spectral radius
If V is a vector space over ℂ, the spectrum of a linear mapping T:V→V is the set
σ(T)={λ∈ℂ: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 eigenvalues. 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
ρ(T)=sup{|λ|:λ∈σ(T)}. |
More generally, the spectrum and spectral radius can be defined for Banach algebras with identity element
: If 𝒜 is a Banach algebra over ℂ with identity element e, the spectrum of an element a∈𝒜 is the set
σ(a)={λ∈ℂ:a-λeis not invertible in𝒜} |
The spectral radius of a is ρ(a)=sup{|λ|:λ∈σ(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 |