Sequentially Right-like properties on Banach spaces

In this paper, we first study the concept of $p$-sequentially Right property, which is the $p$-version of the sequentially Right property. Also, we introduce a new class of subsets of Banach spaces which is called $p$-Right${}^{\ast }$ set and obtain the relationship between p-Right subsets and p-Right${}^{\ast }$ subsets of dual spaces. Furthermore, for $ 1\leq p<q\leq\infty, $weintroducetheconceptsofproperties$ (SR){p,q}$and$ (SR^{\ast}){p,q}$inordertofindaconditionwhicheveryDunford-Pettis$ q $-convergentoperatorisDunford-Pettis$p$-convergent. Finally, we apply these concepts and obtain some characterizations of $p$-Dunford-Pettis relatively compact property of Banach spaces and their dual spaces.