{
  "id": "1805.00663",
  "version": "v1",
  "published": "2018-05-02T08:06:44.000Z",
  "updated": "2018-05-02T08:06:44.000Z",
  "title": "Characterization of continuous endomorphisms in the space of entire functions of a given order",
  "authors": [
    "Takashi Aoki",
    "Ryuichi Ishimura",
    "Yasunori Okada",
    "Daniele C. Struppa",
    "Shofu Uchida"
  ],
  "comment": "13 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "The aim of this paper is to characterize continuous endomorphisms in the space of entire functions of exponential type of order $p>0$. Let $A_p$ denote the space of entire functions of $n$ complex variables $z\\in{\\mathbb C}^n$ of order $p$ of normal type. We consider an endomorphism $F$ in the space, which is considered to be a DFS-space. We show that there is a unique linear differential operator $P$ of infinite order with coefficients in the space which realizes $F$, that is, $Ff=Pf$ holds for any $f\\in A_p$. The coefficients satisfy certain growth conditions and conversely, if a formal differential operator of infinite order with coefficients in $A_p$ satisfy these conditions, then it induces a continuous endomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-02T08:06:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B38",
      "30D15"
    ],
    "keywords": [
      "entire functions",
      "continuous endomorphism",
      "infinite order",
      "unique linear differential operator",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}