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 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?