The commutant of multiplication operators

In this paper we consider the Hilbert space of analytic functions on a plane domain $ \Omega $ and multiplication operators on such spaces induced by functions in $ H^{\infty}(\Omega) $. Recently, K. Zhu has given condition under which the adjoints of multiplication operators on Hilbert spaces of analytic functions belong to the Cowen-Douglas classes. In this paper, we provide some sufficient conditions which give the converse of the main result obtained by K. Zhu. We also characterize the commutant of certain multiplication operators.