{
  "abstract": "Let $A$ be a bounded operator on a Hilbert space and let $u$ be a non-constant Blaschke product, finite or infinite. We prove that if the spectrum of $A$ does not meet the closure of the set of poles of $u$, then $u(A)$ is similar to a contraction if and only if $A$ is. This settles a conjecture of R.G. Douglas. For infinite products $u$, no additional condition on the zeros beyond the Blaschke condition is required. We also show that '$u(A)$ is similar to a contraction if and only if $A$ is' for all non-constant inner functions $u$ whose reflected singularity set, the closure of the set of poles together with the singular support, avoids the spectrum of $A$. We also prove a weighted resolvent version of the theorem, which allows the poles to accumulate on the spectrum to a limited extent; a column criterion for partial products, which is shown to force their norm convergence; and a simultaneous version for commuting tuples of operators tested by separated inner functions sharing one Lyapunov operator. For finite Blaschke products we obtain a polynomial form of the Lyapunov operator, which gives an estimate for the similarity constants. We also characterise the tuples of finite Blaschke products that form a uniform strong test collection: they are exactly the tuples that are injective, with non-vanishing joint derivative, on the closed disc. Finally, we show by examples that the separation hypothesis cannot be replaced by the existence of a bounded $H^\\infty$ calculus, by the existence of a norm radial limit, or by the strong convergence of the partial products, even for Ritt operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01703",
  "author_total": 1,
  "authors": [
    {
      "name": "Catalin Badea"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3m12d0eyntq9j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01703",
  "fulltext_url": null,
  "id": "3m12d0eyntq9j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01703",
  "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"
    ],
    [
      "arxiv",
      "math.OA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3m12d0eyntq9j.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Transfer of similarity to Hilbert space contractions through inner functions: a proof of Douglas's conjecture",
  "title_text": "Transfer of similarity to Hilbert space contractions through inner functions: a proof of Douglas's conjecture",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/3m12d0eyntq9j",
  "venue": "arXiv",
  "versions": []
}