Abstract
The strong Szegö limit theorem describes the asymptotic behavior of determinants of finite Toeplitz matrices. This article is a survey that describes a simple proof of the strong Szegö limit theorem using some observations and results of Bill Helton. A proof of an exact identity for the determinants is also given along with some applications of the theorem and generalizations to other classes of operators.
Mathematics Subject Classification. 47B35, 82B44.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Basor, E.L., Ehrhardt, T.: Determinant computations for some classes of Toeplitz- Hankel matrices, Oper. Matrices 3 (2009), no. 2, 167-186.
Basor, E.L., Ehrhardt, T.: Asymptotic formulas for the determinants of a sum of finite Toeplitz and Hankel matrices, Math. Nachr. 228 (2001), 5-45.
Basor, E.L., Ehrhardt, T.: Asymptotic formulas for the determinants of symmetric Toeplitz plus Hankel matrices, In: Oper. Theory: Adv. Appl., Vol. 135, Birkhauser, Basel 2002, 61–90.
Basor, E.L., Ehrhardt, T: Asymptotics of determinants of Bessel operators, Commun. Math. Phys. 234 (2003), 491-516.
Basor, E.L., Ehrhardt, T., Widom, H.: On the determinant of a certain Wiener-Hopf + Hankel operator, Integral Equations Operator Theory 47, no. 3 (2003), 257-288.
Basor, E.L., Helton, J.W.: A new proof of the Szegö limit theorem and new results for Toeplitz operators with discontinuous symbol, J. Operator Th. 3 (1980) 23-39.
Basor, E.L., Widom, H.: On a Toeplitz determinant identity of Borodin and Ok- ounkov, Integral Equations Operator Theory 37, no. 4, 397-401 (2000).
Böttcher, A.: On the determinant formulas by Borodin, Okounkov, Baik, Deift and Rains, In: Oper. Theory Adv. Appl., Vol. 135, Birkhauser, Basel, 2002, 91-99.
Borodin, A., Okounkov, A.: A Fredholm determinant formula for Toeplitz determinants, Integral Equations Operator Theory 37, no. 4 (2000), 386-396.
Böttcher, A., Silbermann, B.: Analysis of Toeplitz operators, Springer, Berlin 1990.
Case, K.M., Geronimo, J.S.: Scattering theory and polynomials orthogonal on the unit circle. J. Math. Phys. 20, (1979), no. 2, 299-310.
Ehrhardt, T.: A new algebraic approach to the Szegö-Widom limit theorem. Acta Math. Hungar. 99 (2003), no. 3, 233-261.
Helton, J.W., Howe, R.E.: Integral operators: traces, index, and homology, proceed- ings of the conference on operator theory, Lecture Notes in Math., 345 Springer- Verlag, Berlin, (1973) 141-209.
Szegö, G.: Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion, Math. Ann., 76, (1915) 490-503.
Widom, H.: Asymptotic Behavior of Block Toeplitz Matarices and Determinants. II, Adv. in Math. 21, No. 1, (1976), 1–29.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this chapter
Cite this chapter
Basor, E. (2012). A Brief History of the Strong Szegö Limit Theorem. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0411-0_8
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0410-3
Online ISBN: 978-3-0348-0411-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)