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Let Ã°ÂÂ”Â»={zÃ¢ÂˆÂˆÃ¢Â„Â‚:|z|<1} be the open unit disk in the complex plane Ã¢Â„Â‚. Let A2(Ã°ÂÂ”Â») be the space of analytic functions on Ã°ÂÂ”Â» square integrable with respect to the measure dA(z)=(1/ÃÂ€)dxÃ‚Â dy. Given aÃ¢ÂˆÂˆÃ°ÂÂ”Â» and f any measurable function on Ã°ÂÂ”Â», we define the function Caf by Caf(z)=f(ÃÂ•a(z)), where ÃÂ•aÃ¢ÂˆÂˆAut(Ã°ÂÂ”Â»). The map Ca is a composition operator on L2(Ã°ÂÂ”Â»,dA) and A2(Ã°ÂÂ”Â») for all aÃ¢ÂˆÂˆÃ°ÂÂ”Â». Let Ã¢Â„Â’(A2(Ã°ÂÂ”Â»)) be the space of all bounded linear operators from A2(Ã°ÂÂ”Â») into itself. In this article, we have shown that CaSCa=S for all aÃ¢ÂˆÂˆÃ°ÂÂ”Â» if and only if Ã¢ÂˆÂ«Ã°ÂÂ”Â»SÃ‹Âœ(ÃÂ•a(z))dA(a)=SÃ‹Âœ(z), where SÃ¢ÂˆÂˆÃ¢Â„Â’(A2(Ã°ÂÂ”Â»)) and SÃ‹Âœ is the Berezin symbol of S

On a Class of Composition Operators on Bergman Space