This question is a continuation of the discussion which can be found here. From the exterior derivative one constructs an operator $S$ with the property that the graph of $S$ is the (closure of) the image of $d$. More precisely, $S$ acts as follows: $Sh=0$ when $h$ is a harmonic form and $S(\frac{1-\gamma}{2}d\alpha)=\frac{1+\gamma}{2}d\alpha$ for $\omega$ orthogonal to harmonic form (any such $\omega$ can be expressed as $\frac{1-\gamma}{2}d\alpha$).
Is it clear from this description that $S$ is a pseudodifferential operator? How we can compute the symbol of $S$ in terms of the symbol of $d$ (and possibly $\gamma$ provided we know that $\gamma$ is pseudodifferential as well which is not clear for me).
