A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty }:L^{\infty }\left(X\right)\to L¹\left(X\right)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T\circ i_{\infty }:L^{\infty }\left(X\right)\to Y$ is a weakly compact operator. Moreover, we obtain that if T: L¹(X)...

We show that a B-space E has the (CRP) if and only if any dominated operator T from C[0, 1] into E is compact. Hence we apply this result to prove that c0 embeds isomorphically into the B-space of all compact operators from C[0, 1] into an arbitrary B-space E without the (CRP).

Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let $L^{\Phi }\left(X\right)$ be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to $L^{\Phi }\left(X\right)$. In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to $L^{\infty }\left(X\right)$ is $(\tau (L^{\infty }\left(X\right),L¹(X*)),|\left|·\right|{|}_Y)$-compact.