A Glimpse at Hilbert Space Operators (eBook)

Copyright Page 5
Table of Contents 6
Preface 8
Part I Paul Halmos 10
Paul Halmos – Expositor Par Excellence* 12
Introduction 12
The invariant subspace problem 13
Quasitriangularity, quasidiagonality and the Weyl-von Neumann-Voiculescu theorem 14
Subnormal operators and unitary dilations 15
Ergodic theory 17
A brief biography 17
Suggested reading 19
Technical papers by Halmos 19
Expository articles by Halmos 19
Two non-technical books by Halmos 19
Paul Halmos: In His Own Words* 20
On writing 20
Excerpts from: “How to write mathematics”, Enseign. Math. (2) 16 (1970), 123–152. 20
On speaking 22
Excerpts from: “How to talk mathematics”, Notices of AMS 21 (1974), 155– 158. 22
Excerpt from: I Want to Be a Mathematician, p. 401, Springer-Verlag, New York (1985). 24
On exposition 24
Excerpt from: Response from Paul Halmos on winning the Steele Prize for Exposition (1983). 24
On publishing 25
Excerpts from: “Four panel talks on publishing”, American Mathematical Monthly 82 (1975), 14–17. 25
On research 26
Excerpt from: I Want to Be a Mathematician, pp. 321–322, Springer-Verlag, New York (1985). 26
On teaching 27
Excerpt from: “The problem of learning to teach”, American Mathematical Monthly 82 (1975), 466–476. 27
Excerpt from: “The heart of mathematics”, American Mathematical Monthly 87 (1980), 519–524. 28
On mathematics 29
Excerpt from: “Mathematics as a creative art”, American Scientist 56 (1968), 375–389. 29
On pure and applied 32
Excerpt from: “Applied mathematics is bad mathematics”, pp. 9–20, appearing in Mathematics Tomorrow, edited by Lynn Steen, Springer-Verlag, New York (1981). 32
On being a mathematician 34
Excerpt from: I Want to Be a Mathematician, p. 400, Springer-Verlag, New York (1985). 34
Obituary: Paul Halmos, 1916–2006 35
Mathematical Review of “How to Write Mathematics” * 38
Publications of Paul R. Halmos 39
Research and Expository Articles 39
Books 45
Photos 47
Part II Articles 84
What Can Hilbert Spaces Tell Us About Bounded Functions in the Bidisk? 85
1. Introduction 85
2. Realization formula 86
3. Pick problem 88
4. Nevanlinna problem 90
5. Takagi problem 91
6. Interpolating sequences 91
7. Corona problem 94
8. Distinguished and toral varieties 96
9. Extension property 97
10. Conclusion 98
References 98
Dilation Theory Yesterday and Today 102
1. Preface 102
2. Origins 103
3. Positive linear maps on commutative *-algebras 106
4. Subnormality 108
5. Commutative dilation theory 112
6. Completely positivity and Stinespring’s theorem 114
7. Operator spaces, operator systems and extensions 117
8. Spectral sets and higher-dimensional operator theory 119
9. Completely positive maps and endomorphisms 121
Appendix: Brief on Banach *-algebras 123
References 125
Toeplitz Operators 127
Products of Toeplitz operators 128
The spectrum of a Toeplitz operator 129
Subnormal Toeplitz operators 130
The symbol map 132
Compact semi-commutators 133
Remembering Paul Halmos 134
References 134
Dual Algebras and Invariant Subspaces 136
1. Introduction 136
2. An open mapping theorem for bilinear maps 138
3. Hyper-reflexivity and dilations 142
4. Dominating spectrum 147
5. A noncommutative example 150
6. Approximate factorization 152
7. Contractions with isometric functional calculus 159
8. Banach space geometry 162
9. Dominating spectrum in Banach spaces 164
10. Localizable spectrum 167
11. Notes 171
References 173
The State of Subnormal Operators 178
1. Introduction 178
2. Fundamentals of subnormal operators 180
2.1. Proposition. 180
2.2 180
2.3. Proposition. 181
2.4. Theorem. 181
3. The functional calculus 182
3.1. Proposition. 182
3.2. Proposition. 183
3.3. Sarason’s Theorem. 183
3.4. Proposition. 183
3.5. Theorem. 183
3.6. Theorem. 184
4. Invariant subspaces 184
4.1. Theorem. 184
4.2. Theorem. 185
4.3. Theorem. 186
4.4. Theorem. 186
4.5. Problem. 186
5. Bounded point evaluations 186
5.1. Thomson’s Theorem. 188
5.2. Theorem. 188
5.3. Problem. 189
5.4. Problem. 189
5.5. Theorem. 190
5.6. Problem. 190
5.7. Theorem. 190
5.8. Problem. 190
5.9. Problem. 191
5.10. Theorem. 191
5.11. Theorem. 192
5.12. Problem. 193
5.13. Problem. 193
6. Conclusion 193
References 193
Polynomially Hyponormal Operators 196
1. Hyponormal operators 196
2. Linear operators as positive functionals 198
3. k-hyponormality for unilateral weighted shifts 200
4. The case of Toeplitz operators 203
References 205
Essentially Normal Operators 209
1. Introduction 209
2. Weyl–von Neumann theorems 211
3. Essentially normal operators 213
4. Almost commuting matrices 217
References 220
The Operator Fejer-Riesz Theorem 223
1. Introduction 223
2. The operator Fejer-Riesz theorem 225
3. Method of Schur complements 228
4. Spectral factorization 233
5. Multivariable theory 239
6. Noncommutative factorization 244
Appendix: Schur complements 249
References 251
A Halmos Doctrine and Shifts on Hilbert Space 255
1. Introduction 255
2. Halmos’s theorem 256
3. C*-correspondences, tensor algebras and C*-envelopes 260
4. Representations and dilations 265
5. Induced representations and Halmos’s theorem 267
6. Duality and commutants 272
7. Noncommutative function theory 274
References 282
The Behavior of Functions of Operators Under Perturbations 286
1. Introduction 286
2. Double operator integrals 291
3. Multiple operator integrals 294
4. Besov spaces 296
5. Nuclearity of Hankel operators 298
6. Operator Lipschitz and operator differentiable functions. Sufficient conditions 298
7. Operator Lipschitz and operator differentiable functions. Necessary conditions 304
8. Higher-order operator derivatives 306
9. The case of contractions 308
10. Operator Holder–Zygmund functions 312
11. Lifshits–Krein trace formulae 314
12. Koplienko–Neidhardt trace formulae 316
13. Perturbations of class Sp 317
References 320
The Halmos Similarity Problem 324
References 335
Paul Halmos and Invariant Subspaces 339
References 344
Commutant Lifting 348
References 353
Double Cones are Intervals 355
1. Introduction 355
2. The (H2, | · |) model of Minkowski space 355
3. Some applications 357
References 358
Operator Theory: Advances and Applications (OT) 359
Operator Theory: Advances and Applications (OT) 360