Definition:Spectral Radius

Definition

Bounded Linear Operator

Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\C$.

Let $A : X \to X$ be a bounded linear operator.

Let $\map \sigma A$ be the spectrum of $A$.

The spectral radius of $A$ is defined as:

$\ds \size {\map \sigma A} := \sup_{z \mathop \in \map \sigma A} \cmod z$

Banach Algebra

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:

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