{
  "abstract": "We consider the harmonically weighted Dirichlet spaces $D(μ)$ induced by Borel measures which are mutually absolutely continuous with respect to the Lebesgue measure $m$ on the unit circle $\\mathbb T$. Let $H^2$ and $D$ denote the Hardy space and the Dirichlet space on the unit disc $\\mathbb D$, respectively. For any $f\\in H^2$, let $m_f$ denote the measure defined by $dm_f(ζ)=|f(ζ)|^2 dm(ζ)$. Our first result provides a characterization of functions in $D(m_f)$, when $f$ satisfies the condition $|f'(z)|^2=O(\\frac{1}{(1-|z|)^{1-ε}})$ for some $ε>0$. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when $f'$ is bounded, we obtain an explicit description of all singular inner functions in the multiplier algebra of $D(m_f)$ in terms of the radial zero set of $f$. As an application, we prove that if $f',g'$ are bounded and the radial zero sets of $f$ and $g$ are different, then the operators $(M_z, D(m_f))$ and $(M_z, D(m_g))$ are not similar. This generalizes a result of Richter showing that the operators $(M_z, D(m_{z-1}))$ and $(M_z, D)$ are not similar. Then, we classify all invariant subspaces $\\mathcal M$ of $(M_z, D(μ))$ for which $M_z|_{\\mathcal M}$ is similar to $(M_z, D(μ))$, where $μ$ is any measure mutually absolutely continuous with respect to $m$. Finally, we study a special case of the problem of finding all functions $g\\in D$ for which $gD(m_g)$ is a closed invariant subspace of $(M_z, D)$. A key ingredient in many of our results is the Richter-Sundberg formula for the local Dirichlet integral.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16988",
  "author_total": 2,
  "authors": [
    {
      "name": "Soumitra Ghara"
    },
    {
      "name": "Avipsa Patra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6957sd3wz8xxk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16988",
  "fulltext_url": null,
  "id": "6957sd3wz8xxk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16988",
  "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": "W7213411756",
  "pdf_url": "https://arxiv.org/pdf/2609.16988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6957sd3wz8xxk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Inner multipliers, similarity and invariant subspaces for a class of Harmonically weighted Dirichlet spaces",
  "title_text": "Inner multipliers, similarity and invariant subspaces for a class of Harmonically weighted Dirichlet spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/6957sd3wz8xxk",
  "venue": "arXiv",
  "versions": []
}