A study on Dunford-Pettis completely continuous like operators

In this article, the class of all Dunford-Pettis $p$-convergent operators and $p$-Dunford-Pettis relatively compact property on Banach spaces are investigated. Moreover, we give some conditions on Banach spaces $X$ and $Y$ such that the class of bounded linear operators from $X$ to $Y$ and some its subspaces have the $p$-Dunford-Pettis relatively compact property. In addition, if $\Omega$ is a compact Hausdorff space, then we prove that dominated operators from the space of all continuous functions from $K$ to Banach space $X$ (in short $C(\Omega,X)$) taking values in a Banach space with the $p$-$(DPrcP)$ are $p$-convergent when $X$ has the Dunford-Pettis property of order $p.$\ Furthermore, we show that if $T:C(\Omega,X)\rightarrow Y$ is a strongly bounded operator with representing measure $m:\Sigma\rightarrow L(X,Y)$ and $\hat{T}:B(\Omega,X)\rightarrow Y$ is its extension, then $T$ is Dunford-Pettis $p$-convergent if and only if $\hat{T}$ is Dunford-Pettis $p$-convergent.