The famous Hodge Decomposition Theorem can be generalized in the context of elliptic operators. The general decomposition theorem can be found in Theorem 5.5 of Spin Geometry of H. Blaine Lawson and M.L. Michelsohn as follows:
Theorem Let $(E,M,\pi)$ be a vector bundle and $P:\Gamma(E)\to\Gamma(E)$ be a self-adjoint elliptic operator (respect to a suitable inner product in $\Gamma(E)$). Then we have the orthogonal decomposition: $$\Gamma(E)=\ker P\oplus \mathrm{im}\, P.$$
Now, let $(F,M,\pi')$ be another vector bundle. Given $P:\Gamma(E)\to \Gamma(F)$ a (non-elliptic) operator. Then $P\circ P^\star:\Gamma(E)\to \Gamma(F)$ is self-adjoint by construction.
I was wondering if there are any conditions under which we can ensure that the composition $P\circ P^\star$ is elliptic. If so, we would have the following orthogonal decompositions: $$\Gamma(E)=\mathrm{im}\, P\oplus\ker P^\star$$
Thanks in advance.