Functional differential operators and equations:
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| Format: | Book |
| Language: | English |
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Dordrecht [u.a.]
Kluwer
1999
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| Series: | Mathematics and its applications
473 |
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 432 S. |
| ISBN: | 0792356241 |
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| 100 | 1 | |a Kurbatov, Vitalii G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Functional differential operators and equations |c by Vitalii G. Kurbatov |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1999 | |
| 300 | |a XX, 432 S. | ||
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| 490 | 1 | |a Mathematics and its applications |v 473 | |
| 650 | 4 | |a Functional differential equations | |
| 650 | 4 | |a Operator theory | |
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| adam_text | CONTENTS
PREFACE xv
Chapter I. Functional analysis preliminaries 1
1.1. Banach spaces and bounded linear operators 1
1.1.1. Topological spaces 1
1.1.2. Absolutely convergent series in a normed space 3
1.1.3. The completeness of a quotient space 4
1.1.4. The completeness of the space of bounded operators 5
1.1.5. The Banach Steinhaus theorem 5
1.1.6. The Banach theorem 5
1.1.7. The conjugate space 6
1.1.8. The Hanh Banach theorem 6
1.1.9. The second conjugate space 6
1.1.10. The conjugate operator 7
1.1.11. Compact operators 7
1.2. Diagrams 7
1.2.1. Commutative diagrams 7
1.2.2. Operators on subspaces and quotient spaces 8
1.2.3. Short exact sequences 9
1.2.4. The continuation of a diagram 10
1.2.5. The existence of a restriction and a quotient operator 11
1.2.6. The conjugate sequence 11
1.2.7. The exactness of the conjugate of a short exact sequence 11
1.2.8. The isomorphism of the simplest rows 12
1.2.9. The conjugate of a subspace and a quotient space 12
1.2.10. The closedness of the image of the conjugate operator 13
1.2.11. The kernel and the image of the conjugate operator 13
1.3. Lower norms 14
1.3.1. The definition of the lower norms 14
1.3.2. The main properties of the lower norms 14
1.3.3. An equivalent definition of | • |_ 15
1.3.4. The lower norms of the conjugate operator 16
1.3.5. The continuity of the lower norms 18
1.3.6. The lower norms as bounds of invertibility 18
1.3.7. Multiplicative properties of the lower norms 18
1.3.8. Isometric injections and isometric surjections 18
vi CONTENTS
1.3.9. Isometric isomophisms of diagrams 19
1.3.10. The closedness of the image of the pre conjugate operator 19
1.3.11. Fredholm operators 19
1.3.12. The index of a Fredholm operator 20
1.4. Banach algebras 20
1.4.1. Algebras 20
1.4.2. The inverse element 21
1.4.3. The spectrum 22
1.4.4. Full subalgebras 23
1.4.5. The spectrum in a subalgebra 23
1.4.6. Morphisms of algebras 24
1.4.7. Ideals and quotient algebras 24
1.4.8. The radical 25
1.4.9. Holomorphic vector valued functions 26
1.4.10. The spectral mapping theorem 28
1.4.11. The Gel fand transform 29
1.4.12. The Gel fand Naimark theorem 31
1.5. Spaces of continuous and measurable functions 31
1.5.1. The Urysohn theorem 32
1.5.2. The partition of unity 32
1.5.3. Finite dimensional functions 33
1.5.4. The Stone Weierstrass theorem 33
1.5.5. The spaces LP(T), p oo 34
1.5.6. The spaces L^T) and L0(T) 37
1.5.7. The spaces lq(I) 40
1.5.8. The product of measures 40
1.5.9. The isomorphism Lx(T x 5, E) ~Ii(T,Li(5, E)) 42
1.5.10. The Fubini theorem 42
1.5.11. The Riesz Thorin theorem 43
1.5.12. The norm of an integral operator 43
1.6. Infinite matrices 44
