On ordered spaces of polynomials
Let \(K\) be a cone in \(E={\mathbb R}^n\) and \(K^*\) the dual cone. The space \(L(E)\) of all endomorphisms on \(E\) is ordered by the wedge \(\widetilde{K}:=\{T \in L(E) : T(K) \subseteq K \}\). Denote by \(Q_+\) the wedge of all quasimonotone increasing (in the sense of Volkmann) endomorphisms, i.e., \(T \in Q_+\) if it follows from \(x \geq 0\) and \(\varphi(x)=0\) that \(\varphi(Tx) \geq 0\) for all \(x \in E\) and \(\varphi \in K^*\). Then \(Q_{\pm}:=(-Q_+) \cap Q_+\) is a linear subspace of \(L(E)\). Consider \((a_0,\ldots,a_n) \in {\mathbb R}^{n+1}\) as polynomial \(a_0+a_1x+\ldots+a_nx^n=p(x) \in P_n\), where \(P_n\) is ordered by the cone \(K:=\{p \in P_n : p(x)\geq 0,x \in {\mathbb R}\}\). The authors prove that dim\(Q_{\pm}\) is equal to 4 or 3 if \(n \geq 1\) is even or odd, respectively. Moreover a basis is given (Theorem 1). In Theorem 2 they present a general class of operators lying in \(Q_+\).
scientific article; zbMATH DE number 1556783