Commutative algebras of Toeplitz operators on the Bergman space

This book deals with the symbiotic relationship between the theory of function spaces, fractal geometry, and spectral theory of (fractal) pseudodifferential operators as it has emerged quite recently. Most of the presented material is published here for the first time 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 spe.  Read more... 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