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

Part of: Special classes of linear operators Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  23 October 2023

Redouane Nouira Open the ORCID record for Redouane Nouira [Opens in a new window]  and
Belmesnaoui Aqzzouz
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
*
e-mail: rednouira@gmail.com
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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.

Keywords

Dunford–Pettisweak* Dunford–Pettis operatorweakly compact operatorM-weakly compact operatoralmost M-weakly compact operator

MSC classification

Primary: 46B42: Banach lattices
Secondary: 47B60: Operators on ordered spaces 47B65: Positive operators and order-bounded operators
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 12
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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