# Cesaro bounded Operator which is not power bounded

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

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Consider $T = \pmatrix{-1 & 1\cr 0 & -1\cr}$. Then $T^n = \pmatrix{(-1)^n & (-1)^{n+1} n\cr 0 & (-1)^n\cr}$ so $T$ is not power-bounded. But $A_n = \pmatrix{\frac{1-(-1)^n}{2n} & \frac{(-1)^n}{2} + \frac{1-(-1)^n}{4n}\cr 0 & \frac{1-(-1)^n}{2n}\cr}$ so it is cesaro-bounded.

You could replace $-1$ by any $\lambda \ne 1$ with $|\lambda|=1$.

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Thank you for this nice example! –  Matthias Dec 14 '11 at 20:37