{
  "id": "2010.01501",
  "version": "v1",
  "published": "2020-10-04T07:57:07.000Z",
  "updated": "2020-10-04T07:57:07.000Z",
  "title": "Uniqueness of unconditional basis of $H_p(\\mathbb{T})\\oplus\\ell_{2}$ and $H_p(\\mathbb{T})\\oplus\\mathcal{T}^{(2)}$ for $0<p<1$",
  "authors": [
    "Fernando Albiac",
    "Jose L. Ansorena",
    "Przemyslaw Wojtaszczyk"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Our goal in this paper is to advance the state of the art of the topic of uniqueness of unconditional basis. To that end we establish general conditions on a pair $(\\mathbb{X}, \\mathbb{Y})$ formed by a quasi-Banach space $\\mathbb{X}$ and a Banach space $\\mathbb{Y}$ which guarantee that every unconditional basis of their direct sum $\\mathbb{X}\\oplus\\mathbb{Y}$ splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces $H_p(\\mathbb{T}^d) \\oplus\\mathcal{T}^{(2)}$ and $H_p(\\mathbb{T}^d)\\oplus\\ell_2$, for $p\\in(0,1)$ and $d\\in\\mathbb{N}$, have a unique unconditional basis (up to equivalence and permutation).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-04T07:57:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B15",
      "46B20",
      "46B42",
      "46B45",
      "46A16",
      "46A35",
      "46A40",
      "46A45"
    ],
    "keywords": [
      "uniqueness",
      "unique unconditional basis",
      "establish general conditions",
      "direct sum",
      "quasi-banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}