{
  "id": "1710.09853",
  "version": "v1",
  "published": "2017-10-26T18:24:56.000Z",
  "updated": "2017-10-26T18:24:56.000Z",
  "title": "Characterization of Invariant subspaces in the polydisc",
  "authors": [
    "Amit Maji",
    "Aneesh Mundayadan",
    "Jaydeb Sarkar",
    "Sankar T. R"
  ],
  "comment": "22 pages, preliminary version, comments are welcome",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "We give a complete characterization of invariant subspaces for $(M_{z_1}, \\ldots, M_{z_n})$ on the Hardy space $H^2(\\mathbb{D}^n)$ over the unit polydisc $\\mathbb{D}^n$ in $\\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \\ldots, M_{z_n})$ on $H^2(\\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over $\\mathbb{D}^n$, $n > 1$. Our invariant subspace theorem solves the well-known open problem on characterizations of invariant subspaces of the Hardy space over the unit polydisc.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-26T18:24:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A15",
      "47A13",
      "47A80",
      "30H10",
      "30H05",
      "32A10",
      "32A70",
      "46E22",
      "47B32"
    ],
    "keywords": [
      "unit polydisc",
      "invariant subspace theorem",
      "well-known open problem",
      "complete characterization",
      "results hold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}