Integral Equations and Operator Theory - Volume 65

Integr. equ. oper. theory 65 (2009), 1–50 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-50, published o...

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Integr. equ. oper. theory 65 (2009), 1–50 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010001-50, published online August 24, 2009 DOI 10.1007/s00020-009-1709-7

Integral Equations and Operator Theory

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions Vladimir Derkach and Harry Dym Abstract. A class Uκ1 (J) of generalized J-inner mvf’s (matrix valued functions) W (λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p × q mvf’s Sκp×q and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation TW based on W and applied to Sκp×q . Factorization formulas for mvf’s W in a subclass Uκ◦1 (J) 2 of Uκ1 (J) found and then used to parametrize the set Sκp×q ∩ TW [Sκp×q ]. 2 1 +κ2 Applications to bitangential interpolation problems in the class Sκp×q will 1 +κ2 be presented elsewhere. Mathematics Subject Classification (2000). Primary 47A56; Secondary 46C20, 46E22, 47A57, 47B20. Keywords. Linear fractional transformation, generalized Schur class, Kre˘ınLanger factorization, resolvent matrix, Potapov-Ginzburg transform, coprime factorization, reproducing kernel space, associated pair.

1. Introduction Let J be an m × m signature matrix (i.e., J = J ∗ and JJ ∗ = Im ) and let Ω+ ¯ > 0}. An be equal to either D = {λ ∈ C : |λ| < 1} or Π+ = {λ ∈ C : λ + λ m × m mvf (matrix valued function) W (λ) that is meromorphic in Ω+ belongs to the class Uκ (J) of generalized J-inner mvf’s if: (i) the kernel J − W (λ)JW (ω)∗ (1.1) KW ω (λ) = ρω (λ) V. Derkach, Weston Visiting Scholar, wishes to thank the Weizmann Institute of Science for hospitality and support.

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+ has κ negative squares in h+ W × hW (see definition in Subsection 2.1) and ∗ (ii) J − W (µ)JW (µ) = 0 a.e. on the boundary Ω0 of Ω+ ;

where h+ W denotes the domain of holomorphy of W in Ω+ and ( if Ω+ = D; 1 − λω, ρω (λ) = 2π(λ + ω), if Ω+ = Π+ . Thus, in both cases Ω+ = {ω ∈ C : ρω (ω) > 0} and Ω0 = {ω ∈ C : ρω (ω) = 0} is the boundary of Ω+ . Correspondingly we set Ω− = C \ (Ω+ ∪ Ω0 ) = {ω ∈ C : ρω (ω) < 0}.

(1.2)

Most of the other notation that we use will be fairly standard: mvf for matrix valued function, vvf for vector valued function, ker A for the kernel of a matrix A, rng A for its range, σ(A) for its spectrum if A is square, and, if A = A∗ , ν− (A) for the number of negative eigenvalues (counting multiplicities). If f (λ) is a mvf, then ( 1/λ, if Ω+ = D and λ 6= 0; # o ∗ o f (λ) = f (λ ) , where λ = −λ, if Ω+ = Π+ and hf = {λ ∈ C at which f (λ) is holomorphic} and h± f = h f ∩ Ω± . We also set

hf, gist

 Z 2π 1   g(µ)∗ f (µ)dµ,  2π 0 =    R∞ g(iµ)∗ f (iµ)dµ, −∞

if Ω+ = D; if Ω+ = Π+

and e p×q = L 1

 p×q  L1 (Ω0 ), 

H2p×q

{f : (1 + |µ|2 )−1 f ∈ Lp×q (Ω0 )}, 1

if Ω+ = D; if Ω+ = Π+ .

p×q The symbol (resp., H∞ ) stands for the class of p × q mvf’s with entries in the Hardy space H2 (resp., H∞ ); H2p is short for H2p×1 and (H2p )⊥ is the orthogonal complement of H2p in Lp2 with respect to the standard inner product on Ω0 . The class Uκ (J) and reproducing kernel Pontryagin spaces K(W ) with the reproducing kernel KW ω (λ) based on W ∈ Uκ (J) were studied in [4] and [2]. In [36], [12], [11] and [18] mvf’s W ∈ Uκ (J) appear as resolvent matrices of interpolation problems; in [35], [10], [21], [20] and [19] mvf’s W ∈ Uκ (J) were considered as characteristic functions of linear operators in indefinite inner product spaces.

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Let Sκp×q (Ω+ ) denote the generalized Schur class of mvf’s s that are meromorphic in Ω+ and for which the kernel Λsω (λ) =

Ip − s(λ)s(ω)∗ ρω (λ)

(1.3)

+ has κ negative squares on h+ s × Hs (see [32]). A fundamental result of Kre˘ın and Langer [32] guarantees that every generalized Schur function s ∈ Sκp×q := Sκp×q (Ω+ ) admits a pair of coprime factorizations

s(λ) = b` (λ)−1 s` (λ) = sr (λ)br (λ)−1

for λ ∈ h+ s ,

(1.4)

where b` and br are Blaschke–Potapov products of degree κ and sizes p × p and q × q, respectively, and the mvf’s s` and sr both belong to the Schur class S p×q := p×p S0p×q (Ω+ ). The classes of inner and outer mvf’s in S p×p will be denoted Sin and p×p Sout , respectively. In this paper we fix Ip 0 J = jpq = , where p + q = m, 0 −Iq and if W ∈ Uκ (jpq ) is written in block form W = [wij ]2i,j=1 conformally with jpq , then the linear fractional transformation TW [ε] = (w11 ε + w12 )(w21 ε + w22 )−1

(1.5)

Sκp×q 2

is well defined on for all λ ∈ Ω+ except for at most a finite set of points and p×q s = TW [ε] belongs to Sκ0 with κ0 ≤ κ1 + κ2 . The main results of this paper are: (1) A characterization of the set TW [Sκp×q ] = TW [ε] : ε ∈ Sκp×q 2 2 that is formulated in terms of the mvf Ip −s(µ) ∆s (µ) := −s(µ)∗ Iq

a.e. on Ω0

and a κ-dimensional operator Γr that is defined below. when W belongs (2) A parametrization of the intersection of this set with Sκp×q 1 +κ2 to a subclass Uκ◦1 (jpq ) of Uκ1 (jpq ), introduced below in (1.8). Let the mvf s ∈ Sκp×q admit the Kre˘ın-Langer factorizations (1.4) and let H∗ (b` ) := (H2q )⊥ b∗` (H2p )⊥

and H(br ) = H2q br H2q .

Since dim H(br ) = κ, the operator Xr : h 7→ P− sh (h ∈ H(br )), is a finite-dimensional operator of rank at most κ. In fact, as will be shown below, Xr is a 1-1-isomorphism from H(br ) onto H∗ (b` ). Let Γr : g ∈ Lp2 7→ Xr−1 PH∗ (b` ) g ∈ H(br ).

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Theorem 1.1. Let κ1 , κ2 ∈ N ∪ {0}, let W ∈ Uκ1 (jpq ) and let s ∈ Sκp×q admit the 1 +κ2 Kre˘ın-Langer factorizations (1.4). Then s ∈ TW [Sκp×q ] if and only if the following 2 conditions hold: (1) b` −s` f ∈ H2p for every f ∈ K(W ); (2) −s∗r b∗r f ∈ (H2q )⊥ for every f ∈ K(W); 0 Γ∗r (3) {∆s + ∆s ∆s }f, f ≤ hf, f iK(W ) for every f ∈ K(W ). Γr 0 st The description of the set TW [Sκp×q ] given in Theorem 1.1 is a generalization 2 to the indefinite setting of a result from [25]. The proof is based on the theory of the reproducing kernel Pontryagin spaces K(s) and K(W ) associated with the kernels Λsω (λ) and KW ω (λ) developed in [2] and [4], respectively. For the convenience of the reader we review and partially extend the parts of this theory that are needed in this paper in Section 2. In particular, we furnish an indefinite analog of the de Branges-Rovnayk description ([15]) of the space K(s) and a boundary characterization of an indefinite analog D(s) of the de Branges-Rovnyak reproducing kernel Hilbert space. In the definite case the left hand side of (3) in Theorem 1.1 coincides with kf k2D(s) , which clarifies the connection of this result with the setting of the abstract interpolation problem in [30]. For every mvf W ∈ Uκ (jpq ) W = [wij ]2i,j=1 the lower right hand q × q corner w22 (λ) of W (λ) is invertible for all λ ∈ h+ W except for at most κ points. Thus, the Potapov-Ginzburg transform S = P G(W ) of W is defined on h+ W by the formula −1 s11 (λ) s12 (λ) w11 (λ) w12 (λ) Ip 0 S(λ) = := (1.6) s21 (λ) s22 (λ) 0 Iq w21 (λ) w22 (λ) and it belongs to the class Sκm×m (as may be verified by writing ΛSω (λ) in terms of KW ω (λ)). Moreover, since S = P G(W ) =⇒ W = P G(S) ,

(1.7)

the mvf W is of bounded type. Thus, the nontangential limits W (µ) exist a.e. on Ω0 and assumption (ii) in the definition of Uκ (J) makes sense. Let −1 Uκ◦ (jpq ) = {W ∈ Uκ (jpq ) : s21 := −w22 w21

belongs to Sκq×p }.

(1.8)

The Kre˘ın-Langer factorizations of s21 will be written as s21 (λ) := b` (λ)−1 s` (λ) = sr (λ)br (λ)−1

for λ ∈ h+ s ,

(1.9) q×p

where b` , br are Blaschke-Potapov products of degree κ and s` , sr ∈ S ; german fonts are used to emphasize that the factorization is now for a mvf of size q × p. In Theorem 4.6 we shall show that the mvf’s b` s22 and s11 br belong to the classes S p×p and S q×q , respectively. Therefore, they admit inner-outer and outerinner factorizations s11 br = b1 ϕ1 , b` s22 = ϕ2 b2 , (1.10)

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p×p q×q p×p q×q where b1 ∈ Sin , b2 ∈ Sin , ϕ1 ∈ Sout , ϕ2 ∈ Sout . In keeping with the usage in [7] and [8], the pair {b1 , b2 } is called an associated pair of the mvf W ∈ Uκ◦ (jpq ) and denoted {b1 , b2 } ∈ ap(W ). If κ = 0, the formulas in (1.10) reduce to the inner-outer and outer-inner factorizations of s11 and s22 (see [6], [7]). In Theorem 4.12 it will be shown that if W ∈ Uκ◦ (jpq ) and {b1 , b2 } ∈ ap(W ), then there are H∞ mvf’s K, c` , d` , cr , dr , such that the factorizations e Φ, e W = Θ Φ and W = Θ (1.11)

hold over Ω+ and Ω− , respectively, with b1 0 b1 Kb−1 2 e Θ= , Θ= = jpq Θ−# jpq , K # b1 b−1 0 b−1 2 2 # −# br −s# ϕ1 0 cr dr 0 ϕ1 r e Φ= . and Φ= −s` b` 0 ϕ−1 d# c# 0 ϕ# 2 ` ` 2

(1.12) (1.13)

If W ∈ Uκ◦1 (jpq ), then Theorem 1.1 is supplemented by the following param] ∩ Sκp×q in terms of the parameter ε. etrization of the set TW [Sκp×q 2 1 +κ2 Theorem 1.2. Let the mvf W ∈ Uκ◦1 (jpq ), S = P G(W ), let s21 have the Kre˘ınLanger factorizations (1.9), {b1 , b2 } ∈ ap(W ), and let ε ∈ Sκp×q satisfy the as2 sumption e q×q (Iq − s21 ε)−1 |Ω0 ∈ L (1.14) 1 and admit the Kre˘ın-Langer factorizations ε = θ`−1 ε` = εr θr−1 , where θ` , θr are Blaschke-Potapov products of degree κ2 and ε` , εr ∈ S p×q . Then the mvf s = TW [ε] belongs to Sκp×q if and only if the factorizations 1 +κ2 # # θ` w11 + ε` w12 = (θ` br − ε` sr )(b1 ϕ1 )−1 ,

w21 εr + w22 θr = (ϕ2 b2 )−1 (−s` εr + b` θr ) are coprime over Ω+ .

(1.15) (1.16)

The proof of this result is based on the factorizations (1.11)-(1.13) and an application of the Kre˘ın-Langer generalization of Rouche’s theorem from [34]. Cases when assumption (1.14) can be dropped are also discussed. The paper is organized as follows. In Section 2 the basic notions of left and right coprime factorizations are introduced. Their connection with the Kre˘ınLanger factorizations of generalized Schur functions is discussed. The theory of reproducing kernel Pontryagin spaces, associated with a generalized Schur function s from [2] is reviewed and extended. In Section 3 we prove the first main result of the paper: Theorem 1.1, which characterizes the range of the linear fractional transformation TW associated with the mvf W ∈ Uκ1 (jpq ). In Section 4 we obtain factorization formulas for mvf’s W ∈ Uκ◦ (jpq ) and use them to characterize the corresponding reproducing kernel Pontryagin spaces K(W ). A parametrization of the set TW [Sκp×q ] ∩ Sκp×q is given in Theorem 1.2 for W ∈ Uκ◦1 (jpq ). 2 1 +κ2

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2. Preliminaries 2.1. The generalized Schur class Recall that a Hermitian kernel Kω (λ) : Ω × Ω → Cm×m is said to have κ negative squares if for every positive integer n and every choice of ωj ∈ Ω and uj ∈ Cm (j = 1, . . . , n) the matrix

n Kωj (ωk )uj , uk j,k=1 has at most κ negative eigenvalues and for some choice of ω1 , . . . , ωn ∈ Ω and u1 , . . . , un ∈ Cm exactly κ negative eigenvalues. In this case we write sq− K = κ. The class S p×q := S0p×q (Ω+ ) is the usual Schur class. Recall that a mvf s ∈ S p×q is called inner (resp., ∗-inner), if s(µ) is an isometry (resp., a co-isometry) for a.e. µ ∈ Ω0 , that is Iq − s(µ)∗ s(µ) = 0 p×q Sin

(resp., Ip − s(µ)s(µ)∗ = 0),

µ ∈ Ω0 (a.e.).

p×q S∗in )

Let (resp., denote the set of all inner (resp., ∗-inner) mvf’s s ∈ S p×q . An example of an inner square mvf is provided by the finite Blaschke–Potapov product, that in the case of the unit disc (Ω+ = D) is given by b(λ) =

κ Y

bj (λ),

bj (λ) = I − Pj +

j=1

λ − αj Pj , 1−α ¯j λ

(2.1)

where αj ∈ D and Pj is an orthogonal projection in Cp for j = 1, . . . , n. The factor bj is called simple if Pj has rank one. The representation of b(λ) as a product of simple Blaschke-Potapov factors is not unique. However, the number κ of such simple factors is the same for every representation (2.1). It is called the degree of the Blaschke–Potapov product b(λ) [37]. A theorem of Kre˘ın and Langer [32] guarantees that every generalized Schur function s ∈ Sκp×q (Ω+ ) admits a factorization of the form s(λ) = b` (λ)−1 s` (λ)

for λ ∈ h+ s ,

(2.2)

where b` is a Blaschke–Potapov product of degree κ, s` is in the Schur class S p×q (Ω+ ) and ker s` (λ)∗ ∩ ker b` (λ)∗ = {0} for λ ∈ Ω+ . (2.3) The representation (2.2) is called a left Kre˘ın–Langer factorization. The constraint (2.3) can be expressed in the equivalent form rank b` (λ) s` (λ) = p (λ ∈ Ω+ ). (2.4) If αj ∈ D (j = 1, . . . , n) are all the zeros of b` in Ω+ , then the noncancellation condition (2.3) ensures that h+ ın–Langer facs = Ω+ \ {α1 , . . . , αn }. The left Kre˘ torization (2.2) is essentially unique in a sense that b` is defined uniquely up to a left unitary factor V ∈ Cp×p .

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Similarly, every generalized Schur function s ∈ Sκp×q (Ω+ ) admits a right Kre˘ın-Langer factorization s(λ) = sr (λ)br (λ)−1

for λ ∈ h+ s ,

where br is a Blaschke–Potapov product of degree κ, sr ∈ S

(2.5) p×q

(Ω+ ) and

ker sr (λ) ∩ ker br (λ) = {0} for λ ∈ Ω+ .

(2.6)

This condition can be rewritten in the equivalent form rank br (λ)∗ sr (λ)∗ = q (λ ∈ Ω+ ).

(2.7)

Under assumption (2.6) the mvf br is uniquely defined up to a right unitary factor V 0 ∈ Cq×q . p×p Lemma 2.1. A mvf s` ∈ S p×q and a finite Blaschke-Potapov product b` ∈ Sin p×p meet the rank condition (2.4), if and only if there exists a pair of mvf ’s c` ∈ H∞ q×p and d` ∈ H∞ such that

b` (λ)c` (λ) + s` (λ)d` (λ) = Ip

for λ ∈ Ω+ .

(2.8)

Proof. This is a matrix version of the Carleson Corona theorem. The proof is adapted from Fuhrmann [28] who treated the square block case. We sketch the details for the convenience of the reader. Let a(λ) = [a1 (λ) · · · am (λ)] = [b` (λ) s` (λ)] be the p × m matrix with columns ai (λ), i = 1, . . . , m, m = p + q, and let αi1 ...ip (λ) = det[ai1 (λ) · · · aip (λ)] for every choice of positive integers i1 , . . . , ip that meet the constraint 1 ≤ i1 < i2 < · · · < ip ≤ m. Then, since b` (λ) is a finite Blaschke product, there exists a positive number r < 1 such that 1 | det b` (λ)| = | det[a1 (λ) · · · ap (λ)]| ≥ 2 for r ≤ |λ| ≤ 1. Thus, it is readily checked that there exists a δ > 0 such that X |αi1 ...ip (λ)| > δ > 0 i1 ...ip

for every point λ ∈ Ω+ . Therefore, by the scalar Corona theorem [23], there exists a set of functions βi1 ...ip (λ) in H∞ such that X αi1 ...ip (λ)βi1 ...ip (λ) = 1. i1 ...ip

Now expand each of the determinants αi1 ...ip along its ith row and express the resulting equality as ai1 (λ)b1i (λ) + · · · + aim (λ)bmi (λ) = 1 and observe that ak1 (λ)b1i (λ) + · · · + akm (λ)bmi (λ) = 0

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if k 6= i, because this expression may be obtained by replacing the ith row in each of the determinants αi1 ...ip by its k th row. The proof is completed by setting   " # b11 (λ) · · · b1p (λ) c` (λ)   .. =  ... . . d` (λ) bm1 (λ) · · · bmp (λ) A dual statement for Lemma 2.1 is obtained by applying Lemma 2.1 to transposed vvf’s. q×q Lemma 2.2. A mvf sr ∈ S p×q and a finite Blaschke-Potapov product br ∈ Sin q×q meet the rank condition (2.7), if and only if there exists a pair of mvf ’s cr ∈ H∞ q×p and dr ∈ H∞ such that

cr (λ)br (λ) + dr (λ)sr (λ) = Iq

for λ ∈ Ω+ .

(2.9)

The factorization (2.2) is called a left coprime factorization of s if s` and b` satisfy (2.8). Similarly, the factorization (2.5) is called a right coprime factorization of s if sr and br satisfy (2.9). Every vvf h(λ) from H2p (resp., (H2p )⊥ ) has nontangential limits a.e. on the boundary Ω0 . These nontangential limits identify the vvf h uniquely. In what follows we often identify a vvf h ∈ H2p (resp., (H2p )⊥ ) with its boundary values h(µ). ⊥ Let P+ and P− denote the orthogonal projections from Lk2 onto H2k and H2k , respectively, where k is a positive integer that will be understood from the context. The Hilbert spaces H(br ) = H2q br H2q ,

H∗ (b` ) := (H2p )⊥ b∗` (H2p )⊥

(2.10)

and the operators Xr : h ∈ H(br ) 7→ P− sh and X` : h ∈ H∗ (b` ) 7→ P+ s∗ h based on s ∈

Sκp×q

(2.11)

will play an important role.

Lemma 2.3. (cf. [17]) If s ∈ Sκp×q and its two factorizations are given by (2.2) and (2.5), then: (1) The operator X` maps H∗ (b` ) injectively onto H(br ). (2) The operator Xr maps H(br ) injectively onto H∗ (b` ). (3) X` = Xr∗ . Proof. If h ∈ H∗ (b` ), h+ ∈ H2q and f = br h+ , it is readily checked that hP+ s∗ h, f ist = hs∗ h, br h+ ist = hh, sr h+ ist = 0

∀ h+ ∈ H2q ,

i.e., X` maps H∗ (b` ) into H(br ). Therefore, since H∗ (b` ) and H(br ) are finite dimensional spaces of the same dimension, and dim H∗ (b` ) = dim ker X` + dim range X` ,

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it suffices to show that ker X` = {0}. But, if h ∈ H∗ (b` ) and P+ s∗ h = 0, then b` h ∈ H2p and, in view of Lemma 2.1, ξ ξ ist = hb` h, (b` c + s` d) ist ρω ρω ξ ξ = hb` h, s` d ist = hP+ s∗ h, d ist = 0 ρω ρω

ξ ∗ (b` h)(ω) = hb` h,

for every choice of ω ∈ Ω+ , and ξ ∈ Cp . Since b` (ω) 6≡ 0, this implies that h(ω) ≡ 0. Statement (ii) can be obtained by similar calculations, and (iii) is easy. Definition 2.4. Let Γ` : f ∈ Lq2 → X`−1 PH(br ) f ∈ H∗ (b` ) Γr : g ∈

Lp2

→

Xr−1 PH∗ (b` ) g

and

∈ H(br ),

(2.12)

where X` and Xr are defined in formula (2.11). It is readily checked that P+ s∗ Γ` f = PH(br ) f

and P− sΓr g = PH∗ (b` ) g,

f ∈ Lq2 , g ∈ Lp2 .

(2.13)

Remark 2.5. If f1 ∈ H∗ (b` ) and f2 ∈ H(br ), then, in view of Lemma 2.3, P+ s∗ f1 = PH(br ) f ⇐⇒ f1 = Γ` f, and P− sf2 = PH∗ (b` ) f ⇐⇒ f2 = Γr f. Lemma 2.6. The operators Γ` and Γr satisfy the equalities: (1) Γ∗` = Γr ; (2) h(ψΓ` − Γ` ψ)f, gist = 0 for f ∈ H(br ), g ∈ H∗ (b` ) and ψ a scalar inner function. Proof. (1) Let f ∈ Lq2 and g ∈ Lp2 . Then, since X` maps H∗ (b` ) onto H(br ) and X`∗ = Xr ,

hΓ` f, gist = X`−1 PH(br ) f, PH∗ (b` ) g st = PH(br ) f, Xr−1 PH∗ (b` ) g st = hf, Γr gist . (2) If f ∈ H(br ) and g ∈ H∗ (b` ), then h(ψΓ` − Γ` ψ)f, gist = hψΓ` f, P− sΓr gist − hP+ s∗ Γ` f, ψΓr gist = hψΓ` f, sΓr gist − hs∗ Γ` f, ψΓr gist = 0.

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2.2. Reproducing kernel Pontryagin spaces In this subsection we review some facts and notation from [10, 13] on the theory of indefinite inner product spaces for the convenience of the reader. A linear space K equipped with a sesquilinear form h·, ·iK on K × K is called an indefinite inner product space. A subspace F of K is called positive (negative) if hf, f iK > 0 (< 0) for all f ∈ F, f 6= 0. If the full space K is positive and complete with respect to 1/2 the norm kf k = hf, f iK then it is a Hilbert space. An indefinite inner product space (K, h·, ·iK ) is called a Pontryagin space, if it can be decomposed as the orthogonal sum K = K+ ⊕ K−

(2.14)

of a positive subspace K+ which is a Hilbert space and a negative subspace K− of finite dimension. The number ind− K := dim K− is referred to as the negative index of K. The convergence in a Pontryagin space (K, h·, ·iK ) is meant with respect to the Hilbert space norm khk2 = hh+ , h+ iK − hh− , h− iK ,

h = h+ + h− ,

h ± ∈ K± .

(2.15)

It is easily seen that the convergence does not depend on a choice of the decomposition (2.14). A Pontryagin space (K, h·, ·iK ) of Cm -valued functions defined on a subset Ω of C is called a reproducing kernel Pontryagin space if there exists a Hermitian kernel Kω (λ) : Ω × Ω → Cm×m such that (1) for every ω ∈ Ω and every u ∈ Cm the vvf Kω (λ)u belongs to K; (2) for every h ∈ K, ω ∈ Ω and u ∈ Cm the following identity holds hh, Kω uiK = u∗ f (ω).

(2.16)

It is known (see [38]) that for every Hermitian kernel Kω (λ) : Ω × Ω → Cm×m with a finite number of negative squares on Ω × Ω there is a unique Pontryagin space K with reproducing kernel Kω (λ), and that ind− K = sq− K = κ. In the case κ = 0 this fact is due to Aronszajn [5]. Let K, K1 be Pontryagin spaces and let A : K1 → K be a continuous linear operator. The adjoint A∗ : K → K1 is defined by the equality hA∗ h, giK1 = hh, AgiK ,

for all h ∈ K, g ∈ K1 .

The operator A is called a contraction if hAg, AgiK ≤ hg, giK1

for all g ∈ K1 ;

(2.17)

A is called an isometry if equality prevails in (2.17); it is called a coisometry if A∗ : K → K1 is an isometry. A subspace K1 of a Pontryagin space K is said to be contained contractively in K, if the inclusion mapping ı : K1 → K is a contraction, i.e. hg, giK ≤ hg, giK1 for all g ∈ K1 .

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K1 is said to be contained isometrically in K if the inclusion ı : K1 → K is an isometry. A subspace K2 of a Pontryagin space K is said to be complementary to the subspace K1 in the sense of de Branges if: (1) Every h ∈ K can be decomposed as h = h1 + h2 ,

h1 ∈ K1 , h2 ∈ K2 .

(2.18)

(2) The inequality hh, hiK ≤ hh1 , h1 iK1 + hh2 , h2 iK2 (2.19) holds for every decomposition h = h1 + h2 , with h1 ∈ K1 and h2 ∈ K2 . (3) There is a unique decomposition (2.18) such that equality prevails in (2.19). If the subspaces K1 and K2 are complementary in K in the sense of de Branges, then ind− K = ind− K1 + ind− K2 and the overlapping space R = K1 ∩ K2 of K1 and K2 is a Hilbert space with respect to the inner product hh, giR = hh, giK1 + hh, giK2 ,

h, g ∈ R.

(2.20)

The following theorem is due to L. de Branges [14]. Theorem 2.7. Let K, K1 be Pontryagin spaces such that K1 is contained contractively in K. Then there is exactly one subspace K2 of K that is complementary to K1 in the sense of de Branges. Moreover, the following statements are equivalent: (1) K2 is the orthogonal complement of K1 in K; (2) K1 ∩ K2 = {0}; (3) K1 is contained isometrically in K. These notions come into play when the reproducing kernel Kω (λ) of a Pontryagin space is decomposed as a sum of two Hermitian kernels (2) Kω (λ) = K(1) ω (λ) + Kω (λ).

(2.21)

The following theorem is a paraphrase of Theorem 1.5.5 from [2]. (1)

(2)

Theorem 2.8. ([2, Theorem 1.5.5].) Let Kω (λ), Kω (λ), Kω (λ) be Cm×m - valued Hermitian kernels on Ω × Ω with finite negative squares such that (2.21) holds. If K, K1 , K2 are the corresponding reproducing kernel Pontryagin spaces, then sq− K ≤ sq− K(1) + sq− K(2) .

(2.22)

Moreover the following assertions are equivalent: (1) Equality prevails in (2.22); (2) K1 and K2 are contained contractively in K as complementary spaces in the sense of de Branges; (3) the overlapping space R = K1 ∩ K2 is a Hilbert space with respect to the inner product (2.20).

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A multiplicative version of this statement is formulated below for kernels of the form ∗ Kω (λ) = R(λ)K(1) (2.23) ω (λ)R(ω) . Theorem 2.9. (see [2, Theorem 1.5.7].) Let R(λ) be a Cm×n -valued function on Ω, (1) let Kω (λ), Kω (λ) be Cm×m and Cn×n valued Hermitian kernels on Ω × Ω, respectively, with finite negative squares and let K, K1 be the corresponding reproducing kernel Pontryagin spaces. Then sq− K ≤ sq− K(1) .

(2.24)

Equality prevails in (2.24) if and only if the multiplication by R(λ) is a coisometry from K1 to K, whose kernel is a Hilbert space. Next we consider some examples of reproducing kernel Hilbert spaces associated with a mvf s ∈ S p×q (Ω+ ). Example 1. Let H(s) be the reproducing kernel Hilbert space associated with the + kernel Λsω (λ) on H+ s × Hs . The following description of H(s) is due to de Branges and Rovnyak, [15]: A vvf f ∈ H2p belongs to H(s) if and only if α(f ) := sup{kf + sϕk2st − kϕk2st : ϕ ∈ H2q } < ∞.

(2.25)

Moreover, if f ∈ H(s), then kf k2H(s) = α(f ). p×q If s ∈ Sin (Ω+ ), then

H(s) = H2p sH2p

and kf kH(s) = kf kst .

Example 2. Let H∗ (s) be the reproducing kernel Hilbert space associated with the kernel Iq − s# (λ)s# (ω)∗ (2.26) on Ωs# × Ωs# . Lsω (λ) = −ρω (λ) The space H∗ (s) admits the following description: A vvf f ∈ H2q (Ω+ )⊥ belongs to H∗ (s) if and only if β(f ) := sup{kf + s# ϕk2st − kϕk2st : ϕ ∈ H2p (Ω+ )⊥ } < ∞.

(2.27)

If f ∈ H∗ (s), then kf k2H∗ (s) = β(f ). p×q If s ∈ S∗in (Ω+ ), then

H∗ (s) = H2q (Ω+ )⊥ s# H2p (Ω+ )⊥ . In the following example the reproducing kernel space K(s) is negative. Example 3. If b is a Blaschke-Potapov product of degree κ, then s = b−1 ∈ Sκp×p and it follows from Theorem 2.9 and the identity −1

b (λ) = b(λ)−1 (−Λbω (λ))b(ω)−∗ Λω

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−1

that the space K(b−1 ) corresponding to the kernel Λbω (λ) coincides with b−1 H(b) as a set. Since the inner product hf, f iK(b)−1 is negative in K(b−1 ), hf, f iK(b−1 ) = −hbf, bf iH(b) = −hf, f ist ,

f ∈ K(b−1 ),

K(b−1 ) is called the antispace of the Hilbert space b−1 H(b). 2.3. The spaces K(s) and K∗ (s) If s ∈ Sκp×q (Ω+ ), then the reproducing kernel Λsω (λ) of the reproducing kernel Pontryagin space K(s) can be expressed in terms of the right Kre˘in-Langer factorization (2.5) of s ∈ Sκp×q as b−1

Λsω (λ) = Λsωr (λ) + sr (λ)Λωr (λ)sr (ω)∗ = Λsωr (λ) − s(λ)Λbωr (λ)s(ω)∗ ,

(2.28)

which leads to fundamental decomposition of the Pontryagin space K(s). The following theorem extends Theorem 4.2.3 of [2]. Theorem 2.10. Let s ∈ Sκp×q have Kre˘in-Langer factorization (2.5) s = sr b−1 r . Then: (1) The space K(s) admits the orthogonal decomposition K(s) = H(sr ) ⊕ sr K(b−1 r ).

(2.29)

(2) If f ∈ K(s), then f − sΓr f ∈ H(sr ),

sΓr f ∈ sr K(b−1 r ),

(2.30)

f = (f − sΓr f ) + sΓr f,

(2.31)

hf, f iK(s) = kf − sΓr f k2H(sr ) − kΓr f k2st .

(2.32)

and −1 Proof. If f ∈ H(sr ) ∩ sr K(b−1 r ) and f = sr h for some h ∈ K(br ), then, since the factorization (2.5) is coprime, it follows from Lemma 2.2 that there exists a pair q×q q×p of mvf’s c ∈ H∞ , d ∈ H∞ such that

cbr + dsr = Iq .

(2.33)

Hence h = (cbr )h + d(sr h) = c(br h) + df ∈ H2q . q ⊥ Since h ∈ K(b−1 r ) ⊂ (H2 ) this implies that h ≡ 0, and hence f ≡ 0, i.e.,

H(sr ) ∩ sr K(b−1 r ) = {0}. The space K(s) can be identified with the set K of vvf’s f1 (λ) f (λ) = Ip −s(λ) with f1 ∈ H(sr ) and f2 ∈ H(br ) f2 (λ)

(2.34)

(2.35)

endowed with the indefinite inner product hf, f iK = hf1 , f1 iH(sr ) − hf2 , f2 ist ,

(2.36)

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which is correctly defined in view of (2.34). In particular, Λsωr u s p Λω u = Ip −s for every ω ∈ h+ s and u ∈ C Λbωr s(ω)∗ u

(2.37)

and hf, Λsω uiK = hf1 , Λsωr uiH(sr ) − hf2 , Λbωr s(ω)∗ uist = u∗ f1 (ω) − u∗ s(ω)f2 (ω) = u∗ f (ω) for every f ∈ K. Since K is a Pontryagin space with respect to the indefinite inner product (2.36), it coincides with K(s). Next, the decomposition (2.35) implies that P− f = PH∗ (b` ) f for every f ∈ K(s). Therefore, by another application of (2.35), PH∗ (b` ) f = P− f = −P− sf2 = −Xr f2 , i.e., f2 = −Xr−1 PH∗ (b` ) f = −Γr f

belongs to H(br ) for every f ∈ K(s)

and f − sΓr f = f + sf2 = f1 This proves the second statement.

belongs to H(sr ).

Corollary 2.11. The space K(s) consists of vvf ’s f ∈ sup{kf − sΓr f −

sr ϕk2st

−

kϕk2st

H2p

⊕ H∗ (b` ) such that

: ϕ ∈ H2q } < ∞.

(2.38)

Moreover, if f ∈ K(s), then hf, f iK(s) = sup{kf − sΓr f − sr ϕk2st − kϕk2st : ϕ ∈ H2q } − kΓr f k2st .

(2.39)

Let K∗ (s) designate the reproducing kernel Pontryagin space with kernel Lsω (λ) =

Iq − s# (λ)s# (ω)∗ . −ρω (λ)

The left Kre˘ın-Langer factorization (2.2) yields the representation # b` ∗ Lsω (λ) = Lsω` (λ) + s# ` (λ)Lω (λ)s` (ω)

× h− , on h− s# s#

(2.40)

which leads to a dual version of Theorem 2.10: Theorem 2.12. If s ∈ Sκp×q and its left Kre˘in-Langer factorization (2.2) is s = b−1 ` s` , then: (1) the space K∗ (s) admits the fundamental decomposition K∗ (s) = H∗ (s` ) ⊕ s∗` K(b−1 ` );

(2.41)

(2) for every f ∈ K∗ (s) its orthogonal decomposition corresponding to (2.41) takes the form f = (f − s∗ Γ` f ) + s∗ Γ` f, (2.42) −1 ∗ ∗ ∗ where f − s Γ` f ∈ H∗ (s` ), s Γ` f ∈ s` K(b` ) and hf, f iK∗ (s) = kf − s∗ Γ` f k2H∗ (s` ) − kΓ` f k2st .

(2.43)

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Corollary 2.13. The space K∗ (s) consists of vvf ’s f ∈ (H2q )⊥ ⊕ H(br ) such that sup{kf − s∗ Γ` f − s∗` ϕk2st − kϕk2st : ϕ ∈ (H2p )⊥ } − kΓ` f k2st < ∞.

(2.44)

For every f ∈ K∗ (s) the inner product hf, f iK∗ (s) coincides with the left hand side of (2.44). Another decomposition of the reproducing kernel Pontryagin space K(s) can be based on the representation b−1

s` −∗ Λsω (λ) = b−1 + Λω` (λ) ` (λ)Λω (λ)b` (ω)

+ on h+ s × hs .

(2.45)

In view of Theorems 2.8 and 2.9, this leads to the following representation of K(s) −1 K(s) = b−1 ` H(s` ) u K(b` ),

(2.46)

K(b−1 ` )

where H(s` ) is a Hilbert space and is the antispace of the Hilbert space b−1 H(b ). The decomposition (2.46) is not necessarily an orthogonal decomposition ` ` p×q in K(s) (see e.g., [2, p.148] for an example). However, if s` ∈ Sin , then kf kH(s` ) = kf kst and the decomposition (2.46) is orthogonal. Theorem 2.14. If s ∈ Sκp×q and its left Kre˘ın-Langer factorization (2.2) is b−1 ` s` p×q and s` ∈ Sin , then: (1) the decomposition (2.46) of K(s) is orthogonal; (2) f ∈ K(s) if and only if b` f ∈ H2p

and

b∗r s∗ f ∈ (H2q )⊥ .

(2.47)

Proof. By Lemma 2.1, there exists a pair of mvf’s c` and d` with entries in H∞ such that b` c` + s` d` = Ip . Thus, if f ∈ H(s` ) ∩ H(b` ), then hf, f ist = hf, b` c` f ist + hf, s` d` f ist = 0, and, hence, f = 0, i.e., H(s` ) ∩ H(b` ) = {0}. Therefore, the decomposition (2.46) is orthogonal, by Theorem 2.7. If f ∈ K(s) then it follows from (2.46) that f admits the decomposition −1 f = b−1 ` h + b` x,

(2.48)

where h ∈ H(s` ) and x ∈ H(b` ). Therefore b∗r s∗ f = b∗r (s∗` h) + s∗r (b∗` x). p×q p×p Since s` ∈ Sin and b` ∈ Sin one obtains s∗` h ∈ (H2q )⊥ , b∗` x ∈ H∗ (b` ) ⊂ (H2q )⊥ . q Thus b∗r s∗ f ∈ (H2 )⊥ . Conversely, assume that (2.47) holds. Then

s∗ f ∈ br (H2q )⊥ = (H2q )⊥ ⊕ H(br ). In view of (2.13) Γ` (s∗ f ) satisfies the equality P+ (s∗ f − s∗ Γ` (s∗ f )) = 0

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and hence h = b` (f − Γ` (s∗ f )) ∈ H(s` ),

x = b` Γ` (s∗ f ) ∈ H(b` ).

−1 Therefore, f = b−1 ` h + b` x ∈ K(s).

The right Kre˘ın–Langer factorization (2.5) leads to the following decomposition of the reproducing kernel Pontryagin space K∗ (s) (see [2, Theorem 4.2.3]) −1 K∗ (s) = b−# r H∗ (sr ) u K∗ (br ),

where H∗ (sr ) is a Hilbert space and br−# H∗ (br ).

K∗ (b−1 r )

(2.49)

is the antispace of a Hilbert space

Corollary 2.15. If the mvf sr in the Kre˘ın-Langer factorization (2.5) of s ∈ Sκp×q p×q belongs to the class S∗in , then: (1) the decomposition (2.49) of K∗ (s) is orthogonal; (2) f ∈ K∗ (s) if and only if b∗r f ∈ (H2q )⊥ ,

b` sf ∈ H2p .

(2.50)

Combining Theorem 2.14 and Corollary 2.15 one obtains Corollary 2.16. If the mvf s` in the Kre˘ın-Langer factorization (2.2) of a mvf m×m s ∈ Sκm×m belongs to the class Sin , then: K∗ (s) = s∗ K(s). 2.4. The de Branges-Rovnyak space D(s). The symbol A[−1] stands for the Moore-Penrose pseudoinverse of the matrix A,  1 if Ω0 = T  2π dµ, dm(µ) =  dµ, if Ω0 = iR and ∆s (µ) :=

Ip −s(µ)∗

−s(µ) Iq

a.e. on Ω0

for s ∈ Sκp×q .

(2.51)

Definition 2.17. Let a mvf s ∈ Sκp×q admit left and right Kre˘ın-Langer factorizations (2.2) and (2.5). Define D(s) as the set of vvf ’s f (t) = col (f + (t), f − (t)), such that the following conditions hold: (D1) b` f + ∈ H2p ; ⊥ (D2) b∗r f − ∈ H2q ; (D3) f (t) belongs to the range of the matrix ∆s (µ) Ω0 -a.e. and the following integral Z f (µ)∗ ∆s (µ)[−1] f (µ)dm(µ) Ω0

converges.

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The inner product in D(s) is defined by Z 0 ∗ [−1] hf, giD(s) = g(µ) ∆s (µ) + Γ r Ω0

Γ∗r 0

17

f (µ)dm(µ),

(2.52)

where Γr is the operator from Lp2 onto H(br ) that is defined in (2.12). Remark 2.18. If κ = 0, then b` = Ip , br = Iq , Γr = 0 and D(s) coincides with the Hilbert space introduced in [15]. Remark 2.19. If κ > 0, then D(s) is an indefinite inner product space, and, since Γ` = Γ` PH(br ) maps the κ dimensional space H(br ) bijectively onto H∗ (b` ) and 0 Γ` Γ` 0 0 IH(br ) Γ∗` 0 0 Γ∗r e = ∗ = , Γ := 0 Γr 0 Γ` 0 0 PH(br ) IH(br ) 0 PH(br ) e = κ. Therefore, ν− (Γ) ind− D(s) ≤ κ,

(2.53)

[−1] ∆s

is nonnegative. In fact equality prevails in (2.53), as will since the operator be shown in Lemma 2.21. Remark 2.20. Definitions 2.4 and (2.13) imply that if f ∈ D(s), then 0 Γ` Ip − sΓr + Γ` Im + ∆s f= f + f −, Γr 0 Γr Iq − s∗ Γ` where

Ip − sΓr + f ∈ H2m , Γr

(2.54)

Γ` f − ∈ (H2m )⊥ . Iq − s∗ Γ`

(2.55)

Now let s ∈ Sκp×q , and let the mvf Φω be defined on Ω0 by the formula  " #  Ip 1   ∆s , if ω ∈ h+  s ; ∗ ρω s(ω) (2.56) Φω = # ∗  −1 s (ω)  −  ∆s , if ω ∈ hs# ,  Iq ρω and let D− = span {Φβ v : β ∈ h− , v ∈ Cq }, s# (2.57) where span stands for the set of finite linear combinations. Then D± ⊂ D(s). p + − Indeed, if α ∈ h+ s and u ∈ C , then the vvf f = Φα u = col (f (µ), f (µ)), where Ip − s(µ)s(α)∗ s∗ (µ) − s(α)∗ f + (µ) = u and f − (µ) = − u, (2.58) ρα (µ) ρα (µ) p D+ = span {Φα u : α ∈ h+ s , u ∈ C },

− have meromorphic continuations to h+ s and hs , respectively:

f + (λ) =

Ip − s(λ)s(α)∗ u, ρα (λ)

and f − (λ) = −

s# (λ) − s(α)∗ u. ρα (λ)

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It follows from (2.2) and (2.5) that b` f + ∈ H2p ,

− b# ∈ H2q⊥ . r f

Since f (µ) belongs to the range of ∆s (µ), it is readily checked that (D3) also holds p and hence that f = Φα u ∈ D(s), if α ∈ h+ s and u ∈ C . − q If β ∈ hs# and v ∈ C , then the vvf f = Φβ v = col (f + , f − ), where f + (µ) =

s(µ) − s# (β)∗ v, ρβ (µ)

f − (µ) = −

Iq − s(µ)∗ s# (β)∗ v. ρβ (µ)

(2.59)

Similar observations show that f = Φβ v ∈ D(s) for β ∈ h− and v ∈ Cq . s# Lemma 2.21. Let s ∈ Sκp×q (κ ∈ N), and let D+ and D− be the subspaces of D(s) defined by (2.57). Then: (1) span {D+ , D− } is dense in D(s); (2) for every choice of α1 , . . . , αn ∈ h+ s ,

u 1 , . . . , u n ∈ Cp ,

β1 , . . . , βm ∈ h− , s#

v1 , . . . , vm ∈ Cq ,

(2.60)

the Gram matrix of the system of vvf ’s {Φαj uj , Φβk vk , 1 6 j 6 n, 1 6 k 6 m}   hΦαi ui , Φαj uj iD(s) hΦαi ui , Φβl vl iD(s)  G= (2.61) hΦβk vk , Φαj uj iD(s) hΦβk vk , Φβl vl iD(s) is of the form 

u∗j Λαi (αj )ui

 G=  ∗ s(αj ) − s# (βk )∗ uj vk ρβk (αj )

vl∗

 s(αi )∗ − s# (βl ) ui  ραi (βl ) ;  ∗ vl Lβk (βl )vk

(2.62)

(3) the negative index of D(s) is equal κ: ind− D(s) = κ.

(2.63)

p Proof. If f ∈ D(s), α ∈ h+ s and u ∈ C , then, in view of Remark 2.20, 1 0 Γ` u hf, Φα uiD(s) = h(Im + ∆s )ist )f, Γr 0 ρα s(α)∗ u

s(α)∗ u u ist + hΓr f + , ist ρα ρα = u∗ (f + − sΓr f + )(α) + u∗ s(α)(Γr f + )(α) = h(Ip − sΓr )f + ,

= u∗ f + (α).

(2.64)

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Similarly, if β ∈ h− and v ∈ Cq , then, in view of (2.54)–(2.56), s# 1 s# (β)∗ v 0 Γ` hf, Φβ viD(s) = (Im + ∆s )f, − Γr 0 v ρβ st = hf − − s∗ Γ` f − , −

v −s# (β)∗ v ist + hΓ` f − , ist ρβ ρβ

(2.65)

= v ∗ (f − − s# Γ` f − )(β) + v ∗ s# (β)(Γ` f − )(β) = v ∗ f − (β). Thus, if f ∈ D(s) is orthogonal to span {D+ , D− }, then f (µ) = 0

a.e. on Ω0 ,

thanks to (2.64) and (2.65). The entries in the matrix (2.61) are now easily calculated from the entries in the matrix (2.62) with the help of the evaluations (2.64) and (2.65) and formula (2.56). Finally, since s ∈ Sκp×q (Ω+ ) the kernel Λsω (λ) has κ negative squares on h+ s , p and hence there is a choice of αj ∈ h+ s , uj ∈ C (1 ≤ j ≤ n), such that the Gram matrix n hΦαi ui , Φαj uj iD(s) i,j=1 has exactly κ negative eigenvalues. Thus, ind− D(s) ≥ κ. On the other hand, ind− D(s) ≤ κ, by Remark 2.19. This proves (2.63). Lemma 2.22. Let s ∈ Sκp×q (Ω+ ) and let f = col (f + , f − ) ∈ D(s). Then: (1) f + ∈ K(s) and hf + , f + iK(s) ≤ hf, f iD(s) ; (2) f − ∈ K∗ (s) and hf − , f − iK∗ (s) ≤ hf, f iD(s) .

(2.66) (2.67)

Proof. Since f ∈ D(s), there exists a measurable vvf h = col(h1 , h2 ) with components h1 of height p and h2 of height q such that + f (µ) h1 (µ) − s(µ)h2 (µ) f (µ) = − = ∆s (µ)h(µ) = (Ω0 − a.e.). f (µ) −s(µ)∗ h1 (µ) + h2 (µ) The decomposition

Ip 0 0 ∆s (µ) = [I , −s(µ)] + −s(µ)∗ p 0 Iq − s(µ)∗ s(µ) implies that if f ∈ D(s), then (I − s∗ s)1/2 h2 ∈ Lq2 and 0 hf, f iD(s) = h∆s h, hist + h Γr =

kf + k2st

Γ` ∆s h, ∆s hist 0 ∗

(2.68) +

−

+ h(Iq − s s)h2 , h2 ist + 2RhΓr f , f ist .

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Let αϕ (f + ) = kf + − sΓr f + + sr ϕk2st − kΓr f + k2st − kϕk2st for ϕ ∈ H2q . Then kf + − sΓr f + + sr ϕk2st = kf + − sΓr f + k2st + ksr ϕk2st + 2Rhf + − sΓr f + , sr ϕist ,

(2.69)

and, as f − ∈ br (H2q )⊥ , Γr f + ∈ H(br ) and br ϕ ∈ br H2q , hf + − sΓr f + , sr ϕist = hh1 − sh2 , sr ϕist − hsΓr f + , sr ϕist = hs∗ h1 − h2 , br ϕist + h(Iq − s∗ s)h2 , br ϕist − hs∗ sΓr f + , br ϕist

(2.70)

= h(Iq − s∗ s)(h2 + Γr f + ), br ϕist . Therefore, αϕ (f + ) = kf + − sΓr f + k2st − kΓr f + k2st + ksr ϕk2st − kϕk2st + 2Rh(Iq − s∗ s)(h2 + Γr f + ), br ϕist = kf + k2st − h(Iq − s∗ s)Γr f + , Γr f + ist − h(Iq − s∗ s)br ϕ, br ϕist

(2.71)

+ 2Rh(Iq − s∗ s)(h2 + Γr f + ), br ϕist − 2RhsΓr f + , f + ist , and hence, hf, f iD(s) − αϕ (f + ) = h(Iq − s∗ s)h2 , h2 i + h(Iq − s∗ s)br ϕ, br ϕist + h(Iq − s∗ s)Γr f + , Γr f + ist − 2Rh(Iq − s∗ s)(Γr f + + h2 ), br ϕist + 2R hΓr f + , h2 − s∗ h1 ist + hΓr f + , s∗ (h1 − sh2 )ist

(2.72)

= h(Iq − s∗ s)(h2 + Γr f + − br ϕ), (h2 + Γr f + − br ϕ)ist ≥ 0. Therefore, by Theorem 2.10, f + ∈ K(s) and (2.66) holds. The proof of the second set of assertions is similar.

b 2.5. The space D(s) − − + p×q Let s ∈ Sκ (Ω+ ) and let the kernel Dsω (λ) be defined on (h+ s ∪ hs# ) × (hs ∪ hs# ) by the formulas  + Λsω (λ), if (λ, ω) ∈ h+  s × hs , s # ∗ s (λ) − s(ω) Dω (λ) = (2.73) + , if (λ, ω) ∈ h−  # × hs , s −ρω (λ)   s(λ) − s# (ω)∗ − , if (λ, ω) ∈ h+ s s × h s# , Dω (λ) = (2.74) ρω (λ)  − Lsω (λ), if (λ, ω) ∈ h− × h . # # s s

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It will be shown below that the kernel Dsω (λ) has a finite number of negative b squares and that the corresponding reproducing kernel Pontryagin space D(s) is unitarily equivalent to D(s). Theorem 2.23. Let s ∈ Sκp×q . Then: − (1) The kernel Dsω (λ) has a finite number of negative squares on h+ s × h s# ;

(2) The de Branges-Rovnyak space D(s) is unitarily equivalent to the reproducing b kernel space D(s) via the mapping + f b U : f = − 7→ fb ∈ D(s), (2.75) f b where fb|Ω+ is the meromorphic continuation of f + to h+ s , and f |Ω− is the − − ± meromorphic continuation of f to hs# such that f are nontangential limits of fb|Ω± from Ω± . Proof. For every choice of αi , βj , ui , vj as in (2.60) the Gram matrix in (2.61) coincides with the matrix  ∗ s  uj Dαi (αj )ui vl∗ Dsαi (βl )ui . G= u∗j Dsβk (αj )vk vl∗ Dsβk (βl )vk Therefore, the first statement of the theorem is implied by Lemma 2.21. Next, it follows from (2.73) and (2.56) that the restriction of U to the subspace D+ is given by U : Φα u 7→ Dsα u

(2.76)

and maps D+ onto b + := Ds u : α ∈ h+ , u ∈ Cp . D α s Similarly, the formulas (2.73) and (2.41) show that U maps the subspace D− onto b − := Ds v : β ∈ h−# , v ∈ Cq . D β s Moreover, the restriction of U to D+ + D− is isometric by the definition of the kernel Dsω , since hDsωj uj , Dsωk uk iD(s) = u∗k Dsωj (ωk )uj = hΦωj uj , Φωk uk iD(s) b

(2.77)

b+ +D b− for every choice of ωj , uj such as in (2.60). Since the sets D+ + D− and D b are dense in D(s) and D(s), respectively, this proves the second statement.

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3. The class Uκ (jpq ) and the basic theorem 3.1. The class Uκ (J) and the space K(W ). If W ∈ Uκ (J), then assumption (ii) in the definition of Uκ (J) guarantees that W (λ) is invertible in Ω+ except for an isolated set of points. Define W in Ω− by the formula W (λ) = JW # (λ)−1 J = JW (λ◦ )−∗ J

◦ if λ◦ ∈ h+ W and det W (λ ) 6= 0.

(3.1)

Recall [22], [28], [8], that a mvf fe of bounded type in Ω− is said to be a pseudocontinuation of a mvf f of bounded type in Ω+ , if f (ζ) = fe(ζ) a.e. on Ω0 . Since W is of bounded type both in Ω+ and in Ω− , the nontangential limits W± (µ) = ∠ lim {W (λ) : λ ∈ Ω± } λ→µ

exist a.e. on Ω0 ; and assumption (ii) in the definition of Uκ (J) implies that the nontangential limits W± (µ) coincide a.e. in Ω0 , that is W |Ω− is a pseudocontinuation of W |Ω+ . If W (λ) is rational this extension is meromorphic on C. Formula (3.1) implies that W (λ) is holomorphic and invertible in ΩW := hW ∩ hW # .

(3.2)

Let W ∈ Uκ (J) and let K(W ) be the reproducing kernel Pontryagin space W associated with the kernel KW ω (λ). The kernel Kω (λ) extended to ΩW by the equality (3.1) has the same number κ of negative squares. This fact is due to a generalization of the Ginzburg inequality [2, Theorem 2.5.2]. 3.2. Admissibility and linear fractional transformations Definition 3.1. [4] A mvf X(λ) that is meromorphic in Ω+ is said to be (Ω+ , J)κ admissible if the kernel X(λ)JX(ω)∗ ρω (λ) has κ negative squares on h+ X , the domain of holomorphy of X in Ω+ . Lemma 3.2. Let ϕ(λ) and ψ(λ) be p × p and p × q meromorphic mvf ’s in Ω+ and assume that: (i) the mvf X(λ) = ϕ(λ) ψ(λ) is (Ω+ , jpq )κ -admissible; (ii) ker X(λ)∗ = {0} for all λ ∈ h+ X. Then: (1) ϕ(λ) is invertible for all λ ∈ h+ X except for at most κ points; (2) The mvf ε(λ) = −ϕ(λ)−1 ψ(λ) belongs to Sκp×q . ∗ Proof. Let ω1 , . . . , ωn be n distinct points in h+ X such that ϕ(ωj ) uj = 0 for some p ∗ nonzero vectors uj ∈ C , j = 1, . . . , n. Since ker X(λ) = {0} this implies that

vj := ψ(ωj )∗ uj 6= 0

for j = 1, . . . , n ,

(3.3)

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and hence that the matrix n X(ωk )jpq X(ωj )∗ G = u∗k uj ρωj (ωk ) j,k=1 n ∗ ∗ ϕ(ω )ϕ(ω ) − ψ(ω k j k )ψ(ωj ) = u∗k uj ρωj (ωk ) j,k=1 ∗ n vk vj =− ρωj (ωk ) j,k=1

n = − vj /ρωj , vk /ρωk st j,k=1

(3.4)

differs in sign from the Gram matrix of a system {vj /ρωj }nj=1 of linearly independent vectors in H2p . Therefore, ν− (G) = n ≤ κ, since G has at most κ negative eigenvalues, by assumption. Suppose next that ω1 , . . . , ωn and u1 , . . . , un are chosen so that ν− (G) = κ. Then, by perturbing these points slightly, if necessary, one can insure that the matrices ϕ(ω1 ), . . . , ϕ(ωn ) are all invertible, ν− (G) = κ and n Iq − ε(ωk )ε(ωj )∗ ∗ ∗ G = uk ϕ(ωk ) ϕ(ωj ) uj . ρωj (ωk ) j,k=1 The (Ω+ , jpq )κ -admissibility of X implies that ε ∈ Sκp×q .

Remark 3.3. The assumption (ii) in Lemma 3.2 can be relaxed by invoking a version of Leech’s theorem that is valid in Pontryagin spaces; see e.g., [3] for the latter. Let TW [ε] := (w11 (λ)ε(λ) + w12 (λ))(w21 (λ)ε(λ) + w22 (λ))−1 denote the linear fractional transformation of a mvf ε ∈ the block decomposition w11 (λ) w12 (λ) W (λ) = w21 (λ) w22 (λ)

Sκp×q 2

(3.5)

(κ2 ∈ Z+ ) based on (3.6)

of a mvf W ∈ Uκ (jpq ) with blocks w11 (λ) and w22 (λ) of sizes p × p and q × q, respectively. Let ΩW be defined by (3.2) and let Λ = {λ ∈ ΩW ∩ h+ ε : det (w21 (λ)ε(λ) + w22 (λ)) = 0}. The transformation TW [ε] is well defined for λ ∈ (ΩW ∩

h+ ε )

(3.7)

\ Λ.

Lemma 3.4. Let W ∈ Uκ1 (jpq ), ε ∈ Sκp×q and let Λ be defined by (3.7). Then 2 # # Λ = {λ ∈ ΩW ∩ h+ ε : det(w11 (λ) + ε(λ)w12 (λ)) = 0}

(3.8)

and: (1) TW [ε] admits the supplementary representation # # # # (λ)) TW [ε] = (w11 (λ)+ε(λ)w12 (λ))−1 (w21 (λ)+ε(λ)w22

λ ∈ (ΩW ∩h+ ε )\Λ. (3.9)

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(2) Λ consists of at most κ1 + κ2 points and TW [ε] ∈ Sκp×q with κ0 ≤ κ2 + κ1 . 0 (3) The following equalities hold: + + ΩW ∩ h + ε ∩ hs = (ΩW ∩ hε ) \ Λ,

ΩW ∩ h − ∩ h− = (ΩW ∩ h− ) \ Λ◦ , (3.10) ε# s# ε#

Λ◦ = {λ ∈ ΩW ∩ h− : det(w11 (λ)ε(λ) + w12 (λ)) = 0} ε# # # (λ)) = 0}. = {λ ∈ ΩW ∩ h− (λ) + ε(λ)w22 : det(w21 ε#

Proof. (1) The identity [Ip

ε(λ)]W # (λ)jpq W (λ)

ε(λ) = [Ip Iq

ε(λ)]jpq

ε(λ) =0 Iq

implies that for all λ ∈ ΩW ∩ h+ ε # # (λ))(w21 (λ)ε(λ) + w22 (λ)) (λ) + ε(λ)w22 (w21 # # = (w11 (λ) + ε(λ)w12 (λ))(w11 (λ)ε(λ) + w12 (λ)).

(3.11)

Thus, if (w21 (λ)ε(λ) + w22 (λ))η = 0 for some vector η 6= 0, then ξ = (w11 (λ)ε(λ) + w12 (λ))η 6= 0,

(3.12)

since W (λ) is invertible for all λ ∈ ΩW . It follows from (3.11) that # # (w11 (λ) + ε(λ)w12 (λ))ξ = 0.

(3.13)

Therefore, # # Λ ⊆ {λ ∈ ΩW ∩ h+ ε : det(w11 (λ) + ε(λ)w12 (λ)) = 0}.

A similar argument shows that the opposite inclusion is also valid. Therefore, the two sets are equal. The identity (3.9) is immediate from (3.11). (2) Next, let X(λ) = ϕ(λ) ψ(λ) = Ip ε(λ) W # (λ). Then the formula 1 X(λ)jpq X(ω)∗ Ip Ip ε(λ) W # (λ)jpq W # (ω)∗ = ε(ω)∗ ρω (λ) ρω (λ) Ip −∗ = Ip −ε(λ) W (λ)−1 KW (λ)W (ω) + Λεω (λ) ω −ε(ω)∗ implies that X(λ) is (Ω+ , jpq )κ0 admissible with κ0 ≤ κ1 + κ2 , thanks to Theorem 2.8. Clearly, condition (ii) in Lemma 3.2 is also satisfied, and, by Lemma 3.2, # # (λ) + ε(λ)w12 (λ) ϕ(λ) = w11 0 is invertible for all λ ∈ ΩW ∩ h+ ε except for at most κ points, and the mvf s(λ) := p×q 0 TW [ε](λ) given by (3.9) belongs to Sκ0 with κ ≤ κ1 + κ2 . (3) If λ ∈ Λ then it follows from (3.12) and (3.13) that λ is a pole of s. This proves the inclusion + + ΩW ∩ h+ ε ∩ hs ⊆ (ΩW ∩ hε ) \ Λ,

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and hence the first of the equalities in (3.10), since the converse is obvious. The proof of the second one is similar. The last equality is implied by (3.11). The next theorem characterizes the range of the linear fractional transform TW in terms of admissibility; it extends a result of de Branges and Rovnyak to an indefinite setting. Theorem 3.5. Let s ∈ Sκp×q (Ω+ ) and let W ∈ Uκ1 (jpq ) with κ1 ≤ κ. Then s ∈ p×q TW [Sκ−κ1 ] if and only if Ip −s W is (Ω+ , jpq )κ−κ1 - admissible. p×q Proof. If s = TW [ε] for some ε ∈ Sκ−κ and λ ∈ ΩW ∩ h+ ε , then, in view of (3.9), 1 # # −1 # # # # Ip −s(λ) W (λ) = (w11 jpq W (λ) + εw12 ) + εw22 w21 w11 + εw12 # # # −1 (3.14) Ip ε(λ) W (λ)jpq W (λ) = (w11 + εw12 ) # # −1 Ip −ε(λ) . = (w11 + εw12 ) Therefore, the mvf Ip −s(λ) W (λ) is (Ω+ , jpq )κ−κ1 - admissible. Conversely, if the mvf Ip −s(λ) W (λ) = w11 (λ) − s(λ)w21 (λ) w12 (λ) − s(λ)w22 (λ) p×q such is (Ω+ , jpq )κ−κ1 - admissible, then, by Lemma 3.2, there is a mvf ε ∈ Sκ−κ 1 that

w12 (λ) − s(λ)w22 (λ) = −[w11 (λ) − s(λ)w21 (λ)]ε(λ). p×q Thus, s = TW [ε], i.e., s ∈ TW [Sκ−κ ]. 1

3.3. Proof of Theorem 1.1 Since the boundary values s(µ) of a mvf s ∈ Sκp×q exist a.e. on Ω0 , the mvf ∆s (µ) may be defined a.e. on Ω0 by formula (2.51) for such s. Proof.

Necessity. Let s = TW [ε] for some mvf ε ∈ Sκp×q , (κ = κ1 + κ2 ) and let 2  if λ ∈ h+  ∆+ (λ) = Ip −s(λ) , s , (3.15) ∆(λ) =  ∆− (λ) = −s# (λ) Iq , if λ ∈ h− # s

and R(λ) =

 # #  (w11 (λ) + ε(λ)w12 (λ))−1 , 

# # (λ))−1 , (ε# (λ)w21 (λ) + w22

if λ ∈ ΩW ∩ h+ ε \ Λ, (3.16) if λ ∈ ΩW ∩ h− \ Λ◦ . ε#

Then, since ∆+ (λ)W (λ) = R(λ) Ip

−ε(λ)

+ for λ ∈ ΩW ∩ h+ ε ∩ hs

(3.17)

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by (3.14), (3.17) implies that Λsω (λ) = Ip

jpq Ip −s(λ) ρω (λ) −s(ω)∗ (3.18) jpq − W (λ)jpq W (ω)∗ W (λ)jpq W (ω)∗ ∆+ (ω)∗ + ∆+ (λ) ∆+ (ω)∗ = ∆+ (λ) ρω (λ) ρω (λ) ∗ ε ∗ = ∆+ (λ)KW ω (λ)∆+ (ω) + R(λ)Λω (λ)R(ω)

+ for λ ∈ ΩW ∩ h+ ε ∩ hs .

Similarly, (3.5) implies that ∆− (λ) W (λ) = − s# (λ) Iq jpq W (λ) # −1 # # # # # # jpq W (λ) ) = −(ε# w21 + w22 ε w11 + w12 ε# w21 + w22 # # −1 # # # ε Iq W (λ)jpq W (λ) = −(ε w21 + w22 ) # = R(λ) −ε (λ) Iq for λ ∈ ΩW ∩ h− ∩ h− . ε# s#

(3.19)

Therefore, s# (λ) − s(ω)∗ = −s# (λ) −ρω (λ)

Iq

jpq Ip ρω (λ) −s(ω)∗

∗ = ∆− (λ)KW ω (λ)∆+ (ω) + R(λ)

ε# (λ) − ε(ω)∗ R(ω)∗ −ρω (λ)

(3.20)

+ and ω ∈ ΩW ∩ h+ ∩ h− for λ ∈ ΩW ∩ h− ε ∩ hs . s# ε# In much the same way, (3.19) implies that jpq −s# (ω)∗ Lsω (λ) = −s# (λ) Iq Iq ρω (λ) ∗ ε ∗ = ∆− (λ) KW ω (λ)∆− (ω) + R(λ)Lω (λ)R(ω)

(3.21)

for λ, ω ∈ ΩW ∩ h− ∩ h− . Thus, it follows from (2.73), (2.74), (3.18)–(3.21) that ε# s# ∗ ε ∗ Dsω (λ) = ∆(λ)KW ω (λ)∆(ω) + R(λ)Dω (λ)R(ω)

(3.22)

− − + for all λ, ω ∈ ΩW ∩ (h+ ε ∪ hε# ) ∩ (Hs ∪ hs# ). Applying Theorems 2.8 and 2.9 to the identity (3.22) and using the fact that the number sq− Dsω (λ) coincides with the sum of the numbers of negative squares of the kernels on the right hand side of (3.22) one obtains the following decomposition b of the space D(s) b b D(s) = ∆(λ)K(W ) + R(λ)D(ε) (3.23) b where the mapping f 7→ ∆f from K(W ) into D(s) is a contraction. The necessity part of the theorem now follows from Definition 2.17 and Theorem 2.23, since the mapping f 7→ ∆s f from K(W ) into D(s) is the composition of the mapping b b ∆ : K(W ) → D(s) and the unitary mapping U −1 : D(s) → D(s) from (2.75).

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Sufficiency. If conditions (1)–(3) of the theorem hold, then f ∈ K(W ) =⇒ ∆s f ∈ D(s)

and h∆s f, ∆s f iD(s) ≤ hf, f iK(W ) . (3.24) Moreover, in view of Lemma 2.22, Ip −s f ∈ K(s) and h Ip −s f, Ip −s f iK(s) ≤ h∆s f, ∆s f iD(s) ≤ hf, f iK(W ) . (3.25) Therefore, the operator T : f 7→ Ip −s f maps K(W ) into K(s) contractively. To find the adjoint operator T ∗ : K(s) → K(W ), p let α ∈ hW ∩ h+ s , u ∈ C and f ∈ K(W ). Then hf, T ∗ Λsα uiK(W ) = hT f, Λsα uiK(s) = h Ip −s f, Λsα uiK(s) Ip = u∗ Ip −s(α) f (α) = f, KW u . α −s(α)∗ K(W )

Therefore, T ∗ Λsα u = KW α

Ip u −s(α)∗

p (α ∈ hW ∩ h+ s , u ∈ C ).

(3.26)

Since T : K(W ) → K(s) is a contraction, Theorem 1.3.4 of [2] implies that ν− (I − T T ∗ ) = ind− K(s) − ind− K(W ) = κ2 , p and hence that for every choice of αj ∈ hW ∩ h+ s , uj ∈ C and ξj ∈ C (1 ≤ j ≤ n) the form ( ) n D E X Ip Ip s s W W Λαj uj , Λαk uk − Kαj u ,K u ξ ξ¯ −s(αj )∗ j αk −s(αk )∗ k K(W ) j k H(s) j,k=1

(3.27) p has at most κ2 (and for some choice of αj ∈ hW ∩ h+ s , uj ∈ C and ξj ∈ C exactly κ2 ) negative squares. Since (3.27) can be rewritten in the form n X u∗k Λsαj (αk )uj − u∗k Ip j,k=1 n X

=

j,k=1

u∗k Ip

Ip u ξ ξ¯ −s(αj )∗ j j k W (αk )jpq W (αj )∗ Ip −s(αk ) u ξ ξ¯ , −s(αj )∗ j j k ραj (αk ) −s(αk ) KW αj (αk )

it follows that the mvf Ip −s(λ) W (λ) is (Ω+ , jpq )κ2 –admissible. Theorem 3.5 serves to complete the proof.

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4. The resolvent matrix and the class Uκ◦ (jpq ) p×q 4.1. The class Hκ,∞ .

Let G(λ) be a p × q mvf that is meromorphic on Ω+ with a Laurent expansion G(λ) = (λ − λ0 )−k G−k + · · · + (λ − λ0 )−1 G−1 + G0 + · · ·

(4.1)

in a neighborhood of a pole λ0 ∈ Ω+ . The pole multiplicity Mπ (G, λ0 ) is defined by (see [32])   G−k 0   .. Mπ (G, λ0 ) = rank L(G, λ0 ), L(G, λ0 ) =  ... (4.2) . . G−1

...

G−k

The pole multiplicity of G over Ω+ is given by X Mπ (G, λ). Mπ (G, Ω+ ) =

(4.3)

λ∈Ω+

This definition coincides with the pole multiplicity based on the Smith-McMillan representation of G (see [11]). Remark 4.1. For a Blaschke-Potapov product b of the form (2.1) the following statements are equivalent: (1) the degree of the Blaschke-Potapov product b is equal κ; (2) Mπ (b−1 , Ω+ ) = κ; −1

b (3) the kernel Λω (λ) has κ negative squares in Ω+ . p×p The zero multiplicity of a square mvf F ∈ H∞ over Ω+ is defined in [34] as the degree of the maximal left Blaschke-Potapov factor b of F . Remark 4.1 implies p×p with nontrivial determinant that the zero multiplicity of a square mvf F ∈ H∞ −1 is connected with the pole multiplicity of F :

Mζ (F, Ω+ ) = Mπ (F −1 , Ω+ ). p×q The class Hκ,∞ (Ω+ ) consists of p × q mvf’s G of the form G = H + B, where p×q (Ω+ ) B is a rational p × q mvf of pole multiplicity Mπ (B, Ω+ ) ≤ κ and H ∈ H∞ (see [1]). The next lemma is useful for calculating pole multiplicities. r×p q×r Lemma 4.2. Let H1 ∈ H∞ , H2 ∈ H ∞ , and let mvf ’s G1 ∈ Hκp×q , G2 ∈ Hκp×q 1 ,∞ 2 ,∞ have the following Laurent expansions (j)

Gj (λ) =

(j) G−k G−1 (j) + · · · + + G0 + · · · (λ − λ0 )k λ − λ0

(j = 1, 2)

(4.4)

at λ0 ∈ Ω+ and let rng L(G1 , λ0 ) ⊆ rng L(G2 , λ0 )

∀ λ 0 ∈ Ω+ .

(4.5)

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Then Mπ (H1 G1 , Ω+ ) ≤ Mπ (H1 G2 , Ω+ ).

(4.6)

If ker L(G1 , λ0 ) ⊇ ker L(G2 , λ0 )

∀ λ 0 ∈ Ω+ ,

(4.7)

then Mπ (G1 H2 , Ω+ ) ≤ Mπ (G2 H2 , Ω+ ).

(4.8)

Proof. Let the mvf’s Hj (j = 1, 2) have the following expansions at λ0 (j)

(j)

(j)

Hj (λ) = H0 + H1 (λ − λ0 ) + · · · + Hn−1 (λ − λ0 )k−1 + · · · and let



(j)

H  0.  T (Hj , λ0 ) =  ..

(j) Hk−1

0 ..

.

...

(j) H0

   

for j = 1, 2.

Then L(H1 Gj , λ0 ) = T (H1 , λ0 )L(Gj , λ0 )

and L(Gj H2 , λ0 ) = L(Gj , λ0 )T (H2 , λ0 ).

Therefore, in view of (4.5), Mπ (H1 G1 , λ0 ) = rank T (H1 , λ0 )L(G1 , λ0 ) ≤ rank T (H1 , λ0 )L(G2 , λ0 ) = Mπ (H1 G2 , λ0 )

for λ0 ∈ Ω+ ,

which implies (4.6). Next (4.7) yields the inequality Mπ (G1 H2 , λ0 ) = rank L(G1 , λ0 )T (H2 , λ0 ) ≤ rank L(G2 , λ0 )T (H2 , λ0 ) = Mπ (G2 H2 , λ0 )

for λ0 ∈ Ω+ ,

which proves (4.8).

As a corollary we obtain the following generalization of a noncancellation lemma from [18]. q×q p×p p×q . Then and H2 ∈ H∞ , H1 ∈ H∞ Lemma 4.3. Let G ∈ Hκ,∞

Mπ (H1 G, Ω+ ) = Mπ (G, Ω+ ) =⇒ Mπ (H1 GH2 , Ω+ ) = Mπ (GH2 , Ω+ ),

(4.9)

whereas Mπ (GH2 , Ω+ ) = Mπ (G, Ω+ ) =⇒ Mπ (H1 GH2 , Ω+ ) = Mπ (H1 G, Ω+ ).

(4.10)

Proof. Since ker L(H1 G, λ) ⊇ ker L(G, λ) for every point λ ∈ Ω+ , it follows that Mπ (H1 G, λ) = rank L(H1 G, λ) ≤ rank L(G, λ) = Mπ (G, λ)

(4.11)

for every point λ ∈ Ω+ . Therefore, the first equality in (4.9) implies that equality prevails in (4.11) and so too in (4.5) and (4.7) with G1 = H1 G and G2 = G. The second equality in (4.9) then follows easily from Lemma 4.2. The verification of the second assertion is similar.

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It follows from the results of [32], [18] (see also [31, Theorem 5.2]) that every p×q mvf G ∈ Hκ,∞ (Ω+ ) admits factorizations G(λ) = G` (λ)−1 H` (λ) = Hr (λ)Gr (λ)−1 , p×p Sin ,

(4.12)

q×q Sin

where G` ∈ Gr ∈ are Blaschke-Potapov factors of degree Mπ (G, Ω+ )(≤ p×q κ) and H` , Hr ∈ H∞ are such that rank G` (λ) H` (λ) = p for λ ∈ Ω+ , (4.13) # # rank Gr (λ) Hr (λ) = q for λ ∈ Ω− . (4.14) It is easily shown that Lemma 2.1 can be extended to this more general situation. (The details are left to the reader.) p×q p×p q×q Lemma 4.4. Let H` , Hr ∈ H∞ and let G` ∈ H∞ , Gr ∈ H∞ be a pair of mvf ’s −1 p×p −1 such that G` ∈ Hκ,∞ and Gr ∈ Hκ,∞ for some κ ∈ N ∪ {0}. Then: (i) H` and G` meet the rank condition (4.13), if and only if there exists a pair p×p q×p of mvf ’s C` ∈ H∞ and D` ∈ H∞ such that

G` (λ)C` (λ) + H` (λ)D` (λ) = Ip for λ ∈ Ω+ ;

(4.15)

(ii) Hr and Gr meet the rank condition (4.14), if and only if there exists a pair q×p p×p such that and Dr ∈ H∞ of mvf ’s Cr ∈ H∞ Cr (λ)Gr (λ) + Dr (λ)Hr (λ) = Iq for λ ∈ Ω+ .

(4.16)

Pairs of mvf’s H` , G` and Hr , Gr which satisfy the assumptions of Lemma 4.4 and the conditions (4.15) and (4.16) are called left coprime and right coprime, respectively, over Ω+ . Left and right coprime factorizations may be characterized in terms of the pole multiplicities of their factors. p×q p×p q×q Proposition 4.5. Let H` , Hr ∈ H∞ and let G` ∈ H∞ and Gr ∈ H∞ be a pair −1 p×p −1 of mvf ’s such that G` ∈ Hκ,∞ and Gr ∈ Hκ,∞ for some κ ∈ N ∪ {0}. Then: −1 (i) G` and H` are left coprime over Ω+ ⇐⇒ Mπ (G−1 ` H` , Ω+ ) = Mπ (G` , Ω+ ). −1 (ii) Gr and Hr are right coprime over Ω+ ⇐⇒ Mπ (Hr G−1 r , Ω+ ) = Mπ (Gr , Ω+ ).

Proof. Suppose first that H` and G` are left coprime over Ω+ . Then it follows from (4.15) that −1 G−1 ` = G` H` D` + C` and hence, that −1 −1 Mπ (G−1 ` , Ω+ ) = Mπ (G` H` D` , Ω+ ) ≤ Mπ (G` H` , Ω+ ).

The converse inequality is obvious. This proves the implication =⇒ in (i). Next, assume that −1 0 Mπ (G−1 ` H` , Ω+ ) = Mπ (G` , Ω+ ) = κ

for some finite nonnegative integer κ0 and let G−1 and G−1 ` ` H` have the following left coprime factorizations G` (λ)−1 = b` (λ)−1 ϕ` (λ),

(4.17)

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G` (λ)−1 H` (λ) = eb` (λ)−1 ϕ e` (λ),

(4.18)

p×p Sin

0

p×q where b` , eb` ∈ are Blaschke-Potapov factors of degree κ and ϕ` , ϕ e` ∈ H∞ . Then G` admits the left coprime factorization

G` (λ) = ϕ` (λ)−1 b` (λ),

(4.19)

and it follows from the first part of the proof that Mπ (ϕ`−1 , Ω+ ) = Mπ (G` , Ω+ ) = 0. Then as kϕ` (µ)−1 k = kG` (µ)k a.e. on Ω0 , the maximum principle implies that p×p ϕ−1 ` ∈ H∞ . By Lemma 4.2, e e−1 e` , Ω+ ) = Mπ (ϕ Mπ (eb` b−1 e` , Ω+ ) = 0. ` ϕ` , Ω+ ) = Mπ (b` b` ϕ p×p p×p Since ϕ−1 ∈ H∞ , this implies that eb` b−1 ∈ H∞ which shows that b` and eb` ` ` coincide up to a constant right unitary factor. Now it follows from (4.18) and (4.19) that H` = ϕ−1 e` . Since the factoriza` ϕ tion (4.18) is also left coprime over Ω+ , e` (λ) rank G` (λ) H` (λ) = rank ϕ` (λ)−1 b` (λ) ϕ` (λ)−1 ϕ e` (λ) = p for λ ∈ Ω+ . = rank b` (λ) ϕ

This proves the implication ⇐ in (i) and completes the proof of (i). Assertion (ii) follows from (i) by passing to adjoints. In the rational case this statement can be found in [11, Theorem 11.1.4]. 4.2. A class of generalized Schur mvf ’s In this section we study the class of mvf’s s s12 S = 11 ∈ Sκm×m for which s21 ∈ Sκq×p , s21 s22

(4.20)

where the indicated block decomposition of S is conformal with jpq . Theorem 4.6. If S ∈ Sκm×m meets the constraint (4.20) and the Kre˘ın-Langer factorizations of s21 are −1 s21 = b−1 (4.21) ` s` = sr br , q×q p×p then b` s22 ∈ S and s11 br ∈ S . Proof. Let G = [s21

s22 ]

and H =

s11 . s21

Then the kernels ΛG ω (λ) = and ∆H ω (λ)

Iq − G(λ)G(ω)∗ = [0 ρω (λ)

Ip − H(ω)∗ H(λ) = [Ip = ρω (λ)

Iq ]ΛSω (λ)

0 Iq

0]∆Sω (λ)

Ip 0

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q×p have at most κ negative squares in h+ , S ⊂ Ω+ . On the other hand, since s21 ∈ Sκ the formulas s22 (λ)s22 (ω)∗ s21 ΛG ω (λ) = Λω (λ) − ρω (λ) and s11 (ω)∗ s11 (λ) s21 ∆H (λ) = ∆ (λ) − ω ω ρω (λ) H imply that ΛG ω (λ) and ∆ω (λ) have at least κ negative squares. Therefore, G ∈ q×m m×p Sκ , H ∈ Sκ and hence

Mπ (G, Ω+ ) = Mπ (s21 , Ω+ ) = Mπ (H, Ω+ ) = κ.

(4.22)

Consequently, in view of Lemma 4.2 and the factorization (4.21), Mπ (b` G, Ω+ ) = Mπ (b` s21 , Ω+ ) = Mπ (s` , Ω+ ) = 0 and hence b` G = [b` s21

m×q b` s22 ] ∈ H∞ .

Similarly, by Lemma 4.2, Mπ (Hbr , Ω+ ) = Mπ (s21 br , Ω+ ) = Mπ (sr , Ω+ ) = 0. p×p . By the maximum principle, b` s22 ∈ S q×q and s11 br ∈ Therefore, s11 br ∈ H∞ p×p S .

Corollary 4.7. If S ∈ Sκm×m meets the constraint (4.20) and its Kre˘in-Langer factorization is S = B`−1 S` = Sr Br−1 , (4.23) then [0

b` ]B`−1

∈S

q×m

and

Br−1

br ∈ S m×p . 0

Proof. Since rng L(B`−1 , λ) = rng L(S, λ) for all λ ∈ Ω+ , Lemma 4.2 implies that Mπ ([0 b` ]B`−1 , Ω+ ) = Mπ ([0 b` ]S, Ω+ ) = Mπ ([s`

ϕ2 b2 ], Ω+ ) = 0.

q×m Thus, [0 b` ]B`−1 ∈ H∞ , and hence is in S q×m by the maximum principle. Similarly, since ker L(Br−1 , λ) = ker L(S, λ) for all λ ∈ Ω+ , Lemma 4.2 implies that br s11 br −1 br , Ω+ ) = Mπ (S , Ω + ) = Mπ ( , Ω+ ) = 0, Mπ (Br 0 0 sr

which yields the second assertion.

Lemma 4.8. If S ∈ Sκm×m meets theconstraint (4.20) and its Kre˘ın-Langer fac torization is given by (4.23), and if 0 Iq B`−1 h ∈ H2q for some h ∈ H2m , then B`−1 h ∈ H2m .

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Proof. Assume that λ0 ∈ Ω+ is a pole of f = B`−1 h and that f (λ) =

f−1 f−k + ··· + + ··· k (λ − λ0 ) λ − λ0

in a deleted neighborhood of λ0

and h(λ) = h0 + · · · + hk−1 (λ − λ0 )k−1 + · · ·

in a neighborhood of λ0 .

Then   f−k   f =  ...  = L(B`−1 , λ0 )h,



 h0   where h =  ...  . f−1 hk−1 q Since [0 Iq ]f ∈ H2 , it follows that h ∈ ker L( 0 Iq B`−1 , λ0 ). Clearly, ker L( 0 Iq B`−1 , λ0 ) ⊇ ker L(B`−1 , λ0 ). (4.24) The equality (4.22) and Remark 4.1 imply that Mπ ( s21 s22 , Ω+ ) = Mπ (B`−1 , Ω+ ) = κ and, since rank L( s21

s22

, λ0 ) ≤ rank L( 0 Iq B`−1 , λ0 ) ≤ rank L(B`−1 , λ0 ), (4.25)

equality holds in (4.25) and (4.24). Thus, f = L(B`−1 , λ0 )h = 0, and hence B`−1 h ∈ H2m . −1 Theorem 4.9. If s ∈ Sκq×p with Kre˘in-Langer factorizations s = b−1 ` s` = sr br , then: q×q p×q p×p p×q (1) There exists a set of mvf ’s c◦` ∈ H∞ , d◦` ∈ H∞ , c◦r ∈ H∞ and d◦r ∈ H∞ such that b` c◦` + s` d◦` = Iq and c◦r br + d◦r sr = Ip . (4.26) p×q p×p p×q q×q are solutions to and dr ∈ H∞ , cr ∈ H∞ , d ` ∈ H∞ (2) The mvf ’s c` ∈ H∞ the equations in (4.26) if and only if

c` = c◦` + sr ψ

and

d` = d◦` − br ψ

p×q for some ψ ∈ H∞

(4.27)

cr = c◦r + φs`

and

dr = d◦r − φb`

p×q for some φ ∈ H∞

(4.28)

and q×q p×q p×p p×q (3) There exists a set of mvf ’s c` ∈ H∞ , d ` ∈ H∞ , cr ∈ H∞ and dr ∈ H∞ such that cr dr br −d` Ip 0 = . (4.29) −s` b` sr c` 0 Iq

Moreover, this set of mvf ’s satisfies the auxiliary equalities br cr + d` s` = Ip , sr cr − c` s` = 0 and

br dr − d` b` = 0, sr dr + c` b` = Iq .

(4.30) (4.31)

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Proof. The first assertion is immediate from Lemmas 2.1 and 2.2. If also b` c` + s` d` = Iq , then b` (c` − c◦` ) + s` (d` − d◦` ) = 0. ◦ ◦ Thus, if ψ = −b−1 r (d` − d` ), it is readily seen that c` − c` = sr ψ and hence that −c◦r (d` − d◦` ) + d◦r (c` − c◦` ) = c◦r br ψ + d◦r sr ψ = ψ. p×q Therefore, ψ ∈ H∞ . Similar considerations serve to justify the necessity of (4.28). The sufficiency is self-evident. Next, in view of (4.27) and (4.28) the matrix product on the left of (4.29) can be expressed as ◦ cr dr br −d` Ip −φ cr d◦r br −d◦` Ip ψ Ip Z = = , −s` b` sr c` 0 Iq −s` b` sr c◦` 0 Iq 0 Iq

where Z = −φ − c◦r d◦` + d◦r c◦` + ψ. Formula (4.29) is obtained by choosing φ and ψ to make Z = 0. The final set of formulas (4.30) and (4.31) follow by reversing the order of multiplication on the left hand side of formula (4.29). The formulation (4.29) was motivated by the discussion of the Bezout identity, (3) in Section 4.1 of [26]. 4.3. Associated pairs. If W belongs to the class Uκ◦ (jpq ) which is defined by (1.8), then the PotapovGinzburg transform S = P G(W ) meets the constraints imposed in the preceding subsection. In this case, the class of associated pairs of W are defined as the inner factors in the factorizations in (1.10). Theorem 4.10. Let W ∈ Uκ◦ (jpq ), and let {b1 , b2 } ∈ ap(W ). Then W can be expressed in terms of the factors in (1.10) as −∗ ∗ b1 0 ϕ1 0 w11 w12 br −s∗r W = = a.e. in Ω0 . (4.32) w21 w22 −s` b` 0 b−1 0 ϕ−1 2 2 Moreover, Ip − s∗r sr = ϕ∗1 ϕ1

and

Iq − s` s∗` = ϕ2 ϕ∗2

a.e. in Ω0 .

Proof. The asserted identities follow easily from the formulas # w11 br −1 w21 w22 = b−1 −s` ϕ−1 b−1 and # = 2 ϕ2 −sr 1 1 w12 and the fact that S # S = SS # = Im in hS ∩ hS # .

b`

(4.33)

(4.34)

Theorem 4.11. Let W ∈ Uκ◦ (jpq ), let {b1 , b2 } ∈ ap(W ), and let cr , dr , c` and d` be as in Theorem 4.9 (3) for s = s21 and let K = (−w11 d` + w12 c` )(−w21 d` + w22 c` )−1 . Then K belongs to

p×q H∞

(4.35)

and admits the representation K = (−w11 d` + w12 c` )ϕ2 b2 ,

(4.36)

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35

and the dual representation # # K = b1 ϕ1 (cr w21 − dr w22 ).

Moreover, if spondingly

c◦`

and

d◦`

(4.37)

is a second set of mvf ’s such that (4.26) holds and corre-

K ◦ = (−w11 d◦` + w12 c◦` )(−w21 d◦` + w22 c◦` )−1 ,

(4.38)

p×q (b1 ϕ1 )−1 (K − K ◦ )(ϕ2 b2 )−1 ∈ H∞ .

(4.39)

then There is a choice of mvf ’s c` and d` , such that the corresponding mvf K admits a pseudocontinuation of bounded type in Ω− . Proof. The formula S = P G(W ) implies that w11

w12 = s11

s12

Ip w21

0 . w22

Therefore, since (−w21 d` + w22 c` )ϕ2 b2 = Iq by (4.34), it is readily seen that K = (−w11 d` + w12 c` )ϕ2 b2 Ip 0 −d` = s11 s12 ϕ2 b2 = s11 w21 w22 c`

−d` ϕ2 b2 s12 Iq

and hence that −d` ϕ2 b2 K S = . Iq c` ϕ2 b2

(4.40)

Consequently, −d` ϕ2 b2 [0 Iq ]S h = c` ϕ2 b2 h Iq belongs to H2q for every h ∈ H2q . Thus, Lemma 4.8 implies that [Ip

0]Sh = Kh

p×q belongs to H2p for every h ∈ H2q . Therefore, K ∈ H∞ . The identity −d` −d` cr −dr W # jpq W c −d = r =0 r jpq c` c`

implies that # # # # − dr w22 )(−w21 d` + w22 c` ), (cr w11 − dr w12 )(−w11 d` + w12 c` ) = (cr w21

and hence that K admits the dual representation # # # # # −1 # ) (cr w21 − dr w22 ) = b1 ϕ1 (cr w21 − dr w22 ) K = (cr w11 − dr w12

which coincides with (4.37).

(4.41)

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It follows from (4.36), (4.38), (4.34), (4.27) and (4.33) that −(d` − d◦` ) br −# # ◦ # br −sr K − K = w11 w12 ϕ2 b2 = b1 ϕ1 ψϕ2 b2 c` − c◦` sr # # = b1 ϕ−# 1 (br br − sr sr )ψϕ2 b2 = b1 ϕ1 ψϕ2 b2

on Ω0 .

This justifies (4.39). Finally, to verify the last statement, first note that, in view of (1.10), formula (4.36) can be rewritten as K = (−w11 d` + w12 c` )b` s22 . Therefore, since the mvf’s w11 , w12 , b` , s22 admit pseudocontinuations of bounded type in Ω− , it remains only to show that there is a choice of mvf’s c` and d` of the form (4.27) that admit pseudocontinuations to Ω− . If c◦` and d◦` is a a fixed pair of mvf’s that satisfy (4.26) and Ω+ = D, then ◦ b−1 r d ` = ψ + + ψ− , p×q with ψ+ ∈ H2p×q , and ψ− ∈ L∞ ∩ (H2p×q )⊥ , since br is a finite Blaschke-Potapov product. Therefore, p×q p×q ψ+ ∈ L∞ ∩ H2p×q ⊂ H∞ ,

by the maximum principle and hence the particular choice d` = d◦` − br ψ+ = br ψ− admits a pseudocontinuation to Ω− by a matrix version of the Douglas-ShapiroShields condition [22], due to Fuhrmann [29, Theorem 1]. 4.4. Factorization of the resolvent matrix Theorem 4.12. Let W ∈ Uκ◦ (jpq ), {b1 , b2 } ∈ ap(W ) let K be defined as in Theorem 4.11, and let cr , dr , c` and d` be as in Theorem 4.9 (3). Then W admits the factorizations eΦ e in Ω− , W = Θ Φ in Ω+ and W = Θ (4.42) where b1 0 b1 Kb−1 2 e Θ= in Ω+ , Θ = in Ω− , K # b1 b−1 0 b−1 2 2 ϕ1 0 ϕ11 ϕ12 cr dr Φ= = in Ω+ , ϕ21 ϕ22 −s` b` 0 ϕ−1 2 −# # br −s# e11 ϕ e12 ϕ 0 r e= ϕ Φ = 1 in Ω− ϕ e21 ϕ e22 d# c# 0 ϕ# ` ` 2

(4.44)

e # jpq Θ = Φ e # jpq Φ = jpq in Ω+ . Θ

(4.46)

(4.43)

(4.45)

and

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37

Moreover, br −d` ϕ−1 0 1 in Ω+ , sr c` 0 ϕ2 # # 0 s# ϕ1 ` e −1 = cr in Ω− , Φ 0 ϕ−# −d# b# r 2 ` K −1 −d` m×q W −1 b2 = ϕ2 ∈ H ∞ , Iq c` # Ip cr m×p W −1 b = ϕ# (Ω− ) 1 ∈ H∞ K# 1 −d# r Φ−1 =

(4.47)

(4.48)

and S = P G(W ) admits the representations s11 s11 d` ϕ2 b2 + K S= s21 s21 d` ϕ2 b2 + c` ϕ2 b2 b1 ϕ1 cr + b1 ϕ1 dr s21 K + b1 ϕ1 dr s22 = . s21 s22

(4.49)

Proof. The evaluations w11 br + w12 sr = s11 br = b1 ϕ1 , w21 br + w22 sr = (w21 + w22 s21 )br = 0, −1 −w11 d` + w12 c` = Kb−1 2 ϕ2 , −1 −w21 d` + w22 c` = b−1 2 ϕ2

lead easily to the formula br −d` b1 ϕ1 Kb−1 ϕ−1 2 2 W = ; −1 sr c` 0 b−1 2 ϕ2

ϕ i.e., W = Θ 1 0

0 ϕ−1 2

br sr

−d` c`

−1 .

Formula (4.44) for Φ in (4.42) is easily verified with the help of Theorem 4.9. The second factorization formula follows from the first and the observation that W = jpq (W # )−1 jpq ,

e = jpq (Θ# )−1 jpq Θ

e = jpq (Φ# )−1 jpq . and Φ

Moreover, the first formula in (4.49) is equivalent to formula (4.40). The second follows by much the same sort of manipulations: # # # −1 s11 = b1 ϕ1 (cr w11 − dr w12 )(w11 ) = b1 ϕ1 cr + b1 ϕ1 dr s21

and # # # −1 # ) w21 s12 = b1 ϕ1 (cr w11 − dr w12 )(w11 # # + dr s22 ) = K + b1 ϕ1 dr s22 . = b1 ϕ1 (cr w21 − dr w22

Finally, formulas (4.42), (4.43) and (4.47) imply that 0 −d` −1 −1 K −1 −1 0 W b =W Θ =Φ = ϕ2 Iq 2 Iq Iq c`

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and W

−1

# Ip cr −1 e Ip −1 Ip e b =W Θ =Φ = ϕ# 1 0 0 K# 1 −d# r

which serves to justify (4.48)

Corollary 4.13. In the setting of Theorem 4.12, ϕ 0 W H2m = Θ 1 H2m . 0 ϕ−1 2

(4.50)

Proof. Theorem 4.9 and Theorem 4.12 imply that ϕ 0 m×m W = Θ Φ◦ E where Φ◦ = 1 and E ±1 ∈ H∞ . 0 ϕ−1 2 Therefore, W H2m = ΘΦ◦ EH2m ⊆ ΘΦ◦ H2m = ΘΦ◦ EE −1 H2m ⊆ ΘΦ◦ EH2m . Thus, equality must prevail throughout.

Corollary 4.14. Let W ∈ Uκ◦ (jpq ) and the Kre˘ın-Langer factorizations of S = P G(W ) are S(λ) = B`−1 (λ)S` (λ) = Sr (λ)Br−1 (λ), then, in the setting of Theorem 4.11, K − s12 s# m×q # m×p 11 B` b−1 ∈ H , B b1 ∈ H ∞ (Ω− ). ∞ r 2 # −s22 s# 12 − K

(4.51)

Proof. The relations in (4.51) are implied by (4.49), since K − s12 s11 0 −1 m×q B` b2 = −B` d ` ϕ2 − B ` ∈ H∞ −s22 s21 c` ϕ2 and Br#

s# 11 s# − K# 12

b1 = Br#

−s# 21 −s# 22

# # d# r ϕ1 + B r

# c# r ϕ1 0

m×p ∈ H∞ (Ω− ).

Corollary 4.15. If, in the setting of Theorem 4.12, p×p p×q q×p q×q ϕ e11 ∈ H∞ (Ω− ), ϕ e12 ∈ H∞ (Ω− ), ϕ21 ∈ H∞ (Ω+ ) and ϕ22 ∈ H∞ (Ω+ ),

then: (1) the mvf w11

(2) the mvf w21

w12 admits a right coprime factorization over Ω− w11 w12 = b1 ϕ e11 ϕ e12 ; w22 admits a left coprime factorization over Ω+ w21 w22 = b−1 ϕ21 ϕ22 . 2

(4.52)

(4.53)

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4.5. Characterization of K(W ) spaces. The next theorem characterizes K(W ) spaces in terms of the Kre˘ın-Langer factorizations of S = P G(W ). Theorem 4.16. If W ∈ Uκ (jpq ) and the Kre˘ın-Langer factorizations of S = P G(W ) is S(λ) = B` (λ)−1 S` (λ) = Sr (λ)Br (λ)−1 and if h1 , h2 is a pair of measurable vvf ’s on Ω0 of height p and q, respectively, then h = col(h1 , h2 ) ∈ K(W ) if and only if: ∗ Ip −s12 s11 0 m ∗ B` h ∈ H2 and Br h ∈ (H2m )⊥ . 0 −s22 s∗12 −Iq Moreover, in this case D E e ` (S ∗ f ) hh, hiK(W ) = kf k2st − 2< f, Γ ,

(4.54)

st

e ` (S ∗ f ) are defined by the formulas (cf. (2.12)) where f ∈ K(Br ), S ∗ f and Γ ∗ Ip −s12 s11 0 f := h, S ∗ f = h, (4.55) 0 −s22 s∗12 −Iq e ` (S ∗ f ) := X e −1 P+ (S ∗ f ) Γ `

and

e` : g ∈ H∗ (B` ) −→ P+ S ∗ g ∈ H(Br ). X

Proof. The formula for the inverse Potapov-Ginzburg transform −1 Ip −s12 (λ) s11 (λ) 0 W (λ) = 0 −s22 (λ) s21 (λ) −Iq leads to the following representation of the kernel KωW (λ) −1 Im − S(λ)S(ω)∗ Ip Ip −s12 (λ) KωW (λ) = 0 0 −s22 (λ) ρω (λ) This identity implies that the mapping Ip h 7→ f = 0

−s12 (λ) −s22 (λ)

−s12 (ω) −s22 (ω)

−∗ .

h

is an isometry from K(W ) onto K(S) (cf. [7]). Since S` is a square inner mvf, Corollary 2.16 guarantees that the inclusion f ∈ K(S) is equivalent to the inclusion S ∗ f ∈ K∗ (S). Now the first statement of the theorem is implied by Theorem 2.14. To verify formula (4.54), consider the orthogonal decomposition of the vector f ∈ K(S) corresponding to the fundamental decomposition of K(S) (see (2.46)) f = B`−1 y + x,

y ∈ H(S` ), x ∈ H∗ (B` ),

(4.56)

where x ∈ H∗ (B` ) is the unique solution of the equation P+ S`∗ x = P+ S ∗ f (= PH(Br ) S ∗ f ).

(4.57)

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In the notation of Definition 2.4, e ` (S ∗ f ) = X e −1 P+ S ∗ f. x=Γ `

(4.58)

Now (4.56) yields the formula hf, f iK(S) = kf − xk2st − kxk2st = kf k2st − hf, xist − hx, f ist , which is equivalent to (4.54).

e ` = 0 and the statement of Remark 4.17. If κ = 0, then B` = Br = Im , Γ Theorem 4.16 reduces to the characterization of H(W ) spaces given in Theorem 2.4 of [24] and the next theorem reduces to Theorem 2.7 of [24] (see also [9] for another proof of the latter). Theorem 4.18. Let W ∈ Uκ◦ (jpq ), S = [sij ]2i,j=1 be the Potapov-Ginzburg transform p×q of W , let {b1 , b2 } be an associated pair of W , and let the mvf K ∈ H∞ be defined as in Theorem 4.11. Then h = col (h1 , h2 ) ∈ K(W ) if and only if: (1) b` s22 h2 ∈ H2q ; (2) b∗r s∗11 h1 ∈ (H2p )⊥ ; (3) Ip −K h ∈ H2p ; (4) −K ∗ Iq h ∈ (H2q )⊥ . Proof. Necessity. Let h = col(h1 , h2 ) ∈ K(W ). Then by Theorem 4.16 ∗ Ip −s12 s11 0 B` h ∈ H2m , Br∗ h ∈ (H2m )⊥ 0 −s22 s∗12 −Iq and hence, by Corollary 4.7 and the formulas in (1.10), Ip −s12 0 b` h = −b` s22 h2 = −ϕ2 b2 h2 ∈ H2q , 0 −s22

(4.59)

(4.60)

and

b∗r

0

s∗11 s∗12

0 −Iq

h = b∗r s∗11 h1 = ϕ∗1 b∗1 h1 ∈ (H2p )⊥ .

(4.61)

The first condition in (4.59) can be rewritten as h1 − Kh2 K − s12 B` + B` h2 ∈ H2m , 0 −s22 where due to (4.49) B`

K − s12 −s22

h2 = B `

−s11 d` ϕ b h . −s21 d` − c` 2 2 2

Thus, as s m×q B` 11 ∈ H∞ s21

and ϕ2 b2 h2 ∈ H2q ,

(4.62)

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it is readily seen that B`

K − s12 −s22

h2 ∈ H2m .

Therefore,

h1 − Kh2 ∈ H2m , 0 which in view of Lemma 4.8 implies (3). Similarly, since ∗ ∗ ∗ ∗ 0 0 ∗ s11 ∗ s21 dr + cr ∗ ∗ ∗ Br ∗ h = Br ϕ1 b 1 h1 + B r s12 −Iq s∗22 d∗r K∗ B`

0 −Iq

h1 ∈ (H2m )⊥ h2

and ϕ∗1 b∗1 h1 ∈ (H2p )⊥ , it follows readily that 0 0 h1 ∗ Br ∈ (H2m )⊥ , K ∗ −Iq h2 which justifies (4) with the help of a self-evident version of Lemma 4.8. Sufficiency. Let h = col(h1 , h2 ) satisfy assumptions (1)–(4). Then h1 − s12 h2 h1 − Kh2 K − s12 B` = B` + B` h2 ∈ H2m , −s22 h2 0 −s22 thanks to assumptions (1) and (3) and formula (4.62), whereas s∗11 h1 0 s∗11 ∗ ∗ Br∗ = B + B h1 ∈ (H2m )⊥ r r s∗12 h1 − h2 K ∗ h1 − h2 s∗12 − K ∗ by much the same sort of arguments. Thus, Theorem 4.16 guarantees that h ∈ K(W ). 4.6. Description of K(W ) ∩ Lm 2 . In this subsection we supply a description of the space K(W ) ∩ Lm 2 analogous to the description of the space H(W ) ∩ Lm that is presented in Section 5.14 of [8]. 2 The main formulas (4.63)–(4.68) look the same as their counterparts in the Hilbert space setting and the proofs of all but one of them are also the same; only the verification (4.66) requires a different argument. Theorem 4.19. Let W ∈ Uκ◦ (jpq ), {b1 , b2 } ∈ ap(W ), let K be defined by (4.35) and let Γ11 : f ∈ H2q −→ PH(b1 ) Kf, Γ22 : f ∈ H∗ (b2 ) −→ P(H2p )⊥ Kf, and Then:

Γ12 : f ∈ H∗ (b2 ) −→ PH(b1 ) Kf.

u1 : u ∈ H(b ) , 1 1 Γ∗11 u1 Γ22 u2 K(W ) ∩ (H2m )⊥ = : u2 ∈ H∗ (b2 ) , u2 K(W ) ∩ H2m =

m ˙ m ⊥ K(W ) ∩ Lm 2 = (K(W ) ∩ H2 )+(K(W ) ∩ (H2 ) )

(4.63) (4.64) (4.65)

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and hh, hiK(W ) = Moreover, if I h= Γ∗11

Γ22 I

−K h, h Iq st

Ip −K ∗

u,

with

for every h ∈ K(W ) ∩ Lm 2 .

u1 u2

u=

(4.66)

,

u1 ∈ H(b1 )

and

u2 ∈ H∗ (b2 ),

then hh, hiK(W ) = hP u, uist ,

(4.67)

where P =

I − Γ∗11 Γ11 −Γ∗12

−Γ12 I − Γ∗22 Γ22

(4.68)

Proof. Let h = col(h1 , h2 ) ∈ K(W ) ∩ Lm 2 . Then, in view of (1) and (2) of Theorem 4.18 and the identities b` s22 = ϕ2 b2 and s11 br = b1 ϕ1 , it is readily checked p − + that b2 h2 ∈ H2q and b∗1 h1 ∈ (H2p )⊥ . Thus, if h1 = h+ 1 + h1 with h1 ∈ H2 , p q q + − + − ⊥ ⊥ h− 1 ∈ (H2 ) , and h2 = h2 + h2 with h2 ∈ H2 and h2 ∈ (H2 ) , then the condition p ⊥ + b∗1 h1 ∈ (H2p )⊥ =⇒ b∗1 h+ 1 ∈ (H2 ) =⇒ h1 ∈ H(b1 ), whereas the condition q − b2 h2 ∈ H2q =⇒ b2 h− 2 ∈ H2 =⇒ h2 ∈ H∗ (b2 ).

Next, (3) and (4) of Theorem 4.18 imply that p − h− 1 − Kh2 ∈ H2

and

q ⊥ + − K ∗ h+ 1 + h2 ∈ (H2 ) .

Therefore, − − h− 1 = P− Kh2 = Γ22 h2

∗ + ∗ + and h+ 2 = P+ K h1 = Γ11 h1 .

∗ Since h ∈ Lm 2 and S S = Im a.e. on Ω0 ,

2

Ip −s12

Ip

= h ∗

0 −s22

−s 12 st

−s12 Iq

and formula (4.54) can be expressed as Ip −s12 Ip − 2< hh, hiK(W ) = h, h −s∗12 Iq 0 st

h, h st

−s12 −s22

h, x , st

where x ∈ H∗ (B` ) is the unique solution of (4.57). Therefore, since K = s12 + s11 d` ϕ2 b2 by (4.49), Ip −K hh, hiK(W ) = h, h + 2 0 or formally κ = ∞. Using the perturbation techniques from [14, 16, 17, 18] we prove (among other results) in the case of a regular family of type (1.1) (for a definition see Section 1.2 below) the estimates |λκi − λ∞ ub1 lb i | ≤ ≤ 2 (1.4) ∞ 2 κ λi κ ub2 lb hκ [vi∞ − viκ ] ≤ 2 ≤ (1.5) κ 2 κ hκ [vi ] κ ub3 , D ∈ R \ spec(H∞ ), (1.6) Eκ (D) − E∞ (D) ≤ κ and we compute the constants lb and ubi , i = 1, 2, 3 explicitly for several concrete model problems. Further, we also give a formula for determining a critical κ0 such that (1.4)–(1.6) hold for κ ≥ κ0 and we show that the estimates are optimal in the sense that |λκ − λ∞ | lb −1 =1 lim i ∞ i κ→∞ λi κ2 holds.

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Relative Convergence Estimates

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To show that our abstract approach to problems (1.1) does not incur accuracy tradeoffs when applied to concrete problems, we consider several case studies. A prototype for the (less trivial) regular problem is the Arch Model from e.g. [7, Chapter 8.8:3]. In our case study we compute explicit estimates for the asymptotic behavior of the eigenvalues and spectral projections of the low frequency problem as the diameter of the arch goes to zero. The limit of such a family of arches is the so called Curved Rod Model from [20, 33]. On the other hand, Schr¨ odinger (like) operators from [2, 6, 10] are representative for (higher dimensional) operators which have less “well-behaved” spectral asymptotic. More to the point, in the case of the Schroedinger (like) operator from (1.3) we obtain the optimality same statements, but the convergence is of the fractional order O κ12α , α = 12 ( cf. [10, 14] for higher dimensional problems in unbounded domains). These concrete examples determine a framework for presenting our (otherwise) more abstract results. 1.1. Local (resolvent) estimates We approach this analysis by reformulating the convergence problem so that the perturbation framework and the error representation formulae (this is the main constructive feature of our framework) from [11, 14, 16, 17, 18] can be applied as a backbone of our construction. A difference between our approach and the standard results of works like [6, 8, 10, 27] can best be seen when considering a way to compute a constant ub3 for an estimate like (1.6). The standard approach requires a study of the integral (ζ − H∞ )−1 PN(he ) − (ζ − Hκ )−1 dζ, (1.7) C(λ∞ i )

∞ where C(λ∞ i ) is a circle in the resolvent set of Hκ which has λi in its interior and the rest of the spectrum in its exterior. This frequently leads to cumbersome estimation formulae. Thanks to the local character of the error representation formula from [14], we are able to base our theory on a study of the integrals1 ∞ ∞ ∞ −1 ∞ −1 ∞ 2 (vi∞ , H−1 v ) − (v , H v ) = H1/2 (1.8) κ i i ∞ i e Hτ vi dτ, i = 1, . . . , m. κ2

Here He is the operator defined by he in the sense of Kato and m ∈ N is the multiplicity of λ∞ i . The results from [3, 4, 10] show that the integrals (1.8) are better amenable for a quantitative study than are (1.7). In fact, in [3] the authors give necessary and sufficient conditions that guarantee that the righthand side of

> 0 is some κ independent constant. For

2 , where ub (1.8) is dominated by ub/κ further comments on the rate of convergence in (1.8) see Remark 4.5. 1 The

notation (·, ·) and · always refer to the scalar product and the norm of the background 1/2 Hilbert space H. The functions of the operator like He are always meant in the sense of the spectral calculus. By PN(h e ) we generically denote the H orthogonal projection onto the space N(he )

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Due to the difficulties in dealing with a formula like (1.7), typical results from semiclassical analysis from e.g. [8, 29] establish only the fact that the projections converge in a much weaker sense (than is the convergence of spectral projections in norm) without giving information on the speed of convergence as measured by the coupling κ2 . The nearest in spirit to our analysis is the approach of [27]. However, in this work only a particular family of model problems is considered and no estimates for the convergence of Eκ (·) in (unitary invariant) operator norm(s) are presented. Furthermore, the authors do not discuss the radius of convergence of their “asymptotic” expansions. For the geometric theory on the relationship between two projections and the importance of establishing convergence estimates for all unitary invariant operator norms we refer the reader to the seminal works [9, 19]. 1.2. A notion of regularity Let us now make precise what we mean by the regularity of he . In the terminology of [29] a family of the type (1.1) is said to be non-inhibited stiff if he is a closed and positive quadratic form and the subspace N(he ) := {u ∈ Q(he ) : he [u] := he (u, u) = 0}

(1.9)

(of H) is nontrivial. For technical convenience we assume ( without reducing the level of the generality) that hb is positive definite and use Hb and He to denote the self-adjoint operators which are defined in the sense of Kato by hb and he respectively. We identify the regular family of quadratic forms–with structure (1.1)–by requiring that hb and he satisfy a Ladyzhenskaya-Babuˇska-Brezzi type condition 1/2

|(q, He v)| 1 ≥ PN(he ) q, 1/2 k hb [v] v∈Q(he ) sup

q ∈ H,

(1.10) 1/2

−1/2

for some k > 0. The condition (1.10) is equivalent to the claim that R(He Hb ), 1/2 −1/2 the range of the operator He Hb , is closed in H, cf. examples (1.2) and (1.3). The ramifications of the assumption (1.10) will enable us to formulate a new method for studying integrals of the type (1.8) for this class of model problems and thus complement the study of singular obstacle potentials from [3, 4, 10], cf. Remark 4.5. 1.3. An outline of the paper Let us finish the introduction by briefly outlining the structure of the paper. In Section 2 we introduce the notation and present the qualitative convergence framework from [36]. The main approximation results of the paper appear in Section 3. To be more precise in Section 3.1 we review the operator matrix approach to Ritz value estimation from [18, 14]. In Section 3.2 this approach to spectral estimation is specialized to the problems of the large coupling limit. In particular we make precise in which sense these estimates can be considered sharp. We also revisit, in Section 3.2.1, the example from [14] to show how do (1.4)-(1.6) look in practice

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for a non-regular he . In Section 4 we characterize regular perturbations he and give convergence estimates which utilize this additional structural information. In Section 5 we consider a model problem from the elasticity theory and show that its asymptotic behavior is regular. In the last section we put the results in the broader context and give an outlook of further research. At the end we would like to emphasize that our study is distinguished by its constructive character. This can be seen in the fact that we give a general method to compute the constants lb and ubi , i = 1, 2, 3 (as functions of Hκ and vi∞ ) in (1.4)-(1.6). With such a result we give a method to establish both a first order correction for the limit eigenvalue λ∞ i , as well as to assess the quality of this approximation to λκi . The optimality result is a justification for this claim. For other connections between elementary linear algebra and spectral theory we refer the reader to [32].

2. Convergence of non-densely defined quadratic forms In this section we fix the notation and give background information on the previous results which we use. We follow the general notational conventions and the terminology of Kato [21, Chapters VI-VIII]. Minor differences are contained in the following list of notation and terminology. • H is an infinite dimensional Hilbert space, real or complex. • (·, ·) is the scalar product on H, linear in the second argument and anti-linear (when H is complex) in the first. • · = (·, ·)1/2 is the norm on H. spaces H1 and H2 . For any x ∈ • H1 ⊕ H2 is the direct sum of the Hilbert

x1 H1 ⊕ H2 we have x = x1 ⊕ x2 = , with xi ∈ Hi , i = 1, 2. x2 • spec(H), specess (H) is the spectrum and the essential spectrum of H, respectively. • λess (H) is the infimum of the essential spectrum of H. • A ≤ B is an order relation between self-adjoint operators (matrices). It is equivalent to the statement that B − A is positive. • L(H), L(H1 , H2 ) are the spaces of bounded linear operators on H and from H1 to H2 , respectively, equipped with the norm · . • R(X), N(X) are the range and the null space of the linear operator X, respectively. • A† is the generalized inverse of the closed densely defined operator A. If A has closed range, then A† = (A(A∗ A)−1 )∗ is bounded, see [25]. We will extend this notion below to hold for non-densely defined self-adjoint operators. • P , P⊥ are orthogonal projections, P⊥ := I − P . • j(·) is a permutation of N. • diag(M, W ) is the block diagonal operator matrix with the operators M, W on its diagonal. The operators M, W can be both bounded and unbounded.

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•

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The same notation is used to define the diagonal m × m matrix diag(α1 , . . . , αm ), with α1 , . . . , αm on its diagonal. s1 (A) ≥ s2 (A) ≥ · · · are the singular values of the compact operator A, ordered in descending order according to multiplicity. smax (A), smin (A) are the minimal (if it exists) and the maximal singular value of A, respectively. ||| X ||| denotes a unitary invariant or operator cross norm of the operator X. Since ||| · ||| depends only on the singular values of the operator, we do not notationaly distinguish between the norm ||| · ||| on L(H), L(R(P )), L(R(P ), R(P )⊥ ), or such. For details see [31]. tr(X), ||| X |||HS are the trace and the Hilbert-Schmidt norm of the operator X, respecively. ||| X |||HS = tr(X ∗ X), see [31].

As a general policy to simplify the notation we shall always drop indices when there is no danger of confusion. Let us assume that we have a closed, symmetric form h, semibounded from below and with dense domain Q(h) ⊂ H, as given in [21, (VI.1.5)-(VI.1.11), pp. 308-310]. A form h which has a strictly positive lower bound will be called positivedefinite. This is also a small departure from the terminology of [21, Section VI.2, pp. 310]. Such an h defines a self-adjoint and positive definite operator H in the sense of [21, Theorem VI.2.23, pp. 331]. Furthermore, the operator H is densely defined with the domain D(H) ⊂ Q(h) and D(H1/2 ) = Q(h). We also generically assume that H has discrete eigenvalues λ1 (H) ≤ · · · ≤ λm (H) ≤ · · · < λess (H), where we count the eigenvalues according to multiplicity. Another departure from the terminology of Kato is that we use h(ψ, φ) to denote the value of h on ψ, φ ∈ Q(h), but we write h[ψ] := h(ψ, ψ) for the associated quadratic form h[·]. We also emphasize that we use ·∗ to denote the adjoint both in the real as well as in the complex Hilbert space H as is customary in [21, Chapters VI-VIII]. In order to be able to handle the problems of the type (1.1), we shall need to work with operators that are not necessarily densely defined, cf. (1.2) and (1.3). We use the notion of the pseudo inverse of the operator H that is assumed to · ⊂ H (tacitly be self-adjoint in the closure of its domain of definition D(H) assumed to be a non-trivial subspace). A definition from [36] will be used. The defined by pseudo inverse of the operator H is the self-adjoint operator H = R(H) ⊕ D(H)⊥ , D(H) + v) = H−1 u, H(u

u ∈ R(H), v ∈ D(H)⊥ .

is bounded if and only if R(H) is closed = H−1 in R(H) · and H It follows that H · the operator H is obviously self-adjoint, in H. When considered solely in D(H) so we can also use the spectral calculus from [30] to define the generalized inverse,

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which extends the definition from the case of the densely defined operator, as 0, λ = 0, H† = f (H), f (λ) = 1 λ , λ > 0, D(H† ) = u ∈ H : f 2 (λ)d(E(λ)u, u) < ∞ , ⊕ N(H) = D(H† ) and the where E(·) = EH (·)PD(H) . Obviously, we have D(H) † 1/2 u ∈ D(H ), holds. In what follows we shall tacitly drop identity H u = Hu, the notational distinction between the generalized and pseudo inverse. The usual monotonicity properties can be extended to the generalized inverse. In particular 1/2 ). u, u ∈ D(H 1 (2.1) This monotonicity principle is the main ingredient of the proof of the convergence result for (1.1). When dealing with non-densely defined forms this principle can be formulated as follows. Let h1 and h2 be two closed positive definite forms and let H1 and H2 be the self-adjoint operators defined by h1 and h2 in Q(h1 ) and Q(h2 ). We say h1 ≤ h2 when Q(h2 ) ⊂ Q(h1 ) and 1/2

1/2

1/2

1/2†

H1 u ≤ H2 u, u ∈ D(H2 ) ⇔ H2

1/2

1/2†

u ≤ H1

1/2

h1 [u] = H1 u2 ≤ h2 [u] = H2 u2 ,

u ∈ Q(h2 ).

(2.2)

Equivalently, we write H1 ≤ H2 when h1 ≤ h2 . Now, we can write (2.1) as H1 ≤ H2 ⇐⇒ H†2 ≤ H†1 .

(2.3)

Let us define, for non-inhibited (see definition (1.9)) quadratic forms like hκ from (1.1), the domain Q∞ := {u ∈ Q : limκ→∞ hκ [u] < ∞}. Then, according to [30, 36], the symmetric form h∞ (u, v) = lim hκ (u, v), κ→∞

is closed in Q∞

·

u, v ∈ Q∞ ,

and defines a self-adjoint operator H∞ there. Further, H†∞ = s-lim H−1 κ . k→∞

The general framework for a description of families of converging positive definite forms will be the following theorem from [36]. Theorem 2.1. Let sn , hn , un and h∞ be closed symmetric forms in H such that they are all uniformly2 positive definite. 1. If sn ≥ sn+1 ≥ h∞ , where h∞ (u, v) = lim sn (u, v), u, v ∈ Q(sn ), n→∞

then h∞ is closed with Q(h∞ ) = 2 By

n∈N

n∈N

Q(sn )

h∞

and H†∞ = s-limn S†n .

this we mean that they have a uniform positive lower bound.

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2. If un ≤ un+1 ≤ h∞ , where h∞ (u, v) = lim un (u, v),

u, v ∈ Q(h∞ ), then h∞ is closed with domain Q(h∞ ) = f ∈ n∈N Q(un ) : sup un [f ] < ∞ and H†∞ = s-limn U†n . 3. If un and sn are as before and un ≤ hn ≤ sn also holds, then n→∞

h∞ (u, v) = lim hn (u, v), n→∞

u, v ∈ Q(h∞ ),

H†∞ = s-lim H†n . κ→∞

For the families of forms which satisfy the assumptions of Theorem 2.1 the following qualitative convergence result on spectral families has been established in [36]. Theorem 2.2. Let hn be a sequence of positive definite forms that satisfies any of the assumptions of Theorem 2.1 for un , sn or hn . Let there also be a positive definite form s such that hn ≥ s and λe (S) > 0. Then En (D) − E∞ (D) → 0,

D < λe (S), D ∈ spec(H∞ ).

(2.4)

The results like Theorem 2.1 have independently been obtained in [29, 30]. We have opted for Theorem 2.1 since it extensively uses the monotonicity (or “sandwiched” monotonicity) to establish the stability of the converging eigenvalues and this fits neatly into the perturbation framework of [11]. This was the chief source of motivation for the main construction from the PhD thesis [15] (those results appeared later in [14, 16, 17, 18]).

3. A constructive approach to asymptotic eigenvalue/eigenvector estimates Let us reiterate that we use the notion of the constructiveness in this paper in two contexts. First, it should be emphasized that all of our theory is based on the error representation result like (3.8)-(3.9), below. But second, it is also meant to emphasize that in a result like those of the type (1.4)-(1.6) we present a way κ to construct an improvement to the approximation λ∞ i (of the eigenvalue λi ). The constants lb and ubi , i = 1, 2, 3 are explicit functions of the approximation defects ηi (P ), to be defined below and it is the aim of this section to reveal this dependence. 3.1. Background information on the block-diagonal part of the operator/form In this section we review the results from our previous work which we use to prove our first contribution in Section 3.2. A reader who would like to go straight to the new results can do that directly after reading equation (3.1) and Definition 3.1 below.

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In this section we assume that we have a fixed closed symmetric and densely defined form h. We will review the basic spectral properties of the block-diagonal part of h with respect to an orthogonal projection P , R(P ) ⊂ Q(h), as is presented in [16]. In order to simplify the presentation we temporarily suppress (in the notation) the dependence of quantities on H, where there is no danger of confusion. Assuming that R(P ) is finite dimensional we define the block-diagonal part of h by setting hP (u, v) := h(P u, P v) + h(P⊥ u, P⊥ v),

u, v ∈ Q(hP ) := Q(h).

(3.1)

Obviously the form hP is closed and positive definite and so it defines a self-adjoint operator HP in the sense of Kato. We further have (for a proof see [14, 16]): R(H−1 − H−1 P )

is finite dimensional.

|h[u] − hP [u]| < 1. hP [u] u∈Q(h)

ηmax (P ) := sup

(3.2) (3.3)

A first consequence of these two features is the stability of essential spectra, namely Weyl’s theorem gives specess (H) = specess (HP ). Further, we have the estimate–of the same form as (3.3)– for the eigenvalues λi (HP ) and λi (H), i ∈ N which are below the infimum of the essential spectrum λess (H) = λess (HP ) |λi (H) − λi (HP )| < ηmax (P ), λi (HP )

i ∈ N.

(3.4)

The attractiveness of interpreting the form h as a perturbation of its block-diagonal part lies in the fact that spec(HP ) = spec(Ξ) ∪ spec(W) (3.5) where Ξ = (H1/2 P )∗ (H1/2 P )R(P ) is a finite dimensional operator and W is the self-adjoint operator which is defined in R(P⊥ ) by the quadratic form h(P⊥ ·, P⊥ ·). Since spec(Ξ) is computable, we can start building our constructive estimation procedure on this fact. As a convention we will use µ1 ≤ · · · ≤ µdim R(P ) to denote the eigenvalues of Ξ. The numbers µi will be called the Ritz values from the subspace R(P ). In this section we also use the notation λi := λi (H). Let us now assume that dim R(P ) = m ∈ N. To examine the relationship between h and hP in further detail define 1/2  −1 −1 (ψ, H ψ) − (ψ, HP ψ)   | ψ ∈ S, ψ = 1  , ηi (P ) :=  max min −1 (ψ, H ψ) S⊂R(P ), dim(S)=m−i+1

(3.6) for i = 1, . . . , m. It has also been shown in [16] that ηmax (P ) = ηm (P ). Although the perturbation δP (h) := h − hP is in general–for some P , R(P ) ⊂ Q(h)–not −1/2 −1/2 representable by an operator, the quadratic form δPs (h)[·] := h[HP ·]−hP [HP ·]

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can always be represented by the bounded operator block-matrix (with respect to P ⊕ P⊥ = I) 0 Γ∗ s , and (·, δPs (H)·) = δPs (h)[·]. δP (H) = Γ 0 Furthermore, [14, Lemma 2.1] gives that si (Γ) = ηi (P ), i = 1, . . . , m. The analysis of [14] now yields the conclusion that the test space R(P ) can be used to generate good approximation for the eigenvalues λi , i = q, . . . , q + m − 1 when ηm (P ) is smaller than half of the relative gap λq+m − µm µ1 − λq−1 ! . , γq := min λq+m + µm µ1 + λq−1 Ample numerical evidence corroborates that such estimates are robust (with regard to scaling) and sharp. Assume that ηmax (P ) < 12 γq and that dim R(P ) = m, where m is the multiplicity of the eigenvalue λq . Using [14, Theorem 3.3] we conclude that the operator matrix

I − λq Ξ−1 Γ∗ , (3.7) δPs (Hq ) = Γ I − λq W−1 which is the block-matrix representation (with respect to P ⊕ P⊥ = I) of the quadratic form −1/2

δPs (hq )[·] := h(HP

−1/2

·, HP

−1/2

·) − λq (HP

−1/2

·, HP

·),

satisfies dim N(δPs (Hq )) = m and the mechanism of [32, (1.1)-(1.2)]–also known in Linear Algebra as the Wilkinson’s Schur complement trick (see [28, pp. 183] and [14, Theorem 3.3])–allows us to conclude I − λq Ξ−1 = Γ∗ (I − λq W−1 )−1 Γ = Γ∗ Γ + λq Γ∗ W−1/2 (I − λq W−1 )−1 W−1/2 Γ.

(3.8) (3.9)

Identity (3.8) is the basis of the proof of [14, Theorem 3.3] which we now quote. Note that (3.8)-(3.9) also hold for λq which is in a gap of the essential spectrum. Based on the definition (3.6) we now define (for later usage) the approximationdefects for hκ . Definition 3.1. Let the sequence hκ be given and let the orthogonal projection P be such that R(P ) ⊂ Q(hκ ) and dim R(P ) < ∞. We write ηi (κ, P ) for ηi (P ) from (3.6) when applied on hκ . We call ηi (κ, P ) the κ-approximation defects. If we are given a subspace P = R(P ), then we abuse (simplify) the notation and freely write ηi (κ, P) = ηi (κ, P ). Theorem 3.2. Let the discrete eigenvalues of the positive definite operator H be ordered such that λq−1 < λq = λq+m−1 < λq+m . Let R(P ) ⊂ Q(h) be the test ηm (P ) subspace such that dim R(P ) = m and 1−η < γq . Then we have m (P ) m ≤ ηm (P ) ||| diag(ηi (P ))m ||| diag |λq − µi | (3.10) i=1 g µi q,ηm (P ) i=1

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where gq,ζ := max


µ1 (1 − ζ) − (1 + (1 +

ζ 1−ζ )λq−1

ζ 1−ζ )λq−1

,

(1 −

ζ 1−ζ )λq+m − (1 + ζ (1 − 1−ζ )λq+m

61

ζ)µm

"

−µm for q > 1 and we set g1,ζ := g1 := λλm+1 . Here we use diag(αi )m i=1 to denote m+1 +µm the m × m diagonal matrix with scalars αi on its diagonal and ||| · ||| denotes any unitary invariant matrix norm and µi are the Ritz values from R(P ).

In the case in which we do not have explicit information on the multiplicity of λq we have a weaker upper estimate. There is also an accompanying lower estimate which establishes the equivalence of the estimators ηi (P ) and the error. Assuming that H = λ dE(λ) and that we use vi , Hvi = λi vi , vi = 1 to denote eigenvectors and ψi ∈ R(P ), Ξψi = µi ψi , ψi = 1 to denote Ritz vectors, we collect some representative spectral estimates (bases on R(P )) from [14, 18]. Theorem 3.3. Let the discrete eigenvalues of the positive definite operator H be ordered such that λm < λm+1 and let λs1 < λs2 < · · · < λsp be all the elements3 of ηm (P ) −µm spec(H) \ {λ ∈ spec(H) : λ ≥ λm+1 }. If 1−η < λλm+1 , then there exist m (P ) m +µm eigenvectors vi , Hvi = λi vi , vi = 1, and Ritz vectors ψi ∈ R(P ), Ξψi = µi ψi , ψi = 1, such that m λm+1 µm ||| diag((ηi (P ))m i=1 ) ⊕ diag((ηi (P ))i=1 ) ||| , (3.11) ||| E(µm ) − P ||| ≤ λm+1 − µm 1 − ηm (P ) m m m # # µ1 # 2 1 |λi − µi | ηi (P ) ≤ ≤ ηi2 (P ), 2µm i=1 µ min g i s ,η (P ) i m s i i i=1 i=1 i=1,...,p √ ηm (P ) 2λµi vi − ψi ≤ max , λ∈spec(H)\{λi } |λ − µi | 1 − ηm (P ) µi − λi h[ψi − vi ] = vi − ψi 2 + , i = 1, . . . , m. h[vi ] λi

(3.12)

(3.13) (3.14)

$i Here Psi is the orthogonal projection onto the linear span of {ψj : j = k=1 mk +1, $i+1 . . . , k=1 mk } and mi is the multiplicity of the eigenvalue λsi , i = 1, . . . , p. Obviously the identity Ps1 ⊕ Ps2 ⊕ · · · ⊕ Psp = P holds. In the case in which λ1 = λm we µ1 can drop the constant 2µ from the lower estimate. We can also allow for other m cross norms ||| · ||| of the diagonal matrix diag((ηi (P ))m i=1 ) in (3.12). The proof of the estimate for the spectral projection (3.11) can be found in [18], the proof of (3.13) is in [14] and identity (3.14) is well-known. For reader’s convenience let us also point out that the problem of estimating the spectral projections E(I)–where I is some contiguous interval whose boundary points are not 3 We

assume that 1 ≤ s1 < s2 < · · · < sp ≤ m.

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accumulation points of spec(H)– can be seen as a problem in obtaining a robust computable estimate of the Cauchy integral % % % 1 % % % −1 −1 E(I) − P = (ζ − HP ) − (ζ − H) dζ % . (3.15) % % 2π % C(I) By C(I) we denote the circle in the resolvent set of H such that I is in the interior of the associated disc and the rest of the spectrum is outside the disc. However, contrary to intuition, the direct analysis of (3.15) is not the most natural way to obtain computable and robust estimates of E(I)−P . A problem is that, although the integral of the resolvent difference does not depend on the integration path C(I), estimates of it do. Furthermore, the circle is only one of many possible curves which should be taken into account. As an alternative we consider the approach of (weakly formulated) operator equations. Not only are the estimation formulae which are so obtained sharp (see [18, Remark 2.3]), but also the technique allows for a natural consideration of estimates which utilize other operator cross norms ||| · |||. Such results are known as sin Θ theorems in the recognition of the milestone work [9] and have been extensively studied in the computational Linear Algebra literature, see [23, 22] and the references there. We use a recent generalization of those results, which is particularly suitable for an application in the quadratic form setting, see [18]. Remark 3.4. Note that as ηsi (Pi ) → 0 we have λsi+1 − λsi λsi − λsi−1 , gsi ,ηmi (Pi ) → min λsi λsi−1 and min{gsi ,ηmi (Pi ) : i = 1, . . . , p} quantifies the minimal relative gap among the eigenvalues λs1 < λs2 < · · · < λsp . Note that the relative gap gsi ,ηsi (Pi ) distinguishes better between the close eigenvalues than the absolute gap, eg. min{λsi+1 − λsi , λsi −λsi−1 } is an example of an absolute gap. In Theorem 3.3, equivalently as in |λs −λs |

[12, Proposition 2.3], we have that when ηmi (Pi ) < 13 mink=j λsk +λsj , i = 1, . . . , p, j k then 1 3 ≤ . |λs −λs | min gsi ,ηmi (Pi ) mink=j λsk +λsj i=1,...,p k

j

3.2. Estimates for the spectral asymptotic We will now use Theorem 3.3 to obtain convergence rate estimates for (2.4). This is the central result which guaranties the stability of the spectrum of the converging family of forms hκ . Subsequently we will also prove results like (1.4)-(1.6) and use the motivating example of the Schroedinger operator with a singular obstacle potential from [14, Section 4] to show our estimates in action. Although we are working under the assumptions of Theorem 2.1, we assume– in order to be more explicit– that we have the non-inhibited stiff family hκ from (1.1). The form h∞ obviously defines a self-adjoint operator H∞ in N(he ). By H∞ = λE∞ (λ) we denote the spectral representation of H∞ in N(he ). We

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identify E∞ (·) with the product E∞ (·)PN(he ) and write H∞ = λE∞ (λ) for the non-densely defined H∞ in the space H operator. Let I be a contiguous interval in R. Then EI∞ := R(E∞ (I)) is a subspace of Q := Q(hb ). Let now I be such that EI∞ is finite dimensional, then the κ-approximation defect is given by  "1/2 −1 −1 ψ) − (ψ, H ψ) (ψ, H κ E∞ ηi (κ, EI∞ ) :=  max min ψ ∈ S, ψ = 1  , S⊂E∞ , (ψ, H−1 ψ) κ dim(S)=m−i+1

(3.16) † −1 I where H−1 := (H ) = (H ) and i = 1, . . . , dim R(E ). To further I E (I) E (I) ∞ κ ∞ ∞ ∞ E ∞

simplify the notation we set ηi (κ, I) := ηi (κ, EI∞ ). Theorem 2.1 now obviously yields i = 1, . . . , dim R(EI∞ ). lim ηi (κ, I) = 0, κ→∞

A similar construction can be performed in the case in which EI∞ is infinite dimensional. The main features which are lost in this generalization are the easy −1 E∞ (I)), the property ηmax (κ, I) < 1 and computability of spec(Ξ−1 κ ) = spec(HEI ∞ the result on the stability of the essential spectrum. This makes, in general, such method less attractive for practical constructive considerations. Let us first give a quantitative version of Theorem 2.1 which is based on the application of Theorem 3.3. As a notational convenience we use λ∞ 1 ≤ ··· ≤ ∞ κ κ κ ≤ λ and λ ≤ · · · ≤ λ ≤ λ to denote the discrete eigenvalues below the λ∞ ess 1 ess i i infimum of the essential spectrum of the operators H∞ and Hκ respectively. Theorem 3.5. Let Hκ = λ d Eκ (λ) be the operators which are associated with the ∞ family of forms hκ . Take D ∈ R such that λ∞ m < D < λm+1 and set I = −∞, D]. Then ηi (κ, I) < 1, |λκj

− λ∞ j | λ∞ j

i = 1, . . . , m,

≤ ηm (κ, I),

(3.17)

j = 1, . . . , m,

(3.18)

Dλ∞ ηm (κ, I) m Eκ (D) − E∞ (D) ≤ |D − λ∞ | 1 − ηm (κ, I) m

(3.19)

for κ large enough. (For the meaning of the phrase large enough see Remark 3.6.) Proof. Statement (3.17) is a direct consequence of [14, Lemma 2.1]. Let us now remember (3.4). This estimate is the consequence of [16, Theorem 4.5] which, when applied to the form hκ and its E∞ block-diagonal part (hκ )E∞ , yields (1 − ηm (κ, I)) hκ )E∞ ≤ hκ ≤ (1 + ηm (κ, I)) hκ )E∞ . Let Hκ E∞ be the self-adjoint operators which represent the forms hκ )E∞ in the sense of Kato, then λ∞ ∈ spec Hκ E∞ . Set Wκ = Hκ E∞ E∞ (D)⊥ and i † W∞ = H∞ E∞ (D)⊥ then Theorem 2.1 implies that W∞ = s-limκ→∞ Wκ† . By

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the construction of Wκ we have R(E∞ (D)) ⊥ w for any w ∈ D(Wκ ). This implies λ1 (Wκ ) → λ1 (W∞ ) = λ∞ m+1 . On the other hand, since spec( Hκ E ) = {λ ∈ spec(H∞ ) : λ ≤ D} ∪ spec Wκ , ∞

it follows that there is κ0 such that & ∞ ' λm , D ⊂ R \ spec( Hκ E ), ∞

κ > κ0 .

Since ηm (κ, I) → 0 we conclude that for κ > κ0 (here we slightly abuse the noD−λ∞ m holds. Now, the conclusion (3.18) follows tation) the estimate ηm (κ, I) ≤ 12 D+λ∞ m from [16, Theorem 5.2]. Equivalently, the conclusion (3.19) follows from (3.11) and [18, Theorem 3.2]. Remark 3.6. The coupling constant κ0 is large enough when ηm (κ, I)
κ0 . This follows by a similar consideration as in Remark 3.4. A direct application of the results from [18, Section 3] and the results of Theorem 3.5 is the following corollary. Corollary 3.7. Assuming the setting and the notation of the previous theorem we have m m Dλ∞ m ||| diag((ηi (κ, I))i=1 ) ⊕ diag((ηi (κ, I))i=1 ) ||| . ||| Eκ (D) − E∞ (D) |||≤ |D − λ∞ 1 − ηm (κ, I) m| & ' In the case in which I = D− , D+ and λκq−1 < D− ≤ λκq ≤ λκq+m−1 ≤ D+ < λκq+m , κ > κ0 then ) ( ∞ λ1 D− D+ λ∞ ηm (κ, I) m . (3.20) + ∞ Eκ (I) − E∞ (I) ≤ |D+ − λ∞ | |λ − D | 1 − ηm (κ, I) − m 1 An easy comparison with the single operator estimates from Theorem 3.3 reveals that, unlike the spectral family estimate (3.19), the eigenvalue result (3.18) is suboptimal in the asymptotic setting. The problem is that we can not uniformly apply the estimate (3.12) on all the operators Hκ , κ > κ0 since we have no information of the multiplicity of the eigenvalue λκi for all κ > κ0 . We only know the multiplicity of λ∞ i . The only statement which we can make in general is a lower estimate on the convergence rate. A way to solve this multiplicity problem will be presented in Section 3.2.4. For now we only give the following result. Corollary 3.8. Assuming the setting and the notation of Theorem 3.5 we have m m # # λ∞ |λκi − λ∞ 1 2 i | η (κ, I) ≤ . i ∞ 2λ∞ λ m i=1 i i=1

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Furthermore, for each κ > 0 we can chose eigenvectors viκ , Hκ viκ = λκi viκ , viκ = 1 ∞ ∞ and vi∞ , H∞ vi∞ = λ∞ i vi , vi = 1 such that m m # # hκ [viκ − vi∞ ] λ∞ 1 2 . η (κ, I) ≤ i ∞ 2λm i=1 hκ[vi∞ ] i=1

One situation in which we can readily obtain upper estimates like those from Theorem 3.3 is when we know that λ∞ i has multiplicity one. This is frequently a case for the 1D differential operators. Also, the lowest eigenvalue of many Schroedinger operators, like those from [8] have multiplicity one. In what follows we use · A−1 = A−1/2 · to denote the standard A−1 -norm, which is associated to a positive definite operator A. Theorem 3.9. Assume the setting and the notation of Theorem 3.5, and let λ∞ q , q ∈ N be of multiplicity one. Then κ λ∞ q −λq λ∞ lim 2 q ∞ κ→∞ η1 (κ, λq ) hκ [vqκ −vq∞ ] hκ [vqκ ] lim 2 κ→∞ η1 (κ, λ∞ q )

= 1,

(3.21)

= 1.

(3.22)

Proof. By the same argument as above we may assume that we have κ0 such that " ∞ ∞ λ∞ λ∞ 1 q+1 − λq q − λq−1 ∞ , , κ > κ0 . η1 (κ, λq ) ≤ min ∞ λ∞ + λ∞ 3 λ∞ q q+1 + λq q−1 Theorem 3.5 yields that there exist D− , D+ such that 0 < D− < λ∞ q < D+ and λκq−1 < D− < λκq < D+ < λκq+1 ,

κ > κ0 .

(3.23)

According to [14] we conclude that we may apply the error representation formula ∞ (3.9) to the operator Hκ and the test vector vq∞ , such that H∞ vq∞ = λ∞ q vq , vq∞ = 1. To the vector vq∞ we can define the residuum as the functional rκq := ∞ Hκ vq∞ − λ∞ q vq and the identity rκq 2(H

−1 κ )E∞

= (vq∞ , H∞ vq∞ ) η12 (κ, λ∞ q )

can be established by an easy computation. Also note the following identities * + | rκq , v | |hκ (v, vq∞ ) − (hκ )E∞ (v, vq∞ )| κ rq (Hκ )−1 = max = max 1/2 1/2 E∞ v∈Q\{0} (H ) v∈Q\{0} (Hκ )E∞ v κ E∞ v = max

v∈Q\{0} v⊥N(he )

|hκ (v, vq∞ ) − (hκ )E∞ (v, vq∞ )| 1/2

(Hκ )E∞ v

,

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* + where ·, · is the standard duality product. Analogous manipulation and the error representation formula (3.9) yield the conclusion κ λ∞ q −λq λ∞ q η12 (κ, λ∞ q )

−1 −1 κ κ κ (Hκ )−1 λκq ((Hκ )−1 E∞ rq , (I − λq (Hκ )E∞ ) E∞ rq ) =1+ ∞ 2 λq η1 (κ, λ∞ q ) −1/2 = 1 + O (Hκ )E∞ PN(he )⊥ 2 .

Finally, Theorem 2.1 implies (3.21). The conclusion (3.22) follows from (3.14) We would like to emphasize that in this result the monotonicity of the family κ hκ played a role. It is possible to prove the result without the property λ∞ q > λq . The proof is technically more involved and it does not further the understanding of the problem, so we leave it out. This theorem establishes that the estimate–which follows directly from Theorem 3.2–is sharp. We formulate this as the following corollary. Corollary 3.10. Assume the setting of the preceding theorem. Then κ |λ∞ q − λq | ≤ λ∞ q min

3 η12 (κ, λ∞ q )

∞ λ∞ −λ∞ λ∞ q q+1 −λq q−1 ∞ , λ∞ +λ∞ λ∞ +λ q q q+1 q−1

!.

This estimate is sharp in the sense of (3.21). 3.2.1. A concrete example. Let Hκ be the operators which are defined by the family of positive definite forms ∞ ∞ 2 ∂x u∂x v dx + κ uv dx, u, v ∈ H01 (R+ ). (3.24) hκ (u, v) = 0

1

Theorem 2.1 readily yields h∞ (u, v) =

1

∂x u∂x v dx, 0

u, v ∈ H01 [0, 1].

& + Here we have used and H01 (R+ ), R+ := 0, ∞ to denote the standard Sobolev spaces. We also identify the functions from H01 [0, 1] with their extension by zero to the whole of R+ and write H01 [0, 1] ⊂ H01 (R+ ). We also formally write Hκ = −∂xx + κ2 χ[1,∞ and H∞ = −∂xx and chose √ 2 sin(kπx) 0 ≤ x ≤ 1, ui (x) = i∈N (3.25) 0 1 ≤ x, H01 [0, 1]

as a test function(s). A simple computation yields that λκi is a solution of the equation √ √ κ2 − λκ = − λκ cot( λκ ) (3.26)

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and we know that each λκi has the multiplicity one. The quotient 1λ∞ 1 can be 1 represented (for κ → ∞) by a convergent Taylor series (see [26]) κ 1 1 1 1 2 1 4 2 1 λ∞ 1 1 − λ1 − 3 + π + π = 2 + 8 − 10 + · · · . (3.27) λ∞ κ κ2 2! 4! κ3 2! 4! κ4 1 2 Using Green’s functions we also directly compute η12 (κ, λ∞ i ) := 3+κ . For computational details see [15]. By utilizing the information from (3.26) we can establish . , 2 4π 2 =: D(κ) ≤ λ2 (H), κ ≥ 5, (3.28) 1− 3+κ

which leads, in combination with (3.12), to the estimate λ∞ − λκ 10 1 1 2 D(κ) + π 2 2 √ ≤ 1 ∞ 1 ≤ = + O , 3+κ λ1 D(κ) − π 2 3 + κ 3κ κ κ

κ ≥ 5.

(3.29)

Similar sharp results can be obtained for other λ∞ i and using (3.14) for corresponding eigenvectors. We tacitly leave out the details. 3.2.2. A remark on higher dimensional singular obstacle problems. This paradigm has been applied in [15] to operators which are defined both in H 1 (Rn ) as well as in H 1 (O), where O ⊂ Rn is a bounded domain. The only ingredient which is necessary is a result on the behavior of the momenta ∞ † −1 2 f ) − (f, H f ) = H1/2 f ∈ E∞ . (3.30) (f, H−1 κ ∞ e Hτ f dτ, κ2

Estimates of such momenta have been obtained in many places in the literature. We illustrate our point by a consideration of a model problem of the electromagnetic waveguide O × R, where the section O ⊂ R2 is a smooth and connected domain. The material Ω ⊂ O of very large conductivity is compactly immersed in O, which is to say that the closure cl(Ω) is contained in O and that Ω is bounded. The dielectric material is now modeled by U = O \ Ω. Assuming that the boundary of Ω is sufficiently smooth we study the eigenvalue problem for Hκ = − + κ2χΩ . Here, χΩ is the characteristic function of Ω and Hκ is the operator which is defined in the sense of Kato by the quadratic form ∇ψ · ∇φ + κ2 χΩ ψφ, ψ, φ ∈ Q∞ := H01 (O), hκ (ψ, φ) = O

O

where κ ∈ R+ and ∇ is the usual gradient operator on H01 (O), the Sobolev space of functions with zero trace on the boundary ∂O. This problem has been analyzed in [13]. Let us assume that λ∞ m < D < ∞ ∞ λm+1 , for some m ∈ N. We compare P = E∞ (λ∞ ) and Q = E (λ κ κ m m ), where 2 Hκ = λ dEκ (λ) is the spectral integral in L (O) and H∞ = λ dE∞ (λ) is the spectral integral in L2 (U). Since, as has been shown in [13], 1 1 1 ∂vi∞ ∂vi∞ ∞ −1 ∞ ∞ † ∞ + O , (vi , Hκ vi ) − (vi , H∞ vi ) = ∞ κ (λi )2 ∂Ω ∂ν ∂ν κ3/2

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we have coarse eigenvector estimates 1 Dλ∞ m 1 √ + O 3/4 ≤ Qκ − P ≤ ∞ D − λm κ κ min

i=1,...,m

∂vi∞ ∂vi∞ ∂Ω ∂ν ∂ν λ∞ i

hκ [viκ − vi∞ ] 1 ≤ ≤ 2κ hκ [vi∞ ]

4 ∞ λ∞ m+1 −λm ∞ λ∞ +λ m m+1

1 , κ

4 ∞ λ∞ m+1 −λm ∞ λ∞ m+1 +λm

1 √ , κ

(3.31)

(3.32)

which can be improved in a straightforward manner by bringing the factor ∂vi∞ ∂vi∞ into estimates, as has been shown in [13, Section 2.1]. The last in∂Ω ∂ν ∂ν equality in (3.31) and (3.32) hold for κ large enough. The optimal eigenvalue estimate can easily be constructed from Theorem 3.3 and 3.9 and we know that the eigenvector estimate (3.32) is optimal in the sense of (3.14) and (3.22). Remark 3.6 indicates how to assess the radius of convergence of these first order estimate(s). 3.2.3. Remarks on (finite) eigenvalues in gaps of essential spectrum and on general converging families hκ . We have said that the theory can be applied to eigenvalues which are in gaps of the essential spectrum. Since we do not consider any model examples which show such behavior (e.g. operators with periodic boundary conditions) we will only briefly outline a possibility to obtain results like Theorem 3.9 or Theorem 3.5 in this setting. In dealing with the eigenvalues in gaps of the essential spectrum we do not have the safe convergence environment of Theorem 2.1. Instead, we have to have an a priori information that the assumption like (3.23) holds. An example of how to obtain this type of a priori information can be seen in the proof of (3.28). To this end we would like to emphasize that such a type of “precise” result on the separation of the target eigenvalue from the unwanted component of the spectrum is an unavoidable ingredient of all constructive spectral estimates. An assumption like (3.23) is equivalent to requiring that the eigenvalue λq be stable under the perturbation hκ , see [21, chapter VIII.4, pp. 437]. For a characterization of perturbations for which this holds see [24] and references therein. Given such an estimate–i.e. assuming that λq is a stable eigenvalue–the appropriate result from [14] or [18] can be applied to obtain convergence estimates. We also emphasize that the theory of [14, 18] allows for more general spectral intervals I. To be more precise, to establish an estimate like (3.20) the spectrum in I ∩ spec(H∞ ) does not have to be discrete. However, in such a situation we have no guarantee that ηmax (κ, I) < 1 and obtaining computational formulae requires much more technical work. The precise use in a given situation is dependent on the application, but always follows the procedure outlined in Theorems 3.5 and 3.9. In the case in which we consider a general converging family of quadratic forms from [36] we cannot conclude that (H†∞ )E∞ (I) = (Hκ )−1 E∞ (I) , so we have −1 := (H ) in (3.16) instead. If we set to use explicitly computable Ξ−1 κ E∞ (I) E∞ (I) κ κ κ ∞ µi := λi (Ξκ ), then µi substitutes for λi in eigenvalue estimates like (3.18), (3.21),

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whereas the estimates for the spectral projections like remain unchanged, √ κ(3.20) λm+1 µκ m e.g. we have the convergence estimate Qκ − P ≤ |λκ −µκ | √ ηm (κ,P ) . m+1

m

1−ηm (κ,P )

3.2.4. A method to solve the multiplicity problem. A tacit assumption in this semiclassical analysis is that the operator H∞ is a well known object. In order to be able to apply Theorem 3.3 one should establish that there exists κ0 > 0 such that (3.33) λκq−1 < D− < λκq = λκq+m−1 < D+ < λκq+m , for κ > κ0 . However, if m > 1 it is not plausible to expect that (3.33) will hold in general. Instead, we will get a tight cluster of m eigenvalues (counting the eigenvalues according to their multiplicity) that converge to λ∞ q . Since we aim to express the spectral information about Hκ in terms of the spectrum of H∞ we further opt to give specific values for D− and D+ as functions of the gaps in the spectrum of H∞ . Theorem 3.11. Let the eigenvalues of the operator H∞ be ordered such that λ∞ q−1 < ∞ ∞ ∞ λ∞ = λ < λ . Define the measure of the relative separation of λ q q+m q from q+m−1 the rest of the spectrum of H∞ as the number " ∞ ∞ λ∞ λ∞ q+m − λq q − λq−1 ∞ , . γs (λq ) = min ∞ λ∞ + λ∞ λ∞ q q+m + λq q−1 There exists κ0 > 0 such that for κ ≥ κ0 3ηm (κ,λ∞ q )

|λκq+i−1 − λ∞ γc (λ∞ q | m) < ηm (κ, λ∞ , q ) ∞ 3ηm (κ,λ∞ q ) λq 1− ∞

i = 1, . . . , m.

(3.34)

γs (λm )

Proof. Since ηm (κ, λ∞ q ) → 0, an argument analogous to that which led to Theorem 3.5 implies that we can pick κ0 > 0 such that for κ > κ0 1 ∞ ηm (κ, λ∞ (3.35) q ) ≤ γs (λq ) 3 1 ∞ ∞ |λκk − λ∞ k = q, q + 1, . . . , q + m − 1, (3.36) q | ≤ γs (λq )λq , 3 ∞ κ )| > 1 γs (λ∞ |ζ − λk (H (3.37) q )λk (Hκ ), k ∈ {q, . . . , q + m − 1}, ζ ∈ C(λq ). 3 1 ∞ ∞ Here C(λ∞ q ) is the circle in the complex plane with the radius 3 γs (λq )λq and the ∞ center λq . Assume κ > κ0 is fixed, then define the family a(τ ) = (hκ )P + τ δP (hκ ),

τ ∈ C.

(3.38)

This is a holomorphic family of type (B) (for the definition see [21, Chapter VII]). We know that |δP (hκ )[u]| < ηm (κ, λ∞ q )(hκ )P [u],

u ∈ Q,

so [21, Theorem VII-4.9 and (VII-4.45)] imply that the resolvent R(τ, ζ) = (A(τ ) − ζI)−1

(3.39)

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can be represented by a convergent power series in τ for ζ ∈ C(λ∞ q ). The power series for R(τ, ζ) converges for every |τ | < r0 =

1 ηm (κ, λ∞ q )

inf∞

ζ∈C(λq ), λ∈spec((Hκ )P )

1 1 |λ − ζ| = γs (λ∞ q ). ∞ λ ηm (κ, λq ) 3

(3.40)

In particular, assumption (3.35) implies that the series converges for τ = 1. Define ) := − 1 A(τ ) B(τ R(τ, ζ) dζ, 2πi C(λ∞ q ) ) is a holomorphic operator family and there exist m holomorfic functions then B(τ 1 (τ ), . . . , λ m (τ ) are all the nonzero eigenvalues of the operator λi (τ ) such that λ ). Due to the assumptions we have made it follows that for i = 1, . . . , m B(τ 1 ∞ γs (λ∞ |τ | < r0 . q )λq , 3 Cauchy’s integral inequality4 for the coefficients of the Taylor expansion implies, for every i = 1, . . . , m, the estimate i (τ ) − λ∞ | < |λ q

|< |λ i (n)

1 ∞ ∞ 3 γs (λq )λq , r0n

n = 1, 2, . . .

i (τ ) = λ∞ + τ λ (1) + τ 2 λ (2) + τ 3 λ (3) + · · · . This yields where λ q i i i (1) | < i (τ ) − λ∞ − τ λ |λ q i

1 ∞ ∞ 3 γs (λq )λq

r0

|τ |2 ≤ r0 − |τ |

1 ∞ ∞ 3 γs (λq )λq r02

|τ |2 1−

|τ | r0

for |τ | < r0 . In particular for τ = 1 there exists a permutation j(·) such that j (1) = λκ λ i q+i−1 , i = 1, . . . , m. Hence ∞ ∞ |λκq+i−1 − λ∞ q − λji | < ηm (κ, λq )λq (1)

3ηm (κ, λ∞ q ) γc (λ∞ ) q 1−

1 3ηm (κ,λ∞ q ) γs (λ∞ q )

.

