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

CARANDO, D., LASSALLE, S. ZALDUENDO, I.

PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY

Cambridge Univ. Press The Edinburgh Bldg

Lugar: Cambridge England; Año: 2010 vol. 253 p. 609 - 618

0013-0915

It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form$$P(x)=int_K x^k , dmu .$$Thus $xmapsto x^k$ 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 function of bounded type $f$ over $C(K)$ is of the form$$f(x)=int_K h(x) , dmu$$for some $mu$ and holomorphic $hcolon C(K) ightarrow L^1(mu)$ of bounded type.