Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition.
- $\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$
- $\frac{1}{n}\lVert T\rVert^{n}\rightarrow0$, as $n \rightarrow\infty$
For $x \in X$, take $x_{n}=\frac{1}{n} \sum\limits_{i=0}^{n-1}T^{i}x$. If there is a subquence $\{x_{n_{k}}\}$ which has a weak limit $x^{*}$(in the weak topology), prove that $x_{n} $ is convergent to $ x^{*}$, in the norm topology, and $Tx^{*}=x^{*}$
This can been seen as a generalization of the von Neumann Ergodic Theorem for Banach spaces.
Any advice and discussions will be appreciated.