Bochner representable operators on Banach function spaces

Let ((E,Vert cdot Vert _E)) be a Banach function space, (E') the Köthe dual of E and ((X,Vert cdot Vert _X)) be a Banach space. It is shown that every Bochner representable operator (T:Erightarrow X) maps relatively (sigma (E,E'))-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm (Vert cdot Vert _{E'}) on (E') is order continuous, then every Bochner representable operator (T:Erightarrow X) is ((gamma _E,Vert cdot Vert _X))-compact, where (gamma _E) stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.