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On some refinement of the Cauchy–Schwarz inequality. (English) Zbl 1121.47010

This papers deals with operator means in the sense of F. Kubo and T. Ando [Math. Ann. 246, 205–224 (1979; Zbl 0412.47013)]. Denote by # the geometric mean and by \(\sigma\) an arbitrary operator mean acting on the cone of positively semidefinite operators on a Hilbert space. Then the following generalization of the Cauchy–Schwarz inequality \[ \begin{aligned} (A\#B)\otimes(A\#B) & \leq\frac{1}{2}\left[(A\sigma B)\otimes(A\sigma^{\bot}B)+(A\sigma^{\bot}B)\otimes(A\sigma B)\right] \\ & \leq\frac{1}{2}\left[(A\otimes B)+(B\otimes A)\right]\end{aligned} \] holds. Here, \(\sigma^{\bot}\) is the dual of \(\sigma\), defined by \(A\sigma ^{\bot}B=(B^{-1}\sigma A^{-1})^{-1}\) for \(A>0\), \(B>0\). This represents a noncommutative analogue of some well-known inequalities for families of nonnegative numbers.

MSC:

47A63 Linear operator inequalities
26D15 Inequalities for sums, series and integrals

Citations:

Zbl 0412.47013
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References:

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