in Λ, both ∞ k=0 eg ⊗ eh and k=0 fg ⊗ fh converge in the operator norm on H . We first show that ∞

egk ⊗ ehk =

∞

k=0

fgk ⊗ fhk .

(2.3)

k=0

Indeed, from SMφ = Mφ S, we see that for any polynomials P and Q

P (φ)g, Q(φ)h = P (φ)Sg, Q(φ)Sh .

Writing g for Sg and h for Sh, we get

(P Q) ◦ φ(w)g(w)h(w) − (P Q) ◦ φ(w) g (w) h(w) dA(w) = 0,

D

and hence

v φ(w) g(w)h(w) − g (w) h(w) dA(w) = 0,

for any v ∈ C(Ω).

(2.4)

D

Lusin’s Theorem [24, p. 242] shows that for each v ∈ L∞ (Ω), there is a uniformly bounded sequence {vn } in C(Ω) such that {vn } converges in measure to v. Thus by the Dominated Convergence Theorem, (2.4) holds for any v ∈ L∞ (Ω). In particular, for any v ∈ L∞ (V 0 ), (2.4) gives that

v φ(w) g(w)h(w) dA(w) =

φ −1 (V 0 )

From φ −1 (V 0 ) = lation shows that

v φ(w) g (w) h(w) dA(w).

φ −1 (V 0 ) k ρk ( ),

and noticing that φ| : → V 0 is biholomorphic, a simple calcu-


u(z)

∞

2 (gh) ◦ ρk (z)ρk (z) dA(z) =

k=0

u(z)

1227

∞ 2 ( g h) ◦ ρk (z)ρk (z) dA(z) k=0

for each u ∈ L∞ ( ). So ∞

∞ 2 2 (gh) ◦ ρk (z)ρk (z) = ( g h) ◦ ρk (z)ρk (z) ,

k=0

z ∈ .

k=0

That is,

∞ ∞

k k k k eg ⊗ eh Gz , Gz = fg ⊗ fh Gz , Gz , k=0

z ∈ ,

k=0

where Gz is the reproducing kernel of L2a ( ) at z. By the property of Berezin transform, we get ∞

egk ⊗ ehk =

k=0

∞

fgk ⊗ fhk ,

k=0

completing the proof of (2.3). Next we show the existence of W . For each g ∈ Λ, we set Ag : l 2 → H c →

∞

ck egk ,

k=0

and Bg : l 2 → H c →

∞

ck fgk ,

k=0

where c = {ck } is in l 2 . It is easy to check that Ag and Bg are well defined since both {egk } and {fgk } are square-summable sequences from H . Moreover, we have A∗g : H → l 2 p → p, egk , and Bg∗ : H → l 2 p → p, fgk .

1228


Then for any g, h ∈ Λ, we have Ah A∗g p =

∞

∞

k p, egk ehk = eh ⊗ egk p

k=0

=

k=0

∞

k fh ⊗ fgk p = Bh Bg∗ p,

p ∈ H.

k=0

The third equality follows from (2.3). So Ah A∗g = Bh Bg∗ , g, h ∈ Λ, and then it is easy to verify that l i=1

A∗gi pi →

l

Bg∗i pi ,

pi ∈ H and gi ∈ Λ

i=1

is a well-defined isometry from some subspace of l 2 to another, and by a careful observation, we see that the initial space and the final space are equal. Therefore this isometry can be extended to a unitary operator from l 2 onto l 2 , say V . Clearly, we have V A∗g = Bg∗ ,

g ∈ Λ.

Recall that Gw is the reproducing kernel at w of H (w ∈ ), and V A∗g Gw = Bg∗ Gw ,

g ∈ Λ and w ∈ .

That is, V egk (w) = fgk (w) .

(2.5)

Now write the infinite matrix form of V as (ai,j )∞ i,j =1 and put W = (ai,j )∞ i,j =1 . We also denote the corresponding operator by W , and clearly W is also unitary since V is unitary. Then by (2.5), we get W egk (w) = fgk (w) ,

g ∈ Λ, w ∈ ,

(2.6)

as desired. We must show that (2.6) determines an operator W uniquely. To see this, we will show that for each fixed w0 ∈ , the span of all {egk (w0 )} (g ∈ Λ) is dense in l 2 . Suppose conversely that there is a nonzero vector {di } in l 2 such that {egk (w0 )}, {di } = 0, g ∈ Λ, that is,

K. Guo, H. Huang / Journal of Functional Analysis 260 (2011) 1219–1255 ∞

dk ρk (w0 )g ρk (w0 ) .

1229

(2.7)

k=0

On the other hand, Lemma 2.1 shows that for each w0 ∈ D − φ −1 (Eφ ), {ρk (w0 )} is an interpolating sequence. In particular, for each i there is a bounded holomorphic function g satisfying g(ρk (w0 )) = δki . By (2.7), we have di ρi (w0 ) = 0, forcing di = 0 (∀i). This is a contradiction. So the operator W in (2.7) is unique. Step 3. Now we can show (2.1) and (2.2). To prove (2.1), it is equivalent to show that for any h ∈ L2a (D), W πw h, ei = πw (Sh), ei ,

w ∈ D and i = 1, 2, . . . ,

where {ei ; i ∈ Z+ } denotes the usual orthonormal basis of l 2 . From (2.6), the above identity holds on an open subset of D, and hence on D, since both sides of the above are holomorphic functions in D. Therefore (2.1) holds. Without loss of generality, assume that ρ0 (z) = z, and then Uρ0 = I . Denote the first row of the matrix W by {ck } and expanding (2.1) yields Sh(z) =

∞

ck Uρk h(z),

h ∈ L2a (D), z ∈ D.

k=0

For the uniqueness of the coefficients ck , just notice that for any fixed w0 ∈ D − φ −1 (Eφ ), {ρk (w0 )} is an interpolating sequence. Then following the last paragraph in Step 2 shows that ck are unique. The proof is complete. 2 Since any von Neumann algebra is the finite linear span of its unitary operators [11], each operator in V ∗ (φ) has the representation (2.2), whose uniqueness comes from the reasoning in the above Step 2. This is the following. Corollary 2.3. Suppose φ : D → Ω is a holomorphic regular branched covering map, and S ∈ V ∗ (φ). Then there is a unique square-summable sequence {ck } satisfying Sh(z) =

ck Uρk h(z),

h ∈ L2a (D), z ∈ D,

k

where ρk run over G(φ). The following lemma gives elementary analysis for operators in V ∗ (φ). Lemma 2.4. Suppose φ : D → Ω is a holomorphic regular branched covering map, and G(φ) is infinite. For two operators S, T in V ∗ (φ) we write Sh(z) =

∞

ck Uρk h(z)

and T h(z) =

k=0

where {ck } and {ck } belong to l 2 . Then we have

∞ k=0

ck Uρk h(z),

h ∈ L2a (D), z ∈ D,

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S ∗ h(z) =

(1)

∞

ck Uρ∗k h(z),

h ∈ L2a (D), z ∈ D,

(2.8)

k=0

Notice that Uρ∗k = Uρ −1 . k c (2) Define SN = N k k=0 Uρk , then we have ∗ Kz − S ∗ Kz = lim SN Kz − SKz = 0, lim SN

N →∞

∞

ST h(z) =

(3)

a.e. z ∈ D;

N →∞

di Uρi h(z),

z ∈ D, h ∈ L2a (D),

i=0

where

di =

ck cj .

k; ρj ◦ρk =ρi

When G(φ) is finite, similar versions of (1), (2) and (3) are trivial. Proof of Lemma 2.4. (1) Before proving (2.8), we make an observation. For an operator V ∈ V ∗ (φ), by Corollary 2.3 there is a unique vector {dk } ∈ l 2 such that V h(z) =

∞

dk Uρ∗k h(z),

h ∈ L2a (D), z ∈ D.

k=0

For each λ ∈ D − φ −1 (Eφ ), we have V Kλ (w) = V Kλ (w) =

∞

∞

∗ k=0 dk Uρk Kλ (w),

dk ρk (λ)Kρk (λ) (w),

and it follows that

w ∈ D.

k=0

Below we will see that V Kλ =

∞

dk ρk (λ)Kρk (λ) ,

(2.9)

k=0

where the right-hand side converges in L2a (D) and the coefficients of Kρk (λ) are unique. In fact, following the proof of [13, Lemma A] shows that for each λ ∈ D − φ −1 (Eφ ), h →

2 1 − ρk (λ) h ρk (λ)

induces a bounded invertible linear map A from ((φ − φ(λ))L2a (D))⊥ onto l 2 . Denote by {ek }∞ k=0 the standard orthogonal basis of l 2 , and then 2

A∗ ek = 1 − ρk (λ) Kρk (λ) ,

k = 0, 1, . . . .


