Editing Spectrum (functional analysis) (section)

=== Basic properties ===

The spectrum of a bounded operator ''T'' is always a [[closed set|closed]], [[bounded set|bounded]] subset of the [[complex plane]].

If the spectrum were empty, then the [[Resolvent formalism|''resolvent function'']]

:<math>R(\lambda) = (T-\lambda I)^{-1}, \qquad \lambda\in\Complex,</math>

would be defined everywhere on the complex plane and bounded. But it can be shown that the resolvent function ''R'' is [[Holomorphic function|holomorphic]] on its domain. By the vector-valued version of [[Liouville's theorem (complex analysis)|Liouville's theorem]], this function is constant, thus everywhere zero as it is zero at infinity. This would be a contradiction.

The boundedness of the spectrum follows from the [[Neumann series|Neumann series expansion]] in ''λ''; the spectrum ''σ''(''T'') is bounded by ||''T''||. A similar result shows the closedness of the spectrum.

The bound ||''T''|| on the spectrum can be refined somewhat. The ''[[spectral radius]]'', ''r''(''T''), of ''T'' is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum ''σ''(''T'') inside of it, i.e.

:<math>r(T) = \sup \{|\lambda| : \lambda \in \sigma(T)\}.</math>

The '''spectral radius formula''' says<ref>Theorem 3.3.3 of Kadison & Ringrose, 1983, ''Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory'', New York: Academic Press, Inc.</ref> that for any element <math>T</math> of a [[Banach algebra]],
:<math>r(T) = \lim_{n \to \infty} \left\|T^n\right\|^{1/n}.</math>