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.


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


Zbl 0412.47013
Full Text: DOI


[1] Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear algebra appl., 26, 203-241, (1979) · Zbl 0495.15018
[2] Bhatia, R., Matrix analysis, (1997), Springer-Verlag New York
[3] Callebaut, D.K., Generalization of the cauchy – schwarz inequality, J. math. anal. appl., 12, 491-494, (1965) · Zbl 0136.03401
[4] Conway, J.B., A course in functional analysis, (1985), Springer-Verlag New York · Zbl 0558.46001
[5] Daykin, D.E.; Eliezer, C.J.; Carlitz, C., Problem 5563, Amer. math. monthly, 75, 198, (1968), and 76 (1969) 98-100
[6] F. Kubo, Theory of Operator Means, Doctor thesis, Hokkaido Univ., 1991.
[7] Kubo, F.; Ando, T., Means of positive linear operators, Math. ann., 246, 3, 205-224, (1979) · Zbl 0412.47013
[8] Milne, E.A., Note on rosseland’s integral for the stellar absorption coefficient, Monthly notices roy. astronom. soc., 85, 979-984, (1925) · JFM 52.1032.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.