Composition operators on QK spaces

Suppose that ϕ is a nonconstant analytic self-map of the unit disk D. Compactness of composition operator Cϕ between two Möbius spaces QK1 and QK2 is studied. Moreover, necessary and sufficient condition for Cϕ from the Dirichlet space D to a general class of analytic functions F(p,q,s) to be compact is given in terms of the map ϕ.