Power quasinormal operators and the root problem

RESUMO

In this paper, we construct an $n$-power quasinormal operator $T$ such that $T^n$ is not quasinormal for some positive integer $n$, thereby providing a counterexample to \cite[Lemma 3.1]{ko-filomat-2023}. We then investigate the relationships among the $n$-power quasinormality of $T$, the normality of $T^n$, and the quasinormality of $T^n$, and show that these three conditions are equivalent in finite-dimensional spaces. We also provide a new proof that $n$-power quasinormal operators have the single-valued extension property \cite[Theorem 3.2]{ko-filomat-2023}; unlike the original proof, our argument does not rely on \cite[Lemma 3.1]{ko-filomat-2023} and thus closes the gap in the original argument. In addition, for a fixed operator $T$, we characterize all positive integers $n$ for which $T$ is $n$-power quasinormal. As consequences, several results of Sid Ahmed \cite{ahmed-bmaa-2011} are extended. Finally, we prove that every paranormal $n$-power quasinormal operator is quasinormal. Closely related to this, we also give an affirmative answer to the root problem of Stanković and Kubrusly \cite[Question 2.11]{stankovic-afa-2025}.