Compactness and hypercyclicity of co-analytic Toeplitz operators on de Branges-Rovnyak spaces

We study the compactness and the hypercyclicity of Toeplitz operators T ϕ ¯ , b {T_{\bar \varphi ,b}} with co-analytic and bounded symbols on de Branges-Rovnyak spaces ℋ(b). For the compactness of T ϕ ¯ , b {T_{\bar \varphi ,b}} , we will see that the result depends on the boundary spectrum of b. We will prove that there are non trivial compact operators of the form T ϕ ¯ , b {T_{\bar \varphi ,b}} , with ϕ ∈ H ∞ ∩ C(𝕋), if and only if m(σ(b) ∩ 𝕋) = 0. We will also show that, when b is non-extreme, then T ϕ ¯ , b {T_{\bar \varphi ,b}} is hypercyclic if and only if ϕ is non-constant and ϕ(𝔻) ∩ 𝕋 ≠ ∅.