Topics in Operator Theory (eBook)

Title Page 3
Copyright Page 4
Table of Contents 5
The XIXth International Workshop on Operator Theory and its Applications. I 8
Speeches and Reminiscences 11
1. Presentation of book 11
2. Gohberg’s colleagues 14
3. Gohberg’s family 23
3.1. The young years of Israel Gohberg 23
3.2. My father I.C. Gohberg 26
3.3. Dad’s 80th birthday 28
3.4. Family reminiscences 29
3.5. Congratulations Izinka 31
3.6. My grandfather 32
4. To Izia Gohberg on his 80th birthday 33
5. Reminiscences of meetings with Israel Cudicovic Gohberg 34
A Quantitative Estimate for Bounded Point Evaluations in Pt(µ)-spaces 37
1. Introduction 37
2. Thomson’s theorem 38
3. Some auxiliary lemmas 40
4. The proof of Theorem 1.1 42
5. Analytic bounded point evaluations 44
References 45
Weighted Composition Operators on the Bloch Space of a Bounded Homogeneous Domain 47
1. Introduction 47
1.1. Purpose of the paper 48
1.2. Organization of the paper 48
2. The Bloch space 49
3. Weighted composition operators on the Bloch space of D 51
4. Weighted composition operators on the Bloch space of a bounded homogeneous domain 53
5. Special case: The unit ball 58
6. Special case: The unit polydisk 63
7. Weighted composition operators from the Bloch spaces into H8 66
8. Further developments 70
8.1. Isometries 70
8.2. Spectrum 71
8.3. Essential norm 71
References 72
Images of Minimal-vector Sequences Under Weighted Composition Operators on L2(D) 74
1. Introduction and background 74
2. Koenigs and Valiron functions 78
3. Regularity properties of a 80
4. Main results 83
References 85
On Extensions of Indefinite Toeplitz-Kre n-Cotlar Triplets 87
1. Introduction 87
2. Preliminaries 88
3. Toeplitz-Kre n-Cotlar triplets 90
4. Extension result 92
5. Generalized Toeplitz kernels with real parameter 93
References 94
Multivariable Weighted Composition Operators: Lack of Point Spectrum, and Cyclic Vectors 96
1. Introduction 96
2. The single-variable case 98
3. The multivariable case 100
4. Cyclic vectors for Ta, a . Qd 107
4.1. The case a . Q2 108
4.1.1. The case q1 = q2. 108
4.1.2. The case q1 = q2. 116
4.2. Cyclic vectors for Ta, a . Qd, d = 1 116
References 117
Factorization Algorithm for Some Special Non-rational Matrix Functions 119
1. Introduction 119
2. Relations between a generalized factorization of A.(b) and the spectrum of N+(b) 122
3. Relations between the spectrum of the operator N+(b) and a linear system 128
4. Algorithms [AEq] and [AFact] 135
5. Examples 138
5.1. Canonical factorization 138
5.2. Non-canonical factorization 139
References 140
Structured Primal-dual Interior-point Methods for Banded Semidefinite Programming 142
1. Introduction 142
2. Sequentially semi-separable representation (SSS) for banded matrices 147
2.1. Structures of SSS matrices 147
2.2. Numerical rank reduction 152
2.3. SSS matrix operations 155
3. Structured primal-dual interior-point method 160
3.1. Method descriptions 161
3.2. Algorithm analysis 164
3.3. Experiments 166
4. Conclusion and future work 171
References 171
A Note on Semi-Fredholm Hilbert Modules 173
1. Introduction 173
2. Necessary conditions 174
3. Sufficient conditions 176
4. Further comments 179
References 180
The S-recurrence of Schur Parameters of Non-inner Rational Schur Functions 181
0. Introduction 181
1. Preliminaries 186
2. The S-recurrence property of the Schur parameter sequences associated with non-inner rational Schur functions 191
3. Recovering the matrices Lr+1(.) from its first column and the sequence (.j)rj=0 204
4. On the structure of the information matrix Ar+1,0 208
5. Constructing a sequence belonging to Gl2 and having finite rank n0 = r from a section (.j)rj=0 and compatible data [.1,1, (Lj,1)rj=1] 211
References 223
Curvature of Universal Bundles of Banach Algebras 225
1. Introduction 225
2. Algebraic preliminaries 227
2.1. The Grassmannian over a semigroup 227
2.2. The canonical section 228
2.3. Partial isomorphisms and relative inverses 228
2.4. Proper partial isomorphisms 229
2.5. The spaces V (p,A) and Gr(p,A) 230
2.6. The role of the canonical section 231
2.7. The spatial correspondence 233
3. The restricted Banach *-algebra Ares and the space of polarizations 234
3.1. Hilbert modules and their polarizations 234
