Integral operators mapping into the space of bounded analytic functions

We address the problem of studying the boundedness, compactness and weak compactness of the integral operators Tg(f)(z)=∫0zf(ζ)g′(ζ)dζ acting from a Banach space X   into H∞H∞. We obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of X.