A. Böttcher
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Generalized Krein algebras and asymptotics of Toeplitz determinants
A. Böttcher, A. Karlovich, B. Silbermann
MFAT 13 (2007), no. 3, 236-261
236-261
We give a survey on generalized Krein algebras Kα,βp,q and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that K1/2,1/22,2 is a Banach algebra. Subsequently, Widom proved the strong Szego limit theorem for block Toeplitz determinants with symbols in (K1/2,1/22,2)N×N and later two of the authors studied symbols in the generalized Krein algebras (Kα,βp,q)N×N, where λ:=1/p+1/q=α+β and λ=1. We here extend these results to 0<λ<1. The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.