Spectral properties of the adjoint operator and applications

We present some spectral properties of the adjoint operator corresponding to an admissible dilatation vector field and its perturbations. Next, we apply these results via the Nash–Moser function inverse theorem to show that the group $G$ of diffeomorphisms on the Euclidean space $R^n$ which are 1-time flat, close to the identity and of small support acts transitively on the affine space of appropriate perturbations of the dilation vector field $X_o$.