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

Abstract

In this paper, we prove that if $A$ is a Banach $\ast$-subalgebra of $B(H)$, $T$ is a compact Hausdorff space equippedwith a Radon measure $\mu$¦and $\alpha :T\to [0,\mathrm{\infty })$ 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$${\int }_T\alpha (s)d\mu (s){\int }_T\alpha (t)(A_t\circ B_t)d\mu (t)\ge {\int }_T\alpha (t)A_{t}d\mu (t)\circ {\int }_T\alpha (t)B_{t}d\mu (t).$$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.