Functions of operators under perturbations of class Sp

This is a continuation of our paper 2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).