On the Cowen-Douglas Class for Banach Space Operators

In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class if and only if T is unitarily equivalent with the operator M _z on a Hilbert space -valued analytic functions, where M _z denotes the operator of multiplication by the independent variable. The proof involves holomorphic vector bundles and Grauert’s theorem. In this paper we use a theorem by I. Gohberg and L. Rodman [4] to give a more elementary proof of this fact, which also works for Banach space operators.