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

Localization and Compactness in Bergman and Fock spaces