Spectral Theorem
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
and
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a spectral theorem is a result about when a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
or
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
can be
diagonalized In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
(that is, represented as a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s that can be modeled by
multiplication operator In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is, T_f\varphi(x) = f(x) \varphi (x) \quad for all in th ...
s, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
s. See also
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result o ...
for a historical perspective. Examples of operators to which the spectral theorem applies are
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
s or more generally
normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operat ...
s on
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. The spectral theorem also provides a
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
decomposition, called the spectral decomposition, of the underlying vector space on which the operator acts.
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
proved the spectral theorem for
symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about determinants. The spectral theorem as generalized by
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
is today perhaps the most important result of operator theory. This article mainly focuses on the simplest kind of spectral theorem, that for a
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...
operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.


Finite-dimensional case


Hermitian maps and Hermitian matrices

We begin by considering a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
on \mathbb^n (but the following discussion will be adaptable to the more restrictive case of
symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
on \mathbb^n). We consider a Hermitian map on a finite-dimensional
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
endowed with a
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
sesquilinear In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows ...
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
\langle\cdot,\cdot\rangle. The Hermitian condition on A means that for all , : \langle A x, y \rangle = \langle x, A y \rangle. An equivalent condition is that , where is the
Hermitian conjugate In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
of . In the case that is identified with a Hermitian matrix, the matrix of can be identified with its
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
. (If is a
real matrix In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \beg ...
, then this is equivalent to , that is, is a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
.) This condition implies that all eigenvalues of a Hermitian map are real: it is enough to apply it to the case when is an eigenvector. (Recall that an
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of a linear map is a (non-zero) vector such that for some scalar . The value is the corresponding
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
. Moreover, the
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
are roots of the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
.) Theorem. If is Hermitian on , then there exists an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of consisting of eigenvectors of . Each eigenvalue is real. We provide a sketch of a proof for the case where the underlying field of scalars is the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. By the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomial ...
, applied to the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
of , there is at least one eigenvalue and eigenvector . Then since : \lambda_1 \langle e_1, e_1 \rangle = \langle A (e_1), e_1 \rangle = \langle e_1, A(e_1) \rangle = \bar\lambda_1 \langle e_1, e_1 \rangle, we find that is real. Now consider the space , the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of . By Hermiticity, is an
invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General descrip ...
of . Applying the same argument to shows that has an eigenvector . Finite induction then finishes the proof. The spectral theorem holds also for symmetric maps on finite-dimensional real inner product spaces, but the existence of an eigenvector does not follow immediately from the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomial ...
. To prove this, consider as a Hermitian matrix and use the fact that all eigenvalues of a Hermitian matrix are real. The matrix representation of in a basis of eigenvectors is diagonal, and by the construction the proof gives a basis of mutually orthogonal eigenvectors; by choosing them to be unit vectors one obtains an orthonormal basis of eigenvectors. can be written as a linear combination of pairwise orthogonal projections, called its spectral decomposition. Let : V_\lambda = \ be the eigenspace corresponding to an eigenvalue . Note that the definition does not depend on any choice of specific eigenvectors. is the orthogonal direct sum of the spaces where the index ranges over eigenvalues. In other words, if denotes the
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
onto , and are the eigenvalues of , then the spectral decomposition may be written as : A = \lambda_1 P_ + \cdots + \lambda_m P_. If the spectral decomposition of ''A'' is A = \lambda_1 P_1 + \cdots + \lambda_m P_m, then A^2 = (\lambda_1)^2 P_1 + \cdots + (\lambda_m)^2 P_m and \mu A = \mu \lambda_1 P_1 + \cdots + \mu \lambda_m P_m for any scalar \mu. It follows that for any polynomial one has : f(A) = f(\lambda_1) P_1 + \cdots + f(\lambda_m) P_m. The spectral decomposition is a special case of both the
Schur decomposition In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper tri ...
and the
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...
.


Normal matrices

The spectral theorem extends to a more general class of matrices. Let be an operator on a finite-dimensional inner product space. is said to be
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
if . One can show that is normal if and only if it is unitarily diagonalizable. Proof: By the
Schur decomposition In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper tri ...
, we can write any matrix as , where is unitary and is upper-triangular. If is normal, then one sees that . Therefore, must be diagonal since a normal upper triangular matrix is diagonal (see
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
). The converse is obvious. In other words, is normal if and only if there exists a
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...
such that : A = U D U^*, where is a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
. Then, the entries of the diagonal of are the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of . The column vectors of are the eigenvectors of and they are orthonormal. Unlike the Hermitian case, the entries of need not be real.


Compact self-adjoint operators

In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
self-adjoint operators In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
is virtually the same as in the finite-dimensional case. Theorem. Suppose is a compact self-adjoint operator on a (real or complex) Hilbert space . Then there is an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of consisting of eigenvectors of . Each eigenvalue is real. As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector. One cannot rely on determinants to show existence of eigenvalues, but one can use a maximization argument analogous to the variational characterization of eigenvalues. If the compactness assumption is removed, then it is ''not'' true that every self-adjoint operator has eigenvectors.


Bounded self-adjoint operators


Possible absence of eigenvectors

The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space. Such operators may have no eigenvalues: for instance let be the operator of multiplication by on L^2( ,1, that is, : \varphit) = t \varphi(t). \; This operator does not have any eigenvectors ''in'' L^2( ,1, though it does have eigenvectors in a larger space. Namely the
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
\varphi(t)=\delta(t-t_0), where \delta is the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, 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 or any other
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. Thus, the delta-functions are "generalized eigenvectors" of A but not eigenvectors in the usual sense.


