{
  "abstract": "Any holomorphic Lipschitz map $f$, defined on the open unit ball $B_X$ of a complex Banach space $X$ and taking values in a complex Banach space $Y$, can be uniquely and canonically associated with a bounded linear operator $\\widehat f : \\mathcal G_0(B_X) \\longrightarrow \\mathcal G_0(Y)$ between the corresponding holomorphic Lipschitz-free spaces. The aim of this paper is to establish necessary and sufficient conditions for the compactness and weak compactness of such operators. In particular, we obtain a general characterisation of the compactness of $\\widehat f$ in terms of metric properties of its symbol $f$. We then focus on the case where the target space is a subspace of $\\ell_p$, $1\\leq p<\\infty$. When the target space is $\\ell_1$, we show that compactness and weak compactness of $\\widehat f$ are both equivalent to $\\widehat f$ not fixing a copy of $\\ell_1$. Finally, when the target space is $\\ell_p$ with $1<p<\\infty$, we obtain a metric characterisation of the weak compactness of $\\widehat f$. As a consequence, we show that compactness and weak compactness need not coincide, even when the target space is $\\ell_p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26047",
  "author_total": 3,
  "authors": [
    {
      "name": "Frédéric Bayart"
    },
    {
      "name": "Mathis Lemay"
    },
    {
      "name": "Colin Petitjean"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cp16dm1zgfn6n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26047",
  "fulltext_url": null,
  "id": "cp16dm1zgfn6n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26047",
  "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"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7223041012",
  "pdf_url": "https://arxiv.org/pdf/2610.26047",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cp16dm1zgfn6n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26047",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26047"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Compactness and weak compactness of linearized Holomorphic-Lipschitz maps",
  "title_text": "Compactness and weak compactness of linearized Holomorphic-Lipschitz maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/cp16dm1zgfn6n",
  "venue": "arXiv",
  "versions": []
}