 Lefschetz operator on a Hermitian manifold

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.