Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 832686, 14 pages
Research Article

Composition Operator on Bergman-Orlicz Space

1College of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, China
2Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China

Received 19 May 2009; Accepted 22 October 2009

Academic Editor: Shusen Ding

Copyright © 2009 Zhijie Jiang and Guangfu Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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={fH(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=fφ 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.