{
  "id": "math/0412163",
  "version": "v1",
  "published": "2004-12-08T14:52:30.000Z",
  "updated": "2004-12-08T14:52:30.000Z",
  "title": "Multivariable $ρ$-contractions",
  "authors": [
    "Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We suggest a new version of the notion of $\\rho$-dilation ($\\rho>0$) of an $N$-tuple $\\mathbf{A}=(A_1,...,A_N)$ of bounded linear operators on a common Hilbert space. We say that $\\mathbf{A}$ belongs to the class $C_{\\rho,N}$ if $\\mathbf{A}$ admits a $\\rho$-dilation $\\widetilde{\\mathbf{A}}=(\\widetilde{A}_1,...,\\widetilde{A}_N)$ for which $\\zeta\\widetilde{\\mathbf{A}}:=\\zeta_1\\widetilde{A}_1+... +\\zeta_N\\widetilde{A}_N$ is a unitary operator for each $\\zeta:=(\\zeta_1,...,\\zeta_N)$ in the unit torus $\\mathbb{T}^N$. For N=1 this class coincides with the class $C_\\rho$ of B. Sz.-Nagy and C. Foia\\c{s}. We generalize the known descriptions of $C_{\\rho,1}=C_\\rho$ to the case of $C_{\\rho,N}, N>1$, using so-called Agler kernels. Also, the notion of operator radii $w_\\rho, \\rho>0$, is generalized to the case of $N$-tuples of operators, and to the case of bounded (in a certain strong sense) holomorphic operator-valued functions in the open unit polydisk $\\mathbb{D}^N$, with preservation of all the most important their properties. Finally, we show that for each $\\rho>1$ and $N>1$ there exists an $\\mathbf{A}=(A_1,...,A_N)\\in C_{\\rho,N}$ which is not simultaneously similar to any $\\mathbf{T}=(T_1,...,T_N)\\in C_{1,N}$, however if $\\mathbf{A}\\in C_{\\rho,N}$ admits a uniform unitary $\\rho$-dilation then $\\mathbf{A}$ is simultaneously similar to some $\\mathbf{T}\\in C_{1,N}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-12-08T14:52:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "47A20",
      "47A56"
    ],
    "keywords": [
      "contractions",
      "simultaneously similar",
      "open unit polydisk",
      "common hilbert space",
      "multivariable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....12163K"
    }
  }
}