3.2. The restricted Banach *-algebra Ares 235
3.3. The space P of polarizations 235
4. Constructions for the submanifold geometry and bundle theory 236
4.1. Some preliminaries 236
4.2. The tangential extension 237
4.3. Tangential isomorphisms 238
5. The space V. and its geometry 239
5.1. Transversality and the transition map 239
5.2. The connection map V 240
6. The connection form and its curvature 241
6.1. The connection form .. 241
6.2. The curvature form O. 242
7. The universal bundle over . 244
7.1. The Koszul connection 244
8. The T -function 246
8.1. Definition of the T -function 246
8.2. Curvature formulas 247
8.3. Remarks on the operator cross ratio 248
8.4. The connection and curvature forms on VP 249
8.5. Trace class operators and the determinant 249
References 250
A Contractive Operator View on an Inversion Formula of Gohberg-Heinig 253
1. Introduction 253
2. Proof of Theorem 1.2 258
3. Invertibility and proof of Theorem 1.1 261
4. Toeplitz operators 265
5. Toeplitz plus Hankel 266
6. Compressions of a Toeplitz operator 270
6.1. The isometric lifting setting 271
6.2. The model case 272
7. Inverting solutions of Stein equations 275
References 280
A Spectral Weight Matrix for a Discrete Version of Walsh’s Spider 283
1. Birth-and-death processes and orthogonal polynomials 283
2. Recall M.G. Krein 285
3. The first example 287
4. Random walk with an attractive force 287
5. Allowing for a “defect” at the origin 288
6. An assortment of graphs 289
7. Spider or star graphs 290
8. The invariant measure 293
References 293
Norm Inequalities for Composition Operators on Hardy and Weighted Bergman Spaces 295
1. Introduction 295
2. Positive semidefinite matrices 297
3. Norm inequalities 298
4. Open questions 300
References 302
Theory vs. Experiment: Multiplicative Inequalities for the Numerical Radius of Commuting Matrices 303
1. Introduction 303
2. Multiplicative inequalities relative to w(T) S 305
3. Multiplicative inequalities relative to w(T)w(S) 309
References 313
Best Constant Inequalities Involving the Analytic and Co-Analytic Projection 315
1. Introduction 315
2. A best constant inequality involving P+ and P 316
3. Some open problems 323
References 325
Quasi Commutativity of Regular Matrix Polynomials: Resultant and Bezoutian 326
0. Introduction 326
1. Definition of total common multiplicity 329
2. The Bezout matrix for regular matrix polynomials 331
3. The resultant in relation to the Bezout matrix 336
4. Proof of the sufficiency part of Theorem 0.1 339
References 342
Quasidiagonal Extensions of the Reduced Group C*-algebras of Certain Discrete Groups 344
1. Introduction 344
2. Proof of the theorem 346
References 348
Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves 349
1. Introduction 349
2. Preliminaries and main results 351
2.1. Carleson curves 351
2.2. Variable Lebesgue spaces with weights 352
2.3. Boundedness of the Cauchy singular integral operator 352
2.4. Fredholm criterion 353
3. Proof of the boundedness result 356
3.1. Main ingredients 356
4. Proof of the Fredholm criterion for the operator aP + Q 358
4.1. Local representatives 358
4.2. Wiener-Hopf factorization of local representatives 358
4.3. Proof of Theorem 2.2 359
5. Construction of the symbol calculus 360
5.1. Allan-Douglas local principle 360
5.2. Localization 360
5.3. The two projections theorem 361
5.4. Local algebras At and Lt are subject to the two projections theorem 362
References 362
Almost Periodic Polynomial Factorization of Some Triangular Matrix Functions 365
1. Introduction 365
2. Borderline trinomials 367
3. Beyond trinomials 377
References 381
Revisit to a Theorem of Wogen 383
1. Introduction 383
2. Carleson measures 384
3. A new proof for the sufficiency of Wogen’s theorem 386
4. General inducing maps 388
References 391
Survey on the Best Constants in the Theory of One-dimensional Singular Integral Operators 392
1. Introduction 392
2. G is the unit circle 395
3. G coinciding with R or with its connected subset 399
4. A local principle for best constants 403
5. Composed contours. Norms and essential norms 407
6. On the norms of polynomials in S and S* 411
7. Some important inequalities and their applications 414
8. Symmetric symbols and their applications 416
