{
  "abstract": "This paper investigates the domain $\\mathbfΘ_n$ from an operator-theoretic perspective. We establish several characterizations of $\\mathbfΘ_n$-contractions, $\\mathbfΘ_n$-unitaries, and $\\mathbfΘ_n$-isometries, and explore their connections with $Γ_n$- and tetrablock operator tuples, as well as with the corresponding operator classes associated with $\\mathbfΘ_{n+1}$. We prove that every $\\mathbfΘ_n$-contraction admits a canonical decomposition as the direct sum of a $\\mathbfΘ_n$-unitary and a completely non-unitary $\\mathbfΘ_n$-contraction. We further develop the conditional dilation theory for $\\mathbfΘ_n$-contractions and establish necessary conditions for the existence of $\\mathbfΘ_n$-isometric dilations. We show that the conditions in Theorem~\\ref{Conditional Dilation} are not, in general, sufficient by constructing a $\\mathbfΘ_3$-contraction that admits a $\\mathbfΘ_3$-isometric dilation although condition~(2) of Theorem~\\ref{Conditional Dilation} fails. We further study pure $\\mathbfΘ_n$-isometric dilations of $\\mathbfΘ_n$-contractions and establish a Hardy space characterization of such dilations. We show that the resulting $\\mathbfΘ_n$-isometric dilation is minimal. We also study isometric dilations of doubly commuting $\\mathbfΘ_n$-contractions under suitable Hardy space compatibility conditions. Finally, we identify a special class of $\\mathbfΘ_2$-contractions that admits a $\\mathbfΘ_2$-isometric dilation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.03574",
  "author_total": 3,
  "authors": [
    {
      "name": "Aparna Gupta"
    },
    {
      "name": "Avijit Pal"
    },
    {
      "name": "Bhaskar Paul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/737ch41dhcvtg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.03574",
  "fulltext_url": null,
  "id": "737ch41dhcvtg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.03574",
  "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.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7172554382",
  "pdf_url": "https://arxiv.org/pdf/2608.03574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/737ch41dhcvtg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2608.03574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.03574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Dilation Theory and Canonical Decomposition of $\\mathbfΘ_n$-Contractions",
  "title_text": "On the Dilation Theory and Canonical Decomposition of Θ_n-Contractions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/737ch41dhcvtg",
  "venue": "arXiv",
  "versions": []
}