On the density of the additive semigroup generated by a subset of a Hilbert-Schmidt ellipsoid

We obtain a criterion for the density of the additive semigroup generated by a subset of an ellipsoid in a Hilbert space with finite sum of the squared semiaxes. This criterion is then applied to the semigroups generated by the images of sets under holomorphic maps. As a consequence, we obtain a result on the uniform approximation of a holomorphic function on compact subsets by sums of shifts of a single function which generalizes Korevaar's theorem. Bibliography: 7 titles.