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Some results on various types of compactness of weak* Dunford–Pettis operators on Banach lattices

Published online by Cambridge University Press:  23 October 2023

Redouane Nouira*
Affiliation:
Department of Mathematics, Regional Center for Education and Formation (CRMEF), Rabat, Morocco
Belmesnaoui Aqzzouz
Affiliation:
Faculty of Economics, Law and Social Sciences, Mohammed V University of Rabat, B.P. 5295, Sala Aljadida, Morocco e-mail: baqzzouz@hotmail.com

Abstract

We study the relationship between weak* Dunford–Pettis and weakly (resp. M-weakly, order weakly, almost M-weakly, and almost L-weakly) operators on Banach lattices. The following is one of the major results dealing with this matter: If E and F are Banach lattices such that F is Dedekind $\sigma $-complete, then each positive weak* Dunford–Pettis operator $T:E\rightarrow F$ is weakly compact if and only if one of the following assertions is valid: (a) the norms of $E^{\prime }$ and F are order continuous; (b) E is reflexive; and (c) F is reflexive.

Type
Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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