Orthogonally additive polynomials on convolution algebras associated with a compact group

TL;DR

This paper characterizes orthogonally additive, continuous homogeneous polynomials on convolution algebras over compact groups, showing they can be represented via linear maps involving convolution powers, and explores similar results for other algebras.

Contribution

It establishes a unique linear representation for orthogonally additive polynomials on $L^1(G)$ and investigates analogous results for other convolution algebras.

Findings

Existence of a unique continuous linear map representing the polynomial.

Representation of polynomials as linear maps applied to convolution powers.

Extension of results to other convolution algebras like $L^p(G)$ and $C(G)$.

Abstract

Let $G$ be a compact group, let $X$ be a Banach space, and let $P\colon L^1(G)\to X$ be an orthogonally additive, continuous $n$-homogeneous polynomial. Then we show that there exists a unique continuous linear map $\Phi\colon L^1(G)\to X$ such that $P(f)=\Phi \bigl(f\ast\stackrel{n}{\cdots}\ast f \bigr)$ for each $f\in L^1(G)$. We also seek analogues of this result about $L^1(G)$ for various other convolution algebras, including $L^p(G)$, for $1< p\le\infty$, and $C(G)$.