Complete form of Furuta inequality

摘要

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) A(min(1,p}) A(r/2) >= (A(r/2) B-p A(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 > p(0) > 0 and 2p(0) + r >= p > p(0), then (A(r/2) B-p0 A(r/2)) (p+r/p0+r) >= (A(r/2) B-p A(r/2)) (p+r/p+r). This paper aims to show a sharpening of Furuta (1989): Let r >= 0, p(0) > 0 and s = min {p, 2p(0) + min {1, r}}; then (A(r/2) B-p0 A(r/2)) (s+r/p0+r) >= (A(r/2) B-p A(r/2)) (s+r/p+r). We call it the complete form of Furuta inequality because the case p(0) = 1 of it implies the essential part (p > 1) of Furuta inequality for 1+r/8+r is an element of (0, 1] by the famous Lowner-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.