{
  "id": "2004.10649",
  "version": "v1",
  "published": "2020-04-22T15:48:13.000Z",
  "updated": "2020-04-22T15:48:13.000Z",
  "title": "Some stability properties for the Bishop--Phelps--Bollobás property for Lipschitz maps",
  "authors": [
    "Rafael Chiclana",
    "Miguel Martin"
  ],
  "comment": "19 pages. The content of this paper overlaps with the old version of arXiv: arXiv:1901.02956 (arXiv:1901.02956v1, submitted on 9 Jan 2019). Nevertheless, there is no intersection between the present version and the updated one of arXiv:1901.02956 (arXiv: arXiv:1901.02956v2, submitted on 17 Jun 2019)",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the stability behavior of the Bishop-Phelps-Bollob\\'as property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollob\\'as property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very closed to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property from scalar functions to some vector-valued maps, getting some positive results related to the notions of $\\Gamma$-flat operators and $ACK$ structure. We get sharper results for the case of Lipschitz compact maps. The behaviour of the property with respect to absolute sums of the target space is also studied. We also get results similar to the above ones about the density of strongly norm attaining Lipschitz maps and of Lipschitz compact maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-22T15:48:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B04",
      "46B20",
      "26A16",
      "54E50"
    ],
    "keywords": [
      "bishop-phelps-bollobás property",
      "stability properties",
      "lipschitz compact maps",
      "lip-bpb property",
      "bishop-phelps-bollobas property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}