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

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

2008-10-29

arxiv.org/abs/0810.5352v1

Proc. Edinb. Math. Soc. (2) 53 (2010), no. 3, 609-618,

Mathematics

Functional Analysis

Scientific paper

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.

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