A glimpse at the operator Kantorovich inequality

TL;DR

This paper presents a new inequality related to the operator Kantorovich inequality, providing bounds for positive operators under positive linear maps, which enhances understanding of operator inequalities in functional analysis.

Contribution

The paper introduces a novel inequality involving positive operators and positive linear maps, extending the classical Kantorovich inequality with new bounds.

Findings

Established a new upper bound for (A^{-1}) under positive linear maps.

Derived a refined inequality involving positive operators within specified bounds.

Extended the classical Kantorovich inequality to a broader operator setting.

Abstract

We show the following result: Let $A$ be a positive operator satisfying $0<m{{\mathbf{1}}_{\mathcal{H}}}\le A\le M{{\mathbf{1}}_{\mathcal{H}}}$ for some scalars $m,M$ with $m<M$ and $\Phi $ be a normalized positive linear map, then \[\Phi \left( {{A}^{-1}} \right)\le \Phi \left( {{m}^{\frac{A-M{{\mathbf{1}}_{\mathcal{H}}}}{M-m}}}{{M}^{\frac{m{{\mathbf{1}}_{\mathcal{H}}}-A}{M-m}}} \right)\le \frac{{{\left( M+m \right)}^{2}}}{4Mm}\Phi {{\left( A \right)}^{-1}}.\]