Integral Equations and Operator Theory - Volume 62

Integr. equ. oper. theory 62 (2008), 1–10 c 2008 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-10, published o...

Author: I. Gohberg (Chief Editor)

43 downloads 849 Views 8MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Integr. equ. oper. theory 62 (2008), 1–10 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010001-10, published online July 24, 2008 DOI 10.1007/s00020-008-1599-0

Integral Equations and Operator Theory

Weyl’s Theorem for Algebraically Quasi-class A Operators Il Ju An and Young Min Han Abstract. If T or T ∗ is an algebraically quasi-class A operator acting on an infinite dimensional separable Hilbert space then we prove that Weyl’s theorem holds for f (T ) for every f ∈ H(σ(T )), where H(σ(T )) denotes the set of all analytic functions in an open neighborhood of σ(T ). Moreover, if T ∗ is algebraically quasi-class A then a-Weyl’s theorem holds for f (T ). Also, if T or T ∗ is an algebraically quasi-class A operator then we establish that the spectral mapping theorems for the Weyl spectrum and the essential approximate point spectrum of T for every f ∈ H(σ(T )), respectively. Mathematics Subject Classification (2000). Primary 47A10, 47A53; Secondary 47B20. Keywords. Weyl’s theorem, Browder’s theorem, algebraically quasi-class A operator, a-Weyl’s theorem, a-Browder’s theorem, single valued extension property.

1. Introduction Throughout this note let B(H) and K(H) denote, respectively, the algebra of bounded linear operators and the ideal of compact operators acting on an infinite dimensional separable Hilbert space H. If T ∈ B(H) we shall write N (T ) and R(T ) for the null space and range of T . Also, let α(T ) := dimN (T ), β(T ) := dimN (T ∗ ), and let σ(T ), σa (T ), p0 (T ), and σp (T ) denote the spectrum, approximate point spectrum, the set of poles of the resolvent of T , and point spectrum of T , respectively. For T ∈ B(H), the smallest nonnegative integer p such that N (T p ) = N (T p+1 ) is called the ascent of T and denoted by a(T ). If no such integer exists, we set a(T ) = ∞. The smallest nonnegative integer q such that R(T q ) = R(T q+1 ) is called the descent of T and denoted by d(T ). If no such integer exists, we set This research was supported by the Kyung Hee University Research Fund in 2007 (KHU20071605).

2

An and Han

IEOT

d(T ) = ∞. An operator T ∈ B(H) is called Fredholm if it has closed range, finite dimensional null space, and its range has finite co-dimension. The index of a Fredholm operator T ∈ B(H) is given by i(T ) := α(T ) − β(T ). T ∈ B(H) is called Weyl if it is Fredholm of index zero, and Browder if it is Fredholm of finite ascent and descent: equivalently ([11, Theorem 7.9.3]) if T is Fredholm and T − λ is invertible for sufficiently small λ 6= 0 in C. The essential spectrum σe (T ), the Weyl spectrum σw (T ) and the Browder spectrum σb (T ) of T ∈ B(H) are defined by ([11],[12]) σe (T ) := {λ ∈ C : T − λ is not Fredholm}, σw (T ) := {λ ∈ C : T − λ is not Weyl}, and σb (T ) := {λ ∈ C : T − λ is not Browder}, respectively. Evidently σe (T ) ⊆ σw (T ) ⊆ σb (T ) = σe (T ) ∪ acc σ(T ), where we write acc K for the accumulation points of K ⊆ C. If we write iso K = K \ acc K then we let π00 (T ) := {λ ∈ iso σ(T ) : 0 < α(T − λ) < ∞ }, a π00 (T ) := {λ ∈ iso σa (T ) : 0 < α(T − λ) < ∞ },

and p00 (T ) := σ(T ) \ σb (T ). We say that Weyl’s theorem holds for T ∈ B(H) if σ(T ) \ σw (T ) = π00 (T ),

(1.1)

and that Browder’s theorem holds for T ∈ B(H) if σ(T ) \ σw (T ) = p00 (T ).

(1.2)

We consider the sets Φ+ (H) := {T ∈ B(H) : R(T ) is closed and α(T ) < ∞}, Φ− + (H) := {T ∈ B(H) : T ∈ Φ+ (H) and i(T ) ≤ 0}. By definition, σea (T ) := ∩{σa (T + K) : K ∈ K(H)} is the essential approximate point spectrum, and σab (T ) := ∩{σa (T + K) : T K = KT and K ∈ K(H)} is the Browder essential approximate point spectrum. We say that a-Weyl’s theorem holds for T ∈ B(H) if a σa (T ) \ σea (T ) = π00 (T ),

(1.3)

Vol. 62 (2008)

Weyl’s Theorem for Algebraically Quasi-class A Operators

3

and that a-Browder’s theorem holds for T ∈ B(H) if σea (T ) = σab (T ).

(1.4)

It is known ([7],[12],[17]) that if T ∈ B(H) then we have : a-Weyl’s theorem =⇒ Weyl’s theorem =⇒ Browder’s theorem, a-Weyl’s theorem =⇒ a-Browder’s theorem =⇒ Browder’s theorem. In [19] H. Weyl proved that (1.1) holds for hermitian operators. Weyl’s theorem has been extended from hermitian operators to hyponormal and Toeplitz operators ([4]), and to several classes of operators including seminormal operators ([2],[3]). Also, V. Rakoˇcević showed that (1.3) holds for cohyponormal operators ([17]). Recently, B.P. Duggal, I.H. Jeon, and I.H. Kim [9] showed that Weyl’s theorem holds for quasi-class A operators. In this paper, we extend this result to algebraically quasi-class A operators using the local spectral theory.

2. Weyl’s theorem for algebraically quasi-class A operators We say that T ∈ B(H) is a class A operator if |T |2 ≤ |T 2 |, and T is called a quasi-class A operator if T ∗ |T |2 T ≤ T ∗ |T 2 |T. We say that T ∈ B(H) is an algebraically quasi-class A operator if there exists a nonconstant complex polynomial p such that p(T ) is a quasi-class A operator. An operator T ∈ B(H) is called isoloid if every isolated point of σ(T ) is an eigenvalue of T . An operator T ∈ B(H) is called normaloid if r(T ) = ||T ||, where r(T ) is the spectral radius of T . X ∈ B(H) is called a quasiaffinity if it has trivial kernel and dense range. S ∈ B(H) is said to be a quasiaffine transform of T ∈ B(H) (notation: S ≺ T ) if there is a quasiaffinity X ∈ B(H) such that XS = T X. If both S ≺ T and T ≺ S then we say that S and T are quasisimilar. We say that T ∈ B(H) has the single valued extension property (SVEP) if for every open set U of C the only analytic function f : U −→ H which satisfies the equation (T − λ)f (λ) = 0 is the constant function f ≡ 0 on U . In general, the following implications hold: hyponormal =⇒ class A =⇒ quasi-class A =⇒ algebraically quasi-class A. The following facts follow from the above definition and some well known facts about quasi-class A operators. (i) If T ∈ B(H) is algebraically quasi-class A then so is T − λ for each λ ∈ C.

4

An and Han

IEOT

(ii) If T ∈ B(H) is algebraically quasi-class A and M is a closed T -invariant subspace of H then T |M is algebraically quasi-class A. Before we state our main theorem, we need several preliminary results. We begin with the following lemma. The following Lemma 2.1 was proved in ([13]). We give a simple proof here. Lemma 2.1. Suppose T is a quasi-class A operator and it does not have dense range. Then A B T = on H = T (H) ⊕ N (T ∗ ), 0 0 where A := T |T (H) is a class A operator. Proof. Since T does not have dense range, we can represent T as the upper triangular matrix A B T = on H = T (H) ⊕ N (T ∗ ). 0 0 We shall show that A is a class A operator. Since T ∗ |T 2 |T ≥ T ∗ |T |2 T , T ∗ (|T 2 | − |T |2 )T ≥ 0. So < T ∗ (|T 2 | − |T |2 )T x, x >≥ 0 for each x ∈ H, and hence < (|T 2 | − |T |2 )T x, T x >≥ 0 for all x ∈ H. Therefore A is a class A operator. Lemma 2.2. Suppose T is a quasinilpotent algebraically quasi-class A operator. Then T is nilpotent. Proof. We first assume that T is a quasi-class A operator. We consider two cases: Case I: Suppose T has dense range. Then it is a class A operator. But every class A operator is paranormal, hence by [6, Lemma 2.1] T is nilpotent. Case II: Suppose T does not have dense range. Then by Lemma 2.1 we can represent T as the upper triangular matrix A B T = on H = T (H) ⊕ N (T ∗ ), 0 0 where A := T |T (H) is class A. Since T is quasinilpotent, σ(T ) = {0}. But σ(T ) = σ(A) ∪ {0}, hence σ(A) = {0}. Since A is a class A operator, A = 0. Therefore T is nilpotent. Thus if T is a quasinilpotent quasi-class A operator then it is nilpotent. Now we suppose that T is an algebraically quasi-class A operator. Then there exists a nonconstant polynomial p such that p(T ) is a quasi-class A operator. If p(T ) has dense range then p(T ) is a class A operator. Therefore T is an algebraically class A operator, and so it is an algebraically paranormal operator. It follows from [6, Lemma 2.2] that it is nilpotent. If p(T ) does not have a dense range then by Lemma 2.1 we can represent p(T ) as the upper triangular matrix A B p(T ) = on H = p(T )(H) ⊕ N (p(T )∗ ), 0 0 where A := p(T )|p(T )(H) is class A. Since σ(T ) = {0}, σ(p(T )) = p(σ(T )) = {p(0)}. Therefore p(T ) − p(0) is quasinilpotent. But σ(p(T )) = σ(A) ∪ {0}, hence

Vol. 62 (2008)


5

σ(A) ∪ {0} = {p(0)}. So p(0) = 0, and hence p(T ) is quasinilpotent. Since p(T ) is a quasi-class A operator, by the previous argument p(T ) is nilpotent. On the other hand, since p(0) = 0, p(z) = cz m (z − λ1 )(z − λ2 ) · · · (z − λn ) for some natural number m. Therefore p(T ) = cT m (T − λ1 )(T − λ2 ) · · · (T − λn ). Since p(T ) is nilpotent, T is nilpotent. This completes the proof. An operator T ∈ B(H) is called polaroid if iso σ(T ) ⊂ p0 (T ). In general, if T is polaroid then it is isoloid. However, the converse is not true. Consider the following example. Let T ∈ B(l2 ) be defined by 1 1 T (x1 , x2 , x3 , . . .) = ( x2 , x3 , . . .). 2 3 Then T is a compact quasinilpotent operator with α(T ) = 1, and so T is isoloid. However, since T does not have finite ascent, T is not polaroid. In [9] they showed that every quasi-class A operator is isoloid. We can prove more: Lemma 2.3. Suppose T is an algebraically quasi-class A operator. Then T is polaroid. Proof. Suppose T is an algebraically quasi-class A operator. Then p(T ) is quasiclass A for some nonconstant polynomial p. Let λ ∈ iso σ(T ). Using the spectral R 1 (µ − T )−1 dµ, where D is a closed disk of center λ which projection P := 2πi ∂D contains no other points of σ(T ), we can represent T as the direct sum T1 0 T = , where σ(T1 ) = {λ} and σ(T2 ) = σ(T ) \ {λ}. 0 T2 Since T1 is algebraically quasi-class A, T1 − λ is also algebraically quasi-class A. But σ(T1 − λ) = {0}, it follows from Lemma 2.2 that T1 − λ is nilpotent. Therefore T1 − λ has finite ascent and descent. On the other hand, since T2 − λ is invertible, clearly it has finite ascent and descent. Therefore T − λ has finite ascent and descent, and hence λ is a pole of the resolvent of T . Thus λ ∈ iso σ(T ) implies λ ∈ p0 (T ), and so iso σ(T ) ⊂ p0 (T ). Hence T is polaroid. In the following theorem, recall that H(σ(T )) is the space of functions analytic in an open neighborhood of σ(T ). Theorem 2.4. Suppose T or T ∗ is an algebraically quasi-class A operator. Then Weyl’s theorem holds for f (T ) for every f ∈ H(σ(T )). Proof. Suppose that T is algebraically quasi-class A. We first show that Weyl’s theorem holds for T . Let λ ∈ σ(T ) \ σw (T ). Then T − λ is Weyl but not invertible. We claim that λ ∈ ∂σ(T ). Assume to the contrary that λ is an interior point of σ(T ). Then there exists a neighborhood U of λ such that dim N (T − µ) > 0 for all µ ∈ U . It follows from [10, Theorem 10] that T does not have SVEP. On the other hand, since p(T ) is quasi-class A for some nonconstant polynomial p, it follows from [9, Lemma 1.5] and [14, Proposition 1.8] that p(T ) has SVEP. Hence by [15, Theorem 3.3.9], T has SVEP. This is a contradiction. Therefore

6

An and Han

IEOT

λ ∈ ∂σ(T ) \ σw (T ), and it follows from the punctured neighborhood theorem that λ ∈ π00 (TR ). Conversely, suppose that λ ∈ π00 (T ). Using the spectral projection 1 (µ − T )−1 dµ, where D is a closed disk of center λ which contains no P := 2πi ∂D other points of σ(T ), we can represent T as the direct sum T1 0 T = , where σ(T1 ) = {λ} and σ(T2 ) = σ(T ) \ {λ}. 0 T2 Since σ(T1 ) = {λ}, T1 − λ is quasinilpotent. But T is algebraically quasi-class A, hence T1 is also algebraically quasi-class A. It follows from Lemma 2.2 that T1 − λ is nilpotent. Since λ ∈ π00 (T ), T1 − λ is a finite dimensional operator, so T1 − λ is Weyl. Since T2 − λ is invertible, T − λ is Weyl. Thus Weyl’s theorem holds for T . Now we claim that f (σw (T )) = σw (f (T )) for all f ∈ H(σ(T )). Let f ∈ H(σ(T )). Since σw (f (T )) ⊆ f (σw (T )) with no other restriction on T , it suffices to show that f (σw (T )) ⊆ σw (f (T )). Suppose that λ ∈ / σw (f (T )). Then f (T ) − λ is Weyl and f (T ) − λ = c(T − α1 )(T − α2 ) · · · (T − αn )g(T ),

(2.4)

where c, α1 , α2 , . . . , αn ∈ C and g(T ) is invertible. Since the operators in the right side of (2.4) commute, every T − αi is Fredholm. Since T is algebraically quasiclass A, T has SVEP. It follows from [1, Theorem 2.6] that i(T − αi ) ≤ 0 for each i = 1, 2, . . . , n. Therefore λ ∈ / f (σw (T )), and hence f (σw (T )) = σw (f (T )). Now recall ([16, Lemma]) that if T is isoloid then f σ(T ) \ π00 (T ) = σ(f (T )) \ π00 (f (T )) for every f ∈ H(σ(T )). Since T is isoloid by Lemma 2.3 and Weyl’s theorem holds for T , σ(f (T )) \ π00 (f (T )) = f σ(T ) \ π00 (T ) = f (σw (T )) = σw (f (T )), which implies that Weyl’s theorem holds for f (T ). Now suppose that T ∗ is algebraically quasi-class A. We first show that Weyl’s theorem holds for T . Suppose that λ ∈ σ(T ) \ σw (T ). Observe that σ(T ∗ ) = σ(T ) and σw (T ∗ ) = σw (T ). So λ ∈ σ(T ∗ ) \ σw (T ∗ ), and hence λ ∈ π00 (T ∗ ). Therefore λ is an isolated point of σ(T ), and so λ ∈ π00 (T ). Conversely, suppose that λ ∈ π00 (T ). Then λ is an isolated point of σ(T ) and 0 < α(T − λ) < ∞. Since λ is an isolated point of σ(T ∗ ) and T ∗ is algebraically quasi-class A, it follows from Lemma 2.3 that λ ∈ p0 (T ∗ ). Therefore there exists a natural number n0 such that n0 = a(T ∗ − λ) = d(T ∗ − λ). Hence we have H = N ((T ∗ − λ)n0 ) ⊕ R((T ∗ − λ)n0 ) and R((T ∗ − λ)n0 ) is closed. Therefore R((T − λ)n0 ) is closed and H = N ((T ∗ − λ)n0 )⊥ ⊕ R((T ∗ − λ)n0 )⊥ = R((T − λ)n0 ) ⊕ N ((T − λ)n0 ). So λ ∈ p0 (T ), and hence T − λ is Weyl. Consequently, λ ∈ σ(T ) \ σw (T ). Thus Weyl’s theorem holds for T . Now we show that f (σw (T )) = σw (f (T )) for each f ∈ H(σ(T )). Let f ∈ H(σ(T )). To show that f (σw (T )) = σw (f (T )) it is sufficient to show that f (σw (T )) ⊆ σw (f (T )). Suppose that λ ∈ / σw (f (T )). Then f (T ) − λ is Weyl. Since T ∗ is algebraically quasi-class A, it has SVEP. It follows from [1, Theorem 2.8]

Vol. 62 (2008)


7

that i(T − αi ) ≥ 0 for each i = 1, 2, . . . , n. Since 0≤

n X

i(T − αi ) = i(f (T ) − λ) = 0,

i=1

T − αi is Weyl for each i = 1, 2, . . . , n. Hence λ 6∈ f (σw (T )), and so f (σw (T )) ⊆ σw (f (T )). Thus f (σw (T )) = σw (f (T )) for each f ∈ H(σ(T )). Since Weyl’s theorem holds for T and T is isoloid, Weyl’s theorem holds for f (T ) for every f ∈ H(σ(T )). This completes the proof. From the proof of the Theorem 2.4, we obtain the following useful consequence. Corollary 2.5. Suppose T or T ∗ is an algebraically quasi-class A operator. Then σw (f (T )) = f (σw (T )) for every f ∈ H(σ(T )).

3. a-Weyl’s theorem for algebraically quasi-class A operators Let T ∈ B(H). It is well known that the inclusion σea (f (T )) ⊆ f (σea (T )) holds for every f ∈ H(σ(T )) with no restriction on T ([18, Theorem 3.3]). The next theorem shows that the spectral mapping theorem holds for the essential approximate point spectrum for algebraically quasi-class A operators. Theorem 3.1. Suppose T or T ∗ is an algebraically quasi-class A operator. Then σea (f (T )) = f (σea (T )) for every f ∈ H(σ(T )). Proof. Assume first that T is algebraically quasi-class A and let f ∈ H(σ(T )). It suffices to show that f (σea (T )) ⊆ σea (f (T )). Suppose that λ ∈ / σea (f (T )). Then f (T ) − λ ∈ Φ− + (H) and f (T ) − λ = c(T − α1 )(T − α2 ) · · · (T − αn )g(T ),

(3.1)

where c, α1 , α2 , . . . , αn ∈ C, and g(T ) is invertible. Since T is algebraically quasiclass A, it has SVEP. It follows from [1, Theorem 2.6] that i(T − αi ) ≤ 0 for each i = 1, 2, . . . , n. Therefore λ 6∈ f (σea (T )), and hence σea (f (T )) = f (σea (T )). Suppose now that T ∗ is algebraically quasi-class A. Then T ∗ has SVEP, and so by [1, Theorem 2.8] i(T − αi ) ≥ 0 for each i = 1, 2, . . . , n. Since n X 0≤ i(T − αi ) = i(f (T ) − λ) ≤ 0, i=1

T − αi is Weyl for each i = 1, 2, . . . , n. Hence λ 6∈ f (σea (T )), and so σea (f (T )) = f (σea (T )). This completes the proof. Theorem 3.2. Suppose T is an algebraically quasi-class A operator and that S ≺ T . Then a-Browder’s theorem holds for f (S) for every f ∈ H(σ(S)).

8

An and Han

IEOT

Proof. Suppose T is algebraically quasi-class A and that S ≺ T . We first show that S has SVEP. Let U be any open set and let f : U −→ H be any analytic function such that (S − λ)f (λ) = 0 for all λ ∈ U . Since S ≺ T , there exists a quasiaffinity X such that XS = T X. So X(S − λ) = (T − λ)X for all λ ∈ U . Since (S − λ)f (λ) = 0 for all λ ∈ U , 0 = X(S − λ)f (λ) = (T − λ)Xf (λ) for all λ ∈ U . But T is algebraically quasi-class A, hence T has SVEP. Therefore Xf (λ) = 0 for all λ ∈ U . Since X is a quasiaffinity, f (λ) = 0 for all λ ∈ U . Therefore S has SVEP. Now we show that a-Browder’s theorem holds for S. It is well known that σea (S) ⊆ σab (S). Conversely, suppose that λ ∈ σa (S) \ σea (S). Then S − λ ∈ Φ− + (H) and − S − λ is not bounded below. Since S has SVEP and S − λ ∈ Φ+ (H), it follows from [1, Theorem 2.6] that S − λ has finite ascent. Therefore by [18, Theorem 2.1], λ ∈ σa (S) \ σab (S). Thus a-Browder’s theorem holds for S. Let f ∈ H(σ(S)) be arbitrary. Since S has SVEP, it follows from the proof of Theorem 3.1 that σea (f (S)) = f (σea (S)). Therefore σab (f (S)) = f (σab (S)) = f (σea (S)) = σea (f (S)), and hence a-Browder’s theorem holds for f (S).

An operator T ∈ B(H) is called a-isoloid if every isolated point of σa (T ) is an eigenvalue of T . Clearly, if T is a-isoloid then it is isoloid. However, the converse is not true. Consider the following example: let T = U ⊕ Q, where U is the unilateral forward shift on l2 and Q is an injective quasinilpotent operator on l2 , respectively. Then σ(T ) = {λ ∈ C : |λ| ≤ 1} and σa (T ) = {λ ∈ C : |λ| = 1} ∪ {0}. Therefore T is isoloid but not a-isoloid. Suppose that T ∗ is an algebraically quasi-class A operator. Then we can prove more: Theorem 3.3. Suppose T ∗ is an algebraically quasi-class A operator. Then a-Weyl’s theorem holds for f (T ) for every f ∈ H(σ(T )). Proof. Suppose T ∗ is an algebraically quasi-class A operator. We first show that a-Weyl’s theorem holds for T . Suppose that λ ∈ σa (T ) \ σea (T ). Then T − λ is upper semi-Fredholm and i(T − λ) ≤ 0. Since T ∗ is algebraically quasi-class A, T ∗ has SVEP. Therefore by [1, Theorem 2.8] i(T − λ) ≥ 0, and hence T − λ is Weyl. Since T ∗ has SVEP, it follows from [10, Corollary 7] that σ(T ) = σa (T ). Also, a since Weyl’s theorem holds for T by Theorem 2.4, λ ∈ π00 (T ). a Conversely, suppose that λ ∈ π00 (T ). Since T ∗ has SVEP, it follows from [10, Corollary 7] that σ(T ) = σa (T ). Therefore λ is an isolated point of σ(T ), and hence λ is an isolated point of σ(T ∗ ). But T ∗ is algebraically quasi-class A, hence by Lemma 2.3 that λ ∈ p0 (T ∗ ). Therefore there exists a natural number n0 such that n0 = a(T ∗ −λ) = d(T ∗ −λ). Hence we have H = N ((T ∗ −λ)n0 )⊕R((T ∗ −λ)n0 ) and R((T ∗ − λ)n0 ) is closed. Therefore R((T − λ)n0 ) is closed and H = N ((T ∗ − λ)n0 )⊥ ⊕ R((T ∗ − λ)n0 )⊥ = R((T − λ)n0 ) ⊕ N ((T − λ)n0 ). So λ ∈ p0 (T ), and hence T − λ is Weyl. So λ ∈ σa (T ) \ σea (T ). Thus a-Weyl’s theorem holds for T .

Vol. 62 (2008)


9

Now we show that T is a-isoloid. Let λ be an isolated point of σa (T ). Since T ∗ has SVEP, λ is an isolated point of σ(T ). But T ∗ is polaroid, hence T is also polaroid. Therefore it is isoloid, and hence λ ∈ σp (T ). Thus T is a-isoloid. Finally, we shall show that a-Weyl’s theorem holds for f (T ) for every f ∈ H(σ(T )). Let f ∈ H(σ(T )). Since a-Weyl’s theorem holds for T , it satisfies a-Browder’s theorem. Therefore σea (T ) = σab (T ). It follows from Theorem 3.1 that σab (f (T )) = f (σab (T )) = f (σea (T )) = σea (f (T )), and hence a-Browder’s theorem holds for f (T ). So σa (f (T )) \ σea (f (T )) ⊂ a a π00 (f (T )). Conversely, suppose that λ ∈ π00 (f (T )). Then λ is an isolated point of σa (f (T )) and 0 < α(f (T ) − λ) < ∞. Since λ is an isolated point of f (σa (T )), if αi ∈ σa (T ) then αi is an isolated point of σa (T ) by (3.1). Since T is a-isoloid, 0 < α(T − αi ) < ∞ for each i = 1, 2, . . . , n. Since a-Weyl’s theorem holds for T , T − αi is upper semi-Fredholm and i(T − αi ) ≤ 0 for each Pn i = 1, 2, . . . , n. Therefore f (T ) − λ is upper semi-Fredholm and i(f (T ) − λ) = i=1 i(T − αi ) ≤ 0. Hence λ ∈ σa (f (T )) \ σea (f (T )), and so a-Weyl’s theorem holds for f (T ) for each f ∈ H(σ(T )). This completes the proof.

References [1] P. Aiena and O. Monsalve, Operators which do not have the single valued extension property, J. Math. Anal. Appl. 250 (2000), 435–448. [2] S.K. Berberian, An extension of Weyl’s theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273–279. [3] S.K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 (1970), 529–544. [4] L.A. Coburn, Weyl’s theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288. [5] R.E. Curto and Y.M. Han, Weyl’s theorem, a-Weyl’s theorem, and local spectral theory, J. London Math. Soc. (2) 67 (2003), 499–509. [6] R.E. Curto and Y.M. Han, Weyl’s theorem holds for algebraically paranormal operators, Integral Equations Operator Theory 47 (2003), 307–314. [7] S.V. Djordjević and Y.M. Han, Browder’s theorems and spectral continuity, Glasgow Math. J. 42 (2000), 479–486. [8] B.P. Duggal, Polaroid operators satisfying Weyl’s theorem, Linear Algebra Appl. 414 (2006), 271–277. [9] B.P. Duggal, In Ho Jeon, and In Hyoun Kim On Weyl’s theorem for quasi-class A operators, J. Korean Math. Soc. 43 (2006), 899–909. [10] J.K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), 61–69. [11] R.E. Harte, Invertibility and Singularity for Bounded Linear Operators, Dekker, New York, 1988. [12] R.E. Harte and W.Y. Lee, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. 349 (1997), 2115–2124.

10

An and Han

IEOT

[13] In Ho Jeon and In Hyoun Kim, On operators satisfying T ∗ |T 2 |T ≥ T ∗ |T |2 T , Linear Algebra Appl. 418 (2006), 854–862. [14] K.B. Laursen, Operators with finite ascent, Pacific J. Math. 152 (1992), 323–336. [15] K.B. Laursen and M.M. Neumann, An Introduction to Local Spectral Theory, London Mathematical Society Monographs New Series 20, Clarendon Press, Oxford, 2000. [16] W.Y. Lee and S.H. Lee, A spectral mapping theorem for the Weyl spectrum, Glasgow Math. J. 38 (1996), 61–64. [17] V. Rakoˇcević, On the essential approximate point spectrum II, Mat. Vesnik. 36 (1984), 89–97. [18] V. Rakoˇcević, Approximate point spectrum and commuting compact perturbations, Glasgow Math. J. 28 (1986), 193–198. ¨ [19] H. Weyl, Uber beschr¨ ankte quadratische Formen, deren Differenz vollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392. Il Ju An and Young Min Han Department of Mathematics and Research Institute for Basic Sciences Kyung Hee University Seoul 130-701 Korea e-mail: [email protected] e-mail: [email protected] Submitted: October 26, 2006. Revised: February 25, 2008.

Integr. equ. oper. theory 62 (2008), 11–28 0378-620X/010011-18, DOI 10.1007/s00020-008-1613-6 c 2008 Birkh¨ auser Verlag Basel/Switzerland


Metric Properties of Projections in Semi-Hilbertian Spaces M. Laura Arias, Gustavo Corach and M. Celeste Gonzalez To our teacher Mischa Cotlar, in memoriam

Abstract. Several results on norms of projections on a Hilbert space H are extended for the operator seminorm defined by a positive semidefinite operator A ∈ L(H)+ . Mathematics Subject Classification (2000). Primary 46C05; Secondary 47A05, 47A30. Keywords. Projections, semi inner products, compatibility, symmetrizable.

1. Introduction In this paper, H denotes a Hilbert space, L(H) is the algebra of bounded linear operators on H and Q is the subset of L(H) of all projections (i.e. idempotents). Given a closed subspace S of H, QS denotes the subset of Q of all projections with image S. The topology and differential geometry of Q and P = {P ∈Q : P ∗ = P } have been studied in detail in many places [3], [13], [9], [15], [29], [30], [32], [37], [38] and [42]. This paper is devoted to the study of several metrical properties of Q and QS when an additional seminorm is considered on H. Let PS ∈QS denote the unique Hermitian projection with image S. The following properties are well known: (I) For all 0 = Q ∈ Q it holds Q = 1 if and only if Q∗ = Q; (II) For every non trivial Q ∈ Q it holds Q = I − Q; (III) Given closed subspaces S and T of H it holds PS − PT ≤ QS − QT for every QS ∈ QS and QT ∈ QT ; (IV) For all closed subspaces S and T of H it holds PS − PT ≤ 1. Equality holds if and only if PS and PT commute; The authors were supported in part by UBACYT I030, ANPCYT PICT 03-09521, PIP 2188/00.

12

Arias, Corach and Gonzalez

IEOT

(V) For all closed subspaces S and T of H it holds PS − PT = max { PS (I − PT ), PT (I − PS ) }; 1 if θ ∈ [0, π/2] is the angle such that (VI) For every Q ∈Q it holds Q = sin θ cos θ = sup{| ξ, η | : ξ ∈ R(Q), η ∈ N (Q) and ξ = η = 1}. Here R(Q) is the image of the projection Q and N (Q) is its nullspace. Proofs of properties (I), (II) and (IV) can be found in textbooks like [8] and [25]. A proof of property (V) can be found in the book by Akhiezer and Glazman [1]. Property (III) is due to T. Kato [[25], Th. 6.35, p. 58] (see also M. Mbektha [[33], 1.10]). Property (VI) is due to V. Ljance [28]. Proofs of it can be found in the monograph of Gokhberg and Krein [22] and in the papers by V. Ptak [35], J. Steinberg [40], D. Buckholtz [6] and I. Ipsen and C. Meyer [24] (for finite dimensional spaces). The main goal of this paper is to study these properties if we consider an additional seminorm . A , defined by a positive semidefinite operator A ∈ L(H) by ξ2A = Aξ, ξ, ξ ∈ H, and we replace the operator norm in formulas (I) to (VI) by the quantity T A = sup{T ξA : ξA = 1}. Of course, many difficulties arise. For instance, it may happen that T A = +∞ for some T ∈ L(H). Besides, there is no obvious choice for an adjoint operation defined by A. In order to describe our results, we need to introduce a certain relationship between positive operators and closed subspaces called compatibility in the recent literature. We say that a positive semidefinite operator A on H and a closed subspace S of H are compatible if there exists a projection Q ∈ QS such that AQ is Hermitian (or symmetric). This means that Qξ, ηA = ξ, QηA for all ξ, η ∈ H where ξ, ηA = Aξ, η. In this case, it can be proved that H = S +(AS)⊥ and the projection PA,S onto S with nullspace (AS)⊥ S ∩N (A) satisfies APA,S = ∗ A. This operator, PA,S , has similar, but not identical, metric properties like PA,S the orthogonal projection PS . For example, if the pair (A, S) is compatible, then for every ξ ∈ H it holds that (I − PA,S )ξA = dA (ξ, S) = inf{ξ − ηA : η ∈ S}. See [12] for its proof. Under convenient hypothesis of compatibility we are able to prove properties analogous to (I)-(VI) for the operator seminorm . A and a convenient adjoint operation. The subject of operators which are symmetric under a certain inner product is quite old. Papers by M.G. Krein [26] in 1937 and W. T. Reid [36] in 1951, with references to earlier works, studied many spectral properties of the so-called symmetrizable operators. Later, P. Lax [27] and J. Dieudonné [17] studied conditions for the symmetrizability of operators. In more recent times, Z. Sebestyén [39], B.A. Barnes [4], S. Hassi, Z. Sebestyén and H. de Snoo [23] and P. Cojuhari and A. Gheondea [7] have found many interesting results and applications of various notions of symmetrizability. The contents of the paper are the following. In section 2 we collect some facts about Moore-Penrose pseudoinverses, compatibility between positive operators and closed subspaces, and a brief description of a result by R. G. Douglas [19]

Vol. 62 (2008) Metric Properties of Projections in Semi-Hilbertian Spaces

13

which plays a relevant role in this paper. Douglas theorem (sometimes called range inclusion theorem) gives necessary and sufficient conditions for the existence and uniqueness of solution for equations of the type AX = T A, with an additional condition on the range of X. In section 3 we explore the existence of A-adjoints for projections. If a projection Q admits an A-adjoint, then we define Q as the unique solution of the problem AX = Q∗ A, R(X) ⊆ R(A). Properties of Q are described. Sections 4 and 5 contain the main results of the paper, i.e., the extension of properties (I) to (VI) above, as follows (I ) every projection Q such that AQ = Q∗ A = 0 satisfies QA = 1; (II ) equality QA = I − QA holds for any projection Q such that R(Q) ∩ R(A) = {0} and R(I − Q) ∩ R(A) = {0}; (III ) if (A, S), (A, T ) are compatible pairs, then for every QS ∈ QS and QT ∈ QT which admit adjoint respect to , A it holds PA,S − PA,T A ≤ QS − QT A ;

(III ) if S = S1 + S2 and T = T1 + T2 , where S1 , T1 ⊆ R(A) and S2 , T2 ⊆ N (A) and the pairs (A, S1 ) and (A, T1 ) are compatible, then, for every QS ∈ QS ∩ LA (H) and QT ∈ QT ∩ LA (H) it holds PA,S − PA,T A ≤ QS − QT A , A

where L (H) = {T ∈ L(H) : T A < ∞}; (IV ) if A is compatible with the closed subspaces S and T , then PA,S −PA,T A ≤ 1 and equality holds if PA,S commutes with PA,T ; (V ) if A is compatible with the closed subspaces S and T , then PA,S −PA,T A = max{ PA,S (I − PA,T )A , PA,T (I − PA,S )A }; (VI ) if (A, S) and (A, T ) are compatible pairs and S ∩ R(A) = {0}, then it holds 1 , where θA ∈ [0, π/2] is the angle such that cos θA = QS//T A = sin θA sup{| ξ, ηA | : ξ ∈ S, η ∈ T and ξA = ηA = 1}.

2. Preliminaries Throughout H denotes a complex Hilbert space. L(H) is the space of bounded linear operators on H, L(H)+ denotes the cone of all positive operators of L(H), i.e., L(H)+ = {A ∈ L(H) : Aη, η ≥ 0 for all η ∈ H}, Gl(H) is the group of invertible operators of L(H) and Gl(H)+ = Gl(H) ∩ L(H)+ . For every T ∈ L(H), its range is denoted by R(T ), its nullspace by N (T ) and its adjoint by T ∗ . S and T are closed subspaces of H and S T = S ∩ T ⊥ . In this paper, given closed subspaces S, T of H, by L(S, T ) we denote the subspace {T ∈ L(H) : T (S ⊥ ) = . {0} and T (S) ⊆ T }. If H is decomposed as a direct sum H = S + T , where S

14


IEOT

and T are closed subspaces of H, then the unique projection with range S and nullspace T is denoted by QS//T . 2.1. Moore-Penrose pseudoinverse Recall that given T ∈ L(H), the Moore-Penrose inverse of T , denoted by T † , is defined as the unique linear extension of T˜ −1 to D(T † ) := R(T ) + R(T )⊥ with N (T † ) = R(T )⊥ , where T˜ is the isomorphism T |N (T )⊥ : N (T )⊥ −→ R(T ). It holds that T † is the unique solution of the four “Moore-Penrose equations”: T XT = T, XT X = X, XT = PN (T )⊥ and T X = PR(T ) |D(T † ) . T † has closed graph and T † is bounded if and only if R(T ) is closed. Proofs of these facts can be found in many places, e.g. the books [34], [5] and [20]. Observe that, since T † has closed graph, then for every B ∈ L(H) such that R(B) ⊆ D(T † ) it holds that T † B is bounded. In the next proposition we collect without proof some properties of T † that we will need in this work. Proposition 2.1. Let T ∈ L(H). 1. If T = T ∗ , then (T † )∗ = T † . 2. If T ∈ L(H)+ , then T † = (T 1/2 )† (T 1/2 )† . A bounded linear densely defined operator T can be uniquely extended to L(H); its unique extension will be denoted by T . Clearly, T = T . It can be checked ∗ that T = (T ∗ )∗ . Then, as a consequence, T ∗ = T = T ∗ and if T = R∗ R, then ∗ T = R R. 2.2. A-selfadjoint projections and compatibility Any A ∈ L(H)+ defines a positive semidefinite sesquilinear form: , A : H × H → C, ξ, ηA = Aξ, η . 1/2

By . A we denote the seminorm induced by , A , i.e., ξA = ξ, ξA . Observe that ξA = 0 if and only if ξ ∈ N (A). Then . A is a norm if and only if A is an injective operator. Moreover, , A , induces a seminorm on a subset of L(H). Namely, given T ∈ L(H), if there exists a constant c > 0 such that T ωA ≤ cωA for every ω ∈ R(A) it holds T ωA < ∞. ωA ω∈R(A)

T A = sup

ω=0

It is straightforward that T A = sup{| T ξ, ηA | : ξ, η ∈ H and ξA ≤ 1 ηA ≤ 1}. From now on we will denote LA (H) = {T ∈ L(H) : T A < ∞}. It can be seen that LA (H) is not a subalgebra of L(H). In [4] it is proved that if A ∈ L(H)+ is injective, then T ∈ LA (H) if and only if A1/2 T A−1/2 is bounded. In


15

the next proposition we extend this result for a not necessary injective operator A ∈ L(H)+ . Before that we state the next theorem of R. G. Douglas (for its proof see [19] or [21]) which will be used frequently during these notes. Theorem (Douglas). Let A, B ∈ L(H). The following conditions are equivalent: 1. R(B) ⊆ R(A). 2. There exists a positive number λ such that BB ∗ ≤ λAA∗ . 3. There exists C ∈ L(H) such that AC = B. If one of these conditions holds there exists an unique operator D ∈ L(H) such that AD = B and R(D) ⊆ R(A∗ ). Furthermore, N (D) = N (B). Such D is called the reduced solution or Douglas solution of AX = B. Note that if the equation AX = B has solution, then A† B is the reduced solution. Indeed, since R(B) ⊆ R(A) ⊆ D(A† ), A† B ∈ L(H). Moreover, AA† B = PR(A) |D(A† ) B = B and R(A† B) ⊆ R(A). Proposition 2.2. Let A ∈ L(H)+ and T ∈ L(H). Then the following conditions are equivalent: 1. T ∈ LA (H). 2. A1/2 T (A1/2 )† is a bounded linear operator. 3. R(A1/2 T ∗ A1/2 ) ⊆ R(A). Moreover, if one of these conditions holds, then T A = A1/2 T (A1/2 )† . Proof. 1⇒2: If T ∈ LA (H), then there exists c > 0 such that T ωA ≤ cωA for every ω ∈ R(A). Then, for every ξ ∈ D((A1/2 )† ) it holds that A1/2 T (A1/2 )† ξ

= T (A1/2 )† ξA ≤ T A(A1/2 )† ξA ≤ T Aξ.

Therefore, A1/2 T (A1/2 )† is bounded and A1/2 T (A1/2 )† ≤ T A. 2⇒1: Let A1/2 T (A1/2 )† be a bounded linear operator. Then, for every ξ ∈ R(A) we have that T ξA

= T PR(A) ξA = A1/2 T (A1/2 )† A1/2 ξ ≤ A1/2 T (A1/2 )† A1/2 ξ = A1/2 T (A1/2 )† ξA ,

i.e., item 2. holds. Moreover, T A ≤ A1/2 T (A1/2 )† . 2⇔3: It is clear that T ξA ≤ cξA for every ξ ∈ R(A) if and only if A1/2 T ξ ≤ cA1/2 ξ for every ξ ∈ R(A1/2 ), i.e. if and only if A1/2 T A1/2 η ≤ cAη for every η ∈ H. Now, by Douglas theorem, this is equivalent to R(A1/2 T ∗ A1/2 ) ⊆ R(A). By Proposition 2.2, if A ∈ L(H)+ has closed range, then LA (H) = L(H) because (A1/2 )† is bounded. But, as the next example shows, if A has not closed range, then LA (H) L(H) .

16


IEOT

Example 1. Let A ∈ L(H)+ with non closed range and let µ ∈ R(A1/2 ) \ R(A). Then, there exists η ∈ R(A) \ R(A1/2 ) such that µ = A1/2 η. Now, let ξ ∈ R(A1/2 ) and S a closed subspace of H such that H = span{ξ} + span{η} + S. Then, define T : H → H by T ξ = η, T η = η and T (S) = {0}. Thus, T ∈ L(H). Moreover, T ∈Q. / LA (H). In fact, µ = A1/2 η = A1/2 T ξ ∈ R(A1/2 T A1/2 ) Then, T ∗ ∈Q but T ∗ ∈ and µ ∈ / R(A). So, R(A1/2 T A1/2 ) ⊆ R(A), i.e., T ∗ ∈ / LA (H) by Proposition 2.2. A bounded linear operator W ∈ L(H) is called an A-adjoint of T ∈ L(H) if T ξ, ηA = ξ, W ηA

for every ξ, η ∈ H,

or, which is equivalent, if W satisfies the equation AW = T ∗ A. The operator T is said A-selfadjoint if AT = T ∗ A. The existence of an A-adjoint operator is not guaranteed. In fact, by Douglas theorem, T ∈ L(H) admits an A-adjoint operator if and only if R(T ∗ A) ⊆ R(A). We shall denote by LA (H) the subalgebra of L(H) consisting of such operators, i.e, LA (H) = {T ∈ L(H) : R(T ∗ A) ⊆ R(A)}. Again, by Douglas theorem, it is easy to see that LA1/2 (H) = {T ∈ L(H) : ∃ c > 0 T ξA ≤ cξA ∀ ξ ∈ H}. The inclusions LA (H) ⊆ LA1/2 (H) ⊆ LA (H) hold. The first of them was proved in Theorem 5.1 of [23], the second one follows from Proposition 2.2. Observe that these inclusions assure that T A is finite for every T which admits an A-adjoint. If T ∈ LA (H), then there exists a distinguished A-adjoint operator of T , namely, the reduced solution of equation AX = T ∗ A. We denote this operator by T . Therefore T = A† T ∗ A and its main properties are AT = T ∗ A, R(T ) ⊆ R(A) and N (T ) = N (T ∗ A). Observe that if W is an A-adjoint of T , then T = PR(A) W . In [2] we have studied some properties of the operation which are relevant for studying Apartial isometries, i.e. operator which behave as partial isometries with respect to , A . We add now a few properties. Proposition 2.3. Let A ∈ L(H)+ and T ∈ LA (H). Then 1. 2. 3. 4.

1/2

T A = T A = T T A . W A = T A for every W ∈ L(H) which is an A-adjoint of T . If W ∈ LA (H), then T W A = W T A. T ≤ W for every W ∈ L(H) which is an A-adjoint of T . Nevertheless, T is not in general the unique A-adjoint of T that realizes the minimal norm.


17

Proof. ∗ 1. It is easy to check that A1/2 T (A1/2 )† = A1/2 (A† T ∗ A)(A1/2 )† . Then T A

∗

= A1/2 T (A1/2 )† = A1/2 T (A1/2 )† = A1/2 T (A1/2 )† = A1/2 (A† T ∗ A)(A1/2 )† = A1/2 (A† T ∗ A)(A1/2 )† = A1/2 T (A1/2 )† = T A .

On the other hand, T T A

=

A1/2 T T (A1/2 )† = A1/2 A† T ∗ AT (A1/2 )†

=

(A1/2 )† T ∗ AT (A1/2 )† = (A1/2 )† T ∗ AT (A1/2 )†

=

(A1/2 T (A1/2 )† ) (A1/2 T (A1/2 )† ) = A1/2 T (A1/2 )† 2

=

A1/2 T (A1/2 )† 2 = T 2A.

∗

2. If W ∈ L(H) is an A-adjoint operator of T , then W = T + Z, where Z is a solution of the homogeneous equation AX = 0. Then W A = A1/2 W (A1/2 )† = A1/2 (T + Z)(A1/2 )† = A1/2 T (A1/2 )† = T A . 3. Note that T W A

=

(T W ) A = W T A = A1/2 W T (A1/2 )†

=

A1/2 W (A1/2 )† A1/2 T (A1/2 )†

=

A1/2 T (A1/2 )† A1/2 W (A1/2 )†

=

T W A = (W T ) A

=

W T A.

4. Let W ∈ L(H) be an A-adjoint operator of T . Then W = T + Z, where AZ = 0. Let ξ ∈ H with ξ = 1. Since R(T ) ⊆ R(A) and R(Z) ⊆ N (A) we get 2 ξ2 + Zξ2 . Then T ξ2 ≤ W W ξ2 = T ξ and so T ≤ W . Now, let 1 0 2 0 ∈ M2 (R). It is easy to check that T A= ∈ M2 (R)+ and T = 1 1 0 0 1 0 admits A-adjoint operators and that T = . Furthermore, observe that 0 0 the identity matrix I is an A-adjoint of T , T = I = 1 and T = I. Given A ∈ L(H)+ and a closed subspace S, we denote by P(A, S) the set of A-selfadjoint projections with fixed range S: P(A, S) = {Q ∈ QS : AQ = Q∗ A}. With a fixed A ∈ L(H)+ the set P(A, S) can be empty, or have one element (for example if A ∈ Gl(H)+ ) or have infinitely many elements. If P(A, S) = ∅, then the pair (A, S) is said to be compatible. For a fuller treatment on the theory of compatibility see [10], [11], [13] and [31]. Given Q ∈ QS , Q is A-selfadjoint if and only if Qξ, ξA ≥ 0 for all ξ ∈ H. If the pair (A, S) is compatible, the unique element in P(A, S) with nullspace (AS)⊥ N , where N = N (A) ∩ S, is denoted by PA,S . This element has minimal norm in P (A, S). Nevertheless, PA,S is not in

18


IEOT

general the unique Q ∈ P(A, S) that realizes the minimal norm. See [10] Theorem 3.5 for its proof. The next proposition provides a parametrization of P(A, S) and it expresses the element PA,S as the solution of certain Douglas-type equations. For its proof the reader is referred to [11] (section 3), [31] (section 6). Proposition 2.4. Let A ∈ L(H)+ such that the pair (A, S) is compatible and N = N (A) ∩ S. If Q is the reduced solution of the equation (PS APS )X = PS A, then 1. Q = PA,SN . 2. PA,S = PA,SN + PN . 3. P(A, S) is an affine manifold that can be parametrized as P(A, S) = PA,S + L(S ⊥ , N ). In particular, if N = {0}, then P(A, S) = {PA,S }.

3. The A-adjoint operation on projections In this paper, we are mainly interested in how the A-adjoint operation acts on A-adjointable projections. We first notice that there is no obvious notion of selfadjointness: an operator T such that AT = T ∗ A could be named A-Hermitian, but also an operator T ∈ LA (H) such that T = T. We discuss this problem focusing in the set of projections. For this, we consider the following subsets of Q: W = {Q ∈ Q ∩ LA (H) : Q = Q} X = {Q ∈ Q ∩ LA (H) : AQ = Q∗ A} Y = {Q ∈ Q ∩ LA (H) : (Q )2 = Q } Z = Q ∩ LA (H). Proposition 3.1. The next inclusions hold: W X Y = Z. Proof. Let Q ∈ W, then Q = Q. Thus, Q∗ A = AQ = AQ and so Q ∈ X . On 1 1 1 1 the other hand, consider A = ∈ M2 (C)+ and Q = . Then 1 1 0 0 it is easy to check that Q ∈ X , but Q ∈ / W. It is immediate that X ⊆ Z. In 1 1 order to see that this is a strict inclusion consider A = ∈ M2 (C)+ 1 2 1 2 and Q = . Since A is invertible then R(Q∗ A) ⊆ R(A), i.e., Q ∈ Z, 0 0 but Q ∈ / X . Finally, let Q ∈ Z, i.e, Q2 = Q and there exists Q . Let us show that that (Q )2 = Q . Indeed, (Q )2 = A† Q∗ AA† Q∗ A = A† Q∗ PR(A) |D(A† ) Q∗ A = A† (Q∗ )2 A = A† Q∗ A = Q . i.e., Q ∈ Y. The other inclusion is trivial. Proposition 3.2. If Q ∈ P(A, S), then: 1. Q = Q Q = PR(A) Q = PR(A) PA,S is a projection. 2. I − Q ∈ P(A, N (PS A)).


19

Proof. 1. It is sufficient to prove that Q Q is the reduced solution of the equation AX = Q∗ A. In fact, AQ Q = Q∗ AQ = (Q∗ )2 A = Q∗ A and R(Q Q) ⊆ R(Q ) ⊆ R(A). Therefore, Q Q = Q . In order to see that Q = PR(A) PA,S , observe that, by Proposition 2.4, we get Q = PA,S + Z, where Z ∈ L(S ⊥ , N ). Therefore, Q = A† Q∗ A = PR(A) Q = PR(A) (PA,S + Z) = PR(A) PA,S . 2. If Q ∈ P(A, S), then Q is also an A-selfadjoint projection. On the other hand, R(I −Q ) = N (Q ) = N (Q∗ A) = R(AQ)⊥ = R(APS )⊥ = N (PS A). Then I −Q ∈ P(A, N (PS A)). Remarks 3.3. Considering the subsets defined before, it is clear that if the pair (A, S) is compatible, then P(A, S) ⊆ X . On the other hand, P(A, S) ∩ W = ∅ if and only if S ⊆ R(A) and the pair (A, S) is compatible. In fact, if there exists Q ∈ P(A, S) ∩ W, then Q = Q and so S = R(Q) = R(Q ) ⊆ R(A). Conversely, if S ⊆ R(A) and (A, S) is compatible, then PA,S = PR(A) PA,S = PA,S , i.e. PA,S ∈ P(A, S) ∩ W.

4. Identities on the seminorm of projections In this section we generalize several identities on the norm of projections when the seminorm induced by A ∈ L(H)+ is considered. We start by establishing an useful relationship between orthogonal projections and A-selfadjoint projections. Proposition 4.1. Let A ∈ L(H)+ and Q ∈ L(H) such that S = R(Q) is a closed subspace of R(A). 1. If Q ∈ QS ∩ LA (H), then A1/2 Q(A1/2 )† is a projection. 2. The following conditions are equivalent: (a) Q ∈ P(A, S). (b) Q ∈ LA (H) and A1/2 Q(A1/2 )† is an orthogonal projection. If one of these conditions holds, then QA = A1/2 Q(A1/2 )† = 1. Proof. 1. Since Q ∈ QS and S ⊆ R(A) then A1/2 Q(A1/2 )† is a projection. Futhermore, as Q ∈ LA (H), by Proposition 2.2, it holds that A1/2 Q(A1/2 )† is bounded. Therefore A1/2 Q(A1/2 )† is a projection of L(H). 2. Let Q ∈ P(A, S). By item 1. it holds that A1/2 Q(A1/2 )† is a projection. In order to see that (A1/2 Q(A1/2 )† )∗ = A1/2 Q(A1/2 )† , observe that (A1/2 Q(A1/2 )† )∗ = (A1/2 Q(A1/2 )† )∗ ⊃ (A1/2 )† Q∗ A1/2 . Furthermore, since D((A1/2 )† Q∗ A1/2 ) = H, we obtain that (A1/2 Q(A1/2 )† )∗ = (A1/2 )† Q∗ A1/2 = (A1/2 )† Q∗ A1/2 |D((A1/2 )† ) = A1/2 Q(A1/2 )† where the last equality holds since AQ = Q∗ A. Conversely, let A1/2 Q(A1/2 )† be an orthogonal projection. First, it is shown that that Q is a projection. Since, A1/2 Q(A1/2 )† is a projection, then A1/2 Q(A1/2 )†

20


IEOT

is also a projection. Thus, A1/2 Q(A1/2 )† = (A1/2 Q(A1/2 )† )2 = A1/2 Q2 (A1/2 )† . Then, Q(A1/2 )† = Q2 (A1/2 )† , i.e., (Q2 −Q)(A1/2 )† = 0. Hence, R(A) ⊆ N (Q2 −Q), or which is the same R((Q∗ )2 −Q∗ ) ⊆ N (A). Thus, R(((Q∗ )2 −Q∗ )A) ⊆ N (A). On the other hand, since R(Q∗ A) ⊆ R(A), it is easy to prove that R((Q∗ )2 A) ⊆ R(A). So, R(((Q∗ )2 − Q∗ )A) ⊆ R(A). Then, ((Q∗ )2 − Q∗ )A = 0, i.e., AQ2 = AQ and so Q2 = Q. It only remains to show that Q is A-selfadjoint. Now, as A1/2 Q(A1/2 )† is selfadjoint, it holds A1/2 Q(A1/2 )† = (A1/2 Q(A1/2 )† )∗ = (A1/2 Q(A1/2 )† )∗ = (A1/2 )† Q∗ A1/2 . Hence, A1/2 Q(A1/2 )† = (A1/2 )† Q∗ A1/2 |D((A1/2 )† ) and as a consequence, AQPR(A) = PR(A) |D((A1/2 )† ) Q∗ A = Q∗ A. Now, taking adjoints we get Q∗ A = AQ. Hence Q ∈ P(A, S). The equality QA = A1/2 Q(A1/2 )† follows by Proposition 2.2. For the seminorm A , it is not true, in general, that 1 ≤ QA for every Q ∈QS . See example 2 below. Proposition 4.2. Let A ∈ L(H)+ . If S ∩ R(A) = {0}, then 1 ≤ QA for every Q ∈ QS . Proof. If Q ∈ / LA (H), then the assertion is trivial. Now, suppose Q ∈ LA (H). Let A1/2 Qξ A1/2 Q(A1/2 )† η = = 0 = ξ ∈ S ∩ R(A) and η = A1/2 ξ. Then, we get η A1/2 ξ A1/2 ξ = 1. Therefore, QA = A1/2 Q(A1/2 )† ≥ 1. A1/2 ξ In what follows, given A in L(H)+ we shall say that a projection Q is non-trivial for A if AQ = 0. Note that if Q ∈ P(A, S), then QA is finite. Moreover, in the next proposition we show that if Q ∈ P(A, S) is non-trivial for A, then QA = 1. Proposition 4.3. Let A ∈ L(H)+ . If Q ∈ QS is non-trivial for A, then the following conditions are equivalent: 1. Q ∈ P(A, S) (i.e. Q is A-selfadjoint ). 2. QA = 1 and Q ∈ LA (H). Proof. 1 ⇒ 2. If Q ∈ P(A, S), then, by Proposition 3.2, Q Q is a projection. In addition, R(Q Q) ⊆ R(A). Then applying Proposition 4.1 we deduce that A1/2 Q Q(A1/2 )† is an orthogonal projection. Moreover, since Q is non-trivial, R(Q) ⊆ N (A) and so A1/2 Q Q(A1/2 )† = 0. Thus, applying Proposition 2.3, Q2A = Q QA = A1/2 Q Q(A1/2 )† 2 = A1/2 Q Q(A1/2 )† 2 = 1. 2 ⇒ 1. As R(Q∗ A) ⊆ R(A) then Q is a projection whose range is contained in R(A). Then, (A1/2 Q (A1/2 )† )2 = A1/2 Q (A1/2 )† and so A1/2 Q (A1/2 )† is a projection. In addition, as 1 = QA = Q A = A1/2 Q (A1/2 )† , it follows that A1/2 Q (A1/2 )† is an orthogonal projection. On the other hand, since Q = A† Q∗ A we get that A1/2 Q (A1/2 )† = (A1/2 )† Q∗ A1/2 |D((A1/2 )† ) is an orthogonal


21

projection. Hence, it holds (A1/2 )† Q∗ A1/2 |D((A1/2 )† ) = ((A1/2 )† Q∗ A1/2 |D((A1/2 )† ) )∗ and ((A1/2 )† Q∗ A1/2 |D((A1/2 )† ) )∗ ⊃ A1/2 Q(A1/2 )† . As a consequence, we have that (A1/2 )† Q∗ A1/2 |D((A1/2 )† ) = A1/2 Q(A1/2 )† and so A1/2 Q(A1/2 )† is an orthogonal projection. Thus A1/2 Q(A1/2 )† = (A1/2 Q(A1/2 )† )∗ ⊃ (A1/2 )† Q∗ A1/2 . Moreover, since D((A1/2 )† Q∗ A1/2 ) = H then A1/2 Q(A1/2 )† = (A1/2 )† Q∗ A1/2 . In particular, A1/2 Q(A1/2 )† = (A1/2 )† Q∗ A1/2 |D((A1/2 )† ) . So AQ(A1/2 )† = Q∗ A1/2 |D((A1/2 )† ) and then AQ = Q∗ A. Thus Q ∈ P(A, S). Corollary 4.4. Let A ∈ L(H)+ and (A, S) be a compatible pair. If S ∩ R(A) = {0}, then, for every QS ∈ QS it holds PA,S A ≤ QS A .

(4.1)

Proof. Note that PA,S A = 1. Therefore, the assertion follows from Proposition 4.2. In [[25], Th. 6.35, p. 58] T. Kato proved that PS − PT ≤ Q1 − Q2 for every Q1 ∈ QS and Q2 ∈ QT (see also M. Mbekhta [[33], 1.10]) . We shall generalize this property for A-selfadjoint projections and the seminorm induced by A ∈ L(H)+ in three different manners. In Proposition 4.5 the inequality is proved for every QS , QT ∈ LA (H). In order to obtain this inequality for every QS , QT ∈ Q new hypotheses on the subspaces S and T are required (Proposition 4.6, Corollary 4.7). The proof of the next proposition follows the same lines that the proof of [33], Proposition 1.10. Proposition 4.5. Let A ∈ L(H)+ and (A, S), (A, T ) be compatible pairs. Then, for every QS ∈ QS ∩ LA (H) and QT ∈ QT ∩ LA (H) it holds PA,S − PA,T A ≤ QS − QT A . Proof. First observe that QS PA,S = PA,S , PA,S QS = QS , QT PA,T = PA,T and PA,T QT = QT . From this it holds that (I − QS )(PA,S − PA,T ) = (QS − QT )PA,T , (PA,S − PA,T )QS = (I − PA,T )(QS − QT ) and as consequence ((PA,S − PA,T )QS ) = ((I − PA,T )(QS − QT )) . On the other hand, simple computations show that ((I − PA,T )(QS − QT )) = (QS − QT )(I − PA,T ) and ((PA,S − PA,T )QS ) = QS (PA,S − PA,T ). Now, if ξ ∈ H, then it is easy to check that ξ2A + (QS − QS )ξ2A = (I − QS )ξ2A + QS ξ2A .

22


IEOT

Therefore, if η ∈ R(A) and we define ξ = (PA,S − PA,T )η: (PA,S − PA,T )η2A

≤ = = ≤ =

(PA,S − PA,T )η2A + (QS − QS )(PA,S − PA,T )η2A

(I − QS )(PA,S − PA,T )η2A + QS (PA,S − PA,T )η2A (QS − QT )PA,T η2A + (QS − QT )(I − PA,T )η2A QS − QT 2A (PA,T η2A + (I − PA,T )η2A )

QS − QT 2A η2A .

So, PA,S − PA,T A ≤ QS − QT A .

Proposition 4.6. Let A ∈ L(H)+ and S, T ⊆ R(A). If the pairs (A, S) and (A, T ) are compatible, then, for every QS ∈ QS ∩ LA (H) and QT ∈ QT ∩ LA (H) it holds PA,S − PA,T A ≤ QS − QT A .

(4.2)

Proof. Since the subspaces S, T ⊆ R(A), it holds that Q1 = A1/2 QS (A1/2 )† and Q2 = A1/2 QT (A1/2 )† are projections with the same range as A1/2 PA,S (A1/2 )† and A1/2 PA,T (A1/2 )† , respectively. On the other hand, by Proposition 4.1, it holds that A1/2 PA,S (A1/2 )† and A1/2 PA,T (A1/2 )† are orthogonal projections. Therefore, PA,S − PA,T A

= A1/2 (PA,S − PA,T )(A1/2 )† = A1/2 PA,S (A1/2 )† − A1/2 PA,T (A1/2 )† ≤ A1/2 QS (A1/2 )† − A1/2 QT (A1/2 )† = A1/2 QS (A1/2 )† − A1/2 QT (A1/2 )† = QS − QT A

where the inequality holds by [[25], p. 58].

Corollary 4.7. Let A ∈ L(H)+ and S, T ⊆ H such that S = S1 + S2 and T = T1 + T2 , where S1 , T1 ⊆ R(A) and S2 , T2 ⊆ N (A). If the pairs (A, S1 ) and (A, T1 ) are compatible, then, for every QS ∈ QS ∩ LA (H) and QT ∈ QT ∩ LA (H) it holds PA,S − PA,T A ≤ QS − QT A . Proof. Observe that S1 and S2 are orthogonal subspaces, then every projection QS can be decomposed as QS1 + QS2 where QS1 = PS1 QS and QS2 = PS2 QS . Furthermore, since S2 ⊆ N (A) then PA,S = PA,S1 + PS2 . Then, PA,S − PA,T A

= A1/2 (PA,S1 − PA,T1 )(A1/2 )† = A1/2 PA,S1 (A1/2 )† − A1/2 PA,T1 (A1/2 )† ≤ A1/2 QS1 (A1/2 )† − A1/2 QT1 (A1/2 )† = A1/2 QS1 (A1/2 )† − A1/2 QT1 (A1/2 )† = A1/2 (QS1 + QS2 )(A1/2 )† − A1/2 (QT1 + QT2 )(A1/2 )† = QS − QT A .


23

As the next example shows, a naive extension of Kato’s theorem is false. Our results 4.5, 4.6 and 4.7 offer different additional hypothesis which guarantee the conclusion. Example 2. Consider H = R2 , S = span{(1, 1)}, T = span{(−1, 2)} and A = 2 1 ∈ L(R2 )+ . Therefore R(A) = span{(2, 1)} and S does not satisfy 1 1/2 −ξ −1/2(ξ + 1) the condition of Corollary 4.7. Moreover, QT = ,ξ ∈ R 2ξ ξ+1 1/2(1 + ξ) 1/2(1 − ξ) and QS = , ξ ∈ R . It is easy to check that PA,S = − ξ) 1/2(1 + ξ) 1/2(1 2/3 1/3 1/5 −2/5 0 1 and PA,T = . Now, if we take QS = 2/3 1/3 −2/5 4/5 0 1 0 −1/2 and QT = , then QS does not admit an A-adjoint operator, PA,S − 0 1 PA,T A = 1 and QS A = QS − QT A = 0.6. The following lemma shows that in Corollary 4.4, Proposition 4.5, Corollary 4.7 and Proposition 4.10, the elements PA,S and PA,T can be replaced for any element of P(A, S) and P (A, T ) respectively. Lemma 4.8. Let A ∈ L(H)+ . If (A, S) and (A, T ) are compatible pairs, then Q1 − Q2 A = PA,S − PA,T A for every Q1 ∈ P(A, S) and Q2 ∈ P(A, T ). Proof. By Propositions 2.3 and 3.2 it holds that Q1 − Q2 A = Q1 − Q2 A = PR(A) PA,S − PR(A) PA,T A = PA,S − PA,T A . Given a non trivial projection Q in L(H), i.e., one which is different from 0 and I, it holds Q = I −Q. In [41] different proofs of this fact are collected. In the next proposition we generalize this identity for the seminorm induced by A ∈ L(H)+ . The proof we present is similar to the one due to Krainer presented in [41]. Proposition 4.9. Let A ∈ L(H)+ . Therefore, for every Q ∈ QS such that R(Q) ∩ R(A) = {0} and R(I − Q) ∩ R(A) = {0} it holds QA = I − QA . Proof. Observe that by Proposition 4.2, the conditions R(Q) ∩ R(A) = {0} and R(I − Q) ∩ R(A) = {0} imply that QA ≥ 1 and I − QA ≥ 1. Let ξ ∈ H such that ξA = 1. Define η = Qξ and µ = (I − Q)ξ. Then ξ = η + µ. Let us show that QξA ≤ I − QA . If η ∈ N (A), then QξA = 0 and so the inequality holds. If µ ∈ N (A), then QξA = 1 and so the inequality holds. ηA A ˜ = µ µ. Consider η, µ ∈ / N (A) and define ω = η˜ + µ ˜ where η˜ = µ ηA η and µ A 2 2 2 2 2 2 Then ωA = ˜ η A +˜ µA +2Re ˜ η, µ ˜ A = ηA +µA +2Re η, µA = ξA = 1. Therefore, QξA = ηA = ˜ µA = (I − Q)ωA ≤ I − QA . Thus, QA ≤ I − QA . The other inequality holds by symmetry.

24


IEOT

The conditions R(Q) ∩ R(A) = {0} and R(I − Q) ∩ R(A) = {0} in the above Proposition are necessary. Indeed, if Q = PN (A) , then I − Q = PR(A) and so QA = 0 and I − QA = 1. In [1] § 34, properties (IV) and (V) enunciated in the introduction are proved. They where first proved by M. G. Krein, M. A. Krasnoselski and B. Sz.-Nagy. We extend now these facts for A-selfadjoint projections and the operator seminorm induced by A, with convenient compatibility hypothesis. Proposition 4.10. Let A ∈ L(H)+ such that the pairs (A, S) and (A, T ) are compatible. Then: (a) PA,S − PA,T A ≤ 1; (b) If PA,S and PA,T commute, then PA,S − PA,T A = 1; (c) PA,S − PA,T A = max { PA,S (I − PA,T )A , PA,T (I − PA,S )A }. is an A-selfadjoint projection. FurProof. By Proposition 3.1, the element PA,S ) ⊆ R(A). Therefore, by Proposition 4.1, we get that P1 = thermore, R(PA,S A1/2 PA,S (A1/2 )† is an orthogonal projection. Analogously, P2 = A1/2 PA,T (A1/2 )† is an orthogonal projection. By the above remarks,

PA,S − PA,T A

= PA,S − PA,T A = A1/2 (PA,S − PA,T )(A1/2 )† (A1/2 )† − A1/2 PA,T (A1/2 )† = A1/2 PA,S

= P1 − P2 and so, by (IV), PA,S − PA,T A ≤ 1; this proves (a). It is easy to check that if PA,S and PAT commute, then P1 and P2 commute. Therefore, applying (IV), PA,S − PA,T A = P1 − P2 = 1, which proves (b). For the proof of (c) observe that PA,S (I − PA,T )A

=

(I − PA,T ) PA,S A = (PR(A) − PA,T )PA,S A

=

(I − PA,T )PA,S A = A1/2 (I − PA,T )PA,S (A1/2 )†

=

)PA,S (A1/2 )† = (I − P2 )P1 A1/2 (I − PA,T

=

P1 (I − P2 ).

Analogously, PA,T (I − PA,S )A = P2 (I − P1 ). On the other hand, PA,S − PA,T A = P1 − P2 , by the proof of (b). Then the assertion follows applying (V).

5. Angles and seminorm of projections In [28], V. Ljance proved that if H is decomposed as H = S + T , then the norm of the projection QS//T equals 1/ sin θ, where θ ∈ [0, π/2] is the angle between


25

the subspaces S and T introduced by Dixmier in [18]. Proof of this theorem can be found in the papers by Ptak [35], Steinberg [40], Buckholtz [6] and Ipsen and Meyer [24] (for finite dimensional spaces). As a final result, we extend Ljance’s theorem for the A-seminorm, with a convenient definition of angle between subspaces depending on the semi-inner product , A . First, recall that given two closed subspaces S and T of H the Dixmier’s angle between them is the angle θ(S, T ) ∈ [0, π2 ] whose cosine is defined by cos θ(S, T ) = sup{| ξ, η | : ξ ∈ S, η ∈ T and ξ ≤ 1 η ≤ 1}. Note that, even though if S and T are not closed subspaces, then the angle between them can be also defined as above. Moreover, it holds cos θ(S, T ) = cos θ(S, T ). It is well known that cos θ(S, T ) = PS PT (see [16]). Definition 5.1. Let A ∈ L(H)+ . The A-angle between two closed subspaces S and T is the angle θA (S, T ) ∈ [0, π2 ] whose cosine is defined by cos θA (S, T ) = sup{| ξ, ηA | : ξ ∈ S, η ∈ T and ξA ≤ 1 ηA ≤ 1}. Observe that 0 ≤ cos θA (S, T ) ≤ 1. Furthermore, it holds that cos θA (S, T ) = cos θ(A1/2 (S), A1/2 (T )). Proposition 5.2. Let A ∈ L(H)+ . If (A, S) and (A, T ) are compatible pairs, then cos θA (S, T ) = PA,S PA,T A . Proof. cos θA (S, T ) = = = =

sup{| ξ, ηA | : ξ ∈ S, η ∈ T and ξA ≤ 1 ηA ≤ 1} sup{| PA,S ξ, PA,T ηA | : ξ, η ∈ H and ξA ≤ 1 ηA ≤ 1} sup{| ξ, PA,S PA,T ηA | : ξ, η ∈ H and ξA ≤ q ηA ≤ 1}

PA,S PA,T A .

.

Proposition 5.3. Let A ∈ L(H)+ and S, T closed subspaces of H such that S + T = H. If (A, S) and (A, T ) are compatible pairs and S ∩ R(A) = {0}, then for Q = QS//T it holds QA = (1 − PA,T PA,S 2A )−1/2 . Proof. Let ξ ∈ H. Then ξ = PA,T ξ + (I − PA,T )ξ, so Qξ = Q(I − PA,T )ξ and (I − PA,T )ξA ≤ ξA . Therefore, as R(I − PA,T ) = N (PA,T ) = T ⊥A N , where N = T ∩N (A), then QA = Q|T ⊥A N A . Now, consider ξ ∈ (T ⊥A N )∩R(A). Thus PA,T Qξ = PA,T ξ + PA,T (Qξ − ξ) = Qξ − ξ and as a consequence Qξ2A = ξ2A + Qξ − ξ2A = ξ2A + PA,T PA,S Qξ2A . Note that, without loss of generality, ξ2A PA,T PA,S Qξ2A we can consider Qξ ∈ R(A). Then we get that 1 = + and 2 QξA Qξ2A from this −1/2 PA,T PA,S Qξ2A QξA 1− = . Qξ2A ξA

26


IEOT

Now, since QA = Q|T ⊥A N A and PA,T PA,S A = PA,T PA,S |S A the assertion follows. .

Corollary 5.4. Let A ∈ L(H)+ and S, T closed subspaces of H such that S +T = H. If (A, S) and (A, T ) are compatible pairs and S ∩R(A) = {0}, then for every QS//T it holds 1 . QS//T A = sin θA (T , S) The following example shows that the condition S ∩ R(A) = {0} in Proposition 5.3 is not superfluous. 2 1 0 1 2 2 + Example 3. Let H = R , A = ∈ L(R ) and Q = . 1 1/2 0 1 Then S = R(Q) = span{(1, 1)} andT = N (Q) = span{(1, 0)}. Furthermore, 2/3 1/3 1 1/2 and PA,S = . Now, PA,T PA,S A = 1 and PA,T = 2/3 1/3 0 0 QA = 0.6. Acknowledgment We thank the referee for a careful reading, and many useful comments which greatly improved the final form of the paper.

References [1] N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space. Vol. 1, Ungar, New York, 1961. [2] M. L. Arias, G. Corach and M. C. Gonzalez, Partial isometries in semi-Hilbertian spaces. Linear Algebra Appl. 428 (2008), 1460–1475. [3] B. Aupetit, E. Jr Makai, M. Mbekhta and J. Zem´ anek, The connected components of the idempotents in the Calkin algebra, and their liftings, Operator theory and Banach algebras. (Rabat, 1999), Theta, Bucharest, (2003), 23–30. [4] B. A. Barnes, The spectral properties of certain linear operators and their extensions. Proc. Am. Math. Soc. 128 (2000), 1371–1375. [5] A. Ben-Israel and T. N. E. Greville, Generalized inverses. Theory and applications. 2nd Edition, Springer-Verlag, New York, 2003. [6] D. Buckholtz, Hilbert space idempotents and involutions. Proc. Amer. Math. Soc. 128 (2000), no. 5, 1415–1418. [7] P. Cojuhari and A. Gheondea, On lifting of operators to Hilbert spaces induced by positive seladjoint operators. J. Math. Anal. Appl. 304 (2005), 584–598. [8] J. B. Conway, A course in functional analysis. Springer-Verlag, New York, 1985. [9] G. Corach, H. Porta and L. Recht, Differential geometry of systems of projections in Banach algebras. Pacific Journal of Mathematics 143 (1990), 209–228. [10] G. Corach, A. Maestripieri and D. Stojanoff, Schur complements and oblique projections. Acta Sci. Math. 67 (2001), 439–459.


27

[11] G. Corach, A. Maestripieri and D. Stojanoff, Generalized Schur complements and oblique projections. Linear Algebra and its Applications 341 (2002), 259–272. [12] G. Corach, A. L. Maestripieri, D. Stojanoff, Oblique projections and abstract splines. Journal of Aproximation Theory, 117, 2 (2002), 189–206 [13] G. Corach, A. Maestripieri and D. Stojanoff, A classification of projectors. Topological algebras, their applications and related topics, Banach Center Publications 67, Polish Acad. Sci., Warsaw, (2005), 145–160. [14] G. Corach, A. Maestripieri, D. Stojanoff, Projections in operator ranges. Proc. Amer. Math. Soc. 134 (2006), no 3, 765–778. [15] G. Corach, H. Porta, L. Recht, The geometry of spaces of projections in C*-algebras. Advances in Mathematics 101 (1993), 59–77. [16] F. Deutsch, The angle between subspaces in Hilbert space. in Approximation theory, wavelets and applications, S. P. Singh, editor, Kluwer (1995), 107–130. [17] J. Dieudonné, Quasi-hermitian operators. Proc. Inter. Symp. Linear Algebra, Jerusalem (1961), 115–122. ´ [18] J. Dixmier, Etude sur less variétés et le opératerus de Julia avec quelques applications. Bull. Soc. Math. France, 77 (1949) 11–101. [19] R. G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert spaces. Proc. Am. Math. Soc. 17 (1966) 413–416. [20] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer, Dordrecht, 1996. [21] P. A. Fillmore and J. P. Williams, On operator ranges. Advances in Mathemathics 7 (1971), 254–281. [22] I.C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I. 1969. [23] S. Hassi, Z. Sebestyén and H. S. V. De Snoo, On the nonnegative of operator products. Acta Math. Hungar. 109 (2005), 1–14. [24] I. Ipsen and C. Meyer, The angle between complementary subspaces. Amer. Math. Monthly 102 (1995), no 10, 904–911. [25] T. Kato, Perturbation theory of linear operators. Springer, New York, (First edition), 1966. [26] M. G. Krein, Compact linear operators on functional spaces with two norms. Integr. equ. oper. theory 30 (1998), 140–162 (translation from the Ukranian of a paper published in 1937). [27] P. D. Lax, Symmetrizable linear transformations. Comm. Pure Appl. Math. 7 (1954), 633–647. ` Ljance, Certain properties of idempotent operators. (Russian) Teoret. Prikl. [28] V. E. Mat. Vyp. 1 1958 16–22. [29] S. Maeda, On arcs in the space of projections of a C ∗ -algebra. Math Japon. 21 (1976), no 4, 371–374. [30] S. Maeda, Probability measures on projections in von Neumann algebras. Rev. Math. Phys. 1 (1989), no 2–3, 235–290.

28


IEOT

[31] A. Maestripieri and F. Martinez Per´ıas, Decomposition of selfadjoint projections in Krein spaces. Acta Sci. Math. (Szeged) 72 (2006), no. 3–4, 611–638. [32] E. Jr. Makai and J. Zem´ anek, On polynomial connections between projections. Linear Algebra Appl. 126 (1989), 91–94. [33] M. Mbektha, Résolvant généralisé et théorie spectrale. Journal of Operator Theory 21 (1989), 69–105. [34] M. Z. Nashed, Inner, outer, and generalized inverses in Banach and Hilbert spaces. Numer. Funct. Anal. Optim. 9 (1987), 261–325. [35] V. Pt´ ak, Extremal operators and oblique projections. Casopis P est. Mat. 110 (1985), no. 4, 343-350, 413. [36] W. T. Reid, Symmetrizable completely continuous linear transformations in Hilbert space. Duke Math. J. 18 (1951), 41–56. [37] C. E. Rickart, General theory of Banach algebras. Van Nostrand, Princeton, N.J.Toronto-London-New York 1960. [38] T. Sakaue, M. O’uchi and S. Maeda, Connected components of projections of a C ∗ algebra. Math. Japon. 29 (1984), 427–431. [39] Z. Sebestyén, On ranges of adjoint operators in Hilbert space. Acta Sci. Math. 46 (1983), 295–298. [40] J. Steinberg, Oblique projections in Hilbert spaces. Integral Equations Operator Theory 38 (2000), no. 1, 81–119 [41] D. Szyld, The many proofs of an identity on the Norm of oblique projections. Numerical Algorithms 42 (2006), 309–323. [42] J. Zem´ anek, Idempotents in Banach algebras. Bull. London Math. Soc. 11 (1979), 177–183. M. Laura Arias, Gustavo Corach and M. Celeste Gonzalez Instituto Argentino de Matem´ atica Saavedra 15, 3er. Piso 1083 Buenos Aires Argentina e-mail: ml [email protected] [email protected] [email protected] Submitted: November 27, 2007. Revised: July 17, 2008.



Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions Damir Z. Arov and Olof J. Staffans Abstract. Let S(U ; Y ) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain Ω ⊂ C, which is either the open unit disk D or the open right half-plane C+ . In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient under which a function θ ∈ S(U ; Y ) has a bi-inner dilation conditions

θ mapping U1 ⊕ U into Y ⊕ Y1 ; here U1 and Y1 are two more sepΘ = θθ11 21 θ22 arable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions ψr ∈ S(U ; Y1 ) and ψl ∈ S(U1 ; Y ) (namely ψr = θ22 and ψl = θ11 ) satisfying ψr∗ (z)ψr (z) = I − θ∗ (z)θ(z) and ψl (z)ψl∗ (z) = I − θ(z)θ∗ (z) for almost all z ∈ ∂Ω. We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψr of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψl of the second factorization problem exists if and only if the minimal ∗-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.

Mathematics Subject Classification (2000). Primary 47A48, 93B15; Secondary 94C05. Keywords. Darlington method, optimal passive realization, ∗-optimal passive realization, balanced passive realization.

30

Arov and Staffans

IEOT

1. Introduction The well-known Darlington method in passive circuit theory can be used to synthesize a lossy finite passive circuit network with a given frequency characteristic in an impedance, (input) scattering, or transmission (chain scattering) setting. Typically the given frequency characteristic function is a rational matrix function of the appropriate class (positive real, contractive, or J-contractive in the open right half-plane C+ ), and the Darlington synthesis is carried out by first constructing a lossless network and then loading it with a number of resistors. This number is required to be as small as possible. Darlington [16] introduced this method in the context of a one-pole with a given scalar rational positive real characteristic function, which was realized as the impedance of a network. This approach was extended to multi-poles in a scattering setting by Belevitch [14, Chapter 12]. Let S p×q be the class of all contractive holomorphic matrix-valued function of size p × q defined on C+ . In the scattering setting the Darlington method is related to the representation of a rational (usually real) θ ∈ S p×q as one block of a rational bi-inner matrix-valued function θ11 θ Θ= (1) θ21 θ22 of size m × m, where m is required to be as small as possible. If θ is real, the Θ should also be real. If p = q (which is usually the case), then r = m − p is the minimal number of resistors in a circuit whose input scattering matrix is θ. Thus, we obtain a lossy network with input scattering matrix θ by dropping a total of r exterior branches in a lossless network with scattering matrix Θ. Later the Darlington method was extend from finite to infinite networks (with both lumped and distributed parameters). In this setting the given frequency characteristic is no longer rational (see [5, 9]). In order to use the same method to realize a given holomorphic contractive non-rational matrix-valued function θ defined on C+ as the scattering matrix (transfer function) of a conservative or passive linear time-invariant input/state/output system one again needs to solve an extension problem, where θ is embedded as a block of a bi-inner non-rational matrix-valued function Θ of dimension m × m (see [2, 17]). As in [8, 9] we again require m to be as small as possible. Arov [2] and Dewilde [17] discovered that θ ∈ S p×q has a representation (1) with some inner matrix function Θ of size m × m if and only if θ has a meromorphic pseudo-continuation θ− into the left half-plane C− with bounded Nevanlinna characteristic in C− . Recall that a meromorphic function θ− is said to have a bounded Nevanlinna characteristic (or that it is of bounded type) in C− if θ− can be represented as a quotient θ− (z) = a(z)−1 b(z), where a and b are bounded holomorphic functions on C− , a being scalar-valued and b matrix-valued. The statement that θ− is a pseudo-continuation of θ means that for almost all y ∈ R the limits θ(iy) := limx↓0 θ(x + iy) and θ− (iy) := limx↑0 θ− (x + iy) are equal. See, e.g., [18, Section 1] or [23, Sections 4.2 and 6.3] for more detailed discussions of these two notions.

Vol. 62 (2008)

Bi-Inner Dilations

31

At the next stage of generality we replace the matrix-valued function θ ∈ S p×q by an operator-valued function. Let U (the input space) and Y (the output space) be separable Hilbert spaces, and let S(U ; Y ) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping U into Y , defined on C+ . Given a function θ ∈ S(U ; Y ) we look for a bi-inner ; Y ) of θ of the following type. The spaces U and Y are dilations dilation Θ ∈ S(U of U and Y , i.e., U = U1 ⊕ U and Y = Y ⊕ Y1 where U1 and Y1 are two auxiliary separable Hilbert spaces, and Θ has the block matrix form (1), where this time θ11 ∈ S(U1 ; Y ), θ21 ∈ S(U1 ; Y1 ), and θ22 ∈ S(U ; Y1 ), and θ(z) = PY Θ(z)|U ,

(2)

where PY is the orthogonal projection in Y onto Y . We recall that a function ; Y ) is called inner, co-inner, or bi-inner if its almost everywhere defined Θ ∈ S(U boundary values Θ(iy) := limx↓0 Θ(x + iy) are isometric, co-isometric, or unitary, respectively, a.e. on iR. A representation (2) of θ with a bi-inner function Θ is called a D-representation of θ, and the function Θ is called a bi-inner dilation of θ; see [8] and [18]. As we saw above, a function θ ∈ S p×q has a matrix-valued bi-inner dilation if and only if θ has a meromorphic pseudo-continuation θ− into the left half-plane C− with bounded Nevanlinna characteristic. In the case where θ ∈ S(U ; Y ) (with possibly infinite-dimensional U and Y ) the property of θ of having a meromorphic pseudo-continuation θ− into the left half-plane C− with bounded Nevanlinna characteristic is still a sufficient condition for the existence of a bi-inner dilation; see [2, 4, 8], and [18] (we define pseudo-continuation and bounded Nevanlinna characteristic in the same way as above; in particular, we require the denominator a to be a scalar function). However, this condition is no longer necessary, as we will prove below. Neither is it necessary in the case where θ ∈ S p×q , but we allow the and Y to be infinite-dimensional. dilated spaces U If U or Y is infinite-dimensional, then the additional condition that the di and Y should have a minimal dimension is no longer meaningful. lated spaces U The appropriate notion is instead that the dilated function should have minimal losses, a notion which was introduced by Arov in [4, 8]. A bi-inner dilation Θ of θ has minimal losses if the multiplication operator by θ11 is injective on L2 (iR; U ), or equivalently, if the multiplication operator by θ22 , acting on L2 (iR; U1 ), has a dense range. A D-representation with minimal losses is called a D-representation, and in the matrix-valued case this is a standard Darlington representation with minimal dimension described earlier. For a more detailed description of how this condition on minimal losses should be interpreted we refer the reader to [8]. Arov [8] also obtains a necessary and sufficient condition for the existence of a D-representation. An obvious (and well-known) necessary condition for the existence of a biinner dilation of θ ∈ S(U ; U ) is that there must exist two functions ψr ∈ S(U ; Y1 ) and ψl ∈ S(U1 ; Y ) (namely ψr = θ22 and ψl = θ11 ) satisfying ψr∗ (z)ψr (z) = I − θ∗ (z)θ(z), ψl (z)ψl∗ (z) = I − θ(z)θ∗ (z), a.e. on ∂Ω.

(3)

32

Arov and Staffans

IEOT

The main result of this paper is that the converse is also true: the solvability of the two factorization problems (3) in the Schur class of functions is not only necessary, but also sufficient for the existence of a bi-inner dilation. This answers a question posed more than 30 years ago by Douglas and Helton [18, p. 66]. Above we have concentrated on the case where the function θ is defined on the open half-plane C+ . A similar theory is valid when θ is defined on the open unit disc D instead. The latter case is technically slightly simpler, and we shall therefore in the sequel mainly concentrate on the case where θ is defined on D, and only at the very end return to the case where θ is defined on C+ . In particular, below we let S p×q and S(U ; Y ) stand for the class of Schur functions defined on D (instead of being defined on C+ ). Darlington synthesis for time-dependent systems has been studied by Pick [22].

2. Preliminaries B Let U , X, and Y be separable Hilbert spaces, and let [ A C D ] be a quadruple of bounded linear operators mapping X ⊕ U into X ⊕ Y . With these operators we associate the following discrete-time system:

Σ:

x(n + 1) = Ax(n) + Bu(n), y(n) = Cx(n) + Du(n).

(4)

We call A the main operator, B the control operator, C the observation operator, A B ] ; (Y, X, U )]. The transfer function of and D the feedthrough operator of Σ = [[ C D Σ is given by θΣ (z) = zC(I − zA)−1 B + D, 1/z ∈ ρ(A) A B] (where ρ(A) is the resolvent set of A). The system Σ is (scattering) passive if [ C D A B is a contraction from X ⊕ U to X ⊕ Y , and conservative if [ C D ] is unitary. In these cases the transfer function is defined on all of D, and it belongs to S(U ; Y ) (it is a Schur function on D). Conversely, given a function θ ∈ S(U ; Y ), we call the system Σ in (4) a passive or conservative realization of θ if Σ is passive or conservative, and the transfer function of Σ is θ in the sense that

θ(z) = θΣ (z) = zC(I − zA)−1 B + D,

z ∈ D.

The system Σ (and its main operator A) is strongly stable if Ak x → 0 as k → ∞ for all x ∈ X, and it is strongly co-stable if (A∗ )k x → 0 as k → ∞ for all x ∈ X (where both limits are taken in the strong sense). Let Z+ = 0, 1, 2, . . . and Z− = −1, −2, −3, . . .. We define the reachability map of Σ to be the operator B which maps a sequence {u(−k)}∞ k=1 with only finitely many nonzero elements into Bu =

∞ k=1

Ak−1 Bu(−k).

Vol. 62 (2008)

Bi-Inner Dilations

33

The observability map of Σ is the operator which maps x ∈ X into the sequence Cx = {CAk x}∞ k=0 . (In the case of a (scattering) passive system the operator B can be extended to a bounded linear operator on 2 (Z− ; U ), and C is bounded from X to 2 (Z+ ; Y ).) The reachable subspace R is the closure of the range of B, and the unobservable subspace U is the kernel of C. We call Σ controllable if R = X, it is observable if U = {0}, it is minimal if it is both controllable and observable, and it is simple if U ∩ R⊥ = {0}. As first shown by Sz.-Nagy and Foia¸s [25] and Brodski˘ı [15], every B(U ; Y )valued Schur function θ has a simple conservative realization, which is unique up to unitary similarity. It also has a minimal passive realization (as was noticed by Arov [3]). The latter realization isnot unique. Any two minimal passive realizations A B A B ; (Y, X, U ) of θ are pseudo-similar to each Σ = [[ ] ; (Y, X, U )] and Σ = C D

D C

other in the sense that there is a closed (possibly unbounded) injective linear such that operator Q with dense domain D(Q) ⊂ X and dense range R (Q) ⊂ X

B Qx A QA QB x = , x ∈ D(Q), u ∈ U. D u C D u C In particular, R (B) ⊂ D(Q), and A maps D(Q) into itself. We refer the reader to [12] or [24, Section 9.2] for details.1 Among all minimal passive realizations of θ there is one whose norm is the weakest possible one (in the sense that the pseudo-similarity which maps the state space of any other minimal passive realization into the state space of this particular realization is a contraction). It is clearly unique up to unitary similarity. We call this a minimal optimal passive realization, and denote it by Σ◦ = A◦ realization B◦ ; (Y, X , U ) . There is also another minimal passive realization of θ whose ◦ C◦ D◦ norm is the strongest possible We call this realization a minimal ∗-optimal one. • B• one and denote it by Σ• = A C• D• ; (Y, X• , U ) . See [7, 8, 11] for more details on these two realizations (and also for non-minimal versions2 of these two types of realizations, as well as passive realizations in general).

3. The Balanced Passive Realization From the minimal optimal and ∗-optimal realizations of θ ∈ S(U ; Y ) we can construct still another one, the balanced passive realization, by using interpolation. In the control literature this realization is often called the bounded real balanced 1 The

domain of such a pseudo-similarity Q is not unique in general. However, if Q is bounded (or Q−1 is bounded), then the domain (or the range) of Q is the whole space, and in this case it is unique. 2 Every (non-minimal) optimal system is a passive dilation of a minimal optimal system, and every ∗-optimal system is a passive dilation of a minimal ∗-optimal system.

34

Arov and Staffans

IEOT

realization (see, e.g., [20, Section 5]), and it was originally introduced by Opdenacker and Jonckheere [21] in a continuous time impedance setting. The most common among all balanced realization is the Hankel balanced realization, whose infinite-dimensional version was first developed by Young [26]. ◦ B◦ Let Σ◦ = A C◦ D◦ ; (Y, X◦ , U ) be a minimal optimal realization of θ, and let A B Σ• = C• D• ; (Y, X• , U ) be a minimal ∗-optimal realization of θ. These two sys•

•

tems are pseudo-similar. Let Q be the pseudo-similarity mapping X• ⊃ D(Q ) → R (Q ) ⊂ X◦ . Since Σ◦ has the weakest possible norm and Σ• has the strongest possible norm among all minimal passive realizations of θ, Q is a contraction. In particular, as Q is closed, D(Q ) = X• . Let X• = R (Q ) ⊂ X◦ . Then X• is dense in X◦ . We make X• into a Hilbert space by defining x X• = (Q )−1 x X• . Let Q be the operator that we get from Q by interpreting Q as an operator X• → X• (i.e., it is otherwise the same operator as Q, but space is X• instead of A•itsB•range X◦ ). Clearly Q is unitary X• → X• . Let Σ• = C• D• ; (Y, X• , U ) be the system defined by QA• Q−1 QB• A• B• = . C• D• C• Q−1 D• Then Σ• and Σ• are unitarily similar. In particular Σ• is minimal and ∗-optimal. This system can be interpreted as a restriction of Σ◦ in the sense that A• B• x A◦ B◦ x x = , ∈ X• ⊕ U. C• D• u C◦ D◦ u u The pseudo-similarity Q• : X• = D(Q• ) → R (Q• ) ⊂ X◦ between these two systems is simply the contractive embedding operator X• → X◦ , i.e., for all x ∈ X• we have Q• x = x ∈ X◦ and Q• x X◦ = x X◦ ≤ x X• . Let E• ∈ B(X• ) be the Gram operator corresponding to the embedding X• ⊂ X◦ , i.e., E• is the positive self-adjoint injective contraction on X• which is determined by the fact that x, y X◦ = x, E• y X• for all x, y ∈ X• . This operator can be extended to an operator E◦ ∈ B(X◦ ) in the following way. Since E• is a self-adjoint contraction in X• , we have E•3 ≤ E• , and therefore, for all x ∈ X• , E• x 2X◦ = E• x, E• x X◦ = E• x, E•2 x X• = x, E•3 x X• ≤ x, E• x X• = x 2X◦ . Thus, we may interpret E• as a densely defined contraction X◦ → X◦ . By continuity, it can be extended to a contraction E◦ ∈ B(X◦ ). To see that E◦ is self-adjoint in X◦ we argue as follows. For all x, y ∈ X• we have y, E◦ x X◦ = y, E• x X◦ = y, E•2 x X• = E• y, E• x X• = E• y, x X◦ = E◦ y, x X◦ . By continuity, the identity y, E◦ x X◦ = E◦ y, x X◦ must hold for all x, y ∈ X◦ . 1/2 1/2 Thus, E◦ is self-adjoint. We remark that X• = R E◦ (where E◦ is the positive

Vol. 62 (2008)

Bi-Inner Dilations

35

square root of E◦ in X◦ ), and that −1/2

−1/2

x, E◦ y X◦ , x, y ∈ X• . x, y • = E◦ 1/4 1/4 Let X = R E◦ (here E◦ is the positive 4th root of E◦ in X◦ ). This subspace of X◦ becomes a Hilbert space if we equip it with the inner product −1/4

x, y = E◦

−1/4

x, E◦

1/4

1/4

y X◦ = E◦ x, E◦ y X• ,

x, y ∈ X .

With this inner product, we have X• ⊂ X ⊂ X◦ , with contractive and dense embeddings. (The space X is known as the Riesz interpolation space between X◦ and X• with exponent 12 . A full scale of Hilbert spaces parameterized by α ∈ (0, 1) is constructed in [19, p. 142].) Theorem 3.1. Let X• ⊂ X ⊂ X◦ be the spaces defined above. Define the operators A , B , C , and D by A B x A◦ B◦ x x (5) = , ∈ X ⊕ U. C D u C◦ D◦ u u A B A B ∈ B(X Then C ⊕ U ; X ⊕ U ), and D C D ; (Y, X , U ) is a minimal passive realization of θ. We call Σ (and any other system which is unitarily similar to Σ ) a balanced passive realization of θ. The proof of this theorem is based on the following result on Riesz interpolation. Lemma 3.2. Let X• , X◦ , Y• , and Y◦ be four Hilbert spaces, with X• ⊂ X◦ and Y• ⊂ Y◦ (with continuous embeddings). Let X and Y be the Riesz interpolation spaces with exponent 12 between X• and X◦ respectively Y• and Y◦ (constructed as explained above). If A◦ is a contraction from X◦ into Y◦ with the property that A• := A◦|X• is a contraction from X• into Y• , then A := A◦|X is a contraction from X into Y (in particular, the range of A lies in Y ). This lemma is contained in [19, Theorem 9.1, p. 144] (take the exponent to be 12 in that theorem). The case where X◦ = Y◦ and X• = Y• is also found in [24, Lemma 9.5.8]. An even more general version is given in [1, Theorem C.4, p. 283]. There also exponents α ∈ (0, 1) different from 12 are allowed, and our requirements X• ⊂ X◦ and Y• ⊂ Y◦ have been relaxed to the requirement that these subspaces should be compatible in the sense of interpolation theory. If A◦ and A• are just bounded operators rather than contractions, then the conclusion is that A is a bounded operator with A ≤ A• α A◦ 1−α . Proof of Theorem 3.1. We begin by showing that that the ranges of A and B (with domains X respective U ) are contained in X . The latter inclusion follows trivially from the fact that R (B ) = R (B• ) ⊂ X• ⊂ X . The former inclusion is a consequence of Lemma 3.2 with Y◦ = X◦ and Y• = X• . (In addition, we find that A is a contraction on X .)

36

Arov and Staffans

IEOT

We next show that Σ is passive. To do this we use Lemma 3.2 with the following substitutions: A◦ B◦ X◦ → X◦ ⊕ U, X• → X• ⊕ U, Y◦ → X◦ ⊕ Y, Y• → X• ⊕ Y, A◦ → . C◦ D◦ A B According to Lemma 3.2, C D is contractive, hence Σ is passive. To see that Σ is controllable we argue as follows. By construction, the reachability maps B• and B of Σ• respectively Σ have the same range. By assumption, Σ• is controllable, i.e., R (B• ) is dense in X• . Since X• is dense in X , also R (B ) = R (B• ) is dense in X . Thus Σ is controllable. That Σ is observable follows from the fact that the null space of the observability map C of Σ is a subset of the null space of the observability map C◦ of Σ◦ , and the latter is trivial since Σ◦ is observable. That the transfer function of Σ is θ follows from the fact that for all z with |z| > 0 and all u ∈ U (recall that R (B ) = R (B• ) ⊂ X• ) [C (z − A )−1 B + D ]u = [C• (z − A• )−1 B• u + D• ]u = θ(z)u.

4. Inner Dilations A function θ ∈ S(U ; Y ) (on the unit disk D) has almost everywhere defined limits θ(eiϕ ) := limr↑1 θ(reiϕ ) on the unit circle T in the strong sense. By an inner B(U ; Y )-valued function on D we mean a function θ ∈ S(U ; Y ) satisfying θ(z)∗ θ(z) = I for almost all z with |z| = 1. The function θ ∈ S(U ; Y ) is co-inner is θ(z)∗ is inner, and it is bi-inner if it is both inner and co-inner. Definition 4.1. Let θ ∈ S(U ; Y ).

1) By an inner dilation of θ we mean an inner function Θ of the form Θ = θθr , where θr ∈ S(U ; Y1 ) for some Hilbert space Y1 . 2) By dilation of θ we mean a co-inner function Θ of the form Θ = a co-inner θl θ , where θl ∈ S(U1 ; Y ) for some Hilbert space U1 . θ ∈ 3) By a bi-inner dilation of θ we mean a bi-inner function Θ = θθ11 21 θ22 S(U1 ⊕ U ; Y ⊕ Y1 ) for some Hilbert spaces U1 and Y1 .

Not every θ ∈ S(U ; Y ) has an inner, or co-inner, or bi-inner dilation. But obviously, if θ has a bi-inner dilation, then it has both an inner dilation and a coinner dilation. Our main theorem, stated below, says that the converse statement is also true. Theorem 4.2. Let θ ∈ S(U ; Y ). Then θ has a bi-inner dilation if and only if it has both an inner dilation and a co-inner dilation. This is a part of Corollary 4.6 below, which in turn follows from Lemmas 4.3– 4.5 below.

Vol. 62 (2008)

Bi-Inner Dilations

37

Lemma 4.3 (see [8, Proposition 3]). 1) θ has an inner dilation if and only if θ has a minimal passive strongly stable realization. 2) θ has a co-inner dilation if and only if θ has a minimal passive strongly co-stable realization. 3) θ has a bi-inner dilation if and only if θ has a minimal passive realization with is both strongly stable and strongly co-stable. Proof. If we ignore the word “minimal”, then this is [8, Proposition 3]. However, from any non-minimal realization we can always get a minimal one by first replacing the state space by the reachable subspace, restricting A and C to this subspace, and then projecting the state space onto the orthogonal complement of the unobservable subspace. (See [8, 11] for details.) Lemma 4.4. 1) θ has an inner dilation if and only if the minimal optimal passive realization of θ is strongly stable. 2) θ has a co-inner dilation if and only if the minimal ∗-optimal passive realization of θ is strongly co-stable. This lemma could be derived from [6, Theorem 3]. For the convenience of the reader we include a proof. Proof. We prove only 1) and leave the analogous proof of 2) to the reader. Suppose that θ has an inner dilation. Then it has a minimal strongly stable realization Σ. The norm of the minimal optimal passive realization Σ◦ is the weakest one among all passive realizations, and therefore the pseudo-similarity Q which maps the state space X of Σ into the state space X◦ of Σ◦ is a contraction. The image of X under Q is dense in X◦ , and each (autonomous) trajectory which starts in this set is the image of a trajectory of Σ, hence it tends to zero. Since all trajectories in Σ◦ are bounded, this implies that all trajectories of Σ◦ tend to zero. Thus, Σ◦ is strongly stable. The converse statement is trivial. Lemma 4.5. The balanced passive realization of θ is strongly stable if and only if the minimal optimal realization of θ is strongly stable. The balanced passive realization of θ is strongly co-stable if and only if the minimal ∗-optimal realization of θ is strongly co-stable. Proof. It follows from Lemmas 4.3 and 4.4 that the minimal optimal realization Σ◦ is strongly stable whenever the balanced realization Σ is strongly stable, and that the minimal ∗-optimal realization Σ• is strongly co-stable whenever Σ is strongly co-stable.

38

Arov and Staffans

IEOT

Suppose that Σ◦ is strongly stable. Let x ∈ X• . Then for all n = 0, 1, 2, . . ., by the relationships between the different inner products and the Schwartz inequality, An x 2X = E 1/4 An x 2X• = E 1/4 An x, E 1/4 An x X• = An x, E 1/2 An x X• ≤ An x X• E 1/2 An x X• = An x X• An x X◦ . Let n → ∞. Then An x X◦ → 0 whereas An x X• stays bounded. Thus An x X → 0. This being true on a dense subset (and since AX is a contraction), the same statement must be true for all x ∈ X . Thus, Σ is strongly stable, as claimed. That Σ is strongly co-stable whenever Σ• is strongly co-stable is proved in a similar way. Corollary 4.6. Let θ be a B(U ; Y )-valued Schur function on the open unit disk D. Then the following conditions are equivalent. 1) θ has a bi-inner dilation. 2) θ has both an inner dilation and a co-inner dilation. 3) The balanced passive realization of θ is both strongly stable and strongly costable. 4) The minimal optimal passive realization of θ is strongly stable and the minimal ∗-optimal passive realization of θ is strongly co-stable. 5) θ has a minimal passive realization with is both strongly stable and strongly co-stable. This follows from Lemmas 4.3, 4.4, and 4.5 (and from [8, Proposition 3] for the non-minimal version of 5)). By Definition 4.1, θ ∈ S(U ; Y ) has an inner dilation if and only if the factorization problem ψr∗ (z)ψr (z) = I − θ∗ (z)θ(z), a.e. on T, (6) has a solution θr ∈ S(U ; Y1 ) for some Hilbert space Y1 , and θ ∈ S(U ; Y ) has a co-inner dilation if and only if the factorization problem ψl (z)ψl∗ (z) = I − θ(z)θ∗ (z), a.e. on T,

(7)

has a solution θl ∈ S(U1 ; Y ) for some Hilbert space U1 . Both of these problems are special cases of the right or left spectral factorization that we get by replacing the right-hand sides of (6) or (7) by an operator-valued function f which is integrable and strictly positive a.e. on T. In the case of a nonnegative scalar function f the Szegö theorem gives necessary and sufficient conditions for the existence of H 2 -solutions of these spectral factorization problems. Analogous conditions for matrix-valued functions f satisfying f (z) > 0 for almost all z ∈ T are given by the Zasuhin–Krein theorem. Finally, for operator-valued functions f the factorization theorem of Devinatz gives sufficient conditions. All of these conditions have the property that if they hold when we replace f by I − θ∗ θ, then they also hold if we

Vol. 62 (2008)

Bi-Inner Dilations

39

replace f by I − θθ∗ . By combining these criteria with Corollary 4.6 we obtain the following result. Corollary 4.7. 1) A scalar-valued Schur function θ, which is not inner, has a bi-inner dilation if and only if

− ln 1 − |θ(z)| d|z| < ∞. T

2) An n × n matrix-valued Schur function θ which satisfies θ∗ (z)θ(z) < I for almost all z ∈ T has a bi-inner dilation if and only if

− ln det I − θ∗ (z)θ(z) d|z| < ∞. T

3) A function θ ∈ S(U ; Y ) (where U and Y are allowed to be infinite-dimensional) has a bi-inner dilation if

− ln 1 − θ(z) 2 d|z| < ∞. T

Proof. Assertion 1) follows from Corollary 4.6 together with Szegö’s theorem (see [23, p. 110]), which says the following: given a nonnegative function f ∈ L1 (T), the factorization problem |ψ(z)|2 = f (z) a.e. on T,

has a solution ψ in the Hardy class H 2 (D) if and only if T − ln f (z)d|z| < ∞. We 2 take f (z) = 1−|θ(z)|2 , and notice that ln 1−|θ(z)| = ln(1−|θ(z)|)+ln(1+|θ(z)|), where the latter function is essentially bounded, hence T − ln 1 − |θ(z)| d|z| < ∞ if and only if T − ln f (z)d|z| < ∞. Cauchy’s formula and the boundedness of ψ on T imply that ψ must actually be a Schur function. Assertion 2) is proved in a similar way. We replace Szeg¨ o’s theorem by the Zasuhin–Krein theorem (see [25, part c) of Proposition 7.1, p. 227] and also the discussion on [25, p. 236]). According to this theorem, if θ∗ (z)θ(z) < I for almost all z ∈ T, or equivalently, if θ(z)θ∗ (z) < I for almost all z ∈T, then the factorization problem (6) has a H 2 -solution if and only if T − ln det I − θ∗ (z)θ(z) d|z| < ∞, 2 ∗ whereas (7) has -solution if and only if ∗ T − ln det I − θ(z)θ (z) d|z| < ∞. aH ∗ However, det I − θ (z)θ(z) = det I − θ(z)θ (z) , so if one of the two problems has a solution, then so has the other. Again Cauchy’s formula implies that the factors ψr and ψl in (3) will be Schur functions. The proof of assertion 3) is similar to the proof of assertion 2), but this time we use the Devinatz factorization theorem (see, [25, part b) of Proposition 7.1, p. 227]). Example 4.8. Let θ(z) = (3 + z)−1/2 , where we take the branch of the square root which is analytic on C \ (−∞, −3]). This function is analytic on D. It is easy to see that it is contractive, with an absolute value which is bounded away from both zero and 1. Therefore, by part 1) of Corollary 4.7, this function has an operator-valued

40

Arov and Staffans

IEOT

bi-inner dilation (with infinite-dimensional input and output spaces). However, according to Arov [2] and Dewilde [17], it cannot have a matrix-valued bi-inner dilation since it is does not have a pseudo-continuation to C \ D with bounded Nevanlinna characteristic (it has a branch point at −3). The above example also sheds some additional light on the theory about the existence of a D-realization 3 of a given Schur function θ and how this property is related to the existence of other passive realizations of θ with some special properties. According to Arov [8], θ has a D-representation if an only if θ has a realization with minimal losses which is both strongly stable and strongly costable. On the other hand, by [2], [17], a scalar-valued or matrix-valued Schur function θ has a D-representation if and only if θ has a pseudo-continuation to C\D with bounded Nevanlinna characteristic. Thus, the balanced realization of the Schur function in Example 4.8 is both strongly stable and strongly co-stable, but it does not have minimal losses. The function θ in Example 4.8 is a typical example of a scalar Schur function which has a bi-inner dilation but no D-representation.

5. Continuous Time Bi-Inner Dilations Results analogous to those presented in the last two sections are valid also in the case where the given Schur function θ is defined on the open right-half plane C+ (instead of on the open unit disk D). In this case we use the L2 -well-posed continuous time realizations described in, e.g., [13] and [24]. All the proofs remain essentially the same, except for the fact that references to known discrete time results are replaced by references to the analogous continuous time results, all of which are found in [24] (and many of them also in [13]). In particular, the minimal optimal, ∗-optimal and balanced passive realizations are described in [24, Section 11.8]. Alternatively, the continuous time case can be reduced to the discrete time case by means of the Cayley transform.

Acknowledgement. Damir Z. Arov thanks ˚ Abo Akademi for its hospitality and the Academy of Finland for its financial support during his visit to ˚ Abo in August– September 2003. He also gratefully acknowledges the partial financial support by the joint grant UM1-2567 OD 03 from the U.S. Civilian Research and Development Foundation (CRDF) and the Ukrainian Government. Olof J. Staffans gratefully acknowledges the financial support from the Academy of Finland, grant 203991. 3 The

notions of a D-representation and a passive system with minimal losses was explained briefly in the introduction.

Vol. 62 (2008)

Bi-Inner Dilations

41

References [1] Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. [2] Damir Z. Arov, Darlington’s method for dissipative systems, Dokl. Akad. Nauk SSSR. 201 (1971), 559–562, translation in Soviet Physics Dokl. 16 (1971), 954–956. [3]

, Scattering theory with dissipation of energy, Dokl. Akad. Nauk SSSR. 216 (1974), 713–716, translation in Soviet Math. Dokl. 15 (1974), 848–854.

[4]

, Unitary couplings with losses (a theory of scattering with losses), Funkcional. Anal. i Priloˇzen. 8 (1974), 5–22, translation in Funct. Anal. Appl. 8 (1974), 280–294.

[5]

, The realization of a canonical system with dissipative boundary conditions at one end of a segment in terms of the coefficient of dynamic flexibility, Sibir. Mat. Zh. 16 (1975), 440–463, translation in Sib. Math. J. 16 (1975), 335–352.

[6]

, Optimal and stable passive systems, Dokl. Akad. Nauk SSSR. 247 (1979), 265–268, translation in Soviet Math. Dokl. 20 (1979), 676–680.

[7]

, Passive linear stationary dynamic systems, Sibir. Mat. Zh. 20 (1979), 211– 228, translation in Sib. Math. J. 20 (1979), 149–162.

[8]

, Stable dissipative linear stationary dynamical scattering systems, J. Operator Theory 1 (1979), 95–126, translation in [10].

[9]

, Scattering matrix and impedance of a canonical differential system with a dissipative boundary condition whose coefficient is a rational matrix-valued function of the spectral parameter (in Russian), Algebra Anal. 13 (2001), 26–53, translation in St. Petersburg Math. J. 13, No. 4 (2002), 527–547.

[10]

, Stable dissipative linear stationary dynamical scattering systems, pp. 99– 136 in: Interpolation Theory, Systems Theory, and Related Topics. The Harry Dym Anniversary Volume. Operator Theory: Advances and Applications, vol. 134, Birkh¨ auser-Verlag, 2002, English translation of the article in J. Operator Theory 1 (1979), 95–126.

[11] Damir Z. Arov, Marinus A. Kaashoek, and Derk R. Pik, Minimal and optimal linear discrete time-invariant dissipative scattering systems, Integral Equations Operator Theory 29 (1997), 127–154. [12]

, The Kalman–Yakubovich–Popov inequality and infinite dimensional discrete time dissipative systems, J. Operator Theory 55 (2006), 393–438.

[13] Damir Z. Arov and Mark A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996), 1–45. [14] Vitold Belevitch, Classical network theory, Holden-Day, San Francisco, Calif.Cambridge-Amsterdam, 1968. [15] V. M. Brodski˘ı, On operator colligations and their characteristic functions, Soviet Math. Dokl. 12 (1971), 696–700. [16] Sidney Darlington, Synthesis of reactance 4-poles which produce prescribed insertion loss characteristics including special applications to filter design, Journal of Mathematics and Physics 18 (1939), 257–353.

42

Arov and Staffans

IEOT

[17] Patrick Dewilde, Input-output description of roomy systems, SIAM J. Control Optim. 14 (1971), 712–736. [18] Ronald G. Douglas and J. William Helton, Inner diations of analytic matrix functions and Darlington synthesis, Acta Sci. Math (Szeged) 34 (1973), 61–67. [19] Selim G. Kre˘ın and Ju. I. Petunin, Scales of Banach spaces, Uspehi Mathem. Nauk SSSR 21 (1966), 89–168, translation in Russian Math. Surveys 21 (1966), 85–159. [20] Raimund Ober, Balanced parameterization of classes of linear operators, SIAM J. Control Optim. 29 (1991), 1251–1287. [21] Philippe C. Opdenacker and Edmond A. Jonckheere, A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds, IEEE Trans. Circuits Systems 35 (1988), 184–189. [22] Derk R. Pik, Time-variant Darlington synthesis and induced realizations, Internat. J. Appl. Math. Comput. Sci. 6 (2001), 1331–1360. [23] Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford University Press, New York, Oxford, 1985. [24] Olof J. Staffans, Well-posed linear systems, Cambridge University Press, Cambridge and New York, 2005. [25] Béla Sz.-Nagy and Ciprian Foia¸s, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam London, 1970. [26] Nicolas J. Young, Balanced realizations in infinite dimensions, pp. 449–471 in: Operator Theory and Systems. Operator Theory: Advances and Applications, vol. 19, Birkh¨ auser-Verlag, 1986, pp. 449–471. Damir Z. Arov South-Ukrainian Pedagogical University Division of Mathematical Analysis 65020 Odessa Ukraine e-mail: damir [email protected] Olof J. Staffans ˚ Abo Akademi University Department of Mathematics Biskopsgatan 8, Axelia 3 v˚ an FIN-20500 ˚ Abo Finland e-mail: [email protected] Received: April 21, 2004. Revised: February 21, 2005.



Toeplitz plus Hankel Operators with Infinite Index G. Bogveradze and L.P. Castro Abstract. We study Toeplitz plus Hankel operators acting between Lebesgue spaces on the unit circle, and having symbols which contain standard almost periodic discontinuities. Conditions are obtained under which these operators are right-invertible and with infinite kernel dimension, left-invertible and with infinite cokernel dimension or simply not normally solvable. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47A53, 47A68, 42A75. Keywords. Toeplitz operator, Hankel operator, almost periodic function, onesided invertibility, infinite index.

1. Introduction Discontinuities of almost periodic type appeared for the first time in the work of Gohberg and Fel’dman [9, 10, 11] when studying Wiener-Hopf equations. The paper by Coburn and Douglas [5] is also an important mark for the beginning of the study of integral operators with symbols which present such kind of discontinuities. Since then, the consideration of Toeplitz and singular integral operators with symbols and coefficients with discontinuities of almost periodic type were considered by a large number of authors (cf., e.g., [2, 6, 7, 12, 14, 15]). The book of Dybin and Grudsky [8] provides a comprehensive description of the known results for Toeplitz operators with infinite index originated by symbols with almost periodic discontinuities. The present work is devoted to the study of Toeplitz plus Hankel operators (cf. [1, 16]) with a finite number of standard almost periodic discontinuities in their symbols. The operators are acting between L2 spaces on the unit circle. The results (cf. Section 3) provide conditions under which the Toeplitz plus Hankel This research was supported by Funda¸c˜ ao para a Ciência e a Tecnologia (Portugal) through Unidade de Investiga¸c˜ ao Matem´ atica e Aplica¸co ˜es of University of Aveiro.

44

Bogveradze and Castro

IEOT

operators are right-invertible but with infinite kernel dimension or left-invertible but with infinite cokernel dimension or simply not normally solvable. The paper is organized as follows. In Section 2 (which is divided into five subsections) we present the auxiliary notions and known results for Toeplitz operators. In Section 3, new results for Toeplitz plus Hankel operators are proposed which lead to their lateral invertibility although within the case of infinite index. In Section 4 we provide a concrete example which is characterized by the use of the results in Section 3.

2. Auxiliary Notions and Known Results 2.1. Toeplitz plus Hankel Operators Let Γ be a closed rectifiable Jordan curve in the complex plane C. The interior of this curve will be denoted by DΓ+ , and the exterior by DΓ− . Further, Γ0 will stand for the unit circle in the complex plane and in this case we will simply write D± in the place of DΓ±0 . Let SΓ0 for the Cauchy singular integral operator on Γ0 , given by the formula: f (τ )dτ 1 (SΓ0 f )(t) = , t ∈ Γ0 , πi Γ0 τ − t where the integral is understood in the principal value sense. When this operator is acting between Lebesgue spaces L2 (Γ0 ), it induces two complementary projections, namely: 1 1 QΓ0 = (I − SΓ0 ) . PΓ0 = (I + SΓ0 ), 2 2 Consider the following image spaces: PΓ0 (L2 (Γ0 )) =: L2+ (Γ0 ) and QΓ0 (L2 (Γ0 )) =: L2− (Γ0 ). Besides the above introduced spaces we will also make use of the well2 (Γ0 ) which can be isometrically identified with L2± (Γ0 ). known Hardy spaces H± Assume that B is a Banach algebra. Let us agree with the notation GB for the group of all invertible elements from B. Let φ ∈ L∞ (Γ0 ). The Toeplitz operator acting between L2+ (Γ0 ) spaces is given by Tφ = PΓ0 φI : L2+ (Γ0 ) → L2+ (Γ0 ) , (2.1) where φ is called the symbol of the operator and I stands for the identity operator. The Hankel operator is defined by Hφ = PΓ0 φJ : L2+ (Γ0 ) → L2+ (Γ0 ) , where J is a Carleman shift operator which acts by the rule: 1 1 (Jf )(t) = f , t ∈ Γ0 . t t The Toeplitz plus Hankel operator with symbol φ will be denoted by T Hφ , and has therefore the form T Hφ = PΓ0 φ(I + J) : L2+ (Γ0 ) → L2+ (Γ0 ) .

Vol. 62 (2008)

Toeplitz plus Hankel Operators with Infinite Index

45

2.2. Factorization and Fredholm Theory Let T : X → Y be a bounded linear operator acting between Banach spaces. The kernel of T will be denoted by Ker T , and the cokernel of T by Coker T . Assume that the image of T , Im T , is closed (i.e. T is normally solvable). Then, T is said to be Fredholm if both dim Ker T and dim Coker T are finite. The numbers dim Ker T and dim Coker T (the infinite case not excluded) are referred to as the defect numbers of the operator T . Additionally, we say that T is left-invertible or right-invertible if there exist Tl− : Y → X or Tr− : Y → X such that Tl− T = IX or T Tr− = IY , respectively. As usual, in the case when both Tl− and Tr− exist, the operator T is said to be invertible. Consider a function f given on the unit circle: f : Γ0 → C. By the notation f we mean the following new function: f(t) = f (t−1 ), t ∈ Γ0 . We will say that f defined on the unit circle is even if f(t) = f (t), for almost all t ∈ Γ0 . Let us now recall several types of factorizations. Definition 2.1. [8, Section 2.4] A function φ ∈ GL∞ (Γ0 ) admits a generalized factorization with respect to L2 (Γ0 ) if it can be represented in the form φ(t) = φ− (t)tk φ+ (t),

t ∈ Γ0 ,

where k is an integer, called the index of the factorization, and the functions φ± satisfy the following conditions: (1) (φ− )±1 ∈ L2− (Γ0 ) ⊕ C, (φ+ )±1 ∈ L2+ (Γ0 ) , −1 2 (2) the operator φ−1 + SΓ0 φ− I is bounded in L (Γ0 ). The class of functions admitting a generalized factorization will be denoted by F. Definition 2.2. [1, Section 3] A function φ ∈ GL∞ (Γ0 ) is said to admit a weak even asymmetric factorization in L2 (Γ0 ) if it admits a representation φ(t) = φ− (t)tk φe (t) ,

t ∈ Γ0 ,

such that k ∈ Z and 2 2 (i) (1 + t−1 )φ− ∈ H− (Γ0 ), (1 − t−1 )φ−1 − ∈ H− (Γ0 ), 2 −1 2 (ii) |1 − t|φe ∈ Leven(Γ0 ), |1 + t|φe ∈ Leven(Γ0 ), where L2even(Γ0 ) stands for the class of even functions from the space L2 (Γ0 ). The integer k is called the index of the weak even asymmetric factorization. Definition 2.3. [1, Section 3] A function φ ∈ GL∞ (Γ0 ) is said to admit a weak antisymmetric factorization in L2 (Γ0 ) if it admits a representation −1 φ(t) = φ− (t)t2k φ − (t) ,

t ∈ Γ0 ,

such that k ∈ Z and 2 (1 + t−1 )φ− ∈ H− (Γ0 ),

2 (1 − t−1 )φ−1 − ∈ H− (Γ0 ).

Also in here the integer k is called the index of a weak antisymmetric factorization.

46


IEOT

The next proposition relates weak even asymmetric factorizations with weak antisymmetric factorizations. −1 . Proposition 2.4. [1, Proposition 3.2] Let φ ∈ GL∞ (Γ0 ) and consider Φ = φφ

(i) If φ admits a weak even asymmetric factorization, φ = φ− tk φe , then the function Φ admits a weak antisymmetric factorization with the same factor φ− and the same index k; −1 (ii) If Φ admits a weak antisymmetric factorization, Φ = φ t2k φ , then φ ad−

−

mits a weak even asymmetric factorization with the same factor φ− , the same index k and the factor φe := t−k φ−1 − φ. The next two theorems were obtained by Basor and Ehrhardt (cf. [1]), and give an useful invertibility and Fredholm characterization for Toeplitz plus Hankel operators with essentially bounded symbols. Theorem 2.5. [1, Theorem 5.3] Let φ ∈ GL∞ (Γ0 ). The operator T Hφ is invertible if and only if φ admits a weak even asymmetric factorization in L2 (Γ0 ) with index k = 0. Theorem 2.6. [1, Theorem 6.4] Let φ ∈ GL∞ (Γ0 ). The operator T Hφ is a Fredholm operator if and only if φ admits a weak even asymmetric factorization in L2 (Γ0 ). In this case, it holds dim Ker T Hφ = max{0, −k},

dim Coker T Hφ = max{0, k} ,

where k is the index of the weak even asymmetric factorization. The next theorem is a classical result which deals with the Fredholm property for the Toeplitz operators. Theorem 2.7. Let φ ∈ L∞ (Γ0 ). The operator Tφ given by (2.1) is Fredholm in the space L2+ (Γ0 ) if and only if φ ∈ F. We will now turn to the generalized factorizations with infinite index. Definition 2.8. [8, section 2.7] A function φ ∈ GL∞ (Γ0 ) admits a generalized factorization with infinite index in the space L2 (Γ0 ) if it admits a representation φ = ϕh or φ = ϕh−1

(2.2)

where (1) ϕ ∈ F , ∞ (2) h ∈ L∞ + (Γ0 ) ∩ GL (Γ0 ) . The class of functions admitting a generalized factorization with infinite index in L2 (Γ0 ) will be denoted by F∞ . We list here some known important properties of the class F∞ :

Vol. 62 (2008)


47

1. F ⊂ F∞ . Therefore (from this inclusion and Theorem 2.7), it follows that the class F∞ contains the symbols of Fredholm Toeplitz operators. However, in another way, the following condition excludes elements which generate Fredholm operators from this class: for any polynomial u with complex coefficients, (2.3) u/h ∈ L∞ + (Γ0 ). 2. A generalized factorization with infinite index does not enjoy the uniqueness property. 3. Let φ ∈ F∞ and let condition (2.3) be satisfied. Then the function h in (2.2) can be chosen so that ind ϕ = 0. 4. Let φ ∈ F∞ . Then for the function h in (2.2) one can choose an inner function ∞ u (i.e., a function u from the Hardy space H+ and such that |u(t)| = 1 almost everywhere on Γ0 ). The proof of these facts can be found for example in [8, section 2.7]. Theorem 2.9. [8, Theorem 2.6] Assume that φ ∈ F∞ , condition (2.3) is satisfied, and ind ϕ = 0. 1. If φ = ϕh−1 , then the operator Tφ is right-invertible in the space L2+ (Γ0 ), and dim Ker Tφ = ∞. 2. If φ = ϕh, then the operator Tφ is left-invertible in the space L2+ (Γ0 ), and dim Coker Tφ = ∞. 2.3. Almost periodic functions We will define the class AP of almost periodic functions in the following way. A function α of the form n α(x) = cj exp(iλj x) , x∈R, j=1

where λj ∈ R and cj ∈ C, is called an almost periodic polynomial. If we construct the closure of the set of all almost periodic polynomials by using the supremum norm, we will then obtain the class AP of almost periodic functions. Theorem 2.10 (Bohr). Suppose that p ∈ AP and inf |p(x)| > 0 .

x∈R

(2.4)

Then the function arg p(x) can be defined so that arg p(x) = λx + ψ(x) , where λ ∈ R and ψ ∈ AP. Definition 2.11 (Bohr mean motion). Let p ∈ AP and let the condition (2.4) be satisfied. The Bohr mean motion of the function p is defined to be the following real number 1 arg p(x)|− . k(p) = lim →∞ 2

48


IEOT

Let us transfer to the unit circle Γ0 the class of almost periodic functions (introduced above for the real line R), by means of the following operator V : 1+t (V f )(t) = f i . 1−t To denote the almost periodic functions class in the unit circle, we will use the notation APΓ0 . Furthermore, almost periodic polynomials on the circle are of the form: n t+1 a(t) = cj exp λj , λj ∈ R. t−1 j=1 Next, the standard almost periodic discontinuities will be defined for the unit circle. Definition 2.12. [8, section 4.3] A function φ ∈ L∞ (Γ0 ) has a standard almost periodic discontinuity in the point t0 ∈ Γ0 if there exists a function p0 ∈ APΓ0 and a diffeomorphism τ = ω0 (t) of the unit circle Γ0 onto itself, such that ω0 preserves the orientation of Γ0 , ω0 (t0 ) = 1, the function ω0 has a second derivative at t0 , and lim (φ(t) − p0 (ω0 (t))) = 0 , t ∈ Γ0 . (2.5) t→t0

In such a situation we will say that φ has a standard almost periodic discontinuity in the point t0 with characteristics (p0 , ω0 ). Remark 2.13. Assume that φ ∈ L∞ (Γ0 ) has a standard almost periodic discontinuity in the point t0 and let a diffeomorphism ω0 satisfy the conditions in the definition of a standard almost periodic discontinuity. Then, by means of a simple change of variable, the equality (2.5) can be rewritten in the following way: lim [φ(ω0−1 (τ )) − p0 (τ )] = 0,

τ →1

τ ∈ Γ0 .

2.4. Model functions −1 An invertible function h with properties h ∈ L∞ ∈ L∞ (Γ0 ) is called + (Γ0 ) and h a model function on the curve Γ0 . The operator Th−1 , acting in L2+ (Γ0 ), and its kernel Ker Th−1 will be referred to as the model operator and the model subspace in the space L2+ (Γ0 ) generated by the function h, respectively. We say that the model function on the curve Γ0 belongs to the class U if h−1 ∈ L∞ − (Γ0 ). The just described notion of a model function, model operator and model space, can be generalized to the real line, and furthermore for any rectifiable Jordan curve. As an example, take exp(iλx), with λ > 0, and we will obtain a model function for the real line R. In the space L2 (Γ0 ) let us also consider the pair of complementary projections: Ph = hQΓ0 h−1 I, Qh = hPΓ0 h−1 I , and the subspace M(h) = Ph (L2+ (Γ0 )).

Vol. 62 (2008)


49

Proposition 2.14. [8, Proposition 3.4] Let hj ∈ U, j = 1, 2, ..., n. Then h = n h ∈ U and j=1 j n−1

M(h) = M(h1 ) ⊕ h1 M(h2 ) ⊕ . . . ⊕ (

hj )M(hn ) .

j=1

Let ak ∈ C, k = 1, 2, 3, 4, and assume that ∆ = a1 a4 − a2 a3 = 0. Consider the following two fractional linear transformations, which are inverses of one another: a1 t + a2 a4 x − a2 v(t) = . (2.6) , v −1 (x) = a3 t + a4 a1 − a3 x If we apply a fractional linear transformation of the form (2.6) to the model function exp(iλx), with λ > 0, we arrive at the function h0 (t) = exp(φ0 (t − t0 )−1 ), φ0 ∈ C \ {0} ,

(2.7)

which will be considered on the unit circle Γ0 (and t0 ∈ Γ0 ). Proposition 2.15. The function h0 given by (2.7) is a model function on Γ0 if and only if arg φ0 = ηΓ0 (t0 ). The previous proposition is just a particularization of a corresponding result in [8, Proposition 4.2] when passing from the case of simple closed smooth contours to our Γ0 case. Proposition 2.16. [8, Proposition 4.6] Suppose that a diffeomorphism τ = ω0 (t) of the unit circle Γ0 onto itself satisfies the conditions in the definition of a standard almost periodic discontinuity at the point t0 ∈ Γ0 . Then the following representation holds on Γ0 : ω0 (t) + 1 (2.8) φ(t) = exp λ = h0 (t)c0 (t), λ ∈ R , ω0 (t) − 1 where c0 ∈ GC(Γ0 ), h0 ∈ L∞ (Γ0 ) is given by (2.7) with φ0 = 2λ/ω0 (t0 ), and C(Γ0 ) is the usual set of continuous functions on Γ0 . Remark 2.17. (cf. [8, Remark 4.6]) Proposition 2.15 ensures that whenever on Γ0 there exists a function φ that has a standard almost periodic discontinuity in the point t0 , one of the functions h0 given by (2.7) or h−1 0 is a model function on Γ0 . Since here the mapping τ = ω0 (t) preserves the orientation of Γ0 , arg φ0 = ηΓ0 (t0 ) when λ > 0 (cf. (2.8)) and arg φ0 = ηΓ0 (t0 ) − π when λ < 0. 2.5. Functional σt0 and Lateral Invertibility of Toeplitz Operators Let t0 ∈ Γ0 and let the function φ ∈ GL∞ (Γ0 ) be continuous in a neighborhood of t0 , except, possibly, in the point t0 itself. Let us recall the real functional used by Dybin and Grudsky in [8]: δ δ [arg φ(t)] |tt=t = lim (arg φ (t ) − arg φ (t )) δ→0 4 4 where t , t ∈ Γ0 , t ≺ t0 ≺ t , |t − t0 | = |t − t0 | = δ.

σt0 (φ) = lim

δ→0

(2.9)

50


IEOT

The notation t ≺ t0 ≺ t , used above, means that when we are tracing the curve in the positive direction we will meet the point t first, then the point t0 and then the point t . The next proposition establishes a connection between the functional σt0 (φ) and the standard almost periodic discontinuities on Γ0 . Proposition 2.18. [8, Proposition 4.9] Suppose that the diffeomorphism τ = ω0 (t) of the unit circle Γ0 onto itself satisfies the conditions in the definition of a standard almost periodic discontinuity in the point t0 ∈ Γ0 and that p ∈ GAPΓ0 . Then φ(t) = p(ω0 (t)) ∈ GL∞ (Γ0 ), σt0 (φ) exists, and σt0 (φ) = k(p)/|ω0 (t0 )| . A factorization theorem which is crucial for the theory of Toeplitz operators is now stated. Theorem 2.19. [8, Theorem 4.12] Let the function φ ∈ GL∞ (Γ0 ) be continuous on the set Γ0 \ {tj }nj=1 and have standard almost periodic discontinuities in the points tj . Then   n exp(λj (t − tj )−1 ) ϕ(t) , φ(t) =  j=1

with ϕ ∈ F and λj = σtj (φ) tj where the functional σtj (φ) is defined by the formula (2.9) at the point tj . Let us always write the factorization of a function φ in the way of the nondecreasing order of the values of σtj (φ). I.e., we will always assume that σt1 (φ) ≤ σt2 (φ) ≤ . . . ≤ σtn (φ). This is always possible because we can always re-enumerate the points tj to achieve the desired non-decreasing sequence. The next result characterizes the situation of Toeplitz operators with a symbol having a finite number of standard almost periodic discontinuities, and it was our starting point motivation in view to obtain a corresponding description to Toeplitz plus Hankel operators. Theorem 2.20. [8, Theorem 4.13] Suppose that φ ∈ GL∞ (Γ0 ) is continuous on the set Γ0 \ {tj }nj=1 , has standard almost periodic discontinuities in the points tj , and σtj (φ) = 0, 1 ≤ j ≤ n. 1. If σtj (φ) < 0, 1 ≤ j ≤ n, then the operator Tφ is right-invertible in L2+ (Γ0 ) and dim Ker Tφ = ∞, 2. If σtj (φ) > 0, 1 ≤ j ≤ n, then the operator Tφ is left-invertible in L2+ (Γ0 ) and dim Coker Tφ = ∞, 3. If σtj (φ) < 0, 1 ≤ j ≤ m, and σtj (φ) > 0, m + 1 ≤ j ≤ n, then the operator Tφ is not normally solvable in L2+ (Γ0 ) and dim Ker Tφ = dim Coker Tφ = 0.

Vol. 62 (2008)


51

3. Toeplitz plus Hankel Operators with Almost Periodic Discontinuities in their Symbols To achieve the Toeplitz plus Hankel version of Theorem 2.20, we will combine several techniques. In particular, we will make use of operator matrix identities (cf. [3, 4, 13]), and therefore start by recalling the notion of ∆-relation after extension [3]. We say that T is ∆-related after extension to S if there is an auxiliary bounded linear operator acting between Banach spaces T∆ : X1∆ → X2∆ , and bounded invertible operators E and F such that

S 0 T 0 =E F, (3.1) 0 IZ 0 T∆ where Z is an additional Banach space and IZ represents the identity operator in Z. In the particular case where T∆ = IX1∆ : X1∆ → X2∆ = X1∆ is the identity operator, T and S are said to be equivalent after extension operators. From [3] we can derive that T = T Hφ : L2+ (Γ0 ) → L2+ (Γ0 ) is ∆-related 2 2 after extension to the Toeplitz operator S = Tφφ −1 : L+ (Γ0 ) → L+ (Γ0 ), where this relation is given to T∆ = Tφ − Hφ : L2+ (Γ0 ) → L2+ (Γ0 ) in (3.1). Indeed, this can be achieved by performing the following steps: (a) we start by extending 2 2 Tφφ −1 : L+ (Γ0 ) → L+ (Γ0 ) by the identity operator and obtaining its equivalent after extension operator 0 Tφφ Tφφ Tφ 0 −IL2+ (Γ0 ) −1 −1 −1 = =: AB , −IL2+ (Γ0 ) Tφ 0 IL2+ (Γ0 ) IL2+ (Γ0 ) 0 −1 where the last operator B is obviously an invertible operator; (b) we extend the 2 2 2 operator A : [L2+ (Γ0 )] → [L2+ (Γ0 )] to the whole space [L2 (Γ0 )] by using the complementary projection QΓ0 , obtaining therefore the following new operator −1 2 2 0 φφ PΓ0 + QΓ0 : [L2 (Γ0 )] → [L2 (Γ0 )] C := PΓ0 −1 −1 φ which is equivalent after extension to A; (c) the operator C is equivalent to −1 2 2 0 φφ + QΓ0 : [L2 (Γ0 )] → [L2 (Γ0 )] D := PΓ0 −1 −1 φ due to the explicit operator equivalence relation −1 0 φφ QΓ0 D , C = I[L2 (Γ0 )]2 − PΓ0 −1 −1 φ where the paired operator which appears inside the last round brackets is an invertible operator; (d) multiplying D on the right by the invertible operator

1 0 G = I[L2 (Γ0 )]2 φ φ

52


IEOT

it turns out the new operator

1 0 φ φ H = PΓ0 I[L2 (Γ0 )]2 + QΓ0 I[L2 (Γ0 )]2 0 1 φ φ which is therefore equivalent to D; (e) we are now in a position to use the so-called Gohberg-Krupnik-Litvinchuk identity and obtain

1 I J I I H J −J 2 I −J

2 2 PΓ0 φ + PΓ0 φJ + QΓ0 0 = : [L2 (Γ0 )] → [L2 (Γ0 )] ; 0 PΓ0 φ − PΓ0 φJ + QΓ0 (f) we notice here that PΓ0 φ±PΓ0 φJ+QΓ0 = (I+PΓ0 φQΓ0 ±PΓ0 φJQΓ0 )(PΓ0 φPΓ0 ± PΓ0 φJPΓ0 + QΓ0 ), where I + PΓ0 φQΓ0 ± PΓ0 φJQΓ0 are invertible operators with inverses being given by I − PΓ0 φQΓ0 ∓ PΓ0 φJQΓ0 ; (g) it is clear that T Hφ is equivalent after extension to PΓ0 φPΓ0 + PΓ0 φJPΓ0 + QΓ0 : L2 (Γ0 ) → L2 (Γ0 ). Thus, combining the steps (a)–(g), it follows the above mentioned ∆-relation after extension. In view of the main goal of the present section, let us start by considering functions (defined on Γ0 ) which have three standard almost periodic discontinuities, namely in the points t1 , t2 and t3 , and such that t−1 1 = t2 . As we shall see, this is the most representative case, and the general case can be treated in the same manner as to this one. Assume therefore that φ has standard almost periodic discontinuities in the points t1 , t2 , t3 , with characteristics (p1 , ω1 ), (p2 , ω2 ), (p3 , ω3 ). it is clear that φ has the standard almost periodic discontinuities Considering φ, −1 −1 in the points t−1 1 (= t2 ), t2 (= t1 ) and t3 (cf. Remark 2.13). Moreover, it is useful −1 will have standard almost periodic discontinuities in the points to observe that φ −1 t1 , t2 and t3 . −1 . From formula (2.9) we will have: Set Φ := φ φ δ −1 ) = lim −1 (t)) t arg(φ(t)φ σt1 (Φ) = σt1 (φφ t=t δ→0 4 δ δ −1 (t) t = lim [arg φ(t)] |tt=t + lim arg φ t=t δ→0 4 δ→0 4 δ t arg φ(t) = σt1 (φ) − lim t=t δ→0 4 δ1 (t )−1 [arg φ(t)] |t=(t )−1 = σt1 (φ) + lim δ1 →0 4 = σt1 (φ) + σt−1 (φ) , (3.2) 1

where δ1 = |(t ) − = |(t ) − t−1 1 | = |t − t1 | = |t − t1 | = δ. On the other hand, it is also clear that σt−1 (Φ) = σt1 (φ) + σt−1 (φ). Thus, the 1 1 points of symmetric standard almost periodic discontinuities (with respect to the x-axis on the complex plane) fulfill formula (3.2). This is the main reason why we −1

t−1 1 |

−1

Vol. 62 (2008)


53

do not need to treat more than three points of the standard almost periodic discontinuities in order to understand the qualitative result for Toeplitz plus Hankel operators with a finite number of standard almost periodic discontinuities in their symbols. We are now in conditions to present the Toeplitz plus Hankel version of Theorem 2.20 for three points of discontinuity. Theorem 3.1. Suppose that the function φ ∈ GL∞ (Γ0 ) is continuous on the set Γ0 \ {tj }3j=1 , has standard almost periodic discontinuities in the points tj , such that t−1 1 = t2 , and let σtj (φ) = 0, 1 ≤ j ≤ 3. (i) If σt1 (φ) + σt2 (φ) ≤ 0 and σt3 (φ) < 0, then the operator T Hφ is rightinvertible in L2+ (Γ0 ) and dim Ker T Hφ = ∞, (ii) If σt1 (φ)+ σt2 (φ) ≥ 0 and σt3 (φ) > 0, then the operator T Hφ is left-invertible in L2+ (Γ0 ) and dim Coker T Hφ = ∞, (iii) If (σt1 (φ)+σt2 (φ))σt3 (φ) < 0, then the operator T Hφ is not normally solvable in L2+ (Γ0 ) and dim Ker T Hφ = dim Coker T Hφ = 0. −1 . It is clear that Φ can be considered (due Proof. Let us work with Φ := φφ to the invertibility of φ), and also that Φ is invertible in L∞ (Γ0 ). As far as φ has three points of almost periodic discontinuities (namely t1 , t2 and t3 ), then Φ will have four points of almost periodic discontinuities (due to the reason that −1 t−1 1 = t2 ). The discontinuity points of Φ are the following ones: t1 , t2 , t3 and t3 . From formula (3.2), we will have that

σt1 (Φ) = σt1 (φ) + σt−1 (φ) = σt1 (φ) + σt2 (φ) ,

(3.3)

σt2 (Φ) = σt2 (φ) + σt−1 (φ) = σt2 (φ) + σt1 (φ) ,

(3.4)

σt3 (Φ) = σt3 (φ) + σt−1 (φ) = σt3 (φ) ,

(3.5)

σt−1 (Φ) = σt−1 (φ) + σt3 (φ) = σt3 (φ) .

(3.6)

1 2 3

3

3

In the above formulas, the fact that φ is a continuous function in the point t−1 3 was used. Now, employing Theorem 2.19, we can ensure a factorization of the function Φ in the form:   4 Φ(t) =  exp(λj (t − tj )−1 ) ϕ(t) , (3.7) j=1

where ϕ ∈ F. Let us denote h(t) =

4

exp(λj (t − tj )−1 ) .

(3.8)

j=1

We will now deduce propositions (i)–(iii) in the following three points 1–3, respectively: 1. If the conditions in part (i) are satisfied, then we will have that σtj (Φ) ≤ 0, j = 1, 4 (cf. formulas (3.3)–(3.6)). Hence, the function h given by (3.8) belongs

54


IEOT

to L∞ − (Γ0 ). Moreover, relaying on Proposition 2.14 and Remark 2.17, we have that h−1 ∈ U. Using the first part of Theorem 2.20, we can conclude that TΦ is right-invertible. Then, the ∆-relation after extension allows us to state that T Hφ is right-invertible. We are left to deduce that dim Ker T Hφ = ∞. Suppose that dim Ker T Hφ = k < ∞. We will show that in the present situation this is not possible. In the case at hand we would have a Fredholm Toeplitz plus Hankel operator with symbol φ. Thus, by Theorem 2.6, φ admits a weak even asymmetric factorization φ = φ− tk φe , with corresponding properties for φ− and φe . Employing now Proposition 2.4 we will have that Φ admits a weak antisymmetric factorization: −1 . (3.9) Φ = φ t2k φ −

−

This leads to the conclusion that dim Ker TΦ < ∞. In addition, according to Theorem 2.20, dim Ker TΦ = ∞ and we have therefore a contradiction. We would like however to present an alternative and more constructive proof of the just presented contradiction (although much more longer) since it will be useful in the case (iii): besides (3.9), from (3.7) we also have that Φ = ϕ− tm ϕ+ h ,

(3.10)

where ϕ± have the properties as stated in Definition 2.1 and m is integer. From −1 m equalities (3.9) and (3.10), we derive φ− t2k φ − = ϕ− t ϕ+ h and this leads to −1 m−2k φ− φ ϕ− ϕ+ h . − =t

(3.11)

In the last equality, performing the change of variable t → t−1 , we get that −1 2k−m φ ϕ − φ− = t −ϕ + h. Now, taking the inverse of both sides of the last formula, one obtains: −1 −1 m−2k −1 . φ− φ (3.12) ϕ−1 − =t − ϕ+ h −1 −1 −1 . This From formulas (3.11) and (3.12) we have tm−2k ϕ− ϕ+ h = tm−2k ϕ − ϕ+ h leads us to the following equality: −1 −1 ϕ+ ϕ − hh = ϕ− ϕ+ .

(3.13)

To our reasoning, the most important term in the last equality is now h h. There fore, let us understand better the structure of hh. Firstly, let us assume that h h = const. Rewriting formula (3.8) in a more detail way we will have: λ1 λ2 λ3 λ4 h(t) = exp exp exp exp . t − t1 t − t2 t − t3 t − t−1 3 From here, we also have the following identity: −λ1 t22 −λ2 t21 −λ3 t23 −λ4 t−2 3 exp exp exp , h(t) = c1 exp t − t2 t − t1 t − t3 t − t−1 3

Vol. 62 (2008)


y t1 r

γ γ−

55

Γ0

t0 r r t3 γ+

D+ 0

D

−

x r

t−1 3

r t2 Figure 3.1. The unit circle Γ0 intersected with a Jordan curve γ. where c1 is a certain nonzero constant which can be calculated explicitly (in fact, c1 = exp −λ1 t2 − λ2 t1 − λ3 t−1 3 − λ4 t3 ). Performing the multiplication of the last two formulas, one obtains: λ1 − λ2 t21 λ2 − λ1 t22 h(t) h(t) = c1 exp exp t − t1 t − t2 2 λ3 − λ4 t3 λ4 − λ3 t−2 3 · exp exp . t − t3 t − t−1 3 Hence, we have that h(t) h(t) = h1 (t)h2 (t)h3 (t)h4 (t), where λ1 − λ2 t21 λ2 − λ1 t22 h1 (t) = c1 exp , h2 (t) = exp , t − t1 t − t2 λ3 − λ4 t23 λ4 − λ3 t−2 3 h3 (t) = exp , h4 (t) = exp . t − t3 t − t−1 3 ∞ If h1 ∈ L∞ − (Γ0 ), then h2 ∈ L+ (Γ0 ) (because h2 = c2 h1 , where c2 is a certain nonzero constant). Of course, the same holds true for h3 and h4 . At this point, we arrive at the fact that two of the four functions hi , 1 ≤ i ≤ 4, are from the minus class and two of them are from the plus class. Therefore, without loss of generality we can assume that h1 and h3 belong to L∞ − (Γ0 ), and h2 and h4 belong (Γ ). Consequently, we have a decomposition h h = h− h+ , where h− := h1 h3 to L∞ 0 + and h+ := h2 h4 . From the (3.13) we will have: −1 −1 ϕ+ ϕ − h− h+ = ϕ− ϕ+ .

56


IEOT

Let us introduce the notation: Ψ+ := ϕ+ ϕ − h+ , H− := h− , and Ψ− :=

−1 ϕ−1 − ϕ+ .

The last identity can be therefore presented in the following way: H− Ψ + = Ψ − .

(3.14)

We will use now the same reasoning as in the proof of [8, Theorem 4.13, part (3)]. First of all let us observe that Ψ+ ∈ L1+ (Γ0 ) and Ψ− ∈ L1− (Γ0 ). We claim that the functions Ψ± are analytic in the points of the curve Γ0 , except for the set M = {t1 , t3 }. Let us take any point t0 ∈ Γ0 \ M and surround it by a contour γ, such that Dγ+ ∩ M = ∅ and such that the unit circle Γ0 divides the domain Dγ+ into two simply connected domains bounded by closed curves γ+ and γ− with Dγ++ ⊂ D+ and Dγ+− ⊂ D− (cf. Figure 3.1). Let us make use of the function H− (z)Ψ+ (z) if z ∈ D+ , Ψ(z) = Ψ− (z) if z ∈ D− , which is defined on C \ Γ0 and has interior and exterior nontangential limit values almost everywhere on Γ0 , which coincide due to equality (3.14). We will now evaluate the integral Ψ(z)dz = Ψ(z)dz + Ψ(z)dz . γ

γ+

γ−

L1+ (Γ0 ),

one can verify that Ψ+ ∈ L1+ (γ+ ) (by using the definition Since Ψ+ ∈ of the Smirnov space E1 (Γ0 ) = L1+ (Γ0 ); cf., e.g., [8, Section 2.3]). Therefore, Ψ ∈ L1+ (γ+ ) (H− is analytic in a neighborhood of the point t0 ) and the integral along γ+ is equal to zero (cf. [8, Proposition 1.1] for the Γ0 case). Arguing in a similar way, one can also reach to the conclusion that the corresponding integral along γ− is equal to zero. Thus, γ Ψ(z)dz = 0, and the contour γ can be replaced by any closed rectifiable curve contained in Dγ+ . By Morera’s theorem, Ψ is analytic in Dγ+ . Let us consider a neighborhood O(ti ) of any of the points ti = t2 or −1 1 ti = t−1 − = h+ Ψ+ , where ϕ+ ϕ − ∈ L+ (Γ0 ), we see 3 . Due to the identity ϕ+ ϕ −1 1 that h+ Ψ+ ∈ L+ (Γ0 ). However, Ψ+ is analytic in O(ti ), and the function h−1 + (z) grows exponentially when z approaches ti nontangentially, z ∈ D+ . Since the function (t − ti )n exp(−λi (t − ti )−1 ) does not belong to L1+ (Γ0 ) for any choice of positive integer n, we conclude that Ψ+ = 0, identically. This means that Ψ− = 0, identically. From here we infer that ϕ+ or ϕ− must vanish on a set with positive Lebesgue measure, which gives that Φ is not invertible. Therefore, in this case we obtain a contradiction (due to the reason that Φ was taken to be invertible from the beginning). Let us now assume that h h = c1 = const = 0. From (3.13) we get that ϕ− = −1 −1 −1 m = c ϕ . Hence, Φ = c ϕ t ϕ h. Combining this with (3.9) yields φ t2k φ 1 +

−1 c1 ϕ− tm ϕ − h.

1 −

−

−

−

Rearranging the last equality, one obtains: −1 m−2k c1 ϕ− φ−1 h = φ − . − t − ϕ

(3.15)

Vol. 62 (2008)


57

−1 2 2 We have that (1 − t−1 )φ−1 − ∈ H− (Γ0 ) and (1 − t)φ− ∈ H+ (Γ0 ) (cf. Definition 2.2). −1 If we use the multiplication by (1 − t)(1 − t ) in both sides of formula (3.15), then we will obtain: −1 (1 − t)(1 − t−1 )c ϕ φ−1 tm−2k h = (1 − t)(1 − t−1 )φ ϕ . (3.16) 1 − −

−

−

2 − ∈ It is clear that Θ− ∈ H− (Γ0 ), and that Θ Let us denote Θ− := (1 − 2 H+ (Γ0 ). Rewriting formula (3.16) and having in mind the introduced notation, we get: − ϕ (3.17) c2 Θ− ϕ− tm−2k+1 h = Θ − ,

t−1 )φ−1 − .

where c2 = −c1 . Set N := m − 2k + 1. If N ≤ 0, then we have a trivial situation. Therefore, let us assume that N > 0. In this case, we will rewrite the formula (3.17) −1 − ϕ . From the last equality we have that in the following way: c2 Θ− ϕ− tN = Θ −h 1 the right-hand side belongs to L+ (Γ0 ). Therefore, the left-hand side must also belong to L1+ (Γ0 ). This means that tN must “dominate” the term Θ− ϕ− , which in its turn implies that: Θ− ϕ− = b0 + b−1 t−1 + · · · + b−N +ν t−ν + · · · + b−N t−N ,

0≤ν≤N .

In particular, this shows that we will not have terms with less exponent than −N. In addition, the last equality directly implies that ν N − ϕ Θ − = b0 + b−1 t + · · · + b−N +ν t + · · · + b−N t .

From the last three equalities we obtain that: h = c−1 2

b0 + b−1 t + · · · + b−N +ν tν + · · · + b−N tN . b−N + b−N +1 t + · · · + b−N +ν tN −ν + · · · + b0 tN

We are left to observe that h ∈ L∞ − (Γ0 ). Due to its special form (cf. (3.8)), h cannot be represented as a fraction of two polynomial functions (at least near the standard almost periodic discontinuity points). Hence, once again, we arrive at a contradiction. 2. Let the conditions of proposition (ii) be satisfied. Then, σtj (Φ) > 0 for j = 1, 4. Now, by using the argument of passage to the adjoint operator in the last case 1., we can conclude that in the present conditions T Hφ is left-invertible and dim Coker T Hφ = ∞. 3. If the conditions of proposition (iii) are satisfied, then σtj (Φ) will have different signs (cf. formulas (3.3)–(3.6)). Therefore, by the ∆-relation after extension and Theorem 2.20, we will obtain that dim Ker T Hφ + dim Ker(Tφ − Hφ ) = 0, and that dim Coker T Hφ + dim Coker(Tφ − Hφ ) = 0. As far as the dimensions cannot be negative, we will have that both defect numbers of the operator T Hφ must vanish; hence, dim Ker T Hφ = dim Coker T Hφ = 0. We are left to prove that T Hφ is not normally solvable. Let us assume the contrary, i.e. let T Hφ be normally solvable. Then we immediately conclude that T Hφ is invertible, due to the triviality of the defect numbers.

58


IEOT

Hence (by Theorem 2.5) φ admits a weak even asymmetric factorization with index zero: φ = φ− φe , (3.18) where φ− and φe have the appropriate properties (as stated in the Definition 2.2). From (3.18), we obtain that: −1 −1 = φ φ φφ − − .

(3.19)

As far as Φ admits a factorization (cf. (3.7)), we have: Φ = h1 h2 ϕ− ϕ+ tm ,

(3.20) L∞ − (Γ0 )

(h−1 1

where h1 and h2 are exponential type functions such that h1 ∈ ∈ U), ±1 ±1 2 2 (Γ ) (h ∈ U), ϕ ∈ L (Γ ), ϕ ∈ L (Γ ), and m is an integer. h2 ∈ L ∞ 0 2 0 − + + − 0 + Combining (3.19) and (3.20), we obtain that −1 m φ− φ − = h1 h2 ϕ− ϕ+ t .

(3.21)

From the last equality it also follows that −1 −1 −1 −1 m h1 h2 ϕ− ϕ+ tm = h 1 h2 ϕ− ϕ+ t . From here, rearranging the terms of the last equality, one obtains: −1 −1 h h =ϕ ϕ . ϕ ϕ h h +

− 1 1 2 2

+

−

(3.22)

1 As it was shown in the proof of proposition (i), we can factorize the functions h1 h and h2 h2 in the following way (in case that h1 h1 h2 h2 = const): 1 h1 h 2 h2 h

+ = h− 1 h1 , + = h− 2 h2 .

1 h2 h 2 in the convenient way: These equalities allow us to factorize h1 h 1 h2 h 2 = h− h+ , h1 h − −1 + + ∞ ∞ where h− := h− 1 h2 ∈ L− (Γ0 ) (h− ∈ U) and h+ := h1 h2 ∈ L+ (Γ0 ) (h+ ∈ U) are exponential type functions. Recalling formula (3.22) within this notation, we have: −1 −1 ϕ+ ϕ − h+ h− = ϕ+ ϕ− . 1 Let us also introduce the notation: Ψ+ := ϕ+ ϕ − h+ ∈ L+ (Γ0 ), H− := h− and Ψ := ϕ−1 ϕ−1 ∈ L1 (Γ ). We will therefore have: −

+

−

−

0

H− Ψ + = Ψ − . Now we are in a very similar situation as in the proof of proposition (i) of the present theorem. Arguing in a very similar way as in the proof of part 1., we can obtain that Ψ+ = Ψ− = 0, identically. This leads to the conclusion that Φ is not invertible – which is a contradiction. Consequently, in this case T Hφ is not a normally solvable operator. 1 h2 h 2 = const = 0. Let us now consider the case when h1 h

Vol. 62 (2008)


59

Similarly as in the proof of the part 1., we have that ϕ+ = c2 ϕ−1 − , where c2 is a nonzero constant. From the equality (3.21), we have that: −1 −1 m φ− φ − = h1 h2 c2 ϕ− ϕ− t . In a very similar manner as in the part 1., we derive the equality: − ϕ c2 Θ− ϕ− tm+1 h1 h2 = Θ − (with Θ− := (1 − t)φ−1 − ), and may rewriting it in the form: −1 t−m−1 h−1 − = c2 Θ− ϕ− . 1 h2 Θ − ϕ 1 Assume that m > −1. Then, by denoting Ψ+ := h−1 − ∈ L+ (Γ0 ), Ψ− := 1 Θ− ϕ 1 −m−1 −1 ∞ c2 Θ− ϕ− ∈ L− (Γ0 ) and H− := t h2 ∈ L− (Γ0 ), we obtain:

H− Ψ + = Ψ − . This is enough to reach to a contradiction (by arguing in the same way as above). Let us now assume that m ≤ −1. For this case, we will use the notation: −1 1 1 Ψ+ := t−m−1 h−1 − ∈ L+ (Γ0 ), Ψ− := c2 Θ− ϕ− ∈ L− (Γ0 ), and H− := h2 ∈ 1 Θ− ϕ ∞ L− (Γ0 ). Then, also in this case we will obtain a corresponding equality with the appropriate structure H− Ψ + = Ψ − which also leads us to a contradiction. Therefore, we conclude that T Hφ is not normally solvable under the conditions of proposition (iii). We will present in the next theorem the general case of a symbol φ with n ∈ N points of standard almost periodic discontinuities. Theorem 3.2. Suppose that the function φ ∈ GL∞ (Γ0 ) is continuous in the set Γ0 \ {tj }nj=1 , and has standard almost periodic discontinuities at the points tj , 1 ≤ j ≤ n. In addition, assume that σtj (φ) = 0 for all 1, n. (i) If σtj (φ) + σt−1 (φ) = 0 for all j = 1, n, then the operator T Hφ is Fredholm. j

(ii) If σtj (φ) + σt−1 (φ) ≤ 0 for all j = 1, n, and there is at least one index j j for which σtj (φ) + σt−1 (φ) = 0, then the operator T Hφ is right-invertible in j

L2+ (Γ0 ) and dim Ker T Hφ = ∞. (iii) If σtj (φ) + σt−1 (φ) ≥ 0 for all j = 1, n, and there is at least one index j j for which σtj (φ) + σt−1 (φ) = 0, then the operator T Hφ is left-invertible in j

L2+ (Γ0 ) and dim Coker T Hφ = ∞. (iv) If (σtj (φ) + σt−1 (φ))(σtl (φ) + σt−1 (φ)) < 0 for at least two different indices j j l and l, then dim Ker T Hφ = dim Coker T Hφ = 0 and the operator T Hφ is not normally solvable.

60


IEOT

Since the proof of this theorem goes along the same methods as in the proof of Theorem 3.1, we will not present here the corresponding fully detailed proof but just the following sketch of proof. Proof Sketch. For a continuous φ ∈ GL∞ (Γ0 ) in Γ0 \ {tj }nj=1 , and with standard almost periodic discontinuities at the points tj , 1 ≤ j ≤ n, such that σtj (φ) = 0 −1 (as in the previous case of for all 1 ≤ j ≤ n, we will work with Φ := φφ Theorem 3.1). In general, this Φ will have 2n points of standard almost periodic discontinuities. In addition, the formula σtj (Φ) = σtj (φ) + σt−1 (φ) , j

allows us to employ the ∆-relation after extension and to deduce the above stated right and left invertibility properties of T Hφ , as well as the triviality of the kernel and the cokernel of T Hφ , and the Fredholm property of T Hφ (under the corresponding different assumptions). The propositions which deal with the dimensions of the kernel and the cokernel under the assumptions in (ii) and (iii), and also the issues about the normal solvability of T Hφ , are proved in the same manner as in the proof of three discontinuity points case. In fact, e.g., to prove the formula for the dimension of the kernel in case (ii), the method used in the proof of the part (i) of Theorem 3.1 also works here for the situation of n points of standard almost periodic discontinuities. In this situation, instead of the factorization (3.7), we will have   2n −1 exp(λj (t − tj ) ) ϕ(t) , Φ(t) =  j=1

and therefore we have to choose now h(t) =

2n

exp(λj (t − tj )−1 ) .

(3.23)

j=1

Thus, a corresponding reasoning as in the proof of Theorem 3.1 applies also in here. Finally, note that the assumption in propositions (ii) and (iii) which requires that there is at least one index j for which σtj (φ) + σt−1 (φ) = 0 was automatically j fulfilled in Theorem 3.1, and is added in here only for the matter of excluding these cases to fall in the situation of the present proposition (i). Remark 3.3. Note that in the first case of the last theorem we will have that −1 ) has an invertible continuous the Toeplitz operator TΦ (with symbol Φ = φφ symbol, and hence it is a Fredholm operator. As a direct conclusion from the last theorem, if we consider only one point with standard almost periodic discontinuity, we have the following result.

Vol. 62 (2008)


61

Corollary 3.4. Let the function φ ∈ GL∞ (Γ0 ) be continuous on the set Γ0 \ {t0 } and have a standard almost periodic discontinuity at the point t0 with σt0 (φ) = 0. (i) If σt0 (φ) < 0, then the operator T Hφ is right-invertible in L2+ (Γ0 ) and dim Ker T Hφ = ∞. (ii) If σt0 (φ) > 0, then the operator T Hφ is left-invertible in L2+ (Γ0 ) and dim Coker T Hφ = ∞.

4. Examples In this last section we would like to present two simple examples for illustrating some of the above presented theory. As for the first example, let us consider the Toeplitz operator Tρ1 : L2+ (Γ0 ) → 2 L+ (Γ0 ), where i i 1 ρ1 (t) = exp exp exp , t ∈ Γ0 . t−i t+i t−1 From the definition of ρ1 it is clear that it is an invertible element. It is also clear that ρ1 has three points of standard almost periodic discontinuities (namely, the points i, −i and 1). A direct computation allows the conclusion that σi (ρ1 ) = 1 ,

σ−i (ρ1 ) = −1 ,

σ1 (ρ1 ) = 1 .

Hence, Tρ1 is not normally solvable and dim Ker Tρ1 = dim Coker Tρ1 = 0 (cf. Theorem 2.20, part 3). Let us analyze the corresponding Toeplitz plus Hankel operator T Hρ1 : L2+ (Γ0 ) → L2+ (Γ0 ), with symbol ρ1 . Direct computations lead us to the following equalities and an inequality: σi (ρ1 ) + σi−1 (ρ1 ) = σi (ρ1 ) + σ−i (ρ1 ) = 0 , σ−i (ρ1 ) + σ(−i)−1 (ρ1 ) = σ−i (ρ1 ) + σi (ρ1 ) = 0 , σ1 (ρ1 ) + σ(1)−1 (ρ1 ) = 2σ1 (ρ1 ) = 2 > 0 . Applying proposition (iii) of Theorem 3.2, we conclude that T Hρ1 is a leftinvertible operator with infinite cokernel dimension. As a second example, we will consider an adaptation of the first example in which a Toeplitz operator with a particular symbol will be not normally solvable but the Toeplitz plus Hankel operator with the same symbol will be invertible. Let us work with the Toeplitz operator Tρ2 : L2+ (Γ0 ) → L2+ (Γ0 ), where i i ρ2 (t) = exp exp , t ∈ Γ0 . t−i t+i The symbol ρ2 is invertible, and has standard almost periodic discontinuities only at the points i and −i. In particular, we have σi (ρ2 ) = 1 ,

σ−i (ρ2 ) = −1 .

Hence, Tρ2 is not normally solvable (cf. Theorem 2.20, part 3).

(4.1)

62


IEOT

Let us now look for corresponding properties of the Toeplitz plus Hankel operator T Hρ2 : L2+ (Γ0 ) → L2+ (Γ0 ), with symbol ρ2 . It turns out that by using (4.1) and proposition (i) of Theorem 3.2 we conclude that T Hρ2 is a Fredholm operator. In fact, in this particular case, we can even reach to the stronger conclusion that −1 T Hρ2 is invertible. Indeed, ρ2 ρ −1 is invertible (since it is 2 = 1 and therefore T ρ2 ρ2

the identity operator on L2+ (Γ0 )). Thus, the ∆-relation after extension ensures in this case that T Hρ2 is also an invertible operator on L2+ (Γ0 ). Acknowledgement. The authors would like to thank the referees for their valuable remarks.

References [1] E. L. Basor and T. Ehrhardt, Factorization theory for a class of Toeplitz + Hankel operators, J. Oper. Theory 51 (2004), 411–433. [2] F. D. Berkovich and E. M. Konyshkova, On a case of Riemann boundary value problems with infinite index (in Russian), Soobshch. Nauch. Mat. Obs., Rostov-onDon (1968), 158–164. [3] L. P. Castro and F.-O. Speck, Regularity properties and generalized inverses of delta-related operators, Z. Anal. Anwendungen 17 (1998), 577–598. [4] L. P. Castro and F.-O. Speck, Relations between convolution type operators on intervals and on the half-line, Integral Equations Oper. Theory 37 (2000), 169–207. [5] L. A. Coburn and R. G. Douglas, Translation operators on the half-line, Proc. Natl. Acad. Sci. USA 62 (1969), 1010–1013. [6] R. V. Duduchava and A. I. Saginashvili, Convolution integral equations on a half-line with semi-almost-periodic presymbols, Differ. Equations 17 (1981), 207–216. [7] V. B. Dybin and V. N. Gaponenko, The Riemann boundary value problem with quasi-periodic degeneracy of coefficients (English; Russian original), Sov. Math., Dokl. 14 (1973), 1516–1520; translation from Dokl. Akad. Nauk SSSR 212 (1973), 1046–1049. [8] V. Dybin and S. M. Grudsky, Introduction to the Theory of Toeplitz Operators with Infinite Index, Operator Theory: Adv. and Appl. 137. Birkh¨ auser, Basel, 2002. [9] I. Gohberg and I. A. Fel’dman, Wiener-Hopf integro-difference equations (English; Russian original), Sov. Math., Dokl. 9 (1968), 1312–1316; translation from Dokl. Akad. Nauk SSSR 183 (1968), 25–28. [10] I. Gohberg and I. A. Fel’dman, Integro-difference Wiener-Hopf equations, Acta Sci. Math. 30 (1969), 119–137. [11] I. Gohberg and I. A. Fel’dman, Convolution Equations and Projection Methods for their Solution (translated from the Russian), Translations of Mathematical Monographs 41, Providence, R.I.: American Mathematical Society, 1974. [12] S. M. Grudsky and V. B. Dybin, The Riemann boundary value problem with discontinuities of almost periodic type in its coefficient (English; Russian original), Sov. Math., Dokl. 18 (1977), 1383–1387; translation from Dokl. Akad. Nauk SSSR 237 (1977), 21–24.

Vol. 62 (2008)


63

[13] N. Karapetiants and S. Samko, Equations with Involutive Operators, Boston, MA: Birkh¨ auser, 2001. [14] E. M. Konyshkova, The solution of a characteristic singular integral equation with infinite index (Russian), Soobshch. Akad. Nauk Gruz. SSR 65 (1972), 535–538. [15] M. F. Kulagina, Riemann boundary value problem for almost periodic functions, Complex Analysis and Applications, Proc. Int. Conf., Varna/Bulg., 1984, pp. 301– 306. [16] A. P. Nolasco and L. P. Castro, A Duduchava–Saginashvili’s type theory for Wiener– Hopf plus Hankel operators, J. Math. Anal. Appl. 331 (2007), 329–341. G. Bogveradze and L.P. Castro Research Unit “Mathematics and Applications” Department of Mathematics University of Aveiro 3810–193 Aveiro Portugal e-mail: [email protected] [email protected] Submitted: July 24, 2007 Revised: August 5, 2008

Integr. equ. oper. theory 62 (2008), 65–76 c 2008 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010065-12, published online July 24, 2008 DOI 10.1007/s00020-008-1606-5


Regularization for Higher Order Singular Integral Equations L¨ uping Chen∗ and Tongde Zhong Abstract. By means of the definition of Hadamard principal value permutation formula and composition formula for higher order singular integrals developing in [5], the regularization problems for higher order singular integral equations with variable coefficients are discussed, and the higher order singular integral equations with constant coefficient are solved. It is the first time to treat the high order singular integral equations of arbitrary degree in the theory of singular integral equations. Mathematics Subject Classification (2000). 32A25, 32A40, 32C10, 32F20. Keywords. Complex hypersphere, higher order singular integrals, Hadamard principal value, permutation formula, regularization problem.

1. Introduction It is well known that principal values of higher order singular integrals play an important role in the study of the Cauchy problem of hyperbolic partial differential equations and higher order singular integral equations. In 1952, Hadamard defined his principal values of higher order Cauchy integrals on the real line by separating the finite part from the divergent integral on the real line([1]). But this definition hardly extends to the case of several complex variables ([2-4]). In 2002, Tao Qian and Tongde Zhong ([5]) study higher order singular integrals on the hypersphere making use of the Euler radial differential operator D = z1 ∂z∂ 1 + · · · + zn ∂z∂n . They discussed two kinds of kernel under the condition that the density function is holomorphic in a neighbourhood of hypersphere and they obtained two different kinds of permutation formula and composite formula. In the present paper, based on these two different kinds of kernel and two different kinds of permutation formula ∗ Corresponding

author. The project was supported by the Natural Science Foundation of Fujian Province of China (No.S0850029), Innovation Foundation of Xiamen University (No.XDKJCX20063019) and the National Science Foundation of China (No.10771174).

66

Chen and Zhong

IEOT

and composite formula we discuss the higher order singular integral equations with Cauchy kernel on the hypersphere. It is the first time to treat the higher order singular integral equations of arbitrary degree, before which one can consider higher order singular integral equations of degree one only. Let B = {ξ ∈ C n |ξ ξ¯ < 1}, ∂B = {ξ ∈ C n |ξ ξ¯ = 1} be an unit ball and its boundary(hypersphere) in C n , where z = (z1 , . . . , zn ), ξ = (ξ1 , . . . , ξn ), z ξ¯ = z1 ξ¯1 + · · · + zn ξ¯n . Denote the area element of hypersphere S by dS(u) = dσ(ζ). If f (z) is holomorphic in unit ball B and continuous on the boundary ∂B, then the Cauchy formula of f (z) is f (ζ) 1 dσ(ζ), (1.1) f (z) = ωn ∂B (1 − z ζ¯ )n where

ωn =

dσ(ζ) = ∂B

2π n . Γ(n)

When z is a point ξ on the boundary ∂B, that is z = ξ ∈ ∂B, and f (ζ) is a function defined on ∂B, f (ζ) 1 dσ(ζ) (1.2) ωn ∂B (1 − ξ ζ¯ )n is called to be a singular integral on hypersphere S. For simplicity, we omit the coefficient ω1n , and call Hp (z, ζ) =

1 , (1 − z ζ¯ )p

z ∈ B ∪ ∂B, p ∈ R

to be a Cauchy type kernel of degree p. When z = ξ ∈ ∂B, p > n, the singular integral defined by Hp (ξ, ζ) is called to be a singular integral, sometimes we write p = n + k, k ∈ R or p = n + l, l ∈ R, and call k or l as the degree of the higher order singular integral. Tao Qian and Tongde Zhong([5]) use Euler radial differential operator D = z1 ∂z∂ 1 +· · ·+zn ∂z∂n to study higher order singular integral on complex hypersphere. In order to assure that the operator D can be differential with respect to higher order singular integral arbitrary times, assume that f (z) ∈ , where the set of functions defined by = {f : C n → C|f is holomorphic in an annulus−δ < |z| < 1+δ, for some δ > 0}, owing to Hartogs’ Theorem, for n ≥ 2 the space automatically becomes = {f : C n → C|f is holomorphic in B1+δ for some δ > 0}. In C, the class R1 is dense in Lp (∂B), 1 ≤ p < ∞; while in C n , n ≥ 2, the p classes is only dense in H+ (∂B), 1 ≤ p < ∞, the Hardy spaces inside the unit

Vol. 62 (2008)

Regularization for Singular Integral Equations

67

ball, which is defined by p H+,l = {f : B → C n |f is holomorphic in B, and there exist functions

gk ∈ Lp (∂B), k = 0, 1, . . . , l such that Dk f (rz) converges to gk (z), k = 0, 1, . . . , l, in the Lp (∂B) norm, as r → 1 − 0}. In the following when we discuss higher order singular integral we always p . assume that f ∈ or f ∈ H+,l In [5] using two kinds of kernel to study higher order singular integrals, that is kernel Hn+s (z, ζ), s > 0 and kernel Dzr H(z, ζ), r > 0, this two kernels can be expressed each other: Let l ∈ Z + , s > 0, then Dzl Hs (z, ζ) =

l

bkl Hk+s (z, ζ),

(1.3)

k=0

where bkl , k = 0, 1, . . . , l are constants depending on s, l, k, if k = 0, l = 0, then b00 = 1. In [5] the Hadamard principal value of the higher order singular integrals defined by the kernel Hn+l (ξ, ζ) and Dξl Hn (ξ, ζ) are defined, and their PoincaréBertrand permutation formula and composite formula are derived. p Definition 1.1 (Hadamard principal value[5] ). If f ∈ or f ∈ H+,l , then FP Dξl Hn (ξ, ζ)f (ζ)dσ(ζ) = P V Hn (ξ, ζ)Dl f (ζ)dσ(ζ) ∂B

FP ∂B

(1.4)

∂B

Hn+l (ξ, ζ)f (ζ)dσ(ζ) = P V

∂B

Hn (ξ, ζ)

l−1 k=0

(

D + 1)f (ζ)dσ(ζ) (1.5) n+k

where notation FP denotes Hadamard principal value, PV denotes Cauchy principal value. Proposition 1.2 (Poincaré-Bertrand formulas[5] ). Suppose that for almost all η on p , and for almost all ζ on ∂B, φ(ζ, ·) is a function ∂B, φ(·, η) is a function in H+,l p + in or H+,l , where l, k ∈ Z ∪ {0}. Then we have Dξl Hn (ξ, ζ)dσ(ζ) Dζk Hn (ζ, η)φ(ζ, η)dσ(η) ∂B

∂B

1 = (Dζ + Dη )l Dηk φ(ξ, ξ) 4 dσ(η)

+ ∂B

and

∂B

Hn (ξ, ζ)Hn (ζ, η)(Dζ + Dη )l Dηk φ(ζ, η)dσ(ζ)

(1.6)

68

Chen and Zhong

∂B

Hn+l (ξ, ζ)dσ(ζ)

Hn+k (ζ, η)φ(ζ, η)dσ(η)

∂B k−1

l−1

=

IEOT

1 Dζ + Dη Dη + 1) + 1)φ(ξ, ξ) ( ( 4 i=0 n + i n+j i=0 l−1 k−1 Dζ + Dη Dη + + 1) + 1)φ(ζ, η)dσ(ζ), dσ(η) Hn (ξ, ζ)Hn (ζ, η) · ( ( n+i n+j ∂B ∂B i=0 j=0 (1.7)

where when l = 0 or k = 0, then the corresponding product of differential operators reduces to the identity operator. p , Proposition 1.3 (Composite formulas[5] ). Suppose φ is a function in or H+,l+k + where l, k ∈ Z ∪ {0}, then for |r| ≤ l, |s| ≤ k, we have 1 Dξr Hn (ξ, ζ)dσ(ζ) Dζs Hn (ζ, η)φ(η)dσ(η) = Dr+s φ(ξ) (1.8) 4 ∂B ∂B

and

∂B

Hn+l (ξ, ζ)dσ(ζ)

=

∂B

Hn+k (ζ, η)φ(η)dσ(η)

l−1 k−1 D 1 D + 1) + 1)φ(ξ), ( ( 4 i=0 n + i n+j j=0

(1.9)

where when l = 0 or k = 0, then the corresponding product of differentiable operators takes values 1.

