The best constants for operator Lipschitz functions on Schatten classes

Suppose that f is a Lipschitz function on R with parallel to f parallel to(Lip) <= 1. Let A be a bounded self-adjoint operator on a Hilbert space H Let p is an element of (1, infinity) and suppose that x is an element of B (H) is an operator such that the commutator [A, x] is contained in the Schatten class S-p. It is proved by the last two authors, that then also [f (A), x] is an element of S-p and there exists a constant C-p independent of x and f such that parallel to[f(A), x]parallel to(p) <= C-p parallel to[A, x]parallel to(p). The main result of this paper is to give a sharp estimate for C-p in terms of p. Namely, we show that C-p similar to p(2)/p-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 A(1),..., A(n), for which we prove the analogous result. The case described here in the abstract follows as a special case. (C) 2014 Elsevier Inc. All rights reserved.