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

DANIEL CARANDO; SILVIA LASSALLE; IGNACIO ZALDUENDO

PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY

CAMBRIDGE UNIV PRESS

Año: 2010 vol. 53 p. 609 - 618

0013-0915

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