{
  "id": "1511.07632",
  "version": "v1",
  "published": "2015-11-24T10:24:14.000Z",
  "updated": "2015-11-24T10:24:14.000Z",
  "title": "Direct sums and summability of the Szlenk index",
  "authors": [
    "Szymon Draga",
    "Tomasz Kochanek"
  ],
  "comment": "26 pp",
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove that the $c_0$-sum of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums. We also give a formula for the Szlenk power type of the $\\mathfrak{E}$-direct sum of separable spaces provided that $\\mathfrak{E}$ has a shrinking unconditional basis whose dual basis yields an asymptotic $\\ell_p$ structure in $\\mathfrak{E}^\\ast$. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of $c_0$ has power type $1$ but non-summable Szlenk index.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-24T10:24:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B03",
      "46B20"
    ],
    "keywords": [
      "summability",
      "tsirelson direct sum",
      "dual basis yields",
      "general direct sums",
      "szlenk power type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151107632D"
    }
  }
}