Hankel Operators and their Applications

Preface The purpose of this book is to describe the theory of Hankel operators, one of the most important classes of o...

Author: Vladimir Peller

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Preface

The purpose of this book is to describe the theory of Hankel operators, one of the most important classes of operators on spaces of analytic functions. Hankel operators can be defined as operators having infinite Hankel matrices (i.e., matrices with entries depending only on the sum of the coordinates) with respect to some orthonormal basis. Finite matrices with this property were introduced by Hankel, who found interesting algebraic properties of their determinants. One of the first results on infinite Hankel matrices was obtained by Kronecker, who characterized Hankel matrices of finite rank as those whose entries are Taylor coefficients of rational functions. Since then Hankel operators (or matrices) have found numerous applications in classical problems of analysis, such as moment problems, orthogonal polynomials, etc. Hankel operators admit various useful realizations, such as operators on spaces of analytic functions, integral operators on function spaces on (0, ∞), operators on sequence spaces. In 1957 Nehari described the bounded Hankel operators on the sequence space 2 . This description turned out to be very important and started the contemporary period of the study of Hankel operators. We begin the book with introductory Chapter 1, which defines Hankel operators and presents their basic properties. We consider different realizations of Hankel operators and important connections of Hankel operators with the spaces BM O and V M O, Sz.-Nagy–Foais functional model, reproducing kernels of the Hardy class H 2 , moment problems, and Carleson imbedding operators.

vi

Preface

It turns out that for the needs of applications it is also important to consider vectorial Hankel operators, i.e., Hankel operators on spaces of vector functions. We introduce vectorial Hankel operators in Chapter 2, to be used later in the book in control theory, approximation theory, and Wiener–Hopf factorizations (Chapters 11, 13, and 14). In Chapter 3 we introduce another very important class of operators on spaces of analytic functions, the class of Toeplitz operators. They can be defined as operators having infinite matrices with entries depending only on the difference of the coordinates. Though Hankel and Toeplitz operators have quite different properties, Hankel operators play an important role in the study of Toeplitz operators, and vice versa. We also study in Chapter 3 vectorial Toeplitz operators. In Chapter 4 we analyze the singular values of Hankel operators. The main result of the chapter is the Adamyan–Arov–Krein theorem, which shows that the nth singular value of a Hankel operator is the distance to the set of Hankel operators of rank at most n. Chapter 5 deals with parametrization of solutions of the Nehari problem. In other words, we parametrize the symbols of a Hankel (or a vectorial Hankel) operator that belong to the ball in L∞ of a given radius. In Chapter 6 we describe the Hankel operators that belong to Schatten– von Neumann classes S p as those whose symbols belong to certain Besov classes. We consider various applications of this description. In particular we obtain sharp results on rational approximation in the norm of BM O. In this book we study many different applications of Hankel operators (in approximation theory, prediction theory, interpolation problems, control theory, etc). In Chapters 8 and 9 we use Hankel operators to study regularity conditions for stationary processes. Chapter 10 is an introduction to the spectral theory of Hankel operators. We continue the analysis of the spectral problems of Hankel operators in Chapter 12, where we give a complete description of the spectral properties of the self-adjoint Hankel operators. It turns out that not only are Hankel operators used in control theory, but also the theory of Hankel operators can benefit from methods of control theory. In particular, in Chapter 12 the results on spectral properties of self-adjoint Hankel operators are based on balanced linear systems with continuous time and discrete time, a notion borrowed from control theory. Chapter 11 is devoted to applications of Hankel operators in control theory. We consider linear systems with discrete time and continuous time, the problems of robust stabilization, model reduction, and model matching. In Chapter 13 we study hereditary properties of maximizing vectors of vectorial Hankel operators. In other words, for a broad class of function spaces X we prove that if the symbol belongs to X, then all maximizing vectors belong to the same space X. We give several applications of this result. In particular, we use it to obtain hereditary properties of Wiener– Hopf factorizations.

Preface

vii

In Chapter 14 vectorial Hankel operators are used in the theory of approximation by analytic matrix and operator functions. We introduce the important notion of superoptimal approximation and prove the uniqueness of a superoptimal approximation under certain mild conditions on the matrix function. We obtain certain special factorizations and prove inequalities between the singular values of the corresponding Hankel operators and superoptimal singular values of their symbols. This beautiful theory has been developed during the last decade; it demonstrates the importance of vectorial Hankel operators in noncommutative analysis. One of the most beautiful applications of Hankel operators is given in Chapter 15. The last chapter gives a solution to the famous problem of whether a polynomially bounded operator on Hilbert space must be similar to a contraction. This problem remained open for a long time. In particular, it was one of the problems in a famous paper by Paul Halmos called “Ten problems in Hilbert space”. Recently it has been solved in the negative with the help of vectorial Hankel operators. In this book we discuss only classical Hankel operators (i.e., operators with Hankel matrices or, in other words, Hankel operators on the Hardy class H 2 ). For the last 20 years many interesting results have been obtained about various generalizations of Hankel operators (commutators of multiplications and Calder´ on–Zygmund operators, paracommutators, Hankel operators on Bergman spaces, Hankel operators on function spaces on the polydisk, on the unit ball in Cn , on classical domains, etc). However, it is physically impossible to cover such generalizations in one book, and we restrict ourselves to classical Hankel operators. Even under this constraint it is hardly possible to cover all aspects of Hankel operators and their applications (for example, this book does not include applications of Hankel operators in noncommutative geometry, perturbation theory, or asymptotics of Toeplitz determinants). Each chapter ends with Concluding Remarks, where the reader can find references to some results not included here. Theorems, lemmas, and corollaries (as well as displayed formulas) are numbered lexicographically. Within the same chapter Theorem 3.5 means the fifth item of Section 3. To refer to a result from a different chapter, we use three numbers: Lemma 5.5.4 means the fourth item of Section 5 in Chapter 5. Reference to §6 means Section 6 within the same chapter. Reference to §4.3 means Section 3 in Chapter 4. Displayed formulas have an independent numeration. For convenience we add two appendices in which the reader can find necessary information on operator theory and function spaces. Reference to Appendix 2.5 means Section 5 of Appendix 2. I would like to express my deep gratitude to my colleagues with whom I discussed many aspects of Hankel operators and their applications. I am especially grateful to A.B. Aleksandrov, R.B. Alexeev, E.M. Dyn’kin,

viii

Preface

A.Y. Kheifets, S.V. Khrushchëv, S.V. Kislyakov, A.V. Megretskii, N.K. Nikol’skii, S.R. Treil, V.I. Vasyunin, A.L. Volberg, and N.J. Young. Okemos, Michigan May, 2002

Vladimir V. Peller

Notation

Symbols Z, R, C denote the set of integers, real numbers, and complex numbers. def

def

R+ = {x ∈ R : x ≥ 0}, Z+ = {n ∈ Z : n ≥ 0}. def

def

T = {ζ ∈ C : |ζ| = 1}, D = {ζ ∈ C : |ζ| < 1}. z denotes the identity map of C (or a subset of C) onto itself. m is normalized Lebesgue measure on T. m2 is planar Lebesgue measure. ϕ(n) ˆ is the nth Fourier coefficients of ϕ. Lp means the Lp space of functions on T with respect to m, unless otherwise specified. B(X, Y ) for Banach spaces X and Y is the space of bounded linear operadef

tors from X to Y , B(X) = B(X, X). H p is the Hardy class of functions analytic in D (see Appendix 2). If X is a Banach space, Lp (X) and H p (X) denote the X-valued Lp and H p spaces. If E is a subset of R or T, Lp (E) denotes the Lp -space of functions on E; this should not lead to a confusion with the previous notation. H p (C+ ) (H p (C− ), H p (C+ ), or H p (C− )) denotes the Hardy class of functions analytic in the upper (lower, right, or left) half-plane.

x

Notation

If E is a subset of R or T and X is a Banach space, we use the notation Lp (E, X) for the Lp -space of X-valued functions on E. def F is the Fourier transform, (Ff )(t) = R f (s)e−2πist ds, f ∈ L1 (R). span A means the closed linear span of a subset E in a Banach space. 1 is the constant function identically equal to 1. O is the zero in a vector space.

P is the space of trigonometric polynomials of the form cj z j . P + is the space of analytic polynomials of the form cj z j . j≥0

P − is the space of antianalytic polynomials of the form

cj z j .

jε (“p.v.” stands for “principal value”). Indeed,

f (τ ) f (τ ) − f (ζ) 1 p.v. dm(τ ) = dm(τ ) + f (ζ) p.v. dm(τ ). 1 − τ ¯ ζ 1 − τ ¯ ζ 1 − τ¯ζ T T T

12

Chapter 1. An Introduction to Hankel Operators

Clearly,

p.v. T

1 dm(τ ) 1 − τ¯ζ

= = =

1 dm(τ ) 1 − τ¯ T 1 1 + dm(τ ) 1 − τ¯ 1 − τ Im τ >0

1 2 dm(τ ). Re 1 − τ Im τ >0

p.v.

1 It is easy to see that for τ ∈ T \ {1} we have Re 1−τ = 12 , and so

1 1 dm(τ ) = . p.v. ¯ζ 2 T 1−τ

On the other hand, it is easy to verify that for f = z n we have

f (τ ) − f (ζ) 0, n ≥ 0, dm(τ ) = −ζ n , n < 0. 1 − τ¯ζ T Therefore

p.v. T

f (τ ) 1 1 dm(τ ) = −(P− f )(ζ) + f (ζ) = (P+ f )(ζ) − f (ζ), 1 − τ¯ζ 2 2

which implies (1.14). In fact, for any function f in L2 the integral

f (τ ) p.v. dm(τ ) ¯ζ T 1−τ exists for almost all ζ ∈ T and determines a bounded linear operator on L2 (see Stein [2], Ch. II, §4). Hence, (1.14) holds for any function f in L2 and for almost all ζ ∈ T. Obviously, (1.14) implies the following formula for Cϕ :

ϕ(ζ) − ϕ(τ ) f (τ )dm(τ ), f ∈ P. (1.15) Cϕ f = p.v. 1 − τ¯ζ T The following theorem gives a boundedness criterion for Cϕ . Theorem 1.10. Let ϕ ∈ L2 . The commutator Cϕ is bounded on L2 if and only if ϕ ∈ BM O. def

def

def

def

Proof. Put f+ = P+ f , f− = P− f , ϕ+ = P+ ϕ, and ϕ− = P− ϕ. We have Cϕ f

= ϕP+ f − P+ ϕf = ϕ+ f+ + ϕ− f+ − ϕ+ f+ − P+ ϕ− f+ − P+ ϕ+ f− = P− ϕ− f+ − P+ ϕ+ f− = Hϕ− f+ − (Hϕ+ )∗ f− .

It is easy to see now that Cϕ is bounded if and only if both Hankel operators Hϕ− and Hϕ+ are bounded. The result now follows from Theorem 1.3.

2. Hankel Operators and Compressed Shift

13

2. Hankel Operators and Compressed Shift We begin this section with the study of the commutant of compressions of the shift operator on H 2 to its coinvariant subspaces. We prove Sarason’s theorem that describes this commutant in terms of a functional calculus. We use an approach suggested by N.K. Nikol’skii that is based on Hankel operators and the Nehari theorem. Then we establish an important formula that relates functions of such a compression with Hankel operators. We also describe the kernel and the closed range of Hankel operators, the partially isometric Hankel operators, and the Hankel operators with closed ranges. Let ϑ be an inner function. Put Kϑ = H 2 ϑH 2 . By Beurling’s theorem (see Appendix 2.2) any nontrivial invariant subspace of the backward shift operator S ∗ on H 2 coincides with Kϑ for some inner function ϑ. Denote by S[ϑ] the compression of the shift operator S to Kϑ : S[ϑ] f = Pϑ zf,

f ∈ Kθ ,

(2.1)

where Pϑ is the orthogonal projection from H 2 onto Kϑ . It is easy to see that ∗ = S ∗ |Kϑ . S[ϑ]

Since multiplication by ϑ is a unitary operator on L2 , it follows that the orthogonal projection onto ϑH 2 is given by ¯ f → ϑP+ ϑf,

f ∈ L2 .

Therefore ¯ = ϑP− ϑf, ¯ Pϑ f = f − ϑP+ ϑf

f ∈ H 2.

(2.2)

Note that S[ϑ] is the so-called model operator in the Sz.-Nagy–Foias functional model. Any contraction T for which lim T ∗n = O in the strong n→∞ operator topology and rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1 is unitarily equivalent to S[ϑ] for some inner function ϑ (see Sz.-Nagy and Foias [1] and N.K. Nikol’skii [2]). The operator S[ϑ] admits an H ∞ functional calculus. Indeed, given ϕ ∈ H ∞ , we define the operator ϕ(S[ϑ] ) by ϕ(S[ϑ] )f = Pϑ ϕf,

f ∈ Kϑ .

(2.3)

Clearly, this functional calculus is linear. Let us show that it is multiplicative. Let ϕ, ψ ∈ H ∞ . We have ((ϕψ)(S[ϑ] ))f = Pϑ ϕψf = Pϑ ϕPϑ ψf + Pϑ ϕ Qϑ ψf,

(2.4)

14


where Qϑ is the orthogonal projection from H 2 onto ϑH 2 . Since Qϑ ψH 2 ∈ ϑH 2 , it follows that ϕ Qϑ ψf ∈ ϑH 2 and so Pϑ ϕ Qϑ ψf = O. Therefore it follows from (2.4) that (ϕψ)(S[ϑ] ) = ϕ(S[ϑ] )ψ(S[ϑ] ). It follows from (2.4) that ϕ(S[ϑ] ) commutes with S[ϑ] , ϕ ∈ H ∞ , and ϕ(S[ϑ] ) ≤ ϕH ∞ (von Neumann’s inequality). The following theorem, which is due to Sarason, describes the commutant of S[ϑ] . It is a special case of the commutant lifting theorem due to Sz.-Nagy and Foias (see Appendix 1.5). Theorem 2.1. Let T be an operator that commutes with S[ϑ] . Then there exists a function ϕ in H ∞ such that T = ϕ(S[ϑ] ) and T = ϕH ∞ . To prove the theorem, we need a lemma that relates operators in the commutant of S[ϑ] with Hankel operators. Lemma 2.2. Let T be an operator on Kϑ . Consider the operator 2 A : H 2 → H− defined by Af = ϑ¯ T Pϑ f. (2.5) Then T commutes with S[ϑ] if and only if A is a Hankel operator. Proof. A is a Hankel operator if and only if P− zAf = Azf,

f ∈ H 2,

(see (1.13)), which means ¯ Pϑ f = ϑT ¯ Pϑ zf, P− z ϑT

f ∈ H 2,

which in turn is equivalent to ¯ Pϑ f = T Pϑ zf, ϑP− ϑzT

f ∈ H 2.

(2.6)

(2.7)

We have by (2.2) ¯ Pϑ f = Pϑ zT Pϑ f = S[ϑ] T Pϑ f. ϑP− ϑzT Since obviously the left-hand side and the right-hand side of (2.7) are zero for f ∈ ϑH 2 , it follows from (2.1) that (2.6) is equivalent to the equality S[ϑ] T f = T S[ϑ] f,

f ∈ Kϑ .

Proof of Theorem 2.1. By Lemma 2.2, the operator A defined by (2.5) is a Hankel operator. By the Nehari theorem, there exists a function ψ in L∞ such that ψ∞ = A and Hψ = A, i.e., P− ψf = ϑ¯ T Pϑ f, f ∈ H 2 . It follows that P− ψf = O for any f ∈ ϑH 2 . That means that Hψϑ = O. ¯ We have Put ϕ = ψϑ. Clearly, ϕ ∈ H ∞ and ψ = ϑϕ. ¯ f ∈ Kϑ , ϑ¯ T f = P− ϑϕf,


15

so ¯ = Pϑ ϕf = ϕ(S[ϑ] )f, T f = ϑ P− ϑϕf

f ∈ Kϑ .

(2.8)

Obviously, T = A = ψ∞ = ϕ∞ , which completes the proof. Remark. Formula (2.8) implies the following remarkable relation between Hankel operators and functions of model operators. Let ϑ be an inner function, ϕ ∈ H ∞ , then ϕ(S[ϑ] ) = Mϑ Hϑϕ ¯ |Kϑ ,

(2.9)

where Mϑ is multiplication by ϑ. This formula shows that ϕ(S[ϑ] ) has the same metric properties as Hϑϕ ¯ (e.g., compactness, nuclearity, etc.; we shall discuss that later). Formula (2.9) relates the Hankel operators of the form Hϑϕ ¯ with functions of model operators. The following theorem describes the Hankel operators of that form. Theorem 2.3. Let ψ ∈ L∞ . The following are equivalent: (i) Hψ has nontrivial kernel; 2 (ii) Range Hψ is not dense in H− ; ¯ (iii) ψ = ϑϕ for some inner function ϑ and some function ϕ in H ∞ . Proof. (i)⇒(iii). Suppose that Ker Hψ = {O} and Hψ = O. Then Ker Hψ is a nontrivial invariant subspace of the shift operator S (see Corollary 1.9). By Beurling’s theorem (see Appendix 2.2), Ker Hψ = ϑH 2 for some inner function ϑ. So P− ψϑf = O for any f in H 2 . Therefore Hψϑ = O ¯ and so ϕ = ψϑ ∈ H ∞ . Hence, ψ = ϑϕ. ¯ (iii)⇒(ii). For g = z¯ϑ we have ¯ ¯ = (f, z¯ϕ) z¯ϑ) ¯ =0 (Hϑϕ ¯ f, g) = (ϑϕf, for any f ∈ H 2 , and so g ⊥ Range Hψ . 2 ¯ , g ⊥ Range Hψ , and g = O. Then g = z¯h, (ii)⇒(i). Suppose that g ∈ H− 2 2 where h ∈ H . Then for f ∈ H we have ¯ = (f, ψ¯ ¯z h), ¯ 0 = (Hψ f, g) = (ψf, z¯h) ¯z h ¯ ∈ H 2 , i.e., ψh ∈ H 2 , which means that h ∈ Ker Hψ . and so ψ¯ − Let us now describe the kernel and the closure of the range of Hankel ∞ and ϑ is inner. Recall that two operators of the form Hϑϕ ¯ when ϕ ∈ H p H functions are called coprime if they have no common nonconstant inner divisors. Theorem 2.4. Let ϕ ∈ H ∞ and let ϑ be an inner function such that ϕ and ϑ are coprime. Then 2 Ker Hϑϕ ¯ = ϑH

and

2 ¯ 2. clos Range Hϑϕ ¯ = H− ϑH −

16


2 Proof. Obviously, if f ∈ ϑH 2 , then Hϑϕ ¯ f = O. Suppose that f ∈ H 2 2 ¯ and Hϑϕ ¯ f = O. Then ϑϕf ∈ H , and so ϕf ∈ ϑH . Since ϕ and ϑ are coprime, it follows that f is divisible by ϑ, i.e., f ∈ ϑH 2 . 2 ¯ 2 . Let us It follows immediately from (2.9) that Range Hϑϕ ¯ ⊂ H− ϑH − 2 ¯ 2 . Suppose that h ∈ H 2 and show that Range Hϑϕ ¯ is dense in H− ϑH − − 2 h ⊥ Hϑϕ ¯ g for every g ∈ H . Then

¯ ¯ dm, g ∈ H 2 . gϕϑ¯h 0 = (P− ϑϕg, h) = (ϕg, ϑh) = T

1 2 Hence, ϕϑh ¯ ∈ H− . Since ϕ and ϑ are coprime, it follows that ϑh ∈ H− , 2 ¯ i.e., h ∈ ϑH− . The following result shows that the Hankel operators involved in (2.9) form a dense subset in the space of bounded Hankel operators. Theorem 2.5. The set ¯ : ϑ is inner, ϕ ∈ H ∞ } {ϑϕ

is dense in L∞ . Proof. Let E ⊂ T, 0 < mE < 1. Consider the function u in L∞ defined by

−1, ζ ∈ E, u(ζ) = 1, ζ ∈ E. u). Then fn ∈ H ∞ and |fn | is separated away Let fn = 1 + exp n(u + i˜ from zero on T. Indeed, |fn (ζ)| ≥ 1 − e−n for ζ ∈ E and |fn (ζ)| ≥ en − 1 for ζ ∈ E. So fn admits a factorization fn = ϑn hn , where ϑn is an inner function and hn is an outer function invertible in H ∞ . The result now follows from the obvious fact that fn−1 = ϑ¯n h−1 n converges uniformly to the characteristic function of E. Let us now describe the Hankel operators that are partial isometries. Recall that an operator V from a Hilbert space H to a Hilbert space K is called a partial isometry if the restriction of V to (Ker V )⊥ maps isometrically (Ker V )⊥ onto Range V . The subspace (Ker V )⊥ is called the initial space of the partial isometry V while the subspace Range V is called the final space of the partial isometry V . 2 is a partial isometry Theorem 2.6. A Hankel operator from H 2 to H− if and only if it has the form Hϑ¯ for an inner function ϑ. The initial space 2 ¯ 2. ϑH of Hϑ¯ is Kϑ while the final space of Hϑ¯ is H− − Proof. Let us first show that Hϑ¯ is a partial isometry. If f ∈ ϑH 2 , then ¯ ∈ H 2 , and obviously, Hϑ¯f = O. On the other hand, if f ⊥ ϑH 2 , then ϑf − so Hϑ¯f 2 = f 2 . It remains to observe that ¯ 2 H 2 = H 2 ϑH ¯ 2. H ¯(H 2 ϑH 2 ) = ϑH ϑ

−

−

Consider now an arbitrary partially isometric Hankel operator. Since obviously, Hψ z n → 0 for any ψ ∈ L∞ , a Hankel operator cannot be an isometry. Thus a partially isometric Hankel operator must have nontrivial


17

kernel. By Theorem 2.3, it has the form Hϑϕ ¯ , where ϑ is an inner function and ϕ ∈ H ∞ . Since Hϑϕ ¯ ≤ 1, we may assume that ϕ∞ ≤ 1. We may 2 also assume that ϑ and ϕ are coprime. By Theorem 2.4, Ker Hϑϕ ¯ = ϑH . Since Hϑϕ ¯ is a partial isometry, it follows that multiplication by ϕ is an isometric map of Kϑ into itself. In particular, if f ∈ Kϑ , we have

2 |ϕf | dm = |f |2 dm. T

T

Since |f | > 0 almost everywhere on T, it follows that |ϕ| = 1 almost everywhere, i.e., ϕ is an inner function. Let us show that ϕ is constant. Suppose that f is a nonzero function in Kϑ . Then ϕn f ∈ Kϑ for any n ∈ Z+ , i.e., ¯ g) = 0 (ϕn f, ϑg) = (ϕn ϑf, ¯ ∈ H 2 , and so ϕ¯n ϑ¯ z f¯ ∈ H 2 , n ∈ Z+ . Put for any g ∈ H 2 . Hence, ϕn ϑf − 2 n 2 ¯ h = ϑ¯ z f ∈ H . Then ϕ¯ h ∈ H for any n ∈ Z+ , i.e., h is divisible by ϕn , which is impossible unless ϕ is constant. This completes the proof. We conclude this section with a description of the Hankel operators with closed ranges. First we determine when functions of model operators S[ϑ] are invertible. We say that H ∞ functions g1 and g2 satisfy the corona condition if there exists δ > 0 such that |g1 (ζ)| + |g2 (ζ)| ≥ δ,

ζ ∈ D.

By the Carleson corona theorem (see Appendix 2.1), g1 and g2 satisfy the corona condition if and only if there exist functions ψ1 and ψ2 in H ∞ such that ψ1 (ζ)g1 (ζ) + ψ2 (ζ)g2 (ζ) = 1, ζ ∈ D. Theorem 2.7. Let ϑ be an inner function and let ϕ ∈ H ∞ . The operator ϕ(S[ϑ] ) is invertible if and only if ϕ and ϑ satisfy the corona condition. Proof. Suppose that ϕ and ϑ satisfy the corona condition, and u and v are functions in H ∞ such that ϕu + ϑv = 1. Then ϕ(S[ϑ] )u(S[ϑ] ) + ϑ(S[ϑ] )v(S[ϑ] ) = I, and since obviously, ϑ(S[ϑ] ) = O, it follows that ϕ(S[ϑ] )u(S[ϑ] ) = u(S[ϑ] )ϕ(S[ϑ] ) = I, i.e., ϕ(S[ϑ] ) is invertible. Assume now that ϕ(S[ϑ] ) is invertible, and ϕ and ϑ do not satisfy the corona condition. Then there exists a sequence {ζn }n≥0 of points in D such that lim (|ϕ(ζn )| + |ϑ(ζn )|) = 0. n→∞

Define the H ∞ functions ϕn and ϑn by ϕ − ϕ(ζn ) = bn ϕn

and ϑ − ϑ(ζn ) = bn ϑn ,

18


where the elementary Blaschke factor bn is defined by def bn (z) = (z − ζn )(1 − ζ¯n z)−1 .

Consider the normalized reproducing kernels kn defined by kn (z) = (1 − |ζn |2 )1/2 (1 − ζ¯n z)−1 def

(see §6 for more details). Clearly, kn 2 = 1. Finally, we define the functions fn in Kϑ by fn = Pϑ kn ϑn . Let us show that lim inf fn 2 > 0, n→∞

while

lim ϕ(S[ϑ] )fn 2 = 0,

n→∞

which will complete the proof. We have ¯ n ϑn = (ϑ − ϑ(ζ ))k ϑ fn 2 = P− ϑk P − n n n 2 = P−¯bn ϑ¯n kn ϑn 2 = P−¯bn |ϑn |2 kn 2 ≥ P−¯bn kn 2 − P−¯bn (|ϑn |2 − 1)kn 2 .

2

It is easy to see that P−¯bn kn = (1 − |ζn |2 )1/2 P−

1 1 = (1 − |ζn |2 )1/2 , z − ζn z − ζn

and so P−¯bn kn 2 = 1. On the other hand, P−¯bn (|ϑn |2 − 1)kn ≤ (|ϑn |2 − 1)kn = (|ϑ − ϑ(ζn )|2 − 1)kn 2 2 2 ≤ |ϑ − ϑ(ζn )|2 − 1∞ ≤ 3|ϑ(ζn )| → 0, n → ∞, and so the norms of fn are separated away from 0. Finally, we have ¯ ¯ ¯ ϕ(S[ϑ] )fn = P− ϑϕϑP − ϑkn ϑn 2 = P− ϕϑkn ϑn 2 2 = P− (ϑ − ϑ(ζn ))ϕkn ϑn + ϑ(ζn )ϕϑn

2

= P− (ϑ − ϑ(ζn ))ϕkn ϑn = P−¯bn ϕkn |ϑn |2 2 2

≤ P−¯bn ϕkn 2 + P−¯bn ϕkn (|ϑn |2 − 1)2 . Since lim |ϑn |2 − 1∞ = 0, it follows that n→∞

lim P−¯bn ϕkn (|ϑn |2 − 1)2 = 0.

n→∞

3. Hankel Operators of Finite Rank

19

On the other hand, P−¯bn ϕkn ≤ P−¯bn (ϕ − ϕ(ζn ))kn + |ϕ(ζn )| · P−¯bn kn 2 2 2 = P−¯bn ϕn bn kn 2 + |ϕ(ζn )| · ¯bn kn 2 = |ϕ(ζn )| → 0

as

n → ∞.

This completes the proof. Now we are in a position to obtain a description of the Hankel operators with closed ranges. 2 Theorem 2.8. A Hankel operator Hψ has closed range in H− if and ∞ ¯ only if ψ = ϑϕ, where ϕ ∈ H , ϑ is an inner function such that ϕ and ϑ satisfy the corona condition. 2 Proof. Since Hψ z n 2 → 0 as n → ∞, Range Hψ = H− . Thus by ¯ where ϑ is inner and ϕ ∈ H ∞ . Theorem 2.3, we may assume that ψ = ϑϕ, We may also assume that ϑ and ϕ are coprime. By Theorem 2.7, it remains to prove that ϕ(S[ϑ] ) is invertible if and only 2 if Hϑϕ ¯ has closed range in H− . If ϕ(S[ϑ] ) is invertible, it follows immediately from (2.9) that Hϑϕ ¯ has closed range. Suppose now that Hϑϕ ¯ has closed range. By Theorem 2.4, Hϑϕ ¯ is a 2 2 ¯ one-to-one map of Kϑ onto H− ϑH− . Clearly, multiplication by ϑ maps 2 ¯ 2 onto Kϑ . It follows now from (2.9) that ϕ(S[ϑ] ) is invertible. H− ϑH −

3. Hankel Operators of Finite Rank One of the first results about Hankel matrices was a theorem of Kronecker that describes the Hankel matrices of finite rank. Let r = p/q be a rational function where p and q are polynomials. If p/q is in its lowest terms, the degree of r is, by definition, deg r = max{deg p, deg q}, where deg p and deg q are the degrees of the polynomials p and q. It is easy to see that deg r is the sum of the multiplicities of the poles of r (including a possible pole at infinity). We are going to describe the Hankel matrices of finite rank without any assumption on the boundedness of the matrix. We identify sequences of complex numbers with the corresponding formal power series. If a = {aj }j≥0 is a sequence of complex numbers, we associate with it the formal power series a(z) = aj z j . j≥0

20


The space of formal power series forms an algebra with respect to the following multiplication:   m  (ab)(z) = aj bm−j  z m , a = aj z j , b = bj z j . j=0

m≥0

j≥0

It is easy to see that a(z) =

j≥0

aj z j is invertible in the space of formal

j≥0

power series if and only if a0 = 0. Consider the shift operator S and the backward shift operator S ∗ defined on the space of formal power series in the following way: aj z j = aj+1 z j . (Sa)(z) = za(z), S ∗ j≥0

j≥0

Let α = {αj }j≥0 be a sequence of complex numbers that we identify with the corresponding formal power series: α(z) = αj z j . (3.1) j≥0

Theorem 3.1. Let Γα be a Hankel matrix {αj+k }j,k≥0 . Then Γ has finite rank if and only if the power series (3.1) determines a rational function. In this case rank Γα = deg zα(z). Proof. Suppose that rank Γα = n. Then the first n + 1 rows are linearly dependent. That means that there exists a nontrivial family {cj }0≤j≤n of complex numbers (i.e., not all the cj are equal to zero) such that c0 α + c1 S ∗ α + · · · + cn S ∗n α = O.

(3.2)

It is easy to see that S n S ∗k α = S n−k α − S n−k

k−1

αj z j ,

k ≤ n.

(3.3)

j=0

It follows easily from (3.2) and (3.3) that O = Sn

n

ck S ∗k α =

k=0

n

ck S n S ∗k α =

k=0

n k=0

ck S n−k α − p, (3.4)

where p has the form p(z) =

n−1

pj z j .

j=0

Put q(z) =

n j=0

cn−j z j .

(3.5)

3. Hankel Operators of Finite Rank

21

Then p and q are polynomials, and it follows from (3.4) that qα = p. Note that cn = 0, since otherwise the rank of Γ would be less than n. Thus we can divide by q within the space of formal power series. Hence, α(z) = (p/q)(z) is a rational function. Clearly, deg zα(z) ≤ max (deg zp(z), deg q(z)) = n. Conversely, suppose that α(z) = (p/q)(z), where p and q are polynomials such that deg p ≤ n − 1 and deg q ≤ n. Consider the complex numbers cj defined by (3.5). We have n cj S n−j α = p. j=0

Therefore (S ∗ )n

n

cj S n−j α =

j=0

n

cj (S ∗ )j α = O,

j=0

which means that the first n + 1 rows of Γ are linearly dependent. Let m ≤ n be the largest number for which cm = 0. Then (S ∗ )m α is a linear combination of the (S ∗ )j α with j ≤ m − 1: (S ∗ )m α =

m−1

dj (S ∗ )j α,

dj ∈ C.

j=0

Let us show by induction that any row of Γ is a linear combination of the first m rows. Let k > m. We have m−1 dj (S ∗ )k−m+j α. (3.6) (S ∗ )k α = (S ∗ )k−m (S ∗ )m α = j=0

Since k − m + j < k for 0 ≤ j ≤ m − 1, by the inductive hypothesis each of the terms on the right-hand side of (3.6) is a linear combination of the first m rows. Therefore rank Γ ≤ m which completes the proof. Let us reformulate Kronecker’s theorem for Hankel operators Hϕ on H 2 . Corollary 3.2. Let ϕ ∈ L∞ . The Hankel operator Hϕ has finite rank if and only if P− ϕ is a rational function. In this case rank Hϕ = deg P− ϕ.

(3.7)

It is easy to derive this result from Theorem 3.1. However, here we give a different proof, which is also instructive. Another proof. As we have already noticed, Ker Hϕ is an invariant subspace of the shift operator (Corollary 1.9). By Beurling’s theorem, Ker Hϕ = ϑH 2 , where ϑ is an inner function. Clearly, Hϕ has finite rank if and only if dim(H 2 ϑH 2 ) < ∞ and the latter coincides with rank Hϕ . So rank Hϕ < ∞ if and only if ϑ is a finite Blaschke product, and rank Hϕ = deg ϑ (see Appendix 2.2). It follows that rank Hϕ < ∞ if and only if Hϑϕ = O, i.e., ϑϕ ∈ H ∞ for a finite Blaschke product ϑ.

22


It is easy to see that P− ϕ is a rational function of degree n if and only if there exists a Blaschke product ϑ of degree n such that ϑϕ ∈ H ∞ (the zeros of the Blaschke product are the poles of P− ϕ counted with multiplicities). This proves that rank Hϕ < ∞ if and only if P− ϕ is a rational function and that (3.7) holds. Corollary 3.3. Hϕ has finite rank if and only if there exists a finite Blaschke product B such that Bϕ ∈ H ∞ . Note, however, that the first proof of Kronecker’s theorem is universal; it works for any Hankel matrices without any additional assumption, while the second proof assumes that the Hankel operator is bounded on H 2 . Remark. Let Hϕ be a finite rank operator. In this case Hϕ f can be computed explicitly. Indeed, P− ϕ is a rational function. Consider first the case when 1 (P− ϕ)(z) = , λ ∈ D. z−λ We have f (λ) f (λ) f − f (λ) f = P− + = , f ∈ H 2. Hϕ f = P− z−λ z−λ z−λ z−λ (3.8) If (P− ϕ)(z) =

1 , (z − λ)k

λ ∈ D, k ≥ 1,

then f P− (z − λ)k

= P− +

=

f − f (λ) − f (λ)(z − λ) − · · · −

f (k−1) (λ) (k−1)! (z

− λ)k−1

(z − λ)k

k−1

k−1

1 f (j) (λ) 1 f (j) (λ) = k−j j! (z − λ) j! (z − λ)k−j j=0 j=0 k−1 f (ζ) ∂ z−ζ 1 (z, λ), f ∈ H 2 . (k − 1)! ∂ζ k−1

Therefore in the general case (P− ϕ)(z) =

mj n j=1 k=1

we obtain (Hϕ f )(z) =

mj n j=1 k=1

cjk (k − 1)!

∂ k−1

cjk (z − λj )k

f (ζ) z−ζ

∂ζ k−1

(z, λj ),

f ∈ H 2.

4. Interpolation Problems

23

4. Interpolation Problems We consider here certain classical interpolation problems and discuss their connections with Hankel operators.

Nevanlinna–Pick Interpolation Problem Let ζj , 1 ≤ j ≤ n, be distinct points in the unit disc and let wj , 1 ≤ j ≤ n, be complex numbers such that |wj | < 1, 1 ≤ j ≤ n. The problem is to find out under which conditions there exists a function f in H ∞ such that 1 ≤ j ≤ n;

f (ζj ) = wj ,

f H ∞ ≤ 1.

(4.1)

If the problem (4.1) has a solution, then we can ask a question how to describe all solutions of the problem (4.1) and find a solution of minimal norm. We are going to reduce the problem (4.1) to a problem about Hankel operators. Let g be an arbitrary function in H ∞ for which g(ζj ) = wj , 1 ≤ j ≤ n. Such a function can easily be constructed. For example, we can put g=

n j=1

wj Bj , Bj (ζj )

where B is the Blaschke product with zeros ζj , B=

n

bj ,

bj (z) =

j=1

z − ζj , 1 − ζ¯j z

and

B . bj Then a function f in H ∞ interpolates the values wj at ζj if and only if f has the form f = g − Bh, h ∈ H ∞ . Hence the problem (4.1) is solvable if and only if ¯ − h∞ : h ∈ H ∞ } = HBg inf{g − Bh∞ : h ∈ H ∞ } = inf{Bg ¯ ≤1 Bj =

by the Nehari theorem. Later (see §7.1) we shall discuss properties of the solution of minimal norm (it is unique since HBh ¯ has finite rank (see Corollaries 1.6 and 3.3)). We shall also describe the set of solutions in the case it contains at least two different ones (see §5.2). Let us reformulate the condition HBg ¯ ≤ 1 in terms of the interpolation data. Theorem 4.1. The interpolation problem (4.1) is solvable if and only if the matrix

1−w ¯j wk 1 − ζ¯j ζk 1≤j,k≤n

24


is positive semi-definite. 2 Proof. It is easy to see that HBg ¯ BH = O, so HBg ¯ = HBg ¯ KB . Consider the following basis in the (n-dimensional space KB (see §2)): fj (z) =

B(z) , z − ζj

1 ≤ j ≤ n.

Clearly, HBg ¯ ≤ 1 if and only if ∗ HBg ¯ ≤ I. ¯ HBg

(4.2)

Inequality (4.2) means that for any complex numbers cj , 1 ≤ j ≤ n, n

cj c¯k (HBg ¯ fj , HBg ¯ fk ) ≤

j,k=1

n

cj c¯k (fj , fk ).

(4.3)

j,k=1

We have HBg ¯ fj = P−

g g(ζj ) wj = = z − ζj z − ζj z − ζj

(see (3.8)). Therefore

¯k (HBg ¯ fj , HBg ¯ fk ) = wj w Clearly, (fj , fk ) =

B B , z − ζj z − ζk

1 1 , z − ζj z − ζk

=

¯k = wj w

1 1 , z − ζj z − ζk

=

1 . 1 − ζ¯k ζj 1 . 1 − ζ¯k ζj

Thus (4.3) is equivalent to the following inequality: n j,k=1

cj c¯k

1 − wj w ¯k ≥0 1 − ζj ζ¯k

for any complex numbers cj , 1 ≤ j ≤ n. This completes the proof. In a similar way one could consider Nevanlinna–Pick problem with an infinite data set. Indeed, we can reduce the case of infinitely many interpolating points to the case of finitely many points by considering a limit process. It is also possible to consider multiple interpolation problem where not only the values of f at the ζj are prescribed but also its derivatives up to certain orders. We shall consider here a very special case of multiple interpolation problem.

Carath´ eodory–Fej´ er Interpolation Problem Let c0 , c1 , · · · , cn be complex numbers. The problem is to determine whether there exists a function f in H ∞ such that fˆ(j) = cj ,

0 ≤ j ≤ n;

f H ∞ ≤ 1.

(4.4)

5. Compactness of Hankel Operators

Let g =

n

25

cj z j . Clearly, a function f in H ∞ satisfies fˆ(j) = cj , 0 ≤ j ≤ n,

j=0

if and only if f = g + z n+1 h, h ∈ H ∞ . So (4.4) is solvable if and only if inf{g + z n+1 h∞ : h ∈ H ∞ } = Hz¯n+1 g ≤ 1. Theorem 4.2. The interpolation problem (4.4) is solvable if and only if the operator norm of the (n + 1) × (n + 1) matrix   cn−1 . . . c0 cn  cn−1 cn−2 . . . 0     .. .. ..  . .  . . .  . 0 ... 0 c0 is less than or equal to one. Proof. This is an immediate consequence of the Nehari theorem.

5. Compactness of Hankel Operators In this section we shall establish the compactness criterion for Hankel operators which is due to Hartman. Moreover we shall compute the essential norm of a Hankel operator and study the problem of approximation of a Hankel operator by compact Hankel operators. We shall also establish some useful properties of the algebra H ∞ + C and compute the norm of the Hilbert matrix.

The Essential Norm We begin with the computation of the essential norm for a Hankel operator. Recall that the essential norm T e of an operator T from a Hilbert space H1 to a Hilbert space H2 is, by definition, T e = inf{T − K : K is compact}.

(5.1)

To compute the essential norm of a Hankel operator, we have to introduce the space H ∞ + C. Definition. The space H ∞ + C is the set of functions ϕ in L∞ such that ϕ admits a representation ϕ = f + g, where f ∈ H ∞ and g ∈ C(T). Theorem 5.1. The set H ∞ + C is a closed subalgebra of L∞ . To prove the theorem, we need the following elementary lemma, where as usual CA = H ∞ ∩ C(T); see Appendix 2.1. Lemma 5.2. Let ϕ ∈ C(T). Then distL∞ (ϕ, H ∞ ) = distL∞ (ϕ, CA ).

(5.2)

26


Proof. The inequality dist(ϕ, H ∞ ) ≤ distL∞ (ϕ, CA ) is trivial. Let us prove the opposite one. For f ∈ L∞ we consider its harmonic extension to the unit disk and keep the same notation for it. Put fr (ζ) = f (rζ), ζ ∈ D. Let ϕ ∈ C(T), h ∈ H ∞ . We have ϕ − h∞

≥ =

lim (ϕ − h)r ∞ ≥ lim (ϕ − hr ∞ − ϕ − ϕr ∞ )

r→1

r→1

lim ϕ − hr ∞ ≥ distL∞ (ϕ, CA ),

r→1

since ϕ − ϕr ∞ → 0 for continuous ϕ. Proof of Theorem 5.1. Equality (5.2) means exactly that the natural imbedding of C(T)/CA in L∞ /H ∞ is isometric and so C(T)/CA can be considered as a closed subspace of L∞ /H ∞ . Let ρ : L∞ → L∞ /H ∞ be the natural quotient map. It follows that H ∞ + C = ρ−1 (C(T)/CA ) is closed in L∞ . This implies that   H ∞ + C = closL∞  (5.3) z¯n H ∞  . n≥0

It is easy to see that if f and g belong to the right-hand side of (5.3), then so is f g. Hence, H ∞ + C is an algebra. Now we are going to compute the essential norm of a Hankel operator. Theorem 5.3. Let ϕ ∈ L∞ . Then Hϕ e = distL∞ (ϕ, H ∞ + C). 2 Lemma 5.4. Let K be a compact operator from H 2 to H− . Then

lim KS n = 0.

n→∞

Proof of Lemma 5.4. Since any compact operator can be approximated by finite rank operators, it is sufficient to prove the assertion for rank 2 . We have one operators K. Let Kf = (f, ξ)η, ξ ∈ H 2 , η ∈ H− KS n f = (f, S ∗n ξ)η and so KS n = S ∗n ξ2 η2 → 0. Proof of Theorem 5.3. By Corollary 3.2, Hf is compact for any trigonometric polynomial f . Therefore Hf is compact for any f in C(T). Consequently, distL∞ (ϕ, H ∞ + C) =

inf Hϕ − Hf ≥ Hϕ e .

f ∈C(T)

2 On the other hand, for any compact operator K from H 2 to H− we have

Hϕ − K

≥ (Hϕ − K)S n ≥ Hϕ S n − KS n = Hzn ϕ − KS n = distL∞ (ϕ, z¯n H ∞ ) − KS n ≥ distL∞ (ϕ, H ∞ + C) − KS n .


27

Therefore in view of Lemma 5.4, Hϕ e ≥ distL∞ (ϕ, H ∞ + C).

Compactness Criterion Now we can easily obtain the Hartman compactness criterion. Theorem 5.5. Let ϕ ∈ L∞ . The following statements are equivalent: (i) Hϕ is compact; (ii) ϕ ∈ H ∞ + C; (iii) there exists a function ψ in C(T) such that Hϕ = Hψ . Proof. Obviously, (ii)⇔(iii). Suppose that ϕ ∈ H ∞ +C. Then Hϕ e = 0 by Theorem 5.3, which means that Hϕ is compact. Thus (ii)⇒(i). Let us show that (i)⇒(ii). If Hϕ is compact, then by Theorem 5.3, distL∞ (ϕ, H ∞ + C) = 0, which together with Theorem 5.5 yields ϕ ∈ H ∞ + C. Corollary 5.6. Let ϕ ∈ L∞ . Then Hϕ e = inf{Hϕ − Hψ : Hψ is compact}.

(5.4)

In other words, to compute the essential norm of a Hankel operator we can consider on the right-hand side of (5.1) only compact Hankel operators. Corollary 5.7. Let ϕ ∈ H ∞ + C. Then for any ε > 0 there exists a function ψ in C(T) such that Hψ = Hϕ and ψ∞ ≤ Hϕ + ε. Proof. Without loss of generality we can assume that ϕ ∈ C(T). By Theorem 5.3, Hϕ = distL∞ (ϕ, H ∞ ). On the other hand, by Lemma 5.2, distL∞ (ϕ, H ∞ ) = distL∞ (ϕ, CA ). This means that for any ε > 0 there exists a function h ∈ CA such that ϕ − h ≤ Hϕ + ε. Thus ψ = ϕ − h does the job. Remark. However, it is not always possible for a compact Hankel operator to find a continuous symbol whose L∞ -norm is equal to the norm of the operator. Indeed, let α be a real-valued function in C(T) such that a ˜ ∈ C(T), where α ˜ is the harmonic conjugate of α. Put ϕ = z¯eiα˜ . Then α+iα ˜ −α ϕ = z¯e e . Clearly, eα+iα˜ ∈ H ∞ and e−α ∈ C(T). It follows from Theorem 5.1 that ϕ ∈ H ∞ + C and so Hϕ is compact. Let us show that Hϕ = 1. Put 1 h = exp (−α − i˜ α). 2 ¯ 2 = h2 . Hence, Hϕ = ϕ∞ = 1. By z h Then h ∈ H 2 and Hϕ h2 = ¯ Corollary 1.6, ϕ + f ∞ > 1 for any nonzero f in H ∞ . It is also clear that ϕ ∈ C(T). This proves the result. In §1 we have given a boundedness criterion for a Hankel operator Hϕ in terms of P− ϕ. That criterion involves the condition P− ϕ ∈ BM O. We can give a similar compactness criterion if we replace BM O by the space V M O of functions of vanishing mean oscillation (see Appendix 2.5).

28


Theorem 5.8. Let ϕ ∈ L2 . Then Hϕ is compact if and only if P− ϕ ∈ V M O. The result can be derived from Theorem 5.5 in the same way as it has been done in Theorem 1.2 if we use the following description of V M O: V M O = {ξ + η˜ : ξ, η ∈ C(T)} (see Appendix 2.5). Theorem 5.8 allows us to obtain a compactness criterion for the commutators Cϕ defined in §1 (see (1.15)). The following theorem can be deduced from Theorem 5.8 in exactly the same way as Theorem 1.10 has been deduced in §1 from Theorem 1.3. Theorem 5.9. Let ϕ ∈ L2 . The commutator Cϕ is a compact operator on L2 if and only if ϕ ∈ V M O. The Hartman theorem also allows us to obtain a compactness criterion for functions ϕ(S[ϑ] ) of the model operator S[ϑ] (see §2). Theorem 5.10. Let ϑ be an inner function and S[ϑ] the compressed shift on Kϑ . If ϕ ∈ H ∞ , then the following statements are equivalent: (i) ϕ(S[ϑ] ) is compact; ¯ ∈ H ∞ + C; (ii) ϑϕ ¯ ∈ V M O. (iii) P− ϑϕ The result follows directly from Theorems 5.5 and 5.8, and from formula (2.9). Consider now the partial case of this situation when ϑ = B is an interpolating Blaschke product (see Appendix 2.1). In this case KB has an unconditional basis that consists of eigenvectors of S[B] . Namely, let fj (z) = (1 − |ζj |2 )1/2

B , z − ζj

(5.5)

where the ζj , j ≥ 1, are the zeros of B. Then it is easy to see that S[B] fj = ζj fj and ϕ(S[B] )fj = ϕ(ζj )fj for any ϕ ∈ H ∞ . Since {fj }j≥1 is an unconditional basis, there is an invertible operator V on Kϑ such that {V fj }j≥1 is an orthogonal basis of Kϑ (see N.K. Nikol’skii [2], Lect. VI, §3). Therefore ϕ(S[B] ) is compact if and only if ϕ(ζj ) → 0 as j → ∞. This together with Theorem 5.10 implies the following assertion. Theorem 5.11. Let ϕ ∈ H ∞ and let B be an interpolating Blaschke product with zeros {ζj }j≥1 . The following statements are equivalent: (i) ϕ(S[B] ) is compact; ¯ ∈ H ∞ + C; (ii) Bϕ ¯ ∈ V M O; (iii) P− Bϕ (iv) ϕ(ζj ) → 0 as j → ∞. The above facts about the bases of eigenvectors of S[B] can be found in N.K.Nikol’skii [2], Lect. III, §7.


29

Approximation by Compact Hankel Operators In connection with (5.4) a natural question arises of whether the infimum in (5.4) is attained. This is equivalent to the existence of a best approximation from H ∞ + C for any bounded function. This question turns out to be more difficult than in the case of best approximation from H ∞ since the compactness argument does not work here. Another question arising in this connection is whether such a best approximation is unique (if it exists). The following theorem answers these questions. Theorem 5.12. Let ϕ ∈ L∞ \ (H ∞ + C). Then there exist infinitely many different functions f in H ∞ + C such that ϕ − f ∞ = distL∞ (ϕ, H ∞ + C). By Theorems 1.3 and 5.5, Theorem 5.12 is a consequence of the following theorem. Theorem 5.13. Let Γ be a noncompact Hankel operator on H 2 . Then there exist infinitely many different compact Hankel operators Υ such that Γ − Υ = Γe . We establish a more general result. First we need the following lemma. Lemma 5.14. Let H1 and H2 be Hilbert spaces and T : H1 → H2 a bounded operator. If {An }n≥1 is a sequence of operators from H1 to H2 such that An → O and A∗n → O in the strong operator topology, then for any ε > 0 there exists a positive integer N such that T + AN ≤ ε + max(T , T e + AN ). Proof of Lemma 5.14. Clearly, without loss of generality we can assume that H1 = H2 = H. Put ρn = max(T , T e + An ). Suppose that for any n there exists xn ∈ H, xn = 1, such that (T + An )xn > ε + ρn .

(5.6)

By passing to a subsequence, if needed, we may assume that xn → x ∈ H in the weak topology. Let yn = xn − x. Then (T + An )xn = T x + T yn + An yn + An x.

(5.7)

Let Kn be a sequence of finite-dimensional subspaces of H such that Kn ⊂ Kn+1 and Kn is dense in H. Denote by Pn the orthogonal projecn≥1

tion onto Kn and by Qn the orthogonal projection onto Kn⊥ . Then Qn → O in the strong operator topology. Therefore for any compact operator K on H we have Qn K → 0. Since T − K ≥ Qn (T − K) ≥ Qn T − Qn K,

30


it follows that Qn T → T e . Let δ be a positive number that we specify later. Fix a positive integer M such that QM T x < δ, QM x < δ, and QM T < T e + δ. Since T yn → 0 weakly, it follows that QM T yn < δ for large n. Since PM is compact and A∗n → O in the strong operator topology, it follows that PM An → 0 and so PM An yn < δ for large n. We have from (5.7) (T +An )xn = PM T x+QM (T yn +An yn )+An x+QM T x+PM (T yn +An yn ). Since An → O in the strong operator topology, we have An x < δ for large n. Thus for large n (T + An )xn ≤ PM T x + QM (T yn + An yn ) + 4δ.

(5.8)

Since PM T x is orthogonal to QM (T yn + An yn ), we obtain PM T x + QM (T yn + An yn )2

= PM T x2 + QM (T yn + An yn )2 ≤

T 2 x2 + (QM T + An )2 yn 2

≤ ρ2n x2 + (T e + δ + An )2 yn 2 ≤ (ρn + δ)2 (x2 + yn 2 ). It follows from the fact that yn → 0 weakly that PM yn < δ for large n. Hence, x2 + yn 2

= PM x2 + QM x2 + PM yn 2 + QM yn 2 ≤

PM x2 + QM yn 2 + 2δ 2 = PM x + QM yn 2 + 2δ 2

= x − QM x + yn − PM yn 2 + 2δ 2 2

≤ (xn + QM x + PM yn ) + 2δ 2 ≤ (1 + 2δ)2 + 2δ 2 . Thus we have

PM T x + QM (T yn + An yn )2 ≤ (ρn + δ)2 (1 + 2δ)2 + 2δ 2 .

This together with (5.8) yields 1/2 + 4δ (T + An )xn ≤ (ρn + δ) (1 + 2δ)2 + 2δ 2 for large n. Now we can choose δ so small that the right-hand side of the last inequality is less than ρn + ε, which contradicts (5.6). Corollary 5.15. Under the hypotheses of Lemma 5.14, for any ε > 0 there exists a positive integer N such that T + βAN ≤ ε + max(T , T e + βAN ) for any β ∈ [0, 1].


31

Proof. Suppose that for any n there exists βn ∈ [0, 1] such that T + βn An > ε + max(T , T e + βAn ). But βn An → O, (βn An )∗ → O in the strong operator topology, which contradicts Lemma 5.14. Theorem 5.16. Let H1 and H2 be Hilbert spaces, T : H1 → H2 a noncompact bounded operator. Let {Tn }n≥1 be a sequence of compact operators from H1 to H2 such that Tn → T and Tn∗ → T ∗ in the strong operator topology. Then there exist sequences {an }n≥1 and {bn }n≥1 of nonnegative real numbers such that an = bn = 1 n≥1

n≥1

and

where K1 =

T − K1 = T − K2 = T e , an Tn and K2 = bn Tn . Moreover, K1 = K2 .

n≥1

(5.9)

n≥1

Proof. Let us first find {an }n≥1 . Let An = T − Tn . Clearly, lim An = n→∞ lim A∗n = O in the strong operator topology.

n→∞

Let us show that there exist an increasing sequence {nk }k≥1 of positive integers and a sequence {αk }k≥1 of positive real numbers such that ∞

αk = 1,

(5.10)

k=1

m αk Ank = T e − εm ,

(5.11)

k=1

for any m ≥ 1, where εm = 3−m T e . We proceed by induction. Put n1 = 1 and choose α1 > 0 so that α1 A1 = T e − ε1 . Clearly, α1 < 1. Suppose that we have chosen m αk < 1 and (5.11) holds. We n1 , · · · , nm and α1 , · · · , αm such that k=1

can choose nm+1 > nm such that

R + βAnm+1 ≤ εm+1 + max R, Re + βAnm+1 , (5.12)

for any β in [0,1], where R =

m k=1

αk Ank . This is clearly possible by Corol-

lary 5.15. Consider µ(α) = R + αAnm ,

α ≥ 0.

32


m Clearly, µ(α) → ∞ as α → ∞ and µ(0) = αk Ank = T e − εm k=1 by the inductive hypothesis. Therefore there exists αm+1 > 0 such that µ(αm+1 ) = T e − εm+1 . m+1 αk < 1. We have Let us show that k=1

T e −εm+1

m+1 m+1 m+1 = µ(αm+1 ) = αk T − αk Tnk ≥ αk T e , k=1

so

m+1

k=1

k=1

αk < 1.

k=1

Put β = αm+1 in (5.12). Suppose that the first term under the maximum on the right-hand side of (5.12) is greater than the second. Then m+1 m αk Ank ≤ εm+1 + αk Ank . k=1

k=1

Therefore by (5.11) T e − εm+1 ≤ εm+1 + T e − εm , which implies εm ≤ 2εm+1 , which contradicts the definition of εm . Consequently, (5.12) with β = αm+1 turns into m+1 m αk Ank ≤ εm+1 + αk Ank + αm+1 Anm+1 k=1 k=1 m e αk T e + αm+1 Anm+1 . (5.13) = εm+1 + k=1

Let m → ∞, it follows from (5.11) and (5.13) that ∞ ∞ T e = αk Ank ≤ αk T e , k=1

k=1

which implies ∞

αk ≥ 1,

k=1

and since we have already proved that

m

αk < 1, the sequences {αk } and

k=1

{nk } satisfy (5.11) and (5.13). We can define the sequence {an }n≥1 in the following way: ank = αk and an = 0 if n does not coincide with any nk . We have K1 =

∞ n=1

an Tn =

∞ k=1

αk Tnk = T −

∞ k=1

αk Ank ,


33

∞ T − K1 = αk Ank = T e . k=1

It remains to construct {bn }n≥1 . Let T˜ = T − K1 , Tñ = Tn − K1 . Then ˜ Tn → T˜ and Tñ∗ → T˜∗ in the strong operator topology. Let O be a convex neighborhood of T˜ in the strong operator topology whose closure does not contain O. Since Tñ → T˜ strongly, we can assume that Tñ ∈ O for all n. As we have just proved, there exists a sequence {bn }n≥1 of nonnegative ∞ bn = 1 and numbers such that n=1

˜ = T˜e = T e , T˜ − K ∞ ∞ ∞ ˜ ˜ = bn Tn ˜ bn Tn = bn Tn − K1 . Thus K2 = K1 + K where K = n=1

n=1

n=1

˜ is an infinite convex combination of the Tñ , we have satisfies (5.9). Since K ˜ ˜ K ⊂ O. Thus K = O and K1 = K2 . Now we are in a position to deduce Theorem 5.13 from Theorem 5.16. Proof of Theorem 5.13. Let Γ = Hϕ . Let ϕn = ϕ ∗ Kn be the Césaro means of the partial sums of the Fourier series of ϕ (see Appendix 2.1). Then Hϕn is compact. Let us show that Hϕn → Hϕ in the strong operator topology. Let f ∈ H 2 . We have (Hϕ − Hϕn )f 22

= P− (ϕ − ϕn )f 2 ≤ (ϕ − ϕn )f 22

= |ϕ(ζ) − ϕn (ζ)|2 |f (ζ)|2 dm(ζ) → 0, T

since ϕn ∞ ≤ ϕ∞ and ϕn (ζ) → ϕ(ζ) a.e. on T (see Appendix 2.1). 2 Using the fact that Hϕ∗ n g = P+ ϕg, ¯ g ∈ H− , one can prove in the same way ∗ ∗ that Hϕn → Hϕ strongly. It follows from Theorem 5.16 that there are distinct compact Hankel operators Υ0 and Υ1 (infinite convex combinations of Hϕn ) such that Γ − Υ0 = Γ − Υ1 = Γe . It remains to observe that if Υ1 and Υ2 are best compact Hankel approximations to Γ, then so is Υs = (1 − s)Υ0 + sΥ1 with s ∈ [0, 1].

The Local Distance Let f and g be functions in L∞ , λ ∈ T. The local distance distλ (f, g) between f and g at λ is by definition distλ (f, g) = lim (f − g) |{ζ ∈ T : |ζ − λ| < ε} ∞ . ε→0

If K is a subset of L∞ , the local distance from f to K is defined by distλ (f, K) = inf{distλ (f, h) : h ∈ K}.

34


It is easy to see that the function λ → distλ (f, K) is upper semicontinuous and so it attains a maximum on T. Theorem 5.17. Let ϕ ∈ L∞ . Then Hϕ e = max distλ (ϕ, H ∞ ). λ∈T

Proof. Clearly, by Theorem 5.3, we have to show that distL∞ (ϕ, H ∞ + C) = max distλ (ϕ, H ∞ ). λ∈T

It is evident that distλ (ϕ, H ∞ ) ≤ distL∞ (ϕ, H ∞ + C) for any λ ∈ T. Let us show that distL∞ (ϕ, H ∞ + C) ≤ max distλ (ϕ, H ∞ ). λ∈T

(5.14)

Let M be the right-hand side of (5.14) and let ε > 0. For each λ we can find hλ in H ∞ such that distλ (ϕ, hλ ) < M + ε. Since the function ζ → distζ (ϕ, hλ ) is upper semi-continuous, it follows that for any λ there is an open interval Iλ such that distζ (ϕ, hλ ) < M + 2ε for ζ ∈ Iλ . We can choose a finite cover Iλ1 , · · · , Iλn of T. Let wj ∈ C(T), 1 ≤ j ≤ n, be a partition of unity subordinate to the cover {Iλj }, i.e., n supp wj ⊂ Iλj , wj ≥ 0, and wj = 1. Put h =

n j=1

j=1

wj hλj . Since H ∞ + C is an algebra, it follows that

h ∈ H ∞ + C. Let λ ∈ T. We have   n n wj ϕ, wj hλj  distλ (ϕ, h) = distλ  j=1

≤ =

n j=1 n

j=1

distλ (wj ϕ, wj hλj ) =

n

distλ (wj (λ)ϕ, wj (λ)hλj )

j=1 n

wj (λ) distλ (ϕ, hλj ) ≤

j=1

wj (λ)(M + 2ε),

j=1

since distλ (f, hλj ) ≤ M +2ε for λ ∈ Iλj and wj (λ) = 0 if λ ∈ Iλj . Therefore distλ (ϕ, h) ≤ M + 2ε, which proves that distL∞ (ϕ, H ∞ + C) ≤ M . Let us now compute Hϕ e for piecewise continuous functions ϕ. Denote by P C the class of piecewise continuous functions on T. A function ϕ on T is said to belong to P C if for any ζ ∈ T the limits def

ϕ(ζ + ) = lim ϕ(ζeit ), t→0+

ϕ(ζ − ) = lim ϕ(ζeit ) def

t→0−


35

exist. Since we identify functions that differ on a set of zero measure, we can assume that for ϕ ∈ P C ϕ(ζ + ) = ϕ(ζ),

ζ ∈ T.

Denote by κζ (ϕ) the jump of ϕ at ζ: κζ (ϕ) = ϕ(ζ) − ϕ(ζ − ). It is easy to see that for any ε > 0 the set {ζ ∈ T : |κζ (ϕ)| > ε} is finite. Theorem 5.18. Let ϕ ∈ P C. Then Hϕ e =

1 max |κζ (ϕ)|. 2 ζ∈T

(5.15)

Proof. In view of Theorem 5.17 it is sufficient to show that 1 |κζ (ϕ)| = distζ (ϕ, H ∞ ). 2 It is easy to see that distζ (ϕ, H ∞ ) ≤ distζ (ϕ, c) =

1 |κζ (ϕ)|, 2

where c is the constant function equal to 12 (ϕ(ζ) + ϕ(ζ − )). Let us show that 12 |κζ (ϕ)| ≤ distζ (ϕ, H ∞ ). By multiplying ϕ by a constant and adding to it a constant function if needed, we can assume without loss of generality that ϕ(ζ) = 1 and ϕ(ζ − ) = −1. We have to show that distζ (ϕ, H ∞ ) ≥ 1. We can also assume that ζ = 1. Suppose that there exists an f in H ∞ such that dist1 (ϕ, f ) < 1. It follows that there exists δ > 0 such that Re f (eit ) > δ,

0 ≤ t ≤ ε;

Re f (eit ) < −δ,

−ε ≤ t ≤ 0, (5.16)

for some ε > 0. Let h(ζ) = Re f (ζ), ζ ∈ D. Then ˜ f (ζ) = h(ζ) + ih(ζ) + i Im f (0), ˜ is the harmonic conjugate to h (see Appendix 2.1). Then h ˜ is where h bounded on D. We have ˜ iϑ ) = (h ∗ Qr )(eiϑ ), h(re where Qr is the conjugate Poisson kernel (see Appendix 2.1). Let us show ˜ that |h(r)| → ∞ as r → 1. It is easy to see that

h(eit )Qr (t)dt ≤ const . {|t|≥ε}

36


On the other hand, it follows easily from (5.16) that ε

ε sin t 1 it it h(e )Q (e )dt ≥ const δ dt ≥ const · log , r 2 1 − 2r cos t + r 1 − r −ε 0 ˜ is where the constant depends on δ and ε. This contradicts the fact that h ∞ bounded. Thus dist1 (ϕ, H ) = 1 and the proof is completed. Remark. Clearly the right-hand side of (5.15) is equal to distL∞ (ϕ, C(T)). So for ϕ ∈ P C we have distL∞ (ϕ, C(T)) = distL∞ (ϕ, H ∞ + C) = Hϕ e . Now we are in a position to compute the norm and the essential norm of ˆ = 1/n, the Hilbert matrix. Let ψ(eit ) = i(t − π), t ∈ [0, 2π). Then ψ(−n) n = 0, and Hψ has Hilbert matrix. Corollary 5.19. Let ψ be as above. Then Hψ e = Hψ = π. Proof. We have established in §1 that Hϕ ≤ π. Since ϕ ∈ P C, we have Hψ e = π by Theorem 5.18. The result follows now from the obvious inequality Hψ e ≤ Hψ .

A Factorization Theorem We conclude this section with the following elegant property of H ∞ + C. Theorem 5.20. Let f ∈ L∞ . Then there exist a Blaschke product B ¯ and a function ϕ in H ∞ + C such that f = Bϕ. Proof. By Theorem 2.4, there exist sequences {ϑn }n≥1 of inner functions and {gn }n≥1 of H ∞ -functions such that lim f − ϑ¯n gn ∞ = 0. n→∞

By Frostman’s theorem (see Appendix 2.1), each inner function is a uniform limit of Blaschke products. Therefore there exists a sequence of Blaschke products {Bn }n≥1 such that ¯n gn ∞ = 0. lim f − B n→∞

(1)

(2)

(1)

Let us decompose each Bn as follows: Bn = Bn Bn , where Bn is a finite Blaschke product and the product Bn(2) B= n≥1

converges in D. Then

¯ (1) hn ∞ → 0, Bf − B n

where hn = gn

m =n

(2) Bm ∈ H ∞.

6. Hankel Operators and Reproducing Kernels

37

¯n(1) hn ∈ H ∞ + C. Hence, Bf ∈ H ∞ + C by Theorem 5.1. Clearly, B

6. Hankel Operators and Reproducing Kernels In this section we show that to verify the boundedness (or compactness) of a Hankel operator, we can consider the action of the operator on the so-called reproducing kernels of H 2 . Let ζ ∈ D. Put 1 Kζ (z) = ¯ . 1 − ζz Then Kζ ∈ H 2 and for any f ∈ H 2 (f, Kζ ) = f (ζ). Because of this property the functions Kζ are called the reproducing kernel functions (or simply kernel functions) of H 2 . It is easy to see that Kζ 2 = (1 − |ζ|2 )−1/2 . The functions kζ defined by kζ (z) =

(1 − |ζ|2 )1/2 , ¯ 1 − ζz

ζ ∈ D,

are called the normalized kernel functions. Obviously, for any bounded operator A on H 2 the norms Akζ are uniformly bounded. It is easy to see that kζ → O weakly as |ζ| → 1. Therefore Akζ → 0 as |ζ| → 1 for any compact operator A on H 2 . We show in this section that for Hankel operators the converse is also true. We shall work here with the Garsia norm on BM O (see Appendix 2.5). Let Pζ be the Poisson kernel, Pζ (z) =

1 − |ζ|2 2 ¯ 2 = |kζ (z)| . |1 − ζz|

Given ϕ ∈ L2 , we denote by ϕ# its harmonic extension to the unit disc:

ϕ# (ζ) = ϕ(τ )Pζ (τ )dm(τ ), ζ ∈ D. T

The Garsia norm of ϕ is defined by 1/2

2 ϕG = sup |ϕ(τ ) − ϕ# (ζ)| Pζ (τ )dm(τ ) . ζ∈D

T

Then · G is a norm on BM O modulo the constants which is equivalent to the initial norm (see Appendix 2.5).

38


Theorem 6.1. Let ϕ ∈ L2 . The Hankel operator Hϕ is bounded on H if and only if 2

sup Hϕ kζ 2 < ∞.

(6.1)

ζ∈D

Proof. Let ψ = P− ϕ. We have P− ϕ z¯P− ϕ ψ Hϕ Kζ = P− ¯ = P+ 1 − ζ z¯ = P+ z − ζ . 1 − ζz Clearly, P+

ψ ψ − ψ(ζ) 1 ψ − ψ(ζ) = P+ + ψ(ζ)P+ = . z−ζ z−ζ z−ζ z−ζ

Therefore

Hϕ kζ 2

1/2 |ψ(τ ) − ψ(ζ)|2 (1 − |ζ|2 ) dm(τ ) |τ − ζ|2 T

1/2 2 = |ψ(τ ) − ψ(ζ)| Pζ (τ )dm(τ ) . =

T

It follows that sup Hϕ kζ 2 = ψG , ζ∈D

which proves that (6.1) is equivalent to the fact that P− ϕ ∈ BM O, which in turn means that Hϕ is bounded (see Theorem 1.3). To prove the compactness criterion in terms of kernel functions, we need the following characterization of V M O. Let ϕ ∈ L2 . Then ϕ ∈ V M O if and only if

lim |ϕ(τ ) − ϕ(ζ)|2 Pζ (τ )dm(τ ) = 0 |ζ|→1

T

(see Appendix 2.5). Theorem 6.2. Let ϕ ∈ L2 . The Hankel operator Hϕ is compact on H if and only if 2

lim Hϕ kζ = 0.

|ζ|→1

(6.2)

Proof. The same argument as in the proof of Theorem 6.1 shows that (6.2) is equivalent to the condition

|ψ(τ ) − ψ# (ζ)|2 Pζ (τ )dm(τ ) = 0, lim |ζ|→1

T

which means that P− ϕ ∈ V M O, which in turn is equivalent to the compactness of Hϕ (Theorem 5.8).

7. Hankel Operators and Moment Sequences

39

7. Hankel Operators and Moment Sequences In this section we see that Hankel matrices appear naturally in connection with power moment problems. We are going to solve the Hamburger moment problem, and characterize the bounded and compact positive semidefinite Hankel matrices. We also characterize arbitrary bounded and compact Hankel matrices in terms of moment sequences of Carleson (vanishing Carleson) measures. Let {αj }j≥0 be a sequence of complex numbers. The classical Hamburger moment problem is to find a positive measure µ on R satisfying

|t|j dµ(t) < ∞, j ≥ 0, (7.1) R

and such that its moments coincide with the αj , i.e.,

tj dµ(t), j ≥ 0. αj =

(7.2)

R

In this case µ is called a solution of the power moment problem with data {αj }j≥0 . The Hamburger theorem (Theorem 7.1 below) gives a solvability criterion in terms of positive semi-definiteness of the corresponding Hankel matrix {αj+k }j,k≥0 . (The matrix {αj+k }j,k≥0 is called positive semi-definite if αj+k xj x ¯k ≥ 0 (7.3) j,k≥0

for any finitely supported sequence {xj }j≥0 of complex numbers.) Theorem 7.1. Let {αj }j≥0 be a sequence of complex numbers. The Hamburger moment problem with data {αj }j≥0 is solvable if and only if the Hankel matrix {αj+k }j,k≥0 is positive semi-definite. It is very easy to prove the “only if” part. To prove the “if” part we need some facts from the extension theory of symmetric operators. We are going to deal with (possibly unbounded) densely defined operators on Hilbert space. We review briefly some basic notions and facts and refer the reader to Birman and Solomjak [1] for a detailed presentation of the theory. Let H be a Hilbert space. We consider linear transformations (operators) T whose domain D(T ) is a dense linear manifold in H and whose range is a subset of H. The operator T is called closed if its graph {x⊕T x : x ∈ D(T )} is closed in H ⊕ H and it is called closable if the closure of its graph in H ⊕ H is a graph of a linear operator. If T1 and T2 are densely defined linear operators, we say that T2 is an extension of T1 if D(T1 ) ⊂ D(T2 ) and T2 x = T1 x for x ∈ D1 . If T is a densely defined closable linear operator, we define its adjoint operator T ∗ as follows. The domain D(T ∗ ) consists of those vectors y in H for which the linear functional x → (T x, y), x ∈ D(T ), extends to a bounded linear functional. Since T is closable, D(T ∗ ) is dense in H. For y ∈ D(T ∗ ) we define T ∗ y by the equality (T x, y) = (x, T ∗ y).

40


A densely defined linear operator A is called symmetric if A∗ is an extension of A or, in other words, x, y ∈ D(A).

(Ax, y) = (x, Ay),

The operator A is called self-adjoint if A∗ = A, i.e., A is symmetric and D(A∗ ) = D(A). With a symmetric operator operator A we can associate the deficiency indices n− (A) = dim(Range(A + iI))⊥ ,

n+ (A) = dim(Range(A − iI))⊥ .

The symmetric operator A has a self-adjoint extension if and only if n− (A) = n+ (A). If A is a cyclic self-adjoint operator (i.e., there exists a vector x0 such that An x0 ∈ D(A) for all n ∈ Z+ and span{An x0 : n ∈ Z+ } = H), then it follows from the spectral theorem that there exists a positive Borel measure µ on R such that A is unitary equivalent to the operator Aµ of multiplication by the independent variable on L2 (µ), i.e.,

D(Aµ ) = f ∈ L2 (µ) : |f (t)|dµ(t) < ∞ , (Aµ f )(t) = tf (t). R

Moreover, if x0 is such a vector, the unitary equivalence can be chosen to send x0 to the constant function 1 identically equal to 1. We are ready now to prove Theorem 7.1. Proof of Theorem 7.1. It is very easy to prove that if there exists a measure µ satisfying (7.1) and (7.2), then the matrix {αj+k }j,k≥0 satisfies (7.3). Indeed,

αj+k xj x ¯k = xj x ¯k tj+k dµ(t) j,k≥0

j,k≥0

R

2

j xj t dµ(t) ≥ 0. = R j≥0

Let us prove now the sufficiency of (7.3) for the solvability of the moment problem (7.2). Let H0 be the set of polynomials on R. For p, q ∈ H0 define the following pairing: p, q =

m n

pj q¯k αj+k ,

(7.4)

j=0 k=0

where p(t) = p0 + p1 (t) + · · · + pm tm and q(t) = q0 + q1 t + · · · + qn tn . We consider separately two cases. Case 1. The form (7.4) is nondegenerate, i.e., p, p = 0 for any nonzero p ∈ H0 . Let H be the completion of H0 with respect to the inner product (7.4). Then H is a Hilbert space.


41

Consider the operator A0 with domain H0 defined by (A0 p)(t) = tp(t),

p ∈ H0 .

It is easy to see that A is symmetric. We claim that A has a self-adjoint extension. Let J0 be the operator on H0 defined by J0 p = p¯. Clearly, J0 is conjugatelinear and isometric. So it extends by continuity to a conjugate-linear isometry J of H onto itself. It is also easy to see that J Range(A0 + iI) = Range(A0 − iI). Therefore n− (A0 ) = n+ (A0 ). Let A be a self-adjoint extension of A0 . Clearly, span{An 1 : n ∈ Z+ } = H. By the spectral theorem there exist a positive Borel measure on R and a unitary operator U from H to L2 (µ) such that U 1 = 1 and U AU −1 is multiplication by the independent variable on L2 (µ). Obviously, U p = p for any p ∈ H0 . It is easy to see that µ is a solution of our moment problem. Case 2. There exists a nonzero polynomial p for which p, p = 0. We can choose a polynomial p minimal degree with this property. Let n = deg p. We now identify polynomials q and r if q − r, q − r = 0. We can consider now the finite-dimensional Hilbert space H of equivalence classes endowed with inner product ·, · = 0. Let us show that if q is a polynomial such that q, q = 0, then Aq, q = 0. Let m = deg q. Suppose that q(t) = q0 + p1 t + · · · + qm tm and q, q = 0. Let Γ be the Hankel matrix {αj+k }j,k≥0 . Consider the matrix ΓN = {αj+k }0≤j,k≤m+N and the vector   q0  q1     ..   .    m+1+N  x= .  qm  ∈ C  O     .   ..  O Since q, q = 0, it is easy to see that (ΓN x, x) = 0, where (·, ·) is the standard inner product in Cm+1+N . Since the matrix ΓN is positive semidefinite, it follows that ΓN x = O for any N ∈ Z+ . Now it is easy to see that   O  q0     q1     ..   = O, . Γ    qm     O    .. .

42


and so Aq, Aq = 0. Thus we can consider the operator A as a self-adjoint operator on the n-dimensional Hilbert space H. Now the same reasoning as in Case 1 allows us to construct a measure µ (in this case it consists of at most n atoms), which solves our moment problem. Remark. It is easy to see that the deficiency indices n− (A0 ) and n+ (A0 ) are either both equal to 0 or both equal to 1. In the case n− (A0 ) = n+ (A0 ) = 0 there is only one solution of the moment problem, since there is only one self-adjoint extension of A0 . If n− (A0 ) = n+ (A0 ) = 1, there are infinitely many solutions of the moment problem, since there are infinitely many self-adjoint extensions of A0 . Moreover in the last case one can obtain other solutions of the moment problem by considering so-called minimal self-adjoint extensions of A0 in Hilbert spaces larger than H. Let us proceed now to positive semidefinite Hankel matrices Γα = {αj+k }j,k≥0 that are bounded on 2 . We are going to describe such matrices in terms of solutions of the corresponding moment problems. We are going to use the Carleson imbedding theorem, which says that a measure µ on D is a Carleson measure if and only if sup I

where

|µ|(RI ) < ∞, |I|

RI =

ζ ∈I ζ∈D: |ζ|

and

(7.5)

1 − |ζ| ≤ |I| , def

the supremum is taken over all subarcs I of T, and |I| = m(I) (see Appendix 2.1). Note that for µ supported on (−1, 1) condition (7.5) means that |µ|(1 − t, 1) ≤ const · t,

|µ|(−1, −1 + t) ≤ const · t.

Theorem 7.2. Let Γ = {αj+k }j,k≥0 be a nonnegative Hankel matrix. The following statements are equivalent: (i) Γ determines a bounded operator on 2 ; (ii) there exists a positive measure µ on (−1, 1) such that

1 tn dµ(t) αn = −1

and µ is a Carleson measure (as a measure on D); (iii) |αn | ≤ const(1 + n)−1 . Proof. By Hamburger’s theorem there exists a measure µ on R that satisfies (7.1) and (7.2). Suppose that Γ is bounded on 2 . Then αn → 0 as n → ∞ and it is easy to see that this implies that µ is supported on [−1, 1]. It is also easy to see that µ({1}) = µ({−1}) = 0.


43

Let I be the imbedding operator from H 2 to L2 (µ) which is defined on the set of polynomials by If = f (−1, 1). (7.6) ˆ Let f = f (j)z j be a polynomial. It is easy to see that j≥0

(If, If )L2 (µ) = Γα {fˆ(j)}j≥0 , {fˆ(j)}j≥0 . 2

(7.7)

It follows that Γ is bounded on 2 if and only if I is a bounded operator from H 2 to L2 (µ). Thus the equivalence (i)⇔(ii) follows from the Carleson imbedding theorem. Let us show that (ii)⇒(iii). We have µ((t, 1)) ≤ c(1−t) for some constant c. Then

1

1 tn dµ(t) = µ((0, 1)) − ntn−1 µ((0, t))dt 0

0

nt

=

1 n−1

µ((t, 1))dt ≤ c

0

1

ntn−1 (1 − t)dt ≤ 0

c . n+1

0 One can prove in the same way that −1 tn dµ(t) ≤ const(1 + n)−1 . The fact that (iii) implies (i) follows easily from the boundedness of the Hilbert matrix. Indeed, let ξ = {ξj }j≥0 and η = {ηk }k≥0 be finitely supported sequences. We have αj+k ξj η¯k |(Γα ξ, η)| = j,k≥0 |ξj | · |ηk | ≤ πξ2 η2 , ≤ |j + k + 1| j,k≥0

since the norm of the Hilbert matrix is equal to π (see Corollary 5.19). Let us characterize the compact nonnegative Hankel matrices in terms of vanishing Carleson measures (see Appendix 2.1). Theorem 7.3. Let Γ = {αj+k }j,k≥0 be a nonnegative Hankel matrix. The following statements are equivalent: (i) Γ determines a compact operator on 2 ; (ii) there exists a positive measure µ on (−1, 1) such that

1 tn dµ(t) αn = −1

and µ is a vanishing Carleson measure (as a measure in D); (iii) lim αn (1 + n) = 0. n→∞

Proof. We identify H 2 with 2 in the following natural way: f → {fˆ(n)}n≥0 ,

f ∈ H 2.

44


It follows from (7.7) that the operator I defined by (7.6) satisfies Γ[µ] = I ∗ I, and so Γ[µ] is compact if and only if I is, which proves that (i)⇔(ii). Suppose that µ is a vanishing Carleson measure. Then for any ε > 0, µ admits a representation µ = µ1 + µ2 , where µ1 is supported on (−r, r) for some r < 1 and µ2 satisfies µ2 ((1 − r, 1)) ≤ εt,

µ2 ((−1, −1 + r)) ≤ εt.

The proof of the implication (ii)⇒(iii) in Theorem 7.1 gives the estimate 1 2ε n . t dµ2 (t)dt ≤ n+1 −1 It is easy to see that

r −r

tn dµ1 (t) ≤ const rn .

This proves that (ii)⇒(iii). (n) (n) To show that (iii)⇒(i) we can represent Γ[µ] as Γ1 + Γ2 , where (1) Γ1 = αj+k , Γ2 = Γ − Γ1 , j,k≥0

(1)

(1)

= αj for j ≤ n, αj

(n)

has finite rank and (n) the proof of the application (iii)⇒(i) in Theorem 7.1 shows that Γ2 → 0, which completes the proof. Note that (ii) in Theorem 7.3 means that

and αj

lim

t→0+

= 0 for j > n. Then Γ1

µ((1 − t, 1)) µ((−1, −1 + t)) = lim = 0. t→0+ t t

Let now µ be a finite complex measure on D. As in the case of measures on [−1, 1] we can define the Hankel matrix Γ[µ] as follows:

j+k Γ[µ] = ζ dµ(ζ) . D

j,k≥0

As before we can identify 2 with H 2 and consider Γ[µ] as the matrix of an operator on H 2 . It is easy to see that for any polynomials f and g in H 2

¯ (Γ[µ]f, g) = f (ζ)g(ζ)dµ(ζ). (7.8) D

Theorem 7.4. If µ is a complex Carleson measure, then Γ[µ] is bounded on H 2 . If Γ = {αj+k }j,k≥0 is a bounded Hankel matrix, then there exists a complex Carleson measure µ such that

αj = ζ j dµ(ζ). D


45

Proof. If µ is a Carleson measure, then it follows from (7.8) that for any polynomials f and g in H 2

1/2

1/2 ¯ 2 d|µ|(ζ) |(Γ[µ]f, g)| = |f (ζ)|2 d|µ|(ζ) |g(ζ)| D

D

≤ const f H 2 gH 2

by the definition of Carleson measures. So Γ[µ] is bounded. To prove the converse, we need the fact that any function in BM O can be represented as the Poisson balayage of a Carleson measure (see Appendix 2.5). Suppose that {αj+k }j,k≥0 is a bounded matrix on 2 . By Theorem 1.2, def ϕ = αj z j ∈ BM OA. j≥0

Then there exists a Carleson measure µ whose Poisson balayage coincides with ϕ, that is,

Pλ (ζ)dµ(λ) for almost all ζ ∈ T, ϕ(ζ) = D

where Pλ is the Poisson kernel. It is easy to see that

αj = ϕ(j) ˆ = ζ j dµ(ζ), j ≥ 0. D

In a similar way one can prove the following compactness criterion. Theorem 7.5. If µ is a complex vanishing Carleson measure, then Γ[µ] is compact on H 2 . If Γ = {αj+k }j,k≥0 is a compact Hankel matrix, then there exists a complex vanishing Carleson measure µ such that

ζ j dµ(ζ). αj = D

To conclude this section, we evaluate the Hankel matrices Γ[m1−1 ] and Γ[m10 ] , where m1−1 is Lebesgue measure on [−1, 1] and m10 is Lebesgue measure on [0, 1]. It is easy to see that 2

1 j+1 , j is even, j x dx = 0, j is odd. −1 Hence,



Γ[m1−1 ]

1  0  1   3 = 2 0  1   5 .. .

0 1 3

0 1 5

0 .. .

1 3

0 1 5

0 1 7

.. .

0 1 5

0

1 5

0 1 7

1 7

0

0 .. .

.. .

1 9

··· ··· ··· ··· ··· .. .

     .   

(7.9)

46


On the other hand,

1

xj dx = 0

and so



Γ[m10 ]

1

1/2

1 , j+1

1/3

  1/2 1/3 1/4   =  1/3 1/4  ..  1/4 .  .. .. . .

1/4 ..

.

 ··· ..  .        

(7.10)

is the Hilbert matrix (see §1). Corollary 7.6. The matrices on the right-hand sides of (7.9) and (7.10) are bounded positive semi-definite matrices. Proof. This is an immediate consequence of Theorem (7.2).

8. Hankel Operators as Integral Operators on the Semi-Axis In this section we consider integral operators on the space L2 (R+ ) of functions on the semi-axis R+ with kernels depending only on the sum of the variables. We show that such operators are unitarily equivalent to Hankel operators on 2 introduced in §1 and this will allow us to reduce the study of such operators to the study of Hankel operators on 2 . For a function k ∈ L1 (R+ ) consider the operator Γ k on L2 (R+ ),

∞ k(s + t)f (s)ds. (8.1) (Γ k f )(t) = 0

Below we shall see that Γ k is a bounded operator on L2 (R+ ). The definition of Γ k is similar to the definition of Hankel operators as operators with matrices whose entries depend only on the sum of the coordinates (see §1). We are going to define the operator Γ k in a much more general case where k does not necessarily belong to L1 (R+ ). We shall see that Γ k can be bounded for certain distributions. We shall find in this section boundedness, compactness, and finite rank criteria for Γ k , and we establish the unitary equivalence between operators Γ k and Hankel operators on 2 . We call operators Γ k Hankel operators on L2 (R+ ). It will be slightly more convenient for us to deal with the following integral operators. Let q ∈ L1 (R− ). We define the integral operator Gq from

8. Hankel Operators as Integral Operators on the Semi-Axis

L2 (R+ ) to L2 (R− ) by

∞

(Gq f )(t) =

q(t − s)f (s)ds,

t < 0.

47

(8.2)

0

Clearly, (Γ k f )(t) = (Gq f )(−t),

k(t) = q(−t),

t ∈ R+ .

(8.3)

We can assume for convenience that q ∈ L (R) and q(t) = 0 for t > 0, since the right-hand side in (8.2) does not depend on the values of q on R+ . Denote by P+ and P− the operators of multiplication by the characteristic function of R+ and R− , respectively. We can also identify the spaces L2 (R+ ) and L2 (R− ) with subspaces of L2 (R). Then P− and P+ are the orthogonal projections of L2 (R) onto L2 (R− ) and L2 (R+ ) respectively. It is easy to see that Gq can also be defined by 1

Gq f = P− (q ∗ f ),

f ∈ L2 (R+ ),

where the the convolution ϕ ∗ ψ of functions ϕ and ψ on R is defined by

(ϕ ∗ ψ)(t) = ϕ(s)ψ(t − s) ds. R

It is easy to see now that Gq is bounded for q ∈ L1 (R). We have Gq f 2 = P− (q ∗ f )2 ≤ q ∗ f 2 ≤ q1 f 2 ; the last inequality is well known and can be proved elementarily.

Digression. Distributions To define the operators Gq and Γ k in a more general context we need the notion of distributions (see Schwartz [1] for more detailed information). Let D be the space of infinitely differentiable functions on R with compact support. The space D is endowed with the natural topology of inductive limit. Without entering into detail let us mention the following. A sequence {fj }j≥0 converges in D to f if and only if there is a compact set K in R (m) such that supp fj ⊂ K for any j and all derivatives fj converge uniformly on K to the derivatives f (m) , m ≥ 0. A linear operator T from D to a locally convex space X is continuous if and only if for any sequence {fj }j≥0 converging to O in D, the sequence {T fj }j≥0 converges in X to O. For convenience we define the space of distributions D as the space of continuous antilinear functionals on D, i.e., def

¯ < g, f >, < g, λf > = g(λf ) = λ

f ∈ D,

g ∈ D ,

λ ∈ C;

< g, f1 + f2 >=< g, f1 > + < g, f2 >, f1 , f2 ∈ D, g ∈ D . If g is a locally summable function on R, it can be interpreted as a distribution that acts on D as follows:

g(t)f (t)dt, f ∈ D. (8.4) < g, f >= R

48


For g ∈ D , we define its support supp g as the minimal closed set F for which < g, f >= 0 for any f ∈ D with support in R \ F . If Ω is an open set in R, we can consider in a similar way the space D(Ω) of infinitely smooth functions with compact support in Ω and we can define the space of distributions D (Ω). We need the important subclass of D , the space of tempered distributions. Let S be the space of infinitely differentiable functions f on R such that for any m, n ∈ Z+ sup f (m) (t) (1 + |t|)n < ∞. (8.5) t∈R

The topology on S is determined by the system of seminorms on the lefthand side of (8.5). As in the case of the space D we define the space S of tempered distributions as the space of continuous antilinear functionals on S. Note that if g is a measurable function on R such that

|g(t)| dt < ∞ (1 + |t|)n R for some n ∈ Z+ , then g can be considered as a tempered distribution that acts on S as in (8.4). Recall that the Fourier transform Ff of a function f in L1 (R) is defined by

(Ff )(s) = e−2πits f (t)dt. R By Plancherel’s theorem F L1 (R) L2 (R) extends to a unitary operator on L2 (R) and the inverse Fourier transform F −1 = F ∗ is defined on L1 (R) L2 (R) by

(F ∗ f )(t) =

e2πits f (t)ds. R

It is easy to see that F ∗ S = S, which allows us to define the Fourier transform on the space of tempered distributions S by < Fg, f > = < g, F ∗ f >, def

f ∈ S, g ∈ S .

Recall one more elementary identity: F ∗ (f ∗ g) = (F ∗ f )(F ∗ g),

f ∈ L1 (R), g ∈ L2 (R).

(8.6)

Boundedness, Compactness, and Finite Rank Criteria Now we are in a position to define the operators Gq and Γ k in a more general situation. If q ∈ L1 (R), it is an elementary exercise to see that

def ¯ q(s)(f¯◦ ∗ g)(s)ds, f ∈ L2 (R+ ), g ∈ L2 (R− ), (Gq f, g) = (q, f◦ ∗ g) = R (8.7) def

where f◦ (s) = f (−s).


49

Formula (8.7) suggests the following definition of Gq for some distributions q in D (−∞, 0). Put def (Gq f, g) = < q, f¯◦ ∗ g >,

f ∈ D(0, ∞), g ∈ D(−∞, 0) (8.8)

(we identify in a natural way D(0, ∞) and D(−∞, 0) with the subspaces of functions in D with support in (0, ∞) and (−∞, 0), respectively). We say that Gq is bounded on L2 (R+ ) if |(Gq f, g)| ≤ const · f 2 g2 . In this case for f ∈ D(0, ∞) we define Gq f as the function in L2 (R− ) such that (Gq f, g) =< q, f¯◦ ∗ g > for g ∈ D(−∞, 0). We can define now the operator Γ k by (8.3) in the case when the operator Gq is bounded. The identity k(t) = q(−t), t ∈ R, means that < k, f >=< q, f◦ >,

f ∈ D.

The following result characterizes the bounded operators Gq . Theorem 8.1. Let q ∈ D (−∞, 0). The operator Gq defined by (8.8) is bounded on L2 (R+ ) if and only if there exists a function κ ∈ L∞ (R) such that Fκ (−∞, 0) = q. Moreover, Gq = inf κ∞ : κ ∈ L∞ (R), Fκ (−∞, 0) = q . (8.9) (By Fκ (−∞, 0) = q we mean that < Fκ, g >=< q, g > for any g ∈ D(−∞, 0)). To prove Theorem 8.1 we need the notion of Hankel operator on the Hardy class H 2 (C+ ) of functions in the upper half-plane (see Appendix 2.1). Recall that the Paley–Wiener theorem allows us to identify H 2 (C+ ) with the following subspace of L2 (R): H 2 (C+ ) = f ∈ L2 (R) : Ff (−∞, 0) = O . Similarly, the Hardy class H 2 (C− ) of functions analytic in the lower halfplane can be identified with the subspace of functions in L2 (R) whose Fourier transforms vanish on R+ . Clearly, under this identification L2 (R) = H 2 (C+ ) ⊕ H 2 (C− ). Recall that the unitary operator U on L2 (T) defined by 1 (f ◦ ω −1 )(t) (Uf )(t) = √ , t+i π

t ∈ R,

(8.10)

is a unitary operator from L2 (T) onto L2 (R) and U(H 2 ) = H 2 (C+ ) (see Appendix 2.1). Here ω is the conformal map ω of D onto C+ defined by ω(ζ) = i

1+ζ , 1−ζ

ζ ∈ D,

ω −1 (w) =

w−i , w+i

w ∈ C+ .

and

50


ˆ \ clos D onto C− , where C ˆ def Note that ω also maps conformally C = C ∪ ∞. To prove Theorem 8.1 we need the following lemma. Lemma 8.2. Let q be a distribution on (−∞, 0) such that the operator 2 Gq defined by (8.8) is bounded on L2 (R+ ). Then the operator Γ : H 2 → H− defined by Γ = U ∗ F ∗ Gq FU

(8.11)

is a Hankel operator on H 2 . Proof. In view of the commutation relation (1.13) it suffices to show that (P− zΓξ, η) = (Γzξ, η)

(8.12)

2 for ξ in a dense subset of H 2 and η in a dense subset of H− . We have

(Γzξ, η) = (Gq FUzξ, FUη). Put def

F = Uξ,

def

G = Uη,

We have by (8.10) (Uzξ)(t) = F (t)

def

f = FF,

def

g = FG.

1 t−i . = F (t) 1 − 2i t+i t+i

Put α(t) = 2i/(t + i), t ∈ R. Clearly, α ∈ H 2 (C+ ). Let h = Fα ∈ L2 (R+ ). It follows from (8.6) that F ∗ (f ∗ h) = F α. Therefore (Γzξ, η) = (Gq f, g) + (Gq (h ∗ f ), g). ∗¯

¯ and so we can obtain in a similar way Clearly, F h◦ = α ¯ ◦ ∗ g). (P− zΓξ, η) = (Γξ, z¯η) = (Gq FUξ, FU z¯η) = (Gq f, g) + (Gq f, h Therefore to prove (8.12) it is sufficient to show that ¯ ◦ ∗ g) (Gq (h ∗ f ), g) = (Gq f, h

(8.13)

for any functions f ∈ D(0, ∞), g ∈ D(−∞, 0) and any h ∈ L2 (R+ ). In view of (8.8) equality (8.13) is equivalent to the following one: ¯ ◦ ∗ g). (h ∗ f )◦ ∗ g = f¯◦ ∗ (h The latter follows from the following identity, which can be verified elementarily: (h ∗ f )◦ = h◦ ∗ f◦ . Before we proceed to the proof of Theorem 8.1 we define Hankel operators on the Hardy class H 2 (C+ ). Denote by P + and P − the orthogonal projections from L2 (R+ ) onto H 2 (C+ ) and H 2 (C− ). It follows from the Paley–Wiener theorem that P + = F ∗ P+ F,

P − = F ∗ P− F.


51

Let ψ ∈ L∞ (R). The Hankel operator Hψ : H 2 (C+ ) → H 2 (C− ) is defined by Hψ F = P − ψF, F ∈ H 2 (C+ ). Clearly, Hψ is a bounded operator on H 2 (C+ ). def

Lemma 8.3. Let ψ ∈ L∞ (R) and ϕ = ψ ◦ ω ∈ L∞ (T). Then Hϕ = U ∗ Hψ U. 2 . We have Proof. Let f ∈ H 2 , g ∈ H−

(Hϕ f, g) = (ϕf, g); (U ∗ Hψ Uf, g) = (Hψ Uf, Ug) = (ψUf, Ug) = (Uϕf, Ug) = (ϕf, g) by (8.10). Corollary 8.4. Let ψ ∈ L∞ (R). Then Hψ = distL∞ (R) (ψ, H ∞ (C+ )). The next result gives a compactness criterion for Hankel operators Hψ . ˆ is by definition the space of continuous functions on R The space C(R) which have equal limits at ∞ and −∞. Corollary 8.5. Let ψ ∈ L∞ (R). Hψ is compact if and only if ˆ ψ ∈ H ∞ (C+ ) + C(R). Clearly, Corollary 8.5 is an immediate consequence of Lemma 8.3. Now we are in a position to prove Theorem 8.1. Proof of Theorem 8.1. Suppose that Gq is bounded. Then by Lemma 8.2, the operator Γ defined by (8.11) is a bounded Hankel operator on H 2 . So Γ = Hϕ , where ϕ ∈ L∞ (T) and ϕ∞ = Gq . It follows from (8.11) and from Lemma 8.3 that UHϕ U ∗ = Hψ = F ∗ Gq F, where ψ = ϕ ◦ ω −1 . Hence, Gq = FHψ F ∗ . Put r = Fψ ∈ S . Let us show that FHψ F ∗ = Gr . We have def

(FHψ F ∗ f, g)

(ψF ∗ f, F ∗ g) =< ψ, F ∗ f · F ∗ g > = < r, F(F ∗ f · F ∗ g) >=< r, f¯◦ ∗ g >, =

f ∈ D(0, ∞),

(8.14)

g ∈ D(−∞, 0).

Therefore Gq = Gr . Since the set of functions f¯◦ ∗ g,

f ∈ D(0, ∞), g ∈ D(−∞, 0),

is obviously dense in D(−∞, 0), it follows that q = r (−∞, 0) = Fψ (−∞, 0) and Gq = ψ∞ , which proves that Gq is greater than or equal to the right-hand side of (8.9).

52


If q = Fψ (−∞, 0) and ψ ∈ L∞ (R), then the same reasoning proves that Gq = FHψ F ∗ ,

(8.15)

which implies that Gq ≤ ψ∞ = κ∞ . In the case of functions on T we can represent any bounded function ϕ as ϕ = P+ ϕ + P− ϕ and the Hankel operator Hϕ depends only on P− ϕ. However, in the case of functions on R the projections P + and P − are not defined on L∞ (R). Instead we define on L∞ (R) the operators P + and P − by P + ψ = (P+ (ψ ◦ ω)) ◦ ω −1 , def

P − ψ = (P− (ψ ◦ ω)) ◦ ω −1 , def

ψ ∈ L∞ (R).

We shall also use the notation ψ + = P + ψ, ψ − = P − ψ. Clearly, ψ = ψ + + ψ − . Note that P + and P − are unbounded on L∞ (R). Lemma 8.6. Let ψ ∈ L∞ (R). Then supp Fψ + ⊂ [0, ∞) and supp Fψ − ⊂ (−∞, 0]. Proof. Let g ∈ S, supp g ⊂ R− . Let us show that < Fψ + , g >= 0, which would mean that supp Fψ + ⊂ R+ . The proof of the fact that supp Fψ − ⊂ R− is similar. We have

+ + ∗ < Fψ , g >=< ψ , F g >= ψ + (t)(F ∗ g)(t)dt = 0 R

(F ∗ g)

∈ H (C+ ) (see Appendix 2.1). since ψ We can obtain now an analog of Theorem 1.2 in terms of the space BM O(R) of functions on R (see Appendix 2.5). Recall that ψ ∈ BM O if and only if ψ can be represented as ψ = ξ + P + η, for some ξ and η in L∞ (R). As in the case of functions on the unit circle the following result follows easily from Theorem 8.1. Corollary 8.7. Let q ∈ D (−∞, 0). Then Gq is bounded on L2 (R+ ) if and only if q = r(−∞, 0) for some r ∈ S such that supp r ⊂ R− and F ∗ r ∈ BM O(R). Let us now obtain the boundedness criterion for Γ k . Theorem 8.8. Let k ∈ D (0, ∞). The following statements are equivalent: (i) Γ k is bounded on L2 (R+ ); (ii) there exists a function κ in L∞ (R) such that Fκ (0, ∞) = k; (iii) there exists r ∈ S such that k = r(0, ∞), supp r ⊂ R+ , and F ∗ r ∈ BM O(R). If Γ k is bounded, then Γ k = inf{κ∞ : κ ∈ L∞ (R), Fκ R+ = k}. +

1

Proof. It remains to prove that (ii) implies (iii); everything else follows immediately from of Γ k . Let κ ∈ Theorem 8.1 and the definition (0, ∞) = k. Define r by r = FP + κ. By Lemma 8.6, such that Fκ L∞ (R) k = r(0, ∞) and supp r ⊂ R+ . Clearly, F ∗ r = P + κ ∈ BM O(R).


53

The operator Γ k defined by (8.1) is a continual analog of a Hankel matrix. The next result shows that a bounded operator Γ k is unitarily equivalent to a Hankel operator on 2 (i.e., an operator with Hankel matrix). Theorem 8.9. If Γ k is a bounded operator, then the operator U ∗ F ∗ Γ k FU on H 2 has Hankel matrix in the basis {z n }n≥0 . Proof. Let q(t) = k(−t), t ∈ R. Then Gq is bounded and so by (8.15) Gq = FHψ F ∗ for some bounded function ψ on R. Put ϕ = ψ ◦ ω ∈ L∞ (T). def

Let ξn = Uz n , n ∈ Z+ . We have (U ∗ F ∗ Γ k FUz n , z m ) = (Γ k Fξn , Fξm ) = (Gq Fξn , (Fξm )◦ ) = (Gq Fξn , F ∗ ξm ) = (F ∗ Gq Fξn , (F ∗ )2 ξm ) = (Hψ ξn , (ξm )◦ ) . It is easy to see that (ξm )◦ ∈ H 2 (C− ) and so (Hψ ξn , (ξm )◦ ) = (ψξn , (ξm )◦ ). Therefore

1 (t − i)n+m ψ(t) dt (U ∗ F ∗ Γ k FUz n , z m ) = (ψξn , (ξm )◦ ) = − π R (t + i)n+m+2 n+m+1

dt t−i 1 ψ(t) = − π R t+i |t + i|2

= − ϕ(ζ)ζ n+m+1 dm(ζ) = −ϕ(−n ˆ − m − 1), T

which proves the result. Remark. Theorem 8.9 says that Γ k has Hankel matrix in the orthonormal basis {FUz n }n≥0 of the space L2 (R+ ). Using the residue theorem, one can show that √ (FUz n )(t) = −2 πiLn (4πt)e−2πt , t > 0, where the functions Ln are the so-called Lageurre polynomials, 1 s d n Ln (s) = n! e ds (e−s sn ); see Szegö [1]. To proceed to compactness criteria, we need the space V M O(R) (see Appendix 2.5 for the definition). Note that there are several possible definitions of V M O(R) and ours differs from the one in Garnett [1]. The space ˆ V M O(R) consists of functions of the form ξ + P + η with ξ, η ∈ C(R). It is easy to obtain from Corollary 8.5 and formula (8.15) a compactness criterion for Gq . We state instead the following compactness criterion for Γ k. Theorem 8.10. Let k ∈ D (0, ∞). The following statements are equivalent: (i) Γ k is a compact operator on L2 (R+ ); ˆ such that Fκ (0, ∞) = k; (ii) there exists a function κ in C(R) (iii) there exists r ∈ S such that supp r ⊂ R+ , k = r(0, ∞), and F ∗ r ∈ V M O(R).

54


We have started this section with the operators Γ k for k ∈ L1 (R+ ) and observed that such operators are bounded. In fact it follows immediately from Theorem 8.10 that they are compact. Corollary 8.11. Let k ∈ L1 (R+ ). Then Γ k is compact on L2 (R+ ). ˆ by the Riemann–Lebesgue lemma, the result Proof. Since F ∗ k ∈ C(R) follows directly from statement (ii) of Theorem 8.10. Let us now characterize the finite rank operators Γ k . It is easy to see from Lemma 8.3 that Hψ has finite rank if and only if P − ψ is a rational function and rank Hψ = deg P − ψ. This implies the following result. Lemma 8.12. Let Γ k , k ∈ D (0, ∞), be a bounded operator and let r be a tempered distribution such that supp r ⊂ R+ and k = r(0, ∞). Then Γ k has finite rank if and only if F ∗ k is a rational function. In this case rank Γ k = deg F ∗ k. It is easy to see that if supp r ⊂ R+ , then F ∗ r is rational if and only if r admits a representation r(t) =

j −1 n m

cj,l tl eλj t ,

t > 0,

j=1 l=0

where the λj , 1 ≤ j ≤ n, are complex numbers satisfying Re λj < 0. This implies the following finite rank criterion. Theorem 8.13. Let Γ k , k ∈ D (0, ∞), be a bounded operator and let r be a tempered distribution such that supp r ⊂ R+ and k = r|(0, ∞). Then Γ k has finite rank if and only if r(t) =

j −1 n m

cj,l tl eλj t ,

t > 0,

(8.16)

j=1 l=0

where Re λj < 0. If the operator Γ k has finite rank and cj,mj −1 = 0 in (8.16), then n mj . rank Γ k = j=1

The Carleman Operator def

Consider the operator Γ = Γ k on L2 (R+ ), where k(t) = 1/t, t > 0. It is called the Carleman operator. Here we interpret k as an element of D (0, ∞). Let us show that Γ is bounded. By Theorem 8.8 it suffices to find a function χ ∈ L∞ (R) such that Fχ|(0, ∞) = k. Define χ as follows:

πi, t > 0, χ(t) = −πi, t < 0.


Let us prove that

∞

< Fχ, f >= 0

f (t) − f (−t) dt, t

which would imply that Fχ|(0, ∞) = right-hand side belongs to L1 ). Put   πi, −πi, χn (t) =  0,

f ∈ D,

55

(8.17)

k (note that the integrand on the 0 < t < n, −n < t < 0, otherwise,

where n > 0. Clearly, χn → χ in the weak topology σ(D, D ). Thus it remains to prove that < Fχn , f > converges to the right-hand side of (8.17) for any f ∈ D. We have n

0 1 − cos 2πnt −2πits −2πits (Fχn )(t) = πi e ds − e ds = . t 0 −n It follows that

< Fχn , f > =

f (t)(1 − cos 2πnt) dt t

R ∞

=

0

(f (t) − f (−t))(1 − cos 2πnt) dt, t

which tends to the right-hand side of (8.17) by the Riemann–Lebesgue lemma. By Theorem 8.9, Γ has Hankel matrix in the basis {Fξn }n≥0 . Let us find its entries. Put ψ = −χ. Then it follows from (8.14) that Gq = FHψ F ∗ , where q(t) = k(−t), t < 0. By Theorem 8.9, (Γ Fξn , Fξm ) = −ϕ(−n ˆ − m − 1), where ϕ = ψ ◦ ω. Clearly,

ϕ(ζ) = It is easy to see that

πi, Im ζ > 0, −πi, Im ζ < 0.

2 j,

j odd, 0, j even. Thus Γ has the following matrix in the basis {Fξn }n≥0 :   1 0 13 0 15 · · ·  0 1 0 1 0 ···   1 3 1 5 1     3 01 5 01 7 · · ·  2 0 .  1 5 01 7 01 · · ·     5 0 7 0 9 ···  .. .. .. .. .. . . . . . . . . ϕ(j) ˆ =

Clearly, this is the matrix on the right-hand side of (7.9).

(8.18)

56


Theorem 8.14. Let Γ be the Carleman operator. Then Γ = Γ e = π. Proof. Clearly, Γ = Hϕ and Γ e = Hϕ e , where ϕ is defined by (8.18). The result now follows from Theorem 5.18.

Concluding Remarks Hankel matrices appeared in the paper by Hankel [1] who studied determinants of finite Hankel matrices. They were used on numerous occasions in the study of moment problems, orthogonal polynomials (see Akhiezer [1], Szeg¨ o [1]). The modern period of the study of Hankel operators begins with the paper Nehari [1], where the bounded Hankel matrices were characterized (see Theorem 1.1). The Fefferman characterization of BM O found in Fefferman [1] (see also Fefferman and Stein [1]) shed a new light on Hankel operators. Theorem 1.4 is taken from Adamyan, Arov, and Krein [1]. Note that earlier several authors studied the problem of best approximations of continuous functions on T by bounded analytic functions in D. S.Ya. Khavinson [1] proved that for a continuous function on T the best approximation is unique. We also mention here the paper Rogosinski and Shapiro [1] in which the authors rediscovered the results of Khavinson and constructed a function with different best approximations. The example of a function with different best approximations given in §1 is taken from J.L. Walsh [1], Ch. X, §3 (see also Sarason [1]). Commutators Cϕ were studied by many authors, see e.g., Coifman, Rochberg, and Weiss [1], where the authors considered commutators in the case of several variables. We also mention here the paper Treil and Volberg [1], where a fixed point approach to the Nehari problem was given. Theorem 2.1 is due to Sarason [1]. It is a special case of the Sz.-Nagy– Foias commutant lifting theorem (see Appendix 1.5). The approach given in §2 is due to N.K. Nikol’skii (see N.K. Nikol’skii [1]). Formula (2.9) is also due to N.K. Nikol’skii [1]. Theorem 2.5 is a special case of the Douglas– Rudin theorem (Douglas and Rudin [1]); it can be found in Douglas [2]. The description of the partially isometric Hankel operators (Theorem 2.6) is taken from Power [2]. Theorem 2.7 was obtained by Fuhrmann [1], [3]. Theorem 2.8 is due to Clark [3]. The characterization of the finite rank Hankel matrices was obtained in Kronecker [1]. The second proof is taken from Sarason [5]; it was suggested by Axler. The Nevanlinna–Pick interpolation problem was studied independently by Nevanlinna [1] and Pick [1]. The Carathéodory–Fejér problem was studied in Carathéodory and Fejér [1].

Concluding Remarks

57

The Hartman theorem (Theorem 5.5) was obtained in Hartman [1]. Theorem 5.1 was obtained in Sarason [1]. It is amusing that the fact that H ∞ +C is closed in L∞ follows immediately from the Hartman theorem and the fact that the set of compact operators is closed in the space of bounded linear operators. However, in this book we use the Sarason theorem to prove the Hartman theorem. The essential norm of a Hankel operator (see Theorem 5.4) was computed in Adamyan, Arov, and Krein [1]. The equivalence of (i) and (ii) in Theorem 5.10 was established in Sarason [1]. The class V M O was introduced in Sarason [4]. Theorem 5.11 is essentially due to Clark [2]. The results on approximation by compact Hankel operators were obtained in Axler, Berg, Jewell, and Shields [1]. These results give a complete solution to a problem posed in Adamyan, Arov, and Krein [5]. Another solution to the Adamyan–Arov–Krein problem (based on the notion of M -ideal) was given by Luecking [1]. Theorem 5.17 was found in Douglas and Sarason [1], while Theorem 5.18 was obtained in Bonsall and Gillespie [1]. The factorization theorem for L∞ functions (Theorem 5.20) is due to Axler [1]. Theorems 6.1 and 6.2 were published in Bonsall [1], [2], though they were known before to some experts. The Hamburger theorem (Theorem 7.1) was established in Hamburger [1]. The approach based on self-adjoint extensions of symmetric operators can be found in Akhiezer [1]. Our approach is taken from Sarason [6]. Theorems 7.1 and 7.2 are obtained in Widom [3]. Theorems 7.4 and 7.5 can be found in Power [2]. The equivalence of the operators Γ k and the Hankel operators is well known; see Power [2] for some related results and see Devinatz [2] for similar results on Wiener–Hopf operators. The Carleman operator was studied in Carleman [1]. In connection with the material of §8 we would like to offer the following exercise. Exercise. Let k(x) = e−x /x, x > 0. Prove that Γ k is bounded on L2 (R+ ) and is unitarily equivalent to the Hankel operator with Hilbert matrix. Let us mention here the following boundedness criterion for Hankel operators (see Bonsall [4], and Holland and D. Walsh [1]). For ζ ∈ T consider the following polynomials: un,ζ = n−1/2 def

n−1

ζ j zj ,

j=0

Then the following are equivalent: (i) Hϕ is bounded; (ii) sup Hϕ un,ζ < ∞; n>0, ζ∈T

(iii)

sup n>0, ζ∈T

Hϕ vn,ζ < ∞.

vn,ζ = n−1/2 def

2n−1 j=n

ζ j zj .

58


In Bonsall [1] it was shown that for a sequence {aj }j≥0 of nonnegative numbers the Hankel matrix {αj+k }j,k≥0 is bounded if and only if n−1 2 ∞ 1 sup αj+k < ∞. n>0 n j=0 k=0

This is also equivalent to the condition n−1 2 ∞ αjn+k < ∞. sup n>0

j=1

k=0

The fact that thej last inequality for nonnegative αj is equivalent to the condition αj z ∈ BM O is an unpublished theorem of C. Fefferman; its j≥0

proof is contained in Bonsall [1]. We would like to mention here a series of papers by Arocena, Cotlar, and Sadosky (see Arocena [1], Arocena and Cotlar [1], Cotlar and Sadosky [1], and Arocena, Cotlar, and Sadosky [1]). In those papers the authors develop an interesting theory of generalized Toeplitz kernels which generalizes the Nehari theory for Hankel matrices. Let K : Z × Z :→ C be a Toepliz kernel, i.e., K(i + 1, j + 1) = K(i, j), i, j ∈ Z, which is positive semidefinite, i.e., K(i, j)xi x ¯j ≥ 0 i,j∈Z

for any finitely supported sequence {xj }j∈Z of complex numbers. Then by the F. Riesz–Herglotz theorem (see Riesz and Sz.-Nagy [1], §53) there exists a positive Borel measure µ on T such that µ ˆ(n) = K(j + n, j), j ∈ Z. Suppose now that K is a positive semidefinite generalized Toeplitz kernel, i.e., K(i + 1, j + 1) = K(i, j), i, j ∈ Z, i = −1, j = −1. Then there are sequences {kj+ }j∈Z , {kj− }j∈Z , {kj+− }j>0 , and {kj−+ }j 0. Here Z− = Z \ Z+ . − µ µ−+ If µ = is a positive matrix Borel measure on T, then µ−+ µ+ we can put def

k + (j) = µ+ (j), k +− (j) = µ+− (j),

k − (j) = µ− (j),

j ∈ Z+ ,

k −+ (−j) = µ−+ (−j),

j < 0,

Concluding Remarks

59

and obtain a positive semidefinite generalized Toeplitz kernel. We say that this generalized Toeplitz kernel is determined by the matrix µ. Obviously, the measures µ+ and µ− are uniquely determined by the generalized Toeplitz kernel while only the Fourier coefficients µ ˆ−+ (n) with n < 0 are −+ is determined up to a function in the Hardy uniquely determined. Thus µ class H 1 . The main result obtained in the papers by Arocena, Cotlar, and Sadosky mentioned above says that each positive semidefinite generalized Toeplitz kernel is determined by a positive matrix measure. This theorem generalizes the Nehari theorem. Indeed, let {αi+j }i≥0,j≥1 be a bounded Hankel matrix and let c = {αi+j }i≥0,j≥1 . Put

c, j = 0, k + (j) = k − (j) = and k −+ (j) = k +− (−j) = αj , j > 0. 0, j = 0, It is easy to see that the corresponding generalized Toeplitz kernel is positive semi-definite, and so it is determined by a positive matrix measure µ− µ−+ . Since µ ≥ O, it follows that µ−+ is absolutely conµ= µ−+ µ+ tinuous with respect to Lebesgue measure, dµ = ϕdm, ϕ ∈ L∞ , ϕ∞ ≤ c, and ϕ(−j) ˆ = αj , j > 0. The authors also found other interesting applications; in particular, they obtained two weight inequalities for the operator of harmonic conjugation. Let us also mention the Axler–Chang–Sarason–Volberg theorem that characterizes the compact operators of the form Hψ∗ Hϕ , ϕ, ψ ∈ L∞ : the operator Hψ∗ Hϕ is compact if and only if H ∞ [ϕ] ∩ H ∞ [ψ] ⊂ H ∞ + C. Here H ∞ [ϕ] is the smallest closed subalgebra of L∞ that contains ϕ. The part “if” was obtained in Axler, Chang, and Sarason [1], while the part “only if” was obtained in Volberg [1]. We conclude this chapter with the following list of books and survey articles that discuss Hankel operators: Ando [1], Khrushchëv and Peller [2], N.K. Nikol’skii [2], [3], [4], Partington [1], Peetre [2], Peller [23], Peller and Khrushchëv [1], Power [2], Sarason [5], and Zhu [1].

2 Vectorial Hankel Operators

In this chapter we study Hankel operators on spaces of vector functions. We prove in §2 a generalization of the Nehari theorem which describes the bounded block Hankel matrices of the form {Ωj+k }j,k≥0 , where the Ωj are bounded linear operators from a Hilbert space H to another Hilbert space K. The proof is based on a more general result on completing matrix contractions. This result is obtained in §1. Namely, we obtain in §1 a necessary and sufficient condition on Hilbert space operators A, B, and Cfor the ex A B istence a Hilbert space operator Z such that the block matrix C Z is a contraction (i.e., has norm at most 1). Moreover, we describe in §1 all solutions of this completion problem. Note that the results of §2 will be used in Chapter 5 to parametrize all solutions of the Nehari problem. In §3 we give an alternative approach to the problem of boundedness of vectorial Hankel operators. This approach is based on the Sz.-Nagy–Foias commutant lifting theorem. Section 4 is devoted to the description of the compact vectorial Hankel operators. We obtain an analog of the Hartman theorem. In §5 we introduce the important notion of a Blaschke–Potapov product, we give a description of the finite rank vectorial Hankel operators, and we obtain an explicit formula for the rank of such operators. Finally, in §6 we discuss relationships between vectorial Hankel operators and imbedding theorems.

62

Chapter 2. Vectorial Hankel Operators

1. Completing Matrix Contractions This section is devoted to the following problem. Let H, K be Hilbert spaces, A a bounded linear operator on H, B a bounded linear operator from K to H, and C a bounded linear operator from H to K. The problem is to find out under which conditions there exists a bounded linear operator Z on K such that the operator A B QZ = (1.1) C Z on H ⊕ K is a contraction, that is, QZ ≤ 1. It is easy to see that if the problem is solvable, then the operators A A B and (1.2) C from H to H ⊕ K and from H ⊕ K to H, respectively, are contractions. It turns out that the converse is also true. Theorem 1.1. Let H, K be Hilbert spaces, A : H → H, B : K → H, and C : H → K bounded linear operators. There is an operator Z : K → K for which the operator QZ defined by (1.1) is a contraction on H ⊕ K if and only if A A B ≤ 1. (1.3) C ≤ 1 and Another problem we shall consider here is to describe all operators Z on K for which QZ is a contraction. Clearly, the second problem is meaningful only under the condition (1.3). To state the description we need some preliminaries. Lemma 1.2. Let H, H1 , and H2 be Hilbert spaces, and let T : H1 → H and R : H2 → H be bounded linear operators. Then T T ∗ ≤ RR∗ if and only if there exists a contraction Q : H1 → H2 such that T = RQ. Proof. Suppose that T = RQ and Q ≤ 1. We have (T T ∗ x, x)

=

(RQQ∗ R∗ x, x) = (Q∗ R∗ x, Q∗ R∗ x)

= Q∗ R∗ x2 ≤ R∗ x2 = (RR∗ x, x). Conversely, assume that T T ∗ ≤ RR∗ . We define the operator L on Range R∗ as follows: LR∗ x = T ∗ x, x ∈ H2 . The inequality T T ∗ ≤ RR∗ implies that L is well-defined on Range R∗ and L ≤ 1 on Range R∗ . We can extend L by continuity to clos Range R∗ and put L Ker R = L (Range R∗ )⊥ = O. Set Q = L∗ . Clearly T = RQ. For a contraction A : H1 → H2 the defect operator DA is defined on H1 by DA = (I − A∗ A)1/2 . It is also convenient besides DA to consider

1. Completing Matrix Contractions

63

˜ such that D∗ DA = I − A∗ A, where H ˜ is other operators DA : H1 → H A a Hilbert space. In this case DA = V DA for some isometry V defined on clos Range DA . ˜ be Hilbert spaces, A : H → H, B : K → H Lemma 1.3. Let H, K, H ˜ be an operator such linear operators such that A ≤ 1. Let DA∗ : H → H ∗ ∗ ∗ D = I − A A . Then that DA ∗ A A B ≤1 (1.4) ∗ ˜ if and only if B = DA ∗ K for a contraction K : K → H. Proof. It is easy to see that (1.4) is equivalent to the fact that A∗ A B ≤ IH , B∗ ∗ which means that AA∗ + BB ∗ ≤ I or, which is the same, BB ∗ ≤ DA ∗ DA∗ . ∗ By Lemma 1.2, this is equivalent to the fact that B = DA K for some ∗ ˜ contraction K : K → H.

Remark. The conclusion of the lemma is valid if DA∗ = DA∗ = (I − AA∗ )1/2 . It is easy to see that we can choose a contraction K : K → H such that Range K ⊂ clos Range(I − AA∗ ) and B = DA∗ K. Clearly, such a contraction K is unique. ˜ be Hilbert spaces and let A : H → H, Lemma 1.4. Let H, K, H ˜ be an C : H → K be linear operators such that A ≤ 1. Let DA : H → H ∗ ∗ operator such that DA DA = I − A A. Then A (1.5) C ≤1 ˜ → K. if and only if C = LDA for some contraction L : H Proof. The result follows from Lemma 1.3, since (1.5) is equivalent to the inequality ∗ A C ∗ ≤ 1. Remark. As in Lemma 1.3 we can take DA = DA = (I − A∗ A)1/2 . Clearly, one can find a contraction L : H → K such that ⊥ L (Range(I − A∗ A)) = O and C = LDA . As in Lemma 1.3 it is easy to see that such a contraction L is unique. Now we are in a position to state the description of those operators Z : K → K for which the operator QZ defined by (1.1) is a contraction. As we have already observed, the operators in (1.2) are contractions. Therefore (see the Remarks after Lemmas 1.3 and 1.4) there exist unique contractions K : K → H and L : H → K such that Range K ⊂ clos Range(I − AA∗ ),

B = DA∗ K,

(1.6)

64


⊥ L (Range(I − A∗ A)) = O,

C = LDA .

(1.7)

Theorem 1.5. Let H, K be Hilbert spaces, A : H → H, B : K → H, and C : H → K bounded linear operators satisfying (1.3). Let K : K → H and L : H → K be the operators satisfying (1.6) and (1.7). If Z : K → K is a bounded linear operator, then the operator QZ defined by (1.1) is a contraction if and only if Z admits a representation Z = −LA∗ K + DL∗ M DK ,

(1.8)

where M is a contraction on K. Note that we may always assume that M (Range DK )⊥ = O and Range M ⊂ clos Range DL∗ .

(1.9)

If these two conditions are satisfied, then Z determines M uniquely, and so the contractions M satisfying (1.9) parametrize the solutions Z. It is easy to see that Theorem 1.1 follows from Theorem 1.5. Indeed, we can always take M = O. To prove Theorem 1.5, we need one more lemma. Lemma 1.6. Let A, B be as above and let K : K → H be an operator satisfying (1.6). Then the operator DA −A∗ K D(AB) = (1.10) O DK satisfies ∗ D(AB) = IH⊕K − D(AB)

A∗ B∗

A

B

.

Proof. We have ∗ ∗ DA −A∗ K A DA −A∗ K A B + ∗ O DK O DK B ∗ ∗ DA O DA −A K A A B = + ∗ ∗ B −K A DK O DK ∗ 2 ∗ −DA A K A A A∗ B DA + = 2 −K ∗ ADA B∗A B∗B K ∗ AA∗ K + DK −DA A∗ K IH − A∗ A = 2 K ∗ K + DK − K ∗ DA∗ DA∗ K −K ∗ ADA ∗ A A A∗ B + . B∗A B∗B Let us show that DA A∗ = A∗ DA∗ . We have (I −A∗ A)A∗ = A∗ (I −AA∗ ). It follows that ϕ(I − A∗ A)A∗ = A∗ ϕ(I − AA∗ ) for any polynomial ϕ, so the same equality holds for any continuous function ϕ. If we take ϕ(t) = t1/2 , t ≥ 0, we obtain DA A∗ = A∗ DA∗ . Similarly, DA∗ A = ADA .

1. Completing Matrix Contractions

Consequently,

∗ D(AB) D(AB)

=

IH − A∗ A −B ∗ A

−A∗ B I − B∗B

+

+

A∗ B∗

A∗ A B∗A

A

A∗ B B∗B

65

B

=

IH O

O IH

.

Proof of Theorem 1.5. Suppose that QZ ≤ 1. By Lemma 1.2 C Z = X Y D(AB) , where X Y is a contraction from H ⊕ K to K and D(AB) is defined by (1.10). Then C = XDA . Let P be the orthogonal projection from H ˜ = XP . Clearly, C = XD ˜ A and by the onto clos Range(I − A∗ A). Put X ˜ Remark after Lemma 1.4, X = L. It is easy to see that C Z = L Y D(AB) . (1.11) Clearly,

L Y

=

X

Y

P O

O IH

,

which proves that L Y is a contraction. Then by Lemma 1.3, the operator Y admits a representation Y = DL∗ M for a contraction M on K. Formula (1.8) follows now immediately from (1.11). Suppose that Z satisfies (1.8), where M is a contraction on K. Then it is easy to see that    A DA∗ O I O A B I O O  DA −A∗ O   O K  . = C Z O L DL∗ O O M O DK (1.12) The result follows from the fact that all factors on the right-hand side of (1.12) are contractions, which is a consequence of the following lemma. Lemma 1.7. Let T be a contraction on Hilbert space. Then the operator

T DT

DT ∗ −T ∗

is unitary. Proof. We have

=

T DT

DT ∗ −T ∗

T ∗ T + DT2 DT ∗ T − T D T

∗

T DT

DT ∗ −T ∗

T ∗ DT ∗ − DT T ∗ DT2 ∗ + T T ∗

.

66


It has been shown in the proof of Lemma 1.6 that T ∗ DT ∗ = DT T ∗ and T DT = DT ∗ T . Thus ∗ T DT ∗ I O T DT ∗ = . DT −T ∗ DT −T ∗ O I Similarly,

T DT

DT ∗ −T ∗

T DT

DT ∗ −T ∗

∗

=

I O O I

.

The following uniqueness criterion is an immediate consequence of Theorem 1.5. Corollary 1.8. The following statements are equivalent: (i) there is only one operator Z for which QZ ≤ 1; (ii) either DL∗ = O or DK = O; (iii) either K is an isometry or L∗ is an isometry. Remark. The same results are valid if A is an operator from H1 to H2 , B is an operator from K1 to H2 , C is an operator from H1 to K2 , and Z is an operator from K1 to K2 , where H1 , H2 , K1 , K2 are Hilbert spaces. The proof given above works in this more general situation.

2. Bounded Block Hankel Matrices Let H and K be separable Hilbert spaces and Ω = {Ωj }j≥0 a sequence of bounded linear operators from H to K. Consider the block matrix ΓΩ = {Ωj+k }j,k≥0 .

(2.1)

Such matrices are called block Hankel matrices. In the case H = K = C the matrix ΓΩ can be considered as a Hankel matrix with scalar entries. The main aim of this section is to obtain an analog of the Nehari theorem for block Hankel matrices, that is to describe the block Hankel matrices that determine bounded operators from 2 (H) to 2 (K). There are several different approaches to this problem. Here we use the one based on the problem of completing matrix contractions that was considered in §1. In §3 we present another approach based on the Sz.-Nagy–Foias commutant lifting theorem. As in the scalar case we consider a realization of Hankel operators as operators on the Hardy class H 2 (H) of functions with values in a Hilbert space H. We study elementary properties of symbols with minimal norm and maximizing vectors of vectorial Hankel operators. Denote by B(H, K) the space of bounded linear operators from H to K and by L∞ (B(H, K)) the space of bounded weakly measurable B(H, K)-valued functions. In the case H = K we shall use the notation

2. Bounded Block Hankel Matrices

67

B(H) = B(H, H). If Φ ∈ L∞ (B(H, K)), we define the Fourier coefficients ˆ Φ(n), n ∈ Z, of Φ as usual:

ˆ Φ(n) = ζ¯n Φ(ζ)dm(ζ). T

ˆ Clearly, Φ(n) ∈ B(H, K), n ∈ Z. Theorem 2.1. Let {Ωj }j≥0 be a sequence of bounded linear operators from H to K. The block Hankel matrix ΓΩ determines a bounded linear operator from 2 (H) to 2 (K) if and only if there exists a function Φ in L∞ (B(H, K)) such that ˆ Ωn = Φ(n), In this case

n ≥ 0.

(2.2)

ˆ = Ωn , n ≥ 0 . ΓΩ = inf ΦL∞ (B(H,K)) : Φ(n)

As in the scalar case we can consider vectorial Hankel operators as operators on the Hardy class H 2 of vector-valued functions. If H is a separable Hilbert space, the Hardy class H 2 (H) is defined as follows: H 2 (H) = {F ∈ L2 (H) : Fˆ (n) = 0, n < 0}, where L2 (H) is the space of weakly measurable H-valued functions F for which

2 F L2 (H) = F (ζ)2H dm(ζ) < ∞. T

2 (H) = L2 (H) H 2 (H). Let K be another separable Hilbert space Put H− and let Ψ be a function in L2s (B(H, K)) (see Appendix 2.3), i.e.,

Ψ(ζ)x2K dm(ζ) < ∞ for any x ∈ H. (2.3) T

Recall that for functions Ψ satisfying (2.3) the Fourier coefficients ˆ Ψ(n) ∈ B(H, K) are defined by

ˆ Ψ(n)x = ζ¯n Ψ(ζ)x dm(ζ), n ∈ Z, x ∈ H. T

We can also define for such functions Ψ the Hankel operator 2 HΨ : H 2 (H) → H− (K) on the set of polynomials in H 2 (H) by HΨ F = P− ΨF,

F ∈ H 2 (H),

(2.4)

where as in the scalar case we denote by P+ and P− the orthogonal pro2 jections onto H 2 (H) and H− (K). If Ψ is an operator function satisfying (2.3), we denote by P− Ψ and P+ Ψ the functions ((P− Ψ)(ζ)) x = (P− (Ψx)) (ζ),

ζ ∈ T, x ∈ H,

((P+ Ψ)(ζ)) x = (P+ (Ψx)) (ζ),

ζ ∈ T, x ∈ H.

68


We can identify in a natural way 2 (H) with H 2 (H): z k xk . {xk }k≥0 ∈ 2 (H) → k≥0

As in the scalar case it is easy to see that Theorem 2.1 is equivalent to the following statement in which def ˆ H ∞ (B(H, K)) = Φ ∈ L∞ (B(H, K)) : Φ(n) = 0, n < 0 . Theorem 2.2. Let Ψ be a B(H, K)-valued function that satisfies (2.3). Then the operator HΨ defined by (2.4) extends to a bounded operator on H 2 (H) if and only if there exists a function Φ in L∞ (B(H, K)) such that ˆ ˆ Φ(n) = Ψ(n),

n < 0,

and HΨ = distL∞ (Φ, H ∞ (B(H, K)) . Proof of Theorem 2.1. Let Φ ∈ L∞ (B(H, K)) and ΓΩ = {Ωj+k }j,k≥0 be the block Hankel matrix defined by (2.1) and (2.2). Let us show that ΓΩ is a bounded operator from 2 (H) to 2 (K) and ΓΩ ≤ ΦL∞ . Put Ψ(z) = z¯Φ(¯ z ). It is easy to see that ΓΩ is bounded on 2 (H) if and only if HΨ is bounded on H 2 (H) and ΓΩ = HΨ . We have HΨ F L2 (K) = P− ΨF L2 (K) ≤ ΨF L2 (K) ≤ ΨL∞ F H 2 (K) . Hence, HΨ ≤ ΨL∞ = ΦL∞ . To show the converse, we extend the sequence {Ωj }j≥0 to a two-sided sequence {Ωj }j∈Z so that the block matrix Λ = {Ωj+k }j∈Z,k≥0 determines a bounded operator from 2 (H) to 2Z (K), where     2Z (K) = x = {xj }j∈Z : xj ∈ K, x22 (K) = xj 2K < ∞ . Z   j∈Z

Moreover, we show that Λ = ΓΩ . Let us first find Ω−1 . Consider the matrix {Ωj+k }j≥−1,k≥0 where Ω−1 = Z is an unknown operator:   Z Ω0 Ω1 Ω2 · · ·  Ω0 Ω1 Ω2 Ω3 · · ·     Ω1 Ω2 Ω3 Ω4 · · ·  (2.5) .   Ω2 Ω3 Ω4 Ω5 · · ·    .. .. .. .. .. . . . . .

2. Bounded Block Hankel Matrices

Let



Ω1 Ω2 Ω3 .. .

  A= 

C=

Ω0

Ω2 Ω3 Ω4 .. . Ω1

Ω3 Ω4 Ω5 .. . Ω2

··· ··· ··· .. . ···

   , 

   B= 

Ω0 Ω1 Ω2 .. .

69

   , 

.

The matrix in (2.5) can be identified with the matrix Z C . B A Clearly, B

A = ΓΩ ,

C A = ΓΩ .

By Theorem 1.1 and the Remark at the end of §1, there exists an operator Z such that Z C B A = ΓΩ . Put Ω−1 = Z. We have {Ωj+k }j≤−1,k≥0 = ΓΩ . If we continue this process, we can find a sequence of operators {Ωj }j≤−1 such that for any m ∈ Z+ {Ωj+k }j≤−m,k≥0 = ΓΩ . Given m ∈ Z+ , consider the matrix Λm = {Tjk }j∈Z,k≥0 , Tjk = Ωj+k , j ≥ −m, and Tjk = O, j < −m. Clearly, Λ = ΓΩ and the sequence {Λm }m≥1 converges to Λ = {Ωj+k }j∈Z,k≥0 in the weak operator topology. Thus Λ = ΓΩ . We can now consider the block matrix Q = {Ωj+k }j,k∈Z . As above it is easy to see that {Ωj+k }j∈Z,k≥−m = ΓΩ and Q = ΓΩ . Thus Q determines a bounded operator from 2Z (H) to 2Z (K). Let us identify 2Z (H) with L2 (H) and 2Z (K) with L2 (K) in the standard way: {xj }j∈Z → z j xj . j∈Z

We can consider now the operator Q as an operator from L2 (H) to L2 (K). Let x ∈ H and let x be the constant H-valued function on T identically equal to x. Then ζ k Ωk x (Qx)(ζ) = k∈Z

70


and Qx ∈ L2 (K). Since K is separable, we can define a B(H, K)-valued function Φ almost everywhere on T such that Φ(ζ)x = ζ k Ωk x, a.e. k∈Z

It is easy to see that

(QF )(ζ) = Φ(ζ)F ζ¯ ,

a.e.

(2.6)

for any trigonometric polynomial F with coefficients in H. Now let M be multiplication by Φ defined on the set of trigonometric polynomials in L2 (H) by (M F )(ζ) = Φ(ζ)F (ζ). It is easy to see from (2.6) that M extends to a bounded operator from L2 (H) to L2 (K) and its norm is equal to Q. To prove that Φ ∈ L∞ (B(H, K)) and ΦL∞ ≤ Q, it is sufficient to show that ess sup |(Φ(ζ)x, y)| ≤ Q ζ∈T

for any x ∈ H and y ∈ K with x ≤ 1 and y ≤ 1. Put ϕ(ζ) = (Φ(ζ)x, y), ζ ∈ T. Clearly, multiplication by the scalar function ϕ defined on the set of trigonometric polynomial extends to a bounded operator on L2 . It is also obvious that ϕ ∈ L2 , and so multiplication by ϕ is a bounded operator from L2 to L1 . By continuity, multiplication by ϕ is a bounded operator on L2 . It is easy to see now that ϕL∞ ≤ M = Q, which completes the proof. Now let Φ ∈ L∞ (B(H, K)). As in the scalar case it is easy to use a compactness argument to show that there exists a function F ∈ H ∞ (B(H, K)) such that HΦ = Φ − F L∞ (B(H,K)) . In other words, Φ − F is a symbol of HΦ of minimal norm. Suppose that HΦ has a maximizing vector f ∈ H 2 (H), i.e., f is a nonzero vector function in H 2 (H) such that HΦ f H−2 (Cm ) = HΦ · f H 2 (Cn ) . We obtain a result similar to Theorem 1.1.4. However, unlike the scalar case HΦ may have infinitely many symbols of minimal norm (we will discuss this in more detail in Chapter 14). Theorem 2.3. Let Φ ∈ L∞ (H, K) and F ∈ H ∞ (H, K) satisfy Φ − F L∞ (B(H,K)) = HΦ > 0. Suppose that f ∈ H 2 (H) is a maximizing vector of HΦ . Then 2 (K), (Φ − F )(ζ)B(H,K) = HΦ for almost all ζ ∈ T, (Φ − F )f ∈ H− and f (ζ) is a maximizing vector of (Φ − F )(ζ) almost everywhere on T. If g = HΦ −1 HΦ f , then f (ζ)H = g(ζ)K for almost all ζ ∈ T.

(2.7)

3. Hankel Operators and the Commutant Lifting Theorem

71

Proof. Without loss of generality we may assume that HΦ = 1. We have f 2 = HΦ f 2 = P− (Φ − F )f 2 ≤ (Φ − F )f 2 ≤ f 2 . Hence, P− (Φ − F )f 2 = (Φ − F )f 2 = f 2 . 2 (K). Since f (ζ) = O and The first equality means that (Φ − F )f ∈ H− (Φ − F )(ζ) ≤ 1 almost everywhere on T, it follows from the second equality that (Φ−F )(ζ) = 1 and f (ζ) is a maximizing vector of (Φ−F )(ζ) for almost all ζ ∈ T. Therefore g(ζ) is a maximizing vector of (Φ − F )∗ (ζ) for almost all ζ ∈ T. Hence, g(ζ)K = HΦ −1 (Φ − F )∗ (ζ)B(K,H) f (ζ)H = f (ζ)H , which proves (2.7). Remark. We can also consider vectorial Hankel operators on the Hardy class H 2 (C+ , H) of vector-valued functions on the upper half-plane C+ (see §1.8) and integral operators Γ Ξ on spaces of vector functions on R+ (see §1.8). The same reasoning as in §1.8 shows that bonded operators Γ Ξ have block Hankel matrices with respect to the block bases of vector-valued Laguerre polynomials.

3. Hankel Operators and the Commutant Lifting Theorem In this section we give an alternative approach to the boundedness problem for vectorial Hankel operators. We show that Theorem 2.1 can be deduced from the commutant lifting theorem. On the other hand, we show that in an important special case the commutant lifting theorem can be deduced from the vector version of the Nehari theorem (Theorem 2.1). An alternative proof of Theorem 2.1. We give a proof of the nontrivial part of Theorem 2.1. Namely, we prove that if ΓΩ = {Ωj+k }j,k≥0 determines a bounded operator from 2 (H) to 2 (K), then there exists a function ˆ = Ωj , j ∈ Z+ , and ΦL∞ = ΓΩ . Φ ∈ L∞ (B(H, K)) such that Φ(j) We denote by SH the shift operator on 2 (H): SH (x0 , x1 , · · · ) = (O, x0 , x1 , · · · ),

(x0 , x1 , · · · ) ∈ 2 (H).

Similarly, we denote by SK the shift operator on 2 (K). It is easy to see that ∗ ΓΩ = ΓΩ SH SK

(3.1)

and both operators in (3.1) are given by the block matrix {Ωj+k+1 }j,k≥0 . ∗ Equality (3.1) means that the operator ΓΩ intertwines the contractions SK and SH .

72


Consider now the bilateral shift SH on the space 2Z (H) of two-sided sequences: SH {xj }j∈Z = {xj−1 }j∈Z , {xj }j∈Z ∈ 2 (H). Similarly, we can define the bilateral shift SK on the space 2Z (K). We identify in a natural way the spaces 2 (H) and 2 (K) with subspaces of 2Z (H) and 2Z (K). It is easy to see that under these identifications the ∗ is operator SH is a minimal unitary dilation of SH while the operator SK ∗ a minimal unitary dilation of SK . Now we are in a position to apply the commutant lifting theorem (see Appendix 1.5), which implies that there exists an operator R : 2 (H) → 2 (K) such that ∗ SK R = RSH ,

P2 (K) R2 (H) = ΓΩ ,

(3.2) (3.3)

and R = ΓΩ

(3.4)

(as usual P2 (K) is the orthogonal projection onto 2 (K)). It follows easily from (3.2) that R has a block matrix representation in the form {Rj+k }j,k∈Z , where the Rj are bounded linear operators from H to K. It is easy to see from (3.3) that Rj = Ωj , j ∈ Z+ . The rest of the proof is the same as in the proof of Theorem 2.1 given in §2. As in that proof we can find a function Φ in L∞ (B(H, K)) such ˆ that Φ(j) = Rj , j ∈ Z, and ΦL∞ = R. It follows now from (3.4) that ΦL∞ = ΓΩ . We are going to consider now an important special case of the commutant lifting theorem and deduce it from the vector version of the Nehari theorem. Let H be a Hilbert space and let Θ ∈ H ∞ (B(H)) be an operator inner function such that Θ(ζ) is a unitary operator for almost all ζ ∈ T. On the space def KΘ = H 2 (H) ΘH 2 (H) we consider the compression S[Θ] of the shift operator SH (multiplication by z) on H 2 (H) defined by S[Θ] = PΘ SH KΘ , where PΘ is the orthogonal projection onto KΘ . In the case of scalar functions such operators have been considered in §1.2. As in the scalar case it is easy to see that PΘ f = ΘP− Θ∗ f,

f ∈ H 2 (H).

Recall that T is unitary equivalent to an operator S[Θ] for some unitaryvalued operator inner function Θ if and only if T is a C00 -contraction on Hilbert space (i.e., T ≤ 1 and lim T n = lim (T ∗ )n = O in the strong n→∞

operator topology) (see Appendix 1.6).

n→∞

3. Hankel Operators and the Commutant Lifting Theorem

73

The following theorem is a special case of the commutant lifting theorem (see Appendix 1.5). It will be deduced from the vector version of the Nehari theorem (Theorem 2.1). Recall that the case of scalar functions has been considered in §1.2 (Theorem 1.2.1). Theorem 3.1. Let H be a Hilbert space and let Θ ∈ H ∞ (B(H)) be a unitary-valued operator inner function. If R commutes with S[Θ] , then there exists a function Φ ∈ H ∞ (B(H)) such that ΦL∞ = R and Rf = PΘ Φf,

f ∈ KΘ .

The proof of Theorem 3.1 is similar to that of Theorem 1.2.1. 2 (H) by Proof. Define the operator A : H 2 (H) → H− Af = Θ∗ RPΘ f,

f ∈ H 2 (H).

(3.5)

As in Lemma 1.2.2 we prove that A is a Hankel operator. Again, as in the scalar case it is easy to see that A is Hankel if and only if P− zAf = Azf,

f ∈ H 2 (H).

(3.6)

We have P− zAf

= P− zΘ∗ RPΘ f = Θ∗ ΘP− Θ∗ zRPΘ f = Θ∗ PΘ zRPΘ f = Θ∗ S[Θ] RPΘ f.

On the other hand,

Azf = Θ∗ RPΘ zf.

Clearly, for f ∈ ΘH 2 (H) P− zAf = Azf = O. If f ∈ KΘ , then bearing in mind that R and S[Θ] commute, we obtain P− zAf = Θ∗ S[Θ] Rf = Θ∗ RS[Θ] f = Azf, which proves (3.6). By Theorem 2.1, there exists Ψ ∈ L∞ (B(H)) such that ΨL∞ = A 2 (H). Hence, and A = HΨ . It follows from (3.5) that ΘP− ΨH 2 (H) ⊥ H− 2 2 ∞ ΘΨH (H) ⊥ H− (H), and so ΘΨ ∈ H (B(H)). def

Now put Φ = ΘΨ ∈ H ∞ (B(H)). We have Rf = ΘP− Ψf = ΘP− Θ∗ Φf = PΘ Φf,

f ∈ KΘ .

(3.7)

We have ΦL∞ = ΨL∞ = A ≤ R. On the other hand, R ≤ PΘ · ΦL∞ = ΦL∞ . Remark. Formula (3.7) implies that

R = M Θ HΘ ∗ Φ KΘ ,

(3.8)

where MΘ is multiplication by Θ. By (3.5), HΘ∗ Φ ΘH 2 (H) = HΨ ΘH 2 (H) = O. It follows now from (3.8) that the operators R and HΘ∗ Φ have the same metric properties (compactness, singular values, etc.; see §6.9).

74


4. Compact Vectorial Hankel Operators Here we obtain an analog of the Hartman theorem for vectorial Hankel operators. In §4.3 we obtain a formula for the essential norm of a vectorial Hankel operator. Let H and K be Hilbert spaces. Denote by H ∞ (B(H, K)) + C(C(H, K)) the subspace of L∞ (B(H, K)), which consists of functions F that admit a representation F = G+H, where G ∈ H ∞ (B(H, K)) and H is a continuous function with values in the space C(H, K) of compact operators from H to K. Theorem 4.1. Let Φ ∈ L∞ (H, K)). The following statements are equivalent: (i) the Hankel operator HΦ is compact on H 2 (H); (ii) Φ ∈ H ∞ (B(H, K)) + C(C(H, K)); ˆ ˆ (iii) there exists a function Ψ in C(C(H, K)) such that Φ(n) = Ψ(n) for n < 0. It is easy to see that Theorem 4.1 can be reformulated as follows. Theorem 4.2. Let {Ωj }j≥0 be a sequence of bounded linear operators from H to K. Then the block Hankel matrix {Ωj+k }j,k≥0 determines a compact operator from 2 (H) to 2 (K) if and only if there exists a function ˆ Ψ in C(C(H, K)) such that Ψ(n) = Ωn for n ≥ 0. To prove Theorem 4.1, we need a lemma where CA (B(H, K)) denotes the ˆ set of functions Φ ∈ C(B(H, K)) for which Φ(n) = O, n < 0. Lemma 4.3. Let H, K be finite-dimensional Hilbert spaces. Let Φ ∈ C(B(H, K)). Then distL∞ (Φ, H ∞ (B(H, K))) = distL∞ (Φ, CA (B(H, K))) . The proof of the lemma is exactly the same as that of Lemma 1.5.2. Proof of Theorem 4.1. Clearly, (ii)⇔(iii). The implication (iii)⇒(i) is very easy. Indeed, if Ψ is a trigonometric polynomial with coefficients in ˆ − k)}j≥0,k≥1 C(H, K), then HΨ is compact since the block matrix {Ψ(−j corresponding to HΨ has finitely many nonzero entries, each of which is compact. If Ψ ∈ C(C(H, K)), the result follows from the fact that Ψ can be approximated by trigonometric polynomials with values in C(H, K). Suppose now that HΦ is compact. Let {Pn }n≥1 be a sequence of finite rank orthogonal projections on H such that Pn H ⊂ Pn+1 H and H = clos Pn H. Let {Qn }n≥1 be a sequence of projections on K with n≥1

the same properties. Let Φ1 = Q1 ΦP1 , Φn = Qn ΦPn − Qn−1 ΦPn−1 , n ≥ 2. It is easy to see that HΦ1 = Q1 HΦ P1 ,

HΦn = Qn HΦ Pn − Qn−1 HΦ Pn−1 ,

n ≥ 2,

4. Compact Vectorial Hankel Operators

75

where Pn is the orthogonal projection from L2 (H) onto the subspaces of functions with values in Pn H and Qn is the orthogonal projection from L2 (K) onto the subspace of functions with values in Qn K. Since HΦ is compact, it follows that HΦn → 0 as n → ∞. Therefore it is possible to choose sequences {Pn }n≥1 and {Qn }n≥1 such that HΦn < ∞. n≥1

Clearly, HΦ =

HΦ n .

(4.1)

n≥1

We need the following lemma. Lemma 4.4. Let H and K be finite-dimensional Hilbert spaces and 2 let HΦ be a compact Hankel operator from H 2 (H) to H− (K). Then for any ε > 0 there exists a function Ψ ∈ C(B(H, K)) such that HΦ = HΨ and ΨL∞ ≤ HΦ + ε. Let us first complete the proof of Theorem 4.1 and then prove Lemma 4.4. Consider HΦn H 2 (Pn H) as a Hankel operator from H 2 (Pn H) to 2 (Qn K). It follows from Lemma 4.4 that for any ε > 0 there exists a H− function Ψn ∈ C(B(Pn H, Qn K)) such that HΨn = Qn HΦn H 2 (Pn H) and Ψn ∞ ≤ Φn ∞ + ε2−n . Now put Ψn Pn . Ψ= n≥1

Clearly, Ψn Pn ∈ C(C(H, K)) and Ψn Pn ∞ ≤ Φn ∞ + ε. n≥1

(4.2)

n≥1

It follows from (4.1) that HΨ = HΦ , and it follows from (4.2) that Ψ ∈ C(C(H, K)). Proof of Lemma 4.4. We can assume that H = Cm and K = Cd . Then we can interpret Φ as a matrix function {ϕjk }1≤j≤d,1≤k≤m . It is easy to see that HΦ is compact if and only if so are the Hϕjk for all j, k. By the Hartman theorem (Theorem 1.5.5) for each j, k there exists a function ξjk in C(T) such that Hξjk = Hϕjk . Let Ξ = {ξjk }1≤j≤d,1≤k≤m . Then Ξ ∈ C(B(Cm , Cd )) and HΦ = HΞ . By Theorem 2.2 HΞ = distL∞ (Ξ, H ∞ (B(H, K))) . Since Ξ ∈ C(B(Cm , Cd )), it follows from Lemma 4.3 that for any ε > 0 there exists a function Υ ∈ CA (B(H, K)) such that Ξ − Υ∞ ≤ HΦ + ε. def

Now the function Ψ = Ξ − Υ is desired.

76


Remark. It is easy to see that we can choose sequences {Pn }n≥1 and {Qn }n≥1 such that HΦn ≤ HΦ + ε n≥1

for any given positive ε. It follows that if HΦ is a compact vectorial Hankel operator, then for any ε > 0 there exists a function Ψ in C(C(H, K)) such that HΨ = HΦ and Ψ∞ ≤ HΦ + ε.

5. Vectorial Hankel Operators of Finite Rank In this section we obtain an analog of Kronecker’s theorem for vectorial Hankel operators. We shall show that a vectorial Hankel operator has finite rank if and only if it has a rational symbol whose coefficients have finite rank. We shall also obtain a formula for the rank of such operators. However, this formula is considerably more complicated than in the scalar case. First we describe the invariant subspaces of the multiple shift operator (multiplication by z on H 2 (H)) that have finite codimension. To this end we need the notion of a finite Blaschke–Potapov product. Let H be a Hilbert space. A function B in H ∞ (B(H)) is called a finite Blaschke–Potapov product if it admits a factorization B = U B1 B2 · · · Bn , where Bj (z) =

(5.1)

λj − z ¯ j z Pj + (I − Pj ) 1−λ

for some λj in D, orthogonal projections Pj on H, and a unitary operator U . The degree of the Blaschke–Potapov product (5.1) is defined by deg B =

n

rank Pj .

j=1

Lemma 5.1. Let L be an invariant subspace of multiplication by z on H 2 (H). Then L has finite codimension if and only if L = BH 2 (H) for a Blaschke–Potapov product B of finite degree. Moreover, codim L = deg B. Proof. Denote by SH multiplication by z on H 2 (H). Clearly, L⊥ is ∗ an invariant subspace of SH and dim L⊥ = codim L. If dim L⊥ < ∞, ∗ ⊥ ¯ ∈ D. Since Ker(S ∗ − λI) ¯ consists of then SH L has an eigenvalue λ H the functions x ¯ , x ∈ H, 1 − λz

5. Vectorial Hankel Operators of Finite Rank

it follows that Ker

∗ (SH

¯ L⊥ = − λI)

77

x ¯ : x ∈ Hλ , 1 − λz

where Hλ is a finite-dimensional subspace of H. Let z−λ ¯ PHλ + (I − PHλ ), 1 − λz where PHλ is the orthogonal projection onto Hλ . It is easy to see that ∗ ¯ L⊥ = Ker P+ B ∗ , Ker (SH − λI) 1 B1 (z) =

where B ∗1 is multiplication by B1∗ . ∗ whose dimension is Clearly, P+ B ∗1 L⊥ is an invariant subspace of SH ⊥ less than dim L . Therefore we can repeat the above procedure finitely many times and we find that L⊥ = Ker P+ B ∗ , where B is a Blaschke– Potapov product of the form (5.1) and B ∗ is multiplication by B ∗ . Clearly, deg B < ∞. Hence, L = BH 2 (H). Conversely, let B be a Blaschke–Potapov product of the form (5.1). Clearly, multiplication by Bj on H 2 (H) is an isometry and the result will follow from the fact that codim Bj H 2 (H) = rank Pj . Let us show that

H (H) Bj H (H) = 2

2

x ¯ j z : x ∈ Pj H . 1−λ

(5.2)

Indeed, if x ∈ Pj H, F ∈ H 2 (H), we have x x Bj F, = 0, = F, ¯j z z − λj 1−λ 2 since 1/(z − λj ) ∈ H− . Suppose now that G ∈ H 2 (H) and G ⊥ Bj H 2 (H). We have (Bj F, G) = (bPj F, G) + ((I − Pj )F, G) = (F, ¯bPj G + (I − Pj )G) = 0

¯ j z)−1 . Hence, for any F ∈ H 2 (H), where b(z) = (λj − z)(1 − λ ¯bPj G + (I − Pj )G ∈ H 2 (H). Consequently, (I − Pj )G = O, that is, G 2 . takes values in the finite-dimensional space Pj H and ¯bG ∈ H− def 2 2 ¯ Let x ∈ Pj H and let f (z) = (G(z), x). Then f ∈ H and bf ∈ H− . It follows that f ∈ {g ∈ H 2 : H¯b g2 = g2 }.

(5.3)

Since rank H¯b = 1 (see Corollary 1.3.2), the space in (5.3) is one-dimens¯j z ¯ j z , and so f = c/ 1 − λ ional; obviously it contains the function 1/ 1 − λ for some c ∈ C. It is easy to see now that G belongs to the right-hand side of (5.2).

78


The following analog of Corollary 1.3.3 describes the vectorial Hankel operators of finite rank. As above H and K are Hilbert spaces. Theorem 5.2. Let Φ ∈ L∞ (B(H, K)). Then HΦ has finite rank if and only if there exists a Blaschke–Potapov product B in H ∞ (B(H)) of finite degree such that ΦB ∈ H ∞ (B(H, K)). In this case it is possible to choose B so that deg B = rank HΦ . Proof. Let rank HΦ < ∞. Then Ker HΦ is an invariant subspace of SH of codimension rank HΦ . By Lemma 5.1, Ker HΦ = BH 2 (H), where B is a Blaschke–Potapov product such that deg B = rank HΦ . Clearly, HΦB = O, and so ΦB ∈ H ∞ (B(H, K)). Conversely, if ΦB ∈ H ∞ (B(H, K)), then HΦ BH 2 (H) = O, and so Ker HΦ has finite codimension. Remark. It can also be shown that HΦ has finite rank if and only if there exists a Blaschke–Potapov product B1 in H ∞ (B(K)) of finite degree such that B1 Φ ∈ H ∞ (B(H, K)). Moreover, B1 can be chosen so that deg B1 = rank HΦ . Indeed, let Ψ(z) = (Φ(¯ z ))∗ . Then Ψ ∈ L∞ (B(K, H)), and it is easy to see that rank HΦ = rank HΨ . Therefore by Theorem 5.2, ΨB ∈ H ∞ (B(K, H)) for some Blaschke–Potapov product B in H ∞ (B(K)) of finite degree. Let B1 (z) = (B(¯ z ))∗ . It is easy to show that B1 is a Blaschke–Potapov product ∞ in H (B(K)) and deg B1 = deg B. Clearly, B1 Φ ∈ H ∞ (B(H, K)). Theorem 5.2 gives a description of the vectorial Hankel operators of finite rank. However, it is difficult to compute the rank of such an operator by using Theorem 5.2. We conclude this section with a more explicit formula for the rank of a vectorial Hankel operator. To state the result, we need some preliminaries. Let Λ be a finite subset of the unit disc and let Ψ be a rational function of the form Ψ(ζ) =

k(λ) λ∈Λ n=1

Tλ,n , (z − λ)n

Tλ,k(λ) = O,

(5.4)

where the k(λ) are positive integers and Tλ,n ∈ B(H, K), λ ∈ Λ, (j) 1 ≤ n ≤ k(λ). Let us construct by induction on j the operators Tλ,n for 0 ≤ j ≤ k(λ) − n. Put (0)

Tλ,n = Tλ,n ,

(j+1)

Tλ,n

(j)

= Tλ,n PKer T (j)

(j)

+ Tλ,n+1 PRangeT (j)

λ,k(λ)−j

λ,k(λ)−j

∗ ,

(5.5)

where PM denotes the orthogonal projection onto M . Theorem 5.3. Let Φ ∈ L∞ (B(H, K)). Then rank HΦ < ∞ if and only if Ψ = P− Φ admits a representation (5.4), where Λ is a finite set in D and the Tλ,n are finite rank operators from H to K, λ ∈ Λ, 1 ≤ n ≤ k(λ). In

5. Vectorial Hankel Operators of Finite Rank

79

this case rank HΦ =

k(λ)−1 λ∈Λ

(j)

rank Tλ,k(λ)−j ,

(5.6)

j=0

(j)

where the operators Tλ,n are defined by (5.5). 2 Proof. Let S˜ be the compressed shift on H− (K) defined by ˜ = P− zf, Sf

2 f ∈ H− (K).

˜ Let L = Range HΦ . Clearly, L is S-invariant and dim L = rank HΦ . If HΦ has finite rank, there exist a finite set Λ in D and N ∈ Z+ such that L ⊂ span{Ker(S˜ − λI)N : λ ∈ Λ}. Put

k(λ) = min n ∈ Z+ : Ker((S˜ − λI)L)n = Ker((S˜ − λI)L)n+1 .

It is easy to see that

Ker(S˜ − λI)n = span Therefore def

L ⊂ L = span

x : x ∈ K, 1 ≤ m ≤ n . (z − λ)m

x : λ ∈ Λ, x ∈ K, 1 ≤ m ≤ k(λ) . (z − λ)m

Let us now define the operators Tλ,n , λ ∈ Λ, 1 ≤ n ≤ k(λ), by Tλ,n x = (z − λ)n Pλ,n P− (Φx),

x ∈ H,

(5.7)

where Pλ,n is the projection onto the subspace

x : x∈K (z − λ)n along the other subspaces

x : x ∈ K , (z − µ)m

µ ∈ Λ, 1 ≤ m ≤ k(λ),

that is,

x =O (z − λ)m whenever λ = µ or m = n. In (5.7) we identify vectors in K with constant K-valued functions. It is easy to see that the operators Tλ,n have finite rank and Ψ = P− Φ satisfies (5.4). Let ' Lλ = L Ker(S˜ − λI)k(λ) , λ ∈ Λ. Pλ,n

Clearly, rank HΦ =

λ∈Λ

dim Lλ .

80


Therefore formula (5.6) will be established if we prove that

k(λ)−1

dim Lλ =

(j)

rank Tλ,k(λ)−j ,

λ ∈ Λ.

j=0

Assume for simplicity that λ = 0. The general case can be treated similarly; moreover it can be reduced to the case λ = 0 by a conformal change of variable. (j) (j) Let k = k(λ), Tn = T0,n , and Tn = Tλ,n . We have k L0 = P0,m HΦ H 2 (H) = P− Ψ0 H 2 (H) m=1

k−n ( Tk+n x span : 0 ≤ n < k, x ∈ H , (z − λ)m m=1

= where

Ψ0 (z) =

k

z −n Tn .

n=1

Let E = Ker Tk . Then all nonzero functions in P− Ψ0 E ⊥ are polynomials of z −1 of degree k. On the other hand, the space P− Ψ0 H 2 (H) E ⊥ = P− Ψ0 H 2 (E) ⊕ zH 2 (E ⊥ ) consists of polynomials of degree less than k. (Here we identify vectors in E ⊥ with constant E ⊥ -valued functions.) Therefore the above subspaces are linearly independent and so dim P− Ψ0 H 2 (H) = dim P− Ψ0 E ⊥ + dim P− Ψ0 H 2 (E) + zΨ0 H 2 (E ⊥ ) . Clearly, and

dim P− Ψ0 E ⊥ = rank Tk dim P− Ψ0 H 2 (E) + zΨ0 H 2 (E ⊥ ) = dim Ψ1 H 2 (H),

where ˜ E⊥ = Ψ1 = ΨPE + SΨP

k−1

z −n Tn(1) ,

n=1 (1) Tn

(1) T0,n

the operators = being defined by (5.5). Repeating the above procedure k times, we arrive at the formula dim L0 =

k−1

(j)

rank Tk−j .

j=0

Suppose that λ ∈ C and Φ is a rational operator function of the form

k(λ)

Φ=

n=1

Tλ,n + Qλ , (z − λ)n

Tλ,k(λ) = O,

6. Imbedding Theorems

81

where Qλ is analytic at λ. Then the McMillan degree degλ Φ of Ψ at λ can be defined as k(λ)−1 (j) degλ Φ = rank Tλ,k(λ)−j , j=0 (j)

where the Tλ,k(λ)−j are defined by (5.5). Similarly, one can define the McMillan degree deg∞ Φ of Φ at infinity. The McMillan degree deg Φ of Φ is defined by degλ Φ. deg Φ = λ∈C∪{∞}

Theorem 5.3 says that rank HΦ is equal to the McMillan degree of P− Φ.

6. Imbedding Theorems In this section we find relations between vectorial Hankel operators and certain imbedding operators. In particular we shall show that for any Carleson measure µ there exists a bounded Hankel operator HΦ , Φ ∈ L∞ (B(C, H)), such that f L2 (µ) = HΦ f ,

f ∈ H 2.

In other words, the Carleson imbedding operator Iµ : H 2 → L2 (µ) (see Appendix 2.1) can be represented as Iµ = V HΦ for some isometry V . This allows one to reduce the study of metric properties of Iµ (such as the norm, singular values, etc.) to the corresponding properties of Hankel operators. Note that in §1.7 we have already found a relation between scalar Hankel operators and Carleson imbedding operators. We are going to consider more general imbedding operators and relate ¯ −1 them with certain Hankel operators. Recall that kζ (z) = (1−|ζ|2 )(1− ζz) 2 is the normalized reproducing kernel of H . Theorem 6.1. Let H be a Hilbert space, Φ ∈ L2 (H). The Hankel 2 (H) defined on the set of polynomials by operator HΦ : H 2 → H− HΦ f = P− f Φ

(6.1)

is bounded if and only if sup HΦ kζ < ∞. ζ∈D

The proof is the same as that of Theorem 1.6.1. The only difference is that we have to deal with vector-valued functions. Clearly, in our case B(C, H)-valued functions can be identified with H-valued functions. By Theorem 2.2, HΦ is bounded if and only if Φ determines a bounded linear functional on H 1 (H). Therefore Theorem 6.1 is equivalent to the fact that

82


an H-valued function Φ determines a bounded linear functional on H 1 (H) if and only if the Garsia norm

sup Ψ(ζ) − Ψ(τ )2H Pζ (τ )dm(τ ) T

ζ∈D

def

of Ψ = P− Φ is finite, where as before we denote the harmonic extension of Ψ to D by the same symbol Ψ. The proof of this Hilbert-valued version can be obtained in exactly the same way as the corresponding scalar result (see Garnett [1], Ch. VI, §2, where the scalar case is treated). We consider here the following imbedding operators. Let ν be a positive Borel measure on a subset Λ of D and let λ → ϑλ , λ ∈ Λ, be a measurable family of inner functions. As before Pϑλ is the orthogonal projection of H 2 onto Kϑλ = H 2 ϑλ H 2 . Let f ∈ H 2 . Define the function If in the direct integral

Λ

⊕Kϑλ dν(λ)

by (If )(λ) = Pϑλ f,

λ ∈ Λ,

(6.2)

(see Appendix 1.4 for the definition of direct integrals). We show that I = V HΦ for a certain vectorial Hankel operator HΦ and an isometry V . The following generalized imbedding theorem holds. Theorem 6.2. Let ν be a positive measure on a subset Λ of D and let {ϑλ }λ∈Λ be a measurable family of inner functions. The following statements are equivalent: (i) the operator I defined by (6.2) is bounded, i.e.,

Pϑλ f 2 dν(λ) ≤ cf 22 , f ∈ H 2 ; Λ

(ii)

(1 − |ϑλ (ζ)|2 )dν(λ) < ∞;

sup ζ∈D

Λ

(iii)

sup ζ∈D

Λ

Pϑλ kζ 2 dν(λ) < ∞.

Note that the Carleson imbedding operator can be considered as a special case of the above operators (6.2). Indeed, if z−λ 2 ¯ , dµ(λ) = (1 − |λ| )dν(λ), 1 − λz then it is easy to see that

2 Pϑλ f dν(λ) = |f (λ)|2 dµ(λ), f ∈ H 2 . ϑλ (z) =

Λ

Λ

6. Imbedding Theorems

83

In particular, Theorem 6.2 implies the following well-known version of the Carleson imbedding theorem (see N.K. Nikol’skii [2], Lect. VII, §1). Theorem 6.3. Let µ be a positive measure on D. Then the imbedding operator Iµ : H 2 → L2 (µ) is bounded if and only if

1 − |ζ|2 sup ¯ 2 dµ(λ) < ∞. ζ∈D D |1 − ζλ| In other words, µ is a Carleson measure if and only if sup Iµ kζ L2 (µ) < ∞. ζ∈D

To prove Theorem 6.2, we need the following result, which reduces the study of the imbedding operators (6.2) to the study of vectorial Hankel operators. Theorem 6.4. Let ν be a positive measure on a subset Λ of D and let {ϑλ }λ∈Λ be a measurable family of inner functions. For H = L2 (ν) define the H-valued function Φ by (Φ(ζ)) = ϑλ (ζ),

ζ ∈ T, λ ∈ Λ.

Then If = HΦ f for f ∈ H 2 , where I is defined by (6.2) and the Hankel operator 2 HΦ : H 2 → H− (H) is defined by (6.1). Proof. Clearly, Pϑλ f = ϑλ P− ϑ¯λ f = Hϑ¯λ f . It follows that

If = 2

Pϑλ f dν(λ) = 2

Λ

Λ

Hϑ¯λ f 2 dν(λ) = HΦ f 2 .

Proof of Theorem 6.2. Let us show that (ii)⇔(iii). We have Pϑλ kζ 2 = 1 − ϑλ P+ ϑ¯λ kζ 2 = 1 − ϑλ (ζ)ϑkζ 2 = 1 − |ϑλ (ζ)|2 , which proves that (ii)⇔(iii). The fact that (i)⇒(iii) is trivial. The implication (iii)⇒(i) is an immediate consequence of Theorems 6.1 and 6.4. Remark. It can also be proved that I is compact if and only if

Pϑλ kζ 2 dν(λ) = 0. lim |ζ|→1

Λ

84


Concluding Remarks Theorem 1.1 was established in Parrott [1]. Lemma 1.3 is taken from Douglas [1]. Theorem 1.5 in the special case of self-adjoint operators was proved in Krein [1]. A more general case was considered in Arlinskii and Tsekanovskii [1]. In the general case Theorem 1.5 was obtained independently in Shmul’yan and Yanovskaya [1], Arsene and Gheondea [1], and Davis, Kahan, and Weinberger [1]. The proof of Theorem 1.5 given in §1 is suggested by Vasyunin. Theorem 2.1, which is the operator-valued version of the Nehari theorem, was obtained first in Page [1]. The approach given in §2 is due to Adamyan, Arov, and Krein [4]; see also Parrott [1]. Theorem 2.3 is a well-known analog of Theorem 1.1.4. The approach to the operator-valued version of the Nehari theorem, given in §3 is due to Page [1]. We also mention the approach found in Sarason [1] which is based on factorizations of operator functions of class H 1 (S 1 ). The proof Theorem 3.1 based on the operator version of the Nehari theorem was obtined by N.K. Nikol’skii [2]. Formula (3.8) is also due to N.K. Nikol’skii [2]. The compactness criterion obtained in §4 was first given in Page [1]. Theorem 5.2 is well known. Theorem 5.3 is due to Treil [3]. The results of §6 were obtained in Treil [6]. Theorem 4.1 was used in the paper Yafaev [1] to construct a function Φ ∈ H ∞ (B(H)) (H is a Hilbert space) such that Φ is continuous on a nondegenerate closed arc I ⊂ T, Φ(ζ) is compact for ζ ∈ I but Φ(ζ) is not compact for ζ ∈ I. Theorem 2.2 leads naturally to the Nehari problem for operator-valued functions: for Hilbert spaces H and K, and Φ ∈ L∞ (B(H, K)) find a best approximation from H ∞ (B(H, K)), i.e., find an operator function Q ∈ H ∞ (B(H, K)) such that Φ − QL∞ (B(H,K)) = HΦ . A more general problem, very important in control theory (see Ch. 11), is the so-called four block problem.Let H1 , H2 , K1 , and K2 be Hilbert Φ11 Φ12 be an operator function such that spaces, and let Φ = Φ21 Φ22 Φij ∈ L∞ (B(Hj , Ki )), i, j = 1, 2. The four block problem is to find Q ∈ H ∞ (B(H1 , K1 )), which minimizes Φ11 − Q Φ12 . Φ21 Φ22 L∞ (B(H ⊕H ,K ⊕K )) 1

2

1

2

The four block problem reduces to the Nehari problem in the case Φ12 , Φ21 , and Φ22 are the zero functions. Consider now the four block operator 2 (K1 ) ⊕ L2 (K2 ), 4Φ : H 2 (H1 ) ⊕ L2 (H2 ) → H−

Concluding Remarks

85

which plays the same role in the four block problem as the Hankel operators play in the Nehari problem: f1 f1 = P− Φ , f1 ∈ H 2 (H1 ), f2 ∈ L2 (H2 ), 4Φ f2 f2 where P− is the orthogonal projection from L2 (K1 ) ⊕ L2 (K2 ) onto 2 (K1 ) ⊕ L2 (K2 ). The analog of the Nehari theorem is H− Φ11 − Q Φ12 4Φ = inf . Φ21 Φ22 L∞ (B(H ⊕H ,K ⊕K )) Q∈H ∞ (H1 ,K1 ) 1

2

1

2

Note that 4Φ can be compact only if Φ12 , Φ21 , and Φ22 are the zero functions. We refer the reader to Foias and Tannenbaum [1], [2] for the above facts and detailed information on the four block problem.

3 Toeplitz Operators

We study in this chapter another very important class of operators on spaces of analytic functions, the class of Toeplitz operators. Toeplitz operators can be defined as operators with matrices of the form {tj−k }j,k≥0 . In §1 we characterize the bounded Toeplitz operators and study elementary properties of Toeplitz operators on the Hardy class H 2 . We obtain spectral inclusion theorems, and we reduce the problem of the invertibility of an arbitrary Toeplitz operator to the case of Toeplitz operators with unimodular symbols. Then we use the Nehari theorem to obtain necessary and sufficient conditions for left invertibility and right invertibility in the case of unimodular symbols. We obtain similar results for left Fredholmness and right Fredholmness. In §2 we obtain a general invertibility criterion for Toeplitz operators. Although this criterion is not geometric, it turns out to be very useful in obtaining geometric descriptions of the spectra for various classes of Toeplitz operators (see §3). We give geometric descriptions of spectra for several classes of Toeplitz operators in §3. We consider here the classes of Toeplitz operators with real-valued symbols, with continuous symbols, with symbols in H ∞ + C, and the class of Toeplitz operators with piecewise continuous symbols. Section 4 is devoted to Toeplitz operators on spaces of vector functions. We obtain a boundedness criterion and prove general spectral inclusion theorems. Then we study invertibility conditions for Toeplitz operators with isometric-valued symbols, and we conclude §4 with a Fredholmness criterion for Toeplitz operators with matrix-valued symbols of class H ∞ + C.

88

Chapter 3. Toeplitz Operators

In §5 we prove that if a Toeplitz operator with matrix-valued symbol is Fredholm, then its symbol admits a Wiener–Hopf factorization. We characterize the invertible and the Fredholm Toeplitz operators with matrixvalued symbols in terms of such Wiener–Hopf factorizations. We also find formulas for the indices of such factorizations. We conclude this chapter with §6 in which we obtain a criterion for the left invertibility of an operator function Ω in the space of bounded analytic operator functions. We prove that Ω is left invertible if and only if the Toeplitz operator with symbol Ω(¯ z ) is left invertible.

1. Basic Properties We define here Toeplitz operators on the Hardy class H 2 as those which have Toeplitz matrices (i.e., their entries depend only on the difference of the coordinates) in the basis {z n }n≥0 . In this section we study basic properties of Toeplitz operators. We shall see that a special role is played by Toeplitz operators with unimodular symbols. Definition. An operator T : H 2 → H 2 defined on the set of polynomials is called a Toeplitz operator if there is a two-sided sequence of complex numbers {tn }n∈Z such that (T z k , z j ) = tj−k ,

j, k ∈ Z+ .

(1.1)

Theorem 1.1. A Toeplitz operator T defined by (1.1) is bounded on H 2 if and only if there exists a bounded function ϕ on T whose Fourier coefficients coincide with the tj : ϕ(n) ˆ = tn . In this case T = ϕ∞ . Proof. Suppose that T is a bounded Toeplitz operator. For each n ∈ Z we consider the operator Tn on L2 (T) defined by Tn f = z¯n T P+ z n f,

f ∈ L2 .

Clearly, Tn is bounded and Tn ≤ T . If j ≥ −n and k ≥ −n, then (Tn z k , z j ) = (T z n+k , z n+j ) = tj−k . Therefore the sequence {Tn }n≥0 converges weakly to a bounded operator M on L2 . Obviously, M ≤ T and (M z k , z j ) = tj−k for any j, k ∈ Z. It is also clear that M zf = zM f , f ∈ L2 (it suffices to verify this equality for f = z j , j ∈ Z). Let ϕ = M 1 ∈ L2 . It is easy to see that M f = ϕf , for any polynomial f . Clearly, multiplication by ϕ is a bounded operator from L2 to L1 , and it follows that M f = ϕf for any f ∈ L2 . Since M is a bounded operator on L2 , it is easy to see that ϕ ∈ L∞ and ϕ∞ = M (it suffices to apply M to the charcteristic functions of the sets {ζ ∈ T : |ϕ(ζ)| > N }).

1. Basic Properties

89

We have ϕ(n) ˆ = (M 1, z n ) = tn and T f = P+ M f = P+ ϕf for any polynomial f . Consequently, T f 2 ≤ ϕ∞ f 2 = M · f 2 . Thus T ≤ M = ϕ∞ . The converse follows immediately from the equality T f = P+ ϕf (which is sufficient to verify for f = z n , n ≥ 0). Given ϕ ∈ L∞ we define the Toeplitz operator Tϕ on H 2 by Tϕ f = P+ ϕf,

f ∈ H 2.

(1.2)

It follows from Theorem 1.1 that Tϕ = ϕ∞ . The function ϕ is called the symbol of Tϕ . In contrast with Hankel operators the symbol of a Toeplitz operator is uniquely determined by the operator. It is easy to see that Tϕ∗ = Tϕ¯ ,

ϕ ∈ L∞ .

(1.3)

Although the definition of Toeplitz operators (1.2) looks similar to the definition of Hankel operators (see (1.1.6)), the properties of Hankel operators are quite different from those of Toeplitz operators. In particular the Toeplitz operator Tϕ is never compact unless ϕ = O. Indeed, let ϕ ∈ H 2 and Tϕ f = O. Then it is easy to see that Tϕ z n f 2 ≥ Tϕ f 2 ,

n ≥ 0.

On the other hand, the sequence {z f }n≥0 converges weakly to O, which makes it impossible for Tϕ to be compact. However, there are important relations between Hankel and Toeplitz operators and there are useful formulas that relate Hankel operators with Toeplitz ones. We begin with the simplest one. Given ϕ ∈ L∞ , denote by Mϕ multiplication by ϕ on L2 : n

Mϕ f = ϕf,

f ∈ L2 .

Then obviously Mϕ f = Hϕ f + Tϕ f,

f ∈ H 2.

(1.4)

Another formula that will be used frequently is the following: Tϕψ − Tϕ Tψ = Hϕ∗¯ Hψ ,

ϕ, ψ ∈ L∞ .

(1.5)

The proof of (1.5) is straightforward: Tϕψ f − Tϕ Tψ f

= P+ ϕψf − P+ (ϕP+ ψf ) = P+ (ϕP− ψf ) = Hϕ∗¯ Hψ f,

f ∈ H 2.

The Toeplitz operators can also be described as the operators satisfying a certain commutation relation. Theorem 1.2. Let T be an operator on H 2 . Then T is a Toeplitz operator if and only if S ∗ T S = T, where S is multiplication by z on H 2 .

(1.6)

90


Proof. Let j, k ∈ Z+ . Clearly, (1.6) means that (T z k , z j ) = (S ∗ T Sz k , z j ) = (T z k+1 , z j+1 ), that is, T has Toeplitz matrix (see (1.1)). It follows from Theorem 1.5.5 and from (1.5) that the functional calculus ϕ → Tϕ , ϕ ∈ H ∞ + C, is multiplicative modulo the compact operators. But it is not multiplicative. The following assertion characterizes the pairs ϕ, ψ ∈ L∞ for which Tϕψ = Tϕ Tψ . Theorem 1.3. Let ϕ, ψ ∈ L∞ . Then Tϕψ = Tϕ Tψ if and only if ψ ∈ H ∞ or ϕ¯ ∈ H ∞ . Proof. If ψ ∈ H ∞ , then Tϕψ f = P+ ϕψf = Tϕ ψf = Tϕ Tψ f , f ∈ H 2 . If ∗ ϕ¯ ∈ H ∞ , it follows from (1.5) that Tϕψ = Tϕ¯ψ¯ = Tψ¯ Tϕ¯ = Tψ∗ Tϕ∗ = (Tϕ Tψ )∗ , so Tϕψ = Tϕ Tψ . Suppose now that Tϕ Tψ = Tϕψ . It follows from (1.5) that Hϕ∗¯ Hψ = O. Let us show that either Hψ = O or Hϕ¯ = O. We have ¯ = 0, (Hϕ∗¯ Hψ z n , z k ) = (Hψ z n , Hϕ¯ z k ) = (P− z n ψ, P− z k ϕ)

n, k ≥ 0.

Therefore (P+ z¯n f, P+ z¯k g) = (S ∗n f, S ∗k g) = 0,

n, k ≥ 0,

(1.7)

where f = z¯P− ψ ∈ H 2 , g = z¯P− ϕ¯ = S ∗ P+ ϕ ∈ H 2 . If f = O and g = O, put L1 = span{S ∗n f : n ≥ 0}, L2 = span{S ∗k g : k ≥ 0}. The subspaces L1 and L2 are S ∗ -invariant and nontrivial. So by Beurling’s theorem (see Appendix 2.2) there exist nonconstant inner functions ϑ1 and ϑ2 such that L1 = H 2 ϑ1 H 2 , L2 = H 2 ϑ2 H 2 . It follows from (1.7) that L1 ⊥ L2 and so H 2 ϑ1 H 2 ⊂ ϑ2 H 2 . It is easy to see that if τ h ∈ L1 , and τ is inner, then h ∈ L1 . Indeed, (τ h, ϑ1 η) = 0 for every η ∈ H 2 . Put η = τ ξ, ξ ∈ H 2 . Then (τ h, ϑ1 τ ξ) = (h, ϑ1 ξ) = 0 for every ξ ∈ H 2 , which means that h ∈ L1 . Therefore L1 contains an outer function that certainly cannot belong to ϑ2 H 2 . Thus either f = O or g = O, which means that either Hψ = O or Hϕ¯ = O. One of the most important questions in studying Toeplitz operators is to find an invertibility criterion and describe the spectrum. To study such a problem we need the notion of Fredholm operators. Recall (see Appendix 1.2 for more information) that an operator T on Hilbert space H is called Fredholm if it is invertible modulo compact operators. The index of a Fredholm operator T is defined by ind T = dim Ker T − dim Ker T ∗ ,

1. Basic Properties

91

and the essential spectrum σe (A) of a bounded operator A is, by definition, σe (A) = {λ ∈ C : A − λI is not Fredholm}. Clearly, σe (A) ⊂ σ(A), where σ(A) is the spectrum of A. Obviously, the index of an invertible operator is zero. It turns out that for Fredholm Toeplitz operators the converse is also true. Theorem 1.4. Let ϕ be a nonzero function in L∞ . Then either Ker Tϕ = {O} or Ker Tϕ∗ = {O}. 2 2 Proof. Let f ∈ Ker Tϕ and g ∈ Ker Tϕ∗ . Then ϕf ∈ H− and ϕg ¯ ∈ H− . 1 def 1 1 ˆ = 0, j ≥ 0} and ϕ¯f¯g ∈ H− . Let Therefore ϕf g¯ ∈ H− = {ψ ∈ L : ψ(j) 1 ˆ ¯ , which means that h(n) =0 h = ϕf g¯. Then both h and h belong to H− for any n ∈ Z and so h = O. Since ϕ = O, it follows that either f or g must vanish on a set of positive measure, which implies that either f = O or g = O (see Appendix 2.1). Corollary 1.5. A Toeplitz operator Tϕ is invertible if and only if it is Fredholm and ind Tϕ = 0. In the case ϕ ∈ H ∞ the operator Tϕ is simply multiplication by ϕ. Such operators are called analytic Toeplitz operators. The spectrum of such operators can easily be described. Theorem 1.6. Let ϕ ∈ H ∞ . Then σ(Tϕ ) = clos ϕ(D). Proof. Let us show that Tϕ is invertible if and only if ϕ is invertible in H ∞ . Let ψ ∈ H ∞ and ϕψ = 1. Then clearly Tϕ Tψ = Tψ Tϕ = Tϕψ = I. Suppose now that Tϕ is invertible. Then ϕH 2 = H 2 . Consequently, there exists a function ψ in H 2 such that ϕψ = 1. Let g = ϕf , f ∈ H 2 . Then Tϕ−1 g = f = ψg. Since ϕH 2 = H 2 , it follows that Tϕ−1 g = ψg for any g in H 2 . Thus ψ is bounded and so ϕ is invertible in H ∞ . We are going to establish some elementary properties of the spectrum of Toeplitz operators.

Definition. Let ϕ ∈ L∞ . The set R(ϕ) of those λ in C for which the set {ζ ∈ T : |f (ζ) − λ| < ε} has positive Lebesgue measure for any ε > 0 is called the essential range of ϕ. In other words λ ∈ R(ϕ) if and only if ϕ − λ is noninvertible in L∞ . Clearly, R(ϕ) is closed in C. Theorem 1.7. Let ϕ ∈ L∞ . Then R(ϕ) ⊂ σe (Tϕ ). Proof. It suffices to show that if Tϕ is Fredholm, then ϕ is invertible in L∞ . By Theorem 1.4, either Ker Tϕ = {O} or Ker Tϕ∗ = {O}. To be definite, suppose that Ker Tϕ = {O}. Then there exists ε > 0 such that εf 2 ≤ Tϕ f 2 ≤ ϕf 2 , Therefore z n f 2 . ε¯ z n f 2 ≤ ϕ¯

f ∈ H 2.

