A decreasing operator function associated with the Furuta inequality

Abstract

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