Cyclic Vectors in de Branges-Rovnyak Spaces

In this talk we consider the problem of characterizing the cyclic vectors in de Branges-Rovnyak spaces, ℋ(𝑏). Some time will be spent introducing the relevant objects to those unfamiliar. We show that the difficulty lies entirely in understanding the subspace (𝑎𝐻²)^⊥ and give a complete function theoretic description in the case dim (𝑎𝐻²)^⊥ < ∞. This generalizes the case of rational b often discussed in the literature. Most of our attention will be given to the case dim (𝑎𝐻²)^⊥ = ∞, where the problem seems drastically more difficult. We will give sufficient results for cyclicity in the general case and discuss examples. Our methods are based on a decomposition of ℋ(𝑏) analogous to a decomposition often seen in the case of rational 𝑏.