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

TL;DR

This paper explores curvature inequalities for operators in the Cowen-Douglas class, characterizing a subclass of contractions and extending results to tuples of operators, with implications for operator metrics and negative curvature conditions.

Contribution

It introduces a stronger curvature inequality to characterize a specific subclass of contractions and generalizes curvature conditions to commuting operator tuples in several complex variables.

Findings

Characterization of a smaller class of contractions via curvature inequality

Conditions for negative definiteness of the curvature function

Extension of results to tuples of operators in several complex variables

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 $\mathcal K_{S^*}(w)$ 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 $\mathbb C^m$.