Hankel Operators and Their Applications

1. Front Matter

   Pages i-xv

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2. An Introduction to Hankel Operators

   • Vladimir Peller

   Pages 1-59

3. Vectorial Hankel Operators

   • Vladimir Peller

   Pages 61-85

4. Toeplitz Operators

   • Vladimir Peller

   Pages 87-123

5. Singular Values of Hankel Operators

   • Vladimir Peller

   Pages 125-146

6. Parametrization of Solutions of the Nehari Problem

   • Vladimir Peller

   Pages 147-229

7. Hankel Operators and Schatten—von Neumann Classes

   • Vladimir Peller

   Pages 231-301

8. Best Approximation by Analytic and Meromorphic Functions

   • Vladimir Peller

   Pages 303-378

9. An Introduction to Gaussian Spaces

   • Vladimir Peller

   Pages 379-413

10. Regularity Conditions for Stationary Processes

    • Vladimir Peller

    Pages 415-429

11. Spectral Properties of Hankel Operators

    • Vladimir Peller

    Pages 431-451

12. Hankel Operators in Control Theory

    • Vladimir Peller

    Pages 453-488

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

    • Vladimir Peller

    Pages 489-551

14. Wiener—Hopf Factorizations and the Recovery Problem

    • Vladimir Peller

    Pages 553-563

15. Analytic Approximation of Matrix Functions

    • Vladimir Peller

    Pages 565-684

16. Hankel Operators and Similarity to a Contraction

    • Vladimir Peller

    Pages 685-704

17. Back Matter

    Pages 705-784

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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.

• approximation theory
• functional analysis
• operator
• operator theory

From the reviews:

"A transformation on the sequence space l² is called a Hankel operator if its matrix entries depend only on the sum of the indices. … This book gives a systematic account of the tremendous development, the theory underwent in the last half century. … Each chapter ends with Concluding Remarks, providing historical background … . This volume will certainly serve as a standard reference on Hankel operators. It can be warmly recommended to graduate students as well as experts in analysis and operator theory." (L. Kérchy, Acta Scientiarum Mathematicarum, Vol. 71, 2005)

"There have already been several short texts and survey articles written about Hankel operators … . However, this is the first major monograph on the subject, and it covers far more ground. … Indeed, this lengthy book contains a huge amount of interesting material … . this is a very clear and well-written book, which will be a major source of reference for many years to come." (J.R. Partington, Proceedings of the Edinburgh Mathematical Society, Issue 47, 2004)

"The present monograph, comprising of almost 800 pages, is an overwhelmingly concise and lucid exposition … . There are several older good books on Hankel operators … but none of them can probably match this excellent monograph in terms of depth of the treatment and of amount of the included material relevant to Hankel operators and their applications. The book has a very good chance of becoming one of the standard references on Hankel operators for the next decades." (Miroslav Engliš, Zentralblatt MATH, Vol. 1030, 2004)

"The book … is a comprehensive account (704 pages of text, plus two appendices) of Hankel operators and their applications. … The exposition is systematic and carefully written. … This book is an impressive achievement. It is wide in scope … . is an excellent source of information on Hankel operators and their applications. It is a very welcome addition to the mathematical literature, which both experts and nonexperts will surely find extremely useful." (Harry Dym, Mathematical Reviews, 2004e)

"The book under review is devoted to the study of Hankel operators and their applications to a variety of problems. … . The book itself is extremely well written. It is encyclopaedic in its coverage, and clear and crisp in its exposition. The two appendices alone are worth the purchase price … . I would say that any mathematically inclined practitioner of linear control theory would definitely benefit from reading this book." (M. Vidyasagar, IEEE Transactions on Automatic Control, Vol. 51 (2), February, 2006)

“Hankel operators are one of the most important operators that have intrigued the greatest mathematicians and engineers in the last one and a half century. The book is definitely a standard reference work on these operators for the next century to come. … Two appendices on operator theory and function spaces are helpful and informative. The text is accessible to a broad readership. … must for every mathematics library and many researchers and system theorists will like to have a personal copy on their shelf.” (Adhemar Bultheel, Bulletin of the Belgian Mathematical Society, 2007)

• Department of Mathematics, Michigan State University, East Lansing, USA

  Vladimir Peller

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