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Spectral theorem

The spectral theorem is a cornerstone of functional analysis that establishes the diagonalizability of self-adjoint operators on Hilbert spaces via a spectral decomposition.^[1] In finite dimensions, it asserts that every real symmetric matrix or complex Hermitian matrix admits an orthonormal basis of eigenvectors and is thus orthogonally or unitarily diagonalizable, with all eigenvalues real.^[2]^[3] This result extends to infinite-dimensional Hilbert spaces, where a bounded self-adjoint operator T can be represented as T = \int_{\sigma(T)} \lambda \, dE(\lambda), with E a projection-valued measure on the spectrum \sigma(T). The theorem's finite-dimensional version, often traced to early work on symmetric matrices, provides a complete spectral resolution that simplifies computations involving quadratic forms and covariance matrices.^[1] In the unbounded case, it applies to densely defined self-adjoint operators, ensuring a similar integral form over the real line, which is crucial for handling differential operators.^[4] These decompositions guarantee that functions of the operator, such as exponentials used in time evolution, can be defined unambiguously via functional calculus.^[5] Beyond pure mathematics, the spectral theorem underpins key applications in physics and engineering; in quantum mechanics, self-adjoint operators model observables, with eigenvalues representing measurable values and the spectral measure encoding probabilities.^[6] It also facilitates the analysis of vibrations in materials and solutions to partial differential equations, where eigenfunctions form natural bases for expansions.^[7] The theorem's generalizations to normal operators and its role in unitary representations further extend its influence across operator algebras and harmonic analysis.

Finite-dimensional case

Self-adjoint matrices

A self-adjoint matrix, also known as a Hermitian matrix, is a square matrix A that equals its own conjugate transpose, denoted A^*, so A = A^*.^[8] This condition implies that the diagonal entries of A are real numbers, while the off-diagonal entries satisfy a_{ji} = \overline{a_{ij}}, where the bar denotes complex conjugation.^[8] The spectral theorem for self-adjoint matrices states that every self-adjoint matrix has real eigenvalues and is unitarily diagonalizable. Specifically, there exists a unitary matrix U (satisfying U^* U = I) and a real diagonal matrix D such that A = U D U^*.^[4] This decomposition reveals the matrix's eigenvalues on the diagonal of D and an orthonormal basis of eigenvectors as the columns of U.^[4] To establish this theorem, the proof proceeds in steps, often by induction on the matrix dimension. First, all eigenvalues are shown to be real: for any eigenvalue \lambda and corresponding eigenvector x with \|x\| = 1, the inner product satisfies \langle Ax, x \rangle = \lambda \langle x, x \rangle = \lambda, but self-adjointness also gives \langle Ax, x \rangle = \langle x, Ax \rangle = \overline{\lambda}, so \lambda = \overline{\lambda} and \lambda is real.^[4] Next, eigenvectors corresponding to distinct eigenvalues are orthogonal, as \langle Ax, y \rangle = \lambda \langle x, y \rangle and \langle x, Ay \rangle = \mu \langle x, y \rangle imply (\lambda - \mu) \langle x, y \rangle = 0.^[4] The induction step constructs an orthonormal basis of eigenvectors by finding one eigenvalue in the current space, adjoining its eigenvector, and applying the theorem recursively to the orthogonal complement.^[4] Consider the 2×2 self-adjoint matrix

A = \begin{pmatrix} 3 & 2+i \\ 2-i & 1 \end{pmatrix}.

The characteristic polynomial is \det(A - \lambda I) = (\lambda - 3)(\lambda - 1) - |2+i|^2 = \lambda^2 - 4\lambda - 2, with roots \lambda_1 = 2 + \sqrt{6} and \lambda_2 = 2 - \sqrt{6}, both real. The eigenvectors are found by solving (A - \lambda_i I) \mathbf{v}_i = 0 and normalized to form the columns of U, confirming A = U D U^* with D = \operatorname{diag}(\lambda_1, \lambda_2).^[9] The early development of this result traces to Augustin-Louis Cauchy, who in 1829 proved that real symmetric matrices (a special case of self-adjoint matrices over the reals) have real eigenvalues and are orthogonally diagonalizable.

Normal matrices

A matrix A \in \mathbb{C}^{n \times n} is called normal if it commutes with its conjugate transpose, that is, A A^* = A^* A, where A^* denotes the adjoint (conjugate transpose) of A.^[10] This condition generalizes the self-adjoint case, where A = A^*, which forms a special subclass of normal matrices.^[11] The spectral theorem for normal matrices asserts that every normal matrix is unitarily diagonalizable: there exists a unitary matrix U (satisfying U U^* = I) and a diagonal matrix D such that A = U D U^*.^[12] The diagonal entries of D are the eigenvalues of A, which may be complex in general, unlike the real eigenvalues guaranteed for self-adjoint matrices.^[13] This diagonalization occurs over the complex numbers \mathbb{C}, providing an orthonormal basis of eigenvectors for the matrix.^[10] To establish this result, Schur's triangulation theorem first guarantees that any complex matrix is unitarily similar to an upper triangular matrix.^[14] For a normal matrix, the normality condition A A^* = A^* A implies that this triangular form must actually be diagonal, as the off-diagonal entries vanish; this follows from the fact that normal matrices preserve the Euclidean norm of vectors (\|A v\| = \|A^* v\| for all v), which forces the superdiagonal elements to be zero in the Schur form.^[11] Thus, the columns of the unitary matrix from Schur's theorem form an orthonormal basis of eigenvectors.^[14] A concrete example is the 2D rotation matrix by an angle \theta \neq 0, \pi, given by

R = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.

