Some notes on orthogonally additive polynomials

TL;DR

This paper introduces new characterizations of bounded orthogonally additive polynomials on vector lattices, utilizing polynomial identities involving root mean power and geometric mean, and clarifies when orthogonal additivity extends from the positive cone.

Contribution

It provides two novel characterizations of orthogonally additive polynomials and shows that orthogonal additivity on the positive cone implies general orthogonal additivity.

Findings

New characterizations using Kusraeva's polynomial identities

Orthogonal additivity on the positive cone implies full orthogonal additivity

Improves previous characterizations by Buskes and the author

Abstract

We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using separately two polynomial identities of Kusraeva involving the root mean power and the geometric mean. Furthermore, it is shown that a polynomial on a vector lattice is orthogonally additive whenever it is orthogonally additive on the positive cone. These results improve recent characterizations of bounded orthogonally additive polynomials by G. Buskes and the author.