Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions

In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach space of analytic functions on the open unit disk into weighted Banach spaces, determine the operator norm exactly and obtain estimates on the essential norm. We obtain an exact formula of the essential norm when such operators map the Hardy spaces, the weighted Bergman spaces, the Bloch space and the little Bloch space into the Banach space Hμ∞ (where the weight μ is a continuous function). In this general setting we also study such operators when the target space is an α-Bloch space for any α>0 and more generally, any Bloch-type space. As special cases, we derive norm and essential norm approximations for the operators acting on the Hardy spaces and the weighted Bergman spaces. Immediate corollaries are results on certain integral operators.