On some properties of Cowen-Douglas class of operators (Q1709756)

Summary: We will consider multiplication operators on a Hilbert space of analytic functions on a domain \(\Omega \subset \mathbb{C}\). For a bounded analytic function \(\varphi\) on \(\Omega\), we will give necessary and sufficient conditions under which the complement of the essential spectrum of \(M_\varphi\) in \(\varphi (\Omega)\) becomes nonempty and this gives conditions for the adjoint of the multiplication operator \(M_\varphi\) 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.