A Diaz-Metcalf Type Inequality For Positive Linear Maps And Its Applications

We present a Diaz-Metcalf type operator inequality as a reverse Cauchy-Schwarz inequality and then apply it to get some operator versions of Polya-Szego'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 Gruss type inequalities and an operator Ozeki-Izumino-Mori-Seo type inequality. Several applications are included as well.