Subnormal $n$th roots of quasinormal operators are quasinormal

TL;DR

This paper proves that subnormal operators whose nth roots are quasinormal must themselves be quasinormal, answering a specific open question and extending the understanding of operator properties.

Contribution

It establishes that subnormal nth roots of quasinormal operators are necessarily quasinormal, generalizing previous results and answering an open question.

Findings

Subnormal nth roots of quasinormal operators are quasinormal.

Affirmative answer to a previously open question.

Generalization to nth roots beyond squares.

Abstract

In the recent paper, R. E. Curto, S. H. Lee, J. Yoon, asked the following question: Let $A$ be a subnormal operator, and assume that $A^2$ is quasinormal. Does it follow that $A$ is quasinormal? In this paper, we give an affirmative answer to this question. In fact, we prove more general result that subnormal $n$th roots of quasinormal operators are quasinormal.