1231

Thus for any {dk } ∈ l 2 , the series ∞

2

dk 1 − ρk (λ) Kρk (λ)

k=0

converges in L2a (D). On the other hand, by a simple computation, we see |ρk (λ)| =

1−|ρk (λ)|2 , 1−|λ|2

which shows that the right-hand side of (2.9) converges in L2a (D). The uniqueness of the coefficients follows from Lemma 2.1. The above discussion shows that, for a fixed λ ∈ D − φ −1 (Eφ ), the representation (2.9) of V Kλ determines the coefficients dk in the representation of V . Now put V = S ∗ , and then S ∗ h(z) =

∞

dk Uρ∗k h(z),

h ∈ L2a (D), z ∈ D,

k=0

where {dk } ∈ l 2 . To prove (2.8), it suffices to show that di = ci for all i. We rewrite (2.9) as S ∗ Kλ =

∞

dk ρk (λ)Kρk (λ) .

(2.10)

k=0

Since by Lemma 2.1 {ρk (λ)} is an interpolating sequence for H ∞ , then for each i, there is a bounded holomorphic function h such that

h ρk (λ) = δik . Since

S ∗ Kλ , h = Kλ , Sh,

by (2.10) and the representation of Sh we get di = ci . So S ∗ has the representation (2.8), as desired. (2) Set SN = N k=0 ck Uρk . In fact, the above discussion has shown that for each λ ∈ D − φ −1 (Eφ ), ∗ lim SN Kλ − S ∗ Kλ = 0.

(2.11)

N →∞

Furthermore, by (2.8) we can also define (S ∗ )N = place S ∗ with S in (2.11). Then we have lim SN Kλ − SKλ = 0,

N →∞

N

∗ k=0 ck Uρk .

∗ and reNotice (S ∗ )N = SN

λ ∈ D − φ −1 (Eφ ).

Since φ −1 (Eφ ) is discrete (if φ is a holomorphic covering map, then it is empty), then the conclusion follows.

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(3) Since ST is in V ∗ (φ), we can assume ST h(z) =

∞

di Uρi h(z),

h ∈ L2a (D), z ∈ D.

i=0

It remains to determine di . Define TN = N k=0 ck Uρk and it is easy to check that STN h(z) =

∞

(N )

di

Uρi h(z),

h ∈ L2a (D), z ∈ D,

i=0

where (N )

di

=

ck cj .

k; j N, ρj ◦ρk =ρi

Notice that the above is a finite sum. Fix a λ ∈ D − φ −1 (Eφ ). By Lemma 2.1, {ρk−1 (λ)} is an interpolating sequence, and hence for each i, there is a bounded holomorphic function h satisfying

h ρk−1 (λ) = δki .

(2.12)

By the discussion in (2), lim TN Kλ − T Kλ = 0.

N →∞

Thus limN →∞ STN Kλ − ST Kλ = 0, and hence lim STN Kλ , h = ST Kλ , h.

N →∞

In the proof of (1), put V = STN and by (2.9) we have STN Kλ =

(N ) −1 ρk (λ)Kρ −1 (λ) , k

dk

k

and similarly ST Kλ =

dk ρk−1 (λ)Kρ −1 (λ) .

k

k

Combining (2.12) and (2.13) with the above two identities, we have

(N ) −1 lim d ρi (λ) = di ρi−1 (λ). N →∞ i Since ρi ∈ Aut(D), then (ρi−1 ) (λ) = 0, forcing

(2.13)

K. Guo, H. Huang / Journal of Functional Analysis 260 (2011) 1219–1255 (N )

di = lim di N →∞

The proof of the lemma is complete.

=

1233

ck cj .

k; ρj ◦ρk =ρi

2

For each ρ ∈ G(φ), Uρ belongs to V ∗ (φ). Moreover, we have the following. Corollary 2.5. The von Neumann algebra V ∗ (φ) equals the SOT-closure (and WOT-closure) of span{Uρ : ρ ∈ G(φ)}. Proof. It suffices to prove Corollary 2.5 in the case when G(φ) is infinite. To see this, write G(φ) = {ρk }. By von Neumann’s bicommutant theorem, it is enough to show that the commutant of span{Uρk } equals that of V ∗ (φ). For this, for a given operator A commuting with all Uρk , we must show that A commutes with each operator in V ∗ (φ). For any S ∈ V ∗ (φ), assume S has the form in Lemma 2.4 and define SN as there. Notice that A commutes with SN , and hence ∗ ASN Kz , Kz = AKz , SN Kz ,

z ∈ D.

(2.14)

By Lemma 2.4, we have ∗ lim SN Kz − S ∗ Kz = lim SN Kz − SKz = 0,

N →∞

N →∞

a.e. z ∈ D.

Taking limits in (2.14), we get ASKz , Kz = AKz , S ∗ Kz = SAKz , Kz . By the continuity of both sides in variable z, the above holds for every z ∈ D. The property of Berezin transformation yields that AS = SA. The proof is complete. 2 To close this section, we show that there is an ultraweakly continuous faithful trace on V ∗ (φ); that is, we will give an ultraweakly continuous linear map Tr : V ∗ (φ) → C satisfying the following: (1) Tr(I ) = 1; (2) Tr(S ∗ S) 0, and Tr(S ∗ S) = 0 if and only if S = 0; (3) Tr(ST ) = Tr(T S), S, T ∈ V ∗ (φ). We give the construction of Tr as follows. From now on, ρ0 always denote the identity function on D, i.e. ρ0 (z) = z, z ∈ D. By Lemma 2.1 {ρi (0)} is an interpolating sequence, and hence there is a bounded holomorphic function h0 satisfying

h0 ρi (0) = δ0i . Now put Tr(S) = Sh0 , 1,

S ∈ V ∗ (φ),

and then it is easy to see the trace Tr is an ultraweakly continuous linear functional on V ∗ (φ).

1234


To verify the above properties (1), (2) and (3), let us make an observation. For each operator S ∈ V ∗ (φ), there is a unique square-summable sequence {ck } such that Sh(z) =

z ∈ D, h ∈ L2a (D),

ck Uρk h(z),

k

where ρk run over G(φ). It is easy to check that Tr(S) = c0 , and hence Tr(I ) = 1. For any two operators S, T ∈ V ∗ (φ), by Lemma 2.4(3) it is not difficult to verify that Tr(ST ) = Tr(T S). Moreover, by Lemma 2.4 we have S ∗ Sh(z) =

z ∈ D, h ∈ L2a (D),

di Uρi h(z),

i

where di =

ck cj .

k; ρj ◦ρk−1 =ρi

In particular,

Tr S ∗ S = d0 = |ck |2 0.

(2.15)

k

This shows that Tr is a trace on V ∗ (φ). Furthermore, (2.15) immediately gives that if Tr(S ∗ S) = 0, then S ∗ S = 0. Therefore, Tr is a faithful trace. Then applying [11, Corollary 50.13] gives the following. Proposition 2.6. The von Neumann algebra V ∗ (φ) is finite. 3. Abelian V ∗ (φ) In this section, we will completely characterize when the von Neumann algebra V ∗ (φ) is abelian. Before continuing, let us present a fact about holomorphic covering maps. Theorem 3.1 (The Koebe Uniformization Theorem). Given a z0 in D and w0 in Ω, there is a unique holomorphic covering map φ of D onto Ω with φ(z0 ) = w0 and φ (z0 ) > 0, see [18,39]. The above theorem shows that if ψ is another holomorphic covering map from D onto Ω, then there is a ϕ ∈ Aut(D) such that ψ = φ ◦ ϕ. Therefore, it is easy to check that there is a natural spatial isomorphism (and thus a C ∗ isomorphism) between V ∗ (φ) and V ∗ (ψ). The following theorem shows that the community of V ∗ (φ) only depends on the geometric property of Ω rather than on φ.


