engleski

Generalization of the Kantorovich type operator inequalities via grand Furuta inequality

In this note we show the characterization of the $\delta$-order by means of a generalized Kantorovich constant via Grand Furuta inequality, which is an extension result of that from M.Fujii, E.Kamei, Y.Seo, {; ; \it Kantorovich type operator inequalities via grand Furuta inequality}; ; , Sci.\ Math., {; ; \bf 3}; ; (2000), 263--272. Among other, we show the following characterization of the $\delta$-order: Let $A, B$ be positive invertible operators on a Hilbert space $H$ satisfying $M I \geq A \geq m I >0$ and $N I \geq B \geq n I>0$. Then the following statements are mutually equivalent for each $\delta \in [0, 1]$: {; ; \rm (i)}; ; \quad $A^{; ; \delta}; ; \geq B^{; ; \delta}; ; $. {; ; \rm (ii)}; ; \quad $K(n^r, N^r, 1+\frac{; ; p- \delta}; ; {; ; r}; ; , 1+\frac{; ; q- \delta}; ; {; ; r}; ; )A^q \geq B^p$ for all $p > \delta$, $q > \delta$ and $r > \delta$. {; ; \rm (iii)}; ; \quad $\overline{; ; K}; ; (m^r, M^r, 1+\frac{; ; q- \delta}; ; {; ; r}; ; , 1+\frac{; ; p- \delta}; ; {; ; r}; ; )A^q \geq B^p$ for all $p > \delta$, $q > \delta$ and $r > \delta$ where the case $\delta=0$ means the chaotic order $\log A \geq \log B$.

Operator order; chaotic order; Kantorovich inequality; Furuta inequality; grand Furuta inequality

nije evidentirano