PREFACE ......................................................... 1
Chapter 1. INTRODUCTION ......................................... 3
1.1. Operator-valued functions, Laplace transforms and closed
operators .................................................. 3
1.2. C-regularized semigroups and cosine functions ............. 10
1.3. Function spaces ........................................... 21
1.4. Complex powers of operators ............................... 25
1.4.1. Complex powers of densely defined operators ........ 25
1.4.2. Complex powers of non-densely defined operators .... 30
Chapter 2. CONVOLUTED C-SEMIGROUPS AND COSINE FUNCTIONS ........ 43
2.1. Definitions and main structural properties ................ 43
2.2. Exponentially bounded convoluted C-semigroups and cosine
functions ................................................. 75
2.3. Abstract Cauchy problems .................................. 81
2.4. Analytical properties ..................................... 94
2.5. Perturbation theorems .................................... 109
2.6. Convoluted C-groups ...................................... 127
2.7. Spectral characterizations ............................... 145
2.8. Examples and applications ................................ 152
Chapter 3. ABSTRACT CAUCHY PROBLEMS IN THE SPACES OF
OPERATOR VALUED (ULTRA-DISTRIBUTIONS AND
HYPERFUNCTIONS ..................................... 165
3.1. C-Distribution semigroups ................................ 165
3.1.1. Elementary properties of C-distribution
semigroups ........................................ 165
3.1.2. Connections with integrated C-semigroups.
Exponential C-distribution semigroups ............. 168
3.1.3. Dense C-distribution semigroups ................... 174
3.1.4. Chaotic C-distribution semigroups ................. 178
3.2. Various classes of distribution semigroups ............... 188
3.3. Distribution groups ...................................... 206
3.3.1. Introduction and basic properties of
distribution groups ............................... 206
3.3.2. [B0,..., Bn, C0,..., Cn-1] groups ................... 212
3.3.3. Further relations between distribution groups,
local integrated groups and [B0,..., Bn, C0,...,
Cn-1]-groups ....................................... 220
3.4. Distribution cosine functions ............................ 241
3.4.1. Definition and elementary properties .............. 241
3.4.2. Relationship to integrated cosine functions,
convolution equations and local C-regularized
cosine functions .................................. 246
3.4.3. Exponential distribution cosine functions ......... 251
3.4.4. Dense distribution cosine functions ............... 254
3.4.5. Almost-distribution cosine functions, cosine
convolution products and their relations with
distribution cosine functions ..................... 255
3.4.6. Examples .......................................... 264
3.5. Ultradistribution and hyperfunction semigroups ........... 266
3.5.1. The structural properties of ultradistribution
semigroups ........................................ 266
3.5.2. Exponential ultradistribution semigroups .......... 271
3.5.3. Differentiable ultradistribution semigroups ....... 276
3.5.4. Hyperfunction spaces, semigroups and sines ........ 289
3.6. Regularization of ultradistribution semigroups and
sines .................................................... 292
3.6.1. Regularization of Gevrey type ultradistribution
semigroups ........................................ 292
3.6.2. Regularization of ultradistribution semigroups
whose generators possess ultra-polynomially
bounded resolvent ................................. 297
3.6.3. Higher order time-fractional equations.
Regularization of ultradistribution sines ......... 305
APPENDIX ...................................................... 317
Abstract Volterra Equations of Nonscalar Type ................. 317
Bibliography .................................................. 335
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