This matrix is orthogonal, hence normal (R R^* = R^* R = I), but not self-adjoint unless \theta = 0 or \pi, since R^* = R^T equals R only in those cases.^[2] Its eigenvalues are the complex numbers e^{i\theta} and e^{-i\theta}, confirming diagonalization over \mathbb{C} with an orthonormal basis of eigenvectors.^[13]

Spectral decomposition

The spectral decomposition of a self-adjoint matrix A on a finite-dimensional complex inner product space expresses A as a sum of its eigenvalues multiplied by orthogonal projections onto the corresponding eigenspaces. Specifically, if \lambda_1, \dots, \lambda_k are the distinct eigenvalues of A, and P_i is the orthogonal projection onto the eigenspace E_{\lambda_i}, then

A = \sum_{i=1}^k \lambda_i P_i,

where the P_i satisfy P_i P_j = \delta_{ij} P_i and \sum_{i=1}^k P_i = I.^[4] This decomposition arises from the existence of an orthonormal basis of eigenvectors for self-adjoint operators, allowing A to be unitarily diagonalized.^[12] For normal matrices, which commute with their adjoint (A A^* = A^* A), the spectral decomposition takes a similar form, but the eigenvalues \lambda_i may be complex. In this case, there exists a unitary matrix U such that U^* A U = D, where D is diagonal with entries \lambda_i, leading to

A = \sum_{i=1}^n \lambda_i |u_i\rangle \langle u_i|,

with \{u_i\} forming an orthonormal basis of eigenvectors and |u_i\rangle \langle u_i| the rank-one projections.^[11] This extends the self-adjoint case, as self-adjoint matrices are a special class of normal matrices with real eigenvalues.^[5] The spectral decomposition connects to the singular value decomposition (SVD) of arbitrary matrices through the self-adjoint case. For any m \times n matrix A, the SVD is A = U \Sigma V^*, where the singular values \sigma_i (non-negative diagonal entries of \Sigma) are the eigenvalues of the self-adjoint positive semidefinite matrix \sqrt{A^* A}. Thus, applying the spectral theorem to A^* A yields the singular values and right singular vectors V, while left singular vectors U follow from those of A A^*.^[15]^[5] The projections P_i in the decomposition are uniquely determined as the spectral projections E(\{\lambda_i\}), which are the unique orthogonal projections onto the eigenspaces satisfying the resolution of the identity for the discrete spectrum.^[16] This decomposition enables the functional calculus for self-adjoint or normal matrices, where for a function f, f(A) = \sum_i f(\lambda_i) P_i, facilitating computations of powers, exponentials, or other functions by reducing to the diagonal form.^[17]

Compact self-adjoint operators

Discrete spectrum and eigenvalues

A compact self-adjoint operator T on a Hilbert space H is a bounded linear operator that is self-adjoint (T = T^*) and can be approximated in the operator norm by a sequence of finite-rank operators.^[18] Such operators arise naturally in applications like integral equations and play a central role in spectral theory due to their "finite-dimensional-like" behavior in infinite dimensions.^[19] The spectral theorem for compact self-adjoint operators asserts that the spectrum \sigma(T) consists solely of real eigenvalues \{\lambda_n\}, which form a discrete set accumulating only at zero (i.e., |\lambda_n| \to 0 as n \to \infty, assuming they are ordered by decreasing absolute value).^[18] Moreover, H admits an orthonormal basis \{e_n\} consisting of eigenvectors of T, so T e_n = \lambda_n e_n for each n, and the closed linear span of these eigenvectors is all of H (with the kernel corresponding to the eigenvalue 0 if it occurs).^[19] This result extends the finite-dimensional analog, where every self-adjoint matrix is diagonalizable by an orthonormal basis.^[18] In this basis, the operator admits the spectral decomposition

T = \sum_{n=1}^\infty \lambda_n \langle \cdot, e_n \rangle e_n,

where the series converges in the operator norm; if T has finite rank, the sum is finite.^[19] Proofs of the theorem typically rely on finite-dimensional approximations: since T is the norm limit of finite-rank self-adjoint operators T_k, each of which is diagonalizable, perturbation theory for eigenvalues and eigenvectors ensures that the eigenvalues of T are limits of those of the T_k, yielding the discrete spectrum and orthonormal eigenbasis.^[20] A concrete example is the integral operator on L^2[a,b] defined by (Tf)(x) = \int_a^b K(x,y) f(y) \, dy, where K is a continuous symmetric kernel; such operators are compact and self-adjoint, with eigenvalues accumulating at zero corresponding to the smoothness of K.^[18]

