Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek

Preface......Page 6
Contents......Page 7
Curriculum Vitae of M.A. Kaashoek......Page 10
 Research monographs......Page 14
 Papers in professional journals......Page 15
 Edited books and journal issues......Page 27
 Other publications......Page 28
Ph.D. students of M.A. Kaashoek......Page 31
 Marinus A. Kaashoek: Guide, Companion and Friend......Page 32
  Amsterdam......Page 36
  Potchefstroom......Page 38
 My PhD time with Rien Kaashoek......Page 39
 Personal reminiscences: working with Rien......Page 40
 References......Page 41
 Working with Rien Kaashoek......Page 42
 Rien Kaashoek: Mentor, colleague and friend......Page 43
 Introduction......Page 46
 1. Five types of tangent......Page 49
 2. Tangents with a unique extremal Φω......Page 50
 3. Royal tangents......Page 57
 4. Flat tangents......Page 60
 5. Purely balanced tangents......Page 61
 6. Relation to a result of L. Kosiński and W. Zwonek......Page 63
 References......Page 65
Standard versus strict Bounded Real Lemma with infinite-dimensional state space III: The dichotomous and bicausal cases......Page 67
 1. Introduction......Page 68
 2. Dichotomous system theory......Page 73
 3. Bicausal systems......Page 81
 4. Storage functions......Page 88
 5. The available storage and required supply......Page 96
 6. Storage functions for bicausal systems......Page 102
 7. Dichotomous and bicausal bounded real lemmas......Page 107
 8. Bounded real lemma for nonstationary systems with dichotomy......Page 112
 References......Page 115
 1. Introduction......Page 118
  2.1. Terminology, notation and statement of basic results......Page 120
  2.2. Verification via reduction to the upper triangular case......Page 123
  2.3. An algorithm for constructing canonical forms......Page 124
  2.4. An example......Page 126
  3.1. Zero pattern subspaces and algebras......Page 128
  3.3. L-free directed bipartite graphs: definition and related notions......Page 129
  3.4. In-diagrams and out-diagrams......Page 132
  3.5. L-closures......Page 136
 4. L-free directed graphs: canonical forms......Page 137
  5.2. Relationship of the in/out-diagrams to in/out-ultra closures......Page 143
  5.3. L-closures of upper triangular-type partial orders......Page 144
  5.4. Canonical forms......Page 147
 6. Application: counting equivalence classes......Page 148
 7. Concluding remarks and open problems......Page 156
  7.1. L-free directed bipartite graphs: characterization......Page 157
  7.2. Issues for further research......Page 158
 References......Page 159
 1. Introduction......Page 161
 2. Main results......Page 163
 3. An example......Page 166
 4. Proof of the main results......Page 169
 References......Page 183
 1. A personal note......Page 186
  2.1. Wiener–Hopf factorization......Page 187
  2.2. Finite section method......Page 189
 3. Operators on the Bergman space......Page 190
  3.1. Finite section method......Page 191
  3.3. Polynomial collocation......Page 192
  3.4. The Bergman metric......Page 193
  3.5. Analytic element collocation......Page 194
  3.6. Test computations......Page 196
  3.7. General domains......Page 203
 Appendix: Phase plots......Page 204
 References......Page 205
 1. Introduction......Page 208
 2. Approximate identities and VMO......Page 212
  3.1. Fibers over......Page 214
  3.2. Fibers over......Page 216
 4. Localization of PQC OVER QC......Page 218
 References......Page 220
 1. Introduction......Page 221
  2.1. The general quasiseparable structure......Page 224
  2.2. The diagonal plus small rank representation......Page 225
  3.1. The Sturm sequences property for a Hermitian matrix......Page 226
  3.3. The basic eigenvalue algorithm......Page 227
  3.4. The computation of eigenvectors......Page 228
 4. The inverse problem......Page 230
  5.1. The quasiseparable generators of A = (A0)*A0......Page 232
  5.2. Computation in linear time of the standard matrix norm before SVD......Page 233
 6. Numerical experiments......Page 234
   6.1.1. Eigenvalues known in advance.......Page 235
