On some problems related to Berezin symbols

Karaev M.

COMPTES RENDUS MATHEMATIQUE, vol.340, no.10, pp.715-718, 2005 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 340 Issue: 10
  • Publication Date: 2005
  • Doi Number: 10.1016/j.crma.2005.04.021
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.715-718


The following problem was formulated by Zorboska [Proc. Amer. Math. Soc. 13 1 (2003) 793-800]: It is not known if the Berezin symbols of a bounded operator on the Bergman space L-a(2)(D) must have radial hunts almost everywhere oil 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 H-2(D) over the unit disk D. Moreover, we give an alternative proof to the famous theorem of Berling on z-invariant subspaces in the Hardy space H-2(D), using the concepts of reproducing kernels and Berezin symbols. (c) 2005 Academie des sciences. Published by Elsevier SAS. All rights reserved.