Asymptotic periodicity of the iterates of positivity preserving operators

Abstract:

Assume that (A1) $X$ is a real Banach space. (A2) ${X^ + }$ is a closed subset of $X$ with the following properties: (i) if $x \in {X^ + }$, $y \in {X^ + }$, $\alpha \in [0, \infty )$ then $x + y \in {X^ + }$ and $\alpha x \in {X^ + }$; (ii) there exists ${M_0} \in (0, \infty )$ such that for each $x \in X$ there exist ${x_ + } \in {X^ + }$ and ${x_ - } \in {X^ + }$ which satisfy \[ x = {x_ + } - {x_ - },\qquad ||{x_ + }|| \leqslant {M_0}||x||,\qquad ||{x_ - }|| \leqslant {M_0}||x||\] and if $x = {y_ + } - {y_ - }$ for some ${y_ + } \in {X^ + }$, ${y_ - } \in {X^ + }$ then ${y_ + } - {x_ + } \in {X^ + }$; (iii) if $x \in {X^ + }$, $y \in {X^ + }$ then $||x|| \leqslant ||x + y||$. (A3) $B$ is a bounded linear operator on $X$. (A4) $B{X^ + } \subset {X^ + }$. (A5) ${F_0}$ is a nonempty compact subset of $X$ and ${\lim _{n \to \infty }}\operatorname {dist} ({B^n}x, {F_0}) = 0$ whenever $x \in {X^ + }$ and $||x|| = 1$. Then ${B^n}x$ is asymptotically periodic for every $x \in X$. This, and other properties of $B$, are proven in the paper.