SpectralRadius 2002-12-09 15:28:34 2007-08-25 10:09:10 Definition spectrum % 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 If $V$ is a vector space over $\mathbb{C}$, the spectrum of a linear mapping $T:V\rightarrow V$ is the set \[\sigma(T) = \{\lambda\in \mathbb{C}: T-\lambda I \mbox{is not invertible}\},\] where $I$ denotes the identity mapping. If $V$ is finite dimensional, the spectrum of $T$ is precisely the set of its eigenvalues. For infinite dimensional spaces this is not generally true, although it is true that each eigenvalue of $T$ belongs to $\sigma(T)$. The \emph{spectral radius} of $T$ is \[\rho(T) = \sup \{|\lambda|:\lambda\in\sigma(T)\}.\] More generally, the spectrum and spectral radius can be defined for Banach algebras with identity element: If $\mathcal{A}$ is a Banach algebra over $\mathbb{C}$ with identity element $e$, the spectrum of an element $a \in \mathcal{A}$ is the set $$\sigma(a) = \{ \lambda \in \mathbb{C} : a - \lambda e \mbox{is not invertible in} \mathcal{A} \}$$ The spectral radius of $a$ is $\rho(a) = \sup \{| \lambda | : \lambda \in \sigma(a) \}$.