{
  "id": "0810.1192",
  "version": "v2",
  "published": "2008-10-07T13:40:53.000Z",
  "updated": "2008-10-22T12:32:30.000Z",
  "title": "Existence theorems in linear chaos",
  "authors": [
    "S. Shkarin"
  ],
  "categories": [
    "math.FA",
    "math.DS"
  ],
  "abstract": "Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fr\\'echet space $X$ if and only if $X$ is non-isomorphic to the space $\\omega$ of all sequences with coordinatewise convergence topology. It is also shown for any $k\\in\\N$, any separable infinite dimensional Fr\\'echet space $X$ non-isomorphic to $\\omega$ admits a mixing uniformly continuous group $\\{T_t\\}_{t\\in C^n}$ of continuous linear operators and that there is no supercyclic strongly continuous operator semigroup $\\{T_t\\}_{t\\geq 0}$ on $\\omega$. We specify a wide class of Fr\\'echet spaces $X$, including all infinite dimensional Banach spaces with separable dual, such that there is a hypercyclic operator $T$ on $X$ for which the dual operator $T'$ is also hypercyclic. An extension of the Salas theorem on hypercyclicity of a perturbation of the identity by adding a backward weighted shift is presented and its various applications are outlined.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-10-22T12:32:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A16",
      "37A25"
    ],
    "keywords": [
      "infinite dimensional frechet space",
      "existence theorems",
      "linear chaos",
      "dimensional complex frechet space",
      "infinite dimensional complex frechet"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.1192S"
    }
  }
}