{
  "id": "1805.09680",
  "version": "v1",
  "published": "2018-05-23T08:08:29.000Z",
  "updated": "2018-05-23T08:08:29.000Z",
  "title": "Inequalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators",
  "authors": [
    "A. Peperko"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1612.01765, arXiv:1612.01767",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Psi$ and $\\Sigma$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove several refinements of the known inequalities $$\\rho \\left(\\Psi ^{\\left( \\frac{1}{2} \\right)} \\circ \\Sigma ^{\\left( \\frac{1}{2} \\right)} \\right) \\le \\rho (\\Psi \\Sigma) ^{\\frac{1}{2}} \\;\\; \\mathrm{and}\\;\\; \\hat{\\rho} \\left(\\Psi ^{\\left( \\frac{1}{2} \\right)} \\circ \\Sigma ^{\\left( \\frac{1}{2} \\right)} \\right) \\le \\hat{\\rho} (\\Psi \\Sigma) ^{\\frac{1}{2}} $$ for the generalized spectral radius $\\rho$ and the joint spectral radius $\\hat{\\rho}$, where $\\Psi ^{\\left( \\frac{1}{2} \\right)} \\circ \\Sigma ^{\\left( \\frac{1}{2} \\right)} $ denotes the Hadamard (Schur) geometric mean of the sets $\\Psi $ and $\\Sigma$. Furthermore, we prove that analogous inequalities hold also for the generalized essential spectral radius and the joint essential spectral radius in the case when $L$ and its Banach dual $L^*$ have order continuous norms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-23T08:08:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive kernel operators",
      "hadamard geometric mean",
      "bounded sets",
      "generalized spectral",
      "inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}