Composition Operator on Bergman-Orlicz Space

Let [inline-graphic not available: see fulltext] denote the open unit disk in the complex plane and let [inline-graphic not available: see fulltext] denote the normalized area measure on [inline-graphic not available: see fulltext]. For [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext] a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on [inline-graphic not available: see fulltext], the Bergman-Orlicz space [inline-graphic not available: see fulltext] is defined as follows [inline-graphic not available: see fulltext] Let [inline-graphic not available: see fulltext] be an analytic self-map of [inline-graphic not available: see fulltext]. The composition operator [inline-graphic not available: see fulltext] induced by [inline-graphic not available: see fulltext] is defined by [inline-graphic not available: see fulltext] for [inline-graphic not available: see fulltext] analytic in [inline-graphic not available: see fulltext]. We prove that the composition operator [inline-graphic not available: see fulltext] is compact on [inline-graphic not available: see fulltext] if and only if [inline-graphic not available: see fulltext] is compact on [inline-graphic not available: see fulltext], and [inline-graphic not available: see fulltext] has closed range on [inline-graphic not available: see fulltext] if and only if [inline-graphic not available: see fulltext] has closed range on [inline-graphic not available: see fulltext].