Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$).
Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge relative to $A$. That is, the difference $\alpha = A - B$ satisfies $d^*_A \alpha = 0$ and is small in the $C^\infty$-topology.
I'm interested in the operator $d_B \, d^*_A = d_A \, d^*_A + [\alpha \wedge d^*_A \,\cdot ]$ on $2$-forms with values in the adjoint bundle. Of course, for $A = B$ this is just the covariant Laplacian $\triangle_A = d_A \, d^*_A$. So the operator $d_B \, d^*_A$ is elliptic and has vanishing index for close enough connections $A, B$.
Question: Does $\ker d_B \, d^*_A = \ker \triangle_A$ holds so that $d_B \, d^*_A$ induces a bijective operator $B^2_A \to B^2_A$ as it is the case for $\triangle_A$ (here $B^2_A = \mathrm{img} \,d_A$)?
What I have so far are estimates of the form $||\, [\alpha \wedge d^*_A \,\beta ]\,||_{L^p} \leq ||\,\alpha\,||_{W^{1,q}} \, ||\,\beta\,||_{W^{2,p}}$ for the perturbation term. However, I'm unable to convert these estimates into statements about the (co)kernel of $d_B \, d^*_A$. In my situation, $A$ is even a central Yang-Mills connection if this fact changes things.