Spectral subspaces and projection-valued measures

In the absence of (true) eigenvectors, one can look for subspaces consisting of ''almost eigenvectors''. In the above example, for example, where \varphit) = t \varphi(t), \; we might consider the subspace of functions supported on a small interval ,a+\varepsilon/math> inside ,1/math>. This space is invariant under A and for any \varphi in this subspace, A\varphi is very close to a\varphi. In this approach to the spectral theorem, if A is a bounded self-adjoint operator, then one looks for large families of such "spectral subspaces". 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 In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are ...
. One formulation of the spectral theorem expresses the operator as an integral of the coordinate function over the operator's
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
\sigma(A) with respect to a projection-valued measure. : A = \int_ \lambda \, d E_ . When the self-adjoint operator in question is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, 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. The significance of this result is that multiplication operators are in many ways easy to understand. The spectral theorem is the beginning of the vast research area of functional analysis called
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...
; see also the
spectral measure In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his disse ...
. There is also an analogous spectral theorem for bounded
normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operat ...
s on Hilbert spaces. The only difference in the conclusion is that now may be complex-valued.


Direct integrals

There is also a formulation of the spectral theorem in terms of direct integrals. It is similar to the multiplication-operator formulation, but more canonical. Let A be a bounded self-adjoint operator and let \sigma (A) be the spectrum of A. The direct-integral formulation of the spectral theorem associates two quantities to A. First, a measure \mu on \sigma (A), and second, a family of Hilbert spaces \,\,\,\lambda\in\sigma (A). We then form the direct integral Hilbert space \int_\mathbf^\oplus H_\, d \mu(\lambda). The elements of this space are functions (or "sections") s(\lambda),\,\,\lambda\in\sigma(A), such that s(\lambda)\in H_ for all \lambda. The direct-integral version of the spectral theorem may be expressed as follows: The spaces H_ can be thought of as something like "eigenspaces" for A. Note, however, that unless the one-element set has positive measure, the space H_ is not actually a subspace of the direct integral. Thus, the H_'s should be thought of as "generalized eigenspace"—that is, the elements of H_ 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 \lambda\mapsto\lambda.


Cyclic vectors and simple spectrum

A vector \varphi is called a
cyclic vector An operator ''A'' on an (infinite dimensional) Banach space or Hilbert space H has a cyclic vector ''f'' if the vectors ''f'', ''Af'', ''A2f'',... span H. Equivalently, ''f'' is a cyclic vector for ''A'' in case the set of all vectors of the for ...
for A if the vectors \varphi,A\varphi,A^2\varphi,\ldots span a dense subspace of the Hilbert space. Suppose A 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 \mu on the spectrum \sigma(A) of A such that A is unitarily equivalent to the "multiplication by \lambda" operator on L^2(\sigma(A),\mu). This result represents A simultaneously as a multiplication operator ''and'' as a direct integral, since L^2(\sigma(A),\mu) is just a direct integral in which each Hilbert space H_ is just \mathbb. 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 H_'s have dimension one. When this happens, we say that A has "simple spectrum" in the sense of 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 A 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 A 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 In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral the ...
. That is, given a function f defined on the spectrum of A, we wish to define an operator f(A). If f is simply a positive power, f(x)=x^n, then f(A) is just the n\mathrm power of A, A^n. The interesting cases are where f 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. In the direct-integral version, for example, f(A) acts as the "multiplication by f" operator in the direct integral: : (A)s\lambda)=f(\lambda)s(\lambda). That is to say, each space H_ in the direct integral is a (generalized) eigenspace for f(A) with eigenvalue f(\lambda).


General self-adjoint operators

Many important linear operators which occur in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, such as
differential operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
, are unbounded. There is also a spectral theorem for
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
s that applies in these cases. To give an example, every constant-coefficient differential operator is unitarily equivalent to a multiplication operator. Indeed, the unitary operator that implements this equivalence is the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
; the multiplication operator is a type of Fourier multiplier. In general, spectral theorem for self-adjoint operators may take several equivalent forms.See Section 10.1 of Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues.


See also

* *
Spectral theory of compact operators In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space ''H'', the compact operators are the closure of the finite rank operators in the unifo ...
*
Spectral theory of normal C*-algebras In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra \mathcal(H) of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of \mathcal(H). A ...
*
Borel functional calculus In functional analysis, a branch of mathematics, the Borel functional calculus is a '' functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scop ...
*
Spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result o ...
*
Matrix decomposition In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of ...
*
Canonical form In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
* Jordan decomposition, of which the spectral decomposition is a special case. *
Singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...
, a generalisation of spectral theorem to arbitrary matrices. *
Eigendecomposition of a matrix In linear algebra, eigendecomposition is the Matrix factorization, factorization of a matrix (mathematics), matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrix, di ...
*
Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary r ...


Notes


References

*
Sheldon Axler Sheldon Jay Axler (born November 6, 1949, Philadelphia) is an American mathematician and textbook author. He is a professor of mathematics and the Dean of the College of Science and Engineering at San Francisco State University. He graduated fr ...
, ''Linear Algebra Done Right'', Springer Verlag, 1997 * *
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...

"What Does the Spectral Theorem Say?"
''American Mathematical Monthly'', volume 70, number 3 (1963), pages 241–24
Other link
* M. Reed and B. Simon, ''Methods of Mathematical Physics'', vols I–IV, Academic Press 1972. * G. Teschl, ''Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators'', https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009. * {{Spectral theory * Linear algebra Matrix theory Singular value decomposition Theorems in functional analysis Theorems in linear algebra