Inverse of a unbounded operator ?

I’m stumbling over the problem of inversion of a unbounded operator on a Hilbert space. My operator is a perturbation of a Casimir operator acting by double differentiation of a tensor space. It ooks like an elliptic operator, so the inverse is expected to be (at least) bounded, with a “very reasonabe” bound. But I don’t know how to prove that it’s bounded and control its norm at the moment.

If the problem can be solved, then I will obtain the so-called analytic Levi decomposition for any analytic family of vector fields.

Will have to read some books on functional analysis ?!

To think of it, my “near-Casimir” operator always admits a formal inverse, i.e. an inverse in a formal completion of the Hilbert space, whose norm might be infinite. In order to show that it has a true inverse, I just need to prove the following inequality: there exists a positive constant C  such that

$\|\Gamma(x)\| \geq C\|x\|$

for all x. I know that this inequality is true for the unperturbed operator (the true Casimir). But why it remains true for the perturbed operator ? It seems reasonable. But the difficulty here lies in the fact that the true Casimir doesn’t act on my Hilbert space, only the perturbed one acts ! (i.e. the Hilbert space is also modified). OK, the good news is that my Hilbert space is a subspace of the big space where the true Casimir acts (but the true Casimir action doesn’t preserve my Hilbert space).

$x \in H_1 \subset H_2$

$\gamma(x) \in H_2$ but not in $H_1$ in general

$\Gamma(x) \in H_1$

$\Gamma$ is “near” $\gamma$ in some sense, but since both operators are unbounded, I can’t say that their difference has a small norm.