Functions of perturbed n-tuples of commuting self-adjoint operators

Let (A1,…,An) and (B1,…,Bn) be n-tuples of commuting self-adjoint operators on Hilbert space. For functions f on Rn satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of f(A1,…,An)−f(B1,…,Bn) in terms of the corresponding norms of Aj−Bj, 1⩽j⩽n. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for n⩾3, the methods that were used for self-adjoint and normal operators do not work. We propose a new method that works for arbitrary n. We also get sharp estimates for quasicommutators f(A1,…,An)R−Rf(B1,…,Bn) in terms of norms of AjR−RBj, 1⩽j⩽n, for a bounded linear operator R.