{
  "abstract": "We give negative answers to Mbekhta's conjecture \\cite[p.~261]{Mbekhta} and to a related conjecture on pseudo-B-Fredholm operators \\cite[p.~202]{AZO}. A sufficient condition for pseudo-Fredholmness on Banach spaces is established. It yields a characterization on Hilbert spaces in terms of the closedness of $\\cc(T)$, $\\cc(T)+\\qn(T)$ and $\\Ran(T)+\\qn(T)$, as well as characterizations of pseudo-B-Fredholm and pseudo-B-Weyl operators on Banach spaces. A counterexample on a separable Banach space shows that the three closedness conditions do not characterize pseudo-Fredholmness in general. We also construct two operators having the same analytic core, quasinilpotent part, and kernels and ranges of all powers, although only one is pseudo-Fredholm. Moreover, a semi-B-Fredholm operator on a Banach space need not be a direct sum of a semi-Fredholm operator and a nilpotent operator, answering the question raised in \\cite[p.~113]{AznayKato2023}. Finally, the three closedness conditions in the Hilbert-space characterization are shown to be independent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23229",
  "author_total": 2,
  "authors": [
    {
      "name": "Zakariae Aznay"
    },
    {
      "name": "Fadil Chabbabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enmn8pgp2smjk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23229",
  "fulltext_url": null,
  "id": "enmn8pgp2smjk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23229",
  "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": "W7222983762",
  "pdf_url": "https://arxiv.org/pdf/2610.23229",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enmn8pgp2smjk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23229",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23229"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On Mbekhta's conjecture and pseudo-Fredholm operators on Banach and Hilbert spaces",
  "title_text": "On Mbekhta's conjecture and pseudo-Fredholm operators on Banach and Hilbert spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/enmn8pgp2smjk",
  "venue": "arXiv",
  "versions": []
}