On some problems related to Berezin symbols

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 $L_a^2(\mathbf{D})$ 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 $H^2(\mathbf{D})$ 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 $H^2(\mathbf{D})$, 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).