92


Since the set {¯ z n f : f ∈ H 2 , n ≥ 0} is dense in L2 , it follows that εg2 ≤ ϕg2 for any g ∈ L2 , which implies that 1/ϕ ∈ L∞ . Corollary 1.8. Let ϕ ∈ L∞ . Then R(ϕ) ⊂ σ(Tϕ ). Denote by conv E the convex hull of a set E. Theorem 1.9. Let ϕ ∈ L∞ . Then σ(Tϕ ) ⊂ conv R(ϕ). Proof. It is sufficient to prove that if an open half-plane contains R(ϕ), then it also contains σ(Tϕ ). By the linearity of the map ϕ → Tϕ we can assume that the half-plane in question is {ζ ∈ C : Re ζ > 0}. Moreover it is sufficient to show that if R(ϕ) ⊂ {ζ ∈ C : Re ζ > 0}, then Tϕ is invertible. Let us show that there exist ε > 0 and r < 1 such that εR(ϕ) ⊂ {ζ : |1 − ζ| < r}.

(1.8)

Indeed, the discs DR = {ζ : |R − ζ| < R},

R > 0,

cover the half-plane {ζ ∈ C : Re ζ > 0}. Therefore R(ϕ), being compact, is contained in some DR . Then R(ϕ) ⊂ {ζ : |R − ζ| < R − δ} for some δ > 0. Now (1.8) holds with ε = 1/R, r = 1 − δ/R. It follows that εϕ − 1∞ < 1, so Tεϕ − I < 1, which implies that Tεϕ is invertible and so is Tϕ . In the general case the problem of the invertibility of a Toeplitz operator can be reduced to the case of a Toeplitz operator with unimodular symbol. (A function u is called unimodular if |u(ζ)| = 1 almost everywhere on T.) Lemma 1.10. Let ϕ ∈ L∞ . The following statements are equivalent: (i) Tϕ is invertible; (ii) ϕ is invertible in L∞ and Tϕ/h is invertible, where h is an outer function with |h| = |ϕ|; (iii) ϕ is invertible in L∞ and Tϕ/|ϕ| is invertible. Proof. If Tϕ is invertible, then ϕ is invertible in L∞ by Corollary 1.8. By Theorem 1.3, Tϕ = Tϕ/h Th . Since Th is invertible, it follows that (i)⇔(ii). To show that (i)⇔(iii), take an outer function h1 such that |h1 | = |ϕ|−1/2 . By Theorem 1.3, Tϕ/|ϕ| = Th¯1 ϕh1 = Th∗1 Tϕ Th1 . The result follows from the invertibility of Th1 and Th∗1 . Theorem 1.11. Let u be a unimodular function on T. Then Tu is left invertible if and only if distL∞ (u, H ∞ ) < 1. Tu is right invertible if and u, H ∞ ) < 1. only if distL∞ (¯ Proof. It follows from (1.4) that f 2 = Hu f 2 + Tu f 2 ,

f ∈ H 2.

Hence, Tu f 2 ≥ εf 2 for some ε > 0 if and only if Hu < 1. By Theorem 1.1.3 this is equivalent to the fact that dist(u, H ∞ ) < 1.

1. Basic Properties

93

The second assertion of the theorem follows from the equality Tu∗ = Tu¯ .

Theorem 1.12. Let u be a unimodular function on T. If Tu is invertible and h is a function in H ∞ such that u − h∞ < 1, then h is an outer function. Proof. We have ¯h∞ = u − h∞ . I − Tu¯h = 1 − u Therefore Tu¯h = Tu∗ Th is invertible and so is Th , which means by Theorem 1.6 that h is invertible in H ∞ . Theorem 1.13. Let u be a unimodular function on T. Then Tu is invertible if and only if there exists an outer function h such that u − h∞ < 1. Proof. The existence of such an outer function for an invertible Tu follows from Theorems 1.11 and 1.12. Suppose that u − h∞ < 1 for an outer function h. Then h is separated away from zero on T and since h is outer, it follows that h is invertible in H ∞ . We have I − Tu¯ Th < 1, which implies that Tu¯ Th is invertible. Hence, Tu is also invertible. Theorem 1.14. Let u be a unimodular function on T such that Tu is left invertible. Then Tu is invertible if and only if Tz¯u is not left invertible. Proof. If Tu is invertible, then there is a function f in H 2 such that Tu f = 1. Hence, Tz¯u f = O. Suppose now that Tz¯u is not left invertible. That means that inf {Tz¯u f 2 : f 2 = 1} = 0 or, which is the same, inf Tu f − T) u f (0) · 12 : f 2 = 1 = 0. By the hypothesis inf {Tu f 2 : f 2 = 1} > 0. Hence, 1 ∈ clos Tu H 2 = Tu H 2 . Let g be a function in H 2 such that Tu g = 1. From the identity 1 = Tz¯n Tu z n g, n ∈ Z+ , it follows by induction that z n ∈ Tu H 2 , n ∈ Z+ , that is, Tu H 2 = H 2 . We are now going to obtain similar descriptions of Fredholm, left Fredholm, and right Fredholm Toeplitz operators with unimodular symbols. Recall that a bounded linear operator T on Hilbert space is called left Fredholm if there exists a bounded linear operator R such that RT − I is compact and T is called right Fredholm if there is a bounded linear operator R such that T R − I is compact.

94


Theorem 1.15. Let u be a unimodular function on T. Then Tu is left Fredholm if and only if Hu e < 1. Tu is right Fredholm if and only if Hu¯ e < 1. Proof. Clearly, it is sufficient to prove only the first part of the theorem. Suppose that Hu e = distL∞ (u, H ∞ + C) < 1. Then there exists n ∈ Z+ such that distL∞ (z n u, H ∞ ) = distL∞ (u, z¯n H ∞ ) < 1. By Theorem 1.11, the operator Tzn u = Tu Tzn is left invertible. Let R be a bounded linear operator such that RTu Tzn = I. It follows that RTu Tzn Tz¯n = Tz¯n . Clearly, I −Tzn Tz¯n is compact, and so RTu −Tz¯n is compact. Multiplying this operator by Tzn on the left, we find that Tzn RTu − I is compact which proves that Tu is left Fredholm. Suppose now that Tu is left Fredholm. Then there exists a bounded linear operator R such that RTu − I is compact. Hence, dim Ker RTu < ∞. It follows that there exists n ∈ Z+ such that Ker RTzn u = {O}. Since RTzn u − I is compact, it follows that RTzn u is invertible, and so Tzn u is left invertible. By Theorem 1.11, distL∞ (z n u, H ∞ ) < 1. The result follows now from the obvious equality distL∞ (u, H ∞ + C) = distL∞ (z n u, H ∞ + C). Corollary 1.16. Let u be a unimodular function on T. Then Tu is Fredholm if and only if Hu e < 1 and Hu¯ e < 1.

2. A General Invertibility Criterion The following theorem due to Devinatz and Widom gives an invertibility criterion for a Toeplitz operator. Although the criterion is not geometric, we shall see later that it can be used to obtain geometric descriptions of the spectra of certain Toeplitz operators. Theorem 2.1. Let ϕ ∈ L∞ . Then Tϕ is invertible if and only if ϕ is invertible in L∞ and ϕ/|ϕ| admits a representation ϕ/|ϕ| = exp i(˜ α + β + c) ∞

(2.1)

where α and β are real functions in L , β∞ < π/2, and c ∈ R. Recall that α ˜ is the harmonic conjugate of α (see Appendix 2.1). Proof. By Lemma 1.10, we may assume that ϕ is unimodular. Suppose that Tϕ is invertible. Then there exists an outer function h such that 1 − ϕh ¯ ∞ < 1 (see Theorem 1.13). Hence, γπ R(ϕh) ¯ ⊂ ζ ∈ C : | arg ζ| < 2 for some γ < 1. Therefore there exists a real-valued function β such that β∞ < π/2 and ϕh ¯ = |h|e−iβ . Hence, ¯ −1 eiβ = exp(log |h| + β + c) ϕ = |h|h

2. A General Invertibility Criterion

95

def

for some c ∈ R. Clearly, α = log |h| ∈ L∞ . Suppose now that ϕ = exp i(˜ α + β + c). Put ϕ1 = exp i(β + c) and ¯ and so h = exp((α + i˜ α)/2). Then h is invertible in H ∞ , ϕ = ϕ1 h/h Tϕ = T1/h¯ Tϕ1 Th . The operators T1/h¯ and Th are obviously invertible. Since β∞ < π/2, it follows that 0 ∈ conv(R(ϕ1 )). Thus Tϕ1 is invertible by Theorem 1.9. To obtain further results we make use of Zygmund’s theorem, which asserts that exp β˜ ∈ Lp whenever β is a real-valued function satisfying β∞ < π/2p (see Appendix 2.1). Corollary 2.2. Let u be a unimodular function for which Tu is invertible on H 2 . Then there exists an outer function h such that both h and ¯ 1/h belong to H p for some p > 2 and u = h/h. Proof. By Theorem 2.1, u = exp i(˜ α + β + c), where c ∈ R, α, β are real-valued functions in L∞ and β∞ < π/2. Put α ˜ β α β˜ , h2 = exp −i , h = e−ic/2 h1 h2 . h1 = exp − − i 2 2 2 2 ¯ Then u = h/h. It is also clear that h1 is invertible in H ∞ . By Zygmund’s theorem h2 and 1/h2 belong to H p for some p > 2. Let us show that the representation of a unimodular function obtained in Corollary 2.2 is unique. Lemma 2.3. Let h1 and h2 be outer functions such that h1 ∈ H 2 and ¯ 1 /h1 = h ¯ 2 /h2 , then h2 = ch1 for some c ∈ R. 1/h2 ∈ H 2 . If h −1 1 ¯ ¯ −1 Proof. Since h1 h−1 2 ∈ H , it follows from h1 h2 = h1 h2 that h1 /h2 is constant. Corollary 2.4. Suppose that under the hypotheses of Corollary 2.2 the function u has real Fourier coefficients. Then there exists an outer function h with real Fourier coefficients such that h and 1/h belong to H p for some ¯ p > 2 and u = h/h. Proof. The result follows immediately from Corollary 2.2 and Lemma 2.3. ¯ Theorem 2.5. Let u = h/h, where h is an outer function in H 2 . Then Tϕ is invertible if and only if |h|2 admits a representation |h|2 = exp(ξ + η˜), where ξ and η are real functions in L∞ and η∞ < π/2. Proof. Suppose that (2.2) holds with ξ and η as required. Then ξ η˜ η ξ˜ exp exp −i +i 2 2 2 2

(2.2)

96


is an outer function whose modulus coincides with |h| on T. So ξ η˜ ξ˜ η h = τ exp exp +i −i 2 2 2 2 for some unimodular constant τ . Therefore ˜ = exp i(η − ξ˜ + c), ¯ = τ¯ exp i(η − ξ) u = h/h τ where c ∈ R, eic = τ¯/τ . Thus u can be represented as in (2.1) and so Tu is invertible by Theorem 2.1. Suppose now that Tu is invertible. Then u admits a representation (2.1) with α, β ∈ L∞ and β∞ < π/2. Put β˜ α ˜ β α exp − + i , +i h1 = τ exp 2 2 2 2 where τ is a unimodular constant such that τ¯/τ = eic . Then h1 and h−1 1 ¯ 1 /h1 = h/h. ¯ belong to H 2 by Zygmund’s theorem and h By Lemma 2.3, h = γh1 for some constant γ. Then ˜ |h|2 = |γ|2 · |h1 |2 = exp(δ + α − β), def

where δ = log |γ|. Clearly, ξ = δ + α and η = −β satisfy the requirements. Remark. Positive functions that can be represented as exp(ξ + η˜) with ξ and η as in the statement of Theorem 2.5 are said to satisfy the Helson– Szeg¨ o condition. We shall discuss this condition in more detail in connection with prediction theory. It follows from Zygmund’s theorem that if |h|2 satisfies the Helson–Szegö condition, then h ∈ H p and 1/h ∈ H p for some p > 2. Corollary 2.6. Let u = exp i(˜ α + β), where α and β are real-valued functions in C(T). Then Tu is invertible and u can be represented as ¯ u = h/h, where h is an outer function such that h and 1/h belong to H p with any p < ∞. Moreover α ˜ + β = log |h|2 + c for some c ∈ R. Proof. Let q be a trigonometric polynomial such that β − q∞ < π/2p, where 0 < p < ∞. Then α ˜ + β = (α − q˜) + β − q + qˆ(0). So by Theorem 2.1, Tu is invertible. Let α1 = α − q˜, β1 = β − q, d = qˆ(0). Put 1 1 α1 ), h2 = exp (−β˜1 + iβ), h = e−d/2 h1 h2 . h1 = exp (α1 + i˜ 2 2 ¯ Then u = h/h. Clearly, h1 is invertible in H ∞ and by Zygmund’s theorem p h2 , h−1 2 ∈ H . It is easy to see that log |h|2 = α − β˜ − d

3. Spectra of Certain Toeplitz Operators

97

and so ˜+β+c log |h|2 = α for some c ∈ R. Corollary 2.7. If f and g are real functions in V M O such that exp if = exp ig, then f (ζ) − g(ζ) = 2kπ, ζ ∈ T, for some k ∈ Z. Proof. By Corollary 2.6 there exist outer functions h1 and h2 such that ¯ 1 /h1 , exp if = h

¯ 2 /h2 exp ig = h

−1 and h1 , h−1 ∈ H 2 . By Lemma 2.3, h1 = γh2 , γ ∈ C. It follows 1 , h2 , h2 from Corollary 2.6 that

f − g = log |h1 |2 − log |h2 |2 + const = const .

3. Spectra of Certain Toeplitz Operators In this section we find geometric descriptions of spectra of Toeplitz operators with symbols belonging to certain classes. We begin with the class of Toeplitz operators with real symbols. Theorem 3.1. Let ϕ be a real function in L∞ . Then σ(Tϕ ) = σe (Tϕ ) = [ess inf ϕ, ess sup ϕ]. Proof. It follows from Theorems 1.7 and 1.9 that σ(Tϕ ) ⊂ [ess inf ϕ, ess sup ϕ] and the endpoints of this interval belong to σ(Tϕ ). Suppose that λ ∈ (ess inf ϕ, ess sup ϕ) and Tϕ−λ is invertible. Then there exists a function f in H 2 such that Tϕ−λ f = 1. We have (Tϕ−λ f, f z n ) = (1, f z n ) = 0, Therefore

n ≥ 1.

((ϕ − λ)f, f z ) = n

T

(ϕ − λ)|f |2 z¯n dm = 0,

n ≥ 1.

Since ϕ−λ is real-valued, it follows that the Fourier coefficients of (ϕ−λ)|f |2 vanish for n = 0. Therefore (ϕ−λ)|f |2 is constant, which is impossible, since it changes sign on (ess inf ϕ, ess supp ϕ). Thus σ(Tϕ ) = [ess inf ϕ, ess sup ϕ]. Since Tϕ is self-adjoint, it is evident that σe (Tϕ ) = [ess inf ϕ, ess sup ϕ]. Let us proceed to the class of Toeplitz operators with continuous symbols and symbols of class H ∞ + C. Theorem 3.2. Let ϕ ∈ H ∞ + C. Then Tϕ is Fredholm if and only if ϕ is invertible in H ∞ + C.

98


Proof. Let ϕ, ψ ∈ H ∞ + C and ϕψ = 1. Then by (1.5) I − Tϕ Tψ = Hϕ∗¯ Hψ ,

I − Tψ Tϕ = Hψ∗¯ Hϕ .

By the Hartman theorem both operators are compact, which implies that Tϕ is Fredholm. Suppose now that Tϕ is Fredholm. Then 1/ϕ ∈ L∞ by Theorem 1.7 and so ϕ = uη, where η is an outer function invertible in H ∞ and u is a unimodular function. Clearly, Tu = Tϕ Tη−1 is Fredholm. Let n = ind Tu . Let us show that Tzn u is invertible. By Corollary 1.5, it is sufficient to show that it is Fredholm with zero index. If n ≥ 0, then Tzn u = Tu Tzn and ind Tzn u = ind Tu + ind Tzn = n − n = 0.

(3.1)

If n < 0, then Tzn u = Tzn Tu and again (3.1) holds. Thus Tzn u is invertible and so is Tz¯n u¯ = Tz∗n u . By Theorem 1.13, there exists an outer function h in H ∞ such that 1 − z n uh∞ = ¯ znu ¯ − h∞ < 1. Consequently, z n uh is invertible in H ∞ +C, and so ϕ = uh is also invertible in H ∞ + C. It is very easy now to evaluate the spectrum of a Toeplitz operator with symbol in C(T). Let ϕ be a function in C(T) that does not vanish on T. We define the winding number wind ϕ with respect to the origin in the following way. Consider a continuous branch of the argument argϕ of the function t → ϕ(eit ), t ∈ [0, 2π], i.e., ϕ(eit ) = exp(i argϕ (t)), |ϕ(eit )|

argϕ ∈ C([0, 2π]),

t ∈ [0, 2π].

Then

1 argϕ (2π) − argϕ (0) . 2π Theorem 3.3. Let ϕ ∈ C(T). Then σe (Tϕ ) = ϕ(T). For λ ∈ σe (Tϕ ) we have ind(Tϕ − λI) = − wind(ϕ − λ). def

wind ϕ =

Proof. By Theorem 3.2, the operator Tϕ − λI is Fredholm if and only if ϕ − λ is invertible in H ∞ + C. Since ϕ ∈ C(T), the last condition means that ϕ − λ does not vanish on T, which proves that σe (Tϕ ) = ϕ(T). Let λ ∈ σe (Tϕ ). Without loss of generality we can assume that λ = 0. Since ϕ does not vanish on T, the curve t → ϕ(eit ), t ∈ [0, 2π], is homotopic in C \ {0} to the curve t → eint for some integer n. Since the index depends continuously on the operator, it does not change under homotopy. It is also clear that wind ϕ = wind z n = n. Thus ind Tϕ = ind Tzn = −n = − wind z n = − wind ϕ.

Corollary 3.4. Let ϕ ∈ C(T). Then Tϕ is invertible if and only if ϕ = ef for some f ∈ C(T).


99

To obtain an invertibility criterion for Toeplitz operators with symbols in H ∞ + C, we have to define the winding number of an invertible function in H ∞ + C. Let ϕ ∈ H ∞ + C. Consider its harmonic extension to the unit disc, def which we also denote by ϕ. Given r < 1, let ϕr (ζ) = ϕ(rζ), ζ ∈ T. Clearly, ϕr = ϕ ∗ Pr , where Pr is the Poisson kernel. Theorem 3.5. Let ϕ ∈ H ∞ + C. Then ϕ is invertible in H ∞ + C if and only if there exists r0 ∈ (0, 1) such that |ϕ| is separated away from zero on the annulus {ζ : r0 < |ζ| < 1}. In this case wind ϕr is constant for r0 < r < 1. To prove the theorem we need a lemma on the asymptotic multiplicativity of the harmonic extension of functions in H ∞ + C. Lemma 3.6. Let ϕ, ψ ∈ H ∞ + C. Then lim (ϕψ)r − ϕr ψr ∞ = 0.

r→1−

Proof. Let ϕ = f + α, ψ = g + β, where f, g ∈ H ∞ , α, β ∈ C(T). Since (f g)r = fr gr , we have (ϕψ)r − ϕr ψr = (f β)r − fr βr + (gα)r − gr αr + (αβ)r − αr βr . It is obvious that (αβ)r − αr βr ∞ → 0 as r → 1, since α and β can be approximated by trigonometric polynomials and so it is sufficient to consider the case α = z n , β = z m . Then (αβ)r − αr βr ∞

= r|n+m| z m+n − r|n| z n r|m| z m ∞ = |r|n+m| − r|n|+|m| | → 0.

Let us show that (f β)r − fr βr ∞ → 0. The proof of the fact that (gα)r − gr αr ∞ → 0 is similar. It is also clear that we can consider only the case β = z¯n , n > 0, since the trigonometric polynomials are dense in C(T) and for β ∈ H ∞ the situation is trivial. Let f = q + z n h, where q is an analytic polynomial, deg q ≤ n − 1, and h ∈ H ∞ . We have (f β)r − fr βr ∞ ≤ (q¯ z n )r − qr (¯ z n )r ∞ + hr − (z n h)r (¯ z n )r ∞ . Clearly, hr − (z n h)r (¯ z n )r ∞

= hr − rn z n hr · rn z¯n ∞ = (1 − r2n )hr ∞ → 0,

r → 1,

and (q¯ z n )r −qr (¯ z n )r ∞ → 0 since q and z¯n are trigonometric polynomials. Proof of Theorem 3.5. Suppose that ϕ is invertible in H ∞ + C. Then ψ = 1/ϕ ∈ H ∞ + C and by Lemma 3.6, 1 − ϕr ψr ∞ = (ϕψ)r − ϕr ψr ∞ → 0.

100


Since ψr ∞ ≤ ψ∞ , it follows that |ϕ| is separated away from zero on {ζ : r0 < |ζ| < 1} for r0 sufficiently close to 1. Suppose now that |ϕ(ζ)| > δ > 0 for |ζ| > r0 . Then there exist a function f in H ∞ and n ∈ Z+ such that ϕ − z¯n f ∞ = z n ϕ − f ∞ < ε, where 0 < ε < δ/2. By Lemma 3.6, (z n ϕ)r − rn z n ϕr ∞ → 0. Consequently, |f (ζ)| ≥ δ − ε for |ζ| > r1 , where r0 < r1 < 1. Let f = ϑh, where ϑ is an inner function and h is an outer function. It follows that ϑ is a finite Blaschke product and h is invertible in H ∞ . Therefore f is invertible in H ∞ + C. We have z n ϕ/f − 1∞ = (z n ϕ − f )/f ∞ ≤ εf −1 ∞ ≤ ε(δ − ε)−1 < 1. Hence, z n ϕ/f is invertible in H ∞ + C and so is ϕ. The fact that wind ϕr does not depend on r follows from the fact that ϕr depends on r continuously and so wind ϕr is a continuous function on (r0 , 1). Let ϕ be an invertible function in H ∞ + C. Its winding number wind ϕ can be defined as follows. Let r0 be a number in (0, 1) such that inf{|ϕ(ζ)| : |ζ| > r0 } > 0. Then wind ϕ is by definition wind ϕr , where r is an arbitrary number in (r0 , 1). The winding number is well-defined in view of Theorem 3.5. Before we proceed to the description of the spectrum of Toeplitz operators with symbols in H ∞ + C, we compute the index of analytic Toeplitz operators. Lemma 3.7. Let ϕ ∈ H ∞ . Then σe (Tϕ ) = R(ϕ) {λ ∈ C : dim H 2 ϑλ H 2 = ∞}, (3.2) where ϑλ is the inner factor of ϕ − λ. If λ ∈ σe (Tϕ ), then ind(Tϕ − λI) = − deg ϑλ = − wind(ϕ − λ). Proof. By Theorem 3.5 to prove (3.2) it is sufficient to show that if |ϕ| is separated away from zero on T, then Tϕ is Fredholm if and only if the inner part ϑ of ϕ is a finite Blaschke product. Let ϕ = ϑh where h is outer. Since Th is invertible, it follows that Tϕ is Fredholm if and only if so is Tϑ . Since Ker Tϑ∗ = H 2 ϑH 2 , the operator Tϕ is Fredholm if and only if ϑ is a finite Blaschke product, which proves (3.2). Clearly, if Tϕ is Fredholm, then deg ϑ = dim Ker Tϑ∗ = − ind Tϕ = wind ϑ by Theorem 3.3. Now we are in a position to describe the spectrum of Toeplitz operators with H ∞ + C symbols. Theorem 3.8. Let ϕ ∈ H ∞ + C. Then for λ ∈ σe (Tϕ ) ind(Tϕ − λI) = − wind(ϕ − λ).


101

Proof. Clearly, we can assume that λ = 0. Let n = ind Tϕ . Then Tzn ϕ is invertible. Let us show that wind z n ϕ = n + wind ϕ. Indeed, by Lemma 3.6, (z n ϕ)r − r|n| z n ϕr ∞ → 0 as r → 1. Since (z n ϕ)r and r|n| z n ϕr are separated away from zero for r sufficiently close to 1, it follows that wind(z n ϕ)r = wind r|n| z n ϕr = n + wind ϕr Therefore without loss of generality we can assume that n = 0, that is, Tϕ is invertible. Let ε > 0. There exist k ∈ Z+ and f ∈ H ∞ such that ϕ − z¯k f ∞ < ε. If ε is sufficiently small, then Tz¯k f is invertible and 0 = ind Tz¯k f = ind Tf + k. By Lemma 3.7, ind Tf = − wind f . Therefore ind Tz¯k f = − wind z¯k f . It is easy to see that if ε is sufficiently small, then wind ϕ = wind z¯k f = 0, which completes the proof. Let us now introduce the important class of quasicontinuous functions QC = (H ∞ + C) (H ∞ + C). It is easy to see that QC is the maximal self-adjoint subalgebra of H ∞ + C. Lemma 3.9. QC = V M O L∞ . Proof. Let ϕ ∈ QC. Then ϕ ∈ H ∞ + C and P− ϕ ∈ P− C(T) ⊂ V M O. Similarly, P+ ϕ ∈ V M O. Now let ϕ ∈ V M O L∞ . Then ϕ = P+ f + g, where f, g ∈ C(T) (see Appendix 2.5). Since ϕ ∈ L∞ , it follows that P+ f ∈ H ∞ , so ϕ ∈ H ∞ + C. Similarly, ϕ¯ ∈ H ∞ + C. The following theorem, which describes the unimodular functions in QC, will be important in what follows. Theorem 3.10. A unimodular function u belongs to QC if and only if it admits a factorization u = z n exp i(˜ α + β),

(3.3)

where n ∈ Z, and α and β are real functions in C(T). Proof. If u satisfies (3.3), then α) exp(−α + iβ) ∈ H ∞ + C, u = z n exp(α + i˜ since H ∞ + C is an algebra (see Theorem 1.5.1). Similarly, u ¯ ∈ H ∞ + C. Suppose now that u ∈ QC, then u ¯ ∈ QC and so u is invertible in H ∞ +C. By Theorem 3.2, Tu is Fredholm. Let n = − ind Tu and v = z¯n u. Then Tv is invertible. By Theorem 1.13, there exists an outer function h in H ∞ such that v − h∞ = 1 − v¯h∞ < 1. Consequently, v¯h has a logarithm in the Banach algebra H ∞ + C. Therefore there exist an f in C(T) and a g in v h)−1 = v/h = exp(f + g). Hence, H ∞ such that (¯ v = h exp(f + g) = exp(ic + log |h| + ilog |h| + f + g) for some c ∈ R. Since v is unimodular, it follows that log |h|+Re(f +g) = O. Put α = log |h| + Re g = − Re f ∈ C(T ). Since g ∈ H ∞ , we have α ˜ = log |h| + Im g − Im g(0).

102


Putting now β = Im f + c + Im g(0) ∈ C(T), we see that u satisfies (3.3). In Chapter 7 we shall obtain similar results for many other classes of functions. We proceed now to Toeplitz operators with piecewise continuous symbols (see §1.5, where the class P C of piecewise continuous is defined). If ϕ ∈ P C, we can consider the set γ that consists of ϕ(T) and the union of the straight line intervals [ϕ(ζ − ), ϕ(ζ)], where ζ ranges over all jump points. It is easy to find a parametrization of that curve, that is, a function ϕ# in C(T) such that ϕ# (T) = γ. Thus γ = ϕ# (T) = ϕ(T) ∪ [ϕ(ζ − ), ϕ(ζ)], ζ

the union being taken over all jump points ζ. Theorem 3.11. Let ϕ ∈ P C. Then σe (Tϕ ) = ϕ# (T). If λ ∈ σe (Tϕ ), then ind(Tϕ − λI) = − wind(ϕ# − λ). Proof. It suffices to prove that Tϕ is Fredholm if and only if 0 ∈ ϕ# (T) and if Tϕ is Fredholm, then ind Tϕ = − wind ϕ# . Assume that 0 ∈ ϕ# (T). That means that ϕ does not vanish on T and the intervals [ϕ(ζ − ), ϕ(ζ)] do not contain the origin. Without loss of generality we may assume that ϕ is continuous at 1. Clearly, ϕ/|ϕ| ∈ P C and it is possible to find a branch of the argument α of ϕ (i.e., eiα = ϕ/|ϕ|) such that α ∈ P C, α is continuous at the continuity points of ϕ, except for a possible jump at 1, and |α(ζ − ) − α(ζ)| < π, In this case 1 wind ϕ# = 2π

ζ = 1.

lim α(e ) − lim α(e ) . it

t→2π−

it

t→0+

If wind ϕ# = 0, then α has no jump at 1 and so all jumps of α have moduli less than π − δ for some δ > 0. It is easy to see that α admits a representation α = ξ + η, where ξ ∈ C(T) and η∞ ≤ (π −δ)/2. Let q be a trigonometric polynomial such that ξ − q∞ < δ/4 and r = q˜. We have α = η + ξ − q − r˜ + qˆ(0). Clearly, η + ξ − q < π/2 and so by Theorem 2.1, the operator Tϕ/|ϕ| = Texp iα is invertible. z n ϕ)# has zero winding Let n = wind ϕ# = 0 and 0 ∈ ϕ# (T). Then (¯ number and as we have just proved, Tz¯n ϕ is invertible. If n > 0, then

4. Toeplitz Operators on Spaces of Vector Functions

103

Tz¯n ϕ = Tz¯n Tϕ ; if n < 0, then Tz¯n ϕ = Tϕ Tzn . In both cases it follows that Tϕ is Fredholm and 0 = ind Tz¯n f = ind Tz¯n + ind Tϕ = n + ind Tϕ , which implies that ind Tϕ = −n. Suppose now that 0 ∈ ϕ# (T). If 0 ∈ clos ϕ(T), then Tϕ is not Fredholm by Theorem 1.7. If 0 ∈ clos ϕ(T), then 0 ∈ [ϕ(ζ − ), ϕ(ζ)] for some jump point ζ in T. If Tϕ were Fredholm, then so would be any operator Tϕ−λ for sufficiently small λ. Moreover, ind Tϕ−λ = ind Tϕ for all such λ. Let us first take a very small λ0 such that λ0 belongs to only one interval of the form [ϕ(ζ − ), ϕ(ζ)]. Let ε be such a small number that the disc Dε = {ζ : |λ0 − ζ| < ε} does not intersect ϕ(T) and intersects only one interval of the form [ϕ(ζ − ), ϕ(ζ)]. Then as we have already proved, if λ1 and λ2 are in different components of Dε \ [ϕ(ζ − ), ϕ(ζ)], the operators Tϕ−λ1 and Tϕ−λ2 are Fredholm and have different indices, which contradicts the assumption that Tϕ is Fredholm.

4. Toeplitz Operators on Spaces of Vector Functions In this section we study vectorial Toeplitz operators (i.e., Toeplitz operators on spaces of vector functions). Let H and K be separable Hilbert spaces. We identify the spaces H 2 (H) and H 2 (K) with 2 (H) and 2 (K) in the usual way: f → {fˆ(j)}j≥0 .

(4.1)

We define vectorial Toeplitz operators from H 2 (H) to H 2 (K) as operators that with respect to the decompositions (4.1) have block matrices of the form {Ωj−k }j,k≥0 ,

(4.2)

where {Ωj }j∈Z is a two-sided sequence of operators from H to K. Here we obtain basic results on vectorial Toeplitz operators and characterize in the finite-dimensional case the Fredholm Toeplitz operators with symbols in H ∞ + C. In the next section we obtain a characterization of the invertible and Fredholm Toeplitz operators in terms of certain factorizations of their symbols. The following theorem characterizes the bounded vectorial Toeplitz operators. Theorem 4.1. Let {Ωj }j∈Z be a sequence of bounded linear operators from H to K. The block matrix (4.2) determines a bounded operator from

104


H 2 (H) to H 2 (K) if and only if {Ωj }j∈Z is the sequence of the Fourier coefficients of a function Φ ∈ L∞ (B(H, K)). In this case {Ωj−k }j,k≥0 = ΦL∞ (B(H,K)) . Note that, as usual, we identify operators from H 2 (H) to H 2 (K) with their block matrices with respect to the decomposition (4.1). Proof. Denote by T the operator with block matrix (4.2). Suppose that T is a bounded Toeplitz operator from H 2 (H) to H 2 (K). For each n ∈ Z we consider the operator Tn from L2 (H) to L2 (K) defined by Tn f = z¯n T P+ z n f,

f ∈ L2 (H).

Clearly, Tn is bounded and Tn ≤ T . If j ≥ −n and k ≥ −n, then (Tn z k x, z j y) = (T z n+k x, z n+j y) = (Ωj−k x, y), x ∈ H, y ∈ K. Therefore the sequence {Tn }n≥0 converges weakly to a bounded operator M from L2 (H) to L2 (K). Obviously, M ≤ T and (M z k x, z j y) = (Ωj−k x, y) for any j, k ∈ Z. It is also clear that M zf = zM f , f ∈ L2 (it suffices to verify this equality for f = z j x, j ∈ Z, x ∈ H). Let x ∈ H and let x be the constant H-valued function identically equal to x. We have (M x)(ζ) = ζ j Ω−j x. j∈Z

Clearly, M x ∈ L2 (K). We can define a measurable B(H, K)-valued function on T by ζ j Ω−j x, a.e. Φ(ζ)x = ∈Z

It is easy to see that (M f )(ζ) = Φ(ζ)f (ζ),

a.e.

for any trigonometric H-valued polynomial f . In exactly the same way as in the proof of Theorem 2.2.2 it can easily be shown that Φ ∈ L∞ (B(H, K)) and T = ΦL∞ (B(H,K)) . The converse follows immediately from the equality T f = P+ Φf , f ∈ H 2 (H), (which is sufficient to verify for f = z n x, n ≥ 0, x ∈ H). As in the scalar case an operator TΦ can be compact only if Φ = O (the proof is the same). The scalar formula (1.5) is also valid in the vectorial case: if H1 , H2 , H3 are Hilbert spaces, Ψ ∈ L∞ (B(H1 , H2 )), Φ ∈ L∞ (B(H2 , H3 )), then TΦΨ − TΦ TΨ = HΦ∗ ∗ HΨ (the proof is exactly the same). It is also easy to see that TΦ∗ = TΦ∗ ,

Φ ∈ L∞ (B(H, K)).

It follows from (4.3) that TΦΨ = TΦ TΨ

(4.3)


105

if Ψ ∈ H ∞ (B(H1 , H2 )) or Φ ∈ (H ∞ (B(H2 , H3 )))∗ (cf. Theorem 1.3). However, in contrast with the scalar case the converse does not hold. Indeed, let L be a nonzero subspace of H2 such that L = H2 . Suppose that Range Ψ(ζ) ∈ L, a.e. on T and Ker Φ(ζ) ⊂ L, a.e. on T. Then it is easy to see that TΦΨ = TΦ TΨ = O, while Ψ does not have to be in H ∞ (B(H1 , H2 )) as well as Φ does not have to be in (H ∞ (B(H2 , H3 )))∗ . In the case when the corresponding Hilbert spaces H and K are finitedimensional we can identify H with Cn , K with Cm , and B(H, K) with the space Mm,n of m × n matrices. We shall use the notation Mn for Mn,n . Note that the study of spectral problems for Toeplitz operators on spaces of vector functions is much harder than in the scalar case. In particular fundamental Theorem 1.4 does not generalize to the Toeplitz operators with matricial symbols. Indeed, let m = n = 2. Consider the matrix function Φ ∈ L∞ (M2 ) defined by ζ 0 Φ(ζ) = , ζ ∈ T. 0 ζ¯ It is easy to see that TΦ is Fredholm,

O 1 , Ker TΦ∗ = span Ker TΦ = span 1 O (recall that 1 is the function identically equal to 1), and so ind TΦ = 0 though TΦ is noninvertible. Let us obtain some general results. Theorem 4.2. Let H be a Hilbert space and let Φ ∈ L∞ (B(H)). Suppose that TΦ is Fredholm. Then Φ(ζ) is invertible almost everywhere and Φ−1 ∈ L∞ (B(H)). Proof. Assume first that TΦ is left invertible. Then there exists ε > 0 such that εf H 2 (H) ≤ TΦ f H 2 (H) ≤ Φf H 2 (H) ,

f ∈ H 2 (H).

Therefore ε¯ z n f L2 (H) ≤ Φ¯ z n f L2 (H) . Since the set {¯ z n f : f ∈ H 2 (H), n ≥ 0} is dense in L2 (H), it follows that εgL2 (H) ≤ ΦgL2 (H) ,

g ∈ L2 (H).

(4.4)

Let us show that Φ(ζ)xH ≥ εxH , x ∈ H, for almost all ζ ∈ T. Indeed, otherwise there are a positive δ, δ < ε, and a measurable subset ∆ of T of positive measure such that for each ζ ∈ ∆ there exists a nonzero x ∈ H for which Φ(ζ)xH ≤ δxH . Let χ be the characteristic function of [0, δ 2 ]. Clearly, χ((Φ(ζ))∗ Φ(ζ)) = O for ζ ∈ ∆. Denote by M the operator of multiplication by Φ∗ Φ on L2 (H). It is easy to see that χ(M ) = O. Clearly,

106


for f ∈ Range χ(M ) we have Φf L2 (H) ≤ δf L2 (H) , which contradicts (4.4). Let us get rid of the assumption that TΦ is left invertible. Since TΦ is left invertible on a subspace of finite codimension, it follows that TΦ is left invertible on z n H 2 (H) for some n ≥ 0, which is the same as saying that Tzn Φ is left invertible on H 2 (H). Therefore ζ n Φ(ζ)xH ≥ εxH , x ∈ H, a.e. on T, which is equivalent to the inequality Φ(ζ)xH ≥ εxH , x ∈ H, a.e. on T. The same reasoning applied to Φ∗ yields Φ∗ (ζ)xH ≥ κxH , x ∈ H, a.e. on T for some κ > 0. Consequently, Φ(ζ) is invertible almost everywhere on T and Φ−1 ∈ L∞ (B(H)). Note that in the finite-dimensional case for Φ ∈ L∞ (Mn ) the condition −1 Φ ∈ L∞ (Mn ) is equivalent to the condition (det Φ)−1 ∈ L∞ . As in the scalar case the Toeplitz operators with unitary-valued symbols play a special role as one can see from the following lemma. Lemma 4.3. Let H be a Hilbert space and let Φ be a function in L∞ (B(H)) such that Φ−1 ∈ L∞ (B(H)). Then Φ admits a representation Φ = U Ψ, where U is a unitary-valued function in L∞ (B(H)) and Ψ is an invertible function in H ∞ (B(H)). Proof. Consider the subspace L of L2 (H) defined by def

L = {Φg : g ∈ H 2 (H)}. It is easy to see that L is a closed subspace of L2 (H) invariant under multiplication by z. Let us show that it is completely nonreducing (see Appendix 2.3). Indeed, suppose that f is a nonzero function and z¯n f ∈ L for n ≥ 0. Then f = Φh for h ∈ H 2 (H). We can choose m > 0 such that z¯m h ∈ H 2 (H). Then z¯m f ∈ L, which contradicts our assumption. Then L has the form L = U H 2 (K), where K is a Hilbert space and U is a function in L∞ (B(K, H)) such that U (ζ) is isometric for almost all ζ ∈ T (see Appendix 2.3). Since Φ is invertible in L∞ (B(H)), we have H 2 (H) = Φ−1 U H 2 (K) and it is easy to see that U (ζ) maps K onto H for almost all ζ ∈ T. Therefore we can identify K with H after which U becomes a unitary-valued function in L∞ (B(H)). def

Put now Ψ = U ∗ Φ. Clearly, ΨH 2 (H) = H 2 (H), and so Ψ ∈ H ∞ (B(H)). The invertibility of Ψ in H ∞ (B(H)) is obvious. If Φ satisfies the conclusion of Lemma 4.3, then TΦ = TU TΨ , and so TΦ is invertible (Fredholm) if and only if TU is. As in the scalar case the following criterion holds. Theorem 4.4. Let H and K be Hilbert spaces, and let U be a function in L∞ (B(H, K)) such that U (ζ) is isometric for almost all ζ ∈ T. Then TU is a left invertible operator from H 2 (H) to H 2 (K) if and only if distL∞ (B(H,K)) {U, H ∞ (B(H, K))} < 1.


107

Proof. As in the scalar case we have U f 2L2 (K) = TU f 2H 2 (H) + HU f 2H 2 (H) , −

f ∈ H 2 (H).

It follows that TU is left invertible if and only if HU < 1. The result now follows from Theorem 2.2.2. Corollary 4.5. Let H be a Hilbert space and let U be a unitary-valued function in L∞ (B(H)). Then TU is invertible if and only if distL∞ (B(H)) {U, H ∞ (B(H))} < 1 and

distL∞ (B(H)) {U ∗ , H ∞ (B(H))} < 1.

As in §1 we can obtain now a Fredholmness criterion for Toeplitz operators with matricial unitary-valued symbols. Theorem 4.6. Let n be a positive integer and let U be a unitaryvalued function in L∞ (Mn ). The operator TU is Fredholm if and only if HU e < 1 and HU ∗ e < 1. The proof of Theorem 4.6 is exactly the same as the proof of Theorem 1.15. As in Theorem 1.15 one can also state necessary and sufficient conditions for left Fredholmness and right Fredholmness. To conclude this section we obtain a Fredholmness criterion for Toeplitz operators with symbols in (H ∞ + C)(Mn ) and compute their index. Throughout the book we use the following convention. If X is a space of functions, we denote by X(Cn ) the space of n × 1 column functions whose entries belong to X. Sometimes we shall use the notation {fj }1≤j≤n for the column function with entries fj , 1 ≤ j ≤ n. Similarly, we denote by X(Mn,m ) (or X(Mn ) if m = n) the space of m × n matrix functions with entries in X. If it does not lead to a confusion, we can write f ∈ X instead of f ∈ X(Cn ) or Φ ∈ X instead of Φ ∈ X(Mm,n ). The following theorem characterizes the Fredholm Toeplitz operators with symbols in H ∞ + C. Theorem 4.7. Let n be a positive integer and let Φ ∈ (H ∞ +C)(Mn ). The operator TΦ is Fredholm if and only if det Φ is invertible in H ∞ + C. Proof. Suppose that det Φ is invertible in H ∞ + C and Ψ = Φ−1 . Then Ψ ∈ H ∞ + C. By (4.3), I − TΦ TΨ = TΦΨ − TΦ TΨ = HΦ∗ ∗ HΨ ,

∗ I − TΨ TΦ = TΨΦ − TΨ TΦ = HΨ ∗ HΦ ,

and so it follows from the Hartman theorem that both I − TΦ TΨ and I − TΨ TΦ are compact, which means that TΦ is Fredholm. Suppose now that TΦ is Fredholm. It follows that TΦ is left invertible on z m H 2 (Cn ) for some m ≥ 0, which means that the operator Tzm Φ is left invertible. Clearly, it is sufficient to show that z m Φ is invertible in H ∞ +C. Without loss of generality we may assume that m = 0. By Theorem 4.2, Φ−1 ∈ L∞ , and so by Lemma 4.3, Φ admits a representation Φ = U Ψ, where U is a unitary-valued matrix function and Ψ is invertible in H ∞ . We have TΦ = TU TΨ . Since TΨ is invertible, it follows

108


that TU is left invertible and it is sufficient to show that U is invertible in H ∞ + C. By Theorem 4.4, there exists a function Ξ in H ∞ (Mn ) such that U − ΞL∞ < 1. Hence, I − TU ∗ Ξ < 1. It follows that TU ∗ Ξ = TU∗ TΞ is invertible. Since TU is Fredholm, it follows that TΞ is Fredholm. By Theorem 4.2, Ξ is invertible in L∞ , and so by Lemma 4.3, Ξ admits a representation Ξ = ΘG, where Θ is a unitary-valued inner function and G is invertible in H ∞ (Mn ). It remains to show that Θ is invertible in H ∞ + C. Since G is invertible in H ∞ , it follows that TG is invertible, and so TΘ = TΨ TG−1 is Fredholm. Hence, the subspace ΘH 2 (Cn ) of H 2 (Cn ) has finite codimension. By Lemma 2.5.1, Θ is a Blaschke–Potapov product of finite degree that implies that Θ is invertible in H ∞ + C. The following theorem computes the index of a Fredholm Toeplitz operator with symbol in (H ∞ + C)(Mn ) (see §3.3 for the definition of the winding number for scalar functions invertible in H ∞ + C). Theorem 4.8. Let Φ be a function in H ∞ + C(Mn ) such that TΦ is Fredholm. Then ind TΦ = − wind det Φ. Proof. We have lim distL∞ (Mn ) {Φ, z¯m H ∞ (Mn )} = 0,

m→∞

and so for any ε > 0 there exists F ∈ H ∞ (Mn ) such that Φ − z¯m F ∞ < ε. Clearly, if ε is sufficiently small, then Tz¯m F is Fredholm and ind TΦ = ind Tz¯m F . It is also clear that for a sufficiently small ε we have wind det Φ = wind det z¯m F. Hence, it is sufficient to show that ind Tz¯m F = − wind det z¯m F . Since Tz¯m F = Tz¯m In TF , we have ind Tz¯m F = ind Tz¯m In + ind TF = ind TF + mn (recall that In stands for the n × n identity matrix). It is also easy to see that wind det z¯m F = wind det F − mn. Therefore it is sufficient to show that ind TF = − wind det F . Since F is invertible in L∞ , it follows from Lemma 4.3 that F = ΘG, where Θ is an inner function in H ∞ (Mn ) and G is invertible in H ∞ (Mn ). Clearly, TG is invertible, wind det G = 0 (follows from Lemma 3.7). Hence, it suffices to show that ind TΘ = − wind det Θ. Since TΘ is Fredholm, it follows from Lemma 2.5.1 that Θ is a Blaschke– Potapov product of finite degree. Clearly, it is sufficient to consider the case when Θ has degree 1, i.e., Θ(ζ) =

λ−ζ ¯ P + (I − P ), 1 − λζ

5. Wiener–Hopf Factorizations

109

where λ ∈ D and P is an orthogonal projection on Cn of rank 1. Then by Lemma 2.5.1, ind TΘ = −1. On the other hand, it is easy to see that det Θ(ζ) =

λ−ζ ¯ , 1 − λζ

and so wind det Θ = 1.

5. Wiener–Hopf Factorizations of Symbols of Fredholm Toeplitz Operators In this section we show that the symbol of a Fredholm Toeplitz operator on H 2 (Cn ) admits a certain factorization (Wiener–Hopf factorization). This allows us to obtain a general Fredholmness criterion. With a Wiener– Hopf factorization we associate factorization indices. We obtain formulas for the dimension of the kernel of a Fredholm Toeplitz operators and for its index in terms of the factorization indices of its symbol. We also show that if the indices are arranged in the nondecreasing order, they are uniquely determined by the symbol. Let us first characterize the invertible Toeplitz operators. Theorem 5.1. Let Φ be an n×n matrix function in L∞ . The following are equivalent: (i) the Toeplitz operator TΦ is invertible; (ii) Φ admits a factorization Φ = Ψ1 Ψ2 , where Ψ1 and Ψ2 are n × n matrix functions such that both Ψ1 (ζ) and Ψ2 (ζ) are invertible for almost −1 2 all ζ ∈ T; Ψ1 , Ψ2 , Ψ−1 1 , Ψ2 belong to H (Mn ), and the operator R defined 2 n on the set of polynomials in H (C ) by Rf = Ψ−1 2 P+ Ψ1 2

−1

f

(5.1)

n

extends to a bounded operator on H (C ). −1

Note that P+ Ψ1 f is a polynomial for any polynomial f and so R maps polynomials in H 2 (Cn ) into H 2 (Cn ). Proof. Suppose that TΦ is invertible. Then there exist functions fj ∈ H 2 (Cn ) and gj ∈ H 2 (Cn ), 1 ≤ j ≤ n, such that   O  ..   .     ∗ TΦ fj = TΦ gj =   1   .   ..  O (1 is in the jth row). Consider the n × n matrix functions F = f1 · · · fn , G = g1 · · · gn .

110


Clearly, (P+ ΦF )(ζ) = (P+ Φ∗ G)(ζ) = I, Φ∗ G

a.e. on T.

∈H . It follows that ΦF ∈ H and Put Ξ = G∗ ΦF . Then Ξ = G∗ (ΦF ) ∈ H 1 since both G∗ and ΦF are in H 2 . On the other hand, Ξ = (G∗ Φ)F ∈ H 1 since both G∗ Φ = (Φ∗ G)t and F are in H 2 . Therefore Ξ is a constant function. Assume that det Ξ = 0. We have det Ξ = det G(ζ) · det Φ(ζ)F (ζ) for almost all ζ ∈ T. Since the columns g1 (ζ), · · · , gn (ζ) of G(ζ) are linearly independent a.e. on T as well as the columns f1 (ζ) · · · , fn (ζ) of F (ζ), it follows that det Φ(ζ) = 0 a.e. on T. However, this contradicts the invertibility of TΦ (see Theorem 4.2). Hence, det Ξ = 0, and so Ξ is invertible. 2

def

2

def

Put Ψ1 = ΦF , Ψ2 = F −1 . Clearly, Ψ1 Ψ2 = ΦF F −1 = Φ. We have already shown that Ψ1 ∈ H 2 . We have −1 Ψ−1 = Ξ−1 G∗ ∈ H 2 . 1 = (ΦF ) 2 Obviously, Ψ−1 2 = F ∈ H . We have

Ψ2 = F −1 = Ξ−1 G∗ Φ = Ξ−1 (Φ∗ G)∗ ∈ H 2 . −1 2 Suppose now that Ψ1 , Ψ2 , Ψ−1 1 , Ψ2 ∈ H and Φ = Ψ1 Ψ2 . It remains to show that TΦ is invertible if and only if the operator R defined by (5.1) is bounded on H 2 (Cn ). Suppose that R is bounded. In this case it is easy to see that −1 −1 Ψ2 Rf = P+ Ψ1 f for an arbitrary f ∈ H 2 (Cn ) and so P+ Ψ1 f ∈ H 1 (Cn ) −1 for any f ∈ H 2 (Cn ). Hence, Rf = Ψ−1 f for any f ∈ H 2 (Cn ). 2 P+ Ψ1 −1 Let us show that TΦ is invertible and TΦ = R. We have for a polynomial f in H 2 (Cn )

= P+ Ψ1 Ψ2 Ψ−1 2 P+ Ψ1

TΦ Rf

= P+ Ψ1 Ψ1 since obviously, P+ Ψ1 P− Ψ1

−1

−1

−1

f = P+ Ψ1 P+ Ψ1

f − P+ Ψ1 P− Ψ1

−1

−1

f

f = f,

f = O. On the other hand,

RTΦ f = Ψ−1 2 P+ Ψ1

−1

P+ Ψ1 Ψ2 f = Ψ−1 2 P+ Ψ2 f = f,

and so TΦ is invertible and TΦ−1 = R. Suppose now that TΦ is invertible. We have f = TΦ TΦ−1 f = P+ Ψ1 Ψ2 TΦ−1 f = P+ Ψ1 P+ Ψ2 TΦ−1 f. −1

Multiplying by Ψ1

−1 P+ Ψ1 f

on the left and applying P+ , we obtain −1

= P+ Ψ1 P+ Ψ1 P+ Ψ2 TΦ−1 f ∈ (P+ L1 )(Cn ).

Therefore −1

P+ Ψ1 f

−1

−1

= P+ Ψ1 Ψ1 P+ Ψ2 TΦ−1 f − P+ Ψ1 P− Ψ1 P+ Ψ2 TΦ−1 f −1

= P+ Ψ1 Ψ1 P+ Ψ2 TΦ−1 f = Ψ2 TΦ−1 f,


111

−1

since, obviously, P+ Ψ1 P− Ψ1 P+ Ψ2 TΦ−1 f = O. It follows that −1

TΦ−1 f = Ψ−1 2 P+ Ψ1 f,

f ∈ H 2 (Cn ).

−1

−1 2 n 2 n is a Hence, Ψ−1 2 P+ Ψ1 f ∈ H (C ) for any f ∈ H (C ) and R = TΦ 2 n bounded operator on H (C ). Let us now characterize the Fredholm Toeplitz operators in terms of factorizations of their symbols.

Theorem 5.2. Let Φ be an n×n matrix function in L∞ . The following are equivalent: (i) the Toeplitz operator TΦ is Fredholm; (ii) Φ admits a factorization Φ = Ψ1 ΛΨ2 , where Ψ1 and Ψ2 are n × n matrix functions such that both Ψ1 (ζ) and Ψ2 (ζ) are invertible for almost −1 2 all ζ ∈ T; Ψ1 , Ψ2 , Ψ−1 1 , Ψ2 ∈ H (Mn,n ); Λ is a diagonal matrix function of the form  k  O z 1 O ···  O z k2 · · · O    Λ= . (5.2) .. ..  , k1 , · · · , kn ∈ Z; ..  .. . . .  O

O

···

z kn

and the operator B defined on the set of polynomials in H 2 (Cn ) by Bf = Ψ−1 2 P+ ΛP+ Ψ1

−1

f

(5.3)

extends to a bounded operator on H 2 (Cn ). Factorizations of the form Φ = Ψ1 ΛΨ2 , where Ψ1 and Ψ2 are matrix functions invertible in H 2 (Mn,n ) and Λ is of the form (5.2), are called Wiener– Hopf factorizations of Φ. With such a factorization we associate Wiener– Hopf factorization indices (or simply factorization indices) k1 , k2 , · · · , kn . It is easy to see that if Φ has a Wiener–Hopf factorization, we can always find such a factorization that k1 ≤ k2 ≤ · · · ≤ kn . We show in this section that if the factorization indices are arranged in the nondecreasing order, they are uniquely determined by the function Φ itself. To prove Theorem 5.2 we start with the following lemma. Lemma 5.3. Let L be a finite-dimensional subspace of H 2 (Cn ). Then there exist a closed subspace L# of H 2 (Cn ) and analytic polynomials p1 , · · · , pn with zeros in D such that L ∩ L# = {O}, L + L# = H 2 (Cn ), and multiplication by the matrix function P ,   p1 O · · · O  O p2 · · · O    P = . .. . . ..  ,  .. . .  . O O · · · pn maps H 2 (Cn ) isomorphically onto L# .

112


Proof. Let k = dim L. We argue by induction. Let us construct subspaces L0 , L1 , · · · , Lk = L# satisfying the following properties: (1) H 2 (Cn ) = Lj + L, 0 ≤ j ≤ k; (2) dim Lj ∩ L = k − j; (3) Lj = {Pj f : f ∈ H 2 (Cn )}, where Pj is a diagonal matrix function of the form   O p1,j O · · ·  O p2,j · · · O    Pj =  . . ..  , .. ..  .. . .  O

O

···

pn,j

and p1,j , · · · , pn,j are analytic polynomials with zeros in D. Put L0 = H 2 (Cn ). Suppose that we have already constructed Lj , j < k. Let us construct Lj+1 . Let g = {gr }1≤r≤n be a nonzero element in Lj ∩ L. Then there exist m, 1 ≤ m ≤ n, and ζ0 ∈ D such that gm (ζ0 ) = 0. Put def

Lj+1 = {Pj+1 f : f ∈ H 2 (Cn )}, where  O p1,j O · · ·  O p2,j · · · O   .. .. .. .. . . . . def   Pj+1 =  O O · · · (z − ζ 0 )pm,j   . . . . .. .. ..  .. O O ··· O

··· ··· .. . ··· .. . ···

O O .. .



    . O   ..  .  pn,j

Clearly, Lj+1 ⊂ Lj . Let us show that Lj = Lj+1 + span{g}. Indeed, let g = Pj g˘, g˘ ∈ H 2 (Cn ). Let f = Pj f˘, f˘ ∈ H 2 (Cn ), be an arbitrary function in Lj . We have f = Pj (f˘ + λ˘ g − λ˘ g ) = Pj (f˘ + λ˘ g ) − λg, λ ∈ C. We can ˘ now choose λ so that the mth component of (f + λ˘ g )(ζ0 ) is zero. Clearly, Pj (f˘ + λ˘ g ) ∈ Lj+1 , while λg ∈ span{g}. It follows that dim Lj+1 ∩ L = k − j − 1. We have Lj+1 + L = Lj+1 + (span{g} + L) = (Lj+1 + span{g}) + L = Lj + L = L by the inductive hypothesis. def Doing it this way, we can construct Lk . It remains to put L# = Lk . We have to introduce several classes of functions. We denote by R the algebra of rational matrix functions {fjk }1≤j,k≤n such that the poles of the fjk are outside T. We denote by R−1 the set of invertible elements of R. We also consider the class R + H 2 consisting of n × n matrix functions F that can be represented as F = G1 + G2 , where G1 ∈ R and G2 ∈ H 2 (Mn ). It is easy to see that R · H 2 ⊂ R + H 2 , i.e., F G ∈ R + H 2 for any F ∈ R and G ∈ H 2 . We denote by (R + H 2 )−1 the set of functions invertible in R + H 2 .


113

Finally we need the class R∩H 2 , which is a subalgebra of R that consists of matrix functions in R with poles outside clos D. We denote by (R∩H 2 )−1 the set of invertible elements in R ∩ H 2 . To prove Theorem 5.2 we need the following result. Theorem 5.4. Let Ψ be a function of class (R + H 2 )−1 . Then Ψ admits a representation Ψ = Ξ1 ΛΞ2 , where Ξ1 , Ξ2 , Λ are matrix functions −1 2 2 such that Ξ1 , Ξ−1 1 ∈ H , Ξ2 , Ξ2 ∈ R ∩ H , and Λ is a diagonal matrix −1 function of the form (5.2). If Ψ ∈ R , then we can factorize Ψ so that 2 Ξ1 , Ξ−1 1 ∈R∩H . The proof of Theorem 5.4 is based on the following lemma. Lemma 5.5. Let F be an n × n matrix function in H 2 such that det F is not identically equal to 0 and det F (ζ0 ) = 0 for some ζ0 ∈ D. Then F admits a representation F = F1 W , where F1 ∈ H 2 , det F1 (ζ) = (ζ − ζ0 )−1 det F (ζ), and W is a matrix function of the form   1 0 ··· 0 ξ1 0 ··· 0  0 1 ··· 0 0 ··· 0  ξ2    .. .. . . .. .. .. . . ..    . . . . . . . .    0 0 · · · 1 ξk−1 0 · · · 0    (5.4) W (ζ) =    0 0 · · · 0 ζ − ζ0 0 · · · 0   0 0 ··· 0 0 1 ··· 0     . . . . . .. . . ..  . . .. ..  .. .. . .  . 0 0 ··· 0 0 0 ··· 1 for some k ∈ Z, 1 ≤ k ≤ n, and ξ1 , · · · , ξk−1 ∈ C. Proof of Lemma 5.5. Let f1 , · · · , fn ∈ H 2 (Cn ) be the columns of F . Since det Φ(ζ0 ) = 0, there exist k ≤ n and complex numbers ξ1 , · · · , ξk−1 such that fk (ζ0 ) = ξ1 f1 (ζ0 ) + · · · + ξk−1 fk−1 (ζ0 ). Consider the matrix function W defined by (5.4). It is easy ζ = ζ0 the matrix W (ζ) is invertible and  −ξ1 0 ··· 0 1 0 · · · 0 ζ−ζ 0  −ξ2  0 1 · · · 0 ζ−ζ 0 ··· 0 0   .. .. . . .. .. .. . . .  . . . . . .. . .   0 0 · · · 1 −ξk−1 0 · · · 0 ζ−ζ0 W −1 (ζ) =   1  0 0 ··· 0 0 ··· 0 ζ−ζ0   0 0 ··· 0 0 1 ··· 0   . . . . . . . . .. .. .. . . . ...  .. .. 0 0 ··· 0 0 0 ··· 1

(5.5) to see that for         .       

114

Chapter 3. Toeplitz Operators def

Put F1 (ζ) = F (ζ)W −1 (ζ), ζ ∈ T. Clearly, det F1 (ζ) = (ζ−ζ0 )−1 det F (ζ). Let us show that F1 ∈ H 2 . It suffices to show that the kth column of F1 belongs to H 2 (Cn ). It is easy to see that the kth column of F1 is equal to 1 (−ξ1 f1 − · · · − ξk−1 fk−1 + fk ) ∈ H 2 (Cn ) z − ζ0 by (5.5). Proof of Theorem 5.4. Obviously, there exists a scalar analytic polynomial p with zeros in D such that P Ψ ∈ H 2 , where   p O ··· O  O p ··· O    P = . . . . . . ...   .. ..  O

O ···

p

def

Let m = deg p. We have



  Ψ = (P Ψ)  

z¯m O .. .

O z¯m .. .

··· ··· .. .

O O .. .

O

O

···

z¯m

   m −1  (z P ). 

(5.6)

Clearly, P Ψ ∈ H 2 , (P Ψ)−1 ∈ R + H 2 , and (z m P −1 )±1 ∈ R + H 2 . Since Ψ ∈ (R + H 2 )−1 , it follows that det P Ψ has finitely many zeros in D. Let d be the number of zeros of det P Ψ in D (counted with multiplicities). For j = 0, · · · , d we construct by induction factorizations Ψ = Ξ1,j Λj Ξ2,j ,

(5.7)

where Ξ2,j ∈ (R ∩ H 2 )−1 , Λj is a diagonal matrix function of the form 2 (5.2), Ξ1,j ∈ H 2 , Ξ−1 1,j ∈ R + H , and det Ξ1,j has d − j zeros in D. After we do that we simply put Ξ1 = Ξ1,d , Λ = Λd , and Ξ2 = Ξ2,d . For j = 0 we take representation (5.6). Let 0 ≤ j < d and suppose that Ψ admits a representation of the form (5.7) satisfying the desired properties. Let  s  z 1 O ··· O  O z s2 · · · O    Λj =  . .. ..  . ..  .. . . .  O

O

···

z sn

Without loss of generality we may assume that s1 ≥ s2 ≥ · · · ≥ sn (otherwise we can change the order of the basis vectors in Cn and write matrix representations in the new basis). Let us apply Lemma 5.5 to F = Ξ1,j and obtain a representation 2 Ξ1,j = Ξ1,j+1 W , where W is given by (5.4), Ξ1,j+1 ∈ H 2 , Ξ−1 1,j+1 ∈ R + H , and det Ξ1,j+1 has d − j − 1 zeros in D.


We have Ψ = Ξ1,j+1 W Λj Ξ2,j . It is easy where  1 0 ··· 0 ξ1 ζ¯  0 1 ··· 0 ξ2 ζ¯   . . . ..  . . . . ... .  . .  1 ξk−1 ζ¯ def  0 0 · · · W# (ζ) =   0 0 · · · 0 1 − ζ0 ζ¯   0 0 ··· 0 0   .. .. . . .. ..  . . . . . 0 0 ··· 0 0 and



Λj+1

115

to see that W Λj = W# Λj+1 , ··· ··· .. .

0 0 .. .

0 ··· 0 ··· 1 ··· .. . . . . 0 ···

0 0 0 .. .

0 0 .. .

        ,      

ζ ∈ T,

1

z s1  ..  .   O def  =   O  O   .  ..

··· .. . ··· ··· ··· .. .

O .. .

O .. .

O .. .

z sk−1 O O .. .

O z sk +1 O .. .

O O z sk+1 .. .

··· .. . ··· ··· ··· .. .

O

···

O

O

O

···

 O ..  .   0   O  . O   ..  .  z sn

Let V be the matrix function satisfying the equation W# Λj+1 = Λj+1 V . Then V = Λ−1 j+1 W# Λj+1 . It is easy to see that       def  V (ζ) =       

0

ξ1 ζ sk −s1 ξ2 ζ sk −s2 .. .

··· ··· .. .

0 0 .. .

··· ··· ··· .. .

1 0 0 .. .

ξk−1 ζ sk −sk−1 1 − ζζ0 0 .. .

0 ···

0

0

1 0 0 1 .. .. . . 0 0 0 0 0 0 .. .. . .

··· ··· .. .

0 0 .. .

0 ··· 0 ··· 1 ··· .. . . . . 0 ···

0 0 0 .. .

0 0 .. .

       ,      

ζ ∈ T.

1 def

We have Ψ = Ξ1,j+1 Λj+1 Ξ2,j+1 , where Ξ2,j+1 = V Ξ2,j . Clearly, −1 V ∈ R ∩ H 2 and V ∈ R ∩ H 2 , and so Ξ2,j+1 ∈ (R ∩ H 2 )−1 . It is easy to see that if Ψ ∈ R−1 , then the function Ξ1 constructed above 2 satisfies Ξ1 , Ξ−1 1 ∈R∩H . Proof of Theorem 5.2. Suppose that TΦ is Fredholm. Let us first show that Φ admits a factorization of the form Ψ1 ΛΨ2 , where Ψ1 and Ψ2 are 2 matrix functions such that Ψ±1 j ∈ H , j = 1, 2, and Λ is a diagonal matrix function of the form (5.2).

116


Put L = Ker TΦ . We can apply Lemma 5.3 and find a subspace L# and a diagonal polynomial P satisfying the conclusion of Lemma matrix function 5.3. Then Ker TΦ L# = Ker TΦ P H 2 (Cn ) = {O}. Hence, Ker TΦP = {O}. Since P is a diagonal matrix whose diagonal entries can have zeros only in D, it follows that TP is Fredholm, and so TΦP = TΦ TP is a Fredholm operator with trivial kernel. ∗ = TP ∗ Φ∗ and We can proceed in the same way with the operator TΦP find a diagonal polynomial matrix function Q whose diagonal entries can have zeros only in D such that TP ∗ Φ∗ Q is a Fredholm operator with trivial kernel. Clearly, TP ∗ Φ∗ Q maps H 2 (Cn ) onto itself, and so it is invertible. Hence, TQ∗ ΦP = TP∗ ∗ Φ∗ Q is invertible. By Theorem 5.1, there exist matrix functions Υ1 and Υ2 such that ±1 2 ∗ ∗ −1 Υ±1 Υ1 Υ2 P −1 . 1 , Υ2 ∈ H and Q ΦP = Υ1 Υ2 . It follows that Φ = (Q ) −1 t −1 Consider the functions (Q∗ )−1 Υ1 = (Q ) Υ1 and P Υ2 = (Υ2 P −1 )−1 . Clearly, they belong to (R + H 2 )−1 . Therefore by Theorem 5.4 there ex±1 ∈ H 2, ist matrix functions Ξ1 , Ξ2 , Ω1 , Ω2 , Λ1 , Λ2 such that Ξ±1 1 , Ξ2 ±1 ±1 Ω1 , Ω2 ∈ R ∩ H 2 , Λ1 and Λ2 are diagonal matrix functions of the form (5.2), and (Qt )−1 Υ1 = Ξ1 Λ1 Ω1 , P Υ−1 2 = Ξ2 Λ2 Ω2 . Therefore −1 −1 Φ = (Q∗ )−1 Υ1 Υ2 P −1 = Ξ1 Λ1 Ω1 Ω−1 2 Λ2 Ξ2 . −1 It is easy to see that Λ1 Ω1 Ω−1 ∈ R−1 . Applying Theorem 5.4 to 2 Λ2 −1 −1 Λ1 Ω1 Ω−1 , we can find matrix functions ∆1 , Λ, ∆2 such that 2 Λ2 ±1 2 2 ∆±1 1 ∈ R ∩ H , ∆2 ∈ R ∩ H , Λ is of the form (5.2), and −1 −1 −1 −1 Λ1 Ω1 Ω−1 = ∆−1 ∆1 . 2 Λ2 2 Λ

We now have Φ = Ψ1 ΛΨ2 ,

(5.8)

where Ψ1 = Ξ1 ∆1 and Ψ2 = ∆2 Ξ−1 2 . Clearly, Ψ1 and Ψ2 satisfy the requirements of Theorem 5.2. It remains to prove that if Φ ∈ L∞ (Mn,n ) satisfies (5.8) with Ψ1 ,Ψ2 , and Λ are as above, then TΦ is Fredholm if and only if the operator B defined by (5.3) extends to a bounded operator on H 2 . We can represent Λ in the form Λ = Λ− Λ+ , where  d1  m   z¯ O ··· O z 1 O ··· O  O z¯d2 · · ·  O z m2 · · · O  O      = , Λ Λ−=  .  . . ..  , . . +  . . .. .. .. .. ..   ..  .. .  (5.9) mn dn O O ··· z O O · · · z¯ where d1 , · · · , dn ∈ Z+ and m1 , · · · , mn ∈ Z+ , and min{dj , mj } = 0,

1 ≤ j ≤ n.


117

Clearly, kj = mj − dj (see (5.2)). It is easy to see that Λ− Λ+ = Λ+ Λ− . Now we are in a position to complete the proof of Theorem 5.2. −1 Suppose that B is bounded. It is easy to see that Ψ2 Bf = P+ ΛP+ Ψ1 f −1 for any f ∈ H 2 (Cn ). It follows that P+ ΛP+ Ψ1 f ∈ H 1 (Cn ) for any −1 2 n f ∈ H 2 (Cn ). Hence, Bf = Ψ−1 2 P+ ΛP+ Ψ1 f for any f ∈ H (C ). ∗ Let us show that the operators BTΦ − I and B TΦ∗ − I have finite rank, which would imply that TΦ is Fredholm. 2 n Let p be a polynomial in H 2 (Cn ) and f = Ψ−1 2 p ∈ H (C ). We have BTΦ f

−1

=

Ψ−1 2 P+ ΛP+ Ψ1 P+ Ψ1 ΛΨ2 f

=

−1 Ψ−1 2 P+ ΛP+ ΛΨ2 f = Ψ2 P+ ΛP+ Λp.

Consider the linear manifold def

L = {p = {pj }1≤j≤n : gj ∈ P A , z¯dj pj ∈ H 2 } and the subspace def

2 }. M = {p = {pj }1≤j≤n : gj ∈ P A , z¯dj pj ∈ H−

(5.10)

Clearly, dim M = d1 + · · · + dn . Obviously, Λp ∈ H (C ) for any p ∈ L, −1 and so BTΦ Ψ−1 2 p = Ψ2 p for p ∈ L. It follows that BTΦ f = f for any −1 f ∈ clos Ψ2 L. We need the following simple fact. Lemma 5.6. Let Ψ be an n×n function on T such that Ψ ∈ H 2 (Mn,n ) and Ψ−1 ∈ H 2 (Mn,n ). Then Ψ is an outer function, i.e., the set 2

n

def

K = {Ψp : p = {pj }1≤j≤n : pj ∈ P A } is dense in H 2 (Cn ). Proof. Suppose that ϕ ∈ H 2 (Cn ) and ϕ ⊥ K. We have

∗ (Ψ∗ (ζ)ϕ(ζ)) p(ζ)dm(ζ) 0 = (Ψp, ϕ) = T 1 for any polynomial p ∈ H 2 (Cn ). It follows that Ψ∗ ϕ ∈ H− (Cn ). Since 2/3 Ψ−1 ∈ H 2 (Mn,n ), it follows that ϕ = (Ψ∗ )−1 Ψ∗ ϕ ∈ H− (Cn ). Since 2 n 2 n ϕ ∈ L (C ), we have ϕ ∈ H− (C ), which is possible only if ϕ = O. Now we are able to complete the proof of Theorem 5.2. 2 n By Lemma 5.6, Ψ−1 2 (L + M) is dense in H (C ). Since BTΦ f = f on −1 −1 clos Ψ2 L and dim Ψ2 M = d1 + · · · + dn , it follows that

rank(BTΦ − I) ≤ d1 + · · · + dn < ∞. Consider now the operator B ∗ TΦ∗ − I. It is easy to see that B ∗ f = (Ψt1 )−1 P+ ΛP+ (Ψ∗2 )−1 f for any polynomial f . Now the proof of the fact that rank(B ∗ TΦ∗ − I) < ∞ is exactly the same as for the operator BTΦ − I.

118


Finally, suppose that TΦ is Fredholm. Let us show that the operator B defined on the set of polynomials by (5.3) extends to a bounded operator on H 2 (Cn ). Denote by X the space {f = ϕ + P+ ψ : ϕ, ψ ∈ L1 }. We can endow X with the natural norm f X = inf{ϕL1 + ψL1 : f = ϕ + P+ ψ}. Obviously, P+ is a bounded operator on X. Define the operator A on H 2 (Cn ) by −1 Af = P+ ΛP+ Ψ1 f. Clearly, Af = Ψ2 Bf if f is a polynomial in H 2 (Cn ). It is easy to see that A maps H 2 (Cn ) to X(Cn ). We have −1

ATΦ f = P+ ΛP+ Ψ1 P+ Ψ1 ΛΨ2 f = P+ ΛP+ ΛΨ2 f. It is easy to see that

ATΦ Ψ−1 2 p = p,

p ∈ L,

and so ATΦ f = Ψ2 f,

f ∈ clos Ψ−1 2 L.

(5.11)

We can define now Bf for each function f ∈ H 2 (Cn ) by Bf = Ψ−1 2 Af . It follows from (5.11) that BTΦ f = f for any f ∈ clos Ψ−1 L. Since TΦ is 2 2 n Fredholm, it follows that B maps the subspace TΦ clos Ψ−1 L into H (C ). 2 −1 By Lemma 5.6, TΦ clos Ψ2 L has finite codimension. −1 As we have already explained, P+ Ψ1 f is a polynomial for any polynomial f , and so B maps analytic vector polynomials into H 2 (Cn ). Let P m be the space of polynomials {pj }1≤j≤n such that deg pj ≤ m. Since −1 2 n codim TΦ clos Ψ−1 2 L < ∞, it follows that P m + TΦ clos Ψ2 L = H (C ) for 2 n some m ∈ Z+ . Hence, B is a bounded operator on H (C ). Let us now describe the kernel of TΦ end evaluate ind TΦ . Theorem 5.7. Ker TΦ = Ψ−1 2 M, where M is defined by (5.10). Proof of Theorem 5.7. It is easy to see that Ψ−1 2 M ⊂ Ker TΦ . Sup2 pose now that f ∈ Ker TΦ . This means that Ψ1 Λ− Λ+ Ψ2 f ∈ H− (Cn ). −1 2 1 n Since Ψ1 ∈ H (Mn,n ), it follows that Λ− Λ+ Ψ2 f ∈ H− (C ). Hence, Λ+ Ψ2 f = {pj }1≤j≤n , where pj is a polynomial, deg pj < dj if dj > 0 and pj = O if dj = 0. Since min{dj , mj } = 0, 1 ≤ j ≤ n, it follows that Ψ2 f = {pj }1≤j≤n which implies the result. Corollary 5.8. dim Ker TΦ = d1 + · · · + dn ,

ind TΦ = −(k1 + · · · + kn ).

Proof. The first equality follows immediately from Theorem 5.7. If we apply it to TΦ∗ = TΦ∗ , we find that dim Ker TΦ∗ = m1 + · · · + mn which implies the result.

6. Left Invertibility of Bounded Analytic Matrix Functions

119

Corollary 5.9. Suppose that under the above assumptions TΦ is Fredholm. Then TΦ is invertible if and only if Λ is identically equal to I. We conclude this section with the following result. Theorem 5.10. Let Φ ∈ L∞ (Mn,n ) be the symbol of a Fredholm Toeplitz operator on H 2 (Cn ). Suppose that k1 ≤ k2 ≤ · · · ≤ kn are the factorization indices of a Wiener–Hopf factorization of Φ. Then the indices kj , 1 ≤ j ≤ n, are uniquely determined by the function Φ itself. Proof. Let m ∈ Z. Consider the function z m Φ. Clearly, Tzm Φ is Fredholm and z m Φ has a Wiener–Hopf factorization with indices m + k1 , m + k2 , · · · , m + kn . Consider the sequence {Nm }m∈Z defined by Nm = dim Ker Tzm Φ . By Corollary 5.8, Nm = 0,

m ≥ −k1 ,

and N−k1 −1 > 0.

This uniquely determines k1 . Let ν1 = N−k1 −1 . It follows from Corollary 5.8 that ν1 is the number of indices equal to k1 . We have k1 = · · · = kν1 and if ν1 < n, then kν1 +1 > k1 , and so k1 , · · · , kν1 are uniquely determined by the sequence {Nm }m∈Z . It follows from Corollary 5.8 that Nm = −ν1 (k1 + m),

−kν1 +1 ≤ m < −k1 ,

and N−kν1 +1 −1 > −ν1 (k1 − kν1 +1 − 1). This uniquely determines kν1 +1 . Put ν2 = N−kν1 +1 −1 + ν1 (k1 − kν1 +1 − 1). It follows easily from Corollary 5.8 that ν2 is the number of indices equal to kν1 +1 . Clearly, we can continue this process and determine all indices kj , 1 ≤ j ≤ n, from the sequence {Nm }m∈Z .

6. Left Invertibility of Bounded Analytic Matrix Functions The famous Carleson corona theorem (see Appendix 2.1) says that if Ω is a function in H ∞ (Cn ) such that inf Ω(ζ)Cn > 0,

ζ∈D

(6.1)

then there exists a function Ξ in H ∞ (Cn ) such that Ξt (ζ)Ω(ζ) = 1 for all ζ ∈ D. Later this result was generalized to matrix functions; however, it

120


turns out that it does not generalize to the case of infinite matrix functions (see Concluding Remarks). We consider here another condition on a bounded analytic matrix function which is necessary and sufficient for left invertibility. Instead of (6.1) we assume that the Toeplitz operator TΩ is left invertible. Unlike condition (6.1), it also works in the case of infinite matrix functions. In this section we deal with functions taking values in the space Mm,n of m × n matrices. We identify Mm,n with the space of linear operators from Cn to Cm and equip Mm,n with the operator norm. We also admit the case of infinite matrix functions: if n = ∞, Cn is just the sequence space 2 . As usual, In stands for the identity matrix in Mn,n . The following theorem gives a necessary and sufficient condition for the left invertibility of a bonded analytic matrix function. The main tool in the proof is the commutant lifting theorem. Theorem 6.1. Let m and n be positive integers or equal to ∞, and K > 0. Let Ω be a function in H ∞ (Mm,n ). The following are equivalent: (i) there exists a function Ξ in H ∞ (Mn,m ) such that Ξ(ζ)Ω(ζ) = In ,

ζ ∈ D,

and

ΞH ∞ (Mn,m ) ≤ K;

(6.2)

(ii) the Toeplitz operator TΩ satisfies TΩ f H 2 (Cm ) ≥ K −1 f H 2 (Cn ) ,

f ∈ H 2 (Cn ).

(6.3)

Proof. Suppose first that (6.2) holds for a function Ξ in H ∞ (Mn,m ). We have TΞ TΩ = TΞΩ = I. Since TΞ = ΞH ∞ (Mn,m ) ≤ K, it follows that TΩ is left invertible and (6.3) holds. Conversely, assume that (6.3) holds. Consider the Toeplitz operator TΩ : H 2 (Cn ) → H 2 (Cm ). Clearly, ∗ TΩ = TΩ Sn∗ , Sm

(6.4)

where Sn and Sm are multiplications by z on H 2 (Cn ) and H 2 (Cm ). Let L be the range of TΩ . It follows from (6.3) that L is a closed subspace of H 2 (Cm ) and there exists a bounded linear operator Q : L → H 2 (Cn ) such that QTΩ = I and Q ≤ K. It follows easily from (6.4) that L is an ∗ invariant subspace of Sm and ∗ L = Sn∗ Q, QSm and so

(PL Sm L)Q∗ = Q∗ Sn ,

(6.5)

where PL is the orthogonal projection onto L. The operator Sn is isometric on H 2 (Cn). It is easy to see that the operator Sm is an isometric dilation of PL Sm L. The operator Q∗ intertwines Sn and PL Sm L. By the commutant lifting theorem (see Appendix 1.5),

Concluding Remarks

121

there exists a bounded linear operator R from H 2 (Cn ) to H 2 (Cm ) such that R ≤ K, Q∗ = PL RL, (6.6) and Sm R = RSn .

(6.7)

It follows easily from (6.7) that R has a block Toeplitz matrix. By Theorem 4.1, there exists a matrix function Ψ ∈ L∞ (Mm,n ) such that ΨL∞ ≤ K and R = TΨ . It also follows from (6.7) that Rz k H 2 (Cn ) ⊂ z k H 2 (Cm ), ∗ k ∈ Z+ . Hence, Ψ ∈ H ∞ (Mm,n ). Since L is an invariant subspace of Sm , ∗ ∗ it follows that L is also an invariant subspace of R = TΨ , and it follows from (6.6) that Q = PL R∗ L = R∗ L. Therefore I = QTΩ = R∗ TΩ = TΨ∗ TΩ . Hence, Ψ∗ Ω = In . It remains to put Ξ = Ψt . Theorem 6.1 admits an “invariant” reformulation. We need the following notation. If H and K are separable Hilbert spaces and Φ ∈ L∞ (B(H, K)), we denote by Ψ# the function in L∞ (B(H, K)) defined by ¯ Ψ# (ζ) = Ψ(ζ),

ζ ∈ T.

Theorem 6.2. Let H and K be separable Hilbert spaces, K > 0, and let Ω ∈ H ∞ (B(H, K)). The following are equivalent: (i) there exists a function Ξ in H ∞ (B(K, H)) such that Ξ(ζ)Ω(ζ) = I,

ζ ∈ D,

and

ΞH ∞ (B(K,H) ≤ K;

(ii) the Toeplitz operator TΩ# satisfies TΩ# f H 2 (K) ≥ K −1 f H 2 (H) ,

f ∈ H 2 (H).

It is easy to see that Theorem 6.2 is equivalent to Theorem 6.1.

Concluding Remarks The systematic study of spectral properties of Toeplitz operators was started in the paper Brown and Halmos [1] in which Theorem 1.1 was obtained. Theorem 1.4 appeared in Coburn [1]. Theorem 1.6 is due to Wintner [1]. Theorem 1.7 can be found in Douglas [2]; see also Hartman and Wintner [1], where Corollary 1.8 was established. Theorem 1.9 is due to Brown and Halmos [1]. Theorems 1.11–1.13 were found by Widom [1] and Devinatz [1]. Theorem 1.14 was proved by N.K. Nikol’skii [2]. Finally, Theorem 1.15 is taken from Douglas and Sarason [1].

122


The fundamental invertibility criterion, Theorem 2.1, was found in Widom [1] and Devinatz [1]. The remaining results of §2 are well-known; see Peller and Khrushchëv [1]. Theorem 3.1 was found in Hartman and Wintner [1]. Theorem 3.3 was discovered independently by many authors; see Gohberg [1], Mikhlin [1]. Theorems 3.2 and 3.5 are due to Douglas, see Douglas [2]. Theorem 3.10 was obtained in Sarason [3]. Theorem 3.11 was proved in Widom [1] in the case of finitely many jumps and in Devinatz [1] in the general case. Let us also mention the paper Krein [2] in which the theory of Toeplitz operators whose symbols have absolutely convergent Fourier series was developed. Theorems 4.1, 4.2, 4.4, and 4.6 are straightforward generalizations of the corresponding results in the scalar case. Theorem 4.8 is due to Douglas [3]. Theorems 5.1 and 5.2 were obtained in Simonenko [2]; see also Litvinchuk and Spitkovskii [1]. Theorems 5.7 and 5.10 are well-known. Theorem 6.1 was proved in Arveson [3] (for m = 1), Schubert [1], and Sz.-Nagy and Foias [2]. Note that in Fuhrmann [1] the following matricial corona theorem was proved: if Ω is a matrix function in H ∞ (Mm,n ) such that Ω(ζ)xCm ≥ δxCn for any ζ ∈ D and x ∈ Cn , then Ω is left invertible in the space of bounded analytic matrix functions. Vasyunin (see Tolokonnikov [1]) generalized this result to the case n = ∞ (in the case m = 1 and n = ∞ this was done by Tolokonnikov [1] and Rosenblum [4]). However, Treil [5] showed that these results do not generalize to the case of an arbitrary bounded analytic operator function. We also mention here the important and fruitful local theory of Toeplitz operator developed by Simonenko [1]. Another version of local theory was developed by Douglas [2]. A remarkable result by Widom [2] shows that the spectrum of Toeplitz operator is always connected; see also Douglas [2], where it was shown that the essential spectrum of a Toepliz operator is always connected. The spectral structure of self-adjoint Toeplitz operator was completely described in Ismagilov [1] and Rosenblum [3]. Earlier Rosenblum [2] showed that a self-adjoint Toeplitz operator must have absolutely continuous spectral measure. In Peller [7], [14], and [21] estimates of resolvents of Toeplitz operators for certain classes of symbols were obtained. Based on those estimates the existence of nontrivial invariant subspaces as well as some similarity results were found there. Note, however, that Treil [4] has shown that the resolvent of a Toeplitz operator with continuous symbol can grow arbitrarily fast. Interesting similarity theorems were obtained in Clark [4] and Yakubovich [1] for certain classes of Toeplitz operators. Let us mention here the classical Riemann–Hilbert problem. Given func2 tions g, h ∈ L2 , find functions f+ ∈ H 2 and f− ∈ H− such that

f+ = gf− + h.

Concluding Remarks

123

It is easy to see that the study of the Riemann–Hilbert problem is equivalent to the study of equations involving Toeplitz operators. Note that a complete solution of the Riemann–Hilbert problem with data in Λα was obtained for the first time by Gakhov [1]. Finally, we mention the following books and survey articles on Toeplitz operators: Douglas [2], [3], B¨ otcher and Silbermann [1], Nikol’skii [2]-[4], Peller and Khrushchëv [1], and Litvinchuk and Spitkovskii [1].

4 Singular Values of Hankel Operators

In this chapter we study singular values of Hankel operators. The main result of the chapter is the fundamental theorem of Adamyan, Arov, and Krein. This theorem says that if Γ is a Hankel operator, then to evaluate the nth singular value sn (Γ) of Γ, there is no need to consider all operators of rank at most n, sn (Γ) is the distance from Γ to the set of Hankel operators of rank at most n. In §1 we prove the Adamyan–Arov–Krein theorem in the special case when sn (Γ) is greater than the essential norm of Γ. In §2 we reduce the general case to the case treated in §1. In §1 we also prove the uniqueness of the corresponding Hankel approximant of rank at most n under the same condition sn (Γ) > Γe , and we obtain useful formulas for multiplicities of singular values of related Hankel operators. We prove a generalization of the Adamyan–Arov–Krein theorem to the case of vectorial Hankel operators in §3. We also obtain in §3 a formula for the essential norm of vectorial Hankel operators. Finally, in §4 we consider relationships between the singular values of Hu and Hu¯ for unimodular functions u. We obtain a very useful formula that relates these two Hankel operators which leads to interesting results about the singular values of Hu and Hu¯ . We also obtain in §4 similar results for vectorial Hankel operators.

126

Chapter 4. Singular Values of Hankel Operators

1. The Adamyan–Arov–Krein Theorem Recall that for a bounded linear operator T from a Hilbert space H1 to a Hilbert space H2 the singular values sm (T ), m ∈ Z+ , are defined by sm (T ) = inf{T − R : rank R ≤ m}.

(1.1)

Clearly, s0 (T ) = T and sm+1 (T ) ≤ sm (T ). Put s∞ (T ) = lim sm (T ). m→∞

It is easy to see that s∞ (T ) = T e , the essential norm of T . Clearly, T is compact if and only if s∞ (T ) = 0 (see Appendix 1.1). The main aim of this and the next section is to prove that in the case of a Hankel operator T to find sm (T ) we can consider the infimum in (1.1) over only the Hankel operators of rank at most m. 2 , m ≥ 0. Theorem 1.1. Let Γ be a Hankel operator from H 2 to H− Then there exists a Hankel operator Γm of rank at most m such that Γ − Γm = sm (Γ).

(1.2)

Since by Kronecker’s theorem rank Γm ≤ m if and only if Γm has a rational symbol of degree at most m, Theorem 1.1 admits the following ∞ be the set of functions f in L∞ such that P− f is a reformulation. Let H(m) ∞ rational function of degree at most m. Clearly, H(m) can be identified with the set of meromorphic functions in D bounded near T and having at most m poles in D counted with multiplicities. Theorem 1.2. Let ϕ ∈ L∞ , m ∈ Z+ . Then there exists a function ψ ∞ in H(m) such that ϕ − ψ∞ = sm (Hϕ ).

(1.3)

Note that the problem of approximation of an L∞ function on T by ∞ is called the Nehari–Takagi problem. functions in H(m) We shall consider in this section the case when sm (Γ) > s∞ (Γ). The proof in the case sm (Γ) = s∞ (Γ) will be given in the next section. It turns out that in the case sm (Γ) > s∞ (Γ) there exists a unique Hankel operator Γm of rank at most m that satisfies (1.2). In other words, the following theorem holds. Theorem 1.3. Let ϕ ∈ L∞ , m ∈ Z+ . If sm (Hϕ ) > s∞ (Hϕ ), then ∞ there exists a unique ψ in H(m) that satisfies (1.3). Definition. Let T be a bounded linear operator from a Hilbert space H1 to a Hilbert space H2 . If s is a singular value of T and s > s∞ (T ), consider the subspaces Es(+) = {x ∈ H1 : T ∗ T x = s2 x}, (+)

Es(−) = {y ∈ H2 : T T ∗ y = s2 y}.

are called Schmidt vectors of T (or, more precisely, Vectors in Es (−) s-Schmidt vectors of T ). Vectors in Es are called Schmidt vectors of

1. The Adamyan–Arov–Krein Theorem

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T ∗ (s-Schmidt vectors of T ∗ ). Clearly, x ∈ Es if and only if T x ∈ Es . A pair {x, y}, x ∈ H1 , y ∈ H2 , is called a Schmidt pair of T (s-Schmidt pair) if T x = sy and T ∗ y = sx. (+)

(−)

Proof of Theorem 1.1 in the case sm (Γ) > s∞ (Γ). Put s = sm (Γ). If s = Γ, the result is trivial. Assume that s < Γ. Then there exist positive integers k, µ such that k ≤ m ≤ k + µ − 1 and sk−1 (Γ) > sk (Γ) = · · · = sk+µ−1 (Γ) > sk+µ (Γ).

(1.4)

Clearly, it suffices to consider the case m = k. Lemma 1.4. Let {ξ1 , η1 } and {ξ2 , η2 } be s-Schmidt pairs of Γ. Then ξ1 ξ¯2 = η1 η¯2 . To prove the lemma we need the following identity (see formula (1.1.13)): P− (z n Γf ) = Γz n f,

n ∈ Z+ .

(1.5)

Proof of Lemma 1.4. Let n ∈ Z+ . We have ¯ ξ) 1 ξ2 (−n)

1 n 1 (z ξ1 , Γ∗ η2 ) = (Γz n ξ1 , η2 ) s s

=

(z n ξ1 , ξ2 ) =

=

1 (P− z n Γξ1 , η2 ) = (z n η1 , η2 ) = η) 1 η¯2 (−n) s

¯ ¯ by (1.5). Similarly, ξ) ¯2 . 1 ξ2 (n) = η) 1 η¯2 (n), n ∈ Z+ , which implies ξ1 ξ2 = η1 η Corollary 1.5. Let {ξ, η} be an s-Schmidt pair of Γ. Then the function η ϕs = (1.6) ξ is unimodular and does not depend on the choice of {ξ, η}. Proof. Let ξ1 = ξ2 = ξ, η1 = η2 = η in Lemma 1.4. It follows that |ξ|2 = |η|2 and so η/ξ is unimodular for any Schmidt pair {ξ, η}. Let {ξ1 , η1 } and {ξ2 , η2 } be s-Schmidt pairs of Γ. By Lemma 1.4, η1 /ξ1 = ξ¯2 /¯ η2 . Since η2 /ξ2 is unimodular, it follows that η1 /ξ1 = η2 /ξ2 . We resume the proof of Theorem 1.1. Put Γs = Hsϕs , where ϕs is defined by (1.6). Clearly, Γs ≤ s. The result will be established if we show that rank(Γ − Γs ) ≤ k. Let {ξ, η} be an s-Schmidt pair of Γ. Let us show that it is also an s-Schmidt pair of Γs . Indeed η Γs ξ = sP− ξ = sη, ξ

ξ Γ∗s η = sP+ η = sξ. η

Let E+ = {ξ ∈ H 2 : Γ∗ Γξ = s2 ξ},

2 E− = {η ∈ H− : ΓΓ∗ η = s2 η}

128


be the spaces of Schmidt vectors of Γ and Γ∗ . It is easy to see that dim E+ = dim E− = µ. It follows easily from if Γξ = Γs ξ, then Γz n ξ = Γs z n ξ for (1.5) that any n ∈ Z+ . Since Γs E+ = Γ E+ , it follows that Γ and Γs coincide on the S-invariant subspace spanned by E+ , where S is multiplication by z on H 2 . By Beurling’s theorem this subspace has the form ϑH 2 , where ϑ is an inner function (see Appendix 2.2). Denote by Θ multiplication by ϑ. We have ΓΘ = Γs Θ. The proof will be completed if we show that dim(H 2 ϑH 2 ) ≤ k. Put d = dim(H 2 ϑH 2 ). Lemma 1.6. The singular value s of the operator ΓΘ has multiplicity at least d + µ. Note that sj (ΓΘ) ≤ sj (Γ)Θ = sj (Γ) and so s∞ (ΓΘ) < s. It will follow from Lemma 1.6 that d < ∞. Proof of Lemma 1.6. Let τ be an inner divisor of ϑ (i.e., ϑτ −1 ∈ H ∞ ). Let us show that for any ξ ∈ E+ (Γs Θ)∗ (Γs Θ)¯ τ ξ = s2 τ¯ξ ∈ E+ .

(1.7)

Indeed it is easy to see that Γ∗ z¯f¯ = z¯Γf for any f ∈ H 2 . Let J be the transformation on L2 defined by Jf = z¯f¯. It follows that J maps E+ onto ¯ 2. E− . Since E+ ⊂ ϑH 2 , we have E− ⊂ ϑH − −1 ¯ ∗ , where Let ξ ∈ E+ , η = s Γξ ∈ E− . We can represent η as η = ϑη 2 η∗ ∈ H− . We have (Γs Θ)∗ (Γs Θ)¯ τξ

η τ ξ = s(Γs Θ)∗ P− η∗ τ¯ (Γs Θ)∗ sP− ϑ¯ ξ ξ¯ ¯ = s2 τ¯ξ, = s(Γs Θ)∗ η∗ τ¯ = s2 P+ ϑη ∗τ η =

which proves (1.7). Since d = dim(H 2 ϑH 2 ), it follows that for any n < d we can find inner divisors {ϑj }1≤j≤n+1 of ϑ such that ϑn+1 = ϑ, ϑj+1 ϑ−1 ∈ H ∞, j ϑj+1 ϑ−1 j = const, and ϑ1 = const (see Appendix 2.2). Then it follows from (1.7) that the subspace Ej = span{E+ , ϑ¯1 E+ , · · · ϑ¯j E+ },

1 ≤ j ≤ n + 1,

∗

consists of eigenvectors of (ΓΘ) (ΓΘ) corresponding to the eigenvalue s2 . Clearly, E1 \ E+ = ∅ and Ej+1 \ Ej = ∅, 1 ≤ j ≤ n. Therefore dim Ker (ΓΘ)∗ ΓΘ − s2 I ≥ dim En+1 ≥ µ + n + 1. Therefore the left-hand side is equal to ∞ if d = ∞ and is at least µ + d if d < ∞. We can now complete the proof of Theorem 1.1. As we have already observed, sj (ΓΘ) ≤ sj (Γ). Thus if d = ∞, then it follows from Lemma 1.6 that s is a singular value of Γ of infinite multiplicity, and so d < ∞. Again,

1. The Adamyan–Arov–Krein Theorem

129

by Lemma 1.6 we have sk+µ (Γ) < sk+µ−1 (Γ) = · · · = sk (Γ) = sd+µ−1 (ΓΘ) ≤ sd+µ−1 (Γ). Therefore d + µ − 1 < k + µ and so d ≤ k, which completes the proof of Theorem 1.1 in the case sm (Γ) > s∞ (Γ). Remark. Note that d = k. For otherwise if d < k, then it would follow that sk−1 (Γ) = s, which contradicts (1.4). Let Γ = Hϕ , where ϕ ∈ L∞ . Assuming (1.4) and s = sk (Hϕ ), we can consider the function 1 Γξ , (1.8) u = ϕs = · s ξ where ξ ∈ E+ , that is, Hϕ∗ Hϕ ξ = s2 ξ. Then u is a unimodular function and sk (Hϕ ) = ϕ − ψ∞ , where ψ = ϕ − su.

(1.9)

It has been shown in the proof of Theorem 1.1 that rank Γψ ≤ k. Since sk−1 (Hϕ ) > sk (Hϕ ), it follows that rank Γψ = k. Theorem 1.7. Let Γ = Hϕ be a Hankel operator satisfying (1.4) and let u be the unimodular function defined by (1.8). Then dim Ker Tu = 2k + µ. Remark. Clearly, dim Ker Tu is the multiplicity of the singular value 1 of Hu . Proof. Let us first show that dim Ker Tu ≥ 2k + µ. Let ϑ be the greatest common divisor of functions in E+ . Let us prove that for any inner divisor τ of ϑ both τ E+ and τ¯E+ are contained in Ker Tu . Let ξ ∈ E+ . Then Γξ η u= = . sξ ξ Therefore Tu τ ξ = P+ τ η = O, 2 . We since it has been shown in the proof of Theorem 1.1 that ϑE− ⊂ H− have Tu τ¯ξ = P+ τ¯η = O. ∞ Let {ϑj }1≤j≤k be inner divisors of ϑ such that ϑ1 = const, ϑ−1 j ϑj+1 ∈ H , −1 ϑj ϑj+1 = const, 1 ≤ j ≤ k − 1, and ϑk = ϑ. Consider the subspaces (1.10) ϑ¯k E+ , ϑ¯k−1 E+ , · · · , ϑ¯1 E+ , E+ , ϑ1 E+ , · · · , ϑk E+ . Clearly, dim ϑ¯k E+ = dim E+ = µ. Let Fj be the span of the j leftmost subspaces in (1.10), 1 ≤ j ≤ 2k + 1. Then Fj+1 \ Fj = ∅, 1 ≤ j ≤ 2k. It follows that dim F2n+1 ≥ 2k + µ and F2k+µ ⊂ Ker Tu .

130


Suppose now that dim Ker Tu > 2k + µ. It follows that 1 is a singular value of Hu of multiplicity greater than 2k + µ. Therefore s is a singular value of Hϕ−ψ of multiplicity greater than 2k + µ. We have s = s2k+µ (Hϕ−ψ ) ≤ sk+µ (Hϕ ) + sk (Hψ ) (see Appendix 1.1). Since rank Hψ = k, it follows that sk (Hψ ) = 0 and so sk+µ (Hϕ ) ≥ s, which contradicts (1.4). To prove uniqueness, we need the following lemma. Lemma 1.8. Let u be a unimodular function such that ∞ N = dim Ker Tu > 0. Let f be a function in H(N −1) such that P− f = O. Then Hu+f > 1. ∞ Proof. Since f ∈ H(N −1) , there exists a finite Blaschke product B of degree at most N − 1 such that Bf ∈ H ∞ . Let v = Bu. Then ' dim Ker Tv = dim Ker Tu BH 2 > 0. Clearly, Hv g2 = g2 for any g ∈ Ker Tv , and by Theorem 1.4 of Chapter 1, v admits a representation ¯ h v = z¯ϑ¯ , h where ϑ is an inner function and h is an outer function in H 2 . Clearly, Hu h2 = h2 . ¯ Therefore ¯ h. Let us show that Hu h ⊥ Hf h. We have Hu h = z¯ϑ¯B ¯ f h) = (¯ ¯ Bf h) = 0, ¯ h, (Hu h, Hf h) = (¯ z ϑ¯B z ϑ¯h, since Bf ∈ H ∞ . Consequently, Hu+f h2 = h2 + Hf h2 . To complete the proof, it suffices to show that Hf h = O. Suppose that Hf h = O. Then Hf z n h = O and since h is outer, it would follow that Hf = O, which contradicts the assumption P− f = O. Proof of Theorem 1.3. Suppose that there exists a function q in ∞ such that q = ψ and H(k+µ−1) ϕ − q∞ = sk (Hϕ ).

(1.11)

We have ϕ − q = ϕ − ψ + ψ − q. Let u = (ϕ − ψ)/sk (Hϕ ), f = (ψ − q)/sk (Hϕ ). By Theorem 1.7, ∞ ∞ , q ∈ H(k+µ−1) and so dim Ker Tu = 2k + µ. Clearly, ψ ∈ H(k) ∞ ψ − q ∈ H(2k+µ−1) . If P− (ψ − q) = O, it follows from Lemma 1.8 that Hu+f > 1 and so Hϕ−q > sk (Hϕ ), which contradicts (1.11). Thus ψ − q ∈ H ∞ . The Hankel operator Hϕ−ψ attains its norm on the unit ball of H 2 , since for any ξ ∈ E+ Hϕ−ψ ξ2 = sk (Hϕ ) · ξ2 = Hϕ−ψ · ξ2 .

2. The Case sm (Γ) = s∞ (Γ)

131

Therefore if (1.11) holds, we have Hϕ−ψ = ϕ − ψ∞ = ϕ − ψ + (ψ − q)∞ . By Theorem 1.1.4 it follows that ψ − q = O. ∞ Remarks. 1. It follows from (1.9) that the function ψ in H(m) satisfying the equality ϕ − ψ∞ = sm (Hϕ ) is given by

ψ =ϕ−

Hϕ ξ , ξ

(1.12)

where ξ is an arbitrary nonzero function satisfying Hϕ∗ Hϕ = s2m ξ. 2. It will be shown in §7.1 that if ϕ ∈ V M O, then ϕ − ψ ∈ QC and so it will follow from Theorem 1.7 that wind(ϕ − ψ) = −(2k + µ).

2. The Case sm (Γ) = s∞ (Γ) In this section we complete the proof of Theorem 1.1. To this end we ˜ > s∞ (Γ) ˜ and ˜ for which sm (Γ) construct an auxiliary Hankel operator Γ ˜ apply Theorem 1.1 to Γ, which we can do because of the above inequality. ˜ we make use of the one-step extension method, which has To construct Γ been used in Chapter 2 to establish the analog of the Nehari theorem for vectorial Hankel operators. Proof of Theorem 1.1 in the case sm (Γ) = s∞ (Γ). If sm (Γ) = s0 (Γ), the situation is trivial (Γm = O). Assume that sk−1 (Γ) > sk (Γ) = sm (Γ) = s∞ (Γ). It is sufficient to show that there exists a Hankel operator Γk of rank k such that Γ − Γk = sk (Γ). Let ρ be a real number satisfying sk−1 (Γ) > ρ > sk (Γ).

(2.1)

Let us show that if ρ satisfies (2.1), then there exists a Hankel operator Hψ such that rank Hψ ≤ k and Γ − Hψ ≤ ρ. To do that, we reduce the situation to the case already treated in §1. ˜ such that Namely, we construct an auxiliary Hankel operator Γ ˜ − ρ2 I) = {O} ˜∗Γ Ker(Γ

(2.2)

˜ ≥ sk−1 (Γ) ≥ sk (Γ) ˜ ≥ sk (Γ) ≥ sk+1 (Γ). ˜ sk−1 (Γ)

(2.3)

and

132


˜ in the form We are searching for such a Γ f ∈ H 2,

˜ = z¯Γf + (f, h)¯ Γf z,

(2.4)

where H ∈ H 2 . ˜ defined by Let us find out under which conditions on h the operator Γ (2.4) is a Hankel operator. In other words, we have to characterize those h for which ˜ n = P− z n Γ1, ˜ Γz

n ≥ 0.

(2.5)

˜ satisfies (2.5) if and only if Let Γ = Hϕ . It is easy to see that Γ ˆ h(n) = ϕ(−n), ˆ

n > 1.

˜ = Hϕ , then ϕˆ1 (−n) = ϕ(−n If Γ ˆ + 1), n > 1. Let α = ϕˆ1 (−1). Then 1 ¯ + zΓ∗ z¯. h=α ¯ + zP+ z¯ϕ¯ = α

(2.6)

It follows from (2.4) that ˜ ∗ Γf ˜ = Γ∗ Γf + (f, h)h, Γ

f ∈ H 2.

˜ is a perturbation of Γ∗ Γ by a nonnegative rank one operator, ˜∗Γ Therefore Γ which implies (2.3) (see Appendix 1.1). Lemma 2.1. For any ρ ∈ (sk (Γ), sk−1 (Γ)) sufficiently close to sk (Γ) there exists α ∈ C such that (2.2) holds. By Lemma 2.1, if ρ is sufficiently close to sk (Γ), then ρ is a singular ˜ and since (2.3) is satisfied, the only possibility is that ρ = sk (Γ). ˜ value of Γ, ˜ Clearly, s∞ (Γ) = s∞ (Γ) < ρ. So it follows from the part of Theorem 1.1 already treated in §1 that there exists a Hankel operator Hψ1 such that ˜ − Hψ = sk (Γ) ˜ = ρ. As we have already observed, rank Hψ1 = k and Γ 1 ˜ = Hϕ , where ϕˆ1 (n) = ϕ(n Γ ˆ + 1), n ≤ −2, and so 1 ˜ Γ = P− S Γ, where S is multiplication by z on L2 . Put ψ = zψ1 . We have ˜ − P− SHψ ≤ Γ ˜ − Hψ = ρ. Γ − Hψ = P− S Γ 1 1 On the other hand, rank Hψ = rank P− SHψ1 ≤ rank Hψ1 = k. To complete the proof, we can take a sequence {ρn } in (sk (Γ), sk−1 (Γ)) that tends to sk (Γ) and construct a sequence of Hankel operators Hϕn such that rank Hϕn ≤ k and Γ − Hϕn ≤ ρn . Since the operators Hϕn are uniformly bounded, there exists a subsequence weakly convergent to an operator, say Γm . It is easy to see that rank Γm ≤ k and Γm is a Hankel operator. Clearly, Γ − Γm ≤ sk (Γ), which completes the proof.

2. The Case sm (Γ) = s∞ (Γ)

133

Proof of Lemma 2.1. Clearly, (2.2) is equivalent to the fact that there is a nonzero f in H 2 satisfying (ρ2 I − Γ∗ Γ)f = (f, h)h.

(2.7)

Put Rρ = (ρ2 I − Γ∗ Γ)−1 and r(ρ) = (Rρ 1, 1). It is easy to see that there can be at most one ρ in (sk (Γ), sk−1 (Γ)) such that r(ρ) = 0. Indeed, r is differentiable on (sk (Γ), sk−1 (Γ)) and dr(ρ) = −2ρ(Rρ2 1, 1) < 0, dρ

ρ ∈ (sk (Γ), sk−1 (Γ)),

and so r is strictly decreasing on (sk (Γ), sk−1 (Γ)). Lemma 2.2. If r(ρ) = 0, then there exists α ∈ C such that the function h defined by (2.6) satisfies (Rρ h, h) = 1.

(2.8)

Let us first complete the proof of Lemma 2.1. Put f = Rρ h, then f is nonzero and satisfies (2.7). ¯ + g. Proof of Lemma 2.2. Put g = zΓ∗ z¯. Then (2.6) means that h = α It is easy to see that (2.8) is equivalent to the following equation: α(Rρ 1, g)) + (Rρ g, g) = 1. |α|2 r(ρ) + 2 Re (¯ Multiplying both sides of this equation by r(ρ), we see that it is equivalent to the following one: |αr(ρ) + (Rρ 1, g)|2 = r(ρ) + |(Rρ 1, g)|2 − r(ρ)(Rρ g, g).

(2.9)

Let us prove that r(ρ) + |(Rρ 1, g)|2 − r(ρ)(Rρ g, g) =

1 . ρ2

(2.10)

If we do that, we can put for any complex τ of modulus 1 α=

(Rρ 1, g) τ − . ρr(ρ) r(ρ)

Clearly, α satisfies (2.9). Thus to complete the proof, we have to verify (2.10). Let J be the operator on L2 defined by Jf = z¯f¯, f ∈ L2 . Put Γ = SJΓ, where S is multiplication by z on H 2 . Clearly, Γ 1 = Γ1 = g. We are going to establish the identity Γ Rρ 1 = ρ2 r(ρ)Rρ g − ρ2 (Rρ g, 1)Rρ 1.

(2.11)

134


Let us first show that (2.11) implies (2.10). Taking the inner product of both sides of (2.11) with g = Γ 1, we obtain ρ2 r(ρ)(Rρ g, g) − ρ2 |(Rρ g, 1)|2

=

(Γ Rρ 1, Γ 1) = (SJΓRρ 1, SJΓ1)

=

(JΓRρ 1, JΓ1) = (Γ1, ΓRρ 1)

=

(1, Γ∗ ΓRρ 1) = ρ2 r(ρ) − 1,

which proves (2.10). To establish (2.11), we apply to both sides of (2.11) the operator Rρ−1 = ρ2 I − Γ∗ Γ and find that it is equivalent to Rρ−1 Γ Rρ 1 = ρ2 r(ρ)g − ρ2 (Rρ g, 1)1.

(2.12)

Let P0 be the orthogonal projection from H 2 onto the space of constant functions, i.e., P0 f = (f, 1)1. Let us show that (I − P0 )Rρ−1 Γ Rρ 1 = ρ2 r(ρ)g

(2.13)

(Rρ−1 Γ Rρ 1, 1) = −ρ2 (Rρ g, 1),

(2.14)

and

which will imply (2.12). Let us first prove (2.13). It is easy to check that JΓ = Γ∗ J and JΓS = S ∗ Γ∗ J. Hence, (Γ )2 = SJΓSJΓ = SS ∗ Γ∗ Γ = (I − P0 )Γ∗ Γ. It follows that (I − P0 )Rρ−1 = (I − P0 )(ρ2 I − (Γ )2 ).

(2.15)

Hence, (I − P0 )Rρ−1 Γ Rρ 1 = (I − P0 )(ρ2 I − (Γ )2 )Γ Rρ 1 = Γ (ρ2 I − (Γ )2 )Rρ 1 = Γ (I − P0 )(ρ2 I − (Γ )2 )Rρ 1 + Γ P0 (ρ2 − (Γ )2 )Rρ 1. By (2.15) Γ (I − P0 )(ρ2 I − (Γ )2 )Rρ 1 = Γ (I − P0 )1 = O. Thus (I − P0 )Rρ−1 Γ Rρ 1 =

Γ P0 (ρ2 − (Γ )2 )Rρ 1

= ρ2 Γ P0 Rρ 1 = ρ2 r(ρ)Γ 1 = ρ2 r(ρ)g, since P0 Γ = O. It remains to prove (2.14). We have ΓΓ = J(JΓS)JΓ = J(S ∗ Γ∗ J)JΓ = JS ∗ (ρ2 I − Rρ−1 ).

3. Finite Rank Approximation of Vectorial Hankel Operators

135

Consequently, (Rρ−1 Γ Rρ 1, 1)

= ρ2 (Γ Rρ 1, 1) − (Γ∗ ΓΓ Rρ 1, 1) = −(JS ∗ (ρ2 I − Rρ−1 )Rρ 1, Γ1) = −ρ2 (JS ∗ Rρ 1, Γ1) = −ρ2 (JΓ1, S ∗ Rρ 1) = −ρ2 (g, Rρ 1),

since Γ f ⊥ 1 for any f ∈ H 2 , which completes the proof.

3. Finite Rank Approximation of Vectorial Hankel Operators In this section we present another approach to the problem of finite rank approximation of Hankel operators which works in both cases sm (Γ) > s∞ (Γ) and sm (Γ) = s∞ (Γ). Moreover, it also works in the case of vectorial Hankel operators. However, the proof given in §1 for the scalar case gives more information on the best approximant. As an application we find a formula for the essential norm of a vectorial Hankel operator. Let H and K be separable Hilbert spaces. The main result of this section is the following generalization of the Adamyan–Arov–Krein theorem. Theorem 3.1. Let Φ ∈ L∞ (B(H, K)). Then sn (HΦ ) = inf{Φ − Ψ∞ : rank HΨ ≤ n}. Remark. It is easy to see that “infimum” in the statement of Theorem 3.1 can be replaced with “minimum”. Indeed, if {Ψj } is a bounded sequence of functions in L∞ (B(H, K)) such that rank HΨj ≤ n and ˆ j (m) Φ − Ψj ∞ → sn (HΦ ), there exists a subsequence {Ψjk } such that Ψ k ∞ ˆ converges weakly to Ψ(m), m ∈ Z, for some Ψ ∈ L (B(H, K)). It is easy to see that the sequence {HΨjk } converges to HΨ in the weak operator topology, and so rank HΨ ≤ n. Clearly, Φ − Ψ∞ = sn (HΦ ). To prove Theorem 3.1, we make use of the following fact, which will be established after the proof of Theorem 3.1. Theorem 3.2. Let A be a bounded self-adjoint operator and S a bounded operator on a Hilbert space H. Let C = {x ∈ H : (Ax, x) ≥ 0} and let P− = E((−∞, 0)), P+ = E([0, ∞)), where E is the spectral measure of A. Suppose that the following conditions are satisfied: def (a) A− = A |Range P− is invertible; (b) SC ⊂ C; (c) P+ SP− is compact. Then there exists a subspace L of H such that SL ⊂ L and L ⊂ C, and L is a maximal (by inclusion) subspace of H contained in C.

136


Put H− = P− H, H+ = P+ H, and A− = AH− , A+ = AH+ . Denote by M the set of subspaces L of H such that L ⊂ C and P+ L = H+ . Clearly, M = ∅, since H+ ∈ M. To deduce Theorem 3.1 from Theorem 3.2 we need the following lemma. Lemma 3.3. The following assertions hold: (i) if L ∈ M, then there exists a unique operator TL : H+ → H− such that L = {x + TL x : x ∈ H+ };

(3.1)

(ii) the set {TL : L ∈ M} is a convex subset of B(H+ , H− ), which is compact in the weak operator topology; (iii) if L is a subspace of H, L ⊂ C, then L belongs to M if and only if it is a maximal subspace contained in C; (iv) if rank A− < ∞, L is a subspace of H, L ⊂ C, then it belongs to M if and only if codim L = rank A− . First we deduce Theorem 3.1 from Theorem 3.2. Then we prove Lemma 3.3 and, finally, we prove Theorem 3.2. Proof of Theorem 3.1. Let H = H 2 (H), A = s2n (HΦ )I − HΦ∗ HΦ , and S is multiplication by z on H 2 (H). Clearly, in this case C = f ∈ H 2 (H) : HΦ f ≤ sn (HΦ )f . It is easy to see that if f ∈ C, then HΦ SF ≤ HΦ f ≤ sn (HΦ )f = sn (HΦ )Sf . Hence SC ⊂ C. Clearly, rank P− ≤ n, and so conditions (a) and (c) in the statement of Theorem 3.2 are obviously satisfied. By Theorem 3.2 and Lemma 3.3, (iii), (iv) there exists a subspace L ⊂ H 2 (H) such that SL ⊂ L, codim L ≤ n, and L ⊂ C. Clearly, the last inclusion means that HΦ L ≤ sn (HΦ ). (3.2) By Lemma 2.5.1, there exists a Blaschke–Potapov product B of finite degree such that L = BH 2 (H). Therefore HΦ L = HΦ BH 2 (H) = HΦB , and it follows from (3.2) that HΦB ≤ sn (HΦ ). By Theorem 2.2.2, there exists a function Ω in H ∞ (H, K) such that HΦB = ΦB − Ω∞ . Since B takes unitary values on T , we have ΦB − Ω∞ = Φ − ΩB ∗ ∞ . Since L = BH 2 (H) ⊂ Ker HΩB ∗ , it follows that rank HΩB ∗ ≤ codim L ≤ n, and so inf{Φ − Ψ : rank HΨ ≤ n} ≤ Φ − ΩB ∗ ∞ ≤ sn (HΦ ).


137

The opposite inequality sn (HΦ ) ≤ inf{Φ − Ψ∞ : rank HΨ ≤ n} is trivial. Proof of Lemma 3.3. (i) Let x ∈ C. Then x = x+ + x− , where x+ = P+ x, x− = P− x. We have −1 0 ≤ (Ax, x) = (A+ x+ , x+ ) + (A− x− , x− ) ≤ A+ · x+ 2 − A−1 x− 2 . −

Therefore 1/2 x+ . x− ≤ (A+ · A−1 − )

(3.3)

This implies that P+ |L is bounded from below, since it follows from (3.3) that for x ∈ L 2 x2 = x+ 2 + x− 2 ≤ (1 + A+ · A−1 − )x+ .

Thus we can put

TL = P− (P+ L)−1 ,

(3.4)

which is well-defined on H+ . It is easy to check that (3.1) holds. The uniqueness of TL is obvious. (ii) The fact that the set {TL : L ∈ M} is bounded follows from (3.3) and (3.4): 1/2 TL ≤ (A+ · A−1 − ) for any L in M. Let us show that it is convex. Let L1 , L2 ∈ M, λ ∈ (0, 1), def

and Tλ = λTL1 + (1 − λ)TL2 . If x ∈ H+ , we have (A(x + Tλ x), x + Tλ x)

= (A+ x, x) + (A− Tλ x, Tλ x) ≥ (A+ x, x) + min (A− TLj x, TLj x) ≥ 0, j=1,2

since L1 , L2 ∈ M. Consequently, {x + Tλ x : x ∈ H+ } ∈ M for λ ∈ (0, 1). Since for any x+ ∈ H+ , the set {x− ∈ H− : (A(x+ + x− ), x+ + x− ) ≥ 0} is closed, it follows that the set {TL : L ∈ M} is closed in the strong operator topology and since this set is convex, it is also closed in the weak operator topology (see Dunford and Schwartz [1], VI.1.5). Hence, it is weakly compact. (iii) Let L ∈ M. We are going to show that L is a maximal subspace contained in C. Suppose that L is a subspace of H and L ⊂ L ⊂ C. It follows that P+ L = H+ and by (i) L = {x + TL x : x ∈ H+ }. On the other hand, L = {x + TL x : x ∈ H+ }. Since L ⊂ L , it follows that TL = TL and so L = L.

138


Suppose now that L is a maximal subspace of C but P+ L = H+ . Since L is bounded from below, it follows that it has been proved in (i) that P + P+ L is a closed subspace of H+ . Assume first that A+ is invertible. Then A+ P+ L is a closed subspace of H+ and A+ P+ L = H+ . Put L = L + (H+ A+ P+ L). Clearly, L = L. If x ∈ L and f ∈ H+ A+ P+ L, we have (A(x + f ), x + f ) = (A− x− , x− ) + (A+ (x+ + f ), x+ + f ) = (A− x− , x− ) + (A+ x+ , x+ ) + (A+ f, f ) ≥ (Ax, x) ≥ 0. Hence, L ⊂ C, which contradicts the assumption that L is maximal. Let us now consider the general case. Put 1 A(n) = A− P− + A+ + IH+ P+ , n ≥ 1. n

(n)

Then A+ is invertible. Clearly, def

L ⊂ Cn = {x ∈ H : (A(n) x, x) ≥ 0}. By Zorn’s lemma L is contained in a maximal subspace of Cn and since (n) A+ is invertible, we have already proved there exists Tn : H+ → H− such that L ⊂ {x + Tn x : x ∈ H+ } ⊂ Cn . Clearly, sup Tn < ∞. Let T be the weak limit of a subsequence of n

{Tn }n≥1 . Since Cn+1 ⊂ Cn , we have

'

L ⊂ {x + T x : x ∈ H+ } ⊂

Cn = C.

n≥1

Clearly, {x + T x : x ∈ H+ } ∈ M and since L is maximal, it follows that L = {x + T x : x ∈ H+ } ∈ M. (iv) Since codim H+ = rank A− , the result follows from the following equality, which is valid for any bounded operator T : H+ → H− : codim{x + T x : x ∈ H+ } = codim H+ . To prove Theorem 3.2 we need the following fact. Lemma 3.4. Let H1 , H2 be Hilbert spaces and let X be a convex subset of B(H1 , H2 ) that is compact in the weak operator topology. Let α be a map from X to B(H1 ) and β a map from X to B(H1 , H2 ) that satisfy the following conditions: (1) for any X ∈ X there exists Y ∈ X such that Y α(X) = β(X); (2) the map F : X × X → B(H1 , H2 ) defined by F (X, Y ) = Y α(X) − β(X)

(3.5)


139

is jointly continuous in the weak operator topology. Then there exists X0 ∈ X such that X0 α(X0 ) = β(X0 ). Proof of Theorem 3.2. Put H1 = H+ , H2 = H− , X = {TL : L ∈ M}, α(X) = P+ S H+ + P+ SX, β(X) = P− S H+ + P− SX,

X ∈ X, X ∈ X.

By Lemma 3.3, X is convex and weakly compact. Since the operator P+ SP− is compact and P+ SX = P+ SP− X, X ∈ X , it follows that the function F defined by (3.5) satisfies condition (2) of Lemma 3.4. Let us verify condition (1). Since SC ⊂ C and any subspace contained in C is contained in a maximal subspace in C (i.e., a subspace from M), it follows that for any L ∈ M there exists L ∈ M such that SL ⊂ L .

(3.6)

If x ∈ C, L ∈ M, then x ∈ L if and only if TL P+ x = P− x.

(3.7)

Clearly, SL = {Sx + STL x : x ∈ H+ }. It follows from (3.7) that (3.6) is equivalent to the fact that TL (P+ Sx + P+ STL x) = P− Sx + P− STL x

(3.8)

for any x ∈ H+ , which means that condition (1) of Lemma 3.4 is satisfied. It follows from Lemma 3.4 that there exists L ∈ M such that (3.8) holds with L = L, which means that SL ⊂ L. Lemma 3.4 follows trivially from the following fact. Theorem 3.5. Let W be a locally convex Hausdorff space, X a compact convex subset of W , and F a continuous map from X × X to W such that F (x, λy1 +(1−λ)y2 ) = λF (x, y1 )+(1−λ)F (x, y2 ), x, y1 , y2 ∈ X , 0 ≤ λ ≤ 1. If for any x ∈ X there exists y ∈ X such that F (x, y) = O, then there exists x0 ∈ X such that F (x0 , x0 ) = O. Note that if F (x, y) = y − f (x), where f is a continuous map from X to X , then Theorem 3.5 turns into Tikhonov’s fixed point theorem (see Edwards [1], §3.6). Lemma 3.6. Let X be a compact convex subset in a locally convex space and let Q be a closed subset of X × X such that (x, x) ∈ Q for any x ∈ X and the set {y ∈ X : (x, y) ∈ Q} is convex for any x ∈ X . Then there exists x0 ∈ X such that (x0 , y) ∈ Q for any y ∈ X .

140


Let us first deduce Theorem 3.5 from Lemma 3.6. Proof of Theorem 3.5. Let {ρν } be a set of seminorms that determines the topology in W . Put Gν = {x ∈ X : ρν (F (x, x)) = 0} . Since X is compact, it is sufficient to show that for any finite set ν1 , · · · , νn the set n ' Gνk = ∅. k=1

Fix ν1 , · · · , νn and put Q=

(x, y) ∈ X × X :

n

n

ρνk (F (x, y)) ≥

k=1

( ρνk (F (x, x)) .

k=1

It is easy to see that Q satisfies the assumptions of Lemma 3.6. Therefore there exists x0 ∈ X such that (x0 , y) ∈ Q for any y ∈ X . By the assumptions of Theorem 3.5 there exists y0 ∈ X such that F (x0 , y0 ) = O. By the definition of Q n

ρνk (F (x0 , x0 )) ≤

k=1

n

ρνk (F (x0 , y0 )) = 0,

k=1

so ρνk (F (x0 , x0 )) = 0 for any k, 1 ≤ k ≤ n, which means that x0 ∈

n '

Gνk .

k=1

To prove Lemma 3.6, we need one more lemma. Lemma 3.7. Let t1 , t2 , · · · , tn be points in Rn−1 such that their convex hull K = conv{t1 , t2 , · · · , tn } is an (n − 1)-dimensional simplex (i.e., has nonempty interior in Rn−1 ). Let V1 , V2 , · · · , Vn be subsets of K that are open in K. Suppose that for any subset {k1 , k2 , · · · , kl } of {1, 2, · · · , n} l '

Vjs

'

conv {tk1 , tk2 , · · · , tkl } = ∅.

(3.9)

s=1

Then K =

n

Vj .

j=1

Proof. Suppose that K

=

n

Vj . Then there are closed sets

j=1

W1 , W2 , · · · , Wn such that Wj ⊂ Vj , 1 ≤ j ≤ n, and K =

n j=1

Wj . Without


141

loss of generality we can assume that n tj = 0. j=1

It follows that any n − 1 vectors from {t1 , t2 , · · · , tn } are linearly independent. Put dj (t) = dist(t, Wj ), t ∈ K, and Dj (t) = max{s ≥ 0 : t − stj ∈ K}.

(3.10)

Let F be the continuous function on K defined by n 1 F (t) = t − (min{dj (t), Dj (t)}) tj . n j=1 It follows from (3.10) that F (t) ∈ K for any t ∈ K. By Brouwer’s fixed point theorem (see Dunford and Schwartz [1], Ch. V, §12), there exists n Wj = K, it follows that dj (t0 ) = 0 t0 ∈ K such that F (t0 ) = t0 . Since j=1

at least for one j. Since any n − 1 vectors from the set t1 , t2 , · · · , tn are linearly independent, it follows from the equality F (t0 ) = t0 that min{dj (t0 ), Dj (t0 )} = 0

(3.11)

for 1 ≤ j ≤ n. Clearly, dj (t0 ) = 0 if and only if t0 ∈ Wj , while Dj (t0 ) = 0 if and only if t ∈ conv{t1 , · · · , tj−1 , tj+1 , · · · , tn }. Therefore it follows from (3.11) that there exists a subset {k1 , k1 , · · · , kl } of {1, 2, · · · , n} such that l '

Wjs

'

conv {tk1 , tk2 , · · · , tkl } = ∅,

s=1

which contradicts (3.9). Proof of Lemma 3.6. Let y ∈ X . Put def

Qy = {x ∈ X : (x, y) ∈ Q}. We have to verify that Qy = ∅. Since X is compact, it is sufficient to show that

n j=1

y∈X

Qyj = ∅ for any finite set y1 , y2 , · · · , yn in X . The latter is

equivalent to the fact that

n j=1

def

Vyj = X , where Vy = X \ Qy . Clearly, Vy is

open. Let us show that for any y1 , y2 , · · · , yn in X n ' ' Vyj conv{y1 , y2 , · · · , yn } = ∅. j=1

142


Indeed, if x belongs to the above intersection, then yj ∈ {y ∈ X : (x, y) ∈ Q},

1 ≤ j ≤ n,

and so by the hypotheses of Lemma 3.6, (x, x) ∈ Q, which contradicts the assumption that Q contains the diagonal. Let t1 , t2 , · · · , tn ∈ Rn−1 be vectors satisfying the hypotheses of Lemma 3.7. Let τ : K = conv{t1 , t2 , · · · , tn } → conv{y1 , y2 , · · · , yn } ⊂ X def

be the affine map defined by τ tj = yj . Then the sets Vj = τ −1 (Vyj ) satisfy n the hypotheses of Lemma 3.7. It follows that Vj = K, so j=1

conv{y1 , y2 , · · · , yn } ⊂

n

V yj ,

j=1

which completes the proof. To conclude this section, we obtain a formula for the essential norm of a vectorial Hankel operator that is analogous to Theorem 1.5.3. Theorem 3.8. Let H and K be separable Hilbert spaces and let Φ ∈ L∞ ((B(H, K)). Then (3.12) HΦ e = distL∞ Φ, H ∞ (B(H, K)) + C(C(H, K)) . Proof. Clearly, by Theorems 2.2.2 and 2.4.1, (3.12) is equivalent to the fact that HΦ e is equal to the distance from HΦ to the set of compact 2 (K). Clearly, HΦ e is equal vectorial Hankel operators from H 2 (H) to H− to the distance from HΦ to the set of finite rank operators. The result follows now from Theorem 3.1.

4. Relations between Hu and Hu¯ The aim of this section is to obtain a useful formula that relates the Hankel operators Hu and Hu¯ for a unimodular function u. This would allow us under certain circumstances to get information about properties of Hu¯ from properties of Hu . In particular, we will be able to get information about the singular numbers of Hu¯ from the singular values of Hu . Let u be a unimodular function on T. Then Hu¯∗ Hu¯ Tu = Tu Hu∗ Hu .

(4.1)

To prove this formula we make use of formula (3.1.5): Hu¯∗ Hu¯ Tu = (I − Tu Tu¯ )Tu = Tu (I − Tu¯ Tu ) = Tu Hu∗ Hu . To obtain consequences of formula (4.1) we need the notion of polar decomposition. Let T be an operator on a Hilbert space H. Then T admits a representation in the form T = U(T ∗ T )1/2 , where Ker U = Ker T and

4. Relations between Hu and Hu¯

143

U |H Ker U is an isometry onto clos Range T (see Halmos [2], Problem 134). U is called the partially isometric factor of T . Lemma 4.1. Let A and B be self-adjoint operators on Hilbert space and let T be an operator such that AT = T B. Then AU = UB, where U is the partially isometric factor of T . Proof. It follows from AT = T B that T ∗ A = BT ∗ and so T ∗ T B = T ∗ AT = BT ∗ T. Hence, B commutes with any function of the self-adjoint operator T ∗ T , in particular with (T ∗ T )1/2 . We have AU(T ∗ T )1/2 = U(T ∗ T )1/2 B = UB(T ∗ T )1/2 . Multiplying on the right by (T ∗ T )1/2 , we obtain AUT ∗ T = UBT ∗ T . It follows that AU and UB coincide on clos Range T ∗ T . Let us verify that they also coincide on Ker T ∗ T = Ker T , which is the orthogonal complement to clos Range T ∗ T . Since Ker U = Ker T and since Ker T is B-invariant (the latter follows from AT = T B), we see that AU Ker T = UB Ker T = O. Applying Lemma 4.1 to (4.1), we obtain Hu¯∗ Hu¯ Uu = Uu Hu∗ Hu ,

(4.2)

where Uu is the partially isometric factor of Tu . Theorem 4.2. Let u be a unimodular function on T such that clos Tu H 2 = H 2 and E = f ∈ H 2 : Hu∗ Hu f = f . Then Hu¯∗ Hu¯ is unitarily equivalent to Hu∗ Hu H 2 E . Proof. Note that E = Ker Tu = Ker Uu and Uu maps H 2 E isometrically onto H 2 . It follows from (4.2) that Hu¯∗ Hu¯ = Uu Hu∗ Hu Uu∗ = Uu Hu∗ Hu H 2 E Uu∗ . Corollary 4.3. Let u be a unimodular function such that Ker Tu = Ker Tu∗ = {O}. Then Hu¯∗ Hu¯ and Hu∗ Hu are unitarily equivalent. The following assertion allows us to get information about the singular values of Hu from the singular values of Hu¯∗ . Corollary 4.4. Let u be a unimodular function in H ∞ + C such that Tu has dense range. Then the Hankel operator Hu¯ is compact and sk (Hu¯ ) = sk+n (Hu ), where n = dim Ker Tu = dim E.

k ≥ 0,

144


Proof. This follows immediately from Theorem 4.2 and the facts that ∗ Hu Hu H 2 E < 1 and Hu∗ Hu f = f for f ∈ E. In the next corollary S p , 0 < p < ∞, is the Schatten–von Neumann class (see Appendix 1.1). Corollary 4.5. Let u be a unimodular function such that Tu has dense range in H 2 . If Hu is compact, then so is Hu¯ . If Hu ∈ S p , 0 < p < ∞, then Hu¯ ∈ S p and Hu¯ S p < Hu S p . Corollary 4.6. Let u be a unimodular function in H ∞ + C such that distL∞ (¯ u, H ∞ + C) < 1. Then u ∈ QC. Proof. There is an integer n such that distL∞ (z n u ¯, H ∞ ) < 1. Without loss of generality we can assume that n = 0. By Theorem 1.11 of Chapter 1, Tu¯ is left invertible and so Tu H 2 = H 2 . Since Hu is compact by the Hartman theorem, Hu¯ is also compact and again by the Hartman theorem u ¯ ∈ H ∞ + C. Corollary 4.7. Let u be a unimodular function such that dim Ker Tu < ∞ and dim Ker Tu¯ < ∞. Then Hu e = Hu¯ e . Proof. By Theorem 3.1.4, Ker Tu = {O} or Ker Tu¯ = {O}. To be definite, suppose that Ker Tu¯ = {O}. Then Tu has dense range in H 2 and by Theorem 4.2, Hu¯∗ Hu¯ is unitarily equivalent to the restriction of Hu∗ Hu to the orthogonal complement of Ker Tu . By the hypotheses, dim Ker Tu < ∞, and so Hu¯ e = lim sj (Hu¯ ) = lim sj (Hu ) = Hu e . j→∞

j→∞

Corollary 4.8. Let u be a unimodular function such that the Toeplitz operator Tu is Fredholm. Then Hu e = Hu¯ e . Let us now characterize those unimodular functions u for which Tu has dense range in H 2 . Theorem 4.9. Let u be a unimodular function. Then clos Tu H 2 = H 2 if and only if there exist an outer function h in H 2 and an inner function ¯ ϑ such that u = zϑh/h. Proof. Suppose that clos Tu H 2 = H 2 . Therefore Ker Tu¯ = {O}. Let f 2 be a nonzero function in Ker Tu¯ . Then u ¯f ∈ H− . Let g be a function in 2 H such that z¯g¯ = u ¯f . Since |g| = |f | on T, there exist an outer function h in H 2 and inner functions ϑ1 , ϑ2 such that f = ϑ1 h, g = ϑ2 h. Then ¯ u = zϑ1 ϑ2 h/h. ¯ then obviously h ∈ Ker Tu¯ . Hence, clos Tu H 2 = H 2 . If u = zϑh/h, The following sufficient condition for Tu to have a dense range will be very useful.

Concluding Remarks

145

¯ Theorem 4.10. Let u = ϑ¯h/h, where ϑ is an inner function and h is 2 an outer function in H . Then Tu has dense range in H 2 . ¯ = 0 for Proof. Suppose that f ⊥ Tu H 2 and f = O. Then (f, ϑ¯hg/h) any g in H 2 . Let f = τ k, where τ is an inner function and k is an outer ¯ = 0, that is, k(0)h(0) = 0, which is function. Take g = τ ϑh. Then (k, h) impossible since both h and k are outer. Consider now the case of vectorial Hankel operators with unitary-valued symbols. Let H be a separable Hilbert space and let U be a unitary-valued function with values in B(H). Then the following analog of formula (4.1) holds: HU∗ ∗ HU ∗ TU = TU HU∗ HU . This allows us to generalize Theorem 4.2 to the case of Hankel operators with unitary-valued symbols. Theorem 4.11. Let U be a unitary-valued function in L∞ (B(H)) such that clos TU H 2 (H) = H 2 (H) and E = f ∈ H 2 : HU∗ HU f = f . Then HU∗ ∗ HU ∗ is unitarily equivalent to HU∗ HU H 2 E . The proof of Theorem 4.11 is exactly the same as that of Theorem 4.2. It is also easy to see that Corollaries 4.3–4.5 also generalize very easily to the case of unitary-valued operator functions. Let us also state an analog of Corollary 4.8. Corollary 4.12. Let U be a unitary-valued matrix function such that the Toeplitz operator TU is Fredholm. Then HU e = HU ∗ e .

Concluding Remarks The results of §1 and §2 were obtained in Adamyan, Arov, and Krein [3]. Note that earlier Clark [1] proved a special case of the Adamyan– Arov–Krein theorem; namely, he considered the case of self-adjoint Hankel matrices. Later Butz [1] gave another proof of the Adamyan–Arov–Krein theorem. Section 3 follows the paper Treil [3]. Note that the Adamyan–Arov–Krein theorem was generalized earlier to the case of finite matrix functions in Ball and Helton [1]. The results of §4 were obtained in Peller and Khrushchëv [1] for scalar functions; see Peller [17], where the case of operator functions was considered in connection with applications to vectorial stationary processes. Note that Corollary 4.5 for p = 2 was established in Solev [1]. In Le Merdy [1] the following generalization of the Adamyan–Arov–Krein theorem was found. It was shown that if Hϕ is a Hankel operator from H p

146


q to H− , p ≥ 2, 1 < q ≤ 2, then

inf{Hϕ − KB(H p ,H−q ) : rank K ≤ n} ≤ const inf{Hϕ − Hf B(H p ,H−q ) : rank Hf ≤ n}. In Baratchart and Seyfert [1] the authors proved a precise analog of the Adamyan–Arov–Krein theorem for Hankel operators from H q , q ∈ [2, ∞] 2 to H− . They proved that the distance to the set of operators of rank at most n equals the distance to the set of Hankel operators of rank at most n. In other words, def

σn (Hϕ ) = inf{Hϕ − KB(H q ,H 2 ) : rank K ≤ n} is the distance in the norm of Lp from ϕ to the set of meromorphic functions with at most n poles in D, where 1/p + 1/q = 1/2. Moreover, in Baratchart and Seyfert [1] a precise analog of the Adamayan–Arov–Krein formula was obtained for the best meromorphic approximant in terms of “singular vectors” of the Hankel operator Hϕ : H q → H 2 . It was also mentioned in Baratchart and Seyfert [1] that a precise analog of the Adamyan–Arov–Krein theorem can also be obtained for Hankel operators from H q to the quotient space Lr /H r , where 1/p+1/q+(r−1)/r = 1. This allows one to express the numbers σn (Hϕ ) in terms of the distance in Lp to the set of meromorphic functions with at most n poles in D. Here p can be an arbitrary number in [1, ∞). However, in this case no explicit formulas for the best meromorphic approximant in terms of “singular vectors” were found. We also mention here the paper Prokhorov [3] in which the analog of the Adamyan–Arov–Krein theorem for Hankel operators from H q to the quotient space Lr /H r was obtained independently of the paper Baratchart and Seyfert [1].

5 Parametrization of Solutions of the Nehari Problem

2 For a Hankel operator Γ from H 2 to H− and ρ ≥ Γ, we consider in this section the problem of describing all symbols ϕ ∈ L∞ of Γ (i.e., Γ = Hϕ ) which satisfy the inequality ϕ∞ ≤ ρ. If ϕ0 is a symbol of Γ, then as we have seen in §1.1, this problem is equivalent to the problem of finding all approximants f ∈ H ∞ to ϕ0 satisfying ϕ0 − f ∞ ≤ ρ. This problem is called the Nehari problem. If ρ = Γ, a solution ϕ of the Nehari problem (i.e., a symbol ϕ of Γ of norm at most ρ) is called optimal. If ρ > Γ, the solutions of the Nehari problem are called suboptimal). Clearly, the optimal solutions of the Nehari problem are the symbols of Γ of minimal norm. In §1 we present the original approach by Adamyan, Arov, and Krein which is based on studying unitary extensions of isometric operators defined on a subspace of a Hilbert space. We obtain a uniqueness criterion and parametrize all solutions in the case of nonuniqueness. In particular, we show that in the case of nonuniqueness there are solutions of the Nehari problem of constant modulus equal to ρ. In §2 we apply the results of §1 to the Nevanlinna–Pick interpolation problem. In case there are two distinct solutions of the Nevanlinna–Pick problem we show that there are solutions that are inner functions and we parametrize all solutions. Another approach to the Nehari problem (also due to Adamyan, Arov, and Krein) is given in §3. This approach works only in the case ρ > G and it also allows us to treat a more general problem, the so-called Nehari– Takagi problem. This problem can be described as follows. Suppose that Γ is a Hankel operator, ρ > 0, and m ∈ Z+ is such that sm (Γ) < ρ < sm−1 (Γ) if m > 0 and s0 (Γ) < ρ if m = 0. The problem is to describe all functions

148

Chapter 5. Parametrization of Solutions of the Nehari Problem

ψ ∈ L∞ such that Hψ ≤ ρ and rank(Γ − Hψ ) ≤ m. Clearly, this problem is just the Nehari problem if m = 0. We parametrize in §3 all solutions of the Nehari problem. In §4 we consider the Nehari problem for a block Hankel operator Γ and use the method of so-called one-step extensions developed by Adamyan, Arov, and Krein. We parametrize all solutions of the Nehari problem in the case of finite-dimensional spaces (i.e., in the case of Hankel operators with matricial symbols) and ρ > Γ. Finally, in §5 we obtain a parametrization formula in the general case of vectorial Hankel operators. We use an approach that also depends on studying unitary extensions of isometric operators. However, it is different from the original approach by Adamyan, Arov, and Krein presented in §1, and it is based on the notion of the characteristic function of a unitary colligation. Note that we return to the Nehari problem in Chapter 14.

1. Adamyan–Arov–Krein Parametrization in the Scalar Case In this section we use the original approach by Adamyan, Arov, and Krein to the Nehari problem in the scalar case. We start with a Hankel operator 2 and a number ρ ≥ Γ, and consider the corresponding Γ from H 2 to H− Nehari problem. We obtain a criterion for the uniqueness of a solution and find a parametrization formula for the solutions in the case there are at least two distinct solutions (this is always the case if ρ > Γ). Consider the auxiliary operator Aρ on L2 defined by 1 Aρ f = f + (Γf+ + Γ∗ f− ), ρ where def

f+ = P+ f,

def

f− = P− f,

f ∈ H 2.

We introduce a new pseudo-inner product on L2 by (f, g)ρ = (Aρ f, g). Let us show that Aρ ≥ O. We have (Aρ f, f )

2 Re(Γf+ , f− ) ρ f+ 2 + f− 2 − 2f+ · f− ≥ 0.

= f+ 2 + f− 2 + ≥

If ρ > Γ, then (·, ·)ρ is equivalent to the standard inner product on L2 . The pseudo-inner product (·, ·)ρ is nondegenerate on L2 unless ρ = Γ and the norm of Γ is attained on the unit ball of H 2 . Indeed (f, f )ρ = 0 if

1. Adamyan–Arov–Krein Parametrization in the Scalar Case

149

and only if 2 Re(Γf+ , f− ) = 0, ρ

f+ 2 + f− 2 + which is equivalent to

Γf+ = ρf+ ,

Γf+ = −ρf− .

Denote by K the Hilbert space obtained from L2 by factorization and completion with respect to (·, ·)ρ . From now on we denote the inner product in K by (·, ·)K , (f, g)K = (f, g)ρ for f, g ∈ L2 . 2 Let S be multiplication by z on L2 . Clearly, H 2 and H− are isometrically imbedded in K and 2 K = span{H 2 , H− } 2 with respect to (·, ·)K . It is also clear that S maps isometrically z¯H− + H2 2 2 onto H− +zH in the metric of K. Denote by V the closure of this isometric operator. Then the domain DV and the range RV of V are given by

DV = closK {f ∈ L2 : fˆ(−1) = 0}, RV = closK {f ∈ L2 : fˆ(0) = 0}. We consider minimal unitary extensions of V . Let K be imbedded in a Hilbert space H. A unitary operator U on H is called a unitary extension of V if U |DV = V. If in addition to that H = span{U k H : k ∈ Z}, the extension is called minimal. Let U be a minimal unitary extension of V and EU its spectral measure. Consider the operator-valued measure EU on the space K defined by EU (∆) = PE U (∆)|K, ∆ ⊂ T, where P is the orthogonal projection onto K. Define the scalar measure µU on T by dµU = ρ¯ z d(EU 1, z¯)H = ρ¯ z d(EU 1, z¯)K .

(1.1)

Theorem 1.1. For any minimal unitary extension U of V the measure µU is absolutely continuous with respect to Lebesgue measure and the density ψ=

dµU dm

(1.2)

satisfies ψ∞ ≤ ρ

and

Hψ = Γ.

(1.3)

Conversely if ψ satisfies (1.3), then there exists a minimal unitary extension U of V such that ψ is given by (1.2).

150


Theorem 1.1 describes the solutions of the Nehari problem with norm at most ρ in terms of minimal unitary extensions of V . It turns out that the map dµU U → ψ = dm is a one-to-one correspondence between the minimal unitary extensions of V and the solutions of the Nehari problem with norm at most ρ if we identify equivalent extensions. By equivalent extensions we mean the following. Let U1 and U2 be minimal extensions of V on spaces H1 and H2 . They are called equivalent if there is a unitary map W of H1 onto H2 such that W K is the identity and W U1 = U2 W.

(1.4)

It is easy to see that if U1 and U2 are equivalent, then EU1 = EU2 . U The following result shows that the map U → ψ, ψ = dµ dm , is a oneto-one correspondence between the equivalence classes of minimal unitary extensions of V and the solutions of the Nehari problem with norm at most ρ. Theorem 1.2. Let U1 and U2 be minimal unitary extensions of V on H1 and H2 . The following statements are equivalent: (i) U1 is equivalent to U2 ; (ii) EU1 = EU2 ;

(iii) ψ1 = ψ2 , where ψj =

dµUj dm

, j = 1, 2.

Proof of Theorem 1.1. Let U be a minimal unitary extension of V and EU its spectral measure. We have for all k ≥ 0 U k 1 = zk ,

U −k z¯ = z¯k+1 .

Since U is unitary, the vectors U k 1 are pairwise orthogonal for k ∈ Z, and so are U k z¯. Therefore we have for any trigonometric polynomials f and g:

f (ζ)g(ζ)(dEU (ζ)1, z¯)ρ = | (f (U )1, g(U )¯ z )H | T

≤

f (U )1 · g(U )¯ z = f L2 gL2 .

Therefore µU is absolutely continuous with respect to Lebesgue measure U and the density ψ = dµ dm satisfies ψ ∈ L∞ ,

ψL∞ ≤ ρ.

Let k ≥ 1. We have

d ˆ ψ(−k) = (EU (ζ)1, z¯)K dm(ζ) ψ(ζ)ζ k dm(ζ) = ρ ζ k−1 dm T T = ρ(U k−1 1, z¯)H = ρ(z k−1 , z¯)K = ρ(Aρ z k−1 , z¯) = (Γz k−1 , z¯). So Γ = Hψ .


151

Suppose that ψ∞ ≤ ρ and Γ = Hψ . Put ξ=

1 ψ, ρ

κ = 1 − |ξ|2 ,

L2κ = closL2 κ 1/2 L2 ,

Hψ = L2 ⊕ L2κ .

Clearly,

L2κ = f ∈ L2 : f {ζ : |ξ(ζ)| = 1} = O . Define the imbedding I = Iψ of K in Hψ on a dense subset of K by If = (f− + ξf+ ) ⊕ κ 1/2 f+ ,

f ∈ L2 .

Let us show that I extends to an isometric imbedding of K in Hψ . We have (If, Ig)

(f− + ξf+ , g− + ξg+ ) + (κ 1/2 f+ , κ 1/2 g+ ) 1 1 = (f− , g− ) + (f− , ψg+ ) + (ψf+ , g− ) ρ ρ 1 1 2 f (ψf , ψg ) + 1 − |ψ| , g + + + + + ρ2 ρ2 1 1 = (f− , g− ) + (f+ , g+ ) + (f− , P− ψg+ ) + (P− ψf+ , g− ) ρ ρ 1 1 = (f, g) + (f− , Γg+ ) + (Γf+ , g− ) = (f, g)K . ρ ρ Let U be the unitary operator defined on Hψ by =

U (f ⊕ g) = zf ⊕ zg. It is evident that U IDV = IVρ DV . So U is a unitary extension of V under the identification of K with IK. Let us show that U is minimal. Indeed, U n I z¯k = z n−k ⊕ O, U Iz = ξz n

k

n+k

⊕κ

k ≥ 2, n ∈ Z; 1/2 n+k

z

,

k ≥ 0, n ∈ Z.

Clearly, span{U n I z¯k : k ≥ 2, n ∈ Z} ∪ {U n z k : k ≥ 0, n ∈ Z} = Hψ . Let EU be the spectral measure of U . Then EU (∆) is multiplication by χ∆ on Hψ . Therefore

(EU (∆)I1, I z¯)Hψ = ζ ξ(ζ)dm(ζ). ∆

It follows that d d z (EU I1, I z¯)Hψ = ρ¯ (EU 1, z¯)K , dm dm which completes the proof. ψ = ρ¯ z

Proof of Theorem 1.2. If U1 and U2 are equivalent minimal unitary extensions of V and W : H1 → H2 is a unitary map such that W K = IK and (1.4) holds, then for any f, g in K (U1k f, g)H1 = (W ∗ U2k W f, g)H1 = (U2k f, g)H2 .

152


This implies (EU1 (∆)f, g)H1 = (EU2 (∆)f, g)H2 ,

f, g ∈ K,

which proves (i)⇒(ii). The implication (ii)⇒(iii) is trivial. Let us establish (iii)⇒(i). We have from (iii): (EU1 1, z¯)H1 = (EU2 1, z¯)H2 , which implies (U1k 1, z¯)H1 = (U2k 1, z¯)H2 ,

k ∈ Z.

(1.5)

Since U1 and U2 are extensions of V , it follows that U1k 1 = U2k 1 = z k ,

k ≥ 0,

U1−k z¯ = U2−k z¯ = z¯1+k ,

(1.6)

k ≥ 0.

(1.7)

H1 = span{U1k 1, U1k z¯, k ∈ Z},

(1.8)

H2 = span{U2k 1, U2k z¯, k ∈ Z}.

(1.9)

By minimality

Define the operator W : H1 → H2 by W U1k 1 = U2k 1,

W U1k z¯ = U2k z¯,

k ∈ Z.

It follows from (1.8) and (1.9) that to show that W is a unitary operator from H1 to H2 , it is sufficient to prove that (U1k 1, U1m 1)H1 = (U2k 1, U2m 1)H2 ,

(1.10)

(U1k z¯, U1m z¯)H1 = (U2k z¯, U2m z¯)H2 ,

(1.11)

(U1k z¯, U1m 1)H1 = (U2k z¯, U2m 1)H2 .

(1.12)

Clearly, (1.12) follows directly from (1.5). To establish (1.10) and (1.11), we assume (to be definite) that k ≥ m. Then (U1k 1, U1m 1)H1

(U1k , z¯, U1m z¯)H1

=

(U1k−m 1, 1)H1 = (z k−m , 1)K

=

(U2k−m 1, 1)H2 = (U2k 1, U2m 1)H2 ,

=

(¯ z , U m−k z¯)H1 = (¯ z , z¯1+k+m )K

(¯ z , U2m−k z¯)H2 = (U2k z¯, U2m z¯)H2 . Thus W is unitary. It follows from (1.6) and (1.7) that W DV = IDV . The identity W U1 = U2 W follows directly from (1.8), (1.9), and the definition of W . =


153

Uniqueness We are now going to consider the question of uniqueness of solutions ψ of the Nehari problem with ψ∞ ≤ ρ. Define the deficiency indices of the operator V by d1 = dim K DV ,

d2 = dim K RV .

Clearly, 0 ≤ d1 , d2 ≤ 1. Lemma 1.3. d 1 = d2 . Proof. Let J be the (real-linear) operator on L2 defined by Jf = z¯f¯. 2 Clearly, JH 2 = H− , J 2 = I, and Jf 2 = f 2 for any f in L2 . Let us show that 2 . ΓJh = JΓ∗ h, h ∈ H−

Indeed, let Γ = Hϕ , ϕ ∈ L∞ . We have ¯ = JP+ J(ϕ¯ ¯ = JP+ ϕh ΓJh = Hϕ Jh = P− ϕ¯ zh z h) ¯ = JΓ∗ h. In a similar way one can prove that Γ∗ Jh = JΓh,

h ∈ H 2.

It follows that (Jf, Jg)K = (f, g)K ,

f, g ∈ H 2 .

Indeed, let f+ = P+ f , f− = P− f . We have (Jf, Jg)K

1 1 (Aρ Jf, Jg) = (Jf, Jg) + (ΓJf− , Jg) + (Γ∗ Jf+ , Jg) ρ ρ 1 1 ∗ = (f, g) + (JΓ f− , Jg) + (JΓf+ , Jg) = (f, g)K . ρ ρ

=

Since obviously J1 = z¯ and J{f ∈ L2 : fˆ(−1) = 0} = {f ∈ L2 : fˆ(0) = 0}, it follows that JDV = RV , and so DV coincides with K if and only if so does RV . This implies that d1 = d2 . Lemma 1.4. The following statements are equivalent: (i) there is a unique solution ψ of the Nehari problem with ψ∞ ≤ ρ; (ii) d1 = d2 = 0. Proof. Suppose that d1 = d2 = 0. Then DV and RV coincide with K and so V extends by continuity to a unitary operator on K. Clearly, such an extension is unique.

154


If d1 = d2 = 1, then dim K DV = dim K RV = 1. Let g1 ∈ K DV , g2 ∈ K RV , g1 ρ = g2 ρ = 1. Given γ ∈ T we can define a unitary extension U (γ) of V by U (γ) f = f, f ∈ DV ,

U (γ) g1 = γg2 .

(1.13)

Clearly, U (γ) extends by linearity to a unitary extension of V on K and U (γ1 ) = U (γ2 ) for distinct γ1 and γ2 . If ρ > Γ, then (·, ·)ρ is equivalent to the initial inner product in L2 . So z¯ ∈ DV and by Lemma 1.4, there are infinitely many solutions of the Nehari problem. If ρ = Γ and the norm of Γ is attained on the unit ball of H 2 (the latter is fulfilled if Γ is compact), then as we have seen in §1.1, there exists a unique solution ϕ of the Nehari problem and |ϕ(ζ)| = ρ almost everywhere on T. However, we will see in this section that if Γ = ρ and Γ does not attain its norm on the unit ball of H 2 , the Nehari problem may have a unique solution or infinitely many solutions. The following theorem establishes a criterion of the uniqueness of a solution of the Nehari problem. Theorem 1.5. The following statements are equivalent: (i) there is a unique solution ψ of the Nehari problem with ψ∞ ≤ ρ; (ii) lim ((r2 I − Γ∗ Γ)−1 1, 1) = ∞; r→ρ+

(iii) 1 ∈ Range(ρ2 I − Γ∗ Γ)1/2 . We need the following elementary lemma. Lemma 1.6. Let A be a nonnegative operator on a Hilbert space H, def

(f, g)A = (Af, g),

1/2

f A = (f, f )A , f ∈ H.

Given h ∈ H, the linear functional f → (f, h) is continuous in · A if and only if lim (A + sI)−1 h, h < ∞, s→0+

which in turn is equivalent to the fact that h ∈ A1/2 H. Proof of Lemma 1.6. Let h = A1/2 x, x ∈ H. We have |(f, h)| = |(f, A1/2 x)| = |(A1/2 f, x)| ≤ x · A1/2 f = x · |(A1/2 f, A1/2 f )|1/2 = x · |(Af, f )|1/2 = x · fA . Thus the functional f → (f, h) is continuous in · A . On the other hand, if f → (f, h) is a continuous linear functional with norm c, then ((A + sI)−1 h, h) ≤ c (A + sI)−1 hA = c (A(A + sI)−1 h, h)1/2 ≤ c h1/2 A(A + sI)−1 h1/2 ≤ c h, since t(t + s)−1 ≤ 1 for t ≥ 0, and so A(A + sI)−1 ≤ c.


155

Suppose now that lim ((A + sI)−1 h, h) < ∞. Then s→0+

lim (A + sI)−1/2 h < ∞.

s→0+

There exists a sequence {sn }n≥0 of positive numbers such that lim sn = 0 n→∞

and the sequence {(A + sn I)−1/2 h}n≥0 weakly converges to an element x of H. Then obviously, w– lim A1/2 (A + sn I)−1/2 h = A1/2 x. n→∞

On the other hand, it follows easily from spectral theorem that A1/2 (A + sn I)−1/2 h → h in the strong operator topology, which implies that h = A1/2 x. Proof of Theorem 1.5. By Lemmas 1.3 and 1.4, (i) is equivalent to the fact that Rρ = Kρ . The latter means that the linear functional f → fˆ(0) is unbounded in · ρ . By Lemma 1.6 this is true if and only if lim (Aρ + sI)−1 1, 1 = ∞. (1.14) s→0

Let fs = (Aρ + sI)−1 1. We have by the definition of Aρ 1 1 1 = (1 + s)fs + ΓP+ fs + Γ∗ P− fs . ρ ρ Since 1 ∈ H 2 , it follows that 1 (1 + s)P− fs + ΓP+ fs = O ρ

(1.15)

1 1 = (1 + s)P+ fs + Γ∗ P− fs . ρ

(1.16)

and

Substituting P− fs from (1.15) in (1.16), we obtain 1 = (1 + s)P+ fs − and so

1 Γ∗ ΓP+ fs , (1 + s)ρ2

P+ fs = (1 + s)ρ2 (1 + s)2 ρ2 I − Γ∗ Γ)−1 1.

We have (Aρ + sI)−1 1, 1

= =

(fs , 1) = (P+ fs , 1) (1 + s)ρ2 (1 + s)2 ρ2 I − Γ∗ Γ)−1 1, 1 .

Thus (1.14) is equivalent to the fact that lim (1 + s)2 ρ2 I − Γ∗ Γ)−1 1, 1 = ∞, s→0+

which is obviously equivalent to (ii). The fact that (ii) and (iii) are equivalent follows from Lemma 1.6.

156


Let d0 = d1 = 1. Then by Theorem 1.1, the solutions of the Nehari problem are in a one-to-one correspondence with the minimal unitary extensions of V . There can be two different types of extensions. Extensions of the first type are defined on the space K while extensions of the second type are defined on a space larger than K. It is easy to see that all extensions of the first type are described by (1.13); they can be parametrized by the complex numbers of modulus one. First we are going to study solutions of the Nehari problem that correspond to minimal unitary extensions on the space K.

Canonical Functions A solution ψ of the Nehari problem with norm at most ρ is called a ρ-canonical function if the operator V has deficiency indices equal to 1 and the corresponding minimal unitary extension of V is an operator on K. We shall call ρ-canonical functions canonical if this does not lead to confusion. Theorem 1.7. If ψ is a canonical function, then |ψ| = ρ almost everywhere. Proof. As in the proof of Theorem 1.1 put ξ=

1 ψ, ρ

κ = 1 − |ξ|2 ,

L2κ = closL2 κ 1/2 L2 ,

If = (f− + ξf+ ) ⊕ κ 1/2 f+ ,

Hψ = L2 ⊕ L2κ ,

f ∈ L2 .

Then I extends to an isometric imbedding of K in Hψ . Put ˜ = IK, K

H+ = IH 2 ,

2 H− = IH− ,

˜ V = IRV . R

Then 2 ⊕ {O}, H− = H−

H+ = {ξf ⊕ κ 1/2 f : f ∈ H 2 }.

It follows that ˜ = clos(H− + H+ ), K

˜ V = clos(H− + zH+ ). R

˜ V . Let f ⊕ g ∈ Hψ R ˜ V . Since f ⊕ g ⊥ H− , it Let us describe Hψ R 2 follows that f ∈ H . The fact that f ⊕ g ⊥ zH+ means that ¯ + κ 1/2 g, zω) = 0 (f ⊕ g, zξω ⊕ zκ 1/2 ω)Hψ = (f, zξω) + (g, zκ 1/2 ω) = (ξf ¯ + κ 1/2 g ∈ zH 2 . Thus for any ω in H 2 . That is, ξf − ¯ + κ 1/2 g ∈ zH 2 }. ˜ V = {f ⊕ g : f ∈ H 2 , ξf Hψ R −

(1.17)

˜ V ) = 1. Let us show that this is possible By the assumption dim(Hψ R only if κ = 0. This would imply that |ψ| = ρ almost everywhere. ¯ + κ 1/2 g ∈ zH 2 . Suppose ˜ V . Then f ∈ H 2 , η def Let f ⊕ g ∈ Hψ R = ξf − that κ = O. Then there exist a set ∆ of positive measure and a positive number δ such that κ ≥ δ on ∆.


157

It is easy to construct a nonconstant function α in H ∞ such that Im α(ζ) = 0 for ζ ∈ ∆. Indeed, let χ = χ∆ . Put α = exp −(−χ ˜ + iχ)2 . Clearly,

˜2 ) ∈ L∞ . |α| = exp − Re(−χ ˜ + iχ)2 = exp(χ2 − χ

On the other hand, for ζ ∈ ∆

2 α(ζ) = exp χ ˜ (ζ) ∈ R.

So α ∈ H ∞ and Im α(ζ) = 0 for ζ ∈ ∆. We can define now the functions f∗ , η∗ , g∗ by f∗ = αf, g∗ (ζ) =

η∗ = α ¯ η, κ −1/2 (ζ) η∗ (ζ) − ξ(ζ)f∗ (ζ) , ζ ∈ ∆, ζ ∈ ∆.

α(ζ)g(ζ),

2 ˜ ρ . Clearly, f∗ ∈ H 2 , η∗ ∈ zH− , and Let us show that f∗ ⊕ g∗ ∈ Hψ R 2 1/2 ¯ ∗ + κ g∗ = η∗ . Let ζ ∈ ∆. Then g∗ ∈ Lκ . We have to prove that ξf

ξ(ζ)f∗ (ζ) + κ 1/2 (ζ)g∗ (ζ) = ξ(ζ)α(ζ)f (ζ) + κ 1/2 (ζ)α(ζ)g(ζ) = α(ζ)η(ζ) = α(ζ)η(ζ) = η∗ (ζ), since α(ζ) is real for ζ ∈ ∆. Let ζ ∈ ∆. We have ξ(ζ)f∗ (ζ) + κ 1/2 (ζ)g∗ (ζ) =

ξ(ζ)α(ζ)f (ζ) + α(ζ)η(ζ) − ξ(ζ)α(ζ)f (ζ)

= α(ζ)η(ζ) = η∗ (ζ), ˜ V (see (1.17)). which proves that f∗ ⊕ g∗ ∈ Hψ R Clearly, f ⊕ g and f∗ ⊕ g∗ are linear independent, which contradicts the ˜ V = 1. fact that dim Hψ ⊕ R Theorem 1.8. Let ψ be a canonical function. Then ψ admits a representation ¯ ψ = ρ h/h,

(1.18)

where h is an outer function in H 2 . Proof. Let ξ = ψ/ρ. By Theorem 1.7, |ξ(ζ)| = 1 almost everywhere on ˜ V , then T and Hψ = L2 . If f ∈ Hψ R ¯ = η, ξf

2 f ∈ H 2 , η ∈ zH− .

It follows that ξ = f /η. The functions f and η¯ belong to H 2 and have the same moduli. So f and η admit factorizations f = kϑ1 ,

η = k¯ϑ¯2 ,

158


where k is an outer function in H 2 , and ϑ1 and ϑ2 are inner functions. ¯ If ϑ1 ϑ2 = const, then k and ϑ1 ϑ2 k belong to Hψ R ˜V . Then ξ = ϑ1 ϑ2 k/k. Indeed, ¯ = ηk = k¯ϑ¯2 ϑ¯1 ∈ zH 2 , ξk − f ηϑ1 ϑ2 k 2 ξϑ1 ϑ2 k = = k¯ ∈ zH− . f Clearly, k and ϑ1 ϑ2 k are linear independent, which contradicts the fact ˜ is one-dimensional. that Hψ R Thus ϑ1 ϑ2 = c, |c| = 1, which implies ¯ ξ = c k/k. ¯ Let ζ ∈ T, ζ 2 = c¯. Put h = ζk. Clearly, h is an outer function and ξ = h/h. Remark. A representation of ψ in the form (1.18) is unique up to a multiplicative constant. Moreover, if ψ = ρϕ1 /ϕ¯2 , 2

where ϕ1 and ϕ2 are in H , then ϕ1 and ϕ2 are scalar multiples of h. ¯ 1 = ϕ¯2 , so ϕ1 ∈ Hψ R ˜ V , which implies that Indeed, in this case ξϕ ϕ1 /h = const. Since ϕ2 has the same modulus as ϕ1 , it follows that ϕ2 = ϑϕ1 , where ϑ is an inner function. If ϑ = const, then the same ˜ V > 1. reasoning as in the proof of Theorem 1.8 shows that dim Hψ R It follows from Theorem 1.7 that if the Nehari problem has at least two distinct solutions, then there exists a solution with constant modulus. As we have seen in §1.1, in the case ρ = Γ and Γ attains its norm on the unit ball of H 2 there is a unique solution of the Nehari problem and it has constant modulus. The following example shows that in the case of uniqueness the solution does not necessarily have constant modulus. Example. Let ¯ ψ = χϑ, where χ is the characteristic function of I = {eit : −α < t < α},

0 < α < π,

ζ +1 . ζ −1 Let us show that ψ is the only solution of the Nehari problem with ρ = 1 and Γ = Hψ . Suppose that f is a nonzero function in H ∞ such that ψ − f ∞ ≤ 1. Then χ − ϑf ∞ ≤ 1 and so |(ϑf )(ζ) − 1| ≤ 1 for ζ ∈ I. It follows that Re(ϑf )(ζ) ≥ 0 for ζ ∈ I. Since ϑf ∈ H ∞ , the inner factor of ϑf extends analytically across I (see Garnett [1], Ch. II, Prob. 14). However, ϑ does not extend analytically across I. ϑ(ζ) = exp


159

Theorem 1.9. Let ψ be a solution of the Nehari problem for Γ such that |ψ(ζ)| = Γ = ρ almost everywhere on T. Then the Nehari problem for Γ = Hψ has only one solution with norm at most ρ if and only if ψ does not admit a representation ψ = f /¯ g,

f, g ∈ H 2 .

(1.19)

Proof. Let ξ = ψ/ρ. Then κ = 0 a.e. on T. So Hψ = L2 and ¯ ∈ zH 2 }. ˜ V = {f ∈ H 2 : ξf Hψ R −

(1.20)

˜ V and f is a nonzero function in Hψ R ˜ V , then ψ satisfies (1.19) If Hψ = R 1 ¯ 2 with g = ρ ξ f ∈ H− . ˜ V , and so the Conversely, if ψ is represented by (1.19), then f ∈ Hψ R Nehari problem has infinitely many solutions. The following theorem contains simple facts about canonical functions. Theorem 1.10. Let ψ be a solution of the Nehari problem for Γ (i.e., Hψ = Γ). The following assertions hold: (i) If ψ admits a representation ¯ ψ = ρ h/h, where h ∈ H 2 , 1/h ∈ H 2 , then ψ is a canonical function. (ii) If ψ takes values in an arc of the circle {ζ : |ζ| = ρ} of angle less than π, then ψ is a canonical function. ¯ where h ∈ H 2 . Then ψ is a ρ-canonical function with (iii) Let ψ = ρh/h, Hψ < ρ if and only if the Toeplitz operator Th/h i s invertible on H 2 . ¯ ˜ V if and only if Proof. (i). It follows from (1.20) that f ∈ Hψ R f ∈ Ker Tz¯h/h = 1. Obviously, ¯ . Let us show that dim Ker Tz¯h/h ¯ h ∈ Ker Tz¯h/h ¯ . = Tz¯Th/h We have Tz¯h/h ¯ ¯ . Since dim Ker Tz¯ = 1, it remains to prove that ∗ = {O} or, which is equivalent, that Th/h = Th/h¯ has dense range. Ker Th/h ¯ ¯ 2 Let g = 1/h ∈ H . Clearly, g is outer and Th/h¯ = Tg¯/g . Now Tg¯/g has dense range by Theorem 4.4.10, which proves that ˜ V = dim K RV = 1. dim Hψ R Hence, Hψ = Iψ K and so ψ is canonical. (ii). In this case ψ can be represented in the form ψ = ρ exp i(c + α), where c ∈ R and α is a real-valued function, α∞ < π/2. Let 1 α + i(α + c)). h = exp (−˜ 2 Then by Zygmund’s theorem h ∈ H 2 , 1/h ∈ H 2 (see Appendix 2.1). Obvi¯ The result follows from (i). ously, ψ = ρh/h.

160


(iii). If Th/h is invertible, then by Theorem 3.1.11, Hh/h¯ < 1, and so ¯ Hψ < ρ. In this case dim Ker Tz¯h/h = 1 and as in (ii) we can show that ¯ ˜ dim Hψ RV = 1. ˜ V = 1 and Hψ < ρ. Then dim Ker Tz¯h/h Suppose that dim Hψ R = 1. ¯ Therefore Ker Th/h = {O}, since if Th/h ¯ , ¯ ¯ f = O, then h + zf ∈ Ker Tz¯h/h = 1. Hence, Th/h¯ has dense which contradicts the fact that dim Ker Tz¯h/h ¯ range. ¯ Let u = h/h. Then Tu has dense range in H 2 . Indeed, let h = gϑ, where g is outer and ϑ is inner. Then u = ϑ¯2 g¯/g and by Theorem 4.4.10, Tu has dense range in H 2 . It follows now from Corollary 4.4.3 that Hu = Hu¯ . Since Hψ < ρ, it follows that Hu¯ < 1. Hence, Hu < 1, which implies that Tu is invertible (see Theorem 3.1.11). Now we can give an example of a ρ-canonical function ψ for which Γψ = ρ. In this case the Nehari problem has infinitely many solutions and so ψ has different best approximations by bounded analytic functions. In §1.1 we have already constructed an example of a function ϕ in L∞ , which has different best approximations. We have shown here that for that function ϕ, there exists a function f in H ∞ such that ϕ−f ∞ = Γϕ and ϕ − f is a canonical function. The following example produces an explicit canonical function ψ satisfying the above requirements. Example. Let h be an outer function such that h ∈ H 2 , 1/h ∈ H 2 , but |h|2 does not satisfy the Helson–Szegö condition (see §3.2). To produce such a function h, it suffices to pick any positive function w such that w ∈ L1 , 1/w ∈ L1 , but w ∈ L1+ε with any positive ε and to take h to be an outer function with modulus w1/2 (see the remark after the proof of Theorem is noninvertible. 2.5). Then by Theorem 2.2.5, the Toeplitz operator Th/h ¯ By Theorem 4.4.10 and Corollary 4.4.3, Hh/h ¯ = 1. Therefore ¯ = Hh/h would be invertible by Theorem 3.1.11. Hh/h¯ = 1; otherwise Th/h ¯ ¯ is a It follows now from Theorem 1.10, (i) and (iii) that ψ = ρh/h ρ-canonical function and Hψ = ρ.

Parametrization of Canonical Solutions We are going to describe all canonical solutions of the Nehari problem in the case of nonuniqueness. In this case there exists a canonical solution. Let us fix a canonical solution ψ: Hψ = Γ,

ψ∞ = ρ.

It follows from Theorem 1.8 that ψ admits a representation ¯ ψ = ρh/h, where h is an outer function in H 2 .


161

Consider the function G defined by *2 . G = |h|2 + i|h| G is an outer function and Re G(ζ) ≥ O, ζ ∈ D. Let k be the function analytic in D defined by G−1 . (1.21) k= G+1 Clearly, 1+k = G. (1.22) 1−k Since Re G ≥ O, it follows that k ∈ H ∞ and k∞ ≤ 1. The following theorem gives a parametrization of all canonical solutions of the Nehari problem. ¯ be a canonical function of the Nehari Theorem 1.11. Let ψ = ρh/h problem, where h is an outer function in H 2 of norm one and let k be the function defined by (1.21). Then for any w ∈ T the function ϕ defined by ϕ(ζ) = ρ

h(ζ)

·

h(ζ)

1 − k(ζ) 1 − k(ζ)

·

w − k(ζ) 1 − wk(ζ)

(1.23)

is ρ-canonical. Conversely, if ϕ is a ρ-canonical function, then there exists a complex number w of modulus one such that (1.23) holds. Proof. Let ϕ be a canonical function. Let Uϕ and Uψ be the minimal unitary extensions of V that correspond to ϕ and ψ. Let Eϕ = EUϕ = EUϕ ,

Eψ = EUψ = EUψ

be the spectral measures of Uϕ and Uψ . It follows from (1.1) and (1.2) that ϕ − ψ = ρ¯ z We have

(P+ (ϕ − ψ))(λ) =

T

d((Eϕ − Eψ )1, z¯)K . dm ϕ(τ ) − ψ(τ ) dm(τ ), 1 − λ¯ τ

(1.24)

λ ∈ D.

Substituting ϕ − ψ from (1.24), we obtain

d((Eϕ − Eψ )1, z¯)K (λ) (ϕ) (ψ) (P+ (ϕ − ψ))(λ) = ρ = ρ Rλ − Rλ 1, z¯ , τ −λ K T where Rλ = (Uϕ − λI)−1 , Rλ = (Uψ − λI)−1 are the resolvents of Uϕ and Uψ . Since Hϕ = Hψ , it follows that P− ϕ = P− ψ and so for ζ ∈ T (ϕ)

(ψ)

(ϕ − ψ)(ζ)

lim (P+ (ϕ − ψ)) (rζ) (ϕ) (ψ) = ρ lim Rrζ − Rrζ 1, z¯ . =

r→1−

r→1−

K

(1.25)

162


We identify K and Hψ = L2 with the help of the imbedding Iψ (since ψ is canonical, Iψ is onto): Iψ f = f− + ξf+ ,

f ∈ L2 ∩ K,

(1.26)

where ξ = ρ−1 ψ, f− = P− f , f+ = P+ f . To avoid confusion, we introduce the notation ˜ϕ = Iψ Uϕ I −1 ˜ψ = Iψ Uψ I −1 , U U ψ

ψ

˜ϕ is defined in terms of Iψ rather than Iϕ !). (U ¯ It is easily seen from (1.20) that We have ξ = h/h. ˜ V = {λh : λ ∈ C}. L2 R ˜ ∗ h ∈ Hψ D ˜V . It is evident that z¯h = U ψ ˜ψ on L2 D ˜ V . So U ˜ϕ f = zf , f ⊥ D ˜V . ˜ϕ coincides with U The operator U Clearly, ˜ V ) = L2 R ˜V . ˜ϕ (L2 D U ˜ϕ is unitary, there exists a complex number w of modulus one such Since U that ˜ϕ z¯h = wh. ¯ U 2 ˜ϕ acts on L as follows: Therefore U ˜ϕ f = zf + (w ¯ − 1)(f, z¯h)h, U

f ∈ L2 .

(1.27)

Formula (1.25) can be rewritten in the following way. Let ˜ (ϕ) = Iψ R(ϕ) I −1 , R λ λ ψ

˜ (ψ) = Iψ R(ψ) I −1 R λ λ ψ

˜ϕ and U ˜ψ . Clearly, Iψ 1 = ξ, Iψ z¯ = z¯. be the resolvents of U Then ˜ (ϕ) − R ˜ (ψ) ξ, z¯ , ζ ∈ T. R (ϕ − ψ)(ζ) = ρ lim rζ rζ r→1−

(1.28)

Let λ = rζ. We have ˜ (ϕ) − R ˜ (ψ) R λ λ

˜ϕ − λI)−1 − (U ˜ψ − λI)−1 = (U ˜ϕ − λI)(U ˜ϕ − λI)−1 I − (U ˜ψ − λI)−1 = (U (ϕ)

(ψ)

˜ (U ˜ψ − U ˜ϕ )R ˜ . =R λ λ Therefore by (1.27) ˜ (ϕ) − R ˜ (ψ) ξ, z¯h R ˜ (ψ) ξ = (1 − w) ˜ (ϕ) h. ¯ R R λ λ λ λ

(1.29)

(1.30)

Let us show that ˜ (ϕ) h = R λ

˜ (ψ) h R λ . ˜ (ψ) h, z¯h ¯ R 1 − (1 − w) λ

(1.31)


163

We have ˜ (ψ) h, z¯h R ˜ψ − U ˜ (ψ) h ˜ (ϕ) h = R ˜ (ϕ) h − R ˜ (ϕ) U ˜ϕ R ¯ R 1 − (1 − w) λ λ λ λ λ by (1.27). Formula (1.31) follows now from (1.29). Therefore we have from (1.30) ˜ (ψ) ξ, z¯h R ˜ (ψ) h ¯ R (1 − w) λ λ (ϕ) (ψ) ˜ −R ˜ . ξ= R λ λ ˜ (ψ) h, z¯h ¯ R 1 − (1 − w) λ It follows that

˜ (ψ) ˜ (ϕ) − R R λ λ

We have ˜ (ψ) ξ, z¯h R λ

˜ (ψ) h, z¯ R λ

˜ (ψ) h, z¯h R λ

= ρ

T

˜ (ψ) h, z¯ ˜ (ψ) ξ, z¯h R ¯ R (1 − w) λ λ ξ, z¯ = . (ψ) ˜ ¯ Rλ h, z¯h 1 − (1 − w)

h(τ ) τ h(τ ) dm(τ ) = ρ h(τ ) τ − λ

T

τ h(τ ) dm(τ ) = ρh(λ), τ −λ

τ h(τ ) = dm(τ ) = h(λ), T τ −λ

h(τ ) |h(τ )|2 = τ h(τ )dm(τ ) = dm(τ ). τ T τ −λ T 1 − λ¯

Let k be the function defined by (1.21). Then

1 + k(λ) τ +λ |h(τ )|2 dm(τ ). = G(λ) = 1 − k(λ) T τ −λ Since h2 = 1, we have 1 1 − k(λ)

Thus

1 + k(λ) 1 1+ = 2 1 − k(λ)

τ +λ 1 2 2 |h(τ )| + |h(τ )| dm(τ ) = 2 T τ −λ

|h(τ )|2 = dm(τ ). τ T 1 − λ¯

˜ (ψ) )ξ, z¯ ˜ (ϕ) − R (R λ λ

= ρ

2 ¯ (1 − w)(h(λ)) ¯ 1 − (1 − w)/(1 − k(λ))

= ρ

(w − 1)(h(λ))2 (1 − k(λ)) . 1 − wk(λ)

Let ζ ∈ T. It follows from (1.28) that ϕ(ζ) = ρ

h(ζ) h(ζ)

+ρ

(w − 1)(1 − k(ζ))(h(ζ))2 . 1 − wk(ζ)

164


Note that |h(ζ)|2 = Re Therefore ϕ(ζ) = ρ

h(ζ) h(ζ)

1 − |k(ζ)|2 1 + k(ζ) = . 1 − k(ζ) |1 − k(ζ)|2

(w − 1)(1 − k(ζ))(1 − |k(ζ)|2 ) 1+ (1 − wk(ζ))|1 − k(ζ)|2

(1.32) ,

and after elementary transformations we obtain ϕ(ζ) = ρ

h(ζ) 1 − k(ζ) w − k(ζ) . h(ζ) 1 − k(ζ) 1 − wk(ζ)

Suppose now that w ∈ T and ϕ is given by (1.23). Then we can define an extension of V as follows. Let H = L2 and let Iψ be the imbedding of K in L2 given by (1.26). Define the unitary operator U on L2 by ¯ − 1)(f, zh)h, U f = zf + (w

f ∈ L2 .

Then U is a minimal unitary extension of Iψ V Iψ−1 . Let ϕ1 be the solution of the Nehari problem that corresponds to U . Then ϕ1 is a canonical function ˜ϕ . and U = U 1 It follows from the part of the theorem already proved that h 1 − k w − k¯ ϕ1 = ρ ¯ . h 1 − k¯ 1 − wk Thus ϕ1 = ϕ and ϕ is a canonical solution of the Nehari problem. Let ω = (1 − k)h. Then ω is an outer function and formula (1.23) can be rewritten as follows: ω w − k¯ ϕ=ρ , w ∈ T. (1.33) ω ¯ 1 − wk In this parametrization the function k depends on the choice of a canonical function ψ. However, the following result shows that k is determined uniquely by the Hankel operator Γ up to a unimodular constant. Theorem 1.12. Let ψ1 and ψ2 be canonical solutions of the Nehari problem, h1 h2 ψ1 = ρ ¯ , ψ2 = ρ ¯ , h1 h2 2 where h1 and h2 are outer functions in H of norm one, and let k1 and k2 be the functions defined by 1 + kj * 2 = |hj |2 + i|h j| , 1 − kj Then k1 /k2 is a unimodular constant. If ωj = (1 − kj )hj , then ω1 /ω2 is a unimodular constant.

j = 1, 2.

j = 1, 2,


165

Proof. Let U1 and U2 be the minimal unitary extensions of V on the space K that correspond to ψ1 and ψ2 . Then U1 |DV = U2 |DV .

(1.34)

Let x ∈ K DV , x = 1, y = U1 x. Then y ∈ K RV and U2 x = γy

(1.35)

for some unimodular constant γ. Consider the unitary map Iψ1 of K onto L2 given by 1 Iψ1 f = f− + ψ1 f+ , ρ

f ∈ K ∩ L2 .

˜ψ = Iψ U1 I −1 is multiplication by z on L2 . As we Then the operator U 1 1 ψ1 z h1 , where c have seen in the proof of Theorem 1.11, Iψ1 y = ch1 , Iψ1 x = c¯ is a unimodular constant. By the definition of k1 we have

1 + k1 (ζ) τ +ζ = |h1 (τ )|2 dm(τ ). 1 − k1 (ζ) T τ −ζ Since h1 2 = 1, it follows that

1 k1 (ζ) ˜ψ − ζI −1 h1 , h1 = ζ |h1 (τ )|2 dm(τ ) = ζ U 1 1 − k1 (ζ) τ −ζ T ˜ψ − ζI)−1 ch1 , ch1 = ζ (U1 − ζI)−1 y, y . = ζ (U 1 K It is also clear that 1 = 1 − k1 (ζ)

T

τ |h1 (τ )|2 dm(τ ) = U1 (U1 − ζI)−1 y, y K . τ −ζ

(1.36)

If we do the same with ψ2 , we obtain k2 (ζ) 1 − k2 (ζ)

= ζ (U2 − ζI)−1 U2 x, U2 x K = ζ (U2 − ζI)−1 γy, γy K = ζ (U2 − ζI)−1 y, y K .

(1.37)

It is easy to see from (1.34) and (1.35) that (U2 − ζI)−1 f − (U1 − ζI)−1 f = −(U2 − ζI)−1 (U2 − U1 )(U1 − ζI)−1 f = (1 − γ) (U1 − ζI)−1 f, x K (U2 − ζI)−1 y,

f ∈ K.

