On extensions of $J$-skew-symmetric and $J$-isometric operators

TL;DR

This paper proves that certain classes of $J$-skew-symmetric and $J$-isometric operators in Hilbert spaces can be extended to $J$-skew-self-adjoint and $J$-unitary operators, respectively, generalizing previous results.

Contribution

It establishes the existence of $J$-skew-self-adjoint and $J$-unitary extensions for densely defined $J$-skew-symmetric and $J$-isometric operators, extending earlier work by Galindo.

Findings

Every densely defined $J$-skew-symmetric operator has a $J$-skew-self-adjoint extension.

Every $J$-isometric operator with dense domain and range has a $J$-unitary extension.

The method follows and modifies Galindo's approach from 1962.

Abstract

In this paper it is proved that each densely defined $J$-skew-symmetric operator (or each $J$-isometric operator with $\overline{D(A)}=\overline{R(A)}=H$) in a Hilbert space $H$ has a $J$-skew-self-adjoint (respectively $J$-unitary) extension in a Hilbert space $\widetilde H\supseteq H$. We follow the ideas of Galindo in~[A.~Galindo, On the existence of $J$-self-adjoint extensions of $J$-symmetric operators with adjoint, Communications on pure and applied mathematics, Vol. XV, 423-425 (1962)] with necessary modifications.