Curvature inequalities and extremal operators

Abstract

A curvature inequality is established for contractive commuting tuples of operators $\mathbf{T}$ in the Cowen–Douglas class $B_n(\Omega )$ of rank $n$ defined on some bounded domain $\Omega$ in ${\mathbb{C}}^m$. Properties of the extremal operators (that is, the operators which achieve equality) are investigated. Specifically, a substantial part of a well-known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.