{
  "id": "1709.02297",
  "version": "v1",
  "published": "2017-09-07T15:09:26.000Z",
  "updated": "2017-09-07T15:09:26.000Z",
  "title": "Direct sums of finite dimensional $SL^\\infty_n$ spaces",
  "authors": [
    "Richard Lechner"
  ],
  "comment": "29 pages, 2 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "$SL^\\infty$ denotes the space of functions whose square function is in $L^\\infty$, and the subspaces $SL^\\infty_n$, $n\\in\\mathbb{N}$, are the finite dimensional building blocks of $SL^\\infty$. We show that the identity operator $I_{SL^\\infty_n}$ on $SL^\\infty_n$ well factors through operators $T : SL^\\infty_N\\to SL^\\infty_N$ having large diagonal with respect to the standard Haar system. Moreover, we prove that $I_{SL^\\infty_n}$ well factors either through any given operator $T : SL^\\infty_N\\to SL^\\infty_N$, or through $I_{SL^\\infty_N}-T$. Let $X^{(r)}$ denote the direct sum $\\bigl(\\sum_{n\\in\\mathbb{N}_0} SL^\\infty_n\\bigr)_r$, where $1\\leq r \\leq \\infty$. Using Bourgain's localization method, we obtain from the finite dimensional factorization result that for each $1\\leq r\\leq \\infty$, the identity operator $I_{X^{(r)}}$ on $X^{(r)}$ factors either through any given operator $T : X^{(r)}\\to X^{(r)}$, or through $I_{X^{(r)}} - T$. Consequently, the spaces $\\bigl(\\sum_{n\\in\\mathbb{N}_0} SL^\\infty_n\\bigr)_r$, $1\\leq r\\leq \\infty$, are all primary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-07T15:09:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B25",
      "46B26",
      "60G46",
      "46B07"
    ],
    "keywords": [
      "direct sum",
      "identity operator",
      "finite dimensional factorization result",
      "standard haar system",
      "finite dimensional building blocks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}