Spectral theory on the S-spectrum for quaternionic operators

✍ Scribed by Colombo F., Gantner J., Kimsey D.P

• Publisher: Birkhauser
• Year: 2018
• Tongue: English
• Leaves: 357
• Series: Operator Theory 270
• Category: Library

✦ Table of Contents

Preface......Page 6
Contents......Page 8
1.1 What is Quaternionic Spectral Theory?......Page 11
1.2.1 The Discovery of the S-Spectrum......Page 16
1.2.2 Why Did It Take So Long to Understand the S-Spectrum?......Page 18
1.3 The Fueter–Sce–Qian theorem and spectral theories......Page 19
Chapter 2 Slice Hyperholomorphic Functions......Page 21
2.1 Slice Hyperholomorphic Functions......Page 23
2.2 The Fueter Mapping Theorem in Integral Form......Page 43
2.3 Vector-Valued Slice Hyperholomorphic Functions......Page 48
2.4 Comments and Remarks......Page 58
3.1 The S-Spectrum and the S-Resolvent Operators......Page 62
3.2 Definition of the S-Functional Calculus......Page 72
3.3.1 The Left Spectrum σL(T) and the Left Resolvent Operator......Page 78
3.3.2 Power Series Expansions and the S-Resolvent Equation......Page 80
4.1 Algebraic Properties and Riesz Projectors......Page 84
4.2 The Spectral Mapping Theorem and the Composition Rule......Page 91
4.3 Convergence in the S-Resolvent Sense......Page 96
4.4 The Taylor Formula for the S-Functional Calculus......Page 99
4.5 Bounded Operators with Commuting Components......Page 113
4.6 Perturbations of the SC-Resolvent Operators......Page 117
4.7 Some Examples......Page 121
4.8 Comments and Remarks......Page 125
4.8.1 The S-Functional Calculus for n-Tuples of Operators......Page 126
4.8.2 The W-Functional Calculus for Quaternionic Operators......Page 129
Chapter 5 The S-Functional Calculus for Unbounded Operators......Page 134
5.1 The S-Spectrum and the S-Resolvent Operators......Page 135
5.2 Definition of the S-Functional Calculus......Page 139
5.3 Comments and Remarks......Page 145
6.1 The Rational Functional Calculus......Page 146
6.2 The S-Functional Calculus for Operators of Type ω......Page 148
6.3 The H∞-Functional Calculus......Page 151
6.4 Boundedness of the H∞-Functional Calculus......Page 153
6.5.1 Comments on Fractional Diffusion Processes......Page 155
7.1 The F-Resolvent Operators and the F-Functional Calculus......Page 159
7.2 Bounded Perturbations of the F-Resolvent......Page 167
7.3 The F-Resolvent Equations......Page 171
7.4 The Riesz Projectors for the F-Functional Calculus......Page 173
7.5 The Cauchy–Fueter Functional Calculus......Page 176
7.6 Comments and Remarks......Page 179
7.6.1 The F-Functional Calculus for n-Tuples of Operators......Page 180
7.6.2 The Inverse Fueter–Sce Mapping Theorem......Page 182
Chapter 8 The F-Functional Calculus for Unbounded Operators......Page 185
8.1 Relations Between F-Resolvent Operators......Page 186
8.2 The F-Functional Calculus for Unbounded Operators......Page 189
8.3 Comments and Remarks......Page 191
8.3.1 F-Functional Calculus for n-Tuples of Unbounded Operators......Page 192
9.1 Preliminary Results......Page 194
9.2 The S-Spectrum of Some Classes of Operators......Page 199
9.3 The Splitting of a Normal Operator and Consequences......Page 203
9.4 The Continuous Functional Calculus......Page 211
9.5 Comments and Remarks......Page 224
Chapter 10 Spectral Integrals......Page 225
10.1 Spectral Integrals for Bounded Measurable Functions......Page 226
10.2 Spectral Integrals for Unbounded Measurable Functions......Page 231
10.3 Comments and remarks......Page 237
Chapter 11 The Spectral Theorem for Bounded Normal Operators......Page 238
11.1 Construction of the Spectral Measure......Page 239
11.2 The Spectral Theorem and Some Consequences......Page 246
11.3 Comments and Remarks......Page 248
12.1 Some Transformations of Operators......Page 249
12.2 The Spectral Theorem for Unbounded Normal Operators......Page 251
12.3 Some Consequences of the Spectral Theorem......Page 254
12.4 Comments and Remarks......Page 257
13.1 Herglotz's Theorem in the Quaternionic Setting......Page 259
13.2 Preliminaries for the Spectral Resolution......Page 263
13.3 Further Properties of Quaternionic Riesz Projectors......Page 269
13.4 The Spectral Resolution......Page 272
13.5 Comments and Remarks......Page 275
Chapter 14 Spectral Integration in the Quaternionic Setting......Page 276
14.1 Spectral Integrals of Real-Valued Slice Functions......Page 277
14.2 Imaginary Operators......Page 281
14.3 Spectral Systems and Spectral Integrals of Intrinsic Slice Functions......Page 288
14.4 On the Different Approaches to Spectral Integration......Page 298
15.1 The Spectral Decomposition of a Spectral Operator......Page 306
15.2 Canonical Reduction and Intrinsic S-Functional Calculus for Quaternionic Spectral Operators......Page 327
Contents of the Monograph: Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes......Page 339
Index......Page 341
Bibliography......Page 344