A study of reciprocal Dunford-Pettis-like properties on Banach spaces

In this article, we study the relationship between $p$-$(V)$ subsets and p-$V^{\ast }$ subsets of dual spaces. We investigate the Banach space X with the property that adjoint every $p$-convergent operator $T:X\to Y$ is weakly $q$-compact, for every Banach space $Y$. Moreover, we define the notion of $q$-reciprocal Dunford-Pettis$\hat{\ast }$property of order $p$ on Banach spaces and obtain a characterization of Banach spaces with this property. The stability of reciprocal Dunford-Pettis property of order $p$ for the projective tensor product is given.