A well-known example from quantum mechanics is the explanation for the discrete spectral lines and the continuous band in the light emitted by excited atoms of hydrogen.
If T − λ is one-to-one and onto, i.e. bijective, then its inverse is bounded; this follows directly from the open mapping theorem of functional analysis.
One distinguishes three separate cases: So σ(T) is the disjoint union of these three sets,
We will show first that σ(Th) coincides with the essential range of h and then examine its various parts.
Let f be the characteristic function of the measurable set h−1(λ), then by considering two cases, we find
Any λ in the essential range of h that does not have a positive measure preimage is in the continuous spectrum of Th.
In particular, by the spectral theorem, normal operators on a Hilbert space have no residual spectrum.
This space consists of complex valued sequences {xn} such that
So σ(T) lies in the closed unit disk of the complex plane.
and T x = λ x. Consequently, the point spectrum of T contains the open unit disk.
The spectrum of a bounded operator is closed, which implies the unit circle, { |λ| = 1 } ⊂ C, is in σ(T).
For a self-adjoint T ∈ B(H), the Borel functional calculus gives additional ways to break up the spectrum naturally.
For the continuous functional calculus, the key ingredients are the following: The family C(σ(T)) is a Banach algebra when endowed with the uniform norm.
According to the Riesz–Markov–Kakutani representation theorem a unique measure μh on σ(T) exists such that
For a bounded function g that is Borel measurable, define, for a proposed g(T)
According to a refinement of Lebesgue's decomposition theorem, μh can be decomposed into three mutually singular parts:
These subspaces are invariant under T. For example, if h ∈ Hac and k = T h. Let χ be the characteristic function of some Borel set in σ(T), then
This leads to the following definitions: The closure of the eigenvalues is the spectrum of T restricted to Hpp.
Therefore, one can also apply to T the decomposition of the spectrum that was achieved above for bounded operators on a Banach space.
Unlike the Banach space formulation,[clarification needed] the union
When more than one measure appears in the above expression, we see that it is possible for the union of the three types of spectra to not be disjoint.
the decomposition of σ(T) from Borel functional calculus is a refinement of the Banach space case.
The preceding comments can be extended to the unbounded self-adjoint operators since Riesz-Markov holds for locally compact Hausdorff spaces.
In quantum mechanics, observables are (often unbounded) self-adjoint operators and their spectra are the possible outcomes of measurements.
[6] Intuitively one might therefore think that the "discreteness" of the spectrum is intimately related to the corresponding states being "localized".
However, a careful mathematical analysis shows that this is not true in general.
These are free states belonging to the absolutely continuous spectrum.
In the spectral theorem for unbounded self-adjoint operators, these states are referred to as "generalized eigenvectors" of an observable with "generalized eigenvalues" that do not necessarily belong to its spectrum.
Alternatively, if it is insisted that the notion of eigenvectors and eigenvalues survive the passage to the rigorous, one can consider operators on rigged Hilbert spaces.
[8] An example of an observable whose spectrum is purely absolutely continuous is the position operator of a free particle moving on the entire real line.