Trace formulas in higher dimensions

Abstract: The paper establishes the Krein and Koplienko trace formulas for multivariable operator functions on symmetrically normed ideals of bounded operators. This partially answers the question of Aleksandrov, Peller and Potapov on the existence of spectral shift measures for tuples of commuting self-adjoint operators. Results are proved for self-adjoint and maximal dissipative operators. They cover both ideals with normal and singular traces. The admissible function classes considered in the trace formulas include both analytic and non-analytic scalar functions. Results are illustrated with examples.