Composition Operator on Bergman-Orlicz Space

Abstract

Let denote the open unit disk in the complex plane and let denote the normalized area measure on . For and a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on , the Bergman-Orlicz space is defined as follows Let be an analytic self-map of . The composition operator induced by is defined by for analytic in . We prove that the composition operator is compact on if and only if is compact on , and has closed range on if and only if has closed range on .