{
  "id": "2312.08706",
  "version": "v1",
  "published": "2023-12-14T07:47:15.000Z",
  "updated": "2023-12-14T07:47:15.000Z",
  "title": "Lipschitz Estimates and an application to trace formulae",
  "authors": [
    "Tirthankar Bhattacharyya",
    "Arup Chattopadhyay",
    "Saikat Giri",
    "Chandan Pradhan"
  ],
  "comment": "15pp",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this short note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on $\\cir$ and for a pair of unitaries $(U,V)$ with $U-V\\in\\mathcal{S}_{2}(\\hilh)$ (Hilbert-Schmidt class). As a consequence, we obtain the Schatten $2$-Lipschitz estimate $\\|f(U)-f(V)\\|_2\\leq \\|f\\|_{\\lip(\\cir)}\\|U-V\\|_2$ for all Lipschitz functions $f:\\cir\\to\\C$. Moreover, we develop an approach to the operator Lipschitz estimate for a pair of contractions with the assumption that one of them is strict, which significantly extends the function class from results known earlier. More specifically, for each $p\\in(1,\\infty)$ and for every pair of contractions $(T_0,T_1)$ with $\\|T_0\\|<1$, there exists a constant $d_{f, p,T_0}>0$ such that $\\|f(T_1)-f(T_0)\\|_p\\leq d_{f,p, T_0}\\|T_1-T_0\\|_p$ for all Lipschitz functions on $\\cir$. Utilizing our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of contraction pairs featuring Hilbert-Schmidt perturbations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-14T07:47:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A20",
      "47A55",
      "47A56",
      "47B10",
      "42B30",
      "30H10"
    ],
    "keywords": [
      "lipschitz function",
      "krein trace formula applicable",
      "application",
      "contraction pairs featuring hilbert-schmidt perturbations",
      "modified krein trace formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}