## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 63, Number 2 (2019), 193-217.

Dates
Revised: 3 April 2019
First available in Project Euclid: 1 August 2019

https://projecteuclid.org/euclid.ijm/1564646431

Digital Object Identifier
doi:10.1215/00192082-7768711

Mathematical Reviews number (MathSciNet)
MR3987495

Zentralblatt MATH identifier
07088305

#### Citation

Misra, Gadadhar; Reza, Md. Ramiz. Curvature inequalities and extremal operators. Illinois J. Math. 63 (2019), no. 2, 193--217. doi:10.1215/00192082-7768711. https://projecteuclid.org/euclid.ijm/1564646431

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