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 one-to-one correspondence between SHIFT preserving operators and range operators on L2 (G) where Gis a locally compact abelian group.