Eigenfunction expansion

For a compact self-adjoint operator T on a separable Hilbert space H, the spectral theorem guarantees the existence of a countable orthonormal basis \{e_n\}_{n=1}^\infty consisting of eigenvectors with corresponding real eigenvalues \{\lambda_n\}_{n=1}^\infty satisfying \lambda_n \to 0 as n \to \infty. For any f \in H, the action of T admits the eigenfunction expansion

Tf = \sum_{n=1}^\infty \lambda_n \langle f, e_n \rangle e_n,

where the series converges in the norm topology of H. This representation follows from the completeness of the eigenbasis and the fact that T maps H into the closed span of the eigenvectors. If an eigenvalue \lambda_k has multiplicity greater than one, the sum incorporates an orthonormal basis for the associated eigenspace, ensuring the expansion remains valid across degenerate cases.^[18] The convergence of the expansion relies on the properties of the discrete spectrum. The eigenvalues converge absolutely to zero due to the compactness of T, which implies that the only accumulation point of the spectrum is at zero. For the coefficients, Bessel's inequality ensures \sum_{n=1}^\infty |\langle f, e_n \rangle|^2 \leq \|f\|^2, bounding the series terms and guaranteeing norm convergence of the partial sums to Tf. This framework allows for the approximation of Tf by finite-rank projections onto the eigenspaces, with the error controlled by the tail of the eigenvalue sequence.^[18] A key application arises in the context of positive compact integral operators on L^2 spaces. Mercer's theorem states that if K(x,y) is a continuous symmetric positive definite kernel on a compact domain, defining the operator (Tf)(x) = \int K(x,y) f(y) \, dy, then

K(x,y) = \sum_{n=1}^\infty \lambda_n \phi_n(x) \overline{\phi_n(y)},

where \{\phi_n\} are the orthonormal eigenfunctions and \lambda_n > 0 with \lambda_n \to 0, and the series converges absolutely and uniformly. This expansion facilitates the solution of integral equations Tf = g by projecting g onto the eigenbasis, yielding f = \sum (\langle g, \phi_n \rangle / \lambda_n) \phi_n for \lambda_n \neq 0, with convergence in L^2. Such decompositions are foundational for numerical methods and approximation theory in solving Fredholm equations of the second kind.^[18]

Bounded self-adjoint operators

General statement and continuous spectrum

The spectral theorem for bounded self-adjoint operators provides a canonical decomposition that generalizes the finite-dimensional diagonalization of self-adjoint matrices to infinite-dimensional Hilbert spaces. Specifically, for a bounded self-adjoint operator A acting on a separable Hilbert space H, the spectrum \sigma(A) is a closed subset of the real numbers \mathbb{R}, and there exists a unique resolution of the identity, or projection-valued measure \{E(\Delta)\}_{\Delta \subset \mathbb{R}}, such that

A = \int_{\sigma(A)} \lambda \, dE(\lambda),

where the integral is understood in the strong operator topology.^[21] This representation allows A to be expressed as a "continuous sum" of scalar multiples of orthogonal projections, capturing the operator's action across its entire spectrum. The theorem was originally formulated by John von Neumann in his foundational work on quantum mechanics. The spectrum \sigma(A) decomposes into disjoint parts: the point spectrum (eigenvalues), the continuous spectrum, and the singular continuous spectrum (though the latter is often empty or absent in simple cases). The continuous spectrum \sigma_c(A) consists of those \lambda \in \sigma(A) for which the spectral projection E(\{\lambda\}) = 0, implying that \lambda is not an eigenvalue and there are no corresponding eigenvectors in H.^[22] In this regime, \lambda belongs to the spectrum because A - \lambda I fails to be invertible, typically as its range is dense but not closed, leading to approximate eigenvectors that become arbitrarily good in the limit but never exact within the space.^[21] A canonical example of an operator with purely continuous spectrum is the multiplication operator M_x on L^2[0,1], defined by (M_x f)(t) = t f(t) for f \in L^2[0,1]. Here, \sigma(M_x) = [0,1], which is entirely continuous, as there are no eigenvalues: for any \lambda \in [0,1], the equation M_x f = \lambda f forces f(t) = 0 almost everywhere except possibly at t = \lambda, yielding only the zero function in L^2[0,1].^[22] The spectral projections correspond to multiplication by characteristic functions of intervals in [0,1], illustrating how the operator's action is distributed continuously without discrete jumps.^[23] When the spectrum includes a continuous part, the spectral theorem implies that no complete orthonormal basis of eigenvectors exists in H, unlike the compact case where the spectrum is purely point-like and admits such a basis. This follows from the structure of L^2 spaces, where the Riesz-Fischer theorem identifies completeness but precludes a discrete orthonormal basis for operators with continuous multiplicity, necessitating integral representations for expansions.^[21] Thus, vectors in H are decomposed via the continuous measure dE(\lambda), reflecting the operator's "smeared" spectral support.^[17]

