Axler / Rosenthal / Sarason A Glimpse at Hilbert Space Operators

Paul R. Halmos in Memoriam
1. Auflage 2011
ISBN: 978-3-0346-0347-8
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 207, 362 Seiten

Reihe: Operator Theory: Advances and Applications

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.

Autoren/Hrsg.

Axler, Sheldon

Rosenthal, Peter

Sarason, Donald

Inhaltsverzeichnis

1;Copyright Page;5
2;Table of Contents;6
3;Preface;8
4;Part I Paul Halmos;10
4.1;Paul Halmos – Expositor Par Excellence*;12
4.1.1;Introduction;12
4.1.2;The invariant subspace problem;13
4.1.3;Quasitriangularity, quasidiagonality and the Weyl-von Neumann-Voiculescu theorem;14
4.1.4;Subnormal operators and unitary dilations;15
4.1.5;Ergodic theory;17
4.1.6;A brief biography;17
4.1.7;Suggested reading;19
4.1.7.1;Technical papers by Halmos;19
4.1.7.2;Expository articles by Halmos;19
4.1.7.3;Two non-technical books by Halmos;19
4.2;Paul Halmos: In His Own Words*;20
4.2.1;On writing;20
4.2.2;Excerpts from: “How to write mathematics”, Enseign. Math. (2) 16 (1970), 123–152.;20
4.2.3;On speaking;22
4.2.3.1;Excerpts from: “How to talk mathematics”, Notices of AMS 21 (1974), 155– 158.;22
4.2.3.2;Excerpt from: I Want to Be a Mathematician, p. 401, Springer-Verlag, New York (1985).;24
4.2.4;On exposition;24
4.2.4.1;Excerpt from: Response from Paul Halmos on winning the Steele Prize for Exposition (1983).;24
4.2.5;On publishing;25
4.2.5.1;Excerpts from: “Four panel talks on publishing”, American Mathematical Monthly 82 (1975), 14–17.;25
4.2.6;On research;26
4.2.6.1;Excerpt from: I Want to Be a Mathematician, pp. 321–322, Springer-Verlag, New York (1985).;26
4.2.7;On teaching;27
4.2.7.1;Excerpt from: “The problem of learning to teach”, American Mathematical Monthly 82 (1975), 466–476.;27
4.2.7.2;Excerpt from: “The heart of mathematics”, American Mathematical Monthly 87 (1980), 519–524.;28
4.2.8;On mathematics;29
4.2.8.1;Excerpt from: “Mathematics as a creative art”, American Scientist 56 (1968), 375–389.;29
4.2.9;On pure and applied;32
4.2.9.1;Excerpt from: “Applied mathematics is bad mathematics”, pp. 9–20, appearing in Mathematics Tomorrow, edited by Lynn Steen, Springer-Verlag, New York (1981).;32
4.2.10;On being a mathematician;34
4.2.10.1;Excerpt from: I Want to Be a Mathematician, p. 400, Springer-Verlag, New York (1985).;34
4.3;Obituary: Paul Halmos, 1916–2006;35
4.4;Mathematical Review of “How to Write Mathematics” *;38
4.5;Publications of Paul R. Halmos;39
4.5.1;Research and Expository Articles;39
4.5.2;Books;45
4.6;Photos;47
5;Part II Articles;84
5.1;What Can Hilbert Spaces Tell Us About Bounded Functions in the Bidisk?;85
5.1.1;1. Introduction;85
5.1.2;2. Realization formula;86
5.1.3;3. Pick problem;88
5.1.4;4. Nevanlinna problem;90
5.1.5;5. Takagi problem;91
5.1.6;6. Interpolating sequences;91
5.1.7;7. Corona problem;94
5.1.8;8. Distinguished and toral varieties;96
5.1.9;9. Extension property;97
5.1.10;10. Conclusion;98
5.1.11;References;98
5.2;Dilation Theory Yesterday and Today;102
5.2.1;1. Preface;102
5.2.2;2. Origins;103
5.2.3;3. Positive linear maps on commutative *-algebras;106
5.2.4;4. Subnormality;108
5.2.5;5. Commutative dilation theory;112
5.2.6;6. Completely positivity and Stinespring’s theorem;114
5.2.7;7. Operator spaces, operator systems and extensions;117
5.2.8;8. Spectral sets and higher-dimensional operator theory;119
5.2.9;9. Completely positive maps and endomorphisms;121
5.2.10;Appendix: Brief on Banach *-algebras;123
5.2.11;References;125
5.3;Toeplitz Operators;127
5.3.1;Products of Toeplitz operators;128
5.3.2;The spectrum of a Toeplitz operator;129
5.3.3;Subnormal Toeplitz operators;130
5.3.4;The symbol map;132
5.3.5;Compact semi-commutators;133
5.3.6;Remembering Paul Halmos;134
5.3.7;References;134
5.4;Dual Algebras and Invariant Subspaces;136
5.4.1;1. Introduction;136
