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

Abstract

Abstract. We present a Diaz–Metcalf type operator inequality as a reverseCauchy–Schwarz inequality and then apply it to get the operator versions ofPo´lya–Szego¨’s,Greub–Rheinboldt’s, Kantorovich’s,Shisha–Mond’s, Schweitzer’s,Cassels’ and Klamkin–McLenaghan’s inequalities via a unified approach. WealsogivesomeoperatorGru¨sstypeinequalitiesand anoperatorOzeki–Izumino–Mori–Seo type inequality. Several applications are concluded as well. 1. IntroductionThe Cauchy–Schwarz inequality plays an essential role in mathematical in-equalities and its applications. In a semi-inner product space (H ,h·,·i) theCauchy–Schwarz inequality reads as follows|hx,yi| ≤ hx,xi 1/2 hy,yi 1/2 (x,y ∈ H ).There are interesting generalizations of the Cauchy–Schwarz inequality in var-ious frameworks, e.g. finite sums, integrals, isotone functionals, inner productspaces, C ∗ -algebras and Hilbert C ∗ -modules; see [5, 6, 7, 13, 17, 20, 9] and refer-ences therein. There are several reverses of the Cauchy–Schwarz inequality in theliterature: Diaz–Metcalf’s, Po´lya–Szego¨’s, Greub–Rheinboldt’s, Kantorovich’s,Shisha–Mond’s, Ozeki–Izumino–Mori–Seo’s, Schweitzer’s, Cassels’ and Klamkin–McLenaghan’s inequalities.Inspired by the work of J.B. Diaz and F.T. Metcalf [4], we present severalreverse Cauchy–Schwarz type inequalities for positive linear maps. We give aunified treatment of some reverse inequalities of the classical Cauchy–Schwarztype for positive linear maps.Throughout the paper B(H ) stands for the algebra of all bounded linear oper-ators acting on a Hilbert space H . We simply denote by α the scalar multiple αIof the identity operator I ∈ B(H ). For self-adjoint operators A,B the partially