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Positive one-parameter semigroups on ordered Banach spaces. (English) Zbl 0554.47022

In this review there is a description of positive continuous one- parameter semigroups acting on ordered Banach spaces. The review is in two parts.
Firstly there is a discussion of the general structure of ordered Banach spaces, including normality and generation properties of the cones of positive elements, with particular emphasis on monotone properties of the norm. The special cases of Banach lattices, order-unit spaces, and base- norm spaces, are also examined.
Secondly there is an exposition of the theory of positive strongly continuous semigroups on ordered Banach spaces, and positive weak*- continuous semigroups on the dual spaces. Initially there are analogous of the Feller-Miyadera-Phillips and Hille-Yosida theorems on generation of positive semigroups. Subsequently there is an analysis of strict positivity, irreducibility, and spectral properties, in parallel with the Perron-Frobenius theory of positive matrices.

MSC:

47D03 Groups and semigroups of linear operators
47B60 Linear operators on ordered spaces
46A40 Ordered topological linear spaces, vector lattices
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
46B42 Banach lattices
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