A Nakano Carrier Theorem for Polynomials

TL;DR

This paper extends the Nakano Carrier theorem to orthogonally additive polynomials on Banach lattices, providing new characterizations and continuity results using localisation techniques.

Contribution

It introduces a Nakano Carrier theorem for orthogonally additive polynomials, generalizing previous results and offering new characterizations and continuity properties.

Findings

Orthogonal additivity characterized via localisation techniques

Order continuity at one point implies order continuity everywhere for orthogonally additive polynomials

Counterexample shows the result does not hold for regular polynomials in general

Abstract

We use a localisation technique to study orthogonally additive polynomials on Banach lattices. We derive alternative characterisations for orthogonal additivity of polynomials and orthosymmetry of $m$-linear mappings. We prove that an orthogonally additive polynomial which is order continuous at one point is order continuous at every point and we give an example to show that this result does not extend to regular polynomials in general. Finally, we prove a Nakano Carrier theorem for orthogonally additive polynomials, generalising a result of Kusraev.