Subseries convergence and localization properties in sequence spaces supported by ideals in \(\mathcal{P}(\mathbb{N})\) (Q2425338)

The authors consider subseries convergence in sequence spaces. A strictly increasing sequence of positive integers \(\{n_{k}\}\) is lacunary if \(\lim_{k}(n_{k+1}-n_{k})=\infty\) and has density zero if \(\lim_{k}k/n_{k}=0\). Lacunary subseries (zero density subseries) of a given series are those subseries corresponding to lacunary (zero density) indices. A space \(X\) has the lacunary convergence property (LCP) [zero density convergence property (ZCP)] if a series in \(X\) is subseries convergent provided each of its lacunary (zero density) subseries is subseries convergent. Then (LCP) implies (ZCP). Conjecture: If in a sequence space like \(l^{1}\), for example, one considers the subspace \(l^{1}(\mathcal{L})\) consisting of those elements whose supports lie in the ideal \(\mathcal{L}\) generated by the lacunary sets, then \(l^{1}(\mathcal{L})\) should have (ZCP) but fail (LCP). In the paper under review, the authors establish this conjecture for sequence spaces supported by ideals in \(\mathcal{P}(\mathbb{N})\), the power set of \(\mathbb{N}\). The construction of the proof of this conjecture requires an extensive study of ideals in \(\mathcal{P}(\mathbb{N})\) and should be of independent interest. Along the way, there are results about measures, submeasures and generalizations to vector valued sequence spaces and even continuous linear operators.