Editing Spectral theorem (section)

== Bounded self-adjoint operators ==
{{See also|Eigenfunction|Self-adjoint operator#Spectral theorem}}

===Possible absence of eigenvectors===

The next generalization we consider is that of [[Self-adjoint_operator#Bounded_self-adjoint_operators|bounded self-adjoint operators]] on a Hilbert space. Such operators may have no eigenvectors: for instance let {{math|''A''}} be the operator of multiplication by {{math|''t''}} on <math>L^2([0,1])</math>, that is,<ref>{{harvnb|Hall|2013}} Section 6.1</ref>
<math display="block"> [A f](t) = t f(t). </math>

This operator does not have any eigenvectors ''in'' <math>L^2([0,1])</math>, though it does have eigenvectors in a larger space. Namely the [[Distribution (mathematics)|distribution]] <math>f(t)=\delta(t-t_0)</math>, where <math>\delta</math> is the [[Dirac delta function]], is an eigenvector when construed in an appropriate sense. The Dirac delta function is however not a function in the classical sense and does not lie in the Hilbert space {{math|''L''<sup>2</sup>[0, 1]}}. Thus, the delta-functions are "generalized eigenvectors" of <math>A</math> but not eigenvectors in the usual sense. 

===Spectral subspaces and projection-valued measures===

In the absence of (true) eigenvectors, one can look for a "spectral subspace" consisting of an ''almost eigenvector'', i.e, a closed subspace <math>V_E</math> of <math>V</math> associated with a [[Borel set]] <math>E \subset \sigma(A)</math> in the [[Spectrum_(functional_analysis)|spectrum]] of <math>A</math>. This subspace can be thought of as the closed span of generalized eigenvectors for <math>A</math> with eigen''values'' in <math>E</math>.<ref>{{harvnb|Hall|2013}} Theorem 7.2.1</ref> In the above example, where <math> [A f](t) = t f(t), \;</math> we might consider the subspace of functions supported on a small interval <math>[a,a+\varepsilon]</math> inside <math>[0,1]</math>. This space is invariant under <math>A</math> and for any <math>f</math> in this subspace, <math>Af</math> is very close to <math>af</math>. Each subspace, in turn, is encoded by the associated projection operator, and the collection of all the subspaces is then represented by a [[projection-valued measure]]. 

One formulation of the spectral theorem expresses the operator {{math|''A''}} as an integral of the coordinate function over the operator's spectrum <math>\sigma(A)</math> with respect to a projection-valued measure.<ref>{{harvnb|Hall|2013}} Theorem 7.12</ref>
<math display="block"> A = \int_{\sigma(A)} \lambda \, d \pi (\lambda).</math>When the self-adjoint operator in question is [[compact operator|compact]], this version of the spectral theorem reduces to something similar to the finite-dimensional spectral theorem above, except that the operator is expressed as a finite or countably infinite linear combination of projections, that is, the measure consists only of atoms.
===Multiplication operator version===
An alternative formulation of the spectral theorem says that every bounded self-adjoint operator is unitarily equivalent to a multiplication operator, a relatively simple type of operator.{{math theorem
| math_statement = Let <math>A</math>  be a bounded self-adjoint operator on a Hilbert space <math> V </math>.  Then there is a [[measure space]] <math>(X, \Sigma, \mu) </math> and a real-valued [[ess sup|essentially bounded]] measurable function <math>\lambda </math> on <math>X</math> and a [[unitary operator]] <math>U : V \to L^2(X, \mu)</math> such that
<math display="block"> U^* T U = A,</math>
where <math> T </math> is the [[multiplication operator]]:
<math display="block"> [T f](x) = \lambda(x) f(x) </math>
and <math>  \vert  T  \vert </math>  <math> = \vert  \lambda  \vert_\infty </math>.
| name = '''Theorem'''<ref>{{harvnb|Hall|2013}} Theorem 7.20</ref>
}}Multiplication operators are a direct generalization of diagonal matrices. A finite-dimensional Hermitian vector space <math>V</math> may be coordinatized as the space of functions <math>f: B \to \C </math> from a basis <math>B</math> to the complex numbers, so that the <math>B</math>-coordinates of a vector are the values of the corresponding function <math>f</math>. The finite-dimensional spectral theorem for a self-adjoint operator <math>A: V \to V </math> states that there exists an orthonormal basis of eigenvectors <math>B</math>, so that the inner product becomes the [[dot product]] with respect to the <math>B</math>-coordinates: thus <math>V</math> is isomorphic to <math>L^2( B ,\mu )  </math> for the discrete unit measure <math>\mu</math> on <math>B</math>. Also <math>A</math> is unitarily equivalent to the multiplication operator <math>[Tf](v)  =  \lambda(v)  f(v)  </math>, where <math>\lambda(v)</math> is the eigenvalue of <math>v \in B  </math>: that is, <math>A</math> multiplies each <math>B</math>-coordinate by the corresponding eigenvalue <math>\lambda(v)</math>, the action of a diagonal matrix. Finally, the [[operator norm]] <math>|A| = |T| </math> is equal to the magnitude of the largest eigenvector <math>|\lambda|_\infty </math>.

The spectral theorem is the beginning of the vast research area of functional analysis called [[operator theory]]; see also [[spectral measure]].

There is also an analogous spectral theorem for bounded [[Normal operator|normal operators]] on Hilbert spaces. The only difference in the conclusion is that now ''<math>\lambda</math>'' may be complex-valued. 

