Reflexivity for spaces of regular operators on Banach lattices

We prove that if Banach lattices EE and FF are reflexive and each positive linear operator from EE to FF is compact then Lr(E;F){\mathcal L}^r(E;F), the space of all regular linear operators from EE to FF, is reflexive. Conversely, if E∗E^\ast or FF has the bounded regular approximation property then the reflexivity of Lr(E;F){\mathcal L}^r(E;F) implies that each positive linear operator from EE to FF is compact. Analogously we also study the reflexivity for the space of regular multilinear operators on Banach lattices.