Christian Emmel | Searcharxiv

arXiv2024

Operator models for meromorphic functions of bounded type

In this article, we construct operator models for meromorphic functions of bounded type on Krein spaces. This construction is based on certain reproducing kernel Hilbert spaces which are closely related to model spaces. Specifically, we show that each function of bounded type corresponds naturally to a pair of such spaces, extending Helson's representation theorem. This correspondence enables an explicit construction of our model, where the Krein space is a suitable sum of these identified spaces. Additionally, we establish that the representing self-adjoint relations possess a relatively simple structure, proving to be partially fundamentally reducible. Conversely, we show that realizations involving such relations correspond to functions of bounded type.

math.FA↗

arXiv2024

A generalization of Krein`s extension formalism for symmetric relations with deficiency index (1,1)

Let S be a symmetric relation with deficiency index (1,1). In this article, we extend Krein`s resolvent formalism in order to describe all, not necessarily self-adjoint, extensions $S \subset \tilde{A}$ with $\varrho(\tilde{A})\neq \emptyset$. The corresponding $Q$-functions turn out to be quasi-Herglotz functions. We will use their structure to characterize the spectrum of such extensions. Finally, we also provide a model for such an extension on a reproducing kernel Hilbert space when $S$ is simple.

math.FA↗