{
  "id": "1707.09200",
  "version": "v1",
  "published": "2017-07-28T11:54:25.000Z",
  "updated": "2017-07-28T11:54:25.000Z",
  "title": "Entropy numbers in $γ$-Banach spaces",
  "authors": [
    "Thanatkrit Kaewtem"
  ],
  "comment": "11 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $X$ be a quasi-Banach space, $Y$ a $\\gamma$-Banach space $(0<\\gamma \\leq 1)$ and $T$ a bounded linear operator from $X$ into $Y$. In this paper, we prove that the first outer entropy number of $T$ lies between $2^{1-1/\\gamma}\\|T\\|$ and $\\|T\\|$; more precisely, $2^{1-1/\\gamma}\\|T\\| \\leq e_1(T) \\leq \\|T\\|,$ and the constant $2^{1-1/\\gamma}$ is sharp. Moreover, we show that there exist a Banach space $X_0$, a $\\gamma$-Banach space $Y_0$ and a bounded linear operator $T_0:X_0 \\rightarrow Y_0$ such that $0 \\neq e_k(T_0) = 2^{1-1/\\gamma}\\|T_0\\| $ for all positive integers $k.$ Finally, the paper also provides two-sided estimates for entropy numbers of embeddings between finite dimensional symmetric $\\gamma$-Banach spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-28T11:54:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B06",
      "46B45"
    ],
    "keywords": [
      "bounded linear operator",
      "first outer entropy number",
      "finite dimensional symmetric",
      "quasi-banach space",
      "embeddings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}