Complete Continuity of Composition-Differentiation Operators on the Hardy Space H1

Abstract

We study composition-differentiation operators on the Hardy space H1 on the unit disk. We prove that if φ is an analytic self-map of the unit disk such that the composition-differentiation operator induced by φ is bounded on the Hardy space H1, then it is completely continuous. This result is stronger than the similar result for composition operators which says that the composition operator induced by φ is completely continuous if and only if φeiθ<1 almost everywhere on the unit circle.