TITLE: Green's operator of elliptic differential operator
QUESTION [16 upvotes]: Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections of vector bundles $E\rightarrow M$ and $F\rightarrow M$ on a compact Riemannian manifold $(M,g)$ without boundary.
Question:  What is the elliptic operator's associated Green's operator?  
More concretely, $(M,g)=(\mathbb{S}^m,g)$ be the unit $m$-sphere with constant curvature =1 metric $g$, so $\text{Ricc}(g)=g$.  Also let $E=F=S^2\mathbb{S}^m$, the space of $2$-covariant tensors on $\mathbb{S}^m$.  Consider the operator:
\begin{align*}
P:\Gamma(S^2\mathbb{S}^m)&\rightarrow\Gamma(S^2\mathbb{S}^m)\\
h&\mapsto Ph_{ij}=\frac{1}{2}g^{kl}(\nabla_i\nabla_jh_{kl}+\nabla_k\nabla_lh_{ij}).
\end{align*}
The symbol is:
$$
\sigma_P(\xi)h_{ij}=\frac{1}{2}g^{kl}(\xi_i\xi_jh_{kl}+\xi_k\xi_lh_{ij}).
$$
We can show that $P$ is elliptic with index $=0$ and closed image of codimension $=1$.  
Question:  What is the Green's operator of $P$?
Reference request:  A good reference on Green's operators for elliptic partial differential operators would be welcomed.
Addition:  There is mention of Green's operator on pages 157 & 158 of Hamilton's 1982 paper The Inverse Function Theorem of Nash and Moser.  In the context of the above, it goes roughly as follows:
Choose finite dimensional vector spaces $N$ and $M$ and continuous linear maps
$$
j:\Gamma(S^2\mathbb{S}^m)\rightarrow N\qquad\text{and}\qquad i:M\rightarrow\Gamma(S^2\mathbb{S}^m).
$$
Define another map
\begin{align*}
L:\Gamma(S^2\mathbb{S}^m)\times M&\rightarrow\Gamma(S^2\mathbb{S}^m)\times N\\
(h,x)&\mapsto L(h,x)=(Ph+ix,jh),
\end{align*}
which is required to be invertible.  Then Theorem 3.3.3. states that:

The inverse map
$$
L^{-1}:\Gamma(S^2\mathbb{S}^m)\times N\rightarrow\Gamma(S^2\mathbb{S}^m)\times M
$$
is a smooth, tame, and linear.
For each $k\in\Gamma(S^2\mathbb{S}^m)$ there is a unique $h\in\text{kernal}\,j$ such that $Ph-k\in\text{image}\,i$.  The resulting map
\begin{align*}
G:\Gamma(S^2\mathbb{S}^m)&\rightarrow\text{kernal}\,j\\
k&\mapsto Gk=h
\end{align*}
is smooth, tame, and linear, and is called the Green's operator of $P$.

New question:  I'm wondering how to adapt this construction to the map $P$ above.  What would the finite-dimensional vector spaces $N$ and $M$, and the maps $j$ and $i$, be to make this construction work?

REPLY [10 votes]: If it exists, the inverse of an elliptic operator $P$ is its Green's operator. In general, an inverse does not exist, but a parametrix does. A parametrix is an operator $Q$ such that $PQ-I$ and $QP-I$ are compact operators. (I am assuming that $P$ is an elliptic operator on a closed manifold and acts between sections of vector bundles.) The existence of a parametrix is a basic application of pseudodifferential calculus, but it can also be deduced from elliptic estimates. 
The existence of a parametrix implies that $P$ is Fredholm, thus the kernel and cokernel of $P$ are finite-dimensional. Using this, one can make $P$ become part of an invertible operator
$$ \begin{pmatrix} P& R_-\\ R_+& 0\end{pmatrix}^{-1}= \begin{pmatrix}E&E_+\\ E_-&E_{-+}\end{pmatrix}, $$
where $R_-$ has finite-dimensional domain and $R_+$ finite-dimensional range. This setup is presented, with many useful applications given, by Sjöstrand and Zworski in their paper Elementary linear algebra for advanced spectral problems. See, in particular, the section 2.4 on analytic Fredholm theory. The operator $P$ is invertible if and only if the finite-dimensional operator $E_{-+}$ is, and $P^{-1}=E-E_+ E_{-+}^{-1}E_-$. In the context of Theorem 3.3.3 of Hamilton's paper on the inverse function theorem set $P=L(f)$, $R_+=j$, and $R_-=i$. He calls $G(f)=E$ the Green's operator, which it is when he is allowed to ``forget'' his spaces $M$ and $N$. The finite-dimensional spaces $M$ and $N$ can be chosen as, respectively, the cokernel and the kernel of $P$. It may be useful to choose them larger if it is desired to have them idependent of $f$ in $P=L(f)$.
Decomposing with respect to kernel and cokernel is standard in Fredholm theory. In Deane Yang's answer the decomposition is interior relative to the spaces while here, and in Hamilton's paper, it is exterior.