2. Dirac operator and regularization for higher order singular integral equations with variable coefficients Denote the linear space consists of functions which are complex-valued differentiable and its rth order partial derivatives satisfy H¨ older condition on complex hypersphere ∂B by L∗ . The discussion in this section will proceed in L∗ ∩ or p . In the following we will discuss the regularization problem of higher L∗ ∩ H+,k order singular integral equations with Cauchy kernel on complex hypersphere ∂B. Denote Hn (ξ, ζ) = (1 − ξ ζ¯ )−n , ξ ∈ S = ∂B; Hn+r (ξ, ζ) = (1 − ξ ζ¯ )−n−r , ξ ∈ S = ∂B. Define Dξl Hn f = 2

∂B

Dξl Hn (ξ, ζ)f (ζ)dσ(ζ)

(2.1)

Hn+l (ξ, ζ)f (ζ)dσ(ζ)

(2.2)

and Hn+l f = 2 by Hadamard principal value.

∂B

Vol. 62 (2008)


69

Theorem 2.1. Consider the higher order singular integral equations with Cauchy kernel and variable coefficients on complex hypersphere as follows Sk f ≡ af + bDξk Hn f + Dξk Kn f = h

(2.3)

and (2.4) Sk φ ≡ aφ + bHn+k φ + Kn+k φ = h, p p , f ∈ H+,l , where a(ξ), b(ξ) = k0 (ξ, ξ) and h(ξ) all belong to L∗ ∩ or L∗ ∩ H+,k + l, k ∈ Z , 1 < p < ∞. Dξk Kn f = f (ζ)k(ξ, ζ)dσ(ζ), Kn+k φ = φ(ζ)k(ξ, ζ)dσ(ζ), (2.5) ∂B

∂B

their kernels are k(ξ, ζ) = (k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Dξk Hn (ξ, ζ),

(2.6)

and k(ξ, ζ) = (k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn+k (ξ, ζ), 2

k0 (ξ, ζ) ∈ L∗ .

(2.7)

2

If a and b in formula (2.3) and (2.4) all satisfy a −b = 0 on complex hypersphere, then we can transform (2.3) and (2.4) to partial differential integral equations. Remark 2.2. By formula (1.3) it follows that when k = 0, Dξ0 Hn = Hn , Dξ0 Kn f = (k (ξ, ζ) − k0 (ξ, ξ))Hn (ξ, ζ)dσ(ζ), so when k = 0, equation (2.3) and (2.4) are ∂B 0 identified. Proof. (i) When (2.1) is defined by Hadamard principal value and the kernel k(ξ, ζ) is defined by (2.6), by Definition 1.1 of Hadamard principal value, it follows that all the principal value of higher order singular integral can be expressed by the usual Cauchy principal value, now the equation (2.3) can be regarded as a singular integral equation with usual Cauchy principal value, therefore the equation (2.3) can be regularized by the following singular integral operator with usual Cauchy kernel ([2]): 1 p (aψ − bHn ψ), ψ ∈ L∗ ∩ or L∗ ∩ H+,k . (2.8) R0 ψ = 2 a − b2 Applying R0 to both sides of (2.3) from left, and denote the right-hand side of (2.3) by R0 h = g, while the left-hand side becomes 1 R0 Sk f = 2 [a2 f + abDξk Hn f + aDξk Kn f a − b2 − bHn (af ) − bHn (bDζk Hn f ) − bHn (Dζk Kn f )]. (2.9) In the following we write down the higher order singular integrals of every terms in the right hand side of the above formula concretely. By formulas (2.1) and (2.5) it follows that Dξk Hn f

=2 ∂B

Dξk Hn (ξ, ζ)f (ζ)dσ(ζ),

(2.10)

70

Chen and Zhong

Dξk Kn f =

∂B

(k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Dξk Hn (ξ, ζ)f (ζ)dσ(ζ),

IEOT

(2.11)

Hn (af ) = 2

∂B

Hn (ξ, ζ)[a(ζ)f (ζ)]dσ(ζ),

(2.12)

Hn (ξ, ζ)(b(η)Dηk Hn (ζ, η)f ) = b(ξ)Dζk f (ξ) +4

dσ(ζ)

∂B

∂B

Hn (ξ, ζ)Hn (ζ, η)Dζk [b(η)f (ζ)]dσ(η),

(2.13)

(apply Poincaré-Bertrand permutation formula (1.6) in Proposition 1.2) Hn (Dζk Kn f ) =2 Hn (ζ, η)dσ(η) (k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Dζk Hn (ξ, ζ)f (ζ)dσ(ζ) ∂B ∂B =2 dσ(ζ) (k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)Dζk Hn (ζ, η)f (ζ)dσ(η) ∂B

(2.14)

∂B

(apply the permutation formula of a singular integral and an integral of weak singularity on complex hypersphere ([2])). By Definition 1.1 of Hadamard principal value it follows that those higher order singular integrals appearing in the above formulas can be expressed by the usual Cauchy principal value. Moreover by the definition of the operator R0 ((2.8)) it follows that g = R0 h is defined by the usual Cauchy principal value also. After acting by the operator (2.8), equation (2.3) becomes b2 (ξ)Dk f (ξ) a2 (ξ)f (ξ) − 2 2 − b (ξ) a (ξ) − b2 (ξ) 2a(ξ)b(ξ) + 2 Hn (ξ, ζ)Dk f (ζ)dσ(ζ) a (ξ) − b2 (ξ) ∂B a(ξ) + 2 (k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)Dk f (ζ)dσ(ζ) a (ξ) − b2 (ξ) ∂B 2b(ξ) − 2 Hn (ξ, ζ)[a(ζ)f (ζ)]dσ(ζ) a (ξ) − b2 (ξ) ∂B 4b(ξ) − 2 dσ(ζ) Hn (ξ, ζ)Hn (ζ, η)Dζk [b(η)f (ζ)]dσ(η) a (ξ) − b2 (ξ) ∂B ∂B 2b(ξ) − 2 dσ(ζ) (K0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)Hn (ζ, η)Dk f (ζ)dσ(η) a (ξ) − b2 (ξ) ∂B ∂B =g (2.15)

a2 (ξ)

Vol. 62 (2008)


71

or a2 (ξ) b2 (ξ) f (ξ) − Dk f (ξ) a2 (ξ) − b2 (ξ) a2 (ξ) − b2 (ξ) 2b(ξ) − 2 Hn (ξ, ζ)[a(ζ)f (ζ)]dσ(ζ) a (ξ) − b2 (ξ) ∂B 1 Dk f (ζ)l(ξ, ζ)dσ(ζ) = g + 2 a (ξ) − b2 (ξ) ∂B where the kernel

(2.16)

l(ξ, ζ) = 2a(ξ)b(ξ)Hn (ξ, ζ) + a(ξ)k(ξ, ζ) − 4b(ξ) − 2b(ξ) k(ξ, ζ)Hn (ζ, η)dσ(η),

∂B

b(η)Hn (ξ, ζ)Hn (ζ, η)dσ(η) (2.17)

∂B

this is a partial differential integral equation with usual Cauchy kernel ([6]). 1 Operator R∗ = a + bHn is an inverse operator of R0 = a2 −b 2 (a − bHn ), that ∗ ∗ is R0 R0 = I, applying R0 both sides to (2.16) from left we get (2.3), that is (2.3) and (2.16) are equivalent, hence the solution of (2.16) is a solution of (2.3). (ii) Similarly, when (2.2) is defined by Hadamard principal value and the kernel k(ξ, ζ) is defined by (2.7), equation (2.4) can be regarded as a singular integral equation with usual Cauchy principal value, therefore the equation (2.4) can be regularized by the following singular integral operator with usual Cauchy kernel ([2]): R0 ψ =

a2

1 (aψ − bHn ψ), − b2

p ψ ∈ L∗ ∩ or L∗ ∩ H+,k

(2.18)

Applying R0 to both sides of (2.4) from left, and denote the right hand side of (2.4) by R0 h = g, while the left hand side becomes R0 Sk φ =

1 (a2 φ + abHn+k φ + aKn+k φ a2 − b 2 − bHn (aφ) − bHn (bHn+k φ) − bHn Kn+k φ).

(2.19)

write down the higher order singular integrals of every terms in the right hand side of the formula above concretely, by formulas (2.1) and (2.5) it follows that k−1 Dζ Hn+k φ = 2 + 1)φ(ζ)dσ(ζ), (2.20) Hn (ξ, ζ) ( n+i ∂B i=0 Kn+k φ =

∂B

((k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)

j=0

Hn (aφ) = 2

∂B

k−1

Hn (ξ, ζ)(aφ(ζ))dσ(ζ),

(

Dζ + 1)φ(ζ)dσ(ζ), (2.21) n+j (2.22)

72

Chen and Zhong

Hn (bHn+k φ) = 4

= b(ξ)

k−1 j=0

(

∂B

Hn (ζ, η)dσ(η)b(ξ)

∂B

Hn (ξ, ζ)

k−1

(

j=0

Dζ + 1)φ(ζ)dσ(ζ) n+j

D + 1)φ(ξ) n+j

+4

IEOT

dσ(ζ)

∂B

∂B

Hn (ξ, ζ)Hn (ζ, η)b(η)

k−1 j=0

(

Dζ + 1)φ(ζ)dσ(η), n+j

(2.23)

(apply Poincaré-Bertrand permutation formula (1.7) in Proposition 1.2) Hn Kn+k φ = 2 dσ(ζ) (k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)Hn (ζ, η) ∂B k−1

∂B

Dζ · + 1)φ(ζ)dσ(η) ( n+j j=0

(2.24)

(apply the permutation formula of a singular integral and an integral of weak singularity on complex hypersphere ([2])). By definition 1.1 of Hadamard principal value it follows that those higher order singular integrals appearing in the above formulas can be expressed by the usual Cauchy principal value. Moreover by the definition of the operator R0 ((2.18)) it follows that g = R0 h is defined by the usual Cauchy principal value also. After acting by the operator (2.18), equation (2.4) becomes k−1 D b2 (ξ) a2 (ξ) φ(ξ) − + 1)φ(ξ) ( 2 2 2 2 a (ξ) − b (ξ) a (ξ) − b (ξ) j=0 n + j

+

+

2a(ξ)b(ξ) 2 a (ξ) − b2 (ξ) a(ξ) 2 a (ξ) − b2 (ξ)

2b(ξ) − 2 a (ξ) − b2 (ξ) 4b(ξ) − 2 a (ξ) − b2 (ξ) 2b(ξ) − 2 a (ξ) − b2 (ξ) ·

l−1 i=0

(

∂B

∂B

∂B

Hn (ξ, ζ)

k−1 i=0

k−1

(

j=0

Dζ + 1)φ(ζ)dσ(ζ) n+j

Hn (ξ, ζ)(a(ξ)φ(ζ))dσ(ζ)

∂B

∂B

dσ(ζ) ∂B

Dζ + 1)φ(ζ)dσ(ζ) n+i

(k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)

dσ(ζ)

(

∂B

Hn (ξ, ζ)Hn (ζ, η)b(η)

k−1 j=0

(

Dζ + 1)φ(ζ)dσ(η) n+j

(k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)Hn (ζ, η)

Dζ + 1)φ(ζ)dσ(η) = g n+i

(2.25)

Vol. 62 (2008)


73

or k−1 Dξ a2 (ξ) b2 (ξ) φ(ξ) − + 1)φ(ξ) ( a2 (ξ) − b2 (ξ) a2 (ξ) − b2 (ξ) j=0 n + j 2a(ξ)b(ξ) − 2 Hn (ξ, ζ)φ(ζ)dσ(ζ) a (ξ) − b2 (ξ) ∂B k−1 Dζ 1 + 2 + 1)φ(ζ)l(ξ, ζ)dσ(ζ) = g ( a (ξ) − b2 (ξ) ∂B j=0 n + j

where the kernel

(2.26)

l(ξ, ζ) = 2a(ξ)b(ξ)Hn (ξ, ζ) + a(ξ)k(ξ, ζ) − 4b(ξ) − 2b(ξ) k(ξ, ζ)Hn (ζ, η)dσ(η),

∂B

b(η)Hn (ξ, ζ)Hn (ζ, η)dσ(η) (2.27)

∂B

this is a partial differential integral equation with usual Cauchy kernel ([6]). For the same argument, since R∗ = a + bHn is an inverse operator of R0 = 1 ∗ a2 −b2 (a − bHn ), that is R0 R0 = I, so equation (2.4) and equation (2.26) are equivalent. Remark 2.3. Although on complex hypersphere for Dirac operator there exists composite formula of higher order singular integrals (refer to formulas (1.8) and (1.9)), but they are not appearance as an identity operator, that is in space L∗ ∩ p or L∗ ∩ H+,k when l is an integer greater than 0, Dξl Hn Dξl Hn = I, Hn+l Hn+l = I, therefore operator Rl∗ = a + bDξl Hn (or Rl∗ = a + bHn+l ) is not an inverse operator 1 1 l ∗ of Rl = a2 −b 2 (a − bDξ Hn ) (or Rl = a2 −b2 (a − bHn+l )), that is Rl Rl = I, therefore 1 l when l > 0, we can not use operator Rl = a2 −b2 (a−bDξ Hn ) to regularize equation 1 (2.3), or use operator Rl = a2 −b 2 (a − bHn+l ) to regularize equation (2.4). 1 It is worthwhile to remark that the operator R0 = a2 −b 2 (a − bHn ) may be used to regularize the following higher order partial differential integral equation in a simple and direct way.

Sk f ≡ aDξk f + bDξk Hn f + Dξk Kn f = h

(2.28)

Since by Definition of Hadamard principal value, equation (2.28) can be written as Sk f ≡ aDξk f + 2 Hn (ξ, ζ)Dζl f (ζ)dσ(ζ) ∂B +2 (k0 (ξ, ζ) − k0 (ξ, ξ))k+1 Hn (ξ, ζ)Dζl f (ζ)dσ(ζ) = h. (2.29) ∂B

Let Dξk f (ξ) = φ(ξ),

(2.30)

74

Chen and Zhong

IEOT

(2.29) becomes

Sk f ≡ aφ + 2 Hn (ξ, ζ)φ(ζ)dσ(ζ) ∂B +2 (k0 (ξ, ζ) − k(ξ, ξ))k+1 Hn (ξ, ζ)φ(ζ)dσ(ζ) ∂B

= h.

(2.31)

This is an equation with usual Cauchy kernel discussed in [2], which can be 1 regularized by the operator R0 = a2 −b 2 (a−bHn ), that is if acting on the both sides of (2.31) from left by operator R0 , then (2.31) reduces to a Fredholm equation. Once we obtain the solution φ(ξ) of equation (2.31), then by solving the partial differential equation (2.30), we can obtain the solution of (2.28). 1 Similarly, the operator R0 = a2 −b 2 (a − bHn ) also can be used to regularized the following higher order partial differential singular integral equation. Sk f ≡ a

k−1 j=0

(

D + 1)f + bHn+k + Kn+k f = h. n+j

(2.32)

3. Dirac operator and solution of higher order singular integral equations with constant coefficient In this section we discuss the higher order singular integral equations with constant p coefficient, and the discussion will proceed in space L∗ ∩ or L∗ ∩ H+,k . Consider the higher order singular integral equation with Cauchy kernel on complex hypersphere aHn φ + Hn+k φ + Kφ = h, (3.1) p where a is a fixed complex constant, h ∈ L∗ ∩ or L∗ ∩ H+,k is a given function, p ∗ ∗ k(ξ, ζ) ∈ L ∩ or L ∩H+,k is the kernel of K, which is a complex valued function, and satisfy H¨ older condition with respect to the variable ξ and ζ. Firstly, we study the characteristic equation (3.1) when K = 0,

aHn φ + Hn+k φ = h,

(3.2)

by definition of Hadamard principal value (1.5), it follows that it is a partial differential integral equation. Applying operator M = aI − Hn , (3.3) where Iφ = φ

p ) (φ ∈ L∗ ∩ or L∗ ∩ H+,k

to both sides of (3.2) from left, we have aHn Hn φ + Hn Hn+k φ = Hn h.

(3.4)

Vol. 62 (2008)


75

Apply composite formula (1.9), we have k−1 1 1 D aφ + + 1)φ = Hn h, ( 4 4 j=0 n + j

(3.5)

solving this partial differential equation under suitable boundary value conditions we can obtain the unknown function φ uniquely. For the general equation (3.1) we reduce it to an equivalent Fredholm type equation by using the operator M . Indeed, appling the operator M to both sides of the equation and the composite formula (1.9) for l = 0, and using (3.1) to cancel some terms, we obtain k−1 1 D 1 aφ + + 1)φ + Hn Kφ = Hn h. ( 4 4 j=0 n + j

(3.6)

Exchanging the order of integration for the Cauchy singular integral and non-singular integral, we have H(ξ, η)dσ(η) φ(ζ)k(ζ, η)dσ(ζ) Hn Kφ =

∂Bη

∂Bζ

φ(ζ)dσ(ζ)

= ∂Bζ

∂Bη

k(ζ, η)Hn (ξ, η)dσ(η)

it can be shown without much difficulty that ∂Bη k(ζ, η)Hn (ξ, η)dσ(η) satisfies Hölder condition with respect to ζ and ξ ([2]). Therefore, the equation (3.6) is a Fredholm type partial differential integral equation ([6-7]). Equation (3.6) and (3.1) are equivalent. To prove this we need to show that the solution of (3.6) satisfies equation (3.1). Applying the operator M ∗ = aI + Hn

(3.7)

to the both sides of (3.6) and then using the composite formula (1.9) we obtain (3.1). To summarize, we have Theorem 3.1. Under the assumption for the equation (3.1), we have (i) The solution of the characteristic equation (3.2) of (3.1) can be obtained p by solving the equation (3.5) under suituniquely in L∗ ∩ or L∗ ∩ H+,k able boundary value condition. p , the higher order singular integral equation (3.1) is (ii) In L∗ ∩ or L∗ ∩ H+,k equivalent to the Fredholm type partial differential integral equation (3.6). Acknowledgment The authors wish to express their sincere gratitude to the referee for his valuable suggestions to improve this manuscript.

76

Chen and Zhong

IEOT

References [1] J. Hadamard, Lecture on Cauchy’s problem in linear partial differential equations, New York (1952). [2] S. Gong, Integrals of Cauchy type on the ball, International Press Co., New York (1993). [3] Wang Xiaoqin, Singular integrals and analyticity theorems in several complex variables, Doctoral Dissertation, Uppsala University, Sweden (1990). [4] Qian Tao and Zhong Tongde, Transformation formula of higher order integrals, J. Austral. Math. Soc. (Series A), 68, (2000), 155-164. [5] Qian Tao and Zhong Tongde, Hadamard principal value of higher order singular integrals, Chinese Journal of Contemporary Mathematics, 23 (2002), 181-192. [6] J.M. Appell, A.S. Kalitvin and P.P. Zabrejko, Partial Integral Operators and IntegroDifferential Equations, Marcel Dekker, Inc, New York (2000). [7] S.G. Mikhlin, Integral equation and its application, Commercial Press (in Chinese), Shanghai (1957). L¨ uping Chen and Tongde Zhong School of Mathematical Sciences Xiamen University Xiamen, 361005 P.R. of China e-mail: [email protected] [email protected] Submitted: October 14, 2007. Revised: June 6, 2008.



On a Beurling-Arveson Type Theorem for Some Functional Hilbert Spaces and Related Questions M.T. Karaev

Abstract. We prove in terms of Berezin symbols a Beurling-Arveson type theorem for some functional Hilbert space over the unit disc D of the complex plane C. We give also an affirmative answer to a question of Yang [15]. Mathematics Subject Classification (2000). Primary 47B35. Keywords. Functional Hilbert space, submodule, Arveson space, reproducing kernel, Berezin symbol.

1. Introduction The celebrated Beurling theorem [3] describes all invariant subspaces of the shift operator S, Sf (z) = zf (z) , on the Hardy space H 2 (D) over the disc D = {z ∈ C : |z| < 1} of the complex plane C in terms of inner functions. Namely, for each S-invariant subspace E ⊂ H 2 (i.e., SE ⊂ E) there exists an inner function θ ∈ E SE such that E = θH 2 (see [3], [10], [12]). It is easy to see that such a 2 description of E ∈ Lat (S) is equivalent to PE (λ) = |θ (λ)| (∀λ ∈ D) , where PE is the so-called Berezin symbol of the orthogonal projector PE in H 2 (D) onto E defined by PE (λ) :=

kλ kλ PE , kλ H 2 (D) kλ H 2 (D)

(λ ∈ D) ,

78

Karaev

IEOT

1 where kλ (z) := 1−λz is the reproducing kernel of H 2 (D) . In fact, if E = θH 2 (D) for some inner function θ, then Pθ = Tθ Tθ∗ , which implies that

PE (λ)

= =

Tθ Tθ∗ kλ , kλ θ (λ) 2

kλ H 2 (D)

1 2 kλ H 2 (D)

=

θ (λ) 2

kλ H 2 (D)

Tθ kλ , kλ

kλ , Tθ∗ kλ = θ (λ)θ (λ) = |θ (λ)|2

2 for all λ ∈ D. Conversely, if E ∈ Lat (S) and PE (λ) = |θ (λ)| for some inner 2 function θ and for all λ ∈ D, then by considering that PθH 2 (D) (λ) = |θ (λ)| = is one-to-one (see, for instance Englis PE (λ) for all λ ∈ D, and the map A ←→ A [5], Fricain [6] and Yang [15]), we have PE = PθH 2 (D) , and hence E = θH 2 (D) , as desired. In [1] and [2], Arveson studied an important analytic function space Hd2 in the unit ball Bd in complex d-dimensional space Cd , Bd = z = (z1 , z2 , ..., zd ) ∈ Cd : z < 1 ,

(where z denotes the norm associated with the usual inner product in Cd , 2

2

2

2

z = |z1 | + |z2 | + ... + |zd | ; in dimension d = 1 there is a familiar Hardy space ) with the reproducing kernel 1 kλ (z) = 1 − λ1 z1 − λ2 z2 − ... − λd zd and the normalized reproducing kernel kλ := kλ . For every submodule E of H 2 , kλ

d

Arveson [2] proved that there exists a multiplier sequence {φi } of Hd2 such that PE = Mφi Mφ∗i (SOT ) , (1) i≥1

and hence, as we have seen above, this equality means that 2 PE (λ) = |φi (λ)| . i≥1

If E contains a nonzero polynomial, Arveson [2, Theorem E] proved that {φi } is an inner sequence, i.e., for almost all ζ ∈ ∂Bn with respect to the normalized natural measure σ on ∂Bn , 2 |φi (λ)| → 1 i≥1

as λ → ζ non tangentially, or equivalently, PE (λ) → 1 as λ → ζ non tangentially. The general case was proved by Greene, Richter and Sundberg [8]: for every submodule E, any sequence satisfying (1), is inner. Similar results for the Hardy space H 2 Dd over the polydisk Dd = D×...×D (which is a module over the polynomial ring C[z1 , ..., zd ] with module actions defined by multiplication of functions ) and Bergman space H 2 (Bd ) over the unit

Vol. 62 (2008)

On a Beurling-Arveson Type Theorem

79

ball Bn were obtained by Yang [15, Corollary 4.6] and Guo and Yang [9, Theorem 2.1, Corollaries 2.3 and 2.4 ]. (There are the following inclusions of the corresponding Hilbert spaces of analytic functions in the open ball Bd : Hd2 ⊆ H 2 (∂Bd ) ⊆ H 2 (Bd ) ; note that of these function spaces, Hd2 is the only one that does not contain H ∞ , see Arveson [1].) In the present article we study the analogous questions for some general functional Hilbert spaces. We prove a Beurling-Arveson type theorem for such spaces (see Theorem 1 below). We also give a particular affirmative answer to the following question of Yang [15, Question 1]: if E1 , E2 are two submodules in the Hardy module H 2 (D × D) with the reproducing kernels kλE1 (z) , kλE2 (z) , respectively, then does kλE1 (λ) ≤ kλE2 (λ) , ∀ λ ∈ D × D, imply E1 ⊂ E2 ?

2. A Beurling-Arveson type theorem Let H = H (D) be a functional Hilbert space of complex-valued functions on D. By a multiplier of H we mean a complex-valued function f : D → C with the property f · H ⊆ H. The set of multipliers is a complex algebra of functions defined on the unit disk D which contains the constant functions, and if H itself also contains the constant function 1 it follows that every multiplier must belong to H. The algebra of all multipliers of H is denoted M (H) . H ∞ (D) will denote the Banach algebra of all bounded analytic functions f : D → C with norm f ∞ = supz∈D |f (z)| , and a straightforward application of the closed graph theorem shows that every f ∈ M (H) defines a unique bounded operator Mf on H by way of Mf g = f g, g ∈ H. The main result of this article is the following theorem. Theorem 1. Let H = H (D) be a separable functional Hilbert space of complexvalued functions on the unit disk D with the reproducing kernel kλ,H and with the property that H ∞ (D) ⊂ M (H) . Suppose that the shift operator S is an isometry in H. If E ⊂ H is a closed subspace such that SE ⊂ E, then there exist the functions Φi ∈ E SE with Φi H = 1, i = 1, 2, ..., dim (E SE) , such that   dim(ESE) 1 2 PE (λ) =  |Φi (λ)|  , λ ∈ D. (2) 1 − |λ|2 kλ,H 2H i=1 Proof. Assume that E ⊂ H is invariant under S. Let PE be a projection of H on E. Since S is an isometry in H, SPE S ∗ is the projection of H on SE and PE − SPE S ∗ ESE is the projection of H on E SE. Then (PE − SPE S ∗ ) kλ,H = kλ,H , where

80

Karaev

IEOT

ESE kλ,H is the reproducing kernel of the subspace E SE. From this we have ESE E = PE kλ,H − SPE S ∗ kλ,H = kλ,H − z PE λkλ,H kλ,H E E E = kλ,H − λzkλ,H = 1 − λz kλ,H ,

that is

E ESE 1 − λz kλ,H = kλ,H , λ ∈ D.

(3)

ESE Since kλ,H is the reproducing kernel of E SE and the subspace E SE is separable, there exists an orthonormal basis {Φi }i≥1 of E SE such that

dim(ESE) ESE kλ,H =

Φi (λ)Φi (z) , λ ∈ D.

i=1

Then, by considering this equality in (3), we have E 1 − λz kλ,H =

dim(ESE)

Φi (λ)Φi (z) , λ ∈ D,

(4)

i=1

or

dim(ESE)

Φi (λ)Φi (z) 1 − λz dim(ESE) = kλ (z) Φi (λ)Φi (z)

PE kλ,H

i=1

=

i=1

dim(ESE)

= Mkλ

Φi (λ)Φi (z) ,

i=1 1 ∈ H ∞ (D) is the reproducing kernel of the Hardy space where kλ (z) := 1−λz H 2 (D), and by condition H ∞ (D) ⊂ M (H) , the multiplication operator Mkλ is bounded in H. Then we have: dim(ESE) Mkλ Φi (λ)Φi (z) , kλ,H PE kλ,H , kλ,H = i=1

=

dim(ESE)

Φi (λ)Φi (z) , Mk∗λ kλ,H

i=1

=

dim(ESE)

Φi (λ)Φi (z) , kλ (λ)kλ,H

i=1

dim(ESE)

= kλ (λ)

i=1

2

|Φi (λ)| =

dim(ESE)

1 2

1 − |λ|

i=1

|Φi (λ)|2 ,

Vol. 62 (2008)


and thus kλ,H kλ,H 1 PE , = 2 kλ,H H kλ,H H 1 − |λ| kλ,H 2H

81

dim(ESE) 2

|Φi (λ)| , λ ∈ D,

i=1

which proves the theorem.

Following Nordgren and Rosenthal [14], we will say that a functional Hilbert space H (Ω) of complex-valued functions on some set Ω which is contained in some topological space and has a nonempty boundary ∂Ω, with the normalized kλ,H is weakly convergent to zero whenever reproducing kernel kλ,H is standard, if λ tends to the boundary ∂Ω. The following lemma essentially was proved by the author in [11] (see Lemma 2.4 and formula (2 ) in [11]). Lemma 2. Let H = H (Ω) be a standard functional Hilbert space over some set Ω and E ⊂ H be a closed subspace. Then E is a finite codimensional subspace if and only if (5) lim PUE (λ) = 1 λ→∂Ω

for all unitary operators U on H. Below we derive some corollaries of Theorem 1. Corollary 3. Let H = H (Ω) be a standard functional Hilbert space satisfying the conditions of Theorem 1. Then, for every finite codimensional S-invariant subspace E, we have lim

λ→∂Ω

2

1 − |λ|

1

dim(ESE)

kλ,H 2H

2

|Φi (λ)| = 1

i=1

The proof of this corollary is immediate from Theorem 1 and formula (2) by taking in (5) U = IH , where IH is an identity operator in H. Corollary 4. Let H = H (Ω) be a standard functional Hilbert space satisfying the conditions of Theorem 1, and let kλ,H H ≤ kλ H 2 (D) . Then for every Sinvariant subspace E of finite codimension we have

dim(ESE) 2

|Φi (λ)| ≤ 1, λ ∈ D

i=1

and lim

λ→∂Ω

1 − |λ|

2

1

dim(ESE) 2 kλ,H H

i=1

2

|Φi (λ)| = 1.

(6)

82

Karaev

IEOT

Proof. We have from the inequality kλ,H H ≤ kλ H 2 (D) and formula (4) that

dim(ESE) 2

|Φi (λ)|

=

i=1

= = = ≤ ≤

2

1 − |λ|

2

E kλ,H (λ)

E E kλ,H , kλ,H 2 E 2 1 − |λ| kλ,H H 1 − |λ|2 PE kλ,H 2H 2 2 1 − |λ| kλ,H H 2 2 1 − |λ| kλ H 2 (D) = 1,

1 − |λ|

that is

dim(ESE) 2

|Φi (λ)| ≤ 1

i=1

for all λ ∈ D. The claim (6) follows, as before, from Lemma 2.

By considering that the Hardy module H 2 (D × D) and the Arveson space d 2 H := H C are the standard functional Hilbert spaces (see, Guo [4, p.145]), the following straightforward corollary of Lemma 2, improves the results of Yang [15, Corollary 4.6] and Arveson [2, Theorem E]. 2

Corollary 5. (a) If a closed submodule E ⊂ H 2 (D × D) has a finite codimension, then limλ→T2 PE (λ) = 1, where T2 = ∂D × ∂D. (b) Let E be a closed submodule of the Arveson space H 2 which has a finite codimension. Then every sequence φ1 , φ2 , . . . of multipliers satisfying ∗ n Mφn Mφn = PE is an inner sequence, with 2 lim |φn (λ)| = 1 λ→ζ

n

for all ζ ∈ ∂Bd .

3. On a property of submodules of H 2 (D × D) In this section we partially solve a question of Yang posed in [15, Question 1], which was mentioned above in Section 1. Before giving our result, we recall the definition of the so-called distance function of Nikolski [13]. Let Ω be a connected domain of Cn and X be a Banach space of analytic functions on Ω continuously imbedded into the space Hol (Ω) of all such functions in Ω, and let ϕλ , λ ∈ Ω, be the point evaluation functional on X, ϕλ (f ) := f (λ),

Vol. 62 (2008)


83

f ∈ X. Nikolski introduces in [13] (see also Fricain [6]), the distance function, defined in Ω for a closed subspace E ⊂ X, by θE (λ) := sup {|f (λ)| : f ∈ E, f ≤ 1} , λ ∈ Ω. In other words, θE (λ) = ϕλ | E , λ ∈ Ω. When X = H, the functional Hilbert space on Ω with reproducing kernel kλ , it is easy to verify that ϕλ | E = PE kλ and 2 2 kλE (λ) = θE (λ) = kλ PE (λ) , λ ∈ Ω. (7) Recall also that a vector f ∈ E is a cyclic vector for the submodule E of the Hardy space H 2 (Dn ) , where Dn := D× · · · × D, if {pf : p ∈ C[z1 , z2 ]} is dense in E. Let f ∈ H 2 (D × D); then we will write [f ] for the smallest invariant subspace of the operator tuple Mz := (Mz1 , ..., Mzn ) acting on H 2 (Dn ) . Thus, [f ] is the closure of the polynomial multiples of f in H 2 (Dn ) and E is a cyclic submodule if E = [f ] for some f ∈ E. The main result of this section is the following theorem. Theorem 6. If E1 , E2 are two submodules in the Hardy space H 2 (Dn ) with the reproducing kernels kλE1 (z) , kλE2 (z) , respectively, such that E1 = [f0 ] for some f0 ∈ E1 with f0 ≤ 1 (i.e., E1 is a cyclic submodule) and kλE1 (λ) ≤ kλE2 (λ) , ∀λ ∈ Dn , then E1 ⊂ E2 . Proof. It follows from the formula (7) that the inequality kλE1 (λ) ≤ kλE2 (λ) , ∀λ ∈ Dn , is equivalent to θE1 (λ) ≤ θE2 (λ) , ∀λ ∈ Dn . Then we have sup {|g (λ)| : g ∈ E2 , g ≤ 1} = θE2 (λ) ≥ θE1 (λ) ≥ |f (λ)| for each f ∈ E1 with f ≤ 1. Since E2 is a closed subspace, we have sup {|g (λ)| : g ∈ E2 , g ≤ 1} = |Gλ (λ)| for some Gλ ∈ E2 , Gλ ≤ 1. Let us define the function F by the formula F (λ) := Gλ (λ), ∀λ ∈ Dn . Then we have that F ∈ E2 , F ≤ 1 and |F (λ)| ≥ |f (λ)| , ∀λ ∈ Dn . Then by using an argument of the paper [7, Remark 5.2], one easily proves that [f ] ⊆ [F ] . Indeed, we set ϕ = Ff . Since |ϕ (z)| ≤ 1 for all z ∈ Dn , ϕ ∈ H ∞ (Dn ) , and therefore the function ϕ defines a bounded multiplication operator on H 2 (Dn ) , Mϕ f = ϕf and Mϕ = ϕ∞ . Note that for 0 < r < 1 we have ϕr F ∈ [F ] , where ϕr (z) = ϕ (rz) , and ϕr F ≤ F . It follows that ϕr F → f weakly as r → 1, thus f ∈ [F ] and the statement follows. Thus [f ] ⊆ [F ] ⊂ E2 . If we take f as a cyclic vector f0 for E1 , then we have from the last inclusions that E1 ⊂ E2 , which completes the proof. For n = 2, Theorem 6 gives a particular answer to the Yang question [15].

84

Karaev

IEOT

References [1] Arveson, W., Subalgebras of C ∗ -algebras III: Multivariable operator theory, Acta Math. 181(1998), 159 − 228. [2] Arveson, W., The curvature invariant of a Hilbert module over C [z1 , ..., zd ] , J. Reine Angew Math. 522(2000), 173 − 236. [3] Beurling A., On two problems concerning linear tranaformations in Hilbert space, Acta Math. 81(1949), 239 − 253. [4] Chen, X., Guo, K., Analytic Hilbert Modules, Chapman and Hall, London, 2003. [5] Englis, M., Berezin and Berezin-Toeplitz quantizations for general function spaces, Rev. Math. Comput. 19(2006), 385 − 430. [6] Fricain E., Uniqueness theorems for analytic vector-valued functions, J. Math. Sci. (New York) 101(2000), 3193 − 3210; translation from Zap. Nauch. Semin. POMI 247(1997), 242 − 267. [7] Gleason, J., Richter, S., m-Isometric Commuting Tuples of Operators on a Hilbert space, Int. Equat. Oper. Theory 56(2006), 181 − 196. [8] Greene, D., Richter, S., Sundberg, C., The structure of inner multiplications on spoaces with complete Nevanlinna Pick kernels, J. Funct. Anal. 194(2002), 311 − 331. [9] Guo, K., Yang R., The core function of submodule over the bidisc, Ind. Univ. Math. J. 53(2004), 205 − 222. [10] Hoffman K., Banach spaces of analytic functions, Prentice-Hall, 1962. [11] Karaev, M.T., On the Berezin symbol, J. Math. Sci. (New York) 115(2003), 2135 − 2140; translated from Zap. Nauch. Semin. POMI 270(2000), 80 − 89. [12] Nikolski, N.K., Treatise on the shift operator, Springer-Verlag, 1986. [13] Nikolski, N.K., Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann ζ-function, Ann. Inst. Fourier, Grenoble 45(1995), 143 − 159. [14] Nordgren, E., Rosenthal, P., Boundary values of Berezin symbols, Operator Theory: Advances and Applications 73(1994), 362 − 368. [15] Yang R., Beurling’s phenomenon in two variables, Int. Equat. Oper. Theory 48(2004), 411 − 423. M.T. Karaev Suleyman Demirel University Isparta Vocational School Department of Technical Programs 32260 Isparta Turkey e-mail: [email protected] Submitted: January 15, 2008. Revised: June 18, 2008.



Wiener-Hopf Operators with Semi-almost Periodic Matrix Symbols on Weighted Lebesgue Spaces Yu. I. Karlovich and J. Loreto Hernández Abstract. Fredholm criteria and index formulas are established for WienerHopf operators W (a) with semi-almost periodic matrix symbols a on weighted Lebesgue spaces LpN (R+ , w) where 1 < p < ∞, w belongs to a subclass of Muckenhoupt weights and N ∈ N. We also study the invertibility of WienerHopf operators with almost periodic matrix symbols on LpN (R+ , w). In the case N = 1 we also obtain a semi-Fredholm criterion for Wiener-Hopf operators with semi-almost periodic symbols and, for another subclass of weights, a Fredholm criterion for Wiener-Hopf operators with semi-periodic symbols. Mathematics Subject Classification (2000). Primary 47B35; Secondary 42A75, 47A53, 47A68, 47G10. Keywords. Wiener-Hopf operator, almost periodic and semi-almost periodic matrix functions, weighted Lebesgue space, almost periodic factorization, symbol, invertibility, Fredholmness, index.

1. Introduction Given 1 ≤ p ≤ ∞, let Lp (R) be the usual Lebesgue space with norm denoted by · p . A (Lebesgue) measurable function w : R → [0, ∞] is called a weight if w−1 ({0, ∞}) has (Lebesgue) measure zero. For 1 ≤ p < ∞ and a weight w, we denote by Lp (R, w) the weighted Lebesgue space with the norm 1/p p p f p,w := |f (x)| w (x)dx . R

Given N ∈ N, let LpN (R, w) be the Banach space of vector-functions f = (fk )N N 1/p k=1 p with entries fk ∈ Lp (R, w) and the norm f LpN (R,w) = f . If A k p,w k=1 Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.

86

Karlovich and Hern´ andez

IEOT

is a subalgebra of L∞ (R), then AN ×N or [A]N ×N denote the matrix functions a : R → CN ×N whose entries belong to A. For 1 < p < ∞, let Ap (R) denote the set of all weights w on R for which the Cauchy singular integral operator SR given by 1 f (t) (SR f )(x) = lim dt, x ∈ R, (1.1) ε→0 πi R\(x−ε,x+ε) t − x is bounded on the space Lp (R, w). By [16] (also see [11]), w ∈ Ap (R) if and only if 1/p 1/q 1 1 p −q cp,w := sup w (x) dx w (x) dx < ∞, |I| I |I| I I where 1/p + 1/q = 1, I ranges over all bounded intervals I ⊂ R, and |I| means the length of I. The weights w ∈ Ap (R) are called the Muckenhoupt weights. Let F : L2 (R) → L2 (R) denote the Fourier transform, ˆ f (t)eitx dt, x ∈ R. (F f )(x) := f (x) := R

In what follows we assume that 1 < p < ∞ and w ∈ Ap (R). A function a ∈ L∞ (R) is called a Fourier multiplier on Lp (R, w) if the convolution operator W 0 (a) := F −1 aF maps L2 (R) ∩ Lp (R, w) into itself and extends to a bounded operator on Lp (R, w) (notice that L2 (R) ∩ Lp (R, w) is dense in Lp (R, w) if w ∈ Ap (R)). Let Mp,w stand for the set of all Fourier multipliers on Lp (R, w). One can show that Mp,w is a Banach algebra under the norm aMp,w := W 0 (a)B(Lp (R,w)) . Let χ+ be the characteristic function of R+ = (0, ∞). By Lp (R+ , w) we understand the space Lp (R+ , w|R+ ). For a ∈ Mp,w , the Wiener-Hopf operator W (a) is defined on the space Lp (R+ , w) by W (a)f = χ+ W 0 (a)χ+ f, for f ∈ Lp (R+ , w). ˙ = R ∪ {∞} and R = [−∞, +∞], we denote by Cp,w (R) ˙ (resp. Cp,w (R)) For R ˙ (resp. a ∈ C(R)) with the closure in Mp,w of the set of all functions a ∈ C(R) ˙ ˙ finite total variation. Obviously, Cp,w (R) ⊂ C(R), Cp,w (R) ⊂ C(R). To study Wiener-Hopf operators with semi-almost periodic (SAP ) symbols we need the set A0p (R) consisting of all weights w ∈ Ap (R) for which the functions 0 0 eλ : x → eiλx belong to Mp,w for all λ ∈ R. Let w ∈ Ap (R). Then the set AP of all almost periodic polynomials λ∈Λ0 cλ eλ , where cλ ∈ C and Λ0 is a finite subset of R, is contained in Mp,w . We define APp,w as the closure of AP 0 in Mp,w . Clearly, APp,w is a Banach subalgebra of Mp,w . Let SAPp,w denote the smallest closed subalgebra of Mp,w that contains Cp,w (R) and APp,w . It is clear that APp,w ⊂ AP ⊂ L∞ (R),

SAPp,w ⊂ SAP ⊂ L∞ (R).

(1.2)

Let B(X) be the Banach algebra of all bounded linear operators on a Banach space X, and K(X) the closed two-sided ideal of all compact operators in B(X). An

Vol. 62 (2008)

Wiener-Hopf Operators with Oscillating Symbols

87

operator A ∈ B(X) is called normally solvable if Im A is closed in X. An operator A ∈ B(X) is said to be semi-Fredholm if it is normally solvable and at least one of the numbers n(A) := dim Ker A and d(A) := dim(X/Im A) is finite. In that case Ind A := n(A) − d(A). A normally solvable operator A is called Fredholm if n(A) < ∞ and d(A) < ∞, n-normal if n(A) < ∞, d-normal if d(A) < ∞, properly n-normal if n(A) < ∞ and d(A) = ∞, and properly d-normal if d(A) < ∞ and n(A) = ∞. Wiener-Hopf operators with semi-almost periodic symbols on the spaces Lp (R+ ) (1 < p < ∞) were studied by R. V. Duduchava and A. I. Saginashvili [10] (for preceding results on integro-difference operators see [13], [14]). The Fredholm theory of Wiener-Hopf operators with semi-almost periodic matrix symbols on the spaces LpN (R+ ) (1 < p < ∞, N > 1) was constructed by I. M. Spitkovsky and the first author (see [21], [22], [17], [23] and also [3]). The study was based on the concept of almost periodic (AP ) factorization of almost periodic matrix functions (see [3, Section 6.3]). In the present paper we establish Fredholm criteria and index formulas for Wiener-Hopf operators W (a) with matrix symbols a ∈ [SAPp,w ]N ×N on the spaces LpN (R+ , w) where 1 < p < ∞, w ∈ A0p (R) and N ∈ N. We also study the invertibility of Wiener-Hopf operators W (a) with almost periodic matrix symbols on LpN (R+ , w). Appearance of weights leads to serious difficulties in the study of such operators. The main difficulty is related to the fact that the translation operators Uλ given by (Uλ f )(x) = f (x − λ) for λ ∈ R stop to be isometric on the weighted Lebesgue spaces Lp (R, w), and we cannot guarantee, in general, their uniform boundedness. The isometric property of all operators Uλ on all the spaces Lp (R) (1 < p < ∞) allowed earlier to prove the equivalence of the invertibility of WienerHopf operators with almost periodic matrix symbols and the existence of canonical almost periodic factorization for corresponding symbols (see [3, Chapters 9, 19]). Another obstacle is related to the appearance of massive local spectra for W (a). The paper is organized as follows. In Section 2 we recall the definitions of AP factorization and related concepts for almost periodic matrix functions, introduce and characterize the subclass A0p (R) of Muckenhoupt weights related to semi-almost periodic functions being Fourier multipliers on Lp (R, w), expose the Fredholm theory for Wiener-Hopf operators with piecewise continuous symbols on the spaces LpN (R+ , w) and study the Fredholmness of Wiener-Hopf operators with symbols in the Douglas algebras [Dp,w,± ]N ×N generated by matrix functions with ∞ ˙ and H± ∩ Mp,w . entries in Cp,w (R) In Section 3 we establish a weighted analogue of Sarason’s representation of semi-almost periodic (matrix) functions, prove the inverse closedness of the algebras [APp,w ]N ×N and [SAPp,w ]N ×N on the basis of the Bochner-Phillips theorem [1], prove the invertibility of [SAPp,w ]N ×N symbols for semi-Fredholm WienerHopf operators and the invertibility of corresponding almost periodic representatives for Fredholm Wiener-Hopf operators on the space LpN (R+ , w), establish and apply a weighted version of Sarason’s lemma [30].

88


IEOT

Section 4 is devoted to establishing (semi-)Fredholm criteria and index forµ mulas of Wiener-Hopf operators with SAPp,w symbols and semi-periodic (SPp,w ) p symbols on the space L (R+ , w). As a result, we establish a weighted analogue of the Duduchava-Saginashvili theorem [10] (also see [3, Theorem 19.15]) and obtain invertibility criteria for Wiener-Hopf operators with APp,w symbols in terms of mean motions of symbols and, for symbols in AP Wp,w , in terms of their canonical AP Wp,w factorability (see Subsection 4.2). In Sections 5 and 6 we construct Fredholm theories for Wiener-Hopf operators with [SAPp,w ]N ×N symbols on the spaces LpN (R+ , w) under the condition of APp,w factorability of almost periodic representatives of symbols in ±∞ and, respectively, for [SAPp,w ]N ×N symbols with almost periodic representatives in [AP Wp,w ]N ×N provided that logarithms of Muckenhoupt weights in A0p (R) slowly oscillate at ∞ in the sense of [8]. In Section 6, applying results of [18] and [19] on pseudodifferential operators with non-regular symbols, we obtain invertibility and Fredholm criteria for Wiener-Hopf operators with almost periodic and semi-almost periodic matrix symbols, respectively. These criteria are essentially related to the canonical AP W factorability of almost periodic representatives of symbols (see Subsection 2.1).

2. Preliminaries 2.1. Almost periodic matrix functions and their factorization The C ∗ -subalgebra AP of L∞ (R) generated by all the functions eλ (λ ∈ R) is called the algebra of (uniformly) almost periodic functions: AP := alg L∞ (R) {eλ : λ ∈ R}. The C ∗ -algebra SAP := alg L∞ (R) {AP, C(R)} of all semi-almost periodic functions on R is defined as the smallest closed subalgebra of L∞ (R) that contains AP and C(R). Let a ∈ AP. Then there exists the finite number T 1 M (a) := lim a(x)dx, T →+∞ 2T −T which is called the (Bohr) mean value of a, and the Bohr-Fourier spectrum Ω(a) := {λ ∈ R : M (ae−λ ) = 0} of a is at most countable (see [3, Propositions 2.22, 2.23]). Let GA be the group of all invertible elements of a unital algebra A. Theorem 2.1 (Bohr [3, Theorem 2.25]). If a ∈ GAP, then there exist a real number κ(a) and a function b ∈ AP such that a(x) = eiκ(a)x eb(x) for all x ∈ R. For a ∈ GAP, the number κ(a) whose existence is guaranteed by Theorem 2.1 is obviously uniquely determined. It is called the mean motion of a. The number d(a) := eM(b) is called the geometric mean value of the function a. ∞ Let H± be the set of all a ∈ L∞ (R) that are non-tangential limits of bounded ∞ are analytic functions in C± := {z ∈ C : ±Im z > 0}. It is well known that H± ∞ closed subalgebras of L (R). We also consider the Banach subalgebras AP + := algL∞ (R) {eλ : λ ≥ 0}, AP − := algL∞ (R) {eλ : λ ≤ 0} ∞ of L∞ (R). By [3, Corollary 7.7], AP ± = AP ∩H± , and therefore AP − ∩AP + = C.

Vol. 62 (2008)


89

Let AP W be the set of all functions in AP represented in the form a(x) = aj eiλj x , λj ∈ R, aW := |aj | < ∞. j

j

It is easily seen that AP W is a Banach algebra with pointwise algebraic operations and the norm · W . Let AP W + and AP W − stand for the set of all functions a ∈ AP W for which Ω(a) ∈ [0, ∞) and Ω(a) ∈ (−∞, 0], respectively. Obviously, AP W ± are closed subalgebras of AP W. By [3, Definition 8.3], a matrix function a ∈ GAPN ×N admit a right AP factorization if it can be represented in the form a(x) = a− (x)d(x)a+ (x) for all x ∈ R, GAPN±×N

iλ1 x

(2.1)

iλN x

with a± ∈ and d(x) = diag(e ,...,e ), λj ∈ R. Factorization (2.1) is called a right AP W factorization, if a± ∈ GAP WN±×N . A right AP (resp. AP W ) factorization with d(x) = IN is called a canonical right AP (AP W ) factorization. By [3, Theorem 8.2], if a ∈ GAPN ×N admits a right AP factorization a = a− da+ , with d = diag(eλj )N j=1 , then the numbers λ1 , . . . , λN ∈ R are up to their arrangement independent of the particular choice of the factorization and they are called the right AP indices of a. Define κ(a) := (λ1 , . . . , λN ), λ1 ≥ · · · ≥ λN . If a ∈ GAPN ×N has a right AP factorization (2.1) with coinciding right AP indices, then the matrix d(a) = M (a− )M (a+ ) ∈ GCN ×N is independent of the particular choice of the right AP factorization and is called the (right) geometric mean of a. 2.2. Muckenhoupt weights and Fourier multipliers Let 1 < p < ∞ and w ∈ Ap (R). Then each of the sets

λ Ix (p, w) := λ ∈ R : (ξ − x)/(ξ + i) w(ξ) ∈ Ap (R) (x ∈ R),

I∞ (p, w) := λ ∈ R : |ξ + i|−λ w(ξ) ∈ Ap (R) is an open interval of length not greater than 1 which contains the origin: ˙ Ix (p, w) = (−νx− (p, w), 1 − νx+ (p, w)) (x ∈ R)

(2.2)

(2.3)

˙ with 0 < νx− (p, w) ≤ νx+ (p, w) < 1 (see, e.g., [5, Theorem 2.10]). For every x ∈ R, the numbers νx± (p, w) are closely related to the indices of powerlikeness αx (w) and βx (w) of the weight w ∈ Ap (R) (see [2, Section 3.6]): νx− (p, w) = 1/p + αx (w),

νx+ (p, w) = 1/p + βx (w).

˙ By Let P C denote the C ∗ -algebra of all piecewise continuous functions on R. ∞ definition, a ∈ P C if and only if a ∈ L (R) and the one-sided limits a(x ± 0) exist ˙ The set of discontinuities for a ∈ P C is at most countable. at each point x ∈ R. Theorem 2.2 (Stechkin’s inequality). (see, e.g., [3, Theorem 17.1]) If a ∈ P C and a has finite total variation V1 (a), then a ∈ Mp,w and aMp,w ≤ SR B(Lp (R,w)) a∞ + V1 (a) , (2.4) where SR is the Cauchy singular integral operator given by (1.1).

90


IEOT

According to Theorem 2.2, we denote by P Cp,w the closure in Mp,w of the set of all functions a ∈ P C with finite total variation. Obviously, P Cp,w ⊂ P C. Proposition 2.3. Let 1 < p < ∞ and w ∈ Ap (R). For every λ ∈ R, the operator Uλ = F −1 eiλx F : f (x) → (Uλ f )(x) = f (x − λ) f or all x ∈ R, p

is bounded on the space L (R, w) if and only if the function vλ (x) := belongs to the space L∞ (R). In that case Uλ B(Lp (R,w)) = vλ ∞ .

(2.5) w(x+λ) w(x)

(2.6)

Proof. For every λ ∈ R, the operator Uλ is bounded on the space Lp (R, w) if and only if the operator wUλ w−1 I = Uλ vλ I is bounded on the space Lp (R); and their norms coincide. Since the operators Uλ are isometric on the space Lp (R), we obtain Uλ B(Lp (R,w)) = Uλ vλ IB(Lp (R)) = vλ IB(Lp (R)) = vλ ∞ ,

which completes the proof.

By Proposition 2.3, the set A0p (R) introduced in Section 1 can be defined as

A0p (R) = w ∈ Ap (R) : vλ = w(· + λ)/w(·) ∈ L∞ (R) for all λ ∈ R . If 1 < p < ∞ and w ∈ A0p (R), then the operators Uλ = W 0 (eλ ) are bounded on the space Lp (R, w) for all λ ∈ R. Hence, the convolution operators W 0 (a) with symbols a in the Banach subalgebra SAPp,w = alg {APp,w , Cp,w (R)} of Mp,w are also bounded on the space Lp (R, w) if w ∈ A0p (R). According to [24], a weight w ∈ A0p (R) is equivalent to the continuous weight ω ∈ C(R) given by 1/2 ω(x) = exp ln w(x + t) dt , (2.7) −1/2

where the equivalence means that w/ω, ω/w ∈ L∞ (R). Thus, we may without loss of generality assume that w is continuous on R. Moreover, from the equivalence of the weights w and ω it follows that −1 max sup ess |w(x)|, sup ess |w (x)| < ∞ for every T > 0, x∈[−T,T ]

x∈[−T,T ]

which implies that αx (w) = βx (w) = 0 and hence νx± (p, w) = 1/p for all points ˙ except x = ∞. x∈R The following example supply us with a class of weights w ∈ A0p (R) with different indices of powerlikeness α∞ (w) and β∞ (w). Example 2.4. Following [2, Section 3.6] we consider the weight

(δ+ν sin(η log(log |x|))) log |x| e if |x| ≥ e, w(x) = δ if |x| < e, e

Vol. 62 (2008)


91

on R, where the parameters δ, ν, η ∈ R satisfy the relations −1/p < δ − |ν| η 2 + 1 ≤ δ + |ν| η 2 + 1 < 1/q.

(2.8)

Setting h(x) := δ + ν sin(ηx) and calculating the indices of powerlikeness by α∞ (w) = 1 − 2/p − lim sup [h(x) + h (x)], x→+∞

β∞ (w) = 1 − 2/p − lim inf [h(x) + h (x)] x→+∞

according to [20, Section 5], we deduce from [2, Example 2.37] that α∞ (w) = 1 − 2/p − δ − |ν| η 2 + 1, β∞ (w) = 1 − 2/p − δ + |ν| η 2 + 1. (2.9) By (2.8) and (2.9), we −1/p < α∞ (w) ≤ β∞ (w) < 1/q. Passing to the obtain 1−2/p weight ρ(t) = w i 1+t on T and applying the formulas α1 (ρ) = α∞ (w) |1 − t| 1−t and β1 (ρ) = β∞ (w), we infer from [2, Theorem 2.33] that ρ ∈ Ap (T). Hence, w ∈ Ap (R). Furthermore, for |x| > e, we obtain (log w) (x) = |x|−1 νη cos(η log(log |x|)) + δ + ν sin(η log(log |x|)) , and therefore lim|x|→∞ (log w) (x) = 0. Consequently, for every λ ∈ R it follows 0 ˙ that vλ = w(·+λ) w(·) ∈ C(R) and vλ (∞) = 1, which implies that w ∈ Ap (R). Similarly to the case w = 1 considered in [32, Section 3.2], we obtain the following result for weights w ∈ Ap (R). Lemma 2.5. If 1 < p < ∞ and w ∈ Ap (R), then the set Y of all functions ψ ∈ L2 (R) ∩ Lp (R, w) for which F ψ has compact support is dense in Lp (R, w). Proof. As 1 < p < ∞ and w ∈ Ap (R), it follows that w ∈ Lploc (R) and the set C0∞ (R) of all infinitely differentiable functions with compact support is dense in Lp (R, w) (see, e.g., [2, Section 4.2]). Fix f ∈ C0∞ (R) and put fy (x) := f (x − y). Since the functions fy − f have compact support for all y ∈ R and since w ∈ Lploc (R), we deduce from the uniform convergence fy → f as y → 0 that ∆(y) := fy − f p,w → 0 as y → 0. For every function ϕ ∈ L1 (R) with R ϕ(x) dx = 1, we have (f ∗ ϕε )(x) − f (x) = [f (x − y) − f (x)] ϕε (y) dy

(2.10)

(2.11)

R

where ϕε (x) = ε−1 ϕ(x/ε) for x ∈ R and ε > 0. By the Minkowsky inequality for integrals, we infer from (2.11) that p 1/p p f ∗ ϕε − f p,w = f (x − y) − f (x) ϕ (y)dy w (x)dx ε R R 1/p f (x − y) − f (x) p wp (x)dx ≤ |ϕε (y)| dy R R = ∆(y) |ϕε (y)| dy = ∆(εy) |ϕ(y)| dy < ∞. R

R

92


IEOT

Thus, f ∗ ϕε is in Lp (R, w) along with f . Moreover, f ∗ ϕε ∈ L2 (R) ∩ Lp (R, w) because f ∈ L2 (R) and ϕε ∈ L1 (R). Furthermore, from the Lebesgue dominated convergence theorem and (2.10) it follows that lim

ε→0

R

∆(εy) |ϕ(y)| dy = 0, and

therefore f ∗ ϕε − f p,w → 0 as ε → 0. Finally, choosing ϕ ∈ L1 (R) ∩ L2 (R) such that the function F ϕ has compact support, we conclude that the functions f ∗ ϕε belong to the set Y, and hence this set is dense in Lp (R, w). 2.3. Wiener-Hopf operators with piecewise continuous matrix symbols on the spaces LpN (R+ , w) Wiener-Hopf operators with piecewise continuous symbols on the spaces Lp (R+ , w) with 1 < p < ∞ and w ∈ Ap (R) were studied by A. Böttcher and I. M. Spitkovsky [5], [6] (also see [2], [3], [29]). Let ν ∈ (0, 1). The set e2π(x+iν) : x ∈ R is a ray starting at the origin and making the angle 2πν with the positive real half-line. For z1 , z2 ∈ C, the M¨ obius transform Mz1 ,z2 (ζ) := (z2 ζ − z1 )/(ζ − 1) maps 0 to z1 and ∞ to z2 . Thus, A(z1 , z2 ; ν) := Mz1 ,z2 e2π(x+iν) : x ∈ R ∪ z1 , z2 is a circular arc between z1 and z2 , which degenerates to the line segment between z1 and z2 , if ν = 1/2. Finally, given 0 < ν1 ≤ ν2 < 1, we obtain the horn A(z1 , z2 ; ν). H(z1 , z2 ; ν1 , ν2 ) := ν∈[ν1 ,ν2 ]

Theorem 2.6. [3, Theorem 17.7] Let 1 < p < ∞, w ∈ Ap (R), and let νx± (p, w) be defined by (2.2)–(2.3). If a ∈ P Cp,w , then the operator W (a) is Fredholm on the space Lp (R+ , w) if and only if − + ˙ 0∈ / a# p,w (R) := x∈R H a(x − 0), a(x + 0); ν∞ (p, w), ν∞ (p, w) H a(+∞), a(−∞); ν0− (p, w), ν0+ (p, w) . (2.12) If W (a) is Fredholm on Lp (R+ , w), then Ind W (a) = −windp,w a, where windp,w a is the winding number about the origin of the naturally oriented continuous curve − + ˙ := a0p,w (R) A a(x − 0), a(x + 0); (ν (p, w) + ν (p, w))/2 ∞ ∞ x∈R − + A a(+∞), a(−∞); (ν0 (p, w) + ν0 (p, w))/2 . According to [3, Theorem 17.10] we have the following matrix result. Theorem 2.7. Let 1 < p < ∞, w ∈ Ap (R), N ∈ N, and ν0± := ν0± (p, w). For b ∈ [Cp,w (R)]N ×N , the following assertions are equivalent: (i) the operator W (b) is Fredholm on the space LpN (R+ , w); 1 arg ξj ∈ / Z for all ν ∈ [ν0− , ν0+ ] and all (ii) det b(x) = 0 for all x ∈ R and ν + 2π eigenvalues ξj (j = 1, 2, . . . , N ) of the matrix b−1 (+∞)b(−∞);

Vol. 62 (2008)


93

(iii) det b(x) = 0 for all x ∈ R and sp b−1 (+∞)b(−∞) ∩ Σ[1 − ν0+ , 1 − ν0− ] = ∅,

where Σ[1 − ν0+ , 1 − ν0− ] := re2πiϕ : r ∈ [0, ∞], ϕ ∈ [1 − ν0+ , 1 − ν0− ] . If the operator W (b) is Fredholm on the space LpN (R+ , w), then

N 1 Ind W (b) = −ind (det b) + N ν00 − arg ξj , ν00 + (2.13) j=1 2π 1 where ind (det b) := 2π arg det b(x) x∈R is the Cauchy index of det b ∈ C(R) on − + 0 R, ν0 := (ν0 + ν0 )/2, and {x} in (2.13) means the fractional part of x ∈ R. 2.4. Wiener-Hopf operators with symbols in Douglas algebras ∞ Proposition 2.8. If a± ∈ H± ∩ Mp,w and b ∈ Mp,w , then W (a− ba+ ) = W (a− )W (b)W (a+ ). ∞ and b ∈ Proof. As is well known (see, e.g., [3, Proposition 2.17]), if a± ∈ H± ∞ 2 L (R), then for every f ∈ L (R+ ),

W (a− )W (b)W (a+ )f = W (a− ba+ )f.

(2.14)

∞ H±

∩ Mp,w and b ∈ Mp,w because Mp,w ⊂ In particular, (2.14) holds if a± ∈ M2 = L∞ (R). It remains to apply (2.14) to functions f ∈ L2 (R+ ) ∩ Lp (R+ , w) and extend this equality by continuity to all f ∈ Lp (R+ , w). ∞ ∞ For 1 < p < ∞ and w ∈ Ap (R), define Hp,w,± := H± ∩ Mp,w . It is easily seen ∞ ∞ ˙ Hp,w,± that Hp,w,± are closed subalgebras of Mp,w . Let Dp,w,± := alg Cp,w (R), ∞ ˙ and Hp,w,± be the minimal Banach subalgebras of Mp,w that contain Cp,w (R) . p We now prove an L (R+ , w) analogue of the following result by R. G. Douglas [9] (cf. [3, Theorem 2.19]).

Theorem 2.9. Let 1 < p < ∞, w ∈ Ap (R) and N ∈ N. (a) If a± ∈ Dp,w,± and b ∈ Mp,w , then W (a− ba+ ) − W (a− )W (b)W (a+ ) is a compact operator on the space Lp (R+ , w). (b) If a± ∈ [Dp,w,± ]N ×N and det a± is invertible in Dp,w,± , then the operators W (a± ) are Fredholm on the space LpN (R+ , w). Proof. If 1 < p < ∞ and w ∈ Ap (R), then from [3, Lemma 17.13] it follows that ˙ is compact on the space the operator χ+ W 0 (c) − W 0 (c)χ+ I with any c ∈ Cp,w (R) Lp (R, w). Hence, the operator χ+ W 0 (a)χ+ W 0 (b)χ+ I −χ+ W 0 (a)W 0 (b)χ+ I is also ˙ and b ∈ Mp,w or a ∈ Mp,w compact on the space Lp (R, w) whenever a ∈ Cp,w (R) ˙ and b ∈ Cp,w (R). This and Proposition 2.8 imply that the operator W (a− ba+ ) − ∞ ˙ + Hp,w,± W (a− )W (b)W (a+ ) is compact on the space Lp (R+ , w), if a± ∈ Cp,w (R) and b ∈ Mp,w . Hence W a± W (a± ij = ij ) + K, i

j

i

j

94


IEOT

where the sum and product are finite, K is a compact operator on Lp (R+ , w), and ∞ ˙ a± ij ∈ Cp,w (R) + Hp,w,± . The latter implies the assertion of part (a) for b ∈ Mp,w ± ∞ ˙ and a± = i j aij with a± ij ∈ Cp,w (R) + Hp,w,± . Approximating a± ∈ Dp,w,± (n) ± ∞ ˙ in Mp,w by functions a± of the form i j a± ij with aij ∈ Cp,w (R) + Hp,w,± , we obtain the convergent sequence of compact operators (n) (n) (n) (n) Kn = W a− ba+ − W a− W (b)W a+ . (2.15) Passing to limits on both sides of (2.15), we complete the proof of part (a). Part (b) immediately follows from part (a).

If 1 < p < ∞ and w ∈ Ap (R), then Theorem 2.9 implies the following. Corollary 2.10. If a, b ∈ Mp,w and at least one of the functions a, b belongs to ˙ then the operator W (ab)−W (a)W (b) is compact on the space Lp (R+ , w). Cp,w (R), ˙ Let χγ stand for the characteristic function of a set γ. By employing a Cp,w (R) partition of the identity in Mp,w , one can sharpen Corollary 2.10 as follows. ˙ there is an open neighCorollary 2.11. If a, b ∈ Mp,w and if for each point t ∈ R ˙ borhood γ(t) ⊂ R of t such that at least one of the functions χγ(t) a, χγ(t) b belongs ˙ then the operator W (ab) − W (a)W (b) is compact on Lp (R+ , w). to χγ(t) Cp,w (R), Let X be a Banach space. By analogy with [3, p. 56], we call two operators A, B ∈ B(X) weakly Φ-equivalent if either none of them is semi-Fredholm or if both A and B are properly n-normal, or both A and B are properly d-normal, or both A and B are Fredholm. Lemma 2.12. Let 1 < p < ∞, w ∈ Ap (R), N ∈ N. If a, b ∈ [Mp,w \{0}]N ×N and there are matrix functions f± ∈ G[Dp,w,± ]N ×N such that a = f− bf+ , then the operators W (a) and W (b) are weakly Φ-equivalent on the space LpN (R+ , w), and Ind W (a) = Ind W (f− ) + Ind W (b) + Ind W (f+ ). Proof. Theorem 2.9 implies that W (a) = W (f− )W (b)W (f+ ) + K, where the operators W (f± ) are Fredholm and K is a compact operator. It remains to apply general results on (semi-)Fredholm operators (see, e.g., [4], [15]).

3. Wiener-Hopf operators with matrix semi-almost periodic symbols on spaces LpN (R+ , w) 3.1. Representation of semi-almost periodic functions Let V1 (R) ⊂ P C be the Banach algebra of all functions of bounded total variation equipped with the norm f V := f L∞ (R) + V1 (f ). We now establish the following analogue of Sarason’s result [31].

(3.1)

Vol. 62 (2008)


95

Theorem 3.1. Let 1 < p < ∞, w ∈ A0p (R), N ∈ N, and let u be a monotonically increasing real-valued function in C(R) such that u(−∞) = 0 and u(∞) = 1. Then every matrix function a ∈ [SAPp,w ]N ×N can be uniquely represented in the form a = (1 − u)al + uar + a0

(3.2)

˙ N ×N and a0 (∞) = 0. The maps a → al where al , ar ∈ [APp,w ]N ×N , a0 ∈ [Cp,w (R)] and a → ar are (continuous) Banach algebra homomorphisms of [SAPp,w ]N ×N onto [APp,w ]N ×N of norm 1, where a[Mp,w ]N ×N := W 0 (a)B(LpN (R,w)) . Proof. Clearly, it suffices to give the proof only for N = 1. Since u∞ = V1 (u) = 1, we infer from Stechkin’s inequality (2.4) that u ∈ Cp,w (R). ˙ and ψ(∞) = 0, then aψ ∈ Cp,w (R) ˙ and (aψ)(∞) = If a ∈ APp,w , ψ ∈ Cp,w (R) 0. Indeed, we can approximate ψ in Mp,w by continuous functions in V1 (R) vanishing at ∞. Moreover, such functions in V1 (R) we can approximate in the norm of V1 (R) by continuous functions ψn ∈ V1 (R) taking zero values in neighborhoods of ˙ ∩ V1 (R), which implies ∞. Hence, ψ − ψn Mp,w → 0 as n → ∞, and aψn ∈ C(R) in view of the estimate lim aψ − aψn Mp,w ≤ aMp,w lim ψ − ψn Mp,w = 0

n→∞

n→∞

(3.3)

˙ and (aψ)(∞) = 0. that aψ ∈ Cp,w (R) Applying this property, by analogy with [3, Theorem 1.21], one can show that SAPp,w is the closure in Mp,w of the set

˙ a0 (∞) = 0 , (1 − u)al + uar + a0 : al , ar ∈ APp,w , a0 ∈ Cp,w (R), (3.4) which actually is an algebra, and the mappings (1 − u)al + uar + a0 → al and (1 − u)al + uar + a0 → ar are algebraic homomorphisms of (3.4) onto APp,w . It remains to show that the set (3.4) is closed. The assertion will therefore follow as soon as we have proved that if a is of the form (3.2), then al Mp,w ≤ aMp,w , ar Mp,w ≤ aMp,w .

(3.5)

We verify the second inequality of (3.5). Let a = (1−u)al +uar +a0 . Without loss of generality assume that al and ar are almost periodic polynomials and that ˙ has compact support. Moreover, we can also assume that there is a a0 ∈ Cp,w (R) point x0 > 0 such that u(x) = 0 for all x < −x0 and u(x) = 1 for all x > x0 . By [3, Lemma 10.2], there is a sequence hn → +∞ such that lim ar − (ar )hn ∞ = 0,

n→∞

lim ar − (ar )hn ∞ = 0

n→∞

where (ar )hn (x) := ar (x + hn ). Let us show that s-lim ehn W 0 (a)e−hn I = W 0 (ar ) on the space Lp (R, w). n→∞

(3.6)

(3.7)

As ehn W 0 (a)e−hn I = W 0 (ahn ) where ahn (x) := a(x + hn ), we have to prove that W 0 (ahn − ar )f p,w → 0 for every f ∈ Lp (R, w).

(3.8)

96


IEOT

By Lemma 2.5, the set Y of all f ∈ L2 (R) ∩ Lp (R, w) for which F f has compact support is dense in Lp (R, w). Since the operators W 0 (ahn − ar ) are uniformly bounded on the space Lp (R, w), it is sufficient to show (3.8) for f ∈ Y. Fix f ∈ Y. Let χ be the characteristic function of a bounded interval J ⊂ R that contains supp F f . Then W 0 (ahn − ar )f p,w = F −1 (ahn − ar )χF f p,w . For x ∈ J and sufficiently large hn > 0, we get u(x + hn ) = 1, a0 (x + hn ) = 0 and ahn (x) − ar (x) χ(x) = ar (x + hn ) − ar (x) χ(x) =: bn (x)χ(x), whence F −1 (ahn − ar )χF f p,w = F −1 (bn χ)F f p,w ≤ W 0 (bn χ)B(Lp (R,w)) f p,w . By Stechkin’s inequality,

W 0 (bn χ)B(Lp (R,w)) ≤ SR B(Lp (R,w)) bn χ∞ + V1 (bn χ) .

(3.9)

Since bn χ∞ = bn ∞ and V1 (bn χ) ≤ 2bn ∞ + bn ∞ |supp χ|, we infer from (3.6) that the right-hand side of (3.9) tends to zero as hn → ∞. This completes the proof of (3.8) and thus, (3.7). Applying now the Banach-Steinhaus theorem, we infer from (3.7) that 0 W (ar ) p ≤ lim inf ehn W 0 (a)e−hn I p ≤ W 0 (a) p . B(L (R,w))

n→∞

B(L (R,w))

B(L (R,w))

Thus, ar Mp,w ≤ aMp,w . Analogously, we get the first inequality in (3.5).

3.2. Applications of the Bochner-Phillips theorem For 1 < p < ∞, w ∈ A0p (R) and N ∈ N, let Dp,w,N be the closure in B(LpN (R, w)) of the set

0 := dλ Uλ : dλ ∈ L∞ (R), R ⊂ R is a finite set ⊂ B(LpN (R, w)), Dp,w,N 0 N ×N λ∈R0

where the translation operators Uλ are given by (2.5) and have the norm (2.6). We use the norm dL∞ := dIB(LpN (R,w)) for matrix functions d ∈ L∞ N ×N (R). N ×N (R) Lemma 3.2. Given 1 < p < ∞, w ∈ A0p (R) and N ∈ N, for every h ∈ R, define 0 Nh : Dp,w,N → L∞ B= dλ Uλ → dh , N ×N (R), λ∈R0

/ R0 . Then Nh extends to a surjective bounded linear operator where dh = 0 if h ∈ Nh : Dp,w,N → L∞ N ×N (R), where Nh = v−h ∞ if N = 1 and Nh ≤ N v−h ∞ for N > 1. 0 0 Proof. Let N = 1 and Dp,w := Dp,w,1 . First we prove the boundedness of Nh on 0 Dp,w in the case w = 1. Since Nh (B) = N0 (BU−h ), it suffices to show this for h = 0. 0 . We may without Fix a finite set R0 ⊂ R and an operator B = λ∈R0 dλ Uλ ∈ Dp,1 loss of generality assume that 0 ∈ R0 and d0 ∞ = sup essx∈R |d0 (x)| > 0. Fix ε ∈ (0, d0 ∞ ). Then there exists a point x0 ∈ R such that for every neighborhood

Vol. 62 (2008)


97

Vδ (x0 ) := (x0 − δ, x0 + δ) of x0 the set Iδ := x ∈ Vδ (x0 ) : |d0 (x)| ≥ d0 ∞ − ε has a positive Lebesgue measure. Fix now δ > 0 such that for all λ ∈ R0 \ {0}, (x0 − δ + λ, x0 + δ + λ) ∩ (x0 − δ, x0 + δ) = ∅, and put I := Iδ . Hence (I + λ) ∩ I = ∅ for all λ ∈ R0 \ {0}, |I| := mes I > 0 and |d0 (x)| ≥ d0 − ε > 0 for all x ∈ I. Setting d0 (x) := |d0 (x)|/d0 (x) for all x ∈ I, we infer that the characteristic function χI of I satisfies the relations (BχI )(x) χI (x)d0 (x) dx = dλ (x)χI (x − λ) χI (x)d0 (x) dx λ∈R0 R R = d0 (x)χI (x)d0 (x) dx = |d0 (x)| dx. (3.10) R

I

Taking ϕ := χI and ψ := χI d0 , we deduce from the equality (Bϕ)(x) ψ(x) dx R BB(Lp (R)) = sup sup , ϕp ψq 0=ϕ∈Lp (R) 0=ψ∈Lq (R) where 1/p + 1/q = 1, and from the relations (3.10) that (BχI )(x) χI (x)d0 (x) dx |d0 (x)| dx R ≥ d0 ∞ − ε, BB(Lp (R)) ≥ = I χI p χI q |I| which implies that BB(Lp (R)) ≥ d0 ∞ . Consequently, N0 (B)∞ ≤ BB(Lp (R)) because N0 (B) = d0 . Since Nh (B) = N0 (BU−h ) and Uλ B(Lp (R)) = 1 for every λ ∈ R, we infer that Nh (B)∞ ≤ BU−h B(Lp (R)) = BB(Lp(R)) . 0 If w ∈ A0p (R) is arbitrary, along with the operator B = λ∈R0 dλ Uλ ∈ Dp,w −1 −1 0 p we consider the operator wBw I = λ∈R0 dλ v−λ Uλ ∈ Dp,1 ⊂ B(L (R)). By the part already proved, we have −1 ∞ ≤ wBw−1 IB(Lp (R)) = BB(Lp (R,w)) , dh v−h

which implies that −1 ∞ ≤ v−h ∞ BB(Lp (R,w)) . Nh (B)∞ = dh ∞ ≤ v−h ∞ dh v−h

Hence, for every h ∈ R, the operator Nh extends by continuity to all of Dp,w , and Nh B(Dp,w ,L∞ (R)) ≤ v−h ∞ .

(3.11)

On the other hand, if B0 = v−h Uh , then −1 B0 B(Lp (R,w)) = v−h v−h Uh B(Lp (R)) = 1,

and therefore Nh (B0 )∞ = v−h ∞ = v−h ∞ B0 B(Lp (R,w)) .

(3.12)

Finally, from (3.11) and (3.12) it follows that Nh B(Dp,w ,L∞ (R)) = v−h ∞ .

98


IEOT

∞ Let now N > 1. For a matrix function d = (dij )N i,j=1 ∈ LN ×N (R) and a p vector function f = (fk )N older’s inequality we obtain k=1 ∈ LN (R, w), applying H¨ N N p dij (x)fj (x) wp (x)dx df pLp (R,w) = i=1 R j=1 N p N p ≤ N max dij ∞ |fj (x)| wp (x)dx j=1 i,j=1,...,N R N ≤ N max dij p∞ N p/q |fj (x)|p wp (x)dx i,j=1,...,N

= Np

max

i,j=1,...,N

j=1

R

dij p∞ f pLp (R,w) , N

whence dL∞ = dIB(LpN (R,w)) ≤ N N ×N (R)

max

i,j=1,...,N

dij ∞ .

(3.13)

(ij) N 0 Let h ∈ R and B = λ∈R0 dλ Uλ ∈ Dp,w,N , where dλ = (dλ )i,j=1 ∈ ∞ LN ×N (R). It follows from (3.13) and the part already proved for N = 1 that (ij)

Nh (B)L∞ = dh L∞ ≤ N max dh ∞ N ×N (R) N ×N (R) i,j=1,...,N (ij) = N max Nh dλ Uλ λ∈R0 i,j=1,...,N ∞ (ij) ≤ N v−h ∞ max dλ Uλ p λ∈R0 i,j=1,...,N B(L (R,w)) ≤ N v−h ∞ B B(Lp (R,w)) . N

Thus Nh : Dp,w,N →

L∞ N ×N (R)

is bounded and Nh ≤ N v−h ∞ .

W Given 1 < p < ∞, w ∈ A0p (R) and N ∈ N, let VN ×N denote the set of all p operators A ∈ B(LN (R, w)) which can be written in the form a λ U λ , aλ ∈ L ∞ aλ L∞ Uλ B(Lp (R,w)) < ∞, A= N ×N (R), N ×N (R) λ∈R

λ∈R

W where aλ = 0 for at most countably many λ ∈ R. Obviously, VN ×N is a Banach algebra with the norm aλ L∞ Uλ B(Lp (R,w)) . (3.14) AW := N ×N (R) λ∈R

B(LpN (R, w))

instead of E = B(LpN (R)) and literally repeating Setting E := the proof of [3, Theorem 19.10] based on the Bochner-Phillips theorem [1] (also see [3, Theorem 9.5]), we obtain the following. W Theorem 3.3. If 1 < p < ∞, w ∈ A0p (R) and N ∈ N, then the algebra VN ×N is p W inverse closed in B(LN (R, w)), i.e., if an operator A ∈ VN ×N is invertible on the W space LpN (R, w), then the inverse operator A−1 also belongs to VN ×N .

Vol. 62 (2008)


99

W W We also need the Banach subalgebra UN ×N of VN ×N consisting of the opep rators A = λ∈R aλ Uλ ∈ B(LN (R, w)) with constant matrix coefficients aλ ∈ N ×N , where aλ = 0 for at most countably many λ ∈ R and (3.14) takes the form C AW := aλ CN ×N Uλ B(Lp (R,w)) , (3.15) λ∈R

with aλ CN ×N := aλ I

B(Lp N (R,w))

.

W Theorem 3.4. If 1 < p < ∞, w ∈ A0p (R) and N ∈ N, then the algebra UN ×N is p inverse closed in B(LN (R, w)). p W Proof. Let A ∈ UN ×N and suppose that A is invertible on the space LN (R, w). W W −1 W Since UN ×N ⊂ VN ×N , Theorem 3.3 implies that A ∈ VN ×N and therefore A−1 = dλ Uλ , with dλ ∈ L∞ (3.16) N ×N (R), λ∈R A−1 W := dλ L∞ Uλ B(Lp (R,w)) < ∞. N ×N (R) λ∈R

W On the other hand, every operator A ∈ UN ×N commutes with each operator Uh (h ∈ R), and if A is invertible, then its inverse operator A−1 also commutes with all operators Uh . Consequently, for the operator (3.16) and every h ∈ R, we get A−1 = Uh A−1 Uh−1 = Uh dλ Uλ Uh−1 = dλ (x − h)Uλ , λ

whence,

λ

dλ (x)Uλ =

λ

λ

dλ (x − h)Uλ .

By Lemma 3.2, we conclude that dλ (x) = dλ (x − h) for all x, h ∈ R and all λ, W which implies that all dλ are constant matrices in CN ×N . Thus A−1 ∈ UN ×N . Proposition 3.5. If 1 < p < ∞, w ∈ A0p (R) and N ∈ N, then the algebras [APp,w ]N ×N and [SAPp,w ]N ×N are inverse closed in APN ×N and SAPN ×N , respectively, and are therefore inverse closed in L∞ N ×N (R). Proof. For [APp,w ]N ×N , this immediately follows from Theorem 3.4 and (1.2). Applying the inverse closedness of APp,w in AP , one can obtain the inverse closedness of SAPp,w in SAP similarly to [3, Proposition 19.4] with only replacement of APp and SAPp by APp,w and SAPp,w , respectively. Let now a ∈ [SAPp,w ]N ×N . If a ∈ GSAPN ×N , then det a ∈ GSAP. By the inverse closedness of SAPp,w in SAP , it follows that det a ∈ G[SAPp,w ]. Thus a ∈ G[SAPp,w ]N ×N . Proposition 3.5 and Theorem 3.1 immediately imply the following. Corollary 3.6. Let 1 < p < ∞, w ∈ A0p (R) and N ∈ N. If a ∈ G[SAPp,w ]N ×N , then al , ar ∈ G[APp,w ]N ×N .

100


IEOT

3.3. Invertibility of symbols of Wiener-Hopf operators Theorem 3.7. Let 1 < p < ∞, w ∈ A0p (R) and N ∈ N. If a ∈ [SAPp,w ]N ×N and W (a) is semi-Fredholm on LpN (R+ , w), then a ∈ G[SAPp,w ]N ×N . Proof. Modifying a little the proof of this result for w = 1 exposed in [3, Theorem 19.5], we now treat the general case of w ∈ A0p (R). Suppose w = 1. By Proposition 3.5, it suffices to show that a ∈ GSAPN ×N . Assume that a ∈ / GSAPN ×N . Then inf | det a(x)| = 0. We first prove that x∈R

in this case any [Mp,w ]N ×N neighborhood of a contains a b ∈ [SAPp,w ]N ×N such that det b(x) = 0 for all x in some open interval of R. a[Mp,w ]N ×N < δ Given any δ > 0, we can find a ∈ [SAPp,w ]N ×N such that a− and det a(x0 ) = 0 for some x0 ∈ R. Moreover, we may without loss of generality assume that the restriction of a to a closed neighborhood γx0 of x0 is in [C(γx0 ) ∩ V1 (γx0 )]N ×N . For ε > 0, let ϕε be a function ϕε : R → [0, 1] such that ϕε |(x0 −ε,x0 +ε) = 1, ϕε |R\[x0 −2ε,x0 +2ε] = 0 and ϕε is ˙ ∩ V1 (R)]N ×N continuous and monotonous on each side of x0 . Then aϕε ∈ [C(R) for all sufficiently small ε > 0. Put b := a + ϕε ( a(x0 ) − a). Then b ∈ [SAPp,w ]N ×N . If x ∈ (x0 − ε, x0 + ε), then b(x) = a(x0 ) and hence det b(x) = 0. We have N b(x) − a(x) = ϕε (x) a(x0 ) − a(x) =: ϕε (x) cjk (x) j,k=1 ˙ ∩ V1 (R) for all j, k = 1, 2, . . . , N . Applying the relations where ϕε cjk ∈ C(R) lim cjk |(x0 −2ε,x0 +2ε) ∞ = lim V1 cjk |(x0 −2ε,x0 +2ε) = 0, ε→0 ε→0 V1 (ϕε cjk ) ≤ V1 (ϕε )cjk |(x0 −2ε,x0 +2ε) ∞ + ϕε ∞ V1 cjk |(x0 −2ε,x0 +2ε) , we infer that ϕε cjk ∞ → 0 and V1 (ϕε cjk ) → 0 as ε → 0. Hence, by Stechkin’s inequality (2.4), ϕε cjk Mp,w → 0 as ε → 0. Thus, choosing sufficiently small δ > 0 and ε > 0, we conclude from the estimate a − a) + (b − a)[Mp,w ]N ×N ≤ δ + N max ϕε cjk Mp,w b − a[Mp,w ]N ×N = ( j,k

that b − a[Mp,w ]N ×N is as small as desired. a(x0 )d = 0. As det a(x0 ) = 0, there is a nonzero matrix d ∈ CN ×N such that ˙ Since dϕε/2 ∈ [Cp,w (R)]N ×N , from Theorem 2.9 (a) it follows that the operator W (bdϕε/2 ) − W (b)W (dϕε/2 ) ∈ B(LpN (R+ , w)) is compact. Obviously, b(x)dϕε/2 (x) = a(x0 )dϕε/2 (x) = 0

if x ∈ (x0 − ε, x0 + ε),

b(x)dϕε/2 (x) = b(x)d · 0 = 0

if x ∈ / (x0 − ε, x0 + ε).

Consequently, bdϕε/2 ≡ 0, and therefore W (b)W (dϕε/2 ) must be compact. Suppose now that W (a) is n-normal. Then W (b) is also n-normal, and hence, by analogy with [3, Theorem 19.5], the operator W (dϕε/2 ) must be compact on the space LpN (R+ , w) together with W (b)W (dϕε/2 ). As a nonzero Wiener-Hopf

Vol. 62 (2008)


101

operator is never compact, we get a contradiction. Thus W (a) cannot be n-normal. Consideration of adjoints shows that W (a) is not d-normal too. 3.4. Invertibility of the almost periodic representatives Let A and E stand for the smallest closed subalgebras of B(LpN (R, w)) which contain the operators χ± I and the sets {W 0 (a) : a ∈ [APp,w ]N ×N } and {W 0 (a) : a ∈ [SAPp,w ]N ×N }, respectively. Lemma 3.8. If 1 < p < ∞, w ∈ Ap (R), N ∈ N, and K ∈ K(LpN (R, w)), then s-lim ehn Ke−hn I = 0 for every sequence {hn } ⊂ R such that hn → ∞ as n → ∞. n→∞

Proof. By analogy with [3, Lemma 10.1], we conclude that if hn → ∞, then e−hn I converges weakly to zero on Lp (R, w). Indeed, if f ∈ Lp (R, w) and g ∈ Lq (R, w−1 ), then e−ithn f (t)g(t) dt = (f g)ˆ(−hn ), R

1

˙ and (f g)ˆ(∞) = 0, it follows and as f g ∈ L (R) and therefore (f g)ˆ ∈ C(R) that (f g)ˆ(−hn ) → 0. For every compact operator K ∈ B(Lp (R, w)), the weak convergence of e−hn I to zero implies that ehn Ke−hn I → 0 strongly. Modifying the proof of [3, Theorem 19.6], we obtain its weighted analogue. Theorem 3.9. If 1 < p < ∞, w ∈ A0p (R) and N ∈ N, then the maps µr : χ± I → χ± I, W 0 (a) → W 0 (ar ) 0

0

µl : χ± I → χ± I, W (a) → W (al )

(a ∈ [SAPp,w ]N ×N ), (a ∈ [SAPp,w ]N ×N )

extend to (continuous) Banach algebra homomorphisms of E onto A such that µr (A) ≤ Aess , µl (A) ≤ Aess f or all A ∈ E,

where Aess := inf A + KB(LpN (R,w)) : K ∈ K(LpN (R, w)) . Proof. It suffices to consider the case N = 1. We prove the theorem only for µr ; the proof for µl is analogous. Let A = j k Ajk where the sum and the (ordered) products are finite and each Ajk is either of the form χ± I or of the form W 0 (ajk ) = W 0 (1 − u)aljk + uarjk + a0jk , ˙ having compact support in R. Moreover, with aljk , arjk ∈ AP 0 and a0jk ∈ Cp,w (R) we can also assume that there is a point x0 > 0 such that u(x) = 0 for all x < −x0 and u(x) = 1 for all x > x0 . We define µr (Ajk ) (3.17) µr (A) := j

k

0

with µr (χ± I) = χ± I and µr (W (ajk )) = W 0 (arjk ). The theorem will follow as soon as we have shown that µr (A) ≤ A + K for every K ∈ K(Lp (R, w)).

(3.18)

102


IEOT

By [3, Lemma 10.2], there is a sequence hn → +∞ such that (arjk )hn − arjk ∞ → 0, ((arjk ) )hn − (arjk ) ∞ → 0,

(3.19)

the prime denoting the derivative, and (arjk )h (x) = arjk (x + h). Making use of (3.19), we infer from the proof of Theorem 3.1 (see (3.7)) that ehn W 0 (ajk )e−hn I → W 0 (arjk ) = µr (W 0 (ajk )) strongly.

(3.20)

Taking into account (3.17), (3.20) and the relation ehn χ± e−hn I = χ± I = µr (χ± I), we conclude that ehn Ae−hn I → µr (A) strongly. This and Lemma 3.8 imply that ehn (A + K)e−hn I → µr (A) strongly for every K ∈ K(Lp (R, w)). Clearly, the latter gives (3.18). By analogy with [3,Corollary 18.11], taking the operators AW (d) of index zero, where d(x) := diag (x − i)m (x + i)−m , 1, . . . , 1 and m := Ind A, we infer from Theorem 3.9 the following. Corollary 3.10. Let 1 < p < ∞, w ∈ A0p (R) and N ∈ N. If A ∈ E is Fredholm on LpN (R, w), then the operators µl (A) and µr (A) are invertible on LpN (R, w). Corollary 3.10 immediately implies the following two assertions. Corollary 3.11. Let 1 < p < ∞, w ∈ A0p (R) and N ∈ N. If a ∈ [SAPp,w ]N ×N and the operator W (a) is Fredholm on the space LpN (R+ , w), then the operators W (al ) and W (ar ) are invertible on this space. Corollary 3.12. For a ∈ [APp,w ]N ×N , the following assertions are equivalent: (a) W (a) is Fredholm on the space LpN (R+ , w); (b) W (a) is invertible on the space LpN (R+ , w). 3.5. An analogue of Sarason’s lemma and its applications Given 1 < p < ∞ and w ∈ A0p (R), let AP Wp,w be the Banach subalgebra of Mp,w which is composed by the series a = λ aλ eλ with coefficients aλ ∈ C such that W the operators W 0 (a) belongs to the algebra U W := U1×1 ⊂ B(Lp (R, w)). Then according to (3.15) the functions a ∈ AP Wp,w have the norm aW := W 0 (a)W := |aλ | Uλ B(Lp (R,w)) , λ

where the norm Uλ B(Lp (R,w)) = vλ L∞ (R) actually depends on p and w because the function vλ (x) = w(x+λ) is defined for weights w ∈ A0p (R). Note that the w(x) algebras AP Wp,w and U W are commutative. + − Let APp,w (resp. APp,w ) stand for the Mp,w closure of the set of all almost periodic polynomials j aj eλj with λj ≥ 0 (resp. λj ≤ 0). Along with the Ba± ± nach subalgebras APp,w of Mp,w we consider the Banach subalgebras AP Wp,w := ± AP Wp,w ∩ APp,w of AP Wp,w . Clearly, AP 0 ⊂ AP Wp,w ⊂ APp,w ⊂ AP,

± ± AP Wp,w ⊂ APp,w ⊂ AP ± .

We need the next analogue of Sarason’s lemma (cf. [30] and [3, Lemma 3.5]).

Vol. 62 (2008)


103

± Lemma 3.13. If h ∈ APp,w and M (h) = 0, then hv ∈ Dp,w,± for every v ∈ Cp,w (R). ∞ ˙ if ±λ > 0. In particular, eλ v ∈ Cp,w (R) + Hp,w,± + ∞ ˙ Hp,w,+ Proof. Consider the case of h ∈ APp,w . Since Dp,w,+ = alg Cp,w (R), is a + = alg Mp,w {eλ : λ ≥ 0}, we can assume that closed subalgebra of Mp,w and APp,w h = eα with α > 0. According to [3, Lemma 3.5], put

z

1 sin z . 2 − z2 z π 0 The function ϕ is analytic in C, and there is a constant M < ∞ such that Φ(z) :=

ϕ(ζ) dζ,

|ϕ(z)| ≤ M

ϕ(z) := π

e|Im z| for all z ∈ C. 1 + |z|3

Then Φ is also analytic in C, and eiz Φ(z) is a bounded analytic function in C+ . Let σ = Φ|R be the restriction of Φ on R. Then σ ∈ C(R) with σ(±∞) = ±1 (see [3, Lemma 3.5]). Furthermore, since σ is absolutely continuous on R, its total variation can be estimated by ∞ ∞ ∞ dx |σ (x)|dx = |ϕ(x)|dx ≤ M < ∞, V1 (σ) = 1 + |x|3 −∞ −∞ −∞ and then Stechkin’s inequality implies that σ ∈ Cp,w (R). Consequently, every function v ∈ Cp,w (R) can be written in the form v(−∞) + v(+∞) v(+∞) − v(−∞) + σ(αx) + v0 (x), (3.21) 2 2 ˙ and v0 (∞) = 0. By the proof of Theorem 3.1 (see (3.3) for where v0 ∈ Cp,w (R) ˙ Furthermore, as σ ∈ Cp,w (R) a = eα and ψ = v0 ), it follows that eα v0 ∈ Cp,w (R). iαx ∞ ∞ and eα ∈ Mp,w , we infer that e σ(αx) is a function in Hp,w,+ = H+ ∩ Mp,w . iαx ∞ ˙ Finally, because eα v0 ∈ Cp,w (R) and e σ(αx) is in Hp,w,+ , from (3.21) it follows ∞ ˙ + Hp,w,+ that eα v is a function in Cp,w (R) . This implies that hv ∈ Dp,w,+ . v(x) =

Applying Lemma 3.13 we obtain the following Mp,w analogue of [3, Theorem 10.9], which will allow us to reduce studying Wiener-Hopf operators with symbols in [SAPp,w ]N ×N to studying such operators with symbols in [Cp,w (R)]N ×N . Although the scheme of the proof below is similar to that in [3, Theorem 10.9], with natural Mp,w modifications, we will give its complete proof because of subsequent applications in Theorem 3.15. Theorem 3.14. Let a, b ∈ G[SAPp,w ]N ×N and suppose the almost periodic representatives al , bl , ar , br satisfy the relations + ± al = ϕ− with ϕ± l bl ϕl l ∈ G[APp,w ]N ×N , + ± ± ar = ϕ− r br ϕr with ϕr ∈ G[APp,w ]N ×N ,

M (ϕ− l )

=

M (ϕ− r ),

M (ϕ+ l )

=

M (ϕ+ r ).

(3.22)

104


IEOT

Then a = f− bf+ with certain matrix functions f± ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N . + Proof. Put c0 := M (ϕ+ l ) = M (ϕr ) and choose almost periodic matrix polynomials rj eiλj x , ψr+ (x) = c0 + sj eiλj x ψl+ (x) = c0 + j

j

such that λj > 0 for all j, and + −1 ψl+ − ϕ+ [Mp,w ]N ×N , l [Mp,w ]N ×N < q/(ϕl )

(3.23) + −1 [Mp,w ]N ×N ψr+ − ϕ+ r [Mp,w ]N ×N < q/(ϕr ) with some q ∈ 0, 4−1 SR −1 p B(L (R,w)) . Let I := IN be the identity matrix. Put + + −1 −1 v := I + ψl (ϕl ) − I (1 − u) + ψr+ (ϕ+ − I u, (3.24) r ) increasing on R such that where u ∈ Cp,w (R) is a fixed real monotonically function −1 + + −1 u(−∞) = 0 and u(+∞) = 1. As M ψl+ (ϕ+ ) − I = M ψ (ϕ ) − I = 0, we r r l deduce from Lemma 3.13 that + + −1 −1 ψl (ϕl ) − I (1 − u) + ψr+ (ϕ+ − I u ∈ [Dp,w,+ ]N ×N , r ) and, thus, v ∈ [SAPp,w ]N ×N ∩ [Dp,w,+ ]N ×N . By (3.23), + + −1 −1 ψ (ϕ ) − I < q, ψr+ (ϕ+ − I [Mp,w ]N ×N < q, r ) l l [Mp,w ]N ×N whence from (3.24) it follows that v − I[Mp,w ]N ×N ≤ q1 − uMp,w + quMp,w ≤ 4qSR B(Lp (R,w)) < 1 (note that SR B(Lp (R,w)) ≥ 1 because

SR2

= I). This shows that

v ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N . In what follows, we let e−∞ := 0. Put ω(x) := c0 +

j

pj (x)e

iλj x

,

(3.25)

pj (x) :=

(3.26)

sj e−λj /x , for x ≥ 0, rj eλj /x , for x < 0.

(3.27)

Since pj ∈ [C(R)]N ×N ∩ [V1 (R)]N ×N , we deduce from Stechkin’s inequality that pj ∈ [Cp,w (R)]N ×N . Again by Lemma 3.13, ω ∈ [SAPp,w ]N ×N ∩ [Dp,w,+ ]N ×N . ψl+

ψr+

(3.28) ψl+

We infer from (3.23) that and are invertible in As and ψr+ + are almost periodic matrix polynomials, they are even invertible in [AP Wp,w ]N ×N (cf. Theorem 3.4). Thus, j > 0, sj eiλj x , λ sj CN ×N vλ j ∞ < ∞, (ψr+ (x))−1 = c−1 0 + j j jx + −1 −1 iλ rj e , λj > 0, rj CN ×N vλ j ∞ < ∞. (ψl (x)) = c0 + j

+ [APp,w ]N ×N .

j

Vol. 62 (2008)


By analogy with (3.27), we put ω (x) := c−1 0 +

j

pj (x)e

j x iλ

pj (x) :=

,

105

sj e−λj /x , for x ≥ 0, rj eλj /x , for x < 0.

(3.29)

Applying Stechkin’s inequality, we obtain pj [Mp,w ]N ×N ≤ sj CN ×N e−λj /(·) χ+ M + rj CN ×N eλj /(·) χ− M p,w p,w ≤ 2 SR B(Lp (R,w)) sj CN ×N + rj CN ×N , which implies that ω[Mp,w ]N ×N ≤ c−1 0 CN ×N +

j

pj [Mp,w ]N ×N vλ j ∞ < ∞.

(3.30)

Hence, as above, we see that ω ∈ [SAPp,w ]N ×N ∩ [Dp,w,+ ]N ×N . If x > 0, then ω(x) = c0 +

j

ω (x) = c−1 0 +

sj e−λj /x eiλj x = c0 +

j

sj e−λj /x eiλj x = c−1 0 +

j

(3.31)

sj eiλj (x+i/x) = ψr+ (x + i/x),

j

sj eiλj (x+i/x) = [ψr+ (x + i/x)]−1 .

Analogously, for x < 0, ω(x) = ψl+ (x − i/x),

ω (x) = [ψl+ (x − i/x)]−1 .

Consequently, ω ω=ω ω = I on R \ {0}. From (3.28) and (3.31) we therefore get ω ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N .

(3.32)

Combining (3.26) and (3.32) we obtain that g + := v −1 ω ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N . Clearly,

−1 (g + )l = vl−1 ωl = (I + (ψl+ (ϕ+ − I))−1 ψl+ = ϕ+ l ) l ,

−1 − I))−1 ψr+ = ϕ+ (g + )r = vr−1 ωr = (I + (ψr+ (ϕ+ r ) r . Analogously one can construct

g − ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,− ]N ×N such that We now put f− := g −

(3.33)

(3.34)

(3.35)

− − (g − )l = ϕ− (3.36) l , (g )r = ϕr . + −1 − + and f+ := g h a where h := g bg . Then we have

f− bf+ = g − bg + (g − bg + )−1 a = a.

(3.37)

By (3.35), f− ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,− ]N ×N . (3.38) Since g , b, g ∈ G[SAPp,w ]N ×N , it follows that h ∈ G[SAPp,w ]N ×N and thus −

+

f+ ∈ G[SAPp,w ]N ×N .

(3.39)

106


IEOT

We infer from (3.34), (3.36) and (3.22) that + hl = (g − )l bl (g + )l = ϕ− l bl ϕl = al ,

+ hr = (g − )r br (g + )r = ϕ− r br ϕr = ar ,

˙ N ×N and (a − h)(∞) = 0. This implies that and hence a − h ∈ [Cp,w (R)] ˙ N ×N , a−1 h = I − a−1 (a − h) ∈ [Cp,w (R)] ˙ N ×N h−1 a = I + h−1 (a − h) ∈ [Cp,w (R)] and shows that f+ = g + (h−1 a) ∈ G[Dp,w,+ ]N ×N .

(3.40)

Finally, (3.37), (3.38), (3.39) and (3.40) complete the proof.

In the next assertion we construct homotopies of the matrix functions (3.33) and (3.35) to I in G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N (cf. [22, Lemma 3.2]). Theorem 3.15. Let a ∈ G[SAPp,w ]N ×N and suppose the almost periodic representatives al , ar satisfy the relations (3.22) with some bl , br ∈ GCN ×N and + − + M (ϕ− l ) = M (ϕr ) = I, M (ϕl ) = M (ϕr ) = I.

(3.41)

Then there exist matrix functions b ∈ G[Cp,w (R)]N ×N such that b(−∞) = bl , b(+∞) = br and a = g − bg + with certain matrix functions g ± ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N , which are homotopic to the identity matrix I, respectively, within the open sets G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N . Proof. In the proof of Theorem 3.14 we constructed two matrix functions g ± ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N such that the matrix function b := (g − )−1 a(g + )−1 belongs to G[Cp,w (R)]N ×N in view of (3.22). It remains to show that the matrix functions g ± are homotopic to the identity matrix I within the open sets G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N , respectively. Let us prove this for g + (for g − the proof is analogous). Fix θ ∈ [0, 1]. Since vθ := (1 − θ)v + θI ∈ [SAPp,w ]N ×N ∩ [Dp,w,+ ]N ×N and vθ − I[Mp,w ]N ×N = |1 − θ| v − I[Mp,w ]N ×N < 1 due to (3.25), we deduce that vθ ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N . On the other hand, modifying (3.27) and (3.29), we define the matrix functions ωθ (x) := I + pj (x)eiλj x e−λj θ/(1−θ) , j (3.42) pj (x)eiλj x e−λj θ/(1−θ) , ω θ (x) := I + j

where pj , pj ∈ [Cp,w (R)]N ×N . Hence, by Lemma 3.13, the matrix functions x → pj (x)eiλj x e−λj θ/(1−θ) ,

x → pj (x)eiλj x e−λj θ/(1−θ)

Vol. 62 (2008)


107

∞ ˙ + Hp,w,+ belong to [SAPp,w ]N ×N ∩ [Cp,w (R) ]N ×N for every j and every θ ∈ [0, 1]. Since j in ωθ in contrast to ω θ runs through a finite set, we conclude that ∞ ˙ + Hp,w,+ ωθ ∈ [SAPp,w ]N ×N ∩ [Cp,w (R) ]N ×N .

Analogously, ω θ ∈ [SAPp,w ]N ×N ∩ [Dp,w,+ ]N ×N because due to (3.42) and (3.30), pj [Mp,w ]N ×N vλ j ∞ e−λj θ/(1−θ) ωθ [Mp,w ]N ×N ≤ 1 + j ≤1+ pj [Mp,w ]N ×N vλ j ∞ < ∞. j

Since

−1 ωθ (x) = ψr+ x + i/x + iθ/(1−θ) , ω θ (x) = ψr+ x + i/x + iθ/(1−θ) if x > 0, −1 if x < 0, ωθ (x) = ψl+ x − i/x + iθ/(1−θ) , ω θ (x) = ψl+ x − i/x + iθ/(1−θ) and therefore ωθ ω θ = ω θ ωθ = I on R \ {0}, we conclude that ωθ ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N

for all θ ∈ [0, 1].

With g + given by (3.33) we associate the family of matrix functions gθ+ := vθ−1 ωθ ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N ,

(θ ∈ [0, 1]).

vθ−1

Obviously, θ → is a continuous [Mp,w ]N ×N -valued function on [0, 1], and therefore v −1 is homotopic to I in G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N . Since for all λ > 0 the functions θ → e−λθ/(1−θ) are homeomorphisms of [0, 1] onto itself and since the sum in ωθ given by (3.42) is finite, we infer from the estimate pj [Mp,w ]N ×N vλj ∞ e−λj θ/(1−θ) − e−λj θ0 /(1−θ0 ) ωθ − ωθ0 [Mp,w ]N ×N ≤ 1 + j

that θ → ωθ also is a continuous [Mp,w ]N ×N -valued function on [0, 1]. Thus, ω is homotopic to I in G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N too. Hence, gθ+ is a homotopy of g + to I in G[SAPp,w ]N ×N ∩ G[Dp,w,+ ]N ×N .

4. Wiener-Hopf operators with semi-almost periodic symbols on the spaces Lp(R+ , w) 4.1. Wiener-Hopf operators with almost periodic symbols Let us get an Lp (R+ , w) analogue of Theorem 2.28 in [3] established independently by I. Gohberg and I. Feldman [13] and by L. Coburn and R. G. Douglas [7]. Theorem 4.1. Let 1 < p < ∞, w ∈ A0p (R) and a ∈ APp,w \ {0}. (a) If a ∈ / GAPp,w , then W (a) is not semi-Fredholm on the space Lp (R+ , w). (b) If a ∈ GAPp,w and κ(a) > 0, then W (a) is properly n-normal and leftinvertible on the space Lp (R+ , w). (c) If a ∈ GAPp,w and κ(a) < 0, then W (a) is properly d-normal and rightinvertible on the space Lp (R+ , w). (d) If a ∈ GAPp,w and κ(a) = 0, then W (a) is invertible on Lp (R+ , w).

108


IEOT

Proof. Suppose that W (a) is semi-Fredholm on Lp (R+ , w). By Theorem 3.7, a ∈ GSAPp,w , and from Corollary 3.6 it follows that a ∈ GAPp,w . This proves part (a). Assume now that a ∈ GAPp,w . By Theorem there exist κ(a) ∈ R and −iκ(a)x2.1, iκ(a)x b(x) is a locally analytic function e . As ln ae b ∈ AP such that a(x) = e −iκ(a)x of ae ∈ GAPp,w , we infer from [12, §13] that actually b ∈ APp,w . Let p = j rj eλj be an almost periodic polynomial such that b = c + p and ec − 1Mp,w ≤ ecMp,w − 1 < 1.

(4.1)

Write p = p− + p+ where p± are almost periodic polynomials in AP ± such that p− consists of the terms of p in which λj < 0, and p+ contains the terms of p in which λj ≥ 0. We then have a(x) = ep− (x) ec(x) eiκ(a)x ep+ (x) ∞ ∞ where ep± ∈ GHp,w,± and eiκ(a)x ∈ Hp,w,± if ±κ(a) ≥ 0. Thus, by Proposition 2.8, W (a) = W (ep− )W (ec )W eiκ(a)x W (ep+ ) if κ(a) ≥ 0, (4.2) W (a) = W (ep− )W eiκ(a)x W (ec )W (ep+ ) if κ(a) ≤ 0,

where [W (ep± )]−1 = W (e−p± ) and the operator W (ec ) is invertible dueto (4.1). Hence, by (4.2), the required properties of W (a) coincide with those of W eiκ(a)x , which implies parts (b)–(d) as in [3, Theorem 2.28]. 4.2. Invertibility of almost periodic representatives and canonical factorization Given p ∈ (1, ∞) and w ∈ A0p (R), we consider the commutative Banach algebras ± ± ∞ ⊂ AP Wp,w defined in Subsection 3.5. Clearly, AP Wp,w ⊂ Hp,w,± . AP Wp,w Theorem 4.2. If p ∈ (1, ∞), w ∈ A0p (R) and a ∈ AP Wp,w , then the following assertions are equivalent: (a) the operator W (a) is Fredholm on the space Lp (R+ , w); (b) the operator W (a) is invertible on the space Lp (R+ , w); (c) the function a has a canonical right AP Wp,w factorization, that is, ± . a = a− a+ with a± ∈ GAP Wp,w

(4.3)

Proof. The implication (a)⇒(b) follows from Corollary 3.12. (b)⇒(c). If W (a) is invertible on the space Lp (R+ , w), then we infer from Theorem 4.1 that a ∈ GAPp,w and κ(a) = 0. Since the algebra AP Wp,w is similar to the algebra U W = F −1 (AP Wp,w )F , from Theorem 3.4 it follows that the invertibility of a ∈ AP Wp,w in APp,w is equivalent to the invertibility of a in AP Wp,w . Then Theorem 2.1 and [12, §13] imply in view of κ(a) = 0 that a = eb where b ∈ AP Wp,w . Hence, a ∈ AP Wp,w is represented in the form (4.3) where the ± are given by a± = eP± b , and functions a± ∈ GAP Wp,w P+ b = bλ eλ , P− b = bλ eλ for b = bλ eλ ∈ AP Wp,w . λ≥0

λ 0. If c is given by (4.4), then Ker W (c) = {0} and the algebraic complement of Im W (c) in Lp (R+ , w) is infinite-dimensional. ˙ b0 (∞) = 0, and let Lemma 4.5. Let α, β ∈ C, b0 ∈ Cp,w (R), b := α(1 − u)eλ + βueµ + b0 . If b ∈ GSAPp,w and either µ > 0 = λ or λ > 0 = µ, then W (b) is properly n-normal on the space Lp (R+ , w). Proof. Assume, for example, that µ > 0 and λ = 0. Since b ∈ GSAPp,w , from Corollary 3.6 it follows that αβ = 0. Put s1 := α−1 (1 − u) + 4β −1 ue−µ and choose ˙ with s0 (∞) = 0 so that s := s1 +s0 ∈ GSAPp,w . Lemma 3.13 implies s0 ∈ Cp,w (R) ∞ ˙ + Hp,w,± that ue±µ ∈ Cp,w (R) . Hence, by analogy with [3, Lemma 3.8], we infer from Proposition 2.8 and Corollary 2.10 that W (s)W (b) = [W (1 − u)]2 + W (4u2 + δ0 ) + K := A

(4.6)

110


IEOT

where K is a compact operator and ˙ δ0 (∞) = 0. δ0 := βα−1 (1 − u)ueµ + 4αβ −1 u(1 − u)e−µ + s1 b0 + s0 b ∈ Cp,w (R), Applying now [3, Theorem 17.9], we conclude that the operator A is Fredholm on the space Lp (R+ , w) because, for x ∈ R, (1 − u(x))2 + 4u2 (x) + δ0 (x) = s(x)b(x) = 0 and, for x = ∞ and t ∈ H 0, 1; ν0− (p, w), ν0+ (p, w) , 2 1 − u(+∞) (1 − t) + 1 − u(−∞) t + 4u2 (+∞)(1 − t) + 4u2 (−∞)t = t2 + 4(1 − t) = (t − 2)2 = 0. As A is Fredholm, from (4.6) it follows that W (b) is left-Fredholm. Assume that W (b) is even Fredholm. Then Corollary 3.11 implies that W (br ) is invertible on Lp (R+ , w). On the other hand, since µ > 0, Theorem 4.1 implies that W (br ) is properly n-normal and left-invertible, which contradicts the invertibility of W (br ). Thus, W (b) cannot be Fredholm. This shows that W (b) is properly n-normal. Theorem 4.6. Let w ∈ A0p (R). If a ∈ Cp,w (R)\{0} and W (a) is semi-Fredholm on Lp (R+ , w), then W (a) is Fredholm on Lp (R+ , w). Proof. Since the weight w ∈ A0p (R) is equivalent to the continuous weight (2.7) in view of [24], and since therefore ν0± (p, w) = 1/p, we deduce for a ∈ Cp,w (R)\{0} that (2.12) takes the form ˙ a# p,w (R) = a(R) ∪ A a(+∞), a(−∞); 1/p . If W (a) is properly n-normal (resp. properly d-normal) on the space Lp (R+ , w), / then slightly perturbing a one can find a function b ∈ Cp,w (R) such that 0 ∈ # # ˙ ˙ / bp,w (R), Theorem 2.6 bp,w (R) and b − aMp,w is so small as desired. Since 0 ∈ implies that W (b) is Fredholm on the space Lp (R+ , w). On the other hand, from [3, Theorem 2.1] it follows that W (b) remains properly n-normal (resp. properly d-normal) if b − aMp,w is sufficiently small. Thus, we arrive at a contradiction, and hence W (a) is Fredholm. 4.4. Semi-Fredholm theory for Wiener-Hopf operators with semi-almost periodic symbols on Lp (R+ , w) In this section we get an analogue of the Duduchava-Saginashvili theorem [10] for the weighted Lebesgue spaces Lp (R+ , w) with Muckenhoupt weights w ∈ A0p (R) (cf. [3, Theorem 19.15]). Theorem 4.7. Let a ∈ SAPp,w \{0}, 1 < p < ∞ and w ∈ A0p (R). (a) If a ∈ / GSAP, then W (a) is not semi-Fredholm on Lp (R+ , w). (b) If a ∈ GSAP and κ(al )κ(ar ) < 0, then W (a) is not semi-Fredholm on Lp (R+ , w). (c) If a ∈ GSAP, κ(al )κ(ar ) ≥ 0 and κ(al ) + κ(ar ) > 0, then W (a) is properly n-normal on Lp (R+ , w).

Vol. 62 (2008)


111

(d) If a ∈ GSAP, κ(al )κ(ar ) ≥ 0 and κ(al ) + κ(ar ) < 0, then W (a) is properly d-normal on Lp (R+ , w). (e) If a ∈ GSAP, κ(al ) = κ(ar ) = 0 and 0 ∈ / H d(ar ), d(al ); ν0− (p, w), ν0+ (p, w) , then W (a) is Fredholm on Lp (R+ , w). (f) If a ∈ GSAP, κ(al ) = κ(ar ) = 0 and 0 ∈ H d(ar ), d(al ); ν0− (p, w), ν0+ (p, w) , then W (a) is not semi-Fredholm on Lp (R+ , w). Proof. If W (a) is semi-Fredholm on Lp (R+ , w), then from Theorem 3.7 it follows that a ∈ GSAPp,w . This implies that a ∈ GSAP , which proves part (a). By Proposition 3.5, the invertibility in SAP of functions a ∈ SAPp,w is equivalent to their invertibility in SAPp,w . So suppose a ∈ GSAPp,w . By Theorem 3.1, a can be written in the form a = (1 − u)al + uar + a0 with al , ar ∈ GAPp,w , a0 ∈ ˙ and a0 (∞) = 0. From Theorem 2.1 and [12, §13] it follows that Cp,w (R) al = eiκ(al )x d(al )eψl (x) ,

ar = eiκ(ar )x d(ar )eψr (x) ,

(4.7)

with d(al )d(ar ) = 0, ψl , ψr ∈ APp,w and M (ψl ) = M (ψr ) = 0. Put b(x) := (1 − u(x))d(al )eiκ(al )x + u(x)d(ar )eiκ(ar )x + b0 (x),

(4.8)

˙ is chosen so that b0 (∞) = 0 and b ∈ GSAPp,w . We claim that where b0 ∈ Cp,w (R) W (a) and W (b) are weakly Φ-equivalent. Suppose W (a) is properly n-normal on Lp (R+ , w). Then there is an ε > 0 such that W (v) is properly n-normal whenever a − vMp,w < ε. Choose almost ± ± ± ± periodic polynomials p± l , pr ∈ APp,w such that M (pl ) = M (pr ) = 0 and + ψl − p− l − pl Mp,w < δ,

+ ψr − p− r − pr Mp,w < δ,

and consider the function −

v(x) := (1 − u(x))d(al )epl

(x) iκ(al )x p+ l (x)

e

e

−

+

+ u(x)d(ar )epr (x) eiκ(ar )x epr (x) + a0 (x).

Clearly, if δ > 0 is sufficiently small, then a − vMp,w < ε. Hence W (v) is properly n-normal, and Theorem 3.7 yields that v is invertible in SAPp,w together with a. Theorem 3.14 implies that there are functions f± ∈ GSAPp,w ∩ G[Dp,w,± ] such that v = f− bf+ . From Lemma 2.12 we infer that W (b) is properly n-normal. Analogously one can show that W (b) is properly d-normal (resp. Fredholm) if W (a) is properly d-normal (resp. Fredholm). Conversely, assume W (b) is properly n-normal. Then there exists an ε > 0 such that W (v) is properly n-normal if only v(x) := (1 − u(x))d(al )eiκ(al )x ecl (x) + u(x)d(ar )eiκ(ar )x ecr (x) + b0 (x) ˙ and b0 (∞) = 0. with cl , cr ∈ APp,w , cl Mp,w < ε, cr Mp,w < ε, b0 ∈ Cp,w (R) Theorem 3.14 and Lemma 2.12 imply that W (ω) is properly n-normal whenever −

ω(x) := (1 − u(x))d(al )epl + u(x)d(ar )e

p− r (x)

(x) iκ(al )x cl (x) p+ l (x)

e

e

eiκ(ar )x ecr (x) e

e

p+ r (x)

+ ω0 (x)

112


IEOT

± ± where cl , cr ∈ APp,w , cl Mp,w < ε, cr Mp,w < ε, p± l , pr ∈ APp,w are almost ± ˙ periodic polynomials such that M (p± l ) = M (pr ) = 0, and ω0 ∈ Cp,w (R) is chosen so that ω0 (∞) = 0 and ω ∈ GSAPp,w . But given a ∈ GSAPp,w with almost ± ± periodic representatives (4.7), we can put ω0 = a0 and find p± l , pr ∈ APp,w such ± − + ± − that M (pl ) = M (pr ) = 0 and ψl = cl + pl + pl , ψr = cr + pr + p+ r , with cl , cr ∈ APp,w , cl Mp,w < ε, cr Mp,w < ε. Hence, W (a) is properly n-normal. It can be shown analogously that if W (b) is properly d-normal (resp. Fredholm), then W (a) is properly d-normal (resp. Fredholm). This completes the proof of the weak Φ-equivalence of W (a) and W (b). Thus, W (a) is semi-Fredholm if and only if so is W (b). If κ(al ) = κ(ar ) = 0, then b ∈ Cp,w (R), and Theorems 2.6 and 4.6 give parts (e) and (f). ˙ such Since b ∈ GSAPp,w , we deduce from (4.8) that there is a b0 ∈ Cp,w (R) that b0 (∞) = 0 and

b−1 (x) := (1 − u(x))(d(al ))−1 e−iκ(al )x + u(x)(d(ar ))−1 e−iκ(ar )x + b0 (x). (4.9) If κ(al ) > 0 and κ(ar ) > 0, we infer from (4.8), (4.9) and Lemma 3.13 that b±1 ∈ ∞ −1 ˙ Cp,w (R)+H ) is a left regularizer of p,w,± . Then Theorem 2.9(b) implies that W (b W (b), and therefore W (b) is Fredholm or properly n-normal. Corollary 3.11 and Theorem 4.1(b) show that W (b) cannot be Fredholm. Hence, W (b) is properly n-normal. If κ(al ) = 0 and κ(ar ) > 0, we deduce from Lemma 4.5 that W (b) is properly n-normal, which completes the proof of part (c). Part (d) follows from part (c) after passage to adjoints. We are so left with part (b). Let us write (4.8) in the form b(x) := α(1 − u(x))eiλx + βu(x)eiµx + b0 (x)

(4.10)

where α = d(al ), β = d(ar ), λ = κ(al ) and µ = κ(ar ). Let λ > 0 and µ < 0. Assume that − + (p, w), ν∞ (p, w) . (4.11) 0∈ / H α, β; ν∞ Then Lemma 4.3 shows that W (b) is weakly Φ-equivalent to W (c) where c(x) := αχ− (x)eiλx + βχ+ (x)eiµx . Since max{µ, λ} > 0, Lemma 4.4 implies that the operator W (c) has trivial kernel on Lp (R+ , w). Now assume that W (b) is d-normal on Lp (R+ , w). Then W (c) is also d-normal on Lp (R+ , w), and as W (c) has trivial kernel, it follows that W (c) is Fredholm. Hence W (b) is also Fredholm, but that is impossible due to Corollary 3.11 and Theorem 4.1(b). If W (b) is n-normal on Lp (R+ , w), then W (b) is d-normal on Lq (R+ , w−1 ). Repeating the previous arguments with c, q and w−1 in place of c, p and w, we again arrive at a contradiction. Consequently, W (b) cannot be semi-Fredholm if (4.11) holds. Now suppose that − + 0 ∈ H α, β; ν∞ (p, w), ν∞ (p, w) , (4.12)

Vol. 62 (2008)


113

and consider b given by (4.10) with λ > 0 and µ < 0. Since a ∈ GSAPp,w , the numbers α = d(al ) and β = d(ar ) are non-zero. Assume that W (b) is d-normal. Consider the operator W (d) with symbol d ∈ GSAPp,w given by d(x) := α0 (1 − u(x))e−iεx + β0 u(x)e−iεx + d0 (x), ˙ and d0 (∞) = 0. Since where 0 < ε < λ and α0 , β0 ∈ C\{0}, d0 ∈ Cp,w (R) iεx + β0−1 u(x)eiεx + d0 (x) d−1 (x) := α−1 0 (1 − u(x))e

˙ and d0 (∞) = 0, we infer from Lemma 3.13 that d±1 ∈ where d0 ∈ Cp,w (R) ∞ ˙ + Hp,w,∓ . Hence, by part (d), W (d) is properly d-normal. Furthermore, Cp,w (R) W (d)W (b) = W (bd)+K where K is a compact operator, so W (bd) is also properly d-normal. Clearly, b(x)d(x) = αα0 (1 − u(x))ei(λ−ε)x + ββ0 u(x)ei(µ−ε)x + q0 (x) ˙ q0 (∞) = 0, and bd ∈ GSAPp,w . We can choose α0 and β0 where q0 ∈ Cp,w (R), such that − + (p, w), ν∞ (p, w) . (4.13) 0∈ / H αα0 , ββ0 ; ν∞ − + (p, w), ν∞ (p, w) and choose α0 and β0 such that Indeed, put γ := diam H α, β; ν∞ |αα0 | > 2γ and ββ0 = αα0 + β − α. In this case − + diam H αα0 , ββ0 ; ν∞ (p, w), ν∞ (p, w) = γ. (4.14) Since |αα0 | > 2γ, we infer from (4.14) that (4.13) is fulfilled because all points − + (p, w), ν∞ (p, w) satisfy the condition |z| > γ. z ∈ H αα0 , ββ0 ; ν∞ As (4.13) holds, Lemma 4.3 implies that W (bd) is weakly Φ-equivalent to W (c), where c(x) = αα0 χ− (x)ei(λ−ε)x + ββ0 χ+ (x)ei(µ−ε)x and λ − ε > 0, µ − ε < 0. Thus W (c) is properly d-normal along with W (bd). But this is impossible because, by the part of (b) already proved, W (c) cannot be semi-Fredholm. Passing to adjoint operators we conclude that W (b) cannot be n-normal in case (4.12) too. Thus, W (b) cannot be semi-Fredholm. This completes the proof of part (b). According to [3, Definition 3.13], the Cauchy index of any function a ∈ GSAP with κ(al ) = κ(ar ) = 0 is defined by the formula 1 1 2T lim ind a := (arg a)(x) − (arg a)(−x) dx (4.15) T →+∞ 2π T T where the limit exists, is finite, is independent of the particular choice of continuous branch of arg a, and possesses the logarithmic property: for every f1 , f2 ∈ GSAP with almost periodic representatives at ±∞ having zero mean motions, ind (f1 f2 ) = ind f1 + ind f2 .

(4.16)

114


Theorem 4.8. If a ∈ GSAPp,w , κ(al ) = κ(ar ) = 0 and 0∈ / H d(ar ), d(al ); ν0− (p, w), ν0+ (p, w) ,

IEOT

(4.17)

then the operator W (a) is Fredholm and

d(al ) 1 0 arg − ν0 (p, w) + Ind W (a) = −ind a + 2π d(ar ) − + 0 where ν0 (p, w) := ν0 (p, w) + ν0 (p, w) /2. ν00 (p, w)

(4.18)

˙ a0 (∞) = 0, Proof. We have a = (1 − u)al + uar + a0 , where a0 ∈ Cp,w (R), al = d(al )eψl , ar = d(ar )eψr , ψl , ψr ∈ APp,w , and M (ψl ) = M (ψr ) = 0. Without loss of generality assume that u ∈ C(R) is real-valued and monotonous. Consider b = ae−(1−u)ψl −uψr = (1 − u)d(al )eψl + ud(ar )eψr + a0 e−(1−u)ψl −uψr . Clearly, b(−∞) = d(al ), b(+∞) = d(ar ). As a ∈ GSAPp,w and (4.17) holds, we ˙ / b# deduce that b ∈ GCp,w (R) and 0 ∈ p,w (R). For µ ∈ [0, 1], put fµ := beµ[(1−u)ψl +uψr ] . Obviously, f1 = a and f0 = b. Since (fµ )l = d(al )eµψl and (fµ )r = d(ar )eµψr , we see that d((fµ )l ) = d(al ) and d((fµ )r ) = d(ar ). Hence fµ ∈ GSAPp,w and 0∈ / H d((fµ )r ), d((fµ )l ); ν0− (p, w), ν0+ (p, w) . Theorem 4.7(e) implies that the operators W (fµ ) are Fredholm for all µ ∈ [0, 1]. Since the map [0, 1] → B(Lp (R+ , w)), µ → W (fµ ) is continuous according to the estimate µϕ e − eνϕ ≤ eϕ e|µ−ν| ϕ − 1 (µ, ν ∈ [0, 1]), and since the operator index is stable with respect to small perturbations, we get Ind W (a) = Ind W (f1 ) = Ind W (f0 ) = Ind W (b). On the other hand, Theorem 2.7 implies that b(−∞) 1 0 0 Ind W (b) = −ind b + ν0 (p, w) − ν0 (p, w) + arg . 2π b(+∞)

(4.19)

(4.20)

According to [3, p. 66], from the definition of the Cauchy index it follows that 1 ind b = ind a − ind e(1−u)ψl +uψr = ind a − M (Im ψr ) − M (Im ψl ) = ind a. 2π Hence, as b(−∞) = d(al ) and b(+∞) = d(ar ), we get from (4.19) and (4.20) that Ind W (a) is computed by the formula (4.18). Remark 4.9. Since every weight w ∈ A0p (R) can be replaced by an equivalent continuous weight on R according to [24] (also see [25]), we may put ν0± (p, w) = ν00 (p, w) = 1/p in Theorems 4.7 and 4.8.

Vol. 62 (2008)


115

Remark 4.10. We actually use the equality ν0± (p, w) = 1/p in Theorem 4.7 only for the proof of part (f) when we apply Theorem 4.6 proved on the basis of that equality. 4.5. Wiener-Hopf operators with semi-periodic symbols on Lp (R+ , w) For Wiener-Hopf operators with semi-periodic symbols, in contrast to the case of semi-almost periodic symbols, we may consider Muckenhoupt weights for which 0 < ν0− (p, w) ≤ ν0+ (p, w) < 1. Given µ > 0, consider the following subset of Muckenhoupt weights:

Aµp (R) := w ∈ Ap (R) : w(· + λ)/w(·) ∈ L∞ (R) for all λ = nµ, n ∈ Z . (4.21) Let ω ∈ Ap ([0, µ/2)]) and let w1 be the symmetric extension of ω to [−µ/2, 0]. If w is the periodic extension of w1 to R, then w ∈ Ap (R) according to [2, Section 2.4], and w is a periodic function of period µ. Thus w ∈ Aµp (R). n1 cn e2πinx/µ , Let P µ be the set of µ-periodic polynomials of the form n=n 0 µ µ with cn ∈ C. Then Pp,w := closMp,w P is a Banach subalgebra of Mp,w . We define µ µ SPp,w as the smallest closed subalgebra of Mp,w that contains Pp,w and Cp,w (R). µ Theorem 4.11. Let 1 < p < ∞, w ∈ Aµp (R), and a ∈ SPp,w \{0}. Then the operator p µ W (a) is Fredholm on the space L (R+ , w) if and only if a ∈ GSPp,w , κ(al ) = − + κ(ar ) = 0 and 0 ∈ / H d(ar ), d(al ); ν0 (p, w), ν0 (p, w) . If W (a) is Fredholm, then its index is calculated by (4.18). µ Proof. Suppose a ∈ GSPp,w . By Theorem 3.1, a can be written in the form (3.2) µ ˙ and a0 (∞) = 0. Theorem 2.1 implies that with al , ar ∈ GPp,w , a0 ∈ Cp,w (R)

al = eiκ(al )x d(al )eψl (x) ,

ar = eiκ(ar )x d(ar )eψr (x) ,

(4.22)

with d(al )d(ar ) = 0, ψl , ψr ∈ APp,w and M (ψl ) = M (ψr ) = 0. Substituting the µ by the functions al , ar ∈ GC(T) such that functions al , ar ∈ GPp,w 2πix/µ 2πix/µ al e = al (x), = ar (x) (x ∈ R), ar e we deduce that κ(al ), κ(ar ) ∈ (2π/µ)Z. Then, following the proof of Theorem 4.1, µ we infer that actually ψl , ψr ∈ Pp,w . Further, as in the proof of Theorem 4.7, one can show that W (a) is weakly Φ-equivalent to W (b), where b is given by (4.8) and µ ˙ is chosen so that b0 (∞) = 0 and b ∈ GSPp,w b0 ∈ Cp,w (R) . If κ(al ) = κ(ar ) = 0, then b ∈ Cp,w (R), and hence b ∈ GCp,w (R). Since H b(+∞), b(−∞); ν0− (p, w), ν0+ (p, w) = H d(ar ), d(al ); ν0− (p, w), ν0+ (p, w) , Theorem 2.6 implies that if 0 ∈ / H d(ar ), d(al ); ν0− (p, w), ν0+ (p, w) , then the operator W (b) is Fredholm on the space Lp (R+ , w). Hence, by the weak Φ-equivalence of W (a) and W (b), the operator W (a) is also Fredholm on the space Lp (R+ , w). Conversely, assume that W (a) is Fredholm on the space Lp (R+ , w). One can µ µ show as in Proposition 3.5 that the Banach algebras Pp,w and SPp,w are inverse

116


IEOT

µ closed in AP and SAP , respectively. If a ∈ / GPp,w , then a ∈ / GSAP , and Theorem 4.7(a) implies that W (a) is not semi-Fredholm, which is a contradiction. Thus µ . If κ(al ) = 0 or κ(ar ) = 0, then, taking into account Remark 4.10 a ∈ GSPp,w and applying parts (b), (c) and (d) of Theorem 4.7, we again arrive at a contradiction. This implies that κ(al ) = κ(al ) = 0. Since W (a) is weakly Φ-equivalent to W (b) where b := (1 − u)d(al ) + ud(ar ) + b0 ∈ GCp,w (R), and therefore the operator W (b) Lp (R is Fredholm −on the space + , w), we deduce from Theorem 2.6 that + 0 ∈ / H d(ar ), d(al ); ν0 (p, w), ν0 (p, w) . Moreover, Ind W (a) = Ind W (b) where Ind W (b) is calculated by (4.20) due to Theorem 2.7. Finally, (4.20) implies (4.18) because ind b = ind a (see the proof of Theorem 4.8).

5. Fredholm theory for Wiener-Hopf operators with semi-almost periodic matrix symbols: factorable AP asymptotics Definition 5.1. Let 1 < p < ∞ and w ∈ A0p (R). A matrix function a ∈ G[APp,w ]N ×N is said to admit a right APp,w factorization if it can be represented in the form a(x) = a− (x)d(x)a+ (x) ±

a ∈

± G[APp,w ]N ×N ,

for all x ∈ R,

d = diag{eλ1 , . . . , eλN }, λ1 , . . . , λN ∈ R.

(5.1) (5.2)

Put κ(a) := (λ1 , . . . , λN ). A right APp,w factorization with κ(a) = (0, . . . , 0) is referred to as a canonical right APp,w factorization. In this section we will construct the Fredholm theory for Wiener-Hopf operators W (a) ∈ B(LpN (R+ , w)) with symbols a ∈ [SAPp,w ]N ×N under condition that the almost periodic representatives al and ar of a admit right APp,w factorizations. Since any right APp,w factorization is a right AP factorization, we infer from [3, Proposition 8.4] that if a ∈ G[APp,w ]N ×N admits a canonical right APp,w factorization, then the geometric mean d(a) = M (a− )M (a+ ) ∈ CN ×N is independent of the particular choice of the canonical right APp,w factorization of a. Lemma 5.2. If 1 < p < ∞, w ∈ A0p (R), N ∈ N, and a matrix function a ∈ G[APp,w ]N ×N admits a right APp,w factorization (5.1)–(5.2), then the WienerHopf operator W (a) is invertible (resp. left invertible, right invertible) on the space LpN (R+ , w) if and only if λj = 0 (resp. λj ≥ 0, λj ≤ 0) for all j = 1, 2, . . . , N . ± ∞ Proof. Since APp,w ⊂ Hp,w,± , we deduce from Proposition 2.8 that the operator W (a) is invertible (left invertible, right invertible) on the space LpN (R+ , w) if so is the operator W (d). It remains to apply Theorem 4.1 to each operator W (eλj ).

Theorem 5.3. Let 1 < p < ∞, w ∈ A0p (R), N ∈ N, a ∈ [SAPp,w ]N ×N , and suppose al and ar have right APp,w factorizations. Then the operator W (a) is Fredholm on the space LpN (R+ , w) if and only if a ∈ GSAPN ×N , κ(al ) = κ(ar ) = (0, . . . , 0), and one of the following two equivalent conditions is satisfied:

(5.3)

Vol. 62 (2008)


117

1 1 + arg ξj ∈ / Z for all eigenvalues ξj of the matrix d−1 (ar )d(al ); p 2π (ii) sp(d−1 (ar )d(al )) ∩ re2πi/q : r ∈ [0, ∞] = ∅. (i)

If W (a) is Fredholm, then N

Ind W (a) = −ind (det a) +

N − p j=1

1 1 + arg ξj . p 2π

(5.4)

Proof. If W (a) is Fredholm, then a ∈ G[SAPp,w ]N ×N by virtue of Theorem 3.7, and the operators W (al ), W (ar ) are invertible due to Corollary 3.11. By Lemma 5.2, κ(al ) = κ(ar ) = (0, . . . , 0). This proves the necessity of (5.3). Now + − + suppose (5.3) holds. Let al = a− l al and ar = ar ar be canonical right APp,w factorizations. By analogy with [3, Theorem 10.11], we can write + al = ϕ− l bl ϕl ,

+ ar = ϕ− r br ϕr ,

(5.5)

where bl := d(al ) ∈ GCN ×N , br := d(ar ) ∈ GCN ×N and − −1 − ϕ− (al ), l = al M

−1 + + ϕ+ (al )al , l =M

− −1 − (ar ), ϕ− r = ar M

−1 + + ϕ+ (ar )ar . r =M

(5.6)

± ± Obviously, ϕ± l , ϕr ∈ G[APp,w ]N ×N and (3.41) holds. Theorem 3.15 implies that there exist matrix functions b ∈ G[Cp,w (R)]N ×N and

g ± ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N

(5.7)

such that a = g − bg + , b(−∞) = bl = d(al ), b(+∞) = br = d(ar ) and the matrix functions g ± are homotopic to the identity matrix IN in the open sets G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N . Thus, by Lemma 2.12, the operators W (a) and W (b) are weakly Φ-equivalent and, in the case of Fredholmness, Ind W (a) = Ind W (g − ) + Ind W (b) + Ind W (g + ). A0p (R)

(5.8)

According to Remark 4.9, every weight w ∈ can be replaced by an equivalent 0 continuous weight in Ap (R), and then we may take ν0± (p, w) = ν00 (p, w) = 1/p. Hence, as b ∈ G[Cp,w (R)]N ×N , from Theorem 2.7 it follows that condition (i) (equivalently, (ii)) is necessary and sufficient for W (b) to be Fredholm on the space LpN (R+ , w). Since W (a) and W (b) are weakly Φ-equivalent under condition (5.3), we infer that conditions (5.3) and (i) (equivalently, (5.3) and (ii)) are necessary and sufficient for W (a) to be Fredholm on LpN (R+ , w). By Theorem 2.7 and Remark 4.9, N 1 N 1 − + arg ξj Ind W (b) = −ind (det b) + (5.9) p p 2π j=1 where ξj are the eigenvalues of the matrix b−1 (+∞)b(−∞) = d−1 (ar )d(al ).

118


IEOT

According to the proof of Theorem 3.14, matrix functions (5.7) have the form ± g ± = (1−u)gl± +ugr± + g0± where gl± , gr± ∈ G[APp,w ]N ×N , M (gl± ) = M (gr± ) = IN , ˙ N ×N and g± (∞) = 0. Hence, g0± ∈ [Cp,w (R)] 0 det g ± = (1 − u) det gl± + u det gr± + g0± ∈ GSAPp,w ∩ GDp,w,± ± ˙ where det gl± , det gr± ∈ GAPp,w , g0± ∈ Cp,w (R), g0± (∞) = 0, and

M (det gl± ) = det M (gl± ) = 1, M (det gr± ) = det M (gr± ) = 1. ±

±

Therefore, we deduce that det gl± = eψl , det gr± = eψr with ψl± , ψr± ∈ APp,w and M (ψl± ) = M (ψr± ) = 0. By the proof of Theorem 4.8, ind (det g ± ) = ind c± where ±

±

c± := (det g ± )e−(1−u)ψl −uψr ± ± ± ± ˙ = (1 − u)eψl + ueψr + g0± e−(1−u)ψl −uψr ∈ GCp,w (R). Hence, ind (det g ± ) is an integer number along with ind c± . By analogy, it is easily seen that ind (det gθ± ) are integers for all matrix functions in the homotopy families gθ± (θ ∈ [0, 1]) constructed in the proof of Theorem 3.15 and such that g0± = g ± , g1± = IN . Then, due to stability of the Cauchy index (4.15), we obtain ind (det g ± ) = ind (det g0± ) = ind (det g1± ) = ind (det IN ) = 0.

(5.10)

By the logarithmic property (4.16) of the Cauchy index (4.15) and by (5.10), we infer from the equality det a = det g − · det b · det g + that ind (det a) = ind (det g − ) + ind (det b) + ind (det g + ) = ind (det b).

(5.11)

Since the continuous families W (gθ± ) are homotopies of the operators W (g ± ) to the identity operator I in the set of Fredholm operators, we infer from (5.10) that Ind W (g ± ) = Ind I = 0.

(5.12)

Finally, the equalities (5.8)–(5.9) and (5.11)–(5.12) imply (5.4).

6. Fredholm theory for Wiener-Hopf operators with semi-almost periodic matrix symbols: special Muckenhoupt weights 6.1. An invertibility criterion In this section we will consider the weights w ∈ A0p (R) that satisfy the condition (6.1) lim sup ess ln w(x) − ln w(y) = 0. |t|→∞ x,y∈[t, t+1]

Example 2.4 and [2] give plenty of such weights. By [24], any weight w ∈ A0p (R) is equivalent to the weight ω ∈ C(R) given by (2.7). Furthermore, (2.7) implies that 1/2 ln ω(x) − ln ω(y) ≤ ln(w(x + t)) − ln(w(y + t)) dt, −1/2

Vol. 62 (2008)


119

whence it follows due to (6.1) that ω satisfies (6.1) too. Hence we may without loss of generality assume that w ∈ C(R) ∩ A0p (R). Then, for every λ ∈ R, we infer from (6.1) that vλ (x) =

w(x + λ) ˙ ∈ C(R) and w(x)

lim vλ (x) = 1.

|x|→∞

(6.2)

Since vλ ∞ ≥ 1 for all λ ∈ R due to (6.2), we conclude that G[AP Wp,w ]N ×N ⊂ GAP WN ×N ,

± G[AP Wp,w ]N ×N ⊂ GAP WN±×N

for all N ∈ N in view of the relation aλ CN ×N ≤ aλ CN ×N vλ ∞ . λ

λ

Theorem 6.1. Let 1 < p < ∞, N ∈ N, w ∈ A0p (R), and let condition (6.1) hold. If a ∈ [AP Wp,w ]N ×N , then the operator W (a) is invertible on the space LpN (R+ , w) if and only if a admits a canonical right AP W factorization. Proof. Necessity. Let a = λ aλ eλ ∈ [AP Wp,w ]N ×N . Since the operator W (a) is p invertible on the space LN (R+ , w) if and only if the operator A = χ+ W 0 (a) + χ− I is invertible on the space LpN (R, w), and since W χ+ a λ U λ ∈ V N A = (χ+ a0 + χ− )I + ×N , λ=0

W we deduce from Theorem 3.3 that the inverse operator A−1 also belongs to VN ×N . Then, in view of (3.14) and the estimate Uλ B(Lp (R,w)) = vλ ∞ ≥ 1, the mutually inverses operators A and A−1 are bounded on the space LpN (R) too. Hence, the operator W (a) with symbol a ∈ [AP Wp,w ]N ×N ⊂ [AP W ]N ×N is invertible on the space LpN (R+ ). Then, by [3, Corollary 19.11], the matrix function a admits a canonical right AP W factorization a = a− a+ with a± ∈ GAP WN±×N . Sufficiency. We assume without loss of generality that w ∈ C(R) ∩ A0p (R), which implies (6.2). Since a ∈ AP WN ×N and since a admits a canonical right AP W factorization, we infer from [3, Corollary 19.11] that the operator W (a) is invertible on the space LpN (R+ ). Put χh (x) := χ+ (x−h) for all x, h ∈ R. Obviously, for every h > 0 the operator χh W (a)χh I is invertible on the space χh LpN (R+ ) and < ∞. sup (χh W (a)χh I)−1 p h>0

B(χh LN (R+ ))

Hence, for every h > 0 the operator χh w−1 W (a)wχh I is invertible on the space χh LpN (R+ , w) and sup (χh w−1 W (a)wχh I)−1 B(χ Lp (R+ ,w)) < ∞. (6.3) Since a =

h

h>0

λ

N

aλ eλ ∈ [AP Wp,w ]N ×N ⊂ AP WN ×N and since aλ v−λ Uλ , w−1 W (a)wI = λ

from (6.2) it follows that lim χh w−1 W (a)wχh I − χh W (a)χh I h→∞

B(χh Lp N (R+ ,w))

= 0.

(6.4)

120


IEOT

As the operators χh w−1 W (a)wχh I are invertible on the spaces χh LpN (R+ , w), we infer from (6.3) and (6.4) that for a sufficiently large h > 0, the operator χh W (a)χh I is invertible on the space χh LpN (R+ , w). Hence, in view of W (a) = U−h χh W (a)χh Uh , the operator W (a) is invertible on the space LpN (R+ , w). Remark 6.2. If a ∈ [AP Wp,w ]N ×N and W (a) is invertible on LpN (R+ , w) for some p ∈ (1, ∞) and w ∈ A0p (R), then, by the proof of necessity in Theorem 6.1, the operator W (a) is invertible with the same inverses on all the spaces Lr (R, ω) whenever r ∈ (1, ∞), ω ∈ A0r (R) and ω((·) + λ)/ω∞ ≤ vλ ∞ for all λ ∈ R. 6.2. An application of pseudodifferential operators Given a continuous function a : R → C, let Cb (R) := C(R) ∩ L∞ (R) and let cmx (a) := max |a(x + h) − a(x)| : h ∈ [−1, 1] be a local oscillation of a at a point x ∈ R. According to [8, p. 122], a function a ∈ Cb (R) is called slowly oscillating at ∞ if lim cmx (a) = 0. |x|→∞

Let SO(R) be the C ∗ -algebra of all functions f ∈ Cb (R) which are slowly oscillating at ∞. According to [19, Remark 6.4], we also introduce the C ∗ -algebra

(6.5) SO(R2 ) := f ∈ Cb (R2 ) : lim cm(x,y) (f ) = lim cm(x,y) (f ) = 0 x,y→+∞

x,y→−∞

where

cm(x,y) (f ) := max f (x + h1 , y + h2 ) − f (x, y) : h1 , h2 ∈ [−1, 1] .

(6.6)

Let V (R) ⊂ V1 (R) be the Banach algebra of all absolutely continuous functions of bounded total variation on R with the norm f V = f L∞ (R) + f L1 (R) (cf. (3.1)), and let Cb (R2 , V (R)) be the Banach algebra of all bounded continuous V (R)-valued functions (x, y) → a(x, y, ·) on R2 equipped with the norm aCb (R2 ,V (R)) = sup a(x, y, ·)V . x,y∈R

By analogy with (6.6), for b(x, y, µ) ∈ Cb (R2 , V (R)), we define

b(x + h1 , y + h2 , ·) − b(x, y, ·) : h1 , h2 ∈ [−1, 1] . (b) := max cmC (x,y) ∞ Let E2C be the Banach algebra of all functions b ∈ Cb (R2 , V (R)) such that the V (R)-valued functions (x, y) → b(x, y, ·) are uniformly continuous on R2 , lim sup b(x, y, ·) − bh (x, y, ·)V = 0 |h|→0 x,y∈R

where bh (x, y, µ) := b(x, y, µ + h) for all (x, y, µ) ∈ R3 , and lim

x,y→−∞

cmC (x,y) (b) =

lim

x,y→+∞

cmC (x,y) (b) = 0.

Vol. 62 (2008)


121

By [18, Theorem 3.1], the pseudodifferential operator a(x, D) with a symbol a(x, µ) ∈ Cb (R, V (R)) defined for functions f ∈ C0∞ (R) by the iterated integral 1 dµ a(x, µ)ei(x−y)µ f (y)dy, for x ∈ R, [a(x, D)f ](x) = 2π R R extends to a bounded linear operator on every Lebesgue space Lp (R), p ∈ (1, ∞). By [19, Theorem 4.1], if ∂µj ∂yk b(x, y, µ) ∈ Cb (R2 , V (R)) for all k, j ∈ 0, 1, 2, then the pseudodifferential operator B defined for functions f ∈ C0∞ (R) by 1 (Bf )(x) := dµ b(x, y, µ)ei(x−y)µ f (y)dy, for x ∈ R, (6.7) 2π R R extends to a bounded linear operator on every Lebesgue space Lp (R), 1 < p < ∞. Given a weight w ∈ A0p (R) satisfying (6.1), and repeating the procedure (2.7) from [24], we can construct an equivalent weight ω ∈ A0p (R), n times continuously differentiable and satisfying (6.1) for any n ∈ N. Moreover, by (2.7) and (6.1), (ln ω) (x) = ln w(x + 1/2) − ln w(x − 1/2) → 0

as |x| → ∞.

˙ Hence, if w ∈ A0p (R) ∩ C n (R) for an n ∈ Z+ := {0, 1, 2, . . .}, then (ln ω) ∈ C n (R) (j+1) and (ln ω) (∞) = 0 for all j = 0, 1, . . . , n. Thus, we may without loss of ˙ and (ln w) (∞) = 0. generality assume that w ∈ A0p (R) ∩ C 3 (R), (ln w) ∈ C(R) By analogy with [34, Lemma 1.1], we obtain the following. ˙ v (∞) = 0. Lemma 6.3. Let p ∈ (1, ∞), w = ev ∈ A0p (R) ∩ C 1 (R), v ∈ C(R), Then for every η > 0 the functions 1 and γ(x) := tanh(ηx) belong to the Banach algebra Cp,w (R), and the closure in Cp,w (R) of the algebra generated by these two functions coincides with Cp,w (R). Proof. Obviously, the functions 1 and γ are in C(R) ∩ V1 (R) ⊂ Cp,w (R). To prove the second part of the lemma, observe that γ : x → tanh(ηx) is a homeomorphism of R onto [−1, 1]. Consider the operator Qγ : C[−1, 1] → C(R), f → f ◦ γ. Clearly, Qγ is an isometric isomorphism, which also preserves the total variation due to monotonicity of γ. Let C 1 [−1, 1] be the space of one time continuously differentiable functions on [−1, 1], and let Bp,w be the closure in Cp,w (R) of the algebra generated by 1 and γ. First we will show that Qγ (C 1 [−1, 1]) ⊂ Bp,w . Indeed, let f ∈ Qγ (C 1 [−1, 1]), ϕ = Q−1 γ f . Approximating ϕ by a sequence of polynomials ϕm in the norm of C 1 [−1, 1], we see that ϕm approximate ϕ in the norm of V [−1, 1] too. Hence, the sequence fm = Qγ ϕm approximate f by both the norms of C(R) and V1 (R). Therefore, by Stechkin’s inequality, fm is a Cauchy sequence in Mp,w . Because fm ∈ Bp,w , we conclude that f ∈ Bp,w as well. Let us now show that C(R) ∩ V1 (R) ⊂ Bp,w . Fix f ∈ C(R) ∩ V1 (R). By the proof of [34, Lemma 1.1], there exists a sequence of functions fm ∈ Qγ (C 1 [−1, 1]) which approximates f in the norm of C(R) and is uniformly bounded in the norm of V1 (R). By the part already proved, fm ∈ Br,ω for all r ∈ (1, ∞), ω ∈ A0r (R). It

122


IEOT

is clear that f ∈ Mr,ω for such r and ω. On the other hand, fm converges to f in the norm of M2 too. If w ∈ Ap (R), then w1+δ ∈ Ap(1+ε) (R) whenever |ε| and |δ| are sufficiently small (see, e.g., [11, Chapter 6, Corollary 6.10] or [2, Theorem 2.31]). Moreover, if w ∈ A0p (R), then w1+δ ∈ A0p(1+ε) (R) for small |ε| and |δ|. Clearly, if p = 2, then there exist sufficiently small |ε| > 0 and δ > 0 such that −1 1 − 2ε p(1 + ε) − 2 = (1 + δ)−1 where ε > 0 if p ∈ (2, ∞) and ε < 0 if p ∈ (1, 2). Setting now r := p(1 + ε), ω = w1+δ and t := (1 + δ)−1 ∈ (0, 1), we obtain 1 1−t t = + , p 2 r

w = 11−t ω t .

(6.8)

If p = 2 and w = 1, we again get (6.8) for r := p and t := (1 + δ)−1 ∈ (0, 1). By the Stein-Weiss interpolation theorem [33], we infer from (6.8) that 1−t t 0 W (fm − f ) p ≤ W 0 (fm − f )B(L2 (R)) W 0 (fm − f )B(Lr (R,ω)) , B(L (R,w)) which implies, in view of the uniform boundedness of fm − f Mr,ω with respect to m, that limm→∞ fm − f Mp,w = 0. Therefore, f = lim fm belongs to Bp,w together with all fm . Thus, C(R) ∩ V1 (R) ⊂ Bp,w , and hence Bp,w = Cp,w (R). ˙ and v (∞) = 0. Lemma 6.4. Let p ∈ (1, ∞), w = ev ∈ A0p (R) ∩ C 3 (R), v ∈ C(R), −1 If u(µ) := tanh(ηµ) for µ ∈ R where 0 < η < (π/2)v ∞ , then the operator wW 0 (u)w−1 I − W 0 (u)

(6.9)

is compact on the space Lp (R). Proof. By Stechkin’s inequality, u ∈ Cp,w (R), and therefore the convolution operator W 0 (u) = F −1 uF is bounded on the space Lp (R, w). Put v(x) := ln w(x), mv (x, y) := v(x) − v(y) /(x − y). (6.10) ˙ ⊂ SO(R), then by [19, Lemma 8.1] the functions Since v ∈ C 3 (R) and v ∈ C(R) j (x, y) → ∂y mv (x, y) belong to SO(R2 ) for all j = 0, 1, 2, where SO(R2 ) is given by (6.5)–(6.6). Since u(−µ) = −u(µ) and wW 0 (u)w−1 I ∈ B(Lp (R)), for f ∈ C0∞ (R) we get 1 dµ ev(x)−v(y) u(µ)e−i(x−y)µ f (y)dy wW 0 (u)w−1 f (x) = 2π R R 1 =− dµ u(µ)ei(x−y)[µ−imv (x,y)] f (y)dy 2π R R 1 =− dµ u[µ + imv (x, y)]ei(x−y)µ f (y)dy. (6.11) 2π R−imv (x,y) R

Vol. 62 (2008)


123

Since w ∈ A0p (R) and v ∈ SO(R), we deduce that |mv (x, y)| ≤ v ∞ for all x, y ∈ R, and hence |ηmv (x, y)| < π/2, which implies that the function b(x, y, µ) := u[µ + imv (x, y)] = tanh η(µ + imv (x, y)) (6.12) is well defined. As the function u(z) is analytic in the strip z ∈ C : 2η |Im z| < π , representing pseudodifferential operators via oscillatory integrals (see, e.g., [28], [19]), we can replace R − imv (x, y) in (6.11) by R by analogy with [28, Theorem 4.5.3]. It is easily seen that ∂µj ∂yk b(x, y, µ) ∈ E2C ⊂ Cb (R2 , V (R))

(k = 0, 1, 2).

Hence, by (6.11), (6.12) and [19, Theorem 4.1], we get wW 0 (u)w−1 I = −B,

(6.13)

p

where the pseudodifferential operator B ∈ B(L (R)) is given by (6.7). Since mv (x, x) = v (x), we deduce from (6.12) that u (x, µ) := b(x, x, µ) = u[µ + iv (x)] = tanh η(µ + iv (x)) , where the function tanh η(µ + iv (x)) is bounded because 2ηv ∞ < π. Consequently, from [19, Lemma 3.4 and Theorem 4.4] it follows that the operator B−u (x, D) is compact on the space Lp (R). This together with (6.13) shows that the operator wW 0 (u)w−1 I + u (x, D) is compact on the space Lp (R). Since u (x, µ) ∈ C(R, V (R)) and u (x, µ) − u(µ) = 0 for all (x, µ) ∈ ∂(R × R), we infer from [18, Corollary 4.3] that the operator u (x, D) + W 0 (u) is compact on p the space L (R). Hence, the operator (6.9) is also compact on Lp (R). ˙ and v (∞) = 0, Corollary 6.5. If p ∈ (1, ∞), w = ev ∈ A0p (R) ∩ C 1 (R), v ∈ C(R), then the Banach algebra Cp,w (R) is continuously embedded into the Banach algebra Cp (R) and, for every f ∈ Cp,w (R), f Mp ≤ f Mp,w . (6.14) and take γ(x) = tanh(ηx). Put gn := Pn ◦ γ. ObProof. Fix η ∈ 0, (π/2)v −1 ∞ viously, the functions γ and gn (n ∈ N) belong to Cp,w (R) ∩ Cp (R). By Lemma 6.4, the operator wW 0 (γ)w−1 I − W 0 (γ) is compact on the space Lp (R). Hence, for every algebraic polynomial Pn , the operator wW 0 (Pn ◦ γ)w−1 I − W 0 (Pn ◦ γ) is also compact on the space Lp (R). Then we obtain 0 W (gn ) + K p gn = W 0 (gn ) p = inf

Mp

B(L (R))

K∈K(Lp (R))

B(L (R))

wW 0 (gn )w−1 I + K p = infp ≤ wW 0 (gn )w−1 I B(Lp (R)) B(L (R)) K∈K(L (R)) = W 0 (gn ) p = gn . (6.15) B(L (R,w))

Mp,w

By Lemma 6.3, every function f ∈ Cp,w (R) can be approximated in the norm of Cp,w (R) by functions gn . Hence, we deduce from (6.15) that the sequence {gn } converges to f in the norm of Cp (R) too. Thus, the space Cp,w (R) is continuously embedded into Cp (R), and (6.14) holds for all f ∈ Cp,w (R).

124


IEOT

Approximating functions u ∈ Cp,w (R) by the functions gn = Pn ◦ γ in view of Lemma 6.3 and Corollary 6.5, we infer from Lemma 6.4 the following. ˙ and v (∞) = 0, Corollary 6.6. If p ∈ (1, ∞), w = ev ∈ A0p (R) ∩ C 1 (R), v ∈ C(R), then for every u ∈ Cp,w (R) the operator (6.9) is compact on the space Lp (R). 6.3. Fredholm theory Lemma 6.7. Let 1 < p < ∞, N ∈ N, w ∈ A0p (R), and let condition (6.1) hold. If a ∈ [SAPp,w ]N ×N , al , ar ∈ [AP Wp,w ]N ×N and the operator W (a) is Fredholm on the space LpN (R+ ), then the operator W (a) is Fredholm on the space LpN (R+ , w) too, and Ind W (a) on the spaces LpN (R+ ) and LpN (R+ , w) is the same. Proof. As AP Wp,w ⊂ AP W and Cp,w (R) ⊂ C(R), we conclude that a ∈ SAPN ×N and al , ar ∈ AP WN ×N . Then the operator W (a) is bounded on both the spaces LpN (R+ ) and LpN (R+ , w). Since the operator W (a) is Fredholm on the space LpN (R+ ), we infer from the relation χh W (a)χh I = χh W 0 (a)χh I = Uh χ+ W 0 (a)χ+ U−h = Uh W (a)U−h

(6.16)

the operators χh W (a)χh I are Fredholm on the spaces χh LpN (R+ ) for all 0, and their regularizers (χh W (a)χh I)(−1) := Uh [W (a)](−1) Uh−1 satisfy the

that h > condition sup (χh W (a)χh I)(−1) B(χ

p h LN (R+ ))

h>0

≤ [W (a)](−1) B(Lp (R+ )) < ∞. N

(6.17)

By Corollary 6.6, the operator wW 0 (a)w−1 I = (wW 0 (al )w−1 )(wW 0 (u− )w−1 )I +(wW 0 (ar )w−1 )(wW 0 (u+ )w−1 )I + wW 0 (a0 )w−1 I ∈ B(LpN (R)) can be rewritten in the form wW 0 (a)w−1 I = (wW 0 (al )w−1 )W 0 (u− ) + (wW 0 (ar )w−1 )W 0 (u+ ) + W 0 (a0 ) + K (6.18) where K is a compact operator in B(LpN (R)). Further, by (6.1) and (6.4), we get lim χh W (al ) − wW (al )w−1 χh I B(Lp (R)) = 0, N h→∞ (6.19) −1 lim χh W (ar ) − wW (ar )w χh I B(Lp (R)) = 0, h→∞

N

Taking into account (6.18) and (6.19), we deduce that lim χh W 0 (a)χh I − χh wW 0 (a)w−1 χh I B(Lp (R)) = 0. h→∞

N

(6.20)

Hence, according to (6.20) and (6.17), we conclude that for a sufficiently large h > 0 the operator χh wW (a)w−1 χh I = χh wW 0 (a)w−1 χh I is Fredholm on the space χh LpN (R+ ) along with the operator χh W (a)χh I, and Ind (χh wW (a)w−1 χh I) = Ind (χh W (a)χh I).

Vol. 62 (2008)


125

This implies that the operator χh W (a)χh I is Fredholm on the space χh LpN (R+ , w), and Ind (χh W (a)χh I) is the same on the spaces χh LpN (R+ , w) and χh LpN (R+ ). Further, from the relation W (a) = U−h (χh W (a)χh I)Uh it follows that the operator W (a) is Fredholm on the space LpN (R+ , w), and Ind W (a) = Ind (χh W (a)χh I).

(6.21)

Since (6.21) is fulfilled for both the spaces LpN (R+ ) and LpN (R+ , w) due to (6.16), we conclude that Ind W (a) on the spaces LpN (R+ ) and LpN (R+ , w) is the same. Theorem 6.8. Let 1 < p < ∞, N ∈ N, w ∈ A0p (R), and let (6.1) hold. If a ∈ [SAPp,w ]N ×N and al , ar ∈ [AP Wp,w ]N ×N , then the operator W (a) is Fredholm on the space LpN (R+ , w) if and only if the following three conditions are satisfied: (i) a ∈ GSAPN ×N , (ii) al and ar admit canonical right AP W factorizations, 1 1 / Z for all eigenvalues ξj of the matrix d−1 (ar )d(al ). arg ξj ∈ (iii) + p 2π If W (a) is Fredholm, then its index is calculated by the formula (5.4). Proof. Necessity. Let u± := 2−1 (1 ± tanh x) ∈ [Cp,w (R)]N ×N and let a = al u − + ar u + + a0 ,

(6.22)

˙ N ×N , and a0 (∞) = 0. where al , ar ∈ [AP Wp,w ]N ×N , , a0 ∈ [Cp,w (R)] If W (a) is Fredholm, then a ∈ G[SAPp,w ]N ×N by virtue of Theorem 3.7, and W (al ), W (ar ) are invertible on LpN (R+ , w) due to Corollary 3.11. Hence al and ar admit canonical right AP W factorizations in view of Theorem 6.1. This proves the necessity of (i) and (ii). + − + Suppose (i) and (ii) hold. Let al = a− l al and ar = ar ar be canonical right AP W factorizations of al and ar , respectively. Then the matrix functions al and ar can be represented in the form (5.5). From (5.5) it follows that for any ε > 0 ± there are almost periodic matrix polynomials a± a± r ∈ GAP WN ×N such that l , al := a− a+ ar := a− a+ r d(ar ) r , l d(al ) l , W (al ) − W ( al ) B(Lp (R )) < ε, W (ar ) − W ( ar ) B(Lp (R N

+

N

M ( a− a− a+ a+ r ) = M ( r ) = I. l ) = M ( l ) = M (

(6.23) + ))

< ε,

(6.24) (6.25)

Since the operator W (a) = χ+ W 0 (a)χ+ I is Fredholm on the space LpN (R+ , w), we deduce from (6.16) that for every h > 0 the operator χh W (a)χh I is Fredholm on the space χh LpN (R+ , w). Hence, for every h > 0, the operator χh wW (a)w−1 χh I is Fredholm on the space χh LpN (R+ ). By (6.1) and (6.4), along with (6.19) we get lim χh W ( al ) − wW ( al )w−1 χh I B(Lp (R)) = 0, N h→∞ (6.26) lim χh W ( ar ) − wW ( ar )w−1 χh I B(Lp (R)) = 0. h→∞

N

126


IEOT

Then (6.24), (6.19) and (6.26) imply that for every ε > 0 there is an h > 0 such that χh wW (al − al )w−1 χh I B(Lp (R+ )) < 2ε, N (6.27) χh wW (ar − ar )w−1 χh I p < 2ε. B(LN (R+ ))

Along with the matrix function a ∈ [SAPp,w ]N ×N given by (6.22), we introduce the matrix function a := al u − + ar u+ + a0 ∈ [SAPp,w ]N ×N . By (6.27), χh wW (a)w−1 − wW ( a)w−1 χh I B(Lp (R )) < 2ε max u± Mp,w . (6.28) N

+

Since the operator χh wW (a)w χh I is Fredholm on the space χh LpN (R+ ), we can choose ε > 0 in (6.28) so small that the operator χh wW ( a)w−1 χh I is also p Fredholm on the space χh LN (R+ ). Hence the operator χh W ( a)χh I is Fredholm on the space χh LpN (R+ , w), which in view of χh W ( a)χh I = Uh W ( a)Uh−1 imp plies that the operator W ( a) is Fredholm on the space LN (R+ , w). But the matrix functions al , ar admit canonical right AP W factorizations (6.23) with factors ± a± a± a± a± r ∈ GAP WN ×N that are almost periodic matrix polynomials. Then r ∈ l , l , ± ]N ×N . Furthermore, since al , ar ∈ [AP Wp,w ]N ×N ∩ G[AP W ]N ×N , we in[AP Wp,w ar ∈ G[AP Wp,w ]N ×N . Consequently, we deduce from fer from Theorem 3.4 that al , the relations −1

−1 ( a− = d(al ) a+ a−1 l ) l l ,

−1 ( a+ = a−1 a− l ) l l d(al ),

−1 = d(ar ) a+ a−1 ( a− r ) r r ,

−1 ( a+ = a−1 a− r ) r r d(ar )

−1 −1 ± ± , ( a± ∈ [AP Wp,w ]N ×N and therefore a± a± that ( a± r ) r ∈ G[AP Wp,w ]N ×N . l ) l , Thus, we may without loss of generality assume that the original matrix functions al , ar ∈ [AP Wp,w ]N ×N admit canonical right AP W factorizations with ± ± ± factors a± l , ar ∈ G[AP Wp,w ]N ×N ⊂ G[APp,w ]N ×N . In that case + al = ϕ− l d(al )ϕl ,

+ ar = ϕ− r d(ar )ϕr ,

± ± where the matrix functions ϕ± l , ϕr ∈ G[APp,w ]N ×N are given by (5.6) and satisfy (3.41). Hence, by Theorem 3.15, there are matrix functions b ∈ G[Cp,w (R)]N ×N and g± ∈ G[SAPp,w ]N ×N ∩ G[Dp,w,± ]N ×N such that a = g− bg+ and b(−∞) = d(al ), b(+∞) = d(ar ). Then, by Lemma 2.12, the operator W (b) is Fredholm on the space LpN (R+ , w) along with W (a). Finally, since b ∈ G[Cp,w (R)]N ×N , we infer from Theorem 2.7 that condition (iii) also holds. Sufficiency. Since AP Wp,w ⊂ AP W and Cp,w (R) ⊂ C(R), we conclude that a ∈ SAPN ×N and al , ar ∈ AP WN ×N . Therefore, if conditions (i)–(iii) hold, from [3, Corollary 19.17] it follows that the operator W (a) is Fredholm on the space LpN (R+ ). Then, by Lemma 6.7, the operator W (a) is Fredholm on LpN (R+ , w) too. Index. By Lemma 6.7, Ind W (a) is the same on the spaces LpN (R+ , w) and p LN (R+ ). Hence, we infer from [3, Theorem 19.6] that Ind W (a) on the space LpN (R+ , w) is calculated by the formula (5.4).

Vol. 62 (2008)


127

References [1] S. Bochner and R. S. Phillips, Absolutely convergent Fourier expansions for noncommutative normed rings. Ann. of Math. 43 (1942), 409–418. [2] A. B¨ ottcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics 154, Birkh¨ auser, Basel, 1997. [3] A. B¨ ottcher, Yu. I. Karlovich, and I. M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions. Operator Theory: Advances and Applications 131, Birkh¨ auser, Basel, 2002. [4] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators. 2nd edition, Springer, Berlin, 2006. [5] A. B¨ ottcher and I. M. Spitkovsky, Wiener-Hopf integral operators with P C symbols on spaces with Muckenhoupt weight. Revista Matem´ atica Iberoamericana 9 (1993), 257–279. [6] A. B¨ ottcher and I. M. Spitkovsky, Pseudodifferential operators with heavy spectrum. Integral Equations and Operator Theory 19 (1994), 251–269. [7] L. Coburn and R. G. Douglas, Translation operators on the half-line. Proc. Nat. Acad. Sci. USA 62 (1969), 1010–1013. [8] H. O. Cordes, Elliptic Pseudo-Differential Operators - An Abstract Theory. Lect. Notes. in Math. 756, Springer, Berlin, 1979. [9] R. G. Douglas, Toeplitz and Wiener-Hopf operators in H ∞ + C. Bull. Amer. Math. Soc. 74 (1968), 895–899. [10] R. V. Duduchava and A. I. Saginashvili, Convolution integral equations on a half-line with semi-almost-periodic presymbols. Differential Equations 17 (1981), 207–216. [11] J. B. Garnett, Bounded Analytic Functions. Academic Press, New York, 1981. [12] I. M. Gelfand, D. A. Raikov, and G. E. Shilov, Commutative Normed Rings. Fizmatgiz, Moscow, 1960 [Russian]. English transl.: Chelsea, New York, 1964. [13] I. Gohberg and I. Feldman, Wiener-Hopf integro-difference equations. Soviet Math. Dokl. 9 (1968), 1312–1316. [14] I. Gohberg and I. Feldman, Convolution Equations and Projection Methods for Their Solutions. Amer. Math, Soc., Providence, R. I. 1974. Russian original: Nauka, Moscow, 1971. [15] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations. Vols. 1 and 2. Birkh¨ auser, Basel, 1992. Russian original: Shtiintsa, Kishinev, 1973. [16] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227– 251. [17] Yu. I. Karlovich, On the Haseman problem. Demonstratio Mathematica 26 (1993), No. 3–4, 581–595. [18] Yu. I. Karlovich, An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. London Math. Soc. (3) 92 (2006), 713–761. [19] Yu. I. Karlovich, Pseudodifferential operators with compound slowly oscillating symbols. Operator Theory: Advances and Applications 171 (2006), 189–224.

128


IEOT

[20] Yu. I. Karlovich and E. Ram´ırez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces. Integral Equations and Operator Theory 48 (2004), 331-363. [21] Yu. I. Karlovich and I. M. Spitkovskii, On the theory of systems of convolution type equations with semi-almost-periodic symbols in spaces of Bessel potentials. Soviet Math. Dokl. 33 (1986), 180–184. [22] Yu. I. Karlovich and I. M. Spitkovsky, Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type. Math. USSR Izvestiya 34 (1990), 281–316. [23] Yu. I. Karlovich and I. M. Spitkovsky, (Semi)-Fredholmness of convolution operators on the spaces of Bessel potentials. Operator Theory: Advances and Applications 71 (1994), 122–152. [24] V. Petkova, Symbole d’un multiplicateur sur L2ω (R). Bull. Sci. Math. 128 (2004), 391–415. [25] V. Petkova, Wiener-Hopf operators on L2ω (R+ ). Arch. Math. 84 (2005), 311–324. [26] Ch. Pommerenke, Boundary Behaviour of Conformal Maps. Springer, Berlin, 1992. [27] I. I. Privalov, Boundary Properties of Analytic Functions. GITTL, Moscow, 1950 [Russian]. [28] V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory. Operator Theory: Advances and Applications 150, Birkh¨ auser, Basel, 2004. [29] S. Roch and B. Silbermann, Algebras of Convolution Operators and Their Image in the Calkin Algebra. Report R-Math-05/90, Akad. Wiss. DDR, Karl Weierstrass Institut f. Mathematik, Berlin, 1990. [30] D. Sarason, Approximation of piecewise continuous functions by quotients of bounded analytic functions. Canad. J. Math. 24 (1972), 642–657. [31] D. Sarason, Toeplitz operators with semi-almost periodic symbols. Duke Math. J. 44 (1977), 357–364. [32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1970. [33] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ, 1971. [34] I. B. Simonenko and Chin Ngok Min, Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewise Continuous Coefficients. Noetherity. University Press, Rostov on Don, 1986 [Russian]. Yu. I. Karlovich Facultad de Ciencias, Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos, México e-mail: [email protected] J. Loreto Hern´ andez Instituto de Matem´ aticas, Universidad Nacional Aut´ onoma de México Av. Universidad 1001, Col. Chamilpa, C.P. 62210 Cuernavaca, Morelos, México e-mail: [email protected] Submitted: February 26, 2008



Norm Inequalities for Commutators of Self-adjoint Operators Fuad Kittaneh Abstract. Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with a1 ≤ A ≤ a2 and b1 ≤ B ≤ b2 for some real numbers a1 , a2 , b1 , and b2 , then for every unitarily invariant norm |||·|||, |||AX − XB||| ≤ max(a2 − b1 , b2 − a1 ) |||X||| . If, in addition, X is positive, then |||AX − XA||| ≤

1 (a2 − a1 ) |||X ⊕ X||| . 2

These norm inequalities generalize recent related inequalities due to Kittaneh, Bhatia-Kittaneh, and Wang-Du. Mathematics Subject Classification (2000). Primary 47A30; Secondary 47B15, 47B47. Keywords. Commutator, normal operator, self-adjoint operator, positive operator, unitarily invariant norm, norm inequality.

1. Introduction This paper is a continuation of our recent work on norm inequalities for commutators of positive operators that we have begun in [6] and [7], and the related works in [3], [8], and [9]. Let A, B, and X be bounded linear operators on a complex separable Hilbert space H. An operator of the form AX − XA is called a commutator, and that of the form AX − XB is called a generalized commutator. It has been recently shown in [6], and later in [3], that if A and B are positive, then |||AX − XB||| ≤ max(A , B) |||X||| ,

(1.1)

130

Kittaneh

IEOT

where · is the usual operator norm and |||·||| denotes an arbitrary unitarily invariant norm. Moreover, if X is also positive, then 1 A |||X ⊕ X||| , (1.2) 2 X 0 where X ⊕X represents the block-diagonal operator defined on H ⊕ H. 0 X In particular, 1 AX − XA ≤ A X . (1.3) 2 |||AX − XA||| ≤

Among other ideas, the proofs of the inequalities (1.1) and (1.2) given in [6] involve basic properties of unitarily invariant norms and projection dilation. Different proofs of these sharp inequalities, in the finite-dimensional setting, have been recently given in [3], and their connections with norms of derivations, pinchings, and spectral variation have been pointed out. Generalizations of the inequalities (1.1) and (1.3) to normal operators have been given in [8], where applications of these inequalities have been also given. For the usual operator norm, the inequality (1.1), together with some applications, has been also given in [7], and for this norm, generalizations of the inequalities (1.1) and (1.3) have been given in [9]. The main purpose of this paper is to give very simple proofs of norm inequalities for commutators of self-adjoint operators generalizing the inequalities (1.1) and (1.2), from which the corresponding inequalities in [3, Theorem 3] and in [9, Theorem 3] follow as special cases. Our proofs are shorter, much simpler than, and completely different from those given for the special cases in [3] and in [9]. In the sequel, the symbol |||·||| denotes any unitarily invariant (or symmetric) norm. Such a norm is defined on a norm ideal associated with it, which will not be explicitly mentioned for the sake of brevity. Each unitarily invariant norm |||·||| satisfies the submultiplicativity property |||XY Z||| ≤ X |||Y ||| Z and the unitary invariance property |||U Y V ||| = |||Y ||| for all operators X, Y, Z, and all unitary operators U, V. Our analysis in this paper is closely related to that utilized in [8], and depends heavily on the triangle inequality, the submultiplicativity property of unitarily invariant norms, and the fact that for a self-adjoint operator A and for a positive real number a, A ≤ a if and only if −a ≤ A ≤ a, which is also equivalent to the condition that σ(A) ⊆ [−a, a], where σ(A) is the spectrum of A.