With this the proof of the theorem is finished. To see this, note that it (1) are the eigenvalues of the matrix was established in [21, (VII-4.50)], that λ ji Mkp = δP (hκ )(uk , up ), where uk , k = 1, . . . , m form an orthonormal basis for R(E∞ [D− D+ ]). Since δP (hκ )(u, v) = hκ (P⊥ u, P u) + hκ (P u, P⊥ u) = 0, we obtain

(1) λ ji

= 0, i = 1, . . . , m and the conclusion follows.

u, v ∈ R(P ),

Remark 3.12. The estimate of this theorem is optimal in the sense of Corollary 3.8. The upper estimate which has a similar form to (3.12) can be established for ∞ the limit eigenvalues λ∞ 1 ≤ · · · ≤ λm . The the role of the constant from (3.12) is 4 For

further details see [1, Section 8.1.4] and [21, Section II-3].

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∞ taken by the constant γmin (λ∞ m ) := min{γc (λi ) : i = 1, . . . , m}, as is given by the repeated application of Theorem 3.11. We leave out the technical details.

4. Spectral asymptotic in the regular case We now concentrate on the non-inhibited families hκ (u, v) = hb (u, v) + κ2 he (u, v),

u, v ∈ Q := Q(hb ) ⊂ Q(he ), 1/2

(4.1) −1/2

which satisfy the additional regularity assumption that the range of He Hb closed in H. As already mentioned in Section 1.2 this is equivalent to −1/2 †

(H1/2 e Hb

) = k < ∞.

is

(4.2)

With this additional requirement, which has a flavor of Linear Algebra, we can use an adaptation of the Lagrange-Multiplier technique5 to establish an upper estimate for the momenta ∞ −1 † −1 2 (f, Hκ f ) − (f, H∞ f ) = H1/2 f ∈ Q∞ := Q(h∞ ). (4.3) e Hτ f dτ, κ2

The lower estimate for (4.3) follows by an adaptation of the spectral-calculus technique from [4, 5]. With this we prove the optimality of our approach to spectral asymptotic estimation. The following lemmata are the main technical results which are needed to estimate the quantities (4.3). Lemma 4.1. Take f ∈ Q∞

·

† −1 † . Then hκ [H−1 κ f − H∞ f ] = (f, Hκ f ) − (f, H∞ f ).

Proof. The proof is a straightforward computation. Take f ∈ Q∞ hκ [H†∞ f ]

=

·

. Then

(f, H†∞ f )

and we have † −1 −1 † † −1 † hκ [H−1 κ f − H∞ f ] = (f, Hκ f ) − hκ (Hκ f, H∞ f ) − hκ (H∞ f, Hκ f ) + (f, H∞ f ) −1/2 † 1/2 † −1/2 f, H1/2 f) = (f, H−1 κ f ) − (Hκ κ H∞ f ) − (Hκ H∞ f, Hκ

+ (f, H†∞ f ) † = (f, H−1 κ f ) − (f, H∞ f ).

−1/2

f − Hb H†∞ f . If we assume Lemma 4.2. Let f ∈ H be given and set rf := Hb 1/2 −1/2 † 1/2 −1/2 † that (He Hb ) < ∞, then qf = (He Hb ) rf and hb (H†∞ f, v) + (qf , H1/2 e v) = (f, v), Furthermore, rf 2 = 5 The

(f, H−1 b f)

1/2

v ∈ Q.

− (f, H†∞ f ).

use of the Lagrange-Miltiplier technique in this context has been suggested to the author by J. Tambaˇca, Zagreb.

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1/2

Proof. rf ⊥ Hb Q∞ , which can be checked by a direct computation. The operator 1/2 −1/2 B := (He Hb ) has closed range, so H = R(B∗ ) ⊕ N(B) = R(B∗ ) ⊕ Hb Q∞ . 1/2

Therefore, we have rf ∈ R(B∗ ) and so we may write qf := B† rf . A direct computation now shows that † ∗† hb (H†∞ f, v) + (qf , H1/2 e v) = (Hb H∞ f, Hb v) + (B rf , BHb v) 1/2

1/2

1/2

= (Hb H†∞ f + rf , Hb v) = (f, v). 1/2

1/2

The main quantitative theorem about the asymptotic behavior of (4.1) now follows directly. 1/2

−1/2 †

·

Theorem 4.3. Assume k := (He Hb

) < ∞. Then, for f ∈ Q∞ , we have † † k2 (f, H−1 (f, H−1 b f )−(f, H∞ f ) 1 f )−(f, H∞ f ) −1 † ≤ (f, H f ) − (f, H f ) ≤ κ ∞ κ2 κ2 (4.4) and k2 1 2 ηi (1, λ∞ ) ≤ ηi2 (κ, λ∞ ) ≤ 2 ηi2 (0, λ∞ ), i = 1, . . . , m , (4.5) 2 κ κ where m is the multiplicity of the discrete eigenvalue λ∞ (not necessarily below the infimum of the essential spectrum of H∞ ). Proof. For any f ∈ H we have hb (H†∞ f, v) + (qf , H1/2 e v) = (f, v), hb (H−1 κ f, v)

+κ

2

he (H−1 κ f, v)

= (f, v),

v ∈ Q, v ∈ Q,

which implies † 2 −1 1/2 hb (H−1 κ f − H∞ f, v) + κ he (Hκ f, v) = (qf , He v)

and subsequently −1 1/2 κ2 he [H−1 . κ f ] ≤ qf he [Hκ f ] The inequality on the right-hand side of (4.4) follows from Lemma 4.2. To establish the inequality on the left-hand side of (4.4) we start from the identity [4, (22)]. † We combine the integral representation for (f, H−1 κ f ) − (f, H∞ f ) from [4, pp. 41] and [4, (29)] to obtain ∞ 1 −1 † 1/2 −1 f ) − (f, H f ) = (dEHe (λ)H1/2 (f, H−1 κ ∞ e Hb f, He Hb f ) 2 λ2 λ + κ 0 −1 = ((I + κ2 He )−1 H−1 b f, Hb f ) 1 −1 ≥ 2 ((I + He )−1 H−1 b f, Hb f ) κ 1 † = 2 (f, H−1 1 f ) − (f, H∞ f ) . κ

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The conclusion (4.5) for the approximation defects follows directly from the definition (3.16) and the observaton that 1 1 1 ≤ ≤ , f ∈ R E∞ ({λq }) . −1 −1 † (f, H1 f ) (f, Hκ f ) (f, H∞ f ) Example 4.4. We will present this example as an abstract variation on (1.2). Let H be a positive definite operator, let P be a projection, R(P ) ⊂ D(H1/2 ) and let −1/2 1/2 rfκ := Hκ f − Hκ H†∞ f . Consider hκ (u, v) = ((I + κ2 P )H1/2 u, H1/2 v) = hb (u, v) + κ2 he (u, v). Then

−1/2 †

(H1/2 e Hb and (4.4) gives for f ∈

) ≤1

1/2 N(P He )

−1/2 rfκ 2 P⊥ H−1/2 f ) (f, H−1 1 κ f ) − (f, H = ≤ 2. rf 2 κ (H−1/2 f, P H−1/2 f )

(4.6)

† κ 2 −1 Here we have used rf 2 = (f, H−1 b f ) − (f, H∞ f ) and rf = (f, Hκ f ) − † (f, H∞ f ) to simplify the notation. On the other hand, we compute 1 −1/2 H−1 (P⊥ + P )H−1/2 κ =H 1 + κ2 to establish 1 −1/2 P⊥ H−1/2 f ) = (H−1/2 f, P H−1/2 f ). (4.7) (f, H−1 κ f ) − (f, H 1 + κ2 Formulae (4.6) and (4.7) give

rfκ 2 1 1 = ≤ 2, 1 + κ2 rf 2 κ which is a very favorable estimate for κ large. The lower estimate can be computed to be rfκ 2 1 ≤ , 2κ2 rf 2 which is not as sharp as the upper estimate, but newertheless, it is asymptotically optimal. Remark 4.5. A necessary and the sufficient requirement for the upper bound in Theorem 4.3 to hold (with a different constant, but of same order κ−2 ) has been given in [3]. The regularity condition from [3] is less restrictive (see the examples from the introduction to [4]) than the condition (4.2), since it poses less strin1/2 −1/2 . On the other hand the gent conditions on the range of the operator He Hb approach from [3] to evaluating convergence constants appears to be more computationally involved, at least in the case of a regular family hκ . It is important to note that the authors of [3] also presents a condition which guarantees the convergence of order κ−2α , for some α, 0 < α ≤ 1. Their technique in this less

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regular case is based on estimating the relative size of perturbation He in a certain scale of Hilbert spaces. In comparison, our condition (4.2) yields a new variational characterization of the convergence constant in the special case of a regular family hκ . More so, our approach yields a more direct way to compute the convergence estimates for such hκ in a practical situation, cf Section 5. It should be noted that for all of the hκ which satisfy the requirements of [3, Theorem 1] the spectral estimates like (1.4)-(1.6) can be established as outlined in Section 6. Furthermore, in [3, Example 13] the authors show that there are families of forms hκ which only exhibit strong resolvent convergence. This indicates a further advantage of our approach to study a resolvent difference locally, like in (1.8), rather than working with the full resolvent difference which is possibly not converging uniformly. The reference [3] has been communicated by an anonymous referee to the author after the review process of this paper was concluded. The author is very thankful to the anonymous referee for communicating this reference.

5. A model problem from 1D theory of elasticity As an illustration of the applicability of Theorem 4.3, we consider the small frequency problem for the circular arch as described in [7, Chapter 8.8:3] and [29], cf. Figure 1. Let φ : [0, l] → R2 be the middle curve of the arch. We take φ to be the upper part of the circle with the radius R. The arch (the model problem we are considering) will be a thin homogeneous, elastic body of constant cross-section A, whose area is A > 0. The arch will be clamped at one end and free at the other. The strain energy of the arch is given6 by the positive definite form l l u1 v1 u2 v2 u2 + v2 + u1 − v1 − ds, a(u, v) = EI ds + EA R R R R 0 0 (5.1) u, v ∈ Q(a) = {u ∈ H 1 [0, l] × H 2 [0, l] : v(0) = 0, v2 (0) = 0}. Here u = (u1 , u2 ) and v = (v1 , v2 ) are the functions of the curvilinear abscissa s ∈ [0, l], the constant E is the Young modulus of elasticity, the constant A is the area of the cross-section A and the constant I is the moment of inertia of the cross-section A . Let us assume that we have s referent arch with the cross-section area A and the cross-section moment I. We consider the family of rods whose cross-section and the moment of inertia of the cross-section behave like 1 1 Iκ = 4 I = ε4 I. Aκ = 2 A = ε2 A, κ κ We want to study the spectral properties of this family of arches as ε → 0. More general arch models can be treated by analogous procedures. This is a subject for future reports. 6 See

also [35].

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Figure 1. The Curved rod model For some given κ > 0, κ := ε−1 , we write E I l u1 v1 E A l u2 v2 u2 + v2 + u1 − v1 − ds aκ (u, v) = 4 ds + 2 κ R R κ R R 0 0 and use Aκ to denote the operator which is defined by aκ . Since Aκ has only discrete spectrum, we write λi (Aκ ), i ∈ N. After rescaling 1 κ λ , κ4 i we see that λκi are the eigenvalues of the operator Hκ , which is defined by λi (Aκ ) =

hκ (u, v) = hb (u, v) + κ2 he (u, v) l l u1 v1 u2 v2 u2 + v2 + u1 − v1 − ds = EI ds + κ2 EA R R R R 0 0 for u, v ∈ Q(aκ ) = Q(hκ ). Since λκi enable us to describe only the eigenvalues of Aκ for which 1 λi (Aκ ) < ∞, lim κ→∞ κ4 we see where the name “low frequency problem” for the eigenvalue problem for Hκ comes from. The low frequency problem satisfies the conditions of Theorem 2.1, so we conclude that the limiting form is l u1 v1 f2 u2 + v2 + =0 . ds, u, v ∈ f ∈ Q(aκ ), f1 − h∞ (u, v) = EI R R R 0 (5.2)

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In [35] it has shown that (5.2) is the strain energy of the Curved Rod Model and that hκ , κ > 0 are positive definite with Q(hκ ) = {u ∈ H 1 [0, l] × H 2 [0, l] : v(0) = 0, v2 (0) = 0}. Remark 5.1. From (5.2) we can see the significance of the condition f2 = 0. (5.3) R Assume the rod is locally straight. That is to say, assume R → ∞, then (5.3) turns into f1 = 0, a condition of the inextensibility of the middle curve of the straight rod. The fact that f1 − fR2 = 0 is an inextensibility condition for the middle curve of the curved rod can be established by a rigorous differential geometric argument, see [35]. Continuing this heuristic reasoning, we conclude that Curved Rod model describes the transversal vibrations (perpendicular to the middle curve) of the curved rod. Arch Model “couples” the longitudinal vibrations of the rod with the transversal vibrations. The study of finer properties of longitudinal vibrations requires the analysis of the so called “middle frequency problem”, which will not be further considered here. However, since the “middle frequency problem” also falls under the scope of Theorem 2.1 this theory could also be applied in that case. f1 −

5.1. Computational details Based on (5.1) and (5.2) one concludes that the sequence hκ satisfies the assumptions of Theorem 4.3. Here, a word of additional explanation is in order. We have formulated all of our results about the forms hb and he based on the representations 1/2

1/2

hb (u, v) = (Hb u, Hb v), 1/2 he (u, v) = (H1/2 e u, He v).

However, we can represent (see (5.2)) the forms hb and he with the help of the operators Rb : Q(hb ) → Hb and Re : Q(he ) → He . The only assumptions on the operators Rb (and Re ) is that they have a closed range in the auxiliary Hilbert spaces Hb (and He ), cf. [17]. The representation theorem for the nonnegative definite forms implies 1/2

1/2

hb (u, v) = (Hb u, Hb v) = (Rb u, Rb v)Hb ,

(5.4)

he (u, v) =

(5.5)

1/2 (H1/2 e u, He v)

= (Re u, Re v)He ,

where (·, ·)X generically denotes the scalar product in the Hilbert space X . The relations (5.4) and (5.5) imply that there exist isometric isomorphisms Qb : Hb → 1/2 1/2 H and Qe : He → H such that Hb = Qb Rb , He = Qe Re , and in particular 1/2

1/2

(Hb u, Hb v) = (Qb Rb u, Qb Rb v) = (Rb u, Rb v)Hb , 1/2 (H1/2 e u, He v) = (Qe Re u, Qe Re v) = (Re u, Re v)He .

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We also have for u ∈ Q(hb )

77

u1 E I u2 + , R √ u2 1/2 Q−1 E A u1 − e He u = Re u = R 2 and Rb : Q(hb ) → Hb = L [0, l] and Re : Q(he ) → He = L2 [0, l]. Note that Hb is not positive definite but H1 , which is defined by the form h1 = hb + he , is. For the details see [20, 33]. If we were to change the notation we would have to set / hb := h1 . Since this would unnecessarily complicate the exposition we opt not to do so. We show that I + A R2 1/2 −1/2 † (5.6) (He H1 ) ≤ A R2 Q−1 b Hb u = Rb u = 1/2

√

−1/2 †

1/2

for our model problem. Set k := (He H1

1/2

−1/2 ∗

(H1/2 e H1

) . Then

) qf =

|(qf , He v)|

sup v∈Q(hb )

1/2 H1 v

≥

1 PQ∞ qf , k

since −1/2 ∗

−1/2

1/2

1/2

) ) = N( H1 He ) = N(He ) = Q∞ · . (·) 2 −1 For Q−1 e qf ∈ L [0, l] we define v0 = 0 (Qe qf )(s)ds, 0 (an element of Q(hκ )). For general v we have l l & v1 ' 2 v2 2 1/2 1/2 H1 v = E I v1 − ds + E A ds . v2 + R R 0 0 N((H1/2 e H1

Now, set v = v0 and compute

√

1/2 H1 v0

=

E I + E A R2 qf . R

This establishes 1/2

sup v∈Q(hb )

|(qf , He v)| 1/2

H1 v

≥

1/2

|(qf , He v0 )| 1/2

H1 v0

≥

|(Q−1 e qf , Re v0 )L2 | √ E I+E A R2 qf R

=

A R2 qf , I + A R2

which completes the proof of (5.6). 5.2. Quantitative (and qualitative) conclusions The fact (5.6) allows us to apply Theorem 4.3 to obtain precise estimates for the behavior7 of ηi (ε, λ∞ q ). Since H1 and not Hb is the positive definite operator, we will use the rod with the diameter ε0 as a referent configuration. We chose √ ∞ 3 I + A R2 λ∞ sec − λmin ε0 = , (5.7) 2 ∞ 6 AR λsec + λ∞ min 7 We

have tacitly dropped the exponent from ηi (ε−2 , λ∞ q ) in order to simplify the notation.

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∞ where λ∞ sec and λmin are the two lowermost eigenvalues of the Curved Rod model ε and λmin denotes the lowermost eigenvalue of the Arch Rod Model of the rod with diameter ε. Theorems 3.11, 4.3, Remark 3.6 and Corollary 3.10, together with the observation that ηi (ε, λ∞ ) < 1 for any ε > 0, directly imply that ε ∞ λ∞ 4(I + A R2 ) λ∞ sec + λmin min − λmin ≤ ε2 , 0 < ε ≤ ε0 . ∞ 2 ∞ λmin AR λsec − λ∞ min √ 0 ∞ ∞ R2 λsec −λmin Furthermore, if we chose ε1 = 123 I+A A R2 λ∞ +λ∞ , then we obtain sec

min

ε 2 ∞ ∞ 2(I + A R2 )ηi (ε1 , λ∞ λ∞ 2 4(I + A R ) λsec + λmin min ) min − λmin ε2 ≤ ≤ ε ∞ , A R2 λ∞ A R2 λ∞ sec − λmin min

(5.8)

0 < ε ≤ ε1 . If we are only interested in the upper estimate and we assume that ∞ there is m ∈ N such that λ∞ m < λm+1 , then we have ε λ∞ i − λi ≤ λ∞ i

4(I + A R2 ) 2 ε , |λ∞ − λ∞ A R2 i | max min k∞ ∞ i=1,...,m k=i λ k + λi 3

i = 1, . . . , m.

(5.9)

This estimate holds for all ε ≤ ε2 , where ε2 is defined as the first ε for which the righthand side of (5.9) is smaller than 1. Estimate (5.9) can naturally be refined with the use of other ηi (ε2 , λ∞ j ) as given by the framework of Theorem 3.3. The optimality of the estimate is meant in the sense of Theorem 3.9.

6. Conclusion We have presented a constructive approach to spectral asymptotic estimates in the large coupling limit. Although we have concentrated on a use of this results for theoretical considerations from [6, 8, 10, 29], they are expected to be particularly useful in a design of computational procedures for various singularly perturbed spectral problems. This can be illustrated when comparing the numerical procedures for the Arch Model and the Curved Rod Model. It has been shown that the Curved Rod Model is better behaved with respect to the finite element approximations than the Arch Model, see [34]. Furthermore, a qualitative conclusion of Section 5.2 is that when interested in the transversal vibrations only, Arch Model can be ignored (up to the corrections of order ε2 ). For more on the lower dimensional approximations in the theory of elasticity see [7, 20, 29, 33, 35]. In a practical computational setting it is not reasonable to assume that the spectral problem for H∞ will be exactly solvable. We would like to emphasize that in the design of this theory we have not built the requirement of the explicit solvability of H∞ into our results. To be more precise, nowhere in the proofs of Theorems 3.5 and 3.11 or Corollaries 3.7 and 3.8 is it necessary to have R(P ) = E∞ . The only place where this assumption was necessary was to establish that (3.6) and (3.16) define the same approximation defects. Theorem 3.3 and similar results from [14, 18], which are the workhorses of this theory, do not need this

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assumptions. Subsequently, the only limiting factor is the computability of ηi (P ) and the availability of information on the distance of spec(Ξκ ) (from Theorems 3.2 and 3.3) to the unwanted component of the spectrum. With this we hope to have illustrated the advantages and limitations of our theory. Acknowledgement The author would like to thank V. Enss, Aachen for the support and for many helpful discussions during the final preparation of this manuscript. The help of J. Tambaˇca, Zagreb in the proof of the upper estimate in Theorem 4.3 as well as in the proof of estimate (5.6) is gratefully acknowledged. The use of the technique of Lagrange multipliers in this context I have learned and adapted from his papers and notes. The author would particularly like to thank K. Veselić, Hagen for constant support and encouragement and for many useful discussions in this and other contexts. The author is also thankful to the anonymous referee for a very constructive report.

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[13] L. Grubiˇsić. On relative perturbation theory for eigenvalues and eigenvectors of block operator matrices. Proceedings of Applied Mathematics and Mechanics, 7(1):2050001-2050002, 2007. [14] L. Grubiˇsić. On Temple-Kato like inequalities and applications. submitted for a review. Preprint available from http://arxiv.org/abs/math/0511408. [15] L. Grubiˇsić. Ritz value estimates and applications in Mathematical Physics. PhD thesis, Fernuniversit¨ at in Hagen, dissertation.de Verlag im Internet, ISBN: 3-89825998-6, 2005. [16] L. Grubiˇsić. On eigenvalue estimates for nonnegative operators. SIAM J. Matrix Anal. Appl., 28(4):1097-1125, 2006. [17] L. Grubiˇsić and K. Veselić. On Ritz approximations for positive definite operators I (theory). Linear Algebra and its Applications, 417(2-3):397-422, 2006. [18] L. Grubiˇsić and K. Veselić. On weakly formulated Sylvester equation and applications. Integral Equations and Operator Theory, 58(2):175-204, 2007. [19] P. R. Halmos. Two subspaces. Trans. Amer. Math. Soc., 144:381-389, 1969. [20] M. Jurak and J. Tambaˇca. Linear curved rod model. General curve. Math. Models Methods Appl. Sci., 11(7):1237-1252, 2001. [21] T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin, second edition, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. [22] R.-C. Li. A bound on the solution to a structured Sylvester equation with an application to relative perturbation theory. SIAM J. Matrix Anal. Appl., 21(2):440-445 (electronic), 1999. [23] R.-C. Li. Relative perturbation theory. II. Eigenspace and singular subspace variations. SIAM J. Matrix Anal. Appl., 20(2):471-492 (electronic), 1999. ¨ [24] H. Linden. Uber die Stabilit¨ at von Eigenwerten. Math. Ann., 203:215-220, 1973. [25] Z. M. Nashed. Perturbations and approximations for generalized inverses and linear operator equations. In Generalized inverses and applications (Proc. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1973), pages 325-396. Publ. Math. Res. Center Univ. Wisconsin, No. 32. Academic Press, New York, 1976. [26] W. Neuschwenger. Einige Konvergenzaussagen f¨ ur den Schr¨ odinger-Operator mit tiefen Potentialen. Diplom Arbeit, supervised by K. Veselić, Universit¨ at Dortmund, 1979. [27] G. P. Panasenko and E. Pérez. Asymptotic partial decomposition of domain for spectral problems in rod structures. J. Math. Pures Appl. (9), 87(1):1-36, 2007. [28] B. N. Parlett. The symmetric eigenvalue problem. Prentice-Hall Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics. [29] E. S´ anchez-Palencia. Asymptotic and spectral properties of a class of singular-stiff problems. J. Math. Pures Appl. (9), 71(5):379-406, 1992. [30] B. Simon. A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal., 28(3):377-385, 1978. [31] B. Simon. Trace ideals and their applications, volume 35 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1979. [32] J. Sjoestrand and M. Zworski. Elementary linear algebra for advanced spectral problems. Annales de l’Institut Fourier, 57:2095-2141, 2007.

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[33] J. Tambaˇca. One-dimensional approximations of the eigenvalue problem of curved rods. Math. Methods Appl. Sci., 24(12):927-948, 2001. [34] J. Tambaˇca. A numerical method for solving the curved rod model. ZAMM Z. Angew. Math. Mech., 86(3):210-221, 2006. [35] J. Tambaˇca. The evolution model of the curved rod (in Croatian: Evolucijski model zakrivljenog ˇstapa). PhD. Thesis, University of Zagreb, 2000. [36] J. Weidmann. Stetige Abh¨ angigkeit der Eigenwerte und Eigenfunktionen elliptischer Differentialoperatoren vom Gebiet. Math. Scand., 54(1):51-69, 1984. Luka Grubiˇsić Department of Mathematics University of Zagreb Bijeniˇcka cesta 30 10000 Zagreb Croatia e-mail: [email protected] Submitted: February 6, 2008. Revised: August 17, 2009.




Connectedness of Spectra of Toeplitz Operators on Hardy Spaces with Muckenhoupt Weights Over Carleson Curves Alexei Yu. Karlovich and Ilya M. Spitkovsky Abstract. Harold Widom proved in 1966 that the spectrum of a Toeplitz operator T (a) acting on the Hardy space H p (T) over the unit circle T is a connected subset of the complex plane for every bounded measurable symbol a and 1 < p < ∞. In 1972, Ronald Douglas established the connectedness of the essential spectrum of T (a) on H 2 (T). We show that, as was suspected, these results remain valid in the setting of Hardy spaces H p (Γ, w), 1 < p < ∞, with general Muckenhoupt weights w over arbitrary Carleson curves Γ. Mathematics Subject Classification (2000). 47B35. Keywords. Toeplitz operator, Muckenhoupt weight, Carleson curve, Hardy space, spectrum, essential spectrum, index, Pettis integral.

1. Introduction and main results Let Γ be a Jordan curve, that is, a curve in the complex plane C that is homeomorphic to a circle. We suppose that Γ is rectifiable. We equip Γ with the Lebesgue length measure |dτ | and the counter-clockwise orientation. A measurable function w : Γ → [0, ∞] is referred to as a weight if w−1 ({0, ∞}) has measure zero. We denote by Lp (Γ) (1 ≤ p ≤ ∞) the usual Lebesgue spaces of Γ, and given a weight w on Γ, we define Lp (Γ, w) (1 < p < ∞) as the Lebesgue space with the norm Z 1/p kf kLp (Γ,w) := |f (τ )|p wp (τ ) |dτ | . Γ

The first author is partially supported by the grant FCT/FEDER/POCTI/MAT/59972/2004. The second author is partially supported by NSF grant DMS-0456625.

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The Cauchy singular integral of f ∈ L1 (Γ) is defined by Z 1 f (τ ) (Sf )(t) := lim dτ (t ∈ Γ), R→0 πi Γ\Γ(t,R) τ − t where Γ(t, R) := {τ ∈ Γ : |τ − t| < R} for R > 0. The operator S is said to be bounded on Lp (Γ, w) if Lp (Γ, w) ∩ L1 (Γ) is dense p in L (Γ, w) and there exists a constant M > 0 such that kSf kLp (Γ,w) ≤ M kf kLp (Γ,w)

for all f ∈ Lp (Γ, w) ∩ L1 (Γ).

The problem of characterizing the Γ, p, w for which S is bounded on Lp (Γ, w) has been studied by many mathematicians for a long time. Here is the final result. Theorem 1.1 (see e.g. [1, Theorem 4.15]). Let 1 < p < ∞, let Γ be a rectifiable Jordan curve, and let w be a weight on Γ. The operator S is bounded on Lp (Γ, w) if and only if 1/p Z 1/q Z 1 1 p −q w (τ ) |dτ | w (τ ) |dτ | < ∞. (1.1) sup sup R Γ(t,R) t∈Γ R>0 R Γ(t,R) Here and in what follows 1/p + 1/q = 1. The set of all weights w on Γ satisfying (1.1) is usually denoted by Ap (Γ) and referred to as the set of Muckenhoupt weights. An easy application of Hölder’s inequality shows that if (1.1) holds, then |Γ(t, R)| 0

CΓ := sup sup

(1.2)

where, of course, |Γ(t, R)| is the measure of the portion Γ(t, R). Condition (1.2) is a condition solely on the curve Γ, and curves Γ satisfying (1.2) are commonly called Carleson curves. From now on we always suppose that 1 < p < ∞, that Γ is a Carleson Jordan curve, and that w ∈ Ap (Γ). Then S is bounded and one can show that S 2 = I (see [1, Corollary 6.6]). Hence, P := (I + S)/2,

Q := (I − S)/2

are bounded and complementary projections on Lp (Γ, w). We define H p (Γ, w) := P Lp (Γ, w),

p H˙ − (Γ, w) := QLp (Γ, w),

p p (Γ, w) + C, (Γ, w) := H˙ − H−

where C is identified with the set of the constant functions on Γ. Since Im P = p p Ker Q and Im Q = Ker P , the spaces H p (Γ, w), H˙ − (Γ, w), and H− (Γ, w) are closed p p subspaces of the space L (Γ, w). The space H (Γ, w) is called the pth Hardy space of Γ and w. If w ≡ 1, then we will simply write H p (Γ) instead of H p (Γ, w). The Toeplitz operator T (a) induced by a function a ∈ L∞ (Γ) is the bounded operator T (a) : H p (Γ, w) → H p (Γ, w), f 7→ P (af ).

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The function a is in this context referred to as the symbol of the operator T (a). We will study the spectrum of T (a), that is, the set sp T (a) := {λ ∈ C : T (a) − λI is not invertible on H p (Γ, w)} and its essential spectrum, that is, the set spess T (a) := {λ ∈ C : T (a) − λI is not Fredholm on H p (Γ, w)}. Recall that a bounded linear operator A on a Banach space X is said to be Fredholm if its image (= range) Im A is closed and its cokernel Coker A := X/ Im A and its kernel Ker A := {f ∈ X : Af = 0} have finite dimensions. In that case the index of A is defined as the integer Ind A := dim Ker A − dim Coker A. Halmos [6] posed the following as a test question for any theory of invertibility of Toeplitz operators: Is the spectrum of a Toeplitz operator necessarily connected ? Widom gave the affirmative answer for H 2 (T) in [12] and for H p (T), 1 < p < ∞, in [13]. Here T stands for the unit circle. Later on, Douglas modified Widom’s arguments and proved that the essential spectrum of T (a) is connected for H 2 (T) (see [3, Theorem 7.45]). B¨ ottcher and Silbermann [2, Section 2.35] conjectured that the essential spectrum of T (a) is connected for H p (T) with p ∈ (1, ∞). The aim of this paper is to prove this conjecture and to generalize Widom’s and Douglas’ results on connectedness of spectra of Toeplitz operators to the case of Hardy spaces H p (Γ, w) (1 < p < ∞) with Muckenhoupt weights over Carleson curves. The main results of the paper are the following. Theorem 1.2. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a ∈ L∞ (Γ), then the essential spectrum of a Toeplitz operator T (a) on H p (Γ, w) is a connected subset of C. This result together with the Coburn-Simonenko theorem (see Theorem 2.2) implies connectedness of the spectrum. Corollary 1.3. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a ∈ L∞ (Γ), then the spectrum of a Toeplitz operator T (a) on H p (Γ, w) is a connected subset of C. Finally, notice that Theorem 1.2 and Corollary 1.3 are no longer true if a is a matrix-valued symbol. Simply by considering diagonal symbols it is easy to see that for N × N matrix functions a the (essential) spectrum of T (a) may contain from 1 to N connected components. If a is a continuous or piecewise continuous N × N matrix function, then the number of such components is bounded by N . Litvinchuk and one of the authors [10, Notes to Chapter 7] raised the question about the upper bound for the number of connected components for arbitrary matrix-valued symbols. However, as it has been recently shown by Isgur and one of the authors [9], there is no such bound even for triangular 2×2 matrix functions in the setting of H 2 (T).

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2. Outline of the proofs We follow the ideas of Widom’s proof [13] of the connectedness of the spectrum on H p (T) with Douglas’ modifications [3, Sections 7.38-7.45] for the essential spectrum on H 2 (T). Our intention is to present self-contained proofs of Theorem 1.2 and Corollary 1.3, so the presentation has an expository character in some places. 2.1. Two basic theorems on invertibility and Fredholmness of Toeplitz operators We start with two well known results on invertibility and Fredholmness of Toeplitz operators (see e.g. [1, Section 6.6] for the proofs in the general setting given below and [1, Section 6.13] for the historical remarks). The spectrum of a function a ∈ L∞ (Γ) as an element of the Banach algebra ∞ L (Γ) is its essential range R(a) := λ ∈ C : |{t ∈ Γ : |a(t) − λ| < ε}| > 0 for all ε > 0 . Theorem 2.1 (Hartman-Wintner-Simonenko). Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a ∈ L∞ (Γ), then R(a) ⊂ spess T (a). Theorem 2.2 (Coburn-Simonenko). Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a ∈ L∞ (Γ), then T (a) is invertible on H p (Γ, w) if and only if it is Fredholm of index zero on H p (Γ, w). 2.2. Description of auxiliary results Section 3 contains necessary facts from the theory of weighted Hardy spaces over Carleson curves. Further, we state Simonenko’s factorization theorem (Theorem 3.8) and prove as a consequence that T (a) is Fredholm on H p (Γ, w) if and only if T (a−1 ) is Fredholm on H q (Γ, w−1 ). In Section 4 we collect some elements of the theory of vector-valued functions. We start with the definition and some properties of the weak (Pettis) integral. Further, we give the definition and formulate elementary properties of a vectorvalued version of the Riemann-Stieltjes integral. Properties of analytic functions with values in a Banach space X (or X-analytic functions) are the subject of Section 4.3. All these facts can be found in the monograph by Hille and Phillips [8]. Section 4.4 contains properties of analytic functions with values in weighted Lebesgue spaces as well as analytic functions with values in the set of measurable functions. These results are essentially due to Widom [13]. 2.3. Curves lying in the essential resolvent of T (a) cannot disconnect R(a) Having in mind that R(a) is a part of the essential spectrum of T (a) (see Theorem 2.1), we will first show that any curve disjoint with the essential spectrum of T (a) cannot disconnect R(a). We may without loss of generality assume that the origin belongs to the bounded component D+ of C \ Γ. For n ∈ Z, define χn : Γ → C,

χn (τ ) := τ n .

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Assume that a rectifiable Jordan curve C lies in the component ρn (T (a)) of the essential resolvent ρess (T (a)) := C \ spess T (a). Here ρn (T (a)) by definition consists of all λ such that Ind T (a − λ) = n. By using the Coburn-Simonenko theorem (Theorem 2.2), one can show that the operator T [χn (a − λ)] is invertible on H p (Γ, w), so fλ = T −1 [χn (a − λ)]1 is well-defined and Lp (Γ, w)-analytic on ρess (T (a)). Analogously, gλ = T −1 [χ−n /(a − λ)]1 is well-defined and Lq (Γ, w−1 )-analytic on ρess (T (a)) (see Lemma 5.1). In the proof of Lemma 5.2 it will be shown that fλ gλ = 1 and fλ satisfies the differential equation dfλ = P (a − λ)−1 gλ dλ in Lp (Γ, w) whose solution is ! Z λ −1 fλ = fλ0 exp (P) P (a − µ) dµ , (2.1) λ0

R where λ0 is a fixed point of C. Here (P) denotes the weak (Pettis) integral. If one takes the path of integration to be the entire curve C then it can be shown on the basis of (2.1) and the Lusin-Privalov theorem (see Theorem 3.1) that the following result holds. Lemma 2.3. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a ∈ L∞ (Γ) and C is a rectifiable Jordan curve lying in ρess (T (a)), then R(a) lies either entirely inside or entirely outside of C. The proof of this lemma is given in Section 5. We note here only that it is rather technical and makes heavy use of the auxiliary results stated in Sections 3 and 4. 2.4. Analytical continuation inside of a curve lying in the essential resolvent The next logical step is to show that if a rectifiable Jordan curve C is disjoint with spess T (a) and R(a) is not inside of C, then all other parts of spess T (a) are outside of C, too. There is an explicit formula which gives the solution of the equation T [χn (a − λ)]h = k

for λ ∈ ρn (T (a))

(2.2)

(see Lemma 6.1). This formula shows us how to continue h = hλ analytically to the interior of C and this continuation will provide the unique solution of (2.2). So the operator T [χn (a − λ)] is invertible for all λ in the interior of C. Therefore T (a−λ) = T (a)−λI is a Fredholm operator of index n. This implies the following.

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Lemma 2.4. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). Suppose that a ∈ L∞ (Γ) and that C is a rectifiable Jordan curve lying in ρess (T (a)). If R(a) lies entirely outside of C, then spess T (a) lies entirely outside of C. Details of the proof of this result will be provided in Section 6. In course of this proof several auxiliary functions appear whose analyticity, Lp (Γ, w)-analyticity, or Lq (Γ, w−1 )-analyticity must be checked. In these verifications we will use the results gathered in Section 4. 2.5. Proof of Theorem 1.2 Now we are in a position to outline a proof of the main result of this paper. It is well known that the essential spectrum of any bounded linear operator on a Banach space is a compact subset of the complex plane. Assume that spess T (a) is not connected. It is not difficult to show that in this case there exist two nonempty subsets K1 and K2 of spess T (a) such that K1 ∪ K2 = spess T (a) and two rectifiable Jordan curves γ1 ⊂ ρess (T (a)) and γ2 ⊂ ρess (T (a)) surrounding K1 and K2 , respectively, and such that K1 lies entirely outside of γ2 and K2 lies entirely outside of γ1 . From Lemma 2.3 it follows that R(a) lies either entirely inside or entirely outside of each of the curves γ1 and γ2 . Since R(a) ⊂ K1 ∪ K2 by Theorem 2.1, we conclude that either R(a) ⊂ K1 and R(a) ∩ K2 = ∅ or R(a) ⊂ K2 and R(a) ∩ K1 = ∅. On the other hand, Lemma 2.4 implies that spess T (a) lies entirely outside of γ2 in the first case and entirely outside of γ1 in the second case. Therefore K2 is empty in the first case and K1 is empty in the second case, which contradicts the assumption that spess T (a) is not connected. 2.6. Proof of Corollary 1.3 The proof is, in fact, a verbatim repetition of the proof of [3, Corollary 7.46]. Since T (a) − λI = T (a − λ) is a Fredholm operator for λ in each component of the complement of spess T (a), from Theorem 2.2 it follows that sp T (a) = spess T (a) ∪ {λ ∈ C : T (a − λ) is Fredholm with nonzero index}. From this equality and the fact that the index is constant on each connected component of the complement of spess T (a) it follows that sp T (a) is obtained by taking the union of a compact connected set spess T (a) (see Theorem 1.2) and some of the components of its complement. Thus, sp T (a) is connected.

3. Some known facts on Toeplitz operators on Hardy spaces with Muckenhoupt weights over Carleson curves 3.1. Smirnov classes and weighted Hardy spaces Let Γ be a rectifiable Jordan curve in the complex plane C. We denote by D+ and D− the bounded and unbounded component of C \ Γ, respectively. A function analytic on D+ is said to be in the Smirnov class E p (D+ ) (1 ≤ p < ∞) if there

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exists a sequence of rectifiable Jordan curves Γ1 , Γ2 , . . . in D+ approaching the boundary Γ in the sense that Γn eventually surrounds each point of D+ such that Z sup |f (z)|p |dz| < ∞. (3.1) n≥1

Γn

The Smirnov class E p (D− ) (1 ≤ p < ∞) is the set of all analytic functions f : D− ∪{∞} → C for which (3.1) holds with some sequence of curves Γ1 , Γ2 , . . . ⊂ D− tending to the boundary Γ in the sense that every point of D− ∪ {∞} lies outside Γn for some n. One can prove that every function f ∈ E p (D± ) has a non-tangential limit a.e. on Γ and that the boundary function belongs to Lp (Γ). We denote by p E p (Γ) (resp. E− (Γ)) the collection of all boundary functions in E p (D+ ) (resp. E p (D− )). Denote by E˙ p (D− ) the functions in E p (D− ) which vanish at infinity p and by E˙ − (Γ) the boundary functions on Γ of the functions in E˙ p (D− ). Finally, ∞ ∞ 1 put E (Γ) := L∞ (Γ) ∩ E 1 (Γ) and E− (Γ) := L∞ (Γ) ∩ E− (Γ). Theorem 3.1 (Lusin-Privalov). Let Γ be a rectifiable Jordan curve. A function in 1 E± (Γ) vanishes either almost everywhere or almost nowhere on Γ. For a proof, see e.g. [4, Theorem 10.3]. Theorem 3.2. For every rectifiable Jordan curve Γ, Z 1 1 n E (Γ) = f ∈ L (Γ) : f (τ )τ dτ = 0 for all n ∈ Z+ . Γ

A proof is in [4, Theorem 10.4]. The following facts are proved, for instance, in [1], Theorem 6.4 and Corollary 6.8, respectively. 1 (Γ) = {0}. Theorem 3.3. If Γ is a rectifiable Jordan curve, then E 1 (Γ) ∩ E˙ −

Lemma 3.4. If Γ is a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ), then H p (Γ, w) = E 1 (Γ) ∩ Lp (Γ, w), 1 H˙ p (Γ, w) = E˙ − (Γ) ∩ Lp (Γ, w), − p H− (Γ, w)

1 = E− (Γ) ∩ Lp (Γ, w).

We will need the following version of the Hölder inequality for weighted Hardy spaces. It is proved in [1, Lemma 6.11]. Lemma 3.5. Let Γ be a Carleson Jordan curve, 1 < p < ∞, 1/p + 1/q = 1, and w ∈ Ap (Γ). (a) If f ∈ H p (Γ, w) and g ∈ H q (Γ, w−1 ), then f g ∈ E 1 (Γ). p q 1 (b) If f ∈ H− (Γ, w) and g ∈ H− (Γ, w−1 ), then f g ∈ E− (Γ). If, in addition, p q −1 1 ˙ ˙ ˙ f ∈ H− (Γ, w) or g ∈ H− (Γ, w ), then f g ∈ E− (Γ). Let R(Γ) be the set of all rational functions without poles on Γ. From [1, Lemma 4.4] it follows that R(Γ) is dense in Lp (Γ, w). This fact immediately implies the following.

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Lemma 3.6. If Γ is a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ), then the set P R(Γ) of all rational functions with poles only in D− is dense in H p (Γ, w). 3.2. Wiener-Hopf factorization The next result follows immediately from [1, Theorem 6.24]. Proposition 3.7. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). For every n ∈ Z, the operator T (χn ) is Fredholm on H p (Γ, w), and Ind T (χn ) = −n. Recall that GL∞ (Γ) stands for the set of all functions in L∞ (Γ) which are invertible in L∞ (Γ). One says that a ∈ GL∞ (Γ) admits a Wiener-Hopf factorization in Lp (Γ, w) if a can be written in the form a(τ ) = a− (τ )τ κ a+ (τ )

for almost all τ ∈ Γ

where κ is an integer and the functions a± enjoy the following properties: p a− ∈ H− (Γ, w),

q −1 a−1 ), − ∈ H− (Γ, w

a+ ∈ H q (Γ, w−1 ),

p a−1 + ∈ H (Γ, w),

and |a−1 + |w ∈ Ap (Γ). The proof of the following factorization result in its full generality can be found in [1, Theorem 6.32] while its original version goes back to Simonenko (see e.g. [1, Section 6.13] for the history of the subject). Theorem 3.8 (Simonenko). Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a ∈ GL∞ (Γ), then T (a) is Fredholm on H p (Γ, w) if and only if a admits a Wiener-Hopf factorization a = a− χκ a+ in Lp (Γ, w). In that case the integer κ is uniquely determined and Ind T (a) = −κ. This result implies the following analogue of [3, Lemma 7.38]. Corollary 3.9. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). Suppose a ∈ GL∞ (Γ). Then T (a) is Fredholm on H p (Γ, w) if and only if T (a−1 ) is Fredholm on H q (Γ, w−1 ). In this case Ind T (a) = − Ind T (a−1 ). Proof. If T (a) is Fredholm on H p (Γ, w) and κ := − Ind T (a), then, by Theorem 3.8, the function a admits a Wiener-Hopf factorization a = a− χκ a+ in Lp (Γ, w). Put b± := a−1 ± . Then q b− ∈ H − (Γ, w−1 ),

p b−1 − ∈ H− (Γ, w),

b+ ∈ H p (Γ, w),

q −1 b−1 ), + ∈ H (Γ, w

−1 and |b+ |w ∈ Ap (Γ) ⇐⇒ |b−1 ∈ Aq (Γ). Thus a−1 = b− χ−κ b+ is a Wiener-Hopf + |w −1 q −1 factorization of a in L (Γ, w ). Applying Theorem 3.8 again, we conclude that T (a−1 ) is Fredholm on H q (Γ, w−1 ) and Ind T (a−1 ) = κ.

Proposition 3.10. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). ∞ If a− ∈ E− (Γ), a+ ∈ E ∞ (Γ), and b ∈ L∞ (Γ), then T (a− ba+ ) = T (a− )T (b)T (a+ ). For a proof, see [1, Section 6.7].

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4. Elements of the theory of vector-valued functions 4.1. Weak integration of vector-valued functions Let (S, Σ, µ) be a σ-finite measure space and X be a complex Banach space with the dual space X ∗ and the second dual space X ∗∗ . Following [8, Definition 3.5.4], a function x : S → X is said to be weakly measurable on S if the scalar valued functions x∗ (x) are measurable for all x∗ ∈ X ∗ . Theorem 4.1 (see [8, Theorem 3.7.1]). If x : S → X is weakly measurable and if x∗ (x) ∈ L1 (S, Σ, µ) for each x∗ ∈ X ∗ , then there exists x∗∗ ∈ X ∗∗ such that Z x∗∗ (x∗ ) = x∗ [x(σ)] dµ(σ) S

for all x∗ ∈ X ∗ . In general x∗∗ cannot be replaced by an element of X; when such a replacement can be made the integral is called a Pettis integral. More precisely (see [8, Definition 3.7.1]), a function x : S → X is said to be Pettis integrable if there is an element xE for each E ∈ Σ such that Z ∗ x (xE ) = x∗ [x(σ)] dµ(σ) E ∗

∗

for all x ∈ X , where the integral on the right is supposed to exist in the sense of Lebesgue. By definition, Z (P) x(σ) dµ(σ) := xE . E

If X is reflexive, then it is clear from Theorem 4.1 that x : S → X is Pettis integrable if and only if x∗ (x) ∈ L1 (S, Σ, µ) for each x∗ ∈ X ∗ . Theorem 4.2 (see [8, Theorem 3.7.3]). If T is a bounded linear operator on X and x ∈ X is Pettis integrable, then so is T x and Z (P) T x(σ) dµ(σ) = T xE . E

4.2. Strong integration of vector-valued functions The Riemann-Stieltjes integral can be extended to vector-valued functions as follows. Let X be a Banach space, x : [α, β] → X and g : [α, β] → C. We denote the partition α = σ0 ≤ σ1 ≤ · · · ≤ σn = β together with the points τi ∈ [σi−1 , σi ] by π and set |π| = maxi |σi − σi−1 |. Following [8, Definition 3.3.1], let Sπ (x, g) :=

n X i=1

x(τi )[g(σi ) − g(σi−1 )].

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Then if lim|π|→0 Sπ exists in a given topology, one defines this limit to be the generalized Riemann-Stieltjes integral Z β (RS) x(ξ) dg(ξ) α

relative to this topology. The following facts are proved in [8, Theorem 3.3.2 and Corollary 1]. Theorem 4.3. Suppose X is a Banach space. If x : [α, β] → X is a strongly continuous vector-valued function and g : [α, β] → C is a function of bounded variation, then the integral Z β x(ξ) dg(ξ) (RS) α

exists in the norm topology of X. Theorem 4.4. For function pairs of the type considered in Theorem 4.3, we have Z β Z δ Z β x(ξ) dg(ξ) = (RS) x(ξ) dg(ξ) + (RS) x(ξ) dg(ξ), (4.1) (RS) α

α

where α < δ < β, and

Z

(RS)

β

α

δ

x(ξ) dg(ξ)

X

≤

sup kx(ξ)kX

Var g.

(4.2)

α≤ξ≤β

Notice that every parametrization g : [α, β] → γ of a rectifiable curve γ is a continuous function of bounded variation. We will write Z Z β (RS) x(µ) dµ instead of (RS) x(g(ξ)) dg(ξ). γ

α

Theorem 4.5. Suppose X is a Banach space. If γ is a rectifiable curve in the complex plane and x : γ → X is a strongly continuous function, then x is RiemannStieltjes and Pettis integrable and the values of the integrals coincide: Z Z (RS) x(µ) dµ = (P) x(µ) dµ. γ

γ

Proof. Every Riemann-Stieltjes integrable function is Bochner integrable. Every Bochner integrable function is also Pettis integrable and the integrals have the same value (see [8, p. 80]). 4.3. Analytic vector-valued functions Let Ω be a non-empty open set in the complex plane and let X be a Banach space. A function x : Ω → X is said to be X-analytic at λ0 ∈ Ω if there is a disk D(λ0 , δ) = {λ ∈ C : |λ − λ0 | < δ}

(4.3)

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and a sequence {aj }∞ j=0 of elements of X which do not depend on λ and such that for all λ ∈ D(λ0 , δ) the series ∞ X

aj (λ − λ0 )j

(4.4)

j=0

converges to x in the norm of X. If x is X-analytic at each point of Ω, then x is called X-analytic on Ω. A function x : Ω → X is said to be strongly differentiable or X-differentiable in Ω if for each λ0 ∈ Ω the limit 1 x(λ) − x(λ0 ) x0 (λ0 ) := lim λ→λ0 λ − λ0 exists in the norm of X. This limit is called the derivative of x at λ0 . Theorem 4.6 (see e.g. [5, Section I.1]). Let Ω be a non-empty open set in the complex plane and let X be a Banach space. A function x : Ω → X is X-analytic on Ω if and only if it is X-differentiable on Ω. Lemma 4.7. Let X be a Banach space and Y be its closed subspace. If x is Xanalytic on an open set Ω ⊂ C and x(λ) ∈ Y for all λ ∈ Ω, then x0 (λ) ∈ Y for all λ ∈ Ω. The proof is straightforward and is therefore omitted. From [8, Theorems 3.10.1 and 3.11.1] we obtain the following generalization of the Cauchy theorem. Theorem 4.8. Let D be a domain in the complex plane and X be a Banach space. If x : D → X is an X-analytic function, then Z (RS) x(ζ) dζ = 0 C

for every rectifiable Jordan curve C in D such that the interior of C belongs to D. From [8, Theorems 3.10.1 and 3.13.1] we get the following. Theorem 4.9. Let Ω be a domain in the complex plane and C be its boundary. Suppose X is a Banach space and x : Ω ∪ C → X is a strongly continuous function such that sup kx(ζ)kX = M, ζ∈C

If x is X-analytic on Ω, then either kx(ζ)kX = M or kx(ζ)kX < M for all ζ ∈ Ω. We will need the following generalization of Vitali’s theorem to vector-valued functions (see [8, Theorem 3.14.1]). Theorem 4.10. Let X be a Banach space and D be a domain in the complex plane. Let {xn }∞ n=1 be a sequence of X-analytic functions xn : D → X such that

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kxn (ξ)kX ≤ M for all n ∈ N and all ξ ∈ D, where M is an absolute constant. Let there be a set {ξk } ⊂ D having a limit point ξ0 ∈ D such that lim xn (ξk )

n→∞

exists for each k ∈ N. Then lim xn (ξ)

n→∞

exists everywhere in D, the convergence is uniform with respect to ξ on any compact subset of D, and the limit function x is X-analytic on D. 4.4. Analytic functions with values in weighted Lebesgue spaces Let Ω be a non-empty open set in the complex plane and assume that for each λ ∈ Ω there is associated a measurable a.e. finite function fλ : Γ → C. One says that f is analytic on Ω if for each λ0 ∈ Ω there is a disk (4.3) and a sequence {aj }∞ j=0 of measurable functions aj : Γ → C such that for all λ ∈ D(λ0 , δ) the series (4.4) converges a.e. on Γ to fλ . If fλ is given by (4.4), then the derivative of fλ is defined by ∞ X dfλ jaj (λ − λ0 )j−1 = dλ j=0 for all λ ∈ D(λ0 , δ). If λ ∈ D(λ0 , δ0 ) ∩ D(λ1 , δ1 ), then the two possible interpretations of dfλ /dλ agree a.e. Moreover, if fλ is analytic, then dfλ /dλ is analytic, too. Lemma 4.11 (see [13, Lemma 1]). Let Γ be a rectifiable Jordan curve. If f is L1 (Γ)-analytic on an open set Ω ⊂ C, then f is analytic on Ω. The following lemma in the non-weighted case is proved in [13, Lemma 3]. Lemma 4.12. Let Γ be a rectifiable Jordan curve, 1 < p < ∞, and w : Γ → [0, ∞] be a weight such that w ∈ Lp (Γ) and w−1 ∈ Lq (Γ). Let C be a rectifiable Jordan curve contained in a simply connected open set Ω ⊂ C. Suppose f is analytic on Ω and sup kfµ kLp (Γ,w) = M < ∞. (4.5) µ∈C

Then f is Lp (Γ, w)-analytic inside C and for all λ inside C we have kfλ kLp (Γ,w) ≤ M. Proof. The proof is developed by minor adjustments of Widom’s arguments [13, Lemma 3]. First, for the convenience of the reader we repeat arguments from the proof of [13, Lemma 2], which will be needed later. Let λ0 be inside of C and let δ be so small that D(λ0 , δ) is entirely inside of C and fλ =

∞ X j=0

aj (λ − λj )j

(4.6)

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a.e. on Γ for each λ ∈ D(λ0 , δ). Then the series j ∞ X δ aj 2 j=0

95

(4.7)

converges a.e. on Γ and so, by Egoroff’s theorem, there is a set Fε whose complement Γ \ Fε has measure at most ε on which (4.7) converges uniformly. There is a constant N such that for all τ ∈ Fε and all j we have −j δ N. (4.8) |aj (τ )| ≤ 2 Now let λ1 be any point in the disk D(λ0 , δ/2). Then (4.8) shows that for λ ∈ D(λ1 , δ/2 − |λ1 − λ0 |) the series ∞ X

aj (λ − λ0 )j ,

j=0

which converges a.e. to fλ , may be rearranged into a power series in λ − λ1 which converges uniformly for τ ∈ Fε . This shows that f restricted to Fε is L∞ (Fε )analytic on D(λ0 , δ/2). Now we can find a countable set of disks D(λn , δn ), where n ∈ N, of the type just considered and such that Ω=

∞ [

D(λn , δn /2).

n=1

For each n, there is a set Fε,n whose complement Γ \ Fε,n has measure at most ε/2n and such that f restricted to Fε,n is L∞ (Fε,n )-analytic on D(λn , δn /2). But then ∞ \ Eε = Fε,n n=1

has complement Γ \ Eε of measure at most ε and f restricted to Eε is L∞ (Eε )analytic on Ω. Moreover, if we restrict f to Eε , then the series (4.6) converges uniformly on D(λ0 , δ/2). Further we will argue by analogy with the proof of [13, Lemma 3]. Take any function g ∈ L∞ (Γ). Then ! Z Z ∞ X j fλ (τ )g(τ ) dτ = aj (τ )(λ − λ0 ) g(τ ) dτ Eε

Eε

=

j=0

∞ Z X j=0

aj (τ )g(τ ) dτ

(λ − λ0 )j

Eε

for all λ ∈ D(λ0 , δ/2) because of the uniform convergence of (4.6) in D(λ0 , δ/2). On the right-hand side we have a power series with scalar coefficients. By the Cauchy

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estimates for the coefficients of such a power series (see e.g. [7, Section 1.10]), we have −j Z Z δ (4.9) aj (τ )g(τ ) dτ ≤ fλ (τ )g(τ ) dτ . max 2 |λ−λ0 |=δ/2 Eε

Eε

But since f restricted to Eε is L∞ (Eε )-analytic on Ω, Z fλ (τ )g(τ ) dτ Eε

is a complex-valued analytic function on Ω, and so for any λ inside of C we have Z Z . (4.10) ≤ max f (τ )g(τ ) dτ f (τ )g(τ ) dτ µ λ µ∈C Eε

Eε

By H¨ older’s inequality, Z fµ (τ )g(τ ) dτ ≤ kfµ kLp (Γ,w) kgkLq (Γ,w−1 ) .

(4.11)

Eε

Combining (4.5), (4.10), and (4.11), we arrive at Z ≤ M kgkLq (Γ,w−1 ) f (τ )g(τ ) dτ λ

(4.12)

Eε ∞

for all ε > 0 and all g ∈ L (Γ). This inequality and (4.9) give −j Z δ aj (τ )g(τ ) dτ ≤ M kgkLq (Γ,w−1 ) 2

(4.13)

Eε

for all ε > 0 and all g ∈ L∞ (Γ). Passing in (4.12) and (4.13) to the limit as ε → 0, we obtain Z fλ (τ )g(τ ) dτ ≤ M kgkLq (Γ,w−1 ) (4.14) Γ

and

−j Z aj (τ )g(τ ) dτ ≤ δ M kgkLq (Γ,w−1 ) 2

(4.15)

Γ

for all g ∈ L∞ (Γ), respectively. Since L∞ (Γ) is dense in Lq (Γ, w−1 ), we have for any measurable function ϕ on Γ, Z 1 ϕ(τ )g(τ ) dτ : g ∈ L∞ (Γ) \ {0} sup kgkLq (Γ,w−1 ) Γ Z 1 q −1 = sup ϕ(τ )g(τ ) dτ : g ∈ L (Γ, w ) \ {0} = kϕkLp (Γ,w) . −1 kgk q L (Γ,w

)

Γ

Applying this fact to (4.14) with ϕ = fλ and to (4.15) with ϕ = aj , we obtain −j δ kfλ kLp (Γ,w) ≤ M, kaj kLp (Γ,w) ≤ M. 2 The former inequality holds for all λ inside of C. The latter inequality implies that the series (4.6) converges in Lp (Γ, w) for each λ ∈ D(λ0 , δ/2). Thus f is Lp (Γ, w)-analytic inside of C.

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Lemma 4.13. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). Suppose C is a rectifiable Jordan curve contained in a simply connected open set Ω ⊂ C. (a) If Fλ is Lp (Γ, w)-analytic on Ω and Fλ ∈ H p (Γ, w) for all λ ∈ C, then Fλ ∈ H p (Γ, w) for all λ inside of C. (b) If Gλ is Lq (Γ, w−1 )-analytic on Ω and Gλ ∈ H q (Γ, w−1 ) for all λ ∈ C, then Gλ ∈ H q (Γ, w−1 ) for all λ inside of C. Proof. (a) Fix n ∈ Z+ . Since Fλ is Lp (Γ, w)-analytic on Ω, for every λ0 ∈ Ω there p exists a disk D(λ0 , δ) and a sequence {aj }∞ j=0 of elements of L (Γ, w) such that the series ∞ X aj (λ − λ0 )j j=0

converges to Fλ in the norm of Lp (Γ, w) for all λ ∈ D(λ0 , δ). Without loss of generality we can assume that δ is less than the radius of convergence of the series. Hence ∞ X kaj kLp (Γ,w) |λ − λ0 |j < ∞ j=0

for all λ ∈ D(λ0 , δ). Therefore, by Hölder’s inequality, ∞ Z ∞ X X j n n |aj (τ )(λ − λ0 ) τ | |dτ | ≤ kτ kLq (Γ,w−1 ) kaj kLp (Γ,w) |λ − λ0 |j < ∞. j=0

Γ

j=0

This means that we can apply the Lebesgue dominated convergence theorem (see, e.g., [11, Theorem 1.38]): ! Z Z X ∞ n j Fλ (τ )τ dτ = aj (τ )(λ − λ0 ) τ n dτ Γ

Γ

Z = Γ

j=0 ∞ X

! n

j

aj (τ )τ (λ − λ0 )

dτ =

∞ Z X j=0

j=0

aj (τ )τ n dτ

(λ − λ0 )j

Γ

for all λ ∈ D(λ0 , δ). This means that the function Z Φn : Ω → C, λ 7→ Fλ (τ )τ n dτ Γ

is analytic. Since Fλ ∈ H p (Γ, w) for all λ ∈ C, from Theorem 3.2 and Lemma 3.4 it follows that Φn (λ) = 0 for all n ∈ Z+ and all λ ∈ C. By the maximum principle (apply Theorem 4.9 with X = C), Φn (λ) = 0 for all n ∈ Z+ and all λ inside of C. Applying Theorem 3.2 and Lemma 3.4 again, we conclude that Fλ ∈ H p (Γ, w) for all λ inside of C. Part (a) is proved. (b) The proof is the same as for part (a).

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5. Curves in the essential resolvent cannot disconnect the essential range of the symbol 5.1. Solutions of the equations T [χn (a − λ)] = 1 and T [χ−n /(a − λ)] = 1 If a is a function in L∞ (Γ), then the essential resolvent ρess (T (a)) of T (a) is the open set of those λ in C for which T (a − λ) is a Fredholm operator. Put ρn (T (a)) = λ ∈ ρess (T (a)) : Ind T (a − λ) = n . Let λ ∈ ρn (T (a)). From Propositions 3.7 and 3.10 it follows that T [χn (a − λ)] is a Fredholm operator of index zero on H p (Γ, w). By Corollary 3.9, the operator T [χ−n /(a − λ)] is a Fredholm operator of index zero on H q (Γ, w−1 ). From Theorem 2.2 we conclude that the operators T [χn (a − λ)] and T [χ−n /(a − λ)] are invertible on H p (Γ, w) and H q (Γ, w−1 ), respectively. Hence the functions fλ := T −1 [χn (a − λ)]χ0 gλ := T

−1

[χ−n /(a − λ)]χ0

for λ ∈ ρn (T (a)), for λ ∈ ρn (T (a))

are well-defined for every λ ∈ ρess (T (a)). It is clear that fλ ∈ H p (Γ, w) and gλ ∈ H q (Γ, w−1 ). Lemma 5.1. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). Then: (a) The function fλ is Lp (Γ, w)-analytic on ρess (T (a)). (b) The function gλ is Lq (Γ, w−1 )-analytic on ρess (T (a)). Proof. We will prove only part (b). Denote by B := B(H q (Γ, w−1 )) the Banach algebra of all bounded linear operators on H q (Γ, w−1 ). Let λ0 ∈ ρn (T (a)) and 1 R := dist λ0 , ∂ρn (T (a)) . 2 It is clear that T [χ−n /(a − λ)] − T [χ−n /(a − λ0 )] = T [(λ − λ0 )χ−n (a − λ)−1 (a − λ0 )−1 ] (5.1) = (λ − λ0 )T [χ−n (a − λ)−1 (a − λ0 )−1 ]. Since D(λ0 , R) ⊂ ρn (T (a)), from Theorem 2.1 it follows that k(a − λ)−1 kL∞ (Γ) is a continuous function of λ on D(λ0 , R). From (5.1) we get for λ ∈ D(λ0 , R),

T [χ−n /(a − λ)] − T [χ−n /(a − λ0 )] B

≤ |λ − λ0 | kP kB(Lq (Γ,w−1 )) kχ−n (a − λ)−1 (a − λ0 )−1 kL∞ (Γ) ≤ |λ − λ0 | kP kB(Lq (Γ,w−1 )) kχ−n (a − λ0 )−1 kL∞ (Γ) ! ×

sup

−1

k(a − λ)

kL∞ (Γ)

λ∈D(λ0 ,R)

=: |λ − λ0 |Mn . We know already that T [χ−n /(a − λ0 )] is invertible on H q (Γ, w−1 ). If −1 Rn := min R, Mn kT −1 [χ−n /(a − λ0 )]kB

(5.2)

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99

and λ ∈ D(λ0 , Rn ), then (5.2) implies that kT [χ−n /(a − λ)] − T [χ−n /(a − λ0 )]kB ≤ |λ − λ0 |Mn ≤

Rn Mn kT −1 [χ−n /(a − λ0 )]kB ≤ kT −1 [χ−n /(a − λ0 )]k−1 B . kT −1 [χ−n /(a − λ0 )]kB

Hence, for λ ∈ D(λ0 , Rn ), the operator T [χ−n /(a − λ)] is invertible on H q (Γ, w−1 ) and its inverse is given by the Neumann series expansion T −1 [χ−n /(a − λ)] ∞ n X oj = T −1 [χ−n /(a − λ0 )] T [χ−n /(a − λ)] − T [χ−n /(a − λ0 )] j=0

× T −1 [χ−n /(a − λ0 )] ∞ n oj X = (λ − λ0 )j T −1 [χ−n /(a − λ0 )]T [χ−n (a − λ)−1 (a − λ0 )−1 ]

(5.3)

j=0

× T −1 [χ−n /(a − λ0 )]. This series converges absolutely in the norm of B on the disk D(λ0 , Rn ). However, it is not a power series. For λ ∈ D(λ0 , Rn ), put a0 (λ) := Aχ0 , A := T −1 [χ−n /(a − λ0 )],

B(λ) := T [χ−n /(a − λ)],

and j aj (λ) := A[B(λ) − B(λ0 )] Aχ0 = A[B(λ) − B(λ0 )]aj−1 (λ), where j ∈ N. From (5.3) it follows that gλ = T −1 [χ−n /(a − λ)]χ0 =

∞ X

aj (λ).

(5.4)

j=0

Let us show that for every j ∈ Z+ , the function aj (λ) is H q (Γ, w−1 )-differentiable on the disk D(λ0 , Rn ). Fix λ1 ∈ D(λ0 , Rn ). Then B(λ) − B(λ1 ) − T [χ−n (a − λ1 )−2 ] λ − λ1 T [χ−n /(a − λ)] − T [χ−n /(a − λ1 )] − T [χ−n (a − λ1 )−2 ] = λ − λ1 = T [χ−n (a − λ)−1 (a − λ1 )−1 ] − T [χ−n (a − λ1 )−2 ] = T χ−n (a − λ1 )−1 {(a − λ)−1 − (a − λ1 )−1 } = (λ − λ1 )T [χ−n (a − λ)−1 (a − λ1 )−1 ].

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Consequently,

B(λ) − B(λ1 )

−2

− T [χ−n (a − λ1 ) ]

λ − λ1 B ≤ |λ − λ1 | kP kB(Lq (Γ,w−1 )) kχ−n (a − λ1 )−1 kL∞ (Γ) ! ×

sup

k(a − λ)−1 kL∞ (Γ)

,

λ∈D(λ0 ,R)

that is,

B(λ) − B(λ1 )

0

− B (λ ) 1 = O(|λ − λ1 |)

λ − λ1 B where B 0 (λ1 ) = T [χ−n (a − λ1 )−2 ], and therefore

B(λ) − B(λ1 )

= O(1) as

λ − λ1 B

λ → λ1 ,

as

λ → λ1 .

(5.5)

(5.6)

It is obvious that the function a0 (λ) is H q (Γ, w−1 )-differentiable at λ1 . Suppose that aj−1 (λ) is H q (Γ, w−1 )-differentiable at λ1 , that is, there exists a function a0j−1 (λ1 ) ∈ H q (Γ, w−1 ) such that

aj−1 (λ) − aj−1 (λ1 )

0

= o(1) as λ → λ1 . (5.7) − aj−1 (λ1 )

q λ − λ1 H (Γ,w−1 ) Hence kaj−1 (λ) − aj−1 (λ1 )kH q (Γ,w−1 ) = o(1)

as

λ → λ1 .

(5.8)

Put a0j (λ1 ) := AB 0 (λ1 )aj−1 (λ1 ) + A[B(λ1 ) − B(λ0 )]a0j−1 (λ1 ). Then aj (λ) − aj (λ1 ) − a0j (λ1 ) λ − λ1 B(λ) − B(λ1 ) − B 0 (λ1 ) aj−1 (λ1 ) =A λ − λ1 B(λ) − B(λ1 ) +A aj−1 (λ) − aj−1 (λ1 ) λ − λ1 aj−1 (λ) − aj−1 (λ1 ) 0 + A[B(λ1 ) − B(λ0 )] − aj−1 (λ1 ) . λ − λ1 From (5.5)–(5.9) it follows that

aj (λ) − aj (λ1 )

0

− aj (λ1 )

q λ − λ1 H (Γ,w−1 )

(5.9)

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101

B(λ) − B(λ1 )

0

≤ kAkB − B (λ ) 1 kaj−1 (λ1 )kH q (Γ,w−1 )

λ − λ1 B

B(λ) − B(λ1 )

kaj−1 (λ) − aj−1 (λ1 )kH q (Γ,w−1 ) + kAkB

λ − λ1 B

aj−1 (λ) − aj−1 (λ1 )

0

= o(1) + kAkB kB(λ) − B(λ1 )kB − aj−1 (λ1 )

q λ − λ1 H (Γ,w−1 ) as λ → λ1 . This means that aj (λ) is H q (Γ, w−1 )-differentiable at λ1 . Thus, by induction, aj (λ) is H q (Γ, w−1 )-differentiable at every point λ1 ∈ D(λ0 , Rn ). Then every aj (λ) is H q (Γ, w−1 )-analytic on D(λ0 , Rn ) by Theorem 4.6. Put Sm (λ) :=

m X

aj (λ).

j=0

Clearly, Sm (λ) is H q (Γ, w−1 )-analytic on D(λ0 , Rn ) for every m ∈ Z+ . Since the series (5.3) is absolutely convergent in the norm of B on the disk D(λ0 , Rn ), we infer that there exists a constant M > 0 such that kSm (λ)kH q (Γ,w−1 ) ≤ M for all λ ∈ D(λ0 , Rn ) and all m ∈ Z+ . We already know that Sm (λ) converges at every point λ ∈ D(λ0 , Rn ). By Theorem 4.10, gλ =

∞ X

aj (λ) = lim Sm (λ) m→∞

j=0

is H q (Γ, w−1 )-analytic on D(λ0 , Rn ). Hence this function is Lq (Γ, w−1 )-analytic on the essential resolvent ρess (T (a)). Part (b) is proved. The proof of part (a) is similar and is even simpler because T −1 [χn (a − λ)] can be represented as a power series (which is not the case for T −1 [χ−n /(a − λ)], see (5.3)). The following lemma is an analogue of [13, Lemma 6] (see also [3, Corollary 7.41]). Lemma 5.2. Let γ be a rectifiable curve lying in ρess (T (a)) and having initial and terminal points λ0 and λ, respectively. Then Z −1 fλ = fλ0 exp (P) P (a − µ) dµ , (5.10) γ Z gλ = gλ0 exp −(P) P (a − µ)−1 dµ . (5.11) γ

Proof. The proof is developed by analogy with the proof of [13, Lemma 6] (see also [3, Proposition 7.40]). From the definition of fλ and gλ it follows that T [χn (a − λ)]fλ = χ0 ∈ H p (Γ, w),

T [χ−n /(a − λ)]gλ = χ0 ∈ H q (Γ, w−1 ).

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p q Hence there exist uλ ∈ H˙ − (Γ, w) and vλ ∈ H˙ − (Γ, w−1 ) such that

χn fλ (a − λ) = χ0 + uλ ,

(5.12)

χ−n gλ /(a − λ) = χ0 + vλ .

(5.13)

Multiplying these two identities, we obtain fλ gλ = χ0 + uλ + vλ + uλ vλ or fλ gλ − χ0 = uλ + vλ + uλ vλ . From Lemmas 3.4 and 3.5 it follows that the left-hand side of the latter identity 1 belongs to E 1 (Γ), while the right-hand side lies in E˙ − (Γ). Theorem 3.3 implies that uλ + vλ + uλ vλ = 0 and fλ gλ = 1. (5.14) By Lemma 5.1(a), the function fλ is Lp (Γ, w)-analytic. Then from (5.12) it follows that uλ is also Lp (Γ, w)-analytic. Differentiation of both sides of (5.12) yields −χn fλ + χn (a − λ)

duλ dfλ = dλ dλ

and hence dfλ duλ − χ−n . dλ dλ Multiplying both sides of this identity by gλ /(a − λ) and using (5.13)–(5.14), we obtain dfλ duλ 1/(a − λ) = fλ gλ /(a − λ) = gλ − (1 + vλ ) . dλ dλ ∞ p From Theorem 2.1 it follows that 1/(a − λ) ∈ L (Γ) ⊂ L (Γ, w). Then the above identity can be rewritten as fλ = (a − λ)

dfλ duλ = −Q(a − λ)−1 − (1 + vλ ) . (5.15) dλ dλ p Since fλ ∈ H p (Γ, w) and uλ ∈ H˙ − (Γ, w) for all λ ∈ ρess (T (a)), from Lemma 4.7 p p we conclude that dfλ /dλ ∈ H (Γ, w) and duλ /dλ ∈ H˙ − (Γ, w). By Lemma 3.5, 1 (Γ). From Lemma 3.4 we conclude gλ dfλ /dλ ∈ E 1 (Γ) and (1 + vλ )duλ /dλ ∈ E˙ − 1 (Γ). Thus the left-hand side of that P (a − λ)−1 ∈ E 1 (Γ) and Q(a − λ)−1 ∈ E˙ − 1 1 (5.15) belongs to E (Γ), while the right-hand side of (5.15) belongs to E˙ − (Γ). Then from Theorem 3.3 and (5.15) it follows that P (a − λ)−1 − gλ

dfλ . dλ Multiplying the latter identity by fλ and taking into account (5.14), we arrive at P (a − λ)−1 = gλ

dfλ = fλ P (a − λ)−1 . (5.16) dλ Now consider a disk D(λ0 , δ) inside of which we have series representations fλ =

∞ X j=0

aj (λ − λ0 )j ,

P (a − λ)−1 =

∞ X j=0

bj (λ − λ)j .

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For each λ ∈ D(λ0 , δ) the two series converge a.e. on Γ and this implies that for all τ not belonging to some subset N of measure zero of the curve Γ the series converge for all λ ∈ D(λ0 , δ). Let us denote by U (τ, λ) and V (τ, λ) the sums of the two series, respectively. They are defined for τ ∈ Γ \ N and λ ∈ D(λ0 , δ). Equation (5.16) is equivalent to the statement that for each j ≥ 0 the identity (j + 1)aj+1 (τ ) =

j X

am (τ )bj−m (τ )

m=0

holds for almost all τ ∈ Γ. It follows that for all τ not belonging to a set N1 ⊂ Γ of measure zero the above identities hold for all j ≥ 0. Thus ∂ U (τ, λ) = U (τ, λ)V (τ, λ) for all τ ∈ Γ \ (N ∪ N1 ), λ ∈ D(λ0 , δ). ∂λ This is a linear first order ordinary differential equation. For any rectifiable curve γ which lies in D(λ0 , δ) and has initial point λ0 and terminal point λ, one has (see e.g. [7, Section 5.1]) the following solution of the Cauchy problem: Z U (τ, λ) = U (τ, λ0 ) exp V (τ, µ) dµ (5.17) γ

for all τ ∈ Γ \ (N ∪ N1 ). From Theorem 2.1 it follows that (a − µ)−1 : γ → Lp (Γ, w) is continuous in µ. Therefore P (a − µ)−1 : γ → Lp (Γ, w) also is continuous in µ. Since γ is a compact set, kP (a − µ)−1 kLp (Γ,w) ≤ M (γ) for all µ ∈ γ, where M (γ) is a positive constant depending only on γ. Assume that g ∈ Lq (Γ, w−1 ). By H¨ older’s inequality, Z Z Z kP (a − µ)−1 kLp (Γ,w) kgkLq (Γ,w−1 ) dµ P (a − µ)−1 (τ )g(τ ) |dτ | dµ ≤ γ

γ

Γ

≤ kgkLq (Γ,w−1 ) M (γ)|γ| < ∞. Then, by the Fubini theorem (see e.g. [11, Theorem 8.8]), Z Z Z Z V (τ, µ) dµ g(τ ) |dτ | = V (τ, µ)g(τ ) |dτ | dµ Γ γ γ Γ Z Z = P (a − µ)−1 (τ )g(τ ) |dτ | dµ Therefore

Z V (τ, µ) dµ = γ

γ

Γ

Z

P (a − µ)−1 dµ (τ )

(P)

(5.18)

γ

for almost all τ ∈ Γ. Since fλ (τ ) = U (τ, λ) for all τ ∈ Γ \ (N ∪ N1 ), from (5.17)– (5.18) it follows that (5.10) holds for curves γ joining λ0 and λ and lying entirely within the disk D(λ0 , δ). But any rectifiable curve lying in ρess (T (a)) may be obtained by joining finitely many curves of this type, so (5.10) holds in general. Formula (5.11) is an immediate consequence of (5.10) and (5.14).

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5.2. Proof of Lemma 2.3 The proof follows the scheme of the proof of [3, Prpoposition 7.42]. Let us apply Lemma 5.2 with γ = C and observe that by (5.14) the function fλ is almost nowhere zero. Then Z exp (P) P (a − µ)−1 dµ = 1. (5.19) C

For almost each τ ∈ Γ, the integral Z 1 (a(τ ) − µ)−1 dµ 2πi C equals the winding number of the curve C with respect to the point a(τ ). Thus the function f : Γ → C defined by Z 1 f (τ ) := (a(τ ) − µ)−1 dµ (τ ∈ Γ) (5.20) 2πi C is integer-valued. From Theorem 2.1 it follows that (a − µ)−1 : C → Lp (Γ, w) is continuous in µ. Since C is a compact set, k(a − µ)−1 kLp (Γ,w) ≤ M (C) for all µ ∈ C, where M (C) is a positive constant depending only on C. Assume that g ∈ Lq (Γ, w−1 ). By H¨ older’s inequality, Z Z Z k(a − µ)−1 kLp (Γ,w) kgkLq (Γ,w−1 ) dµ (a(τ ) − µ)−1 g(τ ) |dτ | dµ ≤ C

Γ

C

≤ kgkLq (Γ,w−1 ) M (C)|C| < ∞. Hence, application of the Fubini theorem (see e.g. [11, Theorem 8.8]) to the iterated integral on the left-hand side is justified. Therefore, multiplying (5.20) by g(τ ), integrating over Γ and then changing the order of integration, we get Z Z Z 1 −1 f (τ )g(τ ) |dτ | = (a(τ ) − µ) dµ g(τ ) |dτ | 2πi Γ Γ Z Z C 1 = (a(τ ) − µ)−1 g(τ ) |dτ | dµ. 2πi C Γ By definition of the Pettis integral, Z 1 (P) (a − µ)−1 dµ. f= 2πi C Therefore

Z 1 Pf = (P) P (a − µ)−1 dµ (5.21) 2πi C by Theorem 4.2. From (5.19) and (5.21) it follows that exp(2πiP f ) = 1. Thus there p exists an integer number N such that P f = N . Then f ∈ H− (Γ, w). The function f is integer-valued. Hence there is an integer number M and a subset ΓM ⊂ Γ of positive measure such that f − M = 0 for almost all points of ΓM . Further, p 1 (Γ) in view of Lemma 3.4. By Theorem 3.1, f = M a.e. (Γ, w) ⊂ E− f − M ∈ H−

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on Γ. Hence, the winding number of C with respect to R(a) is constant. Thus R(a) lies either entirely inside or entirely outside of C.

6. Analytical continuation of fλ and gλ to any component of C \ ρess (T (a)) which does not contain R(a) 6.1. Relation between fλ , gλ , and the inverse of T [χn (a − λ)] The remainder of the proof consists of showing that we can analytically continue solutions fλ and gλ to any component of the complement of the essential resolvent of T (a) which does not contain R(a). First of all we need to relate the inverse of T [χn (a − λ)] and the functions fλ and gλ . Lemma 6.1. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a ∈ L∞ (Γ), λ ∈ ρn (T (a)), and k ∈ E ∞ (Γ), then hλ = fλ P [χ−n gλ k/(a − λ)] ∈ H p (Γ, w)

(6.1)

and T [χn (a − λ)]hλ = k. Proof. This statement is proved by analogy with [3, Lemma 7.43]. We already know that the operators T [χn (a − λ)] and T [χ−n /(a − λ)] are invertible on H p (Γ, w) and H q (Γ, w−1 ), respectively. Since k ∈ E ∞ (Γ) ⊂ H p (Γ, w), we have hλ := T −1 [χn (a − λ)]k ∈ H p (Γ, w). p q Then there exist ` ∈ H˙ − (Γ, w) and vλ ∈ H˙ − (Γ, w−1 ) such that

χn (a − λ)hλ = k + `,

χ−n gλ /(a − λ) = 1 + vλ .

Multiplying these identities, we obtain hλ gλ = (k + `)(1 + vλ ) = χ−n gλ k/(a − λ) + `(1 + vλ ).

(6.2)

p (Γ, w), H˙ −

Since gλ ∈ H q (Γ, w−1 ), hλ ∈ H p (Γ, w) and ` ∈ from Lemma 3.5 we conclude that hλ gλ ∈ E 1 (Γ) and `(1 + k ∈ E ∞ (Γ), we see that χ−n gλ k/(a − λ) ∈ Lq (Γ, w−1 ). Then

q (Γ, w−1 ), vλ ∈ H˙ − 1 ˙ vλ ) ∈ E− (Γ). Since

P [χ−n gλ k/(a − λ)] ∈ H q (Γ, w−1 ) ⊂ E 1 (Γ), Q[χ−n gλ k/(a − λ)] ∈ H˙ q (Γ, w−1 ) ⊂ E˙ 1 (Γ) −

−

(see Lemma 3.4), and we can rewrite (6.2) as hλ gλ − P [χ−n gλ k/(a − λ)] = Q[χ−n gλ k/(a − λ)] + `(1 + vλ ), where the left-hand side belongs to E 1 (Γ) and the right hand side belongs to 1 E˙ − (Γ). From Theorem 3.3 we conclude that both sides are equal to zero and hence hλ gλ = P [χ−n gλ k/(a − λ)]. Multiplying both sides by fλ and taking into account (5.14), we arrive at (6.1).

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6.2. Analyticity of auxiliary functions In this subsection we will prove various kinds of analyticity of some auxiliary functions. Lemma 6.2. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). If a simply connected open set Ω and a function a ∈ L∞ (Γ) are such that Ω∩R(a) = ∅, then P (a − λ)−1 is Lp (Γ, w)-analytic on Ω as a function of variable λ. Proof. Let λ0 ∈ Ω and let D(λ0 , δ) be a disk lying in Ω. Since D(λ0 , δ) ∩ R(a) = ∅, we conclude that (a − λ0 )−1 ∈ L∞ (Γ) and (a − λ)−1 ∈ L∞ (Γ) for all λ ∈ D(λ0 , δ) and

(a − λ)−1 − (a − λ0 )−1

−2

− (a − λ0 )

λ − λ0 L∞ (Γ) = k(a − λ)−1 (a − λ0 )−1 − (a − λ0 )−2 kL∞ (Γ) ≤ k(a − λ0 )−1 kL∞ (Γ) k(a − λ)−1 − (a − λ0 )−1 kL∞ (Γ)

λ − λ0 −1

= k(a − λ0 ) kL∞ (Γ) (a − λ)(a − λ0 ) L∞ (Γ) ! ≤ k(a − λ0 )−1 k2L∞ (Γ)

sup

k(a − λ)−1 kL∞ (Γ)

|λ − λ0 |.

λ∈D(λ0 ,δ)

Since L∞ (Γ) ⊂ Lp (Γ, w) continuously, from the latter inequality we get

(a − λ)−1 − (a − λ0 )−1

−2

− (a − λ ) = O(|λ − λ0 |) 0

p λ − λ0 L (Γ,w)

as λ → λ0 . The hypothesis w ∈ Ap (Γ) implies that kP kB(Lp (Γ,w)) < ∞ and therefore

P (a − λ)−1 − P (a − λ0 )−1

−2

− P (a − λ0 ) = O(|λ − λ0 |)

λ − λ0 p L (Γ,w)

−1

as λ → λ0 . Then P (a−λ) is L (Γ, w)-differentiable. Thus P (a−λ)−1 is Lp (Γ, w)analytic in view of Theorem 4.6. p

Lemma 6.3. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). Suppose a simply connected open set Ω and a function a ∈ L∞ (Γ) are such that Ω ∩ R(a) = ∅. Suppose k ∈ E ∞ (Γ). (a) If Fλ is Lp (Γ, w)-analytic on Ω and Fλ ∈ H p (Γ, w) for all λ ∈ Ω, then Ψλ := T [χn (a − λ)]Fλ − k p

is L (Γ, w)-analytic on Ω. (b) If Gλ is Lq (Γ, w−1 )-analytic on Ω and Gλ ∈ H q (Γ, w−1 ) for all λ ∈ Ω, then Φλ := T [χ−n /(a − λ)]Gλ − k q

is L (Γ, w

−1

)-analytic on Ω.

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Proof. (b) Fix λ0 ∈ Ω. Since Gλ is Lq (Γ, w−1 )-analytic, from Theorem 4.6 it follows that the derivative G0λ0 exists in the norm of Lq (Γ, w−1 ). From the proof of Lemma 6.2 we know that (a − λ)−1 is L∞ (Γ)-differentiable at the point λ0 and its derivative there equals (a − λ0 )−2 . Since the embedding L∞ (Γ) ⊂ Lq (Γ, w−1 ) is continuous, we conclude that (a − λ)−1 is Lq (Γ, w−1 )-differentiable at λ0 and its derivative is (a − λ0 )−2 . Let gλ = χ−n (a − λ)−1 Gλ . It is easy to see that gλ0 0 = χ−n ((a − λ0 )−1 )0 Gλ0 + χ−n (a − λ0 )−1 G0λ0 = χ−n (a − λ0 )−2 Gλ0 + χ−n (a − λ0 )−1 G0λ0 , that is,

gλ − gλ0 0

= o(1)

λ − λ0 − gλ0 q L (Γ,w−1 )

as

λ → λ0 .

Then, taking into account that kP kB(Lq (Γ,w−1 )) < ∞ in view of the equivalence w ∈ Ap (Γ) ⇐⇒ w−1 ∈ Aq (Γ), we get

Φλ − Φλ0

0

λ − λ0 − P gλ0 q L (Γ,w−1 )

T [χ−n /(a − λ)]Gλ − T [χ−n /(a − λ0 )]Gλ0

0 = − P g λ0

λ − λ0 Lq (Γ,w−1 )

gλ − gλ0 = − gλ0 0

P

q λ − λ0 L (Γ,w−1 )

gλ − gλ0 0 − g = o(1) ≤ kP kB(Lq (Γ,w−1 )) λ0

λ − λ0 q −1 L (Γ,w

q

)

−1

as λ → λ0 . Therefore, Φλ is L (Γ, w )-differentiable. By Theorem 4.6, Φλ is Lq (Γ, w−1 )-analytic on Ω. Part (b) is proved. The proof of part (a) is analogous. Lemma 6.4. Let Γ ba a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). Suppose a simply connected open set Ω and a function a ∈ L∞ (Γ) are such that Ω ∩ R(a) = ∅. If Fλ is Lp (Γ, w)-analytic on Ω, Gλ is Lq (Γ, w−1 )-analytic on Ω, and k ∈ E ∞ (Γ), then Hλ := Fλ P [χ−n Gλ k/(a − λ)] is analytic on Ω. Proof. We will show that Hλ is even L1 (Γ)-analytic on Ω. Fix λ0 ∈ Ω. From the proof of Lemma 6.2 we know that (a − λ)−1 is L∞ (Γ)-differentiable at λ0 and its derivative is (a − λ0 )−2 . Then

(a − λ)−1 − (a − λ0 )−1 −2

− (a − λ0 ) = o(1) as λ → λ0 . (6.3)

λ − λ0 L∞ (Γ)

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From Lq (Γ, w−1 )-analyticity of Gλ we conclude that G0λ0 exists and

Gλ − Gλ0

0

= o(1) as λ → λ0 . − G λ0

λ − λ0 Lq (Γ,w−1 )

(6.4)

Since Gλ is Lq (Γ, w−1 )-differentiable, it is continuous in the norm of Lq (Γ, w−1 ). Then kGλ kLq (Γ,w−1 ) = O(1)

as

λ → λ0 ,

(6.5)

kGλ − Gλ0 kLq (Γ,w−1 ) = o(1)

as

λ → λ0 .

(6.6)

−1

Put ϕλ := χ−n Gλ k(a − λ) ϕ0λ0

. Let us show that

= χ−n G0λ0 k(a − λ)−1 + χ−n Gλ0 k(a − λ0 )−2

in the norm of Lq (Γ, w−1 ). Write Gλ (a − λ)−1 − Gλ0 (a − λ0 )−1 − G0λ0 (a − λ0 )−1 − Gλ0 (a − λ0 )−2 λ − λ0 (a − λ)−1 − (a − λ0 )−1 −2 = Gλ − (a − λ0 ) + (Gλ − Gλ0 )(a − λ0 )−2 λ − λ0 Gλ − Gλ0 0 − Gλ0 (a − λ0 )−1 . + λ − λ0 Taking into account (6.3)–(6.6), from the latter identity we get

ϕλ − ϕλ0

0

− ϕ λ0

λ − λ0 q −1 L (Γ,w

)

≤ kχ−n kkL∞ (Γ)

Gλ (a − λ)−1 − Gλ0 (a − λ0 )−1

0 −1 −2

− Gλ0 (a − λ0 ) − Gλ0 (a − λ0 ) × λ − λ0

Lq (Γ,w−1 )

≤ kχ−n kkL∞ (Γ)

(a − λ)−1 − (a − λ0 )−1 −2

− (a − λ0 ) kGλ kLq (Γ,w−1 ) λ − λ0

L∞ (Γ)

−2

+ kGλ − Gλ0 kLq (Γ,w−1 ) k(a − λ0 ) kL∞ (Γ) !

Gλ − Gλ0

0 + k(a − λ0 )−1 kL∞ (Γ) = o(1)

λ − λ0 − Gλ0 q L (Γ,w−1 ) as λ → λ0 . Since w ∈ Ap (Γ), one has kP kB(Lq (Γ,w−1 )) < ∞ and then

P ϕλ − ϕλ0 − ϕ0λ = o(1) as λ → λ0 . 0

q λ − λ0 L (Γ,w−1 )

(6.7)

On the other hand, Fλ is Lp (Γ, w)-analytic on Ω. According to Theorem 4.6, there exists Fλ0 0 such that

Fλ − Fλ0 0

= o(1) as λ → λ0 . (6.8)

λ − λ0 − Fλ0 p L (Γ,w)

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Further, Fλ is strongly continuous in the norm of Lp (Γ, w). Then kFλ kLp (Γ,w) = O(1)

as λ → λ0 ,

(6.9)

kFλ − Fλ0 kLp (Γ,w) = o(1)

as

λ → λ0 .

(6.10)

As usual, we write Hλ − Hλ0 − Fλ0 0 P ϕλ0 − Fλ0 P ϕ0λ0 λ − λ0 P ϕλ − P ϕλ0 Fλ − Fλ0 0 0 0 = Fλ − P ϕλ0 + (Fλ − Fλ0 )P ϕλ0 + − Fλ0 P ϕλ0 . λ − λ0 λ − λ0 By H¨ older’s inequality,

Hλ − Hλ0 0 0

λ − λ0 − Fλ0 P ϕλ0 − Fλ0 P ϕλ0 1 L (Γ)

P ϕλ − P ϕλ0

0

≤ kFλ kLp (Γ,w) − P ϕλ0 λ − λ0 q

L (Γ,w−1 ) 0 Fλ0 kLp (Γ,w) kP ϕλ0 kLq (Γ,w−1 )

+ kFλ −

Fλ − Fλ0

0

+ − Fλ0 λ − λ0

(6.11)

kP ϕλ0 kLq (Γ,w−1 ) .

Lp (Γ,w)

Combining (6.7)–(6.11), we arrive at

Hλ − Hλ0

0 0

λ − λ0 − Fλ0 P ϕλ0 − Fλ0 P ϕλ0 1 = o(1) L (Γ)

λ → λ0 ,

as

that is, Hλ is L1 (Γ)-analytic on Ω. From Lemma 4.11 we conclude that Hλ is analytic on Ω. Lemma 6.5. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ Ap (Γ). Suppose a simply connected open set Ω and a function a ∈ L∞ (Γ) are such that Ω ∩ R(a) = ∅. Let γ be a rectifiable curve lying in Ω and having initial and terminating points λ0 and λ, respectively. If fλ0 and gλ0 are measurable functions on Γ, then Z Z Fλ = fλ0 exp (P) P (a − µ)−1 dµ , Gλ = gλ0 exp −(P) P (a − µ)−1 dµ γ

γ

are analytic functions on Ω. Proof. By Lemma 6.2, the function P (a − λ)−1 is Lp (Γ, w)-analytic on Ω. Since Ω is simply connected, from Theorem 4.8 it follows that Z (RS) P (a − µ)−1 dµ (6.12) γ

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Rλ R is independent of γ. We will write λ0 instead of γ , where γ is a curve lying in Ω with the starting point λ0 and the terminating point λ. Let us show that Z λ (RS) P (a − µ)−1 dµ λ0

is Lp (Γ, w)-differentiable at λ and Z λ d (RS) P (a − µ)−1 dµ = P (a − λ)−1 . dλ λ0

(6.13)

Indeed, let λ + h be an arbitrary point in an arbitrarily small neighborhood of λ. From (4.1) it follows that Z λ+h Z λ (RS) P (a − µ)−1 dµ − (RS) P (a − µ)−1 dµ λ0 λ0 (6.14) Z λ+h P (a − µ)−1 dµ.

= (RS) λ

Without loss of generality one can take the segment [λ, λ + h] as the path of integration in the integral on the right-hand side of (6.14). It is clear that Z λ+h 1 P (a − λ)−1 dµ. (6.15) P (a − λ)−1 = (RS) h λ Since P (a − µ)−1 is strongly continuous at λ, for any ε > 0 there exists a δ > 0 such that if |µ − λ| < δ, then kP (a − µ)−1 − P (a − λ)−1 kLp (Γ,w) < ε.

(6.16)

If |h| < δ, then taking into account (6.14)–(6.16) and (4.2), we get

! Z λ+h Z λ

1

(RS) P (a − µ)−1 dµ − (RS) P (a − µ)−1 dµ − P (a − λ)−1

h

p λ0 λ0 L (Γ,w)

Z λ+h

1

= (RS) P (a − µ)−1 − P (a − λ)−1 dµ

h

p λ L (Γ,w) ! 1 ≤ sup kP (a − µ)−1 − P (a − λ)−1 kLp (Γ,w) |h| ≤ ε. |h| µ∈[λ,λ+h] This inequality implies (6.13). By Theorem 4.6, the function (6.12) is Lp (Γ, w)analytic on Ω. If P (a − µ)−1 is Riemann-Stieltjes integrable, then it is Pettis integrable and Z Z (RS) P (a − µ)−1 dµ = (P) P (a − µ)−1 dµ γ

γ

by Theorem 4.5. Thus Z (P) γ

P (a − µ)−1 dµ

(6.17)

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is Lp (Γ, w)-analytic on Ω. From the continuity of the embedding Lp (Γ, w) ⊂ L1 (Γ) and Lemma 4.11 we conclude that (6.17) is also analytic on Ω. Then Z exp ±(P) P (a − µ)−1 dµ (6.18) γ

are analytic on Ω as compositions of an entire function exp(z) and analytic functions. Multiplying (6.18) by constants fλ0 and gλ0 , we obtain analytic functions again. 6.3. Proof of Lemma 2.4 The proof is developed by the scheme of the proof of [3, Theorem 7.45]. Suppose Ω is a simply connected open set which contains C and such that any point of Ω not inside of C belongs to ρess (T (a)). Denote by Ωi the set of all points lying inside of C and by Ωo the set of all points of Ω lying outside of C. Since C ∪Ωo ⊂ ρess (T (a)), from Theorem 2.1 we get (C ∪ Ωo ) ∩ R(a) = ∅. By the assumption, Ωi ∩ R(a) = ∅. Thus Ω ∩ R(a) = Ωi ∪ C ∪ Ωo ∩ R(a) = ∅. Our goal is to show that Ωi ∩ spess T (a) = ∅. (6.19) Choose λ0 ∈ C and keep it fixed. Let fλ0 := T −1 [χn (a − λ0 )]χ0 ,

gλ0 := T −1 [χ−n /(a − λ0 )]χ0 .

Let γ be a rectifiable curve with starting point λ0 and terminating point λ and such that γ ⊂ Ω. Define Z Z Fλ = fλ0 exp (P) P (a − µ)−1 dµ , Gλ = gλ0 exp −(P) P (a − µ)−1 dµ γ

γ

for all λ ∈ Ω. These functions are analytic on Ω in view of Lemma 6.5. Since Ωo ⊂ ρess (T (a)), from Lemma 5.2 it follows that Fλ = fλ = T −1 [χn (a − λ)]χ0 ,

Gλ = gλ = T −1 [χ−n /(a − λ)]χ0

(6.20) p

for λ ∈ Ωo ∩ ρn (T (a)). Then from Lemma 5.1 we conclude that Fλ is L (Γ, w)analytic on Ωo and Gλ is Lq (Γ, w−1 )-analytic on Ωo . Let C 0 be an arbitrary rectifiable Jordan curve lying in Ωo and surrounding C. From Lp (Γ, w)-analyticity of Fλ on Ωo and Lq (Γ, w−1 )-analyticity of Gλ on Ωo we deduce that the functions Fλ |C 0 : C 0 → Lp (Γ, w),

Gλ |C 0 : C 0 → Lq (Γ, w−1 )

are continuous. Thus, in view of the compactness of C 0 , sup kFµ kLp (Γ,w) =: Mp < ∞,

µ∈C 0

sup kGµ kLq (Γ,w−1 ) =: Mq < ∞.

µ∈C 0

The functions Fλ and Gλ are analytic inside of C 0 . Therefore, by Lemma 4.12, Fλ is Lp (Γ, w)-analytic inside of C 0 and Gλ is Lq (Γ, w−1 )-analytic inside of C 0 . Thus Fλ is Lp (Γ, w)-analytic (resp. Gλ is Lq (Γ, w−1 )-analytic) on the whole domain Ω.

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From the equalities (6.20) we know that Fλ ∈ H p (Γ, w) and Gλ ∈ H q (Γ, w−1 ) for all λ ∈ Ωo ∩ ρn (T (a)). Then Lemma 4.13 implies that Fλ ∈ H p (Γ, w),

Gλ ∈ H q (Γ, w−1 )

for all λ ∈ Ω.

For λ ∈ Ω, define the function Ψλ := T [χn (a − λ)]Fλ − 1. Since Fλ is Lp (Γ, w)-analytic on Ω, from Lemma 6.3(a) it follows that Ψλ is Lp (Γ, w)-analytic on Ω. From Lemma 5.2 we deduce that Ψλ = 0 for all λ ∈ Ωo . Then kΨλ kLp (Γ,w) = 0 for all λ ∈ C 0 . By the maximum principle (see Theorem 4.9), Ψλ ≡ 0 for all λ inside C 0 . Thus T [χn (a − λ)]Fλ = 1 for all λ ∈ Ω. Analogously, taking into account Lemma 6.3(b), one can show that T [χ−n /(a − λ)]Gλ = 1

for all λ ∈ Ω.

Let λ ∈ Ωi . We will show that T [χn (a − λ)] is invertible. From the latter identity it follows that q χ−n Gλ /(a − λ) ∈ H− (Γ, w−1 ),

Gλ 6= 0.

(6.21)

p

Let h ∈ H (Γ, w) and T [χn (a − λ)]h = 0. Then p χn (a − λ)h ∈ H˙ − (Γ, w).

(6.22)

1 (Γ). On the Combining (6.21)–(6.22) with Lemma 3.5(b), we obtain hGλ ∈ E˙ − 1 other hand, by Lemma 3.5(a), hGλ ∈ E (Γ). From Theorem 3.3 it follows that hGλ = 0 and so h = 0. This means that

T [χn (a − λ)] is one-to-one on H p (Γ, w) for every λ ∈ Ωi .

(6.23)

Let us show that T [χn (a − λ)] is onto for every λ ∈ Ωi . Suppose k ∈ E ∞ (Γ) and put Hλ := Fλ P [χ−n Gλ k/(a − λ)] (λ ∈ Ω). If λ ∈ ρess (T (a)) ∩ Ω, then Fλ = fλ and Gλ = gλ . Therefore, by Lemma 6.1, for every λ ∈ C ∪ Ωo ⊂ ρess (T (a)) ∩ Ω, Hλ = fλ P [χ−n gλ k/(a − λ)] ∈ H p (Γ, w) and T [χn (a − λ)]Hλ = k. Then sup kHµ kLp (Γ,w) = sup kT −1 [χn (a − µ)]kkH p (Γ,w) µ∈C

µ∈C

≤ sup kT −1 [χn (a − µ)]kB(H p (Γ,w)) kkkH p (Γ,w) . µ∈C

From the proof of Lemma 5.1(a) we know that T −1 [χn (a − µ)] can be represented locally as a power series absolutely converging in the norm of B(H p (Γ, w)). This

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implies that T −1 [χn (a−µ)] is continuous in the norm of B(H p (Γ, w)) as a function of variable µ ∈ C. Since C is compact, we conclude that sup kT −1 [χn (a − µ)]kB(H p (Γ,w)) = M < ∞. µ∈C

From Lemma 6.4 we deduce that Hλ is analytic on Ω. Then by Lemma 4.12, the function Hλ is Lp (Γ, w)-analytic on Ωi and kHλ kLp (Γ,w) ≤ M kkkLp (Γ,w)

for all λ ∈ Ωi .

(6.24)

Repeating this argument with an arbitrary rectifiable Jordan curve C 0 ⊂ Ωo surrounding C, we conclude that Hλ is Lp (Γ, w)-analytic on the whole Ω. Since Hλ ∈ H p (Γ, w) for all λ ∈ Ωo , from Lemma 4.13 we obtain that Hλ ∈ H p (Γ, w) for all λ ∈ Ω. By Lemma 6.3(a), the function e λ := T [χn (a − λ)]Hλ − k Ψ e λ kLp (Γ,w) ≡ 0 for is Lp (Γ, w)-analytic on Ω and, arguing as before, we see that kΨ e λ ≡ 0 everywhere on all λ ∈ C ∪ Ωo . By the maximum principle (Theorem 4.9), Ψ Ωi . Thus T [χn (a − λ)]Hλ = k for all λ ∈ Ωi . Let k ∈ H p (Γ, w). Since P R(Γ) is dense in H p (Γ, w) by Lemma 3.6, there ∞ is a sequence {kj }∞ j=1 ⊂ P R(Γ) ⊂ E (Γ) such that kk − kj kH p (Γ,w) → 0 as j ∞ tends to infinity. Then {kj }j=1 is a Cauchy sequence in H p (Γ, w). From (6.24) it follows that the corresponding functions {Hλj }∞ j=1 for fixed λ ∈ Ωi form a Cauchy sequence in H p (Γ, w) and hence converge to a function Hλ ∈ H p (Γ, w) such that T [χn (a − λ)]Hλ = k. Thus T [χn (a − λ)] is onto on H p (Γ, w) for every λ ∈ Ωi .

(6.25)

From (6.23) and (6.25) we deduce that T [χn (a − λ)] is invertible for every λ ∈ Ωi . Then Propositions 3.7 and 3.10 imply that T (a − λ) is Fredholm on H p (Γ, w) and its index is equal to n for every λ ∈ Ωi . Thus λ ∈ / spess T (a), which proves (6.19). Acknowledgment. We would like to thank Harold Widom who kindly agreed to read the preliminary version of the paper and made several valuable suggestions regarding the exposition.

References [1] A. B¨ ottcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Birkh¨ auser, Basel, 1997. [2] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators. 2nd edition. Springer, Berlin, 2006.

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[3] R. G. Douglas, Banach Algebra Techniques in Operator Theory. 2nd edition. Springer, New York, 1998. [4] P. Duren, Theory of H p Spaces. Academic Press, New York, 1970. [5] I. Gohberg, S. Goldberg, and M. Kaashoek, Classes of linear operators. Vol. II. Birkh¨ auser Verlag, Basel, 1990. [6] P. R. Halmos, A glimpse into Hilbert space. In: “A Survey of Modern Mathematics”, Wiley, New York, 1963. [7] E. Hille, Ordinary Differential Equations in the Complex Domain. Wiley, New York, 1976. [8] E. Hille and R. S. Phillips, Functional Analysis and Semigroups. American Mathematical Society, 1957. [9] A. Isgur and I. M. Spitkovsky, On the spectra of some Toeplitz and Wiener-Hopf operators with almost periodic matrix symbols. Operators and Matrices 2 (2008), 371–383. [10] G. S. Litvinchuk and I. M. Spitkovsky, Factorization of Measurable Matrix Functions. Birkh¨ auser Verlag, Basel, 1987. [11] W. Rudin, Real and Complex Analysis. McGraw-Hill, New York, 1986. [12] H. Widom, On the spectrum of a Toeplitz operator. Pacific J. Math. 14 (1964), 365– 375. [13] H. Widom, Toeplitz operators on H p . Pacific J. Math. 19 (1966), 573–582. Alexei Yu. Karlovich Departamento de Matem´ atica Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa Quinta da Torre 2829–516 Caparica Portugal e-mail: [email protected] Ilya M. Spitkovsky Department of Mathematics College of William & Mary Williamsburg, VA, 23187-8795 U.S.A. e-mail: [email protected] Submitted: August 19, 2008. Revised: June 16, 2009.


auser Verlag Basel/Switzerland 0378-620X/010115-15, published online August 24, 2009 DOI 10.1007/s00020-009-1712-z


Hyponormal Toeplitz Operators on the Weighted Bergman Space Yufeng Lu and Yanyue Shi Abstract. Consider ϕ = f + g, where f and g are polynomials, and let Tϕ be the Toeplitz operators with the symbol ϕ. It is known that if Tϕ is hyponormal then |f 0 (z)|2 ≥ |g 0 (z)|2 on the unit circle in the complex plane. In this paper, we show that it is also a necessary and sufficient condition under certain assumptions. Furthermore, we present some necessary conditions for the hyponormality of Tϕ on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47B20. Keywords. Hyponormality, weighted Bergman space, Hankel operator, Toeplitz operator.

1. Introduction Let D denote the open unit disk in the complex plane. For −1 < α < ∞, L2 (D, dAα ) is the space of functions on D which are square integrable with respect to the measure dAα (z) = (α + 1)(1 − |z|2 )α dA(z), where dA denotes the normalized Lebesgue areaR measure on D. L2 (D, dAα ) is a Hilbert space with the inner product hf, giα = D f (z)g(z)dAα . The weighted Bergman space A2α is the closed subspace of L2 (D, dAα ) consisting of analytic functions on D. If α = 0, A20 is the Bergman space. We write A2 =A20 . The reproducing kernel in A2α is given by 1 α Kw (z) = , for z, w ∈ D. (1 − wz)2+α Let Pα be the orthogonal projection from L2 (D, dAα ) onto A2α . For an essential bounded measurable function ϕ on D, the Toeplitz operator Tϕ and the Hankel operator Hϕ with symbol ϕ are defined on A2α by Tϕ (h) = Pα (ϕh) and Hϕ (h) = (I − Pα )(ϕh) for h ∈ A2α . This research is supported by NSFC, Item Number: 10671028.

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A bounded linear operator A on a Hilbert space is said to be hyponormal if its self-commutator A∗ A − AA∗ is a positive operator. In the setting of the classical Hardy space H 2 , C. Cowen[3] gave an elegant characterization of the hyponormality of Toeplitz operator with a bounded measurable symbol on T, where T is the unit circle in the complex plane. It is natural for us to consider the same problem on the Bergman space. However, C. Cowen’s proof does not adapt to the Bergman space, since the multiplication operator Mz is no longer ⊥ an isometry and A2α is much larger than the conjugates of zA2α . The question concerning the characterization of hyponormal Toeplitz operators on the Bergman space still remains open. For the case that ϕ is a continuous function, Yufeng Lu and Chaomei Liu [8] gave a sufficient condition for the hyponormality of Tϕ by Mellin transform on weighted Bergman space. For the case that ϕ = f + g with f, g bounded and analytic, H. Sadraoui[9] gave a necessary and sufficient condition on the Bergman space, which is also true on the weighted Bergman space. Theorem 1.1 ([9]). Let f, g be bounded and analytic in L2 (D, dA). Then the followings are equivalent: (i) Tf +g is hyponormal; (ii) Hg∗ Hg ≤ Hf∗ Hf ; (iii) Hg = CHf , where C is of norm less than or equal to one. In [9], H. Sadraoui also showed that if f, g ∈ H ∞ (D) and f 0 , g 0 ∈ H 2 (D) such that Tf +g is hyponormal, then |f 0 (z)| ≥ |g 0 (z)| almost everywhere on T. Later, P. Ahern and Z. Cuckovic[1] generalized H. Sadraoui’s result by a mean value inequality and Berezin transform. Theorem 1.2 ([1]). Suppose that f, g are holomorphic in D, that ϕ = f + g is bounded in D and that Tϕ is hyponormal, then lim (|f 0 (z)|2 − |g 0 (z)|2 ) ≥ 0 ξ ∈ T.

z→ξ

In particular, if f 0 and g 0 are continuous at ξ ∈ T, then |f 0 (ξ)| ≥ |g 0 (ξ)|. Unfortunately, the condition in above theorem is not sufficient. We will give a counterexample in Section 2. One way to determine the hyponormality of an operator is to study the matrix representation of the self-commutator of the operator. This matrix is positive if and only if the operator is hyponormal. But the matrix is very complex even for the case that ϕ = z p+1 + bz p + cz p , where b, c ∈ C, p > 0 (see [7], Theorem 3.1). Recently, I. S. Hwang [5] studied the hyponormality of Tϕ , where ϕ is a certain trigonometric polynomial. The method is mainly based on dividing A2 N into several orthogonal subspaces {Ki }N i=1 , such that for every p ≥ 0, {Tz p Ki }i=1 are orthogonal. Theorem 1.3 ([5]). Let ϕ(z) = f (z) + g(z), where f (z) = am z m + an z n and g(z) = bm z m + bn z n (0 < m < n). If am an = bm bn , then Tϕ is hyponormal on A2 if and only if both of the following statements hold:

Vol. 65 (2009)

Hyponormal Toeplitz Operators

117

1 1 (i) n+1 (|an |2 − |bn |2 ) ≥ m+1 (|bm |2 − |am |2 ), if |bn | ≤ |an |; (ii) n2 (|bn |2 − |an |2 ) ≤ m2 (|am |2 − |bm |2 ), if |an | ≤ |bn |.

So we see that the hyponormality of the Toeplitz operators on the Bergman space may be very different from the case on the Hardy space. The reader is referred to [4] for details. Moreover, I. S. Hwang and J. Lee [6] obtained the following result without the assumption am an = bm bn . Theorem 1.4 ([6]). Let ϕ(z) = f (z) + g(z), where f (z) = am z m + an z n , g(z) = bm z m + bn z n (0 < m < n) with |an | ≤ |bn |. If Tϕ on A2 is hyponormal, then we have n2 (|bn |2 − |an |2 ) ≤ m2 (|am |2 − |bm |2 ). In this paper, we continue to investigate the hyponormality of Tf +g with f, g ∈ H ∞ (D). In Section 2, we show that if ϕ is as in Theorem 1.4, then Tϕ on A2 is hyponormal if and only if |f 0 (z)|2 ≥ |g 0 (z)|2 on T. In Section 3, we obtain some necessary conditions on the weighted Bergman space under the assumption that ||f ||α = ||g||α .

2. Necessary and sufficient condition It is well known that Toeplitz operators on the Bergman space are related to Hankel operators by the following algebraic relation: Tf g = Tf Tg + Hf∗ Hg , Tgf = Tg Tf +

(2.1)

Hg∗ Hf ,

∞

for every f, g ∈ L (D). If f, g ∈ H ∞ (D), then Tf∗+g Tf +g − Tf +g Tf∗+g = Hf∗ Hf − Hg∗ Hg .

(2.2)

For the purpose of studying the hyponormality of Toeplitz operators, we may pay more attention to the Hankel operators Hf and Hg . Let γk = ||z k ||α , then a straight calculation shows that Z 1 γk2 = (α + 1) |z|2k (1 − |z|2 )α dA(z) 0

Z = (α + 1)

1

tk (1 − t)α dt

0

= (α + 1)B(k + 1, α + 1) = Qk

k!

j=1 (α

+ 1 + j)

≤ 1,

where B(p, q) is the Beta-function. It then follows that for l ≥ 0, %(k) =

γk2 is 2 γk+l

strictly decreasing and γk2 = 1. 2 k→∞ γk+l lim

(2.3)

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The following lemma shows many basic properties of the Hankel operators. Lemma 2.1. Fix m, n ≥ 1, α > −1. Then (a)  m ξ ξk  γ2 m Hzm (z k )(ξ) =  ξ ξ k − 2 k ξ k−m γk−m

if 0 ≤ k < m if

m ≤ k;

2 (b) the functions {Hzm (z k )}∞ k=0 are orthogonal in L (D, dAα ); ∗ k 2 k (c) Hzm Hzm (z )(ξ) = ωmk ξ (k = 0, 1, 2, . . .), where

2 ωmk =

      

(d) (e) (f) (g)

2 γk+m γk2

2 γk+m γk2

−

γk2 2 γk−m

if 0 ≤ k < m if

m ≤ k;

||Hzm (z k )||α = ωmk γk ; hHz∗m Hzn z k+n−m , z k iα > 0, ∀k ≥ max{0, m − n}; If n > m ≥ 1, l ≥ 0, then hHz∗n Hzm z k+l , z k iα = 0, ∀k ≥ 0; 2 the functions {Hzk (z m )}∞ k=1 are orthogonal in L (D, dAα ).

Proof. The proof of (a), (b), (c) and (d) was given in [2]. (e) From (a) we see that  2 γk+n if max{0, m − n} ≤ k < m    hHz∗m Hzn z k+n−m , z k iα = 2 2 γk+n γk−m+n   ) if m ≤ k.  γk2 ( 2 − 2 γk γk−m Thus hHz∗m Hzn z k+n−m , z k iα > 0, ∀k ≥ 0. (f) Note that for n > m and l ≥ 0, hz m z k+l , z n z k iα = hz k+l−m , z n z k iα = 0. Making use of (a) again, a direct computation shows that hHz∗n Hzm z k+l , z k iα = 0. (g) By (f), it is easy to check that hHzk1 z m , Hzk2 z m iα = 0 for k1 6= k2 . For each positive integer p, we denote by Qp (k) the polynomial in one variable k of degree at most p. Lemma 2.2. Fix n, m ≥ 1, l ≥ 0, α > −1, then the following equalities hold: ||Hzm z k ||2α m2 = ; (i) lim k→∞ ||Hz n z k ||2 n2 α ||Hzm z k ||2α (ii) lim = 1. k→∞ ||Hz m z k+l ||2 α Proof. It is clear that when k is big enough, Qm Qm−1 2 γk+m (k + i) (k − i) γk2 2 i=1 . ωmk = − 2 = Qm − Qm−1 i=0 γk2 γk−m (α + k + 1 + j) j=1 j=0 (α + k + 1 − j)

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We claim that m Y

(k + i)

m−1 Y

i=1

(α + k + 1 − j) −

m−1 Y

j=0

(k − i)

i=0

m Y

(α + k + 1 + j)

j=1

is a polynomial in one variable k of degree 2m − 2. For p ≥ 0, let Cp be the coefficient of k p . It is easy to see that C2m = 0. Furthermore, C2m−1 =

m X

i+

i=1

Let ∆ =

Pm

i=1

i+

2C2m−2 =

(α + 1 − j) −

j=0

Pm−1 j=0

m X

m−1 X

(−i) −

i=0

m X

(α + 1 + j) = 0.

j=1

(α + 1 − j) = 2m + mα. Then

i(∆ − i) +

i=1

−

m−1 X

m−1 X

(α + 1 − j)(∆ − (α + 1 − j))

j=0

m X

(α + 1 + j)(∆ − (α + 1 + j)) +

m−1 X

j=1

i(∆ + i)

i=1

= − m2 +

m X

((α + 1 + j)2 − (α + 2 − j)2 ) = 2(α + 1)m2 .

j=1

Thus, there exists a polynomial Q2m−3 (k) such that 2 ωmk =

(α + 1)m2 k 2m−2 + Q2m−3 (k) Qm . j=−m+1 (α + 1 + k + j)

(2.4)

Similarly, there exists a polynomial Q2n−3 (k) such that 2 ωnk =

(α + 1)n2 k 2n−2 + Q2n−3 (k) Qn . j=−n+1 (α + 1 + k + j)

(2.5)

Therefore, by (2.4), (2.5) and Lemma 2.1(d), ||Hzm z k ||2α k→∞ ||Hz n z k ||2 α lim

Qn (α + 1)m2 k 2m−2 + Q2m−3 (k) j=−n+1 (α + 1 + k + j) m2 Q = lim = . m k→∞ (α + 1)n2 k 2n−2 + Q2n−3 (k) n2 j=−m+1 (α + 1 + k + j) Replacing k by k + l in (2.4), we see that 2 ωm(k+l) =

(α + 1)m2 (k + l)2m−2 + Q2m−3 (k + l) Qm j=−m+1 (α + 1 + k + l + j)

e 2m−3 (k) (α + 1)m2 k 2m−2 + Q = Qm , j=−m+1 (α + 1 + k + l + j)

(2.6)

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e 2m−3 (k) = Q2m−3 (k + l) + (α + 1)m2 ((k + l)2m−2 − k 2m−2 ) is a polynomial where Q in one variable k of degree at most 2m − 3. By (2.3), (2.4) and (2.6), we know that ||Hzm z k ||2α ω2 γ 2 = lim 2 mk k2 = 1. k+l 2 k→∞ ||Hz m z k→∞ ω ||α m(k+l) γk+l lim

Lemma 2.3. Fix N , n, m ≥ 1, α > −1, then the following equalities hold: hHz∗m Hzn z k+n−m , z k iα m (i) lim = ; k→∞ ||Hz n z k ||α ||Hz n z k+n−m ||α n hHz∗m Hzn z k+n−m , z k iα mn = 2. (ii) lim k→∞ ||HzN z k+n−m ||2α N Proof. Assume k > m, by the proof of Lemma 2.1 (e), we have 2 hHz∗m Hzn z k+n−m , z k iα /γk+n−m =

2 γk+n

−

γk2

2 2 γk+n−m γk−m Qm−1 (k + n − m + i) (k − i) = Qm i=1 − Qm−1 i=0 . (α + k + n − m + 1 + j) j=1 j=0 (α + k + 1 − j)

Qm

We claim that m Y

(k + n − m + i)

i=1

m−1 Y

(α + k + 1 − j) −

j=0

m−1 Y

(k − i)

i=0

m Y

(α + k + n − m + 1 + j)

j=1

is a polynomial in one variable k of degree 2m − 2. Noting that m X

(n − m + j) +

j=1

m−1 X

(α + 1 − j) =

j=0

m X

(α + 1 + n − m + j) +

j=1

m−1 X

(−j)

j=0

= m(α + 2) + m(n − m), we set ∆ = m(α + 2) + m(n − m). For p ≥ 0, let dp be the coefficient of k p , then d2m = d2m−1 = 0 and 2d2m−2 =

m X

(n − m + j)(∆ − (n − m + j)) +

(α + 1 − j)(∆ − (α + 1 − j))

j=0

j=1

−

m−1 X

m X

(α + 1 + n − m + j)(∆ − (α + 1 + n − m + j) +

=

2

(α + 1 + n − m + j) +

j=1

=

m X

m−1 X

2

j −

j=0

(α + 1)(α + 1 + 2(n − m + j)) +

j=1

=2(α + 1)mn.

j(∆ + j)

j=0

j=1 m X

m−1 X

m X

2

(n − m + j) −

j=1 m−1 X j=0

m−1 X

(α + 1 − j)2

j=0

(α + 1)(−α − 1 + 2j)

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121

Hence there exists a polynomial Q2m−3 (k) such that mn(α + 1)k 2m−2 + Q2m−3 (k) hHz∗m Hzn z k+n−m , z k iα = Qm . Qm−1 2 γk+n−m j=1 (α + k + n − m + 1 + j) j=0 (α + k + 1 − j) Thus hHz∗m Hzn z k+n−m , z k i2α k+n−m ||2 k→∞ ||Hz n z k ||2 α ||Hz n z α ∗ 2 2 k+n−m γk+n−m hHzm Hzn z , z k iα 1 = lim 2 2 ω2 k→∞ γk+n−m ωnk γk2 n(k+n−m) 2 2 γk+n−m mn(α + 1)k 2m−2 + Q2m−3 (k) = lim Q Qm−1 m 2 k→∞ γk j=1 (α + k + n − m + 1 + j) j=0 (α + k + 1 − j) Qn Qn j=−n+1 (α + 1 + k + j) j=−n+1 (α + k + n − m + 1 + j) × e 2n−3 (k) (α + 1)n2 k 2n−2 + Q2n−3 (k) (α + 1)n2 k 2n−2 + Q lim

m2 n2 (α + 1)2 k 4(n+m−1) + Q4n+4m−5 (k) m2 = 2, e 4n+4m−5 (k) k→∞ n4 (α + 1)2 k 4(n+m−1) + Q n

= lim

e 4n+4m−5 (k) are polynomials in one variable k of degree where Q4n+4m−5 (k) and Q at most 4n + 4m − 5. So we get (i). Combining (i) and Lemma 2.2, it is easy to check that hHz∗m Hzn z k+n−m , z k iα lim k→∞ ||HzN z k+n−m ||2α hHz∗m Hzn z k+n−m , z k iα ||Hzn z k+n−m ||α = lim lim k→∞ ||Hz n z k ||α ||Hz n z k+n−m ||α k→∞ ||Hz N z k+n−m ||α ||HzN z k ||α mn ||Hzn z k ||α lim = 2. × lim k→∞ ||Hz N z k ||α k→∞ ||Hz N z k+n−m ||α N Now we are ready to give a necessary condition for the hyponormality of the Toeplitz operator Tϕ on A2α . Theorem 2.4. Let ϕ(z) = f (z) + g(z), where f (z) = am z m + an z n and g(z) = bm z m + bn z n (0 < m < n). For α > −1, if Tϕ is hyponormal on A2α , then the following statements hold: (α + m + 2) · · · (α + n + 1) (i) (|am |2 − |bm |2 ) + (|an |2 − |bn |2 ) ≥ 0; (m + 1)(m + 2) · · · (n − 1)n (ii) m2 (|am |2 − |bm |2 ) + n2 (|an |2 − |bn |2 ) ≥ mn|am an − bm bn |. Proof. Suppose Tϕ is a hyponormal operator. (i) For k ≥ 0, since {Hzm (z k )}∞ m=1 are orthogonal (Lemma 2.1(g)), h(Hf∗ Hf − Hg∗ Hg )z k , z k iα = (|am |2 − |bm |2 )||Hzm z k ||2α + (|an |2 − |bn |2 )||Hzn z k ||2α ≥ 0.

(2.7)

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We can define a function η by η(k) = h(Hf∗ Hf − Hg∗ Hg )z k , z k i1/2 α ≥ 0. In particular,

η 2 (0) kz n k2

(2.8)

≥ 0, which implies that the condition (i) holds.

(ii) Combining the fact that

η 2 (k) kHzn z k k2

≥ 0 and Lemma 2.2(i), we have

m2 (|am |2 − |an |2 ) + (|bm |2 − |bn |2 ) ≥ 0, n2 that is, the statement (ii) holds whenever am an = bm bn . It remains to deal with the case am an 6= bm bn . Let hk (z) = dk z k + dk+n−m z k+n−m , then h(Hf∗ Hf − Hg∗ Hg )hk , hk iα = |dk |2 h(Hf∗ Hf − Hg∗ Hg )z k , z k iα + |dk+n−m |2 h(Hf∗ Hf − Hg∗ Hg )z k+n−m , z k+n−m iα + 2Re dk+n−m dk h(Hf∗ Hf − Hg∗ Hg )z k+n−m , z k iα = |dk+n−m |2 η 2 (k + n − m) + 2Re dk+n−m dk h(am an − bm bn )Hz∗m Hzn z k+n−m , z k iα

+ |dk |2 η 2 (k) + 2Re dk+n−m dk h(am an − bm bn )Hz∗n Hzm z k+n−m , z k iα

= |dk |2 η 2 (k) + |dk+n−m |2 η 2 (k + n − m) + 2Re (am an − bm bn )dk dk+n−m hHz∗m Hzn z k+n−m , z k iα . The third equality follows from (e) and (f) in Lemma 2.1. Since η(k) ≥ 0 and 2 2 η 2 (k) = (|am |2 − |bm |2 )ωmk γk2 + (|an |2 − |bn |2 )ωnk γk2 , 2 2 where ωmk and ωnk are rational polynomials, the equation η 2 (k) = 0 has only finitely many roots. Thus when k is big enough, η(k) > 0. We can choose the certain nonzero dk and dk+n−m such that |dk η(k)| = |dk+n−m η(k + n − m)| and dk dk+n−m (am an − bm bn ) = −|dk dk+n−m (am an − bm bn )|. For example, let dk = 1,

dk+n−m = −

η(k)(am an − bm bn ) . η(k + n − m)|am an − bm bn |

In this case, h(Hf∗ Hf − Hg∗ Hg )hk , hk iα ∗ k+n−m k n = 2|dk dk+n−m | η(k)η(k + n − m) − |am an − bm bn |hHzm Hz z , z iα . From this and the fact that Tϕ is hyponormal, it follows that η(k)η(k + n − m) ≥ |am an − bm bn |hHz∗m Hzn z k+n−m , z k iα ,

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123

or equivalently, η(k)η(k + n − m) |am an − bm bn |hHz∗m Hzn z k+n−m , z k iα ≥ . k+n−m k ||Hzn z ||Hzn z k+n−m ||α ||Hzn z k ||α ||α ||Hzn z ||α Finally, letting k → ∞, by the equality (2.7), Lemma 2.2 and Lemma 2.3 (i), we get the desired inequality. Remark 2.5. Let ϕ be as in the above Theorem, and assume am an = bm bn . Note ∞ ∞ P P that for every h = dk z k ∈ A2α , we have |dk |2 γk2 < ∞, k=0

∞ X

k=0

|dk |2 h(Hf∗ Hf − Hg∗ Hg )z k , z k iα ≤ kHf∗ Hf − Hg∗ Hg kα

∞ X

|dk |2 γk2 < ∞,

k=0

k=0

|

X

dk dl hHz∗m Hzn z l ,

z iα | ≤

X

≤

X

k

l>k

l

|dk dl |kHzn z kα kHzm z k kα

l>k

|dk dl γk+m γl+n |

l>k

≤ ≤

X

|dk dl γk γl |

l>k ∞ X

|dk |2 γk2

∞ 12 X

|dl |2 γl2

21

< ∞.

l=0

k=0

From the proof of Theorem 2.4 and Lemma 2.1(f), it is easy to see that h(Hf∗ Hf − Hg∗ Hg )h, hiα =

∞ X

|dk |2 h(Hf∗ Hf − Hg∗ Hg )z k , z k iα

k=0

+ 2Re

X

dk dl (am an −

bm bn )hHz∗m Hzn z l ,

k

z iα

l>k

=

∞ X

|dk |2 h(Hf∗ Hf − Hg∗ Hg )z k , z k iα .

k=0

Thus, Tϕ is hyponormal on A2α if and only if h(Hf∗ Hf −Hg∗ Hg )z k , z k iα ≥ 0, ∀k ≥ 0. If α = 0, define the function ζ by ζ(k) = decreasing and

||Hzm z k ||2 . Then ζ is strictly ||Hzn z k ||2

m2 n+1 ≤ ζ(k) ≤ . It follows that the inequality 2 n m+1 (|am |2 − |bm |2 )ζ(k) + (|an |2 − |bn |2 ) ≥ 0

holds for any k ≥ 0 if and only if the statements (i) and (ii) in Theorem 1.3 both hold. In view of the equality (2.7), we have also proved Theorem 1.3.

124


Corollary 2.6. Let ϕ(z) = f (z) + g(z), where f (z) =

N P

IEOT N P

ak z k and g(z) =

k=1

bk z k .

k=1

For α > −1, if Tϕ is hyponormal on A2α , then (i)

NP −1

(ii)

m=1 N P

(α + m + 2) · · · (α + N + 1) (|am |2 − |bm |2 ) + (|aN |2 − |bN |2 ) ≥ 0, (m + 1)(m + 2) · · · (N − 1)N P m2 (|am |2 − |bm |2 ) ≥ max | ij(ai aj − bi bj ) |. 1≤l≤N −1 j−i=l

m=1

Proof. As in the proof of Theorem 2.4, we can define the function η by (2.8). Then η 2 (0) the inequality (i) follows from ≥ 0. Let hkl (z) = dk z k + dk+l z k+l and N ||2 ||z α P βkl = (ai aj − bi bj )hHz∗i Hzj z k+l , z k iα . For sufficiently large k, we can choose j−i=l

nonzero dk , dk+l such that |dk η(k)| = |dk+l η(k + l)| and dk dk+l βkl = −|dk dk+l βkl |. For instance, let dk = 1, and  η(k)   if βkl = 0;   η(k + l) dk+l =  η(k)βkl    − if βkl 6= 0. η(k + l)|βkl | Therefore, h(Hf∗ Hf − Hg∗ Hg )hkl , hkl iα = |dk |2 η 2 (k) + |dk+l |2 η 2 (k + l) + 2Re dk dk+l h(Hf∗ Hf − Hg∗ Hg )z k+l , z k iα = |dk |2 η 2 (k) + |dk+l |2 η 2 (k + l) + 2Re dk dk+l βkl

= 2|dk dk+l |η(k)η(k + l) − 2|dk dk+l ||βkl | ≥ 0, i.e., η(k)η(k + l) ≥ ||HzN z k ||α ||HzN z k+l ||α

P (ai aj − bi bj )hHz∗i Hzj z k+l , z k iα j−i=l

||HzN z k ||α ||HzN z k+l ||α

.

Finally, fix l and let k → ∞, then we obtain the inequality (ii).

Our main result now follows. Theorem 2.7. Let ϕ(z) = f (z) + g(z), where f (z) = am z m + an z n , g(z) = bm z m + bn z n (0 < m < n) with |an | ≤ |bn |. Then Tϕ is hyponormal on A2 if and only if |f 0 (z)| ≥ |g 0 (z)| on T. Proof. By Theorem 1.2, we only need to proof the sufficiency. It is clear that max{Re (an am − bn bm )z n−m | z ∈ T} = |an am − bn bm |,

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125

and |f 0 (z)|2 − |g 0 (z)|2 = m2 (|am |2 − |bm |2 ) + n2 (|an |2 − |bn |2 ) + 2mnRe (an am − bn bm )z n−m , for z ∈ T. So |f 0 (z)| ≥ |g 0 (z)| on T if and only if m2 (|am |2 − |bm |2 ) + n2 (|an |2 − |bn |2 ) ≥ 2mn|am an − bm bn |.

Since |bn | ≥ |an |, we have |am | ≥ |bm |. Noting that ζ(k) = decreasing and ζ(k) ≥

(2.9)

||Hzm z k ||2 is strictly ||Hzn z k ||2

m2 , by (2.9) and Lemma 2.2 we have n2 m η 2 (k) ≥ 2 |am an − bm bn |, k 2 n ||Hz z || n

and n η 2 (k) ≥ 2 |am an − bm bn |, ||Hzm z k ||2 m for every k ≥ 0. Therefore, η(k)η(k + n − m) ≥ 2|am an − bm bn |||Hzn z k+n−m ||||Hzm z k || ≥ 2|am an −

bm bn |hHz∗m Hzn z k+n−m ,

(2.10)

k

z i,

and then 2

|dk |2 η 2 (k) +

|dk+n−m | 2 η (k + n − m) 2

(2.11)

1 2 |dk |2 η 2 (k) + |dk+n−m | η 2 (k + n − m) 2 ≥ 2|dk+n−m dk ||am an − bm bn |hHz∗m Hzn z k+n−m , z k i. ≥

For every h = series gent.

∞ P k=0

∞ P n=0

2 2

|dk | η (k) and

dk z k ∈ A2 , we have ∞ P k=0

∞ |d |2 P k < ∞. It follows that the k + 1 n=0

2|dk+n−m dk |hHzn z k+n−m , Hzm z k i are both conver-

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IEOT

Combining the inequalities (2.10) and (2.11), we get hTϕ h, hi =

∞ X

|dk |2 η 2 (k)

k=0

+2

∞ X

Re dk+n−m dk (am an − bm bn ) hHzn z k+n−m , Hzm z k i

k=0

≥

n−m−1 X l=0

+ −

1 2

1 |dl |2 η 2 (l) + |dl+n−m |2 η 2 (l + n − m) 2

∞ X

|dk |2 η 2 (k) + |dk+n−m |2 η 2 (k + n − m)

k=n−m

∞ X

2|dk+n−m dk ||am an − bm bn |hHz∗m Hzn z k+n−m , z k i ≥ 0.

k=0

Remark 2.8. Note that the assumption |an | ≤ |bn | cannot be removed. For example, let f (z) = z m + z n , g(z) = z m with n > 2m > 1. A straight calculation implies that |f 0 (z)|2 − |g 0 (z)|2 = n2 + 2mn Rez n−m ≥ n2 − 2mn > 0. However, the operator Tf +g is not a hyponormal operator. In fact, let h(z) = 1 − q 2n−m+1 n−m n+1 z

2 n+1 (1

−

q

∈ A2 , then a direct computation shows that ||Hf h||2 − ||Hg h||2 =

2n−m+1 n+1 )

< 0.

3. Necessary conditions on A2α with ||f ||α = ||g||α Since the hyponormality of operators is translation invariant, we may assume that f (0) = g(0) = 0. Theorem 3.1. Suppose f, g ∈ zH ∞ (D) such that ||f ||α = ||g||α . If Tf +g is hyponormal on A2α , then Pα (|f |2 − |g|2 ) = 0. Proof. For c ∈ C, the integer k ≥ 0 and h ∈ zH ∞ (D), ||(I − Pα )(h(c + z k ))||2α = ||Hh (z k ) + ch||2α = ||Hh (z k )||2α + |c|2 ||h||2α + 2Re (chHh z k , hiα ). Therefore, by the hypothesis and the equality (2.2), we have ||Hf (c + z k )||2α − ||Hg (c + z k )||2α = ||Hf (z k )||2α − ||Hg (z k )||2α + 2Re(chz k , |f |2 − |g|2 iα ) ≥ 0. Let |c| → ∞, then hPα (|f |2 − |g|2 ), z k iα = h |f |2 − |g|2 , z k iα = 0, k ≥ 0.

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127

For m, k = 1, 2, . . . , N , define a = det m bk

Am,k

bm , ak

and abbreviate An,n by An . I. S. Hwang [5] gave some necessary conditions for the hyponormality of the Toeplitz operators by Am,k . The following corollary shows that Theorem 3.1 is a generalization of Theorem 3.3 in [5]. Corollary 3.2. Let f =

N P

an z n , g =

n=1

is hyponormal on A2α , then  A1,1 A2,2  0 A1,2   0 0  ϑα =   0 0   . .. .  . . 0 0

N P

bn z n with ||f ||α = ||g||α . Suppose Tf +g

n=1

··· A2,3 A1,3

··· ··· ··· .. .

0

..

. ···

···

··· ··· ···

A1,N −1 0

 AN,N  γ 2  1 AN −1,N  γ22     AN −2,N  γ32     .  .. = 0.  .  .  .    2  A2,N  γN −1 2 γN A1,N

Proof. By Theorem 3.1, it is sufficient to show that when ||f ||α = ||g||α , the vector ϑα = 0 if and only if Pα (|f |2 − |g|2 ) = 0. Note that  0 if 0 ≤ k < m,  2 k m α k m γ hz z , Kλ iα = Pα (z z )(λ) = k−m if m ≤ k.  2k λ γk−m It follows that hPα (|f |2 − |g|2 ), Kλα iα =

X

(ak am − bk bm )z k z m , Kλα

α

k,m

=

X

(ak am −

k≥m

=

X k≥m

Theorem 3.3. Suppose f (z) =

N P k=1

Am,k

γ2 bk bm ) 2 k λk−m γk−m

γk2 k−m λ . 2 γk−m

ak z mk , g(z) =

N P

bk z mk (N ≤ +∞), where

k=1

{ak }, {bk } are nonzero in C, and for i, j, k, l ≥ 1, mi − mj 6= ml − mk with (i, j) 6= (l, k). If ||f ||α = ||g||α , then the following statements are equivalent: (a) Tf +g on A2α is hyponormal; (b) Tf +g on A2α is normal; (c) there exists a λ ∈ C with |λ| = 1 such that g(z) = λf (z).

128


IEOT

Proof. Firstly, (b)⇒(a) is trivial. Secondly, assume (c) holds, then Hf∗ Hf = |λ|2 Hf∗ Hf = Hg∗ Hg . Together with this and equality (2.2), we see that Tf +g is normal. So (c)⇒(b). Finally, to show (a)⇒(c), we suppose that Tf +g is a hyponormal operator and ||f ||α = ||g||α . A straight calculation shows that X h(al ak − bl bk )z ml z mk , z m2 −m1 iα h|f |2 − |g|2 , z m2 −m1 iα = l,k

=

X

h(al ak − bl bk )z ml z mk , z m2 −m1 iα

ml − mk = m2 − m1

= h(a2 a1 − b2 b1 )z m2 z m1 , z m2 −m1 iα . The third equality holds since mi − mj 6= m2 − m1 whenever (i, j) 6= (2, 1). Now invoke Theorem 3.1 to conclude that a2 a1 = b2 b1 . Replacing the integer m2 − m1 by ml − mk , we see that al ak = bl bk , ∀ l > k ≥ 1.

(3.1)

In particular, a2 a3 = b2 b3 and a1 a3 = b1 b3 , and therefore |a1 |2 a2 a3 = (a1 a2 )(a1 a3 ) = (b1 b2 )(b1 b3 ) = |b1 |2 b2 b3 . This implies |a1 | = |b1 |. So by (3.1), we have |ak | = |bk | and bk a1 = ak b1 , ∀k ≥ 1. a1 Let λ = , it is easy to check that f (z) = λg(z). b1

References [1] P. Ahern and Z. Cuckovic, A mean value inequality with applications to Bergman space operators, Pacific J. Math. 173(1996), No. 2, 295–305. [2] J. Arazy, S. D. Fisher and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110(1988), No. 6, 989–1053. [3] C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103(1988), No. 3, 809–812. [4] D. R. Farenick and W. Y. Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc. 348(1996), No. 10, 4153–4174. [5] I. S. Hwang, Hyponormal Toeplitz operators on the Bergman space, J. Korean Math. Soc. 42(2005), No. 2, 387–403. [6] I. S. Hwang and J. Lee, Hyponormal Toeplitz operators on the Bergman space. II, Bull. Korean Math. Soc. 44(2007), No. 3, 517–522. [7] K. M. Lewis, Toeplitz operators and hyponormality, Thesis, Purdue University. 1999. [8] Yufeng Lu and Chaomei Liu, Commutativity and hyponormality of Toeplitz operators on the weighted Bergman space, J. Korean Math. Soc. 46(2009), No. 3, 621-642. [9] H. Sadraoui, Hyponormality of Toeplitz operators and composition operators, Thesis, Purdue University. 1992.

Vol. 65 (2009)


Yufeng Lu and Yanyue Shi School of Mathematical Sciences Dalian University of Technology Dalian, 116024 China e-mail: [email protected], [email protected] [email protected] Submitted: December 5, 2008. Revised: April 18, 2009.

129




Common Hypercyclic Vectors and the Hypercyclicity Criterion Rebecca Sanders Abstract. An operator on a separable, infinite dimensional Banach space satisfies the Hypercyclicity Criterion if and only if the associated left multiplication operator is hypercyclic; see [14], [16], [29]. By examining paths of operators where each operator along the path satisfies the criterion, we provide necessary and sufficient conditions for a path of left multiplication operators to have an SOT-dense set of common hypercyclic vectors. As a corollary, we establish a natural sufficient condition for a path of operators to have a common hypercyclic subspace. Mathematics Subject Classification (2000). Primary 47A16; Secondary 47A80, 47B37. Keywords. Hypercyclic operator, hypercyclic vector, unilateral weighted backward shift.

1. Introduction Throughout the present paper, let X be a separable, infinite dimensional Banach space over the scalar field C, and let B(X) denote the algebra of all bounded linear operators on X. An operator T in B(X) is hypercyclic if there is a vector x in X for which its orbit, Orb(T, x) = {T n x : n ≥ 0}, is dense in X. Such a vector x is called a hypercyclic vector for T , and we use the notation HC(T ) to represent the set of hypercyclic vectors for T . An operator T is hypercyclic if and only if its set HC(T ) of hypercyclic vectors is a dense Gδ set; see Kitai [26]. By the Baire Category Theorem, all the operators in a countable T∞ family F = {Tn : n ≥ 1} are hypercyclic if and only if the intersection n=1 HC(Tn ) is a dense Gδ set. However, for an uncountable family F of hypercyclic operators, we cannot apply T the above argument to show the intersection T ∈F HC(T ) of all the hypercyclic vectors forms a dense Gδ set, or is even nonempty. This motivated the study of the existence of vectors that are hypercyclic for each operator in an uncountable family. Throughout the literature, many of these uncountable families of operators

132

Sanders

IEOT

maintain continuity within the family, which leads us to the definition of a path of operators. Formally, a family of operators {Ft ∈ B(X) : t ∈ I}, where I is an interval of real numbers, is a path of operators if the map F : I −→ B(X), given by F (t) = Ft , is continuous with respect to the usual topology T on the interval I and the operator norm topology on B(X). The intersection t∈I HC(Ft )Tis the set of common hypercyclic vectors for the path, and any vector x in the set t∈I HC(Ft ) is a common hypercyclic vector. Several authors have worked on the existence of common hypercyclic vectors. Bayart and Matheron [6], Chan and Sanders [15], and Costakis and Sambarino [19] have established different sufficient conditions for a path of operators to have a dense Gδ set of common hypercyclic vectors. Furthermore, they have provided many natural examples of such paths. Other results on common hypercyclic vectors include the work of Abakumov and Gordon [1], Bayart [4], Bayart and Grivaux [8], Costakis [18], Le´ on and M¨ uller [28], and Conejero, M¨ uller, and Peris [17]. Before we discuss the results in the present paper, we need to turn our attention to the Hypercyclicity Criterion, which is a well known sufficient condition for a single operator to be hypercyclic that was originally established by Kitai [26], and later rediscovered by Genther and Shapiro [23] in a more general setting. The following version of the criterion is due to Bès and Peris [11]: An operator T in B(X) is hypercyclic if there are dense sets X1 , X2 , an increasing sequence (mk )∞ k=1 of positive integers, and maps Sk : X1 −→ X such that (i) for each x ∈ X1 , we have Sk x −→ 0 as k → ∞, (ii) for each x ∈ X2 , we have T mk x −→ 0 as k → ∞, and (iii) for each x ∈ X1 , we have T mk Sk x −→ x as k → ∞. De La Rosa and Read [21] have recently provided an example of a Banach space that admits a hypercyclic operator which fails to satisfy the Hypercyclicity Criterion, while Bayart and Matheron [7] showed some well known Banach spaces also admit such hypercyclic operators. However, prior to these recent results, every known hypercyclic operator satisfied the criterion, and as a result, it has played a vital role in the development of the field of hypercyclicity, and many results involve the criterion. With the introduction of common hypercyclic vectors for paths of operators, one can ask which results about operators that satisfy the criterion can be generalized to paths of operators. We tackle a question of this nature in Section 3. Before this in Section 2, we examine common hypercyclic vectors for paths of operators where each operator along the path satisfies the Hypercyclicity Criterion. While a single operator satisfying the criterion has a dense Gδ set of hypercyclic vectors, we cannot say the same for the set of common hypercyclic vectors of a path of operators where each operator along the path satisfies the criterion. In fact, even assuming each operator satisfies the criterion with respect to the exact same increasing sequence (mk )∞ k=1 of positive integers, and the exact same dense sets X1 , X2 does not guarantee the existence of a single common hypercyclic vector; see Proposition 2.1 below. Thus, when attempting to extend a result about an

Vol. 65 (2009)

Hypercyclicity Criterion

133

operator that satisfies the criterion to a path of operators, it may not be enough to assume each operator along the path satisfies the criterion. Now over the years, several different characterizations of the criterion have emerged. Bès and Peris [11] showed an operator T satisfies the criterion if and only if the set HC(T ⊕ T ) is a dense Gδ set. Their characterization together with LM Furstenberg’s [22] older result that n=1 T is topologically transitive whenever T ⊕ T is topologically transitive leads to the following characterization: An operator T in B(X) satisfies the Hypercyclicity Criterion if and only if for each LM positive integer M , the set HC( n=1 T ) of hypercyclic vectors is a dense Gδ set LM in n=1 X. A more general version of this statement is provided by Bernal and Grosse-Erdmann [9]. With this characterization in mind, it becomes natural to inT LM stead study paths {Ft ∈ B(X) : t ∈ [a, b]} such that the set t∈[a,b] HC( n=1 Ft ) of common hypercyclic vectors is a dense Gδ set for each positive integer M . In this case, we still maintain that each operator along the path satisfies the criterion. In Section 2, we provide a necessary and sufficient condition for the set LM t∈[a,b] HC( n=1 Ft ) of common hypercyclic vectors to be a dense Gδ set for each positive integer M ; see Theorem 2.3 below. Furthermore, for the case when a = b, this condition is equivalent to a condition provided by Bernal and GrosseErdmann [9] for a single operator to satisfy the criterion. Next, there are several classes of operators where being hypercyclic is equivalent to satisfying the Hypercyclicity Criterion. For example, Bès and Peris [11] proved the weighted shift operators are one such class. We conclude Section 2 by extending this result to paths of shifts operators; that is, we show a path {Ft : t ∈ [a, b]} of weighted shift operators has a dense Gδ set of common hypercyclic vectors if and only if for each T LM positive integer M , the set t∈[a,b] HC( n=1 Ft ) is a dense Gδ set; see Corollary 2.4 and Corollary 2.5 below.

T

In Section 3, we extend results about a single operator satisfying the Hypercyclicity Criterion to paths of operators. To begin, the operator algebra B(X) under the strong operator topology, abbreviated SOT for short, is separable, and we say a bounded linear operator L : B(X) −→ B(X) is hypercyclic if there is a vector V in B(X) for which its orbit Orb(L, V ) is SOT-dense in B(X). Chan [14] pioneered the study of hypercyclic operators on the operator algebra by showing that if an operator T on a Hilbert space H satisfies the Hypercyclicity Criterion, then the left multiplication operator LT : B(H) −→ B(H), which is given by LT V = T V , is hypercyclic. This result was generalized to the Banach space case by Chan and Taylor [16]. Mart´ınez and Peris [29] improved the result by showing an operator T in B(X) satisfies the criterion if and only if the associated left multiplication operator LT : B(X) −→ B(X) is hypercyclic. To extend this result to a path of operators first note that a family of operators {Ft ∈ B(X) : t ∈ [a, b]} is a path if and only if the associated family of left multiplication operators {LFt : t ∈ [a, b]} is a path because kLFt − LFs k = kFt − Fs k. Bayart [5] established a sufficient condition for a path of left multiplication operators to have a common hypercyclic

134

Sanders

IEOT

vector. It naturally follows to ask if there exists necessary and sufficient conditions for such a path to have an SOT-dense set of common hypercyclic vectors. Keeping Mart´ınez and Peris’ [29] result and results of Section 2 in mind, we show the path {LFt : t ∈ [a, b]} of left multiplication operators has an SOT-dense set of common hypercyclic vectors if and only if for each positive integer M , the set T LM t∈[a,b] HC( n=1 Ft ) is a dense Gδ set; see Theorem 3.1 below. A hypercyclic manifold for an operator T in B(X) is an infinite dimensional manifold consisting entirely, except the zero vector, of hypercyclic vectors, while a hypercyclic subspace is a closed hypercyclic manifold. Montes [31] showed that if an operator T satisfies the Hypercyclicity Criterion, and there is a closed, infinite dimensional subspace X0 for which kT m xk −→ 0 as m → 0 for each x ∈ X0 , then the operator T has a hypercyclic subspace. Later, by studying the spectral properties of operators that satisfy the criterion, he and León [27] developed several necessary and sufficient conditions for the existence of hypercyclic subspaces. In the area of common hypercyclic subspaces, Bayart [5] and Aron, Bès, León, and Peris [3] have different sufficient conditions for the existence of common hypercyclic subspaces. We conclude Section 3 by using Theorem 3.1 to provide a sufficient condition for a path of operators to have a hypercyclic subspace. Furthermore, the sufficient condition is a natural extension of Montes’ [31] original result; see Corollary 3.5 below.

2. The Hypercyclicity Criterion An operator that satisfies the Hypercyclicity Criterion has a dense Gδ set of hypercyclic vectors. However, a path of operators where each operator along the path satisfies the criterion may fail to have even a single common hypercyclic vector. For example, let {ej : j ≥ 0} be the canonical orthonormal basis of `2 . A bounded linear operator T : `2 −→ `2 is a unilateral weighted backward shift if there is a positive weight sequence {wj : j ≥ 1} such that T ej = wj ej−1 for integers j ≥ 1, and T e0 = 0. Chan and Sanders [15] constructed an example of a path of hypercyclic unilateral weighted backward shifts on `2 that fails to have a common hypercyclic vector. Since every shift along this path satisfies the criterion, see Bès and Peris [11], this is the desired path. However, we in fact have something worse. Proposition 2.1. There is a path of hypercyclic operators on `2 which fails to have a common hypercyclic vector, and yet each operator along the path satisfies the Hypercyclicity Criterion with respect to the exact same increasing sequence (mk )∞ k=1 of positive integers, and the exact same dense sets X1 , X2 . Proof. Let {Ft : t ∈ 0, 1/4]} be the path of hypercyclic unilateral weighted backward shifts on `2 that fails to have a common hypercyclic vector given in the proof of Theorem 4.1 in [15]. That is, for each t ∈ [0, 1/4], the weight sequence (t) {wj : j ≥ 1} of the operator Ft is constructed in the following manner. Set

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135

r0 = s0 = 1 and a0 = 0. For each integer i ≥ 0, inductively set ri+1 = (2 + ai + si )3 ,

(2.1)

si+1 = (i + 1) + (r0 + r1 + · · · + ri+1 ), and

(2.2)

ai+1 = si+1 + (r0 + r1 + · · · + ri+1 ).

(2.3)

For each integer i ≥ 0, define (t) w1+ai

=

1 +t 2

r i

(t)

,

(2.4)

(t)

w2+ai = · · · = w1+ai +si = 2,

(2.5)

(t) w2+ai +si

=2

,

(2.6)

(t) w3+ai +si

= · · · = ws(t) = 1, and i+1

(2.7)

(t) w1+si+1

−(si +r0 +r1 +···+ri+1 )

= ··· =

wa(t) i+1

= 2.

(2.8)

To show that each operator Ft satisfies the criterion with respect to the same increasing sequence (mk )∞ k=1 of positive integers, and the exact same dense sets X1 , X2 , first observe that by (2.1)–(2.8), we have ai+1 (t)

Y

wl

≥ 2−ri

whenever i ≥ 0.

(2.9)

l=1+ai

Thus, for any integers i, j with i ≥ 0 and 0 ≤ j ≤ i, we get 1+aY i +si −j

(t)

wl

=

Y ai

l=1

≥2

(t)

l=1 −r0 −r1

2

(t)

(t)

(t)

w1+ai (w2+ai · · · w1+ai +si −j )

wl

· · · 2−ri−1 2−ri 2si −j

(2.10) by (2.9), (2.4), (2.5)

= 2i−j . Now, for each integer k ≥ 1, let ik = 2k and mk = 1 + aik + sik − k. Set X1 = X2 = 1 ej+1 span{ej : j ≥ 0}, and define the linear map St : X1 −→ `2 by St ej = (t) wj+1

for all integers j ≥ 0. Clearly, Ftmk Stmk x = x for each x ∈ X1 , and kFtmk xk −→ 0 as k → ∞ for each x ∈ X2 . Lastly, observe that for integers j, k with j ≥ 0 and k ≥ j, we have kStmk ej k−1 =

mk Y l=1

=

(t)

(t)

w1 · · · wj

1 (t) w1

mY k +j

1

(t)

wj+l =

(t) · · · wj

(t)

wl

l=1

1+aik +sik −(k−j)

Y l=1

(t)

wl

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Sanders

≥ =

1 (t) w1 (t)

(t)

· · · wj 1

(t)

w1 · · · wj

2ik −(k−j)

IEOT

by (2.10)

2k−j ,

and so kStmk xk −→ 0 as k → ∞ for each x ∈ X1 .

