A note on class $p$-$wA(s,t)$ operators

Abstract

Let $A$ and $B$ be positive operators. In this paper, we shall study the relation between the inequalities
$$(B^{\frac{r}{2}}A^pB^{\frac{r}{2}})^{\frac{qr}{p+r}}\geq B^{rq}\quad \mbox{and}\quad A^{pq}\geq (A^{\frac{p}{2}}B^rA^{\frac{p}{2}})^{\frac{qp}{p+r}}$$
for each $p\geq 0$, $r\geq 0$ and $0<q\leq 1$. Also, we shall show some applications of this result to operator classes.
Moreover, we shall show that if $T$ or $T^*$ belongs to class $p$-$wA(s,t)$ for every $s>0, t>0$ and $0<p\leq 1$ and $S$ is an operator for which $0 \notin\overline{ W(S)}$ and $ST=T^*S$, then $T$ is self-adjoint.