{
  "id": "1501.03267",
  "version": "v1",
  "published": "2015-01-14T07:52:53.000Z",
  "updated": "2015-01-14T07:52:53.000Z",
  "title": "Operator Lipschitz functions on Banach spaces",
  "authors": [
    "Jan Rozendaal",
    "Fedor Sukochev",
    "Anna Tomskova"
  ],
  "comment": "30 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "Let $X$, $Y$ be Banach spaces and let $\\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form \\begin{align*} \\|f(B)S-Sf(A)\\|_{\\mathcal{L}(X,Y)}\\leq \\textrm{const} \\|BS-SA\\|_{\\mathcal{L}(X,Y)} \\end{align*} for a large class of functions $f$, where $A\\in\\mathcal{L}(X)$, $B\\in \\mathcal{L}(Y)$ are scalar type operators and $S\\in \\mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\\ell_{p}$ and $Y=\\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\\mathrm{c}_{0}$. We also obtain results for $p\\geq q$. We study the estimate above in the setting of Banach ideals in $\\mathcal{L}(X,Y)$. The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-14T07:52:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A55",
      "47A56",
      "47B47"
    ],
    "keywords": [
      "operator lipschitz functions",
      "banach spaces",
      "commutator estimates",
      "scalar type operators",
      "bounded linear operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150103267R"
    }
  }
}