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
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
Academic Editor: Manfred H. Moller
Copyright © 2007 Namita Das et al. 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.
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.