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 $\mathcal{H}(b)$ associated to a function $b$ in the closed unit ball of $H^{\mathrm{\infty }}$ and satisfying $\log (1-\mathrm{\mid }b\mathrm{\mid })\in L^1(\mathbb{T})$. 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 [S. Luo, C. Gu, S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, \emph{Adv. Math., $\mathbf{\text{385}}$ (2021), paper No. 107748, 47], 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 $b=(1+I)\mathrm{/}2$, where $I$ is a non-constant inner function such that the associated model space $K_I=\mathcal{H}(I)$ has an orthonormal basis of reproducing kernels.