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The article is devoted to obtaining the first, and second-order Taylorremainder formulas corresponding to multivariable operator functions on normedideals in $\sigma$-finite, semi-finite von Neumann algebra factors. Thisanswers an open question posed by Aleksandrov, Peller, and Potapov \cite[Openproblems]{AlPePo19} concerning the existence of Krein spectral-shift measuresfor pairs of commuting tuples of self-adjoint operators. More precisely, weestablish the Krein trace formulas (first-order) for a class of tuples ofcommuting self-adjoint operators and for a class of tuples of commuting unitaryoperators associated with a broad class of multivariable scalar functions,which includes both analytic and non-analytic functions. Moreover, we alsoestablish the Koplienko trace formula (second-order) for certain pairs oftuples of commuting self-adjoint operators corresponding to multivariablerational scalar functions. Consequently, using dilation theory, we obtain thetrace formulas for a class of tuples of commuting contractions and for a classof tuples of commuting maximal dissipative operators. In addition, we establisha linearization formula for a Dixmier trace applied to perturbed operatorfunctions, which does not typically hold for normal traces. Our work continuesthe study of Skripka's work \cite{Sk15} on multivariable trace formulas andextends the trace formulas obtained by Dykema and Skripka \cite{DySk14} fromsingle variable to multivariable operator functions. In particular, our workcomplements the work of Dykema and Skripka \cite{DySk14} in the context ofmultivariable trace formulas.

Krein and Koplienko trace formulas on normed ideals in several variables