A representation theorem for orthogonally additive polynomials in Riesz spaces

TL;DR

This paper establishes a representation theorem for orthogonally additive polynomials in Riesz spaces, extending recent Banach lattice results to a more general Riesz space setting.

Contribution

It introduces the notion of p-orthosymmetric multilinear forms and proves their equivalence to orthogonal additivity, providing a new representation theorem in Riesz spaces.

Findings

Positive orthogonally additive polynomials are isomorphic to positive linear forms on the n-power of the Riesz space.

The notion of p-orthosymmetric multilinear form is introduced and shown to be equivalent to orthogonal additivity.

The theorem extends the representation of orthogonally additive polynomials from Banach lattices to Riesz spaces.

Abstract

The aim of this article is to prove a representation theorem for orthogonally additive polynomials in the spirit of the recent theorem on representation of orthogonally additive polynomials on Banach lattices but for the setting of Riesz spaces. To this purpose the notion of $p$--orthosymmetric multilinear form is introduced and it is shown to be equivalent to the orthogonally additive property of the corresponding polynomial. Then the space of positive orthogonally additive polynomials on an Archimedean Riesz space taking values on an uniformly complete Archimedean Riesz space is shown to be isomorphic to the space of positive linear forms on the $n$-power in the sense of Boulabiar and Buskes of the original Riesz space.