Compact composition operators on the Bloch space

Abstract:

Necessary and sufficient conditions are given for a composition operator ${C_\phi }f = f{\text {o}}\phi$ to be compact on the Bloch space $\mathcal {B}$ and on the little Bloch space ${\mathcal {B}_0}$. Weakly compact composition operators on ${\mathcal {B}_0}$ are shown to be compact. If $\phi \in {\mathcal {B}_0}$ is a conformal mapping of the unit disk $\mathbb {D}$ into itself whose image $\phi (\mathbb {D})$ approaches the unit circle $\mathbb {T}$ only in a finite number of nontangential cusps, then ${C_\phi }$ is compact on ${\mathcal {B}_0}$. On the other hand if there is a point of $\mathbb {T} \cap \phi (\mathbb {D})$ at which $\phi (\mathbb {D})$ does not have a cusp, then ${C_\phi }$ is not compact.