{
  "id": "2309.12839",
  "version": "v1",
  "published": "2023-09-22T12:53:28.000Z",
  "updated": "2023-09-22T12:53:28.000Z",
  "title": "Invariant subspaces of the direct sum of forward and backward shifts on vector-valued Hardy spaces",
  "authors": [
    "Caixing Gu",
    "Shuaibing Luo"
  ],
  "journal": "J. Funct. Anal. 282 (2022)",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $S_{E}$ be the shift operator on vector-valued Hardy space $H_{E}^{2}.$ Beurling-Lax-Halmos Theorem identifies the invariant subspaces of $S_{E}$ and hence also the invariant subspaces of the backward shift $S_{E}^{\\ast}.$ In this paper, we study the invariant subspaces of $S_{E}\\oplus S_{F}^{\\ast}.$ We establish a one-to-one correspondence between the invariant subspaces of $S_{E}\\oplus S_{F}^{\\ast}$ and a class of invariant subspaces of bilateral shift $B_{E}\\oplus B_{F}$ which were described by Helson and Lowdenslager. As applications, we express invariant subspaces of $S_{E}\\oplus S_{F}^{\\ast}$ as kernels or ranges of mixed Toeplitz operators and Hankel operators with partial isometry-valued symbols. Our approach greatly extends and gives different proofs of the results of C\\^{a}mara and Ross, and Timotin where the case with one dimensional $E$ and $F$ was considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-22T12:53:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector-valued hardy space",
      "backward shift",
      "direct sum",
      "beurling-lax-halmos theorem identifies",
      "express invariant subspaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}