The spinorial operator

Question

Let $(M,g)$ be a spin manifold with spinors $\psi$, I define a spinorial operator $\nabla$ like an operator by the following equation:

$$\nabla (X.\psi)= dX.\psi -X.\nabla (\psi)$$

where $X$ is a vector field and $d$ is a superderivation of the Clifford algebra.

What is the space of spinorial operators?