On the Cowen-Douglas Class for Banach Space Operators

Abstract

In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class \({\mathcal{B}}_n(\Omega)\) if and only if T is unitarily equivalent with the operator M z on a Hilbert space \({\mathcal{H}}\,{\rm of}\,{\mathbb{C}}^n\)-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.