1235

Theorem 3.2. If φ : D → Ω is a holomorphic covering map, then the following are equivalent: (1) the von Neumann algebra V ∗ (φ) is abelian; (2) the fundamental group π1 (Ω) of Ω is abelian; (3) Ω is conformally isomorphic to one of the disk, annuli or the punctured disk. Proof. In fact, up to a conformal isomorphism, disk, annuli and the punctured disk are the only hyperbolic surfaces with abelian fundamental group [30]. So (2) ⇔ (3) is immediate. Now it remains to show (1) ⇔ (2). Recall that G(φ) is the group consisting of those automorphisms ρ of D satisfying φ ◦ ρ = φ. For any ρ and γ in G(φ), we have Uρ Uγ = Uγ ◦ρ . So all Uρ compose a group, denoted by UG(φ) ; and the above identity shows that UG(φ) is abelian if and only if G(φ) is abelian; if and only if π1 (Ω) is abelian since π1 (Ω) is isomorphic to G(φ) as groups [20, Theorem 5.8]. By Corollary 2.5, the von Neumann algebra V ∗ (φ) equals the SOT-closure of span{Uρ : ρ ∈ G(φ)}. So V ∗ (φ) is abelian if and only if UG(φ) is abelian, if and only if π1 (Ω) is abelian. The proof is complete. 2 Recall that, a closed subspace M is called a reducing subspace of Mφ if Mφ M ⊆ M and Mφ∗ M ⊆ M. The following theorem tells us when V ∗ (φ) is nontrivial, or equivalently, when Mφ has a nontrivial reducing subspace. Theorem 3.3. Let φ : D → Ω be a holomorphic covering map. Then the following are equivalent: (1) (2) (3) (4)

V ∗ (φ) is nontrivial; Mφ has a nontrivial reducing subspace; φ is not univalent; Ω is not simply connected.

Recall that a holomorphic function f on a domain U ⊆ C is called univalent if f is injective in U . Proof of Theorem 3.3. Clearly, (1) ⇔ (2). We will show (3) ⇒ (4) ⇒ (1) ⇒ (3) to complete the proof. (3) ⇒ (4). It is well known that a domain is simply connected if and only if its covering map is a conformal isomorphism. Therefore, if φ : D → Ω is not univalent, then φ is not a conformal isomorphism, and hence Ω is not simply connected. (4) ⇒ (1). Now assume that Ω is not simply connected. Equivalently, the fundamental group π1 (Ω) is nontrivial. Since G(φ) is isomorphic to π1 (Ω) as groups, G(φ) contains a nontrivial element ρ, and hence Uρ is a nontrivial unitary operator in V ∗ (φ). (1) ⇒ (3). Assume conversely that φ is univalent. Since φ is also a covering map, this implies that φ is a conformal isomorphism, and hence both π1 (Ω) and G(φ) are trivial. By Corollary 2.5, V ∗ (φ) is trivial. This contradiction shows (1) ⇒ (3). The proof is complete. 2

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Theorem 3.3 also holds in the Hardy space of the disk, which first was noted by Abrahamse [1] and Abrahamse and Ball [2]. Remark 3.4. Under a mild assumption, Theorem 3.3 can be generalized to high dimensional case, see [22] for some consideration of V ∗ (Φ) and W ∗ (Φ). Example 3.5. Let Ω be the punctured disk D − {0}, and let 1+z

φ(z) = e− 1−z . Since φ is the composition of exponent map and the linear fraction map κ(z) = − 1+z 1−z from the unit disk onto the left half plane, this implies that φ is a covering map. Applying Theorem 3.2 and Theorem 3.3 shows that V ∗ (φ) is nontrivial and abelian. Furthermore, by a simple computation, the deck transformation group of φ is

nπi − (nπi + 1)z : n∈Z , κ −1 ◦ hn ◦ κ: n ∈ Z = nπi − 1 + nπiz

where hn (z) = z + 2nπi are automorphisms of the left half plane. λ−φ(z) . By Frostman’s Theorem [16, Theorem 6.4, p. 79], except For λ ∈ D, write ψλ (z) = 1− λ¯ φ(z) for a subset of capacity zero of D, ψλ (z) are infinite Blaschke products. For such Blaschke products, we see that V ∗ (ψλ ) = V ∗ (φ) is abelian. Notice that φ is an inner function. Therefore, on the Hardy space of the disk, V ∗ (φ) is not abelian. So Theorem 3.2 fails in the case of the Hardy space of the disk. For 0 < r < 1, we take annuli Ωr = {z ∈ C: r < |z| < 1}, and let φr denote the covering map from D onto Ωr . As done in [1] and [35], 1+z 1 i ln + . φr (z) = exp ln r π 1−z 2 Applying Theorem 3.2 and Theorem 3.3 shows that V ∗ (φ) is nontrivial and abelian. 1 : j = 1, . . . , n} for some n 2. By Theorem 3.2, V ∗ (φ) is not If we take Ω = D − { j +1 abelian. The following example shows that there do exist interpolating Blaschke products B such that the von Neumann algebras V ∗ (B) are not abelian. Example 3.6. Let E be a relatively closed subset of D − {0} with capacity zero (for the definition of capacity zero, see [10] or [16]). For example, let E be a discrete subset of D − {0} and φ is a holomorphic covering map from D onto D − E. Below we will see that φ is an interpolating Blaschke product. In fact, Theorem 6 in [12] (also see E. Stout [37]) states that if ψ is a holomorphic covering map from D onto a bounded domain Ω, then for each λ ∈ D, the inner part of ψ − ψ(λ) is an interpolating Blaschke product. From this fact and 0 ∈ φ(D), we see that the inner part of φ is an interpolating Blaschke product. Furthermore, in our case, by [10, p. 37] or the proof of Theorem 1.1 in [19], φ is itself an inner function, and hence an interpolating Blaschke product. If E contains at least two points, then the fundamental group of D − E is not abelian. By Theorem 3.2, V ∗ (φ) is not abelian.


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Moreover, we have the following proposition, whose former part easily is deduced by [10,16] and its latter part is a consequence of Theorem 3.2. Proposition 3.7. If φ : D → Ω is a holomorphic covering map, then φ is an inner function if and only if Ω = D − E, where E is a relatively closed subset of D with capacity zero. In this case, V ∗ (φ) is abelian if and only if E contains no more than one point. Proof. First we consider the former part. The “if” part follows directly from the arguments on [10, pp. 37, 38], and the “only if” part easily comes from [16, Theorem 6.6, p. 80]. For the latter part, notice that by Theorem 3.2, V ∗ (φ) is abelian if and only if the fundamental group π1 (Ω) is abelian. Therefore, it suffices to show that if E is a relatively closed subset of D with capacity zero, and E contains at least two points, then π1 (D − E) is not abelian. To complete the proof, notice that a set of capacity zero is of linear measure zero [10] (linear measure is just the 1-dimensional Hausdorff measure, see [34]). Since E has linear measure zero, {|z|: z ∈ E} is a subset of (−1, 1) with linear measure zero. Therefore, there is an r0 (0 < r0 < 1) such that r0 D ∩ E contains at least two points, and r0 T ∩ E is empty. Since {Re z: z ∈ E} has linear measure zero, by omitting a rotation, we can assume that there is an r1 ∈ (−1, 1) and an ε0 > 0 satisfying the following: (1) [r1 − ε0 , r1 + ε0 ] ⊆ (−1, 1); (2) E ∩ {z ∈ D: Re z ∈ [r1 − ε0 , r1 + ε0 ]} is empty; (3) Both {z ∈ r0 D ∩ E: Re z < r1 − ε0 } and {z ∈ r0 D ∩ E: Re z > r1 + ε0 } are not empty. Now take (see Fig. 1) V1 = {z ∈ D − E: Re z < r1 + ε0 } and V2 = {z ∈ D − E: Re z > r1 − ε0 }. Below we will apply Seifert–Van Kampen’s Theorem [20] to show that π1 (D − E) is not abelian. In fact, by (3), the complement of V1 in the Riemann sphere C has at least two components, and one is contained in {z ∈ r0 D: Re z < r1 − ε0 }. It is well known that an open subset X of C is simply connected if and only if both X and its complement in C are connected. Therefore V1 is not simply connected, and hence π1 (V1 ) is nontrivial. Similarly, π1 (V2 ) is nontrivial. On the other hand, V1 ∩ V2 is simply connected and D − E = V1 ∪ V2 . Then by Seifert–Van Kampen’s Theorem [20], π1 (D − E) is isomorphic to the free product of π1 (V1 ) and π1 (V2 ), and hence is not abelian. The proof is complete. 2 Both Theorem 3.2 and Theorem 3.3 can be generalized to the case of holomorphic regular branched covering map. This is the following proposition.

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Fig. 1.

