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| The [[spectrum (functional analysis)|spectrum]] of a [[linear operator]] <math>T</math> that operates on a [[Banach space]] <math>X</math> (a fundamental concept of [[functional analysis]]) consists of all [[scalar (mathematics)|scalars]] <math>\lambda</math> such that the operator <math>T-\lambda</math> does not have a bounded [[inverse function|inverse]] on <math>X</math>. The spectrum has a standard '''decomposition''' into three parts:
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| * a '''point spectrum''', consisting of the [[eigenvalues and eigenvectors|eigenvalues]] of <math>T</math>
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| * a '''continuous spectrum''', consisting of the scalars that are not eigenvalues but make the range of <math>T-\lambda</math> a [[proper subset|proper]] [[dense subset]] of the space;
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| * a '''residual spectrum''', consisting of all other scalars in the spectrum
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| This decomposition is relevant to the study of [[differential equation]]s, and has applications to many branches of science and engineering. A well-known example from [[quantum mechanics]] is the explanation for the [[discrete spectrum (physics)|discrete spectral lines]] and the continuous band in the light emitted by [[excited state|excited]] atoms of [[hydrogen]].
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| == Definitions ==
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| === For bounded Banach space operators ===
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| Let ''X'' be a [[Banach space]], ''L''(''X'') the family of [[bounded operator]]s on ''X'', and ''T'' ∈ ''L''(''X''). By [[spectrum (functional analysis)|definition]], a complex number ''λ'' is in the '''spectrum''' of ''T'', denoted ''σ''(''T''), if ''T'' − ''λ'' does not have an inverse in ''L''(''X'').
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| If ''T'' − ''λ'' is [[injective|one-to-one]] and [[surjective|onto]], then its inverse is bounded; this follows directly from the [[open mapping theorem (functional analysis)|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:
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| #''T'' − ''λ'' is not [[injective]]. That is, there exist two distinct elements ''x'',''y'' in ''X'' such that (''T'' − ''λ'')(''x'') = (''T'' − ''λ'')(''y''). Then ''z'' = ''x'' − ''y'' is a non-zero vector such that ''T''(''z'') = ''λz''. In other words, ''λ'' is an eigenvalue of ''T'' in the sense of [[linear algebra]]. In this case, ''λ'' is said to be in the '''point spectrum''' of ''T'', denoted ''σ''<sub>p</sub>(''T'').
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| #''T'' − ''λ'' is injective, and its [[range (mathematics)|range]] is a [[dense subset]] '' R'' of ''X''; but is not the whole of ''X''. In other words, there exists some element ''x'' in ''X'' such that (''T'' − ''λ'')(''y'') can be as close to ''x'' as desired, with ''y'' in ''X''; but is never equal to ''x''. It can be proved that, in this case, ''T'' − ''λ'' is not bounded below (i.e. it sends far apart elements of ''X'' too close together). Equivalently, the inverse linear operator (''T'' − ''λ'')<sup>−1</sup>, which is defined on the dense subset ''R'', is not a bounded operator, and therefore cannot be extended to the whole of ''X''. Then ''λ'' is said to be in the '''continuous spectrum''', ''σ<sub>c</sub>''(''T''), of ''T''.
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| #''T'' − ''λ'' is injective but does not have dense range. That is, there is some element ''x'' in ''X'' and a neighborhood ''N'' of ''x'' such that (''T'' − ''λ'')(''y'') is never in ''N''. In this case, the map (''T'' − ''λ'') ''x'' → ''x'' may be bounded or unbounded, but in any case does not admit a unique extension to a bounded linear map on all of ''X''. Then ''λ'' is said to be in the '''residual spectrum''' of ''T'', ''σ<sub>r</sub>''(''T'').
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| So ''σ''(''T'') is the disjoint union of these three sets,
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| :<math>\sigma(T) = \sigma_p (T) \cup \sigma_c (T) \cup \sigma_r (T).</math>
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| ==== Spectrum of dual operator ====
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| If ''X*'' is the dual space of ''X'', and ''T*'' : ''X*'' → ''X*'' is the adjoint operator of ''T'', then
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| ''σ''(''T'') = ''σ''(''T*'').
