Primitive cohomology of degree 2 on compact symplectic manifolds

In this paper, we define the generalized Lejmi's $P_J$ operator on a compact almost K\"{a}hler $2n$-manifold. We get that $J$ is $C^\infty$-pure and full if $\dim\ker P_J=b^2-1$. Additionally, we investigate the relationship between $J$-anti-invariant cohomology introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic $4$-manifold.