Spectral theory

 ( Linear Algebra ) 

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 of studies of linear algebra and the solutions of systems of linear equations and their generalizations.

William Arveson (2026). 9780387953007, Springer. . ISBN 9780387953007

The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

Viktor Antonovich Sadovnichiĭ (1991). 9780306110283, Springer. ISBN 9780306110283

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Mathematical background The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous. Hilbert himself was surprised by the unexpected application of this theory, noting that "I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis' without any presentiment that it would later find application to the actual spectrum of physics."

There have been three main ways to formulate spectral theory, each of which find use in different domains. After Hilbert's initial formulation, the later development of abstract and the spectral theory of single on them were well suited to the requirements of physics, exemplified by the work of von Neumann.

The further theory built on this to address in general. This development leads to the Gelfand representation, which covers the commutative case, and further into non-commutative harmonic analysis.

The difference can be seen in making the connection with Fourier analysis. The Fourier transform on the real line is in one sense the spectral theory of derivative as a differential operator. But for that to cover the phenomena one has already to deal with generalized eigenfunctions (for example, by means of a rigged Hilbert space). On the other hand, it is simple to construct a group algebra, the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means of Pontryagin duality.

One can also study the spectral properties of operators on Banach spaces. For example, on Banach spaces have many spectral properties similar to that of matrices.

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Physical background The background in the physics of has been explained in this way:E. Brian Davies, quoted on the King's College London analysis group website

Such physical ideas have nothing to do with the mathematical theory on a technical level, but there are examples of indirect involvement (see for example Mark Kac's question Can you hear the shape of a drum?). Hilbert's adoption of the term "spectrum" has been attributed to an 1897 paper of Wilhelm Wirtinger on Hill differential equation (by Jean Dieudonné), and it was taken up by his students during the first decade of the twentieth century, among them Erhard Schmidt and Hermann Weyl. The conceptual basis for Hilbert space was developed from Hilbert's ideas by Erhard Schmidt and Frigyes Riesz.

Jean-Luc Dorier (2026). 9780792365396, Springer. . ISBN 9780792365396

It was almost twenty years later, when quantum mechanics was formulated in terms of the Schrödinger equation, that the connection was made to atomic spectra; a connection with the mathematical physics of vibration had been suspected before, as remarked by Henri Poincaré, but rejected for simple quantitative reasons, absent an explanation of the Balmer series.Cf. Spectra in mathematics and in physics by Jean Mawhin, p.4 and pp. 10-11. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous, rather than being an object of Hilbert's spectral theory.

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A definition of spectrum

Consider a bounded linear transformation T defined everywhere over a general Banach space. We form the transformation: $$R\_\{\backslash zeta\}\; =\; \backslash left(\; \backslash zeta\; I\; -\; T\; \backslash right)^\{-1\}.$$

Here I is the identity operator and ζ is a complex number. The inverse of an operator T, that is T⁻¹, is defined by: $$T\; T^\{-1\}\; =\; T^\{-1\}\; T\; =\; I.$$

If the inverse exists, T is called regular. If it does not exist, T is called singular.

With these definitions, the resolvent set of T is the set of all complex numbers ζ such that R_ζ exists and is Bounded operator. This set often is denoted as ρ( T). The spectrum of T is the set of all complex numbers ζ such that R_ζ fails to exist or is unbounded. Often the spectrum of T is denoted by σ( T). The function R_ζ for all ζ in ρ( T) (that is, wherever R_ζ exists as a bounded operator) is called the resolvent of T. The spectrum of T is therefore the complement of the resolvent set of T in the complex plane.

Edgar Raymond Lorch (2026). 9780758171566, Textbook Publishers. . ISBN 9780758171566

Every eigenvalue of T belongs to σ( T), but σ( T) may contain non-eigenvalues.

Nicholas Young (1988). 9780521337175, Cambridge University Press. . ISBN 9780521337175

This definition applies to a Banach space, but of course other types of space exist as well; for example, topological vector spaces include Banach spaces, but can be more general.

