Abstract
In this work, we characterize the bounded and the compact weighted composition operators from a large class of Banach spaces X of analytic functions on the open unit disk D into the weighted-type Banach spaces V-n,V-mu, for n >= 3, where given a positive continuous function mu on D, the sequence {V-n,V-mu}(n >= 0) is defined iteratively by f is an element of V-0,V-mu if and only if
||f||(V0,mu) := sup(|z|<1) mu(z)|f(z)| < infinity,
and for n >= 1, f is an element of V-n,V-mu if and only if f ' is an element of V-n-1,V-mu, with norm ||f||(Vn,mu) := |f(0)| + ||f '||(Vn-1,mu), thereby extending known results for the cases n = 0, 1, 2. Under more restrictive conditions, we provide an approximation of the essential norm. We apply our results to the cases when X is a Hardy space and a weighted Bergman space.