A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam (Operator Theory: Advances and Applications Book 207)

The book is a commemorative volume honoring the mathematician Paul R. Halmos (1916-2006), who contributed passionately to mathematics in manifold ways, among them by basic research, by unparalleled mathematical exposition, by unselfish service to the mathematical community, and, not least, by the inspiration others found in his dedication to that community.Halmos made fundamental contributions in several areas of mathematics. This volume emphasises Halmos's contributions to operator theory, his venue for most of his mathematical life. The core of the volume is a series of expository articles by prominent operator theorists providing an overview of how operator theory prospered during the Halmos era, in no small measure thanks to Halmos's leadership and penetrating insights. 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 Paul Richard Halmos, who lived a life of unbounded devotion to mathematics and to the mathematical community, died at the age of 90 on October 2, 2006. This volume is a memorial to Paul by operator theorists he inspired. Paul’sinitial research,beginning with his 1938Ph.D. thesis at the University of Illinois under Joseph Doob, was in probability, ergodic theory, and measure theory. A shift occurred in the 1950s when Paul’s interest in foundations led him to invent a subject he termed algebraic logic, resulting in a succession of papers on that subject appearing between 1954 and 1961, and the book Algebraic Logic, published in 1962. Paul’s ?rst two papers in pure operator theory appeared in 1950. After 1960 Paul’s research focused on Hilbert space operators, a subject he viewed as enc- passing ?nite-dimensional linear algebra. Beyond his research, Paul contributed to mathematics and to its community in manifold ways: as a renowned expositor, as an innovative teacher, as a tireless editor, and through unstinting service to the American Mathematical Society and to the Mathematical Association of America. Much of Paul’s in?uence ?owed at a personal level. Paul had a genuine, uncalculating interest in people; he developed an enormous number of friendships over the years, both with mathematicians and with nonmathematicians. Many of his mathematical friends, including the editors ofthisvolume,whileabsorbingabundantquantitiesofmathematicsatPaul’sknee, learned from his advice and his example what it means to be a mathematician. This volume is a memorial to Paul R. Halmos by operator theorists he inspired. It contains expository articles by prominent operator theorists, photos of Paul and taken by Paul, and three tributes to Paul written shortly after his death in 2006. From the expository articles, this generation of operator theorists and future generations will get a glimpse of many aspects of their subject, and of how Paul enriched and advanced it through his fundamental insights and prescient questions. -- Book Jacket