Analysis on Fock spaces / Kehe Zhu

Auteur: Zhu, Kehe (1961-) - AuteurType de document: Livre numériqueCollection: Graduate texts in mathematics ; 263Langue: anglaisÉditeur: Berlin : Springer, cop. 2012 ISBN: 9781441988003 Note: As explained in the preface, this book is intended to provide a convenient reference to Fock spaces. The book begins with preliminary results about entire functions, lattices in the complex plane, pseudodifferential operators, and the Heisenberg group. The reader who is familiar with these materials may skip this chapter. The second chapter introduces the basics of the Fock space: reproducing kernel, integral representation, duality, complex interpolation, atomic decomposition, translation invariance and a version of the maximum principle. The third chapter is devoted to the study of the Berezin transformation and bounded mean oscillation (BMO) on the complex plane. Since the Berezin transform is closely related to the notion of Carleson measures, the discussion of Fock-Carleson measures is also included in this chapter. The fourth chapter is devoted to the study of characterizations of interpolating and sampling sequences for the Fock space. In the fifth chapter, the author studies zero sets for the Fock space. The remaining chapters (Chapter 6–8) are devoted to the study of three kinds of operators: Toeplitz, small Hankel, and Hankel operators. Each chapter ends with a series of exercises. The material is presented in a pedagogical way. The reference list contains 259 relevant items. This book is well written and it is a good reference for graduate students who are interested in Fock spaces. (zbMath) Sujets MSC: 30H20 Functions of a complex variable -- Spaces and algebras of analytic functions -- Bergman spaces, Fock spaces
30D15 Functions of a complex variable -- Entire and meromorphic functions, and related topics -- Special classes of entire functions and growth estimates
46E20 Functional analysis -- Linear function spaces and their duals -- Hilbert spaces of continuous, differentiable or analytic functions
47B10 Operator theory -- Special classes of linear operators -- Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.)
47B35 Operator theory -- Special classes of linear operators -- Toeplitz operators, Hankel operators, Wiener-Hopf operators
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As explained in the preface, this book is intended to provide a convenient reference to Fock spaces.

The book begins with preliminary results about entire functions, lattices in the complex plane, pseudodifferential operators, and the Heisenberg group. The reader who is familiar with these materials may skip this chapter.

The second chapter introduces the basics of the Fock space: reproducing kernel, integral representation, duality, complex interpolation, atomic decomposition, translation invariance and a version of the maximum principle.

The third chapter is devoted to the study of the Berezin transformation and bounded mean oscillation (BMO) on the complex plane. Since the Berezin transform is closely related to the notion of Carleson measures, the discussion of Fock-Carleson measures is also included in this chapter.

The fourth chapter is devoted to the study of characterizations of interpolating and sampling sequences for the Fock space.

In the fifth chapter, the author studies zero sets for the Fock space.

The remaining chapters (Chapter 6–8) are devoted to the study of three kinds of operators: Toeplitz, small Hankel, and Hankel operators.

Each chapter ends with a series of exercises. The material is presented in a pedagogical way. The reference list contains 259 relevant items. This book is well written and it is a good reference for graduate students who are interested in Fock spaces. (zbMath)

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