References: p-adic Differential Equations by Kiran S. Kedlaya
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
\[ |T|_G = \sup_{x\in G,x\neq 0} \frac{|Tx|}{|x|}. \]
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
\[ |T|_{sp} = \lim_{n\to\infty} |T^n|_G^{1/n}. \]
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.