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Chebyshev type inequalities for Hilbert space operators
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 is a -algebra, is a compact
Hausdorff space equipped with a Radon measure , is a measurable function and are
suitable continuous fields of operators in 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 -algebras to obtain some versions
related to synchronous functions. We also present some Chebyshev type
inequalities involving the singular values of positive matrices.
Several applications are given as well.Comment: 18 pages, to appear in J. Math. Anal. Appl. (JMAA
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