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Operator-valued inner product and operator inequalities. (English) Zbl 1151.47024

For a Hilbert space $H$, the ${C}^{*}$-inner product ${𝐘}^{*}𝐗={\sum }_{j=1}^{n}{Y}_{j}^{*}{X}_{j}$ $\left(𝐗=\left({X}_{j}\right)$, $𝐘=\left({Y}_{j}\right)\in B\left(H,{H}^{n}\right)\right)$ makes $B\left(H,{H}^{n}\right)$ into a Hilbert ${C}^{*}$-module over $B\left(H\right)$. Using this operator-valued inner product, the author of present paper nicely presents operator versions for the Schwarz and Jensen inequalities and gives simple conditions that the equalities hold. Among his results, we mention the following version of the Schwarz inequality: for operators $X$ and $Y$ acting on a Hilbert space, the inequality

${Y}^{*}{X}^{*}YX+{X}^{*}{Y}^{*}XY\le {\left(XY\right)}^{*}XY+{\left(YX\right)}^{*}YX$

holds with equality only when $X$ commutes with $Y$.

##### MSC:
 47A63 Operator inequalities 47A75 Eigenvalue problems (linear operators) 47A80 Tensor products of operators 47A56 Functions whose values are linear operators