Posinormality and the Root Problem

TL;DR

The work broadens the classical nth root problem by embedding posinormal and coposinormal operator classes into the analysis on Hilbert spaces. It proves that under suitable paranormal and posinormal hypotheses, normality of T^n forces T to be normal, and it sharpens this to a separability-free decomposition T=N⊕L with N normal and L nilpotent in many cases. The results extend to k-quasiparanormal operators, showing that T^n normal implies T is normal or decomposable with a nilpotent part of controlled index, and they remove separability restrictions present in earlier treatments. Collectively, the paper enlarges the classes of operators for which the nth root implication holds and provides structural decompositions that clarify how nonnormal roots arise.

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.