Commutative Algebras of Toeplitz Operators on the Bergman Space - Preamble

Preface The book is devoted to the spectral theory of commutative C *-algebras of Toeplitz operators on Bergman spaces, and its applications. For each such commutative algebra we construct a unitary operator which reduces each Toeplitz operator from this algebra to a certain multiplication operator, thus also providing its spectral type representation. This gives us a powerful research tool allowing direct access to the majority of the important properties of the Toeplitz operators studied herein. The presence and exploitation of these spectral type representations forms the basis for an essential part of the results presented in this book. We give a criterion of when the algebras are commutative on each commonly considered weighted Bergman space. For Toeplitz operators generating such com-mutative algebras we describe their boundedness, compactness, and spectral properties. Furthermore, the above commutative algebras serve as model or local cases for a number of problems treated in the book, thus making their solutions possible. We note that in the Bergman space case considered in the book the underlying manifold (the unit disk equipped with the hyperbolic metric) possesses a richer geometric structure in comparison with the Hardy space case (the unit circle). This fact has an important reflection in the presented theory. We mention as well that from the general operator point of view the Toeplitz operators, both on Hardy space and on Bergman space, are compressions of multiplication operators onto certain subspaces, and thus they represent two interesting different models of operators having similar structure. At the same time the presented results show clearly essential differences between the theories for these two species of operators. The book is addressed to a wide audience of mathematicians, from graduate students to researchers, whose primary interests lie in complex analysis and operator theory. The prerequisites for reading this book include a basic knowledge in one-dimensional complex analysis, functional analysis, and operator theory. An acquaintance with some facts of the theory of Banach and C *-algebras will be useful as well. Among various excellent sources which may serve for the preliminary reading we mention, for example, the books The author is greatly indebted to his colleagues Serguei Grudsky, his coauthor in many papers, and Michael Porter who read the manuscript and made many important suggestions essentially improving the book. The author would like to address special words of gratitude to Olga Grudskaia who tragically died in a car accident in February 2004. …