This book is devoted to the spectral theory of commutative C*-algebras of Toeplitz operators on the Bergman space and its applications. For each such commutative algebra there is a unitary operator which reduces Toeplitz operators from this algebra to certain multiplication operators, thus providing their spectral type representations. This yields a powerful research tool giving direct access to the majority of the important properties of the Toeplitz operators studied herein, such as boundedness, compactness, spectral properties, invariant subspaces.

The presence and exploitation of these spectral type representations forms the core for many results presented in the book. Among other results it contains a criterion of when the algebras are commutative on each commonly considered weighted Bergman space together with their explicit descriptions; a systematic study of Toeplitz operators with unbounded symbols; a clarification of the difference between compactness of commutators and semi-commutators of Toeplitz operators; the theory of Toeplitz and related operators with symbols having more than two limit values at boundary points; and a kind of semi-classical analysis of spectral properties of Toeplitz 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.

Zusammenfassung

This unique book is devoted to the detailed study of the recently discovered commutative C*-algebras of Toeplitz operators on the Bergman space over the unit disk. Surprisingly, the key point to understanding their structure and classifying them lies in the hyperbolic geometry of the unit disk. The book develops a number of important problems whose successful solution was made possible and is based on the specific features of the Toeplitz operators from these commutative algebras.

Inhalt
Preliminaries.- Prologue.- Bergman and Poly-Bergman Spaces.- Bergman Type Spaces on the Unit Disk.- Toeplitz Operators with Commutative Symbol Algebras.- Toeplitz Operators on the Unit Disk with Radial Symbols.- Toeplitz Operators on the Upper Half-Plane with Homogeneous Symbols.- Anatomy of the Algebra Generated by Toeplitz Operators with Piece-wise continuous Symbols.- Commuting Toeplitz Operators and Hyperbolic Geometry.- Weighted Bergman Spaces.- Commutative Algebras of Toeplitz Operators.- Dynamics of Properties of Toeplitz Operators with Radial Symbols.- Dynamics of Properties of Toeplitz Operators on the Upper Half-Plane: Parabolic Case.- Dynamics of Properties of Toeplitz Operators on the Upper Half-Plane: Hyperbolic Case.