On the representation of orthogonally additive polynomials in $\ell_p$

TL;DR

This paper provides a new proof demonstrating the isometric isomorphism between the space of orthogonally additive polynomials on _p and certain _{p/p-k} or _ spaces, clarifying their structure.

Contribution

It offers a novel proof of Sundaresan's result on the structure of orthogonally additive polynomials in _p spaces.

Findings

_o(^k_p) is isometrically isomorphic to _{p/p-k} for p>k

_o(^k_p) is isometric to _ for 1pk

The proof simplifies understanding of polynomial spaces in _p

Abstract

We present a new proof of a Sundaresan's result which shows that the space of orthogonally additive polynomials $\mathcal{P}_o(^k\ell_p)$ is isometrically isomorphic to $\ell_{p/p-k}$ if $k<p<\infty$ and to $\ell_\infty$ if $1\leq p\leq k$.