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 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 $\rho(a)=\sup\{|\lambda|:\lambda\in\sigma(a)\}$.

Title spectral radius SpectralRadius 2013-03-22 13:13:58 2013-03-22 13:13:58 Koro (127) Koro (127) 11 Koro (127) Definition msc 58C40 spectrum