Acta Applicandae Mathematica

, Volume 2, Issue 3, pp 221–296

Positive one-parameter semigroups on ordered banach spaces

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

DOI: 10.1007/BF02280855

Cite this article as:
Batty, C.J.K. & Robinson, D.W. Acta Appl Math (1984) 2: 221. doi:10.1007/BF02280855

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)

46A4015A4806F2046L0546L1054C4054C4547B5547D0547D0747B4446L5546L6046B20

Key words

Ordered Banach spacenormal conegenerating conemonotone normRiesz normorderunitBanach latticeC*-algebrahalf-normdissipativeCo-semigroupCo*-semigroupPerron-Frobenius theoryirreducible semigroup

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
  2. 2.Dept. of MathematicsUniversity of EdinburghEdinburghU.K.