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

In three chapters the authors first cover generalizations of the Herglotz representation theorem, von Neumann's inequality and the Sz.-Nagy dilation theorem to multiply connected domains. They describe the fist through third Herglotz representation and provide an application. In the second the describe the computational generation of counterexamples to rational dilation conjecture, including an analysis of the dilation condition for nonsingularly hyperextremal grammians, and in the third they examine arbitrary precision computations of Poisson kernel and Herglotz kernels on multiply connected circle domains. The authors have structured the chapters so readers can use each independently, making the first accessible to function theorists unskilled in operator theory and operator theorists unaccustomed to function theory, the second most likely to be used by experts in operator theory, and the third available to those interested in computational theory of multiply-connected domains who choose not to study the operator theories of the first and second chapters.