Krein and Koplienko trace formulas on normed ideals in several variables

Arup Chattopadhyay, Saikat Giri, Chandan Pradhan

Published 2023-03-23Version 1

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.