Best possibility of the Furuta inequality

Abstract

Let 0 ≤ p , q , r ∈ R , p + 2 r ≤ ( 1 + 2 r ) q 0\le p,q,r\in \Bbb R, p+2r\le (1+2r)q , and 1 ≤ q 1\le q . Furuta (1987) proved that if bounded linear operators A , B ∈ B ( H ) A,B\in B(H) on a Hilbert space H H ( dim ⁡ ( H ) ≥ 2 ) (\dim (H)\ge 2) satisfy 0 ≤ B ≤ A 0\le B\le A , then ( A r B p A r ) 1 / q ≤ A ( p + 2 r ) / q (A^r B^p A^r)^{1/q} \le A^{(p+2r)/q} . In this paper, we prove that the range p + 2 r ≤ ( 1 + 2 r ) q p+2r\le (1+2r)q and 1 ≤ q 1\le q is best possible with respect to the Furuta inequality, that is, if ( 1 + 2 r ) q > p + 2 r (1+2r) q>p+2r or 0 > q > 1 0>q>1 , then there exist A , B ∈ B ( R 2 ) A,B\in B(\Bbb R^2) which satisfy 0 ≤ B ≤ A 0\le B\le A but ( A r B p A r ) 1 / q ≰ A ( p + 2 r ) / q (A^r B^p A^r)^{1/q}\nleq A^{(p+2r)/q} .