Compressed resolvents of selfadjoint contractive extensions with exit and holomorphic operator-functions associated with them

TL;DR

This paper investigates selfadjoint contractive extensions of Hermitian contractions with an exit in a larger space, introducing a geometric approach to analyze associated holomorphic operator functions and their connections to passive system transfer functions.

Contribution

It introduces a new geometric method for characterizing analytic properties of Kren-Ovcharenko type functions and explores compressed resolvents linking to passive system transfer functions.

Findings

New geometric characterization of holomorphic operator functions.

Connections established between compressed resolvents and passive system transfer functions.

Enhanced understanding of analytic properties of selfadjoint extensions.

Abstract

Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space ${\mathfrak H}$ with an exit in some larger Hilbert space ${\mathfrak H}\oplus{\mathcal H}$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Kre\u{i}n-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.