This book is a systematic presentation of the theory of Hankel operators. It covers the many different areas of Hankel operators and presents a broad range of applications, such as approximation theory, prediction theory, and control theory. The author has gathered the various aspects of Hankel operators and presents their applications to other parts of analysis. This book contains numerous recent results which have never before appeared in book form. The author has created a useful reference tool by pulling this material together and unifying it with a consistent notation, in some cases even simplifying the original proofs of theorems. Hankel Operators and their Applications will be used by graduate students as well as by experts in analysis and operator theory and will become the standard reference on Hankel operators.
Vladimir Peller is Professor of Mathematics at Michigan State University. He is a leading researcher in the field of Hankel operators and he has written over 50 papers on operator theory and functional analysis.
Author(s): Vladimir Peller
Series: Springer Monographs in Mathematics
Edition: 1
Publisher: Springer
Year: 2003
Language: English
Pages: 800
Preface......Page 5
Notation......Page 9
Contents......Page 11
1 An Introduction to Hankel Operators......Page 17
1. Bounded Hankel Operators......Page 18
2. Hankel Operators and Compressed Shift......Page 29
3. Hankel Operators of Finite Rank......Page 35
4. Interpolation Problems......Page 39
5. Compactness of Hankel Operators......Page 41
6. Hankel Operators and Reproducing Kernels......Page 53
7. Hankel Operators and Moment Sequences......Page 55
8. Hankel Operators as Integral Operators on the Semi-Axis......Page 62
Concluding Remarks......Page 72
2 Vectorial Hankel Operators......Page 77
1. Completing Matrix Contractions......Page 78
2. Bounded Block Hankel Matrices......Page 82
3. Hankel Operators and the Commutant Lifting Theorem......Page 87
4. Compact Vectorial Hankel Operators......Page 90
5. Vectorial Hankel Operators of Finite Rank......Page 92
6. Imbedding Theorems......Page 97
Concluding Remarks......Page 100
3 Toeplitz Operators......Page 103
1. Basic Properties......Page 104
2. A General Invertibility Criterion......Page 110
3. Spectra of Certain Toeplitz Operators......Page 113
4. Toeplitz Operators on Spaces of Vector Functions......Page 119
5. Wiener…Hopf Factorizations of Symbols of Fredholm Toeplitz Operators......Page 125
6. Left Invertibility of Bounded Analytic Matrix Functions......Page 135
Concluding Remarks......Page 137
4 Singular Values of Hankel Operators......Page 141
1. The Adamyan…Arov…Krein Theorem......Page 142
2. The Case......Page 147
3. Finite Rank Approximation of Vectorial Hankel Operators......Page 151
4. Relations between Hu and Hu......Page 158
Concluding Remarks......Page 161
5 Parametrization of Solutions of the Nehari Problem......Page 163
1. Adamyan…Arov…Krein Parametrization in the Scalar Case......Page 164
2. Parametrization of Solutions of the Nevanlinna…Pick Problem......Page 186
3. Parametrization of Solutions of the Nehari…Takagi Problem......Page 189
4. Parametrization via One-Step Extension......Page 203
5. Parametrization in the General Case......Page 228
Concluding Remarks......Page 244
6 Hankel Operators and Schatten…von Neumann Classes......Page 247
1. Nuclearity of Hankel Operators......Page 248
2. Hankel Operators of Class......Page 255
3. Hankel Operators of Class......Page 259
4. Hankel Operators and Schatten…Lorentz Classes......Page 269
5. Projecting onto the Hankel Matrices......Page 273
6. Rational Approximation......Page 283
7. Other Applications of the Sp Criterion......Page 291
8. Generalized Hankel Matrices......Page 297
9. Generalized Block Hankel Matrices and Vectorial Hankel Operators......Page 308
Concluding Remarks......Page 312
7 Best Approximation by Analytic and Meromorphic Functions......Page 319
1. Function Spaces That Can Be Described in Terms of Rational Approximation in......Page 322
2. Best Approximation in Decent Banach Algebras......Page 328
3. Best Approximation in Spaces without a Norm......Page 332
4. Examples and Counterexamples......Page 334
5. Badly Approximable Functions......Page 354
6. Perturbations of Multiple Singular Values of Hankel Operators......Page 357
7. The Boundedness Problem......Page 360
8. Arguments of Unimodular Functions......Page 371
9. Schmidt Functions of Hankel Operators......Page 372
10. Continuity in the sup-Norm......Page 375
11. Continuity in Decent Banach Spaces......Page 378
