Author: Eiko

Time: 2025-02-02 15:58:11 - 2025-02-02 15:58:11 (UTC)

References: p-adic Differential Equations by Kiran S. Kedlaya

Spectral Radius of Bounded Operators

Let $G$ be any normed abelian group, and $T\in \mathrm{End}(G)$ a bounded operator. The simple operator norm (which is a sub-multiplicative norm) is

$$|T{|}_G=\underset{x\in G,x\ne 0}{sup}\frac{|Tx|}{|x|}.$$

But this is not an invariant under norm equivalence (i.e. $|\cdot {|}_1\asymp |\cdot {|}_2$), so we define the spectral radius of $T$ as the limit

$$|T{|}_{sp}=\underset{n\to \mathrm{\infty }}{lim}|T^n{|}_G^{1/n}.$$

It is clear that $|T{|}_{sp}$ is invariant under norm equivalence. But keep in mind the spectral radius is not a norm or a semi-norm.