166


Therefore k1 (ζ) k2 (ζ) − = ζ (U2 − ζI)−1 y − (U1 − ζI)−1 y, y K 1 − k2 (ζ) 1 − k1 (ζ) = (1 − γ)ζ (U1 − ζI)−1 y, x K (U1 − ζI)−1 y, x K k2 (ζ) (U2 − ζI)−1 y, y K = (1 − γ) 1 − k2 (ζ) k2 (ζ) = (1 − γ) U1 (U1 − ζI)−1 y, y K 1 − k2 (ζ) k2 (ζ) 1 = (1 − γ) 1 − k2 (ζ) 1 − k1 (ζ) by (1.37) and (1.36). This easily implies that k1 (ζ) = γk2 (ζ). Let us now show that ω1 and ω2 differ by a unimodular constant. Since ω1 and ω2 are outer, it is sufficient to show that |ω1 | = |ω2 |. It follows from (1.22) that 1 + kj 1 − |kj |2 |hj |2 = Re = , j = 1, 2, 1 − kj |1 − kj |2 and so |ω1 |2 = |1 − k1 |2 · |h1 |2 = 1 − |k1 |2 = 1 − |k2 |2 = |1 − k2 |2 |h2 |2 = |ω2 |2 .

Parametrization of All Solutions We have proved that each unimodular constant w determines by formula (1.33) a canonical solution of the Nehari problem. The following result claims that if w ranges over the unit ball of H ∞ , formula (1.33) describes all solutions. Theorem 1.13. Let Γ be a Hankel operator, Γ ≤ ρ. Suppose that the set of solutions {ϕ ∈ L∞ : Hϕ = Γ, ϕ ≤ ρ}

(1.38)

¯ be a of the Nehari problem contains two different functions. Let ψ = ρh/h 2 canonical solution, where h is an outer function in H of norm one, let k be the function defined by (1.21), and let ω = (1 − k)h. Then the map w → ϕw = ρ

ω w − k¯ ω ¯ 1 − wk

(1.39)

is an isomorphism from the unit ball of H ∞ onto the set (1.38) of solutions of the Nehari problem. Remark. It is easy to see that the map (1.39) is an isomorphism between the unit ball of L∞ and the ball of radius ρ of L∞ .


167

Proof. Let us first prove that ϕw is a solution of the Nehari problem for any w in the unit ball of H ∞ . Clearly, h 1 − k 1 − k¯ ϕ1 = ρ ¯ = ψ. h 1 − k¯ 1 − k Let w ∈ H ∞ and wH ∞ ≤ 1. We have 1 − k w − k¯ h ϕ1 − ϕ w = ρ ¯ 1 − h 1 − k¯ 1 − wk 1 − k w − k¯ h2 = ρ 2 1− |h| 1 − k¯ 1 − wk = ρ

h2 (1 − w)(1 − |k|2 ) . ¯ |h|2 (1 − k)(1 − wk)

Bearing in mind that |h|2 =

1 − |k|2 |1 − k|2

(see (1.32)), we obtain ϕ1 − ϕw = ρh2

(1 − w)|1 − k|2 (1 − w)(1 − k) = ρh2 . ¯ 1 − wk (1 − k)(1 − wk)

(1.40)

Since w∞ ≤ 1 and k∞ ≤ 1, it follows that Re(1 − wk) ≥ O and so 1 − wk is an outer function (see Appendix 1.2). Obviously, ϕ1 − ϕw ∈ L∞ , which implies that ϕ1 − ϕw ∈ H ∞ . This proves that Hϕw = Hϕ1 , so ϕw is a solution of the Nehari problem. Suppose now that ϕ is a solution of the Nehari problem, that is, Hϕ = Γ and ϕ∞ ≤ ρ. Then as we have noticed above, there is a function w in the def

unit ball of L∞ such that ϕ = ϕw, which means that f = ϕ1 − ϕw ∈ H ∞ . For our convenience we can assume without loss of generality that ρ = 1. We have |1 − f (ζ)ϕ1 (ζ)| = |ϕ1 (ζ) − f (ζ)| = |ϕw(ζ)| ≤ 1,

ζ ∈ T.

Therefore f ϕ1 takes values in {λ ∈ C : Re λ ≥ 0}. Hence, there exists a real-valued function α such that π π − ≤ α(ζ) ≤ , ζ ∈ T, 2 2 and f (ζ)ϕ1 (ζ) = |f (ζ)| exp iα(ζ),

ζ ∈ T.

(1.41)

Since |1 − f (ζ)ϕ1 (ζ)| ≤ 1, it follows elementarily that |f (ζ)| = |f (ζ)ϕ1 (ζ)| ≤ 2 cos α(ζ),

ζ ∈ T.

(1.42)

Put g = exp(˜ α − iα).

(1.43)

168


Then g is an outer function. Consider the harmonic extensions of α and α ˜ to the unit disk; we keep the same notation α and α ˜ for them. Then g(ζ) = exp(˜ α(ζ) − iα(ζ)),

ζ ∈ D.

Clearly,

π π ≤ α(ζ) ≤ , ζ ∈ D. 2 2 Therefore Re g(ζ) ≥ 0 for ζ ∈ D. Let us show that gf ∈ H 1 . We have by (1.42) −

|g(ζ)f (ζ)| ≤ 2|g(ζ)| cos α(ζ) = 2 Re g(ζ),

ζ ∈ T.

The function Re g is a nonnegative harmonic function in D and so

|g(ζ)f (ζ)|dm(ζ) ≤ 2 Re g(ζ)dm(ζ) T T

Re g(rζ)dm(ζ) = 2 Re g(0) ≤ 2 lim inf r→1−

T

(see Appendix 2.1). Being the product of an outer function and a function in H ∞ , the function gf belongs to H 1 . We are going to establish that gf is a positive scalar multiple of h2 . Being in H 1 , the function gf admits a factorization gf = ϑh21 , where ϑ is an inner function and h1 is an outer function in H 2 . Then ϑh21 gf gf ϑh1 = = = . ¯ |h1 |2 |h1 |2 |gf | h1 It follows from (1.41) and (1.43) that gf ϕ¯1 = |gf ϕ¯1 | = |gf |, which implies h ϑh1 = ϕ1 = ¯ . ¯ h1 h However, being a canonical function, ϕ1 admits a unique (up to a constant η2 , η1 , η2 ∈ H 2 (see the Remark factor) representation of the form ϕ1 = η1 /¯ ¯ ¯ Thus after Theorem 1.8). Therefore ϑh1 = ch, h1 = ch. gf = ϑh21 = |c|2 h2 . It follows that Re

h2 (ζ) ≥ 0, f (ζ)

ζ ∈ D,

(1.44)

since Re g(ζ) ≥ 0 in D. Recall that by (1.40) f = ϕ1 − ϕw = h2 which implies h2 (ζ) 1 = f (ζ) 2

(1 − w)(1 − k) , 1 − wk

1 + k(ζ) 1 + w(ζ) + 1 − k(ζ) 1 − w(ζ)

,

ζ ∈ T.

(1.45)


Since

1 + k(ζ) = |h(ζ)|2 , 1 − k(ζ)

Re

169

ζ ∈ T,

we obtain Re

1 + w(ζ) h2 (ζ) 1 − |h(ζ)|2 = ≥ 0, f (ζ) 2 1 − w(ζ)

ζ ∈ T.

(1.46)

Let

h2 (ζ) def 1 , w(ζ) = |f (ζ)|2 , ζ ∈ T. f (ζ) 2 Obviously, w ∈ L1 . However, it follows from (1.44) that the function v on T, being the boundary-value function of the positive harmonic function Re(h2 /f ), also belongs to L1 (see Appendix 2.1). Consider the harmonic extensions of v and w to the unit disk, keeping for them the same notation v and w. We have 1 + k(ζ) 1 , ζ ∈ D. (1.47) w(ζ) = Re 2 1 − k(ζ) def

v(ζ) = Re

It follows from the positivity of Re(h2 /f ) that Re

h2 (ζ) ≥ v(ζ), f (ζ)

ζ ∈ D,

(1.48)

(see Appendix 2.1). Inequality (1.46) means that v(ζ) − w(ζ) ≥ 0, ζ ∈ T, which implies v(ζ) − w(ζ) ≥ 0,

ζ ∈ D,

and together with (1.47) and (1.48) this yields Re

h2 (ζ) 1 1 + k(ζ) − Re ≥ 0, f (ζ) 2 1 − k(ζ)

ζ ∈ D.

(1.49)

Let

h2 (ζ) 1 1 + k(ζ) − . f (ζ) 2 1 − k(ζ) It follows from (1.49) that Re η(ζ) ≥ 0, ζ ∈ D. Therefore η is an outer function and we obtain from (1.45) def

η(ζ) =

1 + w(ζ) , ζ ∈ T. 1 − w(ζ) Hence, the function w analytically extends to D by η(ζ) − 1 , ζ ∈ D. w(ζ) = η(ζ) + 1 η(ζ) =

Since Re η(ζ) ≥ 0, ζ ∈ D, it follows that |w(ζ)| ≤ 1, ζ ∈ D. Remark. We have just proved that formula (1.39) parametrizes all solutions of the Nehari problem when w ranges over the unit ball of H ∞ . Theorem 1.11 claims that to describe the canonical functions, we have to substitute in (1.39) unimodular constants w. It is easy to see that a solution

170


ϕ in (1.39) has constant modulus ρ is and only if w is an inner function. This allows us to construct solutions of the Nehari problem of constant modulus ρ that are not canonical.

2. Parametrization of Solutions of the Nevanlinna–Pick Problem We apply in this section the results of §1 to the Nevanlinna–Pick interpolation problem. We consider here the case of infinitely many distinct points ζj , j ≥ 0, in D. The case of finitely many points can be treated in the same way. In a similar way one can consider the case of multiple interpolating, i.e., interpolating not only values of a function but also values of its derivatives up to a certain order. Let ζj , j ≥ 0, be a sequence of distinct points in D and let wj , j ≥ 0, be complex numbers satisfying |wj | < 1. We are looking for functions f ∈ H ∞ satisfying: f (ζj ) = wj ,

j ∈ Z+ ,

and

f H ∞ ≤ 1.

(2.1)

We consider the case when this Nevanlinna–Pick interpolation problem has at least two distinct solutions. Clearly, in this case the sequence {ζj }j≥0 must satisfy the Blaschke condition (1 − |ζj |) < ∞. j≥0

We show that if the interpolation problem (2.1) has at least two distinct solutions, then there exist inner solutions of the problem (2.1) and we parametrize all solutions of (2.1). We denote by B a Blaschke product with simple zeros at the ζj , j ≥ 0. Suppose that f0 is a function in H ∞ interpolating the values wj at ζj , i.e., f (ζj ) = wj , j ≥ 0. Then all functions g in H ∞ satisfying g(ζj ) = wj , j ≥ 0, have the form g = f0 − Bq, where q ∈ H ∞ . We have g∞ = f0 − Bq∞ = Bf0 − q∞ , and so by the Nehari theorem, min f0 − Bq∞ = HBf0 .

h∈H ∞

It is easy to see that the Nevanlinna–Pick interpolation problem reduces to the Nehari problem. Indeed, let ϕ = Bf0 . and ϕ − q∞ ≤ 1, then f0 − Bq is a solution of (2.1). Clearly, all solutions of (2.1) can be obtained in this way. Thus a necessary and sufficient condition for the problem (2.1) to have a solution is HBf0 ≤ 1. It is easy to see that if HBf0 < 1, then (2.1) has infinitely many distinct solutions.

2. Parametrization of Solutions of the Nevanlinna–Pick Problem

171

However, the problem (2.1) can have distinct solutions even if HBf0 = 1. Indeed, in §1.1 we have constructed an example of a Hankel operator with distinct symbols of minimal norm. We can use the same example now. Recall that in that example ω is the conformal map of D onto the domain

|1 − ζ| 0, and m ∈ Z+ , the Nehari–Takagi ap∞ proximation problem is to approximate ϕ by functions f in H(m) so that ∞ ϕ − f ∞ ≤ ρ. Recall (see §4.1) that H(m) is the set of bounded functions whose antianalytic part is a rational function of degree at most m. By the Adamyan–Arov–Krein theorem (see §4.1 and §4.2), this problem is solvable if and only if sm (Hϕ ) ≤ ρ. We are going to parametrize all solutions of the Nehari–Takagi problem in the suboptimal case (i.e., sm (Hϕ ) < ρ). Moreover, we consider the case when sm (Hϕ ) < ρ < sm−1 (Hϕ ) (sm (Hϕ ) < ρ if m = 0). If m = 0, we deal with analytic approximation and thus we arrive at the Nehari problem. All solutions of the Nehari problem in the case of nonuniqueness have been parametrized in §1. We offer here a different approach that also works for m = 0 and ρ > Hϕ in which case it leads to a parametrization of all solutions of the Nehari problem. Using the Adamyan–Arov–Krein theorem mentioned above, one can reformulate the Nehari–Takagi problem in the following way. Let

174


2 Γ : H 2 → H− be a bounded Hankel operator, ρ > 0, and k ∈ Z+ . The ∞ problem is to parametrize all functions f ∈ H(m) such that Γ − Hf ≤ ρ. We assume in this section that

sm (Γ) < ρ < sm−1 (Γ)

(3.1)

(in the case m = 0 we simply assume that sm (Γ) < ρ). We are going to use the notation introduced in §4.2. Recall that Rρ = (ρ2 I − Γ∗ Γ)−1 , def

def

r(ρ) = (Rρ 1, 1),

and J is the real-linear operator on L2 defined by Jg = z¯g¯. We define the functions pρ and qρ in H 2 by pρ = ρRρ 1,

qρ = zRρ Γ∗ z¯.

(3.2)

We need the following properties of pρ and qρ . Lemma 3.1. Let Γ, ρ, pρ , and qρ be as above. The following holds: qρ , Γpρ = ρJS ∗ qρ = ρ¯

Γqρ = ρJS ∗ pρ ,

(3.3)

and |pρ |2 − |qρ |2 = r(ρ)

on

T.

(3.4)

Proof. Let us show that JΓRρ = Rρ Γ∗ J.

(3.5)

∗

Indeed, it is easy to see that JΓ = Γ J, and so JΓRρ = Γ∗ JRρ . Multiplying this equality by ρ2 I − Γ∗ Γ on the left, we obtain (ρ2 I − Γ∗ Γ)JΓRρ = ρ2 Γ∗ JRρ − Γ∗ ΓΓ∗ JRρ = Γ∗ (ρ2 I − ΓΓ∗ )JRρ . Using the obvious fact that ΓΓ∗ J = JΓ∗ Γ, we have (ρ2 I − Γ∗ Γ)JΓRρ = Γ∗ J(ρ2 I − Γ∗ Γ)Rρ = Γ∗ J, which implies (3.5). Let us now establish (3.3). It follows from (3.5) that Γpρ = ρΓRρ 1 = ρJRρ Γ∗ z¯ = ρJ z¯qρ = ρJS ∗ qρ , since qρ ∈ zH 2 . Obviously, J z¯qρ = q¯ρ . To prove the second equality in (3.3), we observe the following equalities: (ρ2 I − Γ∗ Γ)−1 Γ∗ = Γ∗ (ρ2 I − ΓΓ∗ )−1

(3.6)

J(ρ2 I − ΓΓ∗ )−1 = (ρ2 I − Γ∗ Γ)−1 J.

(3.7)

and To verify (3.6), it suffices to multiply it by ρ2 I − Γ∗ Γ on the left and by ρ2 I − ΓΓ∗ on the right. To verify (3.7), it suffices to multiply it on the right by ρ2 I − ΓΓ∗ .

3. Parametrization of Solutions of the Nehari–Takagi Problem

175

We have Γqρ

=

ΓzRρ Γ∗ z¯ = ΓzΓ∗ (ρ2 I − ΓΓ∗ )−1 z¯ = P− zΓΓ∗ (ρ2 I − ΓΓ∗ )−1 z¯

= −P− z(ρ2 I − ΓΓ∗ )(ρ2 I − ΓΓ∗ )−1 z¯ + ρ2 P− z(ρ2 I − ΓΓ∗ )−1 z¯ = ρ2 J 2 P− z(ρ2 I − ΓΓ∗ )−1 z¯ = ρ2 JS ∗ Rρ J z¯ = ρ2 JS ∗ Rρ 1 = ρJS ∗ pρ . Finally, to prove (3.4), it suffices to show that pρ , (S ∗ )j pρ = qρ , (S ∗ )j qρ , j ≥ 1,

(3.8)

and pρ 22 − qρ 22 = r(ρ). Using (3.3), we obtain ρpρ − 1 = =

(3.9)

ρ2 Rρ − I 1 = ρ2 Rρ − ρ2 I − Γ∗ Γ Rρ 1

Γ∗ ΓRρ 1 = Γ∗ J z¯qρ = Γ∗ q¯ρ .

Hence, for j ≥ 1, ρ(S ∗ )j pρ = (S ∗ )j Γ∗ q¯ρ = Γ∗ z¯j q¯ρ . Thus for j ≥ 1, ρ pρ , (S ∗ )j pρ

=

(pρ , Γ∗ z¯j q¯ρ ) = (Γpρ , z¯j q¯ρ )

qρ , z¯j q¯ρ ) = ρ qρ , (S ∗ )j qρ . = ρ(J z¯qρ , z¯j q¯ρ ) = ρ(¯ It remains to verify (3.9). We have by (3.3), 1 (pρ , pρ ) − (qρ , qρ ) = (pρ , pρ ) − 2 (Γpρ , Γpρ ) ρ 1 2 (ρ I − Γ∗ Γ)pρ , pρ = (Rρ 1, 1) = r(ρ). ρ2 ˜=Γ ˜ (α) defined as in Now let α ∈ C and consider the Hankel operator Γ §4.2: ˜ (α) = Hϕ , Γ α where ϕα is a function such that ϕˆα (j) = ϕ(j ˆ + 1) for j ≤ −2 and ϕˆα (−1) = α. Let us make the assumption that =

r(ρ) = (Rρ 1, 1) = 0. It has been shown in the proof of Lemma 4.2.2 that ρ2 is a simple eigenvalue ˜∗Γ ˜ if α = ατ is given by of the operator Γ ατ =

τ (Rρ 1, zΓ∗ z¯) − , ρr(ρ) r(ρ)

(3.10)

where |τ | = 1. Moreover, the function gτ defined by ¯ τ Rρ 1 + Rρ zΓ∗ z¯ = Rρ (¯ ατ + zΓ∗ z¯) gτ = α

(3.11)

176


˜∗Γ ˜ that corresponds to the eigenvalue ρ2 . is an eigenfunction of Γ Lemma 3.2. The eigenfunction gτ defined by (3.11) satisfies τ ρ2 r(ρ)gτ = pρ + τ qρ .

(3.12)

Proof. By (3.11), τ ρ2 r(ρ)gτ = τ α ¯ τ ρ2 r(ρ)Rρ 1 + τ ρ2 r(ρ)Rρ zΓ∗ z¯. We have from (3.10) τα ¯ τ ρ2 r(ρ) = ρ − ρ2 τ (Rρ zΓ∗ z¯, 1), and so τ ρ2 r(ρ)gτ = ρRρ 1 − ρ2 τ (Rρ zΓ∗ z¯, 1)Rρ 1 + τ ρ2 r(ρ)Rρ zΓ∗ z¯.

(3.13)

By formula (2.11) of Chapter 4, we have ρ2 r(ρ)Rρ zΓ∗ z¯ = ρ2 (Rρ zΓ∗ z¯, 1)Rρ 1 + SJΓRρ 1, and so by (3.5), ρ2 r(ρ)Rρ zΓ∗ z¯ = ρ2 (Rρ zΓ∗ z¯, 1)Rρ 1 + zRρ Γ∗ z¯. It follows now from (3.13) that τ ρ2 r(ρ)gτ

= ρRρ 1 − ρ2 τ (Rρ zΓ∗ z¯, 1)Rρ 1 + ρ2 τ (Rρ zΓ∗ z¯, 1)Rρ 1 + τ zRρ Γ∗ z¯ = ρRρ 1 + τ zRρ Γ∗ z¯,

which proves (3.12). We start with the case m = 0, i.e., we first consider the Nehari problem. We have already obtained in §1 a parametrization formula in this case. Nevertheless, here we give another approach that works if ρ > Γ, and we parametrize all solutions of the Nehari problem in terms of the functions pρ and qρ defined by (3.2). Clearly, in this case r(ρ) = ((ρ2 I −Γ∗ Γ)−1 1, 1) > 0. We need the following lemma. Lemma 3.3. Suppose that ρ > Γ. Then for any υ in the unit ball of H ∞ , the function pρ + υqρ is outer. To prove Lemma 3.3 we need one more lemma. Lemma 3.4. Let p and q be functions in H 2 such that p + q is outer and |p(ζ)|2 − |q(ζ)|2 ≥ δ > 0 for almost all ζ ∈ T. Then p is outer. Proof. We have |p| |p| · |p + q| 2 ≤ ≤ |p|2 ∈ L1 , 2 2 |p + q| |p| − |q| δ and since p + q is outer, it follows that p(p + q)−1 ∈ H 1 . To prove that p is outer, it suffices to show that Re p(p + q)−1 ≥ O on T (see Appendix 2.1). This is obvious, since Re

p+q q |q| = 1 + Re ≥ 1 − ≥ O. p p |p|


177

Proof of Lemma 3.3. Let τ ∈ C and |τ | = 1. Let α = ατ be the number ˜ (α) defined (α) by (3.10) and let Γ be the(α)corresponding Hankel operator. Then ˜ of multiplicity 1 (see formula (2.3) of ˜ = ρ, ρ is a singular value of Γ Γ Chapter 4). It follows from Lemma 3.2 that pρ + τ qρ is a maximizing vector ˜ (α) . It is easy to see that pρ + τ qρ is an outer function, since otherwise of Γ we could divide it by its inner factor and obtain another maximizing vector ˜ (α) , which is impossible since the multiplicity of the singular value ρ is of Γ one. It follows now from Lemma 3.4 and (3.4) that pρ is outer. Finally, suppose that υH ∞ ≤ 1. Since pρ is outer, it follows from (3.4) that υqρ /pρ ∈ H ∞ and υqρ /pρ ∞ ≤ 1. Hence, υqρ Re 1 + ≥ O on T, pρ and so

υqρ pρ + υqρ = pρ pρ is outer which implies that pρ + υqρ is outer. Theorem 3.5. Suppose that ρ > Γ. A function ψ is a solution of the Nehari problem Hψ = Γ, ψ∞ ≤ ρ 1+

if and only if it has the form ψ=ρ

υ p¯ρ + q¯ρ pρ + υqρ

(3.14)

for a function υ in the unit ball of H ∞ . Proof. Let τ be a complex number of modulus 1. Suppose first that ψ is defined by (3.14), where υ is the constant function identically equal to τ . Clearly, ψ∞ = ρ. Let us show that Hψ = Γ. Let α = ατ be the number ˜=Γ ˜ (α) be the corresponding Hankel operator. defined by (3.10) and let Γ ˜ the function Since by Lemma 3.2, pρ + τ qρ is a maximizing vector of Γ, ˜ ρ + τ qρ ) Γ(p p ρ + τ qρ ˜ (see Theorem 1.1.4). Let us show that is a symbol of the Hankel operator Γ ˜ ρ + τ qρ ) = ρ¯ z (¯ qρ + τ p¯ρ ). Γ(p

(3.15)

˜ of multiplicity 1, it follows from Indeed, since ρ is a singular value of Γ Theorem 1.1.4 that ˜ ρ + τ qρ ) = cρ¯ Γ(p z (¯ pρ + τ¯q¯ρ ) = c¯ τ z¯(¯ qρ + τ p¯ρ )

(3.16)

for some complex number c of modulus 1. We have to prove that c¯ τ = 1. Using formulas (2.4) and (2.6) of Chapter 4, we obtain ˜ ρ + τ qρ ) = z¯Γ(pρ + τ qρ ) + (pρ + τ qρ , α ¯ + zΓ∗ z¯)¯ z. Γ(p

178


It follows easily from (3.3) that ˜ ρ + τ qρ ) = ρ¯ Γ(p z (¯ qρ + τ p¯ρ ) + d¯ z for some d ∈ C. It is easy to see from (3.2) that q¯ρ + τ p¯ρ is not a constant function. On the other hand, if we compare the jth Fourier coefficients of ˜ ρ + τ qρ ) for j ≤ −2, we see from (3.16), that c¯ τ = 1. c¯ τ z¯(¯ qρ + τ p¯ρ ) and Γ(p This proves (3.15). We can conclude now that the function q¯ρ + τ p¯ρ ρ¯ z pρ + τ q ρ ˜ i.e., is a symbol of Γ, ˜ = H q¯ρ +τ p¯ρ . Γ ρ¯ z pρ +τ qρ

˜ it follows that Since Γ = ΓS, Γ = Hzρ¯z q¯ρ +τ p¯ρ = Hρ q¯ρ +τ p¯ρ . pρ +τ qρ

pρ +τ qρ

In other words, the function ρ(¯ qρ + τ p¯ρ )(pρ + τ qρ )−1 is a symbol of Γ. Suppose now that υ is in the unit ball of H ∞ . Obviously, for almost all ζ ∈ T the function τ →

q¯ρ (ζ) + τ p¯ρ (ζ) pρ (ζ) + τ qρ (ζ)

(3.17)

maps T onto itself. By (3.4), |qρ (ζ)/pρ (ζ)| ≤ 1 for almost all ζ ∈ T, and so the map (3.17) is a conformal map of D onto itself. Thus the function def

ϕυ = ρ


satisfies ϕυ ∞ ≤ ρ. To show that ϕυ is a symbol of Γ, it suffices to prove that τ p¯ρ + q¯ρ def ϕυ − ϕτ = ϕυ − ρ ∈ H ∞. pρ + τ qρ We obtain from (3.4) ϕυ − ϕτ = ρr(ρ)

υ−τ ∈ H ∞, (pρ + υqρ )(pρ + τ qρ )

(3.18)

since ϕυ − ϕτ ∈ L∞ and the function (pρ + υqρ )(pρ + τ qρ ) is outer by Lemma 3.3. Suppose now that ψ ∈ L∞ , ψ∞ ≤ ρ, and Γ = Hψ . Since (3.17) is a conformal map of D onto itself, it follows that there exists a function υ in the unit ball of L∞ such that υ p¯ρ + q¯ρ . ψ=ρ pρ + υqρ


179

We have to show that υ ∈ H ∞ . Let τ be a complex number of modulus 1. Since Γ = Hϕτ , it follows that υ−τ υ p¯ρ + q¯ρ def − ϕτ = ρr(ρ) = ρr(ρ)κτ ∈ H ∞ . (3.19) ρ pρ + υqρ (pρ + υqρ )(pρ + τ qρ ) def

Let L = {ω ∈ H 1 : υω ∈ H 1 }. Clearly, L is a closed subspace of H 1 that is invariant under multiplication by z. Next, L = {O}, which is a consequence of the following easily verifiable equality: υ(1 − κτ qρ (pρ + τ qρ )) = κτ pρ (pρ + τ qρ ) + τ. (Note that 1 − κτ qρ (pρ + τ qρ ) cannot be the zero function for all τ ∈ T.) Then L has the form L = ϑH 1 for an inner function ϑ (see Appendix 2.2). def

Let h = ϑυ ∈ H ∞ . It is easy to see that h and ϑ are coprime; otherwise if they had a nonconstant common inner factor ϑ# , we would have the inclusion ϑ¯# ϑH 1 ⊂ L, which is impossible. It follows from the definition of κτ (3.19) that κτ (pρ + τ qρ )(ϑpρ + hqρ ) = ϑ(υ − τ ) = h − τ ϑ. It follows that the inner factor (ϑpρ + hqρ )(i) of ϑpρ + hqρ is a divisor of h − τ ϑ for any τ ∈ T. Hence, (ϑpρ + hqρ )(i) is a common divisor of h and ϑ. However, h and ϑ are coprime, and so the function ϑpρ + hqρ is outer. By (3.4), |ϑpρ |2 − |hqρ |2 ≥ |pρ |2 − |qρ |2 = r(ρ) on T. It follows now from Lemma 3.4 that ϑpρ is outer, and so ϑ is constant, ¯ ∈ H ∞. which means that υ = ϑh We proceed now to the case when m > 0 and sm (Γ) < ρ < sm−1 (Γ). Recall that for a function ψ ∈ H 1 we denote by ψ(o) its outer factor and by ψ(i) the inner factor of ψ (see Appendix 2.1). Recall also that for an inner function ϑ its degree deg ϑ is the dimension of H 2 ϑH 2 and deg ϑ < ∞ if and only if ϑ is a finite Blaschke product (see Appendix 2.1). We need the following version of Rouché’s theorem. Lemma 3.6. Let ξ and η be functions in H 1 such that |η(ζ)| ≤ |ξ(ζ)|, ξ ζ ∈ T. Then deg(ξ + η)(i) ≤ deg ξ(i) . If, in addition, ξ+η ∈ L1 , then deg ξ(i) ≤ deg(ξ + η)(i) . Proof. It is easy to see that ξ+η η ∈ H ∞ and ∈ H ∞, ξ(o) ξ(o) and we can replace the functions ξ and η with ξ/ξ(o) and η with η/ξ(o) . Thus without loss of generality we may assume that ξ is inner and η∞ ≤ 1. def

Put ϑ = (ξ + η)(i) . Assume that deg ξ < deg ϑ. Then dim(H 2 ξH 2 ) < dim(H 2 ϑH 2 ). Hence, there exists a nonzero function ω in ξH 2 ∩ (H 2 ϑH 2 ). Then def ¯ ∈ H 2 . On the other hand, ω ϑ¯ ∈ H 2 . ψ = ξω −

180


Consider the Hankel operator Hϑξ ¯ . Clearly, Hϑξ ¯ ≤ 1. On the other hand, ¯ ξω ¯ 2 = P− ϑω ¯ 2 = ϑω ¯ 2 = ψ2 . Hϑξ ¯ ψ2 = P− ϑξ 2 Thus Hϑξ ¯ = 1 and Hϑξ ¯ attains its norm on the unit ball of H . By ¯ of norm 1. However, Theorem 1.1.4, Hϑξ ¯ has only one symbol ϑξ

¯ − (ξ + η)(o) ∞ = ξ − ϑ(ξ + η)(o) ∞ = ξ − (ξ + η)∞ = ξ∞ = 1, ϑξ ¯ and ϑξ ¯ − (ξ + η)(o) are symbols of H ¯ of minimal norm, and so both ϑξ ϑξ and we have got a contradiction. ξ Suppose now that ξ+η ∈ L1 . Assume that deg ξ > deg ϑ. Then ϑ is a def

finite Blaschke product. Let Γ = Hϑξ ¯ . Let us show that Γ < 1. Indeed, since ϑ is a finite Blaschke product, rank Γ < ∞ (see Corollary 1.3.3). If Γ = 1, then there is a nonzero function w ∈ H 2 such that 2 ¯ ϑξw ∈ H− , and so Ker Tϑξ ¯ = {O}. Consider now the Toeplitz operator ∗ 2 Tϑξ ¯ = Tξ¯Tϑ . We have Range Tϑ = ϑH and ¯ = Tξϑ Ker Tξ¯ = Kξ = H 2 ξH 2 . By our assumption, dim Kξ = deg ξ > codim ϑH 2 = deg ϑ, and so Ker Tξϑ ¯ = {O}, which contradicts Theorem 3.1.4. ¯ + ϑη ¯ ∈ H ∞ , and so Obviously, ϑξ Γ = Hϑξ ¯ . ¯ = H−ϑη ¯ and −ϑη ¯ of norm at most 1. Put Hence, Γ has two distinct symbols ϑξ ρ = 1 and apply Theorem 3.5. Thus there exist functions υ and υ# in the unit ball of H 1 such that ¯ = υ# p¯1 + q¯1 . ¯ = υ p¯1 + q¯1 and − ϑη ϑξ p1 + υq1 p1 + υ# q1 Recall that for almost all ζ ∈ T τ p¯1 (ζ) + q¯1 (ζ) p1 (ζ) + τ q1 (ζ) ¯ is a unimodular function, is a conformal map of D onto itself, and since ϑξ it follows that υ is also unimodular, i.e., υ is an inner function. We claim that p1 and q1 are rational functions with poles outside the closed unit disk. Indeed, since rank Γ < ∞, the range of Γ consists of rational functions (see the Remark at the end of §1.3). Then by (3.3), q1 is rational and S ∗ p1 is rational, and so p1 is also rational. Now it follows easily from (3.4) that the function (p1 + υq1 )(p1 + υ# q1 ) is invertible in H ∞ . We have by (3.4) τ →

(ξ + η)(o)

2 2 ¯ − (−ϑη) ¯ = (υ − υ# )(|p1 | − |q1 | ) = ϑξ (p1 + υq1 )(p1 + υ# q1 ) υ − υ# . = r(1) (p1 + υq1 )(p1 + υ# q1 )


181

By Lemma 3.3, the functions p1 + υq1 and p1 + υ# q1 are outer, which implies that υ − υ# is outer. Moreover, υ 1 1 −1 ξ 1 = υ − υ# |υ − υ# | = r(1) |(p1 + υq1 )(p1 + υ# q1 )| ξ + η ∈ L . On the other hand, since |υ# | ≤ |υ| on T, it is easy to see that υ Re ≥ 0 on T, υ − υ# and so the function υ(υ − υ# )−1 is outer, and since υ − υ# is outer, we find that υ is outer. However, υ is inner, and so υ(ζ) = τ almost everywhere on T, where |τ | = 1. Then ¯ = τ p¯1 + q¯1 = τ p1 + τ q1 . ϑξ p1 + τ q1 p1 + τ q1 ¯ is Again, it follows from (3.4) that p1 + τ q1 has no zeros on T, and so ϑξ a rational function, which implies that ξ is a rational function, i.e., ξ is a finite Blaschke product. Since p1 + τ q1 is invertible in H ∞ , it follows that the Toeplitz operators T(p1 +τ q1 )−1 and Tp1 +τ q1 are invertible on H 2 , and so Tϑξ ¯ = τ Tp1 +τ q1 T(p1 +τ q1 )−1 is invertible. We now have 0 = ind Tϑξ ¯ = ind Tϑ ¯ + ind Tξ = deg ϑ − deg ξ, which completes the proof. Corollary 3.7. Suppose that sm (Γ) < ρ < sm−1 (Γ) and r(ρ) > 0. Then deg(pρ )(i) = m. Moreover, for any inner function ϑ and any υ in the unit ball of H ∞ , deg(ϑpρ + υqρ )(i) = deg ϑ + m.

(3.20)

Proof. By (3.4), |ϑpρ |2 − |υqρ |2 ≥ |pρ |2 − |qρ |2 = r(ρ) > 0

on T.

Next, |pρ |(|ϑpρ | + |υqρ |) ϑpρ 1 ϑpρ + υqρ ≤ |ϑpρ |2 − |υqρ |2 ≤ r(ρ) |pρ |(|ϑpρ | + |υqρ |), and so ϑpρ (ϑpρ + υqρ )−1 ∈ L1 . It follows now from Lemma 3.6 that deg(ϑpρ + υqρ )(i) = deg(ϑpρ )(i) = deg ϑ + deg(pρ )(i) . In particular, for any complex number τ of modulus 1 deg(ϑpρ + τ qρ )(i) = deg(pρ )(i) . ˜ = Γ ˜ (α) be the Now let α = ατ be the number defined by (3.10) and Γ ˜ corresponding Hankel operator. Then ρ = sm (Γ) (see §4.2) and by Lemma ˜ which corresponds to the singular 3.2, pρ + τ qρ is a Schmidt function of Γ,

182


value ρ. By the Remark after the proof of Lemma 4.1.6, deg(pρ )(i) = m, which completes the proof. Now we are ready to parametrize the solutions of the Nehari–Takagi problem. We start with the case r(ρ) > 0. Theorem 3.8. Let m be a positive integer. Suppose that sm (Γ) < ρ < sm−1 (Γ) and r(ρ) > 0. A function ψ ∈ L∞ satisfies the conditions rank(Γ − Hψ ) ≤ m

and

ψ∞ ≤ ρ

(3.21)

if and only if it can be represented in the form def

ψ = ϕυ = ρ


for a function υ in the unit ball of H ∞ . Moreover, υ → ϕυ is a one-to-one map of the unit ball of L∞ onto the ball of radius ρ of L∞ . Proof. Obviously, for almost all ζ ∈ T the map τ →

υ p¯ρ (ζ) + q¯ρ (ζ) pρ (ζ) + υqρ (ζ)

(3.22)

maps T onto itself. By (3.4), |qρ (ζ)/pρ (ζ)| < 1 almost everywhere on T, and so the map (3.21) is a conformal map of D onto itself, which implies that υ → ϕυ is a one-to-one map of the unit ball of L∞ onto the ball of radius ρ of L∞ . Let us show that for any complex number τ of modulus 1 the function def

ϕτ = ρ

τ p¯ρ + q¯ρ pρ + τ qρ

satisfies rank(Γ − Hϕτ ) ≤ m. The proof of this fact is similar to the corresponding fact for m = 0 that has been established in the proof of Theorem 3.5. Indeed, let ˜ def Γ(pρ + τ qρ ) κ = , p ρ + τ qρ ˜=Γ ˜ (α) and α = ατ is defined by (3.10). Recall that pρ + τ qρ is a where Γ ˜ that corresponds to the singular value ρ. It has been Schmidt function of Γ ˜ − Hκ ) = m. It has been proved in §4.1 that proved in §4.1 that rank(Γ ˜ corresponding to the singular J maps the space of singular vectors of Γ ˜ ∗ corresponding to the same value ρ onto the space of singular vectors of Γ singular value. Since the multiplicity of the singular value ρ is 1, it follows ˜ ρ + τ qρ ) = const J(pρ + τ qρ ). In the same way as in the proof of that Γ(p Theorem 3.5 we can now show that ˜ ρ + τ qρ ) = ρ¯ z (¯ qρ + τ p¯ρ ). Γ(p

(3.23)


We have rank(Γ − Hϕτ )

˜ − Hκ )S rank(Γ − Hzκ ) = rank (Γ ˜ − Hκ = m. ≤ rank Γ

183

=

(3.24)

Suppose that ψ satisfies (3.21). Then ψ = ϕυ , where υ is in the unit ball of L∞ . Let us show that υ ∈ H ∞ . Let Γ = Hϕ , where ϕ ∈ L∞ . Then there exists an inner function θ such that deg θ ≤ m and θ(ϕ − ϕυ ) ∈ H ∞ .

(3.25)

By Corollary 3.7, for any complex number τ of modulus 1 deg(pρ + τ qρ )(i) = m.

(3.26)

Let us show that pρ + τ qρ ∈ Ker(Γ − Hϕτ ). Indeed, by (3.23), Γ(pρ + τ qρ ) = ρP− (τ p¯ρ + q¯ρ ). On the other hand, it follows from the definition of ϕτ that Hϕτ (pρ + τ qρ ) = ρP− (τ p¯ρ + q¯ρ ). Since the kernel of a Hankel operator is an invariant subspace under multiplication by z, we have (pρ + τ qρ )(i) H 2 ⊂ Ker(Γ − Hϕτ ). It follows now from (3.24) and (3.26) that Ker(Γ − Hϕτ ) = (pρ + τ qρ )(i) H 2 .

(3.27)

We now have from (3.25) and (3.27) hτ = θ(pρ + τ qρ )(i) (ϕυ − ϕτ ) ∈ H ∞ . def

As in (3.18), ϕυ − ϕτ = ρr(ρ)

υ−τ . (pρ + υqρ )(pρ + τ qρ )

Now it is an elementary exercise to verify the following equality: υ ρr(ρ)θ − hτ qρ (pρ + τ qρ )(o) = τ ρr(ρ)θ + hτ pρ (pρ + τ qρ )(o) .

(3.28)

(3.29)

Clearly, ρr(ρ)θ − hτ qρ (pρ + τ qρ )(o) = O, since otherwise we would have pρ = −τ qρ , which contradicts (3.4). As in the proof of Theorem 3.5, consider the closed subspace L of H 1 def defined by L = {ω ∈ H 1 : υω ∈ H 1 }. It follows from (3.29) that L = {O}. Then L has the form L = ϑH 1 for an inner function ϑ (see Appendix 2.2). def

Thus the function g = ϑυ belongs to H ∞ , g∞ ≤ 1, and g and ϑ are coprime. It follows from (3.28) that hτ (pρ + τ qρ )(o) (ϑpρ + gqρ ) = ρr(ρ)θ(g − τ ϑ), and so (ϑpρ +gqρ )(i) is a divisor of θ(g −τ ϑ). Since τ is an arbitrary number of modulus 1 and ϑ is coprime with g, it follows that (ϑpρ + gqρ )(i) is a divisor of θ. Hence, by (3.25), deg(ϑpρ + gqρ )(i) ≤ deg θ ≤ m.

184


On the other hand, by Corollary 3.7, deg(ϑpρ + gqρ )(i) = deg ϑ + m. Thus deg ϑ = 0, and so υ = g/ϑ ∈ H ∞ . Suppose now that υ ∈ H ∞ and υ∞ ≤ 1. Let us show that if τ1 and τ2 are distinct numbers of modulus 1, then pρ + τ1 qρ and pρ + τ2 qρ have no common zeros. Indeed, we have from (3.28) ϕτ1 − ϕτ2

=

ρr(ρ)(τ1 − τ2 ) (pρ + τ1 qρ )(pρ + τ2 qρ )

= ρr(ρ)

(τ1 − τ2 )(pρ + τ1 qρ )(i) (pρ + τ2 qρ )(i) (pρ + τ1 qρ )(o) (pρ + τ2 qρ )(o)

.

−1 Clearly, (pρ + τ1 qρ )−1 (o) (pρ + τ2 qρ )(o) is a bounded outer function, and so (3.30) Ker Hϕτ1 − Hϕτ2 = (pρ + τ1 qρ )(i) (pρ + τ2 qρ )(i) H 2 .

On the other hand, by (3.27), Ker(Γ − Hϕτj ) = (pρ + τj qρ )(i) H 2 , Clearly,

j = 1, 2.

(3.31)

Ker(Γ − Hϕτ1 ) ∩ Ker(Γ − Hϕτ2 ) ⊂ Ker Hϕτ1 − Hϕτ2 ,

and (3.30) and (3.31) imply that (pρ + τ1 qρ )(i) and (pρ + τ2 qρ )(i) have no common zeros. Let ϕ ∈ L∞ and Γ = Hϕ . It follows from (3.27) and (3.28) that P− (ϕ − ϕυ ) is a rational function with poles in the zeros of (pρ + τ qρ )(pρ + υqρ ) counted with multiplicities. However, P− (ϕ − ϕυ ) does not depend on the choice of τ and we have just proved that for distinct τ1 and τ2 the functions (pρ + τ1 qρ ) and (pρ + τ2 qρ ) have no common zeros. Thus P− (ϕ − ϕυ ) is a rational function with poles in the zeros of pρ + υqρ . It follows now from Corollary (3.7) that rank(Γ − Hϕυ ) ≤ m. Consider now the case r(ρ) < 0. Theorem 3.9. Let m be a positive integer. Suppose that sm (Γ) < ρ < sm−1 (Γ) and r(ρ) < 0. A function ψ ∈ L∞ satisfies the conditions rank(Γ − Hψ ) ≤ m

and

ψ∞ ≤ ρ

if and only if it can be represented in the form def

ψ = ϕυ = ρ

p¯ρ + υ q¯ρ υpρ + qρ

for a function υ in the unit ball of H ∞ . Moreover, υ → ϕυ is a one-to-one map of the unit ball of L∞ onto the ball of radius ρ of L∞ .


185

The proof of Theorem 3.9 is the same as the proof of Theorem 3.8. We simply have to interchange the roles of pρ and qρ . In particular, in this case the analog of Corollary 3.7 is the following: deg(ϑqρ + υpρ )(i) = deg ϑ + m for any υ in the unit ball of H ∞ and any inner ϑ. We proceed now to the remaining case r(ρ) = 0. By (3.4), |pρ | = |qρ |. However, much more can be said in this case. It follows from formula (2.11) of Chapter 4 and from (3.5) that −ρ2 (Rρ zΓ∗ z¯, 1)Rρ 1 + ρ2 r(ρ)Rρ zΓ∗ z¯ = zRρ Γ∗ z¯, and since r(ρ) = 0, we have from (3.2) −ρ(Rρ zΓ∗ z¯, 1)pρ = qρ , i.e., pρ and qρ are linear dependent. Thus pρ + τρ qρ = O

(3.32)

for some complex number τρ of modulus 1. Consider now on the interval (sm (Γ), sm−1 (Γ)) the operator function t → Rt = (t2 I − Γ∗ Γ)−1 . Clearly, it is differentiable. It follows that the vector functions t → pt

and t → qt

are also differentiable on (sm (Γ), sm−1 (Γ)). Here by the derivatives we understand the derivatives of the above operator and vector functions as functions of t. It is easy to see that Rt = −2tRt2 .

(3.33)

By (3.3), we have for τ ∈ T Γ(pt + τ qt ) = tJS ∗ qt + tτ JS ∗ pt = tτ JS ∗ (pt + τ qt ). Hence,

Γ(pt + τ qt ) = τ JS ∗ (pt + τ qt ) + τ tJS ∗ (pt + τ qt ). Substituting t = ρ and τ = τρ , we obtain Γ(pρ + τρ qρ ) = ρτρ JS ∗ (pρ + τ qρ ). Now put wρ = pρ + τ qρ .

(3.34)

Γwρ = ρτρ JS ∗ wρ .

(3.35)

Clearly, wρ ∈ H 2 and Let us show that pρ and wρ are linearly independent. Indeed, pρ (0) = ρ(Rρ 1, 1) = r(ρ) = 0. On the other hand, wρ (0) = (wρ , 1) = (pρ + τρ qρ , 1).

186


By (3.2), qt (0) = 0 for any t ∈ (sm (Γ), sm−1 (Γ)), and so qρ (0) = 0. On the other hand, (pρ , 1) = ρ(Rρ 1, 1) = −2ρ2 (Rρ2 1, 1) by (3.33). Thus wρ (0) = −2ρ2 (Rρ 1, Rρ 1) < 0. By (3.3) and (3.32), we have Γpρ = ρJS ∗ qρ = −ρτρ JS ∗ pρ .

(3.36)

Now we obtain from (3.35) and (3.36) Γ(wρ + ispρ ) = ρτρ JS ∗ (wρ + ispρ ),

s ∈ R.

(3.37)

For s ∈ R we find the number α = α(s) ∈ C from the equation ¯ + zΓ∗ z¯) = 0. − ρτρ (J(wρ + ispρ ), 1) + (wρ + ispρ , α

(3.38)

The equation is solvable, since (wρ , 1) = 0 and (pρ , 1) = O. As before, ˜=Γ ˜ (α) defined by consider the Hankel operator Γ ˜ = z¯Γg + (g, α z, Γg ¯ + zΓ∗ z¯)¯

g ∈ H 2.

It follows easily from (3.37) and (3.38) that ˜ ρ + ispρ ) Γ(w

= ρτρ z¯JS ∗ (wρ + ispρ ) + (wρ + ispρ , α ¯ + zΓ∗ z¯) = ρτρ J(wρ + ispρ ) − ρτρ (J(wρ + ispρ ), 1) +

¯ + zΓ∗ z¯) (wρ + ispρ , α

= ρτρ J(wρ + ispρ ). Hence, ˜ ∗ Γ(w ˜ ρ + ispρ ) = ρ2 (wρ + ispρ ). Γ ˜ and since sm (Γ) < ρ, we can prove in the same Consequently, ρ = sm (Γ) way as in the proof of Theorem 3.8 (see (3.27)) that Ker(Γ − Hϕs ) = (wρ + ispρ )(i) H 2 , where (wρ − pρ ) + wρ + pρ w ¯ρ − is¯ pρ = 1+is . wρ + ispρ wρ − pρ + 1−is 1+is (wρ + pρ ) 1−is

ϕs = ρτρ Note that

1 − is 1 + is = 1.

Lemma 3.10. Suppose that r(ρ) = 0. Then the function wρ defined by (3.34) satisfies Re wρ (ζ)pρ (ζ) = −ρRρ 12

for almost all

ζ ∈ T.

4. Parametrization via One-Step Extension

187

Proof. In the proof of Lemma 3.1 we have shown (see (3.8) and(3.9)) that for any t ∈ (sm (Γ), sm−1 (Γ)) the following equalities hold: (pt , (S ∗ )j pt ) − (qt , (S ∗ )j qt ) = 0,

j ≥ 1,

and (pt , pt ) − (qt , qt ) = (Rt 1, 1). Let us differentiate these equalities and put t = ρ. Then using (3.32), we obtain pρ + τρ qρ , (S ∗ )j pρ + pρ , (S ∗ )j (pρ + τ qρ ) = 0, j ≥ 1, and

(pρ + τρ qρ , pρ ) + (pρ , pρ + τ qρ ) = −2ρRρ 12

(see (3.33)). Hence, the jth Fourier coefficient of Re wρ (ζ)pρ (ζ) is zero for j = 0 and −ρRρ 12 for j = 0. Theorem 3.11. Let m be a positive integer. Suppose that sm (Γ) < ρ < sm−1 (Γ) and r(ρ) = 0. A function ψ ∈ L∞ satisfies the conditions rank(Γ − Hψ ) ≤ m and ψ∞ ≤ ρ if and only if it can be represented in the form def

ψ = ϕυ = ρτρ

υ(wρ − pρ ) + wρ + pρ wρ − pρ + υ(wρ + pρ )

for a function υ in the unit ball of H ∞ . Moreover, υ → ϕυ is a one-to-one map of the unit ball of L∞ onto the ball of radius ρ of L∞ . Proof. The fact that υ → ϕυ is a one-to-one map of the unit ball of L∞ onto the ball of radius ρ of L∞ is a simple consequence of the equality |wρ (ζ) − pρ (ζ)|2 − |wρ (ζ) + pρ (ζ)|2 = 4ρRρ 12 > 0, which, in turn, follows immediately from Lemma 3.10. Next, as in Corollary 3.7 one can prove that for any inner function ϑ and any function υ in the unit ball of H ∞ deg(ϑ(wρ − pρ ) + υ(wρ + pρ )(i) = deg ϑ + m. The rest of the proof is the same as the proof of Theorem 3.8.

4. Parametrization via One-Step Extension We consider here the Nehari problem for vectorial Hankel operators. Let H and K be Hilbert spaces and let Γ = {Ωj+k }j,k≥0 be a block Hankel matrix, where the Ωj , j ∈ Z+ , are bounded linear operators from H to K such that Γ determines a bounded linear operator from 2 (H) to 2 (K). Suppose that Γ ≤ ρ. The Nehari problem in this context is to describe ˆ all functions Ξ ∈ L∞ (B(H, K)) such that Ξ(j) = Ωj , j ∈ Z+ . As we have already discussed in §2.2, this problem is equivalent to the problem of

188


describing all extensions {Ωj }j∈Z of the sequence {Ω }j≥0 such that the operator matrix Γ = {Ωj+k }j,k∈Z determines a bounded linear operator from 2Z (H) to 2Z (K) of norm at most ρ. Note that the adjoint operator Γ∗ is also a Hankel operator from 2 (K) to 2 (H) and it has Hankel matrix {Ω∗j+k }j,k≥0 .

One-Step Extension We begin with the problem of one-step extension, i.e., we want to describe the operators Ω : H → K such that the sequence Ω, Ω0 , Ω1 , Ω2 , · · · determines a bounded block Hankel matrix   Ω Ω0 Ω1 · · ·  Ω0 Ω1 Ω2 · · ·   ˜= (4.1) Γ  Ω1 Ω2 Ω3 · · ·    .. .. .. .. . . . . ˜ is a one-step ρ-extension of norm at most ρ. In this case we say that Γ of Γ. We parametrize the set of all such operators Ω. Then we obtain a parametrization of all solutions of the Nehari problem in the case when the spaces H and K are finite-dimensional and Γ < ρ. We consider the natural imbedding JH into 2 (H) defined by def

JH x = (x, O, O, · · · ),

x ∈ H.

Clearly, J∗H (x0 , x1 , x2 , · · · ) = x0 and JH J∗H (x0 , x1 , x2 , · · · ) = (x0 , O, O, · · · ). Similarly, we define the imbedding JK of K into 2 (K). For ρ > Γ we consider the following operators: Rρ = (ρ2 I − Γ∗ Γ)−1 ,

R∗ρ = (ρ2 I − ΓΓ∗ )−1 ,

Gρ = (J∗H Rρ JH )−1/2 ,

G∗ρ = (J∗K R∗ρ JK )−1/2 .

def

def

def

def

Here we use the same notation I for the identity operator on different Hilbert spaces. Sometimes to avoid confusion we write IX for the identity operator on a space X. Recall that SH and SK are shifts on 2 (H) and 2 (K). The following identity characterizes the vectorial Hankel operators: ∗ Γ = ΓSH . SK

(4.2)

˜ defined by (4.1) satisfies It is easy to see that the one-step extension Γ the following equality: ˜ = JK ΩJ∗ + SK Γ + JK J∗ ΓS ∗ . (4.3) Γ H K H


189

The following result parametrizes all one-step ρ-extensions in the case ρ > Γ. Theorem 4.1. Let ρ > Γ. An operator Ω : H → K determines a one ˜ defined by (4.1) if and only if Ω admits a representation step ρ-extension Γ 1 ∗ Ω = −J∗K ΓSH Rρ JH G2ρ + G∗ρ EGρ , (4.4) ρ where E : H → K and E ≤ 1. Proof. Replacing Γ with ρ1 Γ, we can reduce the general case to the case ρ = 1. Let R = R1 , R∗ = R∗1 , G = G1 , and G∗ = G∗1 . Then R = DΓ−2 and R∗ = DΓ−2 ∗ (see the notation in §2.1). Now we can deduce Theorem 4.1 from Theorem 2.1.5. We have Ω C ˜= Γ , B A where



A = ΓSH

Ω1  Ω2 ∗ = SK Γ =  .. .  Ω0   =  Ω1  , .. .

Ω2 Ω3 .. .

 ··· ···  , .. .



B = ΓJH

and C = J∗K Γ =

Ω0

Ω1

···

.

Then B = DA∗ K and C = LDA , where the operators K and L are the ˜ ≤ 1 if and only if Ω same as in Theorem 2.1.5. By Theorem 2.1.5, Γ admits a representation Ω = −LA∗ K + DL∗ EDK , with E ≤ 1. To compute DK , we first observe that 2 DA ∗

Hence,

∗ ∗ = I − ΓSH SH Γ = DΓ2 ∗ + ΓJH J∗H Γ∗ ∗ ∗ −1 DΓ∗ . = DΓ∗ I + DΓ−1 ∗ ΓJH JH Γ DΓ∗

−2 −1 −1 ∗ ∗ −1 −1 −1 DA DΓ∗ . ∗ = DΓ∗ I + DΓ∗ ΓJH JH Γ DΓ∗

−1 Since K = DA ∗ ΓJH , it follows that 2 DK

−2 = I − J∗H Γ∗ DA ∗ ΓJH

−1 ∗ ∗ −1 −1 −1 = I − J∗H Γ∗ DΓ−1 DΓ∗ ΓJH ∗ I + DΓ∗ ΓJH JH Γ DΓ∗ =

−1 (I + JH Γ∗ DΓ−2 = (JH DΓ−2 JH )−1 = G2 . ∗ ΓJH )

Here we have used the following elementary and easily verifiable facts: T ∗ (I + T T ∗ )−1 T = I − (I + T ∗ T )−1

190


and T ∗ (I − T T ∗ )−1 T = (I − T ∗ T )−1 − I

(4.5)

(the operator on the left-hand side is invertible if and only if the operator on the right-hand side is). Similarly, one can show that DL∗ = G∗ . −1 , we have Finally, let us compute LA∗ K. Since L = J∗K ΓDA LA∗ K

−1 ∗ −1 −2 = J∗K ΓDA A DA∗ ΓJH = J∗K ΓA∗ DA ∗ ΓJH

∗ ∗ −1 ∗ ∗ −1 −1 −1 = J∗K ΓSH Γ DΓ∗ I + DΓ−1 DΓ∗ ΓJH ∗ ΓJH JH Γ DΓ∗ ∗ ∗ −2 −1 = J∗K ΓSH Γ DΓ∗ ΓJH (I + J∗H Γ∗ DΓ−2 ∗ ΓJH ) ∗ ∗ = J∗K ΓSH (R − I)JH (J∗H RJH )−1 = J∗K ΓSH RJH G2 , ∗ since, clearly, SH JH = O. Here we have used (4.5) and another elementary identity: (I + T T ∗ )−1 T = T (I + T ∗ T )−1 .

This completes the proof. The set of operators Ω that admit a representation of the form (4.4) is an operator quasiball. Operator quasiballs can be defined as follows. If Λ is a nonnegative operator on K, Π is a nonnegative operator on H, and O is a bounded linear operator from H to K, the set of operators of the form B = {O + ΛEΠ : E ∈ B(H, K), E ≤ 1} is called an operator quasiball. The operator O is called the center of B, Λ is called a left semiradius and Π is called a right semiradius of B. The quasiball B is nontrivial, i.e., it consists of more than one element if and only if both Λ and Π are nonzero operators. Clearly, the semiradii are not uniquely determined by the quasiball. However, if the quasiball is nontrivial, then the left and right semiradii are determined modulo positive scalar constant multiples. It is also easy to see that the center is uniquely determined by the quasiball. ∗ The center of the quasiball in (4.4) is −J∗K ΓSH Rρ JH G2ρ . If we apply formula (4.4) to the operator Γ∗ and take the adjoint, we obtain another expression for the center: −G2∗ρ J∗K R∗ρ SK ΓJH . Hence, ∗ − J∗K ΓSH Rρ JH G2ρ = −G2∗ρ J∗K R∗ρ SK ΓJH

(4.6)

is the center of the quasiball (4.4). B 3 ⊃ · · · is a sequence of operator quasiballs, then If B 1 ⊃ B 2 ⊃ B j is also an operator quasiball. Indeed, let Oj be their intersection j≥0

the center of B j . It can easily be seen that one can choose the semiradii Λj and Πj of B j so that Λj ≥ Λj+1 and Πj ≥ Πj+1 . Let O be the limit in the weak operator topology of a subsequence of {Oj }j≥1 . It can be shown


191

that the sequences {Λj }j≥1 and {Πj }j≥1 converge in the weak operator topology to operators Λ and Π and ' B j = {O + ΛEΠ : E ∈ B(H, K), E ≤ 1}. j≥0

Moreover, it is not hard to see that the sequence {Oj }j≥1 converges in the weak operator topology to O. We refer the reader to Shmul’yan [1] for the facts on operator quasiballs mentioned above. Consider now the case ρ = Γ, i.e., we consider one-step extensions of minimal norm. Let {ρj }j≥1 be a sequence of positive numbers such that ρj > Γ and ρj → Γ as j → ∞. An operator Ω : H → K determines a one-step extension of minimal norm if and only if it belongs to the intersection of the quasiballs

1 def ∗ ∗ 2 B j = −JK ΓSH Rρj JH Gρj + G∗ρj EGρj : EB(H,K) ≤ 1 . ρj −1/2

−1/2

G∗ρj }j≥1 and {ρj Gρj }j≥1 are It is easy to see that the sequences {ρj nonincreasing and we obtain the following result. Theorem 4.2. An operator Ω : H → K determines a one-step extension of minimal norm if and only if it belongs to the operator quasiball with center ∗ − lim J∗K ΓSH Rρ JH G2ρ ρ→ Γ +

and semiradii

1 lim J∗ R∗ρ JK Γ ρ→ Γ + K

and

1 lim J∗ Rρ JH , Γ ρ→ Γ + H the limits being taken in the strong operator topology. Now we are able to describe the operators Γ with a unique one-step extension of minimal norm. Theorem 4.3. Let Γ be a vectorial Hankel operator from 2 (H) to 2 (K). Then it has a unique one-step extension of minimal norm if and only if at least one of the following conditions (i) and (ii) is satisfied: (i) s- lim Gρ = O; ρ→ Γ +

(ii) s- lim ρ→ Γ +

G∗ρ = O.

Moreover, (i) is equivalent to the condition (i ) Range(Γ2 I − Γ∗ Γ)1/2 JH H = {O}, and (ii) is equivalent to the condition (ii ) Range(Γ2 I − ΓΓ∗ )1/2 JK K = {O}. Proof. In view of the above discussion it remains to show that (i)⇔(i ) and (ii)⇔(ii ). Let us establish the equivalence of (i) and (i ). To verify the equivalence of (ii) and (ii ), it suffices to interchange the roles of Γ and Γ∗ .

192


Suppose that (i) is satisfied and x is a vector in H such that JH x ∈ Range(Γ2 I − Γ∗ Γ)1/2 , i.e.,

JH x = (Γ2 I − Γ∗ Γ)1/2 g, g ∈ 2 (H). Then for ρ > Γ we have 2 G−1 = (ρ2 I − Γ∗ Γ)−1 JH x, JH x ρ x 2 = (ρ2 I − Γ∗ Γ)−1/2 (Γ2 I − Γ∗ Γ)1/2 g 2

(H)

→ g2 def

as ρ → Γ+, by the spectral theorem. Thus the family yρ = G−1 ρ x, ρ > Γ, is bounded. Hence, it follows from (i) that Gρ yρ → 0 as ρ → Γ+ in the weak operator topology. However, Gρ yρ = x, which proves that (i)⇒(i ). Conversely, suppose that (i) does not hold. Since Gρ1 ≥ Gρ2 for ρ1 ≥ ρ2 and the operators Gρ are nonnegative, it is well known (see Halmos [2], Problem 120) that there exists a strong limit G of Gρ as ρ → Γ+. Let w ∈ H such that def x = Gw = lim Gρ w = O. ρ→ Γ +

Let us show that −1 G−1 ρ x = Gρ Gw ≤ w. −1 Clearly, G−1 ρ G = GGρ . and so G2ρ ≥ G2 for ρ ≥ Γ.

It is easy to see that Thus

(4.7) G2ρ1

≥

G2ρ2

for ρ1 ≥ ρ2 ,

2 2 −1 −1 2 −1 −1 2 GG−1 ρ v = (G Gρ v, Gρ v) ≤ (Gρ Gρ v, Gρ v) = v ,

which proves (4.7). We have 2 2 (ρ I − Γ∗ Γ)−1 JH x, JH x = G−1 ρ x ≤ const, and it follows from Lemma 5.1.6 that JH x ∈ Range(Γ2 I − Γ∗ Γ)1/2 . Corollary 4.4. Suppose that Γ2 is an eigenvalue of Γ∗ Γ and clos J∗H Ker(Γ2 I − Γ∗ Γ) = H;

(4.8)

then Γ has a unique one-step extension of minimal norm. Proof. Suppose that x ∈ H and JH x = (Γ2 I − Γ∗ Γ)1/2 f for some f ∈ 2 (H). Then JH x ⊥ Ker(Γ2 I − Γ∗ Γ). By (4.8), x = O. Remark. Another sufficient condition for uniqueness can be obtained if we interchange the roles of Γ and Γ∗ . We obtain now some useful identities that will be used in this section. In addition to (4.2) we need the following easily verifiable commutation relation: ∗ ∗ = SK Γ + JK J∗K ΓSH − SK ΓJH J∗H . ΓSH

(4.9)


193

It follows easily from (4.2) and (4.9) that ∗ ∗ (ρ2 I − ΓΓ∗ )SK = SK (ρ2 I − ΓΓ∗ ) + SK ΓJH J∗H Γ∗ − JK J∗K ΓSH Γ .

This implies the following identity: ∗ ∗ Γ R∗ρ . (4.10) SK R∗ρ = R∗ρ SK + R∗ρ SK ΓJH J∗H Γ∗ R∗ρ − R∗ρ JK J∗K ΓSH

If we apply (4.10) to the operator Γ∗ , we obtain ∗ ΓRρ . SH Rρ = Rρ SH + Rρ SH Γ∗ JK J∗K ΓRρ − Rρ JH J∗H Γ∗ SK

(4.11)

We also need the following simple formula: Γ∗ R∗ρ Γ = ρ2 Rρ − I.

(4.12)

To prove (4.12), we observe that Γ∗ R∗ρ Γ(ρ2 I − Γ∗ Γ) = Γ∗ R∗ρ (ρ2 I − ΓΓ∗ )Γ = Γ∗ Γ = ρ2 I − (ρ2 I − Γ∗ Γ). Multiplying this on the right by Rρ , we obtain (4.12). If we apply (4.12) to Γ∗ , we find that ΓRρ Γ∗ = ρ2 R∗ρ − I.

(4.13)

Finally, we need the formula ∗ ˜Γ ˜ ∗ = (JK Ω + SK ΓJH )(Ω∗ J∗K + J∗H Γ∗ SK Γ ) + ΓΓ∗ .

Indeed,

 ˜Γ ˜∗ Γ

  =  

Ω Ω0 Ω1 .. .

Ω0 Ω1 Ω2 .. .

Ω1 Ω2 Ω3 .. .

··· ··· ··· .. .

    

Ω∗ Ω∗0 Ω∗1 .. .

Ω∗0 Ω∗1 Ω∗2 .. .

Ω∗1 Ω∗2 Ω∗3 .. .

(4.14) ··· ··· ··· .. .



 Ω   =  Ω0  Ω∗ .. . 

Ω0  Ω1 + .. . =

Ω1 Ω2 .. .

Ω∗0

···

 ∗ Ω0 ···  Ω∗1 ···   .. .. . .

Ω∗1 Ω∗2 .. .

 ··· ···   .. .

∗ (JK Ω + SK ΓJH )(Ω∗ J∗K + J∗H Γ∗ SK ) + ΓΓ∗ .

Let us introduce now the following important operators. Pρ : Qρ : P∗ρ : Q∗ρ :

H → 2 (H), H → 2 (K), K → 2 (K), K → 2 (H),

Pρ = ρRρ JH Gρ ; Qρ = SK ΓRρ JH Gρ ; P∗ρ = ρR∗ρ JK G∗ρ ; Q∗ρ = SH Γ∗ R∗ρ JK G∗ρ .

    

194


One-Step Canonical Extensions Without loss of generality we may assume that dim H ≤ dim K. Otherwise we could study Γ∗ rather than Γ. If ρ > Γ, we say that a one-step ˜ is canonical if the operator E : H → K in (4.4) is an isometric ρ-extension Γ operator, i.e., E ∗ E = IH . ˜ be a canonical one-step Theorem 4.5. Let ρ > Γ and let Γ ρ-extension of Γ determined by an isometry E in (4.4). Then the following equalities hold: ˜ ρ + Q∗ρ E) = ρ(Qρ + P∗ρ E), Γ(P

˜ ∗ (Qρ + P∗ρ E) = ρ(Pρ + Q∗ρ E). (4.15) Γ

Moreover, ˜ ∗ Γ) ˜ = (Pρ + Q∗ρ E)H, Ker(ρ2 I − Γ and

Γ2 , ρ2

'

˜Γ ˜ ∗ ) ⊃ (Qρ + P∗ρ E)H, Ker(ρ2 I − Γ (4.16)

' ∗ ˜ = Γ2 , ρ2 ˜∗Γ ˜Γ ˜ = ∅. σ Γ σ Γ

Remark. It is easy to see that J∗H (Pρ + Q∗ρ E)H = H. Indeed, it follows from the definition of Pρ , Qρ , P∗ρ , Q∗ρ that J∗H (Pρ + Q∗ρ E)H = J∗H Pρ H = ρJ∗H Rρ JH Gρ H = ρG−1 ρ H = H. Proof of Theorem 4.5. Let us show that for any operator E from H to K the following equalities hold: ∗ Γ(Pρ + Q∗ρ E) = ρSK (Qρ + P∗ρ E);

∗ Γ∗ (Qρ + P∗ρ E) = ρSH (Pρ + Q∗ρ E). (4.17)

Indeed, by the definition of Pρ , Qρ , P∗ρ , Q∗ρ , we have Γ(Pρ + Q∗ρ E)

= ρΓRρ JH Gρ + ΓSH Γ∗ R∗ρ JK G∗ρ E ∗ ∗ Qρ + SK ΓΓ∗ R∗ρ JK G∗ρ E = ρSK ∗ ∗ Qρ + SK ((ΓΓ∗ − ρ2 I) + ρ2 I)R∗ρ JK G∗ρ E = ρSK ∗ ∗ ∗ = ρSK Qρ + SK (−I + ρ2 R∗ρ )JK G∗ρ E = ρSK (Qρ + P∗ρ E),

∗ JK G∗ρ E = O. The second equality in (4.17) can be since obviously, SK proved in exactly the same way. Using (4.3), we obtain

˜ ρ + Q∗ρ E) Γ(P

= JK ΩJ∗H (Pρ + Q∗ρ E) ∗ (Pρ + Q∗ρ E). + SK Γ(Pρ + Q∗ρ E) + JK J∗K ΓSH


195

By (4.17), SK Γ(Pρ + Q∗ρ E)

∗ = ρSK SK (Qρ + P∗ρ E)

= ρ(Qρ + P∗ρ E) − ρJK J∗K (Qρ + P∗ρ E) = ρ(Qρ + P∗ρ E) − ρJK J∗K P∗ρ E = ρ(Qρ + P∗ρ E) − ρ2 JK G−1 ∗ρ E. Next, ∗ ∗ (Pρ + Q∗ρ E) = ρJK J∗K ΓSH Rρ JH Gρ + JK J∗K ΓΓ∗ R∗ρ JK G∗ρ E, JK J∗K ΓSH

and JK J∗K ΓΓ∗ R∗ρ JK G∗ρ E

= JK J∗K ((ΓΓ∗ − ρ2 I) + ρ2 I)R∗ρ JK G∗ρ E = −JK G∗ρ E + ρ2 JK J∗K R∗ρ JK G∗ρ E = −JK G∗ρ E + ρ2 JK G−1 ∗ρ E.

It follows from (4.4) that JK ΩJ∗H (Pρ + Q∗ρ E)

= JK ΩJ∗H Pρ 1 ∗ = −JK J∗K ΓSH Rρ JH G2ρ J∗H Pρ + JK G∗ρ EGρ J∗H Pρ ρ ∗ = −ρJK J∗K ΓSH Rρ JH Gρ + JK G∗ρ E.

This proves the first equality in (4.15). To establish the second equality we ˜∗: use the following analog of (4.3) for Γ ∗ ˜ ∗ = JH Ω∗ J∗K + SH Γ∗ + JH J∗H Γ∗ SK Γ .

We have ˜ ∗ (Qρ + P∗ρ E) Γ

= JH Ω∗ J∗K (Qρ + P∗ρ E) ∗ (Qρ + P∗ρ E). + SH Γ∗ (Qρ + P∗ρ E) + JH J∗H Γ∗ SK

By (4.17), SH Γ∗ (Qρ + P∗ρ E)

∗ = ρSH SH (Pρ + Q∗ρ E)

= ρ(Pρ + Q∗ρ E) − ρJH J∗H (Pρ + Q∗ρ E) = ρ(Pρ + Q∗ρ E) − ρ2 JH J∗H Rρ JH Gρ = ρ(Pρ + Q∗ρ E) − ρ2 JH G−1 ρ . Next, ∗ ∗ (Qρ + P∗ρ E) = JH J∗H Γ∗ ΓRρ JH Gρ + ρJH J∗H Γ∗ SK R∗ρ JK G∗ρ E JH J∗H Γ∗ SK

196


and JH J∗H Γ∗ ΓRρ JH Gρ

= JH J∗H ((Γ∗ Γ − ρ2 I) + ρ2 I)Rρ JH Gρ = −JH Gρ + ρ2 JH J∗H Rρ JH Gρ = −JH Gρ + ρ2 JH G−1 ρ .

Finally, it follows from (4.4) and (4.6) that JH Ω∗ J∗K (Qρ + P∗ρ E)

= JH Ω∗ J∗K P∗ρ E ∗ R∗ρ JK G2∗ρ J∗K R∗ρ JK G∗ρ E = −ρJH J∗H Γ∗ SK

+ JH Gρ E ∗ G∗ρ J∗K R∗ρ JK G∗ρ E ∗ R∗ρ JK G∗ρ E + JH Gρ E ∗ E = −ρJH J∗H Γ∗ SK ∗ R∗ρ JK G∗ρ E + JH Gρ . = −ρJH J∗H Γ∗ SK

This proves the second equality in (4.15). Obviously, (4.15) implies that ˜ ˜ ∗ Γ) (Pρ + Q∗ρ E)H ⊂ Ker(ρ2 I − Γ and ˜Γ ˜ ∗ ). (Qρ + P∗ρ E)H ⊂ Ker(ρ2 I − Γ It follows from the Remark preceding the proof of this theorem that if ˜ ˜ ∗ Γ), (Pρ + Q∗ρ E)H = Ker(ρ2 I − Γ ˜∗Γ ˜ with eigenvalue ρ2 such that then there exists an eigenvector f of Γ ∗ ∗ f∈ / (Pρ + Q∗ρ E)H and JH f = O. Let g = SH f ∈ 2 (H). Then ˜ = Γ ˜ · f = ρg. Γf On the other hand, it is easy to see that ˜ = ΓS ˜ H g = Γg, Γf and so Γg ≥ ρg, which contradicts the assumption Γ < ρ. The second part of (4.16) is obvious. To complete the proof, we need the following elementary result. Lemma 4.6. Let Y and Z be Banach spaces, and let T1 : Y → Z and T2 : Z → Y be bounded linear operators. If λ is a nonzero complex number, then λ belongs to the spectrum of T1 T2 if and only if λ belongs to the spectrum of T2 T1 . Proof. Let λ be a nonzero complex number. It has to be proved that if T1 T2 − λI is invertible, then T2 T1 − λI is. Clearly, without loss of generality we may assume that λ = 1. We claim that T2 (T1 T2 −I)−1 T1 −I is the inverse of T2 T1 − I. Indeed, we have (T2 (T1 T2 −I)−1 T1 −I)(T2 T1 −I) = T2 (T1 T2 −I)−1 (T1 T2 −I)T1 −T2 T1 +I = I


197

and (T2 T1 −I)(T2 (T1 T2 −I)−1 T1 −I) = T2 (T1 T2 −I)(T1 T2 −I)−1 T1 −T2 T1 +I = I. By Lemma 4.6, it is sufficient to show that (Γ2 , ρ2 ) is contained in the ˜Γ ˜∗. resolvent set of Γ Suppose that λ > Γ. We have by (4.14) ˜Γ ˜ ∗ = λ2 I − ΓΓ∗ − (JK Ω + SK ΓJH )(Ω∗ J∗ + J∗ Γ∗ S ∗ ). λ2 I − Γ K H K Multiplying this equality on the left and on the right by (λ2 I − ΓΓ∗ )−1/2 , ˜Γ ˜ ∗ is invertible if and and only if the operator we find that λ2 I − Γ ∗ )(λ2 I − ΓΓ∗ )−1/2 I − (λ2 I − ΓΓ∗ )−1/2 (JK Ω + SK ΓJH )(Ω∗ J∗K + J∗H Γ∗ SK

is invertible. It follows from Lemma 4.6 that this is equivalent to the invertibility of the operator ∗ T (λ) = IH − (Ω∗ J∗K + J∗H Γ∗ SK )R∗λ (JK Ω + SK ΓJH ) def

on H. Let us compute T (ρ). Using (4.4) and (4.6), we obtain ∗ ∗ (Ω∗ J∗K + J∗H Γ∗ SK )R∗ρ JK Ω = −J∗H Γ∗ SK R∗ρ JK G2∗ρ J∗K R∗ρ JK Ω

+

1 ∗ Gρ E ∗ G∗ρ J∗K R∗ρ JK Ω + J∗H Γ∗ SK R∗ρ JK Ω ρ

1 ∗ = −J∗H Γ∗ SK R∗ρ JK Ω + Gρ E ∗ G−1 ∗ρ JK Ω ρ ∗ R∗ρ JK Ω = + J∗H Γ∗ SK

1 Gρ E ∗ G−1 ∗ρ JK Ω ρ

1 2 ∗ = − Gρ E ∗ G−1 ∗ρ JK G∗ρ JK R∗ρ SK ΓJH ρ 1 + 2 Gρ E ∗ G−1 ∗ρ JK G∗ρ EGρ ρ 1 1 = − Gρ E ∗ G∗ρ J∗K R∗ρ SK ΓJH + 2 Gρ E ∗ EGρ ρ ρ and Ω∗ J∗K R∗ρ SK ΓJH

∗ = −J∗H Γ∗ SK R∗ρ JK G2∗ρ J∗K R∗ρ SK ΓJH

+

1 Gρ E ∗ G∗ρ J∗K R∗ρ SK ΓJH . ρ

Since E ∗ E = IH , we have T (ρ)

1 2 ∗ G + J∗H Γ∗ SK R∗ρ JK G2∗ρ J∗K R∗ρ SK ΓJH ρ2 ρ ∗ − J∗H Γ∗ SK R∗ρ SK ΓJH .

= IH −

198


Let us show that T (ρ) = O. By (4.10), we have ∗ ∗ ∗ ∗ ∗ R∗ρ = SK R∗ρ SK + SK R∗ρ SK ΓJH J∗H Γ∗ R∗ρ − SK R∗ρ JK J∗K ΓSH Γ R∗ρ .

Hence, ∗ J∗H Γ∗ SK R∗ρ SK ΓJH

= J∗H Γ∗ R∗ρ ΓJH ∗ − J∗H Γ∗ SK R∗ρ SK ΓJH J∗H Γ∗ R∗ρ ΓJH ∗ ∗ ∗ + J∗H Γ∗ SK R∗ρ JK J∗K ΓSH Γ R∗ρ ΓJH . (4.18)

Using (4.12), we obtain from (4.18) ∗ J∗H Γ∗ SK R∗ρ SK ΓJH

∗ ∗ ∗ = −I + ρ2 G−2 ρ + JH Γ SK R∗ρ SK ΓJH ∗ − ρ2 J∗H Γ∗ SK R∗ρ SK ΓJH G−2 ρ ∗ ∗ − J∗H Γ∗ SK R∗ρ JK J∗K ΓSH JH ∗ ∗ + ρ2 J∗H Γ∗ SK R∗ρ JK J∗K ΓSH Rρ JH .

∗ Since obviously, SH JH = O, it follows that ∗ ∗ ∗ R∗ρ SK ΓJH = −G2ρ + ρ2 I + ρ2 J∗H Γ∗ SK R∗ρ JK J∗K ΓSH Rρ JH G2ρ . ρ2 J∗H Γ∗ SK

Dividing this equality by ρ2 and applying (4.6), we see that T (ρ) = O. Let us show that for λ ∈ (Γ, ρ), (−T (λ)x, x) ≥ δx2 ,

x ∈ H,

(4.19)

for some positive number δ. Indeed, it follows from the spectral theorem that ρ2 − λ2 f 2 , f ∈ 2 (K), λ ∈ (Γ, ρ). (R∗λ − R∗ρ )f, f ≥ λ2 ρ2 Now let f = (JK Ω + SK ΓJH )x. Then (R∗λ − R∗ρ )f, f = T (ρ) − T (λ))x, x = (−T (λ)x, x) and f 2 (K)

= (JK Ω + SK ΓJH )x2 (K) −1/2

= R∗ρ ≥

1/2

R∗ρ (JK Ω + SK ΓJH )x2 (K)

R∗ρ −1 R∗ρ (JK Ω + SK ΓJH )x2 (K) 1/2

1/2

1/2 1/2 = R∗ρ −1 (JK Ω + SK ΓJH )∗ R∗ρ (JK Ω + SK ΓJH )x, x 1/2 1/2 1/2 = R∗ρ −1 (I − T (ρ))x, x = R∗ρ −1 xH , which implies (4.19). Thus T (λ) is invertible for λ ∈ (Γ, ρ).


199

˜ is a canonical one-step ρ-extension It follows from Theorem 4.6 that if Γ ˜ of Γ, then Γ = ρ. Now we can show that a canonical one-step ρ-extension has a unique one-step extension of minimal norm. ˜ be a canonical one step Theorem 4.7. Let ρ > Γ and let Γ ˜ ρ-extension of Γ. Then Γ has a unique one-step ρ-extension. Proof. This is an immediate consequence of Corollary 4.4 and the Remark followed by the proof of Theorem 4.5.

Complete ρ-Extensions If Γ = {Ωj+k }j,k≥0 : 2 (H) → 2 (K) is a vectorial Hankel operator such that Γ ≤ ρ and {Ωj }j∈Z is an extension of the sequence {Ωj }j≥0 such that Ωj ∈ B(H, K), j ∈ Z, and the operator Γ# = {Ωj+k }j,k≥∈Z : 2Z (H) → 2Z (K) has norm at most ρ, we say that Γ# is a complete ρ-extension (or simply ρ-extension) of Γ. It is evident that Γ# ≥ Γ. We have proved in §2.2 that each bounded vectorial Hankel operator Γ has a complete extension of norm Γ. Moreover, in §2.2 a complete extension has been constructed by constructing first a one-step extension, then a one step extension of the one-step extension, etc. It turns out that a complete extension of norm Γ is unique if and only if a one-step extension of norm Γ is unique. Theorem 4.8. Let Γ = {Ωj+k }j,k≥0 be a bounded block Hankel matrix. The following are equivalent: (i) Γ has a unique one-step extension of minimal norm; (ii) Γ has a unique complete extension of minimal norm. Proof. The implication (ii)⇒(i) is obvious. Let us prove that (i)⇒(ii). ˜ of norm It is sufficient to show that if Γ has a unique one-step extension Γ ˜˜ ˜ also has a unique one-step extension of minimal norm. Let Γ Γ, then Γ ˜˜ = Γ. Since Γ = S ∗ Γ, ˜ such that ˜ we have be a one-step extension of Γ Γ K

∗˜ ˜∗Γ ˜−Γ ˜ ∗ JK J∗K Γ ˜≥Γ ˜ ∗ Γ. ˜ ˜ ∗ SK SK Γ=Γ Γ∗ Γ = Γ ˜ −1 , ρ > Γ. Consequently, ˜ ∗ Γ) Hence, (ρ2 I − Γ∗ Γ)−1 ≤ (ρ2 I − Γ ∗ 2 ∗ −1 ˜ −1 JH def ˜ ∗ Γ) ˜ −2 JH ≤ J∗H (ρ2 I − Γ = G G−2 ρ = JH (ρ I − Γ Γ) ρ .

˜ 2ρ ≤ G2ρ . By the Heinz inequality (see Appendix 1.7), It follows that G ˜ Gρ ≤ Gρ . If we interchange the roles of Γ and Γ∗ , we obtain the inequality ˜ −1 JK −1/2 . The result follows ˜ ∗ρ def ˜ ∗ Γ) ˜ ∗ρ ≤ G∗ρ , where G = J∗K (ρ2 I − Γ G now from Theorem 4.3 (see conditions (i) and (ii)). Theorems 4.3 and 4.8 give a necessary and sufficient condition for the uniqueness of a complete extension of minimal norm. Note that in the scalar case the same result has been proved in §5.1 (see Theorem 1.5).

200


We can obtain now an analog of the uniqueness result in Theorem 1.1.4 for vectorial Hankel operators. Corollary 4.9. Suppose that ρ2 is an eigenvalue of Γ∗ Γ and (4.8) holds. Then Γ has a unique extension of minimal norm. Proof. This is an immediate consequence of Corollary 4.4 and Theorem 4.8. Note that another sufficient condition for uniqueness can be obtained if we interchange the roles of Γ and Γ∗ . We now consider a condition stronger than (4.8). Namely, we assume that J∗H Ker(Γ2 I − Γ∗ Γ) is a one-to-one map of the eigenspace Ker(Γ2 I − Γ∗ Γ) of Γ∗ Γ onto H. Let V : H → Ker(Γ2 I − Γ∗ Γ) be the inverse of this operator. We consider V as an operator from H to 2 (H). Let W : H → 2 (K) be the operator defined by 1 W = ΓV. ρ Clearly, 1 ΓV = ρW and Γ∗ W = Γ∗ ΓV = ρV. (4.20) ρ We denote by Γ# the complete extension Γ# of norm Γ, which is unique ˆ for an operator by Corollary 4.9. Then Γ# = {Ωj+k }j,k∈Z , where Ωj = Ξ(j) function Ξ ∈ L∞ (B(H, K)) such that ΞL∞ = Γ (see §2.2). As usual we identify in a natural way the spaces 2 (H) and 2 (K) with the Hardy classes H 2 (H) and H 2 (K). We also identify the spaces 2Z (H) and 2Z (K) with L2 (H) and L2 (K). Under this identification we can write (Γ# f )(ζ) = Ξ(ζ)f ζ¯ , ζ ∈ T, f ∈ L2 (H). We have ˜ · V x, P+ Γ# V x = ΓV x = ρV x = Γ ∗

Hence, ΓV = P+ Γ# V = Γ# V . Similarly, Γ W = Γ# V = ρW

and

x ∈ H.

P+ Γ∗# W

= Γ∗# W . Thus

Γ∗# W = ρV.

(4.21)

Let V x = {Vj x}j≥0 and W x = {Wj x}j≥0 , where Vj : H → H and Wj : H → K are the operators defined by ∗ j Vj = J∗H (SH ) V

∗ j and Wj = J∗K (SK ) W.

The following identity holds: j ∗ j V = W ∗ (SK ) W. V ∗ SH

Indeed, ∗ j W ∗ (SK ) W =

1 ∗ ∗ j 1 j j W (SK ) ΓV = W ∗ ΓSH V = V ∗ SH V. ρ ρ

(4.22)


It follows from the definition of V that V0 = IH . Let V and W be the operator functions defined by Vx= z j Vj x and W x = z j Wj x, j≥0

201

x ∈ H.

j≥0

It is easy to see that (4.20) and (4.21) can be rewritten in the following way: z ) = ρW , P+ Ξ∗ W (¯ z ) = ρV P+ ΞV (¯ and ΞV (¯ z ) = ρW ,

Ξ∗ W (¯ z ) = ρV .

(4.23) For x and the functions vx,y = V (z)V (z)x, y and y∗ in H we consider wx,y = W (z)W (z)x, y on the unit circle. We have

j j V y) = (x, V ∗ SH V y), vˆx,y (j) = V (ζ)x, ζ j V (ζ)y dm(ζ) = (V x, SH

∗

T

for j ≥ 0 and

−j −j −j vˆx,y (j) = ζ V (ζ)x, V (ζ)y dm(ζ) = (SH V x, V y) = (V ∗ SH V x, y), T

for j ≤ 0. Similarly,

∗ j W (ζ)x, ζ j W (ζ)y dm(ζ) = W ∗ (SH ) W x, y , w ˆx,y (j) = T

and w ˆx,y (j) =

T

∗ −j ζ −j W (ζ)x, W (ζ)y dm(ζ) = x, W ∗ (SH ) Wy ,

j ≥ 0,

j ≤ 0.

By (4.22), we obtain w ˆx,y (j) = vˆx,y (−j),

j ∈ Z.

z ), and so This means that wx,y (z) = vx,y (¯ W ∗ (z)W (z) = V ∗ (¯ z )V (¯ z ).

(4.24)

If the space H is finite-dimensional, det V belongs to the Hardy class H 2/ dim H , and since det V (0) = det X0 = 1, it is a nonzero function. Therefore for almost all ζ ∈ T the operator V (ζ) is invertible, and by (4.23), we have Ξ(ζ) = ρW (ζ)V −1 ζ¯ , ζ ∈ T. It now follows from (4.23) and (4.24) that the function ρ1 Ξ takes isometric values almost everywhere on T. Suppose now that Γ is a Hankel operator from 2 (H) to 2 (K) and ˜ be a canoniρ > Γ (we do not assume here that dim H < ∞). Let Γ cal one-step ρ-extension that corresponds to an isometry E : H → K in (4.4). By Theorem 4.5 and the Remark following its statement, J∗H is a ˜ ∗ Γ) ˜ onto H, and formulas (4.15) hold. Now one-to-one map of Ker(ρ2 I − Γ

202


the operators Pρ + Q∗ρ and Qρ + P∗ρ can play the roles of the operators V and W above. Note, however, that in this case (Pρ + Q∗ρ )0 = J∗H (Pρ + Q∗ρ ) def

does not have to be equal to IH , but we do not need it! We have used only the fact that V0 is invertible on H, and the fact that (Pρ + Q∗ρ )0 is invertible on H is an immediate consequence of the definition of Pρ and Q∗ρ . ˜ has a unique complete extension of norm As we have already observed, Γ def ˜ ∗˜ ˜ ρ, which we denote by Γ# . It is easy to see that Γ# = Γ # SH = SK Γ# is a complete ρ-extension of Γ. (Recall that SH and SK are bilateral shifts on 2Z (H) and 2Z (K).) Thus by (4.15), we have ∗ (Qρ + P∗ρ E), Γ# (Pρ + Q∗ρ E) = ρSK

∗ Γ∗# SK (Qρ + P∗ρ E) = ρ(Pρ + Q∗ρ E).

(4.25) We define the operator functions P ρ , Qρ , P ∗ρ , and Q∗ρ by P ρx =

z j (Pρ )j x,

Qρ x =

j≥0

and P ∗ρ x =

z j (P∗ρ )j x,

z j (Qρ )j x,

x ∈ H,

j≥0

Q∗ρ x =

j≥0

z j (Q∗ρ )j x,

x ∈ H,

j≥0

where as in the case of operators V and W above, Pρ x = {(Pρ )j x}j≥0 , P∗ρ x = {(P∗ρ )j x}j≥0 ,

Qρ x = {(Qρ )j x}j≥0 ,

and Q∗ρ x = {(Q∗ρ )j x}j≥0 .

ˆ Let Ξ be an operator function in L∞ (B(H, K)) such that Ξ(j) = Ωj , j ≥ 0. As we have already mentioned, ΞL∞ = ρ and (Γ# f )(z) = Ξ(z)f (¯ z ), f ∈ L2 (H). We obtain from (4.25) z ) + Q∗ρ (¯ z )E = ρ¯ z Qρ (z) + P ∗ρ (z)E . (4.26) Ξ(z) P ρ (¯ and

Ξ∗ (z) = z¯ Qρ (z) + P ∗ρ (z)E = ρ P ρ (¯ z ) + Q∗ρ (¯ z )E .

Again, if dim H < ∞, we obtain from (4.26) the following formula for Ξ: −1 z ) + Q∗ρ (¯ z )E . Ξ(z) = ρ¯ z Qρ (z) + P ∗ρ (z)E (P ρ (¯ Such complete ρ-extensions are called canonical.

(4.27)


203

Properties of the Functions Pρ , P∗ρ , Qρ , and Q∗ρ We establish here some important properties of P ρ , P ∗ρ , Qρ , and Q∗ρ which we are going to use later in this section. Theorem 4.10. Suppose that ρ > Γ. Then the following identities hold for almost all ζ ∈ T: ¯ ¯ P ∗ρ (ζ)P ρ (ζ) − Q∗ρ (ζ)Q ρ (ζ) = IH ;

(4.28)

¯ ∗ρ (ζ) ¯ − Q∗ (ζ)Q (ζ) = IK ; P ∗∗ρ (ζ)P ∗ρ ∗ρ

(4.29)

∗ ¯ ¯ P ∗∗ρ (ζ)Q ρ (ζ) − Q∗ρ (ζ)P ρ (ζ) = O.

(4.30)

Proof. Let us establish the following formulas:

IH , j = 0, j ∗ j Pρ∗ SH Pρ − Q∗ρ (SK ) Qρ = O, j > 0, ∗ j P∗ρ SK P∗ρ

−

∗ j Q∗∗ρ (SH ) Q∗ρ

=

(4.31)

IK , j = 0, O, j > 0,

(4.32)

and j ∗ j ∗ j ∗ ∗ j P∗ρ SK Qρ − Q∗∗ρ (SH ) Pρ = P∗ρ (SK ) Qρ − Q∗∗ρ SH Pρ = O,

j ≥ 0. (4.33)

It follows easily from definition that SK ΓPρ = ρQρ

and

∗ Γ∗ SK Qρ = ρPρ − JH Gρ .

Hence, Pρ∗ = ρ1 (Gρ J∗H + Q∗ρ SK Γ), and so for j > 0 we have j Pρ∗ SH Pρ

=

1 1 j j (Gρ J∗H + Q∗ρ SK Γ)SH Pρ = Q∗ρ SK ΓSH Pρ ρ ρ

=

1 ∗ 1 ∗ j ∗ ∗ j ∗ j ) ΓPρ = Q∗ρ SK SK (SK ) SK ΓPρ = Q∗ρ (SK ) Qρ . Q SK (SK ρ ρ ρ

On the other hand, Pρ∗ Pρ − Q∗ρ Qρ

= Pρ∗ Pρ − =

1 ∗ ∗ P Γ ΓPρ ρ2 ρ

1 ∗ 2 P (ρ I − Γ∗ Γ)Pρ = Gρ J∗H Rρ JH Gρ = IH . ρ2 ρ

This proves (4.31). Using the identities SH Γ∗ P∗ρ = ρQ∗ρ

and

∗ ΓSH Q∗ρ = ρP∗ρ − JK G∗ρ ,

we can prove in the same way (4.32).

204


Let us establish (4.33). We have for j ≥ 0 ∗ j SK Qρ P∗ρ

=

1 1 j j+1 Qρ = Q∗∗ρ SH Γ∗ SK ΓRρ JH Gρ (G∗ρ J∗K + Q∗∗ρ SH Γ∗ )SK ρ ρ

=

1 ∗ 1 ∗ j+1 ∗ ∗ j ∗ Q∗ρ SH (SH ) Γ ΓRρ JH Gρ = Q∗∗ρ (SH ) Γ ΓRρ JH Gρ ρ ρ

=

1 ∗ ∗ j Q (S ∗ )j (Γ∗ Γ − ρ2 I + ρ2 I)Rρ JH Gρ = Q∗∗ρ (SH ) Pρ , ρ ∗ρ H

since obviously, 1 ∗ Q (S ∗ )j JH Gρ = O. ρ ∗ρ H Finally, for j > 0 we obtain j Pρ Q∗∗ρ SH

=

1 ∗ 1 ∗ ∗ j−1 j−1 Pρ = P∗ρ (SK ) ΓPρ P∗ρ ΓSH ρ ρ

=

1 ∗ ∗ j ∗ ∗ j P (S ) SK ΓPρ = P∗ρ (SK ) Qρ , ρ ∗ρ K

which completes the proof of (4.33). Now it is easy to deduce (4.28), (4.29), and (4.30) from (4.31), (4.32), and (4.33). Let us show how to obtain, for example, (4.28). We compare z )Qρ (¯ z ). Since the Fourier coefficients of the functions P ∗ρ (z)P ρ (z) and Q∗ρ (¯ both functions take self-adjoint values, it is sufficient to compare their jth Fourier cofficients with j ≥ 0. Let x, y ∈ H and j > 0. Then

∗ ∗ P ρ (ζ)P ρ (ζ)x, y ζ¯j dm(ζ) = P ρ (ζ)P ρ (ζ)x, ζ j P ρ (ζ)y dm(ζ) T

T

j ∗ j (x, Pρ∗ SH Pρ y) = (x, Q∗ρ (SK ) Qρ y)

∗ Qρ (ζ)Qρ (ζ)x, y ζ j dm(ζ) =

=

T

=

T

j ¯ ¯ ¯ Q∗ρ (ζ)Q ρ (ζ)x, y ζ dm(ζ).

For j = 0 we have

∗ ∗ ¯ ¯ P ρ (ζ)P ρ (ζ)x, y dm(ζ) − Qρ (ζ)Q ρ (ζ)x, y dm(ζ) T

T

=

(Pρ∗ Pρ x, y) − (Q∗ρ Qρ x, y) = (x, y).

Formulas (4.29) and (4.30) can be deduced from (4.32) and (4.33) in a similar way.


205

Suppose now that the spaces H and K are finite-dimensional and dim H ≤ dim K. Consider the block matrix function Uρ on T defined by P ρ (z) Q∗ρ (z) (4.34) Uρ (z) = z ) P ∗ρ (¯ z) Qρ (¯ and the matrix

def

J =

O −IK

IH O

.

(4.35)

It is easy to see that (4.28)–(4.30) mean that Uρ∗ (ζ)JUρ (ζ) = J

for almost all ζ ∈ T.

(4.36)

In other words, the matrix function Uρ is J-unitary. It is easy to see that the matrix function Uρ∗ is also J-unitary. Indeed, by (4.36), Uρ (ζ) = J(Uρ∗ (ζ))−1 J, and so Uρ (ζ)JUρ∗ (ζ) = J(Uρ∗ (ζ))−1 JJUρ∗ (ζ) = J(Uρ∗ (ζ))−1 Uρ∗ (ζ) = J.

(4.37)

Considering all entries in (4.37), we obtain the following identities P ρ (ζ)P ∗ρ (ζ) − Q∗ρ (ζ)Q∗∗ρ (ζ) = IH ;

(4.38)

¯ ∗ (ζ) ¯ − Q (ζ)Q ¯ ∗ (ζ) ¯ = IK ; P ∗ρ (ζ)P ρ ∗ρ ρ

(4.39)

¯ − Q (ζ)P ∗ (ζ) ¯ = O. P ρ (ζ)Q∗ρ (ζ) ∗ρ ∗ρ

(4.40)

It follows from (4.28) and (4.29) (as well as from (4.38) and (4.39)) that the operator functions P ρ (z) and P ∗ρ (¯ z ) are invertible. Moreover, −1 ¯ (ζ) ≤ 1 and P ( ζ) ≤ 1 almost everywhere on T. P −1 ρ ∗ρ Consider now the operator function Xρ on T defined by ∗ z )(P ∗∗ρ )−1 (¯ z) Xρ (z) = P −1 ρ (z)Q∗ρ (z) = Qρ (¯

(4.41)

(see (4.40)). Theorem 4.11. The function Xρ has the following properties: Xρ (0) = O;

Xρ (ζ) < 1,

ζ ∈ D;

and

(1 − Xρ (z))−1 ∈ L1 . (4.42)

The operator functions P ρ , P ∗ρ , Qρ , and Q∗ρ are uniquely determined by Xρ , and the functions P ρ and P ∗ρ satisfy ∞ P −1 ρ ∈H ;

P −1 z) ∈ H ∞. ∗ρ (¯

Proof. The fact that Xρ (0) = O is an immediate consequence of the definition of Q∗ρ . It is easy to see from (4.38) that for almost all ζ ∈ T ∗ −1 (ζ) = 1−P ρ (ζ)−2 < 1. Xρ (ζ)2 = Xρ (ζ)Xρ∗ (ζ) = IH −P −1 ρ (ζ)(P ρ )

Hence, (1 − Xρ (ζ))−1 = (1 + Xρ (ζ))P ρ (ζ)2 ≤ 2P ρ (ζ)2 ,

ζ ∈ T,

206

and so


T

(1 − Xρ (ζ))

−1

dm(ζ) ≤ 2

T

P ρ (ζ)2 dm(ζ) < ∞.

This completes the proof of (4.42). Now let E be an isometry from H to K and let −1 def z Qρ (z) + P ∗ρ (z)E (P ρ (¯ z ) + Q∗ρ (¯ z )E . ΞE (z) = ρ¯

(4.43)

ˆ E (j) = Ωj , j ≥ 0. It follows from (4.28)– We have already shown that Ξ (4.30) that ΞE also admits the following representation: −1 ∗ ΞE (z) = ρ¯ Q∗ρ (¯ z P ∗∗ρ (z) + EQ∗ρ (z) z ) + EP ∗ρ (¯ z) . (4.44) Moreover, if E is an arbitrary contractive operator from H to K (i.e., E ≤ 1), then the operator functions defined by (4.43) and (4.44) coincide. It is easy to see that if E1 and E2 are contractive operators from H to K, and we use (4.43) for ΞE1 and (4.44) for ΞE2 , then the following identity follows from (4.28)–(4.30): −1 −1 ΞE1 (z)−ΞE2 (z) = ρ¯ z (P ∗∗ρ (z)+E2 Q∗ρ (z) (E1 −E2 )(P ρ (¯ z ) + Q∗ρ (¯ z )E1 . (4.45) If E1 and E2 are isometries, then it follows from (4.45) that −1 2 z¯(P ∗∗ρ (z) + E2 Q∗ρ (z) (E1 − E2 ) ∈ H− (B(H, K)) and

−1 2 z ) + Q∗ρ (¯ z )E1 ∈ H− (B(H, K)). z¯(E1 − E2 )(P ρ (¯ If E1 is an arbitrary isometry, we can put E2 = −E1 and we find that −1 2 z ) + Q∗ρ (¯ z )E1 ∈ H− (B(H)). (4.46) z¯(P ρ (¯

On the other hand, given an isometry E2 and an arbitrary vector b in K, we can find an isometry E1 such that b ∈ Range(E1 − E1 ). Thus it is easy to see that −1 2 ∈ H− (B(K)). z¯(P ∗∗ρ (z) + E2 Q∗ρ (z) Note that for finite-dimensional spaces H1 and H2 and a space X of functions on T, we say that an operator function F belongs to X(B(H1 , H2 )) (or simply F ∈ X) if (F a, b) ∈ X for any a ∈ H1 and b ∈ H2 . For an isometry E from H to K we define the function GE by −1 GE (z) = P ρ (z) + Q∗ρ (z)E P ρ (z) − Q∗ρ (z)E . It is easy to see from (4.46) that GE ∈ H 1 . By the definition of Xρ , we have for almost all ζ ∈ T: GE (ζ) = (IH + Xρ (ζ)E)−1 (IH − Xρ (ζ)E). Since Xρ (ζ) < 1 for almost all ζ ∈ T, it is easy to see that GE∗ (ζ) + GE (ζ) > O,

a.e. on T.


207

Taking the Poisson integral, we find that GE∗ (ζ) + GE (ζ) > O,

ζ ∈ D.

(4.47)

Define the operator function Y by Y(ζ) = (IH + GE (ζ))−1 (IH − GE (ζ)),

ζ ∈ D.

It is well known (see e.g., Sz.-Nagy and Foias [1], Ch. IV, §4) and easy to verify that that (4.47) implies that Y(ζ) < 1, ζ ∈ D. It is also clear from the definition of Y that GE (ζ) = (IH + Y(ζ))−1 (IH − Y(ζ)),

ζ ∈ D.

It follows that the boundary values of Y coincide with the function Xρ E almost everywhere on T. Thus Xρ E ∈ H ∞ . Since E is an arbitrary isometry, it follows that Xρ ∈ H ∞ . Next, −1 (IH + Xρ (ζ)E) P ρ (ζ) + Q∗ρ (ζ)E −1 = (IH + P −1 = P −1 ρ (ζ)Q∗ρ (ζ)E) P ρ (ζ) + Q∗ρ (ζ)E ρ (ζ) 2 for almost all ζ ∈ T. It follows that P −1 ρ ∈ H and since it is bounded on −1 −1 ∞ z) ∈ H ∞. T, it follows that P ρ ∈ H . Similarly, P ∗ρ (¯ Finally, by (4.41) and (4.38), we have for almost all ζ ∈ T ∗ ∗ −1 −1 ∗ −1 Xρ (ζ)Xρ∗ (ζ) = P −1 ρ (ζ)Q∗ρ (ζ)Q∗ρ (ζ)(P )ρ (ζ) = IH − P ρ (ζ)(P )ρ (ζ).

Hence, Thus

P −1 ρ

∗ −1 IH − Xρ (ζ)Xρ∗ (ζ) = P −1 ρ (ζ)(P )ρ (ζ),

ζ ∈ T.

is a solution of the factorization problem IH − Xρ Xρ∗ = F F ∗ .

It is easy to see that this factorization problem has a unique solution F such that F ∈ H 2 , F −1 ∈ H 2 , and F (0) > O. Indeed, if F1 and F2 are both solutions of the factorization problems satisfying the above conditions, we have F2−1 F1 = F2∗ (F1∗ )−1 . The left-hand side of this equality belongs to H 1 while the right-hand side belongs to H 1 . Thus F2−1 F1 = C, where C is a constant operator. It follows that F2 CC ∗ F2∗ = F2 F2∗ , and so CC ∗ = I. Since, C = F2−1 (0)F1 (0), and both F1 and F2 are positive, we have CC ∗ = F2−1 (0)F1 (0)F1 (0)F2−1 (0) = I. Hence, F22 (0) = F12 (0), which implies C = I. −1 Thus P −1 ρ is uniquely determined by Xρ . Since Xρ (z) = P ρ (z)Q∗ρ (z), Q∗ρ (z) is also uniquely determined by Xρ . Similarly, it can be shown that IK − Xρ∗ (z)Xρ (z) = P −1 z )(P ∗∗ρ )−1 (¯ z ), ∗ρ (¯ and the functions P ∗ρ and Qρ are uniquely determined by Xρ .

208


Lemma 4.12. The following equality holds: ¯ det P ρ (ζ) = det P ∗∗ρ (ζ),

ζ ∈ D.

(4.48)

Proof. Since, P ρ ∈ H 2 , it follows that det P ρ ∈ H 2/n . Next, ∈ H ∞ , and so (det P ρ )−1 ∈ H ∞ . Hence, det P ρ is an outer function. It follows from the definition of P ρ that det P ρ (0) > 0. Similarly, z ) is an outer function, and it takes a positive value at 0. By det P ∗∗ρ (¯ (4.38) and (4.29), we have for almost all ζ ∈ T P −1 ρ

| det P ρ (ζ)|2

=

det(IH + Q∗ρ (ζ)Q∗∗ρ (ζ))

=

¯ 2. det(IK + Q∗∗ρ (ζ)Q∗ρ (ζ)) = | det P ∗∗ρ (ζ)|

z ) are outer functions with positive valSince both det P ρ (z) and det P ∗∗ρ (¯ ues at 0, we obtain (4.48). Consider now the operator quasiball (4.5). Obviously, the left and right semiradii Λρ and Πρ of this quasi-ball admit the following representations: Λρ = ρ1/2 P ρ (0),

Πρ = ρ1/2 P ∗∗ρ (0).

By (4.48), we have det Λρ = det Πρ . It follows that

lim ρ→ Γ +

Λρ is invertible if and only if

lim ρ→ Γ +

Πρ is invertible.

Parametrization of Complete Extensions Now we are ready to obtain a parametrization formula for complete extensions in the case dim H < ∞, dim K < ∞, and ρ > Γ. As we have already mentioned, the problem of describing all complete ρ-extensions of Γ is equivalent to the problem of describing the function Ξ ∈ L∞ (B(H, K)) satisfying ˆ Ξ(j) = Ωj ,

j ≥ 0,

and ΞL∞ ≤ ρ.

(4.49)

Theorem 4.13. Suppose that dim H < ∞, dim K < ∞, and ρ > Γ. An operator function Ξ satisfies (4.49) if and only if Ξ admits a representation −1 def z Qρ (z) + P ∗ρ (z)E(¯ z ) (P ρ (¯ z ) + Q∗ρ (¯ z )E(¯ z) (4.50) Ξ(z) = ΞE (z) = ρ¯ for an operator function E in the unit ball of H ∞ (B(H, K)). Recall that if E is an isometry from H to K, then ΞE determines a canonical complete ρ-extension. We identify here a contractive operator E from H to K with the constant function identically equal to E. We need the following well-known lemma. Lemma 4.14. Let J be the block matrix defined by (4.35). Suppose that the block matrix A = {Ajk }j,k=1,2 is J-unitary, i.e., A∗ JA = J. Then


209

the operator A11 + A12 E is invertible in B(H) for any E in the unit ball of B(H, K) and the transformation E → (A21 + A22 E)(A11 + A12 E)−1

(4.51)

maps the unit ball of B(H, K) onto itself. Proof. We have

A∗ JA =

=

A∗11 A∗12

−A∗21 −A∗22

A∗11 A11 − A∗21 A21 A∗12 A11 − A∗22 A21

A11 A21

A12 A22

A∗11 A12 − A∗21 A22 A∗12 A12 − A∗22 A22

= J.

Hence, we have the following identities: A∗11 A11 = A∗21 A21 + IH , A∗12 A11 = A∗22 A21 ,

A∗11 A12 = A∗21 A22 , A∗12 A12 = A∗22 A22 − IK .

(4.52)

As we have already observed, the matrix A∗ is also J-unitary, and this means that A11 A∗11 = A12 A∗12 + IH , A21 A∗11 = A22 A∗12 ,

A11 A∗21 = A12 A∗22 , A21 A∗21 = A22 A∗22 − IK .

(4.53)

Let E : H → K and E ≤ 1. By (4.53), A∗11 x2 = x2 + A∗12 x2 ,

a ∈ H.

In particular, A11 is invertible. Hence, if x = O, we have (A11 + A12 E)∗ x

≥

A∗11 x − E ∗ A∗12 x ≥ A∗11 x − A∗12 x

=

A∗11 x2 − A∗12 x2 x2 = ≥ x, ∗ ∗ ∗ A11 x + A12 x A11 x + A∗12 x

and so A11 + A12 E is invertible. Now let E1 be a contractive operator from H to K and E2 = (A21 + A22 E1 )(A11 + A12 E1 )−1 . Let us show that E2 ≤ 1. This is equivalent to the inequality (A21 + A22 E1 )∗ (A21 + A22 E1 )(A11 + A12 E1 )−1 x,(A11 + A12 E1 )−1x ≤ x2 (4.54) for any x ∈ H. Put (A11 + A12 E1 )−1 x = y. Clearly, (4.54) is equivalent to the inequality (A21 + A22 E1 )∗ (A21 + A22 E1 )y, y ≤ (A11 + A12 E1 )∗ (A11 + A12 E1 )y, y

210


for any y ∈ H. Using (4.52), we obtain (A11 + A12 E1 )∗ (A11 + A12 E1 )y, y − (A21 + A22 E1 )∗ (A21 + A22 E1 )y, y = (A∗11 A11 y, y) + (E1∗ A∗12 A12 E1 y, y) + (A∗11 A12 E1 y, y) + (E1∗ A∗12 A11 y, y) − (A∗21 A21 y, y) − (E1∗ A∗22 A22 E1 y, y) − (A∗21 A22 E1 y, y) − (E1∗ A∗22 A21 y, y) = y2 − E1 y2 ≥ 0, which proves (4.22). To show that our transformation maps the unit ball of B(H, K) onto itself, we observe that E1∗ = (−A∗21 + A∗11 E2∗ )(A∗22 − A∗12 E2∗ )−1 , and we have to show that E1∗ ≤ 1 whenever E2∗ ≤ 1. Now consider the matrix IK O J# = . O −IH It follows easily from (4.52) that the matrix A∗22 −A∗12 def A# = −A∗21 A∗11 is J# -unitary, i.e., A∗# J# A# = J# . It remains to apply what we have already proved to the matrix A# . Proof of Theorem 4.13. Since the matrix Uρ (ζ) defined by (4.34) is J-unitary for almost all ζ ∈ T, it follows from Lemma 4.14 that the transformation (4.51) maps the unit ball of L∞ (B(H, K)) onto the ball of radius ρ of the space L∞ (B(H, K)). It remains to show that Ξ = ΞE satisfies (4.49) if and only if E ∈ H ∞ (B(H, K)). Let E1 and E2 be functions in the unit ball of L∞ (B(H, K)). In the same way as in (4.45) for constant matrix functions we can show that the following formula still holds for our contractive matrix functions E1 and E2 : ΞE1 (z) −ΞE2 (z)

−1 −1 z )Q∗ρ (z) (E1 (¯ z ) − E2 (¯ z ))(P ρ (¯ z ) + Q∗ρ (¯ z )E1 (¯ z) . = ρ¯ z (P ∗∗ρ (z) + E2 (¯ Suppose first that Ej L∞ < 1. Then by Theorem 4.11, −1 ∞ = (IH + Xρ E1 )−1 P −1 (P ρ + Q∗ρ E1 ρ ∈H . Similarly,

−1 (P ∗∗ρ (¯ z ) + E2 (z)Q∗ρ (¯ z) ∈ H ∞,

and so ΞE1 (z) − ΞE2 (z) ∈ z¯H ∞ .

(4.55)

Suppose now that E1 and E2 are arbitrary functions in the unit ball of H ∞ (B(H, K)). Since sup ΞrE1 − ΞrE2 L∞ ≤ 2ρ

r∈(0,1)


211

and for almost all ζ ∈ T, lim ΞrE1 (ζ) − ΞrE2 (ζ) = ΞE1 (ζ) − ΞE2 (ζ), r→1−

it follows that (4.55) holds for arbitrary E1 and E2 in the unit ball of H ∞ (B(H, K)). ˆ E (j) = Ωj , j ∈ Z+ , if E is a constant isometry, it Since we know that Ξ follows from (4.55) that the same is true for an arbitrary E in the unit ball of H ∞ (B(H, K)). ˆ E (j) = Ωj , j ∈ Z+ , for some E in the unit ball of Suppose now that Ξ L∞ (B(H, K)). Let us show that E ∈ H ∞ (B(H, K)). Put E1 = E and E2 = O. Then −1 −1 ΞE1 (z) − ΞE2 (z) = ρ¯ z (P ∗∗ρ (z) E(¯ z )(P ρ (¯ z ) + Q∗ρ (¯ z )E(¯ z) ∈ z¯H ∞ . It follows that −1 z ) = ρE(¯ z ) IH + Xρ (¯ z )E(¯ z) ∈ H 1 . (4.56) zP ∗∗ρ (z) ΞE (z) − ΞO (z) P ρ (¯ Therefore

−1 z )E(¯ z ) IH + Xρ (¯ z )E(¯ z) IH − Xρ (¯

2 =IH + zXρ (¯ z )P ∗∗ρ (z) ΞE (z) − ΞO (z) P ρ (¯ z) ∈ H 1. ρ Consequently,

−1 IH − E ∗ (¯ z )Xρ∗ (¯ z) z )Xρ∗ (¯ z) ∈ H 1. IH + E ∗ (¯

In the same way as in the proof of Theorem 4.11, we can deduce that Xρ E ∈ H ∞ . It follows now from (4.56) that E ∈ H ∞ . Note that the proof of Lemma 4.14 yields that the transformation (4.51) maps the set of isometries from H to K onto itself. This implies the following result. Corollary 4.15. Let E be in the unit ball of H ∞ (B(H, K)). The function ρ−1 ΞE takes isometric values, i.e., ρ−2 Ξ∗E (ζ)ΞE (ζ) = IH

a.e. on T

if and only if E is an inner function, i.e., E ∗ (ζ)E(ζ) = IH for almost all ζ ∈ T. To conclude this section, we consider the following version of the Nehari problem. Let Ψ ∈ L∞ (H, K)). Consider the Hankel operator 2 HΨ : H 2 (H) → H− (K). Let ρ > HΨ . We want to parametrize all func∞ tions Φ ∈ L (B(H, K)) such that HΦ = H Ψ

and ΦL∞ ≤ ρ.

(4.57)

This problem easily reduces to the problem we have solved in this secˆ − 1), j ∈ Z+ . Consider the block Hankel matrix tion. Let Ωj = Ψ(−j

212


Γ = {Ωj+k }j,k≥0 . Then Γ = HΨ . Clearly, a function Ξ in L∞ (B(H, K)) satisfies the conditions ˆ Ξ(j) = Ωj ,

j ≥ 0,

and ΞL∞ ≤ ρ

if and only if the function Φ = z¯Ξ(¯ z ) satisfies (4.57). Starting from Γ we can construct the operator functions P ρ , Qρ , P ∗ρ , and Q∗ρ as above and we can deduce from Theorem 4.13 the following parametrization formula for the solutions of the Nehari problem (4.57). Theorem 4.16. Suppose that dim H < ∞, dim K < ∞, Ψ is a function in L∞ (B(H, K)), and ρ > HΨ . A function Φ ∈ L∞ (B(H, K)) satisfies (4.57) if and only if it has the form −1 def Φ = ΦE = ρ Qρ (¯ z ) + P ∗ρ (¯ z )E(z) (P ρ (z) + Q∗ρ (z)E(z) (4.58) for a function E in the unit ball of H ∞ (B(H, K)). Proof. This is an immediate consequence of Theorem 4.13. Note that formula (4.58) can be rewritten in the following way: −1 −1 z ) Xρ∗ (z) + E(z) (IH + Xρ (z)E(z) P ρ (z). ΦE = ρP −1 ∗ρ (¯ This parametrization formula has the same form as the parametrization formula in the scalar case given in Theorem 1.13. Recall that in the scalar case Theorem 1.13 also covers the case ρ = HΨ .

5. Parametrization in the General Case In this section we consider the most general Nehari problem and obtain a parametrization formula for its solutions. Let H and K be separable Hilbert spaces and let Γ be a Hankel oper2 ator from H 2 (H) to H− (K), i.e., Γ = HΨ for some Ψ ∈ L∞ (B(H, K)). For ρ ≥ Γ we consider the Nehari problem of finding all functions Φ ∈ L∞ (B(H, K)) such that HΦ = Γ

and ΦL∞ ≤ ρ.

(5.1)

If ρ > 0, we can multiply Γ by 1/ρ and reduce the general case to the case ρ = 1. Thus we assume that Γ ≤ 1 and we are going to describe operator functions Φ satisfying (5.1) with ρ = 1. Recall that Γ satisfies the following commutation relation: Γ(zf ) = P− zΓf,

f ∈ H 2 (H).

(5.2)

2 2 Consider the space X = H− (K) ⊕ H (H), which consists of pairs f1 2 (K) and f2 ∈ H 2 (H). X is a Hilbert f = such that f1 ∈ H− f2

5. Parametrization in the General Case

213

space with its natural inner product (·, ·)2 . We define a new (semi-)inner product on X by g1 g1 f1 I Γ f1 def , , = . Γ∗ I f2 g2 f2 g2 X 2 Since Γ ≤ 1, it is easy to see that (·, ·)X is a semi-inner product. We identify vectors f and g in X if (f − g, f − g)X = 0. Then (·, ·)X is an inner product on the space of equivalence classes. Finally, we consider the completion of this space with respect to this inner product and denote the resulting Hilbert space by X. We use the notation (·, ·)X for the inner def

1/2

product in X and · X for the norm in X: f X = (f, f )X . We consider the operator V with domain 2 (K) ⊕ zH 2 (H) H−

and range 2 (K) ⊕ H 2 (H) z¯H−

defined by

V

f1 f2

=

z¯f1 z¯f2

.

Then V is an isometric operator in the norm · X . Indeed, let 2 f1 , g1 ∈ H− (K) and f2 , g2 ∈ zH 2 (H). We obtain from (5.2) z¯f1 + Γ¯ z f2 g1 z¯g1 f1 = ,V , V f2 g2 Γ∗ z¯f1 + z¯f2 z¯g2 X X =

(¯ z f1 , z¯g1 ) + (¯ z f2 , z¯g2 )

+

(Γ¯ z f2 , z¯g1 ) + (¯ z f1 , Γ¯ z g2 )

=

(f1 , g1 ) + (f2 , g2 )

+

z f2 , g1 ) + (f1 , P− zΓ¯ z g2 ) (P− zΓ¯

=

(f1 , g1 ) + (f2 , g2 ) + (Γf2 , g1 ) + (f1 , Γg2 ) g1 f1 , . = f2 g2 X Thus V extends to an isometric operator in the norm of X with domain def

2 (K) ⊕ zH 2 (H) DV = closX H−

and range def

2 RV = closX z¯H− (K) ⊕ H 2 (H).

As in §1, we consider unitary extensions U of V on Hilbert spaces Y ⊃ X, i.e., U is a unitary operator on Y such that U DV = V . We are interested only in minimal unitary extensions of V (see §1 for the definition).

214


Consider the natural imbedding I1 of H into X defined by O def I (1) x = , x ∈ H; x here x is the function identically equal to x. We also consider the isometric imbedding I (2) of K into X defined by z¯x I (2) x = , x ∈ K. O Definition. Given a unitary extension U of V , consider the harmonic operator function ΦU on D defined by ∗ def ¯ ∗ )−1 − I I (1) . ΦU (ζ) : H → K, ΦU (ζ) = I (2) U (I − ζU )−1 + (I − ζU The harmonic function ΦU is called the scattering function of the unitary extension U of V . Lemma 5.1. Let U be a unitary extension of V . Then ΦU (ζ)B(H,K) ≤ 1,

ζ ∈ D.

Proof. Let x ∈ H. Let us show that ¯ ∗ )−1 − I I (1) x, I (1) x , x2H = (I − ζU )−1 + (I − ζU Y

Since (I − ζU )−1 =

ζjU j

¯ ∗ )−1 = (I − ζU

and

j≥0

ζ ∈ D. (5.3)

ζ¯j (U ∗ )j ,

ζ ∈ D,

j≥0

and (I (1) x, I (1) x)Y = (x, x)H , to prove (5.3), it suffices to show that (U ∗ )j I (1) x, I (1) x = 0, j > 0. Y

By definition, for j > 0 O O O j = V = = I (1) x. Uj x zj x zj x Hence, ∗ j

(U ) I

(1)

x=

Consequently, for j > 0 we obtain I (U ∗ )j I (1) x, I (1) x = Γ∗ Y =

O zj x Γ I

.

O zj x

O , x 2

(z j x, x)H 2 (H) = 0,

which proves (5.3). Similarly, for y ∈ K, ¯ ∗ )−1 − I U ∗ I (2) y, U ∗ I (2) y , y2K = (I − ζU )−1 + (I − ζU Y

ζ ∈ D. (5.4)


215

Using the following easily verifiable identity ¯ ∗ )−1 (I − ζU )−1 , ¯ ∗ )−1 − I = 1 − |ζ|2 (I − ζU (I − ζU )−1 + (I − ζU we find from (5.3) and (5.4) that for x ∈ H and y ∈ K I ΦU (ζ) y y , I x x Φ∗U (ζ) 2

=(y, y) + (ΦU (ζ)x, y) + (Φ∗U (ζ)y, x) + (x, x) ¯ ∗ )−1 −I I (1) x+U ∗ I (2) y , I (1) x+U ∗ I (2) y = (I −ζU )−1 + (I − ζU Y (1) 2 −1 (1) ∗ (2) −1 ∗ (2) I x+U I y Y ≥ 0, = 1−|ζ| (I −ζU ) I x+U I y ,(I −ζU ) which proves the lemma. Definition. If U is a unitary operator on Hilbert space, the function def z¯j (U ∗ )j + zj U j PU (z) = (I − zU )−1 + (I − z¯U ∗ )−1 − I = j0

Then the coupling U acts in the following way: −1 ζ x ˇ+ (ζ) − x(0) + ζ¯ x ˇ− (ζ) + x(0) x ˇ(ζ) = . U yˇ+ (ζ) ζ −1 yˇ+ (ζ)

(5.21)


227

As above, we put ψ(ζ) =

I (2) O

∗ A

x ˇ(ζ) ˇ2 (ζ) n

ζ ∈ D,

,

and we have (ˇ n2 )+ (ζ) (ˇ n1 )+ (ζ) s11 (ζ) s12 (ζ) = , ψ(ζ) s21 (ζ) s22 (ζ) x We have from (5.21) O

yˇ+ (ζ) (ˇ n1 )+ (ζ)

= yˇ+ (ζ),

ζ ∈ D.

ζ B11 yˇ+ (ζ) + B12 (ˇ n1 )+ (ζ) = yˇ+ (ζ),

ζ ∈ D,

ζ Thus

I

B

ζ ∈ D.

and so n1 )+ (ζ), yˇ+ (ζ) = ζ(I − ζB11 )−1 B12 (ˇ

ζ ∈ D.

On the other hand, we have from (5.21) yˇ+ (ζ) O I B = (ˇ n2 )+ (ζ), (ˇ n1 )+ (ζ)

(5.22)

ζ ∈ D.

Thus n1 )+ (ζ) = (ˇ n2 )+ (ζ), B21 yˇ+ (ζ) + B22 (ˇ

ζ ∈ D.

Substituting (5.22) in (5.23), we obtain (ˇ n2 )+ (ζ) = ζB21 (I − ζB11 )−1 B12 + B22 (ˇ n1 )+ (ζ),

(5.23)

ζ ∈ D,

i.e., n1 )+ (ζ), (ˇ n2 )+ (ζ) = EB (ζ)(ˇ

ζ ∈ D,

(5.24)

where EB is the characteristic function of the colligation B, i.e., EB (ζ) = PN2 B(I − ζPZ B)−1 N1 . It follows from (5.24) that s11 (ζ) s12 (ζ) n1 )+ (ζ) (ˇ n1 )+ (ζ) EB (ζ)(ˇ = , ψ(ζ) s21 (ζ) s22 (ζ) x

ζ ∈ D, (5.25)

and so s11 (ζ)EB (ζ)(ˇ n1 )+ (ζ) + s12 (ζ)x = (ˇ n1 )+ (ζ),

ζ ∈ D.

Thus (ˇ n1 )+ (ζ) = (I − s11 (ζ)EB (ζ))−1 s12 (ζ)x,

ζ ∈ D.

(5.26)

228


Next, we have ψ(ζ)

= = =

I (2) O

I (2)

∗

∗

∗ I (2) U

A

x ˇ(ζ) ˇ2 (ζ) n

ζ −1 x ˇ+ (ζ) − x(0) + ζ¯ x ˇ− (ζ) + x(0)

x ˇ(ζ) yˇ+ (ζ)

,

ζ ∈ D.

ψ(ζ) = ΦU (ζ)x,

ζ ∈ D.

Since x(0) = I (1) x, this implies that

Together with (5.25) and (5.26) this implies that ΦU (ζ)x = s22 (ζ) + s21 (ζ)EB (ζ)(I − s11 (ζ)EB (ζ))−1 x,

ζ ∈ D.

It remains to show that for any E in the unit ball of H ∞ (B(N1 , N2 )) the function Φ defined by (5.17) is the scattering function of some unitary extension U of V . This follows from Theorem 5.7. Indeed, given E, we can find a unitary colligation B : Z ⊕ N1 → Z ⊕ N2 such that E = EB . Then Φ = ΦU , where U is the unitary coupling of A and B. Remark. Clearly, the Nehari problem has a unique solution if and only if N1 = {O} or N2 = {O}.

Concluding Remarks The results of §1 were obtained in Adamyan, Arov, and Krein [2]. Note that the proof of the fact that w ∈ H ∞ (see the proof of Theorem 1.13) was not given in Adamyan, Arov, and Krein [2]. This proof can be found in Ando [1]. The results of §2 were obtained in Nevanlinna [1] by a different method. The example following Theorem 2.1 is taken from Garnett [1], Ch. IV, §4. The results of §3 are due to Adamyan, Arov, and Krein [3]. Note also that in Adamyan, Arov, and Krein [3] the authors applied these results to the following interpolation problem (the Schur–Takagi problem) for the ∞ class H(m) : given points ζ1 , · · · , ζn in D and complex numbers w1 , · · · , w1 , ∞ such that f ∞ ≤ 1 and f (ζj ) = wj . parametrize all functions f ∈ H(m) The results of §4 are taken from Adamyan, Arov, and Krein [4]. Section 5 follows the paper Kheifets [1] where the author presents a method that allows him to parametrize the solutions of the Nehari problem in the general case. Note that Adamyan [1] parametrized all solutions of the Nehari problem for matrix functions in the completely indeterminate case, i.e., in the case when the subspaces DV and RV (see §5) have maximal

Concluding Remarks

229

codimensions. In the unpublished thesis Adamyan [2] the solutions of the Nehari problem were parametrized in the general case of matrix functions.

6 Hankel Operators and Schatten–von Neumann Classes

In this chapter we study Hankel operators that belong to the Schatten– von Neumann class S p , 0 < p < ∞. The main result of the chapter says that Hϕ ∈ S p if and only if the function P− ϕ belongs to the Besov class 1/p Bp (see Appendix 2.6). We prove this result in §1 for p = 1. We give two different approaches. The first approach gives an explicit representation of a Hankel operator in terms of rank one operators while the second approach is less constructive but it allows one to represent a nuclear Hankel operator as an absolutely convergent series of rank one Hankel operators. We also characterize in §1 nuclear Hankel operators of the form Γ[µ] in terms of measures µ in D. In §2 we prove the main result for 1 < p < ∞. We use the result for p = 1 and the Marcinkiewicz interpolation theorem for linear operators. Finally, in §3 we treat the case p < 1. To prove the necessity 1/p of the condition ϕ ∈ Bp , we reduce the estimation of Hankel matrices to the estimation of certain special finite matrices that are normal and whose norms can be computed explicitly. In §4 we use the real interpolation method to describe the Hankel operators of Schatten–Lorentz class S p,q . As an application we obtain a descrip1/p tion of real interpolation spaces between Bp , 0 < p < ∞, and the space V M O. We study properties of projections onto the Hankel matrices in §5. We prove that the natural (averaging projection) P onto the set of Hankel matrices is bounded on S p for 1 < p < ∞ but is unbounded on S 1 and on the set of bounded (or compact) operators. We also show that there are no bounded projections from the space of bounded (or compact) operators

232

Chapter 6. Hankel Operators and Schatten–von Neumann Classes

onto the set of bounded (or compact) Hankel operators. However, it turns out that there exist bounded (weighted averaging) projections from S 1 onto the set of Hankel matrices in S 1 . Finally, we prove in §5 that PS 1 ⊂ S 1,2 and if T belongs to the generalized Matsaev ideal S 2,ω , i.e., (sj (T ))2 j≥0

1+j

< ∞,

then PT is a compact operator. On the other hand, we show in §5 that if {sj } is a nonincreasing sequence of positive numbers such that s2j = ∞, 1+j j≥0

then there exists an operator T such that sj (T ) = sj and PT is not the matrix of a bounded operator. In §6 we apply the main result of the chapter to the study of the rate of rational approximation in terms of properties of functions. We obtain analogs of the Jackson–Bernstein theorem for rational approximation in the norm of BM O. We also obtain some results on rational approximation in 0 )+ . As an application the L∞ norm and in the norm of the Bloch space (B∞ of the nuclearity criterion for Hankel operators we obtain a sharp estimate of the principal part of a meromorphic function with at most n poles in D in terms of its norm in L∞ (T). We consider in §7 several other applications of the main result of this chapter. Namely, we consider functions of the model operators, commutators, integral operators on L2 (R+ ) with kernels depending on the sum of the variables, and Wiener–Hopf operators on finite intervals. In §8 we consider generalized (weighted) Hankel matrices of the form {(1 + j)α (1 + k)β ϕ(j ˆ + k)}j,k≥0 and we obtain boundedness, compactness, and S p criteria for such matrices. Finally, in §9 we study Hankel (and generalized Hankel) operators on spaces of vector functions.

1. Nuclearity of Hankel Operators In this section we describe the trace class Hankel operators. For conˆ + k)}j,k≥0 , where venience with Hankel matrices Γϕ = {ϕ(j we work j ϕ= ϕ(j)z ˆ is a function analytic in the unit disk. We shall treat Hankel j≥0

matrices Γϕ as operators on the space 2 . The following theorem is the main result of this section. Theorem 1.1. Let ϕ be a function analytic in the unit disk. Then the Hankel operator Γϕ belongs to the trace class S 1 if and only if ϕ ∈ B11 .

1. Nuclearity of Hankel Operators

233

Recall that the analytic functions of Besov class B11 admit the following descriptions (see Appendix 2.6): (i)

D

(ii)

|ϕ (ζ)|dm2 (ζ) < ∞;

T T

(iii)

|ϕ(ζτ )−2ϕ(ζ)+ϕ(ζ τ¯)| dm(ζ)dm(τ ) |τ −1|2

< ∞;

2n ϕ∗Wn L1 < ∞ (the polynomials Wn are defined in Appendix

n≥0

2.6). The following description of trace class Hankel operators Hϕ is an immediate consequence of Theorem 1.1. Corollary 1.2. Let ϕ be a function on T of class BM O. Then Hϕ ∈ S 1 if and only if P− ϕ ∈ B11 . There are several different proofs of Theorem 1.1. First we present a proof that gives sharp estimates for the norms Γϕ S 1 from above and from below. Then we give another proof of the sufficiency of the condition ϕ ∈ B11 that allows one to expand a trace class Hankel operator into an absolutely convergent series of rank one Hankel operators. In §5 we give another proof of the necessity of the condition ϕ ∈ B11 . In the proof given below we use the duality between S 1 and the space of bounded linear operators B (see Appendix 1.1). Recall that we consider the following pairing between S 1 and B: T, R = trace T R∗ ,

T ∈ S 1 , R ∈ B.

Proof of Theorem 1.1. Let us first prove that Γϕ ∈ S 1 if ϕ ∈ B11 . We have ϕ= ϕ ∗ Wn n≥0

and

2n ϕ ∗ Wn L1 < ∞

n≥0

(see Appendix 2.6). Clearly, ϕ ∗ Wn is a polynomial of degree at most 2n+1 − 1. The following lemma gives a sharp estimate of the trace norm of a Hankel operator with polynomial symbol. Lemma 1.3. Let f be an analytic polynomial of degree m. Then Γf S 1 ≤ (m + 1)f 1 .

234


Proof of Lemma 1.3. Given ζ ∈ T, we define the elements xζ and yζ of 2 by

j ζ , 0 ≤ j ≤ m, xζ (j) = 0, j > m;

f (ζ)ζ¯k , 0 ≤ k ≤ m, yζ (k) = 0, k > m. Define the rank one operator Aζ on 2 by Aζ x = (x, xζ )yζ , x ∈ 2 . Then Aζ ∈ S 1 and Aζ S 1 = xζ 2 yζ 2 = (m + 1)|f (ζ)|. Let us prove that

Aζ dm(ζ) (1.1) Γf = T

(the function ζ → Aζ is continuous and so the integral can be understood as the limit of integral sums). We have

f (ζ)ζ¯j+k dm(ζ), (Γf ej , ek ) = fˆ(j + k) = T

(Aζ ej , ek ) = f (ζ)ζ¯j ζ¯k ,

j, k ≤ m.

Since fˆ(k) = 0 for k > m, (1.1) holds and

Γf S 1 ≤ Aζ S 1 dm(ζ) ≤ (m + 1) |f (ζ)|dm(ζ). T

T

Let us now complete the proof of the sufficiency of the condition f ∈ B11 . It follows from Lemma 1.3 that Γϕ S 1 ≤ Γϕ∗Wn S 1 ≤ 2n+1 ϕ ∗ Wn L1 . n≥0

n≥0

Suppose now that Γϕ ∈ S 1 . Define the polynomials Qn and Rn , n ≥ 1, by  k ≤ 2n−1 ,   0, ˆ n (k) = 1 − |k − 2n |/2n−1 , 2n−1 ≤ k ≤ 2n + 2n−1 , Q   0 k ≥ 2n + 2n−1 ;  k ≤ 2n ,   0, ˆ n (k) = 1 − |k − 2n − 2n−1 |/2n−1 , 2n ≤ k ≤ 2n+1 , R   0 k ≥ 2n+1 . Clearly, Wn = Qn + 12 Rn , n ≥ 1. Let us show that 22n+1 ϕ ∗ Q2n+1 L1 < ∞. n≥0

(1.2)

2n 2n+1 The inequalities 2 ϕ ∗ Q2n L1 < ∞, 2 ϕ ∗ R2n+1 L1 < ∞, n≥1 n≥0 2n and 2 ϕ ∗ R2n < ∞ can be proved in exactly the same way. n≥1


235

To prove (1.2), we construct an operator B on 2 such that B ≤ 1 and Γϕ , B = 22n f ∗ Q2n+1 L1 . n≥0

Consider the squares Sn = [22n−1 , 22n−1 + 22n − 1] × [22n−1 + 1, 22n−1 + 22n ],

n ≥ 1,

on the plane. Let {ψn }n≥1 be a sequence of functions in L∞ such that ψn L∞ ≤ 1. We define the matrix {bjk }j,k≥0 of B by

bjk

  ψˆn (j + k), (j, k) ∈ Sn , n ≥ 1, = (j, k) ∈ Sn .  0, n≥1

22n−1 .

· 22n−1+1 .

·

2n+1 3·22n−1 . −1 .2

3·22n+1 . −1

·

. .. .. .. ... Ωn .ψˆn(j . +k). .. .... . .. .. 3 · 22n−1.

22n+1+1 .

3 · 22n+1 .

.. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . .. .. .. .. . . . . Ωn+1 . ψˆn+1 (j + k) .. . .. .. . . . . .. .. .. .. . . .. .. .. .. . . . . .. .. .. .. . . .. .. .. .. .. ..

? FIGURE 1.

·

·

·

.

236


Let us show that B ≤ 1. Consider the subspaces Hn = span{ej : 22n−1 ≤ j ≤ 22n−1 + 22n − 1}, Hn = span{ej : 22n−1 + 1 ≤ j ≤ 22n−1 + 22n }. It is easy to see that B=

Pn Γψn Pn ,

n≥1

Pn

where Pn and are the orthogonal projection onto Hn and Hn , respectively. Since the spaces {Hn }n≥1 are pairwise orthogonal as well as the spaces {Hn }n≥1 , we have B = sup Pn Γψn Pn ≤ sup Γψn ≤ sup ψn L∞ ≤ 1. n

n

Let us show that Γϕ , B = where as usual (g, h) =

T

Γϕ , B =

n

22n (Q2n+1 ∗ ϕ, ψn ),

n≥0

f (ζ)h(ζ)dm(ζ), g ∈ L1 , h ∈ L∞ . We have

Γϕ , Pn Γψn Pn

n≥1 2n

=

n≥1

=

2n+1

2 +2

(22n − |j − 22n+1 |)ϕ(j) ˆ ψˆn (j)

j=22n

22n (Q2n+1 ∗ ϕ, ψn ).

n≥1

We can pick now a sequence {ψn }n≥1 such that (Q2n+1 ∗ ϕ, ψn ) = Q2n+1 ∗ ϕL1 . Then Γϕ , B = Hence,

22n Q2n+1 ∗ ϕL1 .

n≥1 2n+1

2

n≥1

Q2n+1 ∗ ϕL1 = 2Γϕ , B ≤ 2Γϕ S 1 < ∞.

Remark. It is easy to see that the above proof gives the following estimates 1 n 2 ϕ ∗ Wn L1 ≤ Γϕ S 1 ≤ 2 2n ϕ ∗ Wn L1 . 6 n≥1

n≥0

To give an alternative proof of the sufficiency of the condition ϕ ∈ B11 in Theorem 1.1, we need the following decomposition theorem for the class (B11 )+ .


237

Theorem 1.4. Let ϕ ∈ (B11 )+ . Then there exist sequences of complex numbers {cn }n≥1 and {λn }n≥1 , λn ∈ D, such that cn , ζ ∈ D, (1.3) ϕ(ζ) = 1 − λn ζ n≥1

n≥1

|cn | < ∞. 1 − |λn |

(1.4)

Proof. Consider the space (b−1 ∞ )+ of functions analytic in D (see Appendix 2.6). Then its dual can naturally be identified with (B11 )+ with respect to the following pairing: fˆ(n)ϕ(n), ˆ (1.5) (f, ϕ) = n≥0

where ϕ ∈ (B11 )+ and f is a polynomial in (b−1 ∞ )+ (see Appendix 2.6). We consider here the following norm on (b−1 ∞ )+ : def

f & = sup |f (ζ)|(1 − |ζ|). ζ∈D

Let X be the space of functions analytic in D that admit a representation (1.3) satisfying (1.4). The norm of a function ϕ in X is, by definition, the infimum of the right-hand side of (1.4) over all representations (1.3). Let ∗ us show that (b−1 ∞ )+ = X isometrically with respect to the pairing (1.5), which would imply that X = (B11 )+ and the proof would be completed. Let ϕ be as in (1.3), f ∈ (b−1 ∞ )+ . We have 1 |(f, ϕ)| = = cn f, cn f (λn ) 1 − λn z n≥0 n≥0 ≤

|cn | · |(f (λn ))| ≤

n≥0

n≥0

|cn | · f & . 1 − |λn |

(1.6)

∗ Hence, ϕ ∈ (b−1 ∗ ≤ ϕX . Let us now show that ∞ )+ and ϕ(b−1 ∞ ) +

f (b−1 = sup{|(f, ϕ)| : ϕX ≤ 1}. ∞ )+

(1.7)

It follows from (1.6) that the left-hand side of (1.7) is less that or equal to the right-hand side. Let us prove the opposite inequality. Let ϕ(z) = 1−|λ| 1−λz . We have ¯ (f, ϕ) = (1 − |λ|)f (λ) and since ϕX ≤ 1, this proves (1.7). Therefore the norm · X coincides on X with the norm · (b−1 ∗ and ∞ ) +

so X is a closed subspace of B11 . The result follows from the fact that the linear combinations of functions of the form (1 − λz)−1 , λ ∈ D, are dense in (B11 )+ .

238


Theorem 1.4 enables us to prove that any nuclear Hankel operator can be expanded into absolutely convergent series of rank one Hankel operators. Theorem 1.5. Let ϕ ∈ (B11 )+ . Then Γϕ admits a representation Γϕ = Γn , n≥1

where the Γn are Hankel operators of rank one and

Γn < ∞.

n≥1

Proof. By Theorem 1.4, ϕ admits a representation (1.3) that satisfies (1.4). Let cn Γn = Γ 1−λ z n

Clearly, rank Γn = 1 (see Theorem 3.1 of Chapter 1). The result follows from the obvious estimate 1 1 . ≤ Γ 1−λz 1 − |λ| As we have mentioned above, Theorem 1.5 gives an alternative proof of the necessity of the condition f ∈ B11 for Γf to be nuclear. In §1.7 we have considered the Hankel operators of the form Γ[µ], where µ is a finite complex measure on D. The following theorem describes a class of measures that produce trace class Hankel operators. Theorem 1.6. Let µ be a finite complex measure on D such that

1 dµ(ζ) < ∞. (1.8) 2 D (1 − |ζ| ) Then Γ[µ] is nuclear. Conversely, if Γ is a nuclear Hankel operator on 2 , there exists a finite complex measure µ satisfying (1.8) such that Γ = Γ[µ]. Note that the proof that (1.8) implies the nuclearity of Γ[µ] is similar to the proof of the sufficiency of the condition ϕ ∈ B11 in Theorem 1.1. Proof. Suppose that µ satisfies (1.8). Let ζ ∈ D. Define the vector fζ ∈ 2 by fζ = (1 − |ζ|2 )1/2 {ζ n }n≥0 . Consider the rank one operator Aζ defined by Aζ x = (x, fζ¯)fζ , x ∈ 2 . Clearly, Aζ = 1, ζ ∈ D. Let us show that Γ[µ] = (1 − |ζ|2 )−1 Aζ dµ(ζ). D

We have

D

2 −1

(Aζ ej , ek )(1 − |ζ| )

dµ(ζ) = D

ζ j+k dµ(ζ) = (Γ[µ]ej , ek ),

standard orthonormal basis of the space 2 . Therefore where {e j }j≥0 is2 the −1 Γ[µ] = (1 − |ζ| ) Aζ dµ(ζ), and so D

Γ[µ]S 1 ≤

2 −1

D

Aζ (1 − |ζ| )

dµ(ζ) = D

(1 − |ζ|2 )−1 dµ(ζ) < ∞.

2. Hankel Operators of Class S p , 1 < p < ∞

239

Suppose now that Γ is a nuclear Hankel operator on 2 . Then by Theorems 1.1 and 1.5, Γ = Γϕ , where ϕ admits a representation (1.3) that satisfies (1.4). Put def µ = cn δλn , n≥1

where δλ is the unit mass at λ. It is easy to see that Γ = Γ[µ] and

1 |cn | dµ(ζ) = < ∞, 2 (1 − |ζ| ) (1 − |λn |2 ) D n≥1

which completes the proof. Consider now positive Hankel matrices of trace class. Theorem 1.7. Let Γ be a positive Hankel operator on 2 . Then Γ ∈ S 1 if and only if Γ = Γ[µ] where µ is a positive measure on (−1, 1) such that

1 (1 − t2 )−1 dµ(t) < ∞. (1.9) −1

Proof. If Γ = Γ[µ] and µ satisfies (1.9), then by Theorem 1.6, Γ ∈ S 1 . Suppose now that Γ is a positive nuclear operator. By Theorem 1.7.2, Γ = Γ[µ], where µ is a positive measure on (−1, 1). We have

1 1 trace Γ[µ] = (Γ[µ]ej , ej ) = t2j dµ(t) = (1 − t2 )−1 dµ(t) < ∞, j≥0

j≥0

−1

−1

which completes the proof.

2. Hankel Operators of Class S p , 1 < p < ∞ In this section we obtain a criterion for a Hankel operator to belong to S p for 1 < p < ∞. We are going to use the nuclearity criterion obtained in §1 and the Marcinkiewicz interpolation theorem. As in §1 we shall work with ˆ + k)}n,k≥0 . operators Γϕ on 2 whose matrix in the standard basis is {ϕ(n As usual we identify operators on 2 with their matrices in the standard basis of 2 . The following theorem is the main result of the section. Theorem 2.1. Let 1 < p < ∞ and let ϕ be a function analytic in D. 1/p Then Γϕ ∈ S p if and only if f ∈ Bp . Corollary 2.2. Let 1 < p < ∞ and let ϕ be a function on T of class 1/p BM O. Then Hϕ ∈ S p if and only if P− ϕ ∈ Bp . We are going to use the following special case of the Marcinkiewicz interpolation theorem: Let (X , µ) be a measure space and let A be a bounded linear operator from ∞ to L∞ (X , µ). Suppose that A has weak type (1,1), i.e., µ{x ∈ X : |(Aξ)(x)| > δ} ≤ const

ξ1 , δ

ξ ∈ 1 , δ > 0.

240


Then the restriction of A to p is a bounded operator from p to Lp (X , µ) for 1 < p < ∞. Consider the closed subspace c0 of ∞ , which consists of sequences converging to zero. The conclusion of the Marcinkiewicz theorem holds if A is a bounded operator from c0 to L∞ (X , µ) and has weak type (1,1). We need the following consequence of the Marcinkiewicz interpolation theorem. Recall that S ∞ is the space of compact operators on Hilbert space. Corollary 2.3. Let V be a bounded linear operator from S ∞ to L∞ (X , µ). Suppose that A has weak type (1,1), i.e., µ{x ∈ X : |(VT )(x)| > δ} ≤ const

T S 1 , δ

T ∈ S 1 , δ > 0.

Then the restriction of V to S p is a bounded operator from S p to Lp (X , µ) for 1 < p < ∞. Proof. Suppose that T ∈ S p , 1 < p < ∞. Let us show that sj (·, fj )gj . VT ∈ Lp (X , µ). Consider the Schmidt expansion of T : T = j≥0

We define the operator A on c0 by A{xj }j≥0

  =V xj (·, fj )gj  . j≥0

Obviously, A is a bounded operator from c0 to L∞ (X , µ) and A has weak type (1,1). Consequently, by the Marcinkiewicz interpolation theorem A is a bounded operator from p to Lp (X , µ). Clearly, VT = A{sj }j≥0 ∈ Lp (X , µ). ∗ 1/p , 1 ≤ p < ∞. To prove Theorem 2.1 we consider the dual spaces Bp ∗ 1/p Recall (see Appendix 2.6) that the dual space Bp , 1 ≤ p < ∞, can + −1/p of functions analytic in D and we be identified with the space Bp +

consider the following pairing (f, g) = fˆ(j)ˆ g (j),

−1/p f ∈ P + , g ∈ Bp . +

j≥0

For an operator T on 2 we define the analytic function WT by (WT )(z) =

n≥0

n

tn z ,

def

tn =

n

(T ej , en−j ),

j=0

where {ej }j≥0 is the standard orthonormal basis of 2 . Lemma 2.4. Let 1 < p ≤ ∞. Then W is a bounded operator from S p −1/p to Bp .

2. Hankel Operators of Class S p , 1 < p < ∞

241

Proof. Consider the measure µ on the Borel subsets of D defined by

1 def dm2 (ζ). µ(E) = (1 − |ζ|)2 E Let V be the following transformation defined on the set of operators on 2 : def (VT )(z) = (1 − |z|)2 (WT ) (z). To prove the lemma it suffices to show that V is a continuous operator from S p to Lp (D, µ), 1 < p ≤ ∞. Indeed, this would mean that

|(WT ) |p (1 − |ζ|)2p−2 dm2 (ζ) < ∞, 1 < p < ∞; D

sup(1 − |ζ|)2 |(WT ) (ζ)| < ∞,

p = ∞.

ζ∈D −1/p

. The latter means exactly that WT ∈ Bp Consider first the case p = ∞. We have to show that WT ∈ (B11 )∗+ whenever T ∈ S ∞ . Let ϕ be a polynomial. Then Γϕ , T = trace Γϕ T ∗ =

n≥0

ϕ(n) ˆ

n

(T ej , en−j ) = (ϕ, WT ).

j=0

Therefore |(ϕ, WT )| ≤ T · Γϕ S 1 ≤ const ·T · ϕB11 . It follows that WT determines a continuous linear functional on (B11 )+ , −1 . Hence, V is a bounded operator from S ∞ which means that WT ∈ B∞ ∞ to L (D, µ). Let us show that V has weak type (1,1), i.e., T S 1 , δ > 0. δ Let T ∈ S 1 . Then WT ∈ H 1 . Indeed, it is sufficient to establish this only for rank one operator T . Let T = (·, ξ)η, where ξ = {ξn }n≥0 , 2 y = {yn }n≥0 ∈ . Consider the functions f and g in H 2 defined by n ¯ ξn z , f (z) = ηn z n ∈ H 2 . Clearly, WT = f g and, consequently, µ{ζ ∈ D : |(VT )(ζ)| > δ} ≤ const

n≥0

n≥0

WT ∈ H 1 . (If we use the fact that every function in H 1 can be represented as a product of two functions in H 2 , we get that WS 1 = H 1 .) We now use the Littlewood–Paley theorem (see Appendix 2.6): 1/2

1 1 2 ϕ∈H ⇒ |ϕ (rζ)| (1 − r)dr dm(ζ) < ∞. T

0

To prove that V has weak type (1,1), it suffices to prove the following assertion: if ψ is a measurable function on [0, 1] and Eδ = {x ∈ [0, 1] : |ψ(x)|(1 − x)2 > δ},

242


then

Eδ

dx const ≤ (1 − x)2 δ

1/2

1

|ψ(x)|2 (1 − x)dx

.

(2.1)

0

Indeed, inequality (2.1) implies that V has weak type (1,1), since by (2.1) µ{ζ ∈ D : (1 − |ζ|)2 (WT ) (ζ)| > δ} 1/2

1 const |(WT ) (rζ)|2 (1 − r)dr dm(ζ), ≤ δ 0 T and the integral on the right-hand side is finite by the Paley–Littlewood theorem. Let us prove (2.1). We have

1 dx 2 2 2 |ψ(x)| (1 − x)dx ≥ |ψ(x)| (1 − x)dx ≥ δ . (1 − x)3 Eδ Eδ 0 Therefore it is sufficient to show that 1/2

dx dx ≤ const 2 3 E (1 − x) E (1 − x)

(2.2)

for an arbitrary measurable subset of [0,1]. Let us transfer the interval (0,1] onto the half-line [1, ∞) by means of the substitution s = 1/(1 − x). It is easy to see that (2.2) is equivalent to the inequality

1/2

ds ≤ const s ds F

F

for every measurable subset of [1, ∞). It is easy to see that

a+1 1 s ds ≥ const s ds = ((a + 1)2 − 1), 2 1 F where a = ds is the Lebesgue measure of F . Hence, F

1/2 s ds

F

≥

1/2

1 a2 a +a ≥√ =√ ds. 2 2 2 F

Thus V has weak type (1,1), and so by Corollary 2.3, V is a bounded operator from S p to Lp (D, µ) for 1 < p ≤ ∞. As we have already mentioned, it follows that W is a bounded operator from S p to Bp−1p , 1 < p ≤ ∞. Proof of Theorem 2.1. Suppose that Γϕ ∈ S p . Let us show that 1/p −1/p ϕ ∈ Bp . By Lemma 2.4, WΓϕ ∈ Bp . This means that p

n p−2 (n + 1)ϕ(n)ζ ˆ (1 − |ζ|) dm2 (ζ) < ∞. D n≥0

3. Hankel Operators of Class S p , 0 < p < 1

Clearly, this is equivalent to the fact that

D

1/p

243

|ϕ (ζ)|(1 − |ζ|)p−2 dm2 (ζ) < ∞.

Hence, ϕ ∈ Bp . 1/p Suppose now that ϕ ∈ Bp . Let us show that Γϕ ∈ S p . Let T be an operator on 2 that has only finitely many nonzero matrix entries. We have T, Γϕ =

ϕ(n) ˆ

n

(T ej , en−j ) = (WT, ϕ).

j=0

n≥0

Therefore |T, Γϕ | ≤ const ·ϕB 1/p WT B −1/p ≤ const ·ϕB 1/p T S p p

p

p

by Lemma 2.4. This implies that Γϕ determines a continuous linear functional on S p , that is, Γϕ ∈ S p .

3. Hankel Operators of Class S p , 0 < p < 1 In this section we describe the Hankel operators of class S p with 0 < p < 1. As in the previous sections for for a function ϕ analytic in ˆ + k)}j,k≥0 , which we the unit disk we denote by Γϕ the Hankel matrix {ϕ(j identify with an operator on 2 . The following theorem is the main result of the section. Theorem 3.1. Suppose that 0 < p < 1 and let ϕ be a function analytic 1/p in D. Then Γϕ ∈ S p if and only if ϕ ∈ Bp . Corollary 3.2. Let 0 < p < 1 and let ϕ be a function on T of class 1/p BM O. Then Hϕ ∈ S p if and only if P− ϕ ∈ Bp . The proof of the sufficiency is based on the same idea as in the case p = 1 (see §1). The only difference is that instead of integrating over the unit circle T we integrate over its discrete subgroups. However, the proof of necessity is much trickier. Unlike the case p ≥ 1, we cannot use duality arguments. Instead we introduce certain operations on matrices that do not increase the S p quasinorm. Using such operations we cut certain “pieces” of Γϕ and then paste them to get a matrix whose singular values can be evaluated explicitly. This will allow us to get lower estimates for Γϕ S p . 1/p In this section we use the following definition of the Besov space Bp ∞ (see 2.6). Let v ∈ C (R) such that v ≥ 0, supp v = [1/2, 2], and Appendix v(x/2n ) = 1 for x ≥ 1. The kernels Vn are defined by n≥0

Vn =

k z k , n ≥ 1, v 2n

k≥0

V0 (z) = z¯ + 1 + z.

244

Chapter 6. Hankel Operators and Schatten–von Neumann Classes 1/p

A function ϕ analytic in D belongs to Bp if and only if 2n ϕ ∗ Vn pp < ∞. n≥0

Recall (see Appendix 1.1) that for 0 < p < 1 the S p quasinorm satisfies the following version of triangle inequality: T1 + T2 pS p ≤ T1 pS p + T2 pS p ,

T1 , T2 ∈ S p .

(3.1)

We start with an auxiliary fact that we need to estimate certain “pieces” of Γϕ . Let F be an infinitely differentiable function on R with compact support and let m be a positive integer. Consider the trigonometric polynomial k zk . Fm (z) = F m k∈Z Recall that FF is the Fourier transform of F , (FF )(t) = F (x)e−2πixt dx. R

Lemma 3.3. Let Gm (t) = Fm (e2πit ) − m(FF )(−mt),

t ∈ [−1/2, 1/2].

Then lim Gm L∞ [−1/2,1/2] mN = 0

m→∞

for all N ∈ Z+ . Proof. Let ψ(x) = F (x/m)e2πixt . Then (Fψ)(y) = m(FF )(m(y − t)). Let us apply the Poisson summation formula for ψ (see Stein and Weiss [1], Ch. VII, §2): ψ(k) = (Fψ)(k). k∈Z

k∈Z

It follows that k e2πikt = m(FF )(−mt) + F m k∈Z

Then Gm =

m(FF )(m(k − t)).

k∈Z, k =0

m(FF )(m(k − t)).

k∈Z, k =0

Since F is infinitely smooth and has compact support, we have cN |(FF )(x)| ≤ (1 + |x|)N for all N ∈ Z+ . Hence, m sup{|FF )(x)| : x ∈ [m(k − 1/2), m(k + 1/2)]} Gm L∞ [−1/2,1/2] ≤ k =0

≤ 2cN

k>0

1 m ≤ const , (1 + m(k − 1/2))N (1 + m/2)N


245

which implies the result. Corollary 3.4. Fm Lp (T) ≤ cF m1−1/p for some constant cF . Proof. By Lemma 3.3 the result follows from the obvious inequality

1/2

|m(FF )(−mt)|p dt ≤ mp−1 |(FF )(t)|p dt. R

−1/2

2

We identify operators on with their matrices in the standard orthonormal basis {ej }j≥0 of 2 . We need the following notion. Definition. Let A = {ajk }j,k≥0 , B = {bjk }j,k≥0 be matrices. The Schur product of A and B is by definition the matrix A B = {ajk bjk }j,k≥0 . If Ω is a bounded subset of R+ × R+ , we denote by PΩ the operator whose matrix is given by

1, (n, k) ∈ Ω, (PΩ ek , en ) = 0, (n, k) ∈ Ω. The following lemma is important both for upper and lower estimates. Lemma 3.5. Let ψ be a polynomial of degree m − 1 and let A ∈ S p . Then Γψ AS p ≤ (2m)1/p−1 ψp AS p .

(3.2)

Proof. It suffices to show that (3.2) holds for operators A of rank one. Indeed, if A is an arbitrary operator, then we can consider its Schmidt expansion and apply (3.2) to each term. Then inequality (3.2) for A follows from (3.1). Let Ax = (x, α)β, x ∈ 2 , where α = {αn }n≥0 , β = {βn }n≥0 . Let ζj = e2πij/(2m) , 0 ≤ j ≤ 2m−1. Define 2 vectors fj and gj , 0 ≤ j ≤ 2m−1, by

βn ζ¯jn , 0 ≤ n < m, ψ(ζj )αk ζjk , 0 ≤ k < m, fj (k) = gj (n) = 0, n ≥ m. 0, k ≥ m; We define the rank one operators Aj , 0 ≤ j ≤ 2m − 1, by Aj x = (x, fj )gj ,

x ∈ 2 .

Let us show that Γψ A =

2m−1 1 Aj . 2m j=0

(3.3)

Indeed, if k ≥ m or n ≥ m, then (Aj ek , en ) = ((Γψ A)ek , en ) = 0. On the other hand, if n < m and k < m, then (Aj ek , en ) = ψ(ζj )αk βn ζ¯n+k j

and ˆ + k). ((Γψ A)ek , en ) = αk βn ψ(n

246


Identity (3.3) now follows from the equality 2m−1 1 ˆ ψ(d) = ψ(ζj )ζ¯jd , 2m j=0

0 ≤ d ≤ 2m − 1,

which is true for every polynomial ψ of degree at most 2m − 1. We have Aj S p ≤ α2 β2 |ψ(ζj )| = |ψ(ζj )| · AS p . It now follows from (3.1) that Γψ ApS p ≤

2m−1 1 p AS p |ψ(ζj )|p . p (2m) j=0

(3.4)

For τ ∈ T we define the polynomial ψτ by ψτ (ζ) = ψ(τ ζ). Let us show that Γψτ AS p = Γψ AS p for all τ ∈ T. Indeed, this follows from the obvious equality Γψτ A = Dτ (Γψ A)Dτ , where Dt is the unitary operator on 2 defined by Dτ en = τ n en (here {en }n≥0 is the standard orthonormal basis of 2 ). To complete the proof, we integrate inequality (3.4) in τ :

Γψ ApS p = Γψτ ApS p dm(τ ) T

≤

2m−1

1 p AS p |ψτ (ζj )|p dm(τ ) (2m)p T j=0

=

2m ApS p ψpp = (2m)1−p ApS p ψpp . (2m)p

Remark. It can also be proved that Γψ AS p ≤ (4m)1/p−1 ψp AS p for an arbitrary trigonometric polynomial ψ of degree at most m − 1 (in ˆ + k)}n,k≥0 ). The proof is exactly the same, except this case Γψ = {ψ(n that it is necessary to consider the set of points {e2πij/(4m) }0≤j≤4m−1 . Corollary 3.6. Let ψ be a polynomial of degree m and suppose that a, b, c, d are integers such that 0 ≤ a < b < m, 0 ≤ c < d < m. Put Ω = [a, b) × [c, d). Then PΩ Γψ S p ≤ ((b − a)(d − c))1/2 (2m)1/p−1 ψp . Proof. We have PΩ x = (x, α)β, x ∈ 2 , where α = β=

b−1 n=a

d−1

en and

n=c

en . Therefore PΩ S p = ((b − a)(d − c))1/2 . The result now follows

immediately from Lemma 3.6 by putting A = PΩ .


247

Corollary 3.7. Let ψ be a polynomial of degree m − 1. Then Γψ S p ≤ 21/p−1 m1/p ψp . Proof. It suffices to let a = c = 0 and b = d = m. To obtain a lower estimate for Γϕ S p we evaluate explicitly the singular values of the following matrix:   c0 c1 · · · cN −2 cN −1  cN −1 c0 · · · cN −3 cN −2     .. . . .. ..  , (3.5) A =  ... . . . .      c2 c3 · · · c0 c1 c1 c2 · · · cN −1 c0 where the cj , 0 ≤ j ≤ N − 1, are complex numbers. Lemma 3.8. Let ψ(z) =

N −1

ck z k and ζj = e2πij/N , 0 ≤ j ≤ N − 1.

k=0

Then



N −1

AS p = 

1/p |ψ(ζj )|p 

.

(3.6)

j=0

Proof. Let xj be the vector with coordinates 1, ζj , ζj2 , · · · , ζjN −1 . It is easy to verify that the vectors xj are pairwise orthogonal and Axj = ψ(ζj )xj . It follows that the matrix A is normal and so (3.6) holds. Remark. Let



c1 c2 .. .

   B=   cN −1 c0

c2 c3 .. .

··· ··· .. .

cN −1 c0 .. .

c0 c1 .. .

c0 c1

··· ···

cN −3 cN −2

cN −2 cN −1

    .  

It is easy to see that BS p = AS p . Consider now the squares Ωn = [2n−3 , 2n+1 ] × [2n−3 , 2n+1 ] in R+ × R+ . We are now in a position to get a lower estimate for PΩn ∗ Γϕ S p . Lemma 3.9. PΩn Γϕ S p ≥ const ·2n/p ϕ ∗ Vn p . Proof. It is not very convenient for us to deal with the polynomial ϕ ∗ Vn since the interval [2n−1 + 1, 2n+1 − 1] on which the Fourier coefficients of Vn are nonzero is too wide for our purpose. To overcome this problem, we consider a representation of the function v, in terms of which 10 the polynomials Vn are defined, in the form v = rs , where rs ∈ C ∞ (R), s=0

248

Chapter 6. Hankel Operators and Schatten–von Neumann Classes (s)

supp rs = [1/2+s/8, 1/2+(s+1)/8]. Consider the polynomials Rn defined by k z k , n ∈ Z+ . rs Rn(s) (z) = 2n k∈Z

Clearly, Vn =

10

(s)

Rn . Obviously, it suffices to prove that

s=0

p 2 ϕ ∗ Rn(s) ≤ const PΩn Γϕ pS p ,

0 ≤ s ≤ 10.

n

p

To be definite, assume that s = 0 (the proof is the same for the remaining def

(0)

values of s) and put Rn = Rn . Clearly, ˆ n = {k : 2n−1 < k < 2n−1 + 2n−2 }. supp R We have ΓRn PΩn Γϕ = PΩn Γϕ∗Rn . Consequently, by Lemma 3.5, PΩn Γϕ∗Rn S p ≤ const ·2n(1−1/p) Rn p PΩn Γf S p . In view of Corollary 3.4, Rn p ≤ const ·2n(1−1/p) , and so PΩn Γϕ∗Rn S p ≤ const PΩn Γϕ S p . Therefore it is sufficient to get a lower estimate for PΩn Γϕ∗Rn S p . def Let f = ϕ ∗ Rn , N = 2n−2 , ck = fˆ(2n−1 + k), k = 0, 1, · · · , N , I = [2n−3 , 2n−2 + 2n−3 ), J1 = (2n−3 , 2n−2 + 2n−3 ], and n−2 n−3 n−1 n−3 +2 ,2 +2 ]. Obviously, J2 = (2 PI×J1 Γf S p ≤ PΩn Γf S p , PI×J2 Γf S p ≤ PΩn Γf S p . Consider the matrices    c1 c1 c2 · · · cN −1 cN c2  c2   c2 c · · · c 0 c3 3 N     ..   . . . . .. . .. .. .. ..  =  .. M1 =  . .     cN −1 cN · · · 0 0   cN −1 0 cN 0 0 0 ··· 0 0 and



0 0 .. .

   M2 =    0 c0 It is easy to see that

0 0 .. .

··· ··· .. .

0 c0 .. .

c0 c1 .. .

c0 c1

··· ···

cN −3 cN −2

cN −2 cN −1

··· ··· .. .

cN −1 0 .. .

··· ···

0 0

    .  

M1 S p = PI×J2 Γf S p ≤ PΩn Γf S p ,

0 0 .. .



     0  0


249

M2 S p = PI×J1 Γf S p ≤ PΩn Γf S p .

I

2+n−3 ,-

.3·22n−3 .·

.·

3·2n−2

2n+1

.

.-

3. 2n−

 

J1

 

3·2n−3. c0   c1 . J2  ..



5·2n−3 . cN 3·2n−2.

Ωn

2n+1.

?

FIGURE 2.

Hence,  c1  c2   ..  .   cN −1 c0

c2 c3 .. .

··· ··· .. .

cN −1 c0 .. .

c0 c1 .. .

c0 c1

··· ···

cN −3 cN −2

cN −2 cN −1

p    p p  = M1 +M2 S p ≤ 2PΩn Γf S p .   Sp

.

250


Since f = z 2

n−1

N −1

cj z j , it follows from Lemma 3.8 and the remark

j=0

after it that N −1

|f (ζj )|p ≤ 2PΩn Γf pS p ,

(3.7)

j=0

where ζj = e2πij/N , 0 ≤ j ≤ N − 1. To get a lower estimate of the righthand side of (3.7) in terms of f p , we argue as in the proof of Lemma 3.5. def

Let τ ∈ T, put fτ (ζ) = f (τ ζ), ζ ∈ T. It is easy to see that PΩn Γfτ = Dτ (PΩn Γf )Dτ , where the unitary operator Dτ is defined in the proof of Lemma 3.5. Hence, PΩn Γfτ S p = PΩn Γf S p ,

τ ∈ T.

Applying inequality (3.7) to fτ and integrating with respect to τ , we obtain 2n−3 ϕpp ≤ PΩn Γf pS p . 1/p

Proof of Theorem 3.1. Assume first that ϕ ∈ Bp . By Corollary 3.7 and inequality (3.1) we have Γϕ∗Vn pS p ≤ 21−p 2n ϕ ∗ Vn pp . Γϕ pS p ≤ n≥0

n≥0

Consequently, Γϕ ∈ S p . 1/p Assume now that Γϕ ∈ S p . We have to show that ϕ ∈ Bp . Fix an integer M > 4 whose choice will be specified later. Without loss of generality we can assume that ϕ(k) ˆ = 0 for k ≤ 2M , since Γϕ is clearly in S p for every polynomial ϕ. Let us observe that it suffices to prove that Γψ pS p ≥ const ψB 1/p p

(3.8)

for every polynomial ψ. Indeed, suppose that q ∈ C ∞ (R), supp q = [−2, 2], and q(t) = 1 for t ∈ [−1, 1]. Let k q Qn = zk , 2n k∈Z

and consider the polynomials ϕ ∗ Qn . It follows from (3.8) that ϕ ∗ Qn B 1/p ≤ const Γϕ∗Qn S p = const Γϕ ΓQn S p . p

In view of the remark after Lemma 3.5 we have Γϕ ΓQn S p ≤ const Qn p 2n(1/p−1) Γϕ S p and, since Qn p ≤ const 2n(1−1/p) (see Corollary 3.4), we obtain ϕ ∗ Qn B 1/p ≤ const . p


251

Therefore n−1

2j ϕ ∗ Vj pp =

j=0

n−1

2j (ϕ ∗ Qn ) ∗ Vj pp ≤ ϕ ∗ Qn p 1/p ≤ const, Bp

j=0 1/p

which implies that ϕ ∈ Bp . To prove inequality (3.8), consider the orthogonal projection Pj onto    / 0 span es : s ∈ 2kM +j−3 , 2kM +j+1 , 0 ≤ j ≤ M − 1.   k≥1

Define the operator T by T =

M −1

⊕Pj Γϕ Pj .

j=0

Obviously, T pS p ≤ M Γϕ pS p . Let us estimate T S p from below. It is M −1 Pj Γϕ Pj pS p . clear that T pS p = j=0

Let (1) def

Tj

=

(2)

PΩkM +j Γϕ

and Tj

(1)

= Pj Γϕ Pj − Tj .

k≥1

Since the intervals [2kM +j−3 , 2kM +j+1 ], k ≥ 1, are pairwise disjoint, it follows that (1) p PΩkM +j Γϕ pS p . Tj = Sp

By Lemma 3.9,

k≥1

(1) p 2kM +j ϕ ∗ VkM +j pp . Tj ≥ const Sp

k≥1

(2) Let us get an upper estimate for Tj

. Put

(k)

2 0 1 / = 2kM +j−3 , 2kM +j+1 × 0, 2(k−1)M +j+1 ,

(k)

2 / 1 0 = 0, 2(k−1)M +j+1 × 2kM +j−3 , 2kM +j+1 .

∆j Υj

It is easy to see that (2)

Tj

=

(2)

P∆(k) Tj

k≥1

Consequently,

Sp

j

+

k≥1

(2)

PΥ(k) Tj . j

p (2) p (2) P∆(k) Tj . Tj ≤ 2 Sp

k≥1

j

Sp

252


p p (2) Obviously, P∆(k) Tj ≤ P∆(k) Γϕ . It is also clear that Sp

j

Sp

j

P∆(k) Γϕ = P∆(k) Γψ , j

j

where ψ =ϕ∗

2

VkM +j+s .

s=−3

Hence, by Corollary 3.6, P∆(k) Γψ pS p ≤ const 2(kM +j+1)p/2 2((k−1)M +j+1)p/2 2(kM +j+3)(1−p) ψpp j

≤ const 2−M p/2 2kM +j ψpp ≤ const 2−M p/2 2kM +j

2

ϕ ∗ VkM +j+s pp .

s=−3

Note that the constants are all independent of M . Therefore M −1

(2) p 2n ϕ ∗ Vn pp . Tj ≤ const 2−M p/2 Sp

j=0

n≥M

Finally, we have T pS p

≥

M −1

M −1 (1) p (2) p Tj − Tj Sp

j=0

≥ c1

Sp

j=0

2n ϕ ∗ Vn pp − c2 2−M p/2

n≥M

2n ϕ ∗ Vn pp .

n≥M

We can choose now M large enough that c2 2−M p/2 ≤ c1 /2. Then we have 1 c1 n T pS p ≥ Γϕ pS p ≥ 2 ϕ ∗ Vn pp . M 2M n≥M

Remark. It is easy to see that the above proof also works for p = 1. It is also easy to modify the proof of necessity to work for 1 < p < ∞. We can now deduce the following property of polynomials. Corollary 3.10. Let n be a positive integer, and let ζj = e2πij/(4n) , 0 ≤ j ≤ 4n − 1. Then for 0 < p ≤ 1 c1 ψpp ≤

4n−1 1 |ψ(ζj )|p ≤ c2 ψpp 4n j=0

for every polynomial ψ of degree n − 1, where the positive constants c1 and c2 do not depend on n.

4. Hankel Operators and Schatten–Lorentz Classes

253

Proof. Put ϕ = z n ψ. We have shown (see Lemma 3.5 and Corollary 3.7) that 4n−1 Γϕ pS p ≤ const |ψ(ζj )|p . j=0

On the other hand, it follows from (3.7) that 4n−1

|ψ(ζj )|p ≤ const Γϕ pS p .

j=0

To complete the proof we note that for polynomials ϕ of the form ϕ = z n ψ, deg ψ ≤ n − 1, we have ϕp 1/p ≤ const nϕpp ≤ const ϕp 1/p , Bp

Bp

which follows easily from the definition of with the Vn .

1/p Bp

in terms of convolutions

4. Hankel Operators and Schatten–Lorentz Classes In this section we study Hankel operators of Schatten–Lorentz classes S p,q . We shall see in the next section that such questions are important for the study of metric properties of the averaging projection onto the Hankel matrices. For 0 < p < ∞ and 0 < q ≤ ∞ the Schatten–Lorentz class S p,q is, by definition, the class of operators T on Hilbert space such that (sn (T ))q (1 + n)q/p−1 < ∞, q < ∞, n≥0

sup (1 + n)1/p sn (T ) < ∞,

q = ∞.

n∈Z+

Clearly, S p,p = S p , S p1 ,q1 ⊂ S p2 ,q2 if p1 < p2 and S p,q1 ⊂ S p,q2 if q1 < q2 . The Schatten–Lorentz classes can be obtained from the Schatten–von Neumann classes S p with the help of interpolation by the real method. Definition. Let (X0 , X1 ) be a quasi-Banach pair, i.e., X0 and X1 are imbedded in a quasi-Banach space X. We define on def

X0 + X1 = {x0 + x1 : x0 ∈ X0 , x1 ∈ X1 } the Peetre K-functional by K(t, x, X0 , X1 )

def

=

inf{x0 X0 + tx1 X1 : x0 ∈ X0 , x1 ∈ X1 }, x ∈ X0 + X1 , t > 0.

254


For 0 < θ < 1, 0 < q ≤ ∞ the real interpolation space (X0 , X1 )θ,q consists of the elements x ∈ X0 + X1 , for which q 1/q ∞ K(t, x, X0 , X1 ) dt def xθ,q = < ∞, q < ∞, θ t t 0 def

xθ,∞ = sup t>0

K(t, x, A0 , A1 ) . tθ

pairs and It is easy to see that if (X0 , X1 ) and (Y0 , Y1 ) are quasi-Banach X0 is a bounded T : X0 ∩ X1 → Y0 + Y1 is a linear operator such that T operator from X0 to X1 and T Y0 is a bounded operator from Y0 to Y1 , then T (X0 , X1 )θ,q is a bounded operator from (X0 , X1 )θ,q to (Y0 , Y1 )θ,q for 0 < θ < 1, 0 < q < ∞. We refer the reader to Bergh and L¨ ofstr¨ om [1] for the theory of interpolation spaces. It is well known that for 0 < p < ∞ there are constants c1 and c2 such that  1/p  1/p n n c1  (sj (T ))p  ≤ K(t, T, S p , S ∞ ) ≤ c2  (sj (T ))p  , j=0 j=0 (4.1) n1/p ≤ t ≤ (n + 1)1/p ,

T ∈ S∞,

which implies that for 0 < p0 < ∞ (S p0 , S ∞ )θ,q = S p,q ,

0 < θ < 1, 0 < q ≤ ∞, p =

p0 1−θ

(4.2)

(see, e.g., Karadzhov [1]). We denote by ΓS p,q the space of Hankel matrices of class S p,q and by ΓS p the space of Hankel matrices of class S p . Theorem 4.1. Let p0 > 0. Then (ΓS p0 , ΓS ∞ )θ,q = ΓS p,q ,

0 < θ < 1, 0 < q ≤ ∞, p =

p0 . 1−θ

Proof. Clearly, it suffices to show that for T ∈ ΓS ∞ K(t, T, S p0 , S ∞ ) ≤ K(t, T, ΓS p0 , ΓS ∞ ) ≤ const K(t, T, S p0 , S ∞ ). The left-hand inequality is obvious. Let us prove the right-hand one. Let n1/p ≤ t ≤ (n + 1)1/p . It follows from Theorem 4.1.1 that there exists a Hankel operator R of rank at most n such that T − R = sn (T ). We have 1/p  n−1 K(t, T, ΓS p0 , ΓS ∞ ) ≤ RS p0 + tT − R =  (sj (R))p  + tsn (T ). j=0

4. Hankel Operators and Schatten–Lorentz Classes

255

It is easy to see that sj (R) ≤ T − R + sj (T ) = sn (T ) + sj (T ) ≤ 2sj (T ). Therefore 1/p  n−1 K(t, T, ΓS p0 , ΓS ∞ ) ≤ 2  (sj (T ))p  + tsn (T ) j=0

≤



const 

n

1/p (sj (T ))p 

.

j=0

It follows now from (4.1) that K(t, T, ΓS p0 , ΓS ∞ ) ≤ const K(t, T, S p0 , S ∞ ). Corollary 4.2. Let p0 > 0. Then (ΓS p0 , ΓS ∞ )θ,q = ΓS p,q , where p = p0 /(1 − θ). Proof. The result follows immediately from Theorem 4.1 and (4.2). Corollary 4.2 implies the following description of the interpolation spaces 1/p between Bp and V M O. Theorem 4.3. Let p0 > 0. Then 0 , V M O)θ,p = Bp1/p , (Bp1/p 0

where p = p0 /(1 − θ). Proof. Since the Riesz projection P+ is bounded both on V M O and 1/p Bp , 0 < p < ∞, it is easy to see that it is sufficient to prove that 0 Bp1/p , V M OA = Bp1/p + . (4.3) 0 + θ,p

Consider the map J : ϕ → Γϕ , ϕ ∈ V M OA. By Theorems 1.1, 2.1, and 1/p 3.1, J is an isomorphism of Bp + onto ΓS p , 0 < p < ∞, and by Hartman’s theorem J is an isomorphism of V M OA onto ΓS ∞ . Consequently, 1/p0 isomorphically onto (ΓS p0 , ΓS ∞ )θ,p . It J maps Bp0 + , V M OA θ,p

now follows from Corollary 4.2 that 0 J Bp1/p , V M OA 0 +

θ,p

= ΓS p ,

p=

p0 , 1−θ

which implies (4.3). Clearly, Corollary 4.2 implies that for 0 < p < ∞, 0 < q ≤ ∞ 0 , V M O θ,q , Γϕ ∈ S p,q , ⇔ ϕ ∈ Bp1/p 0

(4.4)

where p0 is an arbitrary positive number less than p and ϑ = (p − p0 )/p. To 1/p describe the space Bp0 0 , V M O θ,q we make use of the reiteration theorem (see Bergh and Löfstr¨ om [1], Ch. §3.5):

256


Let (X0 , X1 ) be a quasi-Banach pair, 0 < θ0 < θ1 < 1, 0 < q0 , q1 < ∞. Put Y0 = (X0 , X1 )θ0 ,q0 , Y1 = (X0 , X1 )θ1 ,q1 . Then (Y0 , Y1 )θ,q = (X0 , X1 )(1−θ)θ0 +θθ1 ,q . It follows from the reiteration theorem and from (4.4) that 0 1 , Bp1/p , Γϕ ∈ S p,q , ⇐⇒ ϕ ∈ Bp1/p 0 1 θ,q where p0 < p < p1 and 1/p = (1 − θ)/p0 + θ/p1 . 1/p 1/p To describe the space Bp0 0 , Bp1 1 θ,q , we have to introduce the Lorentz spaces. Let (X , µ) be a measure space. The Lorentz space Lp,q (X , µ), 0 0

where f ∗ is the nonincreasing rearrangement of f , i.e., f ∗ is the measurable function on (0, ∞) defined by f ∗ (t) = inf{σ : µ{x ∈ X : |f (x)| > σ} ≥ t}, def

t > 0.

∞

It is well known that for f ∈ L (X , µ) + L (X , µ) p0

1/p0 t ∗ p0 p0 ∞ c1 (f (s)) ds ≤ K(t, f, L , L ) ≤ c2 p0

0

tp0

1/p0 ∗

p0

(f (s)) ds

0

which by the reiteration theorem implies that (Lp0 (X , µ), Lp1 (X , µ))θ,q = Lp,q (X , µ), and Löfstr¨ om [1], §5.2). 1/p = (1 − θ)/p0 + θ/p1 (see Bergh 1/p 1/p We describe the spaces Bp0 0 + , Bp1 1 + . It will be clear that the θ,q 1/p0 1/p1 spaces Bp0 , Bp1 θ,q admit a similar description. Consider the measure space ∞ ⊕(T, 2n m), (X , µ) = n=0

i.e., X is an infinite union of disjoint copies of T and the restriction of µ 1/p to the nth copy is 2n m. Consider the space Bp,q , 0 < p < ∞, 0 < q ≤ ∞, which consists of functions ϕ, which are analytic in D and such that ∞

⊕ ϕ ∗ Vn ∈ Lp,q (X , µ).

n=0

Theorem 4.4. Let ϕ be a function analytic in D and let 0 < p < ∞, 1/p 0 < q ≤ ∞. Then Γϕ ∈ S p,q if and only if ϕ ∈ Bp,q .

