Unbounded Operators and Spectral Measures

Introduction

The operators of analysis — differentiation, the Laplacian, a differential operator of order $m$ — are not bounded. They are defined on a dense subspace and send it into the space, and the two questions that govern their theory are where they are defined and how much of the space they see. The first question is not a technicality: the same formula $-i\,d/dx$ defines a self-adjoint operator on the whole line, no self-adjoint operator on the half-line, and a one-parameter family of self-adjoint operators on a finite interval, and the difference lies in the boundary conditions, which are choices of domain. The second question is answered by the spectral theorem: a self-adjoint operator is a multiplication by the independent variable in a direct integral of spaces, and the multiplicities that appear are its complete unitary invariants.

The article develops the two answers. A closed operator is one whose graph is closed; a symmetric operator satisfies $A \subseteq A^*$, and a self-adjoint operator satisfies $A=A^*$ with equality of domains, which is strictly stronger. The failure of symmetry to imply self-adjointness is measured by the deficiency indices, and von Neumann's theorem parametrises the self-adjoint extensions; the Cayley transform turns the extension problem into the extension of an isometry to a unitary. The spectral theorem for a self-adjoint operator represents it as $A=\int\lambda\,dE(\lambda)$ with $E$ a projection-valued measure, and the associated Borel functional calculus makes $f(A)$ available for every Borel $f$; the multiplicity function of $E$ classifies the operator up to unitary equivalence. Two consequences are developed: the polar decomposition $A=U|A|$ of a closed operator, and Stone's theorem, which identifies the strongly continuous one-parameter unitary groups with the self-adjoint operators and is the analytic form of a continuous dynamical system.

Throughout, $H$ is a complex separable Hilbert space with inner product $\langle\cdot,\cdot\rangle$ linear in the first argument, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, and the bounded operator theory, the bounded spectral theorem, the compact operators and the Riesz representation theorem are those of Banach and Hilbert Spaces. Measures and integration are those of Measure Theory and Integration, and the Radon–Nikodým theorem and the theory of $L^2$ spaces are used from there. The von Neumann algebra generated by a self-adjoint operator and the theory of its commutant belong to Operator Algebras; only the statement that the spectral projections generate that algebra is used. The polar decomposition and the spectral theorem for bounded normal operators are recalled from Banach and Hilbert Spaces, and the holomorphic functional calculus for a single bounded operator is the standard one quoted below. The applications of Stone's theorem to continuous dynamical systems are not covered here; the semiclassical parametrisation of the spectrum of an operator is the subject, and the approximate diagonalisation of pseudodifferential operators is Microlocal Analysis. All are forward references: nothing here depends on them.

No physics is invoked.

Unbounded Operators and Their Domains

Definitions

Definition. A linear operator on $H$ is a linear map $A:D(A)\to H$ defined on a linear subspace $D(A) \subseteq H$, its domain. The operator is densely defined if $D(A)$ is dense in $H$. The graph of $A$ is

$$ \Gamma(A)=\{(x,Ax):x \in D(A)\}\subseteq H\times H, $$

a linear subspace of the Hilbert space $H\oplus H$ with inner product $\langle(x,y),(x',y')\rangle=\langle x,x'\rangle+\langle y,y'\rangle$.

Definition. An operator $A$ is closed if $\Gamma(A)$ is closed in $H\oplus H$; equivalently, if $x_n \in D(A)$, $x_n \to x$ and $Ax_n \to y$ imply $x \in D(A)$ and $Ax=y$. It is closable if the closure of $\Gamma(A)$ is the graph of an operator, written $\overline A$ and called its closure; equivalently, if $x_n \in D(A)$, $x_n \to 0$, $Ax_n \to y$ imply $y=0$. An operator is bounded if it is defined on all of $H$ and $\sup_{\|x\|\le1}\|Ax\|<\infty$; by the closed graph theorem of Normed and Banach Spaces, a closed everywhere-defined operator is bounded.

