Green's boundary relation model in a Krein space

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.