Adjoint of some composition operators on the Dirichlet and Bergman spaces

Let phi be a holomorphic self-map of the unit disk U := {z is an element of C : vertical bar z vertical bar < 1} and the composition operator with symbol is defined by C(phi)f = f omicron phi. In this paper we present formula for the adjoint of composition operators in some Hilbert spaces of analytic functions, in the case that phi is a finite Blaschke product or a rational univalent holomorphic self-map of the unit disk U.