On the class of almost (L) limited operators

In this paper, we introduce and study the class of almost (L) limited operators, these are operators from a Banach space into a Banach lattice whose adjoint carries disjoint (L) and weak* null sequences to norm null ones. We establish some characterizations of this class of operators, and present some connections between this class of operators and almost limited operators. After that, we prove that almost (L) limited operators from a Banach lattice into a σDedekind complete one which are lattice homomorphism are exactly operators whose adjoint carries almost (L) sets into L weakly compact ones. Finally, we derive some characterizations of a σ-Dedekind complete Banach lattice whose dual has order continuous norm.