On a class of composition operators on Bergman space

Abstract

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 ofall 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 ∈ ℒ (A2 (D)) and S̃ is the Berezin symbol of S.