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→L1(μ,X) such that for each weakly compact decomposable set K⊂L1(μ,X) and each ε>0 there is n∈N such that K⊂nS(BZ)+εBL1(μ,X). This answers partially a question posed by Schlüchtermann and Wheeler. Some applications are also given.