Topics in operator theory/ Vol. 1, Operators, matrices and analytic functions

Cover......Page 1
Operator Theory: Advances and Applications 203......Page 3
Topics in Operator Theory, Volume 2: Systems and Mathematical Physics......Page 4
ISBN 9783034601603......Page 5
Table of Contents\r......Page 6
The XIXth International Workshop on Operator Theory and its Applications. II......Page 8
1. Introduction......Page 12
2. Main results......Page 15
3. Examples......Page 19
References......Page 22
1. Introduction......Page 24
2. The 1-D systems/single-variable case......Page 29
2.2. The frequency-domain stabilization and H∞ problem......Page 31
2.3. The state-space approach......Page 33
3. The fractional representation approach to stabilizability and performance......Page 36
3.1. Parametrization of stabilizing controllers in terms of a given stabilizing controller......Page 38
3.2. The Youla-Kucera parametrization......Page 42
3.3. The standard H∞-problem reduced to model matching.......Page 46
3.4. Notes......Page 48
4.1. Multivariable frequency-domain formulation......Page 49
4.2. Multidimensional state-space formulation......Page 52
4.3. Equivalence of frequency-domain and state-space formulations......Page 63
5. Robust control with structured uncertainty: the commutative case......Page 69
5.1. Gain-scheduling in state-space coordinates......Page 70
5.3. Robust control with a hybrid frequency-domain/state-space formulation......Page 72
5.4. Notes......Page 74
6.1. The state-space LFT-model formulation......Page 75
6.2. A noncommutative frequency-domain formulation......Page 80
6.3. Equivalence of state-space noncommutative LFT-model and noncommutative frequency-domain formulation......Page 85
6.4. Notes......Page 89
References......Page 91
1. Introduction......Page 100
2. Preliminaries......Page 102
3. Absence of existence and uniqueness......Page 103
4. Satisfaction of (C1)–(C4)......Page 106
References......Page 108
1. Introduction......Page 110
2. Generalized total variation......Page 112
3. Estimates of eigenvalues......Page 114
4. Optimality of bounds I......Page 117
5. Optimality of bounds II......Page 118
6. The periodic p-Laplacian......Page 121
References......Page 123
1. Introduction......Page 126
2. Notation......Page 129
3. The spectral function ρ0,β......Page 130
4. The transformation operator......Page 133
5. Existence of the transformation operator......Page 136
7. The inverse problem......Page 141
8. The limit-point case......Page 143
9. The string......Page 144
References......Page 146
Introduction......Page 148
1. Basic boundary value problems for Maxwell’s equations......Page 149
2. A fundamental solution to Maxwell’s operator......Page 155
3. Green’s formulae......Page 162
4. Representation of solutions and layer potentials......Page 164
5. The uniqueness of a solution......Page 167
References......Page 174
1. Introduction......Page 176
2. Preliminary results......Page 177
3. Dichotomy and boundedness......Page 179
4. The case of operators acting on Banach spaces......Page 182
References......Page 184
1. Introduction......Page 186
2. Preliminaries and the new model......Page 188
3. Stability analysis......Page 190
4. Stabilization......Page 193
5. Robustness......Page 195
6. Numerical example......Page 200
7. Conclusions and further work......Page 202
References......Page 203
1. Introduction......Page 206
2. Preliminary results......Page 209
3. Fourier representation of the operators LPer±......Page 224
4. Fourier representation for the Hill–Schrodinger operator with Dirichlet boundary conditions......Page 227
5. Localization of spectra......Page 233
6. Conclusion......Page 242
References......Page 244
1. Introduction......Page 248
2. Proof of Theorem 1.1......Page 250
3. Examples......Page 251
4. Bi-quantum channels......Page 254
References......Page 255
1. Introduction......Page 258
2.1. Preliminaries; the functional model of a symmetric operator......Page 261
