Posinormality and the Root Problem

TL;DR

This paper investigates the properties of posinormal and related operators in Hilbert spaces, extending known results about roots and normality, and showing certain conditions under which operators are guaranteed to be normal.

Contribution

It extends three key results on the nth root problem by embedding Hilbert-space operators into posinormal classes, broadening the understanding of operator normality conditions.

Findings

For coposinormal operators, paranormal T with T^n quasinormal implies T is normal.

For posinormal operators, k-quasiparanormal T with T^n normal implies T is normal.

The results hold regardless of the Hilbert space's separability, with non-posinormal T decomposing into a normal and nilpotent part.

Abstract

The paper extends three results regarding the nth root problem by embedding classes of Hilbert-space operators into the class of posinormal operators. For instance, it is shown that (i) for coposinormal operators, if T is paranormal and T^n is quasinormal, then T is normal, and (ii) for posinormal operators, if T is k-quasiparanormal and T^n is normal, then T is normal. Moreover, (iii) it is also shown that the latter result is not conditioned to the separability of the underlying Hilbert space, even if T is not posinormal, where, in such a case, T is the direct sum of a normal operator with a nilpotent one.