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

We study closed subspaces of L-2(X), where (X, mu) is a sigma-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group Gamma 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-2 (X) generated by the action of the unitary representation under consideration on a countable set of functions in L-2 (X). Finally, closed subspaces of L-2(G), for G being an LCA group, that are invariant under translations by elements on a closed subgroup Gamma 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. (C) 2015 Elsevier Inc. All rights reserved.