2. Main results Our first result is a general norm inequality for generalized commutators of normal operators, formulated in terms of the spectral bounds of the Cartesian parts of the normal operators involved. This result yields the promised generalization of the inequality (1.1) to generalized commutators of self-adjoint operators.

Vol. 62 (2008)

Norm Inequalities for Commutators

131

Theorem 2.1. Let S and T be normal operators with the Cartesian decompositions S = A + iC and T = B + iD such that a1 ≤ A ≤ a2 , b1 ≤ B ≤ b2 , c1 ≤ C ≤ c2 , and d1 ≤ D ≤ d2 for some real numbers a1 , a2 , b1 , b2 , c1 , c2 , d1 , and d2 . Then, for every operator X, 1 ( (a2 − a1 )2 + (c2 − c1 )2 + (b2 − b1 )2 + (d2 − d1 )2 |||SX − XT ||| ≤ 2 + (a1 + a2 − b1 − b2 )2 + (c1 + c2 − d1 − d2 )2 ) |||X||| . (2.1) b1 +b2 c1 +c2 d1 +d2 2 Proof. Let a = a1 +a 2 , b = 2 , c= 2 , d = 2 , z = a + ic, and w = b + id. Then, by the triangle inequality and the submultiplicativity property of unitarily invariant norms, we have

|||SX − XT ||| = |||(S − z)X − X(T − w) + (z − w)X||| ≤ (S − z + T − w + |z − w|) |||X||| .

(2.2)

Since S − z and T − w are normal, it follows that (S − z)∗ (S − z) = (A − a)2 + (C − c)2 and (T − w)∗ (T − w) = (B − b)2 + (D − d)2 . 2

So by the fact that for every operator Y , Y = Y ∗ Y , and by the triangle inequality for the usual operator norm, we have 2

2

2

2

2

2

S − z ≤ A − a + C − c

(2.3)

and a2 −a1

T − w ≤ B − b + D − d .

(2.4)

1 ≤ a2 −a 2 , it follows that A − a b2 −b1 c2 −c1 2 , C − c ≤ 2 , and D

1 ≤ a2 −a 2 . Similarly, we 1 − d ≤ d2 −d 2 . In view

≤ A−a Since − 2 conclude that B − b ≤ of this, combining the inequalities (2.2), (2.3), and (2.4), we obtain the desired inequality (2.1). Related to the inequality (2.1), it has been recently shown in [8] that if S, T , a1 , a2 , b1 , b2 , c1 , c2 , d1 , and d2 are as in Theorem 2.1, then for every operator X, |||SX − XT ||| (2.5) 2 2 ≤ (max(a2 , b2 ) − min(a1 , b1 )) + (max(c2 , d2 ) − min(c1 , d1 )) |||X||| . It should be mentioned here that none of the inequalities (2.1) and (2.6) is uniformly better than the other one. This can be easily illustrated by using two-dimensional examples. An important special case in which the inequality (2.6) is not better than the inequality (2.1) is the case where S and T are self-adjoint operators, and so c1 = c2 = d1 = d2 = 0. In this case, it is evident that the coefficient in the righthand side of the inequality (2.6) is not smaller than that in the right-hand side of the inequality (2.1).

132

Kittaneh

IEOT

As an application of Theorem 2.1, we obtain the desired generalization of the inequality (1.1) to generalized commutators of self-adjoint operators, which includes Theorem 3 in [3] and Theorem 3 in [9] as special cases. Corollary 2.1. Let A and B be self-adjoint operators such that a1 ≤ A ≤ a2 and b1 ≤ B ≤ b2 . Then, for every operator X, |||AX − XB||| ≤ max (a2 − b1 , b2 − a1 ) |||X||| .

(2.6)

Proof. Applying Theorem 2.1 with c1 = c2 = d1 = d2 = 0, we have |||AX − XB|||

as required.

1 (|a2 − a1 | + |b2 − b1 | + |a1 + a2 − b1 − b2 |) |||X||| 2 1 (a2 − a1 + b2 − b1 + |a1 + a2 − b1 − b2 |) |||X||| = 2 1 (a2 − b1 + b2 − a1 + |(a2 − b1 ) − (b2 − a1 )|) |||X||| = 2 = max (a2 − b1 , b2 − a1 ) |||X||| , ≤

It should be mentioned here that if we set a1 = b1 = 0, a2 = A, and b2 = B in Corollary 2.1, then we retain the inequality (1.1). The finite-dimensional version of Corollary 2.1 has been given in [3, Theorem 3] using some characterizations of unitarily invariant norms and the concept of spectral variation. Also, the usual operator norm version of Corollary 2.1 has been given in [9, Theorem 3] using some results about norms of derivations. Thus, Corollary 2.1 is a generalization, with a very simple proof, of the corresponding results in [3] and [9]. For positive invertible operators, we have the following improvement of the inequality (1.1), which is a considerable generalization of Corollary 10 in [9]. Corollary 2.2. Let A and B be positive invertible operators. Then, for every operator X, −1 −1 |||X||| . (2.7) |||AX − XB||| ≤ max A − B −1 , B − A−1 Proof. The inequality (2.7) follows from the inequality (2.6) by letting a1 = −1 −1 A , b1 = B −1 −1 , a2 = A, and b2 = B . Our promised generalization of the inequality (1.2) can be stated as follows. Theorem 2.2. Let A be self-adjoint operator such that a1 ≤ A ≤ a2 . Then, for every positive operator X,

a2 − a1 |||AX − XA||| ≤ |||X ⊕ X||| . (2.8) 2

Vol. 62 (2008)


133

Proof. Since a1 ≤ A ≤ a2 , it follows that 0 ≤ A − a1 ≤ a2 − a1 . Now, by the inequality (1.2), we have |||AX − XA||| = |||(A − a1 )X − X(A − a1 )||| 1 A − a1 |||X ⊕ X||| ≤ 2 1 (a2 − a1 ) |||X ⊕ X||| , ≤ 2

as required.

It should be mentioned here that if we set a1 = 0 and a2 = A in Theorem 2.2, then we retain the inequality (1.2). Akin to the inequality (1.1), it has been shown in [6], and later in [3], that if A and B are positive operators, then for every operator X, |||AX − XB||| ≤ X |||A ⊕ B||| ,

(2.9)

A 0 defined on H ⊕ H. 0 B For the usual operator norm, the inequalities (1.1) and (2.9) are the same. The inequality (2.9) is the generalized commutator version of an earlier inequality obtained in [2], which says that if A and B are positive operators, then where A ⊕ B is the block-diagonal operator

|||A − B||| ≤ |||A ⊕ B||| .

(2.10)

For the reader’s convenience, we present an alternative proof of the inequality (2.9). This proof is based on unitary dilation, together with basic properties of unitarily invariant norms. First, assume that X ≤ 1, and let U=

X T

S −X ∗

,

√ √ 1 − XX ∗ and T = 1 − X∗ X. Then U is a unitary operator on A 0 H ⊕ H (see, e.g., [5, pp. 322-323]). If C = , then 0 0 where S =

CU − U C = and so

AX − XA 0 0 0

AX − XA AS −T A 0

,

= P (CU − U C)P,

134

Kittaneh

IEOT

1 0 . Now, by the submultiplicativity and the unitary invariance 0 0 properties of unitarily invariant norms, and by the inequality (2.10), we have AX − XA 0 ≤ |||CU − U C||| 0 0

where P =

=

|||C − U CU ∗ |||

≤ =

|||C ⊕ U CU ∗ ||| |||C ⊕ C|||  A 0 0  0 A 0   0 0 0 0 0 0

=

 0 0   . 0  0

Hence, |||AX − XA||| ≤ |||A ⊕ A||| ,

(2.11)

from which it follows that if A is a positive operator, then for every operator X, |||AX − XA||| ≤ X |||A ⊕ A||| .

(2.12)

The desiredinequality from the inequality (2.12) applied (2.9) now follows A 0 0 X . Here we invoke the properties that to the operators and 0 B X∗ 0 for every operators X, Y , and Z, 0 X X ∗ 0 = X , 0 Y Z 0 = |||Y ⊕ Z||| , and that |||Y ⊕ Y ||| ≤ |||Z ⊕ Z||| if and only if |||Y ||| ≤ |||Z||| . The last two properties follow from the fact that every unitarily invariant norm is a symmetric gauge function of singular values, and the Fan dominance principle (see, e.g., [1, pp. 91-93] or [4, pp. 71, 82]). Finally, we remark that a singular value inequality stronger than the inequality (2.9) has been obtained in [6] using the singular value version of the inequality (2.10) obtained by Zhan [10], and the concept of Cayley transforms. In this inequality, the norms are replaced by the singular values. Our new proof of the inequality (2.9) can be modified to yield the stronger singular value inequality. For the norm inequalities (1.1) and (1.2), it should be mentioned here that while the singular value version of the inequality (1.2) has been obtained in [6], it has been shown there that the singular value version of the inequality (1.1) does not hold.

Vol. 62 (2008)


135

Acknowledgement. The author is grateful to the referee for his suggestions.

References [1] R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. [2] R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. 11 (1990), 272-277. [3] R. Bhatia and F. Kittaneh, Commutators, pinchings, and spectral variation, Oper. Matrices 2 (2008), 143-151. [4] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, 18, Amer. Math. Soc., Providence, RI, 1969. [5] P. R. Halmos, A Hilbert Space Problem Book, second ed., Springer-Verlag, New York, 1982. [6] F. Kittaneh, Inequalities for commutators of positive operators, J. Funct. Anal. 250 (2007), 132-143. [7] F. Kittaneh, Norm inequalities for commutators of positive operators and applications, Math. Z. 258 (2008), 845-849. [8] F. Kittaneh, Norm inequalities for commutators of normal operators, Inequalities and Applications, Internat. Ser. Numer. Math., Vol. 157, Birkh¨ auser Verlag, Basel, 2008. [9] Y.-Q. Wang and H.-K. Du, Norms of commutators of self-adjoint operators, J. Math. Anal. Appl. 342 (2008), 747-751. [10] X. Zhan, On some matrix inequalities, Linear Algebra Appl. 376 (2004), 299-303. Fuad Kittaneh Department of Mathematics University of Jordan Amman Jordan e-mail: [email protected] Submitted: May 12, 2008. Revised: May 18, 2008.



Reduced Minimum Modulus Preserving in Banach Space Ha¨ıkel Skhiri Abstract. Let B(X) be the algebra of all bounded linear operators on a complex Banach space X and γ(T ) be the reduced minimum modulus of operator T ∈ B(X). In this work, we prove that if φ : B(X) −→ B(X), is a surjective linear map such that φ(I) is an invertible operator, then γ(T ) = γ(φ(T )), for every T ∈ B(X), if and only if, either there exist two bijective isometries U1 ∈ B(X) and U2 ∈ B(X) such that φ(T ) = U1 T U2 for every T ∈ B(X), or there exist two bijective isometries V1 ∈ B(X , X) and V2 ∈ B(X, X ) such that φ(T ) = V1 T V2 for every T ∈ B(X). This generalizes for a Banach space the Mbekhta’s theorem [12]. Mathematics Subject Classification (2000). Primary 47B48, 47A10; Secondary 47A30, 46H05. Keywords. Reduced minimum modulus, isometry, linear preservers, generalized spectrum.

1. Terminology and introduction Let X and Y be complex infinite-dimensional Banach spaces, and let B(X, Y) be the space of all bounded linear operators from X to Y with the usual operator norm. If X = Y, we shall use the symbol B(X) instead of B(X, Y). For T ∈ B(X, Y), N(T ) and R(T ) denote the kernel and the range of T, respectively. Let G(X) denote the group of invertible elements of the Banach algebra B(X) (namely, the group of all linear bounded operators T on X such that N(T ) = {0} and R(T ) = X). Throughout this paper, we denote by X the dual space of X and the conjugated of an operator T by T . A bounded operator S ∈ B(X, Y) is called an isometry if S(x) = x, ∀ x ∈ X.

138

Skhiri

IEOT

Recall that the reduced minimum modulus of the operator T ∈ B(X), denoted by γ(T ), is defined to be T (x) γ(T ) = inf :x∈ / N(T ) . (1.1) dist(x, N(T )) We see also γ(T ) = sup a ≥ 0 : T (x) ≥ a dist(x, N(T )) . (1.2) For more details about the reduced minimum modulus see [5, 8, 9]. It is well-know that for all T ∈ B(X), γ(T ) ⇐⇒ R(T ) is closed, γ(T ) = γ(T ). If T ∈ B(X) is an invertible operator, then we have 1 γ(T ) = . T −1

(1.3)

If U ∈ B(X, Y) and V ∈ B(Y, X) are two bijective isometries, it is not difficult to verify that γ(U T V ) = γ(T ) ∀T ∈ B(X). (1.4) Linear preserver problems present one of the most active research topics in matrix theory (see the survey paper [10]). Over the past decades, there has been a considerable interest also to similar questions in infinite-dimension, that is to linear preserver problems on operator algebra (see survey papers [1, 6, 7, 13, 14]). In both cases, the problem is to characterize those linear maps on the algebra in question which leave invariant certain properties of operators such as the spectrum, numerical range, commutativity, invertibility, etc. Recently, in the Hilbert space M. Mbekhta [12] studied surjective linear map from B(H) onto B(H) that preserve the generalized spectrum and he gave a characterization of surjective unital linear map from B(H) onto B(H) that preserve the reduced minimum modulus. In this paper we are interested in the problem of linear problem preserving in Banach space of the generalized spectrum and the reduced minimum modulus using other technics. The paper is organized as follows. In the next section we prove some results related to the problem of linear map preserving generalized spectrum in Banach space. In Section 3 we are interested to the particular linear map T −→ AT B where A and B are two invertible operators. Specifically we show that γ(AT B) = γ(T ) holds for all T ∈ B(X), if and only if, there exist two bijective isometries U and V such that A = λU, B = µV where λ, µ ∈ C∗ with |λµ| = 1. This result is used in Section 4 to prove our main result. More precisely, we prove that if φ : B(X) −→ B(X) is a surjective linear map such that φ(I) is an invertible operator, then γ(T ) = γ(φ(T )) for all T ∈ B(X), if and only if, either there exist two bijective isometries U1 ∈ B(X) and U2 ∈ B(X) such that φ(T ) = U1 T U2 for every T ∈ B(X), or there exist two bijective isometries V1 ∈ B(X , X) and V2 ∈ B(X, X ) such that φ(T ) = V1 T V2 for every T ∈ B(X). This generalizes for a

Vol. 62 (2008)

Reduced Minimum Modulus Preserving in Banach Space

139

Banach space the Mbekhta’s theorem [12] and give a positive answer for Mbekhta’s conjecture [12] under the additional assumption of invertibility of φ(I) even in a Banach space.

2. Linear map preserving generalized spectrum in Banach space We say that T ∈ B(X) is regular if R(T ) is closed and N(T ) ⊆ R(T n ), for every n ∈ N. (see, [11, Definition 2.2]). For every T ∈ B(X) we set reg(T ) = {λ ∈ C : T − λI is regular}. The generalized spectrum of T ∈ B(X), denoted by σg (T ), is the set σg (T ) = {λ ∈ C : λ ∈ reg(T )}. According to [11], for an operator T ∈ B(X), we have the following: σg (T ) = {λ ∈ C : lim γ(T − zI) = 0}.

(2.1)

∂σ(T ) ⊆ σg (T ) ⊆ σ(T ).

(2.2)

z→λ

σg (T ) = σg (T ).

(2.3)

It is clear that if A ∈ B(X) is an invertible operator, then σg (AT A−1 ) = σg (T ), ∀T ∈ B(X).

(2.4)

We shall say that φ : B(X) −→ B(X), is generalized spectrum preserving (resp. compressing) if σg (T ) = σg (φ(T )), (resp. σg (φ(T )) ⊆ σg (T )) for every T ∈ B(X). Remark 2.1. Let A ∈ B(X) be an invertible operator, from the equality (2.4), we deduce that the linear map φA : B(X) −→ B(X), defined by T −→ AT A−1 is generalized spectrum preserving (resp. compressing). Let K be a compact subset of C, we recall that the polynomial convex hull of K is defined by K = {z ∈ C : |P (z)| ≤ sup |P (t)|, for all polynomials P }. K t∈K

First we prove the following simple lemma. Lemma 2.2. Let K be a compact subset of C and Ω be a subset of C such that = K. ∂K ⊆ Ω ⊆ K, then Ω ⊆ K. Moreover, we know that, if P is a Proof. Since Ω ⊆ K, we deduce that Ω polynomial, then P is an analytic function. Thus sup |P (t)| = sup |P (t)|. This t∈K

t∈∂K

= K. ⊆ Ω. Since, ∂K ⊆ Ω we have ∂K Therefore Ω = K. This implies that ∂K completes the proof. The following result generalized in the case of Banach space the Theorem 3.1 of Mbekhta [12].

140

Skhiri

IEOT

Theorem 2.3. Let φ : B(X) −→ B(X) be a surjective linear map. Then the following assumptions are equivalent: (1) φ is generalized spectrum preserving; (2) either (i) there exists an invertible operator A ∈ B(X) such that φ(T ) = AT A−1 for every T ∈ B(X); or (ii) there exists an invertible operator B ∈ B(X , X) such that φ(T ) = BT B −1 for every T ∈ B(X). Proof. (2) implies (1) follows immediately from Remark 2.1. It only remains to establish that (1) implies (2). For all T ∈ B(X), by the relation (2.2), we get ∂σ(φ(T )) ⊆ σg (φ(T )) = σg (T ) ⊆ σ(φ(T )),

(∗)

and ∂σ(T ) ⊆ σg (T ) = σg (φ(T )) ⊆ σ(T ). From (∗), (∗∗) and Lemma 2.2, we deduce that

(∗∗)

)) = σ σ(φ(T g (T ) = σ(T ). Now by [4, Corollary 4.5] (see, also [3, Theorem 3.1]), we know that either there exists an invertible A ∈ B(X) such that φ(T ) = AT A−1 for every T ∈ B(X), or there exists an invertible B ∈ B(X , X) such that φ(T ) = BT B −1 for every T ∈ B(X). This completes the proof. If we suppose in addition that φ is a bijective map, we obtain the following result. Theorem 2.4. Let φ : B(X) −→ B(X) be a bijective linear map. Then the following assumptions are equivalent: (1) φ is generalized spectrum preserving; (2) φ is generalized spectrum compressing; (3) either (i) there exists an invertible operator A ∈ B(X) such that φ(T ) = AT A−1 for every T ∈ B(X); or (ii) there exists an invertible operator B ∈ B(X , X) such that φ(T ) = BT B −1 for every T ∈ B(X). Proof. (1) ⇐⇒ (3) follows immediately from Theorem 2.3. (3) =⇒ (2) is clear, we only need to prove (2) =⇒ (3). For all T ∈ B(X), from the relation (2.2), we obtain ∂σ(φ(T )) ⊆ σg (φ(T )) ⊆ σ(φ(T )),

(∗)

∂σ(T ) ⊆ σg (T ) ⊆ σ(T ).

(∗∗)

and

Vol. 62 (2008)


141

Using (∗), (∗∗) and Lemma 2.2, we get )) = σg σ(φ(T (φ(T )) ⊆ σ g (T ) = σ(T ). From [4, Theorem 4.1], either there exists an invertible A ∈ B(X) such that φ(T ) = AT A−1 for every T ∈ B(X), or there exists an invertible B ∈ B(X , X) such that φ(T ) = BT B −1 for every T ∈ B(X). This completes the proof.

3. Reduced minimum modulus and the map: T −→ AT B Now we fix some more notations. In the sequel we denote by S(X) the set of the unit vectors in X: S(X) = {x ∈ X : x = 1}. Let M be a linear subspace of X. We say that M is of finite codimension if dim(X/M) < +∞. In this case, we define the codimension of M by codimM = dim(X/M). We will use ⊕ to denote the usual direct sum of closed subspaces, i.e., M = M1 ⊕ M2 if the linear space M = M1 + M2 is closed and M1 ∩ M2 = {0}. Throughout, we denote by x ⊗ ϕ the bounded linear operator on X defined for any x ∈ X and ϕ ∈ X by (x ⊗ ϕ)(y) = ϕ(y)x for arbitrary y ∈ X. Note that this operator is of rank one whenever x and ϕ are non-zero, and that every operator of rank one can be written in this form with x and ϕ non-zero. The most important result in this section is Theorem 3.1. This theorem will be used in the next section to prove our main result. Theorem 3.1. Let A ∈ B(X, Y) and B ∈ B(Y, X) be invertible operators. Then the following assumptions are equivalent: (i) AT B = T for every T ∈ G(X); (ii) there exist two bijective isometries U ∈ B(X, Y) and V ∈ B(Y, X) such that A = λU, B = µV where λ, µ ∈ C∗ with |λµ| = 1. Proof. We need only to prove (i)=⇒(ii). It is not difficult to verify the following equivalence AT B = T ∀ T ∈ G(X) (1) AT A−1 = T A−1 B −1 ∀ T ∈ G(X). First, we remark from the equivalence (1) the following equalities 1 = AA−1 B −1 B = A−1 B −1 .

(2)

We claim that A(x) = A × x ∀ x ∈ X. On the contrary, assume that there exists x0 ∈ S(X) such that A(x0 ) < A. Let us denote y0 = A(x0 ) ∈ Y and let M = x0 be the subspace of X spanned by x0 . If ϕ is the linear functional defined on M by ϕ(λx0 ) = λ, for all λ ∈ C, then ϕ = 1. It follows from Hahn-Banach theorem that there exists ϕ ∈ X such that ϕ = ϕ = 1 and the restriction of ϕ to M is ϕ. Let M = Ker(ϕ), since

142

Skhiri

IEOT

is a closed subspace of X. It is routine to verify that ϕ is continuous, we have M M ⊕ M = X. On the other hand, there exists a sequence (αn )n∈N of unit vectors in X such that lim A(αn ) = A. So, applying Hahn-Banach theorem again, we know n→+∞

that there exists ϕ n ∈ X satisfying the following ϕ n = 1 and ϕ n (αn ) = 0. n = Let Mn = αn be the subspace of X spanned by the vector αn and put M Ker( ϕn ). Since ϕ n is continuous, it follows that Mn is a closed subspace of X. n = X. Since codimM = 1 and codimM n = 1, we deduce that Clearly, Mn ⊕ M ∩M n = 1 or 2. codimM ∩M n ) = 2, let zn ∈ M ∩ S(X) such that • Case 1. If codim(M = (M ∩M n ) ⊕ zn . M

(3)

n ∩ S(X) such that For any n ∈ N, we can find βn ∈ M n = (M ∩M n ) ⊕ βn . M

(4)

∩M n ). Let Qn be the From the equality (3), we deduce that X = x0 ⊕ zn ⊕ (M projection onto M ∩ Mn along x0 ⊕ zn . Since M and Mn are closed subspaces of X, we deduce that Qn is a continuous projection (see, [15, Theorem 12.5] or [2, Lemma 2.5.1]). Define the linear functional ψn on x0 ⊕zn by ψn (λx0 +µzn ) = µ, where λ, µ ∈ C. Clearly, we have ψn (x0 ) = 0 and ψn (zn ) = 1. From Hahn-Banach n ∈ X such that ψ n = ψn and the theorem, we know that there exists ψ n to x0 ⊕ zn is ψn . restriction of ψ ∩M n ) = 1, then M =M n . Let Q be the projection onto M • Case 2. If codim(M along x0 . Since M and x0 are closed subspaces of X, we deduce that Q ∈ B(X). Now for n ∈ N∗ , we define the operator Tn by   n ) if codim(M ∩M n) = 2  Tn = αn ⊗ ϕ + 1 Qn + 1 (βn ⊗ ψ n n  T = α ⊗ ϕ + 1Q ∩M n ) = 1.  if codim(M n n n Clearly, we have Tn ∈ B(X), N(Tn ) = {0}, R(Tn ) = X and Tn (x0 ) = αn . So, Tn is an invertible operator. Since αn ∈ S(X) and ϕ = 1, we deduce that αn ⊗ ϕ = 1 for every n ∈ N∗ . Consequently, there exists n0 ∈ N such that 1 A Tn < 1 + − 1 = a, ∀ n ≥ n0 . 3 y0 Moreover, it is not difficult to verify that y A(α ) 0 n = . ATn A−1 y0 y0

Vol. 62 (2008)


This implies that lim ATn A−1 ≥ n→+∞

such that ATn A−1 >

143

1 A > = 1. So, there exists n1 ≥ n0 y0 x0

1 A A − − 1 = b, ∀ n ≥ n1 . y0 3 y0

It is routine to see that b > a. Using (1) and (2), we deduce that b < ATn A−1 = Tn A−1 B −1 ≤ Tn < a. This contradicts the fact that b > a. Consequently, A(x0 ) = y0 = A × x0 . We deduce that A(x) = A × x, ∀ x ∈ X. A A is a bijective isometry. It is clear that if is a bijective A A isometry, then A × B = 1. Indeed, if T = I, from the hypothesis we get 1 B = . A A B B is isometry. Let S = and L = . It is clear that Now, we show that B A B This implies that

AT B = T ∀ T ∈ G(X) ST L = T ∀ T ∈ G(X).

(5)

If either T = S −1 or T = L−1 from the equivalence (5), we deduce that L = S −1 = 1 and 1 = S = L−1 = 1. So, L = L−1 = 1 and it is not difficult to see L(x) = x ∀ x ∈ X. This achieves the proof of Theorem 3.1.

As an application of Theorem 3.1, we have the following corollary. Corollary 3.2. Let A ∈ B(X, Y) and B ∈ B(Y, X) be invertible operators. Then the following assumptions are equivalent: (1) (2) (3) (4) (5)

AT B = T for all T ∈ B(X); AT B = T for all T ∈ G(X); γ(AT B) = γ(T ) for all T ∈ B(X); γ(AT B) = γ(T ) for all T ∈ G(X); there exist two bijective isometries U ∈ B(X, Y) and V ∈ B(Y, X) such that A = λU, B = µV where λ, µ ∈ C∗ with |λµ| = 1.

144

Skhiri

IEOT

Proof. (1) ⇐⇒ (2) ⇐⇒ (5) follows immediately from Theorem 3.1. (3) =⇒ (4) and (5) =⇒ (3) are trivial. We prove that (4) =⇒ (5). Assume (4), from the equality (1.3) we get 1 γ(AT B) = ∀ T ∈ G(X). B −1 T −1 A−1 So, B −1 T −1 A−1 = T −1 for all T ∈ G(X). Now by Theorem 3.1, we deduce that A = λU, B = µV where U ∈ B(X, Y) and V ∈ B(Y, X) are bijective isometries, and λ, µ ∈ C∗ with |λµ| = 1. The proof is completed. The following corollaries are obvious consequences of Corollary 3.2. Corollary 3.3. Let A ∈ B(X, Y) and B ∈ B(Y, X) be invertible operators such that γ(AT B) = γ(T ) ∀ T ∈ G(X), then A × B = 1. Corollary 3.4. Let A ∈ B(X) be an invertible operator then the following assumptions are equivalent: (1) γ(AT A−1 ) = γ(T ) for all T ∈ B(X); (2) γ(AT A−1 ) = γ(T ) for all T ∈ G(X); (3) there exists a bijective isometry U ∈ B(X) and λ ∈ C∗ such that A = λU.

4. Linear map preserving the reduced minimum modulus in Banach space In this section we prove our main result. First of all we start with Lemma 4.1. This lemma will be used in the proof of Theorem 4.2. Lemma 4.1. Let S ∈ B(X) be a left invertible operator and let T ∈ B(X). Then γ(ST ) ≥ γ(S)γ(T ). Proof. Since S is injective, we get N(T ) = N(ST ). It follows for all x ∈ N(ST ) that ST (x) ≥ γ(S) T (x) ≥ γ(S) γ(T ) dist(x, N(T )). This implies that γ(ST ) ≥ γ(S)γ(T ), which completes the proof of this lemma.

The following theorem is our main result. Theorem 4.2. Let φ : B(X) −→ B(X) be a surjective linear map. Suppose that φ(I) is an invertible, then the following assumptions are equivalent:

(i) γ(T ) = γ φ(T ) for every T ∈ B(X);

Vol. 62 (2008)


145

(ii) either (ii1 ) there exist two bijective isometries U1 ∈ B(X) and U2 ∈ B(X) such that φ(T ) = U1 T U2 for every T ∈ B(X); or (ii2 ) there exist two bijective isometries V1 ∈ B(X , X) and V2 ∈ B(X, X ) such that φ(T ) = V1 T V2 for every T ∈ B(X). Proof. Since (ii)=⇒(i) is clear, we need only to prove (i)=⇒(ii). Let φ(I) = S and ψ = S −1 φ. We prove that ψ is generalized spectrum preserving. For z ∈ C and T ∈ B(X), we have γ ψ(T ) − zI = γ S −1 φ(T ) − zS −1 S (1) = γ S −1 (φ(T ) − zS) = γ S −1 φ(T − zI) . Using (1) and Lemma 4.1, we deduce that γ φ(T − zI) 0≤ ≤ γ ψ(T ) − zI ∀ T ∈ B(X). S

Let λ ∈ σg ψ(T ) , then lim γ ψ(T ) − zI = 0. It follows from (2) that z→λ lim γ(T − zI) = lim γ φ(T − zI) = 0. z→λ

(2)

z→λ

This implies that λ ∈ σg (T ) and therefore

σg ψ(T ) ⊆ σg (T ).

(3)

Conversely, let λ ∈ σg (T ), then lim γ(T − zI) = 0. z→λ So, by hypothesis we deduce that lim γ φ(T − zI) = 0. z→λ

On the other hand, we have γ φ(T − zI)

= γ φ(T ) − zS = γ Sψ(T ) − zS = γ S(ψ(T ) − zI) .

Using the same argument as above, we can deduce that lim γ ψ(T − zI) = lim γ ψ(T ) − zI = 0. z→λ z→λ

This implies that λ ∈ σg ψ(T ) and therefore

σg (T ) ⊆ σg ψ(T ) . Now by (3) and (5), we get

σg (T ) = σg ψ(T ) ∀ T ∈ B(X).

(4)

(5)

146

Skhiri

IEOT

From Theorem 2.3, we know that either there exists an invertible A ∈ B(X) such that ψ(T ) = AT A−1 for all T ∈ B(X) or there exists an invertible B ∈ B(X , X) such that ψ(T ) = BT B −1 for all T ∈ B(X). Thus, either φ(T ) = SAT A−1 for all T ∈ B(X) or φ(T ) = SBT B −1 for all T ∈ B(X). The conclusion follows from the Corollary 3.2. This completes the proof of this theorem. The following corollary is an immediate consequence of the previous theorem. Corollary 4.3. Let φ : B(X) −→ B(X) be a surjective unital linear map. Then the following assumptions are equivalent:

(1) γ(T ) = γ φ(T ) for every T ∈ B(X); (2) either (i) there exists a bijective isometry U ∈ B(X) such that φ(T ) = U T U −1 for every T ∈ B(X); or (ii) there exists a bijective isometry V ∈ B(X , X) such that φ(T ) = V T V −1 for every T ∈ B(X). Now, suppose that φ : B(X) −→ B(X) is a bijective map, if we replace the condition (i) in Theorem 4.2 by the following (weaker) condition

γ(T ) ≤ γ φ(T ) ∀ T ∈ B(X), we obtain the following theorem. Theorem 4.4. Let φ : B(X) −→ B(X) be a bijective linear map such that φ(I) is an invertible operator. Suppose that

γ(T ) ≤ γ φ(T ) ∀ T ∈ B(X). Then, either (i) there exist two invertible operators A ∈ B(X) and B ∈ B(X) such that φ(T ) = AT B for every T ∈ B(X); or (ii) there exist two invertible operators C ∈ B(X , X) and D ∈ B(X, X ) such that φ(T ) = CT D for every T ∈ B(X). Proof. Let ψ(T ) = S −1 φ(T ) where S = φ(I). First we prove that ψ is generalized spectrum compressing. Clearly, γ ψ(T ) − zI = γ S −1 φ(T ) − zS −1 S = γ S −1 (φ(T ) − zS) (1) −1 = γ S φ(T − zI) .

Vol. 62 (2008)


By (1) and Lemma 4.1, we deduce that 0≤

γ(T − zI) S

≤ ≤

147

γ φ(T − zI) S γ S −1 φ(T − zI) γ ψ(T ) − zI .

(2)

=

Let λ ∈ σg ψ(T ) , then lim γ ψ(T ) − zI = 0. From (2) it follows that lim γ(T − z→λ z→λ

zI) = 0. This implies that λ ∈ σg (T ) and therefore σg ψ(T ) ⊆ σg (T ). Now, by Theorem 2.4, we know that either there exists an invertible A1 ∈ B(X) for all T ∈ B(X) or there exists an invertible C1 ∈ such that ψ(T ) = A1 T A−1 1 B(X , X) such that ψ(T ) = C1 T C1−1 for all T ∈ B(X). Consequently, either φ(T ) = Sψ(T ) = SA1 T A−1 for all T ∈ B(X) or φ(T ) = Sψ(T ) = SC1 T C1−1 for all 1 T ∈ B(X). This completes the proof. We conclude this paper by the following corollary which is a direct consequence of the previous theorem. Corollary 4.5. Let φ : B(X) −→ B(X) be that

γ(T ) ≤ γ φ(T ) Then, either (1) there exists an invertible operator A every T ∈ B(X); or (2) there exists an invertible operator B for every T ∈ B(X).

a bijective unital linear map. Suppose ∀ T ∈ B(X). ∈ B(X) such that φ(T ) = AT A−1 for

∈ B(X , X) such that φ(T ) = BT B −1

References [1] B. Aupetit, Spectrum-preserving linear mappings between Banach algebras or JordanBanach algebras, J. London Math. Soc. 62 (2000), 917–924. [2] S. R. Caradus, W. E. Plaffenberger and B. Yood, Calkin algebras and algebras of operators on Banach spaces, Marcel Dekker, New York, 1974. [3] J. Cui and J. Hou, Additive maps on standard operator algebras preserving parts of the spectrum, J. Math. Anal. Appl. 282 (2003), 266–278. [4] J. Cui and J. Hou, Linear maps between Banach algebras compressing certain spectral functions, Rocky Mountain J. Math. 34 (2004), 565–584. [5] S. Goldberg, Unbounded Linear operators, McGraw-Hill, New York, 1966. [6] J. C. Hou, Rank-preserving linear maps on B(X ), Sci. China Ser. A 32 (1989), 929– 940. [7] A. A. Jafarian and A. R. Sourour, Spectrum-preserving, linear maps. J. Funct. Anal. 66 (1986), 255–261.

148

Skhiri

IEOT

[8] T. Kato, Perturbation theory for nullity, deficiency, and other quantities of linear operators, J. Anal. Math. 6 (1958), 261–322. [9] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966. [10] C. K. Li and N. K. Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162/164 (1992), 217–235. [11] M. Mbekhta and A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, Acta Sci. Math. (Szeged) 59 (1994), 525–543. [12] M. Mbekhta, Linear maps preserving the generalized spectrum, Extracta Math. 22, (2007), 45–54. ˇ [13] M. Omladi˘c and P. Semrl, Additive mappings preserving operators of rank one. Linear Algebra Appl. 182 (1993), 239–256 . [14] A. R. Sourour, Inversibility preserving linear maps on L(X), Trans. Amer. Math. Soc. 348 (1996), 13–30. [15] A. Taylor, Introduction to functional analysis, Wiley, New York, 1958. Ha¨ıkel Skhiri Faculté des Sciences de Monastir Département de Mathématiques Avenue de l’environnement 5019 Monastir Tunisia e-mail: [email protected] [email protected] Submitted: August 6, 2007.



J-Class Weighted Shifts on the Space of Bounded Sequences of Complex Numbers George Costakis and Antonios Manoussos Abstract. We provide a characterization of J-class and J mix -class unilateral weighted shifts on l∞ (N) in terms of their weight sequences. In contrast to the previously mentioned result we show that a bilateral weighted shift on l∞ (Z) cannot be a J-class operator. Mathematics Subject Classification (2000). Primary 47A16; Secondary 37B99, 54H20. Keywords. Hypercyclic operators, J-class operators, J mix -class operators, unilateral and bilateral weighted shifts.

1. Introduction During the last years the dynamics of linear operators on infinite dimensional spaces has been extensively studied, see the survey articles [4], [7], [8], [9], [10], [12] and the recent book [1]. Let us recall the notion of hypercyclicity. Let X be a separable Banach space and T : X → X be a bounded linear operator. The operator T is said to be hypercyclic provided there exists a vector x ∈ X such that its orbit under T , Orb(T, x) = {T nx : n = 0, 1, 2, . . .}, is dense in X. If X is Banach space (possibly non-separable) and T : X → X is a bounded linear operator then T is called topologically transitive (topologically mixing) if for every pair of non-empty open subsets U, V of X there exists a positive integer n such that T n U ∩ V = ∅ (T m U ∩ V = ∅ for every m ≥ n respectively). It is well known, and easy to prove, that if T is a bounded linear operator acting on separable Banach space X then T is hypercyclic if and only if T is topologically transitive. A first step to understand the dynamics of linear operators is to look at particular operators as for example the weighted shifts. Salas [11] was the first who During this research the second author was fully supported by SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik” at the University of Bielefeld, Germany. He would also like to express his gratitude to Professor H. Abels for his support.

150

Costakis and Manoussos

IEOT

characterized the hypercyclic weighted shifts in terms of their weight sequences. We would like to point out that l∞ (N) and l∞ (Z) do not support hypercyclic operators since they are not separable Banach spaces. In fact they do not support topologically transitive operators as it was shown by Berm´ udez and Kalton in [2]. Recently Bès, Chan and Sanders [3] showed that there exists a weak* hypercyclic weighted shift T on l∞ (N), i.e there exists a vector x ∈ l∞ (N) whose orbit Orb(T, x) is dense in the weak* topology of l∞ (N). In fact they give a characterization of the weak* hypercyclic weighted shifts in terms of their weight sequences. In [5] we studied the dynamics of operators by replacing the orbit of a vector with its extended limit set. To be precise, let T : X → X be a bounded linear operator on a Banach space X (not necessarily separable) and x ∈ X. A vector y belongs to the extended limit set J(x) of x if there exist a strictly increasing sequence of positive integers {kn } and a sequence {xn } ⊂ X such that xn → x and T kn xn → y. If J(x) = X for some non-zero vector x ∈ X then T is called J-class operator. Roughly speaking, the use of the extended limit set “localizes” the notion of hypercyclicity. The last can be justified by the following: J(x) = X if and only if for every open neighborhood U of x and every non-empty open set V ⊂ X there exists a positive integer n such that T n U ∩ V = ∅. The purpose of this paper is to study the dynamical behavior of weighted shifts on the spaces of bounded sequences of complex numbers l∞ (N) and l∞ (Z) through the use of the extended limit sets. Our main result is the following (see Theorem 3.1). Theorem. Let T : l∞ (N) → l∞ (N) be a backward unilateral weighted shift with positive weights (αn )n∈N . The following are equivalent. (i) T is a J-class operator. n (ii) lim αi+j = +∞. inf n→+∞

j≥0

i=1

In particular, if T is a J-class operator then the sequence of weights (αn )n∈N is bounded from below by a positive number and we have the following complete description of the set of J-vectors. {x ∈ l∞ (N) : J(x) = l∞ (N)} = c0 (N), where c0 (N) = {x = (xn )n∈N ∈ l∞ (N) : limn→+∞ xn = 0}. Observe that if T is a J-class backward unilateral weighted shift on l∞ (N) then in view of the above theorem and Salas’ characterization of hypercyclic weighted shifts, see [11], we conclude that T is hypercyclic on lp (N) for every 1 ≤ p < +∞. However, as we show in section 3, the converse is not always true. On the other hand the situation is completely different in the case of bilateral weighted shifts. In particular we show that a bilateral weighted shift on l∞ (Z) cannot be a J-class operator, see Theorem 3.3. In addition, we prove similar results for J mix -class weighted shifts (see Definitions 2.1 and 2.2).

J-Class Weighted Shifts

Vol. 62 (2008)

151

2. Preliminaries Definition 2.1. Let T : X → X be a bounded linear operator on a Banach space X. For every x ∈ X the sets J(x) = {y ∈ X : there exist a strictly increasing sequence of positive integers {kn } and a sequence {xn } ⊂ X such that xn → x and T kn xn → y}, J mix (x) = {y ∈ X : there exists a sequence {xn } ⊂ X such that xn → x and T n xn → y} will be called the extended limit set of x under T and the extended mixing limit set of x under T respectively. Definition 2.2. A bounded linear operator T : X → X acting on a Banach space X will be called a J-class (J mix -class) operator if there exists a non-zero vector x ∈ X such that J(x) = X (J mix (x) = X respectively). Definition 2.3. Let T be a bounded linear operator acting on a Banach space X. A vector x ∈ X will be called a J-vector (J mix -vector) if J(x) = X (J mix (x) = X respectively). Remark 2.4. Observe that (i) an operator T : X → X is topologically transitive if and only if J(x) = X for every x ∈ X, (ii) an operator T : X → X is topologically mixing if and only if J mix (x) = X for every x ∈ X, see [5]. Hence every hypercyclic operator (topologically mixing) is a J-class operator (J mix -class operator). However the converse is not true. To see that consider the operator 3I ⊕ 2B : C ⊕ l2 (N) → C ⊕ l2 (N) where I is the identity map on C and B is the backward shift on the space of square summable sequences l2 (N). Consider mix (0 ⊕ x) = C ⊕ l2 (N). any non-zero vector x ∈ l2 (N). We shall prove that J3I⊕2B 2 Let y ∈ l (N) and λ ∈ C. There exists a sequence {xn } in l2 (N) such that (2B)n xn → y. Define the vectors 3λn ⊕ xn . Then we have 3λn ⊕ xn → 0 ⊕ x and (3I ⊕ 2B)n ( 3λn ⊕ xn ) → λ ⊕ y. Hence 3I ⊕ 2B is a J mix -class operator which is not hypercyclic. In fact it is not even supercyclic, see [6]. Let us also give an example of a backward weighted shift, acting on a nonseparable space, which is a J-class operator but not topologically transitive. Consider the operator 2B : l∞ (N) → l∞ (N) where B is the backward shift and l∞ (N) is the space of bounded sequences. Theorem 3.6 implies that 2B is a J mix -class operator. On the other hand the space l∞ (N) does not support topologically transitive operators, see [2]. The next lemma, which will be of use to us, also appears in [5]. For the convenience of the reader we give its proof.

152


IEOT

Lemma 2.5. Let T : X → X be a bounded linear operator on a Banach space X and {xn }, {yn } be two sequences in X such that xn → x and yn → y for some x, y ∈ X. (i) If yn ∈ J(xn ) for every n = 1, 2, . . ., then y ∈ J(x). (ii) If yn ∈ J mix (xn ) for every n = 1, 2, . . ., then y ∈ J mix (x). Proof. (i) For n = 1 there exists a positive integer k1 such that 1 1 and yk1 − y < . 2 2 ∈ J(xk1 ) we may find a positive integer l1 and z1 ∈ X such that xk1 − x
M

for every j ≥ 0.

i=1

If N = 1 there is nothing to prove. Assume that N > 1. For every j ≥ 0 and since T = supn αn , we have N N −1 αj+1 T ≥ αj+1 αi+j > M. i=2

Proceeding inductively we conclude that αn ≥

M T N −1

J-Class Weighted Shifts

Vol. 62 (2008)

155

for every n ∈ N. Take any positive integer n > N . There exist positive integers pn , vn such that n = N pn + vn and 0 ≤ vn ≤ N − 1. Since (αn )n∈N is bounded from below by T MN −1 it follows that n

αi+j > M pn C

for every j ≥ 0,

i=1

where C = min

M T N −1

N −1

,1 .

From the last and the fact that M > 1 it clearly follows that n αi+j = +∞. inf lim n→+∞

j≥0

i=1

We shall now prove that (ii) implies (i). Fix a vector x = (x1 , x2 , . . .) in l∞ (N) with finite support. There exists a positive integer n0 such that xn = 0 for every n ≥ n0 and inf j≥0 ni=1 αi+j > 0 for every n ≥ n0 . Consider any vector y = (y1 , y2 , . . .) ∈ l∞ (N). We set

y1 y2 y3 yn = x1 , x2 , . . . , xn0 −1 , 0, . . . , 0, n , n , n ,... i=1 αi i=1 αi+1 i=1 αi+2 for every n ≥ n0 , where the 0’s fill all the coordinates from the n0 -th up to n-th position. Then for every n ≥ n0 we have yj+1 y∞ ≤

n yn − x∞ = sup n , inf j≥0 i=1 αi+j j≥0 i=1 αi+j hence yn → x. Observe also that T n yn = y, so y ∈ J(x). Thus T is a J-class operator and this completes the proof that (ii) implies (i). It remains to show that the set of J-vectors is c0 (N). From the proof that (ii) implies (i) we have that if x is a vector with finite support then J(x) = l∞ (N). Since the closure of the set of all vectors with finite support is c0 (N), by Lemma 2.5, we conclude that c0 (N) ⊂ {x ∈ l∞ (N) : J(x) = l∞ (N)}. To prove the converse inclusion, take a vector x such that J(x) = l∞ (N). Consider the zero vector and let be a positive number. There exist positive integers n0 , n1 and a vector yn0 = (yn0 k )k∈N such that yn0 − x∞ < , T n1 yn0 ∞ <

and

n1

αi+j > 1 for every j ≥ 0.

i=1

Hence we have

n 1 αi+j yn0 (n1 +j+1) < i=1

156


IEOT

for every j ≥ 0. The last and the previous bound on the weights imply that 0. Denote by Ω(t) the exterior of B at time t and assume that Ω(t) is a domain with boundary of class C 1,1 . The flow of a viscous incompressible fluid in the exterior of the body B can be described by the Navier–Stokes equation and the equation of continuity in the space-time region {(x, t) ∈ R3 × I; t ∈ I, x ∈ Ω(t)} where I is an interval on the time axis. The disadvantage of this description is the variability of the spatial domain Ω(t). Therefore, many authors use a time-dependent transformation of spatial coordinates which in fact also represents the rotation about the x1 axis such that the body B is fixed and its exterior is just Ω(0) in the new coordinate

170

Farwig and Neustupa

IEOT

system. The system of equations after the transformation has the form ∂t u − ν∆u − ω(e1 × x) · ∇u + ωe1 × u + (u · ∇)u + ∇p

=

f

(1.1)

∇·u =

0

(1.2)

in Ω(0) × I, where e1 is the unit vector oriented in the direction of the x1 -axis. The condition of the adherence of the fluid to the body on the boundary, after the transformation, has the form u(x, t) = ωe1 × x,

x ∈ ∂Ω(0).

(1.3)

In order to simplify the notation, we shall write only Ω instead of Ω(0). Among a series of results on qualitative properties of the system (1.1)–(1.3) and related linear problems, let us mention T. Hishida [15], [16], [17], G. P. Galdi ˇ Neˇcasová [23], [9], [10], R. Farwig, T. Hishida, D. M¨ uller [5], R. Farwig [3], [4], S. ˇ M. Geissert, H. Heck, M. Hieber [11], S. Kraˇcmar, S. Neˇcasová, P. Penel [19], ˇ Neˇcasová, J. Neustupa [7]. R. Farwig, J. Neustupa [6] and R. Farwig, S. We shall use the usual function spaces and notation: ◦ n is the outer normal vector on ∂Ω. ◦ (. , .)0,2 and . 0,2 are the scalar product and the norm in L2 (Ω)3 , respectively. ◦ W01,2 (Ω) is the subspace of the Sobolev space W 1,2 (Ω) consisting of functions vanishing on ∂Ω in the sense of traces. As is well-known, W01,2 (Ω) equals the closure of C0∞ (Ω) in the norm of W 1,2 (Ω). ◦ . k,2 denotes the norm in W k,2 (Ω)3 , k ∈ N. ∞ ◦ C0,σ (Ω) denotes the space of all divergence-free functions from C0∞ (Ω)3 . ∞ ◦ L2σ (Ω) is the closure of C0,σ (Ω) in L2 (Ω)3 . The space L2σ (Ω) can be characterized as the space of all divergence-free (in the sense of distributions) vector functions u from L2 (Ω)3 such that u · n = 0 on ∂Ω in the sense of traces ([8], pp. 111–115). ◦ Πσ denotes the orthogonal projection of L2 (Ω)3 onto L2σ (Ω). Suppose that U∗ is a steady strong solution of the problem (1.1)–(1.3) such that

|∇U∗ | ∈ L3/2 (Ω) ∩ L3 (Ω), ∗| = 0 lim ess sup |U∗ − U∞

R→+∞

(1.4) (1.5)

|x|>R

∗ = (γ, 0, 0), γ ∈ R. The function U := U∗ − U∗ equals ωe × x− (γ, 0, 0) where U∞ 1 ∞ on ∂Ω. Combining this information with the Sobolev inequality, see e.g. [8], p. 31, we can deduce that U satisfies U ∈ Ls (Ω)3 for all 3 ≤ s < +∞. In order to study the behavior of solutions near the steady solution U∗ , we put u = U∗ + v = (γ, 0, 0) + U + v. Then the perturbation v is a solution of the problem given by

Vol. 62 (2008)

On the Spectrum of an Oseen-Type Operator

171

the equations ∂t v − ν∆v − ω(e1 × x) · ∇v + ωe1 × v + γ∂1 v + (U · ∇)v + (v · ∇)U + (v · ∇)v + ∇p = 0, ∇·v = 0 in Ω × I (where I is a time interval) and by the boundary condition v(x, t) = 0

for x ∈ ∂Ω.

This problem can be written in the form of the operator equation ∂t v = Lω γ v + Nv

(1.6)

in L2σ (Ω) where Lω γv

= Aω γ v + Bv,

(1.7)

Aω γv

= Πσ ν∆v + Πσ [ω(e1 × x) · ∇v − ωe1 × v − γ∂1 v],

(1.8)

Bv

= −Πσ [(U · ∇)v + (v · ∇)U],

(1.9)

Nv

= −Πσ (v · ∇)v.

(1.10)

ω The operators Aω γ and Lγ are defined in the same domains ω D(Aω γ ) = D(Lγ ) = v ∈ W 2,2 (Ω)3 ∩ W01,2 (Ω)3 ∩ L2σ (Ω); ω(e1 × x) · ∇v ∈ L2 (Ω)3

which are dense subsets of L2σ (Ω). The information on spectra of the linear operaω tors Aω γ and Lγ plays a fundamental role in studies of the evolution equation (1.6). Whereas the case γ = 0 was treated in detail in our paper [6], here we consider the important case γ = 0. Our main theorem now reads as follows: Theorem 1.1. (i) The essential spectrum σess (Aω γ ) has the form νβ 2 ω σess (Aω ) = Λ := λ = α + iβ + ikω ∈ C; α, β ∈ R, k ∈ Z, α ≤ − , (1.11) γ γ γ2

(ii) (iii) (iv) (v) (vi)

i.e., it consists of an infinite union of equally shifted filled parabolas in the left half-plane of C, see Fig. 1. The operator Aω γ is not normal. If λ is an eigenvalue of Aω γ then Re λ < 0. If the body B (and therefore also the domain Ω) is axially symmetric about ω ω the x1 -axis, then σ(Aω γ ) = σess (Aγ ) = Λγ . ω The operator Lω γ has the same essential spectrum as Aγ . ω ω σ(Lγ ) = σess (Lγ )∪Γ where Γ consists of an at most countable set of isolated ω eigenvalues of Lω γ which can possibly cluster only at points of σess (Lγ ); each of them has finite algebraic multiplicity.

172

Farwig and Neustupa

q q q

IEOT

6 λi

C3 2ω i

Λωγ C2

ωi

C1

0

C0

C0 : λr = −νλ2i /γ 2 λr -

−ω i

C−1 q q q

Fig. 1 : The shape of set Λωγ .

The proof of statements (i)–(iii) and (v), (vi) is given in Section 6. Statement (iv) is proved in Section 5.

2. Preliminaries All function spaces needed in the following are considered to be spaces of complexvalued functions. ω Lemma 2.1. There exists c1 > 0 such that if v ∈ D(Aω γ ) and Aγ v = f , then (2.1) v2,2 + (ωe1 × x) · ∇v0,2 ≤ c1 (γ) f 0,2 + v0,2 . ω Proof. The equation Aω γ v = f means that A0 v = f + Πσ γ ∂1 v. Applying the results from [5] (Ω = R3 ) or from [16] (Ω being an exterior domain in R3 ) to the solution of the equation Aω 0 v = g (with g = f + Πσ γ ∂1 v), we obtain v2,2 + (ωe1 × x) · ∇v0,2 ≤ c1 (0) Aω 0 v0,2 + v0,2 ≤ c1 (0) f 0,2 + |γ| Πσ ∂1 v0,2 + v0,2 .

Interpolating suitably the norm Πσ ∂1 v0,2 between the norms v0,2 and v2,2 , we arrive at (2.1). 2 Lemma 2.2. Aω γ is a closed operator in Lσ (Ω) and its adjoint has the form −ω ∗ (Aω γ ) v = Πσ ν∆v − Πσ [ω(e1 × x) · ∇v − ωe1 × v − γ∂1 v] = A−γ v ω ∗ with D (Aγ ) = D(Aω γ ).

(2.2)

Vol. 62 (2008)


173

Proof. Since the operator Aω 0 is closed (as the generator of a C0 -semigroup, see [11]), we may use a classical perturbation result for closed operators, see T. Kato [18], p. 190, an interpolation estimate of Πσ γ∂1 and Lemma 2.1 to get that Aω γ is closed as well. Moreover, there exists a sufficiently large positive ζ ∈ ρ(Aω γ ). In order to verify the last statement and to solve the equation Aω γ u − ζu = f for 2 given f ∈ Lσ (Ω), we can follow the procedure described e.g. in [11, the proof of Proposition 4.3] (which concerns the case γ = 0), based on splitting the solution to the problem in the whole space R3 and the problem in a bounded domain. Let us denote by Tγω the operator on the right-hand side of (2.2) with −ω ω ω ω ω 2 D(Tγ ) = D(Aω γ ), i.e. Tγ = A−γ . Then Tγ is closed and R(ζI − Tγ ) = Lσ (Ω) if ζ > 0 is sufficiently large. Using integration by parts, we can verify that ω u, Aω γ v 0,2 = Tγ u, v 0,2 ω ω for all u ∈ D(Tγω ) and v ∈ D(Aω γ ). It means that the operators Aγ and Tγ are ω ω ∗ adjoint to each other and Tγ ⊂ (Aγ ) , see T. Kato [18], p. 167. ∗ ω Suppose that u ∈ D((Aω γ ) ). Then there exists w ∈ D(Tγ ) such that [ζI − ∗ ω ω (Aω γ ) ]u = (ζI − Tγ )w. Multiplying both sides of this identity by v ∈ D(Aγ ), we arrive at ω u, (ζI − Aω γ )v 0,2 = w, (ζI − Aγ )v 0,2 . ω As this holds for all v ∈ D(Aω γ ), we get u = w ∈ D(Tγ ) and consequently, ω ∗ ω ω ∗ ω D((Aγ ) ) ⊂ D(Tγ ). Thus, (Aγ ) = Tγ .

Lemma 2.3. If v ∈ D(Aω γ ), then both the terms ω(e1 × x) · ∇v − ωe1 × v and γ∂1 v belong to L2σ (Ω). Proof. It was already shown in [6] that ω(e1 × x) · ∇v − ωe1 × v ∈ L2σ (Ω). ∞ 1,2 (Ω) is dense in D(Aω (Ω)3 ; hence The space C0,σ γ ) in the topology of W ω n ∞ given v ∈ D(Aγ ), there exists a sequence v ∈ C0,σ (Ω) such that vn → v in 1,2 W 1,2 (Ω)3 . Let ψ be a function from Wloc (Ω) such that ∇ψ ∈ L2 (Ω)3 . Then we have γ∂1 v · ∇ψ dx = lim γ∂1 vn · ∇ψ dx n→+∞ Ω Ω = − lim div (γ∂1 vn ) ψ dx = 0. n→+∞

Ω

Thus the function γ∂1 v is orthogonal to the subspace of all gradients in L2 (Ω)3 , which implies that it belongs to L2σ (Ω), see e.g. G. P. Galdi [8], p. 103. Lemma 2.3 enables us to omit the projection Πσ in front of the terms in the brackets on the right-hand side of (1.8) and (2.2). The operator Aω γ can therefore be simplified to Aω γv

=

A00 v + ω(e1 × x) · ∇v − ωe1 × v − γ∂1 v

=

A0γ v + ω(e1 × x) · ∇v − ωe1 × v

(2.3)

174

Farwig and Neustupa

IEOT

where A00 ≡ νΠσ ∆ is the Stokes operator in L2σ (Ω) with domain D(A00 ) = W 2,2 (Ω)3 ∩W01,2 (Ω)3 ∩ L2σ (Ω). The Stokes operator A00 is selfadjoint in L2σ (Ω), see e.g. Y. Giga, H. Sohr [12]. Moreover, A0γ is the usual Oseen operator with the same domain as A00 . By analogy, the adjoint operator to Aω γ can be simplified to ∗ (Aω γ) v

=

A00 v − ω(e1 × x) · ∇v + ωe1 × v + γ∂1 v

=

A0−γ v − ω(e1 × x) · ∇v + ωe1 × v.

(2.4)

Lemma 2.4. The operator B, defined by (1.9), is Aω γ -compact. Proof. We have proved in [6] that the operator B is Aω 0 -compact. The proof in the case of γ = 0 can be done in the same way. The crucial step is an appropriate application of Lemma 2.1, which enables us to deduce that the boundedness of 2 two sequences {φn } and {Aω γ φn } in Lσ (Ω) implies the boundedness of {φn } in W 2,2 (Ω)3 . 2 Lemmas 2.2 and 2.4 imply that the operator Lω γ is closed in Lσ (Ω), see [18], p. 194. It will be further advantageous to work in cylindrical coordinates. We shall denote by x1 , r and ϕ the cylindrical coordinate system whose axis is the x1 -axis such that the angle ϕ is measured from the positive part of the x2 -axis towards the positive part of the x3 -axis. The corresponding cylindrical components of vector functions will be denoted by the indices 1, r and ϕ, e.g. u1 , ur and uϕ . In order to distinguish between the Cartesian and the cylindrical components of vectors, we shall write the Cartesian components in parentheses and the cylindrical components in brackets. Thus, we have (u1 , u2 , u3 ) [u1 , ur , uϕ ]. Using the transformations

ur = u2 cos ϕ + u3 sin ϕ,

u2 = ur cos ϕ − uϕ sin ϕ,

uϕ = −u2 sin ϕ + u3 cos ϕ,

u3 = ur sin ϕ + uϕ cos ϕ,

we can calculate that (ω × x) · ∇u − ω × u = ω ∂ϕ u − (ω × u) = ω ∂ϕ (u1 , u2 , u3 ) − ω (0, −u3 , u2 )   T T 0 u1 = ω ∂ϕ  ur cos ϕ − uϕ sin ϕ  − ω  −ur sin ϕ − uϕ cos ϕ  ur sin ϕ + uϕ cos ϕ ur cos ϕ − uϕ sin ϕ T T   ∂ϕ u1 ∂ϕ u1 = ω  (∂ϕ ur ) cos ϕ − (∂ϕ uϕ ) sin ϕ  ω  ∂ϕ ur  = ω ∂ϕ u1 , ur , uϕ . (∂ϕ ur ) sin ϕ + (∂ϕ uϕ ) cos ϕ ∂ϕ uϕ In the following, the vector function u will be identified with [u1 , ur , uϕ ]; the same holds for other vectors or vector functions. Thus, the relation (2.3) between the 0 operator Aω γ and the Stokes operator A0 can be written in the form 0 Aω γ u = A0 u + ω ∂ϕ u − γ ∂1 u

(2.5)

Vol. 62 (2008)


175

where A00 now stands for the Stokes operator in cylindrical coordinates. If T is a closed linear operator in a Hilbert space H, then we shall use the following notions and notation: ◦ N (T ) is the null space of T , R(T ) is the range and T ∗ is the adjoint operator to T . ◦ nul(T ) is the nullity and def(T ) is the deficiency of T . ◦ ind(T ) = nul(T ) − def(T ) denotes the index of T . ◦ nul (T ) is the approximate nullity and def (T ) is the approximate deficiency of T . ◦ ρ(T ) denotes the resolvent set of T . ◦ σp (T ) is the point spectrum of T , σc (T ) its continuous spectrum and σr (T ) its residual spectrum. ◦ σ(T ) is the whole spectrum of T (= σp (T ) ∪ σc (T ) ∪ σr (T )). ◦ σess (T ) denotes the essential spectrum of T , i.e. the set of λ ∈ C such that nul (T − λI) = def (T − λI) = +∞. ◦ σ c (T ) denotes the set of those λ ∈ C for which there exists a non-compact sequence {un } in the unit sphere in H such that (T − λI)un → 0 for n → +∞. It is equivalent to the equality nul (T − λI) = +∞. ◦ T is said to be normal if T ∗ T = T T ∗ . The definitions of these notions can be found in [18] or in [13], see [6] for the survey of their main properties. For the purposes of this paper, let us recall that σp (T ), σr (T ) and σc (T ) are mutually disjoint, σ(T ), σess (T ) and σ c (T ) are closed c (T ) ⊂ σ(T ). sets in C and σc (T ) ⊂ σess (T ) ⊂ σ

3. The Oseen operator A0γ It is known that the spectra of the Stokes operator A00 satisfy the identities σp (A00 ) = σ(A00 ) =

σr (A00 ) = ∅,

(3.1)

σess (A00 ) = σc (A00 ) = (−∞, 0].

(3.2)

(The residual spectrum of A00 is empty because A00 is selfadjoint. The reasons why the point spectrum is also empty are explained in [6]. The identities σ(A00 ) = σc (A00 ) = (−∞, 0] follow from I. M. Glazman [13] and O. A. Ladyzhenskaya [20].) The spectrum of the Oseen operator A0γ was studied by K. I. Babenko in [1]. Considering the case Ω = R3 and assuming that λ ∈ Λ0γ where Λ0γ = λ = α + iβ ∈ C; α, β ∈ R, α ≤ −νβ 2 /γ 2 , (3.3) K. I. Babenko mentions a construction (based on the Fourier transform) of a noncompact sequence {vn } in the unit sphere in L2σ (Ω) such that (A0γ −λI)vn 0,2 → 0 as n → +∞. Then λ ∈ σess (A0γ ) and consequently Λ0γ ⊂ σess (A0γ ). On the other

176

Farwig and Neustupa

IEOT

hand, the author states that the equation (A0γ − λI)v = f ,

(3.4)

L2σ (Ω),

for Re λ > 0 and f ∈ can be solved by means of a Green’s function of the Dirichlet problem with a reference to F. Odqvist [24] for more details concerning the construction of Green’s function and its estimates. Furthermore, K. I. Babenko emphasizes that it is not difficult to treat the other cases of λ ∈ C − Λ0γ . Thus, he arrives at Propositions 4 and 5 which imply that C − Λ0γ ⊂ ρ(A0γ ). Since the information on the spectrum of the operator A0γ is of fundamental importance, in the following theorem we present a complete proof based on a totally different approach. Theorem 3.1. σ(A0γ ) = σess (A0γ ) = Λ0γ . Proof. I. Let us begin with the inclusion C−Λ0γ ⊂ ρ(A0γ ) to be proved by contradiction. Suppose that λ = α+iβ ∈ (C−Λ0γ )∩σ(A0γ ). Assume that λ ∈ σp (A0γ )∪σc (A0γ ) at first. Then there exists a sequence vn in the unit sphere in L2σ (Ω) such that (A0γ − λI)vn = n −→ 0 n

in L2σ (Ω) as n → +∞. σp (A0γ ).

(3.5) n

This sequence {v } can be constant if λ ∈ We test (3.5) with v (in the 2 L -sense for complex-valued functions) and get that −ν ∇vn 20,2 − λ vn 20,2 − γ (∂1 vn , vn )0,2 = (n , vn )0,2 . n

(3.6)

n

Note that Re (∂1 v , v )0,2 = 0. Next we consider the real and imaginary part of (3.6) and see that ν ∇vn 20,2 = −α vn 20,2 − Re (n , vn )0,2 ≤ −α + n 0,2 . Using (3.5), we observe that α ≤ 0. From (3.6) and (3.7) we obtain 1/2 α 1 ∇vn 0,2 ≤ − + n 0,2 , ν ν as well as

(3.7)

(3.8)

β = β vn 20,2 = −γ Im (∂1 vn , vn )0,2 − Im (n , vn )0,2 so that

(3.9) |β| ≤ γ ∇vn 0,2 + n 0,2 . Inserting the estimate (3.8) into (3.9) we are led to the inequality α 1 1/2 |β| ≤ γ − + n 0,2 + n 0,2 . ν ν As n → +∞, (3.5) implies that |β| ≤ γ −α/ν, i.e. α ≤ νβ 2 /γ 2 . Obviously, this inequality is in contradiction with the assumption λ ∈ C − Λ0γ . Hence λ ∈ C − Λ0γ cannot belong to σp (A0γ ) ∪ σc (A0γ ). Now assume that λ ∈ σr (A0γ ). Then λ belongs to the point spectrum of the adjoint operator (A0γ )∗ ; this leads to the same contradiction as if λ ∈ σp (A0γ ). Thus, λ ∈ ρ(A0γ ) which implies that C−Λ0γ ⊂ ρ(A0γ ).

Vol. 62 (2008)


177

II. Now we will prove that Λ0γ ⊂ σ c (A0γ ). Let λ = α+iβ ∈ (Λ0γ )◦ be given; here (Λ0γ )◦ denotes the interior of Λ0γ , i.e. the set of α + iβ ∈ C such that α < −νβ 2 /γ 2 . The number α can be written in the form α = α1 + α2 where α1 = −νβ 2 /γ 2 and α2 < 0. We shall explicitly define functions vn ∈ L2σ (Ω) such that vn 0,2 = 1 and 0 (Aγ − λI)vn → 0 in L2σ (Ω) as n → +∞, and such that the sequence {vn } does not contain any subsequence, convergent in L2σ (Ω). Let us denote by v1n , vrn and vϕn the cylindrical components of vn . Put v1n (x1 , r, ϕ)

:= 0,

vrn (x1 , r, ϕ)

:= κn U n (x1 ) V n (r) eikϕ , 1 vϕn (x1 , r, ϕ) := − ∂r [r vrn (x1 , r, ϕ)] ik dV n (r) ikϕ 1 e = − κn U n (x1 ) V n (r) + r ik dr where k is an arbitrary, but fixed chosen non-zero integer. Then, obviously, vn satisfies the condition 1 1 ∇ · vn ≡ ∂1 v1n + ∂r (rvrn ) + ∂ϕ vϕn = 0. r r Here the function U n has the form U n (x1 ) := η1n (x1 ) Y (x1 )

(3.10)

where η1n is an infinitely differentiable function on (−∞, +∞) with values in the interval [0, 1], 0 for x1 ≤ −n − n2 and n + n2 ≤ x1 , n η1 (x1 ) = 1 for −n2 ≤ x1 ≤ n2 , and Y (x1 ) = eiax1 . The identity α1 = −νβ 2 /γ 2 guarantees that the characteristic equation νζ 2 − γζ − (α1 + iβ) = 0, corresponding to the equation (3.11) below, has the root ζ1 = ia where a = −β/γ. Thus, the function Y is a bounded non-trivial solution of the ordinary differential equation ν Y (x1 ) − γ Y (x1 ) − (α1 + iβ) Y (x1 ) = 0 in the interval (−∞, +∞). The function V n has the form α2 n n ibr V (r) := η2 (r) e ; b = − ν

(3.11)

(3.12)

where η2n is an infinitely differentiable function on [0, +∞) such that 0 ≤ η2n ≤ 1 and 0 for 0 ≤ r ≤ n and 3n + n2 ≤ r, n η2 (r) = 1 for 2n ≤ r ≤ 2n + n2 .

178

Farwig and Neustupa

IEOT

Both the functions η1 and η2 can be chosen so that their derivatives are of the order 1/n. The definition of V n guarantees that it satisfies ν

d2 n V (r) − α2 V n (r) = 0 dr2

(3.13)

for 2n < r < 2n + n2 . Finally, the constant κn is chosen so that vn 0,2 = 1. Thus, the support of vn is a subset of S n := x = [x1 , r, ϕ] ∈ R3 ; −n − n2 ≤ x1 ≤ n + n2 , n ≤ r ≤ 3n + n2 , 0 ≤ ϕ < 2π . (3.14) Considering the norm of vn , we can observe that for large n the decisive contribution comes from the integral of |vϕn |2 , namely of its part (−1/ik) κn U n r (dV n /dr) eikϕ 2 , on the region Dn := x = [x1 , r, ϕ] ∈ R3 ; −n2 < x1 < n2 , 2n < r < 2n + n2 , 0 < ϕ < 2π . (3.15) The integrals of all other parts on other regions are of a lower order in n. Cal 2 culating the integral of (−1/ik) κn U n r (dV n /dr) eikϕ on the domain Dn , we obtain 2 n2 2n+n2 2π κn n dV n (r) ik U (x1 ) r dr r dϕ dr dx1 0 −n2 2n 2 2n+n2 n 2 dV (r) κ2n n n 2 dr = 2π 2 |U (x1 )| dx1 r3 k dr −n2 2n b2 κ2 (2n + n2 )4 − (2n)4 . = 2π n2 2n2 k 4 Here we have used the equalities η1n (x1 ) = η2n (r) = 1, hence |U n (x1 )| = |V n (r)| = 1 for (x1 , r, ϕ) ∈ Dn . Thus, there exist n0 ∈ N and positive constants c2 and c3 (independent of n) such that c2 c3 ∀ n ∈ N, n ≥ n0 : ≤ κn ≤ 5 . (3.16) 5 n n Now looking at (A0γ − λI)vn , we can omit the projection Πσ in front of the Laplace operator in A0γ vn because ∆vn is divergence-free and has a compact support in Ω. Thus, (A0γ − λI)vn = ν∆vn − γ∂1 vn − λvn . Calculating the norm of this expression in L2σ (Ω), we observe that the contributions coming from Ω − Dn tend to zero as n → +∞ because they represent square roots of integrals of functions bounded by Cκ2n r2 on S n − Dn . Due to (3.16), this contribution is of the order n−1/2 . Concerning the integral on Dn , the decisive part again comes from (ν∆ − γ∂1 − λI)vϕn , namely from (ν∆ − γ∂1 − λI) applied to the term (−1/ik) κn U n r (dV n /dr) eikϕ because of the factor r inside this term. Note that

Vol. 62 (2008)


179

due to (3.11) and (3.13) κn n dV n (r) ikϕ ν∆ − γ∂1 − λI U (x1 ) r e ik dr κ ν ν dV n (r) ikϕ n 2 2 2 n U (x1 ) r e = ν∂1 + ν∂r + ∂r + 2 ∂ϕ − γ ∂1 − λI r r ik dr dV n (r) ikϕ κn n d2 ν d = U (x1 ) ν 2 + − α2 I r e ik dr r dr dr dV n (r) κn νY (x1 ) − γY (x1 ) − [α1 + iβ] Y (x1 ) r eikϕ + ik dr κn νk 2 n dV n (r) ikϕ e − U (x1 ) r 2 ik r dr κn n d d2 V n (r) κn n d2 V n (r) n U (x1 ) r ν U = − α V (r) + (x ) 2ν 2 1 ik dr dr2 ik dr2 κn n κn n ν d dV n (r) νk 2 dV n (r) U (x1 ) r − U (x1 ) + eikϕ ik r dr dr ik r dr νκn ib k 2 ib i(ax1 +br) ikϕ e −3b2 + − e ik r r where in the last step we used the simple forms of the functions U n and V n on Dn , i.e. U n (x1 ) = eiax1 and V n (r) = eibr . Hence 1 n2 2n+n2 2π 2 κ 2 dV n (r) ikϕ n n ν∆ − γ∂1 − λI U e (x ) r r dϕ dr dx 1 1 ik dr 0 −n2 2n 1 n2 2n+n2 2 ≤ C(ν, k, b) κn r dr dx1 =

−n2

2n

12 = C(ν, k, b) κn n (2n + n2 )2 − (2n)2 . The last term tends to zero as n → +∞ due to (3.16). In this way, we prove that (A0γ − λI)vn 0,2 → 0 as n → +∞. The sequence {vn } does not contain any convergent subsequence because the intersection of supports of any infinite family of functions, chosen from {vn }, is empty. Since λ was an arbitrarily chosen number from (Λ0γ )◦ , we have obtained the inclusion (Λ0γ )◦ ⊂ σ c (A0γ ). It means that nul (A0γ − λI) = +∞. Since the operators A0γ and (A0γ )∗ differ only in the sign in front of γ∂1 , we can prove in the same way that nul ((A0γ )∗ −λI) = +∞. It means that def (A0γ −λI) = +∞ and consequently, λ ∈ σess (A0γ ). The essential spectrum is a closed set, hence Λ0γ ⊂ σess (A0γ ). Theorem 3.1 provides an information on the shape of the whole spectrum σ(A0γ ), but it does not specify which numbers λ from σ(A0γ ) belong to σp (A0γ ),

180

Farwig and Neustupa

IEOT

σc (A0γ ) or to σr (A0γ ). We do answer this question in this paper neither for the operator A0γ nor for the more general operator Aω γ. The Oseen operator A0γ generates an analytic semigroup, see T. Miyakawa [22]. Therefore the operator (−A0γ ) is sectorial, see D. Henry [14], p. 20–21. The next theorem states the non-normality of the Oseen operator, which stresses the difference between the Stokes and Oseen operators. Theorem 3.2. The Oseen operator A0γ is not normal. Proof. Constructing a function z ∈ D((A0γ )∗ A0γ ) which is not in D(A0γ (A0γ )∗ ), we show that the domains D((A0γ )∗ A0γ ) and D(A0γ (A0γ )∗ ) do not coincide. Let R > 0 be so large that the body B is contained in the interior of the cube [−R, R]3 . Recall that Ω = R3 − B. Define the set Ωper := x = (x1 , x2 , x3 ) ∈ R3 ; ∃ k, l, m ∈ Z : x1 − kR, x2 − lR, x3 − mR ∈ [−R, R]3 − B . Note that Ωper is a domain in R3 which consists of infinitely many copies of the set [−R, R]3 − B, periodically repeated in directions parallel with the x1 -, x2 - and x3 -axis. We shall use the following function spaces: ◦ (C0∞ )3per is the space of infinitely differentiable vector functions φ in Ωper , R-periodic in the directions of all Cartesian axes and such that the distance between dist (supp φ and B) is positive. ◦ (L2 )3per is the completion of (C0∞ )3per in the norm identical with the L2 -norm on (−R, R)3 −B. The spaces (W01,2 )3per and (W 2,2 )per are defined analogously. ◦ (L2σ )per is the closure of the space of divergence-free functions from (C0∞ )3per in (L2 )3per . Let (A00 )per denote the Stokes operator in (L2σ )per with dense domain D((A00 )per ) = (W 2,2 )3per ∩ (W01,2 )3per ∩ (L2σ )per . Then (A00 )per has a compact resolvent and its spectrum, as well as the spectrum of (A00 )per −γ∂1 (with the same domain), consists of a countable number of isolated eigenvalues with finite multiplicities and negative real parts. Choose an eigenvalue ζ of (A00 )per − γ∂1 and denote by u an associated eigenfunction so that the equation (A00 )per u − γ∂1 u − ζu = 0

(3.17)

is satisfied in Ωper . Let us show, by contradiction, that the eigenfunction u can be chosen so that ∂1 u ≡ 0 on ∂Ωper . Assume the opposite, i.e. that all eigenfunctions v of the operator (A00 )per −γ∂1 , corresponding to the eigenvalue ζ, satisfy ∂1 v ≡ 0 on ∂Ωper . Then, for each of them, there are two possibilities: either ∂1 v ≡ 0 in Ωper which can be easily excluded or ∂1 v is also an eigenfunction of (A00 )per −γ∂1 corresponding to the same eigenvalue ζ. Since the eigenspace of (A00 )per − γ∂1 , generated by all such eigenfunctions, is finite-dimensional, we can choose an eigenfunction u so that γ ∂1 u = µ u

(3.18)

Vol. 62 (2008)


181

with an appropriate constant µ. Since u = 0 on ∂Ωper and equation (3.18) is satisfied in Ωper , the integration of (3.18) on line segments parallel with the x1 -axis and starting from the boundary of Ωper yields that u = 0 on all such line segments. Thus, u vanishes identically in an open subset of Ωper . Now the unique continuation principle, see e.g. R. Leis [21], applied to w = curl u, shows that w ≡ 0 in Ωper . Consequently, u ≡ 0 in Ωper which is impossible because u is an eigenfunction. Since the assumption that ∂1 u ≡ 0 on ∂Ωper leads to a contradiction, we have ∂1 u ≡ 0 on ∂Ωper. Note that ∂Ω ⊂ ∂Ωper and ∂Ωper consists of infinitely many copies of ∂Ω repeated periodically with the period R in the direction of each Cartesian coordinate. Now we multiply the function u by an infinitely-differentiable cut-off function ηR which equals one in the neighborhood of ∂Ω and whose support is contained in (−R, R)3 − B, and correct the product ηR u by an appropriate function UR which guarantees that div (ηR u − UR ) = 0. By these means we can obtain a function z in D((A0γ )∗ A0γ ) which coincides with the function u constructed above in the neighborhood of Ω and equals zero outside (−R, R)3 . The function z satisfies ∂1 z ≡ 0 on ∂Ω. Then z cannot belong to D(A0γ (A0γ )∗ ) because all functions from D((A0γ )∗ A0γ ) ∩ D(A0γ (A0γ )∗ ) satisfy on ∂Ω the conditions z = A00 z + γ∂1 z = A00 z − γ∂1 z = 0, which implies that ∂1 z = 0 on ∂Ω.

4. Axially symmetric domains – decomposition of L2σ (Ω) and of A0γ We shall assume that the domain Ω ⊂ R3 is axially symmetric with respect to the x1 -axis in this section. Let k be an integer. We introduce the following spaces and notation: ◦ L2 (Ω)3k = {v ∈ L2 (Ω)3 ; v = V(x1 , r) eikϕ } ◦ C0∞ (Ω)3k = C0∞ (Ω)3 ∩ L2 (Ω)3k ∞ ∞ ◦ C0,σ (Ω)k = C0∞ (Ω)3k ∩ C0,σ (Ω) ∞ (Ω)k in L2 (Ω)3k ◦ L2σ (Ω)k = the closure of C0,σ ◦ Pk – the orthogonal projection of L2 (Ω)3 onto L2 (Ω)3k ◦ (A0γ )k – the restriction of the operator A0γ to the space L2σ (Ω)k . Obviously, L2 (Ω)3k , k ∈ Z, is a closed subspace of L2 (Ω)3 , and L2σ (Ω)k is a closed subspace of L2σ (Ω). The domain of (A0γ )k equals D(A0γ ) ∩ L2σ (Ω)k . Each function v from L2 (Ω)3 can be uniquely written in the form of a convergent Fourier series – with respect to the variable ϕ – of terms from L2 (Ω)3k , k ∈ Z: +∞ ! v(x1 , r, ϕ) = Vk (x1 , r) eikϕ (4.1) k=−∞

where Vk (x1 , r) =

1 2π

0

2π

v(x1 , r, ϕ) e−ikϕ dϕ.

182

Farwig and Neustupa

IEOT

Thus, we have L2 (Ω)3 = . . .⊕L2 (Ω)3−2 ⊕L2 (Ω)3−1 ⊕L2 (Ω)30 ⊕L2 (Ω)31 ⊕L2 (Ω)32 ⊕. . . We have proved in [6] that Πσ L2 (Ω)3k = L2σ (Ω) ∩ L2 (Ω)3k = L2σ (Ω)k = Pk L2σ (Ω).

(4.2)

The next lemma generalizes some results from [6]. Lemma 4.1. Let k ∈ Z. Then (A0γ )k is a closed operator in L2σ (Ω)k with the dense domain D((A0γ )k ); moreover D((A0γ )k ) = Pk [D(A0γ )], R((A0γ )k ) ⊂ L2σ (Ω)k . Proof. The operator (A0γ )k is closed because it is the restriction of the closed operator A0γ onto a closed subspace of L2σ (Ω). The domain of (A0γ )k is the set of functions from L2σ (Ω)k , that belong to W 2,2 (Ω)3 ∩ W01,2 (Ω)3 . This set contains ∞ C0,σ (Ω), hence it is dense in L2σ (Ω)k . Let v ∈ D(A0γ ) ≡ W 2,2 (Ω)3 ∩ W01,2 (Ω)3 ∩ L2σ (Ω) and let (4.1) be its Fourier expansion in the variable ϕ. Then Vk (x1 , r) eikϕ ≡ Pk v ∈ W 2,2 (Ω)3 , and, due to the axial symmetry of Ω and the boundary condition satisfied by v on ∂Ω, 3 . Using the div v = 0 and the Vk (x1 , r) eikϕ also belongs to W01,2 (Ω) equation k ikϕ 2 3 orthogonality of the functions div V (x , r) e (Ω) (for different k), we in L 1 can prove that div Vk (x1 , r) eikϕ = 0. Hence Vk (x1 , r) eikϕ ∈ L2σ (Ω)k and consequently, Pk [D(A0γ )] ⊂ D((A0γ )k ). On the other hand, if v ∈ D((A0γ )k ), then it belongs to D(A0γ ), and since Pk v = v, it also belongs to L2 (Ω)3k . Hence v ∈ D(A0γ ) ∩ L2 (Ω)3k = D(A0γ ) ∩ L2σ (Ω)k = Pk [D(A0γ )]. If v ∈ D((A0γ )k ), then ∆v, ∂1 v ∈ L2 (Ω)3k , and due to (4.2), A0γ v = νΠσ ∆v − γ∂1 v ∈ L2σ (Ω)k . Hence A0γ is reduced onto L2σ (Ω)k . Lemma 4.2. Let k ∈ Z. Then σ((A0γ )k ) = σess ((A0γ )k ) = Λ0γ where Λ0γ is the parabolic region in C defined by (3.3): Λ0γ = λ = α + iβ ∈ C; α, β ∈ R, α ≤ −νβ 2 /γ 2 . Proof. The operator (A0γ )k is a part of A0γ , hence σ((A0γ )k ) ⊂ σ(A0γ ) = Λ0γ . On the other hand, for λ ∈ (Λ0γ )◦ , we have shown the existence of a noncompact sequence vn ∈ L2σ (Ω) such that vn 0,2 = 1 and (A0γ − λI)vn → 0 in L2σ (Ω) as n → +∞ in the proof of Theorem 3.1, part II. The construction of vn involved the choice of an arbitrary non-zero integer k. An easy examination shows that the functions vn actually belong not only to L2σ (Ω), but to L2σ (Ω)k . Thus, we obtain that (Λ0γ )◦ ⊂ σ c ((A0γ )k ) for k = 0. Using the same arguments as at the end of the proof of Theorem 3.1, we deduce that Λ0γ ⊂ σess ((A0γ )k ) for k = 0. It completes the proof in the case when k = 0. The case k = 0 must be treated separately. Suppose that λ = α + iβ ∈ (Λ0γ )◦ . Let us construct a non-compact sequence {vn } in the unit sphere in L2σ (Ω)0 such that (A0γ − λI)vn → 0 as n → +∞. The requirement that vn ∈ L2σ (Ω)0 means that vn ≡ [v1n , vrn , vϕn ] does not depend on ϕ. Then the condition div vn = 0 says

Vol. 62 (2008)


183

that ∂1 (rv1n ) + ∂r (rvrn ) = 0. This equation is automatically satisfied if vn has the cylindrical components 1 1 vn (x1 , r) = ∂r ψ n (x1 , r), − ∂1 ψ n (x1 , r), 0 . (4.3) r r Put ψ n (x1 , r) = δn U n (x1 ) V n (r) where U n and V n are the same functions as in the proof of Theorem 3.1, i.e. the functions given by (3.10) and (3.12), and where the factor δn must be chosen so that vn 0,2 = 1. Calculating the norm of vn 0,2 , we can observe that for large n the decisive contribution comes from the integral on Dn , see (3.15). The contribution coming from Ω − Dn is of a lower order in powers of n. The cut-off functions η1n and η2n are both equal to1 on Dn . Hence U n (x1 ) = eiax1 , where a = −β/γ, and V n (r) = eibr , where b = −α2 /ν, see the proof of Theorem 3.1. Thus, ψ(x1 , r) = δn ei(ax1 +br) on Dn and n2 2n+n2 2π n2 |v1 | + |vrn |2 dϕ r dr dx1 |vn |2 dx = Dn

=

n2

=

−n2

2n+n2

2π −n2

=

2n n2

0

2n

1 |∂r ψ n |2 + |∂1 ψ n |2 dr dx1 r

2n+n2

1 2 (a + b2 ) dr dx1 r −n2 2n 2π δn2 (a2 + b2 ) 2n2 ln(2n + n2 ) − ln(2n) . 2π δn2

The condition that this tends to 1 as n → +∞ leads to the existence of constants c4 , c5 > 0 and n0 ∈ N such that c c5 √4 ≤ δn ≤ √ . (4.4) ∀ n ∈ N, n ≥ n0 : n ln n n ln n Suppose that n is so large that the support of vn is a subset of Ω. Then, since (A0γ − λI)vn = ν∆vn − γ∂1 vn − λvn , we get ν∆vn − γ∂1 vn − λvn 2 dx (A0γ − λI)vn 20,2 = δn2 S n −Dn ν∆vn − γ∂1 vn − λvn 2 dx + δn2 (4.5) Dn

n

where S is defined by (3.14). The integrands are less than or equal to C/r2 . Thus, the first term in (4.5) can be estimated from above by 2n 3n+n2 n+n2 r dr Cδn2 dx1 + r2 n −n−n2 2n+n2 −n2 n+n2 3n+n2 r dr 2 + Cδn + dx1 r2 −n−n2 n2 n 2n 2 3n + n2 3n + n 2 + Cδ 2n ln + ln . ≤ Cδn2 (2n + 2n2 ) ln n n 2n + n2 n

184

Farwig and Neustupa

IEOT

Using (4.4), we verify that the right-hand side tends to zero as n → +∞. In the second term in (4.5), we use the identity ψ n (x1 , r) = δn ei(ax1 +br) which holds on Dn . Note that the function Y (x1 ) = eiax1 satisfies the differential equation (3.11) and that the function eibr satisfies the differential equation (3.13). Then calculating the expression ν∆vn − γ∂1 vn − λvn we find that |ν∆vn − γ∂1 vn − λvn | ≤ Cδn /r2 . Consequently the second term in (4.5) can be estimated by n2 2n+n2 2π 2n+n2 1 dr 2 2 dϕ r dr dx = Cδ n ≤ Cδn2 . Cδn2 1 n 4 3 r r 2 0 −n 2n 2n Due to (4.4), the right-hand side tends to zero as n → +∞. Hence we have shown that (A0γ − λI)vn 0,2 → 0 as n → +∞. The sequence {vn } is non-compact because the intersection of the supports of any infinite family of functions chosen from {vn } is empty. Since all the functions vn belong to L2σ (Ω)0 , we have proved that λ ∈ σ c ((A0γ )0 ). Applying once again the same arguments as at the end of the proof of Theorem 3.1, we observe that λ ∈ σess ((A0γ )0 ) and due to the closedness of σess ((A0γ )0 ), we obtain the inclusion Λ0γ ⊂ σess ((A0γ )0 ). This completes the proof in the case k = 0. Since −(A0γ )k is, by definition, the reduction of the sectorial operator −A0γ on the space L2σ (Ω)k , −(A0γ )k is a sectorial operator in L2σ (Ω)k .

5. Axially symmetric domains Ω – the operator Aωγ and its decomposition ω 2 Let k ∈ Z. We shall denote by (Aω γ )k the restriction of Aγ to Lσ (Ω)k . The domain ω 0 of (Aγ )k is the same as the domain of (Aγ )k , i.e., 0 2,2 (Ω)3 ∩ W01,2 (Ω)3 ∩ L2σ (Ω)k . D((Aω γ )k ) = D((Aγ )k ) ≡ W

If u ∈ L2σ (Ω)k , then it has the form u(x1 , r, ϕ) = U(x1 , r) eikϕ and ∂ϕ u = i k U eikϕ = i k u. Therefore, (Aω γ )k can be rewritten as 0 0 (Aω γ )k u = (Aγ )k u + ω ∂ϕ u = (Aγ )k u + ikω u.

(5.1)

2 Thus, (Aω γ )k is a closed and densely defined operator in Lσ (Ω)k . The representation ω (5.1) of the operator (Aγ )k and Lemma 4.2 imply that ω 0 σ((Aω γ )k ) = σess ((Aγ )k ) = {λ ∈ C; λ − ikω ∈ Λγ }.

(5.2)

The next lemma provides the information on the spectrum of the full operator Aω γ . It confirms that statement (iv) of Theorem 1.1 is true. ω ω ω Lemma 5.1. σ(Aω γ ) = σess (Aγ ) = Λγ where set Λγ is defined by (1.11): ω Λγ = {λ = α + iβ + ikω ∈ C; α, β ∈ R, k ∈ Z, α ≤ −νβ 2 /γ 2 }.

Vol. 62 (2008)


185

ω ω Proof. Each operator (Aω γ )k , k ∈ Z, is a part of the operator Aγ , hence σess ((Aγ )k ) ω ω ω ω ⊂ σess (Aω γ ). Thus, ∪k∈Z σess ((Aγ )k ) = Λγ ⊂ σess (Aγ ) ⊂ σ(Aγ ). ω It remains to prove the opposite inclusion, i.e. that σ(Aω γ ) ⊂ Λγ or equivaω ω ω lently that (C − Λγ ) ⊂ ρ(Aγ ). Suppose that λ ≡ α + iβ ∈ C − Λγ . We will show 2 2 that the operator Aω γ − λI has a bounded inverse in Lσ (Ω). Let f ∈ Lσ (Ω) with Fourier expansion

f (x1 , r, ϕ) =

+∞ !

fk (x1 , r) eikϕ ,

k=−∞

where fk eikϕ ∈ L2σ (Ω)k , be given. Let us at first solve the equation ((Aω γ )k − λI)wk = fk eikϕ in L2σ (Ω)k . Putting wk = uk eikϕ and using (5.1), we observe that this equation is equivalent with (A0γ )k (uk eikϕ ) − (α + iβ − ikω)(uk eikϕ ) = fk eikϕ .

(5.3)

Due to Lemma 4.2 we have α+iβ −ikω ∈ ρ((A0γ )k ) for all k ∈ Z. Since the operator −(A0γ )k is sectorial, we deduce from resolvent estimates for sectorial operators, see e.g. D. Henry [14], p. 23, that there exists a constant M > 0, independent of k, such that M M fk eikϕ 0,2 = fk 0,2 . (5.4) uk 0,2 = uk eikϕ 0,2 ≤ 1 + |k| 1 + |k| "+∞ "+∞ ikϕ ikϕ Then the series converges in L2σ (Ω) and u = k=−∞ uk e k=−∞ uk e satisfies the estimate +∞ +∞ ! ! M2 2 2 u0,2 = uk 0,2 ≤ fk 20,2 ≤ M 2 f 20,2 . (5.5) (1 + |k|)2 k=−∞

k=−∞

¿From the equation (5.3) and the estimate (5.4), we have # ω # #(Aγ )k (uk eikϕ )# ≤ fk eikϕ 0,2 + |α + iβ| uk eikϕ 0,2 ≤ C fk eikϕ 0,2 0,2 where C is independent of k. Using these inequalities and the closedness of the ω ω operator Aω γ , we deduce that u ∈ D(Aγ ) and (Aγ − λI)u = f . This information, together with (5.5), completes the proof.

6. General exterior domains – the operators Aωγ and Lωγ Using the same procedure as in the proof of Theorem 3.2, we can show that the operator Aω γ is not normal, i.e., that the statement (ii) of Theorem 1.1 is true. If λ = α + iβ is an eigenvalue of Aω γ and v is a corresponding eigenfunction, then, multiplying the equation Aω γ v = λ v by v and integrating on Ω, we obtain the identity −ν ∇v20,2 = α v20,2 ; compare with that part of the proof of Theorem 3.1 which lead to (3.8). This verifies Theorem 1.1 (iii). Let R0 = max {|x|; x ∈ B} and ΩR = Ω ∩ BR (0).

186

Farwig and Neustupa

IEOT

Lemma 6.1. Let λ ∈ σ c (Aω γ ). Then there exists R > R0 and a non-compact sequence {un } in D(Aω ) such that un 0,2 = 1, un = 0 in ΩR and γ n (Aω γ − λI) u −→ 0

in L2σ (Ω)

for n → +∞.

(6.1)

ω Proof. The condition λ ∈ σ c (Aω γ ) means that nul (Aγ − λI) = +∞. Then there n 2 exists an orthonormal sequence {v } in Lσ (Ω) such that n n 2 (Aω γ − λI) v = −→ 0 in Lσ (Ω)

for n → +∞;

(6.2)

the construction of the sequence {vn } is based on Lemma IV.2.3 in [18] and is explained in [6]. Obviously {vn } converges to the zero function weakly in L2σ (Ω). Using (6.2) and the estimate (2.1), we get that the sequence {vn } is bounded in W01,2 (Ω)3 ∩ W 2,2 (Ω)3 . Then there exists a subsequence, again denoted by {vn }, which is weakly convergent to 0 in W01,2 (Ω)3 ∩ W 2,2 (Ω)3 . Moreover, {ω(e1 × x) · ∇vn } converges weakly to 0 in L2σ (Ω). Suppose that R ≥ R0 + 3 is a fixed number. The compact imbedding W 2,2 (ΩR )3 → → W 1,2 (ΩR )3 yields vn −→ 0

strongly in W 1,2 (ΩR )3 .

(6.3)

The first part of (6.2) can be written in the form ν∆vn + ω(e1 × x) · ∇vn − ωe1 × vn − γ∂1 vn − λvn + ∇q n = n

(6.4)

where q n is an appropriate scalar function. It follows from (6.4) and (2.1) that ∇q n → 0 weakly in L2 (Ω)3 . Thus, the functions q n , which are given uniquely up to an additive constant by (6.4), can be chosen so that q n → q ≡ const. strongly in L2 (ΩR ). We may even assume that q = 0. Denote by η an infinitely differentiable cut-off function in Ω such that η(x) =

0 1

if |x| < R − 2, if |x| > R − 1,

and 0 ≤ η(x) ≤ 1 if R−2 ≤ |x| ≤ R−1. Put un = ηvn −Vn where div Vn = ∇η·vn . Although Vn is not given uniquely, the results on solutions of the equation div V = f , see e.g. [2], show that the function Vn can be chosen such that supp Vn ⊂ {x ∈ Ω; R − 3 < |x| < R} and there exist c6 > 0 such that Vn 2,2 = Vn 2,2; ΩR ≤ c6 ∇η · vn 1,2; ΩR −→ 0

(6.5)

as n → +∞. (Here . 2,2; ΩR and . 1,2; ΩR denote the norm in W 2,2 (ΩR )3 and in W 1,2 (ΩR )3 , respectively.) The function un is divergence-free, equals 0 in ΩR−3 , equals vn in Ω − ΩR and belongs to L2 (Ω)3 . Due to the properties of the functions n η and Vn we get un ∈ D(Aω γ ). Obviously u satisfies

Vol. 62 (2008)


187

ν∆un + ω(e1 × x) · ∇un − ωe1 × un − γ∂1 un − λun + ∇(ηq n ) = η ν∆vn + ω(e1 × x) · ∇vn − ωe1 × vn − γ∂1 uv − λvn + 2ν∇η · ∇vn + ν(∆η)vn − ν∆Vn + ω(e1 × x) · ∇η vn − γ (∂1 η) vn − ω(e1 × x) · ∇Vn + ωe1 × Vn + γ∂1 Vn + λVn + ∇(ηq n ) =

ηn + 2ν∇η · ∇vn + ν(∆η)vn − ν∆Vn + ω(e1 × x) · ∇η vn − γ (∂1 η) vn

− ω(e1 × x) · ∇Vn + ωe1 × Vn + γ∂1 Vn + λVn + (∇η)q n n

2

3

n

(6.6) n

where η → 0 in L (Ω) due to (6.2), and ν[2∇η · ∇v + (∆η)v ] → 0 in 2 3 L in ΩR and due to (6.3). The terms (Ω) because ∇ηn and ∆η are supported ω(e1 × x) · ∇η v and γ (∂1 η) vn also tend to zero in in L2 (Ω)3 for the same reasons. Furthermore, all terms involving Vn tend to 0 in L2 (Ω)3 due to (6.5). Finally, (∇η)q n → 0 in L2 (Ω)3 because q n → 0 in L2 (ΩR ) and ∇η is supported in ΩR . Thus, ν∆un + ω(e1 × x) · ∇un − ωe1 × un − γ∂1 un − λun + ∇(ηq n ) −→ 0 in L2 (Ω)3 for n → +∞, and therefore {un } satisfies (6.1). Moreover, we have un 20,2 ≥ |un (x)|2 dx = |vn (x)|2 dx −→ 1 for n → +∞ |x|>R

|x|>R

n

because v 0,2 = 1 and due to (6.3). If we divide each of the functions un by its norm in L2σ (Ω) and denote the new function again by un , we obtain the sequence {un } with all the properties stated in Lemma 6.1. Finally, the orthonormality of {vn } and (6.3) imply the non-compactness of the sequence {un }. $ω the operator which is defined in the same way as Aω , We denote by A γ γ however on the whole space R3 rather than on the exterior domain Ω ⊂ R3 . $ω has all the properties derived in Sections 4 and 5. Obviously, the operator A γ $ω ) = Λω . Lemma 6.2. σ c (Aω c (A γ) = σ γ γ Proof. Suppose that λ ∈ σ c (Aω ). Let R > 0 and {un } be a number and a sequence, respectively, with the properties named in Lemma 6.1. All functions un , extended $ω by zero from Ω to the whole space R3 , belong to the domain of operator A γ . Thus, ω $γ ). (6.1) shows that λ ∈ σ c (A $ω ) then we can use analogous arguments and On the other hand, if λ ∈ σ c (A γ ω prove that λ also belongs to σ c (Aγ ). ω Let us show that σ c (Aω γ ) ⊂ σess (Aγ ); the opposite inclusion is trivial. For λ ∈ ω ∗ we have that nul (Aγ − λI) = +∞. Moreover, λ ∈ σ c ((Aω c (A−ω −γ ), γ) ) = σ ω ∗ ω so that nul ((Aγ ) − λI) = +∞. Hence def (Aγ − λI) = +∞ which shows that λ ∈ σess (Aω γ ), see [18], p. 234. We have thus proved Theorem 1.1 (i). Theorem IV.5.35 in [18] and Lemma 2.4 imply that the essential spectrum of the operator ω Lω γ is the same as σess (Aγ ); therefore it is also given by (1.11). Moreover, since

σ c (Aω γ)

188

Farwig and Neustupa

IEOT

ω ω ind(Lω γ − λI) = 0 in C − σess (Lγ ) and due to Theorem IV.5.31 in [18], C − σess (Lγ ) ω can contain at most countably many eigenvalues λ of Lω , which can cluster only ω ω on the boundary of C − σess (Lω γ ) and 0 < nul(Lγ − λI) = def(Lγ − λI) < +∞ at each of them. This implies Theorem 1.1 (vi).

Acknowledgment The research was supported by the Grant Agency of the Czech Academy of Sciences (grant No. IAA100190612) and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503. It was performed as a part of the cooperation between TU Darmstadt and CTU Prague.

References [1] K. I. Babenko: Spectrum of the linearized problem of flow of a viscous incompressible liquid round a body. Sov. Phys. Dokl. 27, 1 (1982), 25–27. [2] W. Borchers and H. Sohr: On the equations rot v = g and div v = f with zero boundary conditions. Hokkaido Math. J. 19 (1990), 67–87. [3] R. Farwig: An Lp -analysis of viscous fluid flow past a rotating obstacle. Tohoku Math. J. 58 (2005), 129–147. [4] R. Farwig: Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Banach Center Publications 70 (2005), 73–84. [5] R. Farwig, T. Hishida and D. M¨ uller: Lq -theory of a singular “winding” integral operator arising from fluid dynamics. Pacific J. Math. 215 (2004), 297–312. [6] R. Farwig, J. Neustupa: On the spectrum of a Stokes-type operator arising from flow around a rotating body. Manuscripta Mathematica 122 (2007), 419–437. ˇ Neˇcasov´ [7] R. Farwig, S. a and J. Neustupa: On the essential spectrum of a Stokestype operator arising from flow around a rotating body in the Lq -framework. Proc. “Int. Conf. Navier–Stokes Equations and Related Problems”, Kyoto 2006. Publ. RIMS Kyoto Univ., Kokyuroku Bessatsu, 2006, 76–88. [8] G. P. Galdi: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Vol. I, Linear Steady Problems. Springer Tracts in Natural Philosophy 38, 1998. [9] G. P. Galdi: On the motion of a rigid body in a viscous liquid. A mathematical analysis with applications. In: S. Friedlander, D. Serre (eds), Handbook of Mathematical Fluid Mechanics, Elsevier Science, 2002, 653–791. [10] G. P. Galdi: Steady flow of a Navier–Stokes fluid around a rotating obstacle. J. Elasticity 71 (2003), 1–32. [11] M. Geissert, H. Heck and M. Hieber: Lp -theory of the Navier–Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596 (2006), 45–62. [12] Y. Giga and H. Sohr: On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, Sec. IA 36 (1989), 103–130. [13] I. M. Glazman: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators. Moscow 1963 (Russian). English version: Israel Progr. for Sci. Transl., Jerusalem, 1965.

Vol. 62 (2008)


189

[14] D. Henry: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840, Springer-Verlag, Berlin–Heidelberg–New York, 1981. [15] T. Hishida: An existence theorem for the Navier–Stokes flow in the exterior of a rotating obstacle. Arch. Rational Mech. Anal. 150 (1999), 307–348. [16] T. Hishida: The Stokes operator with rotating effect in exterior domains. Analysis 19 (1999), 51–67. [17] T. Hishida: Lq estimates of weak solutions to the stationary Stokes equations around a rotating body. J. Math. Soc. Japan 58 (2006), 743–767. [18] T. Kato: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin–Heidelberg–New York, 1966. ˇ Neˇcasov´ [19] S. Kraˇcmar, S. a and P. Penel: Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations. IASME Transactions 6, 2 (2005), 854–861. [20] O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969. [21] R. Leis: Initial Boundary Value Problems in Mathematical Physics. B. G. Teubner, Chichester–New York, 1986. [22] T. Miyakawa: On nonstationary solutions of the Navier–Stokes equations in an exterior domain. Hiroshima Math. J. 12 (1982), 115–140. ˇ Neˇcasov´ [23] S. a: Asymptotic properties of the steady fall of a body in viscous fluids. Math. Meth. Appl. Sci. 27 (2004), 1969–1995. ¨ [24] F. K. G. Odqvist: Uber die Randwertaufgaben der Hydrodynamik Z¨ aher Fl¨ ussigkeiten. Math. Z. 32 (1930). Reinhard Farwig Darmstadt University of Technology Department of Mathematics Schlossgartenstr. 7 D–64289 Darmstadt Germany e-mail: [email protected] Jiˇr´ı Neustupa Czech Academy of Sciences Mathematical Institute ˇ a 25 Zitn´ 115 67 Praha 1 Czech Republic e-mail: [email protected] Submitted: December 17, 2006. Revised: June 24, 2008.



Existence of Solutions for Second Order Partial Neutral Functional Differential Equations Eduardo Hernández Morales, Hernán R. Henr´ıquez and Mark A. McKibben Abstract. We establish existence of mild solutions for a class of abstract second-order partial neutral functional differential equations with unbounded delay in a Banach space. Mathematics Subject Classification (2000). Primary 35R10; Secondary 34K30, 34K40, 47D09. Keywords. Neutral partial differential equations, cosine function of operators.

1. Introduction In this paper we study existence of mild solutions for a class of abstract secondorder partial neutral functional differential equations with infinite delay of the form d2 (x(t) − g(t, xt )) = Ax(t) + f (t, xt , x (t)), t ∈ I = [0, a], (1.1) dt2 x0 = ϕ ∈ B, (1.2) x (0) =

ξ ∈ X.

(1.3)

Here, A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C(t))t∈R defined on a Banach space X, the history xt : (−∞, 0] → X, θ → x(t + θ), belongs to some abstract phase space B defined axiomatically and f, g are appropriate functions. Motivated by the fact that neutral functional differential equations (abbreviated, NFDE) arise in many areas of applied mathematics, this type of equation has received much attention in recent years. The literature concerning first and second-order ordinary neutral functional differential equations is very extensive. The work of the second author was supported by FONDECYT-CONICYT, Grants 1050314 and 7050034.

192

Hern´ andez Morales, Henr´ıquez and McKibben

IEOT

We refer the reader to the books Hale & Lunel [12], Lakshmikantham, Wen & Zhang [21], Kolmanovskii & Myshkis [20], and the references therein. First-order abstract partial neutral functional differential equations have also been studied by different authors, including Adimy et al. [1, 2, 3], Hale [11], Wu and Xia [26] and Wu [27] for finite delay, and Hern´ andez & Henr´ıquez [14, 15] and Hern´ andez [16] for the unbounded delay case. In addition, some abstract second-order neutral Cauchy problems have been considered recently in the literature, see for instance [5, 6, 7, 22]. However, in these works the authors assume that the operator C(t) is compact for t > 0. In such case, it follows from Travis and Webb [25, p. 557] that the underlying space must be finite dimensional, so that the equations studied in these works are really ordinary NFDE. A neutral differential equation of abstract type similar to those considered in [5, 6, 7, 22] is studied in Hernández & McKibben [17]. To the best of our knowledge, the existence and qualitative properties of solutions for abstract partial functional differential equations described by (1.1)-(1.3) are untreated topics in the literature and this is the main motivation of the present paper. Next, we review some basic concepts, notations and properties needed to establish our results. Throughout this paper, A is the infinitesimal generator of a strongly continuous cosine family (C(t))t∈R of bounded linear operators on the Banach space X. We denote by (S(t))t∈R the sine function associated with (C(t))t∈R , t which is defined by S(t)x = 0 C(s)xds, for x ∈ X, and t ∈ R. In addition, N and ˜ are positive constants such that C(t) ≤ N and S(t) ≤ N ˜ , for every t ∈ I. N In this paper, [D(A)] represents the domain of A endowed with the graph norm given by x A = x + Ax , x ∈ D(A), while E stands for the space formed by the vectors x ∈ X for which C(·)x is of class C 1 on R. We know from [19], that E endowed with the norm xE = x + sup AS(t)x, x ∈ E, 0≤t≤a

is a Banach space. The operator valued function H(t) =

C(t) AS(t)

S(t) C(t)

is a

strongly continuous groupof bounded linear operators on the space E × X gener0 I defined on D(A) × E. It follows from this that ated by the operator A = A

0

AS(t) : E → X is a bounded linear operator and that AS(t)x → 0, t → 0, for each x ∈ E. Furthermore, if x : [0, ∞) → X is locally integrable, then t y(t) = 0 S(t − s)x(s)ds defines an E-valued continuous function. This assertion is a consequence of the fact that t T t t 0 H(t − s) ds = S(t − s)x(s) ds, C(t − s)x(s) ds x(s) 0 0 0 defines an E × X-valued continuous function. In addition, it follows from the definition of the norm in E that a function u : I → E is continuous if, and

Vol. 62 (2008)

Existence of Solutions for Second Order NFDE

193

only if, it is continuous with respect to the norm in X and the set of functions {AS(t)u(·) : t ∈ [0, 1]} is an equicontinuous subset of C(I, X). The existence of solutions for the second-order abstract Cauchy problem x (t) = x(0) =

Ax(t) + h(t), t ∈ I, w, x (0) = z,

(1.4) (1.5)

where h : I → X is an integrable function, is studied in [25]. Similarly, the existence of solutions of semi-linear second-order abstract Cauchy problems has been treated in [24]. We only mention here that the function x(·) given by t S(t − s)h(s)ds, t ∈ I, (1.6) x(t) = C(t)w + S(t)z + 0

is called a mild solution of (1.4)-(1.5), and that when w ∈ E the function x(·) is of class C 1 on I and t C(t − s)h(s) ds, t ∈ I. x (t) = AS(t)w + C(t)z + 0

For additional details on the cosine function theory, we refer the reader to [9, 24, 25]. In this work, B will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm · B and satisfying the following axioms: (A) If x : (−∞, σ + b) → X, b > 0, is continuous on [σ, σ + b) and xσ ∈ B, then for every t ∈ [σ, σ + b), the following conditions hold: (i) xt is in B, (ii) x(t) ≤ H xt B , (iii) xt B ≤ K(t − σ) sup{ x(s) : σ ≤ s ≤ t} + M (t − σ) xσ B , where H > 0 is a constant; K, M : [0, ∞) → [0, ∞), K(·) is continuous, M (·) is locally bounded, and H, K, M are independent of x(·). (A1) For the function x(·) in (A), the function t → xt is continuous from [σ, σ + b) into B. (B) The space B is complete. Example. The phase space Cr × Lp (ρ, X) Let r ≥ 0, 1 ≤ p < ∞ and let ρ : (−∞, −r] → R be a nonnegative measurable function which satisfies the conditions (g-5), (g-6) in the terminology of [18]. In other words, this means that ρ is locally integrable and there exists a non-negative, locally bounded function γ on (−∞, 0] such that ρ(ξ + θ) ≤ γ(ξ)ρ(θ), for all ξ ≤ 0 and θ ∈ (−∞, −r)\Nξ , where Nξ ⊆ (−∞, −r) is a set with Lebesgue measure zero. The space Cr × Lp (ρ, X) consists of all classes of functions ϕ : (−∞, 0] → X such that ϕ is continuous on [−r, 0], Lebesgue-measurable, and ρ ϕ p is Lebesgue integrable on (−∞, −r). The seminorm in Cr × Lp (ρ, X) is defined by −r 1/p ϕB = sup{ϕ(θ) : −r ≤ θ ≤ 0} + ρ(θ)ϕ(θ)p dθ . −∞

194


IEOT

The space B = Cr ×Lp (ρ; X) satisfies axioms (A), (A1), (B). Moreover, when r = 0 1/2 0 , and p = 2, we can take H = 1, M (t) = γ(−t)1/2 and K(t) = 1 + −t ρ(θ) dθ for t ≥ 0. (See [18, Theorem 1.3.8] for details).

The terminology and notations used in this work are standard. In particular, if (Z, · Z ) and (Y, · Y ) are Banach spaces, we indicate by L(Z, Y ) the Banach space of bounded linear operators from Z into Y endowed with the uniform operator topology and we abbreviate this notation to L(Z) whenever Z = Y . Next, C(I, Z) is the space of continuous functions from I into Z endowed with the norm of uniform convergence and Br (x, Z) denotes the closed ball with center at x and radius r in Z. Additionally, for a bounded function ξ : I → Z and 0 ≤ t ≤ a, we will use the notation ξZ,t to mean ξZ,t = sup {ξ(s)Z : s ∈ [0, t]} ,

(1.7)

and we simplify this notation to ξt when no confusion about the space Z arises. Our results are based on fixed point theorems. For completeness, we include them here. Lemma 1.1 (Leray-Schauder Alternative). [10, Theorem 6.5.4]. Let D be a closed convex subset of a Banach space Z and assume that 0 ∈ D. Let G : D → D be a completely continuous map. Then, either the map G has a fixed point in D or the set {z ∈ D : z = λG(z), 0 < λ < 1} is unbounded. Lemma 1.2. [8]. Let Z be a Banach space, Γ1 , Γ2 : Z → Z and Γ = Γ1 +Γ2 . Assume that Γ1 is a contraction and that Γ2 is completely continuous. Then, either the map Γ has a fixed point or the set {z ∈ Z : z = λΓ1 ( λz ) + λΓ2 (z)} is unbounded for 0 < λ < 1. This paper has three sections. In the next Section we discuss the existence of mild solutions for abstract second order neutral systems and, in the Section 3 we consider some applications.

2. Existence Results We initially study the existence of mild solutions for the neutral system d2 [x(t) − g(t, xt )] = dt2 x0 = lim

t→0+

Ax(t) + f (t, xt ), t ∈ I = [0, a],

(2.1)

ϕ ∈ B,

(2.2)

x(t) − g(t, xt ) − C(t)ϕ(0) + g(0, ϕ) t

=

z ∈ X.

In what follows the notation F (b), 0 < b ≤ a, stands for the space F (b) = {u : (−∞, b] → X : u|I ∈ C([0, b], X), u0 = 0},

(2.3)

Vol. 62 (2008)


195

endowed with the sup norm. In addition, we denote by y : (−∞, a] → X the function defined by y0 = ϕ and y(t) = C(t)ϕ(0) + S(t)z, for t ∈ [0, a]. We now introduce our first general assumption. (H1) There exists a Banach space (Y, · Y ) continuously included in X such that AS(t) ∈ L(Y, X), for all t ∈ I, and the function AS(·) : I → L(Y, X) is 1 be positive constants such that x ≤ strongly continuous. We let NY , N 1 , for all t ∈ I. NY xY , for all x ∈ Y, and AS(t)L(Y,X) ≤ N Remark 2.1. It is clear that if assumption (H1) holds, then Y is continuously included in E. To prove this affirmation, we remark that A is closed and C(t)x − t C(s)x = A s S(τ )xdτ for all s ≤ t and every x ∈ X, see [24] for details. As consequence, for x ∈ Y we have that t t C(t)x − x = A S(τ )x dτ = AS(s)x ds, (2.4) 0

0

which implies that the function C(·)x is continuously differentiable and, therefore Y ⊆ E. Moreover, since 1 )xY xE = x + sup AS(t)x ≤ (NY + N 0≤t≤a

when a ≥ 1, we obtain that the inclusion ι : Y → E is continuous in this case. A similar argument using the properties of the sine function shows that ι : Y → E is also continuous for 0 < a < 1. We also observe that the spaces [D(A)] and E satisfy (H1). In some results we will need to ensure that a subset of X is relatively compact. The following remark will be useful. Remark 2.2. Assume that the cosine function C(t) satisfies the condition R(C(t)− I) is closed and dim ker(C(t) − I) < ∞, for some 0 < t ≤ a. If B ⊆ Y is a bounded set with respect to the norm in X such that the set {AS(t)x : 0 ≤ t ≤ a, x ∈ B} is relatively compact in X, then B is relatively compact in X. In fact, by applying (2.4) with x ∈ B, it follows from the mean value theorem for the Bochner integral (see [23, Lemma 2.1.3]) that C(t)x − x ∈ t c ({AS(s)x : 0 ≤ s ≤ t, x ∈ B}), where c(·) denotes the convex hull of a set. Since the set on right-hand side is compact, the assertion is an immediate consequence of the properties of the operator C(t) − I. For this reason we introduce the following condition for the cosine function. (H2) The cosine function C(t) has range R(C(t)−I) closed and dim ker(C(t)−I) < ∞, for some 0 < t ≤ a. Motivated by (1.6), we consider the following concept of mild solutions for (2.1)–(2.3). In this definition, we assume that f, g satisfy appropriate conditions to guarantee that for each function x : (−∞, a] → X that is continuous on [0, a]

196


IEOT

and x0 ∈ B, the functions s → S(t − s)f (s, xs ) and s → AS(t − s)g(s, xs ) are well defined and integrable from [0, t] into X, for all 0 < t ≤ a. Definition 2.3. A function x : (−∞, a] → X is a mild solution of Problem (2.1)– (2.3) if x ∈ C(I, X), satisfies conditions (2.2) and (2.3), and t AS(t − s)g(s, xs ) ds x(t) = C(t)(ϕ(0) − g(0, ϕ)) + S(t)z + g(t, xt ) + 0 t S(t − s)f (s, xs ) ds, t ∈ I. (2.5) + 0

To study this system, we introduce the following assumptions on the functions f and g. (H3) The function g : I × B → X is continuous, g(I × B) ⊂ Y and satisfies these conditions: (i) The function g(t, ·) : B → Y is continuous a.e. t ∈ I and the function g(·, ψ) : I → Y is strongly measurable, for every ψ ∈ B. (ii) There exist an integrable function mg : I → [0, ∞) and a continuous nondecreasing function Wg : [0, ∞) → (0, ∞) such that g(t, ψ)Y ≤ mg (t)Wg (ψB ) ,

(t, ψ) ∈ I × B.

(H4) The function f : I × B → X satisfies these conditions: (i) The function f (t, ·) : B → X is continuous a.e. t ∈ I and f (·, ψ) : I → X is strongly measurable, for every ψ ∈ B. (ii) There exist an integrable function mf : I → [0, ∞) and a continuous nondecreasing function Wf : [0, ∞) → (0, ∞) such that f (t, ψ) ≤ mf (t)Wf ( ψ B ), (t, ψ) ∈ I × B. In what follows we denote W = max{Wg , Wf }. The following two remarks complete the analysis of condition (H3). Remark 2.4. We note that the assumptions (H1) and (H3) are linked to the integrability of the function s → AS(s)g(s, us ). We observe that, except for trivial cases, the operator function s → AS(s) is not integrable over [0, t]. In fact, if we assume that AS(·) ∈ L1 ([0, t]), then from (2.4) it follows that the cosine function is uniformly continuous on [0, t] which, from the results in [25], implies that A is a bounded linear operator on X. On the other hand, if we assume that (H1) and (H3) hold, then from Bochner’s criterion for integrable functions and the estimate 1 mg (s)Wg (Ka x a +Ma ϕ B ), t ∈ I, AS(t − s)g(s, xs ) ≤ N it follows that the function s → AS(s)g(s, us ) is integrable over [0, t), for every t ∈ [0, a]. Remark 2.5. Condition (H3) is satisfied for functions g that arise in functional differential equations. Without specifying the phase space and neglecting for the moment additional technical precisions, we assert that functions of the type

Vol. 62 (2008)


197

0 g(t, ψ) = −∞ Q(s − t)ψ(s) ds, where Q(·) ∈ L(X, Y ) is an appropriate operator valued map, satisfy (H3). We now establish our first existence result. In this statement, we use the notation 1 max yt B . γ = max C(t)g(0, ϕ) + 0≤t≤a Ka 0≤t≤a Theorem 2.6. Assume that assumptions (H1), (H2), (H3) and (H4) are satisfied and that the following conditions hold: (a) The set U (r, t) = {S(t)f (s, ψ) : s ∈ I, ψ ∈ Br (0, B)} is relatively compact in X, for each 0 < t ≤ a and r > 0. (b) For each r > 0, the set V (r) = {AS(ξ)g(s, ψ) : ξ, s ∈ I, ψ ∈ Br (0, B)} is relatively compact in X, the functions t ∈ I → g(t, ut + yt ) ∈ X, for u ∈ Br (0, F (a)), are equicontinuous on I and there exists constant c1 , c2 > 0 such that g(t, ψ) ≤ c1 ψB + c2 , (t, ψ) ∈ I × B. (c) The constant µ = 1 − Ka c1 > 0 and ∞ Ka a ds f (s)] ds. [N1 mg (s) + Nm (2.6) > Ka W (s) µ 0 (c +γ) 2 µ Then there exists a mild solution of Problem (2.1)–(2.3). Proof. We define the map Γ : F (a) → F (a) by (Γu)0 = 0 and t AS(t − s)g(s, us + ys )ds Γu(t) = −C(t)g(0, ϕ) + g(t, ut + yt ) + 0 t S(t − s)f (s, us + ys )ds, t ∈ I. + 0

It follows from Remark 2.4 and the properties of f and g that Γu is well-defined and continuous. To prove that Γ is continuous, let (un )n∈N be a sequence in F (a) and u ∈ F(a) such that un → u in F (a). From axiom A, we infer that uns → us uniformly on I as n → ∞ and that the set U = I × {uns , us : s ∈ I, n ∈ N} is relatively compact in I × B. Thus, g is uniformly continuous on U and then g(s, uns )−g(s, us) → 0 uniformly on I as n → ∞. Now, by a standard application of the Lebesgue dominated convergence Theorem we can complete the proof that Γ is continuous. In order to apply Lemma 1.1, we need to establish an a priori bound for the solutions of the integral equation u = λΓ(u) for λ ∈ (0, 1). To this end, let uλ (·) be a solution of this equation. We introduce the following notation, c3

=

c4

=

βλ (t) =

sup{C(s)g(0, ϕ) : 0 ≤ s ≤ t} 1 sup{ys B : 0 ≤ s ≤ t} Ka sup{uλs + ys B : 0 ≤ s ≤ t}.

198


IEOT

Using assumptions (H3) and (H4) and the axioms of the phase space B, together with the fact that βλ (t) ≤ Ka uλ t +Ka c4 , we have the estimate uλ (t)

t ≤ C(t)g(0, ϕ) + g(t, uλt + yt ) + AS(t − s)g(s, uλs + ys )ds 0 t S(t − s)f (s, uλs + ys ) ds + 0 t 1 mg (s) + N mf (s)]W (uλs + ys B ) ds [N ≤ c1 uλt + yt B + c2 + c3 + 0 t 1 mg (s) + N mf (s)]W (βλ (s)) ds [N ≤ c1 βλ (t) + c2 + c3 + 0

from which it follows that Ka Ka t mf (s)]W (βλ (s)) ds. (c2 + c3 + c4 ) + βλ (t) ≤ [N1 mg (s) + N µ µ 0 Denoting by αλ (t) the right-hand side of the previous inequality and computing αλ (t), we get Ka f (t)]W (αλ (t)). [N1 mg (t) + Nm µ Consequently, upon integrating over [0, t], we obtain αλ (t) Ka a ds f (s)]ds. ≤ [N1 mg (s) + Nm Ka W (s) µ 0 µ (c2 +γ) αλ (t)

≤

Combining this estimate with (2.6), we conclude that {αλ (·) : λ ∈ (0, 1)} is bounded on I and, in turn, that {uλ : λ ∈ (0, 1)} is bounded in F (a). Let r > 0. For each 0 ≤ s ≤ a, the set {g(s, us + ys ) : u ∈ Br (0, F (a))} is bounded. It follows from Remark 2.2 and condition (b) that the set of functions {s → g(s, us + ys ) : u ∈ Br (0, F (a))} is relatively compact in C([0, a], X). Combining this property with [17, Lemma 3.1], we infer that Γ is completely continuous and applying Lemma 1.1, we obtain that Γ has a fixed point u ∈ F(a). Clearly the function x = u + y satisfies (2.5) and the initial condition (2.2). We next show that x(·) also satisfies (2.3). Since S(t) ≤ N t, for all t ≥ 0 and f (s, xs ) is an integrable function, then t 1 t S(t − s)f (s, xs ) ds ≤ N f (s, xs ) ds t 0 0 converges to zero, as t → 0+ . In addition, for δ > 0, we can write t t 1 AS(t − s)g(s, xs ) ds = (I − S(δ)) AS(t − s)g(s, xs ) ds δ 0 0 t 1 S(t − s)AS(δ)g(s, xs ) ds. + δ 0

(2.7)

Vol. 62 (2008)


199

Let r > 0 be such that xs B ≤ r for every s ∈ I. Since AS(t−s)g(s, xs ) ∈ V (r), it follows from the mean value theorem for the Bochner integral ([23, Lemma 2.1.3]) t that 0 AS(t − s)g(s, xs )ds ∈ tc(V (r)), so that 1 t 1 1 (I − S(δ)) AS(t − s)g(s, xs ) ds ∈ (I − S(δ))c(V (r)). δ t 0 δ In view of the fact that (I − 1δ S(δ))x → 0, δ → 0, for each x ∈ X and c(V (r)) is a compact set, we can affirm that (I − 1δ S(δ))x → 0, δ → 0, uniformly for x ∈ c(V (r)). Moreover, there is a positive constant cδ such that AS(δ)g(s, xs ) ≤ cδ for every s ∈ I. Therefore, we can estimate the second term of the right-hand side of (2.7) as 1 t N t N cδ 2 t . S(t − s)AS(δ)g(s, xs ) ds ≤ (t − s)cδ ds ≤ δ 0 δ 0 2δ t From these assertions, it follows that 1t 0 AS(t − s)g(s, xs )ds → 0, as t → 0+ . Consequently, substituting x by y +Γu in (2.3) and using the fact that g(0, ϕ) ∈ E, we have that lim

t→0+

x(t) − g(t, xt ) − C(t)ϕ(0) + g(0, ϕ) t y(t) + Γu(t) − g(t, ut + yt ) − C(t)ϕ(0) + g(0, ϕ) = lim+ t t→0 t 1 = lim AS(t − s)g(s, xs ) ds − (C(t) − I)g(0, ϕ) S(t)z + t→0+ t 0 1 t + lim+ S(t − s)f (s, xs ) ds t→0 t 0 = z,

which completes the proof that x(·) is a mild solution of Problem (2.1)–(2.3). In applications the operator S(t) is often compact. As such, it is useful to establish separately the following immediate consequence of Theorem 2.6. Corollary 2.7. Assume that (H1), (H2), (H3) and (H4), with mf ∈ L∞ (I), are satisfied. If S(t) is compact, for all t ≥ 0, and conditions (b) and (c) of Theorem 2.6 hold, then there exists a mild solution of Problem (2.1)–(2.3). We also obtain existence results when f or g satisfies appropriate Lipschitz conditions. We begin by establishing a result of this type when g(t, ψ) is Lipschitz continuous with respect to ψ. In the next result, we denote t AS(t − s)g(s, ys )ds , γ = max − C(t)g(0, ϕ) + g(t, yt ) + 0≤t≤a

c

=

max{ys B : 0 ≤ s ≤ a}.

0

200


IEOT

Theorem 2.8. Assume that (H1) and (H4) hold, and that f satisfies condition (a) of Theorem 2.6. Suppose, in addition, that g(·, ψ) : I → Y is continuous for every ψ ∈ B, there exists a positive constant Lg such that g(t, ψ1 ) − g(t, ψ2 ) Y

≤

Lg ψ1 − ψ2 B , (t, ψi ) ∈ I × B,

1 ) > 0, and the constant µ = 1 − Ka Lg (NY + aN ∞ Ka a ds > N mf (s) ds. Ka Wf (s) µ 0 µ γ+c

(2.8)

Then there exists a mild solution of Problem (2.1)–(2.3). Proof. Let Γ be defined as in the proof of Theorem 2.6. We decompose Γ in the form Γ = Γ1 + Γ2 , where Γ1 and Γ2 are the maps given by t Γ1 u(t) = −C(t)g(0, ϕ) + g(t, ut + yt ) + AS(t − s)g(s, us + ys )ds, t ∈ I. 0 t S(t − s)f (s, us + ys ) ds, t ∈ I, Γ2 u(t) = 0

It follows from [17, Lemma 3.1] that Γ2 is completely continuous. Moreover, the estimate Γ1 u(t) − Γ1 v(t) ≤

≤

g(t, ut + yt ) − g(t, vt + yt ) t AS(t − s)[g(s, us + ys ) − g(s, vs + ys )] ds + 0 t 1 Lg us − vs B ds NY Lg ut − vt B + N

≤

1 ) u − v Ka Lg (NY + aN

0

shows that Γ1 is a contraction map.

Assume that uλ (·) is a solution of the equation λΓ1 uλ + λΓ2 (u) = u for λ ∈ (0, 1). We can write uλ uλ (t) = −λC(t)g(0, ϕ) + λ g(t, t + yt ) − g(t, yt ) + λg(t, yt ) λ t λ u +λ AS(t − s)[g(s, s + ys ) − g(s, ys )] ds λ 0 t t AS(t − s)g(s, ys ) ds + λ S(t − s)f (s, uλs + ys ) ds +λ 0

0

from which it follows that u (t) λ

≤

γ+ +N

NY Lg uλt B 0

1 Lg +N

0

t

uλs B ds

t

mf (s)Wf (uλs + ys B )ds

Vol. 62 (2008)


and max

0≤s≤t

uλs B

Ka ≤ µ

201

t λ mf (s)Wf (us + ys B ) ds . γ+N 0

Introducing βλ (t) = max{uλs + ys B : 0 ≤ s ≤ t} ≤ max{uλs B : 0 ≤ s ≤ t} + c, we have that βλ (t) ≤

t Ka N Ka γ+c+ mf (s)Wf (βλ (s))ds. µ µ 0

Denoting by αλ (t) the right-hand side of the above inequality, we obtain that αλ (t) ≤ which implies that

αλ (t) Ka µ

γ+c

Ka N mf (t)Wf (αλ )(t) µ

≤

Ka N µ

a 0

mf (s) ds.

Applying the condition (2.8), we deduce that the set {αλ (t) : λ ∈ (0, 1), t ∈ I} is bounded, which, in turn, implies that the set of functions {uλ : λ ∈ (0, 1)} is bounded. Consequently, we conclude from Lemma 1.2 the existence of a fixed point u of Γ. The function x = u + y satisfies (2.2) and (2.5). In order to show that x is a mild solution, it remains to verify that x also satisfies (2.3). We initially estimate 1 t AS(t − s)g(s, xs )ds t 0 1 t N 1 t g(s, xs ) − g(s, ϕ)Y ds + AS(t − s)g(s, ϕ) ds ≤ t 0 t 0 1 t N 1 t Lg xs − ϕB ds + AS(t − s)g(s, ϕ) ds. ≤ t t 0 0 Since {g(s, ϕ) : 0 ≤ s ≤ a} is compact in E, we have that AS(t − s)g(s, ϕ) → 0, as t → 0, uniformly for s ∈ [0, a]. Using also the continuity of the function s → xs , it follows from the above estimate that 1 t AS(t − s)g(s, xs ) ds = 0. lim t→0+ t 0 We complete the proof of this assertion proceeding as in the proof of Theorem 2.6. The proof is complete. In this result, we can replace the hypothesis g(·, ψ) : I → Y continuous for the conditions g(·, ψ) : I → Y integrable and the map AS : I → L(Y, X) Lipschitz continuous, see [4, Proposition 1.3.7]. The following result follows directly from the Theorem 2.8.

202


IEOT

Corollary 2.9. Assume that the conditions (H1) and (H4), with mf ∈ L∞ (I), are satisfied. If S(t) is compact, for all t ≥ 0, g satisfies the hypotheses considered in 1 ) > 0 and (2.8) holds, then there exists a Theorem 2.8, µ = 1 − Ka Lg (NY + aN mild solution of (2.1)–(2.3). The following result is an immediate consequence of the contraction fixed point theorem applied to the map Γ. For the sake of brevity, we omit the proof, Theorem 2.10. Assume that (H1) is satisfied. Suppose, that the functions f (·, ψ) : I → X are integrable, g(·, ψ) : I → Y are continuous for every ψ ∈ B, and that there exist positive constants Lf , Lg such that g(t, ψ1 ) − g(t, ψ2 ) Y f (t, ψ1 ) − f (t, ψ2 ) for all ψ1 , ψ2 ∈ B and + where µ = min{N (2.1)–(2.3).

≤ Lg ψ1 − ψ2 B , ≤ Lf ψ1 − ψ2 B ,

1 ) + aLf µ < 1, Ka Lg (NY + aN aN 2 }.

Then there exists a unique mild solution of Problem

When f (t, ψ) and g(t, ψ) are independent of t, we obtain the following result. Corollary 2.11. Assume that (H1) holds. Suppose, in addition, that the functions f : B → X and g : B → Y satisfy the Lipschitz conditions g(ψ1 ) − g(ψ2 ) Y f (ψ1 ) − f (ψ2 )

≤ Lg ψ1 − ψ2 B , ≤ Lf ψ1 − ψ2 B ,

for all ψ1 , ψ2 ∈ B and certain positive constants Lf , Lg such that K(0)Lg NY < 1. Then there exists a unique mild solution of Problem (2.1)–(2.3) on [0, ∞). Proof. Since K(·) is a continuous function, we can select b > 0 sufficiently small so that 1 2 Kb Lg (NY + bN1 ) + b Lf N < 1. 2 Proceeding as in the proof of Theorem 2.10, we conclude that there exists a unique solution x(·) of Problem (2.1)-(2.3) on [0, b]. On the other hand, let Γ1 be the map defined by replacing y by the function w defined by w0 = xb and w(t) = C(t)x(b), for t ≥ 0, in the definition of Γ. It is not difficult to see from (2.5) that the mild solution x on [b, 2b] can be written in the form x(t + b) = h(t) + u(t), for 0 ≤ t ≤ b, where h(t) is an appropriate function and u is a fixed point of Γ1 . Since Γ1 is a contraction, we obtain that the Problem (2.1)-(2.3) has a unique solution on [0, 2b]. We complete the proof by inductively repeating this argument. We conclude this section with a discussion about the existence of mild solutions for the abstract Cauchy problem (1.1)–(1.3). In order to obtain differentiability of solutions we need to strengthen our assumptions both on the functions f and g, as well as on the space Y . We introduce the following conditions.

Vol. 62 (2008)


203

(H5). The function f : I × B × X → X satisfies these conditions: (i) The function f (t, ·, ·) : B×X → X is continuous a.e. t ∈ I and f (·, ψ, x) : I → X is strongly measurable for every (ψ, x) ∈ B × X. (ii) There exist an integrable function mf : I → [0, ∞) and a continuous nondecreasing function Wf : [0, ∞) → (0, ∞) such that f (t, ψ, x) ≤ mf (t)Wf (ψB + x) , (t, ψ, x) ∈ I × B × X. (H6). Let x : (−∞, a] → X be a function such that x0 = ϕ, x is continuously differentiable on I and x (0) = ξ. Then the function I → X, t → g(t, xt ), is d g(t, xt )|t=0 = η, independent of x. continuously differentiable and dt (H7). We assume that AC(·) : I → L(Y, X) is a strongly continuous map. In 2 = sup0≤t≤a AC(t)L(Y,X) . the sequel, N Remark 2.12. It is clear that if assumption (H7) holds, then Y ⊆ D(A) and (H1) also holds. In addition, in this case,it is well known that if the function h : I → Y is t integrable, then the function t → 0 AS(t− s)h(s) ds is continuously differentiable and t d t AS(t − s)h(s) ds = AC(t − s)h(s) ds. dt 0 0 To study the Problem (1.1)–(1.3) we introduce the following concept of mild solution. Definition 2.13. A function x : (−∞, b] → X, 0 < b ≤ a, is a C 1 -mild solution of (1.1)–(1.3) on [0, b] if x(·) is a function of class C 1 on [0, b] that satisfies conditions (1.2) and (1.3), as well as the integral equation t AS(t − s)g(s, xs )ds x(t) = C(t)(ϕ(0) − g(0, ϕ)) + S(t)(ξ − η) + g(t, xt ) + 0 t S(t − s)f (s, xs , x (s))ds, t ∈ [0, b]. + 0

We next modify our previous notations. We denote by y the function y : (−∞, a] → X such that y0 = ϕ and y(t) = C(t)ϕ(0) + S(t)ξ, for 0 ≤ t ≤ a. We write F (b), 0 < b ≤ a, for the space consisting of functions u : (−∞, b] → X such that u0 = 0, u is continuously differentiable on [0, b] and u (0) = 0. We consider F (b) endowed with the norm u1 = sup0≤t≤b u (t). We denote by d P : F (b) → C(I, X) the function given by P (u)(t) = dt g(t, ut + yt ). The argument used to prove Theorem 2.8 also enables us to obtain the following existence result. For the sake of brevity, we only provide a sketch of the proof. In the next result, t AC(t − s)g(s, ys ), ds, γ = max − AS(t)g(0, ϕ) − C(t)η + P (0)(t) + 0≤t≤a

c

= max{ys B + y (s) : 0 ≤ s ≤ a}.

0

204


IEOT

Theorem 2.14. Assume that (H3), (H5), (H6) and (H7) hold, ϕ(0) ∈ E, and the following conditions hold: (a) For each r > 0, the set {f (t, ut +yt , u (t)+y (t)) : t ∈ I, u ∈ F(a), u1 ≤ r} is relatively compact in X. (b) There exist constants Lg , LP > 0 such that g(t, ψ1 ) − g(t, ψ2 ) Y

≤

P (u)(t) − P (v)(t) ≤

Lg ψ1 − ψ2 B , LP max u (s) − v (s) , 0≤s≤t

2 > 0 and for all t ∈ I, ψ1 , ψ2 ∈ B and u, v ∈ F(a), µ = 1 − LP − 12 a2 Ka Lg N ∞ a (aKa + 1)N ds > mf (s) ds. (2.9) aKa +1 µ 0 γ+c Wf (s) µ Then there exists a C 1 -mild solution of Problem (1.1)–(1.3) on I. Proof. We define the operators Γ, Γ1 , Γ2 on the space F (a) by t Γ1 u(t) = −C(t)g(0, ϕ) − S(t)η + g(t, ut + yt ) + AS(t − s)g(s, us + ys )ds, 0 t S(t − s)f (s, us + ys , u (s) + y (s))ds, Γ2 u(t) = 0

and Γ = Γ1 + Γ2 . Since d Γ1 u(t) = dt d Γ2 u(t) = dt

−AS(t)g(0, ϕ) − C(t)η + P (u)(t) +

t

0

0

t

AC(t − s)g(s, us + ys )ds,

C(t − s)f (s, us + ys , u (s) + y (s))ds,

we have that

1 2 )u − v1 , (2.10) Γ1 u − Γ1 v1 ≤ (LP + a2 Ka Lg N 2 so that Γ1 is a contraction map. Moreover, from [17, Lemma 3.1] we obtain that, d Γ2 u : u ∈ Br (0, F (a))} is relatively compact in C(I). for every r > 0, the set { dt Since Γ2 u(0) = 0, for all u ∈ Br (0, F (b)), we obtain that {Γ2 u : u ∈ Br (0, F (b))} map. is relatively compact in F (b). Therefore, Γ2 is a completely continuous Assume that uλ (·) is a solution of the equation λΓ1 uλ + λΓ2 (u) = u for λ ∈ (0, 1). Then, proceeding as in the proof of Theorem 2.8, we obtain

d λ u (t) dt

≤

so that d 1 max uλ (s) ≤ 0≤s≤t ds µ

d 1 2 max d uλ (s) γ + LP max uλ (s) + a2 Ka Lg N 0≤s≤t ds 0≤s≤t ds 2 t d +N mf (s)Wf (uλs + ys B + uλ (s) + y (s))ds, ds 0 t d λ λ mf (s)Wf (us + ys B + u (s) + y (s))ds . γ+N ds 0

Vol. 62 (2008)


205

Introducing βλ (t) =

max{uλs + ys B +

d λ u (s) + y (s) : 0 ≤ s ≤ t} ds

d λ u (s) : 0 ≤ s ≤ t} + c, ds we can rewrite the preceding estimate as t aKa + 1 βλ (t) ≤ c + mf (s)Wf (βλ (s)) ds . γ+N µ 0 ≤

(aKa + 1) max{

Denoting by αλ (t) the right-hand side of the above inequality, we obtain that αλ (t) ≤ which implies that

αλ (t) aKa +1 γ+c µ

(aKa + 1)N mf (t)Wf (αλ (t)) µ

(aKa + 1)N ds ≤ Wf (s) µ

a

0

mf (s) ds.

We complete the proof as in Theorem 2.8, using the condition (2.9) instead of (2.8). For some applications it is convenient to avoid the global Lipschitz condition on P considered in the previous result. In this case, we obtain existence of local solutions, as follows: Theorem 2.15. Suppose that (H3), (H5), (H6) and (H7) hold, ϕ(0) ∈ E and the following conditions hold: (a) For each r > 0, the set {f (t, ut +yt , u (t)+y (t)) : t ∈ I, u ∈ F(a), u1 ≤ r} is relatively compact in X. (b) There exist Lg > 0 and a continuous function LP : I × [0, ∞) → [0, ∞) with the properties that LP (·, r) is nondecreasing and LP (t, r) → 0, as t → 0, for each r > 0, such that g(t, ψ1 ) − g(t, ψ2 ) Y P (u)(t) − P (v)(t)

≤ Lg ψ1 − ψ2 B , ≤ LP (t, r) max u (s) − v (s) , 0≤s≤t

for all t ∈ I, ψ1 , ψ2 ∈ B, u, v ∈ F(a), with u1 , v1 ≤ r. Then there exists a C 1 -mild solution of Problem (1.1)–(1.3) on [0, b], for some b > 0. Proof. We continue with the notations introduced in the statement of Theorem 2.14. It is clear from conditions (H3), (H5), (H6) and (H7) that Γ is well defined and is continuous from F (a) into F (a). We next show that there exist r > 0 and 0 < b ≤ a such that Γ(Br (0, F (b))) ⊆ Br (0, F (b)). To prove this assertion we

206


IEOT

d estimate dt Γu(t). Proceeding as in the proof of Theorem 2.14, for 0 ≤ t ≤ b and u ∈ F(b) with u1 ≤ r we obtain

d Γ(u)(t) dt ≤

≤

≤

t AC(t − s)[g(s, us + ys ) − g(s, ys )] ds γ + P (u)(t) − P (0)(t) + 0 t mf (s)Wf (us + ys B + u (s) + y (s)) ds +N 0 t max u(τ ) ds γ + LP (t, r) max u (s) + N2 Lg Kb 0≤s≤t 0 0≤τ ≤s t mf (s)Wf (Kb max u(τ ) + u (s) + c) ds +N 0≤τ ≤s 0 t 1 2 mf (s)ds. γ + LP (b, r)r + b N2 Lg Kb r + N Wf ((bKb + 1)r + c) 2 0

If we assume that for each r > 0, there exists a function ur with ur 1 ≤ r and a time tr ∈ [0, b] so that Γur (tr ) > r, then it follows from the above inequality that γ 1 Wf ((bKb + 1)r + c) b 2 1 ≤ + LP (b, r) + N2 Lg Ka b + N mf (s) ds. (2.11) r 2 r 0 It is clear that we can choose r, b > 0 so that γ 1 Wf ((bKb + 1)r + c) b 2 + LP (b, r) + N mf (s) ds < 1, 2 Lg Ka b + N r 2 r 0 which is contrary to (2.11). On the other hand, for r, b selected as it was established previously, we consider Γ defined on Br (0, F (b)). The estimate (2.10) with b instead of a yields 1 2 )u − v1 , Γ1 u − Γ1 v1 ≤ (LP (b, r) + b2 Ka Lg N 2 which shows that Γ1 is a contraction on Br (0, F (b)). Similarly, proceeding as in the proof of Theorem 2.14, we can assert that the map Γ2 is completely continuous. Consequently, Γ is a condensing map on Br (0, F (b)). A result of Sadovskii (see [10]) allows us to establish the existence of a fixed point for Γ and hence, the existence of a mild solution for (1.1)–(1.3). Remark 2.16. The values x(t) of the C 1 -mild solution constructed in Theorem 2.14 and Theorem 2.15 belong to E. To prove this fact, we consider the decomposition x(t) = u(t) + y(t). It follows from the properties of cosine functions that y(t) ∈ E and since u(t) = Γu(t) it remains to prove that Γu(t) ∈ E. Since g is Evalued, using the properties of cosine functions, it is clear that −C(t)g(0, ϕ) − S(t)η + g(t, ut + yt ) ∈ E. Similarly, using the group H we obtain easily that t 0 S(t − s)f (s, us + ys , u (s) + y (s)) ds ∈ E.

Vol. 62 (2008)


207

t Finally, let z = 0 AS(t − s)g(s, us + ys ) ds. We shall show that t t d C(τ )z = AC(τ + t − s)g(s, us + ys )ds − C(τ ) AC(t − s)g(s, us + ys )ds, dτ 0 0 which implies that C(·)z is continuously differentiable and, therefore z ∈ E. To establish this assertion we observe initially that t C(τ )z = [AS(τ + t − s) − AS(τ )C(t − s)]˜ g (s) ds, 0

where we have used g˜(s) = g(s, us + ys ). Hence, we can write 1 (C(τ + h)z − C(τ )z) h t 1 = A[S(τ + h + t − s) − S(τ + t − s)]˜ g (s) ds 0 h t 1 − (S(τ + h) − S(τ )) AC(t − s)˜ g (s) ds. (2.12) h 0 t The second term on the right-hand side converges to −C(τ ) 0 AC(t − s)˜ g (s) ds, as h → 0. Moreover, 1 τ +h+t−s 1 A[S(τ + h + t − s) − S(τ + t − s)]˜ g (s) = AC(α)˜ g (s) dα h h τ +t−s converges to AC(τ + t − s)˜ g (s), as h → 0, and 1 2 ˜ g (s) ≤ N g(s). A[S(τ + h + t − s) − S(τ + t − s)]˜ h Using the Lebesgue dominated convergencetheorem we obtain that the first term t on the right-hand side of (2.12) converges to 0 AC(τ + t − s)g(s, us + ys )ds, which completes the proof. A particular case of Theorem 2.15 is obtained when g is continuously differentiable. To establish this result, we introduce the following condition, where D1 g(t, ψ) denotes the derivative of g(t, ψ) respect to t and D2 g(t, ψ) denotes the Fréchet derivative of g(t, ψ) respect to ψ. (H8). The function g : I × B → X is of class C 1 and there exist a positive constant L1g and a continuous function L2g : I × [0, ∞) → [0, ∞) such that D1 g(t, ψ1 ) − D1 g(t, ψ2 ) ≤

L1g ψ1 − ψ2 B ,

D2 g(t, ψ1 ) − D2 g(t, ψ2 ) ≤

L2g (t, r) ψ1 − ψ2 B ,

for every t ∈ I, r > 0, and ψi ∈ Br (0, B), for i = 1, 2. Lemma 2.17. Assume g satisfies condition (H8), ϕ(0) ∈ E, and the function s → ys t is differentiable at zero with dy dt |t=0 = ψ ∈ B. Then, for every u ∈ F(a) the

208


IEOT

function t → g(t, ut + yt ) is continuously differentiable, the condition (H6) holds d and the map P given by P (u)(t) = dt g(t, ut + yt ) satisfies the condition P (u)(t) − P (v)(t) ≤ LP (t, r) sup u (s) − v (s), 0≤s≤t

for u1 , v1 ≤ r, where LP : I × [0, ∞) → [0, ∞) is a continuous function such that LP (·, r) is nondecreasing for all r ≥ 0. Moreover, if D2 g(0, ϕ) = 0, then LP (t, r) → 0, as t → 0, for each r > 0. Proof. We define w : (−∞, a] → X by w0 = ψ and w(t) = y (t), for t ∈ I. Since ψ(0) = ξ, it follows from the axioms of phase space that wt ∈ B, for t ≥ 0, and that the function I → B, t → wt , is continuous. We shall show that t → yt is d differentiable on I and dt yt = wt . Using the axioms of phase space we obtain yt+h − yt − wt h B yh − ϕ y(s + h) − y(s) − w(s) + M (t) − ψ ≤ K(t) max . 0≤s≤t h h B Since the right-hand side of the above inequality converges to zero, as h → 0+ , we infer that t → yt is continuously differentiable on I. Similarly, we can prove that for each u ∈ F(a) the function t → ut is continuously differentiable and d dt ut = (u )t . On the other hand, it follows from the chain rule that the function t → g(t, ut + yt ) is continuously differentiable and d g(t, ut + yt ) = D1 g(t, ut + yt ) + D2 g(t, ut + yt )((u )t + wt ). dt

(2.13)

Moreover, it is clear that d g(t, ut + yt )|t=0 = D1 g(0, ϕ) + D2 g(0, ϕ)(ψ) dt is independent of u, which proves that (H6) is valid. In order to estimate P (u)(t) − P (v)(t), we point out that ut B ≤ K(t) max u(s) ≤ tK(t) max u (s), 0≤s≤t

0≤s≤t

for all u ∈ F(a). Therefore, ut + yt B ≤ c(r) = bKb r + max0≤t≤b yt B , for all u ∈ F(a) with u1 ≤ r and every 0 ≤ t ≤ b. Similarly, (u )t + wt B ≤ d(r) = Kb r + max0≤t≤b wt B , for all u ∈ F(a) with u1 ≤ r and each 0 ≤ t ≤ b. From these estimates, we see that D2 g(t, ut + yt )

≤ D2 g(t, ut + yt ) − D2 g(t, ϕ) + D2 g(t, ϕ) ≤ L2g (t, c(r))(ut B + yt − ϕB ) + D2 g(t, ϕ),

Vol. 62 (2008)


209

for all u ∈ Br (0, F (a)) and t ∈ [0, b]. Consequently, for u, v ∈ Br (0, F (a)) and t ∈ [0, b] we get P (u)(t) − P (v)(t) ≤ D1 g(t, ut + yt ) − D1 g(t, vt + yt ) +D2 g(t, ut + yt )((u )t + wt ) − D2 g(t, vt + yt )((v )t + wt ) ≤ L1g ut − vt B + [D2 g(t, ut + yt ) − D2 g(t, vt + yt )]((u )t + wt ) +D2 g(t, vt + yt )((u )t − (v )t ) ≤ tL1g K(t) max u (s) − v (s) + L2g (t, c(r))ut − vt B (u )t + wt B 0≤s≤t

+L2g (t, c(r))(vt B

+ yt − ϕB )K(t) max u (s) − v (s) 0≤s≤t

+D2 g(t, ϕ)K(t) max u (s) − v (s) 0≤s≤t

≤

K(t)[tL1g

+ D2 g(t, ϕ)] max u (s) − v (s) 0≤s≤t

+K(t)L2g (t, c(r))[td(r) Hence, defining LP (b, r)

=

+ rtK(t) + yt − ϕB ] max u (s) − v (s). 0≤s≤t

1 Kb bLg + max D2 g(t, ϕ) 0≤t≤b 2 +Kb max Lg (t, c(r)) bd(r) + rbKb + max yt − ϕB 0≤t≤b

0≤t≤b

we see that the stated conditions about P hold.

The reader can see [13, Lemma 1.2] for a condition that ensures the map t → yt be differentiable at t = 0. We can establish the following consequence of Lemma 2.17. Corollary 2.18. Assume that (H3), (H5), (H7) and (H8), with D2 g(0, ϕ) = 0, are satisfied. Further, suppose that ϕ(0) ∈ E, the function s → ys is differentiable at t zero with dy dt |t=0 = ψ ∈ B, and the condition (a) of Theorem 2.15 hold. Then there 1 exists a C -mild solution of Problem (1.1)–(1.3) on [0, b], for some b > 0. Proof. The assertion follows from Lemma 2.17 and Theorem 2.15.

The following example shows that the hypotheses considered in this result are not necessary to obtain existence of mild solutions. Example. Let B = C0 × Lp (ρ, X) be the phase space defined in the Example 1, with p = 1 and ρ = 1. Let Q : X → E be a bounded linear operator, and define 0 g : I × B → E given by g(t, ψ) = Q −t ψ(θ) dθ. It is easy to see that g fulfills the conditions considered in Theorem 2.15, for every ϕ ∈ B. However, we can select ϕ so that the function t → yt is not differentiable.

210


IEOT

We complete this section with an existence result in which f satisfies a Lipschitz condition. Theorem 2.19. Assume that (H3), (H6) and (H7) hold, ϕ(0) ∈ E, and the following conditions hold: (a) The function f (·, ψ, x) is integrable on I, for each ψ ∈ B and x ∈ X, (b) There exist positive constants Lf , Lg and a continuous function LP : I × [0, ∞) → [0, ∞) with the properties that LP (·, r) is nondecreasing, for each r ≥ 0, and LP (t, r) → 0, as t → 0, for each r > 0, such that f (t, ψ1 , x1 ) − f (t, ψ2 , x2 ) ≤ g(t, ψ1 ) − g(t, ψ2 ) Y ≤ P (u)(t) − P (v)(t) ≤

Lf ( ψ1 − ψ2 B + x1 − x2 ), Lg ψ1 − ψ2 B , LP (t, r) max u (s) − v (s) , 0≤s≤t

for all t ∈ I, r > 0, ψi ∈ B, xi ∈ X for i = 1, 2 and u, v ∈ Br (0, F (a)). Then there exists a unique C 1 -mild solution of Problem (1.1)–(1.3) on [0, b], for some b > 0. Proof. We proceed as in the proof of Theorem 2.15. Initially we show that there exist positive constants b, r such that Γ(Br (0, F (b))) ⊆ Br (0, F (b)). To prove this d Γu(t). Indeed, observe that assertion, we estimate dt

d Γ(u)(t) dt d d ≤ [Γ(u)(t) − Γ(0)(t)] + Γ(0)(t) dt dt t ≤ P (u)(t) − P (0)(t) + AC(t − s)[g(s, us + ys ) − g(s, ys )] ds 0 t C(t − s)[f (s, us + ys , u (s) + y (s)) − f (s, ys , y (s))]ds + 0

d + Γ(0)(t) dt 1 2 r + N Lf ( 1 b2 Kb + b)r + d Γ(0)(t), ≤ LP (b, r)r + b2 Kb Lg N 2 2 dt for all t ∈ [0, b] and u1 ≤ r. It is clear from this estimate that we can select positive real numbers b, r such that 1 2 r + N Lf ( 1 b2 Kb + b)r + max d Γ(0)(t) < r, LP (b, r)r + b2 Kb Lg N 2 2 t∈[0,a] dt which establishes the assertion. Moreover, by using the same type of estimation, we can show that Γ is a contraction on Br (0, F (b)). This completes the proof. In the case where f (t, ψ, x) and g(t, ψ) are independent of t, we have the following result.

Vol. 62 (2008)


211

Corollary 2.20. Assume that (H6) and (H7) are satisfied, ϕ(0) ∈ E, and there exist constants Lf , Lg > 0 and 0 < LP < 1 such that f (ψ1 , x1 ) − f (ψ2 , x2 ) ≤ g(ψ1 ) − g(ψ2 ) Y ≤ P (u1 )(t) − P (u2 )(t) ≤

Lf ( ψ1 − ψ2 B + x1 − x2 ), Lg ψ1 − ψ2 B , LP max u1 (s) − u2 (s) , 0≤s≤t

for all (t, ψi , xi ) ∈ [0, b] × B × X, ui ∈ F(b), for i = 1, 2 and every b > 0. Then there exists a unique C 1 -mild solution of Problem (1.1)–(1.3) on [0, ∞). Proof. Proceeding as in the proof of Theorem 2.19 we can show that the map Γ is a contraction on F (b), for some b > 0 sufficiently small. This enables us to conclude that there exists a unique C 1 -mild solution of Problem (1.1)–(1.3) on [0, b]. Moreover, from Remark 2.16 we know that this solution is an E-valued continuous function. Such as in the proof of Corollary 2.11, we can translate the Problem (1.1)–(1.3) in b to obtain the existence of a unique C 1 -mild solution on [0, 2b] and, repeating inductively this argument, we obtain existence on [0, ∞).

3. Applications In this section we consider some applications of the abstract results established in Section 2. We initially study the particular case where the space X is finite dimensional. The literature for neutral systems with x(t) ∈ Rn is extensive. In this case our results are easily applicable. In fact, the operator A is a matrix of order n × n which generates the uniformly continuous cosine func∞ t2n n tion C(t) = cosh (tA1/2 ) = n=1 2n! A , with associated sine function S(t) = 2n+2 1 ∞ t n 1/2 ) A− 2 sinh (tA1/2 ) = n=1 (2n+1)! A . We note that the expressions cosh (tA

and sinh (tA1/2 ) are purely symbolic and do not assume the existence of the square roots of A. We can take Y = X = Rn and conditions (H1), (H2) are automatically satisfied. Moreover, it is obvious that S(t) is compact for t ∈ R and we can estimate C(t) ≤ cosh (tA1/2 ) and S(t) ≤ A−1/2 sinh (tA1/2 ). The next result is an immediate consequence of Theorem 2.6. In this statement we denote γ

= cosh (aA1/2 )(g(0, ϕ) + ϕ(0)) Ma +A−1/2 sinh (aA1/2 ) z + ϕB . Ka

(3.1)

Proposition 3.1. Assume that (H3), (H4) are satisfied, with mf , mg ∈ L∞ (I), and that the following conditions hold: (a) For every r > 0, the set of functions {t → g(t, ut + yt ) ∈ Rn : u ∈ Br (0, F (a))} is equicontinuous on I and there exist constants c1 , c2 > 0 such that g(t, ψ) ≤ c1 ψB + c2 , for every (t, ψ) ∈ I × B

212


IEOT

(b) The constant µ = 1 − Ka c1 > 0 and ∞ a Ka ds −1/2 1/2 > A sinh (aA ) [Amg (s) + mf (s)] ds. Ka W (s) µ 0 µ (c2 +γ) Then there exists a mild solution of (2.1)–(2.3) on [0, a]. Proof. We only observe that the constant γ in the statement of Theorem 2.6 is bounded above by the constant γ defined in (3.1). We next consider a partial neutral differential equation with infinite delay. Initially, we introduce the required technical framework. Let X = L2 ([0, π]) and A : D(A) ⊆ X → X be the linear map defined by Af = f , where D(A) = {f ∈ X : f ∈ X, f (0) = f (π) = 0}. It is well-known that A is the infinitesimal generator of a strongly continuous cosine family C(t) on X. Furthermore, A has a discrete spectrum, the eigenvalues are −n2 , for n ∈ IN , with corresponding eigenvectors

1/2 sin(nτ ), and the following properties hold. zn (τ ) = π2 2 (a) {zn : n ∈ IN } is an orthonormal basis of X and Az = − ∞ n=1 n z, zn zn , for z ∈ D(A). (b) For z ∈ X, C(t)z = ∞ n=1 cos (nt) z, zn zn . It follows from this expression ∞ sin(nt) that S(t)z = n=1 n z, zn zn , which implies that the operator S(t) is compact, for all t ∈ R and that C(t) = S(t) = 1, for all t ∈ R. (c) If Φ is the group of translations on X defined by Φ(t)x(ξ) = x ˜(ξ + t), where x ˜(·) is the extension of x(·) with period 2π, then C(t) = 12 (Φ(t) + Φ(−t)) and A = B 2 , where B is the infinitesimal generator of Φ and E = {x ∈ H 1 (0, π) : x(0) = x(π) = 0} (see [9] for details). In particular, we observe that the inclusion ι : E → X is compact. We consider the partial neutral functional differential equation t π ∂2 b(t − s, ω, τ )u(s, ω)dωds u(t, τ ) − ∂t2 −∞ 0 t ∂2 = u(t, τ ) + a1 (t − s)u(s, τ )ds, ∂τ 2 −∞

(3.2)

for t ∈ I = [0, a], τ ∈ J = [0, π], with boundary conditions u(t, 0) = u(t, π) = 0, t ∈ I,

(3.3)

and initial conditions u(s, τ ) = ϕ(s, τ ),

∂ u(0, τ ) = ξ(τ ), ∂t

s ∈ (−∞, 0], τ ∈ [0, π].

(3.4)

We assume that ϕ ∈ B = Cr × L2 (ρ, X), r = 0, ϕ(0, ·) ∈ H 1 ([0, π]), ξ ∈ X and that the functions a1 (·) and b(·) satisfy the following conditions:

Vol. 62 (2008)


213

2

∂ ∂ (i) The functions b(θ, ω, τ ), ∂τ b(θ, ω, τ ), ∂τ 2 b(θ, ω, τ ) are continuous on (−∞, 0]× I × J, b(θ, ω, π) = b(θ, ω, 0) = 0 for every (θ, ω) ∈ (−∞, 0] × I and 12 2 i π 0 π ∂ b(−s, ω, τ ) 1 Lg = max{ dωdsdτ : i = 0, 1, 2 } < ∞. ∂τ i 0 −∞ 0 ρ(s)

(ii) The function

∂ ∂θ b(θ, ω, τ )

exists and 2 0 π ∂ 1 dω ds < ∞. b(−s, ω, τ ) ∂s −∞ 0 ρ(s) 12 0 a2 (−θ) (iii) The function a1 (·) is continuous and Lf = −∞ 1ρ(θ) dθ < ∞.

Let f, g : B → X be the functions defined by 0 f (ψ)(τ ) = a1 (−s)ψ(s, τ )ds, −∞ 0

g(ψ)(τ )

π

b(−s, ω, τ )ψ(s, ω)dωds.

= −∞

0

Under the above conditions, f and g are bounded linear operators, gL(B,X) ≤ Lg and f L(B,X) ≤ Lf . Moreover, from condition (i) we can prove that g is D(A)valued and that Ag : B → X is a bounded linear operator with g L(B,[D(A)]) ≤ Lg . For this reason, we take Y = [D(A)] and NY = 1. It follows from earlier discussion that if ι : Y → X is the inclusion, then ι(x) ≤ xA and AS(t) L(Y,X)≤ 1. On the other hand, for a function x : (−∞, a] → X such that x0 = ϕ and x is continuous on I, we have that π ∞ π d ∂b(s, ω, τ ) g(xt ) = x(t − s, ω)dωds + b(0, ω, τ )x(t, ω)dω. dt ∂s 0 0 0 In particular, d g(xt )|t=0 = dt

0

∞

0

π

∂b(s, ω, τ ) ϕ(−s, ω)dωds + ∂s

π

b(0, ω, τ )ϕ(0, ω)dω = η, 0

exists and is independent of x. Furthermore, if u is of class C 1 on I with u (0) = 0, the function t → yt is differentiable at t = 0, and since g is a bounded linear map and ϕ ∈ E, we have that P (u)(t) = g(ut + yt ), which implies that P (u)(t) − P (v)(t) = g(ut + yt ) − g(vt + yt ) ≤ NY Lg Ka max u (s) − v (s). 0≤s≤t

We set LP = NY Lg Ka = Lg Ka . By completeness, we study briefly the existence of the derivative of t → yt at zero.

214


IEOT

∂ϕ Lemma 3.2. Assume that ∂ϕ ∂θ (θ, ·) exists a.e. θ ∈ (−∞, 0] and that ∂θ (0, ·) = ξ. If in addition, 2 0 π ϕ(t + θ, τ ) − ϕ(θ, τ ) ∂ϕ − (θ, τ ) dτ dθ → 0, t → 0, ρ(θ) (3.5) t ∂θ −∞ 0

then ϕ (θ) =

∂ϕ ∂θ (θ, ·)

∈ B and

d dt yt |t=0

= ϕ .

Proof. It follows from (3.5) that 2 π ϕ(θ + h, τ ) − ϕ(θ, τ ) ∂ϕ − (θ, τ ) dτ → 0, h → 0, a.e. θ ∈ (−∞, 0] h ∂θ 0 which implies that ϕ (θ) = ∂ϕ ∂θ (θ, ·) ∈ X and ϕ ∈ B. Furthermore, using this property, the definition of y and the semi-norm in B, we have that yt − ϕ − ϕ 2B t C(t)ϕ(0) + S(t)ξ − ϕ(0) − ϕ (0)22 ≤ 2 t −t ϕ(θ + t) − ϕ(θ) − ϕ (θ)22 dθ +2 ρ(θ) t −∞ 0 C(t + θ)ϕ(0) + S(t + θ)ξ − ϕ(θ) − ϕ (θ)22 dθ. ρ(θ) (3.6) +2 t −t

The first term on the right-hand side of (3.6) converges to zero as t → 0 since ϕ(0) ∈ E and ϕ (0) = ξ. The middle term on the right-hand side of (3.6) −t ϕ(θ + t) − ϕ(θ) − ϕ (θ)22 dθ ρ(θ) t −∞ 2 −t π ϕ(t + θ, τ ) − ϕ(θ, τ ) ∂ϕ − (θ, τ ) dτ dθ ρ(θ) = t ∂θ −∞ 0 converges to zero as t → 0 by condition (3.5). On the other hand, the decomposition C(t + θ)ϕ(0) − S(t + θ)ξ − ϕ(θ) t C(t + θ)ϕ(0) − ϕ(0) t + θ = t+θ t ϕ(0) − ϕ(θ) θ S(t + θ)ξ − ϕ(θ) t + θ + + t+θ t θ t − ϕ (θ) is a bounded for −t < θ ≤ 0, shows that θ → C(t+θ)ϕ(0)+S(t+θ)ξ−ϕ(θ) t function with values in X. We infer from this property that the third term on the right-hand side of (3.6) converges to zero as t → 0, which completes the proof. The following result is obtained directly from Corollary 2.11 and Corollary 2.20.

Vol. 62 (2008)


215

Proposition 3.3. Assume that (i)–(iii) hold and that Lg < 1. Then there exists a unique mild solution u(·) of Problem (3.2)–(3.4) on [0, ∞). If, in addition, the function ϕ satisfies the hypotheses of Lemma 3.2, then u(·) is a C 1 -mild solution of (3.2)–(3.4). Proof. Since K(0) = 1 the first assertion is a direct consequence of Corollary 2.11. Similarly, the second assertion is an immediate consequence of Lemma 3.2 and Corollary 2.20. Acknowledgment The authors thank the anonymous referee for helpful suggestions.

References [1] M. Adimy, K. Ezzinbi and M. Laklach, Existence of solutions for a class of partial neutral differential equations. C. R. Acad. Sci. Paris Sir. I Math. 330 (11) (2000), 957–962. [2] M. Adimy, H. Bouzahir and K. Ezzinbi, Existence and stability for some partial neutral functional differential equations with infinite delay, J. Math. Anal. Appl. 294 (2) (2004), 438–461. [3] M. Adimy and K. Ezzinbi, A class of linear partial neutral functional-differential equations with nondense domain. J. Diff. Eqns. 147 (2) (1998), 285–332. [4] W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkh¨ auser Verlag, Basel, 2001. [5] K. Balachandran, D.G. Park and S.M. Anthoni, Existence of solutions of abstractnonlinear second-order neutral functional integrodifferential equations. Comput. Math. Appl. 46 (8–9) (2003), 1313–1324. [6] M. Benchohra, J. Henderson and S.K. Ntouyas, Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces. J. Math. Anal. Appl. 263 (2) (2001), 763–780. [7] M. Benchohra, E. Gatsori and S.K. Ntouyas, Multivalued semilinear neutral functional differential equations with nonconvex-valued right-hand side. Abstr. Appl. Anal. 2004, no. 6, 525–541. [8] T.A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 189 (1998), 23–31. [9] H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, NorthHolland Mathematics Studies, Vol. 108, North-Holland, Amsterdam, 1985. [10] A. Granas and J. Dugundji, Fixed Point Theory. Springer-Verlag, New York, 2003. [11] J.K. Hale, Partial neutral functional-differential equations. Rev. Roumaine Math Pures Appl. 39 (4) (1994), 339–344. [12] J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional-Differential Equations. Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. [13] H.R. Henr´ıquez and C.H. V´ asquez, Differentiability of solutions of second-order functional differential equations with unbounded delay. J. Math. Anal. Appl. 280 (2003), 284–312.

216


IEOT

[14] E. Hern´ andez and H.R. Henr´ıquez, Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay. J. Math. Anal. Appl. 221 (2) (1998), 499–522. [15] E. Hern´ andez and H.R. Henr´ıquez, Existence results for partial neutral functionaldifferential equations with unbounded delay. J. Math. Anal. Appl. 221 (2) (1998), 452–475. [16] E. Hern´ andez, Existence results for partial neutral integrodifferential equations with unbounded delay. J. Math. Anal. Appl. 292 (1) (2004), 194–210. [17] E. Hern´ andez and M.A. McKibben, M. A., “Some comments on: Existence of solutions of abstract nonlinear second-order neutral functional integrodifferential equations, in Comput. Math. Appl. 46 (8–9) (2003)”. Comput. Math. Appl. 50 (5–6) (2005), 655–669. [18] Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations with Infinite Delay. Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991. [19] J. Kisy´ nski, On cosine operator functions and one parameter group of operators, Studia Math. 49 (1972), 93–105. [20] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Acad. Publ., Dordrecht, 1999. [21] V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay. Kluwer Acad. Publ., Dordrecht, 1994. [22] H.J. Lee, J. Park and J.Y. Park, Existence results for second-order neutral functional differential and integrodifferential inclusions in Banach spaces. Electron. J. Differential Equations 96 (2002), 13 pp. (electronic). [23] R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Robert E. Krieger Publ. Co., Florida, 1987. [24] C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungaricae, 32 (1978), 76–96. [25] C.C. Travis and G.F. Webb, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston J. Math. 3 (4) (1977) 555–567. [26] J. Wu and H. Xia, Rotating waves in neutral partial functional-differential equations. J. Dynam. Diff. Eqns. 11 (2) (1999), 209–238. [27] J. Wu and H. Xia, Self-sustained oscillations in a ring array of coupled lossless transmission lines. J. Diff. Eqns. 124 (1) (1996), 247–278.

Eduardo Hern´ andez Morales Departamento de Matem´ atica I.C.M.C. Universidade de S˜ ao Paulo Caixa Postal 668 13560-970 S˜ ao Carlos SP Brazil e-mail: [email protected]

Vol. 62 (2008)


Hern´ an R. Henr´ıquez Departamento de Matem´ atica Universidad de Santiago, USACH Casilla 307 Correo-2, Santiago Chile e-mail: [email protected]. Mark A. McKibben Department of Mathematics and Computer Science Goucher College 1021 Dulaney Valley Road Baltimore, MD 21204 USA e-mail: e-mail: [email protected] Submitted: February 3, 2007. Revised: June 16, 2008.

217



Schatten-von Neumann Hankel Operators on the Bergman Space of Planar Domains Roberto C. Raimondo Abstract. In this paper we study the problem of the joint membership of Hϕ and Hϕ in the Schatten-von Neumann p-class Sp when ϕ ∈ L∞ (Ω) and Ω is a planar domain. We use a result of K. Zhu and the localization near the boundary to solve the problem. Finally, we recover a result of Arazy, Fisher and Peetre on the case Hϕ with ϕ holomorphic. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47B38. Keywords. Hankel operators, Schatten-von Neumann, planar domains, reproducing kernels.

1. Introduction Let Ω be a bounded multiply-connected domain in the complex plane C, whose boundary ∂Ω consists of finitely many simple closed smooth analytic curves γj (j = 1, 2, . . . , n) where γj are positively oriented with respect to Ω and γj ∩ γi = ∅ if i = j. We also assume that γ1 is the boundary of the unbounded component of C\Ω. Let Ω1 be the bounded component of C\γ1 , and Ωj (j = 2, . . . , n) the unbounded component of C\γj , respectively, so that Ω = ∩nj=1 Ωj . For dν = π1 dxdy we consider the usual L2 -space L2 (Ω) = L2 (Ω, dν). The Bergman Space L2a (Ω, dν), consisting of all holomorphic functions which are L2 integrable, is a closed subspace of L2 (Ω, dν) with the inner product given by f, g = f (z)g(z)dν(z) Ω

for any f, g ∈ L2 (Ω, dν). The Bergman Projection is the orthogonal projection P : L2 (Ω, dν) −→ L2a (Ω, dν), and it is well-known that for any f ∈ L2 (Ω, dν) we have f (z)K Ω (w, z)dν(z), P f (w) = Ω

220

Raimondo

IEOT

where K Ω is the Bergman Reproducing kernel of Ω. For ϕ ∈ L∞ (Ω, dν) the Hankel operator Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) is defined by Hϕ = (1 − P )Mϕ where Mϕ is the standard multiplication operator. A simple calculation shows that Hϕ f (z) = (ϕ(z) − ϕ(w))f (w)K Ω (w, z)dν(w) Ω

The problem of finding necessary and sufficient conditions on the function ϕ ∈ L∞ (Ω, dν) for the Hankel operators Hϕ and Hϕ to be in the Schatten-von Neumann p-class has been studied by many authors in many different settings. Recently (see [7]) the author has found a characterization for the Hilbert-Schmidt case (i.e. p = 2). In this paper we prove this more general result. Moreover, using the fact that if ϕ is holomorphic the operator Hϕ is trivial, we are able to recover a result that was proved by Arazy and Fisher and Peetre (see [2]). The paper is organized as follows; in section 2 we transfer our problem from an arbitrary planar domain to a canonical one i.e. a domain whose boundary consists of circles, and we show that it is enough to solve the problem for this special type of planar domains; in section 3 we collect results about the Bergman kernel for a planar domain and useful estimates and the structure of L2a (Ω, dν). In section 4 we prove the main result.

