  We consider 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 review how $Q^*$-convex operators can be used to bound the spectrum of ${\bf
A}$. Then, based on the Cherkaev-Gibiansky transformation and subsequent
developments, that we reformulate, we obtain for non-Hermitian ${\bf B}$ a
Stieltjes type integral representation for the resolvent $(z_0{\bf I}-{\bf
A})^{-1}$. The representation holds in the half plane
$\Re(e^{i\vartheta}z_0)>c$, where $\vartheta$ and $c$ are such that $c{\bf
I}-[e^{i\vartheta}{\bf B}+e^{-i\vartheta}{\bf B}^\dagger]$ is positive definite
(and coercive).