Proposition 3.8. Suppose φ : D → Ω is a holomorphic regular branched covering map. Then Mφ has a nontrivial reducing subspace if and only if φ is not univalent, and V ∗ (φ) is abelian if and only if G(φ) is abelian. Before ending this section, we give the following proposition. Proposition 3.9. Suppose φ : D → Ω is a holomorphic covering map, and Ω is conformally isomorphic to one of the punctured disk or annuli. Then V ∗ (φ) is ∗-isomorphic to L∞ [0, 1]. Proof. Suppose φ : D → Ω is a holomorphic covering map, and Ω is conformally isomorphic to one of the punctured disk or annuli. Since the deck transformation group G(φ) of φ is an infinite cyclic group, we denote its generator by ρ; and by Corollary 2.5, V ∗ (φ) is generated by the unitary operator Uρ . Before continuing, we mention a fact on multiplier. For a separable σ -finite measurable space (X, μ) and a function f in L∞ (X, μ), Mf defines a multiplier on L2 (X, μ). It is known that the von Neumann algebra W ∗ (Mf ) generated by Mf has no minimal projection if and only if W ∗ (Mf ) is ∗-isomorphic to L∞ [0, 1], if and only if Mf has a nonzero eigenvector. Since a normal operator on a separable space is always unitarily equivalent to some Mf on a (necessarily separable) σ -finite measurable space (X, μ) [5], the proof is reduced to showing that Uρ has no nonzero eigenvector. To see this, suppose conversely that h is a nonzero eigenvector of Uρ . Since Uρ is a unitary operator, there is a constant ξ (|ξ | = 1) such that Uρ h = ξ h. Write G(φ) = {ρk }. Since G(φ) is cyclic, it is easy to see that for each k, there is a unimodular constant ξk such that Uρk h = ξk h.

(3.1)


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As done in the proof of Theorem 2.1, we can pick a disk such that all ρk ( ) are pairwise disjoint and k ρk ( ) ⊆ D. Then

∞ h(z)2 dA(z) =

h(w)2 dA(w)

k=0ρ ( ) k

k ρk ( )

=

∞ h ◦ ρk (z)2 ρ (z)2 dA(z) k

k=0

=

∞

|h|2 dA(z).

k=0

The last identity follows from (3.1). Therefore h equals zero constantly, which is a contradiction. So V ∗ (φ) is ∗-isomorphic to L∞ [0, 1]. The proof is complete. 2 4. Type II factors arising from planar domains In this section, we will complete the proof of Theorem 1.1. In the study of V ∗ (φ) and W ∗ (φ), the theory of free groups plays an important role. Roughly speaking, each free group G arises in this way: there is a set E such that any element of G can be written in one and only one way as a product of finitely many elements of E and their inverses (disregarding trivial variations such as st −1 = su−1 ut −1 ). For its definition, see [32] or [25, Chapter 1]. To establish Theorem 1.1, we need some conclusion from algebraic topology, which is implied in [4] (or see [21, Theorem 3.3]), stated as follows. Lemma 4.1. Suppose Ω is a bounded domain in C such that π1 (Ω) is nontrivial. Then π1 (Ω) is a free group on finite or countably many generators. From Theorem 3.2, we see that π1 (Ω) is abelian if and only if Ω is conformally isomorphic to one of the disk, annuli or the punctured disk. Combining this statement with Lemma 4.1 shows that the following are equivalent. 1. Ω is not conformally isomorphic to one of the disk, annuli or the punctured disk; 2. π1 (Ω) is not abelian; 3. π1 (Ω) is a free group with n generators (n 2, allowed as ∞). Therefore, Theorem 1.1 reduces to the following version. Theorem 4.2. Suppose φ : D → Ω is a holomorphic covering map and π1 (Ω) is not abelian, then V ∗ (φ) is a Type II1 factor, and W ∗ (φ) is a Type II∞ factor. Before continuing, let us see an example. 1 Example 4.3. Set Ω = D − { k+1 : 1 k n} (n 2). Let φ : D → Ω be a holomorphic covering map, then as shown in Example 3.6, φ is an interpolating Blaschke product. Applying Theorem 4.2 shows that V ∗ (φ) is a Type II1 factor, and W ∗ (φ) is a Type II∞ factor.

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The above example is related to the Invariant Subspace Problem, which asks that, if T is a bounded linear operator on a separable Hilbert space H , then does there exist a nontrivial subspace M such that T M ⊆ M? In fact, the above problem is equivalent to the problem whether there exists some single operator in the Aℵ0 -class that is saturated [6]. This means that, for an Aℵ0 -operator T , let M and N be two invariant subspaces of T such that N ⊆ M and dim M/N = ∞, is there another invariant subspace L satisfying N L M? It is well known that the Bergman shift Mz is an Aℵ0 -operator. Also, for each φ in Example 4.3, Mφ is in Aℵ0 . The reasoning is as follows. A contraction T in a Hilbert space is called to belong to the class C00 if both {T n } and {T ∗n } converge to zero in the strong operator topology. It is easy to check that Mφ ∈ C00 and that the spectrum of Mφ equals D. On the other hand, [6, Corollary 6.9] states that if T ∈ C00 and σ (T ) = D, then T ∈ Aℵ0 . So Mφ ∈ Aℵ0 . The Bergman shift Mz has trivial reducing subspace lattice. However, for such φ in Example 4.3, V ∗ (φ) is a Type II1 factor. Furthermore, by [27, Corollary 7.1.17], there is an order isomorphism between the interval [0, 1] and the totally ordered set of equivalent classes of projections on a Type II1 factor. Now let M and N be two reducing subspaces of Mφ such that N M, then there always exist uncountably many reducing subspaces L lying strictly between M and N . This would bring some information that is helpful in studying the lattice of invariant subspace for such an Aℵ0 -operator Mφ . Remark. Combing Example 3.5 with Example 4.3, one sees that there exist infinite Blaschke products B such that the V ∗ (B) is abelian or a Type II1 factor. This shows that for an infinite Blaschke product B, the structure of V ∗ (B) is considerably complicated. Now it remains to prove Theorem 4.2. We need several preliminary results. Lemma 4.4. Suppose φ : D → Ω is a holomorphic covering map and π1 (Ω) is a free group of n (n 2) generators, then V ∗ (φ) is a factor. Proof. Let φ : D → Ω be a holomorphic covering map. By [20, Theorem 5.8] or [25, Proposition 1.39], G(φ) is isomorphic to π1 (Ω). Without loss of generality, we assume that G(φ) is a free group on two generators, say, ρ and τ . A general case is similarly considered. To prove that V ∗ (φ) is a factor, it suffices to show that each member in the center Z(V ∗ (φ)) of V ∗ (φ) must be a constant multiple of the identity I . For this, assume T is a unitary operator in Z(V ∗ (φ)). Applying Theorem 2.1 shows that there is a vector {ci } ∈ l 2 such that T h(z) =

∞

ci Uρi h(z),

z ∈ D, h ∈ L2a (D).

i=0

Without loss of generality, assume that ρ0 is the identity e. Since Uρ0 = I , it suffices to show that ci = 0 (i 1). For each k ∈ Z+ , set Λk = {i ∈ Z+ : ci = ck and ci = 0}. Since {ci } ∈ l 2 , Λk is either empty or finite. Indeed, for distinct i, j , we see that either Λi = Λj or Λi ∩ Λj = ∅. Therefore, one can pick a subsequence {Λk } of {Λk } such that each Λk = ∅, and any two of Λk are disjoint, and the union of Λk equals


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{i ∈ Z+ : ci = 0}. Since T ∈ Z(V ∗ (φ)), T commutes with every Uη (η ∈ G(φ)), and thus ∞

ci Uρi Uη h(z) =

i=0

∞

z ∈ D, h ∈ L2a (D).

ci Uη Uρi h(z),

i=0

That is, ∞

ci Uη◦ρi h(z) =

i=0

∞

ci Uρi ◦η h(z),

z ∈ D, h ∈ L2a (D).

i=0

By the uniqueness of the coefficients ci , it is not difficult to see that {η ◦ ρi : i ∈ Λk } = {ρi ◦ η: i ∈ Λk }, and hence

η−1 ◦ ρi ◦ η: i ∈ Λk = {ρi : i ∈ Λk }.

This implies that the finite set Ek {ρi : i ∈ Λk } satisfies η−1 Ek η = Ek for each η ∈ G(φ). To finish the proof, it suffices to show that for each k, either Ek is empty or Ek consists of the identity e. To see this, assume conversely that there is a ρi0 ∈ G(φ) − {e} such that ρi0 ∈ Ek , and we will show that Ek is an infinite set to reach a contradiction. In fact, write ρi0 as a reduced word: kN ρi0 = x1k1 x2k2 · · · xN ,

where each xi equals ρ or τ , xi = xi+1 (1 i N − 1) and each ki is a nonzero integer. Here the multiplication is the composition between members in Aut(D). Without loss of generality, assume x1 = ρ. Then for different positive integers n, m, τ −n ◦ ρi0 ◦ τ n = τ −m ◦ ρi0 ◦ τ m . This shows that Ek is an infinite set, which leads to a contradiction. The above reasoning says that there exists only one Ek which is not empty and consists of e. Therefore, T is a constant multiple of the identity I , and hence V ∗ (φ) is a factor. 2 Suppose G is a group. Two elements a and b of G are called conjugate if there exists an element g in G such that gag −1 = b. Then conjugacy is an equivalence relation on G. The equivalence class containing the element a in G is [a] = {gag −1 : g ∈ G}, called the conjugacy class of a. If every nontrivial conjugacy class of the group G is infinite, then G is called an i.c.c. group. From the proof of Lemma 4.4, we easily get a stronger result. Corollary 4.5. Let φ : D → Ω be a regular branched covering map. Then the following are equivalent: (1) G(φ) is an i.c.c. group; (2) V ∗ (φ) is a factor.

