International Journal of Mathematics and Mathematical Sciences 
Volume 2007 (2007), Article ID 39819, 11 pages
doi:10.1155/2007/39819
Research Article

On a Class of Composition Operators on Bergman Space

Namita Das,1 R. P. Lal,2 and C. K. Mohapatra2

 1P. G. Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar, Orissa 751004, India
2Institute of Mathematics and Applications, 2nd Floor, Surya Kiran Building, Sahid Nagar, Bhubaneswar, Orissa 751007, India
Received 4 May 2006; Revised 14 December 2006; Accepted 15 December 2006

Recommended by Manfred H. Moller

Abstract

Let 𝔻={z∈ℂ:|z|<1} be the open unit disk in the complex plane ℂ. Let A2(𝔻) be the space of analytic functions on 𝔻 square integrable with respect to the measure dA(z)=(1/π)dx dy. Given a∈𝔻 and f any measurable function on 𝔻, we define the function Caf by Caf(z)=f(ϕa(z)), where ϕa∈Aut(𝔻). The map Ca is a composition operator on L2(𝔻,dA) and A2(𝔻) for all a∈𝔻. Let ℒ(A2(𝔻)) be the space of all bounded linear operators from A2(𝔻) into itself. In this article, we have shown that CaSCa=S for all a∈𝔻 if and only if ∫𝔻S˜(ϕa(z))dA(a)=S˜(z), where S∈ℒ(A2(𝔻)) and S˜ is the Berezin symbol of S.