Dmitriĭ Petrovich Zhelobenko (2026). 0821837311, American Mathematical Society. . ISBN-10 0821837311

On the other hand, Banach spaces include , and it is these spaces that find the greatest application and the richest theoretical results.

Edgar Raymond Lorch (2026). 9780758171566, Textbook Publishers. . ISBN 9780758171566

With suitable restrictions, much can be said about the structure of the spectra of transformations in a Hilbert space. In particular, for self-adjoint operators, the spectrum lies on the real line and (in general) is a spectral combination of a point spectrum of discrete eigenvalues and a continuous spectrum.

Edgar Raymond Lorch (2026). 9780758171566, Textbook Publishers. . ISBN 9780758171566

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Spectral theory briefly In functional analysis and linear algebra the spectral theorem establishes conditions under which an operator can be expressed in simple form as a sum of simpler operators. As a full rigorous presentation is not appropriate for this article, we take an approach that avoids much of the rigor and satisfaction of a formal treatment with the aim of being more comprehensible to a non-specialist.

This topic is easiest to describe by introducing the bra–ket notation of Paul Dirac for operators.

PAM Dirac (1981). 9780198520115, Oxford University Press. . ISBN 9780198520115

As an example, a very particular linear operator L might be written as a dyadic product:

Jürgen Audretsch (2026). 9783527406845, Wiley-VCH. ISBN 9783527406845

R. A. Howland (2026). 9780387280592, Birkhäuser. . ISBN 9780387280592

$L\; =\; |\; k\_1\; \backslash rangle\; \backslash langle\; b\_1\; |,$

in terms of the "bra" ⟨₁| and the "ket" |₁⟩. A function is described by a ket as | ⟩. The function defined on the coordinates $(x\_1,\; x\_2,\; x\_3,\; \backslash dots)$ is denoted as

$f(x)=\backslash langle\; x\; |\; f\backslash rangle$

and the magnitude of f by

$\backslash |f\; \backslash |^2\; =\; \backslash langle\; f|\; f\backslash rangle\; =\backslash int\; \backslash langle\; f|\; x\backslash rangle\; \backslash langle\; x\; |\; f\; \backslash rangle\; \backslash ,\; dx\; =\; \backslash int\; f^*(x)\; f(x)\; \backslash ,\; dx$

where the notation (*) denotes a complex conjugate. This inner product choice defines a very specific inner product space, restricting the generality of the arguments that follow.

The effect of L upon a function f is then described as:

$L\; |\; f\backslash rangle\; =\; |\; k\_1\; \backslash rangle\; \backslash langle\; b\_1\; |\; f\; \backslash rangle$

expressing the result that the effect of L on f is to produce a new function $|\; k\_1\; \backslash rangle$ multiplied by the inner product represented by $\backslash langle\; b\_1\; |\; f\; \backslash rangle$.

A more general linear operator L might be expressed as:

$L\; =\; \backslash lambda\_1\; |\; e\_1\backslash rangle\backslash langle\; f\_1|\; +\; \backslash lambda\_2\; |\; e\_2\backslash rangle\; \backslash langle\; f\_2|\; +\; \backslash lambda\_3\; |\; e\_3\backslash rangle\backslash langle\; f\_3|\; +\; \backslash dots\; ,$

where the $\backslash \{\; \backslash ,\; \backslash lambda\_i\; \backslash ,\; \backslash \}$ are scalars and the $\backslash \{\; \backslash ,\; |\; e\_i\; \backslash rangle\; \backslash ,\; \backslash \}$ are a basis and the $\backslash \{\; \backslash ,\; \backslash langle\; f\_i\; |\; \backslash ,\; \backslash \}$ a Dual basis for the space. The relation between the basis and the reciprocal basis is described, in part, by:

$\backslash langle\; f\_i\; |\; e\_j\; \backslash rangle\; =\; \backslash delta\_\{ij\}$

If such a formalism applies, the $\backslash \{\; \backslash ,\; \backslash lambda\_i\; \backslash ,\; \backslash \}$ are eigenvalues of L and the functions $\backslash \{\; \backslash ,\; |\; e\_i\; \backslash rangle\; \backslash ,\; \backslash \}$ are eigenfunctions of L. The eigenvalues are in the spectrum of L.

