{
  "abstract": "Maps $Φ:D\\to D$ of the unit disk preserve frame generator sets for single orbits of normal diagonal operators exactly when they preserve Carleson interpolating sequences bidirectionally and satisfy $1-|Φ(z)|^2\\asymp1-|z|^2$ uniformly. Denote this class by $\\mathcal P$. Equality of frame generator sets forces the boundary estimate, since every frame generator $f=(f_j)$ satisfies $|f_j|^2\\asymp 1-|λ_j|^2$. No continuity, measurability, or analyticity is assumed; Carleson preservation alone remains flexible and, to our knowledge, unclassified in general. Equivalently, normalized Szego kernel Riesz sequences are preserved bidirectionally, with uniformly comparable unnormalized kernel norms at $z$ and $Φ(z)$. Equivalently, we require uniform two-sided Carleson norm bounds for ordinary pushforwards of finite positive discrete measures and bidirectional preservation of pairwise pseudohyperbolic separation. Every $Φ\\in\\mathcal P$ has a canonical bilipschitz radial trace $h_Φ(ζ)=\\lim_{r\\to1^-}Φ(rζ)$ on $T=\\partial D$, with uniform convergence. We give an intrinsic geometric characterization of $\\mathcal K=\\{Ψ\\in\\mathcal P:h_Ψ=\\mathrm{id}\\}$ and prove the unique factorization $Φ=Ψ\\circ E_{h_Φ}$, $Ψ\\in\\mathcal K$, where $E_h(rζ)=rh(ζ)$ and $E_h(0)=0$; thus $\\mathcal P\\cong\\mathcal K\\rtimes\\mathrm{BiLip}(T)$ as monoids. For $m$ orbits, $\\mathcal P_m=\\mathcal P$ for every finite $m\\ge1$, whereas $\\mathrm{Aut}(D)\\subsetneq\\mathcal P_ω\\subsetneq\\mathcal P$. Countable-orbit preservers in $\\mathcal P_ω$ are uniform pseudohyperbolic homeomorphisms. If $φ\\in\\mathrm{Aut}(D)$ and $|Φ(z)-φ(z)|=o(1-|z|)$ uniformly as $|z|\\to1$, then $Φ\\in\\mathcal P_ω$ exactly when $Φ$ is a disk homeomorphism. Two-sided tests characterize $\\mathcal P_ω$. Holomorphic members of these classes are exactly disk automorphisms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.07259",
  "author_total": 1,
  "authors": [
    {
      "name": "Jian Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/055ng8k9eavd3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.07259",
  "fulltext_url": null,
  "id": "055ng8k9eavd3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.07259",
  "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": "W7202146317",
  "pdf_url": "https://arxiv.org/pdf/2608.07259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/055ng8k9eavd3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2608.07259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.07259"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames",
  "title_text": "Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/055ng8k9eavd3",
  "venue": "arXiv",
  "versions": []
}