Abstract

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.