A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam

Cover......Page 1
Operator Theory: Advances and Applications
207......Page 3
A Glimpse at
Hilbert Space
Operators......Page 4
ISBN 9783034603461......Page 5
Table of Contents
......Page 6
Preface......Page 8
Part I:
Paul Halmos......Page 11
Introduction......Page 13
The invariant subspace problem......Page 14
Quasitriangularity, quasidiagonality and the Weyl-von Neumann-Voiculescu theorem......Page 15
Subnormal operators and unitary dilations......Page 16
A brief biography......Page 18
Two non-technical books by Halmos......Page 20
Excerpts from: “How to write mathematics”, Enseign. Math. (2) 16 (1970), 123–152.......Page 21
Excerpts from: “How to talk mathematics”, Notices of AMS 21 (1974), 155– 158.......Page 23
Excerpt from: Response from Paul Halmos on winning the Steele Prize for Exposition (1983).......Page 25
Excerpts from: “Four panel talks on publishing”, American Mathematical Monthly 82 (1975), 14–17.......Page 26
Excerpt from: I Want to Be a Mathematician, pp. 321–322, Springer-Verlag, New York (1985).......Page 27
Excerpt from: “The problem of learning to teach”, American Mathematical Monthly 82 (1975), 466–476.......Page 28
Excerpt from: “The heart of mathematics”, American Mathematical Monthly 87 (1980), 519–524.......Page 29
Excerpt from: “Mathematics as a creative art”, American Scientist 56 (1968), 375–389.......Page 30
Excerpt from: “Applied mathematics is bad mathematics”, pp. 9–20, appearing in Mathematics Tomorrow, edited by Lynn Steen, Springer-Verlag, New York (1981).......Page 33
Excerpt from: I Want to Be a Mathematician, p. 400, Springer-Verlag, New York (1985).......Page 35
Obituary: Paul Halmos, 1916–2006......Page 37
Research and Expository Articles......Page 43
Books......Page 49
Photos......Page 51
Part II:
Articles......Page 89
1. Introduction......Page 91
2. Realization formula......Page 92
3. Pick problem......Page 94
4. Nevanlinna problem......Page 96
6. Interpolating sequences......Page 97
7. Corona problem......Page 100
8. Distinguished and toral varieties......Page 102
9. Extension property......Page 103
References......Page 104
1. Preface......Page 109
2. Origins......Page 110
3. Positive linear maps on commutative ∗-algebras......Page 113
4. Subnormality......Page 115
5. Commutative dilation theory......Page 119
6. Completely positivity and Stinespring’s theorem......Page 121
7. Operator spaces, operator systems and extensions......Page 124
8. Spectral sets and higher-dimensional operator theory......Page 126
9. Completely positive maps and endomorphisms......Page 128
Appendix: Brief on Banach ∗-algebras......Page 130
References......Page 132
Toeplitz Operators......Page 135
Products of Toeplitz operators......Page 136
The spectrum of a Toeplitz operator......Page 137
Subnormal Toeplitz operators......Page 138
The symbol map......Page 140
Compact semi-commutators......Page 141
References......Page 142
1. Introduction......Page 145
2. An open mapping theorem for bilinear maps......Page 147
3. Hyper-reflexivity and dilations......Page 151
4. Dominating spectrum......Page 156
5. A noncommutative example......Page 159
6. Approximate factorization......Page 161
7. Contractions with isometric functional calculus......Page 168
8. Banach space geometry......Page 171
9. Dominating spectrum in Banach spaces......Page 173
10. Localizable spectrum......Page 176
11. Notes......Page 180
References......Page 182
1. Introduction......Page 187
2.2......Page 189
2.4. Theorem.......Page 190
3.1. Proposition.......Page 191
3.5. Theorem.......Page 192
4.1. Theorem.......Page 193
4.2. Theorem.......Page 194
5. Bounded point evaluations......Page 195
5.2. Theorem.......Page 197
5.4. Problem.......Page 198
5.8. Problem.......Page 199
5.10. Theorem.......Page 200
5.11. Theorem.......Page 201
References......Page 202
1. Hyponormal operators......Page 205
2. Linear operators as positive functionals......Page 207
3. k-hyponormality for unilateral weighted shifts......Page 209
4. The case of Toeplitz operators......Page 212
References......Page 214
1. Introduction......Page 219
2. Weyl–von Neumann theorems......Page 221
3. Essentially normal operators......Page 223
4. Almost commuting matrices......Page 227
References......Page 230
1. Introduction......Page 233
2. The operator Fejer-Riesz theorem......Page 235
3. Method of Schur complements......Page 238
4. Spectral factorization......Page 243
5. Multivariable theory......Page 249
6. Noncommutative factorization......Page 254
Appendix: Schur complements......Page 259
References......Page 261
1. Introduction......Page 265
2. Halmos’s theorem......Page 266
3. C∗-correspondences, tensor algebras and C∗-envelopes......Page 270
4. Representations and dilations......Page 275
5. Induced representations and Halmos’s theorem......Page 277
6. Duality and commutants......Page 282
7. Noncommutative function theory......Page 284
References......Page 292
1. Introduction......Page 297
2. Double operator integrals......Page 302
3. Multiple operator integrals......Page 305
4. Besov spaces......Page 307
6. Operator Lipschitz and operator differentiable functions. Sufficient conditions......Page 309
7. Operator Lipschitz and operator differentiable functions. Necessary conditions......Page 315
8. Higher-order operator derivatives......Page 317
9. The case of contractions......Page 319
10. Operator Holder–Zygmund functions......Page 323
11. Lifshits–Krein trace formulae......Page 325
12. Koplienko–Neidhardt trace formulae......Page 327
13. Perturbations of class Sp......Page 328
References......Page 331
The Halmos Similarity Problem......Page 335
References......Page 346
Paul Halmos and Invariant Subspaces......Page 351
References......Page 356
Commutant Lifting......Page 361
References......Page 366
2. The (H2, | · |) model of Minkowski space......Page 369
3. Some applications......Page 371
References......Page 372
Operator Theory: Advances and Applications (OT)......Page 373