Noncommutative Chebyshev inequality involving the Hadamard product

TL;DR

This paper extends the Chebyshev inequality to Hilbert space operators using the Hadamard product, providing new operator inequalities involving integrals over compact spaces and positive operator fields.

Contribution

It introduces novel operator inequalities involving the Hadamard product and the Chebyshev inequality for Hilbert space operators, especially under the synchronous Hadamard property.

Findings

Established operator extensions of Chebyshev inequality

Derived inequalities involving the Hadamard product and integrals

Applied to positive increasing operator fields

Abstract

We present several operator extensions of the Chebyshev inequality for Hilbert space operators. The main version deals with the synchronous Hadamard property for Hilbert space operators. Among other inequalities, it is shown that if ${\mathfrak A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$ as a totaly order set, then \begin{align*} \int_{T} \alpha(s) d\mu(s)\int_{T}\alpha(t)(A_t\circ B_t) d\mu(t)\geq\Big{(}\int_{T}\alpha(t) (A_tm_{r,\alpha} B_t) d\mu(t)\Big{)}\circ\Big{(}\int_{T}\alpha(s) (A_sm_{r,1-\alpha} B_s) d\mu(s)\Big{)}, \end{align*} where $\alpha\in[0,1]$, $r\in[-1,1]$ and $(A_t)_{t\in T}, (B_t)_{t\in T} $ are positive increasing fields in $\mathcal{C}(T,\mathfrak A)$.