Category:Gelfand's Spectral Radius Formula

This category contains pages concerning Gelfand's Spectral Radius Formula:

Bounded Linear Operator

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

Let $\map B X$ be the set of bounded linear operators on $X$.

Let $\norm \cdot_{\map B X}$ denote the operator norm on $\map B X$.

Let $T \in \map B X$.

Let $\size {\map \sigma T}$ be the spectral radius of $T$.

Then:

$\ds \size {\map \sigma T} = \lim_{n \mathop \to \infty} \paren {\norm {T^n}_{\map B X} }^{1/n} = \inf_{n \mathop \in \N_{>0} } \paren {\norm {T^n}_{\map B X} }^{1/n}$

Banach Algebra

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

Let $x \in A$.

Let $\map {r_A} x$ be the spectral radius of $x$ in $A$.

Then, we have:

$\ds \map {r_A} x = \inf_{n \mathop \in \N_{> 0} } \norm {x^n}^{1/n} = \lim_{n \mathop \to \infty} \norm {x^n}^{1/n}$