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Now we are ready to prove Theorem 4.2. Proof of Theorem 4.2. By Lemma 4.4 and Proposition 2.6, V ∗ (φ) is a finite factor. Therefore, by [11, Theorem 48.16], either V ∗ (φ) is Type In or Type II1 . But if it were Type In , it would be ∗-isomorphism to Mn (C) for some n, which is a contradiction to the fact that dim V ∗ (φ) = ∞. So V ∗ (φ) is a Type II1 factor. It remains to prove that W ∗ (φ) is a Type II∞ factor. First we show that W ∗ (φ), as a von Neumann algebra, is infinite. To this aim, it suffices to show that there is an isometry S in W ∗ (φ) which is not unitary. Notice that if we replace φ with φ − φ(0), then neither V ∗ (φ) nor W ∗ (φ) changes. Therefore, we may assume that φ(0) = 0. This gives Range Mφ = φL2a (D) = L2a (D). Let Mφ = U |Mφ | be the polar decomposition of Mφ . Then both U and |Mφ | are in W ∗ (φ). It is easy to check that the initial space of U is L2a (D), and the final space is φL2a (D). Therefore U is a desired isometry. This shows that W ∗ (φ) is infinite. Furthermore, by Corollary 10.1.2 in [27], the commutant of a Type II1 factor is either a Type II1 factor or a Type II∞ factor. Combining this fact with the above discussion shows that W ∗ (φ) is a Type II∞ factor. The proof is complete. 2 Remark. From the latter part of the above proof, we see that for a bounded holomorphic function φ on D, whenever V ∗ (φ) is a Type II1 factor, W ∗ (φ) is always a Type II∞ factor. Also, we notice that when V ∗ (φ) is a Type II1 factor, Mφ is a completely reducible operator. This means that for each nonzero reducing subspace M of Mφ , Mφ |M has a nontrivial reducing subspace [33]. This follows from [27, Corollary 7.1.14], which states that if A is a Type II1 factor on H and p ∈ A is a nontrivial projection, then pAp is a Type II1 factor on pH . Combing the proof of Theorem 4.2 with Corollary 4.5 gives the following proposition. Proposition 4.6. If φ : D → Ω is a holomorphic regular branched covering map, then V ∗ (φ) is a Type II1 factor if and only if G(φ) is an i.c.c. group. Let φi : D → Ω (i = 0, 2) be two holomorphic covering maps, then there exists an automorphism ρ of D such that φ1 ◦ ρ = φ2 . This implies that Uρ∗ Mφ2 Uρ = Mφ2 ◦ρ −1 = Mφ1 , and hence Uρ∗ W ∗ (φ2 )Uρ = W ∗ (φ1 ),

Uρ∗ V ∗ (φ2 )Uρ = V ∗ (φ1 ).

Therefore, omitting a unitary isomorphism, we can assign two von Neumann algebras W ∗ (Ω), V ∗ (Ω) for each Ω; that is, this two von Neumann algebras arise from the domain Ω, independent from the choice of covering maps. Moreover, if G : Ω1 → Ω2 is a conformal isomorphism, and φ : D → Ω1 is a holomorphic covering map, then G ◦ φ : D → Ω2 is a holomorphic covering map, and it is easy to verify that W ∗ (G ◦ φ) = W ∗ (φ),

V ∗ (G ◦ φ) = V ∗ (φ).

This shows that if Ω1 and Ω2 is conformally isomorphic, then W ∗ (Ω1 ) and W ∗ (Ω2 ); V ∗ (Ω1 ) and V ∗ (Ω2 ) are unitarily isomorphic, respectively. For a holomorphic covering map φ : D → Ω, one has G(φ) ∼ = π1 (Ω). Therefore, when Ω is not conformally isomorphic to one of the disk, annuli or the punctured disk, G(φ) is a free group


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on n generators (2 n ∞), that is, G(φ) = Fn . As one knows, L(Fn ) is the von Neumann algebra generated by left regular representation of Fn on l 2 (Fn ), and by Corollary 2.5, V ∗ (φ) is the von Neumann algebra generated by representation Uρ of G(φ) on L2 (D). However, L(Fn ) is not unitarily isomorphic to V ∗ (φ) because L (Fn ) = R(Fn ), a Type II1 factor, but V ∗ (φ) = W ∗ (φ), a Type II∞ factor. Therefore, we ask the following problem: is V ∗ (φ) ∗-isomorphic to L(Fn )? In the next section, we will show that the answer is positive. 5. V ∗ (φ) and free group factors This section will mainly show that for a holomorphic covering map φ : D → Ω, V ∗ (φ) is ∗-isomorphic to the group von Neumann algebra L(π1 (Ω)). Furthermore, we prove that for two bounded domains Ω1 and Ω2 , if π1 (Ω1 ) ∼ = π1 (Ω2 ) and π1 (Ω1 ) is not abelian, then V ∗ (Ω1 ) and ∗ ∗ ∗ V (Ω2 ), W (Ω1 ) and W (Ω2 ) are unitarily isomorphic, respectively. We adopt the notations in [11, Section 43]. For a group G, let R(G) be the WOT-closure of the span of all Ra : l 2 (G) → l 2 (G) (a ∈ G) defined by Ra f (x) = f (xa),

x ∈ G, f ∈ l 2 (G);

and L(G) is the WOT-closure of the span of all La : l 2 (G) → l 2 (G) defined by

La f (x) = f a −1 x ,

x ∈ G, f ∈ l 2 (G).

It is well known that R(G) is unitarily isomorphic to L(G). If G is the free group on n (n 2) generators, then it is an i.c.c. group, and hence by [11, Theorem 53.1], R(G) and L(G) are Type II1 factors. Let us recall some preliminaries of the group von Neumann algebra. As done in [11, pp. 248, 250], for each b ∈ G, let b denote the characterization function of {b}; and let f ∗ (x) = f (x −1 ), x ∈ G. For g, h ∈ l 2 (G), the involution of g and h is defined as follows: g ∗ h(a) =

g(x)h x −1 a ,

a ∈ G.

x∈G

For each f in l 2 (G), we can define a densely-defined operator Lf by setting Lf g = f ∗ g, where g ∈ l 2 (G) satisfies f ∗ g ∈ l 2 (G). It is known that Lf defines a bounded operator on l 2 (G) if and only if f ∗ g ∈ l 2 (G) for every g ∈ l 2 (G); and L(G) consists of all bounded operators Lf [11, p. 250]. Similar results hold for R(G). For any two operators in L(G), say Lg and Lh , we have L∗g = Lg ∗

and Lg Lh = Lg∗h .

(5.1)

To establish Theorem 1.3, we also need the following lemma, which is of independent interest. Lemma 5.1. Suppose G is a countable group and f is in l 2 (G). Then f, La f = 0 for all a = e if and only if f, Ra f = 0 for all a = e, where e is the identity of G.

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Proof. Without loss of generality, we assume that f is nonzero and its norm equals 1. By a simple computation, Lemma 5.1 is equivalent to the following statement: f ∗ ∗ f = e if and only if f ∗ f ∗ = e , and by symmetry, it is reduced to prove that if f ∗ ∗ f = e , then f ∗ f ∗ = e . To see this, we first make the following claim. Claim. If f ∗ ∗ f = e , then Lf is a bounded operator on l 2 (G). As mentioned above, for each h in l 2 (G), we can define a densely-defined operator Lh . Let P be the linear span of {b : b ∈ G}, and then both Lf and Lf ∗ are well defined on P. To prove the above claim, we first show that if f ∗ ∗ f = e , then Lf p, Lf p = p, p, p ∈ P. Since f ∗ ∗ f = e , then by a simple computation Lf b , Lf a = Lf ∗ Lf b , a = b , a ,

a, b ∈ G.

