Let G be an additive subgroup of a normed space X. We say that a point $$x\in X\setminus G$$ is weakly separated (resp. $$\mathcal P$$ -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions:
(ws) if $$Tx\notin\overline{T(G)}$$ , then x is weakly separated from G;
(ps) if $$Tx\notin\overline{T(G)}$$ , then x is $${\mathcal P}$$ -separated from G;
(wp) if Tx is $$\mathcal P$$ -separated from T(G), then x is weakly separated from G.
By $${\mathcal W} {\mathcal S}(X,Y)$$ (resp. $${\mathcal P}{\mathcal S}(X,Y)$$ , $${\mathcal W}{\mathcal P}(X,Y)$$ ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all $$x \in X$$ and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then $${\mathcal W} {\mathcal P}(X,Y)$$ is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then $$T\in {\mathcal P} {\mathcal S}(X,Y)\cap{\mathcal W}{\mathcal P}(X,Y)$$ .