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    Chebyshev type inequalities for Hilbert space operators

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    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 A\mathscr{A} is a Cβˆ—C^*-algebra, TT is a compact Hausdorff space equipped with a Radon measure ΞΌ\mu, Ξ±:Tβ†’[0,+∞)\alpha: T\rightarrow [0, +\infty) is a measurable function and (At)t∈T,(Bt)t∈T(A_t)_{t\in T}, (B_t)_{t\in T} are suitable continuous fields of operators in A{\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 Cβˆ—C^*-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive nΓ—nn\times n matrices. Several applications are given as well.Comment: 18 pages, to appear in J. Math. Anal. Appl. (JMAA
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