Electronic ISBN:  9781470404987 
Product Code:  MEMO/191/892.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 191; 2008; 159 ppMSC: Primary 30; 46; 47;
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

Chapters

1. Generalizations of the Herglotz representation theorem, von Neumann’s inequality and the Sz.Nagy dilation theorem to multiply connected domains

2. The computational generation of counterexamples to the rational dilation conjecture

3. Arbitrary precision computations of the Poisson Kernel and Herglotz Kernels on multiplyconnected circle domains

4. Schwartz kernels on multiply connected domains


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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 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.

Chapters

1. Generalizations of the Herglotz representation theorem, von Neumann’s inequality and the Sz.Nagy dilation theorem to multiply connected domains

2. The computational generation of counterexamples to the rational dilation conjecture

3. Arbitrary precision computations of the Poisson Kernel and Herglotz Kernels on multiplyconnected circle domains

4. Schwartz kernels on multiply connected domains