12. The Recovery Problem in Spaces of Measures and Interpolation by Analytic Functions......Page 387
Concluding Remarks......Page 392
8 An Introduction to Gaussian Spaces......Page 395
1. Gaussian Spaces......Page 396
2. The Fock Space......Page 400
3. Mixing Properties and Regularity Conditions......Page 406
4. Minimality and Basisness......Page 418
5. Scattering Systems and Hankel Operators......Page 421
6. Geometry of Past and Future......Page 424
Concluding Remarks......Page 428
1. Minimality in Spectral Terms......Page 431
2. Angles between Past and Future......Page 433
3. Regularity Conditions in Spectral Terms......Page 436
4. Stronger Regularity Conditions......Page 440
Concluding Remarks......Page 442
10 Spectral Properties of Hankel Operators......Page 447
1. The Essential Spectrum of Hankel Operators with Piecewise Continuous Symbols......Page 449
2. The Carleman Operator......Page 456
3. Quasinilpotent Hankel Operators......Page 459
Concluding Remarks......Page 466
11 Hankel Operators in Control Theory......Page 469
1. Transfer Functions......Page 470
2. Realizations with Discrete Time......Page 473
3. Realizations with Continuous Time......Page 482
4. Model Reduction......Page 488
5. Robust Stabilization......Page 489
6. Coprime Factorization......Page 492
7. Proof of Theorem 5.1......Page 495
8. Parametrization of Stabilizing Controllers......Page 498
9. Solution of the Robust Stabilization Problem. The Model Matching Problem......Page 501
Concluding Remarks......Page 503
12 The Inverse Spectral Problem for Self-Adjoint Hankel Operators......Page 505
1. Necessary Conditions......Page 509
2. Eigenvalues of Hankel Operators......Page 513
3. Linear Systems with Continuous Time and Lyapunov Equations......Page 515
4. Construction of a Linear System with Continuous Time......Page 519
5. The Kernel of......Page 525
6. Proofs of Lemmas 4.1 and 5.2......Page 532
7. Positive Hankel Operators with Multiple Spectrum......Page 534
8. Moduli of Hankel Operators, Past and Future, and the Inverse Problem for Rational Approximation......Page 536
9. Linear Systems with Discrete Time......Page 538
10. Passing to Balanced Linear Systems......Page 542
11. Asymptotic Stability......Page 545
12. The Main Construction......Page 547
13. Proof of Theorem 9.1......Page 552
14. Proofs of Lemmas 13.2 and 13.5......Page 558
15. A Theorem in Perturbation Theory......Page 564
Concluding Remarks......Page 566
13 Wiener…Hopf Factorizations and the Recovery Problem......Page 569
1. The Recovery Problem in R-spaces......Page 570
2. Maximizing Vectors of Vectorial Hankel Operators......Page 571
3. Wiener…Masani Factorizations......Page 573
4. Isometric-Outer Factorizations......Page 576
5. The Recovery Problem and Wiener…Hopf Factorizations of Unitary-Valued Functions......Page 577
6. Wiener…Hopf Factorizations. The General Case......Page 578
Concluding Remarks......Page 579
14 Analytic Approximation of Matrix Functions......Page 581
1. Balanced Matrix Functions......Page 584
2. Parametrization of Best Approximations......Page 590
3. Superoptimal Approximation of ... Matrix Functions......Page 594
4. Superoptimal Approximation of Matrix Functions ... with Small Essential Norm of .........Page 596
5. Thematic Factorizations and Very Badly Approximable Functions......Page 602
6. Admissible and Superoptimal Weights......Page 608
7. Thematic Indices......Page 610
8. Inequalities Involving Hankel and Superoptimal Singular Values......Page 616
9. Invariance of Residual Entries......Page 618
10. Monotone Thematic Factorizations and Invariance of Thematic Indices......Page 623
11. Construction of Superoptimal Approximation and the Corona Problem......Page 631
12. Hereditary Properties of Superoptimal Approximation......Page 634
13. Continuity Properties of Superoptimal Approximation......Page 639
14. Unitary Interpolants of Matrix Functions......Page 652
15. Canonical Factorizations......Page 659
16. Very Badly Approximable Unitary-Valued Functions......Page 677
17. Superoptimal Meromorphic Approximation......Page 678
18. Analytic Approximation of In“nite Matrix Functions......Page 688
19. Back to the Adamyan…Arov…Krein Parametrization......Page 697
Concluding Remarks......Page 698
15 Hankel Operators and Similarity to a Contraction......Page 701
1. Operators ... in the Scalar Case......Page 703
2. Power Bounded Operators......Page 709
3. Counterexamples......Page 710
Concluding Remarks......Page 719
Appendix 1 Operators on Hilbert Space......Page 721
Appendix 2 Summary of Function Spaces......Page 733
References......Page 755
Author Index......Page 792
Subject Index......Page 796
