Dunford–Pettis operators on the space of Bochner integrable functions
Let $(\Omega,\Sigma,\mu)$ be a finite measure space and let $X$ be a real Banach space. Let $L^\Phi(X)$ be the Orlicz–Bochner space defined by a Young function $\Phi$. We study the relationships between Dunford–Pettis operators $T$ from $L^1(X)$ to a Banach space $Y$ and the compactness properties of the operators $T$ restricted to $L^\Phi(X)$. In particular, it is shown that if $X$ is a reflexive Banach space, then a bounded linear operator $T:L^1(X)\to Y$ is Dunford–Pettis if and only if $T$ restricted to $L^\infty(X)$ is $(\tau(L^\infty(X),L^1(X^*)),\|\cdot\|_Y)$-compact.