On positive semigroups
Let \((E,E_+,\|.\|)\) be an ordered Banach space. Assume that the given norm \(\|.\|\) as well as the norm on the dual space are monotonous. Define the associated half norm N on E by \(N(a):=\inf \{\| b\|: b\geq a,\quad b\geq 0\}.\) Under these assumptions the following characterization of the generator A of a positive semigroup is proved: Let A be a densely-defined, closed operator in E. The following assertions are equivalent: (i) A is the generator of a positive \(C_ 0\)-semigroup \((S_ t)\) with bounds \(\| S_ t\|_+(:=\sup \{\| S_ ta\|: a\geq 0,\quad \| a\| \leq 1\})\leq Me^{\omega t}\) for all \(t\geq 0.\) (ii) \(range(I+\alpha A)=E\) and \(N((I+\alpha A)^ na)\geq (1-\alpha \omega)^ nN(a)/M\) for all \(a\in {\mathcal D}(A^ n)\), \(n\geq 1\) and \(\alpha >0\) small enough.
generator of a positive \(C_ 0\)-semigroup
scientific article; zbMATH DE number 3907201