Some integral operators acting on \(H^{\infty}\)

Let \(f\) and \(g\) be analytic on the unit disc \(\mathbb{D}\). The integral operator \(T_g\) is defined by \( T_g f(z) = \int_0^z f(t)g'(t)\,dt\), \(z \in \mathbb{D}\). The problem considered is characterizing those symbols \(g\) for which \(T_g\) acting on \(H^\infty\), the space of bounded analytic functions on \(\mathbb{D}\), is bounded or compact. When the symbol is univalent, these become questions in univalent function theory. The corresponding problems for the companion operator, \( S_g f(z)= \int_0^z f'(t)g(t)\, dt\), acting on \(H^\infty\) are also studied.