{
  "id": "2003.12377",
  "version": "v1",
  "published": "2020-03-27T12:42:09.000Z",
  "updated": "2020-03-27T12:42:09.000Z",
  "title": "Some log and weak majorization inequalities in Euclidean Jordan algebras",
  "authors": [
    "Jiyuan Tao",
    "Juyoung Jeong",
    "Muddappa Gowda"
  ],
  "comment": "18 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In the setting of Euclidean Jordan algebras, we prove majorization inequalities $\\lambda\\big (P_{\\sqrt{a}}(b)\\big )\\underset{log}{\\prec} \\lambda(a)*\\lambda(b)$ when $a,b\\geq 0$, $\\lambda\\big (|P_{a}(b)|\\big )\\underset{w}{\\prec} \\lambda(a^2)*\\lambda(|b|)$ and $\\lambda\\big (|a\\circ b|\\big )\\underset{w}{\\prec} \\lambda(|a|)*\\lambda(|b|)$ for all $a$ and $b$, where $P_u$ and $\\lambda(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$, and $*$ denotes the componentwise product. Extending the second inequality, we show that $\\lambda(|A\\bullet b|)\\underset{w}{\\prec} \\lambda({\\rm{diag}}(A))*\\lambda(|b|)$, where $A$ is a real symmetric positive semidefinite matrix and $A\\,\\bullet\\, b$ is the Schur product of $A$ and $b$. In the form of applications, we prove the generalized H\\\"{o}lder type inequality $||a\\circ b||_p\\leq ||a||_r\\,||b||_s$, where $p,q,r\\in [1,\\infty]$ with $\\frac{1}{p}=\\frac{1}{r}+\\frac{1}{s}$ and compute the norms of Lyapunov transformation $L_a$ and quadratic representation $P_a$ relative to spectral norms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-27T12:42:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A33",
      "17C20",
      "17C55"
    ],
    "keywords": [
      "euclidean jordan algebras",
      "weak majorization inequalities",
      "real symmetric positive semidefinite matrix",
      "quadratic representation",
      "second inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}