On a Class of Composition Operators on Bergman Space

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.