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

TL;DR

This paper extends Krein's extension formalism to describe all extensions of symmetric relations with deficiency index (1,1), characterizing their spectra using quasi-Herglotz functions and providing a model in reproducing kernel Hilbert spaces.

Contribution

It generalizes Krein's resolvent formalism to include non-self-adjoint extensions of symmetric relations with deficiency index (1,1).

Findings

Characterization of spectra via quasi-Herglotz functions.

Extension of Krein's formalism to non-self-adjoint cases.

Model construction on reproducing kernel Hilbert spaces.

Abstract

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.