On cyclicity in de Branges-Rovnyak spaces

We study the problem of characterizing the cyclic vectors in de Branges-Rovnyak spaces. Based on a description of the invariant subspaces we show that the difficulty lies entirely in understanding the subspace $(a{H}^2{)}^{\perp }$ and give a complete function theoretic description of the cyclic vectors in the case $\dim (a{H}^2{)}^{\perp }<\mathrm{\infty }$. Incidentally, this implies analogous results for certain generalized Dirichlet spaces $\mathcal{D}(\mu )$. Most of our attention is directed to the infinite case where we relate the cyclicity problem to describing the exposed points of $H^1$ and provide several sufficient conditions. A necessary condition based on the Aleksandrov-Clark measures of $b$ is also presented.