On extension of regular homogeneous orthogonally additive polynomials

Vladikavkaz Mathematical Journal 2011. Vol. 13. Issue 4.

Abstract:
A homogeneous polynomial is said to be \textit{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
\(\mathcal{P}^{r}_{oa}(^{s}E,F)\) and \(\mathcal{E}(P)\) 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 \(P\in\mathcal{P}^{r}_{oa}(^{s}E,F)\). The following two theorems are the main results of the article. All vector lattices are assumed to be Archimedean.

Theorem 4. 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 \(\widehat{\mathcal{E}}: \mathcal{P}_{oa}^{r}({^s}G,F) \rightarrow\mathcal{P}_{oa}^{r}({^s}E,F)\) (a "simultaneous extension" operator)
such that \(\mathcal{R}_{p}\circ\widehat{\mathcal{E}}=I\), where \(I\) is the identity operator in \(\mathcal{P}^{r}_{oa}({^s}G,F)\).

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 \(\mathcal{E}(P)\) is nonempty for a positive orthogonally additive \(s\)-homogeneous polynomial \(P: E\to F\). A polynomial \(\widehat{P}\in\mathcal{E}(P)\)
is an extreme point of \(\mathcal{E}(P)$\) if and only if \( \inf\big\{\widehat{P}\big(\big|(x^s+u^s)^{\frac1{s}}\big|\big):\ u\in G\big\}=0\quad (x\in E).\)

Keywords: vector lattice, homogeneous polynomial, positive multilinear operator, regular polynomial, orthogonal additivity, extreme extension.

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