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| '''Theorem''' For a bounded operator ''T'', ''σ<sub>r</sub>''(''T'') ⊂ ''σ<sub>p</sub>''(''T*'') ⊂ ''σ<sub>r</sub>''(''T'') ∪ ''σ<sub>p</sub>''(''T'').
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| '''Proof'''
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| The notation <·, ''φ''> will denote an element of ''X*'', i.e. ''x'' → <''x'', ''φ''> is the action of a bounded linear functional ''φ''. Let λ ∈ ''σ<sub>r</sub>''(''T''). So ''Ran''(''T'' - ''λ'') is not dense in ''X''. By the [[Hahn–Banach theorem]], there exists a non-zero ''φ'' ∈ ''X*'' that vanishes on ''Ran''(''T'' - ''λ''). For all ''x'' ∈ ''X'',
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| :<math>\langle (T - \lambda)x, \varphi \rangle = \langle x, (T^* - {\lambda}) \varphi \rangle = 0.</math>
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| Therefore (''T*'' - ''λ'')''φ'' = 0 ∈ ''X*'' and ''λ'' is an eigenvalue of ''T*''. The shows the former inclusion. Next suppose that (''T*'' - ''λ'')''φ'' = 0 where ''φ'' ≠ 0, i.e.
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| :<math>\forall x \in X,\; \langle x, (T^* - \lambda) \varphi \rangle = \langle (T - \lambda) x, \varphi \rangle = 0.</math>
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| If ''Ran''(''T'' − ''λ'') is dense, then ''φ'' must be the zero functional, a contradiction. The claim is proved.
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| In particular, when ''X'' is a [[reflexive Banach space]], ''σ<sub>r</sub>''(''T*'') ⊂ ''σ<sub>p</sub>''(''T**'') = ''σ<sub>p</sub>''(''T'').
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| === For unbounded operators ===
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| The spectrum of an unbounded operator can be divided into three parts in exactly the same way as in the bounded case.
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| === Examples ===
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| ==== Multiplication operator ====
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| Given a σ-finite [[measure space]] (''S'', ''Σ'', ''μ''), consider the Banach space [[Lp space|''L<sup>p</sup>''(''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<sub>h</sub>'' on ''L<sup>p</sup>''(''μ''):
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| :<math>(T_h f)(s) = h(s) \cdot f(s).</math>
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| 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<sub>ε</sub>''(''λ'') under ''h'' has strictly positive measure. We will show first that ''σ''(''T<sub>h</sub>'') coincides with the essential range of ''h'' and then examine its various parts. | |
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| If ''λ'' is not in the essential range of ''h'', take ''ε'' > 0 such that ''h''<sup>−1</sup>(''B<sub>ε</sub>''(''λ'')) has zero measure. The function ''g''(''s'') = 1/(''h''(''s'') − ''λ'') is bounded almost everywhere by 1/''ε''. The multiplication operator ''T<sub>g</sub>'' satisfies
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| ''T''<sub>''g''</sub> · ''T''<sub>''h'' − ''λ''</sub> = ''T''<sub>''h'' − ''λ''</sub> · ''T''<sub>''g''</sub> = ''I''. So ''λ'' does not lie in spectrum of ''T<sub>h</sub>''. On the other hand, if ''λ'' lies in the essential range of ''h'', consider the sequence of sets {''S<sub>n</sub>'' =
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| ''h''<sup>−1</sup>(''B''<sub>1/n</sub>(''λ''))}. Each ''S<sub>n</sub>'' has positive measure. Let ''f<sub>n</sub>'' be the characteristic function of ''S<sub>n</sub>''. We can compute directly
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| :<math>
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| \| (T_h - \lambda) f_n \|_p ^p = \| (h - \lambda) f_n \|_p ^p = \int_{S_n} | h - \lambda \; |^p d \mu
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| \leq \frac{1}{n^p} \; \mu(S_n) = \frac{1}{n^p} \| f_n \|_p ^p.