Therefore for each p ∈ P, we get Lf b , Lf p = b , p,

b ∈ G, p ∈ P,

and hence Lf p, Lf p = p, p, p ∈ P, as desired. Next we will show that for any g ∈ l 2 (G), f ∗ g ∈ l 2 (G). When this is done, we can apply the closed graph theorem to conclude that Lf is a bounded operator. To show f ∗ g ∈ l 2 (G), pick pn ∈ P such that pn − g2 → 0 as n → ∞. This implies that f ∗ pn converges to f ∗ g pointwise, and hence for each finite set F in G, we have f ∗ g(a)2 sup f ∗ pn (a)2 supf ∗ pn 2 2 a∈F

n

n

a∈F

= sup Lf pn 22 = sup pn 22 M < ∞. n

n

Since F is arbitrary, then f ∗ g ∈ l 2 (G), as desired. The proof of the claim is complete. Now Lf is a bounded operator, and then by (5.1) L∗f = Lf ∗ . On the other hand, the proof of [11, Theorem 53.1] shows that for a countable group G, R(G) and L(G), as von Neumann algebras, are finite. This means that for any operator S in L(G) satisfying S ∗ S = I , we must have SS ∗ = I . By a straightforward computation, L∗f Lf = Lf ∗ ∗f = I, and hence Lf L∗f = I. This ensures Lf ∗f ∗ = I , and hence f ∗ f ∗ = e . The proof is complete. 2 From the above proof, we get the following proposition. Proposition 5.2. Suppose G is a countable group and f is in l 2 (G). Then f ∗ f ∗ = e if and only if f ∗ ∗ f = e . In this situation, Lf is a unitary operator. Before continuing, let us make some observations. Given a holomorphic covering map φ : D → Ω, write G(φ) = {ρk }. By the claim in Step 1 of the proof of Theorem 2.2, for any f ∈ l 2 (G(φ)), the formal sum k

f (ρk )Uρ∗k : h(z) →

k

f (ρk )Uρ∗k h(z),

h ∈ L2a (D)


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defines a linear map from L2a (D) to holomorphic functions over D. Notice that for each k ∈ Z+ , Uρ∗k = Uρ −1 , and hence the above map is k

f (ρk )Uρ∗k : h(z) →

k

f (ρk )h ◦ ρk−1 (z) ρk−1 (z),

h ∈ L2a (D).

k

For simplicity, write Θ(f )

f (ρk )Uρ∗k .

k

On the other hand, for any f ∈ l 2 (G(φ)), we identify the formal sum k f (ρk )Lρk with the densely-defined operator Lf in l 2 (G(φ)). This identification is based on the fact that Lx = Lx , x ∈ G(φ). From Corollary 2.3, one sees that for each S ∈ V ∗ (φ), there exists a unique f ∈ l 2 (G(φ)) satisfying S = Θ(f ). We define a linear map Λ on V ∗ (φ) as follows: Λ : Θ(f ) → Lf ; that is, Λ:

k

f (ρk )Uρ∗k →

f (ρk )Lρk .

k

This says that Λ maps each member Θ(f ) in V ∗ (φ) to a densely-defined operator Lf in l 2 (G(φ)). In what follows we will show that these Lf are bounded. Proposition 5.3. The linear map Λ is an injective ∗-homomorphism from V ∗ (φ) to L(G(φ)). Proof. We first show that Λ maps each member in V ∗ (φ) to L(G(φ)). Since any von Neumann algebra is the finite linear span of its unitary operators [11], it suffices to show that if Θ(f ) is a unitary operator, then Lf is bounded, and hence is in L(G(φ)). In fact, by Lemma 2.4 it is easy to check that for any two operators Θ(f1 ) and Θ(f2 ) in V ∗ (φ),

Θ(f1 )∗ = Θ f1∗ ,

(5.2)

Θ(f1 )Θ(f2 ) = Θ(f1 ∗ f2 ).

(5.3)

and

Since Θ(f )∗ Θ(f ) = I, we have

Θ f ∗ ∗ f = Θ f ∗ Θ(f ) = Θ(f )∗ Θ(f ) = I, forcing f ∗ ∗ f = ρ0 , where ρ0 denotes the identity of G(φ). By Proposition 5.2, Lf is a bounded operator in L(G(φ)).

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Furthermore, for any Θ(f1 ), Θ(f2 ) ∈ V ∗ (φ), by (5.1) and (5.2) we get

∗ Λ Θ(f1 )∗ = Λ Θ f1∗ = Lf1∗ = L∗f1 = Λ Θ(f1 ) , and by (5.1) and (5.3),

Λ Θ(f1 )Θ(f2 ) = Λ Θ(f1 ∗ f2 ) = Lf1 ∗f2 = Lf1 Lf2

= Λ Θ(f1 ) Λ Θ(f2 ) . Therefore Λ is ∗-homomorphism from V ∗ (φ) to L(G(φ)), and its injectivity is trivial.

2

Theorem 5.4. The map Λ is a ∗-isomorphism from V ∗ (φ) onto L(G(φ)). Proof. Given two von Neumann algebras A and B, a positive map γ : A → B is called normal if for any increasing net {Aα } in A converging strongly to A, γ (Aα ) converges strongly to γ (A). We will first show that Λ is normal. To see this, notice that by Proposition 5.3 Λ is a ∗-homomorphism, and hence is positive. Now assume that {Θ(fα )} is an increasing net in V ∗ (φ) which converges strongly to Θ(f ). Since Λ is positive, {Λ(Θ(fα ))} is an increasing net satisfying

Λ Θ(fα ) Λ Θ(f ) ; that is, {Lfα } is an increasing net with an upper bound Lf . Therefore {Lfα } converges strongly to some operator Lg in L(G(φ)), which immediately gives lim fα (ρk ) = g(ρk ), α

k = 0, 1, . . . .

(5.4)

On the other hand, by Lemma 2.1 {ρi−1 (0)} is an interpolating sequence, and hence for each k, there is a bounded holomorphic function h satisfying

h ρi−1 (0) = δki .

(5.5)

Since {Θ(fα )} converges strongly to Θ(f ), lim Θ(fα )h, 1 = Θ(f )h, 1 , α

which, combined with (5.5), gives that lim fα (ρk ) = f (ρk ), α

k = 0, 1, . . . .

(5.6)

Then by (5.4) and (5.6), f = g, and hence Λ(Θ(fα )) converges strongly to Λ(Θ(f )). Therefore Λ is normal.


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Since Λ is normal, by [11, Theorem 46.8] Range(Λ) is weak∗ closed, and hence is a von Neumann algebra. Thus Range(Λ) = L(G(φ)), and by Proposition 5.3, Λ is a ∗-isomorphism from V ∗ (φ) to L(G(φ)). The proof is complete. 2 For a holomorphic covering map φ : D → Ω, one has G(φ) ∼ = π1 (Ω). Therefore, Theorem 5.4 shows that V ∗ (Ω) is ∗-isomorphic to L(π1 (Ω)). Then by Lemma 4.1, V ∗ (Ω) is ∗-isomorphic to L(Fn ), where n is the cardinality of generators of π1 (Ω). In particular, if C−Ω has q (q < ∞) bounded components, then n = q. The special case of n = 1 is Proposition 3.8. It is well known that L(G) is a Type II1 factor if and only if G is an i.c.c. group. Therefore, Proposition 4.6 is a direct consequence of Theorem 5.4, whose proof is independent of Section 4. Before continuing, let us see an example, which indicates that the structure of V ∗ (Ω) is very complicated. Example 5.5. Fix an r ∈ (0, 1), and set Ωr = {z ∈ C: r < |z| < 1}. As done in Example 3.5, let φr denote the covering map from D onto Ωr : 1+z 1 i ln + . φr (z) = exp ln r π 1−z 2 Take a w0 ∈ Ωr and put Ω0 = D − φ −1 (w0 ). Now let ϕ denote the covering map from D onto Ω0 . By our convention, V ∗ (ϕ) = V ∗ (Ω0 )

and V ∗ (φr ◦ ϕ) = V ∗ Ωr − {w0 } ,

and hence V ∗ (Ω0 ) is a closed ∗-subalgebra of V ∗ (Ωr − {w0 }). But π1 (Ω0 ) ∼ = F∞ and π1 (Ωr − {w0 }) ∼ = F2 , and hence ∗

V ∗ (Ω0 ) ∼ = L(F∞ ),

∗ V ∗ Ωr − {w0 } ∼ = L(F2 ).

