Almost unbounded L and M-weakly compact operators

In this paper, we introduce and investigate a new class of operators known as almost unbounded L-weakly compact (in shortly, \(_{au}L\)-weakly compact) and almost unbounded M-weakly compact (in shortly, \(_{au}M\)-weakly compact) operators. We explore the lattice properties related to this class and examine their relationships with other established operator classes, such as L-weakly compact operators and almost L-weakly compact operators. We demonstrate that every L-weakly compact operator is an \(_{au}L\)-weakly compact operator, but the reverse implication does not necessarily hold in all cases.