spectral values classification SpectralValuesClassification 2009-03-24 21:57:26 2009-03-27 20:14:12 Definition spectrum of $A-\mu I$ spectrum point spectrum residual spectrum continuous spectrum resolvent set eigenvalues puntual spectrum point spectral value residual spectral value continuous spectral value resolvent set value spectrum eigenvalues vector space topological vector space matrix transformation identity transformation domain dense % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{rem}{Remark} \numberwithin{equation}{section} \title{Spectral points classification}% \author{Fernando Sanz Gamiz}% \begin{defn} 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\footnote{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 $(\lambda -A)$} $(\lambda - A)$ the following definitions apply: \medskip \begin{center} \begin{tabular}{cccc} $(\lambda-A)^{-1}$ & Boundness of $(\lambda-A)^{-1}$ & $R(\lambda-A)$ & Set to which $\lambda$ belongs\\ \hline \hline exists & bounded & dense in X& resolvent set $\rho(A)$\\ \hline exists & unbounded & dense in X & continuous spectrum $C\sigma(A)$\\ \hline exists & bounded or unbounded in X & not dense in X & residual spectrum $R\sigma(A)$\\ \hline not exists & & dense or not dense in X & puntual spectrum $P\sigma(A)$\\ \end{tabular} \end{center} \end{defn} \begin{rem} 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 \emph{spectrum} of the operator A, i.e., $\sigma(A)=C\sigma(A) \cup R\sigma(A) \cup P\sigma(A)$ \end{rem} \bigskip \begin{rem} 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). \end{rem}