Injectivity of powers of the Lefschetz operator ($L^r:{\mathcal{A}}^k\hookrightarrow {\mathcal{A}}^{k+2r}(r+k\le n)$)

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.