We discuss the algebro-geometric initial value problem for the Toda hierarchy
with complex-valued initial data and prove unique solvability globally in time
for a set of initial (Dirichlet divisor) data of full measure. To this effect we
develop a new algorithm for constructing stationary complex-valued
algebro-geometric solutions of the Toda hierarchy, which is of independent
interest as it solves the inverse algebro-geometric spectral problem for
generally non-self-adjoint Jacobi operators, starting from a suitably chosen set
of initial divisors of full measure. Combined with an appropriate first-order
system of differential equations with respect to time (a substitute for the
well-known Dubrovin equations), this yields the construction of global
algebro-geometric solutions of the time-dependent Toda hierarchy. The inherent
non-self-adjointness of the underlying Lax (i.e., Jacobi) operator associated
with complex-valued coefficients for the Toda hierarchy poses a variety of
difficulties that, to the best of our knowledge, are successfully overcome here
for the first time. Our approach is not confined to the Toda hierarchy but
applies generally to $1+1$-dimensional completely integrable discrete soliton
equations.