Let {ej : j ∈ Z} be the canonical orthonormal basis of `2 (Z). A bounded linear operator T : `2 (Z) −→ `2 (Z) is a bilateral weighted shift if there is a positive weight sequence {wj : j ∈ Z} such that T ej = wj ej−1 for each integer j. Chan and Sanders [15] also constructed a path of hypercyclic bilateral weighted shifts that fails to have a common hypercyclic vector. Using techniques similar to those in the proof of Proposition 2.1, one can show every shift along this path also satisfies the criterion with respect to the same sequence (mk )∞ k=1 and the same dense sets X1 , X 2 . The version of the Hypercyclicity Criterion established by Kitai in [26] is stronger than the version stated in the Introduction. The Kitai Criterion states that an operator T in B(X) is hypercyclic whenever there exist dense sets X1 , X2 and a map S : X1 −→ X such that S m → 0 on X1 , T m → 0 on X2 and T S = I on X1 where I is the identity map. Grivaux [25] showed every separable, infinite dimensional Banach space admits a hypercyclic operator that fails to satisfy the Kitai Criterion, while Shkarin [33] showed every such Banach space admits an operator which satisfies the Kitai Criterion. A path of operators consisting entirely of operators that satisfy the Kitai Criterion can still fail to have a common hypercyclic vector. For an example, consider the family of operators {sB ⊕ tB : (s, t) ∈ [2, 3] × [2, 3]}, where B is the unweighted unilateral weighted backward shift on `2 . Borichev [6, Remark 6.3] showed this family of operators fails to have a common hypercyclic vector. Furthermore, each operator in this family satisfies the Kitai Criterion. If we let F : [0, 1] −→ [2, 3] × [2, 3] be the space filling curve, we get the desired path of operators. As stated in the Introduction, instead of examining paths where each operator along the path satisfies the Hypercyclicity Criterion, we examine paths {Ft ∈ B(X) : t ∈ [a, b]} which satisfy the stronger property that for each positive integer LM M , the path of operators { n=1 Ft : t ∈ [a, b]} has a dense Gδ set of common hypercyclic vectors. The set of hypercyclic vectors for an operator T forms a dense Gδ set if and only if the operator T is topologically transitive; that is, for any pair of nonempty open sets U ,V , there is a positive integer m for which T m (U ) ∩ V 6= ∅; see Kitai [26]. Chan and Sanders [15] extended this topological transitivity property by showing a path {Ft ∈ B(X) : t ∈ [a, b]} has a dense Gδ set of common hypercyclic vectors if and only if for any pair of nonempty open sets U ,V , there is a partition P = {a = t0 < · · · < tk = b} of the interval [a, b], positive e ⊆ U such that Ftmi (U e) ⊆ V integers m1 , . . . , mk , and a nonempty open set U whenever 1 ≤ i ≤ k and t ∈ [ti−1 , ti ]. The above transitivity result used on

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LM the path { n=1 Ft : t ∈ [a, b]} combined together with the observation that the LM LM collection { n=1 Un : Un open in X} forms a basis of the topology on n=1 X leads to the following necessary and sufficient condition. Theorem 2.2. Let M be a positive integer, and let {Ft ∈ B(X) : t ∈ [a, b]} be T LM a path of operators. The set t∈[a,b] HC( n=1 Ft ) of common hypercyclic vectors is a dense Gδ set if and only if for any 2M nonempty open sets U1 , . . . , UM and V1 , . . . , VM , there is a partition P = {a = t0 < · · · < tk = b} of the interval [a, b], f1 , . . . , U g positive integers m1 , . . . , mk , and M nonempty open sets U M such that mi f f Un ⊆ Un and Ft (Un ) ⊆ Vn , whenever 1 ≤ n ≤ M , 1 ≤ i ≤ k and t ∈ [ti−1 , ti ]. Returning to the Hypercyclicity Criterion for a single operator, Godefroy and Shapiro [24] proved that an operator T is hypercyclic if for any three nonempty open sets U ,V and W with 0 ∈ W , there is a positive integer m for which T m (U ) ∩ W 6= ∅ and T m (W )∩V 6= ∅. Bernal and Grosse-Erdmann [9] showed this sufficient condition for an operator to be hypercyclic is, in fact, equivalent to the criterion. Using Theorem 2.2, we establish a path version of this result. Theorem 2.3. Let {Ft ∈ B(X) : t ∈ [a, b]} be a path of operators. The set T LM t∈[a,b] HC( n=1 Ft ) of common hypercyclic vectors is a dense Gδ set for each positive integer M if and only if for any positive integer M , and for any 3M nonempty open sets U1 , . . . , UM , V1 , . . . , VM , and W1 , . . . , WM with 0 ∈ Wn for 1 ≤ n ≤ M , there is a partition P = {a = t0 < · · · < tk = b} of the interf1 , . . . , U g val [a, b], positive integers m1 , . . . , mk , and 2M nonempty opens sets U M, f g W1 , . . . , WM such that fn ⊆ Un and Ftmi (U fn ) ⊆ Wn , and (i) U mi g g (ii) Wn ⊆ Wn and Ft (Wn ) ⊆ Vn , whenever 1 ≤ n ≤ M , 1 ≤ i ≤ k, and t ∈ [ti−1 , ti ]. T LM Proof. We prove the forward implication by first assuming t∈[a,b] HC( n=1 Ft ) is a dense Gδ for any positive integer M . By applying Theorem 2.2 with the positive integer 2M and with the nonempty open sets U1 , . . . , UM , V1 , . . . , VM , and W1 , . . . , WM , there is a partition P = {a = t0 < · · · < tk = b} of [a, b], positive f1 , . . . , U g f g integers m1 , m2 , . . . mk , and 2M nonempty open sets U M , W1 , . . . , WM mi g mi f f g such that Un ⊆ Un , Wn ⊆ Wn , and Ft (Un ) ⊆ Wn , Ft (Wn ) ⊆ Vn whenever 1 ≤ n ≤ M , 1 ≤ i ≤ k, and t ∈ [ti−1 , ti ]. For the backward implication, we again use Theorem 2.2. For any positive integer M and any 2M nonempty open sets U1 , . . . , UM and V1 , . . . , VM , choose vectors u1 , . . . , uM , v1 , . . . , vM ∈ X and an > 0 such that B(un , 2) ⊆ Un and B(vn , 2) ⊆ Vn whenever 1 ≤ n ≤ M . By assumption, there is a partition P = {a = t0 < · · · < tk = b} of [a, b], positive integers m1 , . . . , mk , and 2M

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f1 , . . . , U g f g nonempty open sets U M , W1 , . . . , WM such that fn ⊆ B(un , ) U g W n ⊆ B(0, )

and

fn ) ⊆ B(0, ), Ftmi (U

and

g Ftmi (W n)

and

⊆ B(vn , ),

(2.11) (2.12)

whenever 1 ≤ n ≤ M , 1 ≤ i ≤ k and t ∈ [ti−1 , ti ]. For each integer n with fn and w g 1 ≤ n ≤ M , select vectors u fn ∈ U fn ∈ W n . Then whenever 1 ≤ n ≤ M , we have k(f un + w fn ) − un k ≤ kf un − un k + kw fn k < 2,

by (2.11) and (2.12).

In addition, whenever 1 ≤ i ≤ k and t ∈ [ti−1 , ti ], kFtmi (f un + w fn ) − vn k ≤ kFtmi w fn − vn k + kFtmi u fn k < 2,

by (2.11) and (2.12).

Ftmi (f un

Since u fn + w fn ∈ B(un , 2) ⊆ Un and +w fn ) ∈ B(vn , 2) ⊆ Vn , there are g f g f1 , . . . , U M nonempty open sets U M that satisfy Theorem 2.2. The class of weighted shift operators is one of several classes of operators where being hypercyclic is equivalent to satisfying the Hypercyclicity Criterion. From the work above, it naturally follows to ask whether a path {Ft : t ∈ [a, b]} of weighted shift operators has a dense Gδ set of common hypercyclic vectors if and LM only if for each positive integer M , the path { n=1 Ft : t ∈ [a, b]} has a dense Gδ set of common hypercyclic vectors. We first answer this question in the positive for a path of bilateral weighted shifts. Proposition 2.4.T Let {Ft : t ∈ [a, b]} be a path of bilateral weighted shifts on `2 (Z). The set t∈[a,b] HC(Ft ) of common hypercyclic vectors is a dense Gδ set if T LM and only if for each positive integer M , the set t∈[a,b] HC( n=1 Ft ) of common hypercyclic vectors is dense Gδ set. Proof. We clearly only have to establish the forward implication, and we use Theorem 2.3 to do this. Let M be a positive integer, and let U1 , . . . , UM , V1 , . . . , VM , and W1 , . . . , WM be 3M nonempty open sets with 0 ∈ Wn for 1 ≤ n ≤ M . Select an integer N ≥ 1, nonzero vectors u1 , . . . , uM , v1 , . . . , vM ∈ span{ej : |j| ≤ N −1}, and an > 0 such that un ∈ Un , B(vn , ) ⊆ Vn , and B(0, ) ⊆ Wn ,

(2.13)

whenever 1 ≤ n ≤ M . Let C = max{ku1 k, . . . , kuM k, kv1 k, . . . , kvM k}, and select a δ such that δ C< . (2.14) 0 < δ < 1 and 1−δ 4 2N By Ansari HC(FT t ) = HC(Ft ) for each shift Ft , and so by assumption, T [2], we have2N the set t∈[a,b] HC(Ft ) = t∈[a,b] HC(Ft ) is a dense Gδ set. Using Theorem 2.2

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with the path {Ft2N : t ∈ [a, b]}, there is a partition P = {a = t0 < · · · < tk = b} of [a, b], positive integers 2N m1 , . . . , 2N mk , and a nonempty open set O for which X (1 − δ) O⊆B ej , min δ, , (2.15) C |j|≤N −1

and whenever 1 ≤ i ≤ k and t ∈ [ti−1 , ti ], X 2mi N Ft (O) ⊆ B

ej , δ .

(2.16)

|j|≤N −1

The integers 2N m1 , . . . , 2N mk associated with the partition P = {a = t0 < · · · < tk = b} are not necessarily distinct. To simplify our computations, let 2N mλ1 , . . . , 2N mλr be the distinct integers from the list 2N m1 , . . . , 2N mk . For each integer i with 1 ≤ i ≤ r, let Ji be the union of all the closed subintervals from the partition P that correspond to the integer 2N mλi ; that is, [ Ji = {[tλ−1 , tλ ] : 1 ≤ λ ≤ k and mλ = mλi }. Let ni = 2N mλi for 1 ≤ i ≤ r. Note that |ni − ni0 | ≥ 2N whenever i 6= i0 . Since the subsets J1 , . . . , Jr partition the interval [a, b], it suffices to establish Conditions (i) and (ii) in Theorem 2.3 using the partition J1 , . . . , Jr and the distinct integers n1 , . . . , nr . Before we do this, we first establish some inequalities. To this end, let g ∈ O. By (2.15),

2 X X X

g − e |hg, ej i − 1|2 + |hg, ej i|2 = j

|j|≤N −1 |j|≤N −1 |j|≥N (2.17) (1 − δ) , < min δ, C and so, |hg, ej i| > 1 − δ

whenever |j| ≤ N − 1.

(2.18)

(t)

Next, let {wj : j ∈ Z} be the weight sequence of the shift Ft . By (2.16) and by the definitions of the partition J1 , . . . , Jr , the distinct integers n1 , . . . , nr , and the operator Ft , for any integer i with 1 ≤ i ≤ r and any t ∈ Ji , 2

2 ni X Y X

(t) 2 hg, ej+ni i ≤ Ftni g −

w − 1 e (2.19) j < δ , j+l

|j|≤N −1

|j|≤N −1

l=1

and X |j|≥N

|hg, ej+ni i|2

Y ni l=1

(t)

wj+l

2

n i ≤

Ft g −

X |j|≤N −1

2

2 ej

l=1

1−δ |hg, ej+ni i|

by (2.14) and (2.19),

(2.21)

and (1 − δ)

nY i −1

(t)

wj−l ≤ |hg, ej i|

nY i −1

l=0

(t)

wj−l

by (2.18)

l=0

= |hg, e(j−ni )+ni i|

ni Y

(2.22)

(t)

w(j−ni )+l

l=1

0), and bi-parametric scale of function spaces Hβ,p (R ) associated with Jβα . These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces α (Rn ) is given with the aid of a special wavelet–like transform associHβ,p ated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1). Mathematics Subject Classification (2000). Primary 26A33; Secondary 46E35, 42C40. Keywords. Fractional integral, Bessel potential, semigroup, wavelet transforms, Sobolev-type space.

1. Introduction The Bessel potentials J α ϕ, ϕ ∈ Lp (Rn ) and the corresponding spaces Hpα (Rn ) = {f : f = J α ϕ, ϕ ∈ Lp (Rn )} of these potentials were introduced by N. Aronszajn and K. Smith [6], and A.Calder´ on [7]. These potentials play an important role in analysis and its applications; see, e.g., E. M. Stein [23, p. 121–141], S. G. Samko, A. A. Kilbas and O. I.Marichev [20, p. 538–554], T. M. Flett [9], and references therein. Characterization and various properties of Bessel and Riesz potentials, and their diverse generalizations and modifications were studied by many authors; see S. G. Samko [21], [22], A. Almeida and S. G. Samko [5], V. A. Nogin and S. G. Samko [16], V. Kokilashvili, A. Meskhi and S. G. Samko [12], I. A. Aliev and B. Rubin [1], J. Hu and M. Zahle [11], I. A. Aliev and M. Eryigit [4]. More information on this subject can be found in the books of E.M.Stein [23], S. G. Samko [21], S. G. Samko, A. A. Kilbas and O. I. Marichev [20], B. Rubin [17]. The author was supported by the Scientific Research Project Administration Unit of the Akdeniz University and TUBITAK.

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In this paper we define new potentials Jβα ϕ, (α, β > 0), (which we call α (Rn ). bi-parametric potentials), and introduce corresponding function spaces Hβ,p These potentials are formally defined in the Fourier terms by −α/β Jβα ϕ = F −1 (1 + |ξ|β )−α/β F ϕ ≡ E + (−∆)β/2 ϕ, n 2 2 where E is the identity operator and ∆ = 1 ∂ /∂xk is the Laplacian. Biparametric potentials generalize the classical Bessel potentials (for β = 2), and the Flett potentials (for β = 1). For detailed information about the Flett potentials; see T. M. Flett [9, p. 446–447]; S. G. Samko, A. A. Kilbas and O. I. Marichev [20, p. 541–542]; I. A. Aliev, S. Sezer, M. Eryigit [3]. α A characterization of the relevant fractional Sobolev-type spaces Hβ,p (Rn ) is given with the aid of a special wavelet-like transform generated by a so-called β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson semigroup (for β = 1). The paper is organized as follows. Some necessary definitions, notation and auxiliary lemmas are given in Section 2. In Section 3 we introduce a new wavelet transform generated by β-semigroup associated to multiplication by exp(−t|ξ|β ) in terms of the Fourier transform. Such semigroups arise in the context of stable random processes in probability, in pseudodifferential parabolic equations, in integral geometry, and enjoy a number of remarkable properties; see A. Koldobsky [13], N. S. Landkof [14], M. V. Fedorjuk [8], B. Rubin [19], I. A. Aliev, B. Rubin, S. Sezer, S. Uyhan [2]. Section 4 contains the definition and some properties of the biparametric potentials. In this section, using the wavelet-like transform defined in Section 3, we obtain explicit inversion formulas for bi-parametric potentials in the framework of the Lp −theory. Section 5 is devoted to an application of the results α α of Section 4 to the spaces Hβ,p (Rn ) . Although the spaces Hβ,p (Rn ), (1 < p < ∞) α n actually coincide with the classical space H2,p (R ) of Bessel potentials up to equivalence of norms (see Corollary 5.4 below), and various characterizations of the latter space is known (see S. G. Samko [21, p. 184–193]; S. G. Samko, A. A. Kilbas and O. I. Marichev [20, p. 541–542]; B. Rubin [17, p. 260–262]), Theorem 5.5 in Section 5 gives a new characterization of these spaces via wavelet-like transform, introduced in Section 3.

2. Notation and auxiliary lemmas Let Rn be the n-dimensional Euclidean space and Lp ≡ Lp (Rn ) be the standart space of the measurable functions such that 1/p f p = |f (x)|p dx < ∞, (1 ≤ p < ∞). Rn

For technical reasons, we do not consider the case p = ∞ here, although some results of the article extend to the space L∞ substituted for the class C0 ≡ C0 (Rn ) of all continuous functions on Rn vanishing at infinity. The notation f ∞ will

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be used for the sup-norm, i.e., f ∞ = sup |f (x)| . The notation S ≡ S(Rn ) will x∈Rn

denote the Schwartz class of test functions. It is well known that this class is dense in Lp , 1 ≤ p < ∞ and C0 . The Fourier transform of a function f on Rn is defined by ∧

e−ix.ξ f (ξ)dξ,

f (x) ≡ (F f )(x) =

x.ξ = x1 ξ1 + · · · + xn ξn .

Rn

The notation F −1 designates inverse Fourier transform: (F −1 f )(ξ) = (2π)−n (F f )(−ξ). We need the following lemmas. Lemma 2.1. (Lizorkin’s multiplier theorem; see [15], [17, p. 8]). Let 1 < p < ∞. Assume that m(ξ), ξ ∈ Rn and its derivatives ∂1k1 ∂2k2 · · · ∂nkn m(ξ) with kj = 0, 1 are continuous outside of the coordinate hyperplanes and such that sup |ξ1k1 . . . ξnkn (∂1k1 · · · ∂nkn m(ξ))| < ∞. ξ

∧

Then m(ξ) is a p-multiplier, i.e. the linear operator Tm defined by (Tm ϕ) (ξ) = ∧ m(ξ)ϕ (ξ), ϕ ∈ S(Rn ), can be extended to a linear bounded operator acting from Lp to Lp . Lemma 2.2. (on pointwise (a.e.) convergence [24, p. 60]). Let {Tε }ε>0 be a family of linear operators, mapping Lp ≡ Lp (Rn ), 1 ≤ p ≤ ∞ into space of measurable functions on Rn . Define (T ∗ f ) by setting (T ∗ f )(x) = sup |(Tε f )(x)|, x ∈ Rn . ε>0

Suppose that there exists a constant c > 0 and a real number q ≥ 1 such that c f q p meas {x : |(T ∗ f )(x)| > t} ≤ t for all t > 0 and f ∈ Lp . If there exists a dense subset D of Lp such that limε→0 (Tε g)(x) exists and is finite a.e. whenever g ∈ D, then for each f ∈ Lp , limε→0 (Tε f )(x) exists and is finite a.e. The next statement is a special case of Rubin’s Lemma 1.3 from [18]. Lemma 2.3. [18, p. 8] Let µ be a finite Borel measure on R+ , and let kθ (s) = s−1 (I θ+1 µ)(s), where s 1 θ+1 (I µ)(s) = (s − t)θ dµ(t) (2.1) Γ(θ + 1) 0

is the Riemann–Liouville fractional integral of order (θ + 1) of the measure µ. Let further, θ = Re θ ≥ 0, and let µ satisfy the following conditions:

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∞ (i)

tj dµ(t) = 0,

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j = 0, 1, . . . , [θ ] (the integer part of θ ).

0

∞ (ii)

tγ d|µ|(η) < ∞

for some

γ > θ .

1

Then

O(sθ −1 ) if 0 < s < 1, O(s−δ−1 ) if s ≥ 1; δ = min{γ − θ , 1 + [θ ] − θ } > 0. ∞ Furthermore, if µ ˜ (t) = e−tη dµ(η) is the Laplace transform of µ, then kθ (s) =

0

∞ cµ (θ) ≡

kθ (s) ds = 0

∞

0

µ ˜(t) dt. tθ+1

(2.2)

Throughout the paper it will be assumed that cµ (θ) = 0 . The examples of suitable measures µ are given in [1], p. 346-347.

3. β-semigroup and associated wavelet-like transform In this section, for a given real number β > 0, we introduce a semigroup {Wtβ ϕ}t>0 , ϕ ∈ Lp ≡ Lp (Rn ), which we call β-semigroup. The well known Gauss-Weierstrass and Poisson semigroups are obtained from {Wtβ ϕ}t>0 by taking β = 2 and β = 1, respectively. Using this β-semigroup, we define a wavelet-like transform, generated by a signed Borel measure. Definition 3.1. For a given β > 0, the family of integrals (β) (Wt ϕ)(x) = ϕ(x − y)w(β) (y, t) dy, (t > 0, x ∈ Rn )

(3.1)

Rn

is called the β-semigroup, generated by a function ϕ ∈ Lp , (1 ≤ p ≤ ∞). Here w(β) (y, t) = t−n/β G(β) (t−1/β y), (β)

and G

(y) ≡ w

(β)

(t > 0),

(3.2) β

n

(y, 1) is the inverse Fourier transform of exp(−|ξ| ), ξ ∈ R , i.e. β G(β) (y) = (2π)−n eiy.ξ−|ξ| dξ. Rn

Straightforward calculations show that β

F (w(β) (., t))(x) = e−t|x| , (t > 0). The following are remarkable examples of the kernels w

(β)

(3.3) (y, t):


155

t Γ((n + 1)/2) (the Poisson kernel). π (n+1)/2 (|y|2 + t2 )(n+1)/2 2 (b) w(2) (y, t) = (4πt)−n/2 e−|y| /4t (the Gauss-Weierstrass kernel). Note that the β-semigroup (3.1) arises in various contexts of analysis, integral geometry, and probability; see, e.g., [8], [14], [13], [19]. The following lemma contains some properties of kernels w(β) (y, t) and integrals Wtβ ϕ. (Compare with the analogous properties of the classical GaussWeierstrass and Poisson integrals; [17, p. 217-223], [23, p. 61-63] ). (a) w(1) (y, t) =

Lemma 3.2. Let t > 0, y ∈ Rn , and 0 < β < ∞. Then (a) w(β) (y, t) is radial with respect to y ∈ Rn and w(β) (λ1/β y, λt) = λ−n/β w(β) (y, t) , ∀ λ > 0.

(3.4)

(b) w(β) (y, t) is positive provided that 0 < β ≤ 2. (c) If β > 0 is an even integer, then w(β) (y, t) is infinitely smooth and rapidly decreasing as |y| → ∞. Moreover, for any fixed β > 0 and t > 0, w(β) (y, t) = O(|y|−n−β ) as |y| → ∞; therefore, w(β) (y, t) has a decreasing and integrable radial majorant. (d) w(β) (y, t)dy = 1, ∀t > 0, ∀β > 0.

(3.5)

Rn

(e) If 1 ≤ p ≤ ∞, then (β)

where c(β) =

Rn

(f )

Wt ϕp ≤ c(β)ϕp , ∀t > 0, |w

(β)

(3.6)

(y, 1)|dy < ∞. If 0 < β ≤ 2, then c(β) = 1.

(β) sup (Wt ϕ)(x) ≤ c(M ϕ)(x), ϕ ∈ Lp , 1 ≤ p ≤ ∞, t>0

(3.7)

where M ϕ is the well-known Hardy-Littlewood maximal function: 1 |ϕ(y)|dy, (M ϕ)(x) = sup r>0 |B(x, r)| B(x,r)

B(x, r) is the ball of radius r, centered at x ∈ Rn . (g)

(β) sup (Wt ϕ)(x) ≤ ct−n/βp ||ϕ||p ,

x∈Rn

1 ≤ p < ∞.

(3.8)

(h) (the semigroup property) (β)

Wt

(β)

(Wτ(β) ϕ) = Wt+τ ϕ ,

∀ t, τ > 0.

(3.9)

(i) Let ϕ ∈ Lp , 1 ≤ p ≤ ∞, (L∞ ≡ C0 ). Then (β)

lim (Wt

t→0

ϕ)(x) = ϕ(x),

(3.10)

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with the limit being understood in the Lp -norm or pointwise for almost all x ∈ Rn . If ϕ ∈ C0 , the convergence is uniform. Proof. (a) follows from the definition of w(β) (y, t). The positivity of w(β) (y, t) in case of n = 1 can be found in [13, p. 44-45]. The general case n ≥ 1 was investigated by B. Rubin [19], Lemma 7.1. For the statement (c), see [14] when 0 < β ≤ 2, and [8] when 1 < β < ∞. See also [19], [2] where it has been established explicit asymtotic formula: w(β) (y, 1) = cβ |y|−n−β (1 + o(1)), cβ =

2β π −n/2 Γ((n + β)/2) ; β = 2, 4, 6, . . . Γ(−β/2)

The proofs of (d) and (e) are standard. Since by (c) the kernel w(β) (y, t) has a decreasing and integrable radial majorant, (f ) and (i) can be proved by the same way as in Theorem 2 in Stein’s book [23, p. 62-63]. (h) is obvious in terms of Fourier transform if ϕ ∈ S. For an arbitrary ϕ ∈ Lp the result follows by density of S in Lp , (L∞ ≡ C0 ), using the statement (e). The estimate (g) is proved by making use of the Hölder inequality and (3.2). Indeed 1/p (β) |w(β) (y, t)|p dy |(Wt ϕ)(x)| ≤ ϕp = ϕp t−n/β

Rn

|G(β) (t−1/β y)|p dy

Rn

= ct−n/βp ϕp ,

c=

1/p

( we set y = t1/β x, dy = tn/β dx)

1/p (c) |G(β) (x)|p dx < ∞.

Rn

We now define a wavelet-like transform associated with the β-semigroup

(β) {Wt f }t>0 .

(β)

Definition 3.3. Let µ be a finite Borel measure on [0, ∞), ϕ ∈ Lp , and Wt ϕ be the β- semigroup defined as (3.1). A wavelet-like transform associated with (β) β-semigroup Wt ϕ is defined by (Aϕ)(x, η) ≡ (A(β) µ ϕ)(x, η)

(β)

e−tη (Wtη ϕ)(x)dµ(t),

= µ({0})ϕ(x) + (0,∞) n

where x ∈ R , η > 0. Making use of the convention

b (· · · )dµ(t) = a

(· · · )dµ(t) [a,b)

(3.11)


157

and (3.10), it is easy to see that (Aϕ)(x, η) can be written in the form ∞ (Aϕ)(x, η) =

(β)

e−tη (Wtη ϕ)(x)dµ(t).

(3.12)

0

The generalized Minkowski inequality shows that the operator A is well defined for ϕ ∈ Lp , 1 ≤ p ≤ ∞. Indeed, ∞ Aϕ(., η)p ≤

e

−tη

(β) Wtη ϕ)p

∞

(3.6)

d|µ|(t) ≤ c(β)||µ|| ||ϕ||p , µ ≡

0

d|µ|(t) < ∞. 0

4. Bi-parametric potentials and their inverses with the aid of wavelet-like transform In this section we introduce bi-parametric potentials associated with the β-semigroup (3.1). These potentials contain the classical Bessel and Flett potentials as special cases . Definition 4.1. Let 1 ≤ p ≤ ∞, α > 0 and β > 0. A bi-parametric potential of order α of a function ϕ ∈ Lp is defined by (Jβα ϕ)(x)

1 = Γ(α/β)

∞

α

(β)

t β −1 e−t (Wt

ϕ)(x)dt,

(4.1)

0

(β)

(β)

where {Wt ϕ}t≥0 is the β-semigroup defined as (3.1). (As usual, we set W0 the identity operator on Lp ).

= E,

Lemma 4.2. Let 1 ≤ p ≤ ∞ and ϕ ∈ Lp . Then (a) Jβα ϕ is well-defined for all α > 0, β > 0 and Jβα ϕp ≤ c(β)ϕp .

(4.2)

Moreover, if 0 < β ≤ 2, then we can put c(β) = 1. (b) The operator Jβα is a convolution type operator with Fourier multiplier m(ξ) = (1 + |ξ|β )−α/β , ξ ∈ Rn , i.e. ∧

∧

(Jβα ϕ) (ξ) = (1 + |ξ|β )−α/β ϕ (ξ),

∀ϕ∈S.

(4.3)

(c) For a fixed β ∈ (0, ∞), the family {Jβα }α≥0 has the following semigroup property: Jβα1 +α2 ϕ = Jβα1 ◦ Jβα2 ϕ,

(α1 , α2 ≥ 0, Jβ0 = E),

where Jβα1 ◦ Jβα2 ϕ means Jβα1 (Jβα2 ϕ), as usual.

(4.4)

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Proof. The part (a) is a simple consequence of the Minkowski inequality, namely, Jβα ϕp

1 ≤ Γ(α/β)

∞

α

(3.6)

(β)

t β −1 e−t Wt ϕp dt ≤ c(β)ϕp .

0

The part (b) can be easily derived from formulas (3.1) and (3.3). Indeed, using the Fubini theorem, for ϕ ∈ S we have ∧

(Jβα ϕ) (ξ) =

1 Γ(α/β)

∞

α

0

1 = ϕ (ξ) Γ(α/β)

(3.3)

∧

(β)

t β −1 e−t (Wt ∞

∧

ϕ) (ξ) dt

α

β

∧

t β −1 e−t e−t|ξ| dt = (1 + |ξ|β )−α/β ϕ (ξ).

0

Finally, owing to (4.3), part (c) is obvious in Fourier terms for ϕ ∈ S. In the general case, the semigroup property (4.4) follows by the density of the Schwartz class S in Lp , using the boundedness of the operators Jβα , α > 0 on Lp (see (4.2)). Remark 4.3. The identity (4.3) shows that the classical Bessel potentials and Flett potentials are particular cases of bi-parametric potentials Jβα , if one takes β = 2 and β = 1, respectively. Bi-parametric potentials may be interpreted as negative fractional powers of order (−α/β) of the fractional differential operator E + (−∆)β/2 , i.e. formally, Jβα ϕ = (E + (−∆)β/2 )−α/β ϕ,

ϕ ∈ S,

where ∆ is the Laplacian and E is the identity operator. The following theorem shows that explicit inversion formulas for bi-parametric potentials Jβα ϕ can be obtained by making use of the wavelet-like transform (β)

Aϕ ≡ Aµ ϕ defined by (3.12) (compare Theorem 2 in [1]). Theorem 4.4. Let the operator A be defined as (3.12) and Jβα ϕ be the bi-parametric potential of order α > 0 of a function ϕ ∈ Lp , 1 ≤ p < ∞. Suppose that µ is a finite Borel measure on R+ = [0, ∞) satisfying ∞ (a) 1 ∞

(b) 0

tγ d|µ|(t) < ∞ for some γ > α/β,

(4.5)

tk dµ(t) = 0, k = 0, 1, . . . , [α/β], (the integer part of α/β).

(4.6)


159

Then ∞

α AJβ ϕ (x, t)

0

∞

dt t1+α/β

≡ lim

ε→0

α AJβ ϕ (x, t)

ε

dt t1+α/β

= cµ (α/β)ϕ(x),

(4.7)

where cµ (θ) is defined as in (2.2). The limit in (4.7) is to be understood in the Lp -norm and pointwise a.e. for 1 ≤ p < ∞. If f ∈ C0 , the convergence is uniform on Rn . Proof. For f = Jβα ϕ, ϕ ∈ Lp we have (3.12) (Af )(x, t) ≡ AJβα ϕ (x, t) =

∞

(β) e−st Wst Jβα ϕ (x)dµ(s)

0

(being convolution type operators, Wτ(β) commutes with Jβα ) ∞ (β) = e−st Jβα Wst ϕ (x)dµ(s) 0 (4.1)

=

(3.9)

=

=

1 Γ(α/β) 1 Γ(α/β)

1 Γ(α/β)

∞

e−st 

0

∞

∞ dµ(s)

0

α

τ β −1 e−τ

 (β) Wτ(β) Wst ϕ (x)dτ  dµ(s)

0

∞

∞ 0



α (β) τ β −1 e−(τ +st) Wτ +st ϕ (x)dτ

0

∞ α −1 dµ(s) (τ − st)+β e−τ Wτ(β) ϕ (x)dτ,

(4.8)

0

where ab+

=

ab 0

if a > 0 if a ≤ 0

.

We define the following “truncated integrals”: α Dβ,ε f (x) ≡

∞ ε

α

t− β −1 (Af )(x, t)dt, ε > 0.

(4.9)

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By making use of the representation (4.8) for the Af, (f = Jβα ϕ), and the Fubini theorem, we have α Dβ,ε f (x) = 1 = Γ(α/β)

1 Γ(α/β)

∞ e

−τ

∞ t

−α β −1

∞ dt

ε

∞ α −1 dµ(s) (τ − st)+β e−τ Wτ(β) ϕ (x)dτ

0

Wτ(β) ϕ (x)dτ

0

0

τ /ε s

α β −1

τ /s τ αβ −1 α −t dµ(s) t− β −1 dt s ε

0

(replace τ and t by ετ and εt, respectively) 1 = Γ(α/β)

∞

e−ετ

0

τ τ αβ −1 α α −1 (β) β Wετ ϕ (x)dτ s −t dµ(s) t− β −1 dt. s τ /s

0

1

Now the application of the formula (cf.[10], formula No: 3.238(3)) η

α

α

t− β −1 (η − t) β

−1

dt =

1

α Γ(α/β) 1 (η − 1) β , (η > 1) Γ(1 + α/β) η

gives α Dβ,ε f (x) =

∞

e−ετ

0

∞ ≡

  τ α 1 1 (β) Wετ ϕ (x)  (τ − s) β dµ(s) dτ τ Γ(1 + α/β) 0

e−ετ

(β) Wετ ϕ (x)kα/β (τ )dτ,

(4.10)

0

where 1 θ+1 I µ (τ ) , and kθ (τ ) = τ

θ+1 I µ (τ ) =

1 Γ(1 + θ)

τ

(τ − s)θ dµ(s)

0

is the Riemann-Liouville fractional integral of order (θ + 1) of the measure µ (see (2.1)). By Lemma 2.3, the conditions (4.5) and (4.6) yield kα/β ∈ L1 (0, ∞). Denote ∞ cµ (α/β) = kα/β (τ )dτ cf. (2.2) . Then we obtain from (4.10) that 0

α α Dβ,ε Jβ ϕ (x) − cµ (α/β)ϕ(x) =

∞ (β) e−ετ Wετ ϕ (x) − ϕ(x) kα/β (τ )dτ. 0


161

The application of the Minkowski inequality yields ∞ α α (β) Dβ,ε Jβ ϕ − cµ (α/β)ϕ ≤ e−ετ ϕ − ϕ W kα/β (τ ) dτ ετ p p

0

∞ + ϕp (1 − e−ετ ) kα/β (τ ) dτ.

(4.11)

0

From Lemma 3.2 (i) and Lebesgue convergence theorem it follows that α α Dβ,ε Jβ ϕ − cµ (α/β)ϕ → 0 as ε → 0. p

(4.12)

α Setting p = ∞ in (4.11) it is also easy to see that the convergence Dβ,ε Jβα ϕ → cµ (α/β)ϕ is uniform if ϕ ∈ C0 . The proof of the pointwise (a.e.) convergence is based on the maximal function technique. By (4.10) and Lemma 3.2-(f), ∞ α α D J ϕ(x) ≤ sup (Wt(β) ϕ)(x) kα/β (τ ) dτ ≤ c(M ϕ)(x), ∀ > 0. β,ε β t>0

Hence

0

α α Jβ ϕ(x) ≤ c(M ϕ)(x), sup Dβ,ε ε>0

(4.13)

and therefore the maximal operator

α α ϕ(x) −→ sup Dβ,ε Jβ ϕ(x) ε>0

α is weak (p, p), 1 ≤ p < ∞. Since Dβ,ε Jβα ϕ(x) → cµ (α/β)ϕ(x) pointwise (in fact uniformly) as ε → 0 for any ϕ ∈ C0 ∩ Lp (this class is dense in Lp , 1 ≤ p < ∞), it α Jβα ϕ(x) → cµ (α/β)ϕ(x) as ε → 0 for almost all x ∈ Rn , owing to follows that Dβ,ε Lemma 2.2.

5. Lebesgue spaces of bi-parametric potentials and their characterization with the aid of wavelet-like transform In this section we introduce fractional Sobolev-type (or bi-parametric potential type) spaces on Rn and give a characterization of these spaces by making use of (β) the wavelet-like transform A ≡ Aµ defined as in (3.12). Definition 5.1. Let α > 0, β > 0 and 1 < p < ∞. The fractional Sobolev-type α (Rn ) is the image of Lp (Rn ) space (or bi-parametric potential type space) Hβ,p α under operator Jβ , i.e. α Hβ,p = f : f = Jβα ϕ, ϕ ∈ Lp . α The norm of f ∈ Hβ,p is defined by f H α = ϕp . β,p

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α Remark 5.2. When β = 2, the space H2,p coincides with the fractional Lp Sobolev space of Bessel potentials, introduced by Aronszajn and Smith [6], and Calder´ on [7].

The following theorem shows that for a fixed α > 0 and 1 0 actually coincide with the space H2,p up to equivalence of norms. α α Theorem 5.3. Suppose α > 0, 1 < p < ∞ and 0 < β, γ < ∞. Then Hβ,p = Hγ,p α α in the sense that f ∈ Hβ,p if and only if f ∈ Hγ,p .

Proof. It is sufficient to show that c1 Jβα ϕp ≤ Jγα ϕp ≤ c2 Jβα ϕp , ∀ϕ ∈ S.

(5.1)

For ϕ ∈ S we have α ∧ Jβ ϕ (ξ) = (1 + |ξ|β )−α/β ϕ∧ (ξ) ∧ (1 + |ξ|β )−α/β (1 + |ξ|γ )−α/γ ϕ∧ (ξ) = m(ξ) Jγα ϕ (ξ), (5.2) γ −α/γ (1 + |ξ| ) α (1 + |ξ|γ )1/γ , (ξ ∈ Rn , α > 0, 0 < β, γ < ∞). where m(ξ) = (1 + |ξ|β )1/β The straightforward calculations show that the function m(ξ) satisfies all the conditions of Lizorkin multiplier theorem 2.1 ), and therefore, is (Lemma α α a Lp −multiplier. Now the statement J ϕ ≤ c Jγ ϕ follows immediately =

β

p

p

from (5.2). The converse inequality is obtained analogously.

α , (1 0, the space Hβ,p α classical Bessel potential space H2,p up to equivalence of norms. α The following theorem characterizes the space Hβ,p with the aid of the “fracα tional derivatives” Dβ,ε , defined as in (4.9 ). For this we introduce the space α Lα = f : f ∈ L , sup f < ∞ , 1 0

with the norm

α . f Lα = f p + sup Dβ,ε p β,p

α Hβ,p

ε>0

α coincide with the H2,p up to equivalence of norms, Although the space α and some characterization of H2,p is known in terms of suitable hypersingular integrals (see [17], [20], [21]), the following theorem gives a new characterization of these spaces via wavelet-like transform (3.12). Since the choice of the wavelet α measure µ in (3.12) is at our disposal, the diverse “fractional derivatives” Dβ,ε can be constructed in this way.


163

Theorem 5.5. Suppose α > 0, 1 0. Then α Hβ,p = Lα β,p α in the sense that f ∈ Hβ,p ⇐⇒ f ∈ Lα β,p with the equivalent norms, i.e.

c1 f H α ≤ f Lα ≤ c2 f H α . β,p

β,p

β,p

α .Then Proof. We use some ideas from [17, p. 260-262]. Assume that f ∈ Hβ,p α α α f = Jβ ϕ for some ϕ ∈ Lp . Since the family {Dβ,ε Jβ ϕ}ε>0 converges in Lp −norm α α Jβ ϕ ≤ c ϕp . to cµ (α/β)ϕ (see (4.13)), there exists c > 0 such that sup Dβ,ε p

ε>0

Thus

α α α α Jβ ϕp = Jβα ϕp + sup Dβ,ε Jβ ϕp f Lα ≡ f p + sup Dβ,ε β,p

ε>0

ε>0

(4.2)

≤ c1 ϕp + c2 ϕp = c ϕp = c f H α . β,p

α ⊂ Lα . This shows that f Lα ≤ c f H α , and therefore, Hβ,p β,p β,p β,p α Conversely, let f ∈ Lα β,p , that is f ∈ Lp and sup Dβ,ε f < ∞. Let us firstly ε>0

p

show that f = Jβα ψ for some ψ ∈ Lp . Since the Schwartz space S = S(Rn ) is dense in Lp , it suffices to show that (5.3) f, w = Jβα ψ, w for some ψ ∈ Lp and all w ∈ S, where g, w = g(x)w(x)dx, as usual. Rn

We use the following well known equality for the convolution-type operators: h ∗ λ, w = h, λ− ∗ w , λ, w ∈ S, h ∈ S , and λ− (x) = λ(−x) . (5.4) α are the convolution-type operators with the radial Since the operators Jβα and Dβ,ε kernels, it follows from (5.4) that α α a) Jβα g, w = g, Jβα w and b) Dβ,ε g, w = g, Dβ,ε w (5.5)

for all g ∈ Lp and w ∈ S.

α f < ∞ yields that By the Banach-Alaoglu theorem, the condition sup Dβ,ε ε>0

there exist a function ϕ ∈ Lp and a sequence εk → 0, such that α Dβ,εk f, w −→ ϕ, w as εk → 0,

p

(5.6)

for all w ∈ Lq , 1/q + 1/p = 1 (in particular, for all w ∈ S). For this function ϕ ∈ Lp and any Schwartz function w we have α (5.5)−a) (5.6) (5.5)−b) α α = ϕ, Jβα w = lim Dβ,ε f, Jβα w = lim f, Dβ,ε J αw . Jβ ϕ, w k k β εk →0

That is

εk →0

α α J α w , ∀w ∈ S. Jβ ϕ, w = lim f, Dβ,ε k β εk →0

(5.7)

164

Let us show that

Aliev

IEOT

α J α w = f, cµ (α/β)w , lim f, Dβ,ε k β

(5.8)

εk →0

where cµ (θ) is defined as (2.2). The Hölder inequality yields, with 1/q + 1/p = 1 f, Dα J α w − f, cµ (α/β)w = f, Dα J α w − cµ (α/β)w β,εk β β,εk β α ≤ f p Dβ,ε J α w − cµ (α/β)wq . k β From (4.12) it follows that the right hand side of the last expression tends to zero as εk → 0. Hence, (5.8) is true and therefore, by (5.7) we get α Jβ ϕ, w = f, cµ (α/β)w , ∀w ∈ S. The latter equality shows that f = Jβα ψ, where ψ = c−1 µ (α/β)ϕ, ϕ ∈ Lp , and α therefore f ∈ Hβ,p . Furthermore, sup |ϕ, w| , (1/q + 1/p = 1). f H α = ψp = c−1 (5.9) µ (α/β) β,p

wq ≤1

Since wq ≤ 1, we have

α α |ϕ, w| ≤ ϕ − Dβ,ε f, w + Dβ,ε f, w k k α α ≤ ϕ − Dβ,ε f, w + Dβ,ε f . k

k

p

(5.10)

Since the first term in the last expression tends to zero as εk → 0 (see (4.12)), we have from (5.9) and (5.10) that α f H α ≤ c f p + sup Dβ,ε f p = c f Lα . β,p

The proof is completed.

ε>0

β,p

Remark 5.6. It is natural to ask, what is the advantage in the usage of the additional parameter β in the characterization of the Bessel potential space Hpα (Rn ). To answer this question we note that most of known characterizations of the space Hpα (Rn ) (see S.G. Samko [21], B. Rubin [17], S.G. Samko, A.A. Kilbas, and O.I. Marichev [20], and references therein) are given in terms of finite differences, the order of which increases with α. In the “wavelet language” finite differences are replaced by wavelet measures, the number of vanishing moments of which serves as a substitute for the order a finite difference. Apart from purely theoretical reason (e.g., extra flexibility, interpolation between the Gauss-Weierstrass and Poisson integral constructions), the additional parameter β enables us to minimize the number of vanishing moments as much as we please, no matter how big a parameter α . Indeed, owing to (4.6), the bigger β is, the less is the number of vanishing moments. As a result, if we take β > α in (4.6), then [α/β]= 0, and therefore only one vanishing moment ∞ dµ(t) = 0 is sufficient. 0


Example. Suppose the family of integral operators Wtβ ϕ

t>0

165

is defined by (3.1).

Consider the following integral equation ∞ α t β −1 e−t Wtβ ϕ (x)dt = f (x), x ∈ Rn , α > 0, β > 0.

(5.11)

0

Naturally a question arises: under which conditions on f (x), the equation (5.11) is solvable in Lp (Rn ) and how the solution For this we define a wavelet ∞can be found. m k measure µ, for instance, by µ, g ≡ 0 g(t)dµ(t) = k=0 m k (−1) g(k), with a fixed integer m > α/β. It is known that (see [20, p. 116-117 ]) ∞ m m (−1)k k j = 0, j = 0, 1, . . . , m − 1; (we assume 00 = 1). tj dµ(t) ≡ k 0 k=0

According to (3.12), the wavelet-like transform (Af )(x, η) has the form ∞ m m (β) (β) (Af )(x, η) ≡ e−tη Wtη f (x) dµ(t) = (−1)k e−kη Wkη f (x), k 0 k=0 (β) where W0 f (x) = f (x). Further, for the “truncated integrals” (4.9) we have the representation: ∞ α α η − β −1 (Af )(x, η)dη Dβ,ε f (x) = ε

∞ m α m (β) k η − β −1 e−kη Wkη f (x) dη, (ε > 0). = (−1) k ε

(5.12)

k=0

Now by taking into account the statement of Theorem 5.5, we have the following: in order to the equation (5.11) has a solution in Lp (Rn ), (1 0 α Dβ,ε f ε>0 is defined by (5.12). Furthermore, the solution to the equation (5.11) has the form (cf. (4.7)) ∞ α 1 t− β −1 (Af )(x, t) dt, (5.13) ϕ(x) = Γ(α/β)cµ (α/β) 0 where in view of (2.2), cµ (α/β) is defined as ∞ ∞ −α −1 −tη β t e dµ(η) dt = cµ (α/β) ≡ 0

0

∞

0

α

t− β −1 (1 − e−t )m dt = 0.

Remark 5.7. Putting β = α in (4.3) we have ∧

∧

(Jαα ϕ) (ξ) = (1 + |ξ|α )−1 ϕ (ξ). This equality shows that the family of operators 2Jαα , α > 0, where ∞ (α) α (Jα ϕ) (x) = e−t Wt ϕ (x)dt, α > 0 (cf.(4.1)), 0

166

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is an approximative identity as α → 0. It will be interesting to investigate the approximation properties of the family {2Jαα ϕ}α>0 as α → 0. ( Recall that the family {Wtα ϕ}t>0 is an approximative identity as t → 0). Another interesting particular case arises by putting α = β 2 in (4.1). In this case we have ∞ 2 1 β Jβ ϕ (x) = tβ−1 e−t Wtβ ϕ (x)dt. Γ(β) 0

Owing to (4.3),

and formally,

2 ∧ Jββ ϕ (ξ) = (1 + |ξ|β/2 )−β ϕ∧ (ξ), ϕ ∈ S, 2

Jββ ϕ = (E + (−∆)β/2 )−β ϕ. The equality (4.7) with α = β 2 gives an inversion formula for these potential type operators. Acknowledgment The author would like to thank the referees for their valuable comments and suggestions.

References [1] I. A. Aliev and B. Rubin, Wavelet-like transforms for admissible semi-groups; Inversion formulas for potentials and Radon transforms, Journal of Fourier Analysis and Applications, 11, No 3 (2005), 333–352. [2] I. A. Aliev, B. Rubin, S. Sezer, and S Uyhan, Composite Wavelet transforms: Applications and Perspectives, Contemporary Mathematics, AMS, 464(2008), 1–27. [3] I. A. Aliev, S. Sezer, and M. Eryigit, An integral transform associated to the Poisson integral and inversion of Flett potentials, Journal of Mathematical Analysis and Applications, 321(2006), 691–704. [4] I. A. Aliev and M. Eryigit, Inversion of Bessel Potentials with the aid of Wavelet Transforms, Math. Nachr., 242(2002), 27–37. [5] A. Almeida and S. G. Samko, Characterization of Riesz and Bessel Potentials on variable Lebesgue spaces, J. Function Spaces and Applications, 4, No.2 (2006), 1–26. [6] N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11(1961), 385–475. [7] A. P. Calder´ on, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math., 5(1961), 33–49. [8] M. V. Fedoryuk, Asymptotic behavior of the Green function of a pseudodifferential parobolic equation,(Russian), Differentsial’nye Uravneniya, 14, No. 7 (1978), 1296– 1301. [9] T. M. Flett, Temperatures, Bessel potentials and Lipschitz space, Proc. London Math. Soc. 22, No. 3 (1971), 385–451.


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[10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, (Fifth edition), Academic Press, 1994. [11] J. Hu and M. Zahle, Potential spaces on fractals, Stud. Math., 170(2005), 259–281. [12] V. Kokilashvili, A. Meskhi, and S. G. Samko, On the inversion and characterization of the Risz potentials in the weighted Lebesgue Spaces, Memories on Diff. Equations Math. Phys., Razmadze Math. Inst., Tbilisi, 29(2003), 99–106. [13] A. Koldobsky, Fourier Analysis in convex geometry, Math. Surveys and Monographs, AMS, 116(2005) . [14] N. S. Landkof, Several remarks on stable random processes and α-superharmonic functions, (Russian), Mat. Zametki, 14, No. 6 (1973), 901–912. [15] P. I. Lizorkin, Generalized Liouville differentation and functional spaces Lrp (En ). Imbedding theorems, (Russian), Matem. Sbornik, 60 (120)(1963), 325–353. [16] V. A. Nogin and S. G. Samko, Some applications of potentials and approximative inverse operators in multidimensionel fractional calculus, Fractional Calculus and Appl. Analysis, 2, No. 2 (1999), 205–228. [17] B. Rubin, Fractional integrals and potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 82, Longman, Harlow, 1996. [18] B. Rubin, Fractional integrals and wavelet transforms associated with Blaschke-Levy representations on the sphere, Israel J. of Math., 114(1999), 1–27. [19] B. Rubin, Intersection bodies and generalized cosine transforms, Advances in Math., 218(2008), 696–727. [20] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Sci.Publ. London-New York, 1993. [21] S. G. Samko, Hypersingular integrals and their applications, ser.: Analytical Methods and Special Functions, Taylor&Francis, London, 2002. [22] S. G. Samko, Spaces Lα p,r (Rn ) and hypersingular integrals, (in Russian), Studia Math. 61, No. 3 (1977), 193–230. [23] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. [24] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. Ilham A. Aliev Department of Mathematics Education Akdeniz University 07058 Antalya Turkey e-mail: [email protected] Submitted: January 28, 2009. Revised: April 30, 2009.

Integr. equ. oper. theory 65 (2009), 169–193 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020169-25, published online August 3, 2009 DOI 10.1007/s00020-009-1704-z


Almost Periodicity of Parabolic Evolution Equations with Inhomogeneous Boundary Values Mahmoud Baroun, Lahcen Maniar and Roland Schnaubelt Abstract. We show the existence and uniqueness of the (asymptotically) almost periodic solution to parabolic evolution equations with inhomogeneous boundary values on R and R± , if the data are (asymptotically) almost periodic. We assume that the underlying homogeneous problem satisfies the ‘Acquistapace–Terreni’ conditions and has an exponential dichotomy. If there is an exponential dichotomy only on half intervals (−∞, −T ] and [T, ∞), then we obtain a Fredholm alternative of the equation on R in the space of functions being asymptotically almost periodic on R+ and R− . Mathematics Subject Classification (2000). Primary 47D06. Keywords. Asymptotic almost periodicity, Fredholm operators, inhomogeneous evolution equation, evolution family, parabolic initial–boundary value problem, inter- and extrapolation, exponential dichotomy.

1. Introduction In the present paper we study the almost periodicity of the solutions to the parabolic inhomogeneous boundary value problem u (t) = Am (t)u(t) + g(t), B(t)u(t) = h(t),

t ∈ R,

t ∈ R,

(1.1)

for linear operators Am (t) : Z → X and B(t) : Z → Y on Banach spaces Z → X and Y . Typically, Am (t) is an elliptic partial differential operator acting in, say, X = Lp (Ω), and B(t) is a boundary operator mapping Z = Wp2 (Ω) into This work is part of a cooperation project sponsored by DFG (Germany) and CNRST (Morocco). We would like to thank DFG and CNRST for their support. L. M. would also like to thank DAAD (Germany) for its financial support.

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1−1/p

a ‘boundary space’ like Wp (∂Ω), where p ∈ (1, ∞), see Example 5.6. We want to show that the solutions u : R → X of (1.1) inherit the (asymptotic) almost periodicity of the inhomogeneities g : R → X and h : R → Y . Our basic assumptions say that Am (·) and B(·) are (asymptotically) almost periodic in time and that the restrictions A(t) of Am (t) to the kernels of B(t) satisfy the ‘Acquistapace–Terreni’ conditions (2.1) and (2.2). In particular, the operators A(t) are sectorial and they generate a parabolic evolution family U (t, s), t ≥ s, which solves the homogeneous problem (1.1) with g = h = 0, see Section 2. If U has an exponential dichotomy on R, then we show that for each almost periodic g and h there is a unique almost periodic solution of (1.1), see Proposition 5.2. Our main results concern the more complicated case that the evolution family U has exponential dichotomies on (possibly disjoint) time intervals (−∞, −T ] and [T, +∞). Theorem 5.5 then gives a Fredholm alternative for (mild) solutions u of (1.1) in the space AAP ± (R, X) of continuous functions u : R → X being asymptotically almost periodic on R+ and on R− . In fact we prove more detailed results on the Fredholm properties of (1.1), see Theorem 4.7, and we also treat the corresponding inhomogeneous initial/final value problems on R± , see Propositions 5.3 and 5.4. One obtains exponential dichotomies on intervals (−∞, −T ] and [T, +∞) in the asymptotically hyperbolic case where the operators Am (t) and B(t) converge as t → ±∞ and the resulting limit operators A±∞ have no spectrum on iR, see [5], [26], [27]. It should be noted that if the limits at +∞ and −∞ differ, then the operators in (1.1) are asymptotically almost periodic only on R+ and R− separately, so that the space AAP ± (R, X) seems to be a natural setting for our investigations. The asymptotically hyperbolic case can occur if one linearizes a nonlinear problem along a orbit connecting two hyperbolic equilibria, see e.g. [25], and also the references in [18], [22]. To establish our results, we develop a theory for the evolution equation u (t) = Aα−1 (t)u(t) + f (t),

t ∈ R,

(1.2)

t for the operators A(t) and α ∈ (0, 1). in the continuous extrapolation spaces Xα−1 t t contains We recall the definition of Xα−1 in Section 2. Here, we just note that Xα−1 t X and that A(t) can be extended to a sectorial operator Aα−1 (t) in Xα−1 . To relate (1.2) with (1.1), we set f (t) = g(t) + (ωI − Aα−1 (t))D(t)h(t) for the solution operator D(t) : ϕ → v of the corresponding abstract Dirichlet problem (ωI − Am (t))v = 0 and B(t)v = ϕ, where ω ∈ R is large enough. Then (1.1) and (1.2) have the same classical solutions, see e.g. [9] and also Section 5. It is known that the evolution family U (t, s) can be extended to operators s → X, see Section 2. So we can define mild solutions of (1.2) as Uα−1 (t, s) : Xα−1 the functions u ∈ C(R, X) satisfying

u(t) = U (t, s)u(s) + s

t

Uα−1 (t, τ )f (τ ) dτ

(1.3)

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τ for all t ≥ s, where f (τ ) belongs to Xα−1 . In our setting it can be shown that mild solutions essentially coincide with pointwise solutions of (1.2), see [22] and the comments at the beginning of Section 3. We prefer to work with the integral equation (1.3) in order to avoid difficulties with the differential equation (1.2) which lives in possibly time–varying extrapolation spaces.

When treating (1.3) or (1.2), it is crucial to identify suitable function spaces for the inhomogeneity f . To that purpose we consider the multiplication operator A(·) in the space AAP ± (R, X) endowed with the sup–norm. This space pos± sesses the extrapolation spaces AAPα−1 corresponding to A(·). In Section 3 it is shown that the functions in these spaces can be characterized as limits of functions in AAP ± (R, X). Moreover, if the operators A(t) possess constant extrapo± t ∼ lation spaces Xα−1 = AAP ± (R, Xα−1 ). In so far the = Xα−1 , we have AAPα−1 ± ± seem to be natural. For f ∈ AAPα−1 we then set Gα−1 u = f if spaces AAPα−1 ± ± u ∈ AAP (R, X) satisfies (1.3), thus defining a closed operator Gα−1 in AAPα−1 . Its Fredholm properties yield the desired Fredholm alternative for the mild solutions to (1.2) described in Theorems 4.7 and 4.9. This paper combines three lines of research: Fredholm properties of evolution equations on the line, boundary value problems and extrapolation theory, and almost periodic equations in the context of exponential dichotomies. We are not aware of papers on Fredholm properties of inhomogeneous boundary value problems in the framework of almost periodic functions, but we want to recall related previous results. Our reformulation of a boundary value problem as an evolution equation in extrapolation spaces seems to go back to work in boundary control theory, see e.g. [7], [9], [13], [22] for more details and relevant references. Almost periodicity of solutions to autonomous problems is a well studied subject, see e.g. [4]. These results for autonomous problems were partly extended to the case of time periodic A(·), see e.g. [6], [14], [15], [28]. The case of almost periodic A(·) was studied for special classes of parabolic problems in e.g. [16], [19], [24]. For the general case of almost periodic A(·) satisfying the Acquistapace– Terreni conditions we showed in [21] that for each almost periodic f : R → X there is a unique mild solution u of (1.2) for α = 1 provided that the underlying evolution family U (t, s) has an exponential dichotomy on R. (See [17] for the converse implication.) In [21] we also established similar results for asymptotically almost periodic functions f : R+ → X. In our previous paper [22] we treated the Fredholmity of parabolic boundary value problems and of the above operator Gα−1 in the framework of bounded functions (see also [23] for corresponding perturbation theorems). There we generalized the approach of the works [10] and [11] which studied the case of homogeneous boundary values. In [18] one can find a detailed discussion of the literature and the background of such Fredholm theorems as well as a rather complete treatment of (1.2) for α = 1, i.e., for f taking values in X (also in the non parabolic situation).

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After discussing the above mentioned preparations in Section 2 and 3, we prove our main theorems on (1.2) in Section 4. In the last section we then treat (1.1) by means of the results on (1.2) and discuss an example arising in pde.

2. Notations, assumptions, and preliminaries We denote by D(A), N (A), R(A), σ(A), ρ(A) the domain, kernel, range, spectrum and resolvent set of a linear operator A. Moreover, we set R(λ, A) := (λI − A)−1 = (λ − A)−1 for λ ∈ ρ(A), L(X, Y ) is the space of bounded linear operators between Banach spaces X and Y , and L(X) := L(X, X). By c = c(α, . . . ) we designate a generic constant depending on quantities α, · · · . For an unbounded closed interval J, the space of bounded continuous functions f : J → X (vanishing at ±∞) is denoted by Cb (J, X) (by C0 (J, X)). We investigate linear operators A(t), t ∈ R, on a Banach space X subject to the following hypotheses introduced by P. Acquistapace and B. Terreni in [1] and [2]. There are constants ω ∈ R, θ ∈ (π/2, π), K > 0 and µ, ν ∈ (0, 1] such that µ + ν > 1 and K , 1 + |λ| |t − s|µ (A(t) − ω)R(λ, A(t) − ω) [R(ω, A(t)) − R(ω, A(s))] ≤ K |λ|ν λ ∈ ρ(A(t) − ω),

R(λ, A(t) − ω) ≤

(2.1) (2.2)

for all t, s ∈ R and λ ∈ Σθ := {λ ∈ C \ {0} with | arg(λ)| ≤ θ}. (Observe that the domains D(A(t)) are not required to be dense.) Occasionally, we also consider operators A(t) satisfying (2.1) and (2.2) for t and s in an interval J. In this case the results stated below hold in an analogous way. The conditions (2.1) and (2.2) imply that the operators A(·) generate an evolution family U (t, s) for t, s ∈ R with t ≥ s. More precisely, for t > s, the map (t, s) → U (t, s) ∈ L(X) is continuous and continuously differentiable in t, U (t, s) maps X into D(A(t)), and it holds ∂t U (t, s) = A(t)U (t, s). Moreover, U (t, s) and (t − s)A(t)U (t, s) are exponentially bounded. We further have U (t, s)U (s, r) = U (t, r)

and U (t, t) = I

for t ≥ s ≥ r.

Finally, for s ∈ R and x ∈ D(A(s)), the function t → u(t) = U (t, s)x is continuous at t = s and u is the unique solution in C([s, ∞), X)∩C 1 ((s, ∞), X) of the Cauchy problem u(s) = x. u (t) = A(t)u(t), t > s, These facts have been established in [1] and [2], see also [3], [20], [29]. We introduce the inter- and extrapolation spaces for A(t). We refer to [3], [12], [20] for proofs and more details. Let A be a sectorial operator on X (i.e., (2.1) holds with A(t) replaced by A) and α ∈ (0, 1). We define the new norm on D(A) by α xA α := supr>0 r (A − ω)R(r, A − ω)x,

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·A α

and consider the continuous interpolation spaces XαA := D(A) which are Banach spaces endowed with the norms · A . For convenience we further write α A A := x, X := D(A) and x := (ω − A)x. We also need the X0A := X, xA 0 1 1 ˆ A := D(A) of X. Moreover, we define the extrapolation space closed subspace X A ˆ A with respect to the norm xA := R(ω, A)x. as the completion of X X−1 −1 A ˆ A → X−1 . The operator A−1 Then A has a unique continuous extension A−1 : X A satisfies (2.1) in X−1 , it is densely defined, it has the same spectrum as A, and it A being the extension of etA . As above, we generates the semigroup etA−1 on X−1 A A can then define the space Xα−1 := (X−1 )α −1 endowed with the norm A−1 xA = supr>0 rα R(r, A−1 − ω)x. α−1 := xα A A The restriction Aα−1 : XαA → Xα−1 of A−1 is sectorial in Xα−1 with the same A type as A, it has the same spectrum as A, and the semigroup etAα−1 on Xα−1 tA A A is the extension of e . Observe that ω − Aα−1 : Xα → Xα−1 is an isometric isomorphism. We will frequently use the continuous embeddings ˆ A ⊂ X, D(A) → XβA → D((ω − A)α ) → XαA → X (2.3) A A A → D((ω − A−1 )α ) → Xα−1 → X−1 X → Xβ−1

for all 0 < α < β < 1, where the fractional powers are defined as usually. We note A , 0 ≤ α < 1, is the completion with respect to · A that Xα−1 α−1 of each of the A ; in particular of X. spaces in (2.3) which are contained in Xα−1 Given operators A(t), t ∈ R which satisfy (2.1), we set Xαt := XαA(t) ,

A(t)

t Xα−1 := Xα−1 ,

ˆ t := X ˆ A(t) X

for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms. Then the embeddings in (2.3) hold and the norms of the embeddings are uniformly bounded for t ∈ R. For a closed interval J, we further define on E = E(J) := Cb (J, X) the multiplication operator A(·) by (A(·)f )(t) := A(t)f (t)

for all t ∈ J,

D(A(·)) := {f ∈ E : f (t) ∈ D(A(t)) for all t ∈ J, A(·)f ∈ E}. It is clear that the operator A(·) is sectorial. We can thus introduce the spaces Eα := EαA(·) ,

A(·)

Eα−1 := Eα−1 ,

and Eˆ := D(A(·))

for α ∈ [0, 1], where E0 := E and E1 := D(A(·)). We observe that E−1 ⊆ t ˆ t∈J X−1 and that the extrapolated operator A(·)−1 : E −→ E−1 is given by (A(·)−1 f )(t) := A−1 (t)f (t) for t ∈ J and f ∈ E. Further, Eα−1 has the norm f α−1 := sup sup rα R(r, A−1 (s) − ω)f (s), r>0 s∈J

t Xα−1

and we have f (t) ∈ for each t ∈ J if f ∈ Eα−1 . Since R(n, Aα−1 (·)) is the resolvent of the densely defined sectorial operator Aα−1 (·), we have nR(n, Aα−1 (·))f → f in Eα−1 as n → ∞, for each f ∈ Eα−1 and 0 ≤ α < 1.

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The next lemma allows to extend the evolution family U (t, s) to the extrapt olated spaces Xα−1 , see Proposition 2.1 and Remark 3.12 of [22] for the proof. Proposition 2.1. Assume that (2.1) and (2.2) hold and let 1 − µ < α < 1. Then the following assertions hold for s < t ≤ s + t0 and t0 > 0. s (a) The operators U (t, s) have continuous extensions Uα−1 (t, s) : Xα−1 → X satisfying α−β−1 s Uα−1 (t, s)L(Xα−1 , ,Xβt ) ≤ c(α, t0 )(t − s)

(2.4)

and Uα−1 (t, s)x = Uγ−1 (t, s)x for 1 − µ < γ < α < 1, β ∈ [0, 1], and s . x ∈ Xα−1 (b) The map {(t, s) : t > s} (t, s) −→ Uα−1 (t, s)f (s) ∈ X is continuous for f ∈ Eα−1 . Exponential dichotomy is another important tool in our study, cf. [8], [20], [27]. We recall that an evolution family U (·, ·) has an exponential dichotomy on an interval J if there exists a family of projections P (t) ∈ L(X), t ∈ J, being strongly continuous with respect to t, and constants δ, N > 0 such that (a) U (t, s)P (s) = P (t)U (t, s), (s, t), (b) U (t, s) : Q(s)(X) → Q(t)(X) is invertible with the inverse U −δ(t−s) −δ(t−s) (s, t)Q(t) ≤ N e (c) U (t, s)P (s) ≤ N e and U , for all s, t ∈ J with s ≤ t, where Q(t) := I − P (t) is the ‘unstable projection.’ One further defines Green’s function by U (t, s)P (s), t ≥ s, t, s ∈ J, Γ(t, s) = −U(t, s)Q(s), t < s, t, s ∈ J. In the parabolic case one easily obtains regularity results for Green’s function and the dichotomy projections, see e.g. [27, Proposition 3.18]. For instance, if J is bounded from below, then we have A(t)Q(t) ≤ c(η) for all t > η + inf J and ˜ (t − η, t)Q(t). Similarly, it holds each η > 0 since A(t)Q(t) = A(t)U (t, t − η)U A(t)Q(t) ≤ c for all t ∈ J if J is unbounded from below. As a consequence ˆ t and X t for each α ∈ [0, 1] and t ∈ J \ {inf J}. P (t) = I − Q(t) leaves invariant X α In the next proposition (shown in Proposition 2.2 and Remark 3.12 of [22]) we state some properties of Γ(t, s) and Q(t) in extrapolation spaces. We use the convention ±∞ + r = ±∞ for r ∈ R, and we set J = J \ {sup J}, i.e., J = J if J is unbounded from above. Moreover, we write U0 (t, s) := U (t, s), P0 (t) := P (t), and Q0 (t) := Q(t), where X0t = X by definition. Proposition 2.2. Assume that (2.1) and (2.2) hold and that U (t, s) has an exponential dichotomy on an interval J. Let η > 0 and 1 − µ < α ≤ 1. Then the t t operators P (t) and Q(t) have continuous extensions Pα−1 (t) : Xα−1 → Xα−1 and t Qα−1 (t) : Xα−1 → X, respectively, for every t ∈ J ; which are uniformly bounded for t < sup J − η. Moreover, the following assertions hold for t, s ∈ J with t ≥ s. t (a) Qα−1 (t)Xα−1 = Q(t)X;

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(b) Uα−1 (t, s)Pα−1 (s) = Pα−1 (t)Uα−1 (t, s); s t (c) Uα−1 (t, s) : Qα−1 (s)(Xα−1 ) → Qα−1 (t)(Xα−1 ) is invertible with the inverse Uα−1 (s, t); (d) Uα−1 (t, s)Pα−1 (s)x ≤ N (α, η) max{(t − s)α−1 , 1}e−δ(t−s) xsα−1 for x ∈ s and s < t < sup J − η; Xα−1 t α−1 (s, t)Qα−1 (t)x ≤ N (α, η)e−δ(t−s) xt (e) U α−1 for x ∈ Xα−1 and s ≤ t < sup J − η. (f) Let J0 ⊂ J be a closed interval and f ∈ Eα−1 (J0 ). Then P (·)f ∈ Eα−1 (J0 ) and Q(·)f ∈ Cb (J0 , X). Using this proposition, we define Uα−1 (t, s)Pα−1 (s), Γα−1 (t, s) = (t, s)Qα−1 (s), −U

t ≥ s, t, s ∈ J, t < s, t, s ∈ J.

In some results we shall assume that A(·) is asymptotically hyperbolic, i.e., there are two operators A−∞ : D(A−∞ ) → X and A+∞ : D(A+∞ ) → X which satisfy (2.1) and lim R(ω, A(t)) = R(ω, A±∞ )

t→±∞

(in L(X));

σ(A+∞ ) ∩ iR = σ(A−∞ ) ∩ iR = ∅.

(2.5) (2.6)

Under assumptions (2.1), (2.2), (2.5), (2.6), one can show that U (·, ·) has exponential dichotomies on [T , +∞) and (−∞, −T ] for some T ∈ R. We fix a number T ≥ 0 such that T > T .

(2.7)

See [27, Theorem 2.3], as well as [5] and [26] for earlier results under additional assumptions. We further need a result on embeddings of extrapolation spaces which we state in the more general setting of C0 -semigroups, see e.g. [12]. Lemma 2.3. Let A be the generator of a C0 -semigroup T (·) on a Banach space Z. Let Y be an T (·)-invariant closed subspace of Z. Endow Y with the norm of AY is canonically Z and consider the restriction AY of A to Y . Then the space Y−1 A embedded into Z−1 as a closed subspace. Proof. The operator AY generates the semigroup of the restrictions TY (t) ∈ L(Y ) of T (t). By rescaling we may assume that TY (t) ≤ T (t) ≤ ce−t for some > 0 and all t ≥ 0. Observe that then A and AY are invertible and that ∞ ∞ A−1 = T (t)y dt = T (t)y dt = A−1 y Y Y 0

0

for each y ∈ Y . We mostly write A instead of AY , and we endow the extrapolation spaces of A and AY with the norm x−1 = A−1 −1 x. By definition, it holds A = {y = (yn ) + NY : (yn ) = (yn )n∈N ⊂ Y is Cauchy for · −1 }, Y−1

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where NY = {(yn ) ⊂ Y : yn → 0 for · −1 }. We identify y ∈ Y with the element A A (y)n∈N + NY of Y−1 , thus considering Y as a dense subspace of Y−1 . We define the operator A A −→ Z−1 , Φ : Y−1

Φy = (yn ) + NZ ,

A where yn ∈ Y, yn → y in Y−1 .

A If (yn ), (˜ yn ) ⊂ Y converge to y in Y−1 , then yn − yñ → 0 as n → ∞ for · −1 . A such that Φy = 0. Hence, (yn − yñ ) ∈ NZ , and so Φ is well defined. Let y ∈ Y−1 This means that (yn ) ∈ NZ , and hence yn → 0 in · −1 . Therefore (yn ) ∈ NY , and thus y = 0. It is clear that Φ is linear. It is also bounded since A = ΦyZ−1

inf

(zn )∈NZ

(yn − zn )∞ ≤

inf

(zn )∈NY

(yn − zn )∞ = yY−1 A .

A A We have shown that Y−1 → Z−1 with the canonical embedding Φ. To prove that A A such that zj → z the range R(Φ) is closed in Z−1 , we take zj = Φyj ∈ R(Φ) ⊆ Z−1 −1 A in Z−1 as j → ∞. Then A−1 zj =: wj converges in Z to w := A−1 −1 z. We further claim that −1 (2.8) A−1 −1 Φ = (AY )−1 .

−1 −1 Indeed, for x ∈ Y one has A−1 x = A−1 −1 Φx = A Y x = (AY )−1 x. So assertion (2.8) A follows from the density of Y in Y−1 . Equation (2.8) then yields −1 (AY )−1 −1 yj = A−1 zj → w

(in Z).

−1 Since Y is closed in Z and (AY )−1 −1 yj ∈ Y , we obtain (AY )−1 yj → w in Y . As a A consequence, yj converges in Y−1 to y := (AY )−1 w. We conclude that zj = Φyj → A which means that R(Φ) is closed. Φy in Z−1

We further introduce the concept of almost periodicity, see e.g. [4], [19]. Definition 2.4. Let Y be a Banach space. A continuous function g : R → Y is called almost periodic if for every > 0 there exist a set P () ⊆ R and a number () > 0 such that each interval (a, a + ()), a ∈ R, contains an almost period τ = τ ∈ P () and the estimate g(t + τ ) − g(t) ≤ holds for all t ∈ R and τ ∈ P (). The space of almost periodic functions is denoted by AP (R, Y ). We recall that AP (R, Y ) is a closed subspace of the space of bounded and uniformly continuous functions, see [19, Chapter 1]. For a closed unbounded interval J, we also define the space AP (J, Y ) := {g : J → Y : ∃ g˜ ∈ AP (R, Y ) s.t. g˜|J = g} of almost periodic functions on J. We remark that the function g˜ in the above definition is uniquely determined, cf. [4, Proposition 4.7.1]. The following notion is more important for our investigations. Definition 2.5. Let J = [t0 , ∞). A continuous function g : J → Y is called asymptotically almost periodic if for every > 0 there exists a set P () ⊆ J and numbers s(), () > 0 such that each interval (a, a + ()), a ≥ 0, contains an almost period τ = τ ∈ P () and the estimate g(t + τ ) − g(t) ≤ holds for all t ≥ s()

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and τ ∈ P (). The space of asymptotically almost periodic functions is denoted by AAP (J, Y ). Due to e.g. [4, Theorem 4.7.5], these spaces are related by the equality AAP ([t0 , ∞), Y ) = AP ([t0 , ∞), Y ) ⊕ C0 ([t0 , ∞), Y ).

(2.9)

Analogously, we define the asymptotic almost periodicity on J = (−∞, t0 ], and one also has AAP ((−∞, t0 ], Y ) = AP ((−∞, t0 ], Y ) ⊕ C0 ((−∞, t0 ], Y ).

(2.10)

Finally, we recall that M (·)f ∈ (A)AP (J, Y ) if f ∈ (A)AP (J, Y ) and M (·) ∈ (A)AP (J, L(Y )). This follows from the above definitions if one takes into account that we can find common pseudo periods for f and M , cf. [19, p.6].

3. Amost periodicity of parabolic evolution equations In this section of the paper we study the parabolic evolution equation u (t) = Aα−1 (t)u(t) + f (t),

t ∈ J,

(3.1)

where J is an unbounded closed interval, f ∈ Eα−1 (J) and A(t), t ∈ R, are linear operators satisfying the assumptions (2.1) and (2.2). Let U (t, s), t ≥ s, be the evolution family generated by A(t), t ∈ R, and be Uα−1 (t, s), t ≥ s, its extrapolated evolution family defined in Proposition 2.1 for each α ∈ (1 − µ, 1]. A mild solution of (3.1) is a function u ∈ C(J, X) satisfying t Uα−1 (t, τ )f (τ ) dτ, ∀ t ≥ s in J. (3.2) u(t) = U (t, s)u(s) + s

In Proposition 2.6 of [22], we showed that a mild solution actually satisfies (3.1) t pointwise in Xβ−1 for each β ∈ [0, min{ν, α}) and t ∈ J. Conversely, if u ∈ 1 C (J, X) solves (3.1) (and thus u ∈ Eα (J)), then Proposition 2.1(iv) of [23] implies that ∂τ+ U (t, τ )u(τ ) = −Uα−1 (t, τ )Aα−1 (τ )u(τ ) + U (t, τ )u (τ ) = Uα−1 (t, τ )f (τ ) in X for all t > τ . As a result, U (t, t − ε)u(t − ε) − U (t, s)u(s) =

s

t−ε

Uα−1 (t, τ )f (τ ) dτ

for t > t − ε > s. Letting ε → 0, we conclude that u is a mild solution of (3.1). 3.1. Evolution equations on R In this subsection we study the almost periodicity of the solutions to (3.1) on J = R under the following assumptions. (H1) The operators A(t), t ∈ R, satisfy the assumptions (2.1) and (2.2). (H2) The evolution family U generated by A(·) has an exponential dichotomy on R with constants N, δ > 0, projections P (t), t ∈ R, and Green’s function Γ. (H3) R(ω, A(·)) ∈ AP (J, L(X)).

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It is not difficult to verify that then R(λ, A(·)) ∈ AP (J, L(X)) for λ ∈ ω +Σθ ∪{0}. We want to solve (3.2) for f belonging to the space APα−1 (R) which is defined by APα−1 (R) := {f ∈ Eα−1 (R) : ∃ (fn ) ∈ AP (R, X) converging to f in Eα−1 (R)} = {f ∈ E−1 (R) : ∃ (fn ) ∈ AP (R, X) converging to f in Eα−1 (R)} for α ∈ [0, 1]. This space is endowed with the norm of Eα−1 (R). Note that AP0 (R) = AP (R, X). We first characterize the space APα−1 (R). On F := AP (R, X), we define the multiplication operator (A(·)v)(t) := A(t)v(t),

t ∈ R,

D(A(·)) := {v ∈ F : f (t) ∈ D(A(t)) for all t ∈ R, A(·)v ∈ F }. Assumptions (H3) and (2.1) imply that the function R(λ, A(·))v belongs to F for every v ∈ F and λ ∈ ω + Σθ ∪ {0}. Therefore, the operator A(·) is sectorial on F A(·) with the resolvent R(λ, A(·)). We can thus introduce the spaces Fα−1 := Fα−1 for each α ∈ [0, 1), where we set F0 := F and F1 := D(A(·)). Proposition 3.1. Let (2.1) and (H3) hold. We then have Fα−1 ∼ = APα−1 (R) for each α ∈ [0, 1]. Proof. We first note that f Fα−1 = f Eα−1

for all f ∈ F and α ∈ [0, 1].

(3.3)

The embedding F−1 → E−1 holds due to Lemma 2.3. Therefore we obtain Fα−1 = {f ∈ F−1 : ∃ fn ∈ AP (R, X), fn → f in · Fα−1 = · Eα−1 } → {f ∈ E−1 : ∃ fn ∈ AP (R, X), fn → f in · Fα−1 = · Eα−1 } = APα−1 (R). The asserted isomorphy now follows from (3.3).

These spaces are much simpler in the case of constant extrapolation spaces. t 0 ∼ Proposition 3.2. Let (2.1) and (H3) hold. Assume that Xα−1 =: Xα−1 for = Xα−1 some α ∈ [0, 1] and every t ∈ R with uniformly equivalent norms. Then it holds Fα−1 ∼ = AP (R, Xα−1 ). = APα−1 (R) ∼

Proof. Due to the assumptions, the norms of Eα−1 and of Cb (R, Xα−1 ) are equivalent on E, so that Eα−1 ∼ = Cb (R, Xα−1 ). Take f ∈ AP (R, Xα−1 ) → Eα−1 and the sequence fn := nR(n, Aα−1 (·))f for n > ω. We first show that fn ∈ AP (R, X). For that purpose, let x ∈ Xα−1 and take xk ∈ X converging to x in Xα−1 . Due to (H3), t to we have nR(n, A(·))xk ∈ AP (R, X). Since R(n, Aα−1 (t)) is bounded from Xα−1 X uniformly in t (see e.g. [23, (2.8)], we derive that nR(n, Aα−1 (·))x ∈ AP (R, X). The same is true for functions f = φ(·)x, with scalar almost periodic function φ and x ∈ Xα−1 . Since the span of those functions is dense in AP (R, Xα−1 ) by [4, Theorem 4.5.7], it follows that fn ∈ AP (R, X). Observing that fn → f in Eα−1 , we conclude that f ∈ APα−1 (R). For the converse, let f ∈ APα−1 (R)

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∼ Cb (R, Xα−1 ). The continuous embedding and AP (R, X) fn → f in Eα−1 = X → Xα−1 implies that fn ∈ AP (R, Xα−1 ), and hence f ∈ AP (R, Xα−1 ). We state the main result of this subsection. Theorem 3.3. Assume that (H1), (H2) and (H3) hold. Let f ∈ APα−1 (R) for some α ∈ (1 − µ, 1]. Then the evolution equation (3.1) has a unique mild solution u ∈ AP (R, X) given by Γα−1 (t, τ )f (τ ) dτ, t ∈ R. (3.4) u(t) = R

Proof. For f ∈ Eα−1 , one can show that the function u given by (3.4) is a bounded mild solution of (3.1), and that every bounded mild solution is given by (3.4). (See e.g. the remarks after Theorem 3.10 in [22].) This fact shows the uniqueness of bounded mild solutions to (3.1). Take a sequence (fn ) ⊂ AP (R, X) converging to f in Eα−1 . In Theorem 4.5 of [21] we have shown that the functions un (t) = Γ(t, τ )fn (τ ) dτ, t ∈ R, (3.5) R

belongs to AP (R, X). Proposition 2.2 further yields τ τ Γα−1 (t, τ )L(Xα−1 u(t) − un (t) ≤ ,X) fn (τ ) − f (τ )α−1 dτ R

≤ c fn − f Eα−1 ,

t ∈ R.

Therefore un → u in Cb (R, X) as n → ∞, and so u ∈ AP (R, X).

3.2. Forward evolution equations We investigate the parabolic initial value problem u (t) = Aα−1 (t)u(t) + f (t), u(t0 ) = x,

t ≥ t0 ,

(3.6)

under the following assumptions. (H1’) The operators A(t), t > a, satisfy the assumptions (2.1) and (2.2) for t, s > a. (H2’) The evolution family U generated by A(·) has an exponential dichotomy on (a, ∞) with projections P (t), t > a, constants N, δ > 0, and Green’s function Γ. (H3’) R(ω, A(·)) ∈ AAP ([t0 , ∞), L(X)) for some t0 > a. Let now t0 > a, 1 − µ < α ≤ 1, x ∈ D(A(t0 )) and f ∈ Eα−1 ([t0 , ∞)). Assume that (H1’) and (H2’) hold. Then a mild solution of (3.6) is a function u ∈ C([t0 , ∞), X) being a mild solution of the evolution equation in the first line of (3.6) and satisfying u(t0 ) = x. We have shown in [22, Proposition 2.7] that there is a bounded mild function u of (3.6) if and only if ∞ ˜ (t0 , s)Qα−1 (s)f (s) ds. U (3.7) Q(t0 )x = − t0

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In this case the mild solution of (3.6) is uniquely given by t Uα−1 (t, s)Pα−1 (s)f (s) ds u(t) = U (t, t0 )P (t0 )x + −

t0

+∞

˜α−1 (t, s)Qα−1 (s)f (s) ds U +∞ Γα−1 (t, s)f (s) ds, = U (t, t0 )P (t0 )x + t

t0

t ≥ t0 .

(3.8)

We want to study the asymptotic almost periodicity of this solution in the case of an asymptotically almost periodic f . For a closed unbounded interval J = R, we introduce the space AAPα−1 (J) := {f ∈ Eα−1 (J) : ∃ (fn ) ⊆ AAP (J, X), fn → f in Eα−1 (J)}, endowed with the norm of Eα−1 (J). We define the multiplication operator A(·) on AAP (J, X) by (A(·)v)(t) := A(t)v(t),

t ∈ J,

D(A(·)) := {v ∈ AAP (J, X) : v(t) ∈ D(A(t)) ∀ t ∈ J, A(·)v ∈ AAP (J, X)}. Assumption (H3’) and (2.1) imply that the function R(λ, A(·))v belongs to AAP ([t0 , ∞), X) for every v ∈ AAP ([t0 , ∞), X) and λ ∈ ω + Σθ ∪ {0}. Therefore, the operator A(·) is sectorial on AAP ([t0 , ∞), X). We can thus introduce also the A(·) spaces AAP ([t0 , ∞), X)α−1 for α ∈ [0, 1]. These spaces can be characterized as in the previous subsection. Proposition 3.4. Let (2.1) and (H3’) hold. Then we have A(·) AAPα−1 ([t0 , ∞)) ∼ = AAP ([t0 , ∞), X)α−1 . t ∼ for each α ∈ [0, 1]. If, in addition, Xα−1 = Xα−1 with uniform equivalent norms for some 1 − µ < α ≤ 1 and a Banach space Xα−1 , then we obtain ∼ AAP ([t0 , ∞), Xα−1 ). AAPα−1 ([t0 , ∞)) =

We can now prove the main result of this subsection. Theorem 3.5. Let 1 − µ < α ≤ 1. Assume that (H1’), (H2’), and (H3’) hold and that x ∈ D(A(t0 )) and f ∈ AAPα−1 ([t0 , ∞)) satisfy (3.7). Then the unique bounded mild solution u of (3.6) is asymptotically almost periodic. Proof. Let f ∈ AAPα−1 ([t0 , ∞)) and x ∈ X satisfy (3.7). Take a sequence (fn ) ⊂ AAP ([t0 , ∞), X) converging to f in Eα−1 ([t0 , ∞)). Due to [21, Theorem 5.4], the functions ∞ un (t) = U (t, t0 )P (t0 )x + Γ(t, s)fn (s) ds, t ≥ t0 , n ∈ N t0

are asymptotically almost periodic in X (and they are mild solutions of (3.6) for the inhomogeneities fn and the initial values xn = un (t0 )). As in the proof of

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Theorem 3.3, we see that un → u in Cb ([t0 , ∞), X). So we conclude that u ∈ AAP ([t0 , ∞), X). 3.3. Backward evolution equations As a counterpart to the previous subsection, we now study the parabolic final value problem u (t) = Aα−1 (t)u(t) + f (t),

t ≤ t0 ,

u(t0 ) = x.

(3.9)

Mild solutions of (3.9) are defined as in the forward case. We make the following assumptions. (H1”) The operators A(t), t < b, satisfy (2.1) and (2.2) for t, s < b. (H2”) The evolution family U has an exponential dichotomy on (−∞, b) with projections P (t), t < b, constants N, δ > 0, and Green’s function Γ. (H3”) R(ω, A(·)) ∈ AAP ((−∞, t0 ], L(X)) for some t0 < b. Let 1 − µ < α ≤ 1, x ∈ X, and f ∈ Eα−1 ((−∞, t0 ]). We have shown in [22, Proposition 2.8] that there is a unique bounded mild solution u ∈ C((−∞, t0 ], X) of (3.9) on (−∞, t0 ] if and only if t0 Uα−1 (t0 , s)Pα−1 (s)f (s)ds, (3.10) P (t0 )x = −∞

in which case u is given by (t, t0 )Q(t0 )x + u(t) = U −

t

t0

t

−∞

Uα−1 (t, s)Pα−1 (s)f (s) ds

α−1 (t, s)Qα−1 (s)f (s) ds U

(3.11)

for t ≤ t0 . As before, we obtain the asymptotic almost periodicity of this function if f belongs to AAPα−1 ((−∞, t0 ]). We note that the space AAPα−1 ((−∞, t0 ]) can de described as in Proposition 3.4. Theorem 3.6. Assume that (H1”), (H2”), and (H3”) hold. Let x ∈ X and f ∈ AAPα−1 ((−∞, t0 ]) satisfy (3.10). Then the unique bounded mild solution u of (3.9) given by (3.11) belongs to AAP ((−∞, t0 ], X). Proof. Let x and f be as in the assertion. Take a sequence (fn ) in AAP ((−∞, t0 ], X) converging to f in Eα−1 ((−∞, t0 ]). Define the function t0 (t, t0 )Q(t0 )x + Γα−1 (t, s)Qα−1 (s)fn (s) ds un (t) = U −∞

for t ≤ t0 and n ∈ N. Using the same arguments as in [21, Theorem 5.4], we can show that un ∈ AAP ((−∞, t0 ], X) for all n ∈ N. Finally, as in Theorem 3.3 we see that un → u in Cb ((−∞, t0 ], X), so that u ∈ AAP ((−∞, t0 ], X).

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4. Fredholm properties of almost periodic parabolic evolution equations on R Consider a family of operators A(t), t ∈ R, on X satisfying the hypotheses (2.1), (2.2), and (2.7). Again, U (t, s) is the evolution family on X generated by A(·) s for 1 − µ < α ≤ 1. Both families and Uα−1 (t, s) is its extrapolation on Xα−1 have exponential dichotomies on (−∞, −T ] and [T, +∞) for some T ≥ 0 with projections P (·) and Pα−1 (·), respectively. We further assume that (H4) Let T be the number T from (2.7). Then we assume that R(ω, A(·))|[T, ∞) ∈ AAP ([T, ∞), L(X)) and R(ω, A(·))|(−∞, −T ] ∈ AAP ((−∞, −T ], L(X)). We will work on the space AAP ± = AAP ± (R, X) := {f ∈ Cb (R, X) : f |R± ∈ AAP (R± , X)}. of functions being asymptotically almost periodic on R− and R+ , separately. This space is endowed with the sup–norm. The following description of this space turns out to be crucial for our work. Lemma 4.1. Let (2.1) and (H4) hold. We then have AAP ± = {f ∈ Cb (R, X) : f |(−∞, −a] ∈ AAP ((−∞, −a], X), f |[a, ∞) ∈ AAP ([a, ∞), X)} =: F a for each a ≥ 0. Proof. Let a ≥ 0 and f ∈ Cb (R, X) such that f + := f |[a, ∞) = g + + h+ ∈ C0 ([a, ∞), X) ⊕ AP ([a, ∞), X); f − := f |(−∞, −a] = g − + h− ∈ C0 ((−∞, −a], X) ⊕ AP ((−∞, −a], X). It is clear that h+ and h− can be extended to functions in AP (R+ , X) and AP (R− , X) respectively. The functions g˜± := f |R± −h± then belong to C0 (R± , X), i.e, f |R± = g˜± +h± ∈ AAP (R± , X). So we have shown the inclusion F a ⊂ AAP ± . The other inclusion is clear. As in the previous sections we define the multiplication operator A(·) on AAP ± (R, X) by t ∈ R,

(A(·)v)(t) := A(t)v(t), ±

D(A(·)) := {v ∈ AAP (R, X) : f (t) ∈ D(A(t)) ∀ t ∈ R, A(·)v ∈ AAP ± }. Assumption (H4) shows that function R(λ, A(·))f belongs to AAP ± for every f ∈ AAP ± and λ ∈ ω + Σθ ∪ {0}, and thus the operator A(·) is sectorial in AAP ± with the resolvent R(λ, A(·)). So we can define the extrapolation spaces A(·)

± ± = AAPα−1 (R) := (AAP ± (R, X))α−1 AAPα−1

for α ∈ [0, 1],

which are characterized in the following proposition. Proposition 4.2. Let (2.1) and (H4) hold, and let α ∈ [0, 1]. Then we have AAP ± ∼ = {f ∈ Eα−1 (R) : f |[T, ∞) ∈ AAPα−1 ([T, ∞)), α−1

f |(−∞, −T ] ∈ AAPα−1 ((−∞, −T ])}.

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t ∼ Assume that, in addition, Xα−1 = Xα−1 with uniformly equivalent norms for some Banach space Xα−1 and some α ∈ [0, 1]. Then we have AAP ± ∼ = {f ∈ Cb (R, Xα−1 ) : f |[T, ∞) ∈ AAP ([T, ∞), Xα−1 ), α−1

f |(−∞, −T ] ∈ AAP ((−∞, −T ], Xα−1 )}. ± Proof. Due to Lemma 4.1 the space AAP−1 is embedded into E−1 (R). Let f ∈ ± ± AAPα−1 . Then there are fn ∈ AAP converging to f in Eα−1 . The restrictions of fn to (−∞, −T ] and to [T, +∞) converge to the corresponding restrictions of f in Eα−1 ((−∞, −T ]) and Eα−1 ([T, +∞)), respectively. Therefore the restrictions of f belong to AAPα−1 ((−∞, −T ]) and to AAPα−1 ([T, +∞)), respectively, which shows the inclusion ‘⊂’. Let f belong to the space on the right side in the first assertion. The functions fn = nR(n, Aα−1 (·))f then belong to Cb (R, X) for n ≥ ω, and their restrictions belong to AAP ((−∞, −T ], X) and to AAP ([T, +∞), X) (since R(n, Aα−1 (·)) is the resolvent of the respective multiplication operator Aα−1 (·)). Lemma 4.1 thus yields fn ∈ AAP ± . Since fn → f in Eα−1 as n → ∞, the first assertion holds. The second assertion now follows from the results of the previous section. ± As in [22], we define the operator Gα−1 on AAPα−1 (R, X) in the following ± way. A function u ∈ AAP (R, X) belongs to D(Gα−1 ) and Gα−1 u = f if there is ± a function f ∈ AAPα−1 such that (3.2) holds; i.e., t u(t) = U (t, s)u(s) + Uα−1 (t, τ )f (τ ) dτ s

for all t, s ∈ R with t ≥ s. In particular, G0 is defined on AAP ± (R, X) by (3.2), replacing Uα−1 by U . To study the operator Gα−1 , we introduce the stable and unstable subspaces of Uα−1 (·, ·). Definition 4.3. Let t0 ∈ R. We define the stable space at t0 by t0 Xs (t0 ) := {x ∈ Xα−1 :

lim Uα−1 (t, t0 )x = 0},

t→+∞

and the unstable space at t0 by Xu (t0 ) := {x ∈ X : ∃ a mild solution u ∈ C0 ((−∞, t0 ], X) of (3.9) with f = 0}. Observe that the function u in the definition of Xu (t0 ) satisfies u(t) = U (t, s)u(s) for s ≤ t ≤ t0 and u(t0 ) = x, so that Xu (t0 ) ⊂ D(A(t0 )). The following result was shown in [22, Lemma 3.2]. Lemma 4.4. Assume that the assumptions (2.1), (2.2), and (2.7) are satisfied and that 1 − µ < α ≤ 1. Then the following assertions hold. t0 for t0 ≥ T ; (a) Xs (t0 ) = Pα−1 (t0 )Xα−1 (b) Xu (t0 ) = Q(t0 )X for t0 ≤ −T ; (c) Uα−1 (t, t0 )Xs (t0 ) ⊆ Xs (t) for t ≥ t0 in R; (d) U (t, t0 )Xu (t0 ) = Xu (t) for t ≥ t0 in R; t0 (e) Xs (t0 ) is closed in Xα−1 for t0 ∈ R.

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Finally, for technical purposes we introduce the space F T := {f : Cb ((−∞, T ], X) : f |(−∞, −T ] ∈ AAP ((−∞, −T ], X)} T and endow it with the sup norm. The corresponding extrapolation spaces Fα−1 for A(·) are defined as above for α ∈ [0, 1]. − The restrictions G+ α−1 and Gα−1 of Gα−1 to the halflines [T, +∞) and (−∞, T ] are given in a similar way: A function u ∈ AAP ([T, +∞), X) (resp., u ∈ F T ) − belongs to D(G+ α−1 ) (resp., D(Gα−1 )) if there is a function f ∈ AAPα−1 ([T, +∞)) T ) such that (resp., f ∈ Fα−1 t u(t) = U (t, s)u(s) + Uα−1 (t, σ)f (σ)dσ s

holds for all t ≥ s ≥ T (resp., for all s ≤ t ≤ T ). Then we set G+ α−1 u = f and ± G− u = f , respectively. The operators G and G are single valued and α−1 α−1 α−1 closed due to Remarks 2.5 and 3.12 of [22]. As in [10], [11] and [22], we obtain + − right inverses Rα−1 and Rα−1 on AAP ([T, +∞), X) and on F T for G+ α−1 and − Gα−1 , respectively, by setting ∞ t + α−1 (t, s)Qα−1 (s)h(s) ds + U h)(t) = − Uα−1 (t, s)Pα−1 (s)h(s) ds (Rα−1 t

T

for h ∈ AAPα−1 ([T, +∞), X) and t ≥ T , and  T   Γα−1 (t, s)h(s) ds, t ≤ −T,  − (Rα−1 h)(t) = −∞ −T    Uα−1 (t, s)Pα−1 (s)h(s) ds + −∞

t

Uα−1 (t, s)h(s) ds, |t| ≤ T,

−T

T for h ∈ Fα−1 .

Proposition 4.5. Assume that the assumptions (2.1), (2.2), (2.7) and (H4) are satisfied and that 1 − µ < α ≤ 1. Then the following assertions hold. + : AAPα−1 ([T, +∞)) → AAP ([T, +∞), X) is bounded and (a) The operator Rα−1 + + Gα−1 Rα−1 h = h for each h ∈ AAPα−1 ([T, +∞)). − − T (b) The operator Rα−1 : Fα−1 → F T is bounded and G− α−1 Rα−1 h = h for each T h ∈ Fα−1 . ± h(T ) ∈ XεT for all 0 ≤ ε < α. (c) We have Rα−1

Proof. Let h ∈ AAPα−1 ([T, +∞)). In Proposition 3.3 and Remark 3.12 of [22] it + was shown that Rα−1 h is a mild solution of the equation (3.6) for the inhomogene ∞ ˜ (T, s)Qα−1 (s)h(s) ds at t0 = T . Since (3.7) ity h and the initial value x := − T U + holds for h and x, Theorem 3.5 gives the asymptotic almost periodicity of Rα−1 h. + So the operator Rα−1 maps AAPα−1 ([T, +∞)) into AAP ([T, +∞), X), and its boundedness follows from Proposition 2.2 d), e) as in the proof of [22, ProposiT tion 3.3]. Assertion (a) is thus established. To show (b), let h ∈ Fα−1 ((−∞, T ]).

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− Proposition 3.3 and Remark 3.12 of [22] also yield that Rα−1 h is a mild solution of the equation (3.9) with t0 = T and the inhomogeneity h. It is clear that −T h|(−∞, −T ] satisfies (3.11) for x := −∞ Uα−1 (−T, s)Pα−1 (s)f (s) ds. Theorem 3.5 − − then implies that Rα−1 h|(−∞, −T ] ∈ AAP ((−∞, −T ], X) and consequently Rα−1 − T maps Fα−1 into F T . The boundedness of Rα−1 follows again from Proposition 2.2 d),e). The last assertion is a consequence of Propositions 2.1 a) and 2.2 d),e).

We can now describe the range and the kernel of Gα−1 . Proposition 4.6. Assume that (2.1), (2.2), (2.7) and (H4) are satisfied and that ± 1 − µ < α ≤ 1. For f ∈ AAPα−1 we set f + = f |[T, +∞) and f − = f |(−∞, T ]. Then the following assertions hold for Gα−1 . (a) N (G+ α−1 ) = {u ∈ C0 ([T, +∞), X)) : u(t) = U (t, T )x (∀ t ≥ T ), x ∈ ˆ T }; P (T )X − (b) N (Gα−1 ) = {u ∈ C0 ((−∞, T ], X) : u(t) = U (t, s)u(s) (∀ s ≤ t ≤ T ), u(T ) ∈ Xu (T )}; (c) N (Gα−1 ) = {u ∈ C0 (R, X) : u(t) = U (t, s)u(s) (∀ t ≥ s), u(T ) ∈ P (T )X ∩ Xu (T )}; ± + − (d) R(Gα−1 ) = {f ∈ AAPα−1 : Rα−1 f + (T ) − Rα−1 f − (T ) ∈ P (T )X + Xu (T )}, where for f ∈ R(Gα−1 ) a function u ∈ D(Gα−1 ) with Gα−1 u = f is given by (4.1) below; ± + − : Rα−1 f + (T ) − Rα−1 f − (T ) ∈ P (T )X + Xu (T )}, (e) R(Gα−1 ) = {f ∈ AAPα−1 ± where the closure on the left (right) side is taken in AAPα−1 (in X). Proof. The assertions (a), (b) and (c) follow from Proposition 3.5 and Remark 3.12 ˆ T ∩Xu (T ) since Xu (T ) ⊆ D(A(T )). of [22]. We note that P (T )X ∩Xu (T ) = P (T )X ± To show (d), let Gα−1 u = f ∈ AAPα−1 (R) for some u ∈ D(Gα−1 ). Then the − functions f ± belong to R(G+ α−1 ) and to R(Gα−1 ), respectively, because of Proposition 4.2 and (3.2). Proposition 4.5 shows that the functions + f+ v+ = u|[T, +∞) − Rα−1

and

− v− = u|(−∞, T ] − Rα−1 f−

− are contained in the kernels of G+ α−1 and of Gα−1 , respectively. So we obtain + − (Rα−1 f + )(T ) − (Rα−1 f − )(T ) = v− (T ) − v+ (T ) ∈ Xu (T ) + P (T )X ± (R) with by (a) and (b). Conversely, let f ∈ AAPα−1 + − (Rα−1 f + )(T ) − (Rα−1 f − )(T ) = ys + yu ∈ P (T )X + Xu (T ). + − Set x0 := (Rα−1 f + )(T ) − ys = yu + (Rα−1 f − )(T ) and + u+ (t) := −U (t, T )ys + (Rα−1 f + )(t), u(t) := − u− (t) := v˜(t) + (Rα−1 f − )(t),

t ≥ T, t ≤ T,

(4.1)

where v˜ ∈ N (G− ˜(T ) = yu . Observe that u+ (T ) = u− (T ). From α−1 ) such that v ˆ T , so that U (·, T )ys ∈ C0 ([T, ∞), X). Proposition 4.5(c) we deduce ys ∈ P (T )X

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+ Proposition 4.5 shows that Rα−1 f + ∈ AAP ([T, ∞), X), and hence u|[T, ∞) ∈ AAP ([T, ∞), X). We also know from assertion (c) that v˜ ∈ C0 ((−∞, T ], X) and − from Proposition 4.5 that Rα−1 f − ∈ F T . Using also Lemma 4.1, we deduce that u belongs to AAP ± (R, X). Finally, one can check as in the proof of Proposition 3.5 of [22] that Gα−1 u = f . The last assertion can be shown exactly as Proposition 3.5(e) of [22].

Using the above results, we are able to describe the Fredholm properties of the operator Gα−1 in terms of properties of the subspaces Xs (T ) and Xu (T ). The proofs are similar to ones of Theorems 3.6 and 3.10 and Proposition 3.8 of [22] and therefore omitted. Recall that subspaces V and W of a Banach space E are called a semi-Fredholm couple if V + W is closed and if at least one of the dimensions dim(V ∩W ) and codim(V +W ) is finite. The index of (V, W ) is defined by ind(V, W ) := dim(V ∩ W ) − codim(V + W ). If the index is finite, then (V, W ) is a Fredholm couple. Theorem 4.7. Assume that (2.1), (2.2), and (2.7) are satisfied and 1 − µ < α ≤ 1. ± (R). Then the following assertions hold for Gα−1 defined on AAPα−1 ± (a) R(Gα−1 ) is closed in AAPα−1 if and only if P (T )X + Xu (T ) is closed in X. (b) If Gα−1 is injective, then P (T )X ∩ Xu (T ) = {0}. The converse is true if U (T, −T )|Q(−T )(X) is injective, in addition. (c) If Gα−1 is invertible, then P (T )X ⊕ Xu (T ) = X. The converse is true if U (T, −T )|Q(−T )(X) is injective in addition. (d) dim N (Gα−1 ) = dim(P (T )X ∩ Xu (T )) + dim N (U (T, −T )|Q(−T )(X)). We have codim(P (T )X + Xu (T )) = codim R(Gα−1 ), if R(Gα−1 ) is closed in ± AAPα−1 . In particular, Gα−1 is surjective if and only if P (T )X+Xu (T ) = X. (e) If Gα−1 is a semi-Fredholm operator, then (P (T )X, Xu (T )) is a semiFredholm couple, and ind(P (T )X, Xu (T )) ≤ ind Gα−1 . If in addition the kernel of U (T, −T )|Q(−T )(X) is finite dimensional, then

ind(P (T )X, Xu (T )) = ind Gα−1 − dim N (U (T, −T )|Q(−T )(X)).

(4.2)

Conversely, if (P (T )X, Xu (T )) is a semi-Fredholm couple and the kernel of U (T, −T )|Q(−T )(X) is finite dimensional, then Gα−1 is a semi-Fredholm operator and (4.2) holds. Proposition 4.8. Assume that (2.1), (2.2), and (2.7) hold and that 1 − µ < α ≤ 1. Then the closure of R(Gα−1 ) is equal to the space ± s F := {f ∈ AAPα−1 : f (s), v(s)Xα−1 ds = 0 for all v ∈ V}, R

where V is the space of those v ∈ L1 (R, X ∗ ) such that v(s) = Uα−1 (t, s)∗ v(t) for all t ≥ s in R. In the following Fredholm alternative, we restrict ourselves to the asymptotically hyperbolic case. The projections Q±∞ have finite rank if, for instance, the domains D(A±∞ ) are compactly embedded in X.

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Theorem 4.9. Assume that (2.1), (2.2), (2.5) and (2.6) are true, that dim Q±∞ X < ± ∞, and that 1 − µ < α ≤ 1. Let f ∈ AAPα−1 . Then there is a mild solution u ∈ AAP ± (R, X) of (3.1) if and only if s f (s), w(s)Xα−1 ds = 0 R

for each w ∈ L1 (R, X ∗ ) with w(s) = Uα−1 (t, s)∗ w(t) for all t ≥ s in R. The mild solutions u are given by + u(t) = v(t) − U (t, T )ys + (Rα−1 f )(t),

u(t) = v(t) + v˜(t) +

− (Rα−1 f )(t),

± Rα−1

t ≥ T,

t ≤ T,

+ − (Rα−1 f )(T ) − (Rα−1 f )(T )

was defined before Proposition 4.5, = ys + where yu ∈ P (T )X + Xu (T ), v˜ ∈ C0 ((−∞, T ], X) with v˜(T ) = yu and v˜(t) = U (t, s)˜ v (s) for all T ≥ t ≥ s, and v ∈ C0 (R, X) with v(t) = U (t, s)v(s) for all t ≥ s.

5. Non–autonomous parabolic boundary evolution equations In this section we study the non–autonomous forward (resp. backward) parabolic boundary evolution equation u (t) = Am (t)u(t) + g(t), B(t)u(t) = h(t),

t ≥ t0

t ≥ t0

(resp. t ≤ t0 ),

(resp. t ≤ t0 ),

(5.1)

u(t0 ) = u0 , and their variant on the line u (t) = Am (t)u(t) + g(t), B(t)u(t) = h(t),

t ∈ R,

t ∈ R.

(5.2)

Here t0 ∈ R, u0 ∈ X, and the inhomogeneities g and h take values in Banach spaces X and Y , respectively. We assume that the following conditions hold. (A1) There are Banach spaces Z → X and Y such that the operators Am (t) ∈ L(Z, X) and B(t) ∈ L(Z, Y ) are uniformly bounded for t ∈ R and that B(t) : Z → Y is surjective for each t ∈ R. (A2) The operators A(t)u := Am (t)u with domains D(A(t)) := {u ∈ Z : B(t)u = 0}, t ∈ R, satisfy (2.1) and (2.2) with constants ω, θ, K, L, µ, ν. Moreover, the graph norm of A(t) and the norm of Z are equivalent with constants being uniform in t ∈ R. In the typical applications Am (t) is a differential operator with ’maximal’ domain not containing boundary conditions and B(t) are boundary operators. Under the hypotheses (A1) and (A2), there is an evolution family (U (t, s))t≥s solving the problem with homogeneous conditions g = h = 0. Moreover, by [13, Lemma 1.2] there exists the Dirichlet map D(t) for ω − Am (t); i.e., v = D(t)y is the unique solution of the abstract boundary value problem (ω − Am (t))v = 0,

B(t)v = y,

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for each y ∈ Y . (In [13] the density of Z in X was assumed, but this does not play a role in the cited Lemma 1.2.) Let x ∈ X and y ∈ Y be given. The problem (ω − Am (t))v = x,

B(t)v = y,

has the solution v = R(ω, A(t))x + D(t)y. This solution is unique in Z since ω − Am (t) is injective on D(A(t)) = N (B(t)). We further assume that (A3) there is a β ∈ (1−µ, 1] such that Z → Xβt for t ∈ R with uniformly bounded embedding constants and supt∈R D(t)L(Y,Z) < ∞. Lemma 5.1. Assume that assumptions (A1), (A2) and (A3) without (2.2) hold. For a closed unbounded interval J, let Am (·) ∈ AP (J, L(Z, X)) and B(·) ∈ AP (J, L(Z, Y )). Then we have (a) D(·) ∈ AP (J, L(Y, Z)), (b) R(ω, A(·)) ∈ AP (J, L(X, Z)), (c) (ω − A−1 (·))D(·)h ∈ APα−1 (J) for every h ∈ AP (J, Y ) and α ∈ (1 − µ, β). The same results hold if one replaces throughout AP by AAP (if J = R) or by AAP ± (if J = R). Proof. (a) Let y ∈ Y and t, t + τ ∈ J. By the definition of D(t), we have (ω − Am (t))(D(t + τ )y − D(t)y) = (Am (t + τ ) − Am (t))D(t + τ )y =: ϕ(t), B(t)(D(t + τ )y − D(t)y) = −(B(t + τ ) − B(t))D(t + τ )y =: ψ(t), and thus D(t + τ )y − D(t)y = R(ω, A(t))ϕ(t) + D(t)ψ(t). The assumptions now imply that D(t + τ )y − D(t)yZ ≤ c (ϕ(t)X + ψ(t)Y ) ≤ c (Am (t + τ ) − Am (t)L(Z,X) + B(t + τ ) − B(t)L(Z,Y ) ) yY . So the almost periodicity of D(·) follows from that of Am (·) and B(·). (b) For x ∈ X and t, t + τ ∈ J, set y = R(ω, A(t + τ ))x − R(ω, A(t))x ∈ Z. Then we obtain (ω − Am (t))y = (Am (t + τ ) − Am (t))R(ω, A(t + τ ))x =: ϕ1 (t), B(t)y = (B(t) − B(t + τ ))R(ω, A(t + τ ))x =: ψ1 (t). Hence y = R(ω, A(t))ϕ1 (t) + D(t)ψ1 (t), and assertion (b) can now be shown as in (a). (c) Due to (a) and (b), the functions D(·)h and fn := nR(n, A(·))D(·)h are almost periodic in Z, and hence in X, for n > ω. Then A(·)fn = (n2 R(n, A(·)) − n)D(·)h belongs to AP (J, X). Assumptions (2.1) and (A3) imply that fn is uniformly bounded in the norm of Eβ . Since fn → D(·)h in Cb (J, X), we conclude by interpolation that fn → D(·)h in Eα . As a consequence, (ω − A(·))fn → (ω − Aα−1 (·))D(·)h in Eα−1 , whence (c) follows. Similarly one establishes the assertions concerning AAP and AAP ± .

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In order to apply the results from the previous sections to the boundary forward (resp. backward) evolution equation (5.1), we write it as the inhomogeneous Cauchy problem u (t) = A−1 (t)u(t) + f (t),

t ≥ t0

(resp. t ≤ t0 ),

u(t0 ) = u0 ,

(5.3)

setting f := g + (ω − A−1 (·))D(·)h. We also consider the evolution equation u (t) = A−1 (t)u(t) + f (t),

t ∈ R.

(5.4)

In the following we will have f ∈ Eα−1 (J), where we fix the number α ∈ (1 − µ, β) from Lemma 5.1. We note that a function u ∈ C 1 (J, X) with u(t) ∈ Z satisfies (5.1), resp. (5.2), if and only if it satisfies (5.3), resp. (5.4). These facts can be shown as in Proposition 4.2 of [9]. This motivates the following definition. We call a function u ∈ C(J, X) a mild solution of (5.2) and (5.4) on J if the equation t Uα−1 (t, σ)[g(σ) + (ω − A−1 (σ))D(σ)h(σ)] dσ (5.5) u(t) = U (t, s)u(s) + s

holds for all t ≥ s in J. The function u is called a mild solution of (5.1) (resp. (5.3) if in addition u(t0 ) = u0 and J = [t0 , ∞) (resp. J = (−∞, t0 ]). Theorems 3.3, 3.5 and 3.6 and Lemma 5.1 immediately imply three results on the existence of almost periodic mild solutions for (5.2) and (5.1). Proposition 5.2. Assume that (A1)–(A3) hold, that Am (·) ∈ AP (R, L(Z, X)) and B(·) ∈ AP (R, L(Z, Y )), and that U (t, s) has an exponential dichotomy on R. Let g ∈ AP (R, X) and h ∈ AP (R, Y ). Then there is a unique mild solution u ∈ AP (R, X) of the boundary equation (5.2) given by Γα−1 (t, s)[g(s) + (ω − A−1 (s))D(s)h(s)] ds, t ∈ R. u(t) = R

Proposition 5.3. Assume that (A1)–(A3) hold, that Am (·) ∈ AAP ([a, ∞), L(Z, X)), and B(·) ∈ AAP ([a, ∞), L(Z, Y )), and that U (t, s) has an exponential dichotomy on [a, ∞). Let t0 > a, g ∈ AAP ([a, ∞), X), h ∈ AAP ([a, ∞), Y ), and u0 ∈ D(A(t0 )). Then the mild solution u of the equation (5.1) belongs to AAP ([t0 , +∞), X) if and only if +∞ α−1 (t0 , s)Qα−1 (s)[g(s) + (ω − A−1 (s))D(s)h(s)] ds. U Q(t0 )u0 = − t0

In this case u is given by

t

Uα−1 (t, s)Pα−1 (s)[g(s) + (ω − A−1 (s))D(s)h(s)] ds u(t) = U (t, t0 )P (t0 )u0 + t0 ∞ ˜α−1 (t, s)Qα−1 (s)[g(s) + (ω − A−1 (s))D(s)h(s)] ds, t ≥ t0 . U − t

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Proposition 5.4. Let (A1)–(A3) hold, Am (·) ∈ AAP ((−∞, b], L(Z, X)) B(·) ∈ AAP ((−∞, b], L(Z, Y )), and assume that U (t, s) has an exponential dichotomy on (−∞, b]. Let t0 < b, g ∈ AAP ((−∞, b], X), h ∈ AAP ((−∞, b], Y ), and u0 ∈ X. Then there is a mild solution u ∈ AAP ((−∞, t0 ], X) of the equation (5.1) if and only if t0 P (t0 )u0 = Uα−1 (t0 , s)Pα−1 (s)[g(s) + (ω − A−1 (s))D(s)h(s)] ds. −∞

In this case u is given by t0 α−1 (t, s)Qα−1 (s)[g(s) + (ω − A−1 (s))D(s)h(s)] ds U u(t) = U (t, t0 )Q(t0 )u0 − t t Uα−1 (t, s)Pα−1 (s)[g(s) + (ω − A−1 (s))D(s)h(s)] ds, t ≤ t0 . + −∞

Moreover, Theorem 4.9 implies the following Fredholm alternative for the mild solutions of (5.2), where we focus on the asymptotically hyperbolic case. Theorem 5.5. Assume that assumptions (A1)–(A3) hold and that Am (t) → Am (±∞) in L(Z, X) and B(t) → B(±∞) in L(Z, Y ) as t → ±∞. Set A±∞ := Am (±∞)|N (B(±∞)). We suppose that σ(A±∞ ) ∩ iR = ∅ and that the corresponding unstable projections Q±∞ X have finite rank. Let g ∈ AAP ± (R, X) and h ∈ AAP ± (R, Y ). Then there is a mild solution u ∈ AAP ± (R, X) of (5.2) if and only if s f (s), w(s)Xα−1 ds = 0 R

for f := g + (ω − A−1 (·))D(·)h and all w ∈ L1 (R, X ∗ ) with w(s) = Uα−1 (t, s)∗ w(t) for all t ≥ s in R. The mild solutions u are given by + u(t) = v(t) − U (t, T )ys + (Rα−1 f + )(t),

u(t) = v(t) + v˜(t) +

− f − )(t), (Rα−1

t ≥ T,

t ≤ T,

± was defined before Proposition 4.5, f + = f |[T, +∞), f − = where Rα−1 + − f |(−∞, −T ], (Rα−1 f + )(T ) − (Rα−1 f − )(T ) = ys + yu ∈ P (T )X + Xu (T ), v˜ ∈ v (s) for all T ≥ t ≥ s, and C0 ((−∞, T ], X) with v˜(T ) = yu and v˜(t) = U (t, s)˜ v ∈ C0 (R, X) with v(t) = U (t, s)v(s) for all t ≥ s.

Proof. Observe that functions converging at ±∞ belong to AAP ± . So it remains to show that R(ω, A(t)) → R(ω, A±∞ ) in L(X) as t → ±∞. This can be established as Lemma 5.1(b). We conclude with a pde example. One could treat more general problems, in particular systems, cf. [11], and one could weaken the regularity assumptions; but we prefer to keep the example simple.

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Example 5.6. We study the boundary value problem ∂t u(t, x) = A(t, x, D)u(t, x) + g(t, x),

t ∈ R, x ∈ Ω,

t ∈ R, x ∈ ∂Ω,

B(t, x, D)u(t, x) = h(t, x),

(5.6)

on a bounded domain Ω ⊆ Rn with boundary ∂Ω of class C 2 , employing the differential expressions

A(t, x, D) = akl (t, x)∂k ∂l + ak (t, x) ∂k + a0 (t, x), k,l k

bk (t, x) ∂k + b0 (t, x), B(t, x, D) = k

where B(t) is understood in the sense of traces. We require that akl = alk and bk are real–valued, akl , ak , a0 ∈ Cbµ (R, C(Ω)), bk , b0 ∈ Cbµ (R, C 1 (∂Ω)), n

akl (t, x) ξk ξl ≥ η |ξ|2 ,

and

k,l=1

n

bk (t, x)νk (x) ≥ β

k=1

for constants µ ∈ (1/2, 1), β, η > 0 and all ξ ∈ Rn , k, l = 1, · · · , n, t ∈ R, x ∈ Ω older continuous functions.) resp. x ∈ ∂Ω. (Cbµ is the space of bounded, globally H¨ 1−1/p Let p ∈ (1, ∞). We set X = Lp (Ω), Z = Wp2 (Ω), Y = Wp (Ω) (a Slobodeckij space), Am (t)u = A(t, ·, D)u and B(t)u = B(t, ·, D)u for u ∈ Z (in the sense of traces), and A(t) = Am (t)|N (B(t)). The operators A(t), t ∈ R, satisfy (2.1) and (2.2), see [2], [3], [20], or [27, Example 2.9]. Thus A(·) generates an evolution family U (·, ·) on X. It is known that the graph norm of A(t) is uniformly equivalent to the norm of Z, that B(t) : Z → Y is surjective, that Xαt = Wp2α (Ω) with uniformly equivalent norms for α ∈ (1 − µ, 1/2), and that the Dirichlet map D(t) : Z → Y is uniformly bounded for t ∈ R, see e.g. [3, Example IV.2.6.3]. Further let g ∈ AAP ± (R, X) and h ∈ AAP ± (R, Y ). We define mild solutions of (5.6) again by (5.5). We further assume that aα (t, ·) → aα (±∞, ·) in C(Ω)

and

bj (t, ·) → bj (±∞, ·) in C 1 (∂Ω)

as t → ±∞, where α = (k, l) or α = j for k, l = 1, · · · , n and j = 0, · · · , n. As a result, Am (·) ∈ AAP ± (R, L(Z, X)) and B(·) ∈ AAP ± (R, L(Z, Y )). We define the sectorial operators A±∞ in the same way as A(t). As in [11, Example 5.1] one can check that (2.5) holds. Finally we assume that iR ⊂ ρ(A±∞ ). Then the Fredholm alternative stated in Theorem 5.5 holds for mild solutions of (5.6) on X = Lp (Ω) for g ∈ AAP ± (R, X) and h ∈ AAP ± (R, Y ).

References [1] P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova 78 (1987), 47–107. [2] P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations. Differential Integral Equations 1 (1988), 433–457.

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[3] H. Amann, Linear and Quasilinear Parabolic Problems. Volume 1: Abstract Linear Theory. Birkh¨ auser, Basel, 1995. [4] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector Valued Laplace Transforms and Cauchy Problems. Birkh¨ auser, Basel, 2001. [5] C. J. K. Batty and R. Chill, Approximation and asymptotic behaviour of evolution families. Differential Integral Equations 15 (2002), 477–512. [6] C. J. K. Batty, W. Hutter and F. R¨ abiger, Almost periodicity of mild solutions of inhomogeneous Cauchy problems. J. Differential Equations 156 (1999), 309–327. [7] S. Boulite, L. Maniar and M. Moussi, Wellposedness and asymptotic behaviour of non-autonomous boundary Cauchy problems. Forum Math. 18 (2006), 611–638. [8] C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations. American Mathematical Society, Providence (RI), 1999. [9] W. Desch, J. Milota and Schappacher, Least square control problems in nonreflexive spaces. Semigroup Forum 62 (2001), 337–357. [10] D. Di Giorgio and A. Lunardi, On Fredholm properties of Lu = u − A(t)u for paths of sectorial operators. Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 39–59. [11] D. Di Giorgio, A. Lunardi and R. Schnaubelt, Fredholm properties of abstract parabolic operators in Lp spaces on the real line. Proc. London Math. Soc. 91 (2005), 703–737. [12] K.-J. Engel and R. Nagel, One–Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, New York, 2000. [13] G. Greiner, Perturbing the boundary conditions of a generator. Houston J. Math. 13 (1987), 213–229. [14] G. G¨ uhring and F. R¨ abiger, Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations. Abstr. Appl. Anal. 4 (1999), 169–194. [15] G. G¨ uhring, F. R¨ abiger and R. Schnaubelt, A characteristic equation for nonautonomous partial functional differential equations. J. Differential Equations 181 (2002), 439–462. [16] D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer–Verlag, 1981. [17] W. Hutter und F. R¨ abiger, Spectral mapping theorems for evolution semigroups on spaces of almost periodic functions. Quaest. Math. 26 (2003), 191–211. [18] Y. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations. J. Operator Theory 58 (2007), 387–414. [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations. Cambridge University Press, 1982. [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkh¨ auser, Basel, 1995. [21] L. Maniar and R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations. Evolution equations. In: G. Ruiz Goldstein, R. Nagel and S. Romanelli (Eds.), “Recent Contributions to Evolution Equations”, Marcel Dekker, 2003, pp. 299–318.

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[22] L. Maniar and R. Schnaubelt, The Fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions. J. Differential Equations 235 (2007), 308–339. [23] L. Maniar and R. Schnaubelt, Robustness of Fredholm properties of parabolic evolution equations under boundary perturbations. J. London Math. Soc. 77 (2008), 558–580. [24] W. M. Ruess and W. H. Summers, Weak almost periodicity and the strong ergodic limit theorem for periodic evolution systems. J. Funct. Anal. 94 (1990), 177–195. [25] B. Sandstede, Stability of travelling waves. In: B. Fiedler (ed.), “Handbook of Dynamical Systems, vol. 2,” Elsevier, 2002, pp. 983–1055. [26] R. Schnaubelt, Asymptotically autonomous parabolic evolution equations. J. Evol. Equ. 1 (2001), 19–37. [27] R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations. In: M. Iannelli, R. Nagel, S. Piazzera (Eds.), “Functional Analytic Methods for Evolution Equations,” Lecture Notes in Mathematics 1855, Springer–Verlag, Berlin, 2004, pp. 401–472. [28] V˜ u Quˆ oc Ph´ ong, Stability and almost periodicity of trajectories of periodic processes. J. Differential Equations 115 (1995), 402–415. [29] A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II. Funkcial. Ekvac. 33 (1990), 139–150. Mahmoud Baroun Universit¨ at Paderborn Institut f¨ ur Mathematik Warburger Str. 100 33098 Paderborn Germany e-mail: [email protected] Lahcen Maniar Cadi Ayyad University Faculty of Sciences Semlalia 2390 Marrakesh Morocco e-mail: [email protected], [email protected] Roland Schnaubelt Institut f¨ ur Analysis Fakult¨ at f¨ ur Mathematik Universit¨ at Karlsruhe 76128 Karlsruhe Germany e-mail: [email protected] Submitted: December 29, 2008.

Integr. equ. oper. theory 65 (2009), 195–210 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020195-16, published online August 3, 2009 DOI 10.1007/s00020-009-1706-x


Topological Structure of the Set of Weighted Composition Operators on Weighted Bergman Spaces of Infinite Order José Bonet, Mikael Lindström and Elke Wolf Abstract. We consider the topological space of all weighted composition operators on weighted Bergman spaces of infinite order endowed with the operator norm. We show that the set of compact weighted composition operators is path connected. Furthermore, we find conditions to ensure that two weighted composition operators are in the same path connected component if the difference of them is compact. Moreover, we compare the topologies induced by L(H ∞ ) and L(Hv∞ ) on the space of bounded composition operators and give a sufficient condition for a composition operator to be isolated. Mathematics Subject Classification (2000). Primary 47B33; Secondary 47B38. Keywords. Topological structure, weighted composition operator, weighted Bergman space of infinite order.

1. Introduction Let φ and ψ be analytic maps on the open unit disk D of the complex plane C such that φ(D) ⊂ D. These maps define on the space H(D) of analytic functions on D the so-called weighted composition operator ψCφ by (ψCφ )(f ) = ψ(f ◦ φ). Operators of this type have been studied on various spaces of analytic functions. For an introduction we refer the reader to the excellent monographs [24] and [8]. We are interested in weighted composition operators acting in the following setting. We say that a function v which is continuous, strictly positive, bounded and radial (i.e. v(z) = v(|z|) for every z ∈ D) is a weight. We consider operators defined on the weighted Banach spaces of holomorphic functions Hv∞ := {f ∈ H(D); f v = sup v(z)|f (z)| < ∞} z∈D

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and on the smaller spaces Hv0 := {f ∈ Hv∞ ; lim v(z)|f (z)| = 0}, |z|→1

both endowed with the norm .v . Such spaces appear in the study of growth conditions of analytic functions and have been investigated in various articles, see e.g. [14], [1], [2], [18], [17], [3], [4], [5]. The problem of the topological structure of the space of (weighted) composition operators has been considered on several spaces of analytic functions, starting with the paper by MacCluer, Ohno and Zhao [19]; see also [13], [11] and [12]. In [13] Hosokawa and Ohno showed that the set of compact composition operators on the little Bloch space is path connected. Moreover, they also analyzed the case of compact differences of composition operators. In this paper we will prove that the set of compact weighted composition operators on the space Hv∞ is path connected and we investigate the behaviour of pairs of weighted composition operators with a compact difference. We also study the topologies induced by L(H ∞ ) and L(Hv∞ ) on the space of bounded composition operators and obtain a sufficient condition for a composition operator to be isolated.

2. Notation and auxiliary results For notation and more information on composition operators we refer the reader to the excellent monographs [8] and [24]. A radial weight v is called typical if it is non-increasing with respect to |z| and satisfies lim|z|→1 v(z) = 0. The so called associated weights (see [3]) are an important tool to handle problems in the setting of weighted spaces of analytic functions. For a weight v the associated weight v˜ is defined by 1 1 v˜(z) := = , z ∈ D, ∞ sup{|f (z)|; f ∈ Hv , f v ≤ 1} δz Hv∞ where δz denotes the point evaluation of z. By [3] we know that associated weights are continuous, v˜ ≥ v > 0 and that for each z ∈ D we can find fz ∈ Hv∞ , ||fz ||v ≤ 1, 1 . It is well known that Hv˜∞ is isometrically isomorphic to such that |fz (z)| = v˜(z) ∞ Hv . We are especially interested in radial weights which satisfy the following condition which is due to W. Lusky (see [17]) inf k

v(1 − 2−k−1 ) > 0. v(1 − 2−k )

(L1)

The standard weights vp (z) = (1 − |z|2 )p , p > 0, are weights which have (L1). It is known (see [9] and [16]) that the Lusky condition (L1) is equivalent to the following conditions: (U ) There is α > 0 such that

v(z) (1−|z|)α

is increasing near the boundary of D.

(A) There are 0 < r < 1 and 1 < C < ∞ with ρ(p, z) ≤ r.

v(z) v(p)

≤ C for all p, z ∈ D with

Vol. 65 (2009) Topological Structure of Weighted Composition Operators

197

For a detailed proof of this equivalence we refer the reader to [15]. If a weight v satisfies condition (L1), then there is C > 0 such that v ≤ v˜ ≤ Cv on D. See [9]. We also need some geometric data of the unit disk. The pseudohyperbolic metric is given by a−z ρ(z, a) := |σa (z)|, where σa (z) := 1 − az is the automorphism of D which changes 0 and a. Let φ(1) , φ(2) , φ be analytic self-maps of the open unit disk D, ψ (1) , ψ (2) , ψ ∈ H(D) and Cw (X) be the space of all weighted composition operators on the Banach space X. We write ψ (1) Cφ(1) ∼X ψ (2) Cφ(2) if and only if ψ (1) Cφ(1) and ψ (2) Cφ(2) are in the same path component of Cw (X). Furthermore, we put φs (z) := φ(sz) and ψs (z) := ψ(sz) for every 0 ≤ s ≤ 1 and for every z ∈ D as well as (φ, ψ)# (z) :=

ψ(z)v(z) . v˜(φ(z))

Let v be a strictly decreasing, radial and typical weight. Since v is decreasing and |sz| ≤ |z| for every 0 ≤ s ≤ 1 and every z ∈ D, we have #

#

|(φ, ψ)s (z)| := |(φs , ψs ) (z)| =

|ψ(sz)|v(sz) |ψ(sz)|v(z) # ≤ = |(φ, ψ) (sz)| v˜(φ(sz)) v˜(φ(sz))

|ψ(z)|v(z) v ˜(φ(z)) ∞ lim sup|φ(z)|→1 |ψ(z)|v(z) , when ψC : H φ v → v ˜(φ(z)) ||ψCφ ||e = lim sup|z|→1 |ψ(z)|v(z) v ˜(φ(z)) . It is clear

for every 0 ≤ s ≤ 1 and every z ∈ D. Recall that ||ψCφ || = supz∈D

and

the essential norm ||ψCφ ||e =

Hv∞

is bounded. If ψ ∈ Hv0 , then that ∞ ∞ ψs Cφs : Hv → Hv is always compact when s < 1. For these results, see [21], [7]. Finally, let us collect some auxiliary results which will be needed during the paper. The following lemma is taken from [6] Lemma 1 and [15] Lemma 1, see also [9] Lemma 14. Lemma 2.1. Let v be a radial weight on D satisfying (L1) such that v is continuously differentiable respect to |z|. Then there exists a constant C < ∞ such that if f ∈ Hv∞ , then 1 1 , |f (z) − f (p)| ≤ Cf v max ρ(z, p) and v˜(z) v˜(p) |f (z)v(z) − f (p)v(p)| ≤ Cf v ρ(z, p) for all z, p ∈ D. The next result follows from [9] Lemma 14, [15] Lemma 1 and the proof of Lemma 2 in [15]. Lemma 2.2. Let v be a radial weight on D satisfying the Lusky condition (L1) and such that v is continuously differentiable with respect to |z|. Then there are M > 0 and r0 > 0 such that, for each p, z ∈ D with ρ(p, z) ≤ r0 we get (a)

v ˜(z) v ˜(p)

≤ M,

198


(b) 1 −

v ˜(z) v ˜(p)

IEOT

≤ M ρ(z, p).

Let us now estimate the norm of the difference of two weighted composition operators. We have that ||ψ (1) Cφ(1) − ψ (2) Cφ(2) ||L(Hv∞ ) = sup sup |ψ (1) (z)f (φ(1) (z)) − ψ (2) (z)f (φ(2) (z))|v(z). ||f ||v ≤1 z∈D

This can be splitted into two parts by adding and subtracting the following term ψ (1) (z)v(z) f (φ(2) (z))˜ v (φ(2) (z)) and then, by Lemma 2.1, we can deduce v ˜(φ(1) (z)) Lemma 2.3. Let v be a radial weight on D satisfying the Lusky condition (L1) and such that v is continuously differentiable with respect to |z|. Moreover let ψ (1) , ψ (2) ∈ H(D) and φ(1) , φ(2) be analytic self-maps of D. Then there is a constant C < ∞ such that #

||ψ (1) Cφ(1) − ψ (2) Cφ(2) ||L(Hv∞ ) ≤ sup(C|(φ(1) , ψ (1) ) (z)| ρ(φ(1) (z), φ(2) (z)) z∈D

#

+ |(φ(1) , ψ (1) )# (z) − (φ(2) , ψ (2) ) (z)|). Since v is a radial weight, the following theorem can be derived from [15] Theorem 3. See also [20]. Theorem 2.4. Let v be a radial weight such that v is continuously differentiable with respect to |z|, v = v˜ and satisfies the Lusky condition (L1). Let ψ (1) , ψ (2) ∈ Hv0 . If φ(1) , φ(2) : D → D are analytic maps such that ψ (1) Cφ(1) , ψ (2) Cφ(2) : Hv∞ → Hv∞ are bounded, then the operator ψ (1) Cφ(1) − ψ (2) Cφ(2) : Hv∞ → Hv∞ is compact if and only if (1)

|ψ (z)| (1) (z), φ(2) (z)) = 0, (a) lim|z|→1 v(z) v(φ (1) (z)) ρ(φ (2)

|ψ (z)| (b) lim|z|→1 v(z) v(φ ρ(φ(1) (z), φ(2) (z)) = 0, (z)) (2) (1) ψ (z) ψ (2) (z) = 0. (c) lim|z|→1 v(z) v(φ − (1) (z)) v(φ(2) (z))

3. Connectedness of compact weighted composition operators In this section we show that the set of compact weighted composition operators with weights in Hv0 is path connected. Proposition 3.1. Let v be a typical, decreasing weight on D satisfying (L1) such that v is continuously differentiable with respect to |z|, ψ ∈ Hv0 and φ an analytic self-map of D. Moreover, suppose that ψCφ : Hv∞ → Hv∞ is compact. Then the # map [0, 1] → (C(D), .∞ ), s → (φ, ψ)s is continuous.


199

ψ(tz)v(z) Proof. We have to show that if t → s, then I := supz∈D ψ(sz)v(z) − v ˜(φ(sz)) v ˜(φ(tz)) → 0. First, we consider the case s < 1. If t → s and s < 1, we can find 0 < s0 < 1 such that s, t ≤ s0 . Moreover, there is 0 < R < 1 such that |φ(tz)| ≤ R for every z ∈ D and every t ≤ s0 . We have 1 1 + sup v(z) |ψ(sz) − ψ(tz)| − I ≤ sup v(z)|ψ(sz)| v˜(φ(sz)) v˜(φ(tz)) z∈D v˜(φ(tz)) z∈D =: I1 + I2 . Using Lemma 2.1 and the fact that ρ(φ(tz), φ(sz)) ≤ ρ(tz, sz) we obtain, v˜(φ(tz)) − v˜(φ(sz)) I1 ≤ ||ψ||v sup v˜(φ(sz))˜ v (φ(tz)) z∈D ||ψ||v ||ψ||v C v˜(0) ≤ 2 sup |˜ ρ(tz, sz). v (φ(tz)) − v˜(φ(sz))| ≤ v˜ (R) z∈D v˜2 (R) Moreover, with Lemma 2.1, we get I2 ≤

C||ψ||v ρ(tz, sz). v˜(R)

Since s < 1, ρ(tz, sz) ≤ |t−s| 1−s → 0, when t → s, and the claim follows. Next, we treat the case s = 1. Fix ε > 0. Since ψCφ is compact and ψ ∈ Hv0 , there is 0 < r < 1 such that ψ(z)v(z) ε # < for every z ∈ D with |z| > r. |(φ, ψ) (z)| = v˜(φ(z)) 4 Fix r < r < 1. If t >

r r

and |z| > r , then |tz| > r and we have r ε # # |(φ, ψ)t (z)| ≤ |(φ, ψ) (tz)| < for every t > and every |z| > r . 4 r Put now δ1 := 1 − rr . For t > 1 − δ1 we obtain #

#

#

#

sup |(φ, ψ) (z) − (φ, ψ)t (z)| ≤ sup |(φ, ψ) (z)| + sup |(φ, ψ)t (z)|
r

|z|>r

|z|>r

ε . 2

For |z| ≤ r , we can find δ2 > 0 such that #

#

sup |(φ, ψ) (z) − (φ, ψ)t (z)|
1 − δ2 by the first part of the proof. Finally, t > 1 − δ = min{δ1 , δ2 } implies #

#

sup |(φ, ψ) (z) − (φ, ψ)t (z)| < ε. z∈D

Theorem 3.2. Let v be a radial, typical weight satisfying (L1) and ψ (1) , ψ (2) ∈ Hv0 . If the operators ψ (1) Cφ(1) and ψ (2) Cφ(2) are compact, then they belong to the same path component.

200


IEOT

Proof. Fix two analytic self-maps φ(1) , φ(2) of D and ψ (1) , ψ (2) ∈ Hv0 such that the corresponding weighted composition operators ψ (1) Cφ(1) and ψ (2) Cφ(2) are compact on Hv∞ . We want to show that ψ (1) Cφ(1) ∼Hv∞ ψ (1) (0)Cφ(1) (0) ∼Hv∞ ψ (2) (0)Cφ(2) (0) ∼Hv∞ ψ (2) Cφ(2) , where (ψ (i) (0)Cφ(i) (0) )f (z) := ψ (i) (0)f (φ(i) (0)) for every f ∈ Hv∞ and every z ∈ D. First, we note ψ (1) (0)Cφ(1) (0) ∼Hv∞ ψ (2) (0)Cφ(2) (0) . Put ps := (1−s)φ(1) (0)+sφ(2) (0) and qs := (1−s)ψ (1) (0)+sψ (2) (0) for every s ∈ [0, 1]. Then it is easily seen that {qs Cps : s ∈ [0, 1]} is a continuous curve from ψ (1) (0)Cφ(1) (0) to ψ (2) (0)Cφ(2) (0) . Next, we will prove ψ (i) Cφ(i) ∼Hv∞ ψ (i) (0)Cφ(i) (0) for i = 1, 2. By assumption, φ(i) is an analytic self-map of D and ψ (i) ∈ Hv0 such (i) that ψ (i) Cφ(i) is compact on Hv∞ . Then ψs Cφ(i) is also compact for all s ∈ [0, 1]. s We will show that (i)

||ψs(i) Cφ(i) − ψt Cφ(i) ||L(Hv∞ ) → 0 when t → s. s

t

By Lemma 2.3, (i)

||ψs(i) Cφ(i) − ψt Cφ(i) ||L(Hv∞ ) s

≤C

t

(i) # sup |(φ(i) s , ψs ) (z)| z∈D

(i)

(i)

(i) #

(i) (i) # ρ(φ(i) s (z), φt (z)) + ||(φs , ψs ) − (φt , ψt ) ||∞ .

Because of Proposition 3.1, it is enough to show that #

(i)

(i) (i) sup |(φ(i) s , ψs ) (z)|ρ(φs (z), φt (z)) → 0 when t → s. z∈D

(i)

(i) #

If s < 1, then we get that supz∈D |(φs , ψs ) (z)| ≤ ||ψ (i) Cφ(i) ||L(Hv∞ ) < ∞ (i)

(i)

and supz∈D ρ(φs (z), φt (z)) ≤ |t−s| 1−s → 0, as t → s, and we are done. If s = 1, we fix ε > 0. Since ψ (i) Cφ(i) is compact on Hv∞ and ψ (i) ∈ Hv0 , we can find 0 < r < 1 such that (i)

(i)

ρ(φ (tz), φ (z)) ≤ ρ(tz, z) = r(1−t) 1−tr 2

|z|(1−t) 1−t|z|2 ,

|ψ (i) (z)|v(z) v ˜(φ(i) (z))

r. Since (i)

it follows that sup|z|≤r ρ(φ(i) (z), φt (z)) ≤

→ 0, when t → 1. Hence (i)

lim ||ψ (i) Cφ(i) − ψt Cφ(i) ||L(Hv∞ ) ≤ C

t→1

t

ε = ε. C

4. Connectedness and compact differences For the proof of our main result, Theorem 4.2, we need the following crucial estimate.


201

Lemma 4.1. Let v be a radial weight satisfying Lusky’s condition (L1) and such that v˜ is continuously differentiable with respect to |z|. Let φ(1) , φ(2) be analytic self-maps of D, ψ (1) , ψ (2) ∈ H(D) and s ∈ [0, 1]. There are M > 0 and r0 ∈]0, 1[ such that, for each z ∈ D with ρ(φ(1) (z), φ(2) (z)) ≤ r0 we get #

#

|((1 − s)φ(1) + sφ(2) , (1 − s)ψ (1) + sψ (2) ) (z) − (φ(1) , ψ (1) ) (z)| #

#

≤ s M |(φ(1) , ψ (1) ) (z) − (φ(2) , ψ (2) ) (z)| s M ρ(φ(1) (z), φ(2) (z)) (1) (1) # 2 (1) (2) + s M ρ(φ (z), φ (z) . + |(φ , ψ ) (z)| 1 − (1 − s)ρ(φ(1) (z), φ(2) (z)) Proof. Let r0 > 0 and M > 0 be as in Lemma 2.2. First, we notice that ((1 − s)φ(1) + sφ(2) , (1 − s)ψ (1) + sψ (2) )# (z) = (1 − s) +s

# v˜(φ(1) (z)) (φ(1) , ψ (1) ) (z) (1) (2) v˜((1 − s)φ (z) + sφ (z))

# v˜(φ(2) (z)) (φ(2) , ψ (2) ) (z). v˜((1 − s)φ(1) (z) + sφ(2) (z))

Hence #

#

|((1 − s)φ(1) + sφ(2) , (1 − s)ψ (1) + sψ (2) ) (z) − (φ(1) , ψ (1) ) (z)| #

#

≤ |(φ(1) , ψ (1) ) (z) − (φ(2) , ψ (2) ) (z)|

s˜ v (φ(2) (z)) v˜((1 − s)φ(1) (z) + sφ(2) (z))

#

+ |(φ(1) , ψ (1) ) (z)| v˜(φ(2) (z)) v˜(φ(1) (z)) −s × 1 − (1 − s) (1) (2) (1) (2) v˜((1 − s)φ (z) + sφ (z)) v˜((1 − s)φ (z) + sφ (z)) =: I1 + I2 . Assume now that z ∈ D satisfies ρ(φ(1) (z), φ(2) (z)) ≤ r0 . To estimate I1 , observe that the convex combination (1 − s)φ(1) (z) + sφ(2) (z) also satisfies ρ(φ(1) (z), (1 − s)φ(1) (z) + sφ(2) (z)) ≤ r0 . Therefore, by Lemma 2.2, v˜(φ(2) (z)) ≤M v˜((1 − s)φ(1) (z) + sφ(2) (z)) and I1 ≤ sM |(φ(1) , ψ (1) )# (z) − (φ(2) , ψ (2) )# (z)|. Now,

v˜(φ(1) (z)) I2 ≤ sM |(φ(1) , ψ (1) )# (z)| 1 − (1) (2) v˜((1 − s)φ (z) + sφ (z)) v˜(φ(2) (z)) − v˜(φ(1) (z)) (1) (1) # . + s|(φ , ψ ) (z)| v˜((1 − s)φ(1) (z) + sφ(2) (z))

202


IEOT

By Lemma 2.2 and the proof of Lemma 5 in [19], we get v˜(φ(1) (z)) 1 − ≤ M ρ(φ(1) (z), (1 − s)φ(1) (z) + sφ(2) (z)) (1) (2) v˜((1 − s)φ (z) + sφ (z)) ≤M

sρ(φ(1) (z), φ(2) (z)) . 1 − (1 − s)ρ(φ(1) (z), φ(2) (z))

On the other hand, again using Lemma 2.2, (1) v˜(φ(2) (z)) − v˜(φ(1) (z)) v˜(φ(2) (z)) ≤ 1 − v˜(φ (z)) v˜((1 − s)φ(1) (z) + sφ(2) (z)) v˜((1 − s)φ(1) (z) + sφ(2) (z)) (2) v˜(φ (z)) ≤ M 2 ρ(φ(1) (z), φ(2) (z)). The conclusion follows from the estimates.

Theorem 4.2. Let v be a radial, typical weight satisfying (L1) such that v = v˜ and v is continuously differentiable with respect to |z| and ψ (1) , ψ (2) ∈ Hv0 . Asψ (2) Cφ(2) are bounded, lim|z|→1 ρ(φ(1) (z), φ(2) (z)) = 0 and sume that ψ (1) Cφ(1) , (1) ψ (z) ψ (2) (z) that lim|z|→1 v(z) v(φ(1) (z)) − v(φ(2) (z)) = 0. Then the map s → ((1 − s)ψ (1) + sψ (2) )C(1−s)φ(1) +sφ(2) is continuous from [0, 1] into Cw (Hv∞ ). In particular, the operators ψ (1) Cφ(1) and ψ (2) Cφ(2) belong to the same path component.

Proof. Let rs := (1 − s)φ(1) + sφ(2) and qs := (1 − s)ψ (1) + sψ (2) for every s ∈ [0, 1]. We have to show that for every ε > 0 there is δ > 0 such that |s − t| < δ implies qs Crs − qt Crt L(Hv∞ ) < ε. First, by Lemma 2.3 we obtain qs Crs − qt Crt L(Hv∞ ) ≤ C sup |(rs , qs )# (z)|ρ(rs (z), rt (z)) z∈D

+ sup |(rs , qs )# (z) − (rt , qt )# (z)| =: T1 + T2 . z∈D

Now, fix ε > 0. Next, we show that there is C1 > 0 such that sup C|(rs , qs )# (z)| ≤ C1 . z∈D

Fix z ∈ D. If max{|φ(1) (z)|, |φ(2) (z)|} = |φ(1) (z)|, we obtain the following estimate: |(rs , qs )# (z)| ≤ ≤

((1 − s)|ψ (1) (z)| + s|ψ (2) (z)|)v(z) v((1 − s)φ(1) (z) + sφ(2) (z)) ((1 − s)|ψ (1) (z)| + s|ψ (2) (z)|)v(z) . v(φ(1) (z))

Otherwise in case max{|φ(1) (z)|, |φ(2) (z)|} = |φ(2) (z)|, we get similarly |(rs , qs )# (z)| ≤

((1 − s)|ψ (1) (z)| + s|ψ (2) (z)|)v(z) . v(φ(2) (z))


203

Hence

1 1 , sup |(rs , qs )# (z)| ≤ (1 − s) sup v(z)|ψ (1) (z)| max v(φ(1) (z)) v(φ(2) (z)) z∈D z∈D 1 1 (2) + s sup v(z)|ψ (z)| max , 0 such that C sup |(rs , qs )# (z)| ≤ C1 . z∈D

Since we assume lim ρ(φ(1) (z), φ(2) (z)) = 0,

|z|→1

we can find a 1 > R1 > 0 such that ρ(φ(1) (z), φ(2) (z)) ≤ r0 whenever |z| > R1 , and r0 is the constant in Lemma 2.2. Therefore, also supz∈D ρ(φ(1) (z), φ(2) (z)) ≤ λ < 1 and thus the proof of [13] Lemma 4.1 yields that supz∈D ρ(rs (z), rt (z)) → 0 if s → t. Finally, this means that we can find δ1 > 0 such that if |s − t| < δ1 , then ε T1 = C sup |(rs , qs )# (z)|ρ(rs (z), rt (z)) < . 2 z∈D Let us now consider the term T2 . For every z ∈ D with |z| > R1 , so that ρ(φ(1) (z), φ(2) (z)) ≤ r0 , we have by Lemma 4.1 that |(rs , qs )# (z) − (rt , qt )# (z)| ≤ |(rs , qs )# (z) − (φ(1) , ψ (1) )# (z)| + |(rt , qt )# (z) − (φ(1) , ψ (1) )# (z)| #

#

≤ sM |(φ(1) , ψ (1) ) (z) − (φ(2) , ψ (2) ) (z)| # s M ρ(φ(1) (z), φ(2) (z)) 2 (1) (2) + sM ρ(φ (z), φ (z)) + |(φ(1) , ψ (1) ) (z)| 1 − (1 − s)ρ(φ(1) (z), φ(2) (z)) #

#

+ tM |(φ(1) , ψ (1) ) (z) − (φ(2) , ψ (2) ) (z)| # t M ρ(φ(1) (z), φ(2) (z)) 2 (1) (2) ρ(φ (z), φ (z)) + tM + |(φ(1) , ψ (1) ) (z)| 1 − (1 − t)ρ(φ(1) (z), φ(2) (z)) By our assumptions, we can find R2 ≥ R1 > 0 such that if |z| > R2 , then ε sup max{t, s}C|(φ(1) , ψ (1) )# (z) − (φ(2) , ψ (2) )# (z)| < , 10 |z|>R2 (1)

sup |(φ

|z|>R2

R2

IEOT

s M ρ(φ(1) (z), φ(2) (z)) 2 (1) (2) + sM ρ(φ (z), φ (z)) 1 − (1 − s)ρ(φ(1) (z), φ(2) (z))

ε . 10 Moreover, if |z| ≤ R2 , we can find δ2 > 0 such that if |s − t| < δ2 , then ε . sup |(rs , qs )# (z) − (rt , qt )# (z)| < 10 |z|≤R2
R2

×

|z|>R2

s M ρ(φ (z), φ (z)) 2 (1) (2) + s M ρ(φ (z), φ (z)) 1 − (1 − s)ρ(φ(1) (z), φ(2) (z)) (1)

(2)

#

#

+ sup tM |(φ(1) , ψ (1) ) (z) − (φ(2) , ψ (2) ) (z)| |z|>R2

#

+ sup |(φ(1) , ψ (1) ) (z)| |z|>R2

×

t M ρ(φ(1) (z), φ(2) (z)) 2 (1) (2) + t M ρ(φ (z), φ (z)) 1 − (1 − t)ρ(φ(1) (z), φ(2) (z))

ε . 2 The claim follows.
0 and let f ∈ Hv∞ with ||f ||v ≤ 1. Then there are 0 < r0 < 1 and k0 ∈ N such that sup v(z)|f (φ(z)) − f (φk (z))| z∈D v(z) v(z) ρ(φ(z), φk (z)), sup ρ(φ(z), φk (z)) ≤ C max sup ˜(φ(z)) ˜(φk (z)) z∈D v z∈D v v(z) v(z) ≤ C sup ρ(φ(z), φk (z)) + C sup ρ(φ(z), φk (z)) v ˜ (φ(z)) v ˜ (φ(z)) |z|≤r0 |z|>r0 ε ε < + =ε 2 2 for every k ≥ k0 since Cφ is compact on Hv∞ and sup|z|≤r0 ρ(φ(z), φk (z)) → 0 if k → ∞. Thus, Cφk → Cφ in L(Hv∞ ) but Cφk → Cφ in L(H ∞ ). Corollary 5.4. If v is a weight of the form v(z) = (1 − |z|)α , z ∈ D, α > 0, then τ∞ is strictly finer than τv . Proof. By Corollary 5.2, the topology τ∞ is finer than τv . Shapiro [24] pp. 49–50 1−|z| → 0 as constructs an analytic self-map ϕ of D with ϕ∞ = 1 satisfying 1−|ϕ(z)| |z| → 1. We can apply [5, Corollary 3.4] to conclude that Cϕ is compact on Hv∞ . The conclusion follows from Corollary 5.3. Observe that [5, Corollary 3.4] implies that there are the same compact composition operators on all the spaces Hv∞ with v(z) = (1 − |z|)α , z ∈ D, for all α > 0. Remark 5.5. By Corollary 5.3 it is obvious that if Cφ and Cφ1 are path connected in C ⊂ L(Hv∞ ), then they are also path connected in C ⊂ L(H ∞ ), but in general the converse is not true, as the following example shows. We choose φ and φ1 such that φ touches ∂D in 1 and φ1 touches ∂D in −1, φ∞ = φ1 ∞ = 1 1−|z| 1−|z| = lim|z|→1 1−|φ = 0 (see [24] pp. 49-50). Hence Cφ and and lim|z|→1 1−|φ(z)| 1 (z)| Cφ1 are compact and thus in the same path component of C(Hv∞ ). But obviously, supz∈D ρ(φ(z), φ1 (z)) = 1, hence by [19] Theorem 2, Cφ and Cφ1 are not in the same path component of C(H ∞ ). A non negative function g(x) defined on an interval I of the real line is called almost decreasing if there is C > 0 such that, if x, y ∈ I satisfy x < y, then g(y) ≤ Cg(x). Theorem 5.6. Let v and w be typical weights satisfying condition (L1) such that w/v is almost decreasing with respect to |z|. Then τv is finer than τw on the set C0 of composition operators Cϕ such that ϕ(0) = 0.


