A Diaz--Metcalf type inequality for positive linear maps and its applications

TL;DR

This paper introduces a Diaz--Metcalf type inequality for positive linear maps, providing a unified framework to derive various classical inequalities and their operator versions, with multiple applications in operator theory.

Contribution

It presents a new Diaz--Metcalf type inequality for positive linear maps and applies it to unify and extend several classical inequalities in operator theory.

Findings

Derived operator versions of classical inequalities.

Established new operator Gr"uss and Ozeki-Izumino-Mori-Seo type inequalities.

Provided multiple applications demonstrating the utility of the main inequality.

Abstract

We present a Diaz--Metcalf type operator inequality as a reverse Cauchy-Schwarz inequality and then apply it to get the operator versions of P\'{o}lya-Szeg\"{o}'s, Greub-Rheinboldt's, Kantorovich's, Shisha-Mond's, Schweitzer's, Cassels' and Klamkin-McLenaghan's inequalities via a unified approach. We also give some operator Gr\"uss type inequalities and an operator Ozeki-Izumino-Mori-Seo type inequality. Several applications are concluded as well.