Subseries of ℐ $$ \mathrm{\mathcal{I}}$$ -convergent series

Abstract

We say that an ideal \( \mathrm{\mathcal{I}}\) has property (T) if for every \( \mathrm{\mathcal{I}}\)-convergent series \( {\sum}_{n=1}^{\infty }{x}_n \), there exists a set A ∈ \( \mathrm{\mathcal{I}}\) such that ∑n ∈ ℕ\Ax n converges in the usual sense. The main aim of this paper is to focus on several different classes of ideals, such as summable ideals, F σ ideals, and matrix summability ideals, and to show that they do not have the mentioned property.