* An Introduction to Hankel Operators * Vectorial Hankel Operators * Toeplitz Operators * Singular Values of Hankel Operators * Parametrization of Solutions of the Nehari Problem * Hankel Operators and Schatten-von Neumann Classes * Best Approximation by Analytic and Meromorphic Functions * An Introduction to Gaussian Spaces * Regularity Conditions for Stationary Processes * Spectral Properties of Hankel Operators * Hankel Operators in Control Theory * The Inverse Spectral Problem for Self-Adjoint Hankel Operators * Wiener-Hopf Factorizations and the Recovery Problem * Analytic Approximation of Matrix Functions * Hankel Operators and Similarity to a Contraction * Appendix I * Appendix II * References * Author Index * Subject Index
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.
Together with Toeplitz operators Hankel operators form one of the most important classes of operators on spaces of analytic functions. This book contains material which has appeared in scattered papers but never before has it been published in book form. In addition, the author has managed to unify the material and provide a consistent notation. In some cases he has even simplified the original proofs of theorems, as is often possible as a subject matures.