Cauchy-Schwarz norm inequalities for weak*-integrals of operator valued functions

Abstract

For a σ-finite measures μ on Ω and μ-weakly*-measurable families {A t } tεΩ and {B t } tεΩ of Hilbert space operators we have the non-commutative Cauchy-Schwarz inequalities in Schatten p-ideals A formula is presented. for all X∈C p (H) and for all p,q,r≥1 such that 1/q + 1/r = 2/p. If both {A t } t∈Ω and {B t } t∈Ω consists of commuting normal operators, then A formula is presented. Applications include Young's and arithmetic-geometric-logarithmic means inequalities for operators and the mean value theorem for operator monotone functions. © 2004 Elsevier Inc. All rights reserved.