The best constants for operator Lipschitz functions on Schatten classes

TL;DR

This paper provides sharp estimates for the constants in operator Lipschitz inequalities on Schatten classes, showing that the best constant C_p behaves asymptotically like p^2/(p-1), and extends results to multiple operators.

Contribution

It introduces the precise asymptotic behavior of the Lipschitz constant C_p for Schatten class commutators, generalizing to multiple self-adjoint operators.

Findings

C_p asymptotically behaves like p^2/(p-1)

Established sharp bounds for operator Lipschitz inequalities

Extended results to commutators of multiple operators

Abstract

Suppose that f is a Lipschitz function on the real numbers with Lipschitz constant smaller or equal to 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let 1<p<infinity and suppose that x in 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 contained in S_p and there exists a constant C_p independent of x and f such that || [f(A), x] ||_p <= C_p || [A,x] ||_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 ~ 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, for which we prove the analogous result. The case described here in the abstract follows as a special case.