Next we will give the following theorem. Theorem 5.6. Suppose both Ω1 and Ω2 are not conformally isomorphic to one of the disk, annuli and the punctured disk. Then the following are equivalent. ∗

1. V ∗ (Ω1 ) ∼ = V ∗ (Ω2 ); ∗ 2. V (Ω1 ) is unitarily isomorphic to V ∗ (Ω2 ); 3. W ∗ (Ω1 ) is unitarily isomorphic to W ∗ (Ω2 ). Consequently, by Theorem 5.4, if π1 (Ω1 ) ∼ = π1 (Ω2 ), then V ∗ (Ω1 ) is unitarily isomorphic ∗ ∗ to V (Ω2 ), and W (Ω1 ) is unitarily isomorphic to W ∗ (Ω2 ). Remark. If Ω1 is homotopy equivalent to Ω2 , then π1 (Ω1 ) ∼ = π1 (Ω2 ) [29, Theorem 7.24]. Therefore, both W ∗ (Ω) and V ∗ (Ω) only depend on the homotopy class of the domain Ω. We need some preliminary results on Type II1 factors. For a Type II1 factor M, there is always a unique ultraweakly continuous normalized trace tr, which is necessarily faithful. This

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trace gives a GNS construction, and the corresponding Hilbert space is denoted by L2 (M), which contains a dense linear subspace M. By [27, Corollary 7.2.11], M acts on L2 (M) as a von Neumann algebra. In this situation, we say M is in standard form. For example, if G is an i.c.c. group, then both L(G) and R(G) are in standard forms. The following fact about standard form is likely well-known, and here we include a proof. ∗

Lemma 5.7. For two Type II1 factors A and B in standard forms, if A ∼ = B, then A is unitarily isomorphic to B. Proof. Let A and B be two II1 factors in standard forms, and assume that θ : A → B is a ∗-isomorphism. Set trB the unique ultraweakly continuous normalized trace over B, and then tr B ◦ θ is the unique ultraweakly continuous normalized trace over A (for details, see [8,27]). This observation immediately gives a unitary operator u from L2 (A) onto L2 (B), which maps each a in A (A ⊆ L2 (A)) to θ (a). Then it is not difficult to check that ua = θ (a)u,

a ∈ A.

Therefore, as standard forms, A is unitarily isomorphic to B.

2

On the other hand, by a careful verifying for the proof of [27, Theorem 10.1.1], the following is true. Lemma 5.8. (See [27].) Suppose M is a Type II1 factor acting on a separable Hilbert space H, and M is infinite. Then there exists a unitary operator U : H → L2 (M) ⊗ l 2 satisfying U A = (A ⊗ I )U , A ∈ M. Therefore, M is unitarily isomorphic to M|L2 (M) ⊗ I. Now we are ready to prove Theorem 5.6. Proof of Theorem 5.6. Both (2) ⇔ (3) and (2) ⇒ (1) are trivial. We will show (1) ⇒ (2) to complete the proof. ∗ To see this, under the assumption of Theorem 5.6, we assume V ∗ (Ω1 ) ∼ = V ∗ (Ω2 ). Since V ∗ (Ω1 ) is a Type II1 factor and its commutant W ∗ (Ω1 ) is infinite, by Lemma 5.8 V ∗ (Ω1 )

unitarily isomorphic

∼ =

M1 ⊗ I,

(5.7)

where M1 denotes the standard form of V ∗ (Ω1 ), i.e. M1 = V ∗ (Ω1 )|L2 (V ∗ (Ω1 )) . Similarly, V ∗ (Ω2 )


∼ =

M2 ⊗ I,

(5.8) ∗

where M2 denotes the standard form of V ∗ (Ω2 ). Since V ∗ (Ω1 ) ∼ = V ∗ (Ω2 ), by (5.7) and (5.8) ∗

M1 ⊗ I ∼ = M2 ⊗ I, and hence


1249

∗

M1 ∼ = M2 . Then by Lemma 5.7, M1 is unitarily isomorphic to M2 . In view of (5.7) and (5.8), V ∗ (Ω1 ) is unitarily isomorphic to V ∗ (Ω2 ). The proof of (1) ⇒ (2) is complete. Notice that for a holomorphic covering map φ : D → Ω, π1 (Ω) ∼ = G(φ), and by Theorem 5.4, ∗ V ∗ (φ) ∼ = L(π1 (Ω)). This observation immediately gives the remaining part. The proof is complete. 2 Theorem 5.4 and Theorem 5.6 are closely related to an unsolved problem in von Neumann algebras ∗

L(Fn ) ∼ = L(Fm )? for n = m and n, m 2. This problem is thus equivalent to the following: Problem. If π1 (Ω1 ) π1 (Ω2 ), then is V ∗ (Ω1 ) unitarily isomorphic to V ∗ (Ω2 ); or equivalently, is W ∗ (Ω1 ) unitarily isomorphic to W ∗ (Ω2 )? To close this section, we review some results in [3] where some von Neumann algebra closely related to V ∗ (φ) was constructed from Mφ , which act on a vector-valued Hardy space. However, the techniques developed in [3] depend more on the method of vector bundle, and less on the structure of the Bergman space L2a (D); and the ideas in this paper are completely different from that in [3]. As done in [3], let U be the group of all unitary operators on the Bergman space, G be a group, and α : G → U be a representation. Now we set G = G(φ), where φ is a holomorphic covering map from D onto a bounded planar domain Ω. Let H denote H 2 (D) ⊗ L2a (D), the space of L2a (D)-valued Hardy space over the unit disk, and Hα denotes its subspace consisting of all functions f in H satisfying

f ◦ ρ(z) = α(ρ) f (z) ,

z ∈ D, ρ ∈ G.

Observe that the vector-valued Hardy subspace Hα is invariant for Mφ , and this subspace depends on G = G(φ), and hence it depends on φ. Now set Tα = Mφ |Hα , and denote by W ∗ (Tα ) the von Neumann algebra generated by Tα . By [3, Theorem 8], W ∗ (Tα )


∼ =

W ∗ (α) ⊗ B l 2 ,

where W ∗ (α) is the von Neumann algebra generated by {α(ρ): ρ ∈ G}. In particular, if we define α(ρ) = Uρ∗ (ρ ∈ G), then W ∗ (α) = V ∗ (Ω). 6. Type II factors and orbifold domains In this section, we first give some examples of orbifold domain. Then using Proposition 4.6, we give the proof of Theorem 1.5. As a simple application, we consider those regular branched covering maps which are also inner functions. Given a planar bounded domain Ω, and a discrete closed subset Σ of Ω, and a function ν : Ω → {1, 2, . . .}, the triple (Ω, Σ, ν) is called an orbifold domain if ν takes the value

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2 on Σ , and 1 on other points. The function ν is called a ramified function and Σ is called the singular locus of Ω. Given an orbifold domain Ω, from the theory of orbifold Riemann surfaces, there exists a holomorphic regular branched covering φ : D → Ω, and is unique up to conformal isomorphism over D such that the local degree of φ at every point z ∈ D equals ν(φ(z)), where the local degree of φ at z ∈ D is the multiplicity of the zero point z for φ(·) − φ(z). Such a map is called the universal covering. For details, see [30, Appendix E], [36, Theorem 2.3] and [9]. The fundamental group π1 (Ω, Σ, ν) of the orbifold Ω is defined to be the deck transformation group of its universal covering map, that is, π1 (Ω, Σ, ν) = G(φ) = {ρ ∈ Aut(D): φ ◦ ρ = φ}, see [31, Chapter 13] for an equivalent definition. A simple example is the orbifold disk (D, {0}, ν(0) = n). Its universal covering map is given 2kπi by φ(z) = zn : D → D, and its deck transformation group is {ρk (z) = e n z: k = 0, 1, . . . , n − 1} ∼ = Zn . To construct more examples, we need the following Seifert–Van Kampen Theorem from algebraic topology, see [9, Corollary 2.3], [17] and also [28, Theorem 6.8]. Theorem 6.1 (Seifert–Van Kampen Theorem). Let O be an orbifold and O1 , O2 ⊂ O two open suborbifolds such that O1 , O2 and O1 ∩ O2 are connected. If O = O1 ∪ O2 , then π1 O is the amalgamated product, π1 O ∼ = π1 O1 ∗Γ π1 O2 where Γ = π1 (O1 ∩ O2 ). In the next example, all orbifolds and suborbifolds are open, and Γ reduces to a trivial group, then π1 O is isomorphic to π1 O1 ∗ π1 O2 , the free product of π1 O1 and π1 O2 . Here we refer to [26, Section 2.9] for a relatively simple definition of free product of groups Gα , α ∈ I. Due to van der Waerden, the free product α Ga of {Gα : α ∈ I} can be regarded as the set of reduced words, by which we mean either the identity e or the words x1 x2 · · · xn which satisfies that xi and xi+1 (1 i n − 1) do not belong to a same Gα and all xi = e. A free group Fk on k generators can always be regarded as the free product of k groups, with each isomorphic to Z.

