{
  "id": "1011.0262",
  "version": "v1",
  "published": "2010-11-01T07:23:46.000Z",
  "updated": "2010-11-01T07:23:46.000Z",
  "title": "A characterization of compact operators via the non-connectedness of the attractors of a family of IFSs",
  "authors": [
    "Alexandru Mihail",
    "Radu Miculescu"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space X, S and T bounded linear operators from X to X such that \\parallel S \\parallel, \\parallel T \\parallel <1 and w \\in X, let us consider the IFS S_{w}=(X,f_1,f_2), where f_1,f_2:X \\rightarrow X are given by f_1(x)=S(x) and f_2(x)=T(x)+w, for all x \\in X. On one hand we prove that if the operator S is compact, then there exists a family (K_{n})_{n \\in N} of compact subsets of X such that A_{S_{w}} is not connected, for all w \\in H- \\cup K_{n}. One the other hand we prove that if H is an infinite dimensional Hilbert space, then a bounded linear operator S:H \\rightarrow H having the property that \\parallel S \\parallel <1 is compact provided that for every bounded linear operator T:H\\rightarrow H such that \\parallel T \\parallel <1 there exists a sequence (K_{T,n})_{n} of compact subsets of H such that A_{S_{w}} is not connected for all w \\in H- \\cup K_{T,n}. Consequently, given an infinite dimensional Hilbert space H, there exists a complete characterization of the compactness of an operator S:H \\rightarrow H by means of the non-connectedness of the attractors of a family of IFSs related to the given operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-01T07:23:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "47B07",
      "54D05"
    ],
    "keywords": [
      "compact operators",
      "bounded linear operator",
      "infinite dimensional hilbert space",
      "non-connectedness",
      "attractors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.0262M"
    }
  }
}