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arXiv · 2407.07491

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

Christian Emmel
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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.

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Christian Emmel. 2024-07-10. A generalization of Krein`s extension formalism for symmetric relations with deficiency index (1,1). https://arxiv.org/abs/2407.07491

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