Projection-valued measures

In the spectral theorem for a bounded self-adjoint operator A acting on a Hilbert space H, the projection-valued measure (also known as a spectral measure or resolution of the identity) is a function E: \mathcal{B}(\mathbb{R}) \to \mathcal{P}(H), where \mathcal{B}(\mathbb{R}) denotes the Borel \sigma-algebra on \mathbb{R} and \mathcal{P}(H) is the set of orthogonal projections on H. This map assigns to each Borel set \Delta \subseteq \mathbb{R} an orthogonal projection E(\Delta) such that E(\mathbb{R}) = I_H, with I_H the identity operator on H, and satisfies A = \int_{\mathbb{R}} \lambda \, dE(\lambda), where the integral is taken in the strong operator topology.^[24] The projections E(\Delta) are self-adjoint and idempotent by definition, ensuring \|E(\Delta)\| \leq 1 for all \Delta.^[25] The projection-valued measure E possesses several key properties that underpin its role in spectral analysis. It is countably additive: for a countable collection of pairwise disjoint Borel sets \{\Delta_n\}_{n=1}^\infty, E\left(\bigcup_{n=1}^\infty \Delta_n\right) = \sum_{n=1}^\infty E(\Delta_n) in the strong operator topology, with E(\emptyset) = 0.^[24] For any vector f \in [H](/page/H+), the scalar function \Delta \mapsto \langle f, E(\Delta) f \rangle defines a positive finite measure \mu_f on \mathcal{B}(\mathbb{R}), capturing the "spectral distribution" of f with respect to A.^[26] These properties ensure that E behaves analogously to a classical measure but with values in the lattice of projections, facilitating the decomposition of A into its spectral components. The construction of the projection-valued measure E for a bounded self-adjoint A relies on foundational results in operator theory. One standard approach uses Stone's theorem on one-parameter unitary groups: the family \{e^{itA}\}_{t \in \mathbb{R}} forms a strongly continuous unitary group on H, and Stone's theorem guarantees the existence of a unique projection-valued measure E on \mathbb{R} satisfying

e^{itA} = \int_{\mathbb{R}} e^{it\lambda} \, dE(\lambda)

for all t \in \mathbb{R}.^[27] Differentiating formally with respect to t at t=0 (justified by strong continuity) yields iA = \int_{\mathbb{R}} i\lambda \, dE(\lambda), confirming A = \int_{\mathbb{R}} \lambda \, dE(\lambda). An alternative construction employs the Cayley transform U = (A - iI_H)(A + iI_H)^{-1}, which is a unitary operator on H; the spectral measure for U on the unit circle can then be mapped back to a measure on \mathbb{R} via the inverse transform, yielding E.^[17] For each Borel set \Delta \subseteq \mathbb{R}, the spectral subspace H_\Delta = E(\Delta) H (the range of E(\Delta)) is a closed invariant subspace of H under A, known as a reducing subspace: A H_\Delta \subseteq H_\Delta and A (H_\Delta)^\perp \subseteq (H_\Delta)^\perp.^[24] These subspaces provide a direct decomposition of H according to the spectral support of A, with the spectrum of A|_{H_\Delta} contained in the closure of \Delta. The projection-valued measure E is unique for the given operator A, as ensured by the uniqueness clause in Stone's theorem and the bijective correspondence it establishes between self-adjoint operators and such measures.^[27]

Multiplication operator representation

In the context of bounded self-adjoint operators on a Hilbert space H, the spectral theorem provides a concrete realization through unitary equivalence to multiplication operators on suitable L^2 spaces. This representation, often referred to as the multiplication operator form, expresses the operator in terms of pointwise multiplication by a real-valued function, offering an explicit model for its action and spectral properties. Consider a bounded self-adjoint operator A on H. The theorem states that there exists a probability measure space (\Sigma, \mu), a real-valued bounded measurable function \phi: \Sigma \to \mathbb{R}, and a unitary operator U: H \to L^2(\Sigma, \mu) such that U A U^{-1} = M_\phi, where M_\phi denotes the multiplication operator defined by (M_\phi g)(\sigma) = \phi(\sigma) g(\sigma) for g \in L^2(\Sigma, \mu). This equivalence implies that the action of A on vectors in H corresponds directly to scalar multiplication in the transformed space, simplifying the analysis of eigenvalues and spectral projections. In the case where A admits a cyclic vector f \in H, the measure \mu can be taken as the scalar spectral measure \mu(B) = \langle E(B) f, f \rangle for Borel sets B \subseteq \mathbb{R}, where E is the projection-valued measure associated with A, and \phi(\lambda) = \lambda. This isomorphism preserves the spectrum of the operator: the spectrum \sigma(A) coincides with the essential support of \phi with respect to \mu, defined as the smallest closed set outside which \phi is \mu-almost everywhere zero. Thus, the essential range of \phi captures the possible "eigenvalues" in the continuous case, reflecting the distribution of the spectrum across discrete and continuous components. A canonical example arises in quantum mechanics, where the position operator Q on the Hilbert space L^2(\mathbb{R}, dx) acts as multiplication by the coordinate function: (Q \psi)(x) = x \psi(x). Here, Q is already in its spectral multiplication form with \phi(x) = x and Lebesgue measure \mu = dx, and its spectrum is the entire real line \mathbb{R}, corresponding to all possible position measurements. This representation underscores the theorem's role in modeling observables with continuous spectra. The multiplication operator representation was originally developed by John von Neumann in his foundational 1932 paper, where he established the unitary equivalence for self-adjoint operators using integral representations tied to their resolvents.

