  In this paper we study the compactness of operators on the Bergman space of
the unit ball and on very generally weighted Bargmann-Fock spaces in terms of
the behavior of their Berezin transforms and the norms of the operators acting
on reproducing kernels. In particular, in the Bergman space setting we show how
a vanishing Berezin transform combined with certain (integral) growth
conditions on an operator $T$ are sufficient to imply that the operator is
compact. In the weighted Bargmann-Fock space setting we show that the
reproducing kernel thesis for compactness holds for operators satisfying
similar growth conditions. The main results extend the results of Xia and Zheng
to the case of the Bergman space when $1 < p < \infty$, and in the weighted
Bargmann-Fock space setting, our results provide new, more general conditions
that imply the work of Xia and Zheng via a more familiar approach that can also
handle the $1 < p < \infty$ case.
