Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect to some fixed metrics on $M$ and $E$) differential forms with values in $E$ which is naturally a $\Omega^{\bullet}(M)$-module. A connection $\nabla$ on $E$ by definition satisfies $[\nabla, \omega]=d\omega$, where $\omega$ stands for the action of the form $\omega \in \Omega^{\bullet}(M)$ on $H$ and the commutator is graded. So, in particular $\nabla$ satisfies $$[\nabla, d\omega]=0$$

My questions is: Do there exist examples of odd operators $P$ on $H$ which

(1) satisfy that last equation, i.e. $[P, d\omega]=0$ for every $\omega \in \Omega^{\bullet}(M)$ and

(2) which at the same time behave like elliptic operators in the sense that they have compact resolvent.

Or is there any reason to think that such operators do not exist?

By "operator" I mean any densely defined closed operator on $H$, it doesn't need to be a differential operator. A pseudodifferential operator or something even more general will do.