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)= ∫Kxkdμ. Thus, x → xk 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 function of bounded type f over C(K) is of the form f(x)=∫Kh(x)dμ for some μ and holomorphic h : C (K) → L1(μ) of bounded type.