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

Download or read book Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains written by Jim Agler and published by American Mathematical Soc.. This book was released on 2008 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.