A mean ergodic theorem in Banach spaces

Abstract

Let Γ = { U t : t ∈ Λ } ( Λ = Z + − { 0 } or  R + − { 0 } ) \Gamma = \left \{ {{U_t}:t \in \Lambda } \right \}\left ( {\Lambda = {{\mathbf {Z}}^ + } - \left \{ 0 \right \}{\text {or }}{{\mathbf {R}}^ + } - \left \{ 0 \right \}} \right ) be a commuting family of nonexpansive affine operators in a Banach space X X satisfying the following conditions: (i) there is a function M ( x | Γ ) ≥ 0 M\left ( {x\left | \Gamma \right .} \right ) \geq 0 defined on X X such that \[ ‖ U t + s x − U t U s x ‖ ≤ M ( x | Γ ) ( s , t ∈ Λ , x ∈ X ) , \left \| {{U_{t + s}}x - {U_t}{U_s}x} \right \| \leq M\left ( {x\left | \Gamma \right .} \right )\quad \left ( {s,t \in \Lambda ,x \in X} \right ), \] (ii) \[ sup { t − 1 ‖ U t x ‖ : t ∈ Λ , t ≥ 1 } = K ( x | Γ ) > ∞ ( x ∈ X ) . \sup \left \{ {{t^{ - 1}}\left \| {{U_t}x} \right \|:t \in \Lambda ,t \geq 1} \right \} = K\left ( {x\left | \Gamma \right .} \right ) > \infty \left ( {x \in X} \right ). \] Then it is proved that if { t − 1 U t x : t ∈ Λ } \left \{ {{t^{ - 1}}{U_t}x:t \in \Lambda } \right \} is relatively compact for x ∈ X x \in X , the limit X − lim t → ∞ t − 1 U t x = x ¯ X -\lim _{t \to \infty }{t^{ - 1}}{U_t}x = \bar x exists in X X and U t x ¯ = x ¯ ( t ∈ Λ ) \overline {{U_t}x} = \overline x \left ( {t \in \Lambda } \right ) .