发表机构
Shanghai University of Finance and Economics（上海财经大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造反例推翻既有引理，证明有限维空间中n幂拟正规性与T^n正规性及拟正规性等价，给出单值延拓性质的新证明，并肯定回答根问题。
AI 中文摘要
本文构造了一个$n$幂拟正规算子$T$，使得对于某个正整数$n$，$T^n$不是拟正规的，从而为\cite[引理3.1]{ko-filomat-2023}提供了反例。随后，我们研究了$T$的$n$幂拟正规性、$T^n$的正规性以及$T^n$的拟正规性之间的关系，并证明在有限维空间中这三个条件是等价的。我们还给出了$n$幂拟正规算子具有单值延拓性质的新证明\cite[定理3.2]{ko-filomat-2023}；与原始证明不同，我们的论证不依赖于\cite[引理3.1]{ko-filomat-2023}，从而弥补了原始论证中的缺陷。此外，对于固定的算子$T$，我们刻画了使得$T$为$n$幂拟正规的所有正整数$n$。作为推论，Sid Ahmed的若干结果\cite{ahmed-bmaa-2011}得到了推广。最后，我们证明了每个亚正规$n$幂拟正规算子都是拟正规的。与此密切相关，我们还对Stanković和Kubrusly提出的根问题\cite[问题2.11]{stankovic-afa-2025}给出了肯定回答。
英文摘要
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}.
