Hedenmalm H., Korenblum B., Zhu K. Theory of Bergman Spaces

Springer, 2000. — 299p. — (Graduate Texts in Mathematics 199). — ISBN 978-1-461267897.
Теория пространств БергманаFifteen years ago, most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely, yet today the situation has completely changed. For several years, research interest and activity have expanded in this area and there are now rich theories describing the Bergman spaces and their operators. This book is a timely treatment of the theory, written by three of the major players in the field.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. Substantial progress was made by Horowitz and Korenblum, among others, in the areas of zero sets, cyclic vectors, and invariant subspaces. An influential presentation of the situation up to the mid 1970 's was Shields' survey paper "Weighted shift operators and analytic function theory".
The 1980's saw the thriving of operator theoretic studies related to Bergman spaces. The contributors in this period are numerous; their achievements were presented in Zhu's 1990 book "Operator Theory in Function Spaces". The research on Bergman spaces in the 1990 's resulted in several breakthroughs, both function theoretic and operator theoretic. The most notable results in this period include Seip's geometric characterization of sequences of interpolation and sampling, Hedenmalm's discovery of the contractive zero divisors, the relationship between Bergman-inner functions and the biharmonic Green function found by Duren, Khavinson, Shapiro, and Sundberg, and deep results concerning invariant subspaces by Aleman, Borichev, Hedenmalm, Richter, Shimorin, and Sundberg.
Our purpose is to present the latest developments, mostly achieved in the 1990's, in book form. In particular, graduate students and new researchers in the field will have access to the theory from an almost self-contained and readable source.Preface
The Bergman Spaces
The Berezin Transform
A P -Inner Functions
Zero Sets
Interpolation and Sampling
Invariant Subspaces
Cyclicity
Invertible Noncyclic Functions
Logarithmically Subbarmonic Weights