Positivity 12 (2008), 241–268
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020241-28, published online January 11, 2008
On Separation of Points from Additive
Subgroups of Banach Spaces by Continuous
Characters and Positive Definite Functions
Wojciech Banaszczyk and Robert Stegli´nski
Abstract. Let G be an additive subgroup of a normed space X.Wesaythat
apointx ∈ X \ G is weakly separated (resp. 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:
T (G), then x is weakly separated from G;
T (G), then x is P-separated from G;
(wp)ifTx is P-separated from T (G), then x is weakly separated from G.
By WS(X, Y ) (resp. PS(X, Y ), WP(X, Y )) we denote the class of opera-
tors T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all x ∈ 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 .ItisprovedthatifX, Y are Hilbert
spaces, then WP(X, Y )istheclassofHilbert-Schmidtoperators.Itisalso
shown that if T is a Hilbert-to-Banach space operator with ﬁnite -norm, then
T ∈PS(X, Y ) ∩WP(X, Y ).
Mathematics Subject Classiﬁcation (2000). 43A35, 46B20, 47B10.
Keywords. Additive subgroups of Banach spaces, Hilbert-Schmidt operators,
positive definite functions.
1. Introduction. Notation and Terminology
Let G be a (Hausdorﬀ) abelian topological group. By a character of G we mean
a homomorphism of G into the multiplicative group of complex numbers with
modulus 1. The group of continuous characters of G will be denoted by G
A complex-valued function ϕ on G is called positive definite (p.d. for short),
) ≥ 0