===Direct integrals===
There is also a formulation of the spectral theorem in terms of [[Direct integral|direct integrals]]. It is similar to the multiplication-operator formulation, but more canonical.

Let <math>A</math> be a bounded self-adjoint operator and let <math>\sigma (A)</math> be the spectrum of <math>A</math>. The direct-integral formulation of the spectral theorem associates two quantities to <math>A</math>. First, a measure <math>\mu</math> on <math>\sigma (A)</math>, and second, a family of Hilbert spaces <math>\{H_{\lambda}\},\,\,\lambda\in\sigma (A).</math> We then form the direct integral Hilbert space
<math display="block"> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda). </math>
The elements of this space are functions (or "sections") <math>s(\lambda),\,\,\lambda\in\sigma(A),</math> such that <math>s(\lambda)\in H_{\lambda}</math> for all <math>\lambda</math>. 
The direct-integral version of the spectral theorem may be expressed as follows:<ref>{{harvnb|Hall|2013}} Theorem 7.19</ref>
{{math theorem|math_statement= If <math>A</math> is a bounded self-adjoint operator, then <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on <math display="block"> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda) </math>
for some measure <math>\mu</math> and some family <math>\{H_{\lambda}\}</math> of Hilbert spaces. The measure <math>\mu</math> is uniquely determined by <math>A</math> up to measure-theoretic equivalence; that is, any two measure associated to the same <math>A</math> have the same sets of measure zero. The dimensions of the Hilbert spaces <math>H_{\lambda}</math> are uniquely determined by <math>A</math> up to a set of <math>\mu</math>-measure zero.}}

The spaces <math>H_{\lambda}</math> can be thought of as something like "eigenspaces" for <math>A</math>. Note, however, that unless the one-element set <math>\lambda</math> has positive measure, the space <math>H_{\lambda}</math> is not actually a subspace of the direct integral. Thus, the <math>H_{\lambda}</math>'s should be thought of as "generalized eigenspace"—that is, the elements of <math>H_{\lambda}</math> are "eigenvectors" that do not actually belong to the Hilbert space.

Although both the multiplication-operator and direct integral formulations of the spectral theorem express a self-adjoint operator as unitarily equivalent to a multiplication operator, the direct integral approach is more canonical. First, the set over which the direct integral takes place (the spectrum of the operator) is canonical. Second, the function we are multiplying by is canonical in the direct-integral approach: Simply the function <math>\lambda\mapsto\lambda</math>.

===Cyclic vectors and simple spectrum===
A vector <math>\varphi</math> is called a [[cyclic vector]] for <math>A</math> if the vectors <math>\varphi,A\varphi,A^2\varphi,\ldots</math> span a dense subspace of the Hilbert space. Suppose <math>A</math> is a bounded self-adjoint operator for which a cyclic vector exists. In that case, there is no distinction between the direct-integral and multiplication-operator formulations of the spectral theorem. Indeed, in that case, there is a measure <math>\mu</math> on the spectrum <math>\sigma(A)</math> of <math>A</math> such that <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on <math>L^2(\sigma(A),\mu)</math>.<ref>{{harvnb|Hall|2013}} Lemma 8.11</ref> This result represents <math>A</math> simultaneously as a multiplication operator ''and'' as a direct integral, since <math>L^2(\sigma(A),\mu)</math> is just a direct integral in which each Hilbert space <math>H_{\lambda}</math> is just <math>\mathbb{C}</math>.

Not every bounded self-adjoint operator admits a cyclic vector; indeed, by the uniqueness in the direct integral decomposition, this can occur only when all the <math>H_{\lambda}</math>'s have dimension one. When this happens, we say that <math>A</math> has "simple spectrum" in the sense of [[Self-adjoint operator#Spectral multiplicity theory|spectral multiplicity theory]]. That is, a bounded self-adjoint operator that admits a cyclic vector should be thought of as the infinite-dimensional generalization of a self-adjoint matrix with distinct eigenvalues (i.e., each eigenvalue has multiplicity one).

Although not every <math>A</math> admits a cyclic vector, it is easy to see that we can decompose the Hilbert space as a direct sum of invariant subspaces on which <math>A</math> has a cyclic vector. This observation is the key to the proofs of the multiplication-operator and direct-integral forms of the spectral theorem.

===Functional calculus===
One important application of the spectral theorem (in whatever form) is the idea of defining a [[functional calculus]]. That is, given a function <math>f</math> defined on the spectrum of <math>A</math>, we wish to define an operator <math>f(A)</math>. If <math>f</math> is simply a positive power, <math>f(x) = x^n</math>, then <math>f(A)</math> is just the <math>n</math>-th power of <math>A</math>, <math>A^n</math>. The interesting cases are where <math>f</math> is a nonpolynomial function such as a square root or an exponential. Either of the versions of the spectral theorem provides such a functional calculus.<ref>E.g., {{harvnb|Hall|2013}} Definition 7.13</ref> In the direct-integral version, for example, <math>f(A)</math> acts as the "multiplication by <math>f</math>" operator in the direct integral:
<math display="block">[f(A)s](\lambda) = f(\lambda) s(\lambda).</math>
That is to say, each space <math>H_{\lambda}</math> in the direct integral is a (generalized) eigenspace for <math>f(A)</math> with eigenvalue <math>f(\lambda)</math>.