Rapidly convergent series expansions for a class of resolvents

Following advances in the abstract theory of composites, we develop rapidly converging series expansions about z=∞ for the resolvent R(z)=[z I- PQ P]⁻¹ where Q is an orthogonal projection and P is such that P P^ is an orthogonal projection. It is assumed that the spectrum of PQ P lies within the interval [z⁻,z⁺] for some known z⁺≤ 1 and z⁻≥ 0 and that the actions of the projections Q and P P^ 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 Q gets replaced by a projection 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.