Acta Applicandae Mathematica

, Volume 2, Issue 3–4, pp 221–296 | Cite as

Positive one-parameter semigroups on ordered banach spaces

  • Charles J. K. Batty
  • Derek W. Robinson
Article

Abstract

In this review we describe the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces. The review is in two parts.

First we discuss the general structure of ordered Banach spaces and their ordered duals. We examine 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.

Second we develop the theory of positive strongly continuous semigroups on ordered Banach spaces, and positive weak*-continuous semigroups on the dual spaces. Initially we derive analogues of the Feller-Miyadera-Phillips and Hille-Yosida theorems on generation of positive semigroups. Subsequently we analyse strict positivity, irreducibility, and spectral properties, in parallel with the Perron-Frobenius theory of positive matrices.

AMS (MOS) subject classifications (1980)

46A40 15A48 06F20 46L05 46L10 54C40 54C45 47B55 47D05 47D07 47B44 46L55 46L60 46B20 

Key words

Ordered Banach space normal cone generating cone monotone norm Riesz norm orderunit Banach lattice C*-algebra half-norm dissipative Co-semigroup Co*-semigroup Perron-Frobenius theory irreducible semigroup 

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Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • Charles J. K. Batty
    • 1
  • Derek W. Robinson
    • 1
  1. 1.Department of Mathematics, Institute of Advanced StudiesAustralian National UniversityCanberraAustralia

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