On pseudo weakly compact operators of order $ P$

Abstract: In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spaces $ X$ and $ Y $ such that projective tensor and injective tensor products between $ X $ and $ Y $ have the $ p $-sequentially Right property.\ Finally, we introduce two properties for the Banach spaces, namely $ p $-sequentially Right$ ^{\ast} $ and weak $ p $-sequentially Right$ ^{\ast} $ properties and obtain some characterizations of these properties.