Closed Range Composition Operators on the Bloch Space of Bounded Symmetric Domains

Abstract

Let ${\mathbb {B}}_X$ and ${\mathbb {B}}_Y$ be bounded symmetric domains realized as the unit balls of $\hbox {JB}^*$-triples X and Y, respectively. In this paper, we generalize the Landau theorem to holomorphic mappings on ${\mathbb {B}}_X$ using the Schwarz–Pick lemma for holomorphic mappings on ${\mathbb {B}}_X$. Next, we give a necessary condition for the composition operators $C_{\varphi }$ between the Bloch spaces on ${\mathbb {B}}_X$ and ${\mathbb {B}}_Y$ to be bounded below by using a sampling set for the Bloch space, where $\varphi$ is a holomorphic mapping from ${\mathbb {B}}_X$ to ${\mathbb {B}}_Y$. We also obtain other necessary conditions for the composition operators $C_{\varphi }$ between the Bloch spaces in the case ${\mathbb {B}}_Y$ is a complex Hilbert ball ${\mathbb {B}}_H$. We give a sufficient condition for the composition operators $C_{\varphi }$ between the Bloch spaces on ${\mathbb {B}}_X$ and ${\mathbb {B}}_Y$ to be bounded below using a sampling set for the Bloch space. In the case $\dim X=\dim Y<\infty$, we also give another sufficient condition for the composition operators $C_{\varphi }$ between the Bloch spaces on ${\mathbb {B}}_X$ and ${\mathbb {B}}_Y$ to be bounded below.