|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
spectral values classification
|
(Definition)
|
|
Definition 1 Let $X$ a topological vector space and $A: X \supset D_A \longrightarrow X$ a linear transformation with domain $D_A$ . Depending on the properties of 1 $(\lambda - A)$ the following definitions apply:
</>
| $(\lambda-A)^{-1}$ |
Boundness of $(\lambda-A)^{-1}$ |
$R(\lambda-A)$ |
Set to which $\lambda$ belongs |
| exists |
bounded |
dense in X |
resolvent set $\rho(A)$ |
| exists |
unbounded |
dense in X |
continuous spectrum $C\sigma(A)$ |
| exists |
bounded or unbounded in X |
not dense in X |
residual spectrum $R\sigma(A)$ |
| not exists |
|
dense or not dense in X |
puntual spectrum $P\sigma(A)$ |
<>
Remark 1 It is obvious that, if $F$ is the field of possible values for $\lambda$ (usually $F=\mathbb C$ or $F=\mathbb R$ ) then $F=\rho(A) \cup C\sigma(A) \cup R\sigma(A) \cup P\sigma(A)$ , that is, these definitions cover all the possibilities for $\lambda$ . The complement of the resolvent set is called spectrum of the operator A, i.e., $\sigma(A)=C\sigma(A) \cup R\sigma(A) \cup P\sigma(A)$
Remark 2 In the finite dimensional case if $(\lambda-A)^{-1}$ exists it must be bounded, since all finite dimensional linear mappings are bounded. This existence also implies that the range of $(\lambda-A)$ must be the whole X. So, in the finite dimensional case the only spectral values we can encounter are point spectrum values (eigenvalues).
Footnotes
- 1
- the notation $(\lambda -A)$ is to be understood as $\lambda I -A$ with $I$ the identity transformation and $R(\lambda-A)$ is the range of $(\lambda -A)$
|
"spectral values classification" is owned by fernsanz.
|
|
(view preamble | get metadata)
See Also: eigenvalue, spectrum of  , invertible linear transformation
| Other names: |
eigenvalues, spectrum |
| Also defines: |
spectrum, point spectrum, residual spectrum, continuous spectrum, resolvent set, eigenvalues, puntual spectrum, point spectral value, residual spectral value, continuous spectral value, resolvent set value |
| Keywords: |
spectrum, eigenvalues, vector space, topological vector space, matrix, transformation, identity transformation, domain, dense |
This object's parent.
|
|
Cross-references: spectral values, implies, finite dimensional, operator, complement, cover, field, obvious, dense, unbounded, dense in, bounded, definitions, range, identity transformation, properties, domain, linear transformation, topological vector space
There are 100 references to this entry.
This is version 2 of spectral values classification, born on 2009-03-24, modified 2009-03-27.
Object id is 11698, canonical name is SpectralValuesClassification.
Accessed 2102 times total.
Classification:
| AMS MSC: | 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