Direct integral decomposition

The direct integral decomposition provides the canonical representation of a bounded self-adjoint operator on a separable Hilbert space, accommodating arbitrary multiplicities across the spectrum. This formulation, introduced by von Neumann, asserts that for every bounded self-adjoint operator A on a Hilbert space H, there exists a standard measure space (\Sigma, \mu) with \Sigma \subseteq \mathbb{R}, a measurable family of Hilbert spaces \{\mathcal{H}_\lambda\}_{\lambda \in \Sigma}, and a unitary operator U: H \to \int^\oplus_\Sigma \mathcal{H}_\lambda \, d\mu(\lambda) such that U A U^{-1} = \int^\oplus_\Sigma \lambda \, I_\lambda \, d\mu(\lambda), where I_\lambda denotes the identity operator on \mathcal{H}_\lambda and \Sigma = \sigma(A), the spectrum of A.^[28] The fibers \mathcal{H}_\lambda capture the multiplicity of the spectrum at each point \lambda, defined as \dim \mathcal{H}_\lambda, which may be finite, countably infinite, or uncountably infinite and can vary measurably with \lambda. This multiplicity function reflects the dimension of the eigenspaces for discrete points or the "degeneracy" in the continuous spectrum, allowing the decomposition to handle operators where the spectral behavior changes across different parts of the spectrum.^[28] The construction of this direct integral relies on the spectral projection-valued measure E associated with A, which generates a measurable Hilbert bundle over \sigma(A). Specifically, the fibers \mathcal{H}_\lambda are constructed as the ranges of the spectral projections E(\{\lambda\}) for discrete points or via local cyclic decompositions in the continuous case, ensuring the direct integral reproduces the original operator through integration against the identity functions on each fiber. This approach unifies the discrete and continuous cases under a single framework.^[28] As an illustrative example, consider a bounded self-adjoint operator with a simple eigenvalue at \lambda = 0 (multiplicity 1) and continuous spectrum on [1, 2] with multiplicity 2. The direct integral would decompose H into \mathbb{C} \oplus \int^\oplus_{[1,2]} \mathbb{C}^2 \, d\lambda, with the operator acting as multiplication by 0 on the first fiber and by \lambda on the second, highlighting how varying multiplicities are encoded in the fiber dimensions.^[28]

Cyclic vectors and multiplicity

In the context of the spectral theorem for bounded self-adjoint operators, a vector f \in H in a separable Hilbert space H is called a cyclic vector for the operator A if the closed linear span of the set \{ p(A) f : p \text{ [polynomial](/page/Polynomial)} \}, denoted \vee \{ p(A) f : p \text{ [polynomial](/page/Polynomial)} \}, equals the entire space H.^[29] This condition implies that A has simple spectrum, meaning the spectral multiplicity is at most 1, as the operator can be unitarily equivalent to multiplication by the independent variable on L^2(\sigma(A), \mu) for some probability measure \mu supported on the spectrum \sigma(A).^[30] The multiplicity of the spectrum of A is determined through the cyclic decomposition of H. Specifically, H decomposes as an orthogonal direct sum H = \bigoplus_{n=1}^m H_n, where each H_n is a cyclic subspace generated by a cyclic vector for A, and the multiplicity m (possibly infinite) is the minimal number of such copies required to span H; this multiplicity corresponds to the dimension of the fiber in the direct integral representation over the spectrum.^[31] A fundamental theorem states that every bounded self-adjoint operator admits such a decomposition into cyclic subspaces, with the multiplicity function constant on the support of the spectral measure if the spectrum is Lebesgue.^[23] The spectrum of A is simple if and only if there exists a cyclic vector for A, providing a practical criterion to verify spectral simplicity without computing the full decomposition.^[32]

Functional calculus

The Borel functional calculus for a bounded self-adjoint operator A on a Hilbert space H provides a way to define f(A) for any Borel measurable function f: \mathbb{R} \to \mathbb{C}, extending the notion of applying functions to eigenvalues in the finite-dimensional case. Given the spectral projection-valued measure E associated with A by the spectral theorem, such that A = \int_{\mathbb{R}} \lambda \, dE(\lambda), the operator f(A) is defined by the integral

f(A) = \int_{\mathbb{R}} f(\lambda) \, dE(\lambda).

