{
  "abstract": "Let $\\mathcal E$ be a separable Hilbert space, $B\\in H^\\infty_1(L(\\mathcal E))$, and $α>-1$. Associated with the Toeplitz operator $T_B$ on the vector-valued weighted Bergman space $A^2_{α,\\mathcal E}$ are the sub-Bergman spaces $\\mathcal A_α(B)$ and $\\mathcal A_α(B^*)$. In this paper, we prove that \\[ \\mathcal A_α(B)\\supset A^2_{α-1,\\mathcal H(B(0))} \\quad\\text{and}\\quad \\mathcal A_α(B^*)\\supset A^2_{α-1,\\mathcal H(B(0)^*)}. \\] This confirms a conjecture of \\cite{GLM26}. We also construct an infinite-dimensional two-sided inner function $B$ such that $\\|B(0)\\|<1$ and \\[ \\mathcal A_α(B)\\approx \\mathcal A_α(B^*)\\approx A^2_{α-1,\\mathcal E}, \\] but $B$ is not a finite Blaschke-Potapov product. This example shows that the finite-dimensional rigidity theorem of \\cite{GLM26} does not extend to infinite-dimensional coefficient spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34694",
  "author_total": 2,
  "authors": [
    {
      "name": "Shuaibing Luo"
    },
    {
      "name": "Jiming Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3xgcrva8gm434.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34694",
  "fulltext_url": null,
  "id": "3xgcrva8gm434",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34694",
  "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.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7214701507",
  "pdf_url": "https://arxiv.org/pdf/2609.34694",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3xgcrva8gm434.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34694",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34694"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity",
  "title_text": "Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/3xgcrva8gm434",
  "venue": "arXiv",
  "versions": []
}