On Some Properties of Cowen-Douglas Class of Operators

We will consider multiplication operators on a Hilbert space of analytic functions on a domain Omega subset of C. For a bounded analytic function phi on Omega, we will give necessary and sufficient conditions under which the complement of the essential spectrum of M phi in phi(Omega) becomes nonempty and this gives conditions for the adjoint of themultiplication operator M phi 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.