Question

I am finishing up a paper and I would like to be able to quote a theorem that does what is said in the title. To be specific let me introduce some notations: ${\bf F}$ is a local field of charateristic $p>0$, $|\cdot|$ is the absolute value on ${\bf F}$ normalized by requiring that $|\pi|=q^{-1}$ where $\pi$ is a uniformizer for the valutation ring ${\bf A}$ of ${\bf F}$. Now let $X$ be a ${\bf F}$-vector space of finite dimension, $\|\cdot\|$ is a norm on $X$ having the same value group as the normalized absolute value. Given an endomorphism $T:X\rightarrow X$ let $\|T\|$ be the standard operator norm with respect to the norm $\|\cdot\|$. What I would like to have a reference for is the following assertion:

Assume that $T$ has all its eigenvalue in ${\bf F}$, and denote the set of eigenvalues of $T$ by $Sp(T)$, then $ \lim_{d\to\infty} \|T^d\|^{\frac1d}= \max_{\lambda {\in}Sp (T)} |\lambda|$

note: I am not asking for a proof, I just would like to be able to point to some precise reference instead of asking the reader to adapt result obtain in different setting to this situation.

Answer 1

I don't know of a reference off hand, but I don't see any harm in just claiming that the assertion easily follows from the Jordan decomposition of $T$.

After all, if you know that the limit on the left hand side exists you just write $T = D + N$ for some diagonalizable over $\mathbf{F}$ endomorphism $D$, and some nilpotent endomorphism $N$ where $D$ and $N$ commute, and observe that for large $n$,

$$||T^{p^n}||^{p^{-n}} = ||D^{p^n}||^{p^{-n}} = ||D|| = \max_{\lambda \in Sp(T)}|\lambda|.$$

You can also show in this way that the limit exists by being a bit more careful.