Observations on some classes of operators on C(K,X)

Suppose X and Y are Banach spaces,Kis a compact Hausdorff space, Sigma is the sigma-algebra of Borel subsets of K,C(K,X) is the Banach space of allcontinuousX-valued functions (with the supremum norm), and T:C(K,X)-> Yis a strongly bounded operator with representing measure m:Sigma -> L(X, Y).We show that ifT:B(K, X)-> Yis its extension, then T is weak Dunford-Pettis (resp. weak & lowast;Dunford-Pettis, weakp-convergent, weak & lowast;p-convergent) ifand only ifThas the same property.We prove that ifT:C(K, X)-> Yis strongly bounded limited completelycontinuous (resp. limitedp-convergent), thenm(A):X -> Yis limited completely continuous (resp. limitedp-convergent) for eachA is an element of Sigma. We also prove that theabove implications become equivalences whenKis a dispersed compact Hausdorff space.