Some integral operators acting on $H^{\infty }$

Abstract

Let $f$ and $g$ be analytic on the unit disc $\mathbb{D}$. The integral operator $T_g$ is defined by $T_{g}f(z)={\int }_0^{z}f(t)g^{\prime }(t)dt$, $z\in \mathbb{D}$. 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, $S_{g}f(z)={\int }_0^z{f}^{\prime }(t)g(t)dt$, acting on $H^{\infty }$ are also studied.