On Some Properties of Cowen-Douglas Class of Operators

Abstract

We will consider multiplication operators on a Hilbert space of analytic functions on a domainΩ⊂C. For a bounded analytic functionφonΩ, we will give necessary and sufficient conditions under which the complement of the essential spectrum ofMφinφΩbecomes nonempty and this gives conditions for the adjoint of the multiplication operatorMφbelongs 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.