,

5. Projecting onto the Hankel Matrices

Proof. Clearly, to prove the theorem, it suffices to show that 1/p1 0 Bp1/p , Bp1 + = B1/p p,q , 0 +

257

(4.5)

θ,q

where p0 < p < p1 and 1/p = (1 − θ)/p0 + θ/p1 . To prove (4.5), we use a “retract argument”. Consider the linear operator J defined on the space of functions analytic in D by ∞ def Jf = ⊕ ϕ ∗ Vn . n=0

1/p Clearly, J is a bounded operator from Bp0 0 + to Lp0 (X , µ) and a bounded 1/p operator from Bp1 1 + to Lp1 (X , µ). Hence, J is a bounded operator from 1/p 1/p Bp0 0 + , Bp1 1 + to (Lp0 (X , µ), Lp1 (X , µ))θ,q = Lp,q (X , µ), and so θ,q 1/p 1/p 1/p Bp0 0 + , Bp1 1 + ⊂ Bp,q . θ,q 1/p 1/p 1/p . Define the polynoLet us show that Bp,q ⊂ Bp0 0 + , Bp1 1 + θ,q

mials V˘n , n ≥ 0, by def V˘n =

Vn−1 + Vn + Vn+1 , n > 0, n = 0. V0 + V 1 ,

Consider the linear operator ρ on Lp0 (X , µ) + Lp1 (X , µ) defined by ρ ⊕ ϕn = ϕn ∗ V˘n . n≥0

n≥0

1/p Clearly, ρ is a continuous linear operator from Lp0 (X , µ) to Bp0 0 + and 1/p from Lp1 (X , µ) to Bp1 1 + . Hence, ρ is a continuous linear operator from 1/p 1/p (Lp0 (X , µ), Lp1 (X , µ))θ,q = Lp,q (X , µ) to Bp0 0 + , Bp1 1 + . θ,q

It is easy to see that ρJϕ = ϕ for any analytic function ϕ. Suppose that 1/p ϕ ∈ Bp,q , which, by definition, means that Jf ∈ Lp,q (X , µ). As we have 1/p 1/p already observed, it follows that ϕ = ρJϕ ∈ Bp0 0 + , Bp1 1 + . θ,q

Remark. It is easy to see that in a similar way one can describe the inter 1/p 1/p polation spaces Bp0 0 , Bp1 1 θ,q . The only difference is that we have to consider the measure space (Y, ν) = ⊕(T, 2|n| m). Then n∈Z 1/p 1/p ⊕ ϕ ∗ Vn ∈ Lp,q (Y, ν). ϕ ∈ (Bp0 0 , Bp1 1 )θ,q if and only if n∈Z

5. Projecting onto the Hankel Matrices In this section we study continuity properties of the averaging projection P onto the Hankel matrices. We show that it is bounded on S p for

258


1 < p < ∞ and unbounded on S 1 , S ∞ = C, and B(2 ). Then we study the behavior of the averaging projection on S 1 and we show that it maps S 1 to S 1,2 . This allows us to find the best possible condition on the singular values of a linear operator T under which the operator PT is bounded (or compact). However, it turns out that there are projections onto the set of Hankel matrices which are bounded on S 1 while there are no bounded projections on B(2 ) onto the set of Hankel matrices. As before we identify operators on 2 with their matrices in the standard orthonormal basis {ej }j≥0 . Let us define the averaging projection P onto the space of Hankel matrices. Let A = {ajk }j,k≥0 be an arbitrary matrix. The Hankel matrix PA is given by 1 def ajk . PA = {γj+k }j,k≥0 , γm = m+1 j+k=m

It is easy to see that if A belongs to the Hilbert–Schmidt class S 2 , then PA is the orthogonal projection of A onto the space of Hankel matrices in the Hilbert–Schmidt norm. Theorem 5.1. P is a bounded operator on S p for 1 < p < ∞. Proof. In §2 we have defined the operator W on the space of matrices to the space of functions analytic in D by def t m z m , tm = tjk , (WT )(z) = m≥0

j+k=m

where T = {tjk }j,k≥0 . We have shown in Lemma 2.4 that W is a bounded −1/p operator from S p to Bp , 1 < p < ∞. Therefore for T ∈ S p

|(WT )(ζ)|p (1 − |ζ|)p−2 dm2 (ζ) < ∞. (5.1) D

We have PT = Γϕ ,

ϕ=

n≥0

1 ) WT (n)z n . n+1

It now follows from (5.1) that

|ϕ (ζ)|p (1 − |ζ|)p−2 dm2 (ζ) < ∞, D

1/p

i.e., ϕ ∈ Bp . By Theorem 3.1, PT = Γϕ ∈ S p . Theorem 5.2. The operator P is unbounded on S 1 , C, and B(2 ). Proof. Let us first prove that P is unbounded on S 1 . It has been shown in the proof of Lemma 2.4 that WS 1 = H 1 . It follows that PS 1 = {ϕ : ϕ ∈ H 1 }.

(5.2)


259

Let us show that {ϕ : ϕ ∈ H 1 } ⊂ B11 , which would imply that PS 1 ⊂ S 1 . Let {cn }n≥0 ∈ 2 and let n 2−n cn z 2 . ϕ(z) = n≥0

Clearly, ϕ ∈ H 2 ⊂ H 1 . On the other hand, it follows immediately from the description of B11 in terms of convolutions with the polynomials Wn (see Appendix 2.6) that ϕ ∈ B11 if and only if {cn }n≥0 ∈ 1 . Since (C)∗ = S 1 and (S 1 )∗ = B(2 ) (see Appendix 1.1), it now follows by duality that P is unbounded on C and B(2 ). Theorem 5.2 asserts that for an operator T ∈ S 1 the operator PT need not be in S 1 . We are now going to obtain a sharp estimate of the singular values of PT for an arbitrary operator T ∈ S 1 . Theorem 5.3. Let T ∈ S 1 . Then PT ∈ S 1,2 . Recall (see §4) that the ideal S 1,2 consists of the operators A, for which (sj (A))2 (1 + j) < ∞. j≥0

To prove Theorem 5.3 we need an inequality for functions in the Lorentz class L1,2 . Lemma 5.4. Let 1 ≤ q ≤ ∞ and let f1 and f2 be functions in L1,q (X , µ) such that f1 (x)f2 (x) = 0, µ-a.e. Then f1 + f2 1,q ≤ f1 1,q + f2 1,q .

(5.3)

Note that for 1 < q ≤ ∞ the quasinorm · 1,q does not satisfy the triangle inequality. However, the lemma asserts that the triangle inequality holds for disjoint functions. Proof of Lemma 5.4. Let us prove (5.3) for 1 < q < ∞ (for q = ∞ the inequality is obvious). Put def

λf (t) = µ{x : |f (x)| > t}. It is not hard to see that

f 1,q =

∞

1/q (λf (t))q tq−1 dt

,

f ∈ L1,q ,

0

(it is sufficient to verify this equality for step functions f ). Since the functions f1 and f2 are disjoint, it follows that λf1 +f2 = λf1 + λf2 . Consequently,

f1 + f2 1,q ≤ 0

∞

1/q (λf1 (t))q tq−1 dt

+ 0

∞

1/q (λf2 (t))q tq−1 dt

.

260


Proof of Theorem 5.3. Let (X , µ) =

∞

⊕(T, 2n m)

n=0

be the measure space, for which X is the infinite union of disjoint copies of T, the nth copy equipped with the measure 2n m. Theorem 4.4 asserts that Γϕ ∈ S 1,2 if and only if ϕ ∈ B11,2 . Therefore in view of (5.2) it remains to ∞ show that ⊕ ϕ ∗ Vn ∈ L1,2 (X , µ) for any function ϕ such that ϕ ∈ H 1 . n=0

Note that ϕ ∈ B11,2 if and only if ∞

2−n (ϕ ∗ Vn ) ∈ B11,2 .

n=0

Indeed, this follows from the fact that a function ϕ analytic in D belongs 1/p to Bp , 0 < p < ∞, if and only if ∞

2−n (ϕ ∗ Vn ) ∈ Bp1/p

n=0

(see Appendix 2.6) and the fact that B11,2 =

2 B1/2

+

1/2 , B2 + 1

2 ,2

(see

Theorem 4.4). Therefore to prove the theorem it is sufficient to show that for every function ψ in H 1 ∞

⊕ 2−n (ψ ∗ Vn ) ∈ L1,2 (X , µ).

n=0

We are going to use the following characterization of the space H 1 : 1/2

∞ 1 2 ψ∈H ⇐⇒ |(ψ ∗ Vn )(ζ)| dm(ζ) < ∞ T

n=0

(see Stein [1]). In fact, we need only the implication ⇒. Therefore we can reduce Theorem 5.3 to the following assertion: Let {ψn }n≥0 be a sequence of polynomials on T such that

∞ T

1/2 |ψn (ζ)|

2

dm(ζ) < ∞.

n=0

Then ∞ n=0

⊕ 2−n ψn ∈ L1,2 (X , µ).


261

Clearly, it is sufficient to assume that among the functions ψn there are only finitely many nonzero functions and to prove the inequality ∞ −n ⊕ 2 ψn n=0

≤ const

∞

L1,2 (X ,µ)

T

1/2 |ψn (ζ)|

2

dm(ζ). (5.4)

n=0

Let N be a positive integer. It is easy to see that it is sufficient to establish inequality (5.4) for functions ψn that are constant on each of the arcs

2πj 2π(j + 1) Ij = eiϑ : ≤ϑ≤ , 0 ≤ j ≤ N − 1, N N since we can always approximate polynomials by such step function. N −1 ψn,j , where ψn,j is constant on Ij Let 0 ≤ j ≤ N − 1 and let ψn = j=0

and ψn,j is identically equal to 0 on T \ Ij . By Lemma 5.4, ∞ ∞ N −1 ⊕ 2−n ψn ≤ ⊕ 2−n ψn,j . 1,2 1,2 n=0

j=0

L

n=0

L

Therefore to prove inequality (5.4) it is sufficient to assume that for some j all functions ψn are constant on Ij and vanish on T \ Ij . Let ψn (ζ) = cn for ζ ∈ Ij . Without loss of generality we may assume that cn ≥ 0, n ≥ 0, def

and the sequence {2−n cn }n≥0 is nonincreasing. Put δ = mIj . We have ∞ −n ⊕ 2 ψn n=0

= L1,2 (X ,µ)

∞

2−2n c2n δ 2

n=0

≤ const ·δ

∞

t dt (2n −1)δ

1/2 c2n

n=0

=

const

∞ T

1/2

(2n+1 −1)δ

1/2 |ψn (ζ)|

2

dm(ζ),

n=0

which completes the proof. To obtain a sharp sufficient condition for the boundedness of PT we introduce the sequence spaces d(q, ω) and the generalized Matsaev ideals S q,ω . Let 1 ≤ q < ∞. We define the space d(q, ω) of sequences {an }n≥0 such that 1/q  (a∗ )q def n  {an }n≥0 d(q,ω) =  < ∞, 1+n n≥0

262


where {a∗n }n≥0 is the nonincreasing rearrangement of {|an |}n≥0 . Consider now the ideal S q,ω of operators T on Hilbert space such that 1/q  (sn (T ))q def  T S q,ω =  < ∞. 1+n n≥0

Then d(q, ω) and S q,ω are Banach spaces (see Appendix 1.1). It is easy to see that S p ⊂ S q1 ,ω ⊂ S q2 ,ω , 0 < p < ∞, 1 ≤ q1 ≤ q2 . Recall that the dual space (S q,ω )∗ can be identified with respect to the Hermitian pairing T, R = trace T R∗ ,

T ∈ S q,ω ,

R ∈ (S q,ω )∗

with the ideal of operators whose singular values belong to (d(q, ω))∗ (see Appendix 1.1). Here we identify (d(q, ω))∗ with a sequence space with respect to the pairing an¯bn , a = {an }n≥0 ∈ d(q, ω), b = {bn }n≥0 ∈ (d(q, ω))∗ . (a, b) = n≥0

Theorem 5.5. If T ∈ S 2,ω , then PT ∈ S ∞ . We need the following fact. Lemma 5.6. S 1,2 ⊂ (S 2,ω )∗ . Proof of Lemma 5.6. Since T ∈ (S 2,ω )∗

if and only if {sn (T )}n≥0 ∈ (d(2, ω))∗ ,

it suffices to show that (b∗n )2 (1 + n) < ∞ ⇒ {bn }n≥0 ∈ (d(2, ω))∗ . n≥0

Let a = {an }n≥0 be a finitely supported sequence in d(2, ω). We have |an¯bn | = |ank |b∗k |(a, b)| ≤ n≥0

k≥0

for some permutation {nk }k≥0 of the set Z+ . It follows that 1/2  1/2  |an |2 k  |(a, b)| ≤ |ak |b∗k ≤  (b∗k )2 (1 + k)  1+k k≥0



≤ 

k≥0

1/2 

(b∗k )2 (1 + k)

k≥0



k≥0

1/2

|a∗ |2 k  1+k

.

k≥0

Hence, b determines a continuous linear functional on the space d(2, ω), and so b ∈ (d(2, ω))∗ . Remark. It can be shown in the same way that S 1,q ⊂ (S q ,ω )∗ , def

1 < q < ∞, where q = q/(q − 1).


263

Proof of Theorem 5.5. It follows from Theorem 5.3 and Lemma 5.6 that PS 1 ⊂ (S 2,ω )∗ . By duality, PS 2,ω ⊂ B(2 ). Since the set of operators with all but finitely many matrix entries are zero is dense in S 2,ω , it follows that PS 2,ω ⊂ C. Let us now show that Theorem 5.5 is the best possible result. Theorem 5.7. Let {sn }n≥0 be a nonincreasing sequence of positive numbers such that s2n (1 + n)−1 = ∞. n≥0

Then there exists an operator T on 2 such that sn (T ) = sn and PT is unbounded. Proof. We define the matrix {tjk }j,k≥0 of T as follows:   s0 , j = 0, k = 0, sj , 2n ≤ j ≤ 2n+1 − 1, k = 2n + 2n+1 − j, n ≥ 0, tjk =  0, otherwise, (see Fig. 3). Clearly, sj (T ) = sj . Assume that PT is bounded and let PT = Γϕ . Then 2n+1 −1

ϕ(0) ˆ = s0 ,

n

ϕ(2 ˆ +2

n+1

− 1) =

sj j=2n , 2n + 2n+1

n ≥ 0,

and ϕ(m) ˆ = 0 if m > 0 and m = 2n + 2n+1 − 1, n ∈ Z+ . If Γf is bounded, 2 |ϕ(m)| ˆ < ∞. We then by the Nehari theorem, ϕ ∈ BM OA, and so m≥0

have

2 |ϕ(m)| ˆ

= s20 +

m≥0

|ϕ(2 ˆ n + 2n+1 − 1)|2 ≥

n≥0

≥

1 1 9 2n+1 n≥0

1 2 s2n+1 9 n≥0

2n+1 −1

s22n+1 +k ≥

k=0

1 s2k = ∞. 9 k k≥2

Hence, Γϕ is unbounded. Now we are in a position to show that in a sense Theorem 5.3 cannot be improved. Corollary 5.8. Let q < 2. Then PS 1 ⊂ S 1,q . Proof. Assume that PS 1 ⊂ S 1,q . Clearly, it is sufficient to consider the case 1 < q < 2. Then as in Theorem 5.5 it would follow from the Remark after Lemma 5.6 that PT is bounded whenever T ∈ S q ,ω . However, this would contradict Theorem 5.7. Although the natural averaging projection P is unbounded on S 1 , it turns out that there are bounded projections from S 1 onto the subspace of

264

Chapter 6. Hankel Operators and Schatten–von Neumann Classes 2n +1

s0

n

2n+1 + 1

.

·

·

.

2n+2 + 1

-

·

2 .

· 2n+1 .

.

·

·

s2n

s2n+1 −1 s2n+1 · · ·

2n+2 .

s2n+2 −1 ·

·

·

? FIGURE 3.

nuclear Hankel matrices. Recall that nα (α + 1)(α + 2) · · · (α + n) n+α ∼ = Dn(α) = n n! Γ(α + 1) (see Appendix 2.6; here Γ is the gamma function, not a Hankel operator!). 3αβ onto the set of Hankel matrices For α, β > 0 we define the projection P by 3αβ A def P = {γj+k }j,k≥0 ,

γm =

1

α+β+1 Dm j+k=m

(α)

(β)

Dj Dk ajk ,

where A = {ajk }j,k≥0 . 3αβ is a bounded projection on Theorem 5.9. Let α, β > 0. Then P S 1 onto the set of Hankel matrices in S 1 .

.


Proof. It is easy to see that

m

(α)

(β)

(α+β+1)

Dj Dm−j = Dm

j=0

265

. It follows that

3αβ is a projection onto the Hankel matrices. Let us show that P 3α,β is P bounded on S 1 . 3αβ RS ≤ const f 2 g2 Clearly, it suffices to prove the inequality P 1 for an arbitrary rank one operator R = (·, f )g, where f = {fn }n≥0 ∈ 2 , def (α) Dn f¯n z n and g = {gn }n≥0 ∈ 2 . Consider the analytic functions F = def

G = so

n≥0 (β) Dn gn z n .

n≥0

Then F ∈

B2−α

and G ∈

B2−β

(see Appendix 2.6), and

D

|F (ζ)|2 (1 − |ζ|)2α−1 dm2 (ζ) ≤ const f 2

and

D

|G(ζ)|2 (1 − |ζ|)2β−1 dm2 (ζ) ≤ const g2 .

By Hölder’s inequality this implies that

|(F G)(ζ)|(1 − |ζ|)α+β−1 dm2 (ζ) ≤ const f 2 g2 , D

−(α+β)

and so F G ∈ B1 (see Appendix 2.6). It is easy to see that ) 3αβ R = Γϕ , ϕ(z) = G(n)z n = I˜−(α+β+1) (F G) Dn(α+β+1) F P n≥0

(see Appendix 2.6). Therefore ϕ ∈ B11 (see Appendix 2.6) and by Theorem 3αβ R = Γϕ ∈ S 1 . 1.1, P Theorem 5.9 implies the following result. Corollary 5.10. Let α, β > 0 and let F be an analytic in D function −(α+β) . Then there exist sequences of analytic in D functions of class B1 {ϕk }k≥0 and {ψk }k≥0 such that ϕk ∈ B2−α , ψk ∈ B2−β , and F = ϕk ψk , ϕk B2α ψB −β ≤ const F B −(α+β) . k≥0

k≥0

2

1

Proof. Let G = I˜α+β+1 F . Then G ∈ B11 (see Appendix 2.6). By Theorem 1.1, ΓG = (·, f (k) )g (k) , f (k) = {fn(k) }n≥0 ∈ 2 , g (k) = {gn(k) }n≥0 ∈ 2 , k≥0

and

f (k) 2 g (k) 2 = ΓG S 1 ≤ const GB11 ≤ const F B −(α+β) . 1

k≥0

Put ϕk =

n≥0

Dn(α) f¯n(k)

and ψk =

n≥0

Dn(α) gn(k) .

266


Clearly,

ϕk B2α ψB −β ≤ const F B −(α+β) . 2

k≥0

We have 3αβ ΓG = ΓG = P

3αβ (·, f (k) g (k) ) = P

k≥0

which implies that F =

1

ΓI˜α+β+1 (ϕk ψk ) ,

k≥0

ϕk ψk .

k≥0

3αβ . For α, β > 0 Remark. We can slightly modify the definition of P consider the projection Pαβ : (1 + j)α (1 + k)β ajk 0≤j+k≤m def Pαβ A = {γj+k }j,k≥0 , γm = , (1 + j)α (1 + k)β 0≤j+k≤m

where A = {ajk }j,k≥0 . If we replace in the proof of Theorem 5.9 the fractional derivatives I˜s by the fractional derivatives Is (see Appendix 2.6), we can prove that Pαβ is a bounded projection on S 1 onto the nuclear Hankel matrices. However, it turns out that in the case of the space of compact operators or bounded operators the situation is quite different: there are no bounded projections onto the set of Hankel matrices. To prove this we use some results from Banach space geometry. Theorem 5.11. There are no bounded projections from C onto the compact Hankel matrices. Proof. Assume that P is a bounded projection from C onto the compact Hankel matrices. Since the space of compact Hankel matrices is isomorphic to V M OA, it follows that there exists a continuous linear operator A from C onto V M OA. Hence, the operator T ∗ is an isomorphic imbedding of H 1 into S 1 . But this is impossible since every subspace of S 1 either is isomorphic to a Hilbert space or contains a subspace isomorphic to 1 (see Arazy and Lindenstrauss [1]), but H 1 contains subspaces isomorphic to p , 1 < p < 2, (see Kwapie´ n and Pe lczy´ nski [1]). Clearly, p for 1 < p < 2 does not have a subspace isomorphic to 1 since p is reflexive, while 1 is not. It remains to show that p is not isomorphic to a Hilbert space for 1 < p < 2. To prove this we observe the following. Let x1 , x2 , · · · , xn be vectors in a Hilbert space. Then it follows by induction from the parallelogram identity that ± x1 ± x2 ± · · · ± xn 2 = x1 2 + x2 2 + · · · + xn 2 , 2−n where the sum is taken over all combinations of signs. On the other hand, if e1 , e2 , · · · , en are standard basis vectors of p , then ±e1 ±e2 ±· · ·±en 2p = n2/p = n2/p−1 (e1 2p +e2 2p +· · ·+en 2p ), 2−n

6. Rational Approximation

267

which implies that there is no isomorphism from p onto a Hilbert space. Theorem 5.12. There are no bounded projections from B(2 ) onto the bounded Hankel matrices. Proof. Assume that P is a bounded projection from B(2 ) onto the space of bounded Hankel matrices B(Γ). Let M be the symbol operator from def

def

B(Γ) onto BM OA: M Γϕ = ϕ. Put R = M the operator P |C. Consider ∗ 1 1 R∗ : (BM OA)∗ → H S and its restriction R to H ⊂ (BM OA)∗ . Let 1 ∗ 1 1 us show that R H is an isomorphic imbedding of H into S 1 . Suppose that f ∈ H 1 . Let ϕ ∈ V M OA, ϕV M OA = 1, and |(f, ϕ)| ≥ const f H 1 . Since RΓϕ = ϕ, we have |R∗ f, Γϕ | = |(f, ϕ)| ≥ const f H 1 , and since Γϕ ≤ const, it follows that R∗ f ≥ const f H 1 . The rest of the proof repeats the proof of Theorem 5.11.

6. Rational Approximation Classical theorems on polynomial approximation describe classes of smooth functions in terms of the rate of polynomial approximation (in one or another norm). The smoother the function is, the more rapidly its deviations to the set of polynomials of degree n decay. For example, an L∞ function ϕ belongs to Λα , α > 0, if and only if distL∞ {ϕ, P n } ≤ const ·(1 + n)−α , where P n is the set of trigonometric polynomials of degree at most n. This is a classical result due to Bernstein, Jackson, and de la Vallée Poussin; see Akhiezer [2]. However, it turned out that in the case of rational approximation the corresponding problems are considerably more complicated. For a long time there were no sharp results describing classes of functions in terms of the rate of rational approximation. The first sharp result was deduced from the S p criterion for Hankel operators (Theorems 1.1, 2.1). In earlier results there were gaps between “direct” and “inverse” theorems (see the Concluding Remarks to this chapter). 1/p In this section we describe the Besov spaces Bp in terms of the rate of rational approximation in the norm of BM O. Then we obtain an improvement of Grigoryan’s theorem, which estimates the L∞ norm of P− f in terms of f L∞ for functions f such that P− f is a rational function of degree n. As a consequence we obtain a sharp result about rational approximation in the L∞ norm. Using the S p criterion obtained in §§1–3 we obtain sharp Bernstein–S.M. Nikol’skii type inequalities for rational functions of degree n. We also obtain

268

Chapter 6. Hankel Operators and Schatten–von Neumann Classes 1/p

estimates of the Bp norms of rational functions in terms of their norms 0 . This will allow us to conclude this section with a sharp on in B∞ 0 result rational approximation in the norm of the Bloch space B = B∞ . + Denote by Rn , n ≥ 0, the set of rational functions of degree at most n with poles outside T. For f ∈ BM O put def

ρn (f ) = distBM O {f, Rn }. Theorem 6.1. Let ϕ ∈ BM O and 0 < p < ∞. Then {ρn (ϕ)}n≥0 ∈ p 1/p if and only if ϕ ∈ Bp . 1/p

1/p

Proof. We have P+ BM O ⊂ BM O and P+ Bp ⊂ Bp (see Appendix 2.6). Clearly, P+ Rn ⊂ Rn . Therefore it is sufficient to prove the theorem for functions ϕ in BM OA and for functions ϕ ∈ P− BM O. Let us assume that ϕ = P− ϕ, the corresponding result in the case ϕ = P+ ϕ follows by passing to the complex conjugate. It follows from Theorem 4.1.1 that sn (Hϕ ) = inf{Hϕ − Hr : rank Hr ≤ n}. Without loss of generality we may assume that r = P− r. By Corollary 1.3.2, rank Hr ≤ n if and only if r ∈ Rn . Together with Theorem 1.1.3 this yields c1 sn (Hϕ ) ≤ inf{ϕ − rBM O : r ∈ Rn } ≤ c2 sn (Hϕ ) for some positive constants c1 and c2 . The result now follows from Theorems 1.1, 2.1, and 3.1. Denote now by R+ n the set of rational functions of degree at most n with poles outside the closed unit disk and put def

+ ρ+ n (ϕ) = distBM OA {ϕ, Rn }.

Theorem 6.2. Suppose that ϕ ∈ BM OA and 0 < p < ∞. Then 1/p p . {ρ+ n (ϕ)}n≥0 ∈ if and only if ϕ ∈ Bp + Proof. This is an immediate consequence of Theorem 6.1 . We proceed now to an improvement of a theorem of Grigoryan that estimates the P− ϕL∞ in terms of ϕL∞ in the case P− ϕ ∈ Rn . Clearly, the last condition is equivalent to the fact that ϕ is a boundary-value function of a meromorphic function in D which has at most n poles (counted with multiplicities) and is bounded near T. It is not quite obvious that such an estimate exists. If we consider the same question in the case P− ϕ is a polynomial of degree n, it is well known that P− ϕL∞ ≤ const log(1 + n); this follows immediately from the fact that n j z ≤ const log(1 + n) j=0 1 L


269

(see Zygmund [1], Ch. II, §12). Grigoryan’s theorem says that if P− ϕ ∈ Rn , then P− ϕL∞ ≤ const ·n ϕL∞ .

(6.1)

The following result improves the above estimate. The proof is based on the S 1 criterion for Hankel operators (Theorem 1.1). Theorem 6.3. Let n be a positive integer and let ϕ be a function in L∞ such that P− ϕ ∈ Rn . Then P− ϕB11 ≤ const ·n ϕL∞ .

(6.2)

Let us first observe that inequality (6.2) implies (6.1). Indeed, if f ∈ B11 , then 2n f ∗Wn L1 ≤ const f B11 (see Appendix 2.6). It is easy to show n≥0

that ϕL∞ ≤

|fˆ(j)| ≤ const

j≥0

2n f ∗ Wn L1 ,

n≥0

and so (6.2) implies (6.1). Proof of Theorem 6.3. Consider the Hankel operator Hϕ . By the Nehari theorem, Hϕ ≤ ϕL∞ . By Kronecker’s theorem, rank Hϕ ≤ n. Therefore Hϕ S 1 ≤ nHϕ . The result now follows from Theorem 1.1 which says that P− ϕB11 ≤ const Hϕ S 1 . We can now obtain a result on rational approximation in the L∞ norm. For ϕ ∈ L∞ we put d∞ n (ϕ) = distL∞ {ϕ, Rn }, def

n ∈ Z+ .

Theorem 6.4. Let ϕ ∈ L∞ . Then the d∞ (ϕ) decay more rapidly than 1/p n Bp . any power of n if and only if ϕ ∈ p>0

Lemma 6.5. Let r ∈ Rn . Then rL∞ ≤ const ·n rBM O . Proof. Clearly, it is sufficient to prove the inequality in the case r = P− r and in the case r = P+ r. Assume that r = P− r. Let f be the symbol of Hr of minimal norm, i.e., P− r = P− f and f L∞ = Hr (see Corollary 1.1.6). We have P− rL∞

= P− f L∞ ≤ const ·nf L∞ = const ·nHr ≤ const ·nP− rBM O

by (6.1) and Theorem 1.1.3. It is easy to see that Theorem 6.4 is a consequence of the following lemma. Lemma 6.6. Let λ > 1 and let ϕ be a function in L∞ such that ρn (ϕ) ≤ const ·n−λ , n ≥ 0. Then −λ+1 , d∞ n (ϕ) ≤ const ·n

n ≥ 0.

270


Proof. Suppose that rn ∈ R2n and ϕ−rn BM O ≤ const 2−nλ . We have ϕ − rn = (ϕ − rn+j ) − (ϕ − rn+j+1 ) = (rn+j+1 − rn+j ). j≥0

j≥0

Under the hypotheses of the lemma rn+j+1 − rn+j BM O ≤ const 2−(n+j)λ , and since rn+j+1 − rn+j ∈ R2n+j+2 , it follows from Lemma 6.5 that rn+j+1 − rn+j L∞ ≤ const 2−(n+j)(λ−1) . Therefore −n(λ−1) d∞ , 2n (ϕ) ≤ ϕ − rn L∞ const 2

and since the sequence ρ∞ n (ϕ) is nonincreasing, this implies the conclusion of the lemma. Let us proceed now to Bernstein–Nikol’skii type inequalities for rational functions. 0 consists of distributions (see Appendix 2.6). Recall that thespace B∞ 0 Its subspace B∞ + can be identified with the Bloch space B of functions f analytic in D and satisfying sup |f (ζ)|(1 − |ζ|) < ∞. ζ∈D

Theorem 6.7. Let r be a rational function of degree n ≥ 1 with poles outside T and let 0 < p ≤ q ≤ ∞. Then rB 1/p ≤ const ·n1/p rBM O

(6.3)

rB 1/p ≤ const ·n1/p−1/q rB 1/q .

(6.4)

p

and p

q

Proof. Let us first prove (6.3). Clearly, it is sufficient to prove it for 2 of degree n. Consider the Hankel operator Hr . rational functions r in H− By Corollary 1.3.2, rank Hr = n. Hence,  1/p n−1 sj (Hr )p  ≤ n1/p s0 (Hr ) = n1/p Hr . Hr S p =  j=0 (6.5) To prove (6.3), it is sufficient to observe that by Theorems 1.1, 2.1, and 3.1, the left-hand side of (6.5) is equivalent to rB 1/p and by Theorem 1.1.2, p the right-hand side of (6.5) is equivalent to rBM O . 2 , (6.4) follows from Theorems 1.1, 2.1, Similarly, for q < ∞ and r ∈ H− 3.1, and H¨ older’s inequality: 1/p  1/q  n−1 n−1  sj (Hr )p  ≤ n1/p−1/q  sj (Hr )q  . j=0

j=0


271

It remains to prove (6.4) for q = ∞. This time it is more convenient to assume that r ∈ H 2 . We may also assume that r(0) = 0. Consider first the 1/2 case p = 2. For f, g ∈ B2 we put

1/2 (f, g)∗ = f (ζ)g (ζ)dm2 (ζ) and f ∗ = (f, f )∗ . D

1/2

Clearly, (·, ·)∗ is a semi-inner product on B2 and f ∗ = 0 if and only if f is a constant function. It is easy to see that rB 1/2 ≤ const · sup{|(r, ρ)∗ | : ρ ∈ R+ n , ρ∗ = 1}. 2

(6.6)

It follows easily from the description of the dual space to B11 (see Appendix 2.6) that for a rational function ρ ∈ H 2 0 ρB 1 . |(r, ρ)∗ | ≤ const rB∞ 1

Using (6.4) with p = 1 and q = 2, we obtain 0 ρ 1/2 , |(r, ρ)∗ | ≤ const ·n1/2 rB∞ B 2

ρ ∈ R+ n,

and together with (6.6) this yields 0 . rB 1/2 ≤ const ·n1/2 rB∞ 2

It is easy to see now that inequality (6.4) for p < 2 and q = ∞ follows from inequality (6.4) for q = 2 and inequality (6.4) for p = 2 and q = ∞. It remains to prove inequality (6.4) for 2 < p < ∞ and q = ∞. We have

1/p p p−2 |r (ζ)| (1 − |ζ|) dm2 (ζ) rB 1/p ≤ const p

D

=

const

≤

D

1/p p−2 |r (ζ)|2 dm2 (ζ) (sup{|r (ζ)|(1 − |ζ|) : ζ ∈ D}) p

D 2/p

= r

1/p |r (ζ)|2 (|r (ζ)|(1 − |ζ|))p−2 dm2 (ζ)

1/2

B2

(p−2)/p

rB 0

∞

0 ≤ n1/p rB∞

by inequality (6.4) for p = 2 and q = ∞. Remark. Note that inequalities (6.3) and (6.4) are sharp. Indeed, we can consider the function z n ∈ Rn , n ≥ 0. Since L∞ ⊂ BM O ⊂ L2 , c1 ≤ z n BM O ≤ c2 for some constants c1 and c2 . On the other hand, it follows from the de1/p scription of the classes Bp in terms of kernels Vn (see Appendix 2.6) that for 0 < p ≤ ∞ C1 n1/p ≤ z n B 1/p ≤ C2 n1/p , p

where C1 and C2 do not depend on n.

272


We conclude this section with an analog of Theorem 6.1 for rational 0 approximation in the norm of B∞ . For a function ϕ ∈ B we put def

δn (ϕ) = distB{ϕ, R+ n },

n ≥ 0.

Theorem 6.8. Let ϕ ∈ B and let 0 < p < ∞. Then {δn (ϕ)}n≥0 ∈ p 1/p if and only if ϕ ∈ Bp . 1/p

Proof of Theorem 6.8. If ϕ ∈ Bp , it follows from Theorem 6.2 that p p {d+ n (ϕ)}n≥0 ∈ , and since BM OA ⊂ B, it follows that {δn (ϕ)}n≥0 ∈ . 1/p Suppose now that {δn (ϕ)}n≥0 ∈ p . Let us show that ϕ ∈ Bp . Consider first the case 0 < p ≤ 1. For j ∈ Z+ , consider a rational function rj in R+ 2j such that ϕ − rj B ≤ 2δ2j (ϕ). We have ϕ = r1 +

∞

(rj+1 − rj ),

j=1

and it follows from inequality (6.4) with q = ∞ that ϕp 1/p Bp

≤ ≤

r1 p 1/p + Bp

∞

rj+1 − rj p 1/p Bp

j=1

r1 p 1/p + const Bp

 ≤ const r1 pB +  ≤ const r1 pB + ≤ const

r1 pB +

∞

2j rj+1 − rj pB

j=1 ∞



2j (ϕ − rj pB + ϕ − rj+1 pB)

j=1 ∞

 p 2j (δ2j (ϕ)) 

j=1 ∞

p

(δm (ϕ))

.

m=1

The last inequality easily follows from the fact that the sequence {δn (ϕ)}n≥0 is nonincreasing. Suppose now that 1 ≤ p < ∞. We use the following seminorm on 1/p Bp + :

p def ϕ(p) = |ϕ (ζ)|(1 − |ζ|)2 (1 − |ζ|)−2 dm2 (ζ). D

1/p

A function ϕ analytic in D belongs to Bp , 1 ≤ p < ∞, if and only if ϕ(p) < ∞ (see Appendix 2.6).


273

Suppose that rj ∈ R2j and ϕ − rj B ≤ 2δ2j (ϕ). We have rm = r 0 +

m−1

(rj+1 − rj ).

j=0

By inequality (6.4) with q = ∞ we obtain rm (1)

≤

r0 (1) + const

m−1

2j rj − rj+1 B

j=0

≤ const

m−1

2j (δ2j (ϕ) + δ2j+1 (ϕ))

(6.7)

j=0 2 m

≤ const

δk (ϕ),

k=0

since the sequence {δj (ϕ)}j≥0 is nonincreasing. For a function f analytic in D and t > 0 we put N (f, t) = µ{ζ ∈ D : |f (ζ)|(1 − |ζ|)2 > t}, def

def

where the measure µ on D is defined by dµ(ζ) = (1 − |ζ|)−2 dm2 (ζ). Recall that for a function f ∈ B sup |f (ζ)|(1 − |ζ|)2 ≤ const f B, ζ∈D

and so there exists a positive constant M such that N (f, t) = 0 whenever t ≥ 12 M f B. To estimate ϕ(p) we use the following well-known formula (see, e.g., Stein [2], Ch. 1, 1.5):

∞ ϕ (ζ)(1 − |ζ|)2 p dµ(ζ) = p tp−1 N (ϕ, t)dt. (6.8) D

0

Put def

fm = ϕ − rm . We have by (6.8) ϕp(p)

∞

tp−1 N (ϕ, t)dt 0

∞ tp−1 N (ϕ, (1 + M )t)dt. ≤ const

= p

0

To simplify notation we put γm = δ2m (ϕ). If t ≥ γm+1 , then N (ϕ, (1 + M )t) ≤ N (rm , t) + N (fm , M t) ≤ N (rm , t) + N (fm , M γm+1 ) = N (rm , t), since γm+1 ≥ 12 fm B.

274


Hence, ϕp(p)

≤ const

∞

m=0

Using (6.7), we obtain N (rm , t) ≤

1 t

D

γm

tp−1 N (rm , t)dt.

γm+1

|rm (ζ)|(1 − |ζ|)2 dµ(ζ)

1 1 rm (1) ≤ const δj (ϕ). t t j=0 2m

≤ Therefore

γm

 t

p−1

N (rm , t)dt

≤ const 

γm+1

 ≤ const 

2



m

δj (ϕ)

j=0 2 m



γm

tp−2 dt

γm+1

p−1 p−1 . δj (ϕ) γm − γm+1

j=0

Finally, ∞ 2 m

ϕp(p)

≤ const =

const

m=0 j=0 ∞

∞

δj (ϕ)

j=0

≤ const

p−1 p−1 δj (ϕ) γm − γm+1 p−1 p−1 γm − γm+1

{m:2m ≥j} p−1

δj (ϕ) (δj (ϕ))

= const

j=0

∞

p

(δj (ϕ)) ,

j=0

which completes the proof. We conclude this section with the inverse problem for rational approximation in BM OA. By the classical theorem of Bernstein (see Bernstein [1]), for any nonincreasing sequence {cn }n≥0 tending to 0 there exists a continuous function f on [0, 1] such that the distance in C([0, 1]) from f to the set of polynomials of degree at most n is equal to cn . We can ask a similar question about rational approximation in BM OA: is it true that for any nonincreasing sequence {cn }n≥0 there exists ϕ ∈ BM OA such that ρ+ n (ϕ) = cn ,

n ∈ Z+ ?

By the Kronecker theorem and the Adamyan, Arov, and Krein theorem, this question is equivalent to the problem of whether for a given nonincreasing sequence {cn }n≥0 there exists a bounded Hankel operator Γ such that sn (Γ) = cn , n ∈ Z+ . This problem was posed in Khrushchëv and Peller [1]. It will be solved in §12.8 as a consequence of the solution of the problem to describe the

7. Other Applications of the S p Criterion

275

nonnegative operators on Hilbert space that are unitarily equivalent to the moduli of Hankel operators.

7. Other Applications of the S p Criterion In this section we consider other applications of the S p criterion obtained in Sections 1–3.

Functions of the Model Operator We consider functions ϕ(S[ϑ] ) of the model operator S[ϑ] on the space Kϑ = H 2 H 2 (see §1.2). The following theorem gives a necessary and sufficient condition for ϕ(S[ϑ] ) to be in S p . Theorem 7.1. Let 0 < p < ∞, ϕ ∈ H ∞ and let ϑ be an inner function. The following statements are equivalent: (i) ϕ(S[ϑ] ) ∈ S p ; ¯ ∈ Bp1/p . (ii) P− ϑϕ Proof. By formula (2.9) of Chapter 1, the operator ϕ(S[ϑ] ) belongs to S p if and only if Hϑϕ ¯ ∈ S p . Thus the result follows from Theorems 1.1, 2.1, and 3.1. As in §1.5 we consider the special case when the inner function is an interpolating Blaschke product, which we denote by B. As we have mentioned in the discussion before Theorem 1.5.11, the functions fj defined by formula (5.5) of Chapter 1 form an unconditional basis in KB and ϕ(S[B] )fj = ϕ(ζj )fj , ϕ ∈ H ∞ . Again, as we have mentioned before Theorem 1.5.11 the basis {fj }j≥1 is equivalent to an orthogonal basis. Together with Theorem 7.1, this implies the following result. Theorem 7.2. Let 0 < p < ∞, ϕ ∈ H p , and suppose that B is an interpolating Blaschke product with zeros {ζj }j≥1 . The following statements are equivalent: (i) ϕ(S[B] ) ∈ S p ; ¯ ∈ Bp1/p ; (ii) P− Bϕ (iii) {ϕ(ζj )}j≥0 ∈ p .

Commutators We consider here commutators Cϕ on L2 (T) defined for ϕ ∈ BM O by

ϕ(ζ) − ϕ(τ ) Cϕ f = p.v. f (τ )dm(τ ), f ∈ L2 , 1 − τ¯ζ T (see §1.1). By Theorem 1.1.10, Cϕ is bounded on L2 if and only if ϕ ∈ BM O. The following result describes the commutators Cϕ of class S p , 0 < p < ∞.

276


Theorem 7.3. Let ϕ ∈ BM O and let 0 < p < ∞. Then Cϕ ∈ S p if 1/p and only if ϕ ∈ Bp . Proof. It has been shown in the proof of Theorem 1.1.10 that Cϕ = Hϕ (P+ f ) + Hϕ∗¯ (P− f ),

f ∈ L2 .

Thus Cϕ ∈ S p if and only if Hϕ ∈ S p and Hϕ¯ ∈ S p . The result follows now from Theorems 1.1, 2.1, and 3.1.

Integral Operators on L2 (R+ ) We consider here integral operators Γ k on L2 (R+ ) defined by

∞ (Γ k f )(t) = k(s + t)f (s)ds. 0

Here k is a distribution on (0, ∞) such that the operator Γ k is bounded on L2 (R+ ) (see §1.8). In §1.8 we have characterized the bounded, compact, and finite rank operators Γ k . Together with operators Γ k we have considered in §1.8 integral operators Gq : L2 (R+ ) → L2 (R− ) defined by

∞ q(t − s)f (s)ds, t < 0. (Gq f )(t) = 0

Then (Γ k f )(t) = (Gq f )(−t), k(t) = q(−t), t ∈ (0, ∞) (see §1.8). We have proved in §1.8 that if Gq is bounded, then there exists a function ψ in L∞ (R) such that q = Fψ (−∞, 0). Moreover, Gq = FHψ F ∗ , where Hψ is the Hankel operator from H 2 (C+ ) to H 2 (C− ) defined by Hψ F = P − ψF (see §1.8). Next, if ω is the conformal map of D onto C+ defined by 1+ζ ω(ζ) = i , ζ ∈ D, 1−ζ then U ∗ Hψ U = Hϕ , where ϕ = ψ ◦ ω and U is the unitary operator from L2 onto L2 (R) defined by 1 (f ◦ ω −1 )(t) (Uf )(t) = √ . t+i π Thus it follows from Theorems 1.1, 2.1, and 3.1 that Γ k ∈ S p if and only 1/p if P− ϕ ∈ Bp , 0 < p < ∞. Let us rewrite this condition in terms of the function ψ. 1/p We define the Besov space Bp (R), 0 < p < ∞, as the space of distributions ξ on R such that (7.1) 2j ξ ∗ Vj pLp (R) + ξ ∗ V j pLp (R) < ∞. j∈Z


277

def

Here the Vj are functions such that their Fourier transforms υj = FVj have the following properties: 0 / υj ≥ 0, supp υj ⊂ 2j−1 , 2j+1 , x , x > 0, (υj )(x) = (υj−1 ) 2 υj ∈ C ∞ ,

and

(υj )(x) = 1,

x > 0.

j∈Z

We refer the reader to Peetre [1] for more detailed information on Besov 1/p classes. Note that the class Bp (R) does not depend on the choice of the functions Vj . Note also that if a distribution ξ satisfies (7.1), and r is an 1/p arbitrary polynomial, then r + ξ also satisfies (7.1). If ξ ∈ Bp (R), we have the convergence of the series

(ξ ∗ Vj + ξ ∗ V j )

j∈Z

to ξ modulo polynomials. In fact, it can be shown that the series converges modulo constants (see Peetre [1], Ch. 3 and 11). 1/p It is convenient to identify Bp with a space of functions on R that is contained in V M O. It follows from Dyn’kin’s description of Besov classes 1/p (see Appendix 2.6) that if ψ ∈ BM O(R), then P − ψ ∈ Bp (R) if and only 1/p if P− (ψ ◦ ω) ∈ Bp , where P − ψ = (P− (ψ ◦ ω)) ◦ ω −1 , def

ψ ∈ L∞ (R)

(see §1.8). Thus the following theorem follows from Theorems 1.1, 2.1, and 3.1. Theorem 7.4. Let 0 < p < ∞. Suppose that k is a distribution on (0, ∞). The following are equivalent: (i) Γ k ∈ S p ; 1/p (ii) k = Fψ (0, ∞) for a function ψ ∈ Bp (R). Note that (ii) can be reformulated in the following way:

| supp ωj | · F(ωj k)pLp (R) < ∞,

(7.2)

j∈Z

where the ωj are defined above and | supp ωj | means the length of the interval supp ωj . Here we can consider the function ωj k as a function in C ∞ (R) with compact support.

278


Wiener–Hopf Operators on a Finite Interval Let k be a function or a distribution on R. For σ > 0 we consider the integral operator Wσ,k on L2 [−σ, σ] defined by

σ (Wσ,k f )(x) = k(x − y)f (y)dy, x ∈ [−σ, σ]. −σ

Such operators are called Wiener–Hopf operators on a finite interval. We obtain here a necessary and sufficient condition for Wσ,k ∈ S p , 0 < p < ∞. Together with the Wiener–Hopf operator Wσ,k on L2 [−σ, σ] we consider here truncated “Hankel operators” Γ σ,k on L2 [−σ, σ] defined by

σ (Γ σ,k f )(x) = k(x + y)f (y)dy, x ∈ [−σ, σ]. −σ

It is easy to see that Γ σ,k = Wσ,k U , where U is the unitary operator on L2 [−σ, σ] defined by (U f )(x) = f (−x),

x ∈ [−σ, σ].

To state the result, we introduce the functions νj associated with an interval [α, β]. We start with the system of functions ωj defined above. Put x−α νj (x) = ωj 2 , j < 0, β−α νj (x) = νj (α + β − x), νj (x). ν0 (x) = 1 −

j > 0,

j =0

Clearly, 0 ≤ νj (x) ≤ 1, j ∈ Z, supp νj ⊂ [α, (α + β)/2] for j < 0 and supp νj ⊂ [(α + β)/2, β] for j > 0. Theorem 7.5. Let 0 0, and let {νj }j∈Z be the system of functions associated with [−2σ, 2σ]. Then Wσ,k ∈ S p if and only if 2−|j| F(νj k)pLp < ∞. (7.3) j∈Z

Note that (7.3) is equivalent to the condition | supp νj | · F(νj k)pLp < ∞, j∈Z

which is similar to condition (7.2). It is more convenient to work with operators Γ ak on L2 [0, a] defined by

a (Γ ak f )(x) = k(x + y)f (y)dy, x ∈ [0, a]. 0

Γ ak

depends only on the restriction of k to [0, 2a]. It Clearly, the operator is easy to see that Theorem 7.5 is equivalent to the following one.


279

Theorem 7.6. Let 0 0, and let {νj }j∈Z be the system of functions associated with [0, 2a]. Then Γ ak ∈ S p if and only if 2−|j| F(νj k)pLp < ∞. j∈Z

Proof. To prove Theorem 7.6, we reduce the study of the operators Γ ak to the case of operators Γ k on L2 (R+ ). If supp k ⊂ [0, a], it is easy to see that Γ ak ∈ S p if and only if Γ k ∈ S p , since in this case Γ ak is the restriction of Γ k to L2 [0, a] while Γ k L2 [0, ∞) = O. (We identify in a natural way L2 [0, a] with a subspace of L2 (R+ ).) Thus by Theorem 7.4, if supp k ⊂ [0, a], then Γ ak ∈ S p if and only if 2−|j| F(νj k)pLp < ∞. j≤0

The same can be done if supp k ⊂ [a, 2a]. Indeed, in this case Γ ak is unitarily equivalent to the operator Γ ak# , where k# (x) = k(2a − x) (unitary equivalence is provided by the unitary operator V on L2 [0, a] defined by (V f )(x) = f (a−x), f ∈ L2 [0, a]). Clearly, supp k# ⊂ [0, a], and so Γ ak ∈ S p if and only if 2−|j| F(νj k)pLp < ∞. j≥0

To treat the general case, we cut the function k into three pieces. Consider C ∞ functions v1 , v2 , and v3 with compact supports such that v1 (x) + v2 (x) + v3 (x) = 1 for x ∈ [0, 2a], 0 ≤ vj (x) ≤ 1, for j = 1, 2, 3 and x ∈ R, supp v2 = [3a/4, 5a/4], supp v3 = [a, 3a], and v1 (x) = v3 (2a−x). Put def

kj = vj k. We claim that Γ ak ∈ S p if and only if Γ akj ∈ S p for j = 1, 2, 3. This is a consequence of the following more general fact. Lemma 7.7. Let ϕ be a C ∞ function with compact support in (0, ∞) and let 0 < p < ∞. Suppose that R is an integral operator on L2 (R+ ),

u(x, y)f (u)dy, f ∈ L2 (R+ ). (Rf )(x) = R+

Consider the integral operator T on L2 (R+ ) defined by

(T f )(x) = ϕ(x + y)u(x, y)f (u)dy, f ∈ L2 (R+ ). R+

If R ∈ S p , then T ∈ S p . Let us first complete the proof of Theorem 7.6. By Lemma 7.7, Γ ak ∈ S p if and only if Γ akj ∈ S p for j = 1, 2, 3. Clearly, supp k1 ⊂ [0, a] and supp k3 ⊂ [a, 2a]. Thus as we have already observed Γ ak1 ∈ S p ⇐⇒ 2−|j| F(νj k1 )pLp < ∞ (7.4) j≤0

280


and Γ ak3 ∈ S p

⇐⇒

2−|j| F(νj k3 )pLp < ∞.

(7.5)

j≥0

To estimate Γ ak2 , we need the following fact. Lemma 7.8. Let 0 < p < ∞ and let k be a function on R with support in [3a/4, 5a/4]. Then Γ ak ∈ S p if and only if Fk ∈ Lp (R). Let us first complete the proof of the theorem. By Lemma 7.8, Γ ak ∈ S p if and only if F(k2 ) ∈ Lp (R) and and the right-hand sides of (7.4) and (7.5) are finite. Note that for j ≥ 2 we have νj k3 = νj k and for k ≤ −2 we have νj k1 = νj k. It remains to show that 1

F(νj k1 )pLp < ∞

(7.6)

j=−1

if and only if F(ν−1 k1 )pLp

+ F(ν0 k1 )pLp + Fk2 pLp + F(ν0 k3 )pLp + F(ν1 k3 )pLp < ∞.

(7.7)

This is a consequence of the following well-known fact (see Peetre [1]): let ψ be a C ∞ function with compact support in (0, ∞), 0 < α < β < ∞, and let g be a function with support in [α, β]; then F(ψg)Lp ≤ const FgLp ;

(7.8)

the constant may depend on ψ, α, and β but does not depend on g. (Note that (7.8) also follows easily from Theorem 7.4 and Lemma 7.7.) Indeed, it follows from (7.8) that both (7.6) and (7.7) are equivalent to the fact that   1 F k νj  ∈ Lp (R). j=−1

Proof of Lemma 7.7. Suppose that supp ϕ ⊂ [0, T /2]. Put def

v(x, y) = ϕ(x + y)u(x, y). Without loss of generality we may assume that supp u ⊂ [0, T /2] × [0, T /2]. Consider the function φ on T defined by φ e2πix/T = ϕ(x), 0 ≤ x ≤ T. Then ϕ(x + y) =

2πin(x+y)/T ˆ φ(n)e .

n∈Z

Thus T =

n∈Z

ˆ φ(n)M ξn RMηn ,

(7.9)

8. Generalized Hankel Matrices

281

where ξn (x) = e2πinx/T , ηn (y) = e2πiny/T , and Mh is multiplication by h on L2 . Since φ ∈ C ∞ , it follows that p ˆ |φ(n)| 0. Hence, the series (7.9) converges absolutely, and so T ∈ S p . Proof of Lemma 7.8. We consider three integral operators with the same kernel k(x+y). The first one is our Γ ak on L2 [0, a]. The second operator is the operator Γ k on L2 (R+ ). The third one is the integral operator on L2 [a/4, a], which we denote by Γ # k . It is easy to see that a Γ # k S p ≤ Γ k S p ≤ Γ k S p .

Indeed, obviously,

Γ ak = P0a Γ k L2 [0, a]

a 2 a and Γ # k = Pa/4 Γ k L [a/4, a],

where Pαβ is the orthogonal projection onto L2 [α, β]. Since supp Fk is separated away from zero and infinity, it follows from Theorem 7.4 that Γ k S p is equivalent to FkLp . On the other hand, Γ # k S p = Γ k# S p , where k# (x) = k(x + a/4). Hence, for the same reason, Γ k# S p is equivalent to FkLp .

8. Generalized Hankel Matrices In this section we study matrices of the form α β ˆ + k)}j,k≥0 , Γα,β ϕ = {(1 + j) (1 + k) ϕ(j

where ϕ is a function analytic in D, and α and β are real numbers. We obtain boundedness and compactness criteria for certain values of α and β. Then we study the question of when Γα,β ϕ ∈ S p . As usual, here we identify operators on 2 with their matrices in the standard basis {ej }j≥0 of 2 . It is bounded (compact, or belongs to S p ) if and only is easy to see that Γα,β ϕ if the same is true for Γβ,α ϕ . 3 α,β For α > −1 and β > −1 we also consider the matrices Γ defined by ϕ (α) (β) 3 α,β ˆ + k)}j,k≥0 , Γ ϕ = {Dj Dk ϕ(j

where Dn(α)

=

n+α n

=

(α + 1)(α + 2) · · · (α + n) n!

(see Appendix 2.6). Since (α)

c
0. Then Γα,β is the matrix of a bounded ϕ is equivalent operator if and only if ϕ ∈ Λα+β . The operator norm of Γα,β ϕ to ϕΛα+β . Proof. Suppose that ϕ ∈ Λα+β . Let a = {aj }j≥0 and b = {bj }j≥0 be finitely supported sequences in of 2 . We have (1 + j)α (1 + k)β ϕ(j ˆ + k)aj ¯bk (Γα,β ϕ a, b) = j,k≥0

=

=

ϕ(m) ˆ

m≥0

j=0

m

ϕ(m) ˆ

def

(1 + j)α aj (1 + m − j)β ¯bm−j fˆ(j)ˆ g (m − j) =

j=0

m≥0

where f (z) =

m

def

We have f = I−α

ˆ ϕ(m) ˆ h(m) = (ϕ, h),

m≥0

(1 + j)α aj z j , g(z) =

j≥0

def

(1 + j)β ¯bj z j , and h = f g.

j≥0

aj z j , g = I−β

j≥0

f B −α ≤ const a2 , 2

¯ j bj z , and so j≥0

gB −β ≤ const b2 2

(see Appendix 2.6). Therefore

|f (ζ)|2 (1 − |ζ|)2α−1 dm2 (ζ) ≤ const a2 , D

D

|g(ζ)|2 (1 − |ζ|)2β−1 dm2 (ζ) ≤ const b2

(see Appendix 2.6). It follows that

hB −(α+β) ≤ const |h(ζ)|(1 − |ζ|)α+β−1 dm2 (ζ) ≤ const a2 b2 . 1

D

∗ −(α+β) = Λα+β , we obtain Taking into account the fact that B1 |(Γα,β ϕ a, b)| ≤ const ϕΛα+β a2 b2 , is bounded. and so Γα,β ϕ Suppose now that ϕ is a function analytic in D such that the operator is bounded on 2 . We have to show that ϕ ∈ Λα+β . We may assume Γα,β ϕ 3 α,β is bounded. that Γ ϕ


283

Let ψ be an analytic in D function of class B11 . By Theorem 1.1, Γψ ∈ S 1 . We have (see Appendix 1.1) 3 α,β = Γψ , Γ ϕ =

∞ m

(α)

(β)

ˆ Dj Dm−j ψ(m) ϕ(m) ˆ

m=0 j=1 ∞ (α+β+1) ˆ ψ(m)ϕ(m) Dm ˆ m=0

= (ψ, I3−(α+β+1) ϕ).

It follows that I3−(α+β+1) ϕ determines a continuous linear functional on 1 B1 + , and so I3−(α+β+1) ϕ ∈ Λ−1 (see Appendix 2.6). Hence, ϕ ∈ Λα+β (see Appendix 2.6). The following theorem establishes the compactness criterion in the case when α > 0 and β > 0. is compact if and only if Theorem 8.2. Let α, β > 0. Then Γα,β ϕ ϕ ∈ λα+β . To prove the necessity of the condition ϕ ∈ λα+β we use the Schur product of matrices. Recall that in §3 for matrices A = {ajk }j,k≥0 , B = {bjk }j,k≥0 , we have defined the Schur product of A and B by A B = {ajk bjk }j,k≥0 . A matrix A is called a Schur multiplier of the space of bounded operators if Schur multiplication by A is a bounded operator on the space B(2 ). The norm of A in the space of Schur multipliers is denoted by A , def

A = sup{A T : T ≤ 1}. Let µ belong to the space M of finite complex Borel measure on T. µ(j + k)}j,k≥0 . We need the following Denote by Γµ the Hankel matrix {ˆ fact. Theorem 8.3. Let µ be a finite complex Borel measure µ on T. Then Γµ is a Schur multiplier of B(2 ) and Γµ ≤ µM. Proof. Consider first the case when µ = δτ , τ ∈ T, i.e., µ({τ }) = 1, and µ(E) = 0 if E ⊂ T and τ ∈ E. Then µ ˆ(j) = τ¯j , j ∈ Z. Let T be a bounded operator on 2 with matrix {tjk }j,k≥0 . Clearly, Γµ T = Dτ¯ T Dτ¯ , where Dτ¯ is the unitary operator on 2 defined by Dτ¯ ej = τ¯j ej , j ≥ 0. Hence, Γµ T = T . Now let µ be an arbitrary measure in the unit ball of M. By the Krein– Milman theorem, µ is the limit in the weak topology σ(M, C(T )) of a sequence of measures of the form N j=1

λj δτj ,

(8.1)

284


where τj ∈ T and

N

|λj | ≤ 1. Clearly, Γν ≤ 1 for any measure ν of the

j=1

form (8.1). It is also easy to see that if {νj }j≥0 is a sequence of measure converging to µ in the weak topology σ(M, C(T )), then {Γνj }j≥0 converges to Γµ in the weak operator topology, which implies that Γµ ≤ 1. Recall that for a natural number N , KN stands for the Fejér kernel, and KN L1 = 1 (see Appendix 2.1). Corollary 8.4. Let T be a compact operator on 2 . Then lim T − ΓKN T = 0.

(8.2)

N →∞

Proof. Clearly, (8.2) holds for operators T , which can have only finitely many nonzero matrix entries. The result now follows from Theorem 8.3 and the fact that KN L1 = 1. Proof of Theorem 8.2. If ϕ ∈ λα+β , we can approximate ϕ by polynomials in Λα+β . Since the operator Γα,β has finite rank for any polynomial ψ is compact. ψ, it follows now from Theorem 8.1 that Γα,β ϕ Suppose now that Γα,β is compact. By Corollary 8.4, ϕ α,β α,β α,β α,β Γα,β ϕ − ΓKN Γϕ = Γϕ − Γϕ∗KN = Γϕ−ϕ∗KN → 0

as

N → ∞.

It follows from Theorem 8.1 that ϕ−ϕ∗KN Λα → 0 as N → ∞. Hence, ϕ ∈ λα+β . Let us characterize the bounded operators Γα,0 ϕ for α > 0. Theorem 8.5. Let α > 0. Then Γα,0 ϕ is the matrix of a bounded operator if and only if I−α ϕ ∈ BM O. The operator norm of Γα,0 ϕ is equivalent to I−α ϕBM O . Proof. Suppose that I−α ϕ ∈ BM O. Let a = {aj }j≥0 and b = {bj }j≥0 be finitely supported sequences in of 2 . We have (Γα,0 (1 + j)α ϕ(j ˆ + k)aj ¯bk ϕ a, b) = j,k≥0

=

=

where f (z) =

j≥0

m

m≥0

j=0

m

m≥0 def

ϕ(m) ˆ

ϕ(m) ˆ

(1 + j)α aj ¯bm−j fˆ(j)ˆ g (m − j) =

j=0

ˆ ϕ(m) ˆ h(m) = (ϕ, h),

m≥0

¯ j def bj z , and h = f g. (1 + j)α aj z j , g(z) = def

j≥0

We have

1/2 |f (ζ)|2 (1 − |ζ|)2α−1 dm2 (ζ) ≤ const a2 ≤ const a2 D

(see Appendix 2.6). Clearly, gH 2 = b2 .


285

Consider the radial maximal function g (∗) on T defined by def

g (∗) (ζ) = sup |g(rζ)|, 0 0, k > 0, ajk = R 0, otherwise. Then A is a Schur multiplier of B(2 ) and A ≤ µ, where µ is the total variation of µ.

286


Proof. It is easy to see that as in the proof of Theorem 8.3 it is sufficient to consider the case when µ is a point mass at a point v ∈ R, i.e., µ({v}) = 1 and µ(E) = 0 if v ∈ E. In this case ajk = k 2πiv j −2πiv ,

j, k > 0.

#

It is easy to see that A T = P DT D P , where P is the orthogonal projection onto the subspace {H ∈ 2 : (h, e0 ) = 0}, D and D# are multiplication operators defined by Dek = k 2πiv , k > 0, D# ej = j −2πiv , It follows that A T ≤ T .

j > 0.

Proof of Lemma 8.6. Since the entries of M are bounded, it is sufficient 4 = {m to show that the matrix M ˜ jk }j,k≥0 ,

mjk , j, k > 0, m ˜ jk = 0, otherwise, is a Schur multiplier of B(2 ). We have mjk − 1 = F (j, k),

j > 0, k > 0,

where F is a function on R+ × R+ defined by F (x, y) =

(x + y)α − xα − y α , xα + y α

x, y > 0.

Clearly, F (es , et ) = f (s − t),

s, t ∈ R,

where

(et + 1)α − etα − 1 , t ∈ R. etα + 1 It is easy to see that f belongs to the class S (see §1.8). Therefore f is a Fourier transform of an L1 function on R. It follows that there exists a finite complex Borel measure µ on R such that

e−2πitξ dµ(ξ), t ∈ R. f (t) + 1 = def

f (t) =

R

We have mjk

= F (j, k) + 1 = f (log j − log k) + 1

e2πi(log j)ξ e−2πi(log k)ξ dµ(ξ) = R

= k 2πiξ j −2πiξ dµ(ξ), j, k > 0. R

The result now follows from Lemma 8.7. The proof of the following result is similar to the proof of Theorem 8.2. is the matrix of a compact Theorem 8.8. Let α > 0. Then Γα,0 ϕ operator if and only if I−α ϕ ∈ V M O.


287

Let us now characterize the Γα,β of class S p for α, β > max{− 12 , − p1 } ϕ (see Appendix 2.6 for the definition of the Besov classes Bps ). Theorem 8.9. Let 0 max{− 12 , − p1 }, and let 1/p+α+β

ϕ be a function analytic in D. Then Γα,β ϕ ∈ S p if and only if ϕ ∈ Bp

.

One can easily verify that in the case 0 − 12 . The same proof also ϕ works for p = 1 (see the Remark after the proof of Theorem 3.1). Here we present another proof of necessity in the case p = 1 that is of independent interest. we can Proof of necessity for p = 1. As before, instead of Γα,β ϕ α,β α,β 3 3 consider the operator Γϕ . Suppose that Γϕ ∈ S 1 . Let us show that ϕ ∈ B11+α+β . Let ε > 0 and let ψ be an analytic polynomial. Consider the operator 3 3= Γ

Djα+ε Dkβ+ε Djα Dkβ

( ˆ + k) ψ(j

. j,k≥0

It follows easily from Theorem 8.1 that 3 3 ≤ const ψΛ . Γ 2ε We have 3 3 Γ 3 α,β = Γ, ϕ

m≥0

=

 

m

 β+ε  ˆ Djα+ε Dm−j ˆ ψ(m)ϕ(m)

j=0 α+β+2ε+1 ˆ ψ(m)ϕ(m) Dm ˆ = (ψ, I˜−(α+β+2ε+1) ϕ).

m≥0

It follows that 3 3 ≤ const Γ 3 α,β S ψΛ . 3 α,β S Γ |(ψ, I˜−(α+β+2ε+1) ϕ)| ≤ const Γ ϕ ϕ 1 1 2ε Hence, I˜−(α+β+2ε+1) ϕ determines a continuous linear functional on P+ λ2ε , and so I˜−(α+β+2ε+1) ϕ ∈ B1−2ε (see Appendix 2.6). Therefore ϕ ∈ B11+α+β (see Appendix 2.6). We start the proof of Theorem 8.9 for 1 < p < ∞ with sufficiency in the case p = 2. 1/2+α+β

Lemma 8.10. Let ϕ be an analytic in D function of class B2 3 α,β Then Γ ϕ ∈ S2.

.

288


Proof. Obviously, (α) 2 (β) 2 3 α,β 2 Dj Dk Γ = |ϕ(j ˆ + k)|2 ϕ S2 j,k≥0

≤ const

 2 |ϕ(m)| ˆ

m≥0

≤ const



(1 + j)2α (1 + k)2β 

j+k=m 2(α+β)+1

(1 + m)

2 |ϕ(m)| ˆ ≤ const ϕB 1/2+α+β . 2

m≥0

To prove Theorem 8.9 for 1 < p < ∞, we need an interpolation theorem for analytic families of operators. For a Banach pair (X0 , X1 ) (see §4) one can associate the spaces def

X0 + X1 = {x = x0 + x1 : x0 ∈ X0 , x1 ∈ X1 } endowed with the norm def

xX0 +X1 = inf{x0 X0 + x1 X1 : x0 ∈ X0 , x1 ∈ X1 , x0 + x1 = x}, and X0 ∩ X1 endowed with the norm def

xX0 ∩X1 = xX0 + xX1 . For 0 < ϑ < 1 the complex interpolation space (X0 , X1 )[θ] ⊂ X0 + X1 can be defined as follows. Consider the class F(X0 , X1 ) of continuous (X0 + X1 )-valued functions f on {ζ ∈ C : 0 ≤ Re z ≤ 1} that are bounded and analytic in the open strip {ζ ∈ C : 0 < Re ζ < 1} and such that def

f F = sup{f (j + it)Xj : j = 0, 1, t ∈ R} < ∞. We define the complex interpolation space (X0 , X1 )[θ] to be {f (θ) : f ∈ F(X0 , X1 )} and endow it with the natural quotient norm. We need the following result (see Cwikel and Janson [1]). Interpolation theorem for an analytic family of operators. Let (X0 , X1 ) and (Y0 , Y1 ) be Banach pairs. Suppose that at least one of the spaces Y0 or Y1 is separable. Let ζ → Tζ be a bounded function that is defined and continuous in the closed strip {0 ≤ Re ζ ≤ 1}, takes values in the space of bounded linear operators from X0 ∩ X1 to Y0 + Y1 , and is analytic in the open strip {0 < Re ζ < 1}. If Tj+it x ≤ Mj xXj ,

j = 0, 1,

t ∈ R, x ∈ X0 ∩ X1 ,

then Tθ extends to a continuous linear operator from Xθ to Yθ , 0 < θ < 1, of norm at most M01−θ M1θ . In the special case when the function ζ → Tζ is constant this result holds without the separability assumption (Calder´ on’s theorem; see Bergh and Löfstr¨ om [1]). Let us introduce spaces of vector sequences, which will be used in the proof of Theorem 8.9.


289

Definition. Let B be a Banach space. For 1 ≤ p < ∞ and s > 0 we denote by p,s (B) the space of sequences {fn }n≥0 , fn ∈ B, satisfying p (2ns fn B ) < ∞, n≥0

and by cs0 (B) the space of sequences {fn }n≥0 , fn ∈ B, satisfying lim 2ns fn B = 0.

n→∞

Before we proceed to the proof of Theorem 8.9 in the case 1 < p < ∞ we make the following observation. We can consider the generalized Hankel 3 α,β also for complex α and β. It is easy to see that matrices Γ ϕ 3 Re α,Re β M2 , where M1 and M2 are the unitary operators on 2 3 α,β = M1 Γ Γ ϕ ϕ 3 α,β ∈ S p defined by M1 ej = (1 + j)i Im α ej , M2 ej = (1 + j)i Im β ej . Hence, Γ ϕ α,Re β 3 Re if and only if Γ ∈ Sp. ϕ Proof of Theorem 8.9 for 1 < p < ∞. Let us first prove that the 1/p+α+β 3 α,β ∈ S p . Consider first the case is sufficient for Γ condition ϕ ∈ Bp ϕ def

def

def

def

p−2 2 −1/2, α1 > 0, β0 > −1/2, β1 > 0. For 0 ≤ Re ζ ≤ 1 we define the operator Tζ on the set of L1 sequences {fn }n≥0 by def 3 αζ ,βζ Tζ {fn }n≥0 = Γ , ψ def

def

where αζ = α0 + ζ(α1 − α0 ), βζ = β0 + ζ(β1 − β0 ), and ψ =

fn ∗ Wn

n≥0

(see Appendix 2.6 for the definition of the kernels Wn ). Let us verify the hypotheses of the interpolation theorem for analytic 1 +β1 families of operators with X0 = 2,1/2+α0 +β0 (L2 ), X1 = cα (L∞ ), 0 Y0 = S 2 , and Y1 = S ∞ . 1/2+α0 +β0 , and so by Lemma If Re ζ = 0 and {fn }n≥0 ∈ X0 , then ψ ∈ B2 1 +β1 8.10, Tζ {fn }n≥0 ∈ S 2 . If Re ζ = 1 and {fn }n≥0 ∈ X1 , then ψ ∈ λα , 0 and so by Theorem 8.2, we have Tζ {fn }n≥0 ∈ S ∞ . It is easy to see that Tζ {fn }n≥0 S ∞ = TRe ζ {fn }n≥0 S ∞ ≤ T1 {fn }n≥0 S ∞ ≤ const 1 +β1 for 0 ≤ Re ζ ≤ 1 and {fn }n≥0 ∈ cα , since obviously, 0

Tζ {fn }n≥0 = A1 T1 {fn }n≥0 A2 for some operators A1 , A2 satisfying A1 ≤ 1, A2 ≤ 1. Clearly, the function ζ → Tζ is analytic in {ζ : 0 < Re ζ < 1} and the hypotheses of the interpolation theorem for analytic families of operators are satisfied.

290


We need now the following well-known facts: (S 2 , S ∞ )[θ] = S p , θ = p−2 p (see Reed and Simon [1], Ch. IX, §4); 1 +β1 (L∞ ))[θ] (2,1/2+α0 +β0 (L2 ), cα 0

= p,1/p+α+β ((L2 , L∞ )[θ] ) p−2 = p,1/p+α+β (Lp ), θ = p

(see Bergh and Löfstr¨ om [1], §5.3 and §5.1). α,β 3 It follows that Γψ ∈ S P whenever fn ∗ Wn and {fn }n≥0 ∈ p,1/p+α+β . ψ= n≥0 1/p+α+β

. Put fn = ϕ ∗ (Wn−1 + Wn + Wn+1 ). It follows Now let ϕ ∈ Bp easily from the definition of Besov classes in terms of convolutions with Wn that {fn }n≥0 ∈ p,1/p+α+β . It remains to observe that fn ∗ Wn = ϕ. n≥0

The case 1 < p < 2 is easier since we can apply the complex interpolation method for one operator (Calder´ on’s theorem). We define the operator T on the space of L1 sequences {fn }n≥0 by

where ψ =

3 α,β , T {fn }n≥0 = Γ ψ fn ∗ Wn . Since Theorem 8.9 has been already proved for

n≥0

p = 1, it follows that T is a bounded operator from 1,1+α+β (L1 ) to S 1 . Lemma 8.10 implies that T is a bounded operator from 2,1/2+α+β (L2 ) to S 2 . Applying the interpolation theorem for T and θ = 2p−2 p , we find that T 1,1+α+β 2,1/2+α+β 2 (L1 ), (L ))[θ] to (S 1 , S 2 )[θ] . is a bounded operator from ( It is well known that (1,1+α+β (L1 ), 2,1/2+α+β (L2 ))[θ] = p,1/p+α+β (Lp ) (see Bergh and Löfstr¨ om [1], §5.3 and §5.4) and (S 1 , S 2 )[θ] = S p (see Reed and Simon [1], Ch. IX, §4). The rest of the proof is the same as in the case p > 2. 1/p+α+β for It remains to prove the necessity of the condition ϕ ∈ Bp 3 α,β ∈ S p . Let ψ be a polynomial. It follows from Theorem 2.1 that Γ ϕ Γψ S p ≤ const ψB 1/p . We have p

1+α+β ˆ 3 α,β = Γψ , Γ |(ψ, I˜−(1+α+β) ϕ)| = ψ(m)ϕ(m) Dm ˆ ϕ m≥0 α,β α,β Γϕ S p ψB 1/p . ≤ 3 Γϕ S p Γψ S p ≤ 3 p

1/p Hence, I˜−(1+α+β) ϕ determines a continuous linear functional on Bp , and −1/p 1/p+α+β , which is equivalent to the fact that ϕ ∈ Bp so I˜−(1+α+β) ϕ ∈ Bp (see Appendix 2.6).


291

Remark. The restriction min{α, β} > max{− 12 , − p1 } in the statement of Theorem 8.9 is essential. Indeed, suppose that 2 ≤ p < ∞. Without loss of generality we may assume that α ≤ β. It follows from the definition of the Besov spaces (see Appendix 2.6) that z m B 1/p+α+β ≤ const ·m1/p+α+β . p

On the other hand, if α ≤ −1/p, it is easy to verify that  1/p m α,β Γzm =  (1 + j)αp (m − j + 1)βp  S p

j=0

≥ const

mβ , α < −1/p, mβ (log(1 + m))1/p , α = −1/p,

1/p+α+β

does not imply that Γα,β and so the condition ϕ ∈ Bp ϕ ∈ Sp. Suppose now that 0 < p < 2, α ≤ β, and α ≤ −1/2. Let us show that 1/p+α+β the condition ϕ ∈ Bp does not imply that Γα,β ϕ ∈ S 2 . We have  1/2 m α,β 2 Γϕ =  (1 + j)2α (m − j + 1)2β |ϕ(m)| ˆ S2 m≥0 j=0

 1/2  m   2  (1 + m)2β |ϕ(m)| ˆ , α < −1/2,   j=0 ≥ const 1/2  m   2  (1 + m)2β log(1 + m)|ϕ(m)| ˆ , α = −1/2.   j=0

Suppose now that ϕ is a polynomial of the form ϕ = α,β Γϕ ≥ const S2

2N −1

m ϕ(m)z ˆ . Then

j=N +1

N β ϕ2 , N β (log N )1/2 ϕ2 ,

α < −1/2, α = −1/2.

On the other hand, it follows from the definition of the Besov spaces (see Appendix 2.6) that ϕB 1/p+α+β ≤ const ·N 1/p+α+β ϕp . p

1/p+α+β

To prove that the condition ϕ ∈ Bp does not imply that Γα,β ϕ ∈ S2, ∞ it remains to take a nonzero C function F with support in [1, 2], put k zk , ϕ= F N k∈Z

and apply Lemma 3.4.

292


9. Generalized Block Hankel Matrices and Vectorial Hankel Operators In this section we study generalized block Hankel matrices of the form α βˆ Γα,β Φ = {(1 + j) (1 + k) Φ(j + k)}j,k≥0 , where Φ is an analytic function in D that takes values in the space B(H, K) of bounded linear operators from a Hilbert space H to a Hilbert space K. In Chapter 2 we have obtained boundedness and compactness criteria for ΓΦ = Γ0,0 Φ . Here we consider the more general case α, β > 0 and we also study conditions for Γα,β Φ to belong to S p , 0 < p < ∞. Then we obtain some applications. To state the results we need some information about Besov classes of vector functions. Consider first Besov spaces Bps (X), where 1 ≤ p ≤ ∞, s ∈ R, and X is a Banach space. The space Bps (X) can be defined as the space of X-valued ∞ cj z j , distributions f (or X-valued formal trigonometric series f = j=−∞ def fˆ(j) = cj ∈ X) such that 2s|n| Wn ∗ f Lp (X)

n∈Z

∈ p (Z)

(see Appendix 2.6 for the definition of the Wn ). As in the scalar case for a scalar trigonometric polynomial q and a formal X-valued trigonometric series f ∞ def q∗f = qˆ(j)fˆ(j). j=−∞ s We shall use the notation Λs (X) for B∞ (X). The closure of the X-valued trigonometric polynomials in Λs (X) will be denoted by λs (X). in the scalar case it can be proved that for 1 ≤ p < ∞ the dual space As ∗ ∗ Bps (X) can be identified with the space Bp−s (X ) with respect to the following pairing: ∞ (f, g) = (fˆ(j), gˆ(j)), j=−∞ ∗ where f is a trigonometric polynomial in Bps (X) and g ∈ Bp−s (X ). The ∗ space (λs (X)) can be identified with B1−s (X ∗ ) with respect to the same pairing. For s > 0 the space Bps (X) can also be described as the space of X-valued functions f in Lp (X) for which

∆nτ ϕLp (X) dm(τ ) < ∞, p < ∞, (9.1) 1+ps T |1 − τ |

∆nτ f L∞ (X) < ∞, |1 − τ |s τ ∈T,τ =1 sup

p = ∞,

9. Generalized Block Hankel Matrices and Vectorial Hankel Operators

293

def

where n is an integer such that n > s, (∆t f )(ζ) = f (ζτ ) − f (ζ), and def = ∆n+1 t

∆t ∆nt . The proof is the same as in the scalar case. The subspace Bps + (X) = {f ∈ Bps (X) : fˆ(j) = O for j < 0} can be identified in a natural way with the space of X-valued functions analytic in D and satisfying

f (n) (ζ)pLp (X) (1 − |ζ|)(n−s)p−1 dm2 (ζ) < ∞, p < ∞, D (9.2) sup f (n) (ζ)L∞ (X) (1 − |ζ|)n−s < ∞,

p = ∞,

ζ∈D

where n is an integer such that n > s. This can be proved in the same way as in the scalar case. We also need spaces Bps (X) with 0 1/p − 1. The space X here does have to be a Banach space. We assume that X is a p-Banach space, which means that X is a complete quasinormed space with quasinorm · X satisfying x + ypX ≤ xpX + ypX ,

x, y ∈ X.

Clearly, any Banach space is p-Banach. Other examples of p-Banach spaces are Lp (µ), S p . For a p-Banach space X and s > 1/p − 1 the Besov space Bps (X) can be defined as the space of X-valued formal trigonometric series such that ∈ p (Z) 2s|n| Vn ∗ f Lp (X) n∈Z

(see Appendix 2.6 for the definition of the polynomials Vn ). It can be proved that the space Bps (X) consists of the functions f ∈ Lp (X) satisfying (9.1) and the subspace Bps + (X) consists of the X-functions f analytic in D and satisfying (9.2). Let us now state the main results of this section. Theorem 9.1. Let H, K be Hilbert spaces and let Φ be a function analytic in D that takes values in the space B(H, K). Suppose that α > 0 and β > 0. Then Γα,β is a bounded operator if and only if Φ ∈ Λα+β (B(H, K)). Φ Theorem 9.2. Suppose that H, K are Hilbert spaces and let Φ be a B(H, K)-valued function analytic in D. Suppose that α > 0 and β > 0. is a compact operator if and only if Φ ∈ λα+β (S ∞ (H, K)). Then Γα,β Φ Theorem 9.3. Let H, K be Hilbert spaces, 0 max{−1/2, −1/p}. Then Γα,β ∈ S p if and only if Φ 1/p+α+β Φ ∈ Bp (S p (H, K)). Corollary 9.4. Let Φ ∈ L∞ (B(H, K)) and let 0 < p < ∞. The 2 Hankel operator HΦ : H 2 (H) → H− (K) belongs to S p if and only if 1/p P− Φ ∈ Bp (S p ).

294


The proofs of Theorems 9.1–9.3 are similar to the proofs of the corresponding scalar results given in Sections 1, 3, and 8. We give some indications here. Note that the proof of Theorem 2.1 given in §2 does not work in the case of operator-valued functions; we should use the approach given in §8. To prove Theorems 9.1–9.3 we can start with the sufficiency of the condition Φ ∈ B11+α+β (S 1 ) for Γα,β Φ to belong to S 1 . As in the scalar case this follows immediately from the following lemma. Lemma 9.5. Let α > −1/2, β > −1/2, and let Ψ be an analytic polynomial of degree m with coefficients in S 1 (H, K). Then α,β ΓΨ ≤ const ·m1+α+β ΨL1 (S 1 (H,K)) , S1

the constant being independent of Ψ. The proof of the lemma follows the proof of Lemma 1.3. We can consider the operators Aζ : 2 (H) → 2 (K), ζ ∈ T, defined by   m m def (Aζ {xj }j≥0 , {yk }k≥0 ) = Ψ(ζ) (1 + j)α ζ¯j xj , (1 + k)β ζ k yk  . j=0

k=0

It can be shown easily that Aζ S 1

≤

 1/2 1/2 m m Ψ(ζ)S 1  (1 + j)2α  (1 + k)2β j=0

k=0

≤ const Ψ(ζ)S 1 (1 + m)

1+α+β

and Γα,β Ψ

= T

Aζ dm(ζ),

which implies the desired inequality. The next step is to prove Theorem 9.1. The proof is exactly the same as that of Theorem 8.1 in the scalar case. Then we can prove the necessity of the condition Φ ∈ B11+α+β for α,β ΓΦ ∈ S 1 . This can be done in exactly the same way as in Theorem 8.9. To prove Theorem 9.2 we proceed in the same way as in the proof of Theorem 8.2. The sufficiency of the condition Φ ∈ λα+β (S ∞ ) follows immediately from Theorem 9.1. To prove necessity, as in the scalar case we can show that lim Φ − Φ ∗ KN Λα+β = 0. N →∞

It remains to observe that the fact that Γα,β is compact implies that the Φ ˆ Fourier coefficients Φ(j) are compact for all j ∈ Z+ . Therefore the polynomial Φ ∗ KN has compact Fourier coefficients, and so Φ ∗ KN ∈ λα+β (S ∞ ), which implies the result.

9. Generalized Block Hankel Matrices and Vectorial Hankel Operators

295

To prove Theorem 9.3 for 1 < p < ∞ we use the method of interpolation of an analytic family of operators. The proof is the same as in the scalar case (Theorem 8.9) if we use the following facts (see Bergh and Löfstr¨ om, [1], §5.1 and §5.6): 2 +β2 2,1/2+α1 +β1 L2 (S 2 ) , cα (L∞ (S ∞ )) 0 [θ] 2 p,1/p+α+β ∞ (L (S 2 ), L (S ∞ ))[θ] = and

L2 (S 2 ), L∞ (S p ) [θ] = Lp (S 2 , S ∞ )[θ] = Lp (S p )

(see the proof of Theorem 8.9, where α1 , α2 , β1 , β2 , and θ are defined). It remains to prove Theorem 9.3 for 0 < p < 1. The sufficiency of the 1/p+α+β condition Φ ∈ Bp (S p ) can be proved in the same way as in the scalar case (see Theorems 3.1 and 8.9). The proof of necessity is practically the same as in the scalar case. The only point we should mention here is that in the operator case we should consider matrices   C1 · · · CN −2 CN −1 C0  CN −1 C0 · · · CN −3 CN −2    A= , .. .. .. .. ..   . . . . . C1

C2

···

CN −1

C0

where Cj ∈ S p (H, K), 0 ≤ j ≤ N −1. As in the scalar case we can introduce N −1 the polynomial Ψ(z) = Cj z j . We have j=0

ApS p =

N −1

Ψ(ζj )pS p ,

j=o

where ζj = e2πij/N , 0 ≤ j ≤ N − 1. This identity, which can easily be verified:    1 1  ζj   ζj    A x =  .. ..    . . ζjN −1

ζjN −1

is a consequence of the following     Ψ(ζj )x, 

x ∈ H.

Let us proceed now to applications of Theorem 9.3. First we study continuity properties of the averaging projection onto the space of block Hankel matrices. Let H, K be Hilbert spaces. We can identify bounded linear operators from 2 (H) to 2 (K) with block matrices {Ajk }j,k≥0 , where Ajk ∈ B(H, K). On the space of such block matrices we define the averaging projection P onto the space of block Hankel matrices.

296


For A = {Ωjk }j,k≥0 the block Hankel matrix PA is defined by 1 def PA = {Ξj+k }j,k≥0 , Ξm = Ωjk . m+1 j+k=m

Theorem 9.6. Let 1 < p < ∞. The averaging projection P is bounded on S p (H, K). The proof is based on Theorem 9.3 for α = β = 0 and 1 < p < ∞ and it is exactly the same as the proof of Theorem 5.1 in the scalar case. We conclude this section with a theorem on operators commuting with a contraction of class C00 (see §2.3). Recall that a C00 -contraction is unitarily equivalent to a Sz.-Nagy–Foias model operator S[Θ] on KΘ , where Θ is a unitary-valued inner operator function in H ∞ (B(H)), H is a separable Hilbert space, and KΘ = H 2 (H) ΘH 2 (H). The model operator S[Θ] is defined on KΘ by S[Θ] f = PΘ zf, f ∈ H 2 (H), where PΘ is the orthogonal projection from H 2 (H) onto KΘ . By Theorem 2.3.1 and formula (2.3.8), each operator R on KΘ that commutes with S[Θ] has the form Rf = PΘ Φf,

f ∈ KΘ ,

for some Φ ∈ H ∞ (B(H)). Moreover,

R = MΘ HΘ∗ Φ KΘ

and

HΘ∗ Φ ΘH 2 (H) = O

(see the Remark after Theorem 2.3.1). Theorem 9.7. Let Θ, Φ, R be as above. Then R ∈ S p , 0 < p < ∞, 1/p if and only if P− Θ∗ Φ ∈ Bp (S p ). The result follows immediately from Theorem 2.3.1 and the remark after it.

Concluding Remarks The problem to describe the trace class Hankel operators was well known for a long time. It was explicitly stated by Rosenblum [2], Krein [3], and Holland (see Anderson, Barth and Brannan [1]). The problem was solved in Peller [1]; see also Peller [2] and [3]. (Let us mention here some earlier results by Rosenblum and Howland, see Howland [1].) After the paper Peller [1] had appeared several other approaches were found. The approach based on the decomposition of functions in B11 in a series of reproducing kernels was given in Coifman and Rochberg [1] (note that the proof of Theorem 1.4 given in this book differs from the proof given in Coifman and Rochberg

Concluding Remarks

297

[1]). Theorem 1.5 is due to Coifman and Rochberg [1]. Theorem 1.6 is taken from Power [2]. Theorem 1.7 is due to Howland [1]. We mention here another approach to the problem of nuclearity of Hankel operators. It is based on the notion of M¨ obius invariant spaces (see Arazy, Fisher, and Peetre [1]). A Banach space X of functions analytic in D is called M¨ obius invariant if

sup inf ϕ ◦ ω − cX ≤ const . c∈C

obius invariant space which imIt turns out that B11 + is the minimal M¨ plies easily that the condition ϕ ∈ B11 + is sufficient for Γϕ ∈ S 1 (see Arazy, Fisher, and Peetre [1]). The results of §2 had been obtained in Peller [2] and [3]. Later other proofs of the S p criterion for 1 < p < ∞ were found, see Rochberg [1], Peetre and Svensson [1], Arazy, Fisher, and Peetre [1]. 1/p The sufficiency of the condition ϕ ∈ Bp for Γϕ ∈ S p in the case 0 < p < 1 was established in Peller [3]. The necessity of the condition 1/p ϕ ∈ Bp was proved in Peller [8] and Semmes [1] by different methods. The proof given in §3 follows the paper Peller [8]. Another proof is contained in the paper Pekarskii [2], which deals with rational approximation in the norm of BM O (see the discussion of results on rational approximation below). Note that Lemma 3.3 and Corollary 3.4 were obtained in Aleksandrov [1]. Theorem 4.1 was obtained in Peller [3] for p0 ≥ 1 and in Peller [8] for arbitrary p0 > 0. We mention here the paper Janson [1] for further results 1/p in this direction. The spaces Bpq were described in Peller [8] in different terms. Theorems 5.1 and 5.2 were obtained in Peller [2] and [3]. In Peller [9] it was shown that the averaging projection P has weak type (1,1), i.e., PS 1 ⊂ S 1,∞ and P maps the Matsaev ideal S ω = S 1,ω into the ideal of compact operators. Then in Peller [10] those results were improved and Theorems 5.3, 5.5, and 5.7 were proved. A bounded projection on the set of Hankel matrices in S 1 was constructed by A.B. Aleksandrov (see Peller [3]). Theorems 5.11 and 5.12 are due to Kislyakov. Note that Theorems 5.11 and 5.12 are consequences of stronger results found in the paper Kislyakov [1] in which the author obtained lower and upper estimates for the norms of projections onto the subspace of bounded Hankel matrices with {aj+k }j,k≥0 with am = 0 for m > n. Corollary 5.10 was found in Peller [3]. Note that in the case α = β > 0 this was proved earlier [1]. Moreover, in Horowitz it was shown in Horowitz [1] that for any F ∈ B1−2α + there exist ϕ, ψ ∈ B2−α + such that F = ϕψ. However, the technique of Horowitz does not work in the case β = α. Theorem 6.1 was obtained in Peller [3] for 1 ≤ p < ∞. This was the first result in rational approximation in which the “direct theorems” match the

298


“inverse theorems”. There were many earlier results in which there were gaps between the “direct theorems” and the “inverse theorems” (i.e., between necessary conditions and sufficient conditions). We mention here the results of Gonchar [1], Dolzhenko [1] and [2], and Brudnyi [1]. In particular, in Dolzhenko [1] it was shown that if d∞ n (ϕ) < ∞, n≥0

then ϕ ∈ L . Clearly, Theorem 6.1 for p = 1 improves the Dolzhenko theorem, since L∞ ⊂ BM O and B11 ⊂ {ϕ : ϕ ∈ L1 }. Theorem 6.1 for 0 < p < 1 was obtained in Peller [8], Pekarskii [2], and Semmes [1]; all three papers used quite different methods. Later Pekarskii [3] obtained similar results on rational approximation in the norm of Lp , 0 < p ≤ ∞. Note that similar results for rational approximation of functions on a finite interval were obtained in Peller [6]. Theorem 6.4 is taken from Peller [8]. Theorem 6.7 for q < ∞ is an immediate consequence of the above results by Peller, Semmes, and Pekarskii. We also mention here the paper Dyn’kin [3] in which a new approach to such Bernstein–Nikol’skii type inequalities is given as well as new Bernstein–Nikol’skii type inequalities were found. Theorem 6.7 for q = ∞ is due to Semmes [1]. Theorem 6.8 was obtained by Semmes [1]. Grigoryan’s theorem (inequality (6.1)) was established in Grigoryan [1]. Its improvement (Theorem 6.3) was found in Peller [8]. Note that the best possible constant in (6.1) was given by Pekarskii [1]. Theorems 7.1 and 7.2 were proved in Peller [3] and [8] (the equivalence (ii)⇔(iii) is due to Clark [2]). Theorem 7.3 was found in Peller [4]. Theorem 7.4 is due to Peller [3] for 1 ≤ p < ∞, and Peller [8] and Semmes [1] for 0 < p < 1. Theorem 7.5 was established by Rochberg [2] for p ≥ 1 and later by Peller [16] for 0 < p < ∞ (Rochberg’s method works only for p ≥ 1). Theorems 8.1 and 8.2 are taken from Peller [4]. A version of Theorem 8.5 for integral operators on L2 (R) can be found in Janson and Peetre [2]. Volberg informed the author that he also proved Theorem 8.5 but never published his proof. The proof given in §8 was found by the author. Theorem 8.9 for p ≥ 1 was obtained in Rochberg [2] and Peller [4], for p < 1 in Semmes [1] and Peller [8]. The idea to use the interpolation theorem for analytic families of operators is due to Rochberg. The results of §9 can be found in Peller [3], [4], and [8]. We mention here several related results and several applications of the results of this chapter. Jawerth and Milman [1] used extrapolation technique to deduce from Theorem 2.1 a description of Hankel operators in the Matsaev ideal S ω = S 1,ω . Recently Gheorghe [1] studied the problem of when a Hankel operator Hϕ belongs to operator ideals S E , where E is a monotone Riesz–Fischer space (see the definition in Gheorghe [1]). Such ideals S E include the Schatten– von Neumann classes S p as well as the Schatten–Lorentz classes S p,q . Using 1

Concluding Remarks

299

an approach different from the approach given in §4, Gheorghe shows that Hϕ ∈ S E if and only if P− ϕ belongs to the generalized Besov space BE defined in terms of the monotone Riesz–Fischer space E. Bonsall [3] found the following sufficient conditions for {aj+k }j,k≥0 to belong to S 1 : ∞ If lim an = 0 and |an−1 − an | log n < ∞, then {aj+k }j,k≥0 ∈ S 1 ; n→∞

if

lim an

n→∞

n=2

=

0 and

∞

|an−1 − 2an + an+1 |n2

0, Bps , 0 1/p, and on the space F1 of functions with absolutely converging Fourier series. In §8 we describe the arguments of unimodular functions in a broad class of function spaces X. We show that under certain mild conditions on X a unimodular function u belongs to X if and only if it can be represented in the form u = z n eiϕ , where n ∈ Z and ϕ is a real function in X. In §9 we study properties of Schmidt functions of Hankel operators with symbols in a function space X. We prove that under certain natural conditions on X all Schmidt functions must also belong to the same space X. We also describe the Schmidt functions of the compact Hankel operators and the Schmidt functions of the Hankel operators of class S p , 0 < p < ∞. In Sections 10 and 11 we study the continuity problem for the operators A and Am . We prove in §10 that unless ϕ ∈ H ∞ , ϕ is a discontinuity point of the operator A in the sup-norm. We also obtain a similar result for the operators Am . In §11 we consider the continuity problem in the norm of a decent function space X (see §2). We show that A is continuous at ϕ ∈ X in the norm of X if and only if Hϕ is a singular value of Hϕ of multiplicity 1. We also show that in the case of Am (under certain restrictions) ϕ is a continuity point in the norm of X if and only if the singular value sm (Hϕ ) of Hϕ has multiplicity 1. Finally, we show that independently of the multiplicity of sm (Hϕ ) the operator Am is continuous at ϕ as a function from X to Lq , q < ∞. In §12 we consider a recovery problem in spaces of measures. We use the results of §1 to prove that if µ is a complex measure on T such that |ˆ µ(j)|2 j0

j

< ∞.

306

Chapter 7. Best Analytic and Meromorphic Approximation

We obtain similar results for some other spaces of measures. As a corollary 1/p we describe closed subsets E of T such that Bp + E = C(E), where 1 < p < ∞. We conclude this chapter with §13 in which we obtain an analog of the 1/p Fefferman–Stein decomposition for the Besov spaces Bp .

1. Function Spaces That Can Be Described in Terms of Rational Approximation in BM O In this section we introduce the so-called R-spaces, i.e., function spaces that can be described in terms of rational approximation in BM O. We prove that if X is an R-space, then the operator A of best approximation by analytic functions maps X into itself. Moreover, we show that if X is a Banach (or quasi-Banach) R-space, then the operator A is bounded on X. We consider here the following seminorm on BM O: def

ϕ =

inf

f ∈BM OA

ϕ − f L∞ +

inf

f ∈BM OA

ϕ¯ − f L∞ = Hϕ + Hϕ¯ . (1.1)

To define R-spaces, we need several definitions. Recall that Rn is the set of rational functions with poles outside T of degree at most n. For ϕ ∈ BM O we put def

ρn (ϕ) = inf{ϕ − r : r ∈ Rn }. othe space if with any A linear space E of sequences {xj }j≥0 is called a K¨ sequence {xj }j≥0 it contains all sequences {yj }j≥0 satisfying |yj | ≤ |xj |, j ≥ 0. We say that E is a Banach K¨ othe space (quasi-Banach K¨ othe space) if in addition to that E is a Banach (quasi-Banach) space equipped with a norm (quasinorm) · E that has the following property: {xj }j≥0 ∈ E,

|yj | ≤ |xj |, j ≥ 0,

=⇒

{yj }j≥0 E ≤ {xj }j≥0 E .

Definition. A class X of functions on T is said to admit a description in terms of rational approximation in the BM O norm if X ⊂ V M O and there exists a Köthe sequence space E such that f ∈X

⇐⇒

{ρn (f )}n≥0 ∈ E.

For brevity we call such spaces R-spaces. Finally, we say X is a Banach (quasi-Banach) R-space if the corresponding space E is a Banach (quasiBanach) ideal space and C1 inf f − cX ≤ {ρn (f )}n≥0 E ≤ C2 inf f − cX , c∈C

for some positive constants C1 , C2 .

c∈C

f ∈ X,

1. R-spaces

307

It is easy to see that V M O is an R-space. Indeed, it corresponds to the def space E = c0 = {xn }n≥0 : lim xn = 0 . Other important examples of n→∞ 1/p

R-spaces are the Besov spaces Bp , 0 < p < ∞. In this case E = p . This follows from Corollaries 6.1.2, 6.2.2, and 6.3.2, since by Theorem 4.1.1, for any ϕ ∈ BM O ρn (P− ϕ) = sn (Hϕ ),

n ≥ 0.

(1.2)

The (nonlinear) set Rn , n ≥ 0, is also an R-space. The corresponding sequence space E is the space of sequences {xj }j≥0 such that xj = 0 for j ≥ m. The space R of rational functions with poles outside T is also an R-space in which case E is the space of finitely supported sequences. Let us briefly mention some properties of R-spaces. Let X be an R-space. Clearly, ϕ ∈ X if and only if ϕ¯ ∈ X. It is obvious that P+ X ⊂ X. If X is an R-space, E is the corresponding K¨ othe sequence space, ϕ ∈ BM O, and P+ f = O, then it follows from (1.2) that ϕ ∈ X if and only if {sn (Hϕ )}n≥0 ∈ E. Hence, if X is a linear R-space, then a function ϕ ∈ BM O belongs to X if and only if {sn (Hϕ )}n≥0 ∈ E and {sn (Hϕ¯ )}n≥0 ∈ E. Lemma 1.1. Suppose that X is an R-space. The following assertions hold: (i) if f ∈ H ∞ and ϕ ∈ X, then P+ f¯ϕ ∈ X; (ii) if X is a linear R-space, then X ∩ L∞ is an algebra. If X is a (quasi-)Banach R-space, then P+ f¯ϕ ≤ const f ∞ · ϕX , f ∈ H ∞ , ϕ ∈ X, and ϕψX ≤ const(ϕX · ψ∞ + ψX · ϕ∞ ), ϕ, ψ ∈ X ∩ L∞ . Proof. (i). Since obviously P+ f¯ϕ = P+ f¯P+ ϕ, we may assume that def ϕ = P+ ϕ. Let r ∈ Rn . It is easy to see that the function r# = P+ f¯r is also a rational function of class Rn (this function can be computed explicitly in the same way as in the Remark at the end of §3 of Ch. 1). It follows that P+ f¯ϕ − r# = P+ f¯(ϕ − r) = P+ f¯(ϕ − P+ r) ≤ f ∞ ϕ − P+ r . The last inequality is an obvious consequence of the following trivial inequality for the seminorm · defined in (1.1): P+ f¯g ≤ f ∞ g , f ∈ H ∞ , g ∈ BM OA. Therefore ρn (P+ f¯ϕ) ≤ f ∞ ϕ − P+ r ≤ f ∞ ϕ − r , and so ρn (P+ f¯ϕ) ≤ f ∞ ρn (ϕ), which implies that P+ f¯ϕ ∈ X. (ii) Let E be the corresponding K¨ othe sequence space. Let ϕ, ψ ∈ X∩L∞ . Clearly, it is sufficient to prove that P− (ϕψ) ∈ X. Let f ∈ H 2 . We have Hϕψ f = P− ϕψf = P− ϕP+ ψf + P− ϕP− ψf = Hϕ Tψ f + T˘ϕ Hψ f,

308


2 2 2 where T˘ϕ : H− → H− , T˘ϕ g = P− ϕg, g ∈ H− . Clearly, Tψ and T˘ϕ are bounded operators, and so sn (Hϕ Tψ ) ≤ Tψ sn (Hϕ ) and sn (T˘ϕ Hψ ) ≤ T˘ϕ sn (Hψ ). This implies that P− (ϕψ) ∈ X. It is easy to see that if X is a (quasi-)Banach R-space, then the above reasonings yield the desired norm estimates. Another important property of Banach (or quasi-Banach) R-spaces is that they must be M¨ obius invariant, i.e., conformally invariant modulo constants. Lemma 1.2. Let X be a Banach (or quasi-Banach) R-space. Then there is a positive number K such that

inf ϕ ◦ γ − cX ≤ inf ϕ − cX

c∈C

c∈C

(1.3)

for any ϕ ∈ X and any conformal mapping γ : C → C such that γ(T) = T. Proof. The result follows from the fact that (1.3) holds for X = BM O (this follows immediately from the definition (1.1) of the semi-norm · ), and from the obvious facts that Rn + const = Rn and γRn = Rn . Note, however, that the above conditions are not sufficient for a function space X to be an R-space. We shall see in §4 that the space {ϕ : (P+ ϕ) ∈ H 1 , (P+ ϕ) ¯ ∈ H 1 }, which has all the above properties, is not an R-space. Remark. It can easily be deduced from Lemma 1.2 that if X is a Banach (or quasi-Banach) R-space, then X ⊂ Lipω for any modulus of continuity ω. To prove this it is sufficient to consider conformal mappings γ, γ(ζ) = (ζ − α)(1 − α ¯ ζ)−1 , α ∈ D, and take α close to T. Now we are ready to solve the best approximation problem and the recovery problem for unimodular functions. Theorem 1.3. Let u be a unimodular function on T such that P− u ∈ V M O. Suppose that Tu has dense range in H 2 . We have (1) ρn (P+ u) ≤ ρn (P− u), n ∈ Z+ ; (2) if X is an R-space and P− u ∈ X, then P+ u ∈ X; (3) if X is a (quasi-)Banach R-space and P− u ∈ X, then P+ uX ≤ const P− uX . Proof. Clearly, P− u ∈ V M O, and so the Hankel operator Hu is compact. It follows from Corollary 4.4.4 that sn (Hu¯ ) ≤ sn (Hu ), n ∈ Z+ . Therefore by (1.2) ¯) ≤ ρn (P− u). ρn (P+ u) = ρn (P− u Clearly, (2) and (3) follow immediately from (1). Remark. It follows immediately from Corollary 4.4.4 that if the operator Tu is invertible on H 2 , then ρn (P− u) = ρn (P+ u),

n ∈ Z+ .

1. R-spaces

309

For linear R-spaces we can solve another version of the recovery problem. Theorem 1.4. Let X be a linear R-space and let u be a unimodular function on T such that P− u ∈ X. Suppose that the Toeplitz operator Tu on H 2 is Fredholm. Then P+ u ∈ X. Proof. Let n be an integer such that the Toeplitz operator Tzn u is invertible. It follows from Lemma 1.1 that P− z n u ∈ X. By Theorem 1.3, we have P+ z n u ∈ X, and so z n u ∈ X. Again, by Lemma 1.1, u ∈ X. Recall that Tu is Fredholm if u is a unimodular function in V M O. Theorem 1.5. Let ϕ ∈ V M O. Then ρn (Aϕ) ≤ ρn (ϕ), n ∈ Z+ . Moreover, if P− ϕ = O, then ρn (Aϕ) ≤ ρn+1 (ϕ), n ∈ Z+ . Proof. It is easy to see that to prove the desired inequality we may assume that P+ ϕ = O. By Corollary 1.1.5, the function ϕ − Aϕ has the form ¯ ϕ − Aϕ = c¯ z ϑ¯h/h, where c ∈ C, ϑ is an inner function, and h is an outer function in H 2 . Let def ¯ u = z¯ϑ¯h/h. Clearly, ϕ = cP− u and Aϕ = cP+ u. By Theorem 4.4.10, the operator Tu has dense range in H 2 . Clearly, Ker Tu = {O} since h ∈ Ker Tu . It now follows from Corollary 4.4.4 and (1.2) that ¯) = sn (Hu¯ ) ≤ sn+1 (Hu ) = ρn+1 (P− u), ρn (P+ u

n ∈ Z+ ,

which implies the result. Theorem 1.6. If X is an R-space, then AX ⊂ X. If X is a Banach (quasi-Banach) R-space, then A is a bounded operator on X, i.e., AϕX ≤ const ϕX ,

ϕ ∈ X.

Proof. If X is a R-space, then AX ⊂ X by Theorem 1.5. Let us show that if X is a Banach (quasi-Banach) R-space, then A is a bounded operator on X. To estimate, AϕX it is sufficient to assume that P+ ϕ = O. It is easy to see that ˆ inf ϕ − cX + ϕ(0)1 X,

c∈C

ϕ ∈ X,

is an equivalent (quasi-)norm on X. (Recall that 1 is the constant function identically equal to 1.) It follows immediately from Theorem 1.5 and the definition of a Banach (quasi-Banach) R-space that inf Aϕ − cX ≤ const ϕX .

c∈C

On the other hand, ) ) |Aϕ(0)| = |ϕ(0) ˆ − Aϕ(0)| ≤ ϕ − AϕL∞ = Hϕ = s0 (Hϕ ). Hence,

) Aϕ(0)1 X ≤ const s0 (Hϕ )1X ≤ const ϕX . 1/p

As we have already noticed, the spaces Bp , 0 < p < ∞, and V M O are R-spaces, and so Theorems 1.3, 1.4, and 1.6 hold for these spaces. We can obtain the following interesting consequence of the above results.

310

Chapter 7. Best Analytic and Meromorphic Approximation 1/p

Corollary 1.7. Let X = Bp , 1 < p < ∞, or X = V M O. Then there exists a function g ∈ X H 2 such that f − g ∈ L∞ . Proof. Clearly, it is sufficient to put g = Af . 1/p Note that for p ≤ 1 the class Bp is contained in L∞ , and so the above corollary is trivial for such spaces. Let us obtain a corollary of the above results for rational functions. Corollary 1.8. Let f ∈ Rn , n > 0. Then Af ∈ Rn . If, in addition, P− f =

O, then Af ∈ Rn−1 . Proof. The result follows immediately from Theorem 1.5. Corollary 1.8 allows us to obtain an application to the Nevanlinna–Pick interpolation problem. Suppose that {λ1 , · · · , λm } are distinct points of D, {d1 , · · · , dm } are positive integers, and {αjk : 1 ≤ j ≤ m, 0 ≤ k < dj } is a family of complex numbers. We are interested in finding a function f ∈ H ∞ of minimal norm such that f (k) (λj ) = αjk ,

1 ≤ j ≤ m,

0 ≤ k < dj .

(1.4)

Corollary 1.9. Let f be the solution of the interpolation problem (1.4) of minimal norm in H ∞ . Then f = cB, where c ∈ C and B is a Blaschke product of class Rd1 +···+dm −1 . Proof. Lagrange’s interpolation formula gives us a polynomial P of degree at most d1 + · · · + dm − 1 that solves the interpolation problem (1.4). Let B0 be the finite Blaschke product that has at λj a zero of multiplicity dj , 1 ≤ j ≤ m. Then B0 ∈ Rd1 +···+dm . The general solution of (1.4) is f = P + B0 h, h ∈ H ∞ . Clearly, inf P − B0 h∞ = inf∞ B0 P − h∞ .

h∈H ∞

h∈H

(1.5)

We have B0 P ∈ R. Let h be an H ∞ function that minimizes the infimum in (1.5). By Corollary 1.8, h = A(B0 P ) ∈ R. It is easy to see that P − B0 h = cB, where c ∈ C and B is a finite Blaschke product. Consider the Toeplitz operator TBB0 on H 2 . It follows from (1.5) that distL∞ (BB0 , H ∞ ) = 1, and so the kernel of TBB0 is nontrivial (see Theorem 3.1.11). Let µ be the degree of B (the number of zeros of B counted with multiplicities). We have dim Ker TBB0 = ind TB0 TB = ind TB0 + ind TB = d1 + · · · + dm − µ > 0. It follows that B ∈ Rd1 +···+dm −1 . We can consider now the behavior of the operators Am , m ≥ 0, on R-spaces. First, we prove that Am Rk ⊂ Rk . ∞ Theorem 1.10. Let m, k ∈ Z+ and m < k. If ϕ ∈ Rk \ H(m) , then Am ϕ ∈ Rk−1 .

1. R-spaces

311

Proof. Suppose that the singular value sk (Hϕ ) of Hϕ has multiplicity µ and sd (Hϕ ) = · · · = sd+µ−1 (Hϕ ) > sd+µ (Hϕ ),

d ≤ m ≤ d + µ − 1.

def

Clearly, Am ϕ = Ad ϕ and r = P− Am ϕ is a rational function of degree d. Put g = P+ Am ϕ. It is easy to see that g = A(ϕ − r), and so by Corollary 1.8, g is rational. def Consider the function u = ϕ − Am ϕ = ϕ − r − A(ϕ − r). Then u has constant modulus and dim Ker Tu = ind Tu = 2d + µ (see Theorem 4.1.7). Without loss of generality we may assume that the function u is unimodular. Since u is rational, u has the form B1 B2 , where B1 and B2 are finite Blaschke products with disjoint zeros. It is easy to see that deg P− u = deg B1 and deg P+ u = deg B2 . We have ind Tu = ind TB1 +ind TB2 = deg B1 −deg B2 = deg P− u−deg P+ u = 2d+µ. Clearly, deg P− u = deg P− (ϕ − r) ≤ k + d. Hence, deg g

deg P+ u = deg P− u − 2d − µ ≤ deg P− ϕ + deg r − 2d − µ ≤ k + d − 2d − µ = k − d − µ, =

and so deg Am ϕ = deg r + deg g ≤ d + k − d − µ = k − µ ≤ k − 1. Theorem 1.11. Let X be a linear R-space and let m ∈ Z+ . Then Am X ⊂ X. If X is a Banach (or quasi-Banach) R-space, then the operators Am are uniformly bounded on X, i.e., Am ϕX ≤ CϕX ,

ϕ ∈ X,

for a constant C, which does not depend on m. Proof. Let ϕ ∈ X and let rm be the unique function in Rm such that P+ rm = O and Hϕ−rm = sm (Hϕ ) (see Theorems 4.1.2 and 4.1.3). It is easy to see that Am ϕ = rm + A(ϕ − rm ).

(1.6)

It is also easy to see that if X is a linear R-space, then Rn ⊂ X for every n ∈ Z+ . It now follows from Theorem 1.6 that Am ϕ ∈ X. Suppose now that X is a Banach (or quasi-Banach) R-space. By (1.6) and Theorem 1.6, it is sufficient to show that rm X ≤ const ϕX . For 0 ≤ k ≤ m − 1 we have sk (Hrm )

= sk (Hrm −ϕ + Hϕ ) ≤ Hrm −ϕ + sk (Hϕ ) = sm (Hϕ ) + sk (Hϕ ) ≤ 2sk (Hϕ ).

Clearly, for k ≥ m we have sk (Hrm ) = 0.

312


Therefore rm X ≤ const ϕX .

2. Best Approximation in Decent Banach Algebras In this section we consider a class of so-called decent function spaces, i.e., a class of Banach algebras of functions on T that satisfy certain natural conditions (axioms (A1)–(A4)). We prove that the operator A of best approximation by analytic functions acts on decent function spaces and we solve the recovery problem for such Banach algebras. Note, however, that unlike the case of R-spaces the operator A does not have to be bounded on decent function spaces (see §5). In §4 we see examples of many classical decent Banach algebras. Let us introduce here the following notation. For a space X ⊂ L1 of functions on T we put def X+ = {f ∈ X : fˆ(j) = 0, j < 0},

def X− = {f ∈ X : fˆ(j) = 0, j ≥ 0}.

Definition. A space of functions X on T is called decent if X contains the set of trigonometric polynomials P, consists of continuous functions, and satisfies the following axioms: (A1) if f ∈ X, then f¯ ∈ X and P+ f ∈ X; (A2) X is a Banach algebra with respect to pointwise multiplication; (A3) for every ϕ ∈ X the Hankel operator Hϕ is a compact operator from X+ to X− ; (A4) if f ∈ X and f does not vanish on T, then 1/f ∈ X. Note that we do not assume here that the norm in X satisfies the inequality f gX ≤ f X · gX . It is sufficient to assume that f gX ≤ cf X · gX for some constant c. In this case one can introduce an equivalent norm in X for which c = 1. It is easy to see that if X satisfies (A1) and (A2), then for every ϕ ∈ X the Hankel operator Hϕ maps boundedly X+ to X− . Axiom (A3) says that this operator must be a compact operator from X+ to X− . Let us establish two sufficient conditions for (A3). The first condition is almost obvious. Theorem 2.1. Let X be a space of functions on T that satisfies (A1), (A2). Suppose that the trigonometric polynomials are dense in X. Then X satisfies (A3). Proof. Let ϕ ∈ X and let {ϕn } be a sequence of trigonometric polynomials that converges to ϕ. It follows immediately from (A1) and (A2) that

2. Best Approximation in Decent Banach Algebras

313

the sequence Hϕn converges to Hϕ in the norm of the space of bounded operators from X+ to X− . The result follows from the fact that the operators Hϕn have finite rank. The second sufficient condition also works for nonseparable Banach algebras. Theorem 2.2. Let X be a Banach space of functions on T that satisfies (A1) and (A2). Suppose that there exists a Banach space Y such that X+ ⊂ Y ⊂ H ∞ , the inclusion X+ → Y is compact, and Tf¯X+ ⊂ X+ for every f ∈ Y . Then X satisfies (A3). Proof. Let ϕ ∈ X. We want to show that the Hankel operator Hϕ is a compact operator from X+ to X− . Since Hϕ = HP− ϕ , we can assume without loss of generality that ϕ ∈ X− . Since Tf¯X+ ⊂ X+ for every f ∈ Y , it follows that Hϕ f ∈ X− for every f ∈ Y . By the closed graph theorem, Hϕ is a bounded operator from Y to X− . The result follows now from the fact that the inclusion X+ → Y is compact. In §4 we shall see examples of many classical nonseparable Banach spaces that satisfy the hypotheses of Theorem 2.2. Theorem 2.3. Let X be a decent space and let u be a unimodular function on T such that P− u ∈ X. Suppose that at least one of the following two conditions is satisfied: (1) Tu is Fredholm; (2) clos(Tu H 2 ) = H 2 . Then u ∈ X. Theorem 2.4. Let X be a decent space. Then AX ⊂ X. Proof of Theorem 2.3. Since X ⊂ C(T), we have P− u ∈ V M O. We have already solved in the previous section the recovery problem in the space V M O, and so each of the conditions (1) and (2) implies that u ∈ V M O. Then by Theorem 3.3.10 and Corollary 3.2.6, u admits a rep¯ where n ∈ Z and h is an outer function in H 2 such resentation u = z n h/h, that the Toeplitz operator Th/h is invertible on H 2 . By (A2) and (A3), ¯ zX ⊂ X, and so without loss of generality we may assume that n = −1. Then the space Ker Tu = {f ∈ H 2 : Hu f 2 = f 2 } = {λh : λ ∈ C} is one-dimensional. Since Hu = HP− u , it follows from (A3) that the Hankel operator Hu is a compact operator from X+ to X− . Similarly, the operator Hu∗ X− is a compact operator from X− to X+ . Consider the operator R on X+ defined by Rf = Hu∗ Hu f , f ∈ X+ . Then R is a compact operator on X+ . We have X+ ⊂ H 2 . We can naturally imbed the space H 2 in the dual ∗ as follows. Let g ∈ H 2 . We associate with it the linear functional space X+ J (g) on X+ defined by

f (ζ)g(ζ)dm(ζ). f → (f, g) = T

314


Note that J (λ1 g1 + λ2 g2 ) = λ1 J (g1 ) + λ2 J (g2 ), g1 , g2 ∈ H 2 , λ1 , λ2 ∈ C. ∗ The imbedding J allows us to consider H 2 as a subset of X+ . It is easy ∗ ∗ ∗ to see that Rf = Hu Hu f for f ∈ X+ and R g = Hu Hu g for any g ∈ H 2 . Hence, Ker(I − R) ⊂ Ker(I − Hu∗ Hu ) ⊂ Ker(I − R∗ ). Since R is a compact operator, it follows from the Riesz–Schauder theorem (see Yosida [1], Ch. X, §5) that dim Ker(I−R) = dim Ker(I−R∗ ). Therefore Ker(I − R) = Ker(I − Hu∗ Hu ) = Ker(I − R∗ ). Since we know that Ker(I − Hu∗ Hu ) = {λh : λ ∈ C}, it follows that h ∈ X+ . To complete the proof we have to show that h has no ¯ ∈ X. zeros on T, which would imply by (A1), (A2), and (A4) that u = z¯h/h Suppose that h(τ ) = 0 for some τ ∈ T. Consider the linear functional l on X+ defined by ¯ )(τ ), f ∈ X+ . (2.1) l(f ) = (P+ hf Lemma 2.5. R∗ l = l. Proof of Lemma 2.5. Let f ∈ X+ . We have ¯ ∗ Hu f ))(τ ) (R∗ l)(f ) = l(Rf ) = l(Hu∗ Hu f ) = (P+ (hH u by (2.1). Therefore (R∗ l)(f )

¯ + (¯ = (P+ (hP uHu f )))(τ ) ¯ uHu f ))(τ ) − (P+ (hP ¯ − (¯ uHu f )))(τ ). = (P+ (h¯ 2 ¯ −u ¯ −u Since hP ¯ H u f ∈ H− , it follows that P+ (hP ¯Hu f ) = O. Hence, (R∗ l)(f )

=

(P+ (huHu f ))(τ )

=

(P+ (P− (hu)Hu f ))(τ ) + (P+ (P+ (hu)Hu f ))(τ ).

2 Clearly, P+ (hu)Hu f ∈ H− , and so P+ (P+ (hu)Hu f ) = O. We have

(R∗ l)(f )

=

(P+ (Hu hHu f ))(τ ) = (P+ (Hu hP− (uf )))(τ )

=

(P+ (Hu h(uf )))(τ ) − (P+ (Hu hP+ (uf )))(τ ).

Since Hu hP+ (uf ) ∈ H 2 , it follows that (P+ (Hu hP+ (uf )))(τ ) = (Hu h)(τ )(P+ (uf ))(τ ) = 0 ¯ ) = τ¯h(τ ) = 0 by the assumption. Since since (Hu h)(τ ) = (P− z¯h)(τ ¯ u = z¯h/h, we have ¯ ))(τ ) = l(f ). (R∗ l)(f ) = (P+ (Hu h(uf )))(τ ) = (P+ (zhuf ))(τ ) = (P+ (hf Let us now complete the proof of Theorem 2.3. We prove that under the assumption h(τ ) = 0 the function h(1 − τ¯z)−1 , which is analytic in D, must belong to X+ and h h R = . 1 − τ¯z 1 − τ¯z

2. Best Approximation in Decent Banach Algebras

315

We have already proved that Ker(I − R∗ ) ⊂ X+ (here we identify H 2 with a subset of X ∗ ). Therefore the functional l defined by (2.1) must be of the form l(f ) = (f, ψ), where ψ ∈ X+ and Rψ = ψ. Let us show that ψ and h(1 − τ¯z)−1 have the same Taylor coefficients. Let n ∈ Z+ . We have ¯ )= ˆ ψ(n) = l(z n ) = (P+ z n h)(τ

n

ˆ τ n−j . h(j)¯

j=0

On the other hand, h =h 1 − τ¯z

n≥0

τ¯n z n =

n≥0

  n ˆ τ n−j  z n .  h(j)¯ j=0

Clearly, the functions h and h(1 − τ¯z)−1 are linearly independent, which contradicts the fact that dim Ker(I − R) = 1. Proof of Theorem 2.4. Suppose that ϕ ∈ X and P− ϕ = O. Then the function u = Hϕ −1 (ϕ − Aϕ) is unimodular and P− u ∈ X. Since the Toeplitz operator Tu has dense range in H 2 (see the proof of Theorem 1.5), it follows from Theorem 2.3 that u ∈ X. Hence, Aϕ ∈ X. Remark. As in §1 it is easy to prove that if X is a decent function space, then Am X ⊂ X for every m ∈ Z+ . We complete this section with the description of the space of maximal ideals of the Banach algebras X+ and X for decent function spaces X. In other words, we describe all complex homomorphisms on X and X+ . Recall that a complex homomorphism on a Banach algebra is, by definition, a nonzero multiplicative linear functional. If the trigonometric polynomials are dense in X, it is almost obvious that the maximal ideal space of X can be identified naturally with T while the maximal ideal space of X+ can be identified with the closed unit disk. We prove the same result for decent Banach algebras X. Theorem 2.6. Let X be a decent space. Then each complex homomorphism on X has the form f → f (ζ) for some ζ ∈ T, while each complex homomorphism on X+ has the form f → f (ζ) for some ζ ∈ clos D. Proof. Let ω be a complex homomorphism on X that is not a point evaluation at a point of T. Then there are functions f1 , · · · , fn in X such that n inf |fk (ζ)|2 > 0, ω(fk ) = 0, k = 1, · · · , n. ζ∈T

Put f =

n k=1

k=1

|fk |2 ∈ X. Clearly, ω(f ) =

n k=1

not invertible in X, which contradicts (A4).

ω(f¯k )ω(fk ) = 0. Hence, f is

316


Now let ω be a complex homomorphism on X+ that is not of the form f → f (ζ), ζ ∈ clos D. Then there are functions f1 , · · · , fn in X+ such that inf

ζ∈clos D

n

|fk (ζ)|2 > 0,

ω(fk ) = 0, k = 1, · · · , n.

k=1

def

Since X+ ⊂ CA = C(T) ∩ H 2 and all complex homomorphisms on CA are point evaluations at points of clos D, there exist g1 , · · · , gn in CA such that n fk gk = 1. Clearly, we can find analytic polynomials q1 , · · · , qn such that k=1

n inf fk (ζ)qk (ζ) > 0. ζ∈clos D k=1

By (A4), the function h def

ϕk = hqk ∈ X+ . Clearly,

n k=1

n

def

=

−1 fk qk

belongs to X+ . Put

k=1

ϕk fk = 1, and so ω(1) =

which is impossible since ω(1) = 1.

n

ω(ϕk )ω(fk ) = 0,

k=1

3. Best Approximation in Spaces without a Norm The class of decent spaces introduced in the previous section covers a broad class of function spaces (see §4, where many examples of classical functions spaces satisfying (A1)–(A4) are given). However, there are many locally convex function spaces for which it is important to study hereditary properties. In this section we give an approach that allows us to treat such spaces. We introduce the axioms (B1)–(B3), which do not assume any norm and we show that if X is a function space satisfying (B1)–(B3), then AX ⊂ X. In §4 we shall see that many Carleman classes satisfy (B1)–(B3). We consider here linear spaces X (at this point we do not assume any topology on X) that satisfy the following system of axioms: (B1) X is an algebra with respect to pointwise multiplication which contains the trigonometric polynomials and such that if f ∈ X, then f¯ ∈ X and P+ f ∈ X; (B2) there exists a decent function space Y such that X ⊂ Y and Tf¯X+ ⊂ X+ for every f ∈ Y+ ; (B3) if f ∈ X, and inf |f (τ )| > 0, then f −1 ∈ X. τ ∈T

Lemma 3.1. Suppose that X satisfies (B1) and (B2). Let h be an ¯ outer function such that h, h−1 ∈ H 2 . Then P− h/h ∈ X if and only if 2 |h| ∈ X and inf |h(τ )| > 0. τ ∈T

3. Best Approximation in Spaces without a Norm

317

¯ ∈ X. Since h, h−1 ∈ H 2 , it follows that Proof. Suppose that P− h/h 2 2 clos Th/h ¯ H = H (see Theorem 4.4.10). Let Y be the space from (B2). ¯ ∈ X ⊂ Y , it follows from Theorem 2.2 that h/h ¯ ∈ Y . MoreSince P− h/h over, it was shown in the proof of Theorem 2.2 that inf |h(τ )| > 0, and so τ ∈T

h−1 ∈ Y . Let us prove that |h|2 ∈ X. ¯ ∈ X. We have Since zP− z¯g = P+ g¯ for any g ∈ L2 , it follows that P+ h/h h ¯2 ¯ 2 P+ h = T¯ 2 P+ h ∈ X = P+ h P+ |h|2 = P+ ¯ h h ¯ ¯ h h h by (B2). It follows easily from (B1) that P+ z|h|2 ∈ X. Since |h|2 is real, we have P− |h|2 = z¯P+ z|h|2 ∈ X. Suppose now that |h|2 ∈ X and inf |h(τ )| > 0. Then τ ∈T

¯ = P+ 1 |h|2 = T¯ −2 P+ |h|2 ∈ X, P+ h/h h ¯2 h |h|2 ), |h|2 ∈ Y , and Y satisfies (A1), (A2), since h−2 = c|h|− 2 exp(−i log and (A4). Now we can obtain analogs of the results of §2 for spaces satisfying the axioms (B1)–(B3). Theorem 3.2. Let X be a space satisfying (B1)–(B3) and let u be a unimodular function on T such that P− u ∈ X. Suppose that at least one of the following two conditions is satisfied: (1) Tu is Fredholm; (2) clos(Tu H 2 ) = H 2 . Then u ∈ X. Theorem 3.3. Let X be a space satisfying (B1)–(B3). Then AX ⊂ X. Proof of Theorem 3.2. As in Theorem 2.2 each of the conditions (1) and (2) implies that u ∈ V M O. Again by Theorem 3.3.10 and Corollary ¯ 3.2.6, u admits a representation u = z n h/h, where n ∈ Z and h is an outer function in H 2 such that 1/h ∈ H 2 and the Toeplitz operator Th/h ¯ ¯ ∈ X. By Lemma 3.1, we conclude that is invertible on H 2 . Clearly, P− h/h |h|2 ∈ X and inf τ ∈T |h(t)| > 0. It follows from (B3) that |h|−2 ∈ X. Since X ⊂ C(T), we have inf τ ∈T |h(t)|−2 > 0. We can now apply Lemma 3.1 to h−1 and we find that ¯ = P− h−1 /h−1 ∈ X. P− h/h ¯ ∈ X, and so u = z n h/h ¯ ∈ X. . It follows that P+ h/h Theorem 3.3 can be deduced from Theorem 3.2 in the same way as Theorem 2.4 has been deduced from Theorem 2.3 in §2. Remark. As in §1 and §2 it is easy to see that if X satisfies (B1)–(B3), then Am X ⊂ X for any m ∈ Z+ .

318


4. Examples and Counterexamples In this section we give examples of many classical function spaces that are hereditary for the operator A of best approximation by analytic functions as well as examples of spaces that are not hereditary.

R-Spaces 1/p

As we have already noticed in §1, the Besov spaces Bp , 0 < p < ∞, 1/p and the space V M O are R-spaces. Moreover, the spaces V M O and Bp , 1/p 1 ≤ p < ∞, are Banach R-spaces, while the spaces Bp , 0 < p < 1, are 1/p 1/p quasi-Banach R-spaces. It is also easy to see that the spaces Bp0 0,Bp1 1 θ,q considered in §6.4 are also R-spaces. Hence, the operators Am are uniformly bounded on such spaces (see Theorem 1.11). Consider now the space {f ∈ L1 : (P+ f ) ∈ H 1 , (P+ f¯) ∈ H 1 }. As we have mentioned in §1, this space satisfies the necessary conditions to be an R-space, which are given in §1. Let us show now that it is not an R-space. Clearly, it suffices to establish the following result. Theorem 4.1. There is no K¨ othe sequence space E such that ϕ ∈ H 1

⇐⇒

{sn (Γϕ )}n≥0 ∈ E.

Recall that for a function ϕ analytic in D the Hankel operator Γϕ on 2 is the operator with Hankel matrix {ϕ(j ˆ + k)}j,k≥0 . We are going to use results of §6.5 on properties of the averaging projection P onto the space of Hankel matrices. Recall that {ϕ : ϕ ∈ H 1 } = PS 1 .

(4.1)

Let us introduce the following notation. For a sequence {an }n≥0 of complex numbers we denote by {a∗n }n≥0 the nonincreasing rearrangement of the sequence {|an |}n≥0 . We need the following lemma. Lemma 4.2. There exist a subspace L of the space of bounded Hankel operators and a one-to-one mapping V of L onto ∞ such that c1 (VT )∗n ≤ sn (T ) ≤ c2 (VT )∗n ,

T ∈ S ∞ ∩ L,

for some positive constants c1 and c2 . Proof. It is more convenient to work with the Hankel operators Hψ . Let B be an interpolating Blaschke product with zeros {ζn }n≥0 . Define L = {Hf B : f ∈ H ∞ }. It is easy to see that Hf1 B = Hf2 B for f1 , f2 ∈ H ∞ if and only if f1 (ζj ) = f2 (ζj ), j ≥ 0. We define the operator V : L → ∞ by VHf B = {f (ζn )}n≥0 ,

f ∈ H ∞.

Clearly, V is well defined. Since B is an interpolating Blaschke product, the map V is one-to-one and onto.

4. Examples and Counterexamples

319

Consider now the model operator SB (see §1.2). Put gn (z) = (1 − |ζn |2 )1/2

B(z) , z − ζn

n ≥ 0.

As we have noticed in the discussion preceding Theorem 1.5.11, the functions gn form an unconditional basis in the space KB = H 2 BH 2 and f (SB )gn = f (ζn )gn , f ∈ H ∞ , n ≥ 0. The result follows now from the equality sn (Hf B ) = sn (f (SB )), n ≥ 0, (see (1.2.9)). Proof of Theorem 4.1. Suppose that such a K¨ othe space E exists. Consider the space {Γϕ ∈ L : ϕ ∈ H 1 }. By Lemma 4.2, the operator V maps this space one-to-one onto the space {{an }n≥0 : {a∗n }n≥0 ∈ E}. This allows us to introduce on this sequence space a norm that makes it a separable Banach space isomorphic to {Γϕ ∈ L : ϕ ∈ H 1 }. Hence, the space def

S = {T ∈ B(2 ) : {sn (T )}n≥0 ∈ E} is also a separable Banach space (see Gohberg and Krein [2], Ch. III, §3). Let us show that S ⊂ S 1,2 . By Lemma 4.2 it is sufficient to show that T ∈L∩S

=⇒

T ∈ S 1,2 .

However, this follows immediately from Theorem 6.5.3 and from (4.1). We have S ⊂ S 1,2 ⊂ (S 2,ω )∗ and S = S 2,ω , since S 1,2 is not a Banach space (see Appendix 1.1). Consider the dual space S ∗ , which can be identified with an ideal of compact operators on B(2 ) (see Gohberg and Krein [2], Ch. III, §12). Since PS 1 ⊂ S, it follows by duality that PS ∗ ⊂ B(2 ). However, S 2,ω S ∗ , which contradicts Theorem 6.5.7. We show in this section (Theorem 4.5) that {f ∈ L1 : (P+ f ) , (P+ f¯) ∈ H 1 } is a decent space, and so the operators Am , m ∈ Z+ , leave it invariant.

H¨ older Classes It is easy to see that the separable Hölder–Zygmund classes λα , 0 < α < ∞, are decent spaces (see Appendix 2.6). Hence, Aλα ⊂ λa . The fact that the nonseparable H¨ older–Zygmund classes Λα , 0 < α < ∞, are also decent spaces is more complicated. It is obvious that Λα satisfies (A1), (A2), and (A4). To prove that Λα satisfies (A3) we make use of Theorem 2.2. Theorem 4.3. Let 0 < α < ∞. Then Λα satisfies the axiom (A3). Proof. Let X = Λα . By Theorem 2.2, it is sufficient to show that Tf¯X+ ⊂ X+ for any f ∈ H ∞ and the identical inclusion X+ → H ∞ is compact.

320


Let us first show that for any function f in H ∞ the Toeplitz operator Tf¯ maps (Λα )+ into itself. Consider the space (B1−α )+ that consists of functions analytic in D and satisfying

|ϕ(ζ)|(1 − |ζ|)α−1 dm2 (ζ) < ∞. The dual space to the pairing

D −α ∗ (B1 )+

(ϕ, ψ) =

can be identified with the space (Λα )+ with respect

ˆ ϕ(n) ˆ ψ(n),

ϕ ∈ P A,

ψ ∈ (Λα )+ ,

(4.2)

n≥0

(see Appendix 2.6). It is easy to see that the operator Tf¯ is the adjoint of multiplication by f on (B1−α )∗+ , which is obviously bounded. Hence, Tf¯ is bounded on (Λα )+ . It remains to prove that the identical inclusion of (Λα )+ in H ∞ is compact. We prove a stronger result, which we will need later. Let 0 < β < α. We show that the identical inclusion of Λα in Λβ is compact. Let Wn , n ∈ Z, be the trigonometric polynomials defined in Appendix 2.6. Define the finite rank operator Rm : Λα → Λβ , m ∈ Z+ , by Rm f =

m

f ∗ Wn .

n=−m

It is easy to see that f − Rm f Λβ

≤

f ∗ Wn Λβ

|n|≥m

≤

2|n|β f ∗ Wn L∞ ≤ const f Λα

|n|≥m

(see |n|≥m

Appendix 2.6). The result 2|n|(β−α) → 0 as m → ∞.

2|n|(β−α)

|n|≥m

follows

from

the

fact

that

It follows from Theorem 4.3 that Am Λα ⊂ Λα , 0 < α < ∞.

The Space C ∞ The space C ∞ of infinitely differentiable functions is not a Banach space. However, it is easy to see that it is hereditary: Am C ∞ ⊂ C ∞ , m ∈ Z+ . Λα . This follows immediately from the equality C ∞ = α 1/p and q < ∞. Then the trigonometric α α and Bp,q satisfies the axioms (A1)–(A4) (see polynomials are dense in Bp,q


321

Appendix 2.6). The same is true if α = 1/p and q = 1 (see Appendix 2.6). α Therefore for such values of α, p, q the Besov classes Bp,q have the heredity α α property: Am Bp,q ⊂ Bp,q . α Consider now the case α > 1/p and q = ∞. The classes Bp,∞ are nonseparable. We are going to show that they are still decent spaces. It is easy α satisfy (A1), (A2), and (A4) (see Appendix to see that the spaces Bp,∞ 2.6). It remains to verify the axiom (A3). α Theorem 4.4. Let 1 ≤ p < ∞ and α > 1/p. The spaces Bp,∞ satisfy the axiom (A3). α Proof. First of all, Bp,∞ ⊂ Λβ for 0 < β < α − 1/p. (This follows s ⊂ L∞ for s > 1/p; see Appendix immediately from the fact that Bp,∞ 2.6.) As in the proof of Theorem 4.3, we prove that the Toeplitz operators α Tf¯ are bounded on (Bp,∞ )+ for f ∈ H ∞ . −α Consider the space (Bp ,1 )+ that consists of functions ϕ analytic in D and such that

1

(1 − r)α−1 0

T

1/p |ϕ(rζ)|p dm(ζ) dr < ∞.

∗ α Its dual space (Bp−α ,1 )+ can be identified with our space Bp,∞ with respect to the pairing (4.2) (see Appendix 2.6). Since obviously multiplication by f , f ∈ H ∞ , is bounded on (Bp−α ,1 )+ , it follows by duality that Tf¯ is bounded α on (Bp,∞ )+ . α )+ By Theorem 2.2, it remains to show that the identical inclusion of (Bp,∞ α in H ∞ is compact. Since Bp,∞ ⊂ Λβ for 0 < β < α − 1/p, the result follows from the compactness of the identical inclusion of (Λβ )+ in H ∞ , which has been established in the proof of Theorem 4.3. Note that here we have examined all Besov classes Bpα , 1 ≤ p ≤ ∞, α > 0. If α = 1/p and 1 ≤ p < ∞, the space Bpα is an R-space. If α > 1/p and 1 ≤ p ≤ ∞, the space Bpα is decent. However, if p < ∞ and α < 1/p, the space Bpα is not contained in BM O and the operator A is not defined on Bpα . Let us explain briefly why Bpα ⊂ BM O for α < 1/p. Let r > 1 be def

a number satisfying 1/r < δ = 1/p − α. Suppose that Bpα ⊂ BM O. Then Bpα ⊂ Lr . By the Hardy–Littlewood theorem (see Zygmund [1], Ch. XII, §9) the operator Iδ of fractional integration maps Lr to the space λδ−1/r . Since Iδ Bpα = Bpα+δ (see Appendix 2.6), we have Bp1/p = Iδ Bpα ⊂ Iδ Lr ⊂ λδ−1/r . 1/p

obius invariant However, this contradicts the fact that the space Bp is M¨ (see Lemma 1.2 and the Remark after it). Note that the space B11 is an R-space and at the same time it is a decent space.

322


Spaces of Bessel Potentials The spaces Lsp for 1 1/p are decent. Indeed, Lsp forms an algebra for the above choice of indices (see Appendix 2.6). The remaining axioms (A1), (A3), and (A4) are obvious. Therefore Am Lsp ⊂ Lsp for 1 1/p, and m ∈ Z+ .

Spaces of Functions Whose nth Derivatives Belong to V M O, BM O, CA + CA , H ∞ + H ∞ , H 1 + H 1 , or 41 L1 + L Let Z be one of the spaces V M O, BM O, CA + CA , H ∞ + H ∞ , H 1 + H 1 , 41 . Consider the following spaces: or L1 + L def

Z (n) = {f : f (n) ∈ Z},

n = 1, 2, · · · .

We can understand the nth derivative in the distributional sense. Theorem 4.5. Let Z be V M O, BM O, CA + CA , H ∞ + H ∞ , or H + H 1 and let n be a positive integer. Then Z (n) is a decent space. If 41 (n) is also a decent space. n ≥ 2, then the space L1 + L 41 (1) is not an algebra. Moreover, Note, however, that the space L1 + L 41 (1) ⊂ L∞ . this space is not contained in L∞ . Indeed, suppose that L1 + L 1

By the closed graph theorem this inclusion must be bounded. Consider the de la Vallée Poussin kernel Υn , n ∈ Z+ , i.e., Υn is the trigonometric polynomial defined by  1, |j| ≤ n,    |j|−n ˆ n (j) = (4.3) Υ 1 − n , n ≤ |j| ≤ 2n,    0, |j| ≥ 2n. It is easy to see that Υn = 2K2n −Kn , where Kn is the Fejér kernel. Hence, Υn L1 ≤ const, and so P+ Υn L1 +L1 ≤ const. It follows that ˆ Υ(j) j z ≤ const . (1) j≥1 j 1 L1 +L

However, it is easy to see that n ˆ 1 Υ(j) j ≥ z ≥ const · log(1 + n), j j≥1 j j=1

∞

(1)

41

⊂ L∞ . which proves that L1 + L To prove Theorem 4.5 we need the following lemma.


323

Lemma 4.6. Let n ∈ Z+ . The following assertions hold: (i) if f (n) ∈ H ∞ , then (Tz¯k f )(n) ≤ const 2n log(1 + k)f (k) , k > 0; ∞ ∞ (ii) if f (n) ∈ BM OA, then (Tz¯k f )(n)

BM O

≤ const 2n f (k) BM O ,

(iii) if f (n) ∈ H 1 , then (Tz¯k f )(n) ≤ const 2n log(1 + k)f (k) , 1 1 (iv) if f

(n)

∈ P+ L , then (Tz¯k f )(n)

k > 0;

k > 0;

1

≤ const 2n f (k) P+ L1 ,

k > 0. (k) Proof. For a positive integer k consider the sequence γj j≥0 defined by j (k) , j ≥ 0. γj = 1 − j+k (k) (k) Obviously, the sequence γj j≥0 is convex and γ0 = 1. By Polya’s theorem (see Zygmund, Ch. V, Th. 1.5), there exists a probability measure (k) ˆk (j) = γ|j| , j ∈ Z. Let δ be the unit mass at 1 ∈ T. µk on T such that µ Consider the n-fold convolution of the measure δ − µk and denote it by νkn . We have n |j| νˆkn (j) = , j ∈ Z, |j| + k and νkn M(T) ≤ 2n , where M(T) is the space of Borel measures on T. Let f (n) ∈ H ∞ . Then (n) nˆ j j f (j + k)z (Tz¯k f ) ∞ = j≥0 P+ L1

∞

n j nˆ j = (j + k) f (j + k)z j≥0 j + k

∞

≤

n nˆ j+k νk M(T) (j + k) f (j + k)z j≥0

∞

≤ const 2 log(1 + k)f n

(n)

∞ .

The last inequality is a consequence of the well-known estimate for the Dirichlet kernel: k j z ≤ const log(1 + k) j=−k 1 L

324


(see Zygmund [1], Ch. II, §12). To prove (ii) we use the same scheme and the following obvious inequality: Tz¯k f BM OA ≤ const f BM OA , f ∈ BM OA. The proof of (iii) is the same as the proof of (i) while the proof of (iv) is the same as the proof of (ii). We need one more lemma. 41 (1) ⊂ V M O. Lemma 4.7. L1 + L 41 (1) ⊂ BM O. Clearly, it is sufProof. Let us first prove that L1 + L ficient to show that if f is an analytic function in D such that f ∈ P+ L1 , then f ∈ BM OA. Then there exists a function ψ such that ψ ∈ L1 and P+ ψ = f . Since such a function ψ must be in L∞ , it follows that f ∈ BM OA. To complete the proof, it remains to observe that the set of trigonometric 41 (1) . polynomials is dense in L1 + L Proof of Theorem 4.5. (A1) is obvious. To verify that Z (n) is an algebra, it is sufficient to show that (n)

f, g ∈ Z+

=⇒

f g ∈ Z (n)

and P+ f¯g ∈ Z (n) .

(n)

Suppose that f, g ∈ Z+ . Let us first prove that f g ∈ Z (n) . We use the formula n n f (n−1) g + · · · + f g (n−1) + f g (n) . (f g)(n) = f (n) g + 1 n−1 (4.4) Let us show that if ϕ ∈ Z+ and ψ ∈ Z+ , then ϕψ ∈ Z+ . If Z+ is CA , H , V M OA, BM OA, or H 1 , this is obvious, since in this case ϕ and ψ must be in CA , and so ϕψ ∈ CA ⊂ Z+ . Suppose now that Z+ = P+ L1 . By Lemma 4.7, ϕ, ψ ∈ BM OA ⊂ H 2 . Hence, ϕψ ∈ H 1 ⊂ P+ L1 . It follows that the terms f (k) g (n−k) , 1 ≤ k ≤ n − 1, on the right-hand side of (4.4) belong to Z. Let us show that f (n) g and f g (n) are in Z. If Z = CA + CA , Z = H ∞ + H ∞ , or Z = H 1 + H 1 , this is obvious, since f, g ∈ CA . Let Z = V M O. It is sufficient to show that if ϕ ∈ V M OA and ψ ∈ V M OA, then ϕψ ∈ V M OA. It follows by duality from Lemma 4.6, (iii) that z k ϕV M OA ≤ const · log(2 + k)ϕV M OA . ∞

It is also obvious that ψ ∈ Λα for any α < 1. It is well known that Λβ ⊂ F(1 ) for β > 1/2 (Bernshtein’s theorem, see Kahane [1], Ch. II, §6), which implies that ˆ log(2 + k)|ψ(k)| < ∞. k≥0


We have ϕψ =

325

k ˆ ψ(k)z ϕ

k≥0

and k ˆ ψ(k)z ϕ k≥0

≤

z k ϕV M OA ≤ const

k≥0

V M OA

ˆ log(2 + k)|ψ(k)|.

k≥0

For Z = BM O the proof is similar. 41 , n ≥ 2. It is sufficient to show Finally, consider the case Z = L1 + L 1 1 that if ϕ ∈ P+ L and ψ ∈ P+ L , then ϕψ ∈ P+ L1 . By Lemma 4.7, ψ ∈ V M OA, and so ψ ∈ Λa for any α < 1. The rest of the proof is exactly the same as in the case Z = V M O. (n) Let us now prove that P+ f¯g ∈ Z+ . Assume first that Z is V M O, BM O, 41 and n ≥ 2. CA + CA or H ∞ + H ∞ and n ≥ 1, or Z is H 1 + H 1 or L1 + L As we have already seen above, this implies that f ∈ Λα for any α < 1, and so log(2 + k)|fˆ(k)| < ∞, k≥0

P+ f¯g =

fˆ(k)Tz¯k g,

(4.5)

k≥0

and by Lemma 4.6, the series on the right-hand side of (4.5) converges in the norm of Z+ . It remains to consider the case Z = H 1 + H 1 and n = 1. In this case f ∈ H ∞ . Since the trigonometric polynomials are dense in Z it is sufficient to show that P+ f¯gZ (1) ≤ const f H ∞ gZ (1) , +

+

g ∈ P A.

(1)

The dual space to Z+ can naturally be identified with the space {ψ : ψ ∈ BM OA} with respect to the pairing (4.2). This space admits the following description: (1) ∗ ψ ∈ Z+ ⇐⇒ |ψ(ζ)|2 (1 − |ζ|2 )dm2 (ζ) is a Carleson measure (4.6) (this follows from the characterization of BM O in terms of Carleson measures; see Appendix 2.5). We have |(P+ f¯g, ψ)| = |(f¯g, ψ)| = |(g, f ψ)| ≤ const g (1) f ψ Z+

(1)

Z+

∗ ,

(1) ∗ ψ ∈ Z+ .

The result follows now from the fact that multiplication by f is bounded (1) ∗ on Z+ , which is an immediate consequence of (4.6).

326


Let us verify (A4). If Z = V M O, Z = CA + CA , Z = H 1 + H 1 , or 41 , the trigonometric polynomials are dense in the Banach alZ = L1 + L gebra Z (n) , and the result is almost obvious. Indeed, if ω is a complex homomorphism, let λ = ω(z). Then ω(ϕ) = ϕ(λ) for any trigonometric polynomial ϕ. Therefore the linear functional ϕ → ϕ(λ) extends by continuity to Z (n) . Obviously, this implies that λ ∈ T and ω(ϕ) = ϕ(λ) for any ϕ ∈ Z (n) . Hence, all complex homomorphisms on Z (n) are point evaluations at points of T. This implies (A4). Suppose now that Z = BM O or Z = H ∞ + H ∞ . Let f (n) ∈ Z and inf |f (τ )| > 0. Then f ∈ Λa for α < 1 and 1/f ∈ Λa for α < 1. It is easy τ ∈T

to see now that

(n) 1 f (n) = − 2 + g, f f

where g is a function that belongs to Λα for all α < 1. It is sufficient to prove the following fact. Let ϕ ∈ Z and ψ ∈ Λα for def def def all α < 1. Then ϕψ ∈ Z. Put ϕ+ = P+ ϕ, ϕ− = P− ϕ, ψ+ = P+ ψ, and def ψ− = P− ψ. We have ϕψ = ϕ+ ψ+ + ϕ− ψ− + ϕ+ ψ− + ϕ− ψ+ . Let us show that ϕ+ ψ+ ∈ Z and ϕ− ψ+ ∈ Z. The proof that ϕ− ψ− ∈ Z and ϕ+ ψ− ∈ Z are the same. Consider first the case Z = H ∞ +H ∞ . Obviously, ϕ+ ψ+ ∈ H ∞ . It is also clear that ϕ− ψ+ ∈ L∞ , and so it is sufficient to show that P+ ϕ− ψ+ ∈ H ∞ . However, this follows from Lemma 4.6, (i). Consider now the case Z = BM O. When verifying the axiom (A2) we have shown that the product of a BM OA function and a function that belongs to (Λα )+ for some α > 1/2 must be in BM OA. This implies that ϕ+ ψ+ ∈ BM OA. Let us prove now that ϕ− ψ+ ∈ BM O. There exists a function ξ ∈ BM OA such that ϕ− + ξ ∈ L∞ . We have ϕ− ψ+ = (ϕ− + ξ)ψ+ + ξψ+ . Obviously, (ϕ− + ξ)ψ+ ∈ L∞ . We have already proved that the product of a BM OA function and a function in Λα for some α > 1/2 is in BM OA, which implies that ξψ+ ∈ BM OA. It remains to verify (A3). For Z = V M O, Z + CA + CA , Z = H 1 + H 1 , 41 the trigonometric polynomials are dense in Z (n) , and so or Z + L1 + L (A3) follows immediately from Theorem 2.1. To verify (A3) for Z = BM O or Z = H ∞ + H ∞ we make use of Theorem 2.2. Let Z = BM O. Put     1 Y = F + = f ∈ CA : |fˆ(k)| < ∞ .   k≥0

4. Examples and Counterexamples (n)

If f ∈ Y , it follows immediately Lemma 4.6 (ii) that Tf¯Z+

327 (n)

⊂ Z+ . By (n)

Theorem 2.2, it is sufficient to prove that the identical inclusion of Y in Z+ is compact. Let 1/2 < β < α < n. Then BM O(n) ⊂ Λα and Λβ ⊂ F1 . The result follows now from the fact that the inclusion Λα → Λβ is compact (see the proof of Theorem 4.3). Now let Z = H ∞ + H ∞ . Put     log(2 + |k|)|fˆ(k)| < ∞ . Y = f ∈ CA :   k≥0

(n)

(n)

It follows immediately from Lemma 4.7 (i) that Tf¯Z+ ⊂ Z+ for f ∈ Y . (n)

It remains to show that the identical inclusion of Y in Z+ is compact. Again, let 1/2 < β < α < n. Then (H ∞ + H ∞ )(n) ⊂ Λα and Λβ ⊂ Y . As in the previous case the result follows from the fact that the inclusion Λα → Λβ is compact. Theorem 4.5 implies the following interesting corollary. Corollary 4.8. Let n be a positive integer. Then the following assertions hold: (i) if f (n) ∈ CA , then (Af¯)(n) ∈ CA ; (ii) if f (n) ∈ H ∞ , then (Af¯)(n) ∈ H ∞ ; (iii) if f (n) ∈ H 1 , then (Af¯)(n) ∈ H 1 .

The Spaces Fpw Let w = {wn }n≥0 be a sequence of positive numbers and let 1 ≤ p ≤ ∞. Consider the space Fpw of formal trigonometric series fˆ(n)z n f= n∈Z

such that f F pw

 

p ˆ n∈Z |f (n)w|n| | =  sup |fˆ(n)w |, |n|

def

1/p

, p < ∞, p = ∞.

n

We also consider the space Fc0,w that can be defined as the closure of the set of polynomials in F∞ w. It is easy to verify (see Nikol’skii [1]) that under the condition  1/p w|m| p  < ∞, p > 1, (4.7) sup  w|k| w|j| m k+j=m

w|k+j| ≤ const · w|k| w|j| , Fpw

is an algebra (i.e., the space pointwise multiplication).

Fpw

j, k ∈ Z, ∞

⊂L

and

p = 1,

Fpw

(4.8)

is an algebra under

328


If Fpw is an algebra, then inf wn > 0. Indeed, n≥0

w0 = z

−n

· z ≤ const z −n · z n = wn2 . n

Clearly, if lim sup wn1/n = 1,

(4.9)

n→∞

then elements of Fpw ⊂ ACδ for any δ ∈ (0, 1), where ACδ is the space of functions that extend analytically to the annulus {ζ ∈ C : δ < |ζ| < 1/δ}. It is now easy to see that the following theorem holds. Theorem 4.9. Let w = {wn }n≥0 be a sequence of positive numbers satisfying (4.9). The following assertions hold: (i) if 1 < p < ∞ and w satisfies (4.7), then Fpw is a decent space; (ii) if p = 1 and w satisfies (4.8), then F1w is a decent space; (iii) if w satisfies (4.7) with p = 1, then Fc0,w is a decent space. Consider now the nonseparable space F∞ w . It is not obvious for this space that conditions (4.7) and (4.9) imply that the space of maximal ideals of F∞ w can be identified with T. To prove this, we need one more lemma. If X is a Banach space of functions on T, we say that X is homogeneous if f ∈ X, τ ∈ T =⇒ fτ ∈ X and fτ X = f X , def

where fτ (ζ) = (¯ τ ζ), ζ ∈ T. Lemma 4.10. Let X be a homogeneous Banach algebra of functions on T that is invariant under complex conjugation and such that C ∞ (T) ⊂ X ⊂ C(T). Suppose that there exists a homogeneous Banach space L of functions on T such that (i) the trigonometric polynomials are dense in L; (ii) X ⊂ L ⊂ C(T); (iii) f 2 X ≤ const f X f L for every f ∈ X. Then each multiplicative linear functional on X is a point evaluation at some point of T. Proof. Denote by r(f ) the spectral radius of an element f in X. Since r(f ) is the sup-norm of the Gelfand transform of f , it follows that r(f1 + f2 ) ≤ r(f1 ) + r(f2 ),

f1 , f2 ∈ X.

Let X0 be the closure of the set of trigonometric polynomials in X. It is clear that the space of maximal ideals of X0 can be identified naturally with T. Let us show that for every f ∈ X r(f ) = max |f (τ )|. τ ∈T

(4.10)

Indeed, let Kn be the Fejér kernel and fn = f ∗ Kn , n ∈ Z+ . Since X and L are homogeneous, it is easy to see that convolution with Kn on X and L has norm one, and so lim f − fn L = 0

n→∞

and fn X ≤ f X .


329

We have 1/2

r(f ) ≤ r(fn ) + r(f − fn ) ≤ r(fn ) + (f − fn )2 X 1/2

1/2

≤ r(fn ) + const f − fn X f − fn L 1/2

1/2

≤ r(fn ) + const f X f − fn L . Since fn ∈ X0 , clearly, r(fn ) = fn ∞ . Hence, lim r(fn ) = f ∞ , and n→∞

since lim f − fn L = 0, it follows that r(f ) ≤ f ∞ . The opposite n→∞

inequality r(f ) ≥ f ∞ is obvious. Suppose now that ϕ is a complex homomorphism on X that is not a point evaluation at any point of T. Since T is compact, there is a finite collection {f1 , · · · , fn } of elements of X such that inf

τ ∈T

Put f =

n

n

|fk (t)|2 = δ > 0,

ϕ(fk ) = 0,

k = 1, · · · , n.

k=1

|fk |2 ∈ X. Clearly,

k=1

ϕ(f ) =

n

ϕ(fk )ϕ(fk ) = 0.

k=1

Hence, f is not invertible in X. On the other hand, since f ∈ C(T) and 1/f ∈ C(T), there exists a trigonometric polynomial q such that f q − 1L∞ < 1. It follows now from (4.10) that r(f q) < 1, and so f q is invertible in X. Clearly, this implies that f is invertible in X. So we have got a contradiction with the assumption of the existence of a complex homomorphism ϕ that is not a point evaluation at any point of T. Now we are in a position to obtain the following result. Theorem 4.11. Let w = {wn }n≥0 be a nondecreasing sequence of positive numbers which satisfies (4.7) with p = 1 and such that w2n sup < ∞. n wn Then F∞ w is a decent space. Proof. (A1) is obvious. We have already observed that (A2) is satisfied. 1 Let us verify (A3). Let Y = (F1 )+ . To show that F∞ w ⊂ F , it is −1 sufficient to prove that wn < ∞. We have from (4.7) n≥0 m k=0

wm 1 ≤ ≤ const . wk wn−k wk m

k=0

In fact, the identical inclusion of F1 in F∞ w is compact since 1 ˆ(k)z k ∞ f ≤ f → 0 as n → ∞. F w wk 1 |k|≥n |k|≥n F

330


Since the sequence w is nondecreasing, it follows that the norms of the ∞ ∞ Toeplitz operators Tz¯n on F∞ w do not exceed 1, and so Tf¯Fw ⊂ Fw for every f ∈ Y . (A3) follows now from Theorem 2.2. 1 It remains to verify (A4). Let us show that X = F∞ w and L = F satisfy the hypotheses of Lemma 4.10. We have already proved (ii); (i) is obvious. Let us establish (iii). Let f ∈ F∞ w . Then = sup w|n| |f62 (n)|. f 2 F ∞ w n∈Z

Let us first estimate sup w|n| |f62 (n)|. We have n≥0

sup w|n| |f62 (n)|

≤

n≥0

sup wn n≥0

≤ 2 sup wn n≥0

=

2 sup n≥0

|fˆ(k)| · |fˆ(n − k)|

k∈Z

|fˆ(k)| · |fˆ(n − k)|

k≥ n 2

wn wk |fˆ(k)| · |fˆ(n − k)| · wk n

k≥ 2

w2n f F ∞ f F 1 . w n≥0 wn

≤ 2 sup

The estimate of sup w|n| |f62 (n)| is similar. n≤0

Carleman Classes Let E be a a Banach space of functions on T and {Mn }n≥0 an increasing and logarithmically convex sequence with M0 = 1. The Carleman class C(E; {Mn }) consists of infinitely differentiable functions f on T such that f (n) ∈ E, n ∈ Z+ , and (n) f ≤ Cf Qnf · n! · Mn , n ∈ Z+ , E

for some positive constants Cf and Qf . Carleman classes are not Banach spaces. One can introduce in a natural way a locally convex topology of inductive limit on a Carleman class. However, we do not need this. Lemma 4.12. Suppose that E is a Banach algebra of functions on T whose space of maximal ideals can naturally be identified with T. Let f ∈ C(E; {Mn }) and let ϕ be a function analytic in a neighborhood of f (T). Then ϕ ◦ f ∈ C(E; {Mn }). Proof. We use Faa di Bruno’s formula (see Bourbaki [1], Ch. 1, §3, Ex. 7a) (ϕ ◦ f )(n) = (1) m1 (n) mn n! f f ··· . (ϕ(m1 +···+mn ) ◦ f ) m ! · · · m ! 1! n! 1 n m +2m ···+nm =n 1

2

n


331

Since E is a Banach algebra and ϕ is analytic in a neighborhood of f (T), it follows easily from the Riesz–Dunford integral formula (see Gamelin [1], Ch. 1, §5) that ϕ(m) ◦ f E ≤ m!Cϕ Qm ϕ. For brevity we use the notation m = m1 + · · · + mn . Then (1) m1 (n) mn f f (m) ◦ f) ··· (ϕ 1! n! ≤

(1) m1 (n) mn f f (m) ϕ ◦ f E ··· n! 1! E E

E

m1 m m mn ≤ m!Cϕ Qm ϕ Cf Qf M1 · · · Mn . 1/n

Since {Mn }n≥0 is logarithmically convex, the sequence {Mn }n≥1 is nondecreasing. Hence, M1m1 · · · Mnmn

1/2

= M1m1 (M2 )2m2 · · · (Mn1/n )nmn ≤ (Mn1/n )m1 +2m2 +···+nmn = Mn .

Combining all the estimates, we obtain (ϕ ◦ f )(n) E ≤ Cϕ Qnf n!Mn m1 +2m2 +···+nmn

It remains to observe that m1 +2m2 +···+nmn =n; m1 +m2 +···+mn

and

m! (Cf Qϕ )m . m ! · · · m ! 1 2 =n

m! = m1 ! · · · m2 ! =m

n−1 m−1

(4.11)

n n−1 = 2n−1 . m−1

m=1

To prove (4.11), we set in Faa di Bruno’s formula ϕ(t) = tm , f (t) = t(1 − t)−1 and compute the left-hand and right-hand sides at t = 0. Note that under the hypotheses of the lemma the space C(E; {Mn }) is an algebra. This follows from the identity 4f g = (f + g)2 − (f − g)2 . Corollary 4.13. Let 1 < p < ∞ and E = Lp . Then the space C(E; {Mn }) satisfies the conclusion of Lemma 4.12. Proof. Put def Mn+1 , n ∈ Z+ . Mn = M1 Consider the Sobolev space Wp1 = {ϕ ∈ Lp : ϕ ∈ Lp }. It is easy to see that Wp1 is an algebra whose space of maximal ideals can naturally be identified with T. It is also clear that C(Lp ; {Mn }) = C(Wp1 ; {Mn }).

332


The result now follows from Lemma 4.12. The following result shows that if the Mn do not grow terribly rapidly, then C(C(T); {Mn }) is invariant under the Riesz projection. Theorem 4.14. Suppose {Mn }n≥0 is an increasing logarithmically convex sequence such that n Mn+1 ≤ cK K for some c, K > 0. (4.12) Mn Then P+ C(C(T); {Mn }) ⊂ C(C(T); {Mn }). Proof. Let ΥN be the de la Vallée Poussin kernel defined by (4.3). Let g be a function in the Zygmund class Λ1 . It is easy to see from the definition of Λ1 in terms of convolutions with the trigonometric polynomials Wn (see Appendix 2.6) that 1 1 P+ g − ΥN ∗ P+ gL∞ ≤ const · P+ gΛ1 ≤ const · gΛ1 . N N Indeed, to prove this inequality, it is sufficient to consider the case N = 2n and use the elementary identity P+ g − Υ2n ∗ P+ g = Wj ∗ P+ g. j≥n+1

Since ΥN ∗ g is a trigonometric polynomial of degree at most 2N , m j ≤ const log(1 + m), z j=0 1 L

1

and the L -norms of the ΥN are uniformly bounded, it follows that P+ ΥN ∗ gL∞ ≤ const · log(1 + N )ΥN ∗ gL∞ ≤ const · log(1 + N )gL∞ . Hence, for g ∈ C 1 (T) we have P+ gL∞

≤

P+ ΥN ∗ gL∞ + P+ g − ΥN P+ gL∞ 1 ≤ const · log(1 + N )gL∞ + const · g L∞ . N Suppose now that f ∈ C(C(T); {Mn }) and f (n) L∞ ≤ Cf Qnf · n! · Mn . Without loss of generality we may assume that the number K in (4.12) is n an integer. Now let g = f (n) , N = K K . Then P+ f (n) L∞

≤ const ·K n f (n) L∞ + const ·K −K f (n+1) L∞ n

≤ const(K + Qf )n · n! · Mn +

const Qn+1 · (n + 1)! · Mn+1 K −K f

n

≤ const(K + Qf )n · n! · Mn + const Qn+1 · n!(n + 1)Mn f ≤ const(K + Qf )n · n! · Mn , which proves that P+ f ∈ C(C(T); {Mn }).


333

We are now able to prove that under the hypotheses of Theorem 4.14 the Carleman class C(C(T); {Mn }) has the heredity property. Theorem 4.15. Suppose that {Mn }n≥0 is an increasing and logarithmically convex sequence satisfying (4.12). Then the Carleman class C(C(T); {Mn }) satisfies the axioms (B1)–(B3). Proof. By Lemma 4.12 and Theorem 4.14, (B1) and (B3) are satisfied. Let us verify (B2). Put |fˆ(n)| log(2 + |n|) < ∞}. Y = {f : n∈Z

Suppose that ϕ ∈ C(C(T); {Mn }) and ϕ(n) L∞ ≤ Cϕ Qnϕ · n! · Mn . It follows from Lemma 4.6 (i) that for f ∈ Y+ ϕ(n) L∞ ≤ const f Y (2Qϕ )n · n! · Mn . Finally, we prove that for 1 < p < ∞ the Carleman class C(Lp ; {Mn }) also has the hereditary property even without any assumption on the sequence {Mn }n≥0 . Theorem 4.16. Let 1 < p < ∞ and let {Mn }n≥0 be an increasing and logarithmically convex sequence. Then the Carleman class C(Lp ; {Mn }) satisfies the axioms (B1)–(B3). Proof. (B1) and (B3) follows immediately from Corollary 4.13. It is also obvious that (B2) holds with Y = F1 . Consider the important example of Carleman classes. Let 0 < α < ∞. The Gevrey class Gα is the Carleman class C(C(T); {Mn }), where Mn = nn/α . Clearly, this sequence satisfies (7.4.12), and so by Theorem 4.15, Am Gα ⊂ Gα , m ∈ Z+ , α > 0. It can easily be shown that a function f ∈ C(T) belongs to Gα , α > 0, if and only if there exist K, δ > 0 such that |fˆ(n)| ≤ K exp(−δ|n|α/(1+α) ), n ∈ Z.

The Space ACδ For δ ∈ [0, 1) the space ACδ consists of functions on T that extend analytically to the annulus {ζ ∈ C : δ < |ζ| < δ −1 }. It is easy to see that ACδ satisfies (B1) and (B2) (in (B2) we can take Y = F1 ). Therefore Lemma 3.1 holds for X = ACδ . In fact, we can slightly improve Lemma 3.1 for X = ACδ . Theorem 4.17. Let h be an outer function such that h, h−1 ∈ H 2 . ¯ ∈ ACδ if and only if |h|2 ∈ ACδ . Then P− h/h Proof. The theorem follows immediately from Lemma 3.1 and the obvious observation that the conditions |h|2 ∈ ACδ and h−1 ∈ H 2 imply that inf |h(τ )| > 0.

τ ∈T

334


The space ACδ does not satisfy (B3). Indeed, if f ∈ ACδ , f has no zeros on T but the analytic extension of f in {ζ ∈ C : δ < |ζ| < δ −1 } has a zero, then inf |f (τ )| > 0 but f is not invertible in ACδ . τ ∈T ¯ Moreover, A(ACδ ) ⊂ ACδ . Indeed, let h = z − 2/(1 + δ) and u = z¯h/h. Then P− z 2 u = P− z(¯ z − 2/(1 + δ)) · (z − 2/(1 + δ))−1 = 0. Hence, P− u ∈ ACδ . On the other hand, −1 2 2z 2 z− P+ z u = 1 −

∈ ACδ , 1+δ 1+δ since 1 < 2/(1 + δ) < 1/δ. Consequently, AP− u = −P+ u ∈ ACδ .

The Space C(T) The operator A of best approximation does not act on C(T). This has been proved implicitly in §1.5 (see the remark after Corollary 1.5.7). Indeed, let α be a real continuous function on T such that its harmonic conjugate α ˜ is discontinuous. Let u = z¯eiα˜ . Then u ∈ H ∞ + C and distL∞ (u, H ∞ ) = 1. Let u = f + g, where f ∈ C(T) and g ∈ H ∞ . Clearly, distL∞ (f, H ∞ ) = 1 and Af = −g ∈ C(T). It is also easy to show that Am C(T) ⊂ C(T) for any m ∈ Z+ . To show this we can consider the function u = z¯m+1 eiα˜ , observe that Am u = O, and apply the same reasoning as in the case m = 0.

The Spaces C n (T), n ≥ 1 We are going to prove here that the space C n (T) of n times continuously differentiable functions on T does not have the hereditary property. Consider the space H ∞ + C n (T) = {ϕ = f + g : f ∈ H ∞ , g ∈ C n (T)}. def

We need the following fact. Theorem 4.18. The space H ∞ + C n (T) is an algebra with respect to pointwise multiplication on T. Proof. Clearly, it is sufficient to show that the product of an H ∞ function and a function in C n (T) belongs to H ∞ + C n (T). This is equivalent to the following assertion. Let f ∈ H ∞ , g ∈ C n (T), then f¯g ∈ H ∞ + C n (T). Put V M OA(n) = {ϕ ∈ V M OA : ϕ(n) ∈ V M OA}, BM OA(n) = {ϕ ∈ BM OA : ϕ(n) ∈ BM OA} Obviously, V M OA(n) = P+ C n (T). It is easy to see that f¯g ∈ H ∞ + C n (T) if and only if P+ f¯g = P+ (f¯P+ g) ∈ V M OA(n) .


335

Since P+ f¯q is a polynomial for any polynomial q and the polynomials are dense in V M OA(n) , it is sufficient to show that the Toeplitz operator Tf¯ is a bounded operator from V M OA(n) to BM OA(n) . Consider the space I−n H 1 = {ψ (n) : ψ ∈ H 1 }. Clearly, the dual space (I−n H 1 )∗ can be identified with BM OA(n) with respect to the pairing (4.2). The space I−n H 1 admits the following description:

ϕ ∈ I−n H 1

⇐⇒

T

1/2

1

|ϕ(rζ)|2 (1 − r)2n−1 dr 0

dm(ζ) < ∞ (4.13)

(see Appendix 2.6). It follows from (4.13) that multiplication by f ∈ H ∞ is a bounded operator on I−n H 1 . Let ψ be an analytic polynomial in I−n H 1 . We have |(ψ, P+ (f¯P+ g))| = |(ψ, f¯P+ g)| = |(f ψ, P+ g)| ≤ const f ψI−n H 1 P+ gV M OA(n) ≤ const f H ∞ ψI−n H 1 gC n (T) , which proves that P+ (f¯P+ g) determines a continuous linear functional on I−n H 1 , and so P+ (f¯P+ g) ∈ BM OA(n) . Theorem 4.19. Let n be a positive integer. Then AC n (T) ⊂ C n (T). Proof. Let α be a real function in C n (T) such that its harmonic conjugate α ˜ is not in C n (T). Put u = z¯eiα˜ . We have u = z¯eα+iα˜ e−α . Obviously, eα+iα˜ ∈ H ∞ , e−α ∈ C n (T), and so by Theorem 4.18, u ∈ H ∞ +C n (T) but u ∈ C n (T). Clearly, distL∞ (u, H ∞ ) = 1. Let u = f +g, where f ∈ C n (T) and g ∈ H ∞ . Then distL∞ (f, H ∞ ) = 1 and Af = −g. However, g ∈ C n (T), since u ∈ C n (T). As in the case of the space C(T) one can also show that Am C n (T) ⊂ C n (T) for any m ∈ Z+ .

The Space CA + CA The space CA + CA does not have the hereditary property, i.e., A(CA + CA ) ⊂ CA + CA . Clearly, to show this, it is sufficient to prove the following fact. Theorem 4.20. There exists a function ϕ in CA such that the function Aϕ¯ is discontinuous.

336


Proof. Let satisfies

1 3

< γ1 ≤

1 2

and γ2 >

Consider a function ψ on T that

−γ1 log | log t|,

it

ψ(e ) =

1 2.

0 < t ≤ 12 ,

−γ2 log | log |t||, − 12 ≤ t < 0,

and has continuous derivative in T \ {1}. Define the function ϕ ∈ H ∞ by ˜

ϕ = ceψ+iψ , where c > 0. It is easy to see that the function ψ˜ is continuous on T \ {1}. We also have lim |ϕ(ζ)| = lim eψ(ζ) = 0. ζ∈T, ζ→1

ζ∈T, ζ→1

Therefore ϕ is a continuous function on T, and so ϕ ∈ CA . It is easy to see that

1/2 |ϕ(eit )|2 =∞ t 0 while

0

−1/2

|ϕ(eit )|2 < ∞. |t|

(4.14)

(4.15)

Denote by ξ the harmonic extension of |ϕ|2 to the unit disk and by ξ˜ its harmonic conjugate, which is a harmonic function in D. It follows from (4.14) and (4.15) that ˜ =∞ lim ξ(r)

r→1

(4.16)

(see Zygmund [1], Ch. III, Theorem 7.20). Let f = Aϕ. ¯ We claim that f ∈ C(T). Suppose that f ∈ C(T). Choosing the right c in the definition of ϕ, we may assume that distL∞ (f¯, H ∞ ) = 1. Hence, ϕ¯ − f is a continuous unimodular function on T. Since f is continuous, there exists δ > 0 such that 1 def |f (ζ)| ≥ , ζ ∈ Ωδ = {ζ ∈ D : |1 − ζ| < δ}. (4.17) 2 Let f = ϑh, where h = exp(log |f | + ilog |f |) is an outer function and ϑ is an inner function. It follows from (4.17) that ϑ extends analytically across def

the arc Iδ = {ζ ∈ T : |1 − ζ| < δ} and has an analytic logarithm in Ωδ . Then f has an analytic logarithm in Ωδ given by

τ +ζ dm(τ ) + log ϑ(ζ). (4.18) log |f (τ )| log f (ζ) = τ −ζ T Clearly, log ϑ is bounded in Ωδ . Since |ϕ¯ − f | = 1 on T, it follows that |f |2 = 1 + 2 Re(ϕ · f ) − |ϕ|2 .

(4.19)


337

We have for ζ ∈ Iδ 1 log |f (ζ)| = log |f (ζ)|2 2 1 (2 Re(ϕ(ζ)f (ζ)) − |ϕ(ζ)|2 )2 = 2 Re(ϕ(ζ)f (ζ)) − |ϕ(ζ)|2 − 2 2 + χ1 (ζ) 1 = Re(ϕ(ζ)f (ζ)) − |ϕ(ζ)|2 − (Re(ϕ(ζ)f (ζ)))2 + χ2 (ζ) 2 1 1 1 = Re(ϕ(ζ)f (ζ)) − |ϕ(ζ)|2 − |ϕ(ζ)f (ζ)|2 − Re(ϕ(ζ)f (ζ))2 2 2 2 + χ2 (ζ), where χ1 and χ2 are functions on Iδ such that |χj (ζ)| ≤ const |ϕ(ζ)|3 , In view of (4.19) 1 |ϕ(ζ)f (ζ)|2 2

= =

j = 1, 2,

ζ ∈ Iδ .

1 |ϕ(ζ)|2 1 + 2 Re(ϕ(ζ) · f (ζ)) − |ϕ(ζ)|2 2 1 |ϕ(ζ)|2 + χ3 (ζ), 2

and so log |f (ζ)| = Re(ϕ(ζ)f (ζ)) −

1 Re(ϕ(ζ)f (ζ))2 − |ϕ(ζ)|2 + χ4 (ζ), 2 (4.20)

where χ3 and χ4 are functions on Iδ such that |χj (ζ)| ≤ const |ϕ(ζ)|3 ,

j = 3, 4,

ζ ∈ Iδ .

We can extend the above equalities from Iδ to the entire circle T and consider them as the definitions of the functions χj , 1 ≤ j ≤ 4, on T \ Iδ . It is easy to see that the χj are integrable on T and continuous on Iδ . Since γ1 > 13 and γ2 > 13 , it follows that

|χ4 (τ )| dm(τ ) < ∞, T |1 − τ | which implies

χ4 (τ ) τ + r dm(τ ) ≤ const, 0 < r < 1, (4.21) τ −r T (see Zygmund [1], Ch. III, Theorem 7.20). Since both ϕ and f belong to CA , we have for ζ ∈ D

τ +ζ Re(ϕ(τ )f (τ )) dm(τ ) = ϕ(ζ)f (ζ) − Im(ϕ(0)f (0)), τ −ζ T

Re(ϕ(τ )f (τ ))2 T

τ +ζ dm(τ ) = (ϕ(ζ)f (ζ))2 − Im(ϕ(0)f (0))2 . τ −ζ

(4.22)

(4.23)

338


Recall that

T

|ϕ(τ )|2

τ +ζ ˜ dm(τ ) = ξ(ζ) + iξ(ζ), τ −ζ

ζ ∈ D.

(4.24)

It follows now from (4.18) and (4.20) that for ζ ∈ Ωδ

τ +ζ dm(τ ) log f (ζ) = Re(ϕ(τ )f (τ )) τ −ζ T

1 τ +ζ dm(τ ) − Re(ϕ(τ )f (τ ))2 2 T τ −ζ

τ +ζ dm(τ ) − |ϕ(τ )|2 τ −ζ T

τ +ζ dm(τ ) + log ϑ(ζ). + χ4 (τ ) τ −ζ T It is now easy to see from (4.22), (4.23), and (4.24) that for ζ ∈ Ωδ we have 1 ˜ log f (ζ) = ϕ(ζ)f (ζ) − (ϕ(ζ)f (ζ))2 + ξ(ζ) + iξ(ζ) 2

τ +ζ dm(τ ) + log ϑ(ζ) + const . + χ4 (τ ) τ −ζ T It follows now from (4.16) and (4.21) that lim Im log f (r) = ∞,

r→1

which contradicts our assumption that f ∈ CA . Note, however, that for n ≥ 1 the condition ϕ(n) ∈ CA implies (Aϕ) ¯ (n) ∈ CA (see Corollary 4.8).

5. Badly Approximable Functions In this section we find a necessary and sufficient condition for a function f ∈ BM OA to be the best approximant to a function ϕ ∈ V M O. Clearly, f is the best approximant if and only if the error function u = ϕ−f satisfies the condition distL∞ (u, H ∞ ) = uL∞ . Such functions u are called badly approximable. We characterize here the badly approximable functions in V M O. We also obtain similar results for best approximation by meromorphic functions. Recall that by Theorem 1.6, the function f = Aϕ belongs to V M O, and so u ∈ V M O. By Corollary 1.1.6, u has constant modulus and unless ϕ ∈ H ∞ , u must be invertible in H ∞ +C. In §3.3 we have defined the winding number wind u for such functions u and proved that wind u = − ind Tu .

5. Badly Approximable Functions

339

Theorem 5.1. Let u be a nonzero function such that P− u ∈ V M O. Then u is badly approximable if and only if u has constant modulus, u ∈ QC, and wind u < 0. Proof. Suppose that u is badly approximable. Then AP− u = −P+ u and by Theorem 1.6, P+ u ∈ V M O, and so u ∈ QC. By Corollary 1.1.6, u has the form ¯ u = c¯ z ϑ¯h/h, where c ∈ C, ϑ is an inner function, and h is an outer function in H 2 . Thus u has constant modulus. Clearly, h ∈ Ker Tu . By Theorem 3.1.4, Ker Tu∗ = {O}, and so ind Tu > 0, which means that wind u < 0. Suppose now that u has constant modulus and wind u < 0. Without loss of generality we may assume that |u(ζ)| = 1 for almost all ζ ∈ T. Since u is in V M O, the Toeplitz operator Tu is Fredholm and ind Tu = dim Ker Tu = − wind u > 0. So Tu is not left invertible, and by Theorem 3.1.11, distL∞ (u, H ∞ ) = 1, which precisely means that u is badly approximable. Theorem 5.1 can be reformulated as follows. Theorem 5.2. Let ϕ be a function in V M O such that P− ϕ = O and let f be a function in V M OA. Then f = Aϕ if and only if the function ϕ − f has constant modulus on T and wind(ϕ − f ) < 0. We are going to obtain an analog of Theorem 5.2 for best approximation by functions in BM OA(m) . Suppose that ϕ ∈ V M O and the singular value sm (Hϕ ) of Hϕ has multiplicity µ. Then there exists k ∈ Z+ such that sk (Hϕ ) = · · · = sk+µ−1 (Hϕ ) > sk+µ (Hϕ ),

k ≤ m ≤ k + µ − 1. (5.1)

Consider the error function u = ϕ − Am ϕ. It belongs to V M O and − wind u = 2k + µ (see Theorem 4.1.7). Let us prove the converse. We show that if f is a function in def

V M OA(m) = BM OA(m) ∩ V M O and − wind(ϕ − f ) is sufficiently large, then f = Am ϕ. Theorem 5.3. Let m ∈ Z+ , ϕ ∈ V M O, and f ∈ V M OA(m) . Suppose that ϕ − f has nonzero constant modulus on T and def

N = − wind(ϕ − f ) > 2m. Then f = Am ϕ and the singular value sm (Hϕ ) has multiplicity at least N − 2m. Proof. Suppose that g ∈ BM OA(m) and ϕ − g∞ < ϕ − f ∞ .

(5.2)

340


Put u = ϕ − f . Without loss of generality we may assume that u is a unimodular function. We have ϕ − g = u + f − g. By Theorem 5.1, u is a badly approximable function. Consequently, P− (f − g) = O. Clearly, f − g ∈ BM OA(2m) . Since dim Ker Tu = N > 2m, it follows from Lemma 4.1.8 that Hu+f −g > 1. Hence, u + f − g∞ > 1, which contradicts (5.2). Suppose now that the singular value sm (Hϕ ) of Hϕ has multiplicity µ and (5.1) holds. By Theorem 4.1.7, N = 2k+µ. Therefore µ = N −2k ≥ N −2m, which completes the proof. Theorem 5.3 implies the following criterion for a function f to be the best approximation of ϕ by functions in BM OA(k) . Theorem 5.4. Let k ∈ Z+ and ϕ ∈ V M O \ BM OA(k) . Suppose that f ∈ V M OA(k) . Then f = Ak ϕ if and only if |ϕ − f | = const on T

and

def

N = − wind(ϕ − f ) > 2k. (5.3)

If (5.3) holds, then sk (Hϕ ) is a singular value of Hϕ of multiplicity N − 2k. Theorem 5.4 follows immediately from Theorems 5.3 and 4.1.7. Now we are going obtain analogs of the above results for a considerably bigger class of functions. Theorem 5.5. Let u be a function in L∞ such that Hu e < Hu . Then u is badly approximable if and only if ϕ has constant modulus, the Toeplitz operator Tu is Fredholm, and ind Tu > 0. Proof. Suppose that ϕ is badly approximable. Since Hu e < Hu , the Hankel operator Hu attains its norm on the unit ball of H 2 and by Theorem 1.1.6, u has constant modulus and has the form ¯ h u = c¯ zϑ , h where c ∈ C, ϑ is an inner function, and h is an outer function in H 2 . Clearly, we may assume that |c| = 1. Hence, u is a unimodular function. By Theorem 4.4.10, Tu has dense range in H 2 . Consider the subspace E = {f ∈ H 2 : Hu f 2 = f 2 } = Ker Tu . Since Hu e < Hu = 1, it follows that dim E < ∞. By Corollary 4.4.7, Hu e = Hu¯ e , and so by Corollary 3.1.16, Tu is Fredholm. Clearly, h ∈ Ker Tu , which implies that ind Tu > 0. Suppose now that u has constant modulus, Tu is Fredholm, and ind Tu > 0. Without loss of generality we may assume that u is unimodular. Since ind Tu > 0, it follows that Ker Tu = {O}. Let f ∈ Ker Tu . It is easy to see that Hu f 2 = f 2 , and so Hu = 1. Hence, u is badly approximable.

6. Perturbations of Multiple Singular Values of Hankel Operators

341

Similarly, one can state analogs of Theorem 5.3 for functions ϕ satisfying Hϕ e < sm (Hϕ ). We state an analog of Theorem 5.4. Theorem 5.6. Let k ∈ Z+ and ϕ is a function in BM O such that Hϕ e < sk (Hϕ ). Suppose that f ∈ BM OA(k) . Then f is the best approximation of ϕ by functions in BM OA(k) if and only if |ϕ − f | = const,

Tϕ−f

is Fredholm, and

def

N = ind Tϕ−f > 2k. (5.4)

If (5.3) holds, then sk (Hϕ ) is a singular value of Hϕ of multiplicity N − 2k. Proof. By Theorem 4.1.3, ϕ has a unique best approximation by functions in BM OA(k) . Suppose that deg P− f = k and f is the unique best approximation of ϕ. Let µ be the multiplicity of the singular value sk (Hϕ ) of the Hankel operator Hϕ . We have sk (Hϕ ) = · · · = sk+µ−1 (Hϕ ) > sk+µ (Hϕ ).

(5.5)

By Theorem 4.1.7, |ϕ − f | = sk (Hϕ ) almost everywhere on T and dim Ker Tϕ−f = 2k + µ. Let us show that Tϕ−f is Fredholm. Let u = s−1 k (Hϕ )(ϕ − f ). Then u is unimodular, Ker Tu = {O}, and so Ker Tu has dense range. Since the space E = Ker Tu is finite-dimensional, it follows from Corollary 4.4.7 that Hu e = Hu¯ e , and by Corollary 3.1.16, Tu is Fredholm, and so Tϕ−f is Fredholm. Suppose now that (5.4) holds. Put u = ϕ − f . Without loss of generality we may assume that u is unimodular. Suppose that g ∈ BM OA(k) and ϕ − g∞ < 1. We have ϕ − g = u + (f − g) and f − g ∈ BM O(2k) . This contradicts Lemma 4.1.8. Let us prove now that sk (Hϕ ) has multiplicity N − 2k. Let µ be the multiplicity of sk (Hϕ ). Clearly, (5.5) holds. It follows from Theorem 4.1.7 that Ker Tϕ−f = 2k + µ, which proves that µ = N − 2k.

