  Following advances in the abstract theory of composites, we develop rapidly
converging series expansions about $z=\infty$ for the resolvent ${\bf
R}(z)=[z{\bf I}-{\bf P}^\dagger{\bf Q}{\bf P}]^{-1}$ where ${\bf Q}$ is an
orthogonal projection and ${\bf P}$ is such that ${\bf P}{\bf P}^\dagger$ is an
orthogonal projection. It is assumed that the spectrum of ${\bf P}^\dagger{\bf
Q}{\bf P}$ lies within the interval $[z^-,z^+]$ for some known $z^+\leq 1$ and
$z^-\geq 0$ and that the actions of the projections ${\bf Q}$ and ${\bf P}{\bf
P}^\dagger$ are easy to compute. The series converges in the entire $z$-plane
excluding the cut $[z^-,z^+]$. It is obtained using subspace substitution,
where the desired resolvent is tied to a resolvent in a larger space and ${\bf
Q}$ gets replaced by a projection $\underline{{\bf Q}}$ that is no longer
orthogonal. When $z$ is real the rate of convergence of the new method matches
that of the conjugate gradient method.
