{
  "id": "1812.05435",
  "version": "v1",
  "published": "2018-12-13T14:09:30.000Z",
  "updated": "2018-12-13T14:09:30.000Z",
  "title": "An additive formula for multiplicities on reproducing kernel Hilbert spaces",
  "authors": [
    "Arup Chattopadhyay",
    "Jaydeb Sarkar",
    "Srijan Sarkar"
  ],
  "comment": "Preliminary Version, Comments Welcome",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we compute the exact rank of a non-trivial co-doubly commuting submodule of analytic reproducing kernel Hilbert modules over $\\mathbb{C}[z_1,\\ldots,z_n]$. More precisely, let $\\mathcal{H} = \\mathcal{H}_{1} \\otimes \\ldots \\otimes \\mathcal{H}_{n}$ be an analytic reproducing kernel Hilbert module over $\\mathbb{C}[z_1,\\ldots,z_n]$. Let $\\mathcal{S}$ be a co-doubly commuting submodule of $\\mathcal{H}$, that is , \\[ \\mathcal{S} = (\\mathcal{Q}_1 \\otimes \\ldots \\otimes \\mathcal{Q}_n)^{\\bot}, \\] where $\\mathcal{Q}_i$ are quotient modules of $\\mathcal{H}_{i}$. Then our result states that \\[ \\mbox{rank}~\\Big(\\big(\\mathcal{Q}_1\\otimes \\cdots \\otimes \\mathcal{Q}_n\\big)^{\\perp}\\Big) = \\sum_{i=1}^n \\mbox{rank}~\\big(\\mathcal{Q}_i^{\\bot}). \\] This immediately answers an open question in the special case where $\\mathcal{H}_{i} = H^2(\\mathbb{D})$ for all $i \\in \\lbrace 1,\\ldots,n \\rbrace$ posed in [3]. As an immediate consequence we have the following observation: Let $\\mathcal{S}$ be a co-doubly commuting submodule in $H^2(\\mathbb{D}^n)$. Then for $m \\leq n$, rank ($\\mathcal{S})=m$, implies $\\mathcal{S}=\\Theta H^2(\\mathbb{D}^n)$ for some $n-m$ variables inner function $\\Theta\\in H^{\\infty}(\\mathbb{D}^{n-m})$. In particular, for $n=2$ and $m=1$ it positively meets the observations made by R. G. Douglas and R. Yang in [7].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-13T14:09:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "47A15",
      "47A16",
      "47M05",
      "46C99",
      "32A70"
    ],
    "keywords": [
      "reproducing kernel hilbert spaces",
      "analytic reproducing kernel hilbert module",
      "additive formula",
      "co-doubly commuting submodule",
      "multiplicities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}