Chebyshev type inequalities for Hilbert space operators (Q2252098)

If \(a_i,b_i,w_i\), \(i=1,\dots,n\), are positive real numbers with \(a_1\geq a_2\geq\dots\geq a_n\geq 0\) and \(b_1\geq b_2\geq\dots\geq b_n\geq 0\), then Chebyshev's inequality asserts that \[ \left(\sum_{j=1}^{n}w_ja_j\right)\left(\sum_{j=1}^{n}w_jb_j\right)\leq \left(\sum_{j=1}^{n}w_j\right)\left(\sum_{j=1}^{n}w_ja_jb_j\right). \] \textit{J. S. Matharu} and \textit{J. S. Aujla} [JIPAM, J. Inequal. Pure Appl. Math. 10, No. 2, Paper No. 51, 6 p. (2009; Zbl 1170.15009)] proved that, if \(A_j,B_j\) are positive matrices with \(A_1\geq\dots\geq A_n\) and \(B_1\geq\dots\geq B_n\), then \[ \left(\sum_{j=1}^{n}w_jA_j\right)\circ\left(\sum_{j=1}^{n}w_jB_j\right)\leq \left(\sum_{j=1}^{n}w_j\right)\left(\sum_{j=1}^{n}w_j(A_j\circ B_j)\right), \] where \(w_j\) are positive scalars and \(\circ\) means the Hadamard product of matrices. In the present article, the authors present some Chebyshev type inequalities for Hilbert space operators. In particular, as a Hadamard product version, it is shown that \[ \begin{aligned} \left(\int_T\alpha(t)A_td\mu(t)\right)\circ\left(\int_T\alpha(s)B_sd\mu(s)\right) \leq\left(\int_T\alpha(s)d\mu(s)\right)\left(\int_T\alpha(t)(A_t\circ B_t)d\mu(t)\right) \end{aligned} \] in which \((A_t)_t\) and \((B_t)_t\) are suitable continuous fields of operators and the integrals are interpreted as Bochner integrals.