Higher order spectral shift

TL;DR

This paper develops higher order spectral shift functions within operator perturbation theory, generalizing existing formulas and representations for remainders of Taylor approximations in semi-finite von Neumann algebras.

Contribution

It introduces recursive higher order spectral shift functions and extends spectral averaging formulas, broadening the theoretical framework for operator perturbations.

Findings

Constructed higher order spectral shift functions for operator perturbations.

Extended spectral averaging formulas to higher orders.

Represented remainders of Taylor-type approximations recursively.

Abstract

We construct higher order spectral shift functions, extending the perturbation theory results of M. G. Krein and L. S. Koplienko on representations for the remainders of the first and second order Taylor-type approximations of operator functions. The higher order spectral shift functions represent the remainders of higher order Taylor-type approximations; they can be expressed recursively via the lower order (in particular, Krein's and Koplienko's) ones. We also obtain higher order spectral averaging formulas generalizing the Birman-Solomyak spectral averaging formula. The results are obtained in the semi-finite von Neumann algebra setting, with the perturbation taken in the Hilbert-Schmidt class of the algebra.