Sequentially Right-like properties on Banach spaces

Abstract

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, $ we introduce the concepts of properties $ (SR)_{p,q}$ and $ (SR^{\ast})_{p,q}$ in order to find a condition which every Dunford-Pettis $ q $-convergent operator is Dunford-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.