Some remarks concerning operator Lipschitz functions

We consider examples of operator Lipschitz functions $f$ for which the operator Lipschitz seminorm $\|f\|_{\mathrm{OL}(\mathbb{R})}$ coincides with the Lipschitz seminorm $\|f\|_{\mathrm{Lip}(\mathbb{R})}$. In particular, we consider the operator Lipschitz functions such that $f'(0)=\|f\|_{\mathrm{OL}(\mathbb{R})}$. It is well known that every function $f$ whose the derivative $f'$ is positive definite has this property. In the paper it is proved that there are other functions having this property. It is also shown that the identity $|f'(t_0)|=\|f\|_{\mathrm{OL}(\mathbb{R})}$ implies that the derivative of $f$ is continuous at $t_0$. In fact, a more general statement is established concerning commutator Lipschitz functions on a closed subset of the complex plane.