{
  "abstract": "Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition \\[ X=\\K(T)\\oplus \\Hq(T) \\] forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that $T$ is generalized Drazin invertible if and only if \\[ \\dim\\bigl(\\K(T)\\cap \\Hq(T)\\bigr)<\\infty \\quad\\text{and}\\quad \\codim\\bigl(\\K(T)+\\Hq(T)\\bigr)<\\infty. \\] For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18433",
  "author_total": 2,
  "authors": [
    {
      "name": "Zakariae Aznay"
    },
    {
      "name": "Fadil Chabbabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cxg05eevqafx3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18433",
  "fulltext_url": null,
  "id": "cxg05eevqafx3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18433",
  "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",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7221749566",
  "pdf_url": "https://arxiv.org/pdf/2610.18433",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cxg05eevqafx3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18433",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18433"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Algebraic Direct Sums and Generalized Drazin Invertibility",
  "title_text": "Algebraic Direct Sums and Generalized Drazin Invertibility",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/cxg05eevqafx3",
  "venue": "arXiv",
  "versions": []
}