Furuta’s inequality and its mean theoretic approach.

*(English)*Zbl 0734.47008If A, B are positive operators on a Hilbert space it was shown by *T. Furuta* [Proc. Am. Math. Soc. 101, 85-88 (1987; Zbl 0721.47023)] that for $A\ge B\ge 0$,

$${\left({B}^{r}{A}^{p}{B}^{r}\right)}^{1/q}\ge {B}^{(p+2r)/q},$$

for $r\ge 0$, $p\ge 0$, $q\ge 1$ and $(1+2r)q\ge p+2r\xb7$

The author of the present note gives a proof of this based on the theory of means of operators. The method is to prove the inequality for a restricted range of parameter values and to prove a reduction lemma which allows these to change in a prescribed manner.

Reviewer: J.L.R.Webb (Glasgow)