Tensorial and Hadamard Product Integral Inequalities for Convex Functions of Continuous Fields of Operators in Hilbert Spaces

Let H be a Hilbert space and Ω a locally compact Hausdorff space endowed with a Radon measure μ with ∫Ω1dμ(t)=1. In this paper we show among others that, if f is continuous differentiable convex on the open interval I, (Aτ)τ∈Ω is a continuous field of positive operators in B(H) such that Sp(Aτ) ⊂I for each τ∈Ω and B and operator such that Sp(B)⊂I, then we have ∫Ω(f′(Aτ)Aτ)dμ(τ)⊗1-∫Ωf′(Aτ)dμ(τ)⊗B ≥∫Ωf(Aτ)dμ(τ)⊗1-1⊗f(B) ≥(∫ΩAτdμ(τ)⊗1-(1⊗B))(1⊗f′(B)) and the Hadamard product inequality ∫Ω(f′(Aτ)Aτ)dμ(τ)∘1-∫Ωf′(Aτ)dμ(τ)∘B ≥∫Ωf(Aτ)dμ(τ)∘1-1∘f(B) ≥∫ΩAτdμ(τ)∘f′(B)-1∘(f′(B)B).