Editing Spectral radius (section)

===Bounded linear operators===
In the context of a [[bounded linear operator]] {{mvar|A}} on a [[Banach space]], the eigenvalues need to be replaced with the elements of the [[Spectrum of an operator|spectrum of the operator]], i.e. the values <math>\lambda</math> for which <math>A - \lambda I</math> is not bijective. We denote the spectrum by
:<math>\sigma(A) = \left\{ \lambda \in \Complex: A - \lambda I \; \text{is not bijective} \right\}</math> 
The spectral radius is then defined as the supremum of the magnitudes of the elements of the spectrum:
:<math>\rho(A) = \sup_{\lambda \in \sigma(A)} |\lambda|</math>
Gelfand's formula, also known as the spectral radius formula, also holds for bounded linear operators: letting <math>\|\cdot\|</math> denote the [[operator norm]], we have
:<math>\rho(A) = \lim_{k \to \infty}\|A^k\|^{\frac{1}{k}}=\inf_{k\in\mathbb{N}^*} \|A^k\|^{\frac{1}{k}}.</math>

A bounded operator (on a complex Hilbert space) is called a '''spectraloid operator''' if its spectral radius coincides with its [[numerical radius]]. An example of such an operator is a [[normal operator]].