$C$-self-adjoint contractive extensions of $C$-symmetric non-densely defined contractions

TL;DR

This paper investigates $C$-self-adjoint contractive extensions of non-densely defined $C$-symmetric contractions in Hilbert spaces, providing parameterizations and applying results to maximal dissipative extensions.

Contribution

It introduces a comprehensive parameterization of all $C$-self-adjoint contractive extensions for non-densely defined $C$-symmetric contractions and proves a key existence result for maximal dissipative extensions.

Findings

Parameterizations of all such extensions are obtained.

A proof of Glazman's announced result on maximal dissipative $C$-self-adjoint extensions is provided.

The study advances the understanding of $C$-symmetric operator extensions in Hilbert spaces.

Abstract

Given a conjugation (involution) $C$ on a Hilbert space, $C$-self-adjoint contractive extensions of a non-densely defined $C$-symmetric contraction are studied and parameterizations of all such extensions are obtained. As an application, a proof of an announced result of Glazman \cite{Glazman1} on the existence of maximal dissipative $C$-self-adjoint extensions of a densely defined $C$-symmetric dissipative operator is given.