# Sequentially Right-like properties on Banach spaces

## TL;DR

This paper introduces and studies the $p$-sequentially Right property and related concepts in Banach spaces, providing new characterizations and relationships between these properties and operator convergence behaviors.

## Contribution

It defines the $p$-sequentially Right property, introduces $p$-Right$^{ ext{*}}$ sets, and explores their relationships, extending the understanding of operator convergence in Banach spaces.

## Key findings

- Characterization of $p$-Dunford-Pettis relatively compact property.
- Relationships between $p$-Right and $p$-Right$^{ ext{*}}$ sets.
- Conditions for Dunford-Pettis $q$-convergent operators to be $p$-convergent.

## 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.