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

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

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

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

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 GelfandSpectralRadiusTheorem 2013-03-22 13:39:19 2013-03-22 13:39:19 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 9 Andrea Ambrosio (7332) Theorem msc 34L05 spectral radius formula SelfConsistentMatrixNorm