Abstract
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 L a 2 ( D ) 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 H 2 ( D ) 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 H 2 ( D ) , using the concepts of reproducing kernels and Berezin symbols. To cite this article: M.T. Karaev, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
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