  Let N be a nest on a Hilbert space H and AlgN the corresponding nest algebra.
We obtain a characterization of the compact and weakly compact multiplication
operators defined on nest algebras. This characterization leads to a
description of the closed ideal generated by the compact elements of AlgN . We
also show that there is no non-zero weakly compact multiplication operator on
AlgN / (AlgN $\cap$ K(H)).