If f is bounded, then f(A) is a bounded operator with operator norm satisfying \|f(A)\| \leq \|f\|_\infty = \sup_{\lambda \in \mathbb{R}} |f(\lambda)|. This construction is unique and depends continuously on f in the appropriate topology.^[29]^[33] The functional calculus satisfies several key algebraic properties that mirror function algebra on the spectrum. For Borel functions f and g, the composition and multiplication rules hold: f(A) g(A) = (f g)(A), and if h = f \circ g, then h(A) = f(g(A)). Additionally, f(A) commutes with A, since \lambda f(\lambda) = f(\lambda) \lambda almost everywhere with respect to the spectral measure. The identity function gives \mathrm{id}(A) = A, and constant functions yield scalar multiples of the identity operator. These properties ensure that the map f \mapsto f(A) is a ^*-homomorphism from the algebra of bounded Borel functions on (\mathbb{R}to theC^*-[algebra](/page/*-algebra) generated by A$.^[29]^[32] The Borel calculus is constructed by first establishing a continuous functional calculus for bounded continuous functions on the spectrum \sigma(A), using the Stone-Weierstrass theorem to approximate such functions uniformly by polynomials in A. Polynomials in A are defined in the usual way via power series or repeated application, and the density of polynomials in the uniform norm on C(\sigma(A)) allows extension to all continuous functions. This continuous calculus then extends to Borel functions via limits along simple functions or measurable approximations, preserving the integral representation. For operators with compact spectrum, the calculus aligns with the continuous case on C_0(\mathbb{R}) restricted to bounded functions vanishing at infinity.^[17]^[33] Representative applications include defining functions like the square root for positive operators: if A \geq 0, then f(\lambda) = \sqrt{\lambda} yields \sqrt{A}, which is also positive self-adjoint with (\sqrt{A})^2 = A. Another example is the exponential f(\lambda) = e^{-t\lambda} for t > 0, giving e^{-tA}, which generates the analytic semigroup solving the heat equation \partial_t u = -A u. These constructions rely on the boundedness of f to ensure f(A) is well-defined and bounded.^[29]^[34]

Bounded normal operators

Statement for normal operators

A bounded linear operator N on a complex Hilbert space H is called normal if it commutes with its adjoint, that is, NN^* = N^*N.^[17] The spectral theorem for bounded normal operators asserts that for such an N, the spectrum \sigma(N) is a nonempty compact subset of the complex plane \mathbb{C}. There exists a unique projection-valued measure E, also called a spectral measure, defined on the Borel \sigma-algebra of \mathbb{C}, such that for every Borel set \Delta \subset \mathbb{C}, E(\Delta) is an orthogonal projection on H satisfying E(\mathbb{C}) = I and E(\Delta_1) E(\Delta_2) = E(\Delta_1 \cap \Delta_2) for Borel sets \Delta_1, \Delta_2. Moreover, N admits the integral representation

N = \int_{\sigma(N)} \lambda \, dE(\lambda),

where the integral is understood in the strong operator topology, and the support of E is contained in \sigma(N).^[24]^[35] The projections E(\Delta) satisfy \|E(\Delta)\| \leq 1 and resolve the identity, meaning the family \{E(\Delta)\} is a resolution of the identity with values in the orthogonal projections on H. The spectrum \sigma(N) coincides with the smallest closed set K \subset \mathbb{C} such that E(\mathbb{C} \setminus K) = 0.^[17] When N is self-adjoint (hence normal), the spectrum \sigma(N) is real and contained in \mathbb{R}, reducing to the spectral theorem for self-adjoint operators.^[1] In infinite-dimensional Hilbert spaces, normal operators can exhibit complex spectra, including nonreal eigenvalues.

