While Hardy and Bergman spaces are established subjects in the literature on analytic functions, Fock spaces remain virgin territory. This book meets the need for a focused presentation of key results and techniques and includes new and simpler proofs.
Several natural Lp spaces of analytic functions have been widely studied in the past few decades, including Hardy spaces, Bergman spaces, and Fock spaces. The terms "Hardy spaces" and "Bergman spaces" are by now standard and well established. But the term "Fock spaces" is a different story.
Numerous excellent books now exist on the subject of Hardy spaces. Several books about Bergman spaces, including some of the author's, have also appeared in the past few decades. But there has been no book on the market concerning the Fock spaces. The purpose of this book is to fill that void, especially when many results in the subject are complete by now. This book presents important results and techniques summarized in one place, so that new comers, especially graduate students, have a convenient reference to the subject.
This book contains proofs that are new and simpler than the existing ones in the literature. In particular, the book avoids the use of the Heisenberg group, the Fourier transform, and the heat equation. This helps to keep the prerequisites to a minimum. A standard graduate course in each of real analysis, complex analysis, and functional analysis should be sufficient preparation for the reader.
Preface.- Chapter 1. Preliminaries.- Chapter 2. Fock Spaces.- Chapter 3. The Berezin Transform and BMO.- Chapter 4. Interpolating and Sampling Sequences.- Chapter 5. Zero Sets for Fock Spaces.- Chapter 6. Toeplitz Operators.- Chapter 7. Small Hankel Operators.- Chapter 8. Hankel Operators.- References.- Index
From the reviews:
"Excellent books exist in the literature on the theory of Hardy spaces ... but no textbook concerning the theory of Fock spaces has appeared before. The purpose of the author is to fill this gap and provide to any researcher in the field or graduate students the appropriate place to find the results or the bibliographical references needed for their use. ... author succeeds with his goal. ... a great addition to the literature and in the future will become a classic in the field." (Jordi Pau, Mathematical Reviews, January, 2013)
"This book is intended to provide a convenient reference to Fock spaces. ... 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." (Atsushi Yamamori, Zentralblatt MATH, Vol. 1262, 2013)
Über den Autor
Kehe Zhu is a professor of mathematics at the State University of New York at Albany. His research areas include operators on holomorphic function spaces, complex analysis, and operator theory and operator algebras.
Preface.- Chapter 1. Preliminaries.- Chapter 2. Fock Spaces.- Chapter 3. The Berezin Transform and BMO.- Chapter 4. Interpolating and Sampling Sequences.- Chapter 5. Zero Sets for Fock Spaces.- Chapter 6. Toeplitz Operators.- Chapter 7. Small Hankel Operators.- Chapter 8. Hankel Operators.- References.- Index.
Fills the gap in existing literature concerning the natural Lp spaces of analytic functions
First book on the market concerning Fock spaces, summarizing the most important results and techniques in one place, so that new comers, especially graduate students, have a convenient reference to the subject
Features new and simpler proofs than the existing ones in the literature
Includes exercises of various levels at the end of every chapter
Contains an extensive bibliography