{
  "abstract": "For a bounded linear operator $T$ on a complex separable Hilbert space $\\mathcal{H}$ and a vector $x\\in \\mathcal{H}$, let $W_a(T,x)$ be the set of cluster points of the sequence $\\{\\langle|T^n|^{1/n}x,x\\rangle\\}_{n=1}^{\\infty}$. We define the asymptotic numerical range and the asymptotic numerical radius of $T$, respectively, by $W_a(T):=\\bigcup_{\\|x\\|=1}W_a(T,x)$ and $w_a(T):=\\sup W_a(T)$. We prove that $W_a(T,x)$ is a compact interval for every $x\\in\\mathcal{H}$ and that $W_a(T)$ is a bounded interval, where intervals are allowed to be degenerate. Using the asymptotic numerical radius, we show that if there is a nonzero vector $x_0\\in \\mathcal{H}$ with $r(T,x_0)<w_a(T)$, where $r(T,x_0)$ denotes the local spectral radius of $T$ at $x_0$, then the subspaces $\\overline{\\{x\\in\\mathcal{H}:r(T,x)\\le r(T,x_0)\\}}$ and $\\overline{\\operatorname{span}\\{T^n x_0:n\\ge0\\}}$, which are well known to be hyperinvariant and invariant for $T$, respectively, are both nontrivial. We also prove that $w_a(T)=r(T)$ for every hyponormal operator $T$, where $r(T)$ denotes the spectral radius of $T$. Finally, we investigate the connectedness of the set of WOT cluster points of the sequence $\\{|T^n|^{1/n}\\}_{n=1}^\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.21553",
  "author_total": 2,
  "authors": [
    {
      "name": "Youqing Ji"
    },
    {
      "name": "Bangyuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7h0gsnd6jxbc1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.21553",
  "fulltext_url": null,
  "id": "7h0gsnd6jxbc1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.21553",
  "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": "W7204155847",
  "pdf_url": "https://arxiv.org/pdf/2608.21553",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7h0gsnd6jxbc1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2608.21553",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.21553"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Asymptotic Numerical Ranges and Invariant Subspaces of Operators",
  "title_text": "Asymptotic Numerical Ranges and Invariant Subspaces of Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/7h0gsnd6jxbc1",
  "venue": "arXiv",
  "versions": []
}