Editing Spectrum (functional analysis) (section)

== Spectrum of a real operator ==

The definitions of the resolvent and spectrum can be extended to any continuous linear operator <math>T</math> acting on a Banach space <math>X</math> over the real field <math>\mathbb{R}</math> (instead of the complex field <math>\mathbb{C}</math>) via its [[complexification]] <math>T_\mathbb{C}</math>. In this case we define the resolvent set <math>\rho(T)</math> as the set of all <math>\lambda\in\mathbb{C}</math> such that <math>T_\mathbb{C}-\lambda I</math> is invertible as an operator acting on the complexified space <math>X_\mathbb{C}</math>; then we define <math>\sigma(T)=\mathbb{C}\setminus\rho(T)</math>.

=== Real spectrum ===

The ''real spectrum'' of a continuous linear operator <math>T</math> acting on a real Banach space <math>X</math>, denoted <math>\sigma_\mathbb{R}(T)</math>, is defined as the set of all <math>\lambda\in\mathbb{R}</math> for which <math>T-\lambda I</math> fails to be invertible in the real algebra of bounded linear operators acting on <math>X</math>.  In this case we have <math>\sigma(T)\cap\mathbb{R}=\sigma_\mathbb{R}(T)</math>.  Note that the real spectrum may or may not coincide with the complex spectrum. In particular, the real spectrum could be empty.