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Proof. Select a sequence (ψk )k of analytic self-maps of D such that Cψk tends to Cφ in L(Hv∞ ) and ψk (0) = φ(0) = 0. Hence |φ(z)| ≤ |z| and |ψk (z)| ≤ |z| for all z ∈ D and all k. Since w/v is almost decreasing, we can find an M > 0 such that for all z ∈ D and all k, v(z) w(z) v(z) w(z) ≤M and ≤M . w(φ(z)) v(φ(z)) w(ψk (z)) v(ψk (z)) Since the weights v and w are typical and satisfy condition (L1), we can replace the weights in the denominators by the corresponding associated weights in the inequalities above, increasing the constant M . By Proposition 5.1, we conclude ||Cψk − Cφ ||L(Hw∞ ) w(z) w(z) ρ(φ(z), ψk (z)), sup ρ(φ(z), ψk (z)) ≈ max sup ˜ ˜ k (z)) z∈D w(φ(z)) z∈D w(ψ v(z) v(z) ρ(φ(z), ψk (z)), sup ρ(φ(z), ψk (z)) ≤ M max sup ˜(φ(z)) ˜(ψk (z)) z∈D v z∈D v ≤ C M ||Cψk − Cφ ||L(Hv∞ ) ,

from where the conclusion follows.

We conclude this section with a remark about composition operators Cφ ∈ C which are isolated in L(Hv∞ ). To do this fix a typical weight v satisfying condition (L1) and, for ε > 0 set Eφ,ε,v := {(zn )n ⊂ D; |φ(zn )| → 1,

v(zn ) ≥ ε}. v˜(φ(zn ))

Let E φ,ε,v denote the set of limit points of sequences in Eφ,ε,v . Clearly E φ,ε,v is a subset of ∂D. Theorem 5.7. Let v be a typical weight satisfying condition (L1). Moreover let φ be an analytic self-map of D. If there is ε > 0 such that the set E φ,ε,v has Lebesgue measure strictly positive, then Cφ is isolated in the set of composition operators C in L(Hv∞ ). Proof. It is well-known that if φ = φ1 , then the set {z ∈ ∂D; φ(z) = φ1 (z)} is a null set for the Lebesgue measure. By assumption there is ε > 0 such that the set Eφ,ε,v has positive measure. Accordingly, there is a sequence (zn )n ⊂ D, and a point a ∈ E φ,ε,v such that zn → a and φ(zn ) → b ∈ ∂D, but φ1 (zn ) → b. Therefore we can find a subsequence of (zn )n , which we still denote in the same way, such that φ1 (zn ) → c = b. Hence limn→∞ ρ(φ(zn ), φ1 (zn )) = 1. Next, consider hn (z) := ϕφ1 (zn ) (z)fn (z) for every z ∈ D and every n ∈ N, where fn lies in the 1 for every n ∈ N. Thus each hn unit ball of Hv∞ and satisfies |fn (φ(zn ))| = v˜(φ(z n ))

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belongs to the unit ball of Hv∞ and we obtain Cφ − Cφ1 ≥ lim sup v(zn )|hn (φ(zn )) − hn (φ1 (zn ))| n→∞

= lim sup n→∞

Thus, the claim follows.

v(zn ) ρ(φ(zn ), φ1 (zn )) ≥ ε. v˜(φ(zn ))

Acknowledgment The research of José Bonet was partially supported by FEDER and MEC Project MTM 2007-62643 and Generalitat Valenciana project Prometeo/2008/101. Part of the present work was done during a stay of J. Bonet at the University of Paderborn in August 2008. The support of the Alexander von Humboldt Foundation is greatly appreciated.

References [1] K. D. Bierstedt, W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. (Series A) 54 (1993), 70–79. [2] K. D. Bierstedt, J. Bonet, A. Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), 271–297. [3] K. D. Bierstedt, J. Bonet, J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), 137–168. [4] J. Bonet, P. Doma´ nski, M. Lindstr¨ om, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. Math. Bull. 42, no. 2, (1999), 139–148. [5] J. Bonet, P. Doma´ nski, M. Lindstr¨ om, J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 64 (1998), 101–118. [6] J. Bonet, M. Lindstr¨ om, E. Wolf, Differences of composition operators between weighted Banach spaces of holomorphic functions, J. Austral. Math. Soc. Ser. A 84 (2008), 9–20. [7] M. Contreras, A. G. Hernandez-Diaz, Weighted composition operators in weighted Banach spaces of analytic functions, J. Austral. Math. Soc. (Series A) 69 (2000), 41–60. [8] C. Cowen, B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Baca Raton, 1995. [9] P. Doma´ nski, M. Lindstr¨ om, Sets of interpolation and sampling for weighted Banach spaces of holomorphic functions, Ann. Pol. Math. 79, no.3, (2002), 233–264. [10] P. Galindo, M. Lindstr¨ om, Factorization of homomorphisms through H ∞ (D), J. Math. Anal. Appl. 280 (2003), 375–386. [11] T. Hosokawa, K. Izuchi, S. Ohno, Topological structure of the space of weighted composition operators on H ∞ , Integr. equ. oper. theory 53 (2005), 509–526. [12] T. Hosokawa, K. Izuchi, D. Zheng, Isolated points and essntial components of composition operators on H ∞ , Proc. Am. Math. Soc. 130 (2001), no. 6, 1765–1773.


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[13] T. Hosokawa, S. Ohno, Topological structures of the sets of composition operators on the Bloch spaces, J. Math. Anal. Appl. 314 (2006), 736–748. [14] W. Kaballo, Lifting-Probleme f¨ ur H ∞ -Funktionen, Arch. Math. 34 (1980), 540–549. [15] M. Lindstr¨ om, E. Wolf, Essential norm of the difference of weighted composition operators, Monatsh. Math 153 (2008), 133–143. [16] W. Lusky, On the structure of Hv0 (D) and hv0 (D), Math. Nachr. 159(1992), 279– 289. [17] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. 51 (1995), 309–320. [18] W. Lusky, On the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia Math. 175 (2006), no. 1, 19–45. [19] B. MacCluer, S. Ohno, R. Zhao, Topological structure of the space of composition operators on H ∞ , Integr. equ. oper. theory 40 (2001), 481–494. [20] J. S. Manhas, Compact differences of weighted composition operators on weighted Banach spaces of analytic functions, Integr. equ. oper. theory 62 (2008), 419–428. [21] A. Montes-Rodriguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London Math. Soc. 61 (2) (2000), 872–884. [22] P. Nieminen, Compact differences of composition operators on Bloch and Lipschitz spaces, Computational Methods and Function Theory 7 (2007), 325–344. [23] E. Saksman, C. Sundberg, Comparing topologies on the space of composition operators, Recent advances in operator-related function theory, 199–207, Contemp. Math., 393, Amer. Math. Soc., Providence, RI, 2006. [24] J. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. José Bonet Instituto Universitario de Matematica Pura y Aplicada IUMPA-UPV, Edificio ID15(8E), Cubo F, Cuarta Planta Universidad Politécnica de Valencia E-46022 Valencia Spain e-mail: [email protected] Mikael Lindstr¨ om Department of Mathematical Sciences P.O. Box 3000 FIN-90014 University of Oulu Finland e-mail: [email protected] Elke Wolf Institute of Mathematics University of Paderborn D-33095 Paderborn Germany e-mail: [email protected]

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Integr. equ. oper. theory 65 (2009), 211–222 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020211-12, published online September 1, 2009 DOI 10.1007/s00020-009-1715-9


Spectra of Non Power-compact Composition Operators on H∞ Spaces Pablo Galindo and Alejandro Miralles Abstract. We continue the study of spectra of non power-compact composition operators on H ∞ (BE ), E a complex Banach space. We obtain a Juliatype estimate of the growth of the symbol near the sphere for E = C0 (X), thus for c0 or Cn . In particular, the description of the spectra of non powercompact composition operators on H ∞ (Dn ), where Dn denotes the polydisc in Cn , is achieved. Mathematics Subject Classification (2000). Primary 46J10; Secondary 47B38. Keywords. Banach space, composition operator, interpolation, polydisc, spectrum.

1. Introduction and Background The study of composition operators has become a large area of research for the last two decades. Particularly when they act on spaces of analytic functions either of one or many variables. Attention has been paid to topological properties like compactness and alike, structural properties like connected components or isolation, spectral properties, etc. We refer the reader to books like [CM], [S] or [Zu] to get a grasp of the variety of topics and spaces that are involved in the area. In this paper we focus on the description of the spectra of composition operators acting on spaces, actually uniform algebras, of bounded analytic functions of many variables. For background on such functions we refer to [M]. Let E be a complex Banach space with open unit ball BE . Recall that a function f : BE → C is said to be analytic if it is Fréchet differentiable. The space H ∞ (BE ) denotes the set {f : BE → C : f is analytic and bounded}. It is a uniform algebra endowed with the sup-norm f = sup{|f (x)| : x ∈ BE } and it is, obviously, the analogue of the space H ∞ for an arbitrary complex Banach space. Each analytic map ϕ : BE → BE gives rise to a composition operator Supported by Project MTM 2007-064521 (MEC-FEDER. Spain).

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Cϕ : H ∞ (BE ) → H ∞ (BE ) according to Cϕ (f ) = f ◦ ϕ. As usual, we say that Cϕ is power-compact if its n-times self-composition, Cϕn , is compact for some n ∈ N. We will denote by D the open unit disc of C. L. Zheng described the spectrum of some composition operators Cϕ : H ∞ −→ H ∞ in [Zh]: Suppose that ϕ : D → D is a non constant, analytic self-map, not an automorphism and that there exists z0 ∈ D such that ϕ(z0 ) = z0 . Then, the spectrumσ(Cϕ ) coincides with D if Cϕ is not power-compact and σ(Cϕ ) = ϕ (z0 )k : k ∈ N ∪{0, 1} if Cϕ is power-compact. When studying the spectrum of Cϕ we are led to consider the iterates of Cϕ , Cϕn = Cϕn where ϕn = ϕ ◦ . n. . ◦ ϕ is the n-times self composition of ϕ. The behavior of the sequence (ϕn ) is described according to two possibilities: either the image of some iterate ϕm lies strictly inside BE , that is ϕm (BE ) ⊂ rBE for some 0 < r < 1, or for all n ∈ N the closure of ϕn (BE ) in E meets the unit sphere. We shall refer to this second possibility as to the approaching condition. Observe that in case E is finite dimensional the first option leads to Cϕ being power compact. This may fail in the infinite dimensional case: Just consider ϕ(x) = x2 for which ϕn (x) = 2xn , and thus Cϕn is not compact. According to Proposition 3 in [AGL], we have that Cϕ : H ∞ (BE ) −→ H ∞ (BE ) is power-compact if and only if there exists n ∈ N such that ϕn (BE ) is relatively compact and this set lies strictly inside BE . Therefore, under the assumption that ϕ(BE ) is relatively compact, we have that Cϕ is not power-compact if and only if it satisfies the approaching condition. The spectra of power-compact composition operators Cϕ on any H ∞ (BE ) have been completely described by T. W. Gamelin, M. Lindstr¨ om and the first author in [GGL]: it is shown that if Cϕ is power-compact, then ϕ has a fixed point x0 ∈ BE such that ϕ (x0 ) < 1 and the spectrum of Cϕ consists of λ = 0 and λ = 1, together with all possible products λ = λ1 . . . λk , where k ≥ 1 and the λj s are eigenvalues of ϕ (x0 ). The spectrum of non power-compact composition operators Cϕ on H ∞ (BH ), H a complex Hilbert space, is also studied there by these authors. The following is proved: Theorem 1.1. Let H be a complex Hilbert space. Let ϕ : BH −→ BH be an analytic map satisfying ϕ(0) = 0 and ϕ (0) < 1 such that ϕ(BH ) is a relatively compact subset of H. Suppose that ϕ satisfies the approaching condition. Then, the spectrum of Cϕ coincides with the closed unit disk. Recall the following Schwarz’s lemma type inequality as shown in [GGL]: Suppose that ϕ : BE −→ BE satisfies ϕ(0) = 0 and ϕ (0) < 1. Then for each s < 1, there exists a < 1 such that ϕ(x) ≤ ax,

for x ∈ E,

x ≤ s.

(1.1)

Hence, given 0 < r < s < 1, there exists ε > 0 such that 1 − ϕ(x) ≥ 1 + , x ∈ BE , r < x < s. 1 − x Either in the proof of Zheng’s result or that of Theorem 1.1, what is needed in order to find suitable interpolating sequences is that the latter also holds when

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x → 1. In the case of the unit disc D, this estimate is typically obtained by using Julia’s lemma and angular derivatives (see [CM]). In the general case BE , some compactness seems to be necessary: We will say that a subset W ⊂ BE approaches SE compactly if any sequence (xn ) ⊂ W such that xn → 1 has a convergent subsequence. In the Hilbert space setting, the estimate (see [GGL]) reads as: Let ϕ : BH −→ BH be an analytic map satisfying ϕ(0) = 0 and ϕ (0) < 1. Suppose that W ⊂ BH approaches SH compactly. Then, for any 0 < δ < 1, there exists ε > 0 such that 1 − ϕ(x) ≥ 1 + ε, 1 − x

for all x ∈ W such that x ≥ δ.

(1.2)

We will refer to the above type of inequalities (1.2) as Julia-type estimates. What is left in the present setting is to study the spectra of non powercompact composition operators on H ∞ (BE ) for complex Banach spaces other than Hilbert spaces, in particular the n-fold space Cn . We obtain a general result for complex Banach spaces (Theorem 3.6) that covers all the previously known cases. This result is applied to C0 (X) spaces, X a locally compact topological space, since we obtain a Julia-type estimate in such a context. In particular, we are able to complete the description of the spectra of such composition operators acting on H ∞ (Dn ).

2. A Julia-type estimate for C0 (X) spaces In Theorem 3.6 we need to assume that a Julia-type estimate (1.2) for E is satisfied in order to describe the spectrum of Cϕ . As we recalled in the introduction, this estimate holds when dealing with Hilbert spaces. In this section we prove that a Julia-type estimate also holds for C0 (X) spaces. Recall that the pseudohyperbolic distance for H ∞ is given by z−w for all z, w ∈ D. ρ(z, w) = 1 − zw It is well-known that 1 − ρ(z, w)2 =

(1 − |z|2 )(1 − |w|2 ) |1 − zw|2

for all z, w ∈ D.

(2.1)

Recall that the pseudohyperbolic distance for H ∞ (BE ) with E = C0 (X), is given by the following formula (see [AGL2]): ρ(x, y) = sup ρ(x(t), y(t)) t∈X

for all x, y ∈ BE .

(2.2)

It will be useful in our computations to recall that the pseudohyperbolic distance is contractive for analytic self-maps ϕ : BE −→ BE . In addition, we will also need the following calculations.

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Lemma 2.1. The following holds: a) Let {αi : i ∈ I} be a set of real numbers such that 0 < supi∈I αi < 1. Then, 1 1 = . (2.3) sup 2 1 − αi 1 − supi∈I α2i i∈I b) The function h(x) = (1 − x)/(1 + x) is decreasing in [0, 1). Next, we state the section’s result, Theorem 2.2. Let E = C0 (X) and consider an analytic map ϕ : BE −→ BE such that ϕ(0) = 0 and ϕ (0) < 1. Suppose that W ⊂ BE approaches SE compactly. Then, the following Julia-type estimate holds: For every 0 < δ < 1, there exists ε > 0 such that 1 − ϕ(x) ≥ 1 + ε for any x ∈ W such that x ≥ δ. (2.4) 1 − x Proof. Suppose that the estimate (2.4) fails. Then we may choose a sequence (xn ) ⊂ W , xn ≥ δ, such that 1 − ϕ(xn ) → 1. 1 − xn Then limn xn = 1. If not, passing to a subsequence if necessary and bearing in mind (1.1), we would find 0 < a < 1 so that 1 − axn (1 − a)xn (1 − a)δ 1 − ϕ(xn ) ≥ =1+ ≥1+ , 1 − xn 1 − xn 1 − xn 1−δ against our temporary assumption. Consequently, limn ϕ(xn ) = 1 as well. Therefore, also 1 − ϕ(xn )2 lim = 1. n 1 − xn 2 By the assumption on W , we may suppose, passing to a subsequence if necessary, that xn → x0 , for x0 ∈ SE . Fix z ∈ BE . We denote ϕt (x) = ϕ(x)(t) for x ∈ BE and t ∈ X. Then, by (2.2) and (2.1), we have |1 − ϕt (z)ϕt (xn )|2 1 − ϕ(xn )2 1 − ϕ(xn )2 = sup (2.5) 1 − ρ2 (ϕ(z), ϕ(xn )) t∈X 1 − |ϕt (z)|2 1 − |ϕt (xn )|2 and bearing in mind Lemma 2.1 b), we obtain 1 − ||ϕ(z)|| (1 − |ϕt (z)|)2 1 − |ϕt (z)| |1 − ϕt (z)ϕt (xn )|2 ≥ ≥ = 2 2 1 − |ϕt (z)| 1 − |ϕt (z)| 1 + |ϕt (z)| 1 + ||ϕ(z)|| which implies that 1 − ϕ(xn )2 |1 − ϕt (z)ϕt (xn )|2 1 − ϕ(xn )2 1 − ||ϕ(z)|| sup . ≥ sup 1 − |ϕt (z)|2 1 − |ϕt (xn )|2 1 + ||ϕ(z)|| t∈X 1 − |ϕt (xn )|2 t∈X

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Therefore, by (2.5) and using Lemma 2.1 a), we obtain 1 − ||ϕ(z)|| 1 − ϕ(xn )2 1 − ϕ(xn )2 1 − ||ϕ(z)|| ≥ sup . = 2 2 1 − ρ (ϕ(z), ϕ(xn )) 1 + ||ϕ(z)|| t∈X 1 − |ϕt (xn )| 1 + ||ϕ(z)||

(2.6)

On the other hand, using the contractive property of ρ, we have 1 − ϕ(xn )2 1 − ϕ(xn )2 ≤ ≤ 1 − ρ2 (ϕ(z), ϕ(xn )) 1 − ρ2 (z, xn ) 1 − ϕ(xn )2 sup 1 − xn 2 t∈X

|1 − z(t)xn (t)|2 1 − xn 2 1 − |z(t)|2 1 − |xn (t)|2

.

(2.7)

Fix 0 < r < 1 and set z = rx0 . Consider for any n ∈ N the continuous function gn : X −→ R defined by gn (t) =

|1 − rx0 (t)xn (t)|2 . 1 − r2 |x0 (t)|2

It is easy to prove that (gn ) converges uniformly to the function g : X −→ R given by (1 − r|x0 (t)|2 )2 g(t) = 1 − r2 |x0 (t)|2 so passing to a subsequence we can assume that gn (t) ≤ g(t) + ε for all t ∈ X.

(2.8) (1−ru2 )2 1−r 2 u2 .

Consider the continuous function h : [0, 1] −→ R given by h(u) = We have that for ε > 0 there exists 0 < u0 < 1 such that 1−r + ε for any u0 ≤ u < 1. (2.9) h(u) ≤ 1+r Therefore, by (2.8) and (2.9) we have 1−r + 2ε for any t with |x0 (t)| ≥ u0 . gn (t) ≤ 1+r 0 0 Since xn → x0 , given 1−u > 0, there is n0 ∈ N such that |xn (t) − x0 (t)| < 1−u 2 2 for any t ∈ X and n ≥ n0 , so passing to a subsequence if necessary we can consider 0 for any n ∈ N and t ∈ X. that |xn (t)| ≤ |x0 (t)| + 1−u 2 Consider the supremum from inequality (2.7) and pay attention to items with |x0 (t)| < u0 . We have 1 − xn 2 |1 − rx0 (t)xn (t)|2 1 − xn 2 |1 − rx0 (t)xn (t)|2 ≤ 2 2 2 2 2 2 0 1 − r |x0 (t)| 1 − |xn (t)| 1 − r |x0 (t)| 1 − |x0 (t)| + 1−u 2 ≤ ≤

|1 − rx0 (t)xn (t)|2 1 − xn 2 |1 − rx0 (t)xn (t)|2 4(1 − xn 2 ) ≤ 1 − r2 |x0 (t)|2 1 − 1+u0 2 1 − r2 |x0 (t)|2 (1 − u0 )(3 + u0 ) 2 16(1 − xn 2 ) . (1 − u0 )(3 + u0 )(1 − r2 )

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On the other hand, if we pay attention to items with |x0 (t)| ≥ u0 we obtain that 1 − ϕ(xn )2 1 − ρ2 (ϕ(rx0 ), ϕ(xn )) |1 − rx0 (t)xn (t)|2 1 − xn 2 16(1 − xn 2 ) 1−ϕ(xn )2 sup , ≤ 1−xn 2 |x0 (t)|≥u0 1 − |rx0 (t)|2 1 − |xn (t)|2 (1 − u0 )(3 + u0 )(1 − r2 ) 1−ϕ(xn )2 16(1 − xn 2 ) |1 − rx0 (t)xn (t)|2 ≤ . sup , 1−xn 2 |x0 (t)|≥u0 1 − |rx0 (t)|2 (1 − u0 )(3 + u0 )(1 − r2 ) If we let n → ∞ in the last term, we obtain lim sup n→∞

1−r 1 − ϕ(xn )2 ≤ + 2ε. 1 − ρ2 (ϕ(rx0 ), ϕ(xn )) 1+r

(2.10)

Consequently, by (2.6) and (2.10), we obtain 1 − ||ϕ(rx0 )|| (1 − r) + 2ε ≥ 1+r 1 + ||ϕ(rx0 )|| and hence, if ε → 0, we obtain ||ϕ(rx0 )|| ≥ r = ||rx0 ||, a contradiction to (1.1).

3. The spectrum In this section we describe the spectra of non power-compact composition operators on H ∞ (BE ) under the assumption of the validity of a Julia-type estimate. Thus as a consequence of Theorem 2.2, we will be able to describe the spectra of non power-compact composition operators for the n-fold space Cn and c0 . It is in dealing with composition operators Cϕ whose symbol ϕ satisfies the approaching condition where the use of interpolating sequences is crucial. Such use has its root in H. Kamowitz paper [K], where the idea of interpolating iteration sequences already appears. Definition 3.1. A finite or infinite sequence (zk )k≥0 ⊂ BE is an iteration sequence for ϕ if ϕ(zk ) = zk+1 for k ≥ 0. Recall that a sequence (xn ) ⊂ BE is interpolating for H ∞ (BE ) if for any bounded sequence (αn ) ⊂ C, there exists f ∈ H ∞ (BE ) such that f (xn ) = αn for any n ∈ N. If we consider the restriction map R : H ∞ (BE ) −→ ∞ defined by R(f ) = (f (xn )), it is easy that a sequence (xn ) is interpolating for H ∞ (BE ) if and only if there is a map T : ∞ −→ H ∞ (BE ) such that R ◦ T = id∞ . The interpolation constant is defined by M = inf {C > 0 : T (α) ≤ Cα for any α ∈ ∞ } . With the natural changes, the same notions can be considered for finite sequences (xn )N n=0 .

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The following generalization of the Hayman-Newman Theorem [H] to any complex Banach space was obtained in [GM]. It is the key result to obtain interpolating sequences. Theorem 3.2. Let (xn ) be a sequence in BE and suppose that there exists 0 < c < 1 such that 1 − xk+1 < c. (3.1) 1 − xk Then, (xn ) is interpolating for H ∞ (BE ) with interpolation constant M only depending on c. This result allows us to conclude the following: Lemma 3.3. Let E be a complex Banach space and let ϕ : BE −→ BE be analytic such that ϕ(0) = 0 and ϕ (0) < 1. Suppose that there exist δ > 0 and ε > 0 such that 1 − ϕ(x) ≥ 1 + ε, for all x ∈ ϕ(BE ) such that x ≥ δ. (3.2) 1 − x Then, there exists a constant M ≥ 1 which depends only on ε, such that any finite iteration sequence {x0 , x1 , . . . , xN } satisfying x0 ∈ ϕ(BE ) and xN ≥ δ is an interpolating sequence for H ∞ (BE ) with interpolation constant not greater than M. Proof. Let (xk )N k=0 be a finite iteration sequence satisfying x0 ∈ ϕ(BE ) and xN ≥ δ. Recall that xk+1 = ϕ(xk ) for any 0 ≤ k ≤ N − 1 and consider the sequence {xN , xN −1 , . . . , x1 , x0 } . By inequality 3.2, we have that 1 − xk 1 1 − xk = ≤ for any 0 ≤ k ≤ N − 1. 1 − xk+1 1 − ϕ(xk ) 1+ε Therefore, the assumption in Proposition 3.2 is satisfied by the finite sequence {xN , xN −1 , . . . , x0 } (note the reversal of the order). Thus, this sequence is interpolating and its constant of interpolation depends only on ε. In addition, we will need the following lemmas. The first one is an improvement of Lemma 7.17 in [CM]. Lemma 3.4. Let E and F be Banach spaces. Let C : E ⊕ F −→ E ⊕ F be a linear operator which leaves F invariant and for which C|E : E −→ E ⊕ F is a compact operator. If the operator C has the matrix representation

X 0 C= (3.3) Y Z with respect to this decomposition, then σ(C) = σ(X) ∪ σ(Z).

218


Proof. Let λ ∈ / σ(C) and suppose that

Then,

X − λI11 Y

0 Z − λI22

R . V

T U

(C − λI)−1 =

T U

IEOT

R V

=

(3.4)

I11 0

0 I22

(3.5)

which implies that (X − λI11 )T = I11 . If λ = 0, since X : E −→ E is compact, the Fredholm alternative holds, so X − λI11 is surjective if and only if X − λI11 is injective; hence X − λI11 is invertible. If λ = 0, then I11 is a compact operator, so E is finite dimensional, and again X − λI11 is invertible. Thus in any case we obtain that R = 0. This gives that (Z − λI22 )V = I22 . Multiplying the opposite order gives that V (Z − λI22 ) = I22 , so the operator Z − λI22 is invertible and obtain σ(X) ∪ σ(Z) ⊂ σ(C). The converse inclusion is invertible, it is easy to see

X Y

proved as in Lemma 7.17 in [CM]: If X and Z are that −1

0 X −1 0 = , Z −Z −1 Y X −1 Z −1

so C is invertible.

Denote by Pn f the n-th term of the Taylor series at 0 of the analytic function f ∈ H ∞ (BE ). Set, for any m ≥ 1, ∞ Hm (BE ) = {f ∈ H ∞ (BE ) : Pn f = 0 for n = 0, 1, . . . , m − 1} . ∞ In other words, a function in H ∞ (BE ) belongs to Hm (BE ) if the first m − 1 terms of its Taylor series at 0 vanish.

Lemma 3.5. Let ϕ : BE −→ BE be an analytic map such that ϕ(0) = 0. Then Cϕ ∞ leaves invariant the space Hm (BE ) for any m ≥ 1. ∞ Proof. Let f ∈ Hm (BE ) and fix x ∈ BE . It is easy, from the Taylor series expansion of ϕ at 0, that the function g : D −→ C defined by g(λ) = ϕ(λx) satisfies g(λ) = λhx (λ) for a particular analytic function hx which depends on x and λ. ∞ (BE ). We have that Set f ∈ Hm Pn f (ϕ(λx)) = Pn f (λhx (λ)) = λn Pn f (hx (λ)) (f ◦ ϕ)(λx) = n≥m

n≥m

n≥m

and there is no non-null term of degree less than m in this series expansion. Therefore, if n Qn is the Taylor series of

f ◦ ϕ, there must be no non-null term of degree less than m in n Qn (λx) = n λn Qn (x). Thus Qn (x) = 0 for n = ∞ 0, 1, . . . , m − 1 and, therefore, Cϕ (f ) ∈ Hm (BE ).

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Next we prove the main result. The proof below is an evolution of the proof of Theorem 3.4 in [K], later improved by C. Cowen and B. MacCluer [CM] and then used by L. Zheng [Zh] and it is just an adaptation of Theorem 1.1 that is possible thanks to Theorem 3.2 and Lemma 3.3. Theorem 3.6. Let ϕ : BE −→ BE be an analytic map satisfying ϕ(0) = 0 and ϕ (0) < 1 such that ϕ(BE ) is a relatively compact subset of E. Suppose that ϕ satisfies the approaching condition and the following Julia-type estimate: for any 0 < δ < 1, there exists ε > 0 such that 1 − ϕ(x) ≥ 1+ε 1 − x

for all x ∈ ϕ(BE ) such that x ≥ δ.

(3.6)

Then, the spectrum of Cϕ coincides with the closed unit disk D. Proof. Since the spectrum σ(Cϕ ) is a closed subset of D, it is sufficient to prove that D \ {0} ⊂ σ(Cϕ ). Therefore, fix λ ∈ D \ {0}. Denoting by P (<m E) the subspace of polynomials of degree less than m, it ∞ is clear that H ∞ (BE ) is isomorphic to P (<m E) ⊕ Hm (BE ). If ϕ(0) = 0, then the matrix representation of Cϕ with respect to this decomposition has, according to Lemma 3.5, null upper right submatrix as required in Lemma 3.4. Moreover, for every m ∈ N, each norm bounded subset of P (m E) is relatively compact for the compact-open topology by Montel’s theorem. Since ϕ(BE ) is a compact set in E, we conclude that the linear operator Cϕ |P (<m E) : P (<m E) −→ H ∞ (BE ) is compact by Proposition 3 in [AGL]. Therefore the matrix representation of ∞ Cϕ with respect to the decomposition H ∞ (BE ) = P (<m E) ⊕ Hm (BE ), fulfils the assumptions of Lemma 3.4. So, if we let Cm denote the restriction of Cϕ to ∞ (BE ), it will be enough to prove that λ ∈ σ(Cm ) for some m ∈ N, to get Hm λ ∈ σ(Cϕ ) as we aim. Notice that λ ∈ σ(Cm ) if Cm − λI is not invertible and, for this, it is sufficient to show that (Cm −λI)∗ is not bounded from below, so we will prove this assertion. √ Fix δ so that 14 < δ < 1. Since 0 < δ < 1, it follows from inequality (1.1) that there exists 0 < c < 1 satisfying √ (3.7) ϕ(x) ≤ cx, for any x ≤ δ. We can suppose without loss of generality that c is close to 1, so we choose c such √ that c ≥ δ. In addition, we can assume that c = 1/(1 + ε) and get 1 − x ≤ c for any x ≥ δ, 1 − ϕ(x)

x ∈ ϕ(BE )

(3.8)

since we can take ε as close to 0 as we may need and still inequality (3.6) would remain true.

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IEOT

Let (xk )∞ k=0 be an iteration sequence such that x0 > δ. In view of inequality (1.1), there exists 0 < a < 1 such that ϕ(x) ≤ ax

for any x ≤ x0 ,

(3.9)

so we get xk = ϕk (x0 ) ≤ ak x0 and therefore the norms of the elements of the sequence {x0 , x1 , . . . , } decrease to 0. We denote by N = N (x0 ) the largest integer such that xN > δ. Since the approaching condition guarantees that ϕk (BE ) is not contained in the ball δBE for all k ≥ 1, we have that for any arbitrarily large N ∈ N, we can find x0 ∈ BE such that N = N (x0 ). Then, it turns out that (3.10) xN +1 ≤ cxN . √ Indeed, we √ have two possibilities: if xN ≤ δ, this is exactly (3.7); if otherwise xN > δ, inequality (3.10) is also satisfied since, if not, we would have that √ √ xN +1 > cxN ≥ δ δ = δ; a contradiction with the choice of N. Further, for any n ≥ N + 1, we also have that xn+1 ≤ cxn since xn ≤ xN , so we obtain by induction that xN +k ≤ ck xN

for any k ≥ 0.

∞ Notice that for f ∈ Hm (BE ), we have that |f (x)| ≤ ||f ||∞ ||x||m for all x ∈ BE by the Schwarz Lemma (see [M] for instance). Hence, if we choose m so large that cm < |λ|, we obtain for an arbitrary iteration sequence (xk )∞ k=0 ⊂ ϕ(BE ) that: ∞ ∞ ∞ |f (xk )| f ∞ xk m xN m cmk ≤ ≤ f . ∞ |λ|k+1 |λ|k+1 |λ|N +1 |λ|k k=N +1

k=N +1

k=1

∞ (BE ) according to Thus, we may define linear functionals L on Hm

L(f ) =

∞ f (xk ) , λk+1

∞ f ∈ Hm (BE ).

k=0

We claim that their norm can be as large as we want by choosing the appropriate iteration sequence. Firstly, we obtain an estimate for the tail of the series, m ∞ f (xk ) cm ∞ ≤ f ∞ xN , f ∈ Hm (BE ). (3.11) k+1 N +1 λ |λ| |λ| − cm k=N +1

Next we choose an m−homogeneous polynomial P satisfying P = 1 and |P (xN )| = xN m . By Lemma 3.3, there is an interpolation constant M = M (c) and g ∈ H ∞ (BE ) such that g ≤ M , g(xk ) = 0 for 0 ≤ k < N , and g(xN ) = 1. ∞ Then P · g ∈ Hm (BE ) satisfies P · g ≤ M , and using the estimate (3.11) for

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221

P · g, we obtain ∞ ∞ (P · g)(x ) (P · g)(x ) (P · g)(xk ) k N |L(P · g)| = + = λk+1 λN +1 λk+1 k=0 k=N +1 ∞ (P · g)(xN ) (P · g)(xk ) ||xN ||m ||xN ||m M cm − . ≥ − ≥ λN +1 λk+1 |λ|N +1 |λ|N +1 |λ| − cm k=N +1

Since

Mcm |λ|−cm

m

Mc → 0 when m → ∞, we can choose m so that |λ|−c < m Then, since |L(P · g)| ≤ M L and xN > δ > 1/4, we get

1 2.

1 xN m ≥ . (3.12) N +1 m 2|λ| 2 · 4 |λ|N +1 As we have mentioned above, we can form iteration sequences for which N is arbitrarily large, hence by (3.12) for which L is arbitrarily large, as we claimed. ∞ (BE ), Finally, observe that for f ∈ Hm M L ≥ |L(P · g)| ≥

((λI − Cm )∗ (L)) (f ) =λL(f ) − L(Cm (f )) = λL(f ) − L(f ◦ ϕ) =λ

∞ f (xk ) k=0

λk+1

−

∞ f (xk+1 ) k=0

λk+1

= f (x0 ).

Hence (λI − Cm )∗ (L) ≤ 1. This shows that (λI − Cm )∗ is not bounded from below as we wanted. Combining our results we obtain the following extension of Zheng’s result for non power-compact composition operators: Theorem 3.7. Let E = C0 (X) and consider an analytic map ϕ : BE → BE satisfying ϕ(0) = 0 and ϕ (0) < 1 such that ϕ(BE ) is a relatively compact subset of BE . Suppose that ϕ satisfies the approaching condition. Then, σ(Cϕ ) = D. Proof. Apply Theorem 2.2 to W = ϕ(BE ), which approaches SE compactly since ϕ(BE ) is relatively compact and does not lie strictly inside BE . Then, we find the Julia-type estimate necessary to apply Theorem 3.6 and conclude the result. The unit ball of any of the Banach spaces mentioned in the above theorem is homogeneous, that is, there is an analytic automorphism of the ball which maps any given point to the origin. Therefore the fixed point required in the assumptions can be any point in the unit ball. In particular, the above result is valid for c0 and Cn . Hence, we can complete the description of the spectra of composition operators Cϕ : H ∞ (Dn ) −→ H ∞ (Dn ): Corollary 3.8. Let ϕ : Dn −→ Dn an analytic map with a fixed point x0 such that ϕ (x0 ) < 1. If Cϕ : H ∞ (Dn ) −→ H ∞ (Dn ) is not power-compact, then σ(Cϕ ) = D. Acknowledgement. We thank Ted Gamelin and Mikael Lindstr¨ om for the fruitful conversations we had about the topic of this paper.

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IEOT

References R. M. Aron, P. Galindo and M. Lindstr¨ om, Compact homomorphisms between algebras of analytic functions, Studia Math. 123 (1997), 235–247. [AGL2] R. M. Aron, P. Galindo and M. Lindstr¨ om, Connected components in the space of composition operators on H ∞ functions of many variables, Integr. equ. oper. theory 45 (2003), 1–14. [CM] C. Cowen and B. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton (1995). [GGL] P. Galindo, T. W. Gamelin and M. Lindström, Spectra of composition operators on algebras of analytic functions on Banach spaces, Proc. Roy. Soc. Edinburgh A 139 (2009), 107–121. [GM] P. Galindo and A. Miralles, Interpolating sequences for bounded analytic functions, Proc. Amer. Math. Soc. 135 n. 10 (2007), 3225–3231. [H] K. Hoffman, Banach spaces of analytic functions, Dover Publications, New York (1962). [K] H. Kamowitz, The spectra of composition operators on H p , J. Funct. Anal. 18 (1975), 132–150. [M] J. Mujica, Complex Analysis in Banach Spaces. North-Holland Mathematics Studies 120, (1986). [S] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer 1993. [Zh] L. Zheng, The essential norm and spectra of composition operators on H ∞ , Pac. J. Math. 203 n. 2 (2002), 503–510. [Zu] K. Zhu, Operator Theory in Functions Spaces, Second edition. Mathematical Surveys and Monographs, 138. American Mathematical Society, Providence, RI, 2007. [AGL]

Pablo Galindo Departamento de An´ alisis Matem´ atico Universitat de València 46.100 Burjasot, Valencia Spain e-mail: [email protected] Alejandro Miralles Instituto de Matem´ atica Pura y Aplicada Universidad Politécnica de Valencia Edificio 8E, Cubo F, Cuarta Planta 46022 Valencia Spain e-mail: [email protected] Submitted: December 12, 2008.



Krein Orthogonal Entire Matrix Functions and Related Lyapunov Equations: A State Space Approach M. A. Kaashoek, L. Lerer and I. Margulis Abstract. For a class of entire matrix valued functions of exponential type new necessary and sufficient conditions are derived in order that these functions are Krein orthogonal functions. The conditions are stated in terms of certain operator Lyapunov equations. These equations arise by using infinite dimensional state space representations of the entire matrix functions involved. As a corollary, using a recent operator inertia theorem, we give a new proof of the Ellis-Gohberg-Lay theorem which relates the number of zeros of a Krein orthogonal function in the open upper half plane to the number of negative eigenvalues of the corresponding selfadjoint convolution operator. Mathematics Subject Classification (2000). Primary 45E10; Secondary 47A48, 47A62, 33C47, 47B35. Keywords. Krein orthogonal matrix function, entire matrix function, realization, Lyapunov operator equation, inertia, convolution integral operators on a finite interval.

1. Introduction Let K be the convolution integral operator acting on the space Ln1 [0, ω] given by ω (Kf )(t) = k(t − s)f (s)ds, 0 ≤ t ≤ ω. (1.1) 0

[−ω, ω] and is assumed to be hermitian. The kernel function k belongs to Ln×n 1 The latter means that k(t)∗ = k(−t) for each −ω ≤ t ≤ ω. An entire n × n matrix The research of the second author was supported by Israel Science Foundation (Grant No. 121/09).

224

Kaashoek, Lerer and Margulis

function

Φ(λ) = In +

ω

eiλs ϕ(s)∗ ds,

0

λ ∈ C,

IEOT

(1.2)

is called a Krein orthogonal function generated by the operator I − K whenever [0, ω] is a (the) solution of the equation the function ϕ ∈ Ln×n 1 (I − K)ϕ = k+ , or, equivalently,

ϕ(t) −

0

ω

where k+ = k|[0, ω] ,

k(t − s)ϕ(s) ds = k(t),

0 ≤ t ≤ ω.

(1.3)

(1.4)

The symbol I in (1.3) denotes the identity operator on Ln1 [0, ω]. In view of results in [14] (see also [6] or [19]) the solvability of the equation (1.3) is equivalent to invertibility of the operator I − K. Hence, the invertibility of I − K is embedded implicitly in the definition of Krein orthogonal function. In particular, if (1.3) is solvable, then the solution is unique. We also mention that in our definition of a Krein orthogonal function we deviate slightly from the standard notation which does not use the adjoint ∗ under the integral sign; the latter is done here just for convenience. For the scalar case the functions of type (1.2) have been introduced by M.G. Krein [23] in the mid fifties, in the connection with his work on canonical differential systems and related problems. M.G. Krein discovered that in case I − K is strictly positive the functions (1.2) exhibit properties with respect to the real line that are analogous to the properties of Szeg¨ o orthogonal polynomial with respect to the unit circle. In subsequent works of M.G. Krein and H. Langer [24] in 1985 the general case of a non-definite operator I − K was considered and two remarkable problems were solved. The first problem (called the direct problem) is to relate the number of zeros of Φ(λ) in the upper half plane to the spectrum of I − K. The second problem (called the inverse problem) is to find criteria for an entire function of the form Φ(λ) such that it is an orthogonal function generated by I − K for some K as in (1.1). The matrix-valued extension of the direct problem has been obtained in [7], and later in [5]. The inverse problem for Krein orthogonal matrix functions has been solved recently in various ways, namely in [15] using symmetric factorization and recent results about the continuous analogue of the resultant based on [12], and in [18] using entire matrix function equations based on [16] and [17]. To explain the main result of the present paper we review now the solutions of the inverse and direct problem given in [20] for the discrete case. In [20] these solutions are inspired by some relations found in [13] and [25], and are based on the use of the inertia theorems from [3] and [30] for matrix Stein equations. Recall that an n × n matrix polynomial X(λ) = x0 + λx1 + · · · + λk xk is said to be an orthogonal matrix polynomial if there exists a hermitian (k + 1) × (k + 1) block

Vol. 65 (2009) Krein Orthogonal Matrix Functions and Lyapunov Equations

Toeplitz matrix with n × n matrix entries ti−j such that      t1 ··· tk t0 x0 In t−1      t · · · t x 0 k−1   1   0 =  . .  ..    .. . . . .. ..   ..   ..   . . t0 0 t−k t−k+1 · · · xk

225

(1.5)

The inverse problem for matrix polynomials has been solved in many different ways in [20]. For the scalar case see [23] and the comment in Section 3.6 of [9], which does not appear in the English translation [10] of [9]. Here we only mention the following result from [20] which concerns both the direct and the inverse problem (see, also, Theorem 3.1 in [27]). Theorem 1.1. Let X(λ) = x0 + λx1 + · · · + λk xk be an n × n matrix polynomial, with invertible x0 , and put    −1  x0 0 In −x1 x−1 0   −x1 x−1  .. .. In 0    . . ˆX =  KX =  ,  , K .. . ..    · In  . −1 −xk x−1 0 −x x I k 0 n 0 where empty spots denote zero block entries. Then X(λ) is an orthogonal matrix polynomial if and only if x0 is hermitian and the Stein equation   In 0   ∗ S − (KX )∗ SKX = Ex−1 (1.6) E =  . , 0 E ,  ..  0 has a hermitian solution S. More precisely, the following holds. If S is hermitian solution of (1.6), then x0 0 ∗ ˆ ˆX T =K K (1.7) X 0 S k such that (1.5) holds. Conis a hermitian block Toeplitz matrix T = [ti−j ]i,j=0 versely, if X(λ) is an orthogonal matrix polynomial , i.e., (1.5) holds for some k−1 k hermitian block Toeplitz matrix T = [ti−j ]i,j=0 , then S = [ti−j ]i,j=0 is a hermitian solution of (1.6), and substitution of this S into (1.7) yields the given block Toeplitz matrix T .

The main aim of the present paper is to analyze Krein orthogonal functions using a continuous analogue of the above theorem. As expected this requires to replace the matrix Stein equation in Theorem 1.1 by an operator Lyapunov equation. The following theorem is our main result.

226


IEOT

Theorem 1.2. Fix ϕ in Ln×n [0, ω], and let Φ(λ) be the corresponding entire matrix 1 function defined by (1.2). Assume Φ(0) is non-singular, and let V◦× be the operator on Ln2 [0, ω] given by t f (s) ds (V◦× f )(t) = i 0 (1.8)

s ω ϕ(s)∗ i f (r) dr ds, f ∈ Ln2 [0, ω]. − Φ(0)−1 0

0

Furthermore, let τ◦ be the canonical embedding of Cn into the space Ln2 [0, ω], that is, (τ◦ u)(t) = u for each u ∈ Cn and each 0 ≤ t ≤ ω. Then Φ is a Krein orthogonal function if and only if the Lyapunov equation V◦× S◦ − S◦ (V◦× )∗ = iτ◦ Φ(0)−1 Φ(0)−∗ τ◦∗ ,

(1.9)

Ln2 [0, ω],

has an operator solution S◦ , acting on which is selfadjoint and has the additional property that S◦ extends to a bounded linear operator S on Ln1 [0, ω]. More precisly, the following holds. If the selfadjoint operator S◦ on Ln2 [0, ω] is a solution of (1.9) and S◦ extends to a bounded linear operator S on Ln1 [0, ω], then a hermitian k in Ln×n [−ω, ω] such that (1.4) holds is given by 1 (Sϕ)(t) when 0 < t ≤ ω, k(t) = (1.10)

∗ (Sϕ)(−t) when −ω ≤ t > 0. Moreover, in that case S = I − K, where K is the convolution operator acting on Ln1 [0, ω] given by (1.1) with k defined by (1.10). Conversely, let k in Ln×n [−ω, ω] 1 be hermitian and such that (1.4) holds, and let K be the operator defined by (1.1) using the given k. Then S = I − K is a bounded operator on Ln1 [0, ω] and its restriction to Ln2 [0, ω] defines a selfadjoint operator S◦ which is a solution of the Lyaponov equation (1.9). The condition “Φ(0) is non-singular” appearing in the beginning of Theorem 1.2 is needed in the definition of the operator V◦× in (1.8). However, as is well-known (see Chapter 8 in [6]), this condition is always fulfilled if Φ is a Krein orthogonal function. On the other hand, if Φ is given by (1.2) without any further condition on ϕ ∈ Ln×n [0, ω], then clearly it may happen that Φ(0) is singular. 1 To understand better Theorem 1.2 and how it can be used, let us first explain the origin of the operator V◦× in (1.8). For this purpose, we recall (see [15]) that Φ(λ) admits the representation Φ(λ) = Φ(0) + λΓ◦ (I − λV◦ )−1 τ◦ . n

Here τ◦ is the canonical embedding of C into and

0

(1.11)

as defined in Theorem 1.2,

t (V◦ f )(t) = i f (s) ds, 0 ω Γ◦ f = ϕ(s)∗ (V◦ f )(s)ds.

V◦ : Ln2 [0, ω] → Ln2 [0, ω], Γ◦ : Ln2 [0, ω] → Cn ,

Ln2 [0, ω]

0 ≤ t ≤ ω, (1.12) (1.13)


227

Notice that τ◦ and V◦ do not involve Φ; all the information about Φ is contained in the definition of Γ◦ . One refers to the right hand side of (1.11) as a realization or state space representation of Φ(λ); see, e.g., [1] (in [12] the term co-realization is used). Now assume that Φ(0) is non-singular. Then (see Theorem 2.1 in [1]) Φ(λ)−1 = Φ(0)−1 − λΦ(0)−1 Γ◦ (I − λV◦× )−1 τ◦ Φ(0)−1 .

(1.14)

Here V◦× = V◦ − τ◦ Φ(0)−1 Γ◦ , that is, V◦× is precisely the operator on Ln2 [0, ω] given by (1.8). One refers to V◦× as the associate main operator of the realization (1.11), the operator V◦ being the main operator. Note that V◦× is a finite rank perturbation of V◦ , which is a Volterra operator, and hence the non-zero part of the spectrum of V◦× consists of eigenvalues of finite type only (see Chapter 2 of [11]). From the representation (1.14) it follows that λ0 = 0 is a zero of det Φ(λ) is in the spectrum of V◦× (or, equivalently, is an eigenvalue of if and only if λ−1 0 × finite type of V◦ ), and in that case the multiplicity of λ0 as a zero of det Φ(λ) is precisely equal to the algebraic multiplicity of λ−1 0 as an eigenvalue of finite type of V◦× (see [15]). In combination with recent results on inertia of infinite dimensional Lyapunov equations obtained in [28] the connection between the zeros of det Φ(λ) and the non-zero eigenvalues of the operator V◦× (mentioned in the preceding paragraph) turn out be very useful. To illustrate this we present a new proof of the R.L. Ellis– I. Gohberg–D.C. Lay [7] result on the direct problem (see also Chapter 8 in [6]). This result reads as follows. Theorem 1.3. [7] Let Φ in (1.2) be a Krein orthogonal matrix function generated by the convolution integral operator I − K, where K is given by (1.1). Then Φ is non-singular on the real line and the number of zeros of det Φ in the upper half plane, counting multiplicities, is equal to the number of negative eigenvalues of the operator I − K. Let us return to Theorem 1.1. First we mention that an n × n matrix polynomial X(λ) = x0 + λx1 + · · · + λk xk admits a state space representation of a type similar to the one in (1.11), namely X(λ) = x0 + λγ(I − λα)−1 β, where I is the nk × nk  0 In   α=  0 0

identity matrix and    x1  ..  . .  , β =  ..  , In  xk ··· 0

γ = In

0

Hence, if x0 is invertible, then −1 × −1 βx−1 X(λ)−1 = x−1 0 − λx0 γ(Ink − λα ) 0 ,

···

0 .

228

where




−x1 x−1 0  ..  −1 × . α = α − βx0 γ =   · −xk x−1 0

IEOT



In ..

.

  . In  0

Thus the associate main operator α× is presicely the operator KX on Cnk appearing in Theorem 1.1. But then we can apply to the finite dimensional Stein equation (1.6) the Chen-Wimmer inertia theorem [3], [30] (see Theorem 13.2.4 in [26]) to conclude that det X(λ) has no zero on the unit circle and that the number of zeros of det X(λ) inside the unit cicle is equal to the number of negative eigenvalues k−1 of (1.6). In our proof of Theorem 1.3 of the hermitian solution S = [ti−j ]i,j=0 we follow the same line of reasoning, with V◦× in place of KX , with the operator Lyapunov equation (1.9) in place of the finite dimensional Stein equation (1.6), and with the references to the papers [3], [30] being replaced by a reference to [28]. For another application of inertia theory to orthogonal matrix functions, also in a finite matrix setting, we refer to [21]. The paper consists of four sections not counting the present introduction. Section 2 has a preliminary character. In this section we introduce notation that is used throughout the paper and review some known results about convolution integral operator on a finite interval. Section 3 presents a number of auxiliary results which will play an essential role in the two final sections. The proof of Theorem 1.2 is given in Section 4. In Section 5 we use Theorem 1.2 and a recent inertia result from [28], to give a new proof of Theorem 1.3.

2. Preliminaries on convolution operators In this section we present some preliminaries on convolution operators on a finite interval. We begin we some additional notation. Throughout we make use of the following operators: t f (s) ds, 0 ≤ t ≤ ω, (2.1) V : Ln1 [0, ω] → Ln1 [0, ω], (V f )t) = i 0 ω f (s) ds, 0 ≤ t ≤ ω, (2.2) W : Ln1 [0, ω] → Ln1 [0, ω], (W f )t) = −i τ : C n → Ln1 [0, ω], π : Ln1 [0, ω] → C n ,

t

(τ u)(t) = u, 0 ≤ t ≤ ω, ω (πf )(t) = f (s) ds.

(2.3) (2.4)

0

We shall consider the above operators also with Ln2 [0, ω] in place of Ln1 [0, ω]. In that case an additional subscript ◦ is added. The space Ln2 [0, ω] is densely contained in Ln1 [0, ω] and is invariant under V and W . Thus V◦ and W◦ are just the restrictions of the operators V and W to Ln2 [0, ω]. Notice that τ◦∗ = π◦ . We also write K◦ for the restriction of the operator K in (1.1) to Ln2 [0, ω]. Obviously,


229

K◦ will be a selfadjoint operator whenever k is hermitian. Note that the notation used in Theorem 1.2 is consistent with the notation introduced in the present paragraph. The following two propositions are well-known and can be found in Section 1 of [22] or in Chapter 1 of [29] and in [8]. Proposition 2.1. Let K be the convolution integral operator given by (1.1), with k ∈ Ln×n [−ω, ω] being hermitian. Then the operator I − K satisfies the equation 1 ω

V (I − K)f − (I − K)W f (t) = i ( (t) + (s)∗ ) f (s) ds, 0 ≤ t ≤ ω. (2.5) 0

Here f is an arbitrary function in Ln1 [0, ω] and the n × n matrix function is given by t 1 k(s) ds + In , 0 ≤ t ≤ ω. (2.6) (t) = − 2 0 In particular, for each f ∈ Ln1 [0, ω] the function (I−K)W f is absolutely continuous on [0, ω] and its derivative is in Ln1 [0, ω]. Proposition 2.2. Let S◦ be a selfadjoint operator on Ln2 [0, ω] satisfying the equation ω

m(t) + m(s)∗ f (s) ds, 0 ≤ t ≤ ω, (2.7) (V◦ S◦ f − S◦ V◦∗ f )(t) = i 0

where the n × n matrix function m is absolutely continuous on [0, ω] and its derivative is in Ln1 [0, ω]. If, in addition, m(+0) + m(+0)∗ = In , then S◦ = I − K◦ , where K◦ is a convolution integral operator acting on Ln2 [0, ω], ω (K◦ f )(t) = k(t − s)f (s) ds, 0 ≤ t ≤ ω, 0

[−ω, ω], is hermitian, and is given by the kernel function k belongs to Ln×n 1 k(t) = −

d m(t), dt

when 0 < t ≤ ω.

(2.8)

3. Auxiliary results involving Krein orthogonal functions Throughout this section ϕ belongs to Ln×n [0, ω] and k is a hermitian function in 1 [−ω, ω]. With ϕ we associate the n × n entire matrix function Φ given by Ln×n 1 (1.2), and with k we associate the convolution integral operator K on Ln1 [0, ω] given by (1.1). This section has a preparatory character. We present a proposition providing necessary and sufficient conditions in order that Φ is a Krein orthogonal function generated by I − K. As a corollary we prove the well-known fact (see Chapter 8 in [6]) that Φ(0) is non-singular whenever Φ is a Krein orthogonal function.

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We need a number of auxiliary operators. The first two only involve the kernel function k. Put t B : Cn → Ln1 [0, ω], (Bu)(t) = i k(s)u ds, (3.1) 0 ω t ∗ i k(s) ds f (t) dt. (3.2) B : Ln1 [0, ω] → Cn , B f = 0

0

Again, if the operators in (3.1) and (3.2) are considered with Ln2 [0, ω] in place of Ln1 [0, ω], then an additional subscript ◦ is added. Thus B◦ is the operator B viewed as a map into Ln2 [0, ω], and B◦ is just the restriction of B to Ln2 [0, ω]. In that case (B◦ )∗ = B◦ . Using these operators and the operators τ and π defined by (2.3) and (2.4), respectively, (2.5) can be rewritten as V (I − K) − (I − K)W = iτ π − Bπ + τ B . Next we consider the operators

ω

C : Ln∞ [0, ω] → Cn ,

Cf =

C : Cn → Ln1 [0, ω],

C u = ϕ(·)u,

0

ϕ(t)∗ f (t) dt,

f ∈ Ln∞ [0, ω],

u ∈ Cn .

(3.3)

(3.4) (3.5)

Note that these two operators only involve the function ϕ. Since ϕ is an L1 -function and not necessarily an L2 -function, in general, the operator C does not extend to an operator on Ln2 [0, ω]. Let τ and B be the operators defined by (2.3) and (3.1), respectively. Since the ranges of these operators are contained in Ln∞ [0, ω] the products Cτ and CB are well defined. Note that both Cτ and CB are just n×n matrices. These matrices will play an important role in the sequel. We have Φ(0) = In + Cτ,

Φ(0)∗ = In + πC ,

(CB)∗ = B C .

(3.6)

The first two identities follow directly from the definition of Φ in (1.2). The third identity follows directly from the following two identities.

t ω CBu = ϕ(t)∗ i k(s)u ds dt, u ∈ Cn , BC =

0

0

0

ω

∗

t ∗ k(s) u ds ϕ(t) dt, i 0

u ∈ Cn .

The next proposition is the main result of this section. Proposition 3.1. Let B and C be given by (3.2) and (3.4), respectively. Then Φ is the Krein orthogonal function generated by I − K if and only if C(I − K)W f = B f,

f ∈ Ln1 [0, ω].

(3.7)

Proof. Take f ∈ Ln1 [0, ω]. Note that W f ∈ Ln∞ [0, ω]. Since K leaves the space Ln∞ [0, ω] invariant, KW f also belongs to Ln∞ [0, ω]. Hence both CW f and CKW f


231

are well defined. In particular, the left hand side of (3.7) is well defined. We have ω

ω CKW f = ϕ(r)∗ k(r − t)(W f )(t) dt dr 0 ω ω 0

ϕ(r)∗ k(r − t) dr (W f )(t) dt = 0 ω 0 ω ∗

= k(r − t)∗ ϕ(r) dr (W f )(t) dt 0

0

Now use that k is hermitian. Since k(r − t)∗ = k(t − r), it follows that ω ω

∗ CKW f = k(t − r)ϕ(r) dr (W f )(t) dt 0 ω 0

∗ ϕ(t) − (I − K)ϕ)(t) (W f )(t) dt = 0 ω

= CW f − (I − K)ϕ (t)∗ (W f )(t) dt. 0

Hence the vector C(I − K)W f is given by ω

(I − K)ϕ (t)∗ (W f )(t) dt. C(I − K)W f = 0

Next note that Bf =

0

ω

t ∗ i k(s) ds f (t) dt = 0

ω

k(s)∗ (W f )(s) ds.

(3.8)

0

By comparing the above formulas for C(I − K)W f and B f , we see that the equality C(I − K)W f = B f holds if and only if ω ∗ (I − K)ϕ (t) − k(t) (W f )(t) dt = 0. (3.9) 0

Since the range of W is dense in Ln1 [0, ω], we conclude that (3.9) holds for each f ∈ Ln1 [0, ω] if and only if (I − K)ϕ = k+ , where k+ = k[0, ω] . In other words, (3.7) holds if and only if Φ is the Krein orthogonal function generated by I − K. Corollary 3.2. Assume Φ is the Krein orthogonal function generated by I − K, that is, ϕ satisfies the equation (1.3). Then Φ(0) is non-singular. Furthermore,

(3.10) Φ(0)−1 CB − B C Φ(0)−∗ = i In − Φ(0)−1 In − Φ(0)−∗ . Proof. Since ϕ satisfies the integral equation (1.4), we have V (I − K)C = B

and C(I − K)W = B .

(3.11)

Indeed, the first identity in (3.11) is just the identity (I − K)ϕ = k+ written in operator form, while the second identity in (3.11) follows from the previous proposition. Using (3.11) we see that

(3.12) C V (I − K) − (I − K)W C = CB − B C .

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On the other hand, by (3.3),

C V (I − K) − (I − K)W C = iCτ πC − CBπC + Cτ B C = i (Φ(0) − In ) (Φ(0)∗ − In ) ∗

(3.13)

− CBΦ(0) + CB + Φ(0)B C − B C . By subtracting the identities (3.12) and (3.13) we see that i (Φ(0) − In ) (Φ(0)∗ − In ) = CBΦ(0)∗ − Φ(0)B C . n

(3.14)

∗

Now, let u be a vector in C such that u Φ(0) is zero. In order to prove that Φ(0) is non-singular, it suffices to show that u = 0. The fact that u∗ Φ(0) is zero implies that Φ(0)∗ u is zero too. Thus, by multiplying (3.14) from the left by u∗ and from the right by u, we obtain iu∗ u = 0, and hence u = 0. Therefore Φ(0) is non-singular. Finally, since the matrix Φ(0) is non-singular, we can multiply the identity (3.14) from the left by Φ(0)−1 and from the right by Φ(0)−∗ . This yields the identity (3.10). In the remaining part of this section we assume Φ(0) to be non-singular. First we present two identities relating the operator Γ◦ given by (1.13) to the operators C and C in (3.4) and (3.5), respectively. Note that the definition of Γ◦ requires Φ(0) to be non-singular. The following holds: (f ∈ Ln2 [0, ω]),

Γ◦ f = CV f

Γ∗◦ u = W C u

(u ∈ Cn ).

(3.15)

The first identity in (3.15) follows directly from (3.4) and the definition of Γ◦ in (1.13). To get the second identity in (3.15), observe that ω s ∗ ∗ Γ◦ f, u = u ϕ(s) i f (t) dt ds ω0 0 ω ∗ ∗ −i ϕ(s) ds f (t) dt = f, W C u, =u 0

t

which yields the second identity in (3.15). We conclude with two lemmas that will be useful later. In both lemmas Φ(0) is assumed to be non-singular. Recall that the latter assumption is fulfilled whenever Φ is a Krein orthogonal function (cf., the preceding corollary). Lemma 3.3. Let C and C be given by (3.4) and (3.5), respectively, and assume Φ(0) to be non-singular. Then we have the following intertwining relations: Φ(0)−1 Cg = C(ILn∞ [0, ω] + τ◦◦ C)−1 g,

g ∈ Ln∞ [0, ω],

(3.16)

u ∈ Cn .

(3.17)

τ◦◦ Φ(0)−1 u = (ILn∞ [0, ω] + τ◦◦ C)−1 τ◦◦ u, n

Here τ◦◦ is the operator τ viewed as a map from C into

−∗

C Φ(0)

−1

u = (ILn1 [0, ω] + C π)

C,

Φ(0)−∗ πg = π(ILn1 [0, ω] + C π)−1 g,

Ln∞ [0, n

ω]. Furthermore,

u∈C , g ∈ Ln1 [0, ω].

(3.18) (3.19)


233

Proof. Using the definition of τ◦◦ , the first identity in (3.6) can be rewritten as Φ(0) = In + Cτ◦◦ . Since Φ(0) is non-singular, In + Cτ◦◦ is invertible, and hence the same holds true for ILn∞ [0, ω] + τ◦◦ C (see, e.g., formula (3) on page 38 of [11]). Standard calculations then yield Φ(0)−1 = In − C(ILn∞ [0, ω] + τ◦◦ C)−1 τ◦◦ . From this identity it is straightforward to derive (3.16) and (3.17). In a similar way, using the second identity in (3.6), one shows that the operator ILn1 [0, ω] + C π is invertible and that Φ(0)−∗ = In − π(ILn1 [0, ω] + C π)−1 C . Again a straightforward calculation then yields (3.18) and (3.19).

In the final lemma of this section we return to the operator V◦× given by (1.8) in Theorem 1.2. Lemma 3.4. Let C and C be given by (3.4) and (3.5), respectively, and assume Φ(0) to be non-singular. Define

×

V × : Ln1 [0, ω] → Ln1 [0, ω],

V × = V − τ Φ(0)−1 CV,

(3.20)

W × : Ln1 [0, ω] → Ln1 [0, ω],

W × = W − W C Φ(0)−∗ π.

(3.21)

×

Then V and W are restrictions of V × and Furthermore,

well-defined bounded linear operators on Ln1 [0, ω], and the W × to Ln2 [0, ω] coincide with V◦× and (V◦× )∗ , respectively.

V × = (ILn∞ [0, ω] + τ◦◦ C)−1 V,

W × = W (ILn1 [0, ω] + C π).

(3.22)

Here, as in the previous lemma, τ◦◦ is the operator τ viewed as a map from Cn into Ln∞ [0, ω]. Proof. Consider the operators ∆ = τ Φ(0)−1 CV Ln1 [0,

and ∆ = W C Φ(0)−∗ π.

(3.23)

Ln∞ [0,

Notice that V maps ω] into ω] , and thus the product CV is welldefined and acts as an operator from Ln1 [0, ω] into Cn . It follows that ∆ is a well-defined operator on Ln1 [0, ω]. Since V × = V − ∆, the same holds true for V × . From the definiton of C in (3.5) we see that the product W C is well-defined and maps Cn into Ln1 [0, ω]. Thus ∆ is a well-defined operator on Ln1 [0, ω], and hence the same holds true for W × = W − ∆ . Next note that V◦× = V◦ − τ◦ Φ(0)−1 Γ◦ = V◦ − τ◦ Φ(0)−1 CV◦ .

(3.24)

It follows that V◦× is the restriction of V × to Ln2 [0, ω]. Using the second identity in (3.15) and the first identity in (3.24) we see that (V◦× )∗ = W◦ − W C Φ(0)−∗ π◦ , which implies that (V◦× )∗ is the restriction of W × to Ln2 [0, ω].

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Take f in Ln1 [0, ω]. From (3.16) we see that ∆f = τ◦◦ C(ILn∞ [0, ω] + τ◦◦ C)−1 V f = (τ◦◦ C + ILn∞ [0, ω] − ILn∞ [0, ω] )(ILn∞ [0, ω] + τ◦◦ C)−1 V f = V f − (ILn∞ [0, ω] + τ◦◦ C)−1 V f. Thus the first identity in (3.22) holds. Similarly , using (3.18), we get ∆ f = W (ILn1 [0, ω] + C π)−1 C πf = W (ILn1 [0, ω] + C π)−1 (C π + ILn1 [0, ω] − ILn1 [0, ω] ) = W f − W (ILn1 [0, ω] + C π)−1 f.

This yields the second identity in (3.22).

4. Proof of Theorem 1.2 In this section we prove Theorem 1.2. The proof will be split into two parts. Throughout this section ϕ is a given function in Ln×n [0, ω], and Φ(λ) is the cor1 responding entire matrix function defined by (1.2). In the first part Φ is a Krein orthogonal function, and we show that the restriction to Ln2 [0, ω] of the convolution operator I − K corresponding to Φ is a selfadjoint solution of the Lyapunov equation (1.9). The second part concerns the reverse direction. Part 1. Let Φ be a Krein orthogonal function. Recall (see Corollary 3.2 ) that this implies that Φ(0) is non-singular. Thus the operator V◦× in (1.8) is well-defined. Since Φ is a Krein orthogonal function, there exists a hermitian n × n matrix [−ω, ω] such that function k ∈ Ln×n 1 ϕ(t) −

0

ω

k(t − s)ϕ(s)ds = k(t),

0 ≤ t ≤ ω.

Let K be the integral operator on Ln1 [0, ω] corresponding to k as in (1.1), and let K◦ be the restriction of K to Ln2 [0, ω]. Clearly, S◦ := I◦ − K◦ is selfadjoint. We have to prove that S◦ is a solution of the Lyapunov equation (1.9). Let V × and W × be the operators on Ln1 [0, ω] defined by (3.20) and (3.21), respectively. From Lemma 3.4 we know that the operator V◦× defined by (1.8) is the restriction of V × to Ln2 [0, ω]. Furthermore, (V◦× )∗ is the restriction of W × to Ln2 [0, ω]. But then we can use the two identities in (3.22) to show that for each f in Ln2 [0, ω] we have V◦× f = (ILn∞ [0, ω] + τ◦◦ C)−1 V f,

(V◦× )∗ f = W (ILn1 [0, ω] + C π)−1 f.


235

Therefore, again with f ∈ Ln2 [0, ω], we obtain V◦× (I − K◦ )f − (I − K◦ )(V◦× )∗ f = (ILn∞ [0, ω] + τ◦◦ C)−1 V (I − K)f − (I − K)W (ILn1 [0, ω] + C π)−1 f = (ILn∞ [0, ω] + τ◦◦ C)−1 [V (I − K)(ILn1 [0, ω] + C π) − (ILn∞ [0, ω] + τ◦◦ C)(I − K)W ](ILn1 [0, ω] + C π)−1 f. Note that all products are well-defined because (I − K)W maps into Ln∞ [0, ω]. From (3.3) and the identities in (3.11) we get V (I − K)(ILn1 [0, ω] + C π) − (ILn∞ [0, ω] + τ◦◦ C)(I − K)W = iτ π − Bπ + τ B + V (I − K)C π − τ◦◦ C(I − K)W = iτ π = iτ◦◦ π.

Thus, using the identities (3.17) and (3.19), we have V◦× (I − K◦ )f − (I − K◦ )(V◦× )∗ f = (ILn∞ [0, ω] + τ◦◦ C)−1 iτ◦◦ π(ILn1 [0, ω] + C π)−1 f = iτ Φ(0)−1 Φ(0)−∗ πf. Thus S◦ = I◦ − K◦ is a selfadjoint solution of the Lyapunov equation (1.9). Part 2. In this part we assume that Φ(0) is non-singular (so that the operator V◦× in (1.12) is well-defined), and we assume that there exists a selfadjoint operator S◦ on Ln2 [0, ω] such that S◦ is a solution of the Lyapunov equation (1.9) . Furthermore, we assume that S◦ extends to a bounded linear operator S on Ln1 [0, ω]. We have to prove that Φ is a Krein orthogonal function and to produce the corresponding kernel function k. As in the previous part, let V × and W × be the operators defined by the identities (3.20) and (3.21). We know from Lemma 3.4 that the operators V◦× and (V◦× )∗ , appearing in the Lyapunov equation (1.9), are the restrictions to Ln2 [0, ω] of the operators V × and W × , respectively. By assumption, S◦ extends to a bounded linear operator on Ln1 [0, ω], which is denoted by S. Thus all operators appearing in (1.9) extend to bounded linear operators on Ln1 [0, ω], and hence (1.9) can be rewritten in the following equivalent form V × S − SW × = iτ Φ(0)−1 Φ(0)−∗ π.

(4.1)

Using the two identities in (3.22), we see that the preceding equation (4.1) can be rewritten as (ILn∞ [0,ω] + τ◦◦ C)−1 V S − SW (ILn1 [0,ω] + C π)−1 = iτ Φ(0)−1 Φ(0)−∗ π. On the other hand, using (3.17) and (3.19), we see that iτ Φ(0)−1 Φ(0)−∗ π = i(ILn∞ [0,ω] + τ◦◦ C)−1 τ π(ILn1 [0,ω] + C π)−1 .

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Summarizing we have (ILn∞ [0,ω] + τ◦◦ C)−1 V S − SW (ILn1 [0,ω] + C π)−1 = i(ILn∞ [0,ω] + τ◦◦ C)−1 τ π(ILn1 [0,ω] + C π)−1 .

(4.2)

From the above equality we see that SW maps into Ln∞ [0, ω]. In particular, CSW is well-defined. Thus we can multiply both sides of (4.2) from the left by ILn∞ [0,ω] + τ◦◦ C and from the right by ILn1 [0,ω] + C π. This yields V S − SW = iτ π − V SC π + τ CSW.

(4.3)

Put α = V Sϕ. Note that α is absolutely continuous on [0, ω] and for each u ∈ Cn we have V SC u = α(·)u. We claim that ω CSW f = α(t)∗ f (t) dt, f ∈ Ln1 [0, ω]. (4.4) 0

To see this, fix f ∈ Ln1 [0, ω] and let ϕ1 , ϕ2 , . . . be a sequence of continuous n × n matrix functions on [0, ω] such that limi→∞ ϕi = ϕ in the norm of Ln×n [0, ω]. 1 Put αi = V Sϕi , i = 1, 2, . . .. Note that the functions α1 , α2 , . . . are (absolutely) continuous on [0, ω], and limi→∞ αi = α in the supremum norm. Since SW f ∈ Ln∞ [0, ω], we have ω CSW f = ϕ(s)∗ (SW f )(s) ds 0 ω ϕi (s)∗ (SW f )(s) ds = lim SW f, ϕi , = lim i→∞

i→∞

0

Ln2 [0, ω].

Since W f belongs to Ln2 [0, ω], we have where ·, · is the inner product on SW f = S◦ W f . Thus, using the fact that S◦ is selfadjoint, we obtain CSW f = lim S◦ W f, ϕi = lim W f, S◦ ϕi = lim f, V◦ S◦ ϕi i→∞ i→∞ i→∞ ω ω ∗ = lim f, αi = lim αi (t) f (t) dt = α(t)∗ f (t) dt, i→∞

i→∞

0

0

because limi→∞ αi = α in the supremum norm. Formula (4.4) is now proved. We conclude that (4.3) can be rewritten as ω f (s) ds (V Sf − SW f )(t) = i 0 ω (4.5) ω f (s) ds + α(s)∗ f (s) ds, f ∈ Ln1 [0, ω]. − α(t) 0

0

Finally, put 1 1 m(t) = iα(t) + In = i(V Sϕ)(t) + In , 0 ≤ t ≤ ω. (4.6) 2 2 Restricting the identity (4.5) to Ln2 [0, ω] and using m given by (4.6) yields ω

∗ (V◦ S◦ f − S◦ V◦ f )(t) = i m(t) + m(s)∗ f (s) ds, 0 ≤ t ≤ ω. (4.7) 0


237

Note that m is absolutely continuous on [0, ω], because α has this property, and d m(t) = −(Sϕ)(t) a.e. on 0 ≤ t ≤ ω. (4.8) dt Trivially, m(+0) + m(+0)∗ = In . A straightforward application of Proposition 2.2 now shows that S◦ = I −K◦ , where K◦ is the restriction to Ln2 [0, ω] of the operator ω k(t − s)f (s) ds, 0 ≤ t ≤ ω. (Kf )(t) = 0

[−ω, ω], is hermitian, and Moreover the kernel function k belongs to Ln×n 1 d m(t) = (Sϕ)(t), 0 ≤ t ≤ ω. dt In particular, k|[0, ω] = (I − K)ϕ. This shows that Φ is the Krein orthogonal function generated by I − K, and completes the proof of Theorem 1.2. k(t) = −

5. Proof of Theorem 1.3 Let ϕ be in Ln×n [0, ω], and let Φ be the corresponding entire matrix function given 1 by (1.2). Throughout we assume that Φ is a Krein orthogonal matrix function generated by the convolution integral operator I − K, where K is given by (1.1), with the kernel function k ∈ Ln×n [−ω, ω] being hermitian. Our aim is to prove 1 Theorem 1.3. To prove Theorem 1.3 we employ Theorem 1.2 and a recent inertia result proved in [28]. We already know (see Corollary 3.2) that Φ(0) is non-singular. Hence the operator V◦× given by (1.8) is a well-defined operator on Ln2 [0, ω], and we can apply Theorem 1.2 to show that I◦ −K◦ , the restriction of I −K to Ln2 [0, ω], is selfadjoint and satisfies the Lyapunov equation V◦× (I◦ − K◦ ) − (I◦ − K◦ )(V◦× )∗ = iτ◦ Φ(0)−1 Φ(0)−∗ τ◦∗ .

(5.9)

The following proposition summarizes the spectral properties of V◦× , in relation to Φ, as mentioned earlier in the second paragraph after the proof of Theorem 1.2. Proposition 5.1. The non-zero part of the spectrum of V◦× consists of eigenvalues of finite type only. Moreover, λ0 = 0 is a zero of det Φ(λ) if and only if λ−1 0 is an eigenvalue of finite type of V◦× , and in that case the multiplicity of λ0 as a zero of det Φ(λ) is precisely equal to the algebraic multiplicity of λ−1 0 as an eigenvalue of finite type of V◦× . We split the remaining part of this section into two parts. The first part deals with Lyapunov equations and reviews a recent inertia theorem from [28]. The actual proof of Theorem 1.3 is given in the second part. Part 1. In this part A is an operator on a Hilbert space H, and Λ is an operator from an auxiliary Hilbert space H into F . Consider the Lyapunov equation A∗ X + XA = Λ∗ Λ.

(5.10)

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Here the unknown X is an operator on H, and we are particulary interested in solutions X that are selfadjoint. Note the equation (5.10) reduces to our Lyapunov equation (5.9) by taking H = Ln2 [0, ω], F = Cn , A = (−iV◦× )∗ , X = I◦ − K◦ , Λ = Φ(0)−∗ τ◦∗ .

(5.11)

One of the important ingredients in the proof of Theorem 1.3 is a recent generalization of the Carlson–Schneider inertia theorem [2] to the infinite dimensional case, proved in [28] and involving equation (5.10). In this part we review this result and present an additional lemma. We need some additional notation. Given an operator T , acting on a Hilbert space, we write σ(T ) for the spectrum of T . Furthermore, σleft (T ) stands for the part of the spectrum of T in the open left half plane Cleft . Thus σleft (T ) = σ(T ) ∩ Cleft . We are particulary interested in the case when σleft (T ) consists of a finite number of eigenvalues of finite type only. In the latter case we denote by ν(T ) the sum of the algebraic mulitiplicities of the eigenvalues in Cleft . In all other cases, we set ν(T ) = ∞. We are now ready to state the inertia theorem from [28] Theorem 5.2. [28, Theorem 1.6] Let X be a selfadjoint and invertible solution of the Lyapunov equation (5.10) such that ν(X) is finite. If, in addition, A has no eigenvalues on the imaginary axis and σleft (A) is a compact subset of the open left half plane Cleft , then ν(A) is finite and equal to ν(X). For the proof of the above theorem we refer to [28]. The additional lemma needed in the sequel is related to the requirement (appearing in the above theorem) that A has no eigenvalues on the imaginary axis. Lemma 5.3. Let the operator X on H be a selfadjoint solution of (5.10). If, in addition the pair (Λ, A) is observable, that is, ∩k≥0 Ker ΛAk = {0}, then A has no eigenvalues on the imaginary axis. Proof. Assume that Ax = iλx for some x ∈ H and some λ ∈ R. We have to show that x = 0. Since X is a selfadjoint solution of equation (5.10), it follows that 2

Λx = Λ∗ αx, x = XAx, x + Xx, Ax = (iλ − iλ) Xx, x = 0. Hence Λx = 0, i.e., x ∈ Ker Λ. As Ax = iλx, we have Ak x = (iλ)k x, and thus ΛAk x = Λ(iλ)k x = (iλ)k Λx = 0. Hence x ∈ ∩n≥0 Ker ΛAn , and therefore x = 0. So A has no eigenvalue on the imaginary axis. Part 2. In this part we give the proof of Theorem 1.3. We begin with the following proposition which is a corollary to Lemma 5.3. Proposition 5.4. The operator (V◦× )∗ has no real eigenvalue.


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Proof. We apply Lemma 5.3 with A = (−iV◦× )∗ and Λ = Φ(0)−∗ τ◦∗ . Since I◦ − K◦ is a selfadjoint solution of (5.9), we know that X = I◦ − K◦ is a selfadjoint solution of (5.10). Hence it suffices to show that

k Ker Φ(0)−∗ τ◦∗ (−iV◦× )∗ = {0}. (5.12) k≥0

The latter property is equivalent to the requirement that the space Im (V◦× )k τ◦ = Ln2 [0, ω].

(5.13)

k≥0

By definition, the left hand side of (5.13) is the closed linear hull of the spaces Im (V◦× )k τ◦ , where k = 0, 1, 2, . . .. Since V◦× = V◦ − τ◦ Φ(0)−1 Γ◦ , the operator V◦× is a feedback transformation of V◦ , and hence, as is well known (see, e.g., see [4]), we have Im (V◦× )k τ◦ = Im V◦k τ◦ . (5.14) k≥0

k≥0

But τ◦ , V◦ τ◦ , (V◦ )2 τ◦ , . . . forms a sequence of polynomials which is dense in Ln2 [0, ω]. Thus the right side of (5.14) is Ln2 [0, ω]. Hence the same holds true for the left hand side of (5.14). Thus (5.12) is proved, and by Lemma 5.3 we may conclude that (V◦× )∗ has no real eigenvalue. Let us now prove that det Φ(λ) has no real zero. We already know (see Corollary 3.2) that Φ(0) is non-singular. Therefore, take 0 = α ∈ R, and assume det Φ(α) = 0. Then we know from Proposition 5.1 that α−1 is an eigenvalue of V◦× . Since the non-zero part of the spectrum of V◦× consists of eigenvalues of finite type only, α−1 is an eigenvalue of finite type of V◦× . But then α−1 is also an eigenvalue of finite type of (V◦× )∗ . But the latter is impossible by Proposition 5.4 and the fact that α−1 is real. Thus Φ(λ) is non-singular for each real λ. Next, using (1.2) and the Rieman-Lebesgue lemma, we have lim

λ≥0, λ→∞

Φ(λ) = In .

It follows that Φ(λ) has only a finite number of zeros in the open upper half plane. But then, again applying Proposition 5.1, we may conclude that the spectrum of V◦× in open lower half plane consists of a finite number of eigenvalues of finite type. Moreover, the sum of the algebraic multiplicities of these eigenvalues is equal to m+ (Φ), where m+ (Φ) stands for the sum of the multiplicities of the zeros of Φ(λ) in the open upper half plane. Now let us pass from V◦× to (−iV◦× )∗ . We see that the part of the spectrum of (−iV◦× )∗ consists of a finite number of eigenvalues of finite type, and the sum of the algebraic multiplicities of these eigenvalues is equal to m+ (Φ). Thus, using the notation introduced in the paragraph preceding Theorem 5.2, we see that σleft ((−iV◦× )∗ ) is a finite set and ν((−iV◦× )∗ ) = m+ (Φ). The final step in the proof is to apply Theorem 5.2 with A = (−iV◦× )∗ ,

X = I◦ − K◦ ,

Λ = Φ(0)−∗ τ◦∗

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Note that X = I◦ − K◦ is an invertible solution of the Lyapunov equation (5.9). Moreover, since K◦ is compact ν(X) = ν(I◦ − K◦ ) is finite. Proposition 5.4 tells us that A = (−iV◦× )∗ has no eigenvalue on the imaginary axis. Furthermore, by the result of the previous paragraph, σleft (A) = σleft ((−iV◦× )∗ ) is a compact subset (even a finite subset) of the open left half plane Cleft . Thus Theorem 5.2 yields m+ (Φ) = ν((−iV◦× )∗ ) = ν(A) = ν(X) = ν(I◦ − K◦ ). Finally, because of compactness, the spectrum of operator I − K is the same as that of I◦ − K◦ . Hence m+ (Φ) is equal to the number of negative eigenvalues of the operator I − K.

References [1] H. Bart, I. Gohberg, M. A. Kaashoek, and A. C. M. Ran, Factorization of Matrix and Operator Functions: The State Space Method, OT 178. Birkh¨ auser Verlag, Basel, 2008. [2] D. H. Carlson and H. Schneider, Inertia theorems for matrices: the semi-definite case, Bull. Amer. Math. Soc. 68 (1962), 479–483. [3] C. T. Chen, A generalization of the inertia theorem, SIAM J. Appl. Math. 25 (1973), 158–161. [4] R. F. Curtain and H. Zwart. An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics vol. 21, Springer Verlag, New York, 1995. [5] H. Dym, On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares, Comm. Pure Appl. Math. 47 (2) (1994), 207–256. [6] R. L. Ellis and I. Gohberg, Orthogonal systems and convolution operators, OT 140, Birkh¨ auser Verlag, Basel, 2003. [7] R. L. Ellis, I. Gohberg, and D. C. Lay, Distribution of zeros of matrix-valued continuous analogues of orthogonal polynomials, in: Continuous and discrete Fourier transforms, extension problems and Wiener-Hopf equations, OT 58, Birkh¨ auser Verlag, Basel, 1992, 26–70 . [8] I. Gohberg (ed.), Orthogonal Matrix-valued Polynomials and Applications, OT 34, Birkh¨ auser Verlag, Basel, 1988. [9] I. C. Gohberg and I. A. Fel’dman, Convolution equations and projection methods for their solution [in Russian], Nauka, 1971. [10] I. C. Gohberg and I. A. Fel’dman, Convolution equations and projection methods for their solution, Translations of Mathematical Monographs 41, Amer. Math. Soc., Providence, Rhode Island, 1974. [11] I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators, Vol. I, OT 49, Birkh¨ auser Verlag, Basel, 1990. [12] I. Gohberg, I. Haimovici, M. A. Kaashoek, and L. Lerer, The Bezout integral operator: main property and underlying abstract scheme, in: The state space method generalizations and applications, OT 161, Birkh¨ auser Verlag, Basel, 2006, 225–270.


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[13] I. Gohberg and G. Heinig, The inversion of finite Toeplitz matrices consisting of elements of a non-commutative algebra, Rev. Roum. Math. Pures et Apl. 20 (1974), 623–663 (in Russian). [14] I. Gohberg and G. Heinig, On matrix integral operators on a finite interval with kernels depending on the difference of the arguments [in Russian], Rev. Roumaine Math. Pures Appl. 20 (1975), 55–73. [15] I. Gohberg, M. A. Kaashoek, and L. Lerer, The continuous analogue of the resultant and related convolution operators, in: The extended field of operator theory (M.A. Dritschel, ed.), OT 171, Birkh¨ auser Verlag, Basel, 2007, 107–127. [16] I. Gohberg, M. A. Kaashoek, and L. Lerer, Quasi-commutativity of entire matrix functions and the continuous analogue of the resultant, in: Modern operator theory and applications, OT 170, Birkh¨ auser Verlag, Basel, 2007, 101–106. [17] I. Gohberg, M. A. Kaashoek, and L. Lerer, On a class of entire function equations, Linear Algebra and its Applications 425 (2007), 434–442. [18] I. Gohberg, M. A. Kaashoek, and L. Lerer, The inverse problem for Krein orthogonal matrix functions, Journal of Functional Analysis and its Applications 41 (2007), 115– 125. [19] I. Gohberg, M. A. Kaashoek, and F. van Schagen, On inversion of convolution integral operators on a finite interval, in: Operator theoretical methods and applications to mathematical physics, OT 147, Birkh¨ auser Verlag , Basel, 2004, 277–285. [20] I. Gohberg and L. Lerer, Matrix generalizations of M. G. Krein theorems on orthogonal polynomials, in: Orthogonal matrix-valued polynomials and applications, OT 34, Birkh¨ auser Verlag , Basel, 1988, 137–202. [21] G. J. Groenewald and M. A. Kaashoek, A new proof of an Ellis-Gohberg theorem on orthogonal matrix functions related to the Nehari problem, in: Recent Advances in Operator Theory and its Applications. Proceedings IWOTA 2003, OT 160, Birkh¨ auser Verlag, Basel, 2005, 217–232. [22] I. Koltracht, B. Kon, and L. Lerer, Inversion of structured operators, Integral Equations and Operator Theory 20 (1994), 410–480. [23] M. G. Krein, On the location of the roots of polynomials which are orthogonal on the unit circle with respect to an indefinite weight [in Russian], Teor. Funkcii, Funkc. Anal. i Pril. 2 (1966), 131–137. [24] M. G. Krein and H. Langer, On some continuation problems which are closely related to the theory of operators in spaces Πκ . IV. Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), 299–417. [25] P. Lancaster, L. Lerer, and M. Tismenetsky, Factored form of solutions of the equation AX − XB = C in matrices, Lin. Alg. Appl. 62 (1984), 19–49. [26] P. Lancaster and M. Tismenetsky, The Theory of Matrices with Applications, Academic Press, Inc., New York, 1985. [27] L. Lerer and A. C. M. Ran, A new inertia theorem for Stein equations, inertia of invertible hermitian block Toeplitz matrices and matrix orthogonal polynomials, Integral Equations and Operator Theory 47 (2003), 339–360.

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[28] L. Lerer, I. Margulis, and A. C. M. Ran, Inertia theorems based on operator Lyapunov equations, Oper. Matrices 2 (2) (2008), 153–166. [29] L. A. Sakhnovich, Integral equations with difference kernels on finite intervals, OT 84, Birkh¨ auser Verlag, Basel, 1996. [30] H. K. Wimmer, Inertia theorems for matrices, controllability and linear vibrations, Linear Algebra Appl. 8 (1974), 337–343. M. A. Kaashoek Afdeling Wiskunde Faculteit der Exacte Wetenschappen Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: [email protected] L. Lerer and I. Margulis Department of Mathematics Technion – Israel Institute of Technology Haifa 32000 Israel e-mail: [email protected] [email protected] Submitted: June 5, 2009. Revised: June 15, 2009.



Flat Extensions of Nonsingular Moment Matrices Chunji Li and Muneo Ch¯o Abstract. In this paper we consider the truncated complex moment problem suggested by Curto and Fialkow. First, we give another computing proof for a solution of the nonsingular quadratic moment problem. Then we consider the nonsingular quartic moment problem and give some partial solutions. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47B37. Keywords. Truncated complex moment problem, flat extension.

1. Introduction and preliminaries For a given closed subset K ⊆ C and a doubly indexed finite sequence of complex numbers γ : γ00 , γ01 , γ10 , γ02 , γ11 , γ20 , . . . , γ0,2n , γ1,2n−1 , . . . , γ2n−1,1 , γ2n,0 ,

(1.1)

with γ00 > 0 and γji = γ ij , the truncated K-moment problem entails finding a positive Borel measure µ such that γij = z¯i z j dµ (0 ≤ i + j ≤ 2n) and supp µ ⊆ K; (1.2) γ is called a truncated moment sequence (of order 2n) and µ is called a representing measure for γ ([1], [2], [3], [4], [5] and [6]). For n ≥ 1, let m ≡ m(n) := (n + 1)(n + 2)/2. For A ∈ Mm (C) (the set of m × m complex matrices), we denote the successive rows and columns according to the following lexicographic-functional ordering: ¯ Z¯ 2 , Z 3 , ZZ ¯ 2 , Z¯ 2 Z, Z¯ 3 , . . . , Z n , ZZ ¯ n−1 , . . . , Z¯ n−1 Z, Z¯ n . 1 , Z, Z¯ , Z 2 , ZZ, (1)

(2)

(3)

(4)

(n+1)

This research is partially supported by Grant-in-Aid Research No. 20540192.

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For 0 ≤ i + j ≤ n, 0 ≤ l + k ≤ n, we denote the entry in row Z¯ l Z k , column Z¯ i Z j by A(l,k)(i,j) . For the given truncated moment sequence γ in (1.1) and 0 ≤ i, j ≤ n, we define the (i+1)×(j+1) matrix M [i, j] whose entries are the moments of order i+j:   γi,j γi+1,j−1 · · · · · · γi+j,0  γi−1,j+1 γi,j · · · · · · γi+j−1,1    (1.3) M [i, j] :=  , .. .. .. .. ..   . . . . . γ1,i+j−1

γ0,i+j

···

···

γj,i

where we note that M [i, j] has the Toeplitz-like property of being constant on each diagonal; in particular, M [i, i] is a self-adjoint Toeplitz matrix. We now define the moment matrix M (n) ≡ M (n)(γ) via the block decomposition M (n) := (M [i, j])0≤i,j≤n . For example, if n = 1, the quadratic moment problem for γ : γ00 , γ01 , γ10 , γ02 , γ11 , γ20 corresponds to  

γ00 γ01 γ10 M [0, 0] M [0, 1] M (1) = =  γ10 γ11 γ20  , (1.4) M [1, 0] M [1, 1] γ01 γ02 γ11 and if n = 2, the quartic moment problem for γ : γ00 , γ01 , γ10 , γ02 , γ11 , γ20 , γ03 , γ12 , γ21 , γ30 , γ04 , γ13 , γ22 , γ31 , γ40 corresponds to



M (2) = (M [i, j])0≤i,j≤2

   =   

γ00 γ10 γ01 γ20 γ11 γ02

γ01 γ11 γ02 γ21 γ12 γ03

γ10 γ20 γ11 γ30 γ21 γ12

γ02 γ12 γ03 γ22 γ13 γ04

γ11 γ21 γ12 γ31 γ22 γ13

γ20 γ30 γ21 γ40 γ31 γ22

    .   

(1.5)

Let Pn ⊆ C[z, z¯] denote the complex polynomials in z, z¯ of total degree ≤ n. Clearly, dim Pn = m. For p ∈ Pn , p(z, z¯) =

aij z¯i z j ,

p¯(z, z¯) =

0≤i+j≤n

aij z i z¯j ,

0≤i+j≤n

pˆ ≡ (a00 , a01 , a10 , · · · , a0n , · · · , an0 ) ∈ Cm . The basic connection between M (n)(γ) and any representing measure µ is provided by the identity f g¯dµ = (M (n)fˆ, gˆ) (f, g ∈ Pn ); in particular,

(M (n)fˆ, fˆ) =

so M (n) ≥ 0. That is, M (n) is positive.

|f |2 dµ ≥ 0,

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For a positive matrix A, an extension of A is a block matrix of the form

A B ˜ . (1.6) A := B∗ C That A˜ is a flat extension of A means rankA˜ = rankA. It is interesting to find a flat extension M (n + 1) of M (n) because of the following Curto-Fialkow’s main theorem. Theorem 1.1 ([1, Theorem 5.13]). γ as in (1.1) has a rank M (n)-atomic representing measure if and only if M (n) ≥ 0 and M (n) admits a flat extension M (n + 1). In order to construct a flat extension M (n + 1) of M (n), one makes use of Smul’jan’s Theorem. Proposition 1.2 ([1]). For A ≥ 0, the following statements are equivalent: (i) A˜ ≥ 0; (ii) There exists W such that AW = B and C ≥ W ∗ AW . Conjecture 1.3 ([6, Conjecture 2.7]). If M (n) is positive and invertible, then M (n) has a flat extension M (n + 1). If A ≥ 0 and AW = B, i.e., Ran B ⊆ Ran A, there is a unique flat extension of the form (1.6), which we denote by [A; B]. For M (n) ≥ 0, we want to construct a positive flat extension of the form M (n + 1) = [M (n); B(n)]. Let C := (cij )1≤i,j≤n+1 in [M (n); B(n)]. For n = 1, if M (1) > 0 (i.e., M (1) is positive and invertible), then [M (1); B(1)] is a flat extension of M (1) if and only if c11 = c22 , and it is always satisfied ([1, Proposition 6.4]), so Conjecture 1.3 is true for n = 1. In [4], it was shown that there is a positive matrix M (3) but with no representing measure, i.e., Conjecture 1.3 is false for n = 3. But for n = 2, it is still open. We know only from [1] that if M (2) > 0, then [M (2); B(2)] is a flat extension of M (2) if and only if c11 = c22 and c21 = c32 . Curto and Fialkow solved the singular (i.e., det M (2) = 0) quartic moment problem completely ([2, Theorem 1.10], and [3, Theorem 3.1, Theorem 4.1]), but the nonsingular case is still open. Let hij is the cofactor of the i-th row j-th column entry of M (2), i, j = 1, 2, 3, 4, 5, 6. It is easy to see that hji = hij . In last section of [7], the authors consider the quartic moment matrix M (2) as   1 0 0 0 1 0  0 1 0 0 0 0     0 0 1 0 0 0   M (2) =  (1.7)  0 0 0 γ22 γ31 γ40  .    1 0 0 γ13 γ22 γ31  0 0 0 γ04 γ13 γ22 It is shown that if M (2) as in (1.7) is positive and invertible, and the set St := {t ∈ R+ | c ≤ t2 , c − |a|b ≤ t2 − bt ≤ |a|b + c, |a|b + c ≤ t2 + bt}

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is not empty, where a=

h22 h64 h65 , |h64 |2 − |h65 |2 + h66 (h55 − h66 )

b=

h22 2 |h65 |2 , |h64 |2 − |h65 |2 + h66 (h55 − h66 )

c=

h22 2 |h65 |(|h64 |2 − 1) , |h64 |2 − |h65 |2 + h66 (h55 − h66 )

then M (2) as in (1.7) admits a flat extension M (3) and hence a representing measure ([7, Theorem 4.4]). The sufficient condition is very strict, so it is necessary to reconsider the following Nonsingular quartic moment problem. If M (2) is positive and nonsingular, does M (2) have a flat extension M (3)? In Section 2, we give another proof for the nonsingular quadratic moment problem, which Curto and Fialkow already proved in ([1, Proposition 6.4]); in Section 3, we reconsider the nonsingular quartic moment problem, and give some partial solutions. Some of the calculations in this article were obtained through computer experiments using the software tool Scientific Work Place [8].

2. Flat extension of nonsingular quadratic moment matrix We consider the quadratic moment matrix (1.4). Let γ00 = x0 , γ01 = x1 +ix2 , γ10 = x1 − ix2 , γ02 = x3 + ix4 , γ20 = x3 − ix4 and γ11 = x5 . For a positive N × N matrix A, we denote by [A]k the compression of A to the first k rows and columns. By direct calculation of det ([M (1)]1 ) , det ([M (1)]2 ) and det M (1), we have Proposition 2.1. M (1) is positive and invertible if and only if (1) x0 > 0, (2) x0 x5 − x21 − x22 > 0, (3) 4x1 x2 x4 − x0 x23 − x0 x24 + 2x21 x3 + x0 x25 − 2x22 x3 − 2x21 x5 − 2x22 x5 > 0. Theorem 2.2 ([1, Proposition 6.4]). Assume that M (1) > 0. Then M (1) has a flat extension M (2). Proof. Let γ12 = x6 + ix7 , γ21 = x6 − ix7 , γ03 = x8 + ix9 and γ30 = x8 − ix9 . Then

2 d1 c11 − c22 = x0 x5 − x21 − x22 x9 + 2 (x0 x5 − x21 − x22 )

2 d2 p(x0 , . . . , x7 ) 2 2 + x0 x5 − x1 − x2 x8 + + 2 , 2 (x0 x5 − x21 − x22 ) x1 − x0 x5 + x22

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where d1 = 4x1 x3 x4 + 4x1 x2 x6 − 2x0 x3 x7 − 2x0 x4 x6 − 2x1 x4 x5 − 2x2 x3 x5 − 2x2 x23 + 2x2 x24 + 2x21 x7 − 2x22 x7 , d2 = 4x2 x3 x4 − 2x1 x3 x5 − 2x0 x3 x6 − 4x1 x2 x7 + 2x0 x4 x7 + 2x2 x4 x5 + 2x1 x23 − 2x1 x24 + 2x21 x6 − 2x22 x6 , and p(x0 , . . . , x7 ) = a1 x27 + a2 x26 + a3 x6 x7 + a4 x7 + a5 x6 , with a1 = 4x41 − 4x0 x1 x2 x4 − 4x0 x21 x3 − 4x0 x21 x5 + 2x20 x3 x5 + x20 x23 + 4x21 x22 + x20 x24 + x20 x25 , a2 = 4x42 − 4x0 x1 x2 x4 + 4x0 x22 x3 − 4x0 x22 x5 − 2x20 x3 x5 + x20 x23 + 4x21 x22 + x20 x24 + x20 x25 , a3 = 8x0 x1 x2 x5 − 8x1 x32 − 8x31 x2 + 4x0 x21 x4 + 4x0 x22 x4 − 4x20 x4 x5 , a4 = 2x0 x2 x33 − 2x0 x1 x34 − 4x0 x2 x35 + 4x31 x3 x4 − 8x31 x4 x5 + 8x32 x3 x5 − 2x0 x1 x23 x4 + 2x0 x2 x3 x24 + 6x0 x1 x4 x25 − 6x0 x2 x3 x25 − 12x1 x22 x3 x4 + 8x21 x2 x3 x5 − 8x1 x22 x4 x5 + 4x32 x23 + 4x32 x25 − 4x21 x2 x23 + 8x21 x2 x24 + 4x21 x2 x25 , a5 = 6x0 x1 x3 x25 − 4x0 x1 x35 − 2x0 x2 x34 − 8x31 x3 x5 − 4x32 x3 x4 − 8x32 x4 x5 − 2x0 x1 x3 x24 − 2x0 x1 x33 − 2x0 x2 x23 x4 + 12x21 x2 x3 x4 + 6x0 x2 x4 x25 − 8x1 x22 x3 x5 − 8x21 x2 x4 x5 + 4x31 x23 + 4x31 x25 − 4x1 x22 x23 + 8x1 x22 x24 + 4x1 x22 x25 . Now we only need to show that p(x0 , . . . , x7 ) > 0. Claim 1. a1 > 0 and a2 > 0. In fact, 2 x0 x3 − 2x21 2 2 a1 = x0 x5 + + (x0 x4 − 2x1 x2 ) ≥ 0. x0 x0 x3 − 2x21 = 0 and x0 x4 − 2x1 x2 = 0, then If x5 + x0 4x1 x2 x4 − x0 x23 − x0 x24 + 2x21 x3 + x0 x25 − 2x22 x3 − 2x21 x5 − 2x22 x5 = 0. It contradicts to (3) of Proposition 2.1. Thus we have a1 > 0. Similarly, a2 > 0. Let p(x, y) = a1 x2 + a2 y 2 + a3 xy + a4 x + a5 y + a6 , where ai ∈ R (i = 1, . . . , 6). Claim 2. p(x, y) > 0, if (1) a1 > 0,

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(2) 4a1 a2 > a23 , (3) a1 a25 − a3 a4 a5 + a2 a24 + a6 a23 − 4a1 a2 < 0. In fact, ∂p(x, y) = a4 + 2xa1 + ya3 = 0, ∂x

∂p(x, y) = a5 + xa3 + 2ya2 = 0, ∂y

and 4a1 a2 > a23 , we obtain a unique minimizer of p(x, y): a3 a5 − 2a2 a4 a3 a4 − 2a1 a5 , y0 = . x0 = 4a1 a2 − a23 4a1 a2 − a23 If we let p(x0 , y0 ) =

a1 a25 − a3 a4 a5 − 4a1 a2 a6 + a2 a24 + a23 a6 > 0, a23 − 4a1 a2

then p(x, y) ≥ p(x0 , y0 ) > 0. Thus we have Claim 2. Since 2 a23 − 4a1 a2 = (−4) x20 (det M (1)) < 0 and 4 a1 a25 − a3 a4 a5 + a2 a24 + a6 a23 − 4a1 a2 = (−4) det ([M (1)]2 ) (det M (1)) < 0, we have p(x0 , . . . , x7 ) > 0 by Claims 1 and 2. Thus there exist γ12 and γ03 satisfying the condition c11 = c22 .

3. Flat extensions of nonsingular quartic moment matrices In this section, we consider the existence of flat extensions for nonsingular quartic moment matrices. In Section 1, we have known that if M (2) > 0 then [M (2); B(2)] is a flat extension of M (2) if and only if c11 = c22 and c21 = c32 . Note that in B(2), we have ¯ 2 = (γ12 , γ22 , γ13 , γ23 , γ23 , γ14 )T . Z 3 = (γ03 , γ13 , γ04 , γ23 , γ14 , γ05 )T and ZZ Then we get c11 = (γ03 , γ13 , γ04 , γ23 , γ14 , γ05 ) M (2)−1 Z 3 , ¯ 2, c22 = (γ12 , γ22 , γ13 , γ23 , γ23 , γ14 ) M (2)−1 ZZ c21 = (γ12 , γ22 , γ13 , γ23 , γ23 , γ14 ) M (2)−1 Z 3 , ¯ 2. c32 = (γ12 , γ13 , γ22 , γ14 , γ23 , γ23 ) M (2)−1 ZZ We thus have Proposition 3.1 ([7, Proposition 4.1]). Let M (2) > 0. Then M (2) admits a flat extension M (3) if and only if there exist γ23 , γ14 , γ05 ∈ C satisfying (γ03 , γ13 , γ04 , γ23 , γ14 , γ05 ) M (γ03 , γ13 , γ04 , γ23 , γ14 , γ05 )T = (γ12 , γ22 , γ13 , γ23 , γ23 , γ14 ) M (γ12 , γ22 , γ13 , γ23 , γ23 , γ14 )T

(3.1)

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(γ12 , γ22 , γ13 , γ23 , γ23 , γ14 ) M (γ03 , γ13 , γ04 , γ23 , γ14 , γ05 )T = (γ12 , γ13 , γ22 , γ14 , γ23 , γ23 ) M (γ12 , γ22 , γ13 , γ23 , γ23 , γ14 )T ,

249

(3.2)

where M := (det M (2)) M (2)−1 . Observe that the above system consists of two equations in γ23 , γ14 , γ05 . Thus we may put γ23 = 0 for the solution of the above system. In fact, M is the adjoint matrix of moment matrix M (2). From (3.2), we have 2 (h66 γ14 + θ1 )γ05 = h64 γ14 − |γ14 |2 h56 + θ2 γ14 + θ3 γ14 + θ4 ,

(3.3)

where θ1 = γ12 h61 + γ22 h62 + γ13 h63 , θ2 = γ12 (h14 + h61 ) + γ22 (h24 + h63 ) + γ13 (h62 + h34 ) − γ12 h51 − γ22 h52 − γ13 h53 , θ3 = −γ03 h16 − γ13 h26 − γ04 h36 , θ4 = (γ12 h11 + γ13 h12 + γ22 h13 )γ12 + (γ12 h21 + γ22 h23 )γ22 + (γ12 h31 + γ13 h32 + γ22 h33 − γ12 h21 − γ13 h23 )γ13 − (γ12 h11 + γ22 h12 + γ13 h13 )γ03 − (γ12 h31 + γ22 h32 + γ13 h33 )γ04 , and from (3.1) we have h66 |γ05 |2 + (h55 − h66 )|γ14 |2 + 2Re(ν1 γ05 + ν2 γ14 + γ05 γ14 h65 ) + ν3 = 0,

(3.4)

where ν1 = γ03 h61 + γ13 h62 + γ04 h63 , ν2 = γ03 h51 + γ13 h52 + γ04 h53 − γ12 h61 − γ22 h62 + γ13 h63 , 2 )h22 + (|γ04 |2 − |γ13 |2 )h33 ν3 = (|γ03 |2 − |γ12 |2 )h11 + (|γ13 |2 − γ22

+ 2Re[(γ13 γ03 − γ22 γ12 )h21 + (γ04 γ03 − γ13 γ12 )h31 + (γ04 γ13 − γ22 γ13 )h32 ]. Thus, if there exist γ14 , γ05 ∈ C satisfying (3.3) and (3.4), then M (2) admits a flat extension M (3). Now we consider the following example. Example ([7, Example 4.5]). Let  1  0   0 M (2) =   0   1 0

0 1 0 0 0 0

0 0 0 0 1 0 0 4 0 2+i 0 1+i

1 0 0 2−i 4 2+i

0 0 0 1−i 2−i 4

    .   

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It is easy to see that the above matrix M (2) is positive and nonsingular. Here   30 0 0 5 + 3i −14 5 − 3i   0 16 0 0 0 0     0 0 16 0 0 0 .  M =  7 −5 + 3i −i   5 − 3i 0 0  −14 0 0 −5 − 3i 14 −5 + 3i  5 + 3i 0 0 i −5 − 3i 7 We put γ23 = 0, γ14 = x17 + ix18 and γ05 = x19 + ix20 . Then conditions (3.3) and (3.4) are following (3.5) and (3.6), respectively; 6x20 x17 − 10x20 x18 − 10x17 x19 − 6x18 x19 + 7(x217 + x218 + x219 + x220 ) − 224 = 0 (3.5) and

2x17 x18 − 5x217 − 5x218 + 7x20 x18 + 7x19 x17 − 80 = 0 2x217 + 4x218 + 7x20 x17 − 7x19 x18 − 48 = 0.

From (3.6), we have 1 80x17 − 48x18 + 5x317 + 4x318 + 5x17 x218 , x19 = 2 2 7x17 + 7x18 1 x20 = 2 48x17 + 80x18 − 2x317 + 5x318 − 6x17 x218 + 5x217 x18 . 2 7x17 + 7x18 Substituting (3.7) into (3.5), we obtain 2 2 2 2 x17 − 46 + x218 − 52 + 2 (x17 x18 − 5) = 4326.

(3.6)

(3.7)

(3.8)

It is easy to check that (3.8) has a solution. Therefore, M (2) admits a flat extension M (3). But, the result is not clear in [7, Example 4.5], since the set St is empty. Next, we consider the case that of the form  1 0  0 1   0 0 M (2) =   0 0   1 0 0 0

M (1) = I and γ12 = γ03 = 0. Thus M (2) is  0 0 1 0 0 0 0 0   1 0 0 0  . (3.9) 0 γ22 γ31 γ40    0 γ13 γ22 γ31 0 γ04 γ13 γ22

Proposition 3.2 ([7, Proposition 4.2]). Let M (2) be a moment matrix as in (3.9). Then M (2) > 0 if and only if (1) γ22 > 0, (2) γ22 2 − γ22 − |γ13 |2 > 0, 2 (3) γ22 3 − γ22 2 − γ22 (|γ04 |2 + 2|γ13 |2 ) + |γ04 |2 + 2Re(γ04 γ31 ) > 0. We may put γ23 = 0. Then from (3.3) we have γ05 =

2 h64 γ14 − h56 |γ14 |2 + h22 h56 . h66 γ14

(3.10)

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Substituting (3.10) into (3.4) we obtain [|h46 |2 − |h56 |2

+h66 (h55 − h66 )]|γ14 |4 + 2h22 Re(h46 h56 γ14 2 ) −h22 h55 h66 |γ14 |2 + h22 2 |h56 |2 = 0.

(3.11)

Lemma 3.3. Let a1 ∈ R, a2 ∈ C, a3 > 0 and a4 > 0. Then the complex equation 2

a1 |z| + 2Re (a2 z¯) − 2a3 |z| + a4 = 0 has a solution, if a23 − 2a3 |a2 | − a1 a4 ≥ 0. Proof. In fact, let z = reiθ and a2 = ρeiφ . Then the equation

a1 r2 + 2 ρ cos(φ − θ) − a3 r + a4 = 0 has a solution if and only if 2 ∆ = ρ cos (φ − θ) − a3 − a1 a4 ≥ 0. Since ∆ ≥ a23 − 2a3 ρ − a1 a4 , we have our conclusion.

Thus we have the following main theorem. Theorem 3.4. Assume that M (2) as in (3.9) is positive and nonsingular. If 2 2 2 2 h55 h66 − 2 |h56 | ≥ h55 h66 |h46 | |h56 | + 4 |h56 | |h46 | − h266 , (3.12) then M (2) as in (3.9) admits a flat extension M (3). For the above example, we have h55 = 14, h66 = 7, h46 = −i and h56 = −5 + 3i, so 2 h55 h66 − 2 |h56 |2 = 900, and

h55 h66 |h46 | |h56 | + 4 |h56 |2 |h46 |2 − h266 ≈ −5956.6.

Thus the matrix M (2) as in above example satisfies condition (3.12). We consider another example. Example. Let

    M (2) =    

1 0 0 0 1 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 γ22 0 0

1 0 0 0 γ22 0

0 0 0 0 0 γ22

    .   

(3.13)

Then M (2) as in (3.13) is positive and nonsingular if and only if γ22 > 1. In this 2 2 case, it holds h55 = γ22 , h66 = −γ22 + γ22 , h46 = 0 and h56 = 0, so we have 2 2 2 6 h55 h66 − 2 |h56 | = γ22 (γ22 − 1) > 0,

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and

2

h55 h66 |h46 | |h56 | + 4 |h56 |

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2 |h46 | − h266 = 0.

Thus M (2) as in (3.13) satisfies condition (3.12). Hence, by Theorem 3.4, M (2) as in (3.13) admits a flat extension M (3). Finally, we consider the following two cases. Case 1. Let   1 0 0 0 1 0  0 1 0 0 0 γ30     0 0 1 γ03 0 0   M (2) =   0 0 γ30 γ22 0 0     1 0 0 0 γ22 0  0 γ03 0 0 0 γ22

(γ03 = 0) .

(3.14)

Proposition 3.5. M (2) > 0 as in (3.14) if and only if γ22 > max{|γ03 |2 , 1}. Since the coefficient of γ05 in (3.3) is (γ22 − 1) (γ41 − γ03 γ22 ) (γ22 − γ03 γ30 ) . Thus if γ41 = γ03 γ22 and γ14 = γ30 γ22 , then (3.3) holds. Furthermore, if we put γ23 = 0, then (3.4) is equivalent to 2 γ22 γ22 − |γ03 | det ([M (2)]5 ) 2 |γ05 | = . (γ22 − 1) This means there exist γ23 , γ14 , γ05 ∈ C satisfying (3.1) and (3.2). Thus we have the following result. Theorem 3.6. If M (2) as in (3.14) is positive and nonsingular, then M (2) as in (3.14) admits a flat extension M (3). Case 2. Let



   M (2) =    

1 0 0 0 1 0

0 1 0 0 0 γ03

0 0 1 γ30 0 0

0 0 γ03 γ22 0 γ04

1 0 0 0 γ22 0

0 γ30 0 γ40 0 γ22

       

(γ03 γ04 = 0) .

(3.15)

2

Proposition 3.7. M (2) > 0 as in (3.15) if and only if γ22 > max{|γ03 | + |γ04 | , 1}. Since the coefficient of γ05 in (3.3) is (γ22 − 1) (γ41 − γ03 γ22 ) (γ22 − γ03 γ30 ) , if we choose γ41 = γ03 γ22 and γ14 = γ30 γ22 , then 2 2 Left hand side of (3.3) − Right hand side of (3.3) = γ22 γ04 γ30 (γ22 − 1) = 0.

Thus the coefficient of γ05 in (3.3) is not zero. So from (3.3), we obtain 2 2 2 2 2 + γ03 γ22 γ40 + γ30 γ13 γ31 − γ30 γ22 γ04 γ30 γ22 γ14 − γ03 γ40 γ41 − γ14 γ05 = − . (γ03 γ22 − γ41 ) (γ22 − γ03 γ30 )

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Substituting γ05 into (3.4) and put γ23 = 0, we obtain Left hand side of (3.4) − Right hand side of (3.4) 4 2 2 2 a1 |γ14 | + a2 |γ14 | + 2Re a3 γ14 γ41 + 2Re a4 γ14 + 2Re (a5 γ14 ) + a6 = , 2 γ22 − |γ03 | (γ41 − γ03 γ22 ) (γ30 γ22 − γ14 ) where

a1 = − |γ03 |2 − 1 γ22 − |γ04 | − |γ03 |2 γ22 + |γ04 | − |γ03 |2 , 8

4

4

5 6 2 3 a2 = γ22 − γ22 − γ22 |γ03 | − γ22 |γ04 | − 8γ22 |γ03 | 2 2 6 4 2 + 2γ22 (|γ03 |) |γ03 | + 1 − γ22 (|γ03 |) |γ03 |2 − 6 2 2 2 2 4 3 2 −3 |γ03 | − 2 + γ22 + γ22 (|γ03 |) 2 |γ03 | + 1 + |γ04 | γ22 + 2γ22 2 2 2 2 γ22 + |γ04 | − |γ03 | Re γ30 , − 2γ22 |γ03 | − γ22 γ22 − |γ04 | − |γ03 | 4 a3 = γ30 γ22 + |γ03 |6 (γ03 + γ30 γ22 ) + γ22 γ03 |γ04 |2 + γ30 |γ04 |2 4 3 2 (−γ03 − 2γ30 ) + |γ03 | γ22 (−3γ03 − 2γ30 ) − γ30 γ22 + γ22 2 2 2 2 3 + |γ03 | γ22 (3γ03 + 4γ30 ) − γ03 |γ04 | − γ30 γ22 |γ04 | − γ30 γ22 , 2 2 2 2 2 a4 = γ22 γ03 γ03 γ30 − γ30 γ22 − γ03 γ22 + γ03 γ22 |γ04 | − |γ03 | − γ22 , 2 3 |γ03 |8 (γ03 + γ30 ) − γ22 |γ03 |6 (4γ03 + 3γ30 − γ03 γ22 ) a5 = γ22 4 2 2 2 2 2 3 + γ22 |γ03 | 4γ03 γ22 + 3γ30 γ22 − γ03 γ22 − γ30 |γ04 | − γ03 γ22 |γ04 | 2 2 2 2 3 3 2 |γ03 | γ30 |γ04 | − γ03 γ22 − γ03 |γ04 | − γ30 γ22 + 2γ03 γ22 |γ04 | + γ22 2

2

2 + γ22 γ03 (γ22 − 1) (γ22 − |γ04 |) (γ22 + |γ04 |) , 2 2 2 3 |γ04 | − γ22 + |γ03 | a6 = −γ22 (|γ03 |) γ22 + |γ04 | − |γ03 | 2 3 × γ22 − γ22 − |γ04 |2 − |γ03 |4 + γ22 |γ03 |2 + γ22 |γ04 |2 .

The following is needed to show that the complex equation 4 2 2 2 a1 |γ14 | + a2 |γ14 | + 2Re a3 γ14 γ41 + a4 γ14 + a5 γ14 + a6 = 0 has a solution. One of the sufficient condition is a1 a6 < 0. Observe that 2 2 3 γ22 + |γ04 | − |γ03 |2 γ22 − |γ04 | − |γ03 |2 1 − |γ03 |2 a1 a6 = |γ03 |2 γ22 2 2 × |γ03 | γ22 − |γ03 | − (γ22 − 1) (γ22 + |γ04 |) (γ22 − |γ04 |) , 2

2

and it is easy to check γ22 + |γ04 | − |γ03 | = 0 because of γ22 − |γ04 | − |γ03 | > 0. Thus, we have the following result.

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Theorem 3.8. Assume that M (2) as in (3.15) is positive and nonsingular. If 2 2 2 2 2 < 0, (3.16) (1 − |γ03 | ) |γ03 | (γ22 − |γ03 | ) − (γ22 − 1) γ22 − |γ04 | then M (2) as in (3.15) admits a flat extension M (3). Example. Let



 1 0 0 0 1 0  0 1 0 0 0 − 2i    i  0 0 1 0 0  2  . (3.17) M (2) =  i   0 0 − 2 4 0 −i   1 0  0 0 4 0 0 2i 0 i 0 4 M (2) is positive and nonsingular. Furthermore, condition (3.16) is satisfied. In fact, 2115 2 (1 − |γ03 |2 ) |γ03 |2 (γ22 − |γ03 |2 ) − (γ22 − 1) γ22 =− . − |γ04 |2 64 Thus M (2) as in (3.17) admits a flat extension M (3).

References [1] R. Curto and L. Fialkow, Solution of the truncated complex moment problems for flat data, Memoirs Amer. Math. Soc. 568 (1996). , Flat extensions of positive moment matrices: recursively generated rela[2] tions, Memoirs Amer. Math. Soc. 648 (1998). , Solution of the singular quartic moment problem, J. Operator Theory, 48 [3] (2002), 315-354. , Flat extensions of positive moment matrices: Relations in analytic or con[4] jugate terms, Operator Theory: Adv. Appl. 104 (1998), 59-82. [5] R. Curto, L. Fialkow and H. M¨ oller, The extremal truncated moment problem, Integr. equ. oper. theory, 60 (2008), 177-200. [6] L. Fialkow, Positivity, extentions and the truncated complex moment problem, in: Contemporary Mathematics 185, Amer. Math. Soc. (1995), 133-150. [7] C. Li and S. Lee, The quartic moment problem, J. Korean Math. Soc. 42 (2005), No.4, 723-747. [8] MacKichan Software, Inc. Scientific WorkPlace, Version 4.0, MacKichan Software, Inc., 2002. Chunji Li Institute of System Science, College of Sciences, Northeastern University Shenyang 110004, P. R. China e-mail: [email protected] Muneo Ch¯ o Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan e-mail: [email protected] Submitted: April 19, 2008. Revised: June 2, 2008.

Integr. equ. oper. theory 65 (2009), 255–283 © 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020255-29, published online August 3, 2009 DOI 10.1007/s00020-009-1703-0


On the Eigenfunctions of No-Pair Operators in Classical Magnetic Fields Oliver Matte and Edgardo Stockmeyer Dedicated to Hubert Kalf on the occasion of his 65th birthday

Abstract. We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1, ∞) and that there exist infinitely many eigenvalues below 1. (The rest energy of the electron is 1 in our units.) Assuming that the magnetic vector potential is smooth and that all its partial derivatives increase subexponentially, we finally show that an eigenfunction corresponding to an eigenvalue λ < 1 is smooth away from the nucleus and that its partial√derivatives of any order decay pointwise exponentially with any rate a < 1 − λ2 , for λ ∈ [0, 1), and a < 1, for λ < 0. Mathematics Subject Classification (2000). Primary 81Q10; Secondary 47B25. Keywords. Dirac operator, Brown and Ravenhall, no-pair operator, exponential decay, regularity.

1. Introduction The aim of this article is to study the regularity and the pointwise exponential decay of eigenstates of relativistic hydrogenic atoms in exterior magnetic fields which are described in the free picture. The latter model is obtained by restricting the usual Coulomb-Dirac operator with magnetic vector potential, A, to the positive spectral subspace of the magnetic Dirac operator without electrostatic potential. We shall call the resulting operator the no-pair operator, since it belongs to a more general class of models which can be derived by a formal procedure in quantum electrodynamics that neglects pair creation and annihilation processes [31, 32]. If we set A = 0, then the no-pair operator considered here is also known as

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the (one-particle) Brown-Ravenhall or Bethe-Salpeter operator [5, 7]. (Numerous mathematical contributions to the Brown-Ravenhall model are listed in the references at [22].) Although these models have their main applications in the numerical study of relativistic atoms with a large number of electrons [29, 31, 32], they pose some new mathematical problems already in the investigation of hydrogenic atoms. This is due to the fact that both the kinetic and the potential part of the no-pair operator are non-local. There already exist results on the L2 -exponential localization of bound states of (multi-particle) Brown-Ravenhall operators. All of them give, however, suboptimal bounds on the decay rate. The first one has been derived in [3] for a hydrogenic atom and for coupling constants less than 1/2. It has been generalized in [23, 24] to many-electron atoms and to all coupling constants below and including the critical one of the Brown-Ravenhall model determined in [12]. Moreover, systems involving different particle species and their reductions to irreducible representations of the groups of rotation, reflection, and permutation symmetries are studied in [23, 24]. In [22] the present authors study a no-pair model of a many-electron atom which is defined by means of projections including the electrostatic potential as well as perhaps a mean-field and a non-local exchange potential. The main results of [22] are an HVZ theorem, conditions for the existence of infinitely many discrete eigenvalues, and L2 -estimates on the exponential localization of the corresponding eigenvectors. Besides the passage to pointwise exponential bounds on the partial derivatives of eigenstates of the no-pair operator (for a class of magnetic vector potentials whose partial derivatives of any order are allowed to increase subexponentially), the present article includes some further improvements, even in the case A = 0. First, we verify that the rate of exponential decay of an eigenvector of the no-pair operator corresponding to an eigenvalue λ < 1 is not less than any √ 1 − λ2 , λ ∈ [0, 1), a < (λ) := (1.1) 1, λ < 0. This is the same behaviour as it is known for the Chandrasekhar operator [8, 9, 16]. We remark that the Brown-Ravenhall operator is strictly positive [35]. The lowest eigenvalue of the no-pair operator, however, is expected to tend to −∞ as the strength of a constant exterior magnetic field is increased; see [17] for some numerical evidence. Secondly, in order to find a distinguished self-adjoint realization of the no-pair operator we show that the corresponding quadratic form is bounded from below, for all coupling constants less than or equal to the critical one of the Brown-Ravenhall model. This has been known before only in the case A = 0 [12] and all we actually do is to reduce the problem to that special case. (For smaller values of the coupling constant, there exist, however, results on the stability of matter of the second kind in the free picture, where a gauge fixed vector potential is considered as a variable in the minimization. In this situation the field energy is added to the multi-particle Hamiltonian; see [21] and [20] for quantized fields. It is actually important to include the vector potential in the projection determining

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the model for otherwise instability occurs if at least two electrons are considered [14].) Finally, we state conditions ensuring that the essential spectrum of the no-pair operator equals [1, ∞) and that it has infinitely many discrete eigenvalues below 1. As a byproduct of our analysis – roughly speaking, by ignoring√the projections – we find pointwise exponential decay estimates with a rate a < 1 − λ2 for the eigenfunctions of magnetic Coulomb-Dirac operators corresponding to an eigenvalue λ ∈ (−1, 1). Although such bounds are essentially well-known [4, 16, 36] it seems illustrative to include them as a remark here. For a general scheme to study the exponential decay of solutions of an elliptic system of partial differential equations we refer to [27, 28].

2. Definition of the model and main results 2.1. The no-pair operator If energies are measured in units of the rest energy of the electron and lengths in units of one Compton wave length divided by 2π, then the free Dirac operator is given as 3 D0 := −i α · ∇ + β := −i αj ∂xj + β. j=1

Here α = (α1 , α2 , α3 ) and β =: α0 are the usual 4 × 4 hermitian Dirac matrices. They are given as αi = σ1 ⊗ σi , i = 1, 2, 3, and β = σ3 ⊗ ½2 , where σ1 , σ2 , σ3 denote the standard Pauli matrices, and satisfy the Clifford algebra relations {αi , αj } = 2 δij ½,

0 i, j 3.

(2.1)

D0 is a self-adjoint operator in the Hilbert space H := L2 (3 , 4 )

with domain H 1 (3 , 4 ) and its purely absolutely continuous spectrum equals σ(D0 ) = σac (D0 ) = (−∞, −1] ∪ [1, ∞). Moreover, it is well-known [10] that the 3 3 free Dirac operator with magnetic vector potential A ∈ L∞ loc ( , ), DA := D0 + α · A

(2.2)

is essentially self-adjoint on the domain D := C0∞ (3 , 4 ).

(2.3)

We denote its closure again by the symbol DA . Its spectrum is again contained in the union of two half-lines [33], σ(DA ) ⊂ (−∞, −1] ∪ [1, ∞). In order to define the no-pair operator we introduce the spectral projections 1 1 + ½ + sgn(DA ), Λ− (2.4) Λ+ A := E[0,∞) (DA ) = A := ½ − ΛA , 2 2 and a (matrix-valued) potential, V , satisfying the following hypothesis.

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3 4 Hypothesis 1. V ∈ L∞ loc ( \ {0}, L ( )),

V (x) = V (x)∗ ,

x = 0,

V (x) −→ 0,

and there exist γ ∈ (0, 1) and ρ > 0 such that γ , V (x)L (4 ) |x|

|x| → ∞,

0 < |x| < ρ.

(2.5)

(2.6)

The no-pair operator is an operator acting in the projected Hilbert space HA+ := Λ+ AH ,

(2.7)

which on the dense subspace Λ+ A D is given as + BA,V ϕ+ := DA ϕ+ + Λ+ AV ϕ ,

ϕ+ ∈ Λ+ A D.

(2.8)

It is not completely obvious that V Λ+ A ψ is again square-integrable, for every ψ ∈ D. This follows, however, from Lemma 3.6 below. In order to define a distinguished self-adjoint realization of BA,V we shall assume that V satisfies Hypothesis 1 with γ γc , where 2 (2.9) γc := (π/2) + (2/π) is the critical coupling constant of the Brown-Ravenhall model determined in [12]. γ In the case of the atomic Coulomb potential, V (x) = − |x| ½, the coupling constant 2 is given by γ = e Z, where Z ∈ and the square of the electric charge, e2 , is equal to the Sommerfeld fine structure constant in our units, e−2 ≈ 137.037. Since e2 γc ≈ 124.2 the restriction on the strength of the singularities of V imposed in (2.6) with γ < γc or γ γc allows for all nuclear charges up to Z 124. It is shown in [12] that the quadratic form of B0,−γ/|·| is bounded below on Λ+ 0 D, for all γ ∈ [0, γc ], and unbounded below if γ > γc . (Due to a result of [35] one actually has the strictly positive lower bound 1 − γ, for all γ ∈ [0, γc ].) Combining this + with some new technical results on the spectral projections Λ+ A and Λ0 derived in Section 3, we prove the following theorem in Section 4. Theorem 2.1. (i) Assume that V fulfills Hypothesis 1 with γ ∈ (0, γc ) and that 3 3 A ∈ L∞ loc ( , ). Then + + inf ϕ | BA,V ϕ+ : ϕ+ ∈ Λ+ > −∞. (2.10) A D, ϕ = 1 In particular, by the KLMN-theorem, BA,V has a distinguished self-adjoint 1/2 extension with form domain Q(DAH + ) = Λ+ ). A D(|DA | A 3 3 (ii) Assume that V fulfills Hypothesis 1 with γ ∈ (0, γc ] and that A ∈ L∞ loc ( , ) is Lipschitz continuous in some neighbourhood of 0. Then (2.10) holds true also. In particular, BA,V has a self-adjoint Friedrichs extension. The self-adjoint extension of BA,V given by Theorem 2.1 is again denoted by the same symbol. We then have the following result.

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Theorem 2.2. Assume that V fulfills Hypothesis 1 with γ ∈ (0, γc ] and that A ∈ 3 3 L∞ loc ( , ) and let : (−∞, 1) → (0, 1] be given by (1.1). If γ = γc , assume additionally that A is locally Lipschitz continuous. Then σess (BA,V ) ⊂ [1, ∞), and for every eigenvector, φλ , of BA,V corresponding to an eigenvalue λ < 1 and every a < (λ), a|·| e φλ < ∞. H

Proof. Theorem 2.2 is a consequence of Theorems 5.2, 5.3, and 6.1.

In order to derive pointwise decay estimates for all partial derivatives of eigenfunctions we introduce further assumptions on A and V . Hypothesis 2. A ∈ C ∞ (3 , 3 ) and, for all ε > 0 and β ∈ K(ε, β) ∈ (0, ∞) such that |∂xβ A(x)| K(ε, β) eε|x| ,

30 ,

x ∈ 3 .

there is some (2.11)

V ∈ C ( \ {0}, L ( )) fulfills (2.5)& (2.6) and, for all r > 0 and β ∈ there is some C(r, β) ∈ (0, ∞) such that sup ∂xβ V (x) L (4 ) C(r, β). (2.12) ∞

3

4

3 0,

|x|r

We remark that our L2 -exponential bounds on eigenfunctions of BA,V are 3 3 completely independent from the behaviour of A ∈ L∞ loc ( , ) away from the nucleus; see Theorems 5.2&5.3. It seems, however, natural to introduce the condition (2.11) to infer the pointwise bounds of Theorem 2.3 below by means of an induction argument starting from Theorem 2.2. In fact, since we always consider decay rates which are strictly less than (λ) we can borrow a bit of the exponential decay of the eigenfunction φλ to control terms containing a vector potential satisfying (2.11). Theorem 2.3. Assume that A and V fulfill Hypothesis 2 with γ ∈ (0, γc ]. Let φλ be an eigenvector of BA,V corresponding to an eigenvalue λ < 1 and be the function defined in (1.1). Then φλ ∈ C ∞ (3 \ {0}, 4 ) and, for all a < (λ) and β ∈ 30 , we find some C(λ, a, β) ∈ (0, ∞) such that ∀ x ∈ 3 , |x| 1 :

|∂xβ φλ (x)| C(λ, a, β) e−a|x| .

Proof. The statement follows from Theorem 7.1 and the Sobolev embedding theorem. 2.2. The Dirac operator As a remark we state L2 - and pointwise exponential decay estimates for the Dirac operator although they are consequences of the L2 -estimates in [4] and the ellipitic regularity of D0 . In Section 5 we present a proof of the L2 -exponential localization of spectral projections of the Dirac operator since our argument – a new variant

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of one given in [2] which easily extends to the no-pair operator – is particularly simple in this case. 3 3 We assume that V fulfills Hypothesis 1 and that A ∈ L∞ loc ( , ) in what follows. Then it is well-known (and explained in more detail in [30, Proposition 4.3]) that the results of [6, 10, 26] ensure the existence of a distinguished self-adjoint extension, DA,V , of the Dirac operator defined by DA,V ϕ := (D0 + α · A + V ) ϕ,

ϕ ∈ D.

This extension is uniquely determined by the conditions (i) D(DA,V ) ⊂ Hloc (3 , 4 ). (ii) For all ψ ∈ H 1/2 (3 , 4 ) having compact support and all φ ∈ D(DA,V ),

ψ | DA,V φ = |D0 |1/2 ψ sgn(D0 )|D0 |1/2 φ + |X|1/2 ψ U |X|1/2 φ , 1/2

where U |X| is the polar decomposition of X := α · A + V . Standard arguments show that DA,V has the local compactness property and since V drops off to zero at infinity this in turn implies that σess (DA,V ) = σess (DA ) ⊂ (−∞, −1] ∪ [1, ∞).

(2.13)

3 3 Theorem 2.4. Assume that A ∈ L∞ loc ( , ) and that V fulfills Hypothesis 1. Let φλ be a normalized eigenvector of D corresponding to an eigenvalue λ ∈ (−1, 1). A,V √ 2 Then, for all a ∈ (0, 1 − λ ), there is some A-independent constant C(λ, a) ∈ (0, ∞) such that a|·| e φλ C(λ, a). Assume additionally√that A and V fulfill Hypothesis 2. Then φλ ∈ C ∞ (3 \{0}, 4) and, for all a ∈ (0, 1 − λ2 ) and β ∈ 30 , there is some C(λ, a, β) ∈ (0, ∞) such that β ∂x φλ (x) C(λ, a, β) e−a|x| . ∀ x ∈ 3 , |x| 1 : (2.14)

Proof. The assertions follow from Theorem 5.1 and standard arguments using the elliptic regularity of D0 and the Sobolev embedding theorem. (Terms containing derivatives of the vector potential A are dealt with as in (7.7).) 2.3. Examples To complete the picture we state some conditions on A and V which ensure the existence of infinitely many eigenvalues of BA,V (resp. DA,V ) below 1 (resp. in (−1, 1)) and which imply that the essential spectrum covers the whole half-line [1, ∞) (resp. (−∞, −1] ∪ [1, ∞)). The properties of A which are explicitly used in the proofs are stated in the following hypothesis, where BR (y) := x ∈ 3 : |x − y| < R , y ∈ 3 , R > 0. Hypothesis 3. (i) A ∈ C ∞ (3 , 3 ) and, for every λ 1, there exist radii, 1 R1 < R2 < . . ., Rn ∞, and normalized spinors ψ1 (λ), ψ2 (λ), . . . ∈ D such that supp(ψn (λ)) ⊂ 3 \ BRn (0),

lim (DA − λ) ψn (λ) = 0.

n→∞

(2.15)

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(ii) The assumptions of Part (i) are fulfilled and the Weyl sequence {ψn (1)}n∈ has the following additional properties: Its elements have vanishing lower spinor components, ψn (1) = (ψn,1 (1), ψn,2 (1), 0, 0) , n ∈ , there is some δ ∈ (0, 1) such that 2Rn Rn+1 , supp(ψn (1)) ⊂ Rn < |x| < (1 + δ) Rn , (2.16) for all n ∈ , and

DA − 1 ψn (1) = O(1/Rn ),

n → ∞.

(2.17)

Obviously, the vectors ψn (λ) in (2.15) form a Weyl sequence for DA,V and it is easy to see that their projections onto HA+ define a Weyl sequence for BA,V . Under Hypothesis 3(ii) the vectors ψn (1) can be used as test functions in a minimax principle to prove the existence of infinitely many bound states. To give some explicit conditions we recall a result from [15] which provides a large class of examples where Hypotheses 3(i)&(ii) are fulfilled Example ([15]). (i) Suppose that A ∈ C ∞ (3 , 3 ), B = curl A, and set, for x ∈ 3 and ν ∈ , α |α|=ν |∂ B(x)| . ν (x) := 0 (x) := |B(x)|, 1 + |α| 0 such that ρn ∞, the balls Bρn (zn ), n ∈ , are mutually disjoint and sup ν (x) x ∈ Bρn (zn ) −→ 0, n → ∞. Then A fulfills Hypothesis 3(i). This follows directly from the constructions presented in [15]. (ii) Suppose additionally that there is some C ∈ (0, ∞) such that ρn < |zn | C ρn , for all n ∈ , and that either n ∈ , sup |B(x)| : x ∈ Bρn (zn ) C/|zn |2 , or ∀ n ∈ : |B(zn )| 1/C

and

sup ν (x) x ∈ Bρn (zn ) = o(ρ−ν n ).

Then A fulfills Hypothesis 3(ii). This follows by inspecting and adapting the relevant proofs in [15]. Since this procedure is straight-forward but a little bit lengthy we refrain from explaining any detail here. 2 Theorem 2.5. Assume that V fulfills Hypothesis 1 with γ ∈ (0, γc ]. If A fulfills Hypothesis 3 (i) then σess (BA,V ) = [1, ∞). If A fulfills Hypothesis 3 (ii) and if ∃ γ>0

∀ x ∈ 3 \ {0} :

max

v | V (x) v − γ min{1, |x|−1 }, (2.18)

v∈4 : |v|=1

then BA,V has infinitely many eigenvalues below and accumulating at 1. Proof. Theorem 2.5 follows from Theorem 2.2, Lemma 6.2, and Theorem 6.3.

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Theorem 2.6. Assume that V fulfills Hypothesis 1. If A fulfills Hypothesis 3(i), then σess (DA,V ) = (−∞, −1]∪[1, ∞). If A fulfills Hypothesis 3(ii) and V satisfies (2.18), then there exist infinitely many eigenvalues of DA,V in (−1, 1). Proof. In view of (2.13) and since V drops off to zero at infinity the first statement is clear. The second assertion is a special case of [22, Theorem 2.9].

3. Miscellaneous results on spectral projections In order to obtain any of our results on the no-pair operator it is crucial from a technical point of view to have some control on commutators of Λ+ A with multipli+ and Λ . Appropriate estimates cation operators and on the difference between Λ+ 0 A are derived in this section. They are based on the formula (2.4) and the representation of the sign function of a self-adjoint operator, T , acting in some Hilbert space, K , with 0 ∈ (T ) as a strongly convergent Cauchy principal value,

τ dy ψ∈K. (3.1) (T − iy)−1 ψ , sgn(T ) ψ = T |T |−1 ψ = lim τ →∞ −τ π We write RA,V (z) := (DA,V − z)−1 ,

RA (z) := (DA − z)−1 ,

(3.2)

in what follows. Then another frequently used identity is RA, V (z) µ − µ RA,V (z)

= RA, V (z) iα · ∇µ − µ (V − V ) + µ α · (A − A) RA,V (z),

(3.3)

where z ∈ (DA,V ) ∩ (DA, V ). Here we assume that V and V fulfill Hypothesis 1, such that V and µ (V − V ) are bounded, matrix-valued multiplication operators, and µ satisfy and that A, A, ∈ L∞ (3 , 3 ), µ ∈ C ∞ (3 , ), A, A loc (3.4) ∞ < ∞. µ∞ + ∇µ∞ + µ (A − A) (Using the essential self-adjointness of DA, V D , it is actually simpler and sufficient to derive the adjoint of (3.3).) A combination of (2.4), (3.1), and (3.3) yields the µ satisfy (3.4), following formula, where φ, ψ ∈ H and A, A, φ (Λ+ µ − µ Λ+ A) ψ A

τ (3.5)

RA (iy) ψ dy . φ RA (iy) α · i∇µ + µ (A − A) = lim τ →∞ −τ 2π We also recall the identities

|λ| |DA |1/2 RA (iy) ψ 2 dy = dy dEλ (DA )ψ2 = π ψ2 , (3.6) 2 + y2 λ

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3 3 for all ψ ∈ H , A ∈ L∞ loc ( , ), and 3 3 y ∈ , A ∈ L∞ loc ( , ),

RA (iy) = (1 + y 2 )−1/2 ,

v∈ .

α · v L (4 ) = |v|,

3

(3.7) (3.8)

Here (3.8) follows from (2.1). Finally, we need the following crucial estimate stating that RA (z) stays bounded after conjugation with suitable exponential weights, eF , acting as multiplication operators in H . Although it is well-known (see, e.g., [4]), we recall its proof since it determines the exponential decay rates in our main theorems. √ 3 3 2 Lemma 3.1. Let A ∈ L∞ loc ( , ), λ ∈ (−1, 1), y ∈ , a ∈ [0, 1 − λ ), and let F ∈ ∞ 3 C ( , ) have a fixed sign and satisfy |∇F | a. Then λ + iy ∈ (DA + iα · ∇F ), eF RA (λ + iy) e−F = (DA + iα · ∇F + λ + iy)−1D(e−F ) , and

√ 3 1 + y 2 + λ2 + a2 1 + y 2 − λ2 − a2 √ √ 3 1 + λ2 + a2 . 2 1 + y 1 − λ2 − a2

(3.9)

F e RA (λ + iy)e−F

(3.10)

Proof. A straightforward calculation yields, for z = λ + iy, λ ∈ (−1, 1), y ∈ ε > 0, and ϕ ∈ D, 1 eF (DA − z) e−F ϕ 2 + 3ε α · (−i∇ + A) ϕ 2 4ε + 3ε (1 + |z|2 ) ϕ2 + 3ε ϕ |∇F |2 ϕ e−F (DA + z) eF ϕ eF (DA − z) e−F ϕ 2

= α · (−i∇ + A) ϕ + ϕ 1 − z 2 − |∇F |2 ϕ .

,

Together with |∇F | a and z 2 = λ2 − y 2 this implies F e (DA − z) e−F ϕ 2 4ε (b− − 3ε b+) ϕ2 , where b± := 1 + y 2 ± λ2 ± a2 . The optimal choice for ε is ε = b− /(6b+ ). Since D is a core for the closed operator DA + iα · ∇F with domain D(DA ) it follows that (DA + iα · ∇F − z) ψ b− (3b+ )−1/2 ψ, ψ ∈ D(DA ). (3.11) We may replace F, z by −F, z in (3.11), whence Ran(DA + iα · ∇F − z)⊥ = Ker(DA −iα·∇F −z) = {0}. On the other hand we know that Ran(DA +iα·∇F −z) is closed since DA + iα · ∇F − z is closed with a continuous inverse. It follows that z ∈ (DA + iα · ∇F ). We assume that F 0 in the rest of this proof. Let ψ ∈ H . We pick a sequence, {ϕn }n∈ ∈ D , which converges to η := (DA +iα·∇F −z)−1 ψ ∈ D(DA ) with respect to the graph norm of DA . Passing to the limit in RA (z) e−F (DA + iα · ∇F − z) ϕn = e−F ϕn ,

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we obtain RA (z) e−F ψ = e−F η, which implies eF RA (z) e−F = (DA + iα · ∇F − z)−1 . Taking the adjoint we get

∗ = (DA − iα · ∇F − z)−1 . e−F RA (z) eF ⊂ eF RA (z) e−F (3.12) and (3.13) together prove (3.9).

(3.12) (3.13)

To shorten the presentation and since it is sufficient for our applications below we consider only bounded weight functions F in the following Lemma 3.2. Similar estimates have already been derived in [22]. 3 3 ∞ 3 Lemma 3.2. (i) Let A ∈ L∞ loc ( , ), a ∈ [0, 1), χ ∈ C ( , [0, 1]) with ∇χ ∈ ∞ 3 3 ∞ 3 ∞ 3 C0 ( , ), and let F ∈ C ( , ) ∩ L ( , ) have a fixed sign and satisfy |∇F | a. Then √ |DA |1/2 [Λ+ , χ eF ] e−F 6 · a + ∇χ∞ . (3.14) A 2 1 − a2 In particular, √ F F + −F ΛA 1 + 6 · a , where ΛF . (3.15) A := e ΛA e 2 1 − a2 Moreover, √ F 6 e [χ, Λ+ ] eF ∇χ ∞ . (3.16) A 2(1 − a2 ) (ii) Assume additionally that ∇χ = ∇F = 0 in a neighbourhood, U ⊂ 3 , of 0 and let ζ ∈ C0∞ (U, [0, 1]). Then ζ √

a + ∇χ∞ F −F [Λ+ . (3.17) 6 ∇ζ∞ + ζ A∞ A, χ e ] e |·| 1 − a2 √ If a = 0, then the factor 6 in (3.14), (3.16), and (3.17) can be replaced by 1.

Proof. On account of (3.5) we have, for φ ∈ D(|DA |1/2 ) and ψ ∈ H , |DA |1/2 φ [Λ+ , χ eF ] e−F ψ A

dy |DA |1/2 φ RA (iy) iα · (∇χ + χ ∇F ) eF RA (iy) e−F ψ 2π 1/2 ψ2 dy 1/2 √ a + ∇χ∞ 2 |DA |1/2 RA (−iy) φ dy . 6 2 2π(1 − a2 ) 1+y √ √ √ In the last line we have used (3.8) and (3.10) (with λ = 0 and 3 1 + a2 6). F −F Applying (3.6) we conclude that [Λ+ ψ ∈ D(|DA |1/2∗ ) = D(|DA |1/2 ) A, χ e ] e and that (3.14) holds true. The bound (3.16) follows from

F dy −F e φ [χ, Λ+ ] ψ , e RA (−iy)eF φ (eF α · ∇χ) RA (iy) ψ A 2π for all φ, ψ ∈ H , together with (3.7), (3.8), and (3.10).

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In order to prove Part (ii) we first observe that the additional assumption = 0) this permits implies ζ (∇χ+χ ∇F ) = 0. Together with (3.7) and (3.3) (with A 1 3 4 to get, for φ ∈ H ( , ) and ψ ∈ H , ζ | · |−1 φ [Λ+ , χ eF ] e−F ψ A

dy 1 φ ζ RA (iy) iα · (∇χ + χ ∇F ) eF RA (iy) e−F ψ |·| 2π

1 φ R0 (iy) α · {i∇ζ + ζ A} = |·| dy × RA (iy) iα · (∇χ + χ ∇F ) eF RA (iy) e−F ψ 2π √

6 R0 (−iy) | · |−1 ∇ζ∞ + ζ A∞ a + ∇χ∞ φ ψ dy . 2π 1 − a2 1 + y2 By Hardy’s inequality R0 (−iy) | · |−1 = | · |−1 R0 (iy) 2, for all y ∈ . The last statement of this lemma follows from an obvious modification of the proof above. In fact, in the case a = 0 we can always use (3.7) where (3.10) has been applied before. In what follows we set, for any vector-valued function u : 3 → 3 , 3 |ui (x) − ui (y)| ui ∞ + sup . uLip1 := |x − y| x=y i=1 3 3 ∞ 3 Lemma 3.3. Let A ∈ L∞ loc ( , ) and µ ∈ C0 ( , ). Then

|D0 |1/2 (Λ+ µ − µ Λ+ ) |DA |1/2 1 ∇µ∞ + µ A∞ . (3.18) 0 A 2 Assume further that A is Lipschitz continuous in a neighbourhood, U ⊂ 3 , of supp(µ) and let χ ∈ C0∞ (U, [0, 1]) be such that χ µ = µ. Then

√

D0 (Λ+ µ−µ Λ+ ) 1 ∇µLip1 +µ ALip1 3+∇χ∞ +χ A∞ . (3.19) 0 A 2 Proof. In view of (3.5) and (3.6) we have, for ϕ ∈ D(|D0 |1/2 ) and ψ ∈ D(|DA |1/2 ), |D0 |1/2 ϕ (Λ+ µ − µ Λ+ ) |DA |1/2 ψ 0 A

dy |D0 |1/2 ϕ R0 (iy) α · (i∇µ + µ A) RA (iy) |DA |1/2 ψ 2π ∇µ∞ + µ A∞ ϕ ψ. 2 This implies (3.18). In order to prove (3.19) we use i∇µ + µ A = (i∇µ + µ A) χ and (3.3) to write

R0 (iy) α · (i∇µ + µ A) RA (iy) = R0 (iy) α · (i∇µ + µ A) R0 (iy) χ − R0 (iy) α · (i∇µ + µ A) R0 (iy) α · (i∇χ + χ A) RA (iy).

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This identity yields, for all ϕ ∈ D(D0 ) and ψ ∈ H , D0 ϕ (Λ+ µ − µ Λ+ ) ψ 0 A

|D0 |1/2 R0 (−iy) ϕ |D0 |1/2 α · (i∇µ + µ A) |D0 |−1/2 dy × |D0 |1/2 R0 (iy) ψ 2π

D0 R0 (−iy) ∇µ + µ A ∇χ + χ A ϕ ψ dy . = 2π(1 + y 2 )

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(3.20)

Since each matrix entry of M := α·(i∇µ+µ A) is a Lipschitz continuous, compactly supported function and since αi comutes with |D0 |1/2 we readily verify (e.g., by using an explicite integral formula for (1 − ∆)1/4 f 2 [19, Theorem 7.12]; see Appendix A) that 3 |D0 |1/2 (∂i µ + µ Ai ) |D0 |−1/2 |D0 |1/2 M |D0 |−1/2 i=1

√

3 ∇µLip1 + µ ALip1 .

Therefore, (3.19) follows from the above estimates and (3.6).

Remark 3.4. It can easily be read off from the previous proof that √ D0 [Λ+ , µ] 3 ∇µLip1 . (3.21) 0 2 (In fact, if A = 0 then the term in (3.20) is superfluous.) A similar bound has been derived in [25] by means of an explicite formula for the integral kernel of Λ+ 0. 3 3 ∞ 3 ∞ 3 3 Lemma 3.5. Let A ∈ L∞ loc ( , ), χ ∈ C ( , [0, 1]) with ∇χ ∈ C0 ( , ), and ≡ 1 on supp(∇χ). Then let χ ∈ C0∞ (3 , [0, 1]) satisfy χ

DA [Λ+ , χ] 3 χLip1 1 + ∇ χ ∞ + χ (3.22) A∞ . A 2 Proof. Using ∇χ = χ ∇χ and (3.3), we write the term appearing on the right side of (3.5) as

R0 (z) iα · (∇χ) R0 (z) χ RA (z) iα · ∇χ RA (z) = χ χ+χ A) RA (z) − RA (z) iα · ∇χ R0 (z) α · (i∇ + RA (z) α · (i∇ χ−χ A) R0 (z) iα · ∇χ R0 (z) χ . Using this we infer from (3.5) (with φ = DA ϕ) that, for ϕ ∈ D and ψ ∈ H , DA ϕ [Λ+ , χ] ψ A

τ dy lim χ (D0 + α · A) ϕ R0 (z) iα · (∇χ) R0 (z) χ ψ τ →∞ −τ 2π

dy DA RA (−iy) ϕ ∇χ∞ ∇ . χ∞ + χ A∞ ψ + π(1 + y2)

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Applying (3.5) backwards, we thus obtain DA ϕ [Λ+ , χ] ψ A χ+χ A) ϕ [Λ+ χ ϕ D0 [Λ+ ψ + α · (i∇ ψ 0 , χ] χ 0 , χ] χ

+ ∇χ∞ ∇ χ∞ + χ A∞ ϕ ψ. Taking also (3.21) and [Λ+ 0 , χ] ∇χ/2 into account we arrive at the assertion. We close this section by stating another consequence of the resolvent identity (3.3) showing that the no-pair operator BA,V is actually well-defined on Λ+ A D. + 3 3 1 3 4 Lemma 3.6. Let A ∈ L∞ loc ( , ). Then ΛA maps D(DA ) into Hloc ( , ). In + particular, V ΛA ϕ ∈ H , for every ϕ ∈ D(DA ), provided V fulfills Hypothesis 1.

Proof. The identity (3.3) implies, for all ϕ ∈ D(DA ) and χ ∈ C0∞ (3 ), + + χ Λ+ A ϕ = χ RA (0) ΛA DA ϕ = R0 (0) χ − α · (i∇χ + χ A) RA (0) ΛA DA ϕ.

