MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
SOME CHEBYSHEV TYPE INEQUALITIES INVOLVING THE HADAMARD PRODUCT OF HILBERT SPACE OPERATORS
R. Teimourian, A. G. Ghazanfari

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.

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Keywords: Grüss inequality; Chebyshev inequality; operator inequality.

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

Pages:  303--313     

Volume  72 ,  Issue  4 ,  2020