Cyclicity in de Branges–Rovnyak spaces

In this paper, we study the cyclicity problem with respect to the forward shift operator S_b acting on the de Branges–Rovnyak space ℋ (b) associated to a function b in the closed unit ball of H^∞ and satisfying log(1− |b| ∈ L¹(𝕋). We present a characterisation of cyclic vectors for S_b when b is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of S_b in this case, but we provide here an elementary proof. We also study the situation where b has the form b = (1+ I)/2, where I is a non-constant inner function such that the associated model space K_I = ℋ (I) has an orthonormal basis of reproducing kernels.