Reflexivity for spaces of regular operators on Banach lattices

Abstract

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