Abstract. In this paper we discuss various generalizations of the Fock space. We will prove that for a bounded radial function $f$ on ${\msbm C}^n$ the Toeplitz operator $T_f$ is compact on these generalized Fock spaces if and only if its Berezin transform has zero limit.

AMS Subject Classification (1991): 47B35, 47B38

Received February 26, 1998. (Registered under 3314/2009.)