We introduce the concept of the strong Gelfand-Phillips (GP) property in Banach lattices, and we characterize Banach lattices with the strong GP property. Next, by introducing the class of almost limited completely continuous operators from an arbitrary Banach lattice E to another F , we give some properties of them related to some well-known classes of operators and, especially, to the strong GP property of the Banach lattice E.

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By introducing the concepts of order almost Dunford-Pettis and almost weakly limited operators in Banach lattices, we give some properties of them related to some well known classes of operators, such as, order weakly compact, order Dunford-Pettis, weak and almost Dunford-Pettis and weakly limited operators. Then, we characterize Banach lattices E and F on which each operator from E into F that is order almost Dunford-Pettis and weak almost Dunford-Pettis is an almost weakly limited operator.