The best constants for operator Lipschitz functions on Schatten classes

Abstract

Suppose that f is a Lipschitz function on R with ‖f‖Lip≤1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let p∈(1,∞) and suppose that x∈B(H) is an operator such that the commutator [A,x] is contained in the Schatten class Sp. It is proved by the last two authors, that then also [f(A),x]∈Sp and there exists a constant Cp independent of x and f such that‖[f(A),x]‖p≤Cp‖[A,x]‖p. The main result of this paper is to give a sharp estimate for Cp in terms of p. Namely, we show that Cp∼p2p−1. In particular, this gives the best estimates for operator Lipschitz inequalities.We treat this result in a more general setting. This involves commutators of n self-adjoint operators A1,…,An, for which we prove the analogous result. The case described here in the abstract follows as a special case.