Editing Spectrum (functional analysis) (section)

== Spectrum of an unbounded operator ==

One can extend the definition of spectrum to [[unbounded operator]]s on a [[Banach space]] ''X''. These operators are no longer elements in the Banach algebra ''B''(''X'').

===Definition===
Let ''X'' be a Banach space and <math>T:\,D(T)\to X</math> be a [[unbounded operator|linear operator]] defined on domain <math>D(T) \subseteq X</math>.
A complex number ''λ'' is said to be in the '''resolvent set''' (also called '''regular set''') of <math>T</math> if the operator

:<math>T-\lambda I:\,D(T) \to X</math>

has a bounded everywhere-defined inverse, i.e. if there exists a bounded operator

:<math>S :\, X \rightarrow D(T)</math>

such that

:<math>S (T - \lambda I) = I_{D(T)}, \,  (T - \lambda I) S  = I_X.</math>

A complex number ''λ'' is then in the '''spectrum''' if ''λ'' is not in the resolvent set.

For ''λ'' to be in the resolvent (i.e. not in the spectrum), just like in the bounded case, <math>T-\lambda I</math> must be bijective, since it must have a two-sided inverse. As before, if an inverse exists, then its linearity is immediate, but in general it may not be bounded, so this condition must be checked separately.

By the [[closed graph theorem]], boundedness of <math>(T-\lambda I)^{-1}</math> ''does'' follow directly from its existence when ''T'' is [[closed operator|closed]].  Then, just as in the bounded case, a complex number ''λ'' lies in the spectrum of a closed operator ''T'' if and only if <math>T-\lambda I</math> is not bijective.  Note that the class of closed operators includes all bounded operators.

===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>