Noncommutative Functional Calculus

by Colombo, Fabrizio; Sagadini, Irene; Struppa, Daniele C.

• ISBN13: 9783034801096
• ISBN10: 3034801092
• Format: Hardcover
• Copyright: 2011-03-22
• Publisher: Birkhauser

Summary

The book contains a full exposition of a new functional calculus for n-tuples of not necessarily commuting linear operators and of its analog for quaternionic linear operators. These calculi are based on the recently developed theory of hyperholomorphicity for functions with values in a Clifford algebra (the so-called slice monogenic functions) which are carefully described in the book and, in particular, for functions with values in the algebra of quaternions, the so-called slice regular functions.

Table of Contents

Introduction	p. 1
Overview	p. 1
Plan of the book	p. 3
Slice monogenic functions	p. 17
Clifford algebras	p. 17
Slice monogenic functions: definition and properties	p. 23
Power series	p. 33
Cauchy integral formula, I	p. 37
Zeros of slice monogenic functions	p. 42
The slice monogenic product	p. 47
Slice monogenic Cauchy kernel	p. 53
Cauchy integral formula, II	p. 60
Duality Theorems	p. 68
Topological Duality Theorems	p. 73
Notes	p. 76
Functional calculus for n-tuples of operators	p. 81
The S-resolvent operator and the S-spectrum	p. 82
Properties of the S-spectrum	p. 86
The functional calculus	p. 88
Algebraic rules	p. 90
The spectral mapping and the S-spectral radius theorems	p. 93
Projectors	p. 99
Functional calculus for unbounded operators and algebraic properties	p. 101
Notes	p. 108
Quaternionic Functional Calculus	p. 113
Notation and definition of slice regular functions	p. 113
Properties of slice regular functions	p. 117
Representation Formula for slice regular functions	p. 121
The slice regular Cauchy kernel	p. 129
The Cauchy integral formula II	p. 134
Linear bounded quaternionic operators	p. 136
The S-resolvent operator series	p. 138
The S-spectrum and the S-resolvent operators	p. 141
Examples of S-spectra	p. 144
The quaternionic functional calculus	p. 146
Algebraic properties of the quaternionic functional calculus	p. 151
The S-spectral radius	p. 153
The S-spectral mapping and the composition theorems	p. 156
Bounded perturbations of the S-resolvent operator	p. 159
Linear closed quaternionic operators	p. 166
The functional calculus for unbounded operators	p. 173
An application: uniformly continuous quaternionic semigroups	p. 180
Notes	p. 188
Appendix: The Riesz-Dunford functional calculus	p. 201
Vector-valued functions of a complex variable	p. 201
The functional calculus for linear bounded operators	p. 203
The functional calculus for unbounded operators	p. 208
Bibliography	p. 211
Index	p. 219
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