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 Ω 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(Ω, X)) taking values in a Banach space with the p-(DP rcP ) are p-convergent when X has the Dunford-Pettis property of order p. Furthermore, we show that if T : C(Ω, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y ) andT : B(Ω, X) → Y is its extension, then T is Dunford-Pettis p-convergent if and only ifT is Dunford-Pettis p-convergent.Mathematics Subject Classification (2010). 46B20, 46B25,46B28.

A note on some classes of operators on C(K,X)

“…Several authors have found the study of T to be quite helpful. We mention the work of [6], [10], [8], [24], [14], [25], [3], [27], and [23]. In these papers it has been proved that if m is strongly bounded, then T : C(K, X) → Y is Dunford-Pettis, Dieudonné, unconditionally converging, strictly singular, strictly cosingular, weakly precompact, weakly p-compact, limited, pseudo weakly compact, Dunford-Pettis p-convergent if and only if its extension T : B(Σ, X) → Y has the same property.…”