A survey of asymptotics of determinants for structured matrices

TL;DR

This survey reviews methods for deriving asymptotics of determinants of structured matrices, including Toeplitz and Hankel types, with some new results for specific operator classes and matrix symbols.

Contribution

It introduces new asymptotic results for Toeplitz plus Hankel operators and finite sections of functions of Toeplitz operators with matrix-valued symbols.

Findings

Asymptotics for finite Toeplitz matrices with smooth symbols

New results for Toeplitz plus Hankel operators with matrix symbols

First-time presentation of asymptotics for finite sections of functions of Toeplitz operators

Abstract

In this survey we show how to produce asymptotics of determinants of structured matrices using operator theory methods. We describe the asymptotics for finite Toeplitz matrices, finite Toeplitz plus Hankel matrices and generalizations of these, and also finite sections of functions of Toeplitz operators, all with smooth matrix-valued symbols. Many of these results are well-known. However, the results for certain Toeplitz plus Hankel operators with matrix-valued symbol are presented for the first time and the result for the finite sections of functions of Toeplitz operators is new.