{
  "abstract": "For a bounded operator $T$ between Banach spaces, let $d_+(T)$ be its distance to the operators that are not upper semi-Fredholm, and let $\\inn(T)$ be Schechter's quantity $\\inf\\|T|_M\\|$ over infinite-dimensional subspaces $M$. One always has $\\inn(T)\\le d_+(T)$, and González and Martinón showed that the two quantities are not equivalent in general. We prove that $d_+(T)$ is the infimum of the norms of operators on the whole space that extend a restriction of $T$ to some infinite-dimensional subspace. The gap between $\\inn$ and $d_+$ thus becomes an extension problem, and projection constants enter naturally. If the domain is uniformly subprojective with constant $λ$, then $d_+\\leλ\\,\\inn$. In the other direction we prove a quantitative obstruction: a complemented Hilbertian subspace all of whose infinite-dimensional subspaces have projection constant at least $Λ$ produces operators with $d_+/\\inn\\geΛ/2$. Consequently, on the subprojective but not uniformly subprojective space $(\\bigoplus_nL_{p_n})_{\\ell_2}$ with $p_n\\to\\infty$, there is no uniform comparison between $\\inn$ and $d_+$. We conjecture that uniform comparison is equivalent to uniform subprojectivity. Finally, we compute $\\inn$ for inverses and for shears in terms of the quantity $τ(T)=\\sup_M j(T|_M)$, and we use the arbitrary distortability of Hilbert space to show that a rectangular diagonal operator $\\operatorname{diag}(A,B)$ built from two isomorphisms with $\\inn(A)=\\inn(B)=1$ can have arbitrarily small $\\inn$. In contrast, every operator $T$ on a complex Banach space satisfies $\\inn(T)\\leτ(T)$, so every automorphism satisfies $\\inn(T)\\,\\inn(T^{-1})\\le1$ and the same mechanism cannot work for endomorphisms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03835",
  "author_total": 1,
  "authors": [
    {
      "name": "Abdelhalim Azzouz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/691z4bm415d1p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03835",
  "fulltext_url": null,
  "id": "691z4bm415d1p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03835",
  "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.GM"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7220740950",
  "pdf_url": "https://arxiv.org/pdf/2610.03835",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/691z4bm415d1p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.03835",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03835"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Advanced Banach Space Theory",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The upper semi-Fredholm radius, extensions of restrictions, and subprojectivity",
  "title_text": "The upper semi-Fredholm radius, extensions of restrictions, and subprojectivity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/691z4bm415d1p",
  "venue": "arXiv",
  "versions": []
}