A refined polar decomposition for $J$-unitary operators

TL;DR

This paper provides a new characterization of the polar decomposition components for $J$-unitary operators in Hilbert spaces, highlighting differences from known decompositions for symmetric operators and establishing extension results for $J$-imaginary symmetric operators.

Contribution

It introduces a novel structure for the polar decomposition of $J$-unitary operators and proves the existence of $J$-imaginary self-adjoint extensions for certain symmetric operators.

Findings

Characterization of polar decomposition components for $J$-unitary operators

Distinct structure from complex symmetric operators

Existence of $J$-imaginary self-adjoint extensions

Abstract

In this paper, we shall characterize the components of the polar decomposition for an arbitrary $J$-unitary operator in a Hilbert space. This characterization has a quite different structure as that for complex symmetric and complex skew-symmetric operators. It is also shown that for a $J$-imaginary closed symmetric operator in a Hilbert space there exists a $J$-imaginary self-adjoint extension in a possibly larger Hilbert space (a linear operator $A$ in a Hilbert space $H$ is said to be $J$-imaginary if $f\in D(A)$ implies $Jf\in D(A)$ and $AJf = -JAf$, where $J$ is a conjugation on $H$).