BEST POSSIBILITY OF THE FURUTA INEQUALITY

Let 0 less than or equal to p,q,r is an element of R,p + 2r less than or equal to (1 + 2r)q, and 1 less than or equal to q. Furuta (1987) pr oved that if bounded linear operators A, B is an element of B(H) on a Hilbert space H (dim(H) greater than or equal to 2) satisfy 0 less tha n or equal to B less than or equal to A, then (A(r)B(p)A(r))(1/q) less than or equal to A((p+2r)/q). In this paper, eve prove that the range p + 2r less than or equal to (1 + 2r)q and 1 less than or equal to q is best possible with respect to the Furuta inequality, that is, if (1 + 2r)q < p + 2r or 0 < q < 1, then there exist A,B is an element of B (R(2)) which satisfy 0 less than or equal to B less than or equal to A but (A(r)B(p)A(r))(1/q) not less than or equal to A((p+2r)/q).