On $n$th roots of bounded and unbounded quasinormal operators

TL;DR

This paper extends the understanding of quasinormal operators by proving that certain nth roots of bounded and unbounded quasinormal operators are themselves quasinormal, and explores the existence of non-quasinormal roots.

Contribution

It generalizes previous results by showing that class A nth roots of bounded quasinormal operators and subnormal nth roots of unbounded quasinormal operators are quasinormal.

Findings

Class A nth roots of bounded quasinormal operators are quasinormal.

Subnormal nth roots of unbounded quasinormal operators are quasinormal.

Existence of non-quasinormal nth roots for certain quasinormal operators.

Abstract

In a recent paper [9], R. E. Curto, S. H. Lee and J. Yoon asked the following question: Let $T$ be a subnormal operator, and assume that $T^2$ is quasinormal. Does it follow that $T$ is quasinormal?. In [36] we answered this question in the affirmative. In the present paper, we will extend this result in two directions. Namely, we prove that both class A $n$th roots of bounded quasinormal operators and subnormal $n$th roots of unbounded quasinormal operators are quasinormal. We also show that a non-normal quasinormal operator having a quasinormal $n$th root has a non-quasinormal $n$th root.