  This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known
integrable discretization of the Non-linear Schrodinger system) can be
explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian
with respect to both a standard, local Poisson operator J and a new non-local,
skew, almost Poisson operator K, on the appropriate space; (b) can be
recursively generated from a recursion operator R (obtained by composing K and
the inverse of J.) In addition, the proof of these facts relies upon two new
pivotal resolvent identities which suggest a general method for uncovering
bi-Hamiltonian structures for other families of discrete, integrable equations.
