Abstract
Let&#x1D53B;={z∈ℂ:|z|<1}be the open unit disk in the complex planeℂ. LetA2(&#x1D53B;)be the space of analytic functions on&#x1D53B;square integrable with respect to the measuredA(z)=(1/π)dx dy. Givena∈&#x1D53B;andfany measurable function on&#x1D53B;, we define the functionCafbyCaf(z)=f(ϕa(z)), whereϕa∈Aut(&#x1D53B;). The mapCais a composition operator onL2(&#x1D53B;,dA)andA2(&#x1D53B;)for alla∈&#x1D53B;. Letℒ(A2(&#x1D53B;))be the space of all bounded linear operators fromA2(&#x1D53B;)into itself. In this article, we have shown thatCaSCa=Sfor alla∈&#x1D53B;if and only if∫&#x1D53B;S˜(ϕa(z))dA(a)=S˜(z), whereS∈ℒ(A2(&#x1D53B;))andS˜is the Berezin symbol of S.
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