{
  "id": "1807.07423",
  "version": "v1",
  "published": "2018-07-17T19:35:12.000Z",
  "updated": "2018-07-17T19:35:12.000Z",
  "title": "On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials",
  "authors": [
    "Marat V. Markin"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1803.10038",
  "categories": [
    "math.FA"
  ],
  "abstract": "We give a straightforward proof of the non-hypercyclicity of an arbitrary scalar type spectral operator $A$ (bounded or not) in a complex Banach space as well as of the collection $\\left\\{e^{tA}\\right\\}_{t\\ge 0}$ of its exponentials, which, under a certain condition on the spectrum of $A$, coincides with the $C_0$-semigroup generated by it. The spectrum of $A$ lying on the imaginary axis, it is shown that non-hypercyclic is also the generated by it strongly continuous group $\\left\\{e^{tA}\\right\\}_{t\\in {\\mathbb R}}$ of bounded linear operators. As an important particular case, we immediately obtain that of a normal operator $A$ in a complex Hilbert space. From the general results, we infer that, in the complex Hilbert space $L_2({\\mathbb R})$, the anti-self-adjoint differentiation operator $A:=\\dfrac{d}{dx}$ with the domain $D(A):=W_2^1({\\mathbb R})$ is not hypercyclic and neither is the left-translation group generated by it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-17T19:35:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A16",
      "47B40",
      "47B15",
      "47D06",
      "47D60",
      "34G10"
    ],
    "keywords": [
      "complex hilbert space",
      "arbitrary scalar type spectral operator",
      "non-hypercyclicity",
      "exponentials",
      "collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}