Generalized resolvents of isometric operators in Pontryagin spaces

TL;DR

This paper extends the theory of generalized resolvents from standard to non-standard isometric operators in Pontryagin spaces using boundary triplet methods, broadening the understanding of operator theory in indefinite inner product spaces.

Contribution

It provides a description of generalized resolvents for non-standard isometric operators in Pontryagin spaces, expanding previous work limited to standard operators.

Findings

Generalized resolvents for non-standard isometric operators are characterized.

Boundary triplet approach is effectively applied in Pontryagin spaces.

The work generalizes existing resolvent descriptions from Hilbert to Pontryagin spaces.

Abstract

An isometric operator $V$ in a Pontryagin space $\mathcal{H}$ is called standard, if its domain and the range are nondegenerate subspaces in $\mathcal{H}$. Generalized resolvents of standard isometric operators were described in the paper of A. Dijksma, H. Langer, and H. de Snoo in 1990. In the present paper generalized resolvents of non-standard Pontryagin space isometric operators are described. The method of the proof is based on the notion of boundary triplet of isometric operators in Pontryagin spaces. In the Hilbert space setting the notion of boundary triplet for isometric operators was introduced in the paper of M. Malamud and V. Mogilevskii in 2003.