The Schwarz inequality via operator-valued inner product and the geometric operator mean (The deepening of function spaces and its environment)

Abstract

In this papet, by virtue of the Cauchy-Schwarz operator inequality due to J.I. Fujii, we show the covariance-variance operator inequality via the geometric operator mean which differs from Bhatia-Davis's one and estimate the upper bounds. By our formulation: we bhow a Robertson type inequality associated to the conditional expectation on a finite von Nuemann algebra.