We discuss a recently obtained operator theoretic description for when a de Branges-Rovnyak spaces admits a dense subset of functions with differentiable boundary extensions. This description expresses the possibility of such approximations in terms of properties of invariant subspaces of a shift operator on a class of spaces strictly inbetween Hardy and the Bergman spaces. In this way, it connects the de Branges-Rovnyak spaces to the theory of subnormal operators, so called "splitting problems" and also problems of removal of singularities of Cauchy transforms. The talk is based on a joint work with Adem Limani from Lund University, Sweden.