Christian Emmel: Realizations of meromorphic functions of bounded type

Abstract

A realization of a locally analytic function is basically a way of expressing the function in terms of the resolvent of a self-adjoint operator. In the classical Hilbert space setting realizations correspond in an essentially one-to-one way to Herglotz-Nevanlinna functions. Considering more general objects than Hilbert spaces, so called Krein spaces, opens up the possibility to address more general function classes. This thesis is concerned with realizations and related minimality questions for meromorphic functions of bounded type. In the general introduction we review preliminaries from complex analysis and operator theory and give an idea how the main results of the subsequent two articles relate to the already existing theory. In the first included article realizations for meromorphic functions of bounded type are constructed. However, these realizations are not minimal in general. The second article is a first step in addressing this minimality question. There we focus on the well behaved subclass of atomic density functions for which minimal realizations are constructed.

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