  We use shift-invariant subspaces of the Hardy space on the bidisk to provide
an elementary proof of the Agler Decomposition Theorem. We observe that these
shift-invariant subspaces are specific cases of Hilbert spaces that can be
defined from Agler decompositions and analyze the properties of such Hilbert
spaces. We then restrict attention to rational inner functions and show that
the shift-invariant subspaces provide easy proofs of several known results
about decompositions of rational inner functions. We use our analysis to obtain
a result about stable polynomials on the polydisk.
