PlanetMath: spectral radius

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spectral radius

(Definition)

If is a vector space over $\mathbb{C}$ , the spectrum of a linear mapping $T:V\rightarrow V$ is the set

$\displaystyle \sigma(T) = \{\lambda\in \mathbb{C}: T-\lambda I$ is not invertible $\displaystyle \},$

where

denotes the identity mapping. If

is finite dimensional, the spectrum of

is precisely the set of its eigenvalues. For infinite dimensional spaces this is not generally true, although it is true that each eigenvalue of

belongs to $\sigma(T)$ . The spectral radius of

$\displaystyle \rho(T) = \sup \{\vert\lambda\vert:\lambda\in\sigma(T)\}.$

More generally, the spectrum and spectral radius can be defined for Banach algebras with identity element: If $\mathcal{A}$ is a Banach algebra over $\mathbb{C}$ with identity element , the spectrum of an element $a \in \mathcal{A}$ is the set

$\displaystyle \sigma(a) = \{ \lambda \in \mathbb{C} : a - \lambda e$ is not invertible in $\displaystyle \mathcal{A} \}$

The spectral radius of is $\rho(a) = \sup \{\vert \lambda \vert : \lambda \in \sigma(a) \}$ .

"spectral radius" is owned by Koro.

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Also defines:

spectrum

Attachments:: spectrum is a non-empty compact set (Theorem) by asteroid

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Cross-references: identity element, Banach algebras, eigenvalue, infinite dimensional, eigenvalues, finite dimensional, identity mapping, linear mapping, vector space
There are 51 references to this entry.

This is version 8 of spectral radius, born on 2002-12-09, modified 2007-08-25.
Object id is 3703, canonical name is SpectralRadius.
Accessed 11413 times total.

Classification:

AMS MSC:

58C40 (Global analysis, analysis on manifolds :: Calculus on manifolds; nonlinear operators :: Spectral theory; eigenvalue problems)

Pending Errata and Addenda

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