  In this paper we introduce the Schatten class of operators and the Berezin
transform of operators in the quaternionic setting. The first topic is of great
importance in operator theory but it is also necessary to study the second one
because we need the notion of trace class operators, which is a particular case
of the Schatten class. Regarding the Berezin transform, we give the general
definition and properties. Then we concentrate on the setting of weighted
Bergman spaces of slice hyperholomorphic functions. Our results are based on
the S-spectrum for quaternionic operators, which is the notion of spectrum that
appears in the quaternionic version of the spectral theorem and in the
quaternionic S-functional calculus.
