  In this article, we study the relationship between \(p\)-\((V)\) subsets and
p-\(V^*\) subsets of dual spaces. We investigate the Banach space X with the
property that adjoint every \(p\)-convergent operator \(T: X \rightarrow Y\) is
weakly \(q\)-compact, for every Banach space \(Y\). Moreover, we define the
notion of \(q\)-reciprocal Dunford-Pettis\(\^*\)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.
