Higher order spectral shift

Abstract

We construct higher order spectral shift functions, extending the perturbation theory results of M.G. Krein [M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597–626 (in Russian)] and L.S. Koplienko [L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62–71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735–743] 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.