1.6.1. Locally compact abelian groups 45
1.6.2. The Haar measure 46
1.6.3. The spaces Lpq and Cq 48
1.6.4. The matrix representation of operators 50
1.6.5. The norm of a matrix 51
1.6.6. Operators with local memory 52
1.6.7. Operators with uniform memory 53
1.6.8. Operators with summable memory 54
1.6.9. Operators with exponential memory 55
1.6.10. The representation of Cq as a space of sequences 56
1.6.11. The cut in Cq(R) 59
1.6.12. The shift operators 60
CONTENTS vii
1.7. Tensor products 61
1.7.1. Dual families 61
1.7.2. The definition of an algebraic tensor product 61
1.7.3. The definition of a topological tensor product 62
1.7.4. Examples of topological tensor products 64
1.7.5. The conjugate cross norm 66
1.7.6. The Schatten theorem 67
1.7.7. The tensor product of operators 69
1.7.8. The spectral mapping theorem (operator valued variant) 71
1.7.9. Irreducible representations of a Banach algebra 74
1.7.10. The Bochner Phillips theorem 75
1.7.11. The tensor product of a direct sum 78
1.7.12. A remark on the complexification 79
1.8. Conjugate spaces 81
1.8.1. The conjugate of Lpq 81
1.8.2. Coo is * weak dense in L^ 83
1.8.3. The decomposition l oo=l 0® Iq 84
1.8.4. The modulus of a measure 85
1.8.5. Measures with a compact support 87
1.8.6. The decomposition C = C Q@Cfr 88
1.8.7. Absolutely continuous measures 89
1.8.8. The Lebesgue Radon Nikodym theorem 90
1.8.9. Mutually singular measures 90
1.8.10. The Lebesgue decomposition M. = Ms © Moc 92
1.8.11. The Yosida Hewitt theorem 93
1.8.12. The space Mq 95
Chapter II. The initial value problem 97
2.1. The causal invertibility 98
2.1.1. The definition of a causal operator 98
2.1.2. The definition of causal invertibility 101
2.1.3. The monotonicity of causal invertibility 102
2.1.4. The invertibility of a triangular matrix 102
2.1.5. The invertibility on all segments implies causal invertibility 104
2.1.6. Causal invertibility on semi axes 104
2.1.7. The additivity of causal invertibility 105
2.1.8. Local causal invertibility 105
2.1.9. Local ordinary invertibility 106
2.2. The causal spectrum 106
2.2.1. Causal invertibility is a spectral property 106
2.2.2. The definition of the causal spectrum 106
2.2.3. The causal spectrum as the union of ordinary spectra 107
2.2.4. The local causal spectrum 107
2.2.5. The local ordinary spectrum 107
2.2.6. The semi norm G 108
viii CONTENTS
2.2.7. An estimation of the spectral radius 109
2.2.8. The calculation of G(N) for an integral operator N 109
2.2.9. The radical in the algebra of causal operators 110
2.2.10. Compact operators lie in the radical 110
2.2.11. Compact perturbations of the causal spectrum 111
2.2.12. Small perturbations preserve causal invertibility 112
2.3. Spaces of smooth functions and distributions 112
2.3.1. The classical and Lebesgue derivatives 112
2.3.2. The spaces C and Wp 113
2.3.3. The derivative of a product in W£loc 114
2.3.4. The isomorphism U between Cq and Cq, and Wplq and Lpq 115
2.3.5. The distribution derivative 116
2.3.6. The spaces C 1 and Wpqx 119
2.3.7. The equality to zero on (a, b) in C 1 and Wpqr 120
2.3.8. The cut in Cqx 120
2.3.9. The isomorphism U between C~x and Cq, and W~ql and Lvq 122
2.3.10. Atoms in Wpqx 123
2.3.11. The cut in Wpqx 123
2.3.12. The duality between W*q and W~, , 125
2.4. The unique solubility 127
2.4.1. Initial value problem for an abstract Y 127
2.4.2. Initial value problem for Y = Lp 129
2.4.3. Initial value problem for Y C 130
2.4.4. Initial value problem for Y = C~x 131
2.4.5. Initial value problem for Y = W 1 132
2.4.6. The additivity of unique solubility 135
2.5. The evolutionary solubility 136
2.5.1. The definition of evolutionary solubility 136