4. Semi-boundedness of the no-pair operator In the following we show that the quadratic form of BA,V is bounded below on the + dense subspace Λ+ A D ⊂ HA provided that one of the conditions of Theorem 2.1 is fulfilled. To obtain this result we simply compare the models with and without magnetic fields by means of Lemma 3.3. Proof of Theorem 2.1. We pick two cutoff functions µ1 , µ2 ∈ C ∞ (3 , [0, 1]) such that µ1 ≡ 1 in a neighbourhood of 0, µ1 ≡ 0 outside some larger neighbourhood, and µ21 + µ22 = 1. In the case γ = γc we may assume that A is Lipschitz continuous on the support of µ1 by choosing the latter small enough. In view of Hypothesis 1 we may further assume that V −γ/| · |. The following identities are valid on D(DA ), DA = µi DA µi − iα · (∇µi ) µi DA µ2i = i=1,2

=

i=1,2

µi DA µi

− iα · ∇(µ21 + µ22 )/2 =

i=1,2

µi DA µi .

i=1,2

Consequently, we have, for ϕ+ ∈ Λ+ A D, + ϕ+ µi (DA + V ) µi ϕ+ . ϕ BA,V ϕ+ =

(4.1)

i=1,2

A direct application of (3.22) yields + ϕ µ2 (DA + V ) µ2 ϕ+ + + µ2 ϕ+ Λ+ − µ22 V ϕ+ 2 A DA ΛA µ2 ϕ

χ ∞ + χ − C µ2 Lip1 1 + ∇ A∞ ϕ+ 2 ,

(4.2)

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where χ ∈ C0∞ (3 , [0, 1]) equals one in a neighbourhood of supp(∇µ2 ). On account + + 1 3 4 of Lemma 3.6 we further have Λ+ 0 µ1 ϕ ∈ Λ0 H ( , ) ⊂ D(B0,−γc /|·| ), which implies + γ + ϕ µ1 (DA + V ) µ1 ϕ+ µ1 ϕ+ (B0,−γc /|·| ⊕ Λ− 0 ) µ1 ϕ γc + + + (1 − γ/γc ) µ1 ϕ+ Λ+ (4.3) 0 D0 Λ0 µ1 ϕ + 2 + + ϕ µ1 α · A ϕ − + 1 (4.4) − 2γ µ1 ϕ+ Λ+ 0 |·| Λ0 µ1 ϕ

− γ + − + (4.5) + µ1 ϕ Λ0 D0 − |·| Λ0 µ1 ϕ . In order to estimate the terms in (4.4) and (4.5) we write − + + Λ− = (Λ− 0 µ1 ϕ 0 µ1 − µ1 ΛA ) ϕ

and apply Lemma 3.3. The term in (4.5) can be treated by means of (3.18) and Kato’s inequality, | · |−1 (π/2) |∇|. In the case γ = γc , where A is assumed to be Lipschitz continuous on the support of µ1 , the bound (3.19) is available and can be applied together with Hardy’s inequality to estimate the term in (4.4). If γ < γc we apply (3.18) instead and employ the part of the kinetic energy appearing in 1 + 2 (4.3) and Kato’s inequality to control the term ε |·| Λ+ 0 µ1 ϕ in 2γ µ1 ϕ+ Λ+ 1 Λ− µ1 ϕ+ 0 |·|

0

γ2 11/2 (Λ− µ1 − µ1 Λ− ) 2 ϕ+ 2 , 0 A |·| ε for some sufficiently small ε > 0. Combining this with (4.2), we arrive at γ + µ1 ϕ+ (B0,−γc /|·| ⊕ Λ− − C ϕ+ 2 ,

ϕ+ | BA,V ϕ+ 0 ) µ1 ϕ γc + 2 ε |·|11/2 Λ+ 0 µ1 ϕ +

(4.6)

where the constant C ∈ (0, ∞) does not depend on the behaviour of A outside the supports of χ and µ1 and certainly not on ϕ+ ∈ Λ+ A D. Since B0,−γc /|·| is strictly positive [35] this proves the theorem. For later reference we note that the previous proof (recall (4.6) and the choice of supp(µ1 )) implies the following result: Remark 4.1. If V fulfills Hypothesis 1 with γ ∈ [0, γc ] and if A : 3 → 3 is locally Lipschitz continuous then, for every χ ∈ C0∞ (3 , [0, 1]), there is some c ≡ c(χ, A) ∈ (0, ∞) such that + + + ϕ ΛA χ (B0,−γc /|·| ⊕ Λ− ϕ Λ+ 0 ) χ ΛA ϕ A (BA,V + c) ΛA ϕ , for all ϕ ∈ D. Since Λ+ A D is a form core of BA,V this estimate implies that + −1/2 (B + c Λ ) maps HA+ into the form domain of B0,−γc /|·| ⊕ Λ− χ Λ+ A,V 0 and A A (B0,−γ /|·| ⊕ Λ− )1/2 χ Λ+ (BA,V + c Λ+ )−1/2 < ∞. (4.7) 0 c A A L (H + ,H ) A

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5. L2 -exponential localization In this section we derive L2 -exponential localization estimates for spectral projections of the Dirac and no-pair operators. Our proofs are new variants of an idea from [2]. We emphasize that the argument developed in [2] requires no a`-priori knowledge on the spectrum. In particular, one may first prove the exponential localization of the spectral subspace corresponding to some interval I and then infer that the spectrum in I is discrete by means of a simple argument observed in [13]; see Theorem 6.1 below. First, we consider the Dirac operator in which case the assertion of the following theorem is more or less folkloric. Its proof below extends, however, easily to the non-local no-pair operator. For any subset I ⊂ (−1, 1), we introduce the notation (5.1) δ(I) := inf{ 1 − λ2 : λ ∈ I }. Theorem 5.1. Assume that V fulfills Hypothesis 1 with γ ∈ [0, 1) and that A ∈ 3 3 L∞ loc ( , ) and let I ⊂ (−1, 1) be some compact interval. Then, for every a < 3 3 δ(I), there exists a constant C(a, I) ∈ (0, ∞) such that, for all A ∈ L∞ loc ( , ), a|x| e (5.2) EI (DA,V ) C(a, I). Proof. First, we fix a ∈ (0, δ(I)), pick some cut-off function χ ∈ C ∞ (3 , [0, 1]) such that χ(x) = 0, for |x| 1, and χ(x) = 1, for |x| 2, and set χR (x) := χ(x/R), x ∈ 3 , R 1. By the monotone convergence theorem it suffices to show that χ2R eF EI (DA,V ) const(a, R), for some R 1 and all functions F satisfying F ∈ C ∞ (3 , ) ∩ L∞ (3 , ),

F (x) = 0, |x| 1,

F 0,

|∇F | a,

(5.3)

To this end we introduce VR := χR V,

R 1,

and pick some ε > 0 such that it still holds a < δ(Iε ), where Iε := I + (−ε, ε). Choosing R 1 sufficiently large we may assume in the following that every z ∈ Iε + i belongs to the resolvent set of DA + iα · ∇F + VR , for every F satisfying (5.3) (in particular F = 0). Using the notation (3.2), we may further assume that C(a, R) : = sup eF RA,VR (z) e−F : (5.4) 3 3 < ∞. z ∈ Iε + i, A ∈ L∞ loc ( , ), F satisfies (5.3). In fact, since VR → 0, R → ∞, this is a simple consequence of Lemma 3.1. Next, we pick some ω ∈ C0∞ (, [0, 1]) such that ω ≡ 1 on I and supp(ω) ⊂ Iε and preserve the symbol ω to denote an almost analytic extension of ω to a smooth, compactly supported function on the complex plane such that supp(ω) ⊂ Iε + i(−δ, δ) ⊂ (DA,VR + iα · ∇F ),

∂z ω(z) = ON |z|N , N ∈ .

(5.5)

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Here ∂z = 12 (∂ z + i∂ z ) and δ > 0 can be chosen arbitrarily. We have ω(DA,VR ) = 0. By virtue of the Helffer-Sj¨ ostrand formula,

i ∂z ω(z) dz ∧ dz, (T − z)−1 dω(z), dω(z) := − ω(T ) = 2π which holds for every self-adjoint operator T on some Hilbert space (see, e.g., [11]; one could also use a similar formula due to Amrein et al. [1, Theorem 6.1.4(d)] which avoids almost analytic extensions but consists of a sum of integrals over resolvents), we deduce that

χ2R EI (DA,V ) = χ2R ω(DA,V ) − ω(DA,VR ) χ2R EI (DA,V )

= χ2R RA,V (z) − RA,VR (z) χ2R EI (DA,V ) dω(z). Since χ2R (V − VR ) = 0 we infer by means of (3.3) that, for all F satisfying (5.3), χ2R eF EI (DA,V )

= − eF RA,VR (z) e−F (eF iα · ∇χ2R ) RA,V (z) EI (DA,V ) dω(z). On account of (5.3), (5.4), and (5.5) we thus get

4aR ∇χ∞ |dω(z)| χ2R eF EI (DA,V ) C(a, R) e < ∞. 2R |z|

(5.6)

Theorem 5.2. Assume that V fulfills Hypothesis 1 with γ ∈ [0, γc ] and that A ∈ 3 3 L∞ loc ( , ). If γ = γc assume further that A is Lipschitz continuous in some neighbourhood of 0. Let I ⊂ (−1, 1) be some compact interval and a ∈ (0, δ(I)). Then D(ea|x| ) ⊃ Ran(EI (BA,V )) there is some A-independent constant C(a, I) ∈ (0, ∞) such that, for all ζ ∈ C0∞ ({|x| ρ}, [0, 1]) with ζ ≡ 1 in a neighbourhood of 0 (ρ is the parameter appearing in Hypothesis 1), a|x|

e EI (BA,V ) L (H + ,H ) C(a, I) 1 + ∇ζ + ζ A + (1 − ζ) V . A

Proof. We fix some a ∈ (0, δ(I)) and define A,V := BA,V ⊕ DA Λ− , D A

(5.7)

A,V ) Λ+ = EI (BA,V ) ⊕ 0. We choose χR , VR , ε, I, and ω in the same so that EI (D A way as in the proof of Theorem 5.1 and introduce the comparison operator A,VR := DA + Λ+ VR Λ+ . D A A

(5.8)

Then it is clear that A,VR ) Λ+ = 0, ω(D A for all sufficiently large R 1. In particular, writing A,V (z) := (D A,V − z)−1 , R

A,VR (z) := (D A,VR − z)−1 , R

(5.9)

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we deduce the following analogue of (5.6)

+ F A,V (z) − R A,VR (z)χ2R Λ+ EI (D A,V )dω(z). χ2R e EI (DA,V )ΛA = eF χ2R R A Therefore, it suffices to show that, for some sufficiently large R 1, there is some C(a, R) ∈ (0, ∞) such that, for all F satisfying (5.3)

A,VR (z) χ2R − χ2R R A,V (z) C(a, R). sup |z| eF R (5.10) z∈supp(ω)\ To this end we first remark that due to (3.15), (3.16), and VR → 0, R → ∞, we 3 3 find some constant C (a, R) ∈ (0, ∞) such that, for all A ∈ L∞ loc ( , ) and all F satisfying (5.3), χ2R , Λ+ VR Λ+ eF A A (5.11) +

F + Λ VR [χ2R , Λ+ ]eF C (a, R). [χ2R , Λ+ VR Λ−F A ]e A A A Writing χR := 1 − χR we further observe that

F −F F F V χR Λ+ χR , Λ + A e χ2R = (½{|x|2R} e ) V e A e χ2R ,

which together with (3.14) and (3.17) implies

V χR Λ+ eF χ2R C (a, R) ∇ζ + ζ A + (1 − ζ) V , A

(5.12)

for some constant C (a, R) ∈ (0, ∞) which neither depends on A nor ζ. Now, a straightforward computation yields, for ϕ ∈ D and z ∈ \ ,

A,VR (z) − R A,VR − z) ϕ A,V (z) χ2R (D χ2R R A,V (z) iα · ∇χ2R ϕ A,V (z) Λ+ V χR Λ+ χ2R ϕ − R =R A A

(5.13)

A,V (z) [χ2R , Λ+ VR Λ+ ] ϕ. −R A A A,VR − z)D is dense this together with (5.12) implies Since the range of (D

A,VR (z) − R A,V (z) χ2R eF χ2R R

A,VR (z) eF A,V (z) Λ+ V χR Λ+ eF χ2R e−F R (5.14) =R A A

A,VR (z) eF . A,V (z) iα · ∇χ2R eF + [χ2R , Λ+ VR Λ+ ] eF e−F R −R A A + Since Λ+ A VR ΛA → 0, R → ∞, Lemma 3.1 ensures that, for sufficiently large A,VR (z) eF is uniformly bounded, for all z ∈ supp(ω), R 1, the norm of e−F R ∞ 3 3 A ∈ Lloc ( , ), and every F satisfying (5.3). Taking the adjoint of (5.14) and using (5.11) and (5.12) we thus obtain (5.10).

Theorem 5.3. Assume that V fulfills Hypothesis 1 with γ ∈ (0, γc ] and that A ∈ 3 3 L∞ loc ( , ). Assume further that A is locally Lipschitz continuous if γ = γc . Then, for every a ∈ [0, 1), Ran(E(−∞,0) (BA,V )) ⊂ D(ea|·| ) and there is some Aindependent C(a) ∈ (0, ∞) such that, for all ζ ∈ C0∞ ({|x| ρ}, [0, 1]) with ζ ≡ 1 in a neighbourhood of 0, a|·|

e E(−∞,0) (BA,V ) C(a) 1 + ∇ζ + ζ A + (1 − ζ) V . + L (HA ,H )

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Proof. We fix a ∈ [0, 1). It follows from Theorem 5.2 and Theorem 6.1 below that the spectrum of BA,V in (−1, 1) is discrete, σ(BA,V ) ∩ (−1, 1) ⊂ σd (BA,V ).

(5.15)

In particular, we find some e0 ∈ (−1, 0) ∩ (BA,V ) such that E(−∞,0) (BA,V ) = E(−∞,e0 ] (BA,V ) and 1 − a2 − e20 > 0. Using the notation (5.7) and (5.8) we have A,V ) = E(−∞,e ] (D A,V ), E(−∞,0) (D 0

A,VR ) = Λ− , E(−∞,e0 ] (D A

provided R 1 is sufficiently large. Thanks to (3.15) we know that, for fixed R, F F eF Λ+ A (1 − χ2R ) = ΛA e (1 − χ2R ) is uniformly bounded, for all F satisfying (5.3). It thus remains to consider

eF Λ+ A χ2R E(−∞,e0 ] (BA,V ) ⊕ 0

+ = eF Λ+ A χ2R E(−∞,e0 ] (DA,V ) − E(−∞,e0 ] (DA,VR ) χ2R ΛA + 1 ΛA . = eF Λ+ (5.16) A sgn DA,VR − e0 χ2R − χ2R sgn DA,V − e0 2 A,VR − e0 A,V − e0 and D Using (3.1) and (5.9) to represent the sign function of D by a strongly convergent Cauchy principal value and using (5.16) we obtain, for all ψ ∈ H , F +

e Λ χ2R E(−∞,0) (BA,V ) ⊕ 0 ψ A

dy

F F A,V (e0 + iy) Λ+ ψ ΛA e RA,VR (e0 + iy) χ2R − χ2R R A 2π A,V , then we have If δ0 > 0 denotes the distance from e0 to the spectrum of D 2 2 1/2 RA,V (e0 + iy) = (δ0 + y ) . A straight-forward Neumann expansion employing (3.10) and F ΛF A VR ΛA −→ 0,

R → ∞,

further shows that, for every sufficiently large R 1, there is some C (a, R) ∈ 3 3 (0, ∞) such that, for all A ∈ L∞ loc ( , ) and all F satisfying (5.3), F C (a, R) A,VR (e0 + iy) e−F e R , 1 + y2

y ∈ .

Using (3.15), (5.11), (5.12), and (5.14), we thus arrive at F +

e Λ χ2R E(−∞,0) (BA,V )⊕0 ψ C (a, R) 1+∇ζ+ζ A+(1−ζ) V , A for all ψ ∈ H , ψ = 1, where the constant C (a, R) ∈ (0, ∞) neither depends on A nor ζ.

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6. The discrete and essential spectra of BA,V Next, we consider the discrete and essential spectra of BA,V . To start with we prove a theorem we have already referred to in the proof of Theorem 5.3 (to obtain (5.15)) and which completes our proof of Theorem 2.2. It is used to infer the lower bound on the essential spectrum of BA,V from our localization estimates and proved by adapting an argument we learned from [13] to the non-local no-pair operator. Certainly, one could also try to locate the essential spectrum of BA,V by a more direct method without relying on exponential localization estimates. We refer to [18] for recent developments relevant to this question and numerous references. Theorem 6.1. Assume that V fulfills Hypothesis 1 with γ ∈ [0, γc ] and let A ∈ 3 3 L∞ loc ( , ). If γ = γc assume further that A is locally Lipschitz continuous. Let I ⊂ (−∞, 1) be an interval such that Ran(EI (BA,V )) ⊂ D(eε|·| ), for some ε > 0. Then the spectral projection EI (BA,V ) is a compact and, hence, finite rank operator on HA+ . Proof. We pick some cut-off function χ ∈ C ∞ (3 , [0, 1]) such that χ(x) = 1, for |x| 1, and χ(x) = 0, for |x| 2, and set χR (x) := χ(x/R), x ∈ 3 , R 1. Since eε|x| EI (BA,V ) ∈ L (HA+ , H ) and since χR e−ε|x| → e−ε|x| , R → ∞, in the operator norm, it suffices to show that χR EI (BA,V ) = χR e−ε|x| eε|x| EI (BA,V ) is 3 3 compact, for every R 1. First, we show this assuming that A ∈ L∞ loc ( , ) and that V fulfills Hypothesis 1 with γ ∈ [0, γc ). Since DA has the local compactness property we know that χR |DA |−1/2 is compact, for all R 1. It thus remains to show that |DA |1/2 EI (BA,V ) ∈ L (HA+ ), which in turn is readily proved writing −1/2 (BA,V + c)1/2 EI (BA,V ), (6.1) |DA |1/2 EI (BA,V ) = |DA |1/2 Λ+ A (BA,V + c) where c > − inf σ(BA,V ). In fact, by Theorem 2.1 the form domain of BA,V is 1/2 ) and, hence, the operator {· · · } in (6.1) is bounded. Λ+ A D(|DA | Next, we treat the case γ = γc assuming that A is locally Lipschitz continuous. In this case Remark 4.1 is applicable and we may represent χR EI (BA,V ) = χ2R χR Λ+ A EI (BA,V ) as χR EI (BA,V )

−1/2 = (χ2R |D0 |−κ ) |D0 |κ Λ+ 0 B0,−γc /|·| 1/2 −1/2 × B0,−γc /|·| Λ+ (BA,V + c)1/2 EI (BA,V ) 0 χR (BA,V + c) + + χ2R |D0 |−1/2 |D0 |1/2 (χR Λ+ A − Λ0 χR ) EI (BA,V ),

(6.2) (6.3) (6.4)

for some κ ∈ (0, 1/4). We recall from [34] that D(B0,−γc /|·| ) ⊂ D(|D0 |s ), for 1/2 every s ∈ (0, 1/2) and, hence, D(B0,−γc /|·| ) ⊂ D(|D0 |κ ). Therefore, the operator {· · · } in (6.2) is bounded. The operator {· · · } in (6.3) is bounded because of

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Remark 4.1, and the one in curly brackets in (6.4) is bounded according to (3.18). Since χ2R |D0 |−s is compact, for all s > 0, the theorem is proved. In the remaining part of this section we prove Theorem 2.5. Lemma 6.2. Assume that V fulfills Hypothesis 1 with γ ∈ [0, γc ] and that A fulfills Hypothesis 3(i). Let λ ∈ [1, ∞) and let {ψn (λ)}n∈ denote the Weyl sequence appearing in Hypothesis 3(i). Then (BA,V −

Λ+ A ψn (λ) −→ 1,

n → ∞,

(6.5)

ψn (λ) −→ 0,

n → ∞.

(6.6)

λ) Λ+ A

Proof. Since (DA − λ) ψn (λ) → 0 and ψn (λ) = 1, (6.5) follows from the the spectral calculus; see [22, Lemma 6.2]. Next, we pick some ϑ ∈ C ∞ (3 , [0, 1]) such that ϑ(x) = 0, for |x| 1/2, and ϑ(x) = 1, for |x| 1, and set ϑR := ϑ(·/R), R 1. Then ψn (λ) = ϑRn ψn (λ) and, hence, + + V Λ+ A ψn (λ) = ϑRn V ΛA ψn (λ) + V [ΛA , ϑRn ] ψn (λ). + In view of Hypothesis 1 and (3.17) we thus have Λ+ A V ΛA ψn (λ) → 0 and, consequently, (6.6) holds true also.

Theorem 6.3. Assume that V fulfills Hypothesis 1 and (2.18) and that A fulfills Hypothesis 3(ii). Then BA,V has infinitely many eigenvalues below 1 = inf σess (BA,V ). Proof. We construct appropriate trial functions by means of the Weyl sequence {ψn (1)}n∈ of Hypothesis 3(ii). It is shown in [22, Lemma 7.7] that, for every m0 +d d ∈ , there is some n0 ∈ such that the set of vectors {Λ+ A ψn (1)}n=m0 is linearly independent, for all m0 ∈ , m0 n0 . Setting Ψ :=

m 0 +d

cn Λ + A ψn (1),

n=m0

for cm0 , . . . , cm0 +d ∈ , we clearly have

Ψ | (BA,V − 1) Ψ

m 0 +d

+ |cn |2 Λ+ A ψn (1) (DA − 1 + V ) ΛA ψn (1)

(6.7)

n=m0

+

m 0 +d

+ |cn | |cm | Λ+ A ψn (1) (DA − 1 + V ) ΛA ψm (1) .

(6.8)

n,m=m0 n=m

We first comment on the terms in (6.7). Employing the fact that the lower two spinor components of ψn (1) vanish, for all n ∈ , it is shown in [22, Lemma 7.1] that there is some C ∈ (0, ∞) such that + C Rn−2 , n ∈ . (6.9) 0 Λ+ A ψn (1) (DA − 1) ΛA ψn (1)

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Moreover, we find some constant C ∈ (0, ∞) such that, for all n ∈ ,

+ Λ+ − A ψn (1) V ΛA ψn (1)

+ γ Λ ψn (1) 2 + C e−Rn /C . (6.10) A (1 + 2δ) Rn

In fact, since the quadratic form V (x) is negative it clearly suffices to prove (6.10) with V replaced by Vr := ½{|x|1} V . Then its proof is, however, exactly the same + as the one of [22, Lemma 7.3]. (Just replace Λ+ A,V by ΛA there.) The terms in (6.8) are treated in Lemma 6.4 below, where we show that + Λ ψn (1) (DA − 1 + V ) Λ+ ψm (1) = O(Rn−∞ ), m > n, (6.11) A A as n tends to infinity. Combining (6.5) and (6.9)-(6.11) with Hypothesis 1 we find some δ0 > 0 such that

Ψ | (BA,V − 1) Ψ −δ0

m 0 +d

|cn |2 ,

n=m0

for all cm0 , . . . , cm0 +d ∈ , provided m0 ∈ is sufficiently large (depending on d). This implies the assertion of the theorem. Lemma 6.4. Assertion (6.11) holds under the assumptions of Theorem 6.3. Proof. We pick a family of smooth weight functions, {Fk }k, ∈ , such that Fk ≡ 0 on supp(ψk (1)), Fk is constant on {|x| 1} and outside some ball containing supp(ψk (1)) and supp(ψ (1)), ∇Fk ∞ a < 1, and

gk := e−Fk −Fk ∞ C e−a

min{Rk ,R }

,

k, ∈ ,

where a, a ∈ (0, 1) and C ∈ (0, ∞) do not depend on k, ∈ . Such a family exists because of (2.16). We then have + Λ ψn (1) (DA − 1) Λ+ ψm (1) A A −1 , e−Fmn −Fnm ψn (1) eFmn Λ+ e−Fmn (DA − 1) ψm (1) gnm C Rm A

where C ∈ (0, ∞) neither depends on n nor m. In order to treat the term involving V we let {ϑn }n∈ denote the sequence of cut-off functions constructed in the proof of Lemma 6.2. Then (1 − ϑn ) ψn = 0, ϑn V C and, applying (3.17), we find some C ∈ (0, ∞) such that, for all n, m ∈ , ψn (1) Λ+ V Λ+ ψm (1) A A −Fmn −Fmn gnm eFmn Λ+ ψm (1) ϑn V eFmn Λ+ Ae Ae + gnm eFmn Λ+ e−Fmn V [(1 − ϑn ) eFmn , Λ+ ] e−Fmn ψm (1) C gnm . A

A

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7. Pointwise exponential decay To begin with we construct a family of cut-off functions which is used throughout this section. Let θ ∈ C ∞ (, [0, 1]) satisfy θ ≡ 0 on (−∞, 1] and θ ≡ 1 on [2, ∞). For r ∈ (0, 1/2) and R 1, we define χ ≡ χr,R ∈ C0∞ (3 , [0, 1]) by θ(|x|/r), |x| 1 3 χ(x) := χr,R (x) := (7.1) ∀x∈ : θ(3 − |x|/R), |x| > 1.

30 ,

we find some constant

∂xβ χr,R ∞ C(β, r).

(7.2)

Then, for all r ∈ (0, 1/2) and every multi-index β ∈ C(β, r) ∈ (0, ∞) such that ∀R1 :

Furthermore, we set, for r ∈ (0, 1/2) and x ∈ 3 , χ (x) := χ r (x) := θ(4|x|/r),

(7.3)

so that χ = χ χ . We also fix some exponential weight function in what follows. Let κ ∈ C ∞ (, [0, ∞)) satisfy κ ≡ 0 on (−∞, 1], κ(t) = t − 2, for t ∈ [3, ∞), and 0 κ 1 on . Then we define f ∈ C ∞ (3 , [0, ∞)) by f (x) := κ(|x|), x ∈ 3 , so that ∇f 1,

∀ β ∈ 30 , |β| > 1

∃ C(β) ∈ (0, ∞) :

∂xβ f ∞ C(β).

(7.4)

On account of (2.11) we further have ∀ ε > 0, β ∈ 30 ∃ K (ε, β) ∈ (0, ∞) :

β −εf ∂ (e A) x

∞

K (ε, β).

(7.5)

In view of Sobolev’s embedding theorem we shall obtain Theorem 2.3 as an immediate consequence of the following result, where ψk := ψH k ,

ψ ∈ H k := H k (3 , 4 ).

Theorem 7.1. Assume that A and V fulfill Hypothesis 2 with γ ∈ (0, γc ]. Let φλ denote a normalized eigenvector of BA,V corresponding to an eigenvalue λ ∈ (−∞, 1) and let be the function given by (1.1). Then, for all a ∈ [0, (λ)), r ∈ (0, 1/2), R 1, and k ∈ 0 , we have χr,R eaf φλ ∈ H k (3 , 4 ) and we find some C(a, r, k) ∈ (0, ∞) such that (7.6) ∀ R 1 : χr,R eaf φλ k C(a, r, k). Proof. Of course, we prove the assertion by induction in k ∈ 0 . The case k = 0 is follows from Theorem 2.2. So suppose that the assertion holds true for some k ∈ 0 and let β ∈ 30 be some multi-index with length |β| = k. We pick some a ∈ [0, (λ)) and set ε := ((λ) − a)/2, a ˜ := a + ε. Then the induction hypothesis together with (2.12), (7.2), (7.4), and a simple limiting argument implies that χ V ea˜f φλ ∈ H k . Using also Lemma 3.6 and Λ+ A φλ = φλ , we may thus write, for

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every ψ ∈ D, λ χ eaf ∂xβ ψ φλ = χ eaf ∂xβ ψ BA,V φλ af β + = (D0 + α · A + V Λ+ ∂x ψ ΛA φλ A) χ e = D0 ∂xβ ψ χ eaf φλ + ∂xβ ψ (e−εf α · A) (χ ea˜f φλ ) (7.7) β af φλ ) (7.8) − ∂x ψ iα · (∇χ + a χ ∇f ) (e χ β + k k af + ∂x ψ χ R0 (0) Λ0 D0 ( χ V e φλ ) (7.9) β −af −εf − Λ+ ( χ V ea˜f φλ ) (7.10) + ∂x ψ χ (eaf Λ+ 0 )e Ae af (7.11) χ ∂xβ ψ eaf φλ . + (1 − χ )V e−af Λ+ Ae By the induction hypothesis, by (7.5), and by the choice of f , χ, and χ , it is clear that the vectors in the right entries of the scalar products (7.7)-(7.9) belong to H k and that their H k -norms are bounded by constants that do not depend on R 1. Since χ V ea˜f φλ k const(a, r), Lemma 7.4 below implies that the right entry in (7.10) is bounded in H k , uniformly in R 1, too. In order to treat the term in (7.11) we set, for ϕ ∈ D, af ) V e−af Λ+ χϕ U ϕ := Ur,R ϕ : = (1 − χ Ae afR + −afR = V (1 − χ ) e , ΛA e χ ϕ.

Here we are allowed to replace f by some regularized weight function, fR ∈ C ∞ (3 , [0, ∞)) ∩ L∞ (3 , ), satisfying fR (x) = f (x), for |x| 2R, and |∇fR | 1, since 1 − χ and χ vanish outside {|x| 2R}. In view of (3.17) we hence know a-priori that U extends to a bounded operator on H . Moreover, we show in ` Lemma 7.5 below that ∃ C (a, r, k) ∈ (0, ∞) ∀ R 1 :

U ∗ L (H ,H k ) C (a, r, k).

(7.12)

Altogether this implies that the weak derivate (−1)|β| ∂xβ D0 χ eaf φλ exists and belongs to H with H -norm uniformly bounded in R 1. In order to prove Lemmata 7.4 and 7.5 we shall compare eaf RA (iy) e−af with R0 (iy). To this end we have to regularize the difference of these two operators by multiplying it with an exponential damping factor (borrowed from φλ in the previous proof), as the components of A(x) might increase very quickly when |x| gets large. For j, N ∈ 0 , j N + 1, a ∈ [0, 1), and ε ∈ [0, 1 − a), we abbreviate

Aj ≡ Aja,ε,N := A + i a + j ε/(N + 1) ∇f,

DAj := DA + i a + j ε/(N + 1) α · ∇f, RAj (iy) := (DAj − iy)−1 ,

y ∈ .

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Here iy ∈ (DAj ), y ∈ , because of Lemma 3.1, and D(DAj ) = D(DA ), since ∇f is bounded. For n ∈ 0 and T0 , . . . , Tn ∈ L (H ), we further set n

Tj := T0 T1 · · · Tn ,

j=0

0

Tj := 0.

j=1

Lemma 7.2. Assume that A fulfills Hypothesis 2 and let N ∈ ε ∈ [0, 1 − a), and y ∈ . Then the following identity holds true,

RA0 (iy) − R0 (iy) e−εf =

N

(−1)k

k−1

R0 (iy) α · Aj e−εf /(N +1)

0 ,

a ∈ [0, 1),

R0 (iy) e−ε(N +1−k)f /(N +1)

j=0

k=1

+ (−1)N +1

N

R0 (iy) α · Aj e−εf /(N +1)

RAN +1 (iy).

(7.13)

j=0

In particular, there is some C(k, a, ε) ∈ (0, ∞) such that

C(N, a, ε) ∀ y ∈ : RA0 (iy) − R0 (iy) e−εf L (H N ) . 1 + y2

(7.14)

Proof. We write g := ε f /(N +1) and z := iy for short and fix some j ∈ {0, . . . , N }. Using the argument which lead to (3.12) (with DA replaced by DAj and F = g), we check that eg RAj (z) e−g = RAj+1 (z). Now, let ϕ ∈ H . Since D is a core for DA and, hence, also for DAj , we find a sequence, {ψn }n∈ ∈ D , that converges to RAj+1 (z) ϕ ∈ D(DAj+1 ) = D(DAj ) with respect to the graph norm of DAj . Then DAj e−g ψn → DAj e−g RAj+1 (z) ϕ, since DAj e−g ψn = e−g DAj ψn +(iα·∇g) e−g ψn and DAj is closed. Therefore,

RAj (z) − R0 (z) e−g ϕ

= RAj (z) − R0 (z) (DAj − z) RAj (z) e−g ϕ

= RAj (z) − R0 (z) (DAj − z) e−g RAj+1 (z) ϕ

= lim RAj (z) − R0 (z) (D0 + α · Aj − z) e−g ψn n→∞

= − lim R0 (z) (α · Aj e−g ) ψn = −R0 (z) α · Aj e−g RAj+1 (z) ϕ. n→∞

Here the last step is justified according to (2.11). The identity (7.13) now follows from an obvious combination of RAj (z) e−g = R0 (z) e−g ϕ − R0 (z) α · Aj e−g RAj+1 (z), with j = 0, 1, . . . , N . The estimate (7.14) follows from (7.13) and the bounds R0 (iy) L (H ) (1 + y 2 )−1/2 ,

where ∈ 0 ,

RAN +1 (iy) L (H )

R0 (iy) L (H ,H +1 ) 1,

1 + (a + ε)2 , 1 + y 2 1 − (a + ε)2 √ 3

(7.15)

(7.16)

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which is a special case of (3.10), and −ε(N +1−k)f /(N +1) e C (k, , N, ε), L (H ) α · Aj e−εf /(N +1) C (j, , N, ε), L (H )

(7.17) (7.18)

which hold true by construction of f and (2.11).

< ∞ 3 Corollary 7.3. Let χ< −1 , χ ∈ C ( , [0, 1]) satisfy χ−1 (x) = 0, for |x| 1, and < dist(supp(χ), supp(χ−1 )) > 0. Then, for all N ∈ , there is a constant C(N ) ∈ (0, ∞) such that, for all y ∈ ,

χ RA0 (iy) χ< −1

L (H ,H N )

C(N ) . 1 + y2

< < ∞ 3 Proof. We set χ> N +1 := χ, and pick cut-off functions χ0 , . . . , χN ∈ C0 ( , [0, 1]) < < < > such that χj ≡ 1 on the support of χj−1 and such that χj and χj+1 have disjoint < < supports, where χ> j := 1 − χj , j = 0, . . . , N . Since χ−1 (x) = 0, for |x| 1, we < < > have χ−1 = χ−1 e−εf , and, by construction, χ = χ χk , k = 0, . . . , N . Therefore, (7.13) with g := εf /(N + 1) yields > < χRA0 (iy)χ< −1 − χχ0 R0 (iy)χ−1

=

N

(−1)k

0 ,...,k−1 ∈{}

k=1

N +1

+ (−1)

χχ> k

k−1

R0 (iy)χjj α · Aj e−g R0 (iy)χ< −1

(7.19)

j=0

N

χ R0 (iy)α · Aj e−g RAN +1 (iy)χ< −1 . j=0

< Each summand in (7.19) contains at least one factor of the form χ> j+1 R0 (iy) χj , > and since χ< j and χj+1 have disjoint supports we readily verify that

> C(j, M ) 0. Then χ RA0 (iy) ϑ = χ ϑ R0 (iy) + χ RA0 (iy) α · (i∇ϑ − ϑ A0 ) R0 (iy) = χ RA0 (iy) χ< −1 α · (i∇ϑ − ϑ A0 ) R0 (iy), which implies, for ϕ ∈ D, ψ ∈ H , and ψ1 , ψ2 , . . . ∈ D such that ψn → ψ, k ∗ D ϕ U ψ = U Dk ϕ ψ 0 0

dy lim sup D0k ϕ χ RA0 (iy) χ< −1 α · (i∇ϑ − ϑ A0 ) (R0 (iy) V ) ψn 2π n→∞

χ RA0 (iy) χ< CV −1 L (H ,H k ) dy ∇ϑ + ϑ A + ∇f ϕ ψ. In the last line we used that R0 (iy) V extends to a bounded operator on H with a norm bounded uniformly in y ∈ by some CV ∈ (0, ∞). On account of Corollary 7.3 this proves Assertion (7.12).

Appendix A. According to [19, Theorem 7.2] a function ψ belongs to H 1/2 (3 ) if and only if ψ ∈ L2 (3 ) and

|ψ(x) − ψ(y)|2 1 K2 (|x − y|) dx dy (A.1) I(ψ) := 4π 2 3 3 |x − y|2 is finite, where K2 denotes a modified Bessel function. In this case we have I(ψ) = (1 − ∆)1/4 ψ2 − ψ2 . Now, let χ : 3 → be such that χ∞ + Lχ < ∞, where Lχ := sup x=y

|χ(x) − χ(y)| . |x − y|

Estimating 2 2 (χ ψ)(x) − (χ ψ)(y)2 2 |χ(x) − χ(y)| 2 |ψ(x) − ψ(y)| 2|ψ(x)| + 2 |χ(y)| |x − y|2 |x − y|2 |x − y|2

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we obtain, for every ψ ∈ H 1/2 (3 ),

I(χ ψ) 2 L2χ ψ2 K2 (|y|) dy + 2 χ2∞ I(ψ) < ∞, 3 ∞ that is, χ ψ ∈ H 1/2 (3 ), and, using 0 r2 K2 (r) dr = 3π/2, we further obtain (1 − ∆)1/4 χ ψ 2 = I(χ ψ) + χ ψ2 2

3 L2χ ψ2 + χ2∞ 2 (1 − ∆)1/4 ψ − ψ2 . Acknowledgment It is a pleasure to thank Hubert Kalf, Sergey Morozov, and Heinz Siedentop for helpful discussions and useful remarks. E. S. would like to thank the group of Volker Bach at the University of Mainz for their kind hospitality. This work has been partially supported by the DFG (SFB/TR12).

References [1] W. O. Amrein, A.-M. Boutet de Monvel-Berthier, and V. Georgescu, C0 -groups, commutator methods and spectral theory of N -body Hamiltonians. Progress in Mathematics 135, Birkh¨ auser, Basel, 1996. [2] V. Bach, J. Fr¨ ohlich, and I. Sigal, Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137 (1998), 299–395. [3] V. Bach and O. Matte, Exponential decay of eigenfunctions of the Bethe-Salpeter operator. Lett. Math. Phys. 55 (2001), 53–62. [4] A. Berthier and V. Georgescu, On the point spectrum of Dirac operators. J. Funct. Anal. 71 (1987), 309–338. [5] H. A. Bethe and E. E. Salpeter, Quantum mechanics of one- and two-electron atoms. In: Handbuch der Physik, XXXV. Pages 88–436. Edited by S. Fl¨ ugge. Springer, Berlin, 1957. [6] A. M. Boutet de Monvel and R. Purice, A distinguished self-adjoint extension for the Dirac operator with strong local singularities and arbitrary behaviour at infinity. Rep. Math. Phys. 34 (1994), 351–360. [7] G. E. Brown and D. G. Ravenhall, On the interaction of two electrons. Proc. Roy. Soc. London A 208 (1951), 552–559. [8] R. Carmona, Path integrals for relativistic Schr¨ odinger operators. In: Schr¨ odinger operators. Sønderborg, 1988. Pages 65–92. Lecture Notes in Physics 345, Springer, Berlin, 1989. [9] R. Carmona, W. C. Masters, and B. Simon, Relativistic Schr¨ odinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91 (1990), 117–142. [10] P. R. Chernoff, Schr¨ odinger and Dirac operators with singular potentials and hyperbolic equations. Pacific J. Math. 72 (1977), 361–382. [11] M. Dimassi and J. Sj¨ ostrand, Spectral asymptotics in the semi-classical limit. London Math. Soc. Lecture Note Series 268, Cambridge University Press, Cambridge, 1999.

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[12] W. D. Evans, P. Perry, and H. Siedentop, The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Comm. Math. Phys. 178 (1996), 733–746. [13] M. Griesemer, Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210 (2004), 321–340. [14] M. Griesemer and C. Tix, Instability of a pseudo-relativistic model of matter with self-generated magnetic field. J. Math. Phys. 40 (1999), 1780–1791. [15] B. Helffer, J. Nourrigat, and X. P. Wang, Sur le spectre de l’équation de Dirac (dans ´ Ê3 ou Ê2 ) avec champ magnetic. Ann. Sci. Ecole Normale Superieur, 4e Série 22 (1989), 515–533. [16] B. Helffer and B. Parisse, Comparaison entre la d´ ecroissance de fonctions propres pour les opérateurs de Dirac et de Klein-Gordon. Application a ` l’étude de l’effet tunnel. Ann. Inst. Henri Poincaré 60 (1994), 147–187. [17] D. H. Jakubaßa-Amundsen, Variational ground state for relativistic ions in strong magnetic fields. Phys. Rev. A 78 (2008), 062103. [18] Y. Last and B. Simon, The essential spectrum of Schr¨ odinger, Jacobi, and CMV operators. J. d’Analyse Math. 98 (2006), 183–220. [19] E. H. Lieb and M. Loss, Analysis. 2nd Edition, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, Rhode Island, 2001. [20] E. H. Lieb and M. Loss, Stability of a model of relativistic quantum electrodynamics. Comm. Math. Phys. 228 (2002), 561–588. [21] E. H. Lieb, H. Siedentop, and J. P. Solovej, Stability and instability of relativistic electrons in classical electromagnetic fields. J. Statist. Phys. 89 (1997), 37–59. [22] O. Matte and E. Stockmeyer, Spectral theory of no-pair Hamiltonians. Preprint, arXiv:0803.1652, 53 pages, 2008. [23] S. Morozov, Multi-particle Brown-Ravenhall operators in external fields. PhD thesis, Universit¨ at M¨ unchen, 2008. [24] S. Morozov, Exponential decay of eigenfunctions of Brown-Ravenhall operators. Preprint, arXiv:0903.4718, 17 pages, 2009. [25] S. Morozov and S. Vugalter, Stability of atoms in the Brown-Ravenhall model. Ann. Henri Poincaré 7 (2006), 661–687. [26] G. Nenciu, Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms. Comm. Math. Phys. 48 (1976), 235–247. [27] V. S. Rabinovich and S. Roch, Agmon’s type estimates of exponential behavior of solutions of systems of elliptic partial differential equations. Applications to Schr¨ odinger, Moisil-Theodorescu and Dirac operators. Preprint, arXiv:0802.3963, 19 pages, 2008. [28] V. S. Rabinovich and S. Roch, Essential spectrum and exponential decay estimates of solutions of elliptic systems of partial differential equations. Applications to Schr¨ odinger and Dirac operators. Georgian Math. J. 15 (2008), 333–351. [29] M. Reiher and A. Wolf, Relativistic quantum chemistry. Wiley-VCH, Weinheim, 2009. [30] R. Richard and R. Tiedra de Aldecoa, On the spectrum of magnetic Dirac operators with Coulomb-type perturbations. J. Funct. Anal. 250 (2007), 625–641. [31] J. Sucher, Foundations of the relativistic theory of many-electron atoms. Phys. Rev. A 22 (1980), 348–362.

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[32] J. Sucher, Relativistic many-electron Hamiltonians. Physica Scripta 36 (1987), 271– 281. [33] B. Thaller, The Dirac equation. Texts and Monographs in Physics, Springer, BerlinHeidelberg, 1992. [34] C. Tix, Self-adjointness and spectral properties of a pseudo-relativistic Hamiltonian due to Brown and Ravenhall. Preprint, mp arc 97-441, 20 pages, 1997. [35] C. Tix, Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall. Bull. London Math. Soc. 30 (1998), 283–290. [36] X. P. Wang, Puits multiples pour l’opérateur de Dirac. Annales de l’I.H.P., Section A, Physique Théorique, 43 (1985), 269–319. Oliver Matte Institut f¨ ur Mathematik Technische Universit¨ at Clausthal Erzstraße 1 D-38678 Clausthal-Zellerfeld Germany On leave from: Mathematisches Institut Ludwig-Maximilians-Universit¨ at Theresienstraße 39 D-80333 M¨ unchen Germany. e-mail: [email protected] Edgardo Stockmeyer Mathematisches Institut Ludwig-Maximilians-Universit¨ at Theresienstraße 39 D-80333 M¨ unchen Germany. e-mail: [email protected] Submitted: October 31, 2008.



On the Generalized Derivation Induced by Two Projections Dan Popovici and Zoltán Sebestyén Abstract. We prove that P X − XQ = max{P X(1 − Q), (1 − P )XQ}, where P and Q are orthogonal projections on Hilbert spaces H , resp. K and X is a bounded linear operator between K and H , generalizing the well-known Akhiezer-Glazman equality (the case K = H and X = 1H ). Specializing, we obtain results by Z. Boulmaarouf, M. Fernandez Miranda and J.-Ph. Labrouse, T. Kato or S. Maeda. We extend a theorem of Y. Kato and give some sufficient conditions for the equality P X(1 − Q) = (1 − P )XQ. We generalize results by D. Buckholtz, J.J. Koliha and V. Rakoˇcević or S. Maeda and provide several necessary and/or sufficient conditions for the (left, right) invertibility of P X − XQ. Certain counter-examples show the consistency of the theory. Mathematics Subject Classification (2000). Primary 47A05; Secondary 47A30, 46C05, 46C07. Keywords. Generalized derivation, idempotent, orthogonal projection, MoorePenrose inverse, Akhiezer-Glazman norm equality, operator invertibility.

1. Introduction The problem regarding the existence of a bounded idempotent M (acting on a complex Hilbert space H ) determined by a topological direct sum of the form H = L R between two closed subspaces L and R (of H ) in the sense that L is the range and R the kernel of M (M is said to be the idempotent onto L along R) is proved to be equivalent ([5, 6]) to the invertibility of the difference P − Q, or to the fact that the gap between P and 1 − Q (i.e., P + Q − 1; · denotes the operator norm) is strictly less than 1, where P and Q are the (orthogonal) projections with ranges L and R, respectively. In addition, by the

286

Popovici and Sebestyén

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Akhiezer-Glazman norm equality [1, Sect. 34], P + Q − 1 = max{P Q, (1 − P )(1 − Q)},

(1.1)

the last condition can be expressed as P Q < 1 and (1 − P )(1 − Q) < 1. These intreagueing and important problems in operator theory have been extensively discussed in connection to various applications. To give just a few examples we could mention here perturbation theory for linear operators [11], probability theory [18], Fredholm theory [13], complex geometry [20], statistics [27, 28], wavelet theory [24, 25], invariant subspace theory [19] etc. Let L(H , K ) (L(H ) if H = K ) denote the Banach space of all bounded linear operators between complex Hilbert spaces H and K . For A ∈ L(H , K ) let us denote by A∗ the (Hilbert) adjoint, by ker A the kernel and by ran A the range of A. The generalized derivation induced by orthogonal projections P ∈ L(H ) and Q ∈ L(K ) is defined as L(K , H ) X → δP,Q (X) := P X − XQ ∈ L(K , H ). The following formulas [δP,Q (X)]∗ = −δQ,P (X ∗ ) and δP,Q (X) = −δ1−P,1−Q (X), which hold true for every X ∈ L(K , H ), will be frequently used. For the main properties of the generalized derivation induced by two operators the reader is refered, for example, to [2],[9],[21]. It is our aim in this paper to obtain certain results which hold true for any given operator in the range of δP,Q and are similar to or extend the ones presented at the beginning of this section for the difference of two orthogonal projections. The more precise contents of the paper is described in the following. Extending a result of S. Maeda [18] we obtain in Section 2 a formula for the norm of a sum of a finite family {Ti }ni=1 of bounded linear operators satisfying conditions Ti Tj∗ = 0 and Ti∗ Tj = 0 for every i = j. We deduce, in particular, that P X + XQ − X = max{P XQ, (1 − P )X(1 − Q)},

(1.2)

where P and Q are orthogonal projections on H , respectively K and X ∈ L(K , H ), generalizing the Akhiezer-Glazman equality (1.1). We obtain as applications several results due to S. Maeda [16, 18] or Z. Boulmaarouf, M. Fernandez Miranda and J.-Ph. Labrouse [4]. We also extend the theorem of T. Kato [11, Theorem 6.35] which states that the gap between two idempotents is bigger than the gap between the corresponding range projections. S. Maeda showed in [17] that ran P = ran (P Q) if and only if P (1 − Q) < 1. We prove in Section 3 that, more generally, for an operator X ∈ L(K , H ), ran P = ran (P X) if and only if P (X2 − XX ∗ )1/2 < X. We also express the kernel of P X − XQ in terms of the kernels of P, 1 − P, Q and 1 − Q. We provide several necessary and/or sufficient conditions for the (left, right) invertibility of P X − XQ extending results by D. Buckholtz [5, 6], J.J. Koliha and V. Rakoˇcević

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On the Generalized Derivation

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[13, 14, 15] or S. Maeda [17]. Y. Kato noted in [12] (cf. also [11, Theorem 6.34], [26, Theorem 4.33]) that (1.1) can be simplified as P + Q − 1 = P Q = (1 − P )(1 − Q) if P and 1 − Q are interchanged by a symmetry or, in particular, if P − Q is invertible. Some sufficient conditions are given in order to ensure the corresponding simplification in (1.2): P X + XQ − X = P XQ = (1 − P )X(1 − Q). We prove, in the final part, that P X − XQ is invertible if and only if ran [P X(1 − Q)] = ran P, ran [(1 − P )XQ] = ker P , and ran [QX ∗ (1 − P )] (resp. ran [(1 − Q)X ∗ P ]) is dense in ran Q (resp. ker Q). In this case we compute the Moore-Penrose inverses of P X(1 − Q), (1 − P )XQ, QX ∗ (1 − P ) and (1 − Q)X ∗ P . Other conditions for the left (similarly, for the right) invertibility of P X − XQ are obtained: XQ and X(1 − Q) have closed ranges, the sums ker(P X) + ker Q and ker[(1 − P )X] + ran Q are direct, and the sums ker P + ker(QX ∗ ) and ran P + ker[(1 − Q)X ∗ ] are closed. Consequently, if P X − XQ is invertible then the sums ker(P X) + ker Q, ker[(1 − P )X] + ran Q, ker P + ker(QX ∗ ) and ran P + ker[(1 − Q)X ∗ ] are direct and closed. We provide an example showing that the converse is, in general, false. Also, the invertibility of P X − XQ implies that the operators P X, (1−P )X, XQ, X(1−Q), P X(1−Q), (1−P )XQ, QX ∗(1−P ) and (1−Q)X ∗ P have closed ranges. Nevertheless, we prove that in these conditons the range of X is not necessarily closed.

2. Norm equalities and inequalities We start with a general norm equality for the sum of a finite family {Ti }ni=1 of bounded linear operators between Hilbert spaces provided that both {Ti }ni=1 and {Ti∗ }ni=1 have pairwise orthogonal ranges: Lemma 2.1. Let {Ti }ni=1 be a finite family of bounded linear operators between Hilbert spaces H and K such that Ti∗ Tj = 0H and Ti Tj∗ = 0K for every i = j. Then n n Ti = max Ti . (2.1) i=1

i=1

Proof. Let us denote by S the operator H n (h1 , h2 , . . . , hn ) → T1 h1 + T2 h2 + · · · + Tn hn ∈ K and observe that its (Hilbert) adjoint can be computed as K k → (T1∗ k, T2∗ k, . . . , Tn∗ k) ∈ H n .

288


Note that, since Ti∗ Tj = 0H for i = j,  ∗ T1 T1 0 ∗  0 T 2 T2  S ∗ S =  . ..  .. . 0 0

... ... .. . ...

IEOT

  n   = max Ti∗ Ti . i=1  T ∗ Tn 0 0 .. .

n

Also, in view of the equalities Ti Tj∗ = 0K for i = j, n n n n 2

∗ SS ∗ = Ti Ti∗ = Ti Ti = Ti . i=1

i=1

i=1

i=1 ∗

Formula (2.1) follows by the well-known identity SS = S ∗ S.

Remark 2.2. (i) The case n = 2, H = K has been considered (with a different proof) by S. Maeda in [18]. (ii) Let A and B be two complex Hilbert spaces, {Ai }ni=1 ⊂ L(A ) and n {Bi }i=1 ⊂ L(B) two finite families of selfadjoint operators such that Ai Aj = 0A and Bi Bj = 0B for every i = j and let {Xi }ni=1 ⊂ L(B, A ). We use Lemma 2.1 for Ti = Ai Xi Bi , i = 1, 2, . . . , n, to deduce that n n Ai Xi Bi = max Ai Xi Bi . i=1

i=1

(iii) If Xi = λi X, i = 1, 2, . . . , n, where {λi }ni=1 ⊂ T (the unit circle) and X ∈ L(B, A ), then the identity of (ii) becomes n n λi Ai XBi = max Ai XBi . i=1

i=1

We are now in position to extend the Akhiezer-Glazman norm equality (1.1): Theorem 2.3. Let P ∈ L(H ), Q ∈ L(K ) be two orthogonal projections and X ∈ L(K , H ). Then P X + XQ − X = max{P XQ, (1 − P )X(1 − Q)}.

(2.2)

Proof. Let us note that P X + XQ − X = P XQ − (1 − P )X(1 − Q). Formula (2.2) follows by Remark 2.2 (iii) (the case n = 2, λ1 = −λ2 = 1, A1 = 1 − A2 = P, B1 = 1 − B2 = Q). Remark 2.4. (i) An alternate proof of Theorem 2.3 can be obtained via the identity |P X + XQ − X|2 = |P XQ|2 + |(1 − P )X(1 − Q)|2 .

(2.3)

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To this aim, let us observe that, by Maeda’s norm equality (Remark 2.2 (i)), P X + XQ − X2 = |P X + XQ − X|2 = max{|P XQ|2, |(1 − P )X(1 − Q)|2 } 2 = max{P XQ, (1 − P )X(1 − Q)} . (ii) We observe that P XQ, (1 − P )X(1 − Q) ≤ X. Hence, by (2.2), P X + XQ − X ≤ X.

(2.4)

(iii) If X ∈ L(H ) is selfadjoint and Q = 1 − P , then (2.2) becomes P X − XP = (1 − P )XP .

(2.5)

We deduce, in particular, another result of S. Maeda [18]: P Q − QP = (1 − P )QP = (1 − Q)P Q, where P and Q are two orthogfonal projections on H . (iv) Let P, Q and X be as in Theorem 2.3. Then P X − 2P XQ + XQ = P X(1 − Q) + (1 − P )XQ = max{P X(1 − Q), (1 − P )XQ} by Rem. 2.2 (iii) = P X − XQ.

by Thm. 2.3

In particular, for K = H and X = 1H , we get P − 2P Q + Q = P − Q.

(2.6)

If P Q = QP then (2.6) can be re-written in equivalent form as P −Q2 = P −Q. Hence P = Q or P − Q = 1, as in [16]. (v) Let V ∈ L(H , K ) be a partial isometry, P = V ∗ V, Q = V V ∗ and let X ∈ L(K , H ). We firstly note that P X(1 − Q)2 = (1 − Q)X ∗ V ∗ V X(1 − Q) = V X(1 − Q)2 = (V X − XQ)(1 − Q)2 ≤ V X − XQ2. Similarly, (1 − P )XQ ≤ P X − XV . Consequently, by Theorem 2.3, P X − XQ ≤ max{P X − XV , V X − XQ}. In the particular case when K = H and X = 1H we actually have that P − V 2 = V ∗ V − V V ∗ V = (V ∗ − Q)V = (V ∗ − Q)V V ∗ (V − Q) = (V ∗ Q − Q)(QV − Q) = (V ∗ − Q)(V − Q) = V − Q2 . We deduce the following result of S. Maeda [18]: P − Q ≤ P − V = V − Q. (vi) Let M be an idempotent and P an orthogonal projection, both acting on the Hilbert space H . It is shown in [8] that √ 1 √ α + βM + γM ∗ = ( r + s + r − s), 2

290

where


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r = |α|2 + |α + β + γ|2 + (|β|2 + |γ|2 )(M 2 − 1)

and

s = 2|α(α + β + γ) − βγ(M 2 − 1)|.

Hence

1 − 2M = M + M 2 − 1. (2.7) We use Theorem 2.3 for K = H , Q = 1 − P and X = 1 − 2M to obtain that 2M P − P M = P (1 − 2M ) − (1 − 2M )P = max{P (1 − 2M )(1 − P ), (1 − P )(1 − 2M )P } ≤ 1 − 2M .

Therefore, by (2.7),

M 2 − 1 . (2.8) 2 If M is an orthogonal projection Q, then (2.8) is the Maeda estimate [18]: 1 P Q − QP ≤ . 2 (vii) Let M ∈ L(H ) be an idempotent and P ∈ L(H ) the range projection of M (i.e., P is the orthogonal projection onto ran M ). In the special case (of Theorem 2.3) Q = 1 − P and X = M ∗ M formula (2.5) becomes M P − P M ≤

M +

P M ∗ M − M ∗ M P = (1 − P )M ∗ M P . Since P M ∗ M = M, M ∗ M P = M ∗ and (1 − P )M ∗ M P = M ∗ − P we obtain the equality M − P = M − M ∗ , which is due to Z. Boulmaarouf, M. Fernandez Miranda and J.-Ph. Labrouse [4]. The following theorem establishes the connection between the generalized derivation induced by two given idempotents and the generalized derivation induced by their range projections: Theorem 2.5. Let M ∈ L(H ) and N ∈ L(K ) be two idempotents, P and Q the range projections of M and, respectively, N and let X ∈ L(K , H ). Then P X − XQ ≤ max{M X − XN , M ∗X − XN ∗ }. Proof. We observe firstly that (1 − P )XQ = (1 − P )(1 − M )XQ ≤ (1 − M )XQ = XQ − M XQ = XN Q − M XQ = (M X − XN )Q ≤ M X − XN .

(2.9)

Also, (2.9)

P X(1 − Q) = (1 − Q)X ∗ P ≤ N X ∗ − X ∗ M = M ∗ X − XN ∗ . (2.10)

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We finally use (2.9) and (2.10) to obtain, by Theorem 2.3, that P X − XQ = max{(1 − P )XQ, P X(1 − Q)} ≤ max{M X − XN , M ∗X − XN ∗ },

as required.

Remark 2.6. (i) For the case K = H and X = 1H we obtain the following inequality of T. Kato [11, Theorem 6.35]: P − Q ≤ M − N ; (ii) For the case K = H and P = Q, if M is an idempotent on H , P is the range projection of M and X ∈ L(H ), then P X − XP ≤ max{M X − XM , M ∗X − XM ∗ }. In particular, if X is selfadjoint then P X − XP ≤ M X − XM .

3. Invertibility We propose in this section several necessary and/or sufficient conditions for the (left, right) invertibility of any given operator in the range of the generalized derivation induced by orthogonal projections P and Q. Lemma 3.1. Let P be a selfadjoint projection on H and A ∈ L(K , H ) \ {0}. The following conditions are equivalent: (a) ran (P A) = ran P ; ∗ P ) = ran P ; (b) ran (P AA 2 (c) P (A − AA∗ )1/2 < A; (d) P AA∗ P is invertible in P L(H )P ; (e) A2 (1 − P ) + P AA∗ P is invertible; (f ) A2 (1 − P ) + P AA∗ is invertible; (g) there exists M > 0 such that A∗ P h ≥ M P h,

h∈H;

(h) there exists M > 0 such that P AA∗ P h ≥ M P h,

h∈H.

A can be replaced in (c) by any real number K > A and in (e)–(f ) by any positive constant. Proof. (a) ⇒ (c). Let us suppose that P = P AC for a certain nonnull operator C ∈ L(H , K ) (by the well-known factorization criterion due to R.G. Douglas [7]). Then, for every h ∈ H , A2 P h2 = (A2 P − P AA∗ P )h, h + P AA∗ P h, h 2 = (A2 − AA∗ )1/2 P h + A∗ P h2

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2 ≥ (A2 − AA∗ )1/2 P h + C−2 C ∗ A∗ P h2 2 = (A2 − AA∗ )1/2 P h + C−2 P h2 . Consequently, (A2 − AA∗ )1/2 P h2 ≤ (A2 − C−2 )P h2 ≤ (A2 − C−2 )h2 , that is,

h∈H,

(A2 − AA∗ )1/2 P ≤ A2 − C−2 < A.

(c) ⇒ (e). The condition P (A2 − AA∗ )1/2 < A is equivalent to

1 1 ∗ A A P < 1, P − P A A 1 1 ∗ which, since 0 ≤ P − P A A A A P ≤ P ≤ 1, is in turn equivalent to the 1 invertibility of 1 − P + A2 P AA∗ P . −1 (e) ⇒ (d). Easy computations show that P A2 (1 − P ) + P AA∗ P P is the inverse of P AA∗ P in P L(H )P . The implications (d) ⇒ (b) and (b) ⇒ (a) are obvious. The equivalence between (e) and (f ) follows by the well-known fact that, for bounded linear operators X, Y on H , 1 − XY is invertible if and only if 1 − Y X is invertible. More precisely, we take X = P and Y = P 1 − A−2 AA∗ . Finally, (a) and (g), respectively (b) and (h) are equivalent by the same criterion of Douglas. Remark 3.2. (i) S. Maeda proved in [17] that, for two selfadjoint projections p and q of a C ∗ -algebra A with identity, p(1 − q) < 1 if and only if pA = pqA. Lemma 3.1 generalizes Maeda’s result (see the equivalence (a) ⇔ (c) below) as follows: let A be a C ∗ -algebra with identity, p a selfadjoint projection of A and a ∈ A \ {0}. The following conditions are equivalent: (a) (b) (c) (d) (e) (f )

paA = pA; ∗ paa pA2 = pA;∗ 1/2 p(a − aa ) < a; paa∗ p is invertible in pAp; a2 (1 − p) + paa∗ p is invertible; a2 (1 − p) + paa∗ is invertible.

a can be replaced in (c) by any real number K > a and in (e)–(f ) by any positive constant. (ii) If one of the equivalent conditions of Lemma 3.1 holds true, then −1 −1 K 2 K 2 (1 − P ) + P AA∗ = 1 + K 2 (1 − P ) + P AA∗ P P (K 2 − AA∗ )

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and

−1 −1 K 2 K 2 (1 − P ) + P AA∗ P = 1 + P (K 2 − AA∗ )P K 2 (1 − P ) + P AA∗

for any real number K ≥ A.

D. Buckholtz [5, 6] proved that the difference P − Q between orthogonal projections P and Q is invertible if and only if P + Q − 1 < 1. In what follows it is our aim to extend this property in the context of generalized derivations induced by projections. Lemma 3.3. Let P, Q and X be as in Theorem 2.3. Then (i) |P X + XQ − X|2 + |P X − XQ|2 = |XQ|2 + |X(1 − Q)|2 ; (ii) ker(P X − XQ) = (ker Q ∩ X −1 ker P ) ⊕ (ran Q ∩ X −1 ran P ); (iii) |XQ|2 + |X(1 − Q)|2 is invertible if and only if ran (QX ∗ ) = ran Q and ran [(1 − Q)X ∗ ] = ran (1 − Q). In this case |XQ|2 + |X(1 − Q)|2 −1 −1 = min inf Xk2, inf Xk2 . (3.1) k∈ran Q k=1

k∈ker Q k=1

The same kind of conditions can be obtained by changing the roles of P with Q and of X with X ∗ . Proof. The condition (i) follows easily by direct computations. (ii) If k ∈ ker(P X − XQ) then P Xk = P XQk, whence k − Qk ∈ X −1 ker P . Also XQk = P XQk, that is Qk ∈ X −1 ran P . We deduce that k = (k − Qk) + Qk ∈ (ker Q ∩ X −1 ker P ) ⊕ (ran Q ∩ X −1 ran P ). Conversely, if k ∈ ker Q ∩ X −1 ker P then P Xk = XQk = 0. Finally, if k ∈ ran Q ∩ X −1 ran P then P Xk − XQk = Xk − Xk = 0. (iii) The operator |XQ|2 + |X(1 − Q)|2 is invertible if and only if there exists M > 0 such that XQk2 + X(1 − Q)k2 ≥ M k2 ,

k∈K.

This inequality can be rewritten in equivalent form as √ √ XQk ≥ M Qk and X(1 − Q)k ≥ M (1 − Q)k,

(3.2) k∈K.

(3.3)

Consequently, by Lemma 3.1, ran (QX ∗ ) = ran Q

and ran [(1 − Q)X ∗ ] = ker Q.

(3.1) follows in a standard way from (3.2) and (3.3).

Proposition 3.4. Let P, Q and X be as in Theorem 2.3. The following conditions are equivalent: (a) P X − XQ is one-to-one; (b) ran [(1 − Q)X ∗ P ] = ker Q and ran [QX ∗ (1 − P )] = ran Q; (c) (P X + XQ − X)k2 < XQk2 + X(1 − Q)k2 for every k ∈ K , k = 0.

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Proof. Let k ∈ ran Q. Then k ⊥ ran [QX ∗ (1 − P )] if and only if XQk = Xk ∈ ran P , that is k ∈ X −1 ran P . Hence ran Q ran [QX ∗ (1 − P )] = ran Q ∩ X −1 ran P.

(3.4)

If Q is replaced by 1 − Q and P by 1 − P we also obtain ker Q ran [(1 − Q)X ∗ P ] = ker Q ∩ X −1 ker P.

(3.5)

The equivalence between (a) and (b) follows by (3.4), (3.5) and Lemma 3.3 (ii). Let us note that the formula (P X+XQ−X)k2+(P X−XQ)k2 = XQk2 +X(1−Q)k2,

k ∈ K (3.6)

holds true by Lemma 3.3 (i). We deduce that ker(P X − XQ) = {0} (i.e., (a) holds true) if and only if XQk2 + X(1 − Q)k2 − (P X + XQ − X)k2 (= (P X − XQ)k2 ) > 0 for every k ∈ K , k = 0 (i.e., (c) holds true).

Remark 3.5. (i) In the special case K = H and X = 1H the equivalence (a) ⇔ (b) is due to Z. Takeda and T. Turumaru [23], while (a) ⇔ (c) is due to S. Maeda [17]. (ii) We exchange the roles of P and Q and of X and X ∗ to obtain that P X − XQ has dense range if and only if ran [(1 − P )XQ] = ker P

and ran [P X(1 − Q)] = ran P

if and only if (X ∗ P + QX ∗ − X ∗ )h2 < X ∗ P h2 + X ∗ (1 − P )h2 for every h ∈ H , h = 0.

Let us now present some necessary conditions for the left invertibility of P X − XQ: Proposition 3.6. Let P, Q and X be as in Theorem 2.3. If P X − XQ is left invertible, then ran (QX ∗ ) = ran Q, ran [(1 − Q)X ∗ ] = ker Q and P X + XQ − X < max{XQ, X(1 − Q)}. Proof. Since P X − XQ is left invertible, there exists M > 0 such that (P X − XQ)k ≥ M k for every k ∈ K . We observe that, by (3.6), (P X + XQ − X)k2 = XQk2 + X(1 − Q)k2 − (P X − XQ)k2 ≤ |XQ|2 + |X(1 − Q)|2 k2 − M 2 k2 for every k ∈ K . Hence, by Remark 2.2 (iii), P X + XQ − X2 ≤ max{XQ2, X(1 − Q)2 } − M 2 . We deduce that P X + XQ − X2 < max{XQ, X(1 − Q)}.

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The invertibility of |XQ|2 + |X(1 − Q)|2 follows by the inequality |XQ|2 + |X(1 − Q)|2 ≥ |P X − XQ|2 which is a consequence of Lemma 3.3 (i). We deduce that, by Lemma 3.3 (iii), ran (QX ∗ ) = ran Q and ran [(1 − Q)X ∗ ] = ker Q. Conversely, several conditions which are similar to the ones in the conclusion of Proposition 3.6 are sufficient for the left invertibility of P X − XQ: Proposition 3.7. Let P, Q and X be as in Theorem 2.3. If ran (QX ∗ ) = ran Q, ran [(1 − Q)X ∗ ] = ker Q and P X + XQ − X < min inf Xk, inf Xk , k∈ran Q k=1

k∈ker Q k=1

then P X − XQ is left invertible. Proof. Let us firstly note that, by Lemma 3.3 (iii), the operator |XQ|2 +|X(1−Q)|2 is invertible. The left invertibility of P X − XQ follows by the formula (3.6)

(P X − XQ)k2 = XQk2 + X(1 − Q)k2 − (P X + XQ − X)k2 −1 ≥ [|XQ|2 + |X(1 − Q)|2 ]−1 k2 − P X + XQ − X2 k2 (3.1) = min inf Xk, inf Xk k∈ran Q k=1

k∈ker Q k=1

− P X + XQ − X2 k2 ,

which holds true for every k ∈ K .

Let us note that, if we replace Q by 1 − Q, then (2.3) can be expressed as |P X − XQ|2 = |(1 − P )XQ|2 + |P X(1 − Q)|2 .

(3.7)

Also, by (2.4), 0 ≤ |P X − XQ| ≤ X. Hence, P X − XQ is left invertible if and only if |(1 − P )XQ|2 + |P X(1 − Q)|2 < 1, 1 − X2 that is X2 − |(1 − P )XQ|2 − |P X(1 − Q)|2 < X2 . This last inequality can be rewritten in equivalent form as Q[X2 − X ∗ (1 − P )X]Q + (1 − Q)(X2 − X ∗ P X)(1 − Q) < X2 or, by Remark 2.2 (ii), as Q[X2 − X ∗ (1 − P )X]1/2 < X and (1 − Q)(X2 − X ∗ P X)1/2 < X. Consequently, by Lemma 3.1, we obtain:

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Theorem 3.8. Let P, Q and X be as in Theorem 2.3. The following conditions are equivalent: (a) P X − XQ is left invertible; (b) ran [QX ∗ (1 − P )] = ran Q and ran [(1 − Q)X ∗ P ] = ker Q; 2 (c) ran [|(1 − P )XQ|2 ] = ran Q and ran [|P X(1 − Q)| ] =2ker Q; 2 ∗ 1/2 < X and (1−Q)(X −X ∗ P X)1/2 < X; (d) Q[X −X (1−P )X] (e) |(1 − P )XQ|2 is invertible in QL(K )Q and |P X(1 − Q)|2 is invertible in (1 − Q)L(K )(1 − Q); (f ) X2(1 − Q) + |(1 − P )XQ|2 and X2Q + |P X(1 − Q)|2 are invertible; (g) X2(1 − Q) + QX ∗ (1 − P )X and X2Q + (1 − Q)X ∗ P X are invertible; (h) there exists M > 0 such that, for every k ∈ K , (1 − P )XQk ≥ M Qk and P X(1 − Q)k ≥ M (1 − Q)k; (i) there exists M > 0 such that, for every k ∈ K , QX ∗ (1 − P )XQk ≥ M Qk and

(1 − Q)X ∗ P X(1 − Q)k ≥ M (1 − Q)k. X can be replaced in (d) by any real number K > X and in (f )–(g) by any positive constant. Remark 3.9. (i) We exchange the roles of P and Q and of X and X ∗ in Propositions 3.6, 3.7 and Theorem 3.8 to obtain necessary and/or sufficient conditions for the right invertibility, respectively invertibility of P X − XQ. (ii) In the special case K = H and X = 1H the equivalences (b) ⇔ (d) ⇔ (e) are due to S. Maeda [17], (a) ⇔ (d) due to D. Buckholtz [5], while (a) ⇔ (f ) ⇔ (g) is due to J.J. Koliha and V. Rakoˇcević [14] (in the setting of rings). It was noted by Y. Kato [12] (cf. also [11, Theorem 6.34], [26, Theorem 4.33]) that, for two orthogonal projections P and Q, if P + Q − 1 < 1 or, equivalently, if P − Q is invertible, then the Akhiezer-Glazman inequality (1.1) can be expressed as P + Q − 1 = P Q = (1 − P )(1 − Q). In the following it is our aim to discuss the possibility of expressing (2.2) as P X + XQ − X = P XQ = (1 − P )X(1 − Q) provided that P X − XQ is (left, right) invertible: Theorem 3.10. Let P, Q and X be as in Theorem 2.3. If P X − XQ is left invertible and ran P is invariant under X|P X − XQ|−1 X ∗ then P X + XQ − X = P XQ = (1 − P )X(1 − Q).

(3.8)

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Proof. It is well known that the operator V = (P X − XQ)|P X − XQ|−1 is isometric. Let us also note that Q and |P X − XQ|−1 commute since, by (3.7), Q|P X − XQ|2 = QX ∗ (1 − P )XQ = |P X − XQ|2 Q. Also, by hypothesis, (1 − P )X|P X − XQ|−1 X ∗ P = 0. We deduce that V QX ∗ P = (P X − XQ)|P X − XQ|−1 QX ∗ P = (P − 1)XQ|P X − XQ|−1 X ∗ P = (P − 1)X(Q − 1)|P X − XQ|−1 X ∗ P = (1 − P )X(1 − Q)V ∗ . Thus, it holds P XQ = V QX ∗ P = (1 − P )X(1 − Q)V ∗ = V (1 − Q)X ∗ (1 − P ) = (1 − P )X(1 − Q). Finally, (3.8) is a direct consequence of (2.2).

Remark 3.11. (i) If we exchange, as before, the roles of P and Q and of X and X ∗ , we obtain in Theorem 3.10 the same conclusion if P X − XQ is right invertible and ran Q is invariant under X ∗ |(P X − XQ)∗ |−1 X. (ii) If P X − XQ is invertible then the operator V = (P X − XQ)|P X − XQ|−1 introduced in the proof of Theorem 3.10 is unitary. In this case QX ∗ P and (1 − P )X(1 − Q) are unitarily equivalent. (iii) If K = H , P X = XP and QX = XQ then |P X − XQ|2 P = |X|2 |P − Q|2 P = P |X|2 |P − Q|2 = P |P X − XQ|2 . It follows that P and |P X − XQ|−1 commute, so (1 − P )X|P X − XQ|−1 X ∗ P = (1 − P )P X|P X − XQ|−1 X ∗ = 0. If P X = XP and QX = XQ then, obviously, P X − XQ is left (resp. right) invertible if and only if P − Q is invertible and X is left (resp. right) invertible. Theorem 3.10 can be specialized as follows: Let X ∈ L(H ) be left (resp. right) invertible and H1 , H2 be two (closed) subspaces of H which reduce X (i.e., ˙ 2 = H . Then H1 , H2 are invariant under both X and X ∗ ) and such that H1 +H P X + XQ − X = P XQ = (1 − P )X(1 − Q), where P = PH1 and Q = PH2 .

4. Operators with closed ranges As emphasized by Theorem 3.8 and Remark 3.9 (i), P X − XQ is invertible if and only if ran [QX ∗ (1 − P )] = ran Q, ran [(1 − Q)X ∗ P ] = ker Q, ran [P X(1 − Q)] = ran P and ran [(1 − P )XQ] = ker P . Consequently, if P X − XQ is invertible then operators P X(1 − Q), (1 − P )XQ, P X, (1 − P )X, XQ and X(1 − Q) have closed

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ranges. In the following it is our aim to study conditions which ensure that the converse also holds true. Let us firstly remark that, despite our remark above, the invertibility of P X − XQ does not imply that X has closed range: Example 4.1. Let P and Q be two orthogonal projections in L(H ) \ {0, 1} such that P Q − QP = (P − Q)(P + Q − 1) is invertible. Equivalently, P − Q and P + Q − 1 are invertible, that is P − Q < 1 and P + Q − 1 < 1. It follows (see [22]) that P and Q are unitarily equivalent. Hence there exists a unitary operator U on ker P onto ker Q. Let Z be a bounded linear operator on ker P which does not have closed range. Then the operator H h → Y h := U Z(h − P h) ∈ H is clearly linear, bounded and does not have closed range. Moreover, ran Y ⊆ ker Q. Let X = Q + (2Y (P Q − QP )−1 )−1 Y . Then P X − XP = P Q − QP +

1 PY 2Y (P Q − QP )−1

is invertible since P Q − QP is invertible and P Y 1 < . 2Y (P Q − QP )−1 (P Q − QP )−1 We claim that ran Y \ ran Y ⊆ ran X \ ran X. If k ∈ ran Y ∩ ran X then there exist vectors h, hn (n ≥ 0) such that lim Y hn = k = Qh +

n→∞

We deduce that

ker Q lim Y hn − n→∞

1 Y h. 2Y (P Q − QP )−1

1 h = Qh = 0, −1 2Y (P Q − QP )

thus k = (2Y (P Q − QP )−1 )−1 Y h ∈ ran Y . On the other hand, if, for h ∈ H , X ∗ h = Qh +

1 Y ∗h = 0 2Y (P Q − QP )−1

then Qh = 0 and Y ∗ h = 0 (the sum ran Q + ker P being direct). Consequently, ker X ∗ ⊆ ker Y ∗ or, equivalently, ran Y ⊆ ran X. Our claim is proved and X does not have closed range. The following result characterizes the closedness of the range of P X − XQ:

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Proposition 4.2. Let P, Q and X be as in Theorem 2.3. The following conditions are equivalent: (a) P X − XQ has closed range; (b) ran [P X(1 − Q)] and ran [(1 − P )XQ] are closed; (c) ran [(1 − Q)X ∗ P ] and ran [QX ∗ (1 − P )] are closed. P X(1 − Q) (1 − P )XQ Proof. Let T be the operator matrix acting between 0 0 Hilbert spaces K ⊕ K and H ⊕ H . Then ran T = ran [P X(1 − Q)] + ran [(1 − P )XQ] ⊕ {0} and, by (3.7) (with X replaced by X ∗ , P by Q and Q by P ), ran |T ∗ |2 = ran |(P X − XQ)∗ |2 ⊕ {0}. Let us recall that a given operator S has closed range if and only if S ∗ has closed range if and only if SS ∗ has closed range. Thus, P X − XQ has closed range if and only if the sum ran [P X(1 − Q)] + ran [(1 − P )XQ] is closed. We note, in addition, that the subspaces ran [P X(1 − Q)] and ran [(1 − P )XQ] are orthogonal. Consequently, ran [P X(1 − Q)] + ran [(1 − P )XQ] is closed if and only if ran [P X(1 − Q)] and ran [(1 − P )XQ] are closed. We deduce that the conditions (a) and (b) are equivalent. A similar argument used for the operator QX ∗ − X ∗ P (instead of P X − XQ) proves the equivalence between (a) and (c). We can use, for example, Proposition 3.4 ((a) ⇔ (b)), Remark 3.5 (ii) and Proposition 4.2 to deduce other necessary and sufficient conditions for the invertibility of P X − XQ: Theorem 4.3. Let P, Q and X be as in Theorem 2.3. The following conditions are equivalent: (a) P X − XQ is invertible; (b) ran [P X(1−Q)] = ran P, ran [(1−P )XQ] = ker P, ran [QX ∗ (1 − P )] = ran Q and ran [(1 − Q)X ∗ P ] = ker Q; (c) ran [P X(1 − Q)] = ran P, ran [(1 − P )XQ] = ker P, ran [QX ∗ (1 − P )] = ran Q and ran [(1 − Q)X ∗ P ] = ker Q. Remark 4.4. We recall from [3, p. 341] that an operator T (acting on a Hilbert space) with closed range has a Moore-Penrose inverse, i.e., there exists a unique operator T † which satisfies the following four identities: T T † T = T, T † T T † = T † , (T † T )∗ = T † T and (T T † )∗ = T T †. We shall prove in the following that, under the invertibility of P X − XQ, the Moore-Penrose inverse of T = P X(1 − Q) can be computed according to the formulas T † = (1 − Q)(P X − XQ)−1 = (P X − XQ)−1 P. To this aim, let us observe that P X(1 − Q)(P X − XQ)−1 = P, (P X − XQ)−1 P X(1 − Q) = 1 − Q

300


and

(P X − XQ)−1 P = (1 − Q)(P X − XQ)−1 .

Consequently, and

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T T † = P X(1 − Q)(P X − XQ)−1 P = P T † T = (1 − Q)(P X − XQ)−1 P X(1 − Q) = 1 − Q,

so T T †T = P X(1 − Q) = T, (T T † )∗ = P = T T †

T † T T † = (1 − Q)(P X − XQ)−1 P = T † , and (T † T )∗ = 1 − Q = T † T.

Hence, (1 − Q)(P X − XQ)−1 = (P X − XQ)−1 P is the Moore-Penrose inverse of P X(1 − Q). Similarly, [(1 − P )XQ]† = −Q(P X − XQ)−1 = −(P X − XQ)−1 (1 − P ), [QX ∗ (1 − P )]† = −(1 − P )[(P X − XQ)−1 ]∗ = −[(P X − XQ)−1 ]∗ Q and [(1 − Q)X ∗ P ]† = P [(P X − XQ)−1 ]∗ = [(P X − XQ)−1 ]∗ (1 − Q).

We recall the following result due to S. Izumino [10]: let A and B be operators with closed range acting on a Hilbert space. Then AB has closed range if and only if the sum ker A + ran B is closed. Lemma 4.5. Let P, Q and X be as in Theorem 2.3. (i) If P X has closed range and the sum ker(P X) + ker Q is closed and direct then ran [(1 − Q)X ∗ P ] = ker Q. The converse is, in general, false. (ii) The following conditions are equivalent: (a) X(1 − Q) has closed range, the sum ker(P X) + ker Q is direct and the sum ran P + ker[(1 − Q)X ∗ ] is closed; (b) ran [(1 − Q)X ∗ P ] = ker Q. (iii) If P X and X(1 − Q) have closed ranges then ker(P X) + ker Q is closed if and only if ran P + ker[(1 − Q)X ∗ ] is closed. (iv) If the sums ker(P X) + ker Q and ran P + ker[(1 − Q)X ∗ ] are direct then ran [P X(1 − Q)] = ran P if and only if ran [(1 − Q)X ∗ P ] = ker Q. Proof. (i) Since ker(P X) + ker Q is closed, we obtain, on the one hand, that P X(1 − Q) has closed range (by Izumino’s result). Equivalently, the range of (1 − Q)X ∗ P is closed. On the other hand, by (3.5), the sum ker(P X) + ker Q is direct if and only if ran [(1 − Q)X ∗ P ] = ker Q. ∗ Hence ran [(1 − Q)X P ] = ker Q. The following example shows that the converse is, in general, false. Let us consider an orthogonal decomposition of a Hilbert space H as H = H1 ⊕ H2 ⊕ H3

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such that there exists a bounded linear operator T on H3 with non-closed range. If X = 1H1 ⊕ 0H2 ⊕ T , P = PH1 ⊕H3 and Q = PH2 ⊕H3 then X ∗ P = X ∗ and (1 − Q)X ∗ = 1 − Q. Hence ran (X ∗ P ) = ran (X ∗ ) = H1 ⊕ ran T ∗ is not closed, although ran [(1 − Q)X ∗ P ] = ran [(1 − Q)X ∗ ] = ker Q. (ii) As mentioned above, the sum ker(P X) + ker Q is direct if and only if the range of (1 − Q)X ∗ P is dense in ker Q. If the subspaces ran [(1 − Q)X ∗ ] and ran P +ker[(1−Q)X ∗ ] are both closed then, by Izumino’s theorem, (1−Q)X ∗ P has closed range. Hence ran [(1 − Q)X ∗ P ] = ker Q. Conversely, if ran [(1 − Q)X ∗ P ] = ker Q then ran [(1 − Q)X ∗] = ker Q is closed and, by the same result of S. Izumino, the sum ran P + ker[(1 − Q)X ∗ ] is closed. (iii) If P X and X(1 − Q) have closed ranges then, again by the theorem of Izumino, the following conditions are equivalent: 1. ran [(1 − Q)X ∗ P ] is closed; 2. ker(P X) + ker Q is closed; 3. ran P + ker[(1 − Q)X ∗ ] is closed. (iv) follows immediately by (i) and (ii).

Some sufficient conditions for the left invertibility of P X − XQ can be obtained by Lemma 4.5 (i) and Theorem 3.8: Proposition 4.6. Let P, Q and X be as in Theorem 2.3 and consider the condition (P, Q, X)1 : P X has closed range and the sum ker(P X)+ker Q is closed and direct. Then each of the conditions (i) (P, Q, X)1 and (1 − P, 1 − Q, X)1 , (ii) (P, Q, X)1 and ran [QX ∗ (1 − P )] = ran Q, (iii) ran [(1 − Q)X ∗ P ] = ker Q and (1 − P, 1 − Q, X)1 implies that P X − XQ is left invertible. Lemma 4.5 (ii) and Theorem 3.8 provide necessary and sufficient conditions for the left invertibility of P X − XQ: Theorem 4.7. Let P, Q and X be as in Theorem 2.3 and consider the condition (P, Q, X)2 : X(1 − Q) has closed range, the sum ker(P X) + ker Q is direct and the sum ran P + ker[(1 − Q)X ∗ ] is closed. The following conditions are equivalent: (a) (b) (c) (d)

P X − XQ is invertible; (P, Q, X)2 and (1 − P, 1 − Q, X)2 ; (P, Q, X)2 and ran [QX ∗ (1 − P )] = ran Q; ran [(1 − Q)X ∗ P ] = ker Q and (1 − P, 1 − Q, X)2 .

Other conditions, which are equivalent to the invertibility of P X −XQ, follow by Lemma 4.5 (i) and Remark 3.9 (i):

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Theorem 4.8. Let P, Q and X be as in Theorem 2.3. The following conditions are equivalent: (a) P X − XQ is invertible; (b) ran [P X(1−Q)] = ran P, ran [(1 − P )XQ] = ker P, ran [QX ∗ (1−P )] = ran Q and ran [(1 − Q)X ∗ P ] = ker Q; (c) ran [P X(1 − Q)] = ran P, ran [(1 − P )XQ] = ker P, ran [QX ∗ (1 − P )] = ran Q and ran [(1 − Q)X ∗ P ] = ker Q. Proof. Clearly (a) implies both (b) and (c) by Theorem 3.8. Conversely, if (b) holds true then P X and XQ have closed ranges (ran (P X) = ran P and ran (QX ∗ ) = ran Q). The sums ker(P X) + ker Q and ker P + ker(QX ∗ ) are obviously direct (as discussed in the proof of Lemma 4.5 (i)). These sums are also closed in view of the theorem of Izumino. Hence ran [(1 − Q)X ∗ P ] = ker Q and ran [(1 − P )XQ] = ker P (cf. Lemma 4.5 (i)). We deduce, by Remark 3.9 (i), that P X − XQ is invertible. We replace, in (b), P by 1 − P and Q by 1 − Q to obtain that (c) also implies (a). Remark 4.9. (i) We exchange the roles of P and Q and of X and X ∗ in Proposition 4.6 and Theorem 4.7 to obtain necessary and/or sufficient conditions for the right invertibility, respectively invertibility of P X − XQ; (ii) Each of the conditions ran (P1 Y P2 ) = ran P1 , where the triple (P1 , Y, P2 ) belongs to the set {(P, X, 1 − Q), (1 − P, X, Q), (Q, X ∗, 1 − P ), (1 − Q, X ∗ , P )}, can be replaced in Theorems 4.3, 4.7 and 4.8, Lemma 4.5 and Proposition 4.6 by one of the equivalent conditions given by Lemma 3.1 with P replaced by P1 and A by Y P2 . (iii) Our results generalize theorems by D. Buckholtz [5, Theorem 1], J.J. Koliha and V. Rakoˇcević [13, Corollary 3.2], [14, Theorem 6.2], [15, Theorems 4.1,5.1] or S. Maeda [17, Theorem 3]. As observed by our approach above, if P X − XQ is invertible then the sums ker(P X) + ker Q, ker[(1 − P )X] + ran Q, ker P + ker(QX ∗ ) and ran P + ker[(1 − Q)X ∗ ] are closed and direct. They induce bounded idempotents on the corresponding closed subspaces of H . The converse is, in general, false: Example 4.10. Let T be any operator on a Hilbert space H0 which is one-to-one, selfadjoint, but not invertible. Let H := H0 ⊕ H0 and X be the operator defined on H by X(h0 , h1 ) := (T h1 , T h0 ), (h0 , h1 ) ∈ H . X is obviously one-to-one (ker X = ker T ⊕ ker T = {0}), selfadjoint X(h0 , h1 ), (h0 , h1 ) = 2 T h0 , h1 ,

(h0 , h1 ) ∈ H ,

but not invertible (ran X = ran T ⊕ ran T = H ); hence X does not have closed range. If P is the orthogonal projection onto the first component of H then (P X − XP )(h0 , h1 ) = (T h1 , −T h0 ),

(h0 , h1 ) ∈ H .

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It is easy to see that P X − XP is one-to-one (ker(P X − XP ) = ker X = {0}), but not invertible (ran (P X − XP ) = ran X = H ). On the other hand, ker(P X) = ran P and ker[(1 − P )X] = ker P . We deduce that the sums ker(P X) + ker P, ker[(1 − P )X] + ran P, ker P + ker(P X ∗ ) and ran P + ker[(1 − P )X ∗ ] reduce to the orthogonal decomposition ker P ⊕ ran P = H ; hence they are closed and direct. Acknowledgment. The work of the first author has been supported by the Hungarian Scholarship Board.

References [1] N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publ., New York, 1993. [2] M. Barraa, Convexoid and generalized derivations, Linear Alg. Appl. 350(2002), 289– 292. [3] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Series: CMS Books in Mathematics, 2nd ed., Springer Verlag, Berlin, Heidelberg, New York, 2003. [4] Z. Boulmaarouf, M. Fernandez Miranda and. J.-Ph. Labrouse, An algorithmic approach to orthogonal projections and Moore-Penrose inverses, Numer. Funct. Anal. Optim. 18(1997), 55–63. [5] D. Buckholtz, Hilbert space idempotents and involutions, Proc. Amer. Math. Soc. 128(2000), 1415–1418. [6] D. Buckholtz, Inverting the difference of Hilbert space projections, Amer. Math. Monthly 104(1997), 60–61. [7] R.G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc. 17(1966), 413–416. [8] I. Feldman, N. Krupnik and A. Markus, On the norm of polynomials of two adjoint projections, Integral Equations and Operator Theory 14(1991), 70–90. [9] O. Hirzallah and F. Kittaneh, On the chordal transform of Hilbert space operators, Glasgow Math. J. 44(2002), 275–284. [10] S. Izumino, The product of operators with closed range and an extension of the reverse order law , Tohoku Math. J. 34(1982), 43–52. [11] T. Kato, Perturbation Theory for Linear Operators, Reprint of the Corr. 2nd ed., Springer Verlag, Berlin, Heidelberg, New York, 1995. [12] Y. Kato, Some theorems on projections of von Neumann algebras, Math. Japonicae 21(1976), 367–370. [13] J.J. Koliha and V. Rakoˇcević, Fredholm properties of the difference of orthogonal projections in a Hilbert space, Integral Equations and Operator Theory 52(2005), 125–134. [14] J.J. Koliha and V. Rakoˇcević, Invertibility of the difference of idempotents, Linear Multilinear Alg. 50(2002), 285–292.

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[15] J.J. Koliha and V. Rakoˇcević, On the norm of idempotents in C ∗ -algebras, Rocky Mountain J. Math. 34(2004), 685–697. [16] S. Maeda, On arcs in the space of projections of a C ∗ -algebra, Math. Japonicae 21(1976), 371–374. [17] S. Maeda, On the distance between two projections in a C ∗ -algebra, Math. Japonicae 22(1977), 61–65. [18] S. Maeda, Probability measures on projections in von Neumann algebras, Reviews in Math. Physics 1(1990), 235–290. [19] L. Noakes and K.Y. Chung, Invariant subspaces: continuous stability implies smooth stability, Proc. Amer. Math. Soc. 120(1994), 119–126. [20] Z. Pasternak-Winiarski, On the dependence of the orthogonal projector on deformations of the scalar product, Studia Math. 128(1998), 1–17. [21] A. Seddik, The numerical range of elementary operators, Integral Equations and Operator Theory 43(2002), 248–252. [22] B. Sz.-Nagy, Spektraldarstellung linearer Transformationen des Hilbertschen Raumes, Ergeb. Math. Grenzgeb., Springer Verlag, Berlin, 1942. [23] Z. Takeda and T. Turumaru, On the property “position p ”, Math. Japonicae 2(1952), 195–197. [24] W.S. Tang, Oblique multiwavelets in Hilbert spaces, Proc. Amer. Math. Soc. 128(2000), 2017–2031. [25] W.S. Tang, Oblique projections, biorthogonal Riesz bases and multiwavelets in Hilbert spaces, Proc. Amer. Math. Soc. 128(2000), 463–473. [26] J. Weidmann, Linear Operators in Hilbert Spaces, Springer Verlag, Berlin, Heidelberg, New York, 1980. [27] H.K. Wimmer, Canonical angles of unitary spaces and perturbations of direct complements, Linear Alg. Appl. 287(1999), 373–379. [28] H.K. Wimmer, Lipschitz continuity of oblique projections, Proc. Amer. Math. Soc. 128(1999), 873–876. Dan Popovici Department of Mathematics and Computer Science University of the West Timi¸soara Bd. Vasile Pˆ arvan nr. 4 RO-300223 Timi¸soara, Romania e-mail: [email protected] Zolt´ an Sebestyén Department of Applied Analysis Lor´ and E¨ otv¨ os University P´ azm´ any Péter sét´ any 1/C H-1117 Budapest, Hungary e-mail: [email protected] Submitted: November 24, 2008.

Integr. equ. oper. theory 65 (2009), 305-306 0378-620X/030305–2, DOI 10.1007/s00020-009-1730-x c 2009 Birkh¨ auser Verlag Basel/Switzerland


In Memoriam Israel Gohberg (23.8.1928–12.10.2009)

Operator theory has lost one of its cornerstones, and the journal Integral Equations and Operator Theory, IEOT, has lost its founder when Israel Gohberg passed away on October 12, 2009. The gap that Israel Gohberg leaves in mathematics, and in the hearts of his friends and colleagues, cannot be measured on Earth. It is only surpassed by the sorrow in the hearts of his family, in particular, of his beloved wife Bella, his two daughters, his sister, and his grand-children. Time will not close this gap, it will make it more apparent.

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The oeuvre of Israel Gohberg comprises more than 500 scientific publications, including more than 26 monographs many of which have become classics in mathematics. He published with 73 coauthors, supervised more than 40 PhD students and inspired countless mathematicians in the many places he worked at during his life, including Kisinev and Tel Aviv, and at the same time Stony Brook, the Weizman Institute, Calgary, Amsterdam, and Maryland. He was the world leading authority in integral equations, nonselfadjoint operators, spectral theory and factorization of matrix and operator functions, inversion problems for structured matrices, the band method, the state space method, and applications of operator theory in a wide range of areas, in particular, in systems theory and engineering. But it is much more than just beautiful mathematics that we owe to Israel Gohberg. He founded two journals, Matematiceskije Issledovanja in Kisinev and, after emigrating to Israel in 1974, Integral Equations and Operator Theory in 1978; he created the series Operator Theory: Advances and Applications in 1980 reaching 200 volumes next year; he initiated the conference series International Workshops on Operator Theory and its Applications, IWOTA, which have grown ever since their start in 1981; he was heavily involved in the organization of the International Symposia on the Mathematical Theory of Networks and Systems, MTNS, to bring together engineers and mathematicians. The great achievements of Israel Gohberg were appreciated by many prizes and academic awards. In 1970 he was elected Corresponding Member of the Academy of Sciences of the Moldavian SSR and in 1985 Foreign Member of the Royal Netherlands Academy of Arts and Sciences; he was awarded the Landau Prize (1976) and the Rothschild Prize (1986) in Mathematics, the Hans Schneider Prize by the International Linear Algebra Society (1993), an Alexander von Humboldt Prize by the German Humboldt Foundation (1992), the M.G. Krein Prize by the Ukrainian Academy of Sciences (2008), and honorary doctoral degrees by the Technische Hochschule Darmstadt (1997), Technische Universität Wien (2001), Universitatea de Vest din Timisoara (2002), Universitatea de Stat din Moldova, Cisinau (2002), Universitatea de Stat Aleco Russo din Balti, Moldova (2002), Technion, Israel Institute of Technology, Haifa (2008); and he was elected SIAM Fellow (2009). These impressive facts and figures describe one of the most exceptional mathematicians of our time, but how to describe his wide and warm heart, his generosity as a friend, his unforgettable humour, and his extraordinary character and strength to overcome the most difficult times in life? It is impossible. We are left with tears in our eyes realizing what we have lost. But Israel would not want us to be desperate. He would want us to strive for proving elegant mathematical theorems and finding simple examples and interesting applications for our theories, so that his legacy in operator theory grows and flourishes. When Israel suggested me as Editor of IEOT in 2008, I knew that he was entrusting me with something very precious to him. I promised him to do all my very best for the well-being of his journal. Zichrono li’vracha – may your memory be for a blessing, Israel. Christiane Tretter (Editor of IEOT)

Integr. equ. oper. theory 65 (2009), 307–318 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030307-12, published online November 9, 2009 DOI 10.1007/s00020-009-1728-4


Complex Moment Problems and Recursive Relations of Fibonacci Type Rajae Ben Taher and Mustapha Rachidi To Ren´ e Ouzilou

Abstract. The complex moment problem for a sequence γ (2n) = {γij }0≤i,j≤n has been studied by Curto-Fialkow, where positivity and extension properties of the moment matrix M (n) = M (n)(γ) (γ ≡ γ (2n) ) are involved, for guaranteeing the existence of representing measure. But it was showed that positivity and recursiveness are not sufficient in order to have a representing measure for γ. Here we combine our techniques based on the Fibonacci sequences’s properties with some Curto-Fialkow’s results to obtain sufficient conditions for insuring that γ is a truncated moment sequence. We focus ourself on the case when rank M (n) ≤ n + 1, and finally we stretch our exploration to the finite-rank infinite positive moment matrix. Mathematics Subject Classification (2000). Primary 40A25; Secondary 44A60, 45M05 15A48, 47A57. Keywords. Truncated complex moment problem, representing measure, flatness, moment matrix, Fibonacci recursive relation, constructive process.

1. Introduction Let K be a nonempty subset of the complex numbers C, in general K is compact. The K-moment problem for a sequence γp = {γk }0≤k≤p≤+∞ (of R or C) can be started simply as follows. Find a positive Borelean measure µ such that z k dµ(z), (0 ≤ k ≤ p) with supp(µ) ⊂ K. (1.1) γk = K

In view of its fundamental importance in various field of mathematics and applied science, the moment problem has been extensively studied in the literature. For p = +∞; the K-moment problem (1.1) is called the full K-moment problem, in particular the case K = [a; b] (respectively K = [0; +∞[ and K = R) corresponds

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to the well known Hausdorff (respectively Stieltjes and Humburger) moment problem. These aforementioned moment problems have been investigated by various methods and techniques (see [1, 2, 4, 11, 12] for example). Necessary and sufficient conditions on the existence of solutions µ of the problem (1.1) are derived from the density of the space of polynomial functions in C ∞ (K). Moreover, the Hahn-Banach’s Theorem and the Riesz’s Theorem are applied respectively for insuring existence and characterization of these previously mentioned solutions (see [1, 4, 11, 12] for example). When p < +∞ the problem (1.1) is known as the truncated K-moment problem. The standard method for solving the full K-moment problem, using the density property of the space of polynomials in C ∞ (K) is obstructed. Therefore, Curto-Fialkow provided a new method for studying the truncated moment problem of a doubly indexed sequence of C (see [5, 6, 7] for example). Let γ = γ (2n) : γ00 , γ01 , γ10 , . . ., γ02n , . . ., γ2n0 be a doubly indexed sequence of C satisfying γ00 > 0 and γi,j = γj,i . The moment problem for γ consists of finding a positive Borel measure µ such that supp(µ) ⊂ C and z i z j dµ, for every i, j with 0 ≤ i + j ≤ 2n. (1.2) γi,j = C

In (1.2) the notation z means the complex conjugate of z. From this moment problem we can recover the known moment problems, particularly for i = 0 the problem (1.2) can be reduced to (1.1) with K ⊂ C (see [1, 2, 3, 4, 5, 8, 9, 11, 12] for example). When the problem (1.2) admits a solution µ, we say that γ is a truncated moment sequence (of order 2n) and µ its representing measure (see [5, 6, 7]). The moment matrix M (n)(γ), associated to the sequence γ, is defined by M (n)(γ) = (M [i, j])0≤i,j≤n , where M [i, j] is the (i + 1) × (j + 1) block of Toeplitz form (i.e. with constant diagonals) whose first row has entries given by γi,j , γi+1,j−1 , . . ., γi+j,0 and whose first column has entries given by γi,j , γi−1,j+1 , . . ., γ0,i+j (see [5, 6] for more details). In [5, 6, 7] Curto-Fialkow consider an approach to study the existence and the uniqueness of solutions of (1.2), using properties of the moment matrix M (n) ≡ M (n)(γ). Indeed, with the aid of the flat extension of the matrix M (n) (rank-preserving i.e. M (n) can be extended to a positive matrix M (n+1) satisfying rank M (n) = rank M (n+1)), important results are established, particularly those concerning the existence of solutions (see [5, 6, 7]). As shown in the Curto-Fialkow theory, the notion of recursiveness for positive Hankel matrices is released from the flat data, that play an intrinsic role for solving the moment problem. In this paper, we consider the notion of recursiveness emanated from Fibonacci sequences, which will serve as an important tool for the former truncated moment problem (1.2). Our main idea is to proceed as in [3, 8, 9], where the full K-moment problem for r-generalized Fibonacci sequences is related to the truncated one. More precisely, we paraphrase on the truncated complex moment in the case when the moment matrices M (n) are singular (Section 2). On the other hand, among the fallout of our techniques we supply a constructive process to produce the flat extension of moment matrix M (n) using Fibonacci

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sequences relations and significant examples exposed. Moreover, discussion on the Curto-Fialkow’s Theorem and our constructive process is considered (Section 3). For reason of clarity we continue to adopt in the sequel the Curto-Fialkow’s notations.

2. Complex moment problem and recursiveness Recall that the notion of recursiveness and flatness of the moment matrix M (n) ≡ M (n)(γ), associated to a sequence γ = (γij )0≤i+j≤2n , are useful in the CurtoFialkow’s theory for insuring the existence of representing measures, in terms of positivity and extensions properties of M (n) (see [5, 6, 7, 10] for example). The notion of flatness is intimately related to the notion of recursiveness between the columns of the matrix M (n). Indeed, let Pn be the space of com plex polynomials q(z, z) = i,j aij z i z j of total degree ≤ n. Note that m(n) =

. For every q ∈ Pn , we denote by q ≡ (aij ) the assodimC Pn = (n+1)(n+2) 2 ciated coefficients vector with respect to the lexicographic order of the basis i j 2 {z of Pn (that is (1, z, z, z , zz, z 2 , . . ., z n , . . . , z n )). Let p(z, z) = z }0≤i+j≤n i j bij γij . The linear form Λγ permits to bij z z be in P2n and set Λγ (p) = define on the space P2n the sesquilinear form p, qM(n) = M (n) p, q = Λγ (pq) (p, q ∈ P2n ). In fact, the moment matrix M (n) is the unique matrix satisfying the preceding relations. On the other hands, if we consider the lexicographic order 2 n 1, Z, Z, Z 2 , ZZ, Z , . . ., Z n , . . . , Z to denote rows and columns of the moment k i matrix M (n), it follows that the rows Z Z l , columns Z Z j entry of M (n) is i j z , z k z l = Λγ (z i+l z j+k ) = γi+l,j+k . Let CM(n) denote the subequal to M (n)z i space of Cm(n) spanned by {Z Z j }0≤i+j≤n . For every p(z, z) = aij z i z j in Pn , i j the expression p(Z, Z) ∈ CM(n) is defined by p(Z, Z) = aij Z Z = M (n)( p). From [5, Lemma 3.10] it follows that p(Z, Z) = 0 is equivalent to p(Z, Z) = 0. And if µ is a representing measure for γ, we have p(Z, Z) = 0 if and only if supp(µ) ⊂ Z(p) ≡ {z ∈ C; p(z, z) = 0} (see [5, Proposition 3.1]). Hence, if µ is a representing measure and p(Z, Z) = 0, where p, q , pq ∈ Pn , we have (pq)(Z, Z) = 0. Therefore, a necessary condition for representing measure is that the matrix M (n) (or the sequence γ) is recursively generated in the following sense : ”p, q, pq ∈ Pn and p(Z, Z) = 0 implies that (pq)(Z, Z) = 0”. Particularly, for the one dimensional real truncated moment problem, a real sequence β (2n) = {βi+j }0≤i+j≤n has a representing measure supported in R if and only if the Hankel matrix H(n) = (βi+j )0≤i+j≤n is positive and recursively generated with respect to the columns denoted by 1, T, . . . , T n (see [5, Theorem 3.9]). Analogous situation has been obtained for Hausdorff moment problem in [8], where the connection with r-generalized Fibonacci sequences has been brought to light. If H(n) is singular, set r := min{i : T i ∈< 1, T, . . . , T i−1 >}; thus 1 ≤ r ≤ n and there exist unique scalars a0 , . . . , ar−1 such that T r = a0 1 + · · · + ar−1 T r−1 . Further, H(n) ≥ 0 admits a positive extension H(n + 1) if and only if H(n)

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is invertible or H(n) is singular and recursively generated, as a matter of this fact, we have T r+s = a0 T s + · · · + ar−1 T r+s−1 for (0 ≤ s ≤ k − r). Such situation has been also discussed in [3], where it was shown that the polynomial P (z) = z r − a0 z r−1 − · · · − ar−1 , called the minimal polynomial, represents also the characteristic polynomial of the associated r-generalized Fibonacci sequence of coefficients a0 , a1 , . . . , ar−1 . Moreover, properties on the flat extension and subnormality are rephrased in terms of Fibonacci sequences setting. We emphasize here on the fact that Curto-Fialkow’s theory has been of great importance in [3, 8, 9], where a crucial bridge between the truncated moment problem and the linear generalized Fibonacci relations has been established. Our technique for studying recursively generated moment matrices is also released from the Fibonacci sequence properties, and it will serve as the latter notion for the truncated complex moment sequence. In this Section, we first focus on the truncated complex moment problem, especially we are interested in the case when M (n) is singular. Next we discuss the finite rank of infinite moment problem. To this aim we start by the following basic proposition, on the closed relationship between the moment sequence (1.2) and the linear Fibonacci relation. Proposition 2.1. Let {γij }i, j≥0 be a sequence of numbers (in C) with γij = γij . Then, the two following affirmations are equivalent. 1. {γij }i,j≥0 is a moment sequence which owns an r-atomic representing measure µ. 2. The family of sequences {γij }j≥0 (i ≥ 0 fixed) are r-generalized Fibonacci sequences, with the same characteristic polynomial which owns distinct roots r−1 t0 , . . . , tr−1 in C. More precisely, γin = j=0 ρi,j tnj , with ρ0,j ≥ 0 and ρi,j = ρ0,j tj i . Proof. Let {γij }i,j≥0 be a moment sequence which owns r-atomic representing measure µ of finite support. Then, there exists a polynomial P (x) = xr − a0 xr−1 − · · · − ar−1 such that its roots are the atoms of µ. Conversely, by expressing the columns Z n and Z n+1 in terms {γij }i, j≥0 , it turns out that in each row Z i Z j the first index i of {γij }i, j≥0 is fixed. So, we infer that, γin = a0 γin−1 + · · · + ar−1 γin−r−1 for i ≥ r − 1 n which leads us to have γin = r−1 j=0 ρi,j tj , where t0 , . . . , tr−1 are the roots of P and ρi,j are derived from the Vandermonde equation V (t0 , . . . , tr−1 )(ρi,0 , . . . , ρi,r−1 )T i n = (yi0 , . . . , yir−1 )T . Since γi,n = C z i z n dµ = r−1 j=0 ρ0,j tj tj , we obtain that ρi,j = ρ0,j tj

i

for 0 ≤ i ≤ r − 1, 0 ≤ j ≤ r − 1.

Proposition 2.1 is quite useful in the sequel, particularly for the construction of the flat extension M (n + 1) of the moment matrix M (n). Indeed, let M ∈ Mm(n) (C) and CM(n) the space of columns of M ; that is n

CM(n) =< 1, Z, Z, . . . , Z n , . . . , Z > .

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i

i

i

311

i

Consider the subspace CM(n) =< Z , ZZ , Z 2 Z , . . . , Z n Z > (0 ≤ i ≤ n) of CM(n) . As a consequence of Proposition 2.1, we have the following result. Corollary 2.2. Under the preceding data, {γij }i,j≥0 is a moment sequence which owns r-atomic representing measure µ if and only if the sequence of analytic columns {Z i }i≥0 is an r-generalized Fibonacci sequence of characteristic polynomial P (x) = xr −a0 xr−1 −· · ·−ar−1 , whose roots are simple. Moreover, the scalars {ρ0,j }0≤j≤r−1 defined here below are positive. We infer a crucial relation between the notion of flatness and recursiveness in the space CM(n) described as follows. Proposition 2.3. Under the preceding data, let γ = γ (2n) be a sequence of C and M (n) = M (n)(γ) ∈ Mm(n) (C). Let M (n+ 1) ∈ Mm(n+1) (C) an extension of M (n) (not necessary a positive structured extension), then the following assertions are equivalent 1. rank M (n) = rank M (n + 1) i (i) (i) (i) (i) C Z n+1 Z (0 ≤ i ≤ n), 2. dimC CM = dimC En , where En = CM i i i i 3. Z n+1 Z = a0 Z + a1 ZZ + · · · + ar−1 Z n Z , for every 0 ≤ i ≤ n. It turns out that if one of the assertion of Proposition 2.3 is satisfied we obtain the following family of recurrence relations considered in Proposition 2.1, γin+1 = a0 γin + a1 γin−1 + · · · + ar−1 γin−r+1 .

(2.1)

More than our techniques, based on the Fibonacci sequences, that may be applied for studying the finite rank of infinite moment matrices. The main issue here is the following theorem given by Curto-Fialkow in [5], Theorem 2.4. Let M be a finite-rank positive infinite moment matrix. Then M has a unique representing measure which is r-atomic, where r = rank(M ). In this case there exists unique scalars α0 , . . . , αr−1 such that Z r = α0 1 + · · · + αr−1 Z r−1 . Moreover, the unique representing measure for M has support equals to the r distinct roots z0 , . . . , zr−1 of the polynomial P (z) = z r − (α0 z r−1 + · · · + αr−1 ), whose densities ρ0 , . . . , ρr−1 are determined by the Vandermonde equation V (z0 , . . . , zr−1 )(ρ0 , . . . , ρr−1 )T = (γ00 , . . . , γ0,r−1 )T . By combining this latter theorem with Proposition 2.3, we get the following result characterizing the infinite matrices of finite rank. Theorem 2.5. Let M be an infinite matrix of finite rank r = rank(M ). Then the following assertions are equivalent. 1. M is a positive infinite moment matrix. 2. M (r) ≥ 0 and the family of sequences {γij }j≥0 (i ≥ 0) are r-generalized Fibonacci sequences, with the same characteristic polynomial r−1 which owns distinct roots t0 , . . . , tr−1 in C. More precisely, γin = j=0 ρi,j tnj , with ρi,j = ρ0,j tj i

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Results of Theorems 2.4, 2.5 illustrate our motivation to study the moment matrices in terms of recursive relation of Fibonacci type. In the next section we pause to paraphrase the contribution of our approach to define a constructive algorithm for providing the flat extension.

3. Constructive process for the complex moment problem 3.1. Constructive process relied on Fibonacci sequences techniques A part of the intrinsic of the fallout of our techniques is to give a constructive method to determine the flat extension of moment matrix. For reason of clarity, we process by induction; that is, we start our survey in the simple case. Let γ (2) = (γ00 , γ10 , γ01 , γ10 , γ20 , γ11 , γ02 ) then M (1) is given by   γ00 γ01 γ10 M (1) =  γ10 γ11 γ20  . γ01 γ02 γ11 Suppose M (1) ≥ 0 and rank M (1) = 2. Let a, b ∈ C such that γ02 = aγ01 + bγ00 . Assume that the polynomial P (z) = z 2 − az − b owns two distinct roots. We label M (2) the matrix,  2  Z Z 2 ZZ Z 1 Z   γ  00 γ01 γ10 γ02 γ11 γ20    γ  10 γ11 γ20 γ12 γ21 γ30   (3.1) M (2) =  γ01 γ02 γ11 γ03 γ12 γ21  .   γ γ γ γ γ γ 20 21 30 22 13 40    γ11 γ12 γ21 γ13 γ22 γ31  γ02 γ03 γ12 γ04 γ31 γ22 Otherwise, the matrix M (2) can be interpreted asan extension of M (1), and

M (1) B also written under the form M (1) = . Observe that all the new B∗ C coefficients in M (2) are localized in the column Z 2 , more especially; there are γ03 , γ22 , γ13 , γ04 . Employing the recursive relations released from the Fibonacci sequences of characteristic polynomial P (z) = z 2 − az − b, one gets at once the latter coefficients defined by  γ12 = aγ11 + bγ10      γ03 = aγ02 + bγ01 γ13 = aγ12 + bγ11   γ22 = aγ21 + bγ20    γ04 = aγ03 + bγ02 . It turns out that rank M (1) = rank M (2). Thus the measure defined by µ = ρ00 δz0 + ρ01 δz1 with z0 , z1 are the distinct roots of P (z) = z 2 − az − b and V (z0 , z1 )(ρ00 , ρ01 )T = (γ00 , γ01 ) is a representing measure for γ = {γ00 , γ01 , γ10 , γ11 , γ02 , γ20 }.

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Since M (1) ≥ 0, we have thereby µ ≥ 0 in light of Proposition 3.8 of [5]. Proposition 3.1. Suppose M (1) ≥ 0 and rank M (1) = 2. Let γ02 = aγ01 + bγ00 and assume that P (z) = z 2 − az − b owns two distinct roots z0 , z1 . Then, the sequence γ (2) admits a 2-atomic representing measure µ = ρ00 δz0 +ρ01 δz1 such that V (z0 , z1 )(ρ00 , ρ01 )T = (γ00 , γ01 ). Moreover, M (1) admits a flat extension M (2) satisfying,  γ12 = aγ11 + bγ10     γ  12 = aγ11 + bγ10   γ03 = aγ02 + bγ01 γ13 = aγ12 + bγ11     γ22 = aγ21 + bγ20    γ04 = aγ03 + bγ02 . An analogous survey is established for the singular quadratic moment problem, where we manage to have a similar result. Let γ (4) = {γ00 , γ10 , γ01 , γ20 , γ11 , γ02 , γ30 , γ21 , γ12 , γ03 , γ40 , γ31 , γ22 , γ13 , γ04 }. The associated moment matrix is M (2) given by (3.1). Assume that M (2) ≥ 0, rank M (2) = 3 and set   γ03 = aγ02 + bγ01 + cγ00 γ13 = aγ12 + bγ11 + cγ10 ,  γ04 = aγ03 + bγ02 + cγ01 where a, b, c are

in C. Moreover, suppose that γ00 γ01 γ02 γ10 γ11 γ12 = 0 and (γ03 , γ13 , γ04 ) = (0, 0, 0), γ01 γ02 γ03

then the scalar are unique. Therefore, we have the representing 2a, b, c exist and (4) measure µ = i=0 ρ0i δzi of γ , where z0 , z1 , z2 are the roots of the polynomial P (z) = z 3 − az 2 − bz − c and V (z0 , z1 , z2 )(ρ00 , ρ01 , ρ02 )T = (γ00 , γ01 , γ02 ), further those roots are distinct. As a matter of fact, we may define the matrix M (3), extension of M (2), by providing the new coefficients that appear in the column Z 3 . We also use the structure of Fibonacci sequence to express these new coefficients as follows, γ14 γ23 γ33 γ05 γ24 γ15 γ06

= aγ13 + bγ12 + cγ11 = aγ22 + bγ21 + cγ20 = aγ32 + bγ31 + cγ30 = aγ04 + bγ03 + cγ02 . = aγ23 + bγ22 + cγ21 = aγ14 + bγ13 + cγ12 = aγ05 + bγ04 + cγ03

(3.2)

Then, we arrive to construct a positive and recursively generated extension M (3), which allows us to infer that µ is a representing measure for γ (4) .

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Proposition 3.2. Suppose M (2) ≥ 0 and rank M (2) = 3. Let the linear system,   γ03 = aγ02 + bγ01 + cγ00 γ13 = aγ12 + bγ11 + cγ10  γ04 = aγ03 + bγ02 + cγ01 , γ00 γ01 γ02 where we assume that γ10 γ11 γ12 = 0 and (γ03 , γ13 , γ04 ) = (0, 0, 0). Then γ01 γ02 γ03 the scalars a, b, c exist and are unique. Moreover, if z0 , z1 , z2 are the distinct roots 2 of P (z) = z 3 − az 2 − bz − c, then µ = i=0 ρ0i δzi is a representing measure for γ (4) . In this case, the representing measure is unique and M (4), the flat extension of M (3), is given via recursiveness by the relations (3.2). Following the preceding method, we can extend our approach to the general case as follows. Let γ (2n) = {γi,j }0≤i,j≤n and assume that M (n) ≥ 0 with rank M (n) = n + 1. Let a0 , a1 , . . . , an such that  γ0,n+1 = a0 γ0,n + a1 γ0,n−1 + · · · + an γ0,0     γ0,n+2 = a0 γ0,n+1 + a1 γ0,n + · · · + an γ0,1     ..   .     = a γ + a1 γ0,2n−2 + · · · + an γ0,n−1 γ  0,2n 0 0,2n−1  γ1,n+1 = a0 γ1,n + a1 γ1,n−1 + · · · + an γ1,0 (S) :  ..   .     = a γ γ  1,2n−1 0 1,2n−2 + a1 γ1,2n−3 + · · · + an γ1,n−2    ..    .   γn−1,n+1 = a0 γn−1,n + a1 γn−1,n−1 + · · · + an γn−1,0 Suppose that this linear system of equations is compatible and assume also that γ0,n . D = . . γ0,2n−1

γ0,n−1 . . . γ0,2n−2

. . . . .

. γ0,0 . . . . . . . γ0,n

= 0 and (γ0,n+1 , . . . , γ0,2n ) = (0, . . . , 0).

Consider that the roots z0 , z1 , . . . , zn of the polynomial P (z) = z n+1 − a0 z n − a1 z n−1 − a2 z n−2 − · · · − an−1 are distinct. This forces at once that {z0 , z1 , . . . , zn } contains the support of an r-atomic representing measure of the sequence γ (2n) , whose densities are determined from the Vandermonde equation V (z0 , . . . , zn )(ρ0 , . . . , ρn )t = (γ0,0 , . . . , γ0,n ).

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In fact, we show that the column Z n+1 consists of all new coefficients (or their conjugates) that appear in M (n + 1). More explicitly, the column Z (n+1) is defined by Z n+1 = (γ0,n+1 , γ1,n+1 , γ0,n+2 , γ2,n+1 , . . . , γ3,n+1 , . . . , γn,n+1 , γn−1,n+2 , . . . , γ0,2n+2 )t . As matter of this fact; the new coefficients such γi,j with i + j = 2n + 1 or i + j = 2n + 2 are given by utilizing the recursive relations emanated from the Fibonacci sequence’s properties by  γ0,2n+1 = a0 γ0,2n + · · · + an γ0,n     γ   0,2n+2 = a0 γ0,2n+1 + · · · + an γ0,n+1 .. (S) : .     γn,n+1 = a0 γn,n + · · · + an γn,0   γn,n+2 = a0 γn,n+1 + · · · + an γn,1 In this way, M (n + 1) is well determined and we have rank M (n + 1) = rank M (n) Proposition 3.3. Let γ (2n) = {γi,j }0≤i,j≤n and a0 , a1 , . . . , an the coefficients determined from the system S. Assume that D = 0 and (γ0,n+1 , γ1,n+1 , . . . , γ0,2n ) = 0. Then the scalars a0 , a1 , . . . , an exist and are unique. Furthermore, if we suppose that roots z0 , . . . , zn of P (z) = z n+1 − a0 z n − · · · − an are distinct, then n µ = i=0 ρ0,i δzi is a representing measure for γ (2n) . In this case, the representing measure is unique and the new coefficients appeared in M (n+ 1); the flat extension of M (n), is given via recursiveness relations in (S) . Example. Consider the following moment matrix associated to γ (4) ,  2  Z Z 2 ZZ Z 1 Z 1 −1  1 −1  1 1  3 3 3 3  −1 1 1 1   1 1  3 3 3 3 3  1 1 −1 1 1 . M (2) =  1  3 3 3 3 3  1 −1  −1 1 1 1   3  3 3 1 −1  −1  1 1 1 3 3 3 1 −1 −1 1 1 1 3 3 3 A calculation established via Matlab allows us to verify that rank M (2) = 3 and γ00 γ01 γ02 M (2) ≥ 0, yet γ10 γ11 γ12 = 0. Let a,b,c such that γ01 γ02 γ03   γ03 = aγ02 + bγ01 + cγ00 γ13 = aγ12 + bγ11 + cγ10  γ04 = aγ03 + bγ02 + cγ01 . A direct computation leads to a = 1, b = −1, c = 1, thereby applying Proposition 3.3 yields to bring out the unique representing measure given explicitly by µ =

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ρ0,j zj , where (z1 , z2 , z3 ) = (1, i, −i) (i2 = 1) are the roots of P (z) = z 3 +z 2 − z + 1 with ρ0,j = 13 (j = 1, 2, 3) are determined from V (z0 , z1 , z2 )(ρ00 , ρ01 , ρ02 )T = (γ00 , γ01 , γ02 ). j=1

3.2. Extension of constructive process and Curto-Fialkow’s Theorem For the Hausdorff moment problem it was established in [8] that the full moment problem for r-generalized Fibonacci sequences is reduced to the truncated moment problem associated to the initial values of this sequence. Conversely, when the truncated Hausdorff moment problem for γ (n) is solvable such that its positive representing measure is discrete and of finite support µ, then γ (n) owns an rgeneralized Fibonacci sequence extension, where r = rank(γ (n) ). In other words, the sequence of moments of the finite support measure µ gives arise to an rgeneralized Fibonacci sequence, extending by the way the sequence γ (n) . More generally, as shown in the preceding Section 3, the r-generalized Fibonacci sequences are inherent to the truncated complex moment problem. That is, the condition rank M (n) ≤ n+1 does not lose the problem of its generality, as it is justified by the Curto-Fialkow’s Theorem 1.5 of [6]. To illustrate this principle, let us recall here the essential content of this theorem. For a truncated complex sequence γ ≡ γ (2n) , the following statements are equivalent, (1) γ has a representing measure up to order 2(n + 1); (2) γ has a representing measure with moments of all orders ; (3) γ has a compactly supported representing measure ; (4) γ has finitely-atomic representing measure (in this case, there is some representing measure with at most (n + 2)(2n + 3) atoms); (5) M (n)(γ) ≥ 0 and for some k ≥ 0, M (n) admits positive extension M (n + k), which in turn admits a flat extension (i.e. rank-preserving) M (n + k + 1) (in this case, we can take k ≤ 2n2 + 6n + 6). This permits us to recover the underlying notion of r-generalized Fibonacci sequences used in the constructive process of Section 3. More precisely, if rank M (n) ≥ n + 1 it follows from the equivalence of assertions (1), (2) and (5) that there exists k (k ≤ 2n2 + 6n + 6) such that rank M (n + k) = rank M (n + k + 1). Thereby, we employ our process, based on Fibonacci sequence’s properties, to γ = {γij }0≤i,j≤n+k associated with M (n + k), this allows us to capture the notion of Fibonacci sequences arising from the linear columns < 1, Z, Z 2 , ..., Z n+k > of M (n + k).

Acknowledgment The authors would like to express their sincere gratitude to Professor R. Curto for his support and encouragement. They express their thankful to the anonymous referee for useful suggestions that improved this paper and to Professor H. Allouch for some helpful discussions on MatLab.

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Dedication. Owing of his humanity and his scientific generosity, we dedicate this paper to Professor René Ouzilou. We are thankful to him for all his support and knowledge that he has provided to us.

References [1] N. I. Akhiezer, The classsical moment problem and some related questions in analysis, Hafner Publ. Co., New York, 1965. [2] A. Atzmon, A moment problem for positive measures on the unit disc, Pacific Journal of Mathematics, 59 (1975), 317-325. [3] R. Ben Taher; M. Rachidi and E. H. Zerouali, Recursive subnormal completion and the truncated moment problem, Bull. London Math. Soc. 33(4) (2001), 425-432. [4] G. Cassier, Problème des moments sur un compact de Rn et décomposition des polynˆ omes a ` plusieurs variables, J. Funct. Anal. 58 no. 3 (1984), 254-266. [5] R. Curto and L. Fialkow, Solution of the truncated complex moment problem with flat data, Mem. Amer. Math. Soc. no. 568, Amer. Math. Soc., Providence 1996. [6] R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc. no. 648, Amer. Math. Soc., Providence 1998. [7] R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Relations in analytique or conjugate terms, Operator Theory: Advances and Applications 104 (1998), 59-82. [8] B. El Wahbi and M. Rachidi, On r-generalized Fibonacci sequences and Hausdorff moment problem, Fibonacci Quart. 39(1) (2001), 5-11. [9] B. El Wahbi, M. Rachidi and E. H. Zerouali, Recursive relations, Jacobi matrices, moment problems and continued fractions, Pacific Journal of Mathematics 216(1) (2004), 39-50. [10] L. Fialkow, Truncated complex moment problems with zz relations, Integral Equations and Operator theory 45 (2003), 405-435. [11] J. A. Shohat and J. D. Tomzrkin, The moment problems, Amer. Math. Soc. Surveys, 2, Amer. Math. Soc., New York, 1943. [12] B. Simon, The classical moment problem as a self-adjoint finite difference operator, Division of Physics, Mathematics, and Astronomy. California Institue of Tecnology. Pasadena, CA 91125. November 14, 1997.

Rajae Ben Taher Département de Mathématiques et Informatique Faculté des Sciences Université Moulay Ismail B.P. 4010 Beni M’hamed, Méknés Morocco e-mail: [email protected]

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Mustapha Rachidi Académie de Reims Mathematics Section - LEGT - F. Arago 1, Rue F. Arago 51100 Reims France e-mail: [email protected] Submitted: May 20, 2007. Revised: June 29, 2009.

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Integr. equ. oper. theory 65 (2009), 319–344 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030319-26, published online October 22, 2009 DOI 10.1007/s00020-009-1720-z


On the Nature of Ill-Posedness of the Forward-Backward Heat Equation Marina Chugunova, Illya M. Karabash and Sergei G. Pyatkov Abstract. We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in L2 (−π, π). Our method can be applied to a wide range of evolution problems given by P T -symmetric operators. Mathematics Subject Classification (2000). Primary 35P10; Secondary 35Q35, 35K15, 76A20. Keywords. Viscous fluid film, forward-backward diffusion, parabolic equation of mixed type, highly non-self-adjoint differential operator, Riesz basis property, completeness.

1. Introduction Analysis of the dynamics of a thin film of liquid entrained on the inside of a rotating cylinder is of great importance in many applications. For example, when liquid thermosetting plastic is placed inside a rotating mould, the best quality outcome can be achieved if the distribution of the liquid is as uniform as possible. More details on this application can be found in [35]. The same problem arises in the coating of fluorescent light bulbs in which a suspension consisting of a coating solute and a solvent is placed inside a spinning glass tube. The model for the coating is described in [6]. The lubrication approximation is used extensively to study flows in thin films. Under the assumption that the film is thin enough for viscous entrainment to The research of M. Chugunova has been supported by the NSERC Postdoctoral Fellowship. The research of I.M. Karabash has been partially supported by the University of Calgary and PIMS.

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compete with gravity, the time evolution of a thin film of liquid on the inner surface of a cylinder rotating in a gravitational field can be described by the forward-backward heat equation: ht + Lh = 0,

θ ∈ (−π, π),

t ∈ (0, T ),

(1.1)

where Lh = ε ∂θ (sin(θ) hθ ) + hθ ,

h(−π) = h(π),

ε > 0.

(1.2)

The effect of the surface tension is neglected in this linearized model derived by Benilov, O’Brien and Sazonov in [3]. We prove that the Cauchy problem related to this equation h|t=0 = h0 (θ),

h(−π, t) = h(π, t)

(1.3)

does not have a weak solution in the Sobolev sense h(θ, t), even locally in time, if h0 (θ) belongs to the class of finitely smooth functions with supp h0 ∩ (δ, π − δ) = ∅ (see Section 3). The statement above can be roughly understood from the classic theory of parabolic equations that states that regularity of a generalized solution depends on the regularity of the equation coefficients (in our case all coefficients are in C ∞ (−π, π)) and on the time-reversibility of the equation, i.e., simultaneous changes of the time variable t to −t and the space variable x to −x lead to the same partial differential equation. Time-reversibility and infinite regularity generally imply ill-posedness. The physical explanation of this explosive blow-up of solutions is related to a drop of fluid that in the absence of surface tension will be detached from the film in the upper part of cylinder, where the effect of the gravity is the strongest [3, p. 217]. Benilov, O’Brien and Sazonov [3] studied the eigenvalues of the operator L asymptotically, with an application of the modified WKB approximation and numerically, with an application of the analytic continuation method. They came to a very interesting set of hypotheses: all eigenvalues of the operator L are located on the imaginary axis; they are all simple; and the set of eigenfunctions is complete in L2 (−π, π) which is not typical for an ill-posed time-evolution problem. The analysis of the spectral properties of this operator was continued by Chugunova, Pelinovsky [10] and by Davies [12]. In particular, it was shown that if the parameter ε is in the interval 0 < ε < 2, then the operator is well defined in the sense that it admits closure in L2 (−π, π) with non-empty resolvent set. Analyzing a tridiagonal matrix representation of the operator L with respect to the Fourier basis, Davies [12] showed that L admits an orthogonal decomposition with respect to three invariant subspaces H2,0 (D), H2,0 (C \ D), and Ker(L) = {c½, c ∈ C} (see := L Section 4 below) and used this fact to prove that the nontrivial part L 2,0 2,0 H (D) ⊕ H (C \ D) of L has a compact inverse of the Hilbert-Schmidt type. We prove that, actually, the inverse operator is nuclear, i.e., belongs to the trace class S1 (see Proposition 4.7).

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It was proved by Weir [39] that if there exists an eigenvalue λ of the operator L, then µ = i 2λ ε is an eigenvalue of some symmetric operator; hence λ can be only purely imaginary. This elegant proof is based on the continuation of the eigenfunctions into the Hardy space H2 (D) in the unit disk D = {z ∈ C : |z| < 1}. In the preprint [5], it was proved that a wider class of operators possess only purely imaginary eigenvalues. It was shown numerically in [10, 12] that the angle between the subspace spanned by the first N eigenfunctions and the (N + 1)th eigenfunction of the operator L tends to 0 as N tends to infinity. This provides numerical evidence that the eigenfunctions do not form a Riesz basis in L2 (−π, π) because the related projectors are not uniformly bounded. The main result of this paper is an analytical proof of this numerical conjecture. Namely, we show that the operator iL is not similar to a self-adjoint operator and, consequently, its eigenfunctions do not form a Riesz basis (see Section 5). In Section 4, we rigorously prove the completeness of eigenfunctions and the existence of an infinite number of eigenvalues of the operator L. Combining this result with [12, 39], it is easy to see that the sequence of eigenvalues is purely imaginary and accumulates at ±i∞. As a consequence of the linearity, the original Cauchy problem has infinitely many global in time solutions which are linear combinations of harmonics eλn t uλn (x) where λn is a purely imaginary eigenvalue of the operator L and uλn (x) is the related eigenfunction. We wish to thank Weir for informing us about [40], in which the existence of the infinite number of eigenvalues is obtained by a different method. The operator L is J-self-adjoint in the Krein space with J(f (θ)) = f (π − θ) and therefore it belongs to the class of P T -symmetric operators. Interesting development of the spectral theory of P T -symmetric operators which are not similar to self-adjoint ones can be found in [26, 33, 34, 5]. Notation. In the sequel, C1 , C2 , . . . denote constants that may change from line to line but remain independent of the appropriate quantities. We use h(θ) when h is considered as a function of one variable and h(θ, t) when h is considered as a function of two variables. The symbol ½ denotes the function that identically equals 1 for θ ∈ (−π, π). Let T be a linear operator in a Hilbert space H. The following classic notations are used: Dom(T ), Ker(T ), Ran(T ) are the domain, the kernel, and the range of T , respectively; σ(T ) and ρ(T ) denote the spectrum and the resolvent set of T ; σp (T ) stands for the set of eigenvalues of T . We write f (x) g(x) (x → x0 ) if both f /g and g/f are bounded functions in a certain neighborhood of x0 . By D = {z ∈ C : |z| < 1} we denote the open unit disc in C. In Sections 2 and 3, we use the Sobolev spaces W2k (−π, π) and W2k1 ,k2 (Q) defined as in [25, Section I.1.2].

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2. Analysis of the differential equation The main aim of this section is to show that some of the spectral properties of the differential operator explicitly defined by (2.15) and of the operator considered in [12, 39] coincide. We use this fact in Proposition 4.2 to prove that these operators coincide. The latter is crucial for Section 5, where results obtained in Section 3 for the parabolic equation (1.1) are combined with results of [12, 39] to prove Theorem 5.6. One of the most general linear second-order differential equations with periodic coefficients that can be solved by a trigonometric series with a three-term recursion relation between the coefficients was studied by Magnus and Winkler [1] and has the form (A + B cos(2θ)) y + C sin(2θ) y + (D + E cos(2θ))y = 0,

(2.1)

where A, B, C, D, E are constants. Under the additional condition that the coefficient A + B cos(2θ) does not have zeros located on the real axis they studied existence of the periodic solutions to (2.1). In this section we study the basic properties of the differential equation [h](x) = f (x), where the differential expression is given by d dh dh , θ ∈ (−π, π), (2.2) [h] := ε sin(θ) + dθ dθ dθ and use these properties to define the maximal periodic differential operator associated with . This equation can be transformed to the form (2.1), but all singularities are located on the real axis, so the additional condition of Magnus and Winkler is not satisfied. Let f ∈ L2 (−π, π) and ε > 0. Denote I+ = (0, π), I− = (−π, 0). Consider the differential equation [h](x) = f (x) a.e. on (−π, π)

(2.3)

assuming that the function h is such that h, ε sin(θ)h + h ∈ ACloc (I− ∪ I+ ),

(2.4)

i.e., h and ε sin(θ)h + h are absolutely continuous on each closed subinterval of I− ∪ I+ . Lemma 2.1. Let h satisfy (2.4). Then h is a solution of the equation [h](x) = f (x) if and only if h has the form h(θ) = | cot(θ/2)|1/ε

k2± −

θ

f (t)| tan(t/2)|1/ε dt

±π/2

where θ ∈ I± , and k1± , k2± are arbitrary constants. The proof is based on direct calculations.

θ

+ 0

f (t)dt + k1± , (2.5)

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Proposition 2.2. Assume that ε ∈ (0, 2). A function h ∈ L2 (−π, π) satisfies (2.4) and is a solution of the equation [h] = f with f ∈ L2 (−π, π) if and only if h has the form θ t 1/ε θ 1/ε θ f (t)tan dt + f (t)dt + k1± , θ ∈ I± , (2.6) h(θ) = −cot 2 2 0 0 where k1± are arbitrary constants. Proof. Assume that h is an L2 (−π, π)-solution of [h] = f . Then it has the form (2.5). Note that t 1/ε f (t)tan ∈ L1 (0, δ) for any δ ∈ (0, π). 2 Therefore there exist the finite limit θ 1/ε t f (t)tan dt . C1 := lim k2± − θ→+0 2 ±π/2 If C1 = 0, then (2.5) implies that |h(θ)| ≥ |C2 | θ−1/ε for θ > 0 small enough, where C2 > 0. Since ε < 2, we see that h ∈ L2 (0, π). This shows that C1 = 0, and therefore h has the form (2.6) on (0, π). Similarly, one can show that h has the form (2.6) on (−π, 0). Let us prove that any function h of the form (2.6) belongs to L2 (0, π) (the proof for L2 (−π, 0) is the same). It is enough to check that h ∈ L2 (0, δ) and h ∈ L2 (π − δ, π) for sufficiently small δ > 0. For θ ∈ (0, δ), we have 1/2 θ θ t 1/ε 2/ε |f (t)| tan dt ≤ C3 f L2 t dt = C3 f L2 θ1/2+1/ε . 2 0 0 Hence,

θ 1/ε θ t 1/ε |f (t)| tan dt ≤ 2−1/ε C3 f L2 θ1/2 , cot 2 2 0

and we finally see that h ∈ L2 (0, δ). For θ ∈ (π − δ, π), we have 0

θ

t 1/ε |f (t)| tan dt ≤ C4 + 21/ε f L2 2

θ

(2.7)

1/2 −2/ε

(π − t)

dt

π−δ

≤ C5 + 21/ε f L2 (π − θ)1/2−1/ε . Hence, t 1/ε θ 1/ε θ |f (t)| tan dt ≤ C6 (π − θ)1/ε + C7 f L2 (π − θ)1/2 . cot 2 2 0 So h ∈ L2 (π − δ, π).

324


In particular, we have proved that lim h(θ) = k1±

and

θ→±0

lim

θ→±π∓0

h(θ) = k1± +

IEOT

±π

f (t)dt 0

hold for any L2 -solution h. This implies that the condition h is continuous on [−π, π] and periodic

(2.8)

is fulfilled exactly when k1+ = k1−

and f ⊥ ½.

k (−π, π) stand for the subspace of the space W2k (−π, π) Let the symbol W2p consisting of periodic functions, i.e., functions satisfying the conditions

u(i) (π) = u(i) (−π),

i = 0, 1, . . . , k − 1.

The norm in this space coincides with that of the Sobolev space W2k (−π, π). π Proposition 2.3. Let ε ∈ (0, 2), f ∈ L2 (−π, π), and −π f (θ)dθ = 0. Then an L2 -solution of [h] = f satisfies (2.8) if and only if θ θ 1/ε θ t 1/ε h(θ) = −cot f (t)tan dt + f (t)dt + k1 (2.9) 2 2 0 0 for a.a. θ ∈ (−π, π), where k1 is an arbitrary constant. Moreover, any function h of the form (2.9) possesses the following properties: (i) h ∈ AC[−π, π], h ∈ L2 (−π, π), and h L2 ≤ K(|k1 | + f L2 ),

(2.10)

where K is a constant independent of f . (ii) sin(θ)h ∈ AC[−π, π] and (sin(θ)h ) ∈ L2 (−π, π). Proof. To show that h ∈ L2 (−π, π), it is enough to prove that h ∈ L2 (0, δ) for any δ > 0 small enough. Since θ θ t 1 h (θ) = − cot1/ε f (t) tan1/ε dt, θ ∈ (0, δ), ε sin(θ) 2 2 0 it is sufficient to show that δ θ−2−2/ε 0

2

θ

f (t)t 0

1/ε

dt

dθ ≤ C1

0

δ

|f (θ)|2 dθ

for any f ∈ L2 (0, δ). Denote g(t) := f (t)t1/ε . Then (2.11) takes the form 2 δ δ θ −2−2/ε θ g(t)dt dθ ≤ C1 |g(θ)|2 θ−2/ε dθ . 0

0

0

(2.11)

(2.12)

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This is a weighted norm inequality for the Hardy operator. Applying [28] (see also [32] and references therein), we see that 1/2 1/2 δ θ −2−2/ε 2/ε sup t dt t dt < ∞, (2.13) θ∈[0,δ)

θ

0

and therefore (2.12) holds true. It is easy to see that the latter implies (2.10) and statement (i) of the theorem. 1 If f ⊥ ½, then sin(θ) h + h ∈ W2p (−π, π) and, by claim (i), so is sin(θ) h . 1 Introduce the space X2 = {h ∈ W2p (−π, π) : sin(θ)h ∈ W21 (−π, π)} endowed with the quasi-norm

h2X2 = h 2L2 (−π,π) + sin(θ)h 2W1 (−π,π) . 2

by X20 the subspace of X2 comprising the functions h with the propDenote π erty −π h(θ) dθ = 0. In turn, one can get the following statement. Proposition 2.4. The set X20 equipped with the norm · X2 is a Hilbert space. As a consequence of the definitions, we obtain that if h ∈ X2 , then the function sin(θ) h is absolutely continuous (may be after a change on a set of zero measure) and sin(θ) h |θ=0 = sin(θ) h |θ=π = sin(θ) h |θ=−π = 0. L2p (−π, π)

(2.14)

2

Let the symbol stand for the subspace of L (−π, π) comprising the π functions f with the property −π f (θ) dθ = 0. We present below a set of corollaries of Proposition 2.3. We also give the alternative proof of this result using the Galerkin method in Appendix A. Assume 0 < ε < 2 and denote by L the operator acting in L2 (−π, π) and defined by Lf = [f ] for f ∈ Dom(L) := X2 . (2.15) Clearly, Ker(L) = {c½, c ∈ C}.

the restriction of L to L2 (−π, π) = (Ker(L))⊥ , Let us denote by L p := L L2p (−π, π), L

:= Dom(L) ∩ L2p (−π, π). Dom(L)

(2.16)

is an operator It follows from the remark after (2.8) that Ran(L) ⊂ L2p (−π, π). So L 2 in the Hilbert space Lp (−π, π). −1 , let us symmetrize (2.9) as To find the inverse operator L t 1/ε θ 1/ε θ 1/ε θ f (t) tan − tan (2.17) dt + k1 , h(θ) = −cot 2 2 2 0 where θ ∈ (−π, π). Solving the equation (h, ½) = 0, we get π t 1/ε θ 1/ε θ 1/ε θ 1 f (t) tan − tan k1 = dt dθ. cot 2π 2 2 2 0 −π

(2.18)

326


IEOT

−1 f = h with h defined by (2.17)-(2.18). So for f ∈ Ran(L), we have L Corollary 2.5. Assume 0 < ε < 2. (i) The operator L defined by (2.15) is a closed operator in L2 (−π, π) (with the dense domain X2 ). (ii) Its kernel Ker(L) is the one-dimensional subspace of constants {c½, c ∈ C}. (iii) The range Ran(L) of L is the orthogonal complement to Ker(L), i.e., Ran(L) = L2p (−π, π). defined by (2.16) has a compact inverse L −1 . (iv) The operator L 0 2 : X → L (−π, π) is an isomorphism of X 0 onto L2 (−π, π). (v) The operator L 2 p 2 p Proof. (i)–(iii). Let k1 = k1 (f ) be the linear functional defined by (2.18). It is easy to see that k1 is bounded on L2 (−π, π). (2.19) This and Proposition 2.3 imply immediately that Ran(L) = L2p (−π, π). Therefore, −1 ) = L2 (−π, π). It follows from (2.10) and we have Dom(L p h L2 (−π,π) ≥ hL2 (−π,π) ,

h ∈ L2p (−π, π),

is closed and so −1 is a bounded linear operator on L2p (−π, π). Therefore L that L is L. −1 is bounded as an operator (iv)–(v). It follows from (2.10) and (2.19) that L 2 0 from Lp (−π, π) onto X2 . This proves statement (v). Note also that hW21 ≤ 2hX20

for any h ∈ X20 .

−1 is a compact operator in L2 (−π, π). This implies that L p

The adjoint differential operation ∗ is given by d dh dh ∗ [h] := ε sin(θ) − , θ ∈ (−π, π). dθ dθ dθ It is easy to see that ∗ [h] = J [Jh], where (Jh)(θ) = h(π − θ). Proposition 2.6. Let 0 < ε < 2. The adjoint operator L∗ has the form L∗ h = ∗ [h],

Dom(L∗ ) = X2 .

Moreover, L = JL∗ J and so L∗ is unitary equivalent to L. Proof. Integrating by parts and taking into account (2.14), one can see that X2 ⊂ Dom(L∗ ). Let us show that L = JL∗ J and therefore X2 = Dom(L∗ ). Consider the ˜ with respect to operator F defined by F = JLJ. Then F ⊂ L∗ . Since L = 0 ⊕ L the decomposition L2 (−π, π) = {c½, c ∈ C} ⊕ L2p (−π, π), ˜ ∗ and F = 0 ⊕ F˜ with respect to the same decomposition. we see that L∗ = 0 ⊕ L ∗ ∗ ˜ ˜ ˜ ˜ J, ˜ where J˜ := J L2p (−π, π) is a unitary operator Here L = (L) and F := J˜L

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˜ = Ran(F˜ ) = L2 (−π, π) and F˜ −1 is in L2p (−π, π). By Corollary 2.5 (iii), Ran(L) p ˜ ∗ )−1 = (L ˜ −1 )∗ is also a a compact operator on L2p (−π, π). On the other hand, (L 2 −1 ∗ −1 ˜ ) . We see that F˜ −1 = (L ˜ ∗ )−1 compact operator on Lp (−π, π) and F˜ ⊂ (L ∗ and therefore L = F . The last proposition implies that the statements analogous to that of Corollary 2.5 are valid for L∗ .

3. The ill-posedness of the Cauchy problem for the forward-backward heat equation The linearized model of the thin film dynamic (1.1) was derived without taking into account the smoothing effect of the surface tension. It is very natural to expect that a drop of fluid will be detached from the “ceiling” of the rotating cylinder. That can be described as a blow-up and it perfectly fits into the ill-posed nature of the Cauchy problem for the forward-backward heat equation (1.1). The intuition based on the classic theory of backward heat equation, says that global in time classic solutions can exist only for some class of functions analytic in vertical strip (−π, π) with exponentially fast decaying Fourier coefficients. In this section we assume that ε > 0. Let us consider the parabolic problem ht + Lh = 0,

θ ∈ (−π, π),

t ∈ (0, T ),

h|t=0 = h0 (θ), h(−π, t) = h(π, t),

(3.1) (3.2)

where L is understood in the sense of the differential expression (1.2). Note that, after the change of variables θ → −θ, equation (3.1) can be replaced with the equation ut − Lu = 0, where u(θ) = h(−θ). Let Q = (−π, π) × (0, T ). We prove in this section that the problem (3.1)– (3.2) is ill-posed in the classes of finite smoothness. In what follows, the symbol (·, ·) stands for the inner product in the space L2 in the corresponding domain (Q, Q1 , . . .).

1 (−π, π) ∩ C [0, T ]; L2 (−π, π) such Definition 3.1. A function h ∈ L2 0, T ; W2p that π h0 (θ)v(θ, 0) dθ −(h, vt ) − ε ( sin(θ) hθ , vθ ) + (hθ , v) = −π

1 for all v ∈ L2 (0, T ; W2p (−π, π)), vt ∈ L2 (Q), v(θ, T ) = 0, will be called a general1,0 (Q). ized solution to the problem (3.1)–(3.2) from the space W2p

Similar definitions can be found in [25, sect. 1 of chap. 3, sect. 3 of chap. 1]. Theorem 3.2 (Nonexistence). Let h0 ∈ L2 (−π, π). If there exist δ ∈ (0, π/2) and k ∈ N such that h0 ∈ / W2k (δ, π − δ) then the problem (3.1)–(3.2) does not have a 1,0 (Q). generalized solution in W2p

328


IEOT

Proof. Assume that a solution h of the problem exists. Take an arbitrary positive number δ ∈ (0, π/2) and put Q0 = (δ, π − δ) × (0, T /2). We have π −(h, vt ) − ε ( sin(θ) hθ , vθ ) + (hθ , v) = h0 (θ)v(θ, 0) dθ (3.3) −π

for all functions v as in Definition 3.1 such that supp v ⊂ Q0 ∪ { (θ, 0) : θ ∈ (δ, π − δ) }. ˜ τ ) = h(θ, T /2−τ ) Make the change of variables τ = T /2−t. Then the function h(θ, ˜ ˜ is a generalized solution of the equation hτ − Lh = 0 in Q0 and ˜h|τ =T /2 = h0 (θ). (3.4)

˜ ∈ L2 0, T /2; W1 (δ, π − δ) ∩ C [0, T /2]; L2 (δ, π − δ) More exactly, we have h 2 and π−δ ˜ ˜ ˜ −(h, vτ ) + ε ( sin(θ) hθ , vθ ) − (hθ , v) = − h0 (θ) v(θ, T /2) dθ (3.5) δ

for all

v ∈ L2 0, T /2; W21 (δ, π − δ) : vt ∈ L2 (Q0 ), v(θ, 0) = 0, v|θ=δ = v|θ=π−δ = 0.

Now construct a function h1 being a generalized solution to the problem hτ − Lh = 0, h|θ=δ = h|θ=π−δ = 0, h|τ =T /2 = h0 (θ), in the domain Q1 = (δ, π − δ) × (T /2, T ). Using the conventional parabolic theory (see, for instance, [25, Theorem III.4.2], or [14, Theorem 3 of Sect. 7.1.2]), we obtain that a solution of this problem exists and

h1 ∈ L2 0, T /2; W21(δ, π − δ) ∩ C [T /2, T ]; L2(δ, π − δ) . This solution satisfies the integral identity (see the definitions of a generalized solution in [25, Section III.1]). π−δ h0 (θ)v(θ, T /2) dθ (3.6) −(h1 , vτ ) + ε(sin(θ) h1θ , vθ ) − (h1θ , v) = δ 2

(T /2, T ; W21 (δ, π−δ))

2

for all v ∈ L : vτ ∈ L (Q1 ), v(θ, T ) = 0, v|θ=δ = v|θ=π−δ = 0. Put Q2 = (δ, π − δ) × (0, T ). By adding (3.5) and (3.6) we conclude that −(h2 , vτ ) + ε(sin(θ) h2θ , vθ ) − (h2θ , v) = 0 for all

(3.7)

v ∈ L2 0, T ; W21 (δ, π − δ) : vτ ∈ L2 (Q2 ), v(θ, T ) = 0, v(θ, 0) = 0, v|θ=δ = v|θ=π−δ = 0.

˜ for t ∈ (0, T /2) and with h1 for t ∈ (T /2, T ). Here the function h2 coincides with h So the function h2 is a generalized solution of the equation hτ − Lh = 0

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in Q2 in the sense of the integral identity (3.7). Since the coefficients of the operator L are infinitely differentiable, we can apply [25, Theorem III.12.1] in which it is demonstrated that interior smoothness of a generalized solution of the parabolic equation is determined by smoothness of the coefficients of L. By this theorem, we have h2 ∈ C ∞ (Q2 ). Therefore, h2 (θ, T /2) = h1 (θ, T /2) = h0 (θ) ∈ C ∞ (δ, π − δ). Since the quantity δ is arbitrary, we can conclude that h0 (θ) ∈ C ∞ (0, π). It contradicts to the condition of the theorem that h0 ∈ / W2k (δ, π − δ) for some δ > 0. Remark 3.3. In the proof of the theorem, we actually have established the following 1,0 statement: if the problem (3.1)–(3.2) has a generalized solution in W2p (Q) for some initial data h0 ∈ L2 (−π, π), then h0 ∈ C ∞ (0, π). k Definition 3.4. The problem (3.1)–(3.2) is said to be densely solvable in W2p (−π, π) (k ≥ 0 is an integer number) if there exists a dense subset K of this space such that, for any h0 ∈ K, the problem (3.1)–(3.2) has a generalized solution h in the sense of the Definition 3.1.

Theorem 3.5 (Instability). Let the problem (3.1)–(3.2) be densely solvable in the k (−π, π) for some nonnegative integer k, and let K be the corresponding space W2p dense subset. Then there is no a constant c = c(k) > 0 such that, for every generalized solution h to the problem (3.1)–(3.2) with an initial value h0 ∈ K, the estimate k (−π,π) (3.8) hL2 (Q) ≤ ch0 W2p holds. Proof. We use the arguments of Theorem 3.2. Assume to the contrary that (3.8) k holds for some constant c > 0 and all h0 ∈ K. We have that K ⊂ W2p (−π, π) and k k K = W2p (−π, π). Given δ ∈ (0, π/2), find a function h0 ∈ W2p (−π, π) such that h0 ∈ / W2k+1 (δ, π − δ). Construct a sequence h0n ∈ K : h0n − h0 W2k (−π,π) → 0 as n → ∞. In accord with the conditions of the theorem, there exists a generalized solution hn to the problem (3.1)–(3.2). By Remark 3.3, h0n ∈ C ∞ (0, π). As it is easily seen, (3.9) h0n Wk+1 (δ,π−δ) → ∞ as n → ∞. 2

Otherwise, there exists a subsequence h0nk which is bounded in W2k+1 (δ, π − δ) ˜ 0 as k → ∞ in Wk+1 (δ, π − δ) and, as a consequence, we can assume that h0nk → h 2 k+1 k ˜ and W2 (δ, π−δ) weakly for some h0 ∈ W2 (δ, π−δ). Due to the uniqueness of the ˜ 0 = h0 on (δ, π − δ). This contradicts to the fact that h0 ∈ / W2k+1 (δ, π − δ). limit, h Next, we repeat the arguments of the proof of Theorem 3.2. After the change of variables τ = T /2 − t we find that the functions ˜hn = hn (θ, T /2 − τ ) satisfy the integral identity π−δ/2 ˜ nθ , vθ ) − (h ˜ nθ , v) = − ˜ n , vτ ) + ε ( sin(θ) h h0n (θ) v(θ, T /2) dθ (3.10) −(h δ/2

330


IEOT

for all v ∈ L2 0, T /2; W21 (δ/2, π − δ/2) such that vt ∈ L2 (Q0 ), v(θ, 0) = 0, v|θ=δ/2 = v|θ=π−δ/2 = 0. ˜ n are generalized solutions to a parabolic equaThis means that the functions h tion in Q0 in the sense of an integral identity (see [25, Section III.1]). Note that coefficients of our parabolic equation (3.1) are infinitely differentiable, and, by [25, Theorem III.12.1], this solution belongs to any H¨ older space H 2+α,1+α/2 (Q0 ) (see the definition of this space in [25]). We put Q1 = (δ , π − δ ) × (ε0 , T /2),

where

δ/2 < δ < π/2,

0 < ε0 < T /2.

Since our solution can be extended to a generalized solution in the domain Q2 = ñ ∈ (δ/2, π − δ/2) × (0, T ) (see the proof of Theorem 3.2), we can say that h 2+α,1+α/2 H (Q1 ) for all δ > δ/2, ε0 > 0. By [25, Theorem III.8.1], for every ˜ n in C(Q1 ) is estimated by some constant δ > δ/2 and ε0 > 0 the norm of h depending on δ , ε0 > 0, and the norm of ˜hn in L2 (Q0 ). Using this fact and ˜ n in any H¨ older space applying [25, Theorem IV.10.1], we obtain that the norm of h 2+α,1+α/2 H (Q1 ) is estimated by some constant depending on δ , ε0 , α, and the ˜ n L2 (Q ) . Assume that δ < δ and α ≥ k − 1. In view of (3.8) with hn , h0n norm h 0 substituted for h, h0 and the fact that the norms h0n W2k (−π,π) are bounded, we can assume that this constant is independent of n. As a consequence, we have the estimate ˜ n (θ, T /2) k+1 = h0n (θ) k+1 ≤ c(δ, k), (3.11) h W2

(δ,π−δ)

W2

(δ,π−δ)

where the constant c(δ, k) is independent of n. Comparing (3.11) and (3.9), we arrive at a contradiction.

