# A Study on Dunford–Pettis Completely Continuous Like Operators

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## 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.

## Keywords

Dunford–Pettis relatively compact property Dunford–Pettis completely continuous operators Dunford–Pettis*p*-convergent operators

## Mathematics Subject Classification

46B20 46B25 46B28## Notes

### Acknowledgements

This paper is part of the author’s Ph.D. thesis at the University of Isfahan. I would like to thank to Professor Jafar Zafarani and Professor Majid Fakhar, for their insight and expertise, which improved this work.

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