We will consider multiplication operators on a Hilbert space of analytic functions on a domain Ω ⊂ C. For a bounded analytic function on Ω, we will give necessary and sufficient conditions under which the complement of the essential spectrum of in (Ω) becomes nonempty and this gives conditions for the adjoint of the multiplication operator belongs to the Cowen-Douglas class of operators. Also, we characterize the structure of the essential spectrum of a multiplication operator and we determine the commutants of certain multiplication operators. Finally, we investigate the reflexivity of a Cowen-Douglas class operator.