A decreasing operator function associated with the Furuta inequality

Furuta, Takayuki; Wang, Derming

doi:10.1090/S0002-9939-98-04632-2

A decreasing operator function associated with the Furuta inequality
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by Takayuki Furuta and Derming Wang PDF

Proc. Amer. Math. Soc. 126 (1998), 2427-2432 Request permission

Abstract:

Let $A\ge B\ge 0$ with $A>0$ and let $t\in [0,1]$ and $q\ge 0$. As a generalization of a result due to Furuta, it is shown that the operator function \[ G_{p,q,t}(A,B,r,s)=A^{-r/2}\{A^{r/2} (A^{-t/2} B^pA^{-t/2})^s A^{r/2}\}^{(q-t+r)/[(p-t)s+r]}A^{-r/2} \] is decreasing for $r\ge t$ and $s\ge 1$ if $p\ge \max \{q,t\}$. Moreover, if $1\ge p>t$ and $q\ge t$, then $G_{p,q,t}(A,B,r,s)$ is decreasing for $r\ge 0$ and $s\ge \frac {q-t}{p-t}$. The latter result is an extension of an earlier result of Furuta.

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