We investigate shift invariant subspaces of L2(G), where G is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift invariant subspaces of L2(G) in terms of range functions. Finally, we investigate shift preserving operators on locally compact abelian groups. We show that there is a oneto- one correspondence between shift preserving operators and range operators on L2(G) where G is a locally compact abelian group.

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