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

Year 2024, Early Access, 1 - 10

Sever Dragomır

https://doi.org/10.15672/hujms.1362698

Abstract

Let H be a Hilbert space and Ω a locally compact Hausdorff space endowed with a Radon measure μ with ∫_{Ω}1dμ(t)=1. In this paper we show among others that, if f is continuous differentiable convex on the open interval I, (A_{τ})_{τ∈Ω} is a continuous field of positive operators in B(H) such that Sp(A_{τ}) ⊂I for each τ∈Ω and B and operator such that Sp(B)⊂I, then we have

∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)⊗1-∫_{Ω}f′(A_{τ})dμ(τ)⊗B
≥∫_{Ω}f(A_{τ})dμ(τ)⊗1-1⊗f(B)
≥(∫_{Ω}A_{τ}dμ(τ)⊗1-(1⊗B))(1⊗f′(B))

and the Hadamard product inequality

∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)∘1-∫_{Ω}f′(A_{τ})dμ(τ)∘B
≥∫_{Ω}f(A_{τ})dμ(τ)∘1-1∘f(B)
≥∫_{Ω}A_{τ}dμ(τ)∘f′(B)-1∘(f′(B)B).

Keywords

Tensorial product, Hadamard Product, Selfadjoint operators, Convex functions

References

• [1] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26, 203-241, 1979.