Power quasinormal operators and the root problem

@misc{zhou2026power,
  title = {{Power quasinormal operators and the root problem}},
  author = {Jing-Bin Zhou and Shihai Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09775},
  eprint = {2609.09775},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09775},
}

TY  - UNPB
TI  - Power quasinormal operators and the root problem
AU  - Jing-Bin Zhou
AU  - Shihai Yang
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09775
UR  - https://arxiv.org/abs/2609.09775
AB  - 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}.
ER  -