Weak compactness of almost L-weakly and almost M-weakly compact operators

Abstract

In this paper, we investigate conditions on a pair of Banach lattices E and F that tells us when every positive almost L-weakly compact (resp. almost M- weakly compact) operator T : E → F is weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operator T : E → F is almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach lattice E into a Banach space X is almost L-weakly compact, then E is a KB-space or X has the Dunford-Pettis property and the norm of E is order continuous.

Mathematics Subject Classification (2010):

• Primary: 46B07
• Secondary: 46B42
• 47B50

Key words:

• Almost L-weakly compact operator
• almost M-weakly compact operator
• M- weakly compact operator
• L-weakly compact operator
• Banach lattice
• order continuous norm