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

Abstract

The curvature ${\mathcal{K}}_T(w)$ of a contraction $T$ in the Cowen-Douglas class $B_1(\mathbb{D})$ is bounded above by the curvature ${\mathcal{K}}_{S^{\ast }}(w)$ 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 $E_T$ corresponding to the operator $T$ in the Cowen-Douglas class $B_1(\mathbb{D})$ which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the class $B_1(\Omega )$, for a bounded domain $\Omega$ in ${\mathbb{C}}^m$.

14 pages

Cited by in corpus (1)

• Geometry of holomorphic vector bundles and similarity of commuting operator tuples