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# 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 Szego 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.

### Article Information

 Title Generalized Krein algebras and asymptotics of Toeplitz determinants Source Methods Funct. Anal. Topology, Vol. 13 (2007), no. 3, 236-261 MathSciNet MR2356757 Copyright The Author(s) 2007 (CC BY-SA)

### Authors Information

A. Bottcher
Fakultat fur Mathematik, Technische Universitat Chemnitz, D-09107, Chemnitz, Germany

A. Karlovich
Departamento de Matem\'atica, Instituto Superior Tecnico, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal

B. Silbermann
Fakultat fur Mathematik, Technische Universitat Chemnitz, D-09107, Chemnitz, Germany

### Citation Example

A. Böttcher, A. Karlovich, and B. Silbermann, Generalized Krein algebras and asymptotics of Toeplitz determinants, Methods Funct. Anal. Topology 13 (2007), no. 3, 236-261.

### BibTex

@article {MFAT406,
AUTHOR = {Böttcher, A. and Karlovich, A. and Silbermann, B.},
TITLE = {Generalized Krein algebras and asymptotics of Toeplitz determinants},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {13},
YEAR = {2007},
NUMBER = {3},
PAGES = {236-261},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=406},
}