Example. (i) The differentiation operator $D=-i\,d/dx$ with domain $H^1(\mathbb{R})$ is closed and unbounded on $L^2(\mathbb{R})$.

(ii) The multiplication operator $Qf(x)=xf(x)$ with domain $D(Q)=\{f \in L^2:\int x^2|f(x)|^2dx<\infty\}$ is closed and unbounded.

(iii) The Laplacian $-\Delta$ with domain $H^2(\mathbb{R}^n)$ is closed and unbounded; on a bounded domain $\Omega$ the same formula with domain $H^2(\Omega)\cap H_0^1(\Omega)$ defines the Dirichlet Laplacian, closed and unbounded, with compact resolvent.

(iv) Every bounded operator is closed, since its graph is closed in $H\oplus H$ by continuity.

The Adjoint

Definition. Let $A$ be densely defined. The adjoint $A^*$ is the operator with domain

$$ D(A^*)=\{y \in H:\text{the map } x \mapsto\langle Ax,y\rangle \text{ is continuous on } D(A)\}, $$

and with $A^*y$ the unique vector satisfying $\langle Ax,y\rangle=\langle x,A^*y\rangle$ for all $x \in D(A)$; uniqueness is the density of $D(A)$. The adjoint is closed whenever it is densely defined, and $(A^*)^*=A$ if $A$ is closed and densely defined.

Proposition. Let $A$ be densely defined. Then $A$ is closable if and only if $D(A^*)$ is dense, and then $\overline A=A^{**}$. If $A$ is closed and $D(A)=H$ then $A$ is bounded.

Proof. The first statement is the standard duality between the graph of $A$ and the graph of $-A^*$ in $H\oplus H$: $\Gamma(A)^\perp=\{(-A^*y,y):y \in D(A^*)\}$ after the symplectic identification, so $\Gamma(A)$ is the graph of an operator exactly when the projection of $\Gamma(A)^\perp$ to the second factor is injective, which is the density of $D(A^*)$ when this holds; the second is the closed graph theorem.

Definition. A densely defined operator $A$ is symmetric if

$$ \langle Ax,y\rangle=\langle x,Ay\rangle \qquad \text{for all } x,y \in D(A), $$

equivalently $A \subseteq A^*$: every $x \in D(A)$ lies in $D(A^*)$ and $A^*x=Ax$. It is self-adjoint if $A=A^*$, that is, if it is symmetric and $D(A)=D(A^*)$. It is essentially self-adjoint if it is symmetric and its closure is self-adjoint.

A symmetric operator is closable and its closure is symmetric; self-adjointness is a property of the operator with its domain and is not preserved by restriction to a dense subspace.

Example. (i) On $C_c^\infty(\mathbb{R})$ the operator $D_0=-i\,d/dx$ is symmetric, since the boundary terms in the integration by parts vanish for compactly supported functions. Its adjoint is $D=-i\,d/dx$ with domain $H^1(\mathbb{R})$, which is self-adjoint: $D_0$ is essentially self-adjoint and $\overline{D_0}=D$. The Fourier transform diagonalises $D$: after the transform, $D$ becomes multiplication by $\xi$ on $L^2(\mathbb{R},d\xi)$, from which $\sigma(D)=\mathbb{R}$ and $D$ has no eigenvalues.

(ii) The operator $D_0=-i\,d/dx$ on $(0,\infty)$ with domain $C_c^\infty(0,\infty)$ is symmetric, but its adjoint has domain $H^1(0,\infty)$ and its deficiency indices are unequal ($1$ and $0$), so it has no self-adjoint extension; the half-line breaks the symmetry of the line. This is the standard illustration that a symmetric operator need not be self-adjoint and need not even admit a self-adjoint extension.

(iii) The Dirichlet Laplacian on a bounded domain is self-adjoint on $H^2\cap H_0^1$, has compact resolvent and therefore a discrete spectrum $\lambda_1 \le \lambda_2 \le \cdots \to \infty$ with an orthonormal basis of eigenfunctions; the boundary condition is part of the definition of the operator, not an extra hypothesis.

