The Zak transform and the structure of spaces invariant by the action of an LCA group

We study closed subspaces of L2(X)L2(X), where (X,μ)(X,μ) is a σ  -finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group Γ on XX. We provide a complete description for these spaces in terms of range functions and a suitable generalized Zak transform. As an application of our main result, we prove a characterization of frames and Riesz sequences in L2(X)L2(X) generated by the action of the unitary representation under consideration on a countable set of functions in L2(X)L2(X). Finally, closed subspaces of L2(G)L2(G), for G being an LCA group, that are invariant under translations by elements on a closed subgroup Γ of G are studied and characterized. The results we obtain for this case are applicable to cases where those already proven in [5] and [7] are not.