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 $\varrho (\tilde{A})\mathrm{\ne }\mathrm{\varnothing }$. 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.