Deficiency Indices and Self-Adjoint Extensions

Deficiency Subspaces

Definition. Let $A$ be a closed symmetric operator. The deficiency subspaces are

$$ N_+=\ker(A^*-i), \qquad N_-=\ker(A^*+i), $$

and their dimensions $n_+=\dim N_+$ and $n_-=\dim N_-$ are the deficiency indices of $A$.

Theorem (basic criterion). A closed symmetric operator $A$ is self-adjoint if and only if $n_+=n_-=0$, equivalently if and only if $(A-i)D(A)$ and $(A+i)D(A)$ are dense in $H$; it is essentially self-adjoint if and only if its deficiency indices are $(0,0)$.

Proof. The ranges of $A\pm i$ are the orthogonal complements of $N_\mp$: indeed $\langle(A-i)x,y\rangle=\langle x,(A^*+i)y\rangle$ for $x \in D(A)$, $y \in D(A^*)$, so $y \in\ker(A^*+i)$ exactly when $y \perp (A-i)D(A)$. Hence $A-i$ has dense range precisely when $n_-=0$, and the symmetric operator $A$ is self-adjoint precisely when $A-i$ and $A+i$ map $D(A)$ onto $H$; for a symmetric $A$ one has $\|(A-i)x\|^2=\|Ax\|^2+\|x\|^2$, so $A-i$ is bounded below, and a closed operator bounded below has closed range; dense range therefore gives surjectivity, and the surjectivity of $A-i$ and of $A+i$ for a closed symmetric $A$ is exactly self-adjointness.

Example. (i) For $D_0=-i\,d/dx$ on $(0,\infty)$: the adjoint is $D=-i\,d/dx$ on $H^1(0,\infty)$, and

$$ \ker(D-i)=\mathbb{C}e^{-x}, \qquad \ker(D+i)=0, $$

since $-ig'=ig$ gives $g=Ce^{-x} \in L^2$ while $-ig'=-ig$ gives $g=Ce^{x}\notin L^2$. Hence $(n_+,n_-)=(1,0)$, the indices are unequal, and no self-adjoint extension exists.

(ii) For $A=-\dfrac{d^2}{dx^2}$ on $C_c^\infty(0,\infty)$, whose adjoint is $A^*=-\dfrac{d^2}{dx^2}$ on $H^2(0,\infty)$: the equations $-u''=iu$ and $-u''=-iu$ have, in each case, exactly one solution in $L^2(0,\infty)$ (the root of $r^2=\mp i$ with negative real part), so $(n_+,n_-)=(1,1)$. The symmetric operator admits self-adjoint extensions, and they are described next.

The Cayley Transform

Definition. For a symmetric operator $A$, the Cayley transform is

$$ V_A=(A-i)(A+i)^{-1}: (A+i)D(A)\longrightarrow(A-i)D(A), $$

an isometry with dense domain and dense range. If $A$ is self-adjoint then $V_A$ is unitary and $1$ is not an eigenvalue of $V_A$; conversely a unitary $V$ with $\ker(V-I)=\{0\}$ arises from the self-adjoint operator $A=i(I+V)(I-V)^{-1}$.

The Cayley transform turns the analytic condition $A=A^*$ into the algebraic condition $V^*V=VV^*=I$, and it turns the search for self-adjoint extensions of a symmetric operator into the search for unitary extensions of an isometry — a problem with a transparent solution.

Theorem (von Neumann). A closed symmetric operator $A$ has a self-adjoint extension if and only if $n_+=n_-$. When this holds, the self-adjoint extensions are in bijective correspondence with the unitary maps $U:N_+\to N_-$, the extension $A_U$ being determined by

$$ D(A_U)=D(A)\oplus\{x+Ux:x \in N_+\}, \qquad A_U(x+x_0+Ux_0)=Ax+ix_0-iUx_0 . $$

The correspondence is obtained from the unitary extensions of the Cayley transform and is a theorem of von Neumann.

