Adjoints of Composition Operators on Standard Bergman and Dirichlet Spaces on the Unit Disk

It is not known a satisfactory way to compute adjoints of composition operators, yet in classical functional Banach spaces (cf. [3]). If K is the reproducing kernel of a functional Hilbert space H, g ∈ H and the composition operator Cφ is bounded then C∗ φg (z) = 〈g(t),K (z, φ (t))〉H , z ∈ D. In general, although reproducing kernels might be described in series developments, it is not possible to determine C∗ φ in a closed form. In this article we establish some formulae in order to evaluate adjoints of composition operators on standard Bergman and Dirichlet spaces.