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

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 $h:C(K)\to L^1(\mu )$ of bounded type.