Grand Furuta inequality and its variant

Abstract

The grand Furuta inequality (GFI) is understood as follows: If positive operators A and B on a Hilbert space satisfy A B 0,A is invertible and t ∈ (0,1) ,t hen A 1−t+r (A r (A − t B p A − t ) s A r ) 1�t+r (pt)s+r holds for p, s 1a ndr t . In this note, we present a short proof of (GFI) which is done by the usual induction on s and the use of the Furuta inequality. Furthermore we propose another simultaneous extension of the Ando-Hiai and Furuta inequalities: If A B 0,A is invertible and t ∈ (0,1) ,t hen A t � 1�t pt B p A −r+t � 1�t+r (pt)s+r (A ts B p )