{
  "abstract": "In this paper, we establish necessary conditions for the minimality of infinite-dimensional $T_z^*$-invariant subspaces of $H^2(\\mathbb D^2)$. More precisely, we prove that if $M\\subset H^2(\\mathbb D^2)$ is an infinite-dimensional, minimal $T_z^*$-invariant subspace, then, for every $n\\in\\mathbb N_0$, either $\\overline{P_n(M)}=H^2(\\mathbb D)$ or $\\overline{P_n(M)}^\\perp =ϕ_{a,θ}H^2(\\mathbb D)$ where $ϕ_{a,θ}$ is an atomic singular inner function and $P_n$ denotes the orthogonal projection of $H^2(\\mathbb D^2)$ onto $H^2(z)w^n$. As $T_z^*$ is a universal operator on $H^2(\\mathbb D^2)$, this result reduces the Invariant Subspace Problem to the analysis of these two alternatives for infinite-dimensional minimal $T_z^*$-invariant subspaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24327",
  "author_total": 2,
  "authors": [
    {
      "name": "João Marcos R. do Carmo"
    },
    {
      "name": "Marcos S. Ferreira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/beesgtjc208tz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24327",
  "fulltext_url": null,
  "id": "beesgtjc208tz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24327",
  "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": "W7223013901",
  "pdf_url": "https://arxiv.org/pdf/2610.24327",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/beesgtjc208tz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24327",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24327"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Minimality of $T_z^*$-Invariant Subspaces of $H^2(\\mathbb D^2)$",
  "title_text": "Minimality of T_z^*-Invariant Subspaces of H^2(𝔻^2)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/beesgtjc208tz",
  "venue": "arXiv",
  "versions": []
}