Positive One-Parameter Semigroups on Ordered Banach Spaces

Charles J. K. Batty and Derek W. Robinson
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Charles J. K. Batty: Australian National University, Department of Mathematics, Institute of Advanced Studies
Derek W. Robinson: Australian National University, Department of Mathematics, Institute of Advanced Studies

A chapter in Positive Semigroups of Operators, and Applications, 1984, pp 221-296 from Springer

Abstract: 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.

Keywords: Ordered Banach space; normal cone; generating cone; monotone norm; Riesz norm; order-unit; Banach lattice; C*-algebra; half-norm; dissipative; C o-semigroup; $$C_o^* $$ -semigroup; Perron-Frobenius theory; irreducible semigroup (search for similar items in EconPapers)
Date: 1984
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DOI: 10.1007/978-94-009-6484-6_2

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