Question

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

Answer 1

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 $|\lambda|=1$.

Answer 2

I think you can read off from the Jordan canonical form that both conditions are equivalent to the spectral radius being at most one and every eigenvalue of modulus one having algebraic and geometric multiplict the same.

EDIT: As Robert Israel points out in his answer, my answer is wrong.