Continuous orthosymmetric multilinear maps and homogeneous polynomials on Riesz spaces

Abstract

We show that any continuous orthosymmetric multilinear map from an Archimedean Riesz space into a Hausdorff topological vector space is symmetric. Then, we establish a linear representation for continuous orthogonally additive homogeneous polynomials. This representation will be used to introduce and describe a new class of homogeneous polynomials, namely that of polyorthomorphisms. In particular, we prove that, for a Riesz space E and a natural number $$n\ge 2$$ , the space $${{\mathcal{P}}}_{orth}(^nE)$$ of all polyorthomorphisms of degree n is a Riesz space.