# Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains

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*Jim Agler; John Harland; Benjamin J. Raphael*

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.

#### Table of Contents

# Table of Contents

## Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains

- Contents v6 free
- Preface vii8 free
- Chapter 1. Generalizations of the Herglotz Representation Theorem, von Neumann's Inequality and the Sz.-Nagy Dilation Theorem to Multiply Connected Domains 110 free
- Chapter 2. The Computational Generation of Counterexamples to the Rational Dilation Conjecture 6170
- Chapter 3. Arbitrary Precision Computations of the Poisson Kernel and Herglotz Kernels on Multiply-Connected Circle Domains 109118
- Chapter 4. Schwartz Kernels on Multiply Connected Domains 127136
- Appendix A. Convergence Results 139148
- Appendix B. Example Inner Product Computation 155164
- Bibliography 157166