Norm, essential norm and weak compactness of weighted composition operators between dual Banach spaces of analytic functions

In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of - non-necessarily reflexive - Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weighted composition operators from these weighted type spaces into a class of Banach spaces of analytic functions, that includes a large family of conformally invariant spaces like BMOA and analytic Besov spaces.