Good evening!

Let X be a banachspace and T a bounded linear operator on X. The cesaro avearges of T are defined as:

$A_n:=\frac{1}{n} \sum\limits_{j=0}^{n-1}T^j $

We call T cesaro bounded if: $\sup_{n \geq 0}\Vert A_n \Vert<\infty$.

We call T power bounded if: $\sup_{n \geq 0}\Vert T^n \Vert<\infty$.

E. Hille showed in "Remarks on ergodic theorems, Trans. Amer. Math. Soc. 57, 1945, 246-269" that one can find a cesaro bounded Operator in $\mathcal{L}(L_1[0,1])$ which is not power bounded.

**Here is my question**: can this be achieved in a finite dimesional setting?

With best regards,

Matthias