Power Bounded Operator on Finite Dimensional Normed Spaces

Question

Let $(\mathbb{X},||\cdot||_{\mathbb{X}})$ be a finite dimensional normed space (i.e, $(\mathbb{X},||\cdot||_{\mathbb{X}})$ is a banach space).

Let $T: (\mathbb{X},||\cdot||_{\mathbb{X}}) \rightarrow (\mathbb{X},||\cdot||_{\mathbb{X}})$ be a bounded linear operator, that is power bounded.

How to prove that $T$ is uniformly ergodic ?

Answer 1

Hint: Make use of the Jordan normal form of $T$. Power bounded implies that the spectral radius of $T$ is $≤1$ hence all diagonal elements are $≤1$ in absolute value. Check explicitly that the sequence $\frac1n\sum_{k=1}^nT^k$ converges if $T$ is a Jordan block with values $≤1$s on the diagonal.