Editing Spectral theorem (section)

== Unbounded self-adjoint operators ==
Many important linear operators which occur in [[Mathematical analysis|analysis]], such as [[differential operators]], are [[unbounded operator|unbounded]]. There is also a spectral theorem for [[self-adjoint operator]]s that applies in these cases.  To give an example, every constant-coefficient differential operator is unitarily equivalent to a multiplication operator. Indeed, the unitary operator that implements this equivalence is the [[Fourier transform]]; the multiplication operator is a type of [[Multiplier (Fourier analysis)|Fourier multiplier]].

In general, spectral theorem for self-adjoint operators may take several equivalent forms.<ref>See Section 10.1 of {{harvnb|Hall|2013}}</ref> Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues. Specifically, the only reason the multiplication operator <math>A</math> on <math>L^2([0,1])</math> is bounded, is due to the choice of domain <math>[0,1]</math>. The same operator on, e.g., <math>L^2(\mathbb{R})</math> would be unbounded. 

The notion of "generalized eigenvectors" naturally extends to unbounded self-adjoint operators, as they are characterized as [[Probability_amplitude#Normalization|non-normalizable]] eigenvectors. Contrary to the case of [[Spectral_theorem#Spectral_subspaces_and_projection-valued_measures|almost eigenvectors]], however, the eigenvalues can be real or complex and, even if they are real, do not necessarily belong to the spectrum. Though, for self-adjoint operators there always exist a real subset of "generalized eigenvalues" such that the corresponding set of eigenvectors is [[Total_set|complete]].{{sfn|de la Madrid Modino|2001|pp=95-97}}