Complete Form of Furuta Inequality

Let A and B be bounded linear operators on a Hilbert space satisfying A ≥ B ≥ 0. The well-known Furuta inequality is given as follows: Let r ≥ 0 and p > 0; then A r 2 Amin{1,p}A r 2 ≥ (A r 2 BpA r 2 ) min{1,p}+r p+r . In order to give a self-contained proof of it, Furuta (1989) proved that if 1 ≥ r ≥ 0, p > p0 > 0 and 2p0 + r ≥ p > p0, then (A r 2 Bp0A r 2 ) p+r p0+r ≥ (A r 2 BpA r 2 ) p+r p+r . This paper aims to show a sharpening of Furuta (1989): Let r ≥ 0, p0 > 0 and s = min{p, 2p0 + min{1, r}}; then (A r 2 Bp0A r 2 ) s+r p0+r ≥ (A r 2 BpA r 2 ) s+r p+r . We call it the complete form of Furuta inequality because the case p0 = 1 of it implies the essential part (p > 1) of Furuta inequality for 1+r s+r ∈ (0, 1] by the famous Löwner-Heinz inequality. Afterwards, the optimality of the outer exponent of the complete form is considered. Lastly, we give some applications of the complete form to Aluthge transformation.