{
  "id": "2406.13333",
  "version": "v1",
  "published": "2024-06-19T08:34:10.000Z",
  "updated": "2024-06-19T08:34:10.000Z",
  "title": "Functions of unitaries with $\\mathcal{S}^p$-perturbations for non continuously differentiable functions",
  "authors": [
    "Clément Coine"
  ],
  "comment": "31 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Consider a function $f : \\mathbb{T} \\to \\mathbb{C}$, $n$-times differentiable on $\\mathbb{T}$ and such that its $n$th derivative $f^{(n)}$ is bounded but not necessarily continuous. Let $U : \\mathbb{R} \\to \\mathcal{U}(\\mathcal{H})$ be a function taking values in the set of unitary operators on some separable Hilbert space $\\mathcal{H}$. Let $1<p<\\infty$ and let $\\mathcal{S}^p(\\mathcal{H})$ be the Schatten class of order $p$ on $\\mathcal{H}$. If $\\tilde{U}:t\\in\\mathbb{R} \\mapsto U(t)-U(0)$ is $n$-times $\\mathcal{S}^p$-differentiable on $\\mathbb{R}$, we show that the operator valued function $\\varphi : t\\in \\mathbb{R} \\mapsto f(U(t)) - f(U(0)) \\in \\mathcal{S}^p(\\mathcal{H})$ is $n$-times differentiable on $\\mathbb{R}$ as well. This theorem is optimal and extends several results related to the differentiability of functions of unitaries. The derivatives of $\\varphi$ are given in terms of multiple operator integrals and a formula and $\\mathcal{S}^p$-estimates for the Taylor remainders of $\\varphi$ are provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-19T08:34:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B49",
      "47B10",
      "46L52",
      "47A55"
    ],
    "keywords": [
      "non continuously differentiable functions",
      "perturbations",
      "multiple operator integrals",
      "schatten class",
      "times differentiable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}