{
  "id": "1911.02072",
  "version": "v1",
  "published": "2019-11-05T20:24:54.000Z",
  "updated": "2019-11-05T20:24:54.000Z",
  "title": "Weak compactness and fixed point property for affine bi-Lipschitz maps",
  "authors": [
    "Cleon S. Barroso",
    "Valdir Ferreira"
  ],
  "comment": "This is an update for arXiv:1610.05642v2. The new version of the manuscript contains improvements of old results and also brings new ones",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz affine mappings for every $L>1$. It is also proved that if $X$ has Pe\\l czy\\'nski's property $(u)$, then either $C$ is weakly compact, contains an $\\ell_1$-sequence or a $\\mathrm{c}_0$-summing basic sequence. In this case, weak compactness of $C$ is equivalent to the $\\mathcal{G}$-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We also introduce a generalized form of property $(u)$, called {it property $(\\mathfrak{su})$}, and use it to prove that if $X$ has property $(\\mathfrak{su})$ then either $C$ is weakly compact or contains a wide-$(s)$ sequence which is uniformly shift equivalent. In this case, weak compactness in such spaces can also be characterized in terms of the $\\mathcal{G}$-FPP for affine uniformly bi-Lipschitz mappings. It is also proved that every Banach space with a spreading basis has property $(\\mathfrak{su})$, thus property $(\\mathfrak{su})$ is stronger than property $(u)$. These results yield a significant strengthening of an important theorem of Benavides, Jap\\'on-Pineda and Prus published in 2004.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-05T20:24:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fixed point property",
      "weak compactness",
      "affine bi-lipschitz maps",
      "weakly compact",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}