Vector Measures, Integration and Related Topics (eBook)

Table of Contents 6
Preface 9
List of Talks 11
On Mean Ergodic Operators 15
1. Introduction 15
2. Preliminary results 17
3. Mean ergodic results 21
References 33
Fourier Series in Banach spaces andMaximal Regularity 35
0. Introduction 35
1. Vector-valued Fourier series and operator-valuedFourier multipliers 36
2. The Marcinkiewicz multiplier theorem in the general case 44
3. The periodic non-homogeneous problems 45
4. Maximal regularity 46
5. The non-autonomous equations 50
References 52
Spectral Measures on Compacts ofCharacters of a Semigroup 54
1. Introduction 54
2. A Berg-Maserick type theorem 55
2.1. Definitions and notations 55
2.2. The Berg-Maserick type theorem 56
3. An integral representation via spectral measures 57
4. Examples of *-representations 58
5. A construction of the spectral measure 60
6. The Gelfand-Naimark theorem for abelian C*-algebras 61
References 62
On Vector Measures, Uniform Integrabilityand Orlicz Spaces 63
1. Introduction and preliminaries 63
2. The results 65
References 69
The Bohr Radius of a Banach Space 70
1. Introduction and preliminaries 70
References 75
Spaces of Operator-valued Functions Measurable with Respect tothe Strong Operator Topology 76
1. Introduction 76
2. Strong µ-normability of operator-valued functions 78
3. Spaces of operator-valued functions 83
References 89
Defining Limits by Means of Integrals 90
1. Introduction 90
2. Preliminaries 90
3. I-convergence in Riesz spaces 92
4. Applications 95
References 97
A First Return Examinationof Vector-valued Integrals 99
1. Introduction 99
2. Preliminaries 100
3. Bochner integrable functions 101
4. Pettis integrable functions 104
References 107
A Note on Bi-orthomorphisms 108
1. Introduction 108
2. Preliminaries 109
3. Separately disjointness preserving operators 111
References 116
Compactness of Multiplication Operators on Spaces of Integrable Functionswith Respect to a Vector Measure 117
1. Introduction 117
2. Compactness and weak compactness 118
References 121
Some Applications of Nonabsolute Integrals in the Theory of Differential Inclusions in Banach Spaces 122
1. Introduction 122
2. Multivalued integrals 123
3. Results 125
References 130
Equations Involving the Mean ofAlmost Periodic Measures 132
1. Introduction 132
2. Preliminaries 133
3. Properties of the almost periodic functions 135
4. Equations with almost periodic measures and functions 137
References 140
How Summable are Rademacher Series? 141
1. Introduction: a problem on vector measures 141
2. The Rademacher system 142
3. A problem on function spaces 144
4. The Rademacher multiplicator space 145
4.1. The space .(R,X) 145
4.2. The symmetric kernel of .(R,X) 147
4.3. When is .(R,X) rearrangement invariant? 149
4.4. Head and tail behavior 152
5. An open question 153
References 153
Rearrangement Invariant Optimal Domain for Monotone Kernel Operators 155
1. Introduction 155
2. Preliminaries 156
3. R.i. optimal domain for T 157
References 163
The Fubini and Tonelli Theoremsfor Product Local Systems 165
1. Introduction 165
2. Preliminaries 166
3. A convergence theorem for the S1-integral on the real line 167
4. Product local system 169
5. S-integral for a product local system 170
6. The Fubini Theorem for a product local system 172
References 176
A Decomposition of Henstock-Kurzweil-PettisIntegrable Multifunctions 177
Introduction 177
1. Notations and preliminaries 178
2. A decomposition theorem for HKP-integrable multifunctions 181
References 187
Non-commutative Yosida-Hewitt Theorems and Singular Functionals inSymmetric Spaces of t-measurable Operators 189
1. Introduction and preliminaries 189
2. Preliminaries and notation 190
3. Normed spaces of t -measurable operators 193
3.1. Normed M-bimodules 193
3.2. Symmetrically normed M-bimodules and their K¨othe duals 193
3.3. Normal and singular functionals on a normed M-bimodule 195
4. The Yosida-Hewitt decomposition in M-bimodules 196
5. Elements of order-continuous norm and singular functionals 198
