Composition Operator on Bergman-Orlicz Space

Abstract

Let 𝔻 denote the open unit disk in the complex plane and let dA(z) denote the normalized area measure on 𝔻. For α>−1 and Φ a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on [0,∞), the Bergman-Orlicz space LαΦ is defined as follows LαΦ={f∈H(𝔻):∫𝔻Φ(log+|f(z)|)(1−|z|2)αdA(z)<∞}. Let φ be an analytic self-map of 𝔻. The composition operator Cφ induced by φ is defined by Cφf=f∘φ for f analytic in 𝔻. We prove that the composition operator Cφ is compact on LαΦ if and only if Cφ is compact on Aα2, and Cφ has closed range on LαΦ if and only if Cφ has closed range on Aα2.