Structural Properties and Normality Criteria for Subclasses of Normaloid Operators

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.