Conformal Killing $L^2-$forms on complete Riemannian manifolds with nonpositive curvature operator

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^2-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^2-$forms on some complete Riemannian manifolds.