# spectral radius

If $V$ is a vector space^{} over $\u2102$, the spectrum of a linear mapping $T:V\to V$ is the set

$$\sigma (T)=\{\lambda \in \u2102:T-\lambda I\text{is not invertible}\},$$ |

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 $\u2102$ with identity element $e$, the spectrum of an element $a\in \mathcal{A}$ is the set

$$\sigma (a)=\{\lambda \in \u2102:a-\lambda e\text{is not invertible in}\mathcal{A}\}$$ |

The spectral radius of $a$ is $\rho (a)=sup\{|\lambda |:\lambda \in \sigma (a)\}$.

Title | spectral radius |
---|---|

Canonical name | SpectralRadius |

Date of creation | 2013-03-22 13:13:58 |

Last modified on | 2013-03-22 13:13:58 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 11 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 58C40 |

Defines | spectrum |