2.5.2. Evolutionary solubility as complete unique solubility 137
2.5.3. The additivity of evolutionary solubility 138
2.5.4. The decreasing of a segment 138
2.5.5. Local evolutionary solubility 138
2.5.6. Local unique solubility 138
2.5.7. Small perturbations preserve the evolutionary solubility 138
2.5.8. The operator U as a causal isomorphism 138
2.5.9. Equations with internal differentiation 139
2.5.10. Equations with external differentiation 140
2.6. Criteria for evolutionary solubility 140
2.6.1. Multiplication operators 141
2.6.2. Difference operators 142
2.6.3. An application to an equation of neutral type 144
2.6.4. Varying retardation 144
2.6.5. The spectral mapping theorem for the causal spectrum 147
2.6.6. The freezing of coefficients 148
CONTENTS be
2.6.7. An application to an equation of neutral type 149
Chapter III. Stability 150
3.1. Algebraic preliminaries 150
3.1.1. The projectors Pa 150
3.1.2. The representation of Xq as lq(Z,Xvi) 152
3.1.3. The space C^} 153
3.1.4. The operators Qa, Ia, and Ra 154
3.1.5. A representation of solutions of the initial value problem 155
3.2. Input output stability: discrete time 157
3.2.1. The definition of local solubility 157
3.2.2. The extended space 158
3.2.3. The definition of input output stability 159
3.2.4. Stability on semi axes 160
3.2.5. Input output stability and causal invertibility 160
3.2.6. Invertibility in (Zqo,/qo) implies that in (Zo,/o) 161
3.2.7. Input output stability and causal invertibility in (/o,io) 162
3.3. Input output stability: continuous time 162
3.3.1. The definition of local solubility 163
3.3.2. Local spaces 163
3.3.3. The extended space 164
3.3.4. The definition of input output stability 165
3.3.5. Stability on semi axes 166
3.3.6. Input output stability and causal invertibility 166
3.3.7. Invertibility in (A^, Y^) implies that in (Xo, Yo) 168
3.3.8. Input output stability and causal invertibility in (Xo,Yo) 168
3.3.9. Small perturbations preserve input output stability 168
3.3.10. Initial value problem on semi axis 170
3.4. Exponential stability: discrete time 171
3.4.1. Exponential weights and the class e 171
3.4.2. The class e is full 174
3.4.3. The independence of invertibility from q for T € e 175
3.4.4. The case of semi axes 175
3.4.5. The definition of uniform solubility 176
3.4.6. The definition of exponential stability 177
3.4.7. Exponential stability and causal invertibility 177
3.4.8. The independence of stability from q for T £ e 179
3.5. Exponential stability: continuous time 179
3.5.1. Exponential weights on R 179
3.5.2. The class e (continuous time) 182
3.5.3. The operators Qa, Ia, and Ra on weighted spaces 186
3.5.4. The independence of invertibility from q for C € e 187
3.5.5. The definition of uniform solubility 187
3.5.6. Weighted spaces on semi axes 188
3.5.7. Operators on semi axes 189
x CONTENTS
3.5.8. The definition of exponential stability 192
3.5.9. Exponential stability and causal invertibility 192
3.5.10. The independence of stability from q for C € e 195
3.5.11. Small perturbations preserve exponential stability 195
3.5.12. Initial value problem on semi axis 195
3.6. Exponential dichotomy: continuous time 196
3.6.1. The definitions of instability 196
3.6.2. Small perturbations preserve rough instability 197
3.6.3. The definition of exponential dichotomy 198
3.6.4. Dichotomy and invertibility 201
3.6.5. Dichotomy, stability, and instability 207
Chapter IV. Shift invariant operators and equations 209
4.1. Algebras of bounded measures 209
4.1.1. Convolution operators on Cq 209
4.1.2. Shift invariant operators 210
4.1.3. The convolution of measures 211
