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.