{
  "abstract": "We answer negatively a finite-dimensional unitary-model question for complex symmetric operators. More precisely, we show that, for every \\(n\\geq 10\\), not every \\(n\\times n\\) symmetric matrix is unitarily equivalent to a direct sum of truncated Toeplitz operators. In order to do this, we first use semialgebraic dimension, a tool from real algebraic geometry, to prove a general theorem showing that, if \\(\\mathcal X\\) is a semialgebraic family of complex symmetric matrices, then the set of complex symmetric matrices which are unitarily equivalent to an element of \\(\\mathcal X\\) is semialgebraic and has dimension at most $\\dim_{\\mathbb R}\\mathcal X+\\frac{n(n-1)}2.$ We then apply this theorem to show that when $n\\geq 10$ there exist irreducible symmetric $n \\times n$ matrices which are not unitarily equivalent to a truncated Toeplitz operator. Although unitary equivalence is too restrictive, we prove that every finite-dimensional complex symmetric operator is complex-orthogonally equivalent to a coanalytic truncated Toeplitz operator. We also answer positively a refined representation question by showing that whenever a symmetric matrix is unitarily equivalent to a truncated Toeplitz operator, it is a matrix representation of that operator with respect to a conjugation-invariant orthonormal basis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14019",
  "author_total": 1,
  "authors": [
    {
      "name": "Ryan O'Loughlin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e08ppf3eapmrv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14019",
  "fulltext_url": null,
  "id": "e08ppf3eapmrv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14019",
  "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": "W7169105166",
  "pdf_url": "https://arxiv.org/pdf/2607.14019",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e08ppf3eapmrv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.14019",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14019"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices",
  "title_text": "Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/e08ppf3eapmrv",
  "venue": "arXiv",
  "versions": []
}