Compact multiplication operators on nest algebras

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)).