  We obtain rapidly convergent series expansions of resolvents of operators
taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is
a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an
operator that acts locally in real space. Such resolvents arise naturally when
one wants to solve any of the large class of linear physical equations surveyed
in Parts I, II, III, and IV that can be reformulated as problems in the
extended abstract theory of composites. We show how the information about the
spectrum of ${\bf A}$ can be used to greatly improve the convergence rate.
