##
**Composition operators on spaces of analytic functions.**
*(English)*
Zbl 0873.47017

Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. xii, 388 p. (1995).

This book treats a fascinating topic on the borderline between complex function theory, functional analysis, and operator theory: given an analytic function \(\varphi\) on the unit disc (or a somewhat more general domain), describe the “interaction” between the (analytical and geometrical) properties of \(\varphi\) and the (analytical and topological) properties of the composition operator \(C_\varphi(f)= f\circ\varphi\) induced by \(\varphi\). To give an idea of the contents of the book, let us cite from the authors’ well-written Preface.

“The introductory chapter, as its title implies, sets the stage for the remainder of the book by giving the basic definitions, proving a few theorems that hold in very great generality, and posing the basic questions that will be addressed. The second chapter defines the Hardy and Bergman spaces and their generalizations that we will be working in and develops the analytic tools that are not usually covered in basic graduate courses but are needed in the study of composition operators. Readers familiar with this material may wish to skim the chapter to pick up our notation and see what we consider the basic background for our study. The third chapter contains the core material on boundedness and compactness of composition operators and estimates for their norms. Many of these computations are based on estimates arising from Carleson type measure considerations. In general, the emphasis will be on the standard spaces of analytic functions, but in chapters four and five we discuss smaller and larger spaces of analytic functions and illustrate the differences between composition operators on these spaces and the standard spaces. While the majority of the theory develops in parallel ways in one and several variables, some more subtle phenomena specific to the study of compactness and boundedness questions in several variables are investigated in chapter six. In chapter seven, computation of the spectra of composition operators is described. This description is most complete in the case of compact operators. In the case of non-compact operators, the theory is more complete in the cases in which a weighted shift analogy can be used, and less complete when the weighted shift analogy fails. For compact and invertible operators, the spectral theory is developed in one and several variables, but for the more difficult cases, we consider the theory only on one variable Hardy space and its close relatives. It turns out that composition operators are rarely normal, subnormal, or hyponormal; some results concerning such phenomena are described in chapter eight. Chapter nine consists of several sections devoted to less developed parts of the theory, such as results on equivalences, the topological structure of the space of composition operators, and an application of composition operators to a problem in polynomial approximation. After chapter three, the chapters are largely independent of each other, although chapters four and five on small and large spaces are best appreciated as a package. The results of the last three chapters are largely restricted to Hilbert spaces and especially \(H^2(D)\), while the earlier chapters include many results on Banach spaces.”

As the impressive list of references at the end shows, there is a vast literature on composition operators on spaces of analytic functions. Unfortunately, many results are scattered in research papers which sometimes are not easily accessible. It seems therefore very useful to report the state-of-the-art from time to time in book form; this has been done so far in Chapter 10 of the book “Operator theory in function spaces” by K. Zhu [New York (1990; Zbl 0706.47019)], and in the monograph “Composition operators and classical function theory” by J. H. Shapiro [New York (1993; Zbl 0791.30033)]. The present book is without doubt another excellent contribution, written by two of the leading specialists in the field. The choice of the material is convincing and coherent and the presentation is clear and suggestive. Each chapter closes with a list of carefully chosen exercises and some bibliographical remarks. Summarizing, the book may be highly recommended to anybody who is interested in complex analysis, function spaces, operator theory, and spectral theory.

According to their last statement, the authors “hope that others will find the reading of this book to be as stimulating as we have found the writing to be”. The reviewer is deeply convinced that this wish will come true, and that the book will be warmly accepted.

“The introductory chapter, as its title implies, sets the stage for the remainder of the book by giving the basic definitions, proving a few theorems that hold in very great generality, and posing the basic questions that will be addressed. The second chapter defines the Hardy and Bergman spaces and their generalizations that we will be working in and develops the analytic tools that are not usually covered in basic graduate courses but are needed in the study of composition operators. Readers familiar with this material may wish to skim the chapter to pick up our notation and see what we consider the basic background for our study. The third chapter contains the core material on boundedness and compactness of composition operators and estimates for their norms. Many of these computations are based on estimates arising from Carleson type measure considerations. In general, the emphasis will be on the standard spaces of analytic functions, but in chapters four and five we discuss smaller and larger spaces of analytic functions and illustrate the differences between composition operators on these spaces and the standard spaces. While the majority of the theory develops in parallel ways in one and several variables, some more subtle phenomena specific to the study of compactness and boundedness questions in several variables are investigated in chapter six. In chapter seven, computation of the spectra of composition operators is described. This description is most complete in the case of compact operators. In the case of non-compact operators, the theory is more complete in the cases in which a weighted shift analogy can be used, and less complete when the weighted shift analogy fails. For compact and invertible operators, the spectral theory is developed in one and several variables, but for the more difficult cases, we consider the theory only on one variable Hardy space and its close relatives. It turns out that composition operators are rarely normal, subnormal, or hyponormal; some results concerning such phenomena are described in chapter eight. Chapter nine consists of several sections devoted to less developed parts of the theory, such as results on equivalences, the topological structure of the space of composition operators, and an application of composition operators to a problem in polynomial approximation. After chapter three, the chapters are largely independent of each other, although chapters four and five on small and large spaces are best appreciated as a package. The results of the last three chapters are largely restricted to Hilbert spaces and especially \(H^2(D)\), while the earlier chapters include many results on Banach spaces.”

As the impressive list of references at the end shows, there is a vast literature on composition operators on spaces of analytic functions. Unfortunately, many results are scattered in research papers which sometimes are not easily accessible. It seems therefore very useful to report the state-of-the-art from time to time in book form; this has been done so far in Chapter 10 of the book “Operator theory in function spaces” by K. Zhu [New York (1990; Zbl 0706.47019)], and in the monograph “Composition operators and classical function theory” by J. H. Shapiro [New York (1993; Zbl 0791.30033)]. The present book is without doubt another excellent contribution, written by two of the leading specialists in the field. The choice of the material is convincing and coherent and the presentation is clear and suggestive. Each chapter closes with a list of carefully chosen exercises and some bibliographical remarks. Summarizing, the book may be highly recommended to anybody who is interested in complex analysis, function spaces, operator theory, and spectral theory.

According to their last statement, the authors “hope that others will find the reading of this book to be as stimulating as we have found the writing to be”. The reviewer is deeply convinced that this wish will come true, and that the book will be warmly accepted.

Reviewer: J.Appell (Würzburg)

### MSC:

47B38 | Linear operators on function spaces (general) |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

47A10 | Spectrum, resolvent |