Lefschetz operator on a Hermitian manifold ($L\alpha =\omega \wedge \alpha$)

Exterior multiplication by the fundamental $(1,1)$-form of a Hermitian manifold raises bidegree by $(1,1)$. Its pointwise formal adjoint lowers bidegree by $(1,1)$. On a Kähler manifold this is the Lefschetz operator of a Kähler manifold, but its pointwise algebraic properties do not require the form to be closed.

The Hermitian Lefschetz operator has injective powers in the indicated range. The commutator formula for powers of the Lefschetz operator proves this by induction on $r+k$: a form in the kernel can be written as $L\alpha$ using injectivity of the smaller power, and the lower-degree form $\alpha$ is then in the kernel of another power covered by the induction hypothesis.