The Zak Transform and the Structure of Spaces Invariant by the Action of an Lca Group

We study closed subspaces of L(X ), where (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 X . 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 L(X ) generated by the action of the unitary representation under consideration on a countable set of functions in L(X ). Finally, closed subspaces of L(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, 7] are not.