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:

$\ds \map {r_A} x = \sup_{\lambda \in \map {\sigma_A} x} \cmod \lambda$

Sources

• 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $4.5$: The spectrum