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.