{
  "abstract": "We generalise Berger's dilation theorem from numerical ranges to scaled $q$-numerical ranges. Specifically, we prove that if the scaled $q$-numerical range $\\frac{W_q(A)}{q}$ is contained in the closed unit disk, then $A$ admits a unitary $ρ_q$-dilation, where $$ ρ_q=\\max\\left\\{1,\\frac{2|q|}{1+\\sqrt{1-|q|^2}}\\right\\}. $$ As a corollary we show that every disk containing $Ω_q(A)$ is a complete $ρ_q$-spectral set for $A$. At $q=1$, this recovers the classical fact that any disk containing $W(A)$ is a complete $2$-spectral set for $A$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00443",
  "author_total": 3,
  "authors": [
    {
      "name": "Matias Kaunisto"
    },
    {
      "name": "Ryan O'Loughlin"
    },
    {
      "name": "Jani Virtanen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90dhgfmh72c3h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00443",
  "fulltext_url": null,
  "id": "90dhgfmh72c3h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00443",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7216592864",
  "pdf_url": "https://arxiv.org/pdf/2610.00443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90dhgfmh72c3h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.00443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Dilation Theory for Scaled $q$-Numerical Ranges",
  "title_text": "Dilation Theory for Scaled q-Numerical Ranges",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/90dhgfmh72c3h",
  "venue": "arXiv",
  "versions": []
}