Decomposition of spectrum (functional analysis)

The spectrum of a linear operator formula_1 that operates on a Banach space formula_2 (a fundamental concept of functional analysis) consists of all scalars formula_3 such that the operator formula_4 does not have a bounded inverse on formula_2. The spectrum has a standard decomposition into three parts: 
This decomposition is relevant to the study of differential equations, and has applications to many branches of science and engineering. 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.
Decomposition into point spectrum, continuous spectrum, and residual spectrum.
For bounded Banach space operators.
Let "X" be a Banach space, "B"("X") the family of bounded operators on "X", and "T" ∈ "B"("X"). By definition, a complex number "λ" is in the spectrum of "T", denoted "σ"("T"), if "T" − "λ" does not have an inverse in "B"("X").
If "T" − "λ" is one-to-one and onto, then its inverse is bounded; this follows directly from the open mapping theorem of functional analysis. So, "λ" is in the spectrum of "T" if and only if "T" − "λ" is either not one-to-one or not onto. One distinguishes three separate cases:
So "σ"("T") is the disjoint union of these three sets,
For unbounded operators.
The spectrum of an unbounded operator can be divided into three parts in the same way as in the bounded case, but because the operator is not defined everywhere, the definitions of domain, inverse, etc. are more involved.
Examples.
Multiplication operator.
Given a σ-finite measure space ("S", "Σ", "μ"), consider the Banach space "Lp"("μ"). A function "h": "S" → C is called essentially bounded if "h" is bounded "μ"-almost everywhere. An essentially bounded "h" induces a bounded multiplication operator "Th" on "Lp"("μ"):
The operator norm of "T" is the essential supremum of "h". The essential range of "h" is defined in the following way: a complex number "λ" is in the essential range of "h" if for all "ε" > 0, the preimage of the open ball "Bε"("λ") under "h" has strictly positive measure. We will show first that "σ"("Th") coincides with the essential range of "h" and then examine its various parts.
If "λ" is not in the essential range of "h", take "ε" > 0 such that "h"−1("Bε"("λ")) has zero measure. The function "g"("s") = 1/("h"("s") − "λ") is bounded almost everywhere by 1/"ε". The multiplication operator "Tg" satisfies
"T""g" · ("T""h" − "λ") = ("T""h"  − "λ")· "T""g" = "I". So "λ" does not lie in spectrum of "Th". On the other hand, if "λ" lies in the essential range of "h", consider the sequence of sets {"Sn" = 
"h"−1("B"1/n("λ"))}. Each "Sn" has positive measure. Let "fn" be the characteristic function of "Sn". We can compute directly
This shows "Th" − "λ" is not bounded below, therefore not invertible.
If "λ" is such that "μ"( "h"−1({"λ"})) > 0, then "λ" lies in the point spectrum of "Th" as follows. Let "f" be the characteristic function of the measurable set "h"−1("λ"), then by considering two cases, we find
so λ is an eigenvalue of "T""h".
Any "λ" in the essential range of "h" that does not have a positive measure preimage is in the continuous spectrum of "Th". To show this, we must show that "Th" − "λ" has dense range. Given "f" ∈ "Lp"("μ"), again we consider the sequence of sets {"Sn" = "h"−1("B"1/n("λ"))}. Let "gn" be the characteristic function of "S" − "Sn". Define
Direct calculation shows that "fn" ∈ "Lp"("μ"), with formula_13. Then by the dominated convergence theorem,
in the "Lp"("μ") norm.
Therefore, multiplication operators have no residual spectrum. In particular, by the spectral theorem, normal operators on a Hilbert space have no residual spectrum.
Shifts.
In the special case when "S" is the set of natural numbers and "μ" is the counting measure, the corresponding "Lp"("μ") is denoted by l"p". This space consists of complex valued sequences {"xn"} such that
For 1 < "p" < ∞, "l p" is reflexive. Define the left shift "T" : "l p" → "l p" by
"T" is a partial isometry with operator norm 1. So "σ"("T") lies in the closed unit disk of the complex plane.
"T*" is the right shift (or unilateral shift), which is an isometry on "l q", where 1/"p" + 1/"q" = 1:
For "λ" ∈ C with |"λ"| < 1,
and "T x" = "λ x". Consequently, the point spectrum of "T" contains the open unit disk. Now, "T*" has no eigenvalues, i.e. "σp"("T*") is empty. Thus, invoking reflexivity and the theorem given above (that "σp"("T") ⊂ "σr"("T"*) ∪ "σp"("T"*)), we can deduce that the open unit disk lies in the residual spectrum of "T*".
The spectrum of a bounded operator is closed, which implies the unit circle, { |"λ"| = 1 } ⊂ C, is in "σ"("T"). Again by reflexivity of "l p" and the theorem given above (this time, that "σr"("T") ⊂ "σp"("T"*)), we have that "σr"("T")
is also empty. Therefore, for a complex number "λ" with unit norm, one must have "λ" ∈ "σp"("T") or "λ" ∈ "σc"("T"). Now if |"λ"| = 1 and
then
which cannot be in "l p", a contradiction. This means the unit circle must lie in the continuous spectrum of "T".
