Generalized Krein algebras and asymptotics of Toeplitz determinants

TL;DR

This paper surveys generalized Krein algebras and their role in analyzing the asymptotic behavior of Toeplitz determinants, extending classical results to new parameter ranges.

Contribution

It extends the analysis of Toeplitz determinants in generalized Krein algebras to the case where 0<λ<1, broadening the scope of previous theorems.

Findings

Extended Szegő limit theorem to 0<λ<1

Generalized Krein algebras form Banach algebras in new parameter ranges

Established asymptotic formulas for Toeplitz determinants with symbols in these algebras

Abstract

We give a survey on generalized Krein algebras $K_{p,q}^{\alpha,\beta}$ and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that $K_{2,2}^{1/2,1/2}$ is a Banach algebra. Subsequently, Widom proved the strong Szeg\H{o} limit theorem for block Toeplitz determinants with symbols in $(K_{2,2}^{1/2,1/2})_{N\times N}$ and later two of the authors studied symbols in the generalized Krein algebras $(K_{p,q}^{\alpha,\beta})_{N\times N}$, where $\lambda:=1/p+1/q=\alpha+\beta$ and $\lambda=1$. We here extend these results to $0<\lambda<1$. The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.