# spectral values classification

Spectral points classificationFernando Sanz Gamiz

###### 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 of11the notation $(\lambda-A)$ is to be understood as $\lambda I-A$ with $I$ the identity transformation and $R(\lambda-A)$ is the range of $(\lambda-A)$ $(\lambda-A)$ 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).

 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