On some refinement of the Cauchy--Schwarz inequality
This papers deals with operator means in the sense of \textit{F.\,Kubo} and \textit{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.
scientific article; zbMATH DE number 5083613