6. Perturbations of Multiple Singular Values of Hankel Operators In this section we use the results of §5 to study the behavior of multiple singular values of Hankel operators under perturbation of their symbols by rational functions of degree 1 and under multiplication of their symbols by z and z¯. Let s > 0 and let ϕ be a function in V M O such that s is a singular value of the Hankel operator Hϕ of multiplicity µ and suppose that s = sk (Hϕ ) = sk+1 (Hϕ ) = · · · = sk+µ−1 (Hϕ ) > sk+µ (Hϕ ). (6.1)

342


Recall that for a function ϕ satisfying (6.1), Am ϕ = Ak ϕ for def

k ≤ m ≤ k + µ − 1 and r = P− Ak ϕ ∈ Rk , i.e., r is a rational function of degree k (see §4.1). Moreover, r is the only function in Rk+µ−1 for which Hϕ−r = s. Consider first the generic case of perturbation of ϕ by rational fractions with one pole. Theorem 6.1. Let ϕ ∈ V M O. Suppose that s is a singular value of Hϕ of multiplicity µ ≥ 2 and (6.1) holds. Let λ ∈ D, ζ ∈ C be such that ζ = 0 and λ is not a pole of P− Ak ϕ. If ψ = ϕ + ζ/(z − λ), then sk (Hψ ) > s > sk+µ−1 (Hψ ). Moreover, s is a singular value of Hψ if and only µ ≥ 3, in which case its multiplicity is µ − 2 and s = sk+1 (Hψ ) = · · · = sk+µ−2 (Hψ ).

(6.2)

Proof. Let r = P− Ak ϕ. Let us show that sk (Hψ ) > s. Since Hψ is a rank one perturbation of Hϕ , it follows that sk (Hψ ) ≥ sk+1 (Hϕ ) = s. Suppose that sk (Hψ ) = s. Let r1 be the function in Rk such that Hψ−r1 = s. Then sk (Hϕ ) = Hϕ−r2 , where r2 = r1 + ζ/(z − λ) ∈ Rk+1 ⊂ Rk+µ−1 . As we have noticed before the statement of Theorem 6.1, this implies that r2 = r. Let us show that this is impossible. By the assumption, λ is not a pole of ρ. Therefore if r2 = r, then λ is not a pole of r2 . Hence, k > 0 and r1 = r3 − ζ/(z − λ), where r3 ∈ Rk−1 , and so sk (Hϕ ) = Hϕ−r3 , which contradicts the fact that sk−1 (Hϕ ) > sk (Hϕ ). Let us show that sk+µ−1 (Hψ ) < s. Since Hψ is a rank one perturbation of Hϕ , it follows that sk+µ−1 (Hψ ) ≤ sk+µ−2 (Hϕ ) = s. Suppose that sk+µ−1 (Hϕ ) = s. Then sk (Hψ ) > s = sk+1 (Hψ ) = · · · = sk+µ−1 (Hψ ). Let ν be the multiplicity of the singular value s of Hψ . Then ν ≥ µ − 1. By the Adamyan–Arov–Krein theorem there exists a unique rational function ρ in Rk+µ−1 such that Hψ−ρ = s. Let def

ρ1 = r + ζ/(z − λ) ∈ Rk+1 ⊂ Rk+µ−1 . Clearly, Hψ−ρ1 = s. Hence, ρ = ρ1 and ψ − Ak+1 ψ = ϕ − Ak ϕ. By Corollary 1.7, ϕ − Ak ϕ ∈ QC. It follows from Theorem 4.1.7 that ϕ − Ak ϕ has constant modulus and wind(ϕ − Ak ϕ) = −(2k + µ). On the other hand, wind(ψ − Ak+1 ψ) = −(2(k + 1) + ν). However, 2(k + 1) + ν > 2k + µ, which proves that the assumption sk+µ−1 (Hϕ ) = s is false. If µ = 2, it is obvious that s is not a singular value of Hψ . If µ ≥ 3, then (6.2) is an immediate consequence of the fact that Hψ is a rank one perturbation of Hϕ , which completes the proof.

6. Perturbations of Multiple Singular Values of Hankel Operators

343

Corollary 6.2. Suppose that s is a singular value of Hϕ of multiplicity µ and (6.1) holds. Let l ≥ (µ − 1)/2 and let Λ be a set of l points in D none of which is a pole of P− Ak ϕ. Then there exist arbitrarily small numbers ζλ , λ ∈ Λ, such that the Hankel operator Hψ , ψ =ϕ+

λ∈Λ

ζλ , z−λ

satisfies sk (Hψ ) > sk+1 (Hψ ) > · · · > sk+µ−1 (Hψ ) > sk+µ (Hψ ). The result follows easily from Theorem 6.1. Consider now the case when λ is a pole of P− Ak ϕ. Theorem 6.3. Let ϕ ∈ V M O, let s be a singular value of Hϕ of multiplicity µ ≥ 2, and suppose that (6.1) holds. If λ is a pole of P− Ak ϕ, ζ ∈ C, ψ = ϕ + ζ/(z − ζ), then either s is a singular value of Hψ of multiplicity µ and s = sk (Hψ ) = sk+1 (Hψ ) = · · · = sk+µ−1 (Hψ ) > sk+µ (Hψ ) (6.3) or s is a singular value of Hψ of multiplicity µ + 2 and s = sk−1 (Hψ ) = sk (Hψ ) = · · · = sk+µ (Hψ ) > sk+µ+1 (Hψ ). (6.4) Proof. Let r = P− Ak ϕ ∈ Rk . It is clear that under the hypotheses of the theorem r + ζ/(z − λ) ∈ Rk . Since Hψ is a rank one perturbation of Hϕ , it follows that sk (Hψ ) ≥ s. On the other hand, Hψ−(r+ζ/(z−λ)) = s, and so sk (Hψ ) = s. Hence, Ak ψ = Ak ϕ + ζ/(z − λ). Therefore ψ − Ak ψ = ϕ − Ak ϕ, ψ − Ak ψ has constant modulus ψ − Ak ψ ∈ QC, and wind ψ − Ak ψ = −(2k + µ). Consider first the case when r + ζ/(z − λ) ∈ Rk−1 . Then the same reasoning as in the proof of Theorem 6.1 shows that (6.4) holds. Suppose now that r + ζ/(z − λ) ∈ Rk−1 . Then again the same reasoning shows that (6.3) holds. We proceed now to the study of the behavior of multiple singular values of a Hankel operator under multiplication of its symbol by z or z¯. We consider a more general problem when z is replaced by a Blaschke factor bλ , λ ∈ D, z−λ bλ (z) = ¯ . 1 − λz Theorem 6.4. Let ϕ ∈ V M O. Suppose that s is a singular value of Hϕ of multiplicity µ ≥ 2 and (6.1) holds. Let λ ∈ D and ψ = ¯bλ ϕ. Then the following assertions hold:

344


(1) if (P− Ak ϕ)(λ) = 0, then s is a singular value of Hψ of multiplicity µ + 1 and s = sk (Hψ ) = sk+1 (Hψ ) = · · · = sk+µ (Hψ ) > sk+µ+1 (Hψ ); (6.5) (2) if (P− Ak ϕ)(λ) = 0, then s is a singular value of Hψ of multiplicity µ − 1 and s = sk+1 (Hψ ) = sk+2 (Hψ ) = · · · = sk+µ−1 (Hψ ) > sk+µ (Hψ ). (6.6) Proof. Consider the function ¯bλ Ak ϕ. The function ψ − ¯bλ Ak ϕ has constant modulus on T, belongs to QC, and wind ¯bλ (ϕ − Ak ϕ) = −(2k + µ + 1). If (P− Ak ϕ)(λ) = 0, then P−¯bλ Ak ϕ is a rational function of degree k. It follows from Theorem 5.4 that Ak ψ = ¯bλ Ak ϕ and (6.5) holds. If (P− Ak ϕ)(λ) = 0, then P−¯bλ Ak ϕ is a rational function of degree k + 1. Again, it follows from Theorem 5.4 that Ak+1 ψ = ¯bλ Ak ϕ and (6.6) holds. Theorem 6.5. Let ϕ ∈ V M O. Suppose that s is a singular value of Hϕ of multiplicity µ ≥ 2 and (6.1) holds. Let λ ∈ D and ψ = bλ ϕ. Then the following assertions hold: (1) if λ is a pole of P− Ak ϕ, then s is a singular value of Hψ of multiplicity µ + 1 and s = sk−1 (Hψ ) = sk (Hψ ) = · · · = sk+µ−1 (Hψ ) > sk+µ (Hψ ); (6.7) (2) if λ is not a pole of P− Ak ϕ, then s is a singular value of Hψ of multiplicity µ − 1 and s = sk (Hψ ) = sk+2 (Hψ ) = · · · = sk+µ−2 (Hψ ) > sk+µ−1 (Hψ ). (6.8) Proof. Consider the function bλ Ak ϕ. The function ψ − bλ Ak ϕ has constant modulus on T, belongs to QC, and wind bλ (ϕ−Ak ϕ) = −(2k +µ−1). If λ is a pole of P− Ak ϕ, then P− bλ Ak ϕ is a rational function of degree k−1. It follows from Theorem 5.4 that Ak−1 ψ = bλ Ak ϕ and (6.7) holds. If λ is not a pole of P− Ak ϕ, then P− bλ Ak ϕ is a rational function of degree k. Again, it follows from Theorem 5.4 that Ak ψ = bλ Ak ϕ and (6.8) holds.

7. The Boundedness Problem We have seen in §1 that the operator A of best approximation by analytic functions (as well as the operators Am ) is bounded on quasi-Banach 1/p R-spaces. In particular, A is bounded on V M O and the Besov spaces Bp ,

7. The Boundedness Problem

345

0 1/p, and the space F1 of functions with absolutely continuous Fourier series. We also obtain similar results for the operators Am of best approximation by meromorphic functions.

H¨ older and Besov Spaces Theorem 7.1. Let 0 1/p. Then there exists a sequence of functions {ϕn } in Bps such that ϕn Bps ≤ const, but lim Aϕn Bps = ∞.

n→∞

s Recall that B∞ = Λs . For ρ ∈ (0, 1) consider the conformal mapping ωρ from the unit disk D onto the disk {ζ : |1 − ζ| < 1} defined by

ωρ (ζ) = 1 +

ζ −ρ . 1 − ρζ

Define the functions ξρ and ηρ on T by ξρ (ζ) =

ωρ (ζ) − ζ¯2 (|ωρ (ζ) − ζ¯2 | − 1); |ωρ (ζ) − ζ¯2 |

ηρ (ζ) = ζ¯2 + ξρ (ζ). Note that for ρ sufficiently close to 1 the function ωρ (z) − z¯2 is separated away from 0 since ωρ (ζ) is close to 0 when ζ lies outside a small neighborhood of 1. Lemma 7.2. For ρ sufficiently close to 1 the following equality holds: Aηρ = ωρ . Proof. We assume that ρ is sufficiently close to 1, and so the function ωρ (z) − z¯2 is separated away from 0 on T. By Theorem 5.1, it is sufficient to show that ηρ − ωρ has constant modulus on T and negative winding number. We have ωρ (ζ) − ζ¯2 ηρ (ζ) − ωρ (ζ) = ζ¯2 + (|ωρ (ζ) − ζ¯2 | − 1) − ωρ (ζ) |ωρ (ζ) − ζ¯2 | = −

ωρ (ζ) − ζ¯2 . |ωρ (ζ) − ζ¯2 |

346


So ηρ − ωρ is unimodular. Let us show that its winding number is −1. Clearly, wind(ηρ − ωρ )

=

wind(ωρ (z) − z¯2 ) = wind(¯ z 2 (z 2 ωρ (z) − 1))

= −2 + wind(z 2 ωρ (z) − 1) = −2 + wind1 (z 2 ωρ (z)), where wind1 means winding number with respect to 1. Let us now show that wind1 (z 2 ωρ (z)) = 1. Suppose that ρ is sufficiently close to 1. Let θ be a positive number such that cos θ = ρ and τ = eiθ . It is easy to check that ωρ (τ ) = 1 − τ¯. When ζ is moving along T counterclockwise from 1 to τ , ωρ (ζ) is moving counterclockwise along the circle {ζ : |1 − ζ| = 1} from 2 to 1 − τ¯ while ζ 2 lies in a small neighborhood of 1. Therefore ζ 2 ωρ (ζ) is moving along a curve close to the arc [2, 1 − τ¯] of the circle {ζ : |1 − ζ| = 1}. Next, when ζ is moving from τ to τ¯ along T, ωρ (ζ) is moving within a small neighborhood of 0 from 1 − τ¯ to 1 − τ while ζ 2 has modulus 1. Therefore ζ 2 ωρ (ζ) is moving within a small neighborhood of 0. Finally, when ζ is moving along T from τ¯ to 1, ωρ (ζ) is moving along the circle {ζ : |1 − ζ| = 1} from 1 − τ to 2 while ζ 2 lies in a small neighborhood of 1. Therefore ζ 2 ωρ (ζ) is moving along a curve close to the arc [1 − τ, 2] of the circle {ζ : |1 − ζ| = 1}. This shows that wind1 (z 2 ωρ (z)) = 1. To prove Theorem 7.1, it is sufficient to show that ωρ Bps → ∞ as ρ → 1 and ωρ Bps lim = ∞. ρ→1 ξρ B s p Lemma 7.3. There exist positive constants c1 and c2 such that c1 (1 − ρ)1/p−s ≤ ωρ Bps ≤ c2 (1 − ρ)1/p−s . Proof. We use the following (quasi)norm on the space (Bps )+ of functions analytic in D:

1/p (1 − |ζ|)(n−s)p−1 |f (n) (ζ)|p dm2 (ζ) , p < ∞, f Bps = f L∞ + D

and n−s (n) s = f L∞ + sup(1 − |ζ|) |f (ζ)|, f B∞ ζ∈D

where n is an integer greater than s (see Appendix 2.6). It is easy to see that ωρ(n) (ζ) = n!

ρn−1 (1 − ρ2 ) , (1 − ρζ)n+1

n > 0.

Suppose that ρ is sufficiently close to 1 so that ρn−1 ≥ 12 . Consider first the case p = ∞. It is easy to see that sup(1 − |ζ|)n−s |ωρ(n) (ζ)| = sup (1 − r)n−s |ωρ(n) (r)|. ζ∈D

r∈(0,1)

(7.1)


347

Put r = ρ. We have sup(1 − |ζ|)n−s |ωρ(n) (ζ)| ≥ const ζ∈D

(1 − ρ2 )(1 − ρ)n−s ≥ const(1 − ρ)−s . (1 − ρ2 )n+1

To obtain an upper estimate, we assume that ρm+1 ≤ r ≤ ρm , m ∈ Z+ . We have (1 − r)n−s (1 − ρ2 ) (1 − ρr)n+1

(1 − ρm+1 )n−s (1 − ρ2 ) (1 − ρm+1 )n+1 1 − ρ2 ≤ const(1 − ρ)−s . (1 − ρm+1 )1+s

≤ =

Suppose now that 0 < p < ∞. We have

ωρ pBps ≥ (1 − |ζ|)(n−s)p−1 |ωρ(n) (ζ)|p dm2 (ζ), Ω

where Ω = {ζ = reiθ ∈ D : ρ < r < 1, 0 ≤ θ ≤ 1 − ρ}. We have p

1−ρ p (n−s)p−1 ωρ Bps ≥ const (1 − |ζ|) dm2 (ζ) |1 − ρζ|n+1 Ω

≥ const(1 − ρ)−np (1 − |ζ|)(n−s)p−1 dm2 (ζ) Ω

≥ const(1 − ρ)

−np

1−ρ

(1 − ρ)

r(n−s)p−1 dr = const(1 − ρ)1−sp . 0

Let us now obtain an upper estimate for ωρ Bps . It is sufficient to estimate

1/p def (n−s)p−1 (n) p Nρ = (1 − |ζ|) |ωρ (ζ)| dm2 (ζ) . D

We have Nρp

p 1 ≤ const(1 − ρ) (1 − |ζ|) dm2 (ζ) |1 − ρζ|n+1 D

1 1 p (n−s)p−1 (1 − r) dm(ζ) dr. ≤ const(1 − ρ) (n+1)p 0 T |1 − ρrζ|

Let us estimate

p

(n−s)p−1

T

1 dm(ζ) |1 − aζ|q

for 0 < a < 1 and q > 1. Consider the substitution w = (a − ζ)(1 − aζ)−1 . Then ζ = (a − w)(1 − aw)−1 . It is easy to see that

1 1 const dm(ζ) = |1 − aw|q−2 dm(ζ) ≤ . q 2 )q−1 |1 − aζ| (1 − a (1 − a)q−1 T T

348


Therefore

1

(1 − r)(n−s)p−1 dr (1 − ρr)(n+1)p−1

1

(1 − r)(n−s)p−1 dr (1 − ρr)(n+1)p−1

t

(1 − r)(n−s)p−1 dr. (1 − ρr)(n+1)p−1

≤ const ·(1 − ρ)p

Nρp

0

≤ const ·(1 − ρ)p t

+

const ·(1 − ρ)p 0

We have

1

(1 − ρ)p t

(1 − r)(n−s)p−1 dr (1 − ρr)(n+1)p−1

(1 − ρ)p ≤ const (1 − ρ)(n+1)p−1 ≤ const =

1

(1 − r)(n−s)p−1 dr t

(1 − ρ)p (1 − ρ)(n−s)p (1 − ρ)(n+1)p−1

const(1 − ρ)1−sp .

On the other hand,

t (1 − r)(n−s)p−1 p (1 − ρ) dr (n+1)p−1 0 (1 − ρr)

≤ const ·(1 − ρ)

p

t

(1 − r)−(1+s)p dr

0

≤ const ·(1 − ρ)p (1 − ρ)−(1+s)p+1 =

const ·(1 − ρ)1−sp .

This yields the desired upper estimate for ωρ Bps . Lemma 7.4. There exist positive constants c1 and c2 such that c1 (1 − ρ)−n ≤ ωρ(n) L∞ ≤ c2 (1 − ρ)−n ,

ρ ∈ (0, 1),

n ∈ Z+ .

The result follows immediately from (7.1). Lemma 7.5. For ρ sufficiently close to 1 there exists a positive constant c such that ξρ L∞ ≤ c(1 − ρ)1/2 . Proof. As in the proof of Lemma 7.2, τ ∈ T, Im τ > 0, and Re τ = ρ. Suppose that ζ belongs to the arc [¯ τ , τ ] of T. Then it is easy to see that 2 |ωρ (ζ) − ζ¯ | − 1 ≤ |τ 2 − 1| ≤ const(1 − ρ)1/2 . Suppose now that ζ does not belong to the arc [¯ τ , τ ] of T. Then ωρ (ζ) lies on the arc [1 − τ¯, 1 − τ ] of the circle {ζ : |1 − ζ| = 1}. It is easy to see that |ωρ (ζ) − ζ¯2 | − 1 ≤ |τ − 1| ≤ const(1 − ρ)1/2 . Clearly, Theorem 7.1 is an immediate consequence of the following result.


349

Lemma 7.6. There exists a number γ > 1/p − s such that for ρ sufficiently close to 1 ξρ Bps ≤ const ·(1 − ρ)γ . Proof. We represent ξρ as ξρ = fρ (gρ − 1), where def

fρ (ζ) =

ωρ (ζ) − ζ¯2 , |ωρ (ζ) − ζ¯2 |

gρ (ζ) = |ωρ (ζ) − ζ¯2 |.

Since for ρ sufficiently close to 1 the function ωρ (z) − z¯2 is separated away from zero, it follows that fρ Bps ≤ const ·ωρ Bps , where the constant does not depend on ρ. This follows from the description of Bps in terms of pseudoanalytic continuation (see Appendix 2.6). We are going to use the following formula, which can easily be verified by induction: n n (∆n−k (∆nτ ϕψ)(ζ) = ϕ)(τ k ζ)(∆kτ ψ)(ζ). (7.2) τ k k=0

Here ϕ and ψ are functions on T, and ∆τ , τ ∈ R, is the difference operator, (∆τ ϕ)(ζ) = ϕ(τ ζ) − ϕ(ζ),

ζ ∈ T,

(see Appendix 2.6). s Suppose first that p = ∞. Consider the following seminorm on B∞ : ϕs,n = sup τ =1

(∆nτ ϕ)L∞ , |1 − τ |s

where n is an integer greater than s. Since, obviously, ξρ L∞ ≤ const, it is sufficient to estimate ξρ s,n . Let us first estimate gρ s,n . Let δ be a positive number that will be specified later. Suppose that |1 − τ | ≥ (1 − ρ)δ . We have |(∆nτ gρ )(ζ)| ≤ const ·ξρ L∞ (1 − ρ)−δs ≤ const ·(1 − ρ)1/2−δs |1 − τ |s by Lemma 7.5. Suppose now that |1 − τ | < (1 − ρ)δ . Then |(∆nτ gρ )(ζ)| |1 − τ |s

≤ const ·gρ(n) L∞ |1 − τ |n−s ≤ const ·ωρ(n) L∞ |1 − τ |n−s ≤ const ·(1 − ρ)−n (1 − ρ)(n−s)δ = const ·(1 − ρ)δn−δs−n

by Lemma 7.4. Choose now δ such that

1 2

− δs = δn − δd − n. Thus δ = 1 +

gρ s,n ≤ const ·(1 − ρ)−s+1/2−s/2n . Note that

1 2

−

s 2n

> 0.

1 2n .

Then (7.3)

350


Let us now estimate ξρ s,n . We apply (7.2) with ϕ = fρ and ψ = gρ − 1. Let 0 < k < n and let s = s1 + s2 , where 0 < s1 < n − k, 0 < s2 < k. We have fρ )(τ k ζ)(∆kτ (gρ − 1))(ζ)| |(∆n−k τ ≤ const ·fρ s1 ,n−k gρ s2 ,k |1 − τ |s ≤ const ·(1 − ρ)−s1 (1 − ρ)−s2 +1/2−s2 /2k ≤ const ·(1 − ρ)−s+1/2−s2 /2k by (7.3) and Lemma 7.3. Next, if k = 0, then |(∆nτ fρ )(ζ)(gρ − 1)(ζ)| ≤ const ·ωρ Λs ξρ L∞ ≤ const ·(1 − ρ)−s+1/2 |1 − τ |s by Lemmas 7.5 and 7.3. Finally, if k = n, then |fρ (τ n ζ)(∆nt (gρ − 1))(ζ)| ≤ const ·(1 − ρ)−s+1/2−s/2n |1 − τ |s by (7.3), which completes the proof of the lemma for p = ∞. Suppose now that 0 < p < ∞. Consider the following seminorm (quasiseminorm if p < 1) on Bps :

1/p |(∆nτ ϕ)(ζ)|p ϕp,s,n = dm(ζ)dm(τ ) . 1+sp T T |1 − τ | Here n is an integer greater than s. As in the case p = ∞ it is sufficient to estimate ξρ p,s,n . Consider first the case 2 < p < ∞. Let us estimate gρ p,s,n . Let δ be a positive number whose choice will be specified later. Let J1 = {eit : t ∈ [−π, π], |t| ≥ (1 − ρ)δ },

J2 = {eit : |t| < (1 − ρ)δ }.

We have

π

|(∆nτ γρ )(ζ)|p p/2 dm(ζ)dm(τ ) ≤ const ·(1 − ρ) t−1−sp dt 1+sp J1 T |1 − τ | (1−ρ)δ ≤ const ·(1 − ρ)p(1/2−δs) by Lemma 7.5. Next,

(1−ρ)δ |(∆nτ γρ )(ζ)|p (n) dm(ζ)dm(τ ) ≤ const ·ωρ L∞ tnp−sp−1 dt 1+sp J2 T |1 − τ | 0 ≤ const ·(1 − ρ)−np (1 − ρ)δ(np−sp) ≤ const ·(1 − ρ)p(δn−δs−n) . 1 . Let us now choose δ such that p(1/2 − δs) = p(δn − δs − n), i.e., δ = 1 + 2n Then

gρ p,s,n ≤ const ·(1 − ρ)1/2−s−s/2n .

(7.4)


351

s s We can pick now n > 1−2/p . Then 12 − s − 2n > p1 − s. Again, we use formula (7.2). Let 0 < k < n and let s = s1 + s2 , where 0 < s1 < (n − k)(1 − 2/p), 0 < s2 < k(1 − 2/p). We have

T×T

|(∆n−k fρ )(τ k ζ)(∆kτ (gρ − 1))(ζ)|p τ dm(ζ)dm(τ ) ≤ |t|1+sp





T×T

|(∆n−k fρ )(τ k ζ)|2p dm(ζ)dm(τ ) τ |1 − τ |1+2s1 p

T×T

1/2 |(∆kτ gρ )(ζ)|2p dm(ζ)dm(τ )  |1 − τ |1+2s2 p

≤ fρ p2p,s1 n−k gρ p2p,s2 ,k ≤ const(1 − ρ)1/2−ps1 (1 − ρ)µp ≤ const(1 − ρ)γp , by Lemma 7.3 and (7.4), where µ > Let now k = 0. Then

1 2p



T×T

1 2p

− s1 + µ > 1/p − s.

1/p



− s2 and γ =

|(∆nτ fρ )(ζ)((gρ − 1))(ζ)|p dm(ζ)dm(τ ) |1 − τ |1+sp 1

≤ fρ p,s,n ξρ L∞

1

≤ const ·(1 − ρ) p −s (1 − ρ) 2 = const ·(1 − ρ)1/2+1/p−s , by Lemmas 7.5 and 7.3. Finally, let k = n. We have 



T×T

1/p |(fρ )(τ n ζ)(∆nτ (gρ − 1))(ζ)|p dm(ζ)dm(τ ) |1 − τ |1+sp

≤ const ·(1 − ρ)1/2−s−s/2n by (7.4), which completes the proof in the case p > 2. Consider now the case p ≤ 2. First of all, let us observe that if n > s and ρ is sufficiently close to 1, then gρ p,s,n ≤ const ·(1 − ρ)1/p−s .

(7.5)

Indeed, by Lemma 7.3, ωρ − z¯2 Bps ≤ const ·(1 − ρ)1/p−s . Since Bps is an algebra (see Appendix 2.6), it follows that gρ2 Bps ≤ const ·(1 − ρ)1/p−s . To obtain (7.5), it suffices to observe that for ρ sufficiently close to 1, gρ is uniformly separated away from 0 and apply to gρ2 an analytic branch of the square root (see Appendix 2.6).

352


Let now q be a positive number such that qp > 4 and let q = q/(q − 1). We have by H¨ older’s inequality

|(∆nτ gρ )(ζ)|p/2 |(∆nτ gρ )(ζ)|p/2 dm(ζ)dm(τ ) p gρ p,s,n = · |1 − τ | |1 − τ |sp/2 |1 − τ |sp/2 T×T

 ≤

T×T

 ×

T×T

1/q |(∆nτ gρ )(ζ)|pq/2 dm(ζ)dm(τ )  |1 − τ | |1 − τ |spq/2 |(∆nτ gρ )(ζ)|pq /2 |1 − τ |spq /2

1/q dm(ζ)dm(τ )  |1 − τ |

= gρ pq/2,s,n gρ pq /2,s,n ≤ const ·(1 − ρ)γp/2 (1 − ρ)(2/pq −s)p/2 p/2

p/2

2 by (7.5), where γ > pq − s (it is possible to find such a γ since pq > 2 and the lemma has already been proved for p > 2). It follows that

gρ p,s,n ≤ const ·(1 − ρ)(γ+2/pq −s)/2 . Clearly, 1 2

1 2 2 2 γ+ −s > − s + − s = 1/p − s. pq 2 pq pq

Finally, to estimate ξρ p,s,n , we apply formula (7.2) and argue as in the case p > 2. We can now obtain a similar result for the operators Am of best approximation by meromorphic functions. Theorem 7.7. Let 0 1/p. For each m ∈ Z+ there exists a sequence {ϕn } of functions in Bps such that ϕn Bps ≤ const, but lim Am ϕn Bps = ∞.

n→∞

The proof of Theorem 7.7 is almost the same as that of Theorem 7.1. The only difference is that in the definition of ξρ and ηρ we have to replace ζ¯2 with ζ¯m+2 . It can be shown in the same way as in Lemma 7.2 that wind(ηρ − ωρ ) = −(m + 1). Then sj (Hηρ ) = 1, 0 ≤ j ≤ m. Hence, Am ηρ = Aηρ = ωρ . The rest of the proof is exactly the same as for the case m = 0.


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The Space F 1 Let us now proceed to the space F1 of functions with absolutely converging Fourier series. Theorem 7.8. The operator A of best approximation by analytic functions is unbounded on the space F1 . Proof. Let K be a positive integer. We construct a function ϕ ∈ F1 such that AϕF 1 > KϕF 1 . Let r > 1. Consider the function z¯ 1 zj z¯ − r =− + 1− 2 ur = . z−r r r rj j≥0

Let ε > 0. For d = 1, 2, · · · , K we inductively find numbers rd > 1, positive integers Nd and Md > Nd , such that |1 − 1/rd |
1 so that (7.6) holds. Clearly, there exist a positive integer M1 and a trigonometric polynomial q1 of the form (7.7) that satisfies (7.8). Put g1 = q1 and N2 = M1 . We can now find r2 > 1, a positive integer M2 , and a trigonometric polynomial q2 of the form (7.7) that satisfies (7.8). def

Consider g2 = g1 q2 = q1 q2 . We have z + P+ q1 )(−¯ z + P+ q2 ) = z¯2 − z¯P+ q1 + q1 P+ q2 . g2 = (−¯ Clearly, P− g2 = z¯2 and gˆ2 (j) = 0 for 0 ≤ j ≤ K. Let us show that P+ g2 F 1 ≥ 4. Since the sets {j ∈ Z : (¯ z P+ q1 ) 7(j) = 0} and {j ∈ Z : (q1 P+ q2 ) 7(j) = 0} are obviously disjoint, it follows that P+ g2 F 1

= ¯ z P+ q1 F 1 + q1 P+ q2 F 1 ≥ P+ q1 F 1 + |q1 (1)(P+ q2 )(1)| = 4

(the last equality follows easily from (7.7) and (7.8)).

354


Suppose now that we have already found r1 , · · · , rl , N1 , · · · , Nl , M1 , · · · , Ml , trigonometric polynomials q1 , · · · , ql , l < K, such that (7.6)– def

(7.8) hold and the function gl = q1 q2 · · · ql has the following properties: gl (1) = 1,

gˆl (j) = 0

for

0≤j ≤K −l+1

and P+ gl F 1 ≥ 2l.

Let us find ql+1 . Put Nl+1 = max{j ∈ Z : gˆl (j) = 0} + K. Clearly, we can find rl+1 > 1, a positive integer Ml+1 , and a trigonometric polynomial ql+1 such that (7.6)–(7.8) hold. Put gl+1 = gl ql+1 . It is easy to see that P− gl+1 = (−1)l+1 z¯l+1 , gl+1 (1) = 1, and gˆl+1 (j) = 0 for 0 ≤ j ≤ K − l. Then the same reasoning as above shows that P+ gl+1 F 1

= ¯ z P+ gl F 1 + gl P+ ql+1 F 1 ≥ P+ gl F 1 + |gl (1)(P+ ql+1 )(1)| ≥ 2l + 2 = 2(l + 1).

Proceeding in this way we can construct q1 , · · · , qK . Consider the function u = z¯ur1 · · · urK . Clearly, u is a unimodular function and Hu f 2 = f 2 , where f (z) =

K d=1

1 . z − rd def

Therefore Hu = 1, and so AP− u = −P+ u. Now put ϕ = P− u. Put now v = z¯q1 · · · qK . We have qd − urd F 1 < ε, P− vF 1 = 1, and P+ vF 1 ≥ 2K. It is easy to see that if ε is sufficiently small, then AϕF 1 = P+ uF 1 > KP− uF 1 = KϕF 1 . A similar result holds for the operators Am . F1 .

Theorem 7.9. Let m ∈ Z+ . Then the operator Am is unbounded on

The proof is almost the same. The only difference is that we should define u = z¯m+1 ur1 · · · urK . Then Am Pu = −P+ u and it is easy to see that P+ uF 1 = ∞. K→∞ P− uF 1 lim

Finally, we obtain an estimate for AϕF 1 for functions ϕ in BM OA(m) . Theorem 7.10. Let ϕ ∈ BM OA(m) . Then AϕF 1 ≤ const ·mϕF 1 . Proof. Clearly, Aϕ = AP− ϕ + P+ ϕ. Hence, we may assume that P+ ϕ = O. Since B11 ⊂ F1 and the operator A is bounded on B11 , we have AϕF 1 ≤ const ·AϕB11 ≤ const ·ϕB11 .

8. Arguments of Unimodular Functions

355

Next, it follows from Corollary 6.1.2 that ϕB11

≤ const ·Hϕ S 1 ≤ const · rank Hϕ · Hϕ ≤ const ·mϕL∞ ≤ const ·mϕF 1 ,

since ϕ ∈ BM OA(m) .

8. Arguments of Unimodular Functions We have frequently used Sarason’s theorem (Theorem 3.3.10), which asserts that a unimodular function u is in V M O if and only if u admits a representation u = z n eiϕ , where n ∈ Z and ϕ is a real function in V M O. We will need analogs of this result for other function spaces. It is not difficult to obtain similar results for spaces of “smooth” functions (e.g., 1/p Λα ). However, for spaces that contain unbounded functions (such as Bp , 1 < p < ∞) such results are not trivial. To obtain analogs of Sarason’s theorem for such spaces, we are going to use the results on best approximation obtained earlier in this chapter. We consider a linear space X, P ⊂ X ⊂ V M O, of functions on T which satisfies the following axioms: (C1) f ∈ X =⇒ f¯ ∈ X and P+ f ∈ X; (C2) AX ⊂ X; (C3) if f ∈ X L∞ and ϕ is analytic in a neighborhood of f (T), then ϕ ◦ f ∈ X. Remark. Note that (C3) implies that X is an algebra with respect to pointwise multiplication, since 4f g = (f + g)2 − (f − g)2 . Theorem 8.1. Suppose that X is a space of functions on T that satisfies the axioms (C1)–(C3) and let u be a unimodular function in X. Then u admits a representation u = z n eiϕ , where n ∈ Z and ϕ is a real function in X. Proof. Since X ⊂ V M O, the Toeplitz operator Tu is Fredholm. Multiplying u by z n , we can reduce the situation to the case ind Tu = 0, that is, Tu is invertible. In this case distL∞ (u, H ∞ ) < 1 and h = Au is an outer function (see Theorem 3.1.13). By (C2), h ∈ X. Since X L∞ is an algebra, it follows that u ¯h ∈ X. We have 1 − u ¯hL∞ = u − hL∞ < 1, and so the essential range of the function u ¯h is contained in the disk {ζ ∈ C : |1 − ζ| < δ} for some δ < 1. Therefore we can represent the function u ¯h in the form u ¯h = |h| exp(iψ), where ψ is a real-valued function such that ψL∞ < π/2. ¯ ∈ X. It follows Clearly, inf |h(τ )| > 0, and so by (C3), log |h| = 12 log(hh) τ ∈T

from (C1) that log |h| ∈ X. Applying an analytic branch of logarithm to

356


u ¯h, we obtain log u ¯h = log |h| + iψ ∈ X (see (C3)). Therefore ψ ∈ X. By Theorem 3.3.10, u can be represented as u = exp(iβ), where β is a real function in V M O. We have |h| 1=u exp(iψ) = exp i(β − log |h| + ψ + c) h for some c ∈ R. It follows from Corollary 3.2.7 that β − log |h| + ψ = const, and so β ∈ X. It is easy to see that all spaces considered in §4 that are hereditary for A satisfy (C1)–(C3). Remark. If X is a space of functions satisfying (C1)–(C3) and X ⊂ L∞ , then it is easy to see that the converse to Theorem 8.1 holds: if n ∈ Z and ϕ ∈ X, then z n eiϕ ∈ X. However, this is also true for the Besov spaces 1/p Bp , 1 < p < ∞, which contain unbounded functions. Corollary 8.2. Let u be a unimodular function and let 0 < p < ∞. 1/p Then u ∈ Bp if and only if u admits a representation u = z n eiϕ , where 1/p n ∈ Z and ϕ is a real-valued function in Bp . Proof. If p ≤ 1, then Bp ⊂ L∞ , and the result follows from the Remark preceding the lemma. If p > 1, then it follows immediately from 1/p the definition of Bp in terms of the difference operator ∆τ (see Appendix 1/p 1/p 2.6) that f ◦ ϕ ∈ Bp whenever ϕ ∈ Bp and f satisfies a Lipschitz condition |f (ζ1 ) − f (ζ2 )| ≤ const |ζ1 − ζ2 |. 1/p

9. Schmidt Functions of Hankel Operators In this section we study properties of Schmidt functions of Hankel operators whose symbols belong to certain function spaces. In particular we prove that if X is a function space satisfying the axioms (A1)–(A4) or the axioms (B1)–(B3) and ϕ ∈ X, then all Schmidt functions of the Hankel operator Hϕ belong to X. Recall that for ϕ ∈ V M O the Schmidt functions of the compact Hankel operator Hϕ are eigenfunctions of the operator Hϕ∗ Hϕ that correspond to nonzero eigenvalues (see §4.1). Consider first the case when X is a space of functions on T, P ⊂ X ⊂ V M O, that satisfies the following axioms. (C1) f ∈ X =⇒ f¯ ∈ X and P+ f ∈ X; (C2) AX ⊂ X;

9. Schmidt Functions of Hankel Operators

357

(C3 ) if f ∈ X and ϕ is analytic in a neighborhood of f (T), then ϕ ◦ f ∈ X. Recall that axioms (C1) and (C2) have been introduced in §7. The axiom C3 is slightly stronger than (C3). In particular, it implies that X is an algebra with respect to pointwise multiplication. It is easy to see that if X satisfies the axioms (A1)–(A4) or the axioms (B1)–(B3), then X also satisfies (C1), (C2), and (C3 ). Lemma 9.1. Suppose that X satisfies (C1), (C2), and (C3 ). Then ¯ ϕ − Aϕ has the form cz n h/h, where c ∈ C, n is a negative integer, and h is an outer function invertible in X. Proof. Suppose that Aϕ = ϕ. Then ϕ − Aϕ = cu, where c is a nonzero complex number and u is a unimodular function (see Theorem 1.1.4). By (C2), u ∈ X. It follows from Theorem 8.1 that u admits a representation u = z n eiξ , where ξ is a real function in X. Since Aϕ is the best approximation to ϕ, it follows that distL∞ (u, H ∞ ) = Hu = 1, and so the Toeplitz operator Tu is not left invertible, which implies that n < 0 (see Theorem 3.1.11 and Corollary 3.2.6). def Put now h = exp 1 (ξ˜ − iξ) . Then h is an outer function. It follows 2

from (C1) and (C3 ) that h ∈ X, and so h−1 ∈ X. Lemma 9.2. Suppose that X satisfies (C1), (C2), and (C3 ). Let ϕ be a function in X such that P− ϕ = O, and let f be a maximizing vector for the Hankel operator Hϕ on H 2 , i.e., Hϕ f 2 = Hϕ · ϕ2 . Then f ∈ X. Proof. Put v = ϕ − Aϕ. Then Hv = Hϕ and v∞ = Hϕ . By Lemma ¯ 9.1, v = c¯ z n h/h, where n is a positive integer, c ∈ C, and h is an outer function invertible in X. Clearly, {g ∈ H 2 : Hv g2 = Hv · g2 } = Ker Tv . If q is an analytic polynomial of degree at most n − 1, then obviously qh ∈ KerTv . On the other hand, Tv = cTz¯n Th¯ T1/h , and so dim Ker Tv = dim Ker Tz¯n = n. It follows that Ker Tv = {qh : q ∈ P + , deg q ≤ n − 1} ⊂ X, which proves the result. Theorem 9.3. Suppose that X satisfies (C1), (C2), and (C3 ). Let ϕ be a function in X such that P− ϕ = O. Then all Schmidt functions of Hϕ belong to X. Proof. Suppose that Hϕ∗ Hϕ f = s2 f , where f ∈ H 2 and s is a nonzero singular value of Hϕ . If s = s0 (Hϕ ), it follows from Lemma 9.2 that f ∈ X. Assume that s < s0 (Hϕ ). Consider the space E = {ξ ∈ H 2 : Hϕ∗ Hϕ ξ = s2 ξ}.

358


Let d = dim E. We can now choose m > 0 so that s = sm (Hϕ ) and sm−1 (Hϕ ) > sm (Hϕ ) = · · · = sm+d−1 (Hϕ ) > sm+d (Hϕ ). Put g = 1s Hϕ f and ϕs = g/f . Then ϕs is a unimodular function (see Corollary 4.1.5). Clearly, Hϕs f = g, and so Hϕs f 2 = Hϕs f · f 2 , which means that f is a maximizing vector of Hϕs f . It has been proved in the proof of Theorem 4.1.1 that the Hankel operators Hϕ and Hsϕs coincide on a subspace of H 2 of codimension m. Hence, by Kronecker’s theorem P− (ϕ−ϕs ) is a rational function of degree m. Since the rational functions belong to X, it follows that P− ϕs ∈ X, and so by Lemma 9.2, f ∈ X. The same can be proved about Schmidt functions of Hϕ∗ . Indeed, if g is an eigenfunction of Hϕ Hϕ∗ that corresponds to a nonzero eigenvalue s2 , then there exists a function f ∈ H 2 such that Hϕ∗ Hϕ f = s2 f and Hϕ f = sg. It follows from Theorem 9.3 that f ∈ X. This implies that g ∈ X. To complete this section we describe Schmidt functions of compact Hankel operators as well as Schmidt functions of Hankel operators of class S p . Theorem 9.4. Let ϕ be a function in L2 such the Hankel operator Hϕ is compact (or Hϕ ∈ S p , 0 < p < ∞). Suppose that f is a Schmidt function of Hϕ . Then f has the form f = qeψ , where q is an analytic polynomial 1/p and ψ ∈ V M OA (or ψ ∈ Bp + ). 1/p

Proof. Let X = V M O if Hϕ is compact or X = Bp if Hϕ ∈ S p . As in Theorem 9.3 it is sufficient to consider the case when f is a maximizing vector of Hϕ . Consider the function v = ϕ − Aϕ. By Theorem 8.1, v has the form v = c¯ z n eiη , where η is a real function in X. As in Theorem 9.3, n > 0. ¯ Clearly, eiη = h/h, where h = exp( 12 (˜ η − iη)). Then h is an outer function def

and ψ = 12 (˜ η − iη) ∈ X. We have shown in the proof of Theorem 9.3 that f has the form qh, where q is a polynomial of degree at most n − 1. Remark 1. The same can be proved about Schmidt functions of Hϕ∗ . This follows from the fact that the operator J defined in the proof of Lemma 4.1.6 by Jg = z¯g¯ maps the eigenspace of Hϕ∗ Hϕ , which corresponds to a positive number s2 onto the eigenspace of Hϕ Hϕ∗ , which corresponds to the same eigenvalue. Remark 2. Any function f of the form f = qeψ , where q is an analytic 1/p polynomial and ψ ∈ V M OA (or ψ ∈ Bp + ),q is a Schmidt function of a compact Hankel operator (a Hankel operator of class S p ). Indeed, let def ¯ = z¯n exp(−2i Im ψ) ∈ X. Then n = deg q + 1, h = eψ and ϕ = z¯n h/h 1/p

Hϕ is compact (or Hϕ ∈ Bp ) and f is a maximizing vector of Hϕ , since ¯ Hϕ f = z¯n h.

10. Continuity in the sup-Norm

359

Corollary If Hϕ is compact and f is a Schmidt function of Hϕ . 9.5. Then f ∈ H p. p 1) their Schmidt functions always belong to 1/p V M O (or to Bp ). However, for p ≤ 1 this is true. Corollary 9.6. Suppose that Hϕ is a Hankel operator of class S p , 1/p 0 < p ≤ 1. Then the Schmidt functions of Hϕ belong to Bp . 1/p

The result follows from the fact that if ψ ∈ Bp , 0 < p ≤ 1, then 1/p eψ ∈ Bp (see Appendix 2.6).

10. Continuity in the sup-Norm In this section we are going to study continuity properties of the operator A of best approximation by analytic functions in the sup-norm. To be more precise, we consider here the operator A as a nonlinear operator from C(T) to H ∞ . The continuity problem is very important in applications. Suppose we are given a function ϕ ∈ C(T) and we want to evaluate Aϕ. We can approximate ϕ by functions ϕn for which it is easier to evaluate Aϕn (e.g., ϕn can be a polynomial, a rational function, etc.). The problem of whether the functions Am ϕn give a good approximation for Am ϕ reduces to the problem of whether ϕ is a continuity point of the operator A. It is obvious that the zero function is a continuity point of A since ϕ − Aϕ∞ ≤ ϕ∞ , and so Aϕ∞ ≤ 2ϕ∞ . In other words, the operator A : C(T) → H ∞ is bounded. It is also obvious that any function in CA is a continuity point of A since Aϕ = ϕ for ϕ ∈ CA . The main result of this section shows that there are no other continuity points of A in the sup-norm. We also obtain similar results for the operators Am of best approximation by meromorphic functions with at most m poles. Such results look disappointing for practical purposes. However, we shall see in §11 that if X is a decent space and we consider the continuity problem for the operator A : X → X+ in the norm of X, the situation is quite different. Theorem 10.1. Let ϕ be a continuous function on T such that ϕ ∈ CA . Then there exists a sequence ϕn ∈ C(T) such that lim ϕn − ϕ∞ → 0

n→∞

but

lim inf Aϕn − Aϕ∞ > 0. n→∞

Note that it follows from Theorem 10.1 that the operator A defined on H ∞ +C is discontinuous in the L∞ -norm at any point f ∈ (H ∞ +C)\H ∞ .

360


Indeed, if f = ϕ + ψ, where ϕ ∈ C(T) and ψ ∈ H ∞ , then Af = Aϕ + ψ, and since A is discontinuous at ϕ by Theorem 10.1, it follows that it is discontinuous at f . To prove Theorem 10.1, we need the following construction. Let 0 < δ < 1/4. Put 9

8 π 3π def I = 1 + eiθ : θ ∈ , 2 2 and def Ωδ = {ζ ∈ C : dist(ζ, I) < δ}. It is well known (see Goluzin [1], Ch. X, §1) that if ι is a conformal map of D onto Ωδ , then ι extends to a continuous map of clos D onto clos Ωδ . We can always choose such a conformal map ι that ι(1) = −δ. Remark. In fact, we do not need the result that ι extends by continuity. For our purpose we can replace Ωδ with ι({ζ ∈ C : |ζ| < r}, where r is def

sufficiently close to 1 and replace ι with ιr = ι(rz). Let now ρ ∈ (0, 1). Consider the conformal map γρ : D → D defined by γρ (ζ) =

ρ−ζ , 1 − ρζ

ζ ∈ D.

It is easy to see that Re γρ (ζ) ≤ ρ

⇐⇒

Re ζ ≥ ρ.

(10.1)

Proof of Theorem 10.1. Consider separately two cases. Suppose first that Aϕ ∈ C(T). Let {ϕn } be a sequence of trigonometric polynomials that converges to ϕ in the sup-norm. It has been shown in §4 that Aϕn ∈ C ∞ . If the sequence {Aϕn } converged in the sup-norm to Aϕ, it would follow that Aϕ ∈ C(T), which contradicts our assumption. Suppose now that Aϕ ∈ C(T). Put u = ϕ − Aϕ. Then Au = O. Clearly, it is sufficient to show that A is discontinuous at u. Without loss of generality we may assume that the function u is unimodular and u(1) = 1. By Theorem 5.1, u has negative winding number. def Consider the function ψρ = ι ◦ γρ . It is a conformal map of D onto Ωδ . Define the function ξρ by ξρ =

ψρ − u (|ψρ − u| − 1) . |ψρ − u|

Let ρ be so close to 1 that |u(ζ) − 1| ≤ δ and |ι(ζ) + δ| ≤ δ whenever Re ζ ≥ ρ. Let us show that ξρ ∞ ≤ 2δ. Suppose first that Re ζ ≥ ρ. Then |u(ζ) − 1| ≤ δ and |ψρ (ζ) − 1| − 1 ≤ δ, which implies that |ψρ (ζ) − u(ζ)| − 1 ≤ 2δ. Suppose now that Re ζ ≤ ρ. It follows from (10.1) that γρ (ζ) ≥ ρ, and so |ψρ (ζ) + δ| ≤ δ. Since |u(ζ)| = 1, it follows that |ψρ (ζ) − u(ζ)| − 1 ≤ 2δ. This completes the proof of the fact that ξρ ∞ ≤ 2δ.

10. Continuity in the sup-Norm

361

In particular, the above reasoning shows that for ρ sufficiently close to 1 the function |ψρ − u| is separated away from zero, and so ξρ ∈ C(T). To complete the proof of the theorem it is sufficient to show that A(u + ξρ ) = ψρ . Indeed, we can make δ as small as possible. On the other hand, Au = O, while ψρ ∞ ≥ 1. To prove that A(u + ξρ ) = ψρ , we use Theorem 5.1. We show that the function (u + ξρ ) − ψρ has constant modulus and negative winding number. It is easy to see that (u + ξρ ) − ψρ = We have

wind((u + ξρ ) − ψρ ) = wind

u − ψρ . |u − ψρ |

u − ψρ |u − ψρ |

= wind(u − ψρ ),

since |u − ψρ | is separated away from 0. Hence, ¯ψρ ). wind((u + ξρ ) − ψρ ) = wind u + wind(1 − u Since wind u < 0, it is sufficient to show that wind(1 − u ¯ψρ ) = 0. This follows from the fact that the function u ¯ψρ − 1 on T does not take values in R+ = {ζ ∈ C : Im ζ = 0, Re ζ ≥ 0}. Indeed, if Re ζ ≥ ρ, then u ¯(ζ) takes values in a small neighbourhood of 1 while ψρ takes values in ∂Ωρ , and so u ¯(ζ)ψρ (ζ) − 1 ∈ R+ . On the other hand, if Re ζ ≤ ρ, then u ¯(ζ)ψρ (ζ) is within a small neighborhood of the origin, and so u ¯(ζ)ψρ (ζ) − 1 ∈ R+ . This completes the proof of the theorem. We can now prove a similar result for the operators Am of best approximation by meromorphic functions of degree at most m. Again, we consider the operator Am here as an operator from C(T) to L∞ . Theorem 10.2. Let ϕ be a continuous function on T such that ∞ . Then the operator Am : C(T) → L∞ is discontinuous at ϕ. ϕ ∈ H(m) Proof. The proof is almost the same as the proof of Theorem 10.1. If Am ϕ ∈ C(T), we can approximate ϕ by a sequence of trigonometric polynomials {ϕn } and as in the case m = 0 it is obvious that {Am ϕn } does not converge to Am ϕ. If Am ϕ ∈ C(T), we consider the function u = ϕ − Am ϕ. Again, without loss of generality we may assume that u is unimodular and u(1) = 1. Suppose now that the singular value sm (Hϕ ) has multiplicity µ, k ≤ m ≤ k + µ − 1, and sk (Hϕ ) = · · · = sk+µ−1 (Hµ ). Then wind u = −(2k + µ) (see Theorem 4.1.7). We can define the functions ξρ and ψρ in the same way as in the proof of Theorem 10.1. We claim that Am (ϕ + ξρ ) = Am ϕ + ψρ . Indeed, we have already proved that the function ϕ + ξρ − (Am ϕ + ψρ ) = u + ξρ − ψρ

362


is unimodular and its winding number is equal to wind u = −(2k + µ). Clearly, deg P− (Am ϕ + ψρ ) = k. It follows from Theorem 5.3 that Ak (ϕ + ξρ ) = Ak ϕ + ψρ . Consider the singular value sk (Hϕ+ξρ ). Clearly, if k > 0, then sk (Hϕ+ξρ ) < sk−1 (Hϕ+ξρ ). Let ν be the multiplicity of the singular value sk (Hϕ+ξρ ). Then by Theorem 4.1.7, wind(ϕ + ξρ − (Ak ϕ + ψρ )) = 2k + ν, and so µ = ν. Hence, Ak (ϕ + ξρ ) = Am (ϕ + ξρ ), which completes the proof.

11. Continuity in Decent Banach Spaces In this section we study continuity properties of the operator A of best approximation by analytic functions in the norm of a decent space X. As we have already mentioned in the previous section, such continuity problems are very important in applications. We show that if ϕ ∈ X and P− ϕ = O, then ϕ is a continuity point of A if and only if s0 (Hϕ ) is a singular value of Hϕ of multiplicity 1. We also consider the problem of whether a function ϕ in X is a continuity point of the operator Am of best approximations by functions in BM OA(m) . We shall see that the answer to this question depends on the multiplicity of the singular value sm (Hϕ ) of the Hankel operator Hϕ . These results do not seem to be very helpful for practical purposes in case of multiple singular values. To find a practical tool in this case, we study continuity properties of the operators Am as operators from X to Lq , q < ∞, and we show that such operators are continuous everywhere. Let X be a decent space of functions on T (see §2). Then for ϕ ∈ X the Hankel operator Hϕ maps X+ to X− . Recall that the operator Hϕ∗ is an 2 into H 2 and Hϕ∗ g = P+ ϕg. ¯ It follows from (A1)–(A4) operator from H− ∗ that Hϕ Hϕ maps X+ into itself and it is a compact operator on X+ . Lemma 11.1. Let X be a decent space of functions on T and let ϕ ∈ X. Then the operator Hϕ∗ Hϕ has the same spectrum on H 2 and X+ . If λ is a nonzero point of the spectrum, then for every n ≥ 1 {f ∈ X+ : (Hϕ∗ Hϕ − λI)n ϕ = O} = {f ∈ H 2 : (Hϕ∗ Hϕ − λI)n ϕ = O} = {f ∈ H 2 : (Hϕ∗ Hϕ − λI)ϕ = O}. Proof. Since the operator Hϕ∗ Hϕ is compact on both H 2 and X+ , it follows that the spectrum of Hϕ∗ Hϕ on both H 2 and X+ consists of 0 and eigenvalues. Since X+ ⊂ H 2 , it follows that each eigenvector of the operator Hϕ∗ Hϕ on X+ is also an eigenvector of the operator Hϕ∗ Hϕ on H 2 . Theorem 9.3 says that each eigenvector of the operator Hϕ∗ Hϕ on H 2 belongs to X+ . Hence, the operator Hϕ∗ Hϕ has the same spectrum on X+

11. Continuity in Decent Banach Spaces

363

and H 2 . To complete the proof, we note that since the operator Hϕ∗ Hϕ on H 2 is self-adjoint, we have {f ∈ H 2 : (Hϕ∗ Hϕ − λI)n ϕ = O} = {f ∈ H 2 : (Hϕ∗ Hϕ − λI)ϕ = O}, n ≥ 1. Let ϕ ∈ X, m ∈ Z+ . We denote by µm (Hϕ ) the multiplicity of the 2 : singular value of the Hankel operator Hϕ : H 2 → H− µm (ϕ) = dim{f ∈ H 2 : Hϕ∗ Hϕ f = s2m (Hϕ )f }. Theorem 11.2. Let m ∈ Z+ and let X be a decent space. Suppose that ϕ is a function in X such that µm (Hϕ ) = 1. Then ϕ is a continuity point of the operator Am in the norm of X. We need the following lemma. Lemma 11.3. Let X, m, and ϕ satisfy the hypotheses of Theorem 11.2. Let f be an eigenfunction of Hϕ∗ Hϕ with eigenvalue s2m (Hϕ ). Then f has no zeros on T. Proof. Let E be the S-invariant subspace of H 2 spanned by f . It has been shown in the proof of Theorem 4.1.1 (see §4.1) that E has the form bH 2 , where b is a finite Blaschke product of degree m. Then f can be represented as f = bh, where h is an outer function in X+ . It has also been ¯ where c ∈ C. z¯bh, shown in the proof of Theorem 4.1.1 that Hϕ f = c¯ def

Let v = ϕ − Am ϕ. Then v = (Hϕ f )/f (see (4.1.12)). We have v=

¯b h ¯ Hϕ f = c¯ z . f bh

By Theorem 4.1.7, wind v = −(2m + 1). Clearly, wind v = −1 + wind

¯ ¯ ¯b h h + wind = −(1 + 2m) + wind , b h h

and so wind

¯ h = 0. h

def ¯ Then wind u = −1, and so the singular value 1 = s0 (Hu ) Let u = z¯h/h. of the Hankel operator Hu on H 2 has multiplicity 1 and h is a maximizing vector of Hu . If h(τ ) = 0 for some point τ ∈ T, then it has been shown in §2 that the function h/(1 − τ¯z) belongs to X+ and is also a maximizing vector of Hu . This contradicts the fact that the multiplicity of s0 (Hu ) is 1. Therefore h has no zeros on T, which completes the proof of the lemma.

Proof of Theorem 11.2. Let f be an eigenfunction of Hϕ∗ Hϕ with eigenvalue s2m (Hϕ )2 . As we have already mentioned in the proof of Lemma 11.3, Hϕ f Am ϕ = ϕ − . f

364


By Lemma 11.1, f ∈ X. Let J be a positively oriented Jordan curve that surrounds s2m (Hϕ ) and such that there are no points of the spectrum of Hϕ∗ Hϕ on J and the only point of the spectrum of Hϕ∗ Hϕ inside J is s2m (Hϕ ). By Lemma 11.1 the same can be said about the spectrum of the operator Hϕ∗ Hϕ on X+ . Obviously, the operators Hϕ∗ n Hϕn converge to Hϕ∗ Hϕ in the operator norm on H 2 and on X+ . It follows that if n is sufficiently large, then the spectrum of Hϕ∗ n Hϕn (whether on H 2 or X+ ) has no points on J and has one point inside J . Consider now the spectral projections P and Pn on X+ defined by

1 1 ∗ −1 (ζI − Hϕ Hϕ ) dζ, Pn = (ζI − Hϕ∗ n Hϕn )−1 dζ. P= 2πi J 2πi J (11.1) It is easy to see from the above integral formulas that Pn → P in the operator norm. Clearly, for sufficiently large n the operator Pn is a projection onto a one-dimensional subspace. Put fn = Pn f . Then for sufficiently large n, fn = O and Hϕ∗ n Hϕn fn = s2m (Hϕn )fn . It follows from the convergence of Pn to P that fn → f in X+ , and so fn → f in L∞ . It follows now from Lemma 11.3 that for large n the function fn does not vanish on T. Therefore by the axiom (A4) lim 1/fn − 1/f X = 0.

n→∞

Since

Hϕn fn , fn it follows that limn→∞ Am ϕn − Am ϕX = 0. Let us proceed now to the case of a multiple singular value sm (Hϕ ). Suppose that µ = µm (Hϕ ) ≥ 2 and Am ϕn = ϕn −

sk (Hϕ ) = · · · = sk+µ−1 (Hϕ ) > sk+µ (Hϕ ),

k ≤ m ≤ k + µ − 1. (11.2)

Consider first the case m = k + (µ − 1)/2. Theorem 11.4. Let m ∈ Z+ , ϕ ∈ V M O, µ = µm (Hϕ ) ≥ 2, and suppose that (11.2) holds. Assume that m = k + (µ − 1)/2 and Λ is a finite set in D of at least (µ − 1)/2 points that are not poles of P− Ak ϕ. Then (n) there exist sequences ζλ n≥0 , λ ∈ Λ, of nonzero complex numbers such (n)

that lim ζλ n→∞

= 0, λ ∈ Λ, and Am ϕn − Am ϕBM O → 0,

where ϕn = ϕ +

ζ (n) λ . z−λ

λ∈Λ


365

Remark. Note that all functions ϕn belong to a finite-dimensional subspace, and so ϕn − ϕ → O in any norm. On the other hand, if X satisfies (A1)–(A4), then the norm in X is certainly stronger than the norm in BM O. So Theorem 11.4 says that the convergence of ϕn to ϕ in any norm does not guarantee the convergence of Am ϕn to Am ϕ even in the BM O norm. We need the following lemma. Lemma 11.5. Let v, vn be unimodular functions such that the Toeplitz operators Tv and Tvn are Fredholm. If ind Tvn = ind Tv , then vn − vBM O → 0. Proof. Multiplying v and vn by z ind Tv , passing, if necessary, to complex conjugation and to a subsequence, we can assume that ind Tv = 0 and ind Tvn > 0. Then Tv is invertible while Tvn is not left invertible. It follows that Hv < 1 while Hvn = 1 (see Theorem 3.1.11). It remains to observe that the convergence of vn to v in BM O implies that lim Hvn − Hv = 0 n→∞

(see Theorem 1.1.3), which leads to a contradiction.

Proof of Theorem 11.4. By Corollary 6.2, there exist sequences (n) {ζλ }n≥0 tending to 0 such that the multiplicity of the singular value sm (Hϕn ) equals 1. Then the function ϕn − Am ϕn has constant modulus and by Theorem 4.1.7, ind Tϕn −Am ϕn = 2m + 1, while ind Tϕ−Am ϕ = 2k + µ = 2m + 1, since m = k + (µ − 1)/2. Put 1 (ϕ − Am ϕ), v= sm (Hϕ )

vn =

1 (ϕn − Am ϕn ). sm (Hϕn )

Clearly, lim sm (Hϕn ) = sm (Hϕ ), the functions v and vn are unimodular. n→∞ The result follows now from Lemma 11.4. Now we are in a position to describe the continuity points of the operator of best approximation A by analytic functions. Theorem 11.6. Let X be a decent function space and let ϕ be a function in X such that P− ϕ = O. Then ϕ is a continuity point of the operator A in the norm of X if and only if the multiplicity of the singular 2 is 1. value s0 (Hϕ ) of the Hankel operator Hϕ : H 2 → H− The result follows immediately from Theorems 11.2 and 11.4. Let us now proceed to the case m = k+(µ−1)/2. This case is considerably more complicated. We prove that if µ ≥ 2 and X = B11 , X = F1 , or X = λα , α > 0, α ∈ Z, then ϕ is not a continuity point of Am in the norm of X. Let us first consider the case when X = B11 or X = F1 .

366


Theorem 11.7. Suppose that X = B11 or X = F1 . Let m ∈ Z+ , ϕ ∈ X, µ = µm (Hϕ ) ≥ 2, and suppose that (11.2) holds with m = k + (µ − 1)/2. Then there exists a sequence {ϕn } of functions in X converging to ϕ such that Am ϕn − Am ϕBM O → 0

as

n → ∞.

To prove the theorem, we need a lemma. For ρ ∈ (0, 1) consider the conformal map γρ : D → D defined by γρ (ζ) =

ρ−ζ . 1 − ρζ

Lemma 11.8. Let ϕ be a function in X+ such that ϕ(1) = 0. Then lim ϕγρ − ϕX = 0,

(11.3)

ρ→1

where ρ ∈ (0, 1). Proof. Let us first prove the lemma for X = B11 . We have ϕγρ − ϕ = ϕ(γρ − 1). Since X is a Banach algebra and sup γρ X < ∞ (see Lemma ρ

1.2), it is sufficient to prove (11.3) for analytic polynomials ϕ such that ϕ(1) = 0. Let us show that

|(ϕ(γρ − 1)) |dm2 = 0. lim ρ→1

D

It is easy to see that lim γρ (ζ) − 1 = 0,

ρ→1

lim γ ρ (ζ) = 0,

ρ→1

lim γ ρ (ζ) = 0,

ρ→1

ζ ∈ D,

and the convergence is uniform on each set of the form {ζ ∈ D : |1−ζ| ≥ δ}, δ ≥ 0. Clearly, it is sufficient to prove that

|γρ − 1|dm2 = 0, lim |γ ρ |dm2 = 0, lim |ϕγ ρ |dm2 = 0. lim ρ→1

D

ρ→1

D

ρ→1

D

It is very easy to see that the first two limits are zero. Let us show that the third one is zero. Since sup γρ X < ∞, it follows that ρ

sup ρ

D

|γ ρ |dm2 < ∞.

Let ε > 0. Since ϕ(1) = 0, we can choose a positive δ such that

ε |ϕγ ρ |dm2 < . sup 2 ρ {ζ∈D:|1−ζ| 0, α ∈ Z. Theorem 11.9. Let m ∈ Z+ , ϕ be a function in λα , α > 0, α ∈ Z, µ = µm (Hϕ ) ≥ 2, and suppose that (11.2) holds with m = k + (µ − 1)/2. Then there exists a sequence {ϕn } of functions in λα converging to ϕ such that Am ϕn − Am ϕBM O → 0 as n → ∞. To prove Theorem 11.9, we obtain an analog of Lemma 11.8.

368


Lemma 11.10. Let α > 0, α ∈ Z, and let d be the integer satisfying d − 1 < α < d. If f is a function in (λα )+ such that f (s) (1) = 0, 0 ≤ s ≤ d − 1, then lim f γρ − f Λα = 0,

(11.6)

ρ→1

where ρ ∈ (0, 1). To prove Lemma 11.10, we need one more lemma. Lemma 11.11. Let f be a function in Λα , α > 0, such that f (j) (1) = 0, 0 ≤ j ≤ d − 1, where d − 1 < α < d. Then sup f (γρ − 1)Λα < ∞.

ρ∈(0,1)

Proof of Lemma 11.11. Clearly, it is sufficient to prove that sup sup |(f (γρ − 1))(d) (ζ)|(1 − |ζ|)d−α < ∞,

d − 1 < α < d,

ρ∈(0,1) ζ∈D

(see Appendix 2.6). This reduces to the following inequalities: sup sup ρ∈(0,1) ζ∈D

|f (d−j) (ζ)| (1 − ρ)|(1 − |ζ|)d−α < ∞, |1 − ρζ|j+1

1 ≤ j ≤ d, (11.7)

and sup sup |f (d) (ζ)| · |γρ (ζ) − 1|(1 − |ζ|)d−α = 0.

(11.8)

ρ∈(0,1) ζ∈D

Since f (s) (1) = 0, 0 ≤ s ≤ d − 1, it follows that |f (d−j) (ζ)| ≤ const ·|1 − ζ|α−d+j , and so |f (d−j) (ζ)| (1 − ρ)|(1 − |ζ|)d−α ≤ const |1 − ρζ|j+1

|1 − ζ| |1 − ρζ|

j

1 − ρ (1 − |ζ|)d−α · , |1 − ρζ| |1 − ζ|d−α

which easily implies (11.7). Inequality (11.8) follows easily from the fact that |f (d) (ζ)| ≤ const(1 − |ζ|)α−d . Proof of Lemma 11.10. By Lemma 11.11, it is sufficient to show that (11.6) holds for f in a dense subset of the space def

Y = {g ∈ (λα )+ : g (j) (1) = 0, 0 ≤ j ≤ d − 1}. It is easy to see that if f ∈ Y , then we can approximate f by polynomials g in Y . If g is a polynomial in Y , we can consider the polynomial gn = g −cn z n , where cn is chosen so that g (d) (1) = 0, n ≥ 1. It is easy to see that z n Λα ≤ const ·nα and |cn | ≤ const ·n−d , and so lim cn z n λα = 0. n→∞


369

Therefore it is sufficient to prove (11.6) for functions f in C d (T) such that f (j) (1) = 0, 0 ≤ j ≤ d. For such functions f it is easy to verify that lim f (γρ − 1)C d (T) = 0,

ρ→1

which immediately implies (11.6). Proof of Theorem 11.9. Clearly, we can assume that ϕ ∈ (λα )− . Let q be an analytic polynomial of degree at most d−1 such that (ϕ−q)(j) (1) = 0, 0 ≤ j ≤ d − 1. If we apply Lemma 11.10 to the function z d−1 (ϕ − q) in (λα )+ , we find that lim (ϕ − q)(¯ γρ − 1)Λα = 0.

ρ→1

Clearly, Am (ϕ) = q+Am (ϕ−q) and Am ((ϕ−q)¯ γρ +q) = q+Am ((ϕ−q)¯ γρ ). Now put ξ = ϕ − q. The rest of the proof is exactly the same as the proof of Theorem 11.7. As we have already seen, in the case when sm (Hϕ ) is a multiple singular value of Hϕ and lim ϕn − ϕX = 0 we should not expect the convergence n→∞ of Am ϕn to Am ϕ even in the norm of BM O. Let us now see whether we can relax the norm of BM O so that Am ϕn converges to Am ϕ. Theorem 11.12. Suppose that m ∈ Z+ , X is a decent function space and let ϕ ∈ X. If {ϕn } is a sequence of functions in X which converges to ϕ in the norm of X, then for all q < ∞ lim Am ϕn − Am ϕLq = 0.

n→∞

Proof. The case sm (Hϕ ) = 0 is obvious. Assume that sm (Hϕ ) > 0. Let s = sm (Hϕ ), µ = µm (ϕ), and s = sk (Hϕ ) = · · · = sk+µ−1 (Hϕ ). As in the proof of Theorem 11.2 consider a positively oriented Jordan curve J that surrounds s2m (Hϕ ) and such that there are no points of the spectrum of Hϕ∗ Hϕ on J and the only point of the spectrum of Hϕ∗ Hϕ inside J is s2m (Hϕ ). Then for large values of n the points of J are in the resolvent set of Hϕ∗ n Hϕn and there are µ points of the spectrum of Hϕ∗ n Hϕn (counted with multiplicities) inside J . Recall that by Lemma 11.1, the operators Hϕ∗ Hϕ and Hϕ∗ n Hϕn have the same spectra on H 2 and X+ . Consider the spectral projections P and Pn on X+ defined by (11.1). Then limn→∞ Pn = P in the operator norm of X+ . Let fn be an eigenfunction of Hϕ∗ n Hϕn with eigenvalue s2m (Hϕn ). We normalize it by the condition fn X = 1. Clearly, for large values of n, Pfn = O, and so Pfn is an eigenfunction of Hϕ∗ Hϕ with eigenvalue s2m (Hϕ ). Let now f be an arbitrary eigenfunction of Hϕ∗ Hϕ with eigenvalue s2m (Hϕ ). Put 1 1 Hϕ f v= = · (ϕ − Am ϕ). sm (Hϕ ) f sm (Hϕ )

370


Then v is a unimodular function, Hv f L2 = f L2 , and the dimension of the subspace E = {g ∈ H 2 : Hv gL2 = gL2 } is equal to 2k + µ (see §4.1). By the remark after Theorem 2.4, v ∈ X and by Theorem 8.1, v admits a representation ¯ h v = z¯2k+µ , h where h is an outer function in X+ that does not vanish on T. It is easy to see that E = {ph : p ∈ P + , deg p < 2k + µ}. Therefore for each n there exists a polynomial pn in P + such that Pfn = pn h. Let us show that lim (ϕn − Am ϕn ) = ϕ − Am ϕ

n→∞

in Lq .

We have

Hϕn fn Hϕ pn h . , ϕ − Am ϕ = fn pn h It follows from the convergence of Pn to P that ϕn − Am ϕn =

fn − pn hX = Pn fn − Pfn X → 0

as

n → ∞.

Next, since lim ϕn − ϕX = 0, we have n→∞

lim Hϕn fn − Hϕ pn hX = 0,

n→∞

lim (pn h)(Hϕn fn ) − fn Hϕ pn hX = 0.

n→∞

Since C(T) ⊂ X, it follows that there exists a sequence {δn } of positive numbers with zero limit such that fn − pn hL∞ ≤ δn ;

(pn h)(Hϕn fn ) − fn Hϕ pn hL∞ ≤ δn .

Let {εn } be a sequence of positive numbers such that δn lim εn = 0, lim = 0. n→∞ n→∞ ε2 n Put Ωn = {ζ ∈ T : |pn (ζ)h(ζ)| ≤ εn }. We have to show that

(Hϕn fn )(ζ) (Hϕ pn h)(ζ) q dm(ζ) = 0. − lim fn (ζ) n→∞ (pn h)(ζ)

(11.9)

T

It follows easily from

lim n→∞

(11.9) that (Hϕn fn )(ζ) (Hϕ pn h)(ζ) q dm(ζ) = 0. − fn (ζ) (pn h)(ζ) T\Ωn

Since almost everywhere Hϕ pn h pn h = sm (Hϕ ),

Hϕn fn fn = sm (Hϕn ) → sm (Hϕ )

as

n → ∞,

12. The Recovery Problem in Spaces of Measures

371

it follows that the proof will be completed as far as we show that lim m(Ωn ) = 0.

n→∞

Put M = min{|h(ζ)| : ζ ∈ T}. Obviously, εn |pn (ζ)| ≤ , ζ ∈ Ωn . (11.10) M Next, since lim pn h − fn X = 0 and fn X = 1, it follows that for large n→∞ values of n, const pn X ≥ . hX Finally, since pn ranges over the finite-dimensional subspace of polynomials of degree at most 2k + µ − 1, we have pn L∞ ≥ const > 0

(11.11)

for large values of n. The result follows now from (11.10) and (11.11) and the following wellknown fact: Let Ω be a closed subset of T of measure ε and let p be a polynomial of degree N . Then max |p(ζ)| ≤ C(N, ε) max |p(ζ)|, ζ∈T

ζ∈Ω

where C(N, ε) depends only on N and ε (see Gaier [1]; see also Nazarov [1], which contains much stronger results).

12. The Recovery Problem in Spaces of Measures and Interpolation by Analytic Functions In this section we consider applications of best approximation results obtained in §1. First we obtain so-called recovery theorems for measures. As an application of such recovery theorems we obtain an interpolation 1/p theorem for functions in Bp ∩ CA . Let X be a space of distributions on T. The recovery problem for measures with respect to the space X is the problem of whether of whether the following implication holds: µ ∈ M(T),

P− µ ∈ X

=⇒

µ ∈ X.

Here M(T) denotes the space of complex regular Borel measures on T. Let us mention two well-known recovery theorems for measures. The famous F. and M. Riesz theorem can be stated in the following way: µ ∈ M(T),

P− µ ∈ L1 (T)

=⇒

µ ∈ L1 (T).

372


The other classical example is due to Rajchman [1]: µ ∈ M(T),

P− µ ∈ Fc0

=⇒

µ ∈ Fc0 ,

where Fc0 is the space of pseudofunctions, i.e., the space of distributions ξ on T satisfying ˆ = 0. lim |ξ(n)| |n|→∞

In this section we show that the results of §1 on best approximation in 1/p Besov spaces Bp lead to other recovery theorems for measures. −1/p , 1 < p < ∞, is the space of distributions ξ on T for Recall that Bp which 2−n(p−1) ξ ∗ Wn pLP < ∞, n∈Z −1/p

if and only if and an analytic function f in D belongs to Bp

|f (ζ)|p (1 − |ζ|)p−2 dm2 (ζ) < ∞ D

−1/p

(see Appendix 2.6). The space Bp can naturally be identified with ∗ 1/p −1/2 . Note that the space B2 coincides with the space of distriBp butions ξ with finite energy E(ξ) =

n∈Z, n =0

2 ˆ |ξ(n)| < ∞. |n|

The following result is one more recovery theorem for measures. −1/p

Theorem 12.1. Let 1 < p < ∞ and µ ∈ M(T). If P− µ ∈ Bp −1/p . then µ ∈ Bp

,

kernel. Proof. Let ∗0 < r < 1 and µr = Pr ∗ µ, where Pr is the Poisson 1/p −1/p 1/p Since Bp = Bp , there exists a function ϕ ∈ (Bp )− such that ϕB 1/p ≤ 1 p

and P+ µr B −1/p ≤ const zϕdµr . p T

Let f = Aϕ. Then ϕ − f L∞ = distL∞ (ϕ, H ∞ ) ≤ const ϕB 1/p , p

1/p

since Bp

⊂ V M O. By Theorem 1.6, 1/p

f ∈ Bp

and f B 1/p ≤ const ϕB 1/p . p

p


We have P+ µr B −1/p p

373

≤ const zϕdµr T

≤ const z(ϕ − f )dµr + const zf dµr T

T

≤ const ϕ − f L∞ µM(T) +

const zf B 1/p P− µr B −1/p p

p

≤ const ϕB 1/p µM(T) + P− µr B −1/p . p

p

Since ϕB 1/p ≤ 1, it follows that the norms P+ µr B −1/p are uniformly p

p

−1/p Bp .

bounded, and so P+ µ ∈ Let us now state the special case of Theorem 12.1 when p = 2. Corollary 12.2. If µ ∈ M(T), then |ˆ |ˆ µ(−n)|2 µ(n)|2 < ∞ =⇒ < ∞. n n n≥1

n≥1

Corollary 12.3. If µ ∈ M(T) and s is a positive integer, then |ˆ |ˆ µ(−n)|2s µ(n)|2s < ∞ =⇒ < ∞. (12.1) n n n≥1

n≥1

def

Proof of Corollary 12.3. Let ν = µ ∗ µ ∗ · · · ∗ µ. Then .+ , s

|ˆ ν (−n)|2 < ∞. n

n≥1

By Lemma 12.2,

|ˆ ν (n)|2 < ∞, n

n≥1

which means that

|ˆ µ(n)|2s < ∞. n

n≥1

The very interesting problem of whether (12.1) holds for other values of s ≥ 1/2 is still open. The rest of this section is devoted to an application of Theorem 12.1 to an interpolation problem for a space of analytic functions. Let 1 < p < ∞. 1/p Consider the space CA ∩ Bp of functions analytic in D. A closed subset 1/p E of T is called an interpolation set for CA ∩ Bp if for each continuous 1/p function f on E there exists a function ϕ ∈ CA ∩ Bp such that ϕE = f .

374

Chapter 7. Best Analytic and Meromorphic Approximation 1/p

We consider the following norm on Bp : 1/p

1 p p |f (τ ζ) − f (ζ)| dm(ζ)dm(τ ) . f B 1/p = f Lp + 2 p T |1 − τ | T Definition. Let E be a closed subset of T. We define its capacity 1/p capp (E) by def 1/p cap1/p p (E) = inf ϕB 1/p : ϕ ∈ C(T) ∩ Bp , |ϕ| E ≥ 1 . p

def

Let ω(ζ) = min{|ζ|, 1}. Clearly, |ω(ζ1 ) − ω(ζ2 )| ≤ |ζ1 − ζ2 |, and so ω ◦ ϕB 1/p ≤ ϕB 1/p . p

p

It follows that

def 1/p cap1/p p (E) = inf ϕB 1/p : ϕ ∈ C(T) ∩ Bp , 0 ≤ ϕ ≤ 1, ϕ E ≥ 1 . p

1/p

Note that for p = 2 the capacity capp is equivalent to the logarithmic capacity (see Landkof [1]). By analogy with the case p = 2, distributions −1/p in the space Bp are called distributions of finite p-energy. Theorem 12.4. Let 1 < p < ∞ and let E be a closed subset of T. If 1/p 1/p capp (E) = 0, then E is an interpolation set for CA ∩ Bp . Note that it follows from the results of Sj¨ odin [1] that if E is an interpo1/p 1/p lation set for C(T) ∩ Bp (i.e., the restrictions of functions in C(T) ∩ Bp 1/p coincide with C(E)), then capp (E) = 0. This implies that the converse 1/p to Theorem 12.4 also holds, i.e., if E is an interpolation set for CA ∩ Bp , 1/p then capp (E) = 0. We need the following lemma. Lemma 12.5. Let µ be a complex Borel measure on T with finite 1/p p-energy and let E be a closed subset of T such that capp (E) = 0. Then |µ|(E) = 0. Proof. Let us show that f dµ = 0 for any smooth function f on T. Let E 1/p

{ϕn } be a sequence of functions in C(T) ∩ Bp 0 ≤ ϕn ≤ 1,

ϕn |E ≡ 1,

such that

lim ϕn B 1/p = 0.

n→∞

p

It is easy to see that ϕn f B 1/p ≤ const f C 1 (T) ϕn B 1/p . We have p p

ϕn f dµ ≤ const µ −1/p ϕn f 1/p ≤ const µ −1/p f C 1 (T) ϕn 1/p . B B B B T

p

p

Hence,

lim

n→∞

T

ϕn f dµ = 0.

p

p


Since ϕn |E ≡ 1, it follows that

f dµ ≤ ϕn dµ + T

E

and so

T\E

f dµ ≤ E

T\E

375

|f |d|µ|,

|f |d|µ|.

(12.2)

It is easy to see that there exists a sequence {fj } of smooth functions on T such that fj E = f E, lim |fj (τ )| = 0 for τ ∈ T \ E, and fj C(T) ≤ const. j→∞ By Lebesgue’s theorem it follows from (12.2) that f dµ = 0. E 1/p 1/p Proof of Theorem 12.4. Clearly, CA ∩ Bp = C(T) ∩ Bp + . Con 1/p sider the subspace D (diagonal) of the direct product C(T) × Bp + defined by D = {f, f } : f ∈ C(T) ∩ Bp1/p + ⊂ C(T) × Bp1/p + 1/p and define the linear operator J : C(T) ∩ Bp + → D by Jf = {f, f }. 1/p ∗ can be identified naturally with the The dual space C(T) × Bp + −1/p space M(T) × Bp . + Since {z n , z n } ∈ D for n ∈ Z+ , it follows that the annihilator D⊥ consists −1/p for which of the pairs {µ, f } in M(T) × Bp + µ ˆ(n) + fˆ(n) = 0, −1/p

Hence, P+ µ ∈ Bp −1/p

µ ∈ Bp operator

n ∈ Z+ .

for {µ, f } ∈ D⊥ . It follows from Theorem 12.1 that

. By Theorem 12.5, |µ|(E) = 0. Consider now the restriction R : C(T) ∩ (Bp1/p )+ → C(E).

By the Banach theorem R is onto if and only if the (J ∗ )−1 R∗ is operator ∗ an isomorphism onto its range. We can identify C(E) with the space M(E) of complex Borel measures on E. It is easy to see that if ν ∈ M(E), then (J ∗ )−1 R∗ ν = {ν, O} + D⊥ . We have

∗ −1 ∗ ⊥ (J ) R ν = inf ν + µM(T) + ξB −1/p : {µ, ξ} ∈ D ≥ νM(E) , p

since |µ|(E) = 0. Thus (J ∗ )−1 R∗ is an isomorphism onto its range, and so R is onto.

376


13. The Fefferman–Stein Decomposition in Bp1/p In this section we obtain an analog of the Fefferman–Stein decomposition for functions in BM O. Recall that a function f ∈ L1 (T) belongs to BM O, (i.e.,

1 1 def |f − fI |dm < ∞, fI = f dm, sup m(I) I I m(I) I (13.1) the supremum in (13.1) being taken over all arcs I of T) if and only if f can be represented as f = ξ + η˜, where ξ, η ∈ L∞ . Such a representation of f is called the Fefferman–Stein decomposition of f . The following theorem is an analog of the Fefferman–Stein decomposition 1/p for functions in Bp . 1/p

Theorem 13.1. Let 1 < p < ∞. If f ∈ Bp , then f can be represented as f = ξ + η˜, 1/p

where ξ, η ∈ Bp

∩ L∞ . 1/p

Proof. We have P− f ∈ Bp . Let g = A(P− ϕ). Then by Theorem 1.6, 1/p 1/p ˆ g ∈ Bp and ϕ = P− f − g ∈ Bp ∩ L∞ . Since P− ϕ = (ϕ − iϕ˜ − ϕ(0))/2, it follows that 1 P− f = P− ϕ = (ϕ − ϕ(0) ˆ − iϕ). ˜ 2 The function P+ f admits a similar representation.

Concluding Remarks The heredity problem for function spaces was studied first by Shapiro [1], who proved that the space of functions which extend analytically to a neighborhood of the unit circle has the hereditary property. In Carleson and Jacobs [1] it was shown that the classes Λα , α ∈ / Z, have the hereditary property. The heredity problem and the recovery problem for unimodular functions were studied systematically in Peller and Khrushchëv [1]. We also mention here the survey article Kahane [2]. The notion of an R-space was introduced in Peller and Khrushchëv [1]. Theorems 1.3, 1.4, 1.5, and 1.6 were obtained in Peller and Khrushchëv [1]. Theorems 1.10 and 1.11 are published here for the first time. The presentation of §2 follows Peller and Khrushchëv [1]. However, in Peller and Khrushchëv [1] the axiom (A3) is different: the set of trigonometric polynomials is dense in X. The fact that Theorems 2.2 and 2.3 hold under less restrictive Axiom (A3) stated in §2 was proved in Alexeev and Peller [2]. Note that the proof of Theorem 2.3 uses an idea of Lax [1]. In that

Concluding Remarks

377

paper the author considered a self-adjoint operator A on a Hilbert space H and a Banach space X continuously imbedded in H such that AX ⊂ X. Under certain conditions it was shown in Lax [1] that if x is an eigenvector of A, then x ∈ X. Note also that the proof of Theorem 2.3 uses an idea of the paper Adamyan, Arov, and Krein [1] in which the authors considered the recovery problem for the space F1 . Theorem 2.1 was established in s as well as Peller [24]. Note that the nonseparable function spaces Λα , Bp∞ many other nonseparable Banach spaces (see §4) were treated in Peller and Khrushchëv [1] as spaces satisfying the axioms (B1)–(B3) stated in §3. The results of §3 are due to Peller and Khrushchëv [1]. Theorem 4.1 was obtained in Peller [10]. Theorem 4.3 is due to Peller [24]. The fact that for ϕ ∈ Λα the Hankel operator Hϕ is a compact operator from (Λα )+ to (Λα )− was obtained for α ∈ Z in Gohberg and Budyanu [2] by a different method. The proof of this fact given in §4 was found in Peller [24]. It works in a much more general situation. Note that the fact that Tf¯ is a bounded operator on (Λα )+ for any f ∈ H ∞ was observed first in Khavin [1] and Shamoyan [1]. Theorem 4.4 is taken from Peller and Khrushchëv [1]. Theorem 4.5 for Z = V M O,CA + CA , H 1 + H 1 , 41 was established in Peller and Khrushchëv [1]. The nonseparable L1 + L spaces Z (n) with Z = H ∞ + H ∞ and Z = BM O were treated in Peller and Khrushchëv as spaces satisfying the axioms (B1)–(B3). The fact that these nonseparable spaces satisfy (A3) was established in Peller [24]. Theorem 4.9 was obtained in Peller and Khrushchëv [1]. Again, Theorem 4.11 is contained in Peller [24] while in Peller and Khrushchëv [1] the axioms (B1)–(B3) were used to show that F∞ w is hereditary. Theorems 4.14, 4.15, 4.16 and 4.17 are taken from Peller and Khrushchëv [1]. Theorem 4.18 is due to A.B. Aleksandrov (unpublished). Theorem 4.19 is new. The fact that CA + CA is not hereditary was discovered in Papadimitrakis [1]. We also mention the papers Krepkogorskii [1–2], in which the author studies 1/p interpolation spaces between Bp and BM O which are R-spaces, and the recent paper Gheorghe [1], in which the author studies generalized Besov spaces BE associated with a monotone Riesz–Fischer space E (such spaces are also R-spaces). Theorem 5.1 for continuous functions u was obtained in Adamyan, Arov, and Krein [1] and Poreda [1]. Theorems 5.3 and 5.4 are contained in Adamyan, Arov, and Krein [3] and Hayashi, Trefethen, and Gutknekht [1]. Theorems 5.5 can be found in Alexeev and Peller [3], while Theorem 5.6 is published here for the first time. The results of §6 were obtained in Peller [19]. Theorems 7.1 and 7.10 we found in Peller [20], while Theorem 7.8 was established in Papadimitrakis [3]. The results of §8 are due to Peller and Khrushchëv [1], while the results of §9 are taken from Peller [15].

378


Theorem 10.1 was obtained independently by different methods in Merino [1] and Papadimitrakis [2]. In §2 we use the method of Merino. Note that the continuity problem in the L∞ norm for the operator A was posed in the preprint Helton and Schwartz [1], though the results of that preprint contain an error. The results of §11 are taken from Peller [19]; the results for the space F1 were established in Hayashi, Trefethen, and Gutknekht [1]. The results of §12 were obtained in Peller and Khrushchëv [1]. Note that Koosis [2] found another proof of Corollary 12.2. The results of §13 are taken from Peller and Khrushchëv [1]. Note that in Adams and Frazier [1] Fefferman–Stein decomposition was obtained for other spaces of functions. Let us also mention the papers Volberg and Tolokonnikov [1] and Tolokonnikov [2] in which the authors use other methods to find more hereditary function spaces. Finally, we mention here the following results of Chui and Li [1]. They considered the nonlinear operator Gn defined as follows. Let ϕ ∈ C(T). By the Adamyan–Arov–Krein theorem there exists a unique Hankel operator Γn of rank at most n such that Hϕ − Γn = sn (Hϕ ). The function Gn ϕ is by definition the antianalytic symbol of Γn . By Kronecker’s theorem Gn ϕ is a rational function of degree at most n. It was shown in Chui and Li [1] that if sn−1 (Hϕ ) > sn (Hϕ ), then ϕ is a continuity point of Gn in the L∞ norm. On the other hand, the authors gave an example that shows that the restriction sn−1 (Hϕ ) > sn (Hϕ ) cannot be dropped.

8 An Introduction to Gaussian Spaces

In this chapter we give a brief introduction in the theory of Gaussian processes by means of the technique of Fock spaces. This approach, which appeared in quantum field theory, simplifies both Wiener’s approach based on Hermite polynomials in several variables and Itô’s method of stochastic integrals. In §1 we define Gaussian variables and Gaussian subspaces. Then we define stationary Gaussian processes and the spectral measure associated to a stationary Gaussian process. In §2 we construct the (bozon) Fock space of a Hilbert space. Then we define the so-called Vick transform that orthogonalizes the spaces of homogeneous polynomials of Gaussian functions of degree k. This technique allows us to pass from the subspace of Gaussian functions in L2 (Ω, A, P ) to the whole space L2 (Ω, A, P ) over a probability space (Ω, A, P ) and to introduce a probability preserving automorphism τ of (Ω, A, P ) associated with the stationary Gaussian process. In §3 we consider various regularity conditions for stationary processes (i.e., conditions of weak dependence of the future on the past) and their relationships with mixing properties of the measure preserving automorphism τ defined in §2. Later in Chapter 9 we characterize various regularity conditions in terms of the spectral measure of the process. Next, we study in §4 interpolation problems for stationary Gaussian processes. In other words, we consider so-called minimal stationary processes, i.e., processes {Xn }n∈Z such that X0 cannot be predicted by the Xn with n = 0. With each such process we associate the so-called interpolation error

380

Chapter 8. An Introduction to Gaussian Spaces

process. Then we study related properties of basisness and unconditional basisness of stationary processes. Finally, in §5 we relate the theory of stationary processes with Hankel operators. Namely, we show that the product of the orthogonal projections onto the past and the future can be expressed in terms of a Hankel operator whose symbol is the so-called phase function of the process. Then we consider the following geometric problem in the theory of stationary processes. Given subspaces K and L of the Gaussian space, the problem is to find out whether there exists a stationary Gaussian process with past K and future L. We show that this problem essentially reduces to the problem of the description of the operators on Hilbert spaces that are unitarily equivalent to the moduli of Hankel operators. The latter problem will be solved in Chapter 12.

1. Gaussian Spaces Let (Ω, Σ, P ) be a probability space, i.e., Ω is a space of elementary events, Σ is a σ-algebra of subsets of Ω (σ-algebra of events), and P is a probability measure on Σ (i.e., P is a positive measure such that P (Ω) = 1). Events A, B ∈ Σ are called independent if P (A ∩ B) = P (A)P (B). By random variables we understand measurable (real or complex) functions on Ω. Random variables f and g are called independent if the events {ω ∈ Ω : f (ω) ∈ F } and {ω ∈ Ω : g(ω) ∈ G} are independent for any Borel sets F and G. For a random variable f ∈ L1 (Ω, Σ, P ) its mathematical expectation (or its mean) Ef is defined by

f dP. Ef = Ω

A measurable function f : Ω → R is called a Gaussian random variable (or simply Gaussian variable) if its distribution function is Gaussian, i.e.,

s t2 1 exp − 2 dt, P {ω : f (ω) < s} = √ 2σf 2πσf −∞

where σf2 = E|f |2 =

|f |2 dP Ω

is the variance of f . Note that we consider only Gaussian variables with zero mean. For convenience we assume that the zero function on Ω is also a Gaussian variable. If f is a Gaussian variable, it is easy to see that Eeitf = e−t

2

σf2 /2

,

t ∈ R.

(1.1)

1. Gaussian Spaces

381

It follows easily from the Fourier inversion formula that (1.1) implies that f is a Gaussian variable with variance σf2 . A family {fα }α∈A is called Gaussian if its linear span consists of Gaussian variables. If f1 , · · · , fn is a Gaussian family and γ = (γ1 · · · γn )t ∈ Rn , n γk fk has variance then the Gaussian variable f = k=1

Ef 2 =

n

γj γk E(fj fk ) = (Qγ, γ),

j,k=1

where the matrix Q is defined by Q = {E(fj fk )}1≤j,k≤n . Clearly, Q is symmetric and positive definite. It is easy to see that Q is invertible if and only if f1 , · · · , fn are linearly independent. In this case the joint distribution µf1 ,··· ,fn of the family f1 , · · · , fn (i.e., the image of P under the mapping ω → (f1 (ω), · · · , fn (ω))) is absolutely continuous with respect to Lebesgue measure in Rn and it is easy to verify that the density Pf1 ,··· ,fn of µf1 ,··· ,fn is given by 1 1 −1 1/2 (det Q) exp − x, x) . (1.2) Pf1 ,··· ,fn (x) = (Q 2 (2π)n/2 It follows immediately from (1.2) that if g1 and g2 are random variables that belong to the same Gaussian family, then they are independent if and only if they are orthogonal. A closed subspace G of the real space L2R (Ω, Σ, P ) is called Gaussian if it consists of Gaussian variables. The following lemma shows that the closure of the linear span of a Gaussian family is a Gaussian subspace. Lemma 1.1. If {fn }n≥0 is a sequence of Gaussian variables and f = lim fn in L2 (Ω, Σ, P ), then f is a Gaussian variable. n→∞

Proof. Passing to a subsequence, we can assume that f = lim fn almost n→∞

everywhere (almost surely) on Ω. By Lebesgue’s theorem 2 t itf itfn 2 Ee = lim Ee lim σ , = exp − n→∞ 2 n→∞ fn which implies that f is a Gaussian variable with variance lim σf2n . n→∞ It is easy to see that there exist infinite-dimensional Gaussian spaces. Indeed, let (Ωj , Σj , Pj ), j ∈ Z, be a copy of the probability space (R, B, µ), where B the set of Borel subsets of R and µ is the measure on R with density 2 1 √ e−x /2 . 2π : (Ωj , Σj , Pj ). Let gj , j ∈ Z, be Consider the infinite product (Ω, Σ, P ) = j∈Z

the function on (Ω, Σ, P ) defined by gj (· · · , ω−2 , ω−1 , ω0 , ω1 ω2 , · · · ) = ωj .

382


It is easy to see that {gj }j∈Z is a sequence of independent Gaussian variables and the closed linear span of {gj }j∈Z is an infinite-dimensional Gaussian subspace space of L2R (Ω, Σ, P ). We fix a Gaussian subspace G in L2R (Ω, Σ, P ) and we consider only Gaussian variables in G. It is natural to assume that Σ is the smallest σ-algebra with respect to which all functions in G are measurable. Let A and B be two closed subspaces of G, and let ΣA and ΣB be the smallest σ-algebras with respect to which the functions in A and B, respectively, are measurable. We say that the subspaces A and B are independent if every function in A is independent with any function in B. It follows immediately from (1.2) that A and B are independent if and only if they are orthogonal. We are going to use the easily verifiable identity n!

n j=1

zj =

n r=1

(−1)n−r

(zj1 + · · · + zjr )n ,

(1.3)

j1 0. Then g = Φ−1 (χ{ζ0 } ) is a nonzero function in G. Let us apply the operator U , (U f )(ω) = f (τ ω), ω ∈ Ω, f ∈ L2 (Ω, Σ, P ). We have U g = ζ0 g, and so U |g| = |g|. Obviously, the set A = {ω ∈ Ω : |g(ω)| > 1} is invariant under τ and 0 < P (A) < 1. Suppose now that µ is a continuous measure and f is a nonconstant function such that U f = f . By (2.7), there exists a positive integer m and

3. Mixing Properties and Regularity Conditions

391

a nonzero h ∈ L2 (T, µ) s ··· s L2 (T, µ) such that ( Mz ⊗ · · · ⊗ Mz )h = h. .+ , .+ , m

m

Hence, (( Mz ⊗ · · · ⊗ Mz )j h, h) = (h, h), .+ ,

j ∈ Z.

m

In other words,

(ζ1 · · · ζm )j |h(ζ)|2 d(⊗n µ)(ζ) = Tm

Tm

|h(ζ)|2 d(⊗n µ)(ζ),

j ∈ Z, (3.1)

(here ζ = (ζ1 · · · , ζm )). Consider the map Dm : Tm → −∞. (3.7) T

The theorem follows immediately from the Szegö–Kolmogorov alternative (see Appendix 2.4).

Uniform Mixing with Respect to the Past We say that τ is a uniform mixing with respect to the past if lim sup |P (A ∩ τ −n B) − P (A)P (B)| = 0

n→∞ A∈Σ0

for any B ∈ Σ.

Clearly, this condition is equivalent to the following: lim

sup |P (A ∩ B) − P (A)P (B)| = 0

n→−∞ A∈Σn

for any B ∈ Σ. (3.8)

We introduce the following notation: def

ξA = χA − P (A),

A ∈ Σ.

We denote by Pn and P n the orthogonal projections from G onto Gn and Gn .


393

Theorem 3.4. Let {Xn }n∈Z be a stationary Gaussian process. The following are equivalent: (i) (3.8) holds; (ii) Σn = {∅, Ω}; n∈Z (iii) Gn = {O}; n∈Z

(iv) lim P−n = O in the strong operator topology; n→∞

(v) µ is absolutely continuous with respect to m and its Radon–Nikodym derivative w has the form w = |h|2 , where h is an outer function in H 2 . Proof. (i)⇒(ii). Suppose that B ∈ Σn . Then setting A = B in (3.8), n∈Z

we find that (P (B))2 = P (B). (ii)⇒(iii). Suppose that g ∈ Σn . {ω ∈ Ω : |g(ω)| < 1} ∈

Gn and g = O. It follows that

n∈Z

n∈Z def

(iii)⇒(iv). It is easy to see that P = lim P−n is an orthogonal projecn→∞ Gn , and so g = O. tion. If g ∈ Range P, then g ∈ n∈Z

(iv)⇒(i). We have F (Pn ) = EΣn . It is easy to see that lim ( Pn ⊗ · · · ⊗ Pn )(g1 ⊗ · · · ⊗ gm ) = 0, .+ ,

n→∞

m > 0.

m

Hence, lim EΣn f L2 (Ω,Σ,P ) = 0

n→−∞

for any f ∈ L2 (Ω, Σ, P ) that is orthogonal to the constants. Let B ∈ Σ and A ∈ Σn . We have 1/2

1/2

|E(ξA ξB )| = |E(ξA EΣn ξB )| ≤ ξA L2 (Ω,Σ,P ) EΣn ξB L2 (Ω,Σ,P ) . The result follows now from the obvious identity E(ξA ξB ) = P (A ∩ B) − P (A)P (B). (iii)⇔(v). This is an immediate consequence of the Szegö–Kolmogorov alternative (see Appendix 2.4). Clearly, w admit a representation w = |h|2 for an outer function h in H 2 if and only if w satisfies (3.7). Note that if µ = µs + µa , dµa = w dm, and w satisfies (3.7), then it follows from the Szeg¨ o–Kolmogorov alternative that ' Gn = Φ−1 L2 (T, µs ). n∈Z

Stationary processes satisfying condition (iii) are called regular. The function w is called the spectral density of the regular process {Xn }n∈Z .

394


n It is easy to see that the process is regular if and only if G = {O}, n∈Z n which in turn is equivalent to the condition Σ = {∅, Ω}. n∈Z

Uniform Mixing with Respect to the Past and the Future We say that τ is a uniform mixing with respect to the past and the future if sup

lim

n→∞ A∈Σ ,B∈Σ0 0

|P (A ∩ τ −n B) − P (A)P (B)| = 0.

Clearly, this condition is equivalent to the condition sup

lim

n→∞ A∈Σ0 ,B∈Σn

|P (A ∩ B) − P (A)P (B)| = 0.

(3.9)

If A and B are subspaces of G, we denote the cosine of the angle between A and B by cos(A, B), cos(A, B)

= =

sup{|(a, b)| : a = b = 1, a ∈ A, b ∈ B} cor0 (A, B) = PA PB .

We say that the subspaces A and B are at nonzero angle if cos(A, B) < 1. If Ξ1 and Ξ2 are σ-subalgebras of Σ, put def

k(Ξ1 , Ξ2 ) =

sup

|P (A ∩ B) − P (A)P (B)|.

A∈Ξ1 ,B∈Ξ2

Theorem 3.5. Let A and B be subspaces of G, and let ΣA and ΣB be the σ-algebras of events measurable with respect to A and B. Then ) 4k(ΣA , ΣB ) ≤ cos(A, B) =

cos(L2 (Ω, ΣA , P ) C, L2 (Ω, ΣB , P ) C)

≤ sin(2πk(ΣA , ΣB )). Recall that we identify the constant functions on Ω with C. def Proof. We identify L2 (Ω, ΣA , P ) C with F 1 (A) = F (A) C and def

L2 (Ω, ΣB , P ) C with F 1 (B) = F (B) C. Then C, L2 (Ω, ΣB , P ) C) = PF 1 (A) PF 1 (B) . cos(L2 (Ω, ΣA , P ) Clearly, PF 1 (A) PF 1 (B) f = F (PA )F (PB )f = F (PA PB )f, It follows that PF 1 (A) PF 1 (B) ≤ PA PB . The opposite inequality is obvious.

f ∈ F 1 (G).


395

Let us estimate k(ΣA , ΣB ). We have k(ΣA , ΣB ) ≤

|E(ξA ξB )|

sup A∈ΣA ,B∈ΣB

≤ cos(A, B)

sup A∈ΣA ,B∈ΣB

2 1/2 2 1/2 (EξA ) (EξB ) =

1 cos(A, B), 4

2 since it is very easy to see that sup EξA = 1/4. A

Finally, let f ∈ A and g ∈ B be Gaussian variables such that E|f |2 = 1, E|g|2 = 1, and |Ef g| ≥ cos(A, B) − ε, where ε > 0. Put A = {ω ∈ Ω : f (ω) > 0},

B = {ω ∈ Ω : g(ω) > 0}.

It is easy to see that P (A) = P (B) = 1/2. If can be shown by direct computations from (1.2) (with f1 = f and f2 = g) that P (A ∩ B) =

1 1 + arcsin |Ef g|. 4 2π

Hence, P (A ∩ B) − P (A)P (B) =

1 arcsin |Ef g|, 2π

and so cos(A, B) ≤ sin(2πk(ΣA , ΣB )). ) B) > 0. Corollary 3.6. k(ΣA , ΣB ) > 0 if and only if cos(A, ) B) < 1. k(ΣA , ΣB ) < 1/4 if and only if cos(A, Let us return to a stationary Gaussian process {Xn }n∈Z . We introduce the sequence n n ρn = cos(G 0 , G ) = cor0 (G0 , G ),

n ∈ Z+ .

(3.10)

The process {Xn }n∈Z is called completely regular if lim ρn = 0.

n→∞

Theorem 3.7. Let {Xn }n∈Z be a stationary Gaussian process. The following are equivalent: (i) {Xn }n∈Z is completely regular; (ii) τ is a uniform mixing with respect to the past and the future; (iii) lim k(Σ0 , Σn ) = 0. n→∞

This is an immediate consequence of Theorem 3.5. In Chapter 9 we characterize the completely regular processes in terms of their spectral measures.

396


Fast Mixing Here we consider the processes for which the sequence (3.10) decreases rapidly. The process is called completely regular of order α, 0 < α < ∞, if ρn ≤ const(1 + n)−α . By Theorem 3.5, this is equivalent to the fact that k(Σ0 , Σn ) ≤ const(1 + n)−α . Completely regular processes of order α are characterized in Chapter 9 in terms of their spectral measures. We also characterize in Chapter 9 the processes with exponential decay of the ρn . As the limit case we consider there the processes for which Σn is independent of Σ0 for some n ∈ Z+ .

Canonical Correlations of the Past and the Future Let {Xn }n∈Z be a regular stationary Gaussian process. Consider the sequence {corm (G0 , G0 )}m≥0 of canonical correlations of the past G0 and the future G0 . We can consider the class of stationary Gaussian processes for which lim corm (G0 , G0 ) = 0.

m→∞

However, it turns out that these are precisely the completely regular processes. We shall prove this in Chapter 9. We impose now a stronger condition on the sequence of canonical correlations. We say that a regular stationary Gaussian process is p-regular, 0 < p < ∞, if {corm (G0 , G0 )}m≥0 ∈ p .

(3.11)

Clearly, (3.11) is equivalent to the condition P0 P 0 ∈ S p . Note that the operators P0 P 0 and P0 P n differ from each other by a finite rank operator. Hence, if the process is p-regular, then P0 P n ∈ S p for every n ∈ Z+ . Theorem 3.8. A stationary Gaussian process {Xn }n∈Z is p-regular, 0 < p < ∞, if and only if there exists N ∈ Z+ such that EΣ0 EΣn ∈ S p for n ≥ N. We need the following lemma. Lemma 3.9. Let T ∈ S p , 0 < p < ∞. Then for any positive integer n

n ; T

Sp

= T nS p .


Proof. Let T =

397

sj (·, ξj )ηj be the Schmidt expansion of T . It is easy

j≥0

to see that n ;

T =

sj1 · · · sjn (·, ξj1 ⊗ · · · ⊗ ξjn )ηj1 ⊗ · · · ⊗ ηjn

j1 ,··· ,jn ≥0 n < is the Schmidt expansion of T . Hence,  n n ; p p (sj1 · · · sjn )p =  sj  = T pn T = Sp . Sp

j1 ,··· ,jn ≥0

j≥0

Corollary 3.10. If T ∈ S p , 0 < p < ∞, and T S p < 1, then F (T ) ∈ S p and F (T )S p ≤ (1 − T pS p )−1/p . This follows immediately from Lemma 3.9. Proof of Theorem 3.8. Suppose that the process is p-regular. Consider the operators P0 P n , n ∈ Z+ . Since the process is regular, the sequence {P n }n≥0 tends to O in the strong operator topology. Therefore lim P0 P n S p = (P0 P 0 )P n S p = 0.

n→∞

Let m be a positive integer for which P0 P m S p < 1. By Corollary 3.10, F (P0 P m ) = EΣ0 EΣm ∈ S p . Suppose now that EΣ0 EΣm = F (P0 P m ) ∈ S p . Clearly, this implies that P0 P m ∈ S p . It remains to show that the process is regular. Assume the contrary. By the Szeg¨ o–Kolmogorov alternative, there exists a nonzero subspace L of G such that L ⊂ G0 ∩ Gn for every n ∈ Z+ . It follows that P0 P n ≥ 1 for any n ∈ Z+ . Therefore P0 P n s ··· s P0 P n ≥ 1 .+ , j

for any j ∈ Z+ . Hence, EΣ0 EΣn ∈ S p for any n ∈ Z+ . In Chapter 9 we characterize the p-regular processes in terms of their spectral measures. Consider now the case of 2-regular stationary Gaussian processes. Such processes are also called absolutely regular. They admit the following characterization. Theorem 3.11. A stationary Gaussian process is absolutely regular if and only if lim E sup |EΣ0 ξA |2 = 0.

n→∞

(3.12)

A∈Σn

We can understand the supremum in (3.12) as the supremum in the space of measurable functions. It is easy to see that (3.12) is equivalent to the condition lim E sup |EΣ0 ξA |p = 0,

n→∞

A∈Σn

p < ∞.

398


We need the following lemma. Lemma 3.12. Let T be an operator on L2 (Ω, Σ, P ). Then T ∈ S 2 if and only if

sup |(T f )(ω)|2 dP (ω) < ∞. (3.13) Ω f 2 ≤1

Moreover, the integral in (3.13) is equal to T 2S 2 . Proof. If T ∈ S 2 , then Tf =

sj (f, ξj )ηj ,

j

where {ξj } and {η}j are orthonormal systems. We have    sup |(T f )(ω)|2 ≤ sup  |(f, ξj )|2   s2j |ηj (ω)|2 

f 2 ≤1

f 2 ≤1

j



= 



j

s2j |ηj (ω)|2  .

j

Hence,

sup |T f |2 dP =

Ω f 2 ≤1

s2j = T 2S 2 < ∞.

j

Suppose now that (3.13) holds. Let P be a finite rank orthogonal projection. Then T P ∈ S 2 . We have sup |(T Pf )(ω)|2 ≤ sup |(T f )(ω)|2 .

f 2 ≤1

f 2 ≤1

Therefore

T PS 2 ≤

sup |T f |2 dP.

Ω f 2 ≤1

It follows that T ∈ S 2 . Proof of Theorem 3.11. Suppose that EΣ0 EΣm ∈ S 2 for some m ∈ Z+ . Then lim PG0 PGj S 2 = 0 j→∞

(see the proof of Theorem 3.8). Put Tj = EΣ0 EΣj PL2 (Ω)C . By Lemma 3.10, lim Tj S 2 = 0. Applying Lemma 3.12, we find that j→∞

E sup |EΣ0 ξA |2 ≤ Tj 2S 2 ξA L2 ≤ A∈Σj

1 Tj 2S 2 → 0 4

as

j → ∞.


399

Suppose now that (3.12) holds. Let Lm be a finite-dimensional subspace of Gm . Consider the Schmidt expansion of the finite rank operator PG0 PLm : PG0 PLm h =

d

sk (h, fk )gk ,

h ∈ Lm .

k=0

Clearly, we can assume that the Gaussian variables fk and gk are real. Let us show that

sk , k = j, (fk , gj ) = 0, k = j. Indeed, −1 −1 (fk , fj ) = s−1 j (fk , PG0 gj ) = sj (PG0 fk , gj ) = sj (fk , gj ) =

1, k = j, 0, k =

j.

Let pf g , pf , and pg be the densities of the joint distribution of the random variables f0 , · · · , fd , g0 , · · · , gd , f0 , · · · , fd , and g0 , · · · , gd , respectively. We denote by Σ(f ) and Σ(g) the σ-subalgebras of Σ generated by f0 , · · · , fd and g0 , · · · , gd . We have E sup |EΣ0 ξA | A∈Σm

≥ E sup |EΣ0 ξA | A∈Σ(f )

≥

Rd+1

pg (y)

= =

1 2

sup Q⊂Rd+1

Rd+1

Rd+1

Q

pf g (x, y) − pf (x) dx dy pg (y)

+ pf g (x, y) − pf (x)pg (y) dx dy

Rd+1

Rd+1

def

|(pf g (x, y) − pf (x)pg (y)|dx dy = K,

def

where as usual ϕ+ (t) = max{ϕ(t), 0}. Put pf g . ϕ = log pf pg It can be shown by elementary computations that the integral

def M = eiϕ pa pb dx dy satisfies the identity d s2k |M | = 1 + (2i + 1) 1 − s2 . 2

k=0

k

We need the following elementary lemma. Lemma 3.13. Let s ∈ R. Then |eis − 1| ≤ 3|es − 1|.

400


Let us first complete the proof of Theorem 3.11. Since

2K = |eϕ − 1|pf pg dx dy, Rd+1

we have by Lemma 3.13,

|M | ≥ 1 −

Rd+1

≥ 1−3

Rd+1

Rd+1

Rd+1

|eiϕ − 1|pf pg dx dy

Rd+1

|eϕ − 1|pf pg dx dy = 1 − 6K.

Hence, (1 − 6K)−2 ≥ |M |−2 ≥

d

(1 + s2k ) ≥ 1 +

k=0

d

s2k .

k=0

Suppose now that m is such that K < 1/12. Then PG0 PLm S 2 =

d

s2k ≤ const

k=0

for an arbitrary subspace Lm of Gm . It follows that PG0 PGm ∈ S 2 . It remains to show that the process is regular. Assume the contrary. Suppose that A ∈ Σ0 ∩ Σn , n ≥ 0, and 0 0, which contradicts (3.12). Proof of Lemma 3.13. Let s ≥ −2 log 3. We have is e − 1 = e−s/2 sin s/2 ≤ e−s/2 ≤ 3. es − 1 sinh s/2 If s < −2 log 3, then

1 |e − 1| ≤ 2 < 3 1 − 9 is

< 3(1 − es ).

We can consider a condition stronger than (3.12): def

αn = ess sup sup |(EΣ0 ξA )(ω)| → 0 ω∈Ω A∈Σn

as

n → ∞. (3.14)

It turns out, however, that this condition does not distinguish a nontrivial class of processes. Theorem 3.14. If a stationary Gaussian process satisfies (3.14), then there exists n ∈ Z+ such that σ-algebras Σ0 and Σn are independent. Such processes will be described in terms of their spectral measure in Chapter 9. We need the following lemma. Lemma 3.15. |P (A ∩ B) − P (A)P (B)| . sup αn = P (B) A∈Σn , B∈Σ0 , P (B)>0


401

Proof. If A ∈ Σn , B ∈ Σ0 , then |P (A ∩ B) − P (A)P (B)| = |E(χB EΣ0 ξA )| ≤ αn χB L1 = αn P (B). Let us prove the opposite inequality. It is easy to see that αn = ess sup sup (EΣ0 ξA )(ω). ω∈Ω A∈Σn

Pick A ∈ Σ such that n

ess sup (EΣ0 ξA )(ω) > αn − ε ω∈Ω

and put B = {ω ∈ Ω : (EΣ0 ξA )(ω) > αn − ε}. We have P (A ∩ B) = E(χA χB ) = E((EΣ0 χA )χB ) = E((EΣ0 ξA )χB ) + P (A)P (B). Hence, P (A ∩ B) − P (A)P (B) = E((EΣ0 ξA )χB ) ≥ (αn − ε)P (B). Proof of Theorem 3.14. Assume the contrary. Choose n ∈ Z+ such that αn < (2πe)−1/2 . Let f ∈ Gn and g ∈ G0 be real Gaussian variables such that Ef 2 = Eg 2 = 1, and E(f g) = ρ > 0. Let pf , pg , and pf g be the distributions of f , g, and (f, g). Put Bk = {ω ∈ Ω : g(ω) ≥ k/ρ} and A = {ω ∈ Ω : f (ω) ∈ [0, 1]}, where k > 0. Then P (A ∩ Bk ) − P (A)P (Bk )

∞

1

(pf g (x, y) − pf (x)pg (y))dx dy

= k/ρ

1 = 2π

0

∞

−y 2 /2

e k/ρ

0

1

1 (x − ρy)2 = exp − 2 1 − ρ2 1 − ρ2 1

−x2 /2

−e

For y ≥ k/ρ and x ∈ [0, 1] we have 2 1 1 (x − ρy)2 = exp − − e−x /2 2 1 − ρ2 1 − ρ2 1 1 (k − 1)2 1 1 ≥ √ 1 − = exp − + . 2 1 − ρ2 2 e 1 − ρ2 Keeping in mind that 1 P (Bk ) = √ 2π

k/ρ

e−y

2

/2

dy,

dx dy.

402


we obtain αn

|P (A ∩ Bk ) − P (A)P (Bk )| P (Bk ) k 1 1 1 (k − 1)2 1 1 √ sup 1 − = . exp − + = √ 2 2 2 1−ρ 2 2πe k 2πe 1−ρ

≥ sup ≥

We have obtained a contradiction. A stronger condition than p-regularity is p-regularity of order α. A stationary Gaussian process {Xn }n∈Z is called p-regular of order α, 0 < α < ∞, if P0 P n S p ≤ const(1 + n)−α ,

n ∈ Z+ .

In Chapter 9 we describe such processes in spectral terms. We also consider in Chapter 9 processes for which P0 P 0 has finite rank, i.e., only finitely many canonical correlations of the future and the past are nonzero.

4. Minimality and Basisness In this section we study the interpolation property for stationary processes. In other words, for a stationary Gaussian process {Xn }n∈Z we study the problem of when X0 can be predicted by the past G0 and the future G1 . We say that a regular stationary Gaussian process is minimal if X0 ∈ span{Xj : j ∈ Z, j = 0}. We also consider stronger conditions, namely basisness and unconditional basisness. Suppose that {Xn }n∈Z is a minimal stationary process. For n ∈ Z conˆ n of Xn from the Gaussian variables Xj , j = n, sider the best prediction X ˆ i.e., Xn is the projection of Xn onto span{Xj : j = n}. def ˆ n be the error in this prediction. The sequence {Yn }n∈Z Let Yn = Xn − X is called the interpolation error process. Theorem 4.1. Let {Xn }n∈Z be a minimal stationary process. Then {Yn }n∈Z is a stationary process. Proof. Consider the unitary operator V on G introduced in §2 which is ˆj = X ˆ j+1 defined by V Xj = Xj+1 , j ∈ Z. Then it is easy to see that V X and V Yj = Yj+1 , j ∈ Z. It follows that

(Yn+1 , Yk+1 ) = (V Yn , V Yk ) = (Yn , Yk ), and so the process {Yn }n∈Z is stationary.

n, k ∈ Z,

4. Minimality and Basisness

403

In Chapter 9 we describe the minimal processes in spectral terms and we find the spectral density of the interpolation error process in terms of the spectral density of the initial process. We proceed to a condition stronger than minimality. We consider the class of stationary Gaussian processes {Xn }n∈Z for which the sequence {Xn }n∈Z forms a basis in the Hilbert space G. This means that for each f ∈ G there exists a unique sequence {βj }j∈Z of complex numbers such that k f= lim βj Xj . k→∞, m→−∞

j=m

Define the linear functionals λj on G by λj (f ) = βj .

(4.1)

Basisness is equivalent to the important property that the past and the future are at nonzero angle. Theorem 4.2. Let {Xn }n∈Z be a stationary Gaussian process. The following are equivalent: (i) the sequence {Xn }n∈Z forms a basis; (ii) the subspaces G0 and G0 are at nonzero angle; (iii) sup |P (A ∩ B) − P (A)P (B)| < 1/4. A∈Σ0 , B∈Σ0

Proof. (i)⇒(ii). Suppose that the sequence {Xn }n∈Z forms a basis in G. It follows easily from the closed graph theorem that the linear functionals λj defined by (4.1) are continuous and it follows from the Banach–Steinhaus theorem that the norms of the linear operators f →

k

λj (f )Xj

j=m

are uniformly bounded. Hence, the operator f → lim

k→∞

k

λj (f )Xj

j=0

defined on the linear combinations of the Xj extends by continuity to a bounded projection Qf onto the future G0 . It is easy to see that def Ker Qf = G0 . It is also evident that Qp = I − Qf is a projection onto the past G0 . Therefore the angle between G0 and G0 is nonzero. (ii)⇒(i). Suppose now that the angle between G0 and G0 is nonzero. Then G0 + G0 = G and there exists a projection Qf onto G0 such that Ker Qf = G0 . Let V be the “translation” operator on G defined by V Xn = Xn+1 . Then V k Qf V −k is a projection onto Gk , k ∈ Z. Define the linear functionals λj , j ∈ Z, on G λj f = V j Qf V −j f − V j+1 Qf V −j−1 f.

404


It is easy to see that k

λj (f )Xj = V m Qf V −m f − V k+1 Qf V −k−1 f.

j=m

Hence, the linear operators f →

k

λj (f )Xj

j=m

are uniformly bounded, and so {Xj }j∈Z is a basis sequence. (ii)⇔(iii). This is an immediate consequence of Corollary 3.6. Note that if {Xn }n∈Z is a basis, then the process is regular. In Chapter 9 we characterize the stationary processes with basisness property in spectral terms. Consider now the class of stationary processes {Xn }n∈Z for which the sequence {Xn }n∈Z forms an unconditional basis in the Hilbert space G. This means that λj (f )Xj = f, for any f ∈ G, lim E

j∈E

where the limit is taken over all finite subsets E of Z; the finite subsets are ordered by inclusion. The following theorem characterizes the processes with unconditional basisness property. Theorem 4.3. Let {Xn }n∈Z be a stationary process with spectral measure µ. The sequence {Xn }n∈Z forms an unconditional basis if and only if µ is absolutely continuous with respect to Lebesgue measure and its Radon– Nikodym derivative w satisfies the conditions w ∈ L∞ ,

w−1 ∈ L∞ .

(4.2)

Proof. Suppose that µ is absolutely continuous and (4.2) holds. Then as a set L2 (w) coincides with L2 (T, m). It follows that the sequence {z n }n∈Z forms an unconditional basis in L2 (w). Consider now the spectral representation of the process Φ : G → L2 (w), ΦXn = z n , n ∈ Z. Since Φ is a unitary operator, it follows that the sequence {Xn }n∈Z forms an unconditional basis in G. If {Xn }n∈Z is an unconditional basis in G, then {z n }n∈Z is an unconditional basis in the Hilbert space L2 (µ). It is well known that this basis must be equivalent to an orthonormal basis, i.e., for sequences {cj }j∈Z with finitely many nonzero terms the following inequalities hold:  1/2 1/2  |cj |2  ≤ cj z j ≤K |cj |2  k j∈Z 2 j∈Z j∈Z L (µ)

5. Scattering Systems and Hankel Operators

405

for some k, K ∈ (0, ∞) (see N.K. Nikol’skii [2], Lect. VI, Sect. 3). It follows that as a set L2 (µ) coincides with L2 (T, m). It is evident that µ is absolutely continuous and w satisfies (4.2).

5. Scattering Systems and Hankel Operators We consider here regular stationary Gaussian processes {Xn }n∈Z and we express the products PG0 PGm , m ∈ Z+ , in terms of Hankel operators. The symbol of this Hankel operator is a so-called phase function of the process. Then we consider a more general situation. We introduce the notion of a scattering system and define a Hankel operator that corresponds to the scattering system. Its symbol is a so-called scattering function of the scattering system. We associate with a regular stationary Gaussian process a scattering system and show that the corresponding scattering function is a phase function of the process. Theorem 5.1. Let {Xn }n∈Z be a regular stationary Gaussian process with spectral density |h|2 , where h is an outer function in H 2 . Then PG0 PGm = V1∗ Hzm h/h ¯ P+ V2 ,

(5.1)

where V1 and V2 are the unitary operators from G onto L2 (T, m) defined ¯ by V1 f = hΦf and V2 f = z¯m hΦf . Proof. Multiplication by 1/h is a unitary operator of L2 onto L2 (|h|2 ). It is easy to see that it maps H 2 onto H 2 (|h|2 ). Indeed, z n h/h = z n , n ∈ Z+ , and since h is outer, the linear combinations of z n h, n ≥ 0, are dense in ¯ maps H 2 unitarily H 2 . Similarly, one can prove that multiplication by 1/h − 2 2 onto H− (|h| ). Hence, ¯ − hψ ¯ ψ → (1/h)P 2 (|h|2 ) while is the orthogonal projection from L2 (|h|2 ) onto H−

ψ → (z m /h)P+ z¯m hψ is the orthogonal projection from L2 (|h|2 ) onto z m H 2 (|h|2 ). Therefore ¯ ¯ − (z m h/h)P z m hΦf ) , PG PGm f = Φ∗ 1/hP + (¯ 0

which proves (5.1). In particular,

PG0 PG0 = V1∗ Hh/h ¯ P+ V2 .

Definition. Let w be the spectral density of a stationary process. Let h be an outer function in H 2 such that |h|2 = w. The unimodular function ¯ is called a phase function of the process. It is determined by w modulo h/h a unimodular constant factor. We can also consider slightly more general objects, stationary sequences in G, i.e., sequences {Xn }n∈Z such that EXj X k depends only on k −j. The difference between stationary sequences and stationary Gaussian processes

406


is that the Gaussian variables Xn in the case of a stationary Gaussian process are real. Let {Xn }n∈Z be a stationary sequence in G. As in the case of stationary Gaussian processes the sequence {Qn }n∈Z defined by Qn = EXn X 0 is positive definite and by the Riesz-Herglotz theorem there exists a positive regular Borel measure µ on T such that µ ˆ(n) = Qn , n ∈ Z. Among all positive measures µ the ones that correspond to stationary Gaussian processes are invariant under the transformation ζ → ζ¯ of T onto itself. We can define in a similar way the past G0 and the future Gm related to a stationcase of regular ary sequence {Xn }n∈Z . If we can restrict ourselves to the stationary sequences, i.e., stationary sequences for which Gm = {O}, m≥0

then Theorem 5.1 also holds for such stationary sequences. If {Xn }n∈Z is a regular stationary Gaussian sequence, V is the unitary operator on G, defined by V Xn = Xn+1 , then the quadruple (G, V, G0 , G0 ) forms a scattering system. Let us define this notion. Definition. Let H be a Hilbert space and let U be a unitary operator on H. A subspace N+ of H is called outgoing for U if ∞ ' U N+ ⊂ N+ , U n N+ = {O}, and span{U n N+ : n ∈ Z} = H. n=−∞

A subspace N− of H is called incoming for U if ∞ ' U ∗ N− ⊂ N− , U n N− = {O}, and span{U n N− : n ∈ Z} = H. n=−∞

If N+ is an outgoing subspace for U and N− is an incoming subspace for U , the quadruple (H, U, N+ , N− ) is called a scattering system. Suppose that (H, U, N+ , N− ) is a scattering system. Let K+ = N+ U N+ . Then there exists a unitary operator V+ from H onto ∗ = SK+ is multiplication L2 (K+ ) such that V+ N+ = H 2 (K+ ) and V+ U V+ 2 by z on L (K+ ). Indeed, it is easy to see that the spaces U j K+ are pairwise orthogonal and 5 U j K+ , H= j∈Z

and we can define V+ by V+ U j xj = z j xj , j∈Z

xj ∈ K+ .

j∈Z

Similarly, there exists a unitary operator V− of H onto L2 (K− ) (where def

2 ∗ (K− ) and V− U V− = SK− . K− = N− U ∗ N− ) such that V− (N− ) = H− Since the spectral multiplicity of SK+ is equal to the spectral multiplicity of U and the same is true for the spectral multiplicity of SK− , it follows that dim K+ = dim K− , and we can identify both K+ and K− with a Hilbert space K.

5. Scattering Systems and Hankel Operators

407

def

∗ The operator S = V− V+ on L2 (K) is called the abstract scattering operator. It is defined modulo constant unitary factors. Clearly, ∗ ∗ SK = V− U V+ = SK S, SSK = V− V+

i.e., S commutes with SK . Then there exists a unitary-valued function Ξ ∈ L∞ (B(K)) such that (Sf )(ζ) = Ξ(ζ)f (ζ),

f ∈ L2 (K), ζ ∈ T,

(see Appendix 1.4). The operator function Ξ is called a scattering function of the scattering system. Clearly, it is defined modulo constant unitary factors. We have ∗ ∗ ∗ ∗ PN− PN+ = V− P− V− V+ P+ V+ = V− P− SP+ V+ = V− HΞ P + V +

and ∗ ∗ ∗ ∗ P+ V− V+ P+ V+ = V− P+ SP+ V+ = V− TΞ P + V + . PN−⊥ PN+ = V−

Recall that PL is the orthogonal projection onto L. Thus the Hankel operator HΞ and the Toeplitz operator TΞ appear naturally when studying scattering systems. Let us return now to stationary sequences. Consider a regular stationary sequence {Xn }n∈Z in G. Then it is easy to see that the future G0 is an outgoing subspace for the unitary operator V on G defined by V Xn = Xn+1 and the past G0 is an incoming subspace for V . Thus the quadruple (G, V, G0 , G0 ) is a scattering system. We are going to compute now its scattering function. Let w = |h|2 be the spectral density of the process, where h is an outer function. We can identify G with L2 (w) via the unitary map Φ introduced in §4. It is easy to see that the spaces K+ and K− are one-dimensional and

λ λ K+ = : λ∈C and K− = z¯ ¯ : λ ∈ C . h h Indeed, to see this, it is sufficient to consider the unitary operators V1 and V2 from G onto L2 (T, m) introduced in Theorem 5.1 with m = 0 and z : λ ∈ C}. observe that V2 K+ is the space of constants and V1 K− = {λ¯ Let us identify both K+ and K− with C. We can define now the unitary operators V− : and V+ from L2 (w) onto L2 by ¯ V+ ϕ = hϕ and V− ϕ = hϕ. Then the abstract scattering operator S is given by h ∗ Sf = V− V+ f = ¯ f, h ¯ and so the phase function h/h is a scattering function of the scattering system (G, V, G0 , G0 ), and ∗ Hh/h PG0 PG0 = V− ¯ P+ V+ .

408


If we consider the scattering system (G, V, Gm , G0 ), then it is easy to ¯ and show that its scattering function is z m h/h ∗ Hzm h/h PG0 PGm = V− ¯ P+ V+ ,

which is just formula (5.1).

6. Geometry of Past and Future We consider the following geometric problem in this section. Let G be a Gaussian subspace and let A and B be subspaces of G such that clos(A + B) = G. The problem is to determine whether there exists a stationary Gaussian process {Xn }n∈Z such that G0 = A and G0 = B. We show in this section that this geometric problem is closely related to the problem of describing operators unitarily equivalent to moduli of Hankel operators. The latter problem will be solved in Chapter 12, which will allow us to solve the above geometric problem in the case when A and B are at nonzero angle. The spectral representation Φ of a stationary Gaussian process {Xn }n∈Z transforms the Gaussian space G to the space L2 (µ). If the process is deterministic, then G0 = G0 = G, and so the geometry of past and future is trivial. If the process is nondeterministic and dµ = dµs + w dm, then 2 (w) + L2 (µs ). Hence, we can pass ΦG0 = H 2 (w) + L2 (µs ) and ΦG0 = H− ∗ 2 to the orthogonal complement to Φ (L (µs )) and consider the subspaces G0 Φ∗ (L2 (µs )) and G0 Φ∗ (L2 (µs )) of the Gaussian space GΦ∗ (L2 (µs )). This allows us to reduce the initial problem to the case when there exists a regular stationary Gaussian process {Xn }n∈Z such that A is its past and B is its future. If A and B are at nonzero angle we reduce this problem to the problem of description of the operators unitarily equivalent to the moduli of Hankel operators. Suppose now that A and B are subspaces of G such that clos(A + B) = G. We consider the problem of when there exists a stationary sequence {Xn }n∈Z of vectors in G such that G0 = A and G0 = B. We consider the set T of triples (K, L, H), where K and L are subspaces of a complex infinite-dimensional separable Hilbert space H such that clos(K +L) = H. Two triples (K1 , L1 , H1 ) and (K2 , L2 , H2 ) in T are said to be equivalent if there exists a unitary operator W of H1 onto H2 such that WK1 = K2 and WL1 = L2 . We want to find out which triples (K, L, H) are equivalent to a triple of the form (G0 , G0 , G). With each triple t = (K, L, H) ∈ T we associate the operator PL PK PL and the following numbers: n+ (t) = dim K⊥ ∩ L,

n− (t) = dim K ∩ L⊥ ,

where as usual PM is the orthogonal projection onto a subspace M.

6. Geometry of Past and Future

409

Theorem 6.1. Triples t1 = (K1 , L1 , H1 ) and t2 = (K2 , L2 , H2 ) are equivalent if and only if the operators PL1 PK1 PL1 and PL2 PK2 PL2 are unitarily equivalent and n± (t1 ) = n± (t2 ). Proof. It is evident that if t1 and t2 are equivalent, then PL1 PK1 PL1 and PL2 PK2 PL2 are unitarily equivalent and n± (t1 ) = n± (t2 ). Let us establish the converse. Clearly, Kj ∩ Lj = {x ∈ Hj : PLj PKj PLj x = x},

j = 1, 2.

Hence, dim K1 ∩ L1 = K2 ∩ L2 and we can replace tj with (Kj (Kj ∩ Lj ), Lj (Kj ∩ Lj ), Hj (Kj ∩ Lj )),

j = 1, 2.

Thus without loss of generality we may assume that Kj ∩ Lj = {O}, j = 1, 2. Lemma 6.2. If t1 and t2 are equivalent, then and

dim H1 L1 = dim H2 L2

(6.1)

dim Ker(PL1 PK1 PL1 L1 ) = dim Ker(PL2 PK2 PL2 L2 ).

(6.2)

Proof. Let us first prove (6.1). Since Hj = clos(Kj +Lj ), it follows under the assumption Kj ∩ Lj = {O} that dim(Hj Lj ) = dim Kj , j = 1, 2. It is easy to see that dim Kj = dim PLj Kj + dim Kj ∩ L⊥ j = dim PLj Kj + n− (tj ) j = 1, 2. Equality (6.1) follows now from the fact that clos Range PLj PKj PLj = clos PLj Kj , j = 1, 2. Let us now prove (6.2). It is easy to see that Ker PLj PKj PLj = (PLj Kj )⊥ = L⊥ j = 1, 2. j ⊕ (Lj PLj Kj ), Therefore Ker(PLj PKj PLj Lj ) = Lj PLj Kj , j = 1, 2. Clearly, Lj PLj Kj = Lj ∩ Kj⊥ , and so

j = 1, 2,

dim Ker(PLj PKj PLj Lj ) = dim Lj ∩ Kj⊥ = n+ (tj ) j = 1, 2,

which proves (6.2). Let us continue the proof of Theorem 6.1. Let W be a unitary operator from H1 onto H2 such that WPL1 PK1 PL1 = PL2 PK2 PL2 W. Then W Ker PL1 PK1 PL1 = Ker PL2 PK2 PL2 . We can now change the unitary operator W on Ker PL1 PK1 PL1 . Let W# be a unitary operator from H1 onto H2 such that W# x = Wx for x ⊥ Ker PL1 PK1 PL1 , W# L⊥ 1 is an arbi⊥ trary unitary operator from L⊥ 1 onto L2 , and W# (L1 ∩Ker PL1 PK1 PL1 ) = L2 ∩ Ker PL2 PK2 PL2 . By Lemma 6.2, such a unitary operator W# exists. The operator W# can identify H1 with H2 and L1 with L2 . Without loss of generality we may assume now that H1 = H2 = H, L1 = L2 = L, and PL PK1 PL = PL PK2 PL .

410


Under the assumption that L ∩ Kj = {O}, j = 1, 2 we have H = clos(L + PKj L) ⊕ Kj ∩ L⊥ ,

j = 1, 2,

and L ∩ PKj L = {O},

j = 1, 2.

We define a unitary operator U on H as follows. We put Ux= x for x ∈ L. On PK1 L we define U by UPK1 x = PK2 x, x ∈ L. Finally, U K1 ∩ L⊥ is an arbitrary unitary operator from K1 ∩ L⊥ onto K2 ∩ L⊥ . Note first of all that U is well-defined on PK1 L. Indeed, if PK1 x1 = PK1 x2 for x1 , x2 ∈ L, then O = PL PK1 PL (x1 − x2 ) = PL PK2 PL (x1 − x2 ) = PL PK2 (x1 − x2 ). Hence, 0

=

(PL PK2 (x1 − x2 ), (x1 − x2 )) = (PK2 (x1 − x2 ), (x1 − x2 ))

=

(PK2 (x1 − x2 ), PK2 (x1 − x2 )) = PK2 (x1 − x2 )2 ,

and so PK2 x1 = PK2 x2 . It is also clear that U maps PK1 L onto PK2 L. To prove that U is a unitary operator on H it suffices to verify that its restriction to clos(L + PK1 L) is unitary. Let y1 , y2 , z1 , z2 ∈ L. Let v1 = y1 + PK1 z1 , v2 = y2 + PK1 z2 . We have (v1 , v2 )

= (y1 , y2 ) + (y1 , PK1 z2 ) + (PK1 z1 , y2 ) + (PK1 z1 , PK1 z2 ) = (y1 , y2 ) + (PL PK1 PL y1 , z2 ) + (PL PK1 PL z1 , y2 ) + (PL PK1 PL z1 , z2 ) = (y1 , y2 ) + (PL PK2 PL y1 , z2 ) + (PL PK2 PL z1 , y2 ) + (PL PK2 PL z1 , z2 ) = (y1 , y2 ) + (y1 , PK2 z2 ) + (PK2 z1 , y2 ) + (PK2 z1 , PK2 z2 ) =

(y1 + PK2 z1 , y2 + PK2 z2 ) = (Uv1 , Uv2 ).

It is easy to see that U maps H onto itself. It is also clear that U maps L1 onto L2 and K1 onto K2 . Note that for each k, m ∈ Z+ ∪ {∞} and for each self-adjoint operator A on a Hilbert space N such that O ≤ A ≤ I, Ker A = {O}, and Ker(I − A) = {O} there exists a triple t = (K, L, H) ∈ T such that n+ (t) = k, n− (t) = m and PL PK PL L K⊥ is unitarily equivalent to A. Indeed, it is easy to reduce the problem to the case n+ (t) = n− (t) = 0. Consider the operator P on N ⊕ N given by the block matrix A A1/2 (I − A)1/2 P= . A1/2 (I − A)1/2 I −A def

It is easy to see that P is a self-adjoint projection. Put K = Range P, def L = N ⊕ {O} and H = clos(K + L). Clearly, PL PK PL L is unitarily equivalent to A. It is easy to verify that K ∩ L⊥ = K⊥ ∩ L = {O}.

6. Geometry of Past and Future

411

We return now to the triple (G0 , G0 , G) associated with a stationary sequence {Xn }n∈Z in G. Theorem 6.3. Let {Xn }n∈Z in G be a stationary sequence in G. Then n− (G0 , G0 , G) = n+ (G0 , G0 , G).

(6.3)

Proof. Let J be the unitary operator on G defined by n ∈ Z,

JXn = X−n−1 ,

on the linear span of the Xn . Clearly, J extends to a unitary operator on G. It is easy to see that JG0 = G0 and JG0 = G0 . This implies (6.3). Theorem 6.4. Let {Xn }n∈Z in G be a regular stationary sequence in G. Then the operator PG0 PG0 PG0 G0 is unitarily equivalent to the square of the modulus of a Hankel operator on H 2 . Proof. Let h be an outer function in H 2 such that |h|2 is the spectral density of the stationary sequence {Xn }n∈Z . By Theorem 5.1, ∗ PG0 PG0 PG0 G0 = V2∗ Hh/h ¯ V2 , ¯ Hh/h ∗ and so PG0 PG0 PG0 G0 is unitarily equivalent to Hh/h ¯ . ¯ Hh/h 2 It is easy to see that if Γ is a Hankel operator from H 2 to H− , then (a) Γ is noninvertible; and (b) either Ker Γ = {O} or dim Ker Γ = ∞. Indeed, (a) follows from the obvious fact that limn→∞ Γz n = 0. To prove (b), we observe that Ker Γ is invariant under multiplication by z, and so by Beurling’s theorem (see Appendix 2.2) either Ker Γ = {O} or Ker Γ = ϑH 2 for an inner function ϑ. Corollary 6.5. (i) The operator PG0 PG0 PG0 G0 is noninvertible; (ii) Either n+ (G0 , G0 , G) = 0 or n+ (G0 , G0 , G) = ∞. Proof. It is easy to see that (i) follows from Theorem 6.4 and (a) while (ii) follows from Theorem 6.4 and (b). We are going prove the converse of Theorem 6.4 in the case when the subspaces are at nonzero angle. If cos(G , G0 ) < 1, then the correspond0

ing Hankel operator has norm less than one. We consider now a triple (A, B, G) ∈ T with cos(A, B) < 1. Theorem 6.6. Let (A, B, G) ∈ T be a triple such that A and B are at nonzero angle and n− (A, B, G) = n+ (A, B, G). If the operator PB PA PB B is unitarily equivalent to the square of the modulus of a Hankel operator, then there exists a stationary sequence {Xn }n∈Z with G0 = A and G0 = B.

412


Proof. Let ϕ be a function in L∞ such that ϕL∞ < 1 and PB PA PB B is unitarily equivalent to Hϕ∗ Hϕ . It follows that Hϕ < 1. By Theorem 1.1.7, there exists a unimodular function u such that Hu = Hϕ and Hz¯u = 1. By Theorem 3.1.14, the Toeplitz operator Tu is invertible. Then ¯ by Corollary 3.2.2, u can be represented in the form u = h/h, where h is 2 2 w = |h| . an outer function in H . Put ∗ It follows that PB PA PB B is unitarily equivalent to Hh/h ¯ . Since ¯ Hh/h A and B are at nonzero angle and n− (A, B, G) = n+ (A, B, G), we have dim B ⊥ = ∞. Thus PB PA PB is unitarily equivalent to PH 2 (w) PH−2 (w) PH 2 (w) . It is also easy to see that 2 n+ (A, B, G) = dim Ker Hh/h = n+ H− (w), H 2 (w), L2 (w) . ¯ It follows from Theorems 5.1 (A, B, G) is equivalent to now and 6.1 that 2 2 the triple H− (w), H 2 (w), L2 (w) . Clearly, H− (w) is the past and H 2 (w) is the future associated with the stationary sequence {z n }n∈Z . Theorem 6.7. If under the hypotheses of Theorem 6.6 the operator PB PA PB B is unitarily equivalent to the square of the modulus of a selfadjoint Hankel operator, and A and B are Gaussian subspaces of a Gaussian space G, then there exists a stationary Gaussian process {Xn }n∈Z with G0 = A and G0 = B. Proof. The proof is the practically same as the proof of Theorem 6.6. By the Remark following Theorem 1.1.7 and Corollary 3.2.4, we can find an outer function h with real Fourier coefficients such that Hϕ = Hh/h ¯ . Thus for the stationary sequence {z n }n∈Z in L2 (w) the numbers (z j , z k )L2 (w) 2 are real, and so the triple (H− (w), H 2 (w), L2 (w)) is equivalent to a triple 0 (G0 , G , G) associated with a stationary Gaussian process. In §12.8 we describe the operators that are unitarily equivalent to the moduli of Hankel operators. We shall prove there that if A is an operator on Hilbert space such that A ≥ O, A is noninvertible and Ker A is either trivial or infinite-dimensional, then A is unitarily equivalent to the modulus of a Hankel operator. In fact we show in §12.8 that if we can construct a Hankel operator whose modulus is unitarily equivalent to A, we can construct a self-adjoint Hankel operator with the same property. This will lead to a solution of the problem to describe the triples (A, B, G) ∈ T such that A = G0 , B = G0 for a stationary process in G for which G0 and G0 are at nonzero angle.

Concluding Remarks We refer the reader to Lamperti [1] for background in probability theory. Basic facts about stationary Gaussian processes can be found in Rozanov

Concluding Remarks

413

[1], Ibragimov and Rozanov [1], Dym and McKean [1], Gikhman and Skorokhod [1], and Yaglom [1]. The important notion of canonical correlations was introduced in Yaglom [2]. There are several methods of reducing probability problems of Gaussian processes to geometric problems in Gaussian spaces. Wiener [1] elaborated the method of orthogonal expansion in Hermite polynomials of infinitely many variables (see also Masani [1]). Later Segal [1] observed that such expansions are closely related to Fock spaces introduced for the needs of quantum mechanics. The use of Fock spaces turned out to be very fruitful both for quantum field theory and probability theory. More information about Fock spaces can be found in Simon [1], Reed [1], and Nelson [1]. We also recommend the recent book Janson [2]. The presentation of §2 and §3 follows the survey article Peller and Khrushchëv [1]. We mention here the papers Vershik [1–2] in which the technique of Fock spaces was used to study spectral properties of the operator U defined by (2.7). Theorem 3.1 is due to Maruyama [1]. Theorem 3.2 was used in Girsanov [1] to construct a mixing measure preserving automorphism with simple continuous singular spectrum. In connection with nondeterministic processes we mention here completely nondeterministic processes. These are processes with trivial intersection of the past and the future, i.e., G0 ∩ G0 = {O}. Such processes were characterized in spectral terms in Bloomfield, Jewell, and Hayashi [1]. Theorem 3.4 is well-known, see e.g., Ibragimov and Rozanov [1]. Theorem 3.5 was obtained in Kolmogorov and Rozanov [1]. Theorem 3.8 was proved in Peller and Khrushchëv [1]. Theorem 3.11 was published in Ibragimov and Rozanov [1]. The proof given in the book is taken from Peller and Khrushchëv [1]; it was inspired by Dym and McKean [1]. Note that a stationary Gaussian process is absolutely regular if and only if it is informationally regular; the last notion was introduced in Gelfand and Yaglom [1]; see Ibragimov and Rozanov [1], and Dym and McKean [1]. Theorem 3.14 appeared in Ibragimov and Linnik [1]. The notion of a minimal stationary process was introduced in Kolmogorov [1]. Theorem 4.2 is a consequence of the Kolmogorov–Rozanov theorem and well-known facts about bases. Theorem 4.3 is well-known. Formula (5.1) was used systematically in Peller and Khrushchëv [1]. Scattering theory was developed by Lax and Phillips [1]. Hankel and Toeplitz operators associated with scattering systems were considered in Mitter and Avniel [1]. Section 6 follows the paper Khrushchëv and Peller [1], in which the problem of the geometry of past and future was stated. The results on the classification of triples in T can be found in the paper Davis [1] in which there are many other results on pairs of subspaces of a Hilbert space.

9 Regularity Conditions for Stationary Processes

In this chapter we characterize different regularity conditions introduced in the previous chapter in spectral terms. In §1 we characterize the minimal stationary processes and find the spectral density of the interpolation error process in terms of the spectral density of the initial process. In §2 we consider the angles between the past and the future of a stationary process. We characterize the processes with nonzero angles between the past and the future. In the next section we consider various regularity conditions for stationary processes (such as complete regularity, complete regularity of order α, p-regularity, etc.) and we characterize such regularity conditions in spectral terms. Note that the original proofs of these results were quite different for different regularity conditions; some proofs were quite complicated. In Peller and Khrushchëv [1] a single approach to all regularity conditions was found. This approach is based on Hankel operators and the results on best approximation given in Chapter 7 and it simplifies the original proofs. Finally, in §4 we consider several stronger regularity conditions and we also characterize them in spectral terms.

1. Minimality in Spectral Terms In this section we prove the Kolmogorov theorem, which characterizes the minimal stationary process in spectral terms. Given a minimal stationary process we also find the spectral density of the interpolation error process and the canonical correlations between its past and future.

416

Chapter 9. Regularity Conditions for Stationary Processes

Theorem 1.1. Let w be the spectral density of a regular stationary process. Then the process is minimal if and only if w−1 ∈ L1 . Proof. Suppose that the process is minimal. Then 1 ∈ spanL2 (w) {z j : j ∈ Z, j = 0}. Let g be a function in L2 (w) such that

0, j = 0, j ¯ g(ζ)ζ w(ζ)dm(ζ) = 1, j = 0. T

(1.1)

Consider the function gw on T. Clearly, it belongs to L1 . It follows from (1.1) that gw(j) 6 = 0, j = 0, and gw(0) 6 = 1. Hence, g(ζ)w(ζ) = 1, ζ ∈ T, and so g = w−1 . Since g ∈ L2 (w), we have

−1 w dm = g 2 wdm < ∞. T

T

Suppose now that w−1 ∈ L1 . Then obviously, w−1 ∈ L2 (w). It is easy to see that

0, j = 0, (w−1 , z j )L2 (w) = 1, j = 0, and so 1 ∈ spanL2 (w) {z j : j ∈ Z, j = 0}. Since multiplication by z is a unitary operator on L2 (w), it follows that n z ∈ spanL2 (w) {z j : j ∈ Z, j = n}. In §8.4 with a minimal stationary process {Xn }n∈Z we have associated the interpolation error process {Yn }n∈Z . The following theorem evaluates the spectral density of {Yn }n∈Z . Theorem 1.2. Let {Xn }n∈Z be a minimal stationary process with spectral density w. Then the interpolation error process {Yn }n∈Z is also minimal and its spectral density is equal to w−1 . Proof. Applying the spectral representation Φ of the process {Xn }n∈Z (see (8.1.5)), we can identify Xn with the function z n in L2 (w). Put gn = z n w−1 , n ∈ Z. It is easy to see that gn ∈ L2 (w) and

0, n = m, n (z , gm )L2 (w) = 1, n = m. It is easy to see that ΦYn = gn , n ∈ Z. Indeed, gn is orthogonal to z j with j = n and (z n , gn )L2 (w) = (z n , z n )L2 (w) = 1. We have (Yk , Yn )

=

(gk , gm )L2 (w)

) −1 (m − k), = z k w−1 z¯m w−1 w dm = w T

k, m ∈ Z.

It follows that w−1 is the spectral density of the stationary process {Yn }n∈Z . Let us now compare the canonical correlations of the past and the future for both processes {Xn }n∈Z and {Yn }n∈Z . We prove that they are the same.

2. Angles between Past and Future

417

Let w be the spectral density of a minimal process {Xn }n∈Z . Let h be an outer function in H 2 such that |h|2 = w. Consider the phase function ¯ of {Xn }n∈Z (see §8.5). It follows immediately from Theorem 1.2 that h/h ¯ is a phase function of {Yn }n∈Z . h/h Theorem 1.3. Let |h|2 be the spectral density of a minimal stationary process, where h is an outer function in H 2 . Then the moduli of the Hankel operators Hh/h and Hh/h¯ are unitarily equivalent. ¯ ¯ ¯ By Theorem 4.4.10, the Toeplitz Proof. Let u = h/h. Then u ¯ = h/h. operator Tu has dense range. Since the process is minimal, it follows from Theorem 1.1 that 1/h ∈ H 2 . Again, by Theorem 4.4.8 the Toeplitz operator Tu¯ has dense range. Therefore {f ∈ H 2 : Hu∗ Hu f = f } = {O}. By Theorem 4.4.2, the operators Hu∗ Hu and Hu¯∗ Hu¯ are unitarily equivalent, which implies the result. Corollary 1.4. Let {Xn }n∈Z be a minimal stationary process and let {Yn }n∈Z be its interpolation error processes. Then both processes have the same canonical correlations between the past and the future. Proof. By Theorem 8.5.1, the canonical correlations of the past and the future of the process {Xn }n∈Z are the singular values of Hh/h ¯ . Similarly, by Theorems 1.2 and 8.5.1, the canonical correlations of the past and the future of the process {Yn }n∈Z are the singular values of the Hankel operator Hh/h¯ . The result follows now from Theorem 1.3.

2. Angles between Past and Future We consider here the class of stationary processes for which the future from time k and the past are at nonzero angle. We prove that the spectral density w of such a process can be represented as |Q|2 w1 , where Q is a polynomial with zeros on T and w1 is the spectral density of a minimal process for which the past before time zero and the future from time zero are at nonzero angle. This reduction to the case of minimal processes will be used in §3 to describe various regularity conditions in spectral terms. Finally, we prove in this section the Helson–Szeg¨ o theorem, which characterizes the stationary processes with nonzero angle between the past and the future. Theorem 2.1. Let {Xn }n∈Z be a stationary process with spectral density w such that for some k ∈ Z+ its past G0 and the future Gk from time k are at nonzero angle. Then there exist a polynomial Q, deg Q ≤ k, with zeros on T and a minimal stationary process {Xn♠ }n∈Z with spectral density w1 such that w = |Q|2 w1 , and the future and the past of {Xn♠ }n∈Z are at nonzero angle.

418


Proof. Let ρj = PG0 PGj , j ∈ Z+ . Let h be an outer function in H 2 such that |h|2 = w. Denote by u the ¯ phase function h/h. By Theorem 8.5.1, ρj = Hzj u , j ∈ Z+ . It is easy to see that Hz¯u = 1. Indeed, ¯ H 2 = hH 2 . z h Hz¯u hH 2 = ¯ −

−

By the hypotheses, Hzk u < 1. Let m be the minimal nonnegative integer such that Hzm u < 1 but Hzm−1 u = 1. Clearly, such an m exists. Put v = z m u. Then by Theorem 3.1.11, Tv is left invertible while Tz¯v is not left invertible. It follows now from Theorem 3.1.14 that Tv is invertible. ¯ 1 /h1 for some outer function h1 such that h1 ∈ H 2 By Corollary 3.2.2, v = h 2 and 1/h1 ∈ H . We need the following lemma. Lemma 2.2. Let h and h1 be outer functions in H 2 and m ∈ Z+ such that ¯ ¯1 h h = z¯m h h1 and Th¯ 1 /h1 is invertible. Then there exists a polynomial Q of degree m with zeros on T such that h = Qh1 . To prove Lemma 2.2, we need one more lemma. Lemma 2.3. Let Q be a polynomial of degree m with zeros on T. Then ¯ Q = c¯ zm, Q where |c| = 1. Proof of Lemma 2.3. It suffices to consider the case Q = z − τ , τ ∈ T. We have ¯ z¯ − τ¯ Q = = −¯ τ z¯. Q z−τ Proof of Lemma 2.2. Since Th¯ 1 /h1 is invertible, it follows that dim Ker Tz¯m+1 h¯ 1 /h1 = m + 1. We claim that Ker Tz¯m+1 h¯ 1 /h1 = span{z j h1 : 0 ≤ j ≤ m}.

(2.1)

Indeed, it is easy to see that the right-hand side of (2.1) has dimension m + 1 and is contained in the left-hand side. Since Tz¯m+1 h¯ 1 /h1 = Tz¯h/h ¯ , it is easy to see that h ∈ Ker Tz¯m+1 h ¯ 1 /h1 . Hence, there exists a polynomial Q of degree at most m such that h = Qh1 . It remains to prove that deg Q = m and all zeros of Q lie on T. Since h is outer, it follows that Q has no zeros inside D. Let Q = Q1 Q2 , where Q1 and Q2 are polynomials such that Q1 has zeros on T while Q2 has zeros outside clos D. Let j = deg Q1 . We have ¯ ¯1 ¯1 ¯1 Q ¯2 h ¯2 h h Q Q = · · = c¯ zj · , |c| = 1, h Q1 Q2 h1 Q2 h1

2. Angles between Past and Future

419

by Lemma 2.3. We have ind Th/h ¯

=

ind(Tz¯j TQ¯ 2 Th¯ 1 /h1 TQ−1 ) = 2

= j + ind TQ¯ 2 + ind Th¯ 1 /h1 + ind TQ−1 = j, 2

since obviously, TQ¯ 2 and ind TQ−1 are invertible. It follows that j = m, and 2 so Q2 is a constant. Let us complete the proof of Theorem 2.1. By Lemma 2.2, h = Qh1 for a polynomial Q of degree m whose zeros lie on T. Put w1 = |h1 |2 . Then w1−1 ∈ L1 , and so by Theorem 1.1, w1 is the density of a minimal stationary process. Obviously, w = |Q|2 w1 . 2 (w1 ) and H 2 (w1 ) By Theorem 8.5.1, the cosine of the angle between H− is equal to Hh¯ 1 /h1 , which is less than 1, and so w1 is the spectral density of a stationary process for which the past and the future are at nonzero angle. ¯ 1 /h1 of the process Remark 1. Consider the phase function v = h ♠ {Xn }n∈Z with density w1 . The equality u = z¯m v

(2.2)

will be used in §3 to obtain spectral interpretations of regularity conditions. Remark 2. Let us show that the representation w = |Q|2 w1 is unique ˘ are polymodulo multiplicative constants. Indeed, suppose that Q and Q ˘ 2w ˘1 and w1 and w ˘1 are nomials with zeros on T such that |Q|2 w1 = |Q| invertible in L1 . Then |Q|2 w ˘1 = ∈ L1 . 2 ˘ w1 |Q| ˘ divides Q. Similarly, Q divides Q. ˘ Hence, Q Let us now prove the Helson–Szegö theorem, which characterizes the processes with nonzero angle between the past and the future. Theorem 2.4. Let µ be a positive Borel measure on T. The following are equivalent: 2 (µ) and H 2 (µ) are at nonzero angle; (i) the subspaces H− (ii) µ is absolutely continuous with respect to m and its Radon–Nikodym derivative w admits a representation w = exp(ξ + η˜), where ξ and η are real functions in L∞ and ηL∞ < π/2. Proof. Let us show that (i) implies (ii). Since 2 H− (µ) ∩ H 2 (µ) = {O},

by the Szeg¨ o–Kolmogorov alternative (see Appendix 2.4), µ is absolutely continuous with respect to m and its Radon–Nikodym derivative w satisfies log w ∈ L1 .

420


Let h be an outer function in H 2 such that |h|2 = w. By Theorem 8.5.1, Hh/h is invertible (see the ¯ = 1, it follows that Th/h ¯ < 1. Since Hz¯h/h ¯ proof of Theorem 2.1). (ii) follows now from Theorem 3.2.5. Conversely, suppose that (ii) holds. Put ξ + η˜ + iξ˜ − iη h = exp . 2 Then |h|2 = exp(ξ+η˜). By Zygmund’s theorem (see Appendix 2.1), h ∈ H 2 . The result now follows from Theorem 3.2.5. Recall that by Theorem 8.4.2, both (i) and (ii) in the statement of Theorem 2.4 are equivalent to the fact that the sequence {z n }n∈Z forms a basis in L2 (µ). Finally, we obtain a characterization of processes with cos(G , Gk ) < 1. 0

Theorem 2.5. Let µ be a positive Borel measure on T and k ∈ Z+ . The following are equivalent: 2 (µ) and z k H 2 (µ) are at nonzero angle; (i) the subspaces H− (ii) µ is absolutely continuous with respect to m and its Radon–Nikodym derivative w admits a representation w = |Q|2 exp(ξ + η˜), where ξ and η are real functions in L∞ and ηL∞ < π/2 and Q is a polynomial with zeros on T whose degree is at most k. Proof. Suppose that (i) holds. Again, by the Szeg¨ o–Kolmogorov alternative, µ is absolutely continuous with respect to m and w = dµ/dm satisfies log w ∈ L1 . (ii) follows immediately from Theorems 2.1 and 2.4. Conversely, suppose that (ii) holds. Put ξ + η˜ + iξ˜ − iη . h = Q exp 2 By Theorem 2.4 and Lemma 2.3, Hzm h/h ¯ < 1, where m = deg Q, and so 2 m 2 by Theorem 8.5.1, cos(H (µ), z H (µ)) < 1. −

3. Regularity Conditions in Spectral Terms In this section we study various regularity conditions for stationary processes and we characterize the processes satisfying those conditions in spectral terms. First, we reformulate regularity conditions in terms of phase functions of processes. Then we describe the corresponding processes in terms of their spectral densities. We are going to essentially use the results of Chapter 7 (the recovery problem for unimodular functions and the description of arguments of unimodular functions), which in turn are

3. Regularity Conditions in Spectral Terms

421

based on Hankel operators. In the next section we study stronger regularity conditions. Recall that for a stationary Gaussian process with spectral density w regularity coefficients ρj are defined by ρj = PH−2 (w) Pzj H 2 (w) ,

j ∈ Z+ .

As we have already observed, ρj is the cosine of the angle between the past 2 H− (w) and the future z j H 2 (w) from time j. In this section we deal only with regular stationary processes. Let h be def ¯ be the an outer function in H 2 such that |h|2 = w and let u = h/h corresponding phase function. Recall that a process is called completely regular if lim ρj = 0. It is j→∞

called completely regular of order α, α > 0, if ρj ≤ const(1 + j)−α . It is called p-regular, p > 0, if PH−2 (w) PH 2 (w) ∈ S p . Finally, it is called p-regular of order α if PH−2 (w) Pzj H 2 (w) S p ≤ const(1 + j)−α . Theorem 3.1. Let w = |h|2 be the spectral density of a regular stationary process. The following assertions hold. (i) The process is completely regular if and only if P− u ∈ V M O. (ii) The process is completely regular of order α, α > 0, if and only if P− u ∈ Λα . 1/p (iii) The process is p-regular, 0 0, if and only 1/p+α if P− u ∈ Bp∞ . Proof. Let us first prove (i). By Theorem 8.5.1, we have lim ρj

j→∞

= =

lim Hzj u = lim distL∞ (z j u, H ∞ )

j→∞

j→∞

lim distL∞ (u, z¯j H ∞ ) = distL∞ (u, H ∞ + C).

j→∞

Since H ∞ + C is closed in L∞ by Theorem 1.5.1, it follows that the process is completely regular if and only if u ∈ H ∞ + C. By Theorem 1.5.5, this is equivalent to the compactness of Hu , which in turn is equivalent by Theorem 1.5.8 to the condition P− u ∈ V M O. To prove (ii), we note that the process is completely regular of order α if and only if PH−2 (w) Pzj H 2 (w) = distL∞ (z j u, H ∞ ) ≤ const(1 + j)−α ,

j ∈ Z+ .

It follows from (A2.11) of Appendix A2.6 that the last inequality is equivalent to the fact that P− u ∈ Λα . By Theorem 8.5.1, the process is p-regular if and only if Hu ∈ S p . By Theorems 6.1.1, 6.1.2, and 6.1.3, this is equivalent to the condition 1/p P− u ∈ Bp , which proves (iii).

422


Finally, by Theorems 8.5.1, 6.1.1, 6.1.2, and 6.1.3, the process is p-regular of order α if and only if P− z j uB 1/p ≤ const(1 + j)−α . p

It follows easily from (A2.12) of Appendix A2.6 that the last inequality is 1/p+α equivalent to the fact that P− u ∈ Bp∞ . Corollary 3.2. A regular stationary process with spectral density w is completely regular if and only if the operator PH−2 (w) PH 2 (w) is compact.

Proof. We have shown in the proof of Theorem 3.1 that complete regularity is equivalent to the fact that Hu is compact. The result now follows from Theorem 8.5.1. We can now see that all the above regularity conditions can be expressed in terms of a condition of the form P− u ∈ X, where X is a suitable space of functions on X. We introduce a more general notion. For a space X of functions on T we say that a stationary process is X-regular if P− u ∈ X. We restrict ourselves here to the case when X is a decent space (i.e., X 1/p satisfies the axioms (A1)–(A4)), or X = Bp , 0 < p < ∞, or X = V M O. 1/p

Theorem 3.3. Let X be decent function space, or X = Bp , 0 < p < ∞, or X = V M O. A stationary process with spectral density w is X-regular if and only if w admits a representation w = |Q|2 eϕ , where Q is a polynomial with zeros on T and ϕ is a real function in X. Proof. Suppose that the process is X-regular. By Theorem 4.4.10 the Toeplitz operator Tu has dense range in H 2 . Therefore we can solve the recovery problem for the unimodular function u (see Theorems 7.1.4 and 7.2.3), which yields u ∈ X. Clearly, X ⊂ V M O, and so lim PH−2 (w) Pzj H 2 (w) = lim distL∞ (u, z¯j H ∞ ) = 0.

j→∞

j→∞

By Theorem 2.1, there exist a polynomial Q with zeros on T and an outer function h1 invertible H 2 such that h = Qh1 and the Toeplitz operator ¯ 1 /h1 , where m = deg Q. Hence, Th¯ 1 /h1 is invertible. Moreover, u = z¯m h def ¯ v = h1 /h1 ∈ X. By Theorem 7.8.1, v = eiψ , where ψ is a real function in X. By Corollary 3.2.2 and Lemma 3.2.3, ψ˜ − iψ h1 = C exp 2 for some C ∈ R. We have ˜

w = C 2 |Q|2 eψ , which proves the part “only if” since ψ˜ ∈ X. Suppose now that w = |Q|2 eϕ , where Q is a polynomial with zeros on T and ϕ is a real function in X. Put ϕ + iϕ˜ h1 = exp . 2

3. Regularity Conditions in Spectral Terms

423

By Lemma 2.3, ¯1 h , h1 where m = deg Q and c is a unimodular constant. It follows that u ∈ X, z m P− e−iϕ˜ ∈ X (see Corollary 7.8.2). and so P− u = c¯ Now we are in a position to state the main result of this section. Theorem 3.4. Let w be the density of a regular stationary process. The following assertions hold. (i) The process is completely regular if and only if w admits a representation w = |Q|2 eϕ , where Q is a polynomial with zeros on T and ϕ is a real function in V M O. (ii) The process is completely regular of order α, α > 0, if and only if w admits a representation w = |Q|2 eϕ , where Q is a polynomial with zeros on T and ϕ is a real function in Λα . (iii) The process is p-regular, 0 0, if and only if w admits a representation w = |Q|2 eϕ , where Q is a polynomial with zeros 1/p on T and ϕ is a real function in Bp∞ . Theorem 3.4 follows immediately from Theorem 3.3. We can also consider the following regularity conditions: j s ρqj < ∞, s > −1, 1 ≤ q < ∞, (3.1) u = c¯ zm

j≥1

and j≥1

j s PH−2 (w) Pzj H 2 (w) qS p < ∞,

s > −1, 1 ≤ q < ∞, 1 ≤ p < ∞. (3.2)

As before it can be easily shown that a stationary process satisfies (3.1) (s+1)/q while (3.2) is if and only if the phase function u satisfies P− u ∈ B∞q (s+1)/q+1/p equivalent to the condition P− u ∈ Bpq . (s+1)/q (s+1)/q+1/p and X = Bpq We can now apply Theorem 3.3 with X = B∞q and obtain the following result. Theorem 3.5. Let s > −1, 1 ≤ q < ∞, 1 ≤ p < ∞, and let w be the spectral density of a stationary process. The following assertions hold: (i) the process satisfies (3.1) if and only if w = |Q|2 eϕ , where Q is a (s+1)/q ; polynomial with zeros on T and ϕ is a real function in B∞q (ii) the process satisfies (3.1) if and only if w = |Q|2 eϕ , where Q is a (s+1)/q+1/p . polynomial with zeros on T and ϕ is a real function in Bpq Finally, note that in Theorem 3.3 X does not have to be a Banach space. In particular, Theorem 3.3 holds for Carleman classes considered in §7.4. The special case when X is the Gevrey class Gα , 0 < α < ∞, is most

424


interesting. Recall (see §7.4) that a function f ∈ C(T) belongs to Gα if and only if there exist K, δ > 0, such that |fˆ(n)| ≤ K exp(−δ|n|α/(1+α) ),

n ∈ Z.

(3.3)

Theorem 3.3 with X = Gα allows us to obtain the following result. Theorem 3.6. Let 0 < β < 1 and 1 ≤ q < ∞ and let w be the spectral density of a stationary process. The following are equivalent: (i) there exist positive numbers K and δ such that ρj ≤ K exp(−δj β ),

j ∈ Z+ ;

(ii) w admits a representation w = |Q|2 eϕ , where Q is a polynomial with zeros on T and ϕ is a real function in Gβ/(1−β) . Proof. Theorem 3.6 follows immediately from Theorem 3.3 and the following characterization of (Gα )− : P− g ∈ Gα ⇐⇒ ∃K, δ > 0 : distL∞ (z n g, H ∞ ) ≤ Ke−δn

α 1+α

,

n ∈ Z+ . (3.4)

It is easy to see that (3.4) follows from (3.3).

4. Stronger Regularity Conditions In this section we study strong regularity conditions for stationary processes. We consider the class of processes for which the regularity conditions decay exponentially. Then we consider the processes for which the kth canonical correlation of the future and the past is zero. Finally, we consider the processes such that the future from time m is independent on the past. Let us first consider the case of exponential decay of the regularity coefficients ρn defined by (8.3.10). Recall that the space ACδ , 0 < δ < 1, has been defined in §7.4. Theorem 4.1. Let w be the spectral density of a regular stationary processes and let 0 < δ < 1. The following are equivalent: (i) for each γ > δ there exists K > 0 such that ρn ≤ Kγ n , n ∈ Z+ ; (ii) w ∈ ACδ . Proof. It is easy to see that a function f ∈ C(T) belongs to ACδ if and only if for each γ > δ there exists K > 0 such that |fˆ(j)| ≤ Kγ |j| ,

j ∈ Z.

(4.1)

It follows easily from (4.1) that if u ∈ L∞ , then P− u ∈ ACδ if and only if for each γ > δ there exists K > 0 such that distL∞ (z n u, H ∞ ) ≤ Kγ j ,

n ∈ Z+ .

(4.2)

4. Stronger Regularity Conditions

425

Let us first show that |Q|2 ∈ ACδ for any polynomial Q with zeros on T. It is sufficient to consider the case Q = z − λ, λ ∈ T. We have ¯ = (ζ − λ)( 1 − λ), ¯ |Q(ζ)|2 = (ζ − λ)(ζ¯ − λ) ζ

ζ ∈ T,

and so |Q|2 ∈ ACδ . Suppose that (i) holds. Then the process is completely regular and by Theorem 3.4, w = |Q|2 |h|2 , where h is an outer function invertible in H 2 ¯ and Q is a polynomial with zeros on T. Put u = h/h. Clearly, u satisfies (4.2), and so P− u ∈ ACδ . By Theorem 7.4.17, |h|2 ∈ ACδ , and so w = |Q|2 |h|2 ∈ ACδ . Suppose now that w ∈ ACδ .Then w can have only finitely many zeros on T. Let Q be a polynomial with zeros on T such that w = |Q|2 |h|2 for a function h invertible in H ∞ . Clearly, |h|2 ∈ ACδ . By Theorem 7.4.17, ¯ ∈ ACδ and so h/h ¯ satisfies (4.2). Hence, P− h/h ¯ ¯h Q , H ∞) Qh ¯ h = distL∞ (z n−deg Q , H ∞ ) ≤ Kγ n−deg Q , n ≥ deg Q. h Consider now the class of stationary processes for which the kth canonical correlation of the past and the future is zero. Such processes are regular. Let w be the spectral density of a regular process. Clearly, the above condition means that the operator PH−2 (w) PH 2 (w) on L2 (w) has rank at most k. By Theorem 8.5.1, this is equivalent to the fact that the Hankel operator Hh/h ¯ has rank at most k, where h is an outer function in H 2 such that |h|2 = w. ρn

=

distL∞ (z n

Theorem 4.2. Let µ be the spectral measure of a stationary process and let k ∈ Z+ . The following are equivalent: (i) rank PH−2 (w) PH 2 (w) = k; (ii) w is a rational function with poles off T such that deg w = 2k. Proof. Let h be an outer function such that |h|2 = w. Suppose that Hh/h ¯ ¯ has finite rank. By Kronecker’s theorem, the function P− h/h is rational. We have ¯ ¯ h h = P− h2 · P− . P− w = P− |h|2 = P− h2 · h h since h2 ∈ H 1 , it follows that P− w is rational and ¯ h deg P− w ≤ deg P− . h Suppose now that w is rational and has no poles on T. Then h is invertible in H ∞ . We have ¯ 1 1 h 2 2 |h| P |h| P− = P− = P . − − h h2 h2

426


¯ is rational and Since 1/h2 ∈ H ∞ , it follows that P− h/h ¯ h deg P− ≤ deg P− w. h It remains to observe that since w is real, we have deg P− w = deg P+ w. Finally, we consider the class of processes for which the future from time m is independent of the past. Theorem 4.3. Let w be the density of a regular stationary process and let m ∈ Z+ . The following are equivalent: (i) ρm = 0; (ii) w = |Q|2 , where Q is a polynomial of degree at most m. ¯ 2 for λ ∈ D and Proof. Suppose that w = |Q|2 . Since |ζ − λ|2 = |1 − λζ| ζ ∈ T, we may assume without loss of generality that Q has no zeros in D, ¯ ∈ H ∞ . Hence, and so Q is an outer function. It is easy to see that z m Q/Q ρm = 0. Suppose now that ρm = 0. Let h be an outer function such that |h|2 = w. def ¯ ¯ Then P− z m h/h = O. In other words, f = z m h/h ∈ H ∞ . Put def ¯ = hf ∈ H 2 . Obviously, Q is a polynomial of degree at most Q = zmh m and |Q|2 = |h|2 = w.

Concluding Remarks Theorem 1.1 was proved in Kolmogorov [1]. Theorem 1.2 is well known. Theorem 1.3 is contained in Peller and Khrushchëv [1]. Corollary 1.4 was observed in Jewell and Bloomfield [1]. Theorem 2.1 is due to Helson and Sarason [1]. The proof given in §2 is a simplification of the proof given in Peller and Khrushchëv [1]. Theorem 2.4 is the famous Helson–Szegö theorem established in Helson and Szeg¨ o [1]. Theorem 2.4 is a combination of Theorems 2.1 and 2.4; see Helson and Sarason [1]. Note that the Helson–Szeg¨ o condition is equivalent to the Muckenhoupt condition A2 :

1 1 −1 w dm w dm . sup m(I) I m(I) I I In other words, the Riesz projection P+ is a bounded operator on L2 (w) if and only if w satisfies the Muckenhoupt condition A2 ; see Hunt, Muckenhoupt, and Wheeden [1]. The completely regular processes were characterized in Helson and Sarason [1] and later this description was refined in Sarason [2]. Of course, the space V M O was not mentioned in Sarason [2], since it did not exist at that time. After the work of Fefferman [1] and Sarason [4] on BM O and V M O it became possible to state the description of the completely regular

Concluding Remarks

427

stationary processes in terms of V M O. Note that before the work of Fefferman, Ibragimov had found necessary conditions and sufficient conditions of local character (see Ibragimov [1–4]). For some time it was not clear whether the necessary condition found in Ibragimov [4] is sufficient. Sarason [2] showed that this is not the case but Ibragimov’s condition is still of interest. The proof given in §2 is based on the approach by Peller and Khrushchëv [1]. Let us mention here another approach given in Arocena, Cotlar, and Sadosky [1]. The completely regular processes of order α were described by Ibragimov [4]. Note that his proof was technically complicated. The p-regular processes were described in Ibragimov and Solev [1] for p = 2 (see also Solev [1]), Peller and Khrushchëv [1] for 1 ≤ p < ∞, and Peller [8] for 0 < p < 1. The p-regular processes of order α were described in Peller and Khrushchëv [1]. Note that in §3 we use the method of Peller and Khrushchëv [1], which gives a single simplified approach to all regularity conditions. Theorems 3.4 and 3.6 are also taken from Peller and Khrushchëv [1]. Theorem 4.1 is due to Ibragimov [3]. Theorem 4.2 was obtained in Yaglom [2]. Finally, Theorem 4.3 can be found in Ibragimov and Rozanov [1]. The corresponding problems for vectorial stationary processes are considerably more complicated. A sequence   Xn,1   Xn =  ...  , Xn,j ∈ G, Xn,d is called a vectorial stationary Gaussian process if the covariance matrix Qnk = {EXn,j1 Xk,j2 }1≤j1 ,j2 ≤d depends only on n − k. By analogy with the scalar case we can define the past and the future: G0 = span{Xm,j : m < 0, 1 ≤ j ≤ d}, Gn = span{Xm,j : m ≥ n, 1 ≤ j ≤ d}. It is well known (see Rozanov [1]) that there exists a matrix-valued positive ˆ (n−k), n, k ∈ Z. It is called the spectral measure M on T such that Qnk = M measure mof the process. As in the scalar case the process is called regular G = {O}. If the process is regular, the spectral measure M is if n≥0

absolutely continuous with respect to Lebesgue measure and its density W is called the spectral density of the process. For regular processes rank W (ζ) is constant almost everywhere on T (see Rozanov [1]). The process is said to be of full rank if rank W (ζ) = d for almost all ζ ∈ T. The process is regular and has full rank if and only if log det W ∈ L1 ; see Rozanov [1]. Here we discuss only regular processes of full rank. As in the scalar case one can identify the Gaussian space with the weighted space L2 (W ) of Cn -valued function. Under this identification G0

428


becomes H 2 (W ), the closure of the analytic polynomials in L2 (W ) while 2 G0 becomes H− (W ). By analogy with the scalar case we can define completely regular processes, completely regular processes of order α, p-regular processes, p-regular processes of order α, X-regular processes, etc. By the Wiener–Masani theorem (Wiener and Masani [1]; see also Rozanov [1]) there are outer square matrix functions Ψ and Ψ# such that W = ΨΨ∗ = Ψ# Ψ∗# . Consider now the matrix function U = Ψ∗# Ψ−1 . It is unitary-valued and it is called a phase function of the process. It is defined up to constant unitary factors. Again, as in the scalar case PG0 PGm = V1∗ Hzm U P+ V2 , where V1 and V2 are unitary operators (see Peller [17]). It was shown in Peller [26] that for a broad class of function spaces X a process with spectral density W is X-regular if and only if W = Q∗ W1 Q, where Q is a matrix polynomial such that all zeros of det Q lie on T and W1 is a spectral density of a minimal X-regular processes (as in the scalar case minimality is equivalent to the fact that W1−1 is integrable). This reduces the study of completely regular, p-regular, completely regular of order α, p-regular of order α processes to the case of minimal processes (i.e., processes with spectral densities invertible in L1 ). The completely regular processes of order α were described by Ibragimov [5]: a process with spectral density W is a completely regular process of order α if and only if W = Q∗ W1 Q, where Q is a matrix polynomial as above, W1 ∈ Λα (which means that all entries of W1 belong to Λα ) and det W1 does not vanish on T. In Peller [17] and [26] the proof of Ibragimov was significantly simplified. The p-regular processes of order α were described in a similar way in Peller 1/p+α [26] (Λα has to be replaced with Bp∞ ). However, it turns out that the problems to characterize the completely regular and p-regular processes are much more complicated. In Treil and Volberg [2] it was shown that a minimal vectorial process with spectral density W is completely regular if and only if 1/2 1/2

1 1 −1 lim W dm W dm = 1, m(I) I m(I)→0 m(I) I where the limit is taken over subarcs I of T. Note that this characterization does not generalize to the case of p-regular processes. In Peller [26] sufficient conditions of different character were found for complete regularity and p-regularity. The problem to characterize the p-regular processes is still open. In this book we consider only stationary processes with discrete time. The case of continuous time is also very important, but the corresponding problems to characterize regularity conditions in spectral terms are still open. A continuous function t → Xt from R to a Gaussian space G is called a stationary Gaussian process with continuous time if EXt Xs depends only on t − s, t, s ∈ R. By Bochner’s theorem (see Akhiezer and Glazman [1], §70), there exists a finite positive measure µ (the spectral measure of the

Concluding Remarks

process) such that EXt X0 =

R

429

e−itx dµ(x). In a similar way one can define

the subspaces G0 , the past, and Gt , the future from time t, and consider similar regularity conditions. For example, the process is called completely regular if PG0 PGt → 0 as t → ∞ and it is called p-regular if PG0 PGt ∈ S p if t is greater than a certain number t0 . However, such problems are much more complicated and still remain open. We mention here that unlike the case of discrete time it is not true that the process is completely regular if and only if PG0 PGt is compact for large values of t.

10 Spectral Properties of Hankel Operators

This chapter is an introduction to spectral properties of Hankel operators. Here we present certain selected results. Note that we do not include in this book some other known results on spectral properties of Hankel operators, and we give some references at the end of this chapter. Certainly, to speak about spectral properties, we have to deal with operators that map a Hilbert space into itself. Given a function ϕ analytic in D, we consider the Hankel matrix ˆ + k)}j,k≥0 . Γϕ = {ϕ(j

(0.1)

By the Nehari theorem, Γϕ determines a bounded linear operator on 2 if and only if there exists a function ψ in L∞ such that ˆ ψ(k) = ϕ(k), ˆ

k ∈ Z+ ,

(0.2)

and Γϕ = inf ψL∞ ,

(0.3)

the infimum being taken over all functions ψ ∈ L∞ satisfying (0.2). For a function ϕ in L∞ we define the Hankel matrix Γϕ by (0.1). It follows easily from (0.3) that for ϕ ∈ L∞ , ∞ ), Γϕ = distL∞ (ϕ, H−

where ∞ = {f ∈ L∞ : fˆ(k) = 0 for k < 0}. H−

432

Chapter 10. Spectral Properties of Hankel Operators

We identify bounded Hankel matrices with operators on H 2 in the orthonormal basis z k , k ≥ 0. It is easy to see that Γϕ f = P+ ϕJ f,

f ∈ H 2,

(0.4)

where the operator J on L2 is defined by (J g)(z) = g(¯ z ),

g ∈ L2 .

(0.5)

Indeed, it is easy to verify that (P+ ϕJ z j , z k ) = ϕ(j ˆ + k),

j, k ∈ Z+ .

It is easy to verify that Γ∗ϕ = ΓJ ϕ . It follows easily from formula (1.13) of Chapter 1 that S ∗ Γϕ = Γϕ S = ΓS ∗ P+ ϕ ,

(0.6)

where S is unilateral shift, i.e., multiplication by z on H 2 . Let us show that the essential spectrum of a bounded Hankel operator always contains the origin. Lemma 0.1. Let Γϕ be a bounded Hankel operator on H 2 . Then 0 ∈ σe (Γϕ ). Proof. Suppose that 0 ∈ / σe (Γϕ ). Then there exist a bounded linear operator T and a compact operator K such that T Γϕ = I + K. It follows from (0.6) that T Γϕ S n 1 = z n + Kz n = T (S ∗ )n Γϕ 1, where 1 is the constant function identically equal to 1. Clearly, Kz n → 0, since K is compact. It is also easy to see that (S ∗ )n Γϕ 1 → 0. However, z n 2 = 1, and we get a contradiction. In §1 we describe the essential spectrum of Hankel operators with piecewise continuous symbols. We also describe the spectrum of the Hilbert matrix. In §2 we study the Carleman operator introduced in §1.8. We prove that the Carleman operator has Lebesgue spectrum on [0, π] of multiplicity 2. In the last section we show that there are no nonzero nilpotent Hankel operators. On the other hand, we show how to construct nonzero quasinilpotent Hankel operators. More generally, we consider in §3 a class of Hankel operators whose symbols have lacunary Fourier series and we develop a procedure to describe their spectra which leads to the construction of a nonzero quasinilpotent Hankel operator.

1. The Essential Spectrum of Hankel Operators with Symbols in P C

433

1. The Essential Spectrum of Hankel Operators with Piecewise Continuous Symbols In this section we describe the essential spectrum of Hankel operators with piecewise continuous symbols. We also describe the spectrum of the Hilbert matrix. We start with certain special Hankel operators. Let us introduce the following notation: def

T+ = {ζ ∈ T : Im ζ ≥ 0}

def

and T− = {ζ ∈ T : Im ζ < 0}.

Consider the characteristic functions of T+ and T− : def

χ+ = χT+

def

and χ− = χT− .

