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Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. On extension of regular homogeneous orthogonally additive polynomialsVladikavkaz 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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