Green's boundary relation model in a Krein space

TL;DR

This paper generalizes Green's boundary relation model to Krein spaces using isometric relations without standard assumptions, extending various boundary triple concepts and revealing new properties of related operators.

Contribution

It introduces a broad Green's boundary model in Krein spaces that does not rely on traditional domain and range conditions, extending existing boundary triple frameworks.

Findings

Proves main properties of the generalized Green's boundary model.

Discovers new properties of the isometric and unitary relations involved.

Extends various boundary triple concepts to Krein spaces using the new model.

Abstract

Given Krein and Hilbert spaces $\left( \mathcal{K},[.,.] \right)$ and $\left( \mathcal{H}, \left( .,. \right) \right)$, respectively, the concept of the boundary triple $\Pi =(\mathcal{H}, \Gamma _{0}, \Gamma_{1})$ is generalized through the abstract Green's identity for the isometric relation $\Gamma$ between Krein spaces $\left( \mathcal{K}^{2}, \left[ .,.\right]_{\mathcal{K}^{2}} \right) $ and $\left(\mathcal{H}^{2}, \left[ .,.\right]_{\mathcal{H}^{2}} \right) $ without any conditions on $\dom\, \Gamma$ and $\ran\, \Gamma$. This also means that we do not assume the existence of a closed symmetric linear relation $S$ such that $\dom\, \Gamma=S^{+}$, which is a standard assumptions in all previous research of boundary triples. The main properties of such a general Green's boundary model are proven. In the process, some useful properties of the isometric relation $V$ between two Krein spaces $X$ and $Y$ are proven. Additionally, surprising properties of the unitary relation $\Gamma : \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$ and the self-adjoint main transformation $\tilde{A}$ of $\Gamma$ are discovered. Then, two statements about generalized Nevanlinna families are generalized using this Green's boundary model. Furthermore, several previously known boundary triples involving a Hilbert space $\mathcal{K}$ and reduction operator $\Gamma : \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$, such as AB-generalized, B-generalized, ordinary, isometric, unitary, quasi-boundary, and S-generalized boundary triples, have been extended to a Krein space $\mathcal{K}$ and linear relation $\Gamma$ using the Green's boundary model approach.