{
  "abstract": "Rencently, Fang, Hou, Jiang, and Zhang reduced Halmos's third problem to invertible operators with conjugate bi-geometric form. An operator has such a form if, with respect to an orthogonal decomposition, it admits a representation $$ T = \\begin{pmatrix} A & C \\\\ 0 & B \\end{pmatrix}, \\quad A^*, B \\in B_1(O_δ),$$ where $O_δ=\\{z\\in\\mathbb C:|z|<δ\\}$ for some $δ>0$, and $B_1(O_δ)$ denotes the rank-one Cowen--Douglas class. We restrict our attention to the case $B=A^*$ with strictly positive coupling $C$, and settle Halmos's third problem for this class of operators. More precisely,we prove that every such operator $T$ is invertible and that $T^{-1}$ has a nontrivial closed invariant subspace. This result contributes to the study of Halmos's third problem within the conjugate bi-geometric framework.The proof combines monotone solutions of operator Riccati equations with strong operator convergence to construct invariant subspaces for associated Hamiltonian operator matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39605",
  "author_total": 2,
  "authors": [
    {
      "name": "Kui Ji"
    },
    {
      "name": "Zhenyue Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6a7qsj4fp548h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39605",
  "fulltext_url": null,
  "id": "6a7qsj4fp548h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39605",
  "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.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39605",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6a7qsj4fp548h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39605",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39605"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Riccati Equations and the Third Problem of Halmos",
  "title_text": "Riccati Equations and the Third Problem of Halmos",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6a7qsj4fp548h",
  "venue": "arXiv",
  "versions": []
}