{
  "id": "0810.3425",
  "version": "v1",
  "published": "2008-10-19T21:24:51.000Z",
  "updated": "2008-10-19T21:24:51.000Z",
  "title": "On the sum of superoptimal singular values",
  "authors": [
    "Alberto A. Condori"
  ],
  "comment": "24 pages",
  "journal": "Journal of Functional Analysis. Volume 257, Issue 3, 1 August 2009. Pages 659-682",
  "categories": [
    "math.FA"
  ],
  "abstract": "We discuss the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an $m\\times n$ matrix function $\\Phi$ on the unit circle $\\mathbb{T}$, when is there a matrix function $\\Psi_{*}$ in the set $A_{k}^{n,m}$ such that \\int_{\\mathbb{T}}{\\rm trace}(\\Phi(\\zeta)\\Psi_{*}(\\zeta))dm(\\zeta)=\\sup_{\\Psi\\in A_{k}^{n,m}}|\\int_{\\mathbb{T}}{\\rm trace}(\\Phi(\\zeta)\\Psi(\\zeta))dm(\\zeta)|? The set $A_{k}^{n,m}$ is defined by A_{k}^{n,m}={\\Psi\\in H_{0}^{1}: \\|\\Psi\\|_{L^{1}}\\leq 1, {\\rm rank}\\Psi(\\zeta)\\leq k{a.e.}\\zeta\\in T}. We introduce Hankel-type operators on spaces of matrix functions and prove that this problem has a solution if and only if the corresponding Hankel-type operator has a maximizing vector. We also characterize the smallest number $k$ for which \\int_{\\mathbb{T}}{\\rm trace}(\\Phi(\\zeta)\\Psi(\\zeta))dm(\\zeta) equals the sum of all the superoptimal singular values of an admissible matrix function $\\Phi$ for some $\\Psi\\in A_{k}^{n,m}$. Moreover, we provide a representation of any such function $\\Psi$ when $\\Phi$ is an admissible very badly approximable unitary-valued $n\\times n$ matrix function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-19T21:24:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "46E40",
      "47L20"
    ],
    "keywords": [
      "superoptimal singular values",
      "admissible matrix function",
      "extremal problem",
      "unit circle",
      "smallest number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.3425C"
    }
  }
}