{
  "id": "2310.10572",
  "version": "v1",
  "published": "2023-10-16T16:48:51.000Z",
  "updated": "2023-10-16T16:48:51.000Z",
  "title": "Factorization in Haar system Hardy spaces",
  "authors": [
    "Richard Lechner",
    "Thomas Speckhofer"
  ],
  "comment": "42 pages, 2 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "A Haar system Hardy space is the completion of the linear span of the Haar system $(h_I)_I$, either under a rearrangement-invariant norm $\\|\\cdot \\|$ or under the associated square function norm \\begin{equation*} \\Bigl\\| \\sum_Ia_Ih_I \\Bigr\\|_{*} = \\Bigl\\| \\Bigl( \\sum_I a_I^2 h_I^2 \\Bigr)^{1/2} \\Bigr\\|. \\end{equation*} Apart from $L^p$, $1\\le p<\\infty$, the class of these spaces includes all separable rearrangement-invariant function spaces on $[0,1]$ and also the dyadic Hardy space $H^1$. Using a unified and systematic approach, we prove that a Haar system Hardy space $Y$ with $Y\\ne C(\\Delta)$ ($C(\\Delta)$ denotes the continuous functions on the Cantor set) has the following properties, which are closely related to the primariness of $Y$: For every bounded linear operator $T$ on $Y$, the identity $I_Y$ factors either through $T$ or through $I_Y - T$, and if $T$ has large diagonal with respect to the Haar system, then the identity factors through $T$. In particular, we obtain that \\begin{equation*} \\mathcal{M}_Y = \\{ T\\in \\mathcal{B}(Y) : I_Y \\ne ATB\\text{ for all } A, B\\in \\mathcal{B}(Y) \\} \\end{equation*} is the unique maximal ideal of the algebra $\\mathcal{B}(Y)$ of bounded linear operators on $Y$. Moreover, we prove similar factorization results for the spaces $\\ell^p(Y)$, $1\\le p \\leq \\infty$, and use them to show that they are primary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-16T16:48:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B25",
      "46B09",
      "47A68",
      "47L20",
      "46E30",
      "30H10"
    ],
    "keywords": [
      "haar system hardy space",
      "bounded linear operator",
      "unique maximal ideal",
      "separable rearrangement-invariant function spaces",
      "dyadic hardy space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}