# Gelfand spectral radius theorem

For every self-consistent matrix norm, $||\cdot ||$, and every square matrix^{} $\mathbf{A}$ we can write

$$\rho (\mathbf{A})=\underset{n\to \mathrm{\infty}}{lim}{||{\mathbf{A}}^{n}||}^{\frac{1}{n}}.$$ |

Note: $\rho (\mathbf{A})$ denotes the spectral radius of $\mathbf{A}$.

This theorem also generalizes to infinite^{} dimensions^{} and plays an important role in the theory of operator algebras. If $\mathcal{A}$ is a Banach algebra^{} with norm $||\cdot ||$ and $A\in \mathcal{A}$, then we have

$$\rho (\mathbf{A})=\underset{n\to \mathrm{\infty}}{lim}{||{\mathbf{A}}^{n}||}^{\frac{1}{n}}.$$ |

It is worth pointing out that the self-consistency condition which was imposed on the matrix norm is part of the definition of a Banach algebra. A common case of the infinite-dimensional generalization^{} occurs when $\mathcal{A}$ is the algebra of bounded operators^{} on a Hilbert space^{} — the operators may be regarded as an infinite-dimensional generalization of the square matrices.

Title | Gelfand spectral radius theorem |
---|---|

Canonical name | GelfandSpectralRadiusTheorem |

Date of creation | 2013-03-22 13:39:19 |

Last modified on | 2013-03-22 13:39:19 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 9 |

Author | Andrea Ambrosio (7332) |

Entry type | Theorem |

Classification | msc 34L05 |

Synonym | spectral radius formula |

Related topic | SelfConsistentMatrixNorm |