Differences of Composition Operators on the Space of Bounded Analytic Functions in the Polydisc

Abstract

This paper gives some estimates of the essential norm for the difference of composition operators induced by $\phi$ and $\psi$ acting on the space, $H^{\infty }\left({\mathbb{D}}^n\right)$, of bounded analytic functions on the unit polydisc ${\mathbb{D}}^n$, where $\phi$ and $\psi$ are holomorphic self-maps of ${\mathbb{D}}^n$. As a consequence, one obtains conditions in terms of the Carathéodory distance on ${\mathbb{D}}^n$ that characterizes those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators on $H^{\infty }\left({\mathbb{D}}^n\right)$ is compact.