Reflexivity for spaces of regular operators on Banach lattices

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