Orthogonally Additive Sums of Powers of Linear Functionals

Abstract

Let $E$ be a Banach lattice, ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ non-zero scalars and ${\phi }_1,{\phi }_2,\dots ,{\phi }_k$ pairwise independent linear functionals on $E$. We show that if $k<m$ then ${\sum }_{j=1}^k{\lambda }_j{\phi }_j^m$ is orthogonally additive if and only if ${\phi }_j$ or $-{\phi }_j$ is a lattice homomorphism for each $j$, $1\le j\le k$. Moreover, for each $m\ge 2$, we provide an example to show that this result does not extend to the case where $k=m$.