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 $1\le p<q\le \infty ,$ we introduce the concepts of properties $(SR{)}_{p,q}$ and $(S{R}^{\ast }{)}_{p,q}$ 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.

arXiv admin note: substantial text overlap with arXiv:1810.05638