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International Journal of Mathematics and Mathematical Sciences
Volume 2007, Article ID 39819, 11 pages
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.


Let &#x1D53B;={z:|z|<1} be the open unit disk in the complex plane . Let A2(&#x1D53B;) be the space of analytic functions on &#x1D53B; square integrable with respect to the measure dA(z)=(1/π)dx dy. Given a&#x1D53B; and f any measurable function on &#x1D53B;, we define the function Caf by Caf(z)=f(ϕa(z)), where ϕaAut(&#x1D53B;). The map Ca is a composition operator on L2(&#x1D53B;,dA) and A2(&#x1D53B;) for all a&#x1D53B;. Let (A2(&#x1D53B;)) be the space of all bounded linear operators from A2(&#x1D53B;) into itself. In this article, we have shown that CaSCa=S for all a&#x1D53B; if and only if &#x1D53B;S˜(ϕa(z))dA(a)=S˜(z), where S(A2(&#x1D53B;)) and S˜ is the Berezin symbol of S.