Rapidly convergent series expansions for a class of resolvents

TL;DR

The paper develops rapidly convergent series for resolvents of the form $\mathbf{R}(z)=[z\mathbf{I}-\mathbf{P}^{\dagger}\mathbf{Q}\mathbf{P}]^{-1}$ by embedding the problem in a larger space and using subspace substitution with non-orthogonal projections. A central strategy maps the spectral interval $[z^-,z^+]$ to the unit disk via a sequence of fractional-linear transforms, producing series in a variable $v$ that converge quickly for $z$ outside the cut and attain a convergence rate matching conjugate-gradient methods for real $z$. The paper derives a hierarchy of accelerated series, including a projection-specific formulation with rate $\mu_3=|u|$ where $u=(\sqrt{\sigma}-1)/(\sqrt{\sigma}+1)$ and several variants that handle unknown or approximate spectral bounds, including non-orthogonal subspace substitutions that further improve bounds to $\mu_4\le\mu_2$. These expansions underpin efficient resolvent evaluations in problems from the abstract theory of composites, notably conductivity and Schrödinger-type equations, and offer practical pathways for adaptive bound estimation and numerical benchmarking against established iterative methods.

Abstract

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.