{
  "id": "2007.04580",
  "version": "v1",
  "published": "2020-07-09T06:40:33.000Z",
  "updated": "2020-07-09T06:40:33.000Z",
  "title": "New properties of the multivariable $H^\\infty$ functional calculus of sectorial operators",
  "authors": [
    "Olivier Arrigoni",
    "Christian Le Merdy"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper is devoted to the multivariable $H^\\infty$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $(A_1,\\ldots, A_d)$ is such a family, if $A_k$ is $R$-sectorial of $R$-type $\\omega_k\\in(0,\\pi)$, $k=1,\\ldots,d$, and if $(A_1,\\ldots, A_d)$ admits a bounded $H^\\infty(\\Sigma_{\\theta_1}\\times \\cdots\\times\\Sigma_{\\theta_d})$ joint functional calculus for some $\\theta_k\\in (\\omega_k,\\pi)$, then it admits a bounded $H^\\infty(\\Sigma_{\\theta_1}\\times \\cdots\\times\\Sigma_{\\theta_d})$ joint functional calculus for all $\\theta_k\\in (\\omega_k,\\pi)$, $k=1,\\ldots,d$. Second we introduce square functions adapted to the multivariable case and extend to this setting some of the well-known one-variable results relating the boundedness of $H^\\infty$ functional calculus to square function estimates. Third, on $K$-convex reflexive spaces, we establish sharp dilation properties for $d$-tuples $(A_1,\\ldots, A_d)$ which admit a bounded $H^\\infty(\\Sigma_{\\theta_1}\\times \\cdots\\times\\Sigma_{\\theta_d})$ joint functional calculus for some $\\theta_k<\\frac{\\pi}{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-09T06:40:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A60",
      "47A20",
      "47D03"
    ],
    "keywords": [
      "sectorial operators",
      "joint functional calculus",
      "multivariable",
      "establish sharp dilation properties",
      "square function estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}