{
  "id": "1812.10365",
  "version": "v1",
  "published": "2018-12-26T16:37:51.000Z",
  "updated": "2018-12-26T16:37:51.000Z",
  "title": "Generalized frame operator distance problems",
  "authors": [
    "Pedro Massey",
    "Noelia Rios",
    "Demetrio Stojanoff"
  ],
  "comment": "24 pages. There exists text overlap with other arxiv manuscripts in the preliminary sections",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $S\\in\\mathcal{M}_d(\\mathbb{C})^+$ be a positive semidefinite $d\\times d$ complex matrix and let $\\mathbf a=(a_i)_{i\\in\\mathbb{I}_k}\\in \\mathbb{R}_{>0}^k$, indexed by $\\mathbb{I}_k=\\{1,\\ldots,k\\}$, be a $k$-tuple of positive numbers. Let $\\mathbb T_{d}(\\mathbf a )$ denote the set of families $\\mathcal G=\\{g_i\\}_{i\\in\\mathbb{I}_k}\\in (\\mathbb{C}^d)^k$ such that $\\|g_i\\|^2=a_i$, for $i\\in\\mathbb{I}_k$; thus, $\\mathbb T_{d}(\\mathbf a )$ is the product of spheres in $\\mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $\\mathcal{M}_d(\\mathbb{C})$, we consider the generalized frame operator distance function $\\Theta_{( N \\, , \\, S\\, , \\, \\mathbf a)}$ defined on $\\mathbb T_{d}(\\mathbf a )$, given by $$ \\Theta_{( N \\, , \\, S\\, , \\, \\mathbf a)}(\\mathcal G) =N(S-S_{\\mathcal G }) \\quad \\text{where} \\quad S_{\\mathcal G}=\\sum_{i\\in\\mathbb{I}_k} g_i\\,g_i^*\\in\\mathcal{M}_d(\\mathbb{C})^+\\,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $\\mathcal G_0\\in\\mathbb T_{d}(\\mathbf a )$ of $\\Theta_{( N \\, , \\, S\\, , \\, \\mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-26T16:37:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "15A60"
    ],
    "keywords": [
      "generalized frame operator distance problems",
      "convex unitarily invariant norm",
      "local minimizers",
      "generalized frame operator distance function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}