2.2. The functional model for J-self-adjoint extensions of symmetric operators......Page 262
2.3. The Sturm-Liouville case......Page 267
3.1. Point spectrum of the model operator......Page 271
3.2. Essential and discrete spectra of the model operator and of indefinite Sturm-Liouville operators......Page 281
3.3. Non-emptiness of resolvent set for Sturm-Liouville operators......Page 283
4.1. The absence of embedded eigenvalues for the case of infinite-zone potentials......Page 284
4.2. Other applications......Page 287
5. Remarks on indefinite Sturm-Liouville operators with the singular critical point 0......Page 288
6. Discussion......Page 290
Appendix A. Boundary triplets for symmetric operators......Page 293
References......Page 294
1. Introduction......Page 300
2. The AKNS differential operator......Page 303
3. Operators with symmetries......Page 304
4. Square-integrable solutions on a half-line......Page 308
5. Jost solutions......Page 313
6. Bounds on the location of eigenvalues......Page 319
7. Nonexistence of eigenvalues......Page 320
8. Nonexistence of purely imaginary eigenvalues......Page 324
9. Purely imaginary eigenvalues......Page 325
References......Page 332
1. Introduction......Page 336
2. Preliminaries......Page 338
3. Relaxed discrete algebraic scattering systems......Page 341
4. Interpolants of relaxed discrete algebraic scattering systems and forms defined on them......Page 342
5. An extension theorem for bounded forms defined in relaxed discrete algebraic scattering systems......Page 343
5.1. The extension theorem for bounded forms defined in relaxed discrete alge- braic scattering systems generalizes the Cotlar-Sadosky Extension Theorem......Page 349
5.2. The extension theorem for bounded forms defined in relaxed discrete alge- braic scattering systems generalizes the Relaxed Commuting Lifting Theorem......Page 351
References......Page 354
1. Introduction......Page 358
2.1. Prerequisites......Page 361
2.2. Spherical means......Page 362
2.3. Related integral operators......Page 363
2.4. Integral representation formulas......Page 364
3.1. An integral formula......Page 365
3.2. Two lemmas......Page 366
3.3. Proofs of Theorems A and B......Page 367
4.1. Refining Theorem A......Page 368
4.2. Refining Theorem B......Page 369
References......Page 371
1. Introduction......Page 374
2.1. Lipschitz domains and nontangential maximal function......Page 379
2.2. Smoothness spaces in Rn......Page 380
2.3. Smoothness spaces in Lipschitz domains......Page 381
2.4. Smoothness spaces on Lipschitz boundaries......Page 383
3.1. Whitney-Lebesgue, Whitney-Sobolev and Whitney-Besov spaces......Page 387
3.2. Whitney-Hardy and Whitney-Triebel-Lizorkin spaces......Page 388
3.3. Multi-trace theory......Page 389
3.4. Whitney-BMO and Whitney-VMO spaces......Page 393
4.1. The fundamental solution......Page 394
4.2. Definition and nontangential maximal estimates......Page 396
4.3. Jump relations......Page 402
4.4. Estimates on Besov and Triebel-Lizorkin spaces......Page 405
5.1. Definition and nontangential maximal estimates......Page 407
5.3. The conormal derivative......Page 410
5.4. Jump relations for the conormal derivative......Page 414
References......Page 416
1. Introduction......Page 420
2. The spectral expansion method (IEM)......Page 421
3. Numerical properties of the method......Page 426
4. Retrospective of the work with Israel Koltracht......Page 430
5. Subsequent developments......Page 431
6. Summary and conclusions......Page 432
Appendix A......Page 433
References......Page 436
1. Introduction......Page 438
2. Notation and preliminaries......Page 440
3. Regularized perturbation determinants......Page 441
4. The regularized perturbation 2-determinant is a KdV invariant......Page 445
5. Almost conserved quantities......Page 449
6. Applications to spectral theory of the Schrodinger operator......Page 450
7. Appendix: Impedance form of Schr¨odinger operators with singular potentials......Page 453
References......Page 454