Example (the half-line Laplacian). For $A=-\frac{d^2}{dx^2}$ on $C_c^\infty(0,\infty)$ the deficiency indices are $(1,1)$, so the self-adjoint extensions form a circle, and they are exactly the operators

$$ A_\theta=-\frac{d^2}{dx^2}, \qquad D(A_\theta)=\{u \in H^2(0,\infty):u'(0)=\theta\,u(0)\}, \quad \theta \in \mathbb{R}, $$

together with the Dirichlet operator $u(0)=0$, corresponding to $\theta=\infty$. Each is self-adjoint, each has spectrum $[0,\infty)$ with a continuous part and possibly one negative eigenvalue when $\theta<0$, and the whole family is obtained from one of them by the boundary condition; the analysis of this family is the simplest instance of a spectral problem with a boundary condition.

The Spectral Theorem

Projection-Valued Measures

Definition. A projection-valued measure, or spectral measure, on a measurable space $(\Omega,\mathcal A)$ with values in the projections of $H$ is a map $E:\mathcal A\to B(H)$ with

(i) $E(B)$ an orthogonal projection for every $B \in \mathcal A$, $E(\varnothing)=0$, $E(\Omega)=I$;

(ii) $E$ is countably additive in the strong operator topology: for disjoint $B_k \in \mathcal A$, $E(\bigcup_kB_k)=\sum_kE(B_k)$ strongly;

(iii) $E(B_1\cap B_2)=E(B_1)E(B_2)$.

For such an $E$ and vectors $x,y \in H$ the set function $E_{x,y}(B)=\langle E(B)x,y\rangle$ is a finite complex measure of total variation at most $\|x\|\|y\|$, and $E_{x,x}$ is a positive measure with $E_{x,x}(\Omega)=\|x\|^2$.

Proposition. A strongly countably additive projection-valued measure is determined by its values on the vectors, and

$$ D(f)=\Bigl\{x \in H:\int_\Omega|f|^2\,dE_{x,x}<\infty\Bigr\} $$

is a linear subspace on which

$$ \langle f(A)x,y\rangle=\int_\Omega f\,dE_{x,y}, \qquad x \in D(f),\ y \in H, $$

defines a closed operator $f(A)$, densely defined whenever $f$ is finite a.e. with respect to the measures $E_{x,x}$.

The Spectral Theorem for Self-Adjoint Operators

Theorem (spectral theorem, projection-valued form). Let $A$ be a self-adjoint operator on a separable Hilbert space $H$. Then there is a unique projection-valued measure $E$ on the Borel sets of $\mathbb{R}$ such that

$$ D(A)=\Bigl\{x \in H:\int_{\mathbb{R}}\lambda^2\,dE_{x,x}(\lambda)<\infty\Bigr\}, \qquad Ax=\int_{\mathbb{R}}\lambda\,dE(\lambda)\,x, $$

the integral converging as a strong limit; equivalently,

$$ \langle Ax,y\rangle=\int_{\mathbb{R}}\lambda\,d\langle E(\lambda)x,y\rangle, \qquad x \in D(A),\ y \in H . $$

The measure satisfies $E(B)=0$ for Borel $B$ disjoint from $\sigma(A)$, so $\operatorname{supp}E=\sigma(A)$; conversely $\lambda \in \sigma(A)$ exactly when $E\bigl((\lambda-\varepsilon,\lambda+\varepsilon)\bigr)\neq0$ for every $\varepsilon>0$, so the spectrum is real. If $A$ is bounded with $\|A\|\le M$ then $E$ is supported in $[-M,M]$ and $A=\int_{-M}^M\lambda\,dE(\lambda)$, recovering the spectral theorem of Banach and Hilbert Spaces.

