The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars such that the operator does not have a bounded inverse on . 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.
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, i.e. bijective, 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 not one-to-one or not onto. One distinguishes three separate cases:
So σ(T) is the disjoint union of these three sets,
Surjectivity of T − λ | Injectivity of T − λ | ||
---|---|---|---|
Injective and bounded below | Injective but not bounded below | not injective | |
Surjective | Resolvent set ρ(T) | Nonexistent | Point spectrum σ_{p}(T) |
Not surjective but has dense range | Nonexistent | Continuous spectrum σ_{c}(T) | |
Does not have dense range | Residual spectrum σ_{r}(T) |
In addition, when T − λ does not have dense range, whether is injective or not, then λ is said to be in the compression spectrum of T, σ_{cp}(T). The compression spectrum consists of the whole residual spectrum and part of point spectrum.
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.
Given a σ-finite measure space (S, Σ, μ), consider the Banach space L^{p}(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 T_{h} on L^{p}(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 B_{ε}(λ) under h has strictly positive measure. We will show first that σ(T_{h}) 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}(B_{ε}(λ)) has zero measure. The function g(s) = 1/(h(s) − λ) is bounded almost everywhere by 1/ε. The multiplication operator T_{g} satisfies T_{g} · (T_{h} − λ) = (T_{h} − λ) · T_{g} = I. So λ does not lie in spectrum of T_{h}. On the other hand, if λ lies in the essential range of h, consider the sequence of sets {S_{n} = h^{−1}(B_{1/n}(λ))}. Each S_{n} has positive measure. Let f_{n} be the characteristic function of S_{n}. We can compute directly
This shows T_{h} − 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 T_{h} as follows. 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 T_{h}. To show this, we must show that T_{h} − λ has dense range. Given f ∈ L^{p}(μ), again we consider the sequence of sets {S_{n} = h^{−1}(B_{1/n}(λ))}. Let g_{n} be the characteristic function of S − S_{n}. Define
Direct calculation shows that f_{n} ∈ L^{p}(μ), with . Then by the dominated convergence theorem,
Therefore, multiplication operators have no residual spectrum. In particular, by the spectral theorem, normal operators on a Hilbert space have no residual spectrum.
Main article: Shift space |
In the special case when S is the set of natural numbers and μ is the counting measure, the corresponding L^{p}(μ) is denoted by l^{p}. This space consists of complex valued sequences {x_{n}} 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,
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 σ_{r}(T) ⊂ σ_{p}(T*)), we have that σ_{r}(T) is also empty. Therefore, for a complex number λ with unit norm, one must have λ ∈ σ_{p}(T) or λ ∈ σ_{c}(T). Now if |λ| = 1 and
So for the left shift T, σ_{p}(T) is the open unit disk and σ_{c}(T) is the unit circle, whereas for the right shift T*, σ_{r}(T*) is the open unit disk and σ_{c}(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.
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.
Further information: Borel functional calculus |
This subsection briefly sketches the development of this calculus. The idea is to first establish the continuous functional calculus, and then pass to measurable functions via the Riesz–Markov–Kakutani 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
For a fixed h ∈ H, we notice 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)
In the present context, the spectral measures, combined with a result from measure theory, give a decomposition of σ(T).
Let h ∈ H and μ_{h} be its corresponding spectral measure on σ(T) ⊂ R. According to a refinement of Lebesgue's decomposition theorem, μ_{h} can be decomposed into three mutually singular parts:
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
This leads to the following definitions:
The closure of the eigenvalues is the spectrum of T restricted to H_{pp}.^{[3]}^{[nb 1]} So
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,^{[clarification needed]} the union
When T is unitarily equivalent to multiplication by λ on
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.
The pure point spectrum corresponds to bound states in the following way:
A particle is said to be in a bound state if it remains "localized" in a bounded region of space.^{[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.^{[7]} For example, consider the function
This function is normalizable (i.e. ) as
Known as the Basel problem, this series converges to . Yet, increases as , i.e, the state "escapes to infinity". 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 . Dynamical localization is more subtle to define.
Sometimes, when performing quantum mechanical measurements, one encounters "eigenstates" that are not localized, e.g., quantum states that do not lie in L^{2}(R). 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. Also, since the momentum operator is unitarily equivalent to the position operator, via the Fourier transform, it has a purely absolutely continuous spectrum as well.
The singular spectrum correspond to physically impossible outcomes. It was believed for some time that the singular spectrum was something artificial. However, examples as the almost Mathieu operator and random Schrödinger operators have shown, that all types of spectra arise naturally in physics.^{[9]}^{[10]}
Let be a closed operator defined on the domain which is dense in X. Then there is a decomposition of the spectrum of A into a disjoint union,^{[11]}