The Second Order Spectrum and Optimal Convergence

TL;DR

This paper extends the second order relative spectra method from self-adjoint to normal operators, achieving optimal convergence rates for eigenvalues and eigenspaces, with improved accuracy and new convergence results for eigenspaces.

Contribution

It introduces the convergence of eigenspaces into the second order spectra framework and improves eigenvalue convergence rates for normal operators.

Findings

Convergence to eigenspaces is established.

Eigenvalue convergence rate is improved by an order of magnitude.

Method extended from self-adjoint to normal operators.

Abstract

The method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self-adjoint operator. We extend the method to normal operators and find optimal convergence rates for eigenvalues and eigenspaces. The convergence to eigenspaces is new, while the convergence rate for eigenvalues improves on the previous estimate by an order of magnitude.