Unitary equivalence to multiplication operators

A central result in the spectral theorem for bounded normal operators on a separable Hilbert space \mathcal{H} is that every such operator N is unitarily equivalent to a multiplication operator M_\phi on an L^2 space over a complex measure space. Specifically, there exists a standard Borel measure space (X, \mathcal{B}, \mu), a bounded measurable function \phi: X \to \mathbb{C}, and a unitary operator U: \mathcal{H} \to L^2(X, \mu) such that U N U^* f = \phi f for all f \in L^2(X, \mu), with the spectrum \sigma(N) coinciding with the essential range of \phi.^[24]^[17] This representation simplifies the study of N by reducing it to pointwise multiplication, preserving norms and spectral properties. The construction of this equivalence proceeds via the spectral measure E associated with N, which is a projection-valued measure on \mathbb{C} satisfying N = \int_{\mathbb{C}} \lambda \, dE(\lambda). For separable \mathcal{H}, one can select a cyclic vector (if it exists) to generate a scalar spectral measure, leading to an L^2 representation over the spectrum; otherwise, the general case employs a direct integral decomposition over Borel subsets of \sigma(N).^[24]^[35] This approach ensures the unitary U intertwines the actions of N and M_\phi, with \mu derived from the trace of the projections in E.^[17] To account for possible infinite multiplicity, the full representation takes the form of a direct integral \mathcal{H} \cong \int^\oplus_{\sigma(N)} \mathbb{C}^{m(\lambda)} \, d\rho(\lambda), where \rho is a scalar measure on \sigma(N) and m: \sigma(N) \to \mathbb{N} \cup \{\infty\} is the multiplicity function determining the dimension of each fiber. In this decomposition, N acts as multiplication by the identity function \lambda on each fiber \mathbb{C}^{m(\lambda)}, capturing the operator's eigenspace dimensions or generalized eigenspaces across the continuous spectrum.^[36] An illustrative example is the Laurent operator on the Hardy space H^2(\mathbb{T}) of the unit circle, defined as multiplication by a Laurent polynomial \phi(z) = \sum_{k=-n}^m a_k z^k. This operator is normal, and by the spectral theorem, it is unitarily equivalent to multiplication by \phi on L^2(\mathbb{T}, d\theta/2\pi), with spectrum given by the image of \phi on the unit circle.^[37] For the specific case of the bilateral shift (multiplication by z), the spectrum is the unit circle, and the multiplicity is 1 everywhere.^[37] The Fuglede-Kadison determinant provides a key analytic tool arising from this representation, particularly for invertible normal operators in finite von Neumann factors. Defined as \Delta(N) = \exp(\tau(\log |N|)), where \tau is the trace, it leverages the multiplication form to compute \Delta(N) = \exp\left( \int_X \log |\phi| \, d\mu \right), generalizing the classical determinant and facilitating estimates in operator inequalities.^[38] This determinant is multiplicative and unitarily invariant, reflecting the spectral structure.^[39]

Unbounded self-adjoint operators

Statement and domain considerations

In a complex Hilbert space H, an unbounded self-adjoint operator A is defined as a densely defined closed linear operator A: D(A) \to H, where the domain D(A) is a dense subspace of H, the domain of the adjoint coincides with D(A^*) = D(A), and A satisfies \langle Af, g \rangle = \langle f, Ag \rangle for all f, g \in D(A).^[16] This dense domain condition ensures that A can be extended continuously in a weak sense, distinguishing unbounded operators from bounded ones, where the domain is the entire space H.^[40] The spectral theorem for such operators asserts the existence of a unique projection-valued measure E, also called a spectral resolution, defined on the Borel \sigma-algebra of \mathbb{R}, such that the operator A is given by

Af = \int_{\mathbb{R}} \lambda \, dE(\lambda) f

for all f \in D(A), where the integral is understood in the strong sense. The domain D(A) consists precisely of those vectors f \in H for which the integral \int_{\mathbb{R}} |\lambda|^2 \, d\mu_f(\lambda) < \infty, with \mu_f(B) = \langle E(B) f, f \rangle denoting the scalar spectral measure induced by f. This formulation captures the operator's action through a "weighted" L^2 condition with respect to the spectral measure, ensuring the integral converges.^[16]^[23] The spectrum \sigma(A) of A is the closed subset of \mathbb{R} supporting the measure E (i.e., the smallest closed set S \subseteq \mathbb{R} such that E(\mathbb{R} \setminus S) = 0), and it may be unbounded, reflecting the operator's potentially infinite range. Considerations of the essential spectrum involve parts where the spectral projections have infinite-dimensional range or where the operator cannot be approximated by compact perturbations, but the theorem guarantees the spectrum lies on the real line due to self-adjointness.^[16] A canonical example is the momentum operator P = -i \frac{d}{dx} on the Hilbert space L^2(\mathbb{R}), defined on the dense Sobolev domain D(P) = H^1(\mathbb{R}) = \{ f \in L^2(\mathbb{R}) : f' \in L^2(\mathbb{R}) \}, which is self-adjoint and has spectrum \sigma(P) = \mathbb{R}. Here, the spectral measure E corresponds to multiplication by the identity function in the Fourier representation, with domain elements satisfying \int_{\mathbb{R}} |\xi|^2 |\hat{f}(\xi)|^2 \, d\xi < \infty, where \hat{f} is the Fourier transform of f.^[41]^[16] This theorem was originally established by Marshall Stone in 1932, building on his work on strongly continuous one-parameter unitary groups generated by self-adjoint operators.^[42]

Spectral form and resolvent

For an unbounded self-adjoint operator A on a separable Hilbert space \mathcal{H}, the spectral theorem guarantees the existence of a unique projection-valued measure E, supported on the real line \mathbb{R}, such that the domain D(A) is the set of vectors f \in \mathcal{H} satisfying \int_{\mathbb{R}} |\lambda|^2 \, d\|E(\lambda) f\|^2 < \infty, and the action of A on this domain is represented by the spectral integral

Af = \int_{\mathbb{R}} \lambda \, dE(\lambda) f.

