tailieunhanh - Báo cáo hóa học: "Research Article Composition Operator on Bergman-Orlicz Space"

Research Article Composition Operator on Bergman-Orlicz Space | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 832686 14 pages doi 2009 832686 Research Article Composition Operator on Bergman-Orlicz Space Zhijie Jiang1 2 and Guangfu Cao1 1 College of Mathematics and Information Science Guangzhou University Guangzhou Guangdong 510006 China 2 Department of Mathematics Sichuan University of Science and Engineering Zigong Sichuan 643000 China Correspondence should be addressed to Zhijie Jiang matjzj@ Received 19 May 2009 Accepted 22 October 2009 Recommended by Shusen Ding Let D denote the open unit disk in the complex plane and let dA z denote the normalized area measure on D. For a -1 and a twice differentiable nonconstant nondecreasing nonnegative and convex function on 0 to the Bergman-Orlicz space L is defined as follows L f e H D JD log f z 1 - z 2 adA z to . Let y be an analytic self-map of D. The composition operator Cy induced by y is defined by Cyf f o y for f analytic in D. We prove that the composition operator Cy is compact on L if and only if Cy is compact on Aa and Cy has closed range on L if and only if Cy has closed range on A . Copyright 2009 Z. Jiang and G. 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. 1. Introduction Let D be the open unit disk in the complex plane and let y be an analytic self-map of D. The composition operator Cy induced by y is defined by Cyf f o y for f analytic in D. The idea of studying the general properties of composition operators originated from Nordgren 1 . As a sequence of Littlewood s subordinate theorem each y induces a bounded composition operator on the Hardy spaces Hp D for all p 0 p to and the weighted Bergman spaces A for all p 0 p to and for all a -1 a to . Thus boundedness of composition operators on these spaces becomes very clear. Nextly a .