On pseudo weakly compact operators of order $ P$

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.