Some results on almost L-weakly and almost M-weakly compact operators

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.