Appearing as volume 138 in the Mathematical Surveys and Monographs series of the American Mathematical Society, Zhu’s book is an update with considerable improvements and additions over the first edition .
The book is useful to both research mathematicians and graduate students working in operator theory and complex analysis. Any graduate student familiar with the standard graduate courses in real analysis, complex analysis, and functional analysis will be able to read the text and use it for an independent study or research work. The text can even be used in an advanced graduate course.
The text introduces bounded operators on Banach and Hilbert Spaces, as well as compact and Schatten Class operators. It then goes on to talk about Interpolation of Banach Spaces and Integral Operators on Lp Spaces. These are followed by chapters on Bergman, Bloch and Besov Spaces, and the Hardy Space. These spaces are the setting of the main part of the book: the study of the Berezin Transform, Toeplitz and Hankel Operators on the Bergman Space, Hankel Operators on the Hardy Space, and Composition Operators.
The book contains a complete bibliography that can be used for deeper and further study. Each chapter contains detailed proofs as well as “notes” with precise references to the bibliography and a set of exercises, some of which also with bibliographic references.
The text has all the characteristics of a very good book: it is mostly self-contained, has very detailed proofs and exercises, includes most of the updated results, and has an extensive bibliography and references to the development and the current stage of research in most of the included topics. For the young researchers it is also of interest to note the exercises, as well as suggestions for future directions of research.
Many research mathematicians and interested students will find this book not only very interesting, but also very useful.
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.