Noncommutative Chebyshev inequality involving the Hadamard product

arXiv1806.05883MaRDI QIDQ6303054

Sever S. Dragomir, Mojtaba Bakherad

Publication date: 15 June 2018

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 $mathfrakA$ is a $C^{*}$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $mu$ as a totaly order set, then �egin{align*} int_{T} alpha(s) dmu(s)int_{T}alpha(t)(A_tcirc B_t) dmu(t)geqBig{(}int_{T}alpha(t) (A_tm_{r,alpha} B_t) dmu(t)Big{)}circBig{(}int_{T}alpha(s) (A_sm_{r,1-alpha} B_s) dmu(s)Big{)}, end{align*} where $alphain[0,1]$, $rin[-1,1]$ and $(A_t{)}_{tinT},(B_t{)}_{tinT}$ are positive increasing fields in $mathcalC(T,mathfrakA)$.

Mathematics Subject Classification ID

Linear operator inequalities (47A63) Functional calculus for linear operators (47A60)

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