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Best possibility of the Furuta inequality. (English) Zbl 0841.47012

Summary: Let 0p,q,r, p+2r(1+2r)q, and 1q. Furuta proved that if bounded linear operators A,BB(H) on a Hilbert space H (dim(H)2) satisfy 0BA, then (A r B p A r ) 1/q A (p+2r)/q .

In this paper, we prove that the range p+2r(1+2r)q and 1q 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,BB( 2 ) which satisfy 0BA but (A r B p A r ) 1/q ¬A (p+2r)/q .


MSC:
47A63Operator inequalities
47B15Hermitian and normal operators