Acta Math. Univ. Comenianae Vol. LXXXIV, 1 (2015), pp. 97–102 97 THE CLASS OF ORDER-ALMOST LIMITED OPERATORS ON BANACH LATTICES A. EL KADDOURI, K. EL FAHRI, J. H’MICHANE and M. MOUSSA Abstract. We introduce and study the class of order-almost limited operators and derive the following interesting consequence the domination property of this class of operators, a characterization of the property (d). Then, we characterize Banach lattices E and F on which each operator from E into F that is order-almost limited and weak almost limited is an almost limited operator. 1. Introduction Recently, J.X. Chen et al. introduced and studied the class of almost limited sets in Banach lattices [3]. Based on this concept of sets, A. Elbour et al. gave a new class of operators on Banach lattices which are called almost limited operators [7]. Following, the authors introduced the class of weak almost limited operators, that is, operators which send relatively weakly compact sets from Banach spaces onto almost limited sets in Banach lattices [5]. The aim of this paper is to introduce a new class of operators that we call order-almost limited operators and give some interesting applications of this class of operators. This class is bigger than the class of order limited (resp., limited) operators introduced in [6] (resp., in [2]). The article is organized as follows after the introduction section, we give all common notations and definitions of Banach lattice theory in preliminaries section. Then in the first section, we give a characterization of a Banach lattice which have the property (d). In the second section, we show that if E and F are two Banach lattices then each order-almost limited and weak almost limited operator T : E -→ F is an almost limited operator if and only if the norm of E 0 is order continuous or F has the dual Schur property (see Theorem 4.2). 2. Preliminaries Throughout this paper, E, F denote Banach lattices. The positive cone of E is denoted by E + . B X is the closed unit ball of the Banach space X. Received April 12, 2014; revised August 25, 2014. 2010 Mathematics Subject Classification. Primary 46B42, 47B60, 47B65. Key words and phrases. Order-almost limited operator, weak almost limited operator, almost limited operator, dual Schur property, almost limited set, Property (d), order continuous norm.