Estimates and Higher-Order Spectral Shift Measures in Several Variables

TL;DR

This paper extends trace estimates and spectral shift measures to multivariable operator functions under Hilbert-Schmidt perturbations, advancing the perturbation theory for tuples of commuting contractions.

Contribution

It introduces new estimates for higher-order derivatives of multivariable operator functions and establishes the existence of higher-order spectral shift measures in a multivariable context.

Findings

Extended trace estimates for higher-order derivatives.

Derived higher-order spectral shift measures for commuting contractions.

Generalized results from single-variable to multivariable setting.

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 multivariable operator functions with associated scalar functions arising from multivariable 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 \cite{Sk15}, where the estimates were proved for traces of first and second-order derivatives of multivariable operator functions. In the context of the existence of higher-order spectral shift measures, our results extend the relative results of \cite{DySk09, PoSkSu14} from a single-variable to a multivariable setting 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 deference of multivariable analytic functions, which are developed in this paper, along with the spectral theorem of tuples of commuting normal operators.