Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators

Abstract

We study perturbations of functions f(A,B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B∞,11(R2), then we have the following Lipschitz-type estimate in the Schatten–von Neumann norm Sp, 1≤p≤2: ‖f(A1,B1)−f(A2,B2)‖Sp≤const(‖A1−A2‖Sp+‖B1−B2‖Sp). However, the condition f∈B∞,11(R2) does not imply the Lipschitz-type estimate in Sp with p>2. The main tool is Schatten–von Neumann norm estimates for triple operator integrals.