Compactness of operators on generalized Fock spaces

Abstract

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 $C_c^{\infty }({\mathbb{C}}^n)$ symbols, and in the Hilbert space setting show that all compact operators are in the norm closure of the set of such Toeplitz operators.

This paper has been withdrawn since the results in this paper are vastly generalized by those in http://arxiv.org/abs/1305.7475