Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma$$\end{document} is the σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma$$\end{document}-algebra of Borel subsets of K, C(K,X)\documentclass[12pt]{minimal}
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\begin{document}$$C(K,X)$$\end{document} is the Banach space of all continuous X-valued functions (with the supremum norm), and T:C(K,X)→Y\documentclass[12pt]{minimal}
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\begin{document}$$T \colon C(K,X)\to Y$$\end{document} is a strongly bounded operator with representing measure m:Σ→L(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$m \colon \Sigma \to L(X,Y)$$\end{document}.
We show that if T^:B(K,X)→Y\documentclass[12pt]{minimal}
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\begin{document}$$\hat{T} \colon B(K, X) \to Y$$\end{document} is its extension, then T is weak Dunford--Pettis (resp.weak* Dunford--Pettis, weak p-convergent, weak*p-convergent) if and only if T^\documentclass[12pt]{minimal}
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\begin{document}$$\hat{T}$$\end{document} has the same property.
