{
  "id": "2409.14481",
  "version": "v1",
  "published": "2024-09-22T15:03:09.000Z",
  "updated": "2024-09-22T15:03:09.000Z",
  "title": "Typical properties of positive contractions and the invariant subspace problem",
  "authors": [
    "Valentin Gillet"
  ],
  "comment": "22 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we first study some elementary properties of a typical positive contraction on $\\ell_q$ for the Strong Operator Topology and the Strong* Operator Topology. Using these properties we prove that a typical positive contraction on $\\ell_1$ (resp. on $\\ell_2$) has a non-trivial invariant subspace for the Strong Operator Topology (resp. for the Strong Operator Topology and the Strong* Operator Topology). We then focus on the case where $X$ is a Banach space with a basis. We prove that a typical positive contraction on a Banach space with an unconditional basis has no non-trivial closed invariant ideals for the Strong Operator Topology and the Strong* Operator Topology. In particular this shows that when $X = \\ell_q$ with $1 \\leq q < \\infty$, a typical positive contraction $T$ on $X$ for the Strong Operator Topology (resp. for the Strong* Operator Topology when $1 < q < \\infty$) does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion, that is there is no non-zero positive operator in the commutant of $T$ which is quasinilpotent at a non-zero positive vector of $X$. Finally we prove that, for the Strong* Operator Topology, a typical positive contraction on a reflexive Banach space with a monotone basis does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-22T15:03:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "strong operator topology",
      "typical positive contraction",
      "invariant subspace problem",
      "typical properties",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}