Construction of some classes of commutative Banach and <i>C</i><sup>⋆</sup>-algebras of Toeplitz operators

Hassan Ahmad Issa

doi:10.1016/j.crma.2019.04.003

Functional analysis/Complex analysis

Construction of some classes of commutative Banach and C^⋆-algebras of Toeplitz operators
[Une construction de quelques classes d'algèbre commutatives de Banach et de C^⋆ via les opérateurs de Toeplitz]

Présenté par : the Editorial Board

Hassan Ahmad Issa ¹

¹ Lebanese University Section I and L.I.U, Lebanon

Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 389-394.

Résumé
Abstract

Dans cet article, nous construisons des algèbres commutatives générées par les opérateurs de Toeplitz sur l'espace de Segal–Bargmann $H_{s}^{2} (C^{n})$ et sur les espaces true-k-Fock. En utilisant une méthode analogue à celle mise en œuvre sur les algèbres de Banach commutatives construites par N. Vasilevski, nous obtenons une algèbre commutative sur $H_{s}^{2} (C^{n})$ formée uniquement par les opérateurs de Toeplitz, et une formule de composition est aussi obtenue. En utilisant une extension naturelle de la notion d'opérateur de Toeplitz, nous introduisons les opérateurs « true-k-Toeplitz » agissant sur les espaces true-k-Fock. Nous fournissons une $C^{⋆}$ -algèbre commutative générée par ces opérateurs, dont les symboles dépendent des parties réelle et imaginaire de la variable complexe dans un certain sens.

In the present paper, we construct commutative algebras generated by Toeplitz operators on the Segal–Bargmann space $H_{s}^{2} (C^{n})$ and on the true-k-Fock spaces. Analogous to the commutative Banach algebras constructed by N. Vasilevski for the case of the unit ball, we obtain a commutative algebra on $H_{s}^{2} (C^{n})$ formed only by Toeplitz operators and a composition formula is obtained. Employing a natural extension for the notion of Toeplitz operators, we introduce “true-k-Toeplitz operators” acting on the true-k-Fock spaces. We provide a commutative $C^{⋆}$ -algebra generated by such operators, whose symbol depends on the real and imaginary parts of the complex variable in a certain sense.

Reçu le : 2018-10-21
Accepté le : 2019-04-08
Publié le : 2019-04-01

DOI : 10.1016/j.crma.2019.04.003

Affiliations des auteurs :

Hassan Ahmad Issa ¹

¹ Lebanese University Section I and L.I.U, Lebanon

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     author = {Hassan Ahmad Issa},
     title = {Construction of some classes of commutative {Banach} and {\protect\emph{C}\protect\textsuperscript{\ensuremath{\star}}-algebras} of {Toeplitz} operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {389--394},
     publisher = {Elsevier},
     volume = {357},
     number = {4},
     year = {2019},
     doi = {10.1016/j.crma.2019.04.003},
     language = {en},
}

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Hassan Ahmad Issa. Construction of some classes of commutative Banach and C^⋆-algebras of Toeplitz operators. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 389-394. doi : 10.1016/j.crma.2019.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.04.003/

Bibliographie
Cité par

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[2] W. Bauer; N. Vasilevski Commutative Toeplitz Banach algebras on the ball and quasi-nilpotent group action, Integral Equ. Oper. Theory, Volume 72 (2012) no. 2, pp. 223-240

[3] K. Esmeral; N. Vasilevski $C^{⋆}$ -algebra generated by horizontal Toeplitz operators on the Fock space, Bol. Soc. Mat. Mexicana, Volume 22 (2016), pp. 567-582

[4] S. Grudsky; R. Quiroga-Barranco; N. Vasilevski Commutative $C^{⁎}$ -algebras of Toeplitz operators and quantization on the unit disc, J. Funct. Anal., Volume 234 (2006), pp. 1-44

[5] H. Issa Compact Toeplitz operators for weighted Bergman spaces on bounded symmetric domains, Integral Equ. Oper. Theory, Volume 70 (2011), pp. 569-582

[6] H. Issa The Analysis of Toeplitz Operators, Commutative Toeplitz Algebras and Applications to Heat Kernel Constructions, Georg-August Universität, Göttingen, Germany, 2012 (Ph.D. thesis Handle: 11858/00-1735-0000-000D-F066-5)

[7] R. Quiroga-Barranco; A. Sánchez-Nungaray Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian frames, J. Oper. Theory, Volume 71 (2014) no. 1, pp. 199-222

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[9] J. Ramírez-Ortega; A. Sánchez-Nungaray Toeplitz operators with vertical symbols acting on the poly-Bergman spaces of the upper half-plane, Complex Anal. Oper. Theory, Volume 9 (2015) no. 8, pp. 1801-1817

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[15] N. Vasilevski On Toeplitz operators with quasi-radial and pseudo-homogeneous symbols (M. Pereyra; S. Marcantognini; A. Stokolos; W. Urbina, eds.), Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory, vol. 2, Association for Women in Mathematics Series, vol. 5, Springer, Cham, Switzerland, 2017

Cité par Sources :

^☆ The results presented in the article were taken from the author's Ph.D. thesis while at Georg-August University of Göttingen (Germany), back in 2012. The project was supported by an “Emmy-Noether scholarship” of DFG (Deutsche Forschungsgemeinschaft).

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