Functions of compact operators under trace class perturbations

We consider the problem to describe the continuous functions $f$ on the real line $\mathbb R$, for which the difference $f(B)-f(A)$ must be of trace class whenever $A$ and $B$ are compact self-adjoint operators with trace class difference. The main result shows that this happens if and only if the function f is operator Lipshchitz on a certain neighbourhood of zero.