The structure of group preserving operators

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