  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.
