  Necessary and sufficient conditions are presented for a measure to be the
spectral measure of a finite range perturbation of a Jacobi or CMV operator
from a finite gap isospectral torus. The special case of eventually periodic
operators solves an open problem of Simon [25, D.2.7].
  We also solve the inverse resonance problem: it is shown that an operator is
completely determined by the set of its eigenvalues and resonances, and we
provide necessary and sufficient conditions on their configuration for such an
operator to exist.
