  We address the problem of studying the boundedness, compactness and weak
compactness of the integral operators $T_g(f)(z)=\int_0^z
f(\zeta)g'(\zeta)\,d\zeta$ acting from a Banach space $X$ into $H^\infty$. 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 $X$.
