Operators in Cowen-Douglas classes

The paper introduces a new approach to the Cowen-Douglas theory based on th e notion of a spanning holomorphic cross-section. This approach is less geo metric and enables one to obtain several additional results, including one about the similarity of operators in the Cowen-Douglass classes and another about the representation of these operators as the adjoint of multiplicati on by z on certain Hilbert spaces of holomorphic functions.