Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T20:59:13.720Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the positive Schur property

Published online by Cambridge University Press:  18 May 2009

Witold Wnuk
Witold Wnuk
Affiliation:
Mathematical Institute, Polish Academy of Sciences, Poznań Branch, Mielżyńskiego 27/29, 61–725 Poznań, Poland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this note is to characterize those Banach lattices (E, ∥·∥) which have the property:

an operator T: E → c0 is a Dunford-Pettis operator if and only if T is regular (*)

(i.e., T is the difference of two positive operators). Our characterization generalizes a theorem recently proved by Holub [6] and Gretsky and Ostroy [4], who have remarked that the space L1[0, 1] has the property (*). The main result presented here is the following theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 2 , May 1989 , pp. 169 - 172
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Aliprantis, C. and Burkinshaw, O., Locally solid Riesz spaces (Academic Press, 1978).Google Scholar
2.Aliprantis, C. and Burhinshaw, O., Positive operators (Academic Press 1985).Google Scholar
3.Buhvalov, A. V., Locally convex spaces that are generated by weakly compact sets (Russian), Vestnik Leningrad Univ. No. 7 Mat. Meh. Astronom. Vyp. 2 (1973), 11–17.Google Scholar
4.Gretsky, N. E. and Ostroy, J. M., The compact range property and c 0, Glasgow Math. J. 28 (1986), 113–114.CrossRefGoogle Scholar
5.Groenewegen, G. and Meyer-Nieberg, P., An elementary and unified approach to disjoint sequence theorems, Indag. Math. 48 (1986), 313–317.CrossRefGoogle Scholar
6.Holub, J. R., A note on Dunford–Pettis operators, Glasgow Math. J. 29 (1987), 271–273.CrossRefGoogle Scholar
7.Meyer-Nieberg, P., Charakterisierung einiger topologischer und ordungstheoretischer Eigenschaften von Banachverbänden mit Hilfe disjunkter Folgen, Arch. Math. (Basel) 24 (1973), 640–647.CrossRefGoogle Scholar
8.Varshavskaya, E. V. and Chuchaev, I. I., On the Dunford–Pettis and Schur properties in locally convex K-spaces (Russian), Soobšč. Akad. Nauk Gruzin. SSR 120 (1985), 21–23.Google Scholar
9.Zaanen, A. C., Riesz spaces II (North-Holland, 1983).Google Scholar