Asymptotic periodicity of the iterates of positivity preserving operators

Milan Miklavčič

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 ∈ X^{+}, y ∈ X^{+}, α ∈ [0, ∞) then x + y ∈ X^{+} and αx ∈ X^{+}; (ii) there exists M_{0} ∈ (0, ∞) such that for each x ∈ X there exist x_{+} ∈ X^{+}

and x_{−} ∈ X^{+} which satisfy
x = x_{+} − x_{−}, ∥x_{+}∥ ≤ M_{0}∥x∥, ∥x_{−}∥ ≤ M_{0}∥x∥
and if x = y_{+} − y_{−} for some y_{+} ∈ X^{+}, y_{−} ∈ X^{+} then y_{+} − x_{+} ∈ X^{+};

(iii) if x ∈ X^{+}, y ∈ X^{+} then ∥x∥ ≤ ∥x + y∥. (A3) B is a bounded linear operator on X. (A4) BX^{+} ⊂ X^{+}. (A5) F_{0} is a nonempty compact subset of X and lim_{n→∞} dist(B^{n}x, F_{0}) = 0 whenever

x ∈ X^{+} and ∥x∥ = 1. Then B^{n}x is asymptotically periodic for every x ∈ X. This, and other properties

of B, are proven in the paper.