4.1.4. The ideal Mac of absolutely continuous measures 212
4.1.5. The push transform 213
4.1.6. Convolution operators on L 214
4.1.7. The ideal Mc of continuous measures 216
4.1.8. The subalgebra Md of discrete measures 217
4.1.9. The main decomposition 218
4.1.10. Full subalgebras of M 219
4.1.11. The algebra M+ 220
4.1.12. Full subalgebras of M+ 221
4.2. The Fourier transform 221
4.2.1. The dual group 221
4.2.2. The character space of Mac 225
4.2.3. The character space of Md 229
4.2.4. The topology on X(G) 229
4.2.5. The Pontrjagin theorem 230
4.2.6. The invertibility in Md 230
4.2.7. The invertibility in %Cc 230
4.2.8. The character space of Md®ac 231
4.2.9. The invertibility in Md®ac 231
4.2.10. The Kronecker theorem 232
4.2.11. A generalization of the Kronecker theorem 232
4.2.12. A remark on the character space of Msc 235
4.3. The Laplace transform 239
4.3.1. The character space of a semi group 239
4.3.2. The character space of M+c 241
4.3.3. The invertibility in Mtc 242
4.3.4. The character space of M^ 243
4.3.5. The invertibility in M J 244
CONTENTS xi
4.3.6. The character space of M+{Z) 244
4.3.7. The invertibility in M+(Z) 245
4.3.8. The character space of M^ac 245
4.3.9. The invertibility in M^@ac 245
4.3.10. X+ is dense in X+ and in X+ U X+ 246
4.3.11. A remark on the character space of Mfc 247
4.4. Convolution operators 247
4.4.1. Convolution operators on Lp, Lpq, and Cq (scalar case) 247
4.4.2. The ordering in the space of measures 250
4.4.3. Operator valued measures 251
4.4.4. Convolution operators on Lp, Lpq, and Cq (vector case) 258
4.4.5. The conjugate of a convolution operator 260
4.4.6. The norm of TM on Li(G, C) and L^G, C) 261
4.4.7. The preservation of multiplication (scalar case) 261
4.4.8. The preservation of multiplication (operator valued case) 262
4.4.9. The main decomposition (operator valued case) 263
4.4.10. Difference operators 264
4.4.11. Integral operators 264
4.4.12. The algebra M+(R,B) 265
4.5. Invertibility and causal invertibility 267
4.5.1. The invertibility in a subalgebra of operator valued measures 267
4.5.2. The invertibility of measures 268
4.5.3. The causal invertibility of measures 271
4.5.4. A general sufficient criterion 273
4.5.5. The invertibility of operators 273
4.5.6. The causal invertibility of operators 276
4.5.7. The spectrum of a difference operator 277
4.5.8. The causal spectrum of a difference operator 279
4.5.9. Differential operators 281
4.5.10. Sufficient conditions for causal invertibility 283
4.5.11. Periodic functional operators 283
4.5.12. Periodic differential operators 284
Chapter V. Operators with varying coefficients 285
5.1. Multiplication operators 285
5.1.1. Multiplication operators 286
5.1.2. Operators with zero memory 286
5.1.3. A characterization of operators with zero memory 287
5.1.4. The equality of X(lq) and Xz(lq) for a discrete group 287
5.1.5. Operators with zero memory on a non discrete group 287
5.1.6. A representation of operators A € Xz 288
5.1.7. The Bohr compactification 289
5.1.8. The group of isometries of a compact set 289
5.1.9. Almost periodic functions 290
5.1.10. Oscillation invariant operators 291
xii CONTENTS
5.1.11. The equality of X* and Xz 292
5.1.12. The algebras x, xz, and x^, 295
5.2. Difference operators 296
5.2.1. Difference operators with summable memory 296
5.2.2. The equality of S, Sz, and S* 297
5.2.3. The mean value of a character 298
5.2.4. The crossed product S(G,L) 298
5.2.5. The subalgebra S* is full 299
5.2.6. The subalgebra s is full 301
5.2.7. The invertibility of D € s is independent from q 301
5.2.8. The subalgebras Sz(Lpq) and S(Lpq) are full 302
