Rapidly convergent series expansions for a class of resolvents

Abstract

Following advances in the abstract theory of composites, we develop rapidly converging series expansions about $z=\infty$ for the resolvent $\mathbf{R}(z)=[z\mathbf{I}-{\mathbf{P}}^{†}\mathbf{Q}\mathbf{P}{]}^{-1}$ where $\mathbf{Q}$ is an orthogonal projection and $\mathbf{P}$ is such that $\mathbf{P}{\mathbf{P}}^{†}$ is an orthogonal projection. It is assumed that the spectrum of ${\mathbf{P}}^{†}\mathbf{Q}\mathbf{P}$ lies within the interval $[z^{-},z^{+}]$ for some known $z^{+}\le 1$ and $z^{-}\ge 0$ and that the actions of the projections $\mathbf{Q}$ and $\mathbf{P}{\mathbf{P}}^{†}$ 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 $\mathbf{Q}$ gets replaced by a projection $\underline{\mathbf{Q}}$ that is no longer orthogonal. When $z$ is real the rate of convergence of the new method matches that of the conjugate gradient method.

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