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?