We give in terms of Berezin symbols some characterizations for the operators belonging to the Schatten-von Neumann classes sigma(p), 0 < p < +infinity, which were motivated by the questions posed by Nordgren and Rosenthal [7]. Some other problems related with the multiplicative equality for Berezin symbols are also discussed. We also prove in terms of Berezin symbols a Gohberg-Krein type theorem on the weak limit of compact operators.