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 Ca f by Ca f (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 (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 ∫ DS̃(φa(z))dA(a) = S̃(z), where S ∈ (A2(D)) and S̃ is the Berezin symbol of S.