Lebesgue–Bochner spaces, decomposable sets and strong weakly compact generation

Let X be a Banach space and μ a probability measure. We prove that X is strongly reflexive (resp. super-reflexive) generated if, and only if, there exist a reflexive (resp. super-reflexive) Banach space Z and a bounded linear operator S : Z → L 1 ( μ , X ) such that for each weakly compact decomposable set K ⊂ L 1 ( μ , X ) and each ε > 0 there is n ∈ N such that K ⊂ n S ( B Z ) + ε B L 1 ( μ , X ) . This answers partially a question posed by Schlüchtermann and Wheeler. Some applications are also given.