Essential norms of composition operators and Aleksandrov measures

Abstract

culated in terms of the Aleksandrov measures of the inducing holomorphic map. The argument provides a purely functiontheoretic proof of the equivalence of Sarason’s compactness condition for composition operators on L 1 and Shapiro’s compactness condition for composition operators on Hardy spaces. An application is given relating the essential norm to angular derivatives. x1. If is a holomorphic map of the unit disk D into itself, it is a consequence of Littlewood’s subordination principle [5] that composition with induces a bounded operator C on each Hardy space H p . A recurring theme in the study of composition operators has been the search for function theoretic conditions on which guarantee the compactness of C on H p .I t was shown by Shapiro and Taylor [11] that if C is compact on H p for some 0 <p< 1, then C is compact on H p for all 0 <p< 1, and so it is enough to study compactness on H 2 . In this context Shapiro [9] gave an expression for the essential norm of C on H 2 in terms of the Nevanlinna counting function of , thus providing a complete function theoretic characterization of compact composition operators on H 2 .