Infinitely divisible metrics and curvature inequalities for operators in the Cowen-Douglas class

The curvature KT (w) of a contraction T in the Cowen-Douglas class B1(D) is bounded above by the curvature KS∗(w) of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this note, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle ET corresponding to the operator T in the Cowen-Douglas class B1(D) which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the class B1(Ω), for a bounded domain Ω in C.