Editing Spectrum (functional analysis) (section)

===Basic properties===

The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.
If the operator ''T'' is not [[closed linear operator|closed]], then <math>\sigma(T)=\Complex</math>.

The following example indicates that non-closed operators may have empty spectra. Let <math>T</math> denote the differentiation operator on <math>L^2([0,1])</math>, whose domain is defined to be the closure of <math>C^{\infty}_c((0,1])</math> with respect to the <math>H^1</math>-[[Sobolev space]] norm. This space can be characterized as all functions in <math>H^1([0,1])</math> that are zero at <math>t = 0</math>. Then, <math>T - z</math> has trivial kernel on this domain, as any <math>H^1([0,1])</math>-function in its kernel is a constant multiple of <math>e^{zt}</math>, which is zero at <math>t = 0</math> if and only if it is identically zero. Therefore, the complement of the spectrum is all of <math>\mathbb{C}.</math>