SOME CHEBYSHEV TYPE INEQUALITIES INVOLVING THE HADAMARD PRODUCT OF HILBERT SPACE OPERATORS

Abstract

In this paper, we prove that if ${A}$ is a Banach $*$-subalgebra of $B(H)$, $T$ is a compact Hausdorff space equippedwith a Radon measure $\mu$¦and $\alpha:T\rightarrow [0,\infty)$ is a integrable function and $(A_t), (B_t)$ are appropriate integrable fields of operators in ${A}$having the almost synchronous property for the Hadamard product, then$$\int_T\!\alpha(s)d\mu(s)\int_T\!\alpha(t)\big(A_t\circ B_t\big) d\mu(t) \geq \int_T\!\alpha(t)A_td\mu(t)\circ\int_T\!\alpha(t)B_td\mu(t).$$We also introduce a semi-inner product for square integrable fields of operators in a Hilbert spaceand using it, we prove the Schwarz and Chebyshev type inequalities dealing with the Hadamard productand the trace of operators.