{
  "id": "2412.19115",
  "version": "v1",
  "published": "2024-12-26T08:18:50.000Z",
  "updated": "2024-12-26T08:18:50.000Z",
  "title": "A note on disjoint hypercyclicity for invertible bilateral pseudo-shifts on $\\ell^{p}(\\mathbb{Z})$",
  "authors": [
    "SongUng Ri",
    "HyonHui Ju",
    "JinMyong Kim"
  ],
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "We first give a note on disjoint hypercyclicity for invertible bilateral pseudo-shifts on $\\ell^{p}(\\mathbb{Z})$, $1\\leq p <\\infty$. It is already known that if a tuple of bilateral weighted shifts on $\\ell^{p}(\\mathbb{Z})$, $1\\leq p <\\infty$, is disjoint hypercyclic, then non of the weighted shifts is invertible. We show that as for pseudo-shifts which is a generalization of weighted shifts, this fact is not true. We give an example of invertible bilateral pseudo-shifts on $\\ell^{p}(\\mathbb{Z})$, $1\\leq p <\\infty$, which are disjoint hypercyclic and whose inverses are also disjoint hypercyclic. Next we partially answer to the open problem posed by Martin, Menet and Puig (2022)\\cite{MMP22} concerned with disjoint reiteratively hypercyclic, that is, we show that as for the operators on a reflexive Banach space, reiteratively hypercyclic ones are disjoint hypercyclic if and only if they are disjoint reiteratively hypercyclic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-26T08:18:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A16",
      "47B37"
    ],
    "keywords": [
      "invertible bilateral pseudo-shifts",
      "disjoint hypercyclicity",
      "disjoint reiteratively hypercyclic",
      "open problem",
      "bilateral weighted shifts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}