{
  "id": "1805.03277",
  "version": "v1",
  "published": "2018-05-08T20:47:43.000Z",
  "updated": "2018-05-08T20:47:43.000Z",
  "title": "Rank-one perturbations of quasinilpotent operators and the invariant subspace problem",
  "authors": [
    "Adi Tcaciuc"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We show that a bounded quasinilpotent operator $T$ acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator $F$ and a scalar $\\alpha\\in\\mathbb{C}$, $\\alpha\\neq 0$, $\\alpha\\neq 1$, such that $T+F$ and $T+\\alpha F$ are also quasinilpotent. We also prove that for any fixed rank-one operator $F$, almost all perturbations $T+\\alpha F$ have invariant subspaces of infinite dimension and codimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-08T20:47:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A15",
      "47A55"
    ],
    "keywords": [
      "invariant subspace problem",
      "rank-one perturbations",
      "infinite dimensional banach space",
      "bounded quasinilpotent operator",
      "fixed rank-one operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}