Dieudonné operators on the space of Bochner integrable functions
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 $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $X$ and $Y$ be Banach spaces. We study Dieudonné operators $T:L^1(X)\to Y$. Let $i_\infty:L^\infty(X) \to L^1(X)$ stand for the canonical injection. We show that if $X$ is almost reflexive and $T:L^1(X)\to Y$ is a Dieudonné operator, then $T\circ i_\infty:L^\infty(X)\to Y$ is a weakly compact operator. Moreover, we obtain that if $T:L^1(X)\to Y$ is a bounded linear operator and $T\circ i_\infty:L^\infty(X)\to Y$ is weakly compact, then $T$ is a Dieudonné operator.