Structural Properties and Normality Criteria for Subclasses of Normaloid Operators

TL;DR

This work extends normality criteria within the normaloid hierarchy to the broad class of absolute-$(p,r)$-paranormal operators. By exploiting polar decomposition $T=U|T|$, it proves a sharp self-adjointness criterion: $T$ is self-adjoint iff $T$ is absolute-$(p,r)$-paranormal and $U$ is self-adjoint, and it generalizes Ando-type results to show that if $T^n$ is normal (or scalar) then $T$ is normal. The authors also establish compactness and normality implications when powers of $T$ are compact, and provide a comprehensive characterization of quasinormal partial isometries, showing equivalences with absolute-$(p,r)$-paranormality and a simple operator identity. Collectively, these results deepen understanding of how paranormal-like conditions enforce normality and quasinormality in the normaloid framework, with concrete criteria for both operators and partial isometries.

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.

Theorems & Definitions (51)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Proposition 2.1
• Proposition 2.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5