{
  "id": "2107.11603",
  "version": "v1",
  "published": "2021-07-24T13:34:43.000Z",
  "updated": "2021-07-24T13:34:43.000Z",
  "title": "On a generalization of Smiley's theorem",
  "authors": [
    "Xiao Chen",
    "Jian-Jian Jiang",
    "Xiaolin Li"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA",
    "math.OA",
    "math.RA"
  ],
  "abstract": "In this short article, we mainly prove that, for any bounded operator $A$ of the form $S+N$ on a complex Hilbert space, where $S$ is a normal operator and $N$ is a nilpotent operator commuting with $S$, if a bounded operator $B$ lies in the collection of bounded linear operators that are in the $k$-centralizer of every bounded linear operator in the $l$-centralizer of $A$, where $k\\leqslant l$ is two arbitrary positive integers, then $B$ must belong to the operator norm closure of the algebra generated by $A$ and identity operator. This result generalizes a matrix commutator theorem proved by M.\\ F.\\ Smiley. For this aim, Smiley-type operators are defined and studied. Furthermore, we also give a generalization of Fuglede's theorem as a byproduct.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-24T13:34:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B02",
      "47B47",
      "47B40",
      "47B15"
    ],
    "keywords": [
      "smileys theorem",
      "generalization",
      "bounded linear operator",
      "complex hilbert space",
      "bounded operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}