spectral radius

If V is a vector spaceMathworldPlanetmath over , the spectrum of a linear mapping T:VV is the set

σ(T)={λ:T-λIis not invertible},

where I denotes the identity mapping. If V is finite dimensional, the spectrum of T is precisely the set of its eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath. For infinite dimensional spaces this is not generally true, although it is true that each eigenvalue of T belongs to σ(T). The spectral radius of T is


More generally, the spectrum and spectral radius can be defined for Banach algebrasMathworldPlanetmath with identity elementMathworldPlanetmath: If 𝒜 is a Banach algebra over with identity element e, the spectrum of an element a𝒜 is the set

σ(a)={λ:a-λeis not invertible in𝒜}

The spectral radius of a is ρ(a)=sup{|λ|:λσ(a)}.

Title spectral radius
Canonical name SpectralRadius
Date of creation 2013-03-22 13:13:58
Last modified on 2013-03-22 13:13:58
Owner Koro (127)
Last modified by Koro (127)
Numerical id 11
Author Koro (127)
Entry type Definition
Classification msc 58C40
Defines spectrum