The Best Constants for Operator Lipschitz Functions on Schatten Classes

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 ∼ 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 A1, . . . , An, for which we prove the analogous result. The case described here in the abstract follows as a special case.