Orthogonally Additive Sums of Powers of Linear Functionals

TL;DR

This paper characterizes when sums of powers of linear functionals on Banach lattices are orthogonally additive, revealing conditions on the functionals and providing counterexamples for certain cases.

Contribution

It establishes a precise criterion for orthogonal additivity of sums of powers of linear functionals and shows the limitations of this criterion for specific parameter values.

Findings

Orthogonal additivity holds if and only if each functional or its negative is a lattice homomorphism for k<m.

Counterexamples demonstrate the criterion does not extend to the case k=m.

Provides a complete characterization for the case k<m in Banach lattices.

Abstract

Let $E$ be a Banach lattice, $\lambda_1,\lambda_2,\ldots,\lambda_k$ non-zero scalars and $\varphi_1,\varphi_2,\ldots,\varphi_k$ pairwise independent linear functionals on $E$. We show that if $k<m$ then $\sum_{j=1}^k\lambda_j\varphi_j^m$ is orthogonally additive if and only if $\varphi_j$ or $-\varphi_j$ 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 $k=m$.