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Furuta’s inequality and its mean theoretic approach. (English) Zbl 0734.47008

If 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 AB0,

(B r A p B r ) 1/q B (p+2r)/q ,

for r0, p0, q1 and (1+2r)qp+2r·

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.

47A63Operator inequalities
47A60Functional calculus of operators
47A30Operator norms and inequalities