2. Preliminaries Let Ω be the bounded multiply-connected domain given at the beginning of Section 1 i.e. Ω = ∩nj=1 Ωj , where Ω1 is the bounded component of C\γ1 , and Ωj (j = 2, . . . , n) is the unbounded component of C\γj . We use the symbol ∆ to indicate the punctured disk {z ∈ C|0 < |z| < 1}. Let Γ be any one of the domains Ω, ∆, Ωj (j = 2, . . . , n). We call K Γ (z, w) the reproducing kernel of Γ and we use the symbol k Γ (z, w) to indicate the normalized reproducing kernel i.e. k Γ (z, w) = 1 K Γ (z, w)/K Γ (w, w) 2 . ˜ the Berezin transform of A, by For any A ∈ B(L2a (Γ, dν)) we define A, Γ Γ Γ Γ (z)dν(z), ˜ A(w) = Akw , kw = Akw (z)kw Γ

1

Γ where kw (·) = K Γ (·, w)K Γ (w, w)− 2 . If ϕ ∈ L∞ (Γ), then we indicate with the symbol ϕ˜ the Berezin Transform of the associated Toeplitz operator Tϕ and we have Γ ϕ(w) ˜ = ϕ(z)|kw (z)|2 dν(z). Γ

˜ ∞≤ We remind the reader that it is well-known that A˜ ∈ Cb∞ (Γ) and we have ||A|| ||A||B(L2 (Ω)) . Before we state the main results of this paper we anticipate (see Section 4 for exact definition and existence) that we call a ∂-partition for Ω a partition of the unit for Ω where each function is smooth and near each connected component

Vol. 62 (2008)

Schatten-von Neumann Hankel Operators

221

of the boundary there is only one function which is different from zero (note that this implies that the function must be equal to 1). With this definition in mind we can state the main theorem Theorem A. Let ϕ ∈ L∞ (Ω), let ϕj = ϕ·pj where j = 1, . . . , n and P = {pj }j=1,...,n is a ∂-partition for Ω. Then the following are equivalent: 1. The operators Hϕ , Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are in Sp . 2. For any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ωj , dν) → L2 (Ωj , dν) L2a (Ωj , dν) are in Sp . We remind the reader (see [5] or [4], Chapter 15 ) that any bounded multiplyconnected domain whose boundary consists of finitely many simple closed smooth analytic curves, i.e. a regular domain, is conformally equivalent to a canonical bounded multiply connected domain whose boundary consists of finitely many circles. Moreover, it is possible to prove (see [1] and [2]) the following Theorem 2.1. Let Ω be a regular domain and let ψ be a conformal mapping from Ω onto D. Then: 1. The Bergman kernels of Ω and ψ(Ω) = D are related via K D (ψ(z), ψ(w))ψ (z)ψ (w) = K Ω (z, w). 2. The operator Vψ f = ψ · f ◦ ψ is an isometry from L2 (D) onto L2 (Ω). 3. Vψ P D = P Ω Vψ . 4. If ϕ : D → W is conformally onto, then Vϕ Vψ = Vϕ◦ψ . In particular Vψ−1 = Vψ−1 . Now we observe that if A ∈ B(L2a (Ω)) and we define AD ∈ B(L2a (D)) as Vψ−1 AVψ , where ψ is a conformal mapping from Ω onto D, then we can prove the following ˜ Proposition 2.2. A(z) = A˜D (ψ(z)). Proof. We have, by definition, ˜ A(z) = AkzΩ , kzΩ , 1

where kzΩ (·) = KzΩ (·, z)KzΩ (z, z)− 2 . Let us take ψ −1 : D → Ω. Since (JR ψ −1 )(β) = |(ψ −1 ) (β)|2 and there exists ζ ∈ D such that ψ(z) = ζ we obtain

222

Raimondo

IEOT

AkzΩ , kzΩ AkzΩ (w)kzΩ (w)dv(w) Ω AkψΩ−1 (ζ) (ψ −1 (β))kψΩ−1 (ζ) (ψ −1 (β))|(ψ −1 ) (β)|2 dv(β) D ((Vψ−1 AVψ )(Vψ−1 kψΩ−1 (ζ) ))(β)kψΩ−1 (ζ) (ψ −1 (β))(ψ −1 ) (β)dv(β).

˜ A(z) = = = =

D

Since Theorem 2.2 implies that kψΩ−1 (ζ) (ψ −1 (β)) = then

D

and

K D (β, ζ)|(ψ −1 ) (ζ)| (ψ −1 ) (β)(ψ −1 ) (ζ)K D (ζ, ζ)

1 2

=

|(ψ −1 ) (ζ)| (ψ −1 ) (β)(ψ −1 ) (ζ)

· kζD (β)

((Vψ−1 AVψ−1 )(Vψ−1 kζΩ ))(β)kψΩ−1 (ζ) (ψ −1 (β))(ψ −1 ) (β)dβ D

(AD kζD )(β)kζD (β)dv(β)

˜ are the same. It follows that A(z) = A˜D (ψ(z)), and this completes our proof.

Since every conformal map is an open map, we can conclude that it is enough to prove the theorem in the case that the domain is a canonical bounded multiply connected domain whose boundary consists of finitely many circles.

3. The Structure of L2a (Ω) and some Estimates about the Bergman Kernel From now on we will assume that Ω = ∩nj=1 Ωj where Ω1 = {z ∈ C : |z| < 1} and Ωj = {z ∈ C : |z − aj | > rj } for j = 2, . . . , n. Here aj ∈ Ω1 and 0 < rj < 1 with |aj − ak | > rj + rk if j = k and 1 − |aj | > rj . We will indicate with the symbol ∆ the punctured disk Ω1 \{0}. With the symbols K Ωj (z, w), K Ω (z, w), K ∆ (z, w) we denote the Bergman kernel on Ωj , Ω, and ∆ respectively. If we define Lpa (Ω1 ) = {f ∈ Lp (Ω)| f is holomorphic} and Lpa (∆) = {f ∈ Lp (∆)| f is holomorphic}, then we have the following (see [7]) Theorem 3.1. There exists an isomorphism I : L2 (∆) −→ L2 (Ω1 ) such that I(L2a (∆)) = L2a (Ω1 ). Moreover for any p ≥ 2 we have that Lpa (∆) = Lpa (Ω1 ) and, for any (z, w) ∈ ∆2 , the Bergman kernels K ∆ and K Ω1 satisfy the equation K ∆ (z, w) = K Ω1 (z, w).

Vol. 62 (2008)


223

The last theorem has an important application. In fact this result together with the well-known fact that the reproducing kernel of the unit disk is given by K Ω1 (z, w) =

1 (1 − z · w)2

imply that K Ωj (z, w) =

rj2

(rj2 − (z − aj ) · (w − aj ))2 n if j = 2, . . . , n. If we define E Ω = K(z, w) − j=1 K Ωj (z, w) it is possible to prove that (see [7]) Lemma 3.2. 1. E Ω is conjugate symmetric about z and w. For each w ∈ Ω, E Ω (·, w) is conjugate analytic on Ω and E Ω ∈ C ∞ (Ω × Ω). 2. There are neighborhoods Uj of ∂Ωj (j = 1, . . . , n) and a constant C > 0 such that Uj ∩ Uk is empty if j = k and Ω K (z, w) − K Ωj (z, w) < C for z ∈ Ω and w ∈ Uj . 3. E Ω ∈ L∞ (Ω × Ω). 4. For any (z, w) ∈ Gj × Ω ∪ Ω × Gj we have |K Ω (z, w)| < D|K Ωj (z, w)| and |K Ωj (z, w)| < |K Ω (z, w)| + M. 5. For any z ∈ Ω we have K Ωj (z, z) < K Ω (z, z). The next Lemma gives an useful decomposition of the space L2a (Ω), in fact we have (see [7]) Lemma 3.3. For f ∈ L2a (Ω), we can write it uniquely as f (z) =

n

(Pj f )(z) + (P0 f )(z)

j=1

with Pj f ∈ L2a (Ωj ), P0 f ∈ L2a (Ω) ∩ C ∞ (Ω), Pk (Pj f ) = 0 if j = k and moreover, there exists a constant M1 such that, for j = 0, 1, . . . , n, we have ||Pj f ||Ω ≤ ||Pj f ||Ωj ≤ M1 ||f ||Ω . In particular, if f ∈ L2a (Ωi ), then Pi f = f and ||f ||Ωi ≤ M1 ||f ||Ω for i = 1, . . . , n. If {fn } is a bounded sequence in L2a (Ω) and fn → 0 weakly in L2a (Ω), then Pj fn → 0 weakly on L2a (Ωj ) for j = 1, . . . , n and P0 fn → 0 uniformly on Ω.

224

Raimondo

IEOT

4. Multiply-Connected Domains In this section we investigate, with the help of the results established in the previous section, necessary and sufficient conditions on the function ϕ ∈ L∞ (Ω, dν) for the Hankel operators Hϕ and Hϕ to be in the p-th Schatten-von Neumann class Sp . First we collect some well-known facts and definitions about this type of operators. Let us recall that a linear operator A : H1 → H2 is in the p-th Schatten-von Neumann class Sp (H1 , H2 ) if there exists an orthonormal basis {ej } of H1 and an orthonormal basis {fj } of H2 such that ∞ Aek , fk

H2

p < ∞.

k=1

A positive operator A on an Hilbert space H is in the trace class if tr(A) = ∞ j=1 Aej , ej H < ∞ for any orthonormal basis {ej } of H. It is possible to show that tr(A) is independent from the choice of the basis and that an operator A on an Hilbert space H is p-th Schatten-von Neumann class Sp if and only if (A∗ A)p/2 is in the trace class. Moreover it is possible to prove the following Theorem 4.1. Let 1 ≤ p < ∞ and (H1 , ·, ·H1 ) and (H2 , ·, ·H2 ) be two Hilbert spaces. An operator A : H1 → H2 belongs to the p-th Schatten-von Neumann class Sp (H1 , H2 ) if and only if, no ∞ ∞ matter how we choose the orthonormal sequences {en }n=1 ⊂ H1 and {fn }n=1 ⊂ H2 , we have ∞ Aen , fn H2 n=1 ∈ p (N). Before we state and prove the main result of this section we need to give the following Definition 4.2. Let Ω = ∩ni=1 Ωi be a bounded canonical multiply-connected domain. We say that the set of n functions P = {p1 , . . . , pn } is a ∂-partition for Ω if: 1. For every j = 1, . . . , n pj : Ω → [0, 1] is a Lipschitz, C ∞ -function. 2. For every j = 1, . . . , n there exists an open set Wj and an j > 0 such that the Uj = {ζ ∈ Ω : rj < |ζ − aj | < rj + j } is contained in Wj and pj (ζ) = 1 if ζ ∈ Uj , and pj (ζ) = 0 if ζ ∈ Wk and j = k. 3. For any ζ ∈ Ω it holds that n i=1

pi (ζ) = 1.

Vol. 62 (2008)


225

These conditions imply that for any j the function pj is equal to 1 when we are close to ∂Ωj , and pj becomes zero when we are close to ∂Ωk if j = k. We observe that for any j it is possible to extend pj to a function p˜j : Ωj → [0, 1] in such a way that pj (ζ) = p˜j (ζ) ζ ∈Ω and p˜j (ζ) = 0 ζ ∈ Ω. Clearly p˜j ∈ C(Ωj , [0, 1]). We will write pj instead of p˜j when it is clear within what context we are working. Finally, we assume that the pj s are nice smooth functions since this makes things easier and the proofs are more natural. We do not need to work with a more complicated partition since we are only using this partition as a tool to localize near the boundary. The reader should observe that it is always possible to construct a ∂-partition. In fact, by construction, we have Ω = ∩nj=1 Ωj where Ω1 = {ζ ∈ C : |ζ| < 1} and, for j = 2, 3, . . . , n, Ωj = {ζ ∈ C : |ζ − aj | > rj }. Since aj ∈ Ω1 , 0 < rj < 1, and |aj − ai | > rj + ri if i = j and 1 − |aj | > rj , it possible to find a δ > 0 such that 1 − |aj | > rj + δ and |aj − ai | > rj + ri + δ if i = j. If we choose such that 0 < < δ and we define U1

=

{ζ ∈ C : 1 − δ < |ζ| < 1}

V1

=

{ζ ∈ C : 1 − < |ζ| < 1}

and, for j = 2, 3, . . . , n, Uj Vj

= =

{ζ ∈ C : rj + δ > |ζ − aj | > rj } {ζ ∈ C : rj + > |ζ − aj | > rj },

and U0 = Ω − ∪nj=1 Vj then {Ui }i=0,1,...,n is an open cover of Ω. By a well-known theorem (see Chap 1 of [6]) we can construct a smooth partition of unity subordinate to this cover, clearly such a partition is a ∂-partition. Before we state the next Theorem we remind the reader that n KΩ (ζ, z), K Ω (ζ, z) = E Ω (ζ, z) + =1

where E ∈ L∞ (Ω × Ω) and, ∀ = 1, . . . , n, we have KΩ (ζ, z) = K Ω (ζ, z) Ω

∀ζ, z ∈ Ω × Ω,

where K is the reproducing kernel of Ω . If we use the symbol K0Ω to indicate E Ω we can write n K Ω (ζ, z) = KΩ (ζ, z). =0

226

Raimondo

IEOT

Before we prove the next lemma we remind the reader that given a domain Γ and φ ∈ L∞ (Γ) we define, as usual, the mean oscillation of φ to be the following function def 2

2 (Γ) MO (φ, z) = |φ| (z) − φ(z) if there is not ambiguity we will use the symbol MO(φ, z) instead of MO(Γ) (φ, z). Lemma 4.3. Let ϕ ∈ L∞ (Ω), if we define ϕ(w) if w ∈ Ω ϕ (w) = 0 if w ∈ / Ω \Ω where = 1, . . . , n, then the following are equivalent: 1. The inequality p/2

2 |ϕ|2 (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

holds. 2. The inequality   (Ω ) 2 p/2 2 (Ω ) ϕ (z) − ϕ (z)  K Ω (z, z)dv(z) < ∞ Ω

holds for any = 1, . . . , n.

p/2

2 2 Proof. We observe that Ω |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) can be written as n j=1

p/2

2 2 Ω |ϕ| (z) − |ϕ(z)| K (z, z) +

Ω\∪n j=1 Gj

Gj

2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)

where Gj is the same as in Lemma 1. Clearly

p/2

2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω\∪n j=1 Gj

therefore we have

p/2

2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

if and only if n j=1

Gj

p/2

2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞

Vol. 62 (2008)


and this is equivalent that

227

p/2

2 2 (z) − | ϕ(z)| K Ω (z, z)dv(z) < ∞ for any |ϕ| Gj

j = 1, . . . , n. We remind to the reader that given a function φ ∈ L∞ (Ω) we denote with the symbol MO(Ω) (φ, z) the following 2 (Ω) def

2 (Ω) |φ| (z) − φ(Ω) (z) . MO (φ, z) = We claim that, for any = 1, . . . , n, there exists a constant C1, (depending on p) such that p · MO(Ω ) (ϕ , z)p MO(Ω) (ϕ, z)p ≤ C1, and there exists a constant C2, (depending on p) such that MO(Ω ) (ϕ , z)p ≤ C2, · (MO(ϕ, z)p + (|z − a | − r )p ) . p To start we prove that MO(ϕ, z)p ≤ C1, · MO(Ω ) (ϕ , z)p . A simple calculation shows that

p/2 2 Ω 2 2 Ω p/2 p 2 MO(ϕ, z) = |ϕ(u) − ϕ(w)| kz (u) kz (w) dv(u)dv(w) Ω

Ω

and, if we denote with the symbol ϕ : Ω → C the following ϕ(w) if w ∈ Ω ϕ (w) = 0 if w ∈ / Ω \Ω such that we know that there is a constant C 2 2 |ϕ(u) − ϕ(w)|2 kzΩ (u) kzΩ (w) dv(u)dv(w) Ω

Ω

is less than or equal to 2 2 2 2 C ϕ (u) − ϕ (w) kzΩ (u) kzΩ (w) dv(u)dv(w) Ω

Ω

since Ω ⊂ Ω , the definition of ϕ and the fact that on G ⊂ Ω ⊂ Ω we have Ω 2 k (u) ≤ C · k Ω (u)2 and this completes the proof of the first inequality. z z We now prove that MO(Ω ) (ϕ , z)p ≤ C2, · MO(Ω) (ϕ, z)p + (||z − a | − r |)p/2 . To start let us note that 2MO(Ω ) (ϕ , z)2

= Ω

Ω

= Ω

+2 ·

Ω

2 2 2 ϕ (u) − ϕ (w) kzΩ (u) kzΩ (w) dv(u)dv(w)

2 2 2 |ϕ(u) − ϕ(w)| kzΩ (u) kzΩ (w) dv(u)dv(w)

(Ω \Ω)×Ω

ϕ(w) · kzΩ (u) · kzΩ (w)2 dv(u)dv(w).

228

Raimondo

IEOT

We also observe that, see Lemma 1, if z ∈ G we have Ω Ω 1 kz (u) ≤ C · kz (u) + K Ω (z, z) and for z ∈ Ω \Ω there is a constant M such that Ω kz (u) ≤ M · (||z − a | − r |) and this implies that 2MO(Ω ) (ϕ , z)2

(Ω) 2 2 2 2 ≤ C · MO (ϕ, z) + (||z − a | − r |) |ϕ| (z) + (||z − a | − r |) . Since

|ϕ|2 (z) ≤ MO(Ω) (ϕ, z)2 + M · (||z − a | − r |)−1/2

then, if p ≥ 2,

p/2 2p/2 MO (Ω ) (ϕ , z)p ≤ C p/2 · MO(Ω) (ϕ, z)2 + (||z − a | − r |) p/2 ≤ max(1, 2p/2−1 )C p/2 · MO(Ω) (ϕ, z)p + |||z − a | − r ||

and the last inequality implies that, for any p ∈ [2, ∞), we have MO(Ω ) (ϕ , z)p

≤ 2−p/2 max(1, 2p/2−1 ) · C p/2 · MO(Ω) (ϕ, z)p + (|||z − a | − r ||p/2

if we define

p = 2−p/2 max(1, 2p/2−1 ) · C p/2 C2, we obtain the second inequality. To complete the proof we observe that p/2

2 (Ω) 2 I = K Ω (z, z)dv(z) |ϕ| (z) − |ϕ(z)| G

is less than



p C1,

G

 (Ω ) 2 p/2 2 (Ω ) ϕ (z) − ϕ (z)  K Ω (z, z)dv(z)

and since there exists a constant C3 such that   (Ω ) 2 p/2 2 (Ω ) ϕ (z) − ϕ (z)  K Ω (z, z)dv(z) G

=

C3

Ω

  (Ω ) 2 p/2 2 (Ω ) ϕ (z) − ϕ (z)  K Ω (z, z)dv(z)

Vol. 62 (2008)


229

we obtain 

p C3 I < C1,

Ω

 (Ω ) 2 p/2 2 (Ω ) ϕ (z) − ϕ (z)  K Ω (z, z)dv(z).

Finally, we also have for some α ∈ (0, ∞)   (Ω ) 2 p/2 2 (Ω ) (Ω ) ϕ I = (z) − ϕ (z)  K Ω (z, z)dv(z) G

= αC3−1

2 the integral G (||z − ai | − ri |p/2 K Ω (z, z)dv(z) is finite we can conclude that p/2

2 (Ω) 2 I = K Ω (z, z)dv(z) |ϕ| (z) − |ϕ(z)| G

is finite if and only if   (Ω ) 2 p/2 2 (Ω ) ϕ (z) − ϕ (z)  K Ω (z, z)dv(z) Ω

is finite and this completes our proof.

Lemma 4.4. Let ϕ ∈ L∞ (Ω), define ϕ = ϕ · p where j = 1, . . . , n and P is a ∂-partition for Ω and ϕ as in the previous lemma. Then: 1. The inequality Ω

p/2 2 2 K Ω (z, z)dv(z) < ∞ ϕ (z) − ϕ (z)

holds for any = 1, . . . , n.

230

Raimondo

IEOT

2. The inequality p/2

2 2 ϕ (z)| K Ω (z, z)dv(z) < ∞ |ϕ | (z) − | Ω

holds for any = 1, . . . , n. Proof. It is sufficient to observe that the fact that P is a ∂-partition and the definition of ϕ imply that, for any , there a a set U = {ζ ∈ C : r + δ > |ζ − a | > r } ⊂ Ω such that ϕ (z) = ϕ (z) ∀z ∈ U We are now ready to prove the main result. Theorem 4.5. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂-partition for Ω. Then the operators Hϕ , Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are in the Schatten-von Neumann p-class if and only if for = 1, . . . , n the operators Hϕ , Hϕ : L2a (Ω , dν) → L2 (Ω , dν) L2a (Ω , dν) are in the Schatten-von Neumann class. Proof. We know that [3] implies that the operators Hϕ , Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are in the Schatten-von Neumann p-class if and only if p/2

2 |ϕ|2 (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

holds and we know, see Lemma 3 and Lemma 4, that this is equivalent to the fact that p/2 (Ω ) (Ω ) 2 2 ϕ |ϕ | (z) − (z) K Ω (z, z)dv(z) < ∞ Ω

for any = 1, . . . , n and this is equivalent, see [8] , to the fact that Hϕ , Hϕ : L2a (Ω , dν) → L2 (Ω , dν) L2a (Ω , dν) are in the Schatten-von Neumann classes Sp .

Finally, we concentrate our attention to the study of the operator Hϕ when ϕ ∈ Hol (Ω)={h : Ω → C |h is halomorphic } and we assume that sup |ϕ(w)| < ∞. w∈Ω

Vol. 62 (2008)


231

This situation has been already studied by Arazy, Fisher and Peetre. It is worth to notice that our result implies the following Theorem 4.6. Let ϕ ∈ Hol (Ω) ∩ L∞ (Ω) then the following are equivalent: 1. The operator Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) is in Sp . 2. The function ϕ satisfies the following Ω

p/2 (Ω ) (Ω ) 2

2 ϕ (z) − (z) K Ω (z, z)dv(z) < ∞ |ϕ |

for any = 1, . . . , n Proof. It is is enough to observe that Hϕ = 0 if ϕ ∈ Hol (Ω) ∩ L∞ (Ω) and to apply the main Theorem of this paper.

Acknowledgment I wish to thank Professor Donald Sarason for a very helpful discussion.

References [1] J. Arazy, Membership of Hankel Operators on Planar Domains in Unitary Ideals, Analysis at Urbana, vol.1, London Math. Soc. Lecture Notes Ser. 137, Cambridge University Press, 1989, 1–40 [2] J. Arazy and S. Fisher, Hankel Operators on Planar Domains, Constr. Approx. 6 (1990), 113–138 [3] F. Beatrous and S.Y. Li , Trace Ideal Criteria for Operators of Hankel Type, Illinois J. Math. 39 (1995), 723–754. [4] J.B. Conway, Functions of One Complex Variable II, Springer-Verlag, 1991. [5] G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Trans. of Math. Monographs 26, AMS, Providence, R.I., 1969. [6] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974. [7] R.C. Raimondo, Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains, Integral equations and Operator Theory 57 (2007), 425–449. [8] K. Zhu, Schatten class Hankel operators on the Bergman Space of the Unit Ball, Amer. J. Math. 113 (1991), 147–167.

232

Raimondo

Roberto C. Raimondo University of Melbourne 3010 Parkville, Victoria Australia and Department of Statistics Mathematics Group Universit` a degli Studi di Milano-Bicocca Via Bicocca degli Arcimoboldi, 8 20126 Milano Italy e-mail: [email protected] Submitted: March 28, 2007. Revised: June 26, 2008.

IEOT



Hankel Operators on Bergman Spaces of Tube Domains over Symmetric Cones Benoit Florent Sehba Abstract. We present here some criteria for Schatten-Von Neumann class membership for the small Hankel operator on Bergman space A2 (TΩ ), when TΩ is the tube over the symmetric cone Ω. Mathematics Subject Classification (2000). Primary 47B35; Secondary 32A37, 46E22, 47B10. Keywords. Bergman space, Besov space, Hankel operator, Schatten class.

1. Introduction Let Ω be an irreducible symmetric cone in the Euclidean vector space Rn of dimension n, endowed with an inner product (.|.) for which the cone Ω is self-dual. We denote by TΩ = Rn + iΩ the corresponding tube domain in Cn . As in the text [9] , we shall write the rank and determinant associated with the cone by r = rank Ω, and ∆(x) = det x, x ∈ Rn . As usual we shall denote by 2z the natural extension to the complex space Cn = Rn + iRn of the generalized wave operator 2x on the cone Ω: 1 ∂ ), i ∂z which is the differential operator of degree r defined by the equality: 2z = ∆(

1 ∂ )[ei(z|ξ) ] = ∆(ξ)ei(z|ξ) , ξ ∈ Rn . i ∂z We will sometimes use the notation 2 when there is no ambiguity. ∆(

The author would like to thank professor Aline Bonami for helpful advices.

234

Sehba

IEOT

Given 1 ≤ p < ∞ and ν > 2 nr − 1, the weighted Bergman space Apν (TΩ ) of the tube TΩ is the space of analytic functions f on TΩ satisfying the integrability condition ||f ||Apν := (

1

n

|f (x + iy)|p ∆ν−2 r (y)dxdy) p < ∞.

(1.1)

TΩ

When ν = 2 nr , we write A2 (TΩ ) = A22 n (TΩ ). r For 1 ≤ p < ∞, the Besov space B p (TΩ ) of the tube TΩ is the space of holomorphic functions f so that |2n f (x + iy)|p ∆(y)np−2n/r dxdy < ∞. TΩ

In other words, f belongs to B p (TΩ ) if and only if 2n f belongs to Apnp (TΩ ). When p = ∞, we denote the Bloch space of TΩ by B = B ∞ , which is the space of analytic functions f satisfying sup ∆n (z)|2n f (z)| < ∞. z∈TΩ

The weighted Bergman projection Pν is the orthogonal projection from the Hilbert space L2ν (TΩ ) onto its closed subspace A2ν (TΩ ) and it is given by the integral formula n

Bν (z, w)f (w)∆ν−2 r (w)dV (w)

Pν f (z) =

(1.2)

TΩ

where

z−w ) (1.3) i is the weighted Bergman kernel (see [3]) and dV is the Lebesgue measure on TΩ . Let us recall that Bν is a reproducing kernel in A2ν (TΩ ), that is for every f ∈ A2ν (TΩ ) we have the formula: n f (z) = Bν (z, w)f (w)∆ν−2 r (w)dV (w). (1.4) Bν (z, w) = dν ∆−ν (

TΩ

In fact, it is easy to check that for any µ > 2 nr − 1, and any f ∈ A2ν (TΩ ) we still have n Bµ (z, w)f (w)∆µ−2 r (w)dV (w). (1.5) f (z) = TΩ

Again, when ν = 2 nr , we write P = Pν and B = Bν . Let b ∈ L2 (TΩ ) = L2 (TΩ , dV ). The small Hankel operator hb on the Bergman space A2 (TΩ ) with symbol b is defined as hb (f ) = P (bf ) 2

(1.6)

for f ∈ A (TΩ ). The aim of this work is to give criteria for Schatten class membership of Hankel operators on the Bergman space A2 (TΩ ). This problem has been considered in [2], [11] for the case of the unit disc of the complex plane, and in [17] and [18] for bounded symmetric domains. Some earlier works were done in [1], [8], [10], [12] and [13] in various domains including the upper half plane. It is shown in those

Vol. 62 (2008)

Hankel Operators on Bergman Spaces

235

cases that the small Hankel operator is in the Schatten class Sp if and only if its symbol belongs to the corresponding Besov space B p . The idea of the proof in [18] is the use of an appropriate integral operator which carries a lot of information on the small Hankel operator. This idea seems to be the appropriate one in our case also. Let us mention that the same problem for Hardy space of tube domains over symmetric cones was considered in [7] where it is stated that classical result extends to this case at least for 1 ≤ p ≤ 2, this work is extended in [6]. The main tool in the proof of the necessity in [7] is the use of the sampling theorem related to a lattice in TΩ for functions in a Bergman space. We will also take advantage of this idea. We show specially that classical results (see [18] for example) extend to the tube domains over symmetric cones for the range 1 ≤ p ≤ ∞. When the symbol is analytic and 1 ≤ p ≤ ∞, we also obtain criteria in terms of the action of the operator on the reproducing kernel, here, “the reproducing kernel thesis”. This last characterization appears in [15] for the same problem in the case of Hardy space of the unit disc. Our main result can be stated in the following way. Theorem 1.1. Suppose b is analytic in TΩ and 1 ≤ p ≤ ∞. Then the following conditions are equivalent: i) hb ∈ Sp . ii) b ∈ B p . iii) For all integers k > nr − 1, ||hb (2k B)(., z)||p ∆(k+n/r)p−2n/r (z)dV (z) < ∞. TΩ

Here the norm || || is the norm of the Hilbert space on which the operator acts, that is A2 (TΩ ). Remark that Condition iii) is still valid when b is not analytic since it only depends on the operator. One can weaken the conditions on k. We will see, in particular, that it is sufficient to consider k ≥ 0 when p ≥ 2. The optimization of the number of derivatives (here the exponent of 2) in the definition of the Besov space B p is considered in the work [5]. In fact to have a norm on this space, we have to see it as a space of equivalence classes. This last difficulty has no effect here since the operator does not depend on the choice of the representantive of a class. Throughout the paper C, Ck , Cn,k , . . . will denote positive constants depending only on the displayed parameters but not necessarily the same at distinct occurrences.

2. Preliminaries We adopt the following notation n

Bz (.) = B(., z) = d∆−2 r (

.−z ). i

236

Sehba

IEOT

We recall that A2 (TΩ ) is a Hilbert space with reproducing kernel Bz , that is for every f ∈ A2 (TΩ ), < f, Bz >= f (z) with the pairing < f, g >= f (z)g(z)dV (z), TΩ

dV being the Lebesgue measure on TΩ . Let us recall the following useful result. Lemma 2.1. (Lemma 4.12 in [3]) Let α be real, y and t in Ω. The following assertions hold: 1) The integral

Jα (y) =

Rn

|∆−α (

x + iy )|dx i n

converges if and only if α > 2 nr − 1. In this case, Jα (y) = Cα ∆−α+ r (y), where Cα is a constant depending on α. p 2) The function f (z) = ∆−α ( z+it i ), with t ∈ Ω, belongs to Aν (TΩ ) if and only if α > max(

2 nr − 1 ν + nr − 1 , ). p p

In this case, ||f ||Apν = Cα,p ∆−pα+ν (t). n

It follows easily that ||Bz ||A2 = C∆−( r ) (z). Let us recall that the wave operator acts on the reproducing kernel in the following way: 2kz Bα (., z) = Cn,α,k Bα+k (., z) with Bα given by the formula (3)(see [7]). It follows easily that for any f ∈ A2 (TΩ ), m < f, 2m z Bz >= Cm 2 f (z).

In particular, n

m m 1/2 = Cm ∆−(m+ r ) (z). ||2m z Bz ||A2 = | < 2z Bz , 2z Bz > |

For any f ∈ A2 (TΩ ), we denote by f˜, the normalization of f , that is f˜ = f /||f ||A2 (TΩ ) .

(2.1)

Let H1 and H2 be two Hilbert spaces. Let B(H1 , H2 ) and K(H1 , H2 ) denote the spaces of bounded and compact linear operators from H1 to H2 , respectively. It is well known that any operator T ∈ K(H1 , H2 ) has a Schmidt decomposition, that is there exist orthonormal bases {ej } and {σj } of H1 and H2 respectively and a sequence {λj } of complex numbers with λj → 0, such that Tf =

∞ j=0

λj < f, ej > σj ,

f ∈ H1 .

(2.2)

Vol. 62 (2008)


237

For 1 ≤ p < ∞, a compact operator T with such a decomposition belongs to the Schatten-Von Neumann p-class Sp (H1 , H2 ), if and only if ∞ 1 |λj |p ) p < ∞. ||T ||Sp = ( j=0

When T ∈ Sp = Sp (H, H), for any image {ej } of an orthonormal sequence by a bounded operator in H, ∞

| < T ej , ej > |p ≤ ||T ||pSp

j=0

(see [14]). For p = 1, S1 = S1 (H, H) is the trace class and for T ∈ S1 , the trace of T is defined by ∞ < T ej , ej > T r(T ) = j=0

where {ej } is any orthonormal basis of the Hilbert space H. We will denote by S∞ (H) the set of all bounded linear operators on H. Finally, we recall the following covering lemma and the sampling theorem. Lemma 2.2. Given δ ∈ (0, 1), there exists a sequence of points {zj } in TΩ called δ-lattice such that, calling Bj and Bj the Bergman balls with center zj and radius δ and δ/2 respectively, then (i) the balls Bj are pairwise disjoint; (ii) the balls Bj cover TΩ with finite overlapping, i.e. there is an integer N such that each point of TΩ belongs to at most N of these balls.

Proof. See [7]. The above balls have the following properties: dV (z) ≈ dV (z) ≈ Cδ ∆2n/r (zj ). Bj

Bj

We recall that the measure dλ(z) = ∆−2n/r (z)dV (z) is invariant measure on TΩ . Lemma 2.3. (Theorem 5.6 in [3]) Let {zj } be a δ-lattice in TΩ , δ ∈ (0, 1). (i) There exists a positive constant Cδ such that every f ∈ Apν (TΩ ) satisfies |f (zj )|p ∆ν (zj ) ≤ Cδ ||f ||pApν . j

(ii) Conversely, if δ is small, there is a positive constant Cδ such that every f ∈ Apν (TΩ ) satisfies |f (zj )|p ∆ν (zj ). ||f ||pApν ≤ Cδ j

238

Sehba

IEOT

3. Sp criterias for arbitrary operators on A2 (TΩ ) 3.1. Hilbert-Schmidt operators These are operators in S2 . The following result is established for an arbitrary operator defined on A2 (TΩ ) with values in a Hilbert space H. Theorem 3.1. If T ∈ B(A2 (TΩ ), H) then ||T ||2S2 (A2 (TΩ ),H)

= Cn,k TΩ

k B )||2 dλ(z), ||T (2 z z

n

for every integer k ≥ 0, where dλ(z) = ∆−2 r (z)dV (z) is the invariant measure on TΩ . Proof. If {ej } is an orthonormal basis of H, then ∞ ||T (2kz Bz )||2 ∆2k (z)dV (z) = | < T (2kz Bz ), ej > |2 ∆2k (z)dV (z) TΩ

=

TΩ j=0 ∞ j=0

=

∞ j=0

=

TΩ

Cn,k

| < 2kz Bz , T ∗ ej > |2 ∆2k (z)dV (z) |2k T ∗ ej (z)|2 ∆2k (z)dV (z)

TΩ

∞ j=0 ∞

|T ∗ ej (z)|2 dV (z)

TΩ

||T ∗ ej ||2A2

=

Cn,k

=

Cn,k ||T ∗ ||2S2 = Cn,k ||T ||2S2 .

j=0

The fourth equality follows from the fact that 2k is an isometric (up to constant Cn,k ) isomorphism from A2 (TΩ ) onto A22k+2n/r (TΩ ). 3.2. Sp (A2 (TΩ ), H) for p = 2 Lemma 3.2. Suppose that T ∈ B(A2 (TΩ ), H) for any Hilbert space H and k = 0, 1, . . .. Then: i) If T ∈ Sp for 2 < p < ∞ then, p k B )||p dλ(z) ≤ C ||T (2 n,k ||T ||Sp . z z TΩ

ii) If for 1 ≤ p < 2,

TΩ

k B )||p dλ(z) < ∞ ||T (2 z z

Vol. 62 (2008)


then T ∈ Sp . Moreover, ||T ||pSp

≤ Cn,k TΩ

239

k B )||p dλ(z). ||T (2 z z

Proof. First of all, we have by Theorem 3.1 that if T ∈ S1 (A2 (TΩ ), A2 (TΩ )) is a positive operator, then T r(T ) = ||T 1/2 ||2S2 = Cn,k < T (2kz Bz ), 2kz Bz > ∆2k (z)dV (z). TΩ

The result follows since in L2 (TΩ ), we have

||T ||pSp

= T r((T ∗ T )p/2 ) and for any unit vector(see [16])

< T ∗ T f, f >p/2 ≤ < (T ∗ T )p/2 f, f >, if p > 2 and

< (T ∗ T )p/2 f, f >≤ < T ∗ T f, f >p/2

if 1 ≤ p ≤ 2.

4. The case of the small Hankel operators We give in this section some Schatten classes membership criteria for the small Hankel operator on the Bergman space A2 (TΩ ). Let Vk and L be the operators defined on L2 (TΩ ) by n Vk f (z) = ∆2k+2 r (z) B(2k+4 nr ) (z, w)f (w)dV (w), z ∈ TΩ , TΩ

and Lf (z) = ∆n (z)

TΩ

Bn+2 nr (z, w)f (w)dV (w), z ∈ TΩ .

We set τz = B( n2 + nr ) (., z). We have the following lemma. Lemma 4.1. Let be the inner product in L2 (TΩ ). For f ∈ L2 (TΩ ), we have k B ), 2 k B > and Lf (z) = C < h τ˜ , τ˜ >. 1) Vk f (z) = Cn < hf (2 n f z z z z z z 2) P Vk f = P Lf = P f . 3) hf = hP f = hVk f = hLf on A2 (TΩ ). Proof. 1) follows from the definition of Vk and L, Fubini’s theorem and reproducing formulas. Let us show 2). It is not hard to see that the operators Vk and L are bounded on L2 (TΩ ) (see [4] for example). We consider the following function: z − ξ −(n+2n/r) ξ − w )∆ )f (w)∆n (ξ), f ∈ L2 (TΩ ), ( i i and z ∈ TΩ . Using the reproducing formula, we obtain that z−ξ )Lf (ξ) := Gz (ξ), F (ξ, w)dV (w) = ∆−2n/r ( i TΩ Fz (ξ, w) = ∆−2n/r (

240

Sehba

and

F (ξ, w)dV (ξ) = ∆−2n/r (

TΩ

IEOT

z−w )f (w) := Hz (w). i

It is clear that Hz is integrable and so is Gz since the operator L is bounded on L2 (TΩ ). Applying Fubini’s theorem, we obtain P Lf (z) = Gz (ξ)dV (ξ) = Hz (w)dV (w) = P f (z). TΩ

TΩ

The equality P Vk f = P f follows in the same way. The first equality in 3) follows from the definition of the little Hankel operator, Fubini’s theorem and reproducing formulas, the second and the third equalities follow from the first one and 2). Lemma 4.2. If 1 ≤ p ≤ ∞ and b ∈ Lp (TΩ , dλ), then the Hankel operator hb is in the Schatten class Sp . Proof. The case p = ∞ is obvious, it suffices then to show the case p = 1 since the result then follows by interpolation. It is not hard to see that b(w)hfw dλ(w), hb = TΩ n

n

n

where fw (z) = ∆2 r (z)∆2 r (w)∆−4 r ( z−w i ) and hfw is the rank 1 Hankel operator given by n

n

hfw g = ∆2 r (w)∆−2 r (

.−w )g(w) i

with ||hfw ||S1 = ||hfw || = C < ∞. It follows that ||hb ||S1 ≤ ||hfw ||S1 |b(w)|dλ(w) ≤ C TΩ

|b(w)|dλ(w).

TΩ

The proof is complete.

Theorem 4.3. Suppose 1 ≤ p ≤ ∞, and b ∈ L2 (TΩ ). Then the following assertions are equivalent: i) hb is in Sp . ii) For every integer k ≥ 0, Vk b ∈ Lp (TΩ , dλ). Proof. ii)⇒i) follows from Lemma 4.2 and the equality hb = hVk b . Let us show that for 1 ≤ p < ∞, i)⇒ii). Let {zj } be a δ-lattice in TΩ . Using k B ), 2 k B > and Lemma 2.3, we obtain the equality V b(z) = C < h (2 k

n,k

b

z

z

z

z

Vol. 62 (2008)


||Vk b||pLp (TΩ ,dλ)

=

Cn,k TΩ

=

Cn,k TΩ

≈

Cn,k

241

k B ), 2 k B > |p dλ(z) | < hb (2 z z z z | < hb (2kz Bz ), 2kz Bz > |p ∆2k (z)dV (z) n

| < hb (2kz Bzj ), 2kz Bzj > |p ∆2k+2 r (zj )

j

=

Cn,k

p k B ), 2 kB | < hb (2 z zj z zj > |

j

.

.−z k+n/r kB To conclude, it suffices to show that 2 (zj )∆−(k+2n/r) ( i j ) z zj = Ck ∆ 2 is the image of an orthonormal sequence ψj in L (TΩ ) through a bounded linear map Tk : L2 (TΩ ) → L2 (TΩ ). Define Tk : L2 (TΩ ) → L2 (TΩ ) by setting n ∆−(k+2 r ) (z − ξ)ψ(ξ)∆k (ξ)dV (ξ), z ∈ TΩ Tk ψ(z) = Cn,k TΩ

k B , ||ψ || 2 and ψj (z) = Cn,k ∆−n/r (z)χBj (z). Then Tk ψj = 2 j L (TΩ ) = 1 with a z zj good choice of Cn,k . The operator Tk = Ck Pk+2 nr is clearly bounded on L2 (TΩ ) (see [4]). k B . It follows that if h is For p = ∞, we take as test functions f = g = 2 z

bounded on A2 (TΩ ), then

z

z

z

b

k B ), 2 kB > | = C |Vk b(z)| = Cn,k | < hb (2 n,k | < hb (fz ), gz > | < ∞. z z z z So Vk b ∈ Lp (TΩ , dλ) for any 1 ≤ p ≤ ∞. The proof is complete.

Theorem 4.4. Suppose 1 ≤ p ≤ ∞, and b ∈ L2 (TΩ ) is analytic. Then the following assertions are equivalent: i) hb is in Sp . ii) b ∈ B p . Proof. ii)⇒i) follows from Lemma 4.2 and the equalities Lb(z) = ∆n (z)2n b(z) and hb = hLb . Let show that i)⇒ii) for 1 ≤ p < ∞. Let {zj } be a δ-lattice in TΩ . Using the equality Lb(z) = Cn < hb τ˜z , τ˜z > and Lemma 2.3, we have | < hb τ˜z , τ˜z > |p dλ(z) ||2n b||pApnp = ||Lb||pLp (TΩ ,dλ) = Cn TΩ n | < hb (τz ), τz > |p ∆np−2 r (z)dV (z) = Cn j

≈

C

Bj

| < hb (τzj ), τzj > |p ∆np (zj )

j

=

C

j

| < hb τ˜zj , τ˜zj > |p .

242

Sehba

IEOT

To conclude, it suffices to show that τ˜zj is the image of an orthonormal sequence ϕj in L2 (TΩ ) trough a bounded linear map T : L2 (TΩ ) → L2 (TΩ ). Define T : L2 (TΩ ) → L2 (TΩ ) by setting n n n n ∆−( 2 + r ) (z − ξ)ϕ(ξ)∆( 2 − r ) (ξ)dV (ξ), z ∈ TΩ T ϕ(z) = Cn TΩ

and ϕj (z) = Cn ∆−n/r (z)χBj (z). Then T ϕj = τ˜zj , ||ϕj ||L2 (TΩ ) = 1 with a good choice of Cn . The operator T = Cn P( n2 + nr ) is bounded on L2 (TΩ ) (see [4]). The case p = ∞ can be handled easily as in Theorem 4.3 taking as test functions fz = gz = τ˜z . The proof is complete.

5. The reproducing kernel thesis In this section, we give Schatten class criteria for the little Hankel operator on the Bergman space A2 (TΩ ) in terms of the action of the operator on the reproducing kernel. Theorem 5.1. Let b ∈ A2 (TΩ ). Then the following conditions are equivalent: i) hb is bounded on A2 (TΩ ). ii) For every integer k ≥ 0, k B )|| 2 sup ||hb (2 z z A (TΩ ) < ∞.

z∈TΩ

Proof. That i)⇒ii) is obvious. Let show that ii)⇒i). For that, it suffices by Theorem 4.3 to show that ii) implies that supz∈TΩ |Vk b(z)| < ∞. But, we already know that k B ), 2 kB > . V b(z) = C < h (2 k

n,k

b

z

z

z

z

The result follows now since k B )|| 2 ||2 k B || 2 = C k |Vk b(z)| ≤ Cn,k ||hb (2 n,k ||hb (2z Bz )||A2 . z z A z z A

The proof is complete.

Theorem 5.2. Let b ∈ L2 (TΩ ) be analytic and 1 ≤ p < ∞. The following conditions are equivalent: i) hb ∈ Sp . ii) For every integer k > nr − 1, k B )||p dλ(z) < ∞. ||hb (2 z z TΩ

Vol. 62 (2008)


243

Proof. To show that ii)⇒i), it suffices by Theorem 4.3 to prove that ii) implies that Vk b ∈ Lp (TΩ , dλ) and this follows easily from the inequality k B ), 2 kB > | |Vk b(z)| = Cn,k | < hb (2 z z z z k B )|| 2 ||2 k B || 2 ≤ Cn,k ||hb (2 z z A z z A k B )|| 2 . = Cn,k ||hb (2 z z A That i)⇒ii) for 2 ≤ p < ∞, follows from part i) of Lemma 3.2. It remains to prove that i)⇒ii) for the range 1 ≤ p < 2. Let us first show the implication for p = 1. By Theorem 4.3, it suffices to show that if Vk b ∈ L1 (TΩ , dλ) then ii) holds. We recall that hb = hVk b and that the following representation holds: hVk b = Vk b(w)hfw dλ(w), n

TΩ

n

n

where fw (z) = ∆2 r (z)∆2 r (w)∆−4 r ( z−w i ) and hfw is the rank 1 Hankel operator given by n n . − w hfw g = ∆2 r (w)∆−2 r ( )g(w). i It follows that n . − w n z−w k B ) = ∆2 n r (w)∆−2 r ( hfw (2 )∆k+ r (z)∆−(k+2n/r) ( ). z z i i Using Lemma 2.1, we obtain k B )|| = C∆n/r (w)∆k+n/r (z)|∆−(k+2n/r) ( z − w )|. ||hfw (2 z z i It follows using Lemma 2.1 again that k B )||dλ(z) ||hfw (2 z z TΩ z−w )|∆(k+n/r)−2n/r (z)dV (z) |∆−(k+2n/r) ( = C∆n/r (w) i TΩ = Cn,k < ∞. We obtain finally that k B )||dλ(z) ≤ ||hb (2 z z

≤

C

TΩ

|Vk b(w)|(

TΩ

TΩ

k B )||dλ(z))dλ(w) ||hfw (2 z z

|Vk b(w)|dλ(w). TΩ

k B ||, we have by the previNow, considering the sublinear operator H : b → ||hb 2 . . ous and Theorem 4.3 and Theorem 4.4 that H is bounded from B 1 to L1 (TΩ , dλ). We also have by Theorem 3.1 and Theorem 4.4 that H is bounded from B 2 to L2 (TΩ , dλ). We deduce by interpolation that H is bounded from B p to Lp (TΩ , dλ), whenever 1 ≤ p ≤ 2. It follows that we have i)⇒ b ∈ B p ⇒ ii). The proof is complete.

244

Sehba

IEOT

From the first part of the proof of the above theorem, we have the following. Theorem 5.3. Let b ∈ L2 (TΩ ) and 2 ≤ p < ∞. The following conditions are equivalent: i) hb ∈ Sp . ii) For every integer k ≥ 0, TΩ

k B )||p dλ(z) < ∞. ||hb (2 z z

References [1] J. Arazy, Boundedness and compactness of generalized Hankel operators on bounded symmetric domains. J. Funct. Anal. 137 (1996), no. 1, 97–151. [2] J. Arazy, S. Fisher and J. Peetre, Hankel operators on weighted Bergman spaces. Amer. J. Math. 110 (1988), 989–1054. ´ s, C. Nana, M.M. Peloso and F. Ricci, [3] D. B´ ekoll´ e, A. Bonami, G. Garrigo Bergman projectors in tube domains over cones: an analytic and geometric viewpoint. In: Lecture Notes of the Workshop “Classical Analysis, Partial Differential Equations and Applications”, Yaoundé, December 10–15, (2001). Available at http://www.harmonic-analysis.org. ´kolle ´, A. Bonami, M.M. Peloso and F. Ricci, Boundedness of Bergman [4] D. Be projections on tube domains over light cones. Math. Z. 237 (2001), 31–59. ´kolle ´, A. Bonami, G. Garrigo ´ s, F. Ricci and B.F. Sehba, Analytic Besov [5] D. Be spaces and Hardy-type inequalities in tube domains over symmetric cones. Preprint 2007. [6] A. Bonami, S. Grellier, C. Nana and B.F. Sehba, Schatten classes of Hankel operators on tube domains. Preprint 2007. [7] A. Bonami and M. Peloso, Hankel operators on Hardy spaces of two classical domains. Functional analysis, VI (Dubrovnik, 1999), 7–18, Various Publ. Ser. (Aarhus), 45, Univ. Aarhus, Aarhus, 2000. [8] R.R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in Lp . Representation theorems for Hardy spaces. Astérisque 77, 11–66, Soc. Math. France, Paris, 1980. ´ nyi, Analysis on symmetric cones. Clarendon Press, Ox[9] J. Faraut and A. Kora ford, 1994. [10] J.S. Howland, Trace class Hankel operators. Quart. J. Math. (2) 22 (1971), 147– 159. [11] S. Janson, J. Peetre and R. Rochberg, Hankel forms and Fock space. Revista Ibero-Amer. 3 (1987), 61–138. [12] V.V. Peller, Hankel operators of the class Sp and their applications (Rational approximation, Gaussian processes, the problem of majorizing operators). Mat. Sb. 113 (1980), 538–581 (Russian). (English translation: Math. USSR- Sb 41 (1980), 443–479).

Vol. 62 (2008)


245

[13] R. Rochberg, Trace ideal criteria for Hankel operators and commutators. Indiana Univ. Math. J. 31 (1982), no. 6, 913–925. [14] R. Rochberg and S. Semmes, Nearly weakly orthonormal sequences, singular values estimates, and Calderon-Zygmund operators. J. Funct. Analysis 86 (1989), 237–306. [15] M.P. Smith, Testing Schatten class Hankel operators and Carleson embeddings via reproducing kernels. J. London Math. Soc. (2) 71 (2005), no. 1, 172–186. [16] K. Zhu, Operator theory in function spaces. Marcel Dekker, New York, 1990. [17] K. Zhu, Hankel operators on the Bergman space of bounded symmetric domains. Trans. of the Amer. Math. Soc. Vol. 324, no. 2, April 1991. [18] K. Zhu, Holomorphic Besov spaces on bounded symmetric domains. Quart. J. Math. Oxford (2), 46 (1995), 239–256. Benoit Florent Sehba Department of Mathematics University of Glasgow G12 8QW Glasgow United Kingdom e-mail: [email protected] Submitted: March 30, 2007. Revised: June 15, 2008.



On Properties of the ξ-Function in Semi-finite von Neumann Algebras Anna Skripka Abstract. Versions of the Birman-Schwinger principle for (relative) trace class perturbation problems of dissipative operators in a semi-finite von Neumann algebra and self-adjoint operators affiliated with the algebra are obtained and applied in the study of the spectral shift function. Mathematics Subject Classification (2000). Primary 47A55, 47C15; Secondary 46L52. Keywords. Spectral shift function, dissipative operator, semi-finite von Neumann algebra.

1. Introduction The Lifshits-Krein spectral shift function (ξ-function), a fundamental object in the spectral perturbation theory, originally appeared in the physical literature in 1952 (cf. [28]). Since then, a comprehensive mathematical theory for this object has been constructed and a number of applications of the ξ-function for trace class (resolvent) perturbations found. The mathematical ξ-theory was initiated by M. G. Krein in his ingenious paper [26]. On account of further development of the ξ-theory in the trace class setting, one can consult [9, 11, 42, 46] for survey exposition and references. The standard integer-valued dimensions traditionally employed in the study of the ξ-function constrain it to the case when a perturbation (or difference of resolvents) is in the trace class. In this paper we invoke the Murray and von Neumann continuous (also called relative) dimensions [30] that allow the ξ-function to compare the spectra of two operators whose difference (as well as difference of their resolvents) has a non-trivial continuous spectrum. We recall that existence of a dimension function with range the interval [0, 1] or [0, ∞] is an extraordinary property of factors of type II1 or II∞ , respectively. A factor is a von Neumann algebra (∗-algebra of bounded linear operators on a Hilbert space H containing the

248

Skripka

IEOT

identity operator and closed in the weak operator topology) with trivial center. In particular, the algebra B(H) of all bounded operators on H is a factor; it is said to be of type In if dim(H) < ∞ and type I∞ if dim(H) = ∞. Basic facts and definitions on the theory of von Neumann algebras can be found, for example, in [18, 24, 37, 44]. Each von Neumann algebra A is decomposable into a direct integral of factors. A von Neumann algebra is called finite when all factors in its direct integral decomposition are of type In and/or II1 and semi-finite when the factors in its direct integral decomposition are of type In , II1 , I∞ , and/or II∞ . If A acts on a separable Hilbert space, then A is (semi-)finite if and only if it has a faithful normal (semi-)finite trace τ . When A is either a factor of finite type In or II1 or semi-finite type I∞ or II∞ , the dimension function gives rise to a faithful normal finite or semi-finite trace τ , respectively, which is unique up to a normalization. In particular, when A = B(H), the trace τ coincides with the canonical trace. Not every von Neumann algebra admits a faithful normal semi-finite trace, but we restrict our attention to those von Neumann algebras that do. We recall that a trace is a functional τ : A+ → [0, ∞] initially defined on the non-negative elements A+ of A and satisfying τ (λA + µB) = λτ (A) + µτ (B) for A, B ∈ A+ , λ, µ ≥ 0 and τ (C ∗ C) = τ (CC ∗ ) for C ∈ A. A trace is called faithful if it does not annihilate non-zero elements of A+ , normal if τ (Aα ) ↑ τ (A) for each increasing net {Aα } ⊂ A+ converging to A in the strong operator topology, semi-finite if for each A ∈ A+ there is a net of elements with finite traces which increases to A. A trace τ with the properties described above extends uniquely to a functional on the relative τ -trace class ideal L1 (A, τ ) = {A ∈ A : τ (|A|) < ∞}, with extended τ satisfying τ (AB) = τ (BA) for A ∈ L1 (A, τ ), B ∈ A. If τ is finite on the whole A+ (and hence, on A), then it is called finite. Operator algebras, in particular von Neumann algebras, naturally arise in mathematical models of quantum mechanics. For instance, a discrete Laplacian ∆ corresponding to a lattice electron moving in a magnetic field with irrational flux is an element of a II1 factor, whose trace τ evaluated on the spectral projection E∆ ((−∞, λ)) of ∆ gives the value of the integrated density of states for ∆ at point λ ∈ R (cf. [41] and references cited therein). A random Schr¨ odinger operator on a manifold whose integrated density of states is expressible via the trace of a II∞ factor is discussed in [27] (cf. also [15, 40]). Resolvents of some Hamiltonians modelling the quantum Hall effect are known to belong to a semi-finite von Neumann algebra [12] (cf. also [5, 45]). More examples can be traced in [16]. The concept of the ξ-function for pairs of self-adjoint operators in a semifinite von Neumann algebra A was introduced by R. W. Carey and J. D. Pincus in their inventive paper [14]. A natural assumption on the perturbation made in [14] is that it is in the τ -trace class. Application of modern operator algebraic techniques in [3] allowed extension of the ξ-function to the case of unbounded self-adjoint operators affiliated with the algebra A, that is, self-adjoint operators

Vol. 62 (2008)

The ξ-Function in von Neumann Algebras

249

whose spectral projections belong to A. The setting in [3] includes the case of self-adjoint densely defined operators with difference in the standard trace class since those operators are affiliated with a semi-finite von Neumann algebra B(H), for which the τ -trace class coincides with the standard trace class. Study of the ξ-function in the context of von Neumann algebras has led to detection of interesting connections between some well-known concepts. For example, a generalization of the Birman-Solomyak spectral averaging formula obtained in [1, 29] (cf. [10] for the original statement) revealed that the ξ-function and the analytically defined spectral flow [34] (cf. aso [6]) are essentially the same notions [1, 2]. In its turn, this discovery helped to extend the range of situations for which the ξ-function can be defined [2]. Another prominent example is that an analog of the Birman-Schwinger principle for dissipative operators (that is, operators with non-negative imaginary part) in a finite von Neumann algebra obtained in [25] plays a principal role in the proof of a finite von Neumann algebra version of the Birman-Krein formula. We prove an extension of the generalized Birman-Schwinger principle in a gap originally attained for the case when A = B(H) [21] (cf. also [35, 36]). For the statement of the classical Birman-Schwinger principle we refer to [42] (cf. also [7, 39]). In the case of an arbitrary semi-finite algebra A this principle (see Theorem 3.1) states that for a self-adjoint operator H0 affiliated with A and its τ -trace class perturbation factored in the form KM −1 K ∗ , with M = M ∗ a boundedly invertible and K a τ -Hilbert-Schmidt (to be defined in the next section) operators in A, one has the equality of the spectral shift functions (1.1) ξ λ, H0 , H0 + KM −1 K ∗ = ξ 0, M + K ∗ (H0 − λ)−1 K, M , for a.e. λ in the joint resolvent set of the operators H0 and H0 + KM −1 K ∗ . The principle (1.1) admits further generalization (see Theorem 3.1) to the case when λ is in the spectrum of the initial and/or perturbed operator. We derive the relation (1.2) ξ(λ, H0 , H0 + KM −1 K ∗ ) = lim ξ M + K ∗ (H0 − λ − iε)−1 K, M , ε→0+

for a.e. λ ∈ R, with the ξ-index (cf. Definition 2.3), a parameter independent analog of the ξ-function associated with pairs of dissipative operators in A, on the right-hand side. In the case of bounded H0 , the ξ-function ξ(·, H0 , H0 +KM −1 K ∗ ) can also be represented via the traces of diagonal blocks of the boundary values of the Ξ-operators [20] (cf. also [13, 21]) associated with the dissipative operators on the right-hand side of (1.2) (see Theorem 3.3 and Remark 3.4 (ii)). Assuming, in addition, existence of the normal boundary values K ∗ (H0 − λ − i0)−1 K = lim+ K ∗ (H0 − λ − iε)−1 K, ε→0

a.e. λ ∈ R,

in the operator norm, with imaginary parts in the τ -trace class, we relate ξ(λ, H0 , H) to the abstract scattering operator associated with the pair (H0 − λ, H − λ) (see Theorem 4.3). The proof of this relation makes use of an analog of the Birman-Schwinger principle for dissipative operators (see Theorem 3.7), the

250

Skripka

IEOT

representation of the ξ-function via the Ξ-operators, and the de la Harpe-Skandalis determinant [23]. The extended Birman-Schwinger principle (1.2) is applicable in the proof of the Birman-Solomyak spectral averaging formula for non-linear paths of selfadjoint operators affiliated with a semi-finite von Neumann algebra A that are smooth in the norm · 1,∞ [43]. The proof of this formula for linear paths in both classical and broadened settings relied on the theory of double operator integrals [1, 17]. The case of non-linear paths of (bounded) operators in A was treated by the perturbation determinant approach [29], with employment of a generalized Dixmier-Fuglede-Kadison differentiation formula (see Lemma 2.5) and representation of the ξ-function via the ξ-index (see Corollary 2.4). The Birman-Solomyak formula and the principle (1.2) are applicable in proofs of trace inequalities [43].

2. Preliminaries Throughout the paper we assume that A is a semi-finite von Neumann algebra and τ a normal faithful semi-finite trace on it. Let Lp (A, τ ), 1 ≤ p < ∞, denote the 1 τ -trace ideal equipped with norm · p,∞ = · p + · , where A p = [τ (|A|p )] p and A is the operator norm of A. The ideal Lp (A, τ ) can be represented as Lp (A, τ ) ∩ A, where Lp (A, τ ) is the noncommutative Lp space associated with (A, τ ) (cf. [33]). A useful representation for the relative Hilbert-Schmidt ideal is L2 (A, τ ) = {A ∈ A : τ (|A|2 ) < ∞}. Similarly to the case of the standard trace, we have the factorization L1 (A, τ ) = L2 (A, τ ) · L2 (A, τ ). We use letter K to refer to an arbitrary operator in Lp (A, τ ) and M , N to refer to dissipative operators in A. We denote self-adjoint operators by H0 , V , H and unitary ones by U , S. Auxiliary self-adjoint and dissipative operators are denoted by A, B, C, and D. For the set of boundedly invertible dissipative operators in A we reserve the symbol DA . The ξ-function associated with a pair (H0 , H) of self-adjoint operators affiliated with A, where H is a τ -trace class perturbation of H0 , is defined as the function satisfying the statement of the theorem below. Theorem 2.1. ([3, Theorem 3.1]) Let H0 = H0∗ be an operator affiliated with A and V = V ∗ ∈ L1 (A, τ ). Denote H = H0 + V . Then, there exists a unique function ξ(·, H0 , H) ∈ L1 (R) such that for every f ∈ C 1 (R) whose derivative f admits the representation ∞ e−iλt dm(t), λ ∈ R, (2.1) f (λ) = −∞

with a finite (complex) Borel measure m(·) on R, ∞ τ f (H) − f (H0 ) = − ξ(λ, H0 , H)f (λ)dλ. −∞


Vol. 62 (2008)

251

Lemma 2.2. Let H0 be a self-adjoint operator affiliated with A and V a self-adjoint operator in L1 (A, τ ). Set H = H0 + V . Then, for a.e. λ ∈ R, 1 lim+ τ Im log(H − λ + iε) − Im log(H0 − λ + iε) , (2.2) ξ(λ, H0 , H) = π ε→0 where log(·) is the principal branch of the logarithm with the cut along the negative imaginary semi-axis. In addition, for a.e. λ ∈ ρ(H0 ) ∩ ρ(H) ∩ R, ξ(λ, H0 , H) = τ EH−λ (R− ) − EH0 −λ (R− ) , (2.3) where R− = (−∞, 0). − z), Imz > 0, where log(·) Proof. The function λ → log(λ is the principal branch of the logarithm with the cut along (−∞, 0], satisfies (2.1) since ∞ (λ − z)−1 = i exp(it(z − λ))dt. 0

Hence, the Krein trace formula (2.1) guarantees that ∞ ξ(λ, H0 , H) dλ. τ log(H − z) − log(H0 − z) = − λ−z −∞

(2.4)

Therefore, for a.e. λ ∈ R, 1 0 − λ − iε) − λ − iε) − Im log(H ξ(λ, H0 , H) = − lim τ Im log(H π ε→0+ 1 lim τ [Im log(H − λ + iε) − Im log(H0 − λ + iε)]. = π ε→0+ The proof of (2.3) in the case of an arbitrary (semi-finite) A goes along the lines of the proof of (2.3) in the case when A = B(H). Along with the ξ-function, we consider its spectral parameter independent analog for dissipative operators, the ξ-index. Similarly to the case of a finite A [25], we define the ξ-index via the Ξ-operators. We recall that the Ξ-operator Ξ(A) associated with an operator A in DA equals π1 Im log A, where the operator logarithm is provided by the Dunford-Riesz functional calculus (cf. [20, 25]). Whenever A is self-adjoint, Ξ(A) simplifies to the spectral projection EA (R− ). Definition 2.3. The ξ-index associated with a pair (A, B) of operators in DA satisfying B − A ∈ L1 (A, τ ) is defined by ξ(A, B) = τ [Ξ(B) − Ξ(A)]. As a straightforward consequence of Lemma 2.2, we obtain the following relation between the ξ-function and the ξ-index. Corollary 2.4. Let H0 be a self-adjoint operator in A and V a self-adjoint operator in L1 (A, τ ). Set H = H0 + V . Then, for a.e. λ ∈ R, ξ(λ, H0 , H) = lim ξ(H0 − λ + iε, H − λ + iε). ε→0+

252

Skripka

IEOT

In addition, for a.e. λ ∈ ρ(H0 ) ∩ ρ(H) ∩ R, ξ(λ, H0 , H) = ξ(H0 − λ, H − λ). We conclude this section with stating technical lemmata on differentiability and continuity of traces of operators functions. Proofs of these facts are demonstrated in Appendix A. Lemma 2.5. Let Ω be an open subset of C whose boundary consists of a finite number of rectifiable Jordan curves and U an open subset of C or R. Let X, Y : U → A be continuously differentiable in the operator norm functions such that σ(X(z)), σ(Y (z)) ⊂ Ω, for all z ∈ U, and either X ≡ I or X ≡ 0. Assume that the function K = Y − X attains its values in L1 (A, τ ) and is continuously differentiable in the norm · 1,∞ . Then, for an analytic function f with Ω in its domain of analyticity, the function f (Y (·)) − f (X(·)) is differentiable in the norm · 1,∞ and d d τ f (Y (z)) − f (X(z)) = τ f (Y (z)) − f (X(z)) (2.5) dz dz d d (2.6) = τ g(Y (z)) · Y (z) − g(X(z)) · X(z) , dz dz where g(z) =

d dz f (z).

Remark 2.6. Differentiation formulae (2.5)–(2.6) for trace class differences of nontrace class operators is an extension of the well-known differentiation formula for composite functions assuming their values in a finite von Neumann algebra [18, 19]. Lemma 2.7. Let X, Y : K → DA be two functions continuous in the operator norm on a compact set K ⊂ C+ . Assume that 0 ∈ ρ(X(z)) ∩ ρ(Y (z)), for all z ∈ K, and Y − X is continuous on K in the norm · 1,∞ . Then, log(Y (·)) − log(X(·)) is continuous in the norm · 1,∞ .

3. The Birman-Schwinger principle 3.1. The case of self-adjoint operators affiliated with A In this subsection, we prove versions of the Birman-Schwinger principle on the spectra of self-adjoint operators affiliated with the algebra A. Theorem 3.1. Let M be a self-adjoint boundedly invertible operator in A, H0 a self-adjoint operator affiliated with A, and K ∈ L2 (A, τ ). Then, for a.e. λ ∈ R, (3.1) ξ(λ, H0 , H0 + KM −1 K ∗ ) = lim ξ M + K ∗ (H0 − λ − iε)−1 K, M . ε→0+ In addition, for a.e. λ ∈ ρ(H0 ) ∩ ρ H0 + KM −1 K ∗ ∩ R, ξ λ, H0 , H0 + KM −1 K ∗ = ξ M + K ∗ (H0 − λ)−1 K, M . (3.2)


Vol. 62 (2008)

253

Proof. We introduce auxiliary operator-valued functions C+ z → M(z) = M + K ∗ (H0 − z)−1 K, C+ z → N (z) = H0 − z + KM −1 K ∗ ,

(3.3)

for which by direct computations M−1 (z) = M −1 − M −1 K ∗ N −1 (z)KM −1 .

(3.4)

Next, employing Lemma 2.5 yields d τ [log M(z) − log M ] = τ M−1 (z)K ∗ (H0 − z)−2 K . (3.5) dz Using (3.4) entails the representation τ M−1 (z)K ∗ (H0 − z)−2 K (3.6) −1 ∗ = τ M K (H0 − z)−2 K − τ M −1 K ∗ N −1 (z)KM −1 K ∗ (H0 − z)−2 K , which equals τ

KM −1 K ∗ (H0 − z)−2 − τ KM −1 K ∗ N −1 (z) KM −1 K ∗ (H0 − z)−2

(3.7)

due to the cyclicity of τ . Rewriting N (z) according to (3.3) and then applying the second resolvent identity imply (3.8) N −1 (z) KM −1 K ∗ (H0 − z)−1 −1 KM −1 K ∗ (H0 − z)−1 = H0 − z + KM −1 K ∗ −1 = (H0 − z)−1 − H0 − z + KM −1 K ∗ . Equalities (3.6)-(3.8) combined together ensure τ M−1 (z)K ∗ (H0 − z)−2 K

−1 (H0 − z)−1 , = τ KM −1 K ∗ H0 − z + KM −1 K ∗

(3.9)

which, in view of the cyclicity of τ and the second resolvent identity, coincides with −1

τ (H0 − z)−1 KM −1 K ∗ H0 − z + KM −1 K ∗ (3.10)

−1 = τ (H0 − z)−1 − H0 − z + KM −1 K ∗ = τ (H0 − z)−1 − N −1 (z) . Combining (3.5), (3.9), and (3.10) results in d τ [log M(z) − log M ] = −τ (H0 − z + KM −1 K ∗ )−1 − (H0 − z)−1 . dz

(3.11)

254

Skripka

IEOT

By the Krein trace formula (2.1),

−1 τ H0 − z + KM −1 K ∗ − (H0 − z)−1 ∞ ξ λ, H0 , H0 + KM −1 K ∗ = dλ. (λ − z)2 −∞ Combining (3.11) and (3.12) implies ∞ ξ λ, H0 , H0 + KM −1 K ∗ τ [log M(z) − log M ] = − dλ + C, λ−z −∞

(3.12)

(3.13)

with C a constant. Our immediate goal is to show that C = 0. We recall (cf. [20]) that the operator logarithm of A ∈ DA can be represented as the norm-convergent Riemann integral ∞ (3.14) (A + it)−1 − (1 + it)−1 I dt. log A = −i 0

From representation (3.14) one infers that log M(iy) − log M (3.15) ∞ =i (it + M(iy))−1 K ∗ (H0 − iy)−1 K(it + M )−1 dt. 0 Employing widely the estimate (A + iµ)−1 ≤ 1/µ, where A is a dissipative operator and µ > 0, yields that the integral in (3.15) converges in the norm · 1,∞ and τ [log M(iy) − log M ] = O(1/y),

y → ∞,

The asymptotic expansion ∞ ξ(λ, H0 , H) dλ = O(1/y), λ − iy −∞

λ ∈ R.

(3.16)

y → ∞,

along with (3.16) implies that the constant C in (3.13) equals zero. Thus, for a.e. λ ∈ R, 1 ξ λ, H0 , H0 + KM −1 K ∗ = − lim τ [Im log M(λ + iε) − Im log M ] (3.17) π ε→0+ = lim+ ξ M + K ∗ (H0 − λ − iε)−1 K, M . ε→0 On the strength of Lemma 2.7, λ ∈ ρ(H0 ) ∩ ρ H0 + KM −1 K ∗ ∩ R, the for a.e. expression in (3.17) equals ξ M + K ∗ (H0 − λ)−1 K, M . Remark 3.2. Under the hypothesis of Theorem 3.1, one also has the equality ξ(λ, H0 , H0 − KM −1 K ∗ ) = lim+ ξ M, M − K ∗ (H0 − λ + iε)−1 K . ε→0

Applying techniques of [14, Section 3] allows us to carry over the limit in (3.1) to the argument of the ξ-index whenever H0 is bounded.

Vol. 62 (2008)


255

Theorem 3.3. Let M, H0 be self-adjoint operators in A and K ∈ L2 (A, τ ). Assume that M is boundedly invertible. Then, for a.e. λ ∈ R, ξ λ, H0 , H0 + KM −1 K ∗ = τ P+ EM (R− ) − Ξλ P+ + τ P− EM (R− ) − Ξλ P− , (3.18) where Ξλ is given by the limit Ξλ = lim

ε→0+

1 Im log M + K ∗ (H0 − λ − iε)−1 K π

(3.19)

in the strong operator topology, P+ =

1 (I + M ) = EM ([0, ∞)), 2

P− =

1 (I − M ) = EM (R− ). 2

Representation (3.18) initially was derived in [14, (3.23)] for M = M ∗ unitary. We briefly outline the proof of Theorem 3.3 in Appendix A. The details will be omitted since they are analogous to those in the proof of representation [14, (3.23)]. Remark 3.4. (i) If EM (R− ) − Ξλ ∈ L1 (A, τ ) for a.e. λ ∈ R, then equality (3.18) reduces to ξ λ, H0 , H0 + KM −1 K ∗ = τ [EM (R− ) − Ξλ ]. A similar result for a finite A [25, Theorem 3.7 (ii)] follows from the BirmanSchwinger principle for dissipative operators and continuity of the normal finite trace in the strong operator topology. In this case, the limit in (3.19) exists for every λ ∈ R. (ii) When A = B(H), the hypothesis of Theorem 3.3 ensures that, for a.e. λ ∈ R, the boundary values M + K ∗ (H0 − λ − i0)−1 K = lim M + K ∗ (H0 − λ − iε)−1 K ε→0+

exist in the Hilbert-Schmidt norm (but in general not in the trace norm) [31, 32], and hence, 1 Ξλ = Im log M + K ∗ (H0 − λ − i0)−1 K , for a.e. λ ∈ R. π Thus, Ξλ coincides with the Ξ-operator associated with the dissipative operator M + K ∗ (H0 − λ − i0)−1 K. An alternative representation for the ξ-function, ξ λ, H0 + KM −1 K ∗ , H0 = trindex Ξ M + K ∗ (H0 − λ − i0)−1 K , Ξ(M ) , is provided by [21, Theorem 5.3]. We recall that a pair of operators (A, Q), with A bounded and Q an orthogonal projection, has a trindex [21] if there exists an orthogonal projection P in H for which A − P is in the trace class and (P, Q) is a Fredholm pair of orthogonal projections. In this case, trindex(A, Q) = tr(A − P ) + index(P, Q).

256

Skripka

IEOT

3.2. The case of bounded dissipative operators In this subsection, we propagate the results of [25, Theorem 3.3, Corollary 3.5] originally obtained for a finite A to the case of an arbitrary semi-finite A. Basic assumptions of this subsection are collected in the form of a hypothesis. Hypothesis 3.5. Let M, N ∈ DA and K ∈ L2 (A, τ ). Suppose that the operators M − K ∗ N −1 K and N − KM −1 K ∗ are boundedly invertible. ∗ −1 Remark 3.6. The requirement that each of the operators M −∗ K−1N K and −1 ∗ K has bounded inverse is redundant for 0 ∈ ρ M − K N K implies N −KN 0 ∈ ρ N − KN −1 K ∗ and vice versa (cf. [25]).

A symmetry relation for invertible dissipative operators provided by the theorem below can be considered a version of the Birman-Schwinger principle for dissipative operators. Theorem 3.7. Under the assumptions of Hypothesis 3.5, τ log M − K ∗ N −1 K − log M = τ log N − KM −1 K ∗ − log N

(3.20)

and, in particular,

ξ M, M − K ∗ N −1 K = ξ N, N − KM −1 K ∗ .

(3.21)

Next theorem offers a method for computing the ξ-index associated with an off-diagonal perturbation problem for a 2 × 2 operator matrix. Theorem 3.8. Assume Hypothesis 3.5. Suppose, in addition, that K ∈ L1 (A, τ ). Let and T the normal faithful semi-finite M2 denote the space of 2 × 2 scalar matrices trace on the von Neumann algebra A M2 given by A B∗ T = τ (A) + τ (D). C D Then, M T log K

K∗ N

M − log 0

and, in particular,

M ξ 0

0 N

M , K

0 N

K∗ N

= τ log M − K ∗ N −1 K − log M (3.22)

= ξ M, M − K ∗ N −1 K .

Proofs of Theorem 3.1 and Theorem 3.8 rely on the lemma below. Lemma 3.9. Assume Hypothesis 3.5. Let M(z) = M + z − K ∗ (N + z)−1 K,

z ∈ C+ ,

N (z) = N + z − K(M + z)−1 K ∗ ,

z ∈ C+ .

(3.23)

Vol. 62 (2008)


257

Then, d τ [log M(z) − log(M + z)] dz = τ M−1 (z) − (M + z)−1 + τ N −1 (z) − (N + z)−1 ,

(3.24) z ∈ C+ .

Proof. On the strength of Lemma 2.5, d τ [log M(z) − log(M + z)] dz = τ M−1 (z) I + K ∗ (N + z)−2 K − (M + z)−1 = τ M−1 (z) − (M + z)−1 + τ M−1 (z)K ∗ (N + z)−2 K .

(3.25)

Direct computations establish M−1 (z) = (M + z)−1 + (M + z)−1 K ∗ N −1 (z)K(M + z)−1 . Making use of (3.26) entails τ M−1 (z)K ∗ (N + z)−2 K = τ (M + z)−1 K ∗ (N + z)−2 K + τ (M + z)−1 K ∗ N −1 (z)K(M + z)−1 K ∗ (N + z)−2 K , which equals τ K(M + z)−1 K ∗ (N + z)−2 + τ K(M + z)−1 K ∗ N −1 (z) K(M + z)−1 K ∗ (N + z)−2

(3.26)

(3.27)

(3.28)

by the cyclicity of τ . Rewriting N (z) according to (3.23) and then applying the second resolvent identity ensure N −1 (z) K(M + z)−1 K ∗ (N + z)−1 (3.29) −1 ∗ −1 −1 ∗ −1 = N + z − K(M + z) K K(M + z) K (N + z) −1 = N + z − K(M + z)−1 K ∗ − (N + z)−1 . Equalities (3.27)–(3.29) combined together imply τ M−1 (z)K ∗ (N + z)−2 K (3.30)

−1 = τ K(M + z)−1 K ∗ N + z − K(M + z)−1 K ∗ (N + z)−1 , which, by the cyclicity of τ and the second resolvent identity, coincides with −1

τ (N + z)−1 K(M + z)−1 K ∗ N + z − K(M + z)−1 K ∗ (3.31)

−1 = −τ (N + z)−1 − N + z − K(M + z)−1 K ∗ = τ N −1 (z) − (N + z)−1 . Combination of (3.25), (3.30), and (3.31) results in (3.24).

258

Skripka

IEOT

Remark 3.10. Similarly to (3.24), one also has d τ [log N (z) − log(N + z)] dz = τ M−1 (z) − (M + z)−1 + τ N −1 (z) − (N + z)−1 ,

(3.32) z ∈ C+ .

Now we proceed to the proofs of Theorem 3.1 and Theorem 3.8. Proof of Theorem 3.1. Comparing (3.24) and (3.32) induces τ [log M(z) − log(M + z)] = τ [log N (z) − log(N + z)] + C,

z ∈ C+ ,

(3.33)

with C a constant. Utilizing asymptotic expansions τ [log M(iy) − log(M + iy)] = O(1/y),

y → ∞,

and τ [log N (iy) − log(N + iy)] = O(1/y), y → ∞, yields that the constant in (3.33) is zero. Finally, applying Lemma 2.7 guarantees (3.20) and, therefor, (3.21).

∗ M + zI K , z ∈ C+ , is Proof of Theorem 3.8. Note that the matrix K N + zI boundedly invertible and the diagonal entries of the inverse matrix are M−1 (z) and N −1 (z). On the strength of Lemma 2.5, for z ∈ C+ ,

d M + zI K∗ M + zI 0 T log − log K N + zI 0 N + zI dz −1 −1 ∗ M + zI K M + zI 0 =T − 0 N + zI K N + zI −1 = τ M (z) − (M + zI)−1 + τ N −1 (z) − (N + zI)−1 . Applying (3.24) then yields M + zI T log K

K∗ N + zI

M + zI − log 0

= τ [log M(z) − log(M + zI)] + C,

0 N + zI

(3.34)

with C a constant. Comparing asymptotic expansions for the expressions in (3.34) with z = iy, y → ∞, implies that C in (3.34) is zero. Finally, Lemma 2.7 completes the proof of the theorem.

4. The ξ-function on the spectra of operators Applying versions of the Birman-Schwinger principle obtained in Theorem 3.3 and Theorem 3.7, we derive a formula resembling the classical Birman-Krein formula [8] that linked the scattering matrix and the spectral shift function. It is convenient to collect primary assumptions and notations in the form of a hypothesis.

Vol. 62 (2008)


259

Hypothesis 4.1. Suppose that H0 = H0∗ and V = V ∗ are elements in A. Assume that V is factored in the form V = KM −1 K ∗ , where K ∈ L2 (A, τ ) and M = M ∗ ∈ A is a boundedly invertible operator. Denote H = H0 + V . Assume that there exist normal boundary values K ∗ (H0 − λ − i0)−1 K = lim K ∗ (H0 − λ − iε)−1 K, ε→0+

in the operator norm. Suppose that Im K ∗ (H0 − λ − i0)−1 K ∈ L1 (A, τ ),

for a.e. λ ∈ R,

for a.e. λ ∈ R.

(4.1)

(4.2)

Let M(λ) = M + K ∗ (H0 − λ − i0)−1 K, 1 1 P+ = (I + M ) = EM ([0, ∞)), P− = (I − M ) = EM (R− ). 2 2 Remark 4.2. When A = B(H) and τ is the canonical trace on it, (4.1) and (4.2) are satisfied provided K ∈ L2 (A, τ ) (cf. [4, 31, 32]); when τ is finite, requirement (4.2) is redundant. Under Hypothesis 4.1, the statement of Theorem 3.3 can be enriched with additional details. Theorem 4.3. Assume Hypothesis 4.1. Let S(λ) = I − 2i(Im M(λ))1/2 M−1 (λ)(Im M(λ))1/2 .

(4.3)

Then, for a.e. λ ∈ R for which Re M(λ) and M(λ) are boundedly invertible, ξ λ, H0 , H0 + KM −1 K ∗ = τ P+ EM (R− ) − ERe M(λ) (R− ) P+ + τ P− EM (R− ) − ERe M(λ) (R− ) P− 1 τ [arg(S(λ))]. (4.4) − 2π We note that S(λ) given by (4.3) coincides with the Lifshits characteristic function (cf. [22, Section IV.6]) of the dissipative operator M(λ) calculated at the spectral point 0, which is also the abstract scattering operator associated with the pair (H0 − λ, H − λ). Remark 4.4. If under Hypothesis 4.1 the limit lim K ∗ (H0 − λ − iε)−1 K

ε→0+

exists in the operator norm, then M(λ) = M + K ∗ (H0 − λ − i0)−1 K is boundedly invertible. Before passing to a proof of Theorem 4.3, we compare (4.4) with known relations between the ξ-function and S.

260

Skripka

IEOT

Remark 4.5. Whenever ERe M(λ) (R− ) − EM (R− ) ∈ L1 (A, τ ), representation (4.4) specializes to 1 ξ λ, H0 , H0 + KM −1 K ∗ = −ξ(M, Re M(λ)) − τ [arg(S(λ))]. (4.5) 2π Upon multiplying by 2πi and exponentiating on both sides of (4.5), one arrives at detτ (S(λ)) = exp −2πiξ λ, H0 , H0 + KM −1 K ∗ − 2πiξ(M, Re M(λ)) (4.6) in the case of a finite A or

det(S(λ)) = exp −2πiξ λ, H0 , H0 + KM −1 K ∗

(4.7)

in the case when A = B(H). Here detτ (S(λ)) is the de la Harpe-Skandalis determinant [23] of S(λ) (cf. [25]) and det(S(λ)) is the Fredholm determinant of S(λ). Formulae (4.6) and (4.7) were obtained in [25] and [21], respectively. Remark 4.6. Assume Hypothesis 4.1. If, in addition, Re K ∗ (H0 − λ − i0)−1 K ∈ L2 (A, τ ), ∗ Re K (H0 − λ − i0)−1 K , M ∈ L1 (A, τ ), and Re M(λ) is boundedly invertible, then ERe M(λ) (R− ) − EM (R− ) ∈ L1 (A, τ ).

(4.8)

The proof of (4.8) parallels the one of [21, Lemma 3.9]. We need an auxiliary Lemma reducing computation of the ξ-index for a purely imaginary perturbation problem A → A + iB, with B ≥ 0, to the one of the Lifshits characteristic function of the dissipative operator A + iB. This result extends the one of [25, Theorem 4.1] for A of finite type. Lemma 4.7. Let A = A∗ ∈ A and 0 ≤ B = B ∗ ∈ L1 (A, τ ). Assume that 0 ∈ ρ(A) ∩ ρ(A + iB). Then, for a.e. λ ∈ R, 1 ξ(A, A + iB) = τ [arg S], (4.9) 2π S. with S = I − 2iB 1/2 (A + iB)−1 B 1/2 and arg S = Im log Proof. Let H denote a self-adjoint operator B 1/2 A−1 B 1/2 . Theorem 3.7 along with Spectral theorem applied to H implies that 1 ξ(A, A + iB) = ξ (iI, iI − H) = τ [Im log (iI − H) − Im log(iI)] π 1 = τ arctan H . (4.10) π It is straightforward to verify that S = (iI − H)(iI + H)−1 ,

(4.11)

Vol. 62 (2008)


261

which, in particular, implies that S is unitary. For a smooth path of unitaries [0, 1] t → Ut = (iI − tH)(iI + tH)−1

(4.12)

linking the identity I = U0 with S = U1 , we have Ut − I ∈ L1 (A, τ ), and hence by t ∈ L1 (A, τ ), for t ∈ [0, 1] (cf. (A.1)). In the Dunford-Riesz functional calculus, logU − logI particular, the latter implies that arg S = Im logS ∈ L1 (A, τ ). Applying Lemma 2.5 to the functions X(t) = Ut = I − 2itH and Y (t) = I yields

− logI = Im τ [arg S] = Im τ logS

d dt τ logUt − logI 0 dt 1

τ U˙ t Ut−1 dt. = Im 1

(4.13)

0

Repeating almost literally the argument in [25, Theorem 4.1], one obtains that 1

τ U˙ t Ut−1 dt = 2τ [arctan H], Im 0

which along with (4.10) and (4.13) completes the proof of the lemma.

Proof of Theorem 4.3. As one can see from the proof of [14, Theorem 3.1], the limits πΞλ = lim+ Im log M + K ∗ (H0 − λ − iε)−1 K ε→0

exist in the strong operator topology for a.e. λ ∈ R. Next, for a.e. λ ∈ R for which M(λ) is boundedly invertible, (4.1) infers that πΞλ = Im log M(λ). By Theorem 3.3, (4.14) ξ λ, H0 , H0 + KM −1 K ∗ = τ P+ EM (R− ) − Ξλ P+ + τ P− EM (R− ) − Ξλ P− , for a.e. λ ∈ R. By the Dunford-Riesz functional calculus (cf. (A.1)) and assumption (4.2), the operator ERe M(λ) (R− ) − Ξλ is in the τ -trace class. Thus, we trivially obtain the decomposition τ P+ EM (R− ) − Ξλ P+ + τ P− EM (R− ) − Ξλ P− (4.15) = τ P+ EM (R− ) − ERe M(λ) (R− ) P+ + τ P+ ERe M(λ) (R− ) − Ξλ P+ + τ P− EM (R− ) − ERe M(λ) (R− ) P− + τ P− ERe M(λ) (R− ) − Ξλ P− . In view of Lemma 4.7, for a.e. λ ∈ R such that 0 ∈ ρ(Re M(λ)) ∩ ρ(M(λ)), τ P+ ERe M(λ) (R− ) − Ξλ P+ + τ P− ERe M(λ) (R− ) − Ξλ P− 1 = τ ERe M(λ) (R− ) − Ξλ = −ξ Re M(λ), M(λ) = − τ [arg(S(λ))]. (4.16) 2π Combining (4.14)–(4.16) completes the proof of the theorem.

262

Skripka

IEOT

Appendix A. In this appendix, we prove Lemma 2.5 and Lemma 2.7 as well as sketch the proof of Theorem 3.3 stated in the main part of the paper. Proof of Lemma 2.5. Let Γ be a finite union of disjoint oriented Jordan curves contained in the domain of analyticity of f and surrounding Ω exactly once. Then using the second resolvent identity results in 1 (A.1) f (Y (z)) − f (X(z)) = f (λ) (λ − Y (z))−1 − (λ − X(z))−1 dλ 2πi Γ 1 = f (λ) (λ − Y (z))−1 K(z)(λ − X(z))−1 dλ, 2πi Γ with the integrals evaluated in the operator norm. Since the function Γ λ → f (λ)(λ − Y (z))−1 K(z)(λ − X(z))−1 is · 1,∞ continuous, the integral in (A.1) exists in the norm · 1,∞ and, in particular, f (Y (z)) − f (X(z)) ∈ L1 (A, τ ). Let z0 ∈ U. By obvious manipulations, (λ − Y (z))−1 K(z)(λ − X(z))−1 − (λ − Y (z0 ))−1 K(z0 )(λ − X(z0 ))−1 (A.2) −1 −1 −1 −1 = (λ − Y (z)) K(z)(λ − X(z)) − (λ − Y (z0 )) K(z)(λ − X(z)) + (λ − Y (z0 ))−1 K(z)(λ − X(z))−1 − (λ − Y (z0 ))−1 K(z0 )(λ − X(z))−1 + (λ − Y (z0 ))−1 K(z0 )(λ − X(z))−1 − (λ − Y (z0 ))−1 K(z0 )(λ − X(z0 ))−1 . Applying the second resolvent identity repeatedly implies that the expression on the right-hand side of (A.2) equals (λ − Y (z))−1 (Y (z) − Y (z0 ))(λ − Y (z0 ))−1 K(z)(λ − X(z))−1 −1

(K(z) − K(z0 ))(λ − X(z))

−1

K(z0 )(λ − X(z))−1 (X(z) − X(z0 ))(λ − X(z0 ))−1 .

+ (λ − Y (z0 )) + (λ − Y (z0 ))

(A.3)

−1

In view of (A.2) and (A.3), the function f (λ) z → (λ − Y (z))−1 K(z)(λ − X(z))−1 − (λ − Y (z0 ))−1 K(z0 )(λ − X(z0 ))−1 z − z0 is continuous uniformly in λ in the norm · 1,∞ around z0 . Therefore, employing (A.1)-(A.3) provides d f (Y (z)) − f (X(z)) (A.4) dz z=z0 1 = f (λ) (λ − Y (z0 ))−1 Y (z0 )(λ − Y (z0 ))−1 K(z0 )(λ − X(z0 ))−1 2πi Γ + (λ − Y (z0 ))−1 K (z0 )(λ − X(z0 ))−1 dλ + (λ − Y (z0 ))−1 K(z0 )(λ − X(z0 ))−1 X (z0 )(λ − X(z0 ))−1 dλ,


Vol. 62 (2008)

263

with the derivative and integrals evaluated in the norm · 1,∞ . In particular, differentiability of f (Y (z)) − f (X(z)) in the norm · 1,∞ and continuity of τ on L1 (A, τ ) imply (2.5). Applying the second resolvent identity leads to the equalities (λ − Y (z0 ))−1 Y (z0 )(λ − Y (z0 ))−1 K(z0 )(λ − X(z0 ))−1 −1

= (λ − Y (z0 ))

−1

−1

Y (z0 )(λ − Y (z0 ))

− (λ − Y (z0 ))

(A.5)

−1

−1

Y (z0 )(λ − X(z0 ))

and (λ − Y (z0 ))−1 K(z0 )(λ − X(z0 ))−1 X (z0 )(λ − X(z0 ))−1 −1

= (λ − Y (z0 ))

−1

X (z0 )(λ − X(z0 ))

−1

− (λ − X(z0 ))

(A.6)

X (z0 )(λ − X(z0 ))

.

Making use of the equality K (z) = Y (z) − X (z), with the derivatives evaluated in the operator norm, entails (λ − Y (z0 ))−1 K (z0 )(λ − X(z0 ))−1 −1

= (λ − Y (z0 ))

(A.7) −1

Y (z0 )(λ − X(z0 ))

−1

− (λ − Y (z0 ))

−1

X (z0 )(λ − X(z0 ))

.

Adding together (A.5)-(A.7) converts the integrand in (A.4) to f (λ)(λ − Y (z0 ))−1 Y (z0 )(λ − Y (z0 ))−1 −1

− f (λ)(λ − X(z0 ))

(A.8) −1

X (z0 )(λ − X(z0 ))

.

Since X equals either I or 0, (A.8) is identical to f (λ)(λ − Y (z0 ))−1 K (z0 )(λ − Y (z0 ))−1 −2

+ f (λ)(λ − Y (z0 ))

(A.9) −2

X (z0 ) − f (λ)(λ − X(z0 ))

X (z0 ).

Combining (A.4)–(A.9) establishes d f (Y (z)) − f (X(z)) (A.10) dz z=z0 1 = f (λ)(λ − Y (z0 ))−1 K (z0 )(λ − Y (z0 ))−1 dλ 2πi Γ 1 + f (λ) (λ − Y (z0 ))−2 X (z0 ) − (λ − X(z0 ))−2 X (z0 ) dλ, 2πi Γ with the derivative and integrals evaluated in the norm · 1,∞ . Since τ is a continuous linear functional on L1 (A, τ ), (A.10) implies d f (Y (z)) − f (X(z)) (A.11) τ dz z=z0 1 = f (λ)τ (λ − Y (z0 ))−1 K (z0 )(λ − Y (z0 ))−1 dλ 2πi Γ

1 −2 −2 f (λ) (λ − Y (z0 )) X (z0 ) − (λ − X(z0 )) X (z0 ) dλ . +τ 2πi Γ

264

Skripka

IEOT

The cyclicity of the trace assures that the expression on the right-hand side of (A.11) equals

1 τ f (λ)(λ − Y (z0 ))−2 K (z0 )dλ (A.12) 2πi Γ

1 f (λ) (λ − Y (z0 ))−2 X (z0 ) − (λ − X(z0 ))−2 X (z0 ) dλ +τ 2πi Γ

1 =τ f (λ) (λ − Y (z0 ))−2 Y (z0 ) − (λ − X(z0 ))−2 X (z0 ) dλ . 2πi Γ Applying partial integration (cf. [38, Lemma II.3.18]) yields 1 f (λ)(λ − Y (z0 ))−2 Y (z0 )dλ = g(Y (z0 )) · Y (z0 ) 2πi Γ and

1 f (λ)(λ − X(z0 ))−2 X (z0 )dλ = g(X(z0 )) · X (z0 ), 2πi Γ which along with (A.12) proves (2.6).

Proof of Lemma 2.7. Applying the second resolvent identity implies that for any z, z0 ∈ K, h(z) = (λ − Y (z))−1 − (λ − X(z))−1 − (λ − Y (z0 ))−1 − (λ − X(z0 ))−1 = (λ − Y (z))−1 (Y (z) − X(z))(λ − X(z))−1 − (λ − Y (z0 ))−1 (Y (z0 ) − X(z0 ))(λ − X(z0 ))−1 . By the computations in (A.2) and (A.3), the function z → log λ·h(z) is continuous uniformly in λ in the norm · 1,∞ at the point z0 . Therefore, lim log Y (z) − log X(z) − log Y (z0 ) − log X(z0 ) 1,∞ = 0. z→z0

Proof of Theorem 3.3. 1) Obtain representation Im log M + K ∗ (H0 − λ − iε)−1 K − Im log M =

∞

−∞

Ξt − P− dt t − λ − iε

(A.13)

(cf. [14, (3.13)] for the analogous representation in the case of unitary M ). 2) Prove decomposition ∞ ∞ τ P+ (Ξt − P− )P+ Ξt − P− dt = dt τ t − λ − iε −∞ t − λ − iε −∞ ∞ τ P− (Ξt − P− )P− dt, ε > 0, + t − λ − iε −∞ (cf. [14, (3.16), (3.17), Proposition 3.1]). 3) Upon evaluating τ on both sides of (A.13) and then computing the limit as

Vol. 62 (2008)


ε → 0+ , on the strength of Theorem 3.1, one concludes that ξ λ, H0 , H0 + KM −1 K ∗ ∞ Ξt − P− 1 dt = − lim τ π ε→0+ −∞ t − λ − iε = τ P+ (P− − Ξλ )P+ + τ P− (P− − Ξλ )P− , for a.e. λ ∈ R.

265

Acknowledgment The author is thankful to Konstantin A. Makarov for helpful discussions and the anonymous referee for constructive comments which led to improvement in the presentation of the material.

References [1] N. A. Azamov, A. L. Carey, P. G. Dodds, F. A. Sukochev, Operator integrals, spectral shift, and spectral flow, Canad. J. Math., to appear. [2] N. A. Azamov, A. L. Carey, F. A. Sukochev, The spectral shift function and spectral flow, Comm. Math. Phys. 276 (2007), no. 1, 51–91. [3] N. A. Azamov, P. G. Dodds, and F. A. Sukochev, The Krein spectral shift function in semifinite von Neumann algebras, Integr. Equ. Oper. Theory 55 (2006), 347–362. [4] H. Baumg¨ artel, M. Wollenberg, Mathematical Scattering Theory, Birkh¨ auser, Basel, 1983. [5] J. Bellisard, A. van Elst, H. Schulz-Baldes, The non-commutative geometry of the quantum Hall effect, J. Math. Phys. 35 (1994), 5373–5451. [6] M.-T. Benameur, A. L. Carey, J. Phillips, A. Rennie, F. A. Sukochev, K. P. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, Analysis, geometryand topology of elliptic operators, 297–352, World Sci. Publ., Hackensack, NJ, 2006. [7] M. Sh. Birman, On the spectrum of singular boundary-value problems, Mat. Sbornik N.S. 55 (97) (1961), 125–174; English. transl. in AMS Trans. 53 (1966), 23–80. [8] M. Sh. Birman, M. G. Krein, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962), 475–478; English transl. in Soviet Math. Dokl. 3 (1962), 740–744. [9] M. Sh. Birman, A. B. Pushnitski, Spectral shift function, amazing and multifaceted, Integr. Equ. Oper. Theory 30 (1998), no. 2, 191–199. [10] M. Sh. Birman, M. Z. Solomyak, Remarks on the spectral shift function, Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 33–46; English transl. in J. Soviet Math. 3 (1975), no. 4, 408–419. [11] M. Sh. Birman, D. R. Yafaev, The spectral shift function. The work of M. G. Krein and its further development, Algebra i Analiz 4 (1992), no. 5, 1–44; English transl. St. Petersburg Math. J. 4 (1993), 833–870. [12] A. L. Carey, K. C. Hannabuss, V. Mathai, P. McCann, Quantum Hall effect on the hyperbolic plane, Comm. Math. Phys. 190 (1998), 629–673.

266

Skripka

IEOT

[13] R. W. Carey, A unitary invariant for pairs of self-adjoint operators, J. Reine Angew. Math. 283/284 (1976), 294–312. [14] R. W. Carey, J. D. Pincus, Mosaics, principal functions, and mean motion in von Neumann algebras, Acta Math. 138 (1977), no. 3–4, 153–218. [15] L. A. Coburn, R. D. Moyer, I. M. Singer, C ∗ -algebras of almost periodic pseudodifferential operators, Acta Math. 130 (1973), 279-307. [16] A. Connes, Noncommutative geometry, Academic Press, San Diego, 1994. [17] Yu. L. Daletski˘i, S. G. Krein, Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations, (Russian) Voroneˇz. Gos. Univ. Trudy Sem. Funkcional. Anal. 1956 (1956), no. 1, 81–105. [18] J. Dixmier, von Neumann algebras, Translated from the second French edition by F. Jellett, North-Holland, Amsterdam, 1981. [19] B. Fuglede, R. V. Kadison, Determinant theory in finite factors, Ann. of Math. (2) 55 (1952), 520–530. [20] F. Gesztesy, K. A. Makarov, S. N. Naboko, The spectral shift operator, Oper. Theory Adv. Appl. 108 (1999), 59–90, Birkh¨ auser, Basel. [21] F. Gesztesy, K. A. Makarov, The Ξ operator and its relation to Krein’s spectral shift function, J. Anal. Math. 81 (2000), 139–183. [22] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc. Transl. of Math. Monographs 18, Providence, RI, 1969. [23] P. de la Harpe, G. Skandalis, Déterminant associ´ e a ` une trace sur une alg` ebre de Banach, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 241–260. [24] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras Vol. II, Academic Press, Orlando, FL, 1986. [25] V. Kostrykin, K. A. Makarov, A. Skripka, The Birman-Schwinger principle in von Neumann algebras of finite type, J. Funct. Anal. 247 (2007), 492–508. [26] M. G. Krein, On the trace formula in perturbation theory, Mat. Sbornik N.S. 33(75) (1953), 597–626. [27] D. Lenz, N. Peyerimhoff, I. Veselic, Groupoids, von Neumann algebras and the integrated density of states, Math. Phys. Anal. Geom. 10 (2007), 1–41. [28] I. M. Lifshits, On a problem of the theory of perturbations connected with quantum statistics, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 1(47), 171–180. (Russian) [29] K. A. Makarov, A. Skripka, Some applications of the perturbation determinant in finite von Neumann algebras, Canad. J. Math., accepted. [30] F. J. Murray, J. von Neumann, On rings of operators, Ann. of Math. 37 (1936), 116–229. [31] S. N. Naboko, Boundary values of analytic operator functions with a positive imaginary part, J. Sov. Math. 44 (1989), 786–795. [32] S. N. Naboko, Nontangential boundary values of operator-valued R-functions in a half-plane, Leningrad Math. J. 1 (1990), 1255–1278. [33] E. Nelson, Notes on noncommutative integration, J. Funct. Anal. 15 (1974), 103–116. [34] J. Phillips, Spectral flowin type I and II factors - a new approach, Fields Inst. Comm. 17 (1997), 137–153.

Vol. 62 (2008)


267

[35] A. B. Pushnitski, A representation for the spectral shift function in the case of perturbations of fixed sign, Algebra i Analiz 9 (1997), no. 6, 197–213; English transl. in St. Petersburg Math. J. 9 (1998), no. 6, 1181–1194. [36] A. Pushnitski, The spectral shift function and the invariance principle, J. Funct. Anal. 183 (2001), no. 2, 269–320. [37] S. Sakai, C ∗ -algebras and W ∗ -algebras, Ergebn. Math. und ihrer Grenzgeb., 60, Springer-Verlag, New York - Heilderberg, 1971. [38] J. T. Schwartz, W ∗ -Algebras, Notes on Mathematics and its Applications, New York, Gordon and Breach, 1967. [39] J. Schwinger, On the bound states of a given potential, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 122–129. [40] M. A. Shubin, Spectral theory and the index of elliptic operators with almost-periodic coefficients, Russian Math. Surveys, 34 (1979), 109-158. [41] M. A. Shubin, Discrete magnetic Laplacian, Comm. Math. Phys. 164 (1994), no. 2, 259–275. [42] B. Simon, Trace ideals and their applications. Second edition, Mathematical Surveys and Monographs 120, Amer. Math. Soc., Providence, RI, 2005. [43] A. Skripka, Trace inequalities and spectral shift, in preparation. [44] M. Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. [45] J. Xia, Geometric invariants of the quantum hall effect, Commun. Math. Phys. 119 (1988), 29–50. [46] D. R. Yafaev, Mathematical scattering theory, Translated from Russian by J. R. Schulenberger, Amer. Math. Soc., Providence, RI, 1992. Anna Skripka Department of Mathematics Texas A&M University College Station, TX 77843 USA e-mail: [email protected] Submitted: January 4, 2007. Revised: March 31, 2008.

Integr. equ. oper. theory 62 (2008), 269–300 0378-620X/020269-32, DOI 10.1007/s00020-008-1628-z c 2008 Birkh¨ auser Verlag Basel/Switzerland


Boundary Value Problems for the Helmholtz Equation in an Octant Frank-Olme Speck and Ernst Peter Stephan Dedicated to Vladimir G. Maz’ya on the occasion of his 70th birthday

Abstract. We consider a class of boundary value problems for the threedimensional Helmholtz equation that appears in diffraction theory. On the three faces of the octant, which are quadrants, we admit first order boundary conditions with constant coefficients, linear combinations of Dirichlet, Neumann, impedance and/or oblique derivative type. A new variety of surface potentials yields 3 × 3 boundary pseudodifferential operators on the quarterplane R2++ that are equivalent to the operators associated to the boundary value problems in a Sobolev space setting. These operators are analyzed and inverted in particular cases, which gives us the analytical solution of a number of well-posed problems. Mathematics Subject Classification (2000). Primary 35J25; Secondary 45E10, 47A68, 47B35. Keywords. Diffraction theory, Helmholtz equation, boundary value problem, pseudodifferential equation, convolution type operator with symmetry, factorization, invertibility, quarter-plane problem.

1. Introduction Canonical problems of diffraction theory [22, 23, 30] serve various purposes such as the discovery of suitable function(al) spaces including regularity properties and the form of singularities or the finding of convenient potentials for the analytical treatment of further boundary value or interface problems. Here we started a new discussion of “octant problems” trying to carry out a three-dimensional analogue of recent investigation in the two-dimensional setting [7, 9, 33]. As a surprise some problems can be solved instantly. However more general problems need a well organized technical effort in operator theory and function spaces. The key ideas come from operator factorization [38], studies of the quarter-plane problem [11,

270

Speck and Stephan

IEOT

29, 39] and extension and trace theorems in spaces of Bessel potentials in special Lipschitz domains [7]. For convenience, please find some of these results and their extension to the present situation in the appendix. The central problem is the following: Find the solutions (in closed analytical form), u ∈ H s (Ω), s ≥ 1, in the first octant Ω = {x ∈ R3 : xj > 0, j = 1, 2, 3} of the three-dimensional Helmholtz equation 2 u=0 in Ω, (1.1) Lu = ∆ + kΩ where kΩ ∈ C, m(kΩ ) > 0, which satisfy the boundary conditions Bj u = gj

on Qj , j = 1, 2, 3.

(1.2)

3

Here Qj are the quadrants Qj = {x ∈ R : xj = 0, xj+1 > 0, xj+2 > 0}, j = 1, 2, 3 (counting indices of the coordinates xj modulo 3, i.e. x4 = x1 etc.) and the boundary operators have the form Bj = TQj (βj0 I + βj1 D1 + βj2 D2 + βj3 D3 )

(1.3)

with given constant coefficients βjk , partial differential operators Dk = ∂/∂xk and the trace operators TQj which act continuously as defined on the space of weak solutions H1 (Ω) that satisfy (1.1). Moreover we know that Bj are bounded linear operators as mappings 3

Bj : Hs (Ω) = H1 (Ω) ∩ H s (Ω) → H s− 2 (Qj ), s ≥ 1

(1.4)

where the domain in equipped with the H s (Ω) norm topology (see Section 2 of the Appendix). Obviously the boundary value problem (BVP) is well-posed (for some fixed s ≥ 1) if and only if the following operator associated with the BVP is boundedly invertible: 3

3

3

B = (B1 , B2 , B3 ) : Hs (Ω) → H s− 2 (Q1 ) × H s− 2 (Q2 ) × H s− 2 (Q3 ). (1.5) Certainly, if Bj is Dirichlet, the image space may be chosen one order higher, which we shall do then by convention. Hence, besides the given parameters s and k we assume, for beginning, that gj are arbitrarily given in these image spaces: g = (g1 , g2 , g3 ) ∈ Y =

3

3

H s− 2 (Qj ).

(1.6)

j=1

If necessary, we distinguish the restricted operator (1.5) for s > 1 from the basic case s = 1 writing B (s) instead of B. For various BVPs, B will not be normally solvable: Its image is not closed in the topology of Y and sometimes cumbersome compatibility conditions for the data are necessary to guarantee a solution in Hs (Ω). So we will find a dense subspace Y < of the data space Y such that the image normalized operator B < = RstB : Hs (Ω) → Y < is boundedly invertible [31]. The strategy is the following: For Dirichlet or Neumann conditions (also in the mixed case) we simply guess an analytic solution, see Section 2, by experience resulting from the two-dimensional case (however, as a bounded composition of

Vol. 62 (2008)

Boundary Value Problems for the Helmholtz Equation

271

unbounded operators). These formulas, which are surprisingly new, represent particular surface potentials that are good for an ansatz to solve some BVPs or to test if those are also well-posed. So we arrive at the idea of using more general potentials, see Section 3, that are suitable for solving further BVPs, i.e., we try to optimize the ansatz depending on the problem. This leads us to a certain operator calculus, see Section 4, and a discussion of resulting strategies, namely • for reducing the problem to the inversion of a triangular boundary pseudodifferential operator matrix with “nice entries” in the main diagonal, so-called two-dimensional convolution type operators with symmetry (CTOs) • by a “good ansatz” of quarter-plane potentials that comes from the solution of a simpler canonical problem and represents an isomorphism between the ansatz data and the solution space. The inversion of Wiener-Hopf or Toeplitz operators on the quarter-plane in general is a strange and difficult task, see Chapter 8 and 9 (particularly § 9.8) in [3], and, moreover, the present CTOs are a modification of them whose structure was partly uncovered recently in the framework of Toeplitz plus Hankel operators on the sphere [2, 13], and in the one-dimensional analogue on the real line [4, 8]. In the present case, scalar CTOs have the form: T = rΓ Ac : H µ (Γ) → H ν (Γ)

(1.7)

where Γ = R++ is a copy of Qj (Γ = R+ in the case of two-dimensional BVPs leading to one-dimensional CTOs [7, 9]), µ, ν ≥ − 12 , c is a continuous extension operator into H µ = H µ (R2 ), here typically double odd, double even or mixed extension (see Appendix 7.1), A is a translation invariant (pseudodifferential) operator of order µ − ν, and rΓ denotes the restriction of functionals from H µ to H ν (Γ). Nevertheless we find a considerable class of BVPs solvable by the present method in Section 5 and give some examples of efficient solution of particular problems in Section 6. Those might be of interest for tackling in future further classes of boundary value or interface problems such as exterior, multi-media, non-rectangular wedge diffraction problems and variants for the Lamé or Maxwell equations, higher order boundary conditions and so on, see the papers cited before, so far mostly for 2D wave propagation or 3D problems that are independent of one variable. The results of the present paper can also be used to investigate the related question of singular behavior and asymptotic expansion of solutions near the corner, which is interesting and important for applications. However there are also different powerful techniques, see for instance the books of V.G. Mazýa et al. [17, 18, 20] and the vast literature cited therein.

272

Speck and Stephan

IEOT

2. Immediate solution of Dirichlet/Neumann problems Now we present ad hoc solutions of the BVPs (1.1-1.2) with pure Dirichlet or Neumann condition on each face of ∂Ω (cases DDD, NNN, DDN, DNN). The formulas immediately imply well-posedness of these problems in an adequate H 1 (Ω) setting, and consequently allow us to consider them as toplinear mappings (linear homeomorphisms) between the data and the solution spaces. Moreover they reproduce the data if plugged into the BCs and are therefore very convenient as (a first choice of) potential operators for the solution of further problems. Theorem 2.1. The pure Dirichlet problem (1.1-1.2) with g ∈ Y < = H 1/2 (∂Ω) = 3 {(g1 , g2 , g3 ) ∈ j=1 H 1/2 (Qj ) : gj = TQj u , j = 1, 2, 3, for some u ∈ H 1 (Ω)} is uniquely solvable in H 1 (Ω) by oo oo oo u(x) = KD,Q g (x) + KD,Q g (x) + KD,Q g (x) (2.1) 1 1 2 2 3 3 −1 oo = F(ξ,η) →(x2 ,x3 ) exp[−t(ξ, η)x1 ] g1 (ξ, η) −1 oo + F(ξ,η) →(x3 ,x1 ) exp[−t(ξ, η)x2 ] g2 (ξ, η) −1 oo g (ξ, η) , x = (x , x , x ) ∈ Ω exp[−t(ξ, η)x ] + F(ξ,η) 3 3 1 2 3 →(x1 ,x2 ) 1/2

2 ) , (ξ, η) ∈ R2 (with the usual vertical branch cut where t(ξ, η) = (ξ 2 + η 2 − kΩ from kΩ to −kΩ over infinity and t(ξ, η) ≈ (ξ 2 + η 2 )1/2 at +∞). Furthermore oo denotes double odd extension into the plane and F , F −1 , ϕˆ stand for the twodimensional Fourier transformation, its inverse (partially operating on two variables) and a transformed function(al), respectively.

Proof. The double odd extension of the three functions gj into the corresponding planes R2j = {x ∈ R3 : xj = 0} do not belong to the space H 1/2 (R2j ) in general, see ˜ 1/2 (Qj ), i.e., if gj are extendible the Appendix, Lemma 7.5. But they do, if gj ∈ H 1/2 2 ˜ 1/2 (Qj ), by zero into the full plane within H (Rj ). Hence, for g ∈ Y˜ = 3j=1 H the function u given by (2.1) can be verified immediately as an H 1 (Ω) solution of the Dirichlet problem. Y˜ is a dense subspace of the data space Y , however this argument does not help very much to show existence and presentation of the solution for general g ∈ Y . On the other hand, uniqueness of the solution of the Dirichlet problem is known anyway, resulting from partial integration of the Green formula, see [10], and [34], p.188-189. However we have here an independent uniqueness proof: Take the difference w = u1 − u2 of two H 1 solutions of the Dirichlet problem which satisfies the homogeneous problem. Then consider the threefold (!) even extension eee w of w from the octant into R3 (by analogy to the notation of Appendix 7.1). It has jumps of the traces on the hyper-planes xj = 0 which are zero (in the same sense as the traces) and jumps of the normal derivatives on xj = 0 (in the distributional sense) which are zero due to symmetry. These two conditions imply

Vol. 62 (2008)


273

that eee w satisfies the Helmholtz equation throughout the hyper-planes, i.e. in the whole R3 , and therefore vanishes identically (almost everywhere). In order to prove that (2.1) is a solution, we use a different method based upon ideas of [26], Proof of Theorem 3.2 in the lower dimensional case. Let ee g˜2 = g2 − TQ2 KD,Q g 1 1 ee g˜3 = g3 − TQ3 KD,Q g 1 1

(2.2)

where the potential operator is as in (2.1) replacing double odd by double even extension (see Section 1 of the Appendix). Thus, g˜j ∈ H 1/2 (Qj ) and both admit zero (and therefore also odd) extension in x1 - direction within H 1/2 space, particularly eo g˜2 ∈ H 1/2 (R2 ), even in x3 , odd in x1 oe g˜3 ∈ H 1/2 (R2 ), odd in x1 , even in x2

(2.3)

according to the Appendix. Now let eo g˜ g˜ ˜3 = g˜3 − TQ3 KD,Q 2 2

(2.4)

eo ˜ 1/2 (Q3 ) and g˜ is odd in x1 and even in x3 ) which belongs to H (where KD,Q 2 2 admits zero, even and odd extension with respect to both variables within H 1/2 spaces. Following [26], Theorem 3.1, we consider ee eo oo v = KD,Q g + KD,Q g˜ + KD,Q g˜˜ 1 1 2 2 3 3

(2.5)

and find that obviously v ∈ H1 (Ω) and that the boundary conditions can be verified as follows: eo oo TQ1 v = g1 + TQ1 KD,Q g˜˜ = g1 g˜ + TQ1 KD,Q 2 2 3 3 ee oo TQ2 v = TQ2 KD,Q1 g1 + g˜2 + TQ2 KD,Q3 g˜˜3 = g2 TQ3 v = TQ3 Kee g1 + TQ3 Keo g˜2 + g˜˜3 = g3 D,Q1

D,Q2

since, in the first line, the second and third term (of the middle part) are zero due to symmetry (restriction of an odd function in x1 to the plane x1 = 0, and the approximation argument: continuous functions are dense); in the second line, the third part is zero and the first two give g2 according to (2.2); in the last line, the second and third term give g˜3 and, together with the first one, sum up to g3 , also due to (2.2). Summarizing, the Dirichlet problem is solvable by v presented in (2.5) and by u given in (2.1) for a dense subspace Y˜ of data. The solution is unique, i.e. u = v for data in Y˜ . Hence (2.1) extends to a boundedly invertible linear operator acting onto H1 (Ω) as defined on the trace space normed by g = ee g1 + eo g˜2 + oo g˜˜3 with the help of substitutions (2.2), (2.4).

(2.6)

274

Speck and Stephan

IEOT

Remark 2.2. Formula (2.5) represents a decomposition of the resolvent into three bounded operators in contrast to (2.1) that is a composition of the same bounded resolvent into three unbounded operators. It is clear that the three components gj of Dirichlet data on the three quadrants Qj , j=1, 2, 3, cannot be given independently in H 1/2 (Qj ), but must satisfy compatibility conditions resulting from (2.2)–(2.4), and are implicitly incorporated in the definition of the data space Y < given in Theorem 2.1. There are several equivalent forms of these compatibility conditions which can be easily derived, looking a little technical however: 3 Corollary 2.3. For g ∈ j=1 H 1/2 (Qj ), the following assertions are equivalent: 1. g represents the trace of an H 1 (Ω) function (as in Theorem 2.1); 2. the substituted functions g˜2 , g˜3 of (2.2) admit zero extension in x1 -direction ˜3 defined by (2.4) admits zero extension in both variwithin H 1/2 space and g˜ ables, x1 and x2 , within H 1/2 space, in brief 1/2

0,1 g˜2 ∈ HQ2 (R2x1 ,x3 + ), 1/2 0,1 g˜3 ∈ HQ3 (R2x1 ,x2 + ), 1/2 00 g˜ ˜3 ∈ HQ3 (R2x1 ,x2 )

(2.7)

where the first function is defined on the half-plane given by x1 ∈ R, x3 > 0 and supported on the quarter-plane Q2 (etc.); 3. the data differences given by ˜ 1 (x2 , x3 ) = g1 (x2 , x3 ) − g3 (x3 , x2 ), h ˜ h2 (x3 , x1 ) = g2 (x3 , x1 ) − g1 (x1 , x3 ), ˜ 3 (x1 , x2 ) = g3 (x1 , x2 ) − g2 (x2 , x1 ) h

(2.8)

for xj > 0, j = 1, 2, 3, admit zero extension with respect to x3 , x1 , or x2 , respectively, within H 1/2 space, in brief ˜ 1 ∈ H 1/2 (R2 0,3 h x2 +,x3 ), Q1 ˜ 2 ∈ H 1/2 (R2 0,1 h x3 +,x1 ), Q2 1/2 ˜ 0,2 h3 ∈ HQ3 (R2x1 +,x2 ).

(2.9)

Proof. Equivalence of the first two assertions follows from the proof of the foregoing theorem, equivalence of the first and third condition is a consequence of the trace theorem. Corollary 2.4. The operator defined by (2.1) KDDD

= :

oo oo oo KD,Q + KD,Q + KD,Q 1 2 3

Y

1, also given by (2.1), or by (2.5) and substitution, is a toplinear isomorphism, if and only if s ∈]1, 2[, and then invertible by the restricted trace operator (TQ1 , TQ2 , TQ3 )(s)T . Proof. This becomes obvious from the invariance properties of (2.5) and Appendix 7.2. Let us turn to the Neumann problem. The “formal solution” is simply based on analogous symmetry considerations, whilst the space discussion is different. By 3 a certain analogy to Corollary 2.3 we say that g ∈ j=1 H −1/2 (Qj ) satisfies the NNN compatibility condition, if the following data ˜ 1 (x2 , x3 ) = g1 (x2 , x3 ) + g3 (x3 , x2 ), h ˜ h2 (x3 , x1 ) = g2 (x3 , x1 ) + g1 (x1 , x3 ), ˜ 3 (x1 , x2 ) = g3 (x1 , x2 ) + g2 (x2 , x1 ) h

(2.13)

for xj > 0, j = 1, 2, 3, admit zero extension with respect to x3 , x1 , or x2 , respectively, within H −1/2 space, in brief ˜ 1 ∈ H −1/2 (R2 0,3 h x2 +,x3 ), Q1 −1/2 ˜ 0,1 h2 ∈ HQ2 (R2x3 +,x1 ), ˜ 3 ∈ H −1/2 (R2 0,2 h x1 +,x2 ). Q3

(2.14)

For convenience we used the same letters gj , ˜hj here for given and substituted functionals as before in the context of the Dirichlet problem, which is understood as local notation, referring to analogous considerations. Theorem 2.6. The Neumann problem with Bj = TQj Dj and g ∈ 3j=1 H −1/2 (Qj ), cf. (1.1–1.2), is uniquely solvable in H 1 (Ω) if and only if the NNN compatibility

276

Speck and Stephan

IEOT

condition (2.14) is satisfied. In this case the solution is given by ee ee ee u(x) = KN,Q g (x) + KN,Q g (x) + KN,Q g (x) 1 1 2 2 3 3 −1 −1 ee g (ξ, η) ] = F(ξ,η) exp[−t(ξ, η)x 1 1 →(x2 ,x3 ) t(ξ, η) −1 −1 ee g (ξ, η) + F(ξ,η) ] exp[−t(ξ, η)x 2 2 →(x3 ,x1 ) t(ξ, η) −1 −1 ee g (ξ, η) , + F(ξ,η) ] exp[−t(ξ, η)x 3 3 →(x1 ,x2 ) t(ξ, η) x = (x1 , x2 , x3 ) ∈ Ω.

(2.15)

Proof. Again, double even extension is not a bounded operator in H −1/2 spaces, but for data in suitable subspaces (the tilde spaces, e.g.) one can verify easily the formula as to be a solution considering traces and using symmetry arguments. The proof that u belongs to H 1 (Ω) is similar as before, by substitution, taking sums of data instead of differences, cf. [26], p. 282. It comes out that the formula represents a bounded linear operator from Y < = YN 0. II. In the cases j = e , mj = 0 or j = o , mj = 1, Kε is bounded iff ε ∈ [0, 1[. Proof. This follows by elementary estimation and our knowledge about the domains of the extension operators, see Appendix 7.1. Remarks 3.3. (a) If K is strict, the operator B associated to the BVP is toplinear equivalent to a BΨDO T = BK that will be analyzed and optimized by the choice of ψ1 , ψ2 and ψ3 later on. (b) As we shall see, in general it is not evident to recognize the strictness of K and it turns out to be convenient to admit non-strict potentials in certain cases, cf. [9]. (c) Statement II can be generalized and holds also for ε ∈ ] − 1, 0[ , if the image space of K is replaced by a suitable “space of generalized weak solutions” that is just the subspace of H 1+ε (Ω) functions representable in this form. However this extension has no importance here. (s)

(s)

Theorem 3.4. The only strict QPPs of Section 2 are KN N N and KDN N (plus cyclic modifications) for s ∈]1, 2[. All the others, namely (s)

(s)

(s)

(s)

KDDD and KDDN for s ∈ [1, 2[, KDN N and KN N N for s = 1, are only bijective onto Hs (Ω) as defined on a dense subspace X (s)< , say, equipped with a norm corresponding to the compatibility conditions. Proof. This results from Section 2 considering the “pure and mixed” trace operators such as 3 H s−1/2 (Qj ) (3.3) (TQ1 , TQ2 , TQ3 ) : X(s) = Hs (Ω) → j=1

due to the DDD case, and analogous for the others, as to be “image normalized” operators (in the sense of [31]) replacing X (s) by X (s)< , which are bounded inverses of the corresponding potential operators. From the appendix follows that X (s)< is continuously and densely embedded into X (s) . That means: For s = 1 we have no strict potential ansatz from Section 2 (in terms of pure Dirichlet and/or Neumann data on Qj ), in contrast to the 2D case where the mixed DN ansatz is so convenient and successful [7, 9] for all s ∈ [1, 2[ including s = 1 ! This fact is remarkable and makes the analysis of 3D problems sort of rather subtle.

Vol. 62 (2008)


281

4. Representation formulas and equivalence with boundary ΨDOs We study the structure of T . Presumably its rather technical details have deterred people from doing this before. It is convenient to introduce the following notation. Definition 4.1. Let φ be given by φ(ξ, η) = β0 − β1 t(ξ, η) − β2 iξ − β3 iη , (ξ, η) ∈ R2 ,

(4.1)

with constant coefficients βj ∈ C , j = 1, 2, 3. Then we denote by φ∗ , φ∗∗ the functions given by φ∗ (ξ, η) = β0 − β1 iη − β2 t(ξ, η) − β3 iξ , φ∗∗ (ξ, η) = β0 − β1 iξ − β2 iη − β3 t(ξ, η) , (ξ, η) ∈ R2 ,

(4.2)

and φ∗∗∗ = φ , i.e., the ∗ -operation on functions of this form corresponds to a cyclic change of t, iξ, iη (not counting the arguments of t which is symmetric anyway). Beside the two-dimensional Fourier transformation F and its inverse F −1 (see Appendix 7.1) we define the two-dimensional Fourier integral operators (FIOs) F ∗ , F ∗∗ on corresponding domains as follows:

1 F ∗ ϕ(x1 , x2 ) = 2π R R exp[−t(ξ, η)x1 − iξx2 ]ϕ(ξ, η)dξdη, x1 > 0, (4.3)

1 exp[−t(ξ, η)x − iηx ]ϕ(ξ, η)dξdη, x > 0. F ∗∗ ϕ(x1 , x2 ) = 2π 2 1 2 R R Formally they result from F −1 replacing in the exponent iξ by t and iη by iξ, etc. For a general treatise of FIOs see [12, 16]. Further we use the following notation. Let B be given by (1.3)–(1.5) and K by (3.1)–(3.2). The following functions are referred to as the Fourier symbols of B and K (and its variants K(s) and K(s)< ), respectively: φ1 (ξ, η) = β10 − β11 t(ξ, η) − β12 iξ − β13 iη , φ2 (ξ, η) = β20 − β21 iη − β22 t(ξ, η) − β23 iξ , φ3 (ξ, η) = β30 − β31 iξ − β32 iη − β33 t(ξ, η) ,

(4.4)

ψ1 (ξ, η) = α10 − α11 t(ξ, η) − α12 iξ − α13 iη , ψ2 (ξ, η) = α20 − α21 iη − α22 t(ξ, η) − α23 iξ , ψ3 (ξ, η) = α30 − α31 iξ − α32 iη − α33 t(ξ, η) ,

(4.5)

where (ξ, η) ∈ R2 and βjk , αjk are constant coefficients, j, k = 1, 2, 3. Theorem 4.2. Under the previous has the form  r++ F −1 σ11 F 1 T =  r++ F ∗∗ σ21 F 1 r++ F ∗ σ31 F 1

assumptions, the composed operator T = BK r++ F ∗ σ12 F 2 r++ F −1 σ22 F 2 r++ F ∗∗ σ32 F 2

 r++ F ∗∗ σ13 F 3 r++ F ∗ σ23 F 3  r++ F −1 σ33 F 3

(4.6)

282

Speck and Stephan

involving “Fourier symbols”



 ΦT = (σjk ) =  

φ1 ψ1 φ∗∗ 2 ψ1 φ∗ 3 ψ1

IEOT

φ∗ 1 ψ2 φ2 ψ2 φ∗∗ 3 ψ2

φ∗∗ 1 ψ3 φ∗ 2 ψ3 . φ3 ψ3

  . 

(4.7)

Note that for simplicity σjk does not only denote a function, but also a multiplication operator in expressions like (4.6). Proof. We have   B1 T =  B2  K1 B3

K2

K3

3

= (Tjk ) : X =

H 1/2−mk (R2++ ) → Y

(4.8)

k=1

where the matrix entries are given by T11

=

B1 K1 = r++ F −1 σ11 F 1 ,

T12

=

B1 K2 = r++ F ∗ σ12 F 2 ,

T13

=

B1 K3 = r++ F ∗∗ σ13 F 3 ,

(4.9)

etc. due to the following calculations, in which we keep the original names of variables corresponding to Qj , see (1.2) and abbreviate t = t(ξ, η): −1 −1 T11 f1 (x2 , x3 ) = rQ1 F(ξ,η) →(x ,x ) φ1 ψ1 F(x2 ,x3 )→(ξ,η) 1 f1 (x2 , x3 ), 2 3 1 T12 f2 (x3 , x1 ) = rQ1 exp[−iξx3 −tx2 ]φ∗1 ψ2−1 F(x3 ,x1 )→(ξ,η) 2 f2 (x3 , x1 )dξdη, 2π R R 1 −1 T13 f3 (x1 , x2 ) = rQ1 exp[−iηx2 −tx3 ]φ∗∗ 1 ψ3 F(x1 ,x2 )→(ξ,η) 3 f3 (x1 , x2 )dξdη. 2π R R The other six formulas follow by a simultaneous cyclic substitution x1 → x2 → x3 → x1 . The final form (4.8) results by copying Qj , j=1,2,3, onto R2++ , where (x2 , x3 ), (x3 , x1 ) and (x1 , x2 ), respectively, are re-named by (x1 , x2 ).

Corollary 4.3. If K is strict, then B and T are equivalent operators, i.e., they coincide up to homeomorphisms. In this case, the BVP is well-posed iff T is boundedly invertible and the solution is given by u = Kf = KT −1 g.

(4.10)

Corollary 4.4 (Strictness Test). Under the previous assumptions, the potential operator K : X → H1 (Ω) (4.11) is strict, iff the operator Ttest = BtestK is bijective for one (and thus for all) operators Btest associated to a well-posed BVP. Remark 4.5. The results may be modified by replacing the spaces of densities X and data Y by suitable subspaces X (s) and Y (s) or X < and/or Y < (including compatibility conditions), etc., particularly if K or T is not normally solvable.

Vol. 62 (2008)


283

5. Analysis of octant problems The basic idea is indeed an equivalent reduction to an invertible operator matrix as exposed in the following Proposition 5.1. Using the notation of Theorem 4.2, let K be strict, T = BK lower triangular and Tjj be boundedly invertible. Then   −1 0 0 T11   −1 (5.1) T −1 =  ∗ T22 0  ∗

∗

−1 T33

with suitable bounded operators in the places indicated by an asterisk. The BVP is well-posed and explicitly solvable by analytic formulas consisting of compositions of bounded operators. Proof. This is immediately clear for triangular matrices of bounded linear operators yielding conclusions due to the operator relations seen in Figure 1. So the question is to find a suitable strict ansatz such that T is (lower) triangular and the operators in the main diagonal can be inverted explicitly. We shall characterize these BVPs. The point is that the idea does not apply to various interesting cases (see Section 6). Then the question turns to: how can we overcome the difficulties, e.g., by using a non-strict ansatz or modified spaces discovered in an operator normalization as described in [31]? Let us study first the question of an off-diagonal element of T being zero. For convenience, we consider T12 separately either for odd or for even extension with respect to the second variable (and arbitrarily continuous with respect to the first), denoted by 2 = co or 2 = ce , respectively. The function´s arguments ξ, η are dropped sometimes for brevity. Lemma 5.2. Let φj and ψj as before, j = 1, 2, 3, see (4.4)–(4.5) and Definition 3.1 with mj = 1. If 2 = co (i.e. odd extension in x1 -direction and arbitrary continuous extension in x3 -direction), the following assertions are equivalent: • T12 = 0 , • σ12 is even in η , • φ∗1 = cψ2 where c is constant or: ψ2 = α20 − α21 t − α22 iξ and φ1 = β10 − β12 iξ − β13 iη (i.e., β11 = 0) or: ψ2 = −α23 iη (i.e., not 1-regular) and φ1 = −β11 t . If 2 = ce , then the following assertions are equivalent: • T12 = 0 , • σ12 is odd in η , • ψ2 = α20 − α21 t − α22 iξ and φ1 = −β11 t or: φ2 = −α23 iη and φ1 = β10 − β12 iξ − β13 iη . Proof. The Fourier symbol of T12 reads σ12 = φ∗1 /ψ2 , as seen in (4.7), which, in the first case of 2 = co , has to be even in η in order to let T12 disappear, and vice versa, see analogous details in [9], Lemma 3.4. The integrand of the first formula

284

Speck and Stephan

IEOT

of (4.3) with ϕ = σ12 F 2 f2 is then odd with respect to η such that the integral disappears. Equivalence with the third condition is evident from the form of σ12 . The second result follows analogously. To prepare forthcoming calculations, we take note of the corresponding result for T13 without proof. Lemma 5.3. Under the same assumptions on φj and ψj as before, if now 3 = oc (odd extension in x1 and arbitrary continuous extension in x2 -direction), the following assertions are equivalent: • T13 = 0 , • σ13 is even in ξ , • φ∗∗ 1 = cψ3 where c is constant or: ψ3 = α30 − α31 t − α33 iη and φ1 = β10 − β12 iξ − β13 iη (i.e., β11 = 0) or: ψ3 = −α32 iξ (i.e., not 1-regular) and φ1 = −β11 t . If 3 = ec (less important case), then the following assertions are equivalent: • T13 = 0 , • σ13 is odd in ξ , • ψ3 = α30 − α31 t − α33 iη and φ1 = −β11 t or: ψ3 = −α32 iξ and φ1 = β10 − β12 iξ − β13 iη . The criteria for each of the other four elements T21 , T23 , T31 , T32 to vanish follow from the above by a cyclic change of variables. For instance we receive the result for T23 from Lemma 5.2 replacing 2 by 3 , φ1 by φ2 and ψ2 by ψ3 with corresponding coefficients. Now, how are the chances to invert explicitly an operator in the main diagonal (cf. the remarks at the end of the introduction). We give here some constructive results needed later on and start with the case where both φj and ψj are 1-regular putting µ = s − 3/2 = 1/2 − mj + ε = 1/2 − ordBj + ε ∈ ] − 1/2, 1/2[,

(5.2)

cf. (1.6), (1.7) and Definition 3.1. Proposition 5.4. Let Tσ be a (two-dimensional) convolution type operator with symmetry that has the form of the diagonal elements of T , be of zero order and normal type, i.e. Tσ c

= r++ F −1 σF c : H µ (R2++ ) → H µ (R2++ ) = oo , oe , eo or ee , µ ∈] − 1/2, 1/2[ ,

σ(ξ, η)

=

β0 − β1 t(ξ, η) − β2 iξ − β3 iη φ(ξ, η) = , (ξ, η) ∈ R2 , ψ(ξ, η) α0 − α1 t(ξ, η) − α2 iξ − α3 iη

σ

∈

GL∞ (R2 ) .

(5.3)

I. If σ is even with respect to both variables, then Tσ is boundedly invertible by Tσ−1

= r++ F −1 σ −1 F c : H µ (R2++ ) → H µ (R2++ ).

(5.4)

Vol. 62 (2008)


285

In this case α2 = α3 = β2 = β3 = 0 or φ = cψ with c being constant. II. If σ is “ strong minus-factor (of order zero)”, i.e., σ and σ −1 have (bounded) holomorphic extensions in m ξ < 0 for all η ∈ R and in m η < 0 for all ξ ∈ R, then Tσ is boundedly invertible by the same formula as before. In this case either σ is constant or the following conditions are satisfied: (i)

α0 = 0 , α1 = 0 ,

(ii)

(e (α2 /α0 ))2 + (e (α3 /α0 ))2 = 0,

(iii)

α2 = 0 or α3 = 0 or e α2 /α0 < 0 and α2 /α3 ∈ R+ ,

(iv) (v)

β0 , β1 , β2 , β3 satisfy conditions analogous to (i)–(iii), (α2 , α3 ) and (β2 , β3 ) are C-linearly dependent.

Proof. Considering the composition of the operators defined before Tσ−1 Tσ

= r++ F −1 σ −1 F c r++ F −1 σF c : H µ (R2++ ) → H µ (R2++ ) ,

we observe that the middle term c r++ may be omitted according to the invariance properties of Tσ or Tσ−1 , respectively. In the very first case (where c = oo ), F −1 σF maps the subspace of double odd functions of H µ onto the same (as a bounded linear operator), such that c r++ has no effect. In the last case, e.g., c r++ can be omitted in the part r++ F −1 σ −1 F c r++ of the above operator composition. With other words, c r++ can be replaced by I since the difference I − c r++ projects onto a subspace of functions supported on R2 \ R2++ which the operator F −1 σ −1 F leaves invariant as a distributional convolution supported on the closure of R2−− (the dual cone). Similar conclusions hold for the inverse composition Tσ Tσ−1 , since both factors F −1 σF and F −1 σ −1 F are bounded on the space H µ and have the same invariance properties with respect to the image of c r++ that is a bounded projector for each present choice of µ and c . For σ to be a strong minus-factor, provided it is not constant, the numerator φ and the denominator ψ have both the mentioned holomorphy properties. This implies that there are no branches caused by t (which are symmetric), i.e., α1 = β1 = 0. Thinking about the coefficients αj we have α0 = 0 since ψ(ξ, η) = 0 in (ξ, η) = (0, 0). e (α2 /α0 ) and e (α3 /α0 ) cannot vanish simultaneously, if ψ is not constant and ψ(ξ, η) = 0 on R2 . Further the conditions e α2 /α0 < 0 and e α3 /α0 < 0 are necessary for ψ not to vanish in m ξ < 0, η ∈ R and m η < 0, ξ ∈ R. Thus either (j) α2 /α0 = 0 and then α2 = 0, otherwise ψ(ξ, η) vanishes for η = 0 and ξ = −iα0 /α2 ∈ R, or (jj) α3 = 0 by analogy or (jjj) both e α2 /α0 and e α3 /α0 are negative. In this case, the convex sets H1

=

{α2 iξ + α3 iη : m ξ ≤ 0, η ∈ R}

H2

=

{α2 iξ + α3 iη : ξ ∈ R, m η ≤ 0}

do not contain the point α0 (i.e., the denominator of (5.3) does not vanish for m ξ ≤ 0, η ∈ R and for ξ ∈ R, m η ≤ 0) if and only if they are sets with parameters that satisfy the condition in the second line of (iii).

286

Speck and Stephan

IEOT

Analogous conditions hold for the numerator. Finally (v) is responsible for σ ∈ GL∞ (R2 ), particularly for guaranteeing that σ does not increase nor decrease on certain rays in R2 . Corollary 5.5. In case of c = oo or ee , the result extents to larger spaces where µ ∈] − 3/2, 1/2[ or ] − 1/2, 3/2[ , respectively, and, in the other two cases, to non-homogeneous Sobolev spaces: µ = (µ1 , µ2 ) ∈] − 3/2, 1/2[×] − 1/2, 3/2[ and µ ∈] − 1/2, 3/2[×] − 3/2, 1/2[ , respectively (which will not be studied further on in this article). Remark 5.6. The conditions (i)–(iii) in Proposition 5.4 are rather restrictive. Typical examples are 1 − β2 iξ − β3 iη σ(ξ, η) = , (ξ, η) ∈ R2 1 − α2 iξ − α3 iη where e αj < 0, e βj < 0 , and α2 β3 = α3 β2 . And this is obviously the general form of a strong minus-factor of order zero up to constant factors, in the sense of assertion II of the foregoing proposition. However, if we like to follow the idea of studying operators Tσ with symbol σ that is a quotient of an even factor of order 1 and a minus-factor (in a modified sense) or vice versa, we have to take into account that there do not exist any two-dimensional translation invariant operators of order 1 with Fourier symbols of the form of φ or ψ and with this kind of holomorphy properties, but only more complicated, so-called Bessel potential operators for the quarter-plane [11, 36, 39], which are not suitable here according to the form of φ. Therefore it is convenient to modify the space setting as follows. Definition 5.7. Let γ

=

λγ (ξ, η) =

1 = R2++ ∩ ∂B1 (0) , i.e., γj ≥ 0 , γ12 + γ22 = 1, (γ1 , γ2 ) ∈ S++

1 + γ1 iξ + γ2 iη , (ξ, η) ∈ R2 .

The space

µ −1 −ν Hγµ,ν = Λ−ν λγ F H µ (R2 ) (5.5) γ H = F with the norm induced by H µ will be referred to as an anisotropic Sobolev space. Hγµ,ν (Ω) is defined by restriction (as usual), particularly for Ω = R2++ . From now on, operators Aψ = F −1 ψF whose symbols ψ satisfy the conditions (i)–(iii) of Proposition 5.4 will be simply called directional minus-factor of order 1 and their ν-powers directional minus-factor of order ν ∈ R (defined with vertical branch cut in m ξ > 0 for η ∈ R, if γ1 > 0, etc.).

Obviously the function λγ (ξ, η) 1 + γ1 iξ + γ2 iη = , (ξ, η) ∈ R2 λ(ξ, η) 1 + (ξ 2 + η 2 )1/2 1 . is bounded in R2 and the numerator does not vanish in R2 if γ = (γ1 , γ2 ) ∈ S++ ∞ 2 However it does not belong to GL (R ) since the numerator is bounded on the

Vol. 62 (2008)


287

lines γ1 ξ + γ2 η = constant whilst the denominator increases. Therefore we have from (5.5) and (7.3), for µ ∈ R, Hγµ−1,1

⊂ Hµ

Hγµ−1,1 (R2++ ) ⊂ H µ (R2++ ) as proper subspaces. Similarly we see that Hµ H µ (R2++ )

⊂ Hγµ+1,−1 ⊂ Hγµ+1,−1 (R2++ )

are properly embedded. We note that analogous results are valid for fractional orders by interpolation, and that the embeddings are dense as seen by approximation with smooth functions. Proposition 5.8. If Aψ is a directional minus-factor of order −ν ∈ R, then Aψ

1 : H µ → Hγµ,ν , µ, ν ∈ R , γ ∈ S++

r++ Aψ (µ)

: H µ (R2++ ) → Hγµ,ν (R2++ )

(µ,ν) r++ A−1 ψ γ

: Hγµ,ν (R2++ ) → H µ (R2++ )

are homeomorphisms where γ = (−e α2 /α0 , −e α3 /α0 ) (resulting from ψ(ξ, η) = (µ,ν) (α0 − α2 iξ − α3 iη)−ν ), (µ) , and γ denote arbitrary extension operators into H µ and Hγµ,ν , respectively. The inverse of Aψ is Aψ−1 = F −1 ψ −1 F and the other two operators are inverse to each other. Proof. The first mapping property is an interpretation of Definition 5.7: If we have f ∈ H µ and ψ(ξ, η) = (α0 − α2 iξ − α3 iη)−ν , then 2 Aψ f Hγµ,ν = |λνγ ψλµ fˆ|2 dξdη R2 2 ≤ C |λµ fˆ|2 dξdη ≤ C 2 f 2H µ R2

since λνγ (ξ, η)ψ(ξ, η) =

(1 + γ1 iξ + γ2 iη)ν 1 1 + γ1 iξ + γ2 iη = · α3 2 (α0 − α2 iξ − α3 iη)ν a0 1 − α α0 iξ − α0 iη

belongs to GL∞ (R2 ), its modulus is bounded by a constant C > 0. The other two mapping properties result from an idea of Lemma 4.6 in [14], Chapter I: To prove the second property, decompose any function f ∈ H µ (R2 ) as f

= (µ) r++ f + f−

where f− ∈ H µ (R2 \ R2++ ) is supported in the complement of R2++ and find that r++ Aψ f− = 0 since Aψ f− is also supported in the complement of R2++ . This is a consequence of the fact that Aψ is a convolution operator with a distributional kernel supported on the closure of R2−− due to the holomorphy properties of ψ

288

Speck and Stephan

IEOT

and the Paley-Wiener theorem. Therefore r++ Aψ (µ) r++ = r++ Aψ . Similarly we get the third property and the formulas for the inverses are now evident. Proposition 5.9. Under the assumptions of (5.3) in Proposition 5.4: I. Let c = oo , φ be even in both variables and ψ a directional minus-factor of order 1. Then the (image) restricted operator Rst Tσ : H µ (R2++ ) → Hγµ+1,−1 (R2++ ) is invertible iff µ ∈] − 1/2, 1/2[ , i.e., s ∈]1, 2[ . In this case the inverse is given by (Rst Tσ )−1 = r++ F −1 φ−1 F oo r++ F −1 ψF oo . c

(5.6)

ee

II. Let = , φ a directional minus-factor of order 1 and ψ be even. Then the (image) extended operator Ext Tσ : H µ (R2++ ) → Hγµ+1,−1 (R2++ ) is invertible iff µ ∈] − 1/2, 1/2[ , i.e., s ∈]1, 2[ , the inverse being given by (Ext Tσ )−1 = r++ F −1 ψF ee r++ F −1 φ−1 F ee .

(5.7)

Proof. In the first case the formulas of the inverses can be verified as before, taking into account the following mapping properties: r++ F −1 ψF oo

: H µ+1,−1 (R2++ ) → H µ−1 (R2++ )

r++ F −1 φ−1 F oo

: H µ−1 (R2++ ) → H µ (R2++ )

both bounded and bijectively for µ ∈] − 1/2, 1/2[, as well as the invariance properties (of a minus-factor or even factor, resp.): r++ F −1 ψF oo r++ oo r++ F −1 φ−1 F oo

= r++ F −1 ψF = F −1 φ−1 F oo

For µ ≥ 1/2 , Rst Tσ is not bounded, and, for µ ≤ −1/2 , Rst Tσ is not surjective. The proof of II runs analogously. Remark 5.10. The other two cases, replacing oo by ee in part I and vice versa in part II, are not so evident, since the corresponding inverse formulas hold only for µ ∈]1/2, 3/2[ and µ ∈] − 3/2, −1/2[ , respectively, according to the ranges of the extension operators, i.e., for s ∈]2, 3[ and s ∈]0, 1[, respectively. The idea of working with Fredholm operators of index ±1 does not apply in this case (at least not straightforwardly), see [11], since the lifting by two-dimensional Bessel potential operators yield rather complicated symbols. The discussion for a Dirichlet type ansatz (ψ = 1) is similar. We summarize without proof just one case: Corollary 5.11. In case of c = ee , ψ = 1 , and φ given by (4.1), the operators T11 T12

= =

r++ F −1 φF ee , r++ F ∗ φF ee ,

T13

=

r++ F ∗∗ φF ee

are bounded from H µ (R2++ ) to H µ−1 (R2++ ) , if µ ∈] − 1/2, 3/2[. T11 is invertible for µ ∈] − 1/2, 1/2[, if φ is even in both variables and t−1 φ ∈ GL∞ (R2 ). If φ is a

Vol. 62 (2008)


289

directional minus-factor (of order 1), the restricted operator Rst T11 : H µ (R2++ ) → Hγµ,−1 (R2++ ) is invertible for (the same) µ ∈] − 1/2, 1/2[. T12 vanishes, if φ is odd with respect to the second variable, and T13 vanishes, if φ is odd with respect to the first variable. Other cases, e.g. for oo instead of ee , are less important here. Now we are prepared to analyze various classes of examples. Note that the results of this section have to be modified if K is not strict as mentioned in Remark 4.5.

6. Examples First let us look at a BVP that is of particular interest from the physical point of view and played an important role in the context of different geometrical situations such as Sommerfeld half-plane problems [30] and 2D BVPs for quarter-planes [9]. 6.1. The three faces impedance problem Consider the BVP where (1.3) gains the form Bj = TQj (βj0 I + βjj Dj )

(6.1)

with non-vanishing coefficients, so that we may assume βj0 = 1, j = 1, 2, 3. The coefficients of the normal derivatives are often written as βjj = −ipj where pj denote (the same or different) impedance numbers related to the surface conductivity (with e pj > 0) in electro-dynamical problems, for instance. Following the method described before, we find, for a certain QPP ansatz (3.1), the operator T of (4.6) with Fourier symbols in the main diagonal places given by σjj =

φj 1 − pj t = , j = 1, 2, 3, ψj ψj

(6.2)

and in the upper right triangle ΦT shows the symbols σ12 =

φ∗ 1 ψ2

=

1−p1 η ψ2

, σ13 = σ23 =

φ∗∗ 1 ψ3 φ∗ 2 ψ3

= =

1−p1 ξ ψ3 1−p2 η ψ3

, .

Looking at our criteria of Lemma 5.2 (etc.) for vanishing T12 (and T13 , T23 as well), we conclude that an ansatz with double even ψ2 , ψ3 does not help, since σ12 cannot be even in η (etc.) – that is exactly the wrong symmetry for our purposes. Also a directive minus-factor ψ3 does not help as T13 and T23 will not vanish simultaneously. So we end up in this example with the insight that the chances for solving BVPs rigorously (as seen in [9]) will be rather limited. At least the three faces impedance problem remains unsolved at present.

290

Speck and Stephan

IEOT

6.2. Problems solvable by a Neumann type ansatz Let us study the question of whether the strict ansatz of Theorems 2.6 and 3.4 (s) where u = KN N N f, s ∈]1, 2[ , can be successfully employed to solve certain BVPs considering only the case where the symbol matrix (4.7) becomes lower triangular, i.e., T12 = T13 = T23 = 0. Theorem 6.1. For any BVP (1.1)–(1.2) with associated operator B defined by (1.6) (s) the Neumann ansatz u = Kf = KN N N f as given by (2.15)–(2.16) for s ∈]1, 2[ yields a lower triangular operator matrix T = BK, see (4.7), iff the first two conditions (on Q1 and Q2 ) are Neumann (up to constant factors) and the third one is arbitrary. Under these assumptions (βjj = 0 for j = 1, 2 and βjk = 0 for j = 1, 2, k = j as well as s ∈]1, 2[), the composed operator T = BK has the form   0 0 β11 I  0 β22 I 0 T = (6.3) ∗ ee ∗∗ ee −1 ee r++ F σ31 F r++ F σ32 F r++ F σ33 F where the symbols are given below. If φ3 is even or a directive minus-factor of order 1, then T has a bounded inverse of the form (5.1) with details explicitly given in Proposition 5.4, part I, or 5.9, part II. In this case, the BVP is well-posed in Hs for s ∈]1, 2[ and any given data g as described in (1.6). For s = 1 we obtain the same results after image normalization, g ∈ Y < (as carried out in Section 2, or in combination with Proposition 5.9, II, respectively). Proof. We consider the symbol matrix (4.7) and search for T12 = T13 = T23 = 0. Now we have β10 − β11 iη − β12 t(ξ, η) − β13 iξ (6.4) σ12 = −t Looking at the second part of Lemma 5.2 we find T12 = 0 iff φ1 = β11 t, corresponding to a Neumann condition on Q1 . In Lemma 5.3 we find T13 = 0 exactly for the same condition (Neumann condition on Q2 ), which is geometrically understandable since Q3 plays a similar role as Q2 before. Analogous arguments hold for T23 = 0 which implies also T21 = 0. Only the third boundary condition can be chosen differently in order to end up with a lower triangular T matrix. So we conclude that the correct solution of this BVP is equivalent to the inversion of the operator T with Fourier symbol (cf. (4.3)–(4.7))   β11 iη β11 iξ β11 −t −t  β22 iξ β22 iη   β22 ΦT =  (6.5) −t  −t  φ∗ 3 −t

φ∗∗ 3 −t

φ3 −t

that causes zero entries in the above-mentioned places of T since the symbol is odd in ξ or η, respectively. The problem is reduced to a discussion of the operator T33 with Fourier symbol −φ3 /t and normalization for the case s = 1 which can be taken from Proposition 5.4 and Corollary 5.5.

Vol. 62 (2008)


291

6.3. Problems solvable by a Dirichlet type ansatz Now let us briefly tackle BVPs in H1 (Ω) or in Hs (Ω) , s ∈]1, 2[ , by the pure Dirichlet ansatz (2.1) with densities f ∈ X < = H 1/2 (∂Ω) or X (s)< = H 1/2 (∂Ω) ∩ 3 s j=1 TQj H (Ω) , i.e. satisfying compatibility conditions as described in the Corollaries 2.3 - 2.5. Formally we find equivalent operators of the form (4.8) with j = oo , ψj = 1, acting from X (s)< into Y (s) = 3j=1 H s−1/2−mj (R2++ ). For T12 = T13 = 0 we see in Lemma 5.2 and 5.3 (modified for this kind of ansatz) that (6.6) φ∗1 = β10 − β11 iη − β12 t − β13 iξ has to be even in η, i.e. β11 = 0, which implies already that φ∗∗ 1 is even in ξ, see (4.2). Further, for T13 = 0, we get φ∗2 to be even in η, i.e., β22 = 0. If the main diagonal elements of ΦT are double even or directional minus type, we can “formally” invert the matrix operator T ignoring the compatibility conditions and the rest is a question of determining the correct subspaces due to the setting of X (s)< and Y (s) by normalization. We find here several solvable problems, however with some complications in the spaces characterization. Also the mixed BVP with Dirichlet, Neumann and impedance condition, respectively, on the three faces of the octant boundary, can be solved by a DNN ansatz given by Proposition 2.11. One can verify easily that the resulting operator matrix T in Theorem 4.2 is upper triangular. The compatibility conditions are like those in the DNN case due to the same principal part of the boundary operators. 6.4. Spatial derivatives The following observation is a direct extension from the 2D case [9]. Consider BVPs where Bj = TQj (β0 + β1 D1 + β2 D2 + β3 D3 ) , j = 1, 2, 3,

(6.7)

with the same coefficients for j = 1, 2, 3 in contrast to (1.3). I.e., they are composed by the trace on Qj with the same “spatial” 3D directional derivative. The definitions of their Fourier symbols (4.4)and their ∗ -modifications (4.2) yield φ∗1 = φ2 = φ∗∗ 3

(6.8)

etc. Now if the symbols φj are 1-regular and if we put ψj = φj , then ΦT gains the form   1 1 1   (6.9) ΦT =  1 1 1  1 oo

1

1

such that, for j = , we have T = I. The BVP is well-posed for s ∈]1, 2[ and solvable by (3.1) with the present special choice of j and ψj . In case of s = 1 the compatibility condition is obviously satisfied. Namely, if the solution u = Kg is given by (3.1) with the present choice of ψj and k = oo we find the compatibility

292

Speck and Stephan

IEOT

condition h = TQj u ∈ H 1/2 (∂Ω) of Theorem 2.1 and Corollary 2.3 (where h is there denoted by g) in the form   h = TQj Kg = r++ F −1 

1 ψ1 1 ψ1 1 ψ1

1 ψ2 1 ψ2 1 ψ2

1 ψ3 1 ψ3 1 ψ3





 g1  oo  g2  ∈ H 1/2 (∂Ω).  F g3

(6.10)

Surprisingly or not, three arbitrary data gj ∈ H −1/2 (Qj ) yield identical (Dirichlet) traces of a solution of a BVP as described by (6.7). 6.5. The method of companion operators Let us combine the ideas of Sections 6.2 and 6.4. Finally we saw that T contains zero elements, e.g., T12 = 0 when 2 = co and σ12 = 1 according to ψ2 = ϕ∗1 which we call the symbol of the companion (boundary) operator on Q2 due to the boundary operator B1 . So we can achieve zeros in the matrix T by choosing an ansatz that contains in some places symbols of companion operators and in other places symbols such as found in Section 6.2. Here is one of various possible results. Proposition 6.2. Let φj , j = 1, 2, 3, be the Fourier symbols of the boundary operators Bj as given in (4.4), being 1-regular, and let φ2 = φ∗1 . Then the QPP ansatz (3.1) with j = oo and densities ψ1 (so far arbitrary), ψ2 = φ∗1 and ψ3 = φ∗2 yields a lower triangular operator matrix T = BK which is equivalent to B in corresponding spaces, provided the symbols ψj are also 1-regular and K is bijective. If, additionally, ψ1 = φ1 and T33 can be inverted (see Proposition 5.4 till Corollary 5.9), we obtain an inverse of T by formula (5.1). The proof of this result is an evident combination of the previous considerations. The bijectivity of K might be proved by the strictness test of Corollary 4.4, eventually plus normalization. However it is difficult to find relevant examples in this class of BVPs in contrast to the 2D situation. Remark 6.3. 1. Further classes of examples can be treated by mixing up Dirichlet type conditions and ansatzes. 2. Unfortunately the “two faces impedance problem”, with impedance boundary conditions on Q2 and Q3 (see (6.1)) and some other convenient condition on Q1 , is still not directly treatable, since the companions of impedance boundary operators have symbols of the form 1 − pj iξ or 1 − pj iη, which are not 1-regular. It leads us to the consideration of further anisotropic Sobolev spaces, maybe in a future publication. 3. Another possible continuation consists in perturbation arguments regarding operators T with certain small coefficients as to be close to another operator that can be proved to be invertible by the methods presented in this article.

Vol. 62 (2008)


293

7. Appendix The one- or two-dimensional Fourier transformation, respectively, in S, L2 , S are written as: u(x1 ) exp[iξx1 ] dx1 , ξ ∈ R, (7.1) Fx1 →ξ u(x1 ) = R

F(x1 ,x2 )→(ξ,η) u(x1 , x2 ) =

R2

u(x1 , x2 ) exp[iξx1 + iηx2 ] dx1 dx2 ,

(ξ, η) ∈ R2 ,

and for other variables xj replacing the coordinate indices cyclically and maintaining ξ, η. Also partial Fourier transformations and their partial inverses are used in a suggestive notation such as mentioned in (2.1), e.g., u(x1 , x2 , x3 ) = =

−1 F(ξ,η) →(x1 ,x2 ) ϕ(ξ, η, x3 ) 1 ϕ(ξ, η, x3 ) exp[−iξx1 − iηx2 ] dξdη. 2π R2

(7.2)

The usual Bessel potential or fractional Sobolev spaces (sometimes named Beppo Levi or Sobolev-Slobodeckii spaces and, in brief, just Sobolev spaces) H s = H s (Rn ) ⊂ S (Rn ) are defined and normed by 2 (1 + |ξ| 2 )s dξ1 · · · dξn < ∞. u2H s = | u(ξ)| (7.3) Rn

where we also used the n-dimensional Fourier transform u ˆ = F u with variable ξ = (ξ1 , · · · ξn ) for convenience. For open sets Ω ⊂ Rn , let u ∈ H s (Ω) if u is the restriction to Ω of a distribution in H s (Rn ), and let uH s (Ω) = inf uH s (Rn )

(7.4)

s (Ω) if taking the infimum with respect to all extensions in H s ; we write u ∈ H n s the extension by zero, 0 u ∈ S (R ), belongs to H . As pointed out earlier, cf. [26] e.g., for the half-axes Ω = R± = {x ∈ R : ±x > 0}, this space can be identified with the subspace HΩs of H s distributions supported on Ω = clos Ω if and only if s ∈] − 1/2, 1/2[, i.e. in brief s (Ω) = r+ HΩs , H

s (Ω) HΩs = 0 H

for

|s| < 1/2

(7.5)

n

and the same holds for (special) Lipschitz domains Ω ⊂ R [15, 41]. On the other hand the “critical order” spaces admit proper, dense embeddings 1/2 (R± ) ⊂ H 1/2 (R± ), H

−1/2 (R± ) ⊂ H −1/2 (R± ). H

(7.6)

7.1. Some extension theorems Now we provide results about even and odd extension operators and start with the one-dimensional case: H µ = H µ (R), for the sake of clarity. They are the key to the so-called direct approach [8] and not to be found in the standard literature [35, 41]

294

Speck and Stephan

IEOT

since they are not of the type of Fichtenholz, Hestenes etc. (there is another operator in [32] called Nikolsky extension, but only for Wp1 (Ω) and Lipschitz domains Ω). For any µ ∈ R, let H µ,e = {ϕ ∈ H µ : ϕ = Jϕ} ,

H µ,o = {ϕ ∈ H µ : ϕ = −Jϕ}

(7.7)

where Jϕ(x) = ϕ(−x) for ϕ ∈ H µ , µ ≥ 0, and Jϕ(ψ) = ϕ(Jψ) for test functions ψ in the case of µ < 0, respectively. The even and odd extension operators from R+ (e.g.) e : H 1/2 (R+ ) → H 1/2 (R) o : H −1/2 (R+ ) → H −1/2 (R)

(7.8)

are bounded as being defined on L2 (R+ ) by e = (I + J)0 and o = (I − J)0 and on H µ (R+ ) by restriction for µ ∈]0, 1/2] and by continuous extension for µ ∈ [−1/2, 0[. Conversely, an element u ∈ H 1/2 (R+ ) admits an odd extension if 1/2 (R+ ), i.e. it is extendable by zero, and u ∈ H −1/2 (R+ ) has an and only if u ∈ H −1/2 (R+ ) [29]. Let us summarize some precise even extension if and only if u ∈ H results around the fact that even and odd extension operators have a wider domain than zero extension, mentioned in (7.5): Lemma 7.1. The operator of even extension e is a well-defined and bounded operator from H µ (R+ ) into H µ with the image H µ,e if and only if µ ∈] − 1/2, 3/2[. In this case, r+ e = I and e r+ is a bounded projection in H µ with the image H µ,e . Otherwise, e does not map into H µ,e (r ≥ 3/2) or not onto H µ,e (µ ≤ −1/2), respectively. Proof. The first (positive) result is known for µ ∈] − 1/2, 1/2[ where ϕ ∈ H µ (R+ ) can be extended by zero to 0 ϕ ∈ H µ and thus e ϕ = (I + J)0 ϕ ∈ H µ . It is shown for µ = 1/2 by estimation in terms of Sobolev-Slobodeckii norms and (the negative result) for µ = −1/2 by a duality argument [29]. Further it follows for µ µ µ ∈]1/2, 3/2[ by decomposition H µ ∼ = H− ⊕ C ⊕ H+ [6]. The case of µ = 3/2 can be reduced to the case µ = −1/2 by application of the Bessel potential operator I − ∆ which maps H 3/2 onto H −1/2 and H 3/2,e onto H −1/2,e (∆ denoting the Laplace operator). For µ > 3/2, the first derivative of ϕ ∈ H µ,e in zero exists due to Sobolev’s lemma but e H µ (R+ ) contains functions with discontinuous first derivative in zero. For µ < −1/2, the delta functional belongs to H µ,e but not to H µ (R+ ). Note that for negative orders, H µ,e is the closure of L2,e with respect to the H µ norm. Lemma 7.2. The operator of odd extension o is a well-defined and bounded operator from H µ (R+ ) into H µ with image H µ,o if and only if µ ∈] − 3/2, 1/2[. In that case, r+ o = I and o r+ is a bounded projection in H µ with the image H µ,o . Otherwise, o does not map into H µ,o (µ ≥ 1/2) or not onto H µ,o (µ ≤ −3/2), respectively.

Vol. 62 (2008)


295

Proof. We reach the result by using duality arguments in the preceding Lemma similarly as in the central case of µ = −1/2 treated in [29]: We know from Lemma 7.1 that, for µ ∈] − 1/2, 3/2[, the linear operator P = e r+ : H µ → H µ is a µ , and, conversely, these properties imply bounded projection onto H µ,e along H− that µ ∈] − 1/2, 3/2[. P can be written as P = (I + J)1+ · where the multiplication by the characteristic function 1+ of R+ , abbreviating 1+ · = 0 r+ , is bounded only if µ < 1/2, i.e., (I + J)1+ · must be seen as a bounded composition with an unbounded operator for µ ≥ 1/2. µ Thus, I − P = 1+ · (I − J) is a bounded projection onto H− along H µ,e precisely for the same parameters µ ∈] − 1/2, 3/2[. Moreover the transposed operator −µ (I − P ) = (I − J)1+ · is a (bounded) projection in H −µ onto H −µ,o along H− if o and only if −µ ∈] − 3/2, 1/2[. It may be written as (I − P ) = r+ which yields the statement of the lemma. Remark 7.3. In the critical cases, even and odd extension operators are bounded in the previous settings, but their images are only proper dense subspaces of the corresponding target spaces: µ ⊂ H µ,e (R), µ = −1/2 or µ = 3/2, e : H µ (R+ ) → (I + J)H+ µ o µ : H (R+ ) → (I − J)H+ ⊂ H µ,o (R), µ = −3/2 or µ = 1/2.

(7.9)

Now we study the higher dimensional case and start with extension from a half-space Rn+j = {x = (x1 , x2 , ..., xn ) ∈ Rn : xj > 0}, j = 1, ..., n.

(7.10)

ej

Lemma 7.4. The operator of even extension with respect to the variable xj (j = 1, ..., n) is a well-defined and bounded operator ej : H µ (Rn+j ) → H µ (Rn )

(7.11)

with the image Hjµ,e = {ϕ ∈ H µ (R+ ) : ϕ = Jj ϕ} , if and only if µ ∈] − 1/2, 3/2[, where Jj ϕ = ϕ(x1 , ..., xj−1 , −xj , xj+1 , .., xn ). It is then left invertible by restriction. An analogous statement holds for odd extension oj : H µ (Rn+j ) → H µ (Rn )

(7.12)

if and only if µ ∈] − 3/2, 1/2[. Proof. The demonstration results from the previous lemmata by an analogous consideration for functions of n variables. The two-dimensional case was carried out in [29] for the orders µ = ±1/2. Let us consider double extension operators with respect to two variables, for simplicity only in the case n = 2 that is needed here. Lemma 7.5. The operators of even/even, odd/odd, even/odd extension with respect to the first/second variable, respectively, ee , oo , eo : H µ (R++ ) → H µ (R2 )

(7.13)

296

Speck and Stephan

IEOT

are well-defined, bounded and onto the corresponding spaces H µ,ee , H µ,oo , H µ,eo , iff µ ∈] − 1/2, 3/2[, ] − 3/2, 1/2[, ] − 1/2, 1/2[, respectively. Proof. The statement results from writing ee = e1 e2 etc.

Remark 7.6. Admitting non-homogeneous spaces of Bessel potentials [36] in the last case, we have a corresponding result for eo : H (µ1 ,µ2 ) (R++ ) → H (µ1 ,µ2 ) (R2 ), iff µ1 ∈] − 1/2, 3/2[ , µ2 ∈] − 3/2, 1/2[. Remark 7.7. For critical orders (limits of the above µ-intervals) we have density properties of the images corresponding with Remark 7.3. Corollary 7.8. I. For Ω0 = R2+j , j = 1, 2 , or Ω0 = R2++ (and their copies Qj etc.), which are special Lipschitz domains (or manifolds), we have ˜ µ (Ω0 ) iff µ ∈] − 1 , 1 [. H µ (Ω0 ) = H 2 2

(7.14)

˜ µ (Ω0 ) iff II. For µ ∈ [ 12 , 32 [ , we have ϕ ∈ H o ϕ = oo ϕ =

(I + Jj )0 ϕ ∈ H µ (R2 ) for Ω0 = R2+j , (I + J1 )(I + J2 )0 ϕ ∈ H µ (R2 ) for Ω0 = R2++

where Jj denotes the flip with respect to xj , j=1, 2. 7.2. Traces of solutions of the Helmholtz equation and their derivatives Let us shortly touch the question: How is it possible to understand higher order boundary data of H 1 solutions in a direct convenient way? As a matter of fact, trace theorems for solutions of elliptic equations involve more regularity than formulated just for H 1 functions [19]. Thus the general theory of elliptic BVPs on (special) Lipschitz domains answers the question sufficiently. However, let us illuminate the situation for the present setting of BVPs on the present “very special domains”. For u(x) defined for x = (x1 , x2 ) ∈ R × R+ and u0 (x1 ) = u(x1 , 0) we have u ∈ H s (R × R+ ) =⇒ u0 ∈ H s−1/2 (R) , s > 1/2. The trace of a derivative of u makes sense only if s is sufficiently large. But, for a weak solution of the Helmholtz equation we have more: Lemma 7.9. Let s ≥ 1, u ∈ Hs (R × R+ ), see (1.4), and α ∈ N2 . Then the distributional derivative of order α of u can be written as Dα u(x) = ϕα

∈

Fξ−1 →x1 exp [−t(ξ)x2 ] ϕα (ξ) , x ∈ R × R+ , H

s−|α|−1/2

(R).

(7.15)

Vol. 62 (2008)


297

Proof. The representation formula (7.15) for u corresponding to α = 0 and ϕ0 = uˆ0 ∈ S is known [26]. Applying Dα to that formula we get ϕα (ξ) u0

= =

(iξ)α1 (−t)α2 uˆ0 (ξ) , ξ ∈ R, TRx1 u.

An estimate and the density argument complete the proof.

Analogously, using the representation formulas for u ∈ Hs (Ω), s ≥ 1 in terms of Dirichlet data for half- and quarter-spaces in Rn and octants of R3 as well, one can define Tα,Γ u = TΓ Dα u ∈ H s−|α|−1/2 (∂Ω) as a generalized trace of a derivative of a solution to the Helmholtz equation. It coincides with the classical derivatives on the boundary for sufficiently smooth data and acts as a continuous operator on Hs (Ω). However, their relation with the surface potential densities gj in (2.1) is simple only if Ω is a half-space.

Conclusion Quite a number of BVPs have been explicitly solved by the present method. However, in comparison with the 2D case, some of the prominent problems remain open such as the three faces impedance problem. Undetected compatibility conditions can be uncovered systematically, starting from the standard setting of product trace spaces and carrying out the so-called image normalization method. In many cases these compatibility conditions turn out to be necessary and sufficient for well-posedness. It is promising to continue this investigation by admitting other kinds of normalized anisotropic Sobolev spaces in order to enlarge the classes of treatable problems. Further the method may be successfully applied to multi-media problems, e.g., where Helmholtz equations with different wave numbers are considered in different octants and interface conditions are involved. Another interesting continuation of the work consists in finding asymptotic results for the behavior of the solutions near edges and corners, based upon the behavior of Fourier symbols at infinity.

Acknowledgement. The authors would like to acknowledge the financial support of Centro de An´ alise Funcional e Aplica¸c˜ oes of Instituto Superior Técnico, Universidade Técnica de Lisboa, through Programa Operacional “Ciência, Tecnologia, Inova¸c˜ ao” (POCTI) of the Funda¸c˜ ao para a Ciência e a Tecnologia (FCT) and the Institute for Applied Mathematics at the University of Hannover. The work was started in December 2006 during the stay of the second author at Instituto Superior Técnico and mainly finished in May 2007 during a visit of the first author to the Leibniz-Universität Hannover.

298

Speck and Stephan

IEOT

References [1] H. Bart, V.E. Tsekanovskii, Matricial coupling and equivalence after extension. T. Ando et al. (eds.), Operator Theory and Complex Analysis. Operator Theory: Advances and Applications 59 (1992), 143–160, Birkh¨ auser, Basel. [2] E. L. Basor, T. Ehrhardt, On a class of Toeplitz + Hankel operators. New York J. Math. 5 (1999), 1–16. [3] A. B¨ ottcher, B. Silbermann, Analysis of Toeplitz Operators. Springer, Berlin 2006. [4] L.P. Castro, R. Duduchava, F.-O. Speck, Asymmetric factorizations of matrix functions on the real line. The Igor Borisovich Simonenko Anniversary Volume (eds. Y.M. Erusalimskii et al). Operator Theory: Advances and Applications 170 (2006), 53–74, Birkh¨ auser, Basel. [5] L.P. Castro, F.-O. Speck, Regularity properties and generalized inverses of deltarelated operators. Z. Anal. Anwend. 17 (1998), 577–598. [6] L.P. Castro, F.-O. Speck, Relations between convolution type operators on intervals and on the half-line. Integr. Equ. Oper. Theory 37 (2000), 169–207. [7] L.P. Castro, F.-O. Speck, F.S. Teixeira, On a class of wedge diffraction problems posted by Erhard Meister. I. Gohberg et al. (eds.), Operator Theoretical Methods and Applications to Mathematical Physics. Operator Theory: Advances and Applications, 147 (2004), 211–238, Birkh¨ auser, Basel. [8] L.P. Castro, F.-O. Speck, F.S. Teixeira, A direct approach to convolution type operators with symmetry. Math. Nachr. 269-270 (2004), 73–85. [9] L.P. Castro, F.-O. Speck, F.S. Teixeira, Mixed boundary value problems for the Helmholtz equation in a quadrant. Integr. Equ. Oper. Theory 56 (2006), 1–44. [10] M. Costabel, E. Stephan, A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985), 367–413. [11] R. Duduchava, F.-O. Speck, Pseudodifferential operators on compact manifolds with Lipschitz boundary. Math. Nachr. 160 (1993), 149–191. [12] J.J. Duistermaat, L. H¨ ormander, Fourier integral operators. II. Acta Math. 128 (1972), 183–269. [13] T. Ehrhardt, Factorization Theory for Toeplitz plus Hankel Operators and Singular Integral Operators with Flip. Habilitation thesis, TU Chemnitz 2004. ` [14] G.I. Eskin, Boundary value problems for elliptic pseudodifferential equations. Translations of Mathematical Monographs 52. AMS, Providence, R.I., 1981. [15] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, London, 1985. [16] L. H¨ ormander, Fourier integral operators. I. Acta Math. 127 (1971), 79–183. [17] V.A. Kozlov, V.G. Maz’ya, J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities. Mathematical Surveys and Monographs. 52. American Mathematical Society, Providence, RI, 1997. [18] V.A. Kozlov, V.G. Maz’ya, J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs. 85. American Mathematical Society Providence, RI, 2001. [19] J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I. Die Grundlehren der mathematischen Wissenschaften 181. Springer-Verlag, Berlin, 1972.

Vol. 62 (2008)


299

[20] V. Maz’ya, S. Nazarov, B. Plamenevskii, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. I. Operator Theory: Advances and Applications 111. Birkh¨ auser, Basel, 2001. [21] E. Meister, Some solved and unsolved canonical problems of diffraction theory. Differential Equations and their Applications, Equadiff 6, Lect. Notes Math. 1192 (1986), 393–398. [22] E. Meister, Some solved and unsolved canonical problems of diffraction theory. Differential Equations and Mathematical Physics, Lect. Notes Math. 1285 (1987), 320–336. [23] E. Meister, Einige gel¨ oste und ungel¨ oste kanonische Probleme der mathematischen Beugungstheorie. Expo. Math. 5 (1987), 193–237. [24] E. Meister (ed.), Modern mathematical methods in diffraction theory and its applications in engineering. Proceedings of the Sommerfeld’96 workshop. Freudenstadt, Germany. Methoden und Verfahren der Mathematischen Physik. 42. Peter Lang, Frankfurt, Europ. Verlag der Wissenschaften, 1997. [25] E. Meister, F. Penzel, F.-O. Speck, F.S. Teixeira, Two-media scattering problems in a half-space. H. Begehr et al. (eds.), Partial Differential Equations with Real Analysis. Longman Scientific & Technical, Harlow. Pitman Res. Notes Math. Ser. 263 (1992), 122–146. [26] E. Meister, F. Penzel, F.-O. Speck, F.S. Teixeira, Some interior and exterior boundary-value problems for the Helmholtz equation in a quadrant. Proc. R. Soc. Edinb., Sect. A 123 (1993), 275–294. [27] E. Meister, F. Penzel, F.-O. Speck, F.S. Teixeira, Two canonical wedge problems for the Helmholtz equation. Math. Methods Appl. Sci. 17 (1994), 877–899. [28] E. Meister, P.A. Santos, F.S. Teixeira, A Sommerfeld-type diffraction problem with second-order boundary conditions. Z. Angew. Math. Mech. 72 (1992), 621–630. [29] E. Meister, F.-O. Speck, A contribution to the quarter-plane problem in diffraction theory. J. Math. Anal. Appl. 130 (1988), 223–236. [30] E. Meister, F.-O. Speck, Modern Wiener-Hopf methods in diffraction theory. Ordinary and Partial Differential Equations, Vol. II, Proc. 10th Dundee Conf., Pitman Res. Notes Math. Ser. 216 (1989), 130–171. [31] A. Moura Santos, F.-O. Speck, F.S. Teixeira, Minimal normalization of Wiener-Hopf operators in spaces of Bessel potentials. J. Math. Anal. Appl. 225 (1998), 501–531. ´ [32] J. Neˇcas, Les Méthodes Directes en Théorie des Equations Elliptiques. Masson et C ie , Paris, 1967. [33] F. Penzel, F.S. Teixeira, The Helmholtz equation in a quadrant with Robin’s conditions. Math. Methods Appl. Sci. 22 (1999), 201–216. [34] T. von Petersdorff, Boundary integral equations for mixed Dirichlet, Neumann and transmission problems. Math. Methods Appl. Sci. 11 (1989), 185–213. [35] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin, 1996. [36] C. Sadosky, M. Cotlar, On quasi-homogeneous Bessel potential operators. Singular Integrals (ed. A.P. Calder´ on), Amer. Math. Soc., Providence, 1967, 275–287.

