Compact composition operators on Bergman-Orlicz spaces

Abstract

We construct an analytic self-map ' of the unit disk and an Orlicz functionfor which the composition operator of symbol ' is compact on the Hardy-Orlicz space H � , but not on the Bergman-Orlicz space B � . For that, we first prove a Carleson embedding theorem, and then characterize the compact- ness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2. Mathematics Subject Classification. Primary: 47B33 - Secondary: 30D50; 30D55; 46E15