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

We study the compactness and the hypercyclicity of Toeplitz operators T-(phi) over bar ,T-b with co-analytic and bounded symbols on de Branges-Rovnyak spaces H(b). For the compactness of T-(phi) over bar ,T-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-(phi) over bar ,T-b, with phi is an element of H-infinity boolean AND C(T), if and only if m(o(b) boolean AND T) = 0. We will also show that, when b is non-extreme, then T-(phi) over bar ,T-b is hypercyclic if and only if phi is non-constant and phi (D) boolean AND T not equal theta.