By virtue of Furuta inequality, we show some characterizations 
of the spectral order of positive operators on a Hilbert space 
from the viewpoint of the Kantorovich type inequality and the Riccati equation. 
Let $A$ and $B$ be positive operators on a Hilbert space. 
Then the spectral order $A\succeq B$ holds 
if and only if there exist a unique positive contraction $T_{p,u}$ 
such that $T_{p,u}A^{\frac{p+u}{2}}T_{p,u}=A^{\frac{u-p}{4}}B^{p}A^{\frac{u-p}{4}}$ 
for all $p\geq 0$ and $u\geq 0$. This form interpolates the chaotic order, 
$\delta$-order and the usual order continuously.