Gelfand spectral radius theorem

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

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

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

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

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

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 $𝒜$ 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