Composition Operator on Bergman-Orlicz Space

Zhijie Jiang,Guangfu Cao

doi:10.1155/2009/832686

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 .

Highlights

Let D be the open unit disk in the complex plane and let φ be an analytic self-map of D
A natural problem is how to characterize the compactness of composition operators on these spaces, which once was a central problem for mathematicians who were interested in the theory of composition operators
The study of compact composition operators was started by Schwartz, who obtained the first compactness theorem in his thesis 2, showing that the integrability of 1 − |φ| −1 over ∂D implied the compactness of Cφ on Hp

Summary

Introduction

Let D be the open unit disk in the complex plane and let φ be an analytic self-map of D. Shapiro 5 developed relations between the essential norm of Cφ on H2 and the Nevanlinna counting function of φ, and he obtained a nice essential norm formula of Cφ in 1987 As a result, he completely gave a characterization of the compactness of Cφ in terms of the function properties of φ. The study of compactness of composition operators is an important subject on other analytic function spaces, and we have chosen two typical examples above, and for more related materials one can consult 7, 8. Another natural interesting subject is the composition operator with closed range. We are going to give affirmative answers for the proceeding questions

Preliminaries

Compactness

Closed Range