Compactness: Introduction

Abstract

Having established that every composition operator is bounded on H ², we turn to the most natural follow-up question:

Which composition operators are compact?

The property of “boundedness” for composition operators means that each one takes bounded subsets of H ² to bounded subsets. The question above asks us to specify precisely how much the inducing map ϕ has to compress the unit disc into itself in order to insure that the operator Cφ, compresses bounded subsets of H ² into relatively compact ones.