Bernard Friedman (1990). 9780486664446, Dover Publications. . ISBN 9780486664446

Some natural questions are: under what circumstances does this formalism work, and for what operators L are expansions in series of other operators like this possible? Can any function f be expressed in terms of the eigenfunctions (are they a Schauder basis) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite-dimensional spaces and finite-dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and matrix algebra.

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Resolution of the identity This section continues in the rough and ready manner of the above section using the bra–ket notation, and glossing over the many important details of a rigorous treatment.

See discussion in Dirac's book referred to above, and

Milan Vujičić (2026). 9783540746379, Springer. . ISBN 9783540746379

A rigorous mathematical treatment may be found in various references.See, for example, the fundamental text of

John von Neumann (2026). 9780691028934, Princeton University Press. . ISBN 9780691028934

and

Arch W. Naylor, George R. Sell (2026). 038795001X, Springer. . 038795001X

,

Steven Roman (2026). 9780387728285, Springer. . ISBN 9780387728285

,

I︠U︡riĭ Makarovich Berezanskiĭ (1968). 9780821815670, American Mathematical Society. . ISBN 9780821815670

In particular, the dimension n of the space will be finite.

Using the bra–ket notation of the above section, the identity operator may be written as:

$I\; =\; \backslash sum\; \_\{i=1\}\; ^\{n\}\; |\; e\_i\; \backslash rangle\; \backslash langle\; f\_i\; |$

where it is supposed as above that $\backslash \{\; |e\_i\backslash rangle\backslash \}$ are a basis and the $\backslash \{\; \backslash langle\; f\_i\; |\; \backslash \}$ a reciprocal basis for the space satisfying the relation:

$\backslash langle\; f\_i\; |\; e\_j\backslash rangle\; =\; \backslash delta\_\{ij\}\; .$

This expression of the identity operation is called a representation or a resolution of the identity. This formal representation satisfies the basic property of the identity:

$I^k\; =\; I$

valid for every positive integer k.

Applying the resolution of the identity to any function in the space $|\; \backslash psi\; \backslash rangle$, one obtains:

$I\; |\backslash psi\; \backslash rangle\; =\; |\backslash psi\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^\{n\}\; |\; e\_i\; \backslash rangle\; \backslash langle\; f\_i\; |\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^\{n\}\; c\_i\; |\; e\_i\; \backslash rangle$

which is the generalized Fourier expansion of ψ in terms of the basis functions { e_i }. See for example,

Gerald B Folland (2026). 9780821847909, American Mathematical Society. ISBN 9780821847909

Here $c\_i\; =\; \backslash langle\; f\_i\; |\; \backslash psi\; \backslash rangle$.

Given some operator equation of the form:

$O\; |\; \backslash psi\; \backslash rangle\; =\; |\; h\; \backslash rangle$

with h in the space, this equation can be solved in the above basis through the formal manipulations:

$O\; |\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^\{n\}\; c\_i\; \backslash left(\; O\; |\; e\_i\; \backslash rangle\; \backslash right)\; =\; \backslash sum\_\{i=1\}^\{n\}\; |\; e\_i\; \backslash rangle\; \backslash langle\; f\_i\; |\; h\; \backslash rangle\; ,$

$\backslash langle\; f\_j|O|\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^\{n\}\; c\_i\; \backslash langle\; f\_j|\; O\; |\; e\_i\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^\{n\}\; \backslash langle\; f\_j|\; e\_i\; \backslash rangle\; \backslash langle\; f\_i\; |\; h\; \backslash rangle\; =\; \backslash langle\; f\_j\; |\; h\; \backslash rangle,\; \backslash quad\; \backslash forall\; j$

which converts the operator equation to a matrix equation determining the unknown coefficients c_j in terms of the generalized Fourier coefficients $\backslash langle\; f\_j\; |\; h\; \backslash rangle$ of h and the matrix elements $O\_\{ji\}=\; \backslash langle\; f\_j|\; O\; |\; e\_i\; \backslash rangle$ of the operator O.

