Some properties of orthogonally additive homogeneous polynomials on Banach lattices

Let E and F be Banach lattices and let Po(sE,F) stand for the space of all norm bounded orthogonally additive s-homogeneous polynomial from E to F. Denote by Pro(sE,F) the part of Po(sE,F) consisting of the differences of positive polynomials. The main results of the paper read as follows. Theorem 3.4. Let s∈N and (E,∥⋅∥) is a σ-Dedekind complete s-convex Banach lattice. The following are equivalent: (1) Po(sE,F)≡Pro(sE,F) for every AM-space F. (2) Po(sE,c0)=Pro(sE,F) for every AM-space F. (3) Po(sE,c0)=Pro(sE,c0). (4) Po(sE,c0)≡Pro(sE,c0). (5) E is atomic and order continuous. Theorem 4.3.For a pair of Banach lattices E and F the following are equivalent: (1) Pro(sE,F) is a vector lattice and the regular norm ∥⋅∥r on Pro(sE,F) is order continuous. (2) Each positive orthogonally additive s-homogeneous polynomial from E to F is L- and M -weakly compact. Theorem 4.6. Let E and F be Banach lattices. Assume that F has the positive Schur property and E is s-convex for some s∈N. Then the following are equivalent: (1) (Pro(sE,F),∥⋅∥r) is a KB-space. (2) The regular norm ∥⋅∥r on Pro(sE,F) is order continuous. (3) E does not contain any sulattice lattice isomorphc to ls.