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

For a σ-finite measures μ on Ω and μ-weakly∗-measurable families {At}t∈Ω and {Bt}t∈Ω of Hilbert space operators we have the non-commutative Cauchy–Schwarz inequalities in Schatten p-ideals ∫ΩAXB dμp⩽∫ΩA∗∫ΩAA∗ dμq−1A dμ2q X∫ΩB∫ΩB∗B dμr−1B∗ dμ2rpfor all X∈p(H) and for all p,q,r⩾1 such that 1q+1r=2p. If both {At}t∈Ω and {Bt}t∈Ω consists of commuting normal operators, then∫ΩAXB dμ⩽∫ΩA∗A dμ X∫ΩB∗B dμfor all unitarily invariant norms |||•||| and all X∈|||•|||(H). If additionally ∫ΩA∗A dμ⩽I and ∫ΩB∗B dμ⩽I, then I−∫ΩA∗A dμ XI−∫ΩB∗B dμ∈|||•|||(H) andI−∫ΩA∗A dμXI−∫ΩB∗B dμ⩽X−∫ΩAXB dμ.Applications include Youngʹs and arithmetic–geometric–logarithmic means inequalities for operators and the mean value theorem for operator monotone functions.