300

Speck and Stephan

IEOT

[37] A.H. Serbest (ed.), S.R. Cloude (ed.), Direct and inverse electromagnetic scattering. Proceedings of the Workshop in Gebze/Istanbul, Turkey. Pitman Research Notes in Mathematics Series 361. Longman, Harlow, 1996. [38] F.-O. Speck, General Wiener-Hopf Factorization Methods. Pitman, London, 1985. [39] F.-O. Speck, R. Duduchava, Bessel potential operators for the quarter plane. Appl. Anal. 45 (1992), 49–68. [40] H. Triebel, Theory of function spaces. Geest & Portig, Leipzig 1983, and Birkh¨ auser Verlag, Basel, 1983. [41] H. Triebel, Theory of function spaces II. Monographs in Mathematics 84. Birkh¨ auser Verlag, Basel, 1992. Frank-Olme Speck Departamento de Matem´ atica Instituto Superior Técnico U.T.L. Av. Rovisco Pais 1049-001 Lisboa Portugal e-mail: [email protected] Ernst Peter Stephan Institut f¨ ur Angewandte Mathematik Leibniz-Universit¨ at Hannover Welfengarten 1 30167 Hannover Germany e-mail: [email protected] Submitted: November 29, 2007 Revised: July 8, 2008

Integr. equ. oper. theory 62 (2008), 301–349 c 2008 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030301-49, published online October 8, 2008 DOI 10.1007/s00020-008-1626-1


Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from NevanlinnaPick to Abstract Interpolation Problem Joseph A. Ball and Vladimir Bolotnikov Abstract. We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffertype linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman. Mathematics Subject Classification (2000). Primary 47A57; Secondary 47A48, 47B32, 47B50, 32A05. Keywords. Abstract Interpolation Problem, linear-fractional map, Stein equation, left tangential interpolation with operator argument.

1. Introduction A multivariable generalization of the Szegö kernel k(x, y) = (1 − xy)−1 is the positive kernel 1 kd (z, ζ) = 1 − z, ζ on Bd × Bd where Bd = z = (z1 , . . . , zd ) ∈ Cd : z, z < 1 is the unit ball of the d-dimensional Euclidean space Cd . By z, ζ = dj=1 zj ζ j we mean the standard inner product in Cd . The reproducing kernel Hilbert space (RKHS) H(kd ) associated with kd via Aronszajn’s construction [8] is a natural multivariable analogue of the Hardy space H 2 of the unit disk and coincides with H 2 if d = 1. In what

302

Ball and Bolotnikov

IEOT

follows, the symbol L(U, Y) stands for the algebra of bounded linear operators mapping U into Y, and we abbreviate L(U, U) to L(U). For Y an auxiliary Hilbert space, we consider the tensor product Hilbert space HY (kd ) := H(kd ) ⊗ Y whose elements can be viewed as Y-valued functions in H(kd ). Then HY (kd ) can be characterized as follows:     n! · fn 2Y < ∞ . HY (kd ) = f (z) = fn z n : f 2 = (1.1)   |n|! d d n∈Z+

n∈Z+

Here and in what follows, we use standard multivariable notations: for multiintegers n = (n1 , . . . , nd ) ∈ Zd+ and points z = (z1 , . . . , zd ) ∈ Cd we set |n| = n1 + n2 + . . . + nd ,

n! = n1 !n2 ! . . . nd !,

z n = z1n1 z2n2 . . . zdnd .

We will be particularly interested in the space of multipliers Md (U, Y) defined as the space of all L(U, Y)-valued analytic functions z → F (z) on Bd such that the multiplication operator MF : f (z) → F (z)f (z) maps HU (kd ) into HY (kd ). It follows by the closed graph theorem that for every F ∈ Md (U, Y), the operator MF is bounded. We denote by Sd (U, Y) the unit ball of Md (U, Y): Sd (U, Y) = {S ∈ Md (U, Y) : MS op ≤ 1}. We let Mz denote the commuting d-tuple Mz := (Mz1 , . . . , Mzd ) consisting of operators of multiplication by the coordinate functions of Cd on H(kd ) (called the shift (operator-tuple) of HY (kd )), whereas we refer to the commuting d-tuple M∗z := (Mz∗1 , . . . , Mz∗d ) consisting of the adjoints of Mzj ’s (in the metric of H(kd )) as the backward shift. Then the space Md (U, Y) can be characterized as those elements R of L(HU (kd ), HY (kd )) which intertwine the shifts of HU (kd ) and HY (kd ) (i.e., such that Mzj R = RMzj for j = 1, . . . , d); if such an R is a contraction, then R = MS for some S ∈ Sd (U, Y). The simplest generalization of the classical Nevanlinna-Pick interpolation problem for the operator-valued case is: Operator-valued Nevanlinna-Pick problem (see [7, 30, 2, 26]): Given points ζ (i) (i) (i) = (ζ1 , . . . , ζd ) in Bd and operators Xi ∈ L(U, Y) for i = 1, . . . , N , find S ∈ Sd (U, Y) so that S(ζ (i) ) = Xi

for

i = 1, . . . , N.

(1.2)

If S is any multiplier in Md (U, Y), then an easy reproducing-kernel-space computation shows that MS∗ kd,ζ ⊗ y = kd,ζ ⊗ S(ζ)∗ y

for

ζ ∈ Bd and y ∈ Y,

Vol. 62 (2008)

Interpolation for Schur Multipliers

303

where kd,ζ (z) := kd (z, ζ) so that kd,ζ ⊗ y ∈ HY (kd ) and kd,ζ ⊗ S(ζ)∗ y ∈ HU (kd ). Hence, if S ∈ Sd (U, Y) satisfies (1.2), then we see that N IY − Xi Xj∗ yj , yi 1 − ζ (i) , ζ (j) i,j=1 =

N

kd,ζj ⊗ yj 2HY (kd ) − MS∗

j=1

N

kd,ζj ⊗ yj 2HU (kd ) ≥ 0

j=1

for any finite collection y1 , . . . , yN of vectors in Y, and hence we see that a necessary

IY −Xi X ∗ N be positive condition for solutions to exist is that the Pick matrix 1−ζi ,ζj j i,j=1

semidefinite. In view of the fact that the Drury-Arveson kernel kd is a complete Pick kernel, it follows that this necessary condition is also sufficient (see e.g. [2]). A more natural problem for the vector-valued setting is the following more general form of the operator-valued Nevanlinna-Pick problem: Left-tangential Nevanlinna-Pick interpolation problem (LNPP) (see [26, 52, 53]: (i) (i) Given points ζ (i) = (ζ1 , . . . , ζd ) ∈ Bd and vectors ai ∈ Y and ci ∈ U for i = 1, . . . , N , find all functions S ∈ Sd (U, Y) such that a∗i S(ζ (i) ) = c∗i

for

i = 1, . . . , N.

(1.3)

There are various ways to study higher multiplicity versions of LNPP; three of which which we study here we call (1) the Sarason Interpolation Problem (SIP), (2) the Commutant Lifting Problem (CLP), and (3) the strongly stable Operator Argument Problem (ssOAP). For example, for the case of ssOAP, the data set consists of a set of the form (T, E, N ) where T = (T1 , . . . , Td ) is a commutative d-tuple of operators on a state space X satisfying a strong stability hypothesis, E : X → Y and N : X → U are output operators such that the associated observability operators OE,T : X → HY (kd ) and ON,T : X → HU (kd ) make sense, and the interpolation conditions on a Schur multiplier S assume the compact form ∗ ∗ OE,T MS = ON,T .

(1.4)

One way to study these problems is to study instead the corresponding problem on the asymmetric Fock space 2 (Fd ) (where Fd is the free semigroup on d letters)— see [49, 50, 30, 7, 51, 52, 53, 17], and then use the result of [7, 30] (see also [18]) that the Drury-Arveson multiplier space is exactly the image of the free-semigroup algebra after applying a point-evaluation map associated with points in the unit ball Bd ; indeed many of the results concerning this problem were first arrived at in this way. It is also possible to study these problems directly, as in [2, 26, 3, 5, 6]. While the work in [50, 7, 51, 52, 53] relies on Commutant Lifting techniques, that of [30] is based on the original duality/Fejér-Riesz-type factorization approach of Sarason [55], that of [2, 26] on the “lurking isometry” technique, and that of [5, 6] on an adaptation of the Schur algorithm. It is also possible to approach

304

Ball and Bolotnikov

IEOT

these problems by using the grassmannian Kre˘ın-space approach of Ball-Helton (see [35]). A broad survey of multivariable interpolation problems going beyond the context of the Drury-Arveson space is given in [23]. In the present paper, we review the well-known solution criteria for these problems and how all these problems can be considered as mutually equivalent once one understands how to transform an admissible data set of one type to admissible data sets of each of the other two types; for the one-variable case these ideas are addressed in detail in the books [21, 36, 37]. We obtain a chain-matrix type linearfractional parametrization for the set of all solutions of such a problem for the case where the Pick operator is strictly positive definite by adapting the methods of Dym [33] and Potapov [45]. That these ideas can be adapted to this multivariable setting was already observed by the second author in [28]. We mention that recently Popescu (see [53, Theroem 2.3]) has obtained a parametrization formula for the set of all solutions in the context of the noncommutative commutant lifting theorem of Popescu [49] as well as a maximum entropy principle for this context (see Theorem 2.8 there). By the symmetrization technique mentioned above, this result in principle gives a parametrization for the set of solutions of interpolation problems discussed here, once one transforms the interpolation data to commutant-lifting data. We mention that the more complicated bitangential problem is studied in [13, 14] as well as in [16, 15] in a more general setting; when specialized to the lefttangential case, the analysis there gives a Redheffer linear-fractional parametrization for a particular case of the ssOAP-type problem (where it is assumed that the joint spectrum of T is contained in the open unit ball from which it follows in particular that T is strongly stable). It turns out that the problem ssOAP still makes sense without the strongly stable hypothesis; we refer to a problem of this type as a (not necessarily strongly stable) Operator Argument interpolation Problem OAP. The same methodology used here for ssOAP applies equally well to the OAP case; the associated reproducing kernel J,J ) (see Theorem 2.3 below) constructed from the interpolation Hilbert space H(KA data is contained contractively rather than isometrically in the ambient DruryArveson Kre˘ın space. We also introduce here a still more general interpolation problem, called the analytic Abstract Interpolation Problem (aAIP), where the assumption that the observability operators OE,T and ON,T map into the Drury-Arveson space is removed. Instead, in this more general formulation it is only assumed that the OE,T and ON,T map the state space into holomorphic vector-valued functions on the ball. For this setting the formulation (1.4) of the interpolation conditions does not make sense and one uses instead the following formulation: a Schur-multiplier

S is OE,T S said to solve the interpolation problem aAIP if the operator F = [ I −MS ] ON,T maps the state space into the de Branges-Rovnyak space H(KS ) associated with S. We show that the same solution procedure as for the previous cases still applies, J,J ) is no despite the fact that the associated reproducing kernel Hilbert space H(KA

Vol. 62 (2008)


305

longer contained in the Drury-Arveson space. It was shown in [27] that the boundary Nevanlinna-Pick problem on the ball is a particular instance of this problem aAIP which does not fit into the OAP framework. The boundary Nevanlinna-Pick problem for the Drury-Arveson space setting has also been studied in [4, 14]. Our final interpolation problem is a more implicit version of aAIP which we call the Abstract Interpolation Problem (AIP). The problem formulation calls for finding not only a Schur-class function S ∈ Sd (U, Y) but also a map F from the part of the state space specified by the problem data into the de Branges-Rovnyak space H(KS ). When specialized to the single-variable case d = 1, this problem coincides with a particular case (the “left-sided version”) of the Abstract Interpolation Problem introduced by Katsnelson, Kheifets and Yuditskii [40] and further studied in [41, 42, 43, 44, 46, 25]. We show how the approach of Arov-Grossman [10, 11] can be used to obtain a Redheffer-type linear-fractional description for the set of all solution pairs (F, S). This approach has already been used in other several-variable contexts in [13, 14, 15]. The paper is organized as follows. Section 2 collects various preliminary material on Schur multipliers, reproducing kernel Hilbert spaces and linear-fractional maps which will be needed in the sequel. Section 3 studies the three problems SIP, CLP and ssOAP and lays out the well-known solution criteria and the data manipulations showing the mutual equivalences among them. Section 4 extends the theory to the setting of OAP. Finally Section 5 studies the analytic Abstract Interpolation Problem and Section 6 handles the most general version of our problems, the Abstract Interpolation Problem. Section 3 is essentially a review of known material to set the context while Sections 4, 5, 6 present new results on more general types of interpolation problems for the Drury-Arveson Schur-multiplier class.

2. Schur multipliers, reproducing kernel Hilbert spaces, and linear-fractional transformations In this section we collect miscellaneous preliminary results needed for the work in the sequel. The following result appears in [26, 2, 34]. Theorem 2.1. Let S be a L(U, Y)-valued function analytic in Bd . The following are equivalent: 1. S belongs to Sd (U, Y).

IY − S(z)S(ζ)∗ is positive definite on Bd ×Bd or equiv1 − z, ζ alently, there exists an auxiliary Hilbert space H and an analytic L(H, Y)valued function H(z) on Bd so that

2. The kernel KS (z, ζ) =

I − S(z)S(ζ)∗ = H(z)H(ζ)∗ . 1 − z, ζ

(2.1)

306

Ball and Bolotnikov

3. There is a unitary operator U=

A C



A1  . B  . = . D Ad C

 B1 d ..  H H .  → : U Y  Bd D

such that

−1

S(z) = D + C (IH − Z(z)A) where Z(z) =

z1 IH

IEOT

···

zd IH

Z(z)B,

(2.2)

(2.3)

.

(2.4)

The representation (2.3) is called a unitary realization of S ∈ Sd (U, Y). More formally, we shall say that a collection C of the form C = {H, U, Y, U} with U of the form (2.2) is a (d-variable) colligation with connecting operator equal to U and with associated characteristic function equal to S(z) given by (2.3). If U is unitary, we say that the colligation is unitary. We say that the unitary colligation C is closely connected if the smallest subspace H0 ⊂ H such that H0 is reducing for A1 , . . . , Ad and H0 contains Ran C ∗ as well as Ran Bj for j = 1, . . . , d is the whole space H. In case H0 is not the whole space H, then d A0 B0 A|H0 B H0 H0 U0 = = : → C0 D U C|H0 D Y is again unitary and gives rise to the same characteristic function S(z). Hence there is no loss of generality in Theorem 2.1 if we assume that the unitary colligation C with connecting operator U in (2.2) is closely connected. Note that for S of the form (2.3), relation (2.1) holds with H(z) = C (IH − Z(z)A)−1 .

(2.5)

Note also that formulas (2.3) and (2.5) can be written directly in terms of the unitary operator U as follows: ∗ S(z) = PY U (IH⊕U − PH Z(z)PHd U)−1 |U ,

H(z) = PY U (IH⊕U −

(2.6)

−1 ∗ PH Z(z)PHd U) |H ,

(2.7)

where PY and PHd are the orthogonal projections of the space H ⊕ Y onto Y and ∗ Hd , respectively, and PH is the inclusion map of H into H ⊕ U. d

Associated with any S ∈ Sd (U, Y) is the de Branges-Rovnyak space H(KS ), the reproducing kernel Hilbert space with reproducing kernel KS (which is positive by Theorem 2.1). The original characterization of H(KS ), as the space of all functions f ∈ HY (kd ) such that f + Sg2HY (kd ) − g2HU (kd ) < ∞, (2.8) f 2H(KS ) := sup g∈HU (kd )

is due to de Branges and Rovnyak [29] (for the case d = 1). In particular, it follows from (2.8) that f H(KS ) ≥ f HY (kd ) for every f ∈ H(KS ), i.e., that H(KS ) is

Vol. 62 (2008)


307

contained in HY (kd ) contractively. On the other hand, the general complementation theory applied to the contractive operator MS provides the characterization of H(KS ) as the operator range 1

H(KS ) = Ran(I − MS MS∗ ) 2

(2.9)

with the lifted norm 1

(I − MS MS∗ ) 2 f H(KS ) = (I − π)f HY (kd )

(2.10) 1

for all f ∈ HY (kd ) where π here is the orthogonal projection onto Ker(I−MS MS∗ ) 2 . 1 Upon setting f = (I − MS MS∗ ) 2 h in (2.10) we get (I − MS MS∗ )hH(KS ) = (I − MS MS∗ )h, hHY (kd ) .

(2.11)

More complete details concerning the spaces H(KS ) and related matters of realization and the model theory for commutative row contractions can be found in the recent series of papers [18, 19, 20]. We shall also have use of the Beurling-Lax theorem for the Drury-Arveson space. The following definition of inner multiplier is useful. Definition 2.2. A contractive multiplier Θ ∈ Sd (F , Y) is called inner if the multiplication operator MΘ : HF (kd ) → HY (kd ) is a partial isometry. If Θ is inner, then the associated de Branges-Rovnyak space H(KΘ ) is isometrically included in HY (kd ) and H(KΘ ) = HY (kd ) ΘHF (kd ).

(2.12)

Moreover, the orthogonal projection PH(KΘ ) of HY (kd ) onto H(KΘ ) is given by ∗ PH(KΘ ) = IHY (kd ) − MΘ MΘ .

(2.13)

Since the space ΘHF (kd ) is shift invariant (i.e., Mzj -invariant for j = 1, . . . , d), it follows from (2.12) that the space H(KΘ ) is backward shift invariant (i.e., Mz∗j invariant for j = 1, . . . , d). The Beurling-Lax theorem for HY (kd ) (see [12, 48, 18, 20] for the commutative setting and [50, 31, 18] for the noncommutative setting from which the commutative setting can be derived) asserts that any shift invariant closed subspace M of HY (kd ) necessarily has the form ΘHF (kd ) for some inner multiplier Θ ∈ Sd (F , Y); in this situation we say that Θ is a Beurling-Lax representer for the shift-invariant subspace M. Therefore any backward-shift-invariant subspace M of HY (kd ) has the form M = H(KΘ ). It is convenient to introduce the following noncommutative multivariable functional-calculus notation from [18] even though here we are only interested in the commutative setting. We let Fd denote the free semigroup generated by the alphabet consisting of the letters {1, . . . , d}. Elements of Fd are words v = iN · · · i1 where each ik ∈ {1, . . . , d}. Given such a word v = iN · · · i1 ∈ Fd , we let |v| = N denote the length of the word (i.e., the number of letters in v) and we let a(v) ∈ Zd+

308

Ball and Bolotnikov

IEOT

be the abelianization of v, i.e., the d-tuple (n1 , . . . , nd ) of nonnegative integers determined by a(v) = (n1 , . . . , nd ) where

nj = #{k : ik = j} for

j = 1, . . . , d

(where in general #Ξ denotes the cardinality of the set Ξ). If T = (T1 , . . . , Td ) is a d-tuple of Hilbert-space operators and if z = (z1 , . . . , zd ) is a collection of complex variables, we use the standard multivariable notation: T v = TiN · · · Ti1 ,

z v = ziN · · · zi1

if v = iN · · · i1 . If n = (n1 , . . . , nd ) ∈ Zd+ and if T is a commutative operator d-tuple, we write Tn = T1n1 · · · Tdnd ,

z n = z1n1 · · · zdnd .

If v ∈ Fd and T is a commutative operator d-tuple, we then have the following connections between the noncommutative and commutative multivariable functional calculus: Tv = Ta(v) , z v = z a(v) . A useful combinatorial fact is the following: for a given n ∈ Zd+ , |n|! . n! We shall need some J-analogues of results concerning ranges of observability operators and associated reproducing kernel Hilbert spaces given in [18]. Let T = (T1 , · · · , Td ) be a d-tuple of operators in L(X ) and let C ∈ L(X , Y). The pair (C, T) is said to be output-stable if the associated observability operator #{v ∈ Fd : a(v) = n} =

OC,T : x → C(IX − z1 T1 − · · · − zd Td )−1 x = C(IX − Z(z)T )−1 x where

 T1   T =  ...  : X → X d 

and Z(z) = z1 IX

···

zd IX ,

Td maps X into HY (kd ) and is bounded, i.e., OC,T ∈ L(X , HY (kd )). To obtain the Taylor expansion for OC,T x ∈ HY (kd ), we compute: (OC,T x)(z) = C(IX − Z(z)T )−1 x = =C

∞

∞

C(Z(z)T )N x

N =0

Tv z v x = C

N =0 v∈Fd : |v|=N

=C

n∈Zd +

v∈Fd : a(v)=n

Tv z v x

v∈Fd

Tv z v x =

We now introduce the observability gramian ∗ OC,T GC,T := OC,T

|n|! CTn z n x. n! d

n∈Z+

(2.14)

Vol. 62 (2008)


whose representation in terms of strongly convergent power series |n|! Tn∗ C ∗ CTn = GC,T = Tv∗ C ∗ CTv n! d n∈Z+

309

(2.15)

v∈Fd

follows from the power-series expansion (2.14) for the observability operator together with the characterization (1.1) of the HY (kd )-norm. An important property of GC,T is that it satisfies the Stein equation P−

d

Tj∗ P Tj = C ∗ C

(2.16)

j=1

as can be seen by plugging in the series expansion (2.15). It is useful to identify the special case where in addition the commutative d-tuple is strongly stable, i.e., N lim Tn x2X = 0 for all x ∈ X . Tv x2X = lim N →∞ N →∞ n! d v∈Fd : |v|=N

n∈Z+ : |n|=N

(2.17) We will consider an output-stable pair (C, T) where E Y C= : X → N U and T = (T1 , . . . , Td ) is a commutative d-tuple of operators on X . We let IY 0 J= . 0 −IU

(2.18)

(2.19)

In addition, we shall often have use for the operator J ⊗ IHY ⊕U (kd ) acting on HY⊕U (kd ); we shall abuse notation and write this operator also as simply J. We J of the pair (C, T) by define the J-gramian GC,T J ∗ ∗ := O∗E ,T JO[ E ],T = OE,T OE,T − ON,T ON,T = GE,T − GN,T . GC,T [N ] N

(2.20)

J is that it solves the Stein equation An important property of GC,T

P−

d

Tj∗ P Tj = C ∗ JC,

(2.21)

j=1

as follows easily from the fact that GE,T and GN,T satisfy Stein equations of the type (2.16), or by plugging in the infinite series representations J = GE,T − GN,T GC,T = Tv∗ (E ∗ E − N ∗ N ) Tv = Tv∗ C ∗ JCTv v∈Fd J for GC,T .

v∈Fd

(2.22)

310

Ball and Bolotnikov

IEOT

If X is a Hilbert space and X is a selfadjoint operator operator on X , we use the notation (X , X) to denote the space X with the indefinite inner product induced by X: x, yX := Xx, yX . Usually it is assumed that X is invertible, so (X , X) is a Hilbert space if X is positive definite and a Kre˘ın space in general. The following result is an indefinite analogue of Theorem 3.14 from [18]. Theorem 2.3. Let T = (T1 , . . . , Td ) be a commutative d-tuple on X and let C of the form (2.18) be such that the pair (C, T) is output-stable and the J-gramian J P := GC,T given by (2.20) is strictly positive definite on X . Then: 1. The operator OC,T : (X , P ) → (HY⊕U (kd ), J) is a contraction. This operator is isometric if and only if T is strongly stable. 2. If the space M := Ran OC,T is given the lifting norm 2

OC,T x = P x, xX , then M is isometrically equal to the reproducing kernel Hilbert space with P reproducing kernel KC,T given by P KC,T (z, ζ) = C(I − Z(z)T )−1 P −1 (I − T ∗ Z(ζ)∗ )−1 C ∗ . (2.23) B F Xd 3. If a Hilbert space F and an operator : → are such that D U Y ⊕U the operator T B X Xd (2.24) U= : → Y ⊕U C D F ⊕U

satisfies −1 P U 0

−1 P ⊗ Id 0 ∗ U = J 0

0 , J

where

I J= F 0

0 , −IU

(2.25)

P (z, ζ) appearing in (2.23) can be expressed as then the kernel KC,T J,J P KC,T (z, ζ) = KA (z, ζ) :=

J − A(z)JA(ζ)∗ 1 − z, ζ

(2.26)

where A(z) is the characteristic function of the colligation U in (2.24): A(z) = D + C(I − Z(z)T )−1 Z(z)B. If the operators B and D are such that U in (2.24) is subject to P 0 P ⊗ Id 0 U= , U 0 J 0 J

(2.27)

(2.28)

then A(z) is bi-(J, J)-contractive for each z ∈ Bd : A(z)JA(z)∗ ≤ J,

A(z)∗ JA(z) ≤ J

(z ∈ Bd ).

(2.29)

Vol. 62 (2008)


311

One such construction of B, D in (2.24) is to take F = X d−1 ⊕ Y and then to solve the J-Cholesky factorization problem −1 P ⊗ Id 0 T B (2.30) − P −1 T ∗ C ∗ J B ∗ D∗ = 0 J C D B ] injective. with [ D

Proof. To prove (1), we make subsequent use of (2.20), (2.22) and (2.21) to get JOC,T x, OC,T HY ⊕U (kd )

J GC,T x, x X = Tv∗ C ∗ JCTv x, xX =

v∈Fd ∞

=

T (P − v∗

N =0 v∈Fd : |v|=N

Tj∗ P Tj )Tv x, x

j=1



=

d

lim P x, xX −

N →∞

X

Tv∗ P Tv x, x

v∈Fd : |v|=N +1

≤ P x, x

X

with equality in the last step for all x ∈ X if and only if T is strongly stable. Statement (2) follows by standard reproducing kernel Hilbert space considerations; for this we refer the reader to [18]. As for statement (3), assume first that U as in (2.24) has been constructed so as to satisfy (2.25) and that we set A equal to the characteristic function of U as in (2.27). From the J-coisometry property (2.25) of U we read off the relations T P −1 T ∗ + BJB ∗ = P −1 ⊗ Id ,

CP −1 T ∗ + DJB ∗ = 0,

T P −1 C ∗ + BJD∗ = 0,

CP −1 C ∗ + DJD∗ = J.

Then we compute A(z)JA(ζ)∗ = DJD∗ + C(I − Z(z)T )−1 Z(z)BJD∗ + DJB ∗ Z(ζ)∗ (I − T ∗ Z(ζ)∗ )−1 C ∗ + C(I − Z(z)T )−1 Z(z)BJB ∗ Z(ζ)∗ (I − T Z(ζ)∗ )−1 C ∗ = J − CP −1 C ∗ − C(I − Z(z)T )−1 Z(z)T P −1 C ∗ − CP −1 T ∗ Z(ζ)∗ (I − T ∗ Z(ζ)∗ )−1 C ∗ + C(I − Z(z)T )−1 (P −1 ⊗ Id − T P −1 T ∗ )Z(ζ)∗ (I − T ∗ Z(ζ)∗ )−1 C ∗ = J − C(I − Z(z)T )−1 Γ(I − T ∗ Z(ζ)∗ )−1 C ∗

312

Ball and Bolotnikov

IEOT

where Γ is given by Γ

= (I − Z(z)T )P −1 (I − T ∗ Z(ζ)∗ ) + Z(z)T P −1 (I − T ∗ Z(ζ)∗ ) +(I − Z(z)T )P −1T ∗ Z(ζ)∗ + Z(z)(T P −1T ∗ − P −1 ⊗ Id )Z(ζ)∗ = P −1 − Z(z)(P −1 ⊗ Id )Z(ζ)∗ = (1 − z, ζ)P −1

from which (2.26) follows. If B and D are constructed so that U also satisfies (2.28), then we have the additional relations T ∗ (P ⊗ Id )T + C ∗ JC = P,

T ∗ (P ⊗ Id )B + C ∗ JD = 0,

B ∗ (P ⊗ Id )T + D∗ JC = 0,

B ∗ (P ⊗ Id )B + D∗ JD = J.

Then a computation similar to that used in the previous paragraph shows that J − A(w)∗ JA(z) = B ∗ (I − Z(w)∗ T ∗ )−1 (P ⊗ Id − Z(w)∗ P Z(z))(I − T Z(z))−1 B. In particular, taking w = z gives J − A(z)∗ JA(z) = (I − Z(z)∗ T ∗ )Π(z)(I − T Z(z))−1 B where Π(z) = P ⊗ Id − Z(z)∗ P Z(z) = (Id − z ∗ z) ⊗ P is the tensor product of two operators Id −z ∗ z (where here we view positive-definite d z as the row matrix z = z1 · · · zd : C → C) and hence is positive-definite. To construct B, D so that U as in (2.24) satisfies(2.25), proceed as follows. From T the Stein equation (2.21), we see that G := Ran is a uniformly positive subC d 0 ın-space orthogonal space of the Kre˘ın space (X d ⊕Y ⊕U, P ⊗I 0 J ). Hence the Kre˘ [⊥] complement G of G is also a Kre˘ ın space in inner product inherited from the d 0 ) with inertia equal to the complement of the ambient space (X d ⊕ Y ⊕ U, P ⊗I 0 J P ⊗ Id 0 inertia of P with respect to the inertia of on the large space, namely, 0 J P ⊗ Id−1 0 on X d−1 ⊕ Y ⊕ U. Therefore there is with inertia equal to that of 0 J

B B I 0 ⊕Y . If an isometry from (X d−1 ⊕ Y ⊕ U, J) where we set J = X d−1 0 −IU D D is such an isometry, then the orthogonal (with respect to the Kre˘ın-space inner [∗] B B d [⊥] product) projection PG [⊥] of X ⊕ Y ⊕ U onto G is given by PG [⊥] = D D where X [∗] denotes the Kre˘ın-space adjoint of the Hilbert space operator X. For the case of d−1 B X ⊕Y Xd P ⊗ Id 0 : ,J → , , D 0 J U Y ⊕U

Vol. 62 (2008)


B ] is given by the Kre˘ın-space adjoint of [ D [∗] P ⊗ Id B = J B ∗ D∗ 0 D

and hence we have PG [⊥] =

B J B∗ D

D∗

P ⊗ Id 0

0 J

313

0 . J

(2.31)

On the other hand the Kre˘ın-space orthogonal projection of X d ⊕ Y ⊕ U onto [∗] T T T T ] G = Ran is given by PG = where the Kre˘ın-space adjoint of [ C C C C is given by [∗] P ⊗ Id 0 T = P −1 T ∗ C ∗ . C 0 J Therefore we get a second expression for the Kre˘ın-space orthogonal projection PG [⊥] , namely ∗ P ⊗ Id 0 T −1 ∗ T C PG [⊥] = IX d ⊕Y⊕U − PG = I − P . (2.32) C 0 J Equating the two expressions (2.31) and (2.32) for PP [⊥] gives the identity ∗ P ⊗ Id 0 P ⊗ Id 0 ∗ B T −1 ∗ ∗ C D T = J B . I− P 0 J D 0 J C

−1 Multiplication on the right by P 0⊗Id J0 then leaves us with the expression (2.30) as the equation to be solved for B and D to complete the construction. We can B ] is injective; this then guarantees that the resulting always solve (2.30) so that [ D U of the form (2.24) also satisfies (2.28). The hypothesis that (C, T) is output-stable in Theorem 2.3 can be weakened as follows. Rather than assuming that OC,T maps the space X into HY⊕U (kd ), we assume only that OC,T maps X into the space HolY⊕U (Bd ) of (Y ⊕ U)-valued holomorphic functions on Bd . Then there of course is no hope for the validity of statement (1) in Theorem 2.3, but statement (2) holds as stated. For the validity B ] so that U as in of statement (3) all that is required (for the construction of [ D (2.24) satisfies (2.25) and (2.28)) is that P be a positive definite solution of the Stein equation (2.21). We are led to the following result; as the proof is essentially the same as that of Theorem 2.3, we leave the details to the reader. Theorem 2.4. Suppose that (C, T) is an analytic output-pair, i.e., C(I − Z(z)T )−1 x ∈ HolY⊕U (Bd ) for all x ∈ X . Assume also that P ∈ L(X ) is a positive definite solution of the Stein equation (2.21). Then statements (2) and (3) in Theorem 2.3 hold without change.

314

Ball and Bolotnikov

IEOT

Remark 2.5. In the setting of Theorem 2.3, we have J,J H(KA ) ⊂ HY⊕U (kd ).

(2.33)

If we assume in addition that T is strongly stable, then the inclusion (2.33) is isometric. The L(F ⊕U, Y ⊕U)-valued function A in general is not in the multiplier class Md (F ⊕ U, Y ⊕ U) but it is true that A maps constant functions in HF ⊕U (kd ) [∗] (and hence also polynomials) into HY⊕U (kd ). In this case the operator I −MA MA , initially only defined on linear combinations of kernel functions via y y y [∗] I − MA MA = I − MA JMA∗ J : kd (·, ζ) → kd (·, ζ) − A(·)kd (·, ζ)JA(ζ)∗ J u u u extends continuously to the J-orthogonal projection operator mapping HY⊕U (kd ) J,J onto H(KA ). In this situation we say that A is a (J, J)-inner multiplier. In the single-variable case (d = 1), we have J = J and these functions coincide with the strongly regular J-inner functions in the sense of Arov-Dym [9] (see also [24]) which are analytic on the unit disk D = B1 . Remark 2.6. Suppose that M ⊂ HY⊕U (kd ) is a shift-invariant subspace which is J-regular in the sense that the J-orthogonal complement M[⊥]J of M in HY⊕U (kd ) with respect to the J-inner product together with M forms a direct-sum decomposition of HY⊕U (kd ): ˙ [⊥]J . HY⊕U (kd ) = M+M As shown in [20, Theorem 4.8], there exists an output-stable pair (C, T) so that ∗ M = Ker OJC,T . It then follows that M[⊥]J = Ran OC,T . One choice of C : X → Y ⊕ U and T = (T1 , . . . , Td ) ∈ L(X )d is X = M[⊥]J ,

Tj = Mz∗j |M[⊥]J for j = 1, . . . , d,

C : f → f (0).

(Note that M[⊥]J is backward-shift-invariant since M is shift-invariant.) Since M[⊥]J is a Kre˘ın subspace of HY⊕U (kd ), it follows that the J-observability gramian J ∗ P := GC,T = OC,T JOC,T is invertible. Moreover, (C, T) satisfies the Stein equation (4.4). One can then construct a solution U of the form (4.10) which satisfies (2.25) and define A(z) as in (2.27). Then, as in the proof of Theorem (2.3) we see J,J that M[⊥]J is isometrically equal to H(KA ). If it is the case that A is a bounded multiplier between HF ⊕U (kk ) and HY⊕U (kd ), it follows that M = (M[⊥]J )[⊥]J = MA HF ⊕U (kd ) where the multiplication operator MA in addition is a (J, J)-partial isometry. This representation of the shift-invariant subspace M is a Kre˘ın-spaces version of the Beurling-Lax theorem [12, 48, 18, 20] cited above and goes back to [22] for the single-variable case.

Vol. 62 (2008)


315

Suppose that we are given a holomorphic L(F ⊕ U, Y ⊕ U)-valued function A(z) on Bd as in Theorem 2.4 so that A(z) is bi-(J, J)-contractive for each z ∈ Bd . If we decompose A as a block 2 × 2 matrix A11 (z) A12 (z) F Y A(z) = : → A21 (z) A22 (z) U U and recall the conformal decompositions (2.19) and (2.25) for J and J, then we see from the (2, 2)-entry in the inequalities (2.29) that A21 (z)A21 (z)∗ − A22 (z)A22 (z)∗ ≤ −IU , A12 (z)∗ A12 (z) − A22 (z)∗ A22 (z) ≤ −IU from which we get A22 (z)A22 (z)∗ ≥ IU + A21 (z)A21 (z)∗ , A22 (z)∗ A22 (z) ≥ IU + A12 (z)∗ A12 (z) and hence also A22 (z) is invertible and A22 (z)−1 A21 (z) < 1 for each z ∈ Bd .

(2.34)

We conclude that A21 (z)E(z) + A22 (z) = A22 (z)(A22 (z)−1 A21 (z)E(z) + I) is invertible for all z ∈ Bd and E ∈ Sd (U, F ) and hence the linear fractional transform of E TA [E](z) = (A11 (z)E(z) + A12 (z))(A21 (z)E(z) + A22 (z))−1

(2.35)

is a well-defined holomorphic L(U, Y)-valued function on B . We denote by SdJ,J (F ⊕ U, Y ⊕ U) the indefinite Schur class of L(F ⊕ U, Y ⊕ U)-valued functions A J,J (z, ζ) given analytic and bi-(J, J)-contractive on Bd and such that the kernel KA by (3.7) is positive on Bd × Bd . Thus, for an A ∈ SdJ,J (F ⊕ U, Y ⊕ U), the linear fractional map TA given by (2.35) is well-defined on Sd (U, F ). Theorem 2.8 gives a useful characterization of its range. As a first step in this direction, we need the following interpolation result (the Leech theorem for Drury-Arveson-space multipliers) which was established in [3, 14]. d

Theorem 2.7. Let E be a Hilbert space and let a : Bd → L(Y, E)

and

c : Bd → L(U, E)

be operator valued functions defined on Bd . The following are equivalent: 1. There exists a function S ∈ Sd (U, Y) such that a(z)S(z) = c(z)

for all z ∈ Bd .

2. There exist a Hilbert space H and a function R(z) : Bd → L(H, E), such that a(z)a(ζ)∗ − c(z)c(ζ)∗ = R(z)R(ζ)∗ . 1 − z, ζ

(2.36)

316

Ball and Bolotnikov

IEOT

Theorem 2.8. Let A ∈ SdJ,J (F ⊕ U, Y ⊕ U). Then a holomorphic L(U, Y)-valued function S has the form S = TA [E] (2.37) for some E ∈ Sd (U, F ) if and only if S ∈ Sd (U, Y) and the operator y(z) I −MS : → y(z) − S(z)u(z) u(z) J,J maps H(KA ) contractively into the de Branges-Rovnyak space H(KS ).

Proof. The result can be found in the proof of Theorem 3.8 in [28] and appears in [33] for the case d = 1; we include the short proof for completeness. Suppose that J,J S ∈ Sd (U, Y) and that I −MS maps H(KA ) contractively into H(KS ). Then the kernel KS is positive and we have the kernel inequality J,J I I −S(z) KA (z, ζ) KS (z, ζ), −S(ζ)∗ or, what is the same, J − A(z)JA(ζ)∗ I − S(z)S(ζ)∗ I I −S(z) . −S(ζ)∗ 1 − z, ζ 1 − z, ζ Note that

I

−S(z) J

I = I − S(z)S(ζ)∗ . −S(ζ)∗

Hence we can rearrange (2.38) to A(z)JA(ζ)∗ I I −S(z) 0. 1 − z, ζ −S(ζ)∗ If we set

then we get

(2.38)

u(z) −v(z) := I

(2.39)

−S(z) A(z),

u(z)u(ζ)∗ − v(z)v(ζ)∗ 0, 1 − z, ζ

(2.40)

where u(z) = A11 (z) − S(z)A21 (z),

−v(z) = A12 (z) − S(z)A22 (z).

By Theorem 2.7, it follows that there exists E ∈ Sd (U, Y) so that v(z) = u(z)E(z), i.e., −(A12 (z) − S(z)A22 (z)) = (A11 (z) − S(z)A21 (z))E(z) which can be rearranged as S(z)(A21 (z)E(z) + A22 (z)) = A11 (z)E(z) + A12 (z). It now follows that we recover S as S = TA [E]. Conversely, suppose that E ∈ Sd (U, F ) and S = TA [E]. By reversing the steps in the argument above and using that condition (2.40) is necessary as well as sufficient in Theorem 2.36, we arrive at (2.39). We then add KS (z, ζ) to both sides of (2.39)

Vol. 62 (2008)


317

J,J to arrive at (2.38). As KA is a positive kernel by assumption, we conclude that KS is a positive kernel, i.e., that S ∈ Sd (U, Y). Then the inequality (2.38) is the J,J statement that I −MS maps H(KA ) contractively into H(KS ).

In addition to the linear-fractional transformations of chain-matrix form (2.35) as discussed above we shall also have use of linear-fractional transformations of Redheffer these, we suppose that we are given a matrix function

form. To define Σ (z) Σ12 (z) ! ∗ , X ⊕ ∆) ! holomorphic on the ball Bd with values in L(X ⊕ ∆ Σ(z) = Σ11 21 (z) Σ22 (z) ! ∗ , X , ∆. ! We assume that Σ22 (z) < 1 for all z ∈ Bd . for some Hilbert spaces X , ∆ ! ∆ ! ∗ ). From the positivity of Suppose that W is in the Schur-multiplier class Sd (∆, ∗ the kernel KW (z, ζ) = [I − W(z)W(ζ) ]/(1 − z, ζ) we see in particular that W(z) ≤ 1 for each z ∈ Bd and it follows that (I − W(z)Σ22 (z))−1 makes sense ! ∗ )-valued function on Bd . We then can define the associated as a holomorphic L(∆ ! ∆ ! ∗ ) to HolL(X ,X ) (Bd ) by Redheffer linear-fractional map RΣ acting from Sd (∆, RΣ [W] := Σ11 (z) + Σ12 (z)(I − W(z)Σ22 (z))−1 W(z)Σ21 (z).

(2.41)

The following criterion for a given function S to be in the range of RΣ , while less explicit than the criterion in Theorem 2.8, nevertheless is useful in some applications (see Theorem 6.4 below). For this purpose we say that a pair of functions d a ∈ HolL(∆ ! ∗ ,X ) (B ),

d c ∈ HolL(∆,X ) (B ) !

is a Schur-pair if the associated kernel below is positive: a(z)a(ζ)∗ − c(z)c(ζ)∗ 0. 1 − z, ζ

(2.42)

Theorem 2.9. Suppose that we are given Σ11 Σ12 d Σ= ∈ HolL(X ⊕∆ ! ∗ ,X ⊕∆) ! (B ) Σ21 Σ22 with Σ22 (z) < 1 for each z ∈ Bd . Suppose that we are also given an operatorvalued function S ∈ HolL(X ,X ) (Bd ). Then there exists a Schur-class multiplier ! ∆ ! ∗ ) such that S = RΣ [W] if and only if there exists a Schur-pair W ∈ Sd (∆, (a(z), c(z)) so that I c(z) Σ(z) = S(z) a(z) . (2.43) Proof. Suppose that (a(z), c(z)) is a Schur-pair satisfying (2.43). By Theorem 2.7, ! ∆ ! ∗ ) so that there is a Schur-class multiplier W ∈ Sd (∆, c(z) = a(z)W(z).

(2.44)

Then (2.43) can be written as Σ11 (z) + a(z)W(z)Σ21 (z) = S(z) Σ12 (z) + a(z)W(z)Σ22 (z) = a(z).

(2.45)

318

Ball and Bolotnikov

IEOT

From the second of equations (2.45) we can solve for a(z): a(z) = Σ12 (z)(I − W(z)Σ22 (z))−1 .

(2.46)

If we plug this expression into the first of equations (2.45) we get S(z) = Σ11 (z) + Σ12 (z)(I − W(z)Σ22 (z))−1 W(z)Σ21 (z) = RΣ [W](z) as wanted. For the converse direction, given that S = RΣ [W], if we define (a(z), c(z)) by (2.44) and (2.46), then (a, c) is a Schur-pair meeting the criterion (2.43). Remark 2.10. As the solution of the Leech problem appearing in both the proof of Theorem 2.8 and the proof of Theorem 2.9 does not appear to be unique, in general we would expect that S does not uniquely determine E (or W) in the formula (2.37) (or (2.41)). A similar issue appears in the parametrization of all solutions of the commutant lifting problem for the single-variable case [36, 37] and in connection with the more general relaxed commutant lifting problem (see [38, 39]). Remarkably it turns out in this context that one can parametrize the free parameters E which give rise to a given S and that, in the context of (nonrelaxed or “stiff”) commutant lifting, it happens that S does uniquely determine E. It remains to be seen if something similar can be worked out in the multivariable context here.

3. Higher-order left tangential Nevanlinna-Pick interpolation: compact operator-theoretic formulations Suppose that we are given the data {ζ (i) , a∗i , c∗i : i = 1, . . . , N } for a LNPP (i.e., a left-tangential Nevanlinna-Pick interpolation problem with associated interpolation conditions of the form (1.3)), and suppose that we know one solution Ψ ∈ Md (U, Y) (with MΨ not necessarily contractive) of the set of interpolation conditions (1.3) (with Ψ in place of S). We note that the subspace N ⊂ HY (kd ) defined by N = {h ∈ HY (kd ) : a∗i h(ζ (i) ) = 0 for i = 1, . . . , N } is closed and shift-invariant. Hence, by the Beurling-Lax theorem for the DruryArveson space mentioned at the end of Section 2 above, we know that there is an inner multiplier Θ ∈ Sd (F , Y) (for some auxiliary Hilbert space F ) so that N = ΘHF (kd ). If S ∈ Md (U, Y) is another solution of the interpolation conditions (1.3), then we see that MS−Ψ : HU (kd ) → Θ · HF (kd ). It then follows as a consequence of the Leech theorem for multipliers on the DruryArveson space (see [3]) that there is a F ∈ Md (U, F ) so that S has the form S = Ψ + ΘF . With the new data set {Ψ, Θ} (where F is a multiplier in Md (U, Y) and where Θ is an inner multiplier in Sd (F , U), we see that the LNPP can be reformulated as a Sarason interpolation problem:

Vol. 62 (2008)


319

Sarason interpolation problem (SIP): Given Ψ ∈ Md (U, Y) and an inner Θ ∈ Sd (F , U), find all S ∈ Sd (U, Y) such that S(z) = Ψ(z) + Θ(z)F (z)

for some

F ∈ Md (U, F ).

As was first done for the single-variable case in the classical paper [55] of Sarason and then in [50, 53] for both the noncommutative and commutative ball setting, the SIP can be put in more operator-theoretic form as follows. Given the data set {Ψ, Θ} for a Sarason interpolation problem, introduce the subspace M by M = HY (kd ) ΘHY (kd ) and define the operator Φ : HU (kd ) → M by Φ = PM M Ψ . Note that M is backward-shift-invariant and that Φ satisfies the intertwining property: PM Mzj Φ = ΦMzj (j = 1, . . . , d). (3.1) Furthermore, one easily checks that a multiplier S ∈ Md (U, Y) solves the SIP(Ψ, Θ) if and only if MS satisfies the conditions PM M S = Φ

and MS ≤ 1.

From these conditions we read off that necessarily Φ ≤ 1

(3.2)

if SIP(Ψ, Θ) has a solution. As mentioned previously, multipliers MS are characterized as those operators between HU (kd ) and HY (kd ) which intertwine the respective shift operators Mzj ⊗ IU and Mzj ⊗ IY for j = 1, . . . , d. It now follows that the SIP can be reformulated as the following Commutant Lifting Problem: Commutant Lifting Problem (CLP): Given an M∗z -invariant subspace M of HY (kd ) and an operator Φ : HU (kd ) → M subject to (3.1), find an operator R : HU (kd ) → HY (kd ) such that R ≤ 1,

PM R = Φ

and

Mzj R = RMzj

(j = 1, . . . , d),

(3.3)

or equivalently, find a contractive multiplier S ∈ Sd (U, Y) such that PM MS = Φ.

(3.4)

If R is a solution of CLP, then it follows from (3.3) that Φ = PM R ≤ R ≤ 1 and also ΦMzj = PM RMzj = PM Mzj R = PM Mzj PM R = PM Mzj Φ, where the third equality holds due to the backward shift invariance of M. Hence the conditions (3.1) and (3.2) are certainly necessary for the existence of a solution to CLP. That the converse holds is the assertion of the commutant lifting theorem

320

Ball and Bolotnikov

IEOT

(see [55, 36] for the single-variable case and [26] as well as [49] combined with [7, 30] for the present Drury-Arveson space setting). Given a Sarason interpolation problem SIP(Ψ, Θ), we have seen how to pass to a CLP(M, Φ). Conversely, it is possible to pass from a CLP(M, Φ) to a SIP(Ψ, Θ) as follows. Take any Beurling-Lax representer Θ ∈ Sd (F , Y) for M ⊂ HY (kd ) and choose any Ψ ∈ Md (U, Y) (not necessarily contractive) so that Φ = PM MΨ . The fact that such a multiplier Ψ always exists is of course a consequence of the commutant lifting theorem. We next formulate the left-tangential interpolation problem with operator argument and show that the CLP is a particular case of this problem. For an output-stable pair (E, T) (in the sense defined in Section 2) with a commutative d-tuple T we define a left-tangential functional calculus f → (E ∗ f )∧L (T∗ ) on HY (kd ) by (E ∗ f )∧L (T∗ ) = T∗n E ∗ fn if f = fn z n ∈ HY (kd ). (3.5) n∈Zd +

The computation ∗n ∗ T E fn , x n∈Zd +

= X

n∈Zd +

fn , ETn xY

n∈Zd +

n! |n|! n ET x = = f, OE,T xHY (kd ) fn , |n|! n! Y d n∈Z+

shows that the output-stability of the pair (E, T) is exactly what is needed to verify that the infinite series in the definition (3.5) of (E ∗ f )∧L (T∗ ) converges in the weak topology on X . In fact the left-tangential evaluation with operator argument f → (E ∗ f )∧L (T∗ ) amounts to the adjoint of the observability operator: ∗ f (E ∗ f )∧L (T∗ ) = OE,T

for f ∈ HY (kd ).

(3.6)

The evaluation map (3.6) extends to multipliers S ∈ Md (U, Y) by ∗ MS : U → X (E ∗ S)∧L (T∗ ) = OE,T

(3.7)

and suggests the interpolation problem with operator argument OAP(T, E, N ) whose data set consists of a commutative d-tuple T = (T1 , . . . , Td ) and operators E ∈ L(X , Y) and N ∈ L(X , U) such that the pair (E, T) is output stable. Operator Argument interpolation Problem (OAP(T, E, N )): Given the data set {T, E, N } as above, find all S ∈ Sd (U, Y) such that ∗ MS |U = N ∗ . (E ∗ S)∧L (T∗ ) := OE,T

(3.8)

Such problems have been considered in [52, 16, 53, 17] for the commutative and noncommutative setting. In case the d-tuple T = (T1 , . . . , Td ) is strongly stable (see

Vol. 62 (2008)


321

(2.17)), we shall refer to OAP(T, E, N ) as a strongly stable Operator Argument interpolation Problem (ssOAP(T, E, N )). If (ζ (i) , a∗i , c∗i ) (i = 1, . . . , N ) is the data set for a left Nevanlinna-Pick problem LNPP and if we set  (1)   ∗  ∗ ζj a1 c1       . ∗ ∗ ∗ .   .. Tj =  E =  ..  , N =  ...   for j = 1, . . . , d, (N ) a∗N c∗N ζj and if we set T = (T1 , . . . , Td ), then it is easily checked that condition (3.8) amounts to the LNPP interpolation conditions (1.3). Furthermore, as by assumption ζ (k) ∈ Bd for k = 1, . . . , N , one can show that T is strongly stable. Thus we see that any LNPP can be encoded as a strongly stable Operator Argument interpolation Problem ssOAP. More generally, Carathéodory-Fejér-type interpolation problems can be embedded into ssOAP problems—see [16]. We shall see in Theorem 3.3 below that, more generally, ssOAP is exactly equivalent to commutant lifting and Sarason interpolation problems after appropriate transformations of the data sets. First we observe a simple necessary conditions for a problem OAP(T, E, N ) (as well as for a problem ssOAP(T, E, N )) to have a solution. Proposition 3.1. Let (E, T) be an output-stable pair with E ∈ L(X , Y), let S ∈ Md (U, Y) and let N be defined as in (3.8). Then 1. The pair (N, T) is output stable and ∗ ∗ MS = ON,T : HU (kd ) → X . OE,T

(3.9)

∗ ∗ 2. If S ∈ Sd (U, Y), then ON,T ON,T ≤ OE,T OE,T .

Hence, if the problem OAP(T, E, N ) has a solution S ∈ Sd (U, Y), then (N, T) is also output-stable and ∗ ∗ OE,T − ON,T ON,T ≥ 0. P := OE,T

Proof. Let h(z) =

(3.10)

hn z n ∈ HU (kd ). By (3.6) and (3.8),

n∈Z+ d ∗ MS h = (E ∗ Sh)∧L (T∗ ) = OE,T

n,m∈Z+ d

(T∗ )n+m E ∗ Sm hn

(3.11)

322

Ball and Bolotnikov

IEOT

where the latter series converges weakly since the pair (E, T) is output stable and since Sh ∈ HY (kd ). On the other hand, ∗ h = (N ∗ h)∧L (T∗ ) = T∗n N ∗ hn ON,T

=

n∈Z+ d

 T∗n 

n∈Z+ d



T∗m E ∗ Sm  hn =

m∈Z+ d

(T∗ )n+m E ∗ Sm hn

n,m∈Z+ d

where all the series converge weakly, since that in (3.11) does. Since h was picked ∗ arbitrarily in HU (kd ), we get (3.9). The operator ON,T is bounded and therefore the pair (N, T) is output stable. If S ∈ Sd (U, Y), then I − MS MS∗ ≥ 0 and by (3.9) we have for every x ∈ X , 0

≤ (I − MS MS∗ )OE,T x, OE,T xHY (kd ) = OE,T x2HY (kd ) − MS∗ OE,T x2HU (kd ) = OE,T x2HY (kd ) − ON,Tx2HU (kd )

which proves the second statement and completes the proof of the proposition. Corollary 3.2. Conditions (3.8) and (3.9) are equivalent. Proof. Indeed, Proposition 3.1 shows that (3.8) implies (3.9). The converse implication follows upon restricting equality (3.9) to constant functions from HU (kd ): ∗ ∗ OE,T MS u = ON,T u ∗ ∗ MS u = (E ∗ S)∧L (T∗ )u and ON,T |U = N ∗ . and taking into account that OE,T

A convenient way to impose various hypotheses in the formulation of a particular class of interpolation problems is to demand that the data set for the interpolation problem of the class satisfy certain admissibility criteria. In what follows, we call the data set {M, Φ} to be admissible for a CLP if M is a backward shift invariant subspace of HY (kd ) and Φ : HU (kd ) → M satisfies relations (3.1), Furthermore, the collection {T = (T1 , . . . , Td ), E, N } is admissible for an OAP if T is a commutative d-tuple of operators on the Hilbert space X and E : X → Y and N : X → U are such that both (E, T) and (N, T) are output-stable. The same collection will be called admissible for a ssOAP, if in addition, the d-tuple T is strongly stable. Theorem 3.3. Let M be a backward shift invariant subspace of HY (kd ), let Φ ∈ HU (kd ) → M satisfy conditions (3.1) and let T = (T1 , . . . , Td ) ∈ L(M)d , E ∈ L(M, Y) and N ∈ L(M, U) be defined by Tj = Mz∗j |M

(j = 1, . . . , d),

E : f → f (0)

and

N : h → (Φ∗ h)(0).

(3.12)

Then {T, E, N } is an admissible data set for a problem ssOAP(T, E, N ) and a contractive multiplier S solves CLP(M, Φ) if and only if S solves ssOAP(T, E, N ).

Vol. 62 (2008)


323

Conversely, suppose that {T, E, N } is an admissible data set for a problem ssOAP. Set M = Ran OE,T and define Φ : HU (kd ) → M via its adjoint Φ∗ : Φ∗ : OE,T x → ON,T x.

(3.13)

Then {M, Φ} is the admissible data set for a Commutant Lifting Problem and a contractive multiplier S ∈ Sd (U, Y) solves ssOAP(T, E, N ) if and only if S solves CLP(M, Φ). Proof. Let T, E, N be defined as in (3.12). From the fact that the backward-shift d-tuple M∗z = (Mz∗1 , . . . , Mz∗d ) is strongly stable on HY (kd ) we see that T = M∗z |M is strongly stable. Now note that (OE,T h)(z) = h(z) for every h ∈ M, i.e., that the observability operator OE,T acting on an element h ∈ M simply reproduces h and hence can be viewed as the operator of inclusion of M in HY (kd ); in particular, (E, T) is also output-stable. Therefore we have ∗ OE,T OE,T = IM ,

∗ OE,T |M = IM

(3.14)

∗ and PM = OE,T .

(3.15)

and furthermore, ∗ |M⊥ = 0 OE,T

We refer to [18, Section 3] for more details. Next we show that, for operators Tj and N given by (3.12), we have ON,T = Φ∗

(3.16)

from which it will follow immediately that the pair (N, T) is also output-stable. To this end, pick up an arbitrary h(z) = hn z n ∈ M and note that n∈Z+ d ∗ (M∗n z Φ h)(0) =

n! (Φ∗ h)n |n|!

for every n ∈ Zd+ .

∗n ∗ By (3.1), Φ∗ Mz∗j |M = Mz∗j Φ∗ for j = 1, . . . , d and therefore, Φ∗ M∗n z |M = Mz Φ d for every n ∈ Z+ . Now we have

N (I −

d

zj Tj )−1 h

=

|n|! z n N Tn h n! d

n∈Z+

j=1

=

|n|! z n (Φ∗ M∗n z h)(0) n! d

n∈Z+

=

|n|! ∗ z n (M∗n z Φ h)(0) n! d

n∈Z+

=

n∈Zd +

z n (Φ∗ h)n = (Φ∗ h)(z)

324

Ball and Bolotnikov

IEOT

and since h is arbitrary, (3.16) follows. It follows immediately from (3.16) that the pair (N, T) is output-stable and therefore {T, E, N } is an admissible data set for a problem ssOAP. By Corollary 3.2, condition (3.8) is equivalent to ∗ ∗ MS = ON,T =Φ OE,T

which coincides with (3.4) due to (3.15). Conversely, let {T, E, N } be an admissible data set for a problem ssOAP. Since T is strongly stale, M = Ran OE,T is a closed backward-shift-invariant subspace of HY (kd ) and OE,T is an isomorphism of X onto M (see [18]). We define Φ : HU (kd ) → M via its adjoint Φ∗ given by (3.13). Then, for S ∈ Sd (U, Y), from Proposition 3.1 we see that S solves ssOAP(T, E, N ) is equivalent to condition (3.9), or, after taking adjoints, to MS∗ OE,T = ON,T . By definition, this in turn is equivalent to problem CLP(M, Φ).

MS∗ |M

(3.17) ∗

= Φ , i.e., to S solving the

Remark 3.4. We have seen in Proposition 3.1 that a necessary condition for a problem OAP(T, E, N ) to have a solution is that P ≥ 0 where P : X → X is given by (3.10). In the context of a problem ssOAP(T, E, N ), note that the operator Φ given by (3.13) satisfies Φ ≤ 1 if and only if P ≥ 0. It thus follows from the Commutant Lifting Theorem that the condition P ≥ 0 is also sufficient for a problem ssOAP(T, E, N ) to have a solution. In the next section we shall see that the same statement holds for the more general problem OAP(T, E, N ).

4. The general Operator Argument interpolation Problem In this section we present our solution of the general Operator Argument Interpolation Problem, including the parametrization of the set of all solutions for the case where the operator P (3.10) is invertible. As a first step we present several useful reformulations of the problem. The main tool for this analysis is the following well-known Hilbert space result. Proposition 4.1. A Hilbert space operator X X P B∗ : → B A H H is positive semidefinite if and only if A is positive semidefinite and for every x ∈ X , there exists a vector hx ∈ H Ker A such that 1

A 2 hx = Bx

and

1

hx H ≤ P 2 xX .

Vol. 62 (2008)


325

Theorem 4.2. Let {T, E, N } be an admissible data set for the OAP(T, E, N ). Let P : X → X be defined as in (3.10), let S be an L(U, Y)-valued function defined on Bd , and let F S : X → HY (kd ) be the linear map given by F S : x → (OE,T − MS ON,T ) x. The following are equivalent: 1. S is a solution of the OAP(T, E, N ). 2. The operator P (F S )∗ X X P := → : F S I − MS MS∗ HY (kd ) HY (kd )

(4.1)

(4.2)

is positive semidefinite. 3. The following kernel is positive on Bd × Bd :   P G(ζ)∗ (E ∗ − N ∗ S(ζ)∗ ) 0 IY − S(z)S(ζ)∗ (4.3) K(z, ζ) =  (E − S(z)N ) G(z) 1 − z, ζ where −1 G(z) = (I − z1 T1 − · · · − zd Td ) . (4.4) 4. S ∈ Sd (U, Y) and the function F S x belongs to the de Branges-Rovnyak space H(KS ) and satisfies 1

F S xH(KS ) ≤ P 2 xX

for every

x ∈ X.

(4.5)

5. S ∈ Sd (U, Y) and the function F S x belongs H(KS ) and satisfies 1

F S xH(KS ) = P 2 xX

for every

x ∈ X.

(4.6)

Proof. First note, as was observed in (3.17), that S ∈ Sd (U, Y) solves the problem OAP(T, E, N ) if and only MS∗ OE,T = ON,T : X → HU (kd ).

(4.7)

To prove Theorem 4.2, we shall show that (2) ⇐⇒ (3) and that (1) =⇒ (5) =⇒ (4) =⇒ (2) ⇐⇒ (1). (2) ⇐⇒ (3): Simply note that that the identity r x xj Pf, f X ⊕HY (kd ) = K(z (j) , z () ) , y yj X ⊕Y j,=1

holds for every vector f ∈ X ⊕ HY (kd ) of the form r xj f= (xj ∈ X , yj ∈ Y, z (j) ∈ Bd ) kd (· , z (j) )yj j=1

(see [28, Theorem 2.4] for details). (1) =⇒ (5): Assume that S ∈ Sd (U, Y) solves OAP(T, E, N ). Then from (4.7) we see that F S = OE,T − MS ON,T = OE,T − MS MS∗ OE,T = (I − MS MS∗ )OE,T .

326

Ball and Bolotnikov

IEOT

Hence F S x2H(KS ) = (I − MS MS∗ )OE,T x, OE,T xHY (kd ) 1

∗ ∗ = (OE,T OE,T − ON,T ON,T )x, xX = P x, xX = P 2 x2X

for all x ∈ X and (5) follows. (5) =⇒ (4): This is trivial. (4) =⇒ (2): Since S is in Sd (U, Y), the operator A := I − MS MS∗ is positive semidefinite on HY (kd ). Furthermore, F S x belongs to H(KS ) for every x ∈ X 1 which means, due to (2.9) that F S x = (I − MS MS∗ ) 2 hx for some element hx ∈ HY (kd ) which can be chosen to be orthogonal to the Ker (I − MS MS∗ ). Then the norm constraint (4.5) implies & & 1 1 & & = hx HY (kd ) ≤ P 2 xX &(I − MS MS∗ ) 2 hx & H(KS )

and positivity of the operator (4.2) follows by Proposition 4.1. (2) ⇐⇒ (1): Let the operator (4.2) be positive semidefinite. Then the operator I −MS MS∗ is positive semidefinite (equivalently, MS is a contraction) which implies S ∈ Sd (U, Y). By definitions (3.10) and (4.1) we have ∗ ∗ ∗ ∗ OE,T OE,T − ON,T ON,T OE,T − ON,T MS∗ P= ≥ 0. OE,T − MS ON,T I − MS MS∗ By the standard Schur complement argument, the latter inequality is equivalent to   ON,T MS∗ IHU (kd ) ∗ ∗ ∗ ' :=  ON,T  ≥ 0, OE,T OE,T OE,T P MS OE,T IHY (kd ) ' On the other hand, the since P is the Schur complement of the block IHU (kd ) in P. latter inequality holds if and only if the Schur complement of the block IHY (kd ) in ' is positive semidefinite: P ∗ IHU (kd ) ON,T MS MS OE,T ≥ 0. (4.8) − ∗ ∗ ∗ ON,T OE,T OE,T OE,T Now we write (4.8) as IHU (kd ) − MS∗ MS ∗ ∗ ON,T − OE,T MS

ON,T − MS∗ OE,T ≥0 0

∗ MS = ON,T which means that S is a solution of OAP(T, E, N ). and arrive at OE,T Since we have already proved (1) =⇒ (5) =⇒ (4) =⇒ (2), it follows that (1) =⇒ (2). However we think it instructive to include the following direct path: (1) =⇒ (2): If S ∈ Sd (U, Y) is a solution of OAP(T, E, N ), then we know that I − MS MS∗ ≥ 0 and we have the identity (4.7). Recalling also the definition (3.10)

Vol. 62 (2008)


327

of P , we then have ∗ ∗ ∗ OE,T OE,T − OE,T MS MS∗ OE,T OE,T (I − MS MS∗ ) P= I − MS MS∗ (I − MS MS∗ )OE,T ∗ ∗ OE,T I ≥0 = (I − MS MS∗ ) OE,T I and (2) follows. Similarly, since we have already proved (2) =⇒ (1) =⇒ (5) =⇒ (4), we know that (2) =⇒ (4). However, for purposes of more general considerations to come in the next section, we shall need the following direct path: (2) =⇒ (4): If the operator (4.2) is positive semidefinite, then by Proposition 4.1 the operator I − MS MS∗ is positive semidefinite (i.e., MS is a contraction which implies S ∈ Sd (U, Y)) and for every x ∈ X , there exists a function hx ∈ HY (kd ) which is orthogonal to the Ker (I − MS MS∗ ) such that 1

1

(I − MS MS∗ ) 2 hx = F S x and hx HY (kd ) ≤ P 2 xX .

(4.9)

The first relation in (4.9) implies in particular that F S x belongs to Ran(I − 1 MS MS∗ ) 2 or equivalently, to H(KS ), due to characterization (2.9). Furthermore, since hx is orthogonal to Ker (I − MS MS∗ ), we conclude from (2.10) and (4.9) that F S xH(KS )

1

=

(I − MS MS∗ ) 2 hx H(KS )

=

(I − π)hx HY (kd ) = hx HY (kd ) ≤ P 2 xX ,

1

which proves (4.5). As a corollary, we get the following curious reformulation of the CLP(M, Φ). Theorem 4.3. Assume that we are given an admissible data set {M, Φ} for a commutant lifting problem and that the necessary conditions (3.1) and (3.2) are satisfied so that the operator P := IM − ΦΦ∗ is positive semidefinite. Then S ∈ Sd (U, Y) is a solution of the CLP(M, Φ) if and only if for every h ∈ M, the function (4.10) Fh (z) := h(z) − S(z)(Φ∗ h)(z) 1

belongs to the de Branges-Rovnyak space H(KS ) and Fh H(KS ) ≤ P 2 hM . Proof. By Theorem 3.3, a function S ∈ Sd (U, Y) is a solution of the CLP(M, Φ) if and only if it is a solution of the OAP(T, E, N ) with the data given by (3.12) which is equivalent (by Theorem 4.2) to the function (F S h)(z) := (E − S(z)N ) (IM −

d

zj Tj )−1 h

(4.11)

j=1 1

being an element of H(KS ) and satisfying F S hH(KS ) ≤ P 2 hM for every h ∈ X = M. It remains to show that the right hand side expressions in formulas

328

Ball and Bolotnikov

IEOT

(4.11) and (4.10) coincide, i.e., that (E − S(z)N ) (IM −

d

zj Tj )−1 h = h(z) − S(z)(Φ∗ h)(z) for every h ∈ M.

j=1

But this follows from (3.16) and the second equality in (3.14). Reformulation of the problem OAP(T, E, N ) in terms of the operator F S = I −MS O[ E ],T N

mapping (X , P ) contractively into the de Branges-Rovnyak space H(KS ) (condition (4) in Theorem 4.2), when combined with Theorems 2.3 and 2.8, leads immediately the a linear-fractional description of the set of all solutions in case (see (2.20) and (3.10)) is strictly positive definite. the Pick operator P = G JE [ N ],T Theorem 4.4. Let {T, E, N } be an admissible data set for the OAP and let the operator P be defined as in (3.10) be strictly positive. Also let A11 (z) A12 (z) E A(z) = =D+ (I − Z(z)T )−1 Z(z)B N A21 (z) A22 (z) be the (J, J)-inner operator-valued function constructed according to the recipe in Theorem 2.3. Then an L(U, Y)-valued function S is a solution of the problem OAP(T, E, N ) if and only if S can be written in the form −1

S(z) = (A11 (z)E(z) + A12 (z)) (A21 (z)E(z) + A22 (z))

,

(4.12)

for some E ∈ Sd (U, Y ⊕ X ). Moreover the condition P ≥ 0 is both necessary and sufficient for the problem OAP(T, E, N ) to have solutions. d−1

Proof. By condition (4) in Theorem 4.2 we know that S solves OAP(T, E, N ) if and only if S ∈ Sd (U, Y) and the operator OE,T S F := I −MS ON,T maps (X , P ) contractively into the de Branges-Rovnyak space H(KS ). By Theorem 2.3, we know that O[ E ],T is a unitary identification between (X , P ) and N

J,J H(K PE ) = H(KA ). Hence the condition for S to solve OAP(T, E, N ) trans[ N ],T J,J lates to: S ∈ Sd (U, Y) and the operator I −MS maps H(KA ) contractively into H(KS ). By Theorem 2.8, this last condition is equivalent to S = TA [E] for some E ∈ Sd (U, F ). If P is strictly positive definite, it follows in particular that OAP(T, E, N ) has solutions. If we only have P ≥ 0, then via a rescaling the result for the strictly positive definite case implies that, for each δ > 0 there exists solutions Sδ ∈ Sd (U, Y) of the interpolation conditions (3.8) with MSδ ≤ 1 + δ. The existence of a solution

Vol. 62 (2008)


329

S of (3.8) with MS ≤ 1 then follows by a standard weak-∗ compactness argument which makes use of the fact that the operators OE,T and ON,T have range inside the Drury-Arveson spaces HY (kd ) and HU (kd ) respectively. The necessity of the condition P ≥ 0 for the existence of solutions is the content of part (2) of Proposition 3.1.

5. The analytic Abstract Interpolation Problem The very formulation of the problem OAP(T, E, N ) appears to require that the operators OE,T and ON,T be bounded operators from X into HY (kd ) and HU (kd ) respectively. However, upon close inspection, one can see that conditions (2), (3), (4) in Theorem 4.2 make sense if we take P to be any positive semidefinite operator on X and if we only assume that (a) T = (T1 , . . . , Td ) is a not necessarily commutative d-tuple of operators on X and E : X → Y and N : X → U are such that E O[ E ],T : x → (I − Z(z)T )−1 x N N maps X into the space HolY⊕U (Bd ) of holomorphic (Y ⊕ U)-valued functions on Bd . A careful inspection of the proof of Theorem 4.2 (specifically, of steps (2) ⇐⇒ (3), (4) =⇒ (2) and (2) =⇒ (4) (direct path)) shows that the mutual equivalence of conditions (2), (3), (4) continues to hold in this more general situation. This suggests that we use any of these conditions as the definition of a more general interpolation problem. In order to apply Theorems 2.4 and 2.8 to this more general situation, we must also require: (b) P is a positive semidefinite solution of the Stein equation (2.21). This leads to the formulation of the analytic Abstract Interpolation Problem: The analytic Abstract Interpolation Problem (aAIP(T, E, N, P )): Given the data {E, N, T, P } subject to assumptions (a), (b), find all S ∈ Sd (U, Y) such that the function F S x defined as in (4.1) belongs to the de Branges-Rovnyak space H(KS ) and satisfies the norm constraint (4.5). The next Theorem summarizes the observations made above. We note that the proof of the linear-fractional parametrization goes through Theorem 2.4 in place of Theorem 2.3. Theorem 5.1. Let P , T, E and N satisfy assumptions (a), (b). The following are equivalent: 1. S is a solution of the aAIP(E, N, T, P ). 2. The operator P ∈ L(X ⊕ HY (kd )) of the form (4.2) is positive semidefinite. 3. The kernel K(z, ζ) of the form (4.3) is positive on Bd × Bd .

330

Ball and Bolotnikov

IEOT

Moreover, if P is strictly positive definite and if the function A ∈ SdJ,J (F ⊕U, Y ⊕U) is constructed as in Theorem 2.4, then S solves aAIP(E, N, T, P ) if and only if S can be realized in the form (4.12) for a Schur-class function E ∈ Sd (U, F ). Remark 5.2. It is tempting to use a weak-∗ compactness argument as in the proof of Theorem 4.4 to conclude from Theorem 5.1 that the problem aAIP always has a solution (even when P is only positive semidefinite rather than strictly positive definite). However the details of such an argument are not so clear since the observability operators OE,T and ON,T no longer have range in the DruryArveson space. We will see that the problem aAIP(T, E, N, P ) always has solutions based on a different approach whereby we get a description, even in the degenerate case, of the set of all solutions in terms of a Redheffer-type linear-fractional map (see Corollary 6.6 below). By Theorem 4.2, the OAP(T, E, N ) is a particular case of the aAIP(E, N, T, P ) (corresponding to a commutative T, output stable (E, T), (N, T) and the Pick operator P defined in (3.10). One of the special features of this case is expressed by the equivalence (4) ⇔ (5) in Theorem 4.2: for every solution S of the problem, inequality (4.5) implies equality (4.6) (in [41] such problems were called possessing the Parseval equality). We next present another interesting particular case of the aAIP(E, N, T, P ) for which this phenomenon does not take place. (i)

(i)

The boundary Nevanlinna-Pick problem: Given n points t(i) = (t1 , . . . , td ) (i = d 2 1, . . . , n) on the unit sphere Sd = {t = (t1 , . . . , td ) : j=1 |tj | = 1}, given vectors ξi ∈ Y and ηi ∈ U and given numbers γi ≥ 0, (i = 1, . . . , n), find all S ∈ Sd (U, Y) such that IY − S(rt(i) )S(rt(i) )∗ (i) ∗ ξj , ξj ≤ γi (5.1) lim S(rt ) ξj = ηj and lim r→1 r→1 1 − r2 Y for i = 1, . . . , n. This problem has been studied in [4, 27, 14]. From [27, 14] it is known that the problem has a solution if and only if ξi Y = ηi U and the matrix P =

[Pij ]ni,j=1

for

i = 1, . . . , n

(5.2)

with the entries

ξj , ξi Y − ηj , ηi U (i = j) and Pii = γi 1 − t(i) , t(j) is positive semidefinite. It is easily seen that P defined as above satisfies the Stein identity d P− Tj∗ P Tj = E ∗ E − N ∗ N (5.3) Pij =

j=1

where Tj ’s are the diagonal n × n matrices defined by (j) (j) j = 1, . . . , d. Tj = diag t1 , t2 , . . . , t(j) n

Vol. 62 (2008)

and where


E = ξ1

ξ2

. . . ξn ,

N = η1

η2

331

. . . ηn .

Note that the pairs (E, T) and (N, T) are not output stable, but the functions E(I − Z(z)T )−1 and N (I − Z(z)T )−1 are analytic on Bd . Thus the problem aAIP(E, N, T, P ) is well defined. As it was shown in [27], this latter problem is equivalent to the boundary Nevanlinna-Pick problem (5.1). Note that equalities (5.2) are necessary for the Stein equation (5.3) to have a solution. Note also that P is not completely determined by the Stein equation (5.1); its diagonal entries form a piece of information independent of that contained in T, E and N . Finally, note that equality (4.6) for a solution S of the aAIP(E, N, T, P ) corresponds to equalities in the second series of interpolation conditions (5.1). When one considers the boundary Nevanlinna-Pick problem with the inequality in (5.1) replaced by equality, then a necessary and sufficient condition for existence of solutions as well as a description for the set of all solutions is unknown; it is known that P ≥ 0 is necessary and that P > 0 is sufficient for existence of solutions. Remark 5.3. If the tuple T = (T1 , . . . , Td ) is not commutative, then condition (3.8) is not equivalent to (3.9) in general. For example, let     0 1 0 0 0 0 T1 = 0 0 0 , T2 = 0 0 1 , E = 1 0 0 , N = α β γ . 0 0 0 0 0 0 Then −1

(I − z1 T1 − z2 T2 )

 1 = 0 0

z1 1 0

 z1 z2 z2  , 1

so that OE,T x =

x1 + x2 z1 + x3 z1 z2 ,

ON,T x =

x1 α + x2 (αz1 + β) + x3 (αz1 z2 + βz2 + γ) if

  x1 x = x2  ∈ C3 . x3

Using the definition of inner product in H(Kd ) it is readily seen that     αf00 f00 ∗ ∗  αf01 + βf00 f =  f01  and ON,T f = OE,T 1 α f f + βf + γf 11 01 00 2 2 11 ∞ i j if f (z1 , z2 ) = i,j=0 fij z1 z2 ∈ H(k2 ). Condition (3.8) is equivalent to S00 = α,

S01 = β,

S11 = 2γ.

(5.4)

On the other hand, condition (3.9) is equivalent to the conditions (Sh)00 = αh00 ,

(Sh)01 = αh10 + βh00 ,

α 1 (Sh)11 = h11 + βh01 + γh00 , 2 2

332

Ball and Bolotnikov

IEOT

or, in more detail, S00 h00 = αh00 , S00 h01 + S01 h00 = h01 + βh00 , α 1 (S00 h11 + S10 h01 + S01 h10 + S11 h00 ) = h11 + βh01 + γh00 2 2 ∞ i j holding for every h(z1 , z2 ) = i,j=0 hij z1 z2 ∈ H(k2 ). It is easily checked that this is the case if and only if S00 = α,

S10 = S01 = β = 0,

S11 = 2γ.

(5.5)

We conclude that conditions (5.4) and (5.5) are not equivalent.

6. The Abstract Interpolation Problem We are now ready to formulate the Abstract Interpolation Problem AIP based on a data set {D, T, T, E, N } described as follows. We are given a linear space X , a positive semidefinite Hermitian form D on X , Hilbert spaces U and Y, linear operators T, T = (T1 , . . . , Td ) on X , and linear operators N : X → U and E : X → Y. In addition we assume that D(Tx, Tx) +

N x2U

=

d

D(Tj x, Tj x) + Ex2Y

for every x ∈ X .

(6.1)

j=1

Definition 6.1. Suppose that we are given the data set {D, T, T, E, N } for an AIP as in (6.1). We say that the pair (F, S) is a solution of the AIP if S is a Schurclass function Sd (U, Y) and F is a linear mapping from X into H(KS ) such that the two conditions hold: F x2H(KS ) ≤ D(x, x) (F Tx)(z) −

d

for all

x ∈ X,

zj (F Tj x)(z) = (E − S(z)N ) x

(6.2) for all

z ∈ Bd .

(6.3)

j=1

Denote by X0 the Hilbert space equal to the completion of the space of equivalence classes of elements of X (where the zero equivalence class consists of elements x with D(x, x) = 0) in the D-inner product. Note that if (S, F ) solves AIP, then condition (6.2) implies that F has a factorization F0 ◦ π where π is the canonical projection operator π : X → X0 and where F0 : X0 → H(KS ) has F0 ≤ 1. We abuse notation and denote also by T and Tk the operators T and Tk followed by the canonical projection into the equivalence class in X0 ; so T, Tk : X → X0 . Let for short   T1  ..  T =  .  : X → X0d , Z(z) = z1 IX0 · · · zd IX0 . Td

Vol. 62 (2008)


333

If we further identify F0 with the operator-valued holomorphic function z → F0 (z) ∈ L(X0 , Y) defined by F0 (z)x0 = (F0 x0 )(z) then we can rewrite (6.3) in the form F0 (z)T − F0 (z)Z(z)T = E − S(z)N

for all z ∈ Bd .

(6.4)

Note that the import of the hypothesis (6.1) is that there is a well-defined isometry V from d T X0 T X0 DV = Ran ⊂ ⊂ onto RV = Ran (6.5) N E U Y such that

T T V: x→ x N E

for all x ∈ X .

(6.6)

Note also that the definition (6.5) and (6.6) of V is completely determined by the problem data {D, T, T, E, N }. If X is already a Hilbert space and there exists a bounded positive semidefinite operator P ≥ 0 such that D(x, y) = P x, yX for every x, y ∈ X , then identity (6.1) can be written as ∗

T PT −

d

Tj∗ P Tj = E ∗ E − N ∗ N.

k=1

Furthermore, equality (6.4) can be written as F0 (z)(T − Z(z)T )x = (E − S(z)N ) x and if the pencil (T−Z(z)T ) is invertible for every z ∈ Bd , then the latter equation defines F0 uniquely by F0 (z)x = (E − S(z)N ) (T − Z(z)T )−1 x. If furthermore, T = IX , then it is readily seen that the AIP(D, IX , T, E, N ) collapses to the aAIP(E, N, T, P ). However, it can happen that S(z) does not uniquely determine F0 (z) and therefore the problem AIP cannot be reduced to a problem aAIP. The following example illustrates this point. Example 6.2. Consider the following single-variable example. Choose operators T and T on a Hilbert space X so that 1. the pencil T − zT is singular, 2. T ∗ T − T ∗ T is positive semidefinite, and 3. Ran T + Ran T is dense in X One such choice is 0 0 1 0 2 X =C , T = , T = δ 0 0 0

334

Ball and Bolotnikov

IEOT

for some δ > 0. Then choose the operator N : X → U so that N ∗ N = T ∗T − T ∗ T . For the example at hand, we take U = C,

N=

√ 1 − δ2

0 .

We also take the output space Y = C and we set E = 0 : X → Y. We take the Hermitian form D on X to be D(x, x) = x, xX . Then {D, T , T, E, N } is an admissible data set for an AIP in the sense above with d = 1. If we define S ∈ S(U, Y) by S(z) = 0, then H(KS ) = H 2 . We seek solutions of the AIP associated with this data set such that S(z) = 0. Thus we seek F : X = C2 → H(KS ) = H 2 so that (F T x)(z) − z(F T x)(z) = Ex − S(z)N x = 0. The associated operator-valued function z → F0 (z) given by F0 (z)x = (F x)(z) then must satisfy F0 (z)T x − zF0 (z)T x = 0. Expressing F0 (z) as a row matrix F0 (z) = F1 (z) F2 (z) , we arrive at 0 0 1 0 F1 (z) F2 (z) − z F1 (z) F2 (z) =0 δ 0 0 0 which can be rewritten as δF2 (z) − zF1 (z) 0 = 0 Solving gives

F1 (z) F2 (z) = F0 (z) δ

0 . z

where F0 is a free-parameter H 2 -function. Choosing F0 of sufficiently small 2norm then guarantees that the resulting operator MF : X → H 2 is contractive. In particular, such F ’s are not uniquely determined. Thus there are many distinct solution-pair solutions to AIP of the form (F, 0). Our next goal is to show that solutions of a problem AIP(D, T, T, E, N ) correspond to minimal unitary-colligation extensions of the partially defined isometric colligation V in (6.5), (6.6). Here we say that the unitary colligation C with connecting operator U : H ⊕ U → H ⊕ Y is a minimal unitary-colligation extension of V if 1. X0 is a subspace of H, 2. U|DV = V : DV → RV , and 3. the smallest reducing subspace for U contained in H and containing X is the whole space H. Theorem 6.3. Let V be the isometry defined by (6.6) associated with the data of a problem AIP and let ) ( A B C = X0 ⊕ X1 , U, Y, U = C D

Vol. 62 (2008)


335

be a minimal unitary-colligation extension of V. Define S ∈ Sd (U, Y) and F0 : X0 ⊕ X1 → H(KS ) by S(z) = D + C(I − Z(z)A)−1 Z(z)B, −1

F0 (z) = C(I − Z(z)A)

|X0 .

(6.7) (6.8)

Then the pair (S, F0 ) is a solution of AIP. Conversely, every solution of AIP arises in this way. Proof. Let H = X0 ⊕ X1 , let

A U= C

d B H H : → Y D U

be the connecting operator for a minimal unitary-colligation extension of V and let −1 S(z) = D + C (I − Z(z)A) Z(z)B (6.9) be the characteristic function of the colligation C = {H, U, Y, U}. Then S ∈ Sd (U, Y) by Theorem 2.1. Furthermore, let H(KS ) be the associated de BrangesRovnyak space and define F0 : X → H(KS ) by x (F0 x0 )(z) = F0 (z)x0 = C(I − Z(z)A)−1 0 for x0 ∈ X0 . (6.10) 0 Then F0 is a contraction (see e.g. part (2) of Theorem 2.1 in [19]). It remains to check the identity (6.4) which, due to (6.10) is the same as H(z)Tx = H(z)Z(z)T x + Ex − S(z)N x

(6.11)

(with H(z) as in (2.5)). Using the unitary realization (6.9) of S written as S(z) = D + H(z)Z(z)B, we rewrite (6.11) as H(z)T = H(z)Z(z)T + E − [D + H(z)Z(z)B]N

(6.12)

To establish (6.12), we use the identity A B T T = , C D N E or, in more detail, AT + BN = T,

CT + DN = E,

which is true by the hypothesis that U extends V, to see that the right hand side of (6.12) is equal to H(z)Z(z)T + E − DN − H(z)Z(z)BN = H(z)Z(z)T + CT − H(z)Z(z)(T − AT) = CT + H(z)Z(z)AT = H(z)T as wanted.

336

Ball and Bolotnikov

IEOT

We postpone the proof of the converse direction to the proof of Theorem 6.4 where a more general result is proved. We leave as an open question the problem of finding a direct proof of the converse direction. We next introduce the defect spaces d X0 X0

DV and ∆∗ =

RV ∆= U Y ! be another copy of ∆ and ∆ ! ∗ another copy of ∆∗ with unitary identifiand let ∆ cation maps ! and i∗ : ∆∗ → ∆ ! ∗. i: ∆ → ∆ ! ∗ onto RV ⊕ ∆∗ ⊕ ∆ ! by the rule Define a unitary operator U0 from DV ⊕ ∆ ⊕ ∆  if x ∈ DV ,  Vx, i(x) if x ∈ ∆, (6.13) U0 x =  −1 ! ∗. i∗ (x) if x ∈ ∆ d DV X0 RV X0 Identifying with and , we decompose U0 defined by with ∆ U ∆∗ Y (6.13) according to     d  X0 X0 U11 U12 U13 (6.14) U0 =  U21 U22 U23  :  U  →  Y  . !∗ ! U31 U32 0 ∆ ∆ The (3, 3) block in this decomposition is zero, since (by definition (6.13)), for d X 0 ! ∗ , the vector U0 δ!∗ belongs to ∆, which is a subspace of every δ!∗ ∈ ∆ and Y ! in other words P ! U0 | ! = 0 where P ! stands for therefore, is orthogonal to ∆; ∆ ∆∗ ∆ ! onto ∆. ! the orthogonal projection of RV ⊕ ∆∗ ⊕ ∆

The unitary operator U0 is the connecting operator of the unitary colligation ( ) U Y C0 = X0 , (6.15) !∗ , ∆ ! , U0 , ∆

which is called the universal unitary colligation associated with the AIP. According to (2.6), the characteristic function of the colligation C0 defined in (6.15) is given by Σ11 (z) Σ12 (z) Σ(z) = Σ21 (z) Σ22 (z) U U U23 = 22 + 21 (I − Z(z)U11 )−1 Z(z) U12 U13 U32 0 U31 ∗ ∗ −1 = PY⊕∆ |U ⊕∆ ! U0 (I − PX0 Z(z)PX0 Z(z)PX0d U0 ) !∗

(6.16)

Vol. 62 (2008)


337

! ∗ , Y ⊕ ∆) ! by Theorem 2.1. The associated and belongs to the class Sd (U ⊕ ∆ observability operator is given by U21 HΣ (z) = (I − Z(z)U11 )−1 U31 ∗ −1 = PY⊕∆ |X0 . ! U0 (I − PX0 Z(z)PX0d U0 )

(6.17)

By another application of Theorem 2.1 we see that KΣ (z, ζ) :=

I − Σ(z)Σ(ζ)∗ = HΣ (z)HΣ (ζ)∗ . 1 − z, ζ

We shall also need an enlarged colligation     X0 U11  Y C0,e = X0 ,  U  , ! , U0,e =  U21  ∆ !∗ U31 ∆

U11 U21 U31

U12 U22 U32

(6.18)

 U13  U23   0

with associated characteristic function U11 U12 U13 U21 + (I − Z(z)U11 )−1 Z(z) U11 Σe (z) = U31 U32 0 U31 U21 (I − Z(z)U11 )−1 Σ11 (z) Σ12 (z) = U31 (I − Z(z)U11 )−1 Σ21 (z) Σ22 (z)

U12

∗ −1 = PY⊕∆ . ! U0 (I − PX0 Z(z)PX0d U0 )

(6.19)

U13

(6.20)

These are the ingredients for the following parametrization for the set of all solutions of AIP. In particular, solutions of AIP exist for any data set (D, T, T, E, N ) which is admissible (i.e., condition (6.1) is satisfied). Theorem 6.4. Suppose that (D, T, T, E, N } is an admissible data set for a problem AIP. Let U0 be the universal unitary-colligation extension of V given by (6.13) with characteristic function (6.16) and let U0,e be the connecting operator for the enlarged universal unitary colligation C0,e given by (6.19). Then the pair (S(z), F0 (z)) solves the problem AIP if and only if there is a Schur-class multiplier ! ∆ ! ∗ ) such that W ∈ Sd (∆, F0 (z) S(z) = RΣe [W](z), (6.21) i.e., such that S(z) = Σ11 (z) + Σ12 (z)(I − W(z)Σ22 (z))−1 W(z)Σ21 (z), F0 (z) = U21 (I − Z(z)U11 )−1 + Σ12 (z)(I − W(z)Σ22 (z))−1 W(z)U31 (I − Z(z)U11 )−1 .

(6.22)

Proof. We have already seen in Theorem 6.3 that (F0 (z), S(z)) is a solution of AIP whenever S(z) = D + C(I − Z(z)A)−1 B, F0 (z) = C(I − Z(z)A)−1 |X0

(6.23)

338

Ball and Bolotnikov

where

A U= C

IEOT

d B X0 ⊕ X1 X0 ⊕ X1d → : U Y D

is the connecting operator for a unitary-colligation extension of the partially defined isometry V ((6.5) and (6.6)). It is also known (see [10, 11]) that such unitary extensions U are parametrized via a free-parameter closely-connected unitary colligation matrix X1 X1 A1 B1 U1 = : ! → ! C1 D1 ∆ ∆∗ in feedback connection with the universal colligation U0 :     x0 x !0 !1  U : x1  → x u y ! δ!∗ ∈ ∆ ! ∗ so that if and only if there exist δ! ∈ ∆,     x !0 x0 x !1 x1 U0 :  u  →  y  and U1 : ! → ! . δ δ∗ δ!∗ δ! One can solve explicitly for U := FU0 [U1 ] and arrive at   B0 A00 A01 A B A B1  U= = A 11 10 C D C0 C 1 D   U11 + U12 D1 U31 U13 C1 U12 + U13 D1 U32   B1 U31 A1 B1 U32 . =   U21 + U23 D1 U31 U23 C1 U22 + U23 D1 U32

(6.24)

To plug this formula into (6.23) we need to be able to compute the resolvent term (I − Z(z)A)−1 . The following computation is an adaptation of the ideas in [43] where the same result for the case d = 1 is proved (for the closely related AIP problem where the de Branges-Rovnyak space is taken to have two components). We introduce the associated multidimensional system of Fornasini-Marchesini type (see [18])   x(n) = A1 x(n − e1 ) + · · · + Ad x(n − ed ) +B1 u(n − e1 ) + · · · + bBd u(n − ed ) Σ(U) :  y(n) = Cx(n) + Du(n) where we write     A1 X0 ⊕ X1     .. A =  ...  : X0 ⊕ X1 →  , . Ad

X0 ⊕ X1



   B1 X0 ⊕ X1     .. B =  ...  : U →  . . Bd

X0 ⊕ X1

Vol. 62 (2008)


339

We assume an initial condition x(0) = x(0) and we assume zero boundary conditions: x(n) = 0 for n ∈ Zd+ \ {0} with nk = 0 for some k ∈ {1, . . . , d}. Then we define the initial condition/input–output map X0 ⊕ X1 HolX0 ⊕X1 (Bd ) TΣ(U) : → (6.25) HU (kd ) HY (kd ) by

* + (0) x(n)z n x n∈Zd + . TΣ(U) : → n∈Zd u(n) d y(n) +

Then it can be shown (see e.g. [18]) that (I − Z(z)A)−1 TΣ(U) = C(I − Z(z)A)−1

n∈Z

(I − Z(z)A)−1 B . S(z)

This object in turn can be computed as the feedback connection FTΣ(U0 ) [TΣ(U1 ) ] of the state/input—state-trajectory/output map   −1 (I − Z(z)U11 )−1 Z(z) U12 U13 (I− Z(z)U11 )  TΣ(U0 ) =  U21 (I − Z(z)U11 )−1 Σ(z) U31 with the the state/input–state-trajectory/output map as the free-parameter load (I − Z(z)A1 )−1 (I − Z(z)A1 )−1 Z(z)B1 TΣ(U1 ) = C1 (I − Z(z)A1 )−1 W(z) where W(z) = D1 + C1 (I − Z(z)A1 )−1 Z(z)B1 is the characteristic function of the free-parameter unitary colligation C1 , and ! ∆ ! ∗ ). We note therefore is itself a free-parameter Schur-class function in Sd (∆, that (I − W(z)Σ22 (z))−1 makes sense for all z ∈ Bd since both W and Σ22 are Schur multipliers and hence have contractive values on Bd and the value of Σ22 (z) is actually strictly contractive for z ∈ Bd since Σ22 (0) = 0. Hence the feedback connection FTΣ(U0 ) [TΣ(U1 ) ] is well-defined. Moreover, one can see that TΣ(U) = FTΣ(U0 ) [TΣ(U1 ) ] and hence the various matrix entries on the right-hand side of (6.25) can be computed explicitly in terms of matrix entries of U0 and U1 . In particular, one can show that X00 (z) X01 (z) −1 (I − Z(z)A) = X10 (z) X11 (z)

340

Ball and Bolotnikov

IEOT

where X00 (z) = (I − Z(z)U11 )−1 + (I − Z(z)U1 )−1 Z(z)U13 (I − W(z)Σ22 (z))−1 W(z) × U31 (I − Z(z)U11 )−1 , X01 (z) = (I − Z(z)U11 )−1 Z(z)U13 (I − W(z)Σ22 (z))−1 C1 (I − Z(z)A1 )−1 , X10 (z) = (I − Z(z)A1 )−1 B1 (I − Σ22 (z)W(z))−1 U31 (I − Z9z)U11 )−1 , X11 (z) = (I − Z(z)A1 )−1 + (I − Z(z)A1 )−1 Z(z)B1 (I − Σ22 (z)W(z))−1 Σ22 (z) × C1 (I − Z(z)A1 )−1 .

(6.26)

Using (6.24) and (6.26), we compute from (6.23) that S(z) = U22 + U23 D1 U32 + U21 + U23 D1 U33 F0 (z) = U21 + U23 D1 U31

X00 (z) X01 (z) U12 + U13 D1 U32 U23 C1 , X10 (z) X11 (z) B1 U32 X00 (z) U23 C1 X10 (z)

= (U21 + U23 D1 U31 )X00 (z) + U23 C1 X10 (z). After a lengthy but elementary calculation, one can see that these formulas collapse to (6.22) as asserted. Furthermore, a closer look at these formulas reveals that the two equations in (6.22) can be combined into a single matrix equation F0 (z) S(z) = Σe,11 (z) + Σe,12 (z)(I − W(z)Σ22,e (z))−1 W(z)Σe,21 (z) where

Σe,11 (z) Σe,12 (z) Σe,21 (z) Σe,22 (z)   U21 (I − Z(z)U11 )−1 Σ11 (z) Σ12 (z)  =  −1 Σ21 (z) Σ22 (z) U31 (I − Z(z)U11 ) U U22 U23 U + 21 (I − Z(z)U11 )−1 Z(z) U11 = 21 U31 U32 0 U31

Σe (z) =

U12

U13

from which we see that Σe (z) can be viewed as the characteristic function of the colligation C0,e in (6.19). We conclude that the solution (F0 (z), S(z)) of the problem AIP has the compact representation given by (6.21). Conversely, let us suppose that the pair (F0 (z), S(z)) is a solution of AIP. The problem is to show that necessarily F0 (z) S(z) is in the range of the linear! ∆ ! ∗ ). To show that there is a fractional map RΣe acting on the Schur class Sd (∆, ! ∆ ! ∗ ) so that (6.21) holds, by Theorem 2.9 it suffices to produce a Schur W ∈ Sd (∆, pair (a, c) so that I c(z) Σe (z) = F0 (z) S(z) a(z) .

Vol. 62 (2008)


341

Using the last expression for Σe (z) in (6.20), we may rewrite this condition as −1 I c(z) PY⊕∆ = F0 (z) S(z) a(z) ! U0 (I − PX0∗ Z(z)PX0d U0 ) which in turn can be converted to the more linear form ∗ I c(z) PY⊕∆ ! U0 = F0 (z) S(z) a(z) (I − PX0 Z(z)PX0d U0 ).

(6.27)

Let us define analytic operator-valued functions ! ∗ , Y), a : Bd → L(∆ by the formulas

! Y) c : Bd → L(∆,

a

-. , . = F0 (z) Z(z)PX0d U0 .

c

=

!∗ ∆

+ PY U0 |∆ !∗ ,

∗ F0 (z)PX0 U∗0 |∆ ! + S(z) PU U0 |∆ ! .

(6.28) (6.29)

Our goal is to show that (a, c) is a Schur-pair satisfying the condition (6.27). This will then complete the proof of Theorem 6.4. Note that the condition (6.27) must be verified on vectors from the space X0 ⊕ ! ∗ . Recall that X0 ⊕ U has the alternative decomposition U ⊕∆ X0 ⊕ U = DV ⊕ ∆. To verify the validity of (6.27), it suffices to consider the three cases: (1) y ∈ DV , ! ∗. (2) y ∈ ∆, and (3) y ∈ ∆ Case 1: y ∈ DV . By construction a dense subset of DV consists of vectors of the form   Tx y = N x where x ∈ X . 0 By definition we then have   Tx U0 y = Ex . 0 Then condition (6.27) applied to the vector y for this case becomes simply Ex = F0 (z)Tx + S(z)Ex − F0 (z)Z(z)Tx which holds true due the data-admissibility condition (6.4). Note that this case holds automatically independently of the definition of (a, c). δ Step 2: y = with δ ∈ ∆. Note that in this case 0   0 δ U0 =  0 . 0 i(δ) and hence the left-hand side of (6.27) is simply c(z)i(δ).

(6.30)

342

Ball and Bolotnikov

IEOT

On the other hand, the right-hand side is F0 (z)PX0 δ + S(z)PU δ.

(6.31)

The equality of (6.30) with (6.31) amounts to the definition of c(z) in (6.29). ! ∗ . In this case we know that Step 3: y = 0 ⊕ 0 ⊕ δ!∗ with δ!∗ ∈ ∆ −1 i∗ (δ∗ ) U0 y = . 0 Then the left-hand side of (6.27) applied to a vector y of this form gives us PY i−1 (δ!∗ ) ∗ ! I c(z) (6.32) = PY i−1 ∗ (δ∗ ) 0 while the right-hand side gives us   −1 ! −Z(z)PX0d i∗ (δ∗ )  = −F0 (z)Z(z)PX d i−1 F0 (z) S(z) a(z)  (δ!∗ ) + a(z)δ!∗ . 0 0 ∗ ! δ∗ (6.33) Equality of (6.32) with (6.33) collapses to the definition (6.28) of a(z). It remains only to verify that (a, c) defined via (6.28) and (6.29) is a Schur-pair. We use the notation HΣ (z) for the observability operator (6.17) associated with the universal colligation C0 and H(z) for any function giving rise to a factorization of the kernel KS (z, ζ) as in (2.1). We note that a particular consequence of (6.27) is that IY c(z) HΣ (z)x0 = F0 (z)x0 ∈ H(KS ) (6.34) for each x0 ∈ X0 . Furthermore, for every x ∈ X0 , there is a unique gx ∈ H which is orthogonal to Ker H(z) for every z ∈ Bd and such that IY c(z) HΣ (z)x = H(z)gx . Therefore we can define a linear operator Γ : X → H by the rule Γx = gx . Thus, IY c(z) HΣ (z) = H(z)Γ. (6.35) By the definition of the norm in H(KS ), F xH(KS ) = gx H = ΓxH . On the other hand, the operator F : X0 → H(KS ) is contractive by assumption; hence F xH(KS ) ≤ xX0 and Γ is a contraction: ΓxH = F xH(KS ) ≤ xX0 . The next step is to show that the functions a and c defined in (6.28) and (6.29) satisfy a(z)a(ζ)∗ − c(z)c(ζ)∗ = R(z)R(ζ)∗ 1 − z, ζ with 1 (6.36) R(z) = H(z)(I − ΓΓ∗ ) 2

Vol. 62 (2008)


343

from which it will follow that (a, c) is a Schur-pair. Indeed, by (6.34), (2.1), (6.18) and (6.35), a(z)a(ζ)∗ − c(z)c(ζ)∗ 1 − z, ζ IY S(ζ)∗ c(z) S(z) a(z) I − Y a(ζ)∗ c(ζ)∗ IY − S(z)S(ζ)∗ = + 1 − z, ζ 1 − z, ζ ∗ I − Σ(z)Σ(ζ)∗ IY IY − S(z)S(ζ) − IY c(z) = c(ζ)∗ 1 − z, ζ 1 − z, ζ IY = H(z)H(ζ)∗ − IY c(z) HΣ (z)HΣ (ζ)∗ c(ζ)∗ = H(z) (I − ΓΓ∗ ) H(ζ)∗ = R(z)R(ζ)∗ , where R is defined in (6.36) It follows that (a, c) is a Schur-pair and the proof of Theorem 6.4 is complete. We are now in position to complete the proof of Theorem 6.3 Proof of the converse direction in Theorem 6.3. Suppose that (F0 (z), S0 (z)) is a solution of AIP. Then Theorem 6.4 tells us that there is a Schur-class multiplier ! ∆ ! ∗ ) so that S(z) = RΣ [W](z). If we plug in a closely-connected unitary W ∈ Sd (∆, colligation ! ∆ ! ∗ , U1 } C1 = {X1 , ∆, having W as its characteristic function together with the realization C0 for Σ, we arrive at a realization C = {X0 ⊕ X1 , U, Y, U} for S(z) having connecting operator U which is a minimal unitary-colligation extension of V. Moreover, the associated mapping F0 (z) is the restriction of the associated observability operator x → PY U(I − PX∗0 ⊕X1 Z(z)P(X0 ⊕X1 )d U)−1 |X0 ⊕X1 to X0 . Thus every solution of the AIP arises from the procedure given in the statement of Theorem 6.3. Corollary 6.5. Suppose that {T, E, N } is an admissible data set for a problem OAP(T, E, N ) and we set ∗ ∗ OE,T − ON,T ON,T . P = OE,T

Then P is the minimal solution of the Stein equation (2.21), i.e., if P! is a solution of (2.21) with P! ≤ P , then P! = P . Proof. Let P! be a positive semidefinite solution of the Stein equation (2.21) and ∗ ∗ let us assume that P! ≤ P := OE,T OE,T − ON,T ON,T . Then any solution S of the

344

Ball and Bolotnikov

IEOT

aAIP(E, N, T, P!) is also a solution of the aAIP(E, N, T, P ). In other words, for every x ∈ X , the function F S x belongs to H(KS ) and F S x2H(KS ) ≤ P! x, xX ≤ P x, xX . But by Theorem 4.2, F S x2H(KS ) = P x, xX which implies therefore, that P = P! . Corollary 6.6. For any aAIP-admissible data set {T, E, N, P }, aAIP has solutions. Proof. We have already observed that the aAIP is a special form of AIP. Hence the result of Theorem 2.7 that any admissible problem of the type AIP has solutions implies the same for aAIP. We conclude by continuing Example 6.2 to illustrate the general theory. Example 6.2 continued: Note that the partial unitary colligation associated with the AIP given in Example 6.2 is V : DV → RV given by     0 1  → 0 V : √ δ 0 1 − δ2 with

 0  = Ran √ δ 2 1−δ 

DV Therefore,



√ 0  ∆ = Ran 1 − δ2 −δ

 1 0 0

and RV

 0 and ∆∗ = Ran 1 0

The universal colligation U0 takes the form √  0 δ 1 − δ2 0 0 0  0 0 0 U0 =   1 0 0 √ 1 − δ2 −δ 0 that is, U11

0 δ = , 0 0

U21 = 0 0 ,

  1 = Ran 0 . 0

0 1 0 0 0

 0 0 . 1

 0 0  1  0 0

1 √ 0 0 0 0 0 U31 = , U33 = , , U13 = 1 0 0 0 0 1 − δ2 √ 0 1 − δ2 U12 = , U23 = 0 1 . , U22 = 0, U32 = −δ 0

Vol. 62 (2008)


Then Σ(z) = =

=

=

345

Σ11 (z) Σ12 (z) Σ21 (z) Σ22 (z) U21 U22 U23 −1 U12 U13 +z (I − zU11 ) U32 0 U31     √ 0 0 0 0 1 1 − δ2  0 0 0 + z 1  1 zδ 0 √ 0 1 0 −δ 0 0 1 − δ2 0   0 1 √0 2 z 1 − δ 2 0 . √z δ 2 −δ z 1−δ 0

Thus, Σ11 (z) = 0, Σ12 (z) = 0

0 0 1 0

√ z2δ z 1 − δ2 1 , Σ21 (z) = , Σ22 (z) = √ −δ z 1 − δ2

0 . 0

Now we apply Theorem 6.4 to get the linear fractional parametrizations: S(z) = Σ11 (z) + Σ12 (z)W(z)(I − Σ22 (z)W(z))−1 Σ21 (z), −1 √ z2δ 0 z 1 − δ2 √ = 0 1 W(z) I − W(z) (6.37) −δ z 1 − δ2 0 and F0 (z) =

U21 (I − zU11 )−1 +Σ12 (z)(I − W(z)Σ22 (z))−1 W(z)U31 (I − zU11 )−1

= =

Σ12 (z)W(z)(I − Σ22 (z)W(z))−1 U31 (I − zU11 )−1 −1 0 1 √ zδ z 2δ 0 1 W(z) I − √ (6.38) W(z) 1 − δ2 z 1 − δ2 0 0

where W is a free-parameter 2 × 2 matrix valued Schur function. The function −1 z 2δ 0 G11 (z) G12 (z) G(z) = (6.39) := W(z) I − √ W(z) G21 (z) G22 (z) z 1 − δ2 0 belongs to H 2 and in turn, z 2δ W(z) = G(z) I − √ z 1 − δ2

0 G(z) 0

−1

.

The formulas (6.37) and (6.38) can be written in terms of G as follows: / S(z) = z 1 − δ 2 G21 (z) − δG22 (z), √ F0 (z) = G21 (z) zδG21 (z) + 1 − δ 2 G22 (z) .

(6.40)

(6.41) (6.42)

346

Ball and Bolotnikov

IEOT

We conclude that different parameters W may lead via formula (6.37) to the same S and to different F0 ’s via formula (6.38). This phenomenon exhibits explicitly the nonuniqueness of F0 corresponding to the same S. For example, the function S(z) ≡ 0 is a solution corresponding to the parameter W ≡ 0; then formula (6.38) gives F0 = 0 : X → H(KS ) = H 2 . But also we have we have S(z) ≡ 0 via (6.41) whenever √ 1 − δ2 zG21 (z) G22 (z) = (6.43) δ and for this relation in force we have from (6.42) z z (6.44) F0 (z) = G21 (z) δ G21 (z) = G21 (z) 1 δ . To show that there are many F0 ’s corresponding to S(z) ≡ 0, take G21 in H ∞ (rather than in H 2 ) with G21 H ∞ small enough. Then define G22 as in (6.43) and choose G11 (z) and G12 (z) so that G(z) in (6.39) has G∞ still small (much less than one). Then formula (6.40) gives a Schur function W which produces via formulas (6.37) and (6.38) S(z) ≡ 0 and the corresponding F0 of the form (6.44) with the prescribed G21 .

References [1] J. Agler, Some interpolation theorems of Nevanlinna–Pick type, 1988, Preprint. [2] J. Agler and J. E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal., 175 (2000), no. 1, 111–124. [3] D. Alpay, V. Bolotnikov and H.T. Kaptano˘ glu, The Schur algorithm and reproducing kernel Hilbert spaces in the ball, Linear Algebra Appl. 343 (2002), 163-186. [4] D. Alpay and C. Dubi, Boundary interpolation in the ball, Linear Algebra Appl. 340 (2002), 33-54. [5] D. Alpay and C. Dubi, Carathéodory-Fejér interpolation in the ball, Linear Algebra Appl. 359 (2003), 1–19. [6] D. Alpay and C. Dubi, Carathéodory-Fejér interpolation in the ball with mixed derivatives, Linear Algebra Appl. 382 (2004), 117–123. [7] A. Arias and G. Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205–234. [8] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404. [9] D.Z. Arov and H. Dym, Criteria for the strong regularity of J-inner functions and γ-generating matrices, J. Math. Anal. Appl. 280 (2003) no. 2, 387–399. [10] D.Z. Arov and L.Z. Grossman, Scattering matrices in the theory of unitary extensions of isometric operators, Soviet Math. Dokl. 270 (1983), 17–20. [11] D.Z. Arov and L.Z. Grossman, Scattering matrices in the theory of unitary extensions of isometric operators, Math. Nachr. 157 (1992), 105–123. [12] W. Arveson, Subalgebras of C ∗ -algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228.

Vol. 62 (2008)


347

[13] J.A. Ball and V. Bolotnikov, On a bitangential interpolation problem for contractive valued functions on the unit ball, Linear Algebra Appl. 353 (2002), 107–147. [14] J.A. Ball and V. Bolotnikov, A bitangential interpolation problem on the closed unit ball for multipliers of the Arveson space, Integral Equations and Operator Theory 46 (2003), 125-164. [15] J.A. Ball and V. Bolotnikov, Boundary interpolation for contractive-valued functions on circular domains in C n , in Current Trends in Operator Theory and its Applications (Ed. J.A. Ball, J.W. Helton, M. Klaus and L. Rodman) pp. 107-132, OT 149 Birkh¨ auser, Basel, 2004. [16] J.A. Ball and V. Bolotnikov, Nevanlinna-Pick interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function in C n , New York J. Math. 11 (2005), 1-44. [17] J.A. Ball and V. Bolotnikov, Interpolation in the noncommutative Schur-Agler class, J. Operator Theory 58 (2007) no. 1, 83–126. [18] J.A. Ball, V. Bolotnikov and Q. Fang, Multivariable backward-shift invariant subspaces and observability operators, Multidimens. Syst. Signal Process. 18 (2007), no. 4, 191–248. [19] J.A. Ball, V. Bolotnikov and Q. Fang, Transfer-function realization for multipliers of the Arveson space, J. Math. Anal. Appl. 333 (2007), no. 1, 68–92 [20] J.A. Ball, V. Bolotnikov and Q. Fang, Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations, J. Math. Anal. Appl. 341 (2008), 519–539. [21] J.A. Ball, I. Gohberg, and L. Rodman. Interpolation of Rational Matrix Functions, OT 45 Birkh¨ auser, Basel, 1990. [22] J.A. Ball and J.W. Helton, A Beurling-Lax theorem for the Lie group U (m, n) which contains most classical interpolation, J. Operator Theory 9 (1983), no.1, 107-142. [23] J.A. Ball and S. ter Horst, Multivariable operator-valued Nevanlinna-Pick interpolation: a survey, Proceedings IWOTA 2007, Potchefstroom, South Africa, Birkh¨ auser volume to appear. [24] J.A. Ball and M.W. Raney, Discrete-time dichotomous well-posed linear systems and generalized Schur-Nevanlinna-Pick interpolation, Complex Anal. Oper. Theory 1 (2007), no. 1, 1–54. [25] J.A. Ball and T.T. Trent, The abstract interpolation problem and commutant lifting: a coordinate-free approach, in: Operator theory and interpolation (Ed. H. Bercovici and C.I. Foias), Oper. Theory Adv. Appl. 115, Birkh¨ auser, Basel, 2000, pp. 51–83. [26] J.A. Ball, T.T. Trent and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernels Hilbert spaces, in: Operator Theory and Analysis (Ed. H. Bart, I. Gohberg and A.C.M. Ran), Oper. Theory Adv. Appl. 122, Birkh¨ auser, Basel, 2001, pp. 89-138. [27] V. Bolotnikov, A boundary Nevanlinna–Pick problem for a class of analytic matrixvalued functions in the unit ball, Linear Algebra Appl. 346 (2002), 239–260. [28] V. Bolotnikov, Interpolation for multipliers on reproducing kernel Hilbert spaces, Proc. Amer. Math. Soc., 131 (2003), 1373-1383. [29] L. de Branges and J. Rovnyak, S quare summable power series, Holt, Rinehart and Winston, New–York, 1966.

348

Ball and Bolotnikov

IEOT

[30] K.R. Davidson and D. R. Pitts, Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), no. 3, 321– 337. [31] K.R. Davidson and D. R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. 78 (1999) no. 2, 401–430. [32] S.W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc., 68 (1978), 300-304. [33] H. Dym, J-Contractive Matrix Functions, Reproducing Kernel Spaces and Interpolation, CBMS Lecture Notes 71, Amer. Math. Soc., Providence, 1989. [34] J. Eschmeier and M. Putinar, Spherical contractions and interpolation problems on the unit ball, J. Reine Angew. Math. 542 (2002), 219–236. [35] Q. Fang, Multivariable Interpolation Problems, Ph.D. dissertation, Virginia Tech, 2008. [36] C. Foias and A.E. Frazho, The Commutant Lifting Approach to Interpolation Problems, OT 44 Birkh¨ auser, Basel, 1990. [37] C. Foias, A.E. Frazho, I. Gohberg and M.A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems, OT 100, Birkh¨ auser, Basel, 1998. [38] A.E. Frazho, S. ter Horst and M.A. Kaashoek, Coupling and relaxed commutant lifting, Integral Equations and Operator Theory 54 (2006), no.1, 33-67. [39] A.E. Frazho, S. ter Horst and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged), 72 (2006), 299–318. [40] V. Katsnelson, A. Kheifets and P. Yuditskii, An abstract interpolation problem and extension theory of isometric operators, in: Operators in Spaces of Functions and Problems in Function Theory (V.A. Marchenko, ed.), 146, Naukova Dumka, Kiev, 1987, pp. 83–96. English transl. in: Topics in Interpolation Theory (H. Dym, B. Fritzsche, V. Katsnelson and B. Kirstein, eds.), Oper. Theory Adv. Appl. 95, Birkh¨ auser, Basel, 1997, pp. 283–298. [41] A. Kheifets, The Parseval equality in an abstract problem of interpolation, and the union of open systems, Teor. Funktsii Funktsional. Anal. i Prilozhen., 49, 1988, 112– 120, and 50, 1988, 98–103. English transl. in: J. Soviet Math., 49(4), 1990, 114–1120 and 49(6), 1990, 1307–1310. [42] A. Kheifets Scattering matrices and Parseval Equality in Abstract Interpolation Problem, Ph.D. Thesis, 1990, Kharkov State University. [43] A. Kheifets, The abstract interpolation problem and applications, in: Holomorphic spaces (Ed. D. Sarason, S. Axler, J. McCarthy), pp. 351–379, Cambridge Univ. Press, Cambridge, 1998. [44] A. Kheifets and P. Yuditskii, An analysis and extension of V.P. Potapov’s approach to interpolation problems with applications to the generalized bitangential SchurNevanlinna-Pick problem and j-inner-outer factorization, in: Matrix and Operator Valued Functions (Ed. I. Gohberg and L.A. Sakhnovich), Oper. Theory Adv. Appl. 72, Birkh¨ auser, Basel, 1994, pp. 133-161. [45] I.V. Kovalishina and V.P. Potapov, Seven Papers Translated from the Russian, Amer. Math. Soc. Transl. (2) 138, Providence, R.I., 1988. [46] S. Kupin, Lifting theorem as a special case of abstract interpolation problem, Z. Anal. Anwendungen 15 (1996), no. 4, 789–798.

Vol. 62 (2008)


349

[47] S. McCullough, The local de Branges-Rovnyak construction and complete Nevanlinna-Pick kernels, in Algebraic methods in operator theory (Ed. R. Curto and P. E. T. Jorgensen), Birkh¨ auser, Boston, 1994, pp. 15–24. [48] S. McCullough and T.T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), no. 1, 226–249. [49] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), no. 2, 523–536. [50] G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989), no. 1, 51–71. [51] G. Popescu, Interpolation problems in several variables, J. Math. Anal. Appl., 227 (1998), 227–250. [52] G. Popescu, Multivariable Nehari problem and interpolation, J. Funct. Anal. 200 (2003), no. 2, 536–581. [53] G. Popescu, Entropy and multivariable interpolation, Mem. Amer. Math. Soc. 184 (2006), no. 868, [54] P. Quiggin, For which reproducing kernel Hilbert spaces is Pick’s theorem true? Integral Equations and Operator Theory 16 (1993), no. 2, 244–266. [55] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. 127 (1967), 179–203. [56] D. Sarason, Sub-Hardy Hilbert spaces in the unit disk, John Wiley and Sons Inc., New York, 1994. Joseph A. Ball Department of Mathematics Virginia Polytechnic Institute Blacksburg, VA 24061-0123 USA e-mail: [email protected] Vladimir Bolotnikov Department of Mathematics The College of William and Mary Williamsburg, VA 23187-8795 USA e-mail: [email protected] Submitted: October 9, 2007. Revised: June 19, 2008.



Two-Wavelet Localization Operators on Homogeneous Spaces and Their Traces Viorel Catan˘a Abstract. We define two-wavelet localization operators in the setting of homogeneous spaces. We prove that they are in the trace class S1 and give a trace formula for them. Then we show that two-wavelet operators on locally compact and Hausdorff groups endowed with unitary and square-integrable representations, general Daubechies operators and two-wavelet multipliers can be seen as two-wavelet localization operators on appropriate homogeneous spaces. Thus we give a unifying view concerning the three classes of linear operators. We also show that two-wavelet localization operators on R, considered as a homogeneous space, under the action of the affine group U are two-wavelet multipliers Mathematics Subject Classification (2000). Primary 47G10; Secondary 43A85. Keywords. Homogeneous spaces, two-wavelet localization operators, two-wavelet multipliers, trace class.

0. Introduction The aim of this paper is to extend certain results in [6] (see Chapter 25, Proposition 25.3 and Theorem 25.4 and Chapter 26) concerning one-wavelet localization operators in the setting of homogeneous spaces to similar results for two-wavelet localization operators in the same setting. We emphasize the fact that M.W.Wong has first defined one-wavelet localization operators on homogeneous spaces in his book [6]. He also has showed that one-wavelet localization operators on locally compact and Hausdorff groups equipped with unitary and square-integrable representations, Daubechies operators and wavelet multipliers can be seen as one-wavelet operators on homogeneous spaces.Thus Wong has given a unifying view concerning the three classes of linear operators. Following Wong’s point of view we have extended these results from the onewavelet case to the two-wavelet case.

352

Catan˘ a

IEOT

The raison d’être for the extension from one wavelet to two wavelets comes from the extra degree of flexibility in signal analysis and imaging when the localization operators are used as time-varying filters. The paper is organized as follows: in Section 1 we give some preliminary results concerning the boundedness and the adjoints of two-wavelet localization operators on homogeneous spaces. These results will be use later in this paper. We prove that two-wavelet localization operators on homogeneous spaces are in the trace class S1 and give a trace formula for them, in Section 2. In Section 3 we show that two-wavelet localization operators associated to unitary and square-integrable representations of locally compact and Hausdorff groups, general Daubechies operators and two-wavelet multipliers are two-wavelet localization operators on appropriate homogeneous spaces. In this manner we give a unified approach for the three classes of linear operators. The aim of Section 4 is to show that two-wavelet localization operators on R, considered as a homogeneous space under the action of the affine group U are two-wavelet multipliers. We first review some basic concepts of homogeneous spaces (see the book [6], by Wong, the book [1], by Ali, Antoine and Gazeau and the book [4], by Folland) that we neeed for our investigation of the trace class property of two-wavelet localization operators on these spaces. We use the notation and terminology [6], to which we refer for more details. Let Ω be a locally compact and Hausdorff topological space and let G be a locally compact and Hausdorff group. Suppose that there exists a continuous mapping G × Ω → Ω, (g, ω) −→ gω such that: for all g in G, the mapping Ω → Ω, ω −→ gω is a homeomorphism, (gh) ω = g (hω) , g, h ∈ G, ω ∈ Ω, and eω = ω, ω ∈ Ω, where e is the identity element in G. Then we call G a left transformation group on Ω or a group action on Ω. The topological space Ω is called a G-space. Let us suppose that for all ω1 and ω2 in Ω there exists an element g in G such that ω2 = gω1 . Then we call the topological space Ω a homogeneous space and say that the group action G is transitive. We also say that the group G acts transitively on Ω. Let H be a closed subgroup of G and let G/H = {gH : g ∈ G} be the left coset space defined by H. Then we can define a transitive action of G on Ω = G/H by G × Ω → Ω, (g, hH) → (gh) H, g, h ∈ G. Thus Ω = G/H is a homogeneous space. In this context, a mapping is said to be a section on Ω if q (s (ω)) = ω, ω ∈ Ω, where q : G → Ω defined by q (g) = gH, g ∈ G

Vol. 62 (2008)

Two-Wavelet Localization Operators

353

is the canonical surjection of G on Ω = G/H. Let ν be a Borel measure on the locally compact and Hausdorff topological space Ω and let νg the measure on Ω defined by νg (S) = ν (gS) , g ∈ G, for all Borel sets S of Ω. Then we say that the Borel measure ν is a left quasiinvariant measure if ν and νg are equivalent measures on Ω (i.e. there exists a positive constant a such that ν = aνg ). In the following we shall consider only the case Ω = G/H. Let ν be a left quasi-invariant measure on Ω. Let π : G → U (X) be a unitary representation of G on complex Hilbert space X. We respectively denote by (, ) , and ∗ the inner product, the norm of the complex Hilbert space X and the norm in B (X) the C ∗ -algebra of all bounded linear operators on X with respect to the usual compositions of mappings. The group of all unitary operators on X with respect to the usual composition of mappings is denoted by U (X) . Let s : Ω → G be a Borel section. We say that π : G → U (X) is a squareintegrable representation of G on X with respect to H and s if there exists an element ϕ in X such that ϕ = 1 and 2 |(ϕ, π (s (ω)) ϕ)| dν (ω) < ∞. (0.1) Ω

We call ϕ an admissible wavelet for π. Suppose now that there exist two admissible wavelets ϕ and ψ in X for the square-integrable representation π : G → U (X) of G on X with respect to H and s such that 2 |(ϕ, π (s) ψ)| dν (ω) < ∞. (0.2) Ω

Then we define the constant cs,H,ϕ,ψ by cs,H,ϕ,ψ = (ϕ, π (s (ω)) ϕ) (π (s (ω)) ψ, ϕ) dν (ω) < ∞.

(0.3)

Ω

Let F ∈ L1 (Ω) and suppose that cs,H,ϕ,ψ = 0. Then we define the linear operator LF,s,H,ϕ,ψ : X → X by 1 (LF,s,H,ϕ,ψ x, y) = F (ω) (x, π (s (ω)) ϕ) (π (s (ω)) ψ, y) dν (ω) (0.4) cs,H,ϕ,ψ Ω

for all x and y in X. We call the operator defined by (0.4) a two-wavelet localization operator on the homogeneous space Ω.

354

Catan˘ a

IEOT

1. Preliminary Results Proposition 1.1. The adjoint of LF,s,H,ϕ,ψ : X → X is LF ,s,H,ψ,ϕ : X → X. Proof. For all x and y in X it follows immediately from (0.4) 1 F (ω) (x, π (s (ω)) ϕ) (π (s (ω)) ψ, y) dν (ω) (LF,s,H,ϕ,ψ x, y) = cs,H,ϕ,ψ Ω

=

1 cs,H,ϕ,ψ

=

F (ω) (y, π (s (ω)) ψ) (π (s (ω)) ϕ, x) dν (ω) Ω

LF ,s,H,ψ,ϕ y, x = x, LF ,s,H,ψ,ϕ y ,

because it is obvious that cs,H,ϕ,ψ = cs,H,ψ,ϕ . Thus we get L∗F,s,H,ϕ,ψ = LF ,s,H,ψ,ϕ . Proposition 1.2. Let F ∈ L1 (Ω). Then LF,s,H,ϕ,ψ : X → X is a bounded linear operator and 1 F L1 (G) . (1.1) LF,s,H,ϕ,ψ ∗ ≤ |cs,H,ϕ,ψ | Proof. Let x and y be any elements in X. Then using (0.4), Schwartz’ inequality, ϕ = ψ = 1 and the fact that π (g) : X → X is a unitary operator for all g in G (because π : G → U (X) is a unitary representation), we have |(x, π (s (ω)) ϕ) (π (s (ω)) ψ, y)| ≤ x y

(1.2)

Since F ∈ L1 (G), it follows from (0.4) and (1.2) that 1 F L1 (G) x y |(LF,s,H,ϕ,ψ x, y)| ≤ |cs,H,ϕ,ψ |

and the proof is complete. Remark 1.3. Suppose that we have 1 (x, y) = (x, π (s (ω)) ϕ) (π (s (ω)) ψ, y) dν (ω) cs,H,ϕ,ψ

(1.3)

Ω

for all x and y in X. Then we call (1.3) the resolution of the identity formula for the square-integrable representation π : G → U (X) of G on X with respect to H and s corresponding to the admissible wavelets ϕ and ψ. Thus the identity operator I on X can be written as 1 (·, π (s (ω)) ϕ) π (s (ω)) ψdν (ω) . (1.4) I= cs,H,ϕ,ψ Ω

The role of the symbol F : Ω → C is again to localize on the homogeneous space Ω so as to produce a nontrivial bounded operator LF,s,H,ϕ,ψ : X → X. We call

Vol. 62 (2008)


355

such an operator, a two-wavelet localization operator on the homogeneous space Ω.

2. Trace Class Property of Two-Wavelet Localization Operators on Homogeneous Spaces We prove in this section that two-wavelet localization operators on homogeneous spaces are in the trace class S1 and compute their traces. Theorem 2.1. Let F ∈ L1 (Ω). The two-wavelet localization operator LF,s,H,ϕ,ψ : X → X is in the trace class S1 . Proof. Let {ϕk : k = 1, 2, . . . } be an orthonormal basis for X. Then by (0.4), Fubini’s theorem, Schwarz’ inequality and Parseval’s identity, the fact that π : G → U (X) is a unitary representation (that is, π (g) : X → X is a unitary operator for all g in G), and ϕ = ψ = 1, we get ∞

2

LF,s,H,ϕ,ψ ϕk =

k=1

∞

(LF,s,H,ϕ,ψ ϕk , LF,s,H,ϕ,ψ ϕk )

k=1

∞ 1 = F (ω) (ϕk , π (s (ω)) ϕ) (π (s (ω)) ψ, LF,s,H,ϕ,ψ ϕk ) dν (ω) cs,H,ϕ,ψ k=1 Ω 1 = F (ω) L∗F,s,H,ϕ,ψ π (s (ω)) ψ, π (s (ω)) ϕ dν (ω) (2.1) cs,H,ϕ,ψ Ω 1 ≤ |F (ω)| L∗F,s,H,ϕ,ψ ∗ dν (ω) |cs,H,ϕ,ψ | Ω

=

1 |cs,H,ϕ,ψ |

F L1 (Ω) L∗F,s,H,ϕ,ψ ∗ < ∞,

where L∗F,s,H,ϕ,ψ : X → X is the adjoint of LF,s,H,ϕ,ψ : X → X.

By virtue of Proposition 1.1 and Proposition 1.2, L∗F,s,H,ϕ,ψ = LF ,s,H,ψ,ϕ and is a bounded linear operator. So, by (2.1) and the Propsition 2.8 in the book [6], by Wong, LF,s,H,ϕ,ψ : X → X is in the Hilbert-Schmidt class S2 and hence compact. Let {ϕk : k = 1, 2, . . . } and {ψk : k = 1, 2, . . . } be orthonormal sets in X. Then using (0.4), Fubini’s theorem, Schwarz’ inequality, Bessel’s inequality, the fact that π (g) : X → X is a unitary operator for all g in G (because π : G → U (X) is a unitary representation), and ϕ = ψ = 1, we get,

356

Catan˘ a

∞

|(LF,s,H,ϕ,ψ ϕk , ψk )|

k=1

≤

≤ ≤

IEOT

∞

1 |cs,H,ϕ,ψ |

|F (ω)| (ϕk , π (s (ω)) ϕ) | |(π (s (ω)) ψ, ψk )| dν (ω)

k=1 Ω

1 |cs,H,ϕ,ψ |

|F (ω)|

|cs,H,ϕ,ψ |

12

2

|(ϕk , π (s (ω)) ϕ)|

k=1

Ω

1

∞

∞

(2.2)

12

2

|(π (s (ω)) ψ, ψk )|

dν (ω)

k=1

|F (ω)| π (s (ω)) ϕ π (s (ω)) ψ dν (ω) = Ω

1 |cs,H,ϕ,ψ |

F L1 (Ω) < ∞.

Thus, by (2.2) and Proposition 2.5 in the book [6], by Wong, LF,s,H,ϕ,ψ : X → X is in the trace class S1 . Theorem 2.2. The trace tr (LF,s,H,ϕ,ψ ) of the two-wavelet localization operator LF,s,H,ϕ,ψ : X → X is given by (ψ, ϕ) tr (LF,s,H,ϕ,ψ ) = F (ω) dν (ω) (2.3) cs,H,ϕ,ψ Ω

Proof. Let {ϕk : k = 1, 2, . . . } be any basis for X. Then by (0.4), Fubini’s theorem, Parseval’s identity, the fact that π (g) : X → X is a unitary operator for all g in G, and ϕ = ψ = 1, we get ∞ tr (LF,s,H,ϕ,ψ ) = (LF,s,H,ϕ,ψ ϕk , ϕk ) =

= =

=

1 cs,H,ϕ,ψ 1 cs,H,ϕ,ψ 1 cs,H,ϕ,ψ (ψ, ϕ) cs,H,ϕ,ψ

∞

k=1

F (ω) (ϕk , π (s (ω)) ϕ) (π (s (ω)) ψ, ϕk ) dν (ω)

k=1 Ω

F (ω)

∞

(ϕk , π (s (ω)) ϕ) (π (s (ω)) ψ, ϕk ) dν (ω)

k=1

Ω

F (ω) (π (s (ω)) ψ, π (s (ω)) ϕ) dν (ω) Ω

F (ω) dν (ω) , Ω

and the proof is complete. Remark 2.3. If we take ϕ = ψ, then it is obvious that cs,H,ϕ,ψ = cs,H,ϕ and LF,s,H,ϕ,ψ = LF,s,H,ϕ

Vol. 62 (2008)


where

357

cs,H,ϕ =

(ϕ, π (s (ω)) ϕ) (π (s (ω)) ϕ, ϕ) dν (ω) , Ω

and (LF,s,H,ϕx, y) =

1 cs,H,ϕ,

F (ω) (x, π (s (ω)) ϕ) (π (s (ω)) ϕ, y) dν (ω) , Ω

for all x and y in X. (see Chapter 25 in the book, [6], by Wong). Thus, we get respectively from Theorem 2.1 and Theorem 2.2, Proposition 25.3 and Theorem 25.4 in the book [6], by Wong.

3. Classical Examples of Two-Wavelet Localization Operators on Homogeneous Spaces Example 3.1. (Two-wavelet localization operators). Let ϕ, ψ ∈ X be two admissible wavelets for a unitary and square-integrable representation π : G → U (X) of a locally compact and Hausdorff group G on a complex Hilbert space X. Let H = {e}, where e is the identity element in the group G. Then G/H = G, and µ, the left Haar measure carries by G is of course a left invariant measure on the homogeneous space G/H. In this context G/H is the same as G. We can take the section s : G/H → G to be the identity mapping on G. So, the two-wavelet localization operator LF,ϕ,ψ : X → X, defined by 1 (LF,ϕ,ψ x, y) = F (g) (x, π (g) ϕ) (π (g) ψ, y) dµ (g) , cϕ,ψ G

for all x and y in X (see p.84 in the book [6], by Wong) where cϕ,ψ = (ϕ, π (g) ϕ) (π (g) ψ, ϕ) dµ (g) , G

can be seen as a two-wavelet localization operator LF,s,H,ϕ,ψ : X → X on the homogeneous space Ω = G/H. Indeed 1 F (g) (x, π (s (g)) ϕ) (π (s (g)) ψ, ϕ) dµ (g) (LF,s,H,ϕ,ψ x, y) = cs,H,ϕ,ψ G (3.1) 1 = F (g) (x, π (g) ϕ) (π (g) ψ, y) , dµ (g) cs,H,ϕ,ψ G

358

Catan˘ a

IEOT

for all x and y in X, and =

cs,H,ϕ,ψ

(ϕ, π (s (g)) ϕ) (π (s (g)) ψ, ϕ) dµ (g) G

(3.2)

=

(ϕ, π (g) ϕ) (π (g) ψ, ϕ) dµ (g) = cϕ,ψ . G

By (3.1) and (3.2) it follows that two-wavelet localization operators associated to a unitary and square-integrable representation of G on X can be seen so as twowavelet localization operators on homogeneous spaces. Example 3.2. (General Daubechies operators) Let H = {(0, 0, t) ; t ∈ R/2πZ}. Then n n H is a subgroup of the Weyl-Heisenberg group (W H) and (W H) /H is simply n n n the Euclidean space R × R . The Lebesgue measure carries by R × Rn is in fact n a left (and right) Haar measure on the homogeneous space (W H) /H = Rn × Rn . n n n Let s : R ×R → (W H) be the section defined by s (q, p) = (q, p, 0) , (q, p) ∈ Rn × Rn . Let ϕ and ψ be two functions in L2 (Rn ) such that ϕL2 (Rn ) = ψL2 (Rn ) = 1 and (ψ, ϕ)L2 (Rn ) = 0. Then the two-wavelet constant cs,H,ϕ,ψ is equal to cs,H,ϕ,ψ

=

(ϕ, π (s (q, p)) ϕ) (π (s (q, p)) ψ, ϕ) dqdp Rn Rn

=

(ϕ, π (q, p, 0) ϕ) (π (q, p, 0) ψ, ϕ) dqdp

(3.3)

Rn Rn

=

(ϕ, ϕq,p ) (ψq,p , ϕ) dqdp, Rn

Rn

n where π : (W H) → U L2 (Rn ) is defined by (π (q, p, t) u) (x) = ei(px−qp+t) u (x − q) ,

x ∈ Rn ,

n

for all points (q, p, t) in (W H) and all functions u in L2 (Rn ) and ϕq,p (x) = eipx ϕ (x − q) ,

x ∈ Rn ,

for all q and p in Rn . Let F ∈ L1 (Rn × Rn ). Then the two-wavelet localization operator LF,s,H,ϕ,ψ : n L (Rn ) → L2 (Rn ) on the homogeneous space (W H) /H = Rn × Rn is defined 2

Vol. 62 (2008)


by (LF,s,H,ϕ,ψ f, g) = = =

1 cs,H,ϕ,ψ 1 cs,H,ϕ,ψ

1 cs,H,ϕ,ψ

F (q, p) (f, π (s (q, p)) ϕ) (π (s (q, p)) ψ, g) dqdp Rn Rn

F (q, p) (f, π (q, p, 0) ϕ) (π (q, p, 0) ψ, g) dqdp Rn

359

(3.4)

Rn

F (q, p) (f, ϕq,p ) (ψq,p , g) dqdp, Rn Rn

for all f and g in L2 (Rn ) . The general Daubechies operators DF,ϕ,ψ : L2 (Rn ) → L2 (Rn ) on the WeylHeisenberg group, associated to the symbols F ∈ L1 (Rn × Rn ) and the admissible wavelets ϕ and ψ, for the representation n n π : (W H) → U L2 (Rn ) of (W H) on L2 (Rn ) , are defined by 1 (DF,ϕ,ψ, f, g) = F (q, p) (f, π (q, p, t) ϕ) (π (q, p, t, ) ψ, g) dqdpdt cϕ,ψ (W H)n (3.5) 2π F (q, p) (f, ϕq,p ) (ψq,p , g) dqdp, = cϕ,ψ Rn Rn

for all f and g in L2 (Rn ), where cϕ,ψ = (ϕ, π (q, p, t) ϕ) (π (q, p, t) ψ, ϕ) dqdpdt (W H)n

=

(3.6) (ϕ, ϕq,p ) (ψq,p , ϕ) dqdp.

2π Rn

Rn

By (3.3) and (3.6) it follows that cs,H,ϕ,ψ = cϕ,ψ /2π.

(3.7)

So, by (3.4), (3.5) and (3.7) it follows that LF,s,H,ϕ,ψ = DF,ϕ,ψ .

(3.8)

Thus, by (3.8) we can say that general Daubechies operators on the Weyl-Heisenberg group are particular cases of two-wavelet localization operators on appropriate homogeneous spaces. Example 3.3. Let π : Rn → B L2 (Rn ) be the unitary representation of the additive group Rn on L2 (Rn ) defined by (π (ξ) u) (x) = eixξ u (x) , x, ξ ∈ Rn , for all functions u in L2 (Rn ). Let H = {0} be the null subgroup of Rn , where 0 is the additive identity of the group Rn . Then Rn /H = Rn , and the Lebesgue measure on Rn is a left (and right) invariant measure on the homogeneous space

360

Catan˘ a

IEOT

Rn /H, which is simply the same as Rn .Let s : Rn /H → Rn be the identity mapping on Rn which is of course a section. Let ϕ, ψ ∈ L2 (Rn )∩L∞ (Rn )∩L4 (Rn ) such that ϕL2 (Rn ) = ψL2 (Rn ) = 1. Then cs,H,ϕ,ψ = (ϕ, π (s (ξ)) ϕ)L2 (Rn ) (π (s (ξ)) ψ, ϕ)L2 (Rn ) dξ Rn

=

(ϕ, π (ξ) ϕ)L2 (Rn ) (π (ξ) ψ, ϕ)L2 (Rn ) dξ Rn

=

(ξ) dξ = (2π)n (ϕϕ)∧ (ξ) ϕψ ∧ (ξ) dξ (ξ) ϕ ϕ ∗ϕ ∗ψ

Rn

=

∧ n ∧ (2π) (ϕϕ) , ϕψ

L2 (Rn )

Rn n

= (2π)

ϕϕ, ϕψ L2 (Rn ) .

Furthermore, let σ ∈ L1 (Rn ). Then (Lσ,s,H,ϕ,ψ u, v)L2 (Rn ) 1 = σ (ξ) (u, π (s (ξ)) ϕ)L2 (Rn ) (π (s (ξ)) ψ, v)L2 (Rn ) dξ cs,H,ϕ,ψ Rn −1 −n ϕϕ, ϕψ L2 (Rn ) σ (ξ) (u, π (ξ) ϕ)L2 (Rn ) (π (ξ) ψ, v)L2 (Rn ) dξ = (2π)

(3.9)

Rn

−1 = ϕϕ, ϕψ L2 (Rn ) (Pσ,ϕ,ψ u, v)L2 (Rn ) , for all u and v in L2 (Rn ) where Pσ,ϕ,ψ : L2 (Rn ) → L2 (Rn ) is defined by −n σ (ξ) (u, π (ξ) ϕ)L2 (Rn ) (π (ξ) ψ, v)L2 (Rn ) dξ, (Pσ,ϕ,ψ u, v) = (2π)

(3.10)

Rn

for all u and v in L2 (Rn ) (see the paper [5], by Wong and Zhang). By (3.9) and (3.10) it follows that the two-wavelet localization operator Lσ,s,H,ϕ,ψ : L2 (Rn ) → L2 (Rn ) on the homogeneous space Rn /H = Rn is a scalar multiple of the two-wavelet multiplier Pσ,ϕ,ψ : L2 (Rn ) → L2 (Rn ). Remark 3.4. When we take ϕ = ψ we obtain from the Examples 3.1, 3.2, 3.3 respectively the Examples 25.5, 25.6, 25.7 in the Chapter 25 in the book [6], by M.W.Wong.

4. Two-Wavelet Localization Operators and the Affine Group Action on We show in this section that two-wavelet localization operators on R, considered as a homogeneous space under the action of the affine group U are two-wavelet multipliers.

Vol. 62 (2008)


361

Let U = {(b, a) : b ∈ R, a > 0} be the upper half plane. Then we define a binary operation · on U by (b1 , a1 ) · (b2 , a2 ) = (b1 + a1 b2 , a1 a2 ) for all points (b1 , a1 ) and (b2 , a2 ) in U . Then U is a locally compact Hausdorff group on which the left Haar measure is different from the right Haar measure. We call this group the affine group. (see Chapter 18 in the book [6], by Wong). Wong has observed in his book [6] that mapping U × R → R, ((b, a) , x) −→ ax + b represents a transitive group action of U on R. So, R is a homogeneous space under the group action of U . Let H = {(0, a) : a > 0} , which is a closed subgroup of U . Wong has proved that homogeneous space U/H is isomorphic with R as topological group and that Lebesgue measure on R is a left quasi-invariant measure on R, considered as a homogeneous space under the group action of U . (see Chapter 26, Proposition 26.1 and Proposition 26.2 in the book [6], by Wong). Let π : U → U L2 (R) be the unitary representation of U on L2 (R) given by

x−b 1 π ((b, a) u) (x) = √ u , a a for all (b, a) ∈ U , u ∈ L2 (R) and x ∈ R. Let s : R = U/H → U, s (x) = (x, 1), x ∈ R, be a section on R. Let ϕ, ψ ∈ L2 (R) such that ϕL2 (R) = ψL2 (R) = 1 and ϕ, ψ ∈ L4 (R). Observe that relations (0.1) and (0.2) are valid. Then using Plancherel’s theorem, we get cs,H,ϕ,ψ

=

+∞ (ϕ, π (s (x)) ϕ) (π (s (x)) ψ, ϕ) dx −∞

=

+∞ (ϕ, π (x, 1) ϕ) (π (x, 1) ψ, ϕ) dx −∞

=

+∞ (ϕ, T−x ϕ)L2 (R) (T−x ψ, ϕ)L2 (R) dx −∞

=

 +∞   +∞  +∞ 2  eixξ |ϕ (ξ)| dξ   eixξ ψ (ξ) ϕ (ξ) dξ  dx

−∞

−∞

−∞

+∞ ϕ ϕ = (2π) (−x) ψϕ (−x) dx = (2π) ϕ ϕ, ϕ ψ −∞

L2 (R)

,

where (T−x ϕ) (y) = ϕ (y − x) , y ∈ R. Let σ ∈ L1 (R). Then we define the two-wavelet localization operator Lσ,s,H,ϕ,ψ : L2 (R) → L2 (R) on R, considered as a homogeneous space under the

362

Catan˘ a

IEOT

group action of U , by the formula +∞ 1 σ (x) (u, π (s (x)) ϕ)L2 (R) (π (s (x)) ψ, v)L2 (R) dx, (Lσ,s,H,ϕ,ψ u, v)L2 (R) = cs,H,ϕ,ψ −∞

2

for all u and v in L (R). Observe that for all u and v in C0 (R) we can use Plancherel’s theorem to obtain (u, π (s (x)) ϕ)L2 (R) = (u, T−x ϕ)L2 (R) +∞ 1 ixξ 2 = e u (ξ) ϕ (ξ) dξ = (2π) u ϕ (−x) , −∞

and

1 (π (s (x)) ψ, v)L2 (R) = (v, π (s (x)) ψ)L2 (R) = (2π) 2 vψ (−x, )

for all x in R. Then for all u and v in L2 (R), we get (Lσ,s,H,ϕ,ψ u, v)L2 (R)

+∞ σ (x) u ϕ (x) vψ (x) dx

−1 ϕϕ, ϕψ

=

L2 (R)

−∞

+∞ v (x) dx Tσ ϕ u (x) ψ

=

−1 ϕψ ϕϕ,

=

−1 ϕψ ϕϕ,

v Tσ ϕ u , ψ

=

−1 ϕϕ, ϕψ

σ ϕ u, v ψT

=

−1 ϕϕ, ϕ ψ

σ ϕ F −1 ψT F u, v

−∞

L2 (R)

L2 (R)

L2 (R)

L2 (R)

L2 (R)

L2 (R)

where σ (x) = σ (−x), x ∈ R. Therefore the two-wavelet localization operator Lσ,s,H,ϕ,ψ : L2 (R) → L2 (R) −1 2 ϕ ψ 2 Pσ ,ϕ, is unitarily equivalent to the linear operator ϕ ϕ, : L (R) → ψ L (R)

2 2 L2 (R), where Pσ ,ϕ, : L (R) → L (R) is a two-wavelet multiplier associated to ψ and is defined by the symbol σ and the admissible wavelets ϕ, ψ,

Pσ ,ϕ, u, v ψ

L2 (R)

= (2π)−1

+∞ v σ (ξ) (u, τ (ξ) ϕ) L2 (R) τ (ξ) ψ,

−∞ 2

L2 (R)

dξ,

for all functions u and v in L (R) (see [5] by Wong and Zhang) where τ : R → U L2 (R) is the unitary representation of the additive group R on L2 (R) defined by (τ (ξ) u) (x) = eixξ u (x) , x, ξ ∈ R

Vol. 62 (2008)


363

for all functions u in L2 (R). Remark 4.1. When we take ϕ = ψ we obtain from the above considerations the main statement in the Chapter 26 in the book [6], by Wong. Acknowledgment The author would like to thank the referee for some helpful remarks, which have improved the paper.

References [1] S.T. Ali , J.-P. Antoine and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations. Springer-Verlag, 2000. [2] P. Boggiatto and M.W. Wong, Two-wavelet localization operators on Lp (Rn ) for the Weyl-Heisenberg group. Integr. Equ. Oper. Theory 49(2002), 1–10. [3] V. Catan˘ a, Schatten-vonNeumann norm inequalities for two-wavelet localization operators. In Pseudo-Differential Operators: Partial Differential Equations and TimeFrequency Analysis, eds. Luigi Rodino, Bert-Wolfgang Schultze, M.W. Wong, 52 (2007), 265–277. [4] G.B. Folland, A Course in Abstract Harmonic Analysis. CRC Press, 1995. [5] M.W. Wong and Z. Zhang, Traces of two-wavelet multipliers. Integr. Equ. Oper. Theory 42(2002), 498–503. [6] M.W. Wong, Wavelet Transforms and Localization Operators. Birkh¨ auser Verlag, Basel-Boston-Berlin, 2002. [7] ] M.W. Wong and Z. Zhang, Traces of localization operators with two admissible wavelets. ANZIAM J. 44(2003), 17–25. [8] Z. Zhang, Localization Operators and Wavelet Multipliers. Ph.D. Dissertation, York University, 2003. Viorel Catan˘ a University Politehnica of Bucharest Department of Mathematics I Splaiul Independent¸ei 313 060042, Bucharest, Romania e-mail: catana [email protected] Submitted: September 11, 2007. Revised: July 8, 2008.

Integr. equ. oper. theory 62 (2008), 365–382 c 2008 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030365-18, published online October 8, 2008 DOI 10.1007/s00020-008-1629-y


Local Minimal Curves in Homogeneous Reductive Spaces of the Unitary Group of a Finite von Neumann Algebra Eduardo Chiumiento To my family

Abstract. We study the metric geometry of homogeneous reductive spaces of the unitary group of a finite von Neumann algebra with a non complete Riemannian metric. The main result gives an abstract sufficient condition in order that the geodesics of the Levi-Civita connection are locally minimal. Then, we show how this result applies to several examples. Mathematics Subject Classification (2000). Primary 58B20; Secondary 46L10. Keywords. Finite von Neumann algebra, metric geometry, Finsler metric.

1. Introduction The aim of this work is to study local minimal curves in homogeneous reductive spaces of the unitary group of a finite von Neumann algebra, where we define a Riemannian metric in terms of the reductive structure and the trace of the algebra. Let M be a finite von Neumann algebra with a fixed trace τ . Denote by Mah the real Banach subspace of antihermitic elements of M, which identifies with the Lie algebra of the unitary group UM of the algebra. Let L2 (M, τ ) be the completion of M with the norm x2 = τ (x∗ x)1/2 . Consider P an (eventually infinite dimensional) homogeneous reductive space of UM . For each ρ ∈ P, there is defined in the tangent space (T P)ρ the coordinate map Kρ of the reductive structure taking values on Mah . Then, we can introduce a metric: for X ∈ (T P)ρ , Xρ = Kρ (X)2 . In general, this quadratic metric does not induce a complete norm in the tangent spaces. The homogeneous spaces treated here fit in the context of what is usually known as weak Riemannian manifolds (see [15]). Therefore the classical theory of

366

Chiumiento

IEOT

Riemann-Hilbert manifolds is not available, so it makes sense to ask about the local minimality of the geodesics. We give a sufficient condition to demonstrate a result on minimality of geodesics of the Levi-Civita connection in the general setting of homogeneous reductive spaces: τ -orthogonality of the supplements in Mah with the Lie algebras of the structure groups. Metric geometry in homogeneous spaces in the context of operator algebras is an area of current research. We can cite the remarkable works [8], [9] of C. Durán, L. Mata-Lorenzo and L. Recht, where they study the problem of finding minimal curves with a quotient metric induced by the operator norm. In the case of a finite algebra, metric and differentiable properties of several examples are treated in [1], [3], [7] and the references given there. The contents of the paper are the following. Section 2 contains basic facts about homogeneous reductive spaces and the metric. In section 3 we state and prove our main result concerning the minimality of the geodesics. In section 4 we give examples of homogeneous spaces satisfying the orthogonality condition. This condition naturally arises in several homogeneous spaces related to operator algebras: unitary orbits of states, normal elements, spectral measures, ∗homomorphisms, conditionals expectations and partial isometries. Finally, we give an example of an homogeneous space related to some particular C ∗ -dynamical systems.

2. Homogeneous reductive spaces We recall the definition of homogeneous reductive spaces and establish basic facts. For a deeper discussion of homogeneous reductive spaces we refer the reader to [6], [12] and [13]. In this work by a homogeneous reductive space we mean: • A C ∞ Banach manifold P. • A smooth transitive action of the unitary group UM on P. We denote it by Lu ρ, where u ∈ UM , ρ ∈ P. • For each ρ ∈ P the map Πρ : UM −→ P, Πρ (u) = Lu ρ is a principal bundle with structure group Gρ = { u ∈ UM : Lu ρ = ρ } (called the isotropy group of ρ). • There is a smooth distribution of horizontal spaces { Hρ : ρ ∈ P } which are supplements for the Lie algebra of Gρ : Hρ ⊕ Lie(Gρ ) = Mah ,

ρ ∈ P.

These supplements are invariant under the inner action of Gρ : uHρ u∗ = Hρ ,

ρ ∈ P, u ∈ Gρ .

From now on P stands for a homogeneous reductive space. We call a homogeneous reductive space P orthogonal if for each ρ ∈ P the supplements Hρ are τ -orthogonal to Lie(Gρ ) . For brevity, we shall frequently just say that P is orthogonal.

Vol. 62 (2008) Local Minimal Curves in Homogeneous Reductive Spaces

367

Remark 2.1. Let us denote by L2 (Mah , τ ) the real Hilbert space obtained by completion of Mah in the 2-norm. One can easily check that if P is orthogonal, then for all ρ ∈ P the orthogonal projection 2

Pρ : Mah −→ Lie(Gρ )

satisfies Pρ (Mah ) ⊆ Lie(Gρ ). Moreover, the converse is also true in the following sense: if the above projection preserves bounded elements for some ρ ∈ P (and hence for all), then Hρ := (I − Pρ )(Lie(Gρ )), defines a smooth distribution of norm closed supplements of Lie(Gρ ) which are invariant under the inner action of Gρ . Indeed, the distribution is smooth since PLu ρ = Ad(u) ◦ Pρ ◦ Ad(u∗ ), where Ad(u) : M −→ M, Ad(u)(x) = uxu∗ . In order to show that each Hρ is norm closed, let xn be a sequence in Hρ such that xn − x → 0, then (I − Pρ )(x) − xn 2 ≤ x − xn 2 ≤ x − xn → 0. Therefore, (I − Pρ )(x) = lim xn = x, so we obtain x ∈ Hρ . On the other hand, the invariance of the supplements is automatically verified. We have Hρ = { y ∈ Mah : τ (yx) = 0, ∀ x ∈ Lie(Gρ ) }. It is easily seen that u∗ Lie(Gρ )u = Lie(Gρ ), for all u ∈ Gρ . Therefore, we obtain for y ∈ Hρ , x ∈ Lie(Gρ ) , τ ((uyu∗ )x) = τ (y(u∗ xu)) = 0. Thus, uyu∗ ∈ Hρ , and our assertion is proved. Now we introduce the Riemannian metric as follows. The differential at the identity of Πρ gives the map δρ := (dΠρ )1 : Mah −→ (T P)ρ that induces the isomorphism −1 Kρ := δρ : (T P)ρ −→ Hρ . We endow P with an inner product on each tangent space, in order that these maps become isometric isomorphisms, i.e. for X, Y ∈ (T P)ρ ,

X, Y ρ = τ (Kρ (Y )∗ Kρ (X)).

(2.1)

Therefore, Xρ = Kρ (X)2 . It is apparent that this defines an invariant metric under the action of UM . Remark 2.2. Since P is a homogeneous space i.e. a quotient, it would be natural to put in P a quotient metric. One can define it by means of the 2-norm: for X ∈ (T P)ρ , 2

}, Xρ , 2 = inf{ z + y2 : y ∈ Lie(Gρ ) where δρ (z) = X. The infimum is attained in the element z − Pρ (z) ∈ L2 (Mah , τ ), and it belongs to Mah when P is orthogonal. Notice that when P is orthogonal,

368

Chiumiento

IEOT

then Pρ is actually the extension to L2 (Mah , τ ) of the projection I − Kρ ◦ δρ . Then, we have Xρ = Kρ (X)2 = (Kρ ◦ δρ )(z)2 = z − Pρ (z)2 = Xρ , 2 . Thus the quotient metric coincides with the metric given in equation (2.1). Let us recall the definition of the reductive connection ∇k introduced in [13]. Suppose that X, Y are tangent fields in P, then the value of ∇kX Y at ρ ∈ P is characterized by the following equation Kρ (∇kX Y ) = X(Y ) + [Kρ (Y ), Kρ (X)], where X(Y ) indicates the derivative of Y along X. On the other hand, we introduce as in [13] the classifying connection as follows: Kρ (∇cX Y ) = (Kρ ◦ δρ )(X(Y )). When P is orthogonal we shall demonstrate that the mean between ∇k and ∇c is the Levi-Civita connection of the metric defined by equation (2.1), and compute its geodesics. Proposition 2.3. Let P be an orthogonal homogeneous reductive space. Then the Levi-Civita connection of the metric , ρ is given by 1 k (∇ + ∇c ). 2 Moreover, γ(t) = LetKρ (X) ρ is the geodesic adapted to X ∈ (T P)ρ at γ(0) = ρ. ∇=

Proof. It is easy to show that the reductive connection ∇k is compatible with the metric. In order to prove that ∇c is compatible, let us consider X(t), Y (t) two tangent fields along a curve α(t) in P. Since P is orthogonal, I − Pρ is an orthogonal projection in L2 (Mah , τ ) which extends Kρ ◦ δρ . Then c D X ,Y = τ (Kα (Y )∗ (I − Pα )(K˙ α (X)) = τ (Kα (Y )∗ K˙ α (X)). dt α c We can proceed analogously with the term X, DdtY α . Then d d ( X, Y α ) = ( τ (Kα (Y )∗ Kα (X)) ) dt dt = τ (Kα (Y )∗ K˙ α (X)) + τ (K˙ α (Y )∗ Kα (X)) c D X Dc Y ,Y + X, . = dt dt α α Thus ∇c is compatible. In [13] was proved that the connection ∇c has the same geodesics as ∇k , but with opposite torsion. Therefore ∇ = 12 (∇k + ∇c ) is the Levi-Civita connection. The geodesics of the reductive connection, and hence of our Levi-Civita connection, were also computed in [13].


369

We end this section recalling a useful notion. For any curve γ ∈ P (not necessarily a geodesic) with γ(0) = ρ, there is an horizontal lifting Γ to the unitary group UM , which is characterized by the following properties: 1. Πρ (Γ) = γ. 2. Γ(0) = 1. 3. Γ˙ ∈ Hγ Γ. Moreover, it is also characterized as the unique solution of the linear differential equation: ˙ Γ˙ = Kγ (γ)Γ Γ(0) = 1.

3. Geodesics as minimal curves We prove several lemmas in the direction of Theorem 3.8 given at the end of this section. The main idea is to compare the lengths of a curve and its horizontal lifting in different norms, and then use a local convexity result given in [3]. Let us first introduce some notation. The length of a curve γ in P measured with the metric given in (2.1) will be denoted by 1 γ(t) ˙ γ(t) dt. L(γ) = 0

While the length of a curve α in UM measured with the 2-norm will be denoted by 1 α(t) ˙ 2 dt. L2 (α) = 0

Lemma 3.1. Let γ be a piecewise smooth curve in P, and let Γ be its horizontal lifting. Then, L(γ) = L2 (Γ). Proof. Using the definition of horizontal lifting we have: 1 1 ˙ 2 dt = L2 (Γ) = Γ(t) Kγ(t)(γ(t))Γ(t) ˙ 2 dt

0

0

1

= 0

Kγ(t)(γ(t)) ˙ 2 dt =

0

1

γ(t) ˙ γ(t) dt = L(γ).

Now we need to introduce another quotient metric, which is defined in [8] by Xρ , ∞ = inf{ z + y : y ∈ Lie(Gρ ) }, where δρ (z) = X. Note that it is also invariant under the action of UM . The length of a curve γ in P measured with this metric will be indicated by 1 Lq , ∞ (γ) = γ(t) ˙ γ(t), ∞ dt. 0

370

Chiumiento

IEOT

While the length of a curve α in UM measured with the operator norm will be denoted by 1 L∞ (α) = α(t) ˙ dt. 0

The next step is to compare the lengths of a curve and its horizontal lifting with these two metrics. In an orthogonal homogeneous space we have Pρ (Mah ) ⊆ Mah for all ρ ∈ P, then we can restrict Pρ to Mah obtaining an idempotent with range Lie(Gρ ), that we still denote by Pρ . Moreover, we can define this norm continuous idempotent in any homogeneous reductive space since Lie(Gρ ) is closed and complemented in the norm topology. Given ρ ∈ P, set M := I − Pρ . Then, observe that this constant is independent of the choice of ρ. Indeed, one computes I − PLu ρ = I − Ad(u) ◦ Pρ ◦ Ad(u∗ ) = I − Pρ . Lemma 3.2. Let γ be a piecewise smooth curve in P, and let Γ be its horizontal lifting. Then, L∞ (Γ) ≤ M Lq , ∞ (γ). Proof. Fix ρ ∈ P. For z ∈ Hρ such that δρ (z) = X and y ∈ Lie(Gρ ) , we have z = (I − Pρ )(z) ≤ M z + y. Then, z ≤ M δρ (z)ρ , ∞ . Therefore, Kρ (X) ≤ M δρ (Kρ (X))ρ , ∞ = M Xρ , ∞ . Using this inequality, and the fact previously noted that M is independent of the point, one finally obtains 1 1 ˙ dt = L∞ (Γ) = Γ(t) Kγ(t)(γ(t))Γ(t) ˙ dt 0

=

0

0

1

Kγ(t)(γ(t)) ˙ dt ≤ M

0

1

γ(t) ˙ γ(t) , ∞ = M Lq , ∞ (γ).

We need some facts about the geometry of the unitary group UM . The curves δ(t) = uetz , where u ∈ M and z ∈ Mah such that z < π, have minimal length along their paths when one measures lengths with the p-norm, p ≥ 2 (see [1]). Based on this fact, in Theorem 2.1 of [3] it was proved that if F2 denotes the energy functional 1 1 2 F2 (α) = α ˙ 2 dt = τ (α˙ ∗ α) ˙ dt, 0

0

where α is a piecewise smooth curve in UM and αs (t) is a smooth variation of α, i.e. αs (t) ∈ UM , s ∈ (−r, r), t ∈ [0, 1], α0 = α,


then the first variation of the energy functional is given by t=1 1 d 1 d F2 (αs ) = τ (x0 y0 ) − τ ( [x0 ]y0 ) dt, 2 ds dt 0 t=0 s=0

371

(3.1)

d d αs (t) and ys (t) = αs (t)∗ ds αs (t). where xs (t) = αs (t)∗ dt

Our result relies in the main theorem of [3]. Proposition 3.3. (Theorem 4.5 of [3]) Let u0 , u1 , u2 ∈ UM , such that ui − uj < √ √ 2− 2 . Let δ(t) = u1 etz be the minimal geodesic joining u1 and u2 . Then f (s) = 2 dp (u0 , δ(s))p , s ∈ [0, 1] (dp = geodesic distance induced by the p-norm) is a convex function, for p an even integer. We shall use this result when p = 2. Remark 3.4. If we compute the differential at 0 ∈ Mah of the exponential map at ρ, i.e. expρ : Hρ −→ P, expρ (z) = Lez ρ we obtain (d expρ )0 : Hρ −→ (T P)ρ , (d expρ )0 (z) = δρ (z), which is an isomorphism. Therefore by the inverse function theorem there exists r > 0 such that in a ball Br (0) we have that expρ : Br (0) −→ expρ (Br (0)) is a diffeomorphism. Let us write Vr (ρ) = expρ (Br (0)) for short. Then, for each ρ1 ∈ Vr (ρ), there exists a unique geodesic given by expρ (tz) = Letz ρ joining ρ and ρ1 inside Vr (ρ), where z satisfies Lez ρ = ρ1 . Remark 3.5. Now consider the following map F : Hρ ⊕ Lie(Gρ ) −→ UM , F ( (z, x) ) = ez ex . Differentiating one obtains (d F )(0,0) : Mah −→ Mah ,

(d F )(0,0) (z, x) = z + x,

which is an isomorphism. Then there exists a neighborhood V of (0, 0) and an > 0 such that F : V −→ B (1) ∩ UM , is a diffeomorphism. Moreover, we can choose V ⊆ B1 (0) × B2 (0) for 1 , 2 so small as we want if we just adjust . In the remainder of this section we require , 1 , 2 to satisfy: √ √ 2 i) < 2− . 2 ii) i < π9 , for i = 1, 2. iii) 1 < r.

372

Chiumiento

IEOT

Lemma 3.6. Let γ be a piecewise smooth curve in P such that γ(0) = ρ and Lq , ∞ (γ) < /M . Then there exists z ∈ Hρ satisfying: i) Γ(1) = ez ex , where x ∈ Lie(Gρ ). ii) z ∈ Br (0) is unique such that Lez ρ = γ(1). √ iii) The elements 1, ez and Γ(1) lie closer than

√ 2− 2 2

in norm.

Proof. i) If we write d∞ for the geodesic distance in UM with the Finsler metric given by the operator norm, then using the Lemma 3.2 we obtain 1 − Γ(1) ≤ d∞ (1, Γ(1)) ≤ L∞ (Γ) ≤ M Lq , ∞ (γ) < . By Remark 3.5 there exists a unique (z, x) ∈ Hρ ⊕ Lie(Gρ ) such that z < 1 , x < 2 and ez ex = Γ(1). ii) Note that Lez ρ = Lez ex ρ = LΓ(1) ρ = γ(1). Moreover, it is unique because z < 1 < r and expρ is one to one on the ball of radius r. √ √ 2 . On the other hand, our choice iii) We just have shown 1 − Γ(1) < < 2− 2 of 2 implies

√

2− 2 z x e − Γ(1) = 1 − e = 2 − 2 cos(x) < . 2 Analogously, our choice of 1 gives

√

2− 2 z . 1 − e = 2 − 2 cos(z) < 2 Proposition 3.7. Let P be an orthogonal homogeneous reductive space. Consider γ a piecewise smooth curve in P such that γ(0) = ρ and Lq , ∞ (γ) < /M . Then there exists a geodesic δ(t) = Letz ρ satisfying δ(1) = γ(1) and L(δ) ≤ L(γ). Proof. In view of Lemma√3.6 we have γ(1) = Lez ρ, where Γ(1) = ez ex , ez and 1 √ 2 lie at distance less than 2− in norm. Consider now µ(s) = ez esx , s ∈ [0, 1], 2 z the minimal geodesic joining e and Γ(1) = ez ex . Therefore by Proposition 3.3 the function f (s) = d2 (1, µ(s))2 is convex. Claim: f (0) = 0. Note the following: µ(s) − 1 ≤ ez esx − esx + esx − 1 ≤ ez − 1 + ex − 1 √ z z = e − 1 + Γ(1) − e < 2 − 2 < 2. Then the antihermitic logarithm log : {u ∈ UM : u − 1 < 2 } −→ { y ∈ Mah : y < π } is well defined and smooth. Let us call s (t) = et log(µ(s)) , which is a smooth variation of 0 (t) = etz . For each s, these curves are minimal geodesics in UM as a consequence of the inequality log(µ(s)) < π. Therefore, f (s) = L2 (s )2 = log(µ(s))22 = F2 (s ).


373

Following the notation adopted in equation (3.1) one notes that xs (t) = log(µ(s)) is constant with respect to t. Thus, the derivative reduces to

f (0) = 2 τ (zy0 (1)) − τ (zy0 (0)) . d Then observe that y0 (0) = 0 (0)∗ ds (0) = 0. On the other hand, we have to s=0 s d (1) = x, and using that Hρ and Lie(Gρ ) are τ compute y0 (1) = 0 (1)∗ ds s=0 s orthogonal, we finally obtain τ (zy0 (1)) = τ (zx) = 0. Thus our claim is proved. From this fact and the convexity of f we deduce that it has a global minimum at s = 0. Therefore, L2 (0 ) = d2 (1, ez ) = f (0)1/2 ≤ f (1)1/2 = d2 (1, Γ(1)) ≤ L2 (Γ). Calling δ the unique geodesic with the same initial and final endpoints as γ, note that 0 is the horizontal lifting of δ. Then by Lemma 3.1 we can conclude L(δ) = L2 (0 ) ≤ L2 (Γ) = L(γ).

We call a piecewise smooth curve a geodesic polygon if it is a continuous path in P, and consists of geodesics paths glued together. As an immediate consequence of Proposition 3.7 we give the following result. Corollary. Let P be an orthogonal homogeneous reductive space and γ a piecewise smooth curve in P. Then there exists a geodesic polygon ν such that ν(0) = γ(0), ν(1) = γ(1) and L(ν) ≤ L(γ). Proof. Clearly we can suppose that γ is smooth. Consider a partition 0 = t0 < t1 < ... < tn = 1 such that Lq , ∞ (γ|[ ti , ti+1 ] ) < /M . We use Proposition 3.7 to find geodesics δi with the same endpoints as γ|[ ti , ti+1 ] for i = 0, . . . , n − 1 satisfying L(δi ) ≤ L(γ|[ ti , ti+1 ] ). Then the curve ν obtained by gluing the geodesics δi is a geodesic polygon shorter than γ. In the next result we shall take R := min{ r , /2M }. Theorem 3.8. Let P be an orthogonal homogeneous reductive space and ρ ∈ P. Given any ρ1 ∈ VR (ρ), there exists a unique geodesic inside VR (ρ) which has minimal length among all the piecewise smooth curves inside VR (ρ), joining the points ρ and ρ1 . Proof. Let γ be a piecewise smooth curve inside VR (ρ). We can take a partition of the unit interval 0 = t0 < t1 < ... < tn = 1 such that Lq , ∞ (γ|[ ti , ti+1 ] ) < /2M . Therefore, ) = Lq , ∞ (γ ) + Lq , ∞ (γ ) < /2M + /2M = /M. Lq , ∞ (γ [ t0 , t2 ]

[ t0 , t1 ]

[ t1 , t2 ]

374

Chiumiento

IEOT

By Lemma 3.6 there is a unique z2 ∈ Br (0) ⊆ Hρ such that Lez2 ρ = γ(t2 ). Applying Proposition 3.7, the geodesic δ2 (t) = Letz2 ρ satisfies L(δ2 ) ≤ L(γ [ t0 , t2 ] ). On the other hand, since we have γ(t2 ) ∈ VR (ρ), by Remark 3.4 there exists a unique z2 ∈ BR (0) such that Lez2 ρ = γ(t2 ). Using our assumption that R ≤ r, we have z2 = z2 , and then δ2 is the unique geodesic in VR (ρ) joining ρ and γ(t2 ). An easy computation shows that Lq , ∞ (δ2 ) = z2 . Therefore, if we use the symbol # to denote a path obtained by adjoining two consecutive paths, we get ) = Lq , ∞ (δ2 ) + Lq , ∞ (γ ) < z2 + /2M < /M. Lq , ∞ (δ2 #γ [t2 , t3 ]

[t2 , t3 ]

Thus by the same argument as before, there exists a unique geodesic δ3 in VR (ρ) such that L(δ3 ) ≤ L(δ2 #γ [t2 , t3 ] ). Moreover, we can also estimate L(δ3 ) ≤ L(δ2 #γ [t2 , t3 ] ) = L(δ2 ) + L(γ [t2 , t3 ] ) ≤ L(γ [t0 , t2 ] ) + L(γ [t2 , t3 ] ) = L(γ [t0 , t3 ] ). It is clear that we can finish the proof by an inductive argument.

Remark 3.9. Our choice of R > 0 works for any ρ ∈ P. To show this, first observe that R depends only on r, M and . 1. r is independent of the point because the action is isometric. 2. The independence of M was shown before the Lemma 3.2. 3. If we considerer the map F u : HLu ρ ⊕ Lie(GLu ρ ) −→ UM , it is straightforward to verify that F u = F ◦ Ad(u). Then, using that Ad(u) is an isometric isomorphism we obtain that F u is a local diffeomorphism if and only if F is also a local diffeomorphism. Remark 3.10. All the work of this section could be carried out in C ∗ -algebra with a faithful trace. However, as we shall see, in the following examples we need to work in a finite von Neumann algebra to prove the orthogonality condition.

4. Examples We give several examples of orthogonal homogeneous reductive spaces. In these examples, the fundamental step to ensure that Theorem 3.8 holds, consists in proving the orthogonal condition. The following situation frequently arises: we have an algebraic subgroup G (in the sense of [6], [11]) of the Banach-Lie group UM and we want to check that G is actually a Banach-Lie subgroup to give a smooth manifold structure in the quotient UM /G. In view of Theorem 4.18 of [6] we only have to prove that Lie(G) is complemented in Mah . On the other hand, to obtain the minimality of geodesics, we must prove the orthogonality condition. So we can obtain both properties if we check that the orthogonal projection 2

P : L2 (Mah , τ ) −→ Lie(G)


375

satisfies P(Mah ) ⊆ Lie(G). Moreover, as we observe in Remark 2.1, this gives invariant supplements under the inner action of UM , and therefore, a reductive structure in UM /G. 4.1. Unitary orbit of a state This first example is concerned with the unitary orbit of a state. Let SM be the set of normal faithful states of M. Consider the action L : UM × SM −→ SM given by

Lu ϕ = ϕ ◦ Ad(u∗ ), u ∈ UM , ϕ ∈ SM . For ϕ ∈ SM , we denote by Uϕ the unitary orbit of ϕ, i.e. Uϕ = { ϕ ◦ Ad(u∗ ) : u ∈ UM }.

The isotropy group under this action Gϕ = { u ∈ UM : ϕ(uy) = ϕ(yu), ∀ y ∈ M } is an algebraic subgroup of order ≤ 1 of the Lie-Banach group UM . Actually this is verified by taking the polynomials py : M × M −→ C , py ((a, b)) = ϕ(ay) − ϕ(ya) , y ∈ M. The Lie algebra Lie(Gϕ ) = { x ∈ Mah : ϕ(xy) = ϕ(yx), ∀ y ∈ M }, consists in the antihermitic elements of the centralizer of ϕ, which is a von Neumann subalgebra of M. It is a well known fact that in a finite algebra there exists a unique trace invariant conditional expectation onto any von Neumann subalgebra (see for instance [16]). This gives that Lie(Gϕ ) is complemented in Mah , and therefore, the set UM /Gϕ has a smooth manifold structure. Then we endow Uϕ with the manifold structure such that the bijection [u] → ϕ ◦ Ad(u∗ ) is a diffeomorphism. Observe that the hypothesis of Theorem 3.8 is verified because this conditional expectation extends to the orthogonal projection over the respective completions with the 2-norm. Thus, we can conclude that the curve γ(t) = ϕ◦ Ad(e−tz u∗ ) has minimal length among all the curves contained in a neighborhood of ϕ◦Ad(u∗ ) that start at this point. 4.2. Unitary orbit of a normal element Let a be a normal element of M. We can study the unitary orbit of a, that is the set U(a) = { uau∗ : u ∈ UM }. The isotropy group at a of the natural action of UM is given by Ga = { u ∈ UM : ua = au }. Clearly, it is an algebraic subgroup of UM of order ≤ 1. The polynomials are defined by p : M × M −→ M, p((c, d)) = ca − ac. As in the previously example,

376

Chiumiento

IEOT

the Lie algebra of the isotropy group is the antihermitic elements of a von Neumann subalgebra Lie(Ga ) = { x ∈ Mah : xa = ax, } = { a } ∩ Mah . Thus U(a) ∼ = UM /Ga is a smooth manifold, and the geodesics γ(t) = etz ae−tz are locally minimizing. 4.3. Unitary orbit of a spectral measure Let Σ be a σ-algebra of subsets of some set. Let E : Σ −→ M be a spectral measure whose values are selfadjoint projections in M. The unitary orbit of E is U(E) = { uE( . )u∗ : u ∈ UM }. We take as subgroup of order ≤ 1 the isotropy group under the natural action of UM as in the preceding examples, i.e. GE = { u ∈ UM : uE(S) = E(S)u, ∀ S ∈ Σ }. The polynomials are pS : M × M −→ M, pS ((a, b)) = aE(S) − E(S)a. The Lie algebra of this group is Lie(GE ) = { x ∈ Mah : xE(S) = E(S)x, ∀ S ∈ Σ }, which consists of the antihermitic elements of a von Neumann subalgebra. Then we can use Theorem 3.8 to show that in the orthogonal homogeneous reductive space U(E) ∼ = UM /GE the geodesics γ(t) = etz E( . )e−tz are locally minimizing. 4.4. Unitary orbit of a ∗-homomorphism Consider ψ : M −→ M a ∗-homomorphism. Again we take the unitary orbit of ψ, i.e. U(ψ) = { uψ( . )u∗ : u ∈ UM }. The algebraic subgroup of order ≤ 1 is the isotropy at ψ under the natural action of the unitary group Gψ = { u ∈ UM : uψ(y) = ψ(y)u, ∀ y ∈ M }. The polynomials used to prove this fact are py : M × M −→ M, py ((a, b)) = aψ(y) − ψ(y)a. The Lie algebra consists in the antihermitic elements of the von Neumann algebra given by the commutant of ψ(M). Therefore, U(ψ) ∼ = UM /Gψ is an orthogonal homogeneous reductive space where the Theorem 3.8 holds.