If α and β are distinct points on the complex plane, we denote by [α, β] the interval of the straight line that joins α and β. Lemma 1.1.

and

9 8 i σ(Γz¯χ− ) = σe (Γz¯χ− ) = 0, 2 9 8 i σ(Γz¯χ+ ) = σe (Γz¯χ+ ) = 0, − . 2

Proof. Clearly, χ ˆ+ (j) = −χ ˆ− (j) for j = 0, and so Γz¯χ+ = −Γz¯χ− . Thus it is sufficient to prove the result for Γ¯ z χ+ . Using (0.4) and (0.5), we obtain for f ∈ H 2 −Γ2z¯χ+ f

=

Γz¯χ+ Γz¯χ− f = P+ z¯χ+ J (Γz¯χ− f )

= P+ z¯χ+ J (P+ z¯χ− J f ) = P+ z¯χ+ J P+ J J (¯ z χ− J f ) = P+ z¯χ+ zP− z¯zχ+ f = P+ χ+ P− χ+ f = P+ χ+ f − P+ χ+ P+ χ+ f = Tχ+ f − Tχ2+ f, and so Γ2i¯zχ+ = Tχ+ − Tχ2+ . By Theorem 3.3.1, σ(Tχ+ ) = [0, 1]. By the spectral mapping theorem, 8 19 σ Tχ+ − Tχ2+ = 0, . 4

434


It is easy to calculate the Fourier  1  0  1 1  3 Γi¯zχ+ =  0 π 1   5 .. .

coefficients of χ+ and find that  0 13 0 15 · · · 1 0 15 0 · · ·  3  0 15 0 17 · · ·   1 , 0 17 0 · · ·  5  1 1  0 7 0 9 ···  .. .. .. .. . . . . . . .

1 and so Γi¯zχ+ = 2π Γ[m1 ] ; see formula (7.9) of Chapter 1. By Corollary 1.7.6, Γi¯zχ+ is a nonnegative operator, and so 9 8 1 . σ(Γi¯zχ+ ) = iσ(Γz¯χ+ ) = 0, 2

Since Γi¯zχ+ is self-adjoint, it is evident that σe (Γi¯zχ+ ) = [0, 1/2]. Remark. Recall (see §1.8) that the matrix of 2πΓi¯zχ+ is the same as the matrix of the Carleman operator Γ in the basis {Fξn }n≥0 , and so 2πΓi¯zχ+ is unitarily equivalent to the Carleman operator. Thus σ(Γ ) = σe (Γ ) = [0, π]. 1 Let us proceed now to the Hilbert matrix 1+j+k

j,k≥0

. In §1.1 we have

seen that it is equal to Γψ1 , where the function ψ1 is defined by ψ1 (eit ) = ie−it (π − t),

t ∈ [0, 2π).

We introduce the operators Dτ , τ ∈ T, defined by (Dτ f )(ζ) = f (τ ζ),

ζ ∈ T.

For τ ∈ T we consider the piecewise linear function ψτ on T defined by ψτ = Dτ¯ ψ1 . It is easy to see that ψτ has a jump discontinuity at τ and ψτ (τ + ) − ψτ (τ − ) = ψ1 (1+ ) − ψ1 (1− ) = 2πi (see the notation in §1.5). Put def

Clearly, Γτ =

j+k

τ¯ 1+j+k

Γτ = Γψτ ,

j,k≥0

τ ∈ T.

, Γ1 is the Hilbert matrix, and Γ∗τ = Γτ¯ ,

τ ∈ T.

It is also easy to verify that Dτ¯ Γ1 Dτ¯ = Γτ .

(1.1)

Indeed, let f ∈ H . We have 2

Dτ¯ Γ1 Dτ¯ f

= Dτ¯ Γ1 f (¯ τ z) = Dτ¯ P+ ψ1 J (f (¯ τ z)) = Dτ¯ P+ ψ1 f (τ z¯) τ z)f (¯ z ) = P+ ψτ J f = Γτ f. = P+ Dτ¯ (ψ1 f (τ z¯)) = P+ ψ1 (¯


435

We need several elementary properties of the operators Γτ . Lemma 1.2. If τ ∈ T and τ = 1, then Γ1 Dτ Γ1 ∈ S 2 . Proof. We have ∞ τ k zk j Γ1 Dτ Γ1 z = Γ1 1+j+k k=0

∞

=

k=0

where amj =

∞ ∞ τk zm = amj z j , 1 + j + k m=0 1 + k + m m=0

∞ k=0

τk . (1 + j + k)(1 + k + m)

Put def

sk = 1 + τ + τ 2 + · · · + τ k . Clearly, |sk | ≤ 2|1 − τ |−1 and ∞

amj

=

sk − sk−1 1 + (1 + j)(1 + m) (1 + j + k)(1 + k + m) k=1

=

∞

sk

k=0

1 1 − (1 + j + k)(1 + k + m) (2 + j + k)(2 + k + m)

Hence, ∞

2 |1 − τ |

|amj | ≤

k=0

=

1 1 − (1 + j + k)(1 + k + m) (2 + j + k)(2 + k + m)

1 2 · . |1 − τ | (1 + m)(1 + j)

It follows that

|amj |2 < ∞,

m,j∈Z+

and so Γ1 Dτ Γ1 ∈ S 2 . Lemma 1.3. Let τ1 , τ2 ∈ T and τ1 τ2 = 1. Then Γτ1 Γτ2 ∈ S 2 . Proof. We have Γτ1 Γτ2 = Dτ¯1 Γ1 Dτ¯1 Dτ¯2 Γ1 Dτ¯2 = Dτ¯1 Γ1 Dτ1 τ2 Γ1 Dτ¯2 ∈ S 2 by Lemma 1.2. Lemma 1.4. Γ1 + Γ−1 = 2πiΓzχ+ . ∞ , which implies Proof. It is easy to verify that ψ1 + ψ−1 − 2πizχ+ ∈ H− the result. We also need the following general fact.

436


Lemma 1.5. Let T1 and T2 be bounded linear operators on Hilbert space such that the operators T1 T2 and T2 T1 are compact. Then σe (T1 + T2 ) ∪ {0} = σe (T1 ) + σe (T2 ). Proof. Suppose that T1 and T2 act on a Hilbert space H. Let B the algebra of operators on H and let C be the ideal of compact operators. Consider the Calkin algebra B/C. It is a C ∗ -algebra. By the Gelfand–Naimark theorem (see Appendix 1.3), it is isomorphic to a C ∗ -algebra of operators on Hilbert space. Thus it is sufficient to prove that if R1 and R2 are operators on Hilbert space such that R1 R2 = O, then σ(R1 + R2 ) ∪ {0} = σ(R1 ) ∪ σ(R2 ). This is an immediate consequence of the following lemma. Lemma 1.6. Let a and b be elements of a unital algebra A such that ab = ba = 0. Then σ(a + b) ∪ {0} = σ(a) ∪ σ(b). Proof. Let λ be a nonzero complex number and let I be the unit of A. We have (a − λI)(b − λI) = λ2 I − λ(a + b) = λ(λI − (a + b)) = (b − λI)(a − λI). Thus, (λI − (a + b)) is invertible if and only if both (a − λI) and (b − λI) are invertible. On the other hand, since ab = 0, 0 must belong to σ(a) ∪ σ(b). The following result describes the spectrum and the essential spectrum of the Hilbert matrix Γ1 . Theorem 1.7. σ(Γ1 ) = σe (Γ1 ) = σ(Γ−1 ) = σe (Γ−1 ) = [0, π]. Proof. By Lemma 1.3, the operator Γ1 Γ−1 is compact. Clearly, Γ1 is self-adjoint. Since Γ−1 = D−1 Γ1 D−1 and D−1 is a self-adjoint unitary operator, the operators Γ1 and Γ−1 are unitarily equivalent. By Lemma 1.4, Γ1 + Γ−1 = 2πiΓzχ+ , and by Lemma 1.1, σe (Γ1 + Γ−1 ) = [0, π]. By Lemma 1.5, [0, π] = σe (Γ1 + Γ−1 ) = σe (Γ1 ) ∪ σe (Γ−1 ) = σe (Γ1 ), since Γ1 and Γ2 are unitarily equivalent. Clearly, σe (Γ1 ) ⊂ σ(Γ1 ). By Corollary 1.7.6, σ(Γ1 ) ⊂ R+ , and since Γ1 ≤ π (see §1.1), it follows that σ(Γ1 ) = [0, π]. Given a bounded operator T on Hilbert space, we denote by Tˇ the coset of T in the Calkin algebra B/C. We need the following general fact. Lemma 1.8. Let Tj , j ≥ 0, be bounded linear operators on Hilbert space such that the operators Tj Tk , Tk Tj , Tj∗ Tk , and Tk Tj∗ are compact ˇ for j = k. Suppose that the series Tj converges unconditionally in j≥0

the Calkin ˇ algebra B/C. Let T be a bounded linear operator such that Tj . Then Tˇ = j≥0

T e = max Tj e j≥0


and

σe (T ) ∪ {O} =

437

σe (Tj ).

j≥0

Proof. Again, we can apply the Gelfand–Naimark theorem and reduce the lemma to the following fact. Let Rj be bounded linear operators on Hilbert space such that Rj Rk = Rk Rj = Rj∗ Rk = Rk Rj∗ = O for j = k. Suppose that the series Rj converges unconditionally in B and j≥0 Rj . Then R= j≥0

R = max Rj

(1.2)

j≥0

and σ(R) ∪ {O} =

σ(Rj ).

(1.3)

j≥0

Let us first prove (1.2). Clearly, it is sufficient to consider the case of two operators R1 and R2 . Since R1 + R2 2 = (R1∗ + R2∗ )(R1 + R2 ) = R1∗ R1 + R2∗ R2 , we may assume without loss of generality that R1 and R2 are self-adjoint and nonnegative. Then Rj = max{λ : λ ∈ σ(Rj )},

j = 1, 2,

and R1 + R2 = max{λ : λ ∈ σ(R1 + R2 )}. Now (1.2) is a consequence of Lemma 1.6. To prove (1.3), we consider the operators def

QN =

N

Rj

def

and Q# N =

j=1

N

Rj .

j>N

By Lemma 1.6, σ(QN ) ∪ {0} =

N

σ(Rj )

j=1

and σ(R) ∪ {0} = σ(QN ) ∪ σ(Q# N ). On the other hand, by (1.2), σ(Q# N ) ⊂ [−εN , εN ], where def

εN = max Rj → 0 j≥N

as

N → ∞.

This proves (1.3). To proceed to the main results of this section, we need one more lemma.

438


Lemma 1.9. Suppose that τ ∈ T and Im τ = 0. Then σe (αΓτ + βΓτ¯ ) = −σe (αΓτ + βΓτ¯ ) for any α, β ∈ C. Proof. By Lemma 1.3, Γ2τ is compact. Thus by the Gelfand–Naimark theorem it is sufficient to prove the following result. Let T be a bounded linear operator on a Hilbert space H such that T 2 = O. Then σ(αT + βT ∗ ) = −σ(αT + βT ∗ ).

(1.4)

Let L = Ker T . Then T has the following representation with respect to the decomposition H = L ⊕ L⊥ : O Q T = . O O

We have ∗

αT + βT =

O βQ∗

αQ O

.

Now (1.4) is a consequence of the following more general fact. If R1 and R2 are bounded linear operators, then the spectrum of O R1 R2 O is symmetric about the origin. Indeed, this can be seen from the following elementary identity: I O λI R1 −I O −λI R1 = . O −I R2 λI O I R2 −λI Now we are going to evaluate the essential spectrum of Hankel operators with piecewise continuous symbols. Recall that the algebra of piecewise continuous functions P C consists of functions ϕ on T that have both the right limit ϕ(ζ + ) and the left limit ϕ(ζ − ) at each point ζ of T (see §1.5). As before, we assume that ϕ(ζ + ) = ϕ(ζ) for any ζ ∈ T and we denote by κζ (ϕ) the jump of ϕ at ζ: κζ (ϕ) = ϕ(ζ) − ϕ(ζ − ). As we have already mentioned in §1.5, for any ε > 0 the set {ζ ∈ T : |κζ (ϕ)| > ε} is finite. Theorem 1.10. Let ϕ ∈ P C and let def

Λ = {ζ ∈ T : κζ (ϕ) = 0}. Then

ˇζ , ˇϕ = i κζ (ϕ)Γ Γ 2π ζ∈Λ

(1.5)


439

and if the set Λ is infinite, the series on the right-hand side of (1.5) converges unconditionally in the Calkin algebra. Proof. Let us first prove that if Λ is infinite, then the series on the right-hand side of 1.5 converges unconditionally in the Calkin algebra. Let ε > 0. Suppose that Ω is a finite subset of the set T+ ∩ {ζ ∈ T : |κζ (ϕ)| ≤ ε}. Then

ˇ ˇ κ (ϕ) Γ + κ (ϕ) Γ ¯ ¯ ζ ζ ζ ζ ζ∈Ω

≤ 2πε

B/C

by Lemma 1.8, which implies unconditional convergence. ˇ ϕ . Put Let us show that the series converges to Γ Λ0 = {ζ ∈ T : |κζ (ϕ)| ≥ 1} and Λn = {ζ ∈ T : 2−n ≤ |κζ (ϕ)| < 2−n+1 },

n = 1, 2, 3, · · · .

It is easy to see that for each n ∈ Z+ there exists a function fn on T such that fn is continuous on T \ Λn , fn has jump κζ (ϕ) at each point ζ ∈ Λn , and 1 fn ∞ = max |κζ (ϕ)|. 2 ζ∈Λn Clearly,

i κζ (ϕ)ψζ ∈ C(T), 2π ζ∈Λn since ψζ has jump 2πi at ζ. Let f = fn . Then ψ ∈ P C and ϕ−f ∈ C(T). fn −

n≥0

We have ˇϕ Γ

ˇf = = Γ

ˇf Γ n

n≥0

=

i ˇζ = i ˇζ . κζ (ϕ)Γ κζ (ϕ)Γ 2π 2π n≥0 ζ∈Λn

ζ∈Λ

Now we are in a position to describe the essential spectrum of Hankel operators with piecewise continuous symbols. Theorem 1.11. Let ϕ ∈ P C. Then σe (Γϕ ) = 8 9 8 9 8 9 iκ1 (ϕ) iκ−1 (ϕ) i 1/2 i 1/2 0, 0, − (κζ (ϕ)κζ¯(ϕ)) , (κζ (ϕ)κζ¯(ϕ)) . 2 2 2 2 ζ∈T\R

440


Proof. By Lemmas 0.1 and 1.8, and Theorem 1.10, the result follows from the equalities σe (Γ1 ) = σe (Γ−1 ) = [0, π] and

2 1 σe (αΓζ + βΓζ¯) = −π(αβ)1/2 , π(αβ)1/2 ,

(1.6)

ζ ∈ T \ R.

(1.7)

Equality (1.6) is a part of Theorem 1.7. Let us prove (1.7). Put def ˇ2 = Γ ˇ 2¯ = O. We have Γ = αΓζ + βΓζ¯. By Lemma 1.3, Γ ζ ζ ˇζ + βΓ ˇ ζ¯)(αΓ ˇζ + βΓ ˇ ζ¯) = αβ(Γ ˇζ Γ ˇ∗Γ ˇ 2 = (αΓ ˇ ∗ζ + Γ ˇ Γ ζ ζ ). Again, by Lemma 1.3, ˇζ Γ ˇ ∗ )(Γ ˇ ˇ∗Γ ˇ∗ ˇ ˇ ˇ∗ (Γ ζ ζ ζ ) = (Γζ Γζ )(Γζ Γζ ) = O, and so by Lemma 1.5, σe (Γ2 ) = σe (Γ∗ζ Γζ ) ∪ σe (Γζ Γ∗ζ ). Since Γζ = Dζ¯Γ1 Dζ¯, we have Γ∗ζ Γζ = Dζ Γ21 Dζ∗

and

Γζ Γ∗ζ = Dζ∗ Γ21 Dζ ,

and so by Theorem 1.7, σe (Γ2 ) = [0, αβπ 2 ]. Finally, by Lemma 1.9, the essential spectrum of Γ is symmetric about the origin, which implies that 2 1 σe (Γ) = −π(αβ)1/2 , π(αβ)1/2 .

2. The Carleman Operator In §1.8 we have introduced the Carleman operator Γ , which is the integral operator on L2 (R+ ) defined by

f (y) dy (Γ f )(x) = x +y R+ on the dense subset of functions with compact support in (0, ∞). Then Γ is a bounded self-adjoint operator, and we have shown in §1 that σ(Γ ) = σe (Γ ) = [0, π]. We obtain a more precise result in this section. Namely, we prove that Γ has Lebesgue spectrum on [0, π] of multiplicity 2. Theorem 2.1. The Carleman operator Γ is unitarily equivalent to multiplication by the function π x → , x ∈ R, cosh(π 2 x) on L2 (R).

2. The Carleman Operator

441

Proof. Consider the operator V : L2 (R+ ) → L2 (R) defined by √ (V f )(t) = 2et f (e2t ), t ∈ R, f ∈ L2 (R+ ). It is easy to see that V is a unitary operator and log x 1 ∗ , x > 0, (V g)(x) = √ g 2 2x

g ∈ L2 (R).

Let us evaluate the operator V Γ V ∗ on L2 (R). Let g ∈ L2 (R). We have

∞ √1 g log y 2 2y (Γ V ∗ g)(x) = dy, x+y 0

and so ∗

(V Γ V g)(t)

∞ =

et 1 √ g 2t e +y y

log y 2

dy

0

∞ = −∞

∞ = −∞

e2t

2et es g(s)ds + e2s

1 g(s)ds. cosh(t − s)

Thus Γ is unitarily equivalent to convolution on L2 (R) with the function s →

1 , cosh s

and so Γ is unitarily equivalent to multiplication on L2 (R) by the Fourier transform of this function. Thus it suffices to prove the following elementary lemma. Lemma 2.2.

∞ −∞

1 π e−2πizx dz = , cosh z cosh(π 2 x)

x ∈ R.

Proof. We use contour integration. Clearly,

∞ −∞

1 e−2πizx dz = lim N →∞ cosh z

N

−N

1 e−2πizx dz. cosh z

It is easy to see that the function (cosh z)−1 has one simple pole at πi/2 in the rectangle with vertices −N, N, N + πi, −N + πi. Thus by the residue

442


theorem we have 2πi Res πi 2

e−2πiz cosh z

N = −N

1 e−2πizx dz + cosh z

N +πi

1 e−2πizx dz cosh z

N

−N

+πi

+

1 e−2πizx dz + cosh z

−N

−N +πi

N +πi


It is easy to see that N +πi

lim

N →∞

1 e−2πizx dz = 0 cosh z

N

and

−N

−N +πi

1 e−2πizx dz = 0. cosh z

It is also easy to compute the residue of (cosh z)−1 at πi/2: Res πi 2

1 = −i, cosh z

and so

2 e−2πizx = 2πeπ x cosh z Using the substitution z → z + πi, we obtain

2πi Res πi 2

−N

+πi

N +πi

1 e−2πizx dz cosh z

N = −N

1 e−2πi(z+πi)x dz cosh z

2π 2 x

N

= e

−N

Thus

∞ −∞


2

1 2πeπ x π e−2πizx dz = . = cosh z cosh(π 2 x) 1 + e2π2 x

Theorem 2.3. The Carleman operator Γ is unitarily equivalent to multiplication by the independent variable on the space L2 ([0, π], C2 ) of C2 -valued functions on [0, π]. In other words, Theorem 2.3 says that Γ has Lebesgue spectrum on [0, π] of multiplicity 2. Proof. Let π def , x ∈ R. ϕ(x) = cosh(π 2 x)

3. Quasinilpotent Hankel Operators

443

By Theorem 2.1, we have to show that multiplication by ϕ on L2 (R) has Lebesgue spectrum on [0, π] of multiplicity 2. Since the function ϕ is even, it is sufficient to show that multiplication by ϕ on L2 (R+ ) is unitarily equivalent to the operator A of multiplication by the independent variable on L2 [0, π]. Consider the operator U : L2 [0, π] → L2 (R+ ) defined by (U f )(s) = |ϕ (s)|1/2 f (ϕ(s)),

f ∈ L2 [0, π].

It is easy to verify that U is a unitary operator from L2 [0, π] onto L2 (R+ ) and U AU ∗ is multiplication by ϕ on L2 (R+ ). This completes the proof.

3. Quasinilpotent Hankel Operators In this section we construct nonzero quasinilpotent Hankel operators. The existence of such operators is not obvious at all. To be more precise, we consider a class of Hankel operators whose symbols have lacunary Fourier series and find a procedure how to evaluate the spectra of such operators. In particular, we will be able to find among them nonzero quasinilpotent Hankel operators. However, we start this section with the problem of whether there are nonzero nilpotent Hankel operators. Theorem 3.1. Let ϕ be a function in L∞ such that Γϕ is nilpotent. Then Γϕ = O. We introduce the following notation. For a function g in L1 we put g# (z) = g(¯ z ). Proof. Suppose that Γϕ = O. Then Ker Γϕ is a nontrivial invariant subspace of multiplication by z on H 2 , and so by Beurling’s theorem, Ker Γϕ = ϑH 2 , where ϑ is a nonconstant inner function (see Appendix 2.2). We have O = Γϕ ϑf = P+ ϕJ (ϑf ) = P+ ϕ(J ϑ)(J f ) = ΓϕJ ϑ f,

f ∈ H 2.

∞ ¯ It , and so there exists ψ ∈ H ∞ such that ϕJ ϑ = z¯ψ. Hence, ϕJ ϑ ∈ H− ¯ follows that ϕ = z¯ϑ# ψ. We may assume that ϑ# and ψ are coprime, i.e., they have no common nonconstant inner factor. Let f ∈ H 2 . We have ¯ f = P+ z¯ϑ# J ((J ψ)f ¯ ) = P+ z¯ϑ# J (ψ# f ) = Γz¯ϑ ψ# f. Γϕ f = P+ z¯ϑ# ψJ #

It is easy to see that Γz¯ϑ# is a partial isometry with initial space Kϑ = H 2 ϑH 2 and final space Kϑ# = H 2 ϑ# H 2 (cf. Theorem 1.2.5). Indeed, if g ∈ H 2 , then Γz¯ϑ# ϑg = P+ z¯ϑ# J (ϑg) = P+ z¯ϑ# (J ϑ)(J g) = P+ z¯(J g) = O. On the other hand, if g ∈ Kϑ , then Γz¯ϑ# g = P+ z¯ϑ# J g = z¯ϑ# J g,

(3.1)

444


2 2 since J g ⊥ z(J ϑ)H− , and so z¯ϑ# J g ⊥ H− . We have 2 2 z¯ϑ# (H 2 ϑH 2 ) = ϑ# H− H− = H 2 ϑ# H 2 = Kϑ# ,

which together with (3.1) proves that the final space of Γz¯ϑ# is Kϑ# . Since Γϕ is quasinilpotent, it follows that there exists a nonzero function g in Kϑ# such that ψ# g ∈ ϑH 2 . Since ψ# and ϑ are coprime, it follows that ϑ divides g, and so g = ϑh with h ∈ H 2 . Since g ∈ Kϑ# , we have P+ ϑ¯# ϑh = O, which means that Ker Tϑ¯# ϑ = {O}. However, Ker Tϑ¯∗# ϑ = Ker Tϑ# ϑ¯ = Ker T(ϑ¯# ϑ)# = {f# : f ∈ Ker Tϑ¯# ϑ }. Thus both Ker Tϑ¯∗# ϑ and Ker Tϑ¯# ϑ are nontrivial, which contradicts Theorem 3.1.4. It turns out, however, that nonzero quasinilpotent Hankel operators do exist. We are going to prove that the operator with the following Hankel matrix is compact and quasinilpotent:   i 12 0 14 0 · · ·  1 0 1 0 0 ···   2 1 4   0   1 4 0 0 01 · · ·  Γ♥ =  (3.2) .  4 0 0 01 8 · · ·   0 0 0  0 ···  8  .. .. .. .. .. . . . . . . . . We consider a more general situation of Hankel  α0 α1 0 α2 0  α1 0 α2 0 0   0 α2 0 0 0  Γ =  α2 0 0 0 α3   0 0 0 0 α 3  .. .. .. .. .. . . . . .

operators of the form  ··· ···   ···   , ···    ···  .. .

where {αk }k≥0 is a sequence of complex numbers. In other words Γ = {γj+k }j,k≥0 , where

αk , j = 2k − 1, k ∈ Z+ , γj = 0, j = 2k − 1, k ∈ Z+ . We evaluate the norm of Γ and give a certain description of its spectrum. By Paley’s theorem, fˆ(2k − 1) k≥0 : f ∈ H 1 = 2 (see Zygmund [1], Ch.XII, §7). It follows that that

αk z 2

k

−1

∈ BM OA

k≥0

if and only if {αk }k≥0 ∈ 2 . So by Theorem 1.1.2, Γ is a matrix of a bounded operator if and only if {αk }k≥0 ∈ 2 ; moreover, Γ is equivalent to {αk }k≥0 2 . It is also clear that for {αk }k≥0 ∈ 2 the function


αk z 2

k

−1

445

belongs to V M OA, and so Γ is bounded if and only if it is

k≥0

compact. We associate with Γ the sequence {µk }k≥0 defined by 1/2 1 µ0 = 0, µk+1 = µk + 2|αk+1 |2 + µ2k + 4|αk+1 |2 , 2

k ∈ Z+ . (3.3)

The following theorem evaluates the norm of Γ. Theorem 3.2. If {αk }k≥0 ∈ 2 , then the sequence {µk }k≥0 converges and Γ2 = lim µk . k→∞

To describe the spectrum of Γ consider the class Λ of sequences of complex numbers {λj }j≥0 that satisfy the equalities λ0 = α0 ,

(λj − λj−1 )λj = αj2 , j ≥ 1.

(3.4)

Theorem 3.3. Suppose that {αj }j≥0 ∈ . Any sequence {λj }j≥0 in Λ converges. The spectrum σ(Γ) consists of 0 and the limits of such sequences. To prove Theorems 3.2 and 3.3, we consider finite submatrices of Γ. Let Lk be the linear span of the basis vectors ej , j = 0, 1, · · · , 2k − 1, and let Pk be the orthogonal projection from 2 onto Lk . Consider the operator def ˜ k def Γk = Pk ΓLk and identify it with its 2k × 2k matrix. Put Γ = Γk Pk , i.e., Γk O ˜k = Γ . O O 2

˜ k and σ(Γ ˜ k ) = σ(Γk ) ∪ {0}. Clearly, It is easy to see that Γk = Γ αk+1 Jk Γk , Γk+1 = αk+1 Jk O where Jk is the 2k × 2k matrix given  0 0  0 0   Jk =  ... ...   0 1 1 0

by ··· ··· .. .

0 1 .. .

··· ···

0 0

1 0 .. .



   .  0  0

We need the following well-known fact from linear algebra (see Gantmakher [1], Ch. 2, §5.3): Let N be a block matrix of the form A B N= , C D where A and D are square matrices and D is invertible. Then det N = det D det(A − BD−1 C).

(3.5)

446


Proof of Theorem 3.2. Since Jk∗ = Jk and Jk2 is the identity matrix of size 2k × 2k (which we denote by I2k ), we have ∗ ˜ ∗ Jk Γk Γk + |αk+1 |2 I2k αk+1 Γ k Γ∗k+1 Γk+1 = . (3.6) ˜k α ¯ k+1 Jk Γ |αk+1 |2 I2k Applying formula (3.5) to the matrix Γ∗k+1 Γk+1 − λI2k+1 , λ = |αk+1 |2 , we obtain ∗ λ ∗ def 2k det Γk+1 Γk+1 − λI2k+1 = ρ det − Γk Γk + ρI2k , ρ = |αk+1 |2 − λ. ρ (3.7) Since Γ is a bounded operator, we have Γ = lim Γk . Therefore it is k→∞

sufficient to show that µk = Γk 1/2 or, which is the same, that µk is the largest eigenvalue of Γ∗k Γk . We proceed by induction on k. For k = 0 the assertion is obvious. If Γk = O, the assertion is obvious. Otherwise, it follows easily from (3.6) that Γ∗k+1 Γk+1 > |αk+1 |2 . It is easy to see from (3.7) that λ = |αk+1 |2 is an eigenvalue of Γ∗k+1 Γk+1 if and only if ρ2 /λ is an eigenvalue of Γ∗k Γk . Put µ = ρ2 /λ = (|αk+1 |2 − λ)2 /λ. If µ is an eigenvalue of Γ∗k Γk , it generates two eigenvalues of Γ∗k+1 Γk+1 : 1 µ + 2|αk+1 |2 + (µ2 + 4|αk+1 |2 )1/2 2 and 1 µ + 2|αk+1 |2 − (µ2 + 4|αk+1 |2 )1/2 . 2 Clearly, to get the largest eigenvalue of Γ∗k+1 Γk+1 , we have to put µ = µk and choose the first of the above eigenvalues. This proves that µk+1 defined by (3.3) is the largest eigenvalue of Γ∗k+1 Γk+1 . To prove Theorem 3.3 we need the following lemmas. Lemma 3.4. Let Λk be the set of kth terms of sequences in Λ, i.e., Λk = {λk : {λj }j≥0 ∈ Λ}. If {ζj }j≥0 is an arbitrary sequence satisfying ζj ∈ Λj , then it converges if and only if lim ζj = 0 or there exists a sequence {λj }j≥0 ∈ Λ such that j→∞

ζj = λj for sufficiently large j. Lemma 3.5. Let A be a compact operator on Hilbert space and let {Aj }j≥0 be a sequence of bounded linear operators such that lim A − Aj = 0. Then the spectrum σ(A) consists of limits of all conj→∞

vergent sequences {νj }j≥0 such that νj ∈ σ(Aj ). Let us first deduce Theorem 3.3 from Lemmas 3.4 and 3.5.


447

Proof of Theorem 3.3. Since Γ is compact, 0 ∈ σ(Γ). For λ ∈ C \ {0} we apply formula (3.5) to the matrix Γk+1 − λI2k+1 and obtain α2 k det(Γk+1 − λI2k+1 ) = (−λ)2 det Γk − λ − k+1 I2k . λ (3.8) Obviously, 0 ∈ σ(Γk )

if and only if αk = 0.

Together with (3.8) this implies that λ ∈ σ(Γk ) if and only if there exists λ ∈ σ(Γk−1 ) such that (λ − λ )λ = αk2 . Let λ be a nonzero point of spectrum of Γ. Then by Lemma 3.5, there ˜ j ). exists a sequence {νj }j≥0 such that νj → λ as j → ∞ and νj ∈ σ(Γ Since λ = 0 we may assume without loss of generality that νj ∈ σ(Γj ). It now follows from Lemma 3.4 that there exists a sequence {λj }j≥0 in Λ such that λj = νj for sufficiently large j, and so λ = lim λj . j→∞

Conversely, let {λj }j≥0 ∈ Λ. By Lemma 3.4, {λj }j≥0 converges to a point λ ∈ C. As we have already observed, λj ∈ σ(Γj ) and so by Lemma 3.5, λ ∈ σ(Γ). Proof of Lemma 3.4. Let {λj }j≥0 ∈ Λ. Then |λj | ≤ |λj−1 |+|αj |2 /|λj |, j ≥ 1. It follows that |λj | ≤ max{ε, |λj−1 | + |αj |2 /ε}

(3.9)

for any ε > 0. Let us show that either λj → 0 as j → ∞ or |λj | ≥ δ for some δ > 0 for sufficiently large j. To prove this, let us show that if ε > 0 and lim inf |λj | < ε, then j→∞

lim sup |λj | ≤ 2ε. Assume the contrary, i.e., lim inf |λj | < ε and j→∞

j→∞

lim sup |λj | > 2ε for some ε > 0. It follows that for any N ∈ Z+ there exist j→∞

positive integers m and n such that N ≤ m < n, |λm−1 | < ε, |λj | ≥ ε for m ≤ j ≤ n, and |λn | ≥ 2ε. It follows from (3.9) that |λj | ≤ |λj−1 | + |αj |2 /ε, m ≤ j ≤ n. Therefore n 1 |λn | ≤ |λm−1 | + |αj |2 . ε j=m

Since {αk }k≥0 ∈ 2 , we can choose N so large that contradicts the inequality |λn | ≥ 2ε. If |λj | ≥ δ > 0 for large values of j, then by (3.4), |λj − λj−1 | ≤ Therefore {λj }j≥0 converges.

|αj |2 . δ

1 ε

∞ j=m

|αj |2 < ε, which

448


Suppose now that {λj }j≥0 and {νj }j≥0 are sequences in Λ that have nonzero limits. Then for sufficiently large j αj2 |λj−1 − νj−1 | = (λj − νj ) 1 + ≤ |λj − νj |(1 + d|αj |2 ) λj νj for some d > 0. Iterating this inequality, we obtain ∞ |λj−1 − νj−1 | ≤ lim λj − lim νj · (1 + d|αm |2 ) j→∞

j→∞

m=j

(the infinite product on the right-hand side converges since {αj }j≥0 ∈ 2 ). Therefore if lim λj = lim νj , then λj = νj for sufficiently large j. j→∞

j→∞

For ε > 0 we consider the set of sequences {λj }j≥0 in Λ such that sup{|λj | : j ≥ k} ≥ ε for any positive integer k. Let us show that the number of such sequences is finite. Suppose that {λj }j≥0 ∈ Λ is such a sequence. As we have observed in the beginning of the proof, there exist δ > 0 and j0 ∈ Z+ such that |λj | ≥ δ for sufficiently large j. Clearly, |αj | < δ for sufficiently large j. It follows that if j is sufficiently large, then λj is uniquely determined by λj−1 by the conditions (λj − λj−1 )λj = αj2 ,

|λj | ≥ δ.

Hence, there are only finitely many possibilities to get such sequences. Now let{ζj }j≥0 be a converging sequence such that ζj ∈ Λj , j ≥ 0, and lim ζj = 0. Then as we have already proved, there are finitely many j→∞ (s) (1) (m) ∈ Λ, s = 1, · · · , m, such that ζj ∈ λj , · · · , λj sequences λj j≥0 (s) have distinct limits. It for sufficiently large j and the sequences λj j≥0

(s)

follows that there exists an s, 1 ≤ s ≤ m, such that ζj = λj for sufficiently large j. Proof of Lemma 3.5. Since the set of invertible operators is open in the norm topology, it follows that σ(A) contains the set of limits of converging sequences {νj }j≥0 such that νj ∈ σ(Aj ). Let us prove the opposite inclusion. Suppose that λ0 ∈ σ(A). If λ0 is not a limit of any such sequence, then there is a subsequence {Ajk }k≥0 such that dist(λ0 , σ(Ajk )) ≥ δ0 for some positive δ0 . Since A is compact, there exists a δ ∈ (0, δ0 ) such that the circle of radius δ with center at λ0 does not intersect σ(A). For arbitrary vectors x, y we put Ψx,y (ζ) = (x, (A − (λ0 + δζ)I)−1 y), −1 y), Ψ(k) x,y (ζ) = (x, (Ajk − (λ0 + δζ)I) (k)

ζ ∈ T.

Obviously, the functions Ψx,y extend analytically to a neighborhood of the (k) closed unit disk and Ψx,y → Ψx,y uniformly on T. It follows that Ψx,y


449

extends to a function in CA . Obviously, sup{|Ψx,y (ζ)| : ζ ∈ T} ≤ const x · y, which implies that sup{|Ψx,y (ζ)| : |ζ| ≤ 1} ≤ const x · y. This contradicts the fact that λ0 ∈ σ(A). Let us now proceed to the operator Γ♥ defined by (3.2). In other words we consider the operator Γ with 1 , j ≥ 1. 2j Theorem 3.6. Γ♥ is a compact quasinilpotent operator. Proof. It is easy to see by induction that if {λj }j≥0 satisfies (3.4), then λj = 2−j i, and so by Theorem 3.3, σ(Γ# ) = {0}. We have already seen that bounded Hankel operators of this form are always compact. α0 = i,

αj =

Remark. We can consider a more general situation when Γ = {γj+k }j,k≥0 with

αk , j = nk − 1, j ∈ Z+ , γj = 0 j = nk − 1, k ∈ Z+ , where {nk }k≥0 is a sequence of natural numbers such that nk+1 ≥ 2nk , k ≥ 0, and {αk }k≥0 ∈ 2 . It is easy to see that the same results hold and the same proofs also work in this situation, which allows one to construct other quasinilpotent Hankel operators. In particular, it is easy to see that the following Hankel operator is quasinilpotent:   0 i 0 12 0 · · ·  i 0 1 0 0 ···  2    0 1 0 0 0 ···    1 2 1 .   2 0 0 01 4 · · ·   0 0 0 0 ···  4   .. .. .. .. .. . . . . . . . . The following fact is an amusing consequence of the generalization of Theorem 3.2 mentioned in the above remark. Corollary 3.7. Let {nk }k≥0 be a sequence of natural numbers such that nk+1 ≥ 2nk , k ≥ 0, and {ak }k≥0 ∈ 2 . Let ϕ be the function on T defined by αk z nk −1 . ϕ(z) = k≥0

Then inf{ϕ − f ∞ : f ∈ P− BM O, ϕ − f ∈ L∞ } depends only on the moduli of the αk , k ≥ 0. Proof. This is an immediate consequence of Theorem 3.2.

450


Concluding Remarks The essential spectrum of Hankel operators with piecewise continuous symbols was described by Power [1]; see also Power [2]. An important role in the proof is played by the Hilbert matrix whose spectrum was described in Magnus [1] (see Theorem 1.7). The results of §2 are due to Carleman [1]; see also Power [2]. Theorem 3.1 is due to Power [3]. The problem of whether there exist nonzero quasinilpotent Hankel operators was posed in Power [3]. It was solved by Megretskii [1]. His solution is presented in §3. Lemma 3.5 is well known; see Newburg [1]. We mention here the paper Rosenblum [1] in which it was shown that the Hilbert matrix has simple Lebesgue spectrum on the interval [0, π]. In other words, it is unitarily equivalent to multiplication by the independent variable on the space L2 [0, π] (with respect to Lebesgue measure). Interesting results on spectral properties of self-adjoint Hankel operators were obtained by Howland [2–4]. In particular, in Howland [2] the following result was obtained. Suppose that ϕ is a real piecewise continuous function with finitely many jumps and such that it is of class C 2 on the complement of the jump points of f . If ξ is a jump point of f , we put  / 1 0 − |κ (f )|, 1 |κ (f )| , Im ξ = 0,    / 2 ξ 0 2 ξ ξ = ±1, κξ (f ) > 0, 0, 12 κξ (f ) , I(ξ) =   0 /  1 ξ = ±1, κξ (f ) < 0, 2 κξ (f ), 0 , where κξ (f ) = f (ξ + ) − f (ξ − ). It was proved in Howland [2] that the absolutely continuous part of Γf is unitarily equivalent to the orthogonal sum of multiplications by the independent variable on I(ξ). In Howland [3] the author studied integral operators on L2 (R+ ) of the form

∞ ϕ(x)ϕ(y) ¯ (Qf )(x) = dy. x + y 0 In the case when the limits a = lim |ϕ(x)| and b = lim |ϕ(x)| exist it was x→0+

x→∞

shown in Howland [3] under mild conditions on ϕ that Q has no singular continuous part, its absolutely continuous part is the orthogonal sum of multiplications by the independent variable on [0, πa2 ] and [0, πb2 ], and the nonzero eigenvalues can accumulate only at 0. On the other hand, Q is unitarily equivalent to the Hankel operator Γ k , where

∞ k(x) = e−xs |ϕ(x)|2 ds. 0

In Howland [4] the author studied Hankel operators Γ k on L2 (R+ ). He proved that if k is of class C 2 on (0, ∞), the limits a = lim xk(x) and x→0

Concluding Remarks

451

b = lim xk(x) exist, then under certain mild assumptions Γ k has no sinx→∞ gular continuous part and its absolutely continuous part is unitarily equivalent to the orthogonal sum of multiplications by the coordinate function on L2 [0, πa] and L2 [0, πb]. The spectral structure of self-adjoint Hankel operators was completely described in Megretskii, Peller, and Treil [1] and [2]. We present these results in Chapter 12. Finally, we mention here the paper Martinez-Avenda˜ no and Treil [1], in which it was shown that the spectrum of a Hankel operator can be an arbitrary compact set in C that contains the origin.

11 Hankel Operators in Control Theory

This chapter is a brief introduction to control theory written for mathematicians. To be more precise, we consider here several problems in control theory that involve Hankel operators. For readers interested in a more detailed study we recommend the books Doyle [1], Francis [1], Fuhrmann [4], Helton [3], Doyle, Francis, and Tannnenbaum [1], Dahleh and Diaz-Bobillo [1], and Chui and Chen [1]. In §1 we introduce the notion of a plant and its transfer function. In §2 we consider realizations of transfer functions with discrete time while in §3 we consider the case of continuous time. Next, in §4 we study the model reduction problem. The rest of the chapter is devoted to the problem of robust stabilization. In §5 we describe the problem and state a necessary and sufficient condition for a feedback stabilizer to be robust (Theorem 5.1). We obtain in §6 a doubly coprime factorization of rational matrix functions. Section 7 is devoted to the proof of Theorem 5.1 stated in §5. In §8 we obtain a parametrization formula for all feedback stabilizers of a given plant. Finally, in §9 we reduce the problem of robust stabilization to the so-called model matching problem, which in an important special case reduces to the Nehari problem. Note that realizations of linear systems are used essentially in Chapter 12 to solve the inverse spectral problem for self-adjoint Hankel operators.

454

Chapter 11. Hankel Operators in Control Theory

1. Transfer Functions The main object we are going to study in this section is a plant (or a black box, or a system). A plant G with discrete time is a map on a space of (two-sided) sequences (scalar or vector). We say that a plant is single input single output (or briefly, SISO) if it maps scalar sequences into scalar sequences. If G is defined on a space of Cn -valued sequences and maps them to Cm -valued sequences and both m and n are greater than 1, we say that G is a multiple input multiple output (or briefly, MIMO) plant. Certainly, there are single input multiple output and multiple input single output plants. One can also consider plants acting between spaces of sequences that take values in infinite-dimensional spaces (such as 2 ). We interpret the index set Z as discrete time. A plant with continuous time is defined on a space functions (scalar or vector) on R and maps it to another space functions. Similarly, we can consider the case of single input single output or multiple input multiple output. Here we interpret R as continuous time. We are going to represent the plant G with the following diagram:

u

-

y G

-

FIGURE 1. Plant

Here G takes an input sequence u = {un } (or a function u on R in the case of continuous time) into an output sequence y = {yn } (a function y on R, respectively). We consider the following properties of plants: 1. Linearity. By this we simply mean that G is a linear operator. 2. Time invariance. This means that if the input u is shifted by m, then the output y is also shifted by m. 3. Causality. This property means that if two inputs are identical before time m, then the corresponding outputs are also identical before time m. All plants considered here are supposed to possess all three above properties. We consider here one more property that some plants have, and others do not have. 4. Stability. A plant is called stable if it takes inputs of finite energy (which is the norm of u in (scalar or vector) 2 or L2 norm) into outputs of finite energy. In other words this means that G is bounded.

1. Transfer Functions

455

Suppose now that G a SISO plant with discrete time. We can identify sequences {xn }n∈Z with zero past (i.e., such that xn = 0 for n < 0) with one-sided sequences {xn }n∈Z+ . It follows from causality that for such inputs the output sequences also have zero past. Consider the input sequence v = {vn }n≥0 with v0 = 1 and vn = 0 for n > 0. Let {gn }n≥0be the corresponding output. Consider the formal power series G(z) = gn z n . n≥0

It follows from the above properties 1–3 that for any finitely supported sequence u = {un }n≥0 the output sequence y = {yn }n≥0 is given by def def un z n and yˇ(z) = yn z n . (1.1) yˇ(z) = G(z)ˇ u(z), where u ˇ(z) = n≥0

n≥0

In control theory one usually considers only plants G such that     |yn M −n |2 < ∞ G2 ⊂ {yn }n≥0 :   n≥0

for some M > 0. It is easy that this is equivalent to the fact that to see the power series G(z) = gn z n converges in a disk of positive radius, the n≥0 function G(z) = gn z n is called the transfer function of the plant G. n≥0

If G is a stable plant, then multiplication by G extends to a bounded operator on H 2 , and so G is a analytic in D and bounded there, i.e., G ∈ H ∞ . Clearly, the converse is also true. Thus a plant G has the above properties 1–4 if and only if there exists G ∈ H ∞ such that (1.1) holds for any input sequence u = {un }n≥0 ∈ 2 , where y = {yn }n≥0 is the output sequence. Note that in control theory the most important case is when the transfer function is rational. Indeed, very often the input and output sequences are connected with each other by p(S)u = q(S)y, where p and q are polynomials and S is the shift operator on the space of sequences. If this is the case, we have p(z)ˇ u = q(z)ˇ y (z), and so the transfer function of such a plant is the rational function p/q. Note, however, that certain problems in control theory lead to nonrational transfer functions (see the case of continuous time below). If the input sequences and the output sequences take values in Hilbert spaces H and K, the transfer function G is a formal power series whose coefficients are operators from H to K. In particular, if H = Cn and K = Cm , the coefficients of the formal power series G can be interpreted as m × n matrices. As in the case of SISO systems, the system is stable if and only if G ∈ H ∞ (B(H, K)).

456


Consider now the case of SISO systems with continuous time. Again we identify in a natural way L2 (R+ ) with the subspace of L2 of functions vanishing on (−∞, 0). In control theory it is natural to assume that there exists a number σ ∈ R such that the plant G maps L2 (R+ ) into the space L2σ (R+ ),

∞ −σt 2 def 2 Lσ (R+ ) = f : e f (t) dt < ∞ . 0

If f is a function in

its Laplace transform Lf ,

∞ 1 def f (t)e−ζt dt, (Lf )(ζ) = √ 2π 0

L2σ (R+ ),

is a function analytic in the half-plane {ζ ∈ C : Re ζ > σ}. By the Paley– Wiener theorem (see Appendix 2.1), L maps unitarily L2 (R+ ) onto the Hardy class H 2 (C+ ). Note that in control theory it is common to consider Laplace transform rather than Fourier transform and to consider the Hardy classes H 2 (C− ) and H 2 (C+ ) of functions on the left and right half-planes def

def

C− = {ζ ∈ C : Re ζ < 0} and C+ = {ζ ∈ C : Re ζ > 0} rather than the lower and the upper half-planes. It follows immediately from the Paley– Wiener theorem that L maps unitarily L2σ (R+ ) onto the Hardy class H 2 of functions on the half-plane {ζ ∈ C : Re ζ > σ}. Consider now the operator G = LGL−1 as an operator from H 2 (C+ ) to H 2 ({ζ ∈ C : Re ζ > σ}). It is easy to verify that time invariance means that G(e−az ϕ) = e−az Gϕ, ϕ ∈ H 2 (C+ ), a > 0. def

It follows that if ϕ and ψ are linear combinations of functions e−az , a > 0, then ψGϕ = ϕGψ, and so, if ϕ is a linear combination of functions e−az , a > 0, then Gϕ = Gϕ, where

(1.2)

(Ge−z )(ζ) , Re ζ > σ. e−ζ Clearly, the function G is analytic in {ζ ∈ C : Re ζ > σ}. It is called the transfer function of the plant G. Since the set of such functions ϕ is dense in H 2 (C+ ), (1.2) extends to arbitrary functions ϕ in H 2 (C+ ). Now it is easy to see that G is a stable system if and only if the transfer function G is bounded and analytic in C+ . As in the case of discrete time the most important case in control theory is when the transfer function is rational, since very often the input and output functions are related to each other via the formula G(ζ) =

p(D)u = q(D)y,

2. Realizations with Discrete Time

457

where D is the differentiation operator, and p and q are polynomials. If we pass to Laplace transform, we see that in this case the transfer function is rational. Example. Consider now an important example of a time-delay system in which the transfer function is nonrational. Suppose that the output function y depends on the input function u via the formula y (t) + y(t − 1) = u(t). If we pass to Laplace transform, we obtain ζ(Ly)(ζ) + e−ζ (Ly)(ζ) = (Lu)(ζ), and so the transfer function is 1 , z + e−z which is nonrational. As in the case of discrete time one can also consider vector-valued input and output functions and deal with operator-valued (or matrix-valued) transfer functions. As an example of a physical system one can consider a flying airplane. Its position is influenced by control input signals. Certainly, this dependence is nonlinear. However, if we consider only small deviations from a certain equilibrium state, we can approximate nonlinear functions by linear ones and study the resulting linear system.

2. Realizations with Discrete Time In this section we introduce the notion of a realization of a transfer function in the case of discrete time. We define the Hankel operator associated with a given realization (linear system). We introduce the important notion of a balanced realization and we show that a transfer function has balanced realization if and only if it generates a bounded Hankel operator. Finally, we establish the uniqueness of a balanced realization up to equivalence. Let K, E, and H be Hilbert spaces. Suppose that A ∈ B(H), B ∈ B(K, H), C ∈ B(H, E), and D ∈ B(K, E). Consider the linear dynamical system with discrete time

xk+1 = Axk + Buk , (2.1) uk ∈ K, xk ∈ H, and yk ∈ E. yk = Cxk + Duk , This system takes an input signal {uk }k∈Z to the output signal {yk }k∈Z . H is called the state space K is called the input space, and E is called the output space. We use the notation {A, B, C, D} for the linear system (2.1). Suppose that {uk }k∈Z is a finitely supported sequence such that uk = O and xk = O for k < 0. Clearly, yk = O for k < 0 and x0 = O. This linear

458


system determines a plant that has the properties of linearity, causality, and time invariance. Let us compute the transfer function of the system. Consider the power series def def def u ˇ(z) = z j uj , x ˇ(z) = z j xj , and yˇ(z) = z j yj . j≥0

j≥0

j≥0

Now we can rewrite (2.1) as follows: xˇ(z) x(z) + B u ˇ(z), z = Aˇ yˇ(z) = C x ˇ(z) + Dˇ u(z). It follows that for sufficiently small ζ yˇ(ζ) = (ζC(I − ζA)−1 B + D)ˇ u(ζ),

(2.2)

def

i.e., G = zC(I − zA)−1 B + D is the transfer function of the plant, or the transfer function of the system. Let now G be a plant. We say that the linear dynamical system (2.1) is a realization of the plant G with transfer function G (or a realization of the transfer function G) if G = zC(I − zA)−1 B + D. As before, we say that the system (2.1) is single input single output (or SISO) if dim K = dim E = 1. Let us illustrate now how linear dynamical systems lead to Hankel operators. Suppose now that {uk }k∈Z is a finitely supported sequence and we cut the input signal at time 0, i.e., we annihilate uk for k ≥ 0. Suppose also that xk = O when k < min{j : uj = O}. Let us study the behavior of the output signal {yk } for k ≥ 0. Clearly, the yk for k ≥ 0 do not depend on D, and so we may assume that D = O. Consider the formal Laurent series def def z j uj and yˇ(z) = z j yj . u ˇ(z) = j σ}. Let us pass to Laplace transform. We have from (3.1)

ζ(Lx)(ζ) = A (Lx)(ζ) + B (Lu)(ζ) , Re ζ > σ. (Ly)(ζ) = C (Lx)(ζ) + D (Lu)(ζ) , It follows that

(Ly)(ζ) = C(ζI − A)−1 B + D (Lu)(ζ) ,

Re ζ > σ.

Thus C(zI − A)−1 B + D is the transfer function of the system. If G is a plant with continuous time whose transfer function is G, we say that the system (3.1) is a realization of the plant G (or a realization of the transfer function G) if G(ζ) = C(ζI − A)−1 B + D for Re ζ sufficiently large. As usual we say that the system (3.1) is single input single output (or SISO) if dim K = dim E = 1. As in the case of discrete time, the study of systems (3.1) naturally leads to Hankel operators. Suppose that u is smooth and has compact support in (−∞, 0) and x(t) = 0 for t < supp u, i.e., cut the input signal at t = 0, and we study the output signal y for positive values of t. Clearly, if we are interested only in the behavior of the output signal for t > 0, D does not play any role, and we can assume that D = O. Let M be a positive number such that supp u ⊂ (−M, 0). Consider the translations uM and yM of u and y, uM (t) = u(t − M ), yM (t) = y(t − M ), t ∈ R. It is easy to see that

(LyM )(ζ) = C(ζI − A)−1 B (LuM )(ζ) ,

Re ζ > σ.

Consider the operator function Ξ on R defined by Ξ(t) = CetA B. Its Laplace transform can be computed very easily:

∞ 1 1 √ e−ζt CetA Bdt = √ C(ζI − A)−1 B, 2π 0 2π if Re ζ is sufficiently large, i.e., 1 1 (LΞ)(ζ) = √ C(ζI − A)−1 B = √ G(ζ). 2π 2π

468


Thus yM is the convolution of the functions Ξ and uM , i.e.,

s yM (s) = Ce(s−t)A BuM (t)dt, 0

and it is easy to see that

∞ Ce(t+s)A Bu(−t)dt = y(s) = 0

∞

def

Ξ(s + t)v(t)dt = (Γ Ξ v)(s),

s > 0.

0 def

where v(t) = u(−t), t ∈ R. The Hankel operator Γ Ξ is defined on the set of smooth functions with compact support in (0, ∞) (see §1.8 and the Remark at the end of §2.2). The function Ξ is called the impulse response function of the system {A, B, C}. We say that Γ Ξ is the Hankel operator associated with the system {A, B, C}. It does not have to be a bounded operator from L2 (R+ , K) to L2 (R+ , E) (we use this notation for the L2 spaces of K-valued and E-valued functions on R+ ). However, as in the case of discrete time, Γ Ξ is bounded in important special cases. Example. Consider the linear ordinary differential equation with constant coefficients y (t) + ay (t) + by(t) = u(t). Here u is an input function and y is an output function. Let us construct a realization of this SISO system. The input space and the output space are x 1 equal to C, while the state space is C2 . Let x = ∈ X. Consider the x2 following realization   x1 (t) x1 (t) 0 1 0   = + u(t),   −b −a 1  x2 (t) x2 (t)   x1 (t)   .   y(t) = 1 0 x2 (t) It is easy to see that the transfer function is the scalar rational function (z 2 + za + b)−1 . The system is stable if and only if both zeros of this polynomial have negative real parts. As in the case of discrete time, we say that the system (3.1) is controllable if span{etA Bu : u ∈ K, t > 0} = H. It is called observable if {x ∈ H : CetA x = O for any t > 0} = {O}. The system is called minimal if it is both controllable and observable.

3. Realizations with Continuous Time

469

The controllability Gramian Wo and the observability Gramian Wc are defined by

∞

∞ ∗ tA ∗ tA∗ e BB e dt, Wo = etA C ∗ CetA dt Wc = 0

0

if the integrals converge in the weak operator topology. The system is called balanced if both Wo and Wc exist and Wo = Wc . The following result is an analog of Theorems 2.1 and 2.2. Theorem 3.1. Suppose that the series that define the controllability and observability Gramians Wc and Wo of the system (3.1) converge in the weak operator topology. Then the Hankel operator Γ Ξ associated with the system (3.1) is a bounded operator from L2 (R+ , K) to L2 (R+ , E). The re1/2 1/2 1/2 striction of |Γ Ξ | to (Ker Γ Ξ )⊥ is unitarily equivalent to Wo Wc Wo . Proof. The proof of Theorem 3.1 is exactly the same as the proof of Theorems 2.1 and 2.2. Indeed, we can introduce the operators Vc : H → Λ2 (R+ K) and Vo : H → Λ2 (R+ , E), (Vc x)(t) = B ∗ etA

∗

and Vo x = CetA x,

x ∈ H,

t ∈ R+ .

(3.2)

Then as in the case of discrete time it is easy to see that Γ Ξ = Vo∗ Vc . Again, as in the case of discrete time it is easy to see that 1/2 1/2 |Γ Ξ |2 = Uc Wo Wc Wo Uc∗ , where Uc is the partially isometric factor in the polar decomposition of Vc . As in the proof of Theorem 2.2 this implies the result. Suppose now that G is a B(K, E)-valued function analytic in C+ such def

that the limits G(y) = lim G(x + iy) exist almost everywhere in the weak x→0

operator topology and the Hankel-like operator f → PH 2 (C+ ) Gf,

f ∈ H 2 (C− ),

is bounded from H 2 (C− ) to H 2 (C+ ). It turns out that unlike the case of discrete time, it is not always possible to find a balanced realization of G of the form (3.1) with bounded operators A, B, and C (see §12.7). One can consider the conformal map ι : C+ → D, ι(ζ) =

ζ −1 , ζ +1

Re ζ > 0,

and define the function Gd on D by Gd = G ◦ ι−1 . Then as in the case of scalar functions (see §1.8 and the Remark at the end ˆ d (j +k)}j,k≥0 is bounded. By Theorem of §2.2), the block Hankel matrix {G

470


2.5, there exists a balanced realization {Ad , Bd , Cd , Dd } of Gd with discrete time. Suppose that Ad − I is invertible. Consider the following operators: √ −1 (Ad + I), B = 2(Ad − I)−1 Bd , A = (A√ d − I) C = − 2Cd (Ad − I)−1 , D = Dd − Cd (Ad − I)−1 Bd . Then {A, B, C, D} is a balanced realization of G with continuous time. Indeed, let us compute the transfer function of the system {A, B, C, D} with continuous time. Put w = (ζ − 1)(ζ + 1)−1 , ζ ∈ C+ . We have C(ζI − A)−1 B + D

−1 = −2Cd (Ad − I)−2 ζI − (Ad − I)−1 (Ad + I) Bd + D −1 = −2Cd (Ad − I)−1 − (ζ + 1)I + (ζ − 1)Ad Bd + D

= −2(ζ + 1)−1 (I − Ad )−1 (I − wAd )−1 Bd + D = wCd (I − wAd )−1 Bd + Cd (Ad − I)−1 Bd + D = wCd (I − wAd )−1 Bd + Dd = Gd (w). Now let us show that the system {A, B, C, D} is balanced. Consider the operator Vo defined by (3.2) and consider the corresponding operator Vod for the system {Ad , Bd , Cd , Dd } with discrete time: Vod x = {Cd And x}n≥0 ,

x ∈ H.

Consider the operator V from H (H) to H (C+ , H) defined by ζ −1 1 1 , ζ ∈ C+ . (f ◦ ι)(ζ) = π −1/2 f (Vf )(ζ) = π −1/2 ζ +1 ζ +1 ζ +1 2

2

Then V is the unitary operator from H 2 (H) onto H 2 (C+ , H) (see Appendix 2.1, where the similar operator U from H 2 onto H 2 (C+ ) is considered). We now define the unitary operator W : 2 (H) → L2 (R+ , H) by   W{xn }n≥0 = L−1 V  z n xn  . n≥0

Here L is the Laplace transform on the space L2 (R+ , H) of H-valued functions which is defined in the same way as for scalar functions. We claim that Vo = WVod . Indeed, we have (LVo x)(ζ)

= =

∞ 1 √ C e−ζt etA xdt 2π 0 1 √ C(ζI − A)−1 x, x ∈ H, 2π

(3.3)

Re ζ > 0,

3. Realizations with Continuous Time

471

(since Ad is a contraction (see Theorems 2.5 and 2.9), it follows that etA ≤ 1, t > 0, which implies the convergence of the integral). On the other hand, z n Cd And x = Cd (I − zAd )−1 x, x ∈ H. n≥0

Thus (3.3) is equivalent to the equality −1 1 ζ −1 1 √ C(ζI − A)−1 = √ (ζ + 1)−1 Cd I − Ad , ζ +1 π 2π We have 1 √ C(ζI − A)−1 2

Re ζ > 0.

−1 = −Cd (Ad − I)−1 ζI − (Ad − I)−1 (Ad + I) −1 = −Cd − (ζ + 1)I + (ζ − 1)Ad =

−1

(ζ + 1)

Cd

ζ −1 Ad I− ζ +1

−1 ,

which completes the proof of (3.3). Consider now the operator Vc defined by (3.2) and the corresponding operator Vcd for the system {Ad , Bd , Cd , Dd } with discrete time: Vcd x = {Bd∗ (A∗d )n x}n≥0 ,

x ∈ H.

In a similar way it is easy to show that Vc = −WVcd .

(3.4)

Since the system {Ad , Bd , Cd , Dd } is balanced, it follows from (3.3) and (3.4) that the system {A, B, C, D} is balanced. However, if I − Ad is not invertible, the situation is more complicated. In the general case the operator A does not have to be bounded, it is the generator of a contractive semigroup, B is an operator from K to the dual space (D(A∗ ))∗ to the domain of A∗ (note that H is naturally imbedded in (D(A∗ ))∗ ), and C is a densely defined functional on the domain D(A) of A. We are going to consider in this book only realizations with bounded operators A, B, and C. We call such realizations (or such linear systems) proper, while realizations for which the operator I − Ad is not invertible will be referred to as generalized realizations. We refer the reader to Ober and Montgomery-Smith [1], where the general case is considered. Remark. Note that if the transfer function G is a stable rational function, then it admits a proper balanced realization. Indeed, in this case the corresponding function Gd in D is rational and has poles outside the closed unit disc, and so the operator Ad has spectrum in the open unit disk. Thus

472


Ad − I is invertible, which implies that A, B, and C are bounded operators. It follows now from Corollary 2.4 that if G is a stable rational transfer function (i.e., G has no poles in the closed right half-plane), then the dimension of the state space of a balanced realization of G is equal to the McMillan degree of G. Let us prove now that any proper rational matrix function (i.e., such a rational matrix function that has no pole at ∞) has a minimal realization. Clearly, any rational transfer function is proper. Corollary 3.2. Let G be a proper rational matrix function. Then G has a minimal realization. Proof. Let a be a positive number such that G has no poles in the halfplane {ζ ∈ C : Re ζ ≥ a}. Consider the auxiliary transfer function G# defined by G# (ζ) = G(ζ + a). Clearly, G# is a stable rational function, and so it has a proper balanced realization {A, B, C, D}. It is easy to see now that {A + aI, B, C, D} is a minimal realization of G.

4. Model Reduction Suppose that we have a plant that admits a realization in a finite-dimensional state space. Such a system can be designed by engineers if the dimension of the state space is not too large. The dimension of the state space reflects the complexity of the system. The larger the dimension is, the more difficult it is to design the system. Let G be the transfer function of our plant. If it is a stable rational function, it admits a balanced realization such that the dimension of the state space is equal to the McMillan degree of G (see Corollary 2.4 and the Remark at the end of §3). If G is not rational and it has a balanced realization, then the state space is infinite-dimensional. In general, if the dimension of the state space is too large, it would be helpful to design another system that has a realization in a space of smaller dimension and whose properties do not differ much from the properties of the initial system. This is exactly the model reduction problem. Suppose that G is the initial transfer function and G# is the transfer function of the realization designed to model the initial system. How do we measure the deviation of G# from G? To be definite, consider the case of discrete time. If G is a stable transfer function, we can consider the norm of the linear operator that takes the input sequence {un }n∈Z ∈ 2 (K) to the output sequence {yn }n∈Z ∈ 2 (E): sup {yn }n∈Z 2 (E) : {un }n∈Z 2 (K) ≤ 1 . Clearly, this norm is equal to GH ∞ (B(K,E)) . This norm on the space of stable transfer functions is physically well-motivated. We say that systems with transfer functions G and G# have close performances if the norm G − G# H ∞ (B(K,E)) is small.

5. Robust Stabilization

473

Thus we arrive at the following problem. Given a stable transfer function G ∈ H ∞ (K, E) and a positive integer m, find a rational transfer function G# ∈ H ∞ (K, E) such that the McMillan degree of G# is at most m and G − G# H ∞ (B(K,E)) is small. Clearly, this is the problem of rational approximation of bounded analytic operator functions by rational operator functions of McMillan degree at most m. In particular, we can consider the problem of finding inf G − G# H ∞ (B(K,E)) : deg G# ≤ m . (4.1) We can mention here the results of Pekarskii [2] which estimate the rate of decay of the numbers in (4.1) in terms of properties of G (in the scalar case). We also mention here the results of Glover [2], which estimate the deviations in (4.1) in terms of the singular values of the block Hankel ˆ + k)}j,k≥0 . operator ΓG = {G(j Another way to measure the closeness of two transfer functions is to consider the “Hankel norm” on the space of stable transfer functions: def

GH = ΓG ˆ = inf G − QL∞ (B(K,E)) : Q ∈ L∞ (B(K, E)), Q(j) = 0 for j < 0 . Recall (see §2) that the Hankel norm is the supremum of the energy of the future output {yn }n≥0 over the past inputs {un }n 0. A feedback controller K with a proper rational transfer function K stabilizes internally all systems in Gε if and only if it internally stabilizes the system G and K(I + GK)−1 L∞ < ε−1 . To prove Theorem 5.1, we need results on coprime factorization of rational matrix functions, which are given in §6. We prove Theorem 5.1 in §7. In §8 we parametrize all feedback controllers with proper rational transfer functions that internally stabilize the system. Finally, in §9 we reduce the problem of robust stabilization to the Nehari problem.

476


u

+- m u − Ky -

+ v ? + - m

G + ∆G

y

-

− 6 Ky

y

K

FIGURE 3. Robust stabilization

6. Coprime Factorization In this section we obtain results on coprime factorization of rational matrix functions which will be used in §7 and §8. Let G be a matrix rational function. We say that a factorization G = N M −1 is a right coprime factorization over H ∞ (C+ ) if both M M is a and N are rational matrix functions bounded in C+ (certainly, N square matrix function) and the matrix function is left invertM ible in the space of bounded rational matrix functions, i.e., there exists a rational matrix function R in H ∞ (C+ ) such that R

N M

= I.

(A matrix function is said to belong to H ∞ (C+ ) if all its entries belong to H ∞ (C+ ).) Similarly, a factorization G = M −1 # N # is called a left coprime factorization over H ∞ (C+ ) if both M # and matrix func N # are rational tions in H ∞ (C+ ) and the matrix function N # M # is right invertible in the space of bounded rational matrix functions in C+ , i.e., there exists a rational matrix function R in H ∞ (C+ ) such that

N#

M#

R = I.

Finally, we say that G = N M −1 = M −1 # N # is a doubly coprime factor∞ + ization of G over H (C ) if there exist rational matrix functions X, Y ,

6. Coprime Factorization

X # and Y # , in H ∞ (C+ ) such that X # −Y # M −N # M # N

Y X

477

= I.

(6.1)

Similarly, one can introduce the notions of left coprime, right coprime, and doubly coprime factorizations over the space H ∞ of bounded analytic functions in D. Theorem 6.1. Each rational matrix function has doubly coprime factorization over H ∞ (C+ ). Proof. Let G be a rational matrix function. Without loss of generality we may assume that G is proper, i.e., has no pole at infinity (otherwise we can apply a suitable conformal map of C+ onto itself). By Corollary 3.2, there exists a minimal realization {A, B, C, D} of G (see the Remark at the end of §3), i.e., G(z) = C(z − A)−1 B + D, where A, B, C, and D are finite matrices. Consider the corresponding linear system

x (t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t). We are going to use the following fact, which can be found in Gohberg, Lancaster, and Rodman [1], Theorem 6.5.1: if {A, B, C, D} is a controllable realization, then there exists a matrix Ξ such that all eigenvalues of the def matrix A[Ξ] = A + BΞ have negative real parts (Ξ has size such that BΞ is a square matrix). In fact, by choosing a suitable matrix Ξ one can make the eigenvalues of A + BΞ arbitrary. Put C[Ξ] = C + DΞ and define the function v by v(t) = u(t) − Ξx(t), t ∈ R. We have   x (t) = A[Ξ] x(t) + Bv(t), u(t) = Ξx(t) + v(t),  y(t) = C[Ξ] x(t) + Dv(t). Denote by M the transfer function from v to u and by N the transfer function from v to y, i.e., M (z) = Ξ(zI − A[Ξ] )−1 B + I

and N (z) = C[Ξ] (zI − A[Ξ] )−1 B + D.

Clearly, both M and N are in H ∞ (C+ ). Since G is the transfer function from u to y, it follows that GM = N , i.e., G = N M −1 . Similarly, since the realization {A, B, C, D} is observable, by Theorem 6.5.1 of Gohberg, Lancaster, and Rodman [1], there exists a matrix Ω such def

that the matrix A[Ω] = A + ΩC has all eigenvalues in the open left halfplane. Put B [Ω] = B + ΩD and consider the rational matrix functions M # (z) = C(zI − A[Ω] )−1 Ω + I

and N # (z) = C(zI − A[Ω] )−1 B [Ω] + D.

If we pass to the transposed matrix function Gt , we deduce from above that G = M −1 # N #. We now define X, Y , X # , and Y # by X(z) = −C[Ξ] (zI − A[Ξ] )−1 Ω + I,

Y (z) = −Ξ(zI − A[Ξ] )−1 Ω,

478


−1 [Ω] X # (z) = −Ξ zI − A[Ω] B + I,

−1 and Y # (z) = −Ξ zI − A[Ω] Ω.

Let us prove (6.1). The equality N # M = M # N is an obvious consequence of G = N M −1 = M −1 # N # . Let us show that X # M − Y # N = I. We have (X # M − Y # N )(z) = (−Ξ(zI − A[Ω] )−1 B [Ω] + I)(Ξ(zI − A[Ξ] )−1 B + I) + (Ξ(zI − A[Ω] )−1 Ω)(C[Ξ] (zI − A[Ξ] )−1 B + D) = Ξ(zI − A[Ω] )−1 (ΩC[Ξ] − B [Ω] Ξ)(zI − A[Ξ] )−1 B − Ξ(zI − A[Ω] )−1 B [Ω] + Ξ(zI − A[Ξ] )−1 B + I + (Ξ(zI − A[Ω] )−1 Ω)D = Ξ(zI − A[Ω] )−1 (ΩC − BΞ)(zI − A[Ξ] )−1 B − Ξ(zI − A[Ω] )−1 B + Ξ(zI − A[Ξ] )−1 B + I = I Again, passing to the transposed functions, we get M # X − N # Y = I. It remains to show that X # Y = Y # X. We have (X # Y − Y # X)(z)

=

(Ξ(zI − A[Ω] )−1 B [Ω] − I)Ξ(zI − A[Ξ] )−1 Ω

− Ξ(zI − A[Ω] )−1 Ω(C[Ξ] (zI − A[Ξ] )−1 Ω − I) =

Ξ(zI − A[Ω] )−1 (BΞ − ΩC)(zI − A[Ξ] )−1 Ω

− Ξ(zI − A[Ξ] )−1 Ω + Ξ(zI − A[Ω] )−1 Ω = O. We conclude this section with the following elementary result, which will be used in the next section. Theorem 6.2. Let G be a rational matrix function that has no poles on T. Let G = N M −1 = M −1 # N# be right coprime and left coprime factorizations of G over H ∞ . Then rank HG = wind det M = wind det M # . Proof. We prove the equality rank HG = wind det M # . To prove the other equality, it suffices to pass to transposition. Let us show that rank HG = rank HM −1 . #

Indeed, HG = HM −1 N # is the product of HM # and multiplication by N # , # and so rank HG ≤ rank HM −1 . #

7. Proof of Theorem 5.1

479

To prove the opposite inequality, we consider matrix functions X and Y in H ∞ such that −N # Y + M # X = I. Then −GY + X = M −1 # , and so rank HM −1 = rank HGY ≤ rank HG . #

Let B be a Blaschke–Potapov product of degree rank HM −1 such that #

∞ (see Theorem 2.5.2). Clearly, det M −1 M −1 # B ∈H # has no zeros in clos D. It is easy to see from the construction of B given in the proof of Lemma ∞ 2.5.1 that M −1 # B is invertible in H . Thus

0 = wind det M −1 # B = wind det B − wind det M # . The result follows from the fact that wind det B = deg B, which is an immediate consequence of the definition of Blaschke–Potapov products (see §2.5).

7. Proof of Theorem 5.1 For convenience, in this section we assume that we deal with discrete time systems. In other words, a rational transfer function is stable if it belongs to the space H ∞ (i.e., has no poles in clos D). We say that a rational matrix function K internally stabilizes G if the four transfer functions (I+GK)−1 , (I + GK)−1 G, K(I + GK)−1 , and (I + KG)−1 are stable. Clearly, with the help of a conformal map from C+ onto D, we can reduce Theorem 5.1 to the following result. Theorem 7.1. Suppose that G is a rational matrix function and ε > 0. Let G ε be the set of rational matrix functions of the form G + ∆G such that ∆GL∞ (T) ≤ ε, and G and G + ∆G have the same number of poles in clos D (in the sense of McMillan degree). A rational matrix function K internally stabilizes all transfer functions in G ε if and only if it internally stabilizes G and K(I + GK)−1 L∞ < ε−1 . Let ρ be a number such that ρ > 1 and the function I + GK has no zeros and no poles in {ζ ∈ C : 1 < |ζ| ≤ ρ} while the functions G and K have no poles in {ζ ∈ C : 1 < |ζ| ≤ ρ}. Lemma 7.2. K stabilizes G internally if and only if wind det (I + GK)(z/ρ) = − deg P− G(z/ρ) + deg P− K(z/ρ) . (7.1) Clearly, it suffices to prove (7.1) for some ρ > 1 sufficiently close to 1. Recall that deg P− G(z/ρ) = rank HG(z/ρ) and deg P− K(z/ρ) = rank HK(z/ρ) . Proof of Lemma 7.2. Consider a left coprime factorization −1 G = M −1 of # N # of G and a right coprime factorization K = U V

480


K. By changing ρ if necessary, we can assume that the functions M # , N # , U , and V have no poles in the disk {ζ ∈ C : |ζ| ≤ ρ}. We have −1 I + GK = M −1 . # (M # V + N # U )V

Hence,

wind det (I + GK)(z/ρ)

= wind det (M # V + N # U )(z/ρ) − wind det(M # (z/ρ)) − wind det(V (z/ρ)).

By the argument principle, wind det(M # (z/ρ)) is the number of zeros of det(M # (z/ρ)) in {ζ ∈ C : |ζ| ≤ ρ} (counted with multiplicities). By Theorem 6.2, wind det(M # (z/ρ)) = rank HG(z/ρ) and wind det(V (z/ρ)) = rank HK(z/ρ) . Hence, it remains to show that the system is internally stable if and only if wind det (M # V + N # U )(z/ρ) = 0. In other words, this is equiv−1 is stable. (Obviously, alent to that fact that (M # V + N # U )(z/ρ) (M # V + N # U ) is nonsingular if and only if I + GK is.) Note also that −1 (M # V + N # U )−1 is stable if and only if (M # V + N # U )(z/ρ) is stable for some ρ > 1. It is easy to see that the following equality holds K(I + GK)−1 G = I − (I + KG)−1 . Hence, (I + KG)−1 is stable if and only if K(I + GK)−1 G is. We have (I + GK)−1 = V (M # V + N # U )−1 M # ,

(7.2)

(I + GK)−1 G = V (M # V + N # U )−1 N # ,

(7.3)

K(I + GK)−1 = U (M # V + N # U )−1 M # ,

(7.4)

and K(I + GK)−1 G = U (M # V + N # U )−1 N # .

(7.5)

−1

Clearly, if (M # V + N # U ) is stable, then all functions in (7.2)–(7.5) are stable. Conversely, suppose that the functions in (7.2)–(7.5) are stable. Let X, Y , Φ, and Ψ are stable rational matrix functions such that −N # Y + M # X = I

and ΦU + ΨV = I.

We have Ψ(I + GK)−1 + ΦK(I + GK)−1 = (M # V + N # U )−1 M # ∈ H ∞ and Ψ(I + GK)−1 G + ΦK(I + GK)−1 G = (M # V + N # U )−1 N # ∈ H ∞ .


481

Hence, − (M # V + N # U )−1 N # Y + (M # V + N # U )−1 M # X = (M # V + N # U )−1 ∈ H ∞ , which completes the proof. Proof of Theorem 7.1. Let G1 ∈ G ε and G1 = G + ∆G. We have I + G1 K

= I + GK + (∆G)K = I + (∆G)K(I + GK)−1 (I + GK).

(7.6)

Suppose that K internally stabilizes G, ρ > 1, and ρ is sufficiently close to 1. We have wind det (I + G1 K)(z/ρ) = wind det I + (∆G)K(I + GK)−1 (z/ρ) + wind det (I + GK)(z/ρ) . (7.7) By Lemma 7.2, wind det (I + GK)(z/ρ) = − deg P− G(z/ρ) + deg P− K(z/ρ) . The internal stability of the perturbed system with plant G1 and controller K is equivalent to wind det (I + G1 K)(z/ρ) = − deg P− G1 (z/ρ) + deg P− K(z/ρ) . Since G and G1 have the same number of unstable poles, the internal stability of the perturbed system is equivalent to the equality wind det (I + GK)(z/ρ) = wind det (I + G1 K)(z/ρ) . By (7.7), this is equivalent to wind det I + (∆G)K(I + GK)−1 (z/ρ) = 0. −1

(7.8)

−1

GK) L∞ < ε and ∆GL∞ ≤ ε. Clearly, Suppose that K(I +−1 (ζ) is invertible for I + t(∆G)K(I + GK) all ζ ∈ T and all t ∈ [0, 1] on and the function det I + t(∆G)K(I + GK)−1 : T → C \ {0} depends −1 t continuously. It follows that wind det I +t(∆G)K(I +GK) does not depend on t. Thus wind det I + (∆G)K(I + GK)−1 = 0. This implies (7.8) for ρ sufficiently close to 1. Conversely, suppose that K(I + GK)−1 L∞ ≥ ε−1 . Let us find a perdef

turbation ∆G such that G1 = G + ∆G has the same number poles in clos D as G and ∆GL∞ ≤ ε, but the system with plant G1 and controller K is not internally stable. Let ζ0 be a point on T such that def σ = K(I + GK)−1 (ζ0 ) ≥ ε−1 . Let f and g be unit vectors such that K(I + GK)−1 (ζ0 )f = σg. Consider the following constant matrix function ∆G: ∆G(ζ)x = −σ −1 (x, g)f. Clearly, ∆G = σ −1 ≤ ε. It is easy to see that I + (∆G)K(I + GK)−1 (ζ0 ) f = O,

482


and so I + (∆G)K(I + GK)−1 is not invertible at ζ0 . It now follows from (7.6) that the transfer function I + G1 K is unstable.

8. Parametrization of Stabilizing Controllers Now we are ready to parametrize all stabilizing controllers whose transfer functions are proper rational matrix functions (i.e., rational matrix functions that have no pole at infinity). Suppose that the transfer function G of the plant G is a rational matrix function that is strictly proper (i.e., is zero at infinity). By Lemma 6.1, G admits a doubly coprime factorization, i.e., G = N M −1 = M −1 # N #,

N , M , N # , M # ∈ H ∞ (C+ ),

and (6.1) holds for rational matrix functions X, Y , X # , and Y # in H ∞ (C+ ). We are interested in feedback controllers K with proper rational transfer function K that internally stabilizes the system. In this case we say that the transfer function K internally stabilizes the system. Theorem 8.1. A proper rational matrix function K internally stabilizes the system if and only if K = (M Q − Y )(X − N Q)−1 = (X # − QN # )−1 (QM # − Y # ), (8.1) where Q is a matrix function in H ∞ (C+ ). Clearly, the size of the matrix function Q is such that (8.1) makes sense. Lemma 8.2. Suppose that K is a proper rational matrix function and K = U V −1 = V −1 # U# are left and right coprime factorizations of K. The following are equivalent: (i) K stabilizes G; M −U (ii) the matrix function is invertible in H ∞ (C+ ); N V V# U# (iii) the matrix function is invertible in H ∞ (C+ ). −N # M # To prove Lemma 8.2, we need one more lemma. Clearly, our system is equivalent to the system in Fig. 4. We need the following fact. Lemma 8.3. The system in Fig. 4 isinternally stable if and only if u ξ the transfer function from to is stable. v η u ξ Proof. Suppose that the transfer function from to is v η stable. We have y = N ξ + v and w = U η + u, which implies the internal stability of the system in Fig. 3.

8. Parametrization of Stabilizing Controllers

u

+- mw -

M −1

ξ-

N

+ v +- ? m

483

y

-

− 6 y

U

η

V −1

FIGURE 4.

Conversely, suppose that the the transfer function from

u v

to

w y

is stable. Since by (6.1), X # M − Y # N = I, we have ξ = X# M ξ − Y# N ξ. It is clear from Fig. 4 that M ξ = w and N ξ = y − v, and so ξ = X# u − Y# y + Y# v. u Hence, the transfer function from to ξ is stable. The proof of the v u fact that the transfer function from to η is stable is the same. v Proof of Lemma 8.2. Clearly, it suffices to prove the equivalence of (i) and (ii). The equivalence of (i) and (iii) would follow by passing to transposition. M −U Obviously, is invertible in H ∞ (C+ ) if and only if N V M U is invertible in H ∞ (C+ ). Let us first show that the matrix −N V M U function is nonsingular, i.e., its inverse exists as a rational −N V matrix function. We have M U I K M O = . −N V −G I O V

484


M O

O V

is nonsingular. On the other hand, since G is I K strictly proper, the determinant of at ∞ is equal to 1, which −G I I K implies that is also nonsingular. −G I It is easy to see from Fig. 4 that M U ξ u = . −N V η v M O Since is nonsingular, we can write O V −1 ξ M U u = . η −N V v Clearly,

The result now follows from Lemma 8.3. Proof of Theorem 8.1. First of all, it is easy to verify that (M Q − Y )(X − N Q)−1 = (X # − QN # )−1 (QM # − Y # ). Suppose that K is given by (8.1) for some stable matrix function Q. Let us show that K stabilizes the system. We have −1 (I + GK)−1 = I + N M −1 (M Q − Y )(X − N Q)−1 −1 = X − N Q + N M −1 (M Q − Y ))(X − N Q)−1 =

(X − N Q)(X − N M −1 Y )−1

=

−1 (X − N Q)(X − M −1 . # N #Y )

It follows from (6.1) that N # Y = M # X − I. Hence, (I + GK)−1 = (X − N Q)M # ∈ H ∞ (C+ ). It follows that K(I + GK)−1

=

(M Q − Y )(X − N Q)−1 (X − N Q)M #

=

(M Q − Y )M # ∈ H ∞ (C+ )

(8.2)

and ∞ + (I + GK)−1 G = (X − N Q)M # M −1 # N # = (X − N Q)N # ∈ H (C ).

Finally,

−1 (I + KG)−1 = I + (X # − QN # )−1 (QM # − Y # )M −1 # N# −1 = X # −QN # +(QM # −Y # )M −1 (X # −QN # ) # N# −1 (X # − QN # ) = (X # − Y # M −1 # N #)

= (X # − Y # N M −1 )−1 (X # − QN # ).

9. Solution. The Model Matching Problem

485

It follows from (6.1) that Y # N = X # M − I. Hence, (I + KG)−1 = M (X # − QN # ) ∈ H ∞ (C+ ). Thus K makes the system internally stable. Let us establish the converse. Suppose that K stabilizes internally the system. Let K = U V −1 be a right coprime factorization of K. Put R = N# U + M# V . It follows from (6.1) that M −U I −(X # U + Y # V ) X # −Y # . (8.3) = −N # M # O R N V By (6.1) and Lemma 8.2, both matrix functions on the left of (8.3) are invertible in H ∞ (C+ ). It follows that R−1 ∈ H ∞ (C+ ). Put Q = (X # U + Y # V )R−1 ∈ H ∞ (C+ ). Then we have from (8.3) X # −Y # M −U I −QR = . −N # M # N V O R M Y Multiplying this equality on the left by , we find from (6.1) N X M −U M Y I −QR = . N V N X O R Thus

−U V

=

M N

Y X

−QR R

=

(Y − M Q)R (X − N Q)R

,

and so K = U V −1 = (M Q − Y )(X − N Q)−1 , which completes the proof.

9. Solution of the Robust Stabilization Problem. The Model Matching Problem In this section we reduce the robust stabilization problem to the so-called model matching problem. It turns out that in the case G has no poles on the imaginary axis, the model matching problem reduces to the Nehari problem. Consider the general model matching problem given in Fig. 5. Here T1 , T2 , and T3 are given stable plants. The problem is to design a stable plant Q that minimizes the energy of the output z in the worst possible case under the assumption that the energy of the input w is at most 1. In other words, the problem is to design a stable plant Q such that the cascade system T3 QT2 models the performance of T1 as precisely as possible.

486


-

T1 + ? z l − 6

w -

-

T2

-

Q

-

T3

FIGURE 5. Model matching problem

Consider the transfer functions T 1 , T 2 , T 3 , and Q of T1 , T2 , T3 , and Q. Clearly, the model matching problem is to find a stable transfer function Q such that the norm T 1 − T 3 QT 2 H ∞ (C+ ) is small possible. Let T 3 = ΥF and T t2 = ΘG be inner–outer factorizations. Consider now the special case when both Υ and Θ are square matrix functions and the operators of multiplication by F and G map the spaces of bounded analytic vector functions in C+ onto the spaces of bounded analytic vector functions in C+ of the corresponding dimensions. Then we have T 1 − T 3 QT 2 H ∞ (C+ ) = Υ∗ T 1 Θ − F QGt L∞ . It is easy to see that under our assumptions on F and G the set of matrix functions of the form F QGt where Q is an arbitrary matrix function in H ∞ (C+ ) coincides with the space of bounded analytic matrix functions in H ∞ (C+ ) that have the same size as Υ∗ T 1 Θ. Thus by Theorem 2.2.2, inf

Q∈H ∞ (C+ )

T 1 − T 3 QT 2 H ∞ (C+ ) = HΥ∗ T 1 Θ

and in this special case the model matching problem reduces to the Nehari problem. It can be shown that in more general cases the model matching problem reduces to the four block problem considered in the Concluding Remarks to Chapter 2. Let us return now to the robust stabilization problem. By Theorem 7.1, we have to look for rational transfer functions K that internally stabilize the feedback system with plant G such that K(I +GK)−1 L∞ < ε−1 . By (8.2) and Theorem 8.1, this problem reduces to the problem of minimizing the norm (M Q − Y )M # H ∞ (C+ ) over rational matrix functions Q in H ∞ (C+ ). This corresponds to the model matching problem with T1 = M Y M# , T2 = M# , and T3 = M . If G has no poles on the imaginary axis, it is easy to see that T2 and T3

Concluding Remarks

487

satisfy the above assumption, and so the problem of robust stabilization reduces to the Nehari problem. Moreover, in this case the inner factors Υ and Θ of M and M t# are finite Blaschke–Potapov products, and so the problem of robust stabilization reduces to the Nehari problem for the rational matrix function Υ∗ M Y M # Θ. Namely, it reduces to the problem of finding stable rational matrix functions R such that Υ∗ M Y M # Θ − RL∞ < ε−1 .

(9.1)

Clearly, this problem has a solution if and only if HΥ∗ M Y M # Θ < ε−1 . Recall that in §5.4 and §5.5 for any ρ ∈ (HΥ∗ M Y M # Θ , ε−1 ) a parametrization formula has been given for all bounded analytic matrix functions R satisfying Υ∗ M Y M # Θ−RL∞ ≤ ρ. It will be shown in Chapter 14 (Corollary 14.12.4) that there exists a rational matrix function R that minimizes the norm Υ∗ Y M # Θ − RL∞ . Moreover, in Chapter 14 we give an algorithm how to find such an optimal matrix function R.

Concluding Remarks “H ∞ control theory” was initiated by Zames [1–3]. Let us also mention the papers Helton [1] and Tannenbaum [1]. We recommend Francis [1], Helton [3], Doyle [1], Dahleh and Diaz-Bobillo [1], Chui and Chen [1], Doyle, Francis, and Tannenbaum [1], and Desoer and Vidiasagar [1] for an introduction in this theory. We refer the reader to Fuhrmann [4] for an introduction in linear systems and their connections with Hankel operators; see also Balakrishnan [1]. Backward shift realizations of symbols of bounded Hankel operators were constructed in Fuhrmann [2] and Helton [1]; see Fuhrmann [4]. Theorem 2.5 was obtained in Young [1]; its original proof is simplified in §2, the simplification is inspired by Megretskii, Peller, and Treil [2]. Theorems 2.8 and 2.9 are due to Young [1]. For realizations of rational functions see Bart, Gohberg, and Kaashoek [1]. The construction of balanced realizations with continuous time was reduced to the construction of balanced realizations with discrete time in Ober and Montgomery-Smith [1]; see also Salamon [1]. We refer the reader to Glover [1,2], Young [3], and Diaz-Bobillo [1] for more information on the model reduction problem. The problem of robust stabilization is treated in Francis [1], Diaz-Bobillo [1]; see also Young [4]. It is a special case of the so-called standard problem; see Francis [1]. The latter also includes as special cases the tracking problem and the model matching problem; see Francis [1]; see also Foias and Tannenbaum [1] for the weighted sensitivity problem. Coprime factorizations are discussed in detail in Francis [1]. The idea of coprime factorizations over H ∞ is due to Vidyasagar [1]; see also Vidyasagar [2].

488


The proof of Theorem 5.1 given in §7 is taken from Ball, Gohberg, and Rodman [1]. Parametrization formula (8.1) was found in Youla, Jabr, and Bonjiorno [1]; see Francis [1]. The reduction of the standard problem to the model matching problem can be found in Francis [1]. Recall that the robust stabilization problem is a special case of the standard problem. More information about the model matching problem can be found in Doyle, Francis, and Tannenbaum [1].

12 The Inverse Spectral Problem for Self-Adjoint Hankel Operators

In §8.5 we have considered a geometric problem in the theory of stationary Gaussian processes and we have reduced this problem to the problem of the description of the bounded linear operators on Hilbert space that are unitarily equivalent to moduli of Hankel operators. In this chapter we are going to solve the latter problem, which in turn will lead to a solution of the above geometric problem in prediction theory. Moreover, we consider in this chapter a considerably more delicate problem of describing the bounded self-adjoint operators on Hilbert space that are unitarily equivalent to Hankel operators on 2 . In other words, we are going to characterize those self-adjoint operators Γ on a Hilbert space H for which there exists an orthonormal basis {ej }j≥0 such that Γ has Hankel matrix with respect to {ej }j≥0 , i.e., (Γej , ek ) = αj+k , j, k ≥ 0, for some sequence {aj }j≥0 of complex numbers. Clearly, Γ is self-adjoint if and only if αj ∈ R for any j ∈ Z+ . We have already mentioned in §8.5 that a Hankel operator is always noninvertible and its kernel is either trivial or infinite-dimensional. Thus if Γ is unitarily equivalent to a Hankel operator, then Γ is noninvertible and either Ker Γ = {O} or dim Ker Γ = ∞. It turns out, however, that these two necessary conditions are not sufficient for a self-adjoint operator Γ to be unitarily equivalent to a Hankel operator. We have to add another condition. Roughly speaking it says that the spectral measure of Γ is “nearly symmetric” with respect to the origin. We state the third condition in terms of the spectral multiplicity function of Γ. Recall (see Appendix 1.4) that each self-adjoint operator Γ on a separable Hilbert space is unitarily equivalent to multiplication by the coordinate

490

Chapter 12. The Inverse Problem for Self-Adjoint Hankel Operators

function on the von Neumann integral

⊕H(t)dµ(t), where µ is a scalar spectral measure of Γ. The spectral multiplicity function ν of Γ is defined µ-a.e. by ν(t) = dim H(t). Recall also that two self-adjoint operators are unitarily equivalent if and only if they have mutually absolutely continuous scalar spectral measures and their spectral multiplicity functions coincide almost everywhere with respect to their scalar spectral measures (see Appendix 1.4). Suppose now that Γ is a self-adjoint operator on Hilbert space with a scalar spectral measure µ and spectral multiplicity function ν. The third necessary condition is that |ν(t) − ν(−t)| is at most 2 on the absolutely continuous spectrum of Γ and at most 1 on the singular spectrum of Γ. To be more precise, we have to explain what we mean by ν(t) − ν(−t). We define the Borel measure µ# by µ# (∆) = µ(∆) + µ(−∆) for Borel subsets ∆ of R. We can assume that the multiplicity function ν is defined µ# -almost everywhere and if δ is a Borel set such that µ(δ) = 0, then ν(t) = 0 for µ# -almost all t ∈ δ. Indeed, we can consider the measure λ defined by dλ = νdµ. Clearly, λ is absolutely continuous with respect to µ# . We can consider now the Radon–Nikodym density of λ with respect to µ# and assume that ν is equal to this density. Clearly, we have changed the function ν only on the set of zero µ-measure. Now we are in a position to state the main result of the chapter. As usual µa and µs are the absolutely continuous and the singular components of µ. Theorem 0.1. Let Γ be a bounded self-adjoint operator on Hilbert space with a scalar spectral measure µ and spectral multiplicity function ν. Then Γ is unitarily equivalent to a Hankel operator if and only if the following conditions are satisfied: (C1) either Ker Γ = {O} or dim Ker Γ = ∞; (C2) Γ is noninvertible; (C3) |ν(t) − ν(−t)| ≤ 2, µa -a.e. and |ν(t) − ν(−t)| ≤ 1, µs -a.e. Note that (C3) means in particular that if one of the numbers ν(t) and ν(−t) is infinite, then the other one must also be infinite almost everywhere. Clearly, (C1) is equivalent to the condition that (C1 ) ν(0) = 0 or ν(0) = ∞, while (C2) is equivalent to the condition (C2 ) 0 ∈ supp ν. It follows easily from Theorem 0.1 that if K is a self-adjoint operator on Hilbert space such that K ≥ O, K is noninvertible, and Ker K = {O} or


491

dim Ker K = ∞, then K is unitarily equivalent to the modulus of a Hankel operator. Let us now compare the problem to describe the operators unitarily equivalent to the modulus of a Hankel operator with the problem to describe the operators unitarily equivalent to a self-adjoint Hankel operator. If Γ is a self-adjoint operator, νΓ is the spectral multiplicity function of Γ and ν|Γ| is the spectral multiplicity function of |Γ|. Then ν|Γ| (t) = νΓ (t)+νΓ (−t), t > 0, µ# -a.e. Thus if we know that Γ satisfies (C1) and (C2), the problem is how we can distribute ν|Γ| (t) between νΓ (t) and νΓ (−t) in order that Γ be unitarily equivalent to a Hankel operator. Theorem 0.1 gives a solution to this problem. The necessity of conditions (C1)–(C3) will be proved in §1. As we have already observed, we have to prove only the necessity of (C3). It is easy to see that (C3) implies the following inequality for the multiplicities of eigenvalues of a self-adjoint Hankel operator Γ: if λ ∈ R, then | dim Ker(Γ − λI) − dim Ker(Γ + λI)| ≤ 1

(0.1)

(if one of the dimensions is infinite, then the other one must also be infinite). In §2 we give a simple proof of (0.1) for an arbitrary Hankel operator Γ (not necessarily self-adjoint) and an arbitrary λ ∈ C. In §§3–6 we use linear systems with continuous time (see Chapter 11) to construct a Hankel operator with prescribed spectral properties. We prove that if Γ is a self-adjoint operator that satisfies (C1), (C2), and the condition |ν(t) − ν(−t)| ≤ 1,

µ-a.e.,

then there exists a balanced linear system (see Chapter 11) with continuous time such that the Hankel operator associated with it is unitarily equivalent to Γ. We obtain explicit formulas for the operators A, B, C that determine the linear system. The above results do not seem to be satisfactory. However, they allow us to solve completely the problem of characterizing the bounded linear operators that are unitarily equivalent to the moduli of Hankel operators. We show in §7 that if Γ is a positive self-adjoint operator with multiple spectrum, then there exists no proper balanced system with continuous time for which the corresponding Hankel operator is unitarily equivalent to Γ. This shows that in contrast with the problem of describing all possible moduli of Hankel operators, the proper balanced linear systems with continuous time cannot produce enough Hankel operators to solve the problem completely. Recall that we consider here only proper linear systems, i.e., those linear systems that involve bounded operators A, B, C. It is also possible to consider (generalized) linear systems for which A is the generator of a contractive semigroup, B is an operator from C to a Hilbert space that is larger than K, and C is a linear functional defined on a dense subset of K (see

492


Chapter 11). As we have mentioned in Chapter 11, for any bounded Hankel operator Γ on L2 (R+ ) there exists a generalized balanced linear system such that the Hankel operator associated with it coincides with Γ (see Ober and Montgomery-Smith [1] and Salamon [1]). However, it is not clear how to evaluate the spectra of Hankel operators associated with systems that involve unbounded operators A, B, and C. Note that positive Hankel operators with multiple spectra do exist. A classical example of such an operator is the Carleman operator K defined on L2 (R+ ) by

∞ f (t) (Kf )(s) = dt s +t 0 (see §9.2). Other interesting examples of such operators are given by Howland [2–4]. In §8 we show how one can simplify the construction given in §4 to solve the problem to describe the operators unitarily equivalent to the moduli of Hankel operators. We also solve in §8 the geometrical problem on past and future that has been posed in §8.5. In §§9–14 we prove that conditions (C1)–(C3) are sufficient for Γ to be unitarily equivalent to a Hankel operator. We use methods of linear systems with discrete time (see Chapter 11). Namely, we prove that if Γ satisfies (C1)–(C3), then there exists a balanced linear system with discrete time such that the corresponding Hankel operator is unitarily equivalent to Γ. Finally, in §15 we obtain the Aronszain–Donoghue theorem in perturbation theory as a consequence of our construction of linear systems with discrete time and condition (C3) for Hankel operators. We can also consider the problem of spectral characterization of the selfadjoint operators that are unitarily equivalent to block Hankel operators on 2 (Cn ), i.e., operators given by block Hankel matrices {Ωj+k }j,k≥0 , where Ωj ∈ Mn,n . Theorem 0.2. Let Γ be a bounded self-adjoint operator on Hilbert space with a scalar spectral measure µ and spectral multiplicity function ν. Then Γ is unitarily equivalent to a block Hankel operator on 2 (Cn ) if and only if the following conditions are satisfied: (C1) either Ker Γ = {O} or dim Ker Γ = ∞; (C2) Γ is noninvertible; (C3n ) |ν(t) − ν(−t)| ≤ 2n, µa -a.e. and |ν(t) − ν(−t)| ≤ n, µs -a.e. The sufficiency of conditions (C1), (C2), and (C3n ) follows easily from Theorem 0.1 and the fact that if Γ satisfies (C1), (C2), and (C3n ), then it can be represented as an orthogonal sum of n operators each of which satisfies (C1)–(C3). Necessity can be proved by the same method as in the case n = 1 (see §1). Finally, we can consider the case of block Hankel operators on 2 (K), where K is a separable infinite-dimensional Hilbert space.

1. Necessary Conditions

493

Theorem 0.3. Let Γ be a bounded self-adjoint operator on Hilbert space with a scalar spectral measure µ and spectral multiplicity function ν and let K be a separable infinite-dimensional Hilbert space. Then Γ is unitarily equivalent to a block Hankel operator on 2 (K) if and only if the following conditions are satisfied: (C1) either Ker Γ = {O} or dim Ker Γ = ∞; (C2) Γ is noninvertible. In view of the above remark we restrict ourselves in this chapter to the proof of Theorem 0.1.

1. Necessary Conditions The main purpose of this section is to prove that conditions (C1)–(C3) are necessary for a self-adjoint operator Γ to be unitarily equivalent to a Hankel operator on 2 . We have already observed (see §8.5) that (C1) and (C2) are necessary. Thus we prove in this section the following fact. Theorem 1.1. Let Γ be a bounded self-adjoint Hankel operator on 2 with a scalar spectral measure µ and spectral multiplicity function ν. Then |ν(t) − ν(−t)| ≤ 2,

µa -a.e.

|ν(t) − ν(−t)| ≤ 1,

µs -a.e.

and To prove Theorem 1.1, we need the description of the set of operators that intertwine two given self-adjoint operators. Let A1 and A2 be bounded selfadjoint operators on Hilbert spaces and let Q be a bounded linear operator that intertwines A1 and A2 , i.e., QA1 = A2 Q.

(1.1)

We can realize A1 and A2 as multiplications by the independent variables on

⊕H1 (t)dµ1 (t) and ⊕H2 (t)dµ2 (t). Next, we can assume that H1 (t) and H2 (t) are defined (µ1 + µ2 )-almost everywhere and if µ1 (δ) = 0 (µ2 (δ) = 0) for a Borel set δ, then H1 (t) = {O} (H2 (t) = {O}) on δ, (µ1 + µ2 )-almost everywhere. Lemma 1.2. Let Q be a bounded linear operator that satisfies (1.1). Then there exists a (µ1 + µ2 )-measurable bounded operator function q, q(t) : H1 (t) → H2 (t), such that

(Qf )(t) = q(t)f (t),

f∈

⊕H1 (t)dµ1 (t).

494


Proof. Without loss of generality we may assume that there exists a Borel measure µ mutually absolutely continuous with µ1 + µ2 and such that µj is the restriction of µ to the set {t ∈ R : Hj (t) = {O}}, Put def

Hj =

j = 1, 2.

⊕Hj (t)dµj (t) =

and

⊕Hj (t)dµ(t),

j = 1, 2,

H=

⊕H(t)dµ(t),

where H(t) = H1 (t) ⊕ H2 (t). Then H1 and H2 are naturally imbedded in H. Let A be multiplication by the independent variable on H. Consider the operator Q# defined by Q# f = QP1 f, where P1 is the orthogonal projection from H onto H1 . Obviously, Range Q# ⊂ H2 , Q = Q# H1 , and Q# A = AQ# . Then there exists a bounded measurable operator function q# , q# (t) : H(t) → H(t), such that (Q# f )(t) = q# (t)f (t) (see Appendix 1.4). It is easy to see that q# (t)x = O for x ∈ H2 (t) and Range q# (t) ⊂ H2 (t). We can now define q(t) by q(t) = q# (t)H1 (t). Clearly, q(t) is an operator from H1 (t) to H2 (t) and (Qf )(t) = q(t)f (t). Before we proceed to the proof of Theorem 1.1 we adopt the following terminology. If f is a Borel function, ρ is a Borel measure, and ∆ is a Borel set, we say that f is supported on ∆ (ρ is supported on ∆), if f is zero almost everywhere outside ∆ (ρ is zero outside ∆). Proof of Theorem 1.1. Suppose that Γ is unitarily equivalent to a Hankel operator. Then H has an orthonormal basis {ej }j≥0 such that Γ has Hankel matrix in this basis, i.e., (Γej , ek ) = αj+k , j, k ≥ 0, where {αj }j≥0 is a sequence of real numbers (αj ∈ R, since Γ is self-adjoint). Consider the shift operator S on H defined by S xn en = xn en+1 . n≥0

Put α =

∞ j=0

n≥0

αj ej = Γe0 and α∗ =

∞

α ¯ j ej = Γ∗ e0 . It is easy to see that

j=0

the following commutation relations hold: S ∗ Γ = ΓS, SΓf − ΓS ∗ f = (f, e0 )Sα − (f, Sα∗ )e0 ,

(1.2) f ∈ H.

(1.3)

1. Necessary Conditions

495

Let us first prove the inequality |ν(t) − ν(−t)| ≤ 2, µ-a.e. Let f be a function in H such that f (t) ⊥ e0 (t) and f (t) ⊥ (Sα∗ )(t), µ-a.e. Consider the function (S ∗ − S)f . We have Γ(S ∗ − S)f = ΓS ∗ f − ΓSf = SΓf − S ∗ Γf = −(S ∗ − S)Γf (1.4) by (1.2) and (1.3), since f ⊥ e0 and f ⊥ Sα∗ . Let A1 be the restriction of Γ to ˜ = {f ∈ H : f (t) ⊥ e0 (t) and f (t) ⊥ (Sα∗ )(t), H

µ-a.e.}

˜ → H defined by and let A2 = −Γ. Consider the operator Q : H Qf = (S ∗ − S)f . By (1.4), QA1 = A2 Q . Thus by Lemma 1.2, there ˜ → H(−t), such exists a bounded weakly measurable function q, q(t) : H(t) that (Qf )(t) = q(t)f (−t), where ˜ = {x ∈ H(t) : x ⊥ e0 (t) and x ⊥ (Sα∗ )(t)}. H(t) Let us show that Ker q(t) = {O}, µ-a.e. Indeed, assume that there exists a Borel set ∆, µ(∆) > 0, such that Ker q(t) = {O} for any t ∈ ∆. Let Pt be ˜ the orthogonal projection of H(t) onto Ker q(t). Then the function t → Pt is weakly measurable. This follows from the fact that Pt = lim χε (q ∗ (t)q(t)), ε→0

where χε (s) = 0 for s > ε and χε (s) = 1 for 0 ≤ s ≤ ε. Therefore (see Appendix 1.4) the spaces Ker q(t) form a measurable family in a natural way, and we can consider the direct integral

⊕ Ker q(t) dµ(t) = {O}. It is easy to see that {O} =

⊕ Ker q(t) dµ(t) ⊂ Ker Q.

However, it is very easy to verify straightforwardly that Ker(S ∗ −S) = {O}. ˜ Thus q(t) is an injective map from H(t) to H(−t) which implies that ˜ ≥ ν(t) − 2, µ-a.e. ν(−t) = dim H(−t) ≥ dim H(t) Interchanging the roles of t and −t, we obtain |ν(t) − ν(−t)| ≤ 2. Let us now prove that |ν(t) − ν(−t)| ≤ 1 on the singular spectrum. Let ∆s be a Borel set such that µs is supported on ∆s and ∆s has zero Lebesgue measure. Lemma 1.3. Let f be a function supported on ∆s such that f (t) ⊥ e0 (t) and if e0 (t) = 0, then f (t) ⊥ (Sα∗ )(t), µ-a.e. Then ΓS ∗ f = SΓf.

496


Let us first complete the proof of Theorem 1.1 and then prove Lemma 1.3. Put

x ⊥ e0 (t), e0 (t) = O, ˇ H(t) = x ∈ H(t) : , t ∈ ∆s , x ⊥ (Sα∗ )(t), e0 (t) = O and ˇ ˇ = {f : f is supported on ∆s and f (t) ∈ H(t), µ-a.e.}. H def

def

ˇ A2 = −A and consider the operator As above we put A1 = A|H, ˇ Q : H → H defined by Qf = (S ∗ − S)f . By Lemma 1.3, QA1 = A2 Q and by Lemma 1.2, (Qf )(t) = q(t)f (−t), where q is a weakly measurable operator-valued function on ∆s such that ˇ → H(−t) and Ker q(t) = {O}, µ-a.e. As above, it follows that q(t) : H(t) ˇ ≥ ν(t) − 1, ν(−t) = dim H(−t) ≥ dim H(t) which implies that |ν(t) − ν(−t)| ≤ 1,

µs -a.e.

Proof of Lemma 1.3. Put ∆0 = {t ∈ ∆s : e0 (t) = 0} (recall that {ej }j≥0 is an orthonormal basis in which Γ has Hankel matrix). If f is supported on ∆s \∆0 , then f ⊥ Sα∗ and f ⊥ e0 , and it follows from (1.3) that ΓS ∗ f = SΓf . Thus we can assume without loss of generality that f is supported on ∆0 . Let k be a positive integer. Put ∆k = {t ∈ ∆s : 2−k ≤ e0 (t) ≤ 2k , f (t) ≤ 2k }. Clearly ∆0 = ∆k . So it is sufficient to consider the case when f is def

k≥1

supported on ∆k for some k. Let {Ij } be a cover of ∆k by disjoint open intervals. Put Λj = ∆k ∩ Ij (k is fixed), fj = χj f , gj = χj e0 , where χj is the characteristic function of Λj . It follows from (1.2) and (1.3) that Γ(S + S ∗ )f − (S + S ∗ )Γf = ce0

(1.5)

for some c ∈ C. Therefore it is sufficient to show that (Γ(S + S ∗ )f − (S + S ∗ )Γf, e0 ) = 0. We have (Γ(S + S ∗ )f − (S + S ∗ )Γf, e0 ) =

(Γ(S + S ∗ )fj − (S + S ∗ )Γfj , e0 )

j≥1

=

j≥1

1 (Γ(S + S ∗ )fj − (S + S ∗ )Γfj , gj ) (e0 , gj )

2. Eigenvalues of Hankel Operators

in view of (1.5). Clearly,

497

e0 (t)2 dµ(t) ≥ 2−2k µΛj .

(e0 , gj ) = Λj

Let λj ∈ Λj . Put g (j) = Γgj − λj gj , f (j) = Γfj − λj fj . Then g (j) ≤ |Ij | · gj ,

f (j) ≤ |Ij | · fj ,

where |Ij | is the length of Ij . We have (Γ(S + S ∗ )f − (S + S ∗ )Γf ), e0 ≤ 22k

1 |((S + S ∗ )fj , Γgj ) − (Γfj , (S + S ∗ )gj )| µΛj j≥1

= 22k

1 ((S + S ∗ )fj , g (j) ) − (f (j) , (S + S ∗ )gj ) µΛj j≥1

≤ const 22k

|Ij | fj · gj µΛj j≥1

= const 2

2k

|Ij | µΛj j≥1

≤ const 22k

1/2

fj (t) dµ(t) Λj

1/2 gj (t) dµ(t)

2

2

Λj

|Ij | 22k (µΛj )1/2 22k (µΛj )1/2 = const 26k |Ij |. µΛj j≥1

Since m(∆k ) = 0, we can make

j≥1

|Ij | as small as possible.

j≥1

2. Eigenvalues of Hankel Operators As we have noticed in the Introduction to the chapter, it follows from Theorem 1.1 that for any self-adjoint Hankel operator Γ and for any λ ∈ R | dim Ker(Γ − λI) − dim Ker(Γ + λI)| ≤ 1.

(2.1)

The main result of this section shows that (2.1) is also valid for an arbitrary bounded Hankel operator and an arbitrary λ ∈ C. The idea of the proof is the same as in Theorem 1.1, but to include the nonself-adjoint case, we have to replace the inner product by the natural bilinear form on 2 . We assume here that Γ is a bounded Hankel operator on the Hilbert space 2 , {en }n≥0 is the standard orthonormal basis in 2 . Then (Γej , ek ) = αj+k , j, k ≥ 0, where {αj }j≥0 is a sequence of complex numbers.

498


It is convenient to consider the following bilinear form on 2 : x, y = xj yj , where x =

xj ej and y =

j≥0

j≥0

yj ej .

j≥0

Clearly, any Hankel operator Γ is symmetric with respect to this form: Γx, y = x, Γy,

x, y ∈ 2 .

Let S be the shift operator on 2 : Sen = en+1 , n ≥ 0. Clearly, Sx, y = x, S ∗ y, x, y ∈ 2 . Theorem 2.1. Let Γ be a bounded Hankel operator. Then inequality (2.1) holds for any λ ∈ C. Proof. Given κ ∈ C, we put Eκ = Ker(Γ − κI). Let α = αj ej . We j≥0

have S ∗ Γ = ΓS, SΓx − ΓS ∗ x = x, e0 Sα − x, Sαe0 ,

(2.2) x ∈ 2 ,

(2.3)

(compare with (1.2) and (1.3)). Let λ ∈ C. To prove the theorem, we have to show that dim E−λ ≥ dim Eλ − 1.

(2.4)

We consider separately two different cases. aj ej in Eλ with a0 = 0. Let us Case 1. There exists a vector a = j≥0

show that if x ∈ Eλ and x, e0 = 0, then

SΓx = ΓS ∗ x,

(2.5)

which is equivalent in view of (2.2) to Γ(S + S ∗ )x = (S + S ∗ )Γx. It follows from (2.2) and (2.3) that Γ(S + S ∗ )x − (S + S ∗ )Γx = ce0 for some c ∈ C. Since a, e0 = 0 and Γ(S + S ∗ )x − (S + S ∗ )Γx, a = a0 c, it is sufficient to show that Γ(S + S ∗ )x − (S + S ∗ )Γx, a = 0. We have Γ(S + S ∗ )x − (S + S ∗ )Γx, a = (S + S ∗ )x, Γa − (S + S ∗ )Γx, a = λ(S + S ∗ )x, a − λ(S + S ∗ )x, a = 0, since both x and a belong to Eλ .

3. Systems with Continuous Time and Lyapunov Equations

499

Now it is easy to show that if x ∈ Eλ and x, Sα = 0, then (S − S ∗ )x ∈ E−λ . Indeed, Γ(S − S ∗ )x = ΓSx − ΓS ∗ x = S ∗ Γx − SΓx = −(S − S ∗ )Γx = −λ(S − S ∗ )x by (2.2) and (2.5). Since Ker(S − S ∗ ) = {O}, it follows that S − S ∗ is a one-to-one map of {x ∈ Eλ : x, Sα = 0} into E−λ , which proves (2.4). Case 2. For any x ∈ Eλ , x, e0 = 0. In this case it follows directly from (2.2) and (2.3) that if x ∈ Eλ and x, Sα = 0, then (2.5) holds. As in Case 1, this implies that S − S ∗ is a one-to-one map of {x ∈ Eλ : x, Sα = 0} into E−λ .

3. Linear Systems with Continuous Time and Lyapunov Equations In this section we make some preparations to construct a Hankel operator associated with a balanced linear system with continuous time that has prescribed spectral properties. Namely, we establish useful facts on the unitary equivalence (modulo the kernel) of the Hankel operator Γ h corresponding to a balanced system {A, B, C} and certain operators related to the system, and introduce Lyapunov equations. We consider here (proper) balanced linear SISO systems {A, B, C} (see §11.3), where A is a bounded linear operator on a separable Hilbert space K, B : C → K and C : K → C are bounded linear operators. Then there are vectors b and c in K such that Bu = ub,

u ∈ C,

Cx = (x, c),

x ∈ K.

In Chapter 11 we have associated with such a linear system {A, B, C} the Hankel operator Γ h , h = h{A,B,C} , on L2 (R+ ), where h{A,B,C} (t) = CetA B = (etA b, c),

t > 0.

Recall also that with such a system {A, B, C} we associate the observability and controllability Gramians Wc and Wo defined by

∗ Wc = etA BB ∗ etA dt, R+

and Wo =

∗

etA C ∗ CetA dt

R+

if the integrals converge in the weak operator topology. We consider here balanced linear systems (see §11.3), i.e., those minimal systems for which the above integrals converge in the weak operator topology and Wo = Wc . We are going to establish the following result.

500


Theorem 3.1. Let Γ be a self-adjoint operator such that (C1) either Ker Γ = {O} or dim Ker Γ = ∞; (C2) Γ is noninvertible; (C3 ) |ν(t) − ν(−t)| ≤ 1, µ-a.e. Then there exists a balanced linear SISO system {A, B, C} such that the corresponding Hankel operator Γ h is unitarily equivalent to Γ. The proof of Theorem 3.1 will be completed in §5. First we need some auxiliary results on linear systems. Theorem 3.2. Let {A, B, C} be a balanced linear SISO system with continuous time, W = Wo = Wc , and let Γ h be the Hankel operator associated with it. Then Γ h is bounded and the restriction of |Γ h | to (Ker Γ h )⊥ is unitarily equivalent to W . Proof. This is an immediate consequence of Theorem 11.3.1. We consider now linear systems for which the corresponding Hankel operator is self-adjoint. Theorem 3.3. Let {A, B, C} be a balanced linear SISO system such that Γ h is self-adjoint, W = Wo = Wc . Then there exists an operator J on K which is self-adjoint and unitary and such that J W = W J , A∗ = J AJ , and c = J b. Proof. Recall (see the proof of Theorem 11.3.1) that the operators Vc and Vo from K to L2 (R+ ) are defined by ∗

(Vc x)(t) = B ∗ etA x,

(Vo x)(t) = CetA x,

x ∈ K,

and Γ h = Vo Vc∗ . Let Vc = Uc W 1/2 ,

Vo = Uo W 1/2

be the polar decompositions of Vc and Vo (|Vc | = |Vo | = W 1/2 ). We have Ker Vc = Ker Vc∗ Vc = Ker Wc = {O} = Ker Wo = Ker Vo∗ Vc = Ker Vo . Therefore Uc and Uo are isometries. Clearly, Range Uc = clos Range Vc and Range Uo = clos Range Vo . We have Γ h = Uo W Uc∗ . Since Γ h is self-adjoint, it follows Uo W Uc∗ = Uc W Uo∗ , whence, W U = U ∗ W,

(3.1)

where U = Uc∗ Uo . Clearly, Ker Uc∗ = Ker Vc∗ = Ker Γ h . Since Γ ∗h = Γ h , it follows that Γ h = Vc Vo∗ , which implies that Ker Γ h = Ker Vo∗ = Ker Uo∗ . Therefore Uc

3. Systems with Continuous Time and Lyapunov Equations

501

and Uo are isometries with the same range (Ker Γ h )⊥ , which implies that U = Uc∗ Uo is unitary. Let us show that it is also self-adjoint. Lemma 3.4. Let W be a nonnegative operator, Ker W = {O} and let U be a unitary operator such that W U = U ∗ W . Then U is self-adjoint. Proof. Multiplying the equality U ∗ W = W U by U on the left and by ∗ U on the right, we obtain U W = W U ∗ . Therefore U ∗W 2U = W U W U = W 2U ∗U = W 2, whence W 2 U = U W 2 . Since W is positive, it follows that W U = U W , and so U ∗ W = W U = U W. Since W has dense range, it follows that U ∗ = U . Let us now complete the proof of Theorem 3.3. We denote U = Uc∗ Uo by J . It follows from (3.1) that W J = J W . Let us show that A∗ = J AJ . Indeed, by the definition of J , Uo = Uc J and since J W = W J , we have Vo = Vc J . This means that ∗

CetA x = B ∗ etA J x,

x ∈ K,

or, which is the same, ∗

(x, etA c) = (x, J etA b) = (x, J etA J (J b)) = (x, etJAJ J b). Hence, ∗

etA c = etJAJ J b,

t ≥ 0.

(3.2)

Substituting t = 0, we obtain c = J b. Differentiating (3.2), we find that A∗ ∗ and J AJ coincide on the orbit {etA c : t ≥ 0}, which is dense since the system is observable. Therefore A∗ = J AJ , which completes the proof. Theorem 3.5. Let {A, B, C} be a balanced linear SISO system, Wc = Wo = W , and let J be an operator that is self-adjoint and unitary and satisfies the equalities J W = W J , A∗ = J AJ , c = J b. Then the Hankel operator Γ h associated with the system is self-adjoint and Γ h (Ker Γ h )⊥ is unitarily equivalent to W J . ∗ Proof. Since A∗ = J AJ , we have etA = J etA J . The equality c = J b implies C = B ∗ J , which in turn leads to the equality Vc = Vo J . Therefore Γ h = Vo J Vo∗ = Uo W 1/2 J W 1/2 Uo∗ = Uo W J Uo∗ , which proves the result. We are going to use Theorem 3.5 to construct a Hankel operator with given spectral properties modulo the kernel. Namely, let Γ be a self-adjoint operator with a scalar spectral measure µ and spectral multiplicity function ν. Suppose that Γ satisfies conditions (C1), (C2), and (C3 ) of Theorem 3.1. ˜ = Γ (Ker Γ)⊥ . Let J be the operator on (Ker Γ)⊥ that is selfPut Γ ˜ = J |Γ| ˜ = |Γ|J ˜ . We are going to conadjoint and unitary and satisfies Γ ˜ struct a balanced linear system {A, B, C} such that W = |Γ|,

502


A∗ = J AJ , c = J b, and W J = J W . Then by Theorem 3.5, Γ h (Ker Γ h )⊥ ˜ Later we settle the problem with the kernel. is unitarily equivalent to Γ. ˜ Fortunately, to prove this It is not easy to verify directly that W = |Γ|. equality, we do not have to compute Wo and Wc . We are going to verify instead the corresponding Lyapunov equations. We say that an operator A generates an asymptotically stable semigroup if lim etA x = 0 t→∞

for any x. Theorem 3.6. Suppose that A generates an asymptotically stable semigroup on a Hilbert space K and let K be a bounded operator on K. If the integral

∗ def etA KetA dt = W (3.3) R+

converges in the weak operator topology, then W is a unique solution of the following Lyapunov equation A∗ W + W A = −K.

(3.4)

Proof. Let us show that W satisfies (3.4). We have for x, y ∈ K

∗ ∗ ((A∗ W + W A)x, y) = ((A∗ etA KetA + etA KetA A)x, y)dt R+

= R+

d tA∗ tA (e Ke x, y) dt dt

lim (KetA x, etA y) − (Kx, y) = −(Kx, y)

=

t→∞

because of asymptotic stability. Let us now establish the uniqueness of the solution. Suppose A∗ X + XA = −K for some operator X. Let ∆ = W − X. Then A∗ ∆ + ∆A = O. Clearly, d (∆etA x, etA y) dt

=

(∆AetA x, etA y) + (∆etA x, AetA y)

=

((A∗ ∆ + ∆A)etA x, etA y) = 0.

Since lim etA x = 0 for any x ∈ K, it follows that (∆x, y) = 0 for any t→∞ x, y ∈ K. Hence, ∆ = O. The following result shows that if W satisfies the Lyapunov equation (3.4), we can obtain the convergence of the integral (3.3) for free.

4. Construction of a Linear System with Continuous Time

503

Theorem 3.7. Let A be an operator such that etA ≤ M < ∞, t ≥ 0, and let K be a nonnegative operator. If W is a solution of the Lyapunov equation A∗ W + W A = −K, then the integral

∗

etA KetA dt R+

converges in the weak operator topology. Proof. Let x, y ∈ K. We have ∗

∗

(etA KetA x, y) = −(etA (A∗ W + W A)etA x, y) = −

d tA∗ (e W etA x, y). dt

Therefore

T ∗ ∗ (etA KetA x, x)dt = (W x, x) − (eT A W eT A x, x) ≤ W · (M + 1). 0 ∗

Since (etA KetA x, x) ≥ 0 for x ∈ K, it follows that the integral

∞ ∗ (etA KetA x, x)dt 0

converges for any x ∈ K. The result follows from the polarization identity. Theorems 3.6 and 3.7 show that to solve the problem modulo the kernel, it is sufficient to construct a SISO linear system {A, B, C} such that (i) the operators A and A∗ generate asymptotically stable semigroups; ˜ is a solution of the Lyapunov equations (ii) the operator W = |Γ| A∗ W + W A = −C ∗ C,

AW + W A∗ = −BB ∗ ;

(iii) A∗ = J AJ , c = J b. If A∗ = J AJ , then A∗ generates an asymptotically stable semigroup if and only if so does A. It follows easily from (iii) that both Lyapunov equations in (ii) coincide. Therefore it is sufficient to verify the following properties: (i ) A generates an asymptotically stable semigroup; (ii ) A∗ W + W A = −C ∗ C; (iii ) A∗ = J AJ , c = J b.

4. Construction of a Linear System with Continuous Time Let Γ be a self-adjoint operator on Hilbert space that satisfies conditions ˜ = Γ(Ker Γ)⊥ . (C1), (C2), and (C3 ) of Theorem 3.1. As before we put Γ

504


˜ is unitarily equivalent to the operator W of multiThen the operator |Γ| plication by t on the von Neumann integral

K= ⊕E(t)dρ(t). (4.1) σ(W )

˜ Note that to avoid a conflict of Here ρ is a scalar spectral measure of |Γ|. notation, we use in §§4–8 the bold symbol t for the coordinate variable in von Neumann integrals while t is used for the time variable. We can also assume that the spaces E(t) are imbedded in an infinitedimensional space E with an orthonormal basis {ej }j≥1 and E(t) = span{ek : 1 ≤ k < νW (t) + 1},

E(t) = {O}

if νW (t) = 0,

where νW is the spectral multiplicity function of W , νW (t) = dim E(t). Clearly, νW (t) = ν(t) + ν(−t), ρ-a.e., where ν is the spectral multiplicity function of Γ. Recall that ν satisfies condition (C3 ) of Theorem 3.1. Consider the sets σ+ = {t ∈ σ(W ) : ν(t) ≥ ν(−t)}, σ− = {t ∈ σ(W ) : ν(t) < ν(−t)}. We define the operator J on K as multiplication by the operator-valued function J:   1 0 0 ···    0 −1 0 · · ·     J(t) = ξ(t)   0 0 1 ··· ,     .. .. .. . . . . . . where

ξ(t) =

1, t ∈ σ+ , −1, t ∈ σ− .

˜ is unitarily equivalent to J W and we can assume that Clearly, Γ ˜ Γ = JW . Recall that the scalar spectral measure ρ of W is not uniquely defined and we can always replace it with a mutually absolutely continuous measure by multiplying it by a positive weight w in L1 (ρ). Let K0 be the subspace of K that consists of functions f of the form f (t) = ϕ(t)e1 ,

ϕ ∈ L2 (ρ).

K0 can be identified naturally with L2 (ρ). Let A0 be the integral operator on K0 = L2 (ρ) defined by

(A0 f )(s) = k(s, t)f (t)dρ(t), σ(W )

4. Construction of a Linear System with Continuous Time

where

k(s, t) =

−1/(s + t), s, t ∈ σ+

505

or s, t ∈ σ− ,

−1/(s − t), s ∈ σ+ , t ∈ σ− or s ∈ σ− , t ∈ σ+ .

(4.2)

The operator A0 certainly need not be bounded. However, the following lemma allows us to change the measure ρ so that A0 becomes a HilbertSchmidt operator. We extend A0 to K by putting A0 K0⊥ = O. Lemma 4.1. There exists w ∈ L1 (ρ), w > 0, ρ-a.e., such that the integral operator

k(s, t)f (t)d˜ ρ(t) f → σ(W )

is a Hilbert–Schmidt operator on L2 (˜ ρ), where d˜ ρ = wdρ. We postpone the proof of Lemma 4.1 until §6. Let us define the vectors b and c in K by c(t) = e1 ,

b(t) = J(t)c(t) = ξ(t)c(t),

t ∈ σ(W ).

It is easy to see that A∗0 = J A0 J

and A∗0 W + W A0 = −C ∗ C.

(4.3)

K0⊥

However, Ker A0 = is nontrivial except for the case when W has simple spectrum. Therefore A0 does not generate an asymptotically stable semigroup in general. To overcome this obstacle, we perturb A0 by an operator D such that the perturbed operator still satisfies (4.3) but generates an asymptotically stable semigroup. Of course, to get asymptotic stability, it is not sufficient (n) n−1to kill the kernel. Let ak k=1 , n ∈ N ∪ {∞}, be positive numbers such that n−1 k=1

(n)

ak

2
ε. Then (5.5) implies that 1 = ϕ22 = (ϕ, ϕ) ≤ εκ∞ ≤ εf 22 < 1 for a sufficiently small ε. The contradiction obtained proves that Ker Γ h = {O}. Let us show that if Ker Γ h = {O}, then s1 dρ(s) < ∞. It is easy to see that the subspace K = (Ker Γ h )⊥ is invariant under the semigroup of def backward translations {Φs }s≥0 . Let Ψs = Φs K. Since Ker Γ h = Ker Vo∗ , it follows that K = clos Range V0∗ and so by (5.2), Vo etA = Ψt Vo . We need the following lemma.

(5.6)

512


Lemma 5.3. The semigroup {Ψt }t≥0 has bounded generator, i.e., Ψt = etG , t ≥ 0, for a bounded linear operator G on K. Let us first complete the proof of Theorem 5.1 and then prove Lemma 5.3. By Lemma 5.3, Ψt = etG , t ≥ 0. It follows from (5.6) that Vo A = GVo , and so G = Vo AVo−1 (Vo AVo−1 is defined on a dense subset of K and extends by continuity to a bounded operator). Put def R = G + G∗ = Vo AVo−1 + (Vo−1 )∗ A∗ Vo∗ . Multiplying this equality by Vo on the right and Vo∗ on the left and bearing in mind that Vo∗ Vo = W , we obtain W A + A∗ W = Vo∗ RVo . On the other hand, we have from (4.6) W A + A∗ W = −C ∗ C = −(·, c)c. So Vo∗ RVo = −C ∗ C. Hence, c ∈ Range Vo∗ or, which is equivalent, c ∈ Range W 1/2 (see the proof of Theorem 3.2). Clearly, the last condition exactly means that s−1 dρ(s) < ∞. Proof of Lemma 5.3. It is slightly more convenient to consider here the Fourier transform F defined by

1 h(x)e−ixy dx, h ∈ L1 (R). (F h)(y) = √ 2π R To show that our contractive semigroup has bounded generator, we apply the inverse Fourier transform F −1 , which maps L2 (R+ ) onto the Hardy ˇ s = F −1 Φs F and Ψ ˇ s = F −1 Ψs F . Then class H 2 (C+ ). Put Φ ˇ s f = P+ e−s f, Ψ where e−s (t) = e−ist and P+ is the orthogonal projection from L2 (R) onto H 2 (C+ ). Clearly, ˇ ˇs = Φ ˇ s |K, Ψ −1 ˇ where K = F K. ˇ is an invariant subspace of the semigroup {Φ ˇ s }s≥0 , Obviously, K ˇ has the form and so by Beurling’s theorem (see Appendix 2.2), K K ϑ = H 2 (C+ ) ϑH 2 (C+ ), where ϑ is an inner function in C+ . ˇ s }s≥0 has bounded generIt is sufficient to show that the semigroup {Ψ ˇ s }s≥0 is a contractive semigroup, we can consider its cogenator. Since {Ψ erator T defined by ˇ s) , T = lim ϕs (Ψ s→0+

5. The Kernel of Γ h

513

where

λ−1+s . λ−1−s Then T ≤ 1 (see Sz.-Nagy and Foias [1], §3.8). Moreover, if we show ˇ def = (T + I)(T − I)−1 is the that 1 ∈ / σ(T ), then the bounded operator G ˇ ˇ ˇ generator of the semigroup {Ψs }s≥0 , i.e., Ψs = esG , s > 0 (see Sz.-Nagy, Foias [1], §3.8). Let us first evaluate the cogenerator T . Taking into account that for ψ ∈ H ∞ (C+ ) ˇ s )f = P+ ψ(e−s )f, f ∈ K ϑ , ψ(Ψ ˇ s is given by we can conclude that the cogenerator T of the semigroup Ψ the formula T f = P+ ϕf, f ∈ K ϑ , ϕs (λ) =

where ϕ(t) = lim ϕs (e−ist ) = (s + i)/(s − i). s→0+

Let ω be the conformal map of the unit disc D onto the half-plane C+ defined by 1+z ω(z) = i . 1−z def Put ϑ˘ = ϑ ◦ ω. Then ϑ˘ is an inner function in D. The operator U, t−i 1/2 1 −1 1/2 1 (Uf )(t) = π (f ◦ ω )(t) = π f , t ∈ R, t+i t+i t+i 2 2 K ϑ . Moreover, maps unitarily H onto ∗ H (C+ ) and Kϑ˘ onto −1 ∗ 2 U T Uf = S Kϑ˘, where S is backward shift on H (see Appendix 2.1). Thus T is unitarily equivalent to S ∗ Kϑ˘. ˘ (see To complete the proof, we have to show that the spectrum σ(ϑ) ˘ Appendix 2.1) of the inner function ϑ does not contain the point 1, since ˘ 2 ) is equal to σ(ϑ), ˘ i.e., the complex conjugate the spectrum of S ∗ (H 2 ϑH ˘ (see Appendix 2.1). of σ(ϑ) ˘ the If ϑ is an inner function in C+ , we put by definition σ(ϑ) = ω(σ(ϑ)), spectrum of ϑ. Note that σ(ϑ) can contain ∞ (this happens if and only if ˘ 1 ∈ σ(ϑ)). Let us show that σ(ϑ) ⊂ −iσ(A), which would imply that σ(ϑ) is bounded and since ˘ = ω −1 (σ(ϑ)), σ(T ) = σ(ϑ)

it would follow that 1 ∈ / σ(T ), which would complete the proof. We are going to use the notion of pseudocontinuation. As in the case of the unit disk a function f ∈ H 2 (C+ ) is said to have a pseudocontinuation if there exists a meromorphic function g in the Nevanlinna class

g1 def : g1 , g2 ∈ H ∞ (C− ) N (C− ) = g2

514


such that the boundary value of g coincides with the boundary values of f almost everywhere on R (see Appendix 2.2). In this case g is called a pseudocontinuation of f . It is well known that if f ∈ K ϑ , then f has a pseudocontinuation (see Appendix 2.2, the case of functions in C+ reduces very easily to the case of functions in D). We are going to use the following facts (see Appendix 2.1 and Appendix 2.2): (i) if a function f ∈ H 2 (C+ ) has a pseudocontinuation and f extends analytically across an interval I ⊂ R, then its analytic extension coincides with its pseudocontinuation; (ii) if ϑ is an inner function such that R ⊂ σ(ϑ), then ϑ extends analytically to C \ σ(ϑ); (iii) if R ⊂ σ(ϑ) and f ∈ K ϑ , then f extends analytically to C \ σ(ϑ). Again, the above results reduce very easily to the case of functions in D. 7 def = C ∪ {∞}. For Now we are ready to prove that σ(ϑ) ⊂ iσ(A). Put C a function f ∈ K ϑ we consider its pseudocontinuation and denote this pseudocontinuation also by f . We need the following lemma. Lemma 5.4. Let ϑ be an inner function in C+ and let σ be a closed subset of its spectrum σ(ϑ) such that σ = σ(ϑ). Then there exists a nontrivial inner divisor ϑ1 of ϑ (i.e., ϑ/ϑ1 = const) such that every function 7 \ σ belongs to K ϑ . f ∈ K ϑ that extends analytically to C 1 Let us first complete the proof of Lemma 5.3. Suppose that σ(ϑ) ⊂ −iσ(A). Put σ = σ(ϑ) ∩ −iσ(A). Let ϑ1 be the inner function satisfying the conclusion of Lemma 5.4. Consider an arbitrary f = F −1 Vo x, x ∈ K. Since

1 −1 √ CetA xe2πist dt (F Vo x)(s) = 2π R+ =

1 √ ((A + isI)−1 x, c), 2π

Im s > 0,

(5.7)

the function f extends analytically outside the set iσ(A). On the other hand, f ∈ K ϑ and by (iii), it extends analytically to C \ σ(ϑ). Hence, 7 \ σ. By Lemma 5.4, f ∈ K ϑ and since f extends analytically to C 1 {F −1 Vo x : x ∈ K} is dense in K ϑ , it follows that K ϑ ⊂ K ϑ1 which contradicts the fact that ϑ1 is a nontrivial inner divisor of ϑ. Thus σ(ϑ) ⊂ −iσ(A), and as we have already observed, this implies the boundedness of G. ˜ = ω −1 (−iσ(A)) and σ(T ) Remark. It can be shown that σ(T ) = σ(ϑ) must be symmetric about the real line: σ(T ) = σ(T ). Thus σ(A) must be symmetric too: σ(A) = σ(A∗ ). Proof of Lemma 5.4. Recall that K ϑ = H 2 (C+ ) ∩ ϑH 2 (C− ). Consider first the simplest case. Suppose that there exists a point λ ∈ C+ such that def λ ∈ σ(ϑ) \ σ. Then we can put ϑ1 = ϑ/bλ , where bλ is the Blaschke factor

5. The Kernel of Γ h

515

¯ with zero at λ. Let f be a function in K ϑ that extends analytically to λ. 2 We have f = (f /ϑ)ϑ, where f /ϑ ∈ H (C− ). The pseudocontinuation of ϑ is defined naturally by ¯ ϑ(ζ) = 1/ϑ(ζ),

ζ ∈ C− .

Thus the pseudocontinuation of ϑ has a pole at λ, and so (f /ϑ)(λ) = 0. The function 1/bλ can be considered as a Blaschke factor in C− with zero at λ. Since the function f /ϑ belongs to H 2 (C− ) and vanishes at λ, it follows that 1/bλ is a divisor of f /ϑ, and so bλ f f = ∈ H 2 (C− ), ϑ1 ϑ which means that f ∈ K ϑ1 .

7 = R ∪ {∞}. Then there Suppose now that C+ ∪ (σ(ϑ) \ σ) = ∅. Put R 7 exists an open connected subset I of R such that I ∩ σ(ϑ) = ∅ and I is separated from σ. Let τ be a nontrivial inner divisor of ϑ whose spectrum is contained in I and let ϑ1 = ϑ/τ . Let us show that ϑ1 satisfies the conclusion of the lemma. Let f be a function in K ϑ that extends analytically to C \ σ, and so f is analytic in a neighborhood of clos I. Since f ∈ N (C− ), we have f = g1 /g2 , where g1 , g2 ∈ H ∞ (C− ), and g1 and g2 have no nontrivial common inner factor. We need the following fact. def

Lemma 5.5. Let ϕ1 and ϕ2 be bounded analytic functions in D that have no nonconstant common inner factor. If ϕ1 /ϕ2 extends analytically to a neighborhood of a closed arc J of T, then the spectra of the inner components of ϕ1 and ϕ2 do not intersect J. Let us first complete the proof of Lemma 5.4. Applying the conformal map from C− onto D, we find from Lemma 5.5 that the spectra of the inner components of g1 and g2 are separated from I. Let ϑ# and τ # be the inner functions in C− defined by ¯ ϑ# (ζ) = ϑ(ζ),

¯ τ # (ζ) = τ (ζ),

ζ ∈ C− .

Then f /ϑ ∈ H 2 (C− ) and (f /ϑ)(ζ) = f (ζ)ϑ# (ζ), ζ ∈ C− . Since the spectrum of the inner factor of g2 is separated from I, it is easy to see that τ # is a divisor of g1 . It follows that the function f ϑ# /τ # in C− belongs to H 2 (C− ). Clearly, the boundary values of this function coincide with the boundary values of f /ϑ1 , and so f ∈ K ϑ1 . Proof of Lemma 5.5. It is easy to see that the zeros of ϕ1 and ϕ2 in D are separated from J. Thus we can divide ϕ1 and ϕ2 by the corresponding Blaschke product and reduce the situation to the case when ϕ1 and ϕ2 have no zeros in D. Then log |ϕ1 /ϕ2 | is the Poisson integral of a real measure υ on T. Clearly, υ = lim υr r→1

516


in the weak-∗ topology, where

ϕ1 (rζ) dm(ζ). dυr (ζ) = log ϕ2 (rζ)

Since log |ϕ1 /ϕ2 | is smooth in a neighborhood of J in D, it follows that the restriction of υr to J converges in the norm to the restriction of υ to J, and so υ is absolutely continuous on J, which means that the singular measures of the inner components of ϕ1 and ϕ2 are supported outside J.

6. Proofs of Lemmas 4.1 and 5.2 The aim of this section is to prove Lemmas 4.1 and 5.2. It is easy to see that they are easy consequences of the following fact. Lemma 6.1. Let ρ be a finite positive Borel measure on [0, a], a > 0, which has no mass at 0 and let σ+ and σ− be disjoint sets such that σ+ ∪ σ− = supp ρ. Then there exists a weight w ∈ L1 (ρ) that is positive ρ-a.e. and such that the measure ρ˜, d˜ ρ = wdρ, satisfies the following conditions: ρ) with kernel function (4.2) belongs (a) the integral operator A0 on L2 (˜ to the Hilbert–Schmidt class S 2 ; (b) if 0 ∈ supp ρ, then we can find a weight w satisfying (a) and such that

1 d˜ ρ(s) = ∞. s Proof. We begin with (a). In the case when one of the sets σ+ , σ− is empty the result is trivial. Assume that both σ+ and σ− have positive (n) (n) measure. Let σ+ and σ− be compact subsets of σ+ and σ− such that (n) (n+1) (n) (n+1) , σ− ⊂ σ− , n ≥ n + 1, and σ+ ⊂ σ+ (n) (n) lim ρ σ+ \ σ+ = lim ρ σ− \ σ− = 0. n→∞

n→∞

(n)

(n)

(n)

(n)

Clearly, the restriction of k to (σ+ ∪ σ− ) × (σ+ ∪ σ− ) is bounded. (n) (n) Suppose we have already defined w on σ+ ∪ σ− . Let us define it on (n+1) (n+1) ∪ σ− . We can easily do it so that σ+

1 (k(s, t))2 d˜ ρ(s)d˜ ρ(t) < n , 2 ∆n

where (n+1) (n) (n+1) (n+1) (n+1) (n) (n) (n) × σ+ \ σ + ∪ σ − × σ+ ∪ σ − , ∪ σ− ∪ σ− ∆n = σ+

6. Proofs of Lemmas 4.1 and 5.2

517

(n+1) (n+1) (n) (n) \ σ+ ∪ σ− . Doing d˜ ρ = wdρ, and w is positive a.e. on σ+ ∪ σ− in this way, define w on supp ρ. Clearly, the measure ρ˜, d˜ ρ = wdµ, satisfies (a). To prove (b) we consider first the simplest case when one of the sets σ+ , σ− is empty and so k(s, t) = −1/(s + t). Without loss of generality we can assume that supp ρ ⊂ [0, 1]. Let δn = (2−n , 2−n+1 ], n ≥ 1. Consider the increasing sequence {nj }j≥1 of integers such that ρ(δnj ) > 0 and ρ(δn ) = 0 if n = nj for any j. Since 0 ∈ supp ρ, the sequence {nj }j≥1 is infinite. We define the weight w by w(s) = Then

0

1

w(s)dρ(s) s

2−nj , ρ(δnj )j log(j + 1)

s ∈ δnj .

(6.1)

w(s)dρ(s) = s j≥1 δ

=

j≥1

≥

j≥1

=

nj

2−nj ρ(δnj )j log(j + 1)

dρ(s) s

δnj

2−nj ρ(δnj ) · 2nj −1 ρ(δnj )j log(j + 1)

1 1 = ∞. 2 j log(j + 1) j≥1

Thus

1

0

On the other hand,

ρ(s)d˜ ρ(t) = (k(s, t))2 d˜ j,r≥1δ

≤

1≤j 0, λ-a.e., and the operator Mψ U0 (I − Q) is a compact operator from L2 (ρ) to L2 (λ) for any ψ satisfying |ψ| ≤ κ, where Mψ is multiplication by ψ on L2 (λ). As before we identify L2 (ρ) with K0 . The proof of Lemma 13.2 will be given in §14. Lemma 13.3. Suppose that the function ω1 in the definition of J (+) (−) satisfies the inequality | sin ω1 (t)| ≤ κ(t) for t ∈ σ2 ∪ σ2 , where κ is the function from Lemma 13.2. Then σF (A∗ ) ∩ T ⊂ σF (J J ). Proof. Let us verify the hypotheses of Lemma 11.2. Let Kj = {f ∈ K : f (t) = ϕ(t)ej+1 , ϕ ∈ L2 (ρ)},

j ≥ 0.

2 Clearly, > Kj can naturally be identified with a subspace of L (ρ). Then K= Kj . Let {J jk }j,k≥0 be the block matrix representation of the operj≥0

˜ U with respect to the orthogonal decomposition ator > J = U ∗J Kj . It follows from the definition of J that K= j≥0

J 00 = U0∗ (I − Mu )U0 ,

J 01 = U0∗ Mv K1 , 2

J 10 = Mv U0 ,

where Mu and Mv are multiplications on L (λ) by the functions u and v defined by  (+) (−)  1 − cos ω1 (t), t ∈ σ2 ∪ σ2 , u(t) = (+) (−)  0, t∈

σ2 ∪ σ 2 ,  (+) (−)  sin ω1 (t), t ∈ σ2 ∪ σ2 , v(t) = (+) (−)  0, t ∈ σ2 ∪ σ2 .

538


(In the case νW (t) = 1, we have v(t) = 0, and so RangeJ 10 ⊂ K1 .) Since |v(t)| ≤ κ(t), it is easy to see that |u(t)| ≤ κ(t). Consider now the block matrix representation of the operator Q∗ J Q with respect to the same decomposition of K. It is easy to see that   ∗ ∗ Q0 U0 (I − Mu )U0 Q0 Q∗0 U0∗ Mv K1 J 02 · · ·    J 11 J 12 · · ·  Mv U0 Q0   Q∗ J Q =  . J 21 J 22 · · ·  J 20     .. .. .. .. . . . . To verify the hypotheses of Lemma 11.2, it is sufficient to prove that for any gn , hn ∈ K0 such that gn = hn = 1, lim Q0 gn = 1, and n→∞

w- lim gn = w- lim hn = O, n→∞

n→∞

the following equalities hold: lim (Q∗0 U0∗ (I − Mu )U0 Q0 − U0∗ (I − Mu )U0 )gn = 0,

(13.4)

lim (Mv U0 Q0 − Mv U0 )gn = 0,

(13.5)

lim (Q∗0 U0∗ Mv − U0∗ Mv )hn = 0.

(13.6)

n→∞

n→∞

n→∞

Since |v(t)| ≤ κ(t), it follows from Lemma 13.2 that the operator Mv U0 Q0 − Mv U0 = Mv U0 (Q0 − I) is compact which implies (13.5). Similarly, Q∗0 U0∗ Mv − U0∗ Mv is compact, and so (13.6) holds. Let us prove (13.4). We have Q∗0 U0∗ (I − Mu )U0 Q0

− U0∗ (I − Mu )U0 = Q∗0 Q0 − I − (Q∗0 − I)U0∗ Mu U0 − U0∗ Mu U0 (Q0 − I) − (Q∗0 − I)U0∗ Mu U0 (Q0 − I).

By Lemma 13.2, the operators (Q∗0 − I)U0∗ Mu U0 ,

U0∗ Mu U0 (Q0 − I),

and

(Q∗0 − I)U0∗ Mu U0 (Q0 − I)

are compact. Hence, it remains to show that lim Q∗0 Q0 gn − gn = 0. n→∞

Since Q0 ≤ 1 and

lim (Q∗0 Q0 gn , gn ) = lim Q0 gn 2 = 1,

n→∞

we have lim

n→∞

Q∗0 Q0 gn

n→∞

= 1. It now follows from the spectral theorem for

the self-adjoint contraction Q∗0 Q0 that lim Q∗0 Q0 gn − gn = 0, which n→∞

proves (13.4) and completes the proof of the lemma. Now it remains to show that we can choose the functions q0 and ωj so that σF (J J ) is at most countable.


539

Let us briefly explain the idea how to make σF (J J ) at most countable. Suppose for a while that ρ = λ and J is multiplication by J˜ on K. Since both functions J and J˜ take at most countably many values, so does J J˜. It can easily be shown that if the functions ωj satisfy the conditions (1 )– (4 ) of §12, then for each t the operator J(t)J˜(t) is a small perturbation of a block diagonal operator whose diagonal blocks are of the form 0 1 ±1, ± . 1 0 The spectrum of each such block is contained in the set {1, −1, i, −i}, and it ˜ is easy to see from (12.6) that the set σ J(t)J (t) is at most countable t

and can have accumulation points only in the set {1, −1, i, −i}. Since the operator J(t)J˜(t) is unitary for each t, it follows that ˜ σ J(t)J (t) , σ(J J ) ⊂ clos t

and so σ(J J ) is at most countable. However, the operators J and J are multiplications by J and J˜ in different systems of coordinates. Nevertheless, if these two systems of coordinates are “sufficiently close” to each other, we can use similar considerations to prove that σ(J J ) is at most countable. Theorem 13.4. There exist a function q0 in L2 (ρ) that satisfies conditions (1)–(3) of §12 and a function η in L∞ (ρ), η(t) > 0, ρ-a.e., such that if the functions ωj satisfy conditions (1 )–(4 ) of §12 and |ω1 (t)| ≤ η(t) (+) (−) for almost all t ∈ σ2 ∪ σ2 , then σF (J J ) is at most countable. To prove the theorem we represent the operator J as a product of three (+) (−) operators. We define the operator-valued functions J˜2 , J˜2 , and J˜1 on ∆ ∪ δλ ∪ δρ by

(±) J˜2 (t)

=

J˜1 (t) =

  J˜(t), t ∈ σ (±) , 2 (±)  I, t ∈ σ2 ,   J˜(t), t ∈ σ (+) ∪ σ (−) , 2 2

 I,

(+)

t ∈ σ2

(−)

∪ σ2 .

˜ (−) , and J ˜ 1 be multiplications by J˜(+) , J˜(−) , and J˜1 on the ˜ (+) , J Let J 2 2 2 2 (±) ˜ (±) U and J 1 = U ∗ J ˜ 1U . = U ∗J von Neumann integral (12.2). Put J (+)

(−)

2

2

Clearly, J = J2 J 2 J 1 . It is easy to see that K0 is a reducing subspace for J 1 and J 1 K0 = I.

540


For a Borel subset γ of σ(W ) we denote by K(γ) the subspace of K that consists of functions supported on γ and by E(γ) the orthogonal projection onto K(γ) (in other words, E is the spectral measure of W ). Put (+) (−) σ1 = σ(W )\ σ2 ∪ σ2 . We call an operator on K essentially (with respect to the block diagonal (+) (−) decomposition K = K(σ1 ) ⊕ K σ2 ⊕ K σ2 if its block matrix has compact off-diagonal entries. It is easy to see from the definition of J that the operator J 1 is block (+) (−) diagonal. It is also easy to show that the operators J 2 and J 2 are essentially block diagonal if the operators (±) (±) (J 2 − I)E σ(W )\σ2 (13.7) are compact. Therefore in this case J is also essentially block diagonal. It follows easily from the compactness of the operators (13.7) that the operators (±) (±) (±) (±) (±) E σ2 J E σ2 − E σ2 J 2 E σ2 (13.8) and E(σ1 )J E(σ1 ) − E(σ1 )J 1 E(σ1 )

(13.9)

are also compact. The following lemma proves the compactness of the operators (13.7). Lemma 13.5. There exist a function q0 in L2 (ρ) that satisfies conditions (1)–(3) of §12 and a function η in L∞ (ρ), η(t) > 0, ρ-a.e., such that if the functions ωj satisfy conditions (1 )–(4 ) of §12 and |ω1 (t)| ≤ η(t) for (+) (−) almost all t ∈ σ2 ∪σ2 , then the operators (13.8) and (13.9) are compact. The proof of Lemma 13.5 will be given in §14. Proof of Theorem 13.4. Let us choose q0 , η, and ωj satisfying the hypotheses of Lemma 13.5. Then the operator J is essentially block diag (+) (−) onal with respect to the decomposition K = K(σ1 ) ⊕ K σ2 ⊕ K σ2 . Obviously, the operator J is block diagonal. Hence, J J is essentially block diagonal. Therefore, to prove that σF (J J ) is at most countable, it is sufficient to prove that the Fredholm spectrum of each diagonal block is at most countable on the unit circle. Consider the block E(σ1 )J J K(σ1 ) = E(σ1 )J E(σ1 )J K(σ1 ). It follows from the compactness of the operators (13.8) and (13.8) that σF E(σ1 )J J K(σ1 ) = σF E(σ1 )J 1 J K(σ1 ) . But both operators E(σ1 )J 1 E(σ1 ) and E(σ1 )J E(σ1 ) are multiplications on the von Neumann integral (12.1). Thus the reasoning given before the


541

statement of Theorem 13.4 works in this case, and so σ E(σ1 )J J K(σ1 ) is at most countable. The situation with two other blocks is a bit more complicated. Consider for example the block (+) (+) (+) (+) (+) J J K σ2 = E σ2 J E σ2 J K σ2 . E σ2

Again by the compactness of the operators (13.8) and (13.8), we have (+) (+) J J K(σ2 ) = σF E σ2 J 2 J K(σ2 ) . σF E σ 2 (+)

Let J2

be the operator-valued function defined by  (+)  J(t), t ∈ σ2 , (+) J2 (t) = (+)  I, t ∈ σ(W )\σ2 , (+)

(+)

and let J 2

be by J2 on the von Neumann integral (12.1). multiplication (+) (+) (+) (+) (+) J E σ2 = E σ2 J 2 E σ2 , and so Obviously, E σ2 (+) (+) (+) (+) (+) (+) (+) J 2 J E σ2 = E σ2 J 2 E σ2 J E σ2 E σ2 (+) (+) (+) (+) (+) J 2 E σ2 J 2 E σ2 = E σ2 (+) (+) (+) (+) = E σ2 J 2 J 2 E σ2 . We have (+)

(+)

J 2 J2

(+)

˜ = U ∗J 2 UJ2

(+)

(+)

∗ ˜ = U ∗J 2 (U J 2 U )U. (+)

(+)

Since K0 is a reducing subspace of J 2 , it follows from (12.3) that the ˜ (+) = U J (+) U ∗ is multiplication on the von Neumann integral operator J 2 2 (+) (12.2) by the operator-valued function J˜2 defined by (+) J2 (t)ej , 2 ≤ j < νW + 1, t ∈ δρ ∪ ∆, (+) ˜ J2 (t)ej = j = 1, t ∈ δρ ∪ ∆ ∪ δ λ . e1 , ˜ (+) and U J (+) U ∗ are both multiplications on (12.2), Now the operators J 2 2 and so the reasoning given before the statement of Theorem 13.4 works, which shows that (+) (+) ∗ (+) ˜ σ J = σ J 2 J 2 UJ2 U is at most countable. (+) (+) (+) (+) J 2 J 2 K σ2 T is at most countTo show that σ E σ2 able, we need the following lemma.

542


Lemma 13.6. Let Z be a contraction and V a unitary operator on Hilbert space. Then σ(Z ∗ V Z) ∩ T ⊂ σ(V ). Let us first complete the proof of Theorem 13.4. It follows from Lemma 13.6 that ' (+) (+) (+) (+) (+) (+) σ E σ2 J 2 J 2 E σ2 T ⊂ σ J 2 J2 , (+) (+) (+) (+) J 2 J 2 K σ2 T is at most countwhich implies that σ E σ2 (−) (−) able. In a similar way it can be shown that σ E σ2 J J E σ2 T is at most countable, which completes the proof. Proof of Lemma 13.6. Suppose that ζ ∈ σ(Z ∗ V Z) ∩ T and ζ ∈

σ(V ). Since Z ∗ V Z is a contraction and |ζ| = 1, it follows that there exists a sequence of vectors {fn }n≥0 such that fn = 1 and ¯ ∗ V Zfn , fn ) − fn 2 → 0. ¯ ∗ V Z − I)fn , fn ) = ζ(Z ((ζZ (13.10) ¯ Since ζ ∈ σ(V ), we have 1 ∈ σ(ζV ) and it follows from the spectral theorem for unitary operators that there exists a positive ε such that ¯ g, g) ≤ (1 − ε)g2 Re(ζV for any g. Hence, ¯ ∗ V Z − I)fn , fn ) Re((ζZ

=

¯ ∗ V Zfn , fn ) − fn Re ζ(Z

≤ (1 − ε)Zfn 2 − 1 ≤ −ε, which contradicts (13.10). Now we are in a position to prove Theorem 9.1. Proof of Theorem 9.1. We can choose the functions q0 and ωj that satisfy the hypotheses of Theorem 13.4 and Lemma 13.3. It follows from Lemma 13.1 that A∗ has no eigenvalues on T. Next, it follows from Lemma 13.3 that σF (A∗ ) ∩ T ⊂ σF (J J ), and so by Theorem 13.4, the set σF (A∗ ) is at most countable. Consequently, by the stability test (see §11), the operator A∗ is asymptotically stable, which by Lemmas 10.1 and 10.2 imply that T is asymptotically stable. The result now follows from Theorem 9.2.

14. Proofs of Lemmas 13.2 and 13.5 In this section we prove Lemmas 13.2 and 13.5. The following lemma will help us to represent the operator U0 (I − Q0 ) as an integral operator. Recall that the space K0 is reducing for both W 2 and Λ2 . As usual def we identify K0 with L2 (ρ). Clearly, M = W K0 is multiplication by the


543

˜ independent variable on L2 (ρ). Let M be multiplication by the independent ˜ U and variable on L2 (λ). Then |Λ| K0 = U ∗ M ˜ 2 U0 = (·, p)p. (W 2 − Λ2 ) K0 = M 2 − U0∗ M (14.1) Lemma 14.1. Let r = U0 p ∈ L2 (λ). Then for any continuously differentiable function ϕ on supp ρ ∪ supp λ the following equality holds:

U0 ϕ M

2

ϕ(t2 ) − ϕ(s2 ) 2 ˜ − ϕ M U0 f (t) = r(t)p(s)f (s)dρ(s). t2 − s2 (14.2)

Note that Lemma 14.1 is a consequence of the theory of double operator integrals (see Birman and Solomyak [2]). Proof of Lemma 14.1. It follows from (14.1) that ˜ 2 U0 = (·, p)r, U0 M 2 − M which is equivalent to the equality

˜ 2 U f (t) = r(t)p(s)f (s)dρ(s). U M02 − M 0 Clearly,

˜ 4 U0 = U0 M 2 − M ˜ 2 U0 M 2 + M ˜ 2 U0 M 2 − M ˜ 2 U0 , U0 M 4 − M

which is equivalent to

4 4 ˜ U0 M − M U0 f (t) = (s2 + t2 )r(t)p(s)f (s)dρ(s). Similarly, U0 M

2n

2n t − s2n 2n ˜ − M U0 f (t) = r(t)p(s)f (s)dρ(s), t2 − s2

which proves (14.2) for ϕ(t) = t2n . The result now follows from the fact that the set of polynomials of t2 is dense in C 1 [0, W 2 ] and the fact that the Hilbert–Schmidt norm of the integral operator on the right-hand side of (14.2) is at most const · ϕC 1 . Proof of Lemma 13.2. Let ϕ(s) = s1/4 , s > 0. It follows from Lemma 14.1 that

1/2 t − s1/2 ˜ 1/2 U0 )f )(t) = ((U0 M 1/2 − M r(t)p(s)f (s)dρ(s) t2 − s2 (14.3) at least in the case when 0 ∈ supp f . Indeed, in this case we can change ϕ in a small neighborhood of 0 to make it continuously differentiable. On the other hand, I − Q = I − |Λ|1/2 W −1/2 = W 1/2 − |Λ|1/2 W −1/2

544


on the dense subset Range W 1/2 of K. Therefore ˜ 1/2 U0 M −1/2 I − Q0 = M 1/2 − U0∗ M on the range of M 1/2 . It follows from (14.3) that ˜ 1/2 U0 M −1/2 f (t) (U0 (I − Q0 )f )(t) = U0 M 1/2 − M

t1/2 − s1/2 r(t)p(s)s−1/2 f (s)dρ(s) t2 − s2 whenever 0 ∈ supp f . Since p = W q0 , we have

1/2 t − s1/2 (U0 (I − Q0 )f )(t) = r(t)q0 (s)s1/2 f (s)dρ(s) t2 − s2 =

(14.4)

whenever 0 ∈ supp f . Consider first the case q0 ∈ Range W (i.e., Ker Γ = {O}). In this case p ∈ Range W 2 , and so Ker Λ = {O}. Therefore λ({0}) = 0. Consequently, for λ-almost all t the function s →

t1/2 − s1/2 1/2 s q0 (s) t2 − s 2

belongs to L2 (ρ). Therefore there exists a function κ in L∞ (λ) such that κ(t) > 0, λ-a.e., and 2

1/2 t − s1/2 1/2 t2 − s2 s r(t)q0 (s)κ(t) dρ(s)dλ(t) < ∞. It follows that the operator Mϕ U (I − Q0 ) is Hilbert–Schmidt whenever |ϕ(t)| ≤ κ(t), λ-a.e. Consider now the case q0 ∈ Range W . Let us show that in this case (I − Q0 ) is a Hilbert–Schmidt operator. Let q0 = W h, h ∈ L2 (ρ). It follows from (14.4) that

1/2 3/2 t s − s2 h(s)r(t)f (s)dρ(s) (U0 (I − Q0 )f )(t) = t2 − s2 (14.5) whenever 0 ∈ supp f . However, it is easy to see that 1/2 3/2 t s − s2 < ∞, sup t2 − s 2 t,s>0 and so the integral operator on the right-hand side of (14.5) is Hilbert– Schmidt. To prove Lemma 13.5, we need one more lemma. We denote by E0 (γ) the orthogonal projection on L2 (ρ) onto the subspace of functions supported on projection on L2 (λ). Note that E0 is the γ and by E1 (γ) the corresponding spectral measure of W K0 and E1 is the spectral measure of U (|Λ|K0 )U ∗ . As usual, Mψ denotes multiplication by ψ.


545

Lemma 14.2. There exist a function q0 satisfying conditions (1)–(3) (+) (−) of §12 and a function η that is positive ρ-a.e. on σ2 ∪ σ2 such that (±) (±) U0 E0 σ(W )\σ2 (14.6) Mψ E1 σ2 is a compact operator from L2 (ρ) to L2 (λ) for any function ψ satisfying |ψ(t)| ≤ η(t) a.e. Proof. Let us first show that for any disjoint compact sets γ1 and γ2 ,

f (s) r(t)p(s)dρ(s), t ∈ γ2 . (E1 (γ2 )U E0 (γ1 )f ) (t) = s2 − t2 (14.7) γ1 Let ϕ be a continuously differentiable function such that

1, s1/2 ∈ γ1 , ϕ(s) = 0, s1/2 ∈ γ2 . Then by Lemma 14.1,

ϕ(s2 ) − ϕ(t2 ) ˜ 2 )U0 f (t) = U0 ϕ(M 2 ) − ϕ(M r(t)p(s)dρ(s). s2 − t2 ˜ 2 U0 = 0. ˜ 2 U0 f (t) = 0 for t ∈ γ2 . Hence, E1 (γ2 )ϕ M Clearly, ϕ M It is also clear that if f is supported on γ1 , then ϕ(M 2 )f = f . It follows that for f supported on γ1 ,

1 (U0 f )(t) = r(t)p(s)f (s)dρ(s), t ∈ γ2 , s2 − t2 which implies (14.7). Note that if γ1 and γ2 are arbitrary disjoint Borel subsets, then we can approximate them by compact subsets and pass to the limit, which proves that (14.7) holds for f in a dense subset of L2 (ρ). Therefore to prove the lemma, it is sufficient to find a function q0 that (±) satisfies conditions (1)–(3) of §12 and such that for almost all t ∈ σ2 ,

|p(s)|2 dρ(s) < ∞. (14.8) |s2 − t2 | (±)

σ(W )\σ2

(Recall that p = W q0 , and so p(s) = sq0 (s).) Indeed, in this case it follows (±) (±) from (14.7) with γ1 = σ2 and γ2 = σ(W )\σ2 that if η is a positive function such that  

  |p(s)|2 |r(t)|2 (η(t))2  dρ(s)   dλ(t) < ∞, |s2 − t2 | (±)

σ2

(±)

σ(W )\σ2

then the operators (14.6) are Hilbert–Schmidt, provided |ψ| ≤ η.

546


(+) Let βn

(−) βn

,

n≥1

n≥1

, and {γn }n≥1 be increasing sequences of

compact sets such that (±) (±) βn(±) ⊂ σ2 , λ σ2 \βn(±) → 0,

γn ⊂ σ1 ,

ρ(σ1 \γn ) → 0, (14.9)

(+) (−) . Note that since ρ and λ are mutually where σ1 = σ(W )\ σ2 ∪ σ2 (±) (±) absolutely continuous on ∆, we have ρ σ2 \βn → 0. Consider first the case Ker Γ = {O}. In this case we have to choose q0 in Range W such that q0 L2 (ρ) = 1 and q0 (s) = 0, ρ-a.e. (+) (−) We can easily define a positive function q0 on β1 ∪ β1 ∪ γ1 such that

|q0 (s)|2 dρ(s) ≤ 1, s2 (+)

β1

(−)

∪β1

∪γ1

(−)

β1

and

(+)

(+)

t ∈ β1 ,

∪γ1

β1

|p(s)|2 dρ(s) ≤ 1, |s2 − t2 |2 |p(s)|2 dρ(s) ≤ 1, |s2 − t2 |2

(−)

t ∈ β1 .

∪γ1

Then we can proceed by induction. At the nth step we define q0 on (+) (−) βn(+) ∪ βn(−) ∪ γn \ βn−1 ∪ βn−1 ∪ γn−1 so that

(+)

βn

|q0 (s)|2 1 1 dρ(s) ≤1 + + · · · + n−1 , 2 s 2 2

(−)

∪βn

∪γ (n)

(−)

βn

|p(s)|2 1 1 dρ(s) ≤ k−1 + ... + n−1 , 2 2 2 |s − t | 2 2

∪γn (+)

(+)

t ∈ βk \βk−1 , and

(+) βn ∪γn

|p(s)|2 dρ(s) ≤ |s2 − t2 |2

(−)

(here

=

(−) β0

= γ0 = ∅).

1 ≤ k ≤ n,

1 1 + · · · + n−1 , 2k−1 2 (−)

t ∈ βk \βk−1 , (+) β0

(14.10)

1 ≤ k ≤ n,

(14.11)


547

Passing to the limit, we obtain a function q0 such that q0 (s) = 0, ρ-a.e., q0 ∈ Range W and (14.8) holds with p(s) = sq0 (s). The only condition that can be violated is q0 L2 (ρ) = 1 but obviously we can multiply q0 by a suitable constant to achieve this condition. Consider now the case Ker Γ = {O}. We have to construct a function q0 such that q0 L2 (ρ) = 1, q0 (s) = 0, ρ-a.e., q0 ∈ Range W , and (14.8) holds. In this case the operator W is noninvertible. Hence, ρ (0, ε) > 0 for any ε > 0. Therefore the same is true for the restriction of ρ to at least one of (+) (−) the sets σ2 , σ2 , σ1 . To be definite, suppose that (+) >0 ρ (0, ε) ∩ σ2 for any ε > 0. (+) Clearly, we can assume that 0 ∈ σ2 . It is easy to see that there exists (+) (+) a compact subset β1 of σ2 such that (+) >0 ρ (0, ε) ∩ β1 (+)

for any ε > 0. We can now choose a positive function q0 on β1

|q0 (s)|2 dρ(s) = ∞. s2

such that

(+)

β1

(±) Then we can choose increasing sequences βn

n≥1

and {γn }n≥1 of com-

pact sets satisfying (14.9) and proceed by induction as in the first case. (+) (−) At the nth step we can define q0 on βn ∪ βn ∪ γn so that (14.10) and (14.11) hold. Passing to the limit and multiplying, if necessary, by a suitable constant, we obtain a function q0 with desired properties. Proof of Lemma 13.5. Let p and η satisfy the assumptions of Lemma (+) (−) 13.5. Suppose that | sin ω1 (t)| ≤ η(t) for t ∈ σ2 ∪ σ2 . It is easy to see (+) (−) that 1 − cos ω1 (t) ≤ η(t), t ∈ σ2 ∪ σ2 . We have to show that the operators (±) (±) J 2 − I E σ(W )\σ2 are compact. Since

(±)

J2

(±) f =O − I E σ(W )\σ2

for f ∈ K0⊥ , it is sufficient to show that the operators (±) (±) P0 J 2 − I E0 σ(W )\σ2 are compact, whereP0 is the orthonormal projection onto K0 . Let us prove (+) (+) P0 is compact. The proof that the operator J 2 − I E σ(W )\σ2 for the other operator is the same. It is easy to see from the definition of

548


(+) (+) J 2 that for f ∈ K0 the vector J 2 − I f can be represented in the > Kj as decomposition K = j≥0



     (+) J2 −I f =     

U0∗ Mu U0 f Mv U0 f O O .. .

      ,    

where Mu and Mv are multiplications on L2 (λ) by the functions u and v defined by   cos ω1 (t) − 1, t ∈ σ (+) , 2 u(t) = (+)  0, t ∈ σ2 ,   sin ω1 (t), t ∈ σ (+) , 2 v(t) = (+)  0, t ∈ σ2 . (+) (+) and Mv U0 E0 σ(W )\σ2 are comBy Lemma 14.2, Mu U0 E0 σ(W )\σ2 (+) (+) pact operators, which proves that J 2 − I E0 σ(W ) \ σ2 P0 is also compact.

15. A Theorem in Perturbation Theory In this section we find a connection between results on Hankel operators obtained in previous sections and a theorem in perturbation theory. In §§9-14 for an operator Γ satisfying conditions (C1)–(C3) we have constructed an output normal system {T, p, q} such that the Hankel operator associated with this system is unitarily equivalent to Γ. If Ker Γ = {O}, then R = Γ and T is unitarily equivalent to backward shift S ∗ . In this case the vectors {(T ∗ )j q}j≥0 form an orthonormal basis in K and R has Hankel matrix {αj+k }j,k≥0 in this basis. It is easy to see that the operator Λ also has Hankel matrix (namely, {αj+k+1 }j,k≥0 ) in the same basis. So Λ must satisfy the same conditions (C1)–(C3). If Ker Γ = {O}, then it follows from the Remark following Theorem 9.2 that T is unitarily equivalent tothe restriction of S ∗ to (Ker Γ)⊥ . It is easy to see that in this case Λ = S ∗ Γ(Ker Γ)⊥ and the operator S ∗ Γ is a Hankel operator whose kernel is contained in Ker Γ. Therefore in both cases the spectral multiplicity function νΛ of the operator Λ must satisfy (C3).

15. A Theorem in Perturbation Theory

549

A natural question can arise if we look at our definition of Λ. The operator Λ was defined with the help of multiplication by the function J˜. It is easy to see from (12.4) that for t ∈ σ0 ∩ ∆ νΛ (t) − νΛ (−t) = 2.

(15.1)

Hence, if the restriction of the measure λ to the set σ0 ∩ ∆ contains a nonzero singular component, then (15.1) contradicts condition (C3). However, this can never happen: the singular components of the measures ρ and λ are mutually singular and so the restriction of λ to σ0 ∩ ∆ is absolutely continuous. This is a consequencce of the following Aronszajn–Donoghue theorem (see Reed and Simon [1], Sec. XIII.6). Theorem 15.1. Let K be a self-adjoint operator on Hilbert space and let L = K + (·, r)r, where r is a cyclic vector of K. Then the singular components of the spectral measures of L and R are mutually singular. Theorem 15.1 has an elementary proof. However, it is possible to deduce it from the results on Hankel operators obtained in this chapter. Proof of Theorem 15.1. Without loss of generality we can assume that both L and K are positive and invertible. Since K is cyclic, it is unitarily equivalent to multiplication by the independent variable on L2 (ρ), where ρ is a positive Borel measure with compact support in (0, ∞). Let Γ be the orthogonal sum of K 1/2 , −K 1/2 and the zero operator on an infinitedimensional space. Clearly, Γ satisfies (C1)–(C3). Let R = Γ(Ker Γ)⊥ and W = |R|. Then W is unitarily equivalent to K 1/2 ⊕ K 1/2 . We can now put p = r, q0 = W −1 x (in this case W is invertible). Without loss of generality we can assume that q0 L2 (ρ) = 1. We can now define the operators J , J , Λ as above. In our case σ(W ) = σ0 , and so the operator A∗ defined by (10.1) is asymptotically stable for any choice of a cyclic vector q0 of W . (Note that in this special case the proof of the asymptotic stability of A∗ is considerably simpler than in the general case.) Assume that the singular components of the measures ρ and λ are not mutually singular. In this case the restriction of λ to ∆ (see §12) has a nonzero singular component. It follows from (12.4) that νΛ (t) = 2,

νΛ (−t) = 0,

λs -a.e. on ∆.

However, as we have already observed, νΛ must satisfy (C3) and so |νΛ (t) − νΛ (−t)| ≤ 1, which contradicts (15.2).

λs -a.e.,

(15.2)

550


Concluding Remarks The results of this chapter were obtained in Megretskii, Peller, and Treil [2] (see also the announcement in Megretskii, Peller, and Treil [1]). The problem to identify nonnegative operators on Hilbert space that are unitarily equivalent to the moduli of Hankel operators was posed in Khrushchëv and Peller [1]. It was shown in Khrushchëv and Peller [1] that if {cn }n≥0 is a nonincreasing sequence of nonnegative numbers and ε > 0, then there exists a Hankel operator Γ such that (1 − ε)cn ≤ sn (Γ) ≤ (1 + ε)cn ,

n ≥ 0.

The next step toward the solution of the problem posed in Khrushchëv and Peller [1] was made in Treil [1] and [2]. He showed that if a nonnegative operator satisfies (C1) and (C2) and has simple discrete spectrum, then it is unitarily equivalent to the modulus of a Hankel operator. Later Vasyunin and Treil [1] proved that the same is true for nonnegative operators with discrete spectrum satisfying (C1) and (C2). In Ober [1] and [2] a new approach was offered. Namely, Ober obtained the results of Treil and Vasyunin [1] “modulo the kernel” using linear systems with continuous time. In Treil [7] Ober’s approach was developed to solve the problem on the moduli of Hankel operators completely. It turned out, however, that the problem of characterizing the self-adjoint operators unitarily equivalent to Hankel operators is considerably more complicated. This problem was solved in Megretskii, Peller, and Treil [2] and this solution is given in this chapter. Theorem 2.1 was proved in Megretskii, Peller, and Treil [2]. For compact self-adjoint Hankel operators inequality (2.1) was proved earlier in Peller [19] by another method. Note that a weaker inequality was obtained in Clark [1]. Another proof of (2.1) for compact self-adjoint Hankel operators was given in the preprint Helton and Woerdeman [1]. Adamyan informed the author that he had found another proof of (2.1) in the compact case but had not published the proof. In §8 solutions of the problem on the geometry of past and future and the inverse problem for rational approximation in BM OA are given. Both problems were posed in Khrushchëv and Peller [1]. One can pose the problem to characterize the normal operators that are unitarily equivalent to Hankel operators. However, this problem trivially reduces to the problem of characterizing the self-adjoint operators unitarily equivalent to Hankel operators. Indeed, if Γ is a normal Hankel operator, then there exists a complex number τ such that |τ | = 1 and the operator τ Γ is self-adjoint (see Nikol’skii [4], 5.6.5 (b)). Finally, we mention here the following result of Abakumov [1]. He showed that for an arbitrary finite set λ1 , · · · , λn of distinct nonzero complex numbers and for an arbitrary set k1 , · · · , kn of positive integers there exists a finite rank Hankel operator Γ such that its nonzero eigenvalues are precisely

Concluding Remarks

551

λ1 , · · · , λn and the algebraic multiplicity of λj is kj . Recall that the algebraic multiplicity of λ is, by definition, the dimension of Ker(λI − Γ)j . j≥1

13 Wiener–Hopf Factorizations and the Recovery Problem

In Chapter 7 we considered the recovery problem for unimodular functions. It is very important in applications to be able to solve the same problem for unitary-valued matrix functions. Namely, for a unitary-valued function U and a space X of functions on T we consider in this chapter the problem of under which natural assumptions we can conclude that P− U ∈ X

=⇒

U ∈ X.

Recall that for a matrix (or vector) function Φ we use the notation Φ ∈ X if all entries of Φ belong to X and P− Φ is obtained from Φ by applying P− to all entries. In this chapter we find a solution to this recovery problem which is similar to the one given in Chapter 7 for scalar functions. As in the scalar case we consider separately the case of R-spaces introduced in §7.1 and the the case of decent function spaces (i.e., spaces satisfying the axioms (A1)–(A4) stated in §7.2). The case of R-spaces is considered in §1. It turns out that the recovery problem for decent function spaces is closely related to the so-called heredity problem for Wiener–Hopf factorizations. Recall that in §3.5 it has been proved that if a matrix function Φ is the symbol of a Fredholm Toeplitz operator, then Φ admits a Wiener–Hopf factorization. We are going to prove in this section that if X is a decent function space and Φ ∈ X, then the factors in the Wiener–Hopf factorization of Φ belong to the same space X. In fact, we are going to solve the recovery problem and the heredity problem for Wiener–Hopf factorizations simultaneously. The solution will be based on properties of maximizing vectors of vectorial Hankel operators that will be discussed in §2.

554

Chapter 13. Wiener–Hopf Factorizations and the Recovery Problem

1. The Recovery Problem in R-spaces In this section we find a solution to the recovery problem for R-spaces that is similar to the one found in §7.1. Suppose that X is a Banach (quasiBanach) R-space and Φ is an m × n matrix function in X (recall that this means that all entries of Φ are in X). Then unless otherwise stated, by its norm (or quasinorm) ΦX we mean def

ΦX = sup{(Φx, y)X : x ∈ Cn , y ∈ Cm , x = y = 1}. Theorem 1.1. Let X be a linear R-space and let U be an n × n unitary-valued function such that P− U ∈ X. If the Toeplitz operator TU : H 2 (Cn ) → H 2 (Cn ) has dense range, then P+ U ∈ X. If in addition to that X is a (quasi-)Banach R-space, then P+ U X ≤ const P− U X . Proof. The result follows immediately from Theorem 4.4.11. Theorem 1.2. Let X be a linear R-space and let U be an n × n unitary-valued matrix function such that P− U ∈ X. Suppose that the Toeplitz operator TU : H 2 (Cn ) → H 2 (Cn ) is Fredholm. Then P+ U ∈ X. Proof. Since dim Ker TU ∗ < ∞, it follows that there exists m ∈ Z+ such that the restriction of TU ∗ to z m H 2 (Cn ) has trivial kernel. Hence, Ker Tzm U ∗ = {O}. Consequently, the Toeplitz operator Tz¯m U has dense range in H 2 (Cn ). It follows easily from Lemma 7.1.1 that P− z¯m U ∈ X. By Theorem 1.1, we may conclude that z¯m U ∈ X. It follows now from Lemma 7.1.1 that U ∈ X. Corollary 1.3. Let U be an n × n unitary-valued function. Suppose that either TU has dense range in H 2 (Cn ) or TU is a Fredholm operator on H 2 (Cn ). Then the following assertions hold: 1/p 1/p (1) if P− U ∈ Bp , 0 < p < ∞, then U ∈ Bp ; (2) if P− U ∈ V M O, then U ∈ V M O. 1/p

The result follows from the fact that V M O and Bp §7.1.

are R-spaces; see

Theorem 1.4. Let U be an n × n unitary-valued function such that TU has dense range in H 2 (Cn ). If P− U is a rational matrix function, then so is P− U and deg P+ U ≤ deg P− U. Recall that the McMillan degree deg Φ of an operator function Φ is defined in §2.5. Proof. The result follows immediately from Theorems 2.5.3 and 4.4.11.

2. Maximizing Vectors of Vectorial Hankel Operators

555

2. Maximizing Vectors of Vectorial Hankel Operators In this section we consider vectorial Hankel operators whose symbols belong to a decent function space (see §7.2). We study properties of their maximizing vectors and obtain results similar to the results of §7.2 in the scalar case. The proofs in the vector case are also similar to scalar proofs given in §7.2. The following theorems are the main results of this section. Theorem 2.1. Suppose that X is a function space satisfying the axioms (A1)–(A3). Let Φ ∈ X(Mm,n ) and let ϕ ∈ H 2 (Cn ) be a maximizing 2 vector of the Hankel operator HΦ : H 2 (Cn ) → H− (Cm ). Then ϕ ∈ X+ (Cn ). Theorem 2.2. Suppose that X, Φ, and ϕ satisfy the hypotheses of Theorem 2.1. If ϕ(τ ) = 0 for some τ ∈ T, then (1 − τ¯z)−1 ϕ ∈ X+ (Cn ) and (1 − τ¯z)−1 ϕ is also a maximizing vector of HΦ . Recall that X+ = {f ∈ X : fˆ(j) = 0, j < 0} and X− = {f ∈ X : fˆ(j) = 0, j ≥ 0}. Proof of Theorem 2.1. Without loss of generality we may assume that 2 the norm of the Hankel operator HΦ : H 2 (Cn ) → H− (Cm ) is equal to 1. ∗ 2 Consider the self-adjoint operator HΦ HΦ on H (Cn ). It follows from (A1) and (A2) that it maps X+ (Cn ) into itself. Let R be the restriction of HΦ∗ HΦ to X+ (Cn ). By (A3), R is a compact operator on X+ (Cn ). Clearly, X+ (Cn ) ⊂ H 2 (Cn ). We can imbed naturally the space H 2 (Cn ) ∗ in the dual space X+ (Cn ) as follows. Let g ∈ H 2 (Cn ). We associate with it the linear functional J (g) on X+ (Cn ) defined by

f → (f, g) = (f (ζ), g(ζ))Cn dm(ζ). T

Note that J (λ1 g1 + λ2 g2 ) = λ1 J (g1 ) + λ2 J (g2 ), g1 , g2 ∈ H 2 , λ1 , λ2 ∈ C. The imbedding J allows us to identify H 2 (Cn ) with a linear subset of ∗ X+ (Cn ). Since HΦ∗ HΦ is self-adjoint, it is easy to see that R∗ g = HΦ∗ HΦ g for g ∈ H 2 (Cn ). Hence, Ker(I − R) ⊂ Ker(I − HΦ∗ HΦ ) ⊂ Ker(I − R∗ ). Since R is a compact operator, it follows from the Riesz–Schauder theorem (see Yosida [1], Ch. X, §5) that dim Ker(I−R) = dim Ker(I−R∗ ). Therefore Ker(I − R) = Ker(I − HΦ∗ HΦ ) = Ker(I − R∗ ).

(2.1)

Clearly, the subspace Ker(I − HΦ∗ HΦ ) is the space of maximizing vectors of HΦ , and it follows from (2.1) that each maximizing vector of HΦ belongs to X+ . Proof of Theorem 2.2. Suppose now that ϕ is a maximizing vector of HΦ and ϕ(τ ) = 0. Consider the following continuous linear functional ω on

556


X+ (Cn ): ω(f ) = (P+ (ϕ∗ f ))(τ ).

(2.2)

∗

∞

Let us show that R ω = ω. Let F be a matrix function in H (Mm,n ) such def

that Φ − F ∞ = HΦ = 1. Put U = Φ − F . Then by Theorem 2.2.3, HΦ ϕ = U ϕ,

and U ∗ U ϕ = ϕ. (2.3)

(HΦ ϕ)(ζ)Cm = ϕ(ζ)Cn , ζ ∈ T,

We have (R∗ ω)(f ) = ω(Rf ) = ω(HU∗ HU f ) = (P+ (ϕ∗ HU∗ HU f ))(τ ) by (2.2). Therefore (R∗ ω)(f )

= (P+ (ϕ∗ P+ (U ∗ HU f )))(τ ) = (P+ (ϕ∗ U ∗ HU f ))(τ ) − (P+ (ϕ∗ P− (U ∗ HU f )))(τ ).

2 , it follows that P+ (ϕ∗ P− (U ∗ HU f )) = 0. Hence, Since ϕ∗ P− (U ∗ HU f ) ∈ H−

(R∗ ω)(f )

= (P+ (ϕ∗ U ∗ HU f ))(τ ) = (P+ ((P− U ϕ)∗ HU f ))(τ ) + (P+ ((P+ U ϕ)∗ HU f ))(τ ).

Clearly, P+ ((P+ U ϕ)∗ HU f ) = 0, and so (R∗ ω)(f )

= (P+ ((HU ϕ)∗ HU f ))(τ ) = (P+ ((HU ϕ)∗ P− (U f )))(τ ) = (P+ ((HU ϕ)∗ U f ))(τ ) − (P+ ((HU ϕ)∗ P+ (U f )))(τ ).

