Prove: Gravitation operator is invertible. Let $(M,g)$ be a Riemannian manifold and $\Gamma(S^2M)$ the space of symmetric 2-covariant tensors. Define the **gravitation operator** as the map \begin{align*} G:\Gamma(S^2M)&\rightarrow\Gamma(S^2M)\\\ h&\mapsto Gh:=h-\frac{1}{2}(\text{tr}_gh)g, \end{align*} where the metric trace $\text{tr}_gh=g^{pq}h_{pq}$. _Question_ : Is the gravitation operator invertible, and if so how does one show this?

Setting $k=Gh$, see if you can write $\mathrm{tr}_gk$ in terms of $\mathrm{tr}_gh$ (and thus vice versa). You should find a relatively clean expression, and from there it's just simple algebra: suppose $\mathrm{tr}_gh = f(\mathrm{tr}_gk)$ for some $f$. Then $k_{ij}=h_{ij}-\frac12f(\mathrm{tr}_gk)g_{ij}$, so $h_{ij}=k_{ij}+\frac12f(\mathrm{tr}_gk)g_{ij}$.