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A note on the space of all Toeplitz operators

Part of: Spaces and algebras of analytic functions Individual linear operators as elements of algebraic systems Special classes of linear operators Linear function spaces and their duals Real functions: Miscellaneous topics

Published online by Cambridge University Press:  12 November 2024

Michał Jasiczak Open the ORCID record for Michał Jasiczak [Opens in a new window]
Michał Jasiczak*
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61–614 Poznań, Poland
*
e-mail: [email protected]
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Abstract

We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

Keywords

Toeplitz operatorToeplitz algebrareal analytic functionholomorphic functionsymbol maptopology of uniform convergence on bounded sets

MSC classification

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Secondary: 26E05: Real-analytic functions 30H99: None of the above, but in this section 47C05: Operators in algebras 46E10: Topological linear spaces of continuous, differentiable or analytic functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 4 , December 2024 , pp. 1123 - 1140
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Axler, S., Toeplitz operators. In A glimpse at Hilbert space operators, Oper. Theory Adv. Appl., Vol. 207, Birkhäuser Verlag, Basel, 2010, 125–133.CrossRefGoogle Scholar
Bierstedt, K. D., An introduction to locally convex inductive limits. In ICPAM Lecture Notes, World Scientific Publishing Co., Singapore, 1988, 35–133.Google Scholar
Bonet, J., Jornet, D. and Sevilla-Peris, P., Function spaces and operators between them, RSME Springer Ser. 11, Springer, Cham, 2023.Google Scholar
Böttcher, A. and Silbermann, B., Analysis of Toeplitz operators, Springer Monogr. Math., Springer-Verlag, Berlin, 2006.Google Scholar
Domański, P., Notes on real analytic functions and classical operators . In Topics in complex analysis and operator theory, Contemp. Math., Vol. 561, Amer. Math. Soc., Providence, RI, 2012, 3–47.CrossRefGoogle Scholar
Domański, P. and Jasiczak, M., Toeplitz operators on the space of real analytic functions. Fredholm property . Banach J. Math. Anal. 12(2018), 31–67.CrossRefGoogle Scholar
Domański, P. and Langenbruch, M., Representation of multipliers of real analytic functions. Analysis 31(2012), 1001–1026.Google Scholar
Domański, P. and Vogt, D., Linear topological properties of the space of analytic functions on the real line , North-Holland Math. Stud., Vol. 189 , North-Holland Publishing Co., Amsterdam, 2001, 113–132.Google Scholar
Domański, P. and Vogt, D., The space of real-analytic functions has no basis . Studia Math. 142(2000), 187–200.CrossRefGoogle Scholar
Douglas, R. G., Banach algebra techniques in operator theory , Pure and Applied Mathematics, Vol. 49, Academic Press, New York, London, 1972.Google Scholar
Gelfond, A. O., Differenzenrechnung. Hochschulbücher für Mathematik Herausgegeben von H. Grell , Maruhn, K. und Rinow, W., Band 41, VEB Deutscher Verlag der Wissenschaften, Berlin, 1958.Google Scholar
Hörmander, L., On the existence of real analytic solutions to partial differential equations with constant coefficients . Invent. Math. 21(1973), 151–182.CrossRefGoogle Scholar
Jasiczak, M., Coburn-Simonenko theorem and invertibility of Toeplitz operators on the space of real analytic functions . J. Operator Theory 79(2018), 327–344.Google Scholar
Jasiczak, M., Semi-Fredholm Toeplitz operators on the space of real analytic functions . Studia Math. 252(2020), 213–250.CrossRefGoogle Scholar
Jasiczak, M. and Golińska, A., One-sided invertibility of Toeplitz operators on the space of real analytic functions on the real line . Integral Equations Operator Theory 92(2020), Paper No. 6.CrossRefGoogle Scholar
Jasiczak, M, Toeplitz operators on the space of all holomorphic functions on finitely connected domains . Rev. R. Acad. Cienc. Exactas Fś. Nat. Ser. A Mat. RACSAM. 117(2023), no. 1, Paper No. 47. 43 pp.Google Scholar
Jasiczak, M., On the Toeplitz algebra in the case of all entire functions and all functions holomorphic in the unit disc . Complex Anal. Oper. Theory 18(2024), no. 3, Paper No. 44.CrossRefGoogle Scholar
Jasiczak, M. , The symbol map and the Fredholm property of the aggregate Toeplitz operators on the space of all real analytic functions on the real line. submitted.Google Scholar
Köthe, G., Topological vector spaces. I, Grundlehren der Mathematischen Wissenschaften, Vol. 159, Springer-Verlag, New York-Berlin, 1979.Google Scholar
Köthe, G., Topological vector spaces. II , Grundlehren der Mathematischen Wissenschaften, Vol. 237, Springer-Verlag, New York, Inc., New York, 1969.Google Scholar
Markushevich, A. I., Theory of functions of a complex variable. Vol. I, II, III, AMS Chelsea Publishing, American Mathematical Society, Providence, RI, 2011.Google Scholar
Martineau, A., Sur la topologie des espaces de fonctions holomorphes . Math. Ann. 163(1966), 62–88.CrossRefGoogle Scholar
Meise, R. and Vogt, D., Introduction to functional analysis , Oxford Graduate Texts in Mathematics, 2, The Clarendon Press, Oxford University Press, New York, 1997.Google Scholar
Mujica, J., A Banach-Dieudonné theorem for germs of holomorphic functions . J. Funct. Anal. 57(1984), 31–48.CrossRefGoogle Scholar
Nikolski, N., Toeplitz matrices and operators , Cambridge Stud. Adv. Math., Vol. 182, Cambridge University Press, Cambridge, 2020.Google Scholar
Nikolski, N., Operators, functions, and systems: an easy reading. Vol. 1, Math. Surveys Monogr., Vol. 92, American Mathematical Society, Providence, RI, 2002.Google Scholar
Vogt, D., A fundamental system of seminorms for $A(K)$ , unpublished.http://www2.math.uni-wuppertal.de/vogt/preprints/seminorms.pdf (access 01.06.2022).Google Scholar