Since (HU ϕ)∗ P+ (U f ) ∈ H 2 , we have (P+ ((HU ϕ)∗ P+ (U f )))(τ ) = (HU ϕ)∗ (τ )(P+ (U f ))(τ ). By the hypotheses ϕ(τ ) = 0, and so by (2.3), (HU ϕ)(τ ) = 0. Hence, (R∗ ω)(f ) = (P+ ((HU ϕ)∗ U f ))(τ ). By (2.3), (R∗ ω)(f )

=

(P+ ((U ϕ)∗ U f ))(τ ) = (P+ ((U ϕ)∗ U f ))(τ )

=

(P+ (ϕ∗ U ∗ U f ))(τ ) = (P+ (ϕ∗ f ))(τ ) = ω(f ),

which proves (2.2). It follows from (2.2) that ω ∈ Ker(I − R∗ ), and by Theorem 2.1, ω ∈ Ker(I − R). Hence, there must be a function ψ ∈ X+ (Cn ) such that ψ is a maximizing vector of HΦ and

ω(f ) = (f (ζ), ψ(ζ))Cn dm(ζ), f ∈ X+ (Cn ). T

To prove that ψ = (1 − τ¯z)−1 ϕ, we show that these two functions have the same Taylor coefficients. Let k ∈ Z+ , γ ∈ Cn . We have ˆ (ψ(k), γ)Cn = ω(z k γ) = (P+ z k ϕ∗ γ)(τ ) =

k j=0

∗ γ, τ¯k−j (ϕ(j)) ˆ

3. Wiener–Masani Factorizations

557

and so ˆ ψ(k) =

k

τ¯k−j ϕ(j). ˆ

j=0

On the other hand,



(1 − τ¯z)−1 ϕ = 

k≥0

 τ¯k z k  ϕ =

k≥0

 zk 

k

 τ¯k−j ϕ(j) ˆ 

j=0

which completes the proof.

3. Wiener–Masani Factorizations We study in this section hereditary properties of Wiener–Masani factorizations of positive-definite matrix functions. Let W be a positive-definite matrix function in L1 (Mn,n ) such that log det W ∈ L1 . By the Wiener– Masani theorem, W admits a factorization of the form W = F ∗ F,

(3.1)

where F is an outer matrix function in H 2 (Mn,n ). Such factorizations are called Wiener–Masani factorizations. We do not need this result in full generality and we refer the reader to Rozanov [1] for the proof. We prove the following special case of the Wiener–Masani theorem. Theorem 3.1. Let W be a positive definite function in L∞ (Mn,n ) such that W −1 ∈ L∞ (Mn,n ). Then there exists an invertible matrix function F in H ∞ (Mn,n ) such that (3.1) holds. Proof. Consider the subspace W 1/2 H 2 (Cn ). It is easy to see that it is a completely nonreducing invariant invariant subspace of multiplication by z on L2 (Cn ) (see Appendix 2.3). Then W 1/2 admits a factorization in the form W 1/2 = U F , where F is an outer matrix function and U is an isometric-valued matrix function (see Appendix 2.3). Since W 1/2 is invertible in L∞ (Mn,n ), it follows that U is unitary-valued and F is invertible in H ∞ (Mn,n ). We have W = (U F )∗ U F = F ∗ F. We consider the case when W is a positive definite matrix function in X such that W (ζ) is invertible for all ζ in T, where X is a decent space of functions on T. We prove that such a matrix function W admits a Wiener– Masani factorization of the form (3.1) such that F belongs to the same space X. First we establish the uniqueness of a Wiener–Masani factorization modulo a constant unitary factor.

558


Theorem 3.2. Let W be a matrix function in L1 (Mn,n ) such that W = F ∗ F = G∗ G, where F and G are outer functions in H 2 (Mn,n ). Then there exists a constant unitary matrix U such that G = UF . Proof. Put U = W 1/2 F −1 . Then U is unitary-valued. Indeed, U ∗ U = (F ∗ )−1 W F −1 = (F ∗ )−1 F ∗ F F −1 = I n , where I n is the constant matrix function identically equal to the n × n identity matrix In . Similarly, if V = W 1/2 G−1 , then V is unitary-valued. We have W 1/2 = U F = V G. Since F is outer, it follows that clos{W 1/2 q : q is a polynomial in H 2 (Cn )} = U H 2 (Cn ). Similarly, since G is outer, we have clos{W 1/2 q : q is a polynomial in H 2 (Cn )} = V H 2 (Cn ). Hence, the invariant subspaces U H 2 (Cn ) and V H 2 (Cn ) of multiplication by z on L2 (Cn ) coincide. Therefore there exists a unitary matrix Q such that V = U Q (see Appendix 2.3). We have G = V ∗ W 1/2 = Q∗ U ∗ W 1/2 = Q∗ F, which proves the result. Theorem 3.3. Let X be a decent function space and let W be a positive definite n × n matrix function in X such that W (ζ) is invertible for any ζ ∈ T. Then W admits a Wiener–Masani factorization of the form (3.1) with F invertible in X+ (Mn,n ). It is easy to see that if F is a function invertible in X+ (Mn,n ), then F is outer. To prove Theorem 3.3, we need a lemma. Note that if Φ is an n × n matrix function in X, then the Toeplitz operator TΦ maps X+ (Cn ) into itself and can be considered as a bounded operator on X+ (Cn ). Lemma 3.4. Let W be a positive-definite n × n matrix function in X such that W (ζ) is invertible for any ζ ∈ T. Then the Toeplitz operator TW is an invertible operator on X+ (Cn ). Proof of Lemma 3.4. We have ∗ TW TW −1 = I − HW HW −1 ,

(3.2)

2 (Cn ) is a Hankel operator while where HW −1 : H 2 (Cn ) → H− ∗ 2 n 2 n HW : H− (C ) → H (C ) is the adjoint to the Hankel operator HW . It ∗ is easy to see that the operator HW HW −1 maps X+ (Cn ) into itself and we can consider it as a bounded operator on X+ (Cn ). In fact, it is a compact

3. Wiener–Masani Factorizations

559

operator on X+ (Cn ) since by the axiom (A3), the Hankel operator HW −1 is a compact operator from X+ (Cn ) to X− (Cn ). Thus in (3.2) we may assume that both sides of the equation are operators on X+ (Cn ). Let us show that TW TW −1 is an invertible operator on X+ (Cn ). ∗ HW −1 is compact, it suffices to show that Ker TW TW −1 = {O}. Since HW This will be proved if we show that Ker TW = Ker TW −1 = {O}. Let us show that Ker TW = {O}, the proof of the fact that Ker TW −1 = {O} is the same. Suppose that f ∈ Ker TW . We have 0

= =

(TW f, f )L2 (Cn ) = (P+ W f, f )L2 (Cn ) = (W f, P+ f )L2 (Cn )

(W f, f )L2 (Cn ) = (W (ζ)f (ζ), f (ζ))Cn dm(ζ). T

Since W (ζ) is positive definite and invertible, it follows that f = O. It follows now from the invertibility of TW TW −1 that TW maps X+ (Cn ) onto itself. A similar argument shows that ∗ TW −1 TW = I − HW −1 HW

is an invertible operator on XA (Cn ), which implies that the operator TW on X+ (Cn ) has trivial kernel. Proof of Theorem 3.3. Since TW is invertible as an operator on X+ (Cn ), there exist functions gj ∈ X+ (Cn ), 1 ≤ j ≤ n, such that   O  ..   .     TW gj =   1   .   ..  O (1 is in the jth row). Consider the n × n matrix function G = g 1 · · · gn . Clearly, (P+ W G)(ζ) = I,

a.e. on T.

It follows that W G ∈ H . Put Ξ = G∗ W G. Then Ξ ∈ H 1 since both G∗ and W G are in H 2 . Clearly, Ξ∗ = Ξ, and so Ξ is a constant function. Let us prove that det Ξ = 0. We have det Ξ = | det G(ζ)|2 det W (ζ) for ζ ∈ T. Since the columns g1 (ζ), · · · , gn (ζ) of G(ζ) are linearly independent on T, it follows that det G(ζ) = 0 on T. Clearly, det W (z) = 0, ζ ∈ T. Hence, det Ξ = 0, and so Ξ is invertible. def Put F = Ξ1/2 G−1 . Clearly, F, F −1 ∈ X+ . We have 2

F ∗ F = G∗−1 Ξ1/2 Ξ1/2 G−1 = W. The last equality is an immediate consequence of the definition of Ξ.

560


Corollary 3.5. Suppose that W satisfies the hypotheses of Theorem 3.3. If G is an outer n × n matrix function in H 2 such that G∗ G = W , then G, G−1 ∈ X. Proof. The result follows immediately from Theorems 3.3 and 3.2. Corollary 3.6. Suppose that W satisfies the hypotheses of Theorem 3.3. If Ω is an outer n × n matrix function in H 2 such that ΩΩ∗ = W , then Ω, Ω−1 ∈ X. Proof. Consider the transposed matrices: W t = (Ωt )∗ Ωt . Then Ωt is also an outer function since a square matrix function in H 2 is outer if and only if its determinant is a scalar outer function (Appendix 2.3). The result follows now from Corollary 3.5.

4. Isometric-Outer Factorizations In this section we consider representations of matrix functions as a product of an isometric-valued function and an outer function. Suppose that Φ is an n × m matrix function in L2 such that det Φ∗ Φ ∈ L1 . Then it follows from the Wiener–Masani theorem mentioned in the previous section that Φ admits a factorization Φ = U F , where F is an m×m outer matrix function in H 2 and U is an n × m isometric-valued function, i.e., for almost all ζ ∈ T U (ζ)xCm = xCn for any x ∈ Cn . Indeed, if Φ∗ Φ = F ∗ F is a Wiener–Masani factorization of Φ∗ Φ, then it is easy to see that the matrix function U = ΦF −1 is isometric-valued and Φ = U F . Such an isometric-outer factorization is unique modulo a multiplicative unitary constant, which follows from the corresponding uniqueness result for Wiener–Masani factorizations. We consider here matrix functions whose entries belong to a decent function space X and we prove that in this case the factors U and F also belong to X. Theorem 4.1. Let X be a decent function space and let Φ be an n×m matrix function in X such that rank Φ(ζ) = m for all ζ ∈ T. Then Φ admits a factorization Φ = U F, where U is an n × m isometric-valued function in X and F is an m × m outer function such that F, F −1 ∈ X. def

Proof. Let W = Φ∗ Φ. Then W is positive definite and W is invertible in X. By Theorem 3.3, W admits a factorization W = F ∗ F, where F is an invertible function in X. Put U = ΦF −1 . Clearly, U ∈ X. We have U ∗ U = F ∗−1 Φ∗ ΦF −1 = F ∗−1 F ∗ F F −1 ≡ I,

5. The Recovery Problem and Factorizations of Unitary Functions

561

and so U is isometric-valued.

5. The Recovery Problem and Wiener–Hopf Factorizations of Unitary-Valued Functions In this section we consider matrix functions that belong to a decent function space X. We use the results of §2 to solve two problems: the Wiener–Hopf factorization problem for unitary-valued functions and the recovery problem for unitary-valued functions. As in §1 we solve the recovery problem for a unitary-valued unction U under the assumption that the Toeplitz operator TU : H 2 (Cn ) → H 2 (Cn ) is Fredholm or under the assumption that TU has a dense range. Theorem 5.1. Suppose that X is a decent function space. Let U be a unitary-valued matrix function such that the Toeplitz operator TU has dense range in H 2 (Cn ). If P− U ∈ X, then U ∈ X. Theorem 5.2. Suppose that X is a decent function space. Let U be a unitary-valued matrix function such that the Toeplitz operator TU is Fredholm on H 2 (Cn ). If P− U ∈ X, then U ∈ X. If TU is Fredholm, then Tz¯N U maps H 2 (Cn )) onto itself for sufficiently large N . Therefore it is sufficient to prove Theorem 5.1. Let us also show that under the hypotheses of Theorem 5.1 the Toeplitz operator TU : H 2 (Cn ) → H 2 (Cn ) is Fredholm. Since X ⊂ C(T), it follows that P− U ∈ V M O. By Corollary 1.3, U ∈ V M O. Therefore the Hankel operators HU and HU ∗ are compact, and so the fact that TU is Fredholm follows from the formulas I − TU ∗ TU = HU∗ HU ,

I − TU TU ∗ = HU∗ ∗ HU ∗

Now let U be a unitary-valued function such that TU is Fredholm. Then by Theorem 3.5.2, U admits a factorization U = Ψ∗2 ΛΨ1 ,

(5.1)

±1 2 n where Ψ±1 1 , Ψ2 ∈ H (C ) and Λ is a diagonal matrix of the form   d1 z O ··· O  O z d2 · · · O    Λ= . .. ..  , d1 , · · · , dn ∈ Z. ..  .. . . .  dn O O ··· z

It is easy to see that Theorems 5.1 and 5.2 follow immediately from the following theorem, which solves the Wiener–Hopf factorization problem in X for unitary-valued functions. In fact, we obtain an even stronger result since instead of assuming that U ∈ X we assume only that P− U ∈ X.

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Theorem 5.3. Suppose that X is a decent function space. Let U be a unitary-valued function such that P− U ∈ X and the Toeplitz operator TU −1 on H 2 (Cn ) is Fredholm. Then the functions Ψ1 , Ψ−1 1 , Ψ2 , and Ψ2 in (5.1) belong to X. ∈ X. Without loss Proof. Let us first prove that Ψ1 ∈ X and Ψ−1 1 of generality we may assume that all the indices of the factorization (5.1) are negative. Otherwise, we can multiply U by z¯N for a sufficiently large integer N . Let ξ be an arbitrary nonzero vector in Cn . It is easy to see that Ψ−1 1 ξ is a maximizing vector for HU . It follows now from Theorem 2.1 that −1 Ψ−1 1 ξ ∈ X, and so Ψ1 ∈ X. Let us show that Ψ1 ∈ X. Put G = Ψ−1 ∈ X. It is sufficient to show that the matrix G(ζ) is invertible for all ζ ∈ T. Consider Ker TU . It admits the following description:     q1      ..  2 Ker TU = G  .  : qj is a polynomial in H , deg qj < |dj | .     (5.2) qn Indeed, it is easy to see that the right-hand side of (5.2) is contained in the left-hand side and by Theorem 3.5.7, the dimensions of both sides are equal. Assume that G(ζ0 ) is noninvertible for some ζ0 ∈ T. Then G(ζ0 )ξ = O for some nonzero ξ ∈ Cn . It is easy to see from (5.2) that Gξ is a nonzero function in Ker TU . Clearly, a nonzero vector in H 2 (Cn ) is maximizing for HU if and only if it belongs to Ker TU . By Theorem 2.2, the function (1−ζ0 z)−1 Ge is a maximizing vector of HU , and so it belongs to Ker TU . However, it is easy to see that (1 − ζ0 z)−1 Ge does not belong to the right-hand side of (5.2). Let us show that Ψ2 , Ψ−1 2 ∈ X. Since U is unitary-valued, it follows that (U ∗ U )(ζ) = (Ψ∗1 Λ∗ Ψ2 Ψ∗2 ΛΨ1 )(ζ) = I,

ζ ∈ T,

and so ∗ Ψ2 Ψ∗2 = ΛΨ∗−1 Ψ−1 1 1 Λ .

(5.3)

Since Ψ2 is an outer function and the right-hand side of (5.3) is invertible in X, the result follows from Corollary 3.6.

6. Wiener–Hopf Factorizations. The General Case We consider in this section Wiener–Hopf factorizations of matrix functions in decent function spaces X. If Φ is an n × n matrix function in X such that det Φ(ζ) = 0 for any ζ in T, then the Toeplitz operator TΦ on

Concluding Remarks

563

H 2 (Cn ) is Fredholm, and so Φ admits a Wiener–Hopf factorization Φ = Ψ∗2 ΛΨ1 , where form

Ψ1 , Ψ−1 1 ,

Ψ2 , Ψ−1 2 

(6.1)

∈ H and Λ is a diagonal matrix function of the  O ··· O z d2 · · · O   .. ..  , d1 , · · · , dn ∈ Z. .. . . .  2

z d1  O  Λ= .  .. O O · · · z dn We are going to prove in this section that for Φ ∈ X the matrix functions Ψ1 and Ψ2 are invertible functions in X. Theorem 6.1. Let X be a decent function space and let Φ be an n × n matrix function in X such that det Φ(ζ) = 0 for any ζ in T. Suppose that −1 (6.1) is a Wiener–Hopf factorization of Φ. Then Ψ1 , Ψ−1 1 , Ψ2 , Ψ2 ∈ X. Proof. By Theorem 4.1, Φ admits a representation Φ = U Ψ, where U is a unitary-valued function in X and Ψ is an outer function that is invertible in X. Suppose that Φ admits a factorization in the form (6.1). We have U = Ψ∗2 ΛΨ1 Ψ−1 . It is easy to see that Ψ1 Ψ−1 ∈ H 2 and (Ψ1 Ψ−1 )−1 ∈ H 2 . By Theorem 5.3, −1 , (Ψ1 Ψ−1 )−1 ∈ X, and so Ψ1 , Ψ−1 Ψ2 , Ψ−1 2 ∈ X and Ψ1 Ψ 1 ∈ X.

Concluding Remarks The results of §1 were obtained in Peller [24]. Theorems 2.1 and 2.2 were proved in Peller and Young [5] under the assumption that the trigonometric polynomials are dense in X. In Alexeev and Peller [2] it was shown how to generalize the proof to the case of arbitrary decent spaces. The results of §§3–4 are due to Peller [24]. Theorems 5.1 and 5.2 were also found in Peller [24]. The heredity problem for Wiener–Hopf factorizatiions was studied by many authors. Classical results by Plemelj, Muskhelishvili, and Vekua solved the Wiener–Hopf factorization problem in H¨ older classes Λα , 0 < α < 1 (see Muskhelishvili [1] and Vekua [1]). In Gohberg and Krein [1] the Wiener– Hopf factorization problem was solved in the class F1 . In Gohberg [2], and Budyanu and Gohberg [1–2] heredity results for Wiener–Hopf factorizations were obtained for a class of function spaces that is close to the class of decent spaces (see also Clancey and Gohberg [1]). In Peller [24] a new method to obtain heredity results for Wiener–Hopf factorizations was found. It is based on properties of maximizing vectors of Hankel operators. Theorem 5.3 was proved in Peller [24].

14 Analytic Approximation of Matrix Functions

We study in this chapter the problem of approximating an essentially bounded matrix function on T by bounded analytic matrix functions in D. For such a matrix function Φ ∈ L∞ (Mm,n ) its L∞ norm ΦL∞ is, by definition, ΦL∞ = ess sup Φ(ζ)Mm,n . ζ∈T

As usual we denote by H ∞ (Mm,n ) the subspace of L∞ (Mm,n ) that consists of bounded analytic matrix functions in D. Recall that the space Mm,n of m × n matrices is equipped with the operator norm of the space of linear operators from Cn to Cm . We have seen in §1.1 that for a scalar continuous function ϕ on T there exists a unique best approximation f by bounded analytic functions. In this chapter we study the problem finding best approximations for matrix functions. As we have seen in Chapter 11, this problem is very important in applications. In this chapter we call best approximations of Φ by functions F in H ∞ (Mm,n ) optimal solutions of the Nehari problem. Note that this terminology slightly differs from the terminology used in Chapter 5, where Φ − F has been called an optimal solution of the Nehari problem. It is easy to see that unlike the scalar case the continuity (and even any smoothness) of a matrix function Φ does not guarantee the uniqueness of z¯ O an optimal solution of the Nehari problem. Indeed, if Φ = , O O z , H ∞ ) = 1. Clearly, for any then distL∞ (Φ, H ∞ (M2,2 )) = 1, since distL∞ (¯

566

Chapter 14. Analytic Approximation of Matrix Functions

scalar function f ∈ H ∞ such that f ∞ ≤ 1 we have z¯ O O −f ∞ = 1, L O O and so is a best approximation of Φ. However, if we were asked O f to find among the above best approximations the “very best” one, we would probably say that it is the zero matrix function, since it does a better job than just minimizing the L∞ -norm. It seems natural to impose additional constraints on a best approximation and choose among best approximations the “very best”. To do this, we minimize lexicographically the essential suprema of the singular values sj (Φ − F )(ζ) , 0 ≤ j ≤ min{m, n} − 1, of (Φ − F )(ζ), ζ ∈ T, Given a matrix function Φ ∈ L∞ (Mm,n ) we define inductively the sets Ωj , 0 ≤ j ≤ min{m, n} − 1, by ( Ω0 =

F ∈ H ∞ (Mm,n ) : F minimizes ess sup Φ(ζ) − F (ζ)Mm,n

,

ζ∈T

Ωj =

F ∈ Ωj−1

( : F minimizes ess sup sj Φ(ζ) − F (ζ) .

ζ∈T

We put

def tj = ess sup sj Φ(ζ) − F (ζ)

for F ∈ Ωj ,

0 ≤ j ≤ min{m, n} − 1.

ζ∈T

Functions in Ωmin{m,n}−1 are called superoptimal approximations of Φ by bounded analytic matrix functions. The numbers tj = tj (Φ) are called the superoptimal singular values of Φ. Note that the functions in Ω0 are just the best approximations by analytic matrix functions. Clearly, t0 (Φ) = HΦ . It is easy to see that in the above example the zero function is the unique superoptimal approximation. The notion of a superoptimal approximation is important in control theory. In the robust stabilization problem one seeks a controller that stabilizes all functions in an L∞ -norm ball of maximal radius about a nominal transfer function, and this reduces mathematically to a Nehari problem (see Chapter 11). If we choose the superoptimal solution, then we optimize robustness with respect not only to the worst direction of uncertainty (in the space of transfer functions), but also to other directions. We refer the reader to the papers Postlethwaite, Tsai, and Gu [1], Limebeer, Halikias, and Glover [1], Kwakernaak [1], and Foo and Postlethwaite [1] written by engineers for further discussions of superoptimal approximation in connection with control theory. It can be shown easily by a compactness argument that for an arbitrary Φ ∈ L∞ (Mm,n ) there exists a superoptimal approximation by bounded analytic matrix functions. As in the scalar case a superoptimal approximation


567

is not unique in general. However, we will see in this chapter that under the condition Φ ∈ H ∞ + C (or even under a certain weaker condition) the superoptimal approximation is unique. It is easy to see that if Φ is a continuous diagonal n × n matrix function of the form   ϕ1 O · · · O  O ϕ2 · · · O    Φ= . .. ..  , ..  .. . . .  O O · · · ϕn and fj ∈ H ∞ is the best approximation of ϕj , then   f1 O · · · O  O f2 · · · O    F = . .. . . ..   .. . . .  O

O

···

fn

is the unique superoptimal approximation of Φ by bounded analytic matrix functions. In the general case the situation is much more complicated, but we are going to perform a sort of diagonalization too. In §1 we introduce a remarkable class of so-called balanced unitary-valued matrix functions and its subclass of thematic matrix functions, and we study their properties. In §2 we begin the diagonalization procedure and parametrize the optimal solutions of the Nehari problem under the condition HΦ e < HΦ . We prove the uniqueness of a superoptimal approximation in §3 under the condition Φ ∈ H ∞ + C. In §4 we apply another method that allows us to prove uniqueness under the weaker condition on Φ: HΦ e is less than any nonzero superoptimal singular value of Φ. Note that both methods give a construction of the unique superoptimal approximation. Section 5 is devoted to the construction of so-called thematic and partially thematic factorizations. We define thematic indices associated with such factorizations. We introduce the notion of a very badly approximable matrix function and under suitable conditions on the essential norm of the Hankel operator we characterize the badly approximable matrix functions and the very badly approximable matrix functions in terms of such factorizations. In §6 we study so-called admissible and superoptimal matrix weights. They are used in §7 to study thematic indices, which may depend on the choice of a (partial) thematic factorization. However, we show in §7 that the sum of the thematic indices that correspond to the superoptimal singular values equal to a specific number depends only on the matrix function Φ itself rather than on the choice of a factorization. We continue the study of invariance properties of (partial) thematic factorization in Sections 9 and 10. In §9 we prove the invariance of so-called residual entries, while in §10 we introduce monotone (partial) thematic factorizations, and prove

568


that among (partial) thematic factorizations it is always possible to choose a monotone one and the indices of a monotone factorization are uniquely determined by the function Φ itself. Section 8 concerns inequalities that involve superoptimal singular values. In particular, we prove an important inequality between the superoptimal singular values tj (Φ) and the Hankel singular values sj (HΦ ). In §11 we suggest a more constructive approach to find the unique superoptimal approximation. It is given in terms of solutions of corona problems. It is used in §12 to study the hereditary properties of the nonlinear operator of superoptimal approximation as well as in §13 to study continuity properties of this operator. In §14 we study the problem of finding so-called unitary interpolants U of Φ. We are interested in those unitary interpolants U for which the Toeplitz operator TU is Fredholm, in which case we study the indices of Wiener–Hopf factorizations of U . We give a solution to this problem in terms of the superoptimal singular values of Φ and the indices of monotone partial thematic factorizations. Section 15 gives an alternative approach to the construction of superoptimal approximation which is based on so-called canonical factorizations. Unlike thematic factorizations, all factors in canonical factorizations are uniquely determined by Φ modulo constant unitary factors. A special role in that section is played by the very badly approximable unitary-valued functions U such that HU e < 1. In §16 we give a characterization of such unitary-valued functions. In §17 we study the problem of superoptimal approximation by meromorphic matrix functions whose McMillan degree is at most k. Unlike the case of analytic approximation the condition Φ ∈ H ∞ + C does not guarantee that a superoptimal approximation is unique. We prove uniqueness under the additional assumption sk−1 (HΦ ) > sk (HΦ ). In §18 we study the problem of superoptimal analytic approximation of infinite matrix functions. We establish uniqueness under the condition that HΦ is compact. We also prove analogs of many results obtained in previous sections for finite matrix functions. Finally, in §19 we revisit the problem of parametrization of all optimal solutions of the Nehari problem under the condition HΨ e < HΨ and we obtain the Adamyan–Arov–Krein parametrization formula using the method developed in §2.

1. Balanced Matrix Functions We consider here a very important class of matrix functions that will play an essential role in studying approximation problem for matrix functions.

1. Balanced Matrix Functions

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Definition. Let n be a positive integer and let r be an integer such that 0 < r < n. Suppose that Υ is an n × r inner and co-outer matrix function and Θ is an n × (n − r) inner and co-outer matrix function (see Appendix 2.3). If the matrix function V= Υ Θ is unitary-valued, it is called an r-balanced matrix function. If r = 0 or r = n, it is natural to say that an r-balanced matrix function is a constant unitary matrix function. An n × n matrix function V is called balanced if it is r-balanced for some r, 0 ≤ r ≤ n. A special role will be played by 1-balanced matrix functions. We also call them thematic matrix functions. We will see that balanced matrix functions have many nice properties that can justify the term “balanced”. First we show that any inner and co-outer matrix function can be completed to a balanced matrix function. Theorem 1.1. Let n be a positive integer, 0 < r < n, and let Υ be an def n × r inner and co-outer matrix function. Then the subspace L = Ker TΥt has the form L = ΘH 2 (Cn−r ), where Θ is an inner and co-outer n × (n − r) matrix function such that Υ Θ is balanced. Proof. Clearly, the subspace L of H 2 (Cn ) is invariant under multiplication by z. By the Beurling–Lax–Halmos theorem (see Appendix 2.3), L = ΘH 2 (Cl ) for some l ≤ n and an n × l inner matrix function Θ. Let us first prove that Θ is co-outer. Suppose that Θt = OF , where O is an inner matrix function and F is an outer matrix function. Since Θ is inner, it is easy to see that O has size l × l while F has size l × n. It follows that O∗ Θt = F , and so F t = ΘO. Since both Θ and O take isometric values almost everywhere on T, the matrix function F t is inner. Let us show that F t H 2 (Cl ) ⊂ L.

(1.1)

t

First of all, it is easy to see that Υ Θ is the zero matrix function. Now let f ∈ H 2 (Cl ). We have Υt F t f = Υt ΘOf = O, which proves (1.1). It follows from (1.1) that F t H 2 (Cl ) = ΘOH 2 (Cl ) ⊂ ΘH 2 (Cl ). Multiplying the last inclusion by Θ∗ , we have OH 2 (Cl ) ⊂ H 2 (Cl ), which implies that O is a constant unitary matrix function. Let us now prove that l = n − r. First of all, it is evident that the columns of Υ(ζ) are orthogonal to the columns of Θ(ζ) almost everywhere

570


on T. Hence, the matrix function Υ Θ takes isometric values almost everywhere, and so l ≤ n − r. To show that l ≥ n − r, consider the functions PL C, where PL is the orthogonal projection onto L and C is a constant function that we identify with a vector in Cn . Note that C ⊥ L if and only if the vectors f (0) and C are orthogonal in Cn for any f ∈ L. Let us prove that dim{f (0) : f ∈ L} ≥ n − r.

(1.2)

Since Υ is co-outer, it is easy to see that rank Υ(0) = r. Without loss of Υ1 generality we may assume that Υ = , where Υ1 has size (n − r) × r, Υ2 Υ2 has size r × r, and the matrix Υ2 (0) is invertible. Now let K be an arbitrary vector in Cn−r . Put K . f = (det Υ2 (0))−1 det Υ2 −(Υt2 )−1 Υt1 K Clearly, f ∈ H 2 (Cn ). We have Υt f = Υt1 Υt2 f = (det Υ2 (0))−1 det Υ2 (Υt1 K − Υt2 (Υt2 )−1 Υt1 K) = 0, and so f ∈ L. On the other hand, it is easy to see that K f (0) = −(Υt2 (0))−1 Υt1 (0)K and since K is an arbitrary vector in Cn−r , this proves (1.2). We have already observed that {f (0) : f ∈ L} = Cn {C ∈ Cn : PL C = O}, and so it follows from (1.2) that dim{PL C : C ∈ Cn } ≥ n − r. It is easy to see that for C ∈ Cn we have PL C = ΘP+ Θ∗ C = Θ(Θ∗ (0))C. Clearly, Θ(Θ∗ (0))C belongs to the linear span of the columns of Θ. This completes the proof of the fact that l = n − r and proves that Υ Θ is a balanced matrix function. Let us now show that a balanced completion is unique modulo a constant unitary factor. Theorem 1.2. Let Υ Θ1 and Υ Θ2 be r-balanced n × n matrix functions. Then there exists an (n − r) × (n − r) unitary matrix O such that Θ2 = Θ1 O. Proof. Let L = Ker TΥt . It is easy to see that Θ2 H 2 (Cn−r ) ⊂ L = Θ1 H 2 (Cn−r ). It follows (see Appendix 2.3) that Θ2 = Θ1 O for an inner matrix function O. Clearly, O has size (n − r) × (n − r). Hence, Θt2 = Ot Θt1 , Ot is inner, and since Θ1 is co-outer, it follows that O is a unitary constant.


571

The following theorem establishes a very important property of balanced matrix functions. Theorem 1.3. Let V be an n × n balanced matrix function. Then the Toeplitz operator TV on H 2 (Cn ) has dense range and trivial kernel. We need the following lemma. Lemma 1.4. Let G be a co-outer matrix function in H ∞ (Mm,n ) and let f be a function in L2 (Cn ) such that Gf ∈ H 2 (Cm ). Then f ∈ H 2 (Cn ). Proof of Lemma 1.4. Since Gt is an outer function, there exists a sequence of analytic polynomial matrix functions {Qj }j≥0 of size m × n such that {Gt Qj }j≥0 converges in H 2 (Mn ) to I n , where I n is the function identically equal to the identity matrix of size n × n. Then the sequence {Qtj Gf }j≥0 of H 2 (Cn ) functions converges to f in L2 (Cn ), and so f ∈ H 2 (Cn ). Proof of Theorem 1.3. Let V = Υ Θ , where Υ is an n × r inner and co-outer matrix function and Θ is an n × (n − r) inner and cooutermatrix function. Let us first show that Ker TV = {O}. Suppose that f1 f= ∈ Ker TV , where f1 ∈ H 2 (Cr ) and f2 ∈ H 2 (Cn−r ). We have f2 f1 2 Vf = Υ Θ = Υf1 + Θf2 ∈ H− (Cn ). (1.3) f2 Since V is unitary-valued, it follows from (1.3) that 2 Υ∗ (Υf1 + Θf2 ) = f1 ∈ H− (Cr ), 2 which implies that f1 = O and so Θf2 ∈ H− (Cn ). Hence, Θ¯ z f¯2 ∈ H 2 (Cn ). 2 n−r 2 ), which means that f2 ∈ H− (Cn−r ), and By Lemma 1.4, z¯f¯2 ∈ H (C so f2 = O. Let us now prove that Ker TV ∗ = {O}. Let f ∈ Ker TV ∗ . We have ∗ ∗ Υ Υ f ∗ 2 (Cn ). (1.4) V f= ∈ H− f= Θt Θt f 2 Hence, Θt f ∈ H− (Cn−r ). Since both Θt and f are analytic, it follows that t Θ f = O. Therefore the vector f (ζ) is orthogonal in Cn to the columns of Θ(ζ) for almost all ζ ∈ T. Since V is unitary-valued, we have f = Υχ, where χ = Υ∗ f ∈ L2 (Cr ). By Lemma 1.4, χ ∈ H 2 (Cr ). On the other hand, 2 (Cr ), and so χ = O. it follows from (1.4) that Υ∗ Υχ = χ ∈ H− Corollary 1.5. Let V be an n × n balanced matrix function. Then the Toeplitz operator TV on H 2 (Cn ) has dense range and trivial kernel. Proof. To deduce Corollary 1.5 from Theorem 1.3, it is sufficient to rearrange the columns of V . We proceed now to the study of theproperty of analyticity of minors of balanced matrix functions. Let V = Υ Θ be an r-balanced n × n matrix function. We are going to study its minors Vı1 ···ık ,1 ···k of order k, i.e., the determinants of the submatrix of V with rows ı1 , · · · , ık and

572


columns 1 , · · · , k . Here 1 ≤ ı1 < · · · < ık ≤ n and 1 ≤ 1 < · · · < k ≤ n. By a minor of V on the first r columns we mean a minor Vı1 ···ık ,1 ···k with k ≥ r and 1 = 1, · · · , r = r. Similarly, by a minor of V on the last n − r columns we mean a minor Vı1 ···ık ,1 ···k with k ≥ n − r and k−n+r+1 = r + 1, · · · , k = n. Theorem 1.6. Let V be an r-balanced matrix function of size n × n. Then all minors of V on the first r columns are in H ∞ while all minors of V on the last n − r columns are in H ∞ . To prove Theorem 1.6, we are going to use the language of wedge prodn n ucts. If 0 ≤ k ≤ n, we consider the vector space C ∧ · · · ∧ Cn, that - ∧ C .+ k

consists of wedge products of the form C1 ∧ · · · ∧ Ck with C1 , · · · , Ck ∈ Cn . In fact, one can identify the space of such wedge products with the space n of minors of order k of n × k matrices C1 C2 · · · Ck . If k ψ1 , · · · , ψk are vector functions in L∞ (Cn ), we can consider the wedge n n product ψ1 ∧ · · · ∧ ψk as a function with values in C ∧ · · · ∧ Cn,. Then - ∧ C .+ k

n n ∧ · · · ∧ Cn,) if and only if all minors of order ψ1 ∧ · · · ∧ ψk is in H ∞ (C - ∧ C .+

k k of the matrix function ψ1 · · · ψk are in H ∞ . We refer the reader to Gantmakher [1] for more information on wedge products.

Proof. It is easy to see that it is sufficient to prove the theorem for the minors on the first r columns of V . To deduce the second assertion of the theorem, we can consider the matrix function V and rearrange its columns to make it (n − r)-balanced. Denote by Υ1 , · · · , Υr and Θ1 , · · · , Θn−r the columns of Υ and Θ. In the proof of Theorem 1.1 we have observed that for any constant C ∈ Cn we have PL C = ΘΘ∗ (0)C and dim{PL C : C ∈ Cn } = dim{Θ∗ (0)C : C ∈ Cn } = n − r. It follows that there exist C1 , · · · , Cn−r ∈ Cn such that Θj = PL Cj ,

1 ≤ j ≤ n − r.

(1.5)

If 1 ≤ d ≤ n − r and 1 ≤ j1 < j2 < · · · < jd ≤ n − r, we consider the vector function Υ1 ∧ · · · ∧ Υr ∧ Θj1 ∧ · · · ∧ Θjd n whose components are the minors of order r + d of the matrix r+d function Υ1 · · · Υr Θj1 · · · Θjd . It follows from (1.5) that Θj = Cj − PL⊥ Cj ,


573

where PL⊥ is the orthogonal projection onto L⊥ = clos Range TΥ . We have Υ1 ∧ · · · ∧ Υr ∧ Θj1 ∧ · · · ∧ Θjd = Υ1 ∧ · · · ∧ Υr ∧ (C j1 − PL⊥ Cj1 ) ∧ · · · ∧ (C jd − PL⊥ Cjd ). The components of this vector belong to L∞ and can be approximated in L2/d by vector functions of the form Υ1 ∧ · · · ∧ Υr ∧ (C j1 − g j1 ) ∧ · · · ∧ (C jd − g jd ),

(1.6)

where gj1 , · · · , gjd ∈ Range TΥ . Hence, it is sufficient to prove that the components of (1.6) belong to H 2/d . Let gjl = P+ Υfl for fl ∈ H 2 (Cr ), 1 ≤ l ≤ d. We have Υ1 ∧ · · · ∧ Υr ∧ (C j1 − g j1 ) ∧ · · · ∧ (C jd − g jd ) =

Υ1 ∧ · · · ∧ Υr ∧ (C j1 − P+ Υf1 ) ∧ · · · ∧ (C jd − P+ Υfd )

=

Υ1 ∧ · · · ∧ Υr ∧ (C j1 − Υf¯1 + P− Υf1 ) ∧ · · · ∧ (C jd − Υf¯d + P− Υfd ).

Clearly, almost everywhere on T the vectors Υ(ζ)fl (ζ) are linear combinations of Υ1 (ζ), · · · , Υr (ζ). Therefore if we expand the above wedge product using the multilinearity of ∧, all terms containing Υf¯l give zero contribution. Thus we have Υ1 ∧ · · · ∧ Υr ∧ (C j1 − Υf¯1 + P− Υf1 ) ∧ · · · ∧ (C jd − Υf¯d + P− Υfd ) = Υ1 ∧ · · · ∧ Υr ∧ (C j1 + P− Υf1 ) ∧ · · · ∧ (C jd + P− Υfd ) ∈ H 2/d . The following fact is an immediate consequence of Theorem 1.6. Corollary 1.7. Let V be a balanced matrix function. Then det V is a constant function of modulus 1. Finally, we obtain in this section the following result, which will play an important role in this chapter. Theorem 1.8. Let 0 < r ≤ min{m, n} and let V and W t be r-balanced matrix functions of sizes n × n and m × m, respectively. Then ' O O O O ∞ = . W H (Mm,n )V O H ∞ (Mm−r,n−r ) O L∞ (Mm−r,n−r ) (1.7) Proof. Let V =

Υ Θ

,

W =

Ω

Ξ

t

,

where Υ has size n × r and Ω has size m × r. Clearly, (1.7) is equivalent to the following equality: ' O 0 H ∞ (Mm,n ) W ∗ V∗ O L∞ (Mm−r,n−r ) O O = W∗ V ∗. O H ∞ (Mm−r,n−r )

574


Let us show first that the right-hand side of this equality is contained in the left-hand side. Let F ∈ H ∞ (Mm−r,n−r ). We have ∗ O O O O Υ V∗ = W∗ Ω Ξ O F Θt O F =

ΞF Θt ∈ H ∞ (Mm,n ).

Let us now prove the opposite inclusion. Suppose that G ∈ L∞ (Mm−r,n−r ) and O O V ∗ ∈ H ∞ (Mm,n ). W∗ O G We have to prove that G ∈ H ∞ (Mm−r,n−r ). As above one can easily see that O O W∗ V ∗ = ΞGΘt ∈ H ∞ (Mm,n ). O G Since Ξ is co-outer, it follows from Lemma 1.4 that GΘt ∈ H ∞ (Mm−r,n ) (one should apply Lemma 1.4 to each column of GΘt ). Again, by Lemma 1.4, since Θt is outer, it follows that G ∈ H ∞ (Mm−r,n−r ).

2. Parametrization of Best Approximations In this section we consider matrix functions Φ ∈ L∞ (Mm,n ) such that the Hankel operator HΦ : H 2 (Cn ) → H 2 (Cm ) has a maximizing vector. For such matrix functions we obtain a parametrization of the set of best approximations of Φ by bounded analytic matrix functions. Among such matrix functions Φ we characterize the badly approximable matrix functions, i.e., such that ΦL∞ = distL∞ (Φ, H ∞ ). Let us first make a couple of obvious observations. Suppose that Φ ∈ L∞ (Mm,n ) and F ∈ H ∞ (Mm,n ) is a superoptimal approximation of Φ by bounded analytic matrix functions. Then the transposed matrix function F t is a superoptimal approximation of Φt . Moreover, the matrix functions Φ and Φt have the same superoptimal singular values. ¯HΦ f ∈ Next, suppose that P− Φ = O, f ∈ H 2 (Cn ), and g = t−1 0 z H 2 (Cm ). (Recall that the largest superoptimal singular value t0 = t0 (Φ) is equal to HΦ .) Then f is a maximizing vector of HΦ if and only if g is a maximizing vector of HΦt . Indeed, f is a maximizing vector of HΦ if and ¯g¯ is a maximizing vector of HΦ∗ , i.e., g is nonzero and only if t−1 0 HΦ f = z HΦ∗ z¯g¯2 = HΦ∗ · P+ Φ∗ z¯g¯2 = HΦ∗ · g2 . It is easy to see that the last equality is equivalent to the following one: P− Φt g2 = HΦt · g2 , which in turn is equivalent to the fact that g is a maximizing vector of HΦt .

2. Parametrization of Best Approximations

575

Let us first consider the simplest case min{m, n} = 1 and prove the uniqueness of a best approximation. Theorem 2.1. Suppose that Φ is a matrix function in L∞ (Mm,n ) such that HΦ has a maximizing vector. If min{m, n} = 1, then Φ has a unique best approximation by bounded analytic matrix functions. Proof. In view of the above observations it is sufficient to assume that n = 1. Let F ∈ H ∞ (Mm,n ) be a best approximation of Φ. Suppose that f ∈ H 2 is a maximizing vector of HΦ . By Theorem 2.2.3, HΦ f = P− (Φ − F )f = (Φ − F )f , and so

HΦ f . f Clearly, the last equality uniquely determines F . Suppose now that min{m, n} > 1, f is a maximizing vector of HΦ , and g = t−1 ¯HΦ f . Let h be a scalar outer function in H 2 such that 0 z |h(ζ)| = f (ζ)Cn for almost all ζ ∈ T. Let ϑ1 be a greatest common inner divisor of the entries of f , i.e., ϑ1 is a scalar inner function such that all entries of f are divisible by ϑ1 and any other common inner divisor of all entries of f is also a divisor of ϑ1 . Clearly, f admits a factorization of the form Φ−F =

f = ϑ1 hv,

(2.1)

where v is an inner and co-outer column function. By Theorem 2.2.3, g(ζ)Cm = f (ζ)Cn for almost all ζ ∈ T. Hence, g admits the following factorization: g = ϑ2 hw,

(2.2)

where ϑ2 is a scalar inner function and w is an inner and co-outer column function. By Theorem 1.1, the column functions v and w admits thematic (in other words, 1-balanced) completions, i.e., there exist inner and co-outer matrix functions Θ and Ξ such that the matrix functions def def V = v Θ and W t = w Ξ (2.3) are unitary-valued. Note that if n = 1, then f is a scalar function, it admits a factorization of the form f = ϑ1 h, where h is an outer function in H 2 and ϑ is an inner function. In this case v is identically equal to 1 and V is a scalar function identically equal to 1 as well. The same can be said in the case m = 1. Theorem 2.2. Let Φ be a matrix function in L∞ (Mm,n ) such that 2 (Cm ) is nonzero and has a maxthe Hankel operator HΦ : H 2 (Cn ) → H− imizing vector. Suppose that V and W are as above and F is a best approximation of Φ by bounded analytic matrix functions. Then there exists

576


Ψ ∈ L∞ (Mm−1,n−1 ) such that ΨL∞ ≤ t0 and t0 u O Φ − F = W∗ V ∗, (2.4) O Ψ ¯ and h, ϑ1 , and ϑ2 are given by (2.1) and (2.2). where u = z¯ϑ¯1 ϑ¯2 h/h, t0 u O we mean the matrix function If n = 1, by O Ψ t0 u O of size m × 1. If m = 1, then this is the 1 × n matrix function t0 u O . We will always use this way of writing, i.e., if a submatrix has size 0, this means that it does not exist. −1 ¯ = t−1 Proof. By Theorem 2.2.3, z¯g 0 HΦ f = t0 (Φ − F )f . It is easy to see that the upper left entry of the matrix function W (Φ − F )V is ϑ¯2 t ϑ¯1 ϑ¯2 t ϑ¯1 wt (Φ − F )v = g HΦ f g (Φ − F ) f = h h h h |h|2 ϑ¯2 ϑ¯1 t ¯ = t0 z¯ϑ¯1 ϑ¯2 2 = t0 u. = t0 z¯ gg h h h Clearly, (W (Φ − F )V )(ζ)Mm,n ≤ t0 for almost all ζ ∈ T. Since u is a unimodular function, |t0 u(ζ)| = t0 almost everywhere, and so the upper right and the lower left entries of (W (Φ−F )V )(ζ) must be the zero. Hence, W (Φ − F )V has the form t0 u O ∗ W V ∗, O Ψ where Ψ ∈ L∞ (Mm−1,n−1 ). Since V and W are unitary-valued functions and Φ − F L∞ ≤ t0 , it follows that ΦL∞ ≤ t0 . We can parametrize now all best approximations of Φ by bounded analytic matrix functions. We fix a best approximation F and consider the factorization (2.4). Theorem 2.3. Let min{m, n} > 1 and let Φ be a matrix function in L∞ (Mm,n ) such that HΦ is nonzero and has a maximizing vector. Suppose that F is a best approximation of Φ and Φ − F satisfies (2.4). Let Q ∈ H ∞ (Mm,n ). Then Q is a best approximation of Φ if and only if there exists G ∈ H ∞ (Mm−1,n−1 ) such that Ψ − GL∞ ≤ t0 and t0 u O ∗ (2.5) Φ−Q=W V ∗. O Ψ−G Proof. Suppose that Q is a best approximation of Φ by bounded analytic matrix functions. Then by Theorem 2.2, Φ − Q admits a factorization t0 u O Φ − Q = W∗ V ∗, O Ψ◦

2. Parametrization of Best Approximations

577

where u is the same as in (2.4) and Ψ◦ is a function in L∞ (Mm−1,n−1 ) such that Ψ◦ L∞ ≤ t0 . Then O O ∗ Q−F =W V∗ O Ψ − Ψ◦ def

and by Theorem 1.8, G = Ψ − Ψ◦ ∈ H ∞ (Mm−1,n−1 ). Now let G be a matrix function in H ∞ (Mm−1,n−1 ) such that Ψ − GL∞ ≤ t0 . Again, by Theorem 1.8, there exists a matrix function Q ∈ H ∞ (Mm,n ) such that O O V ∗. Q − F = W∗ O G Then (2.5) holds and clearly, Φ − QL∞ = t0 , i.e., Q is a best approximation of Φ. Theorems 2.2 and 2.3 allow one to reduce the problem of finding a superoptimal approximation of Φ to the same problem for the matrix function Ψ, which has size (m − 1) × (n − 1). Indeed, the following result follows immediately from Theorems 2.2 and 2.3. Theorem 2.4. Suppose that Φ satisfies the hypotheses of Theorem 2.2 and (2.4) holds. Then tj (Φ) = tj−1 (Ψ) for j ≥ 1. A function Q ∈ H ∞ (Mm,n ) is a superoptimal approximation of Φ if and only if Φ − Q admits a factorization of the form (2.5) for a superoptimal approximation G of Ψ. Moreover, (2.6) s0 (Φ − Q)(ζ) = t0 , for almost all ζ ∈ T, and

sj (Φ − Q)(ζ) = sj−1 (Ψ − G)(ζ) ,

j ≥ 1,

for almost all

ζ ∈ T. (2.7)

Proof. Suppose that Q is a best approximation of Φ and (2.5) holds. Equalities (2.6) and (2.7) follow immediately from the facts that u is unimodular and ΨL∞ ≤ t0 . This in turn implies that tj (Φ) = tj−1 (Ψ) for j ≥ 1 as well as the rest of the theorem. We have now arrived at the problem of finding a superoptimal approximation of Ψ that has size (m − 1) × (n − 1). If we knew that HΨ also possesses a maximizing vector, we could iterate the above procedure and eventually reduce the problem to the case min{m, n} = 1. In the general case it is not true that under the hypotheses of Theorem 2.2 the Hankel operator HΨ has a maximizing vector. Indeed, we can consider the diagonal matrix function ϕ1 O , Φ= O ϕ2 such that ϕ1 and ϕ2 are scalar functions in L∞ , Hϕ1 > Hϕ2 , Hϕ1 has a maximizing vector but Hϕ2 has no maximizing vector. It is easy to see that in this case Ψ = ϕ2 .

578


However, in certain important cases we can make the conclusion that HΨ must have a maximizing vector that will allow us to construct a superoptimal approximation and prove uniqueness. This will be done in §3 in the case Φ ∈ (H ∞ + C)(Mm,n ) and in §4 in the case when the essential norm of HΦ is less than the smallest nonzero superoptimal singular value of Φ. We can describe now the badly approximable matrix functions Φ for which HΦ has a maximizing vector. As in the scalar case a matrix function Φ ∈ L∞ (Mm,n ) is called badly approximable if distL∞ (Φ, H ∞ (Mm,n )) = ΦL∞ . Theorem 2.5. Suppose that Φ ∈ L∞ (Mm,n ) and HΦ = O. The following are equivalent: (i) Φ is badly approximable and HΦ has a maximizing vector; (ii) Φ admits a factorization of the form tu O ∗ Φ=W V ∗, O Ψ where t > 0, V and W t are thematic matrix functions, ΨL∞ ≤ t, and u is a scalar unimodular function of the form ¯ u = z¯ϑ¯h/h for an inner function ϑ and an outer function h in H 2 . If (ii) holds, then HΦ = t. Proof. If (i) holds, the existence of a desired factorization follows from Theorem 2.2. Conversely, suppose (ii) holds with V and W of the form (2.3). Clearly, ΦL∞ = t, and so HΦ ≤ t. Put f = hv ∈ H 2 (Cn ). We have ∗ tu O v ∗ hv Φf = W O Ψ Θt tu O h = W∗ O Ψ O ¯ t¯ z ϑ¯h 2 ¯w ¯ Ξ ¯ ∈ H− w (Cm ). = = t¯ z ϑ¯h O Hence, HΦ f = Φf , HΦ f L2 (Cm ) = hH 2 = f H 2 (Cn ) . It follows that HΦ = t and f is a maximizing vector of HΦ .

3. Superoptimal Approximation of H ∞ + C Matrix Functions In this section we consider the case when Φ ∈ (H ∞ + C)(Mm,n ). By 2 the Hartman theorem, the Hankel operator HΦ : H 2 (Cn ) → H− (Cm ) is compact, and so it has a maximizing vector. Therefore by Theorem 2.2, if

3. Superoptimal Approximation of H ∞ + C Matrix Functions

579

HΦ = O and F is a best approximation of Φ by bounded analytic matrix functions, then Φ−F admits a factorization of the form (2.4). We show that in that factorization Ψ must belong to (H ∞ +C)(Mm−1,n−1 ), and so we can iterate this procedure and reduce the problem to the case min{m, n} = 1. Uniqueness will follow from Theorem 2.1. To prove that Ψ ∈ (H ∞ + C)(Mm−1,n−1 ), we use Theorem 1.6 on the analyticity of minors. Later in this chapter we consider other methods to prove the uniqueness of a superoptimal approximation. Recall that the maximizing vectors f and g of HΦ and HΦt admit factorizations (2.1) and (2.2), and the unitary-valued matrix function V and W are of the form (2.3). First we prove that the unimodular function u in (2.4) belongs to QC. Theorem 3.1. Let Φ ∈ (H ∞ + C)(Mm,n ) and HΦ = O. If F is a best approximation of Φ by bounded analytic matrix functions and t0 u O Φ − F = W∗ V ∗, (3.1) O Ψ then u ∈ QC. Proof. It can be easily seen from (3.1) that wt (Φ − F )v = t0 u, and so u ∈ H ∞ + C. Hence, P− u ∈ V M O. ¯ ¯ ∈ H 2 , and so h ∈ Ker Tu . Since u = z¯ϑ¯1 ϑ¯2 h/h, we have uh = z¯ϑ¯1 ϑ¯2 h − ∗ By Theorem 3.1.4, Ker Tu = {O}, and so Tu has dense range in H 2 . By Theorem 7.1.4 applied to X = V M O, u ∈ V M O, and so u ∈ QC. Now we are in a position to show that the entry Ψ in (3.1) belongs to H ∞ + C. Theorem 3.2. Suppose that Φ ∈ (H ∞ + C)(Mm,n ), HΦ = O, F is a best approximation of Φ by bounded analytic matrix functions, and (3.1) holds. Then Ψ ∈ (H ∞ + C)(Mm−1,n−1 ). Proof. By (3.1),

W (Φ − F )V =

t0 u O

O Ψ

.

(3.2)

For 2 ≤ j ≤ m and 2 ≤ k ≤ n we consider the 2 × 2 minor of the right-hand side of (3.2) with indices 1j, 1k. Clearly, it is equal to t0 u0 ψj−1,k−1 , where Ψ = {ψst }1≤s≤m−1,1≤t≤n−1 . On the other hand, the same minor of the left-hand side of (3.2) equals W1j,st (Φ − F )st,ικ Vικ,1k , s tr , and apply successively the procedure described in §4. Recall that the sets Ωj are defined in the introduction to Chapter 14. As usual I j denotes the constant matrix function identically equal to the j × j identity matrix Ij and tj = 0 for j ≥ min{m, n}. Theorem 5.1. Let Φ ∈ L∞ (Mm,n ) and 1 ≤ r ≤ min{m, n}. Suppose that HΦ e < tr−1 and tr < tr−1 . Suppose that F is a bounded analytic matrix function in Ωr−1 . Then Φ − F admits a factorization   t0 u0 O ··· O O  O O O  t1 u 1 · · ·    . .. .. ..  V ∗ · · · V ∗ , ∗ .. Φ − F = W0∗ · · · Wr−1  .. . 0 . . .    r−1 (5.1)  O  O · · · tr−1 ur−1 O O

O

···

O

in which the Vj and Wj for j ≥ 1 have the form Ij O Ij O , Wj = ˘ j , Vj = O W O V˘j

Ψ

1 ≤ j ≤ r − 1, (5.2)

5. Thematic Factorizations, Very Badly Approximable Functions

587

˘ t , V0 , V˘j are thematic matrix functions, the uj are unimodular the W0t , W j functions such that Tuj is Fredholm, and ind Tuj > 0, Ψ ∈ L∞ (Mm−r,n−r ),

ΨL∞ ≤ tr−1 ,

and

HΨ < tr−1 . (5.3)

Proof. The result follows by successive application of Theorem 2.2 and Theorem 4.1. Factorizations of the form (5.1) with Ψ satisfying (5.3) are called partial thematic factorizations. With such a factorization we associate the factorization indices (or thematic indices) kj defined by kj = ind Tuj .

(5.4)

The matrix function Ψ is called the residual entry of the partial thematic factorization (5.1). Note that by Theorems 3.1 and 3.2, if Φ ∈ (H ∞ + C)(Mm,n ), then the unimodular functions uj in (5.1) belong to QC, and so in this case we can write kj = − wind uj (see §3.3, where it is explained how to define the winding number of an invertible function in H ∞ + C). Suppose now that HΦ e is less than the smallest nonzero superoptimal singular value of Φ. In this case we can apply the algorithm described in the previous section to find the unique superoptimal approximation Q and obtain a thematic factorization of Φ − Q. To be more precise, we mean the following. Theorem 5.2. Suppose that Φ ∈ L∞ (Mm,n ) and t0 , t1 , · · · , tr−1 are all nonzero superoptimal singular values of Φ. Let Q ∈ H ∞ (Mm,n ) be the superoptimal approximation of Φ by bounded analytic matrix functions. Then Φ − Q admits a factorization of the form   O ··· O O t0 u0  O t1 u 1 · · · O O     . . . ..  V ∗ · · · V ∗ , ∗ .. .. .. Φ − F = W0∗ · · · Wr−1  ..  r−1 . 0 .   (5.5)  O O · · · tr−1 ur−1 O  O

O

···

O

O

˘ t , V0 , V˘j in which the Vj and Wj , 0 ≤ j ≤ r−1, have the form (5.2), the W0t , W j are thematic matrix functions, the uj are unimodular functions such that Tuj is Fredholm, and ind Tuj > 0. Note that the lower right entry of the diagonal matrix function on the right-hand side of (5.5) has size (m − r) × (n − r). As usual if m = r or n = r, this means that the corresponding zero row or column does not exist. Proof. We proceed in exactly the same way as in the proof of Theorem 5.1. We can arrive at the situation when the corresponding lower right entry

588


Ψ of the diagonal matrix function is zero. In this case the best approximation of Ψ is the zero function and the process terminates. Otherwise, we arrive at the situation when m = r or n = r, in which case the process also terminates. In both cases we obtain a factorization of the form (5.5). Factorizations of the form (5.5) are called thematic factorizations. The corresponding thematic indices are defined by (5.4). Theorem 5.2 shows that a very badly approximable function satisfying the assumptions of Theorem 5.2 admits a thematic factorization. The following result says that the converse is also true. Theorem 5.3. Let Φ be a matrix function of the form   s0 u0 O ··· O O  O O O  s1 u1 · · ·    .. . . ..  V ∗ · · · V ∗ , ∗ ∗ . .. .. .. Φ = W0 · · · Wr−1  . 0 .    r−1  O O · · · sr−1 ur−1 O  O O ··· O O ˘ t , V0 , V˘j where the Vj and Wj , 0 ≤ j ≤ r −1, have the form (5.2), the W0t , W j are thematic matrix functions, s0 ≥ s1 ≥ · · · ≥ sr−1 , ¯ j /hj , ϑj being an the uj are unimodular functions of the form uj = z¯ϑ¯j h 2 inner function and hj being an outer function in H . Then Φ is very badly approximable and tj (Φ) = sj , 0 ≤ j ≤ r − 1. Proof. By Theorem 2.5, Φ is badly approximable and s0 = t0 (Φ) = HΦ . The rest of the theorem follows by induction from Theorem 2.4 . In the scalar case there is a geometric description of the badly approximable functions ϕ under the condition Hϕ e < Hϕ (see Theorem 7.5.5). Namely such a function ϕ is badly approximable if and only if f = const and Tϕ is Fredholm with positive index. For ϕ ∈ H ∞ + C the last condition is equivalent to the fact that wind ϕ < 0. Note that ind Tϕ > 0 if and only if Tzϕ has dense range in H 2 . It follows from Theorem 4.9 that if Φ ∈ L∞ (Mm,n ) satisfies the condition that HΦ e is less than the smallest nonzero superoptimal singular value of Φ, then all singular values of Φ(ζ) are constant. Moreover, sj (Φ(ζ)) = tj (Φ),

for almost all ζ ∈ T.

It turns out that if in addition to that m ≤ n and tm−1 (Φ) = 0, then the Toeplitz operator TzΦ has dense range in H 2 (Cm ). Theorem 5.4. Let Φ be a matrix function in L∞ (Mm,n ) such that HΦ e is less than the smallest nonzero superoptimal singular value of Φ. Suppose that Φ is very badly approximable, m ≤ n, and tm−1 (Φ) > 0. Then the Toeplitz operator TzΦ : H 2 (Cn ) → H 2 (Cm ) has dense range.


589

Proof. We argue by induction on m. Suppose that m = 1. Without loss of generality we may assume that HΦ = 1. Let f ∈ H 2 (Cn ) be a maximizing vector of HΦ and let g = z¯HΦ f . By Theorem 2.2.3, Φf = z¯g and |g(ζ)| = Φ(ζ)M1,n f (ζ)Cn = Φ∗ (ζ)Cn f (ζ)Cn = f (ζ)Cn . We are going to use the following elementary fact. Suppose that x, y ∈ Cn and c ∈ C satisfy x∗ y = c, |c| = x · y. Then x∗ = cy−2 y ∗ . ¯ We apply this fact for x = Φ(ζ)∗ , y = f (ζ), and c = ζg(ζ) and obtain Φ = (f ∗ f )−1 z¯gf ∗ .

(5.6)

Consider the following factorizations: f = ϑ1 hv,

g = ϑ2 h,

where ϑ1 and ϑ2 are scalar inner functions, h is a scalar outer function in H 2 , and v is an inner and co-outer function in H ∞ (Cn ). It follows from (5.6) that ¯ h Φ = z¯ϑ¯1 ϑ¯2 v ∗ . (5.7) h ∗ ∗ = {O}. Suppose that χ ∈ Ker TzΦ . Then We have to prove that Ker TzΦ ∗ 2 z¯Φ χ ∈ H− , and it follows from (5.7) that 2 ¯ ∈ H− , ϑ1 ϑ2 (h/h)χv

and so all entries of the column function χv belong to Ker Tϑ1 ϑ2 (h/h) ¯ . By Theorem 4.4.10, Ker Tϑ1 ϑ2 (h/h) ¯ = O. Since v = O, it follows that χ = O. This proves the theorem for m = 1. Suppose now that m > 1. Then, by Theorem 2.2, Φ admits a factorization of the form t0 u O ∗ V ∗, Φ=W O Ψ where V and W t are thematic matrix functions, u is a unimodular function ¯ of the form u = z¯ϑ¯h/h with inner ϑ and outer h ∈ H 2 , and Ψ is a very badly approximable function of size (m − 1) × (n − 1). By Theorems 2.4 and 4.1, HΨ e is less than the smallest nonzero superoptimal singular value of Ψ. Thus by the inductive hypotheses, TzΨ has dense range in H 2 (Cm−1 ). Let t V = v Θ , W = w Ξ . ∗ ∗ We want to show that Ker TzΦ = {O}. Suppose that ξ ∈ Ker TzΦ . Then ∗ 2 n z¯Φ ξ ∈ H− (C ), and so t0 u ¯ O 2 ¯ O W ξ = t0 z¯u ¯wt ξ ∈ H− . v ∗ z¯Φ∗ ξ = v ∗ z¯V W ξ = t0 z¯ u ∗ O Ψ

590


Hence, wt ξ ∈ Ker Tz¯u¯ . Again by Theorem 4.4.10, Ker Tz¯u¯ = {O}, and so wt ξ = O. We have t w O ∗ ∗ ξ= w Ξ ξ = W Wξ = W = ΞΞ∗ ξ. Ξ∗ ξ Ξ∗ Put η = Ξ∗ ξ ∈ L2 (Cn−1 ). Then ξ = Ξη. By Lemma 1.4, η ∈ H 2 (Cn−1 ). We have ¯ O t0 u W Ξη z¯Φ∗ ξ = z¯V O Ψ∗ t0 u ¯ O O = z¯V O Ψ∗ η O = z¯ΘΨ∗ η. = z¯V Ψ∗ η 2 (Cm ). Since Θ is co-outer, it follows By the assumption, z¯ΘΨ∗ η ∈ H− ∗ 2 ∗ easily from Lemma 1.4 that z¯Ψ η ∈ H− (Cm−1 ), i.e., η ∈ Ker TzΨ . By the inductive hypothesis, η = O, and so ξ = O. It is easy to see that the same reasoning proves the following more general fact.

Theorem 5.5. Let Φ be a very badly approximable matrix function in L∞ (Mm,n ) such that HΦ e is less than the smallest nonzero superoptimal singular value of Φ. If f ∈ Ker Tz¯Φ∗ , then Φ∗ f = O. Remark. It will follow from the results of §12 that if Φ ∈ (H ∞ + C)(Mm,n ) in the hypotheses of Theorem 5.4, then TzΦ : H 2 (Cn ) → H 2 (Cm ) is onto. However, unlike the scalar case the hypotheses of Theorem 5.4 together with the condition that all sj (Φ) are constant on T do not guarantee that Φ is very badly approximable. Example. Let e : T → C2 be a continuous function such that e(ζ)C2 = 1, ζ ∈ T. Let 0 < α < 1 and let ¯ 2 + (1 − α)e(ζ)e∗ (ζ)), Φ(ζ) = ζ(αI

ζ ∈ T.

Then it is easy to see that Φ is continuous, s0 (Φ(ζ)) = Φ(ζ)M2,2 = 1,

and s1 (Φ(ζ)) = α.

∗ Let us show that Ker TzΦ = {O}. Suppose that ϕ ∈ Ker TzΦ . Then ∗ 0 = (TzΦ ϕ, ϕ) = (ϕ, zΦϕ) = αϕ22 + (1 − α)e∗ ϕ22 .

Thus ϕ = O, and so Ker TzΦ = {O}. However, for suitable e, Φ is not even badly approximable. Indeed, if Φ is badly approximable and f is a maximizing vector of HΦ , then by Theorem 2.2.3, f (ζ) is a maximizing vector of Φ(ζ) for almost all ζ ∈ T, and so f (ζ) = χ(ζ)e(ζ) almost everywhere on


591

T for some scalar function χ. But we can choose e so that χe ∈ H 2 (C2 ) for any scalar function χ. For example, we can take 1 1 , e= √ ξ 2 where ξ is a continuous unimodular function on T such that Ker Hξ = {O}. Such a function ξ can easily be constructed. For example, if ξ is a continuous nonconstant unimodular function such that ξ(ζ) = 1 for Re ζ ≥ 0 but ξ is not constant. Let us show that Ker Hξ = {O}. Indeed, if ψ ∈ Ker Hξ and ψ = O, then ξψ ∈ H 2 and since ξ is unimodular, the H 2 functions ψ and ξψ have the same moduli on T. Thus they can be factorized as follows: ψ = ϑ1 h,

ξψ = ϑ2 h,

for some outer function h ∈ H 2 and inner functions ϑ1 and ϑ2 . Then ϑ1 (ζ) = ϑ2 (ζ) for Re ζ ≥ 0 but ϑ1 = ϑ2 , which is impossible for H ∞ functions. We can associate with a matrix function Φ ∈ L∞ (Mm,n ) another numerical sequence τj = τj (Φ), 0 ≤ j < min{m, n}: τj = inf{HΦ−G : G ∈ L∞ (Mm,n ), rank g(ζ) ≤ j a.e. on T}. Clearly, τ0 = t0 . Suppose that HΦ e is less than the smallest nonzero superoptimal singular value of Φ. It follows from Theorem 5.2 that τj (Φ) ≤ tj (Φ). Indeed, without loss of generality we may assume that Φ is very badly approximable (otherwise we can subtract from Φ its superoptimal approximation). Then Φ admits a factorization of the form (5.5). Put   t0 u0 O ··· O O ··· O  O O O ··· O  t 1 u1 · · ·    .. .. . .. . . ..  . . .  . . . .  . . .   ∗ ∗ ∗  ∗ O · · · tj−1 uj−1 O · · · O  G = W0 · · · Wr−1  O Vr−1 · · · V0 .  O  O ··· O O ··· O    . . .. . . .  . . .. .. ..  .. . ..  . O O ··· O O ··· O Clearly, rank G(ζ) = j for almost all ζ ∈ T. It is also easy to see that Φ − GL∞ = tj , and so τj ≤ tj . However, the following example shows that it is not true in general that τj = tj . Example. Let Φ=

z¯ O O α¯ z

√1 2 −z √ 2

√z¯ 2 √1 2

,

592


where 0 < α < 1. Clearly, the Φ is represented in the form of a thematic factorization. Hence, t0 (Φ) = 1 and t1 (Φ) = α. Now put √1 √z¯ z¯ O 2 2 . G= −z O O √ √1 2

2

Clearly, rank G(ζ) = 1 for ζ ∈ T. It is easy to see that O O . Φ−G= α¯ z −α √ √ 2

Let

F =

Then

O

−α √ 2

O O

2

∈ H ∞ (M2,2 ).

α HΦ−G ≤ Φ − G − F L∞ = √ < α. 2

6. Admissible and Superoptimal Weights Let W ∈ L∞ (Mn,n ) be a matrix weight, i.e., a bounded matrix function whose values are nonnegative self-adjoint matrices. Given Φ ∈ L∞ (Mm,n ), we say that W is admissible for the Hankel operator HΦ if

def 2 HΦ ξ ≤ (Wξ, ξ) = (W(ζ)ξ(ζ), ξ(ζ))dm(ζ), ξ ∈ H 2 (Cn ). T

If W = cI n for √ c > 0, then it is easy to see that W is admissible if and only if HΦ ≤ c. The following theorem generalizes the matrix version of the Nehari theorem. Theorem 6.1. Suppose that W ∈ L∞ (Mn,n ) is an admissible weight for HΦ , Φ ∈ L∞ (Mm,n ). Then there exists F ∈ H ∞ (Mm,n ) such that (Φ − F )∗ (Φ − F ) ≤ W. In other words Theorem 6.1 says that the Hankel operator HΦ has a symbol Φ# satisfying Φ∗# Φ# ≤ W. In this case we say that Φ# is dominated by the admissible weight W. Proof. Put Wε = W+εI n . By Theorem 13.3.1, there exists an invertible matrix function Gε in H ∞ (Mn,n ) such that G∗ε Gε = Wε . Clearly, the weight Wε is admissible, and so HΦ ξ2 ≤ Gε ξ2 ,

ξ ∈ H 2 (Cn ),

which is equivalent to the fact that HΦ G−1 ε ξ2 ≤ ξ2 ,

ξ ∈ H 2 (Cn ).

6. Admissible and Superoptimal Weights

593

Since Gε is invertible in H ∞ (Mn,n ), HΦ G−1 ε is a bounded Hankel operator. By Theorem 2.2.2, this Hankel operator has symbol Ψε ∈ L∞ (Mm,n ) such that ΨL∞ ≤ 1. Then Φε = ΨGε is a symbol of HΦ and Φ∗ε Φε = G∗ε Ψ∗ε Ψε Gε ≤ G∗ε Gε = Wε = W + εI n . It remains to choose a sequence {εj }j≥0 that converges to 0 and such that the sequence {Φεj }j≥0 converges in the weak-∗ topology to a matrix function, say Φ# . Clearly, Φ# is a symbol of HΦ dominated by W. Definition. Let W be an admissible weight for a Hankel operator HΦ . Consider the numbers def 0 ≤ j ≤ n − 1. s∞ j (W) = ess sup sj W(ζ) , ζ∈T

W is called a superoptimal weight for HΦ if it lexicographically minimizes ∞ ∞ the numbers s∞ 0 (W), s1 (W), · · · , sn−1 (W) among all admissible weights, i.e., ∞ s∞ 0 (W) = min{s0 (V) : V is admissible}, ∞ ∞ s∞ 1 (W) = min{s1 (V) : V is admissible, s0 (V) is minimal possible}, etc.

The following result shows that a superoptimal weight always exists. Theorem 6.2. Let Φ ∈ L∞ (Mm,n ) and let F ∈ H ∞ (Mm,n ) be a superoptimal approximation of Φ. Then (Φ − F )∗ (Φ − F ) is a superoptimal weight for HΦ . Recall that a superoptimal approximation always exists (see the Introduction to this chapter). Proof. Put W = (Φ − F )∗ (Φ − F ). Clearly, sj (W)(ζ) = 0, j ≤ min{m, n}. Suppose W is not superoptimal. Then there exists an admissible weight V such that for some j0 , 0 ≤ j0 ≤ min{m, n} − 1, ∞ s∞ j (V) = sj (W),

0 ≤ j ≤ j0 ,

∞ and s∞ j0 (V) < sj0 (W).

(6.1)

By Theorem 6.1, there exists G ∈ H ∞ (Mm,n ) such that Φ−G is dominated by V, i.e., (Φ − G)∗ (Φ − G) ≤ (Φ − F )∗ (Φ − F ). It follows now from (6.1) that sj (Φ − G)(ζ) ≤ sup sj (Φ − F )(ζ) , 0 ≤ j ≤ j0 , ζ∈T

while

sup sj0 (Φ − G)(ζ) < sup sj0 (Φ − F )(ζ) , ζ∈T

ζ∈T

which contradicts the fact that F is a superoptimal approximation of Φ. The following theorem says that if Φ satisfies the hypotheses of Theorem 4.8, then there is a unique superoptimal weight for HΦ .

594


Theorem 6.3. Let Φ be a matrix function in L∞ (Mm,n ) such that HΦ e is less than the smallest nonzero superoptimal singular value of Φ. Then HΦ has a unique superoptimal weight. Proof. Let F be the unique superoptimal approximation of Φ by bounded def analytic matrix functions. By Theorem 6.2, W = (Φ − F )∗ (Φ − F ) is a superoptimal weight for HΦ . By Theorem 4.9, 2 sj W(ζ) = s∞ ζ ∈ T, 0 ≤ j ≤ n − 1 j (W) = tj (Φ), (if min{m, n} ≤ j ≤ n − 1, we assume that tj (Φ) = 0). Suppose that V is another superoptimal weight. Then sj V(ζ) ≤ t2j (Φ), 0 ≤ j ≤ n − 1. (6.2) By Theorem 6.1, there exists a matrix function G in H ∞ (Mm,n ) such that (Φ − G)∗ (Φ − G) ≤ V. It follows now from (6.2) that sj (Φ − G)(ζ) ≤ tj (Φ), ζ ∈ T, 0 ≤ j ≤ n − 1, and so G is a superoptimal approximation of Φ. Hence, G = F and sj V(ζ) = sj W(ζ) , ζ ∈ T, 0 ≤ j ≤ n − 1. Since both W(ζ) and V(ζ) are positive semi-definite, it follows that V = W.

7. Thematic Indices In §5 we have defined the notion of factorization indices (or thematic indices) of a thematic (or partial thematic) factorization. A natural question arises of whether the thematic indices depend on the choice of a factorization or whether they are uniquely determined by the matrix function Φ itself. 2 z¯ O . It admits the Example. Consider the matrix function Φ = O z¯6 following thematic factorizations: 2 1 O z¯ O 1 O Φ = O 1 O z¯6 O 1     z¯ 1 z5 1 √ √ √ z¯ O −√ 2 2 2 2   =  5 1 1 z¯ √ √z 7 − √ √ O z¯ 2 2 2 2 =

O 1 1 O

z¯6 O

O z¯2

O 1 1 O

.

The superoptimal singular values of Φ are t0 = t1 = 1. The indices of the first factorization are 2, 6; the indices of the second are 1, 7; and the indices

7. Thematic Indices

595

of the third are 6, 2. Note that for all above factorizations the sum of the indices is 8. We show in this section that the sum of the indices of a thematic factorization that correspond to the superoptimal singular values equal to a specific value does not depend on the choice of a thematic factorization. As we have observed in the above example such sums are equal to 8. We also obtain similar results for partial thematic factorizations. In §10 we show that among thematic factorizations one can choose a so-called monotone thematic factorization and we prove in §9 that the indices of a monotone thematic factorization are uniquely determined by the function itself. Let Φ be a matrix function in L∞ (Mm,n ) and let σ0 > σ1 > · · · > σl be all distinct superoptimal singular values of Φ. Suppose that 1 ≤ r ≤ min{m, n}, HΦ e < tr−1 , and tr < tr−1 . Let F ∈ Ωr−1 (see the Introduction to this chapter for the definition of the sets Ωj ). Consider a partial thematic factorization (5.1) of Φ − F . Recall that the thematic indices kj , 0 ≤ j ≤ r − 1, are defined by (5.4). Consider the following numbers: kj , σκ ≥ tr−1 . (7.1) νκ = {j:tj =σκ }

The following theorem is the main result of the section. Theorem 7.1. Suppose that 1 ≤ r ≤ min{m, n}, HΦ e < tr−1 , and tr < tr−1 . The numbers νκ for σκ ≥ tr−1 do not depend on the choice of a partial thematic factorization of Φ − F . In particular, if HΦ e is less than the smallest nonzero superoptimal singular value of Φ and Q is the superoptimal approximation of Φ, then Φ − Q admits a thematic factorization (5.5). In this case the numbers νκ are defined in (7.1) for 0 ≤ κ ≤ l. Theorem 7.2. Suppose that HΦ e is less than the smallest nonzero superoptimal singular value of Φ and let Q ∈ H ∞ (Mm,n ) be the superoptimal approximation of Φ. Then the numbers νκ , 0 ≤ κ ≤ l, do not depend on the choice of a thematic factorization of Φ − Q. Clearly, Theorem 7.2 is a special case of Theorem 7.1. To prove Theorem 7.1, we need some preparations. Suppose that V is an admissible weight for HΦ . We say that a nonzero vector function ξ ∈ H 2 (Cn ) is a maximizing vector for V if HΦ ξ22 = (Vξ, ξ). Lemma 7.3. Let V be an admissible weight for HΦ and let 2 ξ ∈ H 2 (Cn ) be a maximizing vector for V. Then Φξ ∈ H− (Cm ), i.e., HΦ ξ = Φξ.

596


Proof. We have (Vξ, ξ) = HΦ ξ22 = P− Φξ22 ≤ Φξ22 ≤ (Vξ, ξ). It follows that P− ξ2 = Φξ2 , which implies the result. Note that for V = cI n , Lemma 7.3 has been proved in Theorem 2.2.3. Given an admissible weight V for HΦ , we put E(V) = {ξ ∈ H 2 (Cn ) : HΦ ξ22 = (Vξ, ξ)}. It is easy to see that ξ ∈ E(V) if and only if ξ ∈ Ker(TV − HΦ∗ HΦ ), and so E(V) is a closed subspace of H 2 (Cn ). Given σ ≥ 0, we define the function Λσ on R by

s, s ≥ σ2 , Λσ (s) = 2 σ , s < σ2 . Theorem 7.4. Suppose that Φ satisfies the hypotheses of Theorem 7.1. Then for σ ≥ tr−1 kj = dim E Λσ ((Φ − F )∗ (Φ − F )) . (7.2) {j:tj ≥σ}

Clearly, (Φ − F )∗ (Φ − F ) is an admissible weight for HΦ . Let us first deduce Theorem 7.1 from Theorem 7.4. def

Proof of Theorem 7.1. Put W = (Φ − F )∗ (Φ − F ). It follows immediately from (7.2) that ν0 = dim E Λσ0 (W) , ν1 = dim E Λσ1 (W) E Λσ0 (W) , etc. Proof of Theorem 7.4. It is easy to see that E(Λσ (W)) is constant on (σκ+1 , σκ ]. Thus it is sufficient to prove that if σκ ≥ tr−1 , then kj = dim E(Λσκ (W)). {j:tj ≥σκ }

We argue by induction on r. Suppose that r = 1. It is easy to see that Λσ0 (W) = t20 I n . We have t0 u0 O Φ − F = W0∗ V0∗ , O Ψ where V0 and W0t are thematic matrix functions of the form V0 = v Θ , W0t = w Ξ .

(7.3)

By Theorem 1.8 we can change F so that ΨL∞ = t1 < t0 . Thus we may assume that (7.4) holds.

(7.4)

7. Thematic Indices

597

Since ind Tu0 = k0 , it follows that that dim Ker Tu0 = k0 . Let χ ∈ Ker Tu0 and ξ = χv. We have χ t 0 u0 O ∗ (Φ − F )ξ = W0 O Ψ O t 0 u0 χ ¯ ∈ H 2 (Cm ), = W0∗ = t0 u0 χw O since χ ∈ Ker Tu0 . It is easy to see now that ξ ∈ E(Λσ0 (W)), and so dim E(Λσ0 (W)) ≥ k0 . Let us prove the opposite inequality. Suppose that ξ ∈ E(Λσ0 (W)) and ξ = O. Then ξ is a maximizing vector of HΦ . It follows that only the upper entry of the column function V ∗ ξ is nonzero. Denote this upper entry by χ. We have χ ξ = V V ∗ξ = v Θ = χv ∈ H 2 (Cn ). O Since v is co-outer, it follows from Lemma 1.4 that χ ∈ H 2 . Again, ¯ (Φ − F )ξ = t0 u0 χw. Hence, ξ ∈ E(Λσ0 (W)) if and only if χ ∈ Ker Tu0 . It follows that dim E(Λσ0 (W)) ≤ k0 . Suppose now that the theorem is proved for r − 1. We have t 0 u0 O ∗ V0∗ , Φ − F = W0 O Φ◦ where Φ◦ is an (m − 1) × (n − 1) matrix function that has the following partial thematic factorization: I r−2 O O I r−2 ∗ ˘ Φ ◦ = W1 · · · D · · · V˘1∗ , ∗ ˘∗ O W O V˘r−1 r−1 where



t1 u1  ..  D= .  O O

··· .. . ··· ···

O .. . tr−1 ur−1 O

 O ..  .  , O  Ψ

˘ j are defined by (5.1) and (5.2). Put W◦ = Φ∗ Φ◦ . and the uj , V˘j , and W ◦ By the inductive hypothesis, the theorem holds for Φ◦ . Hence, if σκ ≥ tr−1 , then def kj . N = dim E(Λσκ (W◦ )) = {j≥1:tj ≥σκ }

By Lemma 7.3, ξ ∈ E(Λσκ (W◦ )) if and only if 2 (Cm−1 ) Φ◦ ξ ∈ H−

and Φ◦ ξ22 = (Λσ (W◦ )ξ, ξ).

598


Let ξ1 , ξ2 , · · · , ξN be a basis in E(Λσκ (W◦ )) and let ηι = HΦ◦ ξι . By Lemma 4.7, there exist scalar functions χι , 1 ≤ ι ≤ N , in H 2 such that χι 2 ∈ H− (Cm ). W0∗ ηι By Theorem 4.5, the matrix function Θ in (7.3) is left invertible. Let A be a matrix function in H ∞ (Mn−1,n ) such that AΘ = I n−1 . As in the proof of Theorem 4.1 we put ξι# = At ξι + qι v, where qι is a scalar function in H 2 satisfying P+ (t0 u0 qι + t0 u0 v ∗ At ξj ) = χj . Put ηι# = HΦ ξι# . We have t0 u0 ηι# = P− W0∗ O

O Φ◦

V0∗ ξι# = P− W0∗

χι + ωι ηι

,

where as in the proof of Theorem 4.1 ωι = P− (t0 u0 qι + t0 u0 v ∗ At ξι ). As we have explained in the proof of Theorem 4.1 χι + ωι χι + ωι P− W0∗ = W0∗ , ηι ηι and so ηι# = Φξι# . Since the matrix function W is unitary-valued, we have ∗ 2 ∗ t0 u0 O v W ξι# Φξι# 22 = t Θ O Φ◦ 2 ∗ t0 u0 = W O =

t20 u0 (qι

∗

O Φ◦ t

+v A

ξι )22

+

qι + v ∗ At ξι ξι

2 2

Φ◦ ξι 22

= t20 qι + v ∗ At ξι 22 + (Λσκ (W◦ )ξι , ξι ) (the last equality holds because ξι ∈ E(Λσκ (W◦ ))). Consider the weight V defined by 2 t0 O V= . O W◦ Since V ∗ ξι# =

qι + v ∗ At ξι ξι

,

we have Φξι# 22 = (Λσκ (V)V ∗ ξι# , V ∗ ξι# ) = (Λσκ (W)ξι# , ξι# )

(7.5)

7. Thematic Indices

599

(the last equality follows from the definitions of W and V). Since HΦ ξι# = Φξι# , it follows from (7.5) that ξι# ∈ E(Λσκ (W)). We can add now another k0 linear independent vectors of E(Λσκ (W)). Let f1 , · · · , fk0 be a basis of Ker Tu0 . Obviously, fj v ∈ E(Λσκ (W)). Let us # show that the vectors ξ1# , · · · , ξN , f1 v, · · · , fk0 v are linearly independent. N cι ξι# = O, then It is sufficient to prove that if f ∈ Ker Tu0 and f v + ι=1

f = O and cι = 0, 1 ≤ ι ≤ N . We have ∗ t N N f v A ξι + qι ∗ # fv + = + V = O. cι ξι cι O ξι ι=1

ι=1

(7.6)

Since the ξι are linearly independent, it follows that cι = 0, 1 ≤ ι ≤ N , which in turn implies that f = O. This proves that kj ≤ dim E(Λσκ (W)). {j:tj ≥σκ }

Let us prove the opposite inequality. Denote by E0 the set of vectors in E(Λσκ (W)) of the form f v, where f is a scalar function in H 2 . It is easy to see that f v ∈ E0 if and only if f ∈ Ker Tu0 . It remains to show that there are at most kj {j≥1:tj ≥σκ }

vectors ξ˘ι in E(Λσκ (W)) that are linearly independent modulo E0 . Let def η˘ι = HΦ ξ˘ι . By Lemma 7.3, η˘ι = Φξι . Define the functions ξι ∈ L2 (Cn−1 ) and ηι ∈ L2 (Cm−1 ) by γι δι , W η˘ι = . V ∗ ξ˘ι = ξι ηι Since V = v Θ and the ξ˘ι are linearly independent modulo E0 , it follows that the ξι are linearly independent. To complete the proof, it is sufficient to show that ξι ∈ E(Λσκ (W◦ )). Since η˘ι = Φξ˘ι , we have ηι = Φ◦ ξι and δι = t0 u0 γι . It follows from the 2 definitions of V and W that ξι ∈ L2 (Cn−1 ) and ηι ∈ H− (Cm−1 ). We have Λσκ (W)ξ˘ι , ξ˘ι = Λσκ (V ∗ WV )V ∗ ξ˘ι , V ∗ ξ˘ι =

t20 O

O Λσκ (W◦ )

γι ξι

= t20 γι 22 + (Λσκ (W◦ )ξι , ξι ) .

γι , ξr

600


On the other hand, Λσκ (W)ξ˘ι , ξ˘ι = Φξ˘ι 22 = ˘ ηι 22 = ηι 22 + δι 22 = Φ◦ ξι 22 + t20 γι 22 . Therefore (Λσκ (W◦ )ξι , ξι ) = Φ◦ ξι 22 , which implies ξι ∈ E(Λσκ (W◦ )).

8. Inequalities Involving Hankel and Superoptimal Singular Values In this section we prove that under the hypotheses of Theorem 5.1 the superoptimal singular values tj (Φ) for 0 ≤ j ≤ r − 1 are majorized by the singular values sj (HΦ ) of the Hankel operator HΦ . In fact, we prove a considerably stronger result. Assuming that Φ satisfies the hypotheses of Theorem 5.1, we consider the extended t-sequence for Φ: t , · · · , t , t , · · · , t , · · · , tr−1 , · · · , tr−1 -0 .+ ,0 -1 .+ ,1 .+ , k0

k1

(8.1)

kr−1

in which tj repeats kj times. We denote the terms of the extended sequence by t˘0 , t˘1 , · · · , t˘k0 +···+kr−1 −1 . Although the indices kj may depend on the choice of a partial thematic factorization, it follows from Theorem 7.1 that the extended t-sequence is uniquely determined by the matrix function Φ. We show in this section that the terms of the extended t-sequence for Φ are majorized by the singular values of HΦ . In fact, we start this section with even a stronger inequality. Theorem 8.1. Let Φ be a badly approximable matrix function in L∞ (Mm,n ) such that HΦ e < HΦ and suppose that t 0 u0 O ∗ (8.2) Φ=W V ∗, O Ψ where V and W t are thematic matrix functions, u0 is a unimodular function such that Tu0 is Fredholm, and k0 = ind Tu0 > 0. Then sj (HΨ ) ≤ sj+k0 (HΦ ),

j ∈ Z+ .

Proof. Clearly, it is sufficient to prove the following fact. Let L be a subspace of H 2 (Cn−1 ) such that HΨ ξ2 ≥ sξ2 , for every ξ ∈ L, where 0 < s ≤ t0 . Then there exists a subspace M of H 2 (Cn ) such that dim M ≥ dim L + k0 and HΦ g2 ≥ sg2 for every f ∈ M. Let ξι , 1 ≤ ι ≤ N , be a basis in L. Put ηι = ι . By Lemma 4.7, there HΨ ξ χι 2 ∗ 2 ∈ H− exist scalar functions χι in H such that W (Cm ). As in the ηι proof of Theorem 7.4 we define the functions ξι# ∈ H 2 (Cn ) by ξι# = At ξι + qι v,

8. Inequalities Involving Hankel and Superoptimal Singular Values

601

where qι is a scalar function in H 2 satisfying P+ (t0 u0 qι + t0 u0 v ∗ At ξι ) = χι . We can now define M by M = span{ξι# + f v : 1 ≤ ι ≤ N, f ∈ Ker Tu0 }. Let us show that dim M = N + k0 . Since dim Ker Tu0 = k0 , it is sufficient N cι ξι# = O, then x = O and cι = 0, 1 ≤ ι ≤ N . to prove that if xv + ι=1

This follows immediately from (7.6). To complete the proof it remains to show that HΦ g2 ≥ sg2 for g = fv +

N

cι ξι# .

ι=1 N

Let ξ =

cι ξι , η = HΨ ξ, q =

ι=1

We have t 0 u0 W∗ O

N

cι qι , and ξ # =

ι=1

O Ψ

V ∗g

N ι=1

= W∗

t0 u0 O

O Ψ

cι ξι# .

V ∗ (f v + qv + At ξ)

f + q + v ∗ At ξ = W ξ t0 u0 f + t0 u0 q + t0 u0 v ∗ At ξ = W∗ . Ψξ ∗

t0 u0 O

O Ψ

Since f ∈ Ker Tu0 , it follows that t0 u0 f ∗ 2 (Cm ). W ∈ H− O On the other hand, it has been shown in the proof of Theorem 4.1 that t0 u0 q + t0 u0 v ∗ At ξ ∗ 2 W ∈ H− (Cm ). O It follows that HΦ g = W ∗ Therefore We have

t0 u0 f + t0 u0 q + t0 u0 v ∗ At ξ HΨ ξ

.

HΦ g22 = |t0 |2 f + q + v ∗ At ξ22 + η22 .

g22 = V ∗ g22 = f + q + v ∗ At ξ22 + ξ22 . Since s ≤ t0 and η2 ≥ sξ2 , it follows that HΦ g22 ≥ s2 g22 . Now we are in a position to prove that the terms of the extended t-sequence defined by (8.1) are majorized by the singular values of the Hankel operator.

602


Theorem 8.2. Suppose that Φ ∈ L∞ (Mm,n ) , HΦ e < tr−1 , and tr < tr−1 . Then (8.3) t˘j ≤ sj (HΦ ), 0 ≤ j ≤ k0 + k1 + · · · + kr−1 − 1. In particular, if HΦ e is less than the smallest nonzero superoptimal singular value of Φ, then inequality (8.3) holds for all terms of the extended t-sequence that correspond to all nonzero superoptimal singular values of Φ. This is certainly the case if Φ ∈ (H ∞ + C)(Mm,n ). Proof. Without loss of generality we may assume that Φ ∈ Ωr−1 (see the Introduction to this chapter for the definition of the sets Ωj ). We argue by induction on r. Consider the factorization (8.2). Let f ∈ Ker Tu0 . It is easy to see that 2 (Cm ). HΦ f v = t0 u0 f w ∈ H− It follows that HΦ 2 = t0 f v2 , which proves that sj (HΦ ) = t0 ,

0 ≤ j ≤ k0 − 1.

(8.4)

Hence, the result holds for r = 1. By the inductive hypothesis, the result holds for Ψ, and so by Theorem 8.1, t˘j ≤ sj (HΦ ), k0 ≤ j ≤ k0 + k1 + · · · + kr−1 − 1. Together with (8.4) this proves the theorem. Corollary 8.3. Suppose that Φ satisfies the hypotheses of Theorem 8.2. Then tj (Φ) ≤ sj (HΦ ), 0 ≤ j ≤ r − 1.

9. Invariance of Residual Entries We continue in this section studying invariance properties of (partial) thematic factorizations. We show that if a matrix function Φ admits a partial thematic factorization of the form (5.1), then the residual entry Ψ in (5.1) is uniquely determined by the function Φ itself modulo constant unitary factors. In the next section we introduce the notion of a monotone (partial) thematic factorization and we use the results of this section to prove that the indices of a monotone (partial) thematic factorization are uniquely determined by the function Φ itself. Lemma 9.1. Let Φ be an m × n matrix of the form u 0 ∗ Φ=W V ∗, 0 Ψ where m, n ≥ 2, u ∈ C, Ψ ∈ Mm−1,n−1 , and V = v Θ ∈ Mn,n , W = w

Ξ

t

∈ Mm,m

are unitary matrices such that v ∈ Mn,1 and w ∈ Mm,1 . Then Ψ = Ξ∗ ΦΘ.

9. Invariance of Residual Entries

Proof. We have

0 Ξ ΦΘ = Ξ W V ∗Θ Ψ ∗ u 0 v ∗ w Ξ = Ξ Θ 0 Ψ Θt u 0 0 0 Im−1 = = Ψ. 0 Ψ In−1 ∗

∗

∗

603

u 0

Corollary 9.2. Let Φ be an m × n matrix of the form   0 0 ϕ0 0 · · ·  0 ϕ1 · · · 0 0     . . . ..  V ∗ · · · V ∗ , ∗ .. . . . .. Φ = W0∗ · · · Wr−1  ..  r−1 0 .    0 0 · · · ϕr−1 0  0 0 ··· 0 Ψ where r < min{m, n}, ϕ0 , ϕ1 , · · · , ϕr−1 ∈ C, Ij 0 Ij Vj = , Wj = 0 V˘j 0 are unitary matrices such that Vj = v j Θj , Wj = wj

Ξj

t

0 ˘j W

,

0 ≤ j ≤ r − 1,

,

v j ∈ Mn−j,1 , wj ∈ Mm−j,1 . Then Ψ = Ξ∗r−1 · · · Ξ∗1 Ξ∗0 ΦΘ0 Θ1 · · · Θr−1 . Proof. The result follows immediately from Lemma 9.1 by induction. Theorem 9.3. Suppose that a matrix function Φ ∈ L∞ (Mm,n ) admits partial thematic factorizations   t 0 u0 O ··· O O  O O O  t1 u1 · · ·    .. . . ..  V ∗ · · · V ∗ , ∗ ∗ . .. .. .. Φ = W0 · · · Wr−1  . 0 .    r−1  O O · · · tr−1 ur−1 O  O O ··· O Ψ and



t0 u♥ 0 ∗ ∗  ..  . ♥ Φ = W0♥ · · · Wr−1   O O

Then there exist constant U2 ∈ Mm−r,m−r such that

··· .. . ··· ···

O .. .

tr−1 u♥ r−1 O

unitary

Ψ♥

matrices

Ψ♥ = U2 ΨU1 .

 O ∗ ∗ ..  ♥ .  · · · V0♥ .  Vr−1  O U1 ∈ Mn−r,n−r

and

604


Recall that by the definition of a partial thematic factorization, Ψ must satisfy (5.3), and this is very important. Proof. Let

Vj =

and Vj♥

=

Ij O Ij O

where V˘j = v j Θj , ♥ , V˘j♥ = v ♥ Θ j j

O V˘j O V˘ ♥

,

Wj =

,

Wj♥

j

=

Ij O

O ˘ Wj

,

O ˘♥ W

,

j

t , wj Ξj t ♥ = w♥ , Ξ j j

˘j = W ˘♥ W j

Ij O

0 ≤ j ≤ r − 1, 0 ≤ j ≤ r − 1.

def def ♥ ˘ 0 def ˘ ♥ def Here V˘0 = V0 , W = V0 , V˘0♥ = V0♥ , and W 0 = W0 . We need the following lemma. Lemma 9.4. ♥ ♥ 2 n−r Θ0 Θ1 · · · Θr−1 H 2 (Cn−r ) = Θ♥ ) 0 Θ1 · · · Θr−1 H (C

(9.1)

♥ ♥ 2 m−r ). Ξ0 Ξ1 · · · Ξr−1 H 2 (Cm−r ) = Ξ♥ 0 Ξ1 · · · Ξr−1 H (C

(9.2)

and Let us first complete the proof of Theorem 9.3. Consider the inner matrix functions ♥ ♥ Θ = Θ0 Θ1 · · · Θr−1 , Θ♥ = Θ♥ 0 Θ1 · · · Θr−1 and ♥ ♥ Ξ = Ξ0 Ξ1 · · · Ξr−1 , Ξ♥ = Ξ♥ 0 Ξ1 · · · Ξr−1 . By Lemma 9.4, ΘH 2 (Cn−r ) = Θ♥ H 2 (Cn−r ). In this case there exists a constant unitary matrix Q1 ∈ Mn−r,n−r such that Θ♥ = ΘQ1 . Indeed, Θ and Θ♥ determine the same invariant subspace under multiplication by z; see Appendix 2.3. Similarly, there exists a constant unitary matrix Q2 ∈ Mm−r,m−r such that Ξ♥ = ΞQ2 . By Corollary 9.2, Ψ = Ξ∗ ΦΘ, Hence,

Ψ♥ = (Ξ♥ )∗ ΦΘ♥ .

Ψ♥ = Q∗2 Ξ∗ ΦΘQ1 = Q∗2 ΨQ1 .

Proof of Lemma 9.4. It is sufficient to prove (9.1). Indeed, (9.2) follows from (9.1) applied to Φt . It is easy to see that without loss of generality we may assume that ΨL∞ < tr−1 . Indeed, we can subtract from Φ a matrix function in Ωr−1 (see the Introduction to this chapter), and it follows from Theorem 1.8 that the resulting function admits a partial thematic factorization with the same unitary-valued function Vj and Wj , 0 ≤ j ≤ r −1, and residual entry whose

9. Invariance of Residual Entries

L∞ norm is less than tr−1 . It is also easy to see that if ΨL∞ Ψ♥ ∞ must also be less than tr−1 . L Consider the subspace L of H 2 (Cn ) defined by    O      ..        r   .       O t  · · · V1t V0t f =  L = f ∈ H 2 (Cn ) : Vr−1  ∗        .    .    .    ∗

605

< tr−1 , then                   

,

i.e., L consists of vector functions f ∈ H 2 (Cn ) such that the first r comt · · · V1t V0t f are zero. ponents of the vector function Vr−1 We define the real function ρ on R by

x, x ≥ t2r−1 , ρ(x) = 0, x < t2r−1 , 2 n 2 n and consider the operator t M : H (C ) → L (C ) of multiplication by the matrix function ρ Φ Φ : M f = ρ Φt Φ f, f ∈ H 2 (Cn ).

Let us show that L = Ker M. We have



t20  ..  Φt Φ = V0 V1 · · · Vr−1  .  O O

··· .. . ··· ···

(9.3)

O .. .

O .. .

t2r−1 O

O Ψt Ψ

and since Ψt ΨL∞ < t2r−1 , it follows that  2 t0 · · · O  .. . . .. t  . . ρ Φ Φ = V0 V1 · · · Vr−1  .  O · · · t2r−1 O ··· O

   t  Vr−1 · · · V1t V0t , 

 O ..  t .  · · · V1t V0t .  Vr−1 O  O

Since all matrix functions Vj are unitary-valued, this implies (9.3). Thus the subspace L is uniquely determined by the function Φ and does not depend on the choice of a partial thematic factorization. It is easy to see that to complete the proof of Lemma 9.4, it is sufficient to prove the following lemma. Lemma 9.5. L = Θ0 Θ1 · · · Θr−1 H 2 (Cn−r ).

(9.4)

606


Proof. We show by induction on r that (9.4) holds even without the assumption that HΨ < tr−1 (note that this assumption is very important in the proof of (9.3)). Suppose that r = 1. Then    O         ∗    2 n t L = f ∈ H (C ) : V0 f =  .  .   ..        ∗ Obviously, if f ∈ Θ0 H 2 (Cn−1 ), then f ∈ L. Suppose now that f ∈ L. We have O t , g ∈ L2 (Cn−1 ). V0 f = g Then O O = Θ0 g. f = V0 = v 0 Θ0 g g By Lemma 1.4, g ∈ H 2 (Cn−1 ), which proves the result for r = 1. Suppose now that r ≥ 2. By the inductive hypothesis,    O       ..      .  r       O  t  L = Θ0 · · · Θr−2 g : Vr−1 · · · V0t Θ0 · · · Θr−2 g =   ∗         .   .    .    ∗ (here g ∈ H 2 (Cn−r+1 )). It follows from functions that  O ··· O  .. . . .  . . ..   O ··· O t · · · V0t Θ0 · · · Θr−2 =  Vr−2  1 ··· O   . . . . ...  .. O ··· Hence,

1

                  

the definition of thematic matrix      r−1      O   . =   I n−r+1    n−r+1   

  O    t   ∗ v r−1  L = Θ0 · · · Θr−2 g : g ∈ H 2 (Cn−r+1 ), g= . ∗ Θ   .. r−1    ∗

       .    

Since the result has already been proved for r = 1, L = Θ0 · · · Θr−2 g : g ∈ Θr−1 H 2 (Cn−r ) = Θ0 · · · Θr−1 H 2 (Cn−r ).

10. Monotone Thematic Factorizations and Invariance of Indices

607

10. Monotone Thematic Factorizations and Invariance of Thematic Indices z¯2 O has theWe have seen in §7 that the matrix function Φ = O z¯6 matic factorizations with different thematic indices. We have shown in §7 that under the hypotheses of Theorem 5.1 the sum of thematic indices that correspond to all superoptimal singular values equal to a positive specific value t ≥ tr−1 does not depend on the choice of a thematic factorization. In other words, for each positive superoptimal singular value t ≥ tr−1 the number def ν = kj

{j:tj =t}

does not depend on the choice of a (partial) thematic factorization. A natural question arises of whether we can arbitrarily distribute the number ν between the indices kj with tj = t by choosing an appropriate thematic factorization (recall that the kj must be positive integers). In this section we show that the answer to this question is negative. Definition. A (partial) thematic factorization  t0 u0 O ··· O  O u · · · O t 1 1   .. . .. ∗ ∗ . .. .. W0 · · · Wr−1  . .   O O · · · tr−1 ur−1 O O ··· O

O O .. .



   ∗  Vr−1 · · · V0∗  (10.1) O  Ψ

is called monotone if for any positive superoptimal singular value t ≥ tr−1 the thematic indices kl , kl+1 , · · · , ks that correspond to all superoptimal singular values equal to t satisfy kl ≥ kl+1 ≥ · · · ≥ ks .

(10.2)

Here tl , tl+1 , · · · , ts are the superoptimal singular values equal to t. We prove in this section that if a matrix function Φ ∈ L∞ (Mm,n ) satisfies the hypotheses of Theorem 5.1 and F ∈ Ωr−1 , then Φ − F possesses a monotone partial thematic factorization of the form (5.1). We also show that the indices of a monotone partial thematic factorization are uniquely determined by the function Φ itself and do not depend on the choice of a partial thematic factorization. The same results also hold for thematic factorizations when HΦ e is less than the smallest nonzero superoptimal ). singular value of Φ. In particular, this is the case if Φ ∈ (H ∞ +C)(Mm,n z¯2 O If we look at the thematic factorizations for the function Φ = O z¯6 considered at the beginning of §7, we see that only the third thematic factorization is monotone. It will follow from the results of this section that the thematic indices of any monotone thematic factorization must be equal

608


to 6, 2. In particular, there are no thematic factorizations with indices 7, 1. Note that it is important that the indices in (10.2) are arranged in the nonincreasing order. The first two thematic factorizations have different thematic indices 2, 6 and 1, 7 that are arranged in the increasing order. Theorem 10.1. Suppose that Φ ∈ L∞ (Mm,n ) and r ≤ min{m, n} is a positive integer such that the superoptimal singular values of Φ satisfy tr−1 > tr ,

tr−1 > HΦ e .

Then Φ admits a monotone partial thematic factorization of the form (10.1). Proof. Clearly, Hzj Φ = distL∞ (Φ, z¯j H ∞ (Mm,n )), and it is easy to see that lim Hzj Φ = distL∞ Φ, (H ∞ + C)(Mm,n ) = HΦ e < HΦ . j→∞

Put def

ι(HΦ ) = min{j ≥ 0 : Hzj Φ < HΦ }. Obviously, ι(HΦ ) depends only on the Hankel operator HΦ and does not depend on the choice of its symbol. We need three lemmas. Lemma 10.2. Let Φ be a matrix function in L∞ (Mm,n ) such that HΦ e < HΦ . Suppose that tu O Φ = W∗ (10.3) V ∗, O ∆ where V and W t are thematic matrix functions of sizes n × n and m × m, t > 0, ∆L∞ ≤ t, and u is a unimodular function such that Tu is Fredholm. Then ind Tu ≤ ι(HΦ ). Lemma 10.3. Let Φ be a badly approximable matrix function in L∞ (Mm,n ) such that HΦ e < HΦ . Then Φ admits a representation (10.3) with thematic matrix functions V and W t , t = t0 = HΦ , and a unimodular function u such that Tu is Fredholm and ind Tu = ι(HΦ ). Lemma 10.4. Let Φ ∈ L∞ (Mm,n ) be a matrix function of the form u O ∗ V ∗, Φ=W O ∆ where V and W t are thematic matrix functions of sizes n×n and m×m such that the Toeplitz operators TV and TW t are invertible, u is a unimodular function such that Tu is Fredholm, ind Tu = 0, H∆ ≤ 1, and H∆ e < 1. If HΦ < 1, then H∆ < 1. Let us first complete the proof of Theorem 10.1. We argue by induction on r. For r = 1 the result is trivial. Suppose now that r > 1. By Lemma


10.3, Φ admits a representation Φ=W

∗

t0 u0 O

O ∆

609

V ∗,

where V and W t are thematic functions, ∆L∞ ≤ t0 , and u0 is a unimodular function such that Tu0 is Fredholm and ind Tu0 = ι(HΦ ). By Theorem 4.1, H∆ e ≤ HΦ e .

(10.4)

By the inductive hypothesis, ∆ admits a monotone partial thematic factorization of the form I r−2 I r−2 O O ∗ ˘ D · · · V˘1∗ , ∆ = W1 · · · ∗ ˘∗ O W O V˘r−1 r−1 where



··· .. . ··· ···

t1 u1  ..  D= .  O O

 O ..  .  , O  Ψ

O .. . tr−1 ur−1 O

˘ j are as in (5.2). Clearly, t1 = ∆L∞ . If t1 < t0 , then and the V˘j and W it is obvious that the above factorization of ∆ leads to a monotone partial thematic factorization of Φ. Suppose now that t1 = t0 . To prove that the above factorization of ∆ leads to a monotone partial thematic factorization of Φ, we have to establish the inequality ind Tu0 ≥ ind Tu1 . By Lemma 10.2, ι(H∆ ) ≥ ind Tu1 , and it suffices to prove the inequality ι(HΦ ) = ind Tu0 ≥ ι(H∆ ). def

Put ι = ι(HΦ ). We have ι

z Φ=W

∗

t0 z ι u0 O

O zι∆

V ∗.

Clearly, ind Tzι u0 = 0. By the definition of ι, Hzι Φ < HΦ = t0 . It is easy to see that Hzι ∆ e = H∆ e < t0 by (10.4). It follows from Lemma 10.4 that Hzι ∆ < t0 , which means that ι(H∆ ) ≤ ι. Proof of Lemma 10.2. Let k = ind Tu . Clearly, it is sufficient to consider the case k > 0. By Theorem 2.5, Φ is badly approximable and HΦ = t. We have k−1 u O tz V ∗. z k−1 Φ = W ∗ O z k−1 ∆

610


Then wind(z k−1 u) = −1, and again by Theorem 2.5, z k−1 Φ is badly approximable and ΦL∞ = t. Hence, Hzk−1 Φ = z k−1 ΦL∞ = ΦL∞ = t = HΦ , and so ι(HΦ ) ≥ k. def

Proof of Lemma 10.3. Put ι = ι(HΦ ). Then Hzι−1 Φ = HΦ = ΦL∞ = z ι−1 ΦL∞ , and so z ι−1 Φ is badly approximable. Clearly, Hzι−1 Φ e = HΦ e < HΦ = Hzι−1 Φ . By Theorem 2.2, z ι−1 Φ admits a representation tω O ι−1 ∗ z Φ=W V ∗, O ∆ where t = HΦ , ω is a unimodular function such that ind Tω > 0, V and W t are thematic functions, and ∆L∞ ≤ t. Therefore ι−1 O t¯ z ω V ∗. Φ = W∗ O z¯ι−1 ∆ Let u = z¯ι−1 ω. Clearly, ind Tu ≥ ι. Finally, by Lemma 10.2, ind Tu = ι. Proof of Lemma 10.4. Let V = v Θ ,

Wt =

w

Ξ

.

By the remark after Theorem 4.5, there exist A ∈ H ∞ (Mn−1,n ) and B ∈ H ∞ (Mm−1,m ) such that AΘ = I n−1 and BΞ = I m−1 . By Theorem 1.8, without loss of generality we may assume that ∆L∞ ≤ 1. Suppose that H∆ = 1. Since H∆ e < 1, there exists a nonzero func2 tion g ∈ H 2 (Cn−1 ) such that H∆ g2 = g2 . Then ∆g ∈ H− (Cm−1 ) and ∆(ζ)g(ζ)Cm−1 = g(ζ)Cn−1 for almost all ζ ∈ T. Let f = At g + qv, where q is a scalar function in H 2 . We want to find such a q that HΦ f 2 = f 2 . Note that f is a nonzero function since ∗ ∗ t v v A g+q ∗ t (A g + qv) = V f= g Θt and g = 0. We have Φf

= W∗ = =

w

∗ t u O v A g+q g O ∆ uv ∗ At g + uq Ξ ∆g

w(uv ∗ At g + uq) + Ξ∆g.


611

Since W ∗ and V ∗ are unitary-valued and ∆(ζ)g(ζ)Cm−1 = g(ζ)Cn−1 , it follows that Φ(ζ)f (ζ)Cm = f (ζ)Cn . It remains to choose q so that 2 (Cm ). Φf ∈ H− Since W ∗ is a unitary-valued matrix function, we have wt = wwt + ΞΞ∗ . Im = w Ξ Ξ∗ Hence, Ξ = Ξ(BΞ)∗ = ΞΞ∗ B ∗ = (I m − wwt )B ∗ . It follows that Φf

= w(uv ∗ At g + uq) + (I m − wwt )B ∗ ∆g = w(uv ∗ At g + uq − wt B ∗ ∆g) + B ∗ ∆g.

2 (Cm ), and so it suffices to find q ∈ H 2 such that Clearly, B ∗ ∆g ∈ H− 2 uv ∗ At g + uq − wt B ∗ ∆g ∈ H− ,

which is equivalent to the condition Tu q = P+ (wt B ∗ ∆g − uv ∗ At g). The existence of such a q follows from the fact that Tu is Fredholm and ind Tu = 0. Corollary 10.5. Let Φ be a very badly approximable matrix function in L∞ (Mm,n ) such that HΦ e is less than the smallest nonzero superoptimal singular value of Φ. Then Φ admits a monotone thematic factorization. Corollary 10.6. Let Φ be a very badly approximable matrix function in (H ∞ + C)(Mm,n ). Then Φ admits a monotone thematic factorization. We are going to prove now that the indices of a monotone thematic factorization are uniquely determined by the function itself. We need the following lemma. Lemma 10.7. Suppose that a matrix function Φ ∈ L∞ (Mm,n ) admits a factorization of the form   O O tu0 O · · ·  O tu1 · · · O O     .. . . ..  V ∗ · · · V ∗ , ∗ ∗ . .. .. .. Φ = W0 · · · Wr−1  . 0 .    r−1  O O · · · tur−1 O  O O ··· O Ψ where the Vj and Wj are of the form (5.2), HΨ < t, and the uj are unimodular functions such that Tuj is Fredholm and ind Tuj ≤ 0. If HΦ e < t, then HΦ < t. Proof. We argue by induction on r. Let r = 1. We have tu O Φ = W∗ V ∗, O Ψ

612


where V and W t are thematic matrix functions, u is a unimodular function such that Tu is Fredholm, ind Tu ≤ 0, and HΨ < t. It follows from Theorem 1.8 that we may subtract from Ψ a best analytic approximation without changing HΦ , and so we may assume that ΨL∞ < t. Without loss of generality we may also assume that t = 1. Suppose that HΦ = 1. Since HΦ e < 1, there exists a nonzero function f ∈ H 2 (Cn ) such that HΦ f 2 = f 2 . Then Φf 2 = f 2 and since ΨL∞ < 1, it follows that V ∗ f has the form   ∗  O    V ∗f =  .  . (10.5)  ..  O Let v be the first column of V . Equality (10.5) means that for almost all ζ ∈ T the remaining columns of V (ζ) are orthogonal to f (ζ) in Cn . Since V is unitary-valued, it follows that f = ξv for a scalar function ξ ∈ L2 . By Lemma 1.4, ξ ∈ H 2 . Note that f H 2 (Cn ) = ξH 2 . We have   uξ  O    Φf = W ∗  .  = uξw,  ..  O where w is the first column of W . Since f is a maximizing vector of HΦ , 2 2 we have uξw ∈ H− (Cn ). Again, by Lemma 1.4, we find that uξ ∈ H− , i.e., ξ ∈ Ker Tu . However, Tu has trivial kernel since ind Tu ≤ 0. We have obtained a contradiction. Suppose now that r > 1. Again, we may assume that ΨL∞ < t. Let d be a negative integer such that d < ind Tuj , 0 ≤ j ≤ r − 1. Then   d O ··· O O tz u0  O tz d u1 · · · O O     . . . ..  V ∗ · · · V ∗ d ∗ ∗ .. .. .. .. z Φ = W0 · · · Wr−1   r−1 . 0 .    O O  O · · · tz d ur−1 O O ··· O zdΨ t

is a partial thematic factorization of z d Φ. Put I r−2 I r−2 O ∗ ˘ D ∆ = W1 · · · ˘∗ O W O r−1 where



t1 u1  ..  D= .  O O

··· .. . ··· ···

O .. . tr−1 ur−1 O

O V˘ ∗

r−1

 O ..  .  , O  Ψ

· · · V˘1∗ ,


613

˘ j are as in (5.2). Since obviously, Hzd ∆ e = H∆ e for and the V˘j and W any d ∈ Z, it follows from Theorem 4.1 that Hzd ∆ e < t, and so by the inductive hypotheses, H∆ < t. We have tu O Φ = W0∗ V0∗ . O ∆ The result follows now from the case r = 1, which has already been established. Theorem 10.8. Let Φ be a badly approximable function in L∞ (Mm,n ) such that HΦ e < HΦ and let r be the number of superoptimal singular values of Φ equal to t0 = HΦ . Consider a monotone partial thematic factorization of Φ with indices k0 ≥ · · · ≥ kr−1

(10.6)

corresponding to the superoptimal singular values equal to t0 . Let κ ≥ 0. Then kj − κ. dim{f ∈ H 2 (Cn ) : Hzκ Φ f 2 = t0 f 2 } = (10.7) {j∈[0,r−1]:kj >κ} Proof. Let

    Φ=  

t0 u0 O .. .

O t0 u1 .. .

··· ··· .. .

O O .. .

O O

O O

··· ···

t0 ur−1 O

O O .. .



     O  Ψ

be a partial thematic factorization of Φ with indices satisfying (10.6). If κ ≥ k0 , then (10.7) holds by Lemma 10.7. Suppose now that κ < k0 . Let q = max{j ∈ [0, r − 1] : kj > κ}. Clearly, the function z κ Φ admits the following representation:   O ··· O O t0 z κ u0  O O  O t0 z κ u1 · · ·   . . . ..  V ∗ · · · V ∗ , κ ∗ ∗ . .. .. .. .. z Φ = W0 · · · W q  0 .    q κ  O O · · · t0 z uq O  O O ··· O ∆ where ∆ is a matrix function satisfying the hypotheses of Lemma 10.7. By Lemma 10.7, H∆ < t0 . Let R ∈ H ∞ be a matrix function such that ∆ − RL∞ < t0 . It follows easily from Theorem 1.8 by induction on q that if we perturb ∆ by a bounded analytic matrix function, z κ Φ also changes by an analytic matrix function. In particular, we can find a matrix function

614


G ∈ H ∞ such that zκ Φ − G     =W0∗ · · · Wq∗   

t0 z κ u0 O .. .

O t 0 z κ u1 .. .

··· ··· .. .

O O

O O

··· ···

O O .. .



O O .. .

t0 z κ u q O

O ∆−R

   ∗  Vq · · · V0∗ .  

By Theorem 7.4,

dim{f ∈ H 2 (Cn ) : Hzκ Φ−G f 2 = t0 f 2 } =

kj − κ.

{j∈[0,r−1]:kj >κ}

Equality (10.7) follows now from the obvious fact that Hzκ Φ−G = Hzκ Φ . We can now deduce from (10.7) the following result. Theorem 10.9. Suppose that Φ ∈ L∞ (Mm,n ) and q ≤ min{m, n} is a positive integer such that the superoptimal singular values of Φ satisfy tq−1 > tq ,

tq−1 > HΦ e

and Φ admits a monotone partial thematic factorization  Φ=

W0∗

∗ · · · Wq−1

     

t0 u0 O .. .

O t 1 u1 .. .

··· ··· .. .

O O .. .

O O

O O

··· ···

tq−1 uq−1 O

O O .. .



   ∗  Vq−1 · · · V0∗ .  O  ∆

Then the indices of this factorization are uniquely determined by the function Φ itself. Proof. Let r be the number of superoptimal singular values equal to HΦ . Then Φ admits the following partial thematic factorization  Φ=

W0∗

∗ · · · Wr−1

     

t 0 u0 O .. .

O t1 u1 .. .

··· ··· .. .

O O .. .

O O .. .

O O

O O

··· ···

tr−1 ur−1 O

Ψ

    ∗  Vr−1 · · · V0∗ ,  

11. Construction of Superoptimal Approximation, Corona Problem

where ˘ ∗ ··· Ψ=W r 

tr ur  ..  D= .  O O

I q−r−1 O

O ˘∗ W q−1

··· .. . ··· ···

O .. . tq−1 uq−1 O

D

I q−r−1 O

O ∗ V˘q−1

615

· · · V˘r∗ ,

 O ..  .  , O  ∆

˘ j are as in (5.2). and the V˘j and W By Theorem 9.3, Ψ is determined uniquely by Φ modulo constant unitary factors. Hence, it is sufficient to show that the indices k0 , · · · , kr−1 are uniquely determined by Φ. It follows easily from (10.7) that k0 = min κ : dim{f ∈ H 2 (Cn ) : Hzκ Φ f 2 = t0 f 2 } = 0 . Now let d be the number of indices among k0 , · · · , kr−1 that are to equal to k0 . It follows easily from (10.7) that d = dim{f ∈ H 2 (Cn ) : Hzk0 −1 Φ f 2 = t0 f 2 }. Next, if d < r, then it follows from (10.7) that kd = min κ : dim{f ∈ H 2 (Cn ) : Hzκ Φ f 2 = t0 f 2 } = d(k0 − κ) . Similarly, we can determine the multiplicity of the index kd , then the next largest index, etc. Corollary 10.10. Suppose that Φ is very badly approximable function in L∞ (Mm,n ) such that HΦ e is less than the largest nonzero superoptimal singular value of Φ. Then the indices of a monotone thematic factorization of Φ are uniquely determined by Φ. Corollary 10.11. Let Φ be a very badly approximable function in (H ∞ + C)(Mm,n ). Then the indices of a monotone thematic factorization of Φ are uniquely determined by Φ.

11. Construction of Superoptimal Approximation and the Corona Problem In §§2–4 we have given an algorithm to find a superoptimal approximation of a matrix function Φ ∈ L∞ (Mm,n ) that satisfies certain natural conditions. Recall that if min{m, n} > 1, we start with a best approximation F ∈ H ∞ (Mm,n ) and find a factorization t0 u O Φ − F = W∗ V ∗, O Ψ

616


where V and W t are thematic matrix functions of the form V = v Θ , Wt = w Ξ .

(11.1)

Then it has been shown in §2 that the problem of finding the superoptimal approximation of Φ reduces to the problem of finding the superoptimal approximation of Ψ that has a smaller size. However, the problem is that first we have to find a best approximation F and the procedure discussed in §§2–4 does not give any hint how to find such a best approximation F . This seems even more disappointing since we want to prove in §12 that the nonlinear operator of superoptimal approximation has hereditary properties and to do it we have to find a best approximation F that inherits properties of Φ. Fortunately, to use the above scheme, F does not have to be a best approximation. It is sufficient to assume that F ∈ H ∞ (Mm,n ) and satisfies certain equations. We show in this section how to find such a function F by solving certain corona problems. Throughout this section we assume that HΦ e < HΦ . Recall that to find thematic matrix functions V and W t , we start with a maximizing vector f of HΦ and define g = t−1 ¯HΦ f . The vector functions f and g 0 z admit factorizations f = ϑ1 hv, g = ϑ2 hw, where ϑ1 and ϑ2 are scalar inner functions, h is a scalar outer, function in H 2 , and v and w are inner and co-outer column functions. The entry u in ¯ (11.1) must satisfy u = z¯ϑ¯1 ϑ¯2 h/h. If F is a best approximation of Φ, then by Theorem 2.2.3, HΦ f = (Φ − F )f = t0 z¯g,

HΦ∗ z¯g = (Φ − F )∗ z¯g = t0 f ,

and so F f = Φf − t0 z¯g = P+ Φf ,

F ∗ z¯g = Φ∗ z¯g − t0 f = P− Φ∗ z¯g.

Thus F must satisfy the following equations F f = TΦ f ,

F t g = TΦt g.

It follows that F v = ϕ1 ,

F t w = ϕ2 ,

(11.2)

where ¯ ϑ¯1 TΦ f def ϑ2 TΦt g , ϕ2 = . (11.3) h h Clearly, (11.2) implies that ϕ1 ∈ H ∞ (Cm ) and ϕ2 ∈ H ∞ (Cn ). It also follows from (11.2) that def

ϕ1 =

ϕt2 v = wt ϕ1 , since both functions are equal to wt F v.

(11.4)

11. Construction of Superoptimal Approximation, Corona Problem

617

Theorem 11.1. Suppose that Φ ∈ L∞ (Mm,n ) and HΦ e < HΦ . Let F be an arbitrary matrix function in H ∞ (Mm,n ) satisfying (11.2). Then Φ − F satisfies (11.1) for some matrix function Ψ ∈ L∞ (Mm−1,n−1 ). A matrix function G ∈ H ∞ (Mm−1,n−1 ) is a superoptimal approximation of Ψ if and only if F + ΞGΘt is superoptimal approximation of Φ. Proof. We have

W (Φ − F )V

= =

wt Ξ∗

(Φ − F )

wt (Φ − F )v Ξ∗ (Φ − F )v

v

Θ

wt (Φ − F )Θ Ξ∗ (Φ − F )Θ

.

It follows from (11.2) that Ξ∗ (Φ − F )v

=

Ξ∗ Φv − Ξ∗ F v = Ξ∗ Φv − Ξ∗

ϑ¯1 TΦ f h

ϑ¯1 ∗ ϑ¯1 ∗ Ξ (Φf − TΦ f ) = Ξ HΦ f h h ¯ ϑ¯1 ϑ¯1 ϑ¯2 h = t0 Ξ∗ z¯g = t0 Ξ∗ w = O, h h since the matrix function W is unitary-valued. Similarly, wt (Φ − F )Θ = (Θ∗ (Φt − F t )w)t = O. =

Next, wt (Φ − F )v =

¯ ϑ¯1 ϑ¯1 ϑ¯2 h ϑ¯1 t w HΦ f = t0 wt z¯g = t0 = t0 u. h h h

This proves (11.1) with Ψ = Ξ∗ (Φ − F )Θ. The rest of the theorem follows from Theorem 1.8 in the same way as it was done in the proof of Theorem 2.4. Now we are going to find a matrix function F ∈ H ∞ (Mm,n ) that satisfies (11.2). Consider the vector functions v and w. By Theorem 4.3, the Toeplitz operators Tv and Tw are left invertible. By Theorem 3.6.1, the column functions v and w are left invertible in H ∞ . In other words, we can solve the corona problems with data v and w, and find vector functions ψ1 ∈ H ∞ (Cn ) and ψ2 ∈ H ∞ (Cm ) such that ψ1t v = ψ2t w = 1,

(11.5)

where as usual 1 is the function identically equal to 1. Theorem 11.2. Let F = ϕ1 ψ1t + ψ2 ϕt2 − ψ2 ϕt2 vψ1t ,

(11.6)

where ϕ1 and ϕ2 are given by (11.3), and ψ1 and ψ2 satisfy (11.5). Then F ∈ H ∞ (Mm,n ) and F satisfies (11.2).

618


Proof. Clearly, F ∈ H ∞ (Mm,n ). We have F v = ϕ1 ψ1t v + ψ2 ϕt2 v − ψ2 ϕt2 vψ1t v. By (11.5), ϕ1 ψ1t v = ϕ1 . By (11.4) and (11.5), ψ2 ϕt2 v = ψ2 wt ϕ1 = ϕ1 . Finally, ψ2 ϕt2 vψ1t v = ϕ1 ψ1t v = ϕ1 by (11.5). This proves that F v = ϕ1 . Let us prove the second equality in (11.2). We have F t w = ψ1 ϕt1 w + ϕ2 ψ2t w − ψ1 v t ϕ2 ψ2t w. By (11.4) and (11.5), ψ1 ϕt1 w = ψ1 v t ϕ2 = ϕ2 . By (11.5), ϕ2 ψ2t w = ϕ2 . Finally, by (11.5), ψ1 v t ϕ2 ψ2t w = ϕ2 , which completes the proof of the theorem. The procedure described above reduces the problem of finding a superoptimal approximation of Φ to the corresponding problem for Ψ. If HΦ e is less than the smallest nonzero superoptimal singular value of Φ, we can iterate this procedure as we have done in §§3–4 and eventually find the superoptimal approximation of Φ. An advantage of the algorithm given in this section is that it does not require finding a best approximation. Instead, it depends on the solution of certain corona problems and we shall see in the next two sections that this will be beneficial for the study of hereditary properties and continuity properties of the nonlinear operator of superoptimal approximation. We refer the reader to Jones [1], where a constructive solution of the corona problem is given.

12. Hereditary Properties of Superoptimal Approximation Theorem 3.3 enables us to define the nonlinear operator A(m,n) of superoptimal approximation on the class (H ∞ + C)(Mm,n ). Namely, if Φ ∈ (H ∞ + C)(Mm,n ) and Q ∈ H ∞ (Mm,n ) is the superoptimal approximation of Φ by analytic matrix functions, we put A(m,n) Φ = Q. In Chapter 7 we have introduced the notation A for the operator of best approximation by scalar analytic functions. Clearly, A = A(1,1) . We are also going to use the notation A = A(m,n) if this does not lead to confusion. Consider now a space X of functions on T. We study in this section the question for which spaces X the condition Φ ∈ X implies that AΦ ∈ X. (By definition a matrix function belongs to X if all its entries belong to X.) Recall that a function space is called hereditary for the operator A(1,1) of best approximation by analytic functions if A(1,1) X ⊂ X. We say that X is called completely hereditary if A(m,n) X(Mm,n ) ⊂ X(Mm,n ) for all positive integers m and n.

12. Hereditary Properties of Superoptimal Approximation

619

It is convenient to enlarge the domain of the operator A(m,n) to the space V M O(Mm,n ). Indeed, if Φ ∈ (H ∞ + C)(Mm,n ) and F ∈ H ∞ (Mm,n ), then A(m,n) (Φ + F ) = A(m,n) Φ + F . Suppose now that Ψ ∈ V M O(Mm,n ). Then Ψ admits a representation Ψ = Φ + G, where Φ is continuous and G ∈ V M OA(Mm,n ) and we put AΨ = AΦ + G. It is easy to see that the operator of superoptimal approximation is well defined on V M O. Thus we can consider the problem of whether a function space X is completely hereditary as far as X ⊂ V M O. We show in this section that both linear R-spaces (see §7.1) and decent function spaces (see §7.2) are completely hereditary. We also show that A(m,n) is bounded on (quasi-)Banach R-spaces, i.e., there exists cm,n > 0 such that A(m,n) ΦX ≤ cm,n ΦX (see §13.1 where the notation ΦX is explained). We also prove in this section that if Φ ∈ X, then Φ − AΦ has a thematic factorization with all factors in X. Note, however, that we use different methods to treat R-spaces and decent function spaces. Let us start with the following lemma. Lemma 12.1. Suppose that X is a linear R-space or X is a decent function space. Let V be a balanced matrix function of the form V = Υ Θ , where Υ and Θ are inner matrix functions. If Υ ∈ X, then Θ ∈ X. If X is a (quasi-)Banach R-space, then ΘX ≤ const ΥX . Note that in this section constants may depend on the size of matrix functions. Proof. By Theorem 1.3, TV ∗ has dense range. By the assumptions, P− V ∗ ∈ X. The result now follows from Theorem 13.1.1. First we prove that linear R-spaces are completely hereditary. Let Φ be an m × n matrix function in V M O. Then there exists an m × n matrix function F ∈ BM OA such that Φ − F is bounded on T and Φ − F L∞ = HΦ . In other words, Φ − F is a badly approximable matrix function. If P− Φ = O, then Φ − F admits a factorization t0 u O ∗ Φ−F =W V ∗, (12.1) O Ψ where V and W t are thematic matrix functions of the form V = v Θ , Wt = w Ξ .

(12.2)

620


Lemma 12.2. Let X be a linear R-space and let Φ be a matrix function in X. Then u ∈ X, v ∈ X, and w ∈ X. If X is a (quasi-)Banach R-space, then uX ≤ const ΦX ,

vX ≤ const ΦX ,

wX ≤ const ΦX . (12.3)

Proof. Clearly, we may assume that P− Φ = O. It follows from (12.1) that wt (Φ − F )v = t0 u. We have ¯ = P+ w∗ (¯ z Φ − z¯F )v = P+ w∗ z¯Φv = P+ w∗ (P+ z¯Φ)v. t0 P+ z¯u ¯ ∈ X. Hence, It follows from Lemma 7.1.1 that P+ z¯u P− u = z¯P+ z¯u ¯ ∈ X. ¯ By Theorem 2.2, u has the form u = ϑ¯h/h, where ϑ is an inner function 2 and h is an outer function in H . It follows from Theorem 4.4.10 that Tu has dense range in H 2 . Thus by Theorem 7.1.4, u ∈ X. It follows from (12.1) that t0 u O V ∗. W (Φ − F ) = O Ψ Take the first row to get wt (Φ − F ) = t0 uv ∗ . Consequently, ¯v t = zP+ z¯w∗ (Φ − F ) = zP+ z¯w∗ P− Φ. t0 P− uv ∗ = t0 zP+ z¯u Again, it follows from Lemma 7.1.1 that P− uv ∗ ∈ X. Clearly, P+ uv ∗ = Tv∗ u ∈ X ¯ ∈ X, it follows from Lemma by Lemma 7.1.1. Hence uv ∗ ∈ X, and since u 7.1.1 that v ∗ = u ¯uv ∗ ∈ X. Therefore v ∈ X. If we apply the same argument to Φt , we find that w ∈ X. It is easy to see that in case X is a (quasi-)Banach R-space, the above reasonings show that (12.3) holds. Now we can prove that heredity result for linear R-space. Theorem 12.3. Let X be a linear R-space. Then X is completely hereditary. If X is a (quasi-)Banach R-space, then the operators A(m,n) are bounded on X(Mm,n ). Proof. We may assume that Φ ∈ L∞ . Indeed, for any Φ ∈ X we can write Φ = Φ 1 + Φ2

(12.4)

12. Hereditary Properties of Superoptimal Approximation

621

for some Φ1 ∈ X ∩ L∞ and Φ2 ∈ V M OA . It is sufficient to find such a representation for each entry of Φ. For a scalar function ϕ in X we can write ϕ = (ϕ − Aϕ) + Aϕ. The fact that ϕ − Aϕ ∈ X follows from Theorem 7.1.6. If Φ is represented as in (12.4), then clearly, AΦ = AΦ1 + Φ2 . Suppose first that min{m, n} = 1. Without loss of generality we may assume that m ≤ n. Since the scalar case has already been treated in §7.1, consider the case n > 1. If t0 = 0, the result is trivial. Otherwise, let F be the best approximation of Φ (it is unique by Theorem 2.2). Then Φ − F = W ∗ t0 u O (see comments after the statement of Theorem 2.2). By Lemmas 12.1 and 12.2, u and W belong to X, which implies that F ∈ X. Suppose now that d = min{m, n} > 1. We argue by induction on d. Suppose that F ∈ H ∞ (Mm,n ) is the superoptimal approximation of Φ by bounded analytic matrix functions. Then Φ − F admits a factorization (12.1) with very badly approximable matrix function Ψ ∈ L∞ (Mm−1,n−1 ). It follows from (12.1) that t0 u O W (Φ − F )V = . O Ψ If we look at the lower right entries of this equality, we obtain Ψ = Ξ∗ (Φ − F )Θ, and so P− Ψ = z¯(P+ z¯Ψ∗ )∗ = z¯(P+ z¯Θt Φ∗ Ξ)∗ − z¯(P+ z¯Θt F ∗ Ξ)∗ . By Lemma 12.1, Θ and Ξ belong to X, and so by Lemma 7.1.1, z¯(P+ z¯Θt Φ∗ Ξ)∗ ∈ X. Again, since F ∈ H ∞ , it follows from Lemma 7.1.1 that z¯(P+ z¯Θt F ∗ Ξ)∗ , which proves that P− Ψ ∈ X. Since Ψ is very badly approximable, AP− Ψ = −P+ Ψ, and by the inductive hypothesis, AP− Ψ ∈ X, which proves that Ψ ∈ X. It follows now from (12.1) that F ∈ X. It is easy to see that if X is a (quasi-)Banach R-space, then the above reasonings show that F X ≤ const ΦX , i.e., the operator of superoptimal approximation is bounded on X(Mm,n ). Corollary 12.4. Let Φ be a rational matrix function with poles outside T. Then AΦ is also rational. Corollary 12.5. Let Φ be a matrix function in H ∞ + C. Then Φ − AΦ ∈ QC. Proof. We have AΦ = AP− Φ + P+ Φ, and so Φ − AΦ = P− Φ − AP− Φ.

622


Clearly, P− Φ ∈ V M O and since V M O is an R-space, it follows from Theorem 12.3 that AP− Φ ∈ V M O. The result follows from the fact that QC = L∞ ∩ V M O (see Appendix 2.5). We proceed now to the case of decent function spaces. To prove that linear R-spaces X are completely hereditary we have used essentially the following property: ϕ ∈ X+ , f ∈ H ∞ =⇒ P+ f¯ϕ ∈ X. (12.5) Many classical decent spaces (e.g., Hölder spaces, Besov spaces) possess property (12.5). However, there are important decent spaces that do not satisfy (12.5). For example, the space F1 of functions with absolutely converging Fourier series does not satisfy (12.5). Indeed, if (12.5) holds for X = F1 , then for any f ∈ H ∞ there exists a constant c such that P+ f¯ϕF 1 ≤ cϕF 1 . Now it remains to take ϕ = z n to see that f must belong to F1 . To prove that decent spaces are completely hereditary we use properties of maximizing vectors of Hankel operators. Theorem 12.6. Let X be a decent function space. Then X is completely hereditary. Proof. Let Φ be an m × n matrix function in X. We want to prove that AΦ ∈ X. If P− Φ = O, then AΦ = Φ and the result is trivial. Suppose now that P− Φ = O. By Theorem 13.2.2, there is a maximizing vector f of HΦ such that f (ζ) = O for any ζ ∈ T. It admits a factorization of the form f = ϑ1 hv, where ϑ1 is a scalar inner function, h is a scalar outer function, and v is ¯HΦ f . Then g an inner and co-outer column function (see §2). Let g = t−1 0 z is a maximizing vector of HΦt and g admits a factorization of the form g = ϑ2 hw, where ϑ2 is a scalar inner function and w is an inner and co-outer column function (see §2). By Theorem 4.2, ϑ1 and ϑ2 are finite Blaschke products. By the axiom (A4) (see §7.2), ϑ1 and ϑ2 belong to X. Since |h(ζ)| = f (ζ)Cn , it follows that h is invertible in X, and so v = h−1 ϑ¯1 f ∈ X. Clearly, g ∈ X, and so w = h−1 ϑ¯2 g ∈ X. By Lemma 12.1, the thematic matrix functions V and W t defined by (12.2) belong to X. We argue by induction on d = min{m, n}. If d = 1, then the unique best approximation F is determined by (12.1) and, obviously, F ∈ X. Suppose that d > 1. Let F be the matrix function defined in (11.6), i.e., F = ϕ1 ψ1t + ψ2 ϕt2 − ψ2 ϕt2 vψ1t , where def

ϕ1 =

ϑ¯1 TΦ f , h

def

ϕ2 =

ϑ¯2 TΦt g , h

13. Continuity Properties of Superoptimal Approximation

623

and ψ1 and ψ2 are solutions of the corona problems ψ1t v = ψ2t w = 1.

(12.6)

Clearly, ϕ1 and ϕ2 belong to X. It follows from Theorem 7.2.6 (see also its proof) that there exist solutions ψ1 and ψ2 of (12.6) that belong to X+ . It follows that F ∈ X. By Theorem 11.1, Φ − F satisfies (12.1) for some Ψ ∈ L∞ (Mm−1,n−1 ). Since Φ, V, W, F ∈ X, it follows that Ψ ∈ X. By Theorem 11.1, the problem of finding the superoptimal approximation of Φ reduces to the problem of finding the superoptimal approximation of Ψ. The result now follows from the inductive hypothesis. It is easy to see that the proofs of Theorems 12.3 and 12.6 imply the following result. Theorem 12.7. Suppose that X is a linear R-space or X is a decent function space. Let Φ be a matrix function in X. Then Φ − AΦ has a thematic factorization with all factors in X. In fact, it follows easily from the proof of Theorem 12.3 that in the case of a linear R-space X all factors of each thematic factorization of Φ − AΦ are in X. Corollary 12.8. Suppose that Φ is a matrix function in H ∞ + C. Then all factors of each thematic factorization of Φ − AΦ belong to QC.

13. Continuity Properties of Superoptimal Approximation In this section we study the important problem of continuity of the operator of superoptimal approximation A = A(m,n) in the norm of decent function spaces X. As in the scalar case the continuity problem is important in applications since to compute a superoptimal approximation in practice, one has to be sure that a minor deviation of the initial data does not lead to a huge perturbatioon of the result. In other words, one needs continuous dependence of the solution on the initial matrix function. Recall that in the scalar case a function ϕ ∈ X with P− ϕ = O is a continuity point of A in the norm of X if and only if HΦ is a singular value of HΦ of multiplicity 1 (see Theorem 7.11.6). We obtain analogs of that result for the operator of superoptimal approximation. We start with sufficient conditions. First we consider the case of an m×n matrix function Φ such that tmin{m,n}−1 (Φ) = 0. We will see later that in the case tmin{m,n}−1 (Φ) = 0 the result depends on the decent space X. We conclude the section with necessary conditions. Theorem 13.1. Let X be a decent function space and let Φ ∈ X(Mm,n ). Suppose that tmin{m,n}−1 (Φ) = 0. If Φ − AΦ has a thematic factorization

624


with indices k0 = k1 = · · · = kmin{m,n}−1 = 1,

(13.1)

then Φ is a continuity point of the operator A(m,n) of superoptimal approximation in X(Mm,n ). By Theorem 7.2, (13.1) implies that all thematic factorizations of Φ−AΦ have indices equal to 1. To prove Theorem 13.1, we use the construction of the superoptimal approximation based on the corona theorem (see §11). We need several lemmas. Lemma 13.2. Suppose that Φ ∈ X(Mm,n ) and Φ−AΦ has a thematic factorization whose indices kj are equal to 1 whenever tj (Φ) = HΦ . Then all maximizing vectors of HΦ and of HΦt are co-outer and do not vanish on T. Proof. Clearly, it follows from Theorem 7.2 that for each thematic factorization of Φ − AΦ the indices kj are equal to 1 whenever tj (Φ) = HΦ . Let f ∈ H 2 (Cn ) be a maximizing vector of HΦ . We have to prove that f is co-outer and f (ζ) = O for any ζ ∈ T. Put g = t−1 ¯HΦ f . Let us first 0 z show that f and g are co-outer. We have f = ϑ1 hv,

g = ϑ2 hw,

where ϑ1 and ϑ2 are scalar inner functions, and v and w are inner and co-outer vector functions. Consider thematic completions V = v Θ and W t = w Ξ of v and w. Then by Theorem 2.2, F ∈ H ∞ (Mm,n ) the matrix function Φ − F t0 u0 Φ − F = W∗ O

for any best approximation admits a factorization O (13.2) V ∗, Ψ

¯ where ΨL∞ ≤ t0 and u0 = z¯ϑ1 ϑ2 h/h. Then the index k0 = ind Tu0 of this factorization is greater than 1 if ϑ1 or ϑ2 is not constant. Suppose now that f vanishes on T. Without loss of generality we may assume that f (1) = O. Then by Theorem 13.2.2, the function f # = (1 − z)−1 f ∈ X+ (Cn ) is also a maximizing vector of HΦ . Then the outer function h# = h/(z − 1) belongs to X and ¯# ¯ ¯# h h z¯ − 1 h u0 = z¯ϑ1 ϑ2 = z¯ϑ1 ϑ2 = −¯ z 2 ϑ1 ϑ2 , h z − 1 h# h# and again the factorization index k0 that corresponds to the factorization (13.2) is at least 2, which leads to a contradiction. For a matrix function G ∈ X(Mm,n ) we consider the Toeplitz operator TG : H 2 (Cn ) → H 2 (Cm ). Obviously, TG acts as a bounded linear operator from X+ (Cn ) to X+ (Cm ). To avoid confusion, we denote by TGX the Toeplitz operator TG as an operator from X+ (Cn ) to X+ (Cm ). Similarly, X we denote the Hankel operator HG as an operator from X+ (Cn ) by HG


625

∗X ∗ to X− (Cm ) and by HG the operator HG as an operator from X− (Cm ) to n X+ (C ). Lemma 13.3. Let n > 1 and let v be an inner and co-outer vector function in X+ (Cn ). Then 0 is an isolated point of spectrum of the operators Tv¯X TvXt on X+ (Cn ) and Tv¯ Tvt on H 2 (Cn ).

Proof. Let us prove the lemma for the operator Tv¯X TvXt . The proof for X X the operator Tv¯ Tvt is even simpler. Let us first show that 0 ∈ σ(Tv¯ Tvt ). Indeed, let v Υ be a thematic matrix function. By Theorems 13.5.1 and 1.3, Υ ∈ X. Clearly, the columns of Υ belong to Ker TvXt , and so 0 ∈ σ(Tv¯X TvXt ). Consider the operator TvXt Tv¯X . By Lemma 5.4.6, it is sufficient to show that it is invertible. We have TvXt Tv¯X = I − HvXt Hv¯X . The operator HvXt Hv¯X is compact on X+ (Cn ) (see the axiom (A3) in §7.2), and so it is sufficient to show that Ker TvXt Tv¯X = {O}. Let ϕ ∈ Ker TvXt Tv¯X . 2 v L∞ (Cn ) = 1, it follows that v¯ϕ ∈ H− (Cn ). Then Hvt Hv¯ ϕ = ϕ. Since ¯ t 2 n 1 t 2 n 2 Thus ϕv ¯ H (C ) ⊂ zH . Since v is co-outer, v H (C ) is dense in H , and so ϕH ¯ 2 (Cn ) ⊂ zH 1 . Consequently, ϕ = O. For an inner and co-outer function v ∈ H ∞ (Cn ) we denote by Lv the kernel of Tvt and by Pv the orthogonal projection from H 2 (Cn ) onto Lv . X Similarly, we denote by LX ¯ Tv t and v the kernel of Tv t . Clearly, Lv = Ker Tv X X X Lv = Ker Tv¯ Tvt . Consider a simple closed positively oriented Jordan curve γ that lies in the resolvent sets of Tv¯ Tvt and Tv¯X TvXt , encircles zero, but does not wind round any other point of the spectra of Tv¯ Tvt and Tv¯X TvXt . Clearly, ? 1 Pv = (ζI − Tv¯ Tvt )−1 dζ. 2πi γ Consider the projection PvX from X+ (Cn ) onto LX v defined by ? −1 1 ζI − Tv¯X TvXt dζ. PvX = 2πi γ

(13.3)

Obviously, PvX ϕ = Pv ϕ for ϕ ∈ X+ (Cn ). Suppose now that {vj }j≥1 is a sequence of inner and co-outer functions in X+ (Cn ), which converges to v in the norm. Then TvXt Tv¯Xj → TvXt Tv¯X in j

the operator norm of X+ (Cn ). As in the proof of Lemma 13.3, TvXt Tv¯X is invertible, and hence there is a neighborhood O of zero that lies in the resolvent set of TvXt Tv¯X and of TvXt Tv¯Xj for all sufficiently large j. Without j loss of generality we may assume that this holds for all values of j. Choose a simple closed contour γ lying in O and winding round 0. Then 0 is the only point inside or on γ of the spectra of Tv¯j Tvjt and Tv¯Xj TvXt . We can therefore j

define projections Pv , PvX , Pvj , PvXj by integrals as above, all using the

626


same contour γ. It is then easy to see from (13.3) that PvXj → PvX in the operator norm. Lemma 13.4. Let V = v Θ be a thematic matrix function, where v ∈ H ∞ (Cn ) and Θ ∈ H ∞ (Mn,n−1 ). Suppose that {vj }j≥1 is a sequence of inner and co-outer functions that converges to v in X+ (Cn ). def is There exist inner co-outer functions Θj such that Vj = vj Θj thematic and V − Vj X(Mn,n ) → 0. Proof. It has been shown in the proof of Theorem 1.1 that v has a thematic completion Θ such that the columns of Θ have the form Pv C1 , Pv C2 , · · · , Pv Cn−1 , where C1 , C2 , · · · , Cn−1 are constant column functions. By Theorem 1.2, the same is true for all thematic completions. Hence, the columns of Θ have the form Pv Ck for some constants Ck . Consider the subspace of H 2 (Cn ) def

Pv Cn = {Pv C : C ∈ Cn }, where we identify C ∈ Cn with a constant function in H 2 (Cn ). This space has the remarkable property that the pointwise and H 2 inner products coincide on it. That is, if ϕk = Pv Ck , k = 1, 2, where C1 , C2 ∈ Cn , then (ϕ1 , ϕ2 )H 2 (Cn ) = (ϕ1 (ζ), ϕ2 (ζ))Cn for almost all ζ ∈ T. Indeed, Pv Cn = ΘH 2 (Cn−1 ) (see the proof of Theorem 1.1), and for any C ∈ Cn Pv C = ΘP+ Θ∗ C = ΘΘ(0)∗ C. Thus (ϕ1 , ϕ2 )H 2 (Cn ) = (Pv C1 , Pv C2 )H 2 (Cn ) = (ΘΘ(0)∗ C1 , ΘΘ(0)∗ C2 )H 2 (Cn ) = (Θ(0)∗ C1 , Θ(0)∗ C2 )H 2 (Cn−1 ) = (Θ(0)∗ C1 , Θ(0)∗ C2 )Cn−1 = (Θ(ζ)Θ(0)∗ C1 , Θ(ζ)Θ(0)∗ C2 )Cn = (ϕ1 (ζ), ϕ2 (ζ))Cn for almost all ζ ∈ T. It follows that any unit vector in Pv Cn is an inner column function, and any orthonormal sequence (with respect to the inner product of H 2 (Cn )) of vectors in Pv Cn constitutes the columns of an inner function. Now let Pv Ck , 1 ≤ k ≤ n − 1, be the columns of Θ, and consider the functions Pvj C1 , Pvj C2 , · · · , Pvj Cn−1 . Clearly, Pv Ck − Pvj Ck X(Cn ) = PvX Ck − PvXj Ck X(Cn ) → 0 as j → ∞. It follows that for large values of j the inner products (Pvj Ck1 ,Pvj Ck2 )H 2 (Cn ) are small for k1 = k2 and are close to 1 if k1 = k2 . We shall show that the desired Θj can be obtained by orthonormalizing the Pvj Ck . Pick M > 1 such that Pvj Ck X(Cn ) ≤ M , j ≥ 1, 1 ≤ k ≤ n − 1. By the equivalence of norms on finite-dimensional spaces there exists K > 0 such


627

that for any (n − 1) × (n − 1) matrix T = {tkl }, max | tkl | ≤ T ≤ K · max | tkl |

(13.4)

(here T is the operator norm of T on Cn−1 ). Let 0 < ε < 1. Choose j0 such that j ≥ j0 implies ε Pvj Ck − Pv Ck X(Cn ) < , k = 1, · · · , n − 1, 2 and ε | (Pvj Ck , Pvj Cl ) − δkl |< , k, l = 1, · · · , n − 1, 2KnM

(13.5)

(13.6)

(here δkl is the Kronecker symbol). Fix j ≥ j0 and let T : Cn−1 → Pvj Cn be the operator that maps the kth standard basis vector ek of Cn−1 to Pvj Ck . The matrix of the operator T ∗ T on Cn−1 is the Gram matrix (Pvj Ck , Pvj Cl ), and so by (13.4) and (13.6) we have 1 ε T ∗ T − I < < . 2nM 2 By diagonalization, ∗ −1 (T T ) 2 − I < ε . 2nM ∗ − 12 Let {τkl } be the matrix of (T T ) , then ε . | τkl − δkl |< 2nM 1

Let T = U (T ∗ T ) 2 be the polar decomposition of T , where U is a unitary operator from Cn−1 onto Pvj Cn . Then the vectors U e1 , · · · , U en−1 are orthonormal in Pvj Cn . Let Θj be the n × (n − 1) matrix function with columns U e1 , . . . , U en−1 . By the remark above, Θj is inner. By the fact that Pvj Cn ⊂ Lvj , the columns of Θj are pointwise orthogonal to vj . Hence, def Vj = vj Θj is unitary-valued. Furthermore, the jth column U ej of Θj satisfies 1

Pvj Ck − U ek X(Cn ) = T ek − T (T ∗ T )− 2 ek X(Cn )  τ1k   .. = T ek − T e1 · · · T en−1  . τn−1,k

   

X(Cn )

≤ | τ1j | ·T e1 X(Cn ) + · · · + | τn−1,j | ·T en−1 X(Cn ) ≤ (n − 1)

ε ε M< . 2nM 2

628


On combining this inequality with (13.5), we obtain Pv Ck − U ek X(Cn )

≤

Pv Ck − Pvj Ck X(Cn ) + Pvj Ck − U ek X(Cn ) ε ε ≤ + = ε. 2 2 That is, the jth column of Θj tends to the jth column of Θ with respect to the norm of X(Cn ). Hence Vj → V in X(Mn,n ). Finally, it follows from the proof of Theorem 1.1 that Θ is co-outer. We need on more elementary fact. Lemma 13.5. Let E and F be Banach spaces, let T : E → F be a surjective continuous linear mapping, and let x ∈ E, y ∈ F be such that T x = y. For any ε > 0 there exists δ > 0 such that, whenever T# ∈ B(E, F ) and T# − T < δ, the equation T# x = y has a solution x satisfying x − x < ε. Proof. We can suppose x = 1. By the Open Mapping Theorem there exists c > 0 such that the ball of radius c in F is contained in the image under T of the unit ball in E. Then T ∗ l ≥ cl

for all l ∈ F ∗ .

Let δ = c min{1, ε}/2. Suppose T# − T < δ. For any l ∈ F ∗ ∗ T# l

= T ∗ l + (T# − T )∗ l ≥ T ∗ l − T# − T · l c c ≥ cl − l = l. 2 2 Thus T# maps the closed unit ball of E to a superset of the closed ball of radius c/2 in F . Since (T − T# )x ≤ T − T# < δ, it follows that there exists ξ ∈ E such that ξ
0. There exists δ > 0 such that, for any ψ ∈ X+ (Cn ) satisfying ϕ − ψX < δ, there exists η ∈ X+ (Cn ) such that ξ − ηX < ε and ψ t η = 1. Proof. Consider the operator T = TϕXt : X+ (Cn ) → X+ : T χ = ϕt χ. Clearly, T is surjective and T ξ = 1. By Lemma 13.5, there exists δ > 0 such that for an operator T# : X+ (Cn ) → X+ with T − T# < δ there exists a function η ∈ X+ (Cn ) such that ξ − ηX < ε and T# η = 1. Suppose now that ψ ∈ X+ (Cn ) and ϕ − ψX < δ. Then TϕXt − TψXt < δ, and we can put T# = TψXt . Clearly, η is a desired of the corona problem (13.7). Now we are ready to prove Theorem 13.1. Proof of Theorem 13.1. Suppose that n ≤ m. We proceed by induction on n. Let {Φj }j≥1 be a sequence of functions in X such that Φ − Φj X(Mm,n ) → 0. We shall show that some subsequence of {AΦj }j≥1 converges to AΦ in the norm of X: this will suffice to establish the continuity of A at Φ. Let f j be a maximizing vector for the operator HΦj on H 2 (Cn ). We can take it that the norm of f j in X+ (Cn ) is equal to 1: f j X(Cn ) = 1,

HΦj f j H−2 (Cm ) = HΦj · f j H 2 (Cn ) .

Let I be a positively oriented Jordan contour that winds once round the largest eigenvalue t20 of HΦ∗ HΦ , contains no eigenvalues, and encircles no other eigenvalues. HΦXj It is easy to see from axioms (A1)–(A4) that the operators HΦ∗X j ∗X X n converge to HΦ HΦ in the operator norm of X+ (C ). It follows that for HΦXj on I. Let large values of j there are no points of the spectrum of HΦ∗X j ? −1 1 ζI − HΦ∗X HΦX dζ P= 2πi I and ? −1 1 Pj = ζI − HΦ∗X HΦXj dζ. j 2πi I It is easy to see that Pj → P in the operator norm of X+ (Cn ). Clearly, if Pf j is nonzero, it is a maximizing vector of HΦ∗X HΦX and f j − Pf j X(Cn ) = Pj f j − Pf j X(Cn ) → 0,

k → ∞.

The vectors Pf j belong to the finite-dimensional subspace of maximizing vectors of HΦ∗ HΦ . Therefore there exists a convergent subsequence of the sequence {Pf j }j≥0 . Without loss of generality we may assume that the sequence {Pf j }j≥0 converges in X+ (Cn ) to a vector f , which is a maximizing vector of HΦ∗ HΦ . Obviously, f j − f X(Cn ) → 0 as j → ∞. We also need the other Schmidt vectors corresponding to f and f j . We may assume that HΦj = 0 for all j. Let ¯HΦ f , g = t−1 0 z

g j = HΦj −1 z¯HΦj f j

630


in X+ (Cm ). Since HΦj → HΦ in the norm of B(X+ (Cn ), X− (Cm )) and f j → f in X+ (Cn ), it follows that g j → g in X+ (Cm ). By Lemma 13.2, the vector-functions f and g are co-outer and do not vanish on T. Hence, they do not vanish on the closed unit disk clos D. Since convergence in X+ implies uniform convergence on clos D, it follows that for sufficiently large values of j the vector functions f j and g j do not vanish on clos D, and so they are co-outer. Now let us show that Theorem 13.1 holds when n = 1. In this case f and f j are scalar functions in X. By Theorem 2.2.3, |f (ζ)| = g(z)Cm a.e. on T. By continuity, equality holds at all points of T. By Lemma 13.2, f is invertible in X. By virtue of the continuity of inversion in Banach algebras −1 we deduce that {f −1 in X. Again by Theorem 2.2.3, j } converges to f z g, (Φ − AΦ)f = HΦ ¯

(Φj − AΦj )f j = HΦj ¯ zgj ,

and hence Φ − AΦ = HΦ

z¯g , f

Φ(k) − AΦ(k) = HΦ(k)

z¯g j , fj

for sufficiently large j. From these equalities it is clear that AΦj → AΦ in X. Thus the case n = 1 is established. Now consider n > 1 and suppose the theorem is true for n − 1. We prove the induction step by block diagonalization of Φ − AΦ. Let f and g be as above and let h be a scalar outer function such that |h(ζ)| = f (ζ)Cn , ζ ∈ T. Then |h(ζ)| = g(ζ)Cm , ζ ∈ T. We can choose h as follows: h = eρ+iρ˜, where ρ(ζ) = log f (ζ)Cn and ρ˜ is the harmonic conjugate of ρ. It follows easily from (A1)–(A4) that f 2Cn ∈ X. By Lemma 13.2, f 2Cn does not vanish on T, and so its spectrum in the Banach algebra X is a compact interval of the positive real numbers. By the analytic functional calculus, ρ = 12 log f 2Cn ∈ X. By (A1) we also have ρ˜ ∈ X. Thus h = eρ+iρ˜ ∈ X. The above construction also makes it clear that if we define in a similar way the scalar outer functioons hj so that |hj | = f j Cn , then hj converges to h in X. Note also that since |h| = f Cn is bounded away from zero, h is invertible in X and 1/hj → 1/h in X. We have f = hv,

g = hw,

f j = hj v j ,

and g j = hj wj ,

where v, w, v j , and wj are inner and co-outer column functions if j is sufficiently large. Then v j → v and wj → w in X+ as j → ∞. By Theorem 13.4, we can find thematic matrix functions Wt = w Ξ , V = v Θ , Vj = v j Θj , Wjt = wj Ξj


631

such that Vj → V and Wj → W in X. A fortiori, Θj → Θ,

Ξj → Ξ

and

in X(Mn,n−1 ) and X(Mm,m−1 ), respectively. We have explained in §11 that if a matrix function F ∈ H ∞ (Mm,n ) satisfies the equations F f = TΦ f

and F t g = TΦt g,

then Φ − F admits a factorization Φ−F =W

∗

t0 u0 O

O Ψ

(13.8)

V ∗,

¯ where u0 = z¯h/h. Moreover, to find the superoptimal approximation of Φ, it is sufficient to find a superoptimal approximation of Ψ: O t0 u0 ∗ (13.9) V ∗. Φ − AΦ = W O Ψ − AΨ Put

TΦ f TΦt g , ϕ2 = . h h Clearly, ϕ1 ∈ X+ (Cm ) and ϕ2 ∈ X+ (Cn ). As we have observed in §11, the corona problems ϕ1 =

ψ1t v = 1

and ψ2t w = 1

(13.10)

∞

are solvable in H . By Theorem 7.2.6, the corona problems (13.10) have solutions ψ1 ∈ X+ (Cn ) and ψ2 ∈ X+ (Cm ). Finally, by Theorem 11.2, the matrix function F ∈ X+ (Mm,n ) defined by F = ϕ1 ψ1t + ψ2 ϕt2 − ψ2 ϕt2 vψ1t satisfies equations (13.8). Similarly, for each sufficiently large j we can consider the functions TΦ f j TΦt g j ϕ1,j = , ϕ2,j = . hj hj By Lemma 13.6, there exist solutions ψ1,j and ψ2,j of the corona problems t t ψ1,j v j = 1 and ψ2,j wj = 1

such that lim ψ1 − ψ1,j X(Cn ) = 0

j→∞

and

lim ψ2 − ψ2,j X(Cm ) = 0.

j→∞

Put t t + ψ2,j ϕt2,j − ψ2,j ϕt2,j vψ1,j . Fj = ϕ1,j ψ1,j

Clearly, Fj → F in X+ (Mm,n ). We have t0 (Φj )u0,j Φj − Fj = Wj∗ O

O Ψj

V ∗,

632


¯ j /hj . It follows that Ψj → Ψ in where for large values of j, u0,j = z¯h X(Mm−1,n−1 ). Clearly, u0,j → u0 in X. We have O t0 (Φj )u0,j ∗ Φj − AΦj = W V ∗. O Ψj − AΨj (13.11) By the inductive hypothesis A is continuous at Ψ, and hence AΨj → AΨ in X(Mm−1,n−1 ). It follows that AΦj → AΦ in X(Mm,n ). Consider now the case when tmin{m,n}−1 (Φ) = 0. One can see by considering diagonal matrix functions like z¯ O O O that it is important whether the scalar operator A of best approximation is continuous at 0 in X, or equivalently whether A is bounded on X. This is not always so for decent spaces (see §7.7), and so the conclusion of Theorem 13.1 is not true if the condition tmin{m,n}−1 (Φ) = 0 is relaxed. There is one case when this can be done. Theorem 13.7. Let X be the Besov space B11 and let Φ ∈ X(Mm,n ). If Φ − AΦ has a thematic factorization, in which the indices corresponding to nonzero superoptimal singular values are all equal to 1, then Φ is a continuity point of the operator A of superoptimal approximation in X(Mm,n ). Proof. Since B11 is a Banach R-space, the fact that this statement is true in the case Φ = O follows from Theorem 12.3. Recall that X is also a decent function space. Let Φ have superoptimal singular values t0 , . . . , tmin{m,n}−1 . Let r be the number of nonzero superoptimal singular values of Φ: r = inf{j : tj = 0}. We prove the result by induction on r. As in the proof of Theorem 13.1, let {Φj }j≥1 be a sequence of functions in X such that Φ − Φj X(Mm,n ) → 0. If r = 0, then Φ = AΦ ∈ H ∞ (Mm,n ). Since Φj → Φ in X, by Theorem 12.3, A(Φj − Φ) → O, and since Φ ∈ H ∞ (Mm,n ), A(Φj − Φ) = AΦj − Φ. Thus AΦj → Φ = AΦ. Hence, A is continuous at Φ. Now consider r ≥ 1 and suppose the assertion holds for r−1. Since t0 = 0, the compact operator HΦ is not zero, and so HΦ∗ HΦ has finite-dimensional eigenspace corresponding to t20 . We now proceed as in the proof of Theorem 13.1: pick Schmidt vectors f , f j , g, g j , thematic functions V, Vj , W t , Wjt , and the matrix functions F, Fj , Ψ, Ψj exactly as described above. Once again (13.9) and (13.11) hold and the indices corresponding to any nonzero superoptimal singular value in any thematic factorization of Ψ − AΨ are all 1. Moreover, the superoptimal singular values of Ψ are t1 , . . . , tmin{m,n}−1 , so that Ψ has r − 1 nonzero superoptimal singular values. By the inductive hypothesis, AΨj → AΨ in X(Mm−1,n−1 ). Equalities (13.9) and (13.11) now show that AΦj → AΦ in X(Mm,n ) as j → ∞. Thus A is continuous at Φ.


633

We proceed now to necessary conditions. We can prove that the sufficient condition established in Theorem 13.1 is also necessary for n × n matrix functions Φ with tn−1 (Φ) = 0. Theorem 13.8. Let X be a decent function space, let Φ ∈ X(Mn,n ), and suppose that the tn−1 (Φ) = 0. If A is continuous at Φ, then all indices in any thematic factorization of Φ − AΦ are equal to 1. The proof of Theorem 13.8 is based on the following lemma. Lemma 13.9. Let Φ ∈ X(Mn,n ) and let ε > 0. Suppose that tn−1 (Φ) = 0. Then there exists Φ# ∈ X(Mn,n ) such that Φ − Φ# X < ε, tn−1 (Φ# ) = 0, and all n thematic indices of Φ# − AΦ# are equal to 1. First we prove another lemma. Lemma 13.10. Let Φ ∈ X(Mn,n ). Then for any ε > 0 there exists a matrix function Q in X(Mn,n ) such that HΦ < HQ , Φ − QX < ε, and rank(HΦ − HQ ) = 1. Proof. Let f be a maximizing vector of HΦ of norm 1 and let g = HΦ −1 z¯HΦ g. Choose a point ζ ∈ D at which f (ζ) = O and g(ζ) = O. We define the matrix function Q by Q(z)ξ = Φ(z)ξ + δ(z − ζ)−1 (f (ζ))∗ ξ · g(ζ),

ξ ∈ Cn ,

where δ > 0. Clearly, rank(HΦ − HQ ) = 1 and Φ − QX < ε if δ is small enough. We have (HQ f , z¯g) = (HΦ f , z¯g) + δ (z − ζ)−1 ((f (ζ))∗ f ) · g(ζ), z¯g H 2 (Cn ) = HΦ + δ ((f (ζ))∗ f ) · (g t g(ζ) ), kζ H 2 = HΦ + δf (ζ)2Cn g(ζ)2Cn , where kζ (ζ) = (1 − ζz)−1 is the reproducing kernel of H 2 . Since f H 2 = gH 2 = 1, it follows that HQ > HΦ . Proof of Lemma 13.9. Suppose that Q is a matrix function in X(Mm,n ) such that rank(HΦ − HQ ) = 1 and HQ > HΦ (the existence of such a Q follows from Lemma 13.10). Then clearly, s1 (HQ ) ≤ s0 (HΦ ) < s0 (HQ ). Hence, the singular value s0 (HQ ) has multiplicity 1, and it follows from Theorem 7.4 (in the case σ = t0 (Q)) that the index k0 of any thematic factorization of Q − AQ equals 1. We argue by induction on n. If n = 1, we apply Lemma 13.10 and put Φ# = Q. Suppose now that the result holds for n − 1. Applying Lemma 13.10 to Φ, we can find a matrix function Q such that Φ − QX < ε/2 and the index k0 of any thematic factorization of Q − AQ is 1. Consider a partial thematic factorization of Q − AQ of the form t0 (Q)u0 O V ∗, Q − AQ = W ∗ O Ψ

634


where V and W t are thematic matrix functions of the form V = v Θ , Wt = w Ξ and ΨL∞ < t0 (HQ ). Then u0 , V, W, Ψ ∈ X (see §12). By the inductive hypothesis, for any δ > 0 there exists a matrix function Ψ# ∈ X(Mn−1,n−1 ) such that Ψ − Ψ# X < δ and all n − 1 indices of any thematic factorization of Ψ# − AΨ# are equal to 1. Clearly, if δ is sufficiently small, then Ψ# L∞ < t0 (Q). Put t0 (Q)u0 O ∗ V ∗. G = Q − AQ − W O Ψ# If δ is sufficiently small, then obviously, GX < ε/2. We have t0 (Q)u0 O V∗ Q − G − AQ = W ∗ O Ψ# O t0 (Q)u0 = W∗ V ∗ + Ξ(AΨ# )Θt . O Ψ# − AΨ# Hence, Q − G − AQ − Ξ(AΨ# )Θt = W ∗ The matrix function W

∗

t0 (Q)u0 O

O Ψ# − AΨ#

t0 (Q)u0 O

O Ψ# − AΨ#

V ∗.

V∗

is very badly approximable (see §5), and so AQ+Ξ(AΨ# )Θt is the superoptimal approximation of Q−G. Now put Φ# = Q−G. Then Φ−Φ# X < ε and Φ# − AΦ# has a thematic factorization with all n indices equal to 1. Thus all n indices of any thematic factorization of Φ# − AΦ# are equal to 1. Proof of Theorem 13.8. Suppose that a thematic factorization of Φ − AΦ has an index greater than 1. By Corollary 1.7, the determinant of a thematic matrix function is constant and so the function det(Φ − AΦ) has constant modulus and wind det(Φ − AΦ) < −n. By Lemma 13.9, there exists a sequence {Φj }j≥0 of matrix functions in X(Mn,n ) converging to Φ in X and such that all indices of any thematic factorizations of Φj − AΦj are equal to 1. Again, the functions det(Φj − AΦj ) have constant modulus. Clearly, wind det(Φj − AΦj ) = −n. Suppose that lim AΦj − AΦX = 0. j→∞

It follows that lim det(Φj − AΦj ) = det(Φ − AΦ)

j→∞

in X,

which is impossible by Lemma 7.11.5. Remark. The proof shows a slightly stronger result. If tn−1 (Φ) = 0 and the mapping A as a mapping from X(Mn,n ) to BM O(Mn,n ) is continuous, then all indices of any thematic factorization of Φ − AΦ are equal to 1.


635

We proceed now to the case when an m × n matrix function Φ has zero superoptimal singular value tmin{m,n}−1 . As we have already mentioned the result depends on the boundedness of the scalar best approximation operator A on X. Theorem 13.11. Suppose that X is one of the Besov spaces Bps , s > 1/p, or the H¨ older–Zygmund spaces λα , Λα , α > 0, or X = F1 is the space of functions with absolutely converging Fourier series. If Φ ∈ X(Mm,n ) and tmin{m,n}−1 = 0, then A is discontinuous at Φ in the norm of X. Proof. It is shown in §7.7 that A is unbounded on these spaces. We can suppose that m ≤ n. Suppose that r is the smallest integer for which tr = 0. Consider first the case r = 0. Then Φ ∈ X+ and so AΦ = Φ. Clearly, it is sufficient to show that Φ is discontinuous at O. Since the scalar operator A of best approximation is unbounded on X, there exists a sequence {ϕj }j≥0 of scalar functions in X such that ϕj X → 0 and Aϕj X ≥ const. Consider the matrix functions Φj defined by   ϕj O · · · O  O O ··· O    Φj =  . ..  . .. . . .  . . .  . O Clearly, Φj X → 0,

   AΦj =  

Aϕj O .. . O

O ···

O ··· O ··· .. . . . . O ···

O

O O .. .

   , 

O

and so AΦj X ≥ const, which proves that A is discontinuous at O. Suppose now that r ≥ 1. Consider a thematic factorisation   O O t0 u0 · · ·  .. .. ..  ..  . ∗ ∗ . . .  Φ − AΦ = W0∗ · · · Wr−1 · · · V0∗ .  Vr−1    O · · · tr−1 ur−1 O O ··· O O Again, since A is unbounded on X, we may pick a sequence {ψj }j≥0 of scalar functions X such that ψj X → 0 but Aψj X ≥ const. Let   O O O t 0 u0 · · ·  .. .. .. ..  ..  . . . . .    ∗ ∗ ∗ V · · · V0∗ . Φj = AΦ + W0 · · · Wr−1  O · · · tr−1 ur−1 O O    r−1  O ··· O ψj O  O ··· O O O

636


Clearly, Φ − Φj X → 0 as j → ∞. If we solve the superoptimal analytic approximation problem for Φj by successive diagonalization, then for the first r stages it proceeds exactly as for Φ. It follows that Φj − AΦj



t 0 u0  ..  .  ∗ =W0∗ · · · Wr−1  O   O O Thus

··· .. . ··· ··· ···

O .. .

O .. .

tr−1 ur−1 O O

O ψj − Aψj O



O ···  .. . .  . .  ∗ AΦ − AΦj = W0∗ · · · Wr−1  O ···   O ··· O ···

O .. .

O .. .

O O O Aψj O O

 O ..  .   ∗ · · · V0∗ . V O   r−1 O  O  O ..  .   ∗ V · · · V0∗ . O   r−1 O  O

Since Aψj X ≥ 1, it cannot be true that AΦj → AΦ in X. Thus A is discontinuous on X at Φ.

14. Unitary Interpolants of Matrix Functions Let Φ be a matrix function in L∞ (Mn,n ). We study in this section the problem of finding unitary-valued symbols of the Hankel operator HΦ . In other words, we want to find n × n unitary-valued functions U such that ˆ (j) = Φ(j), ˆ U

j < 0.

(14.1)

A unitary-valued function U satisfying (14.1) is called a unitary interpolant of Φ. (It would be more appropriate to call it a unitary extrapolant.) Clearly, (14.1) means that HΦ = HU and so for Φ to have a unitary interpolant it is necessary that (14.2) HΦ = distL∞ Φ, H ∞ (Mn,n ) ≤ 1. We are interested in this section in unitary interpolants U of Φ such that the Toeplitz operator TU on H 2 (Cn ) is Fredholm. By Theorem 3.4.6, TU is Fredholm if and only if HU e < 1 and HU ∗ e < 1. Hence, for Φ to have a unitary interpolant U with Fredholm TU it is necessary that HΦ e = distL∞ Φ, (H ∞ + C)(Mn,n ) < 1. (14.3) Clearly, if U is a unitary interpolant of matrix function Φ satisfying (14.3), then TU is Fredholm if and only if HU ∗ e < 1.

(14.4)

14. Unitary Interpolants of Matrix Functions

637

We show in this section that if X is an R-space or X is a decent functiion space and Φ ∈ X, then all unitary interpolants U of Φ that satisfy (14.4) also belong to X. If TU is Fredholm, the unitary-valued function U admits, by Theorem 3.5.2, a Wiener–Hopf factorization of the form  d0  z O ··· O  O z d1 · · · O    Ψ = Q∗2  .  Q1 , . .. .. ..  ..  . . O

O

···

z dn−1

where d0 , · · · , dn−1 ∈ Z are the Wiener–Hopf indices, and Q1 and Q2 are functions invertible in H 2 (Mn,n ). Here we start enumeration with 0 for technical reasons. Moreover, we can always assume that the indices are arranged in the nondecreasing order: d0 ≤ d1 ≤ · · · ≤ dn−1 in which case the indices dj are uniquely determined by the function Ψ (see Theorem 3.5.10). Let us now state the main results of the section. By Theorem 10.1, if Φ satisfies conditions (14.2) and (14.3) and Q ∈ H ∞ (Mn,n ) is a best approximation of Φ by bounded analytic matrix functions, then Φ − Q admits a monotone partial thematic factorization of the form   O O u0 O · · ·  O u1 · · · O O     .. . . ..  V ∗ · · · V ∗ , ∗ ∗ . .. .. .. Φ − Q = W0 · · · Wr−1  . 0 .    r−1  O O · · · ur−1 O  O O ··· O Ψ where ΨL∞ ≤ 1 and HΨ < 1 (here r is the number of superoptimal singular values of Φ equal to 1; it may certainly happen that r = 0, in which case Ψ = Φ − Q). We denote by kj , 0 ≤ j ≤ r − 1, the thematic indices of the above factorization. Recall that by Theorem 10.9, the indices kj are uniquely determined by Φ. Theorem 14.1. Let Φ be a matrix function in L∞ (Mn,n ) such that HΦ e < 1. Then Φ has a unitary interpolant U satisfying HU ∗ e < 1 if and only if HΦ ≤ 1. If U is a unitary interpolant of a matrix function Φ and conditions (14.2), (14.3), and (14.4) hold, we denote by dj , 0 ≤ j ≤ n − 1, the Wiener–Hopf factorization indices of U arranged in the nondecreasing order: d0 ≤ d1 ≤ · · · ≤ dn−1 . Recall that the indices dj are uniquely determined by the function U .

638


Theorem 14.2. Let Φ be a matrix function in L∞ (Mn,n ) such that HΦ ≤ 1 and HΦ e < 1. Let r be the number of superoptimal singular values of Φ equal to 1. Then each unitary interpolant U of Φ satisfying HU ∗ e < 1 has precisely r negative Wiener–Hopf indices. Moreover, dj = −kj , 0 ≤ j ≤ r − 1. In particular, Theorem 14.2 says that the negative Wiener–Hopf indices of a unitary interpolant U of Φ that satisfies (14.4) are uniquely determined by Φ. Theorem 14.3. Let Φ and r satisfy the hypotheses of Theorem 14.2. Then for any sequence of integers {dj }r≤j≤n−1 satisfying 0 ≤ dr ≤ dr+1 ≤ · · · ≤ dn−1 there exists a unitary interpolant U of Φ such that HU ∗ e < 1 and the nonnegative Wiener–Hopf factorization indices of U are dr , dr+1 , · · · , dn−1 . Note that Theorem 14.1 follows immediately from Theorem 14.3. Theorem 14.4. Let Φ satisfy the hypotheses of Theorem 14.2. Then Φ has a unique unitary interpolant if and only if all superoptimal singular values of Φ are equal to 1. Proof of Theorem 14.2. Suppose that U is a unitary interpolant of Φ that satisfies (14.4). Put G = U − Φ ∈ H ∞ (Mn,n ). Let κ ≥ 0. Let us show that if f ∈ H 2 (Cn ), then Hzκ Φ f 2 = f 2 if and only if f ∈ Ker Tzκ U . 2 Indeed, if f ∈ Ker Tzκ U , then z κ U f ∈ H− (Cn ). It follows that Hzκ Φ f 2 = Hzκ (Φ+G) f 2 = P− (z κ U f )2 = z κ U f 2 = f 2 . Conversely. suppose that Hzκ Φ f 2 = f 2 . Then Hzκ U f 2 = f 2 . 2 Hence, z κ U f ∈ H− (Cn ), and so f ∈ Ker Tzκ U . It follows from Corollary 3.5.8 applied to the function z κ U that −dj − κ. dim Ker Tzκ U = {j∈[0,n−1]:−dj >κ}

By Theorem 10.8,

dim{f ∈ H 2 (Cn ) : Hzκ Φ f 2 = t0 f 2 } =

kj − κ.

{j∈[0,r−1]:kj >κ}

Hence,

{j∈[0,n−1]:−dj >κ}

−dj − κ =

{j∈[0,r−1]:kj >κ}

kj − κ,

κ ≥ 0. (14.5)

It is easy to see from (14.5) that U has r negative Wiener–Hopf factorization indices and dj = −kj for 0 ≤ j ≤ r − 1. To prove Theorem 14.3, we need two lemmas.


639

Lemma 14.5. Let Ψ ∈ L∞ (Mm,m ) and HΨ < 1. Then for any nonnegative integers dj , 0 ≤ j ≤ m − 1, there exists a unitary interpolant U of Ψ that admits a representation   O u0 O · · ·  O u1 · · · O    ∗ ∗ U = W0∗ · · · Wm−1  ..  Vm−1 · · · V0∗ , .. . . ..  .  . . . (14.6) O O · · · um−1 where

Wj =

Ij O

O ˘j W

,

Vj =

Ij O

O V˘j

,

1 ≤ j ≤ m − 1,

˘ t are thematic matrix functions, and u0 , · · · , um−1 are uniV0 , W0t , V˘j , W j modular functions such that the Toeplitz operators Tuj are Fredholm and ind Tuj = −dj ,

0 ≤ j ≤ m − 1.

Proof of Lemma 14.5. We argue by induction on m. Assume first that m = 1. Let ψ be a scalar function in L∞ such that Hψ < 1. Without loss of generality we may assume that ψ∞ < 1. Consider the function z¯d0 +1 ψ. Clearly, Hz¯d0 +1 ψ < 1. It is easy to see that there exists c ∈ R such that Hz¯d0 +1 ψ+c¯z = 1. Since Hc¯z has finite rank, it is easy to see that Hz¯d0 +1 ψ+c¯z e = Hz¯d0 +1 ψ e < 1 = Hz¯d0 +1 ψ+c¯z . It follows from Theorems 1.1.4 and 7.5.5 that z¯d0 +1 ψ +c¯ z has a unique best z−g approximation g by H ∞ functions, the error function u = z¯d0 +1 ψ + c¯ is unimodular, Tu is Fredholm, and ind Tu > 0. On the other hand, distL∞ (zu, H ∞ ) = Hz¯d0 ψ+c−zg = Hz¯d0 ψ < 1, and so zu is not badly approximable. Hence, ind Tzu ≤ 0. It follows that ind Tu = 1. We have z d0 +1 u = ψ + cz d0 − z d0 +1 g. Put u0 = z d0 +1 u. Then ψ − u0 = z d0 +1 g − cz d0 ∈ H ∞ . Hence, u0 is a unitary interpolant of ψ and ind Tu0 = −d0 − 1 + ind Tu = −d0 . Suppose now that the lemma is proved for (m − 1) × (m − 1) matrix functions. By Theorem 1.8, without loss of generality we may assume that ΨL∞ < 1. Then Hz¯d0 +1 Ψ < 1. As in the scalar case, there exists c ∈ R such that Hz¯d0 +1 Ψ+c¯zIm = 1. Clearly, Hz¯d0 +1 Ψ+c¯zIm e < 1.

640


Let G be a best approximation of z¯d0 +1 Ψ + c¯ z Im by H ∞ matrix functions. d0 +1 By Theorems 2.2 and 4.1, z¯ Ψ + c¯ z Im − G admits a representation u O z Im − G = W ∗ (14.7) z¯d0 +1 Ψ + c¯ V ∗, O Υ where V and W t are thematic matrices, ΥL∞ ≤ 1, u is a unimodular function such that Tu is Fredholm, and ind Tu > 0. It is easy to see that Hz¯d0 +1 Ψ+c¯zIm −G e < 1. By Theorem 4.1, HΥ e < 1. Let us show that ind Tu = 1. Suppose that ind Tu > 1. Then zu O z(¯ z d0 +1 Ψ + c¯ V∗ z Im − G) = W ∗ O zΥ is still badly approximable (see Theorem 2.5). Hence, 1

= z(¯ z d0 +1 Ψ + c¯ z Im − G)L∞ = Hz¯d0 Ψ+cIm −zG = Hz¯d0 Ψ ≤ ¯ z d0 ΨL∞ < 1.

We have got a contradiction. Multiplying both sides of (14.7) by z d0 +1 , we obtain d +1 O z 0 u V ∗. Ψ + cz d0 Im − z d0 +1 G = W ∗ O z d0 +1 Υ Put u0 = z d0 +1 u. Clearly, ind Tu0 = −d0 − 1 + ind Tu = −d0 . Let us show that Hzd0 +1 Υ < 1. Consider the following factorization: zu O d0 ∗ V ∗. z¯ Ψ + cIm − zG = W O zΥ Clearly, Hz¯d0 Ψ+cIm −zG = Hz¯d0 Ψ < 1 and HzΥ e = HΥ e < 1. By Lemma 10.4, HzΥ < 1. Hence, Hzd0 +1 Υ ≤ HzΥ < 1. We can now apply the inductive hypothesis to z d0 +1 Υ. Finally, by Lemma 1.8, there exists a function F ∈ H ∞ (Mm,m ) such Ψ − F admits a desired representation. Lemma 14.6. Let U be an n × n matrix function of the form (14.6), where the Vj and Wj are as in Lemma 14.5, the uj are unimodular functions such that the operators Tuj are Fredholm whose indices are arbitrary integers. If HU e < 1, then HU ∗ e < 1. Proof of Lemma 14.6. Since HU e = Hzl U e ,

HU ∗ e = Hzl U ∗ e

for any l ∈ Z, we may assume without loss of generality that ind Tuj > 0, 0 ≤ j ≤ n − 1. In this case (14.6) is a thematic factorization of U . By Theorem 5.4, the Toeplitz operator TU has dense range in H 2 (Cn ). The result now follows from Theorem 4.4.11. Proof of Theorem 14.3. If r = n and Q is a best approximation of Φ by bounded analytic matrix functions, then Q is a superoptimal approximation


641

of Φ. It follows from (14.3) that Φ − Q admits a thematic factorization (see Theorem 5.2), and so U = Φ − Q is a unitary interpolant of Φ. By Lemma 14.6, U satisfies (14.4). Suppose now that r < n and Q is a best approximation of Φ by bounded analytic matrix functions. Then Φ − Q admits a thematic factorization of the form   O O u0 · · ·  .. .. ..  ..  . ∗ . . .  Φ − Q = W0∗ · · · Wr−1  V ∗ · · · V0∗ ,   O · · · ur−1 O  r−1 O ··· O Ψ where

Wj =

Ij O

O ˘ Wj

,

O V˘j

Ij O

Vj =

,

1 ≤ j ≤ r − 1,

˘ t are thematic matrix functions, ΨL∞ ≤ 1, HΨ < 1, V0 , W0t , V˘j , W j u0 , · · · , ur−1 are unimodular functions such that the Toeplitz operators Tuj are Fredholm, and ind Tu0 ≥ ind Tu1 ≥ · · · ≥ ind Tur−1 > 0 (see Theorem 10.1). We can now apply Lemma 14.5 to Ψ and find a matrix function G ∈ H ∞ (Mn−r,n−r ) such that I n−1−r I n−1−r O O ∗ ˘ Ψ − G = Wr · · · · · · V˘r∗ , ˘ n−1 D O W O V˘n−1 where



··· .. . ···

ur  .. D= . O

O .. .