References 417
Gantmacher–Krein Theorem for 2-totally Nonnegative Operators in Ideal Spaces 421
1. Introduction 421
2. Ideal spaces. Basic definitions and statements 424
3. Tensor and exterior squares of ideal spaces 425
4. Tensor and exterior squares of linear operators in ideal spaces 427
5. Spectrum of the tensor square of linear operators in ideal spaces 428
6. Spectrum of the exterior square of linear operators in ideal spaces 431
7. Generalization of the Gantmacher–Krein theorems in the case of 2-totally nonnegative operators in ideal spaces 433
References 435
Conditions for Linear Dependence of Two Operators 437
1. Introduction 437
2. Main results 438
3. Proof of Theorem 2.2: t = 0 440
4. Proof of Theorem 2.2: t = p/2 446
5. Proof of Theorem 2.2: 0 < t <
6. Linear dependence in terms of trace functionals 456
References 459
Matrix Inequalities and Twisted Inner Products 461
1. Introduction 461
2. Multilinear functions, contractions, main theorem 463
3. Additional inner products, more inequalities 469
4. Indication of additional applications, concluding remarks 474
References 475
The Spectrum of a Composition Operator and Calderon’s Complex Interpolation 477
1. Introduction 477
2. Preliminaries 478
2.1. The weighted Dirichlet spaces 478
2.2. The analytic Besov spaces 479
2.3. Automorphisms of the disk and their composition operators 480
3. Interpolating spectra 482
4. Spectra of composition operators with automorphic symbol 486
4.1. The weighted Dirichlet spaces 486
4.2. The analytic Besov spaces 489
5. A non-automorphic example 489
References 492
Almost Periodic Factorization of 2 × 2 Triangular Matrix Functions: New Cases of Off Diagonal Spectrum 494
1. Introduction 494
2. Portuguese transformation 497
3. Quadrinomial off diagonal entry 498
4. .-admissibility of sets 502
5. Main results 504
6. On the geometric mean computations 508
7. Off diagonal trinomials 510
References 511
Infinite Hankel Block Matrices, Extremal Problems 513
1. Introduction 513
2. Extremal problem 515
References 519
On Compactness of Operators in Variable Exponent Lebesgue Spaces 520
1. Introduction 520
2. Preliminaries 521
3. Two general results on compactness of operators 522
3.1. Dominated compactness theorem 522
3.2. Compactness interpolation theorem 523
4. Compactness of an integral operator with integrable almost decreasing radial dominant of the kernel in the case |O| < 8
4.1. Non-weighted case 525
4.2. Weighted case 525
5. The case O = Rn: compactness of convolution type operators with coefficients vanishing at infinity 528
References 529
Extension to an Invertible Matrix in Convolution Algebras of Measures Supported in [0,+8) 532
1. Introduction 532
1.1. Relevance of the Hermiteness of M+ in control theory 534
2. Preliminaries 535
3. Contractibility of X(M+) 538
4. Hermite-ness of some subalgebras of M+ 539
References 541
The Invariant Subspace Problem via Composition Operators-redux 542
1. Introduction 542
2. Background material 544
2.1. Disc automorphisms 544
2.2. Spectra of hyperbolic-automorphic composition operators 545
2.3. Poisson kernel estimates 547
3. Main results 548
4. Complements and comments 554
4.1. Non-canonical hyperbolic automorphisms 554
4.2. The Nordgren-Rosenthal-Wintrobe Theorem 555
4.3. Cyclicity 556
References 557
On Norms of Completely Positive Maps 558
References 561
Some Exponential Inequalities for Semisimple Lie Groups 562
1. Introduction 562
2. Preliminaries 563
3. A pre-order of Kostant and some order relations 567
4. Extension of Araki’s result 572
References 574
Parabolic Quasi-radial Quasi-homogeneous Symbols and Commutative Algebras of Toeplitz Operators 576
1. Introduction 576
2. Preliminaries 577
3. Parabolic quasi-radial symbols 582
4. Commutativity results 586
References 591
Algebraic Aspects of the Paving and Feichtinger Conjectures 592
1. Introduction 592
2. Paving Laurent operators and frame theory 594
3. Convolution and Segal algebras 597
4. Gabor systems 599
References 600
Dominating the Commutator 602
1. Introduction 602
2. Appearances and other deceits 603
3. Bounding the problem 606
4. Fundamental examples 608
5. The constants for general p 611
6. Pinpoints 616
7. The border v2 620
8. Open questions 621
References 623