   6.1.3. Eigenvalues of Hermitian band matrices.......Page 236
  6.2. Results for singular values......Page 237
 7. Conclusions and future work......Page 238
 References......Page 239
 1. Introduction......Page 241
 2. Preliminaries......Page 242
 3. Inner-outer factorization......Page 243
 4. An auxiliary lemma......Page 248
 5. Proof of the inner-outer factorization......Page 249
 References......Page 253
 1. Introduction......Page 255
 2. On a special kind of Schur complement......Page 258
 3. On a restricted extension problem for a finite Hankel non-negative definite extendable sequence......Page 263
 4. On equivalence classes of truncated matricial moment problems of type MP[R; (sj)2n j=0, ≤]......Page 266
 5. On truncated matricial [α, ∞)-Stieltjes moment problems......Page 269
 6. On a restricted extension problem for a finite α-Stieltjes right-sided non-negative definite extendable sequence......Page 270
 7. On equivalence classes of truncated matricial moment problems of type MP[[α, ∞); (sj)m j=0, ≤]......Page 274
 References......Page 277
 1. Introduction......Page 279
 2. Basic properties of Tω......Page 284
 3. Intermezzo: Division with remainder by a polynomial in Hp......Page 286
 4. Fredholm properties of Tω for ω ∈ Rat(T)......Page 290
 5. Fredholm properties of Tω: General case......Page 295
 6. Matrix representation......Page 300
 7. Examples......Page 304
 References......Page 307
 1. Introduction......Page 309
  2.1. Complex case......Page 312
  2.2. Real case......Page 315
  3.1. Eigenvalue one......Page 318
  3.3. Eigenvalues not ±1......Page 322
  4.2. Complex unimodular eigenvalues......Page 323
   4.2.1. The case when n is odd......Page 325
   4.2.2. The case when......Page 327
 5. The complex H-symplectic case......Page 328
 References......Page 329
 1. Introduction......Page 331
 2. Properties of the Mellin transformation......Page 332
  3.1. Boundedness......Page 342
  3.2. Fredholmness......Page 350
 References......Page 364
 1. Introduction......Page 366
  Problem 2.1 (Problems of system theory of rational systems).......Page 368
  3.1. Example Glycolysis in Yeast......Page 369
  3.2. A Rational System of an Economic Control Problem......Page 374
  4.1. Concepts......Page 376
 5. Canonical Forms of Rational Systems......Page 380
 6. Decompositions and Subclasses of Rational Systems......Page 384
  6.1. A Line Network of Rational Control Systems......Page 385
  6.2. A Ring Network of Rational Systems......Page 386
 7. System Approximation......Page 388
 8. Control Synthesis of Rational Systems......Page 390
 9. Computer Algebra for Rational Systems......Page 392
 10. Concluding Remarks......Page 394
 References......Page 395
 1. Introduction......Page 399
 2. A brief survey of recent results......Page 402
 3. Some interesting proof techniques......Page 410
 4. Open problems......Page 417
 5. A new result......Page 419
 References......Page 423
 1. Introduction......Page 426
 2. GBDT of discrete skew-selfadjoint Dirac systems......Page 427
 3. Explicit solutions of the corresponding non-stationary systems......Page 433
 References......Page 434
 1. Introduction......Page 436
 2. Review of some known results......Page 438
 3. Direct extensions......Page 443
 4. Proofs......Page 445
 5. Equivalence after extension......Page 447
 6. The case X = Y and A = I......Page 449
  7.2. Equivalence after extension (EAE)......Page 450
  7.4. Delta-relations......Page 451
  7.6. Schur coupling (SC)......Page 452
 8. Open problems......Page 453
 References......Page 454
 1. Introduction......Page 457
 2. Proof of Main Result with Consequences......Page 460
 3. Semidefinite Toeplitz Completions......Page 468
 4. The Singularity Degree of Some Toeplitz Cycles......Page 470
 References......Page 475
 1. Introduction......Page 478
 2. Preliminaries......Page 481
 3. Tm-invariant symbols......Page 482
 4. Toeplitz operator action and algebras......Page 487
 5. C*-algebras......Page 490
 6. C*-algebras of Toeplitz operators on the Fock space F2(Cn)......Page 497
 References......Page 499