{
  "id": "1604.06364",
  "version": "v1",
  "published": "2016-04-21T15:53:09.000Z",
  "updated": "2016-04-21T15:53:09.000Z",
  "title": "On polynomial $n$-tuples of commuting isometries",
  "authors": [
    "Edward J. Timko"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We extend some of the results of Agler, Knese, and McCarthy [1] to $n$-tuples of commuting isometries for $n>2$. Let $\\mathbb{V}=(V_1,\\dots,V_n)$ be an $n$-tuple of a commuting isometries on a Hilbert space and let Ann$(\\mathbb{V})$ denote the set of all $n$-variable polynomials $p$ such that $p(\\mathbb{V})=0$. When Ann$(\\mathbb{V})$ defines an affine algebraic variety of dimension 1 and $\\mathbb{V}$ is completely non-unitary, we show that $\\mathbb{V}$ decomposes as a direct sum of $n$-tuples $\\mathbb{W}=(W_1,\\dots,W_n)$ with the property that, for each $i=1,\\dots,n$, $W_i$ is either a shift or a scalar multiple of the identity. If $\\mathbb{V}$ is a cyclic $n$-tuple of commuting shifts, then we show that $\\mathbb{V}$ is determined by Ann$(\\mathbb{V})$ up to near unitary equivalence, as defined in [1].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-21T15:53:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "commuting isometries",
      "affine algebraic variety",
      "hilbert space",
      "direct sum",
      "scalar multiple"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}