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We introduce and study the notion of almost L-Dunford-Pettis sets in Banach lattices and we give some characterizations of it in terms of sequences. As an application, we establish new properties of almost Dunford-Pettis completely continuous operators. Finally, by introducing the concept of aL-Dunford-Pettis property in Banach lattices, we investigate the weak compactness of almost Dunford-Pettis completely continuous operator.

In this paper, we introduce and study the concept of L-Dunford-Pettis sets and L-Dunford-Pettis property in Banach spaces. Next, we give a characterization of the L-Dunford-Pettis property with respect to some well-known geometric properties of Banach spaces. Finally, some complementability of operators on Banach spaces with the L-Dunford-Pettis property are also investigated.

In this paper, we study the notion of V-sets in Banach spaces and Banach lattices, and we give some characterizations of it in terms of sequences. As an application, we establish new properties of unconditionally converging operators and 1-Schur property in Banach lattices. Next, by introducing the concept of the property (V LD) in Banach spaces, we investigate the Dunford-Pettis completely continuous property of unconditionally converging operator. Finally, we derive the relationships between the property (V LD) and the relatively compact Dunford-Pettis property (resp., the Pelczynski's property (V )), and we deduce some examples of Banach spaces with the property (V LD).У цiй роботi ми вивчаємо поняття V-множин у банахових просторах та банахових ґратках та даємо деякi їхнi характеристики у термiнах послiдовностей. Як застосування, ми встановлюємо новi властивостi безумовно збiжних операторiв i властивiсть 1-Шура в банахових ґратках. Далi, вводячи поняття (V LD) властивостi у банахових просторах, ми дослiджуємо властивiсть цiлковитої неперервностi Данфорда-Петтiса безумовно збiжного оператора. Нарештi, ми виводимо зв'язки мiж (V LD) властивiстю i властивiстю вiдносної компактностi Данфорда-Петтiса (вiдповiдно, властивiстю Пельчинського (V )), i виводимо деякi приклади банахових просторiв iз (V LD) властивiстю. 2020 Mathematics Subject Classification. 46A40, 46B40.