{
  "id": "1311.7575",
  "version": "v1",
  "published": "2013-11-29T14:24:53.000Z",
  "updated": "2013-11-29T14:24:53.000Z",
  "title": "Lipschitz $\\left(\\mathfrak{m}^L\\left(s;q\\right),p\\right)$ and $\\left(p,\\mathfrak{m}^L\\left(s;q\\right)\\right)-$summing maps",
  "authors": [
    "Manaf Adnan Salah"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Building upon the linear version of mixed summable sequences in arbitrary Banach spaces of A. Pietsch, we introduce a nonlinear version of his concept and study its properties. Extending previous work of J. D. Farmer, W. B. Johnson and J. A. Ch\\'avez-Dom\\'inguez, we define Lipschitz $\\left(\\mathfrak{m}^L\\left(s;q\\right),p\\right)$ and Lipschitz $\\left(p,\\mathfrak{m}^L\\left(s;q\\right)\\right)-$summing maps and establish inclusion theorems, composition theorems and several characterizations. Furthermore, we prove that the classes of Lipschitz $\\left(r,\\mathfrak{m}^L\\left(r;r\\right)\\right)-$summing maps with $0<r<1$ coincide. We obtain that every Lipschitz map is Lipschitz $\\left(p,\\mathfrak{m}^L\\left(s;q\\right)\\right)-$summing map with $1\\leq s< p$ and $0<q\\leq s$ and discuss a sufficient condition for a Lipschitz composition formula as in the linear case of A. Pietsch. Moreover, we discuss a counterexample of the nonlinear composition formula, thus solving a problem by J. D. Farmer and W. B. Johnson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-29T14:24:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47L20",
      "47B10"
    ],
    "keywords": [
      "summing map",
      "arbitrary banach spaces",
      "lipschitz composition formula",
      "nonlinear composition formula",
      "lipschitz map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.7575A"
    }
  }
}