In this article, we characterize the classes \(S_{p}^{w}\) classes in terms of the boundary behavior of Berezin symbols of the weighted shift operators on the Dirichlet space \({\mathcal {D}}\left( {\mathbb {D}}\right) \) . Also, we give a result related with the asymptotic multiplicative property of the Berezin symbol of any operator on reproducing kernel Hilbert space. Moreover, by using Berezin symbols, we give some condition for st- \(\left( DL\right) \) convergence of a series of complex numbers.
