A survey of asymptotics of determinants for structured matrices

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.