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| </math>
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| This shows ''T<sub>h</sub>'' − ''λ'' is not bounded below, therefore not invertible. | |
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| If ''λ'' is such that ''μ''( ''h''<sup>−1</sup>({''λ''})) > 0, then ''λ'' lies in the point spectrum of ''T<sub>h</sub>'': Pick an open ball ''B<sub>ε</sub>''(''λ'') that contains only ''λ'' from the essential range. Let ''f'' be the characteristic function of ''h''<sup>−1</sup>(''B<sub>ε</sub>''(''λ'')), then
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| :<math>\forall s \in S, \; (T_h f)(s) = \lambda f(s).</math>
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| Any ''λ'' in the essential range of ''h'' that does not have a positive measure preimage is in the continuous spectrum or in the resolvent spectrum of ''T<sub>h</sub>''. To show this is to show that ''T<sub>h</sub>'' − ''λ'' has dense range for all such ''λ''. Given ''f'' ∈ ''L<sup>p</sup>''(''μ''), again we consider the sequence of sets {''S<sub>n</sub>'' = ''h''<sup>−1</sup>(''B''<sub>1/n</sub>(''λ''))}. Let ''g<sub>n</sub>'' be the characteristic function of ''S'' − ''S<sub>n</sub>''. Define
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| :<math>f_n(s) = \frac{1}{ h(s) - \lambda} \cdot g_n(s) \cdot f(s).</math>
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| Direct calculation shows that ''f<sub>n</sub>'' ∈ ''L<sup>p</sup>''(''μ''), and, by the [[dominated convergence theorem]],
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| :<math>(T_h - \lambda) f_n \rightarrow f</math>
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| in the ''L<sup>p</sup>''(''μ'') norm.
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| Therefore multiplication operators have no residual spectrum. In particular, by the [[spectral theorem]], [[normal operator]]s on a Hilbert space have no residual spectrum.
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| ==== Shifts ====
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| In the special case when ''S'' is the set of natural numbers and ''μ'' is the counting measure, the corresponding ''L<sup>p</sup>''(''μ'') is denoted by l<sup>''p''</sup>. This space consists of complex valued sequences {''x<sub>n</sub>''} such that
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| :<math>\sum_{n \geq 0} | x_n |^p < \infty.</math>
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| For 1 < ''p'' < ∞, ''l <sup>p</sup>'' is reflexive. Define the left shift ''T'' : ''l <sup>p</sup>'' → ''l <sup>p</sup>'' by
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| :<math>T(x_1, x_2, x_3, \dots) = (x_2, x_3, x_4, \dots).</math>
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| ''T'' is a [[partial isometry]] with operator norm 1. So ''σ''(''T'') lies in the closed unit disk of the complex plane.
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| ''T*'' is the right shift (or [[unilateral shift]]), which is an isometry, on ''l <sup>q</sup>'' where 1/''p'' + 1/''q'' = 1:
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| :<math>T^*(x_1, x_2, x_3, \dots) = (0, x_1, x_2, \dots).</math>
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| For ''λ'' ∈ '''C''' with |''λ''| < 1,
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| :<math>x = (1, \lambda, \lambda ^2, \dots) \in l^p</math>
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| and ''T x'' = ''λ x''. Consequently the point spectrum of ''T'' contains the open unit disk. Invoking reflexivity and the theorem given above, we can deduce that the open unit disk lies in the residual spectrum of ''T*''.
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| The spectrum of a bounded operator is closed, which implies the unit circle, { |''λ''| = 1 } ⊂ '''C''', is in ''σ''(''T''). Also, ''T*'' has no eigenvalues, i.e. ''σ<sub>p</sub>''(''T*'') is empty. Again by reflexivity of ''l <sup>p</sup>'' and the theorem given above, we have that ''σ<sub>r</sub>''(''T'')
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| is also empty. Therefore, for a complex number ''λ'' with unit norm, one must have ''λ'' ∈ ''σ<sub>p</sub>''(''T'') or ''λ'' ∈ ''σ<sub>c</sub>''(''T''). Now if |''λ''| = 1 and
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| :<math>T x = \lambda x, \; i.e. \; (x_2, x_3, x_4, \dots) = \lambda (x_1, x_2, x_3, \dots),</math>
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| then
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| :<math>x = x_1 (1, \lambda, \lambda^2, \dots),</math> | |
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| which can not be in ''l <sup>p</sup>'', a contradiction. This means the unit circle must be the continuous spectrum of ''T''.
