{
  "id": "2209.05875",
  "version": "v1",
  "published": "2022-09-13T10:51:49.000Z",
  "updated": "2022-09-13T10:51:49.000Z",
  "title": "A geometric approach to inequalities for the Hilbert--Schmidt norm",
  "authors": [
    "Ali Zamani"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We define angle $\\Theta_{X,Y}$ between non-zero Hilbert--Schmidt operators $X$ and $Y$ by $\\cos\\Theta_{_{X,Y}} = \\frac{{\\rm Re}{\\rm Tr}(Y^*X)}{{\\|X\\|}_{_2}{\\|Y\\|}_{_2}}$, and give some of its essentially properties. It is shown, among other things, that \\begin{align*} \\big|\\cos\\Theta_{_{X,Y}}\\big|\\leq \\min\\left\\{\\sqrt{\\cos\\Theta_{_{|X^*|,|Y^*|}}}, \\sqrt{\\cos\\Theta_{_{|X|,|Y|}}}\\right\\}. \\end{align*} It enables us to provide alternative proof of some well-known inequalities for the Hilbert--Schmidt norm. In particular, we apply this inequality to prove Lee's conjecture [Linear Algebra Appl. 433 (2010), no.~3, 580--584] as follows \\begin{align*} {\\big\\|X + Y\\big\\|}_{_2} \\leq \\sqrt{\\frac{\\sqrt{2} + 1}{2}}\\,{\\big\\|\\,|X| + |Y|\\,\\big\\|}_{_2}. \\end{align*} A numerical example is presented to show the constant $\\sqrt{\\frac{\\sqrt{2} + 1}{2}}$ is smallest possible. Other related inequalities for the Hilbert--Schmidt norm are also considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-13T10:51:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A63",
      "47A30",
      "47B10"
    ],
    "keywords": [
      "hilbert-schmidt norm",
      "geometric approach",
      "inequality",
      "non-zero hilbert-schmidt operators",
      "linear algebra appl"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}