no. 11
Functional analysis
On the singular values of compact composition operators
[Sur les valeurs singulières des opérateurs de composition compacts]

Présenté par : Gilles Pisier

El-Fallah, Omar1 ; El Ibbaoui, Mohamed1
1 Laboratoire Analyse et Applications – URAC/03, Mohammed V University in Rabat, B.P. 1014, Rabat, Morocco
Comptes Rendus. Mathématique, Tome 354 (2016) no. 11, pp. 1087-1091.

Soit μ une mesure de Borel positive sur le disque unité et soit Tμ l'opérateur de Toeplitz associé à μ sur un espace de Bergman standard. Pour une fonction positive h satisfaisant des conditions de convexité, nous donnons des bornes inférieures et supérieures de la trace de h(Tμ). Ceci nous permet d'obtenir quelques estimations asymptotiques des valeurs propres de Tμ. Nous appliquons ces résultats pour les opérateurs de composition et donnons ensuite quelques exemples concrets.

Let μ be a positive Borel measure on the unit disc and let Tμ be the associated Toeplitz operator on a standard Bergman space. Under some convexity conditions on a positive function h, we give an upper and lower bounds of the trace of h(Tμ). As consequence, we give some asymptotic estimates of eigenvalues of Tμ. We also apply these results to composition operators and give some concrete examples.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.09.012
El-Fallah, Omar 1 ; El Ibbaoui, Mohamed 1

1 Laboratoire Analyse et Applications – URAC/03, Mohammed V University in Rabat, B.P. 1014, Rabat, Morocco 
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El-Fallah, Omar; El Ibbaoui, Mohamed. On the singular values of compact composition operators. Comptes Rendus. Mathématique, Tome 354 (2016) no. 11, pp. 1087-1091. doi : 10.1016/j.crma.2016.09.012. http://www.numdam.org/articles/10.1016/j.crma.2016.09.012/

[1] El-Fallah, O.; Mahzouli, H.; Marrich, I.; Naqos, H. Asymptotic behavior of eigenvalues of Toeplitz operators on the weighted analytic spaces, J. Funct. Anal., Volume 270 (2016) no. 12, pp. 4614-4630

[2] El-Fallah, O.; El Ibbaoui, M.; Naqos, H. Composition operators with univalent symbol in Schatten classes, J. Funct. Anal., Volume 266 (2014) no. 3, pp. 1547-1564

[3] El-Fallah, O.; Kellay, K.; Shabankhah, M.; Youssf, H. Level sets and composition operators on the Dirichlet space, J. Funct. Anal., Volume 260 (2011), pp. 1721-1733

[4] Hastings, W.W. A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc., Volume 52 (1975) no. 1, pp. 237-241

[5] Kellay, K.; Lefèvre, P. Compact composition operators on weighted Hilbert spaces of analytic functions, J. Math. Anal. Appl., Volume 386 (2012) no. 2, pp. 718-727

[6] Lefèvre, P.; Li, D.; Queffélec, H.; Rodríguez-Piazza, L. Some examples of compact composition operators on H2, J. Funct. Anal., Volume 255 (2008) no. 11, pp. 3098-3124

[7] Luecking, D. Trace ideal criteria for Toeplitz operators, J. Funct. Anal., Volume 73 (1987), pp. 345-368

[8] Pau, J.; Pérez, P.A. Composition operators acting on weighted Dirichlet spaces, J. Math. Anal. Appl., Volume 401 (2013) no. 2, pp. 682-694

[9] Queffélec, H.; Seip, K. Decay rates for approximation numbers of composition operators, J. Anal. Math., Volume 125 (2015) no. 1, pp. 371-399

[10] Shapiro, J.H. Composition Operators and Classical Function Theory, Springer Verlag, New York, 1993

[11] Zhu, K. Schatten class composition operators on weighted Bergman spaces of the disk, J. Operator Theory, Volume 46 (2001), pp. 173-181

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☆ Research partially supported by “Hassan II Academy of Science and Technology”.