Some conditions implying normality of operators

TL;DR

This paper establishes conditions under which certain classes of operators are normal, focusing on properties of their polar decomposition, hyponormality, and spectral conditions of the partial isometry.

Contribution

It provides new criteria linking hyponormality and spectral properties to the normality of operators, including conditions involving the Aluthge transform.

Findings

Operators with specific hyponormality conditions are normal if their partial isometry satisfies certain powers.

Normality of an operator can be characterized by the normality of its Aluthge transform under spectral conditions.

Spectral containment in an open semicircle implies equivalence of normality between the operator and its Aluthge transform.

Abstract

Let $T\in\mathbb{B}(\mathscr{H})$ and $T=U|T|$ be its polar decomposition. We proved that (i) if $T$ is log-hyponormal or $p$-hyponormal and $U^n=U^\ast$ for some $n$, then $T$ is normal; (ii) if the spectrum of $U$ is contained in some open semicircle, then $T$ is normal if and only if so is its Aluthge transform $\widetilde{T}=|T|^{1\over2}U|T|^{1\over2}$.