4. The completeness property for the operator L In this section we restrict the parameter to the interval 0 < ε < 2. In Proposition 4.2, we show that the operator L defined by (2.15) coincides with the closed operator introduced in [12]. The main aim of this section is to prove that the system of all eigenvectors and generalized eigenvectors of the operator L is complete in L2 (−π, π). In particular, this implies that L has an infinite number of eigenvalues. Denote by H2,0 (D) and H2,0 (C \ D) the subspaces of the Hardy spaces H2 (D) and H2 (C \ D), respectively, that are orthogonal to the function ½ (for basic facts on H2 (D) and H2 (C \ D) see e.g. [15, Section 2.1]). In the sequel, we use the standard identification of the function u(z) ∈ H2 (D) with the function u(eiθ ) := limr1 u(reiθ ), which belongs to L2 (−π, π), and also use the similar agreement for u(z) ∈ H2 (C \ D). Then H2 (D) and H2 (C \ D) are the subspaces of L2 (−π, π) and H2 (D) ∩ H2 (C \ D) = {c½, c ∈ C}. In these terms, the space L2 (−π, π) admits the orthogonal decomposition L2 (−π, π) = H2,0 (D) ⊕ {c½, c ∈ C} ⊕ H2,0 (C \ D).

(4.1)

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Define the operator Lfin in the Hilbert space L2 (−π, π) by Lfin h := [h], Dom(Lfin ) = Pfin , where is the differential expression defined in Section 2 and Pfin is the set of finite trigonometric polynomials h(θ) = (2π)−1/2

N

N < ∞,

vn einθ ,

vn ∈ C.

n=−N

It is easy to see that L∗fin is densely defined, and hence the closure Lmin := Lfin 2

exists as an operator in L (−π, π). be its restriction defined Let L be the operator defined by (2.15) and let L by (2.16). Below we give a proof for Lmin = L using the results of [12]. This proof allows us to use the orthogonal decomposition of L obtained in [12] (see also [10]). Proposition 4.1 (Theorems 11 and 13 in [12]). (i) The operator Lmin admits the orthogonal decomposition Lmin = L− ⊕ 0 ⊕ L+ with respect to (4.1). (ii) The operators L± are invertible, and their inverses L−1 ± are Hilbert-Schmidt operators. −1 are unitary equivalent. (iii) L−1 + and (−L− ) be the operators defined by (2.15) and (2.16), reProposition 4.2. Let L and L = L− ⊕ L+ . spectively. Then L = Lmin and L Proof. By Corollary 2.5 (i), L is a closed extension of Lfin . Hence Lmin ⊂ L. Let us show that L = Lmin . −1 By Proposition 4.1, the operator L−1 + ⊕ L− is compact and is acting on 2 Lp (−π, π). So σ(L− ⊕ L+ ) is at most countable. Then Ran(L− ⊕ L+ − λI) = L2p (−π, π)

for any

λ ∈ ρ(L− ⊕ L+ ).

(4.2)

−1 acting on L2p (−π, π) is compact. So L By Corollary 2.5 (iv), the operator L possesses the same properties, that is, σ(L) is at most countable and (4.2) holds for L. (which is equivalent to Lmin L). Then (4.2) and Assume that L+ ⊕ L− L whenever λ ∈ ρ(L)∩ρ(L the analogous equation for L imply that λ ∈ σp (L) + ⊕L− ), a contradiction. Definition 4.3 (see e.g. [16]). By Sp , 0 < p < ∞, we denote the class of all bounded linear operators A acting on a Hilbert space H for which  1/p ∞

|A|p :=  spj (A) 0 is a certain constant.

Remark 4.6. This fact was obtained in [12, the proof of Theorem 11]. The authors thank E.B. Davies for communicating the proof of Lemma 4.5 to us. −1 ∈ Sp for any −1 is nuclear. More precisely, L Proposition 4.7. The operator L p > 2/3. 2 Proof. Since L−1 + en L2 =

∞ m=1

∞

n=1

|ρm,n |2 , Lemma 4.5 shows that

p L−1 + en L2 ≤ C3

∞

n−3p/2 .

n=1

So the Gohberg-Markus criterion (Theorem 4.4) implies L−1 ∈ Sp for any + −1 −1 due 2/3 2/3 and so does L to Propositions 4.1 (ii) and 4.2.

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−1 ∈ Sp for p > 1 can be Although the weaker result that the operator L obtained directly from the factorization of the operator L found by Chugunova −1 actually belongs to the class and Strauss in [11], the fact that the operator L of nuclear operators S1 is crucial for the proof of the completeness property (see below). Following [16, Section IV.4], we call an operator T acting in a Hilbert space H dissipative if Im(T f, f ) ≥ 0 for all f ∈ Dom(T ). (4.6) Proposition 4.8. The operators L+ , (−L+ )−1 , −L− and L−1 − are dissipative. Proof. Using the tridiagonal matrix representation to the Fourier basis {einθ }∞ 1 (see [10, 12]), we get  1 −ε 0  ε 2 −3ε   0 3ε 3 A+ = (an,m )∞ 1 =   0 0 6ε  .. .. .. . . . ε n(n − 1), 2 This representation implies that an,n = n,

an−1,n =

of A+ (= −iL+) with respect 0 0 −6ε 4 .. .

··· ··· ··· ··· .. .

ε an,n+1 = − n(n + 1), 2

Im(L+ h, h) = Re(A+ h, h) ≥ 0

    ,  

(4.7)

n = 1, 2, . . . .

(4.8)

for all h ∈ H2,0 (D) ∩ Pfin . Since L = Lfin , one gets (4.8) for all h ∈ Dom(L+ ), i.e., L+ is dissipative. Substituting h = (L+ )−1 f into (4.8), we see that so is (−L+ )−1 . Proposition 4.1 (iii) completes the proof. Theorem 4.9 (Lidskii, see e.g. [16, Theorem V.2.3]). If the dissipative operator A acting on a Hilbert space H belongs to the class S1 , then its system of all eigenvectors and generalized eigenvectors is complete in H. Now the main result of this section can be obtained using Propositions 4.1, 4.7, 4.8, and Lidskii’s theorem. Theorem 4.10. The operator L has infinitely many eigenvalues. The system of its eigenvectors and generalized eigenvectors is complete in L2 (−π, π). Proof. By Propositions 4.7, 4.8, and Lidskii’s theorem, the systems of all eigenvec−1 −1 are complete tors and generalized eigenvectors of the operators L−1 + , L− , and L −1 ) = {0}, all in H2,0 (D), H2,0 (C \ D), and L2p (−π, π), respectively. Since Ker(L −1 corresponding to its eigenvalues αn are generalized generalized eigenspaces of L corresponding to eigenvalues λn = 1/αn . Since all eigenvalues of eigenspaces of L −1 and −1 have finite algebraic multiplicities, we see that L the compact operator L L have infinitely many eigenvalues and that standard arguments imply complete (and, consequently, for eigenvectors of L). (In ness property for eigenvectors of L

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are simple, so, Section 5 (see Proposition 5.5), it is shown that all eigenvalues of L −1 actually, L has no generalized eigenvectors.) Remark 4.11. Theorem 4.3 was announced in [10] with arguments partially based on numerical results. The rigorous proof given above uses essentially Proposition 4.7, and so uses the results of [12]. The fact that the operator L has an infinite number of eigenvalues was obtained independently and by a different method in [40].

5. Purely imaginary eigenvalues and the Riesz basis property In this section we prove that if 0 < ε < 2, then the set of eigenvectors {un }∞ 1 of the operator L normalized by un L2 = 1 does not form an unconditional basis in L2 (−π, π). Recall that a basis is called unconditional if it remains a basis for any ordering of the elements. By the well-known theorem of Lorch (see e.g. [16, Theorem VI.2.2]), {un }∞ 1 is an unconditional basis if and only if there exists a bounded and boundedly invertible operator T such that {T un }∞ 1 is a complete orthonormal set in L2 (−π, π) or, in other words, {un }∞ is a Riesz basis. 1 Recall that L+ is an operator in the Hilbert space H2,0 (D) = H2 (D) {c½, c ∈ C}. We identify the function u(z) ∈ H2 (D) with u(eiθ ) ∈ L2 (−π, π). Note that u ⊥ ½ is equivalent to

u(0) = 0.

2,0

So the last equality holds for all u ∈ H (D). Let u, f ∈ H2 (D). Consider the restrictions u and f of the functions u and f on the interval [0, 1) ⊂ D, u(x) = u(x),

f(x) = f (x) for

x ∈ [0, 1).

(5.1)

The following proposition was essentially obtained in [39] (formally for f = λu, see [39, Lemma 2.1 and the proof of Theorem 2.3]). For the sake of completeness, we give another proof that avoids the use of the Fourier representation of L and so is closer to the main approach of the present paper. Proposition 5.1 ([39]). Let 0 < ε < 2. Assume that u ∈ H2,0 (D), u ∈ Dom(L+ ), and Lu = f . Then the restrictions u, f defined by (5.1) satisfy the equation 2i f(x) for all x ∈ (0, 1) , ε where b[·] is the Sturm-Liouville differential expression defined by b[u](x) = −

1 b[u] = − (pu ) , w p(x) = (1 − x)1+1/ε (x + 1)1−1/ε , w(x) = x−1 (1 − x)1/ε (x + 1)−1/ε .

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Proof. First, assume additionally that u belongs to Pfin ∩ H2,0 (D), i.e., u(z) is a iθ ) iθ = ieiθ du polynomial such that u(0) = 0. Then so does f . Since du(e dθ dz (e ), we see that f (eiθ ) = [u](eiθ ) 2 du du iθ du iθ 2iθ d u iθ (e ) − e = ε sin(θ) −e (e ) + ε cos(θ)ieiθ (eiθ ) + ieiθ (eiθ ). 2 dz dz dz dz Since both sides are polynomials in z = eiθ , we get iε 2 d2 u(z) du(z) (z − 1)z (5.2) + i(εz 2 + z) 2 dz 2 dz for all z ∈ C. In particular, for z = x ∈ (0, 1), the last equality takes the form f(x) = iε 2 b[u]. Now, for arbitrary u, f ∈ H2,0 (D) such that u ∈ Dom(L+ ) and Lu = f , ∞ 2,0 consider sequences {fn }∞ (D) such that 1 and {un }1 in Pfin ∩ H f (z) =

lim f − fn L2 = 0,

n→∞

lim u − un L2

n→∞

and L+ un := fn ,

n ∈ N.

Note that such sequences exist since L = Lfin . We complete the proof substituting the polynomials fn and un into (5.2) and passing to the limit. It was noticed in [39], that Proposition 5.1 implies that if u(eiθ ) is an eigenfunction of the operator L+ , L+ u = λu, then the restriction u is a solution of a Sturm-Liouville eigenvalue problem with real-values coefficients. Indeed, b[u](x) = µu(x),

where µ = −2iλ/ε.

(5.3)

2

Consider the weighted space L ((0, 1); w), where the weight function w is defined as in Proposition 5.1. Let Bmax be an operator in L2 ((0, 1); w) associated with the differential expression b[·] and defined on its maximal domain, Bmax u = b[u] and Dom(Bmax ) = {u ∈ L2 ((0, 1); w) : u, u ∈ ACloc (0, 1), b[u] ∈ L2 ((0, 1); w)}. Note that all points of the interval (0, 1) are regular for the differential expression b, however the endpoints 0 and 1 are singular (1 is singular since p−1 ∈ / L1 (1/2, 1)). Proposition 5.2. Let ε > 0. (i) b is in the limit-point case at 0, (ii) b is in the limit-point case at 1 exactly when ε ≤ 1. (iii) Bmax is self-adjoint in L2 ((0, 1); w) exactly when 0 < ε ≤ 1. Proof. (i). Clearly, ½ is a solution of b[u] = 0 and ½ ∈ L2 ((0, 1/2); w). Weyl’s alternative (see e.g. [38, Theorem 5.6]) completes the proof. (ii). The general solution of b[u] = 0 on (0, 1) is x 1 u(x) = k1 ds + k2 , k1 , k2 ∈ C. p(s) 1/2

336


If k1 = 0,

u(x) k1

x

1/2

(1 − s)−1−1/ε ds k1 (1 − x)−1/ε ,

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x → 1 − 0.

Hence all solutions of b[u] = 0 belong to L2 ((1/2, 1); w) if and only if ε > 1. (iii) follows from (i) and (ii).

Proposition 5.3. If u(eiθ ) ∈ H2,0 (D), then the function u defined by (5.1) belongs to L2 ((0, 1); w). Proof. By (5), we have u(0) = 0. Since u is analytic at 0, we see that u ∈ 2 L ((0, 1/2); w). The measure w(x)dx on [1/2, 1) induces a measure M (S) := w(x)dx on D. For any sector S∩[ 1 ,1) 2

S = {reiθ : 1 − l ≤ r < 1, |θ − θ0 | < l}, we have

M (S) ≤ 2

1

max{1/2,1−l}

l ∈ (0, 1),

dx ≤ 2l

since maxx∈[1/2,1) w(x) ≤ 2. So M (·) is a Carleson measure (see e.g. [15, Sec. 4.3]). Therefore, 1 |u(x)|2 w(x)dx ≤ C1 u2H2,0 , 1/2

where C1 is a constant independent of u. This completes the proof.

The following result was essentially obtained by John Weir in [39, Theorem 2.3] (see also [5] for a more general result). The proof of [39, Theorem 2.3] was given under additional condition 1/ε ∈ Z, but John Weir informed us that the case 1/ε ∈ Z can also be treated by similar arguments. For the sake of completeness of the paper, we give below a proof for all ε ∈ (0, 2), which is different, but is based on the same idea of Weir (Proposition 5.1). We use the theory of Hardy spaces and this makes our proof simpler than that of [39, Theorem 2.3], where Frobenius theory and certain delicate estimates obtained in [12] were used. Proposition 5.4 ([39], see also [5]). Let ε ∈ (0, 2). Then all eigenvalues of the operator A+ (= −iL+ ) are real and positive. Proof. Let u(eiθ ) ∈ H2,0 (D) and L+ u = λu. By Proposition 5.3, u ∈ L2 ((0, 1); w). Proposition 5.1 implies that b[u] = µu with µ = −2iλ/ε, and therefore u ∈ Dom(Bmax ). Let us consider two cases: ε ∈ (0, 1] and ε ∈ (1, 2). In the case ε ≤ 1, Proposition 5.2 (iii) makes the proof simple. Since u ∈ Dom(Bmax ), µ is an eigenvalue of the nonnegative self-adjoint operator Bmax . Thus, µ ≥ 0. If 1 < ε < 2, then the proof requires additional analysis. By Proposition 2.3, g(eiθ ) := du (eiθ ) ∈ L2 (−π, π). It is easy to see from the representation dθ∞ iθ −1/2 inθ u(e ) = (2π) that g(eiθ ) ∈ H2 (D) (on the other hand, the latter n=1 vn e

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follows from [12, Theorem 16]) and g(eiθ ) = lim g(reiθ ), where g(z) = z du(z) dz , r→1−0

z ∈ D. By [15, Problem II.5 (a)], |g(x)| ≤ gL2 (1 − |x|2 )−1/2 for x ∈ (0, 1) and therefore, for x ∈ ( 1/2 , 1 ), du(x) −1 2 −1/2 ≤ C1 (1 − x)−1/2 . (5.4) dx ≤ gL2 |x| (1 − |x| ) By [38, Theorem 5.8 (ii)], the operator B½ defined by B½ u := b[u] on the domain Dom(B½ ) := {u ∈ Dom(Bmax ) : [½, u]1 = 0},

[½, u]1 := lim p(x) u (x), x→1−0

is self-adjoint. Note that the limit [½, u]1 exists for any u ∈ Dom(Bmax ) due to [38, Theorem 3.10]). It follows from (5.4) that, for any eigenvector u(eiθ ) of L+ , its restriction u belongs to Dom(B½ ). Indeed, it was shown above that u ∈ Dom(Bmax ). On the other hand, it follows from (5.4) that [½, u]1 = lim (1 − x)1+1/ε (x + 1)1−1/ε u (x) = 0. x→1−0

So µ is an eigenvalue of the operator B½ = B½∗ . x It follows from (5.4) that u(x) = u(1/2) + 1/2 u (t)dt has a finite limit as x → 1 − 0 (this fact also follows from [12, Theorem 16]). Therefore, 1 (B½ u, u)L2 ((0,1);w) = − (p(x)u (x)) u(x)dx = 0

1

0

p(x)|u (x)|2 dx − lim p(x)u (x)u(x) = x→1−0

1 0

p(x)|u (x)|2 dx ≥ 0.

Thus, B½ ≥ 0 and therefore µ ≥ 0. Finally, note that Ker(L+ ) = {0} and therefore µ = 0.

Proposition 5.5. Let ε ∈ (0, 2). Then all eigenvalues of the operator L+ are simple. Proof. Let u(eiθ ) ∈ H2,0 (D) be an eigenvector of the operator L+ , L+ u = λu. Assume that there exists a non-zero vector u1 (eiθ ) ∈ H2,0 (D) such that (L+ − λI)u1 = u. Consider the restrictions u, u1 of the functions u and u1 on the interval [0, 1) ⊂ D. By Proposition 5.3, u, u1 ∈ L2 ((0, 1); w). Proposition 5.1 shows that 2i λ 2i u with µ = − . (5.5) ε ε In the case ε ∈ (0, 1], (5.5) immediately implies that u1 is a generalized eigenvector of the self-adjoint operator Bmax . This is not possible. Consider the case ε ∈ (1, 2). Since u1 ∈ Dom(L), arguing as in the proof of Proposition 5.4, we see that u1 ∈ Dom(B½ ) and therefore u1 is a generalized eigenvector of the operator B½ = B½∗ . This is also a contradiction. b[u1 ] − µ u1 = −

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Theorem 5.6. Let ε ∈ (0, 2) and let {un }∞ 1 be a maximal system of linearly independent eigenvectors of L. We assume that un are normalized by un L2 = 1. Then 2 2 {un }∞ 1 is complete in L (−π, π), but does not form a Riesz basis in L (−π, π). Proof. The first statement follows directly from Proposition 5.5 and Theorem 4.10. We should prove the second one. Assume that the set {un }∞ 1 forms a Riesz basis in L2 (−π, π). Then Proposition 5.4 implies that iL is similar to a certain self-adjoint operator Q. That is, there exists a bounded and boundedly invertible operator S such that S Dom(Q) = Dom(L) and iL = SQS −1 . The spectral theorem for a self-adjoint operator implies that, for arbitrary u0 ∈ Dom(L)(= X2 ), the problem ut + Lu = 0,

u |t=0 = u0 ,

t ∈ R,

has a unique solution u(·, t) in the sense of [22, Definition I.1.1 and Eq. (I.1.2)] (such solutions are sometimes called strong solutions). Moreover, this solution has the form u(·, t) = Se−itQ S −1 u0 (·). Therefore, for any T > 0, 1 u ∈ C([0, T ]; X2 ) ⊂ C([0, T ]; W2p (−π, π)),

ut (·, t) ∈ C([0, T ]; L2 (−π, π)),

and Lu(·, t) ∈ C([0, T ]; L2 (−π, π)).

It is easy to see that u is a generalized solution of (3.1)–(3.2) in the sense of Definition 3.1. Since e−itQ is a unitary operator, u(·, t)L2 (−π,π) ≤ S S −1 u0L2 (−π,π) ,

t ∈ R.

Hence, for any T > 0, we have T u(·, t)2L2 (−π,π) dt ≤ CT u0 2L2 (−π,π) , 0

where C = S2 S −1 2 < ∞. The latter contradicts Theorem 3.5 since Dom(L) 1 (−π, π) and u0 L2 (−π,π) ≤ u0 W2p k (−π,π) for any k ≥ 0. is dense in W2p We would like to note that the linear partial differential equation (1.1) is an interesting example when the nature of explosive blow-up and instability of solutions has its roots not in the location of the eigenvalues but in geometric properties of the eigenfunctions.

6. Further discussion When the eigenfunctions related to neutrally stable eigenvalues of some linearized problem form a complete set, the representation of a solution of the nonlinear problem as a series of these eigenfunctions is one of the general approaches to the nonlinear stability problem. The lack of a basis property of the eigenfunction set is an obstacle for the applicability of this particular method. Due to the ill-posed nature of the forward-backward heat equation all eigenmodes are linearly unstable [3] and it is common to use the smoothing effect of the surface tension to stabilize them. The lubrication approximation that takes

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into account the influence of the capillarity effects and/or surface tension leads to the initial value problem for the fourth order nonlinear partial differential equation described in [8]. Some stability properties of its linearization were studied in [2, 4, 7]. The authors came to the conclusion that almost all modes, except some first ones, become stable even if the surface tension is relatively weak. We would also like to mention that the main assumption of the parameter range |ε| < 2 comes naturally from the theory of mixed type equations. For the case when |ε| > 2 the domain of the operator changes completely, so one needs to introduce a different kind of boundary conditions. Let us consider the equation k(x, t) utt + α(t, x) ut + ∆u = 0, x ∈ Ω, t > 0

(6.1)

where the coefficient k(x, t) can change sign in the domain where the operator is considered. So equation (6.1) is an equation of the mixed type, i.e. it is of the same type as the well-known Tricomi equation. On the lateral boundary of the cylinder Ω × (0, T ) we pose the Dirichlet boundary condition and there are two additional boundary conditions on the lower and upper base of the cylinder: u|t=T = 0, ut |S + = 0, ut |S − = 0, +

S = {(0, x) : k(x, 0) > 0}, S − = {(T, x) : k(x, T ) < 0}. This boundary value problem and closely related problems were studied by many authors (see, for instance, [36, 37]). It was demonstrated that the condition kt ≥ δ0 > 0 ∀(x, t), 2 where δ0 is a positive constant, ensures the existence of generalized solutions to the above-described boundary value problem. Stronger conditions of the type α−

|kt |(2k − 1) ≥ δ0 > 0 ∀(x, t) 2 ensure existence of smooth solutions and uniqueness of generalized solutions. The existence of solutions of non-linear forward-backward heat equations was studied by Hollig [19] and by Pyatkov [30]. Amongst the most recent results devoted to the nonlinear forward-backward parabolic problems we would like to mention the Kuznecov papers [23, 24]. Note also that Eq. (1.1) can be written as a second order equation with a nonnegative characteristic form (see e.g. [21, 29]) or as an abstract kinetic equation (see e.g. [13, 18, 20, 31] and references therein). However boundary value problems arising in these theories are different. One more difficulty in application of the general theory appears because of the strong degeneracy of the coefficient sin(θ) at points 0 and π. In particular, the theory of abstract kinetic equations requires writing the differential expression L in the J -self-adjoint form with respect to J : f (x) → sgn(x)f (x). The latter leads to the operator α−

d d sgn(θ) | tan θ/2|1/ε | sin(θ)| dθ | tan θ/2|1/ε dθ

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in the weighted Hilbert space L2 (−π, π); | tan θ/2|1/ε (see [10, Appendix A]). The boundary conditions associated with this operator and hence its spectral properties are completely different from that of the periodic operator L acting on L2 (−π, π).

Appendix A. Proof of Corollary 2.5 using the Galerkin method The second proof of Corollary 2.5. Let {ωj }∞ j=1 be a basis for the Hilbert space π 2 (−π, π) : −π h(θ) dθ = 0}. Find functions ϕj such that ϕj = ωj , H = {h ∈ W2p π ϕ (θ) dθ = 0. We look for an approximate solution to equation (2.3) in the −π j form n

hn = cjn ϕj , j=1

where the constants cjn are determined from the system of algebraic equations (Lhn , ωj ) = (f, ωj ), j = 1, 2, . . . , n,

π

(A.1)

(the brackets denote the inner product in L2 (−π, π), i.e., (h, v) = −π h(θ)v(θ) dθ). Multiplying (A.1) by cjn and summing the equalities obtained, we arrive at the relation (Lhn , hn ) = (f, hn ). Integrating by parts we derive the estimate hn L2 (−π,π) ≤ cf L2 (−π,π) ,

(A.2)

where c is a constant independent of n. This estimate implies that the system (A.1) is solvable (we can refer, for instance, to [27, lemma 4.3 of ch. 1]). Note that there exists a constant c1 independent of n such that hn L2 (−π,π) ≤ c1 hn L2 (−π,π)

(A.3)

From (A.2), (A.3) we conclude that there π exists a subsequence hnk and a function h ∈ W21 (−π, π), h(−π) = h(π) and −π h(θ) dθ = 0, such that hnk → h in L2 (−π, π),

hnk → h weakly in L2 (−π, π).

(A.4)

Let us multiply (A.1) with n = nk by constants αj (1 ≤ j ≤ m ≤ nk ) and sum the results. Fix m assuming that nk ≥ m. We infer m

−ε(sin(θ) hnk , ω ) + (hnk , ω) = (f (θ), ω), ω = αj ωj . j=1

Passing to the limit as k → ∞ we arrive at the equality

−ε(sin(θ) h , ω ) + (h , ω) = (f (θ), ω), ω =

m

j=1

αj ωj .

(A.5)

α ω are dense in H and thus (A.5) holds The functions ω of the form ω = m π j=1 j j for all functions in H. Since −π f (θ) dθ = 0, we can see that (A.5) also holds

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for all functions of the form ω + c (c is an arbitrary constant) and therefore for 2 all functions in W2p (−π, π). In particular, it holds for ω ∈ C0∞ (−π, π). From the definition of the generalized derivative (in the Sobolev sense) we have that there exist the generalized derivative (sin(θ) h ) and ε(sin(θ) h ) = (f − h ) ∈ L2 (−π, π). Thereby, sin(θ) h ∈ W21 (−π, π). Integrating by parts in (A.5) we obtain that the ˜ is equation (2.3) is satisfied almost everywhere on (−π, π). We have proved that L 0 2 0 1 an isomorphism of X2 onto Lp (−π, π). Since X2 ⊂ W2 (−π, π) and the embedding ˜ −1 : L2p (−π, π) → L2p (−π, π) is W21 (−π, π) ⊂ L2 (−π, π) is compact, the operator L compact. The remaining assertions are more or less obvious. Remark A.1. We can take the set {sin(jθ), cos(jθ)}∞ j=1 rather than an abstract basis {ωj }. ˜ −1 : L2p (−π, π) → L2p (−π, π) Remark A.2. The compactness of the operator L ˜ implies that the spectrum of L and the operator L itself is discrete with the only accumulation point ∞. The fact that the operator L has no real spectrum can be easily proved by integrating by parts. Indeed, assume the contrary that there exist λ ∈ R such that Lh = λh, h ∈ X20 . Let us multiply this equation by h and integrate the result over (−π, π). Integrating by parts and taking the real part, we arrive at the inequality hθ 2L2 (−π,π) ≤ 0 which yields h ≡ 0. Remark A.3. It is possible to prove the πfollowing statement. Let 1−ε(k +1/2) > 0. k Then for every f ∈ W2p (−π, π) with −π f (θ) dθ = 0 there exists a unique solution k+1 of the equation (2.3) such that h ∈ W2p (−π, π), sin(θ) h(k+1) ∈ W21 (−π, π). We do not need this result, so the proof is omitted. Acknowledgment The authors thank A. Burchard and E.B. Davies for useful comments and discussions. The authors are grateful to the organizers of the 7th workshop “Operator Theory in Krein Spaces and Spectral Analysis” for the hospitality of Technische Universit¨ at Berlin. I.M. Karabash would like to thank P. Binding for the hospitality of the University of Calgary. Finally, the authors would like to thank the anonymous referee for the detailed analysis of the paper and for valuable suggestions on improving the way of presentation and the language.

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[24] I. V. Kuznetsov, Entropy solutions to a second-order forward-backward parabolic equation, Dokl. Akad. Nauk 404 (2005), 443–445 (Russian); English translation in Dokl. Math. 72 (2005), 716–717. [25] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23, AMS, Providence, 1968. [26] H. Langer, C. Tretter, A Krein space approach to PT-symmetry, Czechoslovak J. Phys. 54 (2004), 1113–1120. [27] J. L. Lions, Quelques Méthodes de Résolution des Problemes aux Limites Nonlin´ eaire. Dunod, Gauthier-Villars, Paris, 1969. [28] B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 34 (1972), 31–38. [29] O. A. Ole˘ınik, E. V. Radkeviˇc, Second Order Equations with Nonnegative Characteristic Form., Plenum Press, 1973. [30] S. G. Pyatkov, Solvability of initial-boundary value problems for a nonlinear parabolic equation with changing time direction, Preprint, 16, Novosibirsk, Institute of Mathematics (1987) (Russian). [31] S. G. Pyatkov, Operator Theory. Nonclassical Problems. Utrecht, VSP 2002. [32] E. Sawyer, Weighted Lebesgue and Lorentz Norm Inequalities for the Hardy Operator, Trans. Amer. Math. Soc. 281 (1984), 329–337. [33] K. C. Shin, On the reality of the eigenvalues for a class of PT - symmetric oscillators, Comm. Math. Phys. 229 (2002), 543–564. [34] K. C. Shin, Eigenvalues of PT -symmetric oscillators with polynomial potentials, J. Phys. A 38 (2005), 6147–6166. [35] J. L. Throne, J. Gianchandani, Reactive rotational molding, Polym. Eng. Sci. 20 (1980), 899–919. [36] V. N. Vragov, Boundary Value Problems for Nonclassical Equations of Mathematical Physics. Novosibirsk State University, Novosibirsk, 1983 (Russian). [37] V. N. Vragov, A. I. Kozhanov, S. G. Pyatkov, S. N. Glazatov, On the theory of nonclassical equations of mathematical physics, Conditionally Well-Posed Problems. TVP/TSP, Utrecht (1993), 299–321. [38] J. Weidmann, Spectral theory of ordinary differential operators. Lecture Notes in Mathematics 1258, Springer Verlag, Berlin, 1987. [39] J. Weir, An Indefinite Convection-Diffusion Operator With Real Spectrum, Applied Mathematics Letters 22 (2009), 280–283; see also preprint arXiv:0711.1371v1 [math.SP] (2007). [40] J. Weir, Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator, Preprint, arXiv:0801.4959v2 [math.SP] (2008). Marina Chugunova University of Toronto 40 St. George Str. Toronto, Ontario M5S 2E4 Canada e-mail: [email protected]

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Illya M. Karabash Department of Math and Stat University of Calgary 2500 University Drive NW Calgary, Alberta T2N 1N4 Canada and Institute of Applied Mathematics and Mechanics R. Luxemburg str. 74 Donetsk 83114 Ukraine e-mail: [email protected] [email protected] Sergei G. Pyatkov Department of Math. University of Hanty-Mansiisk Chekhov st. 16 628012 Hanty-Mansiisk Russia e-mail: s [email protected] [email protected] Submitted: September 5, 2008. Revised: August 11, 2009.

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Block Matrix Operators and Weak Hyponormalities George Exner, Il Bong Jung and Mi Ryeong Lee Abstract. We introduce a new model of a block matrix operator M (α, β) induced by two sequences α and β and characterize its p-hyponormality. The model may be viewed as arising from the composition operator CT on 2 := L2 (N0 ) induced by a measurable transformation T on the set of nonnegl+ ative integers N0 with point mass measure. Composition operator techniques may then be used to treat the p-hyponormality of M (α, β). Finally, we apply our results to obtain examples of these operators showing the p-hyponormal classes are distinct. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47B38. Keywords. p-hyponormal operator, composition operator, conditional expectation.

1. Introduction and preliminaries Let H be a separable, infinite dimensional complex Hilbert space and let L(H) be the algebra of all bounded linear operators on H. An operator T ∈ L(H) is said to be p-hyponormal if (T ∗ T )p ≥ (T T ∗ )p , p ∈ (0, ∞). If p = 1, T is hyponormal and if p = 12 , T is semi-hyponormal ([15]). Further, T is said to be ∞-hyponormal if it is p-hyponormal for all p > 0 ([13]). The L¨ owner-Heinz inequality implies that every p-hyponormal operator is q-hyponormal for q ≤ p and many operator theorists have studied properties of operators in those classes, such as spectral theory, operator inequalities, and invariant subspaces, etc. (cf. [1], [4], [6], [7], [10]). Also, the study of the gap between subnormality and hyponormality has been of considerable recent interest. These studies include p-hyponormality: JungLee-Park constructed examples induced by some block matrix operators in [9] and [8], showing these classes of operators are sufficient to show the p-hyponormal classes are distinct with respect to positive real numbers p. In [1] and [2] BurnapJung-Lambert discussed some composition operator models CT on L2 , that also

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show the p-hyponormal classes are distinct. Nonetheless, new examples for the p-hyponormal classes are needed. A key tool in [1] and [2] was the use of the notion of conditional expectations for studying p-hyponormality of CT , and this will also be the main tool of this note. Here is some terminology and notation for conditional expectation. Let (X, F , µ) be a σ-finite measure space and let T : X → X be a transformation such that T −1 F ⊂ F and µ◦T −1 µ. It is assumed that the Radon-Nikodym derivative h = dµ◦T −1 /dµ is in L∞ . The composition operator CT acting on L2 := L2 (X, F , µ) is defined by CT f = f ◦ T . The condition h ∈ L∞ assures that CT is bounded. And we denote Ef = E(f |T −1 F ) for the conditional expectation of f with respect to T −1 F . (We refer the reader to [2] and [14] for the definition of conditional expectation. Some useful results will come as well from [11], [2], and [5].) In our situation, the following computation for conditional expectation will be enough: if A is the purely atomic σ-subalgebra of F generated by the measurable partition of X into sets of positive measure {Ak }k≥0 , then

E(f |A) =

∞ k=0

1 µ(Ak )

Ak

f (x)dµ(x) χAk .

(1.1)

The interested reader can find a more extensive list of properties for conditional expectations in [2] and [14]. This paper consists of four sections. In Section 2, we construct a block matrix operator induced by two sequences α and β, which will later be used to show the classes of p-hyponormal operators are distinct with respect to p > 0. A block matrix operator M (α, β) induced by two sequences α and β provides in turn a measurable transformation T on N0 with point mass measure on N0 , and 2 its corresponding composition operator CT on l+ is equivalent to M (α, β). In Section 3, we characterize block matrix operators M (α, β) for p-hyponormality and construct a useful form for some examples. In Section 4, we discuss a “flatness” property of p - hyponormality for these block matrix operators M (α, β): under some natural conditions, ∞-hyponormality of M (α, β) is equivalent to any[some] p-hyponormality. As well, we give various examples that show the p-hyponormality of one of these operators is limited, in ways perhaps surprising, by the dimensions of the block. Finally, we give examples to show the classes of p-hyponormal operators are distinct.

2. Relationships (n)

Let α := {ai }

1≤i≤r 0≤n R. We thus have λ + (T1 k)(x) = 0 for every |x| > R. Similarly, by using the Riemann-Lebesgue lemma of the Fourier transform F it is possible to prove that λ + γ2 (x)(F h)(x) = 0 for every x outside a finite interval. By Proposition 2.2 and the Riemann-Lebesgue Lemma, the two functions (T1 k), γ2 (F h) are continuous on R and vanish at infinity. It follows that there exist x1 , x2 ∈ R such that M0 := |(T1 k)(x1 )| ≥ |(T1 k)(x)|, N0 := |γ2 (x2 )(F h)(x2 )| ≥ |γ2 (x)(F h)(x)| for evey x ∈ R. Hence, if |λ| > max{M0 , N0 }, then we have λ + γ2 (x)(F h)(x) = 0 and λ + (T1 k)(x) = 0 for every x ∈ R. Comparison. a) By the use of T1 , T2 and their inverse transforms as presented in Examples 4.1, 4.2, and 4.3, we have a new approach (apart from the use of the Fourier transform) for the solution of linear partial differential equations. b) There is an approach to integral equations of convolution type by using an appropriate convolution and the Wiener-Lèvy theorem as e.g. in [8, 11, 16, 17]. However, that approach usually includes only sufficient conditions (no necessary conditions) for the solvability of the equations and obtains the solutions in implicit (not explicit) form. By means of the normally solvable conditions we are able to state necessary and sufficient conditions of the equations (4.3), (4.7), and give the solutions in explicit form by using the convolutions (3.1) and (3.8). Acknowledgment The authors would like thank Professor Reinhard Mennicken for a careful proofreading of the manuscript and some corrections and changes of the English usage, and the referee for helpful suggestions.

References [1] H. Bateman and A. Erdelyi, Tables of integral transforms. MC Graw-Hill, New YorkToronto-London, 1954. [2] S. Bochner and K. Chandrasekharan, Fourier transforms. Princeton Uni. Press, 1949. [3] L. E. Britvina, Generalized convolutions for the Hankel transform and related integral operators. Math. Nachr., 280 (2007), no. 9-10, p. 962–970. [4] B. T. Giang, N. V. Mau, and N. M. Tuan, Convolutions for the Fourier transforms with geometric variables and applications. Math. Nachr. (accepted). [5] B. T. Giang and N. M. Tuan, Generalized onvolutions for the integral transforms of Fourier type and applications. Fract. Calc. Appl. Anal., 12 (2009), no. 3 (to appear). [6] B. T. Giang and N. M. Tuan, Generalized convolutions for the Fourier integral transforms and applications. Journal of Siberian Federal Univ., 1 (2008), no. 4, p. 371–379. [7] I. S. Gohberg and I. A. Feldman, Convolution equations and projection methods for their solutions. Nauka, Moscow, 1971 (in Russian). [8] H. Hochstadt, Integral equations. John Wiley & Sons, N. Y., 1973.

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[9] L. H¨ ormander, The Analysis of Linear Partial Differential Operators I. SpringerVerlag, Berlin, 1983. [10] V. A. Kakichev, On the convolution for integral transforms. Izv. ANBSSR, Ser. Fiz. Mat., no. 2, p. 48–57, 1967 (in Russian). [11] V. A. Kakichev, N. X. Thao, and V. K. Tuan, On the generalized convolutions for Fourier cosine and sine transforms. East-West Jour. Math., 1 (1998), no. 1, p. 85–90. [12] M. A. Naimark, Normed Rings. P. Noordhoff Ltd., Groningen, Netherlands, 1959. [13] W. Rudin, Functional Analysis. McGraw-Hill, N. Y., 1991. [14] S. W. Smith, Digital Signal Processing: A Practical Guide for Engineers and Scientists. ISBN 0-7506-7444-X (e-book), 2002. [15] I. Sneddon, Fourier transforms. McGraw-Hill, New York-Toronto-London, 1951. [16] N. X. Thao and N. M. Khoa, On the generalized convolution with a weight function for the Fourier sine and cosine transforms. Integral Transforms Spec. Funct., 17 (2006), no. 9, p. 673–685. [17] N. X. Thao, V. K. Tuan, and N. T. Hong, Generalized convolution transforms and Toeplitz plus Hankel integral equation. Frac. Calc. App. Anal., 11 (2008), no. 2, p. 153–174. [18] E. C. Titchmarsh, Introduction to the theory of Fourier integrals. Third edition. Chelsea, New York, 1986. [19] J. N. Tsitsiklis and B. C. Levy, Integral Equations and Resolvents of Toeplitz plus Hankel Kernels. Technical Report LIDS-P-1170, Laboratory for Information and Decision Systems, M.I.T., December 1981. [20] V. K. Tuan, Integral transform of Fourier type in a new class of functions. Dokl. Akad. Nauk BSSR, 29 (1985), no. 7, p. 584–587 (in Russian). [21] N. M. Tuan and B. T. Giang, Inversion theorems and the unitary of the integral transforms of Fourier type. Integ. Transform and Spec. Func. (accepted). [22] N. M. Tuan and P. D. Tuan, Generalized convolutions relative to the Hartley transforms with applications. Sci. Math. Jpn., 70 (2009), p. 77–89 (e2009, p. 351-363). [23] V. S. Vladimirov, Generalized Functions in Mathematical Physics. Mir, Moscow, 1979. Bui Thi Giang Department of Basic Science, Institute of Cryptography Science 141 Chien Thang Str., Thanh Tri Dist. Hanoi, Vietnam Nguyen Van Mau and Nguyen Minh Tuan Department of Mathematical Analysis University of Hanoi, 334 Nguyen Trai Str. Hanoi, Vietnam e-mail: [email protected] [email protected] Submitted: October 23, 2007. Revised: September 12, 2009.



Joint Hyponormality of Rational Toeplitz Pairs In Sung Hwang and Woo Young Lee Abstract. We characterize hyponormal “rational” Toeplitz pairs which are pairs of Toeplitz operators whose symbols are rational functions in L∞ . The main result of this article is as follows. If T = (Tφ , Tψ ) is a hyponormal rational Toeplitz pair then φ − βψ ∈ H 2 for some constant β; in other words, their co-analytic parts necessarily coincide up to a constant multiple. As a corollary we get a complete characterization of hyponormal rational Toeplitz pairs. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47B35, 47A13. Keywords. Hyponormal, jointly hyponormal, Toeplitz operators, Hankel operators, bounded type symbols, rational symbols.

1. Introduction In the monograph [CL], R. Curto and the second named author characterized the joint hyponormality of Toeplitz pairs which are pairs of Toeplitz operators with trigonometric polynomial symbols. We here obtain a complete characterization of hyponormal rational Toeplitz pairs which are pairs of Toeplitz operators whose symbols are rational functions in L∞ . The Bram-Halmos criterion on subnormality ([Br]) states that an operator T on a Hilbert space H is subnormal if and only if i,j (T i xj , T j xi ) ≥ 0 for all finite collections x0 , x1 , . . . , xk ∈ H. It is easy to see that this is equivalent to the following positivity test:   I T∗ ... T ∗k T T ∗ T . . . T ∗k T    (all k ≥ 1). (1.1)  .. .. ..  ≥ 0 ..  . . . .  Tk

T ∗T k

. . . T ∗k T k

The second author was supported by a grant (KRF-2008-314-C00014) from the Korea Research Foundation.

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In 1988, the notion of “joint hyponormality” (for the general case of n-tuples of operators) was first formally introduced by A. Athavale [At]. He conceived joint hyponormality as a notion at least as strong as requiring that the linear span of the operator coordinates consist of hyponormal operators, the latter notion being called weak joint hyponormality. Subnormality, joint hyponormality, and weak joint hyponormality have been studied by A. Athavale [At], J. Conway and W. Szymanski [CS], R. Curto [Cu], R. Curto, P. Muhly, and J. Xia [CMX], R. Douglas, V. Paulsen, and K. Yan [DPY], R. Douglas and K. Yan [DY], D. Farenick and R. McEachin [FM], C. Gu [Gu], S. McCullough and V. Paulsen [McCP], D. Xia [Xi], R. Curto and the second named author [CL], and others. Joint hyponormality originated from questions about commuting normal extensions of commuting operators, and it has also been considered with an aim at understanding the gap between hyponormality and subnormality for single operators. To date, much of the research on joint hyponormality has dealt with commuting tuples of hyponormal operators. The study of jointly hyponormal Toeplitz operators started from D. Farenick and R. McEachin [FM]. They studied operators that form hyponormal pairs in the presence of the unilateral shift. Since the unilateral shift is a Toeplitz operator on the Hardy space of the unit circle, one can ask whether the results in [FM] extend to Toeplitz pairs, that is, pairs whose coordinates are Toeplitz operators on the Hardy space of the unit circle. R. Curto and the second named author [CL] gave a complete characterization of hyponormal trigonometric Toeplitz pairs which are pairs of Toeplitz operators with trigonometric polynomial symbols. C. Gu [Gu] studied the joint hyponormality of Toeplitz pairs whose coordinates have the same co-analytic parts. The purpose of this article is to provide a complete characterization of hyponormal rational Toeplitz pairs. Let H be a complex Hilbert space and let B(H) be the algebra of bounded linear operators acting on H. For A, B ∈ B(H), we let [A, B] := AB − BA; [A, B] is the commutator of A and B. Given an n-tuple T = (T1 , . . . , Tn ) of operators on H, we let [T∗ , T] ∈ B(H ⊕ · · · ⊕ H) denote the self-commutator of T, defined by   ∗ [T1 , T1 ] [T2∗ , T1 ] . . . [Tn∗ , T1 ]  [T1∗ , T2 ] [T2∗ , T2 ] . . . [Tn∗ , T2 ]    ∗ [T , T] :=  . .. .. .. ..   . . . . ∗ ∗ ∗ [T1 , Tn ] [T2 , Tn ] . . . [Tn , Tn ] (This definition of self-commutator for n-tuples of operators on a Hilbert space was introduced by A. Athavale in [At].) By analogy with the case n = 1, we shall say ([At], [CMX]) that T is jointly hyponormal (or simply, hyponormal) if [T∗ , T] is a positive operator on H ⊕ · · · ⊕ H. Clearly, the hyponormality of an n-tuple requires as a necessary condition that every coordinate in the tuple be hyponormal. Let T = R/2πZ be the unit circle. The Hilbert space L2 ≡ L2 (T) has a canonical orthonormal basis given by the trigonometric functions en (z) = z n , for all n ∈ Z, and the Hardy space H 2 ≡ H 2 (T) is the closed linear span of

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{en : n = 0, 1, . . .}. An element f ∈ L2 is said to be analytic if f ∈ H 2 , and co-analytic if f ∈ L2 H 2 . If P denotes the orthogonal projection of L2 onto H 2 , then for every φ ∈ L∞ ≡ L∞ (T), the operator Tφ on H 2 defined by Tφ g := P (φg)

(g ∈ H 2 )

is called the Toeplitz operator with symbol φ. If φ is a trigonometric polynomial of n the form φ(z) = N n=−m an z , where a−m and aN are nonzero, then the nonnegative integers m and N denote the co-analytic and analytic degrees of φ, respectively. If φ ∈ L∞ , write φ+ ≡ P (φ) ∈ H 2

and φ− ≡ (I − P )(φ) ∈ zH 2 .

Thus we can write φ = φ− + φ+ . D. Farenick and R. MaEachin [FM] showed that if U is the unilateral shift on H 2 then the hyponormality of (U, T ) implies that T is necessarily a Toeplitz operator. Furthermore they proved that if φ = φ− +φ+ ∈ L∞ and if ψ = φ− + U ∗ φ+ ∈ L∞ then (U, Tφ ) is hyponormal if and only if the single Toeplitz operator Tψ is hyponormal. R. Curto and the second-named author [CL] have studied the hyponormality of T = (Tφ , Tψ ) when both symbols φ and ψ are trigonometric polynomials. In [CL], a complete characterization of hyponormal Toeplitz pairs in this case was given. The core of the main result in [CL] is that the hyponormality of T = (Tφ , Tψ ) (where φ and ψ are trigonometric polynomials) forces that the co-analytic parts of φ and ψ necessarily coincide up to a constant multiple, that is, φ− = βψ−

for some β ∈ C.

(1.2)

C. Gu [Gu] gave a characterization of hyponormal Toeplitz pairs T = (Tφ , Tψ ) under the constraint (1.2) on the symbol - the assumption of equal co-analytic parts. In this article we show that if φ and ψ are rational functions in L∞ , then the condition “symbols have equal co-analytic parts” is indeed necessary for the hyponormality of the pair T = (Tφ , Tψ ): this follows a spirit of the main result in [CL]. Thus we get a characterization of hyponormal “rational” Toeplitz pairs. A key step for the proof of the main result is accomplished by a direct and careful analysis on the self-commutator of the pair. The organization of the article is as follows. In Section 2 we introduce basic facts about Toeplitz operators and Hankel operators. In Section 3 we provide auxiliary lemmas to be used in proving the main results. Section 4 is devoted to prove the main results. Observe that if (T1 , T2 ) is hyponormal, then so is (T1 − λ1 , T2 − λ2 ) for every ˆ ˆ λ1 , λ2 ∈ C. Thus if φ, ψ ∈ L∞ have Fourier coefficients φ(n), ψ(n) for every n ∈ Z, respectively, then the hyponormality of (Tψ , Tφ ) is independent of the particular ˆ ˆ values of φ(0) and ψ(0). Therefore, throughout the article, we will assume that ˆ the 0-th coefficient φ(0) of the given symbol φ of a Toeplitz operator is zero.

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2. Preliminaries A bounded linear operator A is called hyponormal if its self-commutator [A∗ , A] := A∗ A−AA∗ is positive (semidefinite). Normal Toeplitz operators were characterized by a property of their symbols in the early 1960’s by A. Brown and P. Halmos [BH] and 25 years passed before the exact nature of the relationship between the symbol φ ∈ L∞ and the positivity of the self-commutator [Tφ∗ , Tφ ] was understood (via Cowen’s theorem [Co]). We shall employ an equivalent variant of Cowen’s theorem [Co], that was first proposed by Nakazi and Takahashi [NT]. Cowen’s Theorem. For φ ∈ L∞ , write E(φ) := {k ∈ H ∞ : ||k||∞ ≤ 1 and φ − kφ ∈ H ∞ } . Then Tφ is hyponormal if and only if E(φ) is nonempty. Cowen’s method is to recast the operator-theoretic problem of hyponormality for Toeplitz operators into the problem of finding a solution with specified properties to a certain functional equation involving the operator’s symbol. This approach has been put to use in many works to study hyponormal Toeplitz operators on the Hardy space of the unit circle. Let J be the unitary operator on L2 defined by J(f )(z) = zf (z). For φ ∈ L∞ , the operator on H 2 defined by Hφ f := J(I − P )(φf ) (f ∈ H 2 ) is called the Hankel operator Hφ with symbol φ. If we define the function v by v(z) := v(z), then Hφ can be viewed as the operator on H 2 defined by zuv, φ = Hφ u, v for all v ∈ H ∞ . We write H02 := {zf : f ∈ H 2 } and L = {f : f ∈ L} for L ⊂ L2 (T). We write, for an inner function θ, H(θ) := H 2 θH 2 . The following is a basic connection between Hankel and Toeplitz operators ([Ni]): (i) Tφψ − Tφ Tψ = Hφ∗ Hψ (φ, ψ ∈ L∞ ); (ii) Hφ Th = Hφh = Th∗ Hφ (h ∈ H ∞ , φ ∈ L∞ ). From this we can see that if k ∈ E(φ) then [Tφ∗ , Tφ ] = Hφ∗ Hφ − Hφ∗ Hφ = Hφ∗ Hφ − Hk∗ φ Hk φ = Hφ∗ (1 − Tk Tk∗ )Hφ .

(2.1)

We here observe that if T = (Tφ , Tψ ) then the self-commutator of T can be expressed as: ∗

[Tφ , Tφ ] [Tψ∗ , Tφ ] ∗ [T , T] = [Tφ∗ , Tψ ] [Tψ∗ , Tψ ] ∗ Hφ Hφ+ − Hφ∗ Hφ− Hφ∗ Hψ+ − Hψ∗ Hφ− + − + − = . Hψ∗ Hφ+ − Hφ∗ Hψ− Hψ∗ Hψ+ − Hψ∗ Hψ− +

−

+

−

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A function φ ∈ L∞ is said to be of bounded type (or in the Nevanlinna class) if there are functions ψ1 , ψ2 ∈ H ∞ (D) such that φ(z) =

ψ1 (z) ψ2 (z)

for almost all z in T. Evidently, rational functions in L∞ are of bounded type. We recall ([Ab, Lemma 6]) that if Tφ is hyponormal and φ is not analytic then φ is of bounded type if and only if φ is of bounded type. In [Ab, Lemma 3], it was also shown that φ is of bounded type ⇐⇒ ker Hφ = {0} ⇐⇒ φ = θb, where θ is an inner function, b ∈ H ∞ , and θ and b are relatively prime, i.e., there does not exist a nonconstant inner function ω such that θ = ωθ0 and b = ωb0 for some θ0 .b0 ∈ H ∞ . Thus if f ∈ H 2 is such that f is of bounded type and f (0) = 0 then we can write f = θb, where θ is an inner function and b ∈ H(θ) satisfies that b and θ are coprime. In particular, we can easily show that f = θb is a rational function ⇐⇒ θ is a finite Blaschke product.

(2.2)

Since Tz Hφ = Hφ Tz it follows from Beurling’s theorem that ker Hφ− = θ0 H 2 and ker Hφ+ = θ+ H 2 for some inner functions θ0 , θ+ . If Tφ is hyponormal and φ is not analytic then by (2.1), ||Hφ+ f || ≥ ||Hφ− f || for all f ∈ H 2 , so that θ+ H 2 = ker Hφ+ ⊂ ker Hφ− = θ0 H 2 ,

(2.3)

which implies that θ0 divides θ+ , i.e., θ+ = θ0 θ1 for some inner function θ1 . Thus if φ = φ− + φ+ is of bounded type, φ+ (0) = φ− (0) = 0, and Tφ is hyponormal then we can write φ+ = θ0 θ1 a ¯

and φ− = θ0¯b,

where a ∈ H(θ0 θ1 ) and b ∈ H(θ0 ).

(2.4)

On the other hand, as in (2.3), the hyponormality of Toeplitz pairs is also related to the kernels of Hankel operators involved with the analytic and co-analytic parts of the symbol. Indeed it was shown ([Gu, Lemma 6.2]) that if neither φ nor ψ is analytic and if (Tφ , Tψ )is hyponormal, then ker (Hφ+ ) ⊂ ker (Hψ− ) and ker (Hψ+ ) ⊂ ker (Hφ− ).

(2.5)

Throughout the article, whenever writing f = θa ∈ H 2 (where θ is an inner function and a ∈ H(θ)), we will assume that θ and a are relatively prime.

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3. Auxiliary Lemmas To prove the main result we need several auxiliary lemmas. The first two lemmas are technical lemmas about Hankel operators. If M is a closed subspace of L2 , we write PM for the orthogonal projection of L2 onto M. Lemma 1. If θ0 and θ1 are inner functions, then Hθ0 θ1 = Tθ1 Hθ0 + Hθ0 θ1 Pθ0 H 2 . Proof. We first claim that Hθ0 θ1 |H(θ0 ) = Tθ1 Hθ0 |H(θ0 ) . Indeed, if f ∈ H(θ0 ) then

θ0 f ∈ H02 , and hence zθ0 f ∈ H 2 . Thus

Hθ0 θ1 f = J(I − P )(θ0 θ1 f ) = J(θ0 θ1 f ) = z θ0 θ1 f = Tθ1 (z θ0 f) = Tθ1 Hθ0 f. But since ker Tθ1 Hθ0 = ker Hθ0 = θ0 H 2 , it follows that Hθ0 θ1 = Hθ0 θ1 PH(θ0 ) + Hθ0 θ1 Pθ0 H 2 = Tθ1 Hθ0 + Hθ0 θ1 Pθ0 H 2 .

Lemma 2. If θ0 and θ1 are inner functions, then Hθ0 θ1 Pθ0 H 2 = PH(θ1 ) Hθ0 θ1 .

Proof. For f ∈ H 2 , write f = f1 + f2 + f3 , where f1 ∈ H(θ0 ), f2 ∈ θ0 θ1 H 2 and f3 ∈ H(θ0 θ1 ) H(θ0 ). Thus f3 = θ0 h for some h ∈ H 2 . Observe that f3 = θ0 h ∈ H(θ0 θ1 ) ⇐⇒ θ0 θ1 θ0 h ∈ H02 ⇐⇒ h ∈ H(θ1 ). Therefore we have that Hθ0 θ1 Pθ0 H 2 f = Hθ0 θ1 θ0 h = J(I − P )(θ0 θ1 θ0 h) = J(θ1 h) = z θ1 h. On the other hand, we have that PH(θ1 ) Hθ0 θ1 f = PH(θ1 ) Hθ0 θ1 (f1 + f3 ) = PH(θ1 ) J(I − P )(θ0 θ1 f1 + θ1 h) = PH(θ1 ) J(θ0 θ1 f1 + θ1 h) h) = PH(θ1 ) (z θ0 θ1 f1 ) + PH(θ1 ) (z θ1 h, = z θ1 where the last equality comes from the following observation: f1 ∈ H(θ0 ) =⇒ z θ0 f1 ∈ H 2 =⇒ z θ0 θ1 f1 ∈ θ1 H 2 =⇒ PH(θ1 ) (z θ0 θ1 f1 ) = 0. This completes the proof.

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For a notational convenience we adopt the following notation: If φ = φ− + φ+ ∈ L∞ and if θ1 and θ2 are inner functions, we write φθ1 ,θ2 := PH02 (θ1 φ− ) + PH02 (θ2 φ+ ). Thus we have that (φθ1 ,θ2 )+ = PH02 (θ2 φ+ ) and (φθ1 ,θ2 )− = PH02 (θ1 φ− ). We also abbreviate φθ ≡ φθ,θ . Lemma 3. Let φ = φ− + φ+ ∈ L∞ be of the form φ+ = θ0 θ1 a

and

φ− = θ0 b,

where θ1 and θ2 are inner functions, and a ∈ H(θ0 θ1 ) and b ∈ H(θ0 ). Then we have: (i) Suppose θ is a factor of θ1 and ω is a factor of θ0 . If Tφ is hyponormal then Tφω,θ is hyponormal. Further if ω = 1 then the converse is also true. (ii) If θ is an arbitrary inner function and Tφ is hyponormal then Tφθ is hyponormal. Proof. (i) Write θ1 = θ∆1 and θ0 = ω∆0 . If Tφ is hyponormal then by Cowen’s theorem there exist a function k ∈ H ∞ with ||k||∞ ≤ 1 and a function h ∈ H 2 for which θ0 b − kθ0 θ1 a = h, that is, ka = θ1 (b − θ0 h). Since a and θ1 are relatively prime, it follows that k = θ1 ζ for some ζ ∈ H ∞ . If we put k1 := ∆1 ζω then ||k1 ||∞ ≤ 1 and ∆0 b − k1 θ0 ∆1 a ∈ ωH 2 , so that PH02 (∆0 b) − k1 PH02 (θ0 ∆1 a) ∈ H 2 , which implies k1 ∈ E(φω,θ ), that is, Tω,θ is hyponormal. For the second assertion, observe that if k ∈ E(φ1,θ ) then kθ ∈ E(φ). (ii) Note that if h ∈ H ∞ and φ ∈ L∞ then HP

H2 0

Thus,

(hφ)

[Tφ∗θ , Tφθ ] = H ∗

= HP (hφ) = Hφh = Hφ Th .

PH 2 (θφ+ ) 0

= =

HP

2 H0

(θφ+ )

− H∗

PH 2 (θφ− )

∗ ∗ Hθφ Hθφ+ − Hθφ Hθφ− + − ∗ ∗ ∗ Tθ (Hφ Hφ+ − Hφ Hφ− )Tθ + − ∗ ∗ Tθ [Tφ , Tφ ]Tθ ,

0

HP

2 H0

(θφ− )

= which implies that if Tφ is hyponormal then so is Tφθ .

The converse of Lemma 3(ii) is not true in general: indeed, a straightforward calculation shows that if φ(z) = z −2 + 3z −1 + 2z + 2z 2 then Tφ is not hyponormal, whereas Tφz is hyponormal. Lemma 4. Let φ = φ− + φ+ ∈ L∞ and ψ = ψ− + ψ+ ∈ L∞ be of the form φ+ = θ0 θ1 a, φ− = θ0 b, ψ+ = θ2 θ3 c and ψ− = θ2 d, where a ∈ H(θ0 θ1 ), b ∈ H(θ0 ), c ∈ H(θ2 θ3 ), and d ∈ H(θ2 ). Suppose θ is an inner function. If T = (Tφ , Tψ ) is hyponormal then Tθ = (Tφθ , Tψθ ) is hyponormal.

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Proof. From the same argument as in the proof of Lemma 3(ii), we have that [T∗θ , Tθ ]

[Tφ∗θ , Tφθ ] [Tψ∗θ , Tφθ ] = [Tφ∗ , Tψθ ] [Tψ∗θ , Tψθ ] ∗θ ∗ Tθ Hφ Hφ+ Tθ − Tθ∗ Hφ∗ Hφ− Tθ + − = Tθ∗ Hψ∗ Hφ+ Tθ − Tθ∗ Hφ∗ Hψ− Tθ + − ∗

Tθ 0 0 Tθ ∗ , T] = [T , 0 Tθ∗ 0 Tθ

Tθ∗ Hφ∗ Hψ+ Tθ − Tθ∗ Hψ∗ Hφ− Tθ + − Tθ∗ Hψ∗ Hψ+ Tθ − Tθ∗ Hψ∗ Hψ− Tθ +

−

which implies that if T is hyponormal then so is Tθ .

If one coordinate of the Toeplitz pair has an analytic symbol then the hyponormality of the pair can be determined by the hyponormality of a single Toeplitz operator (cf. [CL, Theorem 1.10]; [Gu, Theorem 4.1]). Lemma 5. If φ ∈ H ∞ is such that φ = θa for a ∈ H(θ) and ψ = ψ− + ψ+ ∈ L∞ is arbitrary then T = (Tφ , Tψ ) is hyponormal if and only if Tψ1,θ is hyponormal. Proof. See [Gu, Theorem 4.1].

4. Main Results Our main theorem is as follows: Theorem 1. Let φ and ψ be rational functions in L∞ . If T = (Tφ , Tψ ) is hyponormal then φ − βψ ∈ H 2 for some constant β. Proof. In view of (2.2) and (2.4), we can write φ+ = θ0 θ1 a, φ− = θ0 b, ψ+ = θ2 θ3 c and ψ− = θ2 d, where the θj are finite Blaschke products, a ∈ H(θ0 θ1 ), b ∈ H(θ0 ), c ∈ H(θ2 θ3 ) and d ∈ H(θ2 ). We split the proof into four steps. STEP 1: If θ is the greatest common inner factor of θ1 and θ3 , then T1θ = (Tφ1,θ , Tψ1,θ ) is hyponormal =⇒ T = (Tφ , Tψ ) is hyponormal.

(4.1)

Proof. We first claim that [Tψ∗ , Tφ ] = [Tψ∗1,θ , Tφ1,θ ] + Hφ∗ PH(θ) Hψ + . +

(4.2)

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Write θ1 = θδ1 and θ3 = θδ3 for some inner functions δ1 , δ3 . Then we have that Hφ∗ Hψ+ +

= Hθ∗0 θ1 a Hθ2 θ3 c

∗ = Hθ 0 θ 1 T a Hθ 2 θ 3 T c ∗

TθHθ2 δ3 Tc + Hθ2 θ3 Pθ2 δ3 H 2 Tc = TθHθ0 δ1 Ta + Hθ0 θ1 Pθ0 δ1 H 2 Ta

(4.3)

= Ta∗ Hθ∗ δ T TθHθ2 δ3 Tc + Ta∗ Hθ∗ δ T Hθ2 θ3 Pθ2 δ3 H 2 Tc 0 1

0 1

θ

θ

+ Ta∗ Pθ0 δ1 H 2 Hθ∗0 θ1 TθHθ2 δ3 Tc + Ta∗ Pθ0 δ1 H 2 Hθ∗0 θ1 Hθ2 θ3 Pθ2 δ3 H 2 Tc ,

where the third equality follows from Lemma 1. Observe that Ta∗ Hθ∗ δ T TθHθ2 δ3 Tc = Ta∗ Hθ∗ δ Hθ2 δ3 Tc = Hθ∗ δ a Hθ2 δ3 c 0 1

θ

∗ = H(φ

0 1

1,θ )+

0 1

(4.4)

H(ψ1,θ ) . +

For each h ∈ H 2 , if we write h1 := PH(θ) h then

1 = 0, T Hθ2 θ3 (θ2 δ3 h) = T J(I − P )(θ2 θ3 θ2 δ3 h) = T J(θh1 ) = T z θh θ

θ

θ

θ

which implies T Hθ2 θ3 Pθ2 δ3 H 2 = 0, and hence θ

Ta∗ Hθ∗ δ T Hθ2 θ3 Pθ2 δ3 H 2 Tc = 0. 0 1

(4.5)

θ

Similarly, we have that T Hθ0 θ1 Pθ0 δ1 H 2 = 0, and hence Pθ0 δ1 H 2 Hθ∗ θ Tθ = 0. Thereθ 0 1 fore we have that Ta∗ Pθ0 δ1 H 2 Hθ∗0 θ1 TθHθ2 δ3 Tc = 0. (4.6) On the other hand, by Lemma 2, we have that Ta∗ Pθ0 δ1 H 2 Hθ∗0 θ1 Hθ2 θ3 Pθ2 δ3 H 2 Tc = Ta∗ Hθ∗0 θ1 PH(θ) PH(θ) Hθ 2 θ 3 T c = Hφ∗+ PH(θ) Hψ + .

Hence by (4.3)-(4.7), it follows that [Tψ∗ , Tφ ] = Hφ∗ Hψ+ − Hψ∗ Hφ− +

∗ = H(φ

−

1,θ )+

∗ H(ψ1,θ ) + Hφ∗+ PH(θ) Hψ+ − H(ψ +

= [Tψ∗1,θ , Tφ1,θ ] + Hφ∗+ PH(θ) Hψ + ,

1,θ )−

H(φ1,θ )

which proves (4.2). Applying (4.2) to [Tφ∗ , Tφ ] and [Tψ∗ , Tψ ] gives that [T∗ , T] = [T∗1,θ , T1,θ ] + V, where

V =

Hφ∗ PH(θ) Hφ + + Hψ∗ PH(θ) Hφ + +

Hφ∗ PH(θ) Hψ + + Hψ∗ PH(θ) Hψ + +

.

−

(4.7)

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But since V =

∗ (PH(θ) Hφ + ) ∗ (PH(θ) Hψ + )

0 0

PH(θ) Hφ + 0

PH(θ) Hψ + 0

≥ 0,

it follows that if T1θ is hyponormal then T is hyponormal. STEP 2: If θ is the greatest common inner factor of θ1 and θ3 and if ω is the greatest common inner factor of θ0 and θ2 then T = (Tφ , Tψ ) is hyponormal =⇒

Tω 1,θ = (T(φω )1,θ , T(ψω )1,θ ) is hyponormal.

Proof. By Lemma 4, Tω = (Tφω , Tψω ) is hyponormal. Thus without loss of generality we may assume that θ0 and θ2 are relatively prime. Also by (2.5) we have θ0 θ1 H 2 ⊂ θ2 H 2

and θ2 θ3 H 2 ⊂ θ0 H 2 .

Thus θ1 = θ2 1 and θ3 = θ0 3 for some inner functions 1 , 3 . But since θ0 and θ2 are relatively prime, it follows that δ1 = θ2 ∆1

and δ3 = θ0 ∆3

form some inner functions ∆1 , ∆3 .

(4.8)

Now let a1 := PH(θ0 δ1 ) a and c1 := PH(θ0 δ3 ) c. Then we have that [T∗1θ , T1θ ]

[Tφ∗1,θ , Tφ1,θ ] [Tψ∗1,θ , Tφ1,θ ] = [Tφ∗1,θ , Tψ1,θ ] [Tψ∗1,θ , Tψ1,θ ] ∗ Hφ θ Hφ+ θ − Hφ∗ Hφ− Hφ∗ θ Hψ+ θ − Hψ∗ Hφ− + − + − = Hψ∗ θ Hφ+ θ − Hφ∗ Hψ− Hψ∗ θ Hψ+ θ − Hψ∗ Hψ− + − + − ∗ ∗ ∗ Hθ δ a Hθ0 δ1 a1 − Hθ b Hθ0 b Hθ δ a Hθ2 δ3 c1 − Hθ∗ d Hθ0 b 0 1 1 0 0 1 1 2 = . Hθ∗ δ c Hθ0 δ1 a1 − Hθ∗ b Hθ2 d Hθ∗ δ c Hθ2 δ3 c1 − Hθ∗ d Hθ2 d 2 3 1

0

2 3 1

2

Observe that ker [Tφ∗1,θ , Tφ1,θ ] ⊃ θ0 δ1 H 2 and hence ran [Tφ∗1,θ , Tφ1,θ ] ⊂ H(θ0 δ1 ). Thus [Tφ∗1,θ , Tφ1,θ ] has the following matrix representation: [Tφ∗1,θ , Tφ1,θ ] =

∗ Hθ δ

0 1 a1

Hθ0 δ1 a1 − Hθ∗ b Hθ0 b 0 0

0 H(θ0 δ1 ) H(θ0 δ1 ) −→ , : θ0 δ 1 H 2 θ0 δ 1 H 2 0

where the left-upper corner will be, without loss in simplifying the notation, understood as the restriction of Hθ∗ δ a Hθ0 δ1 a1 − Hθ∗ b Hθ0 b to H(θ0 δ1 ). By similar 0 1 1 0 arguments for [Tψ∗1,θ , Tφ1,θ ], [Tφ∗1,θ , Tψ1,θ ] and [Tψ∗1,θ , Tψ1,θ ] using (4.8), we can see that the positivity of [T1θ , T1θ ] is equivalent to the positivity of the restriction,

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say E, of it to H(θ0 θ1 ) ⊕ H(θ2 θ3 ). Note that E can be written as:  ∗ Hθ δ a Hθ0 δ1 a1 − Hθ∗ b Hθ0 b 0 Hθ∗ δ a Hθ2 δ3 c1 − Hθ∗ d Hθ0 b 0 1 1 0 0 1 1 2  0 0 0 E= H ∗ H − Hθ∗ b Hθ2 d 0 Hθ∗ δ c Hθ2 δ3 c1 − Hθ∗ d Hθ2 d θ2 δ3 c1 θ0 δ1 a1 0 2 3 1 2 0 0 0     H(θ0 δ1 ) H(θ0 δ1 ) 2  H(θ0 θ1 ) ∩ θ0 δ1 H 2   −→ H(θ0 θ1 ) ∩ θ0 δ1 H  , :     H(θ2 δ3 ) H(θ2 δ3 ) H(θ2 θ3 ) ∩ θ2 δ3 H 2 H(θ2 θ3 ) ∩ θ2 δ3 H 2

397

 0 0  0 0

(4.9)

where each entry of E should be understood as a restriction to a suitable subspace. Let Aa := PH(θ0 θ1 ) Ma |H(θ0 θ1 ) and Ac := PH(θ2 θ3 ) Mc |H(θ2 θ3 ) , where Mx is a multiplication operator with symbol x. We now argue that Aa and Ac are invertible. To see this, suppose Aa f = 0 for some f ∈ H(θ0 θ1 ). Then PH(θ0 θ1 ) (af ) = 0 and hence af ∈ θ0 θ1 H 2 . Since the inner part of a and θ0 θ1 are relatively prime, it follows that f ∈ θ0 θ1 H 2 . So f ∈ θ0 θ1 H 2 ∩H(θ0 θ1 ) = {0}, which implies that Aa is one-one. But since Aa is a finite dimensional operator (because θ0 θ1 is a finite Blaschke product), it follows that Aa is invertible. Similarly, Ac is also invertible. Observe that PH(θ0 δ1 ) Aa |H(θ0 θ1 )∩θ0 δ1 H 2 = PH(θ0 δ1 ) Ma |H(θ0 θ1 )∩θ0 δ1 H 2 = 0. Thus Aa has the following matrix representation:

H(θ0 δ1 ) H(θ0 δ1 ) a1 0 : −→ . Aa = a2 a3 H(θ0 θ1 ) ∩ θ0 δ1 H 2 H(θ0 θ1 ) ∩ θ0 δ1 H 2 Since a1 = PH(θ0 δ1 ) Ma |H(θ0 δ1 ) , we can see, by the same argument as for Aa , that Thus the inverse of Aa is lower a1 is invertible. As a result, a3 is also invertible. a−1 0 −1 1 triangular; in fact, Aa = −a−1 a a−1 a−1 . Similarly, the inverse of Ac is also lower triangular. Write −1 Aa G := 0

2 1

3

0 A−1 c

:

3

H(θ0 θ1 ) H(θ0 θ1 ) −→ . H(θ2 θ3 ) H(θ2 θ3 )

Since G is invertible, it follows from (4.9) that [T∗1θ , T1θ ] is positive ⇐⇒ D := G∗ EG ≥ 0. −1 Since A−1 a and Ac  d1 0 d2 0 0 0 D= d3 0 d4 0 0 0

are both lower triangular, D should be of the form      0 H(θ0 δ1 ) H(θ0 δ1 ) 2 2   0 : H(θ0 θ1 ) ∩ θ0 δ1 H −→ H(θ0 θ1 ) ∩ θ0 δ1 H  . (4.10)      0 H(θ2 δ3 ) H(θ2 δ3 ) 2 2 0 H(θ2 θ3 ) ∩ θ2 δ3 H H(θ2 θ3 ) ∩ θ2 δ3 H

On the other hand, we know that [T∗1θ , T1θ ] = [T∗ , T] − V,

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where, for a notational convenience, each term will be understood as a restriction to H(θ0 θ1 ) ⊕ H(θ2 θ3 ) (Note that H(θ0 θ1 ) ⊕ H(θ2 θ3 ) reduces [T∗ , T], [T∗1θ , T1θ ] and V ). Thus V can be viewed as: ∗

Hφ∗ PH(θ) Hφ PH(θ) Hφ + Hψ + H(θ0 θ1 ) H(θ0 θ1 ) + + : −→ V = H(θ2 θ3 ) H(θ2 θ3 ) Hψ∗ PH(θ) Hψ∗ PH(θ) Hφ + Hψ + +

+

Observe that C := G∗ V G −1

∗ ∗ Hφ∗ PH(θ) Hφ PH(θ) Hφ + Hψ + 0 A−1 Aa a + + = ∗ ∗ 0 A−1 0 H P H H P H c φ ψ H( θ) H( θ) + + ψ+ ψ+ −1 ∗ ∗ −1 −1 ∗ ∗ −1 Aa Hφ PH(θ) H A A H PH(θ) Hψ+ Ac a a φ + φ + + = ∗ ∗ ∗ ∗ −1 −1 . A−1 Hψ PH(θ) A−1 Hψ PH(θ) Hφ+ Aa Hψ+ Ac c c +

Since

0 A−1 c

(4.11)

+

∗

∗ ∗ −1 −1 −1 H H P H A = H A P A A−1 a φ+ a θ0 θ1 a a θ0 θ1 a a H(θ) φ+ H(θ)

and

−1 Hθ0 θ1 a A−1 a = Hθ0 θ1 Ta Aa

−1 = Hθ0 θ1 P Ma PH(θ0 θ1 ) Ma |H(θ0 θ1 )

−1 = Hθ0 θ1 PH(θ0 θ1 ) Ma |H(θ0 θ1 ) PH(θ0 θ1 ) Ma |H(θ0 θ1 )

= Hθ 0 θ 1 , it follows that ∗

−1 Hφ∗ PH(θ) A−1 Hφ+ Aa a +

= Hθ∗0 θ1 PH(θ) Hθ 0 θ 1

= Hθ∗ θ Hθ0 θ1 Pθ0 δ1 H 2 0 1

= PH(θ0 θ1 ) Pθ0 δ1 H 2

(4.12)

(by Lemma 2)

(since

Hζ∗ Hζ

= PH(ζ) for an inner function ζ)

= PH(θ0 θ1 )∩θ0 δ1 H 2 . Similarly, we also have that ∗

∗

−1 −1 −1 ∗ A−1 Hφ∗ PH(θ) Hθ∗ θ a PH(θ) Hψ+ Ac = Aa Hθ2 θ3 c Ac = Hθ θ PH(θ) Hθ 2 θ 3 . a +

0 1

0 1

Thus, by Lemma 2, we get ∗

−1 PH(θ0 δ1 ) A−1 Hφ∗ PH(θ) Hψ+ Ac PH(θ2 δ3 ) a +

= PH(θ0 δ1 ) Hθ∗0 θ1 PH(θ) Hθ2 θ3 PH(θ2 δ3 )

= PH(θ0 δ1 ) Hθ∗ θ Hθ2 θ3 Pθ2 δ3 H 2 PH(θ2 δ3 ) 0 1

= 0.

(4.13)

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Similarly, we also have that ∗

−1 A−1 Hψ∗ PH(θ) Hψ+ Ac = PH(θ2 θ3 )∩θ2 δ3 H 2 . c +

Therefore  0 0 C = 0 ∗

by (4.11)-(4.14), C should be of the form      0 0 ∗ H(θ0 δ1 ) H(θ0 δ1 ) 2 2   1 ∗ ∗ : H(θ0 θ1 ) ∩ θ0 δ1 H −→ H(θ0 θ1 ) ∩ θ0 δ1 H  .    ∗ 0 0  H(θ2 δ3 ) H(θ2 δ3 ) ∗ 0 1 H(θ2 θ3 ) ∩ θ2 δ3 H 2 H(θ2 θ3 ) ∩ θ2 δ3 H 2

(4.14)

(4.15)

In particular, we have that PH(θ0 δ1 )⊕H(θ2 δ3 ) C PH(θ0 δ1 )⊕H(θ2 δ3 ) = 0.

(4.16)

Observe, by (4.10), that T1θ = (Tφ1θ , Tψ1θ ) is hyponormal if and only if the fol is positive: lowing operator matrix D

d1 d2 H(θ0 δ1 ) H(θ0 δ1 ) D := −→ . (4.17) : H(θ2 δ3 ) H(θ2 δ3 ) d3 d4 But since by (4.10), (4.11), (4.16), and (4.17),

= PH(θ δ )⊕H(θ δ ) G∗ [T∗ , T] − V G PH(θ δ )⊕H(θ δ ) D 0 1 2 3 0 1 2 3

= PH(θ0 δ1 )⊕H(θ2 δ3 ) G∗ [T∗ , T] G − C PH(θ0 δ1 )⊕H(θ2 δ3 ) = PH(θ0 δ1 )⊕H(θ2 δ3 ) G∗ [T∗ , T] G PH(θ0 δ1 )⊕H(θ2 δ3 ) ≥ 0, it follows that T1θ is hyponormal and this proves (4.1). STEP 3: We claim that θ2 = ξθ0

for some nonzero ξ ∈ C.

(4.18)

Proof. In view of STEP 1, STEP 2, and Lemma 4, we may assume that θ1 and θ3 are relatively prime and θ0 and θ2 are relatively prime. Thus for (4.18) it suffices to show that θ0 and θ2 are constant. Since T = (Tφ , Tψ ) is hyponormal it follows from again Lemma 4 that Tθ0 = (Tφθ0 , Tψθ0 ) is hyponormal. Observe that (φθ0 )+ = PH02 (θ0 θ0 θ1 a) = PH02 (θ1 a) = θ1 PH(θ1 ) (a), (φθ0 )− = PH02 (θ0 θ0 b) = 0, (ψθ0 )+ = PH02 (θ0 θ2 θ3 c), (ψθ0 )− = PH02 (θ0 θ2 d). Thus φθ0 is analytic. Therefore by Lemma 5, Tω is hyponormal, where

ω = PH02 θ1 (ψθ0 )+ + (ψθ0 )− = PH02 θ1 PH02 (θ0 θ2 θ3 c) + PH02 (θ0 θ2 d).

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Since T = (Tφ , Tψ ) is hyponormal we have, by (2.5), that θ0 θ1 H 2 ⊂ θ2 H 2 and θ2 θ3 H 2 ⊂ θ0 H 2 . Since θ0 and θ2 are relatively prime we can see that θ1 = θ2 ∆1 and θ3 = θ0 ∆3 for some inner functions ∆1 and ∆3 . Observe that PH02 (θ0 θ2 θ3 c) = PH02 (θ2 ∆3 c) = θ2 ∆3 c3 , where c3 := PH(θ2 ∆3 ) (c). Since the inner part of ∆1 c3 and ∆3 are relatively prime (because c and θ3 are relatively prime) we have that ω+ = PH02 (θ1 θ2 ∆3 c3 ) = PH02 (∆1 ∆3 c3 ) = ∆3 PH(∆3 ) (∆1 c3 ), where ∆3 and PH(∆3 ) (∆1 c3 ) are relatively prime. Since θ0 d and θ2 are relatively prime we also have that ω− = PH02 (θ0 θ2 d) = θ2 PH(θ2 ) (θ0 d), where θ2 and PH(θ2 ) (θ0 d) are relatively prime. Therefore the hyponormality of Tω forces that ∆3 = θ2 ζ3 for some inner function ζ3 . Therefore θ3 = θ0 ∆3 = θ0 θ2 ζ3 and hence θ2 is a common inner factor of θ1 and θ3 . But since θ1 and θ3 are relatively prime, we must have that θ2 is a constant. Interchanging the roles of φ and ψ in the above argument gives that θ0 is also a constant. This proves (4.18). We write Z(θ) for the set of all zeros in D of the inner function θ. STEP 4: We conclude that φ− = βψ− ,

i.e.,

φ − βψ ∈ H 2

for some constant β.

Proof. Suppose T = (Tφ , Tψ ) is hyponormal. Then Tφ−βψ is hyponormal for all β ∈ C. In view of STEP 3, we may assume that θ2 = ξθ0 for some ξ ∈ C. Observe that φ − αψ = θ0 θ1 θ3 θ3 a − ξα θ1 c + θ0 b − ξαd . We want to show that b = β0 d for some β0 ∈ C. Assume to the contrary that b = βd for any β ∈ C. Then we can see that (Tφ−βψ , Tψ ) is hyponormal (cf. [Gu, z−β Lemma 5.1]). Let β ∈ Z(θ0 ), i.e., θ0 = bβ θβ with bβ := 1−βz and some finite Blaschke product θβ . Since θ0 and d are relatively prime, it follows that d(β) = 0. Write b(β) . α := ξ d(β) Then bβ (b − ξαd) ∈ H 2 . Thus we have that

(φ − αψ)− = θβ bβ (b − ξαd) . But since (Tφ−αψ , Tψ ) is hyponormal, applying STEP 3 with (Tφ−αψ , Tψ ) in place of (Tφ , Tψ ) gives that θ0 is an inner factor of θβ up to a unitary constant, a contradiction. Hence b = β0 d for some β ∈ C, and hence φ− = βψ− for some β ∈ C. This completes the proof of Theorem 1. The following corollary is a complete characterization of hyponormal rational Toeplitz pairs.

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Corollary 2. Let φ = φ− + φ+ ∈ L∞ and ψ = ψ− + ψ+ ∈ L∞ be of the form φ+ = θ0 θ1 a, φ− = θ0 b, ψ+ = θ2 θ3 c and ψ− = θ2 d, where the θj are finite Blaschke products, a ∈ H(θ0 θ1 ), b ∈ H(θ0 ), c ∈ H(θ2 θ3 ), and d ∈ H(θ2 ). Then T = (Tφ , Tψ ) is hyponormal if and only if there exists a constant β such that (i) φ− = βψ− ; (ii) θ3 a − βθ1 c ∈ θθ0 H 2 ; (iii) Tψ is hyponormal, where ψ := P (θθ0 θ1 c) + θ2 d. Here θ is the greatest common inner factor of θ1 and θ3 . Proof. We first assume that θ1 and θ3 are relatively prime. Now suppose T = (Tφ , Tψ ) is hyponormal. By Theorem 1, θ0 = θ2 (up to a unitary constant) and φ− = βψ− for some β ∈ C. Thus φ − βψ = θ0 θ1 θ3 (θ3 a − βθ1 c) ∈ z H 2 . We want to show that θ0 (θ3 a − βθ1 c) ∈ H 2 . Assume θ0 (θ3 a − βθ1 c)∈ / H 2. Then there exists a nonconstant factor ζ0 of θ0 such that φ − βψ = ζ0 θ1 θ3 θ0 ζ0 θ3 a − βθ1 c , where θ0 ζ0 θ3 a − βθ1 c ∈ H(ζ0 θ1 θ3 ). By Lemma 5, Tψ is hyponormal, where ψ = ψ1,ζ0 θ1 θ3 = PH02 (θ0 ζ0 θ1 c) + θ0 d. Therefore we have that ker H = θ0 H 2 ⊆ θ0 ζ0 H 2 ⊂ ker H , ψ−

ψ+

which is a contradiction (see (2.3)). Thus θ0 (θ3 a − βθ1 c)∈ H 2 . Therefore, θ3 a − βθ1 c ∈ θ0 H 2 . In particular, φ − βψ = θ1 θ3 θ0 (θ3 a − βθ1 c) ∈ zH 2 , where θ0 (θ3 a − βθ1 c) ∈ H(θ1 θ3 ). Therefore by Lemma 5, Tψ is hyponormal, where ψ := P (θ1 θ3 ψ+ ) + ψ− = P (θ0 θ1 c) + θ2 d = ψ1,θ1 θ3 + c

for a constant c.

The converse is obtained by reversing the above argument. The proof for the general case can be accomplished by passing to (4.1) with the assumption that θ is the greatest common inner factor of θ1 and θ3 . By comparison with the cases of trigonometric Toeplitz pairs, we are tempted to guess θ1 = θ3 in the criterion of Corollary 2. If this were true then we would conclude that if ψ is a trigonometric polynomial and φ is an arbitrary rational symbol then the hyponormality of T = (Tφ , Tψ ) forces φ to be a trigonometric polynomial. However this is not the case. For example, if ψ(z) = 16 z −1 − z and z− 1

φ(z) = 16 z −1 + zB ( 13 B + 23 ), where B = 1− 12z then φ and ψ satisfy all three 2 conditions in Corollary 2, so that T = (Tφ , Tψ ) is hyponormal even though φ is not a trigonometric polynomial.

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References [Ab]

M. B. Abrahamse, Sunormal Toeplitz operators and functions of bounded type, Duke Math. J. 43 (1976), 597–604.

[At]

A. Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc. 103 (1988), 417–423.

[Br]

J. Bram, Sunormal operators, Duke Math. J. 22 (1955), 75–94.

[BH]

A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89–102.

[Con]

J. B. Conway, The Theory of Subnormal Operators, Math. Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, 1991.

[CS]

J. B. Conway and W. Szymanski, Linear combination of hyponormal operators, Rocky Mountain J. Math. 18 (1988), 695–705.

[Co]

C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), 809–812.

[Cu]

R. E. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Operator Theory: Operator Algebras and Applications (Durham, NH, 1988) (W. B. Arveson and R. G. Douglas, eds.), Proc. Sympos. Pure Math., vol. 51, Part II, Amer. Math. Soc., Providence, (1990), 69–91.

[CL]

R. E. Curto and W. Y. Lee, Joint hyponormality of Toeplitz pairs, Memoirs Amer. Math. Soc. 150, No 712 (2001).

[CMX]

R. E. Curto, P. S. Muhly and J. Xia, Hyponormal pairs of commuting operators, Contributions to Operator Theory and Its Applications (Mesa, AZ, 1987) (I. Gohberg, J. W. Helton and L. Rodman, eds.), Operator Theory: Advances and Applications, vol. 35, Birkh¨ auser, Basel–Boston, (1998), 1–22.

[DPY]

R. G. Douglas, V. I. Paulsen, and K. Yan, Operator theory and algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 20 (1989), 67–71.

[DY]

R. G. Douglas and K. Yan, A multi-variable Berger-Shaw theorem, J. Operator Theory 27 (1992), 205–217.

[FM]

D. R. Farenick and R. McEachin, Toeplitz operators hyponormal with the unilateral shift, Integral Equations and Operator Theory 22 (1995), 273–280.

[Gu]

C. Gu, On a class of jointly hyponormal Toeplitz operators, Trans. Amer. Math. Soc. 354 (2002), 3275–3298.

[McCP] S. McCullough and V. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187–195. [NT]

T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc., 338 (1993), 753–769.

[Ni]

N. K. Nikolskii, Treatise on the shift operator, Springer, New York, 1986.

[Xi]

D. Xia, On the semi-hyponormal n-tuple of operators, Integral Equations and Operator Theory 6 (1983), 879–898.

Vol. 65 (2009)


In Sung Hwang Department of Mathematics Sungkyunkwan University Suwon 440-746 Korea e-mail: [email protected] Woo Young Lee Department of Mathematics Seoul National University Seoul 151-742 Korea e-mail: [email protected] Submitted: January 11, 2007. Revised: February 2, 2009.

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Integr. equ. oper. theory 65 (2009), 405–414 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030405-10, published online October 22, 2009 DOI 10.1007/s00020-009-1725-7


The Characteristics of Expansivity of Two Variables Weighted Shift Chunji Li and Junde Wu Abstract. Recently, Curto and Yoon considered a two variables weighted shift W and constructed three different families of commuting pairs of subnormal operators such that each family can be used to answer a conjecture of Curto, Muhly and Xia negatively. In this paper, first, we show that the k-expansivity, k-hyperexpansivity and completely hyperexpansivity of W can be characterized completely only by its moments. Next, we present several interesting examples to give out the applications of these results. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47B39. Keywords. Weighted shift, (k1 , k2 )-expansive, (k1 , k2 )-hyperexpansive, completely hyperexpansive.

1. Introduction and preliminaries Let H be a separable infinite dimensional complex Hilbert space, L(H) the algebra of all bounded linear operators on H, Z+ the set {0, 1, 2, . . .}, Zn+ the direct product of Z+ with itself n times. For simplicity, e sake, we only consider n = 2. For p = (p1 , p2 ) ∈ Z2+ , denote |p| = p1 + p2 . If p =(p1 , p2 ), n= (n1, n2 ), and p1 ≤ n1 , n2 n n1 . If T = = p2 ≤ n2 , then denote p ≤ n. For p ≤ n, let p p1 p2 (T1 , T2 ), denote Tn = T1n1 T2n2 and T∗ = (T1∗ , T2∗ ), where T1 , T2 ∈ L(H). If T = (T1 , T2 ) and T1 T2 = T2 T1 , then for each n = (n1 , n2 ) ∈ Z2+ and ˆ0 = (0, 0), denote n |p| Θn (T) = (−1) T∗p Tp p ˆ 0≤p≤n (1.1) n1 n2 = (−1)p1 +p2 T2∗p2 T1∗p1 T1p1 T2p2 . p1 p2 0≤p1 ≤n1 , 0≤p2 ≤n2

This project is supported by the natural science fund of China (10471124).

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Recall that S ∈ L(H) is said to be subnormal if there exist a Hilbert space K containing H and a normal operator N ∈ L(K) such that N H ⊆ H and N |H = S ([1]). The famous Jim Agler’s criterion for subnormality is ([2]): T ∈ L(H) is a subnormal contraction iff for each n ∈ Z+ , n p Θn (T ) = (−1) (1.2) T ∗p T p ≥ 0. p 0≤p≤n

As the antithetical notion of a subnormal operators, Athavale in [3] first considered the class of completely hyperexpansive operators, that is, an operator T in L(H) is said to be a completely hyperexpansive if for each n ∈ Z+ , n ≥ 1, n p (−1) (1.3) T ∗p T p ≤ 0. Θn (T ) = p 0≤p≤n

Moreover, Athavale and Sholapurkar in [4] studied the following completely hyperexpanisve operator tuples: For T = (T1 , T2 ) with T1 T2 = T2 T1 and for all ˆ 0 = n = (n1 , n2 ) ∈ Z2+ , n |p| Θn (T) = (−1) T∗p Tp p ˆ 0≤p≤n (1.4) n1 n2 p +p T2∗p2 T1∗p1 T1p1 T2p2 ≤ 0. = (−1) 1 2 p1 p2 0≤p1 ≤n1 , 0≤p2 ≤n2

The generalization of completely hyperexpansive of operator tuples are the following k-expansivity and k-hyperexpansivity of operator tuples ([5]): Definition 1.1. For T = (T1 , T2 ) with T1 T2 = T2 T1 and ˆ0 = k = (k1 , k2 ) ∈ Z2+ , (i) T is (k1 , k2 )-expansive if Θ(k1 ,k2 ) (T) ≤ 0; (ii) T is (k1 , k2 )-hyperexpansive if Θ(n1 ,n2 ) (T) ≤ 0 for all ˆ0 = (n1 , n2 ) ≤ (k1 , k2 ) .

2. Main results and proofs Let {en }n∈Z+ be an orthonormal basis of H and {αn }n∈Z+ a positive real number sequence. Then a weighted shift operator T on H with the weight sequence {αn }n∈Z+ is defined through the relations T en = αn e(n+1) (n ∈ Z+ ) and denoted by T : {αn } ([6]). Recently, Curto and Yoon in [7] considered the following weighted shift and constructed three different families of commuting pairs of subnormal operators such that each family can be used to answer negatively a conjecture of Curto, Muhly and Xia in [8]: Let H = 2 Z2+ be the Hilbert space of square-summable complex sequences indexed by Z2+ , E := {e(i,j) }(i,j)∈Z2+ be the canonical orthonormal basis of 2 Z2+ and {αk }k∈Z2+ , {βk }k∈Z2+ ∈ ∞ Z2+ be two double-indexed positive

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The Characteristics of Expansivity

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bounded sequences. Let 1 := (1, 0), 2 := (0, 1). The two variables weighted shift W = (W1 , W2 ) is defined by W1 ek := αk ek+1 , W2 ek := βk ek+2 . ⇐⇒ βk+1 αk = αk+2 βk W1 W2 = W2 W1 The matrix forms of W1 and W2 with respect to E are  0 0 0 0 ···  0 0 0 0 ···    α(0,0) 0 0 0 ···   0 0 0 0 ···    0 α 0 0 · ·· (0,1)   W1 =  0 0 α(1,0) 0 ···   0 0 0 0 ···   ..  0 . 0 0 α(0,2)   .. .. .. . . .. ..  . . .  .. .. .. .. .. . . . . .  0 0 0 0 ···  β(0,0) 0 0 0 ···    0 0 0 0 ···   0 β(0,1) 0 0 ···   0 β(1,0) 0 ···  0 W2 =   0 0 0 0 ···   ..  0 . 0 0 β(0,2)   . . . . . .. .. .. ..  ..  .. .. .. .. .. . . . . .

∀k ∈ Z2+ . ··· ··· ··· ··· ··· ··· ···

(2.1)



          ,      ···   ..  .   .. .  ··· ···    ···   ···    ···  . ···    ···   ..  .   .. .

Given k = (k1 , k2 ) ∈ Z2+ , the moment of (α, β) of order k is γk ≡ γk (α, β)  1    α2 · · · α2 (0,0) (k1 −1,0) = 2 2 β(0,0) · · · β(0,k  2 −1)   2 2 2 α(0,0) · · · α2(k1 −1,0) · β(k · · · β(k 1 ,0) 1 ,k2 −1)

if if if if

k1 k1 k1 k1

= k2 = 0, ≥ 1 and k2 = 0, = 0 and k2 ≥ 1, ≥ 1 and k2 ≥ 1.

In this paper, first, we show that the k-expansive, k-hyperexpansive and complete hyperexpansive of W can be characterized completely only by its moments. Now, we prove the following interesting result:

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Proposition 2.1. For every n ∈ Z2+ , the n-th order moment γn of W is n W 1 W n2 e(0,0) 2 . 1 2 Proof. It is trivial for n1 = n2 = 0. Since 1 −1 αk+j1 ek+n1 1 , W1n1 ek = Πnj=0 2 −1 βk+j2 ek+n2 2 , W2n2 ek = Πnj=0 if n1 ≥ 1 and n2 ≥ 1, then we have W1n1 W2n2 e(0,0) = W2n2 W1n1 e(0,0) = W2n2 α(0,0) · · · α(n1 −1,0) en1 1 = α(0,0) · · · α(n1 −1,0) W2n2 e(n1 ,0) = α(0,0) · · · α(n1 −1,0) · β(n1 ,0) · · · β(n1 ,n2 −1) e(n1 ,n2 ) . If n1 = 0 and n2 ≥ 1, then we have W2n2 e(0,0) = β(0,0) · · · β(0,n2 −1) e(0,n2 ) . If n1 ≥ 1 and n2 = 0, then

W1n1 e(0,0) = α(0,0) · · · α(n1 −1,0) e(n1 ,0) .

Thus, we proved the conclusion. Our main results are: Theorem 2.2. W = (W1 , W2 ) is k = (k1 , k2 )-expansive iff for each m ∈ Z2+ , γm+p k |p| k Am := (−1) ≤ 0. p γm ˆ 0≤p≤k

Proof. If we denote (p1 ) xn = α2n α2n+1 · · · α2n+(p1 −1)1

and 2 2 yn(p2 ) = βn2 βn+ · · · βn+(p , 2 2 −1)2

then by direct computation, we have (p )

(p )

(p )

(p )

(p )

(p )

(p )

(p )

(p )

(p )

(p )

(p )

1 1 1 1 1 1 1 , x(0,1) , x(1,0) , x(0,2) , x(1,1) , x(2,0) , x(0,3) , . . .), W1∗p1 W1p1 = diag(x(0,0)

(p )

(p )

1 1 1 1 1 1 1 W2∗p2 W2p2 = diag(y(0,0) , y(0,1) , y(1,0) , y(0,2) , y(1,1) , y(2,0) , y(0,3) , . . .),

and Λ := W2∗p2 W1∗p1 W1p1 W2p2 (p1 ) (p ) (p1 ) (p ) (p1 ) (p ) (p1 ) (p ) = diag x(0,p y 2 , x(0,p y 2 , x(1,p y 2 , x(0,p y 2 ,··· . 2 ) (0,0) 2 +1) (0,1) 2 ) (1,0) 2 +2) (0,2)

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Thus, we have

Θ(k1 ,k2 ) (W) =

|p|

(−1)

ˆ 0≤p≤k

=

k p

p1 +p2

(−1)

0≤p1 ≤k1 , 0≤p2 ≤k2

=

p1 +p2

(−1)

0≤p1 ≤k1 , 0≤p2 ≤k2

409

W∗p Wp k1 p1 k1 p1

k2 p2 k2 p2

W2∗p2 W1∗p1 W1p1 W2p2

Λ.

Note that (p )

1 2 2 y (p2 ) = α2m+p2 2 · · · α2m+(p1 −1)1 +p2 2 βm · · · βm+(p = xm+p 2 2 m 2 −1)2

γm+p , γm

we have Θ(k1 ,k2 ) (W)

=

 p1 +p2

(−1)

0≤p1 ≤k1 , 0≤p2 ≤k2

    =  

(k ,k )

1 2 A(0,0)

0 0 .. .

0 (k ,k )

1 2 A(0,1)

0 .. .

k1 p1

k2 p2

0 0 (k ,k )

1 2 A(1,0) .. .

     ···

γ(p1 ,p2 ) γ(0,0)



0

0

γ(p1 ,p2 +1) γ(0,1)

0 .. .

0 .. .

0 0 γ(p1 +1,p2 ) γ(1,0)

.. .

 ··· ···    ···   .. .

 ···   . ···   .. .

The theorem is proved. (n ,n2 )

Corollary 2.3. W = (W1 , W2 ) is (k1 , k2 )-hyperexpansive iff Am 1 0 = n = (n1 , n2 ) ∈ Z2+ with (n1 , n2 ) ≤ (k1 , k2 ) . m ∈ Z2+ and each ˆ

≤ 0 for each

(k ,k )

Corollary 2.4. W = (W1 , W2 ) is completely hyperexpansive iff Am1 2 ≤ 0 for each m ∈ Z2+ and each ˆ 0 = k = (k1 , k2 ) ∈ Z2+ with ˆ0 = k = (k1 , k2 ) ∈ Z2+ .

3. Several Examples In this section, we present several examples to give out the applications of above results. 1 +n2 Proposition 3.1. Let W = (W1 , W2 ) and α(n1 ,n2 ) = β(n1 ,n2 ) = 2+n 1+n1 +n2 . Then W is completely hyperexpansive.

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Proof. (1) If k1 ≥ 1 and k2 = 0, then γ(m1 +p1 ,m2 ) k1 p1 (k1 ,0) Am = (−1) p1 γ(m1 ,m2 ) 0≤p1 ≤k1 1 k1 p1 (p1 + 1 + m1 + m2 ) = (−1) p1 1 + m1 + m2 0≤p1 ≤k1 1 k1 p (−1) 1 = p1 , p1 1 + m1 + m2 0≤p1 ≤k1

(1,0)

(k ,0)

thus, we have Am = − 1+m11 +m2 < 0 and Am1 = 0 for each k1 = 2, 3, · · · . (2) If k1 = 0 and k2 ≥ 1, then γ(m1 ,m2 +p2 ) k2 p2 2) A(0,k = (−1) m p2 γ(m1 ,m2 ) 0≤p2 ≤k2 1 k2 p2 = (−1) (p2 + 1 + m1 + m2 ) p2 1 + m1 + m2 0≤p2 ≤k2 1 k2 p2 (−1) = p2 , p2 1 + m1 + m2 0≤p2 ≤k2

(0,1)

(0,k )

thus, we have Am = − 1+m11 +m2 < 0 and Am 2 = 0 for each k2 = 2, 3, · · · . (3) If k1 ≥ 1 and k2 ≥ 1, then γ(m1 +p1 ,m2 +p2 ) k1 k2 p1 p2 (k1 ,k2 ) Am = (−1) (−1) p1 p2 γ(m1 ,m2 ) 0≤p1 ≤k1 0≤p2 ≤k2

1 = 1 + m1 + m2

p1

(−1)

p2

(−1)

0≤p1 ≤k1 0≤p2 ≤k2

k1 p1

k2 p2

× (p1 + p2 + 1 + m1 + m2 )

1 k1 p1 = (−1) p1 1 + m1 + m2 0≤p1 ≤k1   k2 p × (−1) 2 (p1 + p2 + 1 + m1 + m2 ) p2 0≤p2 ≤k2   1 k1  k2 p p (−1) 1 (−1) 2 p2  = 0. = p1 p2 1 + m1 + m2 0≤p1 ≤k1

0≤p2 ≤k2

(k ,k ) Therefore, Am1 2 ≤ 0 for each m ∈ Z2+ and each (k1 , k2 ) ∈ Z2+ − ˆ0 . It follows from Corollary 2.4 that W is completely hyperexpansive.

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Remark 3.2. Let a > 1 and α(n1 ,n2 ) = β(n1 ,n2 ) =

1 a

if n1 ≥ 1 or n2 = 0, if n1 = 0 and n2 ≥ 1;

1 a

if n1 = 0 or n2 ≥ 1, if n1 ≥ 1 and n2 = 0.

411

Then (W1 , W2 ) is not completely hyperexpansive. In fact, (1,1)

2 2 A(0,0) = 1 − β(0,0) − α2(0,0) + α2(0,1) β(0,0) = a2 − 1 > 0.

Remark 3.2 showed that although W1 and W2 are completely hyperexpansive, but W = (W1 , W2 ) may be not completely hyperexpansive. Example 3.3. Let x, y > 1 and (see   1 x α(n1 ,n2 ) =  y   1 x β(n1 ,n2 ) =  y

Figure 1) if n1 ≥ 1, if n1 = 0 and n2 = 0, if n1 = 0 and n2 ≥ 1; if n2 ≥ 1, if n1 = 0 and n2 = 0, if n1 ≥ 1 and n2 = 0.

Figure 1: Weight diagram of two variables weighted shift in Example 3.3

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(k ,k )

Since Am1 2 = 0 if m = (m1 , m2 ) with m1 ≥ 1 and m2 ≥ 1, so we only (k1 ,k2 ) (k1 ,k2 ) (k1 ,k2 ) need to consider A(0,0) , A(0,1) , A(1,0) . We have (k ,0)

(k ,0)

(k ,0)

(0,k ) A(0,1)2

(0,k ) A(1,0)2

1 1 1 A(0,0) = 1 − x2 , A(0,1) = 1 − y 2 , A(1,0) = 0,

(0,k ) A(0,0)2

2

=1−x ,

= 0,

if k1 ≥ 1, 2

= 1 − y , if k2 ≥ 1,

moreover, if k1 ≥ 1, k2 ≥ 1, then (k ,k )

(k ,k )

(k ,k )

1 2 1 2 1 2 = 1 − 2x2 + x2 y 2 , A(0,1) = 0, A(1,0) = 0. A(0,0)

Thus, (W1 , W2 ) is completely hyperexpansive iff 1 − x2 ≤ 0, 1 − y 2 ≤ 0 and 1 − 2x2 + x2 y 2 ≤ 0. (See Figure 2.)

Figure 2: Graph of the region of completely hyperexpansivity in Example 3.3

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413

Example 3.4 ([7, Proposition 2.10]). Consider the two variables weighted shift given by Figure 3,

Figure 3: Weight diagram of two variables weighted shift in Example 3.4 (k ,k )

where x, y > 0 and a ≥ 1. Since Am1 2 = 0 if m = (m1 , m2 ) with m1 ≥ 1 and (k1 ,k2 ) (k1 ,k2 ) (k1 ,k2 ) , A(0,1) , A(1,0) . m2 ≥ 1, so we only need to consider A(0,0) We have (k ,0)

(k ,0)

(k ,0)

1 1 1 A(0,0) = 1 − x2 , A(0,1) = 1 − a2 , A(1,0) =0

(0,k )

A(0,0)2 = 1 − y 2 ,

(0,k )

A(0,1)2 = 0,

(0,k )

A(1,0)2 = 1 −

ay 2 x

if k1 ≥ 1, if k2 ≥ 1,

moreover, if k1 ≥ 1, k2 ≥ 1, then (k1 ,k2 ) (k1 ,k2 ) (k1 ,k2 ) = 1 − x2 − y 2 1 − a2 , A(0,1) = 0, A(1,0) = 0. A(0,0) Thus, we have (1) if a = 1, then (W1 , W2 ) is completely hyperexpansive iff y ≥ x ≥ 1; (2) if a > 1, then (W1 , W2 ) is completely hyperexpansive iff a ≤ x and x x2 −1 ≤ y ≤ a a2 −1 .

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References [1] J. Conway, The theory of subnormal operators, Amer. Math. Soc., Providence, RI, 1991. [2] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), 203–217. [3] A. Athavale, On completely hyperexpansive operators, Proc. Amer. Math. Soc. 124 (1996), 3745–3752. [4] A. Athavale and V. Sholapurkar, Completely hyperexpansive operator tuples, Positivity 3 (1999), 245–257. [5] G. Exner, I. Jung and C. Li, On k-hyperexpansive operators, J. Math. Anal. Appl. 323 (2006), 569–582. [6] A. Shields, Weighted shift operators and analytic function theory, Topics in Operator Theory, Math. Surveys No.13, Amer. Math. Soc., Providence, RI, 1974, 49–128. [7] R. Curto and J. Yoon, Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc. 358 (2006), No.11, 5139–5159. [8] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, Oper. Theory Adv. Appl. 35 (1988), 1–22. Chunji Li Institute of System Science College of Sciences Northeastern University Shenyang 110004 P. R. China e-mail: [email protected] Junde Wu Department of Mathematics Zhejiang University Hangzhou 310027 P. R. China. e-mail: [email protected] Submitted: August 8, 2007. Revised: January 11, 2009.

Integr. equ. oper. theory 65 (2009), 415–448 © 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030415-34, published online November 9, 2009 DOI 10.1007/s00020-009-1726-6


Spectral Approximation and Index for Convolution Type Operators on Cones on Lp(R2) H. Mascarenhas and B. Silbermann Abstract. We consider an algebra of operator sequences containing, among others, the approximation sequences to convolution type operators on cones acting on Lp (R2 ), with 1 < p < ∞. To each operator sequence (An ) we associate a family of operators Wx (An ) ∈ L(Lp (R2 )) parametrized by x in some index set. When all Wx (An ) are Fredholm, the so-called approximation numbers of An have the α-splitting property with α being the sum of the kernel dimensions of Wx (An ). Moreover, the sum of the indices of Wx (An ) is zero. We also show that the index of some composed convolution-like operators is zero. Results on the convergence of the -pseudospectrum, norms of inverses and condition numbers are also obtained. Mathematics Subject Classification (2000). Primary 45E10; Secondary 47A53, 47L80, 15A29. Keywords. Convolution operators, pseudospectrum, Fredholm sequences, index.

1. Introduction

Let u ∈ L1 R2 and λ ∈ C. We denote by C (a) the convolution operator defined by C(a) : Lp (R2 ) → Lp (R2 ) g → λg (t) +

R2

u(t − s)g(s)ds,

where a is given by a(x) = λ + (F (u)) (x) with F being the Fourier transform on R2 . The function a, usually called the symbol of the operator C(a), is continuous in R2 and tends to λ at infinity. It is

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p 2 well known that C(a) ∈ L L R , the Banach algebra of all bounded linear p 2 operators in L R , and its invertibility is equivalent to the invertibility of the symbol a. The set of all such symbols (1.1) W R2 := a = λ + F (u) : u ∈ L1 R2 , λ ∈ C 2 ∞ is a subalgebra of L R closed for the norm aW = |λ| + uL1 and is the socalled Wiener algebra. By a cone K with vertex at the origin, we mean an angular sector in R2 , i.e K := {teiθ : t ∈ [0, +∞[ , θ ∈ Γ} (1.2) where Γ is a closed connected subset of unit circle T containing at least two points. We define the convolution operator on K as the restriction of C(a) to a cone K extended by the unity to the complement of K and we denote it by CK (a) := χK C(a)χK I + (1 − χK ) I, where χK is the characteristic function of K. In 1967, Simonenko proved that CK (a) is Fredholm if and only if a is invertible and if CK (a) is Fredholm then its index is zero [25] . It remains an open question to know if every Fredholm operator CK (a), with a in the Wiener algebra, is invertible. When K is a half-space Gohberg and Goldstein showed that this result is true [9]. For some special symbols the answer is also true [2, 13, 16]. We give a contribution to this problem by presenting an asymptotic formula for the kernel dimension of CK (a) providing it is Fredholm. We consider an algebra E of approximate sequences containing in particular approximate sequences to operators of the type χK AχK + (1 − χK )I with A in the Banach algebra A0 := alg{C (a) , f I : f ∈ C(R2 ), a ∈ W (R2 )},

(1.3)

where C(R2 ) is set of the continuous functions in R2 which have finite limit in every direction (see precise definition in section 2). Our main aim is to describe the so called Fredholm sequences in E and their properties. We mean by a Fredholm sequence a sequence which is regularizable by some special ideal D consisting of sequences of compact type. The interest of Fredholm sequences has at least three aspects: It is a generalization of the stability property, they can be used to construct generalized inverses to some sequences of interest [23], and last but not least Fredholm sequences can be used in some cases to obtain the index and an asymptotic formula for the kernel dimension of some Fredholm operators. Using C ∗ algebras techniques, it was proved in [14], for p = 2, that Fredholm sequences in E possess some splitting property of the singular values (that is, 1 the points in the spectrum of (A∗n An ) 2 ) as n → ∞. Here we prove, for 1 < p < ∞, the analogous result by defining approximation numbers in the context of infinite dimensional Banach spaces and using them instead of singular values. We further prove an index formula for the operators associated to each Fredholm sequence. In [19] results concerning some theory of Banach algebras consisting of

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structured matrix sequences (in which the structure is hidden in the sequence, not in the entries separately) are obtained. Here we adapt this model, although the approximations which occur in this paper are not finite matrices. To achieve our aim we start in section 2 by introducing a Banach algebra B containing sequences with special structure and which contains E. We first study the stability problem for sequences in E by help of some localization procedure and limit operators techniques. A correspondence of a sequence (An ) ∈ E and a set of operators Wx (An ) ∈ L(Lp (R2 )), with x in some index set X is established. We reprove, in particular, earlier stability results of A. Kozak [11] and add some new features which lead to the convergence of condition numbers and -pseudospectrum. These results are given in section 3. The method we use goes back to S. Roch [17], who used it in a different setting, and to our earlier paper [14]. Notice that some results of the recent paper of [15] are also contained in ours, as will be explained at the end of subsection 3.4.1 We start section 4 by defining Fredholm sequences and approximations numbers. In subsection 4.3 we show that if a sequence (An ) is Fredholm then the sum of the indices of Wx (An ) is zero and the approximation numbers of (An ) have the α-splitting property with α being the sum of the kernel dimensions of Wx (An ) . A second localization level is applied in subsection 4.4 to describe all Fredholm sequences belonging to E. It is shown that Fredholm sequences are those for which Wx (An ) is Fredholm for every x ∈ X. In the last subsection we use the main results to deduce that for some Fredholm convolution type operators its index is zero and an asymptotic formula for the kernel dimension is computed. Notice that we consider the above mentioned operators on Lp R2 for simplicity. Although, one can extend the results without difficulties to related operators acting on Lp (Rn ). Moreover, the system case can also be considered by the methods of this paper.

2. Localization

∞ Let F be the set of all sequences (An )n=0 of operators in L Lp (R2 ) such that sup An < ∞. Endowed with the usual pointwise operations and the norm (An ) = sup An , F is a Banach algebra. The set G of the sequences in F tending in norm to zero is a closed ideal of F and plays an important role due to Kozak’s Theorem ([6], Proposition 2.20). This theorem states that a sequence (An ) in F is stable (i.e. An is invertible for n large enough and the norms of their inverses are uniformly bounded) if and only if the coset (An ) + G is invertible in F G = F /G. Thus, stability is an invertibility problem. There are no effective tools to study invertibility in F G because of its generality. The way out is to introduce a suitable subalgebra of F G which serves our aim and which is subject to localization via Allan-Douglas local principle. Let D be the open unit disc in R2 . It is easy to see that ξ : D → R2 , x x → 1−|x| is a homeomorphism. We denote by C(R2 ) the set of all continuous

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functions f on R2 for which f ◦ ξ admits a continuous extension onto the closed disc D. Provided with pointwise operations and the supremum norm C(R2 ) forms a commutative C ∗ -algebra isomorphic to C(D). Its maximal ideal space R2 can be viewed as the compactification (with the Gelfand topology) of R2 with the circle T of infinitely distant points; and a sequence hn ∈ R2 converges to θ∞ ∈ R2 \R2 if ξ −1 (hn ) converges to some eiθ with θ ∈ [0, 2π[. We write f (θ∞ ) := f ◦ ξ(eiθ ), where the second function is the extension of f ◦ ξ . For each function ϕ ∈ C(R2 ), we associate the sequence (ϕn I) given by the expanded functions ϕn (t) = ϕ nt . Clearly, this sequence belongs to F and (ϕn I) = ϕ∞ . For x ∈ R2 , let Vx : Lp (R2 ) → Lp (R2 ) , f (t) → f (t − x) be the usual shift operator in the x-direction. We denote by B the set of all sequences (An ) of F for which both limits s-lim V−nx An Vnx

and

s-lim V−nx A∗n Vnx

exist for every x ∈ R2 , and lim An ϕn I − ϕn An = 0,

n→∞

for every ϕ ∈ C(R2 ).

It is not hard to check that B is a Banach subalgebra of F . Observe that the commutator (An )(ϕn I) − (ϕn I)(An ) belongs to G for every ϕ ∈ C(R2 ) which shows that we can apply localization to study B/G. In order to characterize the algebra B, let us consider the following map, for x ∈ R2 : Wx : B → L Lp (R2 ) (An ) → s-lim V−nx An Vnx . Taking into account the definition of B and the algebraic properties of the strong limits, the map Wx is well-defined and is a homomorphism. Besides, since Vnx is an isometry and Wx (I) = I we deduce from the Banach-Steinhaus Theorem that Wx = 1. We note that the expression Wx (An ) will always refer to Wx applied to the sequence (An ) ∈ B and Wx (A) will denote Wx applied to the constant sequence. Throughout this work we deal with several quotient algebras and homomorphisms defined on them. In order to simplify notation, if A, B are Banach algebras, J ⊂ A a closed ideal of A and H : A/J → B an homomorphism, we write H(a) in place of H(a + J). Proposition 2.1. ([14], Proposition 3.1) Let x ∈ R2 , f, ϕ ∈ C(R2 ), a ∈ W (R2 ) and (Gn ) ∈ G. Then the following holds: (i) (ii) (iii) (iv)

s-lim V−nx C(a)Vnx = C(a). s-lim V−nx f Vnx = f (x∞ )I if x = 0 where f (x∞ ) = limn→∞ f (nx). s-lim V−nx ϕn Vnx = ϕ(x). s-lim V−nx Gn Vnx = 0.

Theorem 2.2. The algebra B G = B/G is an inverse closed subalgebra of F G .

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Proof. Let (An ) + G ∈ B G be invertible. Then, there exists a bounded sequence of operators Bn in Lp R2 , such that Bn An = I + Gn

(2.1)

Gn

(2.2)

An Bn = I +

where (Gn ) , (Gn ) ∈ G. We first prove that Wx (An ) = s-lim V−nx An Vnx is an invertible operator. If (An ) is stable then (V−nx An Vnx ) is also stable, i.e. V−nx An Vnx is invertible for n large enough and the norms of the inverses are uniformly bounded. Therefore, there exists C > 0 such that V−nx An Vnx h ≥ Ch for every h ∈ Lp R2 , and consequently, Wx (An ) has closed image and its kernel dimension is zero. Similar arguments for (A∗n ) show that Wx∗ (An ) has kernel dimension zero and thus Wx (An ) is invertible. Let us now prove that s-lim V−nx Bn Vnx = Wx−1 (An ) and s-lim V−nx Bn∗ Vnx = −1 (Wx (An ))∗ for every x ∈ R2 . From (2.1), we have V−nx Bn Vnx V−nx An Vnx = I + V−nx Gn Vnx which is equivalent to V−nx Bn Vnx Wx (An ) = I + V−nx Gn Vnx − V−nx Bn Vnx (V−nx An Vnx − Wx (An )). Multiplying both sides by Wx−1 (An ) and taking into account that V−nx An Vnx − Wx (An ) and V−nx Gn Vnx converge strongly to zero, it follows that V−nx Bn Vnx is equal to Wx−1 (An ) plus a sequence strongly convergent to zero. Thus, the strong limit of V−nx Bn Vnx exists and equals Wx−1 (An ). Similararguments show that the ∗ strong limit of V−nx Bn∗ Vnx exists and equals Wx−1 (An ) . Finally, let us prove that (Bn ϕn I − ϕn Bn ) ∈ G. From (2.1) and (2.2), we can write I = An Bn − Gn = Bn An − Gn and then obtain Bn ϕn I − ϕn Bn = Bn ϕn (An Bn − Gn ) − (Bn An − Gn ) ϕn Bn = Bn ϕn An Bn − Bn ϕn Gn − Bn An ϕn Bn + Gn ϕn Bn = Bn (ϕn An − An ϕn )Bn − Bn ϕn Gn + Gn ϕn Bn , where the two last sequences (Bn ϕn Gn ) and (Gn ϕn Bn ) belong to G. Once (An ) belongs to B, then (ϕn An − An ϕn ) in also in G and this completes the proof. From Theorem 2.2, we see that a sequence (An ) ∈ B is stable if and only if An + G is invertible in B G . By construction, the set C G = {(ϕn I) + G : ϕ ∈ C(R2 )} is contained in the center of B G and it can be easily seen that it is a C ∗ -algebra. Hence, we can use localization to study invertibility in the algebra B G . Proposition 2.3. The algebra C G is isometric isomorphic to C(R2 ).

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Proof. It is clear that for every ϕ ∈ C(R2 ), we have ϕ∞ = (ϕn I)B ≥ (ϕn I) + GBG . To prove the statement we just need to show the reverse inequality. By Proposition 2.1-(iv), the homomorphism Wx maps G onto zero. Let us then define the analogous homomorphism x : B G → L(Lp (R2 )), (An ) + G → Wx (An ) W (2.3) x = 1 and due to on the quotient algebra B G . For every x ∈ R2 , we have W Proposition 2.1-iii) Wx ((ϕn I) + G) = ϕ(x)I . Thus, ϕ∞ = sup |ϕ(x)| ≤ (ϕn I) + GBG . x∈R2

Since C G C(R2 ) we identify the maximal ideal space of C G with R2 . For each x ∈ R2 the maximal ideal in C G is given by Ix = {(ϕn ) I + G : ϕ (x) = 0}, and for each θ ∈ [0, 2π[ the maximal ideal is Iθ = {(ϕn I) + G : ϕ (θ∞ ) = 0}. For x ∈ R2 , let Jx be the smallest closed ideal in B G generated by Ix and let φx : B G → B G /Jx be the canonical quotient map. Due to the Allan-Douglas Principle we know that (An ) + G ∈ B G is invertible if and only if φx ((An ) + G) is invertible in B G /Jx for every x in R2 .

3. Finite sections of convolution type operators on cones Let us now introduce a set Ω ⊂ R2 which will be needed to define the finite sections of convolution type operators on cones. We say Kx is a cone with vertex x if it is a set of the type K + x = {x + t : t ∈ K} where K is a cone at the origin (see Equation (1.2)). We will use the notation nX, where n is a positive integer and X ⊂ R2 is a set, to denote the expanded set {nt : t ∈ X}. Let Ω be a closed bounded set of R2 containing the origin, so that the strong limit of χnΩ I is not zero. We also assume that for each point x ∈ ∂Ω there exists a cone Kx , neighborhoods U and V of x and a C 1 -diffeomorphism ρ : U → V, such that ρ(x) = x,

ρ (x) = I

ρ(U ∩ Ω) = V ∩ Kx .

If 0 ∈ ∂Ω we require moreover that the associated diffeomorphism ρ is the identity. Observe that the cone Kx is uniquely defined for each x ∈ ∂Ω. The next proposition states some convergence results that will be useful in the sequel. Proposition 3.1. ([14], Proposition 3.1 and Proposition 3.2) Let x ∈ R2 , H be a half-space with 0 ∈ ∂H and T ∈ K(Lp (R2 )). Then,   χKx0 I, if x ∈ ∂Ω, (i) s-lim V−nx χnΩ Vnx = I, if x ∈ int Ω,   0, if x ∈ R2 \Ω, 0 where, for x ∈ ∂Ω, Kx := {t − x : t ∈ Kx } is the cone shifted to the origin;

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(ii) s-lim V−nx χnKx Vnx = χKx0 I, for x ∈ ∂Ω;   χH I, if x ∈ ∂H, (iii) s-lim V−nx χH Vnx = I, if x ∈ int H,   0, if x ∈ R2 \H; (iv) s-lim V−nx T Vnx = 0 for x = 0. Let E be the smallest closed subalgebra of F generated by • (χnΩ AχnΩ I + (1 − χnΩ ) I) with A in A0 , • (Gn ) with (Gn ) ∈ G, • (Vnx T V−nx ) with T ∈ K(Lp (R2 )) and x ∈ ∂Ω ∪ {0}, where A0 is the Banach algebra defined in (1.3) and K denotes the set of all compact operators on Lp (R2 ). Note that E contains the finite sections of the operators χK AχK I+(1 − χK ) I, with A in A0 , relative to the sequence of projections (χnΩ I). We do not know whether E G is inverse closed. However from Proposition 2.1, Proposition 3.1 and commutations relations ([14], Proposition 3.2), we deduce that the generators of E are in B. That is, E G = E/G is a closed subalgebra of the inverse closed algebra B G . Now, our main task is to describe the local subalgebras φx (E G ) of B G /Jx . 3.1. Local algebras for points not at the boundary For x ∈ / ∂Ω, the local algebras φx (E G ) can easily be identified. Proposition 3.2. ([14], Proposition 3.4) Let x ∈ R2 , y ∈ R2 , f ∈ C(R2 ) and T ∈ K. Then, f (x∞ )φx (I), if x ∈ R2 \ {0}, (i) φx (f I) = f (x)φx (I), if x ∈ R2 \R2 , where f (x∞ ) = limn→∞ f (nx); (ii) φx (χnΩ I) = 0 if x ∈ R2 \Ω and φx (χnΩ I) = φx (I) if x ∈ int Ω; (iii) φx (Vny T V−ny ) = 0 if x = y. It is convenient to define the homomorphisms Wx in the local-algebras B G /Jx . x (Jx ) = 0, where W x is defined in (2.3). From proposition 2.1-(iii) it follows W Then, the map B G /Jx → L Lp (R2 ) wx : ((An ) + G) + Jx → s-lim V−nx An Vnx is a well defined homomorphism with wx = 1. For x ∈ / ∂Ω, the local algebras φx (E G ) can be completely described by the homomorphisms wx . Let A00 be the closure of all convolution operators with symbol in the Wiener algebra, i.e the Banach algebra A00 = alg{C(a) : a ∈ W R2 } (3.1)

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Proposition 3.3. Let x ∈ R2 , A00 be the algebra defined 2 in (3.1) and A1 be the Banach algebra generated by all C(a) with a ∈ W R , all f I with f ∈ C(R2 ) and all compact operators in Lp (R2 ). The local algebra φx (E G ) is isometrically isomorphic to the Banach algebra of operators A00 if x ∈ int Ω and x = 0, (i) Lx := A1 if x = 0 and 0 ∈ int Ω, (ii) Lx := CI if x ∈ R2 \Ω. Proof. (i) Let x ∈ int Ω. From Proposition 3.1 and Proposition 3.2, we deduce that φx (An ) = φx (Wx (An ))

(3.2)

for the generators of φx (E G ) and so it is also true for every element of E G . Since φx (Wx (An )) ≤ Wx (An )BG ≤ Wx (An )L(Lp ) and wx ≤ 1, then wx is an isometry. To show that its image is closed, note that for every L ∈ Lx , we have L = wx (φx (L)). (ii) If x ∈ R2 \Ω then, due to Proposition 3.2, φx (χnΩ AχnΩ I + (1 − χnΩ ) I) = φx (I) and φx (Vny T V−ny ) = 0, y ∈ ∂Ω ∪ {0}. Thus φx (E G ) is generated by φx (I) and therefore is isomorphic to C. 3.2. Local algebras at the boundary points When x ∈ ∂Ω, the situation is more involved. Let us define, for each x ∈ ∂Ω, the Banach algebra Fx given by Fx := alg{χnKx AχnKx I + (1 − χnKx )I, (Gn ) , (Vnx T V−nx )}, where A belongs to A0 defined in (1.3) , (Gn ) ∈ G and T ∈ K(Lp (R2 )). We will construct an isometry hρ from B G /Jx onto itself which maps φx (E G ) onto φx (FxG ) and we will see that the second one is easier to describe. Recall that to each x ∈ ∂Ω we associated a C 1 -diffeomorphism ρ : U → V, where U and V are neighborhoods of x and in case 0 ∈ ∂Ω, ρ ≡ I. Thus, if x = 0 and 0 ∈ ∂Ω then χnΩ χnU I = χK0 χnU I which means that φx (χnΩ I) = φx (χK0 I) and the algebras φx (E G ) and φx (FxG ) are the same. So, from now on x is a fixed point in ∂Ω\{0}. (−1)

Let Tn and Tn the isometry hρ .

be the following operators, which will help us to construct

Tn : Lp (R2 ) → ImχnU I ⊂ Lp (R2 ) g nρ nt if t ∈ nU, g → (Tn g)(t) = 0 if t ∈ / nU.

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and Tn(−1) : Lp (R2 ) → ImχnV I ⊂ Lp (R2 )

g nρ−1 nt if t ∈ nV, (−1) g → Tn g (t) = 0 if t ∈ / nV. Proposition 3.4. Let Jρ(t) denote the Jacobian matrix of ρ at t and similarly for (−1) are bounded linear operators satisfying ρ−1 . The operators Tn and Tn (a) (b) (c) (d)

(−1)

Tn ≤ supt∈V |Jρ−1 (t) | and Tn ≤ supt∈U |Jρ (t) |; (−1) (−1) = χnU I and Tn Tn = χnV I (locally invertible); Tn Tn (−1) (−1) χnU Tn χnV I = Tn and χnV Tn χnU I = Tn ; (−1) ∗ The sequences V−ny Tn Vny , and V−ny Tn Vny converge strongly for y ∈ U (−1) ∗ and the sequences V−ny Tn Vny and V−ny Tn Vny converge strongly for y ∈ V .

Proof. (a) Let g ∈ Lp (R2 ), then p p g nρ t dt Tn g = n nU s p = |g(s)|p Jρ−1 ds n nV ≤ sup |Jρ−1 (s)|p |g(s)|p ds. s∈V

So,

nV

Tn g ≤ sup |Jρ−1 (s)|gLp s∈V

(−1)

Analogously, one shows the bound for the operator Tn . (b) and (c) are immediate. (d) Let y ∈ U and Fρ (y) : Lp (R2 ) → Lp (R2 ) be the linear bounded operator given by g(t) → g(ρ (y)t). We claim that s-lim V−ny Tn Vny = Fρ (y) if ρ(y) = y (note that if y = x it gives the identity map) and s-lim V−ny Tn Vny = 0 if ρ(y) = y. Let g ∈ Lp (R2 ) be a continuous function with compact support (it is enough to consider functions from a dense set of Lp (R2 )). We have p t + y − ny − g(ρ (y)t) dt V−ny Tn Vny g−Fρ (y) gpLp(R2 ) = χnU−ny (t) g nρ n 2 R

where χnU−ny is the characteristic function of nU − ny = {nt − ny, t ∈ U }. Let hn (t) = χnU−ny (t)g nρ( nt + y) − ny −g(ρ (y)t). Since the open set U −y contains the origin, then for a fixed t ∈ R2 , there is n0 ∈ N such that for n ≥ n0 , t ∈ nU − ny and thus, t + y − ny − g(ρ (y)t), if n ≥ n0 . hn (t) = g nρ n The function ρ is differentiable in U , i.e, ρ(t + y) = ρ(y) + ρ (y)(t) + δ(t)t, where limt→0 δ(t) = 0. Since we assumed ρ(y) = y then

424


ρ

and it follows that

t +y n

t t t = ρ(y) + ρ (y) + δ n n n t t t = y + ρ (y) + δ n n n

t hn (t) = g ρ (y)t + δ t − g(ρ (y)t), n

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(3.3)

if n ≥ n0 .

Once g is continuous, we have lim hn (t) = 0 for each t ∈ R2 , thus by the Lebesgue dominated convergence theorem lim V−ny Tn Vny g − Fρ (y) gpLp(R2 ) = 0. Suppose now ρ(y) = y. We have p χnU−ny (t)g nρ t + y − ny dt V−ny Tn Vny gpLp (R2 ) = n R2 Defining hn (t) = χnU−ny (t)g(nρ( nt + y) − ny), it follows for fixed t ∈ R2 that there is n0 ∈ N such that for n ≥ n0 , hn (t) = g(nρ( nt + y) − ny). Since limn→+∞ |n(ρ( nt + y) − y)| = +∞, then hn (t) → 0 when n → +∞ because g has compact support. Thus, s-lim V−ny Tn Vny = 0. (−1) One shows analogously that V−ny Tn Vny also converges strongly for every y ∈V. Concerning the adjoint sequences, we start by noting that Tn∗ : Lq (R2 ) → ImχnV I ⊂ Lq (R2 ) g nρ−1 nt |Jρ−1 ( nt )| if t ∈ nV, ∗ g → (Tn g)(t) = 0 if t ∈ / nV. is the adjoint operator of Tn , with 1p + 1q = 1. Now, define for y ∈ V the linear bounded operator Fρ (y) : Lp (R2 ) → Lp (R2 ) , g(t) → g(ρ (y)t)|Jρ−1 (y)|. One can use the same arguments to prove that s-lim V−ny Tn∗ Vny = Fρ (y) if ρ(y) = y (in case y = x it gives the identity map) and s-lim V−ny Tn∗ Vny = 0 if ρ(y) = y . Just note that when ρ(y) = y, the functions hn are, for n large enough, given by t t + y − ny Jρ−1 + x − g(ρ (y)t)|Jρ−1 (y)| hn (t) = g nρ n n and so applying (3.3) one gets lim hn (t) = 0 for each t ∈ R2 . Analogously, we have that

∗ Tn(−1) : Lq (R2 ) → ImχnU I ⊂ Lq (R2 ) g nρ−1 nt |Jρ−1 ( nt )| if t ∈ nU, ∗ g → (Tn g)(t) = 0 if t ∈ / nU.

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is the adjoint operator of Tn for every y ∈ U .

425

(−1) ∗ and that the strong limits of V−ny Tn Vny exit

Let Z ⊂ U ∩V and W be an open ball centered in x such that W ⊂ Z∩ρ−1 (Z). In order to construct the isometry hρ we define the Banach subalgebras of B given by BW = {χnW An χnW I : (An ) ∈ B} and Bρ(W ) = {χρ(W ) An χρ(W ) I : (An ) ∈ B}. Proposition 3.5. The map Hρ : BW → Bρ(W ) (An ) → (Tn(−1) An Tn ) is a well defined isomorphism and satisfies the following: (a) If S ⊂ W then Hρ (χnS I) = χnρ(S) I. (b) The ideal G is invariant under Hρ , i.e Hρ (G) ⊂ G. (c) Let (An ) ∈ BW . If the coset An + G belongs to Jx then Hρ (An ) + G also belongs to Jx . (−1)

Proof. Simple calculations show Tn

(−1)

An Tn = χρ(W ) Tn

(−1) Tn An Tn

An Tn χρ(W ) I for (An ) ∈

BW , it remains to prove that is in B to have a well defined map. For y∈ / ρ(W ), we have s-lim V−ny χρ(W ) Vny = 0 and so s-lim V−ny Hρ (An )Vny = 0. If y ∈ ρ(W ) then, by construction of W , y ∈ U ∩ V and from Proposition 3.4-(d) (−1) we conclude s-lim V−ny Tn Vny and s-lim V−ny Tn Vny exist, as well as its adjoints and so it also exits s-lim V−ny Hρ (An )Vny and its adjoint. Suppose now that ϕ ∈ C(R2 ), we have to show that Hρ (An ) commutes with ϕn I up to a sequence tending in norm to zero, i.e there exists (Gn ) ∈ G such that Tn(−1) An Tn ϕn = ϕn Tn(−1) An Tn + Gn .

(3.4)

R2 ,

then Tn ϕn I = (ϕ◦ ρ˜)n Tn . Observe first that if ρ˜ is a continuous extension of ρ to Indeed, for g ∈ Lp (R2 ) t t t (Tn ϕn g)(t) = Tn ϕ g(t) = χnU (t) ϕ ρ g nρ n n n = [χnU (ϕ ◦ ρ˜)n Tn g](t) = [(ϕ ◦ ρ˜)n Tn g](t). (−1) −1 ) T (−1) holds, where ρ −1 is a conAnalogously, the equality Tn ϕn I = (ϕ ◦ ρ n n −1 tinuous extension of ρ to R2 . Thus,

Tn(−1) An Tn ϕn I = Tn(−1) An (ϕ ◦ ρ˜)n Tn . Using the fact that there exists (Fn ) ∈ G such that An (ϕ ◦ ρ˜)n = (ϕ ◦ ρ˜)n An + Fn , it follows that Tn(−1) An (ϕ ◦ ρ˜)n Tn = Tn(−1) [(ϕ ◦ ρ˜)n An + Fn ] Tn −1 ) T (−1) + T (−1) F T = (ϕ ◦ ρ˜ ◦ ρ n n n n n

= ϕn Tn(−1) + Tn(−1) Fn Tn .

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(−1)

Now, let Gn = Tn Fn Tn and we obtain equality (3.4) finishing the proof that Hρ is well defined. It is clear that Hρ is linear. Since for (An ), (Bn ) ∈ BW , Hρ (An Bn ) = Hρ (An χnW Bn ) = Tn(−1) An χnW Bn Tn = Tn(−1) An Tn Tn(−1) χnW Bn Tn = Hρ (An )Hρ (Bn ), Hρ is a homomorphism. It is not hard to check that its inverse Hρ−1 : Bρ(W ) → BW (−1)

is the map that sends An to Tn An Tn . (a) Let g be a function in Lp (Rp ) and S ⊂ W. Then we have −1 t −1 (Hρ (χnS ) g) (t) = Tn χnS Tn g (t) = Tn χnS (t)g nρ n = χnρ(S) (t)g(t). (b) This is immediate. (c) Since Hρ is an isomorphism, it is enough to show, for ϕ ∈ C(R2 ) with ϕ(x) = 0 there exist φ ∈ C(R2 ) with φ(x) = 0 such that Hρ (χnW ϕn χnW I) = −1 ) χ χρ(nW ) φn χρ(nW ) I. In fact, Hρ (χnW ϕn χnW I) = χρ(nW ) (ϕ ◦ ρ n ρ(nW ) I, where −1 −1 is a continuous extension of ρ −1 and we get the to R2 , thus choose φ = ϕ ◦ ρ ρ

result.

G G G Let BW and Bρ(W ) be the subalgebras of B analogous to BW and Bρ(W ) and

ρ : BG → BG H W ρ(W )

(3.5)

ρ (B G ∩ Jx ) = be the map analogous to Hρ . From Proposition 3.5-(c), one has H W G Bρ(W ) ∩Jx . Every coset ((An ) + G)+Jx is the same coset as ((χnW An χnW ) + G)+ Jx , thus we can now define a map between the local algebras by hρ :

B G /Jx

→ B G /Jx

(3.6)

((An ) + G) + Jx → ((Hρ (An )) + G) + Jx .

(3.7)

Taking into account Proposition 3.5, hρ is an isomorphism, having h−1 ρ :

B G /Jx

→ B G /Jx

((An ) + G) + Jx → ( Tn An Tn(−1) + G) + Jx

as its inverse homomorphism. Note that hρ does not have to be the identity. For example hρ (χn(U∩Ω) I) is equal to the coset ( χn(V ∩Kx ) I + G) + Jx that is in general different from the coset ( χn(U∩Ω) I + G) + Jx . Proposition 3.6. The algebras φx (E G ) and φx (FxG ) are isometrically isomorphic under the isomorphism hρ .

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Proof. Since hρ is an isomorphism of B/Jx onto itself, one has only to check that hρ maps the generators of φx (E G ) onto the generators of φx (FxG ). Notice that the cosets φx (χnΩ I), φx (χnΩ χnW I) and φx (χn(Ω∩W ) I) are the same and from Proposition 3.5-(a), hρ (φx (χn(Ω∩W ) I)) = φx (χnρ(Ω∩W ) I). From the facts φx (χnV I) = φx (χnU I) = φx (I), W ⊂ U , and ρ is a diffeomorphism such that ρ(Ω ∩ U ) = Kx ∩ V , we get φx (χnρ(Ω∩W ) I) = φx (χnρ(Ω) χnρ(W ) I) = φx (χnρ(Ω) χnρ(U) I) = φx (χnρ(Ω∩U) I) = φx (χn(Kx ∩V ) I) = φx (χnKx I). Since hρ maps C(a) onto itself, (see [14], Proposition 3.18), we have hρ (φx (χnΩ AχnΩ I + (1 − χnΩ )I)) = φx (χnKx AχnKx I + (1 − χnKx )I), for A ∈ A0 . We are left to prove hρ (φx (Vnx KV−nx )) = φx (Vnx KV−nx ), where K is a compact operator. Actually, it is enough if we prove Vnx KV−nx − Tn(−1) Vnx KV−nx Tn → 0.

(3.8)

Multiplying both sides by the isometries V−nx and Vnx , one has Vnx KV−nx − Tn(−1) Vnx KV−nx Tn = K − V−nx Tn(−1) Vnx KV−nx Tn Vnx = K − KV−nx Tn Vnx + KV−nx Tn Vnx − V−nx Tn(−1) Vnx KV−nx Tn Vnx

≤ K(I − V−nx Tn Vnx ) + I − V−nx Tn(−1) Vnx KV−nx Tn Vnx . Since V−nx Tn Vnx is a bounded sequence and by Proposition 3.4-(d) both sequences (−1) V−nx Tn∗ Vnx and V−nx Tn Vnx converge strongly to identity, it follows the conver(−1) gence of the sequences K(I − V−nx Tn Vnx ) and (I − V−nx Tn Vnx )K in norm to zero. To prove that hρ is isometric, we know from Proposition 3.8-(a) that hρ ≤ sup Jρ−1 (s) sup |Jρ (s)| . s∈V

s∈U

Given ε > 0, we may assume without loss of generality that U and V are chosen so that hρ ≤ (1 + ε)2 .Since ε is arbitrarily, we have hρ = 1. Using the same = 1. Thus, hρ is an isometry. arguments one also has h−1 ρ Proposition 3.7. Let x ∈ ∂Ω, and A0 and A00 be the algebras defined in (1.3) and (3.1), respectively. Consider the following Banach algebra of operators: alg χKx0 AχKx0 I + 1 − χKx0 I, T : T ∈ K, A ∈ A00 , if x ∈ ∂Ω\ {0} , Lx := alg χKx0 AχKx0 I + 1 − χKx0 I, T : T ∈ K, A ∈ A0 , if x = 0 ∈ ∂Ω. The local algebra φx (E G ) is isometrically isomorphic to Lx .

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Proof. Suppose x ∈ ∂Ω\ {0}. From Proposition 3.6 it is enough to prove that wx restricted to φx (FxG ) is an isometric isomorphism onto Lx . The generators of φx (FxG ) are of the type φx (χnKx AχnKx I + (1 − χnKx )I) with A ∈ A00 or of the type φx (Vnx T V−nx ). The first ones can be rewritten as (3.9) φx Vnx (χKx0 AχKx0 I + (1 − χKx0 )I)V−nx , because A is a shift invariant operator and χnKx I = Vnx χKx0 V−nx , thus φx (An ) = φx (Vnx Wx (An )V−nx )

(3.10)

FxG .

for the generators of Since φx and Wx are continuous homomorphisms the equality (3.10) is true for every element of FxG . Since wx = 1 and φx (Vnx Wx (An )V−nx ) ≤ Vnx Wx (An )V−nx ≤ Wx (An ), we conclude that wx is an isometry. Moreover, the image of wx is closed, because for every L ∈ Lx , we have L = wx (φx (Vnx LV−nx )) with φx (Vnx LV−nx ) in φx (FxG ). If x = 0 and 0 ∈ ∂Ω, then φ0 (E G ) = φ0 (FxG ) is generated by the constant sequences ((χK0 AχK0 I + (1 − χK0 )I) + G) + Jx

and ((T ) + G) + Jx

with A ∈ A0 and T ∈ K. Now, using the same arguments as in the case x ∈ ∂Ω\ {0} we have φ0 (An ) = φ0 (W0 (An )) for every (An ) ∈ E and therefore w0 maps φ0 (E G ) onto L0 . 3.3. Stability for sequences in E The following theorem describes stability of (An ) ∈ E in terms of the operators Wx (An ). Theorem 3.8. Let (An ) ∈ E. The sequence (An ) is stable if and only if Wx (An ) is invertible in L(Lp (R2 )) for every x ∈ ∂Ω ∪ {0} . Proof. If (An ) is stable then it is clear that all operators Wx (An ) are invertible in L(Lp (R2 )) with x ∈ ∂Ω ∪ {0} . Suppose now, Wx (An ) is invertible for every x ∈ ∂Ω ∪ {0} . (i) If x ∈ ∂Ω\{0}, it follows, from the proof of the Proposition 3.7, that φx (An ) = h−1 ρ (φx (Vnx Wx (An )V−nx )).

(3.11)

Since for every (An ) ∈ E the sequence (Vnx Wx (An )V−nx ) belong to the inverse closed algebra B and, hρ defined in (3.6) maps B G /Jx isomorphically onto itself, it follows that if Wx (An ) is invertible in L(Lp (R2 )) then φx (An ) is invertible in B G /Jx . (ii) For x = 0 the equality φ0 (An ) = φ0 (W0 (An )) holds, and since (W0 (An )) belongs to the inverse closed algebra B, we have that φ0 (An ) is invertible providing W0 (An ) is invertible. (iii) Suppose x ∈ int Ω\{0}. Let y be a point on the intersection of the boundary of Ω with the half line that starts at the origin and passes through x and choose z to be any point in the interior of Ky0 . For the generating sequences

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(An ) of E, we have Wx (An ) = Wz (Wy (An )) and φx (An ) = φx (Wx (An )), therefore the equalities hold for every sequence in E. Thus from the invertibility of Wy (An ) it follows the invertibility of Wx (An ) and consequently that of φx (An ) in B G /Jx . (iv) If x ∈ R2 \Ω then φx (An ) = λφx (I) where λ does not depend on x. Now choose some y ∈ ∂Ω; by hypothesis Wy (An ) is invertible and so φy (An ) is also invertible. From Allan-Douglas principle (see [18], Proposition 2.3.17) there exits a neighborhood N of y such that for every t ∈ N the coset φt (An ) is invertible; but for t ∈ N ∩ ext Ω, φt (An ) = λφt (I), and so λ = 0. Now putting together these four cases we have that φx (An ) is invertible for every x ∈ R2 and applying the Allan Douglas principle we finish the proof. We now give examples of equivalent conditions to the stability of a sequence, using the last theorem and previous results. Example. Let 0 ∈ ∂Ω and Aij ∈ A0 , where A0 is the algebra defined in (1.3). The sequence m l [χnΩ Aij χnΩ I + (1 − χnΩ ) I] i=1 j=1

is stable if and only if the following conditions are satisfied: m l (i) j=1 [χK0 Aij χK0 I + (1 − χK0 )I] is invertible. i=1 m l (ii) i=1 j=1 χKx0 Wx (Aij ) χKx0 I + 1 − χKx0 I is invertible for every x in ∂Ω\{0}. Example. Let Ω be the closed unit disc. Assume A = C (a) + f I, with a ∈ W (R2 ) and f ∈ C(R2 ). The sequence (χnΩ AχnΩ I + (1 − χnΩ ) I) is stable if and only if A is invertible. Example. Let 0 ∈ ∂Ω, ∂Ω be a smooth set except at zero and Ω ⊂ K0 . Let A = C (a) + f I. The sequence χnΩ AχnΩ I + (1 − χnΩ ) I is stable if and only if χK0 AχK0 I + (1 − χK0 )I is invertible. Notice that the result in the last example was already proved in [11] when f is a constant. 3.4. A symbol map for E In Theorem 3.8 necessary and sufficient conditions for the stability of a sequence (An ) ∈ E are given. What can we also say about other asymptotic properties of (An ) such as the convergence of the norms of An , condition numbers or the behavior of the spectrum and ε-pseudospectrum? Having these questions in mind, we shall construct an isometric isomorphism from E G onto an algebra of operator valued functions and we shall describe asymptotic properties of (An ).

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Let S be the set of all bounded functions defined on ∂Ω ∪ {0} with values in L Lp (R2 ) . Endowed with pointwise sum and product, and with the supremum norm (Ax ) = supx∈∂Ω∪{0} Ax , S is a Banach algebra. It is easy to check that the map sym :

B/G

→S

(An ) + G → (s-lim V−nx An Vnx )x∈∂Ω∪{0} is a homomorphism. Noting that the norm of (An ) + G in B/G is also given by (An )B/G = lim sup An , it follows from the Banach-Steinhaus Theorem, that sym(An ) ≤ lim inf An ≤ lim sup An which implies that sym ≤ 1. We will now define a topological property of the algebras that have central C ∗ -subalgebras which will be useful to prove that sym is isometric when restricted to E G . So, let A be a Banach algebra with identity and C a closed C ∗ -subalgebra of the center of A which contains the identity. By the Gelfand-Naimark Theorem, C is isomorphic to the algebra of continuous functions on the maximal ideal space of C; therefore an element of C will be called a function. Definition 3.9. We say that A is a KMS-algebra with respect to C if for every A ∈ A and ϕ, ψ ∈ C with disjoint supports A(ϕ + ψ) ≤ max (Aϕ , Aψ) . The importance of KMS-algebras is the relation between the norm of an element of A with the norms of the corresponding local elements. More precisely A is KMS if and only if (3.12) A = max φx (A) , x∈MC

where φx (A) is the local element associated to the Allan-Douglas localization. For a proof see ([5], Theorem 5.3). Proposition 3.10. The algebra B G = B/G is a KMS-algebra with respect to C G . Proof. We need to show that if ϕ and ψ are functions in C(R2 ) whose supports are disjoint, then (An (ϕn + ψn )I)BG ≤ max((An ϕn I)BG , (An ψn I)BG ),

(3.13)

G

for every (An ) ∈ B. The norm in B is given by inf (Gn )∈G sup An + Gn and it is not difficult to see that it coincides with lim sup An . If we define Yn = An ϕn , Zn = An ψn , N = supp ϕ and M = supp ψ then the inequality (3.13) can be written as lim sup Yn χnN + Zn χnM ≤ max (lim sup Yn , lim sup Zn ) . p

(3.14)

2

Given g ∈ L (R ), we have (χnN Yn χnN + χnM Zn χnM )gpLp = χnN Yn χnN gp + χnM Zn χnM gp

≤ max (Yn , Zn )p χnN gp + max (Yn , Zn )p χnM gp ≤ max (Yn , Zn )p gp .

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Thus, χnN Yn χnN + χnM Zn χnM L(Lp ) ≤ max (lim sup Yn , lim sup Zn ) . Since the last inequality is true for every n ∈ N, lim sup χnN Yn χnN + χnM Zn χnM L(Lp ) ≤ max (lim sup Yn , lim sup Zn ) . Now observe that (An ϕn − ϕn An ) belongs to the ideal G, ϕn χnN = ϕn and ψn χnM = ψn . So, Yn χnN − χnN Yn χnN

and Zn χnM − χnM Zn χnM

also belong to this ideal. Thus, lim sup Yn χnN + Zn χnM = lim sup χnN Yn χnN + χnM Zn χnM

and we get the assertion (3.14).

The following theorem shows that algebraic and topological properties of (An ) can be translated in terms of the analogous properties of (Wx (An ))x∈∂Ω∪{0} . Theorem 3.11. The restriction of sym to E G is an isometric isomorphism onto the image. Moreover, for every (An ) ∈ E, the sequence An converges and lim An =

max

x∈∂Ω∪{0}

Wx (An ) .

Proof. Let (An ) be a sequence in E then, from Proposition 3.10, we know that (An )B/G = lim sup An = max φx (An ) x∈R2

and we claim that max φx (An ) ≤

x∈R2

sup x∈∂Ω∪{0}

Wx (An ) ,

which implies that sym is an isometry. If x ∈ R2 \Ω then φx (An ) = λφx (I) and λ do not depend on x. In particular if x ∈ R2 \Ω, φx (An ) = Wx (An ). Let y ∈ ∂Ω ∩ [0, x] and z in the exterior of Ky0 . Then it is easy to check that Wx (An ) = Wz (Wy (An )), for every (An ) of E. Thus, φx (An ) = Wx (An ) ≤ Wy (An ) . If x ∈ int Ω\{0} and y is a point on the intersection of the boundary of Ω with the half line that starts at the origin and passes through x, from the proof of Theorem 3.8-(iii), we have that φx (An ) ≤ Wx (An ) = Wy (Wz (An )) ≤ Wy (An ) . For x = 0 the inequality φx (An ) ≤ Wx (An ) holds (see the proof of Theorem 3.8-(ii)). If x ∈ ∂Ω\{0}, then from equality (3.11) and hρ = 1 we deduce that φx (An ) ≤ Wx (An ). All these inequalities together lead to max φx (An ) ≤

x∈R2


Wx (An ) ,

(3.15)

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which proves that sym is an isometry and so, an isomorphism onto its image. Thus, we can write max φx (An ) =

x∈R2


Wx (An ) .

Moreover from the inequalities above we deduce that for each x ∈ R2 there exists y ∈ ∂Ω ∪ {0} such that φx (An ) ≤ Wy (An ), therefore it is not possible to have Wx (An ) < maxx∈R2 φx (An ) for every x ∈ ∂Ω ∪ {0}, which means that the supremum on the second side of (3.15) is attained. Now, observe that W (An ) = W (Ank ) for every subsequence (Ank ) of (An ), which means lim sup An = lim An , and therefore lim An =

max

x∈∂Ω∪{0}

Wx (An ) .

For an invertible operator A in a Banach algebra, its condition number is defined as the real number cond A = AA−1 . It is clear that cond A is always greater than or equal to one. As a consequence of Theorem 3.11 we obtain Corollary 3.12. If (An ) ∈ E is stable then the condition numbers of An converge and −1 lim An A−1 n = max Wx (An ) max Wx (An ). x∈∂Ω∪0

x∈∂Ω∪0

Proof. Suppose (An ) ∈ E is stable . From Theorem 3.11, lim An = max Wx (An ). x∈∂Ω∪0

We claim the formula is also true for the inverse of (An ), i.e. −1 lim A−1 n = max Wx (An ). x∈∂Ω∪0

(3.16)

From equalities (3.2) and (3.11), it is easy to check that, if x ∈ ∂Ω, φx (Wx−1 (An )), −1 = φx (An ) −1 h−1 (V W (A )V ) , if x∈ / ∂Ω. φ x nx x n −nx ρ Now, following the same reasoning as in the proof of Theorem 3.11, together with the result that says if (An ) is stable then Wx (An ) = s-lim V−nx An Vnx is an invert−1 −1 ible operator with s-lim V−nx A−1 n Vnx = Wx (An ) and such that Wx (An ) ≤ −1 lim inf V−nx An Vnx (see proof of Theorem 2.2), we obtain the claim. Corollary 3.13. Let An = χnΩ C(a)χnΩ I + (1 − χnΩ )I, with a ∈ W (R2 ). Then lim An = max{1, C(a)} ≤ max{1, aW }.

