Chebyshev type inequalities for Hilbert space operators

Mohammad Sal Moslehian, Mojtaba Bakherad

Published 2014-01-08, updated 2014-05-29Version 2

We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$, $\alpha: T\rightarrow [0, +\infty)$ is a measurable function and $(A_t)_{t\in T}, (B_t)_{t\in T}$ are suitable continuous fields of operators in ${\mathscr A}$ having the synchronous Hadamard property, then \begin{align*} \int_{T} \alpha(s) d\mu(s)\int_{T}\alpha(t)(A_t\circ B_t) d\mu(t)\geq\left(\int_{T}\alpha(t) A_t d\mu(t)\right)\circ\left(\int_{T}\alpha(s) B_s d\mu(s)\right). \end{align*} We apply states on $C^*$-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive $n\times n$ matrices. Several applications are given as well.