On a Class of Composition Operators on Bergman Space

Let 𝔻 = { z ∈ â„‚ : | z | < 1 } be the open unit disk in the complex plane â„‚ . Let A 2 ( 𝔻 ) be the space of analytic functions on 𝔻 square integrable with respect to the measure d A ( z ) = ( 1 / Ï€ ) d x  d y . Given a ∈ 𝔻 and f any measurable function on 𝔻 , we define the function C a f by C a f ( z ) = f ( Ï• a ( z ) ) , where Ï• a ∈ A u t ( 𝔻 ) . The map C a is a composition operator on L 2 ( 𝔻 , d A ) and A 2 ( 𝔻 ) for all a ∈ 𝔻 . Let â„’ ( A 2 ( 𝔻 ) ) be the space of all bounded linear operators from A 2 ( 𝔻 ) into itself. In this article, we have shown that C a S C a = S for all a ∈ 𝔻 if and only if ∫ 𝔻 S Ëœ ( Ï• a ( z ) ) d A ( a ) = S Ëœ ( z ) , where S ∈ â„’ ( A 2 ( 𝔻 ) ) and S Ëœ is the Berezin symbol of S.