The role of spectral theory arises in establishing the nature and existence of the basis and the reciprocal basis. In particular, the basis might consist of the eigenfunctions of some linear operator L:

$L\; |\; e\_i\; \backslash rangle\; =\; \backslash lambda\_i\; |\; e\_i\; \backslash rangle\; \backslash ,\; ;$

with the {  λ_i } the eigenvalues of L from the spectrum of L. Then the resolution of the identity above provides the dyad expansion of L:

$LI\; =\; L\; =\; \backslash sum\_\{i=1\}^\{n\}\; L\; |\; e\_i\; \backslash rangle\; \backslash langle\; f\_i|\; =\; \backslash sum\_\{i=1\}^\{n\}\; \backslash lambda\; \_i\; |\; e\_i\; \backslash rangle\; \backslash langle\; f\_i\; |\; .$

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Resolvent operator Using spectral theory, the resolvent operator R:

$R\; =\; (\backslash lambda\; I\; -\; L)^\{-1\},\backslash ,$

can be evaluated in terms of the eigenfunctions and eigenvalues of L, and the Green's function corresponding to L can be found.

Applying R to some arbitrary function in the space, say $\backslash varphi$,

$R\; |\backslash varphi\; \backslash rangle\; =\; (\backslash lambda\; I\; -\; L)^\{-1\}\; |\backslash varphi\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{1\}\{\backslash lambda-\; \backslash lambda\_i\}\; |e\_i\; \backslash rangle\; \backslash langle\; f\_i\; |\; \backslash varphi\; \backslash rangle.$

This function has poles in the complex λ-plane at each eigenvalue of L. Thus, using the calculus of residues:

$\backslash frac\{1\}\{2\backslash pi\; i\; \}\; \backslash oint\_C\; R\; |\backslash varphi\; \backslash rangle\; d\; \backslash lambda\; =\; -\backslash sum\_\{i=1\}^n\; |e\_i\; \backslash rangle\; \backslash langle\; f\_i\; |\; \backslash varphi\; \backslash rangle\; =\; -|\backslash varphi\; \backslash rangle,$

where the line integral is over a contour C that includes all the eigenvalues of L.

Suppose our functions are defined over some coordinates { x_j}, that is:

$\backslash langle\; x|\; \backslash varphi\; \backslash rangle\; =\; \backslash varphi\; (x\_1,\; x\_2,\; ...).$

Introducing the notation

$\backslash langle\; x\; ,\; y\; \backslash rangle\; =\; \backslash delta\; (x-y),$

where δ(x − y) = δ(x₁ − y₁, x₂ − y₂, x₃ − y₃, ...) is the Dirac delta function,

we can write

$\backslash langle\; x,\; \backslash varphi\; \backslash rangle\; =\; \backslash int\; \backslash langle\; x\; ,\; y\; \backslash rangle\; \backslash langle\; y,\; \backslash varphi\; \backslash rangle\; dy.$

Then:

$\backslash begin\{align\}$

\left\langle x, \frac{1}{2\pi i } \oint_C \frac{\varphi}{\lambda I - L} d \lambda\right\rangle &= \frac{1}{2\pi i }\oint_C d \lambda \left \langle x, \frac{\varphi}{\lambda I - L} \right \rangle\\ &= \frac{1}{2\pi i } \oint_C d \lambda \int dy \left \langle x, \frac{y}{\lambda I - L} \right \rangle \langle y, \varphi \rangle \end{align}

The function G(x, y; λ) defined by:

