On extension of regular homogeneous orthogonally additive polynomials

Z. A. Kusraeva

Vladikavkazskij matematičeskij žurnal, Tome 13 (2011) no. 4, pp. 28-34

Résumé

A homogeneous polynomial is said to be positive if the generating symmetric multilinear operator is positive and regular if it is representable as the difference of two positive polynomials. A polynomial $P$ is orthogonally additive if $P(x+y)=P(x)+P(y)$ for disjoint $x$ and $y$. Let $\mathscr P^r_\mathrm{oa}(^sE,F)$ and $\mathscr E(P)$ stand for the sets of all regular $s$-homogeneous orthogonally additive polynomials from $E$ to $F$ and of all positive orthogonally additive $s$-homogeneous extensions of a positive polynomial $P\in\mathscr P^r_\mathrm{oa}(^sE,F)$. The following two theorems are the main results of the article. All vector lattices are assumed to be Archimedean. Theorem 4. {\it Let $G$ be a majorizing sublattice of a vector lattice $E$ and $F$ be a Dedekind complete vector lattice. Then there exists an order continuous lattice homomorphism $\widehat{\mathscr E}\colon\mathscr P_\mathrm{oa}^r(^sG,F)\to\mathscr P_\mathrm{oa}^r(^sE,F)$ (a “simultaneous extension” operator) such that $\mathscr R_p\circ\widehat{\mathscr E}=I$, where $I$ is the identity operator in $\mathscr P^r_\mathrm{oa}(^sG,F)$.} Theorem 6. Let $E,F$ and $G$ be vector lattices with $F$ Dedekind complete, $E$ and $G$ uniformly complete, $G$ sublattice of $E$. Assume that the set $\mathscr E(P)$ is nonempty for a positive orthogonally additive $s$-homogeneous polynomial $P\colon E\to F$. A polynomial $\widehat P\in\mathscr E(P)$ is an extreme point of $\mathscr E(P)$ if and only if $$ \inf\big\{\widehat P\big(\big|(x^s+u^s)^{\frac1s}\big|\big)\colon u\in G\big\}=0\quad(x\in E).$$