The positive fixed points of Banach lattices

Abstract:

Let $Z$ be a Banach lattice endowed with positive cone $C$ and an order-continuous norm $|| \cdot ||$. Let $G$ be a left-amenable semigroup of positive linear endomorphisms of $Z$. Then the positive fixed points ${C_0}$ of $Z$ under $G$ form a lattice cone, and their linear span ${Z_0}$ is a Banach lattice under an order-continuous norm $|| \cdot |{|_0}$ which agrees with $|| \cdot ||$ on ${C_0}$. A counterexample shows that under the given conditions ${Z_0}$ need not contain all the fixed points of $Z$ under $G$, and need not be a sublattice of $(Z,C)$. The paper concludes with a discussion of some related results.