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**Theory of Bergman spaces.**
*(English)*
Zbl 0955.32003

Graduate Texts in Mathematics. 199. New York, NY: Springer. ix, 286 p. (2000).

A function \(f\) that is analytic in the unit disk \(D\) is said to belong to the (weighted) Bergman space \(A^p_\alpha\) (where \(1<p<\infty\), \(-1<\alpha<\infty\)) provided \(f\) is in \(L^p(D, dA_\alpha)\), where \(dA_\alpha=(\alpha +1)(1-|z |^2)^\alpha dA(z)\). The theory of Bergman spaces experienced three main phases of development during the last three decades.

The early 1970’s marked the beginning of function theoretic studies in these spaces. An influential presentation of the situation up to the mid 1970’s was A. L. Shields’ survey paper “Weighted shift operators and analytic function theory” [Topics Oper. Theory, Math. Surveys 13, 49-128 (1974; Zbl 0303.47021)].

The 1980’s saw the thriving of operator theoretic studies related to Bergman spaces. The achievements of this period were presented in K. Zhu’s book [Operator Theory in Function Spaces, Marcel Dekker, New York (1990; Zbl 0706.47019)].

The research on Bergman spaces in the 1990’s resulted in several breakthroughs, both function theoretic and operator theoretic. This book, by three major contributors to the field, is about the latest developments, mostly achieved in the 1990’s.

The book consists of nine chapters. Each chapter ends with a section called Notes and another called Exercises and Further Results. The former sections contain brief historical comments and direct the reader to the orginal sources for the material in the text. The latter ones contain a series of exercises accompanied by references or detailed suggestions. They range from the routine to the challenging. It would be quite suitable for graduate students in the field.

Topics by chapter are as follows. Chapter 1, Bergman spaces, introduces the Bergman spaces and concentrates on the general aspects of these spaces, such as Bergman kernel, Bergman projection and dual space of the Bergman spaces for small exponents. Chapter 2, The Berezin transform, shows how it is related to harmonic functions, Carleson-type measure and a BMO type space. Chapter 3, \(A^p\)-inner functions, introduces the notion of \(A^p_\alpha\)-inner functions and discusses several related topics such as extremal problem, the expansive multiplier properties of \(A^p\)-inner functions, the inner-outer factorization theorem for \(A^p\) and an analogue of Carathéodory-Schur theorem. Chapter 4, Zero sets, studies the zero sets of functions in several Bergman-type spaces, obtains sharp necessary conditions for a sequence to be a zero set for \(A^p_\alpha\), and sharp sufficient conditions as well. Chapter 5, Interpolation and sampling, contains the characterization of interpolation sequences in terms of an upper density and the characterization of sampling sequences in terms of a lower density. Chapter 6, Invariant subspaces, treats several problems related to invariant subspaces of Bergman spaces, e.g., an analogue to the classical Beurling’s theorem on invariant subspaces of the Hardy space. Chapter 7, Cyclicity, concerns the problem of characterizing the cyclic vectors in Bergman spaces \(A^p\). Chapter 8, Invertible noncyclic functions, contains the construction of invertible functions in \(A^p\) that are not cyclic there. Chapter 9, Logarithmically subharmonic weights, studies the weighted Bergman spaces for weights that are logarithmically subharmonic and reproduce for the origin.

The early 1970’s marked the beginning of function theoretic studies in these spaces. An influential presentation of the situation up to the mid 1970’s was A. L. Shields’ survey paper “Weighted shift operators and analytic function theory” [Topics Oper. Theory, Math. Surveys 13, 49-128 (1974; Zbl 0303.47021)].

The 1980’s saw the thriving of operator theoretic studies related to Bergman spaces. The achievements of this period were presented in K. Zhu’s book [Operator Theory in Function Spaces, Marcel Dekker, New York (1990; Zbl 0706.47019)].

The research on Bergman spaces in the 1990’s resulted in several breakthroughs, both function theoretic and operator theoretic. This book, by three major contributors to the field, is about the latest developments, mostly achieved in the 1990’s.

The book consists of nine chapters. Each chapter ends with a section called Notes and another called Exercises and Further Results. The former sections contain brief historical comments and direct the reader to the orginal sources for the material in the text. The latter ones contain a series of exercises accompanied by references or detailed suggestions. They range from the routine to the challenging. It would be quite suitable for graduate students in the field.

Topics by chapter are as follows. Chapter 1, Bergman spaces, introduces the Bergman spaces and concentrates on the general aspects of these spaces, such as Bergman kernel, Bergman projection and dual space of the Bergman spaces for small exponents. Chapter 2, The Berezin transform, shows how it is related to harmonic functions, Carleson-type measure and a BMO type space. Chapter 3, \(A^p\)-inner functions, introduces the notion of \(A^p_\alpha\)-inner functions and discusses several related topics such as extremal problem, the expansive multiplier properties of \(A^p\)-inner functions, the inner-outer factorization theorem for \(A^p\) and an analogue of Carathéodory-Schur theorem. Chapter 4, Zero sets, studies the zero sets of functions in several Bergman-type spaces, obtains sharp necessary conditions for a sequence to be a zero set for \(A^p_\alpha\), and sharp sufficient conditions as well. Chapter 5, Interpolation and sampling, contains the characterization of interpolation sequences in terms of an upper density and the characterization of sampling sequences in terms of a lower density. Chapter 6, Invariant subspaces, treats several problems related to invariant subspaces of Bergman spaces, e.g., an analogue to the classical Beurling’s theorem on invariant subspaces of the Hardy space. Chapter 7, Cyclicity, concerns the problem of characterizing the cyclic vectors in Bergman spaces \(A^p\). Chapter 8, Invertible noncyclic functions, contains the construction of invertible functions in \(A^p\) that are not cyclic there. Chapter 9, Logarithmically subharmonic weights, studies the weighted Bergman spaces for weights that are logarithmically subharmonic and reproduce for the origin.

Reviewer: Lou Zengjian (Canberra)

### MSC:

32A36 | Bergman spaces of functions in several complex variables |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

47A15 | Invariant subspaces of linear operators |