This integral form diagonalizes A with respect to the spectral measure E, allowing the operator to be understood as multiplication by the identity function on a suitable L^2 space. The projection-valued measure E is countably additive and satisfies E(\mathbb{R}) = I, the identity operator on \mathcal{H}.^[41]^[17] The resolvent operator R(\zeta, A) = (A - \zeta I)^{-1}, defined for \zeta \in \mathbb{C} \setminus \sigma(A), admits an integral representation in terms of the spectral measure:

R(\zeta, A) = \int_{\mathbb{R}} \frac{1}{\lambda - \zeta} \, dE(\lambda).

This expression shows that R(\zeta, A) is a bounded operator analytic in the resolvent set \rho(A) = \mathbb{C} \setminus \sigma(A), with the spectrum \sigma(A) consisting precisely of those points where the resolvent fails to exist as a bounded operator; specifically, \sigma(A) is the closure of the support of E. The resolvent integral converges in the strong operator topology for \Im \zeta \neq 0 and provides a means to recover the spectral measure via limits, such as Stone's formula for the density of states.^[40]^[43] Stone's theorem establishes a deep connection between self-adjoint operators and one-parameter unitary groups, stating that A is self-adjoint if and only if there exists a strongly continuous group of unitaries \{ U(t) \}_{t \in \mathbb{R}} on \mathcal{H} generated by A (in the sense that U(t) = e^{itA} for bounded functions), satisfying

U(t) = \int_{\mathbb{R}} e^{it\lambda} \, dE(\lambda)

for all t \in \mathbb{R}. This representation links the time evolution in quantum mechanics to the spectral decomposition, with the generator A determining the frequencies in the integral.^[44] A concrete illustration arises in the quantum harmonic oscillator, where the self-adjoint Hamiltonian H = -\frac{d^2}{dx^2} + x^2 on L^2(\mathbb{R}) has discrete spectrum \sigma(H) = \{2n + 1 \mid n = 0, 1, 2, \dots \} and orthonormal eigenbasis \{\psi_n\} consisting of Hermite functions. The spectral measure is E = \sum_{n=0}^\infty E(\{2n+1\}) with E(\{2n+1\}) = |\psi_n\rangle \langle \psi_n|, yielding the resolvent

R(\zeta, H) = \sum_{n=0}^\infty \frac{1}{2n + 1 - \zeta} |\psi_n\rangle \langle \psi_n|

for \zeta \notin \sigma(H), which explicitly resolves (H - \zeta I)^{-1} via the discrete spectral decomposition.^[45]^[46]

Functional calculus for unbounded operators

The Borel functional calculus provides a framework for defining functions of unbounded self-adjoint operators on a Hilbert space, extending the spectral theorem to construct new operators from Borel measurable functions. For an unbounded self-adjoint operator A with spectral resolution of the identity E, given a Borel measurable function f: \mathbb{R} \to \mathbb{C}, the operator f(A) is defined by the integral representation

f(A) x = \int_{\mathbb{R}} f(\lambda) \, dE(\lambda) x

for all x in the domain

D(f(A)) = \left\{ x \in H \ \middle|\ \int_{\mathbb{R}} |f(\lambda)|^2 \, d \langle E(\lambda) x, x \rangle < \infty \right\},

where H is the Hilbert space and \langle \cdot, \cdot \rangle denotes the inner product. This domain is dense in H provided f is not zero almost everywhere with respect to the spectral measure induced by E. The properties of this calculus mirror those for bounded operators but account for potential unboundedness when f is unbounded. Specifically, if f is real-valued and Borel measurable, then f(A) is self-adjoint on D(f(A)); the calculus is multiplicative, meaning f(A) g(A) = (f g)(A) where defined; and it commutes with A under suitable conditions on f. Moreover, the resolvent operators fit into this framework via the Herglotz representation: for \operatorname{Im} z > 0, the resolvent R(z, A) = (A - z I)^{-1} satisfies

R(z, A) = \int_{\mathbb{R}} \frac{1}{\lambda - z} \, dE(\lambda),

which is an H^\infty-function in the upper half-plane with positive imaginary part, enabling analytic continuation and integral representations central to the calculus.^[47] A key aspect of the spectral theorem's completeness is the ability to define inverses and pseudo-inverses through this calculus, such as |A|^{-1} via f(\lambda) = 1/|\lambda| for \lambda \neq 0, with domain consisting of vectors where the corresponding integral is finite; this operator is bounded if 0 is not an eigenvalue of A. An illustrative example is the Gaussian semigroup e^{-t A^2} for t > 0, defined by f(\lambda) = e^{-t \lambda^2}, which generates solutions to parabolic partial differential equations like the heat equation u_t + A^2 u = 0 with initial data in suitable Sobolev spaces, where A might represent the Laplacian on a domain.

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