6. A vector-valued Yosida-Hewitt theorem 199
References 203
Ideals of Subseries Convergenceand Copies of c0 in Banach Spaces 205
References 209
On Operator-valued Measurable Functions 211
1. Introduction 211
2. Measurable operator-valued functions 212
2.1. Strongly p-integrable functions 213
2.2. Classes of (operator-valued) integral multiplier functions 215
2.3. (p, q)-integral functions 216
2.4. A new class of operator-valued functions 217
References 220
Logarithms of Invertible Isometries, Spectral Decompositions and Ergodic Multipliers 221
1. Introduction 221
2. Logarithms of measures and translations 222
3. Logarithms of invertible isometries 225
4. Trigonometrically well-bounded operators 228
5. Trigonometrically well-bounded operators on super-reflexivespaces and norm growth of iterates 230
References 234
Norms Related to Binomial Series 236
0. Introduction 236
1. Generalities 237
2. l2-norm 238
3. l3-norm 240
4. H as an operator 244
5. Further ideal properties 245
References 248
Vector-valued Extension of Linear Operators, and Tb Theorems 249
1. Introduction 249
2. Review of the proof of the Tb theorem 252
3. Probabilistic approach to the paraproduct 253
References 258
Some Recent Applicationsof Bilinear Integration 259
1. Introduction 259
2. Bilinear integration in tensor products 261
3. Random evolutions 263
4. Scattering theory 265
5. Bilinear integration with respect to white noise 269
References 273
A Complete Classification of ShortSymmetric-antisymmetric Multiwavelets 274
1. Introduction 274
2. Multiwavelets 275
3. Multiwavelets with three taps 277
4. Multiwavelets with four taps 280
References 286
On the Range of a Vector Measure 287
1. Introduction 287
2. Preliminaries 288
3. Main results 289
References 294
Measure and Integration: Characterizationof the New Maximal Contents and Measures 295
1. The relevant properties of contents and measures 296
1.1 Lemma. 296
1.2 Remark. 297
1.3 Proposition. 297
1.4 Theorem. 297
1.5 Example. 298
2. The characterization theorems 299
2.1 Inner Remark. 299
2.2 Continuation. 299
2.3 Outer Remark. 299
2.4 Outer Characterization Theorem. 300
2.5 Remark. 300
3. Another inner characterization theorem 300
3.1 Theorem. 301
3.2 Theorem. 302
3.3 Inner Characterization Theorem. 302
4. Application to the inner measure constructions 303
4.1 Example. 304
References 304
Vector Measures of Bounded .-variationand Stochastic Integrals 305
1. Introduction 305
2. Vector measures of bounded .-variation 306
References 313
Does a Compact Operator Admit a MaximalDomain for its Compact Linear Extension? 314
1. Introduction 314
2. Proof of Theorem 1.1 315
3. Weakly compact linear extension 322
References 323
A Note on R-boundedness in Bidual Spaces 324
References 326
Salem Sets in the p-adics, the Fourier Restriction Phenomenon and Optimal Extension of the Hausdorff-Young Inequality 327
1. Introduction 327
2. p-adic Salem sets and the L2-Fourier restriction phenomenon 328
3. Optimal extension of the Hausdorff-Young inequality in Zp 334
References 337
L-embedded Banach Spaces anda Weak Version of Phillips Lemma 339
References 343
When is the Space of Compact Range MeasuresComplemented in the Space of All Vector-valued Measures? 345
1. Introduction 345
1.1. Complementability 345
1.2. Notation 347
1.3. Main result 348
2. Proof of Theorem 1.1: the separable case 349
2.1. Recalling why c0 is not complemented in l8 349
2.2. Identifying measures and operators 350
2.3. Rademacher functions 351
2.4. Walsh system 353
3. Proof in the non separable case 356
References 358
When is the Optimal Domain of a Positive Linear Operator a Weighted L1-space? 360
1. Introduction 360
2. The L0 case 361
3. The Banach function space case 363
4. Examples 366
References 368
Liapounoff Convexity-type Theorems 369
1. Introduction 369
2. Definitions and notation 370
3. Example 371
4. Non-negative scalar measures 372
5. Vector measures 373
6. Liapounoff convexity-type theorems 375
References 377
List of Participants 379