[Quelques problèmes liés aux symboles de Berezin]
Le problème suivant est formulé par Zorboska [Proc. Amer. Math. Soc. 131 (2003) 793–800] : les symboles de Berezin d'un opérateur borné sur l'espace de Bergman ont-ils nécessairement des limites radiales presque partout sur le cercle unité ? Dans cet article, nous donnons une réponse négative à cette question en exhibant une classe concrète d'opérateurs diagonaux pour lesquels une telle limite n'existe en aucun point du cerle unité. Nous obtenons un résultat semblable dans le cas des espaces de Hardy sur le dique unité D. De plus nous donnons une nouvelle preuve, utilisant les notions de noyaux reproduisants et de symboles de Berezin, du célèbre théorème de Beurling concernant les sous-espaces z-invariants de .
The following problem was formulated by Zorboska [Proc. Amer. Math. Soc. 131 (2003) 793–800]: It is not known if the Berezin symbols of a bounded operator on the Bergman space must have radial limits almost everywhere on the unit circle. In this Note we solve this problem in the negative, showing that there is a concrete class of diagonal operators for which the Berezin symbol does not have radial boundary values anywhere on the unit circle. A similar result is also obtained in case of the Hardy space over the unit disk D. Moreover, we give an alternative proof to the famous theorem of Beurling on z-invariant subspaces in the Hardy space , using the concepts of reproducing kernels and Berezin symbols.
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@article{CRMATH_2005__340_10_715_0,
author = {Mubariz T. Karaev},
title = {On some problems related to {Berezin} symbols},
journal = {Comptes Rendus. Math\'ematique},
pages = {715--718},
publisher = {Elsevier},
volume = {340},
number = {10},
year = {2005},
doi = {10.1016/j.crma.2005.04.021},
language = {en},
}
Mubariz T. Karaev. On some problems related to Berezin symbols. Comptes Rendus. Mathématique, Volume 340 (2005) no. 10, pp. 715-718. doi : 10.1016/j.crma.2005.04.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.021/
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