The structure of group preserving operators

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of $L^2(\mathfrak{S})$ where $\mathfrak{S}$ is a second countable LCA group. The subspaces where the operators act are invariant under the action of a group $\Gamma$ which is a semi-direct product of a uniform lattice of $\mathfrak{S}$ with a discrete group of automorphisms. This class includes the crystal groups which are important in applications as models for images. The operators are assumed to be $\Gamma$ preserving. i.e. they commute with the action of $\Gamma$. In particular we obtain a spectral decomposition for these operators. This generalizes recent results on shift-preserving operators acting on lattice invariant subspaces where $\mathfrak{S}$ is the Euclidean space.