Abstract:
It has been recently shown that if K is a sesqui-analytic scalar valued non-negative definite kernel on a domain Ω in Cm, then the function (K2∂i∂¯jlogK)mi,j=1, is also a non-negative definite kernel on Ω. In this paper, we discuss two consequences of this result. The first one strengthens the curvature inequality for operators in the Cowen-Douglas class B1(Ω) while the second one gives a relationship of the reproducing kernel of a submodule of certain Hilbert modules with the curvature of the associated quotient module.