5.4.2;2. An open mapping theorem for bilinear maps;138
5.4.3;3. Hyper-reflexivity and dilations;142
5.4.4;4. Dominating spectrum;147
5.4.5;5. A noncommutative example;150
5.4.6;6. Approximate factorization;152
5.4.7;7. Contractions with isometric functional calculus;159
5.4.8;8. Banach space geometry;162
5.4.9;9. Dominating spectrum in Banach spaces;164
5.4.10;10. Localizable spectrum;167
5.4.11;11. Notes;171
5.4.12;References;173
5.5;The State of Subnormal Operators;178
5.5.1;1. Introduction;178
5.5.2;2. Fundamentals of subnormal operators;180
5.5.2.1;2.1. Proposition.;180
5.5.2.2;2.2;180
5.5.2.3;2.3. Proposition.;181
5.5.2.4;2.4. Theorem.;181
5.5.3;3. The functional calculus;182
5.5.3.1;3.1. Proposition.;182
5.5.3.2;3.2. Proposition.;183
5.5.3.3;3.3. Sarason’s Theorem.;183
5.5.3.4;3.4. Proposition.;183
5.5.3.5;3.5. Theorem.;183
5.5.3.6;3.6. Theorem.;184
5.5.4;4. Invariant subspaces;184
5.5.4.1;4.1. Theorem.;184
5.5.4.2;4.2. Theorem.;185
5.5.4.3;4.3. Theorem.;186
5.5.4.4;4.4. Theorem.;186
5.5.4.5;4.5. Problem.;186
5.5.5;5. Bounded point evaluations;186
5.5.5.1;5.1. Thomson’s Theorem.;188
5.5.5.2;5.2. Theorem.;188
5.5.5.3;5.3. Problem.;189
5.5.5.4;5.4. Problem.;189
5.5.5.5;5.5. Theorem.;190
5.5.5.6;5.6. Problem.;190
5.5.5.7;5.7. Theorem.;190
5.5.5.8;5.8. Problem.;190
5.5.5.9;5.9. Problem.;191
5.5.5.10;5.10. Theorem.;191
5.5.5.11;5.11. Theorem.;192
5.5.5.12;5.12. Problem.;193
5.5.5.13;5.13. Problem.;193
5.5.6;6. Conclusion;193
5.5.7;References;193
5.6;Polynomially Hyponormal Operators;196
5.6.1;1. Hyponormal operators;196
5.6.2;2. Linear operators as positive functionals;198
5.6.3;3. k-hyponormality for unilateral weighted shifts;200
5.6.4;4. The case of Toeplitz operators;203
5.6.5;References;205
5.7;Essentially Normal Operators;209
5.7.1;1. Introduction;209
5.7.2;2. Weyl–von Neumann theorems;211
5.7.3;3. Essentially normal operators;213
5.7.4;4. Almost commuting matrices;217
5.7.5;References;220
5.8;The Operator Fejer-Riesz Theorem;223
5.8.1;1. Introduction;223
5.8.2;2. The operator Fejer-Riesz theorem;225
5.8.3;3. Method of Schur complements;228
5.8.4;4. Spectral factorization;233
5.8.5;5. Multivariable theory;239
5.8.6;6. Noncommutative factorization;244
5.8.7;Appendix: Schur complements;249
5.8.8;References;251
5.9;A Halmos Doctrine and Shifts on Hilbert Space;255
5.9.1;1. Introduction;255
5.9.2;2. Halmos’s theorem;256
5.9.3;3. C*-correspondences, tensor algebras and C*-envelopes;260
5.9.4;4. Representations and dilations;265
5.9.5;5. Induced representations and Halmos’s theorem;267
5.9.6;6. Duality and commutants;272
5.9.7;7. Noncommutative function theory;274
5.9.8;References;282
5.10;The Behavior of Functions of Operators Under Perturbations;286
5.10.1;1. Introduction;286
5.10.2;2. Double operator integrals;291
5.10.3;3. Multiple operator integrals;294
5.10.4;4. Besov spaces;296
5.10.5;5. Nuclearity of Hankel operators;298
5.10.6;6. Operator Lipschitz and operator differentiable functions. Sufficient conditions;298
5.10.7;7. Operator Lipschitz and operator differentiable functions. Necessary conditions;304
5.10.8;8. Higher-order operator derivatives;306
5.10.9;9. The case of contractions;308
5.10.10;10. Operator Holder–Zygmund functions;312
5.10.11;11. Lifshits–Krein trace formulae;314
5.10.12;12. Koplienko–Neidhardt trace formulae;316
5.10.13;13. Perturbations of class Sp;317
5.10.14;References;320
5.11;The Halmos Similarity Problem;324
5.11.1;References;335
5.12;Paul Halmos and Invariant Subspaces;339
5.12.1;References;344
5.13;Commutant Lifting;348
5.13.1;References;353
5.14;Double Cones are Intervals;355
5.14.1;1. Introduction;355
5.14.2;2. The (H2, | · |) model of Minkowski space;355
5.14.3;3. Some applications;357
5.14.4;References;358
6;Operator Theory: Advances and Applications (OT);359
7;Operator Theory: Advances and Applications (OT);360