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Proof. We have Wx (An ) = χKx0 C(a)χKx0 I + 1 − χKx0 I, and therefore lim An = max χKx0 C(a)χKx0 I + (1 − χKx0 )I x∈∂Ω∪{0} = max{1, χKx0 C(a)χKx0 I }. Since for any interior pointy of Kx0 , we have C(a) = Wy (χKx0 C(a)χKx0 I) and Wy = 1, then C(a) ≤ χKx0 C(a)χKx0 I . The reverse inequality holds obviously. 3.4.1. Convergence of the pseudospectrum. As we saw, Theorem 3.11 and its corollaries established some relations between properties of a sequence (An ) in E for large n, and its associated set of “limit operators” Wx (An ). We will now consider the spectrum of An for n large enough. In particular, what is the relation between the spectrum of the sequence (An ) and the one of Wx (An )? To be precise, let us first introduce two limits of a sequence of subsets of the complex plane. Given a sequence of sets Mn ⊂ C, we define limes superior (resp. limes inferior) and denoted by lim sup Mn , (resp. lim inf Mn ) to be the set of all complex numbers λ which are the limit (resp. partial limit) of a sequence λn ∈ Mn . From Theorem 3.8, we easily deduce that σ(Wx (An )). (3.17) lim sup σ(An ) ⊂ n→∞

x∈∂Ω∪{0}

But in general these two sets do not coincide, (see for instance, the example given in [4], Theorem 1.3, in the context of Wiener-Hopf operators). However, if we consider the ε-pseudospectrum, which we now define, instead of the spectrum, we will see that the results are much more satisfactory. Definition 3.14. For ε > 0, the ε-pseudospectrum of an operator A ∈ Lp (R2 ) is defined to be the set 1 σε (A) = {λ ∈ C : A − λI is not invertible or (A − λI)−1 ≥ }. ε It is well known that ε-pseudospectrum of an operator is a compact set, which clearly contains the spectrum. Theorem 3.15. Let (An ) ∈ E and ε > 0. The ε-pseudospectrum of (An ) converges with respect to the Hausdorff distance and σε (Wx (An )). lim inf σε (An ) = lim sup σε (An ) = x∈∂Ω∪{0}

Proof. The proof of the second equality uses a non-trivial result due to Daniluk in the context of operators defined in a Hilbert space, and which was later proved also for operators in Lp (X, dµ), where (X, dµ) is a measure space, and 1 < p < ∞ ([4], Theorem 5.1). This interesting result, which can be viewed as a maximum principle, states that when A is a bounded operator in Lp (X, dµ) such that A−λI is invertible for all λ in some open subset U of the complex plane, and (A − λI)−1 ≤ M,

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then actually (A − λI)−1 < M, for all λ ∈ U. Using this maximum principle and Theorem 3.11, the proof of σε (Wx (An )) (3.18) lim sup σε (An ) = x∈∂Ω∪{0}

proceeds in analogy to that of Theorem 5.2 in [4]. Suppose now lim inf σε (An ) = lim sup σε (An ), i.e there is λ ∈ lim sup σε (An ) and λ is not the limit of any sequence λn ∈ σε (An ). Then, there is a subsequence (Aη(n) ) of (An ) such that λ ∈ / lim sup σε (Aη(n) ), for some strictly increasing sequence η : N → N. Define Bη as the algebra of sequences of the form (Aη(n) )n∈N such that (An ) is in B. In the same way, for a fixed η, we can define the algebra Eη , the ideal Gη and the homomorphism Wxη : Bη → L Lp (R2 ) as the corresponding analogues to E, G and Wx , respectively. It not hard to check that Theorem 3.11 and equality (3.18) also holds by substituting (An ) by its subsequence (Aη(n) ). The crucial point is that Wxη (Aη(n) ) := s-lim V−η(n)x Aη(n) Vη(n)x = s-lim V−nx An Vnx = Wx (An ). Thus, lim sup σε (Aη(n) ) =

x∈∂Ω∪{0}

σε (Wx (Aη(n) )) =

σε (Wx (An )),

x∈∂Ω∪{0}

showing that λ ∈ lim sup σε (Aη(n) ), which implies lim inf σε (An ) = lim sup σε (An ). This equality is equivalent to the convergence of σε (An ) with respect to the Hausdorff metric of non empty compact subsets of C (see [10], Proposition 3.6) . Consider now the algebra E0 ⊂ E generated by (χnΩ AχnΩ I + (1 − χnΩ )I) with A ∈ A00 (see (3.1)) and let Ω be a polygon with vertex at the origin. If y is any point in one of the edges of the polygon Ω, which is not a vertex, and x is a vertex of that edge, then Wy (An ) = Wy−x (Wx (An )) for every (An ) ∈ E0 . Thus, Wy (An ) ≤ Wx (An ). Furthermore, if Wx (An ) is invertible then Wy (An ) is also invertible and Wy−1 (An ) = Wy−x (Wx−1 (An )) ≤ Wx−1 (An ). This implies, using Corollary 3.12 and Theorem 3.15, the following result which was proved, with other methods, by Maximenko [15]. Corollary 3.16. Let (An ) ∈ E0 . Suppose Ω is a polygon with vertex at the origin and X is the set of its vertices. Then, −1 (i) lim An A−1 n ) maxx∈X Wx (An ) if (An ) is stable; n = maxx∈X Wx (A (ii) lim inf σε (An ) = lim sup σε (An ) = x∈X σε (Wx (An )).

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4. Fredholm sequences, approximation numbers and the splitting property 4.1. Fredholm sequences In order to define Fredholm sequences we start to introduce an ideal related with some sequences of compact operators. Let D ⊂ B be the smallest closed ideal of B containing the sequences converging to zero in norm and the sequences (Vnx T V−nx ) for every compact operator T on Lp (R2 ) and every x ∈ ∂Ω ∪ {0}. Proposition 4.1. The ideal D is the closure of the following set m Vnxi Ti V−nxi + Gn : Ti ∈ K, (Gn ) ∈ G, xi ∈ ∂Ω ∪ {0}

(4.1)

i=1

Proof. It is clear that the sum of elements of the set (4.1) are of the same type. Let us now prove that the set is a right ideal. Let (An ) be any sequence in B and consider a sequence of the type (Vnx T V−nx ) for some x ∈ ∂Ω ∪ {0} and T a compact operator. We have Vnx T V−nx An = Vnx T (V−nx An Vnx − Wx (An ))V−nx + Vnx T Wx (An )V−nx . Since T is compact and s-lim [V−nx An Vnx − Wx (An )] = 0, the sequence Gn = Vnx T (V−nx An Vnx − Wx (An ))V−nx converges to zero in norm. Take now R = T Wx (An ) which is a compact operator and it follows that Vnx T V−nx An = Vnx RV−nx + Gn . (4.2) Following the same ideas we obtain that (4.1) is also a left ideal.

We know that a linear bounded operator is Fredholm if it is invertible up to an operator in the ideal of compact operators. Similarly, we say that (An ) ∈ B is a Fredholm sequence if (An ) + D is invertible in B/D. Notice that this definition depends on B and D. Actually, there is also a more general notion of Fredholm sequence in the large algebra F . A sequence (An ) belonging to F is called Fredholm if it is regularizable with respect to the ideal of all compact sequences. This ideal is defined as follows: it is the smallest closed two-sided ideal in F which contains all sequences (Kn ) of almost uniformly bounded rank, that is lim sup rankKn < ∞. It can be proved that for sequences in E these two notions coincide. This proof involves new ideas which will be part of a forthcoming paper. An application of the special lifting Theorem of S. Roch and B. Silbermann ([10], Theorem 5.37b)) leads to the following proposition which we will give the proof for the convenience of the reader.

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Proposition 4.2. If (An ) ∈ B is a Fredholm sequence then the operator Wx (An ) is Fredholm for every x ∈ ∂Ω ∪ {0} and only a finite number of these operators are non-invertible. Proof. Suppose (An ) ∈ B is a Fredholm sequence, then there are (Bn ) ∈ B and (Jn ) ∈ D such that An Bn = I + Jn . m From Proposition 4.1 there exists Jn ∈ D such that Jn − Jn = i=1 Vnxi Ti V−nxi + n = Bn (I + Jn )−1 , which is in B (see TheoGn and Jn < 12 . Now, defining B rem 2.2), one has m n = (I + Jn )(I + Jn )−1 = I + An B Vnxi Ti V−nxi (I + Jn )−1 + Gn (I + Jn )−1 , i=1

which due to (4.2) can be rewritten as m n = I + n An B Vnxi Ri V−nxi + G

(4.3)

i=1

n ) ∈ G. Applying Wx to both sides of (4.3) we obtain with Ri ∈ K and (G n ) = I + Ri or Wx (An )Wx (B n ) = I depending on x belonging Wx (An )Wx (B or not belonging to {x1 , . . . , xn }, respectively. Thus, Wx (An ) is left invertible for every x, except for x ∈ {x1 , . . . , xn } which is left Fredholm. Analogously, we show the right invertibility. 4.2. Approximation numbers Let X be a Banach space and L(X) the algebra of linear bounded operators on X. We make the convention that if codim F is not finite, then codim F ≥ α for any integer α. Definition 4.3. (Approximation numbers)1 Let X be a Banach space and α ∈ N or α ∈ {1, . . . , dim X}, depending on X has non-finite or finite dimension, respectively. The αth approximation number of an operator A in L(X) is given by sα (A) := inf {A − F : F ∈ L(X) and codim F ≥ α} , where codim F = dim X/ImF . Clearly, for every α one has 0 ≤ sα (A) ≤ sα+1 (A) ≤ A. Notice that if X is a Hilbert space of finite dimension n, then the approximation numbers coincide with the singular values of the matrix A (see [1]), i.e. √ {sα (A), α = 1, . . . , n} = { λ : λ ∈ σ(A∗ A)}. Although if X is a Hilbert space of infinite dimension this is not true. Take for instance the Toeplitz operator T (eiθ ) defined on the Hardy space H 2 (T), where T 1 Notice

that this name is used in the literature for other (related) numbers.

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is the unit circle. It has codimension one and thus s1 (T (eiθ )) = 0 and zero is not a singular value since T ∗ (eiθ )T (eiθ ) = I . Proposition 4.4. Let α ∈ N or α ∈ {1, . . . , dim X} and A, B ∈ L(X) then, (1) sα (AB) ≤ B sα (A) for every α; (2) if A is invertible then s1 (A) ≥ A1−1 ; (3) if A is Fredholm of index zero, then s1 (A) = · · · = sα (A) = 0 and sα+1 (A) is positive, where α = dim ker A. Proof. (1) Since ImF B ⊂ ImF then codim F B ≥ codim F , thus sα (AB) ≤ inf {AB − F B : codim F ≥ α} ≤ B sα (A). (2) If A is invertible and F is such that codim F ≥ 1, then I − A−1 F can not be less than 1, otherwise F would be invertible. Thus, I − A−1 F ≥ 1 , i.e. A−1 (A − F ) ≥ 1 and so A−1 (A − F ) ≥ 1 for every F with codimension greater than or equal to 1. Therefore 1 . s1 (A) ≥ A−1 (3) Suppose that A is a Fredholm operator of index zero with kernel dimension α. Then codim A = α and from the definition of approximate numbers s1 (A) = · · · = sα (A) = 0. Suppose now that sα+1 (A) = 0. Then there exists a sequence of operators Fn ∈ L(X) such that codim Fn ≥ α + 1 and lim A − Fn = 0.

n→∞

Thus, for n large enough, Fn are Fredholm operators with the same index as A, i.e, for n large enough dim ker Fn = codim Fn ≥ α + 1. But due to the upper semicontinuity of the kernel dimension, there is a neighborhood of A where all operators have kernel dimension less than or equal to that of A, which contradicts sα+1 (A) = 0. 4.3. Splitting property of Fredholm sequences Definition 4.5. We say that (An ) ∈ B has the α-splitting property if the approximation numbers of An satisfy the conditions lim sα (An ) = 0 and

lim inf sα+1 (An ) > 0.

Theorem 4.6. Let (An ) ∈ B be a Fredholm sequence such that ind(An ) = 0 for n large enough. Then (An ) has the α-splitting property, with dim ker Wx (An ) . α= x∈∂Ω∪{0}

Moreover,

Proof. Let β =

x∈∂Ω

ind Wx (An ) = 0.

x∈∂Ω∪{0}

dim coker Wx (An ). We will prove that

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(a) sα (An ) → 0, with α = x∈∂Ω dim ker Wx (An ); (b) there exists c > 0 such that for n large enough sβ+1 (An ) ≥ c; (c) α = β, which means that ind Wx (An ) = 0. (a) If (An ) is Fredholm, then from Propositions (4.1) and (4.2) there exists (Bn ) ∈ B such that m Bn An = I + Vnxi Ti V−nxi + Gn , Ti ∈ K and (Gn ) → 0. (4.4) i=1 α

i Let αi = dim ker Wxi (An ), so that α = α1 + · · · + αm . Let {ei,l }l=1 be a basis of ker Wxi (An ) consisting of norm one vectors. For each (i, l) with 1 ≤ i ≤ m and 1 ≤ l ≤ αi , define the sequence eni,l ∈ Lp R2 by eni,l = Vnxi (ei,l ). For each n ∈ N, define the vector space V n = span en1,1 , . . . , enm,αm

and for each (i, l) with 1 ≤ i ≤ m and 1 ≤ l ≤ αi , we define the functional n fi,l :Vn →C

given by

n fi,l

αi m

xj,v enj,v

= xi,l .

j=1 v=1

Due to Lemma 6.1 in [19], there exists n0 ∈ N and C > 0 such that, for n ≥ n0 , m αi

|xi,l | ≤ C

i=1 v=1

m αi

xi,l eni,l .

i=1 l=1

n Thus, for n ≥ n0 , we have by the Hahn-Banach Theorem that fi,l can be extended 2 p to a linear continuous functional to the whole space L R in such a way that n fi,l ≤ C. The operator Sn : Lp R2 → Lp R2 given by

Sn (x) =

αi m

n fi,l (x)eni,l ,

i=1 l=1

is a projection on the space V n of dimension α, for n ≥ n0 . We claim that An Sn → 0 when n → +∞. Let x ∈ Lp R2 . Then αi αi m m n n n An Sn xLp (R2 ) = An fi,l (x)ei,l ≤ |fi,l (x)| An eni,l i=1 l=1

≤ CxLp (R2 )

αi m

i=1 l=1

An eni,l

i=1 l=1

≤ CxLp (R2 )

αi m i=1 l=1

V−nxi An Vnxi ei,l .

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Due to s-limV−nxi An Vnxi = Wxi (An ) and ei,l ∈ ker Wxi (An ), we then obtain that V−nxi An Vnxi ei,l → 0 for all 1 ≤ i ≤ m and 1 ≤ l ≤ αi proving the claim. Since ind An = 0 for n large enough, by assumption, and codim(I − Sn ) = α then codim An (I − Sn ) ≥ α. From the definition, sα (An ) is the distance of An to the set of all operators whose codimension is greater than or equal to α. Thus, for n large enough, sα (An ) ≤ An − An (I − Sn )

(4.5)

= An Sn → 0. (b) If (An ) is Fredholm then it is not difficult to check that (Cn ) ∈ B and (Gn ) ∈ G can be chosen such that An Cn = I −

l i=1

Vnxi I − PIm Wxi (An ) V−nxi + Gn .

Notice that defining βi = codim Wxi (An ), we have that

dim Im Vnxi I − PIm Wxi (An ) V−nxi = codim PIm Wxi (An ) = βi and dim Im

l i=1

l

Vnxi I − PIm Wxi (An ) V−nxi ≤ βi = β. i=1

Since (I + Gn ) is stable there exists d > 0 such that (I + Gn )−1 ≤

1 d

for n large enough, i.e by Proposition 4.4, we have d ≤ (I + Gn )−1 −1 ≤ s1 (I + Gn ) = inf {I + Gn − F : codim F ≥ 1} l

≤ inf I + Gn − Vnxi I − PIm Wxi (An ) V−nxi − F : codim F ≥ β + 1 i=1

≤ inf {An Cn − F : codim F ≥ β + 1} = sβ+1 (An Cn ) ≤ Cn sβ+1 (An ) ≤ Csβ+1 (An ), with C = sup Cn . (c) To simplify the notation define α(An ) :=

m i=1

dim ker Wxi (An )

and

β(An ) :=

m i=1

dim coker Wxi (An ) .

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From equation (4.4), we have, for i = 1, . . . , m, that Wxi (Bn )Wxi (An ) = I + Ti and therefore ind (Wxi (Bn ) Wxi (An )) = 0. Thus, 0= =

m i=1 m

ind (Wxi (Bn ) Wxi (An ))

(4.6)

(ind (Wxi (Bn )) + ind (Wxi (An )))

i=1

= α(Bn ) − β(Bn ) + α(An ) − β(An ) Since An is Fredholm of index zero, for n large enough, we deduce from equation (4.4) that Bn has also index zero for n large enough. Thus from (a) and (b) it follows that and β(Bn ) ≥ α(Bn ). β(An ) ≥ α(An ) But, this together with equation (4.6) implies that α(An ) = β(An ) and α(Bn ) = β(Bn ). 4.4. Description of Fredholm sequences in E In this section we characterize the Fredholm sequences (An ) ∈ E, and prove that the main theorem 4.6 can be applied provided Wx (An ) is Fredholm for every x ∈ ∂Ω ∪ {0}. Lemma 4.7. Let A0 be the algebra defined in (1.3) and M be the smallest closed ideal of A0 generated by all operators C(a), with a = F (u) and u ∈ L1 (R2 ). The algebra A0 decomposes as ˙ A0 = alg{C (a) , f I : f ∈ C(R2 )} = f I +M Proof. Clearly the second algebra is dense in A0 , thus it remains to check that it is closed. We claim that f I ≤ f I + M for every M ∈ M and f ∈ C(R2 ). Given > 0, there exists a bounded set U ⊂ R2 such that f IL(Lp) − ≤ f χU IL(Lp ) = f χU I + KL(Lp )/K .

(4.7)

The last equality is a standard result valid for every bounded function on a compact set. Since C(a)χU I, with a = F (u) and u ∈ L1 (R2 ) is a compact operator for every bounded set U , ([18], Theorem 3.2.2), it follows that M χU I is a compact operator for every M ∈ M. Thus, f χU I + KL(Lp )/K ≤ (f + M )χU IL(Lp ) ≤ f + M L(Lp) .

(4.8)

According to inequalities (4.7) and (4.8), and assuming that is arbitrary, we get the claim. Now, if (fn I + Mn ) is a Cauchy sequence, then (fn I) is also a Cauchy and therefore (fn I) converges to some f ∈ C(R2 ). We, then also deduce that (Mn ) is Cauchy and, due to the closeness of M, it converges to some M ∈ M. Thus lim (fn I + Mn ) = f I + M . The next proposition shows, in conjugation with Theorem 4.6, that every Fredholm sequence from E has the splitting property.

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Proposition 4.8. If (An ) ∈ E is a Fredholm sequence then, for n large enough, the operators An are Fredholm with index zero. Proof. Observe that if (An ) is a Fredholm sequence then for n large enough the operators An are Fredholm. Let us first prove the statement for the non-closed (but dense) subalgebra E0 of E generated by the same generators of E. Due to lemma 4.7 a sequence in E0 is the product of sequences of the type An = χnΩ (f + M )χnΩ I + λ(1 − χnΩ )I + Gn +

m

Vnxi Ti V−nxi

(4.9)

i=1

with M ∈ M, (Gn ) ∈ G and Ti ∈ K. Suppose (An ) is a Fredholm sequence, then (Bn ) = (χnΩ (f + M )χnΩ I + λ(1 − χnΩ )I) is also a Fredholm sequence and therefore for n large enough Bn is a Fredholm operator. Once χnΩ M χnΩ I is a compact operator (see [18], Theorem 3.2.2) , it is clear that χnΩ f χnΩ I + λ(1 − χnΩ )I is a Fredholm operator too for n large enough. Moreover, since it is a multiplication by a piecewise continuous function, it is also invertible. We can look to An as the sum mof χnΩ f χnΩ I + λ(1 − χnΩ )I + Gn with the compact operators χnΩ M χnΩ I + i=1 Vnxi Ti V−nxi . Since (Gn ) ∈ G, there is n0 such that 1 1 Gn < max , , n > n0 . λ inf x∈K0 |f (x)| Thus, χnΩ f I + λ(1 − χnΩ )I + Gn is invertible for every n > n0 , because Gn
1 and p = 2. In the present paper we are interested in the factorization of operators from B ∗ into B with a finite number of negative squares. We have the following result. Theorem 3.4. Let A ∈ L(B ∗ , B) be self-adjoint and with κ negative squares. Then there is a Pontryagin space P of negative index κ and an operator T ∈ L(P, B) such that the factorization (3.5) A = T T [∗] holds and is minimal in the sense that ker T = {0}. Proof. We endow the range of the operator A with the form [Ab∗ , Ac∗ ] = b∗ , Ac∗ B . Then, since A is self-adjoint, the form [·, ·] is well-defined and Hermitian. Furthermore, for every choice of b∗1 , . . . , b∗n ∈ B ∗ the Hermitian matrix

n n

[Ab∗i , Ab∗j ] i,j=1 = b∗i , Ab∗j B i,j=1 has at most κ negative eigenvalues. In particular, if n ∈ N and b∗1 , . . . , b∗n ∈ B ∗ are such that the Hermitian man trix (b∗i , Ab∗j B )i,j=1 has exactly κ negative eigenvalues with the corresponding orthogonal eigenvectors v1 , . . . , vκ , then we set   Ab∗1   F = v∗  ...  , 1 ≤ ≤ κ, Ab∗n

κ and observe that the diagonal matrix − [Fi , Fj ] i,j=1 is positive definite. Thus M = span{F : 1 ≤ ≤ κ} is a κ-dimensional Hilbert space with respect to the inner product −[·, ·]. It follows that ran A[−]M is a pre-Hilbert space with respect to the inner product [·, ·]. We denote by M[⊥] the completion of ran A[−]M and by P the Pontryagin space P = M[⊥] [+]M. Then we consider the operator Q : B ∗ −→ P defined by Qb∗ = Ab∗ ,

b∗ ∈ B ∗ .

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Note that ran Q = ran A is dense in P. In view of Proposition 2.1, Q is continuous, ∗ since for every sequence (b∗n )∞ n=0 in B converging to a limit b∗ we have lim [Qb∗n , Qb∗n ]P = lim [Ab∗n , Ab∗n ]P = lim b∗n , Ab∗n B

n→∞

n→∞

n→∞

= b∗ , Ab∗ B = [Ab∗ , Ab∗ ]P = [Qb∗ , Qb∗ ]P and lim [Qb∗n , Ac∗ ]P = lim [Ab∗n , Ac∗ ]P = lim b∗n , Ac∗ B

n→∞

n→∞

n→∞

= b∗ , Ac∗ B = [Ab∗ , Ac∗ ]P = [Qb∗ , Ac∗ ]P

∀c∗ ∈ B ∗ .

Now the adjoint operator Q[∗] : P −→ B ∗∗ has the following properties: ker Q[∗] = {0} and, since A is self-adjoint, Q[∗] (Ab∗ ), c∗ B∗ = [Ab∗ , Qc∗ ]P = b∗ , Ac∗ B =

c∗ , Ab∗ ∗B

(3.6)

= τ (Ab∗ ), c∗

B∗

∗

∀b∗ , c∗ ∈ B ,

(3.7)

∗∗

where τ is the natural isometric injection of B into B . It follows that ran Q[∗] ⊆ ran τ, because Q[∗] is continuous, ran A is dense in P and ran τ is closed in B ∗∗ . Therefore, there is T ∈ L(P, B) such that Q[∗] = τ T. But then ker T = {0} and T [∗] = Q; moreover, in view of (3.6), T (Ab∗ ) = Ab∗ ,

b∗ ∈ B ∗ ,

which implies (3.5).

4. Reproducing kernel Pontryagin spaces of functions with values in a Banach space Our first goal is to obtain an analogue of Theorem 2.5 in the setting of Banach space valued functions. We start with the definitions. Definition 4.1. An L(B ∗ , B)-valued kernel K(z, w) defined for z, w in a set Ω is called Hermitian if b∗ , K(z, w)c∗ B = (c∗ , K(w, z)b∗ B )∗ ,

∀z, w ∈ Ω, b∗ , c∗ ∈ B ∗ .

In this case the kernel K(z, w) is said to have κ negative squares in Ω if for any positive integer n, any set of n points w1 , w2 , . . . , wn ∈ Ω and any collection of n n vectors b∗1 , . . . , b∗n ∈ B ∗ the Hermitian matrix (b∗i , K(wi , wj )b∗j B )i,j=1 has at most κ negative eigenvalues, and at least one such matrix has exactly κ negative eigenvalues. The kernel is said to be positive if κ = 0. A similar definition can be given for L(B, B ∗)-valued kernels. We will need it for κ = 0 and only in the last section; see Definition 7.1.

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Definition 4.2. Let B be a Banach space and let K be a Krein space of B-valued functions defined in a set Ω. An L(B ∗ , B)-valued kernel K(z, w) defined for z, w in a set Ω is said to be a reproducing kernel of the Krein space K if for every b∗ ∈ B ∗ , w ∈ Ω and f ∈ K we have: 1. the function z → K(z, w)b∗ , z ∈ Ω, belongs to K; 2. the reproducing kernel property [K(·, w)b∗ , f ]K = b∗ , f (w)B holds. We note that, just as in the case of functions with values in a Krein space (see Remark 2.4), a reproducing kernel of a Krein space of Banach-valued functions is unique and Hermitian. Theorem 4.3. Let K(z, w) be a L(B ∗ , B)-valued Hermitian kernel with κ negative squares in a set Ω. Then there is a unique Krein space P(K) of B-valued functions defined in Ω with the reproducing kernel K(z, w). Moreover, the space P(K) is a Pontryagin space of negative index κ. Proof. The proof goes through exactly as in the case of Hilbert-valued functions. First let us show the existence of the Pontryagin space P(K). We endow the space V = span{Kw b∗ : w ∈ Ω, b∗ ∈ B ∗ } with the linear form [·, ·] defined by [Kz b∗ , Kw c∗ ] = b∗ , K(z, w)c∗ B . ∗

Since the L(B , B)-valued function K(z, w) is Hermitian, the form [·, ·] is welldefined and also Hermitian. Furthermore, for every choice of f1 , . . . , fm ∈ V there are n ∈ N, w1 , . . . , wn ∈ Ω, b∗1 , . . . , b∗n ∈ B ∗ and v1 , . . . , vm ∈ Cn such that   Kw1 b∗1   .. f = v∗   , 1 ≤ ≤ m. . Kwn b∗n Therefore, the Hermitian matrix

∗

n

m

b∗i , K(wi , wj )b∗j B i,j=1 v1 [fi , fj ] i,j=1 = v1 · · · vm

···

vm

has at most κ negative eigenvalues. In particular, if n ∈ N, w1 , . . . , wn ∈ Ω and b∗1 , . . . , b∗n ∈ B ∗ are such that n the Hermitian matrix (b∗i , K(wi , wj )b∗j B )i,j=1 has exactly κ negative eigenvalues with the corresponding orthogonal eigenvectors v1 , . . . , vκ then we set   Kw1 b∗1   .. F = v∗   , 1 ≤ ≤ κ, . Kwn b∗n


459

κ

and observe that the diagonal matrix − [Fi , Fj ] i,j=1 is positive definite. Thus M = span{F : 1 ≤ ≤ κ} is a κ-dimensional Hilbert space with respect to the inner product −[·, ·]. It follows that V[−]M is a pre-Hilbert space with respect to the inner product [·, ·]. It remains to denote by M[⊥] the completion of V[−]M and to set P(K) = M[⊥] [+]M. In order to show the uniqueness of P(K), let us assume that K is a Krein space of B-valued functions with the reproducing kernel K(z, w). Then V is a dense linear subspace of K and the restriction of the indefinite inner product [·, ·]K to V coincides with the form [·, ·] defined above. It follows that K is a Pontryagin space of negative index κ with a fundamental decomposition K = (K[−]M)[+]M. Finally, since the positive subspace V[−]M is dense in the Hilbert space K[−]M, the latter must coincide with M[⊥] defined above, which means that K = P(K). In the sequel we consider analytic functions with values in a Banach space. Recall that for such functions, weak analyticity and strong analyticity are equivalent; see [25, pp. 189-190]. We now introduce the class of functions to be studied in this paper. Let B1 and B2 be two Banach spaces and let J1 ∈ L(B1∗ , B1 ) and J2 ∈ ∗ L(B2 , B2 ) be self-adjoint operators with the same number ν− (J1 ) = ν− (J2 ) = ν < ∞ of negative squares. Given an L(B1 , B2 )-valued function S(z) analytic in some open subset Ω one may consider the kernel J2 − S(z)J1 S(w)∗ . (4.1) 1 − zw∗ The kernel (4.1) is the Banach space version of the kernel introduced in Definition 2.7 (see also Remark 2.8). However, in view of Theorem 3.4, there exists a Pontryagin space P1 with the negative index ν and an 1-to-1 operator T1 ∈ L(P1 , B1 ) [∗] such that J1 = T1 T1 . Then (4.1) takes form J2 − (S(z)T1 )(S(w)T1 )[∗] 1 − zw∗ and it is clear that the kernel (4.1) is determined by the function S(z)T1 rather than by S(z) itself. Thus we propose the following definition. Definition 4.4. Let B be a Banach space and let an operator J ∈ L(B ∗ , B) be self-adjoint with a finite number of negative squares. Let P be a Pontryagin space of negative index equal to ν− (J). We denote by Sκ (P, B, J) the class of L(P, B)valued functions S such that:

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1. S(z) is analytic in some open subset Ω of D, and 2. the kernel J − S(z)S(w)[∗] KS (z, w) = 1 − zw∗ has κ negative squares in Ω.

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(4.2)

In the sequel we shall refer to a function S ∈ Sκ (P, B, J) as a generalized Schur function and to the Pontryagin space P(S) of B-valued functions with the reproducing kernel KS (z, w) given by (4.2) as the de Branges-Rovnyak space associated with the generalized Schur function S(z). Remark 4.5. An equivalent form of the kernel (4.2) would be ∗ S (z, w) = J − S(z)J0 S(w) , K ∗ 1 − zw

where J0 is a fundamental symmetry of the Pontryagin space P and the adjoint is taken with respect to the corresponding positive inner product; see Remark 2.8.

5. A representation theorem for generalized Schur functions Throughout this section we assume that a Banach space B, a self-adjoint operator J ∈ L(B ∗ , B) with a finite number of negative squares and a Pontryagin space P of negative index equal to ν− (J) are given. Furthermore, we choose and fix a minimal factorization J = T T [∗] via a Pontryagin space PJ of negative index ν− (J) (which exists by Theorem 3.4). We are going to relate the Schur classes Sκ (P, B, J) introduced in Section 4 (see Definition 4.4) to the previously studied classes Sκ (P1 , P2 ) mentioned in Section 2 (see Definition 2.7). Theorem 5.1. Let S ∈ Sκ (P, B, J) and suppose that the Schur function S(z) is analytic in a connected open set Ω ⊆ D, where the kernel KS (z, w) has κ negative squares. Then there exists a generalized Schur function σ ∈ Sκ (P, PJ ), analytic in Ω and such that S(z) = T σ(z), z ∈ Ω. (5.1) Proof. The proof employs the ideas of [3]; in particular, we shall use the notion of linear relations. Recall that, given two vector spaces a linear relation is by definition a linear subspace of their Cartesian product. For instance the graph of an operator is a linear relation, but not every linear relation is of this form. We shall proceed in several steps. Step 1. Without loss of generality, 0 ∈ Ω. Choose a point z0 ∈ Ω and consider the M¨ obius transformation z − z0 . µ(z) = 1 − zz0∗


461

Then the connected open set µ(Ω) contains the origin and, since KS◦µ−1 (z, w) = (1 − |z0 |2 )

KS (µ−1 (z), µ−1 (w)) , (1 + zz0∗)(1 + w∗ z0 )

z, w ∈ µ(Ω),

the kernel KS◦µ−1 (z, w) has κ negative squares in µ(Ω). Similarly, if σ ∈ Sκ (P, PJ ) then σ ◦ µ ∈ Sκ (P, PJ ) because Kσ◦µ = (1 − |z0 |2 )

Kσ (µ(z), µ(w)) . (1 − zz0∗)(1 − w∗ z0 )

Thus we can assume, without loss of generality, that the set Ω contains the origin. Step 2. Construction of an isometry R from P(S) ⊕ PJ into P(S) ⊕ P. Consider the linear relation R in (P(S) ⊕ PJ , P(S) ⊕ P) spanned by    KS (·, w) − KS (·, 0)       b + K (·, 0)c ∗ S ∗     KS (·, w)b∗  w∗   ,     S(w)[∗] − S(0)[∗]    T [∗] c∗   [∗]   b + S(0) c ∗ ∗ ∗ w as w runs through Ω \ {0}, and b∗ and c∗ through B ∗ . Note that dom R is dense in P(S) ⊕ PJ . Indeed, ran T [∗] is dense in PJ since ker T = {0}. Furthermore, since KS (z, w) is analytic with respect to z and w∗ at (0, 0), for every b∗ , c∗ ∈ B ∗ and w0 ∈ Ω we have lim [KS (·, w)b∗ , KS (·, w)b∗ ]P(S) = lim b∗ , KS (w, w)b∗ B

w→0

w→0

= b∗ , KS (0, 0)b∗ B = [KS (·, 0)b∗ , KS (·, 0)b∗ ]P(S) and lim [KS (·, w)b∗ , KS (·, w0 )c∗ ]P(S) = lim b∗ , KS (w, w0 )c∗ B

w→0

w→0

= b∗ , KS (0, w0 )c∗ B = [KS (·, 0)b∗ , KS (·, w0 )c∗ ]P(S) . Hence, in view of Proposition 2.1, for every b∗ ∈ B ∗ the mapping w → KS (·, w)b∗ from Ω to P(S) is continuous at w = 0. But then for every f ∈ P(S) and b∗ ∈ B ∗ the mapping w → b∗ , f (w)B is also continuous at w = 0 and it follows that span{KS (·, w)b∗ : w ∈ Ω \ {0}, b∗ ∈ B ∗ } is dense in P(S). Our next claim is that the relation R is isometric. Indeed, for every b∗ , c∗ ∈ B ∗ and w ∈ Ω \ {0} we have S(w)[∗] − S(0)[∗] KS (·, w) − KS (·, 0) [∗] b , K (·, 0)c + b , S(0) c ∗ S ∗ ∗ ∗ w∗ w∗ P(S) P =

1 1 b∗ , (KS (w, 0) − KS (0, 0))c∗ B + ∗ b∗ , (S(w) − S(0))S(0)[∗] c∗ B = 0. w∗ w

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Furthermore, for every b∗1 , b∗2 ∈ B ∗ and every w1 , w2 ∈ Ω \ {0} we have KS (·, w1 ) − KS (·, 0) KS (·, w2 ) − KS (·, 0) b , b ∗1 ∗2 w1∗ w2∗ P(S) S(w1 )[∗] − S(0)[∗] S(w2 )[∗] − S(0)[∗] + b∗1 , b∗2 w1∗ w2∗ P 1 = ∗ b∗1 , (KS (w1 , w2 ) − KS (0, w2 ) − KS (w1 , 0) + KS (0, 0))b∗2 B w1 w2 1 + ∗ b∗1 , (S(w1 ) − S(0))(S(w2 )[∗] − S(0)[∗] )b∗2 B w1 w2 = b∗1 , KS (w1 , w2 )b∗2 B = [KS (·, w1 )b∗1 , KS (·, w2 )b∗2 ]P(S) . Finally, for every c∗1 , c∗2 ∈ B ∗ and every w1 , w2 ∈ Ω \ {0} we have [KS (·, 0)c∗1 , KS (·, 0)c∗2 ]P(S) + [S(0)[∗] c∗1 , S(0)[∗] c∗2 ]P = c∗1 , (KS (0, 0) + S(0)S(0)[∗] )c∗2 B = c∗1 , Jc∗2 B = [T [∗] c∗1 , T [∗] c∗2 ]PJ . Now we can take advantage of a theorem of Shmulyan (see [4, Theorem 1.4.1, p. 27]) to conclude that, since P(S) ⊕ PJ and P(S) ⊕ P are Pontryagin spaces with the same negative index, the linear relation R extends as the graph of an isometry R : P(S) ⊕ PJ −→ P(S) ⊕ P. Step 3. Properties of R . Since the everywhere defined operator R, constructed at the previous step, is isometric, it is also continuous. Let us consider the adjoint (coisometric) operator: A B [∗] . R = C D We have the following identities: [∗]

h(z) − h(0) , z T Ch = h(0),

(Ah)(z) =

S(z) − S(0) u, z T Du = S(0)u,

(Bu)(z) =

(5.2) (5.3)

where u ∈ P, h ∈ P(S) and z ∈ Ω \ {0}. Indeed, for every b∗ ∈ B ∗ , w ∈ Ω \ {0}, h ∈ P(S) and u ∈ P we have b∗ , (Ah)(w)B = [A[∗] (KS (·, w)b∗ ), h]P(S) 1 = ∗ [(KS (·, w) − KS (·, 0))b∗ , h]P(S) = w b∗ , (Bu)(w)B = [B [∗] (KS (·, w)b∗ ), u]P 1 = ∗ [(S(w)[∗] − S(0)[∗] )b∗ , u]P = w

h(w) − h(0) b∗ , ; w B

S(w) − S(0) u b∗ , ; w B

b∗ , T ChB = [KS (·, 0)b∗ , h]P(S) = b∗ , h(0)B ;


463

b∗ , T DuB = [S(0)[∗] b∗ , u]P = b∗ , S(0)uB . We also note that T C(IP(S) − zA)−1 h = h(z)

(5.4)

for every h ∈ P(S) and z in a neighborhood of 0 (where (IP(S) −zA)−1 is analytic). Indeed, if z = 0 this follows immediately from (5.3). If z = 0, we denote f = (IP(S) − zA)−1 h. Then, in view of (5.2), z (f (w) − f (0)), w ∈ Ω \ {0}. w Setting w = z in this last identity and taking into account (5.3), we obtain h(w) = f (w) −

h(z) = f (0) = T Cf = T C(IP(S) − zA)−1 h. Step 4. Construction of σ(z). Let us define now −1

B. σ(z) = D + zC IP(S) − zA Then σ(z) is analytic in a neighborhood of 0 and, since C(IP(S) − zA)−1 (IP(S) − wA)−[∗] C [∗] + σ(z)σ(w)[∗]

= zC(IP(S) − zA)−1 IPJ R[∗] R wC(IP(S) − wA)−1

IPJ

[∗]

= zw∗ C(IP(S) − zA)−1 (IP(S) − wA)−[∗] C [∗] + IPJ , we have Kσ (z, w) = C(IP(S) − zA)−1 (IP(S) − wA)−[∗] C [∗] . Hence for any u, v ∈ PJ [Kσ (z, w)u, v]PJ = [(IP(S) − zA)−[∗] C [∗] u, (IP(S) − wA)−[∗] C [∗] v]P(S) and, since the negative index of P(S) is equal to κ, Kσ (z, w) has at most κ negative squares. Thus σ ∈ Sκ (P, PJ ) with some κ ≤ κ. In particular, σ(z) extends to the whole of D as a meromorphic function. Next we observe that, as follows from (5.2), (5.3) and (5.4), for every u ∈ P and z in a neighborhood of 0 we have −1

Bu = S(0)u + (S(z) − S(0))u = S(z)u. T σ(z)u = T Du + zT C IP(S) − zA Hence S(z) = T σ(z) in a neighborhood of 0 and, by analytic extension, at all points in Ω where σ(z) is analytic (which, in view of the injectivity of T , is the whole of Ω). Thus we obtain (5.1). But then KS (z, w) =

T T [∗] − (T σ(z))(T σ(w))[∗] = T Kσ (z, w)T [∗] , 1 − zw∗

z, w ∈ Ω.

(5.5)

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Since the kernel KS (z, w) has κ negative squares in Ω, there exist b∗1 , . . . , b∗n ∈ B ∗ and z1 , . . . , zn ∈ Ω such that the Hermitian matrix n

Xn = (b∗i , KS (zi , zj )b∗j B )i,j=1 has κ negative eigenvalues. Since, in view of (5.5), n Xn = [T [∗] b∗i , Kσ (wi , wj )T [∗] b∗j ]PJ

i,j=1

,

we conclude that Kσ (z, w) has (at least and, therefore, precisely) κ negative squares in Ω. Thus σ ∈ Sκ (P, PJ ). Remark 5.2. Theorem 5.1 implies that a generalized Schur function S ∈ Sκ (P, B, J) can be extended to the whole of D as a function of bounded type via the relation (5.1). Note that, since the operator T is injective, the domain of analyticity of S(z) in D coincides with that of σ(z). Theorem 5.3. Let σ ∈ Sκ (P, PJ ) and let S(z) be an L(P, B)-valued function of bounded type, defined by S(z) = T σ(z). (5.6) Let Ω denote the domain of analyticity of S(z) (and of σ(z)) in D. Then: 1. S ∈ Sκ (P, B, J) and, moreover, the kernel KS (z, w) has κ negative squares in any non-empty subregion of Ω; 2. The formula (MT h)(z) = T (h(z)),

h ∈ P(σ), z ∈ Ω,

(5.7)

defines a unitary operator MT from P(σ) onto P(S). In particular, the elements of P(S) are analytic in Ω. Proof. Since KS (z, w) = T Kσ (z, w)T [∗] , z, w ∈ Ω, (5.8) the kernel KS (z, w) has at most κ negative squares in Ω. In particular, if Ω is an open connected subset of Ω, then KS (z, w) has κ ≤ κ negative squares in Ω . But then, in view of Theorem 5.1 and the fact that T is injective, σ ∈ Sκ (P, PJ ). Hence κ = κ (see (2.7)), which means that in every non-empty subregion of Ω the kernel KS (z, w) has (at least and, therefore, precisely) κ negative squares. This proves Statement (1). In order to prove Statement (2), consider the linear relation M = span{(kσ (·, w)T [∗] b∗ , KS (·, w)b∗ ) : w ∈ Ω, b∗ ∈ B ∗ } in (P(σ), P(S)). Clearly, ran M is dense in P(S). Moreover, since ran T [∗] is dense in PJ , dom M is dense in P(σ), as well. Furthermore, for every b∗ , c∗ ∈ B ∗ and z, w ∈ Ω we have [KS (·, z)b∗ , KS (·, w)c∗ ]P(S) = b∗ , KS (z, w)c∗ B = b∗ , T kσ (z, w)T [∗] c∗ B = [T [∗] b∗ , kσ (z, w)T [∗] c∗ ]PJ = [kσ (·, z)T [∗] b∗ , kσ (·, w)T [∗] c∗ ]P(σ) .


465

Thus the relation M is isometric. It extends as the graph of an isometry M which maps P(σ) onto P(S) (hence M is unitary). In view of (5.8), for h ∈ dom M we have (M h)(z) = T (h(z)),

z ∈ Ω.

Moreover, since dom M is dense in P(σ), the above formula is valid for every h ∈ P(σ). Indeed, if (hn )∞ n=0 is a sequence in dom M converging to h ∈ P(σ) then for every u ∈ PJ and z ∈ Ω(S) lim [u, hn (z)]PJ = lim [kσ (·, z)u, hn ]P(σ) = [kσ (·, z)u, h]P(σ) = [u, h(z)]PJ .

n→∞

n→∞

In particular, for every b∗ ∈ B ∗ b∗ , T (h(z))B = [T [∗] b∗ , h(z)]PJ = lim [T [∗] b∗ , hn (z)]PJ = lim b∗ , T (hn (z))B . n→∞

n→∞

On the other hand, by continuity of M , lim b∗ , T (hn (z))B = lim b∗ , (M hn )(z)B = lim KS (·, z)b∗ , M hn P(S)

n→∞

n→∞

n→∞

= KS (·, z)b∗ , M hP(S) = b∗ , (M h)(z)B . Thus b∗ , (M h)(z)B = b∗ , T (h(z))B

∀b∗ ∈ B ∗ , h ∈ P(σ), z ∈ Ω(S)

and M = MT as defined in (5.7). It follows that a function f (z) belongs to the space P(S) if and only if it has the form f (z) = T (h(z)), z ∈ Ω(S), for some h ∈ Ω(σ). Since the elements of P(σ) are analytic in Ω (see Remark 2.6), so are the elements of P(S).

6. A Beurling-Lax type theorem At Step 3 of the proof of Theorem 5.1 we have constructed an operator A of a special form; see (5.2). This operator, acting on functions analytic at the origin, is usually referred to as the backward-shift operator and denoted by R0 : f (z) − f (0) . (6.1) (R0 f )(z) = z The objective of this section is to characterize the de Branges-Rovnyak spaces of functions, analytic at the origin and with values in a Banach space, in terms of R0 -invariance. The corresponding result in the case of Pontryagin space-valued functions is well-known; see for example [4, Theorems 2.2.1, 3.1.2]. In order to generalize this result to the Banach space setting, we shall make use of Theorems 5.1 and 5.3 which were proved in the previous section.

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In what follows, we assume that a Banach space B and a self-adjoint operator J ∈ L(B ∗ , B) with a finite number of negative squares are given. As in Section 5, we choose and fix a minimal factorization J = T T [∗] via a Pontryagin space PJ of negative index ν− (J). Proposition 6.1. Let P be a Pontryagin space with the negative index ν− (J) and let S ∈ Sκ (P, B, J) be analytic at the origin. Then the space P(S) is R0 -invariant and for every f ∈ P(S) the inequality [R0 f, R0 f ]P(S) ≤ [f, f ]P(S) − [h0 , h0 ]PJ ,

(6.2)

where h0 is the uniquely determined element of PJ such that f (0) = T h0 , holds true. Proof. First, in view of Theorems 5.1 and 5.3, every element f ∈ P(S) is analytic at 0 and takes values in ran T . Thus R0 f and h0 in (6.2) are well defined. Second, if σ ∈ Sκ (P, PJ ) is as in Theorem 5.1 then (see for instance [4, Theorem 2.2.1 p. 49]) the space P(σ) is R0 -invariant and the inequality [R0 h, R0 h]P(σ) ≤ [h, h]P(σ) − [h(0), h(0)]PJ

∀h ∈ P(σ)

(6.3)

holds true. Third, if MT ∈ L(P(σ), P(S)) is the unitary operator mentioned in Theorem 5.3 (see (5.7)) then R0 MT h = MT R0 h

∀h ∈ P(σ)

and hence the space P(S) is R0 -invariant. Moreover, for every f ∈ P(S) we obtain [∗]

[∗]

[R0 f, R0 f ]P(S) =[R0 MT MT f, R0 MT MT f ]P(S) [∗]

[∗]

[∗]

[∗]

= [MT R0 MT f, MT R0 MT f ]P(S) = [R0 MT f, R0 MT f ]P(σ) [∗]

[∗]

[∗]

[∗]

≤ [MT f, MT f ]P(σ) − [(MT f )(0), (MT f )(0)]PJ = [f, f ]P(S) − [h0 , h0 ]PJ .

A converse result may be formulated as follows: Theorem 6.2. Let B be a Banach space and let P0 be a reproducing kernel Pontryagin space of negative index κ whose elements are B-valued functions analytic in a neighborhood of the origin. Assume that: 1. The space P0 is R0 -invariant. 2. For every f ∈ P0 f (0) ∈ ran T . 3. For every f ∈ P0 the inequality [R0 f, R0 f ]P0 ≤ [f, f ]P0 − [h0 , h0 ]PJ , where h0 is the uniquely determined element of PJ such that f (0) = T h0 , holds true.

(6.4)


467

Then there exist a Pontryagin space P of negative index ν− (J) and a function S ∈ Sκ (P, B, J) such that P0 = P(S). Proof. First, let us denote by K(z, w) the L(B ∗ , B)-valued reproducing kernel of the Pontryagin space P0 and by Ω a neighborhood of the origin, where the elements of P0 are analytic. We shall proceed in several steps. Step 1. Continuity of R0 . According to the assumptions of the theorem, the operator R0 is defined everywhere in P0 . We shall show that R0 is, moreover, continuous in P0 . Indeed, let (fn )∞ n=0 be a sequence in P0 which converges to an element f ∈ P0 . Assume that the sequence R0 fn also converges in P0 , to an element g, say. Then for every b∗ ∈ B ∗ and w ∈ Ω \ {0} b∗ , g(w)B = [K(·, w)b∗ , g]P0 = lim [K(·, w)b∗ , R0 fn ]P0 = lim b∗ , (R0 fn )(w)B n→∞

n→∞

1 1 = ∗ lim b∗ , fn (w) − fn (0)B = ∗ lim [(K(·, w) − K(·, 0))b∗ , fn ]P0 w n→∞ w n→∞ 1 1 = ∗ [(K(·, w) − K(·, 0))b∗ , f ]P0 = ∗ b∗ , fn (w) − fn (0)B w w = b∗ , (R0 f )(w)B . Thus the functions g and R0 f coincide in Ω \ {0} and, since both are analytic at the origin, in Ω : g = R0 f. According to the closed graph theorem, this implies that R0 ∈ L(P0 ). In particular, for z in a circular neighborhood of 0 (which, without loss of generality, coincides with Ω) the operator IP0 − zR0 is invertible and the inverse depends analytically on z. Step 2. A reproducing kernel Pontryagin space of PJ -valued functions. We define an operator C : P0 −→ B by Cf = f (0),

f ∈ P0 .

Using the closed graph theorem and the reproducing kernel structure of P0 in the same way as at the previous step of the proof, we observe that C ∈ L(P0 , B). Let us choose an arbitrary f ∈ P0 . Then, exactly as at Step 3 of the proof of Theorem 5.1 (see (5.4)), C(IP(S) − zR0 )−1 f = f (z),

z ∈ Ω.

(6.5)

Therefore, K(z, w) = C(IP0 − zR0 )−1 (IP0 − w∗ R0 )−1 C [∗] . But, since ran C ⊆ ran T and ker T = {0}, there is C1 ∈ L(P0 , PJ ) such that [∗]

C = T C1 (the continuity of the a priori everywhere defined operator C1 follows from the closed graph theorem and the injectivity of T ).

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Let us define K1 (z, w) = C1 (IP0 − zR0 )−1 (IP0 − w∗ R0 )−1 C1 , [∗]

[∗]

z, w ∈ Ω.

Then T K1(z, w)T [∗] = K(z, w) and, using the same reasoning as at Step 4 of the proof of Theorem 5.1, the kernels K(z, w) and K1 (z, w) have the same number κ of negative squares in Ω. We shall denote by P1 the associated reproducing kernel Pontryagin space of PJ -valued functions, which are, in view of Remark 2.6, analytic in Ω. Step 3. Conclusion. As in the proof of Statement (2) of Theorem 5.3, the formula (MT h)(z) = T (h(z)),

z ∈ Ω,

defines a unitary operator MT : P1 −→ P0 . But R0 MT = MT R0 and, since the space P0 is R0 -invariant, so is P1 . Note now that (6.4) implies [R0 h, R0 h]P1 ≤ [h, h]P1 − [h(0), h(0)]PJ and hence the space P1 satisfies the requirements of [4, Theorem 3.1.2 pp. 85-86]. Therefore, there is a Pontryagin space P of the same negative index as PJ and a function σ ∈ Sκ (P, PJ ) such that P1 = P(σ). But then the function S(z) = T (σ(z)), is an element of Sκ (P2 , B, J) and, since

z∈Ω

KS (z, w) = T Kσ (z, w)T [∗] = T K1 (z, w)T [∗] = K(z, w), we conclude that P(S) = P0 .

7. The case B1 = B2 and the Cayley transform We now outline the connections with [7]. We first need a definition, which is the counterpart of Definition 4.1 for L(B, B ∗ )-valued kernels. The paper [7] deals only with positive kernels, and the following definition will be used here only when the number of negative squares is equal to 0. Definition 7.1. An L(B, B ∗ )-valued kernel K(z, w) defined for z, w in a set Ω is called Hermitian if K(z, w)b, cB = (K(w, z)c, bB )∗ ,

∀z, w ∈ Ω, b, c ∈ B.

In this case the kernel K(z, w) is said to have κ negative squares in Ω if for any positive integer n, any set of n points w1 , w2 , . . . , wn ∈ Ω and any collection of


469

n vectors b1 , . . . , bn ∈ B the Hermitian matrix (K(wi , wj )bj , bi B )ni,j=1 has at most κ negative eigenvalues, and at least one such matrix has exactly κ negative eigenvalues. The kernel K(z, w) is said to be positive if κ = 0. In particular, an L(B ∗ , B ∗∗ )-valued kernel K(z, w) defined for z, w in a set Ω is Hermitian if K(z, w)b∗ , c∗ B∗ = (K(w, z)c∗ , b∗ B∗ )∗ ,

∀z, w ∈ Ω, b∗ , c∗ ∈ B ∗ ,

and positive in Ω if for any positive integer n, any set of n points w1 , w2 , . . . , wn ∈ Ω and any collection of n vectors b∗1 , . . . , b∗n ∈ B ∗ the Hermitian matrix n (K(wi , wj )b∗j , b∗i B∗ )i,j=1 is non-negative. def.

We now assume that in formula (4.1) we have B1 = B2 = B, and that the def.

kernel (4.1) is positive, and we set J1 = J2 = J . We will assume that IB + S(z) in invertible in an open subset, say Ω, of the open unit disk, and define the L(B, B)valued function Φ(z) = (IB − S(z))(IB + S(z))−1 . It is seen that Φ(z)J + JΦ(z)∗ −1 KS (z, w) = 2(IB + Φ(z)) (7.1) (IB + Φ(w))−∗ , 1 − zw∗ and thus the L(B ∗ , B)-valued kernel Φ(z)J + JΦ(z)∗ 1 − zw∗ is positive (in the sense of Definition 4.1 with κ = 0). K1 (z, w) =

(7.2)

In the following proposition, τ denotes the canonical injection of B into B ∗∗ , whose definition was recalled in (3.1). Proposition 7.2. The L(B ∗ , B ∗∗ )-valued kernel τ (Φ(z)J) + (Φ(z)J)∗ (7.3) 1 − zw∗ is positive in Ω ⊂ D if and only if the L(B ∗ , B)-valued kernel (7.2) is positive there. K2 (z, w) =

Proof. We first note that ∗

τ (Φ(z)Jb∗ , c∗ B∗ = (c∗ , Φ(z)Jb∗ B ) ,

(7.4)

∗

where z ∈ D and b∗ , c∗ ∈ B . Furthermore, since J is self-adjoint, and using (3.1), we have b∗ , Jc∗ B = J ∗ b∗ , c∗ B∗ = (c∗ , Jb∗ B )∗ = τ Jb∗ , c∗ B∗ , and thus J ∗ = τ J. It follows that ∗

(Φ(z)J)∗ b∗ , c∗ B∗ = τ JΦ(z)∗ b∗ , c∗ B∗ = (c∗ , JΦ(z)∗ b∗ B ) .

(7.5)

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Using (7.4) and (7.5) we have K2 (z, w)b∗ , c∗ B∗ = (c∗ , K1 (z, w)b∗ B )∗ ,

∀z ∈ D

and ∀b∗ , c∗ ∈ B ∗ ,

and the result follows.

(7.6)

Therefore, the L(B ∗ , B)-valued function z → Φ(z)J is a Carathéodory function as defined and studied in [7, §6] (see also [8] for a related study). It also follows from the realization formula proved in [7] that Φ has an analytic extension to all of D. That same formula allows to extend Φ to C \ {z ; |z| = 1} in such a way that the kernel (7.3) stays positive there.

References [1] D. Alpay. Some remarks on reproducing kernel Kre˘ın spaces. Rocky Mountain J. Math., 21:1189–1205, 1991. [2] D. Alpay. Algorithme de Schur, espaces a ` noyau reproduisant et th´ eorie des systèmes, volume 6 of Panoramas et Synthèses. Société Mathématique de France, Paris, 1998. [3] D. Alpay, A. Dijksma, and J. Rovnyak. A theorem of Beurling–Lax type for Hilbert spaces of functions analytic in the ball. Integral Equations and Operator Theory, 47:251–274, 2003. [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] D. Alpay and H. Dym. On applications of reproducing kernel spaces to the Schur algorithm and rational J-unitary factorization. In I. Gohberg, editor, I. Schur methods in operator theory and signal processing, volume 18 of Operator Theory: Advances and Applications, 89–159. Birkh¨ auser Verlag, Basel, 1986. [6] D. Alpay and H. Dym. On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains. In Operator theory and complex analysis (Sapporo, 1991), volume 59 of Oper. Theory: Advances and Applications, 30–77. Birkh¨ auser Verlag, Basel, 1992. [7] D. Alpay, O. Timoshenko, and D. Volok. Carathéodory functions in the Banach space setting. Linear Algebra Appl., 425:700–713, 2007. [8] D. Alpay, O. Timoshenko, and D. Volok. Carathéodory-Fejér interpolation and related topics in locally convex spaces. Linear Algebra Appl. In press. [9] N. Aronszajn. Quadratic forms on vector spaces. In Proc. Internat. Sympos. Linear Spaces, 29–87, Jerusalem, 1960. [10] T. Ya. Azizov and I.S. Iohvidov. Foundations of the theory of linear operators in spaces with indefinite metric. Nauka, Moscow, 1986. (Russian). English translation: Linear operators in spaces with an indefinite metric. John Wiley, New York, 1989. [11] J. Bogn´ ar. Indefinite inner product spaces. Springer–Verlag, Berlin, 1974.


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[12] L. de Branges and J. Rovnyak. Canonical models in quantum scattering theory. In C. Wilcox, editor, Perturbation theory and its applications in quantum mechanics, 295–392. Wiley, New York, 1966. [13] L. de Branges and J. Rovnyak. Square summable power series. Holt, Rinehart and Winston, New York, 1966. [14] W.F. Donoghue. Monotone matrix functions and analytic continuation, volume 207 of Die Grundlehren der mathematischen Wissennschaften. Springer–Verlag, 1974. [15] M. Dritschel and J. Rovnyak. Extensions theorems for contractions on Kre˘ın spaces, volume 47 of Operator Theory: Advances and Applications, 221–305. Birkh¨ auser Verlag, Basel, 1990. [16] D. Drivaliaris and N. Yannakakis. Hilbert space structure and positive operators. J. Math. Anal. Appl., 305(2):560–565, 2005. [17] J. G´ orniak and A. Weron. Aronszajn-Kolmogorov type theorems for positive definite kernels in locally convex spaces. Studia Math., 69(3):235–246, 1980/81. [18] I.S. Iohvidov, M.G. Kre˘ın, and H. Langer. Introduction to the spectral theory of operators in spaces with an indefinite metric. Akademie–Verlag, Berlin, 1982. ¨ [19] M.G. Kre˘ın and H. Langer. Uber die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume Πk . In Hilbert space operators and operator algebras (Proc. Int. Conf. Tihany, 1970), 353–399. North– Holland, Amsterdam, 1972. Colloquia Math. Soc. J´ anos Bolyai. [20] M.G. Kre˘ın and H. Langer. Some propositions on analytic matrix functions related to the theory of operators in the space πk . Acta Sci. Math., 43:181–205, 1981. [21] A. Makagon. Remark on the extrapolation of Banach space valued stationary processes. In Probability theory on vector spaces, II (Proc. Second Internat. Conf., Blazejewko, 1979), volume 828 of Lecture Notes in Math., 196–207. Springer, Berlin, 1980. [22] P. Masani. Dilations as propagators of Hilbertian varieties. SIAM J. Math. Anal., 9:414–456, 1978. [23] A.G. Miamee and H. Salehi. Necessary and sufficient conditions for factorability of nonnegative operator-valued functions on Banach space. Proc. Amer. Math. Soc., 46:43–50, 1974. [24] A.G. Miamee and H. Salehi. On the factorization of a nonnegative operator valued function. In Probability theory on vector spaces (Proc. Conf., Trzebieszowice, 1977), volume 656 of Lecture Notes in Math., 129–137. Springer, Berlin, 1978. [25] M. Reed and B. Simon. Methods of modern mathematical physics. I. Functional analysis. Academic Press, New York, 1972. [26] L. Schwartz. Sous espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants). J. Analyse Math., 13:115–256, 1964. [27] P. Sorjonen. Pontryagin Ra¨ ume mit einem reproduzierenden Kern. Ann. Acad. Fenn. Ser. A. I, 1–30, 1973. [28] N. N. Vakhania. Probability distributions on linear spaces. North-Holland Publishing Co., New York, 1981. Translated from the Russian by I. I. Kotlarski, North-Holland Series in Probability and Applied Mathematics.

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Daniel Alpay, Olga Timoshenko, Prasad Vegulla Dept. of Mathematics Ben Gurion University of the Negev P.O. Box 653 Be’er Sheva 84105 Israel e-mail: [email protected] [email protected] [email protected] Dan Volok Dept. of Mathematics Kansas State University Manhattan, Kansas USA e-mail: [email protected] Submitted: March 3, 2009. Revised: June 16, 2009.

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Integr. equ. oper. theory 65 (2009), 473–484 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040473-12, published online August 3, 2009 DOI 10.1007/s00020-009-1708-8


Almost Invariant Half-Spaces of Operators on Banach Spaces George Androulakis, Alexey I. Popov, Adi Tcaciuc and Vladimir G. Troitsky Abstract. We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on an infinite-dimensional Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T (Y ). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on p (1 p < ∞) or c0 . Mathematics Subject Classification (2000). Primary 47A15. Keywords. Half-space, invariant subspace, weighted shift operator.

1. Introduction Throughout the paper, X is a Banach space and by L(X) we denote the set of all (bounded linear) operators on X. By a “subspace” of a Banach space we always mean a “closed subspace”. Given a sequence (xn ) in X, we write [xn ] for the closed linear span of (xn ). Definition 1.1. A subspace Y of a Banach space X is called a half-space if it is both of infinite dimension and of infinite codimension in X. Definition 1.2. If T ∈ L(X) and Y is a subspace of X, then Y is called almost invariant under T , or T -almost invariant, if there exists a finite dimensional subspace F of X such that T (Y ) ⊆ Y + F . In this work, the following question will be referred to as the almost invariant half-space problem: Does every operator on an infinite-dimensional Banach space have an almost invariant half-space? Observe that every subspace of X that is not The third and the fourth authors were supported by NSERC.

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a half-space is clearly almost invariant under any operator. Also, note that the almost invariant half-space problem is not weaker than the well known invariant subspace problem, because in the latter the invariant subspaces are not required to be half-spaces. The natural question whether the usual unilateral right shift operator acting on a Hilbert space has almost invariant half-spaces has an affirmative answer. Moreover, it is known that this operator has even invariant half-spaces. Indeed, by [6, Corollary 3.15], this operator has an invariant subspace with infinitedimensional orthogonal complement (thus the invariant subspace is of infinite codimension). It is not hard to see that the space exhibited in the proof of this statement is in fact infinite dimensional. It is natural to consider Donoghue operators as candidates for counterexamples to the almost invariant half-space problem, as their invariant subspaces are few and well understood. Recall that a Donoghue operator D ∈ L(2 ) is an operator defined by De0 = 0,

Dei = wi ei−1 ,

i ∈ N,

where (wi ) is a sequence of non-zero complex numbers such that |wi | is monotone decreasing and in 2 . It is known that if D is a Donoghue operator then D has only invariant subspaces of finite dimension and D∗ has only invariant subspaces of finite codimension (see [6, Theorem 4.12]). Hence neither D nor D∗ have invariant half-spaces. In Section 3 we will employ the tools of Section 2 to show that, nevertheless, every Donoghue operator has almost invariant half-spaces. We do not know whether the operators constructed by Enflo [3] and Read [7] have almost invariant half-spaces. The following result explains how almost invariant half-spaces of operators are related to invariant subspaces of perturbed operators. Proposition 1.3. Let T ∈ L(X) and H ⊆ X be a half-space. Then H is almost invariant under T if and only if H is invariant under T + K for some finite rank operator K. Proof. Suppose that T has an almost invariant half-space H. Let F be a subspace of the smallest dimension satisfying the condition in Definition 1.2. Then we have H ∩ F = {0}. Define P : H + F → F by P (h + f ) = f . Since P is a finite rank operator, we can extend it to a finite rank operator on X using Hahn-Banach theorem. That is, there exists P : X → F such that P|H+F = P . Define K : X → X by K := −PT . Clearly K has finite rank and for any h ∈ H we have T h = h + f for some h ∈ H and f ∈ F , so that (T + K)(h) = T h − PT h = h + f − P(h + f ) = h + f − f = h Therefore, (T + K)H ⊆ H, which shows that T + K has an invariant half-space. Conversely, from (T + K)(H) ⊆ H it follows immediately that T (H) ⊆ H + K(H),so that H is an almost invariant half space for T .

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Finally we would like to point out that if an operator has almost invariant half-spaces, then so does its adjoint. For that we will need two simple lemmas. The proof of the first lemma is elementary. Lemma 1.4. Let X be a Banach space and Y be a subspace of X. Then Y is infinite codimensional if and only if Y ⊥ is of infinite dimension. Thus Y is a half-space if and only if both Y and Y ⊥ are of infinite dimension. Lemma 1.5. A subspace Y of X is a half-space if and only if Y ⊥ is a half-space in X ∗ . Proof. Suppose Y is a half-space. By Lemma 1.4, Y ⊥ must be infinite-dimensional. Also (Y ⊥ )⊥ ⊇ j(Y ) where j : X → X ∗∗ denotes the natural embedding. Thus (Y ⊥ )⊥ is infinite dimensional. Now Lemma 1.4 yields that Y ⊥ is a half-space. Let’s assume that Y ⊥ is a half-space. Since Y ⊥ is infinite codimensional Y must be infinite-dimensional (see, e.g. [1, Theorem 5.110]). On the other hand, since Y ⊥ is infinite dimensional, by Lemma 1.4 we obtain that Y is of infinite codimension, thus a half-space. Remark 1.6. The statement dual to that of Lemma 1.5 is not true in general. That is, if Z is a half-space in X ∗ then Z⊥ need not be a half-space. For example, c0 is a half-space in ∞ while (c0 )⊥ = {0} ⊆ 1 is not. Proposition 1.7. Let T be an operator on a Banach space X. If T has an almost invariant half-space then so does its adjoint T ∗ . Proof. Let Y be a half-space in X such that Y is almost invariant under T , and F be a finite-dimensional subspace of X of smallest dimension such that T Y ⊆ Y +F . Then Y ∩ F = {0}. Thus there exists a subspace W of X such that W + F = X, W ∩ F = {0}, and Y ⊆ W . In particular, W ⊥ is finite dimensional. Denote Z = (Y + F )⊥ . By Lemma 1.5, Z is a half-space in X ∗ . For every z ∈ Z and y ∈ Y we have y, T ∗ z = T y, z = 0 since T y ∈ Y + F . Therefore T ∗ Z ⊆ Y ⊥ . To finish the proof, it suffices to show that Y ⊥ = Z + W ⊥ . Indeed, by the definition of Z we have that Z ⊆ Y ⊥ . Also since Y ⊆ W we have W ⊥ ⊆ Y ⊥ . Thus Z + W ⊥ ⊆ Y ⊥ . On the other hand, since F is finite dimensional and F ∩ W = {0}, we may choose a basis (fi ) of F with biorthogonal functionals (fi∗ ) such that fi∗ ∈ W ⊥ . Since Y ⊆ that fi∗ ∈ Y ⊥ . Thus, W ∗we have ∗ ⊥ ∗ ∗ if x is an arbitrary element of Y then x − i x (fi )fi ∈ (Y + F )⊥ = Z, and therefore x∗ ∈ Z + W ⊥ .

2. Basic tools All Banach spaces in Sections 2, 3 and 4 are assumed to be complex. For a subset A of C, we will write A−1 = λ1 : λ ∈ A, λ = 0 . For a Banach space X and T ∈ L(X), we will use symbols σ(T ) for the spectrum of T , r(T ) for the spectral

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radius of T , and ρ(T ) for the resolvent set of T . For a nonzero vector e ∈ X and λ ∈ ρ(T )−1 , define a vector h(λ, e) in X by −1 h(λ, e) := λ−1 I − T (e). Note that if |λ|
0. (iii) There is a vector whose orbit is a minimal sequence. Then T has an almost invariant half-space. Proof. Let e ∈ X be such that (T i e)∞ = T i e. Then i=0 is minimal. For each i put x i n ∗ ∗ for each k, the biorthogonal functional xk defined on span xi by xk i=0 αi xi = αk is bounded. Let rk = x∗k . (ci ) be a sequence of positive real numbers as Let ∞ in Lemma 3.1, so that βk := i=0 ci ri+k < +∞ for every k. By making ci ’s even √ smaller, if necessary, we may assume that i ci → 0. ∞ i Consider a function F : C → C defined by F (z) F is = i=0 ci z . Evidently, entire. Observe that we may assume that the set z ∈ C : F (z) = 0 is infinite. Indeed, by the Picard Theorem there exists a negative real number d such that the set z ∈ C : F (z) = d is infinite. Now replace c0 with c0 − d. This doesn’t affect our other assumptions on the sequence (ci ). Fix a sequence of distinct complex numbers (λn ) such that F (λn ) = 0 for every n. Since F is non-constant, the sequence (λn ) has no accumulation points. Hence, |λn | → +∞. Note that (ii) can be restated as follows: ρ(T )−1 has a connected component C such that 0 ∈ C and C contains a neighbourhood of ∞. Thus by passing to a subsequence of λn ’s and relabeling, if necessary, we can assume that λn ∈ C for all n. Observe that the condition λn ∈ ρ(T )−1 for every n implies that h(λn , e) is defined for each n. Put Y = [h(λn , e)]∞ n=1 . Then Y is almost invariant under T by Lemma 2.1 and dim Y = ∞ by Lemma 2.3. We will prove that Y is actually a half-space by constructing a sequence of linearly independent functionals (fn ) such that every fn annihilates Y . For every k = 0, 1, . . . , put Fk (z) = z k F (z). Let’s write Fk (z) in the form of ∞ (k) Taylor series, Fk (z) = i=0 ci z i . Then 0 if i < k, and (k) ci = ci−k if i k. (k)

i Define a functional fk on span{T i e}∞ i=0 via fk (T e) = ci . Since T has no eigenvalues, the orbit of T is linearly independent thus fk is well-defined. We will

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show now that fk is bounded. Let x ∈ span{T i e}∞ i=0 , then x = some n, so that n n

(k) |fk (x)| = fk x∗i (x)T i e

x∗i ci x

i=0

i=0

=

n

n

i=0

x∗i (x)T i e for

∞

ri ci−k x ri ci−k x = βk x ,

i=k

i=k

so that fk βk . Hence, fk can be extended by continuity to a bounded functional on [T i e]∞ i=1 , and then by the Hahn-Banach Theorem to a bounded functional on all of X. Now we show that each fk annihilates Y . Fix k. Recall that for each λ ∈ ∞ 1 −1 λi T i e. Therefore ρ(T ) such that |λ| < r(T ) we have h(λ, e) = λ i=0

fk h(λ, e) = fk λ

∞ i=0

λi T i e = λ

∞ i=0

(k)

λi ci

= λFk (λ) = λk+1 F (λ).

1 for every λ ∈ C such that |λ| < r(T ) (recall 0 ∈ C). The map λ → h(λ, e) and, there by the prinfore, the map λ → fk h(λ, e) , is analytic on the set ρ(T )−1. Therefore, λk+1 F (λ) ciple of uniqueness of analytic function, the functions e) and fk h(λ, k+1 must agree on C. Since λn ∈ C for all n, we have fk h(λn , e) = λn F (λn ) = 0 for all n. Thus, Y is annihilated by every fk . . Observe that fk = 0 It is left to prove the linear independence of {fk }∞ k=1 N −1 for all k since fk (T i e) = 0 for i k. Suppose that fN = k=M ak fk with aM = 0. N −1 However fN (T M e) = 0 by definition of fN while k=M ak fk (T M e) = aM c0 = 0, contradiction.

Remark 3.3. Note that condition (ii) of Theorem 3.2 is satisfied by many important classes of operators. For example, it is satisfied if σ(T ) is finite (in particular, if T is quasinilpotent) or if 0 belongs in the unbounded component of ρ(T ). Corollary 3.4. Suppose that X = p (1 p < ∞) or c0 and T ∈ L(X) is a weighted right shift operator with weights converging to zero but not equal to zero. Then both T and T ∗ have almost invariant half-spaces. Proof. It can be easily verified that T is quasinilpotent. Clearly, T has no eigenvalues, and the orbit of e1 is evidently a minimal sequence. By Theorem 3.2 and Remark 3.3, T has almost invariant half-spaces. Finally, Proposition 1.7 yields almost invariant half-spaces for T ∗ . The following statement is a special case of Corollary 3.4. Corollary 3.5. If D is a Donoghue operator then both D and D∗ have almost invariant half-spaces.

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Recall that a subset D of C is called a cone if D is closed under addition and multiplication by positive scalars. Remark 3.6. Condition (ii) in Theorem 3.2 can be weakened as follows: instead of requiring that ρ(T ) contains a punctured disk centered at zero, we may only require that it contains a non-trivial sector of this disk, i.e., the intersection of the punctured disk with a non-empty open cone. Equivalently, ρ(T )−1 has a connected component C such that 0∈ C, and there exists an open cone D in C and M > 0 such that z ∈ D : |z| M ⊆ C. Indeed, suppose that such D and M exist. Choose ν in this cone with |ν| = M . Also, suppose that, as in the proof of Theorem 3.2, we have already found a sequence (λm ) of zeros of F . The set {M |λλm }∞ has m | m=0 an accumulation point, say µ. By passing to a subsequence, we may assume that M |λλm → µ. Note that the spectrum of µν T is obtained by rotating the spectrum m| −1 has a connected component C such that 0 ∈ C and there of T . Thus ρ µν T exists anopen cone D in C which contains a neighborhood of µ and z ∈ D : |z| M ⊆ C . Thus by passing to a subsequence of (λm ) we can assume that λm ∈ C for all m. Replace in the proof T with µν T . Note that this doesn’t affect the assumptions on the operator and the definitions of ci ’s, F , and λm ’s. Finally, multiplying an operator by a non-zero number does not affect its almost invariant half-spaces. Note that every operator T with σ(T ) ⊆ R satisfies this weaker version of condition (ii). In particular, it is satisfied by self-adjoint operators on Hilbert spaces. Corollary 3.7. Suppose that T ∈ L(X) such that T has no eigenvectors, σ(T ) ⊆ R, and there is a vector whose orbit is a minimal sequence. Then T has an almost invariant half-space.

4. Non-quasinilpotent operators In this section we will modify the argument of Theorem 3.2 to extend its statement to another class of operators having non-zero spectral is a standard fact radius. It that if (xi ) is a minimal sequence then x1∗ = dist xn , [xi ]i=n for every n. n

Theorem 4.1. Let X be a Banach space and T ∈ L(X) be an operator with r(T ) 1 having no eigenvectors. Let e ∈ X; put xn = T n e for n ∈ N. If (xn ) is a minimal ∞ x∗ sequence and n=1 nn < ∞ then T has an almost invariant half-space. Proof. Let D stand for the unit disk in C. For a sequence (λn ) ⊂ D such that ∞ 1 − |λn | < ∞. n=1

(3)

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The corresponding Blaschke product is defined by ∞ |λn | λn − z B(z) = . λn 1 − λn z n=1

(4)

It is well known that B is a bounded analytic function on D with zeros exactly at (λn ). According to [5, Theorem 2] we can choose a sequence (λn ) ⊂ D satisfying (3) 1 n! (n) such that B n!(0) = O n+1 . Thus B (n) (0) = O n+1 . For m ∈ N set Fm (z) = z m B(z). Obviously the functions (Fm ) are linearly independent. It follows from Fm (z) = z m B(z) = that

(n) (0) = Fm

∞ ∞ B (n) (0) n+m B (n−m) (0) n z z = n! (n − m)! n=m n=0

0 n! (n−m) (0) (n−m)! B

n! C n−m+1

for n < m, for n m.

(5)

Put Y = [h(λn , e)]∞ n=1 . By Lemma 2.1, Y is almost invariant under T and dim Y = ∞ by Lemma 2.3. As in the proof of Theorem 3.2, we will show that under the conditions of the Theorem 4.1 there is a sequence of linearly independent functionals annihilating Y . F (n) (0)

Define a linear functional fm on span{xn } by fm (xn ) = mn! . Since T has no eigenvectors, the orbit of T is linearly independent, so fm is well defined. αn xn ∈ Let’s prove that fm is bounded for every m ∈ N. Take any x := span{xn }. Using (5), we obtain (n) |x∗ (x)| Fm (0) |αn | n fm (x) = αn =C C n! n−m+1 n−m+1 nm

C x

It suffices to show that

nm x∗ n nm n−m+1

nm

x∗n

n−m+1

.

< ∞. Note that

m(n − m + 1) = (m − 1)(n − m) + n n whenever n m, so that ∞

∞ ∞

x∗n

x∗n

x∗n

=m m t

m∈∪Λν

dt1 dt2 dt3 . t 1 t2 t 3

(1.3)

By certain reduction processes, it will be shown that we only have S to deal with the case when Λ4 is contained in a single edge E of N(Λ4 ) and rank [ ν Λν ] ≤ 2 . In this case, the behavior of H(PΛ , ξ) turns out to be quite simple. Let ! X m SΛ (ξ, t) = exp i < ξ, am > t . m∈∪Λν

2 See

section 3 below for the precise descriptions of these notions.

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S If all vectors in ν Λν have even entries at least in one fixed coordinate, say, the first coordinate, then H(PΛ , ξ) = 0 because Z dt1 =0 (1.4) SΛ (ξ, t1 , t2 , t3 ) t1 1 = x1 y1 + · · · + xd yd

for the inner products on Zn , Rd , respectively. As usual, the notation A . B for two scalar expressions A, B will mean A ≤ CB for some positive constant C independent of A, B and A ≈ B will mean A . B and B . A .

2. Preliminaries Under the same setting as in the definition of multiple Hilbert transforms (1.1), we consider the multiparameter maximal function Z r1 Z rn 1 MPΛ f (x) = sup ··· |f (x − PΛ (t))| dt (2.1) r1 ,...,rn >0 r1 · · · rn −r1 −rn defined for each locally integrable function f on Rd .

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Theorem 2.1. For 1 < p ≤ ∞ , MPΛ is a bounded operator from Lp (Rd ) into itself and there exists a bound Cp depending only on p, n, d and the degrees of the vector polynomial PΛ such that k MPΛ f kLp (Rd ) ≤ Cp kf kLp (Rd ) . Remark 2.1. This result can be proved by combining a theorem of Ricci and Stein ([8], Theorem 7.1) and the so-called lifting argument (see Chapter 11 of [9]). Take a function ψ ∈ Cc∞ ([−2, 2]) such that 0 ≤ ψ ≤ 1 and ψ(u) = 1 for |u| ≤ 1/2 . Put η(u) = ψ(u) − ψ(2u) . Given an integer k and α, β, γ ∈ {1, . . . , n} , we consider the measures Pkα,β and Lγk defined in terms of Fourier transforms ξα α,β Pk b(ξ) = ψ , (Lγk )b(ξ) = η 2k ξγ . (2.2) 2k ξ β Lemma 2.1. Suppose that mk , qj ∈ Zn for k = 1, . . . , M , j = 1, . . . , N with rank [q1 , . . . , qN ] = n . Given αk , βk , γj ∈ {1, . . . , n} , define α1 ,β1 αM ,βM PJ = PJ·m ∗ · · · ∗ PJ·m 1 M

1 N LJ = LγJ·q ∗ · · · ∗ LγJ·q 1 N

and

(2.3)

for each J ∈ Zn . Then

X J∈Zn

!1/2

|PJ ∗ LJ ∗ f |

2

. kf kLp (Rd ) .

Lp (Rd )

Proof. It suffices to deal with the sum over Zn+ . With (rJ (t)) denoting the Rademacher functions of product form, it is plain to observe that 1/2 

p

X Z

p

X

2

 rJ (t) PJ ∗ LJ ∗ f Lp (Rd ) dt ≈ |P ∗ L ∗ f | J J

p d

J∈Zn +

L (R )

U J∈Zn +

where U = [0, 1]n . Consider the symbol X m(ξ) = rJ (t) PJ ∗ LJ b(ξ) . J∈Zn +

Using the full rank condition for the qj and the support conditions, it can be shown that m satisfies C` ∂` m(ξ , . . . , ξ ) 1 d ≤ ∂ν · · · ∂ν |ξ | · · · |ξν` | ν 1 1 ` for every ` = 1, . . . , d , where 1 ≤ ν1 < · · · < ν` ≤ d . Thus the desired conclusion follows from the multi-parameter Marcinkiewicz multiplier theorem.

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Lemma 2.2. Let (σJ )J∈Zn be a sequence of positive measures on Rd with the following properties:

(i) σJ ∗ f L1 (Rd ) . kf kL1 (Rd ) (J ∈ Zn )

. kf kLp0 (Rd ) (ii) sup |σJ ∗ f |

J∈Zn

Lp0 (Rd )

for some 1 < p0 ≤ 2 . Then !1/2

X

2

|σJ ∗ fJ |

Lp1 (Rd )

J∈Zn

.

X J∈Zn

!1/2

|fJ |

2

Lp1 (Rd )

for p1 determined by 1/p1 ≤ 1/2 (1 + 1/p0 ) . Proof. For 1 ≤ p, q ≤ ∞ , consider the operator T defined by T [(fJ )] = (σJ ∗ fJ ) on the mixed-norm spaces Lp (`q ). The condition (i) implies that T maps L1 (`1 ) boundedly into itself. The condition (ii) and the positivity of each σJ imply that T maps Lp0 (`∞ ) boundedly into itself. It follows from the vector-valued Riesz-Thorin interpolation theorem that T maps Lp1 (`2 ) boundedly into itself. Lemma 2.3. Given positive reals α1 , . . . , α` and β1 , . . . , β` , suppose that ∆ is a nonnegative mathematical expression satisfying (i) (ii)

∆ . 2α1 n1 +···+α` n` and ∆ . min 2−βi ni : i = 1, . . . , `

for each (n1 , . . . , n` ) ∈ Z` . Then there exists a positive constant depending on the αi , the βi and ` such that ∆ . 2− (|n1 |+···+|n` |) .

(2.4)

Proof. Let max {|αi ni | : i = 1, . . . , ` } = αν |nν | . If nν > 0 , then (2.4) follows from (ii). In the case nν ≤ 0 , if αi ni ≤ αν |nν | /2` for all i = 1, . . . , ` , then (2.4) follows from (i). Otherwise, it is a simple consequence of (ii).

3. Geometry of Newton Polyhedrons In this section we make precise the notions of vertices, edges and faces of Newton polyhedra and classify types of vertices with a view to carrying out our reduction processes effectively. 3.1. Algebraic Descriptions For A ⊂ R3 , we denote by Ch [A] the convex hull of A, the smallest convex set containing A. Given a finite set {q1 , . . . , qm } , its convex hull is given by X m m X Ch [q1 , . . . , qm ] = λ j qj : 0 ≤ λ j ≤ 1 , λj = 1 , j=1

j=1

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the set of all convex combinations of q1 , . . . , qm . If m = 2 , it is the line segment joining v1 , v2 . If m ≥ 3 and rank [v2 − v1 , . . . , vm − v1 ] = 2 , it is a finite polygon. If m ≥ 4 and rank [v2 − v1 , . . . , vm − v1 ] = 3 , it represents a finite polyhedron. For µ, ν ∈ {1, 2, 3} , we denote by Xµ , Xµν the positive xµ -axis of R3 containing the origin and the upper-right quarter xµ xν -plane, respectively, X µ = { α eµ : α ≥ 0 } ,

Xµν = { α eµ + β eν : α ≥ 0 , β ≥ 0 } .

For A ⊂ R3 , we denote by A◦ , ∂A the interior and the boundary of A with respect to the usual topology on R3 or the affine plane or line containing A. Vertices, edges and faces of N(Ω) are special types of subsets of ∂N(Ω). In what follows, we describe them from the convexity point of view. Definition 3.1. A point q ∈ Ω is called a vertex of N(Ω) if it can not be written as a convex combination of two other points of N(Ω), that is, q∈ / Ch [x, y]◦ = { (1 − λ)x + λy : 0 < λ < 1 } for any two distinct points x, y ∈ N(Ω) \ {q} . The set of all vertices of N(Ω) will be denoted by V(Ω). A face appears as a finite or an infinite polygon in the following forms: (F1) F = Ch [q1 , . . . , qN ] , (F2) F = Ch [q1 , q2 ] + Xµ or Ch [q1 , . . . , qN ] + Xµν . Definition 3.2. A set F ⊂ ∂N(Ω) is called a face of N(Ω) if it is given as one of the forms (F1), (F2) with q1 , . . . , qN ∈ V(Ω) and satisfies Ch F ∪ {x} ∩ N(Ω)◦ 6= ∅ for any x ∈ N(Ω) \ F . The set of all faces of N(Ω) will be denoted by F(Ω). Remark 3.1. Observe that for F ∈ F(Ω), Ch F ∪ {x} is the three dimensional polyhedron contained in N(Ω) for all x ∈ N(Ω) \ F . Considering the usual topology on Π containing a face F , it is evident that ∂F consists of line segments or half-lines parallel to the coordinate axes given in the form (E1) (E2)

E = Ch [q1 , q2 ] for some q1 , q2 ∈ V(Ω)

E = q + Xµ for some q ∈ V(Ω) and µ ∈ {1, 2, 3} .

Definition 3.3. A set E is called an edge of N(Ω) if it is given as one of the forms (E1), (E2) and there exists a face F with E ⊂ ∂F . The set of all edges of N(Ω) will be denoted by E(Ω).

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3.2. Projection Given any point p ∈ Ω \ ∂Ω , the convexity of N(Ω) implies that the line segment Ch [o, p] joining the origin o and p meets the boundary of N(Ω) at a unique point. It is this geometric property that will be used decisively in our reduction process (R1) of eliminating the interior points of N(Ω). We shall denote such a unique point by R∂N(Ω) (p) and call it the projection of p into ∂N(Ω). To be precise, we have Ch [o, p] ∩ ∂N(Ω) if p ∈ Ω \ ∂N(Ω) , R∂N(Ω) (p) = p if p ∈ ∂N(Ω) . Unless p ∈ V(Ω) , there exists a unique face F or an edge E whose interior contains R∂N(Ω) (p). It follows from Definitions 3.2 and 3.3 that we can write R∂N(Ω) (p) =

N X i=1

where αi > 0 with

PN

i=1

αi qi +

3 X

βν eν

( qi ∈ V(Ω) )

(3.1)

ν=1

αi = 1 and βν ≥ 0 .

3.3. Types of Vertices In performing the reduction processes (R2) and (R3), one basic principle will be to eliminate a point q ∈ Λ4 ∩ N(Λ4 ) only when all of vertices and edges of the newly-formed Newton polyhedron N(Λ4 \ {q}) are even. As it is clear that N(Λ4 \ {q}) = N(Λ4 ) if q ∈ / V(Λ4 ) , we just need to sort out vertices according to this principle of elimination. Definition 3.4. For each p ∈ Ω \ V(Ω) , let   F ∩ V(Ω) if R∂N(Ω) (p) ∈ F ◦ for F ∈ F(Ω) or E ∩ V(Ω) if R∂N(Ω) (p) ∈ E ◦ for E ∈ E(Ω) or GΩ (p) =  V if R∂N(Ω) (p) = V ∈ V(Ω) . We define two complementary subsets G(Ω), I(Ω) of V(Ω) by [ G(Ω) = GΩ (p) : p ∈ Ω \ V(Ω) , I(Ω) = V(Ω) \ G(Ω) and call each element of G(Ω) a generating vertex of Ω and each element of I(Ω) an isolated vertex of Ω. We now show that the evenness of vertices and edges is invariant under elimination of an isolated vertex. We begin with the following: Lemma 3.1. Let Ω0 = Ω \ {p} where p ∈ Ω \ G(Ω) . Then V(Ω0 ) ⊂ V(Ω) . Proof. Assume that there exists a point q ∈ V(Ω0 ) \ V(Ω) . Let us write GΩ (q) = { q1 , . . . , qN }, where q1 , . . . , qN ∈ V(Ω) . Since p ∈ Ω \ G(Ω), p ∈ / GΩ (q) . Thus { q1 , . . . , qN } = GΩ (q) ⊂ Ω \ {p} = Ω0 . (3.2)

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Figure 1. A Newton polyhedron having generating vertices p1 , . . . , p5 and an isolated vertex p. Observe that after eliminating p, two edges p3 p4 and p3 p5 are newly formed and that new edges do not contain any interior point. In view of Definition 3.4 and the projection expression (3.1), there exists some 0 ≤ < 1 such that q(1 − ) = R∂N(Ω) (q) =

N X i=1

αi qi +

3 X

βν eν

(3.3)

ν=1

PN where αi > 0 with i=1 αi = 1 and βν ≥ 0 . From (3.2) and (3.3), q(1 − ) ∈ 0 N(Ω ) . But the fact q ∈ V(Ω0 ) and q(1 − ) ∈ N(Ω0 ) with > 0 is impossible. If = 0 , then q ∈ V(Ω0 ) in (3.3) is expressed as a convex sum of more than two elements of Ω0 or q = q1 ∈ V(Ω) . First case is a contradiction to definition 3.1 and the second is not true because of the assumption q ∈ V(Ω0 ) \ V(Ω) . Therefore V(Ω0 ) \ V(Ω) = ∅ .

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Lemma 3.2. Suppose that Ω is a finite subset of Z3+ with N(Ω)◦ ∩ Ω = ∅ . Let Ω0 = Ω \ {p} where p ∈ Ω \ G(Ω). If E ∈ E(Ω0 ) \ E(Ω) , then E ◦ ∩ Ω0 = ∅ . For the sake of a visualized understanding of Lemma 3.2, we give a simple picture of Newton polyhedron having 5 generating vertices and a single isolated vertex in Figure 1. Proof. Assume E ∈ E(Ω0 ) and E ◦ ∩ Ω0 6= ∅ . We shall derive a contradiction by showing that E ∈ E(Ω) . For this purpose, we prove Claims 1 and 2. Claim 1. There exists a face F ∈ F(Ω) such that E ⊂ F . Proof of Claim 1. To see this, we shall exploit the following property of Newton polyhedron N(Ω) which results from its convexity. (P) If two vectors q1 , q2 ∈ N(Ω) are not contained in a face of N(Ω), then Ch[q1 , q2 ]◦ ⊂ N(Ω)◦ . Consider first the finite edge E = Ch[q1 , q2 ] with q1 , q2 ∈ V(Ω0 ) . Let q ∈ ◦ E ∩ Ω0 . We may write q = (1 − λ)q1 + λq2 with 0 < λ < 1 . By Lemma 3.1, V(Ω0 ) ⊂ V(Ω) so that q1 , q2 ∈ V(Ω) . Then there exists a face F ∈ F(Ω) such that q1 , q2 ∈ F . If not, then by the above property (P), q ∈ N(Ω)◦ ∩ Ω , which is a contradiction to the assumption N(Ω)◦ ∩ Ω = ∅ . Therefore E ⊂ F by the convexity of the face F . Next consider the infinite case E = q1 + Xµ with q1 ∈ V(Ω0 ) ⊂ V(Ω). Let q = q1 + s0 eµ ∈ E ◦ ∩ Ω0 . Then in the same way as above, there exists a face F ∈ F(Ω) such that Ch[q1 , q2 ] ⊂ F with any q2 = q1 +seµ ∈ E and s > s0 . Thus E ⊂ F . Claim 2. E ⊂ ∂F where F is chosen as in Claim 1. Proof of Claim 2. To prove this, it suffices to show that if F ◦ ∩ E 6= ∅ , then F ∈ F(Ω0 ) because if it were true, then E ⊂ F ∈ F(Ω0 ) and E ∈ E(Ω0 ) would imply E ⊂ ∂F . Assume q ∈ F ◦ ∩E . Since F ∈ F(Ω) , we have GΩ (q) = F ∩V(Ω) . The hypothesis shows p ∈ / GΩ (q) and so GΩ (q) ⊂ Ω0 . Consequently, F ⊂ N(Ω0 ) due to the convexity of N(Ω0 ). We next assert that F ⊂ ∂N(Ω0 ) . If not valid, then there exists some r ∈ F ∩ N(Ω0 )◦ . From N(Ω0 )◦ ⊂ N(Ω)◦ , we then have r ∈ N(Ω)◦ , a contradiction to the fact F ⊂ ∂N(Ω) . Finally, let us prove that F ∈ F(Ω0 ) . Upon assuming the contrary, from the definition 3.2, there exists x ∈ N(Ω0 ) \ F satisfying Ch [F ∪ {x}] ⊂ ∂N(Ω0 ) . This can not happen because the facts F ∈ F(Ω) and x ∈ [N(Ω0 ) \ F ] ⊂ [N(Ω) \ F ] imply that Ch [F ∪ {x}] must be the three dimensional polyhedron (see Remark 3.1). This is a contradiction with Ch [F ∪ {x}] ⊂ ∂N(Ω0 ) . Thus we conclude F ∈ F(Ω0 ) and Claim 2 is completely verified. Therefore E ⊂ ∂F and E is of the form Ch[q1 , q2 ] or q1 + Xµ where q1 , q2 ∈ V(Ω) and F ∈ F(Ω). Thus from the definition 3.3, E ∈ E(Ω), which is a contradiction to our hypothesis. Therefore E ◦ ∩ Ω0 = ∅. The following is what we primarily aim to show:

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Theorem 3.1. Suppose that N(Ω)◦ ∩ Ω = ∅ and all of vertices and edges in N(Ω) are even. Let Ω0 = Ω \ {p} with p ∈ Ω \ G(Ω). Then N(Ω0 )◦ ∩ Ω0 = ∅ and all of vertices and edges in N(Ω0 ) are also even. Proof. That N(Ω0 )◦ ∩ Ω0 = ∅ follows from N(Ω0 )◦ ⊂ N(Ω)◦ . That all of vertices in N(Ω0 ) are even follows from V(Ω0 ) ⊂ V(Ω) , Lemma 3.1. It remains to prove that all of edges in N(Ω0 ) are even. Take any E ∈ E(Ω0 ) . If E ∈ E(Ω) , then it is even by the hypothesis so let us assume E ∈ / E(Ω) . According to Lemma 3.2, E ◦ ∩ Ω0 = ∅ . Consider first the case E = Ch[q1 , q2 ] with q1 , q2 ∈ V(Ω0 ) . Since q1 and q2 are even vertices, at least two coordinates of each vertex q1 and q2 are even numbers. Thus {q1 , q2 } is even in column and it suffices to check that {q1 , q2 } ∪ {eν } is even in column when rank [q1 , q2 , eν ] ≤ 2 . Without loss of generality we may setν = 3. If it is not even in column, then the only possible form of the matrix Q = q1 , q2 , e3 , in terms of evenness or oddness of its entries, is given by either     even odd even odd even even Q =  even odd even or Q =  odd even even . 0 0 1 0 0 1 In either case, we note that det (Q) 6= 0 as the product of two odd numbers is always odd while the product of two even numbers is always even. Consequently, rank [Q] = 3 , leading to a contradiction. Thus we conclude that { q1 , q2 , e3 } is even in column so E is an even edge. In the case E = q + Xµ with q ∈ V(Ω0 ) , as E ∩ Ω0 = {q} , our analysis is reduced to the case of vertices so E is even.

4. Dyadic Decomposition and Overall Reduction Scheme Choose a smooth function ψ ∈ Cc∞ ([−2, 2]) such that 0 ≤ ψ ≤ 1 and ψ(u) = 1 for |u| ≤ 1/2 . Put η(u) = ψ(u) − ψ(2u) and h(u) = η(u)/u for u 6= 0 . Given coefficients am = (a1m , . . . , a4m )’s of PΛ for each J ∈ Z3+ we define ! Z X −J·m m d Λ exp i 2 < ξ, am > t h(t1 )h(t2 )h(t3 ) dt HJ (ξ) = R3

m∈∪Λν

so that H PΛ f =

X

HJΛ ∗ f

(4.1)

J∈Z3+

where the sum is taken only over Z3+ due to the local nature of HPΛ . As for the overall reduction scheme, our ultimate goal is to reduce our L2 analysis of HJΛ 0 associated with Λ = (Λ1 , Λ2 , Λ3 , Λ4 ) to the one of HJΛ associated with Λ0 = 0 0 0 0 (Λ1 , Λ2 , Λ3 , Λ4 ) having the following properties: 0 (1) Λ Sν ⊂0 Λν for ν = 1, 2, 3, 4. (Recall that Λν = {eν } for 3 ν = 1, 2, 3.) (2) Λν is contained in a two-dimensional subspace of Z+ . (3) Λ04 is contained in a single edge of N(Λ04 ).

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(4) X

d d Λ0 (ξ) ≤ C HJΛ (ξ) − H J

uniformly in ξ

J∈B⊂Z3+

where B depends on Λ0 and the union of all possible B’s equals to Z3+ . The properties (1) through (4), combined with Theorem 3.1, will play the key role in proving the sufficiency of our Main Theorem. In achieving our goal, we shall eliminate certain points from the Newton polyhedron N(Λ4 ) one by one in accordance with three aforementioned processes (R1), (R2) and (R3). In passing from one stage to another, we shall use the following basic estimates over and over again. Lemma 4.1. There S exists δ > 0 depending only on the maximum order of all multi-indices in Λν such that n o [ −δ d : m∈ Λν . (4.2) HJΛ (ξ) . min 2−J·m hξ, am i Given p ∈ Λν , let Λ0ν = Λν \ {p} and Λ0µ = Λµ for µ 6= ν . Then −J·p d d Λ0 hξ, ap i . HJΛ (ξ) − H J (ξ) . 2

(4.3)

Proof. The first estimate follows from a multidimensional version of van der Corput lemma and the second is a simple consequence of the mean value theorem.

5. Elimination of Interior Points The purpose of this section is to show that every interior of N(Λ4 ) can be eliminated without altering Lp behavior of HPΛ in the sense of Theorem 5.1 below. The key idea of eliminating an interior point p is to express a projection R∂N(Λ4 ) (p) as a linear combination of generating vertices in GΛ4 (p) and to apply the basic estimates of Lemma 4.1. Lemma 5.1. Let p ∈ N(Λ4 )◦ ∩ Λ4 . Then there exists a uniform constant c > 0 over J = (j1 , j2 , j3 ) ∈ Z3+ such that

Λ(p) ∗f . 2−c (j1 +j2 +j3 ) kf kLp (R4 ) (5.1)

HJΛ − HJ Lp (R4 )

for every 1 < p < ∞ , where Λ(p) = (Λ1 , Λ2 , Λ3 , Λ4 \ {p}) . Proof. Consider the projection of p into ∂N(Λ4 ) given in the general form R∂N(Λ4 ) (p) =

N X i=1

αi qi +

3 X

βν eν

ν=1

P in which {q1 , . . . , qN } = GΛ4 (p) ⊂ V(Λ4 ) and αi = 1 , αi > 0 , βν ≥ 0 . Being p an interior point, R∂N(Λ4 ) (p) = (r1 , r2 , r3 ) with ri > 0 for each i and p = (1 + ) R∂N(Λ4 ) (p)

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Figure 2. An interior point p, its projection R∂N(Λ4 ) (p) and its generators q1 , q2 , q3 .

for some > 0 (see Figure 2). Using these two expressions of projection, we obtain −J·p 2 ξ4 ≤

N Y −J·q αi i 2 ξ4

! 2− (r1 j1 +r2 j2 +r3 j3 )

i=1

P

where we used the fact αi = 1 and J · eν = jν ≥ 0 for each ν. It follows from the estimate (4.3) of Lemma 4.1 that Λ(p) b (ξ) . HJΛ − HJ

N Y −J·q αi i 2 ξ4

! 2− (r1 j1 +r2 j2 +r3 j3 ) .

(5.2)

On the other hand, the estimate (4.2) of Lemma 4.1 gives n o −δ Λ(p) b (ξ) . min 2−J·qi ξ4 : i = 1, . . . , N . HJΛ − HJ

(5.3)

i=1

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For fixed ξ, adopting a similar argument as in the proof of Lemma 2.3, it is not hard to show that the estimates (5.2) and (5.3) imply Λ(p) b (ξ) . 2−γ (j1 +j2 +j3 ) HJΛ − HJ for some

γ > 0 , which proves (5.1) in the case p = 2 . In addition, it is trivial to see HJΛ ∗ f L1 . kf kL1 . Now the inequality (5.1) for p 6= 2 follows from interpolating L1 and L2 estimates and duality. Repeating Lemma 5.1 to every interior point and summing over Z3+ , we are eventually end up with Theorem 5.1. Let Λ04 be any subset of Λ4 satisfying N(Λ04 )◦ ∩ Λ04 = ∅ and put Λ0 = (Λ1 , Λ2 , Λ3 , Λ04 ) . Then for every 1 < p < ∞

X

X

Λ0 Λ

. (5.4) HJ ∗ f . kf kLp (R4 ) + HJ ∗ f

J∈Z3+

Lp (R4 )

J∈Z3+

Lp (R4 )

6. Reduction to One Plane On account of Theorem 5.1, we may now assume that Λ4 is completely contained p in the boundary of N(Λ4 ). The S purpose of this section is to reduce further our L analysis of HPΛ to the case Λν contained on the one plane passing through the origin. Lemma 6.1. Given any Ων ⊂ Z3+ , ν = 1, 2, 3, 4 , consider [ { pθ : θ = 1, . . . , r } ⊂ Ων . Define τ : {1, . . . , r} → {1, 2, 3, 4} by τ (θ) = ν if pθ ∈ Ων . Let J ∈ Z3+ . Then there exist measures PJθ , θ = 1, . . . , r, on R4 such that (i) each PJθ is a finite sum of terms given in the form (2.3) of Lemma 2.1, cθ is a function supported on the set (ii) each P J ξ : 2−J·pθ ξτ (θ) . 2−J·pσ ξτ (σ) for all σ = 1, . . . , r P cθ and PJ (ξ) = 1 for ξ 6= 0 . Proof. As in Lemma 2.1, we choose a function ψ ∈ Cc∞ ([−2, 2]) such that 0 ≤ ψ ≤ 1 and ψ(u) = 1 for |u| ≤ 1/2 . Observe that Y ξτ (σ) ξτ (σ) 1= ψ + 1−ψ 2J·(pσ −pθ ) ξτ (θ) 2J·(pσ −pθ ) ξτ (θ) θ 0 , αθ,ρ = 1 and βi ≥ 0 . Let us write ∆θ = {1, . . . , Nθ } for θ = 1, . . . , ` and ∆ = { (θ, ρ) : θ = 1, . . . , ` , ρ ∈ ∆θ } . Evidently, [ G(Ων ) = { qθ,ρ : (θ, ρ) ∈ ∆ } . (6.3) With the same function ψ as in Lemma 2.1 or Lemma 6.1, recall that η(u) = ψ(u) − ψ(2u) and that the Littlewood-Paley operators Lγk over Z on any space Rd were defined by the symbols ∧

(Lγk ) (ξ) = η(2k ξγ ) ,

γ = 1, . . . , d

N1 +···+N`

(k ∈ Z) .

Let us write each element of Z as the vector n = (n1 , . . . , n` ) with θ θ θ Nθ n = (n1 , . . . , nNθ ) ∈ Z for θ = 1, . . . , ` . We also write each element of Zr

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as m = (m1 , . . . , mr ) . We shall now consider specific extensions of LittlewoodPaley operators over ZN1 +···+N` and Zr on R4 defined in the following manner. We first consider ∧ θ τ (θ) L−J·qθ,ρ −nθ (ξ) = η 2−J·qθ,ρ −nρ ξτ (θ) for (θ, ρ) ∈ ∆ , ρ ∧ τ (k) L−J·pk −mk (ξ) = η 2−J·pk −mk ξτ (k) for k = 1, . . . , r with a view to restricting frequency variables as θ 2−J·qθ,ρ ξτ (θ) ≈ 2nρ for (σ, ρ) ∈ ∆ and 2−J·pk ξτ (k) ≈ 2mk for k = 1, . . . , r on the support of these functions. Given any functions Aλ , λ ∈ Γ , where Γ denotes a finite index set, we introduce the notation ~λ∈Γ Aλ = Aλ1 ∗ · · · ∗ AΛN

if Γ = {λ1 , . . . , λN } .

With this notation, we define τ (θ)

LqJ,n = ~(θ,ρ)∈∆ LJ·qθ,σ −nθ

ρ

τ (k)

and LpJ,m = ~rk=1 LJ·pk −mk .

We now have a Littlewood-Paley decomposition X LqJ,n ∗ LpJ,m ∗ f = f where N = N1 + · · · + N` .

(6.4)

n∈ZN , m∈Zr

Let us focus on proving (6.1). Owing to (6.4), it is sufficient to prove that there exists a uniform constant c > 0 over n ∈ ZN , m ∈ Zr such that

2 1/2

X Ω

Ω(s) q p s

∗ PJ ∗ LJ,n ∗ LJ,m ∗ f HJ − H J

J∈Z3+

Lp (R4 )

. 2−c (|n|+|m|) kf kLp (R4 ) .

(6.5) LqJ,n

LpJ,m

Proof of (6.5). On the support of the Fourier transform of ∗ , the van der Corput type estimate (4.2) gives n o −δ θ d ≈ 2−δ nρ : (θ, ρ) ∈ ∆ (6.6) HJΩ (ξ) . min 2−J·qθ,ρ ξτ (θ) o n −δ d ≈ 2−δ mk : k = 1, . . . , r . (6.7) HJΩ (ξ) . min 2−J·pk ξτ (k) cs , we observe that the In view of the support condition (ii) of Lemma 6.1 for P J \ Ω(s) estimates (6.6) and (6.7) are also valid for HJ (ξ) . In addition, the mean value theorem estimate (4.3) gives d −J·p \ Ω(s) s H Ω (ξ) − H . 2 (ξ) ξτ (s) J J . min 2−J·pk ξτ (k) ≈ 2mk : k = 1, . . . , r . (6.8)

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In view of the expression (6.2), observe that Nθ Nθ Y Y αθ,ρ θ ≈ 2αθ,ρ nρ 2−J·pθ ξτ (θ) ≤ 2−J·qθ,ρ ξτ (θ) ρ=1

ρ=1

for all θ = 1, . . . , ` . This combined with (6.8) implies that P αθ,ρ d θ \ Ω(s) . 2 (θ,ρ)∈∆ ` nρ . H Ω (ξ) − H (ξ) J J

(6.9)

Applying now Lemma 2.3 with the estimates (6.6) and (6.9), we have d \ Ω(s) H Ω (ξ) − H (ξ) . 2−a |n| for some a > 0 . J J In the same way, applying Lemma 2.3 with (6.7) and (6.8), we have d \ Ω(s) . 2−b |m| for some b > 0 . H Ω (ξ) − H (ξ) J J Consequently, we are led to the estimate d \ Ω(s) . 2−c (|n|+|m|) H Ω (ξ) − H (ξ) J J

for some

c > 0.

(6.10)

As we see from the hypothesis of Theorem 6.1 and (6.3) that {qθ,ρ : (θ, ρ) ∈ ∆ } ∪ { pθ : θ = 1, . . . , r } =

4 [

Ων

ν=1

has three linearly independent vectors, we obtain the following Littlewood-Paley inequality as in Lemma 2.1

X 1

q p κ 2 2

|PJ ∗ LJ,n ∗ LJ,m ∗ f |

J∈Z3+

.k f kLp (R4 ) .

(6.11)

Lp (R4 )

By the estimates (6.10) and (6.11), we obtain the desired inequality (6.5) for p = 2. Applying a standard bootstrap argument making use of Theorem 2.1, Lemmas 2.1 and 2.2, we obtain (6.5) for the other values of p. The proof is now complete. Since we start with Λν = { eν } for ν = 1, 2, 3, we may assume the Ων (θ) are mutually S disjoint. Thus we may repeat Theorem 6.1 with the initial sets Ων (θ) if rank [ Ων (θ)] = 3 . We continue this procedure until the union of resulting sets has rank less than or equal to 2. In summary, with a slight abuse of notations and meanings, we eventually obtain the following: Corollary 6.1. Suppose that N(Λ4 )◦ ∩ Λ4 = ∅ and all of vertices and edges of N(Λ4 ) are even. Then there exist a finite family A = { Λ0 = (Λ01 , Λ02 , Λ03 , Λ04 ) : Λ0ν ⊂ Λν for each ν } having the following properties:

504


(1) (2) (3) (4)

S For each Λ0 ∈ A , rank [ Λ0ν ] ≤ 2 . For each Λ0 ∈ A , N(Λ04 )◦ ∩ Λ04 = ∅. For each Λ0 ∈ A , all of vertices and edges For every 1 < p < ∞ ,

X X

Λ

HJ ∗ f

p 4 . kf kLp (R4 ) +

L (R )

J∈Z3+

Λ0 ∈A

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of N(Λ04 ) are even.

X

Λ0

HJ ∗ PJ ∗ f

J∈Z3+

Lp (R4 )

where (PJ )J∈Z3+ depending on each Λ0 is a sequence of functions given in the form of (2.3).

7. Reduction to One Line In view of Corollary 6.1, it is now enough to deal with the Λν satisfying Λ4 ⊂ S ∂N(Λ4 ) and Λν ⊂ Π for some plane Π passing through the origin. We may d Λ assume that Π is not a coordinate plane since the symbols H J (ξ) = 0 in such a case. A simple geometric inspection reveals that there exist a finite number of line segments or half-lines, say, G1 , . . . , GN , with the following properties3 : S (i) Λ4 ⊂ Gi ⊂ Π and each Gi has at least two points of Λ4 . (ii) Gi ∈ E(Λ4 ) for i = 1, . . . , N − 1 . (iii) GN ⊂ F for some F ∈ F(Λ4 ) of the form F = q + Xµν . (iv) Any pair Gi , Gj are not contained on one face of N(Λ4 ). (See Figure 3 for an illustration of this situation.) The purpose of this section is to reduce our proof to one of those Gi . Let us remind that what we need to prove is

X

Λ

HJ ∗ PJ ∗ f

p 4 . kf kLp (R4 ) (1 < p < ∞)

J∈Z3+

L (R )

in which (PJ )J∈Z3+ is a sequence of functions given in the form of (2.3). SN Lemma 7.1. Let Λ4 ⊂ i=1 Gi , N ≥ 2 , where the Gi are line segments or halflines satisfying the properties (i) through (iv) described as above. Then there exist p1 , p2 ∈ Λ4 \ G(Λ4 ) such that (1 + ) p0 ∈ p1 p2

for some

p0 ∈ Λ4 , > 0 .

G◦i

(7.1)

Proof. Let us call each Gi occupied if ∩ Λ4 6= ∅ . In terms of the number of occupied Gi ’s, there are three cases to consider. Case 1. At least two occupied line segments or half-lines, say, G1 , G2 : Choose an interior point pi from each Gi for i = 1, 2 . Let p0 be an end point of G1 , a vertex of N(Λ4 ), and consider the ray L that emanates from the origin and passes through p0 . Since Π is a plane passing through the origin, L ⊂ Π . 3 Of

course, one can verify this rigorously by using the definitions of section 3.

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S4 Figure 3. A Newton polyhedron with Λ4 ⊂ i=1 Gi ⊂ Π. Here G1 , G2 , G3 are edges and G4 is a line segment contained in the face F . In the language of Lemma 7.1, all of the Gi but G3 are occupied. In Lemma 7.2, we eliminate the points p1 , p2 with the aid of p0 . Since p1 and p2 are on two different edges of the convex polygon, L must meet the line segment p1 p2 ⊂ Π at a unique point in the sense as described in (7.1). Case 2. Only one occupied line segment or half-line, say, G1 : In this case, we note that every point in Λ4 \ G1 is an isolated vertex of N(Λ4 ). Choose p1 ∈ G◦1 and p2 ∈ Λ4 \ G1 . Take p0 ∈ G1 ∩ V(Λ4 ) as an end point of G1 . Then (7.1) holds by the same reasoning as in the Case 1. Case 3. No occupied line segments or half-lines: In this case, all points in Λ4 are isolated vertices of N(Λ4 ). Choose p1 ∈ G1 ∩ Λ4 and p2 ∈ Λ4 \ G1 with p2 ∈ G2 , say. Selecting p0 from G1 ∩ Λ4 or G2 ∩ Λ4 in such a way that the ray L through the origin and p0 meets the line segment p1 p2 , we get (7.1).

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Lemma 7.2. Let B be an arbitrary subset of Z3+ . Suppose that p1 , p2 are two distinct points of Λ4 , not lying on the same coordinate plane, such that there exists a point p0 ∈ Λ4 satisfying (1 + ) p0 ∈ p1 p2 for some > 0 . Then we may decompose B into B = B1 ∪ B2 so that for 1 0 where Λ(i) = (Λ1 , Λ2 , Λ3 , Λ4 \ {pi }) , i = 1, 2 . Proof. Assume first that (1 + ) p0 is an interior point of p1 p2 , that is, (1 + ) p0 = (1 − λ) p1 + λ p2

(0 < λ < 1) .

Note that each coordinate of p0 is positive because p1 , p2 do not lie on the same coordinate plane. Let B1 = { J ∈ B : J · p1 ≥ J · p2 } and B2 = B \ B1 . When J ∈ Bi , we have 1−λ −J·p2 λ 2−J·(1+) p0 = 2−J·p1 2 ≥ 2−J·pi (i = 1, 2) . Writing p0 = (r1 , r2 , r3 ) with ri > 0 , we have 2−J·(1+) p0 = 2−J·p0 2−(r1 j1 +r2 j2 +r3 j3 ) . By the estimates (4.2) and (4.3), we obtain that for J ∈ Bi n ∧ o HJΛ − H Λ(i) (ξ) . min 2−J·p0 ξ4 −δ , 2−(r1 j1 +r2 j2 +r3 j3 ) 2−J·p0 ξ4 J . 2−a (j1 +j2 +j3 ) for some a > 0 . This implies the stated inequalities for p = 2 . Using the inter polation between p = 2 and the trivial fact HJΛ ∗ PJ ∗ f Lp . kf kLp for any p > 1, we obtain Lemma 5.1. Next we consider the cases when (1 + ) p0 coincides with one of p1 , p2 . By symmetry, it is enough to deal with the case (1 + ) p0 = p1 . In this case, we shall use the fact that each coordinate of p0 + p2 is positive. Put B1 = { J ∈ B : (1 + /2)J · p0 ≥ J · p2 }

and B2 = B \ B1 .

Let J ∈ B1 . Choose α with 0 < α ≤ /(6 + 3) . Then 2 J · p1 = (1 + )J · p0 = J · p0 + 1 + J · p0 3 3 J · p2 ≥ J · p0 + 1 + 3 6 + 3 ≥ αJ · (p0 + p2 ) + 1 + − α J · p2 6 + 3 ≥ αJ · (p0 + p2 ) + J · p2 .

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It follows from (4.2) and (4.3) that for each J ∈ B1 ∧ n o HJΛ − H Λ(1) (ξ) . min 2−J·p2 ξ4 −δ , 2−αJ·(p0 +p2 ) 2−J·p2 ξ4 J . 2−b1 (j1 +j2 +j3 )

(7.2)

for some b1 > 0 . For J ∈ B2 , we choose β satisfying 0 < β < /(4 + ) . Then J · p2 = βJ · p2 + (1 − β)J · p2 ≥ βJ · p2 + (1 − β) 1 + J · p0 2 h i = βJ · (p0 + p2 ) + (1 − β) 1 + − β J · p0 2 ≥ βJ · (p0 + p2 ) + J · p0 . Thus, (4.2) and (4.3) yield that for each J ∈ B2 n ∧ o HJΛ − H Λ(2) (ξ) . min 2−J·p0 ξ4 −δ , 2−βJ·(p0 +p2 ) 2−J·p0 ξ4 J . 2−b2 (j1 +j2 +j3 )

(7.3)

for some b2 > 0 . By the same reasonings as in the previous case, the desired inequalities follow from (7.2) and (7.3). If Λ4 (i) in Lemma 7.2 has more than two line segments or half-lines, we apply again Lemmas 7.1 and 7.2 with the initial set Λ(i). If it has a single line segment or half-line, we stop. Repeating this algorithm finitely many times, we eventually get the following: Theorem 7.1. Given Λ with Λ4 satisfying (i) through (iv), there exist Λ4 (i) ⊂ Λ4 , Bi ⊂ Z3+ , i = 1, . . . , M , with the following properties: (1) Each Λ4 (i) is contained in a single line segment or half-line. S (2) Bi = Z3+ . (3) With Λ(i) = (Λ1 , Λ2 , Λ3 , Λ4 (i)) , for every 1 < p < ∞ ,

M X X

X Λ

Λ(i)

p 4 H ∗ P ∗ f . . kf k + H ∗ P ∗ f J J L (R ) J J

J∈Z3+

Lp (R4 )

i=1

J∈Bi

Lp (R4 )

8. Final Proof of Sufficiency In view of Theorem 7.1,P we now come to a stage that we only have to deal with the operators of type f 7→ J∈B⊂Z3 HJΛ ∗ PJ ∗ f where B ⊂ Z3+ and the Λν satisfy + the following: S4 (i) rank [ ν=1 Λν ] ≤ 2 . S3 (ii) ν=1 Λν ⊂ {e1 , e2 , e3 } and Λ4 ⊂ G where G is a line segment. (iii) Every vertex and edge of N(Λ4 ) is even.

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Note that G is either an edge of N(Λ4 ) or a non-edge. If it is a non-edge, then it can be proved easily that G is contained in some face F of N(Λ4 ) parallel to a coordinate plane. (See Figure 4 in which G4 is a non-edge line segment contained in the face F of the form F = q + X12 .) In this section we finish proving the sufficiency of our main theorem by treating each case separately. 8.1. Edge Case Suppose that G is an even edge of N(Λ4 ) and Λ4 ⊂ G . We recall from section 4 that Z d Λ HJ (ξ) = SΛJ (ξ, t) h(t1 )h(t2 )h(t3 ) dt where R3 ! X J −J·m m SΛ (ξ, t) = exp i 2 < ξ, am > t . m∈∪Λν

As we pointed S out in (3) of Remark 1.2, the conditions (i), (ii) and (iii) imply that all vectors in [ Λν ] have even entries in at least one fixed coordinate. Therefore d 4 Λ H J (ξ) = 0 for all ξ ∈ R . 8.2. Non-Edge Case Suppose now that G is a non-edge line segment contained in some face F of N(Λ4 ) parallel to a coordinate plane. Owing to the conditions (i), (ii), it is evident that F has the form F = q + Xµν with a vertex q = (q1 , q2 , q3 ) and µ, ν ∈ {1, 2, 3} . We may assume F = q + X12 . We shall need the following two lemmas that will reduce our proof to the case of vertex q. Lemma 8.1. Suppose that F ∈ F(Λ4 ) given in the form F = q+X12 with q3 > 0 . If p ∈ F ◦ ∩ Λ4 , then

X Λ Λ(p)

(8.1) H − H ∗ P ∗ f J J J

p 4 . kf kLp (R4 )

J∈Z3+

L (R )

for every 1 0 . −J·q We claim that if 2 ξ4 ≈ 2n for an integer n, then there exists some c > 0 such that ∧ HJΛ − H Λ(p) (ξ) . 2−c (j1 +j2 ) 2−c|n| . (8.2) J To see this, we observe that the expression for p implies −J·p −J·q −(αj +βj ) 1 2 2 ξ 4 = 2 ξ4 2 ≈ 2−(αj1 +βj2 ) 2n .

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Combining with the estimates (4.3) and (4.4), we obtain ∧ n o HJΛ − H Λ(p) (ξ) . min 2−J·p ξ4 , 2−J·q ξ4 −δ J . min 2−(αj1 +βj2 ) 2n , 2−δn the last of which yields the claim (8.2). Next we claim that for fixed j1 , j2 ,

X

Λ(p) Λ 4

H − H ∗ P ∗ L ∗ f J J −J·q−n J

. 2−c (j1 +j2 +|n|) k f kLp (R4 ) .

Lp (R4 )

j3 ∈Z+

(8.3) Indeed, the estimate (8.2), Plancherel’s theorem and the orthogonality of the Littlewood-Paley operator L4−J·q−n in j3 give (8.3) in the case p = 2 . For p 6= 2 , (8.3) follows from Lemma 2.1, Lemma 2.2 and interpolations. Summing the estimate (8.3) over (j1 , j2 ) ∈ Z2+ and over n ∈ Z , we obtain the desired inequality (8.1). Lemma 8.2. Suppose that Λ1 = Λ2 = ∅ , Λ3 = {e3 } and that Λ4 is a subset of ◦ X12 ∩ Z3+ given in the form Λ4 = {q} ∪ { (1 + i ) q : i = 1, . . . , N

with each

i > 0 }

◦ X12

for some N ≥ 2 and a fixed point q ∈ . If p ∈ Λ4 \ {q} , then

X

Λ(p) Λ

H − H ∗ P ∗ f J J J

p 4 . kf kLp (R4 )

(8.4)

L (R )

J∈Z3+

for every 1 0 , q1 > 0 , q2 > 0 , q3 = 0 .

It suffices that the estimate (8.3) holds in the present case. We claim that to prove when 2−j3 ξ4 ≈ 2n with n ∈ Z , ∧ HJΛ − H Λ(p) (ξ) . 2−c (j1 +j2 ) 2−c|n| (8.5) J for some c > 0 . Note that 2−J·p ξ4 = 2−J·q ξ4 2−(q1 j1 +q2 j2 ) . It follows from (4.3) and (4.4) that ∧ n o HJΛ − H Λ(p) (ξ) . min 2−J·q ξ4 2−(q1 j1 +q2 j2 ) , 2−J·q ξ4 −δ J . 2−a (j1 +j2 )

(8.6)

for some a > 0 . From the fact Λ1 ∪ Λ2 ∪ Λ4 ⊂ X12 and Λ3 = {e3 } , we have Z P ∧ i 2−J·m hξ,am itm Λ(e3 ) HJ (ξ) = e m∈∪ν6=3 Λν h(t1 )h(t2 )h(t3 ) dt = 0 . R3

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Exploiting this cancellation property, the mean value theorem yields d \ Λ(p) Λ HJ (ξ) + HJ (ξ) . 2−j3 ξ4 . By the van der Corput type estimate (4.4), we also have −δ \ d Λ(p) HJΛ (ξ) + HJ (ξ) . 2−j3 ξ4 . Therefore, we obtain ∧ n o HJΛ − H Λ(p) (ξ) . min 2−j3 ξ4 , 2−j3 ξ4 −δ . 2−b|n| J

(8.7)

for some b > 0 . These two estimates (8.6) and (8.7) imply the claim (8.5). Arguing as in the proof of Lemma 8.1 the estimate (8.5) now implies the targeted estimate (8.3) and we are done. With the aid of the above two lemmas, we now finish our proof. (1) If G ⊂ F = q + X12 with q3 > 0 , then we apply Lemma 8.1 repeatedly until there remains only a single even vertex, that is, Λ4 = {q} . In this reduced case, the even vertex condition combined with the rank condition implies d 4 Λ that H J (ξ) = 0 for all ξ ∈ R . (2) If G ⊂ F = q + X12 with q3 = 0 and Λ3 = ∅ , then Z SΛJ (ξ, t) h(t3 ) dt3 = 0 d 4 Λ since SΛJ (ξ, t) is independent of t3 . Thus H J (ξ) = 0 for all ξ ∈ R . (3) If G ⊂ F = q + X12 with q3 = 0 and Λ3 = {e3 } , then Λ1 = Λ2 = ∅ S4 because ν=1 Λν ⊂ Π , a plane passing through the origin. Applying Lemma 8.2 finitely many times, we only have to consider the case when Λ4 = {q} , a single even vertex, as in the case (1). The proof of sufficiency (i) of our main theorem is now complete.

9. Necessity In this section we establish the necessity part of our main theorem. 9.1. Restriction to an Edge or a Vertex Given a non-empty set Ω ⊂ Λ4 , PΩ denotes the polynomial X n l PΩ (t) = am,n,l tm 1 t 2 t3 . (m,n,l)∈Ω∩Λ4

We consider the measure µPΩ defined by Z ∞Z ∞Z µPΩ (φ) = p.v. −∞

−∞

∞

−∞

φ(t, PΩ (t))

dt1 dt2 dt3 t1 t 2 t 3

(9.1)

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for each Schwartz function φ on R4 , where Let RPΩ stand for the operator RPΩ f = µPΩ ∗ f . It is the global triple Hilbert transform restricted to PΩ and given by Z ∞Z ∞Z ∞ dt1 dt2 dt3 f (x1 − t1 , x2 − t2 , x3 − t3 , x4 − PΩ (t)) . RPΩ f (x) = p.v. t1 t2 t3 −∞ −∞ −∞ (9.2) Lemma 9.1. Suppose that Ω ⊂ ∂N(Λ4 ) ∩ Λ4 and there exist positive constants a, b, c, d satisfying (i)

(a, b, c) · (m, n, l) = d

for all

(m, n, l) ∈ Ω ,

(ii)

(a, b, c) · (m, n, l) > d

for all

(m, n, l) ∈ Λ4 \ Ω .

Given 1 < p < ∞ , if HPΛ in Main Theorem is bounded in Lp (R4 ), then RPΩ is also bounded in Lp (R4 ) with the operator norm not exceeding kHPΛ kLp (R4 )7→Lp (R4 ) . Remark 9.1. The conditions (i) and (ii) mean that Ω lies on the hyperplane Π : ax + by + cz = d and every other point of Λ4 lies above Π. In other words, Π is a supporting hyperplane of N(Λ4 ) containing Ω. Since N(Λ4 ) is a convex set in the first octant, these geometric conditions always hold if Ω happens to be a vertex or a finite edge (so not parallel to the coordinate axes) of N(Λ4 ). The proof of Lemma 9.1 is based on a dilation argument. Proof. For δ > 0 , we let Bδ = (t1 , t2 , t3 ) ∈ R3 : |t1 | < δ −a , |t2 | < δ −b , |t3 | < δ −c and consider the measure µδ given by Z dt1 dt2 dt3 µδ (φ) = p.v. φ(t, Pδ (t)) t1 t 2 t 3 Bδ

(9.3)

for each Schwartz function φ on R4 , where Pδ (t) = δ −d PΛ4 (δ a t1 , δ b t2 , δ c t3 ) . Using the geometric conditions (i) and (ii) on Ω, we see that µδ → µPΩ as δ → 0 in the sense that (9.4) lim µδ (φ) − µPΩ (φ) = 0 δ→0

for each Schwartz function φ on R4 . We omit the proof of (9.4) since it can be proved by adapting the similar arguments in [3] and [6]. For each fixed δ > 0 , we now consider the convolution RδPΛ f = µδ ∗ f . Then Lemma 9.1 follows from the dilation invariant property such as (HPΛ fδ )δ−1 (x) = RδPΛ f (x) (x ∈ R4 ) , where gδ (x1 , x2 , x3 , x4 ) = g δ −a x1 , δ −b x2 , δ −c x3 , δ −d x4 for any function g de fined on R4 .

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9.2. The Case When N(Λ4 ) Contains a Non-even Vertex Theorem 9.1. Suppose that N(Λ4 ) contains a non-even vertex. Then HPΛ is unbounded in Lp (R4 ) for any 1 < p < ∞ . Proof. Let (m0 , n0 , l0 ) ∈ V(Λ4 ) be non-even and put Ω = {(m0 , n0 , l0 )} . By Definition 1.2, at least two entries of (m0 , n0 , l0 ) are odd. It follows from a theorem of Ricci and Stein [8] that RPΩ is unbounded in Lp (R4 ) for any 1 < p < ∞ . By Lemma 9.1 and Remark 9.1, so is HPΛ . Note that Theorem 9.1 can also be proved by a simplified version of our argument in this section. 9.3. The Case When N(Λ4 ) Contains a Non-even Edge On account of Theorem 9.1, it suffices to consider now the case when N(Λ4 ) contains a non-even edge E each end-point of which is an even vertex of N(Λ4 ). The following lemma shows that such an edge must be a finite edge. Lemma 9.2. If E ∈ E(Λ4 ) is not even and each end-point of E is an even vertex of N(Λ4 ), then it must be a finite line-segment, not parallel to any coordinate axis, given in the form E = q1 q 2

for some even

q1 , q2 ∈ V(Λ4 ) .

Proof. We claim that if E is given in the form E = (m0 , n0 , l0 )+Xν for some even (m0 , n0 , l0 ) ∈ V(Λ4 ) and ν ∈ {1, 2, 3} , then E is even. We may assume ν = 3 so that E ∩ Λ4 = { (m0 , n0 , li ) : i = 0, . . . , N } for some N ≥ 1 . Consider the case m0 = 0 or n0 = 0. Let m0 = 0. If n0 = 0, then E contained on X3 . Thus (E ∩ Λ4 ) ∪ {eν } is even in column because it is contained on the Xν,3 . Hence E is an even edge. Let m0 = 0 and n0 6= 0. Then E ∩ Λ4 is contained X2,3 . Thus rank [(E ∩ Λ4 ) ∪ {e1 }] = 3 and (E ∩ Λ4 ) ∪ {e2 } ⊂ X2,3 and (E ∩ Λ4 ) ∪ {e3 } ⊂ X3 . Hence E is an even edge. It suffices to consider only the case m0 6= 0 and n0 = 6 0. Since (m0 , n0 , l0 ) is even vertex, at least one of m0 , n0 is even, which means that E ∩ Λ4 is even in column. Evidently, we have rank [(E ∩ Λ4 ) ∪ {e1 }] = rank [(E ∩ Λ4 ) ∪ {e2 }] = 3. Although rank [(E ∩Λ4 )∪{e3 }] = 2 , the set (E ∩Λ4 )∪{e3 } is even in column from the fact that m0 or n0 is an even number. Therefore we conclude from Definition 1.2 that E is an even edge. According to (2) of Remark 1.2, an edge E of N(Λ4 ) is not even if either (i) E ∩ Λ4 is not even in column or (ii) (E ∩ Λ4 ) ∪ {eν } is not even in column when E lies on a plane passing through the origin and eν . We start with the case (i).

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9.4. The Case When E ∩ Λ4 Is Not Even in Column Lemma 9.3. Let E be a finite edge of N(Λ4 ) whose end points are even vertices. If E ∩ Λ4 is not even in column, then one of the following three cases holds: Case 1. E ∩ Λ4 = Eooo ∪ Eeee Case 2. E ∩ Λ4 = Eooo ∪ Eeeo or E = Eooo ∪ Eeoe or E = Eooo ∪ Eoee Case 3. E ∩ Λ4 = Eooe ∪ Eeeo or E = Eoeo ∪ Eeoe or E = Eeoo ∪ Eoee . Here we used the notations Eooo through Eeee to classify the subsets of E ∩Λ4 according to the evenness or oddness of the components of their elements. For example, Eoee is the set Eoee = {m ∈ E ∩ Λ4 : m = (odd, even, even)}. Proof. Let E ∩ Λ4 = {m1 , . . . , mN }. Since E is contained on the one line segment, it follows that rank [m2 − m1 , . . . , mN − m1 ] ≤ 1.

(9.5)

We also note that from the even vertex condition of the end points Eeee ∪ Eeeo ∪ Eeoe ∪ Eoee 6= ∅.

(9.6)

Case 1. Eooo 6= ∅ and Eeee 6= ∅. E ∩ Λ4 = Eooo ∪ Eeee as any union of other form breaks down (9.5). Let m1 = (even, even, even) and m2 = (odd, odd, odd) in (9.5). Then we can check that for any choice of m3 6= (odd, odd, odd), (even, even, even), rank [m2 − m1 , m3 − m1 ] = 2 . Case 2. Eooo 6= ∅ and Eeee = ∅. Even vertex condition (9.6) enables us to write E ∩ Λ4 = Eooo ∪ Eeeo or E ∩ Λ4 = Eooo ∪ Eeoe or E ∩ Λ4 = Eooo ∪ Eoee as any union of other forms breaks down (9.5). Case 3. Eooo = ∅. • Eooe ∪ Eoeo ∪ Eeoo 6= ∅. Let Eooe 6= ∅. By the condition that E is not even in column, the last column has at least one odd entry. Thus we have Eeeo ∪ Eeoo ∪ Eoeo 6= ∅. We can show that Eeoo ∪ Eoeo = ∅. If not, (9.6) and the fact Eooe 6= ∅ break (9.5). Thus Eeeo 6= ∅. Hence E ∩ Λ4 = Eooe ∪ Eeeo because any union of other form breaks (9.5). If Eoeo 6= ∅, then E ∩ Λ4 = Eoeo ∪ Eeoe similarly. If Eeoo 6= ∅, then E ∩ Λ4 = Eeoo ∪ Eoee similarly. • Eooe ∪ Eoeo ∪ Eooe = ∅. Since E is not even in column, Eoee 6= ∅, Eeoe 6= ∅ and Eeeo 6= ∅, which breaks down (9.5).

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Let (m1 , n1 , l1 ), (m2 , n2 , l2 ) be the end-points of an edge E. Since we work with E a finite edge of N(Λ4 ), not parallel to the coordinate axis, we observe that two vertices (m1 , n1 , l1 ), (m2 , n2 , l2 ) do not lie on a line passing through the origin. With no loss of generality, hence, we may assume ∆ = m1 n2 − m2 n1 6= 0 .

(9.7)

Lemma 9.4. Let E be a finite edge of N(Λ4 ) with its end points even vertices (m1 , n1 , l1 ) and (m2 , n2 , l2 ) satisfying (9.7). Let Z Z dt1 dt2 (9.8) I1 , 2 (ξ, t3 ) = exp (iξPE (t1 , t2 , t3 )) t1 t2 |t2 | 0, Q(t1 , t2 , 1, +) > |Q(t1 , t2 , 1, −)| . 2

Thus the equality (9.27) does not hold for the Schwartz function f (x) = e−x . This is a contradiction. Hence we proved (9.22) for the case E = Eooe ∪ Eeeo . Proof of (9.22) for E = Eooo ∪ Eeee . Next consider E = Eooo ∪ Eeee in Case 1 of Lemma 9.3. For this case, GP (ξ, t3 ) is Z ∞Z ∞ dt1 dt2 (9.28) 4 exp (iξV (t1 , t2 , t3 , +)) − exp (iξV (t1 , t2 , t3 , −)) t 1 t2 0 0

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where V (t1 , t2 , t3 , +) =

X

n l am,n,l tm 1 t2 t3 +

(m,n,l)∈Eeee

V (t1 , t2 , t3 , −) =

X

X

n l am,n,l tm 1 t2 t3

(m,n,l)∈Eooo

n l am,n,l tm 1 t 2 t3

X

−

(m,n,l)∈Eeee

n l am,n,l tm 1 t 2 t3 .

(m,n,l)∈Eooo

Thus for any Schwartz function f , from (9.23), Z ∞ GP (ξ)fb(ξ)dξ 0= −∞ Z ∞Z ∞ dt1 dt2 f V (t1 , t2 , 1, +) −f V (t1 , t2 , 1, −) =4 , t 1 t2 0 0 2

which does not hold for f (x) = e−x where all coefficients am,n,l ’s are positive. Therefore we proved (9.22) for E = Eooo ∪ Eeee , Case 1 of Lemma 9.3. Proof of (9.22) for E = Eoee ∪ Eooo . It now remains to consider Case 2 of Lemma 9.3. In view of similarity, we shall only deal with the case E = Eoee ∪ Eooo . By the vanishing assumption 0 = GP (ξ) = GP (ξ, t3 ) and (9.17), for all real frequency variables ξ’s and for all real coefficients am,n,l ’s Z ∞Z ∞ dt1 dt2 GP (ξ, t3 ) = exp iξS(t1 , t2 , t3 , +) − exp iξS(t1 , t2 , t3 , −) =0 t1 t 2 −∞ −∞ where S(t1 , t2 , t3 , +) =

X

n l am,n,l tm 1 t2 t3 +

(m,n,l)∈Eoee

S(t1 , t2 , t3 , −) =

X

X

n l am,n,l tm 1 t2 t 3

(m,n,l)∈Eooo

n l am,n,l tm 1 t 2 t3

−

(m,n,l)∈Eoee

X

n l am,n,l tm 1 t 2 t3 .

(m,n,l)∈Eooo

We set in the above ( am,n,l =

ξ1 ξ ξ2 ξ

for (m, n, l) ∈ Eoee for (m, n, l) ∈ Eooo .

By using GP (ξ, t3 ) = GP (ξ, 1) and regarding ξ1 , ξ2 as two frequency variables, we express GP (ξ, t3 ) as Z ∞Z ∞ dt1 dt2 =0 KP (ξ1 , ξ2 ) = U (ξ1 , ξ2 , t1 , t2 ) − U (ξ1 , −ξ2 , t1 , t2 ) t 1 t2 −∞ −∞ where  U (ξ1 , ξ2 , t1 , t2 ) = exp iξ1

 X (m,n)∈Eoe



n tm exp iξ2 1 t2

 X

n tm . 1 t2

(m,n)∈Eoo

Here Eoe = {(m, n) : (m, n, l) ∈ Eoee } and Eoo = {(m, n) : (m, n, l) ∈ Eooo }.

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Lemma 9.7. For any 1 , 2 and ξ1 , ξ2 , Z Z dt1 dt2 U (ξ1 , ξ2 , t1 , t2 ) ≤ C t 1 t2

521

s 1+

B

|ξ2 | |ξ1 |

! (9.29)

where B = {(t1 , t2 ) : 1 < |t2 | < 1/1 , 2 < |t2 | < 1/2 }. Assume that Lemma 9.7 is true. We shall prove this lemma in section 9.6. Let f1 and f2 be two Schwartz functions defined on R. Then we have from (9.23), Z 0 = KP (ξ1 , ξ2 )fb1 (ξ1 )fb2 (ξ2 )dξ Z Z U (ξ1 , ξ2 , t1 , t2 ) − U (ξ1 , −ξ2 , t1 , t2 ) b b f1 (ξ1 )f2 (ξ2 )dt dξ = lim →0 B t1 t2 Z Z U (ξ1 , ξ2 , t1 , t2 ) − U (ξ1 , −ξ2 , t1 , t2 ) b = lim f1 (ξ1 )fb2 (ξ2 )dt dξ →0 t1 t2 B Z Z Z U (ξ1 , ξ2 , t1 , t2 ) − U (ξ1 , −ξ2 , t1 , t2 ) b = lim f1 (ξ1 )fb2 (ξ2 )dξ dt. (9.30) →0 B t1 t2 where we write dξ = dξ1 dξ2 and dt = dt1 dt2 . Third equality follows from the dominated convergence theorem. In applying the dominated convergence theorem, we used Lemma 9.7 for claiming the integrand with respect to dξ integral is dominated by s ! |ξ2 | b C 1+ f1 (ξ1 )fb2 (ξ2 ) |ξ1 | which is an integrable function on R2 . The last equality follows from the Fubini theorem. By using the Fourier inversion formula on (9.30),     Z ∞Z ∞ X X n n f1  tm f2  tm 1 t2 1 t2 −∞

−∞

(m,n)∈Eoe

(m,n)∈Eoo

 −f1 

 X



n tm f2 − 1 t2

(m,n)∈Eoe

 X

n tm 1 t2

(m,n)∈Eoo

dt1 dt2 = 0. t1 t 2

Choose f1 as a positive even Schwartz function and f2 as an odd Schwartz function positive on [0, ∞). Then the above integral is     Z ∞Z ∞ X X n n  dt1 dt2 . 0= f1  tm f2  tm 1 t2 1 t2 t1 t2 0 0 (m,n)∈Eoe

(m,n)∈Eoo

However we can easily observe that this integral is strictly greater than 0, since all the values of f1 and f2 above are positive.

522


9.6. Proof of Lemma 9.7 For each J = (j1 , j2 ) ∈ Z2 , we define UJ (ξ1 , ξ2 , t1 , t2 ) by    X X n exp iξ1 2−J·(m,n) tm exp iξ2 1 t2 (m,n)∈Eoe

IEOT

 n 2−J·(m,n) tm . 1 t2

(m,n)∈Eoo

We make a dyadic decomposition, Z Z TJ (ξ1 , ξ2 ) = UJ (ξ1 , ξ2 , t1 , t2 )h(t1 )h(t2 )dt1 dt2 where the function h is given as in section 4. In proving (9.29), it suffices to prove that for any Z ⊂ Z2 , s ! X |ξ2 | (9.31) TJ (ξ1 , ξ2 ) ≤ C 1 + |ξ1 | J∈Z

Proof of (9.31). Remind that (m1 , n1 , l1 ) and (m2 , n2 , l2 ) are two even vertices on the edge E. Thus they are contained in Eoee . Since (m, n, l) ∈ Eooo is contained in the interior of the edge E, for any (m, n) ∈ Eoo , −J·(m,n) |ξ2 | −J·(m1 ,n1 ) c1 −J·(m2 ,n2 ) c2 2 ξ2 ≤ 2 ξ1 2 ξ1 (9.32) |ξ1 | where c1 , c2 > 0 and c1 + c2 = 1. Let   Z Z X n SJ (ξ1 , ξ2 ) = exp iξ1 2−J·(m,n) tm h(t1 )h(t2 )dt1 dt2 . 1 t2 (m,n)∈Eoe

By using the fact that (m1 , n1 ), (m2 , n2 ) ∈ Eoe and van der Corput’s lemma in Lemma 4.1 n −δ o −δ |TJ (ξ1 , ξ2 )| + |SJ (ξ1 , ξ2 )| ≤ C min 2−J·(m1 ,n1 ) ξ1 , 2−J·(m2 ,n2 ) ξ1 (9.33) By using the mean value theorem in Lemma 4.1 and (9.32), for some positive c1 , c2 , |ξ2 | −J·(m1 ,n1 ) c1 −J·(m2 ,n2 ) c2 ξ1 2 ξ1 , 1 |TJ (ξ1 , ξ2 ) − SJ (ξ1 , ξ2 )| ≤ C min 2 |ξ1 | s |ξ2 | −J·(m1 ,n1 ) c1 /2 −J·(m2 ,n2 ) c2 /2 ≤C ξ1 ξ1 (9.34) 2 2 |ξ1 | Restrict frequency variables, −J·(m1 ,n1 ) ξ1 ≈ 2k1 and 2−J·(m2 ,n2 ) ξ2 ≈ 2k2 . 2 Hence by using (9.33), (9.34) and Lemma 2.3, for ξ1 , ξ2 satisfying (9.35), s ! |ξ2 | 2−c(|k1 |+|k2 |) . |TJ (ξ1 , ξ2 ) − SJ (ξ1 , ξ2 )| ≤ C 1 + |ξ1 |

(9.35)

(9.36)

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523

In view of (9.7), for each fixed k1 , k2 and ξ1 , ξ2 , there are at most finitely many J’s satisfying (9.35). Hence s ! X |ξ2 | 2−c(|k1 |+|k2 |) . |TJ (ξ1 , ξ2 ) − SJ (ξ1 , ξ2 )| ≤ C 1 + |ξ1 | J⊂Z

Summing over all (k1 , k2 ) ∈ Z2 , s X

|TJ (ξ1 , ξ2 ) − SJ (ξ1 , ξ2 )| ≤ C

1+

J⊂Z

|ξ2 | |ξ1 |

! .

(9.37)

By using Lemma 9.4, X SJ (ξ1 , ξ2 ) ≤ C. J⊂Z

This combined with (9.37) completes the proof of (9.31).

9.7. The Case When (E ∩ Λ4 ) ∪ {e3 } Is Not Even in Column Lemma 9.8. Let E be a finite edge of N(Λ4 ). Suppose that the end-points of E are even vertices and E ∩ Λ4 is even in column but (E ∩ Λ4 ) ∪ {e3 } is not even in column with rank [(E ∩ Λ4 ) ∪ {e3 }] = 2 . Then E ∩ Λ4 = Eooe ∪ Eeee Proof. Let E ∩ Λ4 = {m1 , . . . , mN }. Then rank [(0, 0, 1), m1 , . . . , mN ] = 2 .

(9.38)

By the condition of Lemma 9.8 that E ∩ Λ4 is even in column but (E ∩ Λ4 ) ∪ {e3 } is not even in column, we see that every third component of m1 , . . . , mN is an even number. Since each of the first two columns for the matrix [m1 , . . . , mN ] has odd entry, Eoee ∪ Eeoe ∪ Eooe 6= ∅. If at least two among Eoee , Eeoe and Eooe are nonempty sets, then (9.38) does not hold. Thus only Eooe 6= ∅, since each of the first two columns of the matrix [m1 , . . . , mN ] contains odd entry. Also Eeee 6= ∅ by the even vertex condition of the end points. Finally any union of other form with {e3 } ∪ Eooe ∪ Eeee breaks (9.38). The rank condition of Lemma 9.8 says that E lies on a plane of the form y = γx for some γ > 0 . Thus we may index E ∩ Λ4 = { (mi , ni , li ) : ni = γmi , i = 1, . . . , N } . Let (m1 , n1 , l1 ) and (m2 , n2 , l2 ) represent the end-points of E so that (mi , ni , li ) = (1 − λi )(m1 , n1 , l1 ) + λi (m2 , n2 , l2 ) for some 0 ≤ λi ≤ 1 , i = 1, . . . , N . In addition, we can observe that ∆ = m1 l2 − m2 l1 6= 0 .

(9.39)

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Lemma 9.9. Let E be a finite edge of N(Λ4). Suppose that the end-points (m1 , n1 , l1 ) and (m2 , n2 , l2 ) of E are even vertices satisfying (9.39) and that E ∩ Λ4 is even in column but (E ∩ Λ4 ) ∪ {e3 } is not even in column with rank [(E ∩ Λ4 ) ∪ {e3 }] = 2 . We set Z Z dt1 dt3 , I1 , 3 (ξ3 , ξ4 , t2 ) = exp i (ξ3 t3 + ξ4 PE (t1 , t2 , t3 )) t 1 t3 |t3 | 0, x ∈ X, f ∈ Y with x ≤ 1, f ≤ 1. Then there exists an nj ∈ N such that n1j < 2 and |f (F x) − f (Fnj x)| < 2 . Then |f (T x) − f (F x)| ≤ |f (T x) − f (Fnj x)| + |f (Fnj x) − f (F x)| 1 + < d + . ≤ T − Fnj + < d + 2 nj 2 Since this holds for all x ∈ X, f ∈ Y with x ≤ 1, f ≤ 1 and > 0 is arbitrary, we get T − F ≤ d. On the other hand, since rank(F ) ≤ k − 1, we have T − F ≥ d. Hence T − F = d. From this, the particular case is obvious. The following example shows that the conclusion of the Lemma 2.4 does not hold if the space Y is not reflexive. Example. Let Tn : 1 → 1 be defined by Tn x = x(1)en ,

x = (x(1), x(2), . . .) ∈ 1 ,

n ∈ N.

Vol. 65 (2009)

Approximation Numbers of Operators

535

Clearly Tn ∈ BL(1 ) with rank(Tn ) = 1 and Tn = 1 for all n ∈ N. We claim that {Tn } does not have a subsequence which converges to an operator with respect to the weak operator topology. To see this, suppose there exist a subsequence {Tnk } w of {Tn } and an operator T ∈ BL(1 ) such that Tnk x −→ T x for each x ∈ 1 as k → ∞. Then f (Tnk x) → f (T x) as k → ∞ for all f ∈ (1 ) and x ∈ 1 . In particular, f (enk ) = f (Tnk e1 ) → f (T e1 ) as k → ∞ holds also for f ∈ (1 ) defined by ∞ f (x) = (−1)k x(nk ), x ∈ 1 . k=1

k

But then f (enk ) = (−1) , giving a contradiction to the convergence of f (enk ) and hence to the existence of such a subsequence {Tnk }. Remark 2.6. Note that the sequence {Tn } in the preceding example converges to 0 in the weak* operator topology if 1 is regarded as the dual space of c00 or c0 with respect to the norm · ∞ . To see this, let J denote the canonical isometry from 1 → (c0 ) , defined by ∞ (Jx)(y) = x(j)y(j), x ∈ 1 , y ∈ c0 . j=1 −1

Then JTn J ∈ BL((c0 ) ). Now, for g ∈ (c0 ) , let y ∈ 1 be such that J(y) = g. Then for x ∈ X, (JTn J −1 g)(x) = (JTn y)(x) = y(1)(Jen )(x) = y(1)x(n) → 0 wo

∗

as n → ∞.

wo

0 as n → ∞. Thus JTn J −1 −→ 0 as n → ∞. It can be seen that JTn J −1 −→ The following proposition shows that, for Lemma 2.4 to hold, reflexivity of Y is not only sufficient but also necessary. Proposition 2.7. If Y is a non-reflexive space, then there exists a uniformly bounded sequence {Tn } of operators in BL(X, Y ) such that rank(Tn ) ≤ 1 for all n ∈ N and {Tn } does not have any subsequence which converges in the weak operator topology. Proof. Suppose Y is not reflexive. Then, by Eberlein’s theorem (cf. [4], Theorem 16.5), there exists a bounded sequence {un } in Y which does not have a weakly convergent subsequence. Let a ∈ X with a = 1. By a consequence of the Hahn-Banach Extension Theorem, there exists a functional g ∈ X of norm 1 such that g(a) = a = 1 ([5], Corollary 5.6). We define Tn : X → Y by Tn x = g(x)un for x ∈ X. Then for each n ∈ N, Tn ∈ BL(X, Y ) is of rank 1 and the sequence {Tn} is bounded. We claim that {Tn } does not have any subsequence which converges in the weak operator topology. Suppose there exist a subsequence {Tnj } of {Tn } and an operator T ∈ w BL(X, Y ) such that Tnj x −→ T x as j → ∞, for all x ∈ X. This gives f (unj ) = f (Tnj a) → f (T a) as j → ∞, for every f ∈ Y . But this gives a weakly convergent subsequence of {un }, contradicting the choice of {un }.

536

Deepesh, Kulkarni and Nair

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Now, we prove the main theorem of this section which generalizes Theorem 1.3 to operators in BL(X, Y ), when X is separable and Y is reflexive. Theorem 2.8. Let X and Y be normed linear spaces, T ∈ BL(X, Y ), and {Pn } and {Qn } be sequences of operators in BL(X) and BL(Y ) respectively such that Pn Qn ≤ 1 for all n ∈ N. Then sk (Qn T Pn ) ≤ sk (T ) for all k ∈ N and n ∈ N. wo Further, if X is separable, Y is reflexive and Tn := Qn T Pn −→ T as n → ∞, then for each k ∈ N, lim sk (Tn ) = sk (T ).

n→∞

Proof. Fix k ∈ N and denote d := sk (T ) and dn := sk (Tn ). For > 0, let F ∈ BL(X, Y ) be such that rank(F ) ≤ k − 1 and T − F < d + . Then for any n ∈ N, Qn T Pn − Qn F Pn ≤ Qn T − F Pn ≤ T − F < d + and rank(Qn F Pn ) ≤ k−1. Hence dn < d+ for all n ∈ N. Since > 0 was arbitrary, we obtain supn sk (Tn ) ≤ sk (T ). This proves the first part of the conclusion. wo Next, let X be separable, Y be reflexive and Tn := Qn T Pn −→ T as n → ∞. Then the conclusion holds trivially if d = 0. Assume d > 0 and dn → d. Then there exists an > 0 such that dn < d − for infinitely many n. Hence there exist operators Fnj ∈ BL(X, Y ) such that rank(Fnj ) ≤ k − 1 and Qnj T Pnj − Fnj < d − for all j ∈ N. Thus Fnj ≤ Fnj − Qnj T Pnj + Qnj T Pnj < d + T ∀ j ∈ N. Hence by Lemma 2.4, there exist an operator F ∈ BL(X, Y ) with rank(F ) ≤ k − 1 and a subsequence {Fnj } of {Fnj } such that for each x ∈ X and f ∈ Y , |f (F x) − f (Fnj x)| → 0 as → ∞. Now let x ∈ X, f ∈ Y be such that x ≤ 1 and f ≤ 1. Then |f (T x) − f (F x)| ≤ |f (T x) − f (Qnj T Pnj x)| + |f (Qnj T Pnj x) − f (Fnj x)| + |f (Fnj x) − f (F x)| Note that for each , |f (Qnj T Pnj x) − f (Fnj x)| ≤ Qnj T Pnj − Fnj < d − , whereas the terms |f (T x) − f (Qnj T Pnj x)| and |f (Fnj x) − f (F x)| can be made less than /3 by choosing sufficiently large. Hence |f (T x) − f (F x)| ≤ d − /3. Since this holds for each x ∈ X and f ∈ Y with x ≤ 1 and f ≤ 1, we have T − F ≤ d − /3 so that d ≤ d − /3. Thus we arrive at a contradiction. Hence dn → d as n → ∞. Corollary 2.9. Let X and Y be as in Lemma 2.4 and T ∈ BL(X, Y ). Let {Pn } w and {Qn } be as in Theorem 2.8. Further, if Pn x → x and Qn y −→ y as n → ∞ for every x ∈ X and y ∈ Y , then sk (Tn ) → sk (T ) as n → ∞, for each k ∈ N.

