Estimates and higher-order spectral shift measures in several variables

Abstract

In recent years, higher-order trace formulas of operator functions have attracted considerable attention to a large part of the perturbation theory community. In this direction, we prove estimates for traces of higher-order derivatives of multivariate operator functions with associated scalar functions arising from multivariate analytic function space and, as a consequence, derive higher-order spectral shift measures for pairs of tuples of commuting contractions under Hilbert-Schmidt perturbations. These results substantially extend the main results of [26], where the estimates were proved for traces of first and second-order derivatives of multivariate operator functions. In the context of the existence of higher-order spectral shift measures, our results extend the relative results of [6,20] from a single-variable to a multivariate case under Hilbert-Schmidt perturbations. Our results rely crucially on heavy uses of explicit expressions of higher-order derivatives of operator functions and estimates of the divided difference of multivariate analytic functions, which are developed in this paper, along with the spectral theorem of tuple of commuting normal operators. In conclusion, we explore the significance of our results and provide relevant examples.