{
  "id": "2203.10265",
  "version": "v1",
  "published": "2022-03-19T07:52:34.000Z",
  "updated": "2022-03-19T07:52:34.000Z",
  "title": "Extreme points of the unit ball of $\\mathcal{L}(X)_w^*$ and best approximation in $\\mathcal{L}(X)_w$",
  "authors": [
    "Arpita Mal"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the geometry of $\\mathcal{L}(X)_w,$ the space of all bounded linear operators on a Banach space $X,$ endowed with the numerical radius norm, whenever the numerical radius defines a norm. We obtain the form of the extreme points of the unit ball of the dual space of $\\mathcal{L}(X)_w.$ Using this structure, we explore Birkhoff-James orthogonality, best approximation and deduce distance formula in $\\mathcal{L}(X)_w.$ A special attention is given to the case of operators satisfying a notion of smoothness. Finally, we obtain an equivalence between Birkhoff-James orthogonality in $\\mathcal{L}(X)_w$ and that in $X.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-19T07:52:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B28",
      "47A12"
    ],
    "keywords": [
      "best approximation",
      "unit ball",
      "extreme points",
      "birkhoff-james orthogonality",
      "deduce distance formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}