I would like to better understand the relationship between self-adjointness, pseudodifferential operators, completeness, and positivity in the following context.
Let $D$ be an elliptic differential operator acting on a Riemannian manifold $M$ (or a bundle over it). Consider $D$ as an unbounded, densely defined symmetric operator on $L^2(M)$, with initial domain $C_c^\infty(M)$.
If the Riemannian metric on $M$ was complete, then $D$ would be essentially self-adjoint. But if the metric is not complete, then my understanding is that $D$ could fail to have a self-adjoint extension.
However, I would like to look at this from the point of view of pseudodifferential operators, which is where my confusion lies.
I'm wondering if the following assertions are correct:
Even though $M$ is non-compact and not complete, there exists a pseudodifferential operator $(D+i)^{-1}$ of order $-1$ defined using symbols.
$(D+i)^{-1}$ does not necessarily extend to a bounded operator on $L^2(M)$ because $M$ is non-compact and not complete.
Additionally, I would like to know if the following is true:
- If we know that $D\geq 0$, then it has a self-adjoint extension by Friedrichs' extension theorem. Then the pseudodifferential operator $(D+i)^{-1}$ would indeed extend to an operator in $\mathcal{B}(L^2(M))$.
I'm a bit suspicious of 3, because it seems to imply that $D$ would then be essentially self-adjoint in that case.