A mean ergodic theorem in Banach spaces

Abstract:

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