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2003 | Buch

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Hankel Operators and Their Applications

verfasst von: Vladimir Peller

Verlag: Springer New York

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Professional Book Archive

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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 func tions. Hankel operators can be defined as operators having infinite Hankel matrices (i. e. , matrices with entries depending only on the sum of the co ordinates) 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 Han kel matrices was obtained by Kronecker, who characterized Hankel matri ces 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 (0,00), 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 realiza tions of Hankel operators and important connections of Hankel operators with the spaces BMa and V MO, Sz. -Nagy-Foais functional model, re producing kernels of the Hardy class H2, moment problems, and Carleson imbedding operators.

Inhaltsverzeichnis

Frontmatter

1. An Introduction to Hankel Operators

Abstract

In this introductory chapter we define the Hankel operators and study their basic properties. We introduce in §1 the class of Hankel operators as operators with matrices of the form $ {\left\{ {{\alpha _{i}} + k} \right\}_{{j,k}}} \geqslant 0$ and consider different realizations of such operators. One of the most important realization is the Hankel operators H_φ, from the Hardy class $ H_{-}^{2}\mathop{ = }\limits^{{def}} {L^{2}} \odot {H^{2}} $. We prove the fundamental Nehari theorem, which describes the bounded Hankel operators, and we discuss the problem of finding symbols of minimal norm. We introduce the important Hilbert matrix, prove its boundedness, and estimate its norm. Then we study Hankel operators with unimodular symbols. We conclude §1 with the study of commutators of multiplication operators with the Riesz projection on L² and reduce the study of such commutators to the study of Hankel operators.

Vladimir Peller

2. Vectorial Hankel Operators

Abstract

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 $ {\left\{ {{\Omega _{j}}_{{ + k}}} \right\}_{{j,k}}} \geqslant 0$, where the Ω_j 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 C for the ex- istence a Hilbert space operator Z such that the block matrix $ \left( {\begin{array}{*{20}{c}} A & B \\ C & Z \\\end{array} } \right)$ 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.

Vladimir Peller

3. Toeplitz Operators

Abstract

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 $ {\left\{ {{t_{{j - k}}}} \right\}_{{j,k}}} \geqslant 0$.

Vladimir Peller

4. Singular Values of Hankel Operators

Abstract

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 s_n(Γ) of r, there is no need to consider all operators of rank at most n, s_n(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 s_n (Γ) is greater than the essential norm of r. 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 s_n(Γ) > ||Γ||e, 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.

Vladimir Peller

5. Parametrization of Solutions of the Nehari Problem

Abstract

For a Hankel operator Γ from H² to H² and p ≥ ||Γ||, we consider in this section the problem of describing all symbols φ ∈ L^∞ of Γ (i.e., Γ = H_φ) which satisfy the inequality ||φ||∞ ≤ p. If φ₀ is a symbol of F, 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||_∞. ≤ p. This problem is called the Nehari problem. If p = ||Γ||, a solution yo of the Nehari problem (i.e., a symbol φ of Γ of norm at most φ is called optimal. If p > ||Γ||, the solutions of the Nehari problem are called suboptimal). Clearly, the optimal solutions of the Nehari problem are the symbols of F of minimal norm.

Vladimir Peller

6. Hankel Operators and Schatten—von Neumann Classes

Abstract

In this chapter we study Hankel operators that belong to the Schattenvon 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 B_p^1/p (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 of the condition φ ∈ B _p ^1/p , 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.

Vladimir Peller

7. Best Approximation by Analytic and Meromorphic Functions

Abstract

Let φ be a function on 𝕋 of class BMO. As we have already discussed in Chapter 1, there exists a function f ∈ BMOA such that φ — f ∈L^∞(𝕋) and

$$\left\| {\varphi - f} \right\|{L^{\infty }} = \left\| {{H_{\varphi }}} \right\| $$

(0.1)

Such a function f is called a best approximation to φ by analytic functions in L^∞. We have already seen in §1.1 that in general a best approximation is not unique (see Theorem 5.1.5, which gives a necessary and sufficient condition for uniqueness in terms of the Hankel operator Hφ).

Vladimir Peller

8. An Introduction to Gaussian Spaces

Abstract

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 Ith’s method of stochastic integrals.

Vladimir Peller

9. Regularity Conditions for Stationary Processes

Abstract

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 a, 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 Khrushch6v [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.

Vladimir Peller

10. Spectral Properties of Hankel Operators

Abstract

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.

Vladimir Peller

11. Hankel Operators in Control Theory

Abstract

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].

Vladimir Peller

12. The Inverse Spectral Problem for Self-Adjoint Hankel Operators

Abstract

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.

Vladimir Peller

13. Wiener—Hopf Factorizations and the Recovery Problem

Abstract

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

$$ P\_U \in X \Rightarrow U \in X $$

Vladimir Peller

14. Analytic Approximation of Matrix Functions

Abstract

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^∞(𝕄_m,n) its L^∞ norm ||Φ||L^∞ is, by definition,

$$ {\left\| \Phi \right\|_{{L\infty }}} = ess\mathop{{\sup }}\limits_{{\zeta \in T}} {\left\| {\Phi \left( \zeta \right)} \right\|_{{{M_{{m,n}}}}}} $$

(1)

Vladimir Peller

15. Hankel Operators and Similarity to a Contraction

Abstract

In this chapter Hankel operators are used to solve the problem of whether each polynomially bounded operator on Hilbert space is similar to a con-traction. This problem has a long history.

Vladimir Peller

Backmatter

Titel: Hankel Operators and Their Applications
verfasst von: Vladimir Peller
Verlag: Springer New York
Electronic ISBN: 978-0-387-21681-2
Print ISBN: 978-1-4419-3050-7
DOI: https://doi.org/10.1007/978-0-387-21681-2

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. An Introduction to Hankel Operators

2. Vectorial Hankel Operators

3. Toeplitz Operators

4. Singular Values of Hankel Operators

5. Parametrization of Solutions of the Nehari Problem

6. Hankel Operators and Schatten—von Neumann Classes

7. Best Approximation by Analytic and Meromorphic Functions

8. An Introduction to Gaussian Spaces

9. Regularity Conditions for Stationary Processes

10. Spectral Properties of Hankel Operators

11. Hankel Operators in Control Theory

12. The Inverse Spectral Problem for Self-Adjoint Hankel Operators

13. Wiener—Hopf Factorizations and the Recovery Problem

14. Analytic Approximation of Matrix Functions

15. Hankel Operators and Similarity to a Contraction

Backmatter