{
  "id": "1712.10289",
  "version": "v1",
  "published": "2017-12-29T17:44:32.000Z",
  "updated": "2017-12-29T17:44:32.000Z",
  "title": "$\\mathcal{S}^2$-differentiability and extension of the Koplienko trace formula",
  "authors": [
    "Clément Coine",
    "Christian Le Merdy",
    "Anna Skripka",
    "Fedor Sukochev"
  ],
  "comment": "28 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $A$ be a selfadjoint operator in a separable Hilbert space, $K$ a selfadjoint Hilbert-Schmidt operator, and $f\\in C^n(\\mathbb{R})$. We establish that $\\varphi(t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $\\mathbb{R}$ in the Hilbert-Schmidt norm, provided either $A$ is bounded or the derivatives $f^{(i)}$, $i=1,\\ldots,n$, are bounded. As an application, we extend the Koplienko trace formula \\begin{equation*} \\mathrm{Tr}\\Big(f(A+K) - f(A) - \\dfrac{d}{dt}f(A+tK)_{|t=0}\\Big) = \\int_{\\mathbb{R}} f\"(t) \\eta(t) \\, dt \\end{equation*} from the Besov class $B_{\\infty1}^2(\\mathbb{R})$ to functions $f$ for which the divided difference $f^{[2]}$ admits a certain Hilbert space factorization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-29T17:44:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B49",
      "47A55",
      "46L52"
    ],
    "keywords": [
      "koplienko trace formula",
      "differentiability",
      "selfadjoint hilbert-schmidt operator",
      "hilbert space factorization",
      "separable hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}