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Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains
Jim Agler, University of California, San Diego, La Jolla, CA, John Harland, Carlsbad, CA, and Benjamin J. Raphael, University of California, San Diego, La Jolla, CA

Memoirs of the American Mathematical Society
2008; 159 pp; softcover
Volume: 191
ISBN-10: 0-8218-4046-0
ISBN-13: 978-0-8218-4046-7
List Price: US$75
Individual Members: US$45
Institutional Members: US$60
Order Code: MEMO/191/892
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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

  • Generalizations of the Herglotz representation theorem, von Neumann's inequality and the Sz.-Nagy dilation theorem to multiply connected domains
  • The computational generation of counterexamples to the rational dilation conjecture
  • Arbitrary precision computations of the Poisson kernel and Herglotz kernels on multiply-connected circle domains
  • Schwartz kernels on multiply connected domains
  • Appendix A. Convergence results
  • Appendix B. Example inner product computation
  • Bibliography
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