This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator.

Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator of degree $r$ with coefficients $f \in \Gamma(L(J^r E, F))$, where $J^r E$ is the $r$-th jet bundle of $E$. Given coefficients $f_0$, let $Z^\pm$ be finite-dimensional vector spaces, $T^+: Z^+ \to \Gamma(F)$ and $T^-: \Gamma(E) \to Z^-$ be continuous linear maps such that $$\begin{pmatrix}T_{f_0} & T^+ \\ T^- & 0\end{pmatrix}: \Gamma(E) \times Z^+ \to \Gamma(F) \times Z^- $$ is invertible (for example, we may choose $Z^- = \ker T_{f_0}$ and $Z^+ = \mathrm{coker} \, T_{f_0}$ with $T^\pm$ being the canonical projection/injection). In Theorem II.3.3.3 (p. 157f) of Hamilton's work on the Nash-Moser inverse function theorem it is claimed (and attributed to "standard Fredholm theory") that there exists an open neighborhood $U$ of $f_0$ in $\Gamma(L(J^r E, F))$ such that $$\begin{pmatrix}T_{f} & T^+ \\ T^- & 0\end{pmatrix}$$ is invertible for all $f \in U$.

Side question:

Do you know a reference for the existence of such an open neighborhood $U$?

Real question:

What is the proper generalization of this picture using extended invertible maps to elliptic complexes $\Gamma(E) \overset{T_f}{\to} \Gamma(F) \overset{S_g}{\to} \Gamma(G)$?

The obvious guess is that one goes over to an extended sequence $$\Gamma(E) \times Z^+ \to \Gamma(F) \times H \to \Gamma(G) \times Z^-,$$ which is exact. Is this worked-out somewhere? (I feel like this is a basic statement in elliptic theory but I couldn't find a reference for both questions.)

all$f$, for which $\tilde{L}(f)$ is an iso, is open (in the same way as when proving that the subspace of Fredholm operators is open in bounded operators). $\endgroup$ – user_1789 Oct 3 '18 at 21:29