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

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 Pólya-Szegö'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 Grüss type inequalities and an operator Ozeki-Izumino-Mori-Seo type inequality. Several applications are concluded as well.

arXiv comment12 pages, to appear in Electronic Journal of Linear Algebra (ELA)