377

4.5. Partial isometries This example is concerned with partial isometries in M. The set of partial isometries is I = { v ∈ M : v ∗ v is a projection }. We can give an action of the unitary group UM × UM of M × M on I by acting both on the initial and the final spaces. The action is given by L : (UM × UM ) × I −→ I,

L(u,w) v = uvw∗ .

This action is locally transitive, i.e. if two partial isometries lie closer than 1/2 in the operator norm, then they are conjugate by a pair of unitaries. In [2] was proved that each connected component, which as a consequence of local transitivity coincides with an orbit, is a homogeneous space of UM ×UM and a C ∞ submanifold of M. Hence, by Remark 2.1, to give a reductive structure we just have to prove the orthogonality condition. Fix v ∈ I, we shall study its orbit O(v). The isotropy group at v is Gv = { (u, w) ∈ UM × UM : uv = vw }. Note that if (u, w) ∈ Gv , then u commutes with the final projection vv ∗ and w commutes with the initial projection v ∗ v. One can compute the Lie algebra of this group Lie(Gv ) = { (x, y) ∈ Mah × Mah : xv = vy } ∗

x11 v x11 v 0 0 : x11 , x22 , y22 antihermitic }, ={ , 0 x22 0 y22 where the matrix decomposition is respect to vv ∗ in the first coordinate and to v ∗ v in the second coordinate. Note that in this case the Lie algebra does not consist of the antihermitic elements of a von Neumann algebra. In a finite von Neumann algebra, the orbits have the following particular property. Claim: Let M be a finite von Neumann algebra, then O(v) = O(v ∗ v). In particular, there is a projection in each orbit. To prove our claim, consider the set of partial isometries with fixed initial space. In other words, if p is a projection, we look at the set Ip = { v ∈ M : v ∗ v = p }. First let us demonstrate that { up : u ∈ UM } = Ip . One inclusion is trivial, for the other take v ∈ Ip , and let q = vv ∗ , which is a projection that is equivalent with p. Since M is finite, there exists u ∈ UM such that uqu∗ = p. Note that the element 1 − p + uv is unitary, and therefore, the element w = u∗ (1 − p) + v is also unitary. Finally, we obtain wp = vp = v. Then our claim follows easily. Indeed, since v ∈ Iv∗ v = { uv ∗ v : u ∈ UM }, there is u ∈ UM such that v = u(v ∗ v) = u(v ∗ v)1. Thus, we obtain O(v) = O(v ∗ v).

378

Chiumiento

IEOT

As a corollary of the above claim, it suffices to study the isotropy at a projection p. In this case, the expression of the Lie algebra reduces to

x11 0 x11 0 : x11 , x22 , y22 antihermitic }, Lie(Gp ) = { , 0 x22 0 y22 where the matrix decompositions are both respect to the projection p. Let us define a real bounded projection onto Lie(Gp ) by P : Mah × Mah −→ Lie(Gp ) , y11 y12 x11 x12 → , ∗ −x∗12 x22 −y12 y22

x11 +y11 2

0

0 x22

,

x11 +y11 2

0

0 y22

Observe that the kernel of this projection is w c12 −w d12 ker(P) = { , : w, c12 , d12 antihermitic }. −c∗12 0 −d∗12 0 Using the trace τ of M, we can define a finite trace in M × M by τ (x) + τ (y) , x, y ∈ M. 2 This gives an inner product on M × M such that ker(P) is orthogonal to Lie(Gp ). Indeed, for (x, y) ∈ Lie(Gp ) , (c, d) ∈ ker(P), one obtains τ˜((x, y)) =

2˜ τ ((x, y)(c, d)) = −τ (xc) − τ (yd) 0 w c12 x11 = −τ ( 0 x22 −c∗12 0 x11 0 −w d12 − τ( 0 y22 −d∗12 0 x11 w x11 w 0 ) + τ( = −τ ( 0 0 0

)− ) 0 ) = 0. 0

By the orthogonality of ker(P) with its range, P extends to the real orthogonal 2

projection onto Lie(Gp ) . Thus, we obtain that O(v) = O(v ∗ v) is an orthogonal homogeneous reductive space. Therefore the geodesics γ(t) = etz1 uvw∗ e−tz2 have minimal length among all the curves inside a neighborhood of uvw∗ . 4.6. Unitary orbit of a conditional expectation Let N be a von Neumann subalgebra of M and E : M −→ N the unique trace invariant conditional expectation. Our next example is about the unitary orbit of E. For a treatment of geometric properties of this example in a more general setting than finite algebras, we refer the reader to [4] and [5]. Let us define an action of UM on the algebra B(M) of bounded operators on M by L : UM × B(M) −→ B(M), Lu T = Ad(u) ◦ T ◦ Ad(u∗ ).


379

Consider the unitary orbit of E with this action U(E) = { Lu E : u ∈ M}. The isotropy group at E is usually called the normalizer of E, NE = { u ∈ UM : E(uyu∗ ) = uE(y)u∗ , y ∈ M }. Let us show that NE is an algebraic subgroup of UM of order ≤ 2. For each y ∈ M define the following bilinear maps ψy : (M × M) × (M × M) −→ M, ψy ((a, b), (a , b )) = E(ayb ) − aE(y)b . Then take the polynomials py ((a, b)) = ψy ((a, b), (a, b)) = E(ayb) − aE(y)b. In [5] was proved that for any faithful normal conditional expectation, its unitary orbit is a homogeneous reductive space of UM . In the finite algebra case we shall prove the orthogonality condition restricting to the unique trace invariant conditional expectation E. The arguments involved are adapted from Proposition 4.5 in [5], to this easier case. The Lie algebra of NE is the kernel of the differential of the natural fibration ΠE : UM −→ U(E),

ΠE (u) = Lu E.

In [5] was pointed out that Lie(NE ) = ker((dΠE )1 ) = (N + ME ) ∩ Mah , where ME is the von Neumann subalgebra of M given by ME = { x ∈ N ∩ M : E(xy) = E(yx), ∀ y ∈ M }. Let us call F : M −→ ME the unique conditional expectation such that τ ◦F = τ . Denote by Z(M) the center of M. Note that E(ME ) = Z(N ), then we have a conditional expectation E ◦ F : M −→ Z(N ) satisfying τ ◦ (E ◦ F ) = τ ◦ F = τ . Therefore there exist three orthogonal projections e, f , g in L2 (M, τ ), respectively associated with E, F , E ◦ F such that ef = g. Thus we obtain ef = f e. Let us call ∆ = E + F − EF which a projection onto ME +N satisfying ∆(Mah ) ⊆ Mah . Then ∆|Mah projects onto (ME + N ) ∩ Mah = Lie(NE ) and extends to the orthogonal projection 2

∆|Mah : L2 (Mah , τ ) −→ Lie(NE ) since e commutes with f . Thus Theorem 3.8 applies.

,

380

Chiumiento

IEOT

4.7. An homogeneous space related to C ∗ -dynamical systems This example justifies in part Remark 3.10. Let A be a C ∗ -algebra with a faithful trace τ and (A, G, α) a C ∗ -dynamical system. This means that G is a locally compact group and α is a continuous homomorphism of G into the group Aut(A) of ∗-automorphisms of A equipped with the pointwise convergence topology. Let us assume that G is compact and α is trace invariant, in the sense that τ (αt (x)) = τ (x) for all t ∈ G and x ∈ A. Consider the C ∗ -subalgebra of A given by AG = { x ∈ A : αt (x) = x, ∀ t ∈ G }. Denoting by µ the left Haar normalized measure on G, we can define E : A −→ AG , E(x) = αt (x) dµ(t). G

It is apparent that E is a norm one projection, and therefore a conditional expectation. We can easily show that E is trace invariant: τ (E(x)) = τ (αt (x)) dµ(t) = τ (x) dµ(t) = τ (x). G

G

Then, the Lie algebra of the unitary group UAG which identifies with AG ah is closed and complemented. In proving that UAG is actually a Lie-Banach subgroup of UA , it remains to show that UAG is endowed with a structure of Banach-Lie group whose underlying topology is the norm topology of UA . By Proposition 4.4 of [6] it suffices to find an open neighborhood V of 0 ∈ Aah and an open neighborhood W of 1 ∈ UA such that the exponential map exp |V : V −→ W is a diffeomorphism and exp(V ∩ AG ah ) = W ∩ UAG .

(4.1)

Since the exponential map is the usual exponentiation of operators to prove the equation (4.1) we only have to note that if u = ex ∈ UAG with u close to 1, x ∈ Aah , then x ∈ AG ah . This follows since x = log(u) =

∞

(−1)n+1

n=1

(u − 1)n , n

then αt (x) =

∞

(−1)n+1

n=1

(u − 1)n = x. n

Thus UAG is a Lie-Banach subgroup of UA , so we can consider the homogeneous space UA /UAG . Moreover, this is an orthogonal homogeneous reductive space since E is trace invariant.


381

Acknowledgment The author wishes to express his gratitude to Professor Esteban Andruchow for suggesting the problem and for many helpful conversations during the preparation of the paper.

References [1] E. Andruchow, L. Recht, Grassmannians of a finite algebra in the strong operator topology. Internat. J. Math. 17 (2006), no. 4, 477-491. [2] E. Andruchow, G. Corach, M. Mbekhta, On the geometry of generalized inverses. Mathematische Nachrichten 278 (2005), no. 7-8, 756-770. [3] E. Andruchow, L. Recht, Geometry of unitaries in a finite algebra: Variations formulas and convexity. Int. J. Math. (to appear). [4] M. Argerami, D. Stojanoff, The Weyl group and the normalizer of a conditional expectation. Integral Equations and Operator Theory 34 (1999), no. 2, 165-186. [5] M. Argerami, D. Stojanoff, Orbits of conditional expectations. Illinois J.Math. 45 (2001), no. 1, 243-263. [6] D. Beltita, Smooth homogeneous structures in operator theory. Chapman and Hall/CRC, Monographs and Surveys in Pure and Applied Mathematics 137, 2006. [7] C. Dur´ an, L. Mata-Lorenzo, L. Recht, Natural variational problems in the Grassmann manifold of a C ∗ -algebra with trace. Adv. Math. 154 (2000), no. 1, 196-228. [8] C. Dur´ an, L. Mata-Lorenzo, L. Recht, Metric geometry in homogeneous spaces of the unitary group of a C ∗ -algebra. Part I: Minimal curves. Adv. Math. 184 (2004), no. 2, 342-366. [9] C. Dur´ an, L. Mata-Lorenzo, L. Recht, Metric geometry in homogeneous spaces of the unitary group of a C ∗ -algebra. Part II: Geodesics joining fixed endpoints. Integral Equations and Operator Theory 53 (2005), no. 1, 33-50. [10] P. Halmos, J. McLaughlin, Partials isometries. Pacific J. Math. 13 (1963), no. 2, 585-596. [11] L.A. Harris, W. Kaup, Linear algebraic groups in infinite dimensions. Illinois J. Math. 21 (1977), no. 3, 666-674. [12] S. Kobayashi, K. Nomizu, Foundations of differential geometry. Vol. I. Reprint of the 1969 original, Wiley Classics Library, John Wiley & Sons, 1996. [13] L. Mata-Lorenzo, L. Recht, Infinite-dimensional homogeneus reductive spaces. Acta Cient. Venezolana 43 (1992), 76-90. [14] G. Pedersen, C*-algebras and their automorphism groups. Academic Press, 1979. [15] A. Stacey, Variations on a theme: Riemannian geometry in infinite dimensions. Algebraic topology special session, BMC 2007. [16] M. Takesaki, Theory of Operator Algebras I. Springer-Verlag, 1979.

382 Eduardo Chiumiento Dto. de Matem´ atica FCE-UNLP Calles 50 y 115 (1900) La Plata Argentina e-mail: [email protected] Submitted: November 30, 2007. Revised: June 15, 2008.

Chiumiento

IEOT



Covariant Representations for Matrix-valued Transfer Operators Dorin Ervin Dutkay and Kjetil Røysland Abstract. Motivated by the multivariate wavelet theory, and by the spectral theory of transfer operators, we construct an abstract affine structure and a multiresolution associated to a matrix-valued weight. We describe the one-toone correspondence between the commutant of this structure and the fixed points of the transfer operator. We show how the covariant representation can be realized on Rn if the weight satisfies some low-pass condition. Mathematics Subject Classification (2000). 37C40, 37A55, 42C40. Keywords. Transfer operator, C ∗ -algebra, harmonic map, cocycle, martingale.

1. Introduction Several themes are involved in this paper: the “covariant representations” in the title has been a central construct from the theory of operator algebras since the 1950s, and they have played a key role in numerous applications since. One of these more recent applications is to a class of wavelets called “frequency localized” wavelets. The “transfer operators” in the title refers to a construction with origins in probabilistic path models from physics and ergodic theory. One of our aims here is to point out some connections between the two areas, and to show how operator algebraic ideas and representations throw new light on a classical theme. Since several ideas are involved, readers from one area may look for pointers to the other. We begin with a brief guide to the literature: One use of operator algebras (specifically, C ∗ -algebras) is to the construction of representations. Initially [31], the focus was on groups, but the notion of covariance from physics (see e.g., [10, 35]) suggested crossed products of groups which act by automorphisms on C ∗ -algebras ([14, 38]). Since the pioneering paper by Stinespring [36], a preferred approach (e.g., [1]) to constructing representations begins with a positive operator

384

Dutkay and Røysland

IEOT

valued mapping, and it was Stinespring’s insight that identified the correct “positivity” as complete positivity. But, at the same time, notions of positivity are central in a variety of probabilistic path models, beginning with Doeblin [18], see also [12]. It is now also a key tool in ergodic theory, [33, 37]. As a result, Doeblin’s operator has subsequently taken on a variety of other incarnations and it is currently known as “the transfer” operator, the Ruelle operator, or the Perron-FrobeniusRuelle operator [6]. The name Ruelle is from its use in statistical mechanics as pioneered by David Ruelle, see [35, 6]. Of a more recent vintage are applications to wavelets [16], i.e., special and computational bases in Hilbert space constructed from a class of unitary representation of certain discrete groups of affine transformations. It was realized (e.g., [9]) that there are transfer operators RW for wavelets, that the solution to a spectral problem for RW yields wavelet representations; and moreover that these representations come with a useful covariance. Hence the circle closes with the positivity question from [36], albeit in a different guise. It is intriguing that a variant of the operators RW have now also found use in quantum error correction codes, see [11, 29]. All the versions of transfer operators involve hierarchical processes with branching, and probabilities assigned by a weight function. Our paper focuses on RW in the wavelet context; and we demonstrate that for many wavelets, the weight function must take the form of an operator transformation X → mXm∗ where m is a fixed matrix valued function and where m∗ denotes the adjoint operator, in this case transpose-conjugate. This matrix version of RW is necessary for understanding wavelet constructions associated to wavelet sets [15], and more generally to non-MRA wavelets, [3]. (MRA stands for multiresolution analysis [16].) An orthogonal wavelet is a function ψ ∈ L2 (R) such that {2j/2 ψ(2j ·−k) | j, k ∈ Z} is an orthonormal basis in L2 (R). The main technique used to construct wavelets is by multiresolutions. For this one needs a low-pass filter m0 , i.e., a 2π-periodic Lipschitz continuous function that satisfies the low-pass condition and the QMF condition: √ m0 (0) = 2, 2 x + 2π 1 x 2 = 1, (x ∈ R). m0 + m0 2 2 2 Then, from the low-pass filter, one constructs the Fourier transform of the scaling function

∞ m0 2xn √ , (x ∈ R). ϕ(x) ˆ = 2 n=1 Finally, the wavelet is constructed from the scaling function using the formula x x 1 x ˆ ψ(x) = √ e i 2 m0 + π ϕˆ , (x ∈ R). 2 2 2 It turns out that the low-pass condition and the QMF condition, while necessary, are not always sufficient to obtain an orthonormal wavelet. One of the extra

Vol. 62 (2008)

Covariant Representations

385

conditions on m0 that guarantees the orthogonality of the wavelet was given by W. Lawton [30]. Here the transfer operator was introduced in the study of wavelets. In this context, the transfer operator is defined on 2π-periodic functions by: Rm0 f (x) =

1 x x x + 2π 2 x + 2π (|m0 ( )|2 f ( ) + |m0 ( )| f ( )), 2 2 2 2 2

(x ∈ R).

Lawton’s condition states that the wavelet is orthogonal if and only if the only continuous functions h with Rm0 h = h are the constants. When this condition is not satisfied, the resulting wavelet still has an interesting property, namely it generates a Parseval frame. Having this, the theory of dilations of Parseval frames due to D. Han and D. Larson [26] can be used. The wavelet Parseval frame is the projection of an orthonormal basis in a bigger space. The problem then was if this orthonormal basis has a similar wavelet structure, i.e., if it is generated by the application of unitary dilation and translation operators U and T that satisfy the commutation relation U T U −1 = T 2 (resembling the relation between the dilation and the translation operators in L2 (R)). The answer to this question is positive and it is given in [7]. It turns out that each fixed point of the transfer operator Rm0 will give rise to such a wavelet structure, and we call this a covariant representation. Putting together these representations, one obtains the covariant representation which has an orthonormal wavelet in a bigger Hilbert space whose projection onto L2 (R) is the Parseval wavelet frame constructed in the classical wavelet theory. This is one of the wonderful uses of covariant representations. Since these orthonormal wavelet bases live in a bigger Hilbert space, i.e., one that contains L2 (R) as a subspace, Han and Larson coined the term “super-wavelets”. Another interesting application of covariant representation is the computation of the peripheral spectrum for the transfer operator: the fixed points of Rm0 are in one-to-one correspondence with the commutant of the covariant representation. Often this commutant can be explicitly computed, hence the eigenspace of Rm0 can be obtained from that. The spectral properties of transfer operators play an important role in the ergodic analysis of discrete dynamical system (see [6]). Why matrix-valued transfer operators? It is known (see [16]) that not all wavelets in L2 (R) can be constructed from a multiresolution. However there are generalizations of this that will do the job: each orthogonal wavelet can be constructed from a generalized multiresolution analysis, a notion introduced by Baggett et al. [5]. The construction requires some matricial low-pass filters. Also multiwavelet theory, in which one uses more than one function to generate the basis, requires matricial filters. But many times the resulting wavelet is only a Parseval frame, not an orthogonal basis (see [4]). There is no analogue for the Lawton condition in the case of matricial low-pass filters. The main reason for this is the impossibility to give a good generalization of the notion of zeros for the fixed points of the transfer operator, a notion which plays an essential role in the scalar case. We believe that the covariant representations can provide a way around this, and we can analyze the orthogonality of wavelets

386


IEOT

and scaling functions constructed from matricial low-pass filters through a study of the associated covariant representations. More generally, a transfer operator, also called Ruelle operator, is associated to a finite-to-one continuous endomorphism on a compact metric space r : X → X and a weight function W : X → [0, ∞), and it is defined by W (y)f (y) RW f (x) = r(y)=x

for functions f on X. Transfer operators have been extensively used in the analysis of discrete dynamical systems [6] and in wavelet theory [9]. In multivariate wavelet theory (see for example [27] for details) one has an expansive n × n integer matrix A, i.e., all eigenvalues λ have |λ| > 1, and a multiresolution structure on L2 (Rn ), i.e., a sequence of subspaces {Vj }j∈Z of L2 (Rn ) such that (i) Vj ⊂ Vj+1 , for all j; (ii) ∪j Vj is dense in L2 (Rn ); (iii) ∩j Vj = {0}; (iv) f ∈ Vj if and only if f ((AT )−1 ·) ∈ Vj−1 ; (v) There exist ϕ1 , . . . , ϕd ∈ V0 such that {ϕk (· − j) | k ∈ {1, . . . , d}, j ∈ Zn } forms an orthonormal basis for V0 . The functions ϕ1 , . . . , ϕd are called scaling functions, and their Fourier transforms satisfy the following scaling equation: ϕî (x) =

d

q −1/2 mji (A−1 x)ϕˆj (A−1 x),

(x ∈ Rn , i ∈ {1, . . . , d}),

j=1

where q := | det A| and mji are some Zn -periodic functions on Rn . The orthogonality of the translates of ϕi implies the following QMF equation: 1 m∗ (y)m(y) = 1, (x ∈ Rn /Zn ), q d Ay=x mod Z

where m is the d×d matrix (mij )di,j=1 . When the translates of the scaling functions are not necessarily orthogonal, one still obtains the following relation: if we denote by hij (x) := ϕî (x + k)ϕˆj (x + k), (x ∈ Rn /Zn ), k∈Zn

then the matrix h = (hij )di,j=1 satisfies the following property: 1 m∗ (y)h(y)m(y) = h(x), (x ∈ Rn /Zn ), Rh(x) := q n

(1.1)

Ay=x mod Z

i.e., h is a fixed point for the matrix-valued transfer operator R. The fixed points of a transfer operator are also called harmonic functions for this operator. Thus

Vol. 62 (2008)


387

the orthogonality properties of the scaling functions are directly related to the spectral properties of the transfer operator R. This motivates our study of the harmonic functions for a matrix-valued transfer operator. The one-dimensional case (numbers instead of matrices) was studied in [9, 20, 19]. These results were then extended in [21, 22], by replacing the map x → Ax mod Zn on the torus Tn /Zn , by some expansive endomorphism r on a metric space. Here we are interested in the case when the weights defining the transfer operator are matrices, just as in equation (1.1). We keep a higher level of generality because of possible applications outside wavelet theory, in areas such as dynamical systems or fractals (see [21, 22]). However, for clarity, the reader should always have the main example in mind, where r : x → Ax mod Zn on the torus Tn . In [22] it was shown that, whenever a pair (m, h) is given, with h ≥ 0 and Rh = h, one can construct a covariant structure on some Hilbert space H, i.e., a unitary U , a representation π of continuous functions on X, and some scaling functions ϕ1 , . . . , ϕd ∈ H such that U π(f )U ∗ = π(f ◦ r), (f ∈ C(X)), ϕi , π(f )ϕj = f hij dµ, (f ∈ C(X)), X

U ϕi =

d

π(mji )ϕj ,

(i ∈ {1, . . . , d}),

(1.2)

j=1

{U −n π(f )ϕi | n ∈ Z, i ∈ {1, . . . , d}, f ∈ C(X)} is dense in H,

(1.3)

where µ is a strongly invariant measure on X (see (2.1)). Moreover, this construction is unique up to isomorphism. Thus, from the “filter” m and the “harmonic function” h, one can construct a multiresolution structure, similar to the one used in wavelet theory (see [9]). There are several uses for such a structure: one can analyze the peripheral spectrum of the transfer operator [20], construct super-wavelet bases on spaces bigger than L2 (R) or on some fractal spaces [7, 23], or use the rich algebraic and analytic structure of this multiresolution to perform computations needed in the harmonic analysis of fractal measures [21]. This covariant representation was analyzed in more detail in the one-dimensional case d = 1 in [22, 21]. The main tool used there was the introduction of some random-walk measures Px following an idea of Conze and Raugi [13]. In our case m and h are matrix-valued. We will use the language of vector bundles because we are interested also in continuous harmonic functions and projective multiresolution analyses (see [24, 32]). We will describe the covariant representation in the multivariable case using some positive-matrix valued measures (see Theorem 3.2, Propositions 5.1 and 5.3), thus extending previous results from [22].

388


IEOT

In Section 2 we introduce the setup and the main definitions and assumptions on the transfer operator R associated to a matrix-valued weight h. Then in Section 3 we show how the covariant representation can be constructed on the solenoid of r. In Section 4 we show that the operators that commute with this covariant representation are in one-to-one correspondence with the harmonic functions of R (Theorem 4.1). Then the set of harmonic functions inherits a C ∗ -algebra structure from the commutant, and a more intrinsic description of the multiplication is given in Theorem 4.3 and Remark 4.4. The covariant structure can be described in terms of some operator-valued measures Px . We do this in Section 5. With the aid of these measures we can give another form to the correspondence between harmonic maps and operators in the commutant, which can be identified now with cocycles (Proposition 5.6, Theorem 5.5). In the case when the filter m satisfies a low-pass condition (in this case, the E(l)-condition as defined in [27]), we can realize the covariant representation in the more familiar environment of L2 (Rn ). The space Rn has a natural embedding in the solenoid, the measures Px are atomic and, the measure of the atoms are directly related to the solutions of the refinement equation, i.e., the scaling functions. Our results extend some ideas from [25, 17, 28] to the matricial case.

2. Definitions and preliminaries The dynamical system. Let X be a compact Hausdorff space with a surjective and finite to one continuous map r : X → X. Moreover, let µ be a regular measure on µ that is strongly r-invariant, i.e. 1 f dµ = f (s)dµ(t), (2.1) −1 (t) rs=t X X #r for every f ∈ C(X). Main example. The main example we have in mind, is the following: Let G ⊂ Rn be a discrete subgroup such that Rn /G is compact, i.e., G is a full-rank lattice and Rn /G Tn =: X. Moreover, let A ∈ GL(Rn ) be strictly expansive and such that AG ⊂ G. Let p : Rn → X denote the quotient map and define a map r : X → X as r(p(x)) = p(Ax). This is a | det(A)|-folded normal covering map. Finally, we let µ be the Haar measure on Tn . The transfer operator. Let ρ : ξ → X be a d-dimensional complex vector bundle over X with a Hermitian metric on ξ, i.e., a continuous map ·, · : ξ × ξ → C that restricts to an inner-product on each fiber. Such a map always exist when X is compact, [2, 1.3.1]. Let S be the set of continuous sections in ξ. Then s1 (x), s2 (x) dµ(x), s1 , s2 → X

Vol. 62 (2008)


389

defines an inner-product on S. Let K denote the Hilbert space obtained by the completion with respect to the corresponding norm. Then K is the Hilbert space of L2 -sections in ξ with respect to the measure µ. The C(X)-product on the sections in ξ gives a representation of C(X) on K by pointwise multiplication, κ : C(X) → B(K) such that (κ(f )s)(x) = f (x)s(x) for every x ∈ X. Every bundlemap on ξ commutes with this representation. Remark 2.1. Let us consider the von Neumann algebra κ(C(X)) generated by this representation. Since the bundle is locally trivial around any point in X, we can take U1 , . . . , Ur open sets in X, with bundle isomorphisms φj : ξ|Uj → Uj ×Cd . If V ∈ κ(C(X)) there exists V (x) ∈ C such that x → φj (x)−1 V (x)φj (x) is ∞ an L (Uj , µ|Uj ) function and φj (V s(x)) = φj (V (x)s(x)), µ-a.e. If V ∈ κ(C(X)) there exists V (x) ∈ EndC (ξ|x ) such that the map x → φj (x)−1 V (x)φj (x) is contained in Md (C) ⊗ L∞ (Uj , ν|Uj ) and φj (V s)(x)) = φj (V (x)s(x)),

µ-a.e.

Let r∗ ξ denote the pull-back of ξ along r (see [2, 1.1]). The space of sections ˜ denote in r ξ is endowed with the pull-back Hermitian metric from ξ. Let K the L2 -sections in this bundle with respect to this metric and µ. Moreover, let ˜ denote the corresponding representation of C(X) by pointwise κ ˜ : C(X) → B(K) multiplication. The weight (or the filter if we use wavelet terminology) that is used to define ˜ K) such that m˜ the transfer operator is in our case an operator m ∈ B(K, κ(f ) = κ(f )m for every f ∈ C(X). As in Remark 2.1, m is a pointwise multiplication by a linear map between the fibers ξrx and ξx , i.e., there exist a unique m(x) ∈ HomC (ξ|rx , ξ|x ) such that φj (m(x, v)) = φj (m(x)v) for every x ∈ Uj and v ∈ ξrx . ∗

Definition 2.2. Let M = κ(C(X)) . As we have seen before, the space M consists in bounded measurable bundle maps on ξ. We define the transfer operator on M by 1 (Rf )(x) = m∗ (y)f (y)m(y), (x ∈ X). −1 #r (x) ry=x Assumptions on the filter m. Throughout the paper we will assume that m satisfies the following conditions: ˜ K) is injective (i) m ∈ B(K, (ii) supk Rk < ∞ (iii) There exists an h ∈ End(ξ) such that Rh = h and h ≥ 0 (here we mean h non-negative as an operator). Remark 2.3. The first condition (i) implies that m(x) is invertible µ-a.e. Note that if there exists an h ∈ M such that Rh = h and h ≥ c1 for some c > 0 (here 1 is the identity bundle map on ξ), then (ii) holds automatically, see [24].

390


IEOT

Definition 2.4. A function f ∈ M such that Rf = f is called harmonic with respect to the transfer operator R. We denote by H the set of all bounded harmonic functions.

3. The covariant representation associated to a matricial weight and a harmonic bundle map We mentioned in the introduction that for some choices of the filter m0 such as the “stretched Haar filter” (see [16]) the scaling function and wavelet constructed in L2 (R) are not orthogonal,i.e., they do not have orthogonal translates. Thus classical wavelet theory breaks down if we want to have orthogonal solutions for the prescribed scaling equation. Even deeper problems appear when the filter does not satisfy the low-pass condition, since the infinite product defining the scaling function from m0 is 0. Fractal spaces may occur (see [23]). We will show that we always have a solution to a given scaling equation in some Hilbert space with a covariant representation. The Hilbert space depends sensitively on the filter m and the correlation function h. We construct now the affine structure associated to the matricial filter m and the positive harmonic function h. The covariant representation is a Hilbert space endowed with an affine structure given by a unitary U , which takes the place of the dilation operator, and a representation π of C(X), which takes the place of the representation generated by translations. In this Hilbert space one has several vectors ϕ1 , . . . , ϕn that satisfy the scaling equation with the prescribed filter m as in (1.2), and having correlation function h as in (1.3). The ground space. Consider the projective system Xo

Xo

r

r

Xo

···

and let X∞ be its projective limit, with projections θn : X∞ → X, (n ≥ 0), and a homeomorphism rˆ : X∞ → X∞ such that the following diagrams are commutative for all n ≥ 0: rˆ / X∞ E X∞ EE θn−1 E EE θn EE θn " /X X r

We have the following identification of X∞ X∞ = {(x0 , x1 , x2 , . . . ) ∈ X × X × . . . |rxj+1 = xj , for all j ≥ 0} θk (x0 , x1 , . . . ) = xk ,

rˆ(x0 , x1 , . . . ) = (rx0 , x0 , x1 , . . . ).

The space X∞ is a compact Hausdorff space with the topology generated by the inverse images of the open sets in X with respect to the maps θn , n ≥ 0.

Vol. 62 (2008)


391

The Hilbert space. Let Sk denote the vector space of continuous sections in θ0∗ ξ that depend only on the first k + 1 coordinates x0 , x1 , . . . , xk . We see that Sk is formed by s ◦ θk for sections in ξ, s : X → ξ, i.e., maps of the form x → (ˆ r−k (˜ x), s ◦ θk (˜ x)),

(˜ x ∈ X).

We define the inner-product ·, · k : Sk × Sk → C by 1 m(k) (y)f (y), h(y)m(k) (y)g(y) dµ(x), f ◦ θk , g ◦ θk k := −k (x) X #r k r y=x

where m(k) (x) := m(x)m(rx) . . . m(rk−1 x). We have the following compatibility relation between these inner-products:

f ◦ θl , g ◦ θl k+l = f ◦ rk ◦ θk+l , g ◦ rk ◦ θk+l k+l 1 = m(k+l) (y)f (rk y), h(y)m(k+l) (y)g(rk y) dµ(x) −(k+l) (x) X #r k+l r y=x 1 = m(l) (y)f (y), Rk (h)(y)m(l) (y)g(y) dµ(x) −l X #r (x) l r y=x

= f ◦ θl , g ◦ θl l

(since Rk h = h).

This shows that the restriction ·, · k+l |Sl ×Sl = ·, · l . We obtain a sesquilinear form · , · on ∪k Sk , i.e., a (possibly degenerate) inner-product. Let H denote the Hilbert space completion with respect to this inner-product. An application of the Stone-Weierstrass theorem gives that ∪k Sk is dense in the space of continuous sections in θ0∗ ξ. Moreover, the continuous sections in θ0∗ ξ are contained in H by the following argument. Lemma 3.1. If {sk }k∈N is a sequence of sections in ξ such that {sk ◦ θk }k≥0 converges uniformly to a section s : X∞ → θ0∗ ξ then s, s = limk sk ◦ θk , sk ◦ θk k . Proof. sk+l ◦ θk+l − sk ◦ θk 2 1 = −(k+l) (x) #r X · sk+l (y)−sk ◦ rl (y), m(k+l)∗ (y)h(y)m(k+l) (y)(sk+l (y)−sk ◦ rl (y)) dµ(x) r k+l y=x

≤ sk+l ◦ θk+l −sk ◦

θk 2∞

X

1 #r−(k+l) (x)

m(k+l)∗ (y)h(y)m(k+l) (y)dµ(x)

r k+l y=x

≤ sk+l ◦ θk+l − sk ◦ θk 2∞ dRk+l h∞ , and the inequality follows since i Hi∗ Hi ≤ i Tr(Hi∗ Hi ) = Tr( i Hi∗ Hi ) last ≤ d i Hi∗ Hi for every finite sequence of matrices Hi .

392


IEOT

Next we define the covariant representation and the multiresolution on the Hilbert space H. Theorem 3.2. Let Pk denote the orthogonal projection in B(H) onto the closed subspace Hk generated by Sk . Define π : C(X) → B(H), U ∈ B(H),

(π(f )s)(x) = f ◦ θ0 (x)s(x),

(f ∈ C(X), s ∈ H, x ∈ X∞ ); ˜ (U s)(x) = m ◦ θ0 (x)s ◦ rˆ(x), (s ∈ H, x ∈ X).

The following relations hold: (i) U is a unitary such that U π(f )U ∗ = π(f ◦ r) for all f ∈ C(X). (ii) P0 ∈ π(C(X)) . (iii) Pk ≤ Pk+1 for every k ∈ N. (iv) limk Pk s = s for every s ∈ H. (v) U Pk+1 U ∗ = Pk for every k ∈ N. Proof. (ii) and (iii) follow directly from the definition. (iv) follows from the fact that ∪k Sk is dense in the sections of θ0∗ ξ. We must prove (i). First, let f, g ∈ S, then m ◦ θ0 f ◦ θk ◦ rˆ, m ◦ θ0 g ◦ θk ◦ rˆ = (m ◦ rk−1 f ) ◦ θk−1 , (m ◦ rk−1 g) ◦ θk−1 1 = −(k−1) (x) X #r (k−1) · m (y)m(rk−1 y)f (y), h(y)m(k−1) (y)m(rk−1 y)g(y) dµ(x) r k−1 y=x

1

= X

#r−(k−1) (x)

m(k) (y)f (y), h(y)m(k) (y)g(y) dµ(x)

r k x=y

= f ◦ θk , g ◦ θk This shows that U is isometric on ∪k Sk . To see that U is surjective on H, we will show that U Hk+1 = Hk , (with Hk := Pk H) which by (iv) implies that U is surjective. If s ∈ Hk+1 , then U s = m ◦ θ0 s ◦ rˆ ∈ Hk . If we can show that {(m ◦ rk s) ◦ θk |s measurable section in ξ} is dense in Hk , we are done. Recall that m(x) is invertible µ-a.e. Take s ∈ S. Suppose, {Vl }l∈N is a decreasing family of open sets such that {x ∈ X| det m ◦ rk (x) = 0} ⊂ Vl for every l ∈ N and liml µ(Vl ) = 0. (Note also that by (2.1) the measure µ is invariant under r, i.e., µ(r−1 (E)) = µ(E) for all measurable sets E). For every l ∈ N, there exists a measurable section sl in ξ, such that sl is 0 on Vl , and m ◦ rk (x)sl (x) = s(x) for x ∈ X \ Vl . Then liml m ◦ rk sl = s, µ-a.e. and by the dominated convergence theorem, liml U sl ◦ θk+1 = s ◦ θk , in H, i.e., s ◦ θk is in the closure of U Sk . Finally (v) follows from the relation U Hk+1 = Hk and (i). Definition 3.3. We denote by C ∗ (X, r, m, h) the C ∗ -algebra generated by U and π(f ), f ∈ C(X), and we call it the covariant representation associated to m and h.

Vol. 62 (2008)


393

4. Harmonic and measurable bundle maps In this section we establish the one-to-one correspondence between harmonic functions and operators in the commutant C ∗ (X, r, m, h) . We begin with a result on self-adjoint elements, and we require that the harmonic function h0 be dominated by h, in the sense that |h0 | ≤ ch for some constant c > 0. In the case when h is bounded away from 0, i.e., h ≥ C1 for some C > 0, then this domination condition is automatically satisfied and we can extend the correspondence to non-self-adjoint elements (Corollary 4.2). Theorem 4.1. (i) [Harmonic maps to operators in the commutant] If h0 ∈ M, h∗0 = h0 , Rh0 = h0 and there exists a positive number c ≥ 0 such that |h0 | ≤ ch, there exists a unique selfadjoint operator A in the commutant C ∗ (X, r, m, h) such that f ◦ θ0 , Ag ◦ θ0 =

f , h0 g ,

(f, g ∈ S).

X

We denote this operator by Ah0 . Moreover in this case 1 m(k) (y)f (y), h0 (y)m(k) (y)g(y) µ(dx), f ◦ θk , Ag ◦ θk = −k (x) X #r k r y=x

for every k ∈ N and f, g ∈ S. (ii) [Operators in the commutant to harmonic maps] Conversely, define the operator T ∈ B(P0 H, K) such that T s ◦ θ0 = hs for every s ∈ S. If A is a selfadjoint operator in the commutant C ∗ (X, r, m, h) then hA := T P0 AP0 T ∗ ∈ M is a harmonic function, i.e., RhA = hA , and |hA | ≤ ch for some constant c > 0. Moreover, the correspondences described in (i) and (ii) are inverses to each other, i.e., hAh0 = h0 , AhA = A. Proof. (i) Let Bk denote the sesquilinear form on Sk , given by 1 m(k) (y)f (y), h0 (y)m(k) (y)g(y) µ(dx). Bk (f ◦ θk , g ◦ θk ) = −k #r (x) X k r y=x

Using the computation for ·, · k , we see that Bk |Sk−1 ×Sk−1 = Bk−1 . By the boundedness assumption on h0 , we have −ch ≤ h0 ≤ ch. This shows that −c f ◦ θk , f ◦ θk ≤ Bk (f ◦ θk , f ◦ θk ) ≤ c f ◦ θk , f ◦ θk . Since h0 is self-adjoint this implies that we obtain a bounded sesquilinear map B on H that restricts to Bk on Sk . Let A denote the bounded operator on H such that B(f, g) = f, Ag for every f, g ∈ H. The computation that showed that U was isometric with respect to ·, · applies here too, and it shows that B(f, g) = B(U f, U g) for every f, g ∈ H. Now f, U Ag = U ∗ f, Ag = B(U ∗ f, g) = B(f, U g) = f, AU g ,

394


IEOT

i.e., AU = U A. Moreover, by a direct computation, we have B(f, π(a)g) = B(π(a∗ )f, g) for every a ∈ C(X), f, g ∈ H. Using this, we obtain f, Aπ(a)g = B(f, π(a)g) = B(π(a∗ )f, g) = π(a∗ )f, Ag , i.e., π(a)A = Aπ(a) for every a ∈ C(X). Finally, since U is unitary, U ∗ A = U −1 A = AU −1 = AU ∗ and A commutes with every operator in C ∗ (X, r, m, h). The uniqueness follows from the fact that ∪k Sk is dense in H. (ii) First, a simple computation shows that T ∗ s = s ◦ θ0 for s ∈ K. Then s1 (x), (T aT ∗ )(x)s2 (x) dµ(x) = s1 ◦ θ0 , as2 ◦ θ0 . X

for every a ∈ π(C(X)) ∩ B(P0 H). If A commutes with π(f ) for all f ∈ C(X) then, since P0 commutes also with π(f ), it follows that P0 AP0 commutes with π(f ) so T P0 AP0 T ∗ commutes with κ(f ) for all f ∈ C(X). Therefore T P0 AP0 T ∗ ∈ κ(C(X)) , and we can use Remark 2.1. Moreover, if A ∈ C ∗ (X, r, m, h) , it commutes also with U , and then, using the strong invariance of the measure µ, s1 , R(T P0 AP0 T ∗ )s2 dµ X 1 = m(y)s1 (ry), (T P0 AP0 T ∗ )(y)m(y)s2 (ry) dµ(x) −1 x ry=x X #r = m(x)s1 (rx), (T P0 AP0 T ∗ )(x)m(x)s2 (rx) dµ(x) = U s1 ◦ θ0 , P0 AP0 U s2 ◦ θ0 X

= U s1 ◦ θ0 , AU s2 ◦ θ0 = s1 ◦ θ0 , As2 ◦ θ0 = s1 ◦ θ0 , P0 AP0 s2 ◦ θ0 = s1 , T P0 AP0 T ∗ s2 dµ, X

i.e., T P0 AP0 T ∗ is a fixed point for R. Since A is bounded and self-adjoint, it follows that s1 (x) , hA (x)s1 (x) dµ(x)| = | s1 ◦ θ0 , As1 ◦ θ0 | | X ≤ A s1 ◦ θ0 , s1 ◦ θ0 = A s1 (x) , h(x)s1 (x) dµ(x), X

which implies that −Ah ≤ hA ≤ Ah,, for µ-a.e. x ∈ X. To prove that the correspondences are inverses to each other we compute: s1 , T P0 AP0 T ∗ s2 dµ = s1 ◦ θ0 , P0 AP0 s2 ◦ θ0 = s1 ◦ θ0 , As2 ◦ θ0 X = s1 , h0 s2 dµ, X

for every s1 , s2 ∈ E, i.e., h0 = T P0 AP0 T ∗ and A = Ah0 .

Vol. 62 (2008)


395

Corollary 4.2. Assume in addition that h ≥ c1 for some constant c > 0. Let T ∈ B(P0 H, K) such that T s ◦ θ0 = hs for every s ∈ S. The map A → T P0 AP0 T ∗ yields a bijection between the commutant C ∗ (X, r, m, h) and the bounded harmonic functions H := {f ∈ M | Rf = f }. Proof. Since h ≥ c1, the boundedness condition |h0 | ≤ const h is equivalent to the essential boundedness of h0 , therefore it is automatically satisfied. Theorem 4.1 gives us the bijection between self-adjoint elements. But every element A in the commutant can be written as a linear combination of self-adjoint elements: A = (A + A∗ )/2 + i(A − A∗ )/2i. Similarly for the harmonic functions. This gives the bijection. Corollary 4.2 shows that the set of bounded harmonic maps has also multiplicative structure, the one induced from multiplication of operators via the given bijection. The following theorem gives an alternative, more intrinsic description of this multiplicative structure. It is also a generalization of a result from [34]. Theorem 4.3. Suppose h ≥ c1 µ-a.e., for some constant c > 0. If h1 , h2 ∈ H then h1 ∗ h2 (x) := lim Rk (h1 h−1 h2 )(x) k

exists µ a.e and h1 ∗ h2 ∈ H. The map A → T P0 AP0 T ∗ defines a ∗-isomorphism from C ∗ (X, r, m, h) to H, when H is equipped with the product ∗ and the ordinary involution. ˜ ˜ is completely positive Proof. We define R(b) = h−1/2 R(h1/2 bh1/2 )h−1/2 . Now R and unital (see [24]). The Kadison-Schwarz inequality [8, II.6.8.14] for completely positive and unital maps yields ˜ −1/2 bh−1/2 )∗ R(h ˜ −1/2 bh−1/2 )h1/2 R(b)∗ h−1 R(b) = h1/2 R(h ˜ −1/2 b∗ h−1 bh−1/2 )h1/2 = R(b∗ h−1 b) ≤ h1/2 R(h This implies that R(h∗i h−1 hi ) ≥ R(h∗i )h−1 R(hi ) = h∗i h−1 hi , i ∈ {1, 2}, so Rk (h∗i h−1 hi ) is a positive and increasing sequence in M. Since supk Rk < ∞, the sequence is uniformly bounded, so Rk (h∗i h−1 hi ) converges pointwise µ-a.e. to some element hi ∗ hi in M. And it is easy to see that R(hi ∗ hi ) = hi ∗ hi . 3 Note that h1 h−1 h2 = 14 k=0 i−k (h∗1 + ik h2 )∗ h−1 (h∗1 + ik h2 ). It follows that limk Rk (h1 h−1 h2 )(x) exists µ-a.e. for arbitrary h1 , h2 ∈ H. If A1 , A2 ∈ C ∗ (X, r, m, h) , hi := T P0 Ai P0 T ∗ , h3 := T P0 A1 A2 P0 T ∗ , then, since

s1 ◦ θ0 , Ai s2 ◦ θ0 = s1 (x) , hi (x)s2 (x) dµ(x) = s1 ◦ θ0 , (h−1 hi ) ◦ θ0 s2 ◦ θ0 , X

we see that P0 Ai P0 is an operator of multiplication by hi h−1 ◦ θ0 .

396


IEOT

Let a := h1 h−1 h2 − h3 . Then (Pk A1 Pk A2 Pk − Pk A1 A2 Pk )f ◦ θ0 2 = (P0 A1 P0 A2 P0 − P0 A1 A2 P0 )U k f ◦ θ0 2 = h−1 (h1 h−1 h2 − h3 )m(k) f ◦ rk , hh−1 (h1 h−1 h2 − h3 )m(k) f ◦ rk dµ X = f, Rk (a∗ h−1 a)f dµ, X

since U is unitary, Pk = U −k P0 U k , and µ is strongly invariant. Since Pk converges to the identity strongly as k increases, we obtain that limk X f, Rk (a∗ h−1 a)f dµ = 0 for every f ∈ K. But this implies that there exists a subsequence such that limj Rkj (a∗ h−1 a)(x) = 0 µ-a.e. The Kadison-Schwarz inequality implies that Rk (a)∗ h−1 Rk (a) ≤ Rk (a∗ h−1 a). Then Rkj (a) converges µ-a.e. to 0, so Rkj (h1 h−1 h2 ) converges µ-a.e. to h3 . We already know that Rk (h1 h−1 h2 ) converges µ-a.e. to h1 ∗ h2 , so h1 ∗ h2 (x) = h3 (x), µ-a.e. Remark 4.4. We proved in [24], that in the case when r is obtained by applying a certain covering projection to an expansive automorphism, and m and h are Lipschitz, with h ≥ c1, then there is a C ∗ -algebra structure on the continuous harmonic functions. The multiplication is constructed as follows: due to the quasicompactness of the transfer operator R (restricted to Lipschitz functions), the uniform limit n−1 1 j R f, T1 (f ) := lim n→∞ n j=0 exists for every continuous f and defines a continuous harmonic function. The product of two continuous harmonic functions is defined by (h1 , h2 ) → T1 (h1 h−1 h2 ). Theorem 4.3 shows then that this product coincides with h1 ∗ h2 . In particular, if h1 , h2 are continuous harmonic functions, and Ah1 , Ah2 are the associated operators in the commutant, then Ah1 Ah2 = Ah1 ∗h2 is also associated to a continuous harmonic function h1 ∗ h2 = T1 (h1 h−1 h2 ). The next corollary shows how the covariant representation can be decomposed using projections in the algebra H. Corollary 4.5. Suppose h1 , . . . , hl form a family of mutually orthogonal projections in H, i.e., hi ∗ hi = h∗i = hi , 1 ≤ i ≤ l,

hi ∗ hj = δi,j , 1 ≤ i, j ≤ l.

Let p1 , . . . , pl denote the corresponding orthogonal projections in C ∗ (X, r, m, h) . Moreover, let (Hi , ·, · i ) denote the Hilbert space, associated to hi , in Theorem 3.2 with the representation πi : C(X) → B(Hi ), unitary Ui and increasing family of projections {Pki }k .

Vol. 62 (2008)


397

Then pi H = Hi , pi s1 , pi s2 = s1 , s2 i for every s1 , s2 sections in θ0∗ ξ. We obtain an isometry J : ⊕li=1 Hi → H,

J(s1 , . . . , sl ) = p1 s1 + . . . pl sl ,

such that (i) π(a)J = J(⊕li=1 πi (a)) (ii) U J = J(⊕li=1 Ui ) l If i=1 hi = h, then J is a unitary.

5. Cocycles In this section we give an alternative description of the Hilbert space H and of the operators in the commutant in terms of some matrix valued measures Px on the solenoid X∞ . In the scalar case, the measures Px are random walk measures with variable coefficients. Each point in the solenoid X∞ can be constructed in the following way. Pick a point x0 ∈ X. Then, since rx1 = x0 , one has to make a choice x1 out of the finitely many roots in r−1 x0 . The QMF equation amounts to 1 |m0 (y)|2 = 1. −1 #r x ry=x Therefore |m0 (y)|2 /#r−1 (y) can be interpreted as the probability of transition from x to its root y. At the next step one makes a transition from x1 to its root x2 with probability given by |m0 (x2 )|2 /#r−1 x1 . And so on. The measure Px is the path measure obtained in this way. In our matricial case, the measures Px will be operator valued, but the idea stays the same. However some complications appear because we are dealing with a non-commutative situation. We will also see that the operators in the commutant of the covariant representation are in fact multiplication operators by matrix valued functions. This will enable us to give a more concrete form of the correspondence between harmonic maps and the operators in the commutant. The result resembles the Poisson-FatouPrivalov theorems in harmonic analysis: the harmonic map is the integral on the boundary of the operator in the commutant, and the operator is a radial limit of the harmonic map. In our case the boundary is the solenoid X∞ . First, by applying the Trace we will convert the matrix measures into a scalar measure and a matrix valued Radon-Nykodim derivative. Let Ck ⊂ C(X∞ ) be the set of continuous functions on X∞ that only depend on the k + 1 first coordinates. Then Tr(Rk (hf )(x))dµ(x) f ◦ θk → X

defines a positive linear functional τk ∈ Ck∗ such that the compatibility condition τk+1 |Ck = τk holds.

398


IEOT

Since supk Rk < ∞ and ∪k Ck is dense in C(X∞ ), there exists a positive functional τ ∈ C(X∞ )∗ such that τ |Ck = τk for every k. Let µ ˆ be themeasure on X∞ , provided by the Riezs representation theorem such that τ (f ) = X∞ f dˆ µ ˆ -a.e. x ∈ X∞ Proposition 5.1. (i) There exist a positive ∆(x) ∈ EndC (θ0∗ ξ|x ), for µ such that ∆(·) ∈ L∞ (X∞ , µ ˆ ) and s1 (x), ∆(x)s2 (x) dˆ µ = s1 , s2 , X∞

for every pair of sections s1 , s2 : X∞ → θ0∗ ξ. ˆ-a.e. (ii) If A is an operator in the commutant C ∗ (X, r, m, h) then for µ x ∈ X∞ , there exists A(x) ∈ EndC (θ0∗ ξ|x ), such that As(x) = A(x)s(x), µ ˆ -a.e., and ∆(x)m ◦ θ0 (x)A(ˆ r (x)) = ∆(x)A(x)m ◦ θ0 (x), µ ˆ -a.e. (5.1) Also, conversely, any such esentially bounded matrix-valued function x → A(x), satisfying (5.1), defines an operator in the commutant C ∗ (X, r, m, h) . Proof. Denote L2 (θ0∗ ξ, µ ˆ) denote the L2 -sections in the bundle θ0∗ ξ with respect to the measure µ ˆ. Note that if s is a section in ξ, then 1 m(k) (y)s(y), h(y)m(k) (y)s(y) dµ s ◦ θk , s ◦ θk = −k (x) #r X r k y=x 1 m(k)∗ (y)h(y)m(k) (y)s(y), s(y) dµ(x) ≤ −k (x) #r X r k y=x 1 Tr(m(k)∗ (y)h(y)m(k) (y))s(y), s(y) dµ(x) ≤ −k #r (x) X k r y=x

= τk (s ◦ θk , s ◦ θk ) = s ◦ θk , s ◦ θk dˆ µ, X∞ 2 ∗ ˆ ) such that so there exists a unique and positive bounded operator on L (θ0 ξ, µ ∗ s (x), (∆s )(x) dˆ µ = s , s , for every pair of sections s , s : X 1 2 1 2 1 2 ∞ → θ0 ξ. X∞ ∗ The C(X∞ ) module structure on the section in θ0 ξ gives us representations ˆ ) as follows: Every f ∈ C(X∞ ) gives a multiplication of C(X∞ ) on H and L2 (θ0∗ ξ, µ operator Mf ∈ B(H) such that (Mf s)(x) = f (x)s(x). We see that Mf∗ = Mf as ˆ), so operators on both H and L2 (θ0∗ ξ, µ s1 (x), (∆Mf s2 )(x) dˆ µ = s1 , Mf s2 = Mf s1 , s2 X∞ Mf s1 (x), ∆s2 (x) dˆ µ= s1 (x), Mf ∆s2 (x) dˆ µ, = X∞

X∞

and therefore Mf ∆ = ∆Mf . This implies that ∆s(x) = ∆(x)s(x) for a ∆(x) ∈ End(θ0∗ ξ|x ) as described, for µ ˆ -a.e. x ∈ X.

Vol. 62 (2008)


399

(ii) First note that C ∗ (X, r, R, h) ⊂ C(X∞ ) . This is because U −1 π(f )U = Mf ◦θ1 , so if an operator commutes with U and π then it must commute with all multiplications by functions which depend only on finitely many coordinates. But these functions are dense in C(X∞ ) and we obtain the inclusion. From this, we see that there exist A(x) ∈ θ0∗ ξ|x such that As(x) = A(x)s(x), µ ˆ -a.e. Moreover, U A = AU implies the relation (5.1). The converse follows by a computation to prove that A defined from x → A(x) commutes with U and π(f ) for all f ∈ C(X). Definition 5.2. Because of the relation (5.1), if A is an operator in the commutant C ∗ (X, r, m, h) then we call A a cocycle. In the next proposition we will define the positive-matrix-valued measures Px which can be used to represent the inner-product on the Hilbert space H. Proposition 5.3. For each x ∈ X, let Ωx = {y ∈ X∞ |θ0 (y) = x}. There exists a positive operator-valued measure Px from the Borel sets in Ωx to EndC (ξ|x ) such that Ωx f ◦ θk dPx = Rk (f h)(x), for every bounded measurable function f on X, and s1 (y), dPx (y)s2 (y) dµ(x) = s1 , s2 , X

Ωx

for every pair of sections s1 , s2 : X∞ → θ0∗ ξ. Proof. Note that si (y) ∈ θ0∗ ξ|x for every y ∈ Ωx . Let Ck,x ⊂ C(Ωx ) denote the functions that only depend on the first k + 1 variables. Moreover, define σkx : C(Ωx ) → End(ξ|x ) by σkx (f ◦ θk ) = Rk (f h)(x) x We see that σk defines a positive bounded operator from Ck,x to Md (C). Moreover x |Ck = σkx holds. And, since supk Rk < ∞ and the compatibility condition σk+1 ∪k Ck,x ⊂ C(Ωx ) is dense, we see that there exists a positive and bounded linear map Px : C(Ωx ) → End(ξ|x ) for almost every x ∈ X, such that σ x |Ck,x = σkx . A matrix computation implies that s1 ◦ θk (y), dPx (y)s2 ◦ θk (y) dµ(x) = s1 ◦ θk , s2 ◦ θk , X

Ωx

for every pair of sections s1 , s2 : X → ξ. The last claim follows from this by a density argument. The next Lemma can be obtained by direct computation. Lemma 5.4. For x ∈ X, let µ ˆx be the measure on Ωx defined by: k f ◦ θk dˆ µx = Tr(R (f h)(x)) = Tr( f ◦ θk dPx ). Ωx

Ωx

Then, for all bounded measurable functions on X∞ f dˆ µ= f dˆ µx dµ(x). X∞

X

Ωx

400


IEOT

If Cω1 ,...,ωn is the set of points (x, x1 , x2 , . . . ) ∈ Ωx such that x1 = ω1 , . . . , xn = ωn , then ∗ 1 Px (Cω1 ,...,ωn ) = ˆx (Cω1 ,...,ωn ) (m(n) hm(n) )(ωn ), µ #r−n (x) ∗ 1 Tr (m(n) hm(n) )(ωn ) . = −n #r (x) Having defined the matrix valued measures Px , the correspondence between cocyles and harmonic functions in Theorem 4.1(ii) can be given now in terms of a matrix valued conditional expectation: Proposition 5.5. Let A ∈ C ∗ (X, r, m, h) and A(x) ∈ End(θ0∗ ξ|x ) such that As(x) = A(x)s(x), µ ˆ-a.e., for every section s. We have the following identity: T P0 AP0 T ∗ (x) = dPx (y)A(y), i.e., x →

Ωx

Ωx

dPx (y)A(y) is a fixed point for R.

Proof. If s1 , s2 : X → ξ are sections, then s1 (x), (T P0 AP0 T ∗ )(x)s2 (x) dµ(x) = s1 ◦ θ0 , P0 AP0 s2 ◦ θ0 X = s1 ◦ θ0 , As2 ◦ θ0 = s1 (x), dPx (y)A(y)s2 (x) dµ(x). X

Ωx

For the inverse correspondence in Theorem 4.1(i), from harmonic functions to cocycles, we have the following result: Theorem 5.6. Assume in addition that h ≥ c1, µ-a.e., for some constant c > 0. Let h0 be a bounded harmonic function and let A be the corresponding cocycle as in Theorem 4.1(i) and Proposition 5.1. Then ∗

(m(k) h0 m(k) ) ◦ θk = ∆A, ∗ k→∞ Tr((m(k) hm(k) ) ◦ θk ) lim

pointwise µ ˆ -a.e. Proof. For f bounded measurable function on X∞ let Ek (f ) denote the conditional expectation onto the functions that depend only on the first k+1 coordinates, with respect to the measure µ ˆ . If F is a matrix-valued function on X∞ , then Ek (F ) is the matrix-valued function obtained by applying Ek to each component. −1 Let Ak := (m(k) h−1 h0 m(k) ) ◦ θk . (Recall that m(x) is invertible for µ-a.e. x ∈ X). Then f ◦ θk , Ak g ◦ θk = f ◦ θk , Ag ◦ θk = f ◦ θk , Pk APk g ◦ θk .

Vol. 62 (2008)


Take fk = f ◦ θk , gk = g ◦ θk . fk , Ek+1 (∆)Ak+1 gk dˆ µx dµ(x) = X

Ωx

X

401

fk , ∆Ak+1 gk dˆ µx dµ(x)

Ωx

= fk , Ak+1 g = fk , Pk+1 APk+1 gk = fk , Agk = fk , Ak gk = fk , ∆Ak gk dˆ µx dµ(x) = fk , Ek (∆)Ak gk dˆ µx dµ(x). X

Ωx

X

Ωx

So for µ-a.e. x ∈ X, Ek (Ek+1 (∆)Ak+1 )(x, ·) = (Ek (∆)Ak )(x, ·), µ ˆx -a.e., for all k ≥ 1. Therefore the sequence {(Ek (∆)Ak )(x, ·)}k is a martingale. By Doob’s martingale convergence theorem, Ek (∆)Ak (x, ·) converges pointwise µ ˆ x -a.e. to ∆A(x, ·). Now we compute Ek (∆)(x, ·). We have fk , Ek (∆)gk dˆ µx = fk , ∆gk dˆ µx Ωx

Ωx

=

1 #r−k (x)

∗ f (y) , m(k) h(y)m(k) (y)g(y) . (5.2)

r k y=x

Let Cω1 ,...,ωk be the cylinder of points in Ωx that start with ω1 , . . . , ωk . Let fk and gk be supported on Cω1 ,...,ωk . Then we obtain from (5.2) that 1 (k) ∗ k Ek (∆)(x, ω1 , . . . , ωk )ˆ m µx (Cω1 ,...,ωk ) = hm (ωk ). #r−k (x) But, with Lemma 5.4,

µ ˆx (Cω1 ,...,ωk ) = Tr (Px (Cω1 ,...,ωk )) = Tr Thus

1 #r−k (x)

∗ (m(k) hm(k) )(ωk ) .

∗

Ek (∆) =

(m(k) hm(k) ) ◦ θk ∗ Tr((m(k) hm(k) ) ◦ θk )

Then

(5.3)

∗

Ek (∆)Ak = This proves the theorem.

(m(k) h0 m(k) ) ◦ θk . ∗ Tr((m(k) hm(k) ) ◦ θk )

Remark 5.7. In the scalar case, the terms involving m(n) will disappear and we reobtain the results from [22]. Remark 5.8. An ergodic limit for low-pass filters. Our limit theorem can be used to obtain an interesting limit result. The matrices enable us to use a trick to compare two different measures. Let X = T1 , r(z) = z 2 . Denote by ψ1 , ψ2 the inverse branches of r, ψ1 (eiθ ) = iθ/2 , ψ1 (eiθ ) = ei(θ+2π)/2 , for θ ∈ [0, 2π). Then the solenoid X∞ is measurably e isomorphic to X × Ω, where Ω := {1, 2}N, because a point (z0 , z1 , . . . ) ∈ X∞ consists of z0 ∈ T1 and a choice of the inverse branches ω1 , ω2 , . . . .

402


IEOT

√ Let m1 (z) = (1 + z)/ 2 (or any low-pass filter that gives orthogonal scaling functions in L2 (R)), and m2 (z) = 1. Let h1 = h2 = 1. Then the covariant representation for (m1 , h1 ) is L2 (R) so the measure Px1 on the solenoid is supported on sequences ω ∈ Ω such that there exists n0 such that ωn = 0 for all n ≥ n0 or ωn = 1 for all n ≥ n0 (see [21], or Section 6). The covariant representation for (m2 , h2 ) is on the solenoid X∞ with the Haar measure µH , and the measures Px2 are the Bernoulli measures on Ω where {0} and {1} have equal probabilities 1/2 (see also [21] and [23]). 1 m1 0 Px 0 Let m := . Then the measure Px is clearly , i.e., 0 m2 0 Px2 for f ∈ C(X∞ ), f dPx1 dµ(x) 0 X∞ f d Px = . 2 0 X∞ X∞ f dPx dµ(x) The trace of this measure is µ ˆx = Px1 +Px2 . As we explained before (see also Section 1 6), the measure Px is atomic, and let us denote the support of this measure by Rx . On the other hand, Px2 is the Bernoulli measure, so it has no atoms, therefore Rx has Px2 -measure zero. Then it is easy to see that

dPx1 dˆ µx

dP 2

= χRx and dˆµxx = χΩ\Rx . Then dPx χRx 0 ∆= = . 0 χΩ\Rx dˆ µx

Note that this shows that ∆(x) is singular everywhere. We will remark that this implies an interesting phenomenon, which occurs for any low-pass filter that gives orthogonal scaling functions in L2 (R). First, as shown in [21], since Px1 is supported in on Rx , it follows that for any ω outside Rx one has (k)

|m1 (ψωk . . . ψω1 x))|2 = Px1 ({ω}) = 0. k→∞ 2k lim

However, from the convergence Theorem 5.6, we have that for µ-a.e. x ∈ T1 and for µ ˆx -a.e. ω: ∗

m(k) m(k) (ψωk · · · ψω1 x) = ∆(x, ω). ∗ k→∞ Tr(m(k) m(k) (ψω · · · ψω x)) 1 k lim

This implies that for µ ˆx a.e. ω ∈ Ω \ Rx (k)

0 = lim

k→∞

|m1 (ψωk · · · ψω1 x)|2 (k)

|m1 (ψωk · · · ψω1 x)|2 + 1

,

ˆx -a.e. ω, so we obtain the much stronger so limk→∞ m(k) (ψωk · · · ψω1 x) = 0 for µ limit for Px2 -a.e. ω (and recall that Px2 is the Bernoulli measure): lim m(k) (ψωk · · · ψω1 x) = 0.

k→∞

Vol. 62 (2008)


403

6. Low-pass filters √ In the scalar case, if the filter m0 satisfies a low-pass condition m0 (1) = 2, the classical wavelet theory shows that the scaling equation has a solution in L2 (R). Of course the solution might be a non-orthogonal scaling function, and then there are super-wavelet constructions (see [7]) that will give orthogonal solutions. functions. The point we want to make is that, when a low-pass condition is satisfied, the resulting covariant representation can be realized on L2 (R), as in classical wavelet theory, or in a direct sum of copies of L2 (R), as in the super-wavelet theory developed in [7]. In the matricial case, the low-pass condition is replaced by the E(l)-condition introduced in [27]. Our covariant representations are on the solenoid X∞ , but we show that when a low-pass condition is satisfied the measures Px are supported on an embedding of R in the solenoid, and Px is directly related to the scaling functions. ˜ is a complete metric We recall now the setup from [24]. We assume that X ˜ space with an isometric covering space group action of a group G such that X/G =: ˜ → X ˜ is a strictly expansive X is compact. Moreover, we assume that r˜ : X homeomorphism and there exists an endomorphism A ∈ End(G) such that r˜g = (Ag)˜ r and AG is a normal subgroup of index q. Since r˜ is expansive it has a fixed point x ˜0 . ˜ → X denote the quotient covering map and define r : X → X as Let p : X x0 ). Let µ be a strongly invariant measure on X and r(p(x)) = p(˜ r x). Let x0 := p(˜ ˜ obtained by lifting the measure µ by the covering map p (see µ ˜ the measure on X also (6.1)). We assume that the bundle ξ over X is a Lipschitz continuous bundle, and that p∗ ξ is trivial. ˜ = Rn . The Example 6.1. The main example is the one used in wavelet theory: X n n n group G = Z acts on R by translations: g, x → x + g, (x ∈ R , g ∈ Zn ). Define the map r˜(x) = Ax, where A is an n × n expansive integer matrix. The quotient Rn /Zn can be identified with the torus Tn , p : Rn → Rn /Zn is the quotient map, and let r(x) = Ax mod Zn for x ∈ Tn . The fixed point of r˜ is x ˜0 = 0, and x0 ) = 0. The bundle ξ over Tn is Tn × Cd , and p∗ ξ = Rn × Cd . x0 := p(˜ We let m ∈ Md (Lip1 (X)). Moreover, we assume that • R1 = 1; √ • The matrix m(x0 )/ q satisfies the E(l) condition (according to [27]), i.e., 1 √ is the only eigenvalue of m(x0 )/ q with absolute value greater than or equal to 1, and its algebraic and geometric multiplicity are both equal to l ≥ 1. Let E1 denote the eigenspace corresponding to the eigenvalue 1. Recall that S is the set of continuous sections in the bundle ξ.

404


˜ Let Ξ := {f ∈ Cb (X)| with the inner-product:

2 g∈G |f |

ζ, η =

IEOT

◦ g ∈ C(X)}. Ξ is a C(X)-Hilbert module

(ζη) ◦ g.

g∈G

˜ µ This module inner-product is related to the ordinary inner-product from L2 (X, ˜), by ζηd˜ µ= ζ, η dµ, (ζ, η ∈ Ξ). (6.1) ˜ X

X

˜ ∈ B(L2 (X, ˜ µ Let U ˜ )) denote the unitary operator defined by ˜ f = q 1/2 f ◦ r˜, and , U ˜l := U ˜ ⊕ ···⊕ U ˜. U l times ˜ µ Define the representation π ˜l : C(X) → B(⊕lj=1 L2 (X, ˜)) by π ˜l (a)(f1 , . . . , fl )(x) = (a(px)f1 (x), . . . , a(px)fl (x)), ˜ µ ˜)). (a ∈ C(X), f1 , . . . , fl ∈ L2 (X, We will need to define some “scaling functions”. These will be fixed points of a refinement operator obtained as the limits of the iterates of this refinement operator. To initialize the iteration, we fix an f ∈ Ξ such that: • • •

f, f = 1 ( g∈G |f (gx) − f (gy)|2 )1/2 ≤ Dd(x, y) f (gx0 ) = 0 for every g = 1.

Let s1 , . . . , sl ∈ S be Lipschitz sections in ξ such that s1 (x0 ), . . . , sl (x0 ) form an orthonormal basis for E1 . As in [24], we define the starting points for the cascade algorithm Wj ∈ HomC(X) (S, Ξ) by Wj s = sj , s f and the refinement operator M : HomC(X) (S, Ξ) → HomC(X) (S, Ξ), ˜ −1 W ms ◦ r, for W ∈ HomC(X) (S, Ξ), s ∈ S. (M W )s = U Proposition 6.2. [24] (i) The following limit exists and is uniform on compact sets: P(˜ x) := lim q −k/2 m(k) (p(˜ r−k x˜)), k→∞

˜ (˜ x ∈ X).

Also, P(˜ x0 ) is the projection onto the eigenspace E1 . (ii) For each j ∈ {1, . . . , l} and for s ∈ S, {M k Wj s}k≥1 converges uniformly x) = on compact sets to Wj s, and this defines Wj ∈ HomC(X) (S, Ξ), Wj s(˜ ˜ Moreover M Wj = Wj . x)s(p˜ x) , s ∈ S, x ˜ ∈ X. sj (x0 ) , P(˜

Vol. 62 (2008)


405

(iii) For each j ∈ {1, . . . , l} the map hj := Wj∗ Wj defines a minimal projection in the algebra of continuous harmonic functions Hc (hj is a projection also in the algebra of bounded measurable harmonic functions H but it is not necessarily minimal). ˜ P ∗ (gx)sj (x0 ) , s(px) P ∗ (gx)sj (x0 ), (s ∈ S, x ∈ X). hj (px)s(px) = g∈G

Moreover these projections are mutually orthogonal in this algebra. l Theorem 6.3. Let h := j=1 hj and let H denote Hilbert space of the covariant repl ˜ µ resentation obtained from m and h. There exists a unitary J : H → j=1 L2 (X, ˜) such that ˜l J = JU (i) U l (ii) Js ◦ θ0 = j=1 Wj s for every s ∈ S. (iii) π ˜l (a)J = Jπ(a) for every a ∈ C(X). ˜ −k Wj s. Jk is an ˜ µ) by Jk U −k s ◦ θ0 = U Proof. Define Jk : Hk → j L2 (X, j isometry because ˜ −k Wj s, U ˜ −k Wj s L2 Jk U −k s ◦ θ0 , Jk U −k s ◦ θ0 L2 = U =

Wj s, Wj s L2 =

j

s,

= X

j

j

Wj∗ Wj s dµ

Wj s, Wj s dµ

X

s , hs dµ = s ◦ θ0 , s ◦ θ0

=

j

X

= U −k s ◦ θ0 , U −k s ◦ θ0 ˜ −1 Wj ms ◦ r, Moreover, since Wj = M Wj = U Jk U −(k−1) s ◦ θ0 = Jk U −k U s ◦ θ0 ˜ −k Wj ms ◦ r = ⊕j U ˜ −(k−1) Wj s = Jk−1 U −(k−1) s ◦ θ0 = ⊕j U ˜ µ so Jk |Hk−1 = Jk−1 . This gives us an isometric map J : H → j L2 (X, ˜), and the intertwining properties of J are checked by a direct computation. 2 ˜ ˜ := ∪k JHk ⊂ Let H ˜) and let Q denote the orthogonal projection j L (X, µ ˜ ˜ onto this space. We have Ul Q = QUl , π ˜l (a)Q = Q˜ πl (a) for every a ∈ C(X). k ˜ −k π ˜ We have that U ˜ (a) U is a multiplication by the matrix that has a◦p◦ r˜−k l l l on the diagonal. Since r˜ is expansive, an application of the Stone-Weierstrass theo˜ Since Q commutes rem shows that {a◦p◦ r˜−k | a ∈ C(X), k ≥ 0} is dense in Cc (X). k ˜ −k π ˜ with all operators of the form U ˜ (a) U , this implies that Q commutes with l l l ∞ ˜ ∞ ˜ L (X, µ ˜) ⊗ Il . Since L (X, µ ˜ ) is a maximal abelian subalgebra, it follows that Q ˜ µ corresponds to a pointwise multiplication by a map in L∞ (X, ˜) ⊗ Ml (C). Since Q is a projection, this map is projection valued. Moreover, Q(x) = Q(˜ r x) µ ˜-a.e., ˜l Q. ˜l = U since QU

406


IEOT

Consider now ϕi := ⊕nj=1 Wj si for i ∈ {1, . . . , l}. We have with Proposition 6.2, ϕi (˜ x0 ) = ei , the canonical vectors in Cl . Then, using the continuity of Wj , we have that for x ˜ in a neighborhood of x ˜0 , {ϕi (˜ x) | i ∈ {1, . . . , l}} forms a basis for ˜ Qϕi = ϕi so Q(x)ϕi (x) = ϕi (x) for all x ∈ X. ˜ But then, Q(x) Cl . Since ϕi ∈ H, must be the identity in a neighborhood of x˜0 , and since Q(x) = Q(˜ r x), and r˜ is ˜ Thus Q = 1 and H ˜ expansive, we obtain that Q(x) is the identity for all x ∈ X. 2 ˜ is the entire space j L (X, µ ˜ ). ˜ → X∞ , î(x) = (p(˜ Proposition 6.4. The map î : X r−k x))∞ k=1 is a continuous bijection onto the set of sequences (zk )k ∈ X∞ with limk→∞ zk = x0 . For all ˜ ∩ Ωpx = î(Gx). p(x) ∈ X, î(X) Let Px be the measures associated to h = j hj as in Proposition 5.3. Then Ppx is atomic and supported on î(Gx), and Ppx ({î(gx)}) = P(gx)∗ P(gx),

˜ (x ∈ X).

Proof. Since r˜ is expansive, the sequence r˜−k x converges to the fixed point x ˜0 , so −k for k large r˜ x is in some neighborhood where p is injective. This implies that î is injective. x0 ) = x0 . To see that The continuity of î is clear, and p(˜ r−k x) converges to p(˜ î is onto the given set, take some sequence (zk )k in X∞ such that zk converges ˜0 such that the restriction of p to V is a to x0 . Take a neighborhood V of x homeomorphism onto the neighborhood p(V ) of x0 . Take a smaller neighborhood U ⊂ V of x ˜0 such that r˜(U ) ⊂ V . For k large zk is in p(U ). So zk = p(xk ) for some xk ∈ U . Since r(zk+1 ) = zk it follows that r˜(xk+1 ) = gxk for some g ∈ G. But as both r˜(xk+1 ) and xk are in V , it follows that g must be 1. So xk+1 = r˜−1 xk for k large, bigger than some k0 . Then if define x := r˜−k0 xk0 , we have î(x) = (zk )k . If î(y) is in Ωpx then py = px so y = gx for some g ∈ G. Let us check the Px -measure of the atoms. We have r−j x), 0 ≤ j ≤ k}) Ppx ({î(gx)}) = lim Ppx ({(zn )n | zj = p(˜ k→∞

∗

= lim q −k m(k) (p˜ r−k x)h(p˜ r−k x)m(k) (p˜ r−k x) k→∞

= P(x)∗ h(x0 )P(x) = P(x)∗ P(x), and we used the fact from [24] that the range of P(x) is contained in E1 and h(x0 ) = j hj (x0 ) is the projection onto E1 . Take the cylinder Cpx,z1 ,...,zn of sequences in Ωpx that start with px, z1 , . . . , zn . If we add all the atoms in this cylinder, we obtain, Ppx (î(gx)) = P(x + g)∗ P(x + g). (6.2) g,î(gx)∈Cpx,z1 ,...zn

g,î(gx)∈Cpx,z1 ,...zn

Vol. 62 (2008)


407

On the other hand , for any section s ∈ S, and with the notation pxn := zn , for ˜ using the formula for h in Proposition 6.2, some xn ∈ X, ∗ s(z0 ) , Ppx (Cpx,z1 ,...,zn )s(z0 ) = s(z0 ) , q −n m(n) (zn )h(zn )m(n) (zn )s(z0 ) q −n m(n) (zn )s(z0 ) , P ∗ (gxn )sj (x0 ) , m(n) (zn )s(z0 ) P ∗ (gxn )sj (x0 ) = j

g∈G

g

j

2 = P ∗ (gxn )sj (x0 ) , q −n/2 m(n) (zn )s(z0 ) , and since {sj (x0 )}j is an orthonormal basis for E1 and the range of P(x) is contained in E1 , = P(gxn )q −n/2 m(n) (zn )s(z0 )2 . g

But, from the definition of P, we have P(gxn )q −n/2 m(n) (zn ) = P(g x) for some g ∈ G with î(g x) ∈ Cpx,z1 ,...,zn , and we obtain further P(g x)s(z0 )2 . = g ,î(g x)∈Cpx,z1 ,...,zn

Comparing with (6.2), this shows that the sum of the atoms is equal to the measure of the cylinder and thus the measures Ppx are supported on these atoms. Remark 6.5. Scaling functions. To obtain old-fashioned scaling functions as described in the introduction, let us consider the case when we are dealing with the Example 6.1, used in the regular wavelet theory (but the arguments below work also in the more general case we described in this section), and let us consider the case when ξ is the trivial vector bundle X × Cd = Tn × Cd . Then we can take the canonical sections in ξ, ci (x) = ei , x ∈ Tn , i ∈ {1, . . . , d}, where ei are the canonical vectors in Cd . Then, define for each j ∈ {1, . . . , l}, ϕji := Wj ci , (i ∈ {1, . . . , d}). Then ϕji is in Ξ so it is a function in L2 (Rd ) (according to (6.1)). Also, since M Wj = Wj , we have d d ˜ ϕj = U ˜ M Wj ci = Wj mci ◦ r˜ = Wj ( U m c ) = π ˜ (mki )ϕjk . ki k i k=1

k=1

(Here π ˜ (s)f (x) = s(px)f (x), s ∈ C(T ), f ∈ L (R ), x ∈ R ). Also, for f ∈ C(Tn ), ϕji , π(f )ϕji = ϕji f ϕji dx = f ϕji , ϕji dµ = f Wj ci , Wj ci dµ n Tn Tn R

∗ = f ci , Wj Wj ci dµ = f ci , hj ci dµ = f (hj )ii dµ. n

Tn

2

Tn

n

n

Tn

408


IEOT

Thus, ϕj1 , . . . , ϕjn form a multi-scaling function in L2 (Rn ), with filter m and correlation matrix hj . We can put together all these multi-scaling functions and define ϕi := (ϕ1i , . . . , ϕli ) ∈ ⊕lj=1 L2 (Rn ),

(i ∈ {1, . . . , d}).

Then we still have the same scaling equation, but now in ⊕lj=1 L2 (Rn ): ˜l ϕi = U

d

πl (mki )ϕk ,

(i ∈ {1, . . . , d}),

k=1

and the correlation matrix for ϕ1 , . . . , ϕd is the harmonic function h, i.e., ϕi , πl (f )ϕi = f hii dµ, (f ∈ C(Tn )). Tn

In Theorem 6.3, we see that if s = (s1 , . . . , sd ), then d d d Js ◦ θ0 = ⊕lj=1 Wj s = ⊕lj=1 Wj ( si ci ) = ⊕lj=1 ( π ˜ (si )ϕji ) = π ˜l (si )ϕi . i=1

i=1

i=1

This implies, with Theorem 6.3, that the linear span of ˜ j πl (f )ϕk | j ∈ Z, f ∈ C(Tn ), k ∈ {1, . . . , d}} {U l is dense in ⊕lj=1 L2 (Rn ). Thus, ϕ1 , . . . , ϕd form a “super”-multi-scaling function for the bigger(“super-”)space ⊕lj=1 L2 (Rn ). Acknowledgment We would like to thank professor Palle Jorgensen for his suggestions and for offering us a wider view on the subject, which we included in the introduction.

References [1] William B. Arveson. Subalgebras of C ∗ -algebras. Acta Math., 123:141–224, 1969. [2] M. F. Atiyah. K-theory. Advanced Book Classics. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, second edition, 1989. Notes by D. W. Anderson. [3] Lawrence W. Baggett, Jennifer E. Courter, and Kathy D. Merrill. The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ). Appl. Comput. Harmon. Anal., 13(3):201–223, 2002. [4] Lawrence W. Baggett, Palle E. T. Jorgensen, K. D. Merrill, and J. A. Packer. Construction of Parseval wavelets from redundant filter systems. J. Math. Phys., 46(8):083502, 28, 2005. [5] Lawrence W. Baggett, Herbert A. Medina, and Kathy D. Merrill. Generalized multiresolution analyses and a construction procedure for all wavelet sets in Rn . J. Fourier Anal. Appl., 5(6):563–573, 1999.

Vol. 62 (2008)


409

[6] Viviane Baladi. Positive transfer operators and decay of correlations, volume 16 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co. Inc., River Edge, NJ, 2000. [7] Stefan Bildea, Dorin Ervin Dutkay, and Gabriel Picioroaga. MRA super-wavelets. New York J. Math., 11:1–19, 2005 (electronic). [8] B. Blackadar. Operator algebras, volume 122 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2006. Theory of C ∗ -algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III. [9] Ola Bratteli and Palle E. T. Jorgensen. Wavelets through a looking glass. Applied and Numerical Harmonic Analysis. Birkh¨ auser Boston Inc., Boston, MA, 2002. The world of the spectrum. [10] Ola Bratteli and Derek W. Robinson. Operator algebras and quantum statistical mechanics. 1. Texts and Monographs in Physics. Springer-Verlag, New York, second edition, 1987. C ∗ - and W ∗ -algebras, symmetry groups, decomposition of states. ˙ [11] Man-Duen Choi, David W. Kribs, and Karol Zyczkowski. Quantum error correcting codes from the compression formalism. Rep. Math. Phys., 58(1):77–91, 2006. [12] Harry Cohn. On a paper by Doeblin on nonhomogeneous Markov chains. Adv. in Appl. Probab., 13(2):388–401, 1981. [13] Jean-Pierre Conze and Albert Raugi. Fonctions harmoniques pour un opérateur de transition et applications. Bull. Soc. Math. France, 118(3):273–310, 1990. [14] Ra´ ul E. Curto, Paul S. Muhly, and Dana P. Williams. Cross products of strongly Morita equivalent C ∗ -algebras. Proc. Amer. Math. Soc., 90(4):528–530, 1984. [15] Xingde Dai and David R. Larson. Wandering vectors for unitary systems and orthogonal wavelets. Mem. Amer. Math. Soc., 134(640):viii+68, 1998. [16] Ingrid Daubechies. Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. [17] V. Dobrić, R. Gundy, and P. Hitczenko. Characterizations of orthonormal scale functions: a probabilistic approach. J. Geom. Anal., 10(3):417–434, 2000. [18] Wolfgang Doeblin. Sur l’équation de Kolmogoroff. C. R. Acad. Sci. Paris, 210:365– 367, 1940. [19] Dorin Ervin Dutkay. The spectrum of the wavelet Galerkin operator. Integral Equations and Operator Theory, 50(4):477–487, 2004. [20] Dorin Ervin Dutkay. The wavelet Galerkin operator. J. Operator Theory, 51(1):49– 70, 2004. [21] Dorin Ervin Dutkay and Palle E. T. Jorgensen. Iterated function systems, Ruelle operators, and invariant projective measures. Math. Comp., 75(256):1931–1970, 2006 (electronic). [22] Dorin Ervin Dutkay and Palle E. T. Jorgensen. Martingales, endomorphisms, and covariant systems of operators in Hilbert space. to appear in Journal of Operator Theory, http://arxiv.org/abs/math.CA/0407330, 2006. [23] Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelets on fractals. Rev. Mat. Iberoamericana, 22(1):131–180, 2006.

410


IEOT

[24] Dorin Ervin Dutkay and Kjetil Røysland. The algebra of harmonic functions for a matrix-valued transfer operator. preprint 2006, http://arxiv.org/abs/math.FA/0611539, 2006. [25] Richard F. Gundy. Low-pass filters, martingales, and multiresolution analyses. Appl. Comput. Harmon. Anal., 9(2):204–219, 2000. [26] Deguang Han and David R. Larson. Frames, bases and group representations. Mem. Amer. Math. Soc., 147(697):x+94, 2000. [27] Qingtang Jiang and Zuowei Shen. On existence and weak stability of matrix refinable functions. Constr. Approx., 15(3):337–353, 1999. [28] Palle E. T. Jorgensen. Analysis and probability: wavelets, signals, fractals, volume 234 of Graduate Texts in Mathematics. Springer, New York, 2006. [29] David W. Kribs. A brief introduction to operator quantum error correction. In Operator theory, operator algebras, and applications, volume 414 of Contemp. Math., pages 27–34. Amer. Math. Soc., Providence, RI, 2006. [30] Wayne M. Lawton. Necessary and sufficient conditions for constructing orthonormal wavelet bases. J. Math. Phys., 32(1):57–61, 1991. [31] George W. Mackey. Unitary group representations in physics, probability, and number theory, volume 55 of Mathematics Lecture Note Series. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1978. [32] Judith A. Packer and Marc A. Rieffel. Projective multi-resolution analyses for L2 (R2 ). J. Fourier Anal. Appl., 10(5):439–464, 2004. [33] Karl Petersen. Ergodic theory, volume 2 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1983. [34] Albert Raugi. Théorie spectrale d’un opérateur de transition sur un espace métrique compact. In Fascicule de probabilit´ es, volume 1994 of Publ. Inst. Rech. Math. Rennes, page 21. Univ. Rennes I, Rennes, 1994. [35] David Ruelle. Thermodynamic formalism. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2004. The mathematical structures of equilibrium statistical mechanics. [36] W. Forrest Stinespring. Positive functions on C ∗ -algebras. Proc. Amer. Math. Soc., 6:211–216, 1955. [37] Peter Walters. An introduction to ergodic theory, volume 79 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982. [38] Dana P. Williams. Transformation group C ∗ -algebras with Hausdorff spectrum. Illinois J. Math., 26(2):317–321, 1982. Dorin Ervin Dutkay University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, U.S.A. e-mail: [email protected] Kjetil Røysland University of Oslo, Department of Mathematics, PO Box 1053, Blindern, NO-0316 Oslo, Norway e-mail: [email protected] Submitted: August 14, 2007. Revised: July 8, 2008.



A Class of Operators on Weighted Bergman Spaces Miyeon Kwon and Zhijian Wu Abstract. This paper studies the connection between an analytic function ϕ on the unit disk D and a certain type of operators induced by ϕ on weighted Bergman spaces Aα of analytic functions on D. The general theme is the interplay between smoothness conditions on ϕ and the Sp -norms of the operators induced by ϕ. Mathematics Subject Classification (2000). Primary 47B10; Secondary 47G10. Keywords. Besov space, Schatten-von Neumann class of operators, weighted Bergman spaces.

1. Introduction Let D be the unit disk of the complex plane and dAα (z) = (α + 1)(1 − |z|2 )α dA(z), where dA(z) is the normalized area measure and α > −1. The weighted Bergman space Aα is the subspace of all analytic functions in L2 (dAα ), which is a Hilbert space with the inner product f (z)g(z)dAα (z), f, g ∈ L2 (dAα ). f, g = D

For an analytic function ϕ on D, we define an operator Vϕ on Aα by ϕ(w)f (zw) Vϕ (f )(z) = dAα (w), f ∈ Aα . 2+α D (1 − wz) As one can see, the operator Vϕ is a modified version of the reduced Hankel operator Hϕ on Aα ( see [2] or [3] ) defined by ϕ(w)f (w) Hϕ (f )(z) = dAα (w), f ∈ Aα . (1 − wz)2+α D

412

Kwon and Wu

IEOT

α It is clear that Vϕ is a linear operator on A and its matrix representation Γ(n+2+α) z n , n = 0, 1, 2, . . . is with respect to the usual bases en (z) = Γ(α+2)Γ(n+1) Vϕ (m, n) = Vϕ (en ), em = (Vϕ en )(z)em (z)dAα (z) D 1/2 −3/2 Γ(m−n+2+α) Γ(n+2+α) Γ(m+2+α) Γ(n+1) Γ(m+1) Γ(m−n+1) ϕm , if m ≥ n; = 0, if m < n.

For α = −1 (or equivalently considering a similar definition of Vϕ on the Hardy space), this matrix becomes     

ϕ0 ϕ1 ϕ2 .. .

0 ϕ1 ϕ2 .. .

0 0 ϕ2 .. .

... ... ... .. .

   . 

Such a matrix can be viewed as the discrete x version of the Volterra operator + 2 Q+ on L (R ) defined by Q (f )(x) = ψ(x) + ψ ψ 0 f (y)dy. The systematic study of the operator Q+ can be found in [1]. One can also find results about the matrix ψ operator in [5]. In this paper, we determine the condition for Vϕ to be bounded, compact, or in Schatten-von Neumann class and study more properties of Vϕ . Throughout this paper, for two positive functions a and b on a set X we denote a b if there is a positive constant C such that a(x) ≤ Cb(x) for all x ∈ X and a b if a b and b a.

2. Properties of Vϕ We denote the set of bounded linear operators on a Hilbert Space H by S∞ (H). Recall that the n-th singular number of a bounded linear operator T on a Hilbert space H is sn (T ) = inf{T − R : rank(R) ≤ n − 1}. For 0 < p < ∞, the Schatten-von Neumann class Sp (H) is defined by def

Sp (H) = {T ∈ S∞ (H) : T Sp = {sn (T ) : n ≥ 1}p < ∞}. Let T = {T (m, n)} be a matrix representation of an operator T. For j, k ≥ 0, define the submatrix Dj,k (T ) by Dj,k (T )(m, n) = T (m, n) for (m, n) ∈ [2j+k − 2, 2j+k+1 − 3] × [2j − 1, 2j+1 − 2] and 0 elsewhere.

∞ For ϕ(z) = m=0 ϕm z m and s ∈ R define a sequence Rs (ϕ) by Rs (ϕ)(n) = 2

ns

2n+1 −3 k=2n −2

|ϕk |2

1/2

,

n = 1, 2, . . . .

Vol. 62 (2008)

A Class of Operators on Weighted Bergman Spaces

413

The following theorems determine the Sp (Aα ) membership of Vϕ with respect to the smoothness of ϕ. The main idea of the proof is to decompose the matrix of Vϕ into submatrices {Dj,k (Vϕ ) : j, k ≥ 0} of finite rank and to estimate the Sp −norm of Vϕ using those of the submatrices. ∞

Theorem 2.1. Suppose ϕ =

ϕn z n is analytic on D and 1 < p ≤ ∞. Then

n=0

Vϕ ∈ Sp (Aα ) if and only if R 12 (ϕ) ∈ p . Proof. Suppose that Vϕ ∈ Sp . Let e(z) = 2

−j/2

2j+1 −2

en (z), where en (z) =

n=2j −1

Γ(n + 2 + α) z n . Then e2 = D e(z)e(z)dAα (z) = 1 and applying StirΓ(α + 2)Γ(n + 1) ling’s formula, Dj,2 (Vϕ )e, Dj,2 (Vϕ )e = 2

−j

2j+1 −2

Dj,2 (Vϕ )en , Dj,2 (Vϕ )en˜

n,˜ n=2j −1

2

j

2j+3 −3 m=2j+2 −2

2 |ϕm |2 = R 12 (ϕ)(j + 2) .

From the fact that Dj,2 (Vϕ )e, Dj,2 (Vϕ )e ≤ Dj,2 (Vϕ )2 ≤ Dj,2 (Vϕ )2Sp for all p > 0, we have Vϕ Sp ≥ Dj,2 (Vϕ )Sp = {Dj,2 (Vϕ )Sp : j ≥ 0}p j≥0

{R 21 (ϕ)(j + 2) : j ≥ 0}p .

It concludes that for 0 < p ≤ ∞, Vϕ ∈ Sp implies R 12 (ϕ) ∈ p .

∞ To show the converse, we note that the Gamma function Γ(x) = 0 e−t tx−1 dt 1 and the Beta function B(x, y) = 0 rx−1 (1 − r)y−1 dr satisfy B(x, y) = Γ(x)Γ(y) Γ(x+y) . We then have Γ(m − n + 2 + α) Γ(m − n + 1)

Since (1 − r)−n =

1 (n−1)!

=

[α]+2 Γ(m − n + 2 + α) (m − n + s) Γ(m − n + 3 + [α]) s=1

=

[α]+2 B(1 + [α] − α, m − n + 2 + α)) (m − n + s). Γ(1 + [α] − α) s=1

∞

l=0 (l

+ n − 1) · · · (l + 1)rl ,

414

Kwon and Wu

IEOT

B(1 + [α] − α, m − n + 2 + α) 1 r[α]−α (1 − r)m−n+1+α dr = 0

= =

1 (n − 1)! ∞ l=0

1

r[α]−α (1 − r)m+1+α

0

∞

(l + n − 1) · · · (l + 1)rl dr

l=0

Γ(l + n) B(l + 1 + [α] − α, m + 2 + α). Γ(n)Γ(l + 1)

We define a matrix operator Al , l = 0, 1, 2, . . . as Al (m, n) =

Γ(n + 2 + α) 1/2 Γ(m + 2 + α) −3/2 [α]+2 (m − n + s) Γ(n + 1) Γ(m + 1) s=1

Γ(l + n) B(l + 1 + [α] − α, m + 2 + α) · ϕm . Γ(1 + [α] − α) Γ(n)Γ(l + 1) ∞

∞ Then Dj,k (Vϕ ) = l=0 Dj,k (Al ), k ≥ 1 and Dj,0 (Vϕ ) = T l=0 Dj,0 (Al ) , where T is the lower triangular projection (see [1] or [4]). Since any finite matrix of the form {f (m)g(n)}m,n≥0 has rank 1, Dj,k (Al ) is of rank less than or equal to 3[α]+2 . Therefore for p > 0, ·

3−

[α]+2 2

Dj,k (Al )S2 ≤ Dj,k (Al )Sp ≤ 3

[α]+2 p

Dj,k (Al )S2

and it follows from Stirling’s formula that 2

Dj,k (Al )S2

2−3(α+1)(j+k) 2(α+1)j 22([α]+2)(j+k) 22(j+1)l+j (Γ(l + 1))−2 2 · B(l + 1 + [α] − α, 2j+k + α)

2j+k+1 −3

|ϕm |2

m=2j+k −2 −k(α+2) 2(j+k)(1+[α]−α) 2(j+1)l

2 2 2 (Γ(l + 1))−2 2 2 · B(l + 1 + [α] − α, 2j+k + α) R 12 (ϕ)(j + k) . Therefore, for k ≥ 1 Dj,k (Vϕ )Sp

≤

∞

Dj,k (Al )Sp

l=0

2 ·

Dj,k (Al )S2

l=0

−k(2+α) 2

∞

∞

2(j+k)(1+[α]−α) R 12 (ϕ)(j + k)

2(j+1)l (Γ(l + 1))−1 B(l + 1 + [α] − α, 2j+k + α) .

l=0

By the fact that B(n, x) n−x for sufficiently large n and Dj,k (Vϕ )Sp 2

−k(1+α) 2

R 12 (ϕ)(j + k)

∞ 2l l=0

l!

2

∞

−k(1+α) 2

2l l=0 l!

= e2 , we have

R 12 (ϕ)(j + k).

Vol. 62 (2008)

A Class of Operators on Weighted Bergman Spaces

415

For the case k = 0, it was shown in [1] that the lower triangular projection T is bounded in Sp for 1 < p ≤ ∞. Hence

Dj,0 (Vϕ )Sp

∞

Dj,0 (Al )Sp ≤

l=0

∞

Dj,0 (Al )Sp .

l=0

Applying the same argument in the case k ≥ 1, we have Dj,0 (Vϕ )Sp R 12 (ϕ)(j).

We now note that Vϕ = k≥1 D (V ) + j≥1 Dj,0 (Vϕ ), and both j,k ϕ j≥0

j≥0 Dj,k (Vϕ ) and j≥1 Dj,0 (Vϕ ) are diagonal block matrices. Therefore, Vϕ Sp ≤ Dj,k (Vϕ )Sp + Dj,0 (Vϕ )Sp j≥0

k≥1

j≥1

{Dj,k (Vϕ )Sp : j ≥ 0}p +

k≥1

2

{Dj,0(Vϕ )Sp : j ≥ 1}p

j≥1 −k(α+1) 2

k≥1

{R 21 (ϕ)(j + k) : j ≥ 0}p + {R 21 (ϕ)(j) : j ≥ 0}p

{R 21 (ϕ)(j) : j ≥ 0}p .

This completes the proof. Theorem 2.2. Suppose ϕ =

ϕn z n is analytic on D. Then Vϕ is compact if and

n≥0

only if lim R 12 (ϕ)(n) = 0. n→∞

Proof.

In a similar manner in Theorem 2.1, if Vϕ is compact, the diagonal submatrix j≥0 Dj,2 (Vϕ ) is also compact and hence limj→∞ Dj,2 (Vϕ ) = 0. It follows that limn→∞ R 12 (ϕ)(n) = 0.

Conversely, if limn→∞ R 12 (ϕ)(n) = 0, j≥0 Dj,0 (Vϕ ) is compact since

0≤j 0, then Hv∞ (D) = H ∞ (D) with an |z|→1

equivalent norm and Hv0 (D) = {0}. Also, if lim v(z) = 0, then we call v a typical weight. For given weight v, we define

|z|→1

v˜ (z) = (sup {|f (z)| : f ∈ Hv∞ (D) , f v ≤ 1})

−1

=

1 , δz v

where δz : Hv∞ (D) → C is the evaluation linear functional. It is easy to see that v˜ is also a weight with v (z) ≤ v˜ (z) and f v ≤ 1 if and only if f v˜ ≤ 1. Thus Hv∞ (D) = Hv˜∞ (D) and Hv0 (D) = Hv˜0 (D) isometrically. A weight v is called essential if there exists a constant β > 0 such thatv(z) ≤ v˜ (z) ≤ βv(z), for each 2

z ∈ D. For instance, if we define vp (z) = 1 − |z|

p

, 0 < p < ∞, for each z ∈ D,

then each vp is an essential and typical weight. Also, for each non-zero f ∈ H (D), −1 if we define vf (z) = M (f, r) , for z ∈ D, where M (f, r) = sup {|f (z)| : |z| = r}, then vf is a weight satisfying v˜f = vf . For more details on these weights and

Vol. 62 (2008) Compact Differences of Weighted Composition Operators

421

related weighted spaces of analytic functions we refer to ([3], [4], [9], [10], [15], [16]). The weighted Banach spaces of analytic functions are isometrically isomorphic to the weighted Bloch spaces which are defined as follows

Bv∞ (D) =

f ∈ H(D) : f v = sup v(z) |f (z)| < ∞

and Bv0 (D) =

z∈D

f ∈ Bv∞ (D) :

lim v (z) |f (z)| = 0 .

|z|→1

If we identify functions that differ by a constant, then Bv∞ (D) and Bv0 (D) ∞ become Banach spaces under the norm v . The space is isometric to Bv (D)

Hv∞ (D) by f −→ f . For example, if we take v(z) =

2

1 − |z|

α

, α > 0, then

we have the Bloch type spaces. In particular, for α = 1, the spaces Bv∞ (D) and Bv0 (D) are the usual Bloch space and the little Bloch space, respectively. For more information on these spaces, we refer to [29].

3. Compact Differences of Weighted Composition Operators In this section we shall give necessary and sufficient conditions for a difference of two weighted composition operators to be compact on Hv∞ (D) and at the same time show that conditions of compactness and complete continuity are equivalent. Also, as an application we give a characterization of compactness of differences of composition operators on weighted Bloch spaces. Let T be a bounded linear operator on a Banach space. Then T is said to be compact if T maps every bounded set into relatively compact set. Also, T is said to be completely continuous if T maps every weakly convergent sequence into a norm convergent sequence. To deal with the compactness we need the following lemma. The proof of the lemma is similar to the case of H 2 and for details we refer to [25, Section 2.4]. Lemma 1. Let v be a weight on D and let φ ∈ S (D) and θ ∈ H (D). Then the operator Wφ,θ is compact on Hv∞ (D) if and only if Wφ,θ (fn )v → 0 for every bounded sequence {fn } in Hv∞ (D) such that fn → 0 uniformly on every compact subset of D. We need the following theorem which is needed for proving our main result. For details (see [22]). Theorem 1. Let v and w be weights on D. Then: (i) The operator Wφ,θ : Hv∞ (D) → Hw∞ (D) is bounded if and only if sup z∈D

|θ (z)| w (z) < ∞. v˜ (φ (z))

422

Manhas

IEOT

(ii) The operator Wφ,θ : Hv∞ (D) → Hw∞ (D) is compact if and only if θ ∈ Hw∞ w (z) |θ (z)| = 0. and lim sup r→1 |φ(z)|>r v ˜ (φ (z)) When v is essential, v˜ can be replaced by v. Our result involve the pseudo-hyperbolic metric. For z, w ∈ D, the pseudohyperbolic distance between z and w is given by z−w . ρ (z, w) = 1 − z¯w For more information on the pseudo-hyperbolic metric, we refer to [11]. Our main result is the following: Theorem 2. Let v and w be weights on D such that v is essential. Let φ1 , φ2 ∈ S (D) and let θ, π ∈ H (D) be such that Wφ1 ,θ , Wφ2 ,π : Hv∞ (D) → Hw∞ (D) are weighted composition operators. Then the following are equivalent: (a) Wφ1 ,θ − Wφ2,π : Hv∞ (D) → Hw∞ (D) is compact. (b) Wφ1 ,θ − Wφ2,π : Hv∞ (D) → Hw∞ (D) is completely continuous. w (zn ) |θ (zn )| ρ (φ1 (zn ) , φ2 (zn )) = 0, for every i = 1, 2; (c) (i) lim |φi (zn )|→1 v (φi (zn )) w (zn ) |π (zn )| ρ (φ1 (zn ) , φ2 (zn )) = 0, for i = 1, 2; (ii) lim v (φi (zn )) |φi (zn )|→1 w (zn ) |θ (zn ) − π (zn )| = 0, for i = 1, 2. (iii) lim v (φi (zn )) |φi (zn )|→1 Proof. Clearly (a) implies (b). Now assume that (b) is true. We shall establish part (c). To prove part c(i), suppose that c(i) is not true. Then we may assume w (zn ) |θ (zn )| lim ρ (φ1 (zn ) , φ2 (zn )) = β > 0. Further, we may assume |φ1 (zn )|→1 v (φ1 (zn )) that {φ1 (zn )} is interpolating for H ∞ (D). Then Corollary 23 of [7] implies that {φ1 (zn )} is also interpolating for Hv∞ (D). Thus by Lemma 20 of [7], there exists {hn } ⊂ Hv∞ (D) such that 1 , if n = k v(φ1 (zn )) hn (φ1 (zk )) = 0 , if n = k. and

∞

v (z) |hn (z)| ≤ M < ∞.

n=1

This shows that {hn } is bounded and hn → 0 weakly in Hv∞ (D). Now, we define gn (z) =

z − φ2 (zn ) 1 − φ2 (zn )z

hn (z) , for all z ∈ D.

Vol. 62 (2008) Compact Differences of Weighted Composition Operators ∞

Then still {gn ⊂ Hv∞ (D) and gn → 0 weakly. Since

423

v (z) |gn (z)| ≤ M . Clearly {gn } is bounded and

n=1 Wφ1 ,θ − Wφ2 ,π

is completely continuous, we have

(Wφ1 ,θ − Wφ2 ,π ) (gn )w → 0, as n → ∞.

(1.1)

But (Wφ1 ,θ − Wφ2 ,π ) (gn )w ≥ w (zn ) |θ(zn )gn (φ1 (zn )) − π (zn ) gn (φ2 (zn ))| φ (z ) − φ (z ) 1 n 2 n = w (zn ) |θ (zn )| |hn (φ1 (zn ))| 1 − φ2 (zn )φ1 (zn ) w (zn ) |θ (zn )| φ1 (zn ) − φ2 (zn ) = . v (φ1 (zn )) 1 − φ2 (zn )φ1 (zn ) That is, w (zn ) |θ (zn )| ρ (φ1 (zn ) , φ2 (zn )) |φ1 (zn )|→1 v (φ1 (zn )) ≥ β > 0,

lim (Wφ1 ,θ − Wφ2 ,π ) (gn )w ≥

n→∞

lim

which contradicts (1.1). This proves that lim

|φ1 (zn )|→1

w(zn ) |θ (zn )| ρ (φ1 (zn ) , φ2 (zn )) = 0. v (φ1 (zn ))

Similarly, we can show that lim

|φ2 (zn )|→1

w(zn ) |θ (zn )| ρ (φ1 (zn ) , φ2 (zn )) = 0. v (φ2 (zn ))

This proves part c(i). Also, the same arguments of c(i) can be used to prove part c(ii). Now, we shall prove part c(iii). Suppose that part c(iii) does not holds. Then we may assume that lim

|φ1 (zn )|→1

Now, if

lim

|φ1 (zn )|→1

w (zn ) |θ (zn ) − π(zn )| = δ = 0. v (φ1 (zn ))

(1.2)

ρ (φ1 (zn ) , φ2 (zn )) = α > 0, then from part c(i) and part c(ii),

it follows that 0 0, (Wφ1 ,θ − Wφ2 ,π ) (fn )w > ε > 0, for every n. Then there exists a sequence {zn } in D such that w(zn ) |θ (zn ) fn (φ1 (zn ) − π(zn )fn (φ2 (zn ))| > ε,

(1.5)

for every n. Further, it implies that max {|φ1 (zn )| , |φ2 (zn )|} → 1, as n → ∞. We may assume that |φ1 (zn )| → 1 and φ2 (zn ) → z0 , for some complex number z0 . Also, we may assume that ρ (φ1 (zn ) , φ2 (zn )) → s, as n → ∞.

(1.6)

Suppose that s > 0. Now, if |z0 | = 1, then by c(i) and c(ii), we have lim

w(zn ) |θ (zn )| ρ (φ1 (zn ) , φ2 (zn )) = 0, for i = 1, 2 v (φi (zn ))

lim

w(zn ) |π (zn )| ρ (φ1 (zn ) , φ2 (zn )) = 0, for i = 1, 2. v (φi (zn ))

|φi (zn )|→1

and |φi (zn )|→1

From (1.5), it follows that w(zn ) |θ (zn )| |v (φ1 (zn )) fn (φ1 (zn ))| v (φ1 (zn )) w(zn ) |π (zn )| + |fn (φ2 (zn )) v (φ2 (zn ))| v (φ2 (zn )) w(zn ) |θ (zn )| w(zn ) |π (zn )| + ≤ fn v . v (φ1 (zn )) v (φ2 (zn ))

0 0.

Then the operator Sψ is bounded on Lp (Rn ) for all p ∈ (1, ∞).

430

Sato

IEOT

This is due to Benedek, Calderón and Panzone [3]. Fan-Sato [9] proved the following result, which substantially relaxes the conditions imposed on ψ in Theorem A. Theorem B. Suppose that the function ψ satisfies the following conditions: (1) B (ψ) < ∞ for some > 0, where B (ψ) = |x|≥1 |ψ(x)| |x| dx; 1/u (2) Du (ψ) < ∞ for some u > 1, where Du (ψ) = |x|≤1 |ψ(x)|u dx ;

(3) |ψ(x)| ≤ h(|x|)Ω(x ) for all x ∈ Rn \ {0}, where x = x/|x|, for some nonnegative functions h on (0, ∞) and Ω on S n−1 (the unit sphere in Rn ) such that (a) h(r) is non-increasing on (0, ∞) and h(|x|) ∈ L1 (Rn ), (b) Ω ∈ Lq (S n−1 ) for some q, 1 < q ≤ ∞. Then Sψ is bounded on Lp (Rn ) for all p ∈ (1, ∞). For the rest of this note we assume that ψ is compactly supported and the support is contained in the unit ball {|x| ≤ 1}, for the sake of simplicity. We shall prove Lp estimates for Sψ that are useful in extrapolation arguments to obtain a minimum condition on ψ for Lp boundedness of Sψ . Theorem 1. Suppose that |ψ(x)| ≤ h(|x|)Ω(x ) for all x ∈ Rn \ {0}, where h is a non-negative, non-increasing function on (0, ∞) with support in (0, 1] and Ω is a non-negative function on S n−1 . We assume that h(|x|) ∈ L1 (Rn ), Ω ∈ L1 (S n−1 ) and ψ ∈ Lq (Rn ) for some q > 1. Put mψ (x) = h(|x|)Ω(x ). Then, we have Sψ (f )p ≤ Cp (q/(q − 1))1/2 (ψq + mψ 1 ) f p for all p ∈ (1, ∞), where the constant Cp is independent of q, ψ, h and Ω; and q/(q − 1) = 1 when q = ∞. Theorem 2. Suppose that ψ ∈ Lq (Rn ) for some q > 1. Then, we have 1/2

Sψ (f )p ≤ Cp (q/(q − 1))

ψq f p

for all p ∈ [2, ∞) with a constant Cp independent of q and ψ. We are interested in the Lp estimates for Sψ (f ) of Theorems 1 and 2 when q is near 1. Let L(log L)1/2 (S n−1 ) be the class of the functions Ω on S n−1 satisfying |Ω(θ)| [log(2 + |Ω(θ)|)]1/2 dσ(θ) < ∞, S n−1

where dσ denotes the Lebesgue surface measure on S n−1 . The class L(log L)1/2 (Rn ) of functions on Rn is defined similarly. Using Theorems 1 and 2 and applying extrapolation, we can prove the following two results. Corollary 1. Let Ω ∈ L(log L)1/2 (S n−1 ) and Ω ≥ 0. Suppose that |ψ(x)| ≤ h(|x|)Ω(x ) for all x ∈ Rn \ {0}, where h is as in Theorem 1. We further assume that h(|x|) ∈ Lq (Rn ) for some q > 1. Then, Sψ is bounded on Lp (Rn ) for all p ∈ (1, ∞).

Vol. 62 (2008)

Littlewood-Paley Functions and Extrapolation

431

Corollary 2. Suppose that ψ is in L(log L)1/2 (Rn ). Then Sψ is bounded on Lp (Rn ) for all p ∈ [2, ∞). When ψ ∈ Lq (Rn ) for some q > 1, Lp boundedness of Sψ for p ∈ [2, ∞) was proved in [9]. Corollary 2 is an improvement over the result. See [4] and also Remark 2 of [9] for Lp boundedness of Sψ for p < 2. Let ψ(x) = |1 − |x|2 |α−1 Ω(x ) if 0 < |x| < 1, and ψ(x) = 0 if |x| > 1, where we assume that α > 0, Ω ∈ L(log L)1/2 (S n−1 ) and S n−1 Ω dσ = 0. Then, we can see that Corollary 2 applies to the operator Sψ . Let ψ(x) = |x|−n+1 Ω(x )χ(0,1] (|x|) for x ∈ Rn \ {0}, where Ω ∈ L1 (S n−1 ), S n−1 Ω dσ = 0. Here χE denotes the characteristic function of a set E. Then, the Littlewood-Paley function Sψ (f ) coincides with the Marcinkiewicz integral µΩ (f ) in Stein [14] (see also Hörmander [10]). As an application of Theorem 1 we have the following result: Corollary 3. If Ω ∈ Lq (S n−1 ) for some q > 1, then 1/2

µΩ (f )p ≤ Cp (q/(q − 1))

Ωq f p

for p ∈ (1, ∞), where the constant Cp is independent of q and Ω. Here Ωq denotes the norm of Ω in Lq (S n−1 ). Also, as a consequence of Corollary 1, we have the following result of AlSalman, Al-Qassem, Cheng and Pan [1]. Theorem C . If Ω ∈ L(log L)1/2 (S n−1 ), then µΩ is bounded on Lp (Rn ) for all p ∈ (1, ∞). The case p = 2 of Theorem C is due to Walsh [17]. In Section 2 we shall prove Theorem 1. We adapt the method of [6] involving the Littlewood-Paley theory for the case of square function operators. We apply a Littlewood-Paley decomposition adapted to a suitable lacunary sequence depending on q for which ψ ∈ Lq (Rn ). See [2] and also [13] for the method of appropriately choosing a lacunary sequence. Theorem 2 is proved in Section 3 by applying the Littlewood-Paley theory in like manner. In proving Theorems 1 and 2, basic estimates and key observations of [11], [9] will be used. Also, to prove Theorem 2, we apply an induction argument similar to the one used in [13] to get sharp Lp estimates for singular Radon transforms. Applying an extrapolation method (see, e.g., Zygmund [18, Chap. XII, pp. 119–120]), we shall prove Corollary 1 by Theorem 1 in Section 4. Similarly, Corollary 2 follows from Theorem 2. Throughout this note, the letter C will be used to denote non-negative constants which may be different in different occurrences.

432

Sato

IEOT

2. Proof of Theorem 1 For k ∈ Z (the set of integers) and ρ ≥ 2, let Tk be an operator mapping functions on Rn to H-valued functions on Rn defined as (Tk (f )(x)) (t) = (ψt ∗ f )(x)χ[1,ρ) (ρ−k t), where H is the Hilbert space L2 ((0, ∞), dt/t). Note that 1/2 k+1 ρ 2 dt |ψt ∗ f (x)| . |Tk (f )(x)|H = t ρk Lemma 1. Let ψ be as in Theorem 1. Then we have 1/2 1/2

1/2 1/2 2 1/2 2 |T (f )| ≤ C ψ m (log ρ) |f | k k H s ψ 1 k 1 k k s

s

for all s ∈ (1, ∞), where mψ is as in Theorem 1 and the constant Cs is independent of ρ, ψ, h and Ω. Proof. Define a maximal function Mψ (f )(x) = sup |ψt ∗ f (x)| . t>0

Since the method of rotations implies Mψ (f )r ≤ Cr mψ 1 f r for all r > 1, arguing as in the proofs of Lemmas 1 and 2 of [9] and checking the constants in the arguments, we can obtain Lemma 1. Lemma 2. Let ψ be as in Theorem 2. Then ρk+1 2 dt ˆ ≤ C(log ρ)ψ2q min(1, |ρk+1 ξ|, |ρk ξ|−1 )1/(2q ) , ψ(tξ) t k ρ where the constant C is independent of ρ ≥ 2, q > 1 and ψ. Here q = q/(q − 1), and gˆ denotes the Fourier transform of g; we also write gˆ = F(g). Proof. First, a direct computation implies that ρ ˆ k 2 ψ(tρ ξ) dt/t ≤ (log ρ)ψ21 ≤ C(log ρ)ψ2q .

(2.1)

1

Next, using Lemmas 2 and 3 of [11], we see that ρ

ˆ k 2 ψ(tρ ξ) dt/t ≤ 1

0≤m≤(log ρ)/ log 2

≤

0≤m≤(log ρ)/ log 2

1

2

ˆ m k 2 ψ(t2 ρ ξ) dt/t

−1/(2q ) Cψ2q 2m ρk ξ

−1/(2q ) ≤ C(log ρ)ψ2q ρk ξ .

(2.2)

Vol. 62 (2008)


433

ˆ Finally, by the proof of Lemma 1 of [11], we have |ψ(ξ)| ≤ C|ξ|ψ1 , and hence ρ ρ 2 ˆ k |tρk ξ|2 dt/t ≤ Cψ21 |ρk+1 ξ|2 ψ(tρ ξ) dt/t ≤ Cψ21 (2.3) 1 1 2 k+1 2 ≤ C(log ρ)ψq |ρ ξ| . Combining (2.1)–(2.3), we have the conclusion, since

min(1, |ρk+1 ξ|2 , |ρk ξ|−1/(2q ) ) ≤ min(1, |ρk+1 ξ|, |ρk ξ|−1 )1/(2q ) .

∞ We choose a sequence {Ψk }∞ −∞ of non-negative functions in C (R) such that

supp(Ψk ) ⊂ [ρ−k−1 , ρ−k+1 ], Ψk (t) = 1 for t > 0. k

We further assume that |(d/dt)j Ψk (t)| ≤ cj |t|−j

(j = 1, 2, . . . ),

where the constants cj are independent of ρ and k. This is feasible since ρ ≥ 2. Define Fourier multiplier operators ∆j as F(∆j (f ))(ξ) = Ψj (|ξ|)fˆ(ξ) for j ∈ Z and decompose (ψt ∗ f )(x) as (ψt ∗ f )(x) = j∈Z Fj (x, t), where

∆j+k (ψt ∗ f )(x)χ[ρk ,ρk+1 ) (t). Fj (x, t) = Then, we have Sψ (f )(x) ≤

k∈Z

Uj (f )(x), where 1/2 ∞ 1/2

2 dt 2 |Fj (x, t)| = |Tk (∆j+k (f ))(x)|H . Uj (f )(x) = t 0 j∈Z

k

Let Ej = {ρ−1−j ≤ |ξ| ≤ ρ1−j }. Then by the Plancherel theorem and Lemma 2 we have

ρk+1 2 dt 2 dx |∆j+k (ψt ∗ f ) (x)| Uj (f )2 = t n ρk k∈Z R k+1 2 dt ρ

ˆ 2 ˆ ≤ C f (ξ) dξ ψ(tξ) t Ej+k ρk k∈Z 2

C(log ρ)ψ2q min(1, |ρk+1 ξ|, |ρk ξ|−1 )1/(2q ) fˆ(ξ) dξ ≤ k∈Z

Ej+k

≤ C(log ρ)ψ2q min(1, ρ−|j|+2 )1/(2q )

k∈Z

≤ C(log ρ)ψ2q min(1, ρ−|j|+2 )1/(2q ) f 22 ,

Ej+k

ˆ 2 f (ξ) dξ

434

Sato

IEOT

where the last inequality is valid since the sets Ej are finitely overlapping. Thus we have

Uj (f )2 ≤ C(log ρ)1/2 ψq min(1, ρ−|j|+2 )1/(4q ) f 2.

(2.4)

We now use the Littlewood-Paley inequality 1/2

2 ≤ cp f p , |∆k (f )| k

(2.5)

p

where 1 < p < ∞ and the constant cp is independent of ρ. By (2.5) and Lemma 1 we see that 1/2

2 Uj (f )s = |Tk (∆j+k (f ))|H k∈Z s 1/2

(2.6) 1/2 1/2 1/2 2 ≤ Cψ1 mψ 1 (log ρ) |∆j+k (f )| k∈Z s

1/2

1/2

≤ Cψ1 mψ 1 (log ρ)1/2 f s for s ∈ (1, ∞). Interpolating between (2.4) and (2.6), we get η 1−η 1/2 1/2 Uj (f )p ≤ C(log ρ)1/2 ψq min(1, ρ−|j|+2 )1/(4q ) ψ1 mψ 1 f p (1−η)/2

≤ C(log ρ)1/2 ψ(1+η)/2 mψ 1 q

min(1, ρ−|j|+2 )η/(4q ) f p

for some η ∈ (0, 1] depending on p, where 1 < p < ∞. Thus

Sψ (f )p ≤ Uj (f )p j∈Z

 

(1−η)/2 ≤C min(1, ρ−|j|+2 )η/(4q )  (log ρ)1/2 ψ(1+η)/2 mψ 1 f p q j∈Z

(1−η)/2

≤ C(1 − ρ−η/(4q ) )−1 (log ρ)1/2 ψ(1+η)/2 mψ 1 q

f p

≤ C(1 − ρ−η/(4q ) )−1 (log ρ)1/2 (ψq + mψ 1 ) f p ,

where the last inequality follows from Young’s inequality. Taking ρ = 2q , we get the conclusion of Theorem 1, since

1/2

(1 − ρ−η/(4q ) )−1 (log ρ)1/2 = (1 − 2−η/4 )−1 ((q log 2)/(q − 1))

if ρ = 2q .

Vol. 62 (2008)


435

3. Proof of Theorem 2 Let ψ be as in Theorem 2 and ρ ≥ 2. Lemma 3. Let θ ∈ (0, 1). Suppose 2 ≤ s < 2(1 + θ)/θ and put r = (s/2) . Then we have 1/2 1/2

−1/r

2 1/2 −θ/(4q ) 2 , |Tk (fk )|H |fk | ≤ Cs (log ρ) ψq 1 − ρ k k s

s

where the constant Cs is independent of ψ, q > 1 and ρ. To prove this we need the following: ρ (ρ) Lemma 4. Let p(x) = 1 |ψt (x)| dt/t. Define a maximal function Nψ (f )(x) = supk∈Z |(f ∗ pρk )(x)|. Let θ ∈ (0, 1). Then, for p > 1 + θ we have −2/p (ρ) Nψ (f )p ≤ Cp (log ρ)ψq 1 − ρ−θ/(4q ) f p , where the constant Cp is independent of ψ, q > 1 and ρ. Proof. Let ϕ ∈ C ∞ (Rn ) be compactly supported and satisfy ϕ(0) ˆ = 1, ϕ ≥ 0. Define Φ(x) = p(x) − (log ρ)ψ1 ϕ(x). Then Φρk 1 ≤ C(log ρ)ψq ,

(3.1)

F(Φρk )(ξ) ≤ C(log ρ)ψq min |ρk+1 ξ|, |ρk ξ|−1 1/(4q ) .

(3.2)

Note that p1 = (log ρ)ψ1 and |ˆ p(ξ)| ≤ (log ρ)1/2

ρ

1

|F(|ψ|)(tξ)|2 dt/t

1/2 .

Using these observations and arguing as in the proof of Lemma 2, we can prove the estimates (3.1), (3.2). (ρ) We see that Nψ (f )∞ ≤ C(log ρ)ψq f ∞ , since pρk 1 ≤ C(log ρ)ψq . So, Lemma 4 follows by interpolation if we prove it for p ∈ (1 + θ, 2]. Define ∞ 1/2

2 Φρk ∗ f (x) g(f )(x) = . k=−∞

Then (ρ)

Nψ (f ) ≤ g(f ) + (log ρ)ψ1 sup |ϕt ∗ f | t>0

≤ g(f ) + C(log ρ)ψ1 M (f ),

(3.3)

where M (f ) denotes the Hardy-Littlewood maximal function. Thus, it suffices to prove g(f )p ≤ CAB 2/p f p for p ∈ (1 + θ, 2], where A = (log ρ)ψq and

436

Sato

IEOT

−1 B = 1 − ρ−θ/(4q ) . Recalling a well-known property of Rademacher’s functions, we can see that this follows from V (f )p ≤ CAB 2/p f p for p ∈ (1 + θ, 2] with a constant C independent of , where V (f ) = = {k }, k = 1 or −1. To prove (3.4) we need the following.

(3.4) k k Φρk

∗ f,

Lemma 5. We define a sequence {pj }∞ 1 by p1 = 2 and 1/pj+1 = 1/2 + (1 − θ)/(2pj ) for j ≥ 1. Then, for j ≥ 1 we have V (f )pj ≤ Cj AB 2/pj f pj ,

(3.5)

where Cj is independent of ψ, q > 1, ρ and . We note that {pj } is decreasing and converges to 1 + θ. Proof of Lemma 5. Let Vj (f ) =

∞

k Φρk ∗ ∆j+k (f ) ,

k=−∞

where the operators ∆j are as in the proof of Theorem 1. Then, V (f ) = j Vj (f ). Using Plancherel’s theorem and the estimates (3.1), (3.2) as in the proof of (2.4), we have 1/(4q ) Vj (f )2 ≤ CA min 1, ρ−|j|+2 f 2 . (3.6) It follows that V (f )2 ≤ j Vj (f )2 ≤ CABf 2 . This proves (3.5) for j = 1. We now assume (3.5) for j = s and prove it for j = s + 1. It will complete the proof of Lemma 5 by induction. First, by (3.3) we see that sup |Φρk | ∗ f ≤ sup pρk ∗ |f | + (log ρ)ψ1 sup (ϕt ∗ |f |) k

t>0

k

≤ g(|f |) + 2(log ρ)ψ1 sup (ϕt ∗ |f |) t>0

≤ g(|f |) + C(log ρ)ψ1 M (f ). From this we see that

sup |Φρk | ∗ f k

ps

≤ CAB 2/ps f ps ,

(3.7)

since our assumption implies g(f )ps ≤ CAB 2/ps f ps . Let 1/v − 1/2 = 1/(2ps ). Then, by the proof of Lemma in [6, p. 544] and the estimates (3.1), (3.7), we have the vector valued inequality 1/2 1/2

2 1/ps 2 . |Φρk ∗ gk | |gk | (3.8) ≤ CAB k k v

v

Vol. 62 (2008)


437

For the sake of completeness, here we give a proof of (3.8). By duality it suffices to prove (3.8) with v in place of v. Note that ps = (v /2) . Take a non-negative u ∈ Lps such that ups ≤ 1 and

2 |Φρk ∗ gk | u dx, I= k

where

1/2 2

. I = |Φρk ∗ gk |2 k v Since |Φρk ∗ gk |2 ≤ Φ1 |Φρk | ∗ |gk |2 ,

˜ ρk | ∗ u dx, I ≤ Φ1 |Φρk | ∗ |gk |2 u dx ≤ Φ1 |gk |2 sup |Φ k

k

k

˜ where Φ(x) = Φ(−x). Thus, applying Hölder’s inequality, by (3.1), (3.7) we have 1/2 1/2

2

2 2 2 2/ps 2 ˜ I ≤ Φ1 |gk | |gk | sup |Φρk | ∗ u ≤ CA B k k ps k v

v

as claimed. Also, from the Littlewood-Paley theory it follows that 1/2

2 , Vj (f )p ≤ cp |Φρk ∗ ∆j+k (f )| k

(3.9)

p

where 1 < p < ∞ and cp is independent of ρ. By (2.5), (3.8) and (3.9) we have Vj (f )v ≤ CAB 1/ps f v .

(3.10)

Since 1/ps+1 = θ/2 + (1 − θ)/v, interpolating between (3.6) and (3.10), we have Vj (f )ps+1 ≤ CAB (1−θ)/ps min 1, ρ−θ(|j|−2)/(4q ) f ps+1 . Therefore, V (f )ps+1 ≤

Vj (f )ps+1 ≤ CAB (1−θ)/ps (1 − ρ−θ/(4q ) )−1 f ps+1

j

≤ CAB 2/ps+1 f ps+1 . This proves (3.5) for j = s + 1, which completes the proof of Lemma 5.

We now prove (3.4) for p ∈ (1 + θ, 2]. Let {pj }∞ 1 be as in Lemma 5. Then we have pN +1 < p ≤ pN for some N . Thus, interpolating between the estimates in (3.5) for j = N and j = N + 1, we have (3.4). This completes the proof of Lemma 4.

438

Sato

IEOT

Proof of Lemma 3. From the proof of Lemma 1 in [9], we see that 1/2 2

2 ˜ (ρ) (g) dx, |T (f )| ≤ Cψ (3.11) |fk |2 N k k H 1 ψ k k s (ρ) ˜ (f ) = supk∈Z p˜ρk ∗ f with p˜(x) = p(−x), and g is a non-negative where N ψ function in Lr satisfying gr ≤ 1, r = (s/2) . The range of s in Lemma 3 implies that r > 1 + θ. Therefore, by H¨ older’s inequality and Lemma 4 it follows that 1/2 2

2 ˜ (ρ) 2 N ˜ (ρ) (g)r |fk | |fk | Nψ (g) dx ≤ C ψ k k s 1/2 2 −2/r

. ≤ C(log ρ)ψq 1 − ρ−θ/(4q ) |fk |2 k s

Combining this with (3.11), we can get the conclusion of Lemma 3.

Now we can give a proof of Theorem 2. Given p ∈ [2, ∞), let s, θ ∈ (0, 1) be such that 2 ≤ p < s < 2(1 + θ)/θ. Let Uj be as in the proof of Theorem 1. Then, by Lemma 3 and (2.5) we see that −1/r Uj (f )s ≤ C(log ρ)1/2 ψq 1 − ρ−θ/(4q ) f s , where r = (s/2) . Interpolating between this estimate and (2.4), we have −(1−η)/r Uj (f )p ≤ C(log ρ)1/2 ψq min(1, ρ−|j|+2 )η/(4q ) 1 − ρ−θ/(4q ) f p for some η ∈ (0, 1] (we note that (2.4) holds under the assumptions of ψ in Theorem 2). Thus, arguing as in the proof of Theorem 1, we get −1 −(1−η)/r 1 − ρ−θ/(4q ) f p . Sψ (f )p ≤ C(log ρ)1/2 ψq 1 − ρ−η/(4q )

If we put ρ = 2q , we get the desired estimate in Theorem 2.

4. Proof of Corollary 1 Fix p ∈ (1, ∞) and f with f p ≤ 1. We write H(x) = h(|x|), where h is as in Corollary 1. Put R(ψ) = Sψ (f )p . Let Fk = {θ ∈ S n−1 : 2k−1 < |Ω(θ)| ≤ 2k } for k ≥ 2 and F1 = {θ ∈ S n−1 : |Ω(θ)| ≤ 2}. Let Ωk (θ) = Ω(θ)χFk (θ) for k ≥ 1. We define Ek = {x ∈ B(0, 1) : x = 0, x ∈ Fk } for k = 1, 2, 3, . . . , where B(0, 1) = {|x| ≤ 1}. We decompose ψ as ψ = ∞ k=1 ψk , where ψk = ψχEk − |B(0, 1)|−1 ψ dx χB(0,1) . Ek

Vol. 62 (2008)

We note that


439

ψk dx = 0 and |ψk (x)| ≤ h∗ (|x|)Ω∗k (x ) for x ∈ Rn \ {0}, where

h∗ (|x|) = [h(|x|) + CH1 ] χ(0,1] (|x|),

Ω∗k (x ) = Ωk (x ) + Ωk 1 .

1/r

We see that Ω∗k r ≤ C2k ek for 1 < r < ∞, where ek = σ(Fk ), for k ≥ 1, and that H ∗ q ≤ CHq , where H ∗ (x) = h∗ (|x|). Let mψk (x) = h∗ (|x|)Ω∗k (x ) for k ≥ 1. Then, applying Theorem 1 and using subadditivity of R(ψ), we have

Sψ (f )p = R(ψ) ≤ R(ψk ) k≥1

≤ C(q/(q − 1))1/2

k 1, and φ : Bs (H) → Bs (H) a bijective map with the property that A ∼ B ⇐⇒ φ(A) ∼ φ(B),

A, B ∈ Bs (H).

Then there exist c ∈ {−1, 1}, an operator C ∈ Bs (H), and an invertible bounded linear or conjugate-linear operator T : H → H such that φ(A) = cT AT ∗ + C for every A ∈ Bs (H). In particular, every bijective map preserving comaparability in both directions either preserves order in both directions or satisfies A ≤ B ⇐⇒ φ(B) ≤ φ(A), A, B ∈ Bs (H). Our goal in this paper is not just to improve the Moln´ ar’s result, but also to provide an elementary self-contained proof of this statement. Namely, when proving his theorem on order preserving maps, Moln´ ar used several deep results including Rothaus’s theorem [9] on the automatic linearity of bijective maps between closed convex cones preserving order in both directions, Vigier’s theorem [7, Theorem 4.1.1], and the Kadison’s well-known structural theorem for bijective linear positive unital maps on C ∗ -algebras [3, Corollary 5]. We will achieve the simplification by establishing a connection between comparability preserving maps and adjacency preserving maps, then by solving the two-dimensional case using elementary linear algebra techniques, and finally by reducing the general case to the two-dimensional case. In spite of being completely elementary, our proof is rather short.

2. Comparability preservers and adjacency preservers For any pair A, B ∈ Bs (H) define d(A, B) = rank (A − B) if A − B is a finite rank operator and d(A, B) = ∞ otherwise. We say that A, B ∈ Bs (H) are adjacent if d(A, B) = 1, that is, if B is a rank one perturbation of A. Lemma 2.1. Let H be a Hilbert space, dim H ≥ 2, and A, B ∈ Bs (H). Then the following are equivalent: 1. A and B are adjacent, 2. A ∼ B and there exists C ∈ Bs (H) \ {A, B} such that for every T ∈ Bs (H) satisfying T ∼ A and T ∼ B we have T ∼ C. Proof. In the proof we will need the following simple fact that can be easily verified. Let P ∈ Bs (H) be a projection of rank one and assume that S ∈ Bs (H) satisfies 0 ≤ S ≤ P . Then S = tP for some real t, 0 ≤ t ≤ 1.

Vol. 62 (2008)

Comparability Preserving Maps

443

Let us now assume that A and B are adjacent. Then B = A + tP for some nonzero real number t and some rank one projection P . Interchanging A and B, if necessary, we may, and will assume that t > 0. Further, replacing A and B by A − A and B − A, respectively, we see that there is no loss of generality in assuming that A = 0 and B = tP with t > 0. Then A ∼ B, because A ≤ B. Choose C = (t/2)P . Then C = A and C = B. Let T ∈ Bs (H) be any operator satisfying T ∼ 0 = A and T ∼ tP = B. It is clear that if T ≥ B = tP then T ≥ C and if T ≤ 0 = A, then T ≤ C. In the only remaining case A = 0 ≤ T ≤ tP we have T = sP for some s, 0 ≤ s ≤ t. Clearly, T ∼ C in this last case as well. When proving the converse we will again use the fact that none of the two conditions appearing in our lemma is affected if we replace A and B by A − D and B − D, respectively, or if we interchange A and B. Here, D is any self-adjoint bounded linear operator on H. So, assume that A and B are not adjacent. Assume further that the second condition is satisfied and let C be the operator appearing in this condition. Then, in particular, A ∼ B, C ∼ A and C ∼ B. With no loss of generality we may assume that A ≥ B. We have three possibilities. If C ≥ A, then because C = A, we have C = A + R for some R ≥ 0, R = 0. It is easy to find S ∈ Bs (H) such that S ≥ 0 and S ∼ R. Indeed, there is a unit vector x such that Rx, x = s > 0. Let P be the rank one projection onto the linear span of x and let Q be any rank one projection orthogonal to P . If y is a unit vector spanning the range of Q and t a positive real number satisfying t > Ry, y, then S = (s/2)P + tQ is positive and S ∼ R. For T = A + S we have T ≥ A ≥ B and T ∼ C, a contradiction. Similarly, the case when C ≤ B cannot occur. It remains to consider the case when B ≤ C ≤ A. We may, and will assume that C = 0. Since 0 = B and 0 = A and because A and B are not adjacent, we have two possibilities; either A is a positive rank one operator, B is a negative rank one operator and A and B are linearly independent, or at least one of A and B has at least two-dimensional image. In the first case it is an elementary linear algebra exercise to find a rank two operator T ∈ Bs (H) such that B ≤ T ≤ A, but T is neither positive, nor negative, and hence, T ∼ C, a contradiction. In the second case we may, and will assume that A has at least two-dimensional image. By the spectral theorem for self-adjoint operators we can find a rank two projection P and a positive real number t such that A ≥ A1 = tP . We can also find a rank one projection Q and a negative real number s such that B ≤ B1 = sQ. It follows that there is a rank one positive operator A2 linearly independent of B1 such that A1 ≥ A2 . As before there exists a rank two operator T ∈ Bs (H) such that B ≤ B1 ≤ T ≤ A2 ≤ A, but T is neither positive, nor negative, a contradiction. For us the following simple consequence will be important. Corollary 2.2. Let H be a Hilbert space, dim H ≥ 2, and φ : Bs (H) → Bs (H) a bijective map preserving comparability in both directions. Then for every pair A, B ∈ Bs (H) the operators φ(A) and φ(B) are adjacent if and only if A and B are adjacent. Also, d(φ(A), φ(B)) = 2 if and only if d(A, B) = 2.

444

ˇ Semrl

IEOT

Proof. The first statement is a straightforward consequene of the previous lemma. To prove the second one it is enough to show that d(A, B) = 2 yields d(φ(A), φ(B)) = 2 because φ−1 has the same properties as φ. So, assume that A = B + R with rank R = 2. Every rank two operator is a sum of two rank one operators, and thus, there exists C ∈ Bs (H) such that A and C are adjacent and C and B are adjacent. It follows that φ(A) and φ(C) are adjacent and φ(C) and φ(B) are adjacent. If T and S are finite rank operators, then it follows from Im (T + S) ⊂ Im T + Im S that rank (T +S) ≤ rank T +rank S. Thus, φ(A)−φ(B) = (φ(A)−φ(C))+(φ(C)−φ(B)) is of rank at most two. Because φ is bijective and preserves adjacency in both directions we have d(φ(A), φ(B)) = 2, as desired. Let us add here one short remark that will be used later. From the above arguments we conclude that if rank (T + S) = rank T + rank S, then the image of T + S is a direct sum of the images of T and S. In particular, in this case both the image of T and the image of S are subspaces of the image of T + S.

3. 2 × 2 case In this section we will prove Theorem 1.1 in the case when dim H = 2. In this special case we can identify Bs (H) with H2 , the set of all 2 × 2 hermitian matrices. So, assume that φ : H2 → H2 is a bijective map preserving comparability in both directions. By the previous section we know that d(φ(A), φ(B)) = d(A, B) for every pair A, B ∈ H2 . Let us just mention here that the structure of adjacency preserving maps on H2 is known. The most general structural result for such maps can be found in [2]. The proof in [2] depends heavily on a non-trivial geometrical result [4]. Other proofs of structural results for adjacency preservers are known. But one of our goals is to find a short elementary self-contained proof of our main result and therefore, we will give here a simple linear algebraic proof of the twodimensional case using both adjacency and comparability preserving property. It is based on some ideas from [8]. Our first simple observation is that there is no loss of generality in assuming that φ(0) = 0. Indeed, we may replace the map φ by A → φ(A) − φ(0), A ∈ H2 . Before we proceed we make the following trivial remark. If U, V ≥ 0 ∈ H2 are linearly independent rank one matrices, then we can find an invertible R ∈ H2 such that RU R∗ = E11 and RV R∗ = E22 . Here, Eij , 1 ≤ i, j ≤ 2, denote the members of the standard basis of the space of all 2 × 2 matrices. Indeed, we can write U = xx∗ and V = yy ∗ for some nonzero linearly independent column vectors x and y and then we can take R to be the invertible 2×2 matrix defined by Rx = e1 and Ry = e2 , where e1 and e2 form the standard basis of the space of 2×1 matrices. The next step is to prove that if P, Q ∈ H2 are positive linearly independent matrices of rank one then there exist a positive real number a and bijective functions f, g : R → R such that φ(tP + sQ) = f (t)φ(P ) + g(s)φ(Q),

t, s ∈ R,

Vol. 62 (2008)


445

where f (t) = g(t) > 0 for t > 0, and f (t) = −af (−t) and g(t) = −a−1 g(−t) for t < 0. So, let P, Q be as above. Then there exists an invertible R ∈ H2 such that P = RE11 R∗ and Q = RE22 R∗ . Clearly, φ(P ) and φ(Q) are of rank one. Further, we may assume with no loss of generality that φ(P ) ≥ 0, since otherwise we can consider the map A → −φ(A), A ∈ H2 , instead of the map φ. Since P ∼ Q, we have φ(P ) ∼ φ(Q). It follows that φ(Q) ≥ 0, as well. Thus, there exists an invertible S ∈ H2 such that Sφ(P )S ∗ = E11 and Sφ(Q)S ∗ = E22 . Replacing φ by A → Sφ(RSR∗ )S ∗ , A ∈ H2 , we may assume that φ(E11 ) = E11 and φ(E22 ) = E22 , and we have to prove that φ(tE11 + sE22 ) = f (t)E11 + g(s)E22 ,

t, s ∈ R,

where f, g : R → R are as above. It is obvious that if A, R ∈ H2 are two matrices with rank R = 1, then {A + tR : t ∈ R} is a set having the property that every two different members are adjacent. Moreover, if V ⊂ H2 is a maximal set of matrices with the property that every two different members are adjacent and if A is any member of V, then there exists a rank one matrix R ∈ H2 such that V = {A+tR : t ∈ R}. To see this consider the set of pairwise adjacent matrices {B − A : B ∈ V}, observe that 0 belongs to this set, and thus all nonzero members must be of rank one, and finally use the obvious fact that two distinct rank one hermitian matrices are adjacent if and only if they are linearly dependent. It follows that φ maps every “line”, that is, the set of the form {A + tR : t ∈ R}, where A ∈ H2 is any matrix and R is any rank one matrix, bijectively onto some line. In particular, there exist bijective functions f, g, h : R → R such that φ(tE11 ) = f (t)E11 , φ(tE22 ) = g(t)E22 , and φ(E11 + tE22 ) = E11 + h(t)R for every real number t. Here, R ∈ H2 is some rank one projection. Hence, s s(1 − s)eiϕ R= s(1 − s)e−iϕ 1−s for some real s, 0 ≤ s ≤ 1, and some real ϕ. We will show that s = 0. If s = 1, then R = E11 , and consequently, φ maps both lines RE11 and E11 + RE22 onto the linear span of E11 , which contradicts the bijectivity assumption. If s = 1 and s = 0, then because g is bijective and g(0) = 0, there exists a nonzero p ∈ R such that g(p) = (1 − s)/s. The matrices pE22 and E11 + pE22 are adjacent, and so are their images. It follows that the matrix 1 + h(p)s h(p) s(1 − s)eiϕ φ(E11 + pE22 ) − φ(pE22 ) = h(p) s(1 − s)e−iϕ h(p) − h(p)s − 1−s s

446

ˇ Semrl

IEOT

is of rank one. Consequently, its determinant must be zero. A straightforward computation shows that 1−s = 0, s a contradiction. Hence, s = 0, and therefore, R = E22 . We apply the fact that tE22 and E11 + tE22 are adjacent for every real t once more to conclude that h = g. In the same way we see that φ(tE11 + E22 ) = f (t)E11 + E22 , t ∈ R. In the sequel we will need the following statement. Let A, B ∈ H2 . Then det(A − B) = 0 ⇐⇒ det(φ(A) − φ(B)) = 0. Indeed, det(A − B) = 0 if and only if A = B or A is adjacent to B. Denote the matrix zE12 + zE21 by Az , z ∈ C. Assume that z = 0. We have det(tE11 − Az ) = 0 for every real t, and consequently, det(sE11 − φ(Az )) = 0 for all s ∈ R. It follows that the (2, 2)-entry of φ(Az ) is zero, and similarly, the (1, 1)-entry of this matrix must be zero. As φ−1 has the same properties as φ, we conclude that there exists a bijective function k : C → C such that φ(Az ) = Ak(z) , z ∈ C. Let t ∈ R and z ∈ C. Then det(tE11 + E22 − Az ) = 0 if and only if det(E11 + tE22 −Az ) = 0 if and only if t = |z|2 and this holds if and only if det(φ(tE11 +E22 )− φ(Az )) = 0 and this is then further equivalent to det(φ(E11 + tE22 ) − φ(Az )) = 0. It follows that f (t) ≥ 0 and t > 0, then φ √ f (t) = g(t) for all t ≥ 0. Moreover, if maps the set {Az : |z| = t} bijectively onto the set {Aw : |w| = f (t)}. Let D = tE11 +sE22 with ts > 0. Then d(D, Az ) = 1 for every complex z with |z|2 = ts. Hence, d(φ(D), Aw ) = 1 for every complex w with |w|2 = f (ts). It follows that φ(D) is diagonal. It is adjacent to both f (t)E11 and g(s)E22 . Consequently, φ(D) = f (t)E11 + g(s)E22 . Moreover, applying the fact that d(φ(D), Aw ) = 1 for every complex w with |w|2 = f (ts) we see that f (ts) = f (t)g(s) for all t, s ∈ R such that ts ≥ 0 (in the case when ts = 0 we use the fact that f (0) = g(0) = 0). In particular, if t, s > 0, then f (t)f (s) = f (ts) = f (−t)g(−s). As f is bijective and f ((0, ∞)) = (0, ∞) we have f (−t) < 0. Similarly, g(−s) < 0. It follows that there exists a positive real number a such that f (−t) = −af (t) and g(−s) = −a−1 g(s) = −a−1 f (s) for all t, s > 0. Let now D = tE11 + sE22 with ts < 0. We will prove that we have φ(D) = f (t)E11 + g(s)E22 in this case as well. All we need to do is to observe that the line through tE11 and tE11 + E22 contains D. This line is mapped into a line through the points f (t)E11 and f (t)E11 + E22 . Hence, φ(D) is a diagonal matrix adjacent to both f (t)E11 and g(s)E22 , and hence, φ(D) = f (t)E11 + g(s)E22 . Of course, this equality is valid also when ts = 0. This completes the proof of the first step. Let now P , Q, and R be three pairwise linearly independent positive rank one matrices. By the first step we have φ(tP ) = f (t)φ(P ), φ(tQ) = g(t)φ(Q), and φ(tR) = h(t)φ(R), t ∈ R, for some real bijective functions f , g, and h. Moreover, f (t) = g(t) = h(t) for every t ≥ 0, and there exist positive real numbers a, b, c such that 1 f (t) = −af (−t) and g(t) = − f (−t), t < 0, a

Vol. 62 (2008)


447

1 f (t) = −bf (−t) and h(t) = − f (−t), b

t < 0,

and

1 g(t) = −cf (−t) and h(t) = − f (−t), t < 0. c It follows that a = b, c = a1 , and c = b. Consequently, a = b = c = 1. Hence, we have improved the first step by showing that actually f (t) = g(t) and f (ts) = f (t)f (s) for all t, s ∈ R. After making once again all simplifications done in the first step we have φ(tE11 + sE22 ) = f (t)E11 + f (s)E22 ,

t, s ∈ R,

and φ ({Az : |z| = t}) = {Az : |z| = f (t)}. Moreover, we have φ(tA) = f (t)φ(A) for every real t and every matrix A of rank one. We will prove that this holds true also for invertible matrices A ∈ H2 . First observe that if A, B ∈ H2 are any invertible matrices with the property that for every rank one matrix R ∈ H2 the matrix AR is idempotent if and only BR is idempotent, then A = B. Indeed, the rank one matrix AR is idempotent if and only if its trace tr (AR) is equal to one. By linearity of the trace we have tr (AR) = tr (BR), or equivalently, tr ((A − B)R) = 0 for every rank one R ∈ H2 . It follows easily that A − B = 0, as desired. Next we show that if A, B ∈ H2 are any invertible matrices with the property that for every rank one matrix R ∈ H2 we have det(A − R) = 0 ⇐⇒ det(B − R) = 0, then A = B. Note that for an arbitrary rank one matrix R we have det(A − R) = 0 if and only if I − A−1 R is singular and because A−1 R is of rank one, this is true if and only if A−1 R is an idempotent. Hence, by the previous observation, our assumption yields that A−1 = B −1 which clearly implies the desired equality A = B. Thus, in order to prove that φ(tA) = f (t)φ(A) for every real t and every invertible A ∈ H2 we first observe that there is no loss of generality in assuming that t = 0 and then we have to show that for every rank one matrix R ∈ H2 we have det(φ(tA) − R) = 0 ⇐⇒ det(f (t)φ(A) − R) = 0. Let R be any rank one matrix. Then there exists a rank one matrix T with φ(T ) = R. We know that det(f (t)φ(A) − R) = 0 if and only if det(φ(A) − f (t−1 )φ(T )) = 0, which is equivalent to det(φ(A) − φ(t−1 T )) = 0, and this is further equivalent to det(A − t−1 T ) = 0. The last condition holds true if and only if det(tA − T ) = 0 and this holds true if and only if det(φ(tA) − R) = 0. Hence, φ(tA) = f (t)φ(A) for every real t and every A ∈ H2 . We know that φ ({Az : |z| = 1}) = {Az : |z| = 1}. In particular, φ(E12 +E21 ) = eiϕ E12 +e−iϕ E21 for some real number ϕ. Replacing φ by A → U φ(A)U ∗ , A ∈ H2 , where U = E11 +eiϕ E22 , we may, and will assume that φ(E12 + E21 ) = E12 + E21 , and consequently, φ(tE12 + tE21 ) = f (t)E12 + f (t)E21 , t ∈ R. Let t, s be any nonzero real numbers and set A = tE11 + sE12 + sE21 . Then d(pE11 , A) = 2 for every nonzero real p, and thus, d(qE11 , φ(A)) = 2 for every

448

ˇ Semrl

IEOT

nonzero real q. It follows that the (2, 2)-entry of φ(A) is zero. Applying first the 2 fact that A is adjacent to s(E12 + E21 ), and then to − st E22 we conlclude that φ(tE11 + sE12 + sE21 ) = f (t)E11 + f (s)E12 + f (s)E21 , t, s ∈ R. Similarly, φ(tE22 + sE12 + sE21 ) = f (t)E22 + f (s)E12 + f (s)E21 , t, s ∈ R. If t, s > 0, then (t + s)2 E11 + tE12 + tE21 is adjacent to −E22 − sE12 − sE21 , and consequently, f (t + s)2 = (f (t) + f (s))2 . It follows that f (t + s) = f (t) + f (s) for all t, s > 0. It is well-known that a nonzero additive multiplicative function f : (0, ∞) → (0, ∞) is the identity. Hence, f (t) = t, t > 0, and consequently, f (t) = t for all real numbers t. Let A = tE11 + sE22 + zE12 + zE21 be any 2 × 2 hermitian matrix and denote φ(A) = pE11 + qE22 + wE12 + wE21 . Let B = xE11 + yE22 , x, y ∈ R. We know that det(A − B) = 0 ⇐⇒ det(φ(A) − φ(B)) = 0, or equivalently, xy − sx − ty + (st − |z|2 ) = 0 ⇐⇒ xy − qx − py + (pq − |w|2 ) = 0,

x, y ∈ R.

It follows easily that p = t, q = s, and |w| = |z|. In the next step we will show that φ(A) = A for every A ∈ H2 with real entries. Let t, s, p be real numbers. We already know that φ(tE11 + p(E12 + E21 ) + sE22 ) = tE11 + zE12 + zE21 + sE22 for some z ∈ C with |z| = |p|. Applying the fact that tE11 + p(E12 + E21 )+ sE22 is adjacent to tE11 + p(E12 + E21 ) we conclude that z = p, as desired. We know that iE12 − iE21 is mapped into √ zE12 + zE21 for some complex z with |z| = 1. Because iE12 − iE21 is adjacent to 2I + E12 + E21 we have z = ±i. We have identified self-adjoint operators on the two-dimensional Hilbert space 2 2 C2 with hermitian matrices in the usual way. If T : C → C is a bijective z z conjugate-linear map defined by T = , and A : C2 → C2 is a self-adjoint w w operator and if we denote by the same symbol A the corresponding hermitian matrix, then the matrix corresponding to the self-adjoint operator T AT ∗ is At . Thus, after replacing φ by A → T φ(A)T ∗ , if necessary, we may, and will assume that φ(iE12 − iE21 ) = iE12 − iE21 . It follows that φ(itE12 − itE21 ) = itE12 − itE21 , t ∈ R. Let now A = tE11 + zE12 + zE21 + sE22 be an arbitrary hermitian 2 × 2 matrix. We want to show that φ(A) = A. This is true if z is real. So, assume z ∈ R. We know that φ(A) = tE11 + wE12 + wE21 + sE22 , where |w| = |z|. Using the fact that for every matrix B ∈ H2 with real entries we have det(A − B) = 0 ⇐⇒ det(φ(A) − B) = 0, we can easily deduce that either w = z, or w = z. If ts > |z|2 , then from det(A − B) = 0 ⇐⇒ det(φ(A) − B) = 0, where B is any matrix of the form ip(E12 − E21 ), p ∈ R, we conclude that z = w, that is, we have φ(A) = A for every A ∈ H2 satisfying det A > 0. If A = tE11 + zE12 + zE21 + sE22 with t = 0, then take B = tE11 + zE12 + zE21 + s1 E22 with s1 t > |z|2 and observe that det(A − B) = 0. It follows that det(φ(A) − B) = 0, and thus, w = z in this case as well. So, φ(A) = A for every A with at least one nonzero diagonal entry.

Vol. 62 (2008)


449

It follows trivially that φ(A) = A for all A ∈ H2 . This completes the proof in the two-dimensional case.

4. Reduction to the 2 × 2 case In the proof of our main theorem we will need the following simple lemma. Denote by Bs1+ (H) the set of all positive R ∈ Bs (H) of rank one. Lemma 4.1. Let H be a Hilbert space and A, B ∈ Bs (H) invertible positive operators. Then the following are equivalent: 1. A = B, 2. {R ∈ Bs1+ (H) : R ≤ A} = {R ∈ Bs1+ (H) : R ≤ B}. Proof. All we need to do is to prove that the second condition yields the first one. So, assume that positive invertible A and B satisfy this condition. Let R be a positive rank one operator. Then R = tP for some projection of rank one and some real t > 0. Let x be a unit vector that spans the image of P . Note that A ≥ R if and only if A − R ≥ 0 which is equivalent to I − A−1/2 (tP )A−1/2 ≥ 0. This last inequality is satisfied if and only if tA−1/2 x, A−1/2 x ≤ 1, or equivalently, A−1 x, x ≤ 1/t. Hence, the second condition can be reformulated as A−1 x, x ≤ 1/t ⇐⇒ B −1 x, x ≤ 1/t. Here, x is any unit vector and t any positive real number. It follows that A−1 x, x = B −1 x, x for every x ∈ H. Thus, A−1 = B −1 , and consequently, A = B, as desired. Let us assume that H, Bs (H), and φ : Bs (H) → Bs (H) are as in Theorem 1.1. We may, and will assume that φ(0) = 0. We will first prove that for every projection P ∈ Bs (H) of rank two there exists a projection Q ∈ Bs (H) of rank two such that φ maps P Bs (H)P bijectively onto QBs (H)Q. Of course, we can then apply the fact that both P Bs (H)P and QBs (H)Q are isomorphic to H2 and the result from the previous section to describe the form of the restriction of φ to P Bs (H)P . We know that φ(P ) is an operator of rank two comparable to zero. So, we have two possibilities and we will consider just one of them, say the one when φ(P ) ≤ 0. Let Q be a projection of rank two onto the image of φ(P ), that is, we have Qφ(P )Q = φ(P ). We will prove that φ−1 maps QBs (H)Q into P Bs (H)P . One can then use the same argument (even slightly more simple) to prove that φ maps P Bs (H)P into QBs (H)Q. Thus, we have to show that φ−1 (A) ∈ P Bs (H)P

450

ˇ Semrl

IEOT

for every A ∈ QBs (H)Q. We will first consider the case when A is of rank one. Hence A = tR for some projection R ∈ QBs (H)Q of rank one and some nonzero real t. We consider the continuous map s → φ(P ) + sR, s ∈ R. When s = 0, both nonzero eigenvalues of φ(P ) + sR are negative. If s is large enough, then by the continuity of spectra of finite rank operators, at least one nonzero eigenvalue of φ(P ) + sR is positive. Applying the continuity of the spectra once more, we see that there exists s ∈ R \ {0} such that φ(P ) − (−sR) has rank one. We fix this s. Thus, φ(P ) and −sR are adjacent, and so are P and φ−1 (−sR). Consequently, P is a sum of φ−1 (−sR) and some self-adjoint operator of rank one. By the remark at the end of Section 2 we see that the image of φ−1 (−sR) in contained in the image of P . Hence, φ−1 (−sR) ∈ P Bs (H)P . If t = −s, we are done. If not, then φ−1 (−sR) and φ−1 (tR) are adjacent rank one operators, and consequently, they are linearly dependent. So, φ−1 (A) ∈ P Bs (H)P in this case as well. Now, let A ∈ QBs (H)Q and rank A = 2. Then A = tR + sT for some nonzero real numbers t, s and rank one projections R, T ∈ QBs (H)Q. We know that φ−1 (tR) and φ−1 (sT ) are rank one operators belonging to P Bs (H)P that are linearly independent (because d(φ−1 (tR), φ−1 (sT )) = 2). Moreover, they are both adjacent to φ−1 (A). So, φ−1 (A) is a sum of φ−1 (tR) and some rank one operator, and thus, the image of φ−1 (tR) is contained in the image of φ−1 (A). Similarly, the image of φ−1 (sT ) is contained in the image of φ−1 (A). As the image of φ−1 (A) is two-dimensional and contains linearly independent one-dimensional images of φ−1 (tR) and φ−1 (sT ) that are contained in the image of P , we have φ−1 (A) ∈ P Bs (H)P in this case as well. A simple consequence of the first step of the proof is that we have either φ(R) ≥ 0 for every projection R of rank one, or φ(R) ≤ 0 for every projection R of rank one. Indeed, if R, Q are two projections of rank one, then there is a projection P of rank two such that P RP = R and P QP = Q. The restriction of φ to P Bs (H)P has the form as described in the previous section. In particular, it is either positive, or negative. Thus, either φ(R) ≥ 0 and φ(Q) ≥ 0, or φ(R) ≤ 0 and φ(Q) ≤ 0. We may, and will assume that φ(R) ≥ 0 for every projection R of rank one (otherwise, replace φ by −φ). As the restriction of φ to P Bs (H)P is real-linear for every two-dimensional projection P we conclude that φ maps the set of all positive rank one operators bijectively onto itself and the same is true for the set of all negative rank one operators. Next we show that for every A ∈ Bs (H) we have A ≥ 0 if and only if φ(A) ≥ 0. To check this it is enough to verify that A ≥ 0 if and only if A ∼ R for every negative rank one operator R ∈ Bs (H). Indeed, assume that A ∼ R for every negative rank one operator R ∈ Bs (H). We have to show that A ≥ 0. It is enough to see that A ≥ R for every negative rank one operator R ∈ Bs (H). Assume on the contrary that A ≤ S for some negative rank one operator S. If a real number t is large enough, then A ≤ tS, and therefore, tS ≤ A ≤ S. It follows that A is a negative rank one operator. But then we can find a negative rank one operator R such that A ∼ R, a contradiction.

Vol. 62 (2008)


451

We have proved that if ψ : Bs (H) → Bs (H) is any bijective map preserving comparability in both directions and satisfying ψ(0) = 0, then either ψ(A) ≥ 0 ⇐⇒ A ≥ 0

and ψ(A) ≤ 0 ⇐⇒ A ≤ 0,

or ψ(A) ≤ 0 ⇐⇒ A ≥ 0 and ψ(A) ≥ 0 ⇐⇒ A ≤ 0. Take any A ∈ Bs (H) and apply the above observation to the map ψ : Bs (H) → Bs (H) defined by ψ(X) = φ(X + A) − φ(A), X ∈ Bs (H), to conclude that either φ maps the set of all B ∈ Bs (H) satisfying B ≥ A bijectively onto the set of all C ∈ Bs (H) satisfying C ≥ φ(A) and φ maps the set of all B ∈ Bs (H) satisfying B ≤ A bijectively onto the set of all C ∈ Bs (H) satisfying C ≤ φ(A); or φ maps the set of all B ∈ Bs (H) satisfying B ≥ A bijectively onto the set of all C ∈ Bs (H) satisfying C ≤ φ(A) and φ maps the set of all B ∈ Bs (H) satisfying B ≤ A bijectively onto the set of all C ∈ Bs (H) satisfying C ≥ φ(A). If A ≥ 0, then φ(A) ≥ 0, and the set of all C ∈ Bs (H) satisfying C ≥ φ(A) contains only positive operators, while the set of all B ∈ Bs (H) satisfying B ≤ A contains all negative operators. Hence, the set of all B ∈ Bs (H) satisfying B ≤ A cannot be mapped by φ onto the set of all C ∈ Bs (H) satisfying C ≥ φ(A). It follows that 0 ≤ A ≤ B yields φ(A) ≤ φ(B). In the next step we will show that there exists a positive invertible A ∈ Bs (H) such that φ(A) is also invertible (and of course, positive). Assume that this is not true and let B ∈ Bs (H) be any positive operator. There exists a positive invertible A such that B ≤ A, and consequently, φ(B) ≤ φ(A). As φ(A) is singular, the operator φ(B) must be singular as well. This contradicts the fact that φ maps the set of positive operators bijectively onto itself. Hence, there is an invertible A ≥ 0 such that φ(A) is invertible. Replacing φ by X → φ(A)−1/2 φ(A1/2 XA1/2 )φ(A)−1/2 , X ∈ Bs (H), we may assume that φ(I) = I. A rank one positive operator R is a projection of rank one if and only if for every non-negative real number t we have tR ≤ I ⇐⇒ t ≤ 1. Hence, φ maps the set of projections of rank one onto itself. A positive operator Q of rank two is a projection if and only if Q ≤ I and if R is any rank two operator satisfying Q ≤ R ≤ I then R = Q. It follows that φ maps the set of projections of rank two onto itself. Let R and Q be orthogonal projections of rank one. Then P = R + Q is a projection of rank two. We know that the restriction of φ to P Bs (H)P is reallinear. Hence, φ(P ) = φ(R) + φ(Q). It is well-known and easy to verify that the sum of two rank one projections is a projection if and only if they are orthogonal. Hence φ as well as φ−1 map orthogonal projections of rank one into orthogonal projections of rank one. Let {Pα : α ∈ J} be a maximal subset of pairwise orthogonal projections of rank one. Let eα be a unit vector in the image of Pα for every α ∈ J. Then {eα : α ∈ J} is a complete orthonormal set in H. Denote Qα = φ(Pα ) for every α ∈ J. Clearly, {Qα : α ∈ J} is a maximal set of pairwise orthogonal projections of rank one. Let {fα : α ∈ J} be the corresponding complete orthonormal set.

ˇ Semrl

452

IEOT

Replacing the map φ by the map A → U φ(A)U ∗ , where U : H → H is a unitary operator defined by U fα = eα , α ∈ J, we may, and will assume that φ(Pα ) = Pα , α ∈ J. Choose and fix two different indices α, β ∈ J and consider the set of all operators from Bs (H) that map the linear span of {eα , eβ } into itself and are equal to zero on the orthogonal complement of this two-dimensional subspace. We know that this set of operators is invariant under φ. Moreover, this set of operators can be identified with H2 in a natural way. With this identification we have φ(E11 ) = E11 and φ(E22 ) = E22 . It follows that either φ(λE12 + λE21 ) = τ λE12 + τ λE21 , λ ∈ C, for some τ ∈ C of modulus one, or φ(λE12 + λE21 ) = τ λE12 + τ λE21 , λ ∈ C, for some τ ∈ C of modulus one. Let us consider only the case when H is an infinite-dimensional separable Hilbert space (the proof in the finite-dimensional case is easier and if J is not countable, we can complete the proof using minor and obvious technical modifications). So, we have the countable maximal set of pairwise orthogonal projections {Pn : n = 1, 2, . . .} and the corresponding countable complete orthonormal set {e1 , e2 , . . .}. For every positive integer n > 1 we have a complex number τn of modulus one corresponding to the pair e1 , en as above. Replacing φ once more by a map of the form A → V φ(A)V ∗ , where V is a unitary operator of the form V ek = µk ek , k = 1, 2, . . ., (here, each µk is an appropriate complex number of modulus one) we may, and will assume that τn = 1, n = 1, 2, . . . Hence, if n is any positive integer ≥ 2 and P any projection of rank one onto the subspace spanned by λe1 + µen , |λ|2 + |µ|2 = 1, then either φ(P ) = P , or φ(P ) is a projection of rank one onto the subspace spanned by λe1 + µen . Q be a projection of rank one onto the one-dimensional subspace spanned Let ∞ ∞ by k=1 λk ek , k=1 |λk |2 = 1, and assume that λk = 0 for every positive integer k. We may assume with no loss of generality that λ1 is real. As Q is not orthogonal to Pn , n = 1, 2, . . ., the same is true for φ(Q). Hence,φ(Q) is a rank one orthogonal ∞ projection onto a subspace spanned by some vector k=1 µk ek , where we may, and will assume that µ1 = λ1 and all the µk ’s are non-zero. The rank one projection Q is orthogonal to the rank one projection T onto the linear span of e1 −λ1 λ−1 2 e2 . Using the previous paragraph and the fact that φ maps orthogonal projections of rank one into orthogonal projections of rank one, we see that either µ2 = λ2 , or µ2 = λ2 . We use the same trick with an arbitrary positive integer instead of 2 to conclude that there exists a subset K ⊂ {2, 3, . . .} such that every projection ∞ Q of2 rank one onto the one-dimensional subspace spanned by ∞ k=1 λk ek , k=1 |λk | = 1, λ1 ∈ R, λk = 0 for every k = 1, 2, . . ., is mapped into a rank one projection spanned by the vector λk ek + λk ek . λ1 e1 + k∈K

k∈{2,3,...}\K

We will prove that either K = {2, 3, . . .} or K = ∅. Assume this is not the case, say 2 ∈ K and 3 ∈ K. Then we can find nonzero complex numbers b2 , a3 , b3

Vol. 62 (2008)


453

such that b2 + a3 b3 = 1 and b2 + a3 b3 = 1. We can further find ∞ nonzero real ∞ numbers a1 , a4 , a5 , . . . , b1 , b4 , b5 , . . . such that k=1 |ak |2 < ∞, k=1 |bk |2 < ∞, and a1 b1 + a4 b4 + a5 b5 + . . . = −1. Let R1 and R2 be rank one projections onto linear spans of a1 e1 + e2 + a3 e3 + a4 e4 + a5 e5 + . . . and ∞ k=1 bk ek , respectively. Then it is straightforward to see that R1 and R2 are orthogonal, while φ(R1 ) and φ(R2 ) are not, a contradiction. If K = ∅, then we replace φ by the map A

→ Jφ(A)J, where J is a conjugate∞ ∞ linear operator defined by J ( k=1 λk ek ) = k=1 λk ek . Hence, we may assume that K = {2, 3, . . .}. In other words, if P is any projection of rank one onto the linear span of a vector whose all coordinates with respect to the orthonormal basis {e1 , e2 , . . .} are non-zero, then φ(P ) = P . Applying the orthogonality preserving property and the fact that φ(Pn ) = Pn , n = 1, 2, . . ., we see that if Q is a projection of rank one onto the onedimensional subspace spanned by ∞ k=1 λk ek , and φ(Q) ∞is a projection of rank one onto the one-dimensional subspace spanned by k=1 µk ek , then for every positive integer k we have λk = 0 ⇐⇒ µk = 0. Using the same arguments as above we see that φ(Q) = Q in this case as well. It follows that φ(R) = R for every operator of rank one. Let A ≥ 0 be invertible. Then there exists a positive real number c such that A ≥ cP for every rank one projection P . It follows that φ(A) ≥ cP for every rank one projection P , and consequently, φ(A) is invertible as well. Let R ≥ 0 be a rank one operator. We know that R ≤ A if and only if R = φ(R) ≤ φ(A). By Lemma 4.1, φ(A) = A. Let A ∈ Bs (H) be an arbitrary operator. Choose a real number a such that A + aI is positive and invertible. Define ψa : Bs (H) → Bs (H) by ψa (X) = φ(X − aI) − φ(−aI), X ∈ Bs (H). Obviously, ψa satisfies all assumptions of Theorem 1.1. Moreover, ψa (0) = 0. Hence, by what we have already proved, there exist a bijective bounded linear or conjugate-linear map Sa : H → H and c ∈ {−1, 1} such that ψa (X) = cSa XSa∗ whenever X ≥ 0 is invertible. On the other hand, we have φ(X − aI) = X − aI whenever X − aI is positive and invertible. In particular, ctSa XSa∗ = ψa (tX) = tX − aI − φ(−aI) whenever both X and X − aI are positive invertible and t ≥ 1. Sending t to infinity we get cSa XSa∗ = X for every such X. It follows that c = 1 and Sa is the identity operator multiplied by a complex number of modulus one, and consequently, ψa (X) = X whenever X ≥ 0 is invertible. Hence φ(X) = φ((X + aI) − aI) = ψa (X + aI) + φ(−aI) = X + aI + φ(−aI) whenever X + aI is positive and invertible. If we choose X in such a way that both X and X + aI are positive invertible we get from φ(X) = X that aI + φ(−aI) = 0. Thus, φ(X) = X whenever X + aI is positive and invertible. In particular, φ(A) = A. This completes the proof.

454

ˇ Semrl

IEOT

References [1] G. Cassinelli, E. De Vito, P.J. Lahti, and A. Levrero, Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry transformations. Rev. Math. Phys. 9 (1997), 921–941. ˇ [2] W.-l. Huang and P. Semrl, Adjacency preserving maps on hermitian matrices. to appear in Canad. J. Math. [3] R.V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math. 56 (1952), 494–503. [4] J. Lester, A physical characterization of conformal transformations of Minkowski spacetime. Ann. Discrete Math. 18 (1983), 567–574. [5] L. Moln´ ar, Order-automorphisms of the set of bounded observables. J. Math. Phys. 42 (2001), 5904–5909. [6] L. Moln´ ar, Selected preserver problems on algebraic structures of linear operators and on function spaces. Springer-Verlag, 2007. [7] G.J. Murphy, C ∗ -algebras and operator theory. Academic Press, 1990. ˇ [8] H. Radjavi and P. Semrl, A short proof of Hua’s fundamental theorem of the geometry of hermitian matrices. Expositiones Math. 21 (2003), 83–93. [9] O.S. Rothaus, Order isomorphisms of cones. Proc. Amer. Math. Soc. 17 (1966), 1284– 1288. ˇ Peter Semrl Faculty of Mathematics and Physics University of Ljubljana Jadranska 19 SI-1000 Ljubljana Slovenia e-mail: [email protected] Submitted: November 29, 2007 Revised: July 1, 2008


auser Verlag Basel/Switzerland 0378-620X/040455-9, published online November 10, 2008 DOI 10.1007/s00020-008-1635-0


A Characterization of Schur Multipliers Between Character-Automorphic Hardy Spaces Daniel Alpay and Mamadou Mboup Abstract. We give a new characterization of character-automorphic Hardy spaces of order 2 and of their contractive multipliers in terms of de Branges Rovnyak spaces. Keys tools in our arguments are analytic extension and a factorization result for matrix-valued analytic functions due to Leech. Mathematics Subject Classification (2000). Primary 30F35, 46E22; Secondary 30B40. Keywords. Character-automorphic functions, Hardy spaces, de Branges Rovnyak spaces, Schur multipliers.

1. Introduction Let Γ be a Fuchsian group of Möbius transformations of the unit disk D = {z ∈ C; |z| < 1} onto itself. For 1 6 p 6 ∞ and for any character α of Γ, we consider the spaces Hpα = {f ∈ Hp | f ◦ γ = α(γ)f, ∀ γ ∈ Γ} . These spaces are called character-automorphic Hardy spaces. A characterization of such spaces in terms of Poincaré theta series may be found in [15], [18], [13], [9]. In particular, Pommerenke showed in [15] that the series b0 (z) X γ 0 (z) f (z) = 0 α(γ)θ(γ(z))h(γ(z)) (1.1) b0 (z) γ(z) γ∈Γ

defines a bounded linear operator from the classical Hardy space Hp (D) into the subspace Hpα (D). In the present paper we restrict ourselves to the case p = 2. We first give a characterization of the character-automorphic Hardy space H2α (D) in terms of an associated de Branges Rovnyak space of functions analytic in the open unit disk;

456

Alpay and Mboup

IEOT

see Theorem 3.2. We also characterize the contractive multipliers between H2βα (D) and H2α (D), where α and β two given characters; see Theorem 4.2. Our method is mainly based on analytic extension of positive kernels and factorization results from Nevanlinna-Pick interpolation theory.

2. A review on character-automorphic Hardy spaces 2.1. Fuchsian groups and automorphic functions Let G be a group of linear fractional transformations, T (z) = az+b cz+d , ad − bc = 1, in the complex plane and let ι denotes the identity transformation. Two points z and z 0 in C are said to be congruent with respect to G, if z 0 = T (z) for some T ∈ G and T 6= ι. Two regions R, R0 ⊂ C are said to be G-congruent or G-equivalent if there exists a transformation T 6= ι which sends R to R0 . A region R which does not contain any two G-congruent points and such that the neighborhood of any point on the boundary contains G-congruent points of R is called a fundamental region for G. A properly discontinuous group is a group G having a fundamental region [10]. This amounts to saying that the identity transformation is isolated. Definition 2.1. A Fuchsian group is a properly discontinuous group each of whose transformation maps D, T and C\D onto themselves. A Fuchsian group Γ is said to be of convergence type (see e.g. [15]) if X X 0 1 − |γ(z)|2 = 1 − |z|2 |γ (z)| < ∞ z ∈ D. γ∈Γ

γ∈Γ

Then, the Green’s function [15] of Γ with respect to a point ξ ∈ D is defined as the Blaschke product Y γ(ξ) − z |γ(ξ)| bξ (z) = . (2.1) 1 − γ(ξ)z γ(ξ) γ∈Γ

It satisfies bξ (ϕ(z)) = µξ (ϕ)bξ (z),

∀ϕ ∈ Γ,

(2.2)

where µξ is the character of Γ associated with bξ (z). A function satisfying the relation (2.2) is said to be character-automorphic with respect to Γ while a Γperiodic function, as for example |bξ (z)| = |bξ (ϕ(z))|, is called automorphic with respect to Γ. 2.2. Spaces of character-automorphic functions We now briefly mention the main properties pertaining to spaces of characterautomorphic functions. The materials presented here are essentially borrowed from b be the dual group of Γ, i.e. the group of [15] and [19] (see also [11, 18]). Let Γ

Vol. 62 (2008)

Schur Multipliers in Character-Automorphic Hardy Spaces

457

b associate the subspaces (unimodular) characters. For an arbitrary character α ∈ Γ, of the classical space L2 (T) Lα 2 = {f ∈ L2 | f ◦ γ = α(γ)f, ∀γ ∈ Γ}, \ α H2 (D) = Lα H2 (D). 2 Let Γ be a Fuchsian group without elliptic and parabolic elements. We say that Γ is of Widom type if, and only if, the derivative of b0 (z) is of bounded characteristic. In α this case, Widom [20] has shown that the space H∞ is not trivial for any character b and we have α∈Γ Theorem 2.2 (Pommerenke [15]). Let Γ be of Widom type and let θ(z) be the inner factor of b00 (z). If α is any character of Γ and if h(z) is in Hp (D), 1 6 p 6 ∞, then the function defined by (1.1) is in Hpα (D) and kf kp 6 khkp ,

f (0) = θ(0)h(0).

