spectral values classification

Spectral points classificationFernando Sanz Gamiz

Definition 1.

Let X a topological vector spaceMathworldPlanetmath and A:XDAX a linear transformation with domain DA. Depending on the properties of11the notation (λ-A) is to be understood as λI-A with I the identity transformation and R(λ-A) is the range of (λ-A) (λ-A) the following definitions apply:

(λ-A)-1 Boundness of (λ-A)-1 R(λ-A) Set to which λ belongs
exists bounded dense in X resolvent set ρ(A)
exists unbounded dense in X continuous spectrum Cσ(A)
exists bounded or unbounded in X not dense in X residual spectrum Rσ(A)
not exists dense or not dense in X puntual spectrum Pσ(A)
Remark 1.

It is obvious that, if F is the field of possible values for λ (usually F= or F=) then F=ρ(A)Cσ(A)Rσ(A)Pσ(A), that is, these definitions cover all the possibilities for λ. The complement of the resolvent set is called spectrum of the operator A, i.e., σ(A)=Cσ(A)Rσ(A)Pσ(A)

Remark 2.

In the finite dimensional case if (λ-A)-1 exists it must be bounded, since all finite dimensional linear mappings are bounded. This existence also implies that the range of (λ-A) must be the whole X. So, in the finite dimensional case the only spectral values we can encounter are point spectrum values (eigenvalues).

Title spectral values classification
Canonical name SpectralValuesClassification
Date of creation 2013-03-22 18:52:01
Last modified on 2013-03-22 18:52:01
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 5
Author fernsanz (8869)
Entry type Definition
Classification msc 15A18
Synonym eigenvalues
Synonym spectrum
Related topic Eigenvalue
Related topic SpectrumOfAMuI
Related topic InvertibleLinearTransformation
Defines spectrum
Defines point spectrum
Defines residual spectrum
Defines continuous spectrum
Defines resolvent set
Defines eigenvalues
Defines puntual spectrum
Defines point spectral value
Defines residual spectral value
Defines continuous spectral value
Defines resolvent set value