Sampling sets and closed range composition operators on the Bloch space

Abstract

We give a necessary and sufficient condition for a composition operator C ϕ C_{\phi } on the Bloch space to have closed range. We show that when ϕ \phi is univalent, it is sufficient to consider the action of C ϕ C_{\phi } on the set of Möbius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.