{
  "id": "1812.00164",
  "version": "v1",
  "published": "2018-12-01T07:16:43.000Z",
  "updated": "2018-12-01T07:16:43.000Z",
  "title": "Parallelism in Hilbert $K(\\mathcal{H})$-modules",
  "authors": [
    "M. Mohammadi Gohari",
    "M. Amyari"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $(\\mathcal{H}, [\\cdot, \\cdot ])$ be a Hilbert space and $K(\\mathcal{H})$ be the $C^*$-algebra of compact operators on $\\mathcal{H}$. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert $K(\\mathcal{H})$-module $\\mathcal{E}$ by employing the minimal projections on $\\mathcal{H}$. Let $T,S\\in \\mathcal{L(\\mathcal{E})}$. We show that $T \\| S$ if and only if there exists a sequence of basic vectors $\\{x_n\\}^{\\xi_n}$ in $\\mathcal{E}$ such that $\\lim_n [\\langle Tx_n, Sx_n \\rangle \\xi_n, \\xi_n ] = \\lambda\\| T\\| \\| S\\|$ for some $\\lambda \\in \\mathbb{T}$. In addition, we give some equivalence assertions about the norm-parallelism of \"compact\" operators on a Hilbert $C^*$-module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-01T07:16:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L08",
      "47A30",
      "47B10",
      "47B47"
    ],
    "keywords": [
      "compact operators",
      "norm-parallelism",
      "hilbert space",
      "minimal projections",
      "basic vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}