{
  "abstract": "Let $T\\in\\mathcal{B}(X)$, where $X$ is a complex Banach space. We introduce the spaces $\\mathcal{I}_s(T)$, defined by independently chosen preimages under the powers of $T$ with exponential growth bounds, and study algebraic radial decompositions involving $\\mathcal{H}_r(T)$, $\\mathcal{K}_s(T)$, and $\\mathcal{I}_s(T)$. We prove that every algebraic decomposition $X=\\mathcal{H}_r(T)\\oplus\\mathcal{I}_s(T)$ is topological and that $\\mathcal{I}_s(T)=\\mathcal{K}_s(T)$. This removes the closedness assumption in the radial decomposition theorem of Koliha and Poon. We also prove that, for $0\\leq a\\leq b$, the closedness of $\\mathcal{I}_b(T)$ and $\\mathcal{H}_a(T)+\\mathcal{I}_b(T)$ implies $\\mathcal{I}_b(T)=\\mathcal{K}_b(T)$. We establish finite-defect and circular spectral criteria and characterize the Cantor--Bendixson derivatives of the spectrum, Browder spectrum, and Drazin spectrum in terms of radial decompositions and quotient restrictions. As applications, we obtain new characterizations of generalized Drazin, generalized Drazin--Riesz, generalized Drazin--meromorphic, and $g_z$-invertible operators in terms of radial subspaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06831",
  "author_total": 3,
  "authors": [
    {
      "name": "Zakariae Aznay"
    },
    {
      "name": "Fadil Chabbabi"
    },
    {
      "name": "Hassan Zariouh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6j6179ek1fna.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06831",
  "fulltext_url": null,
  "id": "a6j6179ek1fna",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06831",
  "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"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06831",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6j6179ek1fna.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06831",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06831"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Independent Backward Preimages and Algebraic Radial Decompositions of Operators",
  "title_text": "Independent Backward Preimages and Algebraic Radial Decompositions of Operators",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/a6j6179ek1fna",
  "venue": "arXiv",
  "versions": []
}