Positive one-parameter semigroups on ordered banach spaces

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.

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Batty, C.J.K., Robinson, D.W. Positive one-parameter semigroups on ordered banach spaces. Acta Appl Math 2, 221–296 (1984). https://doi.org/10.1007/BF02280855

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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
  • C o-semigroup
  • C *o -semigroup
  • Perron-Frobenius theory
  • irreducible semigroup