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537

Proof. For x ∈ X and f ∈ Y , |f (Qn T Pn x) − f (T x)| ≤ |f (Qn T Pn x) − f (Qn T x)| + |f (Qn T x) − f (T x)| ≤ f Qn T Pnx − x + |f (Qn (T x)) − f (T x)|, which tends to 0 as n → ∞, since Pn x → x and f (Qn y) → f (y) as n → ∞, for wo every x ∈ X, y ∈ Y and f ∈ Y . This implies Tn −→ T as n → ∞. Hence the result follows from Theorem 2.8. In Theorem 2.8, the operators Pn or Qn need not be projections. Example. For n = 2, 3, . . ., consider Pn : 2 → 2 defined by Pn (x(1), x(2), . . .) = (x(1), . . . , x(n−2), x(n), x(n−1), 0, . . .), (x(1), x(2), . . .) ∈ 2 . Then Pn ∈ BL(2 ) with Pn = 1 and Pn x → x as n → ∞, for all x ∈ X. Now for X = Y = 2 , Pn = Qn satisfy all the assumptions of Theorem 2.8. It is clear that Pn is not a projection operator on 2 .

3. Approximation under the duality assumption In view of Proposition 2.7 we know that the conclusion in Lemma 2.4 holds if and only if the codomain Y is reflexive. Now we prove a result, namely Lemma 3.1, analogous to Lemma 2.4 for the case when Y is not necessarily reflexive, but is the dual space of a separable normed linear space. Here the convergence in weak operator topology is also weakened by convergence in the weak* operator topology. The arguments in the proof of Lemma 3.1 and the subsequent corollary are similar to that of Lemma 2.4 and Corollary 2.5 with weak sense of convergence replaced by weak* sense of convergence. However, for the sake of completeness of the exposition, we supply a detailed proof. Lemma 3.1. Let X be a separable normed linear space, Y be the dual space of a separable normed linear space. Let k ∈ N and {Tn } be a uniformly bounded sequence in BL(X, Y ) with rank(Tn ) ≤ k for all n ∈ N. Then there exist an operator T ∈ BL(X, Y ) with rank(T ) ≤ k and a subsequence {Tn } of {Tn } such wo∗

that Tn −→ T as → ∞. Proof. Since rank(Tn ) =: kn ≤ k for all n ∈ N, by Proposition 2.3, we can write Tn x =

kn i=1

(n)

(n)

ψi (x)wi ,

(n)

x ∈ X, n ∈ N, (n)

(n)

n where {wi }ki=1 forms a basis of range of Tn with wi ≤ 1 and ψi ∈ X with (n) ψi ≤ M , for some M > 0 and for all n ∈ N, i = 1, 2, . . . , k. Here also if kn < k (n) (n) for some n, then taking wj = 0 and ψj = 0 for j > kn , we can write

Tn x =

k i=1

(n)

(n)

ψi (x)wi ,

x ∈ X, n ∈ N.

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By hypothesis, X is separable and Y = Z , for some separable normed linear space (n) (n) Z. Therefore, for each i, the bounded sequences {wi } in Z and {ψi } in X have weak* convergent subsequences. Thus, it follows by considering subsequences that there exist wi ∈ Y and ψi ∈ X , i = 1, 2, . . . , k such that (n )

ψi

(x) → ψi (x),

(n )

wi

(z) → wi (z)

as → ∞ for every x ∈ X, z ∈ Z. Define T : X → Y by Tx =

k

ψi (x)wi ,

x ∈ X.

i=1

Then T ∈ BL(X, Y ), rank(T ) ≤ k and for each x ∈ X, z ∈ Z, we have (Tn x)(z) =

k j=1

(n )

gj

(n )

(x)wj

(z) →

k

gj (x)wj (z) = T x(z),

j=1

as → ∞. The above operator T satisfies the requirements in the lemma.

Corollary 3.2. Let X and Y be as in Lemma 3.1, T ∈ BL(X, Y ) and k ∈ N. Then there exists an operator F ∈ BL(X, Y ) with rank(F ) ≤ k − 1 such that T − F = sk (T ). In particular, sk (T ) = 0 if and only if rank(T ) ≤ k − 1. Proof. Let sk (T ) = d. For all n ∈ N, there exist Fn ∈ BL(X, Y ) such that rank(Fn ) ≤ k − 1 and T − Fn < d + n1 . Thus Fn ≤ T + d + 1 for all n ∈ N. Hence by Lemma 3.1, there exist an operator F ∈ BL(X, Y ) with rank(F ) ≤ k − 1 wo∗

and a subsequence {Fnj } of {Fn } such that Fnj −→ F as j → ∞. Now let > 0, x ∈ X, z ∈ Z with x ≤ 1, z ≤ 1. Then there exists an nj ∈ N such that n1j < 2 and |(F x)(z) − (Fnj x)(z)| < 2 . Then |(T x)(z) − (F x)(z)| ≤ |(T x)(z) − (Fnj x)(z)| + |(Fnj x)(z) − (F x)(z)| 1 + < d + . ≤ T − Fnj + < d + 2 nj 2 Since this holds for all x ∈ X, z ∈ Z with x ≤ 1, z ≤ 1, and > 0 is arbitrary, we get T − F ≤ d. On the other hand since rank(F ) ≤ k − 1, T − F ≥ d. Hence T − F = d. If sk (T ) = 0, then T = F from the above and hence rank(T ) ≤ k − 1. Using Lemma 3.1 we prove a theorem analogous to Theorem 2.8 for operators in BL(X, Y ), where X and Y are as in Lemma 3.1. Theorem 3.3. Let X and Y be as in Lemma 3.1 and T ∈ BL(X, Y ). Let {Pn } and {Qn } be sequences of operators in BL(X) and BL(Y ) respectively such that wo∗

Pn Qn ≤ 1 for each n ∈ N. If Tn := Qn T Pn −→ T as n → ∞, then for each k ∈ N, lim sk (Tn ) = sk (T ). n→∞

Vol. 65 (2009)


539

Proof. Let k ∈ N and denote d := sk (T ) and dn := sk (Tn ). From Theorem 2.8 we have supn sk (Tn ) ≤ sk (T ). The conclusion holds trivially if d = 0. So assume d > 0 and dn → d. Then there exist an > 0 and infinitely many n such that dn < d − . Hence there exist operators Fnj ∈ BL(X, Y ) with rank(Fnj ) ≤ k − 1 such that Qnj T Pnj − Fnj < d − for all j ∈ N. Thus Fnj ≤ Fnj − Qnj T Pnj + Qnj T Pnj < d + T ∀ j ∈ N. Hence by Lemma 3.1, there exist an operator F ∈ BL(X, Y ) with rank (F ) ≤ k −1 and a subsequence {Fnj } of {Fnj } such that for each x ∈ X, z ∈ Z, |(F x)(z) − (Fnj x)(z)| → 0 as → ∞. Now, let x ∈ X, z ∈ Z be such that x ≤ 1 and z ≤ 1. Then |(T x)(z) − (F x)(z)| ≤ |(T x)(z) − (Qnj T Pnj x)(z)| + |(Qnj T Pnj x)(z) − (Fnj x)(z)| + |(Fnj x)(z) − (F x)(z)|. Note that for each , |(Qnj T Pnj x)(z) − (Fnj x)(z)| ≤ Qnj T Pnj − Fnj < d − , whereas the terms |(T x)(z) − (Qnj T Pnj x)(z)| and |(Fnj x)(z) − (F x)(z)| can be made less than /3 by choosing sufficiently large. Hence |(T x)(z) − (F x)(z)| < d − /3. Since this holds for each x ∈ X and z ∈ Z with x ≤ 1 and z ≤ 1, we have T − F ≤ d − /3 so that d ≤ d − /3. Thus we arrive at a contradiction. Hence dn → d as n → ∞. Corollary 3.4. Let X and Y be as in Lemma 3.1 and T ∈ BL(X, Y ). Let {Pn } w∗

and {Qn } be as in Theorem 3.3 such that Pn x → x and Qn y −→ y as n → ∞ for all x ∈ X, y ∈ Y . Then for each k ∈ N, lim sk (Tn ) = sk (T ).

n→∞

Proof. Let x ∈ X and z ∈ Z. Then |(Qn T Pn x)(z) − (T x)(z)| ≤ |(Qn T Pn x)(z) − (Qn T x)(z)| + |(Qn T x)(z) − (T x)(z)| ≤ Qn T Pnx − x z + |(Qn (T x))(z) − (T x)(z)|, which tends to 0 as n → ∞, since Pn x → x and (Qn y)(z) → y(z) as n → ∞, for wo∗

every x ∈ X, y ∈ Y and z ∈ Z. This implies Tn −→ T as n → ∞. Hence the result follows from Theorem 3.3. Remark 3.5. We observe that the conclusion in Theorem 3.3 follows even if we take Y to be linearly isometric with the dual of a separable space Z and assume wo∗

JTn −→ JT as n → ∞, where J denotes the linear isometry from Y to Z .

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Further, we observe that if {Tn } is a sequence in BL(X, Y ) and T ∈ BL(X, Y ), wo∗

wo

then Tn −→ T if and only if Tn −→ T . This is seen as follows: wo

Tn −→ T

⇐⇒

f (Tn x) → f (T x)

∀f ∈ Y , x ∈ X

⇐⇒

(Tn f )(x) → (T f )(x)

∀ f ∈ Y , x ∈ X

⇐⇒ ⇐⇒

w

∗

Tn f −→ T f wo

∗

Tn −→ T

∀f ∈ Y as n → ∞ wo

Also, if X is reflexive, then Tn → T in the strong sense implies Tn −→ T . To see this, let J denote the canonical isometry from X onto X and φ ∈ X . Let x ∈ X be such that Jx = φ. Then for every f ∈ Y , |φ(Tn f ) − φ(T f )| = |(Jx)(Tn f ) − (Jx)(T f )| = |(Tn f )(x) − (T f )(x)| = |f (Tn x) − f (T x)| ≤ f Tnx − T x. wo

Hence, if Tn x − T x → 0 as n → ∞ for every x ∈ X, then Tn −→ T as n → ∞. In view of Remark 3.5 together with Theorems 3.3 and 2.8, we obtain the following corollary. Corollary 3.6. Let T ∈ BL(X, Y ) and Tn := Qn T Pn for all n ∈ N, where {Pn } and {Qn } are sequences of operators in BL(X) and BL(Y ) respectively such that Pn Qn ≤ 1 for all n ∈ N. Then lim sk (Tn ) = sk (T ) if any of the following n→∞

holds:

wo

(i) X and Y are separable and Tn −→ T as n → ∞. wo (ii) X is reflexive, Y is separable and Tn −→ T as n → ∞.

4. Approximation under the compactness assumption In [7], Pietsch discussed the question whether the equality sk (T ) = sk (T ) holds for every T ∈ BL(X, Y ) and k ∈ N. In [3], Hutton proved that this is true for compact operators T in BL(X, Y ) and gave an example to show that this equality need not be true for a non-compact operator. We now make use of the above equality of approximation numbers for compact operators to derive the following results using Corollary 3.6. Theorem 4.1. Let T ∈ BL(X, Y ) and Tn := Qn T Pn for all n ∈ N, where {Pn } and {Qn } be sequences of operators in BL(X) and BL(Y ) respectively such that Pn Qn ≤ 1 for all n ∈ N. Suppose X, Y and Tn satisfy any of the conditions (i) and (ii) of Corollary 3.6. Then we have the following: (a) If Tn is compact for every n ∈ N, then sk (T ) = sk (T ) ⇐⇒

lim sk (Tn ) = sk (T ).

n→∞

(b) If T is compact, then lim sk (Tn ) = sk (T ). n→∞

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Proof. Part (a) follows from Corollary 3.6 by making use of the fact that sk (Tn ) = sk (Tn ) whenever Tn is compact, and (b) is a consequence of (a) by using the equality sk (T ) = sk (T ) whenever T is compact. Theorem 4.2. Let X, Y, Tn be as in Theorem 4.1 and T be a compact operator from X to Y . If, in addition, Pn x → x and Qn y → y as n → ∞ for all x ∈ X, y ∈ Y and X and Y satisfy any of the conditions (i) and (ii) of Corollary 3.6, then for each k ∈ N, sk (Tn ) → sk (T ) as n → ∞. Proof. Since Pn x → x and Qn y → y as n → ∞ for all x ∈ X, y ∈ Y , we obtain wo that Tn → T in strong sense. This gives Tn −→ T and the result follows from Theorem 4.1 under the condition (i) of Corollary 3.6. When X is reflexive, from wo Remark 3.5, we have Tn −→ T whenever Tn → T in strong sense, and this gives the result under the condition (ii) of Corollary 3.6. The following corollary extends the result in [3] so as to include non-compact operators in certain cases. Corollary 4.3. Let X, Y, T and Tn be as in Corollary 3.6. Then for each k ∈ N, sk (T ) = sk (T ) if any of the following holds: (a) X and Y are reflexive and separable and Tn → T in the strong sense as n → ∞. (b) X and Y are separable, Y is the dual space of some normed linear space and wo Tn −→ T as n → ∞. (c) X and Y are as in (a) or (b) and Pn x → x and Qn y → y as n → ∞ for all x ∈ X and y ∈ Y . wo

Proof. From Remark 3.5, Tn → T in the strong sense implies Tn −→ T and wo Tn −→ T as n → ∞, whenever X is reflexive. Now, under the assumption in (a), the equality sk (T ) = sk (T ) follows from Theorem 4.1 and 2.8, due to the wo

wo∗

reflexivity and separability of X and Y . Since Tn −→ T implies Tn −→ T and wo∗ Tn −→ T as n → ∞, the result follows from Theorem 3.3 and 4.1, under the assumption in (b). The assumptions in (c) gives the strong convergence of Tn to T . Hence the result follows from (a) and (b) in this particular case. In [3], Hutton proved the following proposition to establish that sk (T ) need not be equal to sk (T ) for a general non-compact operator T . Proposition 4.4. (cf. [3], Proposition 2.3) If IE : 1 → c0 and IF : 1 → ∞ are the natural injections, then sk (IE ) = 1 for each k ∈ N and sk (IF ) = 1/2 for each k ∈ {2, 3, . . .}. Note that the transpose of the natural injection IE : 1 → c0 is linearly isometric with IF : 1 → ∞ . Now, we make use of Proposition 4.4 to establish that {sk (Qn T Pn )} need not converge to sk (T ) for a non-compact operator T ∈ BL(X, Y ) if the codomain is not the dual space of some separable space. To see this, let T = IE and let Pn ∈ BL(1 ) and Qn ∈ BL(c0 ) be the projection

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operators defined by (x(1), x(2), . . .) → (x(1), x(2), . . . , x(n), 0, 0, . . .) on 1 and c0 , respectively. Then Tn := Qn T Pn satisfy all the assumptions in Corollary 3.6(i). By Proposition 4.4, sk (T ) = sk (T ). Hence, Theorem 4.1(a) shows that sk (Qn T Pn ) → sk (T ) as n → ∞. Remark 4.5. From Corollary 4.3(b) and Proposition 4.4, it follows that c0 is a not linearly isometric with the dual of any normed linear space. Remark 4.6. We would like to remark that there are spaces X and Y admitting the sequences (Pn ) and (Qn ) of operators in BL(X) and BL(Y ), respectively, satisfying the conditions of Corollary 4.3. For example, if Y = p with 1 < p < ∞, which is a reflexive space, then for any separable normed linear space X and for any T ∈ BL(X, Y ), we have sk (T ) = sk (T ), for in this case we may take Pn = I for all n ∈ N and Qn x = (x(1), x(2), . . . , x(n), 0, 0, . . .),

x = (x(1), x(2), . . .) ∈ p .

Remark 4.7. Since the strict inclusion map from X to Y is used in Proposition 4.4 for getting the counter examples, it is of interest to see if the conclusions in Theorem 3.3 and 4.1 hold if X and Y are the same normed linear spaces, by removing the additional assumptions on the codomain. This still remains an open question.

References [1] A. B¨ ottcher, A. V. Chithra and M. N. N. Namboodiri, Approximation of approximation numbers by truncation. Integr. equ. oper. theory 39 (2001), 387–395. [2] A. B¨ ottcher and S. M. Grudsky, Toeplitz matrices, asymptotic linear algebra, and functional analysis. Hindustan Book Agency, New Delhi, 2000. [3] C. V. Hutton, On the approximation numbers of an operator and its adjoint. Math. Ann. 210 (1974), 277–280. [4] B. V. Limaye, Functional analysis Second edition. New Age, New Delhi, 1996. [5] M. T. Nair, Functional analysis: A First Course. Prentice-Hall of India, New Delhi, 2002. [6] A. Pietsch, Operator ideals. North-Holland, Amsterdam, 1980. [7] A. Pietsch, s-numbers of operators in Banach spaces. Studia Math. 51 (1974), 201–223. [8] E. Schock, On projection methods for linear equations of the second kind. J. Math. Anal. Appl. 45 (1974), 293–299. K. P. Deepesh, S. H. Kulkarni and M. T. Nair Department of Mathematics, IIT Madras, Chennai 600 036, India e-mail: [email protected] [email protected] [email protected] Submitted: March 21, 2009. Revised: May 4, 2009.

Integr. equ. oper. theory 65 (2009), 543–550 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040543-8, published online October 22, 2009 DOI 10.1007/s00020-009-1724-8


Compact Operators that Commute with a Contraction K. Kellay and M. Zarrabi Abstract. Let T be a C0 –contraction on a separable Hilbert space. We assume that IH − T ∗ T is compact. For a function f holomorphic in the unit disk D and continuous on D, we show that f (T ) is compact if and only if f vanishes on σ(T ) ∩ T, where σ(T ) is the spectrum of T and T the unit circle. If f is just a bounded holomorphic function on D, we prove that f (T ) is compact if and only if lim T n f (T ) = 0. n→∞

Mathematics Subject Classification (2000). Primary 47B05; Secondary 30H05. Keywords. Compact operators, essentially unitary, commutant.

1. Introduction Let H be a separable Hilbert space, and L(H) the space of all bounded operators on H. For T ∈ L(H), we denote by σ(T ) the spectrum of T . The Hardy space H∞ is the set of all bounded and holomorphic functions on D. A contraction T on H is called a C0 –contraction (or in class C0 ) if it is completely nonunitary and there exists a nonzero function θ ∈ H∞ such that θ(T ) = 0. A contraction T is said essentially unitary if IH − T ∗ T is compact, where IH is the identity map on H. Let T be a C0 –contraction on H, and let H∞ (T ) = {f (T ) : f ∈ H∞ } be the subspace of the commutant {T } = {A ∈ L(H) : AT = T A} obtained from the Nagy–Foias functional calculus. In this note we study the question of when H∞ (T ) contains a nonzero compact operator. B. Sz–Nagy [12] proved that {T } contains always a nonzero compact operator, but there exists a C0 –contraction T such that zero is the unique compact operator contained in H∞ (T ). Nordgreen [15] proved that if T is an essentially unitary C0 –contraction then H∞ (T ) contains a nonzero compact operator. There are also results about the existence of smooth operators The research of the authors was supported in part by ANR–Dynop and ANR–Frab.

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(finite rank, Schatten–von Neuman operators) in H∞ (T ) (see [17]). It is also shown in Atzmon’s paper [2], that if T is a cyclic completely nonunitary contraction such that σ(T ) = {1} and √ (1) log T −n = O( n), n → ∞, then T − IH is compact. Let A(D) be the usual disc algebra, i.e. the space of all functions which are holomorphic on D and continuous on D. In section 2 we study the compactness of f (T ) when f is in the disk algebra. We show (Corollary 2.3), that, if f ∈ A(D) and if T is a C0 –contraction which is essentially unitary, then f (T ) is compact if and only if f vanishes on σ(T ) ∩ T. The main tool used in the proof of this result is the Beurling-Rudin theorem about the characterization of the closed ideals of A(D) . We show also for a large class of C0 –contractions that the condition “ T is essentially unitary” is necessary in the above result (Proposition 2.5) . As a consequence, we obtain that if T is a contraction that is annihilated by a nonzero function in A(D) and if T is cyclic (or, more generally, of finite multiplicity) then f (T ) is compact whenever f ∈ A(D) and f vanishes on σ(T ) ∩ T. We notice that an invertible contraction with spectrum reduced to a single point and satisfying condition (1) is necessarily annihilated by a nonzero function in A(D) (see [1]). In section 3, we are interested in the compactness of f (T ) when f ∈ H∞ . With the help of the corona theorem, we show (Theorem 3.4) that if T is an essentially unitary C0 -contraction, then f (T ) (f ∈ H∞ ) is compact if and only if lim T n f (T ) = 0. We obtain in particular that if lim f (rz) = 0 for every n→∞

z ∈ σ(T ) ∩ T, then f (T ) is compact.

r→1−

2. Compactness of f(T) with f in the disk algebra Let T be a contraction on H. We will introduce some definitions and results we will need later. We call λ ∈ σ(T ) a normal eigenvalue if it is an isolated point of σ(T ) and if the corresponding Riesz projection has finite rank. We denote by σnp (T ) the set of all normal eigenvalues of T . The weakly continuous spectrum of T is defined by σwc (T ) = σ(T ) \ σnp (T ) (see [14], p. 113). Let us suppose that T is essentially unitary and D \ σ(T ) = ∅. There exists a unitary operator U and a compact operator K such that T = U +K and then we have σwc (T ) = σwc (U ) ⊂ T (see [5], [7] Theorem 5.3, p. 23 and [14] p. 115). It follows from the above observation that if D \ σ(T ) = ∅ then T is essentially unitary if and only if T ∗ is essentially unitary too. Let I be a closed ideal of A(D) . We denote by SI the inner factor of I, that is the greatest inner common divisor of all nonzero functions in I (see [8] p. {ζ ∈ T : f (ζ) = 0} and J (E) = {f ∈ A(D) : f|E = 0}, 85). We set Z(I) = f ∈I

for E ⊂ T. We shall need the Beurling-Rudin theorem [16] (see also [8] p. 85) about the structure of closed ideals of A(D) , which states that every closed ideal

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I ⊂ A(D) has the form

545

I = SI H∞ ∩ J Z(I) .

Theorem 2.1. Let T be essentially unitary and D \ σ(T ) = ∅. If f ∈ A(D) and f = 0 on σ(T ) ∩ T then f (T ) is compact. For the proof of this theorem we need the following lemma. Lemma 2.2. Let T1 , T2 be two contractions on H such that T1 − T2 is compact and f ∈ A(D) . Then f (T1 ) is compact if and only if f (T2 ) is compact too. Proof. There exists a sequence (Pn )n of polynomials such that f − Pn ∞ → 0, where · ∞ is the supremum norm on T. For every n, Pn (T2 ) − Pn (T1 ) is compact. By the von Neumann inequality, we have (f − Pn )(Ti ) ≤ f − Pn ∞ , i = 1 or 2. So (f − Pn )(Ti ) → 0 and f (T2 ) − f (T1 ) = lim Pn (T2 ) − Pn (T1 ) . n→+∞

Thus f (T2 ) − f (T1 ) is compact.

Proof of Theorem 2.1. Without loss of generality, we may assume that σ(T ) ∩ T is of Lebesgue measure zero. We set I = {f ∈ A(D) : f (T ) compact}; I is a closed ideal of A(D) . We have to prove that SI = 1 and Z(I) ⊂ σ(T ) ∩ T. As observed above, we have T = U + K, where U is unitary and K is compact. Moreover, we have σwc (U ) = σwc (T ) ⊂ σ(T ) ∩ T ([14] p. 115), and since σnp (U ) is countable, we see that σ(U ) is a subset of T of Lebesgue measure zero. By Fatou theorem ([8] p. 80), there exists a nonzero outer function f ∈ A(D) which vanishes exactly on σ(U ). Since U is unitary we have f (U ) = 0. By Lemma 2.2, f (T ) is compact. This shows that SI = 1 and Z(I) ⊂ σ(U ). We shall now show that Z(I) ⊂ σwc (U ). Let λ ∈ σnp (U ); λ is an isolated point in σ(U ) and Ker (U −λIH ) is of finite dimension. There exists f ∈ A(D) with f (λ) = 0 and f|σ(U)\{λ} = 0. Since (z − λ)f (z) = 0 for every z ∈ σ(U ), and since U is unitary, (U − λIH )f (U ) = 0 and f (U )(H) ⊂ Ker (U − λIH ). So f (U ) is of finite rank, thus f (U ) is compact and by Lemma 2.2, f (T ) is compact. Hence λ ∈ Z(I). We deduce that Z(I) ⊂ σwc (U ) ⊂ σ(T ) ∩ T, which finishes the proof. Corollary 2.3. Let T be an essentially unitary C0 –contraction and let f ∈ A(D) . Then f (T ) is compact if and only if f = 0 on σ(T ) ∩ T. Proof. It follows from Theorem 2.1 that if f vanishes on σ(T ) ∩ T then f (T ) is compact. Let now f ∈ A(D) such that f (T ) be compact. Let BT denote a maximal commutative Banach algebra that contains IH and T . We have σ(T ) = σBT (T ), where σBT (T ) is the spectrum of T in BT . Let λ ∈ σ(T ) ∩ T. There exists a character χλ on BT such that χλ (T ) = λ and we have |f (λ)| = |λn f (λ)| = |χλ (T n f (T ))| ≤ T n f (T ).

(2)

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Since T is in class C0 , T n x → 0 whenever x ∈ H, (see [11] Proposition III.4.1). Thus for every compact set C ⊂ H, lim sup T n x = 0.

n→∞ x∈C

For C = f (T )(B), where B = {x ∈ H : x ≤ 1}, we get lim T nf (T ) = 0. Then it follows from (2) that f (λ) = 0.

n→∞

Let T ∈ L(H). The spectral multiplicity of T is the cardinal number given by the formula µT = inf card L, where card L is the cardinal of L and where the infimum is taken over all nonempty sets L ⊂ H such that span{T nL; n ≥ 0} is dense in H. Notice that µT = 1 means that T is cyclic. Corollary 2.4. Let T be a contraction on H with µT < +∞. Assume that there exists a nonzero function ϕ ∈ A(D) such that ϕ(T ) = 0. Then f (T ) is compact for every function f ∈ A(D) that vanishes on σ(T ) ∩ T. Proof. There exists two orthogonal Hilbert subspaces Hu and H0 that are invariant by T , such that H = Hu ⊕ H0 , Tu = T|Hu is unitary and T0 = T|H0 is completely nonunitary (see [11], Theorem 3.2, p. 9 or [13], p. 7). Then T0 is clearly in class C0 and we have µT0 < +∞. By Proposition 4.3 of [4], IH0 − T0∗ T0 is compact. Let f ∈ A(D) , with f|σ(T )∩T = 0. Since σ(T0 ) ⊂ σ(T ), it follows from Theorem 2.1 that f (T0 ) is compact. Now, since Tu is unitary and σ(Tu ) ⊂ σ(T ) ∩ T, we get f (Tu ) = 0. Thus f (T ) is compact. Remark. Let T be a cyclic contraction satisfying condition (1) and with finite 2 of [1], there exists analytic spectrum, σ(T ) = {λ1 , · · · , λn } ⊂ T. By Theorem function f = n≥0 an z n , f = 0, such that n |an | < +∞ and f (T ) = 0. Then, it follows from Corollary 2.4 that (T − λ1 IH ) · · · (T − λn IH ) is compact. Thus we obtain a new proof of Corollary 4.3 of [2], mentioned in the introduction. Now we conclude this section by showing that the hypothesis ”essentially unitary“ in Theorem 2.1 and Corollary 2.3 is necessary for a large class of contractions. Let us first make some observations. An operator T ∈ L(H) is called essentially normal if T T ∗ − T ∗ T is compact, see [5]. Notice that if T is a C0 – contraction which is essentially unitary then T ∗ is essentially unitary too. Hence T is essentially normal since IH − T ∗ T and IH − T T ∗ are both compacts. Proposition 2.5. Let T ∈ L(H) be a C0 –contraction which is essentially normal and such that σ(T ) ∩ T is of Lebesgue measure zero. Assume that f (T ) is compact for every f ∈ A(D) vanishing on σ(T ) ∩ T. Then T is essentially unitary. Proof. Let K(H) be the ideal of compact operators on H and π : L(H) → L(H)/K(H) be the canonical surjection. The essential spectrum σess (T ) of T is defined as the spectrum of π(T ) in the Banach algebra L(H)/K(H). By Fatou theorem [8], there exists a non zero outer function f ∈ A(D) such that f|σ(T )∩T = 0.

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By hypothesis f (T ) is compact. Let λ ∈ D, the functions z − λ and f have no common zero in D. So there exists two functions g1 and g2 in A(D) such that (z − λ)g1 + f g2 = 1. Thus (T − λIH )g1 (T ) + f (T )g2 (T ) = IH , which shows that π(T ) − λπ(IH ) is invertible in L(H)/K(H). Hence σess (T ) ⊂ σ(T ) ∩ T. By RudinCarleson-Bishop theorem (see [8] p. 81), there exists a function h ∈ A(D) such that z = h(z), z ∈ σ(T ) ∩ T. Since π(T ) is a normal element in the C ∗ –algebra L(H)/K(H), we get π(T )∗ = h(π(T )). On the other hand we have 1 − h(z)z = 0 on σ(T ) ∩ T, which implies that π(IH ) − π(T )∗ π(T ) = π(IH ) − h(π(T ))π(T ) = 0. Therefore IH − T ∗ T is compact.

3. The case of f(T) for f ∈ H∞ In this section we are interested in the compactness of f (T ) when f ∈ H∞ . The spectrum of an inner function θ is defined by σ(θ) = clos θ−1 (0) ∪ supp µ, where µ is the singular measure associated to the singular part of θ and supp µ is the closed support of µ (see [13], p. 63). Notice that for a C0 –contraction T on H, there exists a minimal inner function mT that annihilates T , i.e mT (T ) = 0, and we have σ(T ) = σ(mT ), (see [11, 13]). As a consequence of Corollary 2.3 we prove the following result which was first established by Moore–Nordgren in [9], Theorem 1. The proof given in [9] uses a result of Muhly [10]. Here we gie a simple proof. Lemma 3.1. Let T be an essentially unitary C0 –contraction on H, and let θ be an inner function that divides mT (i.e mT /θ ∈ H∞ ) and such that σ(θ) ∩ T is of Lebesgue measure zero. Let ψ ∈ A(D) be such that ψ|σ(θ)∩T = 0. If φ = ψmT /θ, then φ(T ) is compact. In particular, the commutant {T } contains a nonzero compact operator. Proof. Let Θ = mT /θ and T1 = T |H1 be the restriction of T to H1 := Θ(T )H; T1 is a C0 –contraction with mT1 = θ. Moreover IH1 − T1∗ T1 = PH1 (IH − T ∗ T )|H1 is compact where PH1 is the orthogonal projection from H onto H1 . By Corollary 2.3, ψ(T1 ) is compact and thus φ(T ) = ψ(T )Θ(T ) = ψ(T1 )Θ(T ) is also compact. Lemma 3.2. Let T be an essentially unitary C0 –contraction on H, and let θ be an inner function that divides mT and such that σ(θ) ∩ T is of Lebesgue measure zero. Let f ∈ H∞ be such that lim T n f (T ) = 0. If φ = f mT /θ, then φ(T ) is compact.

n→+∞

Proof. By the Rudin-Carleson-Bishop theorem, for every nonnegative integers n, there exists hn ∈ A(D) such that z n = hn (z), z ∈ σ(θ) ∩ T and hn ∞ = 1, where .∞ is the supremum norm on T (see [8] p. 81). We have, for every n,

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n 1 − z n hn (z) = 0, z ∈ σ(θ) ∩ T. Then, by Lemma 3.1, (I − T h (T )) m /θ (T ) H n T n is compact. So φ(T ) − T f (T )hn (T ) mT /θ (T ) is also compact. Since T n f (T )hn (T ) mT /θ (T ) ≤ T nf (T ) −→ 0, we deduce that φ(T ) is compact.

We need the following lemma about inner functions, which is actually contained in the proof of the main result of [15]. For the sake of completeness we include its proof. Lemma 3.3. Let Θ be an inner function. There exists a sequence (θn )n of inner functions such that for each n, θn divides Θ, σ(θn ) ∩ T is of Lebesgue measure zero and for every z ∈ D, lim θn (z) = Θ(z). n→+∞

Proof. Let Bn be the Blaschke product constructed with the zeros of Θ contained in the disk {|z| ≤ 1 − 1/n}, each zero of Θ repeated according to its multiplicity. Let ν be the singular measure defining the singular part of Θ. There exists F ⊂ T of Lebesgue measure zero such that ν(F ) = ν(T). There exists a sequence (Kn )n of compact subsets of F such that lim ν(Kn ) = ν(F ). For every n, let νn be the n→∞

measure on T defined by νn (E) = ν(E ∩ Kn ). Denote by Sn the singular inner function associated to the measure νn . We only need now to take θn = Bn Sn . We are now able to prove the main result of this section. Theorem 3.4. Let T be an essentially unitary C0 –contraction on H. Let f ∈ H∞ . Then the following assertions are equivalent. (1) lim T n f (T ) = 0, n→∞

(2) f (T ) is compact. Proof. (1) ⇒ (2) : Let Θ = mT and let (θn )n be the sequence of inner functions given by Lemma 3.3. For every n, we set ϕn = mT /θn . Since (ϕn )n is a bounded sequence in H∞ and ϕn (z) −→ 1 (z ∈ D), (ϕn )n converges to 1 uniformly on compact subsets of D. Then, for every k, there exists a nonnegative integer nk such that |ϕnk (z)| ≥ e−1 for |z| ≤ k/(k + 1). Clearly the sequence (nk )k may be chosen to be strictly increasing. Moreover for |z| ≥ k/(k + 1), we have |z k | ≥ e−1 . So e−1 ≤ |z k | + |ϕnk (z)| ≤ 2, z ∈ D. By he corona theorem ([13], p. 66), there exists two functions h1,k and h2,k in H∞ such that z k h1,k + ϕnk h2,k = 1 and |h1,k |, |h2,k | ≤ C, where C is an absolute constant. Thus we get T k f (T )h1,k (T ) + f (T )ϕnk (T )h2,k (T ) = f (T ), and

T k f (T )h1,k (T ) ≤ CT k f (T ) −→ 0.

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Consequently, f (T ) = lim f (T )ϕnk (T )h2,k (T ) in the L(H) norm. Finally f (T ) is k→∞

compact since by Lemma 3.2, for every k, f (T )ϕnk (T )h2,k (T ) is compact. (2) ⇒ (1) : see the proof of Corollary 2.3.

As in Corollary 2.4, Theorem 3.4 holds for a C0 -contraction with µT < +∞. Let T be a contraction on H. It is shown by Esterle, Strouse and Zouakia in [6], that if f ∈ A(D) , then lim T n f (T ) = 0 if and only if f vanishes on n→∞

σ(T ) ∩ T. So Theorem 3.4 implies Corollary 2.3. Now, if T is completely non unitary, Bercovici showed in [3] that if f ∈ H∞ and lim f (rz) = 0, for every r→1−

z ∈ σ(T ) ∩ T, then lim T n f (T ) = 0. So it follows immediately from this fact n→∞ and Theorem 3.4 the following result. Corollary 3.5. Let T be an essentially unitary C0 –contraction on H. Let f ∈ H∞ . If for every z ∈ σ(T ) ∩ T, lim f (rz) = 0, then f (T ) is compact. r→1−

References [1] A. Atzmon, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144 (1980), 27–63. [2] A. Atzmon, Unicellular and nonunicellular dissipative operators. Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 45–54. [3] H. Bercovici, On the iterates of a completely nonunitary contraction, in: Topics in Operator Theory: Ernest D. Hellinger memorial volume, Operator Theory Adv. Appl. 48, Birkh¨ auser, Basel, (1990), 185–188. [4] H. Bercovici; D. Voiculescu, Tensor operations on characteristic functions of C0 contractions. Acta Sci. Math. (Szeged) 39 (1977), no. 3-4, 205–231 [5] L. G. Brown; R. G. Douglas; P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of C ∗ –algebras. Proceedings of a Conference on Operator Theory, 58–128. Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973. [6] J. Esterle; E. Strouse; F. Zouakia, Theorems of Katznelson-Tzafriri type for contractions. J. Funct. Anal. 94 (1990), no. 2, 273–287. [7] I.C. Gohberg; M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, American Mathematical Society, 1969. [8] K. Hoffman, Banach spaces of analytic functions. Prentice-Hall, Inc., Englewood Cliffs, N. J. 1962 [9] B. Moore; E. Nordgren, On transitive algebras containing C0 operators. Indiana Univ. Math. J. 24 (1974/75), 777–784. [10] P. Muhly, Compact operators in the commutant of a contraction, J. Funct. Anal. 8 (1971) 197–224 [11] B. Sz.-Nagy; C. Foias, Analyse harmonique des opérateurs de l’espace de Hilbert, Masson, 1967. [12] B. Sz.-Nagy, On a property of operators of class C0 , Acta Sci. Math. (Szeged) 36 (1974), 219-220.

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[13] N. K. Nikolski, Treatise on the Shift Operator, Springer, Heidelberg, 1986. [14] N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2. Model operators and systems. Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI, 2002. [15] E. A. Nordgren, Compact operators in the algebra generated by essentially unitary C0 operators, Proc. Amer. Math. Soc. 51 (1975), 159–162. [16] W. Rudin, The closed ideals in an algebra of analytic functions, Canadian J. Math., 9 (1957), 426-434. [17] P. Vitse, Smooth operators in the commutant of a contraction. Studia Math. 155 (2003), no. 3, 241–263. K. Kellay CMI LATP UMR–CNRS 6632 Université de Provence 39, rue F. Joliot-Curie 13453 Marseille cedex 13 France e-mail: [email protected] M. Zarrabi IMB UMR–CNRS 5352 Université de Bordeaux 351, cours de la Libération 33405 Talence cedex France e-mail: [email protected] Submitted: June 6, 2009. Revised: June 18, 2009.

Integr. equ. oper. theory 65 (2009), 551–572 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040551-22, published online June 5, 2009 DOI 10.1007/s00020-009-1691-0


Weak Operator Topology, Operator Ranges and Operator Equations via Kolmogorov Widths M. I. Ostrovskii and V. S. Shulman Abstract. Let K be an absolutely convex infinite-dimensional compact in a Banach space X . The set of all bounded linear operators T on X satisfying T K ⊃ K is denoted by G(K). Our starting point is the study of the closure W G(K) of G(K) in the weak operator topology. We prove that W G(K) contains the algebra of all operators leaving lin(K) invariant. More precise results are obtained in terms of the Kolmogorov n-widths of the compact K. The obtained results are used in the study of operator ranges and operator equations. Mathematics Subject Classification (2000). Primary 47A05; Secondary 41A46, 47A30, 47A62. Keywords. Banach space, bounded linear operator, Hilbert space, Kolmogorov width, operator equation, operator range, strong operator topology, weak operator topology.

1. Introduction Let K be a subset in a Banach space X . We say (with some abuse of the language) that an operator D ∈ L(X ) covers K, if DK ⊃ K. The set of all operators covering K will be denoted by G(K). It is a semigroup with a unit since the identity operator is in G(K). It is easy to check that if K is compact then G(K) is closed in the norm topology and, moreover, sequentially closed in the weak operator topology (WOT). It is somewhat surprising that for each absolutely convex infinitedimensional compact K the WOT-closure of G(K) is much larger than G(K) itself, and in many cases it coincides with the algebra L(X ) of all operators on X . Our aim is to understand: how much freedom has an operator which is obliged to cover a given compact? In a simplest form the question is: “How large is G(K)?”. We answer this question describing the WOT-closure W G(K) of G(K) as well as its closure in the ultra-weak topology (for the case of Hilbert spaces). These results are obtained in Sections 2–3 for the Banach spaces, and in more detailed form

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in Section 4 for Hilbert spaces; they are formulated in terms of Kolmogorov’s n-widths of K. In Section 5 we consider a more general object: the set G(K1 , K2 ) of all operators T which have the property T K1 ⊃ K2 where K1 , K2 are fixed convex compacts in Hilbert spaces. In further sections we apply the obtained results for study of some related subjects: operator ranges (Section 7), operator equations of the form XAY = B (Section 8) and operators with the property AXx ≥ Ax for all x ∈ H where A is a given operator on a Hilbert space H. Some applications of the obtained results to the theory of quadratic operator inequalities and operator fractional linear relations will be presented in a subsequent work. In fact our interest to the semigroups G(K) was initially motivated by these applications; the relations to other topics became clear for us in the process of the study. Notation. Our terminology and notation of Banach space theory follows [10]. Our definitions of the standard topologies on spaces of operators follow [3, Chapter VI]. Let X , Y be Banach spaces. We denote the closed unit ball of a Banach space Y by BY , and the norm closure of a set M ⊂ Y by M . We denote the set of bounded linear operators from Y to X by L(Y, X ). We write L(X ) for L(X , X ). The identity operator in L(X ) will be denoted by I. Throughout the paper we denote by lin(K) the linear span of a set K, and by VK the closed subspace spanned by K, that is, VK = lin(K). We denote by AK the algebra of all operators for which VK is an invariant subspace. It is clear that AK is closed in the WOT. Remark on related work. Coverings of compacts by sets of the form R(BZ ) where Z is a Banach space and R ∈ L(Z, X ) have been studied by many authors, see [1], [2], and [6]. However, the main foci of these papers are different. In all of the mentioned papers additional conditions are imposed on Z, or on R, or on both of them, and the main problem is: whether such R exist? In the context of the present paper existence is immediate, while for us (as it was mentioned above) the main question is: “How large is the set of such operators?”. We finish the introduction by showing that for non-convex compacts K the semigroup G(K) can be trivial: Example. There exists a compact K in an infinite-dimensional separable Hilbert space H such that the only element of G(K) is the identity operator. ∞ Proof. Let {en }∞ n=1 be an orthonormal basis in H; {αn }n=1 be a sequence of real ∞ numbers satisfying αn > 0 and limn→∞ αn+1 /α n = 0; and {βn }n=1 be a sequence 1 of distinct numbers in the open interval 2 , 1 . The compact K is defined by ∞

∞

K = {0} ∪ {αn en }n=1 ∪ {αn βn en }n=1 .

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Assume that there exists D ∈ L(H) such that D(K) ⊃ K and D is not the identity operator. Let M = {n ∈ N : Den = en }. Since {en }∞ n=1 is a basis in H and D is not the identity operator, the set M is nonempty. We introduce the → following oriented graph with the vertex set M . There is an oriented edge − nm starting at n ∈ M and ending at m ∈ M (n can be equal to m) if and only if one of the following equalities holds: D(αm em ) = αn en ,

D(βm αm em ) = αn en ,

(1) D(βm αm em ) = βn αn en . 1 Important observation. Since the numbers {βn }∞ n=1 ⊂ 2 , 1 are distinct, the number of edges starting at n is at least 2 for each n ∈ M , while there is at most one edge ending at m ∈ M . D(αm em ) = βn αn en ,

An immediate consequence of this observation is that there are infinitely → with n < m, that is, infinitely many pairs (n, m), n < m, many oriented edges − nm for which one of equalities from (1) holds. Taking into account the conditions satisfied by {αn } and {βn }, we get a contradiction with the boundedness of D. This example shows that in the general case there is a very strong dependence of the size of the semigroup G(K) on the geometry of K. To relax this dependence we restrict our attention to absolutely convex compacts K.

2. AK ⊂ W G(K) Theorem 2.1. Let K be an absolutely convex infinite-dimensional compact. Then AK ⊂ W G(K). Remark 2.2. Theorem 2.1 is no longer true for finite-dimensional compacts. In fact, if K is absolutely convex finite-dimensional compact, then A ∈ G(K) implies that A leaves VK invariant. Since VK is finite-dimensional, the condition AK ⊃ K passes to operators from the WOT-closure. Thus W G(K) is a proper subset of AK . Let F be a subset of X ∗ . We use the notation F⊥ for the pre-annihilator of F , that is, F⊥ := {x ∈ X : ∀f ∈ F f (x) = 0}. Lemma 2.3. Let K be an absolutely convex infinite-dimensional compact in a Banach space X . For each finite-dimensional subspace F ⊂ X ∗ , each finite-dimensional subspace Y ⊂ X , and an arbitrary linear mapping N : Y → X satisfying N (Y ∩ VK ) ⊂ lin(K), there is D ∈ G(K) satisfying the condition Dx − N x ∈ F⊥ ∀x ∈ Y.

(2)

Proof. Note first of all that it suffices to prove the lemma under an additional assumption that Y ⊂ VK . Indeed, suppose that it is done, then in the general case we choose a complement Y1 of Y ∩ VK in Y and choose a complement X1 of Y1 in X that contains VK . By our assumption there is an operator D ∈ L(X1 ) with

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DK ⊃ K and Dx − N x ∈ F⊥ ∩ X1 ∀x ∈ Y ∩ VK . It remains to extend D to X setting Dx = N x for x ∈ Y1 . So we assume that Y ⊂ VK . For brevity denote lin(K) by Z. Let P : X → X be a projection of finite rank, such that P X ⊃ Y + N Y, (I − P )X ⊂ F⊥ and dim(P Z) ≥ dim F + dim Y. The last condition can be satisfied since K is finite-dimensional. The conditions N (Y ∩ VK ) ⊂ lin(K) and Y ⊂ VK imply that the subspace N Y is contained in Z ∩ P X . The space ker(P ) ∩ Z has finite codimension in Z. Therefore there exists a complement L of ker(P ) ∩ Z in Z such that L ⊃ N Y. We have P L = P Z and L ∩ (I − P )X = {0}. Since the subspace (I − P )X has finite codimension in X , we can find a subspace M ⊃ L, which is a complement of (I − P )X in X . Let QM : X → M be the projection onto M with the kernel ker(P ) = (I − P )X , and let M0 be the complement of L in M . Since Z = (ker P ∩ Z) ⊕ L and QM (L) = L, we have QM (Z) = L ⊂ Z. To introduce an operator D ∈ L(X ) it suffices to determine its action on ker P and on M . We do it in the following way: (a) The restriction of D to ker P is a multiple λIker P of the identity operator, where λ is chosen in such a way that (I − QM )K ⊂ λ2 K (such a choice is possible because (I − QM )K is a compact subset of Z = ∪n∈N nK). (b) The restriction of D to M is defined in three ‘pieces’: • D|M0 = 0. • Now we define the restriction of D to QM (Y). Observe that QM (Y) ⊂ L. This follows from Y ⊂ VK ⊂ L ⊕ (I − P )X . In addition QM |Y is an isomorphism, because Y ⊂ P X , and P X and M are complements of the same subspace. Because of this, the operator D|QM (Y) given by D(QM (x)) = N x + αS(QM (x)) for x ∈ Y is well-defined, where α ∈ R and S is an isomorphism of QM (Y) into F⊥ ∩ L. Such isomorphisms exist because dim L ≥ dim(P L) = dim(P Z), and we assumed that dim(P Z) ≥ dim F + dim Y. Now we choose α to be so large that the image of K ∩ QM (Y) covers a ‘large’ multiple of the intersection of QM (K) with the space onto which it maps. This is possible because zero has non-empty interior in K ∩ QM (Y) and QM (K) is compact. • We define D on the complement of QM (Y) in L as a ‘dilation’ operator onto some complement of the D(QM (Y)) in L. The number α and the dilation are selected in such a way that D(K ∩ L) ⊃ 2QM (K).

(3)

To see that it is possible recall that QM (K) ⊂ L and, since L is finite-dimensional, the set K ∩ L contains a multiple of the unit ball of L. It remains to verify that D satisfies the conditions (2) and DK ⊃ K.

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Condition (2). Let x ∈ Y, then x = QM x + (I − QM )x. Therefore Dx = N x + αS(QM (x)) + λ(I − QM )x. Let f ∈ F . We get: f (Dx) = f (N x) + αf (S(QM (x))) + λf (I − QM )x = f (N x), where we use the following facts: (a) The image of S is in F⊥ ; (b) (I − QM )X = (I − P )X ⊂ F⊥ . Condition DK ⊃ K. Let x ∈ K. Then x = QM x + (I − QM )x. The condition (3) implies that there exists v ∈ 12 (L ∩ K) such that Dv = QM x. The choice of λ implies that w = λ1 (I − QM )x satisfies w ∈ 12 K. Let z = v + w. It is clear that z ∈ K. We need to show that Dz = x. We have 1 Dz = Dv + Dw = Dv + D (I − QM )x λ 1 = QM x + λ (I − QM )x = x. λ (We use the fact that (I − QM )X ⊂ ker P .)

Proof of Theorem 2.1. Let T ∈ AK , and U = {E ∈ L(X ) : ∀i ∈ {1, . . . , n} |fi (Exi )| < ε} be a WOT-neighborhood of 0 in L(X ), where n ∈ N, ε > 0, {fi }ni=1 ∈ X ∗ and {xi }ni=1 ∈ X . We need to show that T + U contains an operator from G(K) for each choice of n, ε, fi , and xi . Let F = lin({fi }ni=1 ) and Y = lin({xi }ni=1 ). Let Y1 = Y ∩ VK . Since T ∈ AK , we have T (Y1 ) ⊂ VK . Since VK = Z, we can find a “slight perturbation” T of T satisfying T(Y1 ) ⊂ Z. In particular, we can find such T in T + 12 U. It remains to show that T + 12 U contains an operator D from G(K). It is clear that each operator S satisfying ∀x ∈ Y

Sx − Tx ∈ F⊥

is in T + 12 U. Now the existence of the desired operator D is an immediate consequence of Lemma 2.3 applied to N = T. Corollary 2.4. If VK = X , then W G(K) = L(X ).

3. Application of Kolmogorov n-widths to estimates of the ‘size’ of W G(K) from above We are going to use the notion of Kolmogorov n-width. In this respect we follow the terminology and notation of the book [17, Chapter II]. Let Z be a subset of a Banach space X and x ∈ X . The distance from x to Z is defined as E(x, Z) = inf{||x − z|| : z ∈ Z}.

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Definition 3.1. Let K be a subset of a Banach space X , n ∈ N ∪ {0}. The Kolmogorov n-width of K is given by dn (K) = inf sup E(x, Xn ), Xn x∈K

where the infimum is over all n-dimensional subspaces. Lemma 3.2. Let K and K0 be two subsets in a Banach space X and D ∈ L(X ) be such that D(K0 ) ⊃ K. Then dn (K) ≤ ||D||dn (K0 ) for all n ∈ N ∪ {0}. Proof. Let Z ⊂ X be an n-dimensional subspace. Then DZ ⊂ X is a subspace of dimension ≤ n and E(Dx, DZ) ≤ ||D||E(x, Z). The conclusion follows. Lemma 3.3. Let K be a bounded subset in a Banach space X . If K0 = K ∩ L, where L is a closed linear subspace in X which does not contain K, then there exists a constant 0 < C < ∞ such that dn (K0 ) ≤ Cdn+1 (K) for all n ∈ N ∪ {0}. Proof. It is well-known (see [17, p. 10]) that a bounded set K is compact if and only if limn→∞ dn (K) = 0. Therefore it suffices to consider the case when K is compact. It is clearly enough to consider the case when L is a subspace of codimension 1. Let L = ker ν where ν ∈ X ∗ . We may assume without loss of generality that the norm of the restriction of ν to linK satisfies ||ν|lin(K) || = 1. For each n ∈ N ∪ {0} let Ln ⊂ X be a subspace of dimension n satisfying supx∈K E(x, Ln ) ≤ 2dn (K). First we show that there exists N ∈ N such that ||ν|LM || > 12 for all M ≥ N . Let 0 < ε < 1 and let xi ∈ K and scalars ai (i = 1, . . . , k) be such that the vectorh = i ai xi satisfies ||h|| = 1 and ν(h) > 1 − ε. Let δ > 0 be such that δ||ν|| |ai | < ε. Let N be such that for M ≥ N we have dM (K) < δ/2. Then for M≥ N there exist yi ∈ LM such that ||xi − yi || < δ. Therefore the vector g := i ai yi satisfies ||ν|| · ||g − h|| < ε and g ∈ LM . Choosing appropriate ε and δ we get ||ν|LM || > 12 . Let M ≥ N and let LM,0 = LM ∩ ker ν. Let x ∈ K0 . We are going to show that E(x, LM,0 ) < (2||ν|| + 1)2dM (K). By the definition of LM there is y ∈ LM such that ||x − y|| ≤ 2dM (K). Since ν(x) = 0, we have |ν(y)| ≤ ||ν||2dM (K). Since ||ν|LM || > 12 , we conclude that E(y, LM,0 ) < 4||ν||dM (K). Therefore E(x, LM,0 ) < (2||ν|| + 1)2dM (K). It is clear that dim LM,0 = M − 1. Thus for M ≥ N we have dM−1 (K0 ) < (2||ν|| + 1)2dM (K). The conclusion follows. Definition 3.4. Let {an } be a non-increasing sequence of positive numbers satisfying limn→∞ an = 0. We say that {an } is lacunary if an+1 lim inf = 0. (4) n→∞ an Lemma 3.5. If the sequence {dn (K)}∞ n=1 is lacunary, then G(K) ⊂ AK . Proof. Let R ∈ L(X ) be such that RVK is not contained in VK . We have to show that RK does not contain K. Assume the contrary. It follows from our assumption that R−1 (VK ) is a proper subspace of VK and R(K0 ) ⊃ K where K0 = K ∩ R−1 (VK ) is a proper section of K.

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By Lemma 3.2 we get dn (K) ≤ ||R||dn (K0 ) for all n ∈ N∪{0}. By Lemma 3.3 we get dn (K0 ) ≤ Cdn+1 (K) for some 0 < C < ∞ (which depends on K and K0 , but not on n) and all n ∈ N ∪ {0}. We get dn+1 (K) ≥ (C||R||)−1 dn (K), hence the sequence {dn (K)}∞ n=1 is not lacunary. We get a contradiction. Remark 3.6. The assumptions of convexity and symmetry of K are not needed in Lemmas 3.2, 3.3, and 3.5. Combining Theorem 2.1 and Lemma 3.5 we get Theorem 3.7. If an absolutely convex compact K is such that the sequence {dn (K)} is lacunary, then W G(K) = AK .

4. Covering of ellipsoids 4.1. s-numbers Now we restrict our attention to the Hilbert space case, that is, we consider sets K of the form A(BH ) where A is an infinite-dimensional bounded compact operator from a Hilbert space H to a Hilbert space H1 . Such sets are called ellipsoids. Note. We continue using the Banach space theory notation and terminology. In particular, unless explicitly stated otherwise, by A∗ we mean the Banach-spacetheoretical conjugate operator. It does not seem that anything will be gained if we introduce Hilbert-space duality, but it can cause some confusion when we apply Banach space case results for Hilbert spaces. Remark 4.1. Many of the results below are true for A(BH ) with non-compact A and usually the corresponding proofs are much simpler. We restrict our attention to the compact case. Definition 4.2. (See [8, Chapter II, §2]) The eigenvalues of the operator (E ∗ E)1/2 (where E ∗ is the conjugate in the Hilbert space sense) are called the s-numbers of the operator E. Notation: {sn (E)}∞ n=1 . With this notation we have the following equalities for n-widths: dn (A(BH )) = sn+1 (A) (see [8, Theorem 2.2, p. 31]). For ellipsoids we have a converse to the Lemma 3.2. Lemma 4.3. If K0 , K are ellipsoids in Hilbert spaces H1 , H2 , respectively, and dn (K) ≤ Cdn (K0 ) for some C > 0 and all n ∈ N ∪ {0}, then there is an operator D ∈ L(H1 , H2 ) such that DK0 ⊃ K and D ≤ C.

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Proof. The result follows from the so-called Schmidt expansion of a compact operator (see [8, p. 28]), which implies that ∞ K = A(BH ) = αn sn (A)hn : {αn }∞ n=1 ∈ 2 , n=1

{hn }∞ n=1 is an orthonormal sequence} and

K0 = B(BH ) =

∞

αn sn (B)gn : {αn }∞ n=1 ∈ 2 ,

n=1

{gn }∞ n=1 is an orthonormal sequence} . It is easy to see that there is a bounded linear operator D which maps gn onto Chn , and that this operator satisfies the conditions D(K0 ) ⊃ K, D ≤ C. Remark 4.4. The proof of Lemma 4.3 shows that the desired operator D can be constructed as an operator whose restriction to VK0 is a multiple of a suitable chosen bijective isometry between VK0 and VK , extended to H1 in an arbitrary way. Known results on s-numbers imply the following lemma. Lemma 4.5. Let K be an ellipsoid in a Hilbert space H such that {dn (K)}∞ n=0 is not lacunary. Let K0 be the intersection of K with a closed linear subspace of finite codimension. Then there exists δ > 0 such that dn (K0 ) ≥ δdn (K) and a bounded linear operator Q : lin(K0 ) → lin(K) satisfying Q(K0 ) ⊃ K. Proof. Let A : H → H be a compact operator satisfying K = A(BH ). The sequence {dn (K0 )}∞ n=0 is the sequence of s-numbers of a restriction of A to a subspace of finite codimension. This sequence is, in turn, the sequence of s-numbers of an operator of the form A + G, where G is an operator of finite rank. It is known [8, Corollary 2.1, p. 29] that sn (A + G) ≥ sn+r (A), where r is the rank of G. Combining this inequality with the assumption that the sequence {sn (A)}∞ n=1 is not lacunary, we get the desired inequality. The last statement of the lemma follows from Lemma 4.3. 4.2. WOT Theorem 4.6. If H is a Hilbert space, K ⊂ H is an ellipsoid and the sequence {dn (K)} is not lacunary, then W G(K) = L(H). The definition of WOT shows that to prove Theorem 4.6 it suffices to prove the following lemma. Lemma 4.7. Let K be an ellipsoid in a Hilbert space H. Suppose that the sequence {dn (K)} is non-lacunary. Then for each finite-dimensional subspace Y ⊂ H and each linear mapping N : Y → H, there is an operator D satisfying conditions: Dy = N y for all y ∈ Y, and DK ⊃ K.

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Proof. Let Z = Y ⊥ and K0 = K ∩ Z. By Lemma 4.5 there is an operator E from Z to H with EK0 ⊃ K. Extend it to an operator D : H → H setting Dy = N y on Y. Remark 4.8. One can see from the proof of Lemma 4.7 that under the stated conditions the closure of G(K) in the strong operator topology coincides with L(H). Corollary 4.9. Let K be an ellipsoid in a Hilbert space H. Then: (1) W G(K) = AK if the sequence {dn (K)}∞ n=0 is lacunary. (2) W G(K) = L(H) if the sequence {dn (K)}∞ n=0 is not lacunary. 4.3. Ultra-weak topology It turns out that Theorem 4.6 remains true if we replace closure in the weak operator topology, by a closure in a stronger topology, usually called ultra-weak topology. This topology on L(H) is defined as the weak∗ topology corresponding to the duality L(H) = (C1 (H))∗ , where C1 (H) is the space of nuclear operators. (Necessary background can be found in [18, Chapter II], unfortunately the terminology and notation there is different, the ultra-weak topology is called σ-weak topology, see [18, p. 67]). Ultra-weak and strong operator topologies are incomparable, for this reason our next result does not follow from Remark 4.8. Theorem 4.10. If K is an ellipsoid in a Hilbert space H and the sequence {dn (K)} is not lacunary, then the ultra-weak closure of G(K) coincides with L(H). Proof. Let {Ti }m i=1 be a finite collection of operators in C1 (H) and R ∈ L(H). It suffices to show that there is D ∈ L(H) satisfying tr(DTi ) = tr(RTi ) for i = 1, . . . , m and DK ⊃ K.

(5)

It is clear that we may assume that the operators Ti are linearly independent. Lemma 4.11. If {Ti }m i=1 are linearly independent, then there exists a finite rank projection P ∈ L(H) such that the mapping ω : L(H) → Rm given by

m

ω(U ) = {tr(U P Ti )}i=1 is surjective. Proof. We have to prove that there is a finite rank projection P such that the operators P Ti are linearly independent (in this case the mapping ω will be surjective). Using induction we may suppose that P0 T1 , ..., P0 Tm−1 are linearly independent for some P0 . Consider the set M0 of those finite rank projections P which commute with P0 and satisfy imP ⊃ imP0 . We claim that there exists P ∈ M0 such that P T1 , . . . , P Tm are linearly independent. Assume contrary, then for each P ∈ M0 , one can find λ1 (P ), ..., λm−1 (P ) ∈ C m−1 satisfying P Tm = k=1 λk (P )P Tk (using the definition of M0 it is easy to get a contradiction if P T1 , . . . , P Tm−1 are linearly dependent). Our next step is to show

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that the numbers {λk (P )}m−1 k=1 do not depend on P . In fact, for any P1 , P2 ∈ M0 we have k<m (λk (P1 ) − λk (P2 ))P0 Tk = 0. So let {λk }m−1 k=1 be such that λk (P ) = λk for all P ∈ M0 . Then the operator T = Tm − k<m λk Tk has the property that P T = 0 for all P ∈ M0 . It is easy to see that this implies T = 0. We get a contradiction with the linear independence of {Tk }m k=1 . We complete the proof of the theorem by showing the existence of D satisfying (5). 1. We define D on ker P as in Theorem 4.6. This definition implies that the condition DK ⊃ K is satisfied. 2. To show that the condition tr(DTi ) = tr(RTi ), i = 1, . . . , m, is satisfied it suffices to show the existence of U ∈ L(H) satisfying tr((U P + D(I − P ))Ti ) = tr(RTi ) ∀i = 1, . . . , m.

(6)

Since the condition (6) can be rewritten as {tr(U P Ti )}m i=1 = {tr((R − D(I − P ))Ti )}m i=1 , where the right-hand side does not depend on U , and the vectors m {tr(U P Ti )}m i=1 , U ∈ L(H) cover (by Lemma 4.11) the whole space R , the existence of U satisfying (6) follows. Remark 4.12. It would be interesting to prove an analogue of Theorem 2.1 for the ultra-weak topology.

5. Two ellipsoids Let H1 , H2 be two infinite-dimensional separable Hilbert spaces. We consider two ellipsoids, K1 ⊂ H1 , K2 ⊂ H2 and introduce the set G(K1 , K2 ) := {T ∈ L(H1 , H2 ) : T K1 ⊃ K2 }.

(7)

We are interested in the description of the WOT-closure of G(K1 , K2 ) which we denote by W G(K1 , K2 ). As in the case of one ellipsoid, the description depends on the behavior of sequences of Kolmogorov n-widths. We start with some simple but useful observations. It is easy to see that G(K2 )G(K1 , K2 )G(K1 ) ⊂ G(K1 , K2 ). Using this inclusion and elementary properties of WOT we get W G(K2 )W G(K1 , K2 )W G(K1 ) ⊂ W G(K1 , K2 ).

(8)

Lemmas 3.2 and 4.3 imply that the set G(K1 , K2 ) is non-empty if and only if dn (K2 ) = O(dn (K1 )).

(9)

From now on till the end of this section we assume that (9) is satisfied. Observation 5.1. By Remark 4.4, condition (9) implies that there is an onto isometry M : VK1 → VK2 and a number α ∈ R+ such that αM (K1 ) ⊃ K2 . Consider decompositions H1 = VK1 ⊕ R1 and H2 = VK2 ⊕ R2 . Let A1 ∈ L(VK1 ), B1 ∈ L(R1 , H1 ), A2 ∈ L(VK2 ), B2 ∈ L(R2 , H2 ), and C : R1 → H2 . Combining

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Theorem 2.1 with (8) we get that the composition (A2 ⊕ B2 )(αM ⊕ C)(A1 ⊕ B1 ) is in W G(K1 , K2 ), where Ai ⊕ Bi : VKi ⊕ Ri → Hi , i = 1, 2. To state our results on the description of W G(K1 , K2 ) we need the following definitions. Definition 5.2. The k th left shift of a sequence {an }∞ n=0 (k ≥ 0) is the sequence {an+k }∞ . n=0 ∞ Definition 5.3. Let {an }∞ n=0 and {bn }n=0 be sequences of non-negative numbers. ∞ We say that {an }∞ majorizes {b } n n=0 if there is 0 < C < ∞ such that bn ≤ Can n=0 for all n = 0, 1, 2, . . . .

The following theorem is the main result of this section: Theorem 5.4. Let K1 and K2 be infinite-dimensional ellipsoids in Hilbert spaces H1 and H2 . Assume that (9) holds. Then (A) If all left shifts of the sequence {dn (K1 )} majorize the sequence {dn (K2 )}, then W G(K1 , K2 ) = L(H1 , H2 ). (B) If the k th left shift of {dn (K1 )} majorizes the sequence {dn (K2 )}, but the (k+1)th left shift does not (such cases are clearly possible), then W G(K1 , K2 ) is the set of those operators T ∈ L(H1 , H2 ) for which the image of the space T (VK1 ) in the quotient space H2 /VK2 is at most k-dimensional. Proof. (A) Observe that to show W G(K1 , K2 ) = L(H1 , H2 ) it suffices to find, for an arbitrary finite-dimensional subspace Y ∈ H1 and an arbitrary operator N : Y → H2 , an operator D ∈ L(H1 , H2 ) satisfying the conditions: D|Y = N |Y and D(K1 ) ⊃ K2 . (This condition implies that G(K1 , K2 ) is dense in L(H1 , H2 ) even in the strong operator topology.) We find such an operator D in the following way: let Y ⊥ be an orthogonal complement of Y. The argument of Lemma 4.5 shows that the sequence {dn (K1 ∩ Y ⊥ )} majorizes some left shift of the sequence {dn (K1 )} and thus, by our assumption, majorizes the sequence {dn (K2 )}. By Lemma 4.3 there is a continuous linear operator Y : Y ⊥ → H2 such that Y (K1 ∩ Y ⊥ ) ⊃ K2 . We let D|Y ⊥ = Y and D|Y = N . It is clear that D has the desired properties. (B) Suppose that the k th left shift of {dn (K1 )} majorizes {dn (K2 )}. Let T ∈ L(H1 , H2 ) be such that the dimension of the image of the space T (VK1 ) in the quotient space H2 /VK2 is ≤ k. We show that T ∈ W G(K1 , K2 ). Let F be a finite subset of H2∗ and Y be a finite subset of H1 . It suffices to show that there exists D ∈ G(K1 , K2 ) satisfying f (Dy) = f (T y) for each y ∈ Y and each f ∈ F . With this in mind, we may assume that F and Y are finitedimensional subspaces. Also, we may assume that Y is a subspace of VK1 , because we may let the restriction of D to the orthogonal complement of VK1 be the same as the restriction of T . We decompose F as FO ⊕ FV , where FO = F ∩ VK⊥2 . It is easy to check that the assumption on T implies that (T ∗ FO )⊥ ∩ VK1 is of codimension at most k (if it is of codimension ≥ k + 1, then we can find k + 1 vectors xi ∈ VK1 and

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k + 1 functionals x∗j in FO such that T ∗ x∗j (xi ) = δi,j , but then x∗j (T xi ) = δij shows that {T xi } is a family of k + 1 vectors whose images in H2 /VK2 are linearly independent, contrary to our assumption). Now we decompose Y = Y1 ⊕ Y2 , where Y1 = Y ∩ (T ∗ FO )⊥ . We let D|Y2 = T |Y2 . Our next step is to find a suitable definition of the restriction of D to (T ∗ FO )⊥ ∩ VK1 . To this end we need the following modification of Lemma 4.5, which can be proved using the same argument and Remark 4.4. Lemma 5.5. Let K1 and K2 be ellipsoids in Hilbert spaces H1 and H2 , respectively. ∞ Suppose that the k th left shift of {dn (K1 )}∞ n=0 majorizes {dn (K2 )}n=0 and that K0 is the intersection of K1 with a subspace of H1 of codimension k. Then there exists an operator B : VK0 → VK2 such that B(K0 ) ⊃ K2 and B is a multiple of a bijective linear isometry of VK0 and VK2 . Applying Lemma 5.5 we find an operator B : ((T ∗ FO )⊥ ∩ VK1 ) → VK2 which satisfies B((T ∗ FO )⊥ ∩ K1 ) ⊃ K2 and is a multiple of a bijective isometry. Now we modify B using Lemma 2.3, which we apply for X = VK2 , K = K2 , Y = BY1 , N = T B −1 |BY1 , and F (which is denoted in the same way in this proof). We denote the operator obtained as a result of the application of Lemma 2.3 by H. We let D|(T ∗ FO )⊥ ∩VK1 = HB. This formula defines D on Y1 , and this definition is such that D|Y1 = T |Y1 . We extend D to the rest of the space H1 arbitrarily. It is clear that D satisfies all the assumptions. Thus T ∈ W G(K1 , K2 ). Now we suppose that the (k + 1)th left shift of {dn (K1 )} does not majorize {dn (K2 )} and show that if T is an operator for which T (VK1 ) contains k + 1 vectors whose images in the quotient space H2 /VK2 are linearly independent, then T ∈ / W G(K1 , K2 ). Using the standard argument we find ε > 0, v1 , . . . , vk+1 ∈ VK1 , and functionals f1 , . . . , fk+1 ∈ H2∗ , such that any D ∈ L(H1 , H2 ) satisfying |fj (Dvi − T vi )| < ε, i, j = 1, . . . , k + 1, satisfies the condition: D(VK1 ) contains k + 1 vectors whose images in H2 /VK2 are linearly independent. It remains to show that such operators D cannot satisfy DK1 ⊃ K2 . In fact the condition about k + 1 linearly independent vectors implies that D−1 (VK2 ) ∩ VK1 is a subspace of VK1 of codimension at least k + 1. Therefore K2 is covered by a section K0 of K1 of codimension k + 1. On the other hand, by Lemma 3.3, the sequence of n-widths of K0 is majorized by the (k + 1)th left shift of {dn (K1 )}∞ n=1 . By Lemma 3.2, we get a contradiction with our assumption. Corollary 5.6. If {dn (K1 )}∞ n=0 is non-lacunary and the condition (9) is satisfied, then W G(K1 , K2 ) = L(H1 , H2 ). In fact, if {dn (K1 )}∞ n=0 is non-lacunary, it is majorized by each of its left shifts, and hence the assumption of Theorem 5.4(A) is satisfied. Remark 5.7. In the case where {dn (K1 )}∞ n=0 is lacunary both the situation described in Theorem 5.4(A) and the situation described in Theorem 5.4(B) can occur.

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Similarly to the case of one compact we introduce AK1 ,K2 := {T ∈ L(H1 , H2 ) : T VK1 ⊂ VK2 }. The following is a special case of Theorem 5.4(B) corresponding to the case k = 0: Corollary 5.8. Let K1 , K2 be ellipsoids with lim inf

dn+1 (K1 ) = 0. dn (K2 )

(10)

Then W G(K1 , K2 ) = AK1 ,K2 . Remark 5.9. Note that the combination of the assumption (9) and the condition ∞ (10) imply that the sequences {dn (K1 )}∞ n=1 and {dn (K2 )}n=1 are both lacunary. Indeed, dk (K2 ) ≤ Cdk (K1 ) implies dn+1 (K1 ) dn+1 (K1 ) dn+1 (K1 ) dn+1 (K2 ) ≥ and ≥ . dn (K2 ) Cdn (K1 ) dn (K2 ) Cdn (K2 ) ∞ Therefore (10) implies that {dn (K1 )}∞ n=1 and {dn (K2 )}n=1 are lacunary.

Analysis of all possible cases in Theorem 5.4 implies also the following: Corollary 5.10. If K1 and K2 are ellipsoids for which VKi = Hi for i = 1, 2, and (9) is satisfied, then W G(K1 , K2 ) = L(H1 , H2 ).

6. Covering with compact operators Here we discuss the problem of covering an ellipsoid K2 by the image of an ellipsoid K1 via a compact operator. Let CG(K1 , K2 ) be the set of all compact operators T satisfying the condition T K1 ⊃ K2 . Let us begin with an analogue of Lemma 3.2. Note that the widths dn (K) of a compact subset K in a Banach space X can change if we consider K as a subset of a subspace Y ⊂ X that contains K. Let us denote by dñ (K) the n-width of K considered as a subset of VK (recall that VK = linK, so we choose the minimal subspace and obtain maximal widths). Lemma 6.1. Let X and Y be Banach spaces, K be a compact set in X and T : X → Y be a compact operator. Then dñ (T K)/dñ (K) → 0 as n → ∞. Proof. We may assume that X = VK . By the definition of dñ , for each n ∈ N ∪ {0} and 0 < ε < 1, there exists an n-dimensional subspace Xn ⊂ X such that K ⊂ Xn + dn (K)(1 + ε)BX .

(11)

T K ⊂ T Xn + dn (K)(1 + ε)T BX .

(12)

Therefore Now we show that for each δ > 0 there is N ∈ N such that T BX ⊂ T Xn + δBY for n ≥ N.

(13)

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In fact, since T BX is compact, it has a finite δ/3-net {yi }ti=1 ⊂ T BX . Since T BX ⊂ T VK , the vectors yi can be arbitrarily well approximated by linear combinations of vectors from T K. Let M be the maximum absolute sum of coefficients of a selection of such δ/3-approximating linear combinations. Let N be such that δ for n ≥ N we have dn (K) ≤ 6M||T || , and let us show that (13) holds. We need to show that for all y ∈ T BX we have dist(y, T Xn ) ≤ δ. s Let j ∈ {1, . . . , t} be such that ||y − y || < δ/3, and let j i=1 αi T xi be such s s that xi ∈ K, i=1 |αi | ≤ M , and ||yj − i=1 αi T xi || < δ/3 . By (11), we have dist(xi , Xn ) ≤ dn (K)(1 + ε). Therefore s s δ δ dist( · (1 + ε) < . αi T xi , T Xn ) ≤ |αi |||T ||dn (K)(1 + ε) ≤ M ||T || · 6M ||T || 3 i=1 i=1 Thus dist(y, T Xn ) < δ. If we combine (12) and (13) we get dn (T K) ≤ (1 + ε)δdn (K) for n ≥ N . Since 0 < ε < 1 and δ > 0 can be chosen arbitrarily, the statement follows. Remark 6.2. Note that if X is a Hilbert space, then dñ (K) = dn (K). Indeed, in this case we may assume that Xn ⊂ VK . Such subspace can be found as the orthogonal projection to VK of any subspace Xn satisfying (11). It should be mentioned that in the case where both X and Y are Hilbert spaces a simpler proof of Lemma 6.1 is known, see [7, Lemma 1]. Now we find criteria of non-emptiness of CG(K1 , K2 ) for ellipsoids K1 and K2 . The result can be considered as an analogue of Lemmas 3.2 and 4.3. Lemma 6.3. Let K1 and K2 be ellipsoids in Hilbert spaces H1 and H2 , respectively. There is a compact operator T satisfying T K1 ⊃ K2 if and only if dn (K2 ) = o(dn (K1 )).

(14)

Proof. If there is a compact operator T with T K1 ⊃ K2 then (14) follows from Lemma 6.1 and Remark 6.2. Conversely, if (14) holds, then the existence of a compact operator T follows from the argument of Lemma 4.3. If the condition (14) is satisfied we say: the sequence {dn (K1 )}∞ n=1 strictly majorizes {dn (K2 )}∞ . n=1 Let us define by W CG(K1 , K2 ) the WOT-closure of CG(K1 , K2 ). We have the following analogue of Theorem 5.4: Theorem 6.4. (A) If all left shifts of the sequence {dn (K1 )} strictly majorize the sequence {dn (K2 )}, then W CG(K1 , K2 ) = L(H1 , H2 ). (B) If the k th left shift of {dn (K1 )} strictly majorizes the sequence {dn (K2 )}, but the (k+1)th left shift does not (such cases are clearly possible), then W CG(K1 , K2 ) is the set of those operators T ∈ L(H1 , H2 ) for which the image of the space T (VK1 ) in the quotient space H2 /VK2 is at most k-dimensional. The proof is a straightforward modification of the proof of Theorem 5.4 and we omit it.

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7. Operator ranges In this section by a Hilbert space we mean a separable infinite-dimensional Hilbert space. An operator range is the image of a Hilbert space H1 under a bounded operator A : H1 → H2 . Operator ranges are actively studied, see [2], [4], [9], [13], and references therein. The purpose of this section is to use the results of the previous section to classify operator ranges. Our results complement the classification of operator ranges presented in [4, Section 2]. We restrict our attention to images of compact operators of infinite rank. The set A(BH1 ) will be called a generating ellipsoid of the operator range AH1 . The same operator range is the image of infinitely many different operators, therefore a generating ellipsoid of an operator range is not uniquely determined. However, the Baire category theorem implies that if K1 and K2 are generating ellipsoids of the same operator range, then cK1 ⊂ K2 ⊂ CK1 for some 0 < c ≤ C < ∞. We say that two sequences of positive numbers are equivalent if each of them majorizes the other. The observation above implies that the equivalence class of the sequence of n-widths {dn (K)}∞ n=0 of a generating ellipsoid of Y is uniquely determined by an operator range Y. We denote this equivalence class of sequences by d(Y). It is clear that a sequence is lacunary if and only if all of sequences equivalent to it are lacunary. It is also clear that left shifts of equivalence classes of sequences are well-defined as well as the conditions like d(Y1 ) majorizes d(Y2 ). Therefore the following notions are well-defined for operator ranges: (i) Y is lacunary; (ii) Y1 majorizes Y2 . We say that an operator range Y ⊂ H is dense if Y = H. Results of Section 5 on covering of one ellipsoid by another have immediate corollaries for operator ranges. Let A1 : H → H1 and A2 : H → H2 be compact operators of infinite rank and Yi = Ai H. Let R(Y1 , Y2 ) denote the set of all operators T satisfying (15) T Y1 ⊃ Y2 . We write R(Y) instead of R(Y, Y). The WOT-closure of R(Y1 , Y2 ) will be denoted by WR(Y1 , Y2 ). Corollary 7.1. Suppose that Y1 majorizes Y2 . Then (i) If all left shifts of d(Y1 ) majorize d(Y2 ), then WR(Y1 , Y2 ) = L(H1 , H2 ). (ii) Let k be a non-negative integer. If the k th left shift of d(Y1 ) majorizes d(Y2 ), but the (k + 1)th left shift does not, then WR(Y1 , Y2 ) is the set of those operators T for which the image of T Y1 in the quotient space H2 /Y2 has dimension ≤ k. In particular, if k = 0, we get: if the first left shift of d(Y1 ) does not majorize d(Y2 ), then WR(Y1 , Y2 )Y1 ⊂ Y2 . (iii) If Y1 is non-lacunary, then WR(Y1 , Y2 ) = L(H1 , H2 ). (iv) If Y1 and Y2 are dense, then WR(Y1 , Y2 ) = L(H1 , H2 ). Proof. To derive (i)-(iv) from Theorem 5.4 and its corollaries we need two observations: • R(Y1 , Y2 ) contains G(K1 , K2 ) for any pair of generating ellipsoids.

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• If T ∈ R(Y1 , Y2 ) then T K1 ⊃ K2 for some pair of generating ellipsoids. The first observation immediately implies (i), (iii), (iv), and “estimates from below” in (iv). The second observation shows that for “estimates from above” in (ii) we can use the same argument as in Section 5. One of the systematically studied objects in the theory of invariant subspaces, see [5, 14, 15, 16], is the algebra A(Y) of all operators that preserve invariant a given operator range Y. It is known, see [16, Theorem 1], that if Y is dense, then the WOT-closure WA(Y) of A(Y) coincides with L(H). It follows easily that in general WA(Y) consists of all operators that preserve the closure Y of Y. An operator algebra A is called full if it contains the inverses of all invertible operators in A. We call A weakly full if for each invertible operator T ∈ A, the operator T −1 belongs to the WOT-closure of A. Our next result shows that for algebras of the form A(Y) this property depends on d(Y). Corollary 7.2. (i) If the closure Y of an operator range Y ⊂ H has finite codimension in H, then the algebra A(Y) is weakly full. (ii) If Y is not lacunary and codim(Y) = ∞, then A(Y) is not weakly full. (iii) If Y is lacunary, then A(Y) is weakly full. Proof. (i) If T preserves Y then T Y ⊂ Y. If T is invertible, then it maps a complement of Y onto a complement of T (Y). If Y has finite codimension, this implies T Y = Y. Hence T −1 Y = Y, and T −1 is in the WOT-closure of A(Y). (ii) Let K be a generating ellipsoid of Y. Choose a nonzero vector y ∈ Y and let K0 = K ∩y ⊥ . By Lemma 4.5, the sequences {dn (K)} and {dn (K0 )} are equivalent. Using Observation 5.1 we find an operator D : VK0 → VK which satisfies D(K0 ) ⊃ K and is a (nonzero) multiple of an isometry. Since Y has infinite codimension, we can extend D to an invertible operator D : H → H. Observe that D(Y ∩ y ⊥ ) = Y, / WA(Y). On the other hand, the inclusion therefore D(y) ∈ / Y, and thus D ∈ D(K0 ) ⊃ K implies D−1 ∈ A(Y). (iii) If T ∈ A(Y) is invertible, then T −1 ∈ R(Y). Since Y is lacunary, applying Corollary 7.1 we conclude that T −1 preserves Y. Therefore T −1 ∈ WA(Y).

8. Bilinear operator equations One of the popular topics in operator theory is the study of linear operator equations XA = B and AX = B. We consider here a “bilinear operator equation” XAY = B,

(16)

where operators A, B are given. Its solution is a pair (X, Y ) of operators. We denote the set of all such solutions by S(A, B). For simplicity we restrict our attention to the case when all operators act on a fixed separable Hilbert space H. Such a pair (X, Y ) can be found if we fix one of the operators (say X) and solve the obtained linear equation (which has more than one solution in the degenerate cases only). So the study of the question “how many solutions does equation (16) have?” reduces

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to the study of the set of all first components, that is, the set of those X for which (X, Y ) is a solution for some Y . Let us denote this set by U (A, B). Corollary 8.1.

(i) The equation is solvable if and only if sn (B) = O(sn (A)).

(17)

(ii) Suppose that condition (17) holds. If operators A, B have dense ranges, or if the range of A is non-lacunary, then U (A, B) is WOT-dense in L(H). (iii) If the range of operator B is not dense and the condition sn (B) = O(sn+1 (A))

(18)

does not hold, then U (A, B) is not WOT-dense in L(H). Proof. Clearly X ∈ U (A, B) if and only if the equation (16) is solvable with respect to Y . This is equivalent to the inclusion XAH ⊃ BH. It remains to apply Corollary 7.1. If an operator A is not compact then the set is WOT-dense in L(H). Formally this is not a special case of Corollary 8.1(ii) because s-numbers are usually defined for compact operators only, but the proof in this case along the same lines is even simpler. In the rest of the section we prove that this result can be considerably strengthened: if A is not compact then S(A, B) itself is dense in L(H) × L(H) with respect to the weak (and even strong) operator topology. Lemma 8.2. For any two linearly independent families (x1 , ..., xn ), (y1 , ..., ym ) of vectors in H, two arbitrary families (x1 , ..., xn ), (y1 , ..., ym ) of vectors in H, and a number > 0, there is an invertible operator V with the properties V xi − xi < , V −1 yj − yj < . Proof. One can choose systems z1 , ..., zn and w1 , ..., wm close to (x1 , ..., xn ) and, respectively, (y1 , ..., ym ) in such a way that both systems (x1 , ..., xn , w1 , ..., wm ) and (y1 , ..., ym , z1 , ..., zn ) are linearly independent. Let us define an operator T between their linear spans by T xi = zi , T wj = yj . It is injective and therefore can be extended to an invertible operator on a finite-dimensional space containing these systems. Clearly an invertible operator on a finite-dimensional subspace can be extended to an invertible operator on the whole space (take the direct sum with the identity operator). We denote the group of all invertible operators on H by G(H). Note. In this section A∗ denotes the Hilbert space conjugate of an operator A. Lemma 8.3. If an operator X has dense image and an operator Y has trivial kernel, then the set ΓX,Y = {(XV −1 , V Y ) : V ∈ G(H)} is dense in L(H) × L(H) with respect to the strong operator topology (SOT).

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Proof. Let a system (x1 , ..., xn ), (y1 , ..., ym ), (x1 , ..., xn ), (y1 , ..., ym ) and > 0 be given as above. The system x ˜i = Y xi is linearly independent since ker Y = 0. Since XH is dense, there are zj with Xzj − yj < /2. Take 0 < δ < 2||X|| and choose an invertible operator V as in Lemma 8.2 for the system (˜ x1 , ..., x ñ ), (y1 , ..., ym ), (x1 , ..., xn ), (z1 , ..., zm ) and δ. The obtained inequalities imply that ΓX,Y is SOT-dense in L(H) × L(H).

Any solution (X, Y ) of the equation XY = B will be called a factorization of an operator B. Proposition 8.4. For each operator B in an infinite-dimensional Hilbert space H, the set P(B) of all its factorizations is SOT-dense in L(H) × L(H). Proof. Let H = H1 ⊕ H2 where H1 and H2 are of the same dimension as H. Let U1 and U2 be isometries with the ranges H1 and H2 , respectively. Then U1∗ and U2∗ isometrically map H1 and H2 , respectively, onto H, also U1∗ H2 = {0} and U2∗ H1 = {0}. We set Y = U1 and X = BU1∗ + U2∗ . Since XY = BU1∗ U1 + U2∗ U1 = B, we have (X, Y ) ∈ P(B), and therefore (XV −1 , V Y ) ∈ P(B) for each V ∈ G(H). It follows easily from the definition of operators X, Y that XH = H and ker(Y ) = 0. Applying Lemma 8.3 we conclude that P(B) is SOT-dense in L(H) × L(H). Let us write A B if the set S(A, B) of all solutions of (16) is SOT-dense s in L(H) × L(H). For brevity, we will denote by E the closure of a subset E of L(H) × L(H) with respect to the product of SOT-topologies. Lemma 8.5. If A B and B C, then A C. Proof. If (X, Y ) ∈ S(A, B) and (X1 , Y1 ) ∈ S(B, C), then (XX1 , Y1 Y ) ∈ S(A, C). s Taking (X1 , Y1 ) → (I, I) we get that (X, Y ) ∈ S(A, C) . Hence L(H) × L(H) ⊂ s S(A, C) and A C. We proved in Proposition 8.4 that I C for all C. So our aim is to show that A I for each non-compact A. Lemma 8.6. If P is a projection of infinite rank, then P I. Proof. Let U be an isometry with U U ∗ = P . Then (U ∗ , U ) ∈ S(P, I). Hence (V U ∗ , U V −1 ) ∈ S(P, I) for each V ∈ G(H). s

It follows that S(P, I) contains all pairs (M, N ) with N H ⊂ P H, M (I − P ) = 0. s Hence for each (X, Y ) ∈ L(H)×L(H), the pair (XP, P Y ) belongs to S(P, I) . Choose a net (Xλ , Yλ ) in S(P, I) with (Xλ , Yλ ) → (XP, P Y ) in SOT, then (Xλ + X(I − P ), Yλ + (I − P )Y ) ∈ S(P, I) (indeed (Xλ + X(I − P ))P (Yλ + (I − P )Y ) = Xλ P Yλ = 1). Since (Xλ + X(I − P ), Yλ + (I − P )Y ) → (X, Y ) we get that s (X, Y ) ∈ S(P, I) . The proof of the next lemma is immediate.

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Lemma 8.7. (i) If (X, Y ) ∈ S(F1 AF2 , I), then (XF1 , F2 Y ) ∈ S(A, I). In particular (ii) If F1 AF2 I, ker(F1 ) = 0 and F2 H = H then A I. Lemma 8.8. Let A = 0 ⊕ A1 , where A1 acts on infinite-dimensional space and is invertible. Then A I. then F is invertible and F A is a projection of infinite Proof. Let F = I ⊕ A−1 1 rank. Hence F A 1, by Lemma 8.6. Using Lemma 8.7(ii), we get that A 1. Lemma 8.9. If A ≥ 0 and A is not compact, then A I. Proof. For each ε > 0, let Pε = I − Q, where Q is the spectral projection of A corresponding to the interval (0, ε). Then Pε A is of the form 0 ⊕ B, where B is invertible and, for sufficiently small ε, non-compact. Hence Pε A I. By s Lemma 8.7, L(H)Pε × L(H) ⊂ S(A, I) . Since Pε → I when ε → 0, we get that A I. Theorem 8.10. If A is non-compact, then the set of all solutions of the equation (16) is SOT-dense in L(H) × L(H) for each B. Proof. It suffices to show that A I. Suppose firstly that the operator U in the polar decomposition A = U T of A is an isometry. The operator AU ∗ = U T U ∗ is non-negative and non-compact. By Lemma 8.9, AU ∗ I. Since U ∗ H = H, by Lemma 8.7(ii), we have A I. If U is a coisometry, then U ∗ A = T is a positive non-compact operator. By Lemma 8.9, T I, and since ker(U ∗ ) = 0, by Lemma 8.7(ii), we get A I.

9. A-expanding operators In operator theory, especially in dealing with interpolation problems, one often needs to consider Hilbert (or Banach) spaces with two norms and study operators with special properties with respect to these norms. The main purpose of this section is to show that Kolmogorov n-widths can be used to describe WOT-closures of some sets of operators given by conditions of this kind. Our interest to such conditions is inspired by the theory of linear fractional relations, see [11] and [12]. Let X be a Banach space and A ∈ L(X ) be a compact operator with an infinite-dimensional range. It determines a semi-norm xA = Ax on X . We consider the set E(A) of all operators R that increase this semi-norm: RxA ≥ xA for each x ∈ X , that is, E(A) := {R ∈ L(X ) : ||ARx|| ≥ ||Ax|| ∀x ∈ X }.

(19)

It turns out that the problem of description of E(A) is a dual version of the problem considered in previous sections: the following dual characterization of E(A) relates it with covering operators. Lemma 9.1. Let a Banach space X be reflexive. An operator R ∈ L(X ) satisfies R ∈ E(A) if and only if R∗ ∈ G(K), where K = A∗ (BX ∗ ).

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Proof. Assume that R∗ ∈ G(K), that is, R∗ K ⊃ K. Then ||ARx|| = sup |f (ARx)| = sup |(R∗ A∗ f )(x)| ≥ sup |(A∗ f )(x)| = ||Ax||, f ∈BX ∗

f ∈BX ∗

f ∈BX ∗

(20) for each x ∈ X. Thus R ∈ E(A). Conversely, if R∗ ∈ / G(K), then there is f ∈ K \ R∗ K. The set R∗ K is weakly closed. By the Hahn–Banach theorem and reflexivity of X there is x ∈ X with |f (x)| > supg∈R∗ K |g(x)| = ARx. Since |f (x)| ≤ Ax we obtain that R∈ / E(A). We denote the WOT-closure of E(A) by WE(A). Corollary 9.2. Let X be a reflexive Banach space, A an operator on X . Then {R∗ : R ∈ WE(A)} = W G(K), where K = A∗ (BX ∗ ). Proof. Since X is reflexive the map R → R∗ from L(X ) to L(X ∗ ) is bicontinuous in the WOT-topologies. Hence the result follows from Lemma 9.1. Corollary 9.3. Let X be reflexive. If A ∈ L(X ) is such that the sequence {dn (A∗ (BX ∗ ))}∞ n=0 is lacunary, then WE(A) is contained in the set of all operators for which ker A is an invariant subspace. Proof. Follows immediately from Lemma 3.5 if we take into account the observation that ker A is an invariant subspace of R if and only if A∗ X ∗ is an invariant subspace of R∗ (that is, if and only if R∗ ∈ AK ). Applying Theorem 2.1, we obtain the converse inclusion: Corollary 9.4. The set of all operators preserving ker A is contained in WE(A). If ker A = {0}, then WE(A) = L(X ). Applying Theorem 4.6, we get Corollary 9.5. If X is a separable Hilbert space and A ∈ L(X ) is such that the sequence of s-numbers of A is not lacunary, then WE(A) = L(X ). We can summarize Hilbert-space-case results in the following way: Theorem 9.6 (A complete classification in the Hilbert space case). Let X be a separable Hilbert space. (i) If the sequence of s-numbers of A is not lacunary, then WE(A) = L(X ). (ii) If the sequence of s-numbers of A is lacunary, then WE(A) coincides with the set of operators for which ker A is an invariant subspace. Finally, using Theorem 4.10 we obtain a result on the ultra-weak closure of WE(A): Corollary 9.7. Let A ∈ L(H) be such that its sequence of s-numbers is not lacunary. Then the closure of the set (19) in the ultra-weak topology coincides with L(H).

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Acknowledgment The authors would like to thank Heydar Radjavi for a helpful discussion and his interest in our work.

References [1] P. G. Casazza, H. Jarchow, Self-induced compactness in Banach spaces, Proc. Roy. Soc. Edinburgh, Sect. A 126 (1996), no. 2, 355–362. [2] R. W. Cross, M. I. Ostrovskii, V. V. Shevchik, Operator ranges in Banach spaces. I, Math. Nachr., 173 (1995), 91–114. [3] N. Dunford, J. T. Schwartz, Linear Operators. Part I: General Theory, New York, Interscience Publishers, 1958. [4] P. A. Fillmore, J. P. Williams, On operator ranges, Advances in Math., 7 (1971), 254–281. [5] C. Foia¸s, Invariant para-closed subspaces, Indiana Univ. Math. J., 21 (1971/72), 887–906. [6] V. P. Fonf, W. B. Johnson, A. M. Plichko, V. V. Shevchik, Covering a compact set in a Banach space by an operator range of a Banach space with basis, Trans. Amer. Math. Soc., 358 (2006), no. 4, 1421–1434. [7] C. K. Fong, H. Radjavi, On ideals and Lie ideals of compact operators, Math. Ann., 262 (1983), no. 1, 23–28. [8] I. C. Gohberg, M. G. Kre˘ın, 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. [9] D. Hadwin, V. I. Paulsen, Two reformulations of Kadison’s similarity problem, J. Operator Theory, 55 (2006), 3–16. [10] W. B. Johnson, J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, in: Handbook of the geometry of Banach spaces (W. B. Johnson, J. Lindenstrauss, Eds.) Vol. 1, Elsevier, Amsterdam, 2001, pp. 1–84. [11] V. A. Khatskevich, M. I. Ostrovskii, V. S. Shulman, Linear fractional relations for Hilbert space operators, Math. Nachr., 279 (2006), 875–890. [12] V. A. Khatskevich, M. I. Ostrovskii, V. S. Shulman, Quadratic inequalities for Hilbert space operators, Integral Equations Operator Theory, 59 (2007), 19–34. [13] W. E. Longstaff, Small transitive families of dense operator ranges, Integral Equations Operator Theory, 45 (2003), 343–350. [14] W. E. Longstaff, P. Rosenthal, On operator algebras and operator ranges, Integral Equations Operator Theory, 9 (1986), 820–830. [15] E. Nordgren, M. Radjabalipour, H. Radjavi, P. Rosenthal, Algebras intertwining compact operators, Acta Sci. Math. (Szeged), 39 (1977), no. 1-2, 115–119. [16] E. Nordgren, M. Radjabalipour, H. Radjavi, P. Rosenthal, On invariant operator ranges, Trans. Amer. Math. Soc., 251 (1979), 389–398. [17] A. Pinkus, n-widths in the approximation theory, Springer-Verlag, Berlin, 1985. [18] M. Takesaki, Theory of operator algebras, vol. I, Springer-Verlag, New York, 1979.

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M. I. Ostrovskii Department of Mathematics and Computer Science St. John’s University 8000 Utopia Parkway Queens, NY 11439 USA e-mail: [email protected] V. S. Shulman Department of Mathematics Vologda State Technical University 15 Lenina street Vologda 160000 Russia e-mail: shulman [email protected] Submitted: February 18, 2009.

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A Note on Aluthge Transforms of Complex Symmetric Operators and Applications Xiaohuan Wang and Zongsheng Gao Abstract. In this paper, we reprove that: (i) the Aluthge transform of a com1 1 plex symmetric operator T = |T | 2 U |T | 2 is complex symmetric, (ii) if T is a ∗ are unitarily equivalent. And complex symmetric operator, then (T)∗ and T ∗) we also prove that: (iii) if T is a complex symmetric operator, then (T s,t ∗ and (Tt,s ) are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal. Mathematics Subject Classification (2000). Primary 47A05; Secondary 47B20. Keywords. Complex symmetric operator, Aluthge transform, unitary equivalence, normal operator, p-hyponormal operator, class wA(s, t), w-hyponormal operator.

1. Introduction Throughout this paper, H denotes a separable complex Hilbert space and T a bounded linear operator on H. If T = U |T | denotes the polar decomposition of T , then the Aluthge transform of T is defined to be the operator Ts,t = |T |s U |T |t for s, t > 0. This transformation arose in the study of p-hyponormal operators [1, 7] and has since been studied in many different contexts [2, 3, 5, 15, 16, 19]. S. R. Garcia and M. Putinar have proven interesting results for complex symmetric operators. They show that the class of complex symmetric operators is surprisingly large and includes the normal operators, the Hankel operators, the compressed Toeplitz operators, and many integral operators. We refer the reader to [8, 10, 11] for a further details. Other recent articles concerning complex symmetric operators include [12, 17]. Before proceeding, let us briefly introduce some terminology and lemmas: This work was supported by National Natural Science Fund of China (10771011) and National Key Basic Research Project of China (2005CB321902).

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Definition 1.1. A conjugation is a conjugate-linear operator C: H → H, which is both involutive (C 2 = I) and isometric. Definition 1.2. We say that a bounded linear operator T : H → H is C-symmetric if T = CT ∗ C and complex symmetric if there exists a conjugation C with respect to which T is C-symmetric. Lemma 1.3 ([11]). If T = U |T | is the polar decomposition of a C-symmetric operator T , then T = CJ|T | where J is a partial conjugation, supported on cl(ran|T |), which commutes with |T |. In particular, U = CU ∗ C and |T | = CU |T |U ∗ C. Remark 1.4. To be specific, we say that an antilinear operator J is a partial conjugation if J restricts to a conjugation on (kerJ)⊥ . Note that a partial conjugation J can always be extended to a conjugation J˜ on the entire space H by forming the internal direct sum J˜ = J ⊕ J where J is any partial conjugation with support ker J. So in this paper, without special mention, we always say J is a conjugation on all H. Lemma 1.5 ([11]). If C and J are conjugations on a Hilbert space H, then U = CJ is a unitary operator. Moreover, U is both C-symmetric and J-symmetric. Recently, S. R. Garcia [9] proved that the Aluthge transform of a complex 1 1 symmetric operator T = |T | 2 U |T | 2 is complex symmetric. And he also presented an open question: Question. If T is complex symmetric and 0 < ε ≤ 12 , is it necessarily the case that the Aluthge transform Tε = |T |ε U |T |1−ε is also complex symmetric? In this paper, we study the relationship between the Aluthge transform Ts,t = |T |s U |T |t for s, t > 0 and the class of complex symmetric operators. We obtain the result that the Aluthge transform Ts,t of a complex symmetric operator T is also complex symmetric when s = t. Then in Section 3, we give some applications to complex symmetric operators.

2. Aluthge Transforms and Complex Symmetric Operators In this section, we study the relationship between the Aluthge transform Ts,t and the class of complex symmetric operators. Then we give some examples to interpret our theorem. Theorem 2.1. If T is a complex symmetric operator, then Ts,t = J(Tt,s )∗ J ∗ ) C. Ts,t = C (T s,t

(2.1) (2.2)

i.e., ∗) ∼ ∗ (T s,t = (Tt,s ) where ∼ = denotes unitary equivalence.

(2.3)

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Proof. If T is complex symmetric, by Lemma 1.3 and Lemma 1.5, we may write T = CJ|T | where J is a conjugation on H which commutes with |T | and CU = U ∗ C = J. Indeed the equality of (2.1) and (2.2) will immediately imply (2.3) since CJ is unitary and (CJ)∗ = JC. Since (U ∗ C)(CU )|T |s = |T |s , by Lemma 1.3, we have |T |2 = (CU |T |U ∗ C)2 = CU |T |2 U ∗ C. It follows that fn (|T |2 ) = fn ((CU |T |U ∗ C)2 ) = CU fn (|T |2 )U ∗ C s

for any polynomial fn (t) with real coefficients. Take fn (t) → t 2 , then |T |s = CU |T |s U ∗ C. Then we obtain Ts,t = |T |s U |T |t = (CU |T |s U ∗ C)U (CU |T |t U ∗ C) = CU |T |s U ∗ U ∗ CCU |T |t U ∗ C = CU |T |s U ∗ |T |t U ∗ C = CU (Tt,s )∗ U ∗ C = J(Tt,s )∗ J. Since

T ∗ = |T |U ∗ = U ∗ U |T |U ∗ = U ∗ |T ∗ |

and

s

s

|T |s = CU |T |s U ∗ C = C(U |T |2 U ∗ ) 2 C = C(T T ∗ ) 2 C = C|T ∗ |s C,

we have Ts,t = |T |s U |T |t = (C|T ∗ |s C)U (C|T ∗ |t C) = C|T ∗ |s U ∗ CC|T ∗ |t C = C|T ∗ |s U ∗ |T ∗ |t C ∗ ) C. = C (T s,t Then we obtain (2.1) and (2.2).

By Theorem 2.1, we easily obtain the following result: Corollary 2.2. If T is a complex symmetric operator, then the Aluthge transform Tt,t = |T |t U |T |t is complex symmetric. In other words, if T = CT ∗ C for some conjugation C, then there exists a conjugation J such that Tt,t = J(Tt,t )∗ J.

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Apparently, Theorem 2.1 implies [Theorem 2, 9] when s = t = lary 2.2 implies [Theorem 1, 9] when s = t = 12 . The following example illustrates Theorem 2.1.

1 2.

Corol-

Example. Let S denote the unilateral shift: S(a0 , a1 , . . .) = (0, a0 , a1 , . . .) on H = 2 (N). Both S and its adjoint S ∗ (a0 , a1 , . . .) = (a1 , a2 , . . .) commute with the canonical conjugation C(a0 , a1 , . . .) = (a0 , a1 , . . .) on H. The operator T = S ∗ ⊕ S on H ⊕ H is complex symmetric with respect to the conjugation 0 C C= C 0 on H ⊕ H, and C 2 = I. For ∗ S 0 T = , 0 S

J=

0 CS ∗

CS , 0

P |T | = 0

0 , I

where P denotes the orthogonal projection P (a0 , a1 , a2 , . . .) = (0, a1 , a2 , . . .). We have T = CJ|T |. J is a partial conjugation supported on ran|T |, and J|T | = |T |J. For T = U |T |, the partial isometry U = CJ in the polar decomposition of T is ∗ 0 S . U= 0 S Since

∗ PS P 0 , 0 S ∗ 0 PS P = CP CS 0

Ts,t = |T |s U |T |t = J(Tt,s )∗ J =

CSS ∗ CS ∗ 0

0 , S

we have Ts,t = J(Tt,s )∗ J. Since

∗ S PS |T | = U |T |U = 0

I 0 0 = , 0 P SS ∗ S 0 ∗ s ∗ ∗ t ∗) , = |T | U |T | = (T s,t 0 P S∗P ∗ ∗ 0 PS P 0 ∗ ) C = CP S P C C (T = , s,t 0 CSC 0 S ∗

∗

∗ ) C. Since we have Ts,t = C (T s,t ∗ S P SP S ∗ ∗ U (Tt,s ) U = 0

0 SS ∗ S ∗

S = 0

∗) ∼ ∗ we have (T s,t = (Tt,s ) . The following example illustrates Corollary 2.2.

0 P S ∗P

,

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Example. Consider the 3 × 3 matrix T in H = C3 defined by   0 1 1 T = 0 0 1 . 0 0 0 Let T = U |T | be polar decomposition of T . Then U and |T | turn out to be as follows:     0 0 0 0 √25 √15 2 1 −1 √2  , |T | = 0 √5 √5  . U = 0 √ 5 5 0 √15 √35 0 0 0 Then T is complex symmetric with respect to C(z1 , z2 , z3 ) = (z3 , z2 , z1 ). By Lemma 1.3 and Lemma 1.5, we may write T = CJ|T | where J is a partial conjugation on H which commutes with |T | and CU = U ∗ C = J. First we consider the case s = t. For s = t = 1, we have   0 0 0 2 Ts,t = |T |U |T | = 0 0 √5  0 0 √15 and

 0 ∗ ∗ ∗ ∗ ∗ 2  J(Ts,t ) J = J|T |U |T |J = CU |T |U |T |U C = CU (T ) C = 0 0

0 0 0

0



√2  . 5 √1 5

Then we have Ts,t = J(Ts,t )∗ J, i.e., Ts,t is complex symmetric. Next we consider the case s = t. For s = 2, t = 1, then we have   0 0 0 Ts,t = |T |2 U |T | = 0 0 1 , 0 0 1   0 0 0 J(Ts,t )∗ J = CU |T |U ∗ |T |2 T ∗ C = 0 25 56  . 0 15 53 Apparently, we have Ts,t = J(Ts,t )∗ J. However, this inequality does not imply that T is not a complex symmetric operator. There may exist some other conjugation J such that Ts,t = J (Ts,t )∗ J holds.

3. Some Applications On Complex Symmetric Operators Let T be a bounded linear operator on a Hilbert space H and p > 0. T is called a phyponormal operator if (T ∗ T )p ≥ (T T ∗ )p . In particular, T is called a hyponormal operator if p = 1 and a semi-hyponormal operator if p = 12 . Every p-hyponormal operator is q-hyponormal if 0 < q < p. There exists a q-hyponormal operator which is not p-hyponormal if 0 < q < p. The related topics and basic properties of

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p-hyponormal operators have been studied by many researchers, for example [1, 6, 13, 18]. Aluthge and Wang [4] introduced w-hyponormal operators defined via the Aluthge transform as follows: An operator T is w-hyponormal if and only if 1

1

1

|T | ≥ (|T | 2 |T ∗ ||T | 2 ) 2

1

1

1

and (|T ∗ | 2 |T ||T ∗ | 2 ) 2 ≥ |T ∗ |,

if and only if

|T| ≥ |T | ≥ |(T)∗ |. The definition of w-hyponormality clearly implies that every p-hyponormal operator is w-hyponormal. As a generalization of w-hyponormality, Ito [14] introduced class wA(s, t) as follows: An operator T belongs to class wA(s, t) for s > 0 and t > 0 if and only if t

(|T ∗ |t |T |2s |T ∗ |t ) s+t ≥ |T ∗ |2t if and only if

|Ts,t | s+t ≥ |T |2t 2t

s

and |T |2s ≥ (|T |s |T ∗ |2t |T |s ) s+t , 2s and |T |2s ≥ |(Ts,t )∗ | s+t .

Lemma 3.1. If a complex symmetric operator T is p-hyponormal, then T is normal. Proof. Assume that T is C-symmetric. Since T is p-hyponormal, we have ((T ∗ T )p − (T T ∗ )p )x, x ≥ 0

(3.1)

for any x ∈ H. Since C is isometric, we have Cx, C((T ∗ T )p − (T T ∗)p )x ≥ 0 for any x ∈ H. Since C(T ∗ T )p = (T T ∗ )p C, we obtain Cx, ((T T ∗ )p − (T ∗ T )p )Cx ≥ 0 for any x ∈ H, i.e.,

y, ((T T ∗)p − (T ∗ T )p )y ≥ 0 for any y ∈ H. Then we have ((T T ∗ )p − (T ∗ T )p )y, y ≥ 0

(3.2)

for any y ∈ H. By (3.1) and (3.2), it follows that (T ∗ T )p = (T T ∗)p whence T ∗ T = T T ∗ and T is normal. Theorem 3.2. If T is a complex symmetric operator, then the following are equivalent: (i) T is w-hyponormal, (ii) T belongs to class wA(t, t), (iii) T is normal, (iv) C and |T | commute (i.e., |T | is also C-symmetric).

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Proof. By the definition of w-hyponormality, we have that an operator T is whyponormal if and only if T belongs to class wA( 12 , 12 ). Then it suffices to prove (ii)=⇒(iii)=⇒(iv)=⇒(i). (ii)=⇒(iii). If an operator T belongs to class wA(t, t), by the definition of class wA(t, t), we have the Aluthge transform Tt,t = |T |t U |T |t is semi-hyponormal. Since T is complex symmetric, by Corollary 2.2, we have Tt,t is a complex symmetric operator. Applying Lemma 3.1, we have Tt,t is normal, i.e., T is normal. (iii)=⇒(iv). Since T is complex symmetric, we may write T = CJ|T | where J is a conjugation on H which commutes with |T |. For T is a normal operator, we have |T |2 = C|T |2 C. Then we obtain C|T | = |T |C. (iv)=⇒(i). Since C|T | = |T |C, we have |T | = |T ∗ |. Then T is w-hyponormal. Remark 3.3. Note that every normal operator belongs to class wA(t, t). The operators belonging to class wA(t, t) are not necessarily normal. However, by Theorem 3.2, we obtain that every complex symmetric operator which belongs to class wA(t, t) is normal. Acknowledgment We would like to express our cordial gratitude to the referee for his/her valuable advice and suggestions, especially for the simplified proof of Lemma 3.1.

References [1] A. Aluthge, On p-hyponormal operators, 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), no. 3-4, 281–292. [3] J. Antezana, P. Massey and D. Stojanoff, λ-Aluthge transforms and Schatten ideals, Linear Algebra Appl. 405 (2005), 177–199. [4] A. Aluthge and D. Wang, w-Hyponormal operators, Integral Equations and Operator Theory 36 (2000), 1–10. [5] T. Ando and T. Yamazaki, The iterated Aluthge transforms of a 2-by-2 matrix converge, Linear Algebra Appl. 375 (2003), 299–309. [6] M. Ch¯ o and M. Itoh, Putnam’s inequality for p-hyponormal operators, Proc. Amer. Math. Soc. 123 (1995), 2435–2440. [7] T. Furuta and M. Yanagida, Further extensions of Aluthge transformation on phyponormal operators, Integral Equations and Operator Theory 29 (1997), no. 1, 122–125. [8] S. R. Garcia, Conjugation and Clark Operators, Contemp. Math. 393 (2006), 67–112. [9] S. R. Garcia, Aluthge transforms of complex symmetric operators, Integral Equations and Operator Theory 60 (2008), 357–367. [10] S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), 1285–1315.

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[11] S. R. Garcia and M. Putinar, Complex symmetric operators and applications , Trans. Amer. Math. Soc. 359 (2007), 3913–3931. [12] T. M. Gilbreath and W. R. Wogen, Remarks on the structure of complex symmetric operators, Integral Equations and Operator Theory 59 (2007), 585–590. [13] T. Huruya, A note on p-hyponormal operators, Proc. Amer. Math. Soc. 125 (1997), 3617–3624. [14] M. Ito, Some classes of operators associated with generalized Aluthge transformation, SUT J. Math. 35 (1999), 149–165. [15] I. B. Jung, E. Ko and C. Pearcy, Aluthge transforms of operators, Integral Equations and Operator Theory 37 (2000), no. 4, 437–448. [16] K. Tanahashi, On log-hyponormal operators, Integral Equations and Operator Theory 34 (1999), 364–372. [17] J. Tener, Unitary equivalence to a complex symmetric matrix, J. Math. Anal. Appl. 341 (2008), 640–648. [18] D. Xia, Spectral theory of hyponormal operators, Birkh¨ auser Verlag, Boston, 1983. [19] T. Yamazaki, An expression of spectral radius via Aluthge transformation, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1131–1137. [20] J. Yuan and Z. Gao, Complete form of Furuta inequality, Proc. Amer. Math. Soc. 136 (2008), 2859–2867. Xiaohuan Wang and Zongsheng Gao LMIB and Department of Mathematics Beihang University Beijing 100083 China e-mail: [email protected] [email protected] Submitted: June 14, 2009. Revised: July 9, 2009.

Integr. equ. oper. theory 65 (2009), 581–591 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040581-11, published online August 24, 2009 DOI 10.1007/s00020-009-1713-y


Numerical Ranges of Radial Toeplitz Operators on Bergman Space Kuo Zhong Wang and Pei Yuan Wu Abstract. A Toeplitz operator Tφ with symbol φ in L∞ (D) on the Bergman space A2 (D), where D denotes the open unit disc, is radial if φ(z) = φ(|z|) a.e. on D. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls of analytic images of D and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand, Toeplitz operators Tφ with φ harmonic on D and continuous on D and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not. Mathematics Subject Classification (2000). Primary 47A12; Secondary 47B35. Keywords. Numerical range, radial Toeplitz operator, Bergman space, convexoid operator.

The Bergman space A2 (D) of the open unit disc D in the plane consists of analytic functions f : D → C which are square-integrable with respect to the area measure dA. It is a Hilbert space under the inner product f, g = f (z)g(z)dA(z) for f, g ∈ A2 (D), D

and has the orthonormal basis {en }∞ n=0 , where n+1 n en (z) = z for z ∈ D. π For any (essentially) bounded function φ on D, the Toeplitz operator Tφ with symbol φ is the operator on A2 (D) defined by Tφ f = P (φf )

for f ∈ A2 (D),

where P denotes the (orthogonal) projection from L2 (D) onto A2 (D). Tφ or φ is said to be radial if φ(z) = φ(|z|) for almost all z in D. Such operators have been investigated intensively in recent years (cf. [8, 4, 9]). The purpose of this paper is to study their numerical ranges.

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Kuo Zhong Wang and Pei Yuan Wu

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Recall that the numerical range of an operator A on the Hilbert space H is the set W (A) = {Ax, x : x ∈ H, x = 1}, where ·, · and · denote the inner product and its associated norm in H. The numerical range is always convex. For other properties of the numerical range, the reader may consult [6, Chapter 22] and [5]. We start with the general Toeplitz operators. The next proposition is all we can say about their numerical ranges at the present time. Proposition 1. If φ is a nonconstant function in L∞ (D), then W (Tφ ) is contained in the relative interior of the convex hull of the essential range of φ. The essential range Rφ of a function φ in L∞ (D) is the set of complex numbers u for which {z ∈ D : |φ(z) − u| < } has (strictly) positive area measure for every > 0, the convex hull R∧ of a subset R of the plane is the smallest convex set containing R, and the relative interior, Rel Int , of a (nonempty nonsingleton) convex subset is its interior relative to the affine subspace generated by it. Note that, in Proposition 1, W (Tφ ) is in general not equal to the asserted relative interior as the following example shows. Example 2. If

φ(z) =

1 0

if |z| ≤ 1/2, if 1/2 < |z| < 1,

then Tφ has the matrix representation diag (1/4, 1/16, . . . , 1/22(n+1), . . .) relative 2 to the standard basis {en }∞ n=0 of A (D). Hence W (Tφ ) = (0, 1/4], which is not equal to the relative interior (0, 1) of the convex hull of Rφ = {0, 1}. Note also that the spectrum σ(Tφ ) of Tφ is equal to {1/22(n+1) : n ≥ 0} ∪ {0}, which is not contained in Rφ . Proof of Proposition 1. Let Mφ be the multiplication operator Mφ f = φf on L2 (D). Since Tφ dilates to Mφ , we have W (Tφ ) ⊆ W (Mφ ) = Rφ∧ (cf. [6, Problems 81 and 216]). Assume that W (Tφ ) is not contained in the relative interior of Rφ∧ . Then we can find a real θ and a unit vector f in A2 (D) such that ∧ TRe (eiθ φ) f, f = max RRe (eiθ φ) ≡ a.

Hence MRe (eiθ φ) f, f = max W (MRe (eiθ φ) ) = a, from which we infer that (Re (eiθ φ))f = af . The analyticity of the nonzero f implies that the set {z ∈ D : Re (eiθ φ(z)) = a} has area measure zero. Hence Re (eiθ φ) = a a.e. on D. This says that the essential range of φ is contained in a line. Repeating the above arguments with Im (eiθ φ) replacing Re (eiθ φ) yields that φ is constant, contradicting our assumption. Thus we must have W (Tφ ) ⊆ Rel Int Rφ∧ .

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If the symbol φ of a Toeplitz operator Tφ on A2 (D) is (complex-valued) harmonic, then W (Tφ ) has been considered by Thukral [10]. The next result, though not stated explicitly, is essentially due to him. Proposition 3. If φ is a nonconstant harmonic function in L∞ (D), then W (Tφ ) equals the relative interior of Rφ∧ . Proof. By Proposition 1, W (Tφ ) is contained in the relative interior of Rφ∧ . If they ∧ are not equal, then W (TRe (eiθ (φ+c)) ) Rel Int RRe (eiθ (φ+c)) for some real θ and complex c, which is in contradiction to [10, Lemma 1 and Theorem 2]. The assertion in the preceding proposition is analogous to the corresponding one for Toeplitz operators on the Hardy space (cf. [7]). An operator A is said to be convexoid if W (A) = σ(A)∧ . Note that it is unknown whether σ(Tφ ) = Rφ holds for harmonic φ in L∞ (D). If this is indeed the case, then we would have the convexoidity of Toeplitz operators with harmonic symbols. The following result is a partial confirmation of this. Proposition 4. If φ is harmonic on D and continuous on D, then Tφ is convexoid. Proof. For a continuous φ on D, it is known that σe (Tφ ), the essential spectrum of Tφ , equals φ(∂D) (cf. [1, Corollary 10]). Hence φ(∂D) ⊆ σ(Tφ ). Next we show that every extreme point of Rφ∧ = φ(D)∧ is in φ(∂D). Indeed, if z0 is an extreme point of φ(D)∧ , then it is in ∂φ(D). Let the real θ0 and r0 and the complex c0 be such that φ0 ≡ eiθ0 φ + c0 satisfies φ0 (D) ⊆ {z ∈ C : |z| ≤ r0 } and |eiθ0 z0 + c0 | = r0 . Then φ0 is harmonic on D, continuous on D and eiθ0 z0 + c0 in ∂φ0 (D) satisfies |eiθ0 z0 +c0 | = max |φ0 (D)|. The maximum modulus principle says that eiθ0 z0 +c0 = φ0 (u0 ) for some u0 in ∂D. Hence z0 = φ(u0 ) is in φ(∂D). Therefore, the Krein– Milman theorem implies that Rφ∧ = φ(D)∧ ⊆ φ(∂D)∧ ⊆ σ(Tφ )∧ . This, together with Proposition 1 or 3, yields W (Tφ ) ⊆ σ(Tφ )∧ . Since σ(Tφ )∧ ⊆ W (Tφ ) always holds (cf. [6, Problem 214]), the convexoidity of Tφ follows. We now consider the main topic of this paper: radial Toeplitz operators. The following characterization of such operators is known in the literature (cf. [9, p. 631]). Proposition 5. Let φ be a function in L∞ (D). Then Tφ has a diagonal matrix 2 representation relative to the standard basis {en }∞ n=0 of A (D) if and only if φ is radial. In this case, the asserted matrix representation of Tφ is 1 diag (2(n + 1) r2n+1 φ(r) dr. 0

The next corollary is an easy consequence. Corollary 6. Let φ be a radial function in L∞ (D). If φ(1− ) ≡ limr→1− φ(r) exists, then Tφ is the sum of the scalar operator φ(1− )I and a compact operator.

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By Proposition 4, certain Toeplitz operators with harmonic symbols are convexoid. The same is true for radial Toeplitz operators since they are normal and normal operators are convexoid (cf. [6, Problem 216]). The next theorem gives examples of nonconvexoid Toeplitz operators. Theorem 7. If φ is a radial continuous function on D with φ(1) = 0 and ψ is a function in H ∞ with ψ(0) = 0, then Tφψ is compact and quasinilpotent. If, in addition, φψ is nonzero, then Tφψ is not convexoid. An operator A is quasinilpotent if its spectrum σ(A) is the singleton {0}. Note that the preceding theorem implies that a Toeplitz operator with continuous symbol may not be convexoid. ∞ Proof of Theorem 7. Let ψ(z) = k=1 ak z k on D. For m, n ≥ 0, we have (n + 1)(m + 1) φ(z)ψ(z)z n z¯m dA(z) bmn ≡Tφψ en , em = π D ∞ (n + 1)(m + 1) ak φ(z)z n+k z¯m dA(z) = π D k=1 2π 1 ∞ (n + 1)(m + 1) = ak rn+k+m+1 φ(r)dr ei(n+k−m)θ dθ π 0 0 k=1  1  2 (n + 1)(m + 1)am−n r2m+1 φ(r)dr if m > n, = 0  0 otherwise. Thus A = [bmn ]∞ m,n=0 , the matrix representation of Tφψ relative to the standard basis {en }∞ n=0 , is lower triangular with zero diagonals. For each j ≥ 0, let Aj be the matrix obtained from A by replacing the bmn ’s with m > j by 0. Since φ is radial with φ(1) = 0, the Toeplitz operator Tφ is compact (cf. [8]). The same is true for Tφψ = Tφ Tψ . Hence Aj converges to A in norm and σ(A) is totally disconnected. It follows that σ(Aj ) converges to σ(A) in the Hausdorff metric (cf. [3, Corollary 3.4]). Because σ(Aj ) = {0} for all j, we conclude that A is quasinilpotent and hence so is Tφψ . If φψ is nonzero, then W (Tφψ ) = {0} = σ(Tφψ )∧ , that is, Tφψ is not convexoid. In the following, we show that many commonly seen convex subsets of the plane are numerical ranges of radial Toeplitz operators. This we start with intervals on the real line. Proposition 8. If φ is a real-valued radial function in L∞ (D), then

1 W (Tφ ) = inf λn , sup λn , where λn = 2(n + 1) r2n+1 φ(r) dr for n ≥ 0. n≥0

n≥0

0

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If, in addition, φ(r) is (almost) nonconstant and increasing (resp., decreasing) in r on [0, 1), then 1

1 − − rφ(r) dr, φ(1 ) resp. φ(1 ), 2 rφ(r) dr . W (Tφ ) = 2 0

0

An example of decreasing φ(r) is given in Example 2. Proof of Proposition 8. The first assertion is an easy consequence of Proposition 5 and the fact that normal operators are convexoid [6, Problem 216]. Now assume that φ(r) is increasing in r. By the change of variable s = r2n+2 , we have 1

λn =

0

φ(s1/(2n+2) ) ds ≥

1

0

φ(s1/(2n) ) ds = λn−1

for n ≥ 1. Moreover, if here the equality holds for some n ≥ 1, then φ(s1/(2n+2) ) = φ(s1/(2n) ) a.e. or φ(r) = φ(r(n+1)/n ) a.e. on [0, 1), which is impossible since φ(r) is nonconstant and increasing. Hence the λn ’s are strictly increasing in n. Our assertion for W (Tφ ) follows immediately. Analogous arguments apply to decreasing φ(r). Some of our later results on the numerical ranges of radial Toeplitz operators are proved based on the construction for the essential spectrum due to Grudsky and Vasilevski [4]. These we summarize briefly below. For any real t = 0, let  1  (ln |z|−2 )it if z ∈ D and z = 0, (1) φt (z) = Γ(1 + it)  0 if z = 0, where Γ(·) denotes the usual Gamma function. Then it was shown that the cor1 responding λn ≡ 2(n + 1) 0 r2n+1 φ(r)dr, n ≥ 0, is given by (n + 1)−it (cf. [4, Example 4]). Thus we can derive that W (Tφt ) = D ∪ {(n + 1)−it : n ≥ 0} for t = 0, W (TIm φ1 ) = (−1, 1) and W (TIm φ1 +iIm φπ ) = (−1, 1) × (−1, 1). Theorem 9. Any finite line segment in the plane is the numerical range of some radial Toeplitz operator. Proof. We may assume that the finite line segment I is on the real line. If I = [a, b) (resp., (a, b]), then it is the numerical range of Tφ , where 4a − 3b if |z| ≤ 1/2, φ(z) = b if 1/2 < |z| < 1 (resp.,

φ(z) =

4b − 3a a

(cf. Example 2 or Proposition 8).

if |z| ≤ 1/2, if 1/2 < |z| < 1)

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If I = (a, b), then I = W (Tφ ) for a−b a+b Im φ1 + , 2 2 where φ1 is the radial function given in (1). √ Finally, consider I = [a, b]. Let r1 and r2 be such that 1/ 2 < r1 < 1/21/4 and 21/4 r1 < r2 < 1. If  if 0 ≤ |z| ≤ r1 ,  1 −1 if r1 < |z| ≤ r2 , φ(z) =  0 if r2 < |z| < 1, 1 1 then 2 0 rφ(r)dr = 2r12 − r22 > 0, 4 0 r3 φ(r) dr = 2r14 − r24 < 0, and 1 2(n + 1) r2n+1 φ(r) dr = 2r12n+2 − r22n+2 −→ 0 as n → ∞. φ=−

0

Thus W (Tφ ) is some closed interval [c, d] with c < 0 < d. If ψ=

bc − ad a−b φ+ , c−d c−d

then W (Tψ ) = [a, b], completing the proof.

For convex sets in the plane with nonempty interior, we make use of the radial functions φt in (1) to prove the following theorem. Theorem 10. For any function f analytic on an open set containing D, there is a radial function φ in L∞ (D) such that σe (Tφ ) = f (∂D) and W (Tφm ) = f (D)∧ , where φm (z) = |z|2m φ(z) for z ∈ D, for all m ≥ 1. ∞ k on D. By our assumption on f , we have α ≡ Proof. Let f (z) = k=0 ak z lim supk→∞ |ak |1/k < 1. Let 0 < t < −(2/π) ln α and  ∞  ak  (ln |z|−2 )itk if z ∈ D and z = 0, φ(z) = Γ(1 + itk) k=0   0 if z = 0. Then

lim sup k→∞

1/k ak = lim sup |ak |1/k lim |Γ(1 + itk)|−1/k k→∞ Γ(1 + itk) k→∞ = α lim (2π)−1/(2k) |e−itk(−1/k) ||(itk)(itk+(1/2))(−1/k) | k→∞

= α lim |e(−it−(1/(2k)))(ln(tk)+i(π/2)) | k→∞

= α lim e−(1/(2k)) ln(tk)+t(π/2) k→∞ t(π/2)

= αe

< 1,

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where the second equality is a consequence of Stirling’s formula lim

|z|→∞ −π+<arg z0

(cf. [2, p. 253, Section 34D]). This shows that the radius of convergence of the power series k (ak /Γ(1 + itk))z k is bigger than 1. Hence φ is a radial function in L∞ (D). For n ≥ 0, we have 1 1 ∞ ak (ln r−2 )itk dr r2n+1 φ(r) dr = 2(n + 1) r2n+1 2(n + 1) Γ(1 + itk) 0 0 k=0 1 ∞ ak = (ln s−1/(n+1) )itk ds ( letting s = r2(n+1) ) Γ(1 + itk) 0 =

k=0 ∞

ak (n + 1)−itk = f ((n + 1)−it ).

k=0

Since the set {(n + 1)−it : n ≥ 0} is dense in ∂D, we obtain σe (Tφ ) = f (∂D). For the numerical range, we may assume that f (0) = 0. This is because if f˜(z) = f (z) − a0 and f˜(D)∧ = W (Tψ ) for some radial ψ in L∞ (D), then f (D)∧ = f˜(D)∧ + a0 = W (Tψ ) + a0 = W (Tψ+a0 ). A computation as above with φ replaced by φm yields that 1 n+1 f ((n + m + 1)−it ), r2n+1 φm (r)dr = 2(n + 1) n + m + 1 0

n ≥ 0.

Since 0 is in f (D) and {(n + m + 1)−it : n ≥ 0} is dense in ∂D, the convexity of f (D)∧ implies that W (Tφm ) = f (D)∧ . Corollary 11. Any open elliptic disc is the numerical range of some radial Toeplitz operator. Proof. If E is an open elliptic disc, then let ψ(z) = (aRe z + bIm z + c) + i(uRe z + vIm z + w), where a, b, c, u, v and w are real with av = bu, be an affine transformation which maps D onto E. Theorem 10 says that D = W (Tφ ) for some radial function φ in L∞ (D). If η = ψ ◦ φ, then η is radial in L∞ (D) and W (Tη ) = W (ψ(Tφ )) = ψ(W (Tφ )) = ψ(D) = E.

The proof of Theorem 10 can be combined with the arguments for [4, Corollary 3.10] to yield the following proposition, whose proof we omit. Proposition 12. For any polynomial p (resp., trigonometric polynomial q), the convex set p((−1, 1) × (−1, 1))∧ (resp., Int q(∂D)∧ ) is the numerical range of some radial Toeplitz operator.

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Note that if the assertion in Theorem 10 is true for any function analytic on D and continuous on D, then, in view of the Riemann mapping theorem, every nonempty bounded open convex subset of the plane is the numerical range of some radial Toeplitz operator. Unable to prove this, we show that at least an asymptotic version of it is indeed true. Proposition 13. Let be a nonempty bounded open convex subset of the plane. Then there is a sequence of radial functions φn in L∞ (D) such that W (Tφn ) is open for all n and increases to . Proof. By the Riemann mapping theorem, there is an analytic function f (z) = ∞ k injectively. For each n ≥ 0, let Dn = k=0 ak z on D which maps D onto n {nz/(n + 1) : z ∈ D} and fn (z) = k=0 ak z k . Since the boundaries ∂f (Dn ) are compact and pairwise disjoint, we have dn ≡ dist (∂f (Dn ), ∂f (Dn+1 )) > 0 for all n. Let {kn }∞ n=1 be a (strictly) increasing sequence such that dn−1 dn sup{|fkn (z) − f (z)| : z ∈ Dn } < min , , n ≥ 1. 2 2 Since f (Dn ) increases to , from the construction of the kn ’s, we derive that fkn (Dn ) also increases to . By Theorem 10, each fkn (Dn )∧ is the numerical range of some radial Toeplitz operator Tφn . We conclude that W (Tφn ) is open and increases to . Finally, we come to closed polygonal regions. Theorem 14. Any compact convex polygonal region is the numerical range of some radial Toeplitz operator. To prove this, we need the following lemma. Lemma 15. For any > 0 and complex numbers λ0 , λ1 , . . . , λn−1 , there is an integer m ≥ n − 1 and a function f of the form k+1 k+1 k+1 1 1 1 + a1 + · · · + am m+1 , k ≥ 0, (2) f (k) = a0 2 4 2 such that f (k) = λk for 0 ≤ k ≤ n − 1 and |f (k)| ≤ for all k ≥ n. Proof. Let Am , m ≥ 1, denote the (m + 1)-by-(m + 1) Vandermonde-type matrix [1/2i+j+1 ]m i,j=0 . Since the determinant of Am equals the nonzero m 1 1 1 − , 2i+1 2j+1 2i+1 i=0 0≤i<j≤m

m Am is invertible. Let A−1 m = [bij ]i,j=0 . Here the entries bij depend on m (for the sake of simplicity, we don’t add further indices to them). For a large m (to be

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determined later), let ai = n−1 j=0 bij λj , 0 ≤ i ≤ m, and let f be defined as in (2). If a and c denote the (m + 1)-vectors   λ0    ..  a0  .     a1   λn−1     , and   ..    .   0   .  am  ..  0 respectively, then a = A−1 m c. Hence Am a = c, which is the same as λk if 0 ≤ k ≤ n − 1, f (k) = 0 if n ≤ k ≤ m. We now check that |f (k)| can be made arbitrarily small for any k > m. Indeed, we have |f (k)| ≤ ≤

m

1 k+1 |ai | i+1 2 i=0

m n−1 i=0

1 m+1 |bij ||λj | i+1 2 j=0

m 1 m+1 = |λj | |bij | i+1 . 2 j=0 i=0 n−1

To proceed further, we show that m+1 m 1 |bij | i+1 −→ 0 2 i=0

as m → ∞

for any j, 0 ≤ j ≤ n − 1. Let the (Vandermonde interpolation) polynomial m 1 x − 2l+1 pi (x) = 1 1 , − 2l+1 l=0 2i+1

0 ≤ i ≤ m,

l=i

be expanded as (1/2i+1 ) m 1

2i+1

j=0

m

j j=0 cij x .

cij

1 2l+1

Then

j

= pi

1 2l+1

= δil ,

0 ≤ l ≤ m,

which shows that cij = bij for all i and j. Moreover, for each fixed i, the bij ’s have (j) alternating signs. This can be seen by computing the higher-order derivatives pi

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(j)

of pi and noting that bij = 2i+1 pi (0)/j!. Hence m+1 m 1 |bij | i+1 2 i=0 m+1 j m m 1 1 j ≤ 2 bij − 2 2i+1 i=0 j=0 m+1 m 1 1 = 2j 2i+1 pi − 2 2i+1 i=0 m+1 m 1 m 21 + 2l+1 1 j i+1 =2 2 1 1 2i+1 2i+1 − 2l+1 i=0 l=0 l=i

= 2j

m 1 m 21 + 2l+1 1 − 2i−l i=0 l=0 l=i

−1 ∞ m m−1 1 1 + l+1 (1 − 2−l ) 2 2 i=0 l=0 l=1 −1 m−1 ∞ 1 1 ≤ 2j + (1 − 2−l ) (m + 1) −→ 0 2 4

≤ 2j

as m → ∞.

l=1

Hence for a large m ≥ n − 1 we have |f (k)| ≤ for all k ≥ n.

We now proceed to prove Theorem 14. Proof of Theorem 14. Let be a compact convex polygonal region with n (≥ 3) vertices λ0 , λ1 , . . . , λn−1 . We may assume that 0 is in its interior. Let > 0 be such that the circular disc {z ∈ D : |z| ≤ } is contained in , and let φ(z) =

m

ai χ[0,1/√2i+1 ] (|z|)

for z ∈ D,

i=0

where m and the ai ’s are as given in Lemma 15. Then φ is a radial function in L∞ (D) and 1/√2i+1 1 m r2k+1 φ(r)dr = ai 2(k + 1) r2k+1 dr 2(k + 1) 0

=

i=0 m i=0

0

ai

1 k+1 = f (k), 2i+1

k ≥ 0,

by (2). Since Lemma 15 says that f (k) = λk for 0 ≤ k ≤ n−1 and |f (k)| ≤ for all k ≥ n, the convex hull of the f (k)’s equals , that is, W (Tφ ) = as required.

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We conclude this paper by asking which (nonempty bounded convex) subset of the plane is the numerical range of a radial Toeplitz operator. One constraint is that it can have at most countably many extreme points as is the case for any normal operator (on a separable Hilbert space). In particular, is every nonempty bounded open convex subset the numerical range of some radial Tφ ? This seems to be quite plausible although we don’t know how to prove it at present. Acknowledgements. This research was conducted while the first author was a postdoctor supported by the National Science Council (NSC) of the Republic of China. The second author was also supported by the NSC under NSC 96-2115-M-009-013MY3 and by the MOE-ATU project.

References [1] S. Axler, J. B. Conway and G. McDonald, Toeplitz operators on Bergman spaces, Canad. J. Math. 34 (1982), 466–483. [2] R. P. Boas, Invitation to Complex Analysis, Random House, New York, 1987. [3] J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integral Equations and Operator Theory 2 (1979), 174–198. [4] S. Grudsky and N. Vasilevski, Bergman–Toeplitz operators: radial component influence, Integral Equations and Operator Theory 40 (2001), 16–33. [5] K. E. Gustafson and D. K. M. Rao, Numerical Range. The Field of Values of Linear Operators and Matrices, Springer, New York, 1997. [6] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982. [7] E. M. Klein, The numerical range of a Toeplitz operator, Proc. Amer. Math. Soc. 35 (1972), 101–103. [8] B. Korenblum and K. Zhu, An application of Tauberian theorems to Toeplitz operators, J. Operator Theory 33 (1995), 353–361. [9] D. Su´ arez, The eigenvalues of limits of radial Toeplitz operators, Bull. London Math. Soc. 40 (2008), 631–641. [10] J. K. Thukral, The numerical range of a Toeplitz operator with harmonic symbol, J. Operator Theory 34 (1995), 213–216. Kuo Zhong Wang and Pei Yuan Wu Department of Applied Mathematics National Chiao Tung University Hsinchu 300, Taiwan e-mail: [email protected] [email protected] Submitted: February 26, 2009. Revised: April 29, 2009.

Integr. equ. oper. theory 65 (2009), 593–604 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040593-12, published online June 26, 2009 DOI 10.1007/s00020-009-1693-y


An Interior Inverse Problem for Discontinuous Boundary-Value Problems Chuan-Fu Yang Abstract. In this work, we consider an inverse problem for the Sturm-Liouville equation with an interior discontinuity, and show that the potential function can be uniquely determined by a set of values of eigenfunctions at some interior point and parts of two spectra. Mathematics Subject Classification (2000). Primary 34B24; Secondary 47E05, 65L09. Keywords. Sturm-Liouville operator, discontinuous condition, inverse problem, interior spectral data.

1. Introduction The inverse spectral analysis is an important research topic in mathematical physics. The main task of inverse spectral analysis involves the reconstruction of a linear operator from its spectral characteristics (e.g., see [1, 9, 19, 20, 27]). The inverse problem of a regular Sturm-Liouville operator was first studied by Ambarzumyan in 1929 [2]. From then on, Ambarzumyan’s result has been generalized to various versions. We consider the following Sturm-Liouville problem −y (x) + q(x)y(x) = λy(x), 0 < x < π,

(1.1)

y (0) − hy(0) = 0, y (π) + Hy(π) = 0, (1.2) π π π π + 0 = a1 y − 0 , y + 0 = a−1 − 0 + a2 y − 0 , (1.3) y 1 y 2 2 2 2 2 with discontinuity conditions (1.3) at the point π2 . The problem (1.1), (1.2) and (1.3), denoted by B = B(q, h, H), is called a boundary value problem of the Sturm-Liouville equation with the discontinuity conditions at π2 . Here λ is the spectral parameter, q(x), h, H, a1 , a2 are real, q(x) ∈ L2 (0, π), and a1 > 0. Usually, problems of this kind are related to discontinuous and nonsmooth properties of π

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a medium (see [3, 12, 16]). The problem B has been studied in many papers [8, 12, 16, 30, 33]. In general, to reconstruct the potential function q on the whole interval, it is enough to use two spectra of the problem B with two different boundary conditions, respectively (see [33]). The paper [26] studied an inverse problem of the regular Sturm-Liouville operator by using interior spectral data: to reconstruct the differential operator from some known eigenvalues and some information on eigenfunctions at some interior point. In [31] the authors gave a uniqueness theorem for the Sturm-Liouville operator with discontinuous conditions depending on spectral data of the following kind: one spectrum and some information on eigenfunctions at the mid-point of the interval (0, π). In this work, we study the inverse problem of reconstructing the SturmLiouville operator with discontinuous conditions from spectral data of the following kind: parts of two spectra and some information on the eigenfunctions at some interior point b ∈ ( π2 , π). The technique we used is similar to those used in [14, 28].

2. Main Result It is well known that the boundary value problem B is self-adjoint and all the spectra of the problem B, λn , n = 0, 1, 2, . . . , are real and simple. The sequence {λn }n≥0 satisfies the classical asymptotic form [29, 33, 34] 1 λn = n + O . (2.1) n = B( We consider another boundary value problem B q , h, H) of the same form but with a different coefficient q. In the following we will agree that if a certain symbol δ denotes an object related to B, then δ will denote an analogous object related to B. The eigenvalues and the corresponding eigenfunctions of the problem B are denoted by λn and yn (x), n ∈ N,respectively. Let y(x) and z(x) be continuously differentiable functions on [0, π2 ) ( π2 , π]. Define the Lagrange bi-linear form [y, z](x) := y(x)z (x) − y (x)z(x). Note that if y(x) and z(x) satisfy the conditions (1.3), then a direct calculation yields π π [y, z] + 0 = [y, z] −0 . (2.2) 2 2 In the case b = second spectrum.

π 2,

the uniqueness of q(x) can be obtained from a part of the

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Let l(n), r(n) be sequences of natural numbers with the properties n (1 + 1,n ), 0 < σ1 ≤ 1, 1,n → 0, (2.3) l(n) = σ1 n (1 + 2,n ), 0 < σ2 ≤ 1, 2,n → 0; (2.4) r(n) = σ2 and let µn be the eigenvalues of the problem B(q, h, H1 ), H1 = H, H1 ∈ R. Now we state the main result of this work. Theorem 2.1. Let l(n), r(n) and b ∈ ( π2 , π) be such that σ1 > If for any n ∈ N, n , λn = λ

2b 2b − 1, σ2 > 2 − . π π

µl(n) = µ l(n) ,

(b) yr(n)

yr(n) (b)

=

(b) yr(n)

yr(n) (b)

,

then q(x) = q(x) for almost all x ∈ [0, π]. Remark 2.2. For the problem B without discontinuous conditions, i.e., in the case where a1 = 1 and a2 = 0 for the conditions (1.3), Theorem 2.1 was already proved in [26].

3. Proof First, we recall a result on the product of eigenfunctions as following [30]. Let y(x, λ) be the solution of the equation (1.1) satisfying the initial conditions y(0) = 1, y (0) = h and the conditions (1.3). Denote y(x, λ) = u1 (x, λ) for 0 ≤ x < π2 , and u2 (x, λ) for π2 < x ≤ π. It is well known that u1 (x, λ) satisfies a Volterra integral equation of the second kind, and u1 (x, λ) and u2 (x, λ) are entire functions of λ.√ Let k = λ. There exists a bounded function K(x, t) such that 1 x 1 1 K(x, t) cos(2kt)dt (3.1) y(x, λ) y (x, λ) = + cos(2kx) + 2 2 2 0 for 0 ≤ x < π2 , and π y(x, λ) y (x, λ) = A1 + A2 cos(2kx) + A3 cos 2k(x − ) 2 (3.2) 1 x + A4 cos 2k(x − π) + K(x, t) cos(2kt)dt 2 0 π for 2 < x ≤ π, where A1 = A3 =

1 a21 − a−2 π π 1 u1 ( − 0, λ) u1 ( − 0, λ), + 2 2a1 2 2 2 a21 − a−2 1 , 4

A2 =

2 (a1 + a−1 1 ) , 8

A4 =

2 (a1 − a−1 1 ) . 8

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√ √ √ Let λ = σ + iτ . Using the inequalities | sin λx| < e|τ |x and | cos λx| < e|τ |x , we obtain |y(x, λ) y (x, λ)| ≤ M1 e2b|τ | for 0 ≤ x ≤ b ≤

π 2,

(3.3)

and

|y(x, λ) y (x, λ)| ≤

1 + M2 eπ|τ | + A2 e2x|τ | + A3 e(2x−π)|τ | 2a21 + A4 e(2π−2x)|τ | + M3 e2x|τ |

(3.4)

≤ M4 e2b|τ | for

π 2

≤ x ≤ b ≤ π, where M1 , M2 , M3 and M4 are some positive constants. To prove Theorem 2.1, we need the following lemma.

Lemma 3.1. Let m(n) be a sequence of natural numbers such that n m(n) = (1 + n ), 0 < σ ≤ 1, n → 0. σ

(3.5)

(i) Let b ∈ (0, π2 ) satisfy σ>

2b . π

If for any n ∈ N, (b) ym(n)

m(n) , λm(n) = λ

ym(n) (b)

=

(b) ym(n)

ym(n) (b)

,

then q(x) = q(x) for almost all x ∈ [0, b]. (ii) Let b ∈ ( π2 , π) satisfy σ >2−

2b . π

If for any n ∈ N, (b) ym(n)

m(n) , λm(n) = λ

ym(n) (b)

=

(b) ym(n)

ym(n) (b)

,

then q(x) = q(x) for almost all x ∈ [b, π]. Proof. (i) Let −y (x) + q(x)y(x) = λy(x)

(3.6)

with the initial conditions y(0) = 1, y (0) = h and y (x) = λ y (x) − y (x) + q(x) with the initial conditions y(0) = 1, y (0) = h.

(3.7)

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Multiplying (3.6) and (3.7) by y(x) and y(x), respectively, taking the difference of the resulting equations and integrating the latter over [0, b], we obtain b G(k) := [ q (x) − q(x)]y(x, λ) y (x, λ)dx (3.8) 0 √ b = [ y (x, λ)y(x, λ) − y(x, λ)y (x, λ)]0 , k = λ. From the assumption ym(n) (b)

ym(n) (b)

=

ym(n) (b)

ym(n) (b)

,

together with the initial-value condition at 0, it follows that G λm(n) = 0, n ∈ N. Next, we will show that G(k) ≡ 0 on the whole k-plane. From (3.3) one has |G(k)| ≤ Ce2br| sin θ| (3.9) √ for some positive constant C, k = λ = reiθ . Moreover, we see that the entire function G(k) is a function of exponential type ≤ 2b. Define the indicator of function G(k) by h(θ) = lim sup r→∞

ln |G(reiθ )| . r

(3.10)

√ √ Since | λ| = r| sin θ|, θ = arg λ, from (3.9) and (3.10) one obtains the following estimate: h(θ) ≤ 2b| sin θ|.

(3.11)

It is known [18] that for any entire function G(k) of exponential type, not identically zero, one has 2π 1 n(r) ≤ lim inf h(θ)dθ, (3.12) r→∞ r 2π 0 where n(r) is the number of zeros of G(k) in the disk |k| ≤ r. By (3.11), 2π b 2π 4b 1 h(θ)dθ ≤ | sin θ|dθ = . 2π 0 π 0 π

(3.13)

From the assumption and the known asymptotic expression (2.1) of the eigenvalues λn , for the number of zeros of G(k) in the disk |k| ≤ r we have the estimate

n(r) ≥ 2 1 = 2σr[1 + o(1)], r → ∞. (3.14) n 2 σ (1+O(1/n ))

IEOT

2b π,

4b 1 n(r) ≥ 2σ > ≥ lim n→∞ r π 2π

0

2π

h(θ)dθ.

(3.15)

The inequalities (3.12) and (3.15) imply that G(k) ≡ 0 on the whole k-plane. Define Q(x) = q(x) − q(x).

(3.16)

Finally, we show that Q(x) = 0 for almost all x ∈ [0, b]. From (3.1) and (3.8), we obtain x b b Q(x) [1 + cos(2kx)] dx + Q(x) K(x, t) cos(2kt)dt dx = 0 0

0

0

on the whole k-plane, i.e.,

b b Q(x)dx + cos(2kt) Q(t) + 0

0

b t

Q(x)K(x, t)dx dt = 0.

(3.17)

Letting k → ∞ for real k in (3.17), the Riemann-Lebesgue Lemma shows that b Q(x)dx = 0 0

and

b 0

cos(2kt) Q(t) +

hence

Q(t) +

t

t

b

Q(x)K(x, t)dx dt = 0,

b

Q(x)K(x, t)dx = 0, 0 < t < b,

since the functions {cos(2kt), k = 0, 1, 2, . . . } form an orthogonal basis of L2 [0, b]. But this equation is a homogeneous Volterra integral equation and has only the zero solution, thus Q(x) = 0, 0 < x < b, i.e., q(x) = q(x) for almost all x ∈ [0, b]. (ii) To prove that q(x) = q(x) for almost all x ∈ [b, π], we will consider the supplementary problem B −y (x) + q1 (x)y(x) = λy(x), q1 (x) = q(π − x), 0 < x < π, y (0) − Hy(0) = 0, y (π) + hy(π) = 0, π π π π + 0 = a−1 − 0 , y + 0 = a − 0 + a − 0 . y y a y y 1 1 2 1 2 2 2 2 2 A direct calculation implies that yn (x) := yn (π − x) is the solution to the supple and yn (π − b) = yn (b). Note that π − b ∈ (0, π ). Thus, the mentary problem B 2 in the case (i) are still satisfied. assumption conditions for B π

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Repeating the above arguments we have Q(π − x) = 0 for almost all x ∈ [0, π − b], i.e., q(x) = q(x) for almost all x ∈ [b, π]. Now we can give the proof of the main theorem in this work. Proof of Theorem 2.1 From r(n) , λr(n) = λ

(b) yr(n)

yr(n) (b)

=

(b) yr(n)

yr(n) (b)

,

where r(n) satisfies (2.3) and σ2 > 2 − 2b (x) for π , Lemma 3.1 implies that q(x) = q almost all x ∈ [b, π]. Thus, it needs to be proved that q(x) = q(x) for almost all x ∈ [0, b]. In the case b ∈ ( π2 , π), the equation (3.8) should be replaced by b G(k) : = [ q (x) − q(x)]y(x, λ) y (x, λ)dx 0 π −0 (3.18) = [ y (x, λ)y(x, λ) − y(x, λ)y (x, λ)] 02 + [ y (x, λ)y(x, λ) − y(x, λ)y (x, λ)] bπ2 +0 . We will finish the remainder of the proof by three steps. Step 1 : To show that G(k) ≡ 0 on the whole k-plane. Since the eigenfunctions yn (x) and yn (x) satisfy the same boundary condition at π and q(x) = q(x) for almost all x ∈ [b, π], we have yn (x) = αn yn (x), n ∈ N,

(3.19)

on [b, π], where αn are constants. From (2.2), (3.18) and (3.19), we obtain that G( λn ) = 0, n ∈ N, and, in the same way,

√ G( µln ) = 0, n ∈ N.

√ √ Let kn = λn , sn = µn . The kn and sn satisfy the asymptotic expression (2.1). Counting the number of kn and sn located in the disc of radius r, we have 1 1 + 2r 1 + O n2 of kn ’s and

1 1 + 2rσ1 1 + O n2

of sn ’s. Thus, the total number of kn and sn in the disc is 1 n(r) = 2 + 2 r(σ1 + 1) + O , n2

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and n(r) = 2(σ1 + 1). r→∞ r lim

As in Lemma 3.1, considering the condition σ1 > 2b π −1 we will show that inequality (3.12) does not hold. Repeating the proof of Lemma 3.1 we can show that G(k) ≡ 0 on the whole k-plane. Step 2 : To obtain an integral equation (3.23) and the forms (3.28) and (3.29) of F (t). Substituting the expressions (3.1) and (3.2) of y(x, λ) y (x, λ) into (3.18), we obtain x π2 Q(x)[1 + cos(2kx) + K(x, t) cos(2kt)dt]dx 0

+

b π 2

0

Q(x)[2A1 + 2A2 cos(2kx) + 2A3 cos 2k(x −

+ 2A4 cos 2k(x − π) +

0

π ) 2

(3.20)

x

K(x, t) cos(2kt)dt]dx = 0.

Letting k → ∞ in (3.20), by the Riemann-Lebesgue Lemma b π2 Q(x)dx + 2A1 Q(x)dx = 0 π 2

0

and

π 2

0

Q(x)[cos(2kx) + +

b π 2

x

0

K(x, t) cos(2kt)dt]dx

Q(x)[2A2 cos(2kx) + 2A3 cos 2k(x −

+ 2A4 cos 2k(x − π) +

0

π ) 2

x

K(x, t) cos(2kt)dt]dx = 0.

By changing the integral variables, we obtain an equation of the form b b K(x, t)Q(x)dx cos(2kt)dt = 0 F (t) + 0

(3.21)

(3.22)

t

for all k. Since the functions {cos(2kt), k = 0, 1, 2, . . . } form an orthogonal basis of L2 [0, b], we have b K(x, t)Q(x)dx = 0, t ∈ [0, b]. (3.23) F (t) + t

The form of F (t) will help us to obtain that Q(x) = 0 in [0, b].

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First, we consider the terms with K(x, t) in (3.21). Since K(x, t) is bounded on (x, t) ∈ [0, π] × [0, π] and Q(x) is integrable on [0, π], by the Fubini’s Theorem, π2 x b x Q(x) K(x, t) cos(2kt)dtdx + Q(x) K(x, t) cos(2kt)dtdx π 0 0 0 2 (3.24) b

b

= 0

t

K(x, t)Q(x)dx cos(2kt)dt.

Second, we consider the remainder terms in (3.21). Specifically, we have π2 b Q(x) cos(2kx)dx + 2A2 Q(x) cos(2kx)dx 0

π 2

b

= 0

cos(2kt)dt, Q(t)

where = Q(t)

b π 2

and

(3.25)

t ∈ [0, π2 ] t ∈ [ π2 , b],

Q(t), 2A2 Q(t),

b− π2 π π dx = cos(2kt)dt, 2A3 Q(x) cos 2k x − 2A3 Q t + 2 2 0

b π 2

2A4 Q(x) cos 2k(x − π)dx =

π 2

π−b

2A4 Q(π − t) cos(2kt)dt.

(3.26)

(3.27)

The equations (3.24)–(3.27) imply that F (t) in (3.23) has the following form. If π > b ≥ 3π 4 ,  t ∈ [0, π − b], Q(t) + 2A3 Q(t + π2 ),    Q(t) + 2A3 Q(t + π2 ) + 2A4 Q(π − t), t ∈ [π − b, b − π2 ], F (t) = (3.28) Q(t) + 2A4 Q(π − t), t ∈ [b − π2 , π2 ],    2A2 Q(t), t ∈ [ π2 , b]; and if

π 2

b ≥ 3π 4 , Q(x) = 0 in [0, b]. In the case 2 < b < 4 , the proof is similar and it follows that Q(x) = 0 for almost all x ∈ [0, b]. From the above arguments, we obtain q(x) = q(x) for almost all x ∈ [0, b]. Consequently, q(x) = q(x) for almost all x ∈ [0, π]. This completes the proof of the theorem. Acknowledgment The author acknowledges helpful comments and suggestions from the referees. The author would like to thank Professor Christiane Tretter of Mathematical Institute, University of Bern, Switzerland, for giving some corrections in language and layout, and Professor Hong-you Wu of Department of Mathematics, Northern Illinois University, USA, for discussions related to some topics of spectral analysis of differential operators. This work was supported by a Grant-in-Aid for Scientific Research from Nanjing University of Science and Technology (AB 96240).

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