Riesz decomposition property implies asymptotic periodicity of positive and constrictive operators

Abstract

Consider a linear and positive operator T {\mathbf {T}} acting on an ordered, F F -normed linear space X {\mathbf {X}} . Assume that there exists an open neighborhood U ∋ 0 {\mathbf {U}} \ni {\mathbf {0}} such that the trajectory { T n ( x ) } \left \{ {{{\mathbf {T}}^n}({\mathbf {x}})} \right \} is attracted to a compact set F U {{\mathbf {F}}_{\mathbf {U}}} whenever x {\mathbf {x}} is taken from U {\mathbf {U}} and that the positive cone X + {{\mathbf {X}}_ + } is closed, proper, and reproducing. It is shown that if ( X , X + ) ({\mathbf {X}},{{\mathbf {X}}_ + }) has the Riesz Decomposition Property then T {\mathbf {T}} has asymptotically periodic iterates.