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 $T_g(f)(z)=\int_0^z f(\zeta)g’(\zeta),d\zeta$actingfromaBanachspace$X$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 $X$.