5.2.9. The algebras D, Dz, and D* 302
5.2.10. The invertibility of D e S(Lpq) is independent from p and q 303
5.2.11. The subgroup generated by the set of shifts 303
5.2.12. Difference operators with exponential memory 304
5.3. Smoothing operators 307
5.3.1. Universal operators with summable memory 307
5.3.2. Smoothing operators with summable memory 309
5.3.3. Integral operators with bounded kernels 309
5.3.4. An integral representation for N € N^ 310
5.3.5. The invertibility of N £ Nqo is independent from p and q 311
5.3.6. The subalgebra N^, is full 312
5.3.7. Universal operators with exponential memory 312
5.3.8. Smoothing operators with exponential memory 313
5.3.9. The integral representation for M € M^ 314
5.3.10. The subalgebra M^ is full 314
5.4. Integral operators 315
5.4.1. The class Ni of kernels 315
5.4.2. The multiplication in Ni 316
5.4.3. Operators of the class Nx 317
5.4.4. The algebra N x of operators 318
5.4.5. Nqo is dense in Ni 318
5.4.6. Noo is an ideal inj^i 319
5.4.7. The subalgebra Nx is fuU_ 319
5.4.8. The invertibility of N € Ni is independent from p and q 320
5.4.9. The algebra Mx of kernels 320
5.4.10. Moo is dense in M^ 321
5.4.11. Moo is an ideal in_Mi 321
5.4.12. The subalgebra Mi is full 322
5.5. Operators with locally fading memory 322
5.5.1. Directions on a Banach space 322
5.5.2. The classes t/ and t 324
5.5.3. A two diagonal representation for T € t/ (lq) 326
5.5.4. The subalgebra t{lq) is full 327
CONTENTS xiii
5.5.5. The isomorphism t(Zoo) =± t{l0) 328
5.5.6. The equivalence of the invertibility in t(l0) and t(Zoo) 330
5.5.7. Consistent directions 330
5.5.8. The Fourier transform maps L into Co 330
5.5.9. Compactly supported sets 330
5.5.10. ^ bounded sets 331
5.5.11. Differential operators 332
5.6. Operators with continuous coefficients 333
5.6.1. Operators with uniformly continuous coefficients 333
5.6.2. Operators with continuous coefficients 334
5.6.3. The subalgebra C is full 335
5.6.4. The conjugate of an operator of the class C 335
5.6.5. The isomorphisms C(Lpoo) ~ C(Lp0) and C(Coo) ~ C(C0) 336
5.6.6. An equivalent representation of a convolution 336
5.6.7. The invariant subspaces 336
5.6.8. The main example: S(Cg) C C(Cq) 339
5.6.9. The isomorphism C(Looq) ~ C(Cq) 340
5.6.10. The invertibility of D € S{Cq) 340
5.6.11. The equivalence of the lower norms on Looq and Cq 341
Chapter VI. Differential difference equations 343
6.1. The local Fredholm property 343
6.1.1. Locally compact operators 344
6.1.2. The ideal k 345
6.1.3. Locally Fredholm operators 346
6.1.4. The operator T# 347
6.1.5. The conjugate of a locally compact operator 347
6.1.6. The conjugate of a locally Fredholm operator 347
6.1.7. The lower norms of T* 347
6.1.8. The lower norm on a subspace 347
6.1.9. Rapidly oscillating characters 348
6.1.10. An estimate of |D#|+ 349
6.1.11. Difference operators can not be locally Fredholm 353
6.2. The solubility for derivatives 355
6.2.1. Functional equations 355
6.2.2. The operator I/ 1 belongs to k 355
6.2.3. Bounded solution problem 356
6.2.4. Periodic solution problem 358
6.2.5. Elliptic equations with shifts 359
6.2.6. The representation of D G D as a series 361
6.2.7. Input output stability and causal invertibility for C € t 361
6.2.8. Stability 362
6.3. The independence from the choice of a norm 366
6.3.1. The case of invertibility with D, B € s 366
6.3.2. The case of stability with D, B € s 367
xiv CONTENTS
6.3.3. The case of invertibility with D, B G C 367
6.3.4. The case of stability with D, B G C 368
6.3.5. The case of invertibility with D G S and B eV 368
6.3.6. The case of stability with D G S and B G V 368
