Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes

Preface......Page 6
Contents......Page 8
Chapter 1 Introduction......Page 10
 1.1 Theoretical aspects......Page 12
 1.2 Applications to fractional diffusion processes......Page 18
 1.3 On quaternionic spectral theories......Page 23
 2.1 Slice hyperholomorphic functions......Page 25
 2.2 The S-functional calculus for bounded operators......Page 39
 2.3 Bounded operators with commuting components......Page 50
Chapter 3 The direct approach to the S-functional calculus......Page 53
 3.1 Properties of the S-spectrum of a closed operator......Page 54
 3.2 The S-resolvent of a closed operator......Page 63
 3.3 Closed operators with commuting components......Page 75
 3.4 The S-functional calculus and its properties......Page 78
 3.5 The product rule and polynomials in T......Page 89
 3.6 The spectral mapping theorem......Page 97
 3.7 Spectral sets and projections onto invariant subspaces......Page 101
 3.8 The special roles of intrinsic functions and the left multiplication......Page 107
Chapter 4 The Quaternionic Evolution Operator......Page 113
 4.1 Uniformly continuous quaternionic semigroups......Page 114
 4.2 Strongly continuous quaternionic semigroups......Page 125
 4.3 Strongly continuous groups......Page 138
 5.1 A series expansion of the S-resolvent operator......Page 140
 5.2 The class of operators A(T) and some properties......Page 142
  5.3 Perturbation of the generator......Page 146
 5.4 Comparison with the complex setting......Page 152
 5.5 An application......Page 155
Chapter 6 The Phillips functional calculus......Page 158
 6.1 Preliminaries on quaternionic measure theory......Page 159
 6.2 Functions of the generator of a strongly continuous group......Page 165
 6.3 Comparison with the S-Functional Calculus......Page 169
 6.4 The Inversion of the Operator f(T)......Page 176
 7.1 The S-functional calculus for sectorial operators......Page 180
 7.2 The H-infinity-functional calculus......Page 189
 7.3 The composition rule......Page 196
 7.4 Extensions according to spectral conditions......Page 203
 7.5 The spectral mapping theorem......Page 205
Chapter 8 Fractional powers of quaternionic linear operators......Page 219
 8.1 A direct approach to fractional powers of invertible sectorial operators with negative exponent......Page 220
 8.2 Fractional powers via the H∞-functional calculus......Page 237
  8.2.1 Fractional powers with negative real part......Page 246
 8.3 Kato's Formula for the S-Resolvents......Page 247
Chapter 9 The fractional heat equation using quaternionic techniques......Page 257
 9.1 Spectral properties of the Nabla operator......Page 258
 9.2 A relation with the fractional heat equation......Page 261
 9.3 An example with non-constant coefficients......Page 270
 10.1 New fractional diffusion problems......Page 272
 10.2 The S-spectrum approach to fractional diffusion processes......Page 275
 10.3 Fractional Fourier's law in a Hilbert space......Page 277
 10.4 Concluding Remarks......Page 288
Chapter 11 Historical notes and References......Page 290
 11.1 Theory of slice hyperholomorphic functions......Page 291
  Function theory......Page 292
  Slice hyperholomorphic Schur Analysis......Page 293
  The Phillips functional calculus and semigroups......Page 294
  The F-functional calculus......Page 295
  The spectral theorem on the S-spectrum......Page 296
  Slice hyperholomorphic Schur analysis......Page 298
  Spectral theory on the S-spectrum for quaternionic operators......Page 300
  Noncommutative Functional Calculus......Page 301
  Quaternionic Approximation......Page 302
  Regular Functions of a Quaternionic Variable......Page 303
  Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators......Page 304
Chapter 12 Appendix: Principles of functional Analysis......Page 306
Bibliography......Page 312
Index......Page 326