{
  "id": "1311.2270",
  "version": "v1",
  "published": "2013-11-10T13:25:34.000Z",
  "updated": "2013-11-10T13:25:34.000Z",
  "title": "On linear operators with s-nuclear adjoints: $0< s \\le 1$",
  "authors": [
    "O. I. Reinov"
  ],
  "comment": "11 pages, AMS TeX",
  "categories": [
    "math.FA"
  ],
  "abstract": "If $ s\\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$ to a Banach space $ Y$ and if one of the spaces $ X^*$ or $ Y^{***}$ has the approximation property of order $s,$ $AP_s,$ then the operator $ T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $ T: Z_0^{**}\\to Z_0$ so that $Z_0^{**}$ has a Schauder basis, $ Z_0^{***}$ has the $AP_s$ for every $s\\in (0,r)$ and $T^*$ is $r$-nuclear.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-10T13:25:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear operator",
      "s-nuclear adjoints",
      "banach space",
      "schauder basis",
      "approximation property"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.2270R"
    }
  }
}