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

It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form $P(x)=\int_K x^k dμ.$ Thus $x\mapsto x^k$ factors all orthogonally additive polynomials through some linear form $μ$. 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 $f(x)=\int_K h(x) dμ$ for some $μ$ and holomorphic $h\colon C(K) \to L^1(μ)$ of bounded type.

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