Compact linear combination of composition operators on Bergman-Orlicz spaces

Abstract

Let Ψ be an Orlicz function, and let T denote any finite linear combination of composition operators. Motivated by the work of Choe, Koo, and Wang on the compactness of linear combinations of composition operators on weighted Bergman spaces, we completely characterize the compactness of T on the Bergman-Orlicz space AΨ(D). Unlike the classical weighted Bergman space setting, the precise analysis on AΨ(D) depends on the growth behavior of Ψ and the associated Luxemburg norm. Hence, the power-type estimates available in weighted Bergman spaces cannot be applied directly. To overcome this difficulty, under suitable growth assumptions on Ψ and using the Julia-Caratheodory type function theory and Carleson type measure technique, respectively, we establish two necessary and sufficient compactness characterizations for T on AΨ(D).