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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 908682, 8 pages
http://dx.doi.org/10.1155/2013/908682
Research Article

Bounded Domains of Generalized Riesz Methods with the Hahn Property

Department of Mathematics, Tallinn University, Narva Maantee 25, 10120 Tallinn, Estonia

Received 29 May 2013; Revised 28 August 2013; Accepted 4 September 2013

Academic Editor: Antonio S. Granero

Copyright © 2013 Maria Zeltser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 is solid and 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 to be the bounded summability domain of a regular Riesz matrix or a regular nonnegative Hausdorff matrix , then this assumption alone guarantees that has the Hahn property. For any (infinite) matrix the Hahn property of its bounded summability domain is related to the strongly nonatomic property of the density defined by . We will find a simple necessary and sufficient condition for the density defined by the generalized Riesz matrix to be strongly nonatomic. This condition appears also to be sufficient for the bounded summability domain of to have the Hahn property.