Hankel Operators and Their Applications
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This book is a systematic presentation of the theory of Hankel operators. It covers the many different areas of Hankel operators and presents a broad range of applications, such as approximation theory, prediction theory, and control theory. The author has gathered the various aspects of Hankel operators and presents their applications to other parts of analysis. This book contains numerous recent results which have never before appeared in book form. The author has created a useful reference tool by pulling this material together and unifying it with a consistent notation, in some cases even simplifying the original proofs of theorems. Hankel Operators and their Applications will be used by graduate students as well as by experts in analysis and operator theory and will become the standard reference on Hankel operators. Vladimir Peller is Professor of Mathematics at Michigan State University. He is a leading researcher in the field of Hankel operators and he has written over 50 papers on operator theory and functional analysis.
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.
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