Editing Spectrum (functional analysis) (section)

===Definition===
Let <math>T</math> be a [[bounded linear operator]] acting on a Banach space <math>X</math> over the complex scalar field <math>\mathbb{C}</math>, and <math>I</math> be the [[identity operator]] on <math>X</math>.  The '''spectrum''' of <math>T</math> is the set of all <math>\lambda \in \mathbb{C}</math> for which the operator <math>T-\lambda I</math> does not have an inverse that is a bounded linear operator.

Since <math>T-\lambda I</math> is a linear operator, the inverse is linear if it exists; and, by the [[bounded inverse theorem]], it is bounded.  Therefore, the spectrum consists precisely of those scalars <math>\lambda</math> for which <math>T-\lambda I</math> is not [[bijective]].

The spectrum of a given operator <math>T</math> is often denoted <math>\sigma(T)</math>, and its complement, the [[resolvent set]], is denoted <math>\rho(T) = \mathbb{C} \setminus \sigma(T)</math>. (<math>\rho(T)</math> is sometimes used to denote the spectral radius of <math>T</math>)