Let $\rho(T)=\lim_{n->\infty}||T^n||^{1/n}$, with $T$ being an element of a Banach algebra. Then we know that the largest extent of the spectrum of $T$ (spectral radius) is $$\max\{|\lambda|:\lambda \in \sigma(T)\}=\rho(T)$$
By smallest extent I'm thinking of $\min\{|\lambda|:\lambda \in \sigma(T)\}$, but I'm not sure...
Because, when I use that notion, then we get
\begin{align} \min\{|\lambda|:\lambda \in \sigma(T)\}=&\max\{|1/\lambda|:\lambda \in \sigma(T)\}\\ =&\max\{|\lambda|:\lambda \in \sigma(T^{-1})\}\\ =&\rho(T^{-1}) \end{align}
I don't get $\rho(T^{-1})^{-1}$