Memoirs of the American Mathematical Society 2008; 159 pp; softcover Volume: 191 ISBN10: 0821840460 ISBN13: 9780821840467 List Price: US$75 Individual Members: US$45 Institutional Members: US$60 Order Code: MEMO/191/892
 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 multiplyconnected 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 multiplyconnected domains. These conditions form the basis for the theoretical development of a computational procedure for probing a wellknown 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 multiplyconnected circle domains
 Schwartz kernels on multiply connected domains
 Appendix A. Convergence results
 Appendix B. Example inner product computation
 Bibliography
