Geometric similarity invariants of Cowen-Douglas operators

Abstract

In 1978, M. J. Cowen and R.G. Douglas introduced a class of operators $B_n(\Omega )$ (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such an operator. They gave a complete set of unitary invariants in terms of the curvature and its covariant derivatives. At the same time they asked whether one can use geometric ideas to find a complete set of similarity invariants of Cowen-Douglas operators. We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely determine the similarity orbit of a norm dense class of Cowen-Douglas operators. As an application we show that uncountably many (non-similar) strongly irreducible operators in $B_n(\mathbb{D})$ can be constructed from a given operator in $B_1(\mathbb{D})$. We also characterize a class of strongly irreducible weakly homogeneous operators in $B_n(\mathbb{D})$.