WEAK COMPACTNESS OF ALMOST L-WEAKLY AND ALMOST M-WEAKLY COMPACT OPERATORS

In this paper, we investigate conditions on a pair of Banach latticesEandFthat tells us when every positive almost L-weakly compact (resp. almost M- weakly compact) operatorT:E -> Fis weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operatorT:E -> Fis almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach latticeEinto a Banach spaceXis almost L-weakly compact, thenEis a KB-space orXhas the Dunford-Pettis property and the norm ofEis order continuous.

Automated summary

This paper investigates conditions on a pair of Banach lattices E and F that determine when a positive almost L-weakly compact (or almost M-weakly compact) operator T: E -> F is weakly compact. It also presents some necessary conditions for determining when every weakly compact operator T: E -> F is almost M-weakly compact (or almost L-weakly compact). Furthermore, it proves 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.