An Extension of Order Bounded Operators

Let E be a normed lattice and an order dense majorizing sublattice of a vector lattice E^t. We extend the norm of E to E^t, denoted by \Vert.\Vert_t. The pair (E^t,\Vert.\Vert_t) forms a normed lattice and preserves certain lattices and topological properties whenever these properties hold in E.As a consequence, every positive linear operator defined on a normed lattice E has a linear extension to E^t. This manuscript provides an explicit formula for these extensions. The extended operator T^t is a lattice homomorphism from E^t into F, and it belongs to \mathcal{L}_n(E^t,F) whenever ۰\leq T\in \mathcal{L}_n(E,F) and T(x\wedge y)=Tx \wedge Ty for all ۰\leq x,y\in E. Furthermore, if T\in \mathcal{L}_b(E,F) and certain lattice and topological properties hold for T, then T^t\in \mathcal{L}_b(E^t,F) will also preserve these properties.

Keywords:

Riesz space , order convergence , unbounded order convergence