Abstract

In this paper, we prove that if ${A}$ is a Banach $*$-subalgebra of $B(H)$, $T$ is a compact Hausdorff space equipped with 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 space and using it, we prove the Schwarz and Chebyshev type inequalities dealing with the Hadamard product and the trace of operators.

Keywords: Grüss inequality; Chebyshev inequality; operator inequality.

MSC: 26D10, 26D15, 46C50, 46G12

Pages:  303$-$313     

Volume  72 ,  Issue  4 ,  2020