  Let $A_{i}\ (i=1, 2, ..., k)$ be bounded linear operators on a Hilbert space.
This paper aims to show characterizations of operator order $A_{k}\geq
A_{k-1}\geq...\geq A_{2}\geq A_{1}>0$ in terms of operator inequalities.
Afterwards, an application of the characterizations is given to operator
equalities due to Douglas's majorization and factorization theorem.
