{
  "id": "1810.10445",
  "version": "v1",
  "published": "2018-10-24T15:17:32.000Z",
  "updated": "2018-10-24T15:17:32.000Z",
  "title": "Numerical radius parallelism of Hilbert space operators",
  "authors": [
    "Marzieh Mehrazin",
    "Maryam Amyari",
    "Ali Zamani"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we introduce a new type of parallelism for bounded linear operators on a Hilbert space $\\big(\\mathscr{H}, \\langle \\cdot ,\\cdot \\rangle\\big)$ based on numerical radius. More precisely, we consider operators $T$ and $S$ which satisfy $\\omega(T + \\lambda S) = \\omega(T)+\\omega(S)$ for some complex unit $\\lambda$. We show that $T \\parallel_{\\omega} S$ if and only if there exists a sequence of unit vectors $\\{x_n\\}$ in $\\mathscr{H}$ such that \\begin{align*} \\lim_{n\\rightarrow\\infty} \\big|\\langle Tx_n, x_n\\rangle\\langle Sx_n, x_n\\rangle\\big| = \\omega(T)\\omega(S). \\end{align*} We then apply it to give some applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-24T15:17:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "47L05",
      "47A12"
    ],
    "keywords": [
      "hilbert space operators",
      "numerical radius parallelism",
      "bounded linear operators",
      "complex unit",
      "unit vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}