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In this paper, we introduce the property (h) on Banach lattices and present its characterization in terms of disjoint sequences. Then, an example is given to show that an order-to-norm continuous operator may not be σ -order continuous. Suppose T : E → F is an order-bounded operator from Dedekind σ -complete Banach lattice E into Dedekind complete Banach lattice F. We prove that T is σ -order-to-norm continuous if and only if T is both order weakly compact and σ -order continuous. In addition, if E can be represented as an ideal of L 0 (μ) , where (Ω , Σ , μ) is a σ -finite measure space, then T is σ -order-to-norm continuous if and only if T is order-to-norm continuous. As applications, we extend Wickstead's results on the order continuity of norms on E and E ′ . [ABSTRACT FROM AUTHOR]

Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. For applications, we introduce so-called statistical-unbounded convergence and use these convergences to describe KB-spaces and reflexive Banach lattices. [ABSTRACT FROM AUTHOR]

Recently, continuous functionals for unbounded order (norm, weak and weak*) in Banach lattices were studied. In this paper, we study the continuous operators with respect to unbounded convergences. We first investigate the approximation property of continuous operators for unbounded convergence. Then we show some characterizations of the continuity of the continuous operators for u o , u n , u a w and u a w * -convergence. Based on these results, we discuss the order-weakly compact operators on Banach lattices. [ABSTRACT FROM AUTHOR]

We characterize Banach lattices for which each Dunford–Pettis operator (or weak Dunford–Pettis) is unbounded absolute weak Dunford–Pettis and conversely. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the class of almost weak* Dunford-Pettis operators and give a characterization of this class of operators. We study its relation with the classes of weak* Dunford-Pettis operators and almost Dunford-Pettis operators, and its relation with the closely related classes of almost limited operators and L-weakly compact operators. [ABSTRACT FROM AUTHOR]

Let E and F be Banach lattices, T, K : E → F be such that 0 ≤ T ≤ K and T is either a lattice homomorphism or almost interval-preserving. In this paper we will deduce that (1) If K is AM-compact then T also is AM-compact; (2) If either E′ or F has an order continuous norm and K is compact, then T is compact as well; (3) If K is weakly compact then so is T. Some related results are also obtained. [ABSTRACT FROM AUTHOR]