Let's consider a linear functional $A$ from smooth objects to smooth ones. It is first order operator in the sense that it extends to be a map from $W^{k+1,p}$ to $W^{k,p}$. Assume that we have $L^2$ inner product defined. My question is

(1) the adjoint $A^*$ is still a map from smooth objects to smooth objects and extends to the a map from $W^{k,p}$ to $W^{k-1,p}$.

(2) When is $A^*\circ A$ invertible as a map from $W^{k+1,p}$ to $W^{k-1,p}$? The condition i have in mind is the injectivity of $A$.

(3) Can we obtain some estimate for the inverse?

We don't have explicit formula for $A$. So please try to answer these question in the framework of functional analysis. The background of this question is that I try to generalize some obvious facts in finite dimensional geometry to Banach space.