A study on Dunford-Pettis completely continuous like operators

Abstract

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.