Abstract
It is well-known that for Hilbert space linear operators $0 \leq A$ and $0 \leq C$, inequality $C \leq A$ does not imply $C^2 \leq A^2.$ We introduce a strong order relation $0 \leq B \lll A$, which guarantees that $C^2 \leq B^{1/2}AB^{1/2}$ text for all $0 \leq C \leq B,$ and that $C^2 \leq A^2$ when $B$ commutes with $A$. Connections of this approach with the arithmetic-geometric mean inequality of Bhatia-Kittaneh as well as the Kantorovich constant of $A$ are mentioned.
Citation
Tsuyoshi Ando. "Square inequality and strong order relation." Adv. Oper. Theory 1 (1) 1 - 7, Autumn 2016. https://doi.org/10.22034/aot.1610.1035
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