Riesz representation theorems for orthogonally additive functions

LASSALLE, S.

Madrid, España

Conferencia; XI Function Theory on infinite dimensional spaces; 2009

Universidad Complutense de Madrid

A function $f$ is said to  be orthogonally additive if $f(x+y)=f(x)+f(y)$ whenever $x$ and $y$ are mutually orthogonal. In this talk we review the Riesz representation theorem known for $n$-homogeneous orthogonally additive polynomials  $P$ over $C(K)$. We look at this result as a linearitazion theorem via the mapping   $x -> x^n$ which factors all  orthogonally additive polynomials through some linear form $mu$. Then, 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 $C(K)$ admits a representation via some regular measrure some $mu$ and a holomorphic function $h: C(K) -> L^1(mu)$ of bounded type.