Curvature inequalities for operators of the Cowen–Douglas class

Abstract

Let T be an operator tuple in the Cowen-Douglas class B (n) (Omega) for Omega aS, C (m) . The kernels Ker(T - w) (l) , for w a Omega, l = 1, 2, center dot center dot center dot, define Hermitian vector bundles E (T) (l) over Omega. We prove certain negativity of the curvature of E (T) (l) . We also study the relation between certain curvature inequality and the contractive property of T when Omega is a planar domain.