So for the left shift "T", "σp"("T") is the open unit disk and "σc"("T") is the unit circle, whereas for the right shift "T*", "σr"("T*") is the open unit disk and "σc"("T*") is the unit circle.
For "p" = 1, one can perform a similar analysis. The results will not be exactly the same, since reflexivity no longer holds.
Self-adjoint operators on Hilbert space.
Hilbert spaces are Banach spaces, so the above discussion applies to bounded operators on Hilbert spaces as well. A subtle point concerns the spectrum of "T"*. For a Banach space, "T"* denotes the transpose and "σ"("T*") = "σ"("T"). For a Hilbert space, "T"* normally denotes the adjoint of an operator "T" ∈ "B"("H"), not the transpose, and "σ"("T*") is not "σ"("T") but rather its image under complex conjugation.
For a self adjoint "T" ∈ "B"("H"), the Borel functional calculus gives additional ways to break up the spectrum naturally.
Borel functional calculus.
This subsection briefly sketches the development of this calculus. The idea is to first establish the continuous functional calculus then pass to measurable functions via the Riesz-Markov representation theorem. 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. So the mapping
is an isometric homomorphism from a dense subset of "C"("σ"("T")) to "B"("H"). Extending the mapping by continuity gives "f"("T") for "f" ∈ C("σ"("T")): let "Pn" be polynomials such that "Pn" → "f" uniformly and define "f"("T") = lim "Pn"("T"). This is the continuous functional calculus.
For a fixed "h" ∈ "H", we notice that
is a positive linear functional on "C"("σ"("T")). According to the Riesz-Markov representation theorem that there exists a unique measure "μh" on "σ"("T") such that
This measure is sometimes called the spectral measure associated to h. The spectral measures can be used to extend the continuous functional calculus to bounded Borel functions. For a bounded function "g" that is Borel measurable, define, for a proposed "g"("T")
Via the polarization identity, one can recover (since "H" is assumed to be complex)
and therefore "g"("T") "h" for arbitrary "h".
In the present context, the spectral measures, combined with a result from measure theory, give a decomposition of "σ"("T").
Decomposition into absolutely continuous, singular continuous, and pure point.
Let "h" ∈ "H" and "μh" be its corresponding spectral measure on "σ"("T") ⊂ R. According to a refinement of Lebesgue's decomposition theorem, "μh" can be decomposed into three mutually singular parts:
where "μ"ac is absolutely continuous with respect to the Lebesgue measure, "μ"sc is singular with respect to the Lebesgue measure and atomless,
and "μ"pp is a pure point measure.
All three types of measures are invariant under linear operations. Let "H"ac be the subspace consisting of vectors whose spectral measures are absolutely continuous with respect to the Lebesgue measure. Define "H"pp and "H"sc in analogous fashion. These subspaces are invariant under "T". For example, if "h" ∈ "H"ac and "k" = "T h". Let "χ" be the characteristic function of some Borel set in "σ"("T"), then
So
and "k" ∈ "H"ac. Furthermore, applying the spectral theorem gives
This leads to the following definitions:
The closure of the eigenvalues is the spectrum of "T" restricted to "H"pp. So
Comparison.
A bounded self adjoint operator on Hilbert space is, a fortiori, a bounded operator on a Banach space. 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, the union
need not be disjoint. It is disjoint when the operator "T" is of uniform multiplicity, say "m", i.e. if "T" is unitarily equivalent to multiplication by "λ" on the direct sum
for some Borel measures formula_34. 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. If "λ" ∈ "σac"("T") ∩ "σpp"("T"), "λ" is sometimes called an eigenvalue "embedded" in the absolutely continuous spectrum.
When "T" is unitarily equivalent to multiplication by "λ" on
the decomposition of "σ"("T") from Borel functional calculus is a refinement of the Banach space case.
Physics.
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 self adjoint operators, often not bounded, and their spectra are the possible outcomes of measurements. Absolutely continuous spectrum of a physical observable corresponds to free states of a system, while the pure point spectrum corresponds to bound states. The singular spectrum correspond to physically impossible outcomes. An example of a quantum mechanical observable which has purely continuous spectrum is the position operator of a free particle moving on a line. Its spectrum is the entire real line. Also, since the momentum operator is unitarily equivalent to the position operator, via the Fourier transform, they have the same spectrum.
Intuition may induce one to say 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. The following formula_36
is an element of formula_37 and increasing as formula_38.
However, the phenomena of Anderson localization and dynamical localization describe, when the eigenfunctions are localized in a physical sense. Anderson Localization means that eigenfunctions decay exponentially as formula_40. Dynamical localization is more subtle to define.
Sometimes, when performing physical quantum mechanical calculations, one encounters "eigenvectors" that do not lie in "L"2(R), i.e. wave functions that are not localized. These are the free states of the system. As stated above, in the mathematical formulation, the free states correspond to the absolutely continuous 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.
It was believed for some time that singular spectrum is something artificial. However, examples as the almost Mathieu operator and random Schrödinger operators have shown, that all types of spectra arise naturally in physics.
Decomposition into essential spectrum and discrete spectrum.
Let formula_41 be a closed operator defined on the domain formula_42 which is dense in "X". Then there is a decomposition of the spectrum of "A" into a disjoint union,
where