Ebook: Introduction to the Theory of Toeplitz Operators with Infinite Index

This book is devoted to Toeplitz and singular integral operators with symbols that have discontinuities of the oscillating type. Criteria for the normal solvability of such operators are established and several methods for describing the kernel and image spaces of the operators are presented. The approach is based on the idea of modelling discontinuities with an "infinite index" by appropriate inner functions, especially by infinite Blaschke products. The corresponding techniques have been elaborated by the authors during the last two decades, and they are applicable to both symbols with slowly and rapidly increasing arguments. Moreover, the book reveals exciting connections between invariant subspaces of the shift operator, bases in Banach spaces, and various classes of entire and meromorphic functions. The book aims at making advanced topics accessible to a broad readership. It is addressed to graduate and postgraduate students and to mathematicians interested in functional analysis, the theory of functions of a complex variable, or mathematical physics.

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This book is devoted to Toeplitz and singular integral operators with symbols that have discontinuities of the oscillating type. Criteria for the normal solvability of such operators are established and several methods for describing the kernel and image spaces of the operators are presented. The approach is based on the idea of modelling discontinuities with an "infinite index" by appropriate inner functions, especially by infinite Blaschke products. The corresponding techniques have been elaborated by the authors during the last two decades, and they are applicable to both symbols with slowly and rapidly increasing arguments. Moreover, the book reveals exciting connections between invariant subspaces of the shift operator, bases in Banach spaces, and various classes of entire and meromorphic functions. The book aims at making advanced topics accessible to a broad readership. It is addressed to graduate and postgraduate students and to mathematicians interested in functional analysis, the theory of functions of a complex variable, or mathematical physics.
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-8
Examples of Toeplitz Operators with Infinite Index....Pages 9-31
Factorization and Invertibility....Pages 33-69
Model Subspaces....Pages 71-109
Toeplitz Operators with Oscillating Symbols....Pages 111-174
Generalized Factorization of u-periodic Functions and Matrix Functions....Pages 175-225
Toeplitz Operators Whose Symbols Have Zeros....Pages 227-263
Back Matter....Pages 265-300

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This book is devoted to Toeplitz and singular integral operators with symbols that have discontinuities of the oscillating type. Criteria for the normal solvability of such operators are established and several methods for describing the kernel and image spaces of the operators are presented. The approach is based on the idea of modelling discontinuities with an "infinite index" by appropriate inner functions, especially by infinite Blaschke products. The corresponding techniques have been elaborated by the authors during the last two decades, and they are applicable to both symbols with slowly and rapidly increasing arguments. Moreover, the book reveals exciting connections between invariant subspaces of the shift operator, bases in Banach spaces, and various classes of entire and meromorphic functions. The book aims at making advanced topics accessible to a broad readership. It is addressed to graduate and postgraduate students and to mathematicians interested in functional analysis, the theory of functions of a complex variable, or mathematical physics.
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-8
Examples of Toeplitz Operators with Infinite Index....Pages 9-31
Factorization and Invertibility....Pages 33-69
Model Subspaces....Pages 71-109
Toeplitz Operators with Oscillating Symbols....Pages 111-174
Generalized Factorization of u-periodic Functions and Matrix Functions....Pages 175-225
Toeplitz Operators Whose Symbols Have Zeros....Pages 227-263
Back Matter....Pages 265-300
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