Regular Operators between Banach Lattices

Abstract

If X and Y are Banach lattices then there are several spaces of linear operators between them that may be studied. $$ \mathcal{L}^r $$ (X, Y) is the space of all norm bounded operators from X into Y. There is no reason to expect there to be any connection between the order structure of X and Y and that of $$ \mathcal{L} $$ (X, Y). $$ \mathcal{L}^r $$ (X, Y) is the space of regular operators, i.e., the linear span of the positive operators. This at least has the merit that when it is ordered by the cone of positive operators then that cone is generating. $$ \mathcal{L}^b $$ (X, Y) is the space of order bounded operators, which are those that map order bounded sets in X to order bounded sets in Y. We always have $$ \mathcal{L}^r (X,Y) \subseteq \mathcal{L}^b (X,Y) \subseteq \mathcal{L}(X,Y) $$ and both inclusions may be proper.