{
  "id": "2208.04812",
  "version": "v1",
  "published": "2022-08-09T14:40:56.000Z",
  "updated": "2022-08-09T14:40:56.000Z",
  "title": "On the adjoint of the unbounded operators $p(T)$, $TT^*$, and $T^*T$, and certain applications to $nth$ roots of unbounded operators",
  "authors": [
    "Mohammed Hichem Mortad"
  ],
  "comment": "30 pages",
  "categories": [
    "math.FA",
    "math.SP"
  ],
  "abstract": "In this paper, we are mainly concerned with conditions under which $[p(T)]^*=\\overline{p}(T^*)$, where $p(z)$ is a one-variable complex polynomial, and $T$ is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the identities $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable operator. A particular interest will be paid to the equation $T^*T=p(T)$ and its variants. Also, we have certain results concerning $nth$ roots of classes of normal and nonnormal (unbounded) operators. Some further consequences and counterexamples accompany our results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-09T14:40:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unbounded operators",
      "applications",
      "one-variable complex polynomial",
      "linear operator",
      "counterexamples accompany"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}