Generalized resolvents of isometric operators in 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.