**Martijn Caspers, Stephen Montgomery-Smith, Denis Potapov and Fedor Sukochev, The best constants for operator Lipschitz functions on Schatten classes.**
*Journal of Functional analysis, 267 (2014), no. 10, 3557-3579.*
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 \(\mathcal H\). Let \(p \in (1,\infty)\) and suppose that \(x\) in \(B(\mathcal H)\) is an operator such that the commutator \([A,x]\) is contained in the Schatten class \(\mathcal S_p\). It is proved by the last two authors, that then also \([f(A),x]\) is contained in \(\mathcal S_p\) and there exists a constant \(C_p\) independent of \(x\) and \(f\) such that \({\|[f(A),x]\|}_p \le C_p {\|[A,x]\|}_p\). The main result of this paper is to give a sharp estimate for \(C_p\) in terms of \(p\). Namely, we show that \(C_p \sim p^2/(p-1)\). 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.
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