Bounded modules, extremal problems, and a curvature inequality

Abstract

In this paper we study certain finite dimensional Hilbert modules over the function algebra A(Ω), Ω ⊆ Cⁿ. These modules appear as localizations of a Cowen-Douglas operator. We show that these modules are always bounded, where the bound is related to the solution of an extremal problem. In particular, we obtain necessary sufficient conditions for such a module to be contractive. We apply the above results to produce an example of a contractive module over A(B²), which is not completely contractive.