Riesz decomposition property implies asymptotic periodicity of positive and constrictive operators

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