Compact composition operators on the Bloch space

Abstract

Necessary and sufficient conditions are given for a composition operator C ϕ f = f o ϕ {C_\phi }f = f{\text {o}}\phi to be compact on the Bloch space B \mathcal {B} and on the little Bloch space B 0 {\mathcal {B}_0} . Weakly compact composition operators on B 0 {\mathcal {B}_0} are shown to be compact. If ϕ ∈ B 0 \phi \in {\mathcal {B}_0} is a conformal mapping of the unit disk D \mathbb {D} into itself whose image ϕ ( D ) \phi (\mathbb {D}) approaches the unit circle T \mathbb {T} only in a finite number of nontangential cusps, then C ϕ {C_\phi } is compact on B 0 {\mathcal {B}_0} . On the other hand if there is a point of T ∩ ϕ ( D ) \mathbb {T} \cap \phi (\mathbb {D}) at which ϕ ( D ) \phi (\mathbb {D}) does not have a cusp, then C ϕ {C_\phi } is not compact.