Commutative Algebras of Toeplitz Operators on the Bergman Space (Operator Theory: Advances and Applications, 185) - Hardcover

Vasilevski, Nikolai

Hardcover

ISBN 10: 3764387254 ISBN 13: 9783764387259

Publisher: Birkhäuser, 2008

Synopsis

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.

About this title

From the Back Cover

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.

 

 

Review

From the reviews: “This is a massive, detailed, and comprehensive book on the theory of Toeplitz operators and Toeplitz algebras ... . The book is quite easy to read and understand. Computations as well are explicit and detailed. ... Included Highlights of the chapters in addition to Preface and Introduction will be helpful for the readers to decide and read this book properly and effectively. Furthermore, Bibliographical Remarks for items in the Bibliography will also be helpful for further reading.” (Takahiro Sudo, Zentralblatt MATH, Vol. 1168, 2009)

Edition details

• Publisher: Birkhäuser
• Publication date: 2008
• ISBN 10: 3764387254
• ISBN 13: 9783764387259
• Binding: Hardcover
• Language: English
• Number of pages: 447