6.3.7. The case of invertibility with D, B G S(Cq) 369
6.3.8. The case of stability with D, B G S(Cq) 369
6.3.9. The case of invertibility with D G S1 and Bet 369
6.3.10. The case of stability with D G S1 and B G t 371
6.3.11. The case of invertibility with D G Sl{C) and B G S(C) 371
6.3.12. The case of stability with D G S^C) and B G S(C) 371
6.4. Green s function 372
6.4.1. The integral representation for solutions 372
6.4.2. The integral representation for derivative 373
6.4.3. Distributions of two variables 375
6.4.4. The partial derivatives ^f and ^ 376
6.4.5. The conjugate equation 378
6.4.6. The fundamental solution 380
6.4.7. The case of D G S^C) and B G S(C) 381
6.5. Almost periodic operators 382
6.5.1. Almost periodic operators 382
6.5.2. The subalgebra BAP is full 385
6.5.3. More about the algebra D* 385
6.5.4. More about the algebra d^, 386
6.5.5. The equality tAP = dAp 386
6.5.6. The ideal h 387
6.5.7. The case of Hap (/,(Zn)) 390
6.5.8. The case of hAp(L) 394
6.5.9. Almost periodic difference operators 396
6.5.10. The case of DAP 397
6.5.11. The case of DAP + Kap 400
6.5.12. The case of a differential difference operator 402
Comments 403
Bibliography 411
Index 429
Notation Index 433
|
| any_adam_object | 1 |
| author | Kurbatov, Vitalii G. |
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012797634 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:33:49Z |
| institution | BVB |
| isbn | 0792356241 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008702889 |
| oclc_num | 40698550 |
| open_access_boolean | |
| owner | DE-703 DE-634 |
| owner_facet | DE-703 DE-634 |
| physical | XX, 432 S. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Kurbatov, Vitalii G. Verfasser aut Functional differential operators and equations by Vitalii G. Kurbatov Dordrecht [u.a.] Kluwer 1999 XX, 432 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 473 Functional differential equations Operator theory Operatortheorie (DE-588)4075665-8 gnd rswk-swf Funktional-Differentialgleichung (DE-588)4155668-9 gnd rswk-swf Funktional-Differentialgleichung (DE-588)4155668-9 s Operatortheorie (DE-588)4075665-8 s DE-604 Mathematics and its applications 473 (DE-604)BV008163334 473 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008702889&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kurbatov, Vitalii G. Functional differential operators and equations Mathematics and its applications Functional differential equations Operator theory Operatortheorie (DE-588)4075665-8 gnd Funktional-Differentialgleichung (DE-588)4155668-9 gnd |
| subject_GND | (DE-588)4075665-8 (DE-588)4155668-9 |
| title | Functional differential operators and equations |
| title_auth | Functional differential operators and equations |
| title_exact_search | Functional differential operators and equations |
| title_full | Functional differential operators and equations by Vitalii G. Kurbatov |
| title_fullStr | Functional differential operators and equations by Vitalii G. Kurbatov |
| title_full_unstemmed | Functional differential operators and equations by Vitalii G. Kurbatov |
| title_short | Functional differential operators and equations |
| title_sort | functional differential operators and equations |
| topic | Functional differential equations Operator theory Operatortheorie (DE-588)4075665-8 gnd Funktional-Differentialgleichung (DE-588)4155668-9 gnd |
| topic_facet | Functional differential equations Operator theory Operatortheorie Funktional-Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008702889&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT kurbatovvitaliig functionaldifferentialoperatorsandequations |