Compactness of operators on generalized Fock spaces

For a very general class of weighted Fock spaces on ${\mathbb{C}}^n$, we give necessary and sufficient conditions for a Toeplitz operator with a (not necessarily positive) measure symbol to be compact. Furthermore, we show that all compact operators are in the norm closure of the algebra generated by Toeplitz operators with $C_c^{\mathrm{\infty }}({\mathbb{C}}^n)$ symbols, and in the Hilbert space setting show that all compact operators are in the norm closure of the set of such Toeplitz operators.