{
  "abstract": "We call an $n$-tuple of $2 \\times 2$ matrices a ``\\textit{$2 \\times 2$ matrix $Γ_n$-contraction}'' if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in $Γ_n$. In this short note on $Γ_n$-contraction, we produce two families of $2 \\times 2$ matrix $Γ_n$-contractions whose fundamental operators do not satisfy the commutativity conditions and \\begin{equation*} \\begin{aligned} F_iF^*_{n-j} - F_jF^*_{n-i} = F^*_{n-j}F_i - F^*_{n-i}F_j, \\quad 1 \\le i, j \\le n-1, \\end{aligned} \\end{equation*} which shows that the commutativity as well as the conditions on fundamental operators mentioned above are sufficient, but not necessary, for the existence of $Γ_n$-isometric dilation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39711",
  "author_total": 1,
  "authors": [
    {
      "name": "Bhaskar Paul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ebvktrsvqvhmk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39711",
  "fulltext_url": null,
  "id": "ebvktrsvqvhmk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39711",
  "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": "W7215128589",
  "pdf_url": "https://arxiv.org/pdf/2609.39711",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ebvktrsvqvhmk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39711",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39711"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Families of $Γ_n$-Contractions Having Non-commutative Fundamental Operators",
  "title_text": "The Families of Γ_n-Contractions Having Non-commutative Fundamental Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/ebvktrsvqvhmk",
  "venue": "arXiv",
  "versions": []
}