$\backslash begin\{align\}$

G(x, y; \lambda) &= \left \langle x, \frac{y}{\lambda I - L} \right \rangle \\ &= \sum_{i=1}^n \sum_{j=1}^n \langle x, e_i \rangle \left \langle f_i, \frac{e_j}{\lambda I - L} \right \rangle \langle f_j , y\rangle \\ &= \sum_{i=1}^n \frac{\langle x, e_i \rangle \langle f_i , y\rangle }{\lambda - \lambda_i} \\ &= \sum_{i=1}^n \frac{e_i (x) f_i^*(y) }{\lambda - \lambda_i}, \end{align}

is called the Green's function for operator L, and satisfies:

$\backslash frac\{1\}\{2\backslash pi\; i\; \}\backslash oint\_C\; G(x,y;\backslash lambda)\; \backslash ,\; d\; \backslash lambda\; =\; -\backslash sum\_\{i=1\}^n\; \backslash langle\; x,\; e\_i\; \backslash rangle\; \backslash langle\; f\_i\; ,\; y\backslash rangle\; =\; -\backslash langle\; x,\; y\backslash rangle\; =\; -\backslash delta\; (x-y).$

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Operator equations Consider the operator equation:

$(O-\backslash lambda\; I\; )\; |\backslash psi\; \backslash rangle\; =\; |h\; \backslash rangle;$

in terms of coordinates:

$\backslash int\; \backslash langle\; x,\; (O-\backslash lambda\; I)y\; \backslash rangle\; \backslash langle\; y,\; \backslash psi\; \backslash rangle\; \backslash ,\; dy\; =\; h(x).$

A particular case is λ = 0.

The Green's function of the previous section is:

$\backslash langle\; y,\; G(\backslash lambda)\; z\backslash rangle\; =\; \backslash left\; \backslash langle\; y,\; (O-\backslash lambda\; I)^\{-1\}\; z\; \backslash right\; \backslash rangle\; =\; G(y,\; z;\; \backslash lambda),$

and satisfies:

$\backslash int\; \backslash langle\; x,\; (O\; -\; \backslash lambda\; I)\; y\; \backslash rangle\; \backslash langle\; y,\; G(\backslash lambda)\; z\; \backslash rangle\; \backslash ,\; dy\; =\; \backslash int\; \backslash langle\; x,\; (O-\backslash lambda\; I)\; y\; \backslash rangle\; \backslash left\; \backslash langle\; y,\; (O-\backslash lambda\; I)^\{-1\}\; z\; \backslash right\; \backslash rangle\; \backslash ,\; dy\; =\; \backslash langle\; x\; ,\; z\; \backslash rangle\; =\; \backslash delta\; (x-z).$

Using this Green's function property:

$\backslash int\; \backslash langle\; x,\; (O-\backslash lambda\; I)\; y\; \backslash rangle\; G(y,\; z;\; \backslash lambda\; )\; \backslash ,\; dy\; =\; \backslash delta\; (x-z).$

Then, multiplying both sides of this equation by h( z) and integrating:

$\backslash int\; dz\; \backslash ,\; h(z)\; \backslash int\; dy\; \backslash ,\; \backslash langle\; x,\; (O-\backslash lambda\; I)y\; \backslash rangle\; G(y,\; z;\; \backslash lambda)=\backslash int\; dy\; \backslash ,\; \backslash langle\; x,\; (O-\backslash lambda\; I)\; y\; \backslash rangle\; \backslash int\; dz\; \backslash ,\; h(z)G(y,\; z;\; \backslash lambda)\; =\; h(x),$

which suggests the solution is:

$\backslash psi(x)\; =\; \backslash int\; h(z)\; G(x,\; z;\; \backslash lambda)\; \backslash ,\; dz.$

That is, the function ψ( x) satisfying the operator equation is found if we can find the spectrum of O, and construct G, for example by using:

$G(x,\; z;\; \backslash lambda)\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{e\_i\; (x)\; f\_i^*(z)\}\{\backslash lambda\; -\; \backslash lambda\_i\}.$

There are many other ways to find G, of course.

