On the Commutant of Certain Operators in the Bergman Space

We study the commutant of analytic multiplication operator Mz2 on the weighted Bergman space A2α. According to a result of Zhu [Reducing subspaces for a class of multiplication operators, J. London Math. Soc. (2) 62 (2000), no.2, 553-568], a bounded linear operator T defined on the standard Bergman space A2 commutes with Mz2 if and only if there exist two bounded analytic functions ϕ and ψ such that Tf=ϕfe+ψfo/z where f=fe+fo is the even-odd decomposition of f. We shall prove that this statement holds true in the weighted Bergman space as well.