The Poincaré series [14] in (1.1) thus defines, in particular, a projection: P α : θH2 (D) → H2α (D). An important property of the space H2α (D) that we will need is that point evaluation f 7→ f (ξ), ξ ∈ D is a bounded linear functional. The space therefore admits a reproducing kernel k α : hf (z), k α (z, ξ)iHα2 (D) = f (ξ), ξ ∈ D, with k α (z, ξ) ∈ H2α (D),

for all f ∈ H2α (D)

∀ ξ ∈ D.

Since H2α (D) 6= {const}, we have k α (ξ, ξ) = kk α (·, ξ)k2Hα (D) > 0 for every ξ ∈ D. 2 In the sequel, the Green’s function b0 (z) with respect to 0, will be denoted by b(z) for short. Let Γ be a group of Widom type and let E be associated1 to it in such a way that C\E be equivalent to the Riemann surface D/Γ, obtained by identifying Γcongruent points. Then, there exists a universal covering map z : D → C\E ' D/Γ such that • z maps D conformally onto C\E, • z is automorphic with respect to Γ: z ◦ γ = z, ∀ γ ∈ Γ, • and z(z1 ) = z(z2 ) ⇒ ∃γ ∈ Γ | z1 = γ(z2 ). In particular, z maps one-to-one the normal fundamental domain of Γ with respect to the origin, F = {z ∈ D : |γ 0 (z)| < 1

for all γ ∈ Γ, γ 6= ι}

(2.3)

conformally onto some sub-domain of C\E. We assume that z is normalized so that (zb)(0) is real and positive. In all the sequel, the character associated to the Green’s function b(z) will be denoted by µ. The starting point of the next section is the following result: 1 See

[21] for an example of a construction of a group Γ associated with a finite union of disjoint arcs of the unit circle.

458

Alpay and Mboup

IEOT

Lemma 2.3 ([12]). The reproducing kernel for the space H2α (D) has the form ∗ αµ (ω,0) kαµ (z,0) α k (ω, 0)∗ − k b(ω) k α (z, 0) b(z) (2.4) k α (z, ω) = c(α) z(z) − z(ω)∗ where c(α) =

z(0)b(0) > 0. k αµ (0, 0)

(2.5)

3. An associated de Branges-Rovnyak space In this section we give a characterization of the space H2α (D) in terms of an associated de Branges Rovnyak space. To begin, let Ω+ = {z ∈ D ; Im z(z) > 0} . Setting k αµ (z, 0) α + ik (z, 0) , b(z) αµ p k (z, 0) α α − ik (z, 0) B (z) = c(α) b(z) p A (z) = c(α) α

we can rewrite the reproducing kernel k α as k α (z, ω) =

Aα (z) 1 − Sα (z)Sα (w)∗ Aα (w)∗ 1 − iz(z) 1 − σ(z)σ(w)∗ 1 + iz(w)∗

(3.1)

1+iz(z) where Sα (z) = B α (z)/Aα (z) and σ(z) = 1−iz(z) . We note that the functions Aα (z) α and B (z) are character-automorphic with the same character α while Sα and σ are automorphic functions. From now on, the notation f ν (z) will means that the b being the function f ν (z) is character-automorphic with the superscript ν ∈ Γ associated character, and the notation fν (z) will stand for a function depending on the character ν (automorphic or not).

Proposition 3.1. There exists a Schur function Sα such that Sα (z) = Sα (σ(z)). Proof. Since the kernel k α (z, ω) is positive in D, and hence in Ω+ , it is clear ∗ α (z)Sα (w) that 1−S 1−σ(z)σ(w)∗ is also positive in Ω+ . Now, observe that the function σ maps Ω+ ∩ F into some subset ∆ ⊂ D and this mapping is one-to-one. Let ς be given by: (ς ◦ σ)(z) = z, ∀z ∈ Ω+ ∩ F (in particular this will also hold for any region fα be defined on ∆ by: congruent to Ω+ ∩ F) and let the function S fα (λ) = (Sα ◦ ς)(λ), ∀λ ∈ ∆. S Then it comes that the kernel fα (λ)S fα (µ)∗ 1−S 1 − λµ∗

Vol. 62 (2008)


459

is positive on ∆. Now, this implies (see for instance [1, Theorem 2.6.5]) the exfα (λ), analytic and contractive in all D. We istence of a unique extension of S subsequently call Sα (λ) this extension, and denote by H(Sα ) the reproducing kernel Hilbert space with reproducing kernel KSα (λ, µ) =

1 − Sα (λ)Sα (µ)∗ . 1 − λµ∗

By construction, the equality Sα (z) = Sα (σ(z)) holds for all z ∈ Ω+ ∩ F. Since Sα (z) is analytic in D, it must also hold for all D. In connection with the previous proposition and the next theorem, we recall that reproducing kernel Hilbert spaces H(S ) of functions analytic in the open unit disk and with a reproducing kernel of the form 1 − S (λ)S (µ)∗ 1 − λµ∗ were introduced and studied by de Branges and Rovnyak; see [5, Appendix], [6]. We also refer the reader to [8] and [1] for more information on these and on related spaces. Theorem 3.2. The character-automorphic Hardy space H2α (D) can be described as Aα (z) α H2 (D) = F (z) = f (σ(z)) ; f ∈ H(Sα ) (3.2) 1 − iz(z) with the norm kF kHα2 (D) = kf kH(Sα ) . Proof. Recall that the map which to F ∈ H2α (D) associates its restriction F |Ω+ to Ω+ is an isometry from H2α (D) onto the reproducing kernel Hilbert space with reproducing k α (z, ω) defined by (2.4), where z, ω are now restricted to Ω+ . We denote this last space by H2α (D) Ω+ . By Proposition 3.1 and using (3.1) we see α

A (z) that the operator of multiplication by 1−iz(z) is an isometry from the reproducing kernel Hilbert space H with reproducing kernel

1 − Sα (σ(z))Sα (σ(w)∗ 1 − σ(z)σ(w)∗ onto H2α (D) Ω+ . Furthermore, the composition map by σ is an isometry from the de Branges Rovnyak space H(Sα ) onto H. We have that H = {f ◦ σ ; f ∈ H(Sα )} , with norm kf ◦ σkH = kf kH(Sα ) , as follows from the equalities f (σ(ω)) = hf (·), KSα (·, σ(ω))iH(Sα ) = hf ◦ σ(·), KSα (σ(·), σ(ω))iH .

460

Alpay and Mboup

IEOT

Thus the restrictions of the elements of H2α (D) to Ω+ are of the form as in (3.2) for z restricted to Ω+ . By analytic extension, the elements of H2α (D) have the same form in the whole of D.

4. Schur multipliers A Schur function is a function analytic and contractive in the open unit disk. Equivalently, it is a function s such that the operator of multiplication by s is a contraction from the classical Hardy of the open unit disk into itself. This last definition is our starting point to define character-automorphic Schur multipliers. Definition 4.1. A character-automorphic function sβ (z), with character β, will be called a Schur multiplier if the operator of multiplication by sβ (z) is a contraction from H2βα (D) into H2α (D). Equivalently, the character-automorphic function sβ (z) is a Schur multiplier if and only if the kernel Ksαβ (z, w) = k α (z, w) − sβ (z)sβ (w)∗ k βα (z, w)

(4.1)

is positive in D. The kernel Ksβ (z, w) is in particular positive in Ω+ , and we will consider it in Ω+ . We note that in view of [7, Lemma 2, p. 142], [4, Theorem 1.1.4, p. 10], the positivity of the analytic kernel Ksβ on Ω+ implies its positivity on D. Theorem 4.2. A character-automorphic function sβ is a Schur multiplier if and only if there exists a C2×2 –matrix valued Schur function Σ(z) such that sβ (z) =

Aα (z) Aβα (z)

Σ12 (σ(z)) , 1 − Sβα (z)Σ22 (σ(z))

Sα (z) = Σ11 (σ(z)) +

Σ12 (σ(z))Sβα (z)Σ21 (σ(z)) 1 − Sβα (z)Σ22 (σ(z))

(4.2) .

(4.3)

Proof. The positivity of the kernel (4.1) in Ω+ is equivalent to the positivity in Ω+ of the kernel KSα (σ(z), σ(ω)) − T (z)T (ω)∗ KSβα (σ(z), σ(ω)), where T (z) =

Aβα (z) β s (z). Aα (z)

As in the proof of Proposition 3.1, we note that the function σ maps Ω+ ∩ F into some subset ∆ ⊂ D and this mapping is one-to-one, and consider again the function ς defined by: (ς ◦ σ)(z) = z, ∀z ∈ Ω+ ∩ F. The kernel KSα (λ, µ) − T (ς(λ))T (ς(µ))∗ KSβα (λ, µ)

(4.4)

Vol. 62 (2008)


461

is positive in ∆. We now show that T ◦ ς admits an analytic extension to D. To that purpose we consider the linear relation in H(Sα ) × H(Sβα ) spanned by the elements of the form (KSα (·, ω), T (ς(ω))∗ KSβα (·, ω)),

ω ∈ ∆.

It is densely defined and it is contractive because of the positivity of the kernel e We note (4.4) in ∆. It is therefore the graph of a densely defined contraction X. its extension to H(Sα ) by X. For ω ∈ ∆ and f ∈ H(Sα ) we have (X ∗ f )(ω) = hX ∗ f, KSα (·, ω)iH(Sα ) = hf, T (ς(ω))∗ KSβα (·, ω)iH(Sα ) = T (ς(ω))f (ω). Let f0 (λ) = KSβα (λ, ω0 ) where ω0 ∈ D. We have T (ς(λ)) =

(X ∗ f0 )(λ) , f0 (λ)

λ ∈ ∆.

It follows that T ◦ ς has an analytic extension to D, which we will denote by R. Thus the kernel (4.5) KSα (λ, µ) − R(λ)R(µ)∗ KSβα (λ, µ), is analytic in λ and µ∗ in D. Therefore it is still positive in D; see [7, Lemma 2, p. 142], [4, Theorem 1.1.4, p. 10]. By [2, Theorem 11.1, p. 61], a necessary and sufficient condition for the kernel (4.5) to be positive is that there exists a C2×2 – matrix valued Schur function Σ(λ) such that R(λ) =

Σ12 (λ) , 1 − Sβα (λ)Σ22 (λ)

Sα (λ) = Σ11 (λ) +

Σ12 (λ)Sβα (λ)Σ21 (λ) 1 − Sβα (λ)Σ22 (λ)

(4.6) .

(4.7)

The above mentioned result from [2] stems from rewriting the positive kernel (4.5) as A(λ)A(µ)∗ − B(λ)B(µ)∗ 1 − λµ∗ where A(λ) = 1 R(λ)Sβα (λ) , B(λ) = Sα (λ) R(λ) , and using a factorization result known as Leech’s theorem, which insures the existence of a C2×2 -valued Schur function Σ such that B(z) = A(z)Σ(z), from which (4.6) follows. This unpublished result of R.B. Leech has been proved using the commutant lifting theorem by M. Rosenblum; see [16, Theorem 2, p. 134] and [17, Example 1,

462

Alpay and Mboup

IEOT

p. 107]. Further discussions and applications can also be found in [3]. It can also be proved using tangential Nevanlinna-Pick interpolation and Montel’s theorem. Finally we replace in (4.6) λ by σ(z) where z ∈ Ω+ ∩ F. We obtain the formulas in the statement of the theorem for z ∈ Ω+ ∩ F, and hence for z ∈ D by analytic extension. Acknowledgments The first author thanks the Earl Katz family for endowing the chair which supports his research. The second author thanks the Center for Advanced Studies in Mathematics (CASM) of the Department of Mathematics of Ben-Gurion University which supported his stay at Ben-Gurion University.

References [1] D. Alpay. Algorithme de Schur, espaces ` a noyau reproduisant et théorie des systèmes, volume 6 of Panoramas et Synthèses. Société Mathématique de France, Paris, 1998. [2] D. Alpay and V. Bolotnikov. On tangential interpolation in reproducing kernel Hilbert space modules and applications. In H. Dym, B. Fritzsche, V. Katsnelson, and B. Kirstein, editors, Topics in interpolation theory, volume 95 of Operator Theory: Advances and Applications, pages 37–68. Birkh¨ auser Verlag, Basel, 1997. [3] D. Alpay, P. Dewilde, and H. Dym. On the existence and construction of solutions to the partial lossless inverse scattering problem with applications to estimation theory. IEEE Trans. Inform. Theory, 35:1184–1205, 1989. [4] D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo. Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, volume 96 of Operator Theory: Advances and Applications. Birkh¨ auser Verlag, Basel, 1997. [5] L. de Branges and J. Rovnyak. Canonical models in quantum scattering theory. In C. Wilcox, editor, Perturbation theory and its applications in quantum mechanics, pages 295–392. Wiley, New York, 1966. [6] L. de Branges and J. Rovnyak. Square summable power series. Holt, Rinehart and Winston, New York, 1966. [7] W.F. Donoghue. Monotone matrix functions and analytic continuation, volume 207 of Die Grundlehren der mathematischen Wissennschaften. Springer–Verlag, 1974. [8] H. Dym. J–contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989. [9] C.J. Earle and A. Marden. On Poincaré series with application to H p spaces on bordered Riemann surfaces. Illinois J. Math., 13:202–219, 1969. [10] L.R. Ford. Automorphic functions. Chelsea, 2nd edition, 1951. [11] M. Hasumi. Hardy classes on infinitely connected Riemann surfaces, volume 1027 of Lecture notes in mathematics. Springer–Verlag, 1983.

Vol. 62 (2008)


463

[12] S. Kupin and P. Yuditskii. Analogs of the Nehari and Sarason theorems for characterautomorphic functions and some related questions. In Topics in interpolation theory (Leipzig, 1994), volume 95 of Operator Theory: Advances and Applications, pages 373–390. Birkh¨ auser Verlag, Basel, 1997. [13] T.A. Metzger and K.V. Rajeswara Rao. Approximation of Fuchsian groups and automorphic forms of dimension −2. Indiana Univ. Math. J., 21:937–949, 1971/72. [14] H. Poincaré. Sur l’uniformisation des fonctions analytiques. Acta Math., 31(1):1–63, 1908. [15] Ch. Pommerenke. On the Green’s function of Fuchsian groups. Ann. Acad. Sci. Fenn. Ser. A I Math., 2:409–427, 1976. [16] M. Rosenblum. A corona theorem for countably many functions. Integral Equations and Operator Theory, 3:125–137, 1980. [17] M. Rosenblum and J. Rovnyak. Hardy classes and operator theory. Birkh¨ auser Verlag, Basel, 1985. [18] M.V. Samokhin. Some classical problems in the theory of analytic functions in domains of Parreau-Widom type. Mat. Sb., 182(6):892–910, 1991. [19] M. Sodin and P. Yuditskii. Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of characterautomorphic functions. J. Geom. Anal., 7(3):387–435, 1997. [20] H. Widom. Hp sections of vector bundles over Riemann surfaces. Ann. of Math. (2), 94:304–324, 1971. [21] P. Yuditskii. Two remarks on Fuchsian groups of Widom type. In Operator theory, system theory and related topics (Beer-Sheva/Rehovot, 1997), volume 123 of Operator Theory: Advances and Applications, pages 527–537. Birkh¨ auser, Basel, 2001. Daniel Alpay Department of Mathematics Ben-Gurion University of the Negev POB 653, Beer-Sheva 84105 Israel e-mail: [email protected] Mamadou Mboup UFR Math-Info Université Paris Descartes 45 rue des Saints-Pères 75270 Paris Cedex 06 France e-mail: [email protected] Submitted: March 10, 2008.

Integr. equ. oper. theory 62 (2008), 465–488 0378-620X/040465-24, DOI 10.1007/s00020-008-1637-y c 2008 Birkh¨

auser Verlag Basel/Switzerland


Convergence of the Iterated Aluthge Transform Sequence for Diagonalizable Matrices II: λ -Aluthge Transform Jorge Antezana, Enrique Pujals and Demetrio Stojanoff Abstract. Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U |T |. Then the λ-Aluthge transform is defined by ∆λ (T ) = |T |λ U |T |1−λ . Let ∆n λ (T ) denote the n-times iterated Aluthge transform of T , n ∈ N. We prove that the sequence {∆n λ (T )}n∈N converges for every r × r diagonalizable matrix T . We show regularity results for the two parameter map (λ, T ) 7→ ∞ ∆∞ λ (T ), and we study for which matrices the map (0, 1) 3 λ 7→ ∆λ (T ) is constant. Mathematics Subject Classification (2000). Primary 37D10; Secondary 15A60. Keywords. Aluthge transform, stable manifold theorem, similarity orbit, polar decomposition.

1. Introduction Let H be a complex Hilbert space, and let L(H) be the algebra of bounded linear operators on H. Given T ∈ L(H), consider its (left) polar decomposition T = U |T |. In order to study the relationship among p-hyponormal operators, Aluthge introduced in [1] the transformation ∆1/2 (·) : L(H) → L(H) defined by ∆1/2 (T ) = |T |1/2 U |T |1/2 . Later on, this transformation, now called Aluthge transform, was also studied in other contexts by several authors, such as Jung, Ko and Pearcy [14] and [15], The first and third author were partially supported by CONICET (PIP 4463/96), Universidad de La Plata (UNLP 11 X472) and ANPCYT (PICT03-09521). The second author was partially supported by CNPq.

466

Antezana, Pujals and Stojanoff

IEOT

Ando [2], Ando and Yamazaki [3], Yamazaki [20], Okubo [16] and Wu [19] among others. In this paper, given λ ∈ (0, 1) and T ∈ L(H), we study the so-called λ-Aluthge transform of T defined by λ

1−λ

∆λ (T ) = |T | U |T |

.

This notion has already been considered by Okubo in [16] (see also [4] and [13]). We denote by ∆nλ (T ) the n-times iterated λ-Aluthge transform of T , i.e. ∆0λ (T ) = T ; and ∆nλ (T ) = ∆λ ∆n−1 (T ) n ∈ N. (1.1) λ In a previous paper [5], we showed that the iterates of the usual Aluthge transform ∆n1/2 (T ) converge to a normal matrix ∆∞ 1/2 (T ) for every diagonalizable matrix T ∈ Mr (C) (of any size). We also proved in [5] the smoothness of the map T 7→ ∆∞ 1/2 (T ) when it is restricted to a similarity orbit, or to the (open and dense) set ∗ Dr (C) of invertible r × r matrices with r different eigenvalues. The key idea was to use a dynamical systems approach to the Aluthge transform, thought as acting on the similarity orbit of a diagonal invertible matrix. Recently, Huajun Huang and Tin-Yau Tam [13] showed, with another approach, that the iterates of every λ-Aluthge transform ∆nλ (T ) converge, for every matrix T ∈ Mr (C) with all its eigenvalues of different moduli. In this paper, we study the general case of λ-Aluthge transforms by means of a dynamical systems approach. On one side, this allows us to generalize Huajun Huang and Tin-Yau Tam result to every diagonalizable matrix T ∈ Mr (C). On the other side, it allows us to get some regularity results for the two parameter n map (λ, T ) 7→ ∆∞ λ (T ) = limn∈N ∆λ (T ). Now we briefly describe the dynamical point of view of the problem: For every λ ∈ (0, 1) and any invertible matrix T , it holds that ∆λ (T ) = |T |λ T |T |−λ . So the λ-Aluthge transform of T belongs to the similarity orbit of T . This suggests that we can study the Aluthge transform restricted to the similarity orbit of some invertible operator. From that point of view, the diagonalizable case has a better dynamical behavior, as detailed in [5]. If T is diagonalizable, the similarity orbit of T coincides with the similarity orbit of some diagonal operator D, which we denote S (D). The unitary orbit of D, denoted by U (D), is a compact submanifold of S (D) that consists of all normal matrices in S (D). Hence U (D) is fixed by the Aluthge transform and, as it was shown in [4], all the limit points of the sequence {∆nλ (T )}n∈N belong to U (D). As it was shown in [5] for λ = 1/2, we show that for any N ∈ U (D) there is a s local submanifold Wλ,N transversal to U (D) characterized as the set the matrices (near N ) that converges with an exponential rate to N by the iteration of the λ-Aluthge transform. Moreover, the union of these submanifolds form an open neighborhood of U (D) (see Corollary 3.2.2). Thus, since the sequence {∆nλ (T )}n∈N goes toward U (D), for some n0 large enough the sequence of iterated Aluthge transforms gets into this open neighborhood and converges exponentially.

Vol. 62 (2008)

Convergence of the Iterated λ-Aluthge Transform Sequence

467

These results follow from the classical arguments of stable manifolds (first introduced independently by Hadamard and Perron, see Theorem 2.1.3; for details and general results about the stable manifold theorem, see [11] or the Appendix of [5]). In order to use the stable manifold theorem, we show previously that the derivative of the λ-Aluthge transform in any N ∈ U (D) has two invariant complementary directions, one tangent to U (D), and other one transversal to it where the derivative is a contraction (see Theorem 3.2.1). Using these techniques we prove that ∆nλ (T ) −−−−→ ∆∞ λ (T ) ∈ U(r), for every r × r diagonalizable matrix n→∞

T . We also prove that the two parameter map ∆∞ (λ, T ) = ∆∞ λ (T ) is of class C ∞ , when restricted to (0, 1) × Dr∗ (C) and to (0, 1) × S (D) for a diagonal matrix D ∈ Mr (C). We also study for which matrices T the map (0, 1) 3 λ 7→ RT (λ) = ∆∞ λ (T ) is constant. Some partial results are obtained, in particular that RT is not constant for most diagonalizable matrices T . We also show that RT is constant for every T in the similarity orbit of a diagonal matrix D if σ(D) = {d1 , d2 } with |d1 | = |d2 |. We state and discuss the following conjecture: this is the unique case where RT is constant for every T in the similarity orbit of D. The paper has a structure very similar to [5] because, at any step of the dynamical systems argument, we need to state several results which differ slightly of those results of [5]. The proofs of these results are omitted or just sketched. The paper is organized as follows: in Section 2, we collect several preliminary definitions and results about the stable manifold theorem, about the geometry of similarity and unitary orbits, and about known results on λ-Aluthge transforms. In Section 3, we prove the convergence results. In Section 4 we study the smoothness of the two parameter map (λ, T ) 7→ ∆∞ λ (T ) and we study the behavior of the limit function (T ) with respect to the parameter λ. The basic tool for these results, in order ∆∞ λ to apply the stable manifold theorem to the similarity orbit of a diagonal matrix, is the mentioned Theorem 3.2.1, whose proof, rather technical, is done in Section 5.

2. Preliminaries In this paper Mr (C) denotes the algebra of complex r × r matrices, Gl r (C) the group of all invertible elements of Mr (C), U(r) the group of unitary operators, and Mhr (C) (resp. Mah r (C) ) denotes the real algebra of Hermitian (resp. antiHermitian) matrices. Given T ∈ Mr (C), R(T ) denotes the range or image of T , ker T the null space of T , σ(T ) the spectrum of T , rk T = dim R(T ) the rank of T , tr(T ) the trace of T , and T ∗ the adjoint of T . If v ∈ Cr , we denote by diag(v) ∈ Mr (C) the diagonal matrix with v in its diagonal. We consider Mr (C) as a real Hilbert space with the inner product defined by hA, Bi = Re tr(B ∗ A) . The norm induced by this inner product is the so-called Frobenius norm, denoted by k · k2 .

468


IEOT

On the other hand, let M be a manifold. By means of T M we denote the tangent bundle of M and by means of Tx M we denote the tangent space at the point x ∈ M . Given a function f ∈ C r (M ), where r = 1, . . . , ∞, Tx f (v) denotes the derivative of f at the point x ∈ M applied to the vector v ∈ Tx M . 2.1. Stable manifold theorem In this section we state the stable manifold theorem for an invariant set of a smooth endomorphism (see 2.1.3 below). We refer to [5] for more details on these type of results. Let M be a smooth Riemann manifold and N ⊆ M a submanifold (not necessarily compact). Throughout this subsection TN M denotes the tangent bundle of M restricted to N . Definition 2.1.1. A C r pre-lamination indexed by N is a continuous choice of a C r embedded disc Bx through each x ∈ N . Continuity means that N is covered by open sets U in which x → Bx is given by Bx = σ(x)((−ε, ε)k ) where σ : U ∩ N → Embr ((−ε, ε)k , M ) is a continuous section. Note that the set Embr ((−ε, ε)k , M ) is a C r fiber bundle over M whose projection is β → β(0). Thus σ(x)(0) = x. If the sections mentioned above are C s , 1 ≤ s ≤ r, we say that the C r pre-lamination is of class C s . Definition 2.1.2. Let f be a smooth endomorphism of M , ρ > 0, and suppose that f |N is a homeomorphism. Then, N is ρ-pseudo hyperbolic for f if there exist two smooth subbundles of TN M , denoted by E s and F, such that TN M = E s ⊕ F; TN = F; both E s and F are T f -invariant; T f restricted to F is an automorphism, which expand it by a factor greater than ρ; 5. Tx f : Exs → Efs(x) has norm lower than ρ.

1. 2. 3. 4.

In this case, the stable manifold theorem assures that for any point x ∈ N it is possible to find an f -invariant submanifold transversal to N tangent to E s and characterized as the set of points with trajectories asymptotic to the trajectory of x. We shall state the following particular version of this theorem. For a proof, see Theorem 2.1.4, Corollary 2.1.5 and Remark 2.1.6 of [5]. Theorem 2.1.3 (Stable manifold theorem for a submanifold of fixed points). Let f be a C r endomorphism of M with a ρ-pseudo hyperbolic submanifold N with ρ < 1. Assume that any point p in N is a fixed point. Then, there is a f -invariant C r -pre-lamination W s : N → Embr ((−1, 1)k , M ) of class C r such that, for every x ∈ N: 1. W s (x)(0) = x. 2. Wxs = W s (x)((−1, 1)k ) is tangent to Exs at every x ∈ N .

Vol. 62 (2008)


469

3. There exists an open neighborhood U of x (open relative to M ) such that o n (2.1) Wxs ∩ U = y ∈ U : dist(x, f n (y)) < dist(x, y) ρn . 4. If y ∈ N , x 6= y, then Wxs ∩ Wys = ∅. 5. There exists γ > 0 such that [ B(x, γ) ⊂ Wxs . x∈N

This implies that

[

Wxs

contains an open neighborhood W(N ) of N in M .

x∈N

6. The map p : W(N ) → N

given by

p(a) = x

if

a ∈ Wxs (x),

(2.2)

r

is well defined and it is of class C . 2.2. Similarity orbit of a diagonal matrix In this subsection we recall some facts about the similarity orbit of a diagonal matrix. Definition 2.2.1. Let D ∈ Mr (C) be diagonal. The similarity orbit of D is the set S (D) = { SDS −1 : S ∈ Gl r (C) }. On the other hand, U (D) = { U DU ∗ : U ∈ U(r) } denotes the unitary orbit of D. We donote by πD : Gl r (C) → S (D) ⊆ Mr (C) the C ∞ map defined by πD (S) = SDS −1 . With the same name we note its restriction to the unitary group: πD : U(r) → U (D). Proposition 2.2.2. The similarity orbit S (D) is a C ∞ submanifold of Mr (C), and the projection πD : Gl r (C) → S (D) becomes a submersion. Moreover, U (D) is a compact submanifold of S (D), which consists of the normal elements of S (D), and πD : U(r) → U (D) is a submersion. In particular, the maps πD have C ∞ (similarity and unitary) local cross sections. For every N = U DU ∗ ∈ U (D), it is well known (and easy to see) that TN S (D) = TI (πN )(Mr (C) ) = {[A, N ] = AN − N A : A ∈ Mr (C)}. In particular TD S (D) = {AD − DA : A ∈ Mr (C)} = {X ∈ Mr (C) : Xij = 0 for every (i, j) such that di = dj }.

(2.3)

Note that, TN S (D) = {[A, N ] = AN − N A : A ∈ Mr (C)} = {(U BU ∗ )U DU ∗ − U DU ∗ (U BU ∗ ) : B ∈ Mr (C)} = {U [B, D]U ∗ = BD − DB : B ∈ Mr (C)} = U TD S (D) U ∗ . (2.4)

470


IEOT

∗ On the other hand, since TI U(r) = Mah r (C) = {A ∈ Mr (C) : A = −A} , we obtain ah TD U (D) = TI (πD )(Mah r (C) ) = {[A, D] = AD − DA : A ∈ Mr (C)} ∗ TN U (D) = {[A, N ] = AN − N A : A ∈ Mah r (C)} = U TD U (D) U .

and (2.5)

Finally, along this paper we shall consider on S (D) (and in U (D)) the Riemannian structure inherited from Mr (C) (using the usual inner product on their tangent spaces). For S, T ∈ S (D), we denote by dist(S, T ) the Riemannian distance between S and T (in S (D) ). Observe that, for every U ∈ U(r), one has that U S (D) U ∗ = S (D) and the map T 7→ U T U ∗ is isometric, on S (D), with respect to the Riemannian metric as well as with respect to the k · k2 metric of Mr (C). 2.3. λ-Aluthge transforms Definition 2.3.1. Let T ∈ Mr (C), and suppose that T = U |T | is the polar decomposition of T . Let λ ∈ (0, 1). Then, we define the λ-Aluthge transform of T in the following way: λ

1−λ

∆λ (T ) = |T | U |T |

We denote by ∆nλ (T ) the n-times iterated λ-Aluthge transform of T , i.e. ∆0λ (T ) = T and ∆nλ (T ) = ∆λ ∆n−1 (T ) n ∈ N. λ The following proposition contains some properties of λ-Aluthge transforms which follow easily from its definition. Proposition 2.3.2. Let T ∈ Mr (C) and λ ∈ (0, 1). Then: 1. ∆λ (cT ) = c∆λ (T ) for every c ∈ C. 2. ∆λ (V T V ∗ ) = V ∆λ (T ) V ∗ for every V ∈ U(r). 3. If T = T1 ⊕ T2 then ∆λ (T ) = ∆λ (T1 ) ⊕ ∆λ (T2 ). 4. k∆λ (T ) k2 6 kT k2 . 5. T and ∆λ (T ) have the same characteristic polynomial. 6. In particular, σ (∆λ (T )) = σ (T ). The following theorem states the regularity properties of λ-Aluthge transforms. Theorem 2.3.3. The λ-Aluthge transform is continuous in Mr (C). Moreover, the map (0, 1) × Gl r (C) 3 (λ, T ) 7→ ∆λ (T ) is of class C ∞ . Proof. The continuity part was proved in [4] (see also [10]). If T ∈ Gl r (C), then ∆λ (T ) = |T |λ T |T |−λ ,

λ ∈ (0, 1).

This clearly implies regularity, since the map (0, 1) × Gl r (C) 3 (λ, T ) 7−→ |T |λ = (T ∗ T )λ/2 = exp(λ/2 log T ∗ T ) is of class C ∞ . The following result is proved in [4]:

Vol. 62 (2008)


471

Proposition 2.3.4. Given T ∈ Mr (C) and λ ∈ (0, 1), the limit points of the sequence {∆nλ (T )}n∈N are normal. Moreover, if L is a limit point, then σ (L) = σ (T ) with the same algebraic multiplicity. In particular, for each λ ∈ (0, 1), one has that ∆λ (T ) = T if and only if T is normal. Finally, we mention a result concerning the Jordan structure of Aluthge transforms proved in [4]. We need the following definitions. Definition 2.3.5. Let T ∈ Mr (C) and µ ∈ C. We denote 1. m(T, µ) the algebraic multiplicity of µ for T , i.e. the maximum n ∈ N such that (x − µ)n divides the characteristic polynomial of T . 2. m0 (T, µ) = dim ker(T − µI), the geometric multiplicity of µ. Observe that Proposition 2.3.2 says that m(T, µ) = m(∆λ (T ) , µ) for every µ ∈ C. Proposition 2.3.6. Let T ∈ Mr (C). 1. If 0 ∈ σ (T ), then there exists n ∈ N such that m(T, 0) = m0 (∆nλ (T ) , 0) = dim ker(∆nλ (T )). 2. For every µ ∈ σ(T ), m0 (T, µ) 6 m0 (∆λ (T ) , µ). Observe that this implies that, if T is diagonalizable (i.e. m0 (T, µ) = m(T, µ) for every µ), then also ∆λ (T ) is diagonalizable. Remark 2.3.7. Let T ∈ Gl r (C) with polar decomposition T = U |T |. The Duggal or 1-Aluthge transform of T is ∆1 (T ) = |T |U . It is easy to see that the map ∆1 : Gl r (C) → Gl r (C) is continuous and that ∆1 (T ) = T if and only if T is normal. Observe that U ∈ U(r), so that ∆1 (T ) = U T U ∗ , and the distance of ∆1 (T ) to the normal matrices is the same as the distance of T to the normal matrices. All these facts imply that the iterated Duggal transforms ∆n1 (T ) can not converge, unless T is normal.

3. Convergence In this section, we prove the convergence of iterated λ- Aluthge transforms for every diagonalizable matrix and λ ∈ (0, 1). Throughout the next subsections, a diagonal matrix D = diag(d1 , . . . , dn ) ∈ Gl r (C) is fixed. For every j ∈ {1, . . . , n}, let dj = e iθj |dj | be the polar decomposition of dj , where θj ∈ [0, 2π]. 3.1. Reduction to the invertible case We claim that the proof of the convergence of iterated λ- Aluthge transforms can be reduced to the invertible case. Indeed, let T ∈ Mr (C) be a a diagonalizable matrix with polar decomposition T = U |T |. As R(T ) is a (oblique) complement of ker T = ker |T |λ and R(U |T |1−λ ) = R(T ), it holds that λ

1−λ

R(∆λ (T )) = R(|T | U |T |

) = R(|T |).

472


IEOT

On the other hand, it is easy to see that ker ∆λ (T ) = ker |T |1−λ = ker |T |, which is orthogonal to R(|T |). Therefore T1 0 ker T ⊥ ∆λ (T ) = , (3.1) 0 0 ker T where T1 is invertible and diagonalizable on ker T ⊥ . Then, by Proposition 2.3.2, n−1 ∆λ (T1 ) 0 ker T ⊥ ∆nλ (T ) = , for every n ∈ N. 0 0 ker T This shows that the convergence of {∆nλ (T )}n∈N is equivalent to the convergence of {∆nλ (T1 )}n∈N . 3.2. Main Theorem After reduction to the invertible case, the key tool, which allows to use the stable manifold theorem 2.1.3, is Theorem 3.2.1 below. The proof of this theorem is rather long and technical. For this reason, we postpone it until Section 5, and we continue in this section with its consequences. Theorem 3.2.1. Let λ ∈ (0, 1). The λ-Aluthge transform ∆λ (·) : S (D) → S (D) is s a C ∞ map, and for every N ∈ U (D), there exists a subspace EN,λ in the tangent space TN S (D) such that s 1. TN S (D) = EN,λ ⊕ TN U (D); s 2. Both EN,λ and TN U (D) are TN ∆λ -invariant;

≤k < 1, where T ∆ | 3. s D,λ

N λ EN,λ |dj |1−λ |di |λ + |di |1−λ |dj |λ i(θj −θi ) , max |λ e − 1 + 1| < 1; kD,λ = max max θi 6=θj |di | + |dj | |di |6=|dj | s s 4. If U ∈ U(r) satisfies N = U DU ∗ , then EN,λ = U (ED,λ )U ∗ . s In particular, the map U (D) 3 N 7→ EN,λ is smooth. This fact can be formulated s in terms of the projections PN,λ onto EN,λ parallel to TN U (D), N ∈ U (D).

Proof. See Section 5.

Corollary 3.2.2. Let D = diag(d1 , . . . , dr ) ∈ Mr (C) be an invertible diagonal mas trix and λ ∈ (0, 1). Let EN,λ and kD as in Theorem 3.2.1. Then, in S (D) there ∞ exists a ∆λ -invariant C -pre-lamination {WN,λ }N ∈U (D) of class C ∞ such that, for every N ∈ U (D), 1. WN,λ is a C ∞ submanifold of S (D). s . 2. TN WN,λ = EN,λ 3. If kD < ρ < 1, then dist(∆nλ (T ) − N ) ≤ dist(T, N )ρn , for every T ∈ WN,λ . 4. If N1 6= N2 then WN1 ,λ ∩ WN2 ,λ = ∅. 5. There exists an open subset [ W(D) of S (D) such that a. U (D) ⊆ W(D) ⊆ WN,λ and N ∈U (D)

Vol. 62 (2008)


473

b. the map p : W(D) → U (D), defined by p(T ) = N if T ∈ WN,λ , is of class C ∞ . Proof. By Theorem 3.2.1, for every kD < ρ < 1, U (D) is ρ-pseudo hyperbolic for ∆λ (see Definition 2.1.2), and it consists of fixed points. Thus, by Theorem 2.1.3, we get a C ∞ and ∆λ -invariant pre-lamination of class C ∞ , {WN,λ }N ∈U (D) which satisfies all the properties of the statement. 3.3. Convergence for fixed λ Using the previous results, we can apply exactly the same techniques as in our previous work [5], and obtain for every λ ∈ (0, 1) the same results about ∆λ (·) as those obtained for the classical Aluthge transform ∆1/2 (·). We state these properties in the following theorem. The basic idea is to apply Proposition 2.3.4 in order to assure that the iterations go into the open set W(D), where the smooth projection p can be used. Although the proof of this theorem is omitted to avoid repetitions, Proposition 4.1.2 below gives a detailed proof. Theorem 3.3.1. Let λ ∈ (0, 1): 1. Given a diagonalizable matrix T ∈ Mr (C), the sequence {∆nλ (T )}n∈N converges and its limit will be denoted by ∆∞ λ (T ). 2. Let D ∈ Mr (C) be diagonal. Then the sequence {∆nλ }n∈N , restricted to the similarity orbit S (D), converges uniformly on compact sets to the map ∆∞ λ : ∞ S (D) → U (D), which is of class C ∞ . In particular, ∆∞ is a C retraction λ from S (D) onto U (D). Remark 3.3.2. Let D ∈ Mr (C) be diagonal. For every N ∈ U (D) and λ ∈ (0, 1), denote + WN,λ = T ∈ S (D) : ∆∞ λ (T ) = N . + ∞ Since T 7→ ∆∞ retraction from S (D) onto U (D), each WN,λ is a C ∞ λ (T ) is a C submanifold of S (D); and S (D) stands as the (disjoint) union of these sheets. On + the other hand, the submanifolds WN,λ are prolongations of the sheets WN,λ of Corollary 3.2.2. Indeed, for every N ∈ U (D) and λ ∈ (0, 1), + WN,λ ⊆ WN,λ

and

+ W(D) ∩ WN,λ = W(D) ∩ WN,λ ,

+ s by Corollary 3.2.2. Then TN WN,λ = TN WN,λ = EN,λ .

4. Regularity properties of ∆∞ λ (T ) As in Section 3, we fix D = diag(d1 , . . . , dr ) ∈ Gl r (C). Observe that, using the continuity of ∆λ (·) with respect to λ (Thm. 2.3.3), and the fact that the convergence of its iterations to the limit map ∆∞ λ (·) is uniform on compact subsets of S (D) (for each λ), one can show that the map (0, 1) × S (D) 3 (λ, T ) 7−→ ∆∞ λ (T )

474


IEOT

is continuous. The purpose of this section is to prove that this map is smooth. Firstly we shall analyze the invertible case. Then, in Subsection 4.2 we shall see that the invertibility hypothesis on D can be dropped. 4.1. On the orbit S (D) Denote by SL (D) = (0, 1) × S (D) and UL (D) = (0, 1) × U (D). Consider the map ∆ : SL (D) → SL (D) given by ∆(λ, T ) = (λ, ∆λ (T ) ), (λ, T ) ∈ SL (D) . (4.1) Remark 4.1.1. Using Theorems 2.3.3, 3.2.1 and 3.3.1 and Proposition 2.3.4, one can deduce easily the following properties: 1. SL (D) is a C ∞ manifold, and UL (D) is a submanifold of SL (D). 2. The map ∆ is of class C ∞ . 3. The submanifold UL (D) coincides with the set of all fixed points of ∆. 4. For every (λ, T ) ∈ SL (D), it holds that ∆n (λ, T ) = (λ, ∆nλ (T ) ) −−−−→ (λ, ∆∞ λ (T ) ). n→∞

(4.2)

5. For every N ∈ U (D) and λ ∈ (0, 1), one can describe the tangent spaces as T(λ,N ) UL (D) = TN U (D) ⊕ R,

and

s

s T(λ,N ) SL (D) = TN S (D) ⊕ R = EN,λ ⊕ TN U (D) ⊕ R = EN,λ ⊕ T(λ,N ) UL (D) , s where the spaces EN,λ are those of Theorem 3.2.1. s 6. By Remark 5.2.2, the projections QN,λ = PN,λ + PR onto EN,λ parallel to T(λ,N ) UL (D), given by the above decomposition, satisfy that the map

(0, 1) × U (D) 3 (λ, N ) 7−→ QN,λ ∞

is of class C . Proposition 4.1.2. The iterates ∆n of the map ∆ : SL (D) → SL (D) defined in Eq. (4.1) satisfy ∆n (λ, T ) −−−−→ ∆∞ (λ, T )

for every

n→∞

(λ, T ) ∈ SL (D) ,

where ∆∞ : SL (D) → UL (D) is a C ∞ retraction. Proof. Fix (λ0 , T0 ) ∈ SL (D). Let 0 < λ1 < λ0 < λ2 < 1, and consider the submaniflods So (D) = (λ1 , λ2 ) × S (D) ⊆ SL (D)

and

Uo (D) = (λ1 , λ2 ) × U (D) .

Observe that So (D) is open in SL (D), it is ∆ invariant, and its fixed points coincide with Uo (D), which is also open in UL (D). Fix ρ ∈ (0, 1) such that the constants (of Theorem 3.2.1) kD,λ < ρ for every λ ∈ (λ1 , λ2 ). Observe that, for every (λ, N ) ∈ Uo (D), s T(λ,N ) ∆ s = TN ∆λ s ⊆ EN,λ ⊆ T(λ,N ) SL (D) . E

N,λ

E

N,λ

Using this fact, and items 5 and 6 of Remark 4.1.1, one can assure that Uo (D) is ρ-pseudo hyperbolic for ∆ : So (D) → So (D) (see Definition 2.1.2) consisting of

Vol. 62 (2008)


475

fixed points. Thus, by Theorem 2.1.3, we get a C ∞ and ∆-invariant pre-lamination {W(λ,N ) }(λ,N )∈Uo (D) of class C ∞ such that, for every (λ, N ) ∈ Uo (D), 1. W(λ,N ) is a C ∞ submanifold of So (D). s 2. T(λ,N ) W(λ,N ) = E(λ,N . ) n 3. dist(∆ (µ, T ) − (λ, N ) ) ≤ ρn dist( (µ, T ), (λ, N ) ), for every (µ, T ) ∈ W(λ,N ) . Observe that this implies that, if (µ, T ) ∈ W(λ,N ) , then µ = λ.

(4.3)

4. If (µ1 , N1 ) 6= (µ2 , N2 ) then W(µ1 ,N1 ) ∩ W(µ2 ,N2 ) = ∅. 5. There exists an open subset[ W(D) of So (D) such that a. Uo (D) ⊆ W(D) ⊆ W(λ,N ) . (λ,N )∈U (D)

b. The map p : W(D) → Uo (D), defined by p(λ, T ) = (λ, N )

(λ, T ) ∈ W(λ,N ) ,

if ∞

is well defined and of class C . By item 2 of Remark 4.1.1, the map ∆ and its iterations are C ∞ functions. Hence, there exist an open set U ⊆ So (D) and k ∈ N such that (λ0 , T0 ) ∈ U and ∆k (U) ⊆ W(D). By the regularity the projection p, one can deduce that the map So (D) 3 (λ, T ) 7−→ p(∆k (λ, T ) ) is of class C ∞ on U. Now, if ∆k (λ, T ) ∈ W(λ,N ) then p(∆k (λ, T ) ) = N = lim ∆n (λ, T ), n→∞

by item 3. Hence ∆

∞

is well defined and of class C ∞ .

Theorem 4.1.3. Let D ∈ Dr (C) be invertible. Then the map ∆∞ : (0, 1) × S (D) → U (D)

given by

∆∞ (λ, T ) = ∆∞ λ (T ) ,

for every (λ, T ) ∈ SL (D), is of class C ∞ . Proof. Let ∆∞ : SL (D) → UL (D) be the C ∞ retraction of Proposition 4.1.2. By Eq. (4.2), ∆∞ (λ, T ) = (λ, ∆∞ (λ, T ) )

for every

This implies that also the map ∆∞ is of class C ∞ .

(λ, T ) ∈ So (D) .

Remark 4.1.4. Observe that Proposition 4.1.2 and Eq. (4.2), before taking the limit, also show Theorem 3.3.1, whose proof was omitted. On the other hand, they also show that the map ∆∞ : SL (D) → UL (D) given by ∆∞ (λ, T ) = (λ, ∆∞ λ (T )) is a C ∞ retraction from SL (D) onto UL (D). Therefore, for every (λ, N ) ∈ UL (D), + ∞ −1 the set W(λ,N (λ, N ) is a smooth submanifold of SL (D). Observe that, ) = (∆ ) + + by Remark 3.3.2, Eq. (4.2) and (4.3), we have that W(λ,N ) = {λ} × Wλ,N and + + W(λ,N ) = W(D) ∩ W(λ,N ) = W(D) ∩ {λ} × Wλ,N = {λ} × Wλ,N ,

476


IEOT

+ where Wλ,N are the sheets described in Remark 3.3.2, and Wλ,N are those sheets which appear in Corollary 3.2.2. This can be described as the fact that “the submanifolds Wλ,N move smoothly with λ”.

4.2. The non-invertible case As in Section 3, all the results of this section still hold if the diagonal matrix D ∈ Mr (C) is not invertible. Indeed, suppose that rk D = s < r, and fix T ∈ S (D) and λ ∈ (0, 1). By Eq. (3.1), T1 0 ker T ⊥ ∆λ (T ) = , 0 0 ker T where T1 is invertible and diagonalizable on ker T ⊥ . The same happens for every matrix S ∈ S (D). Denote by P = Pker T and U (P ) = {U P U ∗ : U ∈ U(r)} its unitary orbit. Consider the map Q : S (D) → U (P )

given by

Q(S) = Pker S = Pker ∆λ (S) ,

S ∈ S (D) .

This map takes values in U (P ) because rk Q(S) = rk P = s for every S ∈ S (D). Also, Q is of class C ∞ , since S 7→ ∆λ (S) is smooth, and ∆λ (S) 7→ Pker ∆λ (S) is polynomial. By Proposition 2.2.2, there exist an open set W ⊆ U (P ) which contains P , and a C ∞ local cross section σ : W → U(r), in the sense that σ(R)P σ(R)∗ = R for every R ∈ W. Let V ⊆ S (D) be an open set such that T ∈ V and Q(V) ⊆ W. Denote by η = σ ◦ Q : V → U(r). Then η is also of class C ∞ . So, for every λ ∈ (0, 1) and S ∈ V, there exists γ(S, λ) ∈ L(ker T ⊥ ) such that γ(S, λ) 0 ker T ⊥ η(S)∗ ∆λ (S) η(S) = and γ(S, λ) ∈ S (T1 ) ⊆ L(ker T ⊥ ). 0 0 ker T Therefore, using that ∆λ (U T U ∗ ) = U ∆λ (T ) U ∗ for every U ∈ U(r), we obtain ∞ ∆λ (γ(S, λ) ) 0 ker T ⊥ ∆∞ (S) = Ad ◦ , for every S ∈ W. λ η(S) 0 0 ker T Then the regularity of (λ, S) 7→ ∆∞ λ (S) can be deduced from the regularity of the maps (0, 1) × V 3 (λ, S) 7→ γ(S, λ) and (0, 1) × S (T1 ) 3 (λ, A) 7→ ∆∞ λ (A). Hence, the reduction to the invertible case is proved. 4.3. Different eigenvalues Let Dr∗ (C) be the set of diagonalizable and invertible matrices in Mr (C) with r different eigenvalues (i.e. every eigenvalue has algebraic multiplicity equal to one). Observe that Dr∗ (C) is an open dense subset of Mr (C) and it is invariant by the Aluthge transform. Theorem 4.3.1. The map ∆∞ : (0, 1) × Dr∗ (C) → U (D) given by ∗ ∆∞ (λ, T ) = ∆∞ λ (T ) , (λ, T ) ∈ (0, 1) × Dr (C)

is of class C ∞ . Proof. It follows from a straightforward combination of the techniques of Subsection 3.2 of [5] and those of the previous sections. We omit the details.

Vol. 62 (2008)


477

4.4. The map λ 7→ ∆∞ λ (T ) for fixed T Definition 4.4.1. Let T ∈ Mr (C) be a diagonalizable. We call RT : (0, 1) → Mr (C) the map given by RT (λ) = ∆∞ λ (T ), for λ ∈ (0, 1). The following question arises naturally: Is the map RT constant for every diagonalizable matrix T ∈ Mr (C)? Numerical examples show that the question has 3 0 a negative answer in general. Indeed, taking the 2 × 2 matrix T = , −2 1 numerical computations show that 2.2273 0.97380 1.37162 −0.77790 ∼ ∼ RT (0.3) = and RT (0.7) = . 0.97380 1.7726 −0.77790 2.62838 Nevertheless, it was proved in [4, Thm. 4.9] that, if D = diag(d1 , d2 ) ∈ M2 (C) and |d1 | = |d2 |, then RT is constant for every T ∈ S (D). Our next result shows that, for a diagonal matrix D = diag(d1 , . . . , dr ) ∈ Mr (C), this may happen only if |d1 | = . . . = |dr |. Proposition 4.4.2. Let D ∈ Mr (C) be diagonal and invertible. If D has two eigenvalues with distinct moduli, then there exists T ∈ S (D) such that the map RT is not constant. The proof of Proposition 4.4.2 follows directly from the next two Lemmas. But let us first make some comments about this problem: Remark 4.4.3. Despite Proposition 4.4.2, given D = diag (d1 , . . . , dr ) ∈ Mr (C) with |di | 6= |dj | for some i, j ∈ {1, . . . , r}, the similarity orbit S (D) may contain (non-normal) matrices T for which the map RT is constant. In fact, consider the following example: let D = diag (1, −1, 2) ∈ M3 (C), take D1 = diag (1, −1), T1 0 T1 ∈ S (D1 ) and T = ∈ S (D), then 0 2 ∞ ∆λ (T1 ) 0 RT1 (λ) 0 RT (λ) = ∆∞ (T ) = = , for every λ ∈ (0, 1). λ 0 2 0 2 But the map RT1 is constant by [4, Thm 4.9] (see also Proposition 4.4.6 below). Observe that this example includes several cases where T ∈ / U (D) (otherwise, the map would be trivially constant). On the other hand, Proposition 4.4.2 is not longer true if D is not invertible. This fact can be immediately tested by taking D with σ (D) = {0, 1} (i.e., if D is a projection). Lemma 4.4.4. Let D ∈ Mr (C) be diagonal and invertible. If the map RT is constant for every T ∈ S (D), then the distribution of subspaces s (0, 1) 3 λ 7−→ EN,λ

given by Theorem 3.2.1 must be constant for every N ∈ U (D). Proof. Using the notations of Remark 3.3.2 and Corollary 3.2.2, each submanifold + WN,λ consist of those matrices T such that RT (λ) = ∆∞ λ (T ) = N . But if all the

478


IEOT

+ maps ∆∞ λ (·) are the same on S (D), then the submanifolds WN,λ must agree for s different lambdas. Finally observe that, by Remark 3.3.2 and Corollary 3.2.2, EN,λ + is the tangent space of WN,λ at N , for every N ∈ U (D) and every λ ∈ (0, 1).

Lemma 4.4.5. Let D = diag (d1 , . . . , dn ) ∈ Mr (C) be diagonal and invertible. Then the following conditions are equivalent: s 1. The distribution of subspaces (0, 1) 3 λ 7−→ EN,λ given by Theorem 3.2.1 is constant for every N ∈ U (D). 2. |di | = |dj | for every 1 ≤ i, j ≤ r. ⊥ s Moreover, in this case, EN,λ = TD U (D) for every λ ∈ (0, 1). Proof. Since the proof uses several results and notation from Section 5, we postpone it until that section. See Remark 5.2.1. Now, the natural question is, what happens if D has all its eigenvalues of the same modulus? We first study a particular case: Proposition 4.4.6. Let D ∈ Mr (C) be diagonal such that σ(D) = {d1 , d2 } with |d1 | = |d2 | . Then the map RT is constant for every T ∈ S (D). Proof. Let T ∈ S (D). Denote Si = ker(T − di I) for i = 1, 2, and by Q the oblique projection onto S1 given by the decomposition Cr = S1 ⊕ S2 . Then T = d1 Q + d2 (I − Q). Assume that k = dim S2 ≥ n = dim S1 . In matrix terms, there exists A ∈ L(S1⊥ , S1 ) such that S1 S1 I A d1 I (d1 − d2 )A Q= and T = . 0 0 0 d2 I S1⊥ S1⊥ We can assume that S1 = {x ∈ Cr : xi = 0 for i > n}, by a unitary conjugation, which commutes with ∆∞ λ . In this case, A ∈ Mn,k (C). By the decomposition in singular values of B = (d1 − d2 )A, there exist U ∈ U(n) and V ∈ U(k) such that U BV ∗ = Σ(B), where Σ(B) = (Σ0 (B), 0) ∈ Mn,k (R) with Σ0 (B) = diag (s1 (B), . . . , sn (B) ) ∈ Mn (R), and we add a n × (k − n) block of zeros on the right. If W = U ⊕ V ∈ U(r), then   n n C d1 In Σ0 (B) 0 d1 I Σ(B) C   Cn 0 d I 0 WTW∗ = = . 2 n 0 d2 I Ck 0 0 d2 Ik−n Ck−n Moreover, there exists a permutation matrix S ∈ U(r) which rearranges the entries of W T W ∗ in such a way that n M d1 si (B) M = SW T W ∗ S ∗ = ⊕ d2 Ik−n . 0 d2 i=1

As before, it suffices to show that the map RM is constant. But now Proposition 2.3.6 assures that n M si (B) ∞ ∞ d1 RM (λ) = ∆λ (M ) = ∆λ ⊕ d2 Ik−n for every λ ∈ (0, 1). 0 d2 i=1

Vol. 62 (2008)


Finally, it was proved in [4, Thm. 4.9] that each map λ 7→ ∆∞ λ

d1 0

constant.

479

si (B) is d2

Remark 4.4.7. The case D = D∗ in Proposition 4.4.6, is particularly interesting, because in this case all iterated λ-Aluthge transforms (and the limit) can be explicitly computed for every E ∈ S (D), even in the infinite dimensional case: Let H be a Hilbert space and D ∈ Lsa (H) such that σ(D) = {1, −1}. Observe that if E ∈ S (D), then E 2 = I. The geometry of S (D) has been widely studied (see [17], [6] and [7]). Given E ∈ S (D), observe that, if L = |E|, then |E ∗ | = L−1 . Therefore, if E = RL is the polar decomposition of E, with R ∈ U(r), then RL = E = E −1 = L−1 R∗ . But the right polar decomposition is E = |E ∗ |R (with the same R), so that R = R−1 = R∗ is a unitary reflection and RL = L−1 R. Moreover, R ∈ U (D), because E = RL = L−1/2 RL1/2 . The map E 7→ R is the retraction p : S (D) → U (D) deeply studied in [18]. We claim that p = ∆1/2 (·) = ∆∞ λ (·) for every λ ∈ (0, 1). Indeed, observe that ∆1/2 (E) = L1/2 RL1/2 = L1/2 L−1/2 R = R and, for every λ ∈ (0, 1), ∆λ (E) = Lλ RL1−λ = RL1−2λ , so that ∆2λ (E) = 2 RL(1−2λ) , since ∆λ (E) = RL1−2λ is the polar decomposition of ∆λ (E). Inductively, for every n ∈ N, n

∆nλ (E) = RL(1−2λ) −−−−→ R = ∆∞ λ (E) , n→∞

since |1 − 2λ| < 1. Then the map λ 7→ RE (λ) = ∆∞ λ (E) is constant, but the rate of convergence is very different for each λ, being slower when λ tends to 0 or 1. + On the other hand, the sheets WR,λ of Remark 3.3.2 can be characterized as + WR,λ = { RM : M ∈ Gl (H)+

and

RM = M −1 R }.

The geometry of these hyperbolic manifolds is also deeply studied in [7]. If D has all its eigenvalues of the same modulus and σ (D) has more than two elements, we do not have an answer to the above question, but we have made several computational experiments. In all the tested examples, the map RT fails to be constant for some T in the orbit, even if D satisfies some algebraic condition such as D3 = I. This suggests the following conjecture: Conjecture. Let D ∈ Mr (C) be diagonal and invertible. Then the map RT is constant for every T ∈ S (D) if and only if σ(D) = {d1 , d2 } with |d1 | = |d2 |. Remark 4.4.8. Observe that it would be sufficient to consider the 3 × 3 case, because we can use a similar reduction to the one used in Remark 4.4.3. Example 4.4.9. Let a, b, c ∈ R+ such that abc = 1, and let T be the 3 × 3 matrix defined by     0 0 1 a 0 0 T = 1 0 0 · 0 b 0 = U |T |. 0 1 0 0 0 c

480


IEOT

Computing its spectrum, one shows that T ∈ S (U ). On the other hand, since U is a permutation matrix, for every diagonal matrix D ∈ M3 (C) both U DU ∗ and U ∗ DU are also diagonal matrices. In particular, |T ∗ | = U |T |U ∗ is diagonal and commutes with |T |. Let λ ∈ (0, 1). Then λ 1−λ ∆λ (T ) = |T | U |T | = U U ∗ (|T |λ |T ∗ |1−λ )U = U ∆λ (T ) , where the last equality holds by the uniqueness of the polar decomposition. Note that ∆λ (T ) is diagonal with det ∆λ (T ) = 1. An inductive argument shows that ∆nλ (T ) = U Dn for every n ∈ N, where each Dn is a positive diagonal matrix. As T ∈ S (U ), then ∆∞ −−−→ I and ∆∞ λ (T ) ∈ U (U ) ⊆ U(r). Hence Dn − λ (T ) = U . n→∞

The same happens for any λ ∈ (0, 1), therefore, the function RT (λ) is constant. This example does not contradicts the conjecture, because all the matrices studied satisfy that |T | and |T ∗ | commute, so that they are not dense in S (U ).

5. The proof of Theorem 3.2.1 As in Section 3, in this section we fix an invertible diagonal matrix D ∈ Mr (C) whose diagonal entries are denoted by (d1 , . . . , dn ). For every j ∈ {1, . . . , n}, let dj = e iθj |dj | be the polar decomposition of dj , where θj ∈ [0, 2π]. 5.1. Matricial characterization of TN ∆λ Definition 5.1.1. Given A, B ∈ Mr (C), A ◦ B denotes their Hadamard product, that is, if A = (Aij ) and B = (Bij ), then (A ◦ B)ij = Aij Bij . With respect to this product, each matrix A ∈ Mr (C) induces an operator ΨA on Mr (C) defined by ΨA (B) = A ◦ B, B ∈ Mr (C). Remark 5.1.2. In what follows, we shall state several definitions and results taken from Section 4 of our previous work [5]: 1. By Eq. (2.3), the tangent space TD S (D) consists on those matrices X ∈ Mr (C) such that Xij = 0 if di = dj . Then TD S (D) reduces the operator ΨA , for every A ∈ Mr (C). This is the reason why, from now on, we shall consider all these operators as acting on TD S (D). Restricted in this way, it holds that kΨA k = sup{kA ◦ Bk2 : B ∈ TD S (D) and kBk2 = 1} = max |Aij |, di 6=dj

since ΨA is a diagonal operator on the Hilbert space (TD S (D) , k · k2 ). 2. Let PRe and PIm be the projections defined on TD S (D) by PRe (B) =

B + B∗ 2

and

PIm (B) =

B − B∗ , 2

B ∈ TD S (D) .

That is, PRe (resp. PIm ) is the restriction to TD S (D) of the orthogonal projection onto the subspace of Hermitian (resp. anti-Hermitian) matrices.

Vol. 62 (2008)


481

∗ 3. Observe that, for every K ∈ Mah r (C) (i.e., such that K = −K) and B ∈ Mr (C) it holds that

K ◦ PRe (B) = PIm (K ◦ B)

and

K ◦ PIm (B) = PRe (K ◦ B).

(5.1)

4. Denote by QD the orthogonal projection from TD S (D) onto (TD U (D))⊥ . 5. Let J, K ∈ Mr (C) be the matrices defined by ( |dj − di | sgn(j − i) if di 6= dj and Kij = 0 if di = dj ( −1 (dj − di )Kij if di 6= dj Jij = 1 if di = dj for 1 ≤ i, j ≤ r. Then (a) for every A ∈ Mr (C), AD − DA = J ◦ K ◦ A; (b) it holds that QD = ΨJ PIm Ψ−1 J ; ∗ h (c) if H ∈ Mr (C) (i.e., if H = H), then QD ΨH = ΨH QD . 6. Let λ ∈ (0, 1) and N ∈ U (D) and let QN be the orthogonal projection from ⊥ TN S (D) onto TN U (D) . Then TN ∆λ has the 2 × 2 matrix decomposition A1N (λ) 0 QN TN ∆λ = , (5.2) A2N (λ) I I − QN because TN ∆λ acts as the identity on TN U (D). 7. Let A ∈ Mr (C) and let γ : R → L(H)+ be the curve defined by ∗ ∗ ∗ γ(t) = etA De−tA etA De−tA = e−tA D∗ etA etA De−tA . If R, T + and T − ∈ Mr (C) are defined by Rij = 2d¯i dj , Tij+ = |di |2 + |dj |2 and Tij− = |dj |2 − |di |2 , i, j ∈ Ir , then γ 0 (0) = (R − T + ) ◦ PRe (A) + T − ◦ PIm (A).

(5.3)

The following classical result, proved by Daleki˘ıi and Kre˘ın in 1951 ([8] and [9]) will be useful in the sequel (see also the book [12]). Theorem 5.1.3. Let I, J ⊆ R be open intervals and let γ : I → Mhr (C) be a C 1 curve such that σ(γ(t) ) ⊆ J for every t ∈ I. Let f : J → R be a C 1 map. Consider now the curve f · γ : I → Mhr (C) given by f · γ(t) = f γ(t) , t ∈ I (functional calculus). Suppose that γ(t0 ) = diag (a1 , . . . , ar ) for some t0 ∈ I. Then (f · γ)0 (t0 ) = Mf ◦ γ 0 (t0 ), where Mf ∈ Mn (R) is defined by  f (aj ) − f (ai )    aj − ai (Mf )ij =    0 f (ai )

if ai 6= aj , if ai = aj

for

i, j ∈ Ir .

482


IEOT

Corollary 5.1.4. Let A and γ be as in 7 of Remark 5.1.2, and let λ ∈ (0, 1). Then (γ λ/2 )0 (0) = Mλ/2 ◦ γ 0 (0), where Mλ/2 ∈ Mr (C) is the matrix given by   |d |λ − |di |λ   j 2 if |di | = 6 |dj |   |dj | − |di |2 , i, j ∈ Ir . (5.4) (Mλ/2 )ij =    λ   |di |λ−2 if |di | = |dj | 2 λ Proof. Apply Theorem 5.1.3 to f (t) = t 2 . Use that γ(0) = diag |d1 |2 , . . . , |dr |2 . ⊥ Recall that QD denotes the orthogonal projection from TD S (D) onto TD U (D) . Proposition 5.1.5. Fix λ ∈ (0, 1). Then there exists a matrix H(λ) ∈ Mr (C) such that QD TD ∆λ QD = QD ΨPRe H(λ) QD and (I − QD ) TD ∆λ QD = (I − QD ) ΨPIm H(λ) QD . The entries of H1 (λ) = PRe H(λ) are the following: for every 1 ≤ i, j ≤ r, denote by Aij = |dj |2−λ |di |λ −|di |2−λ |dj |λ and Bij = |dj |1+λ |di |1−λ −|di |1+λ |dj |1−λ . Then  i(θ −θ )  Aij + e j i Bij if |d | = |dj |, i 6 2 |dj | − |di |2 (5.5) H1 (λ)ij =  λ ei(θj −θi ) − 1 + 1 if |di | = |dj |. The proof of this proposition follows the same steps as the proof of [5, Prop. 4.1.5], but using now Corollary 5.1.4, and items 5 and 7 of Remark 5.1.2. For this reason we shall give only and sketched version, pointed out the main differences and the technical difficulties that appear when we loose the symmetry of the case λ = 1/2. Sketch of proof. Fix X = AD − DA ∈ TD S (D), for some A ∈ Mr (C). Then d tA −tA TD ∆λ (X) = . ∆λ e De dt t=0 ∗ ∗ ∗ Let γ(t) = etA De−tA etA De−tA = e−tA D∗ etA etA De−tA . In terms of γ, we can write the curve ∆λ etA De−tA in the following way ∆λ etA De−tA = γ λ/2 (t)(etA De−tA )γ −λ/2 (t). Since γ λ/2 γ −λ/2 = I, then (γ −λ/2 )0 (0) = −γ −λ/2 (0) (γ λ/2 )0 (0) γ −λ/2 (0). Using this identity, easy computations show that TD ∆λ (X) = (γ λ/2 )0 (0) D − D (γ λ/2 )0 (0) |D|−λ + |D|λ (AD − DA)|D|−λ . If we define the matrices L, N ∈ Mr (C) by Nij = |dj |−λ and Lij = |di |λ |dj |−λ and take J, K ∈ Mr (C) as in 5 of Remark 5.1.2. Then TD ∆λ (X) = N ◦ (J ◦ K ◦ (γ λ/2 )0 (0)) + L ◦ (J ◦ K ◦ A).

Vol. 62 (2008)


483

Using Corollary 5.1.4 and 7 of Remark 5.1.2, we get h i TD ∆λ (AD − DA) = N ◦ J ◦ K ◦ Mλ/2 ◦ (R − T + ) ◦ PRe (A) + T − ◦ PIm (A) + L ◦ J ◦ K ◦ A. where R, T + and T − are the matrices defined in 7 of Remark 5.1.2, and Mλ/2 is the matrix defined in Eq. (5.4) of Corollary 5.1.4. Now, we shall express TD ∆λ (AD − DA) in terms of AD − DA = J ◦ K ◦ A. Following the same steps as in Proposition 4.1.5 of [5], we arrive at the formula TD ∆λ (AD − DA) = ΨH(λ) QD + (I − QD ) (AD − DA), ah where H(λ) = Mλ/2 ◦N ◦(R−T + )+L. Given X ∈ R(QD ), then Ψ−1 J (X) ∈ Mr (C) and QD TD ∆λ QD (X) = QD (H(λ) ◦ X) = (ΨJ PIm Ψ−1 J )(H(λ) ◦ X) = J ◦ PIm (H(λ) ◦ Ψ−1 J X) 1 −1 ∗ = J ◦ H(λ) ◦ Ψ−1 J (X) + H(λ) ◦ ΨJ (X) 2 = J ◦ PRe H(λ) ◦ Ψ−1 J (X) = PRe H(λ) ◦ X = QD ΨPRe H(λ) (X). Analogously, one shows that (I − QD ) TD ∆ QD (X) = (I − QD )ΨPIm H(λ) (X). In order to prove Eq. (5.5), recall that H(λ) = Mλ/2 ◦ N ◦ (R − T + ) + L. Hence, H(λ)ij = (Mλ/2 )ij |dj |−λ 2d¯i dj − (|di |2 + |dj |2 ) + |di |λ |dj |−λ . Suppose that

|di | = 6 |dj |. Straightforward computations, using Corollary 5.1.4, show that 1 2|dj |2−λ |di |λ + 2d¯i dj − 2d¯i dj |di |λ |dj |−λ − |di |2 − |dj |2 PRe H(λ)ij = 2 |dj |2 − |di |2 1 2|di |2−λ |dj |λ + 2d¯i dj − 2d¯i dj |dj |λ |di |−λ − |dj |2 − |di |2 + 2 |di |2 − |dj |2 =

Aij + ei(θj −θi ) Bij , |dj |2 − |di |2

where Aij and Bij are those of the statement. If |di | = |dj |, then λ H(λ)ij = |di |λ−2 |di |2−λ 2 ei(θj −θi ) − 1 + 1 = λ ei(θj −θi ) − 1 + 1. 2 So that H(λ)ij = H(λ)ji = PRe H(λ)ij = H1 (λ)ij = λ ei(θj −θi ) − 1 + 1.

Remark 5.1.6. Using the notations of Proposition 5.1.5, let H2 (λ) = PIm H(λ). 1. If |di | = |dj |, as we observed at the end of the proof of Proposition 5.1.5, H(λ)ij = H(λ)ji . Hence, H2 (λ)ij = 0.

484


IEOT

|dj | 6= 1 and α = ei(θj −θi ) . Then one can show that |di | i h H2 (λ)ij = (a − a−1 )−1 a1−λ + aλ−1 + α ( 2 − a1−λ − aλ−1 ) − a − a−1 h i while H1 (λ)ij = (a − a−1 )−1 a1−λ − aλ−1 + α(aλ − a−λ ) .

2. If |di | = 6 |dj |, denote a =

Corollary 5.1.7. Given N ∈ U (D), consider the matrix decomposition A1N (λ) 0 QN TN ∆λ = , A2N (λ) I I − QN as in Eq. (5.2). Then kA1N (λ)k ≤ kD,λ , where |dj |1−λ |di |λ + |di |1−λ |dj |λ i(θj −θi ) kD,λ = max max , max |λe + 1 − λ| < 1. θi 6=θj |di | + |dj | |di |6=|dj | Proof. As in the proof of Corollary 4.1.6 of [5], it holds that kA1N (λ)k = kA1D (λ)k for every N ∈ U (D). On the other hand, by Proposition 5.1.5 and its notations, we get that kA1D (λ)k ≤ kΨH1 (λ) k = max |H1 (λ)ij |. di 6=dj

If |di | 6= |dj |, denote a =

|dj | 6= 1. As in Proposition 5.1.5, we denote |di |

Aij = |dj |2−λ |di |λ − |di |2−λ |dj |λ = |dj | |di | (a1−λ − aλ−1 )

and

Bij = |dj |1+λ |di |1−λ − |di |1+λ |dj |1−λ = |dj | |di | (aλ − a−λ ). Observe that Aij and Bij have the same sign. So |Aij + Bij | = ±(Aij + Bij ) and Aij − ei(θj −θi ) Bij Aij + Bij |H1 (λ)ij | = ≤ |dj |2 − |di |2 |dj |2 − |di |2 |dj | − |di | 1−λ λ 1−λ λ = |d | |d | + |d | |d | j i i j |dj |2 − |di |2 =

|dj |1−λ |di |λ + |di |1−λ |dj |λ . |dj | + |di |

This quantity is strictly less than one (as observed in [13]) because |dj | + |di | − dj |1−λ |di |λ + |di |1−λ |dj |λ = |dj |λ − |di |λ |dj |1−λ − |di |1−λ > 0. On the other hand, if |di | = |dj | but θi 6= θj , using the triangle inequality we obtain that |H1 (λ)ij | = λ ei(θj −θi ) − 1 + 1 = λ ei(θj −θi ) + (1 − λ) < 1. In consequence, the bound for kA1N (λ)k is proved.

Vol. 62 (2008)


485

5.2. The proof Now, we shall restate and prove Theorem 3.2.1: Theorem. The λ-Aluthge transform ∆λ (·) : S (D) → S (D) is a C ∞ map, and for s every N ∈ U (D), there exists a subspace EN,λ in the tangent space TN S (D) such that s 1. TN S (D) = EN,λ ⊕ TN U (D); s 2. both EN,λ and TN U (D) are TN ∆λ -invariant;

3. TN ∆λ E s ≤ kD,λ < 1, where kD,λ is the constant of Corollary 5.1.7; N,λ

s s 4. if U ∈ U(r) satisfies N = U DU ∗ , then EN,λ = U (ED,λ )U ∗ . s In particular, the map U (D) 3 N 7→ EN,λ is smooth. This fact can be formulated s in terms of the projections PN,λ onto EN,λ parallel to TN U (D), N ∈ U (D).

The proof of this theorem follows exactly the same steps as the proof of Theorem 3.1.1 of [5], but now using Corollary 5.1.7, item 5 of Remark 5.1.2, and Proposition 5.1.5 of this work. For this reason, we shall only give a sketch of the proof. Sketch of proof. Fix N = U DU ∗ ∈ U (D). By the statement and the notations of Corollary 5.1.7, kA1N (λ)k < 1. So the operator I − A1N (λ) acting on R(QN ) is s be the subspace defined by invertible. Let EN,λ y s EN,λ = : y ∈ R(QN ) , −A2N (λ)(I − A1N (λ) )−1 y Now, following the same steps as in the proof of Theorem 3.1.1 of [5] (with minor changes in order to adapt it to our case) we can see that the following properties hold: • The operator PN,λ ∈ L(TN S (D) ) given by the matrix I 0 QN (5.6) PN,λ = −A2N (λ)(I − A1N (λ) )−1 0 I − QN s is the projection onto EN,λ parallel to TN U (D). Therefore we have the identity s TN U (D) = EN,λ ⊕ TN U (D). • Since TN ∆λ = AdU (TD ∆λ )Ad−1 , then U

PN,λ = AdU (PD,λ )Ad−1 U s

s

∗

and

PN,λ (TN ∆λ ) = (TN ∆λ )PN,λ .

s both EN,λ and

(5.7)

• EN,λ = U (ED,λ )U and TN U (D) are invariant for TN ∆λ .

• TN ∆λ E s = TD ∆λ E s . So it suffices to show item 3 for N = D. D,λ N,λ y s • Let Y = ∈ ED,λ , for some y ∈ R(QD ). Then −A2D (λ)(I − A1D (λ) )−1 y

2 2 k(TD ∆λ ) Y k22 = kA1D (λ)yk2 + A2D (λ)y − A2D (λ)(I − A1D (λ) )−1 y 2

2 2 2 ≤ kD,λ kyk2 + −A2D (λ)A1D (λ)(I − A1D (λ) )−1 y . 2

486


IEOT

where the inequality holds because kA1D (λ)k ≤ kD,λ , by Corollary 5.1.7. • By item 5 of Remark 5.1.2 and Proposition 5.1.5, we obtain that

−A (λ)A (λ)(I − A (λ) )−1 y 2 ≤ k 2 −A (λ)(I − A (λ) )−1 y 2 . 2D 1D 2D 1D 1D D,λ 2 2

2 2

−A (λ)(I − A (λ) )−1 y 2 = k 2 kY k2 . • k(TD ∆λ ) Y k22 ≤ kD,λ kyk22 + kD,λ 2D 1D 2 D,λ 2 Therefore we have that TN ∆λ E s has norm lower or equal to kD,λ . N,λ

s The smoothness of the map U (D) 3 N 7→ EN,λ follows from the existence of C ∞ local cross sections for the map πD : U(r) → U (D), stated in Proposition 2.2.2. For example, if σD : V → U(r) is such a section near D, then by Eq. (5.7),

PN,λ = AdσD (N ) PD,λ AdσD (N )∗ ,

N ∈ V.

(5.8)

This completes the proof.

Remark 5.2.1. Using the notations and the proof of the Theorem, one can see that the significative parts of the projections PN,λ are (I − QN )PN,λ QN = −A2N (λ)(I − A1N (λ) )−1 = −(I − QN )TN ∆λ QN QN − QN TN ∆λ QN

−1

.

When N = D, by 5 of Remark 5.1.2 and Proposition 5.1.5, and using that the −1 matrices {1 − H1 (λ)ij }ij and { 1 − H1 (λ)ij }ij are selfadjoint, we have that −1 (I − QD )PD,λ QD = −(I − QD )ΨH2 (λ) QD QD − QD ΨH1 (λ) QD −1 QD (5.9) = −(I − QD )ΨH2 (λ) I − ΨH1 (λ) = (I − QD )ΨG(λ) QD , −1 . Now, where G(λ) ∈ Mah r (C) has entries G(λ)ij = −H2 (λ)ij (1 − H1 (λ)ij ) using Remark 5.1.6, we have the following properties, which have been announced in Proposition 4.4.5:

1. If |di | = |dj |, then G(λ)ij = 0. 2. Suppose that all the eigenvalues of D have the same moduli. Then, using ⊥ s Eq. (5.6) and Eq. (5.7), we get that PN,λ = QN (i.e., EN,λ = TN U (D) ) for every λ ∈ (0, 1) and every N ∈ U (D). |dj | 3. If |di | = 6 |dj |, a = 6= 1 and β = ei(θj −θi ) , then |di | −Gij (λ) =

a1−λ + aλ−1 + β ( 2 − a1−λ − aλ−1 ) − a − a−1 −z(λ) = . b(λ) a − a−1 − (a1−λ − aλ−1 ) − β(aλ − a−λ )

(5.10)

Observe that b(λ) −−−→ 0, while z(λ) −−−→ β(a + a−1 − 2) 6= 0. Therefore, λ→0

we have that G(λ)ij −−−→ ∞. λ→0

λ→0

Vol. 62 (2008)


487

4. Suppose that D has at least two eigenvalues di and dj such that |di | 6= |dj |. Then, by the description QD = ΨJ PIm Ψ−1 J given in 5 of Remark 5.1.2, and ah the fact that G(λ) ∈ Mr (C) (which implies that PRe ΨG(λ) = ΨG(λ) PIm ), we have that −1 (I − QD )ΨG(λ) QD = ΨJ PRe ΨG(λ) PIm Ψ−1 J = ΨJ ΨG(λ) PIm ΨJ .

Since G ∈ Mah r (C) , then kΨG(λ) k = kΨG(λ) PIm k. Hence kPD,λ k ≥ k(I − QD )PD,λ QD k = kΨG(λ) k ≥ |G(λ)ij | −−−→ ∞. λ→0

Then the map λ 7→ PD,λ can not be constant. Remark 5.2.2. Note that, using Eq. (5.8), Remark 5.2.1 (particularly Eqs. (5.9) and (5.10) ), and the notations of the Theorem, we can conclude that the map (0, 1) × U (D) 3 (λ, N ) 7−→ AdσD (N ) PD,λ AdσD (N )∗ = PN,λ is of class C ∞ . Another way to prove it is using that the map (λ, N ) 7→ TN ∆λ is smooth, and then to apply Eqs. (5.2) and (5.6).

References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations and Operator Theory 13 (1990), 307-315. [2] T. Ando, Aluthge transforms and the convex hull of the eigenvalues of a matrix, Linear Multilinear Algebra 52 (2004), 281-292. [3] T. Ando and T. Yamazaki, The iterated Aluthge transforms of a 2-by-2 matrix converge, Linear Algebra Appl. 375 (2003), 299-309. [4] J. Antezana, P. Massey and D. Stojanoff, λ-Aluthge transforms and Schatten ideals, Linear Algebra Appl. 405 (2005), 177-199. [5] J. Antezana, E. Pujals and D. Stojanoff, Convergence of the iterated Aluthge transform sequence for diagonalizable matrices, Advances in Math. 216 (2007), no. 1, 255–278. [6] G. Corach, H. Porta and L. Recht, The geometry of spaces of 1 projections in C∗ algebras, Adv. Math. 101 (1993), 59-77. [7] G. Corach, H. Porta and L. Recht, The geometry of spaces of selfadjoint invertible elements of a C∗ -algebra, Integral Equations and Operator Theory 16 (1993), 771794. [8] J. L. Dalecki˘ı, S. G. Kre˘ın, Formulas of differentiation according to a parameter of functions of Hermitian operators. (Russian) Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 13-16. [9] J. L. Dalecki˘ı, S. G. Kre˘ın, Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations. (Russian) Voroneˇz. Gos. Univ. Trudy Sem. Funkcional. Anal. 1956 (1956), no. 1, 81-105. [10] K. Dykema and H. Schultz, On Aluthge Transforms: continuity properties and Brown measure, preprint. Los Alamos preprint version in the site arXiv:math/0512197v3 [math.OA]

488


IEOT

[11] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. [12] R. Horn and C. Johnson Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. [13] Huajun Huang and Tin-Yau Tam, On the convergence of the Aluthge sequence, Oper. Matrices 1 (2007), 121-141. [14] I. Jung, E. Ko and C. Pearcy, Aluthge transform of operators, Integral Equations Operator Theory 37 (2000), 437-448. [15] I. Jung, E. Ko and C. Pearcy, The iterated Aluthge transform of an operator, Integral Equations and Operator Theory 45 (2003), 375-387. [16] K. Okubo, On weakly unitarily invariant norm and the Aluthge transformation, Linear Algebra and Appl. 371 (2003), 369-375. [17] H. Porta and L. Recht, Minimality of geodesics in Grassmann manifolds, Proc. Amer. Math. Soc. 100 (1987), 464-466. [18] H. Porta and L. Recht, Variational and convexity properties of families of involutions, Integral Equations and Operator Theory 21 (1995), 243-253. [19] P. Y. Wu, Numerical range of Aluthge transform of operator, Linear Algebra and Appl. 357 (2002), 295-298. [20] T. Yamazaki, An expression of the spectral radius via Aluthge transformation, Proc. Amer. Math. Soc. 130 (2002), 1131-1137. Jorge Antezana Depto. de Matem´ atica, FCE-UNLP and IAM-CONICET Saavedra 15, 3er Piso CP 1083 Buenos Aires Argentina e-mail: [email protected] Enrique Pujals Instituto Nacional de Matem´ atica Pura y Aplicada (IMPA) Dona Castorina 110 cep 22460-320 Rio de Janeiro Brasil e-mail: [email protected] Demetrio Stojanoff Depto. de Matem´ atica, FCE-UNLP and IAM-CONICET Saavedra 15, 3er Piso CP 1083 Buenos Aires Argentina e-mail: [email protected] Submitted: May 30, 2007.


auser Verlag Basel/Switzerland 0378-620X/040489-27, published online November 10, 2008 DOI 10.1007/s00020-008-1634-1


PT -Symmetric Waveguides Denis Borisov and David Krejˇciˇr´ık Abstract. We introduce a planar waveguide of constant width with nonHermitian PT -symmetric Robin boundary conditions. We study the spectrum of this system in the regime when the boundary coupling function is a compactly supported perturbation of a homogeneous coupling. We prove that the essential spectrum is positive and independent of such perturbation, and that the residual spectrum is empty. Assuming that the perturbation is small in the supremum norm, we show that it gives rise to real weakly-coupled eigenvalues converging to the threshold of the essential spectrum. We derive sufficient conditions for these eigenvalues to exist or to be absent. Moreover, we construct the leading terms of the asymptotic expansions of these eigenvalues and the associated eigenfunctions. Mathematics Subject Classification (2000). 35P15, 35J05, 47B44, 47B99. Keywords. Non-self-adjointness, J-self-adjointness, waveguides, PT -symmetry, Robin boundary conditions, Robin Laplacian, eigenvalue and eigenfunction asymptotics, essential spectrum, reality of the spectrum.

1. Introduction There are two kinds of motivations for the present work. The first one is due to the growing interest in spectral theory of non-self-adjoint operators. It is traditionally relevant to the study of dissipative processes, resonances if one uses the mathematical tool of complex scaling, and many others. The most recent and conceptually new application is based on the potential quantum-mechanical interpretation of non-Hermitian Hamiltonians which have real spectra and are invariant under a simultaneous P-parity and T -time reversal. For more information on the subject, we refer to the pioneering work [3] and especially to the recent review [2] with many references. The other motivation is due to the interesting phenomena of the existence of bound states in quantum-waveguide systems intensively studied for almost two decades. Here we refer to the pioneering work [12] and to the reviews [10, 21]. In

490

Borisov and Krejˇciˇr´ık

IEOT

these models the Hamiltonian is self-adjoint and the bound states – often without classical interpretations – correspond to an electron trapped inside the waveguide. In this paper we unify these two fields of mathematical physics by considering a quantum waveguide modelled by a non-Hermitian PT -symmetric Hamiltonian. Our main interest is to develop a spectral theory for the Hamiltonian and demonstrate the existence of eigenvalues outside the essential spectrum. For non-selfadjoint operators the location of the various essential spectra is often as much as one can realistically hope for in the absence of the powerful tools available when the operators are self-adjoint, notably the spectral theorem and minimax principle. In the present paper we overcome this difficulty by using perturbation methods to study the point spectrum in the weak-coupling regime. In certain situations we are also able to prove that the total spectrum is real. Let us now briefly recall the notion of PT -symmetry. If the underlying Hilbert space of a Hamiltonian H is the usual realization of square integrable functions L2 (Rn ), the PT -symmetry invariance can be stated in terms of the commutator relation (PT )H = H(PT ) , (1.1) where the parity and time reversal operators are defined by (Pψ)(x) := ψ(−x) and T ψ := ψ, respectively. In most of the PT -symmetric examples H is the Schr¨ odinger operator −∆ + V with a potential V satisfying (1.1), so that H ∗ = T HT where H ∗ denotes the adjoint of H. This property is known as the T self-adjointness of H in the mathematical literature [11], and it is not limited to PT -symmetric Schr¨ odinger operators. More generally, given any linear operator H in an abstract Hilbert space H, we understand the PT -symmetry property as a special case of the J-self-adjointness of H: H ∗ = JHJ ,

(1.2)

where J is a conjugation operator, i.e., ∀φ, ψ ∈ H ,

(Jφ, Jψ)H = (ψ, φ)H ,

J 2ψ = ψ .

This setting seems to be adequate for a rigorous formulation of PT -symmetric problems, and alternative to that based on Krein spaces [22, 24]. The nice feature of the property (1.2) is that H “is not too far” from the class of self-adjoint operators. In particular, the eigenvalues are found to be real for many PT -symmetric Hamiltonians [28, 9, 22, 8, 26, 7, 20]. However, the situation is much less studied in the case when the resolvent of H is not compact. The spectral analysis of non-self-adjoint operators is more difficult than in the self-adjoint case, partly because the residual spectrum is in general not empty for the former. One of the goals of the present paper is to point out that the existence of this part of spectrum is always ruled out by (1.2): Fact. Let H be a densely defined closed linear operator in a Hilbert space satisfying (1.2). Then the residual spectrum of H is empty.

Vol. 62 (2008)

PT -Symmetric Waveguides

491

The proof follows easily by noticing that the kernels of H − λ and H ∗ − λ have the same dimension [11, Lem. III.5.4] and by the the general fact that the orthogonal complement of the range of a densely defined closed operator in a Hilbert space is equal to the kernel of its adjoint. The above result is probably not well known in the PT -symmetry community. We continue with an informal presentation of our model and main spectral results obtained in this paper. The rigorous and more detailed statements are postponed until the next section because they require a number of technical definitions. The Hamiltonian we consider in this paper acts as the Laplacian in the Hilbert space of square integrable functions over a straight planar strip and the nonHermiticity enters through PT -symmetric boundary conditions only. The boundary conditions are of Robin type but with imaginary coupling. The PT -symmetric invariance then implies that we actually deal with an electromagnetic waveguide with radiation/dissipative boundary conditions. In fact, the one-dimensional spectral problem in the waveguide cross-section has been studied recently in [20] (see also [19]) and our model can be viewed as a two-dimensional extension of the former. Schr¨ odinger-type operators with similar non-Hermitian boundary conditions were studied previously by Kaiser, Neidhardt and Rehberg [17, 16, 15]. In their papers, motivated by the needs of semiconductor physics, the configuration space is a bounded domain and the boundary coupling function is such that the Hamiltonian is a dissipative operator. The latter excludes the PT -symmetric models of [20] and the present paper. The T -self-adjointness property (1.2) of our Hamiltonian is proved in Section 3. If the boundary coupling function is constant, the spectral problem can be solved by separation of variables and we find that the spectrum is purely essential, given by a positive semibounded interval (cf Section 4). In Section 5 we prove that the essential spectrum is stable under compactly supported perturbations of the coupling function. Consequently, the essential spectrum is always real in our setting, however, it exhibits important differences as regards similar self-adjoint problems. Namely, it becomes as a set independent of the value of the coupling function at infinity when the latter overpasses certain critical value. In Section 6 we study the point spectrum. We focus on the existence of eigenvalues emerging from the threshold of the essential spectrum in the limit when the compactly supported perturbation of the coupling function tends to zero in the supremum norm. It turns out that the weakly-coupled eigenvalues may or may not exist, depending on mean values of the local perturbation. In the case when the point spectrum exists, we derive asymptotic expansions of the eigenvalues and the associated eigenfunctions. Because of the singular nature of the PT -symmetric interaction, our example is probably the simplest non-trivial, multidimensional PT -symmetric model whatsoever for which both the point and essential spectra exist. We hope that the present work will stimulate more research effort in the direction of spectral

492


IEOT

and scattering properties of the present and other non-Hermitian PT -symmetric operators.

2. Main results Given a positive number d, we write I := (0, d) and consider an infinite straight strip Ω := R × I. We split the variables consistently by writing x = (x1 , x2 ) with x1 ∈ R and x2 ∈ I. Let α be a bounded real-valued function on R; occasionally we shall denote by the same symbol the function x 7→ α(x1 ) on Ω. The object of our interest is the operator in the Hilbert space L2 (Ω) which acts as the Laplacian and satisfies the following PT -symmetric boundary conditions: ∂2 Ψ + i α Ψ = 0

on ∂Ω .

(2.1)

More precisely, we introduce Hα Ψ := −∆Ψ ,

Ψ ∈ D(Hα ) := Ψ ∈ W22 (Ω) | Ψ satisfies (2.1) ,

(2.2)

where the action of Hα should be understood in the distributional sense and (2.1) should be understood in the sense of traces [1]. In Section 3 we show that Hα is well defined in the sense that it is an m-sectorial operator and that its adjoint is easy to identify: 1 (R). Then Hα is an m-sectorial operator in L2 (Ω) Theorem 2.1. Let α ∈ W∞ satisfying Hα∗ = H−α . (2.3)

Of course, Hα is not self-adjoint unless α vanishes identically (in this case H0 is the Neumann Laplacian in L2 (Ω)). However, Hα is T -self-adjoint, i.e., it satisfies (1.2) with J being the complex conjugation T : Ψ 7→ Ψ. Indeed, Hα satisfies the relation (2.3) and it is easy to see that H−α = T Hα T .

(2.4)

This reflects the PT -symmetry (1.1) of our problem, with P being defined by (PΨ)(x) := Ψ(x1 , d − x2 ). An important property of an operator H in a Hilbert space H being msectorial is that it is closed. Then, in particular, the spectrum σ(H) is well defined as the set of complex points z such that H − z is not bijective as the operator from D(H) to H. Furthermore, its spectrum is contained in a sector of complex numbers z such that | arg(z − γ)| 6 θ with some γ ∈ R and θ ∈ [0, π/2). In our case, however, we are able to establish a stronger result n o √ σ(Hα ) ⊆ Ξα := z ∈ C : Re z > 0 , | Im z| 6 2 kαkL∞ (R) Re z . (2.5) This follows directly from Lemma 3.1 on which the proof of Theorem 2.1 is based (cf the end of Section 3 for more details). Consequently, the resolvent set ρ(Hα ) := C \ σ(Hα ) contains the complement of Ξα and we have the bound k(Hα − z)−1 k ≤ 1/dist(z, ∂Ξα )

for all z ∈ C \ Ξα ,

(2.6)

Vol. 62 (2008)


493

where k · k denotes the operator norm in L2 (Ω). Given a closed operator H in a Hilbert space H, we use the following decomposition of the spectrum σ(H): Definition. The point spectrum σp (H) equals the set of points λ such that H − λ is not injective. The essential spectrum σe (H) equals the set of points λ such that H − λ is not Fredholm. Finally, the residual spectrum σr (H) equals the set of points λ such that H − λ is injective but the range of H − λ is not dense in H. Remark 2.1. 1. The reader is warned that various other types of essential spectra of non-self-adjoint operators are used in the literature; cf [11, Chapt. IX] for five distinct definitions and a detailed description of their properties. Among them we choose that of Wolf [27], which is in general larger than that of Kato [18, Sec. IV.5.6] based on violating the semi-Fredholm property. (Recall that a closed operator in a Hilbert space is called Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional, while it is called semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional.) However, since our operator Hα is T -self-adjoint, the majority of the different definitions coincide [11, Thm IX.1.6], in particular the two above, and that is why we use the common notation σe (·) in this paper. Then our choice also coincides with the definition of “continuous spectrum” as used for instance in the Glazman’s book [14]. 2. We indeed have the decomposition (cf [14, Sec. I.1.1]) σ(H) = σp (H) ∪ σe (H) ∪ σr (H) , but note that there might be intersections on the right-hand side. In particular, σe (H) contains eigenvalues of infinite geometric multiplicity. 3. On the other hand, the definitions of point and residual spectra are standard and they form disjoint subsets of σ(H). Recalling the general fact [18, Sec. V.3.1] that the orthogonal complement of the range of a densely defined closed operator in a Hilbert space is equal to the kernel of its adjoint, we obtain the following characterization of the residual spectrum in terms of the point spectrum of the operator and its adjoint: ¯ ∈ σp (H ∗ ) & λ 6∈ σp (H) . σr (H) = λ ∈ C | λ (2.7) The T -self-adjointness of Hα immediately implies: Corollary 2.1. Suppose the hypothesis of Theorem 2.1. Then σr (Hα ) = ∅ . Proof. We repeat the proof sketched in Introduction. Since Hα is T -self-adjoint, it is easy to see that λ is an eigenvalue of Hα (with eigenfunction Ψ) if, and only if, ¯ is an eigenvalue of H ∗ (with eigenfunction Ψ). It is then clear from the general λ α identity (2.7) that the residual spectrum of Hα must be empty.

494


IEOT

The case of uniform boundary conditions, i.e. when α equals identically a constant α0 , can be solved by separation of variables (cf Section 4). We find σ(Hα0 ) = σe (Hα0 ) = [µ20 , +∞) , where the threshold

µ20 ,

(2.8)

with the notation ( α0 if |α0 | 6 π/d , µ0 = π/d if |α0 | > π/d ,

(2.9)

denotes the bottom of the spectrum of the “transverse” operator −∆Iα0 ψ := −ψ 00 , ψ ∈ D(−∆Iα0 ) := ψ ∈ W22 (I) | ψ 0 + iα0 ψ = 0

at ∂I .

(2.10)

The operator −∆Iα0 was studied in [20]. Its spectrum is purely discrete and real: σ(−∆Iα0 ) = {µ2j }∞ j=0 , where µ0 has been introduced in (2.9), ( α0 if |α0 | > π/d , µ1 = π/d if |α0 | 6 π/d ,

and

µj := πj/d

(2.11)

for j > 2 .

Making the hypothesis α0 d/π 6∈ Z\{0} ,

(2.12)

−∆Iα0

the eigenvalues of are simple and the corresponding set of eigenfunctions {ψj }∞ can be chosen as j=0 α0 ψj (x2 ) := cos(µj x2 ) − i sin(µj x2 ) . (2.13) µj We refer to Section 4.1 for more results about the operator −∆Iα0 . Let us now turn to the non-trivial case of variable coupling function α. Among a variety of possible situations, in this paper we restrict the considerations to local perturbations of the uniform case. Namely, we always assume that the difference α − α0 is compactly supported. First of all, in Section 5 we show that the essential component of the spectrum of Hα is stable under the local perturbation of the uniform case: 1 Theorem 2.2. Let α − α0 ∈ C0 (R) ∩ W∞ (R) with α0 ∈ R. Then

σe (Hα ) = [µ20 , +∞) . Notice that the essential spectrum as a set is independent of α0 as long as |α0 | > π/d. This is a consequence of the fact that our Hamiltonian is not Hermitian. On the other hand, it follows that the essential spectrum is real. Recall that the residual spectrum is always empty due to Corollary 2.1. We do not have the proof of the reality for the point spectrum, except for the particular case treated in the next statement:

Vol. 62 (2008)


495

1 Theorem 2.3. Let α ∈ C0 (R) ∩ W∞ (R) be an odd function. Then

σp (Hα ) ⊂ R . Summing up, under the hypotheses of this theorem the total spectrum is real (and in fact non-negative due to (2.5)). The next part of our results concerns the behavior of the point spectrum of Hα under a small perturbation of α0 . Namely, we consider the local perturbation of the form α(x1 ) = α0 + ε β(x1 ) , (2.14) where β ∈ C02 (R) and ε is a small positive parameter. In accordance with Theorem 2.2, in this case the essential spectrum of Hα coincides with [µ20 , +∞), and this is also the spectrum of Hα0 . Our main interest is focused on the existence and asymptotic behavior of the eigenvalues emerging from the threshold µ20 due to the perturbation of Hα0 by εβ. First we show that the asymtotically Neumann case is in some sense exceptional: Theorem 2.4. Suppose α0 = 0. Let α be given by (2.14), where β ∈ C02 (R). Then the operator Hα has no eigenvalues converging to µ20 as ε → +0. The problem of existence of the weakly-coupled eigenvalues is more subtle as long as α0 6= 0. To present our results in this case, we introduce an auxiliary sequence of functions vj : R → R by Z  1  |x1 − t1 |β(t1 ) dt1 if j = 0 ,  −2  R Z √ 2 2 (2.15) vj (x1 ) := 1  q e− µj −µ0 |x1 −t1 | β(t1 ) dt1 if j > 1 .    2 µ2 − µ2 R 0 j R Denoting hf i = R f (x1 ) dx1 for any f ∈ L1 (R), we introduce a constant τ , depending on β, d and α0 , by  ∞  2α0 X µ2j hβvj i α0 d + jπ π  2  2α hβv i + tan if |α0 | < ,  0 0 2 2  d j=1 µj − µ0 2 d  τ := ∞  X µ22j hβv2j i 2α0 π 2 cot α20 d 8π 2 π    hβv i + if |α0 | > . 1  2 2 2 2  (µ1 − µ0 )d3 (µ1 − µ0 )d4 j=1 µ22j − µ21 d It will be shown in Section 6.3 that the series converge. Finally, we denote Ωa := Ω ∩ {x : |x1 | < a} for any positive a. Now we are in a position to state our main results about the point spectrum. Theorem 2.5. Suppose |α0 | < π/d. Let α be given by (2.14), where β ∈ C02 (R). 1. If α0 hβi < 0, there exists the unique eigenvalue λε of Hα converging to µ20 as ε → +0. This eigenvalue is simple and real, and satisfies the asymptotic formula λε = µ20 − ε2 α02 hβi2 + 2ε3 α0 τ hβi + O(ε4 ) .

496


IEOT

The associated eigenfunction Ψε can be chosen so that it satisfies the asymptotics Ψε (x) = ψ0 (x2 ) + O(ε) (2.16) 2 in W2 (Ωa ) for each a > 0, and behaves at infinity as √ 2 √ 2 (2.17) Ψε (x) = e− µ0 −λε |x1 | ψ0 (x2 ) + O(e− µ0 −λε |x1 | ) , |x1 | → +∞ . 2. If α0 hβi > 0, the operator Hα has no eigenvalues converging to µ20 as ε → +0. 3. If hβi = 0, and τ > 0, there exists the unique eigenvalue λε of Hα converging to µ20 as ε → +0. This eigenvalue is simple and real, and satisfies the asymptotics λε = µ20 − ε4 τ 2 + O(ε5 ) . (2.18) The associated eigenfunction can be chosen so that the relations (2.16) and (2.17) hold true. 4. If hβi = 0, and τ < 0, the operator Hα has no eigenvalues converging to µ20 as ε → +0. Theorem 2.6. Suppose |α0 | > π/d and (2.12). Let α be given by (2.14) where β ∈ C02 (R). 1. If τ > 0, there exists the unique eigenvalue λε of Hα converging to µ0 as ε → +0, it is simple and real, and satisfies the asymptotics (2.18). The associated eigenfunction can be chosen so that it obeys (2.16) and (2.17). 2. If τ < 0, the operator Hα has no eigenvalues converging to µ20 as ε → +0. In accordance with Theorem 2.5, in the case |α0 | < π/d the existence of a weakly-coupled eigenvalue is determined by the sign of the constant α0 and that of the mean value of β. In the language of Schrödinger operators (treating α as a singular potential), it means that a given non-trivial β plays the role of an effective interaction, attractive or repulsive depending upon the sign of α0 . It is instructive to compare this situation with a self-adjoint waveguide [6], where a similar effective interaction is induced by a local deformation of the boundary. If the boundary is deformed “outward in the mean”, a weakly-coupled bound state exists, while it is absent if the deformation is “inward-pointing in the mean”. As usual, the critical situation hβi = 0 is much harder to treat. In our case, one has to check the sign of τ to decide whether a weakly-coupled bound state exists. However, it can be difficult to sum up the series in the definition of τ . This is why in our next statement we provide a sufficient condition guaranteeing that τ > 0. Proposition 2.1. Suppose 0 < |α0 | < π/d. Let α be given by (2.14) where β(x1 ) = e = 0, l > 0. If βe (x1 /l), βe ∈ C02 (R), hβi

Z

2

4 cot α20 d µ21 d2 d2 e 2 e 1 ) dt1

sign(· − t1 )β(t + + kβkL2 (R) , > 2

l α0 d (µ21 − µ20 )2 16π 2 48 L2 (R) R

then τ > 0.

Vol. 62 (2008)


497

The meaning of this proposition is that for each positive |α0 | < π/d the perturbation εβ in the critical regime hβi = 0 produces a weakly-coupled eigenvalue near the threshold of the essential spectrum provided that the support of β is wide enough. This is in perfect agreement with the critical situation of [6]; according to higher-order asymptotics derived in [5], here the weakly-coupled bound state exists if, and only if, the critical boundary deformation is smeared enough. In the case |α0 | > π/d a sufficient condition guaranteeing τ > 0 is given in Proposition 2.2. Suppose |α0 | > π/d and (2.12). Let α be given by (2.14) where β ∈ C02 (R). Let m be the maximal positive integer such that µ2m < |α0 |. If α0 hβv1 i cot

m 4 X µ22j hβv2j i α0 d , > 2 d j=1 µ21 − µ22j

(2.19)

then τ > 0. In Section 6.6 we will show that the inequality (2.19) makes sense. Namely, it will be proved that there exists β such that this inequality holds true, provided that α0 is close enough to µ2 but greater than this value. Remark 2.2. It is useful to make the hypothesis (2.12), since it implies that the “transverse” eigenfunctions (2.13) form a basis (cf (4.1)) and makes it therefore possible to obtain a relatively simple decomposition of the resolvent of Hα0 (cf Lemma 4.3). However, it is rather a technical hypothesis for many of the spectral results (e.g., Theorem 2.2). On the other hand, it seems that the hypothesis is rather crucial for the statement of Theorem 2.6 and Proposition 2.2. If the hypothesis (2.12) is omitted and α0 = π`/d, with ` ∈ Z \ {0}, the threshold of the essential spectrum is π 2 /d2 . This point corresponds to a simple eigenvalue of the “transverse” operator −∆Iα0 only if |`| > 1, while it is a double eigenvalue if |`| = 1. Under the hypothesis of Theorems 2.5 and 2.6, the threshold is always a simple eigenvalue of −∆Iα0 , and the proof of the theorems actually employs some sort of “non-degenerate” perturbation theory. In view of this, we conjecture that in the degenerate case |`| = 1 two simple eigenvalues (possibly forming a complex conjugate pair) or one double (real) eigenvalue can emerge from the threshold of the essential spectrum for a suitable choice of β, while in the case |`| > 1 there can be at most one simple emerging eigenvalue. The question on the asymptotic behaviour of these eigenvalues constitutes an interesting open problem.

3. Definition of the operator In this section we prove Theorem 2.1. Our method is based on the theory of sectorial sesquilinear forms [18, Sec. VI].

498


IEOT

In the beginning we assume only that α is bounded. Let hα be the sesquilinear form defined in L2 (Ω) by the domain D(hα ) := W21 (Ω) and the prescription hα := h1α + ih2α with Z h1α (Ψ, Φ) := ∇Ψ(x) · ∇Φ(x) dx , ZΩ Z 2 hα (Ψ, Φ) := α(x1 ) Ψ(x1 , d) Φ(x1 , d) dx1 − α(x1 ) Ψ(x1 , 0) Φ(x1 , 0) dx1 , R

R

for any Ψ, Φ ∈ D(hα ). Here the dot denotes the scalar product in R2 and the boundary terms should be understood in the sense of traces [1]. We write hα [Ψ] := hα (Ψ, Ψ) for the associated quadratic form, and similarly for h1α and h2α . Clearly, hα is densely defined. It is also clear that the real part h1α is a densely defined, symmetric, positive, closed sesquilinear form (it is associated to the selfadjoint Neumann Laplacian in L2 (Ω)). Of course, hα itself is not symmetric unless α vanishes identically; however, it can be shown that it is sectorial and closed. To see it, one can use the perturbation result [18, Thm. VI.1.33] stating that the sum of a sectorial closed form with a relatively bounded form is sectorial and closed provided the relative bound is less than one. In our case, the imaginary part h2α plays the role of the small perturbation of h1α by virtue of the following result. Lemma 3.1. Let α ∈ L∞ (R). Then h2α is relatively bounded with respect to h1α , with p 2 hα [Ψ] 6 2 kαkL∞ (R) kΨkL (Ω) h1α [Ψ] 6 δ h1α [Ψ] + δ −1 kαk2 ∞ kΨk2 L (R) L2 (Ω) 2 for all Ψ ∈ W21 (Ω) and any positive number δ. Proof. By density [1, Thm. 3.18], it is sufficient to prove the inequality for restrictions to Ω of functions Ψ in C0∞ (R2 ). Then we have Z 2 2 hα [Ψ] = α(x1 ) ∂|Ψ(x)| dx 6 2 kαkL∞ (R) kΨkL (Ω) k∂2 ΨkL (Ω) , 2 2 ∂x2 Ω which gives the first inequality after applying k∂2 ΨkL2 (Ω) 6 k∇ΨkL2 (Ω) . The second inequality then follows at once by means of the Cauchy inequality with δ. In view of the above properties, Theorem VI.1.33 in [18], and the first representation theorem [18, Thm. VI.2.1], there exists the unique m-sectorial opera˜ α in L2 (Ω) such that hα (Ψ, Φ) = (H ˜ α Ψ, Φ) for all Ψ ∈ D(H ˜ α ) ⊂ D(hα ) and tor H Φ ∈ D(hα ), where ˜ α ) = Ψ ∈ W21 (Ω)| ∃F ∈ L2 (Ω), ∀Φ ∈ W21 (Ω), hα (Ψ, Φ) = (F, Φ)L (Ω) . D(H 2 By integration by parts, it is easy to check that if Ψ ∈ D(Hα ), it follows that ˜ α ) with F = −∆Ψ. That is, H ˜ α is an extension of Hα as defined in (2.2). Ψ ∈ D(H ˜ α in order to prove Theorem 2.1. However, It remains to show that actually Hα = H the other inclusion holds as a direct consequence of the representation theorem and the following result.


Vol. 62 (2008)

499

1 Lemma 3.2. Let α ∈ W∞ (R). For each F ∈ L2 (Ω), a generalized solution Ψ to the problem −∆Ψ = F in Ω , (3.1) ∂2 Ψ + i α Ψ = 0 on ∂Ω ,

belongs to D(Hα ). Proof. For any function Ψ ∈ W21 (Ω), we introduce the difference quotient Ψδ (x) :=

Ψ(x1 + δ, x2 ) − Ψ(x) , δ

where δ is a small real number. By standard arguments [23, Ch. III, Sec. 3.4, Thm. 3], the estimate kΨδ kL2 (Ω) 6 kΨkW21 (Ω)

(3.2)

holds true for all δ small enough. If Ψ is a generalized solution to (3.1), then Ψδ is a generalized solution to the problem

−∆Ψδ = Fδ ∂2 Ψδ + i α Ψδ = g

in Ω , on ∂Ω ,

where g denotes the trace of the function x 7→ −iαδ (x1 )Ψ(x1 + δ, x2 ) to the boundary ∂Ω. Using the “integration-by-parts” formula for the difference quotients, (Fδ , Φ)L2 (Ω) = −(F, Φ−δ )L2 (Ω) , the integral identity corresponding to the weak formulation of the boundary value problem for Ψδ can be written as follows Z hα (Ψδ , Φ) = −(F, Φ−δ )L2 (Ω) − i R

αδ (x1 ) Ψ(x1 + δ, d) Φ(x1 , d) dx1 Z + i αδ (x1 ) Ψ(x1 + δ, 0) Φ(x1 , 0) dx1 , R

where Φ ∈ W21 (Ω) is arbitrary. Letting Φ = Ψδ , and using the embedding of W21 (Ω) in L2 (∂Ω), the boundedness of αδ , Lemma 3.1 and (3.2), the above identity yields kΨδ kW21 (Ω) 6 C , where the constant C is independent of δ. Employing this estimate and proceeding as in the proof of Item b) of Theorem 3 in [23, Ch. III, Sec. 3.4], one can show easily that ∂1 Ψ ∈ W21 (Ω). Hence, ∂11 Ψ ∈ L2 (Ω) and ∂12 Ψ ∈ L2 (Ω). If follows from standard elliptic regularity theorems (see, e.g., [23, Ch. IV, 2 Sec. 2.2]) that Ψ ∈ W2,loc (Ω). Hence, the first of the equations in (3.1) holds true a.e. in Ω. Thus, ∂22 Ψ = −F − ∂11 Ψ ∈ L2 (Ω), and therefore Ψ ∈ W22 (Ω).

500


IEOT

It remains to check the boundary condition for Ψ. Integrating by parts, one has (F, Φ)L2 (Ω) =hα (Ψ, Φ) = (−∆Ψ, Φ)L2 (Ω) Z + ∂2 Ψ(x1 , d) + i α(x1 ) Ψ(x1 , d) Φ(x1 , d) dx1 ZR − ∂2 Ψ(x1 , 0) + i α(x1 ) Ψ(x1 , 0) Φ(x1 , 0) dx1 R

W21 (Ω).

for any Φ ∈ This implies the boundary conditions because −∆Ψ = F a.e. in Ω and Φ is arbitrary. Summing up the results of this section, we get 1 ˜ α = Hα . Proposition 3.1. Let α ∈ W∞ (R). Then H

Theorem 2.1 follows as a corollary of this proposition. In particular, the latter implies that Hα is m-sectorial. Moreover, by the first representation theorem, we ˜ α∗ is simply obtained as the operator associated with know that the adjoint H ∗ hα = h−α . This together with Proposition 3.1 proves (2.3). Let us finally comment on the results (2.5) and (2.6). As a direct consequence ˜ α ), of the first inequality of Lemma 3.2, we get that the numerical range of Hα (= H defined as the set of all complex numbers (Hα Ψ, Ψ)L2 (Ω) where Ψ changes over all Ψ ∈ D(Hα ) with kΨkL2 (Ω) = 1, is contained in the set Ξα . Hence, in view of general results about numerical range (cf [18, Sec. V.3.2]), the exterior of the numerical range of Hα is a connected set, and one indeed has (2.5) and (2.6).

4. The unperturbed waveguide In this section we consider the case of uniform boundary conditions in the sense that α is supposed to be identically equal to a constant α0 ∈ R. We prove the spectral result (2.8) by using the fact that Hα0 can be decomposed into a sum of the “longitudinal” operator −∆R , i.e. the self-adjoint Laplacian in L2 (R), and the “transversal” operator −∆Iα0 defined in (2.10). 4.1. The transversal operator We summarize here some of the results established in [20] and refer to that reference for more details. The adjoint of −∆Iα0 is simply obtained by the replacement α0 7→ −α0 , i.e., (−∆Iα0 )∗ = −∆I−α0 . Consequently, (−∆Iα0 )∗ has the same spectrum (2.11) and the corresponding set of eigenfunctions {φj }∞ j=0 can be chosen as φj (x2 ) := Aj ψj (x2 ) ,


Vol. 62 (2008)

501

where {ψj }∞ j=0 have been introduced in (2.13) and Aj are normalization constants. Choosing Aj0 :=

2iα0 , 1 − exp (−2iα0 d)

Aj1 :=

(µ21

2µ21 , − α02 )d

Aj :=

(µ2j

2µ2j , − α02 )d

where j > 2, (j0 , j1 ) = (0, 1) if |α0 | < π/d and (j0 , j1 ) = (1, 0) if |α0 | > π/d (if α0 = 0, the fraction in the definition of Aj0 should be understood as the expression obtained after taking the limit α0 → 0), we have the biorthonormality relations ∀j, k ∈ N,

(ψj , φk )L2 (I) = δjk

together with the biorthonormal-basis-type expansion (cf [20, Prop. 4]) ∀ψ ∈ L2 (I) ,

ψ=

∞ X

(ψ, φj )L2 (I) ψj .

(4.1)

j=0

Let us show that (4.1) can be extended to L2 (Ω). Lemma 4.1. For any Ψ ∈ L2 (Ω), the identity Ψ(x) =

∞ X

Ψj (x1 ) ψj (x2 )

Ψj (x1 ) := Ψ(x1 , ·), φj

with

L2 (I)

j=0

holds true in the sense of L2 (Ω)-norm. Proof. In view of (4.1), the series converges to Ψ in L2 (I) for almost every x1 ∈ R. We use the dominated convergence theorem to prove that the convergence actually holds in the norm of L2 (Ω). To do so, it is sufficient to check that the L2 (I)-norm of the partial sums can be uniformly estimated by a function from L2 (R). Let us introduce χD (x2 ) := sin(πjx2 /d) and χN /d) for j√ j (x2 ) := cos(πjx p p 2 N ∞ N ∞ j > 1, and χ0 (x2 ) := 1/ 2. We recall that { 2/d χD } and { 2/d χ j j=1 j }j=0 form complete orthonormal families in L2 (I). Expressing ψj in terms of χN j and D χj for j > 2, and using the orthonormality, we have (n > 2) n

X

2

Ψj (x1 ) ψj

j=2

L2 (I)

n

2

X

6 d Ψj (x1 ) χN j

=d

L2 (I)

j=2 n X

|Ψj (x1 )|2 + d α02

j=2

6d 1+

n

2

X

+ d α02 Ψj (x1 ) χD j /µj

L2 (I)

j=2

n X

|Ψj (x1 )|2 /µ2j

j=2

α02 µ22

Next, writing (j > 2) r d α0 D N Ψj = Aj Ψj − i Ψj 2 µj

X ∞

|Ψj (x1 )|2 .

(4.2)

j=2

with

Ψ]j (x1 ) := Ψ(x1 , ·), χ]j

L2 (I)

,

502


IEOT

noticing that |Aj | 6 c (valid for all j > 0) where c is a constant depending uniquely D on |α0 | and d, and using the Parseval identities for χN j and χj , we obtain ! ∞ ∞ X X α02 D 2 2 2 N 2 |Ψj (x1 )| 6 c d |Ψj | + 2 |Ψj | µj j=2 j=2 α2 (4.3) 6 c2 d 1 + 20 kΨ(x1 , ·)k2L2 (I) . µ2 At the same time, using just the estimates |ψj |2 6 (1 + α02 /µ2j ) valid for all j > 0, we readily get 1

X

α2

6 2 c d 1 + 20 kΨ(x1 , ·)kL2 (I) . Ψj (x1 ) ψj (4.4)

µ0 L2 (I) j=0 Summing up, n

X

Ψj (x1 ) ψj

j=0

L2 (I)

6 C kΨ(x1 , ·)kL2 (I) ∈ L2 (R) ,

where C is a constant independent of n, and the usage of the dominated convergence theorem is justified. 4.2. Spectrum of the unperturbed waveguide First we show that the spectrum of Hα0 is purely essential. Since the residual spectrum is always empty due to Corollary 2.1, it is enough to show that there are no eigenvalues. Proposition 4.1. Let α0 ∈ R satisfy (2.12). Then σp (Hα0 ) = ∅ . Proof. Suppose that Hα0 possesses an eigenvalue λ with eigenfunction Ψ. Multiplying the eigenvalue equation with φj and integrating over I, we arrive at the equations −Ψ00j = (λ − µ2j )Ψj in R, j > 0, where Ψj are the coefficients of Lemma 4.1. Since Ψj ∈ L2 (R) due to Fubini’s theorem, each of the equations has just a trivial solution. This together with Lemma 4.1 yields Ψ = 0, a contradiction. That is, the point spectrum of Hα0 is empty. Remark 4.1. Regardless of the technical hypothesis (2.12), the set of isolated eigenvalues of Hα0 is always empty. This follows from Proposition 4.1 and the fact that α0 7→ Hα0 forms a holomorphic family of operators [18, Sec. VII.4]. It is well known that the spectrum of the “longitudinal” operator −∆R is also purely essential and equal to the semi-axis [0, +∞). In view of the separation of variables, it is reasonable to expect that the (essential) spectrum of Hα0 will

Vol. 62 (2008)


503

be given by that semi-axis shifted by the first eigenvalue of −∆Iα0 . First we show that the resulting interval indeed belongs to the spectrum of Hα0 . Lemma 4.2. Let α0 ∈ R. Then [µ20 , +∞) ⊆ σe (Hα0 ). Proof. Since the spectrum of Hα0 is purely essential, it can be characterized by means of singular sequences [11, Thm. IX.1.3] (it is in fact an equivalent definition of another type of essential spectrum, which is in general intermediate between the essential spectra due to Wolf and Kato, and therefore coinciding with them in our case, cf Remark 2.1). That is, λ ∈ σe (Hα0 ) if, and only if, there exists a sequence {un }∞ n=1 ⊂ D(Hα0 ) such that kun kL2 (Ω) = 1, un * 0 and R kHα0 un − λun kL2 (Ω) → 0 as n → ∞. Let {ϕn }∞ n=1 be a singular sequence of −∆ corresponding to a given z ∈ [0, +∞). Then it is easy to verify that un defined by un (x) := ϕn (x1 )ψ0 (x2 )/kψ0 kL2 (I) forms a singular sequence of Hα0 corresponding to z + µ20 . To get the opposite inclusion, we employ the fact that the biorthonormalbasis-type relations (4.1) are available. This enables us to decompose the resolvent of Hα0 into the transverse biorthonormal-basis. Lemma 4.3. Let α0 ∈ R satisfy (2.12). Then C \ [µ20 , +∞) ⊆ ρ(Hα0 ) and for any z ∈ C \ [µ20 , +∞) we have (Hα0 − z)−1 =

∞ X

(−∆R + µ2j − z)−1 Bj .

j=0

Here Bj is a bounded operator on L2 (Ω) defined by (Bj Ψ)(x) := Ψ(x1 , ·), φj L2 (I) ψj (x2 ) ,

Ψ ∈ L2 (Ω) ,

and (−∆R +µ2j −z)−1 abbreviates (−∆R +µ2j −z)−1 ⊗1 on L2 (R)⊗L2 (I) ' L2 (Ω). Proof. Put z ∈ C \ [µ20 , +∞). For every Ψ ∈ L2 (Ω) and all j > 0 we denote Uj := (−∆R + µ2j − z)−1 Ψj ∈ L2 (R), where Ψj are defined in Lemma 4.1. It is clear that kΨj kL2 (R) kΨj kL (R) kΨj kL2 (R) 6C , kUj0 kL2 (R) 6 C p 2 , kUj kL2 (R) 6 j2 + 1 dist z, [µ2j , +∞) j2 + 1 (4.5) where C is a constant depending uniquely on |α0 |, d and z; the second inequality follows from the identity kUj0 k2L2 (R) +(µ2j −z)kUj k2L2 (R) = (Ψj , Uj )L2 (R) . Using (4.5) and estimates of the type (4.4), it is readily seen that each function Rj : x 7→ Uj (x1 )ψj (x2 ) belongs to W21 (Ω). We will show that it is the case for their infinite sum too. Firstly, a consecutive use of (4.2), Pnthe first inequality of (4.5) and (4.3) together with Fubini’s theorem implies k j=2 Rj kL2 (Ω) 6 K kΨkL2 (Ω) , where K is a constant independent of n > 2. Secondly, a similar estimate for the partial sum of ∂1 Rj can be obtained in the same way, provided that one uses the second inequality of (4.5) now. Finally, since the derivative of ψj as well can be expressed

504


IEOT

D in terms of χN j and χj introduced in the proof of Lemma 4.1, a consecutive use of the estimates of the type (4.2) and the first inequality of (4.5) together with Fubini’s theorem implies n

X

2

∂2 Rj

j=2

L2 (Ω)

6d

n X

(α02 + µ2j )kUj k2L2 (R) 6 d C 2

j=2

n X α02 + µ2j kΨj k2L2 (R) ; 2 + 1)2 (j j=2

here the fraction in the upper bound forms a bounded sequence in j, so that we may continue to estimate as above P∞ using (4.3) together with Fubini’s theorem again. Summing up, the series j=0 Rj converges in W21 (Ω) to a function R and ˜ kΨkL (Ω) , kRkW21 (Ω) 6 K 2 ˜ depends uniquely on |α0 |, d and z. Employing this fact and the definition where K of Uj , one can check easily that R satisfies the identity hα0 (R, Φ) − z(R, Φ)L2 (Ω) = (Ψ, Φ)L2 (Ω) for all Φ ∈ W21 (Ω). It implies that R ∈ D(Hα0 ) and (Hα0 − z)R = Ψ, i.e., R = (Hα0 − z)−1 Ψ. Lemmata 4.2 and 4.3 yield Proposition 4.2. Let α0 ∈ R. Then σe (Hα0 ) = [µ20 , +∞) . Proof. In view of the lemmata, the result holds for every α0 ∈ R except for a discrete set of points complementary to the hypothesis (2.12). However, these points can be included by noticing that α0 7→ Hα0 forms a holomorphic family of operators (cf Remark 4.1). Proposition 4.2 and Proposition 4.1 together with Remark 4.1 imply that the spectrum of Hα0 is real and (2.8) holds true for every α0 ∈ R. Remark 4.2. It follows from (2.8) that the spectrum of Hα0 is equal to the sum of the spectra of −∆R and −∆Iα0 . This result could alternatively be obtained by using a general theorem about the spectrum of tensor products [25, Thm. XIII.35] and the fact that the one-dimensional operators generate bounded holomorphic semigroups. However, we do not use this way of proof since Lemma 4.3 is employed not only in the proof of (2.8) but also in the proofs of Theorems 2.5 and 2.6.

5. Stability of the essential spectrum In this section we show that the essential spectrum is stable under a compactly supported perturbation of the boundary conditions. In fact, we will establish a stronger result, namely that the difference of the resolvents of Hα and Hα0 is a compact operator. As an auxiliary result, we shall need the following lemma.

Vol. 62 (2008)


505

Lemma 5.1. Let α0 ∈ R and ϕ ∈ L2 (∂Ω). There exist positive constants c and C, depending on d and |α0 |, such that any solution Ψ ∈ W21 (Ω) of the boundary value problem ( (−∆ − z)Ψ = 0 in Ω , (5.1) (∂2 + iα0 )Ψ = ϕ on ∂Ω , with any z 6 −c, satisfies the estimate kΨkW21 (Ω) 6 C kϕkL2 (∂Ω) .

(5.2)

Proof. Multiplying the first equation of (5.1) by Ψ and integrating over Ω, we arrive at the identity Z Z 2 2 2 (5.3) k∇Ψk − z kΨk + iα0 ν2 |Ψ| − ν2 ϕΨ = 0 , ∂Ω

∂Ω

where ν2 denotes the second component of the outward unit normal vector to ∂Ω. Using the Schwarz and Cauchy inequalities together with |ν2 | = 1, we have Z Z 2 2 ν2 |Ψ| = ∂2 |Ψ| = 2 |Re (Ψ, ∂2 Ψ)| 6 δ −1 kΨk2 + δ k∇Ψk2 , Ω Z∂Ω ν2 ϕΨ 6 δ −1 kϕkL2 (∂Ω) + δ kΨkL2 (∂Ω) , 2 ∂Ω

with any δ ∈ (0, 1). Here kΨkL2 (∂Ω) 6 C kΨkW21 (Ω) , where C is the constant coming from the embedding of W21 (Ω) in L2 (∂Ω) (depending only on d in our case). Choosing now sufficiently small δ and sufficiently large negative z, it is clear that (5.3) can be cast into the inequality (5.2). Now we are in a position to prove 1 (R) with α0 ∈ R. Then Proposition 5.1. Let α − α0 ∈ C0 (R) ∩ W∞

(Hα − z)−1 − (Hα0 − z)−1

is compact in L2 (Ω)

for any z ∈ ρ(Hα ) ∩ ρ(Hα0 ). Proof. It is enough to prove the result for one z in the intersection of the resolvent sets of Hα and Hα0 , and we can assume that the one is negative (since the operators are m-accretive). Given Φ ∈ L2 (Ω), let Ψ := (Hα −z)−1 Φ−(Hα0 −z)−1 Φ. It is easy to check that Ψ is the unique solution to (5.1) with ϕ := −i(α − α0 )T (Hα − z)−1 Φ, where T denotes the trace operator from W22 (Ω) ⊃ D(Hα ) to W21 (∂Ω). By virtue of Lemma 5.1, it is therefore enough to show that (α−α0 )T (Hα −z)−1 is a compact operator from L2 (Ω) to L2 (∂Ω). However, this property follows from the fact that W21 (∂Ω) is compactly embedded in L2 (ω) for every bounded subset ω of ∂Ω, due to the Rellich-Kondrachov theorem [1, Sec. VI]. Corollary 5.2. Suppose the hypothesis of Proposition 5.1. Then σe (Hα ) = [µ20 , +∞) .

506


IEOT

Proof. Our definition of essential spectrum is indeed stable under relatively compact perturbations [11, Thm. IX.2.4].

6. Point spectrum In this section we prove Theorems 2.3–2.6 and Propositions 2.1–2.2. In the proofs of Theorems 2.4–2.6 we follow the main ideas of [13]. Throughout this section we assume that the identity (2.14) and the assumption (2.12) hold true. 6.1. Proof of Theorem 2.3 Any eigenvalue of infinite geometric multiplicity belongs to the essential spectrum which is real by Theorem 2.2. Let λ be an eigenvalue of Hα of finite geometric multiplicity, and Ψ be an associated eigenfunction. Using that α is of compact support, it is easy to check that x 7→ Ψ(−x1 , d − x2 ) is an eigenfunction associated with λ, too. The geometric multiplicity of λ being finite, we conclude that at least one of the eigenfunction associated with λ satisfies |Ψ(x)| = |Ψ(−x1 , d − x2 )|. Taking this identity into account, integrating by parts and using the hypothesis that α is odd, we obtain λ kΨk2L2 (Ω) = h1α [Ψ] + ih2α [Ψ] = k∇Ψk2L2 (Ω) , which implies that λ is real.

6.2. Auxiliary results Let a function F ∈ L2 (Ω) be such that supp F ⊆ Ωb for fixed b > 0. We consider the boundary value problem ( −∆U = (µ20 − k 2 )U + F in Ω , (6.1) (∂2 + iα0 ) U = 0 on ∂Ω , where the parameter k ∈ C ranges in a small neighbourhood of zero. The problem can be solved by separation of variables justified in Lemma 4.3 whenever k 2 6∈ (−∞, 0]. Moreover, it is possible to extend the solution of (6.1) analytically with respect to k. Namely, the following statement is valid. Lemma 6.1. For all small k ∈ C there exists the unique solution to (6.1) satisfying √ 2 2 2 U (x; k) = c± (k)e−k|x1 | ψ0 (x2 ) + O e− Re µ1 −µ0 +k |x1 | , (6.2) in the limit x1 → ±∞, where c± (k) are constants. The mapping T1 (k) defined as T1 (k)F := U is a bounded linear operator from L2 (Ωb ) into W22 (Ωa ) for each a > 0. This operator is meromorphic with respect to k and has the simple pole at zero, Z ψ0 T1 (k)F = F (x)φ0 (x2 ) dx + T2 (k)F, 2k Ω


Vol. 62 (2008)

507

where for each a > 0 the operator T2 (k) : L2 (Ωb ) → W22 (Ωa ) is linear, bounded b := T2 (0)F is the and holomorphic with respect to k small enough. The function U unique solution to the problem  b = µ20 U b + F in Ω, b = 0 on ∂Ω,  −∆U (∂2 + iα0 ) U   Z √ b (x) = − ψ0 (x2 ) |x1 − t1 |F (t)φ0 (t2 ) dt + O e− µ21 −µ20 |x1 | , |x1 | → +∞,  U  2  Ω

(6.3) given by the formula b (x) = U

∞ X

bj (x1 )ψj (x2 ) U

(6.4)

j=0

with Z  1   |x1 − t1 |F (t)φ0 (t2 ) dt −    2 Ω Z bj (x1 ) := √ 2 2 U 1   q e− µj −µ0 |x1 −t1 | F (t)φj (t2 ) dt    2 µ2 − µ2 0 Ω j

if

j = 0,

if

j > 1.

The lemma is proved in the same way as Lemma 3.1 in [4]. Let Mε be the operator of multiplication by the function x 7→ e−iεβ(x1 )x2 . It is straightforward to check that Hα is unitarily equivalent to the operator Mε−1 Hα Mε = Hα0 − εLε , where ∂ ∂ − 2iβ(x1 ) − εβ 2 (x1 ) + iβ 00 (x1 )x2 + εβ 02 (x1 )x22 . ∂x1 ∂x2 We observe that the coefficients of Lε are compactly supported and that their supports are bounded uniformly in ε. It follows that the eigenvalue equation for Hα is equivalent to Lε = −2iβ 0 (x1 )x2

Hα0 U = λU + εLε U, where an eigenfunction Ψ of Hα satisfies Ψ = Mε U . It can be rewritten as ( −∆U = (µ20 − k 2 )U + εLε U in Ω , (∂2 + iα0 )U = 0

on

∂Ω ,

(6.5)

where we have replaced λ by µ20 − k 2 . Now, let λ be an eigenvalue for Hα close to µ20 . As xp 1 → ±∞, the solution U to (6.5) satisfies the asymptotic formula (6.2), where k = µ20 − λ and the branch of the root is specified by the requirement Re k > 0. Such restriction guarantees that the function U together with their derivatives decays exponentially at infinity and thus belongs to W22 (Ω). Hence, the set of all k for which the problem (6.5), (6.2) has a nontrivial solution includes the set of all values of k related to the eigenvalues of Hα by the relation λ = µ20 − k 2 . Thus, it is sufficient to find all

508


IEOT

small k for which a nontrivial solution to (6.5), (6.2) exists and to check whether the solution belongs to W22 (Ω). If it does, the corresponding number λ = µ20 − k 2 is an eigenvalue of Hα . We introduce the numbers Z Z 1 1 k1 (ε) := φ0 (x2 )(Lε ψ0 )(x) dx , k2 (ε) := φ0 (x2 ) Lε T2 (0)Lε ψ0 (x) dx . 2 2 Ω

Ω

Basing on Lemma 6.1 and arguing in the same way as in [13, Sec. 2] one can prove easily the following statement (see also [4, Sec. 4]). Lemma 6.2. There exists the unique function ε 7→ k(ε) converging to zero as ε → +0 for which the problem (6.5), (6.2) has a nontrivial solution. It satisfies the asymptotics k(ε) = εk1 (ε) + ε2 k2 (ε) + O(ε3 ). The associated nontrivial solution to (6.5), (6.2) is unique up to a multiplicative constant and can be chosen so that it obeys (6.2) with c± (k(ε)) = 1 + O(ε),

ε → +0,

(6.6)

as well as U (x; ε) = ψ0 (x2 ) + O(ε) in

W22 (Ωa )

for each fixed a > 0.

6.3. Proof of Theorems 2.5 and 2.6 It follows from Lemma 6.2 that there is at most one simple eigenvalue of Hα converging to µ20 as ε → +0. A sufficient condition guaranteeing the existence of such eigenvalue is the inequality Re k1 (ε) + εk2 (ε) > C(ε) ε2 , C(ε) → +∞, ε → +0, (6.7) that is implied by (6.2), (6.6), the definition of the operator Mε and the assumption on β. The sufficient condition of the absence of the eigenvalue is the opposite inequality Re k1 (ε) + εk2 (ε) 6 −C(ε) ε2 , C(ε) → +∞, ε → +0. (6.8) Thus, we just need to calculate the numbers k1 and k2 to prove the theorems. It is easy to compute the coefficient k1 , ( −α0 hβi + k10 (0) ε if |α0 | < π/d , k1 (ε) = (6.9) k10 (0) ε if |α0 | > π/d , where k10 (0) := −

1 2

Z Ω

ψ0 (x2 )φ0 (x2 ) β 2 (x1 ) + β 02 (x1 ) x22 dx .


Vol. 62 (2008)

509

It is more complicated technically to calculate k2 . This coefficient depends on ε as well; to prove the theorem we need the leading term of its asymptotics as ε → +0. We begin the calculations by observing an obvious identity, Lε U = L0 U + O(ε) , where ∂U (x) ∂U (x) (L0 U )(x) := − 2iβ 0 (x1 )x2 − 2iβ(x1 ) − iβ 00 (x1 )x2 U (x) ∂x1 ∂x2 = i β(x1 )x2 ∆U (x) − ∆ β(x1 )x2 U (x) , 2 which is valid for each U ∈ W2,loc (Ω) in L2 (Ωa ), if a is large enough and independent of ε. Thus, Z 1 k2 (ε) = k2 (0) + O(ε), where k2 (0) = φ0 (x2 ) L0 T2 (0)L0 ψ0 (x) dx. 2 Ω

b := T2 (0)Fb and Fb := L0 ψ0 . Taking into account the problem (6.3) We denote U b and integrating by parts, we obtain for U Z i b (x) − ∆ β(x1 )x2 U b (x) dx k2 (0) = φ0 (x2 ) β(x1 )x2 ∆U 2 Ω Z Z i b (x) dx − i φ0 (x2 )(∆ + µ20 )β(x1 )x2 U φ0 (x2 )β(x1 )x2 Fb(x) dx =− 2 2 Ω Ω iE i Z iD h b (·, d) − φ0 (0)U b (·, 0) − =− β φ0 (d)U φ0 (x2 )β(x1 )x2 Fb(x) dx . 2 2 Ω

(6.10) The last term on the right-hand side of this identity is calculated by integration by parts, Z i φ0 (x2 )β(x1 )x2 Fb(x) dx − 2 Ω Z 1 =− φ0 (x2 )β(x1 )x2 2β(x1 )ψ00 (x2 ) + β 00 (x1 )x2 ψ0 (x2 ) dx 2 Ω Z Z 0 1 1 =− β 2 (x1 )x2 φ0 ψ0 (x2 ) dx + x22 β 02 (x1 )φ0 (x2 )ψ0 (x2 ) dx 2 2 Ω

φ0 (d)ψ0 (d)d 2 1 hβ i + =− 2 2

Ω

Z Ω

φ0 (x2 )ψ0 (x2 ) β 2 (x1 ) + β 02 (x1 )x22 dx .

510


IEOT

This formula, (6.9) and (6.10) yield ( −εα0 hβi + ε2 K + O(ε3 ) k1 (ε) + εk2 (ε) = ε2 K + O(ε3 )

if |α0 | < π/d , if |α0 | > π/d ,

(6.11)

where K := −

iE iD h b (·, d) − φ0 (0)U b (·, 0) − φ0 (d)ψ0 (d)d hβ 2 i. β φ0 (d)U 2 2

b as Thus, it remains to calculate K. In order to do it, we construct the function U the series (6.4). Case |α0 | < π/d : Using the identity Fb(x) = −2iβ(x1 )ψ00 (x2 ) − iβ 00 (x1 )x2 ψ0 (x2 ) , one can check that Fb(x) = −2α0 β(x1 )ψ0 (x2 ) − iβ 00 (x1 )

∞ X

cj ψj (x2 ) ,

j=0

( if j = 0 , bj (x1 ) = ic0β(x1 ) − 2α0 v0 (x1 ) U icj β(x1 ) − (µ2j − µ20 )vj (x1 ) if j > 1 , R where cj := I x2 ψ0 (x2 )φj (x2 )dx2 and the functions vj were introduced in (2.15). bj and (6.4) into (6.11), we arrive at the folSubstituting now the formulae for U lowing chain of identities ∞ ix2 =d 1h X cj ψj (x2 )φ0 (x2 ) hβ 2 i + 2α02 hβv0 i 2 j=0 x2 =0 ∞ i eiα0 d − (−1)j µ2j φ (d)ψ0 (d)d 2 2α0 X 2 hβ i − hβvj i − 0 d j=1 eiα0 d + (−1)j (µj − µ20 ) 2 ix2 =d 1h = x2 ψ0 (x2 )φ0 (x2 ) hβ 2 i + 2α02 hβv0 i 2 x2 =0 ∞ 2α0 X µ2j hβvj i α0 d + πj φ (d)ψ0 (d)d 2 + tan − 0 hβ i , d j=1 µ2j − µ20 2 2

K=

where the last expression coincides with τ for |α0 | < π/d.


Vol. 62 (2008)

511

Case |α0 | > π/d : Following the same scheme as above, we arrive at Fb(x) = − iβ 00 (x1 )

∞ X

cj ψj (x2 ) −

j=0

−

4α0 ψ1 (x2 )β(x1 ) 1 − e−iα0 d

∞ X µ22j ψ2j (x2 ) 2i 2 β(x1 ) , (µ0 − µ21 ) d (µ22j − µ21 )(µ22j − µ20 ) j=1

 ic0 β(x1 ) if j = 0 ,     2α0  if j = 1 , bj (x1 ) = ic1 β(x1 ) − 1 − e−iα0 d v1 (x1) U 2 j   2iµj 1 + (−1)   vj (xj ) if j > 2 , icj β(x1 ) + (µ2j − µ21 )d and check that K = τ for |α0 | > π/d. The series in the formulae for τ converge since the functions vj satisfy −vj00 + (µ2j − µ20 )vj = β

in R ,

(6.12)

and by [18, Ch. V, §3.5, Formula (3.16)] (j > 1) |hβvj i| 6 kβkL2 (R) kvkL2 (R) 6

kβk2L2 (R) µ2j − µ20

.

(6.13)

Summing up, ( k1 (ε) + εk2 (ε) =

− εα0 hβi + ε2 τ + O(ε3 ) 2

3

ε τ + O(ε )

if |α0 | < π/d , if |α0 | > π/d .

All the statements of the theorems – except for the reality of the eigenvalue – follow from these formulae, the identity λ = µ20 − k 2 , the inequalities (6.7) and (6.8), Lemma 6.2, the asymptotics (6.2) for U , and the definition of the operator Mε . Let us show that λε is necessarily real as ε → 0+. Let Ψε be the eigenfunction associated with the eigenvalue λε . It is easy to check that the function x 7→ Ψε (x1 , d − x2 ) is an eigenfunction of Hα associated with the eigenvalue λε . This eigenvalue converges to µ20 as ε → +0. By the uniqueness of such eigenvalue we obtain λε = λε that completes the proof. 6.4. Proof of Theorem 2.4 We employ here the same argument as in the previous proof. The formula for k(ε) in the case α0 = 0 can be obtained from that for |α0 | < π/d by passing to the limit α0 → 0. It leads us to the relation k(ε) = ε2 τ + O(ε3 )

with

τ =−

∞ X 4hβv2j+1 i j=0

µ2j+1 d2

.

512


IEOT

To prove the theorem it is sufficient to show that τ < 0. Indeed, the equation (6.12) implies that for j > 1 hβvj i = kvj0 k2L2 (R) + (µ2j − µ20 )kvj k2L2 (R) > 0 .

(6.14)

Thus, τ < 0.

6.5. Proof of Proposition 2.1 Since hβi = 0, the function v0 is constant at infinity. Hence, by the equation (6.12) for v0 , hβv0 i = kv00 k2L2 (R) . At the same time, it follows from (6.13) and (6.14) that for j > 1, 0 < hβvj i
2α02 kv00 k2L2 (R) − =

2α0 cot d

α0 d 2

 kβk2L2 (R) 

µ21 (µ21 − µ20 )2

+

∞ X

µ22j+1



(µ22j+1 − µ21 )2



j=1

Z

2

α02 l3 e 1 ) dt1

sign(· − t1 )β(t

2 L2 (R) R

2α0 l cot α20 d − d

µ21 d2 d2 + + 2 2 (µ1 − µ0 )2 16π 2 48

e 2 kβk L2 (R) ,

where the resulting expression is positive under the hypothesis.

6.6. Proof of Proposition 2.2 Using (6.14), we obtain τ>

m X µ22j hβv2j i 2α0 π 2 cot α20 d 8π 2 hβv i + , 1 2 2 2 2 3 4 (µ1 − µ0 )d (µ1 − µ0 )d j=1 µ22j − µ21

where the right-hand side is non-negative under the hypothesis (2.19).

Let us show that the inequality (2.19) can be achieved if α0 → µ2 + 0. In this case m = 1, and it is sufficient to check that α0 hβv1 i cot

16π 2 hβv2 i 16π 2 hβv2 i α0 d > 2 = . 2 (µ1 − µ22 )d3 (α02 − µ22 )d3

It follows from the definition of v1 that it satisfies the asymptotic formula v1 (x1 ) = v2 (x1 ) + (µ2 − α0 ) vb(x1 ) + O (µ2 − α0 )2 in L2 (R)-norm, where the function vb is given by √ Z √ ( 3µ0 |x1 − t1 | + 1)e− 3µ0 |x1 −t1 | √ vb(x1 ) := β(t1 ) dt1 , 3 3µ20 R

(6.15)


Vol. 62 (2008)

513

and satisfies the equation −b v 00 + 3µ20 vb = 4µ0 v2

in R.

We multiply this equation by v2 and integrate by parts over R taking into account the equation (6.12) for v2 , 4µ0 kv2 k2L2 (R) = hβb v i. Hence, hβv1 i = hβv2 i + 4(µ2 − α0 )µ0 kv2 k2L2 (R) + O (µ2 − α0 )2 . Employing this identity, we write the asymptotic expansions for the both sides of (6.15) as α0 → µ2 + 0, and obtain 2 4πhβv2 i 8π 2 α0 d 2 + = hβv2 i − 2 kv2 kL2 (R) + O(µ2 − α0 ) , α0 hβv1 i cot 2 (α0 − µ2 )d2 d d 4πhβv2 i hβv2 i 16π 2 hβv2 i = − + O(µ2 − α0 ) . (α02 − µ22 )d3 (α0 − µ2 )d2 d Thus, to satisfy (6.15), it is sufficient to check that 3hβv2 i −

16π 2 kv2 k2L2 (R) > 0 , d2

which is in view of (6.14) equivalent to 3kv20 k2L2 (R) >

7π 2 kv2 k2L2 (R) . d2

It is clear that there exists a function v ∈ C0∞ (R) for which this inequality is valid. Letting v2 := v and β := −v 00 + 3µ20 v, we conclude that there exists β such that the inequality (2.19) holds true, if α0 is sufficiently close to µ2 and greater than this number. Acknowledgments The authors are grateful to Miloslav Znojil for many valuable discussions. They also thank the referee for helpful remarks and suggestions. The work was partially supported by the Czech Academy of Sciences and its Grant Agency within the projects IRP AV0Z10480505 and A100480501, and by the project LC06002 of the Ministry of Education, Youth and Sports of the Czech Republic. D.B. was also supported by RFBR (07-01-00037), by Marie Curie International Fellowship within 6th European Community Framework Programme (MIF1-CT-2005-006254); in addition he gratefully acknowledges the support from Deligne 2004 Balzan Prize in mathematics and the grant of the President of Russian Federation for young scientists and their supervisors (MK-964.2008.1) and for Leading Scientific Schools (NSh-2215.2008.1). D.K. was also supported by FCT, Portugal, through the grant SFRH/BPD/11457/2002.

514


IEOT

References [1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. [2] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (2007), 947–1018. [3] C. M. Bender and P. N. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), no. 24, 5243–5246. [4] D. Borisov, Discrete spectrum of a pair of non-symmetric waveguides coupled by a window, Sbornik Mathematics 197 (2006), no. 4, 475–504. [5] D. Borisov, P. Exner, R. Gadyl’shin, and D. Krejˇciˇr´ık, Bound states in weakly deformed strips and layers, Ann. H. Poincaré 2 (2002), 553–572. [6] W. Bulla, F. Gesztesy, W. Renger, and B. Simon, Weakly coupled bound states in quantum waveguides, Proc. Amer. Math. Soc. 125 (1997), 1487–1495. [7] E. Caliceti, F. Cannata, and S. Graffi, Perturbation theory of PT -symmetric Hamiltonians, J. Phys. A 39 (2006), 10019–10027. [8] E. Caliceti, S. Graffi, and J. Sj¨ ostrand, Spectra of PT-symmetric operators and perturbation theory, J. Phys. A 38 (2005), 185–193. [9] P. Dorey, C. Dunning, and R. Tateo, Spectral equivalences, Bethe ansatz equations, and reality properties in PT -symmetric quantum mechanics, J. Phys. A 34 (2001), 5679–5704. [10] P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73–102. [11] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford University Press, New York, 1987. ˇ [12] P. Exner and P. Seba, Bound states in curved quantum waveguides, J. Math. Phys. 30 (1989), 2574–2580. [13] R. Gadyl’shin, On regular and singular perturbation of acoustic and quantum waveguides, Comptes Rendus Mechanique 332 (2004), no. 8, 647–652. [14] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Israel Program for Scientific Translations, Jerusalem, 1965. [15] H.-Ch. Kaiser, H. Neidhardt, and J. Rehberg, Density and current of a dissipative Schr¨ odinger operator, J. Math. Phys. 43 (2002), 5325–5350. [16]

, Macroscopic current induced boundary conditions for Schr¨ odinger-type operators, Integr. Equ. Oper. Theory 45 (2003), 39–63.

[17]

, On one-dimensional dissipative Schr¨ odinger-type operators, their dilation and eigenfunction expansions, Math. Nachr. 252 (2003), 51–69.

[18] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966. [19] D. Krejˇciˇr´ık, Calculation of the metric in the Hilbert space of a PT -symmetric model via the spectral theorem, J. Phys. A: Math. Theor. 41 (2008), Article ID 244012. [20] D. Krejˇciˇr´ık, H B´ıla, and M. Znojil, Closed formula for the metric in the Hilbert space of a PT -symmetric model, J. Phys. A 39 (2006), 10143–10153. [21] D. Krejˇciˇr´ık and J. Kˇr´ıˇz, On the spectrum of curved quantum waveguides, Publ. RIMS, Kyoto University 41 (2005), no. 3, 757–791.

Vol. 62 (2008)


515

[22] H. Langer and Ch. Tretter, A Krein space approach to PT-symmetry, Czech. J. Phys. 54 (2004), 1113–1120. [23] V. P. Mikhailov, Partial differential equations, Nauka, Moscow, 1976, in Russian. [24] A. Mostafazadeh, Krein-space formulation of PT symmetry, CPT -inner products, and pseudo-Hermiticity, Czech. J. Phys. 56 (2006), 919–933. [25] M. Reed and B. Simon, Methods of modern mathematical physics, IV. Analysis of operators, Academic Press, New York, 1978. [26] K. C. Shin, On the reality of the eigenvalues for a class of PT -symmetric oscillators, Commun. Math. Phys. 229 (2002), 543–564. [27] F. Wolf, On the essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math. 12 (1959), 211–228. [28] M. Znojil, PT -symmetric square well, Phys. Lett. A 285 (2001), 7–10. Denis Borisov Department of Physics and Mathematics Bashkir State Pedagogical University October rev. st. 3a 450000, Ufa Russia e-mail: [email protected] David Krejˇciˇr´ık Department of Theoretical Physics Nuclear Physics Institute Academy of Sciences ˇ z 25068 Reˇ Czech Republic e-mail: [email protected] Submitted: November 20, 2007. Revised: August 27, 2008.

Integr. equ. oper. theory 62 (2008), 517–540 0378-620X/040517-24, DOI 10.1007/s00020-008-1640-3 c 2008 Birkh¨



Scattering Operators for Matrix Zakharov-Shabat Systems Francesco Demontis and Cornelis van der Mee Abstract. In this article the scattering matrix pertaining to the defocusing matrix Zakharov-Shabat system on the line is related to the scattering operator arising from time-dependent scattering theory. Further, the scattering data allowing for a unique retrieval of the potential in the defocusing matrix Zakharov-Shabat system are characterized. Mathematics Subject Classification (2000). Primary 34A55, 81U20; Secondary 81U40. Keywords. Zakharov-Shabat system, scattering operator, scattering matrix.

1. Introduction Consider the matrix Zakharov-Shabat system dX (x, λ) − V (x)X(x, λ) = λX(x, λ), x ∈ R, (1) −iJ dx where 0n×n iq(x) In 0n×m J= , V (x) = , (2) 0m×n −Im ∓iq(x)† 0m×m Ip is the identity matrix of order p, the dagger stands for the conjugate transpose, and the entries of q(x) belong to L1 (R). The plus sign in (2) occurs in the focusing case and the minus sign in the defocusing case. Equation (1) has been studied extensively. We mention the original articles by Zakharov and Shabat [25] (n = m = 1) and Manakov [15] (n = 1 and m = 2) and in particular [1, 2, 3], where also some of the applications are discussed. For the applications to fiber optics we The research leading to this article was supported in part by the Italian Ministery of Education and Research (MIUR) under PRIN grant no. 2006017542-003, and by INdAM. One of the authors (CvdM) wishes to express his gratitude for the hospitality of the Department of Mathematics of the University of Texas at Arlington during the Spring Semester of 2008 when part of the paper was written.

518

Demontis and van der Mee

IEOT

refer to [13, 21]. Equation (1) can also be viewed as a so-called canonical system (cf. [20, 5] and references therein). As in [4, 22, 8], for λ ∈ R we define the Jost solution from the left,1 Fl (x, λ), and the Jost solution from the right, Fr (x, λ), as the (n + m) × (n + m) matrix solutions of (1) that satisfy the boundary conditions Fl (x, λ) = eiλJx [In+m + o(1)], iλJx

Fr (x, λ) = e

[In+m + o(1)],

x → +∞,

(3a)

x → −∞.

(3b)

Actually, the Jost solutions follow from the Volterra integral equations Z ∞ Fl (x, λ) = eiλJx − iJ dy eiλJ(x−y) V (y)Fl (y, λ), Zxx iλJx Fr (x, λ) = e + iJ dy eiλJ(x−y) V (y)Fr (y, λ).

(4a) (4b)

−∞

Then these two Jost solutions satisfy Fl (x, λ) = eiλJx [al (λ) + o(1)], iλJx

Fr (x, λ) = e

[ar (λ) + o(1)],

x → −∞,

(5a)

x → +∞,

(5b)

where al (λ) and ar (λ) are called transition matrices. It is easily seen that al (λ) and ar (λ) are each other’s inverses, while al (λ)−1 = Jal (λ)† J, ar (λ)−1 = Jar (λ)† J, −1

al (λ)

†

= al (λ) ,

−1

ar (λ)

†

= ar (λ) ,

defocusing case, focusing case.

Moreover, since (1) is a first order system, we have Fl (x, λ) = Fr (x, λ)al (λ),

Fr (x, λ) = Fl (x, λ)ar (λ).

(6)

Mr (x, λ) = Fr (x, λ)e−iλJx ,

(7)

Introducing the Faddeev functions Ml (x, λ) = Fl (x, λ)e−iλJx ,

we obtain from (4) the Volterra integral equations Z ∞ Ml (x, λ) = In+m − iJ dy eiλJ(x−y) V (y)Ml (y, λ)eiλJ(y−x) , x Z x Mr (x, λ) = In+m + iJ dy eiλJ(x−y) V (y)Mr (y, λ)eiλJ(y−x) .

(8a) (8b)

−∞

Defining the modified Faddeev functions ˙ m×m + Mr (x, λ) 0n×n +I ˙ m , m+ (x, λ) = Ml (x, λ) In +0 ˙ m×m + Ml (x, λ) 0n×n +I ˙ m , m− (x, λ) = Mr (x, λ) In +0

(9a) (9b)

˙ denotes the direct sum of the square matrices A and B, we see that where A+B m+ (x, λ) is analytic in λ ∈ C+ and continuous in λ ∈ C+ and m− (x, λ) is analytic 1 In

[3] the term “Jost function” is used for the (n+m)×n and (n+m)×m submatrices composed of the first n and last m columns, respectively.

Vol. 62 (2008)

Scattering Operators for Matrix Zakharov-Shabat Systems

519

in λ ∈ C− and continuous in λ ∈ C− . The corresponding modified Jost solutions are then defined by f+ (x, λ) = m+ (x, λ)eiλJx ,

f− (x, λ) = m− (x, λ)eiλJx .

(10)

The modified Faddeev functions are related by the scattering matrix S(λ) by means of the Riemann-Hilbert problem Tl (λ) −R(λ) m− (x, λ) = m+ (x, λ)JS(λ)J = m+ (x, λ) , (11) −L(λ) Tr (λ) where S(λ) is unitary in the defocusing case and J-unitary in the focusing case. The n × n matrix Tl (λ) and the m × m matrix Tr (λ) are called transmission coefficients, while the n × m matrix R(λ) and the m × n matrix L(λ) are called reflection coefficients. In [4, 3, 22] the scattering coefficients were introduced ad hoc to create an operational inverse scattering theory without relating them to time dependent scattering theory [14, 19, 24, 23]. In this article we introduce the scattering operator as in time dependent scattering theory by S = Ω+ (Ω− )† , where Ω± are the Moeller wave operators. In the defocusing case, where the free Hamiltonian H0 = −iJ(d/dx) and the full Hamiltonian H = H0 − V are both selfadjoint on the direct sum of n + m copies of L2 (R), we then prove the existence and asymptotic completeness of Ω± and hence the unitarity of S for potentials with entries in L1 (R). Applying the unitary equivalence by the (modified) Fourier transform F, we then go on to prove that FSF−1 is the multiplication by a unitary matrix function S(λ) of order n + m. We then proceed to identify S(λ) with the scattering matrix S(λ) given by (11) and defined in [4] in terms of the transition matrices. Finally, we briefly touch on the Marchenko integral equation method for recovering the potential from one of the reflection coefficients (See [4, 3, 8] in the defocusing case, [22, 3] in the focusing case without bound states, and [8, 3] in the focusing case) and characterize the scattering data that lead to a defocusing L1 -potential. We note that a full characterization of the scattering data for the Schr¨ odinger equation on the line is known [16, 18] as is a characterization of the scattering data for the matrix Zakharov-Shabat system on the half-line [17]. Let us discuss the contents of the various sections. In Sec. 2 we derive expressions for the resolvent and the spectral decomposition of the full Hamiltonian. Section 3 contains the results on the wave operators, the scattering operator, and the scattering matrix. We also prove the absolute continuity of the full Hamiltonian. In Sec. 4 we review the necessary Marchenko theory, prove boundedness, compactness, and continuous dependence of the Marchenko integral operator on x, and solve the characterization problem. In Appendix A we discuss the precise definition of the full Hamiltonian −iJ(d/dx) − V .

520


IEOT

We now introduce some notations. By C+ and C− we denote the open upper and lower complex half-planes. We put C± = C± ∪ R. The orthogonal direct sum of p copies of L2 (R) is written as Hp . Let Hps denote the orthogonal direct sum of p copies of the Sobolev space H s (R) of measurable functions φ on R whose Fourier transforms φˆ satisfy Z

∞

kφk2,s =

2 ˆ dξ (1 + ξ 2 )s/2 |φ(ξ)|

1/2 .

−∞

The `1 -direct sum of p copies of L1 (R) is written as Lp . By L1p we denote the def

`1 -direct sum of p copies of the weighted space L1 = L1 (R; (1 + |x|)dx). The corresponding spaces of p × q matrices of vectors in the same space are denoted by s Hp×q , Hp×q , Lp×q , and L1p×q , respectively. Throughout this article we partition square matrices H of order n + m as follows: H1 H2 H= , H3 H4 where H1 is n × n, H2 is n × m, H3 is m × n, and H4 is m × m. This partitioning will be applied in particular to Jost solutions, Faddeev matrices, and transition matrices.

2. Resolvent of the Full Hamiltonian In this section we derive an expression for the resolvent operator (λ − H)−1 for λ ∈ C \ R and apply it to find the spectral decomposition of H. Let φ ∈ Hn+m and λ ∈ C \ R. In order to find the resolvent (λ − H)−1 , we need to determine Ψ(·, λ) ∈ Hn+m such that (H − λ)Ψ = −iJ

d Ψ(x, λ) − V (x)Ψ(x, λ) − λΨ(x, λ) = −φ(x). dx

Letting λ ∈ C+ and writing Ψ(x, λ) = f+ (x, λ)Φ(x, λ), we get d d −iJf+ (x, λ) Φ(x, λ)+ −iJ f+ (x, λ)−V (x)f+ (x, λ)−λf+ (x, λ) Φ(x, λ) dx dx {z } | =0(n+m)×(n+m)

= −φ(x), and hence d Φ(x, λ) = −if+ (x, λ)−1 Jφ(x) = −iJg+ (x, λ)φ(x), dx

Vol. 62 (2008)


521

where g+ (x, λ) = Jf+ (x, λ)−1 J. Thus, writing g+ (x, λ) = e−iλJx n+ (x, λ) we get Z x Z x dy [g+ (y, λ)φ(y)]up , dy e−iλy [n+ (y, λ)φ(y)]up = −i Φup (x, λ) = −i −∞ −∞ Z ∞ Z ∞ dn iλy dn Φ (x, λ) = −i dy e [n+ (y, λ)φ(y)] = −i dy [g+ (y, λ)φ(y)]dn , x

x

where Φ (x, λ) = In 0n×m Φ(x, λ) and Φdn (x, λ) = 0m×n Im Φ(x, λ). Analogously, letting λ ∈ C− and writing Ψ(x, λ) = f− (x, λ)Φ(x, λ), we get d d −iJf− (x, λ) Φ(x, λ)+ −iJ f− (x, λ)−V (x)f− (x, λ)−λf− (x, λ) Φ(x, λ) dx dx {z } | up

=0(n+m)×(n+m)

= −φ(x), and hence d Φ(x, λ) = −if− (x, λ)−1 Jφ(x) = −iJg− (x, λ)φ(x), dx where g− (x, λ) = Jf− (x, λ)−1 J. Thus, writing g− (x, λ) = e−iλJx n− (x, λ) we get Z ∞ Z ∞ up −iλy up Φ (x, λ) = +i dy e [n− (y, λ)φ(y)] = +i dy [g− (y, λ)φ(y)]up , x x Z x Z x Φdn (x, λ) = +i dy eiλy [n− (y, λ)φ(y)]dn = +i dy [g− (y, λ)φ(y)]dn . −∞

−∞

Consequently, [(λ − H)−1 φ](x) =

Z

∞

dy G(x, y; λ)φ(y),

(12)

−∞

where the Green’s function G(x, y; λ) is given by  −i f+ (x, λ) 21 (I + J)g+ (y, λ),    −i f (x, λ) 1 (I − J)g (y, λ), + + 2 G(x, y; λ) = 1  (I + J)g +i f (x, λ) − − (y, λ),  2   +i f− (x, λ) 21 (I − J)g− (y, λ),

Im λ > 0, Im λ > 0, Im λ < 0, Im λ < 0,

x > y, x < y, x < y, x > y.

Now recall that for λ ∈ R 1 1 f+ (x, λ) = Fl (x, λ) (I + J) + Fr (x, λ) (I − J) 2 2 1 1 In = Fl (x, λ) (I + J) + ar (λ) (I − J) = Fl (x, λ) 0m×n 2 2 We now easily verify that for λ ∈ R In −1 g+ (x, λ) = Jf+ (x, λ) J = 0m×n

(13)

ar2 (λ) . ar4 (λ)

ar2 (λ)ar4 (λ)−1 JFl (x, λ)−1 J. ar4 (λ)−1

522


IEOT

In the same way we get for λ ∈ R 1 1 f− (x, λ) = Fr (x, λ) (I + J) + Fl (x, λ) (I − J) 2 2 1 1 In al2 (λ) = Fr (x, λ) (I + J) + al (λ) (I − J) = Fr (x, λ) 0m×n al4 (λ) 2 2 In al2 (λ) ar1 (λ) 0n×m = Fl (x, λ)ar (λ) = Fl (x, λ) , 0m×n al4 (λ) ar3 (λ) Im where we have used that ar (λ)al (λ) = In+m . Thus for λ ∈ R In al2 (λ)al4 (λ)−1 −1 g− (x, λ) = Jf− (x, λ) J = JFr (x, λ)−1 J 0m×n al4 (λ)−1 ar1 (λ)−1 0n×m = JFl (x, λ)−1 J. ar3 (λ)ar1 (λ)−1 Im By taking the adjoints we get for λ ∈ R In 0n×m In 0n×m † † f+ (x, λ) = Fl (x, λ) = JFl (x, λ)−1 J, −al3 (λ) al4 (λ) ar2 (λ)† ar4 (λ)† al1 (λ) −al2 (λ) ar1 (λ)† ar3 (λ)† † † f− (x, λ) = Fl (x, λ) = JFl (x, λ)−1 J. 0m×n Im 0m×n Im Thus for λ ∈ R we have In ar2 (λ)ar4 (λ)−1 al1 (λ)−1 al1 (λ)−1 al2 (λ) g+ (x, λ) = f− (x, λ)† 0m×n ar4 (λ)−1 0m×n Im al1 (λ)−1 0n×m = f (x, λ)† , (14a) 0m×n ar4 (λ)−1 − ar1 (λ)−1 0n×m In 0n×m f (x, λ)† g− (x, λ) = al3 (λ)ar1 (λ)−1 Im al4 (λ)−1 al3 (λ) al4 (λ)−1 + ar1 (λ)−1 0n×m = f (x, λ)† . (14b) 0m×n al4 (λ)−1 + Using estimates as derived in [7] for the Schrödinger equation on the line, we prove the following Proposition 1. Suppose the entries of V (x) belong to L1 . Then 1 + max(0, −x) , 1 + |x| 1 + max(0, x) , kMr (x, λ) − In+m k ≤ const. 1 + |x| uniformly in λ ∈ R. kMl (x, λ) − In+m k ≤ const.

Proof. From (8a) we easily derive by iteration Z Z ∞ kMl (x, λ)k ≤ 1 + dy kV (y)kkMl (y, λ)k ≤ exp x

∞

x

(15a) (15b)

dy kV (y)k ≤ ekV k1 .

Vol. 62 (2008)


523

Next, we apply (8a) again and get kMl (x, λ) − In+m k ≤ ekV k1

Z

∞

dy kV (y)k x

Z 1 + max(0, −x) ∞ dy (1 + |y|)kV (y)k 1 + |x| x 1 + max(0, −x) , ≤ kV kL1 ekV k1 1 + |x|

≤ ekV k1

which proves (15a). The estimate (15b) follows from (8b) in an analogous way.

3. Wave Operators Consider the free Hamiltonian H0 and the full Hamiltonian H defined by d d , H = H0 − V = −iJ − V, dx dx defined on dense domains in Hn+m . Here we assume V to have its entries in L1 (R). Then H0 is an absolutely continuous selfadjoint operator with domain 1 and spectrum R. In the defocusing case H is a selfadjoint operator with Hn+m essential spectrum R. Note that iH0 and iH generate the strongly continuous unitary groups {eitH0 }t∈R and {eitH }t∈R on Hn+m (cf. [14]). We first prove the following elementary result [8]. It has a well-known analog in the case of the Schr¨ odinger equation [14, Sec. 5.3]. H0 = −iJ

Proposition 2. Let λ ∈ C \ R and let W have its entries in L2 (R). Then W (λ − H0 )−1 is a Hilbert-Schmidt operator on Hn+m . Moreover, if W1 and W2 have their entries in L2 (R), then W1 (λ − H0 )−1 W2 is a Hilbert-Schmidt operator on Hn+m . Proof. It suffices to prove that, for T = −i(d/dx) defined on L2 (R) with domain H 1 (R) and for W ∈ L2 (R), the operator W (λ − T )−1 is Hilbert-Schmidt. Indeed, letting F stand for the Fourier transform, we have Z ∞ ˆ (ξ − η) 1 W −1 −1 ˆ ˆ √ (FW (λ − T ) F φ)(ξ) = dη φ(η). λ+ξ 2π −∞ Hence, FW (λ − T )−1 F −1 is an integral operator on L2 (R) with square integrable kernel and hence of Hilbert-Schmidt type. The second part is immediate from  Z x   −iW (x) dy eiλ(x−y) W2 (y)φ(y), Im λ > 0,  1 −1 Z−∞ [W1 (λ − H0 ) W2 φ](x) = ∞   +iW (x) dy e−iλ(y−x) W2 (y)φ(y), Im λ < 0,  1 x

which shows W1 (λ − H0 )−1 W2 to be a matrix of integral operators on L2 (R) with square integrable kernels.

524


IEOT

Corollary 3. Let the potential V have its entries in L1 (R) ∩ L2 (R). Then the domains of the free Hamiltonian H0 and the full Hamiltonian H coincide. ˜ ± by In the defocusing case we now define the wave operators Ω ˜ ± φ = lim eiτ H e−iτ H0 φ, Ω τ →±∞

(16)

˜± where the limit is taken in the norm of Hn+m . If the limits in (16) exist, then Ω maps the domain of H0 into the domain of H and ˜ ± H0 = H Ω ˜ ± on D(H0 ). Ω (17) By the same token, in the defocusing case we define the wave operators Ω± φ = lim eiτ H0 e−iτ H Pac (H)φ, τ →±∞

(18)

where Pac (H) is the orthogonal projection onto the absolutely continuous subspace of H. If the limits in (18) exist, then Ω± maps the domain of H into the domain of H0 and Ω± H = H0 Ω± on D(H), (19) while ˜ ± = (Ω± )† . Ω For the basic theory of wave operators we refer to [14, 19, 23, 24]. We summarize the above results in the following Theorem 4. Let the entries of V (x) belong to L1 (R). Then the wave operators Ω± defined by (16) exist and have the absolutely continuous subspace of H as their range. Moreover, the scattering operator ˜− S = Ω+ Ω (20) is unitary. Proof. Since V has its entries in L1 (R), we can write V = V1 V2 , where V1 and V2 both have their entries in L2 (R). Put W (λ) = I + V2 (λ − H0 )−1 V1 ,

λ ∈ C \ R.

Then, according to the second part of Proposition 2, for nonreal λ the operator W (λ) is a Hilbert-Schmidt perturbation of the identity. Then for λ ∈ C \ R we apply Proposition 2 and derive that (λ − H)−1 − (λ − H0 )−1 = −(λ − H0 )−1 V1 W (λ)−1 V2 (λ − H0 )−1 †  = − V1 (λ − H0 )−1  W (λ)−1 V2 (λ − H0 )−1 | {z } | {z } | {z } Hilbert-Schmidt

bounded Hilbert-Schmidt

is a trace class operator. We refer to Appendix A for details on the precise definition ˜ ± defined by of H. According to [23, Theorem 22.19], the wave operators Ω± and Ω ˜ ˜ ± = (Ω± )† , (17) and (18) exist. Then Ω± and Ω± are partial isometries such that Ω where Ω± has full range and the absolutely continuous subspace of H as its cokernel

Vol. 62 (2008)


525

(see [23, Theorem 21.3], [14, Lemma X 4.11]). Moreover, S as defined in (20) is a unitary operator on Hn+m . Equation (16) implies that the wave operators Ω± satisfy the intertwining relations (17) and (19). As a result, the scattering operator S leaves invariant the domain of H0 and commutes with H0 . For φ ∈ Hn+m , put Z ∞ ˘ φ(λ) = (Fφ)(λ) = dx e−iλJx φ(x). (21a) −∞

Then ˘ φ(x) = (F−1 φ)(x) =

1 2π

Z

∞

˘ dλ eiλJx φ(λ).

(21b)

−∞

1 For φ ∈ Hn+m we get

˘ ˘ ψ(λ) = −iJ φ˘0 (λ) = λφ(λ). where ψ = H0 φ. Thus 1 φ ∈ D(H0 ) = Hn+m .

(FH0 φ)(λ) = λ(Fφ)(λ), −1

(22) −1

Since this operator commutes with FSF , the unitary operator FSF coincides with the operator of multiplication by an (almost everywhere existing) unitary scattering matrix [23, Theorem 21.15], S(λ) say. In other words, ˆ ˆ (FSF−1 φ)(λ) = S(λ)φ(λ),

φˆ ∈ Hn+m .

(23)

Let us now introduce the transformation G that diagonalizes the full Hamiltonian H. Lemma 5. Suppose the entries of V (x) belong to L1 ∩ L2 (R). Then the linear operators Gl and Gr defined by Z ∞ ˙ l4 (λ)−1 ]Fr (x, λ)† φ(x), (Gl φ)(λ) = dx [al1 (λ)−1 +a (24a) −∞ Z ∞ ˙ r4 (λ)−1 ]Fl (x, λ)† φ(x), (Gr φ)(λ) = dx [ar1 (λ)−1 +a (24b) −∞

are bounded on Hn+m . Moreover, Gl and Gr are boundedly invertible on Hn+m and their inverses are given by Z ∞ 1 ˆ ˆ (G−1 φ)(x) = dλ Fl (x, λ)φ(λ), (24c) l 2π −∞ Z ∞ 1 ˆ ˆ (G−1 φ)(x) = dλ Fr (x, λ)φ(λ). (24d) r 2π −∞ Further, Gl and Gr diagonalize the Hamiltonian H in the sense that for each φ ∈ D(H) (Gl Hφ)(λ) = λ(Gl φ)(λ),

(25a)

(Gr Hφ)(λ) = λ(Gr φ)(λ).

(25b)

526


IEOT

Proof. Proposition 1 implies that Fl (x, λ) = eiλJx + o((1 + |x|)−1 ),

x → +∞,

= eiλJx al (λ) + o((1 + |x|)−1 ),

x → −∞,

1

where the entries of V (x) are assumed to be in L for the right-hand sides to be true. Therefore, G−1 = 2πF−1 (Π+ + [al (λ)]Π− ) + K, l where Π± are the restrictions of vectors in Hn+m to R± and K is a bounded linear operator on Hn+m . The boundedness of K follows from the estimate Z ∞ kφ(x)k ≤ π const.kφk. kKφk ≤ const. dx √ 1 + x2 −∞ Consequently, G−1 itself is bounded on Hn+m . The boundedness proofs for G−1 r , l Gl , and Gr are similar, but depart from (4b) and the adjoints of (4). It is now immediate from the two versions of (28) that the bounded linear operators defined by (24a) and (24b) are the inverses of those defined by (24c) and (24d). Since def ˙ l4 (λ)−1 ]Fr (x, λ)† = [al1 (λ)−1 +a ˙ l4 (λ)−1 ]JFr (x, λ)−1 J (26) G(x, λ) = [al1 (λ)−1 +a satisfies the adjoint matrix Zakharov-Shabat system ∂G (x, λ)J − G(x, λ)V (x) = λG(x, λ), i ∂x 1 we easily verify that for φ ∈ D(H) = D(H0 ) = Hn+m

(27)

∞

λ(Gl φ)(λ) = [iG(x, λ)Jφ(x)]x=−∞ Z ∞ + dx G(x, λ) (−iJφ0 (x) − V (x)φ(x)) −∞

= (Gl Hφ)(λ), which implies (25a). In the same way we prove (25b).

We now prove that the full Hamiltonian H, like the free Hamiltonian H0 , is absolutely continuous. Theorem 6. Suppose the entries of V (x) belong to L1 ∩ L2 (R). Then the Hamiltonian operator H is absolutely continuous. Proof. We divide the proof into three parts. 1. Let σ denote the resolution of the identity of the selfadjoint operator H, where σ(E) is an orthogonal projection on Hn+m for each real Borel set E. Then for each pair of real numbers a, b with a < b and each φ ∈ Hn+m we have Z b σ((a, b)) + σ([a, b]) 1 φ = lim+ dτ (λ − iε − H)−1 −(λ + iε − H)−1 φ, 2 ε→0 2πi a where the limit is taken in the strong sense [14]. Thus if a and/or b is an eigenvalue of H, which results in a nonzero eigenprojection σ({a}) and/or σ({b}), then these

Vol. 62 (2008)


527

eigenprojections are only taken account of with weight 12 . Let us now apply this identity. Using (12) and (13) we take the limit as ε → 0+ and get

Z b Z ∞ σ((a, b)) + σ([a, b]) dy J (x, y; ζ)φ(y), dζ φ (x) = 2 −∞ a

where J (x, y; ζ) =    1 f+ (x, ζ) 1 (I + J)g+ (y, ζ) + f− (x, ζ) 1 (I − J)g− (y, ζ) , 2 2 2π = 1   f+ (x, ζ) 12 (I − J)g+ (y, ζ) + f− (x, ζ) 12 (I + J)g− (y, ζ) , 2π

x > y, x < y.

