Positive fixed points of lattices under semigroups of positive linear operators

Abstract

Let Z be a Banach lattice endowed with positive cone C and an order-continuous norm [...]. Let G be a semigroup of positive linear endomorphisms of Z. We seek conditions on G sufficient to ensure that the positive fixed points Co of Z under G form a lattice cone, and that their linear span Zo is a Banach lattice under an order-continuous norm [...] which agress with [...] on Co, although we do not require that Zo contain all the fixed points of Z under G, nor that Zo be a sublattice of (Z,C). We give a simple embedding construction which allows such results to be read off directly from appropriate fixed point theorems. In particular, we show that left-reversibility of G (a weaker condition than left-amenability) suffices. Results of this kind find application in statistical physics and elsewhere. [see PDF of report for correct notation]