Gelfand spectral radius theorem

For every self-consistent matrix norm, ||||, and every square matrixMathworldPlanetmath 𝐀 we can write


Note: ρ(𝐀) denotes the spectral radius of 𝐀.

This theorem also generalizes to infiniteMathworldPlanetmath dimensionsPlanetmathPlanetmathPlanetmath and plays an important role in the theory of operator algebras. If 𝒜 is a Banach algebraMathworldPlanetmath with norm |||| and A𝒜, then we have


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 generalizationPlanetmathPlanetmath occurs when 𝒜 is the algebra of bounded operatorsMathworldPlanetmathPlanetmath on a Hilbert spaceMathworldPlanetmath — 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