Green's boundary relation model in a Krein space

• Published in: arXiv.org 2024-07
• Main Author: Borogovac, Muhamed
• Format: Article
• Language: English
• Subjects: Hilbert space
• ISSN: 2331-8422

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.