Curvature Inequalities and Extremal Operators

TL;DR

This paper establishes a curvature inequality for certain operator tuples, investigates extremal operators achieving equality, and addresses a notable question by R. G. Douglas in the context of the unit disc.

Contribution

It introduces a new curvature inequality for contractive commuting operator tuples and provides insights into extremal operators, answering a key question in the field.

Findings

Proved a curvature inequality for operator tuples in the Cowen-Douglas class.

Analyzed properties of extremal operators that attain equality.

Answered a significant question by R. G. Douglas regarding these operators.

Abstract

A curvature inequality is established for contractive commuting tuples of operators in the Cowen-Douglas class of rank n. 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.