On T-sequences and characterized subgroups

Let X be a compact metrizable abelian group and u={un} be a sequence in its dual group X^∧. Set su(X)={x:(un,x)→1} and . Let G be a subgroup of X. We prove that G=su(X) for some u iff it can be represented as some dually closed subgroup Gu of . In particular, su(X) is polishable. Let u={un} be a T-sequence. Denote by the group X^∧ equipped with the finest group topology in which un→0. It is proved that and . We also prove that the group generated by a Kronecker set cannot be characterized.