On the positivity of semigroups of operators (Q2713586)

Let \(E\) be a real Hausdorff topological vector space, \(K\) a (non-empty, closed, convex) cone in \(E\), and consider the initial value problem NEWLINE\[NEWLINEu(0)=x, \qquad u'=Au,\tag \(*\) NEWLINE\]NEWLINE where \(A:D\to E\) is a linear operator defined on \(D\subseteq E\). The author shows that if \((*)\) has a solution \(u:[0,T) \to K\) for each \(x\in D\cap K\) (\(T\) may depend on \(x\)), then \(A\) is quasimonotone increasing (i.e., \(\phi (Ax)\geq 0\) for any \(x\in D\cap K\) and any \(\phi \in E^*\) such that \(\phi \geq 0\) on \(K\) and \(\phi (x)=0\)); and, conversely, if \(A\) is quasimonotone increasing, \(D\cap \text{Int} K\neq \emptyset \), and \(x\in D\cap K\), then any solution to \((*)\) has values in \(K\). Further, if \(E\) is a Banach space, \(K\) is a solid normal cone in \(E\) containing no straight line, \(D\) is dense in \(E\), \(A\) is quasimonotone increasing and \((*)\) has a solution \(u:[0,\infty)\to D\) for any \(x\in D\), then this solution is unique and the corresponding operators \(U(t):x\mapsto u(t)\) form a \(C_0\)-semigroup.