2. Let us now write J (x, y; ζ) in a different way. Using (9) and (10) we write (14) in the form    1 Fl (x, ζ)[al1 (ζ)−1 +a ˙ l4 (ζ)−1 ]Fr (y, ζ)† , x > y, 2π J (x, y; ζ) = 1  ˙ r4 (ζ)−1 ]Fl (y, ζ)† , x < y.  Fr (x, ζ)[ar1 (ζ)−1 +a 2π Using (6) and the J-unitarity of the transition matrices, we compute for x > y ˙ l4 (ζ)−1 ]Fr (y, ζ)† 2πJ (x, y; ζ) = Fl (x, ζ)[al1 (ζ)−1 +a ˙ l4 (ζ)−1 ]ar (λ)† Fl (x, λ)† = Fr (x, ζ)al (ζ)[al1 (ζ)−1 +a ˙ l4 (ζ)−1 ]Jar (ζ)−1 JFl (x, ζ)† = Fr (x, ζ)al (ζ)[al1 (ζ)−1 +a ˙ l4 (ζ)−1 ]Jal (ζ)JFl (x, ζ)† = Fr (x, ζ)al (ζ)[al1 (ζ)−1 +a = Fr (x, ζ)[(al1 (ζ) − al2 (ζ)al4 (ζ)−1 al3 (ζ)) ˙ l4 (ζ) − al3 (ζ)al1 (ζ)−1 al2 (ζ))]Fl (x, ζ)† +(a ˙ r4 (ζ)−1 ]Fl (x, ζ)† , = Fr (x, ζ)[ar1 (ζ)−1 +a where we have used al (ζ)ar (ζ) = In+m at the last step. Consequently, 1 ˙ l4 (ζ)−1 ]Fr (y, ζ)† Fl (x, ζ)[al1 (ζ)−1 +a 2π 1 ˙ r4 (ζ)−1 ]Fl (y, ζ)† , = Fr (x, ζ)[ar1 (ζ)−1 +a 2π

J (x, y; ζ) =

irrespective of whether x > y or x < y. 3. Equations (24) imply that Z ∞ 1 1 Fl (x, ζ)(Gl φ)(ζ) = Fr (x, ζ)(Gr φ)(ζ), dy J (x, y; ζ)φ(y) = 2π 2π −∞

(28)

528


IEOT

where Gl and Gr are bounded operators. Hence, for φ, ψ ∈ Hn+m we have Z ∞ Z ∞ dx dy hC(x, y; ζ)φ(y), ψ(x)i −∞ −∞    1 h(Gl φ)(ζ), [ar1 (ζ)+a ˙ r4 (ζ)](Gr ψ)(ζ)i, = 2π 1  ˙ l4 (ζ)](Gl ψ)(ζ)i,  h(Gr φ)(ζ), [al1 (ζ)+a 2π which implies the boundedness of the integral operator J (ζ) with integral kernel J (x, y; ζ) for each ζ ∈ R. Consequently, H is absolutely continuous with σ 0 (ζ) = J (ζ), as claimed. We now relate the scattering operator S to the scattering matrix Tl (λ) R(λ) S(λ) = , L(λ) Tr (λ) where Tl (λ) = al1 (λ)−1 ,

R(λ) = ar2 (λ)ar4 (λ)−1

L(λ) = al3 (λ)al1 (λ)−1 ,

Tr (λ) = ar4 (λ)−1 .

Here we have defined the scattering coefficients as in [4, 22, 8]. We also have the alternative expressions R(λ) = −al1 (λ)−1 al2 (λ),

L(λ) = −ar4 (λ)−1 ar3 (λ).

Recall that in the defocusing case the scattering matrix S(λ) is unitary. Using the scattering matrix we can write (28) in the form In R(λ) J (x, y; λ) = Fl (x, λ) Fl (x, λ)† R(λ)† Im In L(λ)† = Fr (x, λ) Fr (x, λ)† . L(λ) Im To prove that S(λ) = S(λ) for each λ ∈ R, we follow the path taken in [23] fot the Schr¨ odinger equation on the line. Theorem 7. Let the entries of V (x) belong to L1 ∩ L2 (R). Then the Fourier transformed scattering operator FSF−1 coincides with the operator of premultiplication by the unitary (n + m) × (n + m) matrix function S(λ). Proof. Equations (17), (22), and (25) imply that on F[D(H0 )] (22)

(17)

(25a)

˜ ± F−1 [λ] = Gl Ω ˜ ± H0 F−1 = Gl H Ω ˜ ± F−1 = [λ]Gl Ω ˜ ± F−1 , Gl Ω where [λ] is the operator of multiplication by the independent variable. Analogously, (19), (22), and (25a) imply that on G[D(H)] (25a)

(19)

(22)

−1 FΩ± G−1 = FH0 Ω± G−1 = [λ]FΩ± G−1 l [λ] = FΩ± HGl l l .

Vol. 62 (2008)


529

˜ ± F−1 and FΩ± G−1 are operators of premultiplication by Consequently, both Gl Ω l an (n+m)×(n+m) matrix function [23, Theorem 21.15]. Hence there exist matrix ˜ ± (λ) and w± (λ) such that functions w ˆ ˆ ˜ ± F−1 φ)(λ) ˜ ± (λ)φ(λ), (Gl Ω =w

ˆ ˆ (FΩ± G−1 l φ)(λ) = w ± (λ)φ(λ),

(29)

where φˆ ∈ Hn+m . Then using (20) and (23) it follows that ˜ − (λ), S(λ) = w+ (λ)w

λ ∈ R.

(30)

In the above derivation we could have employed Gr instead of Gl . ˜ ± (λ). Writing G(x, λ) defined by (26) in the form Let us now compute w G(x, λ) = e−iλJx N (x, λ),

(31)

we obtain from (27) the Volterra integral equations Z ∞ iλJx −iλJx dy eiλJ(x−y) N (y, λ)V (y)eiλJ(y−x) J, N (x, λ) = e N+∞ (λ)e +i x

(32a) N (x, λ) = eiλJx N−∞ (λ)e−iλJx − i

Z

x

dy eiλJ(x−y) N (y, λ)V (y)eiλJ(y−x) J,

−∞

(32b) where ˙ l4 (λ)−1 ]ar (λ)† = [al1 (λ)−1 +a ˙ l4 (λ)−1 ]Jar (λ)−1 J N+∞ (λ) = [al1 (λ)−1 +a In −al1 (λ)−1 al2 (λ) = , (33a) −al4 (λ)−1 al3 (λ) Im ˙ l4 (λ)−1 . N−∞ (λ) = al1 (λ)−1 +a (33b) We then compute (29)

ˆ ˆ ˜ ± F−1 φ)(λ) ˜ ± (λ)φ(λ) w = (Gl Ω (16)

=

ˆ lim (Gl eiτ H e−iτ H0 F−1 φ)(λ)

τ →±∞

(22)

ˆ lim eiλτ (Gl F−1 e−i[ξ]τ φ)(λ) Z ∞ (24) ˆ = lim eiλτ dx G(x, λ)(F−1 e−i[ξ]τ φ)(x) τ →±∞ −∞ Z ∞ Z ∞ 1 (21b) ˆ dξ eiξJx e−iξτ φ(ξ) = lim eiλτ dx G(x, λ) τ →±∞ 2π −∞ −∞ Z ∞ Z ∞ 1 ˆ = lim dξ ei(λ−ξ)τ dx G(x, λ)eiξJx φ(ξ) τ →±∞ −∞ 2π −∞ Z ∞ Z ∞ 1 (31) ˆ = lim dξ ei(λ−ξ)τ dx e−iλJx N (x, λ)eiξJx φ(ξ), τ →±∞ −∞ 2π −∞ =

τ →±∞

530


IEOT

where we have changed the order of integration at the penultimate transition. Substituting (32) we get Z ∞ Z ∞ 1 ˆ ˜ ± (λ)φ(λ) dx N+∞ (λ)ei(ξ−λ)Jx dξ ei(λ−ξ)τ w = lim τ →±∞ 2π −∞ 0 Z 0 Z ∞ i(ξ−λ)Jx i(ξ−λ)Jx ˆ + dx N−∞ (λ)e − dx Φ(x, λ)Je φ(ξ), −∞

where

−∞

 Z x   dy e−iλJy N (y, λ)V (y)eiλJy , x < 0, +i −∞ Z Φ(x, λ) = ∞   dy e−iλJy N (y, λ)V (y)eiλJy , x > 0, −i x

is continuous in 0 6= x ∈ R, vanishes as x → ±∞, and satisfies [cf. (32)] Φ(0+ , λ) − Φ(0− , λ) = N+∞ (λ) − N−∞ (λ). Moreover, for potentials V whose entries belong to L1 , the entries of Φ(·, λ) belong to L1 (R) for each λ ∈ R. When taking the limit as τ → ±∞, the terms involving N±∞ (λ) either lead to a delta function integration or vanish and the term involving Φ(x, λ) vanishes. As a result, 1 1 ˆ ˆ ˜ ± (λ)φ(λ) = N±∞ (λ) (I + J) + N∓∞ (λ) (I − J) φ(λ). w (34) 2 2 Next, we compute (29) ˆ ˆ = (FΩ± G−1 w± (λ)φ(λ) l φ)(λ) (18)

ˆ lim (Feiτ H0 e−iτ H Pac (H)G−1 l φ)(λ)

=

τ →±∞

ˆ lim (Feiτ H0 e−iτ H G−1 l φ)(λ)

=

τ →±∞

(22)

−i[ξ]τ ˆ lim eiλτ (FG−1 φ)(λ) l e Z ∞ (21a) −i[ξ]τ ˆ = lim eiλτ dx e−iλJx (G−1 φ)(x) l e τ →±∞ −∞ Z ∞ Z ∞ 1 (25) ˆ = lim eiλτ dξ Fl (x, ξ)e−iξτ φ(ξ) dx e−iλJx τ →±∞ 2π −∞ −∞ Z ∞ Z ∞ 1 i(λ−ξ)τ −iλJx ˆ dx e Fl (x, ξ) φ(ξ) = lim dξ e τ →±∞ −∞ 2π −∞

=

(13)

=

τ →±∞

Z

∞

lim

τ →±∞

i(λ−ξ)τ

dξ e −∞

1 2π

Z

∞ −iλJx

dx e

iξJx

Ml (x, ξ)e

ˆ φ(ξ),

−∞

where we have used the absolute continuity [Lemma 6] at the third transition and changed the order of integration at the penultimate transition. Substituting (8)

Vol. 62 (2008)


531

we get ˆ w± (λ)φ(λ) = lim

τ →±∞

Z

1 2π

Z

∞

dξ ei(λ−ξ)τ

−∞

+

dx e

dx ei(ξ−λ)Jx M+∞ (ξ)

0

0 i(ξ−λ)Jx

∞

Z Z

∞

M−∞ (ξ) −

−∞

ˆ ˜ dx J Φ(x, ξ) φ(ξ),

−∞

where Ml (x, λ) → M±∞ (λ) as x → ±∞ and  Z x   dy e−iλJy V (y)M (y, λ)eiλJy , x < 0, −i ˜ Z−∞ Φ(x, λ) = ∞   dy e−iλJy V (y)M (y, λ)eiλJy , x > 0. +i x

As above, we obtain for potentials V with entries in L1 1 1 ˆ ˆ (I + J)M±∞ (λ) + (I − J)M∓∞ (λ) φ(λ), w± (λ)φ(λ) = 2 2

(35)

where M+∞ (λ) = In+m and M−∞ (λ) = al (λ). Let us compute S(λ): (30)

˜ − (λ) S(λ) = w+ (λ)w (34) In 0n×m ˜ − (λ) = w al3 (λ) al4 (λ) (33) In 0n×m al1 (λ)−1 −al1 (λ)−1 al2 (λ) = al3 (λ) al4 (λ) 0m×n Im −1 −1 al1 (λ) −al1 (λ) al2 (λ) = al3 (λ)al4 (λ)−1 al4 (λ) − al3 (λ)al1 (λ)−1 al2 (λ) al1 (λ)−1 ar2 (λ)ar4 (λ)−1 = al3 (λ)al4 (λ)−1 ar4 (λ)−1 al1 (λ)−1 ar2 (λ)ar4 (λ)−1 Tl (λ) R(λ) = = = S(λ), al3 (λ)al1 (λ)−1 ar4 (λ)−1 L(λ) Tr (λ) which concludes the proof.

4. Marchenko Theory In this section we prove the unique solvability of the Marchenko integral equations that lead to the solution of the inverse scattering problem of determining the potential in the defocusing matrix Zakharov-Shabat system from one of the reflection coefficients. The information obtained will then be used to solve the characterization problem of describing the scattering data leading to a unique determination of a potential having its entries in L1 (R).

532


IEOT

4.1. From Reflection Coefficient to Scattering Matrix We first summarize some of the properties of the scattering coefficients, referring to [4, Theorem 3.1] and [8, Proposition 3.13 and the two lines above its statement] for the proof. Proposition 8. In the defocusing case the scattering matrix S(λ) is unitary, i.e., S(λ)−1 = S(λ)† ,

λ ∈ R.

Further, the reflection and transmission matrices are continuous in λ ∈ R, while as λ → ±∞ the reflection coefficients vanish and the transmission coefficients tend to the identity. Moreover, the transmission coefficients Tl (λ) and Tr (λ) are continuous in λ ∈ C+ and analytic in λ ∈ C+ , while sup kTl (λ)k > 0,

sup kTr (λ)k > 0.

λ∈C+

(36a)

λ∈C+

The reflection coefficients R(λ) and L(λ) satisfy the inequalities sup kR(λ)k < 1,

sup kL(λ)k < 1.

λ∈R

λ∈R

(36b)

In order to construct the scattering matrix from one of the reflection coefficients, we define the so-called Wiener algebra W q denote of all q × q matrix functions of the form [9, 10, 11] Z ∞ Z(λ) = Z∞ + dα z(α)eiλα , (37) −∞

where z(α) is a q × q matrix function whose entries belong to L1 (R) and Z∞ = Z(±∞). Then W q is a Banach algebra with unit element endowed with the norm Z ∞ kZ(λ)kW q = kZ∞ k + dα kz(α)k. −∞

The analogous Banach algebra of q × r matrix functions is denoted as W q×r , so q×r that W q×q = W q . By W+ we denote the closed subalgebra of W q×r consisting of those Z(λ) of the type (37) for which z(α) is supported on the positive real line. q q×q We write W+ instead of W+ . It is important to recall the following result [4, 8]. Theorem 9. The coefficients al (λ) and ar (λ) are elements of W n+m . Now it is simple to show that in the defocusing case the scattering data consist of just one reflection coefficient, either R(λ) or L(λ), while the other reflection coefficient and the transmission coefficients can computed in the process. Indeed, using the unitarity of S(λ) we first determine the unique matrix functions Tl (λ) n×n and Tr (λ) such that Tl is an invertible element of W+ with Tl (±∞) = In , Tr

Vol. 62 (2008)


533

m×m is an invertible element of W+ with Tr (±∞) = Im , and the following two equations are true:

Tl (λ) Tl (λ)† = In − R(λ) R(λ)† , †

λ ∈ R,

†

Tr (λ) Tr (λ) = Im − R(λ) R(λ),

λ ∈ R.

These factorization problems have a unique solution, as a result of the following [12, 10, 6] Theorem 10. Let F ∈ L1 (R; Cp×p ) be such that Z ∞ ˆ (λ) = Ip + W dt eiλt F (t) −∞

is positive and selfadjoint for λ ∈ R. Then there exist unique functions F+ ∈ L1 (R+ ; Cp×p ) and G+ ∈ L1 (R+ ; Cp×p ) such that Z ˆ (λ) = Ip + W

∞

Z dt eiλt F+ (t) Ip +

∞

† Z −iλt dt e G+ (t) Ip +

0

Z ˆ W (λ) = Ip +

∞

† dt e−iλt F+ (t) ,

0

0

∞ iλt

dt e

G+ (t) ,

0

while Z ∞ iλt det Ip + dt e F+ (t) 6= 0, 0 Z ∞ det Ip + dt eiλt G+ (t) 6= 0,

λ∈ C+ , λ∈ C+ .

0

Finally we define L(λ) by L(λ) = −Tr (λ) R(λ)† [Tl (λ)† ]−1 ,

λ ∈ R.

On the other hand, given L ∈ W m×n satisfying the second of (36b), we first determine the unique matrix functions Tl (λ) and Tr (λ) such that Tl is an invertible n×n m×m element of W+ with Tl (±∞) = In , Tr is an invertible element of W+ with Tr (±∞) = Im , and the following two equations are true: Tl† (λ) Tl (λ) = In − L† (λ) L(λ),

λ ∈ R,

Tr (λ) Tr (λ)† = Im − L(λ) L† (λ),

λ ∈ R.

By Theorem 10, these factorization problems again have a unique solution. We then define R(λ) = −Tl (λ)L(λ)† [Tr (λ)† ]−1 ,

λ ∈ R.

534


IEOT

4.2. Coupled and uncoupled Marchenko equation It is well-known [4, 8] that the Marchenko integral equation can be written in two different and equivalent forms. In the defocusing case the coupled Marchenko equations2 are given by Z ∞ K1 (x, α) = − dβ K2 (x, β) Ω(α + β + 2x)† , (38a) 0 Z ∞ K2 (x, α) = −Ω(α + 2x) − dβ K1 (x, β) Ω(α + β + 2x), (38b) 0

where Ω(α) =

1 2π

∞

Z

dλ R(λ)e−iαλ ,

Z

∞

dα Ω(α)eiαλ .

R(λ) =

−∞

(39)

−∞

Substituting (38a) into (38b), we obtain the uncoupled Marchenko equation K2 (x, α) + Ω(α + 2x) Z ∞ Z ∞ − dβ K2 (x, β) dγ Ω(β + γ + 2x)† Ω(α + γ + 2x) = 0n×m . 0

(40)

0

Formally we can write the adjoint of the uncoupled Marchenko equation (40) as I − Ω† Ω K2† = −Ω† . The potential q(x) is obtained from the solution of the Marchenko equation (40) as follows: q(x) = 2K2 (x, 0+ ).

(41)

In the same way we get3 Z ∞ K3 (x, α) = − dβ K4 (x, β) Ω(α + ˜β − 2x) 0 Z ∞ † ˜ ˜ + β − 2x)† , K4 (x, α) = −Ω(α − 2x) − dβ K3 (x, β) Ω(α

(42a) (42b)

0

where 1 ˜ Ω(α) = 2π

Z

∞ −iαλ

dλ L(λ)e

Z ,

∞

L(λ) =

−∞

iαλ ˜ dα Ω(α)e .

−∞

Substituting (42a) into (42b), we obtain the uncoupled Marchenko equation ˜ − 2x)† K4 (x, α) + Ω(α Z ∞ Z ∞ ˜ + γ − 2x)Ω(α ˜ + γ − 2x)† = 0n×m . − dβ K4 (x, β) dγ Ω(β (43) 0

0

The potential q(x) is obtained from the solution of the Marchenko equation (43) as follows: q(x) = −2K4 (x, 0+ ). 2 In 3 In

[4, 8] the notations K1 = Bl1 and K2 = Bl2 were used. [4, 8] the notations K3 = Br1 and K4 = Br2 were used.

Vol. 62 (2008)


535

To prove the unique solvability of (40) we need the following two elementary results. A proof of Lemma 12 can be found in [11]. Lemma 11. Let Ω belong to L1 (R+ ). Then the operator KΩ defined by Z ∞ dβ Ω(α + β)b(β) (KΩ b) (α) =

(44)

0

is bounded on L2 (R+ ) and satisfies the norm estimate ˆ ∞ ≤ kΩk1 , kKΩ k ≤ kΩk ˆ denotes the Fourier transform of Ω. where Ω ˆ ∞ Proof. The estimate kKΩ k ≤ kΩk1 is immediate, while the estimate kKΩ k ≤ kΩk follows using the commutative diagram L2 (R+ ) −−−−−−−−−−−−−−−−→ L2 (R) −−−−→ L2 (R) F imbedding plus inversion    ˆ KΩ y y[Ω] L2 (R+ )

←−−−−−−−−−−−−− orthogonal projection

L2 (R) ←−−−− L2 (R) F −1

where the first step involves the imbedding-plus-inversion f (β) 7→ f (−β) and the ˆ third step multiplication by Ω. Lemma 12. Let Ω belong to L1 (R+ ). Then for 1 ≤ p < +∞ the operator KΩ defined by (44) is compact on Lp (R+ ) and has the norm estimate kKΩ k ≤ kΩk1 . Using Lemma 11, Lemma 12, and Proposition 8, it is easy to prove the following Theorem 13. For each x ∈ R and 1 ≤ p < +∞ the Marchenko equations (38) are uniquely solvable for K1 (x, ·) and K2 (x, ·) with entries in Lp (R+ ). ˆ Proof. In the defocusing case Ω(λ) = R(λ) and (36b) is true. Then the integral operators appearing in (38) have an operator norm bounded above by kRk∞ , which is strictly less than 1. The unique solvability of (38) now follows also in the other Lp -spaces as a result of the compactness of the integral operators involved. Indeed, putting TΩ = I − KΩ and X = L2 (R+ ) ∩ Lp (R+ ) endowed with the sum of the L2 and Lp norms, we first prove the compactness of KΩ on X. Next, since TΩ is Fredholm of index zero on Lp (R+ ) and X and invertible on L2 (Ω) and X is continuously and densely imbedded in both L2 (Ω) and Lp (Ω), the invertibility of TΩ is also true on X and Lp (R+ ).

536


IEOT

4.3. Characterization problem The characterization problem can be described as follows: Give necessary and sufficient conditions for an n × m matrix function R(λ) to be the right reflection coefficient of a defocusing matrix Zakharov-Shabat system (1) whose potential q(x) has its entries in L1 (R). In this subsection we shall solve this characterization problem. A similar characterization problem can be formulated and solved in terms of the left reflection coefficient L(λ). In fact, the solutions of the direct and inverse scattering problems for (1) provide a 1,1-corresponding between potentials q(x) with entries in L1 (R) and a suitable class of n × m matrix functions R(λ) or m × n matrix functions L(λ) on the line, as depicted in the following diagram:

1

q(x) with entries in L (R)

direct scattering problem −−−−−−−−−−−−−−−−−−−−→ ←−−−−−−−−−−−−−−−−−−−−− inverse scattering problem

R(λ) or L(λ)

The solution of the characterization problem for the Schrödinger equation on the line is well-known [16, 18]. As far as we know, no solution of the characterization problem for the matrix Zakharov-Shabat system has been published. In the defocusing case on the half-line Melik-Adamjan [17] has given a complete characterization of the Jost solution as scattering data to retrieve an L1 -potential. Theorem 14. In the defocusing case it is possible to determine a unique potential q(x) with entries in L1 (R) from the right reflection coefficient R(λ) if and only if the following conditions are satisfied: 1) supλ∈R kR(λ)k < 1, and 2) the n × m matrix function Ω(α) given by (39) has its entries in L1 (R). A similar characterization result holds in the case of a left reflection coefficient L(λ) as scattering data. Proof. It is well-known [4, 8] that a defocusing matrix Zakharov-Shabat system (1) with L1 -potential has a right reflection coefficient R(λ) satisfying conditions 1)-2) of Theorem 14. To prove the converse, we assume to have an n × m matrix function R(λ) with properties 1)-2). We then prove uniquely solvable the Marchenko equations (38) for K1 (x, ·) and K2 (x, ·) with entries in L1 (R+ ). We then consider the integral (x) operator KΩ as a function of x ∈ R. Then for b(α) with entries in Lp (R+ ) we have Z ∞ (x1 ) (x2 ) k[KΩ − KΩ ]bkp ≤ kbkp dβ kΩ(β + 2x1 ) − Ω(β + 2x2 )k, 0

Vol. 62 (2008)


537

(x)

so that KΩ depends continuously on x ∈ R in the operator norm. Moreover, its operator norm Z ∞ Z ∞ (x) kKΩ k ≤ dβ kΩ(β + 2x)k = dγ kΩ(γ)k → 0, x → +∞. 0

2x

Since (38) are uniquely solvable for K1 (x, ·) and K2 (x, ·) with entries in L1 (R+ ), we see that for any x0 ∈ R we have Z ∞ def C(x0 ) = sup dβ kK1 (x, β)k < +∞. x≥x0

0

Defining q(x) by (41), i.e., by Z q(x) = −2Ω(2x) − 2

∞

dβ K1 (x, β)Ω(β + 2x),

(45)

0

by integrating (45) we get for each x0 ∈ R Z ∞ dx kq(x)k ≤ (1 + C(x0 ))kΩk1 < +∞, x0

which proves that any right tail of the potential obtained has L1 entries. Next, we apply the same argument to the identity Z ∞ † ˜ ˜ − 2x)† , q(x) = 2Ω(−2x) + 2 dβ K1 (x, β)Ω(β

(46)

0

which follows directly from (42). Arguing that Z ∞ def dβ kK3 (x, β)k < +∞, D(x0 ) = sup x≤x0

we now get Z

0

x0

dx kq(x)k ≤ (1 + D(x0 ))kΩk1 < +∞, −∞

which proves that any left tail of the potential obtained has L1 entries. As a result, the potential as a whole has L1 entries.

Appendix A. Definition of the Full Hamiltonian Write V = V1 V2 as the product of the two matrix functions V1 and V2 with L2 entries. Then for λ ∈ C \ R W (λ) = I + V2 (λ − H0 )−1 V1

(47)

is a Hilbert-Schmidt perturbation of the identity, so that W (λ) is invertible except possibly on a discrete subset of C \ R. Now put def

R(λ) = (λ − H0 )−1 − (λ − H0 )−1 V1 W (λ)−1 V2 (λ − H0 )−1 ,

(48)

which implies R(λ)† = (λ − H0 )−1 − (λ − H0 )−1 V2† [W (λ)† ]−1 V1† (λ − H0 )−1 .

(49)

538


IEOT

Then for nonreal λ the operators R(λ) and R(λ)† both have a zero kernel, because (λ − H0 )−1 does. Hence, R(λ) has a zero kernel and a dense range. Next, we show that R(λ) satisfies the resolvent identity. Indeed, R(λ)R(ζ) = (λ − H0 )−1 (ζ − H0 )−1 − (λ − H0 )−1 (ζ − H0 )−1 V1 W (ζ)−1 V2 (ζ − H0 )−1 − (λ − H0 )−1 V1 W (λ)−1 V2 (λ − H0 )−1 (ζ − H0 )−1 + (λ − H0 )−1 V1 W (λ)−1 V2 (λ − H0 )−1 (ζ − H0 )−1 V1 W (ζ)−1 V2 (ζ − H0 )−1 . Multiplying each term (called I, II, III, and IV) by (ζ − λ) and using theresolvent identity (ζ − λ)(λ − H0 )−1 (ζ − H0 )−1 = (λ − H0 )−1 − (ζ − H0 )−1 , we obtain (ζ − λ)R(λ)R(ζ) = (λ − H0 )−1 − (ζ − H0 )−1 − {(λ − H0 )−1 − (ζ − H0 )−1 }V1 W (ζ)−1 V2 (ζ − H0 )−1 − (λ − H0 )−1 V1 W (λ)−1 V2 {(λ − H0 )−1 − (ζ − H0 )−1 } − (λ − H0 )−1 V1 W (λ)−1 V2 (ζ − H0 )−1 + (λ − H0 )−1 V1 W (ζ)−1 V2 (ζ − H0 )−1 , where the last two terms occur by writing W (λ)−1 {V2 (λ − H0 )−1 V1 − V2 (ζ − H0 )−1 V1 }W (ζ)−1 as the difference of two terms. Letting the terms in the righthand side be called Ia, Ib, IIa, IIb, IIIa, IIIb, IVa, and IVb, we see that IVaand IIIb cancel out and IVb and IIa cancel out. At the end, we get (ζ − λ)R(λ)R(ζ) = R(λ) − R(ζ), which is the resolvent identity. Thus there exists a closed and densely defined linear ˜ such that operator H ˜ −1 , R(λ) = (λ − H)

λ ∈ C \ R.

˜ −1 − (λ − H0 )−1 is a trace class operator. Clearly, (λ − H) To derive “mixed” resolvent identities, we first derive from (48) and (49) with the help of (47) V2 R(λ) = V2 (λ − H0 )−1 − [W (λ) − I]W (λ)−1 V2 (λ − H0 )−1 = W (λ)−1 V2 (λ − H0 )−1 , V1† R(λ)† = V1† (λ − H0 )−1 − [W (λ)† − I][W (λ)† ]−1 V1† (λ − H0 )−1 = [W (λ)† ]−1 V1† (λ − H0 )−1 , which are Hilbert-Schmidt operators [cf. Proposition 2] except possibly on a discrete subset of C \ R. These identities in turn imply

Vol. 62 (2008)


539

(λ − H0 )−1 − (λ − H0 )−1 V1 V2 R(λ) = (λ − H0 )−1 − (λ − H0 )−1 V1 W (λ)−1 V2 (λ − H0 )−1 = R(λ), (λ − H0 )−1 − (λ − H0 )−1 V2† V1† R(λ)† = (λ − H0 )−1 − (λ − H0 )−1 V2† [W (λ)† ]−1 V1† (λ − H0 )−1 = R(λ)† , which again imply (λ − H0 )−1 − [(λ − H0 )−1 V1 ][V2 R(λ)] = R(λ), −1

(λ − H0 )

−1

− [R(λ)V1 ][V2 (λ − H0 )

] = R(λ).

(50a) (50b)

˜ is a selfadjoint operator Equations (50) imply that R(λ)† = R(λ) and hence that H ˜ on Hn+m . Moreover, H is an extension of H0 −V (which can in principle be defined 2 on the dense domain Hn+m , but not as a closed operator). We therefore define the full Hamiltonian H as follows: ˜ H = H. ˜ has the same domain as If the entries of V belong to L1 (R) ∩ L2 (R), then H = H 1 H0 , namely Hn+m .

References [1] M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. [2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Stud. Appl. Math. 53, 249–315 (1974). [3] M.J. Ablowitz, B. Prinari, and A.D. Trubatch, Discrete and Continuous Nonlinear Schr¨ odinger Systems, Cambridge University Press, Cambridge, 2004. [4] T. Aktosun, M. Klaus, and C. van der Mee, Direct and inverse scattering for selfadjoint Hamiltonian systems on the line, Integral Equations and Operator Theory 38, 129–171 (2000). [5] D.Z. Arov and H. Dym, Strongly regular J-inner matrix-valued functions and inverse problems for canonical systems. In: M.A. Kaashoek, S. Seatzu, and C. van der Mee (eds.), Recent Advances in Operator Theory and its Applications, Birkh¨ auser OT 160, Basel and Boston, 101–160 (2005). [6] K.F. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Birkh¨ auser OT 3, Basel and Boston, 1981. [7] P. Deift and E. Trubowitz, Inverse scattering on the line, Commun. Pure Appl. Math. 32, 121–251 (1979). [8] F. Demontis, Direct and Inverse Scattering of the Matrix Zakharov-Shabat System, Ph.D. thesis, University of Cagliari, Italy, 2007. [9] I. Gelfand, D. Raikov, and G. Shilov, Commutative Normed Rings, AMS Chelsea Publ., Amer. Math. Soc., Providence, RI, 1991.

540


IEOT

[10] I.C. Gohberg and I.A. Feldman, Convolution Equations and Projection Methods for their Solution, Transl. Math. Monographs 41, Amer. Math. Soc., Providence, RI, 1974. [11] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Vol. I, Birkh¨ auser OT 49, Basel, 1990. [12] I.C. Gohberg and M.G. Krein, Systems of integral equations on a half line with kernels depending on the difference of arguments, Amer. Math. Soc. Transl. (2)14, 217–287 (1960). [13] A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers, Third ed., Springer Series in Photonics 9, Springer, Berlin, 2002. [14] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966. [15] S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 38, 248–253 (1974). [16] V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkh¨ auser OT 22, Basel and Boston, 1986; also: Naukova Dumka, Kiev, 1977 (in Russian). ` Melik-Adamyan, On a class of canonical differential operators, Soviet J. Con[17] F.E. temporary Math. Anal. 24, 48–69 (1989); also: Izv. Akad. Nauk Armyan. SSR Ser. Mat. 24, 570–592, 620 (1989) [Russian]. [18] A. Melin, Operator methods for inverse scattering on the real line, Commun. Part. Diff. Eqs. 10, 677–766 (1985). [19] M. Reed and B. Simon, Methods of Modern Mathematical Physics. III. Scattering Theory, Academic Press, New York, 1979. [20] L.A. Sakhnovich, Spectral Theory of Canonical Differential Systems. Method of Operator Identities, Birkh¨ auser OT 107, Basel and Boston, 1999. [21] J.K. Shaw, Mathematical Principles of Optical Fiber Communications, CBMS-NSF Regional Conference Series 76, SIAM, Philadelphia, 2004. [22] C. van der Mee Direct and inverse scattering for skewselfadjoint Hamiltonian systems. In: J.A. Ball, J.W. Helton, M. Klaus, and L. Rodman (eds.), Current Trends in Operator Theory and its Applications, Birkh¨ auser OT 149, Basel and Boston, 2004, pp. 407–439. [23] J. Weidmann, Lineare Operatoren in Hilbertr¨ aumen, Teil II: Anwendungen, Teubner, Stuttgart, 2003. [24] D.R. Yafaev, Mathematical Scattering Theory, Transl. Math. Monographs 105, Amer. Math. Soc., Providence, RI, 1992. [25] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34, 62–69 (1972). Francesco Demontis and Cornelis van der Mee Dip. Matematica e Informatica, Universit` a di Cagliari Viale Merello 92, 09123 Cagliari, Italy e-mail: [email protected] [email protected] Submitted: March 27, 2008. Revised: November 10, 2008.


auser Verlag Basel/Switzerland 0378-620X/040541-13, published online November 17, 2008 DOI 10.1007/s00020-008-1636-z


Eigenfunction Expansions for Schrödinger Operators on Metric Graphs Daniel Lenz, Carsten Schubert and Peter Stollmann Abstract. We construct an expansion in generalized eigenfunctions for Schr¨ odinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. Mathematics Subject Classification (2000). Primary 47A70; Secondary 35J10, 81Q10. Keywords. Eigenfunction expansion, metric graphs, Schr¨ odinger operator.

1. Introduction Expansion in generalized eigenfunctions is a topic that dates back to Fourier’s work, at least. A classical reference is Berezanskii’s monograph [2]. Motivated by examples from Mathematical Physics there has been a steady development involving new models. One trigger of more recent results is the importance of generalized eigenfunction expansions in the discussion of random models. See [4, 11, 22] and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, [10] (see [1, 10, 15, 16, 17, 7, 12, 13, 14] for recent results on quantum graphs). There the authors consider a rather special class of metric graphs, due to the random model they have in mind. We point out, however that part of their discussion is rather abstract and pretty much equivalent to what had been obtained in [4]. As was pointed out in [3], the Dirichlet form framework of the latter article applies to a class of quantum graphs with Kirchhoff boundary conditions. The point of the present paper is to establish an expansion in generalized eigenfunctions under somewhat minimal conditions. This means we require just the usual conditions necessary to define the operators in question. These conditions essentially amount to providing a continuous embedding from the form domain of the operator to the Sobolev space W 1,2 of the graph. More concretely, we allow for general boundary conditions, unboundedness of the (locally finite) vertex degree

542

Lenz, Schubert and Stollmann

IEOT

function, loops, multiple edges and edges of infinite lengths. However, we require a uniform lower bound on the length of the edges. To the best of our knowledge, this framework contains all classes of models that have been considered so far. Our discussion is intrinsic and does not require an embedding of the metric graph into an ambient space. As far as methods are concerned, we rely on the results from [18] rather than the approach of [2] that had been used in [10]. However, this is mostly a question of habit. In either approach a main point is to establish certain trace class properties of auxiliary functions. Here, we can rely upon one-dimensional techniques for quantum graphs. An extra asset is that we are able to establish pointwise properties of generalized eigenfunctions. Our paper is structured as follows: In Section 2 we set up model and notation, define metric graphs and introduce the kind of boundary conditions we allow. Moreover, we check the necessary operator theoretic input for the Poerschke-StolzWeidmann method for constructing generalized eigenfunctions. In Section 3 we discuss the notion of generalized eigenfunctions and explore pointwise properties in the quantum graph case. It turns out that in this case generalized eigenfunctions have versions that satisfy the boundary conditions at the vertices. In Section 4 we present the necessary material from [18]. The application to the quantum graph case comes in Section 5 that contains our main results, Theorem 5.1 and Corollary 5.4. The former deals with quantum graphs and the latter includes additional perturbations by a potential that is uniformly locally square integrable.

2. Metric graphs and the associated operators In this section we introduce metric graphs and the associated operators. The basic idea is that a metric graph consists of line segments – edges – that are glued together at vertices. In contrast to combinatorial graphs, these line segments are taken seriously as differential structures and in fact one is interested in the Laplacian on the union of the line segments. To get a self-adjoint operator one has to specify boundary conditions at the vertices. Our discussion of the unperturbed operator associated to a quantum graph in this section relies on the cited works of Kostrykin & Schrader, [12], Kuchment, [16], and the second named author, [21]. In particular, the subsequent discussion up to Lemma 2.3 can essentially be found in [16]. Definition 2.1. A metric graph is Γ = (E, V, i, j) where • E (edges) is a countable family of open intervals (0, l(e)) and V (vertices) is a countable set. • i : E → V defines the initial point of an edge and j : {e ∈ E|l(e) < ∞} → V the end point for edges of finite length. S We let Xe := {e} × e, X = XΓ = V ∪ e∈E Xe and Xe := Xe ∪ {i(e), j(e)}.

Vol. 62 (2008)

Eigenfunction Expansion

543

Note that Xe is basically just the interval (0, l(e)), the first component is added to force the Xe ’s to be mutually disjoint. The topology on X will be such that the mapping πe : Xe → (0, l(e)), (e, t) 7→ t extends to a homeomorphism again denoted by πe : Xe → (0, l(e)) that satisfies πe (i(e)) = 0 and πe (j(e)) = l(e) (the latter in case that l(e) < ∞). A piece of the form I = πe−1 (J) with an edge e and an interval J ⊂ (0, l(e)) is called an edge segment. The length of the edge segment is the length of J. Edge segments will play a role, when we discuss local properties of functions. While we allow multiple edges and loops, we will assume finiteness of each single vertex degree dv , v ∈ V , i. e. (F) dv := |{(0, e) : v = i(e)} ∪ {(l(e), e) : v = j(e)}| < ∞. To define a metric structure on X we then proceed as follows: we say that p = (x1 , x2 , . . . , xN ) ∈ X N is a good polygon if for every k ∈ {1, ..., N − 1} there is a unique edge e ∈ E such that {xk , xk+1 } ⊂ Xe . Using the usual distance on [0, l(e)] we get a distance d on Xe and use it do define l(p) =

N −1 X

d(xk , xk+1 ).

k=1

Since multiple edges are allowed, we needed to restrict our attention to good polygons to exclude the case that {xk , xk+1 } are joined by edges of different length. Given connectedness of the graph and (F), a metric on X is given by d(x, y) := inf{l(p) | p a good polygon with x1 = x and xN = y}. In fact, symmetry and triangle inequality are evident and the separation of points follows from the finiteness. Clearly, with the topology induced by that metric, X is a locally compact, separable metric space. If X is not connected, we can do the above procedure on any connected component. We will assume a lower bound on the length of the edges: (LB) There exists a u > 0 with l(e) ≥ u for all e ∈ E. We will now turn to the relevant Hilbert spaces and operators. We define M M M L2 (X) := L2 (e), W 1,2 (X) := W 1,2 (e), W 2,2 (X) := W 2,2 (e). e∈E

e∈E 2

1,2

2,2

e∈E

Here, of course, L (e) (W (e), W (e)) consists of functions ue on e = (0, l(e)). In the sequel we will view those families u = (ue )e∈E ∈ L2 (X) rather as functions defined on X. Note that W 1,2 (X) and W 2,2 (X) are sometimes referred to as decoupled or maximal Sobolev spaces, see e.g. [9, 19]. Other Sobolev spaces can also be found in the literature. For our purpose, the above definitions seem to be the most convenient ones. Consider a > 0 and recall that h ∈ W 1,2 (0, a) is continuous and h(0) := limx→0+ h(x) exists and satisfies 2 |h(0)|2 ≤ khk2L2 (0,a) + akh0 k2L2 (0,a) (2.1) a

544


IEOT

by standard Sobolev type theorems. Consider now an edge e and u ∈ W 1,2 (e). Then the limit u(0) := limt→0 u(t) exists, as well as u(l(e)) := limt→l(e) u(t) and (2.1) holds (with the obvious modifications). Similarly, for an edge e and u ∈ W 2,2 (e) the limits u0 (0) := limt→0 u0 (t) and u0 (l(e)) := − limt→l(e) u0 (t) exist. Here, we have introduced a sign. This makes our definition of the derivative canonical, i. e. independent of the choice of orientation of the edge. For f ∈ W 1,2 (X) and each vertex v we gather the boundary values of fe over all edges e adjacent to v in a vector f (v). More precisely, let Ev := {(0, e) : v = i(e)} ∪ {(l(e), e) : v = j(e)} denote the set of outgoing and incoming edges adjacent to v and define f (v) := (fe (t))(t,e)∈Ev ∈ CEv . Similarly, for f ∈ W 2,2 (X) we further gather the boundary values of fe0 (t) over all edges e adjacent to v in a vector f 0 (v) ∈ CEv . Note that for each loop at a vertex v there are two entries in the vectors f (v) and f 0 (v). These boundary values of functions will play a crucial role when we discuss the concept of boundary condition. Definition 2.2. A boundary condition is given by a pair (L, P ) consisting of a family L = (Lv )v∈V of self-adjoint operators Lv : CEv −→ CEv and a family P = (Pv ) of projections Pv : CEv −→ CEv . We will assume the following upper bound on (Lv )v∈V : (UB) There exists an S > 0 with kL+ v k ≤ S for any v ∈ V , where the + denotes the positive part of a self-adjoint operator. Given a metric graph satisfying (F) and(LB) and a boundary condition satisfying (UB), we obtain from (2.1) by a direct calculation that X 4S kf kL2 (X) + 2Sεkf 0 kL2 (X) (2.2) hLv f (v), f (v)i ≤ ε v∈V

1,2

for any f ∈ W (X) and any ε > 0 with ε ≤ u. Given a boundary condition (L, P ) we define the form s0 := sL,P by D(s0 ) := {f ∈ W 1,2 (X) : Pv f (v) = 0 for all v ∈ V }, X X Z l(e) hLv f (v), g(v)i. s0 (f, g) := fe0 (t)g 0e (t)dt − e∈E

0

v∈V

By (2.2) we easily see that for C > 0 large enough 1 kf kW 1,2 (X) (2.3) 2 for any f ∈ D(s0 ). This shows that s0 is bounded below and closed. Hence, there exists an associated self-adjoint operator. This operator is denoted by H0 := HL,P . It can be explicitly characterized by s0 (f, f ) + C(f, f ) ≥

D(H0 ) := {f ∈ W 2,2 (X) : Pv f (v) = 0 and Lv f (v) + (1 − Pv )f 0 (v) = 0 for all v ∈ V }, (H0 f )e := −fe00 for all e ∈ E.

Vol. 62 (2008)


545

We will assume the following setting: (S) Γ is a metric graph satisfying (F) and (LB) with associated space X. (L, P ) is a boundary condition satisfying (UB). The induced form is denoted by s0 and the corresponding operator by H0 = HL,P . 1

Lemma 2.3. Assume (S). Then (H0 +C)− 2 provides a continuous map from L2 (X) to L∞ (X) for sufficiently large C > 0. 1

Proof. As H0 is bounded below, (H0 + C)− 2 provides a bounded map from L2 (X) to the form domain equipped with the form norm k·ks0 for sufficiently large C > 0. By (2.3), the form domain (with the form norm) is continuously embedded into W 1,2 (X). By (2.1), W 1,2 (X) is continuously embedded in L∞ (X). Putting this together we obtain the statement. Lemma 2.4. Assume (S). Then {f ∈ D(H0 ) : supp f compact} is a core for H0 . Proof. Choose f ∈ D(H0 ). We have to find fn ∈ D(H0 ) with compact support and fn −→ f and H0 fn −→ H0 f . We will provide fn = ψn f with suitable cutoff functions ψn . We will assume without loss of generality that X is connected (otherwise we will have to perform the process simultaneously on each connected component). Choose x ∈ X. For n ∈ N let Bn = B(x, n) be the ball around x with radius n. Construct ψn = (ψn,e )e∈E with ψn |B(x;n−2u) ≡ 1,

ψn |B(x;n+2u)c ≡ 0

(2.4)

by distinguishing three cases: For edges e with both ends i(e) and j(e) contained in Bn set ψn,e ≡ 1. For edges e with both ends i(e) and j(e) contained in the complement of Bn set ψn,e ≡ 0. For edges e with one endpoint, say i(e) ∈ Bn and j(e) ∈ Bnc we choose ψn,e two times continuously differentiable on e, ψn,e ≡ 1 on a suitable neighborhood of i(e), ψn,e ≡ 0 on a suitable neighborhood of j(e) such that ψn,e and its first two derivatives are bounded by (1 + 4/u)2 . This is possible, since the length of the edges is bounded below by u. Since in this way ψn is constant in the neighborhood of any vertex, smooth and bounded the functions fn := ψn f belong to D(H0 ) for every n ∈ N. By (2.4) we conclude fn → f in L2 (X) as n → ∞. Similarly, H0 (ψn f ) = −ψn f 00 − 2ψn0 f 0 − ψn00 f → H0 f, as ψn0 , ψn00 are uniformly bounded and supported on B(x; n + 2u) \ B(x; n − 2u). Remark 2.5. Let us shortly discuss the necessity of conditions of the form (LB) and (UB) in our context. Our aim is to show (2.3), i. e. that the identity is continuous as a map from the form domain with W 1,2 norm to the form domain with form norm.

546


IEOT

As we allow for rather general boundary conditions and do not assume any connectedness, we need a pointwise estimate on the boundary values of a function on an edge in terms of the corresponding W 1,2 (e) norm. In this respect, the Sobolev estimate (2.1) is essentially optimal. More precisely, testing with the constant function on an interval of finite length shows that the factor 1/a can not be avoided. In particular, (2.3) fails for a graph consisting of countably infinite disjoint edges with lengths going to zero and a δ-boundary condition (corresponding to Lv being the identity) on one of the vertices of each edge. In this sense, a condition of the form (LB) seems unavoidable. Similarly, given (LB), we need a bound of the form (UB) to bound the boundary terms. In particular, (2.3) fails for a graph consisting of countably infinite disjoint edges with lengths one and boundary conditions of the form cv L with cv going to infinity.

3. A word on locality Let a locally compact space X with a measure dx be given. Let L2loc (X) be the space of functions on X whose restrictions to compact sets are square integrable. Let L2comp (X) be the set of functions in L2 (X) which have compact support. The usual inner product can be “extended” to give a map (again denoted by h·, ·i) Z L2comp (X) × L2loc (X) −→ C, hf, gi := f (x)g(x)dx. Definition 3.1. Let X be a topological space with a measure dx. Let H be an operator on X which is local i. e. Hf has compact support whenever f has and D(H) ∩ Lcomp is a core for H. A nontrivial function φ on X is called a generalized eigenfunction for H corresponding to λ if it belongs to L2loc (X) and satisfies hHf, φi = λhf, φi

(3.1)

for any f ∈ D(H) with compact support. Remark 3.2. Here, hHf, φi is defined in the sense discussed at the beginning of the section. The inner product hf, φi is defined in the same way. The condition on the core of H is not necessary to state the definition. However, it is only this condition that makes the definition a sensible one. The question arises to which extent a generalized eigenfunction is locally a good function. We say that φ ∈ L2loc (X) is locally in W 2,2 if the restriction φI belongs to W 2,2 (I) for any compact edge segment. In particular, φe ∈ W 2,2 (e) for every edge of finite length. Note that L2loc (X)-functions belong to L2 of any edge of finite length. Here is one answer in the case of quantum graphs: Lemma 3.3. Assume (S). If φ is a generalized eigenfunction for H0 , then φ is locally in W 2,2 and admits a version that satisfies the boundary condition at any vertex.

Vol. 62 (2008)


547

Proof. To check that φ belongs locally to W 2,2 is suffices to consider f ∈ D(H0 ) with compact support contained in an edge and apply (3.1). This gives −φ00 = λφ so that φ belongs locally to W 2,2 , since φ ∈ L2loc (X) by our definition of generalized eigenfunction. To check that φ satisfies the boundary condition at a vertex v, it suffices to consider f ∈ D(H0 ) supported on a neighborhood of v and apply (3.1). In fact, let f ∈ D(H0 ) with fe ≡ 0 for all edges e not adjacent to v. Then we get hf, λφi = hH0 f, φi = h−f 00 , φi; integration by parts and the condition on the support of f give (with the evident notation for the inner product in CEv ) . . . = hf, −φ00 i + hf 0 (v), φ(v)i − hf (v), φ0 (v)i = hf, λφi + hf 0 (v), φ(v)i − hf (v), φ0 (v)i as the second weak derivative of φ is −λφ. Therefore, hf 0 (v), φ(v)i = hf (v), φ0 (v)i for every choice of f ∈ D(H0 ). Splitting the scalar products in the parts living in the images of Pv and 1 − Pv gives hPv f 0 (v), Pv φ(v)i + h(1 − Pv )f 0 (v), (1 − Pv )φ(v)i = hPv f (v), Pv φ0 (v)i + h(1 − Pv )f (v), (1 − Pv )φ0 (v)i. Choosing f ∈ D(H0 ) with arbitrary Pv f 0 (v) and (1 − Pv )f (v) = 0 (granting (1 − Pv )f 0 (v) = 0), we see that Pv φ(v) has to be equal to zero. If we use the boundary condition for f , the last equation can be transformed to hPv f 0 (v), Pv φ(v)i = h(1 − Pv )f (v), Lv φ(v) + (1 − Pv )φ0 (v)i. Taking an f with arbitrary (1 − Pv )f (v), we conclude that Lv φ(v) + (1 − Pv )φ0 (v) also equals zero, thus giving the boundary condition for φ.

4. Expansion in generalized eigenfunctions: general framework In this section we discuss the expansion in generalized eigenfunctions of a selfadjoint operator. We follow the work of Poerschke, Stolz and Weidmann [18]. This will be used to provide an expansion for metric graphs in a spirit similar to the considerations of [4] for Dirichlet forms. Note that in [10] a different approach has been used. However, an important point in both the different methods is to establish suitable trace class properties for operators constructed from H. In that respect, the analysis of [4, 10] is similar. Actually, the case of quantum graphs is rather easy as far as trace class properties are concerned, as we have a locally one-dimensional situation at hand.

548


IEOT

Let a Hilbert space (H, h·, ·i) and a self-adjoint operator T ≥ 1 in H be given. We will define the following two auxiliary Hilbert spaces: H+ := H+ (T ) := D(T ), hx, yi+ := hT x, T yi and H− as completion of H with respect to the scalar product hx, yi− := hT −1 x, T −1 yi. Thus, the inner product on H can be naturally extended to give a map h·, ·i : H+ × H− −→ C. Let N be a positive integer or infinity, H a self-adjoint operator in H and µ a spectral measure for H. A sequence of subsets Mj ⊂ R, such that Mj ⊃ Mj+1 together with a unitary map U MN U = (Uj ) : H → L2 (Mj , dµ) j=1

is said to be an ordered spectral representation of H if U φ(H) = Mφ U, for every measurable function φ on R. Theorem 4.1 (Theorem 1 of Section 3 in [18]). Let H, T , H+ , H− be as above. Let µ be a spectral measure for H and U an ordered spectral representation. Let γ : R −→ C be continuous and bounded with |γ| > 0 on σ(H) such that γ(H)T −1 is a Hilbert-Schmidt operator. Then there are measurable functions φj : Mj → H− , λ 7→ φj,λ for j = 1, . . . , N such that the following properties hold: (i) Uj f (λ) = hf, φj (λ)i for L f ∈ H+ and µ-a. e. λ ∈ Mj . (ii) For every g = (gj ) ∈ j L2 (Mj , dµ) U −1 g =

lim

N X

n→N,E→∞

Z gj (λ)φj,λ dµ(λ)

j=1 Mj ∩[−E,E]

and, for every f ∈ H, f=

lim

n→N,E→∞

N X

Z (Uj f )(λ)dµ(λ).

j=1 Mj ∩[−E,E]

(iii) For f ∈ {g ∈ D(H) ∩ H+ | Hg ∈ H+ } and µ-a. e. λ ∈ Mj hHf, φj,λ i = λhf, φj,λ i.

(4.1)

If the functions φj,λ fulfill (i) and (ii) of the theorem, we will speak of a Fourier type expansion. If the set {g ∈ D(H) ∩ H+ | Hg ∈ H+ } is a core for H, we speak of an expansion in generalized eigenfunctions. We will apply the previous theorem to the Hilbert Space H = L2 (X) and the operator H0 , where X is a quantum graph satisfying (F), (LB) and (UB) as discussed in Section 2. As T we then use the operator T := Mw of multiplication with a suitable weight function w, i. e. a continuous map w : X −→ [1, ∞).

Vol. 62 (2008)


549

5. The main theorem Theorem 5.1. Assume (S). Let µ be a spectral measure for H0 . Let w : X → [1, ∞) be continuous with w−1 ∈ L2 (X). Then there exists a Fourier type expansion (φj ) for H0 , such that for µ-a. e. λ ∈ σ(H0 ) the function φj,λ is a generalized eigenfunction of H0 for λ with w−1 φj,λ ∈ L2 . Proof. We will apply the abstract result of the previous section. Let γ be the function γ(t) = (C + t)−1/2 . As T choose multiplication with w. Then, γ(H0 ) is a bounded map from L2 (X) to L∞ (X) by Lemma 2.3. This, together with the assumption on w easily shows that the operator T −1 γ(H0 ) has an L2 kernel and is therefore a Hilbert-Schmidt operator. Thus, its adjoint operator γ(H0 )T −1 is a Hilbert-Schmidt operator as well. We can therefore apply the result of the previous section. This gives a Fourier type expansion. By definition of T any function in H− is locally in L2 . Moreover, hH0 f, φj,λ i = λhf, φj,λ i holds µ-a. e. (in λ) for f ∈ Dw := {g ∈ D(H0 )| wg, wH0 g ∈ L2 (X)}. As w is continuous and H0 f has compact support whenever f has compact support by definition of H0 , the set Dw obviously contains D(H0 ) ∩ L2comp (X). Thus, the functions φj,λ are generalized eigenfunctions in the sense of Section 3. This finishes the proof. We denote by m the measure induced on X by the Lebesgue measure on the edges Xe , pulled back via πe . Remark 5.2 (A weight function). Assume that X is connected and define, for > 0, 1+ w(x) = m Bd(x,x0 )+1 (x0 ) . Clearly, w is continuous and w ≥ 1. To see that w−1 ∈ L2 (X), it suffices to consider the case that Γ is infinite. In this case, m(Br (x0 )) ≥ r for every x0 ∈ X and r > 0 by construction of the metric. We consider the volume of the annuli Bn (x0 ) \ Bn−1 (x0 ). For x in this annulus we obviously have that w(x) ≥ m(Bn (x0 ))1+ . Hence, suppressing the x0 in the notation of the balls, Z Z Z −1 2 −2 |w | dx ≤ w dx + w−2 dx + . . . X

B1 \B0

Z

B2 \B1 −2−2

≤

m(B1 ) B1 \B0

≤

∞ X i=1

where we used m(Bn ) ≥ n.

i−1−2 < ∞,

Z dx + B2 \B1

m(B2 )−2−2 dx + . . .

550


IEOT

Schr¨ odinger operators Now we show that our main result can be extended to Schrödinger operators on metric graphs. Here, we treat a rather simple case. More singular perturbations will be considered elsewhere. In the following proposition we gather some operator theoretic results for potential perturbations of the operators H0 = HL,P for a quantum graph satisfying assumption (S). For a general background, we refer the reader to [20], Section X. 2 as well as [8], §5 and §6. Q We are going to consider the class of potentials V ∈ L2 (e) with e

M := MV := sup{kVI k2 : I edge segment with length between u and 2u} < ∞. This class will be denoted by L2loc,u (X). Proposition 5.3. Assume (S) and let V ∈ L2loc,u (X). Then we have: (i) V is infinitesimally small with respect to H0 . In particular, H = H0 + V is self-adjoint on D(H0 ). 1 (ii) (H + C)− 2 provides a continuous map from L2 (X) to L∞ (X) for sufficiently large C > 0. (iii) {f ∈ D(H0 ) : supp f compact} is a core for H. (iv) If φ is a generalized eigenfunction for H, then φ is locally in W 2,2 and admits a version that satisfies the boundary condition at any vertex. Proof. (i) Let a > 0 be arbitrary. Assume w.l.o.g. that a ≤ u. We now decompose the edges of the graph into edge segments, which are disjoint up to their boundary and have length between a and 2a. Then any point of the graph belongs to such an edge segment I. Accordingly, our usual Sobolev estimate (2.1) gives a 0 2 4 kf |I k2 + kf |I k22 . (5.1) 2 a Note that we pick up an extra factor of 2 compared to estimate (2.1) as the point may not lie at the boundary of I (in which case we only have an interval of length a/2 at our disposal). Recall the estimate kf |I k2∞ ≤

kf k2W 1,2 ≤ 2s0 (f, f ) + Ckf k22 . Summing over all I of our decomposition we obtain X kV f k22 = k(V f )|I k22 I

≤

X

kV |I k22 kf |I k2∞

I

≤ M2

X I

kf |I k2∞

(5.2)

Vol. 62 (2008)


(5.1) ≤ M 2

551

X a I

4 kf 0 |I k22 + kf |I k22 2 a

4 a ≤ M 2 kf k2W 1,2 + M 2 kf k22 2 a Ca 4 (5.2) ≤ M 2 as0 (f, f ) + M 2 kf k22 + M 2 kf k22 2 a = M 2 as0 (f, f ) + C(a)kf k22 , where Ca

4 . 2 a As s0 (f, f ) ≤ kH0 f kkf k ≤ kH0 f k2 + kf k2 , we obtain C(a) = M 2

+

kV f k2 ≤ M 2 akH0 f k2 + (C(a) + M 2 a)kf k22 . As a > 0 is arbitrary, self-adjointness of H and (iii) both follow from the KatoRellich theorem, cf [20], Theorem X. 12. (ii) It follows from (i) that V is also form small with respect to H0 , see [8] and [20] so that the form norm of H0 and H are equivalent. Hence (ii) follows from Lemma 2.3 above. (iv) For every compact edge segment I we get that the restriction φI of φ to I satisfies φ00I = VI φI − λφI in the weak sense. Since φI ∈ L2 (I) for every compact I and VI ∈ L2 (I), we get that φI ∈ L1 . In particular, φ0I admits a continuous version so that φI ∈ C(I). Since VI ∈ L2 (I) this gives that φ is locally in W 2,2 . The rest of the argument can be taken from the proof of Lemma 3.3 with the obvious rewording. This gives the following analog of Theorem 5.1 for Schrödinger operators: Corollary 5.4. Assume (S) and let V ∈ L2loc,u (X). Let µ be a spectral measure for H. Let w : X → [1, ∞) be continuous with w−1 ∈ L2 (X). Then there exists a Fourier type expansion (φj ) for H, such that for µ-a. e. λ ∈ σ(H0 ) the function φj,λ is a generalized eigenfunction of H for λ with w−1 φj,λ ∈ L2 .

References [1] M. Aizenman, R. Sims and S. Warzel: Absolutely continuous spectra of quantum tree graphs with weak disorder. Comm. Math. Phys., 264(2):371–389, 2006. [2] Y. M. Berezanskii: Expansion in Eigenfunctions of Self-Adjoint Operators. Transl. of Math. Mon., Vol. 17, Am. Math. Soc., Providence 1968. [3] A. Boutet de Monvel, D. Lenz and P. Stollmann: Sch’nol’s theorem for strongly local forms. Israel J., to appear. [4] A. Boutet de Monvel and P. Stollmann: Eigenfunction expansions for generators of Dirichlet forms. J. Reine Angew. Math., 561:131–144, 2003.

552


IEOT

[5] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon: Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics. Springer, Berlin, 1987. [6] E. B. Davies: Spectral theory and differential operators. Cambridge University Press, Cambridge, 1995. [7] P. Exner: A duality between Schr¨ odinger operators on graphs and certain Jacobi matrices. Ann. Inst. H. Poincaré Phys. Théor., 66(4):359–371, 1997. [8] W. G. Faris: Self-adjoint operators. Springer, Berlin, 1975. [9] M. J. Gruber, M. Helm and I. Veselic: Optimal Wegner estimates for random Schr¨ odinger operators on metric graphs. Analysis on Graphs and Its Applications. Proceedings of Symposia in Pure Mathematics, volume 77, 2008. [10] P. D. Hislop and O. Post: Anderson localization for radial tree-like random quantum graphs. http://www.arxiv.org/abs/math-ph/0611022, 2006. [11] A. Klein, A. Koines and M. Seifert: Generalized eigenfunctions for waves in inhomogeneous media. (Special issue dedicated to the memory of I. E. Segal), J. Funct. Anal., 190(1):255–291, 2002. [12] V. Kostrykin and R. Schrader: Kirchhoff’s rule for quantum wires. J. Phys. A, 32(4):595–630, 1999. [13] V. Kostrykin and R. Schrader: Kirchhoff’s rule for quantum wires. II. The inverse problem with possible applications to quantum computers. Fortschr. Phys., 48(8):703–716, 2000. [14] V. Kostrykin and R. Schrader: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. In Quantum graphs and their applications. Proceedings of an AMS-IMS-SIAM joint summer research conference on quantum graphs and their applications, Snowbird, UT, USA, June 19–23, 2005, volume 415 of Contemporary Mathematics, pages 201–225. Providence, RI: American Mathematical Society (AMS), 2006. [15] P. Kuchment: Graph models for waves in thin structures. Waves Random Media, 12(4):R1–R24, 2002. [16] P. Kuchment: Quantum graphs. I. Some basic structures. Waves Random Media, 14(1):S107–S128, 2004. Special section on quantum graphs. [17] P. Kuchment: Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A, 38(22):4887–4900, 2005. [18] T. Poerschke, G. Stolz and J. Weidmann: Expansions in Generalized Eigenfunctions of Selfadjoint Operators. Math. Z., 202(3):397–408, 1989. [19] O. Post: Spectral analysis of metric graphs and related spaces. http://arxiv.org/abs/0712.1507, 2007. [20] M. Reed and B. Simon: Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness. Academic Press, San Diego, 1975. [21] C. Schubert: Laplace-Operatoren auf Quantengraphen, Diploma Thesis, Chemnitz University of Technology. http://www.tu-chemnitz.de/∼carst/DA Schubert.pdf, 2006. [22] P. Stollmann: Caught by disorder: A Course on Bound States in Random Media, volume 20 of Progress in Mathematical Physics. Birkh¨ auser, 2001.

Vol. 62 (2008)


Daniel Lenz Fakult¨ at f¨ ur Mathematik Technische Universit¨ at Chemnitz 09107 Chemnitz Germany Current address: Department of Mathematics Rice University P. O. Box 1892 Houston, TX 77251 USA e-mail: [email protected] Carsten Schubert Fakult¨ at f¨ ur Mathematik Technische Universit¨ at Chemnitz 09107 Chemnitz Germany e-mail: [email protected] Peter Stollmann Fakult¨ at f¨ ur Mathematik Technische Universit¨ at Chemnitz 09107 Chemnitz Germany e-mail: [email protected] Submitted: January 9, 2008. Revised: September 10, 2008.

553




On the Fedosov-H¨ ormander Formula for Differential Operators Patrick J. Rabier

Abstract. The Fedosov-H¨ ormander formula gives the Fredholm index of some pseudodifferential operators of order 0 on L2 . It is well known that it can be used to calculate the index of elliptic systems under the assumption that, among other things, the coefficients are smooth and their partial derivatives of all orders satisfy specific asymptotic conditions at infinity. We prove that the formula remains valid when the coefficients are only C 1 and bounded and have vanishing oscillation at infinity. In turn, this generalization is used to obtain a nonstandard invariance property of the index as well as various sufficient conditions for the index to be 0, when the coefficients are merely continuous and bounded with vanishing oscillation. Mathematics Subject Classification (2000). 47A53, 35J30. Keywords. Elliptic system, Fredholm index.

1. Introduction Let P (x, ∂) denote the linear differential system P (x, ∂) :=

m X

Aα (x)∂ α ,

(1.1)

|α|=0 |α|

where Aα ∈ C ∞ (RN , L(Cr )) and ∂ α := ∂xα1∂···∂xαN . It has been known for long 1 N that if the coefficients Aα are bounded and lim ∂ β Aα (x) = 0,

|x|→∞

(1.2)

556

Rabier

IEOT

r for every multi-index β 6= 0, then P (x, ∂) is Fredholm from W m,2 := W m,2 (RN ) r to L2 := L2 (RN ) if and only if its symbol σP (x, ξ) :=

m X

Aα (x)i|α| ξ α ∈ L(Cr ),

(1.3)

|α|=0

is invertible when |x|2 + |ξ|2 ≥ R2 for some R ≥ 0 and there is a constant C > 0 such that m

(1 + |ξ|2 ) 2 ||σP (x, ξ)−1 || ≤ C whenever |x|2 + |ξ|2 ≥ R2 .

(1.4)

This is independently due to Seeley [17] (unpublished) and Taylor [19]. Even though both address only the scalar case r = 1, their approach can easily be extended to systems, but it does not provide the value of the index. (For a special class of pseudodifferential systems on RN , a characterization of the index via Brouwer’s degree was given by Seeley [16, Appendix, Theorem 3].) A decade or so later, by sharpening a method of Fedosov [7] and when (1.2) is replaced by the stronger condition ∂ β Aα (x) = O |x|−|β| as |x| → ∞, (1.5) Hörmander [10] obtained the formula N Z ∧2N −1 (N − 1)! i , T r σP−1 dσP index P (x, ∂) = − 2π (2N − 1)! ∂B

(1.6)

where B is any open ball in RN × RN about the origin with sufficiently large radius such that σP is invertible outside B, the orientation of RN × RN is given ∧2N −1 by dx1 ∧ dξ1 · · · ∧ dxN ∧ dξN and σP−1 dσP is evaluated as a matrix product upon replacing ordinary multiplication by the exterior product (and T r is the matrix trace). In [10], the formula is given for the index of a pseudodifferential operator P (x, ∂)w different from P (x, ∂), but its validity for P (x, ∂) is explained1 in the comments following Theorem 19.3.1’ in [11, p. 221]. It was immediately conjectured by Bott and Seeley in [1, p. 245] that (1.6) should still be true under the weaker assumption (1.2). This was proved by Schrohe [15] (apparently unaware of [1]) in 1992, for a class of pseudodifferential operators large enough to accommodate condition (1.2). In this paper, we first give a very different and arguably more accessible proof that (1.2) suffices for the validity of the formula (1.6). This proof is specifically for differential operators and makes no explicit use of pseudodifferential theory (Theorem 3.5). It also avoids some technicalities skipped in [15]. For example, the fact that the right-hand side of (1.6) is always an integer, regardless of whether it coincides with the index of P (x, ∂). This is true, but not obvious (as incidentally pointed out in [1]): The justification relies on nontrivial properties of the de Rham cohomology of GL(Cr ), most notably the Bott periodicity theorem in the form 1A

further clarification is needed; we shall return to this point later.

Vol. 62 (2008)

Fedosov-H¨ ormander Formula

557

given in [1, p. 237]. In contrast, the proof of Theorem 3.5 does not involve any cohomological consideration. Next, based on the generalized formula for the index and an approximation argument, we show that the formula (1.6) remains true when the coefficients Aα are C 1 and bounded and (1.2) holds when |β| = 1. Even less is actually needed (Theorem 4.1). This is used to obtain a nonstandard t-independence property of the index of families (Pt (x, ∂))t∈[0,1] . This property differs from the classical homotopy invariance of the index insofar as the continuity of t ∈ [0, 1] 7→ Pt (x, ∂) ∈ L(W m,2 , L2 ) is not required (Theorem 4.4) and indeed is not true in some applications (Remark 4.6). In Section 5, we give sufficient conditions for P (x, ∂) to have index 0 when the coefficients Aα are merely continuous and bounded with vanishing oscillation at infinity, a case when the formula (1.6) is not directly applicable. Attention is confined to results that cannot be obtained by more elementary arguments. For example, we show that the index is always 0 if r < N (Theorem 5.1) or if (1.4) holds with R = 0 (Theorem 5.2). The proof of the latter provides an example when the invariance property of Section 4, but not the classical homotopy invariance, is applicable. In particular, the index is 0 when the coefficients Aα depend only upon N − 1 among the N variables x1 , ..., xN (Corollary 5.4). Other conditions are given in Theorem 5.6. From the p-independence of the index of P (x, ∂) (see [13]), everything remains true when P (x, ∂) acts from W m,p to Lp for any p ∈ (1, ∞) and also in the weighted setting of [14]. Such generalizations are useful when P (x, ∂) arises as the linearization of a nonlinear operator and neither p = 2 nor the smoothness of the coefficients is a realistic assumption. In that regard, it should be kept in mind that Fredholmness is crucial, especially when the index is 0, to investigate local or global bifurcation issues, or existence questions via degree theory. This is the main reason why Section 5 was incorporated. The results of this paper could also be extended to a suitable class of pseudodifferential operators with nonsmooth symbols, but doing so would of course be in contradiction with the goal of giving an exposition as widely accessible as possible. As in [15], the basic idea to generalize the Fedosov-Hörmander formula is to use a suitable homotopy to reduce the problem to the case when (1.5) holds, so that the original formula is applicable. However, instead of using pseudodifferential methods for that purpose, we rely on the Banach algebra approach to PDEs developed by H. O. Cordes and collaborators throughout the sixties and seventies. Although very much overshadowed by the classical theory of pseudodifferential operators, this approach has significant value -notably from the perspective of nonlinear analysis- due to the weak smoothness requirements about the coefficients. All the necessary background is discussed in the next section. We need only one prerequisite (Theorem 2.2), which is easy to understand without any particular

558

Rabier

IEOT

expertise after a short introduction. Its (known) proof does involve a substantial amount of Banach algebra machinery, but none of this machinery is needed to obtain the generalized Fedosov-Hörmander formula in Theorem 3.5. The paper is completed by two short appendices. The first one is intended to clarify what may be seen as an ambiguity: The formula given in [10], [11] is for pseudodifferential operators of order 0 on L2 and therefore can only be used m after replacing P (x, ∂) by (say) P (x, ∂)(Ir ⊗ (1 − ∆))− 2 . Of course, this operator has the same index as P (x, ∂), but then the formula for the index involves the m symbol (1 + |ξ|2 )− 2 σP (x, ξ) rather than σP (x, ξ). There are several ways to show that this change of symbol is inconsequential, even though we were unable to find an explicit statement to that effect. The quickest argument is to use the homotopy invariance of the right-hand side of (1.6), which follows from Stokes’ theorem and the remark that the form T r (Z −1 dZ)∧2N −1 is closed on GL(Cr ), but a justification involving only a straightforward direct calculation is given in Appendix 1. The second appendix shows that under the most general conditions considered in this paper about the coefficients Aα , the operator P (x, ∂) in (1.1) cannot be Fredholm if mr is odd and N ≥ 2 (this is false if N = 1). This is only used in Remark 5.7 but definitely worth knowing and, once again, it is a feature that we have not been able to find in the literature, even though it is very unlikely to be new.

2. Notation and background If f : RN → C is continuous and x ∈ RN , define the modulus of continuity of f at x by mcf (x) := max |f (y) − f (x)|. (2.1) |y−x|≤1

We set O0 (RN ) := {f ∈ Cb (RN ) : lim mcf (x) = 0}, |x|→∞

(2.2)

where Cb (RN ) denotes the space of complex-valued continuous bounded functions on RN . The above definition is unchanged if |y − x| ≤ 1 is replaced by |y − x| ≤ h for any h > 0 in (2.1). The space O0 (RN ) is a C ∗ -algebra with unity (the involution being simply complex conjugation). It turns out that the maximal ideal space of O0 (RN ), henceforth denoted by PN , is a compact topological space and that RN is open and dense in PN . Aside from compactness, the key property of PN -quite reminisˇ cent of the Stone-Cech compactification- is that every f ∈ O0 (RN ) has a unique continuous extension to PN (for details, see Cordes [6, Lemma 1.5, p. 134], where O0 (RN ) is called CM (RN )). Therefore, the same extension property is true when f : RN → L(Cr ) ' Cr×r and the r2 scalar entries of f are in O0 (RN ). The correr×r sponding space will be denoted by O0 (RN ) . It is customary to say that the N r×r functions in O0 (R ) have vanishing oscillation at infinity.

Vol. 62 (2008)


559

Later on, we shall also need the space N ∞ N N β O∞ 0 (R ) := {f ∈ C (R ) ∩ Cb (R ) : lim ∂ f (x) = 0, ∀|β| ≥ 1} |x|→∞

(2.3)

N r×r and the corresponding space O∞ of L(Cr )-valued functions. With this 0 (R ) N r×r notation, condition (1.2) is simply that Aα ∈ O∞ for 0 ≤ |α| ≤ m. It is 0 (R ) ∞ N r×r N r×r readily checked that O0 (R ) ⊂ O0 (R ) . Now, denote by τ : RN → RN the mapping 1

τ (ξ) := (1 + |ξ|2 )− 2 ξ,

(2.4)

which is a homeomorphism of RN onto the open unit ball B1 of RN and set T(RN ) := {g ∈ Cb (RN ) : g ◦ τ −1 ∈ C(B 1 )}, where g ◦ τ −1 ∈ C(B 1 ) means that g ◦ τ −1 has a continuous extension, necessarily unique, to B 1 . It is plain that T(RN ) is a C ∗ -subalgebra of Cb (RN ). Once again, the maximal ideal space of T(RN ), henceforth denoted by BN , is another compactification of RN containing RN as an open and dense subset and every g ∈ T(RN ) has a unique continuous extension to BN ([6, Lemma 1.5, p. 134]). Obviously, N τ ∈ T(RN ) , so that τ has a unique continuous extension τb to BN . It is readily verified that τb is in fact a homeomorphism of BN to B 1 . [If M is a maximal ideal of T(RN ), then M = {g ∈ T(RN ) : gb(ξ) = 0} for a unique ξ ∈ BN . On the other hand, {g ◦ τ −1 : g ∈ M} is a maximal ideal of C(B 1 ), so that there is a unique x ∈ B 1 such that {g ◦ τ −1 : g ∈ M} = {h ∈ C(B 1 ) : h(x) = 0}. The mapping M 7→ {g ◦ τ −1 : g ∈ M} is a homeomorphism betwen the maximal ideal space of T(RN ) and the maximal ideal space of C(B 1 ), whence ξ ∈ BN 7→ x ∈ B 1 is also a homeomorphism. If ξ ∈ RN , then x = τ (ξ) is obvious. Thus, by the continuity of τb and the denseness of RN in BN , x = τb(ξ) for every x ∈ B 1 . This shows that τb is the homeomorphism ξ ∈ BN 7→ x ∈ B 1 .] Since RN × RN is open in PN × BN , the set MN := PN × BN \ RN × RN is compact. It will henceforth be referred to as “symbol space”. Upon setting δPN := PN \RN and δ BN := BN \RN ,

(2.5)

MN := δPN × RN ∪ PN × δBN .

(2.6)

it follows that For future use, note that τb(δBN ) = SN −1

(2.7)

(unit sphere of RN ) and indeed, from the above, τb is a homeomorphism of δBN onto SN −1 .

560

Rabier

IEOT

Remark 2.1. That τb is a homeomorphism of δBN onto SN −1 explains why MN is N N N sometimes defined by δP × R ∪ P × SN −1 in the literature. It may also help to mention that when the coefficients Aα have limits when |x| → ∞ (hence have vanishing oscillation at infinity), then PN may be replaced by the familiar one-point N N N N N e e compactification R of R , so that M becomes {ω} × R ∪ R × SN −1 , where ω denotes the point at infinity. This definition is frequently used in the early work. Now that we have a symbol space MN , we can associate a symbol sbP defined on MN with every differential operator P (x, ∂) in (1.1) with coefficients Aα ∈ r×r O0 (RN ) . To do this, observe first that for (x, ξ) ∈ RN × RN , m

sP (x, ξ) := (1 + |ξ|2 )− 2 σP (x, ξ) =

m X

(1 − |τ (ξ)|2 )

m−|α| 2

i|α| Aα (x)τ (ξ)α , (2.8)

|α|=0

where σP (x, ξ) and τ (ξ) are given by (1.3) and (2.4), respectively. cα to PN and τ by Upon replacing Aα by its unique continuous extension A N N τb, we obtain a continuous extension sbP of sP to P × B . The restriction of sbP to MN is, by definition, the desired symbol of P (x, ∂). Explicitly (see (2.5), (2.6) and (2.7)): ( P m−|α| m 2 cα (x)τ (ξ)α if (x, ξ) ∈ δPN × RN , 2 i|α| A |α|=0 (1 − |τ (ξ)| ) sbP (x, ξ) := P m c i τ (ξ)α if (x, ξ) ∈ PN × δBN . |α|=m Aα (x)b (2.9) From the above definition, sbP (x, ξ) ∈ L(Cr ) for every (x, ξ) ∈ MN . The main result on the topic is Theorem 2.2. (i) P (x, ∂) : W m,2 → L2 is Fredholm if and only if sbP (x, ξ) ∈ GL(Cr ) for every (x, ξ) ∈ MN (i.e., sbP (MN ) ⊂ GL(Cr )). (ii) If Q(x, ∂) is a differential operator satisfying the same assumptions as P (x, ∂) and the symbols sbP and sbQ are homotopic in C(MN , GL(Cr )), then index P (x, ∂) = index Q(x, ∂). When r = 1, part (i) of Theorem 2.2 is (the case s = 0 of) [6, Theorem 2.4, p. 138]. It is helpful to notice that by the compactness of MN , the condition sbP (MN ) ⊂ GL(Cr ) is equivalent to (1.4). See Remark 2.3 below. Thus, when the “vanishing oscillation” assumption is replaced by the more restrictive condition (1.2), part (i) of Theorem 2.2 coincides with the classical results mentioned in the Introduction. Even though systems are not discussed in [6], it is notorious that they do not present any significant new difficulty. This can be explained by the fact that the invertibility of sbP (x, ξ) is equivalent to the non-vanishing of det sbP (x, ξ), which, roughly speaking, reduces the problem to the scalar case (although there are a few e N (see Remark 2.1), the result for technicalities). When PN can be replaced by R systems is in Cordes [4, Theorem 7] and Cordes and Herman [5, Theorem 9]. The

Vol. 62 (2008)


561

more general case of Theorem 2.2 is treated by Sun [18], but the paper is hard to find. Part (ii) of Theorem 2.2 is a truncated version of the stronger statement: (ii0 ) P (x, ∂) and Q(x, ∂) are homotopic as Fredholm operators if and only if their symbols are homotopic in C(MN , GL(Cr )). However, this formulation requires explaining how (generalized) pseudodifferential operators fit within the Banach algebra approach2 , which is irrelevant for our purposes. At any rate, there is a well established procedure to prove (ii0 ) once part (i) of Theorem 2.2 is available. For instance, this is done in an abstract setting in Breuer and Cordes [2, Theorem 3, p. 320] when r = 1. As pointed out in Cordes and Herman [5, Theorem 10], it is straightforward to modify the argument when r > 1. Remark 2.3. Since the equivalence between (1.4) and sbP (MN ) ⊂ GL(Cr ) will be N used repeatedly, we include a proof. If (1.4) holds, det s−1 P is bounded in (R × N R )\B where B is the open ball about the origin with radius R, so that det sP is bounded away from 0 in (RN × RN )\B. By continuity, det sbP remains bounded away from 0 in (PN × BN )\B. Since B is a compact subset of RN × RN , the set (PN × BN )\B is a neighborhood of MN in PN × BN . Thus, det sbP does not vanish on MN , i.e., sbP (MN ) ⊂ GL(Cr ). Conversely, if sbP (MN ) ⊂ GL(Cr ), then det sbP is bounded away from 0 on MN since MN is compact (and sbP is continuous). By continuity, det sbP is bounded away from 0 in some open neighborhood V of MN in PN × BN . Since (PN × BN )\V is compact and does not intersect MN , it is a compact subset of RN ×RN and thus contained in an open ball B about the origin. From the above, det sp is bounded away from 0 in (RN × RN )\B, which, since the coefficients of sP are bounded (see (2.8)), implies (1.4) with R being the radius of B.

3. A generalization of the Fedosov-H¨ ormander formula In this section, it is assumed that the coefficients Aα of P (x, ∂) in (1.1) are in N r×r O∞ (see (2.3)) and that (1.4) holds. The constant R > 0 is the one 0 (R ) involved in the latter condition and is chosen once and for all. From (1.4) and (2.8), m X m−|α| (1 − |η|2 ) 2 i|α| Aα (x)η α |α|=0

is invertible for every x ∈ RN such that |x| ≥ R and every η ∈ RN with |η| < 1. By Remark 2.3 and (2.9), this remains true when x ∈ δPN and also when x ∈ PN 2 Those may have to be involved in the homotopy between P (x, ∂) and Q(x, ∂), which may not consist entirely of differential operators.

562

Rabier

IEOT

cα . Thus, and |η| = 1 (use (2.7)), upon replacing Aα by A m X

(1 − |η|2 )

m−|α| 2

cα (x)η α ∈ GL(Cr ), i|α| A

∀x ∈ PN \BR , ∀η ∈ B 1 ,

(3.1)

|α|=0

where BR = RB1 and X cα (x)η α ∈ GL(Cr ), im A

∀x ∈ PN , ∀η ∈ SN −1 .

(3.2)

|α|=m

Now, let θ ∈ C0∞ (RN ) be such that 0 ≤ θ ≤ 1, θ(x) = 1 if |x| ≤ R and N N b θ(x) = 0 if |x| ≥ R + 1. Evidently, θ ∈ O∞ 0 (R ) and the extension θ of θ to P N N vanishes on δP . For t ∈ [0, 1], x ∈ P and η ∈ B 1 , set b z(t, x, η) := (1 − t + tθ(x))η, so that |z(t, x, η)| ≤ 1. Next, set h(t, x, η) :=

m X

(1 − |z(t, x, η)|2 )

m−|α| 2

cα (x)z(t, x, η)α . i|α| A

(3.3)

|α|=0

Since |z(t, x, η)| ≤ 1, we infer from (3.1) that h(t, x, η) is invertible when t ∈ [0, 1], x ∈ PN \BR and η ∈ B 1 . In addition, sincePz(t, x, η) = η when |x| < R (because θ(x) = 1), (3.3) shows that h(t, x, η) = im |α|=m Aα (x)η α when |x| < R and |η| = 1. Hence, by (3.2), h(t, x, η) remains invertible when |x| < R and η ∈ SN −1 . From the above, h is continuous on [0, 1] × PN × B 1 and h(t, x, η) ∈ GL(Cr ) when (t, x, η) ∈ [0, 1] × (δPN × B1 ) ∪ (PN × SN −1 ) . Let now Φ : RN → RN be any C ∞ function such that Φ(x) = x if |x| ≤ R + 1, |Φ(x)| ≥ R + 1 if |x| > R + 1 and lim|x|→∞ Dk Φ(x) = 0 in Lk (RN ; RN ) (k-linear mappings) for every k ≥ 1. Let PΦ (x, ∂) denote the operator obtained from P by N r×r replacing the coefficients Aα by Φ∗ Aα := Aα ◦ Φ. Clearly, Φ∗ Aα ∈ O∞ . 0 (R ) Let σPΦ (x, ξ) be given by (1.3) when P is replaced by PΦ , so that σPΦ (x, ξ) = σP (Φ(x), ξ). Since Φ(x) = x when |x| ≤ R and |Φ(x)| ≥ R if |x| > R, it follows that |Φ(x)|2 + |ξ|2 ≥ R2 whenever |x|2 + |ξ|2 ≥ R2 and so, by (1.4), m

(1 + |ξ|2 ) 2 ||σPΦ (x, ξ)−1 || ≤ C whenever |x|2 + |ξ|2 ≥ R2 .

(3.4)

As a result, we may repeat the construction of h above with P (x, ∂) replaced by N r PΦ (x, ∂). This yields a continuous mapping hΦ : [0, 1] × P × B 1 → L(C ) such that hΦ (t, x, η) ∈ GL(Cr ) when (t, x, η) ∈ [0, 1] × (δPN × B1 ) ∪ ( PN × SN −1 ) . When t = 1, h(1, x, η) =

m X |α|=0

2 b (1 − θ(x)|η| )

m−|α| 2

b |α| η α cα (x)θ(x) i|α| A

Vol. 62 (2008)

and hΦ (1, x, η) =


m X

2 b (1 − θ(x)|η| )

m−|α| 2

563

∗ A (x)θ(x) b |α| η α . \ i|α| Φ α

|α|=0

If |x| < R + 1, then Φ(x) = x and so h(1, x, η) = hΦ (1, x, η) for every η ∈ B 1 . b b0 (x) and On the other hand, if |x| ≥ R + 1, then θ(x) = 0, so that h(1, x, η) = A ∗ \ \ hΦ (1, x, η) = Φ A0 (x) = A0 ◦ Φ(x). Since |Φ(x)| ≥ R + 1 whenever |x| ≥ R + 1, this shows that h(1, x, η) = hΦ (1, x, η) for every x ∈ PN \BR+1 and η ∈ B 1 if A0 (x) is constant outside BR+1 (because the extension of both A0 and A0 ◦ Φ to δPN is also equal to that constant). If so, h(1, ·) = hΦ (1, ·) on PN × B 1 and the mapping h(2t, ·) if t ∈ [0, 1/2], H(t, ·) = hΦ (2 − 2t, ·) if t ∈ [1/2, 1], is a homotopy from h(0, ·) to hΦ (0, ·) in L(Cr ) such that H(t, x, η) ∈ GL(Cr ) when (t, x, η) ∈ [0, 1] × (δPN × B1 ) ∪ (PN × SN −1 ) . As a result, H(t, x, τb(ξ)) with τ (ξ) from (2.4) is a homotopy from sbP to sbPΦ (see (2.9)), which is invertible on [0, 1] × MN (recall (2.6) and (2.7)). Thus, by Theorem 2.2 (ii), we obtain: Lemma 3.1. If A0 (x) is independent of x for |x| ≥ R + 1, then index P (x, ∂) = index PΦ (x, ∂). The next lemma makes it possible to reduce the problem to the case resolved in Lemma 3.1. e0 : RN → GL(Cr ) such that A e0 (x) = A0 (x) Lemma 3.2. There is a C ∞ mapping A when |x| ≥ R + 1. Proof. By (1.4) with ξ = 0, it already follows that A0 (x) is invertible when |x| ≥ R. To prove the lemma, it suffices to show that A0|∂BR is homotopic to a constant operator in GL(Cr ). Indeed, if so, A0|∂BR can be extended to B R as a continuous e0 with values in GL(Cr ). Thus, by setting A e0 (x) := A0 (x) for |x| > R, mapping A we obtain a mapping with the desired properties, except that it is continuous on e0 can be smoothed out in such a way RN but not C ∞ . However, it is plain that A r that it still has P values in GL(C ) and coincides with A0 when |x| ≥ R + 1. By (3.2), |α|=m Aα (x)η α is invertible for every η ∈ SN −1 and every x ∈ RN . In particular, A(m,0,...,0) (x) is invertible for every x ∈ RN . Since by (1.4) m (1 + |ξ|2 )− 2 σP (x, ξ) is invertible for every ξ ∈ RN when |x| = R, we obtain a homotopy from A0|∂BR to A(m,0,...,0)|∂BR in GL(Cr ) by choosing ξ = (tan π2 t, 0, ..., 0) and t ∈ [0, 1] (with of course tan π2 := ∞). Since A(m,0,...,0) (x) is invertible for every x ∈ B R , it follows that A(m,0,...,0)|∂BR is homotopic (in GL(Cr )) to A(m,0,...,0) (0). Thus, the same thing is true of A0|∂BR by concatenation of homotopies. Remark 3.3. The above proof shows that P (x, ∂) : W m,2 → L2 cannot be Fredholm if its coefficients have vanishing oscillation at infinity and the restriction of

564

Rabier

IEOT

A0 to a sphere of arbitrarily large radius in RN is not homotopic to a constant in GL(Cr ). With Lemma 3.2, we may now remove the extra requirement that A0 (x) is independent of x for |x| ≥ R + 1 in Lemma 3.1. Lemma 3.4. index P (x, ∂) = index PΦ (x, ∂). e0 be given by Lemma 3.2. Since A e0 (x) = A0 (x) when |x| ≥ R and Proof. Let A r×r N e0 ∈ O∞ (RN ) r×r . Furthermore, since A0 ∈ O∞ , it is obvious that A 0 (R ) 0 e0 (x)−1 is A0 (x)−1 is uniformly bounded for |x| ≥ R by (1.4), it follows that A N e0 is an isomoruniformly bounded for x ∈ R . As a result, the multiplication by A e0 (x)−1 P (x, ∂) are simultaneously phism of L2 , so that P (x, ∂) and Q(x, ∂) := A Fredholm and have the same index. e−1 Aα ∈ O∞ (RN ) r×r Now, Q(x, ∂) is a differential operator with coefficients A 0 0 N r×r because O∞ is an algebra and because the uniform boundedness of 0 (R ) N r×r −1 N r×r e e e−1 ∈ O∞ . Also, A0 (x) together with A0 ∈ O∞ imply A 0 (R ) 0 (R ) 0 e0 (x)−1 σP (x, ξ), so that Q(x, ∂) satisfies (1.4) with the same R (but σQ (x, ξ) = A e0 . Since the zeroth order coefficient A e−1 A0 a different C) by the boundedness of A 0 of Q(x, ∂) is Ir for |x| ≥ R + 1, it follows from Lemma 3.1 that indexQ(x, ∂) = e0 (Φ(x)))−1 PΦ (x, ∂) and the multiplication by index QΦ (x, ∂). But QΦ (x, ∂) = (A e0 ◦Φ is an isomorphism of L2 from the analogous property for A e0 , so that QΦ (x, ∂) A and PΦ (x, ∂) are simultaneously Fredholm and have the same index. This completes the proof. N r×r Theorem 3.5. In (1.1), suppose that Aα ∈ O∞ and that condition (1.4) 0 (R ) holds. Then, P (x, ∂) : W m,2 → L2 is Fredholm and index P (x, ∂) is given by the Fedosov-H¨ ormander formula (1.6), where B is any open ball about the origin in RN × RN such that σP is invertible in the exterior of B. Proof. In Lemma 3.4, Φ : RN → RN is any C ∞ function such that Φ(x) = x for |x| ≤ R + 1, |Φ(x)| ≥ R + 1 if |x| > R + 1 and lim|x|→∞ Dk Φ(x) = 0 in Lk (RN ; RN ) for every k ≥ 1. In particular, we may choose Φ to have the form x Φ(x) := g(|x|) , |x| where g : [0, ∞) → [0, ∞) is a smooth nondecreasing function such that g(s) = s is s ≤ R + 1 and g(s) = R + 2 if s ≥ R + 2. If so, not only Φ(x) satisfies all the x for |x| ≥ R + 2, so that the required conditions but, in addition, Φ(x) = (R + 2) |x| ∗ coefficients Φ Aα of PΦ satisfy the condition ∂ β (Φ∗ Aα )(x) = O |x|−|β| as |x| → ∞, needed to use the Fedosov-H¨ ormander formula (1.6) with PΦ (x, ∂). Recall that in that formula, B denotes any open ball about the origin in RN × RN such that σPΦ

Vol. 62 (2008)


565

is invertible in the exterior of B. By (3.4), B can be chosen to have radius R. Thus, N Z ∧2N −1 i (N − 1)! index PΦ (x, ∂) = − T r σP−1 dσ . P Φ Φ 2π (2N − 1)! ∂B Evidently, the integrand depends only upon the restriction of σPΦ to ∂B. Since |x| ≤ R for every point (x, ξ) ∈ ∂B, it follows that Φ(x) = x and hence that σPΦ (x, ξ) = σP (x, ξ) when (x, ξ) ∈ ∂B. Thus, N Z ∧2N −1 (N − 1)! i T r σP−1 dσP , index PΦ (x, ∂) = − 2π (2N − 1)! ∂B and so, by Lemma 3.4, index P (x, ∂) = −

i 2π

N

(N − 1)! (2N − 1)!

Z Tr ∂B

σP−1 dσP

∧2N −1

.

If now B 0 is another open ball about the origin such that σP is invertible in the exterior of B 0 , then it is readily checked that (1.4) holds with R replaced by the radius R0 of B 0 (after modifying C). Thus, B can be replaced by B 0 in the above formula.

4. Invariance of the index First, we generalize Theorem 3.5 to the C 1 case. r×r Theorem 4.1. In (1.1), suppose that Aα ∈ O0 (RN ) and that Aα is C 1 for 0 ≤ |α| ≤ m. If also condition (1.4) holds, then P (x, ∂) : W m,2 → L2 is Fredholm and the Fedosov-H¨ ormander formula (1.6) is valid. Proof. By Remark 2.3, (1.4) is equivalent to sbP (MN ) ⊂ GL(Cr ), so that P (x, ∂) : W m,2 → L2 is Fredholm by part (i) of Theorem 2.2. Let (ϕk ) ⊂ C0∞ (RN ) be a mollifying sequence. Specifically, ϕk (x) := k N ϕ(kx) R with ϕ ∈ C0∞ (RN ), Suppϕ ⊂ B1 and RN ϕ = 1. It is straightforward to check that (k) (k) N r×r Aα := ϕk ∗ Aα ∈ O∞ and that Aα → Aα uniformly on RN as k → ∞ 0 (R ) (for a proof, see [6, Lemma 10.4, p. 122]). If P (k) (x, ∂) denotes the corresponding sequence of differential operators, it follows that P (k) (x, ∂) → P (x, ∂) in L(W m,2 , L2 ). Thus, P (k) (x, ∂) is Fredholm and index P (k) (x, ∂) = index P (x, ∂) for k large enough by the local constancy of the index. Since the coefficients of sP are uniformly bounded on RN × RN (see (2.8)), (1.4) amounts to saying that det sP is bounded away from 0 in (RN × RN )\B, where B is the ball about the origin with radius R. By the uniform convergence (k) of Aα to Aα on RN , the same thing is true of det sP (k) for k large enough (recall that τ (ξ) ∈ B1 in (2.8)). In turn, this means that (1.4) holds (with another C but the same R) upon replacing P (x, ∂) by P (k) (x, ∂) for k large enough.

566

Rabier

IEOT

Together with Theorem 3.5, it follows from the above not only that the formula (1.6) gives the index of P (k) (x, ∂) for k large enough, but that the same ball B can be used for every such k. Thus N Z ∧2N −1 (N − 1)! i T r σP−1(k) dσP (k) , (4.1) index P (x, ∂) = − 2π (2N − 1)! ∂B for k large enough. (k) Since Aα is C 1 , the partial derivatives ∂Aα /∂xi tend ∂Aα/∂xi uniformly to ∧2N −1 −1 N on the compact subsets of R as k → ∞. Thus, T r σP (k) dσP (k) → ∧2N −1 −1 T r σP dσP uniformly on ∂B as k → ∞, which yields formula (1.6) for index P (x, ∂) by taking the limit in (4.1). From the choice of B above, σP is invertible in the exterior of B. If B 0 is another open ball about the origin such that σP is invertible in the complement of B 0 , the argument at the end of the proof of Theorem 3.5 can be repeated to show that B can be replaced by B 0 in (4.1). r×r r×r Note that Aα ∈ O0 (RN ) provided that Aα ∈ Cb (RN ) is C 1 and that (1.2) holds when |β| = 1. This can be relaxed to lim|x|→∞ ||∂Aα /∂xi ||L1 (B(x,1)) = 0 r×r for 1 ≤ i ≤ N, which makes it is easy to find examples when Aα ∈ O0 (RN ) but the first order partial derivatives of Aα are unbounded. Remark 4.2. Theorem 4.1 cannot be obtained without Theorem 3.5, because the r×r coefficients Aα ∈ O0 (RN ) cannot always be uniformly approximated by C ∞ coefficients satisfying (1.5). We shall now use Theorem 4.1 to prove the t-independence of the index of families of differential operators Pt (x, ∂) when t ∈ [0, 1], under assumptions that do not require that t ∈ [0, 1] 7→ Pt (x, ∂) ∈ L(W m,2 , L2 ) is continuous. We begin with a special case. r×r Lemma 4.3. For |α| ≤ m, let Aα ∈ Cb ([0, 1] × RN ) be such that r×r (i) Aα (t, ·) ∈ O0 (RN ) for every t ∈ [0, 1], (ii) the partial derivatives ∂Aα /∂xi exist and are continuous on [0, 1] × RN for every 1 ≤ i ≤ N. Pm Let Pt (x, ∂) denote the differential operator |α|=0 Aα (t, x)∂ α and suppose that there are R ≥ 0 and C > 0 such that σPt (x, ξ) is invertible for every t ∈ [0, 1] when |x|2 + |ξ|2 ≥ R2 and that m

(1 + |ξ|2 ) 2 ||σPt (x, ξ)−1 || ≤ C whenever |x|2 + |ξ|2 ≥ R2 . Then, Pt (x, ∂) : W m,2 → L2 is Fredholm for every t ∈ [0, 1] and index Pt (x, ∂) is independent of t.

Vol. 62 (2008)


567

Proof. With no loss of generality, assume R > 0. By Theorem 4.1, Pt (x, ∂) is Fredholm with index N Z ∧2N −1 i (N − 1)! index Pt (x, ∂) = − T r σP−1 dσ , P t t 2π (2N − 1)! ∂B where B is the open ball about the origin with t-independent radius R. The hypotheses of the lemma also ensure that the right-hand side depends continuously upon t ∈ [0, 1], so that it is constant. The next theorem is Lemma 4.3 without condition (ii). r×r Theorem 4.4. For |α| ≤ m, let Aα ∈ Cb ([0, 1] × RN ) be such that Aα (t, ·) ∈ r×r O0 (RN ) for every t ∈ [0, 1]. Let Pt (x, ∂) denote the differential operator Pm α |α|=0 Aα (t, x)∂ and suppose that there are R ≥ 0 and C > 0 such that σPt (x, ξ) is invertible for every t ∈ [0, 1] when |x|2 + |ξ|2 ≥ R2 and that m

(1 + |ξ|2 ) 2 ||σPt (x, ξ)−1 || ≤ C whenever |x|2 + |ξ|2 ≥ R2 and t ∈ [0, 1].

(4.2)

Then, Pt (x, ∂) : W m,2 → L2 is Fredholm for every t ∈ [0, 1] and index Pt (x, ∂) is independent of t. Proof. Extend the coefficients Aα (t, x) to R × RN by continuity and let ηk : RN → (0, ∞) be a sequence of functions such that lim|x|→∞ ηk (x) = 0 and that ηk → 0 uniformly on RN . By the denseness of C 1 functions in the strong (Whitney) topology of C 0 (R × RN ) (see e.g. [9, p. 44 and p. 59]), there are (k) (k) 1 N r×r such that Aα (t, x) − Aα (t, x) < ηk (x) for functions Aα ∈ C (R × R ) every (t, x) ∈ R × RN and hence for every (t, x) ∈ [0, 1] × RN . In particular, r×r (k) . Aα ∈ Cb ([0, 1] × RN ) r×r (k) Since lim|x|→∞ ηk (x) = 0, it is clear that Aα (t, ·) ∈ O0 (RN ) when (k) N t ∈ [0, 1] and, since ηk → 0 uniformly on R , it is equally clear that Aα → Aα uniformly on [0, 1] × RN . Thus, by the arguments of the proof of Theorem 4.1, (4.2) holds (with another C but the same R) when Pt (x, ∂) is replaced by the (k) (k) operator Pt (x, ∂) with coefficients Aα (t, x) if k is large enough. Furthermore, (k) Pt (x, ∂) → Pt (x, ∂) in L(W m,2 , L2 ) for every t ∈ [0, 1], so that we may as(k) sume with no loss of generality that index P0 (x, ∂) = index P0 (x, ∂) and index (k) P1 (x, ∂) = index P1 (x, ∂). (k) (k) (k) Now, Lemma 4.3 for Pt (x, ∂) yields index P0 (x, ∂) = index P1 (x, ∂), whence index P0 (x, ∂) = index P1 (x, ∂). To complete the proof, replace Aα (t, x) by Aα (st, x) for any s ∈ [0, 1] (which does not affect the hypotheses) to get index P0 (x, ∂) = index Ps (x, ∂). r×r Since Aα (t, ·) ∈ O0 (RN ) , Aα (t, ·) is uniformly continuous on RN but this does not imply uniform continuity on [0, 1] × RN . If the Aα are uniformly continuous on [0, 1] × RN , then Pt (x, ∂) is continuous with respect to t in L(W m,2 , L2 )

568

Rabier

IEOT

and Theorem 4.4 follows from the standard homotopy invariance of the index. An example when this continuity is not true occurs in the proof of Corollary 4.5 below (see Remark 4.6). It is also instructive to put Theorem 4.4 in the perspective of part (ii) of Theorem 2.2, ensuring that P0 (x, ∂) and P1 (x, ∂) of the above proof have the same index if their symbols sbP0 and sbP1 (see (2.8)) are homotopic in C(MN , GL(Cr )). The only obvious candidate for a homotopy is given by m X m−|α| cα (t, x)τ (ξ)α , (1 − |τ (ξ)|2 ) 2 i|α| A |α|=0

cα (t, x) refers to the unique continuous extension of Aα (t, ·) where for t ∈ [0, 1], A N to P . It is of course hopeless to expect this formula to define a homotopy cα (t, x) is jointly continuous in (t, x) ∈ in C(MN , GL(Cr )) unless the function A N [0, 1] × P . If so, it is uniformly continuous since [0, 1] × PN is compact, so that its restriction Aα to [0, 1] × RN is also uniformly continuous. Thus, the approach via Theorem 2.2 also leads to the standard case when uniform continuity is assumed. Pm α Corollary 4.5. Let Q(x, ∂) := be another differential operator |α|=1 Bα (x)∂ r×r with coefficients Bα ∈ O0 (RN ) . Suppose that both σP and σQ satisfy (1.4) with the same R ≥ 0. Suppose also that Aα (x) = Bα (x) for |x| ≤ R and every 0 ≤ |α| ≤ m. Then, index P (x, ∂) = index Q(x, ∂). Proof. Set Aα (t, x) := Aα ((1 − t)x + tΦ(x)) where Φ(x) = x if |x| ≤ R and x Φ(x) = R |x| if |x| > R. r×r r×r If t ∈ [0, 1), Aα (t, ·) ∈ O0 (RN ) follows from Aα ∈ O0 (RN ) and the remarks that Φ is bounded (by R) and, as noted at the beginning of Section 2, that the definition of O0 (RN ) is unchanged when |y − x| ≤ 1 in (2.1) is replaced by |y − x| ≤ h for any chosen h > 0. More precisely, it is first easily seen that r×r r×r Bα (λ·) ∈ O0 (RN ) whenever λ ∈ R and Bα ∈ O0 (RN ) . Thus, using t this with λ = 1 − t and Bα (x) = Aα (x + 1−t Φ(x)), it suffices to prove that Bα ∈ r×r O0 (RN ) . To see this, write Bα (y)−Bα (x) = Bα (y)−Aα(x)+Aα (x)−B α (x). t Then, lim|x|→∞ |Aα (x) − Bα (x)| = lim|x|→∞ Aα (x) − Aα x + 1−t Φ(x) = 0 tR t since tΦ(x) 1−t ≤ 1−t := h1 . Next, if |y − x| ≤ 1, then y + 1−t Φ(y) − x ≤ 1 + tR 1−t

:= h2 . Hence, max|y−x|≤1 |Bα (y) − Aα (x)| ≤ max|z−x|≤h2 |A(z) − A(x)| and so lim|x|→∞ max|y−x|≤1 |Bα (y) − Aα (x)| = 0. N r×r If t = 1, that can be seen as follows: If |x| > 1 and Aα (1, ·) ∈ O0 (R ) Ry Rx 2R|y−x| 2R |y − x| ≤ 1, then |y| − |x| ≤ |y| ≤ |x|−1 . Thus, for every ε > 0, |Aα (Φ(y)) − Aα (Φ(x))| < ε if |x| is large enough and |y − x| ≤ 1 by the uniform continuity of Aα on the ball B R . Hence, lim|x|→∞ max|y−x|≤1 |Aα (Φ(y)) − Aα (Φ(x))| = 0. Next, |(1 − t)x + tΦ(x)| = |x| if |x| ≤ R and |(1 − t)x + tΦ(x)| ≥ R if |x| ≥ R (and t ∈ [0, 1]). As a result, |(1 − t)x + tΦ(x)|2 + |ξ|2 ≥ R2 whenever

Vol. 62 (2008)


569

|x|2 +|ξ|2 ≥ R2 , so that (4.2) follows from (1.4) after replacing x by (1−t)x+tΦ(x). Thus, index P (x, ∂) = index P (Φ(x), ∂) by Theorem 4.4. By the same arguments, index Q(x, ∂) = index Q(Φ(x), ∂) and Q(Φ(x), ∂) = P (Φ(x), ∂) since |Φ(x)| ≤ R and Aα (x) = Bα (x) for |x| ≤ R. This shows that index P (x, ∂) = index Q(x, ∂). If also the coefficients Aα and Bα are C 1 in Corollary 4.5, the result follows directly from Theorem 4.1. Remark 4.6. In the proof of Corollary 4.5, Theorem 4.4 cannot be replaced by the classical homotopy invariance of the index. Indeed, the operators Pt (x, ∂) with coefficients Aα (t, x) of that proof do not depend continuously upon t in L(W m,2 , L2 ) (discontinuity at t = 1 is easily corroborated on simple examples). In fact, Aα is not uniformly continuous on [0, 1] × RN unless Aα is constant.

5. Operators of index 0 We now use the results of the previous sections to obtain a few sufficient conditions for P (x, ∂) to have index 0. For brevity, we do not include corollaries obtained by standard arguments (compact perturbation or classical homotopy). r×r Theorem 5.1. In (1.1), suppose that Aα ∈ O0 (RN ) . If also condition (1.4) holds and r < N, then P (x, ∂) : W m,2 → L2 is Fredholm of index 0. Proof. By the approximation argument of the proof of Theorem 4.1 (where the C 1 N r×r assumption is not used), it is not restrictive to assume that Aα ∈ O∞ , 0 (R ) so that, by Theorem 3.5, index P (x, ∂) is given by (1.6). That index P (x, ∂) = 0 ∧2N −1 follows from Stokes’ theorem because T r σP−1 dσP is an exact form on ∂B when r < N. The exactness is well known but fully explained in [13, Section 3]. A special case of Theorem 5.1 arises when r = 1 (scalar operators) and N ≥ 2. If r = 1, N ≥ 2 and the coefficients Aα have limits at infinity, it has been known for a long time that the index is 0 ([4, Theorem 6], [8]). Simple examples when N = 1 (and less simple ones if N > 1; see for instance [3]) show that the index d + arctan x has index 1. need not be 0 if r ≥ N : If r = N = 1, the operator dx In light of Theorem 5.1 and since (1.4) is necessary for Fredholmness by Remark 2.3 and Theorem 2.2, the other conditions for index 0 discussed below have value only when r ≥ N, even though this will not be assumed explicitly. r×r Theorem 5.2. In (1.1), suppose that Aα ∈ O0 (RN ) and that condition (1.4) holds with R = 0. Then P (x, ∂) : W m,2 → L2 is Fredholm of index 0. Proof. Since (1.4) holds with R = 0, the hypotheses of Corollary 4.5 are satisfied with the choice Q(x, ∂) = P (0, ∂) with constant coefficients. Since Fredholm differential operators with constant coefficients are isomorphisms (as is well known and easily seen by Fourier transform), they have index 0.

570

Rabier

IEOT

If in addition the coefficients Aα are C 1 , Theorem 5.2 also follows from Theorem 4.1 by letting the radius R > 0 of the ball B tend to 0 in (1.6): Since there is no singularity at the origin, the boundary integral tends to 0. Remark 5.3. In Theorem 5.2, it suffices that (1.4) holds outside a ball B of radius R for some R > 0 and that σP |∂B is homotopic to a constant in C(∂B, GL(Cr )) (hence σP can be continuously deformed into an invertible family of operators inside the ball B). However, this cannot be proved in full generality within the framework of differential operators and we shall not provide the details here. r×r Corollary 5.4. In (1.1), suppose that Aα ∈ O0 (RN −1 ) (so that Aα is independent of xN ). If also condition (1.4) holds, then P (x, ∂) : W m,2 → L2 is Fredholm of index 0. Proof. If the coefficients Aα do not depend upon xN and (1.4) holds, then it holds for every (x1 , . . . , xN −1 ) and every ξ ∈ RN (just choose xN = R). Therefore, (1.4) actually holds with R = 0 and Theorem 5.2 is applicable. Another sufficient condition for index 0 will be derived from the next lemma. r×r Lemma 5.5. In (1.1), suppose that Aα ∈ O0 (RN ) and that condition (1.4) holds. Define PT (x, ∂) :=

m X |α|=0

ATα (x)∂ α and Pe(x, ∂) :=

m X

(−1)|α| Aα (x)∂ α .

(5.1)

|α|=0

Then, PT (x, ∂) and Pe(x, ∂) are Fredholm from W m,2 to L2 with −index P (x, ∂) if N is even, (i) index PT (x, ∂) = index P (x, ∂) if N is odd, index P (x, ∂) if N is even, (ii) index Pe(x, ∂) = −index P (x, ∂) if N is odd. Proof. (i) Clearly, σPT (x, ξ) = σP (x, ξ)T . Thus, σPT satisfies (1.4) and PT (x, ∂) is Fredholm. As in the proof of Theorem 5.1, it is not restrictive to assume Aα ∈ N r×r O∞ and then index P (x, ∂) and index PT (x, ∂) are both given by the 0 (R ) formula (1.6). If Ω1 and Ω2 are matrix forms, then ΩT1 ∧ ΩT2 = (Ω2 ∧ Ω1 )T if either Ω1 or Ω2 has even order and ΩT1 ∧ ΩT2 = −(Ω2 ∧ Ω1 )T if both have odd order. Since T σP−1 dσPT = dσP σP−1 , an induction argument yields T T ∧2N −1 −1 ∧2N −1 N −1 σP−1 dσ = (−1) dσ σ , P P T P T ∧2N −1 ∧2N −1 so that T r σP−1 dσPT = (−1)N −1 T r dσP σP−1 . To prove the T claim, it suffices to show that ∧2N −1 ∧2N −1 T r dσP σP−1 = T r σP−1 dσP . (5.2)

Vol. 62 (2008)


571

This is more easily seen by keeping in mind that since σP−1 is a 0-form, the products dσP σP−1 and σP−1 dσP are also dσP ∧ σP−1 and σP−1 ∧ dσP , respectively. Thus, by the associativity of the exterior product, ∧2N −1 dσP σP−1 = dσP ∧ (σP−1 dσP )∧2N −2 ∧ σP−1 and it is readily verified that, since σP−1 is a 0-form T r dσP ∧ (σP−1 dσP )∧2N −2 ∧ σP−1 ∧2N −1 = T r σP−1 ∧ dσP ∧ (σP−1 dσP )∧2N −2 = T r σP−1 dσP . This proves (5.2). (ii) From the definition of Pe(x, ∂), it follows that σPe (x, ξ) = σP (x, ξ). Thus, σPe satisfies (1.4), so that Pe(x, ∂) is Fredholm. Once again, we may assume Aα ∈ N r×r with no loss of generality, so that indexP (x, ∂) and index Pe(x, ∂) O∞ 0 (R ) ∧2N −1 −1 are both given by the formula (1.6). It is obvious that T r σPe dσPe = ∧2N −1 ∧2N −1 R T r σP−1 dσP . By (1.6), ∂B T r σP−1 dσP is always real if N is ∧2N −1 R equals even and imaginary if N is odd. As a result, ∂B T r σP−1 dσP ∧2N −1 ∧2N −1 R R −1 −1 if N is if N is even and − ∂B T r σP dσP T r σP dσP ∂B odd. This completes the proof. r×r Theorem 5.6. In (1.1), suppose that Aα ∈ O0 (RN ) and that condition (1.4) holds. Assume also that for 0 ≤ |α| ≤ m and |x| < R with R ≥ 0 from (1.4): (i) Aα (x) = ATα (x) and N is even, or (ii) Aα (x) = (−1)|α| Aα (x) and N is odd, or T (iii) Aα (x) = (−1)|α| A∗α (x) = (−1)|α| Aα (x) . Then P (x, ∂) : W m,2 → L2 is Fredholm of index 0. Proof. If R = 0, the conditions (i), (ii) and (iii) are all vacuous and the result coincides with Theorem 5.2. From now on, we assume R > 0. If so, by continuity, the condition required in (i), (ii) or (iii) actually holds for |x| ≤ R. Then, cases (i) and (ii) follow from Lemma 5.5 and Corollary 4.5 with Q(x, ∂) = PT (x, ∂) and Q(x, ∂) = Pe(x, ∂), respectively. In case (iii), use once again Corollary 4.5, but with Q(x, ∂) = P∗ (x, ∂) := PeT (x, ∂), so that index PeT (x, ∂) = −index P (x, ∂) regardless of N being odd or even. In all cases, the assumption that σP and σQ satisfy (1.4) with the same R, required in Corollary 4.5, is satisfied since σPT = σP T , σPe (x, ξ) = σ P and σP∗ = σP ∗ .

572

Rabier

IEOT

Remark 5.7. When R > 0, the hypotheses of Theorem 5.6 can only hold if mr is even. If N ≥ 2, this follows from Appendix 2 (and remains true when R = 0). If N = 1, then case (i) does not occur. In cases (ii) or (iii), just note that det σP (x, ξ) is real for |x| < R, hence for |x| ≤ R since R > 0, and all ξ ∈ R. Since det σP (x, ξ) is a polynomial of degree mr in ξ, it must vanish at some point ξx for every x ∈ R 2 with |x| ≤ R if mr is odd. In particular, det σP (R, ξR ) = 0. Since R2 + ξR ≥ R2 , this contradicts (1.4). By the p-independence of the index ([13]), all the results of this paper remain true when W m,2 and L2 are replaced by W m,p and Lp , respectively, for any p ∈ (1, ∞). They also remain true in the weighted spaces introduced in [14] since the weights do not change the index of P (x, ∂) (only the scalar case r = 1 is discussed in [14], but the weight-independence of the index remains true, with the same proof, when r > 1).

Appendix 1: A clarification As mentioned in the Introduction, the proof of formula (1.6) under condition (1.5) is given with σP replaced by sP in the right-hand side, where sP (x, ξ) = m λ(ξ)σP (x, ξ) with λ(ξ) := (1 + |ξ|2 )− 2 . We shall now clarify this point by showing that the right-hand side of (1.6) is not affected by this change. First, (λσP )−1 d(λσP ) = (λ−1 dλ)Ir + σP−1 dσP . Because of the diagonal structure of (λ−1 dλ)Ir , it follows that ((λ−1 dλ)Ir )∧2 = 0 (this is of course false for ∧2 ∧2 general matrix 1-forms), which yields (λσP )−1 d(λσP ) = σP−1 dσP . Thus, ∧2N −2 ∧2N −1 (λσP )−1 d(λσP ) = (λσP )−1 d(λσP ) ∧ σP−1 dσP ∧2N −2 = (λ−1 dλ)Ir + σP−1 dσP ∧ σP−1 dσP ∧2N −2 ∧2N −1 = (λ−1 dλ) ∧ σP−1 dσP + σP−1 dσP . Accordingly, ∧2N −1 ∧2N −2 T r λσP )−1 d(λσP ) = (λ−1 dλ) ∧ T r σP−1 dσP ∧2N −1 + T r σP−1 dσP . Now, if Ω1 and Ω2 are matrix forms of odd order, then T r(Ω1 ∧ Ω2 ) = ∧2N −3 when −T r(Ω2 ∧ Ω1 ). By using this = σP−1dσP and Ω2 = σP−1 dσP with Ω1∧2N ∧2N −1 −2 −1 −1 N ≥ 2, it follows that T r σP dσP = 0. Thus, T r σP dσP = ∧2N −1 −1 T r (λσP ) d(λσP ) if N ≥ 2, which of course suffices to prove the claim. If N = 1, then ∂B in (1.6) is a circle S and T r (λσP )−1 d(λσP ) = rλ−1 dλ + R R −1 −1 T r(σP−1 P ). Thus, R dσ−1 R the relation S T r (λσP ) d(λσP ) = S T r(σP dσP ) follows from S λ dλ = S d(ln λ) = 0.

Vol. 62 (2008)


573

Appendix 2: Necessary condition for Fredholmness r×r We give a quick proof that if N ≥ 2 and Aα ∈ O0 (RN ) , then P (x, ∂) in (1.1) cannot be Fredholm from W m,2 to L2 unless mr is even. Recall that by Remark 2.3 and Theorem 2.2, (1.4) is necessary for Fredholmness. Assume first that r = 1 (scalar case). It is clear that (1.4) implies that P (x, ∂) is elliptic and it is well known that ellipticity alone implies that m is even if N ≥ 3 ([12, p. 121]). Suppose now that N = 2. Given any x ∈ R2 , it follows from (1.4) that for t ∈ [0, 1], the equation in ξ ∈ R2 X X im Aα (x)ξ α + t Aα (x)i|α| ξ α = 0, |α|=m

|α|<m

has ξ with |ξ| ≥ ρP x for some ρx > 0 large enough. Thus, both mappings P no solution |α| α m α 2 2 A (x)i ξ and i |α|≤m α |α|=m Aα (x)ξ from R to C = R have the same Brouwer degree (at 0) relative to the ball B(0, ρx ) in R2 . If m is odd, the latter mapping is odd in ξ, so that its degree is odd by Borsuk’s theorem. It follows that P |α| α |α|≤m Aα (x)i ξ = 0 has a solution ξ for every x. However, this contradicts condition (1.4) if |x| ≥ R. The above shows that m is even when r = 1 and N ≥ 2 and P (x, ∂) is Fredholm. For arbitrary r, we have already noted that (1.4) amounts to saying that det sP is bounded away from 0 in (RN × RN )\B where B is the open ball mr about the origin with radius R. Now, det sP (x, ξ) = (1 + |ξ|2 )− 2 det σP (x, ξ), and an expansion of det σP (x, ξ) shows that det σP (x, ξ) = σQ (x, ξ) where Q(x, ∂) is a scalar differential operator of order mr and coefficients in O0 (RN ). In other words, (1.4) amounts to the same condition for the scalar operator Q(x, ∂). Thus, mr is even if N ≥ 2 by the first part of the argument.

References [1] R. Bott, and R. T. Seeley, Some remarks on the paper of Callias: “Axial anomalies and index theorems on open spaces” [Comm. Math. Phys. 62 (1978) 213-234], Comm. Math. Phys. 62 (1978) 235-245. [2] M. Breuer and H. O. Cordes, On Banach algebras with σ-symbol, J. Math. Mech. 13 (1964) 313-324. [3] C. J. Callias, Axial anomalies and index theorems on open spaces, Comm. Math. Phys. 62 (1978) 213-234. [4] H. O. Cordes, The algebra of singular integral operators in Rn , J. Math. Mech. 14 (1965) 1007-1032. [5] H. O. Cordes and E. A. Herman, Gel’fand theory of pseudo differential operators, Amer. J. Math. 90 (1968) 681-717. [6] H. O. Cordes, Elliptic pseudo-differential operators-An abstract theory, Lecture Notes in Mathematics, vol. 756 , Springer-Verlag, Berlin 1979. [7] B. V. Fedosov, A direct proof of the formula for the index of an elliptic system in Euclidean space (Russian) Funkcional. Anal. i Prilozhen. 4 (1970) 83-84.

574

Rabier

IEOT

[8] I. C. Gohberg, On the theory of multidimensional singular equations, Soviet Math. 1 (1960) 960. [9] M. W. Hirsch, Differential topology, Springer-Verlag, New York 1988. [10] L. H¨ ormander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979) 360-444. [11] L. H¨ ormander, The Analysis of Linear Partial Differential Operators III, Springer Verlag, Berlin 1985. [12] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris 1968. [13] P. J. Rabier, On the index and spectrum of differential operators on RN , Proc. Amer. Math. Soc. 135 (2007) 3875-3885. [14] P. J. Rabier, Decay transference and Fredholmness of differential operators in weighted Sobolev spaces, Differ. Integral Eqs (to appear). [15] E. Schrohe, Spectral Invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces, Ann. Global Anal. Geom. 10 (1992) 237-254. [16] R. T. Seeley, Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965) 167-204. [17] R. T. Seeley, Fredholm differential operators on Rn , preprint, Battelle Institute 1969. [18] S. H. Sun, A Banach algebra approach to the Fredholm theory of pseudodifferential operators, Sci. Sinica, Series A, 27 (1984) 337-344. [19] M. Taylor, Gelfand theory of pseudo-differential operators and hypoelliptic operators, Trans. Amer. Math. Soc. 153 (1971) 495-510. Patrick J. Rabier Department of mathematics University of Pittsburgh Pittsburgh, PA 15260 USA e-mail: [email protected] Submitted: May 19, 2008.

Integr. equ. oper. theory 62 (2008), 575–577 0378-620X/040575-3, DOI 10.1007/s00020-008-1638-x c 2008 Birkh¨



Errata to: “Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols” Patrik Wahlberg Abstract. In our article “Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols”, Integr. equ. oper. theory 59 (1) (2007), 99–128, Example 4.4 is false. We present a corrected example. Mathematics Subject Classification (2000). Primary 47G30, 42B35; Secondary 47B38, 35S99. Keywords. Errata, time-frequency analysis, vector-valued modulation spaces, localization operators, pseudo-differential operators.

Example 4.4 on page 120 is unfortunately false. We apologize for any inconvenience 0 this may have caused. In the example we define the map a0 : lr (Z) 7→ lr (Z), X 0 0 a0 (β) = βk |βk |r −2 kβkl2−r ek . r0 k

It is isometric but (of course) not a linear transformation, as we have claimed and built the example upon. The text below replaces Example 4.4 and the preceding paragraph and gives a correct example. Correction of Example 4.4 and the preceding paragraph Corollary 4.3 is not true if we replace the modulation spaces between which Ag,γ acts by L2 spaces. The following example shows that Ag,γ is not necessarily a a bounded from L2 (Rd , B1 ) to L2 (Rd , B2 ), uniformly over a ∈ L∞ (R2d , L(B1 , B2 )). This phenomenon can not occur in the theory of scalar-valued modulation spaces since L2 = M 2 then. Example 4.4. Let the dimension d = 1, B1 = l1 (N), B2 = l∞ (N). Note that B1 has Fourier type 1 [35]. Pick a nonnegative window function g ∈ Cc∞ (R) such that g(x) = 1 when x ∈ [−1/2, 3/2].

576

Wahlberg

IEOT

Let an integer n > 0 be given. We pick f ∈ Cc∞ (R) with supp f ⊆ (0, 1/n) and kf kL2 = 1, and define the vector-valued function F (x) = f (x), f (x − 1/n), . . . , f (x − (n − 1)/n), 0, . . . . We observe that the non-overlapping supports of {f (· − j/n)}n−1 j=0 gives ! 2 Z n−1 Z n−1 X X kF k2L2 (R,l1 ) = |f (x − j/n)| dx = |f (x − j/n)|2 dx = n. R

R j=0

j=0

For the Fourier transform we have !2 Z n−1 X 2 −2πiξj/n b kFbkL2 (R,l1 ) = |f (ξ)e | dξ = n2 kfbk2L2 = n2 . R

j=0

n−1 We define the matrix-valued function a(z)j,k j,k=0 of z ∈ R2 by    Vg fj (z)Vg fk (z) if |Vg fj (z)||Vg fk (z)| > 0, a(z)j,k = |Vg fj (z)||Vg fk (z)|  0 otherwise, where we denote fj (x) = f (x − j/n), j = 0, . . . , n − 1. Then ka(z)kL(l1 ,l∞ ) ≤ 1 for all z ∈ R2 , i.e. kakL∞ (R2 ,L(l1 ,l∞ )) ≤ 1. Now we have Z ZZ g,g F (x), Aa F (x) l2 dx = Vg F (z), a(z)Vg F (z) l2 dz R2

R

n−1 X

ZZ = R2

j,k=0 n−1 X

ZZ = R2

= R2 1/2

!2 |Vg fj (z)|

dz !2

\ |T t gfj (ξ)|

dξ dt

j=0

Z

≥ −1/2

Vg fj (z)Vg fk (z) dz |Vg fj (z)||Vg fk (z)|

j=0 n−1 X

ZZ

Z

Vg fj (z)Vg fk (z)

R

n−1 X

!2 \ |T t gfj (ξ)|

! dξ dt.

j=0

The function Tt g(x) = 1 for x ∈ [0, 1] if |t| ≤ 1/2, and therefore Tt gfj = fj for all j = 0, . . . , n − 1. It follows that !2 Z Z 1/2 Z n−1 X g,g b F (x), A F (x) 2 dx ≥ |fj (ξ)| dξ dt a

R

l

−1/2

R

j=0

= kFbk2L2 (R,l1 ) = n2 = nkF k2L2 (R,l1 ) .

Vol. 62 (2008)

Errata

577

Since l1 is isometrically embedded in (l∞ )0 , we finally obtain, using Lemma 3.5, Z g,g g,g G(x), Aa F (x) l∞ dx kAa F kL2 (R,l∞ ) ≥ sup kGkL2 (R,(l∞ )0 ) =1 R Z F (x) ≥ , Ag,g dx ≥ nkF kL2 (R,l1 ) . a F (x) R kF kL2 (R,l1 ) l2 It follows that there is no bound of the form kAg,g a kL(L2 (R,l1 ),L2 (R,l∞ )) ≤ CkakL∞ (R2 ,L(l1 ,l∞ )) . Finally we take the opportunity to correct three other, minor errors in the paper. • Page 106, line –4. “. . . the tensor product rule f ⊗ 1 = 1 ⊗ f . . . ”. Correction: . . . a Fubini type theorem for the tensor product f ⊗ 1. . . p,q • Page 109, line –13. “. . . and hence Mm (Rd , B) are Banach spaces for all 1 ≤ p, q ≤ ∞.” Correction: . . . and hence M p,q (Rd , B) are Banach spaces for all 1 ≤ p, q ≤ ∞. The fact that also the weighted modulation spaces p,q (Rd , B) are complete follows from Lemma 3.7 and the proof of [22, TheMm orem 11.3.5 (a)]. • Page 113, line 6. χ1×Kj should be 1 ⊗ χKj . Patrik Wahlberg School of Electrical Engineering & Computer Science The University of Newcastle Callaghan, NSW 2308 Australia e-mail: [email protected] Submitted: September 30, 2008.


auser Verlag Basel/Switzerland 0378-620X/040579-6, published online July 23, 2008 DOI 10.1007/s00020-008-1608-3


Transitivity of Jordan Algebras of Linear Operators: On Two Questions by Gr¨ unenfelder, Omladiˇc and Radjavi Mikhail Chebotar, Wen-Fong Ke and Victor Lomonosov Abstract. Let A be a Jordan algebra of linear operators on a vector space over a field of characteristic different from 2. In this short note, we show that (1) if A is 2-transitive, then it is dense, and (2) if A is n-transitive, n ≥ 1, then a nonzero Jordan ideal of A is also n-transitive. These answer two questions posed by Gr¨ unenfelder, Omladiˇc and Radjavi. Mathematics Subject Classification (2000). Primary 47A15; Secondary 17C50. Keywords. Jordan algebra, n-transitivity.

1. Introduction Let L = L(V ) be the algebra of all linear operators on a vector space V over the field F . A subset S ⊆ L is said to be n-transitive (n ≥ 1) if for any linearly independent set {x1 , . . . , xn } of V and any elements y1 , . . . , yn ∈ V , there is an S ∈ S such that Sxi = yi for i = 1, . . . , n. If S is transitive for all n ≥ 1, it is called dense. There are nice results on dense associative algebras of linear operators in L. Burnside’s theorem states that if V is finite-dimensional and F is algebraically closed, then the only transitive associative subalgebra of L is L itself. Jacobson [5] showed that if S is an associative subalgebra of L and S is 2-transitive, then S is dense. In 1993 Gr¨ unenfelder, Omladiˇc and Radjavi [3] studied transitive nonassociative algebras, and obtained the Jordan analogs of the Burnside’s and Jacobson’s theorems. By a Jordan algebra of linear operators over a field F of characteristic The second author was partially supported by National Science Council, Taiwan, grant #0952811-M-006-005.

580

Chebotar, Ke and Lomonosov

IEOT

not 2 we mean a linear subspace A of S such that for all A, B ∈ A, it holds that A ◦ B = AB + BA ∈ A. In [3], the following theorems were proved. Theorem 1.1 ([3, Theorem 2.1]). Let F be any formally real closed field. Then, the only transitive Jordan algebra of Sn (F ), the symmetric n × n matrices over F, is Sn (F ) itself. Theorem 1.2 ([3, Theorem 2.4]). Let F be an algebraically closed field, and let A be a transitive Jordan algebra of n × n matrices over F . Then either A = Mn (F ) or there exists an invertible matrix T such that T −1 AT = Sn (F ). Theorem 1.3 ([3, Theorem 1.3]). Let A be a Jordan algebra of finite rank operators on a vector space V If A is 2-transitive, then it is dense, i.e. n-transitive for all n > 1. Theorem 1.4 ([3, Theorem 3.2]). Every Jordan ideal J 6= 0 of an (n + 1)-transitive Jordan algebra A of operators on a vector space V is n-transitive, n > 1. Two questions were raised in [3, p. 346] aiming at parts untouched by Theorems 1.3 and 1.4: Question 1. Is there an n-transitive Jordan algebra A with a Jordan ideal A 6= 0 which is not n-transitive? Question 2. Is there an n-transitive Jordan algebra which is not (n + 1)transitive for any n ≥ 2? The purpose of this paper is to complete the unfinished task left by Gr¨ unenfelder, Omladiˇc and Radjavi. Namely, we are going to answer the two questions stated above. The answers to both questions are negative. We will first prove Theorem 1.5. Let A be a Jordan algebra of linear operators on a vector space V over the field of characteristic different from 2. If A is 2-transitive, then A is dense. Thus Question 2 would be answered. It is interesting to note that there are 1-transitive Jordan algebras which are not dense. The Jordan algebra Sn (C) of symmetric n × n matrices, n ≥ 2, over the complex number field C serves as a nice counterexample (see [3, Theorem 1.2 and Corollary 2.5] for details). Partial answer to Question 1 is already provided by Theorems 1.4 and 1.5. But we will prove the following result to have it covered completely. Theorem 1.6. Let A be an n-transitive Jordan algebra of linear operators on a vector space V over the field of characteristic different from 2, where n ≥ 1. If J is a nonzero Jordan ideal of A, then J is also n-transitive. Related problems for Lie algebras can be found in [1] and [4]. Also, some very interesting results were obtained recently for more general situation of transitive spaces of operators, and the paper by Davidson, Marcoux and Radjavi [2] provides a comprehensive study of algebraic and topological transitivity for linear spaces of operators.

Vol. 62 (2008)

Transitivity of Jordan Algebras

581

2. The results Throughout this section F is a field of characteristic not 2, V a vector space over F , L = L(V ) the algebra of all linear operators on V , and A a Jordan algebra of operators from L. Note that for A, B ∈ A, we have A2 = 21 (A ◦ A) ∈ A and ABA = 12 ((A ◦ B) ◦ A − A2 ◦ B) ∈ A. We start with the following auxiliary lemma. Lemma 2.1. Let S be a n-transitive linear subspace of L, n ≥ 2, and let x1 , . . . , xn+1 ∈ V , be linearly independent. Then there exists an S ∈ S such that Sxi = 0 for all i = 3, . . . , n + 1, and Sx1 ∈ / span{Sx2 }

(2.1)

Proof. Assume the contrary, i.e. T x1 ∈ span{T x2 } for any T ∈ S with T x3 = · · · = T xn = 0. Suppose that there exists a T1 ∈ S such that T1 x1 = T1 x3 = . . . = T1 xn+1 = 0 and T1 x2 = u 6= 0. Let w ∈ V be a vector linearly independent with u. It exists since S is n-transitive with n ≥ 2. By the n-transitivity of S, there exists some T2 ∈ S such that T2 xi = 0 for i ≥ 3 and T2 x1 = w. By our assumption, T2 x1 ∈ span{T2 x2 }, and so T2 x2 = αw for some nonzero α ∈ F . We have S = T1 + T2 ∈ S satisfies (2.1), a contradiction. Therefore, for any T ∈ S, if T x1 = T x3 = . . . = T xn+1 = 0, then readily T x2 = 0. Let S1 , S2 ∈ S with S1 xi = S2 xi = 0 for all i ≥ 3 while S1 x1 6= 0 and S2 x1 6= 0. By our assumption, there are λ1 , λ2 ∈ F such that S1 x1 = λ1 S1 x2 and S2 x1 = λ2 S2 x2 . Now, it may be that S1 x1 and S2 x1 are linearly independent. In this situation, we set S = S1 + S2 . Then −1 Sxi = 0 for i = 3, . . . , n + 1, Sx1 = S1 x1 + S2 x1 , and Sx2 = λ−1 1 S1 x1 + λ2 S2 x1 .

We see that if λ1 6= λ2 , then Sx1 and Sx2 are linearly independent, which cannot be. Hence we have if S1 x1 and S2 x1 are linearly independent, then λ1 = λ2 .

(2.2)

Or, it may be that S1 x1 and S2 x1 are linearly dependent. We argue that λ1 = λ2 as well. Assume that λ1 6= λ2 . Note that we cannot have any T ∈ S and µ ∈ F with T xi = 0 for i = 3, . . . , n + 1, and T x1 = µT x2 6= 0, such that T x1 and S1 x1 are linearly independent (hence T x1 and S2 x1 are linearly independent as well), otherwise a contradiction that λ1 = µ = λ2 would arise by (2.2). Now, this means that for all T ∈ S, T xi = 0 for all i ≥ 3 implies T x1 ∈ span{S1 x1 }. But this contradicts the fact that S is n-transitive. Therefore, we have if S1 x1 and S2 x1 are linearly dependent, then λ1 = λ2 .

(2.3)

From (2.2) and (2.3) we conclude that there is a λ ∈ F such that if S ∈ S with Sxi = 0 for i ≥ 3 and Sx1 6= 0, then Sx1 = λSx2 . By above if Sx1 = Sx3 = . . . = Sxn+1 = 0, then Sx2 = 0. But then for every S ∈ S with Sxi = 0 for all i ≥ 3, we have S(x1 − λx2 ) = 0. Again, as S is n-transitive, this cannot happen. Therefore the lemma holds.

582


IEOT

We are ready to prove Theorem 1.5. Proof of Theorem 1.5. The theorem will be proved if we show that the n-transitivity of A with n ≥ 2 implies that A is (n + 1)-transitive. Let x1 , . . . , xn+1 ∈ V be linearly independent, and y1 , . . . , yn+1 ∈ V . Notice that if we can find Tj ∈ A (j = 1, . . . , n + 1) such that Tj xi = 0 for i 6= j and Tj xj = yj , then T = T1 + · · · + Tn+1 will do the required job that T xi = yi for i = 1, 2, . . . , n + 1. And to achieve this goal, it suffices to show that for any y ∈ V , there is some T ∈ A such that T xi = 0 for i ≥ 2 and T x1 = y. Let y ∈ V . Certainly we may assume that y is nonzero, or the zero operator will do the job. Let A ∈ A be such that Ax1 = y and Axi = 0 for i ≥ 3. If Ax1 and Ax2 are linearly independent, we may take B ∈ A such that BAx1 = x1 and BAx2 = 0, and put T = ABA ∈ A. Then T x1 = ABAx1 = Ax1 = y and T xi = ABAxi = 0 for all i ≥ 2, and we are done. So we assume that Ax2 = λAx1 = λy for some λ ∈ F . Assume that if S ∈ A with Sx2 6= 0 and Sxi = 0 for i = 3, . . . , n + 1, then Sx1 6= 0. By Lemma 2.1 there is an S such that Sxi = 0 for i = 3, . . . , n + 1, and Sx2 ∈ / span{Sx1 }. Thus Sx1 and Sx2 are linearly independent. Let L ∈ A be such that LSx1 = 0 and LSx2 = x2 . Then SLS ∈ A and SLSx1 = 0, SLSx2 = Sx2 6= 0, and SLSxi = 0 for i = 3, . . . , n + 1, a contradiction. Therefore, there is an S ∈ A with Sx1 = Sxi = 0 for i = 3, . . . , n + 1 and Sx2 = v 6= 0, and we fix it for what follows. In the case that v and y are linearly independent, we set C = A + S. Then Cx1 = Ax1 + Sx1 = y, Cx2 = Ax2 + Sx2 = λy + v, and Axi = 0 for i ≥ 3. Since Cx1 and Cx2 are linearly independent, we can find D ∈ A with DCx1 = x1 and DCx2 = 0. Then T = CDC ∈ A will do the job. Finally, assume that v = µy for some nonzero µ ∈ F , and set T = A − µλ S. Then T x1 = Ax1 − µλ Sx1 = y, T x2 = Ax2 − µλ Sx2 = λy − µλ v = 0, and T xi = 0 for i ≥ 3. This completes the proof. Remark 2.2. We note that the above proof is valid for any 2-transitive space A of linear operators on a vector space over any field satisfying ABA ∈ A for all A, B ∈ A. As a corollary to Theorem 1.5 and Theorem 1.4 (or Corollary 3.3 of [3]), we have Corollary 2.3. If A is n-transitive Jordan algebra of linear operators on a vector space V with n ≥ 2, and J is a nonzero Jordan ideal of A, then J is also ntransitive. Moreover, both A and J are dense. Thus, it remains to treat the 1-transitivity case in order to prove Theorem 1.6, which we shall present in the following. Note that if A ∈ A and B ∈ J , then ABA = 12 ((A ◦ B) ◦ A − A2 ◦ B) ∈ J . Theorem 2.4. Let A be a 1-transitive Jordan algebra of linear operators on a vector space V , and J a nonzero Jordan ideal of A. Then J is also 1-transitive.

Vol. 62 (2008)

Transitivity of Jordan Algebras

583

Proof. Let V0 = J V . We first show that V0 = V . We claim that V0 is a subspace of V . First of all, if v0 ∈ V0 , then there is a u0 ∈ V and an S ∈ J such that Su0 = v0 . Thus, for any λ ∈ F , λv0 = λSu0 = S(λu0 ) ∈ V0 . Now, take v1 , v2 ∈ V0 . If either v1 = 0 or v2 = 0, then v1 + v2 ∈ V0 . So we assume that both v1 and v2 are nonzero. Let u1 , u2 ∈ V \{0} and T1 , T2 ∈ J be such that T1 u1 = v1 and T2 u2 = v2 . In the case that T2 u1 = v 6= 0, we may pick T ∈ A such that T v = u2 . Then (T1 + T2 T T2 ) ∈ J and (T1 + T2 T T2 )u1 = v1 + v2 . On the other hand, in the case that T2 u1 = 0, we may pick T ∈ A such that T u1 = u2 . Then (T1 + T2 ◦ T ) ∈ J and (T1 + T2 ◦ T )u1 = v1 + v2 . Hence we see that, in both cases, v1 + v2 ∈ V0 . Therefore V0 is indeed a subspace of V . Pick an arbitrary w ∈ V . Let S ∈ J be nonzero, and u, v ∈ V such that Su = v 6= 0. Let T ∈ A be such that T v = w. Then (ST + T S)u = S(T u) + w. From ST + T S ∈ J , it follows that w = (ST + T S)u − S(T u) ∈ V0 . Therefore V0 = V . Now, let u, v ∈ V with u 6= 0. We want to find some R ∈ J such that Ru = v. Let S ∈ J and w ∈ V be such that Sw = v. If Su = 0, we can simply pick T ∈ A with T u = w and put R = ST + T S ∈ J to get Ru = v. If Su = v 0 6= 0, we can pick T ∈ A such that T v 0 = w, and put R = ST S ∈ J to get Ru = v. Therefore, J is 1-transitive.

References [1] M. Chebotar, V. Lomonosov, On a question by Gr¨ unenfelder, Omladiˇc and Radjavi. Linear Algebra Appl. 426 (2007), 368–370. [2] K.R. Davidson, L.W. Marcoux, H. Radjavi, Transitive spaces of operators. Integral Equations and Operator Theory. To appear. [3] L. Gr¨ unenfelder, M. Olmladiˇc, H. Radjavi, Jordan analogs of the Burnside and Jacobson density theorems. Pacific J. Math. 161 (1993), 335–346. [4] L. Gr¨ unenfelder, M. Olmladiˇc, H. Radjavi, Transitive action of Lie algebra. J. Pure Appl. Algebra 199 (2005), 87–93. [5] N. Jacobson, Structure of rings. Amer. Math. Soc. Colloq. Publ. vol 37, Amer. Math. Soc., Providence, RI, 1984.

Mikhail Chebotar Department of Mathematical Sciences Kent State University Kent, Ohio 44242 USA e-mail: [email protected]

584


Wen-Fong Ke Department of Mathematics National Cheng Kung University and National Center for Theoretical Sciences (South) Tainan 701 Taiwan e-mail: [email protected] Victor Lomonosov Department of Mathematical Sciences Kent State University Kent, Ohio 44242 USA e-mail: [email protected] Submitted: May 15, 2008.

IEOT


auser Verlag Basel/Switzerland 0378-620X/040585-5, published online October 8, 2008 DOI 10.1007/s00020-008-1625-2


Non-supercyclicity of Volterra Convolution and Related Operators S. P. Eveson Abstract. We show that if V α (α > 0) is the Riemann-Liouville fractional integration operator and T is an invertible operator on L2 (0, 1) which commutes with V , then T V α is not supercyclic on L2 (0, 1); in particular, many Volterra convolution operators are not supercyclic. The technique is based on an argument used by Gallardo-Gutiérrez and Montes-Rodr´ıguez to show that V is not supercyclic. Keywords. Supercyclic, Volterra convolution, fractional integration operator.

1. Introduction and Notation Eva A. Gallardo-Gutiérrez and Alfonso Montes-Rodr´ıguez show in [3] that if V is the Volterra operator defined on L2 (0, 1) by Z t (V f )(t) = f (s) ds 0

then V is not supercyclic, which is to say that the set {λV n f : n ∈ N, λ ∈ R} is not dense in L2 (0, 1) for any choice of f ∈ L2 (0, 1). In this note we adapt their method to show that members of a much larger class of operators are also not supercyclic. The concrete examples given below are all of the form Z t (Vk f )(t) = k(t − s)f (s) ds. 0

The results obtained are very closely related to some of those in [1, Section 7], in particular to the first part of Corollary 7.2, which states that V α (I + W ) is not supercyclic on Lp (0, 1), where W is a quasi-nilpotent operator commuting with V . In comparison with the sophisticated techniques used in [1], the arguments below are rather simple.

586

Eveson

IEOT

For n ∈ N, the nth iterate of the Volterra operator V is given by Z t 1 (V n f )(t) = (t − s)n−1 f (s) ds. Γ(n) 0 Its adjoint is given by ((V ∗ )n g)(s) =

1 Γ(n)

Z

1

(t − s)n−1 g(t) dt.

(1)

s n

Note that, while the nth derivative of V f is f , the nth derivative of (V ∗ )n g is (−1)n g. The above formula for V n also makes sense for positive, non-integer values of n; in this case, it is known as the Riemann-Liouville fractional integration operator. Throughout this note, the commutativity hypotheses are all stated in the form “T commutes with V .” Under this hypothesis, T also commutes with every polynomial in V , which is to say every operator VP where P is a polynomial. Since polynomials are dense in L1 (0, 1), T commutes with every operator Vk , where k ∈ L1 (0, 1); in particular, with V α for any α > 0. For much more on the commutant of V , see [6] and [1].

2. Results To avoid obscuring the argument with details, we present the simplest case in full detail, closely following [3, Section 3]. Theorem 1. Suppose T is an invertible, bounded operator on L2 (0, 1) such that V T = T V . Then T V is not supercyclic. Proof. Suppose for a contradiction that f0 is a supercyclic vector for T V = V T on L2 (0, 1), so the set A := {λ(V T )n f0 : λ ∈ R, n ∈ N} is dense in L2 (0, 1). Since T is invertible on L2 (0, 1), T (A) is also dense in L2 (0, 1). Now consider the space X := {φ ∈ C([0, 1]) : φ(0) = 0} with the uniform norm k · k∞ . Since V maps L2 (0, 1) continuously into a dense subspace of X and T (A) is dense in L2 (0, 1), (V T )(A) is dense in X. The set {φ ∈ X : φ(1/2) = 0} is not dense in X, so cannot contain (V T )(A); we thus have ((V T )n+1 f0 )(1/2) 6= 0 for some n ∈ N. Let f = (V T )n+1 f0 ; since any iterate of a supercylic vector is also supercyclic, f is a supercyclic vector for T V on L2 (0, 1); by construction, f is continuous and f (1/2) 6= 0. Let Pñ represent the shifted Legendre polynomial of degree n; that is, Pñ (t) = Pn (2t − 1) where Pn is an ordinary Legendre polynomial of degree n. The usual formula for kPn k2 rescales to give kPñ k2 = (2n + 1)−1/2

(2)

Vol. 62 (2008)

Non-supercyclicity of Volterra Operators

587

and Rodrigues formula rescales to (−1)n dn n Pñ (t) = t (1 − t)n . n! dtn If we let hn (t) = tn (1 − t)n then, since hn and its first n − 1 derivatives are all zero at 1, we can invert this to give (V ∗ )n Pñ = hn /n! (3) R1 2 We also have 0 hn = B(n + 1, n + 1) = (n!) /(2n + 1)! and note that, when rescaled to have integral 1, hn is a positive summability kernel centred at 1/2; we thus have Z 1 (n!)2 f (1/2) (n → ∞). (4) f (t)hn (t) dt ∼ (2n + 1)! 0 Choose a ∈ (0, 1) such that 4kT kkT −1 ka < 1, and let g = χ[0,a] , so kgk2 = a1/2 . It follows from (1) that 1 ((V ∗ )n g)(s) = (a − s)n χ[0,a] (s) n! and hence that k(V ∗ )n gk2 = an+1/2 (2n + 1)−1/2 /n! (5) We now calculate using (3), (2) and (4) that D D E E (T V )n f, (T ∗ )−n Pñ f, (V ∗ )n Pñ n k(T V ) f k2 ≥ = k(T ∗ )−n Pñ k2 k(T ∗ )−n Pñ k2 ≥

|hf, hn /n!i| (2n + 1)1/2 n!|f (1/2)| ∼ kT −1 kn (2n + 1)! kT −1 kn kPñ k2

(n → ∞)

(6)

We also see from (5) that |h(T V )n f, gi| ≤ kT kn kf k2 k(V ∗ )n gk2 =

kT n kkf k2 an+1/2 (2n + 1)1/2 n!

(7)

We can now combine (7) and (6) to give |h(T V )n f, gi| kT kn kf k2 an+1/2 kT −1 kn (2n + 1)! . k(T V )n f k2 kgk2 n!(2n + 1)1/2 a1/2 (2n + 1)1/2 n!|f (1/2)|

(n → ∞)

where an . bn (n → ∞) means that lim supn→∞ an /bn ≤ 1. Stirling’s formula shows that the right-hand side is asymptotically equal to (4kT kkT −1 ka)n kf k2 |f (1/2)|(nπ)1/2 which tends to 0 as n → ∞ because 4kT kkT −1 ka < 1. It now follows from the Angle Criterion [5] that f is not a supercyclic vector for g. Example 1. if k ∈ AC[0, 1] and k(0) 6= 0 then the Volterra operator Vk , as defined in the introduction, can be factorised in the form Vk = (k(0)I + Vk0 )V . The above result applies since Vk0 is quasi-nilpotent; thus, Vk is not supercyclic.

588

Eveson

IEOT

Extension to integer powers of V follows by straightforward modifications to the above argument. Theorem 2. Suppose T is an invertible, bounded operator on L2 (0, 1) such that V T = T V and m ∈ N. Then T V m is not supercyclic. Example 2. Suppose k ∈ C (m−1) [0, 1], k (j) (0) = 0 (0 ≤ j ≤ m − 2), k (m−1) (0) 6= 0 and k (m−1) ∈ AC[0, 1]. Then Vk = (k (m−1) (0)I + Vk(m) )V m , so Vk is not supercyclic. Note that any function which can be represented as a power series on [0, 1] falls into this category. Finally, with a little more work, we can include fractional powers of the Volterra operator, as defined in the introduction. Theorem 3. Suppose T is an invertible, bounded operator on L2 (0, 1) such that V T = T V and α > 0. Then T V α is not supercyclic. Proof. The central point in the argument above is the Rodrigues formula for Legendre polynomials in the following form: if hn (t) = tn (1 − t)n , then (V ∗ )n Pñ is a certain multiple of hn . If we replace V by V α , Pn by a Legendre function of the first kind of (fractional) degree αn and (1 − t)n tn by (1 − t)αn tαn , then the argument works in almost exactly the same way (see [8, Sections 5.7 and 10.4] or [4, Table 17.1 and Section 19] for details of the fractional calculus Rodrigues formula). The only change needed is that the formula kPn k22 = 2/(2n + 1) is not true for fractional indices; instead, we can use the Mehler-Dirichlet formula [2, p. 159, formula (27)] to observe that 1 Z θ cos((ν + 1/2)t) p dt |Pν (cos(θ))| = √ π 2 0 cos(t) − cos(θ) Z θ 1 1 p dt = P−1/2 (cos(θ)) ≤ √ π 2 0 cos(t) − cos(θ) so kPν k2 ≤ kP−1/2 k2 for all ν (note that P−1/2 is in L2 (−1, 1) since it is continuous on (−1, 1] and has only a logarithmic singularity at −1). This has the effect of multiplying the final estimate by O(n1/2 ); since it tends to zero at geometric speed, this has no effect on the conclusion. Example 3. Any operator of the form (cI + Vk )V α with c 6= 0, k ∈ L1 (0, 1) and α > 0 is not supercyclic on L2 (0, 1). Remark. Montes-Rodr´ıguez and Shkarin [7, 2.1–2.4] have established that, as well as failing to be supercyclic, the Volterra operator is not weakly supercyclic on Lp (0, 1); that is, the set {λV n f : n ∈ N, λ ∈ R} is not dense in the weak topology on Lp (0, 1) for any choice of f ∈ Lp (0, 1). Their proof is also based on the strategy in [3]; it can be adapted in a similar way to show that T V α is not weakly supercyclic on L2 (0, 1) for any α > 0 and invertible

Vol. 62 (2008)

Non-supercyclicity of Volterra Operators

589

T commuting with V . I should like to thank the referee for pointing out this fact, and for several other helpful remarks.

References [1] Sergio Bermudo, Alfonso Montes-Rodr´ıguez, Stanislav Shkarin, Orbits of operators commuting with the Volterra operator, J. Math. Pures Appl. 89 (2008), 145–173. [2] A. Erdelyi, W. Magnus, F. Oberhettinger amd F.G. Tricomi, Higher Transcendental Functions vol. I, McGraw-Hill, 1953. [3] Eva A. Gallardo-Gutiérrez and Alfonso Montes-Rodr´ıguez, The Volterra operator is not supercyclic, Integral Equations and Operator Theory 50 (2) (2004), 211–216. [4] J. L. Lavoie, T. J. Osler and R. Tremblay, Fractional Derivatives and Special Functions, SIAM Review 18 (2) (1976), 240–268. [5] Alfonso Montes-Rodr´ıguez and Héctor Salas, Supercyclic subspaces: spectral theory and weighted shifts Adv. Math. 163 (1) (2001), 74–134. [6] Alfonso Montes-Rodr´ıguez and Stanislav A. Shkarin, New results on a classical operator, Contemp. Math. 393 (2006), 139–157. [7] Alfonso Montes-Rodr´ıguez and Stanislav A. Shkarin, Non-Weakly Supercyclic Operators, J. Operator Theory 58 (1) (2007), 39–62. [8] Keith B. Oldham and Jerome Spanier, The Fractional Calculus, Academic Press, 1974. S. P. Eveson Department of Mathematics University of York Heslington York YO10 5DD England e-mail: [email protected] Submitted: September 11, 2007. Revised: July 11, 2008.


auser Verlag Basel/Switzerland 0378-620X/040591-4, published online October 8, 2008 DOI 10.1007/s00020-008-1615-4


Modular Lattice for Co-Operators Yun-Su Kim To my parents

Abstract. We prove modularity of the lattice Lat(T ) of closed invariant subspaces for a C0 -operator T with property (P ). Keywords. C0 -Operators, property (P ), modular lattice, quasiaffinity.

1. Preliminaries In this paper, we will work on the lattice Lat(T ) of closed invariant subspaces for a C0 -operator T which was first studied in detail by B.Sz.-Nagy and C. Foias [4]. For a (bounded linear) operator Ti (i = 1, 2) on a separable Hilbert space Hi , that is, Ti ∈ L(Hi ), if X ∈ {A ∈ L(H1 , H2 ) : AT1 = T2 A}, then we define a function X∗ : Lat(T1 ) → Lat(T2 ) as following: X∗ (M ) = (XM )− . The operator X is said to be a lattice-isomorphism if X∗ is a bijection of Lat(T1 ) onto Lat(T2 ). It is well known that a (separable) Hilbert space H is finite-dimensional if and only if every operator X ∈ L(H), with the property ker(X) = {0}, also satisfies ker(X ∗ ) = {0}. The following definition is a natural extension of finite dimensionality. Definition 1.1. An operator T ∈ L(H) is said to have property (P) if every operator X ∈ {T }0 with the property that ker(X) = {0} is a quasiaffinity, i.e., ker(X ∗ ) = ker(X) = {0}. From the fact that the commutant {0}0 of zero operator on H coincides with L(H), we can see that H is finite-dimensional if and only if the zero operator on H has property (P ).

592

Kim

IEOT

2. Main Results We will consider a sufficient condition for Lat(T ) of a C0 -operator T to be modular. Proposition 2.1. ([1], Lemma 7.1.20) Let Ti ∈ L(Hi ) (i = 1, 2) and X ∈ {A ∈ L(H1 , H2 ) : AT1 = T2 A}. Then, the mapping X∗ is onto Lat(T2 ) if and only if (X ∗ )∗ is one-to-one on Lat(T2∗ ). Corollary 2.2. Under the same assumption as Proposition 2.1, the mapping X∗ is one-to-one on Lat(T1 ) if and only if (X ∗ )∗ is onto Lat(T1∗ ). Proof. Since XT1 = T2 X, T1∗ X ∗ = X ∗ T2∗ . By Proposition 2.1, (X ∗ )∗ is onto Lat(T1∗ ) if and only if (X ∗∗ )∗ = X∗ is one-to-one on Lat(T1 ). From Proposition 2.1 and Corollary 2.2, we obtain the following result. Corollary 2.3. Under the same assumption as Proposition 2.1, X is a latticeisomorphism if and only if X ∗ is a lattice-isomorphism. Proposition 2.4. ([1], Proposition 7.1.21) Assume that T1 ∈ L(H1 ) and T2 ∈ L(H2 ) are two quasisimilar operators of class C0 , and X ∈ {A ∈ L(H1 , H2 ) : AT1 = T2 A} is an injection. If T1 has property (P), then X is a lattice-isomorphism. Corollary 2.5. Assume that T1 ∈ L(H1 ) and T2 ∈ L(H2 ) are two quasisimilar operators of class C0 , and X ∈ {A ∈ L(H1 , H2 ) : AT1 = T2 A} is a quasiaffinity. If T2 has property (P), then X is a lattice-isomorphism. Proof. Since XT1 = T2 X, X ∗ T2∗ = T1∗ X ∗ . Since T2 has property (P ), so does T2∗ . Since X ∗ is a quasiaffinity, by Proposition 2.4, X ∗ is a lattice-isomorphism. From Corollary 2.3, it is proven that X is a lattice-isomorphism. Corollary 2.6. Suppose that Ti ∈ L(Hi ) (i = 1, 2) is a C0 -operator and T1 has property (P ). If X ∈ {A ∈ L(H1 , H2 ) : AT1 = T2 A} and X is a quasiaffinity, then X is a lattice-isomorphism. Proof. Since X is a quasiaffinity, T1 and T2 are quasisimilar. By Proposition 2.4, it is proven. Proposition 2.7. ([1], Proposition 2.4.3) Let T ∈ L(H) be a completely nonunitary contraction, and M be an invariant subspace for T . If T1 X T = (2.1) 0 T2 is the triangularization of T with respect to the decomposition H = M ⊕ (H M ), then T is of class C0 if and only if T1 and T2 are operators of class C0 . Proposition 2.8. [1], Corollary 7.1.17) Let T ∈ L(H) be an operator of class C0 , M be an invariant subspace for T , and T1 X T = (2.2) 0 T2

Vol. 62 (2008)

Modular Lattice for Co -Operators

593

be the triangularization of T with respect to the decomposition H = M ⊕ (H M ). Then T has property (P ) if and only if T1 and T2 have property (P ). Theorem 2.9. Let T ∈ L(H) be an operator of class C0 with property (P ). Then, for Mi (i = 1, 2, 3) in Lat(T ) such that M3 ⊂ M1 , (M1 ∩ M2 ) ∨ M3 = M1 ∩ (M2 ∨ M3 ),

(2.3)

that is, Lat(T ) is a modular lattice. Proof. Suppose that an operator T ∈ L(H) has property (P ) and let Mi (i = 1, 2, 3) be an invariant subspace for T such that M3 ⊂ M1 . Then evidently, (M1 ∩ M2 ) ∨ M3 ⊂ M1 ∩ (M2 ∨ M3 ).

(2.4)

Let Ti = T |Mi (i = 1, 2, 3). Define a linear transformation X : M2 ⊕ M3 → M2 ∨ M3 by X(a2 ⊕ a3 ) = a2 + a3 for a2 ∈ M2 and a3 ∈ M3 . Let Y = X|(ker X)⊥ . Then, clearly, kY k ≤ 2. Since M2 ∨M3 is generated by {a2 +a3 : a2 ∈ M2 and a3 ∈ M3 }, X has dense range. Thus, Y also has dense range. Therefore, Y is a quasiaffinity. By definition of Ti (i = 1, 2, 3), X(T2 ⊕ T3 ) = (T |M2 ∨ M3 )X, and Y (T2 ⊕ T3 )|(ker X)⊥ = (T |M2 ∨ M3 )Y. By Proposition 2.7, T2 ⊕ T3 and T |M2 ∨ M3 are of class C0 and since T has property (P ), by Proposition 2.8, we conclude that T |M2 ∨ M3 also has Property (P ). By Corollary 2.5, Y is a lattice-isomorphism. Since Y∗ is onto, X∗ : Lat(T2 ⊕ T3 ) → Lat(T |M2 ∨ M3 ) is also onto. Let M = {a2 ⊕ a3 ∈ M2 ⊕ M3 : a2 + a3 ∈ M1 }.

(2.5)

−1

Since M = X (M1 ), M is a closed subspace of M2 ⊕M3 . Evidently, M is invariant for T2 ⊕ T3 . From equation (2.5), we conclude that M = (M1 ∩ M2 ) ⊕ M3 .

(2.6)

Since X −1 (M1 ∩(M2 ∨M3 )) = {a2 ⊕a3 ∈ M2 ⊕ M3 : a2 +a3 ∈ M1 ∩ (M2 ∨ M3 )} = {a2 ⊕ a3 ∈ M2 ⊕ M3 : a2 + a3 ∈ M1 }, X −1 (M1 ∩ (M2 ∨ M3 )) = M Since X∗ is onto, X∗ M = (XM )− = M1 ∩ (M2 ∨ M3 ).

(2.7)

By equation (2.6) and definition of X, X∗ M = (XM )− ⊂ (M1 ∩ M2 ) ∨ M3 .

(2.8)

594

Kim

IEOT

From (2.7) and (2.8), we conclude that M1 ∩ (M2 ∨ M3 ) ⊂ (M1 ∩ M2 ) ∨ M3 . By (2.4) and (2.9), it is proven.

(2.9)

Acknowledgment The author would like to express her gratitude to Professor Hari Bercovici for giving her some useful comments.

References [1] H. Bercovici, Operator theory and arithmetic in H ∞ , Amer. Math. Soc., Providence, Rhode Island, 1988. [2] H. Bercovici, C0 -Fredholm operators, II, Acta Sci. Math. (Szeged) 42 (1980), 3–42. [3] P.R. Halmos, A Hilbert Space Problem Book, D. Van Nostrand Company, Princeton, N.Y., 1967. [4] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, NorthHolland, Amsterdam, 1970. Yun-Su Kim Department of Mathematics Indiana University 107 S. Indiana Ave. Bloomington, IN 47405-7000 USA e-mail: [email protected] Submitted: September 6, 2007. Revised: July 14, 2008.




Non-Operator Reflexive Subspace Lattices Kamila Kli´s-Garlicka and Vladimir M¨ uller Abstract. In [2] various types of closedness of subspace lattices were studied. In particular, the authors defined operator reflexivity which can be regarded as a one-point closedness of the lattice. They asked if all subspace lattices are operator reflexive. In this work we give an example that the answer is negative. Mathematics Subject Classification (2000). Primary 47D15; Secondary 47B20. Keywords. Operator reflexive, subspace lattice.

Let H be a Hilbert space. By B(H) we denote the algebra of all bounded linear operators on H and by P(H) the lattice of all orthogonal projections on H. A SOT-closed sublattice of P(H), containing the trivial projections 0 and I is called a subspace lattice. Recall that for any set S of operators the operator-reflexive hull of S is defined as ref S = {A ∈ B(H) : Ax ∈ Sx for all x ∈ H}. It was proved in [2] that if L is a subspace lattice then ref L = {P ∈ P(H) : P x ∈ Lx for all x ∈ H}. Recall after [2] that a projection lattice L is called operator reflexive (or 1-closed) if L = ref L. In [2] authors proved that some important classes of subspaces lattices (for example compact or reflexive in usual sense) are operator reflexive, and that all subspace lattices are algebraically reflexive, but they asked if all subspace lattices are operator reflexive. Here we intend to prove that it is not so. Let M be a subspace of a Hilbert space H. We denote by PM the orthogonal projection onto M . Let M, L ⊂ H be subspaces. Write δ(M, L) = sup{dist {x, L} : x ∈ M, kxk ≤ 1}. ˆ Denote by δ(M, L) = max{δ(M, L), δ(L, M )} the gap between M and L. It is ˆ ˆ well-known, see [1], p. 197, that δ(M, L) = kPM − PL k. Moreover, if δ(M, L) < 1 then dim M = dim L. The second author was supported by grant no. 201/06/0128 of GA CR.

596

Kli´s-Garlicka and M¨ uller

IEOT

Lemma 1. Let H be a finite-dimensional Hilbert space, M, L ⊂ H subspaces, dim M = dim L, dim H = 2 dim M . Let ε > 0. Then there exists a subspace ˆ 0 , M ) ≤ ε and M 0 ∩ L = {0}. M 0 ⊂ H such that δ(M Proof. We may assume that ε < 1. We have dim H = dim M + dim L = dim(M ∩ L) + dim(M + L). Hence dim(M ∩ L) = dim(M + L)⊥ . Let V : M ∩ L → (M + L)⊥ be a surjective isometry. Let M 0 = (I + εV )(M ∩ L) ⊕ (M (M ∩ L)). Clearly M 0 is a subspace and dim M 0 = dim M . Suppose that u ∈ M 0 ∩ L. Then u = (I + εV )x + y for some x ∈ M ∩ L and y ∈ M (M ∩ L). We have u − x − y ∈ M + L and εV x ⊥ (M + L), so εV x = 0 = u − x − y. Thus x = 0 and u = y. Hence y ∈ M ∩ L, and so y = 0 and u = 0. Consequently, M 0 ∩ L = {0}. Suppose that u ∈ M , kuk = 1. Then u = x + y for some x ∈ M ∩ L and y ∈ M (M ∩ L) with kxk2 + kyk2 = kuk2 = 1. Then dist {u, M 0 } ≤ ku − (I + εV )x − yk = kεV xk ≤ ε. Hence δ(M, M 0 ) ≤ ε. Conversely, let v ∈ M 0 , kvk = 1. Then v = (I + εV )x + y for some x ∈ M ∩ L and y ∈ M (M ∩ L). Since εV x ⊥ y, we have k(I + εV )xk ≤ 1. Since εV x ⊥ x, we have kxk ≤ 1. Thus dist {v, M } ≤ kv − (x + y)k = kεV xk ≤ ε ˆ 0 , M ) ≤ ε. and so δ(M

Lemma 2. Let H be a finite-dimensional Hilbert space, dim H = 2n and let M1 , . . . , Mk , L ⊂ H be n-dimensional subspaces, let ε > 0. Then there exists a ˆ 0 , L) ≤ ε and L0 ∩ Mi = {0} (i = 1, . . . , k). subspace L0 ⊂ H such that δ(L Proof. We prove the statement by induction on k. For k = 1 the statement was proved in Lemma 1. Suppose that the statement is true for some k − 1 and let M1 , . . . , Mk , L, ε be given. By the induction assumption, there exists a subspace L00 ⊂ H such that ˆ L00 ) ≤ ε/2 and L00 ∩ Mi = {0} (i = 1, . . . , k − 1). δ(L, By a compactness argument, there exists δ > 0 such that dist {x, L00 } ≥ δ whenever 1 ≤ i ≤ k − 1, x ∈ Mi , kxk = 1. By Lemma 1, there exists L0 ⊂ H such ˆ 0 , L00 ) ≤ min{ε/2, δ/2, } and L0 ∩ Mk = {0}. that δ(L ˆ 0 , L} = kPL0 − PL k ≤ kPL0 − PL00 k + kPL00 − PL k ≤ ε. We have δ{L We show that L0 ∩ Mi = {0} (i = 1, . . . , k − 1). Fix i ∈ {1, . . . , k − 1} and suppose that there exists x ∈ L0 ∩ Mi , kxk = 1. Then there exists x0 ∈ L00 ˆ 0 , L00 ) ≤ δ/2, a contradiction with the definition of δ. Hence with kx0 − xk ≤ δ(L L0 ∩ Mi = {0} (i = 1, . . . , k). Let HWbe the Hilbert space with an orthonormal basis e1 , e2 , . . . For k ∈ N let Hk = {e1 , . . . , ek }. Denote by SH the unit sphere in H. Fix a sequence (xn , yn )∞ n=1 dense in SH × SH such that for each n ∈ N the vectors xn , yn are

Vol. 62 (2008)

Non-Operator Reflexive Subspace Lattices

597

linearly independent and hxn , yn i = 6 0. Moreover, S we may assume that all the vectors xn , yn have finite support, i.e., xn , yn ∈ Hk for each n ∈ N. k∈N

Fix a sequence (tn )∞ n=1 ⊂ (0, 1) consisting of mutually distinct numbers. Lemma 3. There exist subspaces Mn ⊂ H (n ∈ N) such that: (i) Mn ∩ Mm = {0} (m, n ∈ N, m 6= n); (ii) Mn ∨ Mm = H (m, n ∈ N, m 6= n); (iii) kPMn xn − hxn , yn iyn k ≤ 1/n; (iv) there is a constant c > 0 such that for all m, n ∈ N, m 6= n, max kPMn ej − PMm ej k ≥ c;

j=1,2,3

(v) there is an increasing sequence of positive integers (kn )∞ n=1 such that each Mn can be written as _ Mn = Fn ⊕ {e2j+1 + tn e2j+2 : j ≥ kn }, where Fn ⊂ H2kn is a kn -dimensional subspace. Proof. We construct the subspaces Mn by induction on n. Let n ∈ N and suppose that the subspaces W M1 , . . . , Mn−1 satisfying (i)–(v) have already been constructed. Let Ln = {xn , yn }. By assumption, dim Ln = 2. Fix jn ∈ {1, 2, 3} such that dist {ejn , Ln } = max dist {ei , Ln }. i=1,2,3

Clearly there is a constant c > 0 such that max dist {ei , L} ≥ 4c for each 2i=1,2,3

dimensional subspace L ⊂ H. Hence dist {ejn , Ln } ≥ 4c. Let un = Fix kn > max{kn−1 , 2} such that xn , yn ∈ H2kn −1 . Since un =

PL⊥ ejn

kPL⊥ ejn k . n ejn −PLn ejn kPL⊥ ejn k and n

n

thus un ∈ H2kn −1 . ejn ∈ Ln + L0n , W Let L0n = {un , e2kn }. Then dim L0n = 2 and L0n ⊥ Ln . Let Fn0 be any kn dimensional subspace of H2kn such that yn ∈ Fn0 , un + e2kn ∈ Fn0 and dim(H2kn (Ln + L0n )) ∩ Fn0 = kn − 2. For s = 1, . . . , n − 1 let Es ⊂ H2kn be defined by _ Es = Fs ⊕ {e2j+1 + ts e2j+2 : ks ≤ j < kn }. By Lemma 2 for the subspaces E1 , . . . , En−1 , Fn0 there exists a subspace Fn ⊂ H2kn ˆ n , F 0 } < min{ 1 , c}. Note such that Fn ∩ Es = {0} (s = 1, . . . , n − 1) and δ{F n n that this implies that dim F = k and F ∨ E = H (s = 1, . . . , n − 1). n n n s 2k n W Let Mn = Fn ⊕ {e2j+1 + tn e2j+2 : j ≥ kn }. We show that Mn satisfies (i)–(v). Condition (v) follows from the definition. Since tm 6= tn for m < n, we have Mm ∩ Mn = {0} and Mm ∨ Mn = H. ˆ n , F 0 } ≤ 1 . Hence We have PFn0 xn = hxn , yn iyn and kPFn − PFn0 k = δ{F n n kPMn xn − hxn , yn iyn k = kPFn xn − PFn0 xn k ≤

1 . n

598

Kli´s-Garlicka and M¨ uller

IEOT

Let Q be the orthogonal projection onto the 1-dimensional subspace generated by e2kn . Let m < n. We have PMm ejn ∈ H2km , and so QPMm ejn = 0. Furthermore kQPMn ejn k = kQPFn ejn k ≥ kQPFn0 ejn k − kQ(PFn0 − PFn )ejn k ˆ 0 , Fn } ≥ kQPF 0 ej k − c ≥ kQPFn0 ejn k − δ{F n n n and kQPFn0 ejn k = kQPLn ∩Fn0 ejn + QPL0n ∩Fn0 ejn k = kQPL0n ∩Fn0 ejn k u +e = kQPL0n ∩Fn0 (un · kPL⊥ ejn k)k ≥ 4c · kQPL0n ∩Fn0 un k = 4c · kQ n 2 2kn k = 2c. n Hence kPMn ejn − PMm ejn k ≥ kQPMn ejn − QPMm ejn k ≥ 2c − c = c.

Theorem 4. There exists a strongly closed lattice L ⊂ P(H) which is not operator reflexive. Proof. Let Mn be the subspaces constructed in the previous lemma. Let L = {0, I, PMn : n ∈ N}. Clearly L is a lattice and L 6= P(H). We show that L is strongly closed. It is sufficient to show that the set {PMn : n ∈ N} is strongly closed. Let P ∈ P(H), P ∈ {PMn : n ∈ N}−SOT . Let c > 0 be the number from the previous lemma. Let x ∈ H. Then there exists n(x) ∈ N such that c kPMn(x) x − P xk < 2 and c kPMn(x) ej − P ej k < (j = 1, 2, 3). 2 Moreover, n(x) is determined uniquely and is independent of the choice of x ∈ H. Indeed, let y ∈ H and let n(y) ∈ N satisfies c kPMn(y) x − P xk < 2 and c kPMn(y) ej − P ej k < (j = 1, 2, 3). 2 For j = 1, 2, 3 we have kPMn(x) ej − PMn(y) ej k ≤ kPMn(x) ej − P ej k + kP ej − PMn(y) ej k < c. Hence n(x) = n(y). Furthemore, PMn(x) x = P x. Indeed, for each δ ∈ (0, 2c ) there exists r ∈ N such that kPMr x − P xk < δ and kPMr ej − P ej k
0 was arbitrary, we have PMn(x) x = P x and P = PMn(x) . Hence L is closed in the strong operator topology. On the other hand, the operator-reflexive hull of L is the whole lattice P(H). To see this, let P ∈ P(H) and x ∈ H, kxk = 1. If P x = 0 then obviously Px P x ∈ {Qx : Q ∈ L}− . Let P x 6= 0 and y = kP xk . Then there is a sequence (nk ) such that nk → ∞, xnk → x and ynk → y. Thus P x = hx, yiy = lim hxnk , ynk iynk = lim PMnk xnk = k→∞

k→∞

= lim PMnk x ∈ {Qx : Q ∈ L}− , k→∞

and so P is in the operator-reflexive hull of L.

References [1] T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin-Heidelberg-New York 1976. [2] V.S. Shulman and I. Todorov, On Subspace Lattices. I. Closedness type properties and tensor products, Integr. Equ. Oper. Theory 52 (2005), 561-579. Kamila Kli´s-Garlicka Institute of Mathematics University of Agriculture Al.Mickiewicza 24/28 30-059 Krak´ ow Poland e-mail: [email protected] Vladimir M¨ uller Institute of Mathematics Academy of Sciences of the Czech Republic ˇ Zitna 25 115 67, Praha 1 Czech Republic e-mail: [email protected] Submitted: July 24, 2008.

Integral Equations and Operator Theory - Volume 62

Integral Equations and Operator Theory - Volume 70

Integral Equations and Operator Theory - Volume 68

Integral Equations and Operator Theory - Volume 65

Integral Equations and Operator Theory - Volume 67

Integral Equations and Operator Theory - Volume 69

Integral Equations and Operator Theory - Volume 66

Integral Equations and Operator Theory - Volume 58

Integral Equations and Operator Theory - Volume 61

Integral Equations and Operator Theory - Volume 63

Integral Equations and Operator Theory - Volume 57

Integral Equations and Operator Theory - Volume 59

Integral Equations and Operator Theory - Volume 64

Integral Equations and Applications

Numerical analysis for integral and related operator equations

Differential and integral equations

Integral equations and applications

Differential and Integral Equations

Linear integral equations: theory and technique

Differential and Integral Equations