  We study the Poisson structure associated to the defocusing Ablowitz-Ladik
equation from a functional-analytical point of view, by reexpressing the
Poisson bracket in terms of the associated Caratheodory function. Using this
expression, we are able to introduce a family of compatible Poisson brackets
which form a multi-Hamiltonian structure for the Ablowitz-Ladik equation.
Furthermore, we show using some of these new Poisson brackets that the
Geronimus relations between orthogonal polynomials on the unit circle and those
on the interval define an algebraic and symplectic mapping between the
Ablowitz-Ladik and Toda hierarchies.
