Best possibility of the Furuta inequality

Tanahashi, Kôtarô

doi:10.1090/S0002-9939-96-03055-9

Best possibility of the Furuta inequality
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by Kôtarô Tanahashi PDF

Proc. Amer. Math. Soc. 124 (1996), 141-146 Request permission

Abstract:

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

References

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