Suppose that $A$ is a unitary connection on a Hermittian differentiable vector bundle $E$ over a Kahler manifold $X$, then we have operators $$\bar{\partial}_A: \Omega_{X}^{p,q}(E)\to \Omega_{X}^{p,q+1}(E)$$ $${\partial}_A: \Omega_{X}^{p,q}(E)\to \Omega_{X}^{p+1,q}(E)$$ And we have the Kahler identities: $$\bar{\partial}_{A}^{*}=-i[\partial_A,\Lambda]$$ $${\partial}_{A}^{*}=i[\bar{\partial}_A,\Lambda]$$

Is this result also true for almost Kahler manifolds? Does anyone know a good reference for it? By definition, an almost Kahler manifold is a manifold endowed with $(g,J,\omega)$, where $g$ is a Riemannian metric, $\omega$ is a symplectic form, and $J$ is an almost complex structure, and they satisfy the compatibility condition $$g(X,Y)=\omega(X,JY)$$