{
  "abstract": "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}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09775",
  "author_total": 2,
  "authors": [
    {
      "name": "Jing-Bin Zhou"
    },
    {
      "name": "Shihai Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1razq2wam9qdz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09775",
  "fulltext_url": null,
  "id": "1razq2wam9qdz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09775",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7212161814",
  "pdf_url": "https://arxiv.org/pdf/2609.09775",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1razq2wam9qdz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09775",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09775"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Power quasinormal operators and the root problem",
  "title_text": "Power quasinormal operators and the root problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/1razq2wam9qdz",
  "venue": "arXiv",
  "versions": []
}