On The Absolute Value of Unbounded Operators

TL;DR

This paper explores fundamental properties of unbounded operators, establishing conditions under which certain operator relations hold and characterizing self-adjointness and invertibility for unbounded normal operators.

Contribution

It provides new characterizations of self-adjointness and invertibility in the context of unbounded operators, extending classical operator theory results.

Findings

Characterization of self-adjointness for unbounded operators

Criteria for invertibility of unbounded normal operators

Conditions under which operator relations like $|AB|=|A||B|$ hold

Abstract

The primary purpose of the present paper is to investigate when relations of the types $|AB|=|A||B|$, $|A\pm B|\leq |A|+|B|$, $||A|-|B||\leq |A\pm B|$ and $|\overline{\text{Re} A}|\leq |A|$ (among others) hold in an unbounded operator setting. As interesting consequences, we obtain a characterization of (unbounded) self-adjointness as well as a characterization of invertibility for the class of unbounded normal operators.