The paper contains some applications of the notion of (L) sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order (L)-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an (L) sets. As a sequence characterization of such operators, we see that an operator T : X → E from a Banach space into a Banach lattice is order (L)-Dunford-Pettis, if and only if |T (xn)| → 0 for σ(E, E′ ) for every weakly null sequence (xn) ⊂ X. We also investigate relationships between order (L)-DunfordPettis, AM-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator T : E → F between Banach lattices is Dunford-Pettis whenever it is both order (L)-Dunford-Pettis and weak* Dunford-Pettis, if and only if E has the Schur property or the norm of F is order continuous.
Let $\mathcal B_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty $. Let $\mathcal B_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty $. Define $\mathcal A^n_c$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\mathcal B_c$. Similarly with $\mathcal A^n_r$ from $\mathcal B_r$. A type of integral is defined on distributions in $\mathcal A^n_c$ and $\mathcal A^n_r$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb N$, the spaces $\mathcal A^n_c$ and $\mathcal A^n_r$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\mathcal B_c$ and $\mathcal B_r$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\mathcal A_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\mathcal A_r^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.