Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415076 | Journal of Functional Analysis | 2014 | 23 Pages |
Suppose that f is a Lipschitz function on R with âfâLipâ¤1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let pâ(1,â) and suppose that xâB(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]âSp and there exists a constant Cp independent of x and f such thatâ[f(A),x]âpâ¤Cpâ[A,x]âp. The main result of this paper is to give a sharp estimate for Cp in terms of p. Namely, we show that Cpâ¼p2pâ1. 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,â¦,An, for which we prove the analogous result. The case described here in the abstract follows as a special case.