{
  "id": "2003.12855",
  "version": "v1",
  "published": "2020-03-28T17:53:50.000Z",
  "updated": "2020-03-28T17:53:50.000Z",
  "title": "Geometry of nowhere vanishing, point separating sub-algebras of $\\mathcal{H}ol(Γ\\cup\\text{Int}(Γ))$ and zeros of Holomorphic functions",
  "authors": [
    "Babhrubahan Bose",
    "Saikat Roy",
    "Debmalya Sain"
  ],
  "comment": "15 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study $ \\mathcal{H}ol(\\Gamma\\cup\\text{Int}(\\Gamma)) $, the normed algebra of all holomorphic functions defined on some simply connected neighbourhood of a simple closed curve $\\Gamma$ in $\\mathbb{C} $, equipped with the supremum norm on $ \\Gamma $. We explore the geometry of nowhere vanishing, point separating sub-algebras of $ \\mathcal{H}ol(\\Gamma \\cup \\text{Int}(\\Gamma)) $. We characterize the extreme points and the exposed points of the unit balls of the said sub-algebras. We also characterize the smoothness of an element in these sub-algebras by using Birkhoff-James orthogonality techniques. As a culmination of our study, we assimilate the geometry of the aforesaid sub-algebras with some classical concepts of complex analysis and establish a connection between Birkhoff-James orthogonality and zeros of holomorphic functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-28T17:53:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "32A60"
    ],
    "keywords": [
      "point separating sub-algebras",
      "holomorphic functions",
      "birkhoff-james orthogonality techniques",
      "extreme points",
      "unit balls"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}