{
  "id": "2303.13298",
  "version": "v1",
  "published": "2023-03-23T14:30:13.000Z",
  "updated": "2023-03-23T14:30:13.000Z",
  "title": "Krein and Koplienko trace formulas on normed ideals in several variables",
  "authors": [
    "Arup Chattopadhyay",
    "Saikat Giri",
    "Chandan Pradhan"
  ],
  "comment": "35 pages, Initial Draft",
  "categories": [
    "math.FA"
  ],
  "abstract": "The article is devoted to obtaining the first, and second-order Taylor remainder formulas corresponding to multivariable operator functions on normed ideals in $\\sigma$-finite, semi-finite von Neumann algebra factors. More precisely, we establish the Krein trace formulas (first-order) for a class of tuples of commuting self-adjoint operators and for a class of tuples of commuting unitary operators associated with a broad class of multivariable scalar functions, which includes both analytic and non-analytic functions. Moreover, we also establish the Koplienko trace formula (second-order) for certain pairs of tuples of commuting self-adjoint operators corresponding to multivariable rational scalar functions. Consequently, using dilation theory, we obtain the trace formulas for a class of tuples of commuting contractions and for a class of tuples of commuting maximal dissipative operators. In addition, we establish a linearization formula for a Dixmier trace applied to perturbed operator functions, which does not typically hold for normal traces. Our work continues the study of Skripka's work \\cite{Sk15} on multivariable trace formulas and extends the trace formulas obtained by Dykema and Skripka \\cite{DySk14} from single variable to multivariable operator functions. In particular, our work complements the work of Dykema and Skripka \\cite{DySk14} in the context of multivariable trace formulas.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-23T14:30:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A55"
    ],
    "keywords": [
      "koplienko trace formula",
      "normed ideals",
      "taylor remainder formulas corresponding",
      "multivariable trace formulas",
      "multivariable operator functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}