  We study the positivity properties of the curvature operator for holomorphic
Hermitian vector bundles. We obtain new characterization of semi-positive
curvature operators for $(n,q)$ and $(p,n)$-forms by L2-estimates. The
characterization of Nakano semi-positivity by $L^2$-estimate is already known.
Applying our results, we give new characterizations of Nakano semi-negativity.
