Orthogonally additive holomorphic functions of bounded type over $C(K)$

TL;DR

The paper demonstrates that orthogonally additive holomorphic functions of bounded type over C(K) can be represented via integrals involving a holomorphic function h and a measure, extending known linearization results beyond homogeneous polynomials.

Contribution

It extends the linearization of orthogonally additive functions from homogeneous polynomials to more general holomorphic functions of bounded type.

Findings

Representation of orthogonally additive holomorphic functions as integrals involving a holomorphic h

No linearization exists for non-homogeneous orthogonally additive functions

Extension of known polynomial results to broader class of holomorphic functions

Abstract

It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form $$ P(x)=\int_K x^k d\mu . $$ Thus $x\mapsto x^k$ factors all orthogonally additive polynomials through some linear form $\mu$. We show that no such linearization is possible without homogeneity. However, we also show that every orthogonally additive holomorphic functions of bounded type $f$ over $C(K)$ is of the form $$ f(x)=\int_K h(x) d\mu $$ for some $\mu$ and holomorphic $h\colon C(K) \to L^1(\mu)$ of bounded type.