arXiv:1308.3263v1 [math.FA] 14 Aug 2013 A short proof of the Arendt-Chernoff-Kato theorem Sergiy Koshkin Department of Computer and Mathematical Sciences University of Houston-Downtown One Main Street Houston, TX 77002 e-mail: koshkins@uhd.edu Abstract We give a short new proof of the Arendt-Chernoff-Kato theorem, which characterizes gener- ators of positive C 0 semigroups in order unit spaces. The proof avoids half-norms and subdif- ferentials, and is based on a sufficient condition for an operator to have positive inverse, which is new even for matrices. Keywords: ordered Banach space, positive cone, order unit, positive off-diagonal, C 0 semi- group Let A be a matrix with non-negative off-diagonal entries, such matrices are called positive off- diagonal, and consider the equation Ax = −z, where z is a vector with strictly positive entries. A classical result states that if this equation has a positive solution x then −A -1 exists and has non-negative entries [4, 23.1]. As shown below, the result generalizes to operators in order unit spaces. Moreover, it turns out that negative invertibility follows from positive solvability of Ax = −z even if A is not positive off-diagonal, but has a weaker property that we call somewhere positivity. Unlike positivity off-diagonal, somewhere positivity makes sense even for operators between different spaces. Based on this generalization we give a simple proof of the Arendt- Chernoff-Kato theorem without using half-norms or subdifferentials as in [2]. Recall that the theorem characterizes generators of positive C 0 semigroups on order unit spaces as densely defined positive off-diagonal linear operators that satisfy a range condition. In addition to simplifying the proof we replace the positive off-diagonal property with an a priori weaker one, which is easier to verify. Let X be a real Banach space partially ordered by a closed proper cone X + with non-empty interior, int X + and ∂X + denote the interior and the boundary of X + respectively. If X + is also normal, i.e. order intervals are norm bounded, then X is called an order unit space. In such spaces any element e ∈ int X + is called an order unit and defines an equivalent norm ||x|| e := inf {λ> 0 |− λe ≤ x ≤ λe} on X [3, A.2.7]. The dual cone X *+ := {ϕ ∈ X * |〈ϕ,x〉≥ 0 for all x ∈ X + } then is also closed, proper and normal in the dual norm. An element x ∈ X + is quasi-interior if 〈ϕ,x〉 > 0 for all ϕ ∈ X *+ \{0}. By a theorem of Krein, in order unit spaces any quasi-interior element is an order unit [3, A.2.10], but such elements may exist even if int X + = ∅. Let X be an order unit space and Y be an ordered Banach space. Definition 1. A densely defined linear operator A : D A → Y with domain D A ⊆ X is called somewhere positive if for every x ∈D A ∩ ∂X + there is ψ ∈ Y *+ \{0} such that 〈ψ,Ax〉≥ 0. 1