Consider the algebra of exterior forms $\Lambda T^*M$ on an even dimensional $n-$manifold $M$. We can form an operator $\sigma=\bigoplus_{k=1}^ni^{k(n-k)}*_k:\Lambda T^*M\rightarrow\Lambda T^*M$ (where $*_k$ is the Hodge star restricted to $\Lambda^k T^*M$). This operator is know as the $\textit{signature operator}$ and satisfies $\sigma^2=1$.
In index theory (particularly in regards to the Hirzebruch signature theorem) it is commonly stated that this gives a $\mathbb{Z}_2-$grading on $\Lambda T^*M$ by decomposing into $\pm1$ eigenspaces of $\sigma$. However, I cannot see how this makes sense away from the restriction to $\Lambda^{n/2} T^*M$. Given an arbitrary $\alpha\in\Lambda T^*M$ it does not follow that $\sigma\alpha=\pm\alpha$ unless either $\alpha\in\Lambda^{n/2} T^*M$ or $\alpha$ is in the $\textbf{vector space}$ $\Lambda^p T^*M\oplus\Lambda^{n-p} T^*M$.
Any help reconciling this grading would be much appreciated.
Edit: See the MO post.