An Extension of Order Bounded Operators

Let e be a normed lattice and an g-order dense majorizing sublattice of a vector lattice et. we extend the norm of e to et, denoted by ∥.∥t. the pair (et, ∥.∥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 et. this manuscript provides an explicit formula for these extensions. the extended operator tt is a lattice homomorphism from et into f, and it belongs to ln(et, f) whenever 0 ≤ t ∈ ln(e, f) and t(x ∧ y) = tx ∧ ty for all 0 ≤ x, y ∈ e. furthermore, if t ∈ lb(e, f) and certain lattice and topological properties hold for t, then tt ∈ lb(et, f) will also preserve these properties.