Abstract

A classic application of Wiener–Hopf is the computation of the UL factorisation of Toeplitz operators. I’ll review numerical approaches to computing this factorisation, both based on the Wiener–Hopf representation and based on a simple numerical linear algebra interpretation, leading to numerical techniques that also work for finite-dimensional perturbations of Toeplitz operators. Going beyond this classical setting are the (infinite) QR and QL factorisations, the latter having the same feature as the UL factorisation that one needs to kick off the factorisation from the bottom-right of the matrix, which in this case is at infinity! Extensions to block Toeplitz operators and their perturbation will further be discussed. Applications include exploring the continuous spectrum of Toeplitz operators, computing orthogonal polynomials, and topological insulators with defects.