Some Remarks Concerning Operator Lipschitz Functions

We consider examples of operator Lipschitz functions f for which the operator Lipschitz seminorm ‖f‖OL(ℝ) coincides with the Lipschitz seminorm ‖f‖Lip(ℝ). In particular, we consider the operator Lipschitz functions f such that |f ' (0)| = ‖f‖OL(ℝ). It is well known that f has this property if its derivative f′ is positive definite. It is proved in this paper that there are other functions having this property. It is also shown that the identity |f ' (t0)| = ‖f‖OL(ℝ) implies the continuity of f at t0. In fact, a more general statement is established concerning commutator Lipschitz functions on a closed subset of the complex plane.