Classical function theory, operator dilation theory, and machine computation on multiply-connected domains / Jim Agler, John Harland, Benjamin J. Raphael.

Format:

Book

Author/Creator:

Agler, Jim, author.

Harland, John, 1959- author.

Raphael, Benjamin J., 1974- author.

Series:

Memoirs of the American Mathematical Society ; no. 892.

Memoirs of the American Mathematical Society, 0065-9266 ; number 892

Language:

English

Subjects (All):

Geometric function theory.

Operator theory.

Dilation theory (Operator theory).

Functional analysis.

Analytic functions.

Physical Description:

1 online resource (176 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

This work begins with the presentation of generalizations of the classical Herglotz Representation Theorem for holomorphic functions with positive real part on the unit disc to functions with positive real part defined on multiply-connected domains. The generalized Herglotz kernels that appear in these representation theorems are then exploited to evolve new conditions for spectral set and rational dilation conditions over multiply-connected domains. These conditions form the basis for the theoretical development of a computational procedure for probing a well-known unsolved problem in operator theory, the so called rational dilation conjecture. Arbitrary precision algorithms for computing the Herglotz kernels on circled domains are presented and analyzed. These algorithms permit an effective implementation of the computational procedure which results in a machine generated counterexample to the rational dilation conjecture.

Contents:

""Contents""; ""Preface""; ""Chapter 1. Generalizations of the Herglotz Representation Theorem, von Neumann's Inequality and the Sz.-Nagy Dilation Theorem to Multiply Connected Domains""; ""1.1. Introduction""; ""1.2. Preliminaries""; ""1.3. The First Herglotz Representation""; ""1.4. The Second Herglotz Representation""; ""1.5. The Third Herglotz Representation""; ""1.6. The Herglotz Representations and Operator Theory""; ""1.7. An Application""; ""Chapter 2. The Computational Generation of Counterexamples to the Rational Dilation Conjecture""; ""2.1. Introduction""

""2.2. Mathematical Preliminaries""""2.3. Analysis of the Dilation Condition for Nonsingularly Hyperextremal Grammians""; ""2.4. Analysis of Dilation Extremal Grammians""; ""2.5. Algorithms""; ""2.6. A Computational Counterexample""; ""2.7. Plausibility Arguments""; ""Chapter 3. Arbitrary Precision Computations of the Poisson Kernel and Herglotz Kernels on Multiply-Connected Circle Domains""; ""3.1. Introduction""; ""3.2. Computation of the Functions""; ""3.3. Results""; ""Chapter 4. Schwartz Kernels on Multiply Connected Domains""; ""Appendix A. Convergence Results""

""Appendix B. Example Inner Product Computation""""Bibliography""

Notes:

"Volume 191, number 892 (second of 5 numbers)."

Includes bibliographical references (pages 157-159).

Description based on print version record.

ISBN:

1-4704-0498-2