Generalized Krein algebras and asymptotics of Toeplitz determinants

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 Szegő 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.

27 pages