On $C$-Symmetric and $C$-Self-adjoint Unbounded Operators on Hilbert Space

TL;DR

This paper characterizes $C$-self-adjoint unbounded operators on Hilbert spaces, providing criteria based on quasi-analytic vectors and polar decompositions, and describes their extensions.

Contribution

It offers a comprehensive description of $C$-self-adjoint extensions and new criteria for $C$-self-adjointness in unbounded operators.

Findings

Characterization of all $C$-self-adjoint extensions.

A $C$-self-adjointness criterion using quasi-analytic vectors.

Characterization of $C$-self-adjoint operators via polar decompositions.

Abstract

Let $C$ be a conjugation on a Hilbert space $\mathcal{H}$. A densely defined linear operator $A$ on $\mathcal{H}$ is called $C$-symmetric if $CAC\subseteq A^*$ and $C$-self-adjoint if $CAC=A^*$. Our main results describe all $C$-self-adjoint extensions of $A$ on $\mathcal{H}$. Further, we prove a $C$-self-adjointness criterion based on quasi-analytic vectors and we characterize $C$-self-adjoint operators in terms of their polar decompositions.