## A Glimpse at Hilbert Space Operators: Paul R. Halmos in MemoriamSheldon Axler, Peter Rosenthal, Donald Sarason 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. |

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### Contents

3 | |

11 | |

Paul Halmos 19162006 | 27 |

Mathematical Review of How to Write Mathematics | 30 |

Publications of Paul R Halmos | 33 |

Photos | 41 |

Part II Articles | 78 |

What Can Hilbert Spaces Tell Us About Bounded Functions in the Bidisk? | 81 |

Polynomially Hyponormal Operators | 195 |

Essentially Normal Operators | 209 |

The Operator FejerRiesz Theorem | 223 |

A Halmos Doctrine and Shifts on Hilbert Space | 255 |

The Behavior of Functions of Operators Under Perturbations | 286 |

The Halmos Similarity Problem | 325 |

Paul Halmos and Invariant Subspaces | 341 |

Commutant Lifting | 350 |

Dilation Theory Yesterday and Today | 98 |

Toeplitz Operators | 125 |

Dual Algebras and Invariant Subspaces | 134 |

The State of Subnormal Operators | 177 |

Double Cones are Intervals | 359 |

Advances and Applications OT | 363 |

Advances and Applications OT | 364 |

### Other editions - View all

A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam Sheldon Axler,Peter Rosenthal,Donald Sarason No preview available - 2012 |

A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam Sheldon Axler,Peter Rosenthal,Donald Sarason No preview available - 2010 |

### Common terms and phrases

2-hyponormal Amer Anal analytic functions analytic polynomials Arveson Banach algebra Banach space Bergman Bergman space bounded point evaluations C∗-algebras C∗-correspondence commuting completely positive const defined denote Equations Operator Theory exists factorization finite-dimensional Foias formula fully matricial Funct function f function theory functional calculus Halmos’s Hardy space Hilbert space homomorphism hyponormal operators implies inequality Integral Equations Operator interpolation isometry isomorphism kernel Lemma linear map linear operators Math mathematicians mathematics matrix Neumann noncommutative nontrivial invariant subspaces norm operator algebras operator Lipschitz Operator Theory operator-valued operators on Hilbert orthogonal P.R. Halmos paper Paul Halmos Paul’s perturbation positive linear map problem Proc Proposition proved representation satisfies scalar Schur self-adjoint operators self-commutator semigroup sequence space H spectral theorem spectrum subnormal operators Sz.-Nagy T+(E tensor Theorem 5.1 Toeplitz operators trace class unilateral shift unit circle unitary operators vectors weighted shifts