A comparative study for the spectral properties of Toeplitz and Hankel operators

Year 2024, Volume: 53 Issue: 3, 690 - 703, 27.06.2024

Ayşe Güven Sarıhan

https://doi.org/10.15672/hujms.1241656

Abstract

In this introductory review, we study Hankel and Toeplitz operators considering them as acting on certain spaces of analytic functions, namely Hardy spaces and compare their spectral properties such as their compactness criteria. In contrast to Toeplitz operators, the symbol of a Hankel operator is not uniquely determined by the operator. We also connect Toeplitz operators with Fredholm operators and give some of the most beautiful properties of Toeplitz operators such as the essential spectrum of Toeplitz operator with continuous symbol and the index of Toeplitz operator introducing Fredholm operators firstly.

Keywords

Hankel operators, Toeplitz operators, Fredholm operators

References

• [1] P. Ahern, E.H. Youssfi and K. Zhu, Compactness of Hankel operators on Hardy- Sobolev spaces of the polydisk, J. Operator Theory 61 (2), 301-312, 2009.
• [2] A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Berlin, 1990.
• [3] E.B. Davies, Linear Operators and Their Spectra, Cambridge University Press, 2007.
• [4] H. Edelsbrunner, Geometry and Topology for Mesh Generation, Cambridge University Press, 2001.
• [5] D.E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
• [6] M. Engliš and G. Zhang, Hankel operators and the Dixmier trace on the Hardy space, Journal of the London Mathematical Society 94 (2), 337-356, 2016.
• [7] R.E. Harte, W.Y. Lee and L.L. Littlejohn, On generalized Riesz points, J. Operator Theory 47, 187-196, 2002.
• [8] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, 1962.
• [9] J.K. Hunter and B. Nachtergaele, Applied Analysis, World Scientific, 2001.
• [10] Y. Katznelson, An introduction to harmonic analysis, third ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.
• [11] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley and Sons, Canada, 1989.
• [12] J.R. Partington, An Introduction to Hankel Operators, Cambridge University Press, 1988.
• [13] V.V. Peller, Hankel Operators and Their Applications, Springer, 2003.
• [14] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1970.
• [15] Y. Soykan, Fonksiyonel analiz: Çözümlü alıştırmaları, Nobel Yayın Dağıtım, 2012.
• [16] Y. Soykan, Fonksiyonel analiz, Nobel akademik yayıncılık, 2016.