Noncommutative functional calculus : theory and applications of slice hyperholomorphic functions / Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa.

Format:

Book

Author/Creator:

Colombo, Fabrizio.

Contributor:

Sabadini, Irene, 1965-

Struppa, Daniele Carlo, 1955-

Series:

Progress in mathematics (Boston, Mass.) ; v. 289.

Progress in mathematics ; 289

Language:

English

Subjects (All):

Functional analysis.

Functions of complex variables.

Operator theory.

Physical Description:

vi, 221 pages : illustrations ; 24 cm.

Place of Publication:

Basel : Birkhäuser, [2011]

Summary:

This book presents a functional calculus for n-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra: the so-called slice monogenic functions which are carefully described in the book. In the case of functions with values in the algebra of quaternions these functions are named slice regular functions.

Except for the appendix and the introduction all results are new and appear for the first time organized in a monograph. The material has been carefully prepared to be as self-contained as possible. The intended audience consists of researchers, graduate and postgraduate students interested in operator theory, spectral theory, hypercomplex analysis, and mathematical physics. Book jacket.

Contents:

1 Introduction 1

1.1 Overview 1

1.2 Plan of the book 3

2 Slice monogenic functions 17

2.1 Clifford algebras 17

2.2 Slice monogenic functions: definition and properties 23

2.3 Power series 33

2.4 Cauchy integral formula, I 37

2.5 Zeros of slice monogenic functions 42

2.6 The slice monogenic product 47

2.7 Slice monogenic Cauchy kernel 53

2.8 Cauchy integral formula, II 60

2.9 Duality Theorems 68

2.10 Topological Duality Theorems 73

2.11 Notes 76

3 Functional calculus for n-tuples of operators 81

3.1 The S-resolvent operator and the S-spectrum 82

3.2 Properties of the S-spectrum 86

3.3 The functional calculus 88

3.4 Algebraic rules 90

3.5 The spectral mapping and the S-spectral radius theorems 93

3.6 Projectors 99

3.7 Functional calculus for unbounded operators and algebraic properties 101

3.8 Notes 108

4 Quaternionic Functional Calculus 113

4.1 Notation and definition of slice regular functions 113

4.2 Properties of slice regular functions 117

4.3 Representation Formula for slice regular functions 121

4.4 The slice regular Cauchy kernel 129

4.5 The Cauchy integral formula II 134

4.6 Linear bounded quaternionic operators 136

4.7 The S-resolvent operator series 138

4.8 The S-spectrum and the S-resolvent operators 141

4.9 Examples of S-spectra 144

4.10 The quaternionic functional calculus 146

4.11 Algebraic properties of the quaternionic functional calculus 151

4.12 The S-spectral radius 153

4.13 The S-spectral mapping and the composition theorems 156

4.14 Bounded perturbations of the S-resolvent operator 159

4.15 Linear closed quaternionic operators 166

4.16 The functional calculus for unbounded operators 173

4.17 An application: uniformly continuous quaternionic semigroups 180

4.18 Notes 188

5 Appendix: The Riesz-Dunford functional calculus 201

5.1 Vector-valued functions of a complex variable 201

5.2 The functional calculus for linear bounded operators 203

5.3 The functional calculus for unbounded operators 208.

Notes:

Includes bibliographical references (pages 211-217) and index.

ISBN:

3034801092

9783034801096

OCLC:

706784508