{
  "abstract": "We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound $\\sup_{n\\ge1}\\|I-T^n\\|\\le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd power of $T$ a Ritt operator. The sharp unconditional threshold for $T$ itself is $\\sqrt3$. If $Φ_*(q)$ denotes the optimal universal Ritt resolvent bound below this threshold, then $Φ_*(q)=2/(\\sqrt3-q)+O(1)$ as $q\\uparrow\\sqrt3$. At every higher peripheral threshold the optimal finite-peripheral resolvent bound has reciprocal order of growth. We also obtain constructive bounds for general weighted Wiener symbols. For each $1<q<2$, an angular escape invariant characterises exactly the sampling sequences for which a displacement bound by $q$, together with the peripheral spectral condition, forces the Ritt property. We compute this invariant for asymptotically geometric sequences. At the endpoint $q=1$, suitable phase conditions imply the full displacement bound without assuming power boundedness; normal contractions admit an exact scalar criterion. Finally, we establish quantitative Hilbert-space and $L^p$ resolvent estimates, a characterisation of operators with odd Ritt powers on uniformly convex spaces, and strict norm gaps for operator-valued disc-algebra functions, with explicit matrix-polynomial bounds from Fejér--Riesz factorisation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27458",
  "author_total": 1,
  "authors": [
    {
      "name": "Catalin Badea"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jf5e2pkbyg51.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27458",
  "fulltext_url": null,
  "id": "7jf5e2pkbyg51",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27458",
  "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.DS"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7214244816",
  "pdf_url": "https://arxiv.org/pdf/2609.27458",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jf5e2pkbyg51.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27458",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27458"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Iterates of Ritt operators close to the identity",
  "title_text": "Iterates of Ritt operators close to the identity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/7jf5e2pkbyg51",
  "venue": "arXiv",
  "versions": []
}