{
  "abstract": "We study spectral-set constants associated with the algebraic numerical range in unital Banach algebras. If $Γ_n$ denotes the universal constant for algebraic elements of degree at most $n$, then \\[ Γ_1=1, \\qquad 2n-1\\leqslantΓ_n<\\infty\\qquad(n\\geqslant2). \\] Thus the algebraic numerical range is a spectral set with a constant depending only on the algebraic degree. In degree two we obtain \\[ 3\\leqslantΓ_2 \\leqslant\\sqrt{1+(2\\mathrm e-1)^2}<4.55. \\] For unital $C^*$-algebras we prove a gap theorem: the algebra-level constant is one precisely in the commutative case, whereas every non-commutative algebra has constant at least two. Moreover, the constant of the Jiang--Su algebra is exactly the universal Crouzeix constant. In the opposite direction, we construct a norm-one operator with a contractive polynomial calculus on the unit disc but infinite numerical-range spectral constant, and we show that the canonical left shift on every spreading combinatorial space in a broad class has infinite constant. These results settle the three questions posed by Blazhko, Homza, Schwenninger, de Vries, and Wojtylak.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18963",
  "author_total": 1,
  "authors": [
    {
      "name": "Tomasz Kania"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dyq2t8v7bt5ck.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18963",
  "fulltext_url": null,
  "id": "dyq2t8v7bt5ck",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18963",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7213529342",
  "pdf_url": "https://arxiv.org/pdf/2609.18963",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dyq2t8v7bt5ck.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18963",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18963"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral constants for algebraic numerical ranges",
  "title_text": "Spectral constants for algebraic numerical ranges",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/dyq2t8v7bt5ck",
  "venue": "arXiv",
  "versions": []
}