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| For the right shift ''T*'', ''σ<sub>r</sub>''(''T*'') is the open unit disk and ''σ<sub>c</sub>''(''T*'') is the unit circle.
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| For ''p'' = 1, one can perform a similar analysis. The results will not be exactly the same, since reflexivity no longer holds.
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| == Self adjoint operators on Hilbert space ==
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| [[Hilbert space]]s are Banach spaces, so the above discussion applies to bounded operators on Hilbert spaces as well, although possible differences may arise from the adjoint operation on operators. For example, let ''H'' be a Hilbert space and ''T'' ∈ ''L''(''H''), ''σ''(''T*'') is not ''σ''(''T'') but rather its image under complex conjugation.
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| For a self adjoint ''T'' ∈ ''L''(''H''), the [[Borel functional calculus]] gives additional ways to break up the spectrum naturally.
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| === Borel functional calculus ===
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| {{ Further2|[[Borel functional calculus]]}}
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| 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 representation theorem|Riesz-Markov representation theorem]]. For the continuous functional calculus, the key ingredients are the following: | |
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| :1. If ''T'' is self adjoint, then for any polynomial ''P'', the operator norm
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| ::<math>\| P(T) \| = \sup_{\lambda \in \sigma(T)} |P(\lambda)|.</math>
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| :2. The [[Stone-Weierstrass theorem]], which gives that the family of polynomials (with complex coefficients), is dense in ''C''(''σ''(''T'')), the continuous functions on ''σ''(''T'').
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| The family ''C''(''σ''(''T'')) is a [[Banach algebra]] when endowed with the uniform norm. So the mapping
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| :<math>P \rightarrow P(T)</math>
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| is an isometric homomorphism from a dense subset of ''C''(''σ''(''T'')) to ''L''(''H''). Extending the mapping by continuity gives ''f''(''T'') for ''f'' ∈ C(''σ''(''T'')): let ''P<sub>n</sub>'' be polynomials such that ''P<sub>n</sub>'' → ''f'' uniformly and define ''f''(''T'') = lim ''P<sub>n</sub>''(''T''). This is the continuous functional calculus.
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| For a fixed ''h'' ∈ ''H'', we notice that
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| :<math>f \rightarrow \langle h, f(T) h \rangle</math>
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| is a positive linear functional on ''C''(''σ''(''T'')). According to the Riesz-Markov representation theorem that there exists a unique measure ''μ<sub>h</sub>'' on ''σ''(''T'') such that
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| :<math>\int_{\sigma(T)} f \, d \mu_h = \langle h, f(T) h \rangle.</math>
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| 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'')
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| :<math>\int_{\sigma(T)} g \, d \mu_h = \langle h, g(T) h \rangle.</math>
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| Via the [[polarization identity]], one can recover (since ''H'' is assumed to be complex)
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| :<math>\langle k, g(T) h \rangle.</math>
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| and therefore ''g''(''T'') ''h'' for arbitrary ''h''.
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| In the present context, the spectral measures, combined with a result from measure theory, give a decomposition of ''σ''(''T'').
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| === Decomposing the spectrum ===
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| Let ''h'' ∈ ''H'' and ''μ<sub>h</sub>'' be its corresponding spectral measure on ''σ''(''T'') ⊂ '''R'''. According to a refinement of [[Lebesgue's decomposition theorem]], ''μ<sub>h</sub>'' can be decomposed into three mutually singular parts:
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| :<math>\, \mu = \mu_{\mathrm{ac}} + \mu_{\mathrm{sc}} + \mu_{\mathrm{pp}}</math>
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| where ''μ''<sub>ac</sub> is absolutely continuous with respect to the Lebesgue measure, and ''μ''<sub>sc</sub> is singular with respect to the Lebesgue measure,
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| and ''μ''<sub>pp</sub> is a pure point measure. | |
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| All three types of measures are invariant under linear operations. Let ''H''<sub>ac</sub> be the subspace consisting of vectors whose spectral measures are absolutely continuous with respect to the [[Lebesgue measure]]. Define ''H''<sub>pp</sub> and ''H''<sub>sc</sub> in analogous fashion. These subspaces are invariant under ''T''. For example, if ''h'' ∈ ''H''<sub>ac</sub> and ''k'' = ''T h''. Let ''χ'' be the characteristic function of some Borel set in ''σ''(''T''), then
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| :<math>
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| \langle k, \chi(T) k \rangle = \int_{\sigma(T)} \chi(\lambda) \cdot \lambda^2 d \mu_{h}(\lambda) = \int_{\sigma(T)} \chi(\lambda) \; d \mu_k(\lambda).
