  A seminal result of Agler proves that the natural de Branges-Rovnyak kernel
function associated to a bounded analytic function on the bidisk can be
decomposed into two shift-invariant pieces. Agler's decomposition is
non-constructive-a problem remedied by work of Ball-Sadosky-Vinnikov, which
uses scattering systems to produce Agler decompositions through concrete
Hilbert space geometry. This method, while constructive, so far has not
revealed the rich structure shown to be present for special classes of
functions-inner and rational inner functions. In this paper, we show that most
of the important structure present in these special cases extends to general
bounded analytic functions. We give characterizations of all Agler
decompositions, we prove the existence of coisometric transfer function
realizations with natural state spaces, and we characterize when Schur
functions on the bidisk possess analytic extensions past the boundary in terms
of associated Hilbert spaces.