  ,

un−1

and the are thematic matrix functions, ur , · · · , un−1 are unimodular functions such that the operators Tuj are Fredholm and ˘t V˘j , W j

ind Tuj = −dj ,

r ≤ j ≤ n − 1.

Using Theorem 1.8, we can inductively find a matrix function F ∈ H ∞ (Mn,n ) such that Φ − F admits a factorization   O u0 · · ·  .. ..  V ∗ · · · V ∗ , ∗ .. Φ − F = W0∗ · · · Wn−1  . . 0 .  n−1 O where

Wj =

Ij O

O ˘j W

,

Vj =

··· Ij O

un−1 O V˘j

,

r ≤ j ≤ n − 1.

642


Clearly, Φ − F is a unitary-valued function. By Lemma 14.6, it satisfies (14.4). To prove that the Wiener–Hopf factorization indices of Φ − F are equal to − ind Tu0 , − ind Tu1 , · · · , − ind Tun−1 , it is sufficient to apply Theorem 14.2 to the matrix function z¯dn−1 +1 (Φ−F ). Proof of Theorem 14.4. Suppose that all superoptimal singular values of Φ are equal to 1. Let U be a unitary interpolant of Φ. Clearly, Φ − U is the superoptimal approximation of Φ, which is unique because of conditions (14.2) and (14.3). If Φ has superoptimal singular values less than 1, then by Theorem 14.3, Φ has infinitely many unitary interpolants satisfying (14.4). We proceed now to the heredity problem for unitary interpolants. Suppose that the initial matrix function Φ belongs to a certain function space X. We study the question of whether it is possible to obtain results similar to Theorems 14.1–14.4 for unitary interpolants that belong to the same class X. We obtain such results for the linear R-spaces and the decent function spaces. Recall that if U is a unitary-valued function in V M O, then HU and HU ∗ are compact, and so TU is Fredholm. Clearly, it is sufficient to prove an analog of Theorem 14.3. Theorem 14.7. Suppose that X is either a linear R-space or a decent function space Let Φ be a matrix function in X(Mn,n ) such that HΦ ≤ 1. Let r be the number of superoptimal singular values of Φ equal to 1. Then for any integers {dj }r≤j≤n−1 such that 0 ≤ dr ≤ dr+1 ≤ · · · ≤ dn−1 there exists a unitary interpolant U ∈ X(Mn,n ) whose nonnegative Wiener– Hopf factorization indices are dr , dr+1 , · · · , dn−1 . Note that any unitary interpolant U ∈ X(Mn,n ) must satisfy (14.4), and so by Theorem 14.2, U must have precisely r negative Wiener–Hopf factorization indices that are uniquely determined by Φ. Proof of Theorem 14.7. By Theorems 14.2 and 14.3, it is sufficient to prove that under the hypotheses of the theorem any unitary interpolant U of Φ that satisfies (14.4) must belong to X. If U is a unitary interpolant of Φ that satisfies (14.4), then HU e = 0, and so TU is Fredholm. Hence, the desired result follows immediately from Theorems 13.1.2 and 13.5.2. The following special case of Theorem 14.7 is most important. Theorem 14.8. Let Φ be a matrix function in (H ∞ + C)(Mn,n ) such that HΦ ≤ 1. Let r be the number of superoptimal singular values of Φ equal to 1. Then for any integers {dj }r≤j≤n−1 such that 0 ≤ dr ≤ dr+1 ≤ · · · ≤ dn−1

15. Canonical Factorizations

643

there exists a unitary interpolant U ∈ QC(Mn,n ) whose nonnegative Wiener–Hopf factorization indices are dr , dr+1 , · · · , dn−1 . Proof. Let X = V M O. Then X is an R-space. The condition Φ ∈ H ∞ + C implies P− Φ ∈ X. Let U be a unitary interpolant of Φ that satisfies the conclusion of Theorem 14.3. By Theorem 14.8, U ∈ V M O. The result follows now from the identity QC = V M O ∩ L∞ (see Appendix 2.5).

15. Canonical Factorizations In this section we modify the notion of a thematic factorization and introduce so-called canonical factorizations. This allows us to obtain another method of finding the superoptimal approximation for matrix functions satisfying the same sufficient conditions. Unlike thematic factorizations, the factors in canonical factorizations are uniquely determined modulo constant unitary factors. We also consider partial canonical factorizations of badly approximable functions and establish for them the same invariance properties. We characterize the badly approximable matrix functions in terms of partial canonical factorizations and the very badly approximable matrix functions in terms of canonical factorizations. Finally, we obtain in this section heredity results for canonical and partial canonical factorizations. To obtain a thematic factorization, we have started with a maximizing vector of the corresponding Hankel operators. To obtain a canonical factorization, we consider all maximizing vectors of the same Hankel operator. Theorem 15.1. Suppose that Φ ∈ L∞ (Mm,n ) and HΦ e < HΦ . Let M be the minimal invariant subspace of multiplication by z on H 2 (Cn ) that contains all maximizing vectors of HΦ . Then M = ΥH 2 (Cr ),

(15.1)

where r is the number of superoptimal singular values of Φ equal to HΦ , and Υ is an inner and co-outer n × r matrix function. Proof. Consider first the case m = n. Without loss of generality we may assume that HΦ = 1. It follows from Theorems 14.1 and 14.2 that there exists a unitary interpolant U of Φ such that the Toeplitz operator TU is Fredholm and each such unitary interpolant has precisely r negative Wiener–Hopf indices. Consider a Wiener–Hopf factorization of U  d1  z O ··· O  O z d2 · · · O   −1 ∗ (15.2) U = Q2  . . ..  Q1 , . .. ..  .. .  O O · · · z dn

644


where Q1 and Q2 are matrix functions invertible in H 2 (Mn,n ), and d1 ≤ d2 ≤ · · · ≤ dn . Since U has r negative Wiener–Hopf indices, we have d1 ≤ · · · ≤ dr < 0 ≤ dr+1 ≤ · · · ≤ dn . Clearly, HΦ = HU . It is also easy to see that a nonzero function f ∈ H 2 (Cn ) is a maximizing vector of HΦ if an only if f ∈ Ker TU . It is easy to see from (15.2) that     q1          ..        .           q r   : qj ∈ P + , deg qj < −dj , 1 ≤ j ≤ r . Ker TU = Q1   O         (15.3)    .    .       .     O (Recall that P + is the set of analytic polynomials.) Since M is the minimal invariant subspace of multiplication by z that contains Ker TU , it follows from (15.3) that     q1          ..        .           q r   . ∈ P , 1 ≤ j ≤ r : q M = closH 2 (Cn ) Q1  j +       O  (15.4)     .        ..        O Since Q1 (ζ) is an invertible matrix for all ζ ∈ D, it follows easily from (15.4) that dim{f (ζ) : f ∈ M} = r for all ζ ∈ D. Therefore the z-invariant subspace M has the form M = ΥH 2 (Cr ), where Υ is an n × r inner matrix function (see Appendix 2.3). It remains to prove that Υ is co-outer. Denote by Q♥ the matrix function obtained from Q1 by deleting the last n−r columns. It is easy to see that Υ is an inner part of Q♥ . Let Q♥ = ΥF , where F is an r × r outer matrix function. by deleting the Denote by Q♠ the matrix function obtained from Q−1 1 last n − r rows. Clearly, Q♠ (ζ)Q♥ (ζ) = Ir for almost all ζ ∈ T. We have I r = Q♠ Q♥ = Q♠ ΥF, and so I r = F t Υt Qt♠ . t t t Both F and Υ Q♠ are r × r matrix functions. Hence, I r = Υt Qt♠ F t . It follows that Υt is outer, and so Υ is co-outer. Consider now the case m < n. Let Φ# be the matrix function obtained from Φ by adding n−m zero rows. It is easy to see that the Hankel operators


645

HΦ and HΦ# have the same maximizing vectors. This reduces the problem to the case m = n. Finally, assume that m > n. Let Φ be the matrix function obtained from Φ by adding m − n zero columns. It is easy to see that f is a maximizing vector of HΦ if and only if it can be obtained from a maximizing vector of HΦ by adding m − n zero coordinates. Let M be the minimal invariant subspace of multiplication by z on H 2 (Cm ) that contains all maximizing vectors of HΦ . Clearly, the number of superoptimal singular values of Φ equal to 1 is still r. Therefore there exists an m × r inner and co-outer matrix function Υ such that M = Υ H 2 (Cr ). It is easy to see that the last m − n rows of Υ are zero. Denote by Υ the matrix function obtained from Υ by deleting the last m − n zero rows. Obviously, Υ is an inner and co-outer n × r matrix function and M = ΥH 2 (Cr ). We need the following result. Lemma 15.2. Suppose that Φ satisfies the hypotheses of Theorem 15.1 and M is given by (15.1). If ΦL∞ = HΦ and f is a nonzero vector function in M, then f (ζ) is a maximizing vector of Φ(ζ) for almost all ζ ∈ T. Proof. By Theorem 2.2.3, if f is a maximizing vector of HΦ , then f (ζ) is a maximizing vector of Φ for almost all ζ ∈ T and Φ(ζ)Mm,n = HΦ almost everywhere. Without loss of generality we may assume that HΦ = 1. Let L be the set of vector functions of the form q1 g1 + · · · + qM gM , where qj ∈ P + and the gj are maximizing vectors of HΦ . By definition, M is the norm closure of L. Since the gj (ζ) are maximizing vectors of Φ(ζ) for almost all ζ ∈ T, it follows that for g ∈ L, g(ζ) is a maximizing vector of Φ(ζ) almost everywhere on T. Let {fj } be a sequence of vector functions in L that converges to f ∈ M in H 2 (Cn ). Clearly,

2 Φ(ζ)f (ζ)Cm dm(ζ) = lim Φ(ζ)fj (ζ)2Cm dm(ζ) j→∞ T T

fj (ζ)2Cm dm(ζ) = lim j→∞ T

f (ζ)2Cm dm(ζ), = T

and since obviously, Φ(ζ)f (ζ)Cm ≤ f (ζ)Cm almost everywhere on T, it follows that f (ζ) is a maximizing vector of Φ(ζ) for almost all ζ ∈ T. As we have observed in §2, a function f ∈ H 2 (Cn ) is a maximizing vector of HΦ if and only if g = z¯HΦ f is a maximizing vector of HΦt . It is easy to see that the matrix functions Φ and Φt have the same superoptimal

646


singular values. Let N be the minimal invariant subspace of multiplication by z on H 2 (Cm ) that contains all maximizing vectors of HΦt . By Theorem 15.1, there exists an inner and co-outer matrix function Ω ∈ H ∞ (Mm,r ) such that N = ΩH 2 (Cr ). By Theorem 1.1, Υ and Ω have balanced completions, i.e., there exist inner and co-outer matrix functions Θ ∈ H ∞ (Mn,n−r ) and Ξ ∈ H ∞ (Mm,m−r ) such that def def V = Υ Θ and W t = Ω Ξ (15.5) are unitary-valued matrix functions. Theorem 15.3. Let Φ ∈ L∞ (Mm,n ) and HΦ e < t0 = HΦ . Let r be the number of superoptimal singular values of Φ equal to t0 . Suppose that F is a best approximation of Φ by analytic matrix functions. Then Φ − F admits a factorization of the form t0 U O Φ − F = W∗ V ∗, (15.6) O Ψ where V and W are given by (15.5), U is an r × r unitary-valued very badly approximable matrix function such that HU e < 1, and Ψ is a matrix-function in L∞ (Mm−r,n−r ) such that ΨL∞ ≤ t0 and HΨ = tr (Φ) < HΦ . Moreover, U is uniquely determined by the choice of Υ and Ω and does not depend on the choice of F . Proof. Without loss of generality we may assume that HΦ = 1. It follows from Lemma 15.2 that the columns of Υ(ζ) are maximizing vectors of Φ(ζ) − F (ζ) for almost all ζ ∈ T. Similarly, the columns of Ω(ζ) are maximizing vectors of Φt (ζ) − F t (ζ) almost everywhere on T. We need two elementary lemmas. Lemma 15.4. Let A ∈ Mm,n and A = 1. Suppose that v1 , · · · , vr is an orthonormal family of maximizing vectors of A and w1 , · · · , wr is an orthonormal family of maximizing vectors of At . Then t w 1 · · · wr A v1 · · · vr is a unitary matrix. Lemma 15.5. Let A be a matrix in Mm,n such that A = 1 and A has the form A11 A12 , A= A21 A22 where A11 is a unitary matrix. Then A12 and A21 are the zero matrices. Both lemmas are obvious. Let us complete the proof of Theorem 15.3. Consider the matrix function U X def = W(Φ − F )V. Y Ψ


647

Here U ∈ L∞ (Mr,r ), X ∈ L∞ (Mr,n−r ), Y ∈ L∞ (Mm−r,r ), and ∞ Ψ ∈ L (Mm−r,n−r ). If we apply Lemma 15.4 to the matrices (Φ − F )(ζ), ζ ∈ T, and the columns of Υ(ζ) and Ω(ζ), we see U = Ωt (Φ − F )Υ is unitary-valued. By Lemma 15.5, X and Y are the zero matrix functions, which proves (15.6). Let us prove that HU e < 1. Since U = Ωt (Φ − F )Υ, we have HU e

(15.7)

distL∞ U, (H ∞ + C)(Mr,r ) = distL∞ Ωt ΦΥ, (H ∞ + C)(Mr,r ) ≤ distL∞ Φ, (H ∞ + C)(Mm,n ) = HΦ e < 1. =

Now it is time to show that U is very badly approximable. Denote by L the minimal invariant subspace of multiplication by z that contains all maximizing vectors of HU . Suppose that f is a maximizing vector of HΦ . Then f = Υϕ for some ϕ ∈ H 2 (Cr ). Clearly, HΦ f is a maximizing vector of HΦ∗ . By Theorem 2.2.3, HΦ f = (Φ − F )f . Hence, z¯Φ − F f¯ is a maximizing vector of HΦt . Therefore z¯Φ − F f¯ ∈ ΩH 2 (Cr ), and so 2 Ωt (Φ − F )f = Ωt (Φ − F )Υϕ = U ϕ ∈ H− (Cr ).

It follows that ϕ is a maximizing vector of HU and HU = 1. Therefore ΥH 2 (Cr ) ⊂ ΥL. Hence, L = H 2 (Cr ), and by Theorem 15.1, t0 (U ) = · · · = tr−1 (U ) = 1. It follows that U is very badly approximable. Hence, the zero matrix function is the only best approximation of U by analytic matrix functions. This uniqueness property together with (15.7) implies that U does not depend on the choice of the best approximation F . It is evident from (15.6) that ΨL∞ ≤ 1. It remains to prove that HΨ = tr (Φ). Suppose that F$ is another best approximation of Φ by bounded analytic matrix functions. Then as we have already proved, Φ − F$ can be represented as U O Φ − F$ = W ∗ V ∗, O Ψ$ where Ψ$ is a matrix function in L∞ (Mm−r,n−r ) such that Ψ$ L∞ ≤ 1. Clearly, sj ((Φ − F$ )(ζ)) = 1, 0 ≤ j ≤ r − 1, and sr ((Φ − F$ )(ζ)) = Ψ$ (ζ)Mm−r,n−r for almost all ζ ∈ T.

648


By Theorem 1.8, a matrix function G ∈ H ∞ (Mm,n ) is a best approximation of Φ if and only if there exists Q ∈ H ∞ (Mm−r,n−r ) such that Ψ − QL∞ ≤ 1 and U O Φ − G = W∗ V ∗. O Ψ−Q This proves that HΨ = tr (Φ). Corollary 15.6. Under the hypotheses of Theorem 15.3 the Toeplitz operator TU on H 2 (Cr ) is Fredholm. Proof. By Theorem 5.4, TzU has dense range in H 2 (Cr ). By Theorem ∗ 4.4.11, Hz¯∗U ∗ Hz¯U ∗ is unitarily equivalent to the restriction of HzU HzU to the subspace {f ∈ H 2 (Cr ) : HzU f 2 = f 2 }. Since HU e < 1, this subspace is finite-dimensional, and so HU ∗ e = Hz¯U ∗ e = lim sj (Hz¯U ∗ ) = lim sj (HzU ) = HzU e = HU e . j→∞

j→∞

The result now follows from Theorem 3.4.6. Factorizations of the form (15.6) with Ψ satisfying ΨL∞ ≤ t0

and HΨ < t0

form a special class of partial canonical factorizations. The matrix function Ψ is called the residual entry of the partial canonical factorization. The notion of a partial canonical factorization in the general case will be defined later in this section. The following theorem together with Theorem 15.3 gives a characterization of the badly approximable matrix functions Φ satisfying the condition HΦ e < HΦ . Theorem 15.7. Let Φ ∈ L∞ (Mm,n ) and HΦ e < HΦ . Suppose that Φ admits a representation of the form σU O ∗ Φ=W V ∗, O Ψ where σ > 0, V and W t are r-balanced matrix functions, U is a very badly approximable unitary-valued r×r matrix function such that HU e < 1, and ΨL∞ ≤ σ. Then Φ is badly approximable and t0 (Φ) = · · · = tr−1 (Φ) = σ. Proof. Suppose that V and W are given by (15.5). Let ϕ ∈ H 2 (Cr ) be a maximizing vector of HU . Then it is easy to see that HΦ Υϕ2 = σΥϕ, while HΦ ≤ ΦL∞ = σ. Hence, HΦ = σ, Υϕ is a maximizing vector of HΦ , and Φ is badly approximable. Since U is very badly approximable, it follows from Theorem 15.1 that the minimal invariant subspace of multiplication by z on H 2 (Cr ) that contains


649

all maximizing vectors of HU is the space H 2 (Cr ) itself. Let M be the minimal invariant subspace of multiplication by z on H 2 (Cn ) that contains all maximizing vectors of HΦ . Since Υ is co-outer, it follows that the matrix Υ(ζ) has rank r for all ζ ∈ D. Hence, dim{f (ζ) : f ∈ M} ≥ r

for all ζ ∈ D.

It follows now from Theorem 15.1 that t0 (Φ) = · · · = tr−1 (Φ). We also need the following version of the converse to Theorem 15.3. Theorem 15.8. Let Φ ∈ L∞ (Mm,n ) and HΦ e < HΦ . Suppose that Φ admits a representation of the form σU O Φ = W∗ V ∗, O Ψ where σ > 0, V and W t are r-balanced matrix functions of the form (15.5), U is a very badly approximable unitary-valued r × r matrix function such that HU e < 1, and HΨ < σ. Then ΥH 2 (Cr ) is the minimal invariant subspace of multiplication by z on H 2 (Cn ) that contains all maximizing vectors of HΦ and ΩH 2 (Cr ) is the minimal invariant subspace of multiplication by z on H 2 (Cm ) that contains all maximizing vectors of HΦt . Proof. By Theorem 1.8, we may assume without loss of generality that ΨL∞ < s. We need the following lemma. Lemma 15.9. Suppose that Φ satisfies the hypotheses of Theorem 15.8. A function f in H 2 (Cn ) is a maximizing vector of HΦ if and only if f = Υg, where g ∈ H 2 (Cr ) and g is a maximizing vector of HU . Let us first complete the proof of Theorem 15.8. Since U is very badly approximable, we have t0 (U ) = · · · = tr−1 (U ) = 1. By Theorem 15.1, the minimal invariant subspace of multiplication by z on H 2 (Cr ) that contains all maximizing vectors of HU is H 2 (Cr ). It now follows from Lemma 15.9 that the minimal invariant subspace of multiplication by z on H 2 (Cn ) that contains all maximizing vectors of HΦ is ΥH 2 (Cr ). To complete the proof, we can apply this result to the matrix function Φt . Proof of Lemma 15.9. First of all, by Theorem 15.7, HΦ = σ. Without loss of generality we may assume that σ = 1. It has been proved in the proof of Theorem 15.7 that if g is a maximizing vector of HU , then Υg is a maximizing vector of HΦ . Suppose now that f is a maximizing vector of HΦ . We have U O Φf = W ∗ V ∗f O Ψ ∗ U O Υ f = W∗ Θt f O Ψ U Υ∗ f . = W∗ ΨΘt f

650


Since W ∗ is unitary-valued and ΨL∞ < 1, it follows that Θt f = 0. Put g = Υ∗ f ∈ L2 (Cr ). We have ∗ Υ∗ f Υ f = VV ∗ f = V = ΥΥ∗ f = Υg. f = Θ Υ Θt 0 Since Υ is co-outer, it follows from Lemma 1.4 that g ∈ H 2 (Cr ). We have Ug U Υ∗ Υg ΩU g ∗ . = Φf = W = Ω Ξ O O O 2 (Cr ), Clearly, f is a maximizing vector of HΦ if and only if ΩU g ∈ H− which is equivalent to the condition z¯ΩU g ∈ H 2 (Cr ). Since Ωt is outer, it follows from Lemma 1.4 that z¯U g ∈ H 2 (Cr ), which is equivalent to the 2 fact that U g ∈ H− (Cr ). But the latter just means that g ∈ Ker TU , and so g is a maximizing vector of HU . Suppose now that the matrix function Ψ in the factorization (15.6) also satisfies the condition HΨ e < HΨ . Then we can continue this process, find a best analytic approximation G of Ψ and factorize Ψ − G as in (15.6). If we are able to continue this diagonalization process until the very end, we construct the unique superoptimal approximation Q of Φ and obtain a canonical factorization of Φ − Q. Therefore we need an estimate of HΨ e . If r = 1, then by Theorem 4.1, HΨ e ≤ HΦ e . We want to obtain the same inequality for an arbitrary r. We could try to generalize the proof of Theorem 4.1 to the case of an arbitrary r. However, we are going to choose another way. We would like to deduce this result for an arbitrary r from the corresponding result in the case r = 1. Suppose that Φ and F satisfy the hypotheses of Theorem 15.3. Then Φ − F admits a factorization (15.6). On the other hand,q it follows from Theorem 5.1 that Φ − F admits a partial thematic factorization of the form   O ··· O O t 0 u0  O t0 u1 · · · O O     .. . . ..  V ∗ · · · V ∗ , ∗ ∗ . .. .. .. Φ − F = W0 · · · Wr−1  . 0 .    r−1 (15.8)  O O · · · t0 ur−1 O  O O ··· O ∆

where ∆L∞ ≤ t0 Ij Vj = O

and H∆ < t0 , Ij O , W = j V˘j O

O ˘j W

,

0 ≤ j ≤ r − 1,

˘ t , and the uj are unimodular funcwith thematic matrix functions V˘j , W j tions such that the Toeplitz operators Tuj are Fredholm with ind Tuj > 0. ˘ 0 = W0 . For j = 0 we assume that V˘0 = V0 and W


Suppose that V˘j = v j

Θj

,

˘t= W j

wj

Ξj

,

651

0 ≤ j ≤ r − 1, (15.9)

where v j , Θj , wj , Ξj are inner and co-outer. Recall that V and W have the form (15.5). Theorem 15.10. Under the above hypotheses there exist constant unitary matrices U1 ∈ Mn−r,n−r and U2 ∈ Mm−r,m−r such that Θ = Θ0 Θ1 · · · Θr−1 U1

(15.10)

Ξ = Ξ0 Ξ1 · · · Ξr−1 U2 .

(15.11)

and Proof. It follows from Theorem 1.8 that if we replace F with another best approximation G, the matrix function Φ − G will still admit factorizations of the forms (15.6) and (15.8) with the same matrix functions V, W, Vj , Wj . Hence, we may assume that F ∈ Ωr (see the Introduction to this chapter). Then the matrix functions Ψ in (15.6) and ∆ in (15.8) satisfy the inequalities ΨL∞ < t0 and ∆L∞ < t0 . Define the function ρ : R → R by

t, t ≥ t20 , ρ(t) = 0, t < t20 . Consider the operator M : H 2 (Cn) → L2 (Cn ) of multiplication by the matrix function ρ (Φ − F )t (Φ − F ) . It was proved in Lemmas 9.4 and 9.5 that Ker M = Θ0 Θ1 · · · Θr−1 H 2 (Cn−r ).

(15.12)

On the other hand, it follows from (15.6) that 2 O t0 I r Vt (Φ − F )t (Φ − F ) = V O Ψt Ψ and since Ψt ΨL∞ < t20 , we have 2 2 t0 I r O t0 I r Vt = V ρ (Φ − F )t (Φ − F ) = V O O O

O O

Υt Θ∗

.

By Theorem 1.1, Ker TΥt = ΘH 2 (Cn−r ). It is easy to see now that Ker M = ΘH 2 (Cn−r ). Together with (15.12) this yields Θ0 Θ1 · · · Θr−1 H 2 (Cn−r ) = ΘH 2 (Cn−r ), which means that both inner functions Θ and Θ0 Θ1 · · · Θr−1 determine the same invariant subspace of multiplication by z on H 2 (Cn ). Therefore there exists a constant unitary function U1 such that (15.10) holds (see Appendix 2.3). To prove (15.11), we can apply (15.10) to (Φ − F )t .

652


Corollary 15.11. Let Ψ and ∆ be the matrix functions in the factorizations (15.6) and (15.8). Then ∆ = U2 ΨUt1 ,

(15.13)

where U1 and U2 are unitary matrices from (15.10) and (15.11). Proof. By Corollary 9.2, ∆ = Ξ∗r−1 · · · Ξ∗1 Ξ∗0 (Φ − F )Θ0 Θ1 · · · Θr−1 . By Theorem 15.10, ∆ = U2 Ξ∗ (Φ − F )ΘUt1 . On the other hand, it is easy to see from (15.6) that Ψ = Ξ∗ (Φ − F )Θ,

(15.14)

which implies (15.13). Now we are in a position to estimate HΨ e for the residual entry Ψ in the factorization (15.6). Theorem 15.12. Let Φ be a function in L∞ (Mm,n ) such that HΦ e < HΦ . Then for the residual entry Ψ in the partial canonical factorization (15.6) the following inequality holds: HΨ e ≤ HΦ e . Proof. Iterating Theorem 4.1, we find that H∆ e ≤ HΦ e . The result now follows from (15.13). Consider now the unitary-valued matrix function U in the partial canonical factorization (15.6). By Corollary 15.6, the Toeplitz operator TU is Fredholm. We are now going to evaluate the index of TU in terms of the indices kj = ind Tuj ,

0 ≤ j ≤ r − 1.

of the partial thematic factorization (15.8). Theorem 15.13. Let Φ ∈ L∞ (Mm,n ) and HΦ e < HΦ . Then the entry U of of the diagonal block matrix function in the partial canonical factorization (15.6) satisfies ind TU = dim Ker TU = k0 + k1 + · · · + kr−1 . Proof. Without loss of generality we may assume that HΦ = 1. By Theorem 15.3, U is very badly approximable, HU e < HU = 1. Therefore by Theorem 5.4, the Toeplitz operator TzU has dense range in H 2 (Cr ). Hence, Ker TU∗ = {O}, and so ind TU = dim Ker TU . By Theorem 7.4, dim{f ∈ H 2 (Cn ) : HΦ f 2 = f 2 } = k0 + k1 + · · · + kr−1 (15.15)


653

Let us show that the left-hand side of (15.15) is equal to dim Ker TU . Indeed, if g ∈ H 2 (Cr ), then g ∈ Ker TU if and only if g is a maximizing vector of HU . By Lemma 15.9, dim{g ∈ H 2 (Cr ) : HU g2 = g2 } = dim{f ∈ H 2 (Cn ) : HΦ f 2 = f 2 }, which proves the result. We obtain now an inequality between the singular values of HΦ and the singular values of HΨ that generalizes Theorem 8.1. Theorem 15.14. Let Φ ∈ L∞ (Mm,n ) and HΦ e < HΦ . Suppose that F ∈ H ∞ (Mm,n ) is a best approximation of Φ by bounded analytic functions and Φ − F is represented by the partial canonical factorization (15.6). Then sj (HΨ ) ≤ sk+j (HΦ ),

j ≥ 0,

(15.16)

def

where k = ind TU . Proof.

Consider the partial thematic factorization (15.8). Let

def

kj = ind Tuj , 0 ≤ j ≤ r − 1, be the indices of this factorization. We have t 0 u0 O ∗ V0∗ , (15.17) Φ − F = W0 O Φ[1] where Φ[1] is given by the partial thematic factorization I r−2 I r−2 O O [1] ∗ ˘ D · · · V˘1∗ Φ = W1 · · · ∗ ˘∗ O W O V˘r−1 r−1 and



t 0 u1  ..  D= .  O O

··· .. . ··· ···

 O ..  .  . O  ∆

O .. . t0 ur−1 O

Now we can apply Theorem 8.1 to the factorization (15.17) and find that sj (HΦ[1] ) ≤ sk0 +j (HΦ ),

j ≥ 0.

Then we can apply Theorem 8.1 to the above partial thematic factorization of Φ[1] , etc. After applying Theorem 8.1 r times we obtain the inequality sj (H∆ ) ≤ sk0 +···+kr−1 +j (HΦ ),

j ≥ 0.

The result now follows from Corollary 15.11 and Theorem 15.13. Now we are going to prove that under the hypotheses of Theorem 15.1 the Toeplitz operators TV , TV t , TW , and TW t are invertible. Then we deduce that the matrix functions Υ, Θ, Ω, and Ξ are left invertible in H ∞ . Recall that in the case r = 1 those facts have been proved in §4.

654


Theorem 15.15. Let Φ ∈ L∞ (Mm,n ) and suppose that HΦ e < HΦ . Let V and W t be the r-balanced matrix functions in the partial canonical factorization (15.6). Then the Toeplitz operators TV , TV t , TW , and TW t are invertible. Proof. As in the case of r = 1 it follows from Theorem 1.3 that HΥ = HΘ (see Lemma 4.6) and as in the case r = 1 to prove that TV and TV t are invertible, it is sufficient to show that HΘ < 1 (see Lemma 4.6). Consider the partial thematic factorization (15.8). By Lemma 4.6, the Toeplitz operators TV˘j are invertible for 0 ≤ j ≤ r − 1. Since the V˘j are 1-balanced, this is equivalent to the fact that HΘj < 1, 0 ≤ j ≤ r − 1 (see the proof of Lemma 4.6). By Theorem 15.10, Θ = Θ0 · · · Θr−1 U for some constant unitary matrix U. Clearly, to show that HΘ < 1, it is sufficient to prove the following lemma. Lemma 15.16. Let Θ1 be an n × k inner matrix function and let Θ2 be a k × l inner matrix function such that HΘ1 < 1 and HΘ2 < 1. Let Θ = Θ1 Θ2 . Then HΘ < 1. Proof of Lemma 15.16. Let σ < 1 be a positive number such that HΘ1 < σ and HΘ2 < σ. Let f ∈ H 2 (Cl ). We have HΘ f 22

= P− Θ1 Θ2 f 22 = P− Θ1 P− Θ2 f + P− Θ1 P+ Θ2 f 22 = Θ1 P− Θ2 f + P− Θ1 P+ Θ2 f 22 = P− Θ2 f + Θ1t P− Θ1 P+ Θ2 f 22 .

We claim that the functions P− Θ2 f and Θ1t P− Θ1 P+ Θ2 f are orthogonal. 2 Indeed, let ϕ ∈ H− (Ck ) and ψ ∈ H 2 (Ck ). We have (ϕ, Θ1t P− Θ1 ψ) = (Θ1 ϕ, P− Θ1 ψ) = (Θ1 ϕ, Θ1 ψ) = (ϕ, ψ) = 0, since Θ1 takes isometric values almost everywhere on T. It follows that HΘ f 22

= P− Θ2 f 22 + Θ1t P− Θ1 P+ Θ2 f 22 ≤

P− Θ2 f 22 + P− Θ1 P+ Θ2 f 22

= P− Θ2 f 22 + HΘ1 P+ Θ2 f 22 ≤

P− Θ2 f 22 + σ 2 P+ Θ2 f 22

= σ 2 (P− Θ2 f 22 + P+ Θ2 f 22 ) + (1 − σ 2 )P− Θ2 f 22 = σ 2 Θ2 f 22 + (1 − σ 2 )HΘ2 f 22 ≤ σ 2 f 22 + σ 2 (1 − σ 2 )f 22 = (2σ 2 − σ 4 )f 22 .


655

The result now follows from the trivial inequality 2σ 2 − σ 4 < 1. Corollary 15.17. Under the hypotheses of Theorem 15.15 the matrix functions Υ, Θ, Ω, and Ξ are left invertible in H ∞ . Proof. We have shown in the proof of Theorem 15.15 that HΥ < 1 and HΘ < 1. By Theorem 3.4.4, the Toeplitz operators TΥ and TΘ are left invertible. It now follows from Theorem 3.6.1 that Υ and Θ are left invertible in H ∞ . To prove the left invertibility of Ω and Ξ, we can replace Φ with Φt . Let us proceed now to the construction of a canonical factorization for very badly approximable matrix functions. Given Φ ∈ L∞ (Mm,n ), consider the sequence {tj } of its superoptimal singular values. Suppose that t0 = · · · = tr1 −1 > tr1 = · · · = tr2 −1 > · · · > trι−1 = · · · = trι −1 (15.18) are all nonzero superoptimal singular values of Φ, i.e., t0 has multiplicity r1 and trj has multiplicity rj+1 − rj , 1 ≤ j ≤ ι − 1. By Theorem 15.3, if HΦ e < HΦ and F0 is a best approximation of Φ by a bounded analytic matrix function, then Φ − F0 admits a factorization t0 U0 O V0∗ , Φ − F0 = W0∗ O Φ[1] where V0 and W0t are r1 -balanced matrix functions, U0 is a very badly approximable unitary-valued matrix function such that HU0 e < 1, and Φ[1] is a matrix function in L∞ (Mm−r1 ,n−r1 ) such that HΦ[1] = tr1 . By Theorem 15.12, HΦ[1] e ≤ HΦ e . If tr1 is still greater than HΦ e , we can apply Theorem 15.3 to Φ[1] and find that for a best approximation G1 of Φ[1] the matrix function Φ[1] − G1 admits a factorization tr1 U1 O [1] ∗ ˘ Φ − G1 = W1 V˘1∗ , O Φ[2] ˘ t are (r2 −r1 )-balanced matrix functions, U1 is a very badly where V˘1 and W 1 approximable unitary-valued matrix function of size (r2 − r1 ) × (r2 − r1 ) such that HU1 e < 1, and Φ[2] is a matrix function in L∞ (Mm−r2 ,n−r2 ) such that HΦ[2] = tr2 . We can now apply Theorem 1.8 and find a matrix function F1 ∈ H ∞ (Mm,n ) such that   O O t0 U0 tr1 U1 O  V1∗ V0∗ , Φ − F1 = W0∗ W1∗  O O O Φ[2] where

V1 =

I r1 O

O V˘1

and W1 =

I r1 O

O ˘1 W

.

656


If tr2 is still greater than HΦ e , we can continue this process and apply Theorem 15.3 to Φ[2] . Suppose now that trd−1 > HΦ e , 2 ≤ d ≤ ι. Then continuing the above process and applying Theorem 1.8, we can find a function F ∈ H ∞ (Mm,n ) such that Φ − F admits a factorization Φ−F     ∗ =W0∗ · · · Wd−1   

t0 U0 O .. .

O t r1 U 1 .. .

··· ··· .. .

O O .. .

O O

O O

··· ···

trd−1 Ud−1 O

O O .. .



   ∗  Vd−1 · · · V0∗ ,  (15.19) O  Φ[d]

where the Uj are (rj+1 − rj ) × (rj+1 − rj ) very badly approximable unitaryvalued functions such that HUj e < 1, I rj O I rj O Vj = = and W j ˘ j , 1 ≤ j ≤ d − 1, O V˘j O W (15.20) ˘ t are (rj+1 − rj )-balanced matrix functions, and Φ is a matrix V˘j and W j function satisfying [d] Φ ∞ ≤ tr , and HΦ[d] < tr . (15.21) d−1 d−1 L Factorizations of the form (15.19) with Vj and Wj of the form (15.20) and Φ[d] satisfying (15.21) are called partial canonical factorizations. The matrix function Φ[d] is called the residual entry of the partial canonical factorization (15.19). Finally, if HΦ e is less than the smallest nonzero superoptimal singular value trι −1 , then we can complete this process and construct the unique superoptimal approximation of Φ by bounded analytic matrix functions. This proves the following theorem. Theorem 15.18. Let Φ ∈ L∞ (Mm,n ) and suppose that the nonzero superoptimal singular values of Φ satisfy (15.18). If HΦ e < trι −1 and F is the unique superoptimal approximation of Φ by bounded analytic functions, then Φ − F admits a factorization   t0 U0 O ··· O O  O O O  tr1 U1 · · ·   . . . ..  V ∗ · · · V ∗ , ∗ ∗  . .. .. .. Φ − F = W0 · · · Wι−1  .. 0 .    ι−1 (15.22)  O O · · · trι−1 Uι−1 O  O O ··· O O where the Vj , Wj , and Uj are as above. Note that the lower right entry of the diagonal matrix function on the right-hand side of (15.22) is the zero matrix function of size (m − rι ) × (n − rι ). Here it may happen that m − rι or n − rι can be


657

zero, which would mean that the corresponding rows or columns do not exist. Clearly, the left-hand side of (15.22) is a very badly approximable matrix function. Factorizations of the form (15.22) are called canonical factorizations of very badly approximable matrix functions. The following result shows that the right-hand side of (15.22) is always a very badly approximable matrix function. Theorem 15.19. Let Φ be a matrix function in L∞ (Mm,n ) such that HΦ e < HΦ , ι is a positive integer, r1 , · · · , rι are positive integers satisfying r 1 < r 2 < · · · < rι and σ0 > σ1 > · · · > σι−1 > 0. Suppose that Φ admits a factorization  O ··· σ0 U0  O σ U · ·· 1 1  . . ∗ ∗  . .. .. Φ = W0 · · · Wι−1  ..   O O ··· O O ···

O O .. .

O O .. .



   ∗  Vι−1 · · · V0∗ ,  O  O

σι−1 Uι−1 O

in which the Uj , Vj , and Wj are as above. Then Φ is very badly approximable and the superoptimal singular values of Φ are given by   σ0 , κ < r1 , σj , rj ≤ κ < rj+1 , tκ (Φ) =  0, κ ≥ rι . Proof. Let V0 =

Υ Θ

and W0 =

Ω

Ξ

t

,

where Υ, Θ, Ω, and Ξ are inner and co-outer matrix functions. By Theorem 15.8, the minimal invariant subspace of multiplication by z on H 2 (Cn ) that contains all maximizing vectors of HΦ is ΥH 2 (Cr1 ) and the minimal invariant subspace of multiplication by z on H 2 (Cm ) that contains all maximizing vectors of HΦt is ΩH 2 (Cr1 ). By Theorem 15.7, Φ is badly approximable and t0 (Φ) = · · · = tr1 −1 = σ0 . By Theorem 1.8, we can reduce our problem to the function I rι−1 −r1 O O ˘ 1∗ · · · I rι−1 −r1 D · · · V˘1∗ , W ∗ ˘∗ O W O V˘ι−1 ι−1

658


where



σ1 U1  ..  D= .  O O

··· .. . ··· ···

 O ..  .  . O  O

O .. . σι−1 Uι−1 O

This function is also represented by a canonical factorization that makes it possible to continue this process and prove that Φ is very badly approximable and the superoptimal singular values of Φ satisfy the desired equality. We can now demonstrate an advantage of canonical factorizations over thematic factorizations. Namely, we show that canonical factorizations possess certain invariance properties, i.e., the matrix functions Uj in the canonical factorization (15.22) are uniquely determined modulo unitary constant factors. Moreover, if V0 = Υ0 Θ0 , W0t = Ω0 Ξ0 , ˘ t = Ωj Ξj , 1 ≤ j ≤ ι − 1, V˘j = Υj Θj , W j ˘ j are given by (15.20), then the matrix functions and the V˘j and W Υj , Θj , Ωj , and Ξj are also uniquely determined modulo constant unitary factors. We start with partial canonical factorizations of the form (15.6). Suppose that a matrix function Φ in L∞ (Mm,n ) satisfies HΦ e < HΦ and admits partial canonical factorizations σU 0 ∗ (15.23) Φ= Ω Ξ Υ Θ 0 Ψ and Φ=

Ω◦

Ξ◦

σ◦ U ◦ 0

0 Ψ◦

Υ◦

Θ◦

∗

, (15.24)

where ΨL∞ ≤ σ, HΨ < σ, Ψ◦ L∞ ≤ σ ◦ , HΨ◦ < σ ◦ , U is an r × r very badly approximable unitary-valued function such that HU e < 1, U ◦ is an r◦ × r◦ very badly approximable unitary-valued function such that t are r-balanced matrix functions, HU ◦ e < 1, Υ Θ and Ω Ξ ◦ t ◦ ◦ ◦ ◦ and Ω Ξ and Υ Θ are r -balanced matrix functions. Theorem 15.20. Let Φ be a badly approximable function in L∞ (Mm,n ) such that HΦ e < HΦ . Suppose that Φ admits factorizations (15.23) and (15.24). Then r = r◦ , σ = σ ◦ , and there exist unitary matrices V# , V ∈ Mr,r , U# ∈ Mn−r,n−r , U ∈ Mm−r,m−r such that Υ◦ = ΥV# ,

Ω◦ = ΩV ,

(15.25)

Θ◦ = ΘU# ,

Ξ◦ = ΞU ,

(15.26)


659

and U ◦ = (V )t U V# ,

Ψ◦ = (U )∗ ΨU# .

(15.27)

◦

Proof. By Theorem 15.7, σ = σ = HΦ . Next, by Theorem 15.8, the minimal invariant subspace of multiplication by z on H 2 (Cn ) that contains all maximizing vectors of HΦ is equal to ΥH 2 (Cr ) and at the same time ◦ it is equal to Υ◦ H 2 (Cr ). It follows that r = r◦ and there exists a unitary matrix V# ∈ Mr,r such that Υ◦ = ΥV# . Applying the same reasoning to Φt , we find a unitary matrix function V ∈ Mr,r such that Ω◦ = ΩV , which proves (15.25). By Theorem 1.1, ΘH 2 (Cn−r ) = Ker TΥt

and

Θ◦ H 2 (Cn−r ) = Ker T(Υ◦ )t .

By (15.25), Ker TΥt = Ker T(Υ◦ )t , which implies that there exists a unitary matrix U# ∈ Mn−r,n−r such that Θ◦ = ΘU# . Applying the same reasoning to Φt , we find a unitary matrix U ∈ Mm−r,m−r such that Ξ◦ = ΞU , which proves (15.26). By (15.7), σU = Ωt ΦΥ and σU ◦ = (Ω◦ )t ΦΥ◦ . This implies the first equality in (15.27). Finally, by (15.14), Ψ = Ξ∗ ΦΘ

and

Ψ◦ = Ξ◦ ∗ ΦΘ◦ ,

which completes the proof of (15.27). We can obtain similar results for arbitrary partial canonical factorizations and canonical factorizations. Let us consider in detail the following special case. Suppose that the matrix functions Ψ and Ψ◦ in (15.23) and (15.24) admit the following partial canonical factorizations: σ1 U1 0 ∗ Ψ = Ω1 Ξ1 (15.28) Υ1 Θ1 0 ∆ and Ψ◦ =

Ω◦1

Ξ◦1

σ1◦ U1◦ 0

0 ∆◦

Υ◦1

Θ◦1

∗

, (15.29)

where ∆L∞ ≤ σ1 , H∆ < σ1 and ∆◦ L∞ < σ1◦ , H∆◦ < σ1◦ . By Theorem 15.20, Ψ◦ = (U )∗ ΨU# . It now follows from (15.29) that σ◦ U ◦ 0 ∗ 1 1 ◦ ◦ , Ψ = U Ω1 U Ξ1 U# Υ◦1 U# Θ◦1 ◦ 0 ∆ (15.30) which is another partial canonical factorization of Ψ. We can now compare the factorizations (15.28) and (15.30). By Theorem # 15.20, σ1◦ = σ1 and there exist unitary matrices V# 1 , V1 , U1 , U1 such that Υ◦1 = (U# )t Υ1 V# 1 ,

Ω◦1 = (U )t Ω1 V1 ,

660


Θ◦1 = (U# )∗ Θ1 U# 1 ,

Ξ◦1 = (U )∗ Ξ1 U1 ,

and U1◦ = (V1 )t U1 V# 1 ,

∆◦ = (U1 )∗ ∆U1 # .

It is easy to see that the same results hold in the case of arbitrary partial canonical factorizations as well as arbitrary canonical factorizations. To conclude this section, we proceed to the heredity problem for (partial) canonical factorizations. We prove that if X is a linear R-space or X is a decent function space and the initial matrix function Φ belongs to X, then all factors in its (partial) canonical factorizations also belong to X. Theorem 15.21. Suppose that X is a linear R-space or X is a decent function space. Let Φ be a bounded m × n matrix function such that P− Φ is a nonzero matrix function in X(Mm,n ). If F ∈ H ∞ (Mm,n ) is a best approximation of Φ and Φ − F admits a partial canonical factorization t0 U O ∗ , Φ−F = Ω Ξ Υ Θ O Ψ then Υ, Θ, Ω, Ξ, U, P− Ψ ∈ X. Proof. Assume without loss of generality that t0 = 1. By Theorem 1.8, if we replace a best approximating function F with any other best approximation, we do not change P− Ψ. Thus we may assume that F is the unique superoptimal approximation of Φ by bounded analytic matrix functions. By Theorem 12.7, Φ − F belongs to X. Let us first prove that Θ ∈ X. Consider a partial thematic factorization of Φ − F of the form (15.8). Then the matrix functions Vj given by (15.9) belong to X (see Lemma 12.1 and the proofs of Theorems 12.3 and 12.6). In particular, the inner matrix functions Θj in (15.9) belong to X. Since X ∩ L∞ is an algebra, it now follows from (15.10) that Θ ∈X. Consider now the unitary-valued matrix function V = Υ Θ . By Theorem 1.3, the Toeplitz operator TV has dense range in H 2 (Cn ). Therefore by Theorems 13.1.1 and 13.5.1, V ∈ X, and so Υ ∈ X. If we apply the above reasoning to Φt , we prove that Ξ ∈ X and Ω ∈ X. It follows now from (15.7) that U ∈ X. Finally, it follows from (15.14) that P− Ψ ∈ X. Remark. It can be shown that if X is a linear R-space, then the X-norms of Υ, Θ, Ω, Ξ, U, P− Ψ can be estimated in terms of the X-norm of P− Φ. Clearly, it follows from Theorem 15.21 that the same result holds for arbitrary partial canonical factorizations. In particular, the following theorem holds. Theorem 15.22. Let Φ ∈ L∞ (Mm,n ) and F is the superoptimal approximation of Φ by bounded analytic matrix functions. If (15.22) is a canonical factorization of Φ − F , then all factors on the right-hand side of (15.22) belong to X.

16. Very Badly Approximable Unitary-Valued Functions

661

Now consider separately the important case X = V M O, which is an R-space. Theorem 15.23. Let Φ ∈ (H ∞ + C)(Mm,n ) and P− Φ = 0. If F is a best approximation of Φ by bounded analytic functions and Φ − F admits a partial canonical factorization (15.6), then V, W, U ∈ QC and Ψ ∈ H ∞ +C. Theorem 15.23 follows immediately from Theorem 15.21 if we put X = V M O.

16. Very Badly Approximable Unitary-Valued Functions We have seen in the previous section that an important role is played by the very badly approximable unitary-valued matrix functions U satisfying the condition HU e < 1. In this section we obtain a characterization of this class and study some properties of such functions. Recall that by Corollary 15.6, for such a matrix function U the Toeplitz operator TU is Fredholm. Theorem 16.1. Let U be an n × n unitary-valued matrix function such that HU e < 1. The following are equivalent: (i) U is very badly approximable; (ii) the Toeplitz operator TzU : H 2 (Cn ) → H 2 (Cn ) has dense range in H 2 (Cn ); (iii) the Toeplitz operator Tz¯U ∗ : H 2 (Cn ) → H 2 (Cn ) has trivial kernel; (iv) the Toeplitz operator TU is Fredholm and the indices of a Wiener–Hopf factorization of U are negative. Proof. Clearly (ii) and (iii) are equivalent. Next, by Theorem 5.4, (i) implies (ii). Let us show that (ii) implies (iv). Clearly, it follows from (ii) that TU has dense range. It follows easily Theorem 4.4.11 that HU e = HU ∗ e and by Theorem 3.4.6, the operator TU is Fredholm. Consider a Wiener–Hopf factorization of U :  d1  z O ··· O  O z d2 · · · O   ∗ (16.1) U = Ψ2  . . ..  Ψ1 , . .. ..  .. .  O O · · · z dn ±1 2 where Ψ±1 1 , Ψ2 ∈ H (Mn,n ). By Corollary 3.5.8, (iii) is equivalent to the fact that

dj < 0,

1 ≤ j ≤ n.

(16.2)

662


It remains to prove that (iv) implies (i). Clearly, U is a unitary interpolant of itself and since all the indices dj are negative, it follows from Theorem 14.2 that t0 (Φ) = · · · = tn−1 (Φ) = 1, and so U is very badly approximable. Corollary 16.2. Let X be a decent function space. Suppose that U is an n × n unitary-valued function in X. Then U is very badly approximable ±1 if and only if U admits a factorization (16.1) such that Ψ±1 1 ∈ X, Ψ2 ∈ X, and (16.2) holds. Proof. The result follows immediately from Theorems 16.1 and 13.6.1. Corollary 16.3. Let U be a unitary-valued very badly approximable function satisfying HU e < 1 and let k = −(d1 + · · · + dn ), where the dj are the indices of a Wiener–Hopf factorization 16.1 of U . Then sj (HU ∗ ) = sj+k (HU ),

j ∈ Z+ .

Proof. It follows from Theorem 16.1 that TU has dense range. Then by Theorem 4.4.11, sj (HU ∗ ) = sj+d (HU ), j ∈ Z+ , where d = dim{f ∈ H 2 (Cn ) : HU f 2 = f 2 }. Finally, it follows from Theorem 7.4 that d = k.

17. Superoptimal Meromorphic Approximation In this section we study the problem of superoptimal approximation of a matrix function on T by meromorphic matrix functions of degree at most ∞ (m, n) the class of bounded k, k ∈ Z+ . Given k ∈ Z+ , we denote by H(k) Mm,n -valued functions Q on T such that the McMillan degree of P− Q is at ∞ (m, n) if and only if rank HQ ≤ k (see §2.5). most k. Recall that Q ∈ H(k) ∞ ∞ (m, n) if this does not lead to confusion. We also write H(k) instead of H(k) ∞ (m, n)As we have seen in §11.4, the problem of approximation by H(k) functions is important in applications to the problem of model reduction. ∞ As in the scalar case the problem of approximation by H(k) (m, n) functions is called the Nehari–Takagi problem. (k) For Φ ∈ L∞ (Mm,n ) and k ∈ Z+ we define the sets Ωj , 0 ≤ j ≤ min{m, n} − 1, by ( (k)

Ω0 =

∞ Q ∈ H(k) (m, n) : Q minimizes ess sup Φ(ζ) − Q(ζ)Mm,n ζ∈T

(k) Ωj

=

Q∈

(k) Ωj−1

( : Q minimizes ess sup sj Φ(ζ) − Q(ζ) .

ζ∈T

,

17. Superoptimal Meromorphic Approximation

We put (k)

tj

663

def (k) = tj (Φ) = ess sup sj Φ(ζ) − Q(ζ) ζ∈T

(k) Ωj

(0)

and 0 ≤ j ≤ min{m, n} − 1. Note that the sets Ωj coincide for Q ∈ with the sets Ωj defined in the introduction to this chapter. ∞ (m, n) is called a superoptimal approximation A matrix function F ∈ H(k) of Φ by meromorphic matrix functions of degree at most k (or a superop(k) timal solution of the Nehari–Takagi problem) if F ∈ Ωmin{m,n}−1 . (k)

Note that the matrix functions in Ω0 are precisely the best approximations of Φ by meromorphic matrix functions of degree at most k. By (k) Theorem 4.3.1 and the remark after it, Ω0 = ∅ and HΦ −HQ = sk (HΦ ) (k) for any Q ∈ Ω0 . It turns out, however, that unlike the case of analytic approximation the condition Φ ∈ (H ∞ + C)(Mm,n ) (and even any smoothness condition on Φ) does not guarantee the uniqueness of a superoptimal approximation ∞ (m, n). Indeed, suppose that k = 1 and consider the by functions in H(k) matrix function z¯ O Φ= . O z¯ (1)

(1)

It is easy to see that t0 = 1 and t1 = 0. It is also clear that the functions O O z¯ O F1 = and F2 = O z¯ O O ∞ (2, 2). In fact, it is are superoptimal approximations by functions in H(1) easy to see that any function of the form z¯P with P a rank one projection on C2 is a superoptimal approximation of Φ. We show in this section that we still have uniqueness under the additional assumption sk (HΦ ) < sk−1 (HΦ ). We obtain this result not only for H ∞ +C matrix functions but also for matrix functions Φ with a sufficiently small norm HΦ e . We also obtain in this section a parametrization formula for ∞ all best approximations by matrix functions in H(k) .

Theorem 17.1. Let k be a positive integer and let Φ be a matrix function in L∞ (Mm,n ) such that HΦ e is less than the smallest nonzero (k) number among the tj , 0 ≤ j ≤ min{m, n} − 1. If sk (HΦ ) < sk−1 (HΦ ), then there exists a unique superoptimal approximation F of Φ by functions ∞ in H(k) (m, n). Moreover, for this F (k) sj (Φ − F )(ζ) = tj , 0 ≤ j ≤ min{m, n} − 1, ζ ∈ T. (17.1) We need the following elementary fact. Lemma 17.2. Let H1 and H2 be Hilbert spaces and let T be a bounded linear operator from H1 to H2 such that sk−1 (T ) > sk (T ) > T e . Let R be an operator of rank k such that T − R = sk (T ). Then any Schmidt

664


vector of T corresponding to the singular value sk (T ) belongs to Ker R and is a maximizing vector of T − R. Proof. Let ξ be a unit Schmidt vector corresponding to sk (T ) and let η = (sk (T ))−1 T ξ. Let ξ0 , · · · , ξk−1 be orthonormal Schmidt vectors of T corresponding to the singular values s0 (T ), · · · , sk−1 (T ). Put def

def

def

ηj = (sj (T ))−1 T ξj , 0 ≤ j ≤ k − 1. Let ξk = ξ and ηk = η. Clearly, ξ0 , · · · , ξk and η0 , · · · , ηk are orthonormal families. Denote by P the orthogonal projection from H2 onto span{η0 , · · · , ηk }. Consider the operator P R span{ξ0 , · · · , ξk } whose rank is at most k. Then there exists a unit vector x=

k

aj ξj

j=0

that belongs to Ker P R. It follows that P T x = P (T − R)x, and so P T x ≤ sk (T ). We have PTx =

k

aj sj (T )ηj

and P T x2 =

j=0

k

sj (T )2 |aj |2 ,

j=0

and so P T x2 > sk (T )2

k

|aj |2 > 1

j=0

unless a0 = · · · = ak−1 = 0. Thus x = ak ξk with |ak | = 1. Since P Rx = O, Rx is orthogonal to span{η0 , · · · , ηk }. If Rx = O, then sk (T )2 ≥ T − Rx2 = T x2 + Rx2 > T x2 = s2k (T ). Thus Rx = Rξ = O, which proves the result. Proof of Theorem 17.1. Let Q be any best approximation of Φ by matrix functions in H(k) . As we have observed above, def

(k)

Φ − QL∞ = sk = sk (HΦ ) = t0 . ∞ . Clearly, the only Q that If sk = 0, then rank HΦ ≤ k, and so Φ ∈ H(k) minimizes Φ−QL∞ is Φ and the theorem holds in this case. Suppose that sk > 0. By the hypotheses, HΦ e < sk , and so HΦ has a unit Schmidt vector f that corresponds to sk . Let us prove that

HΦ f = (Φ − Q)f .

(17.2)

By Lemma 17.2, f is a maximizing vector of HΦ−Q = HΦ − HQ , and so sk

= HΦ−Q = HΦ−Q f = P− (Φ − Q)f ≤ (Φ − Q)f ≤ Φ − QL∞ f = sk .

It follows that P− (Φ − Q)f = (Φ − Q)f ,


665

and since by Lemma 17.2, f ∈ Ker HQ , it follows that (Φ − Q)f = P− (Φ − Q)f = HΦ−Q f = HΦ f − HQ f = HΦ f , which proves (17.2). Put

¯HΦ f ∈ H 2 (Cm ). g = s−1 k z Since HΦ f = HΦ−Q f , it follows from Theorem 2.2.3 that f (ζ)Cn = g(ζ)Cm almost everywhere on T. Hence, f and g admit factorizations f = ϑ1 hv,

g = ϑ2 hw,

(17.3)

where ϑ1 and ϑ2 are scalar inner functions, h is a scalar outer function in H 2 , and v and w are inner and co-outer column functions. (k) Denote by E the set of matrix functions of the form Φ−Q where Q ∈ Ω0 . (k) By (17.1), Ef = HΦ f = sk z¯g = t0 z¯g for any E ∈ E. It follows from (17.3) that (k)

Ev = t0 u0 w,

E ∈ E,

(17.4)

¯ where u0 = z¯ϑ¯1 ϑ¯2 h/h. If n = 1, then v is a nonzero scalar function in H ∞ and we have E = t0 v −1 u0 w, (k)

which uniquely determines E and proves the constancy of the singular values. By considering Φt , we obtain the result for m = 1. This proves the theorem in the case min{m, n} = 1. Now consider the case min{m, n} > 1 and suppose that the result holds for any lesser value of min{m, n}. By Theorem 1.1, there exist thematic completions (17.5) V = v Θ and W t = w Ξ . It is easily seen from (17.4) that the matrix function W EV has the form (k) t0 u0 ∗ W EV = , ∗ ∗ (k)

where V and W are defined by (17.5). Clearly, W EV L∞ ≤ t0 , u0 is a scalar unimodular function, and so by Theorem 15.5, W EV has the form (k) t0 u0 O W EV = , (17.6) O Ψ where Ψ ∈ L∞ (Mm−1,n−1 ) and ΨL∞ ≤ t0 . Denote now by E˘ the set of all matrix functions Φ − Q with ∞ ˘ Q ∈ H(k) (m, n) such that W (Φ − Q)V is of the form (17.6). Clearly, E ⊂ E. ∞ (m, n), the error function E# = Φ − Q# It is easy to see that for Q# ∈ H(k) belongs to E˘ if and only if (k)

(k)

E# f = t0 z¯g,

g t E# = t0 z¯f ∗ . (k)

666


˘ Then Fix some E# = Φ − Q# ∈ E. g t Q# = g t Φ − t0 z¯f ∗ .

(k)

(k)

Q# f = Φf − t0 z¯g, For any E = Φ − Q ∈ E we have

W EV = W E# V + W (Q# V − QV ). Suppose that

(17.7)

(k) t0 u0 O W E# V = . O Ψ# Since W EV and W E# V have the same first column and W is unitaryvalued, it follows that QV and Q# V have the same first column, say X. Let QV = X N and Q# V = X N# . (17.8)

It follows from (17.7) that (k) (k) t0 u 0 t0 u0 O = O Ψ O

O Ψ#

+

O W (N# − N )

.

The matrix functions N that can appear in (17.8) are precisely those N ∈ L∞ (Mm−1,n−1 ) satisfying ∞ (i) X N ∈ H(k) (m, n)V ; O O + W (N# − N ) and (ii) W (N# − N ) ∈ L∞ (Mm−1,n−1 ) Ψ# has L∞ norm at most t0 . We wish to find N that minimizes lexicographically the sequence (17.9) ess sup sj W EV (ζ) , 0 ≤ j ≤ min{m, n} − 1. (k)

ζ∈T

We have

(k) s0 W EV (ζ) = t0 ,

and ess sup sj W EV (ζ) = ess sup sj−1 ζ∈T

ζ∈T

for j ≥ 1. It follows from (ii) that W (N# − N ) =

ζ ∈ T,

O Ψ# (ζ) O

(17.10)

+ (W (N# − N ))(ζ)

Ξ∗ (N# − N )

(see (17.5)), and so to minimize (17.9) lexicographically, we have to minimize lexicographically the sequence ess sup sj (Ψ# + Ξ∗ (N# − N ))(ζ) , 0 ≤ j ≤ min{m, n} − 2. ζ∈T

Now we are going to parametrize the matrix functions N satisfying (i).


667

∞ Lemma 17.3. Let X ∈ H(k) (m, n)v. Then there exist K ∈ H ∞ (Mm,n ) and an m × m ∞Blaschke–Potapov product B of degree l ≤ k such that X N ∈ H(k) (m, n)V if and only if ∞ (m, n − 1)). N ∈ B ∗ (KΘ + H(k−l)

(17.11)

Proof. Pick an m × m Blaschke–Potapov product B of minimal degree such that BX ∈ H ∞ (Mm,n )v. Let l be the degree of B. Then l ≤ k. Pick any K ∈ H ∞ (Mm,n ) such that BX = Kv. Suppose that N is of the form (17.11), i.e., N = B ∗ (KΘ + Υ∗ G), where Υ is an m × m Blaschke–Potapov product of degree at most k − l and G ∈ H ∞ (Mm,n−1 ). Then ΥB X N = (17.12) ΥKv ΥKΘ + G t ∞ = (ΥK + GΘ ) v Θ ∈ H (Mm,n )V. ∞ (m, n)V . Hence, X N ∈ H(k) ∞ Conversely, suppose that X N ∈ H(k) (m, n)V . Then there exist an m × m Blaschke–Potapov product O of degree at most k and a matrix function Z ∈ H ∞ (Mm,n ) such that X N = O∗ ZV. (17.13) Consideration of the first column of this equality yields OX = Zv. By the choice of B, O admits a factorization O = ΥB for some Blaschke–Potapov product Υ of degree at most k − l. We have (Z − ΥK)v = OX − ΥKv = ΥBX − ΥKv = O by the definition of K. It follows from Lemma 1.4 that Z − ΥK = GΘt for some G ∈ H ∞ (Mm,n−1 ). Consideration of the second block column in (17.13) yields ZΘ = ON = ΥBN, whence ΥBN = (ΥK + GΘt )Θ = ΥKΘ + G. Thus N = B ∗ (KΘ + Υ∗ G), and so N is of the form (17.11). Note that if N is parametrized as in the lemma by N = B ∗ (KΘ + Υ∗ G), then it follows from (17.12) that B X N = (K + Υ∗ GΘt )V, and by (17.8), Q = B ∗ (K + Υ∗ GΘt ) = B ∗ (K + DΘt ), def

∞ . where D = Υ∗ G ∈ H(k−l) Lemma 17.3 enables us to write N and N# in the form

N = B ∗ (KΘ + D),

N# = B ∗ (KΘ + D# )

(17.14)

668


∞ with D, D# ∈ H(k−l) (m, n − 1). We can thus express conditions (i) and (ii) as follows: ∞ (i ) N = B ∗ (KΘ + D), D ∈ H(k−l) (m, n − 1), (ii ) wt B ∗ (D# − D) = O and Ψ# + Ξ∗ B ∗ (D# − D) has L∞ norm at (k) most t0 . ∞ Here D# is a fixed element of H(k−l) (m, n − 1). We have to minimize lexicographically the sequence ess sup sj (Ψ# + Ξ∗ B ∗ (D# − D))(ζ) , 0 ≤ j ≤ min{m, n} − 2. ζ∈T

Note that if this sequence is minimized lexicographically, the norm condition in (ii ) automatically holds. ∞ Now we are going to parametrize those D ∈ H(k−l) (m, n − 1) that satisfy (ii ). Let us first reformulate the first condition in (ii ). Define the m × 1 column function y by y = (adj B)t w. Since B is an inner matrix function, it follows that the matrix function (adj B)t is also inner, and so the column function y is inner as well. Multiplying the first condition in (ii ) by the scalar Blaschke product det B, we find that it is equivalent to the condition y t (D# − D) = O. Let τ be a greatest common def

inner divisor of the entries of y. Then the column function x = τ¯y is an inner and co-outer column function. Clearly, the first condition in (ii ) is equivalent to xt (D# − D) = O. By Theorem 1.1, x admits a thematic completion x Σ . ∞ Lemma 17.4. Let D# ∈ H(r) (m, n − 1) and let x Σ be an m × m thematic matrix function. There exist Y ∈ H ∞ (Mm,n−1 ) and an (n − 1) × (n − 1) Blaschke–Potapov product Λ of degree q ≤ r such that for ∞ (m, n − 1), D satisfies D ∈ H(r) if and only if

xt (D# − D) = O

(17.15)

∞ D ∈ Y + ΣH(r−q) (m − 1, n − 1) Λ∗ .

(17.16)

Proof. Let Λ be a Blaschke–Potapov product of minimal degree such that xt D# Λ ∈ xt H ∞ (Mm,n−1 ). Let q be the degree of Λ. Clearly, q ≤ r. We have xt D# = xt Y Λ∗

for some

Y ∈ H ∞ (Mm,n−1 ).

Suppose that (17.16) holds, i.e., D = (Y + ΣL)Λ∗

for some

∞ L ∈ H(r−q) (m − 1, n − 1).

Then xt (D# − D) = −xt ΣLΛ∗ = O


669

since x Σ is unitary-valued. Conversely, suppose that (17.15) holds. Pick an (n−1)×(n−1) Blaschke– Potapov product Ω of degree at most r such that DΩ ∈ H ∞ (Mm,n−1 ). Then xt D# Ω ∈ H ∞ (M1,n−1 ), and so Ω = Λ∆ for some Blaschke–Potapov product ∆ of degree at most r − q. Since xt (D − Y Λ∗ ) = O, it follows that DΛ − Y = ΣL for some L ∈ L∞ (Mm−1.n−1 ). Then DΩ = DΛ∆ = Y ∆ + ΣL∆ ∈ H ∞ (Mm,n−1 ),

(17.17)

D = (Y + ΣL)Λ∗ .

(17.18)

and so ∞

It follows from (17.17) that ΣL∆ ∈ H (Mm,n−1 ). Since Σ is co-outer, it follows from Lemma 1.4 that L∆ ∈ H ∞ (Mm−1,n−1 ), and so ∞ L ∈ H(r−q) (m − 1, n − 1). The result now follows from (17.18). Consider now the m × (m − 1) inner and co-outer matrix function Σ. It is uniquely determined by x modulo a left constant unitary factor. Let us construct such a matrix function Σ. Clearly, the matrix function y Σ is unitary-valued if and only if x Σ is. The matrix function (adj B)t W = (adj B)t w Ξ = y (adj B ∗ )Ξ = y (det B)BΞ is clearly unitary-valued, and so therefore is y BΞ . Consider the inner-outer factorization of (BΞ)t : (BΞ)t = Πt Σt , where Π is an (m−1)×(m−1) inner matrix function and Σ is an m×(m−1) inner and co-outer matrix function. Clearly, y Σ is unitary-valued, and so x Σ is thematic. We have BΞ = ΣΠ. Thus Π = Σ∗ BΞ.

(17.19)

By Lemma 17.4, there exist Y ∈ H ∞ (Mm,n−1 ) and a Blaschke–Potapov product Λ of degree q ≤ k − l such that any matrix function D satisfying (ii ) can be written as D = (Y + ΣL)Λ∗ ∞ for L ∈ H(k−l−q) (m − 1, n − 1). In particular, D# = (Y + ΣL# )Λ∗ ∞ for some L# ∈ H(k−l−q) (m−1, n−1). We have to minimize lexicographically the sequence ess sup sj (Ψ# + Ξ∗ B ∗ (D# − D))(ζ) , 0 ≤ j ≤ min{m, n} − 2. ζ∈T

Clearly, by (17.19), Ψ# + Ξ∗ B ∗ (D# − D) = Ψ# + Ξ∗ B ∗ Σ(L# − L)Λ∗ = Ψ# + Π∗ (L# − L)Λ∗ ,

670


and so the problem reduces to the problem to minimize lexicographically the sequence ess sup sj (ΠΨ# Λ + L# − L)(ζ) , 0 ≤ j ≤ min{m, n} − 2, ζ∈T ∞ (m − 1, n − 1). It is easy to see that Q is a suas L varies over H(k−l−q) ∞ peroptimal approximation of Φ by meromorphic matrix functions in H(k) if and only if L is a superoptimal approximation of def

Φ1 = ΠΨ# Λ + L# ∞ by meromorphic matrix functions in H(k−l−q) . Let us show that this new approximation problem satisfies the hypotheses of Theorem 17.1 so that we can invoke the inductive hypotheses. Clearly, (k) (k−l−q) (Φ1 ), 1 ≤ j ≤ min{m, n} − 1. To verify the assumption tj (Φ) = tj−1 on the essential norm of the Hankel operator, it is sufficient to show that

HΦ1 e ≤ HΦ e .

(17.20)

Since rank HL# < ∞, it follows that HΦ1 e = HΠΨ# Λ e . It is easy to see that HΠΨ

#Λ

= HΠΨ ΛH 2 (Mn−1,n−1 ) # e e

and since the subspace ΛH 2 (Mn−1,n−1 ) has finite codimension in H 2 (Mn−1,n−1 ), it follows that HΠΨ Λ = HΠΨ . # # e e It is also clear that HΠΨ = H(ΠΨ )t = HΨt Πt , # e # e e # and we can apply the above reasoning again to prove that HΨt Πt = HΨt e = HΨ e . #

e

#

#

Next, rank HQ# < ∞, and so

HΦ e = HΦ−Q# e .

Finally, by Theorem 4.1,

HΨ ≤ HΦ−Q , # e # e

which completes the proof of (17.20). To apply the inductive hypotheses, it is sufficient to show that either k − l − q = 0 (in which case uniqueness is a consequence of Theorem 4.8) or sk−l−q−1 (HΦ1 ) > sk−l−q (HΦ1 ). Suppose that both are false, i.e., l + q < k and sk−l−q−1 (HΦ1 ) = sk−l−q (HΦ1 ).

(17.21)


671

∞ Then there exists L ∈ H(k−l−q−1) , which minimizes Φ1 − LL∞ over ∞ Q1 ∈ H(k−l−q) . Thus L determines via (17.14) and (17.19) the matrix function

Q = B ∗ (K + (Y + ΣL)Λ∗ Θt ), ∞ . Since which is a best approximation of Φ by matrix functions in H(k) ∞ L ∈ H(k−l−q−1) , and B and Λ are of degree l and q, respectively, it follows ∞ that Q ∈ H(k−1) . Thus ∞ ∞ (m, n) = distL∞ Φ, H(k) (m, n) , distL∞ Φ, H(k−1)

and so sk−1 (HΦ ) = sk (HΦ ), which contradicts the hypotheses of the theorem. We have proved that either k − l − q = 0 or (17.21) holds. This allows us to apply the inductive hypothesis to Φ1 . Formula (17.1) also follows easily by induction (see (17.10)). This completes the proof. The following result is an important consequence of Theorem 17.1. Theorem 17.5. Let k be a positive integer and let Φ be a matrix function in (H ∞ + C)(Mm,n ). If sk (HΦ ) < sk−1 (HΦ ), then there exists ∞ (m, n). a unique superoptimal approximation F of Φ by functions in H(k) Moreover, for this F (k) sj (Φ − F )(ζ) = tj , 0 ≤ j ≤ min{m, n} − 1, for almost all ζ ∈ T. Proof. Clearly, under the hypotheses of the theorem HΦ e = 0 and the result immediately follows from Theorem 17.1. We conclude this section with a parametrization formula for the set ∞ (m, n) : Φ − QL∞ = sk (HΦ )} Ω0 = {Q ∈ H(k) (k)

of optimal solutions of the Nehari–Takagi problem. Theorem 17.6. Let k and Φ satisfy the hypotheses of Theorem 17.1 and suppose that m, n ≥ 2. There exist matrix functions K ∈ H ∞ (Mm,n ), Y ∈ H ∞ (Mm,n−1 ), an m × m Blaschke–Potapov product B of degree l, an (n−1)×(n−1) Blaschke–Potapov product Λ of degree q such that l +q ≤ k, inner and co-outer functions Θ ∈ H ∞ (Mn,n−1 ), Σ ∈ H ∞ (Mm,m−1 ), and a matrix function Φ1 ∈ L∞ (Mm−1,n−1 ) such that the set of best approxima∞ tions of Φ by matrix functions in H(k) (m, n) is equal to

B ∗ (K + (Y + ΣL)Λ∗ Θt ) :

∞ L ∈ H(k−l−q) (m − 1, n − 1), Φ1 − LL∞ ≤ sk (HΦ )

.

The proof is contained in the proof of Theorem 17.1. The matrix functions K, Y , B, L, Θ, Σ, Φ1 are defined in the proof of Theorem 17.1.

672


18. Analytic Approximation of Infinite Matrix Functions In this section we study the problem of analytic approximation of infinite matrix functions. We assume that the matrices have size ∞×∞. Obviously, the cases of m × ∞ or ∞ × m matrix functions reduce to the case of ∞ × ∞ matrix functions. As usual we identify infinite matrices with operators on the sequence space 2 and we consider matrix functions that take values in the space B of bounded linear operators on 2 . We denote by C the space of compact operators on 2 . We consider here the class L∞ (B) of weakly measurable matrix functions Φ that take values in B and we equip L∞ (B) with the norm def

ΦL∞ (B) = ess sup Φ(ζ)B . ζ∈T

As in the case of finite matrix functions we consider the problem of approximation of Φ by functions in the space H ∞ (B) of operator-valued analytic functions. Note that instead of 2 we can consider an arbitrary separable infinitedimensional Hilbert space H and study the problem of approximation of functions in L∞ (B(H)) by functions in H ∞ (B(H)). However, we want to fix an orthonormal basis in H, which makes it convenient to consider the space 2 with the standard orthonormal basis. To define the notion of a superoptimal approximation for infinite matrix functions we have to consider the infinite sequence of sets Ωj , j ∈ Z+ , defined by Ω0 = {Q ∈ H ∞ (B) : F minimizes ess sup Φ(ζ) − F (ζ)B }, ζ∈T

Ωj =

Q ∈ Ωj−1

( : F minimizes ess sup sj Φ(ζ) − F (ζ) .

ζ∈T

The sequence {tj }j≥0 = {tj (Φ)}j≥0 of superoptimal singular values is defined by def tj = ess sup sj Φ(ζ) − Q(ζ) for Q ∈ Ωj , j ∈ Z+ . ζ∈T

We say that a matrix function F ∈ H ∞ (B) is a superoptimal approximation of Φ by bounded analytic operator functions if ' F ∈ Ωj . j≥0

We prove in this section the uniqueness of a superoptimal approximation 2 2 under the condition that the Hankel operator HΦ : H 2 (2 ) → H− ( ) is ∞ compact, which is equivalent to the condition that Φ ∈ H (B) + C(C); see Theorem 2.4.1. We apply a method that can also be used to obtain an alternative proof of Theorems 3.3 and 3.4. We also obtain results on

18. Analytic Approximation of Infinite Matrix Functions

673

thematic indices similar to the results of §7 for finite matrix functions and inequalities involving superoptimal singular values similar to those obtained in §8. The following theorem is the main result of the section. Theorem 18.1. Let Φ be a function in H ∞ (B) + C(C). Then there exists a unique superoptimal approximation F ∈ H ∞ (B). Moreover, with such F sj (Φ − F )(ζ) = tj (Φ), for almost all ζ ∈ T, j ∈ Z+ . Proof. Suppose that HΦ = 0. As in the case of finite matrix functions we start with a maximizing vector f of HΦ and factorize it: f = ϑ1 hv,

(18.1)

where ϑ1 is a scalar inner function, h is a scalar outer function in H 2 , and v is an inner and co-outer column function. Consider the maximizing vector g of HΦt defined by g = HΦ −1 z¯HΦ f .

(18.2)

As in the case of finite matrix functions (see §2) it can be shown that g admits a factorization g = ϑ2 hw, where ϑ2 is a scalar inner function, h is the same outer function, and w is an inner and co-outer column function. We say that a matrix function V ∈ L∞ (B) is a thematic matrix function if V (ζ) is a unitary operator for almost all ζ ∈ T and it has the form v Θ , where v is an inner and co-outer column function and Θ is an inner and co-outer function in H ∞ (B). First, we prove that as in the case of finite matrices (see §1) an inner and co-outer matrix function has a thematic completion. Consider the subspace L = Ker Tvt of H 2 (2 ). It is invariant under multiplication by, z and so by the Beurling–Lax–Halmos theorem (see Appendix 2.3), it has the form L = ΘH 2 (K), where K = 2 or K = Cm for some m ∈ Z+ . Lemma 18.2. Let v = {v j }j≥0 be an inner column and let dimKer Θ∗ (ζ) = 1 for almost all ζ ∈ T, L = Ker Tvt = ΘH 2 (K). Then and so the matrix function v Θ is unitary-valued. Proof. Let us first show that dim Ker Θ∗ (ζ) ≥ 1. Consider the matrix function ¯ Θ . U= v It is easy to see that U (ζ) is isometric a.e. on T. It follows that the columns ¯ (ζ) a.e. on T. Therefore ϕ(ζ) of Θ(ζ) are orthogonal to v ¯ ∈ Ker Θ∗ (ζ) a.e., ∗ which proves that dim Ker Θ (ζ) ≥ 1.

674


To show that dim Ker Θ∗ (ζ) ≤ 1 we assume without loss of generality that v 0 = O. Consider the matrix function   −v 1 −v 2 −v 3 · · ·  v0 O O ···     O O ···  v0 G= .  O O v0 · · ·    .. .. .. .. . . . . It is easy to see that GH 2 (2 ) ⊂ L = ΘH 2 (K). The proof will be completed if we show that dim Ker Gt (ζ) ≤ 1 a.e. on T. Assume that c = {cj }j≥0 ∈ Ker Gt (ζ). We have      O −v 1 (ζ) v 0 (ζ) c0 O O ···     −v 2 (ζ)  O v 0 (ζ) O ···     c1   O  = .    −v 3 (ζ)   O O v 0 (ζ) · · ·   c2   O    (18.3) .. .. .. .. .. .. .. . . . . . . . If v 0 (ζ) = 0, we have from (18.3) cj =

v j (ζ) c0 , v 0 (ζ)

j ≥ 1,

which proves that the cj with j ≥ 1 are uniquely determined by c0 , and so dim Ker Gt (ζ) ≤ 1. Lemma 18.3. Under the hypothesis of Lemma 18.2 the function Θ is co-outer. Proof. Assume the contrary. Then Θt = OG, where G is an outer function and O is an inner function . It follows that Θ = G t Ot . Multiplying this equality by (Ot )∗ on the right, we find that G t = Θ(Ot )∗ . Since (Ot )∗ (ζ) is isometric a.e. on T, it follows that G t is inner. Consider the space G t H 2 (2 ). Let us show that it is contained in Ker Tvt . Note first that ¯ (ζ) Θ(ζ)c ⊥ v

a.e. on T for any c ∈ 2 .

(18.4)

Indeed this follows from the fact that Θg ⊂ Ker Tϕt , where g(ζ) ≡ c. Now let f ∈ H 2 (2 ). We have ¯ (ζ))2 = Θ(ζ)(Ot )∗ (ζ)f (ζ), v ¯ (ζ) 2 = 0 (G t (ζ)f (ζ), v by (18.4). It follows that G t H 2 (2 ) = Θ(Ot )∗ H 2 (2 ) ⊂ ΘH 2 (2 ). Multiplying the last inclusion on the left by Θ∗ , we obtain (Ot )∗ H 2 (2 ) ⊂ H 2 (2 ). Clearly, this implies that Ot is a constant unitary matrix. Lemmas 18.2 and 18.3 immediately imply the following fact.


675

Corollary 18.4. Let v be an inner and co-outer function in H ∞ (2 ). Then there exists an inner and co-outer matrix function Θ ∈ H ∞ (B) such that the matrix function v Θ is thematic. Theorem 18.5. Let V be a thematic matrix function in L∞ (B). Then the Toeplitz operator TV on H 2 (2 ) has dense range and trivial kernel. The proof of this theorem is exactly the same as the proof of Theorem 1.3. Theorem 18.6. Let V be a thematic matrix function in L∞ (B). Then the operators HV∗ HV and HV∗ ∗ HV ∗ on H 2 (2 ) are unitarily equivalent. Proof. This is an immediate consequence of Theorems 18.5 and 4.4.11. Corollary 18.7. Let V = v Θ be a thematic matrix function. Suppose that the Hankel operator Hv¯ is compact. Then the Toeplitz operator TV is invertible. Proof. Clearly, the condition that Hv¯ is compact is equivalent to the condition that HV ∗ is compact. It follows from Theorem 18.6 that HV is compact. We have TV∗ TV = I − HV∗ HV

and TV∗ ∗ TV ∗ = I − HV∗ ∗ HV ∗ ,

and so TV is Fredholm. The result now follows now Theorem 18.5. Theorem 18.8. Let V and W t be thematic matrix functions in L∞ (B). Then ' O O O O ∞ = . W H (B)V O L∞ (B) O H ∞ (B) The proof of Theorem 18.8 is exactly the same as the proof of Theorem 1.8 for finite matrix functions. To prove Theorem 18.1, we take inner and co-outer column functions v and w defined in (18.1) and (18.2) and construct thematic completions V = v Θ and W t = w Ξ , (18.5) which exist by Corollary 18.4. Let Q ∈ H ∞ (B) be an arbitrary best approximation of Φ by bounded analytic functions. Then Φ − Q admits a factorization t0 u0 O ∗ V ∗, (18.6) Φ−Q=W O Ψ def ¯ and Ψ is a function in L∞ (B) such that ΨL∞ ≤ t0 . where u0 = z¯ϑ¯1 ϑ¯2 h/h This can be proved in exactly the same way as in §2 for finite matrix functions. Moreover, as in the case of finite matrix functions the problem of finding a superoptimal approximation of Φ reduces via Theorem 18.8 to the problem of finding a superoptimal approximation of Ψ (see §2).

676


To be able to continue this process, we have to be sure that HΨ has a maximizing vector. This would be true if we knew that HΨ is compact. Fortunately, we can prove this. Theorem 18.9. Let Φ ∈ L∞ (B) satisfy the hypotheses of Theorem 18.1. Let Ψ be as in (18.6). Then the Hankel operator HΨ is compact. We need the following analog of scalar Theorem 1.5.1. Lemma 18.10. The set H ∞ (B) + C(C) is a closed subalgebra of L (B). ∞

Proof. The fact that H ∞ (B) + C(C) is closed in L∞ (B) is an immediate consequence of Theorem 2.4.1. To prove that H ∞ (B) + C(C) is an algebra, it is sufficient to show that for C-valued trigonometric polynomials G1 and G2 and for D1 , D2 ∈ H ∞ (B) the function (G1 + D1 )(G2 + D2 ) belongs to H ∞ (B) + C(C). We have (G1 + D1 )(G2 + D2 ) = G1 G2 + G1 D2 + D1 G2 + D1 D2 . Clearly, G1 G2 ∈ C(C), D1 D2 ∈ H ∞ (B). It is also easy to see that P− G1 D2 is a C-valued trigonometric polynomial and P+ G1 D2 ∈ H ∞ (B). The same can be said about G2 D1 . Proof of Theorem 18.9. The fact that u0 ∈ QC can be proved in exactly the same way as in the case of finite matrices (see Theorem 3.1). Let us show that the Hankel operator Hv∗ is compact. It follows easily from (18.6) that u ¯0 wt (Φ − Q) = t0 v ∗ . The compactness of Hv∗ now follows from Lemma 18.10. This is equivalent to the fact that HV ∗ is compact. By Theorem 18.6, the Hankel HV is compact. If we apply the above reasoning to Φt , we find that HW is compact. Thus both V and W belong to H ∞ (B) + C(C). By (18.6), we have t0 u0 O W (Φ − Q)V = . O Ψ By Lemma 18.10, the left-hand side of this equality belongs to H ∞ (B) + C(C). Hence, Ψ ∈ H ∞ (B) + C(C). Theorem 18.9 allows us to apply the above procedure to Ψ and iterate this process. Let r be a positive integer. If tr−1 (Φ) = 0, the process stops and we get a unique superoptimal approximation. Otherwise, we can take an arbitrary Q ∈ Ωr−1 . Then the function Φ − Q admits a factorization O O I r−1 I r−1 ∗ D · · · V0∗ , Φ − Q = W0 · · · ∗ ∗ O Wr−1 O Vr−1 (18.7)


where

    D=  

t0 u0 O .. .

O t1 u1 .. .

··· ··· .. .

O O .. .

O O .. .

O O

O O

··· ···

tr−1 ur−1 O

O Ψ(r)

677

    .  

(18.8)

t Here V0 , · · · , Vr−1 , W0t , · · · , Wr−1 are thematic matrix functions, u0 , · · · , ur−1 are unimodular functions in QC with negative winding numbers, Ψr ∈ L∞ (B) and Ψr L∞ ≤ tr−1 . Moreover, the functions u0 , · · · , ur−1 , V0 , · · · , Vr−1 , W0 , · · · , Wr−1 can be chosen the same for any Q ∈ Ωr−1 . Let us first observe that a superoptimal approximation of Φ exists. Indeed, let Qr ∈ Ωr . Then {Qr }r≥0 is a bounded sequence in H ∞ (B) and we can find a subsequence {Qrk }k≥0 such that {Qrk (ζ)}k≥0 converges for any ζ ∈ D in the weak operator topology to F (ζ) for some F ∈ H ∞ (B). It is easy to see that F is a superoptimal approximation of Φ. To prove that the superoptimal approximation is unique, it is sufficient to show that

lim tj (Φ) = 0.

(18.9)

j→∞

Indeed, suppose that F1 and F2 are superoptimal approximations of Φ. Then for each positive integer r the matrix functions Φ − F1 and Φ − F2 admit factorizations of the form (18.7), i.e., for i = 1, 2 we have O O I r−1 I r−1 D · · · V0∗ , Φ − Fi = Φ − Q = W0∗ · · · i ∗ ∗ O Wm−1 O Vm−1 where



t0 u0 O .. .

O t1 u1 .. .

··· ··· .. .

O O .. .

O O .. .

O O

O O

··· ···

tr−1 ur−1 O

O (r) Ψi

I r−1 O

O

   Di =    It follows that F1 − F2 =

W0∗

···

∗ Wm−1

(D2 − D1 )

    .  

O

I r−1 O

∗ Vm−1

· · · V0∗ ,

and so F1 − F2 L∞

= D1 − D2 L∞  O ··· O O .. ..  .. . .  . . . =  .  O · · · O O O · · · O Ψ(r) − Ψ(r) 1 2

    

L∞

≤ 2tr−1 .

678


Since r is arbitrary, it follows from (18.9) that F1 = F2 . Thus the uniqueness of a superoptimal approximation will be established as soon as we prove (18.9). To prove (18.9), we obtain an inequality between the superoptimal singular values tj (Φ) and the singular values sj (HΦ ) as has been done in §8. First, we need the following fact. Theorem 18.11. The matrix function Θ and Ξ in (18.5) are left invertible in H ∞ , i.e., there exist matrix functions A, B ∈ H ∞ (B) such that A(ζ)Θ(ζ) = I

and

B(ζ)Ξ(ζ) = I,

ζ ∈ D.

The result can be derived from Corollary 18.7 in exactly the same way as it has been done in the proof of Theorem 4.5 in the case of finite matrix functions. We can introduce the thematic indices associated with the factorization (18.7): def

kj = − wind uj = ind Tuj ,

0 ≤ j ≤ r − 1,

where the uj are diagonal entries in (18.8). As in the case of finite matrix functions, kj > 0. Now suppose that tr < tr−1 in the factorization (18.7). Such factorizations are called partial thematic factorizations. It is easy to see that as in the case of finite matrix functions thematic indices of a partial thematic factorization may depend on the choice of a factorization (see §7). However, the following result holds. Theorem 18.12. Let Φ satisfy the hypotheses of Theorem 18.1, let t ≥ tr−1 > tr , and let the kj , 0 ≤ j ≤ r − 1, be the thematic indices of a partial thematic factorization (18.7). Then the numbers kj j:tj =t

do not depend on the choice of a partial thematic factorization. The following theorem is an analog of Theorem 8.1. Theorem 18.13. Let Φ satisfy the hypotheses of Theorem 18.1 and let Ψ be the matrix function in the factorization (18.6). Then sj (HΨ ) ≤ sj+k0 (HΦ ),

j ∈ Z+ .

Theorems 18.12 and 18.13 can be deduced from Theorem 18.11 in the same way as it has been done in the proof of Theorems 7.1 and 8.1 in the case of finite matrix functions. We can consider now the extended t-sequence associated with a partial thematic factorization (18.7): t , · · · , t , t , · · · , t , · · · , tr−1 , · · · , tr−1 -0 .+ ,0 -1 .+ ,1 .+ , k0

k1

kr−1


679

in which tj repeats kj times. As in §8 we denote the terms of the extended sequence by t˘0 , t˘1 , · · · , t˘k0 +···+kr−1 −1 . It follows easily from Theorem 18.12 that the terms of the extended t-sequence are uniquely determined by Φ and do not depend on the choice of a partial thematic factorization. Theorem 18.14. Let Φ satisfy the hypotheses of Theorem 18.1. Then the terms of the extended t-sequence associated with a partial thematic factorization (18.6) satisfy t˘j ≤ sj (HΦ ),

0 ≤ j ≤ k0 + k1 + · · · + kr−1 − 1.

Theorem 18.14 can be deduced easily from Theorem 18.13 in exactly the same way as Theorem 8.2 has been deduced from Theorem 8.1. Corollary 18.15. Under the hypotheses of Theorem 18.1 tj (Φ) ≤ sj (HΦ ),

j ∈ Z+ .

Corollary 18.15 follows immediately from Theorem 18.14. Now we are able to complete the proof of Theorem 18.1. Since HΦ is compact, sj (HΦ ) → 0 as j → ∞, and so by Corollary 18.15, (18.9) holds, which completes the proof of the uniqueness of the superoptimal approximation F. The fact that sj (Φ − F )(ζ) = tj almost everywhere on T follows immediately from factorization formula (18.7). Remark. As in §10 we can introduce the notion of a monotone partial thematic factorization and prove that it is always possible to find a monotone partial thematic factorization of the form (18.6) and the thematic indices of a monotone partial thematic factorization are uniquely determined by the matrix function Φ. We proceed now to the construction of a thematic factorization. Let Φ satisfy the hypotheses of Theorem 18.1. We can consider the infinite process of construction for each positive integer r a factorization (18.6). Doing so, we obtain a sequence of badly approximable unimodular functions {uj }j≥0 , and sequences of thematic matrix functions {Vj }j≥0 and {Wjt }j≥0 such that for each positive integer r a factorization of the form (18.6) holds. Consider the infinite products O 1 O I r−1 ∗ ∗ ··· ··· W = W0 ∗ O Wr−1 O W1∗ (18.10) and

V = V0

1 O O V1

···

I r−1 O

O Vr−1

··· .

(18.11)

If we identify functions in L∞ (B) with multiplication operators on L2 (2 ), it is easy to see that both infinite products converge in the strong operator

680


topology and define unitary-valued functions V and W . Indeed, it is sufficient to verify the convergence on the vector functions {fj }j≥0 with only finitely many nonzero entries, and for such vector functions the convergence is obvious. Theorem 18.16. Let Φ satisfy the hypotheses of Theorem 18.1 and let F be the unique superoptimal approximation of Φ by bounded analytic matrix functions. Then Φ − F admits a factorization Φ − F = W ∗ DV ∗ , where V and W are defined in (18.10) and (18.11), and D is the diagonal matrix function defined by   t0 u0 O O ···  O O ···  t1 u1   D= O . (18.12) O t u ···  2 2   .. .. .. .. . . . . Proof. For each positive integer r the function Φ − F admits a factorization of the form (18.6). We can now pass to the limit. The result follows from the strong convergence of the infinite products (18.10) and (18.11) and the obvious fact that the sequence of functions   O ··· O O t0 u0  O t1 u1 · · · O O     .. .. . ..  .. ..  .  . . .    O O  O · · · tr−1 ur−1 O O ··· O Ψ(r) converges to D in L∞ (B). We conclude this chapter with a heredity result for the nonlinear operator of superoptimal approximation by infinite bounded analytic matrix functions. Recall that in §12 among other results it has been shown that if a finite matrix function Φ belongs to V M O and F is its superoptimal approximation, then F also belongs to V M O. This is equivalent to the fact that if HΦ is compact, then H(Φ−F )∗ is also compact, or in other words, (Φ−F )∗ ∈ H ∞ +C. We prove that the same result holds for infinite matrix functions. Theorem 18.17. Let Φ satisfy the hypotheses of Theorem 18.1 and let F be the unique superoptimal approximation of Φ by bounded analytic matrix functions. Then (Φ − F )∗ ∈ H ∞ (B) + C(C). Proof. Consider the sequence of functions O I r−1 I r−1 Rj = W0∗ · · · D j ∗ O Wj−1 O

O

∗ Vj−1

· · · V0∗ ,

j ≥ 1,

19. Back to the Adamyan–Arov–Krein Parametrization

where

    Dj =   

t0 u0 O .. .

O t 1 u1 .. .

··· ··· .. .

O O .. .

O O .. .

O O

O O

··· ···

tj−1 uj−1 O

681



   .  O  O

Obviously, Rj = W ∗ Dj V ∗ and so Φ − F − Rj L∞ ≤ tj → 0

as

j → ∞.

(18.13)

Since the functions Vj and Wj belong to H ∞ (B) + C(C) (see the proof of Theorem 18.9), it follows that HRj∗ is compact for each j. By (18.13), lim H(Φ−F )∗ − HRj∗ = 0,

j→∞

which implies the compactness of H(Φ−F )∗ .

19. Back to the Adamyan–Arov–Krein Parametrization In this section we use the results of §15 to obtain a version of the Adamyan–Arov–Krein parametrization of solutions of the Nehari problem 2 (Cm ) in the case Γ = ρ and for a Hankel operator Γ : H 2 (Cn ) → H− Γe < Γ. We reduce this case to the case Γ < ρ treated in §5.4 (see Theorem 5.4.16). Though we have parametrized in §5.5 the solutions of the Nehari problem in the most general case, we present here an alternative approach that works in this case and that is a straightforward application of partial canonical factorizations of badly approximable matrix functions. Let Ψ ∈ L∞ (Mm,n ) such that Γ = HΨ . We are looking for solutions of the Nehari problem with ρ = HΨ . Clearly, this is equivalent to the problem of parametrization of all best approximations of Ψ by analytic matrix functions in the L∞ norm. We assume that HΨ e < HΨ . Let tj , 0 ≤ j ≤ min{m, n} − 1, be the superoptimal singular values of Ψ. If tmin{m,n}−1 = t0 , it follows from the results of §15 that Ψ has a unique best approximation by analytic matrix functions. Suppose now that tmin{m,n}−1 < t0 = ρ. Then by Theorem 15.3, for any best approximation F ∈ H ∞ (Mm,n ) the matrix function Ψ − F admits a partial canonical factorization ρU O Ψ − F = W∗ V ∗, O Ψ(1)

682


where Ψ(1) ∈ L∞ (Mm−r,n−r ) and HΨ(1) < ρ. Here r is the number of superoptimal singular values of Ψ equal to t0 and and W t = Ω Ξ V= Υ Θ are r-balanced unitary-valued functions. By Theorem 1.8, Φ is a symbol of Γ with Φ ≤ ρ if and only if there exists a symbol Φ(1) of HΨ(1) such that Φ(1) ≤ ρ and ρU O ∗ (19.1) Φ=W V ∗. O Φ(1) By Theorem 5.4.16, the set of such matrix functions Φ(1) admits the following parametrization: −1 z ) + P ∗ρ (¯ z )E(z) (P ρ (z) + Q∗ρ (z)E(z) , (19.2) Φ(1) = ρ Qρ (¯ where E is in the unit ball of H ∞ (Mm−r,n−r ), and the functions P ρ , Qρ , P ∗ρ , and Q∗ρ have been defined in §5.4. Thus we obtain the following version of the Adamyan–Arov–Krein parametrization. Theorem 19.1. Suppose that Ψ ∈ L∞ (Mm,n ), HΨ e < HΨ = ρ, and r is the number of the superoptimal singular values of Ψ equal to ρ. Then under the above notation, Φ is a symbol of HΨ with Φ = ρ if and only if Φ admits a representation −1 t z ) + P ∗ρ (¯ z )E(z) (P ρ (z) + Q∗ρ (z)E(z) Θ Φ = ρ ΩU Υ∗ + Ξ Qρ (¯ with E is in the unit ball of H ∞ (Mm−r,n−r ). Proof. The result follows straightforwardly from (19.1) and (19.2).

Concluding Remarks The notion of a superoptimal approximation by analytic matrix functions was introduced in Young [2], where a uniqueness result for a class of H ∞ +C functions satisfying a certain additional assumption was stated. However, the proof given in Young [2] contains an error. Note also that earlier in Davis [2] a similar notion had been used for the problem of completing matrix contractions (see §2.1). The uniqueness of a superoptimal analytic approximation for H ∞ + C functions (Theorem 3.3) was established in Peller and Young [1]. Later Treil [8] obtained another proof of the same result. The results of §1 are taken from Alexeev and Peller [3]. The existence and uniqueness (modulo a constant unitary factor) of a balanced completion was established in Vasyunin [1]. Note that in the case of thematic matrix functions (i.e., in the case r = 1) the results of §1 had been obtained earlier in Peller and Young [1]. The results of §2 and §3 can be found in Peller and Young [1].

Concluding Remarks

683

The results of §4 were obtained in Peller and Treil [2]; they deal with noncompact Hankel operators, which requires a considerably more sophisticated technique developed in Peller and Treil [2]. Note that the method of constructing the vectors ξj# in the proof of Theorem 4.1 is based on the technique developed in Peller and Young [1] and [2]. In fact, in Peller and Treil [2] a more general problem was solved. Namely, similar results were obtained in Peller and Treil [2] for superoptimal solutions of the four block problem (see Concluding Remarks to Chapter 2). The results of §5 were found in Peller and Young [1] in the case of functions in H ∞ + C and in Peller and Treil [2] in the case when HΦ e is not too large. The notions of thematic factorizations and thematic indices were introduced in Peller and Young [1]. The notions of admissible and superoptimal weights were given in Treil [8]. The results of §6 were also established in Treil [8]. The fact that thematic indices may depend on the choice of a thematic factorization was observed in Peller and Young [1]. Theorem 7.1 was obtained in Peller and Young [2] in the case of H ∞ + C functions. The notion of an extended t-sequence was introduced in Peller and Young [2]. Theorem 8.2 was proved in Peller and Young [2] in the case of H ∞ + C functions. The stronger result, Theorem 8.1, was found in Peller and Treil [2]. The results of §9 and §10 were obtained in Alexeev and Peller [1]. The results of §11 were found in Peller and Young [3] (see also Peller and Young [5]). Note that in Peller and Young [3] constructive algorithms are discussed to find the superoptimal approximants. The results of §12 were established in Peller and Young [1] for R-spaces and for decent spaces under the assumption (12.5). Later in Peller [24] the same results were proved for arbitrary decent spaces. We also mention here the paper Peller [25] in which hereditary properties are discussed for the four block problem. The continuity results were obtained in Peller and Young [5] and Peller [24]. Section 14 follows Alexeev and Peller [2]. In Dym and Gohberg [1] the authors considered the problem of finding unitary interpolants and describing their Wiener–Hopf indices for functions that belong to a function space that satisfies axioms similar to our axioms (A1)–(A4) of decent spaces. They stated the results that the negative Wiener–Hopf indices are uniquely determined by the function while the nonnegative indices can be arbitrary. Note, however, that the reasoning in Dym and Gohberg [1] contains a gap. Later in Dym and Gohberg [2] the problem of unitary interpolation was studied in a more general situation. Another approach to this problem was given by Ball [1]. In Alexeev and Peller [2] a new method was found that allows one to write explicit formulas for the negative Wiener–Hopf indices of unitary interpolants in terms of the thematic indices of monotone thematic factorizations and express the number of negative Wiener–Hopf indices as the number of superoptimal singular values equal to 1.

684


The results of §15 and §16 are taken from Alexeev and Peller [3]. Theorem 17.5 was proved first in Treil [8] by a different method. The proof given in §17 is more constructive; it was found in Peller and Young [4]. Theorem 17.1 is published here for the first time. Theorem 18.1 was proved in Treil [8] in a different way. The more constructive proof given in §18 was found in Peller [22]. The proof of Lemma 18.2 was suggested by Vasyunin. Theorems 18.9, 18.16, and 18.17 were obtained in Peller [22]. Theorems 18.12, 18.13, and 18.14 were found in Peller and Treil [1] (to be more precise, the statement of Theorem 18.13 is added in proof in Peller and Treil [1]; its proof is the same as the proof given in Peller and Treil [2] for finite matrix functions). The material of §19 is published here for the first time. We also mention here the paper Woerdeman [1] in which the author studied superoptimal approximation of finite block matrices by triangular block matrices and found an analog of thematic factorizations. The results of Woerdeman generalize the results of Davis [2] for 2 × 2 block matrices.

15 Hankel Operators and Similarity to a Contraction

In this chapter Hankel operators are used to solve the problem of whether each polynomially bounded operator on Hilbert space is similar to a contraction. This problem has a long history. Sz.-Nagy [1] showed that if T is an invertible operator on Hilbert space such that sup T n < ∞, then T is similar to a unitary operator, i.e., there n∈Z

exists an invertible operator V such that V T V −1 is unitary. Sz.-Nagy posed a similar problem for operators T (not necessarily invertible) satisfying the condition sup T n < ∞. n∈Z+

Such operators are called power bounded operators. The problem was whether each power bounded operator is similar to a contraction. Recall that an operator R on Hilbert space is called a contraction if R ≤ 1. It turned out, however, that there are power bounded operators that are not similar to contractions. The first example of such an operator was constructed by Foguel [1]. Other examples of power bounded operators not similar to contractions were found in Davie [1], Peller [5], and Bo˙zeiko [1]. After the appearance of Foguel’s counter-example it has become natural to try to impose a stronger condition on an operator under which the operator would have to be similar to a contraction. Von Neumann [1] proved that if T is a contraction on Hilbert space, then for any analytic polynomial ϕ ϕ(T ) ≤ ϕ∞ = max |ϕ(ζ)|, |ζ|≤1

686

Chapter 15. Hankel Operators and Similarity to a Contraction def

where as usual ϕ(T ) =

j ϕ(j)T ˆ . There are many different proofs of

j≥0

von Neumann’s inequality, see e.g., Sz.-Nagy and Foias [1], where the proof based on Sz.-Nagy’s theorem on unitary dilations is given. It follows from von Neumann’s inequality that if T is similar to a contraction, then T is polynomially bounded, i.e., ϕ(T ) ≤ const ϕ∞ for any analytic polynomial ϕ. Lebow [1] showed that the operator constructed by Foguel is not polynomially bounded. Halmos [3] posed the question of whether each polynomially bounded operator is similar to a contraction. Later Arveson [1–2] observed that the Sz.-Nagy dilation theorem implies that an operator similar to a contraction is not only polynomially bounded but also completely polynomially bounded, i.e.,   ϕ11 (T ) ϕ12 (T ) · · · ϕ1n (T )  ϕ21 (T ) ϕ22 (T ) · · · ϕ2n (T )      .. .. .. ..   . . . . ϕn1 (T ) ϕn2 (T ) · · · ϕnn (T )   ϕ11 (ζ) ϕ12 (ζ) · · · ϕ1n (ζ)  ϕ21 (ζ) ϕ22 (ζ) · · · ϕ2n (ζ)    ≤ c · max   .. .. .. ..  |ζ|≤1  . . . . ϕn1 (ζ) ϕn2 (ζ) · · · ϕnn (ζ) for any positive integer n and any polynomial matrix {ϕjk }1≤j,k≤n with a constant c not depending on n. Paulsen [1] showed that the converse is also true, i.e., complete polynomial boundedness implies similarity to a contraction. However, the problem of whether polynomial boundedness implies similarity to a contraction remained open. In Peller [5] the following operator was introduced. Let ψ be a function analytic in the unit disk D. Consider the operator Rψ on 2 ⊕ 2 defined by ∗ S Γψ , (0.1) Rψ = O S where S is the shift operator on 2 and Γψ is the Hankel operator on 2 with ˆ + k)}j,k≥0 in the standard basis of 2 . By the Nehari theorem, matrix {ψ(j Rψ is bounded if and only if ψ ∈ BM OA. Such operators were used in Peller [5] to construct power bounded operators that are not polynomially bounded (see §2). The operators Rψ were considered independently by Foias and Williams (see Carlson, Clark, Foias, and Williams [1]). The reason the operators Rψ do a nice job here is that one can very easily compute functions of such operators (see §1).

1. Operators Rψ in the Scalar Case

687

The hope was to find a function ψ such that Rψ is polynomially bounded but not similar to a contraction. It was shown in Peller [11] that Rψ is polynomially bounded if ψ ∈ BM OA. Then Bourgain [1] showed that under the condition ψ ∈ BM OA the operator Rψ is similar to a contraction. Finally, it was proved by Aleksandrov and Peller [1] that Rψ is polynomially bounded if and only if ψ ∈ BM OA, and so Rψ is polynomially bounded if and only if it is similar to a contraction. However, it was not the end of the story. Pisier [2] considered the operators Rψ on the space 2 (H) ⊕ 2 (H) of functions taking values in a Hilbert space H in which case S is the shift operator on 2 (H), ψ is a B(H)-valued function, and Γψ is the operator on 2 (H) with block Hankel ˆ + k)}j,k≥0 . He showed that there exists such a function ψ for matrix {ψ(j which Rψ is polynomially bounded but not similar to a contraction. In §1 we characterize the polynomially bounded operators Rψ in the scalar case and prove that Rψ is polynomially bounded if and only if it is similar to a contraction. In §2 we describe the class of power bounded operators Rψ and we show that among them there are operators that are not polynomially bounded. Finally, in §3 we construct polynomially bounded operators Rψ with operator-valued ψ that are polynomially bounded but not similar to a contraction.

1. Operators Rψ in the Scalar Case The main result of this section describes the polynomially bounded operators Rψ and says that Rψ is polynomially bounded if and only if it is similar to a contraction. To prove this result we need a so-called weak factorization theorem for the class of functions that consists of the derivatives of H 1 functions. First we prove an elementary lemma. It says how to compute functions of operators Rψ . In fact, one of the most important features of the operators Rψ is that the polynomials of Rψ can be evaluated explicitly. Recall that the operators Rψ are defined by (0.1). As usual we identify the spaces 2 and H 2 in the natural way: {cn }n≥0 ←→ cn z n . n≥0 ∗

The operators S and S are defined on the space of functions analytic in D by f (z) − f (0) . Sf (z) = zf (z), (S ∗ f )(z) = z Lemma 1.1. Let ψ ∈ BM OA and let ϕ be an analytic polynomial. Then ∗ S Γϕ (S ∗ )ψ ϕ(Rψ ) = . (1.1) O S

688

Chapter 15. Hankel Operators and Similarity to a Contraction

Proof. It is sufficient to prove the result in the case when ϕ = z n . Using the formula S ∗ Γψ = Γψ S = ΓS ∗ ψ , one can easily verify (1.1) by induction. The following theorem is the main result of this section. Theorem 1.2. Let ψ be a function analytic in D. The following are equivalent: (i) Rψ is polynomially bounded; (ii) Rψ is similar to a contraction; (iii) ψ ∈ BM OA. To prove Theorem 1.2, we obtain a so-called weak factorization theorem. For 1 ≤ p ≤ ∞ consider the class Xp = {f : f ∈ H p } def

and we endow Xp with the norm def

gXp = f H p ,

f = g,

f (0) = 0.

We also consider the subspace x∞ = {f : f ∈ CA } def

of X∞ . It is well known (see Appendix 2.6) that the class Xp admits the following description: p/2

1 p 2 g∈X ⇐⇒ |g(rζ)| (1 − r)dr dm(ζ) < ∞. (1.2) T

0

The following weak factorization theorem will be used only for p = 1, but it is more natural to prove it for 1 ≤ p < ∞. Theorem 1.3. Let 1 ≤ p < ∞ and let g be a function analytic in D. Then g ∈ Xp if and only if there are functions ξm ∈ X∞ and ηm ∈ H p , 1 ≤ m ≤ 4, such that g=

4

ξm ηm .

(1.3)

m=1

Proof. Let us first prove that for ξ ∈ X∞ and η ∈ H p we have ξη ∈ Xp . Let ξ = ψ , where ψ ∈ H ∞ . We have ξη = ψ η = (ψη) − ψη . Clearly, ψη ∈ H p , and so (ψη) ∈ Xp . It is obvious from (1.2) that ψη ∈ Xp . Now let g be an arbitrary function in Xp . Consider first the case p > 1. Then we are able to represent g in the form g = ξ1 η1 + ξ2 η2 , with ξm ∈ X∞ and ηm ∈ H p , m = 1, 2.


689

Let f be a function in H p such that f = g and f (0) = 0. Then Im f is the harmonic conjugate of Re f . Clearly, we can represent f as f = f1 − f2 , where fj ∈ H p and Re fj ≥ 0 on T, j = 1, 2. Indeed, if def

(Re f1 )(ζ) = (Re f )+ (ζ) = max{0, Re f (ζ)} and Im f1 is the harmonic conjugate of Re f1 , then f1 ∈ H p , since 1 < p < ∞. Hence it is sufficient to show that if f ∈ H p and Re f ≥ 0, then there def def exist ξ ∈ X∞ and η ∈ H p such that f = ξη. Put ξ = (f i ) , η = −if · f −i . i ∞ Since Re f ≥ 0, it follows that f is invertible in H , and so ξ ∈ X∞ , η ∈ H p . The equality f = ξη is obvious. Consider now the case p = 1. Let f be a function in H 1 such that f = g. Then f = uv, where u, v ∈ H 2 . We have already shown that there exist functions ξm ∈ X∞ and τm ∈ H 2 , 1 ≤ m ≤ 4, such that u = ξ1 τ1 + ξ2 τ2 ,

v = ξ3 τ3 + ξ4 τ4 .

Clearly, (1.3) holds with η1 = τ1 v,

η2 = τ2 v,

η3 = τ3 u η4 = τ4 u.

Remark. It is easy to see that gXp is equivalent to inf

4

ξm X∞ ηm H p ,

m=1

where the infimum is taken over all function ξm and ηm satisfying (1.3). Consider now the space X p of functions f analytic in D, which admit a representation ξm ηm , (1.4) f= m≥0 ∞

where ξm ∈ x , ηm ∈ H , and ξm x∞ ηm H p < ∞. p

(1.5)

m≥0

For f in X p we define f X p to be the infimum of the sum in (1.5) over all representations of the form (1.4). Theorem 1.4. Let 1 ≤ p < ∞. Then Xp = X p and the norms in Xp and X p are equivalent. Proof. It follows from Theorem 1.3 that X p ⊂ Xp . Let us show that X ⊂ X p. For a function f analytic in D and for 0 < r < 1 we denote by fr the function defined by fr (ζ) = f (rζ). Let g ∈ Xp . Then by the remark following Theorem 1.3 there exist functions ξm ∈ X∞ and ηm ∈ H p , p

690


1 ≤ m ≤ 4, such that (1.3) holds and 4

ξm X∞ ηm H p ≤ const gXp .

m=1

We have gr =

4

(ξm )r (ηm )r .

m=1

Clearly, (ξm )r ∈ x∞ , (ξm )r x∞ ≤ ξm x∞ , and (ηm )r H p ≤ ηm H p . Therefore gr X p ≤ const gXp . The result follows from the fact that g − gr Xp → 0 as r → 1. Proof of Theorem 1.2. (i)⇔(iii). It follows from Lemma 1.1 that Rψ is polynomially bounded if and only if Γϕ (S ∗ )ψ ≤ const ϕ∞ for any analytic polynomial ϕ. By the Nehari theorem this is equivalent to the fact that ϕ (S ∗ )ψBM OA ≤ const ϕ∞ , which in turn is equivalent to the inequality |g, ϕ (S ∗ )ψ| ≤ const ϕ∞ gH 1 for any analytic polynomials ϕ and g, where the pairing def u ˆ(n)ˆ v (n), u ∈ H 1 , v ∈ BM OA, u, v =

(1.6)

n≥0

is defined at least in the case when u is a polynomial and extends by continuity. We have g, ϕ (S ∗ )ψ = ϕ (S)g, ψ = ϕ g, ψ.

(1.7)

1 ∗

Since (H ) = BM OA with respect to the pairing (1.6), it follows that (X1 )∗ = {h : h ∈ BM OA} with respect to the same pairing. Suppose that ψ ∈ BM OA. We have |ϕ g, ψ| ≤ const ·(ψ BM OA + |ψ(0)|) · ϕ gX1 .

(1.8)

By Theorem 1.3, ϕ gX1 ≤ const ϕH ∞ gH 1 .

(1.9)

It follows now from (1.7), (1.8), and (1.9) that ϕ(Rψ ) ≤ const ϕH ∞ . Suppose now that Rψ is polynomially bounded. Then it follows from (1.7) that |ϕ g, ψ| ≤ const ϕ∞ gH 1


691

for any polynomials ϕ and g. Therefore the same inequality holds for arbitrary ϕ ∈ x∞ and g ∈ H 1 . By Theorem 1.4, it follows that |h, ψ| ≤ const gX1 for an arbitrary function h in X1 . Hence, ψ ∈ (X1 )∗ , and so ψ ∈ BM OA. Clearly, the implication (ii)⇒(i) in Theorem 1.2 is obvious. The remaining implication (iii)⇒(ii) is a consequence of the following theorem. Theorem 1.5. Let ψ be a function analytic in D such that ψ ∈ BM OA. Then Rψ is similar to the operator S ⊕ S ∗ on H 2 ⊕ H 2 . We need some preparation to prove Theorem 1.5. Lemma 1.6. Let f and g be functions in H 2 . Then f g ∈ X1 . Proof. By (1.2), we have to show that

T

1

|(f g)(rζ)|2 (1 − r)dr

1/2 dm(ζ) < ∞.

0

Consider the radial maximal function g (∗) on T defined by def

g (∗) (ζ) = sup |g(rζ)|, 0 0,

j≥n

which proves (iv). It remains to prove that (iv)⇒(iii). Consider the operator D of differentiation, which is defined on the dense subset of polynomials in H 2 by Df = f . Let us show that ΓΞ D extends to a bounded linear operator from H 2 to H 2 (K). Indeed, (ΓΞ D)∗ ΓΞ D = D∗ Γ∗Ξ ΓΞ D and it follows from that (ΓΞ D)∗ ΓΞ D has diagonal matrix with diag (3.4) 2 2 |αj | , · · · , m |αj |2 , · · · , and so it is bounded. onal entries 0, j≥0

j≥m−1

Let V be the operator on H 2 (K) ⊕ H 2 defined by V =

I O

−ΓΞ D I

.

It is straightforward to see that V is invertible and V −1 =

I ΓΞ D O I

.

3. Counterexamples

We have V RΞ V

−1

= = = =

I −ΓΞ D O I

∗ SK

ΓΞ

O

S

∗ SK

∗ ΓΞ + S K ΓΞ D − ΓΞ DS

O

S

∗ SK O ∗ SK O

I O

ΓΞ D I

697

ΓΞ + ΓΞ SD − ΓΞ DS S ∗ ΓΞ (I + SD − DS) SK = S O

O S

,

since obviously, I + SD − DS = O. Now we consider the operators RΨ with Ψ taking values in B(H), where H is an infinite-dimensional Hilbert space. Let us first find a condition sufficient for polynomial boundedness. Suppose that {Yj }j≥0 is a sequence of bounded linear operators on H such that  1/2  λj Yj |λj |2  (3.6) ≤ j≥0 j≥0 for any finitely supported sequence {λj }j≥0 . Theorem 3.4. Let {Yj }j≥0 be a sequence of operators on H satisfying (3.6) and let {αj }j≥0 be a sequence of complex numbers satisfying (3.5). If Ψ is the operator-valued function defined by αj z j Yj , Ψ(z) = j≥0

then the operator RΨ is polynomially bounded. We need the following lemma. Lemma 3.5. Let H1 , H2 , K1 , and K2 are Hilbert spaces and let Q be a bounded linear operator from B(H1 , H2 ) to B(K1 , K2 ). If {Ωj }j≥0 is a sequence in B(H1 , H2 ) such that {Ωj+k }j,k≥0 determines a bounded linear operator from 2 (H1 ) to 2 (H2 ), then {QΩj+k }j,k≥0 determines a bounded linear operator from 2 (K1 ) to 2 (K2 ) and {QΩj+k }j,k≥0 ≤ Q · {Ωj+k }j,k≥0 . Proof. By Theorem 2.2.2, there exists a function Φ ∈ L∞ (B(H1 , H2 )) ˆ such that Φ(j) = Ωj and ΦL∞ (B(H1 ,H2 )) = {Ωj+k }j,k≥0 . Consider the function Φ♥ defined by Φ♥ (ζ) = QΦ(ζ),

ζ ∈ T.

ˆ ♥ (j) = QΦ(j), ˆ Clearly, Φ j ∈ Z+ , and Φ♥ L∞ (B(K1 ,K2 )) ≤ Q · ΦL∞ (B(H1 ,H2 )) .

698


Proof of Theorem 3.4. Let K be the infinite-dimensional Hilbert space from Theorem 3.2. We apply Lemma 3.5 with H1 = C, H2 = K, αj ej = αj Yj . By K1 = K2 = H. The operator Q is defined by Q j≥0

j≥0

(3.6),Q ≤ 1.To prove that RΨ is polynomially bounded, we have to show that Γϕ (S ∗ )Ψ ≤ const ϕ∞ for any polynomial ϕ. Thisis an immediate consequence of Lemma 3.5 and the inequality Γϕ (S ∗ )Ξ ≤ const ϕ∞ guaranteed by Theorem 3.3. Among operator functions Ψ satisfying the hypotheses of Theorem 3.4 we are going to find those for which RΨ is not similar to a contraction. We are going to use CAR sequences (i.e., sequences satisfying the canonical anticommutation relations) of bounded linear operators {Xj }j≥0 on Hilbert space satisfying I, j = k, ∗ ∗ Xj Xk + Xk Xj = O, Xj Xk + Xk Xj = j, k ∈ Z+ . (3.7) O, j = k, We show later that such sequences do exist. First, we prove the following important property of CAR sequences. Theorem 3.6. Let {Xj }j≥0 be a sequence of Hilbert space operators satisfying (3.7). Then  1/2  αj Xj |αj |2  . = j≥0 j≥0 Proof. Clearly, we may assume that {αj }j≥0 is a finitely supported sequence. Let X = αj Xj . It follows from (3.7) that j≥0



X 2 = O and X ∗ X + XX ∗ = 

 |αj |2  I.

j≥0

We have X ∗ XX ∗ X = X4 , and since X ∗ X ∗ XX = O, we obtain X4

= X ∗ (XX ∗ )X + X ∗ (X ∗ X)X      ∗   = |αj |2  I  X |αj |2  X2 . X = j≥0 j≥0

We are going to prove the following result. Theorem 3.7. Let {Xj }j≥0 be a CAR system and let {αj }j≥0 be a sequence of complex numbers such that j 2 |αj |2 = ∞. (3.8) j≥0

3. Counterexamples

Let Ψ=

699

αj z j Xj ,

j≥0

where {Xj }j≥0 is a CAR system. Suppose that ΓΨ is a bounded operator. Then RΨ is not similar to a contraction. Let us first deduce Theorem 3.1 from Theorem 3.7. Proof of Theorem 3.1. Let {αj }j≥0 be a sequence of complex numbers satisfying (3.5) and (3.8). Note that such sequences do exist. For example, we can take αj = (j + 1)−3/2 . Then by Theorems 3.6 and 3.4, RΨ is polynomially bounded while according to Theorem 3.7, it is not similar to a contraction. Now we are going to construct a CAR sequence. We are going to use Hilbert tensor products (see §8.2) and tensor products of operators on Hilbert spaces. Lemma 3.8. There exists a sequence {Xj }0≤j≤n−1 of operators on n C2 satisfying (3.7). n Proof. As usual, we identify operators on C2 with matrices of size 2n × 2n . Consider the following 2 × 2 matrices: 1 0 0 0 1 0 A= , B= , and I2 = . 0 −1 1 0 0 1 Now we can define the Xj by Xj = A ⊗ · · · ⊗ A ⊗B ⊗ I2 ⊗ · · · ⊗ I2 , .+ , .+ , j

0 ≤ j ≤ n − 1.

(3.9)

n−j−1

Clearly, the Xj can be considered as matrices of size 2n × 2n . Let us verify (3.7) for j, k ≤ n − 1. The fact that Xj2 = O follows immediately from the equality B 2 = O. We have Xj∗ = A ⊗ · · · ⊗ A ⊗B ∗ ⊗ I2 ⊗ · · · ⊗ I2 , .+ , .+ , j

0 ≤ j ≤ n − 1.

n−j−1

Then Xj Xj∗ = I2 ⊗ · · · ⊗ I2 ⊗BB ∗ ⊗ I2 ⊗ · · · ⊗ I2 , .+ , .+ , j

0 ≤ j ≤ n − 1,

n−j−1

and Xj∗ Xj = I2 ⊗ · · · ⊗ I2 ⊗BB ∗ ⊗ I2 ⊗ · · · ⊗ I2 , .+ , .+ , j

Xj Xj∗

0 ≤ j ≤ n − 1.

n−j−1

Xj∗ Xj

+ = I is a consequence of the obvious equality The equality BB ∗ + BB ∗ = I2 . Now let j = k. The equality Xj Xk + Xk Xj = O follows immediately from the equality AB + BA = O. Finally, the fact that Xj Xk∗ + Xk∗ Xj = O is an immediate consequence of of the equality AB ∗ + B ∗ A = O.

700


The following result shows that any CAR sequences of length n are in a sense isomorphic. ♠ Lemma 3.9. Let X0 , · · · , Xn−1 and X0♠ , · · · , Xn−1 be CAR sequences ♠ ∗ of of operators on Hilbert space. Let A and A be the C -algebras generated by the Xj and by the Xj♠ , respectively. Then there exists a ∗-isomorphism of A onto A♠ that takes Xj to Xj♠ , 0 ≤ j ≤ n − 1. We need another lemma. Lemma 3.10. Let {Xj } be a CAR system and let j1 , · · · , jm be a sequence of disjoint nonnegative integers. Then Xj1 · · · Xjm = O. Proof. Let Z = Xj2 · · · Xjm = O. Suppose that Xj1 Z = O. Then Xj∗1 Xj1 Z = O and Xj1 Xj∗1 Z = (−1)m−1 Xj∗1 ZXj1 = O. Hence,

Z = (Xj1 Xj∗1 + Xj∗1 Xj1 )Z = O, and the result follows by induction. Proof of Lemma 3.9. We define by induction finite subsets ∆j , j = 0, 1 · · · , n, of A. Put ∆0 = I, ∆j = ∆j−1 ∪ ∆j−1 Xj ∪ Xj∗ ∆j−1 ∪ Xj∗ ∆j−1 Xj ,

j ≥ 1.

It is easy to see that the sets ∆j are finite and A is the linear span of ∆n . If we prove that ∆n is linearly independent, we can construct in the ♠ same way the subset ∆♠ n of A , establish the natural one-to-one correspondence between the elements of ∆n and ∆♠ n , and see that this one-to-one correspondence extends to a ∗-isomorphism of A onto A♠ . We prove by induction the following fact that implies the linear independence of ∆n : if 0 ≤ m < j1 < · · · < jk , then the set Xj∗k · · · Xj∗1 ∆m Xj1 · · · Xjk is linearly independent. The case m = 0 follows from Lemma 3.10. Suppose that we have already proved this for all numbers less than or equal to m and let m + 1 < j1 < · · · < jk . Put Z = Xj1 · · · Xjk . Let D0 , D1 , D2 , and D3 be elements of ∆m such that ∗ ∗ Z ∗ D0 Z + Z ∗ D1 Xm+1 Z + Z ∗ Xm+1 D2 Z + Z ∗ Xm+1 D3 Xm+1 Z = O. (3.10)

We have to prove that D0 = D1 = D2 = D3 = O. ∗ ∗ Since ZXm+1 = (−1)k Xm+1 Z and Xm+1 Z ∗ = (−1)k Z ∗ Xm+1 , we can ∗ on the left and find multiply (3.10) by Xm+1 on the right and by Xm+1 that ∗ Z ∗ Xm+1 D0 Xm+1 Z = O. ∗ By the inductive hypotheses, D0 = O. Now multiplying (3.10) by Xm+1 on the left, we obtain ∗ D1 Xm+1 Z = O, Z ∗ Xm+1 which again implies that D1 = O. Next, multiplying (3.10) by Xm+1 on the right, we find that D3 = O, which in turn implies that D4 = O.

3. Counterexamples

701

Now we are in a position to prove that there exists an infinite CAR sequence. Theorem 3.11. There exists a sequence {Xj }j≥0 of bounded linear operators on Hilbert space that satisfy (3.7). Proof. It follows from Lemma 3.9 that there exists a sequence of finitedimensional C ∗ -algebras ∗-isometrically imbedded in each other: A1 ⊂ A2 ⊂ A3 · · · and such that An is generated by a CAR sequence X0 , X1 , · · · , Xn−1 . This gives us an infinite CAR sequence {Xj }j≥0 . We need the following important property of CAR sequences. Theorem 3.12. Let {Xj }j≥0 be a CAR sequence. Then for every finitely supported sequence {αj }j≥0 the following inequality holds: 1 |αj | ≤ α X ⊗ X |αj |. (3.11) j j j ≤ 2 j≥0

j≥0

j≥0

Proof. Clearly, the right inequality is obvious. To establish the left one, we may assume that we deal with a finite CAR sequence X0 , · · · , Xn−1 . By Lemma 3.9, it is sufficient to prove this inequality for the sequence defined by (3.9). Then we can identify Xj ⊗ Xj with (A ⊗ A) ⊗ · · · ⊗ (A ⊗ A) ⊗(B ⊗ B) ⊗ I4 ⊗ · · · ⊗ I4 . .+ , .+ , n−j−1

j

Let e1 , e2 be the standard orthonormal basis of C2 . Consider the unit vectors 1 xj = √ (e1 ⊗ e1 + ωj e2 ⊗ e2 ) 2 in C2 ⊗ C2 , where ωj ∈ C and αj ω ¯ j = |αj |. Clearly, xj = 1. Let def

x = x0 ⊗ · · · ⊗ xn−1 . It is easy to see that A ⊗ A(ek ⊗ ek ) = ek ⊗ ek , k = 1, 2, and B ⊗ B is a rank one partial isometry that takes e1 ⊗ e2 to e2 ⊗ e2 . Now it is easy to verify that   n−1 n−1 n−1 αj ω ¯j 1  αj (Xj ⊗ Xj )x, x = |αj |. = 2 2 j=0 j=0 j=0 To prove Theorem 3.7, we show that under its hypotheses the operator RΨ is not completely polynomially bounded (see the introduction to this chapter). Thus we need the easy part of the Arveson–Paulsen theorem (which is due to Arveson) stated in the introduction. We prove it here. Theorem 3.13. Let T be an operator on Hilbert space similar to a contraction. Then T is completely polynomially bounded.

702


Recall that we identify here the space Mn,n of n × n matrices with the space of linear operators on Cn . For a Hilbert space H we identify the tensor product B(H) ⊗ Mn,n with the space of n × n operator matrices {Tjk }j,k≥0 , Tjk ∈ B(H). Given a matrix A = {ajk }j,k≥0 ∈ Mn,n , and a bounded linear operator T on H, we identify A ⊗ T with the operator matrix {ajk T }j,k≥0 . Proof of Theorem 3.13. It is easy to see from the definition of complete polynomial boundedness that we have to prove the following inequality: m m j j Bj ⊗ T ≤ const · max ζ Bj (3.12) |ζ|≤1 j=0 j=0 Mn,n ⊗B(H)

Mn,n

for any matrices B0 , · · · , Bm ∈ Mn,n . It is evident that it suffices to prove the result for a contraction T on a Hilbert space H. We use the Sz.-Nagy theorem on unitary dilations (see Appendix 1.5). Thus there exist a Hilbert space K such that H ⊂ K and a unitary operator U on K such that T j = PH U j H for any j ∈ Z+ , where PH is the orthogonal projection onto H. Then Bj ⊗ T j = PCn ⊗H Bj ⊗ U j , where PCn ⊗H is the orthogonal projection from Cn ⊗ K onto Cn ⊗ H. Thus it suffices to prove (3.12) for unitary operators. The spectral theorem for unitary operators allows us to realize a unitary operator as multiplication by z on an L2 space of vector functions (see Appendix 1.4). Thus the operm m Bj ⊗ U j becomes multiplication by the matrix function z j Bj , ator j=0

j=0

which proves (3.12) for T replaced with U . Now we are able to prove Theorem 3.7.

Proof of Theorem 3.7. Suppose that RΨ is similar to a contraction. Then RΨ must satisfy (3.12). Since (S ∗ )j jΓ(S ∗ )j−1 Ψ j , RΨ = O Sj it follows that m jBj ⊗ Γ(S ∗ )j−1 Ψ j=0

Mn,n ⊗B(H)

m j ≤ const · max ζ Bj |ζ|≤1 j=0

(3.13)

Mn,n

for any matrices Bj ∈ Mn,n . We consider now only the (0, 0) entry of Γ(S ∗ )j−1 Ψ which is αj Xj . Thus it follows from (3.13) that m m j jαj Bj ⊗ Xj ≤ const · max ζ Bj (3.14) |ζ|≤1 j=0 j=0 Mn,n ⊗B(H)

Mn,n

for any matrices Bj ∈ Mn,n . By Lemmas 3.8 and 3.9, we may assume that the operators Xj in (3.14) are realized as matrices in M2m+1 ,2m+1 . Now we

Concluding Remarks

put n = 2m+1 and Bj = j α ¯ j Xj in (3.14) and we obtain m m 2 2 j j |α | X ⊗ X ≤ const · max j α ¯ ζ X j j j j j |ζ|≤1 j=0 j=0 Mn,n ⊗B(H)

703

. Mn,n

It follows now from Theorems 3.6 and 3.12 that  1/2 m m j 2 |αj |2 ≤ const  j 2 |αj |2  , j=0

and so

j=0 m

j 2 |αj |2 ≤ const,

j=0

which contradicts (3.8).

Concluding Remarks The operators RΨ were introduced in Peller [5]; see also Carlson, Clark, Foias, and Williams [1]. The implication (iii)⇒(i) of Theorem 1.2 was proved in Peller [11]. The implication (iii)⇒(ii) was obtained in Bourgain [1]. Bourgain’s proof is based on Paulsen’s theorem on similarity to a contraction. The proof of the implication (iii)⇒(ii) given in §1 is due to Stafney [1]. The implication (i)⇒(iii) was obtained in Aleksandrov and Peller [1]. The weak factorization Theorems 1.3 and 1.4 were found in Aleksandrov and Peller [1]. Note that the paper Aleksandrov and Peller [1] also contains weak factorization theorems for other function spaces. Theorems 1.5 and 1.7 are due to Stafney [1]. Note that before the results of Aleksandrov and Peller [2] Paulsen observed that Rψ is similar to a contraction if and only if the matrix ˆ + k)}j,k≥0 determines a bounded linear operator on 2 (un{(j − k)ψ(j published). It can be obtained from the results of Janson and Peetre [2] that this matrix determines a bounded operator if and only if ψ ∈ BM OA (Janson and Peetre considered integral operators, but their technique also works for matrices). Thus these two results give another proof of the fact that Rψ is similar to a contraction if and only if ψ ∈ BM OA. Note that Petrovic observed (unpublished) that a result of Stafney [1] together with the observation by Paulsen mentioned above also implies that Rψ is similar to a contraction if and only if ψ ∈ BM OA. We also mention here the paper Ferguson [1], in which she gave another proof of Bourgain’s result that the condition ψ ∈ BM OA implies similarity to a contraction. The results of §2 were obtained in Peller [5]. The main result of §3, Theorem 3.1, was obtained in Pisier [2]. Namely, Pisier proved that if {Xj }j≥0 is a CAR system and {αj }j≥0 is a sequence

704


of complex numbers satisfying (3.5) and 3.8 and Ψ =

αj z j Xj , then RΨ

j≥0

is polynomially bounded but not similar to a contraction. The proof given in Pisier [2] is rather complicated and involves martingales. In §3 we give the proof of Pisier’s result that was found by Davidson and Paulsen [1]. We refer the reader to Pisier [1] for information about CAR sequences. We also mention the paper Kislyakov [2] in which a proof of Pisier’s result was given that uses a technique of singular integrals. Another proof of Pisier’s result was given in J.E. McCarthy [1].

Appendix 1 Operators on Hilbert Space

We collect in this appendix necessary information on linear operators on Hilbert space. We give here almost no proofs and we give references for more detailed information.

1. Singular Values and Operator Ideals Let T be a bounded linear operator from a Hilbert space H to a Hilbert space K. The singular values sn (T ), n ∈ Z+ , of T are defined by def

sn (T ) = inf{T − K : K : H → K, rank K ≤ n}, where rank K stands for the rank of K. Clearly, the sequence {sn (T )}n≥0 is nonincreasing and its limit def

s∞ (T ) = lim sn (T ) n→∞

is equal to the essential norm of T , which is by definition T e = inf{T − K : K ∈ C(H, K)}, where C(H, K) is the space of compact operators from H to K. It is easy to see that if T1 and T2 are operators from H to K, then sm+n (T1 + T2 ) ≤ sn (T1 ) + sm (T2 ),

m, n ∈ Z+ .

If A, T , and B are bounded linear Hilbert space operators such that the product AT B makes sense, then it can easily be seen that sn (AT B) ≤ Asn (T )B,

n ∈ Z+ .

706

Appendix 1. Operators on Hilbert Space

If T is a self-adjoint operator and D is a nonnegative rank one operator, and T# = T + D, then sn (T# ) ≥ sn (T ) ≥ sn+1 (T# ) ≥ sn+1 (T )

for any n ∈ Z+ .

The operator T is compact if and only if lim sn (T ) = 0. If T is a n→∞ compact operator from H to K, it admits a Schmidt expansion sn (T )(x, fn )gn , x ∈ H, Tx = n≥0

where {fn }n≥0 is an orthonormal sequence in H and {gn }n≥0 is an orthonormal sequence in K. Note that the sum is finite if and only if T has finite rank. If 0 < p < ∞, we denote by S p (H, K) the Schatten–von Neumann class of operators T from H to K such that  1/p def sn (T )p  < ∞. T S p =  n≥0

If this does not lead to a confusion, we write T ∈ S p instead of T ∈ S p (H, K). If 1 ≤ p < ∞, the space S p = S p (H, K) is a Banach space with norm · S p . We refer the reader to Gohberg and Krein [2] or Simon [2] for the proof of the triangle inequality in S p . Together with C(H, K) we use the notation S ∞ (H, K) for the space of compact operators from H to K. If p < 1, the space S p is not a Banach space. However, the following triangle inequality is very useful for p < 1. Theorem A1.1. Let 0 < p < 1 and A, B ∈ S p . Then A + BpS p ≤ ApS p + BpS p .

(A1.1)

Moreover, A + BpS p = ApS p + BpS p if and only if A∗ B = AB ∗ = O. Since it is not easy to find the proof of this theorem in monographs, we give a proof here that is due to A.B. Aleksandrov (private communication); apparently, it is published here for the first time. We refer the reader to Rotfel’d [1] and C.A.McCarthy [1] for other proofs. s0 λ0 and s = are distinct vectors. Suppose Suppose that λ = λ1 s1 also that |λ0 | ≥ |λ1 | and |s0 | ≥ |s1 |. Consider the function f on (0, ∞) defined by f (q) = λq − sq .

(A1.2)

We need the following fact whose proof is an easy exercise. Lemma A1.2. The function f can have at most one zero on (0, ∞). If it has a zero, it changes sign at the zero.


707

Lemma A1.3. Let 0 < p < 1 and let A and B be rank one operators. Suppose that A ≥ B and A + BpS p ≥ ApS p + BpS p . Then s0 (A + B) = A and s1 (A + B) = B and, in particular, A + BpS p = ApS p + BpS p . Proof. Assume that A ≥ B. Let A = λ0 (·, x0 )y0 , B = λ1 (·, x1 )y1 , where x0 , x1 ∈ H, y0 , y1 ∈ K, x0 = x1 = y0 = y1 = 1, λ0 = A, λ1 = B. Let s0 = s0 (A + B) and s1 = s1 (A + B). Put λ0 s0 λ= , s= . λ1 s1 By the assumptions, sp ≥ λp . By the triangle inequality in S 1 , s1 ≤ λ1 . Assume now that λ = s. It follows now from Lemma A1.2 that the function f defined by (A1.2) has one zero on [p, 1]. Clearly, f (q) < 0 for any q > 1. It follows that λ0 > s0 . Since rank A = 1, it follows that s1 ≤ λ1 . We have obtained a contradiction. Corollary A1.4. If A and B are rank one operators, then A + BpS p ≤ ApS p + BpS p . Lemma A1.5. Suppose that A and B are rank one operators such that A ≥ B. The following are equivalent: (i) s0 (A + B) = A and s1 (A + B) = B; (ii) A∗ B = O and AB ∗ = O. Proof. It is sufficient to prove that (i) implies (ii). Again, let Ax = λ0 (x, x0 )y0 , Bx = λ1 (x, x1 )y1 , x ∈ H, where x0 , x1 ∈ H, y0 , y1 ∈ K, x0 = x1 = y0 = y1 = 1, λ0 = A, λ1 = B. Clearly, we may assume that λ1 > 0. We have 1/2 |(x, x1 )| s1 (A + B) ≤ λ1 sup = λ1 1 − |(x0 , x1 )|2 . x x⊥x0 It follows that (x0 , x1 ) = 0. To prove that (y0 , y1 ) = 0, we can apply the above reasoning to the operators A∗ and B ∗ . Corollary A1.6. Let A and B be rank one operators. Then A + BpS p = ApS p + BpS p if and only if A∗ B = O and AB ∗ = O. Lemma A1.7. Let A : H → K be an operator of rank at most N and 0 < p < 1. Then   N N   Aj pS p : A = Aj ; rank Aj ≤ 1 . ApS p = inf   j=1 j=1 (A1.3) Proof. Clearly, it is sufficient to consider on the right-hand side of (A1.3) only those Aj that satisfy Ker A ⊂ Ker Aj and Range Aj ⊂ Range A. The

708


set of such operators is finite-dimensional, and so the infimum on the righthand side is attained. N Let A = Aj be a representation that minimizes the infimum in (A1.3). j=1

Clearly, it is sufficient to show that A∗j Ak = O and Aj A∗k = O. Suppose that A∗j Ak = O or Aj A∗k = O for some j and k. By Lemma A1.5, one can represent Aj + Ak as the sum of rank one operators B1 and B2 such that B1 pS p + B2 pS p < A1 pS p + A2 pS p which contradicts the fact that the representation A =

N

Aj realizes the

j=1

infimum in (A1.3).

Proof of Theorem A1.1. Let us prove (A1.1). Clearly, we can assume that A and B are finite rank operators. Let A=

N

Aj ,

B=

j=1

M

Bk ,

j=k

where the Aj and Bk are of rank one. Then A+B =

N

Aj +

j=1

M

Bk

k=1

and by Corollary A1.4, A + BpS p ≤

N

Aj pS p +

j=1

M

Bk pS p = ApS p + BpS p .

k=1

the conditions A∗ B = O and AB ∗ = O imply that + BpS p . A + BpS p = ApS p + BpS p . Consider the Schmidt B: A= λj (·, xj )yj , B = µk (·, uk )vk .

It is obvious that A + BpS p = ApS p Suppose now that expansions of A and

j

Then A + BpS p =

k

j

|λj |p +

|µk |p .

k

It is sufficient to show xj ⊥ uk and yj ⊥ vk for all j and k. If this is not true for some j0 and k0 , then by Lemma A1.5, we can represent the operator λj (·, xj0 )yj0 + µk (·, uk0 )vk0 as the sum of A1 and B1 such that λj (·, xj0 )yj0 + µk (·, uk0 )vk0 pS p = A1 + B1 pS p < |λj0 |p + |µk0 |p .


Hence, A + BpS p

≤

A1 + B1 pS p +

j =j0

< |λj0 | + |µk0 | + p

p

j =j0

=

ApS p

+

|λj |p +

709

|µj |p

k =k0

|λj | + p

|µj |p

k =k0

BpS p ,

which contradicts the assumption. Operators of class S 1 are also called nuclear operators or operators of trace class. The last term reflects the fact that on the space S 1 of operators on a Hilbert space H we can introduce the important functional trace. Suppose that {ej }j≥0 is an orthonormal basis in H. Put trace T = (T ej , ej ), T ∈ S 1 . j≥0

Then trace is a linear functional on S 1 and | trace T | ≤ T S 1 . Moreover, trace T does not depend on the choice of the orthonormal basis {ej }j≥0 . If 1 < p < ∞, the dual space S ∗p = S p (H, K)∗ can be identified with the space S p = S p (H, K) with respect to the pairing T, R = trace T R∗ ,

T ∈ Sp,

R ∈ S p .

(A1.4)

With respect to the same pairing we can identify S ∗1 with the space B of bounded linear operators and we can identify the dual space C ∗ to the space of compact operators with the space S 1 . The space S 2 is also called the Hilbert–Schmidt class. It is a Hilbert space with respect to the inner product (A1.4). If H = L2 (X , µ) and K = L2 (Y, ν), an operator T ∈ B(H, K) belongs to S 2 if and only if there exists a function k ∈ L2 (X × Y, µ × ν) such that

(T f )(y) = k(x, y)f (x)dµ(x). X

Consider now the ideals S q,ω , 1 ≤ q < ∞, of operators T such that 1/q  q (sn (T ))  def T S q,ω =  < ∞. 1+n n≥0

It is easy to see that S p ⊂ S q,ω for any p < ∞ and any q ∈ [1, ∞). The space S q,ω is a Banach space with norm · S q,ω . This follows from the fact that the sequence space d(q, w) that consists of sequences {an }n≥0 such that 1/q  (a∗ )q def n  {an }n≥0 d(q,w) =  1+n n≥0

710


is a Banach space with norm · d(q,w) (see Lindenstrauss and Tzafriri [1], Ch. 4) and the results of Ch. III, §3, of Gohberg and Krein [2]. Here we denote by {a∗n }n≥0 the nonincreasing rearrangement of {|an |}n≥0 . Note also that the dual space (S q,ω )∗ with respect to the pairing (A1.4) consists of operators T such that the sequence {sn (T )}n≥0 belongs to the sequence space (d(q, w))∗ , which is by definition the dual space to d(q, w) with respect to the pairing (a, b) = an¯bn , a = {an }n≥0 ∈ d(q, w), b = {bn }n≥0 ∈ (d(q, w))∗ . n≥0 def

The space S ω = S 1,ω is called the Matsaev ideal. For 0 < p < ∞ and 0 < q ≤ ∞ we define the Schatten–Lorentz class S p,q as the class of operators T on Hilbert space such that 1/q  def T S p,q =  (sn (T ))q (1 + n)q/p−1  < ∞, q < ∞, n≥0 def

T S p,∞ = sup (1 + n)1/p sn (T ) < ∞,

q = ∞.

n∈Z+

It can be easily verified that S p,p = S p , S p1 ,q1 ⊂ S p2 ,q2 , if p1 < p2 and S p,q1 ⊂ S p,q2 , if q1 < q2 . We need the following fact. Theorem A1.8. Suppose that 1 < q ≤ ∞. Then the space S 1,q is not normable. This fact is well-known. However, it is difficult to find a reference. Thus we sketch the proof. Sketch of the proof. Suppose that {en }n≥0 is an orthonormal basis in a Hilbert space H. Then the subspace of S 1,q of operators T of the form Tx = λn (x, en )en , x ∈ H, is isomorphic to the Lorentz sequence space n≥0

1,q . Thus it suffices to show that 1,q is not normable. Suppose that · is a norm on 1,q that is equivalent to the standard seminorm on 1,q . We define the operators TN on 1,q by    x0 + · · · + xN −1  x0 + · · · + xN −1 ,··· , , 0, 0, · · ·  TN (x0 , x1 , · · · ) =   . N N .+ , N

It is easy to see that the operators TN are uniformly bounded on 1,q . Suppose that x = {xn }n≥0 ∈ 1,q , xn ≥ 0 and x = {xn }n≥0 ∈ / 1 . It is easy to verify that TN x ≥ const

N −1 n=0

xn → ∞

as

N → ∞.


711

We refer the reader to Gohberg and Krein [2] and Simon [2] for more detailed information on singular values and operator ideals.

2. Fredholm Operators and the Calkin Algebra Let H be a Hilbert space, B(H) the algebra of bounded linear operators on H, and C(H) the ideal of compact operators on H. The quotient algebra B(H)/C(H) = B/C is called the Calkin algebra. An operator T in B(H) is called Fredholm if its image in the Calkin algebra is invertible. In other words, T is Fredholm if there exists R ∈ B(H) such that the operators RT − I and T R − I are compact. It is well known that T is Fredholm if and only if Range T is closed in H, dim Ker T < ∞, and dim Ker T ∗ < ∞. For a Fredholm operator T the index ind T is defined by ind T = dim Ker T − dim Ker T ∗ . If T1 and T2 are Fredholm operators, then T1 T2 is also Fredholm and ind T1 T2 = ind T1 + ind T2 . If T is Fredholm, K is compact, then T + K is Fredholm and ind(T + K) = ind T . If T is Fredholm, then there exists ε > 0 such that any perturbation T − K of T of norm less than ε is also Fredholm and ind(T − K) = ind T . The essential spectrum σe (T ) of T ∈ B(H) is by definition the spectrum of the image of T in the Calkin algebra, i.e., σe (T ) = {λ ∈ C : T − λI is not Fredholm}. Clearly, σe (T ) is a nonempty compact subset of C that is contained in σ(T ). We refer the reader to Douglas [2] for proofs and more detailed information.

3. The Gelfand–Naimark Theorem Let A be an algebra with an involution, i.e., A is equipped with a map x → x∗ , x ∈ A, of A into itself that satisfies the following properties: (x + y)∗ = x∗ + y ∗ , x, y ∈ A, ¯ (λx)∗ = λx, x ∈ A, λ ∈ C, (xy)∗ = y ∗ x∗ , ∗∗

x

= x,

x, y ∈ A,

x ∈ A.

A is called a C ∗ -algebra if in addition to that A is a Banach algebra and the involution satisfies x∗ x = x2 ,

x ∈ A.

712


The algebra B(H) of bounded linear operators on a Hilbert space H is a C ∗ -algebra with involution T → T ∗ , T ∈ B(H). Clearly, any closed selfadjoint subalgebra of B(H) is also a C ∗ -algebra. The following fundamental result due to Gelfand and Naimark asserts that an arbitrary C ∗ -algebra is ∗-isometrically isomorphic to such a subalgebra. Theorem A1.9. Let A be a C ∗ -algebra with unit. Then there exist a Hilbert space H and an isometric homomorphism U of A into B(H) such that Ux∗ = (Ux)∗ , x ∈ A. Note here that if A is an arbitrary C ∗ -algebra, the Hilbert space H does not have to be separable. Consider now the Calkin algebra B/C. It is a C ∗ -algebra with involution T → T ∗ , T ∈ B/C. It follows from Theorem A1.9 that B/C is ∗-isometrically isomorphic to a self-adjoint subalgebra of the algebra of bounded linear operators on Hilbert space. We refer the reader to Arveson [4] for the proof of the Gelfand–Naimark theorem and for more information on C ∗ -algebras.

4. The von Neumann Integral Let µ be a finite positive Borel measure on R and let {H(t)}t∈R be a measurable family of Hilbert spaces. That means that we are given an at most countable set Ω of functions f such that f (t) ∈ H(t), µ-a.e., span{f (t) : f ∈ Ω} = H(t) for µ-almost all t and the function t → (f1 (t), f2 (t))H(t) is µ-measurable for any f1 , f2 ∈ Ω. A function g with values g(t) in H(t) is called measurable if the scalar-valued function t → (f (t), g(t))H(t) is measurable for any f ∈ Ω. The von Neumann integral (direct integral) ⊕H(t)dµ(t) consists of measurable functions f, f (t) ∈ H(t), such that 1/2

2 < ∞. f = f (t)H(t) dµ(t) If f, g ∈

⊕H(t)dµ(t), then their inner product is defined by

(f, g) = (f (t), g(t))dµ(t).

By von Neumann’s theorem, each self-adjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the independent variable on a direct integral ⊕H(t)dµ(t):

(Af )(t) = tf (t), f ∈ ⊕H(t)dµ(t). (A1.5)


713

Without loss of generality we can assume that H(t) = 0, µ-almost everywhere. In this case µ is called a scalar spectral measure of A. The spectral multiplicity function νA of the operator A is defined µ-almost everywhere by νA (t) = dim H(t). It is well known that self-adjoint operators A1 and A2 are unitarily equivalent if and only if their scalar spectral measures are mutually absolutely continuous and νA1 = νA2 almost everywhere. If A is a self-adjoint operator with scalar spectral measure µ and spectral multiplicity function νA , then A is unitarily equivalent to multiplication by the independent variable on

⊕H(t)dµ(t), where µ is a scalar spectral measure of A, the H(t) are embedded in a Hilbert space E with basis {ej }j≥1 , and H(t) = span{ej : 1 ≤ j < νA (t) + 1}. In this case a function g with values g(t) ∈ H(t) is measurable if and only if the scalar function t → (g(t), ej ) is measurable for any j ≥ 1. If Ψ is a bounded operator-valued function defined µ-almost everywhere and such that Ψ(t) ∈ B(H(t)), µ-a.e., we say that Ψ is measurable if Ψf is measurable for any function f ∈ ⊕H(t)dµ(t). The following result describes the bounded linear operators that commute with the operator (A1.5). Theorem A1.10. Let T be a bounded linear operator on ⊕H(t)dµ(t) that commutes with the operator (A1.5). Then there exists a bounded measurable operator function Ψ with values Ψ(t) ∈ B(H(t)), µ-a.e., such that (T f )(t) = Ψ(t)f (t),

µ-a.e.

Note that similar results also hold for unitary operators and normal operators. We refer the reader to Birman and Solomyak [1], Ch. 7, for proofs and for more detailed information.

5. Unitary Dilations and Commutant Lifting Let H be a Hilbert space and let T be a contraction on H, i.e., T ≤ 1. A remarkable theorem by Sz.-Nagy (see Sz.-Nagy–Foias [1], Ch. I, §4) asserts that there exists a Hilbert space K and a unitary operator U on K such that H ⊂ K and T n = PH U n H, n ≥ 0, (A1.6) where PH is the orthogonal projection onto H. A unitary operator satisfying (A1.6) is called a unitary dilation of T . Note that earlier Halmos [1] proved

714


the existence of a unitary operator U satisfying (A1.6) with n = 1. U is called a minimal unitary dilation if span{U n x : x ∈ H, n ∈ Z} = K. If we consider the subspace K+ of K defined by K+ = span{U n x : x ∈ H, n ∈ Z+ }, def it is easy to see that U K+ ⊂ K+ and V = U K+ is an isometry. It is also easy to see that T n = PH V n H, n ≥ 0, i.e., V is an isometric dilation of T . In fact, V is a minimal isometric dilation of T in the natural sense. It is very easy to construct an isometric dilation. Indeed, consider the natural imbedding of H in 2 (H): x → (x, O, O, · · · ), x ∈ H. Define the operator V on 2 (H) by V (x0 , x1 , x2 , · · · ) = T x0 , (I − T ∗ T )1/2 x0 , x1 , x2 , · · · . It is easy to verify that V is an isometric dilation of T . To construct a unitary dilation of T , it suffices to construct a unitary extension of an isometric operator, which can be done using the so-called Kolmogorov– Wold decomposition of an isometry (see Sz.-Nagy–Foias [1], Ch. I, §1). We are going to state a very important result, the commutant lifting theorem due to Sz.-Nagy and Foias. Theorem A1.11. Let T1 and T2 be contractions on Hilbert spaces H1 and H2 and let X ∈ B(H1 , H2 ) such that XT1 = T2 X. Suppose that U1 and U2 are minimal unitary dilations of T1 and T2 on Hilbert spaces K1 and K2 and let K1+ ⊂ K1 and K2+ ⊂ K2 be the corresponding subspaces of minimal isometric dilations of T1 and T2 . Then there exists an operator Y ∈ B(K1 , K2 ) such that Y U1 = U2 Y, X = PH2 Y H1 , Y K1+ ⊂ K2+ , and Y = X. It is easy to see that the requirement of the minimality of unitary dilations is not important and the case of arbitrary unitary dilations can easily be reduced to the case of minimal unitary dilations. We refer the reader to Sz.-Nagy and Foias [1], Ch. II, §2. We also recommend Foias and Frazho [1] for comprehensive information on commutant lifting. See also Sarason [7], where an approach due to Arocena [1] is presented that is based on ideas of Adamyan, Arov, and Krein.

6. Sz.-Nagy–Foias Functional Model Let T be a completely nonunitary contraction on a Hilbert space K, i.e., T is a contraction and T has no nonzero invariant subspace H ⊂ K such


715

that T H is a unitary operator on H. The defect operators DT and DT ∗ are defined by DT = (I − T ∗ T )1/2

and DT ∗ = (I − T T ∗ )1/2 .

Consider the defect subspaces def

DT = clos DT K

def

and DT ∗ = clos DT ∗ K.

Since T (I − T ∗ T ) = (I − T T ∗ )T , it follows that T DT = DT ∗ T , and so T DT ⊂ DT ∗ . The characteristic function ΘT of T is the analytic B(DT , DT ∗ )-valued operator function defined by ΘT (z) = − T + zDT ∗ (I − zT ∗ )−1 DT DT . Then ΘT is a purely contractive function, i.e., ΘT (ζ)xDT ∗ < xDT

for any ζ ∈ D

and any nonzero x ∈ DT .

Suppose now that D1 and D2 are Hilbert spaces and Θ is a purely contractive functions with values in B(D1 , D2 ). Consider the Hilbert space def

L = H 2 (D2 ) ⊕ clos Range ∆, where ∆ is multiplication by (I − Θ∗ Θ)1/2 on L2 (D1 ) (see Appendix 2.3 for the definition of the Hardy space H 2 (H) of H-valued functions). We define the operator T on the Hilbert space def

K = L {Θf ⊕ ∆f : f ∈ H 2 (D1 )} by T (f1 ⊕ f2 ) = PK (zf1 ⊕ zf2 ),

f1 ⊕ f2 ∈ K,

(A1.7)

where PK is the orthogonal projection onto K. Note that the operator f → Θf ⊕ ∆f on H 2 (D1 ) is an isometric imbedding of H 2 (D1 ) in L, and so {Θf ⊕ ∆f : f ∈ H 2 (D1 )} is a closed subspace of L. Then T is a completely nonunitary contraction. The operator V on L defined by V (f1 ⊕ f2 ) = zf1 ⊕ zf2 . is a minimal isometric dilation of T . By the Sz.-Nagy–Foias theorem, the characteristic function of the contraction T defined by (A1.7) coincides with Θ modulo constant unitary factors. The representation of a contraction T in the form (A1.7) is called the Sz.-Nagy–Foias functional model. If Θ is an inner operator function (see Appendix 2.3), the functional model looks simpler. Indeed, in this case ∆ = O, L = H 2 (D2 ), def

K = KΘ = H 2 (D2 ) Θ(D1 ), and T f = PKΘ zf,

f ∈ KΘ .

716


A completely nonunitary contraction T on a Hilbert space H has inner characteristic function if and only if lim T n x = 0

n→∞

for any x ∈ H.

Such contractions are called C0∗ -contractions. Note also that the characteristic function ΘT is unitary-valued (i.e., Θ is inner and Θ(ζ)Θ∗ (ζ) = I for almost all ζ ∈ T) if and only if T is a C00 -contraction, i.e., both T and T ∗ are C0∗ -contractions. For more information on functional models see Sz.-Nagy and Foias [1], and Nikol’skii and Vasyunin [1].

7. The Heinz Inequality Let A and B be self-adjoint operators on a Hilbert space H such that O ≤ A ≤ B, i.e., 0 ≤ (Ax, x) ≤ (Bx, x) for any x ∈ H. The Heinz inequality asserts that for any α ∈ (0, 1) Aα ≤ B α . We refer the reader to Birman and Solomyak [1], Ch. 10, §4, for the proof.

Appendix 2 Summary of Function Spaces

In this appendix we gather necessary information on function classes and give references for proofs and more details.

1. Hardy Classes H p spaces. Let 0 < p < ∞. The Hardy class H p consists of functions f analytic in D and such that

1/p def f p = sup |f (rζ)|p dm(ζ) < ∞. (A2.1) 0 0, then n f ∈ Bps+1/p + ⇐⇒ (S ∗ )2 f B 1/p ≤ const 2−ns , n ∈ Z+ . (A2.12) p

The proof of (A2.12) can be found in Peller and Khrushchëv [1], Lemma 4.4. Let us now describe the dual spaces to Besov spaces. Suppose that q < ∞. Then p q s ∗ , q = , (Bpq ) = Bp−s 1 ≤ p ≤ ∞, 1 ≤ q < ∞, s ∈ R, p = q , 1−p 1−q with respect to the pairing ˆ (f, ϕ) = fˆ(j)ϕ(j),

f ∈ P, ϕ ∈ Bp−s q ,

(A2.13)

j∈Z

i.e., the left-hand side is defined at least on the set of trigonometric polynos by continuity. Note that we take mials and extends to the whole space Bpq the complex conjugate of ϕ(j) ˆ for convenience; we could certainly write ϕ(j) ˆ instead of ϕ(j) ˆ in (A2.13). We also note that (bsp∞ )∗ = Bp−s 1,

1 ≤ p ≤ ∞, s ∈ R,

with respect to the same pairing (A2.13). In particular, (λs )∗ = B1−s ,

s ∈ R.

Appendix 2. Summary of Function Spaces

733

The Riesz projections P+ and P− defined by (A2.3) can also be defined in the same way on the space of distributions. It is easy to see from the s , 1 ≤ p ≤ ∞, definition (A2.9) that P+ is a bounded linear projection on Bpq s s 1 ≤ q ≤ ∞, s ∈ R, onto the subspace (Bpq )+ of analytic functions in Bpq . This subspace admits the following description. Let f be a function analytic s )+ if and only if in D. Then f ∈ (Bpq

1 (1 − r)q(n−s)−1 fr(n) qp dt < ∞, q < ∞, (A2.14) 0

sup (1 − r)n−s fr(n) p < ∞,

q = ∞,

0 0,    j≥0 j   fˆ(j) j I˜α f = α < 0, (−α) z ,   j≥0 Dj    f, α = 0. ˆ f (j)z j and Here f = j≥0 (α) def Dj =

j+α j

=

(α + 1)(α + 2) · · · (α + j) . j!

It is well known that (α) jα ≤ const(1 + j)−1 , D − j Γ(α + 1)

α > −1,

j ∈ Z+ ,

where Γ is the Γ function; see Zygmund [1], Ch. 3, §1. The last inequality together with (A2.15) implies that s s+α )+ = (Bpq )+ , 1 ≤ p, q ≤ ∞, α, s ∈ R. I˜α (Bpq

734


s The Besov classes Bpq , s > 0, admit an interesting description due to Dyn’kin in terms of pseudoanalytic continuation. A function f belongs s if and only if there exists a function ψ on the annulus to Bpq {ζ ∈ C : 1/2 < |ζ| < 2} such that

2

q/p |1 − r|q(1−s)−1 |ψ(rζ)|p dm(ζ) dr < ∞,

T

1/2

sup (1 − r)

1−s

1/p < ∞, |ψ(rζ)| dm(ζ)

and def

q = ∞, (A2.17)

p

T

1/2 1/p − 1 the classes Bps also admit a description in terms of pseudoanalytic continuation. However, this description has to be modified slightly. A function f belongs to Bps if and only if it admits a representation (A2.18) for a function ψ satisfying

1 − |ζ|p(1−s)−1 (ψ∗ (ζ))p dm2 (ζ) < ∞, {1/2 1, the boundary values of f on T coincide with f almost everywhere, and p

∂f 1 − |ζ|p(1−s)−1 (ζ) dm2 (ζ) < ∞. ∂ z¯ ∗ {1/2 1, the description of Bps in terms of pseudoanalytic continuation shows that if s > 1/p − 1, f ∈ Bps , and ϕ is a function analytic in a neighborhood of f (T), then ϕ ◦ f ∈ Bps and the seminorm of ϕ ◦ f in Bps modulo the constants admits the same estimate as in the


737

Banach case. Thus if f1 , f2 ∈ Bps ∩ L∞ , then f1 f2 ∈ Bps . In particular, for s ≥ 1/p the space Bps is an algebra with respect to pointwise multiplication. A similar description in terms of pseudoanalytic continuation can be applied for Besov classes Bps (R) of functions on R, s > 1/p − 1. We deal in 1/p

this book only with classes Bp (R), which are defined in §6.7. It follows 1/p from this description that if g ∈ BM O(R), then P − g ∈ Bp (R) if and 1/p only if P− (ψ ◦ ω) ∈ Bp , where 1+ζ , ζ ∈ D, ω(ζ) = i 1−ζ and P − g = (P− (g ◦ ω)) ◦ ω −1 (see Dyn’kin [2]). We refer the reader for more information on Besov classes to Peetre [1], Triebel [1], [2], Bergh and L¨ ofstr¨ om [1], S.M. Nikol’skii [1], Oswald [1], and for the theory of pseudoanalytic continuation to Dyn’kin [1] and [2]. The spaces Lsp of Bessel potentials, 1 < p < ∞, s ∈ R, are defined by def

Lsp = Is (Lp ). If s is a positive integer, Lsp is a classical Sobolev space. It is well known that Bps ⊂ Lsp for 1 ≤ p ≤ 2 and Lsp ⊂ Bps for 2 ≤ p < ∞. The spaces Lsp for s > 0 can also be described in terms of pseudoanalytic continuation. We do not state this description and refer the reader to Dyn’kin [1]. Using that description, one can show that if f1 and f2 are bounded functions in Lsp , s > 0, then f1 f2 ∈ Lsp . In particular, Lsp is a Banach algebra with respect to pointwise multiplication if s > 1/p. We refer the reader to Triebel [1], S.M. Nikol’skii [1], and Dyn’kin [1] for more information. We need the following description of the spaces I−α H p = {f : Iα f ∈ H p }, of analytic functions: ϕ ∈ I−α H p

⇐⇒

T

α > 0,

1 ≤ p < ∞, p/2

1

|ϕ(rζ)|2 (1 − r)2α−1 dr 0

(see Zygmund [1] and Triebel [2]).

dm(ζ) < ∞

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Author Index

Abakumov E.V. 550 Adams D.R. 378 Adamyan V.M. 56, 57, 84, 145, 147, 148, 228, 229, 377, 714 Akhiezer N.I. 56, 57, 267, 428 Aleksandrov A.B. vii, 297, 300, 377, 687, 703, 706 Alexeev R.B. vii, 376, 378, 563, 682–684 Anderson J.M. 296 Ando T. 59, 228 Arazy J. 266, 297, 301 Arlinskii Yu.M. 84 Arocena R. 58, 59, 427, 714 Arov D.Z. 56, 57, 84, 145, 147, 148, 228, 377, 714 Arsene G. 84 Arveson W.B. 122, 686, 712 Avniel Y.B. 413 Axler S. 56, 57, 59 Balakrishnan A.V. 487 Ball J.A. 145, 488, 683 Baratchart L. 145 Bart H. 487

Barth K.F. 296 Barton T.J. 301 Berg I.D. 57 Bergh J. 255, 256, 288, 290, 737 Bernstein S.N. 267, 274 Birman M.S. 39, 301, 713, 716 Bloomfield P. 413, 426 Bonjiorno Jr. J.J. 488 Bonsall F.F. 57, 58, 299 Bötcher A. 123, 301 Bourbaki N. 330 Bourgain J. 687, 703 Boˇzejko M. 685 Brannan D.A. 296 Brown A. 121 Brudnyi Yu.A. 298 Budyanu M.S. 377, 563 Butz J.R. 145 Carathéodory C. 56 Carleman T. 57, 450 Carleson L. 376 Carlson J.F. 686, 703 Chang S.-Y.A. 59 Chen G. 453, 487

Author Index

Chui C.K. 378, 453, 487 Clancey K. 563 Clark D.N. 56, 57, 122, 145, 298, 550, 686, 703 Coburn L.A. 121 Coifman R. 56, 296, 297 Connes 301 Cotlar M. 58, 59, 427 Cwikel M. 288 Dahleh M.A. 453, 487 Davidson K.R. 704 Davie A.M. 685 Davis C. 84, 413, 682, 684 de la Vallée Poussin 267 Desoer C.A. 487 Devinatz A. 57, 121, 122 Diaz-Bobillo I.J. 453, 487 Dolzhenko E.P. 298 Douglas R.G. 56, 57, 84, 121–123, 711 Doyle J.C. 453, 487, 488 Dunford N. 137, 141 Duren P. 724 Dym H. 413, 683, 728 Dyn’kin E.M. vii, 298, 734, 737 Fefferman C. 5, 56, 426, 427, 729 Fejér L. 56 Ferguson S.H. 703 Fisher S.D. 297 Foguel S.R. 685, 686 Foias C. 13, 84, 122, 487, 506, 508, 512, 686, 703, 713, 714, 716 Foo Y.K. 566 Francis B.A. 453, 487, 488 Frazier M. 378 Frazho A.E. 714 Friedman Y. 301 Fuhrmann P. 56, 122, 453, 487 Gaier D. 371 Gakhov F.D. 123 Gamelin T.W. 331 Gantmakher F.R. 445 Garnett J.B. 53, 82, 228, 724, 731

777

Gelfand I.M. 413 Gheondea A. 84 Gheorghe 298, 299, 377 Gikhman I.I. 389. 413 Gillespie T.A. 57 Girsanov I.V. 413 Glazman I.M. 428 Glover K. 473, 487, 566 Gohberg I. 122, 319, 377, 477, 487, 488, 563, 683, 706, 710, 711 Goluzin G.M. 9, 360 Gonchar A.A. 298 Grigoryan L.D. 298 Gu D.W. 566 Gutknekht M.H. 377 Halikias G.D. 566 Halmos P.R. 121, 192, 686 Hamburger H. 57 Hankel H. 56 Hartman P. 57, 122 Hayashi E. 377, 413 Helson H. 426, 726 Helton J.W. 145, 378, 453, 487, 550 Hoffman K. 724, 728 Holland F. 57, 296 Horowitz C. 297 Howland J. 296, 297, 450 Hunt R.A. 426 Ibragimov I.A. 413, 427, 428 Ismagilov R.S. 122 Ivanov O.A. 301 Jabr H.A. 488 Jackson D. 267, 388 Jacobs S. 376 Janson S. 288, 297–300, 413, 703 Jawerth B. 298 Jewell N. 57, 413, 426 John F. 729 Kaashoek M.A. 487 Kahan W.M. 84 Kahane J.-P. 324, 376

778

Author Index

Karadzhov 254 Khavin V.P. 377 Khavinson S.Ya. 56 Kheifets A.Y. vii, 228 Khrushchëv S.V. vii, 59, 122, 123, 145, 274, 376–378, 413, 426, 427, 550, 732 Kislyakov S.V. vii, 297, 704 Kolmogorov A .N. 413, 426 Koosis P. 378, 724, 731 Krein M.G. 56, 57, 84, 122, 145, 147, 148, 228, 296, 319, 377, 563, 706, 710, 711, 714 Krepkogorskii V.L 377 Kronecker L. 56 Kwakernaak H. 566 Kwapie´ n S. 266

Mitter S.K. 413 Montgomery-Smith S. 471, 487, 492 Muckenhoupt B. 426 Muskhelishvili N.I. 563

Lamperti J. 412 Lancaster P. 477 Landkof N.S. 374 Lax P. 376, 377, 413 Lebow A. 686 Le Merdy C. 145 Li X. 378 Limebeer D.J.N. 566 Lindenstrauss J. 266, 710 Linnik Yu.V. 413 Litvinchuk G.S. 122, 123 L¨ ofstr¨ om J. 255, 256, 288, 290, 737 Luecking D. 57

Ober R.J. 471, 487, 492, 550 Oswald P. 737

Magnus W. 450 Martinez-Avenda˜ no R.A. 451 Maruyama G. 413 Masani P. 413, 428 McCarthy C.A. 706 McCarthy J.E. 704 McKean H.P. 413, 728 Megretskii A.V. vii, 450, 451, 487, 550 Merino O. 378 Mikhlin S.G. 122 Milman M. 298

Nazarov F.L. 371 Nehari Z. 3, 56 Nelson E. 413 Neri U. 731 Newburg J.D. 450 Nevanlinna R. 56, 228 Nikol’skii N.K. vii, 13, 56, 59, 84, 121, 123, 327, 405, 550, 716, 724, 726, 728 Nikol’skii S.M. 737 Nirenberg L. 729

Page L. 84 Papadimitrakis M. 377, 378 Parfënov O.G. 300 Parrott S. 84 Partington J. 59 Paulsen V.I. 686, 704 Peetre J. 59, 277, 297–299, 703, 737 Pekarskii A.A. 297, 298, 473 Pe lczy´ nski A. 266 Peller V.V. 59, 122, 123, 145, 274, 296–298, 300, 301, 376–378, 413, 426–428, 451, 487, 550, 563, 682–687, 703, 732 Petrovic S. 703 Phillips R.S. 413 Pick G. 56 Pisier G. 687, 703, 704 Pitt L.D. 300 Plemelj J. 563 Poreda S.J. 377 Postlethwaite I. 566 Power S.C. 56, 57, 59, 297, 450 Prokhorov V.A. 146, 300

Author Index

Rajchman A. 372 Reed M. 290, 413, 549 Riesz F. 58, 382 Rochberg R. 56, 296–299 Rodman L. 477, 488 Rogosinski 56 Rosenblum 122, 296, 450, 724 Rotfel’d S.Yu. 706 Rozanov Yu.A. 412, 413, 427, 428, 557 Rudin W. 56 Sadosky 58, 59, 427 Salamon D. 487, 492 Sarason D. 14, 56, 57, 59, 84, 121, 122, 426, 427, 714, 729 Schubert J. 122 Schwartz D.F. 378 Schwartz J. 137, 141 Schwartz L. 47 Segal I.E. 413 Semmes S. 297, 298 Seyfert F. 146 Shamoyan F.A. 377 Shapiro H.S. 56, 376 Shields A. 57 Shmul’yan Yu.L. 84, 191 Silbermann B. 123, 301 Simon B. 290, 300, 413, 549, 706 Simonenko I.B. 122 Sj¨ odin T. 374 Skorokhod A.V. 389, 413 Solev V.N. 145, 427 Solomyak M.Z. 39, 301, 713, 716 Spitkovskii I.M. 122, 123 Stafney J. 703 Stein E.M. 12, 56, 244, 260 Svensson E. 297 Szegö G. 53, 56, 426 Sz.-Nagy B. 13, 58, 122, 382, 506, 508, 513, 685, 686, 713, 714, 716 Tannenbaum A. 84, 453, 487, 488 Tolokonnikov 122, 378, 724

779

Trefethen L.N. 377 Treil S.R. vii, 56, 84, 122, 145, 428, 451, 487, 550, 683, 684 Triebel H. 737 Tsafriri L. 710 Tsai M.C. 566 Tsekanovskii E.R. 84 Upmeier H. 300 Vasyunin V.I. vii, 84, 122, 550, 682, 684, 716 Vekua N.P. 563 Vershik A.M. 413 Vidyasagar M. 487 Volberg A. vii, 56, 59, 298, 301, 378, 428 von Neumann J. 685 Wallstén R. 300 Walsh D. 57, 299 Walsh J.L. 56, 299 Weinberger H.F. 84 Weiss G. 56, 244 Wheeden R.L. 426 Widom H. 57, 121, 122 Wiener N. 413, 428 Williams J.P. 686, 703 Wintner A. 121, 122 Woerdeman H.J. 550, 684 Wolff T. 724 Yafaev D.R. 84 Yaglom A.M. 413, 427 Yakubocich 122 Yanovskaya R.N. 84 Yosida K. 314 Youla D.C. 488 Young N.J. vii, 487, 563, 682, 683 Zames G. 487 Zhu K. 59 Zygmund A. 269, 321, 323, 324, 336, 444, 694, 719, 733, 737

Subject Index

Adamyan–Arov–Krein theorem 126 absolutely regular processes 397 admissible weight 592 Aronszajn–Donoghue theorem 549 asymptotically stable operator 522 asymptotically stable semigroup 502 averaging projection 258, 295

Blaschke product 721 Bloch space 270, 733 block Hankel matrix 66 BM O 5, 729 BM OA 5, 729 Bochner theorem 428 bounded mean oscillation 729 brothers Riesz theorem 718

backward shift realization 461 badly approximable function 305, 338 badly approximable matrix function 578 balanced unitary-valued matrix function 569 Besov classes 231, 233, 731 Bessel potentials spaces 322 best approximation by analytic functions 7, 303 best approximation by meromorphic functions 304 Beurling theorem 725 Blaschke–Potapov product 76 Blaschke conditiion 721

C ∗ -algebra 711 Calder´ on interpolation theorem 288 Calkin algebra 436, 711 canonical correlations 383, 396 canonical factorization 657 partial 656 canonical function 156 capacity 374 CAR sequence 698 Carathéodory–Fejér problem 24 Carleman classes 330 Carleman operator 54, 434, 440, 492 Carleson imbedding theorem 42, 83, 723 Carleson measure 42, 83, 723

Subject Index

vanishing 43, 723 characteristic function of a colligation 224 characteristic function of a contraction 225, 715 closed linear operator 39 cogenerator of a semi-group 512 colligation 217 simple 217 commutant lifting theorem 14, 72, 120, 714 commutator 11 complete ρ-extension 199 completely hereditary function space 618 completely nondeterministic processes 413 completely nonreducing invariant subspace 106, 727 completely polynomially bounded operator 686 completely regular stationary process 395, 421, 428 of order α 396, 421, 428 compressed shift 13, 72 conditional expectation 388 contraction 685, 713 completely nonunitary 714 controllability Gramian 459, 469, 499, 523 co-outer operator function 727 coprime factorization over H ∞ 476 left 476 right 476 coprime functions 15 corona condition 17 corona problem 617 corona theorem 17, 119, 724 coupling 218 covariance sequence 382 de la Vallée Poussin kernel 322 decent function space 312, 555, 558, 619 defect operators 62, 715

781

defect subspaces 715 deficiency indices 40, 153 degree of a rational function 19 distribution 47, 731 tempered 48 doubly coprime factorization over H ∞ 476 ergodicity 390 essential norm 25, 705 essential range 91 essential spectrum 91, 433 extended t-sequence 600, 678 Faa di Bruno formula 330 feedback controller 473 robust 475 feedback system 474 internally stable 475 Fefferman–Stein decomposition 376 Fejér kernel 719 Fock space 385 four block operator 84 four block problem 84 Fredholm operator 90, 711 Fredholm spectrum 530 Frostman theorem 722 future 390, 521 Garsia norm 37 Gaussian variable 381 Gaussian subspace 381 Gelfand–Naimark theorem 436, 711 generalized Matsaev ideals 261 generator of a semigroup 512 Gevrey class 423 H ∞ + C 25 ∞ H(m) 126 Hamburger moment problem 39 Hamburger theorem 39 Hankel matrix 3 generalized 281 Hankel operator 3 of order n 299 vectorial 66

782

Subject Index

Hankel operator on L2 (R+ ) 46 Hardy class 717 harmonic conjugate 720 Hartman theorem 27 Heinz inequality 716 Helson–Szegö condition 96 Helson–Szegö theorem 419 hereditary function space 303, 618 heredity problem 303 Hermite polynomials 387 Hilbert matrix 6, 36, 434 Hilbert–Schmidt class 709 Hilbert tensor product 384, 699 H¨ older–Zygmund spaces 731 impulse response function 468 impulse response sequence 459 index of a Fredholm operator 90, 711 independent random variables 381 inner function 721 singular 721 inner operator functon 726 inner-outer factorization 721, 727 intput 454, 457, 467 input space 457, 467 interpolating Blaschke product 724 interpolating sequence 724 interpolation error process 402 interpolation set 373 J-unitary matrix 205 J-unitary matrix function 205 Kolmogorov theorem 415, 720 K¨ othe space 306 Kronecker theorem 19 Laguerre polynomials 53 Laplace transform 456 left Fredholm operator 93 linear system 454, 457, 466 balanced 459, 469, 499, 527 controllable 459, 468 minimal 459, 468 observable 459, 468

output normal 462, 523 with continuous time 466, 499 with discrete time 457, 523 Littlewood–Paley theorem 241 local distance 33 Lorentz spaces 256 Lyapunov equations 502 Mm,n 105 majorization property 300 Marcinkiewicz theorem 240 mathematical expectation 380 Matsaev ideal 710 maximal function (radial) 719 McMillan degree 81 MIMO 454 minimal stationary process 402 mixing 391 uniform with respect to the past 392 uniform with respect to the past and the future 394 M¨ obius invariant spaces 297, 308 model matching problem 485 model reduction problem 472 Muckenhoupt condition 426 Nehari problem 7, 147, 565 Nehari–Takagi problem 126, 173, 662 Nehari Theorem 3 Nevanlinna–Pick problem 23, 170, 310 nonsingular rational matrix function 474 nuclear operators 709 observability Gramian 459, 469, 499, 523 one-step extension 188 canonical 194 operator Lipschitz functions 300 operator of best approximation 303 operator of superoptimal approximation 631 operator quasiball 190

Subject Index

optimal solution 147, 565 outer function 721 outer operator function 727 output 454, 457, 467 output space 457, 467 Paley theorem 444, 694 Paley–Wiener theorem 49 partial isometry 16 past 390, 521 p-Banach space 293 P C 35, 102 plant 454 causal 454 stable 454 time-invariant 454 Peetre K-functional 253 phase function 405, 428 piecewise continuous functions 35, 102 Poisson balayage 45, 730 Poisson kernel 719 conjugate 719 Poisson summation formula 244 Poisson transform 215 polar decomposition 142 polynomially bounded operator 686 power bounded operator 685 p-regular stationary process 396, 421, 428 of order α 402, 421, 428 proper rational matrix function 472 pseudoanalytic continuation 734 pseudocontinuation 513, 725 QC 101, 729 quasicontinuous functions 101, 729 random variable 380 r-balanced matrix function 569 real interpolation spaces 254 realization 458, 467 backward shift 461 balanced 459, 469, 499, 526 generalized 471 output normal 462

783

proper 471, 491 restricted backward shift 461 recovery problem for measures 371 recovery problem for unimodular functions 303, 309, 312 recovery problem for unitary-valued functions 553 regular stationary process 393, 427 reproducing kernel 37, 81 residual entry 587, 656 Riemann–Hilbert problem 122 Riesz–Herglotz theorem 58, 382, 406 Riesz projections 720 Riesz–Schauder theorem 314 Riesz theorem 720 right Fredholm operator 93 robust stabilization 475 R-space 306, 554, 619 Sarason theorem 14 scalar spectral measure 713 scattering function of a colligation 220 scattering function of a unitary extension 214 scattering system 407 Schatten–Lorentz classes 253, 710 Schatten–von Neumann classes 231, 239, 244, 706 Schmidt expansion 706 Schmidt pair 127 Schmidt vector 126 Schur multiplier 283 Schur product 245, 283 semiradii 190 shift operator 725 bilateral 725 unilateral 725 similarity to a contraction 685 singular values 126, 705 SISO 454, 458 spectral density of a stationary process 393, 427

784

Subject Index

spectral measure of a stationary process 383, 427 spectral multiplicity function 713 spectrum of an inner function 513, 722 stability test 508, 530 standard problem 487 state space 457, 467 stationary Gaussian process 382 vectorial 427 of full rank 427 with continuous time 428 strictly proper rational matrix function 474 suboptimal solution 147 superoptimal approximation by analytic matrix functions 566 superoptimal approximation by analytic operator functions 672 superoptimal approximation by meromorphic matrix functions 663 superoptimal singular values 566, 672 superoptimal solution of the Nehari problem 566 superoptimal solution of the Nehari– Takagi problem 663 superoptimal weight 593 symbol of a Hankel operator 6 symbol of a Toeplitz operator 89 symmetric operator 40 Szegö–Kolmogorov alternative 393, 728 Sz.-Nagy–Foias functional model 13, 715

Toeplitz operator 89 analytic 91 vectorial 103 trace 709 trace class 232, 709 transfer function 455, 458, 467

thematic factorization 588, 679 monotone 607 partial 587, 678 monotone 607 thematic indices 587, 678 thematic matrix function 569, 673 Toeplitz kernel 58 generalized 58 Toeplitz matrix 87

Zygmund class 731 Zygmund theorem 95, 720

unimodular function 10, 98, 142, 303 unitary dilation of a colligation 223 unitary dilation of a contraction 713 unitary extension 149, 213 unitary interpolant 636 vanishing mean oscillation 728 variance 380 very badly approximable matrix function 586 Vick transform 386 V M O 28, 728 V M OA 729 von Neumann integral 490, 712 wedge products 572 Wiener–Hopf factorization 111 Wiener–Hopf factorization indices 111 Wiener–Hopf operator on a finite interval 278 Wiener–Masani factorization 557 Wiener–Masani theorem 428, 557 winding number 98 of an invertible function in H ∞ + C 100

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An introduction to Hankel operators

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An Introduction to Hankel Operators (London Mathematical Society Student Texts)

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