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| </math> | |
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| So
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| :<math>\lambda^2 d \mu_{h} = d \mu_{k}\,</math>
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| and ''k'' ∈ ''H''<sub>ac</sub>. Furthermore, applying the spectral theorem gives
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| :<math>H = H_{\mathrm{ac}} \oplus H_{\mathrm{sc}} \oplus H_{\mathrm{pp}}.</math>
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| This leads to the following definitions:
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| #The spectrum of ''T'' restricted to ''H''<sub>ac</sub> is called the '''absolutely continuous spectrum''' of ''T'', ''σ''<sub>ac</sub>(''T'').
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| #The spectrum of ''T'' restricted to ''H''<sub>sc</sub> is called its '''singular spectrum''', ''σ''<sub>sc</sub>(''T'').
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| #The set of eigenvalues of ''T'' are called the '''pure point spectrum''' of ''T'', ''σ''<sub>pp</sub>(''T'').
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| The closure of the eigenvalues is the spectrum of ''T'' restricted to ''H''<sub>pp</sub>. So
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| :<math>\sigma(T) = \sigma_{\mathrm{ac}}(T) \cup \sigma_{\mathrm{sc}}(T) \cup {\bar \sigma_{\mathrm{pp}}(T)}.</math>
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| === Comparison ===
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| A bounded self adjoint operator on Hilbert space is a bounded operator on a Banach space.
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| Unlike the Banach space formulation, the union
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| :<math>\sigma(T) = {\bar \sigma_{\mathrm{pp}}(T)} \cup \sigma_{\mathrm{ac}}(T) \cup \sigma_{\mathrm{sc}}(T).</math>
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| 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
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| :<math>\oplus _{i = 1} ^m L^2(\mathbb{R}, \mu_i)</math>
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| for some Borel measures <math>\mu_i</math>. 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'' ∈ ''σ<sub>ac</sub>''(''T'') ∩ ''σ<sub>pp</sub>''(''T''), ''λ'' is sometimes called an eigenvalue ''embedded'' in the absolutely continuous spectrum.
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| When ''T'' is unitarily equivalent to multiplication by ''λ'' on
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| :<math>L^2(\mathbb{R}, \mu),</math>
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| the decomposition of ''σ''(''T'') from Borel functional calculus is a refinement of the Banach space case.
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| === Physics ===
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| The preceding comments can be extended to the unbounded self-adjoint operators since Riesz-Markov holds for [[locally compact]] [[Hausdorff space]]s.
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| In [[mathematical formulation of quantum mechanics|quantum mechanics]], observables are, not necessarily bounded, [[self adjoint operator]]s 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 state]]s. 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.
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| 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 <math>f</math>
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| is an element of <math>L^2(\mathbb{R})</math> and increasing as <math>x \to \infty</math>. | |
| :<math> f(x) = \begin{cases} n & \text{if }x \in \left[n, n+\frac{1}{n^4}\right], \\ 0 & \text{else.} \end{cases} </math>
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| 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 <math> x \to \infty </math>. Dynamical localization is more subtle to define.
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| Sometimes, when performing physical quantum mechanical calculations, one encounters "eigenvectors" that do not lie in ''L''<sup>2</sup>('''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 space]]s.
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| It was believed for some time that singular spectrum is something artificial. However, examples as the [[almost Mathieu operator]] and [[random Schrödinger operator]]s have shown, that all types of spectra arise naturally in physics. | |
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| == See also ==
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| * [[Essential spectrum]], spectrum of an operator modulo compact perturbations.
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| == References ==
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| * N. Dunford and J.T. Schwartz, ''Linear Operators, Part I: General Theory'', Interscience, 1958.
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| * M. Reed and B. Simon, ''Methods of Modern Mathematical Physics I: Functional Analysis'', Academic Press, 1972.
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| [[Category:Spectral theory]]
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