spectral theorem
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theory \PMlinkescapephrasestate \PMlinkescapephrasetheory \PMlinkescapephraseseries \PMlinkescapephrasebounded^{} \PMlinkescapephraseunbounded^{}
1 Introduction
The spectral theorem^{} is series of results in functional analysis^{} that explore conditions for operators^{} in Hilbert spaces^{} to be diagonalizable (in some appropriate sense). These results can also describe how the diagonalization takes , mainly by analyzing how the operator acts in the underlying Hilbert space.
Roughly speaking, the spectral theorems state that normal operators (or selfadjoint operators) are diagonalizable and can be expressed as a sum or, more generally, as an integral of projections^{}. More specifically, a normal (or selfadjoint^{}) operator $T$ is unitarily equivalent^{} to a multiplication operator in some ${L}^{2}$space (http://planetmath.org/L2SpacesAreHilbertSpaces) and we can to it a spectral measure (or a resolution of identity) whose integration gives is $T$.
There is a wide of spectral theorems, each one with its own , applicable to many classes of normal and selfadjoint operators.
2 Motivation
We explore here two to motivate the spectral theorem. The first is by recalling the finitedimensional case, corresponding to the well known result in linear algebra, the spectral theorem for Hermitian matrices (http://planetmath.org/SpectralTheoremForHermitianMatrices) (or the selfadjoint analog). The second motivation comes from the ${C}^{*}$algebra^{} (http://planetmath.org/CAlgebra) theory, by regarding a normal (or selfadjoint) operator as a continuous function^{}.
2.0.1 Finitedimensional case
Suppose $T$ is a selfadjoint operator in a finitedimensional Hilbert space $H$. An important fact about selfadjoint operators (not just in finitedimensional spaces) is the following:
Fact 1  If $V\mathrm{\subseteq}H$ is an invariant subspace by $T$, then so it is ${V}^{\mathrm{\u27c2}}$, the orthogonal complement^{} of $V$.
Proof: Let $x\in V$ and $y\in {V}^{\u27c2}$. Then $\u27e8x,Ty\u27e9=\u27e8Tx,y\u27e9=0$, where the last term is zero because $V$ is invariant^{} by $T$, i.e. $Tx\in V$. But this proves that $Ty\in {V}^{\u27c2}$. $\mathrm{\square}$
In finite dimensions^{} it is known that every linear transformation has, at least, one eigenvector^{}. Of course, the subspace^{} generated by an eigenvector is always invariant.
Let ${v}_{1}$ be an eigenvector of $T$ and ${V}_{1}$ the subspace generated by it. For selfadjoint transformations, Fact 1 above says that ${V}_{1}^{\u27c2}$ is also invariant by $T$. Thus, by restriction^{} (http://planetmath.org/RestrictionOfAFunction), we have a selfadjoint operator $T:{V}_{1}^{\u27c2}\u27f6{V}_{1}^{\u27c2}$ and we could again find an eigenvector and repeat the same . Thus, we are decomposing $H$ as a direct sum of orthogonal^{} onedimensional subspaces $H={V}_{1}\oplus \mathrm{\dots}\oplus {V}_{n}$, and the operator $T$ can be expressed as a sum
$$T=\sum _{i=1}^{n}{\lambda}_{i}{P}_{i}$$ 
where each ${\lambda}_{i}$ is the eigenvalue^{} associated with the eigenvector ${v}_{i}$ and each ${P}_{i}$ is the orthogonal projection onto the subspace ${V}_{i}$.
This is exactly the process of diagonalization of a selfadjoint matrix.
For normal operators it is more subtle as Fact 1 is no longer true. The idea to overpass this is that eigenvectors of normal operators are always orthogonal to each other (see this entry (http://planetmath.org/EigenvaluesOfNormalOperators)).
2.0.2 ${C}^{*}$algebras
Suppose $T$ is a selfadjoint operator in some Hilbert space $H$. The closed *algebra generated by $T$ and the identity operator^{} is a commutative ${C}^{*}$algebra (http://planetmath.org/CAlgebra), which we denote by ${C}^{*}(T)$. Hence, the GelfandNaimark theorem^{} (http://planetmath.org/GelfandTransform) and the continuous functional calculus provide an isomorphism^{}
${C}^{*}(T)\cong C(\sigma (T))$ 
where $\sigma (T)$ stands for the spectrum of $T$ and $C(\sigma (T))$ is the ${C}^{*}$algebra of continuous functions $\sigma (T)\to \u2102$.
Recall that the spectrum of a selfadjoint operator is a always a compact subset of $\mathbb{R}$. Thus, we can think of $T$ as a continuous function defined in a subset of $\mathbb{R}$.
It is a well known fact from measure theory that every continuous function $f:X\u27f6\u2102$ can be approximated by linear combinations^{} of characteristic functions^{}. With some additional effort it can be seen that each continuous function $f$ is in fact a (vector valued) integral of characteristic functions
$f={\displaystyle {\int}_{X}}f\mathit{d}\chi $ 
where $\chi $ is the vector measure of characteristic functions $\chi (A):={\chi}_{A}$.
We now notice that characteristic functions in $\sigma (T)$ are not continuous in general. Hence, they may not have a correspondent in the ${C}^{*}(T)$, the ${C}^{*}$algebra generated by $T$ and the identity^{}. The fact is that they do have a correspondent in the von Neumann algebra^{} generated by $T$. Informally, this is the same as saying that characteristic functions belong to ${L}^{\mathrm{\infty}}(\sigma (T))$ rather then $C(\sigma (T))$.
The correspondent operators in the von Neumann algebra generated by $T$ must be projections (since characteristic functions are projections in ${L}^{\mathrm{\infty}}$), and similarly, $T$ can be approximated by linear combinations of projections and can, in fact, be expressed as an integral of projections:
$T={\displaystyle {\int}_{\sigma (T)}}\lambda \mathit{d}P(\lambda )$ 
where $P(\lambda )$ is a resolution of identity for $T$ (or a projection valued measure, when $T$ is a normal operator).
3 Spectral Theorems
Here we list a series of spectral theorems, applicable to different classes of normal or selfadjoint operators.
3.0.1 Normal Operators

•
spectral theorem for Hermitean matrices (http://planetmath.org/SpectralTheoremForHermitianMatrices)
This is the finitedimensional case. It is often referred to as ”the spectral theorem”, especially in linear algebra.

•
spectral theorem for bounded (http://planetmath.org/OperatorNorm) normal operators in separable Hilbert spaces

•
spectral theorem for bounded normal operators in inseparable Hilbert spaces

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spectral theorem for compact (http://planetmath.org/CompactOperator) normal operators

•
spectral theorem for unitary operators

•
spectral theorem for unbounded (http://planetmath.org/OperatorNorm) normal operators
3.0.2 Selfadjoint Operators

•
spectral theorem for selfadjoint matrices

•
spectral theorem for bounded selfadjoint operators in separable Hilbert spaces

•
spectral theorem for bounded selfadjoint operators in inseparable Hilbert spaces

•
spectral theorem for selfadjoint operators

•
spectral theorem for unbounded selfadjoint operators
Title  spectral theorem 

Canonical name  SpectralTheorem1 
Date of creation  20130322 18:04:30 
Last modified on  20130322 18:04:30 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  12 
Author  asteroid (17536) 
Entry type  Feature 
Classification  msc 47B15 
Classification  msc 47A15 
Classification  msc 47A10 
Classification  msc 46C99 
Classification  msc 15A18 