Compact composition operators on 𝐿¹

Abstract

The composition operator induced by a holomorphic self-map of the unit disc is compact on L 1 {L^1} of the unit circle if and only if it is compact on the Hardy space H 2 {H^2} of the disc. This answers a question posed by Donald Sarason: it proves that Sarason’s integral condition characterizing compactness on L 1 {L^1} is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on H 2 {H^2} .