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:


1. 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.
2. 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?



Green's operator of elliptic differential operator
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](http://aif.cedram.org/item?id=AIF_2007__57_7_2095_0). 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*.


