Some properties of the space of regular operators on atomic Banach lattices

Abstract

Let \({{\mathcal L}^r(E,X)}\) denote the space of regular linear operators from a Banach lattice E to a Banach lattice X. In this paper, we show that if E is a separable atomic Banach lattice, then \({{\mathcal L}^r(E, X)}\) is reflexive if and only if both E and X are reflexive and each positive linear operator from E to X is compact; moreover, if E is a separable atomic Banach lattice such that E and E* are order continuous, then \({{\mathcal L}^r(E, X)}\) has the Radon–Nikodym property (respectively, is a KB-space) if and only if X has the Radon–Nikodym property (respectively, is a KB-space) and each positive linear operator from E to X is compact.