Bounded Domains of Generalized Riesz Methods with the Hahn Property

Abstract

In 2002 Bennett et al. started the investigation to which extent sequence spaces are determined by the sequences of 0s and 1s that they contain. In this relation they defined three types of Hahn properties for sequence spaces: the Hahn property, separable Hahn property, and matrix Hahn property. In general all these three properties are pairwise distinct. If a sequence space E is solid and (0,1ℕ∩E)β=Eβ=ℓ1 then the two last properties coincide. We will show that even on these additional assumptions the separable Hahn property and the Hahn property still do not coincide. However if we assume E to be the bounded summability domain of a regular Riesz matrix Rp or a regular nonnegative Hausdorff matrix Hp, then this assumption alone guarantees that E has the Hahn property. For any (infinite) matrix A the Hahn property of its bounded summability domain is related to the strongly nonatomic property of the density dA defined by A. We will find a simple necessary and sufficient condition for the density dA defined by the generalized Riesz matrix Rp,m to be strongly nonatomic. This condition appears also to be sufficient for the bounded summability domain of Rp,m to have the Hahn property.