{
  "abstract": "Let $\\mathcal{H}$ be the separable, infinite-dimensional complex Hilbert space. In the first paper of this series, we introduced, by means of the Rosenblum operators, the shift representation operators $K_x=\\sum_{n=0}^{\\infty}T^nx\\otimes e_n$, which relate an operator $T$ with $r(T)<1$ to the unilateral shift $S$ and reveal its hidden analytic structure. In this paper we replace the pair $(H^2,S)$ by the Sobolev disk algebra $R(\\mathbb{D})$ and the multiplication operator $M_z$, whose canonical left inverse $B$ plays the role of $S^*$, and study the two-parameter family $K_{x,y}=\\sum_{n=0}^{\\infty}T^nx\\otimes B^{*n}y$. We prove that $T\\in B(R(\\mathbb{D}))$ with $r(T)<1$ is intransitive whenever there exist a nonzero vector $g$ and a nontrivial $M_z$-invariant manifold $\\mathcal{M}$ such that $f(T)g$ has a zero in the closed unit disk for every $f\\in\\mathcal{M}$; the proof explicitly constructs a nontrivial invariant subspace of $T$. We further introduce the ideal property and show that every intransitive operator on $\\mathcal{H}$ is unitarily equivalent to an operator on $R(\\mathbb{D})$ with this property, so that the Invariant Subspace Problem for arbitrary operators reduces to a problem about a concrete function algebra. Applications include a characterization of intransitivity through the rationality of the generating function $\\sum_{n=0}^{\\infty}\\langle T^nξ,η\\rangle z^n$ and new results on Pearcy's problem. Our main application is Halmos's third problem, which has remained open for more than fifty years: we reduce it to invertible operators with conjugate bi-geometric form, and we prove that $T^{-1}$ is intransitive for every invertible $T$ admitting a nontrivial projection $P$ with $PTP=TP$ and $PT(I-P)$ of rank one. Finally, we exhibit tridiagonal operators that are intransitive and admit infinite decreasing chains of invariant subspaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11202",
  "author_total": 4,
  "authors": [
    {
      "name": "Junsheng Fang"
    },
    {
      "name": "Bingzhe Hou"
    },
    {
      "name": "Chunlan Jiang"
    },
    {
      "name": "Yuanhang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bs3b256ts5gpj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11202",
  "fulltext_url": null,
  "id": "bs3b256ts5gpj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11202",
  "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": "W7221066837",
  "pdf_url": "https://arxiv.org/pdf/2610.11202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bs3b256ts5gpj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The invariant subspace problem and Rosenblum operators II",
  "title_text": "The invariant subspace problem and Rosenblum operators II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/bs3b256ts5gpj",
  "venue": "arXiv",
  "versions": []
}