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

Curvature operator of holomorphic vector bundles and $L^2$-estimate condition for $(n,q)$ and $(p,n)$-forms