∗

Example 6.2. Take distinct points in D, Σ = {w1 , w2 , . . . , wn } (1 n < ∞), and give positive integers m1 , m2 , . . . , mn . Define ν(wk ) = mk for k = 1, . . . , n. Applying Seifert–Van Kampen Theorem shows π1 (D, Σ, ν) ∼ = Zm 1 ∗ Zm 2 · · · ∗ Zm n . More generally, put n = ∞, and then by the idea in [30, Problem E-4, p. 258], π1 (D, Σ, ν) ∼ =

∗k Zm . k

The above observations show that for an orbifold domain (D, Σ, υ), π1 (D, Σ, υ) is finite if and only if Σ contains at most one point. Since each holomorphic regular branched covering from D onto D can be realized by a universal covering of some (D, Σ, υ), it is easy to see that if a finite Blaschke product B is a regular branched covering, then B(z) = cϕαk (z) for some α ∈ D and |c| = 1, where


ϕα (z) =

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α−z 1 − αz

is the Möbius map. This is a generalization of Lemma 4.2 in [23]. Before proving our main theorem, we need to establish a lemma, which may be known. Lemma 6.3. For free product of groups, we have the following: (1) The free product G of groups in {Gα : α ∈ I} is an i.c.c. group if at least 3 groups Gα are nontrivial; (2) For two integers m, n 2, if max(m, n) 3, then Zm ∗ Zn is an i.c.c. group. Also, Z ∗ Zn (n 2) is an i.c.c. group; (3) Z2 ∗Z2 is not an i.c.c. group. In fact, it has infinitely many nontrivial finite conjugacy classes. Usually, Z2 ∗ Z2 is called the infinite dihedral group. Proof of Lemma 6.3. The following discussion is based on reduced words. (1) Without loss of generality, let G be the free product of three nontrivial groups G1 , G2 and G3 . For each u ∈ G − {e}, let u = x1 x2 · · · xk (k 1) be the reduced form of u. There must be a group, say G3 , such that x1 ∈ / G3 and xk ∈ / G3 . Pick ρ ∈ G3 and τ ∈ G1 satisfying ρ = e and τ = e, and it is easy to check that ρuρ −1 ,

(τρ)u(τρ)−1 ,

(ρτρ)u(ρτρ)−1 ,

(τρτρ)u(τρτρ)−1 ,

...

is an infinite sequence of words, no two of which are equal. Since these words are in the conjugacy class of u, the arbitrariness of u shows that G is an i.c.c. group. (2) First we prove the former statement. Assume G1 ∼ = Zm and G2 ∼ = Zn with m 3 and n 2, and denote by ρ and τ the generators of G1 and G2 respectively. Now assume u = x1 x2 · · · xk (u = e) is a reduced word. We must show that the conjugacy class of u is infinity. There are several cases under consideration. i) Both x1 and xk are in G1 or G2 . First assume that x1 and xk are in G1 . Then τ uτ −1 ,

(ρτ )u(ρτ )−1 ,

(τρτ )u(τρτ )−1 ,

...

is an infinite sequence whose members are in the conjugacy of u. The case that both x1 and xk are in G2 is similar. ii) There is an i such that x1 = ρ i (1 i m − 1) and xk ∈ G2 . Then it is easy to see that there is an i (1 i m − 1) such that ρ i ρ i = e. Therefore u = ρ i uρ m−i is a word in the conjugacy class of u and it is suffices to consider the conjugacy class of u . This is case i) since both ρ i ρ i and ρ m−i are nontrivial elements in G1 . So the conjugacy class of u is infinite, as desired. iii) There is an i such that xk = ρ i (1 i m − 1) and x1 ∈ G2 . This case is similar to case ii). A similar argument shows that Z ∗ Zn (n 2) is an i.c.c. group. (3) As done in (2), write G1 ∼ = Z2 and G2 ∼ = Z2 , and assume ρ and τ are the generators of G1 and G2 , respectively. We have ρ2 = e

and τ 2 = e,

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from which it is easy to check that for each positive integer n, {(ρτ )n , (τρ)n } is a conjugacy class in G1 ∗ G2 , and no two of which are same. The proof is complete. 2 Now we are ready to establish the following theorem. Theorem 6.4. Let (Ω, ν) be an orbifold domain with the singular locus Σ , and suppose φ : D → (Ω, ν) is a holomorphic universal covering. Then V ∗ (φ) is a Type II1 factor with the following exceptions: (1) Σ is empty, and Ω is conformally isomorphic to one of the disk, annuli or the punctured disk. In this case, φ is a covering map. (2) Σ = {w0 } is a singleton and Ω is conformally isomorphic to the unit disk D. In this case, φ = ψ ◦ ρ n , where ψ is a conformal map from D onto Ω, and ρ ∈ Aut(D), n = ν(w0 ). (3) Σ = {w0 , w1 } with ν(w0 ) = ν(w1 ) = 2 and Ω is conformally isomorphic to the unit disk D. In this case, V ∗ (φ) is a finite von Neumann algebra whose center is infinite dimensional. Whenever V ∗ (φ) is a Type II1 factor, W ∗ (φ) always is a Type II∞ factor. It is clear that V ∗ (φ) is abelian only in situations (1) and (2). This is a clearer version of Proposition 3.8. Proof of Theorem 6.4. For simplicity, we write π1 (Ω, ν) for π1 (Ω, Σ, ν), and so on. We shall first discuss when the fundamental group π1 (Ω, ν) is an i.c.c. group. Recall that π1 (Ω, ν) is defined to be the deck transformation group of φ. For this, it is better to get some instructive ideas of the existence for φ from [30, Problem E-4, p. 258] and [36, pp. 423, 424], where the arguments and ideas give a simple fact: let w be a point in the singular locus Σ, Σ = Σ − {w}, and define ν on Ω such that ν |Σ = ν and ν|Ω−Σ = 1. Then (Ω, ν ) is also an orbifold domain and

π1 (Ω, ν) ∼ = Zν(w) ∗ π1 Ω, ν . Notice that if Σ is empty, then (Ω, ν ) is just a usual domain and π1 (Ω, ν ) is nothing but π1 (Ω). If there are at least k points in Σ , say w1 , . . . , wk , then using the above fact k times shows that there is a ramified function ν such that π1 (Ω, ν) ∼ =

∗i Zν(w ) i

∗ π1 Ω, ν ,

(6.1)

where i ranges from 1 to k. Therefore, if number of points in Σ satisfies Σ 3, then π1 (Ω, ν) is the free product of at least 3 nontrivial groups, and by Theorem 6.4 π1 (Ω, ν) is an i.c.c. group. So it remains to deal with the following three cases: Σ = 0, 1 and 2. Case I: Σ is empty. In this situation, φ is a holomorphic covering map, which has been done by Theorems 4.2 and 3.1, and the only exception is stated as in (1). Case II: Σ = 1. Suppose Σ = {w1 } and n = ν(w1 ), and then (6.1) is reduced to π1 (Ω, ν) ∼ = Zn ∗ π1 (Ω),

n 2.


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In this situation, if π1 (Ω) is trivial, then as shown in Example 6.1, we reach exception as stated as in (2). Otherwise, π1 (Ω) is a free group with m generators (1 m ∞). Therefore π1 (Ω) is isomorphic to the free product of m integer groups Z [26, pp. 87, 88]. Applying Theorem 6.4(1) and (2) shows that π1 (Ω, ν) is an i.c.c. group. Case III: Σ = 2. In this situation, either π1 (Ω) is trivial or not. If π1 (Ω) is nontrivial, then combining Theorem 6.4(1) with (6.1) shows that π1 (Ω, ν) is an i.c.c. group. If π1 (Ω) is trivial, then by Theorem 6.4(3), π1 (Ω, ν) is not an i.c.c. group only if π1 (Ω, ν) = Z2 ∗ Z2 . This happens only in the situation as stated in exception (3). Furthermore, by the proof of Lemma 4.4, V ∗ (φ) is a von Neumann algebra whose center is infinite dimensional. By Proposition 2.6, V ∗ (φ) is finite. Therefore, except for situations (1), (2) and (3) in Theorem 6.4, π1 (Ω, ν) is an i.c.c. group. Since the holomorphic universal covering φ is always a regular branched covering map, the former part of Theorem 6.4 follows from Proposition 4.6. By the latter part of the proof of Theorem 4.2, we see that whenever V ∗ (φ) is a Type II1 factor, W ∗ (φ) always is a Type II∞ factor. The proof is complete. 2 Remark 6.5. By Proposition 2.7 or Proposition 2.8 in [9], the group π1 (Ω, Σ, ν) is isomorphic to

∗

w∈Σ

Zν(w) ∗ π1 (Ω).

Notice that by Lemma 4.1, π1 (Ω) is either trivial or isomorphic to the free product of m integer groups Z (1 m ∞). Below we give an example for situation (3) in Theorem 6.4, which is suggested by Professor W. Qiu. Example 6.6. For fixed r ∈ (0, 1), set Ωr = x + iy:

x2 (r + 1r )2

and Σ = {−1, 1}, and ν(−1) = ν(1) = 2. Consider the Zhukovski function: 1 1 z+ , f (z) = 2 z

+

y2 (r − 1r )2

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