Definition:Spectral Radius/Banach Algebra

Definition
Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.

Let $x \in A$.

Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.

We define the spectral radius $\map {r_A} x$ of $x$ in $A$ by:
 * $\ds \map {r_A} x = \sup_{\lambda \in \map {\sigma_A} x} \cmod \lambda$