[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.
Accepté le :
Publié le :
@article{CRMATH_2005__340_10_715_0,
author = {Karaev, Mubariz T.},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2005.04.021/}
}
TY - JOUR AU - Karaev, Mubariz T. TI - On some problems related to Berezin symbols JO - Comptes Rendus. Mathématique PY - 2005 SP - 715 EP - 718 VL - 340 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2005.04.021/ DO - 10.1016/j.crma.2005.04.021 LA - en ID - CRMATH_2005__340_10_715_0 ER -
Karaev, Mubariz T. On some problems related to Berezin symbols. Comptes Rendus. Mathématique, Tome 340 (2005) no. 10, pp. 715-718. doi : 10.1016/j.crma.2005.04.021. http://www.numdam.org/articles/10.1016/j.crma.2005.04.021/
[1] The Berezin transform on the Toeplitz algebra, Studia Math., Volume 127 (1998), pp. 113-136
[2] Toeplitz operators on the Segal–Bergman space, Trans. Amer. Math. Soc., Volume 301 (1987), pp. 813-829
[3] On two problems concerning linear transformations in Hilbert space, Acta Math., Volume 81 (1949), pp. 239-255
[4] Uniqueness theorems for analytic vector-valued functions, J. Math. Sci. (New York), Volume 101 (2000), pp. 3193-3210 (Translation from Zap. Nauch. Semin. POMI, 247, 1997, pp. 242-267)
[5] Toeplitz operators in n-dimensions, Integral Equations Operator Theory, Volume 7 (1984), pp. 145-205
[6] The core function of submodules over the bidisk, Indiana Univ. Math. J., Volume 53 (2004), pp. 205-222
[7] A Hilbert Space Problem Book, Springer-Verlag, New York, 1982
[8] Berezin symbols and Schatten–von Neumann classes, Math. Notes, Volume 72 (2002), pp. 185-192 (Translated from Mat. Zametki, 72, 2002, pp. 207-215)
[9] Functional analysis proofs of Abel's theorems, Proc. Amer. Math. Soc., Volume 132 (2004), pp. 2327-2329
[10] Some results related with statistical convergence and Berezin symbols, J. Math. Anal. Appl., Volume 299 (2004), pp. 333-340
[11] Beurling's phenomenon in two variables, Integral Equations Operator Theory, Volume 48 (2004), pp. 411-423
[12] Operator Theory in Function Spaces, Marcel Dekker, 1990
[13] Berezin transforms and radial operators, Proc. Amer. Math. Soc., Volume 131 (2003), pp. 793-800
Cité par Sources :
