Structural Properties and Normality Criteria for Subclasses of Normaloid Operators

TL;DR

This paper explores the structural and normality properties of certain classes of bounded linear operators on Hilbert spaces, extending existing theorems and providing new characterizations within the normaloid hierarchy.

Contribution

It introduces new normality criteria for absolute-$(p,r)$-paranormal operators and extends Ando's Theorem to this class, offering novel insights into operator normality and compactness.

Findings

Operators with polar decomposition are self-adjoint under specific conditions.

Extension of Ando's Theorem to absolute-$(p,r)$-paranormal operators.

Characterizations of quasinormal partial isometries within the normaloid hierarchy.

Abstract

We investigate structural properties and normality criteria for certain classes of bounded linear operators on a Hilbert space. We show that an operator $T$ with polar decomposition $T = U|T|$ is self-adjoint if and only if $T$ is absolute-$(p,r)$-paranormal and the partial isometry $U$ is self-adjoint. Extending Ando's Theorem, we prove that if $T$ is absolute-$(p,r)$-paranormal and $T^n$ is normal for some $n \in \mathbb{N}$, then $T$ itself is normal. We further show that if $T$ is absolute-$(p,r)$-paranormal and $T^2$ is compact, then $T$ is a compact normal operator. Finally, we obtain several characterizations of quasinormal partial isometries within the normaloid hierarchy.