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 withdomain $D_{A}$ . Depending on the properties of¹¹the notation $(\\lambda-A)$ is to beunderstood as $\\lambda I-A$ with $I$ the identity transformation and $R(\\lambda-A)$ is the rangeof $(\\lambda-A)$ $(\\lambda-A)$ the following definitions apply:

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, thesedefinitions 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 finitedimensional 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 wecan encounter are point spectrum values (eigenvalues).

单词	SpectralValuesClassification
释义	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 withdomain $D_{A}$ . Depending on the properties of¹¹the notation $(\\lambda-A)$ is to beunderstood as $\\lambda I-A$ with $I$ the identity transformation and $R(\\lambda-A)$ is the rangeof $(\\lambda-A)$ $(\\lambda-A)$ the following definitions apply: 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, thesedefinitions 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 finitedimensional 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 wecan encounter are point spectrum values (eigenvalues).
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