For example, see

and See the articles on Green's functions and on Fredholm integral equations. It must be kept in mind that the above mathematics is purely formal, and a rigorous treatment involves some pretty sophisticated mathematics, including a good background knowledge of functional analysis, Hilbert spaces, distributions and so forth. Consult these articles and the references for more detail.

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Spectral theorem and Rayleigh quotient Optimization problems may be the most useful examples about the combinatorial significance of the eigenvalues and eigenvectors in symmetric matrices, especially for the Rayleigh quotient with respect to a matrix M.

Theorem Let M be a symmetric matrix and let x be the non-zero vector that maximizes the Rayleigh quotient with respect to M. Then, x is an eigenvector of M with eigenvalue equal to the Rayleigh quotient. Moreover, this eigenvalue is the largest eigenvalue of  M.

Proof Assume the spectral theorem. Let the eigenvalues of M be $\backslash lambda\_1\; \backslash leq\; \backslash lambda\_2\; \backslash leq\; \backslash cdots\; \backslash leq\; \backslash lambda\_n$. Since the $\backslash \{v\_i\backslash \}$ form an orthonormal basis, any vector x can be expressed in this basis as

$x\; =\; \backslash sum\_i\; v\_i^T\; x\; v\_i$

The way to prove this formula is pretty easy. Namely,

$\backslash begin\{align\}$

v_j^T \sum_i v_i^T x v_i = {} & \sum_{i} v_i^{T} x v_j^{T} v_i \\4pt = {} & (v_j^T x ) v_j^T v_j \\4pt = {} & v_j^T x \end{align} evaluate the Rayleigh quotient with respect to x:

$\backslash begin\{align\}$

x^T M x = {} & \left(\sum_i (v_i^T x) v_i\right)^T M \left(\sum_j (v_j^T x) v_j\right) \\4pt = {} & \left(\sum_i (v_i^T x) v_i^T\right) \left(\sum_j (v_j^T x) v_j\lambda_j \right) \\4pt = {} & \sum_{i,j} (v_i^T x) v_i^T(v_j^T x) v_j\lambda_j \\4pt = {} & \sum_j (v_j^T x)(v_j^T x)\lambda_j \\4pt = {} & \sum_{j} (v_j^T x)^2\lambda_j\le\lambda_n \sum_j (v_j^T x)^2 \\4pt = {} & \lambda_n x^T x, \end{align} where we used Parseval's identity in the last line. Finally we obtain that

$\backslash frac\{x^T\; M\; x\}\{x^T\; x\}\backslash le\; \backslash lambda\_n$

so the Rayleigh quotient is always less than $\backslash lambda\_n$.Spielman, Daniel A. "Lecture Notes on Spectral Graph Theory" Yale University (2012) http://cs.yale.edu/homes/spielman/561/ .

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See also

• Functions of operators, Operator theory
• Least-squares spectral analysis
• Riesz projector
• Self-adjoint operator
• Spectrum (functional analysis), Resolvent formalism, Decomposition of spectrum (functional analysis)
• Spectral radius, Spectrum of an operator, Spectral theorem
• Spectral theory of compact operators
• Spectral theory of normal C*-algebras
• Sturm–Liouville theory, Integral equations, Fredholm theory
• , Isospectral operators, Completeness
• Spectral geometry
• Spectral graph theory
• List of functional analysis topics

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Notes

• Edward Brian Davies (1996). 9780521587105, Cambridge University Press. . ISBN 9780521587105
• (1988). 9780471608479, Wiley. . ISBN 9780471608479
• (1988). 9780471608462, Wiley. . ISBN 9780471608462
• Sadri Hassani (1999). 9780387985794, Springer. ISBN 9780387985794
• Arch W. Naylor, George R. Sell (2026). 038795001X, Springer. 038795001X
• Gerald Teschl (2026). 9780821846605, American Mathematical Society. . ISBN 9780821846605
• Valter Moretti (2026). 9783319707051, Springer. . ISBN 9783319707051

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External links

• A Short History of Operator Theory by Evans M. Harrell II

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