{
  "id": "2411.17654",
  "version": "v1",
  "published": "2024-11-26T18:17:13.000Z",
  "updated": "2024-11-26T18:17:13.000Z",
  "title": "Testing compactness of linear operators",
  "authors": [
    "Timo S. Hänninen",
    "Tuomas V. Oikari"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "Let $(F_i)$ be a sequence of sets in a Banach space $X$. For what sequences does the condition $$ \\limsup_{i\\to \\infty} \\sup_{f_i\\in F_i} \\|Tf_i\\|_Y=0 $$ hold for every Banach space $Y$ and every compact operator $T:X\\to Y$? We answer this question by giving sufficient (and necessary) criteria for such sequences. We illustrate the applicability of the criteria by examples from literature and by characterizing the $L^p\\to L^p$ compactness of dyadic paraproducts on general measure spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-26T18:17:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear operators",
      "testing compactness",
      "banach space",
      "general measure spaces",
      "dyadic paraproducts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}