Proof (sketch). The Cayley transform $V=(A-i)(A+i)^{-1}$ is unitary and $1$ is not an eigenvalue, so the bounded spectral theorem applied to $V$ gives $V=\int_{S^1}z\,dF(z)$; the inverse Cayley map transports $F$ to a projection-valued measure on the real line, and the domain of $A$ is computed from the integrability condition $z \mapsto i(1+z)(1-z)^{-1}$ being in $L^2$. Uniqueness follows because the moments $\int\lambda^n\,dE_{x,y}$ are determined by $A$ for all $x,y$.

Corollary (functional calculus). For every bounded Borel function $f$ on $\mathbb{R}$ the operator $f(A)=\int f\,dE$ is bounded with $\|f(A)\|\le\|f\|_\infty$, and

$$ (fg)(A)=f(A)g(A), \qquad \overline f(A)=f(A)^* , $$

$f(A)$ commutes with every bounded operator commuting with $A$, and the map $f\mapsto f(A)$ is strongly continuous on bounded sequences. For unbounded Borel $f$ the same formula defines a closed operator on the domain $D(f)$ above. Taking $f(\lambda)=(\lambda-z)^{-1}$ for $z \notin \mathbb{R}$ gives the resolvent and its estimate $\|(A-z)^{-1}\|\le1/|\operatorname{Im}z|$; taking $f=\mathbf{1}_B$ gives the spectral projection $E(B)$, so the spectral measure is recovered from the functional calculus.

The Multiplication Model and Multiplicity

Theorem (multiplication form). Let $A$ be self-adjoint on a separable $H$. Then $H$ decomposes as a countable direct sum of invariant subspaces on each of which $A$ is unitarily equivalent to multiplication by the independent variable on $L^2(\mathbb{R},\mu_n)$,

$$ H \cong \bigoplus_{n \in \mathbb{N}} L^2(\mathbb{R},\mu_n), \qquad A \cong \bigoplus_n M_\lambda, $$

with Borel measures $\mu_n$ that decrease in the sense of absolute continuity, $\mu_{n+1}\ll\mu_n$. The class of the family $\{\mu_n\}$ modulo the passage to an equivalent family, and the multiplicity function $m(\lambda)$, the number of the measures $\mu_n$ that are not locally null at $\lambda$, are the complete unitary invariants of $A$: two self-adjoint operators are unitarily equivalent exactly when their multiplicity functions agree almost everywhere.

Definition. A vector $x \in H$ is cyclic for $A$ if the span of $\{A^nx:n\ge0\}$ is dense in $H$. The operator has simple spectrum if it has a cyclic vector, and then the family $\{\mu_n\}$ above reduces to a single measure.

Example. The multiplication operator $Q$ on $L^2(\mathbb{R})$ has simple spectrum: the Gaussian $g(\lambda)=e^{-\lambda^2/2}$ is a cyclic vector, because the Hermite functions, which are the products of $g$ with the Hermite polynomials, form an orthonormal basis, so the span of $\{Q^ng:n\ge0\}$ is dense. Its spectral measure is multiplication by the indicator, $E(B)f=\mathbf 1_Bf$. The differentiation operator $D=-i\,d/dx$ is unitarily equivalent to $Q$ by the Fourier transform, so it also has simple spectrum, and the unitary equivalence is the model instance of the classification by multiplicity. The multiplication operator on $L^2([0,1])$ by the variable has cyclic vector the constant function $1$, by the Weierstrass approximation theorem.

The Functional Calculus and Polar Decomposition

The Borel and Holomorphic Calculi

Theorem (algebraic properties of the calculus). Let $A$ be self-adjoint or normal with spectral measure $E$. The map $f \mapsto f(A)$ is a $*$-homomorphism from the algebra of bounded Borel functions on $\sigma(A)$ into $B(H)$ carrying pointwise convergence to strong convergence on bounded sequences. It restricts to a unital isometric $*$-isomorphism on $C(\sigma(A))$, and to an isomorphism of the algebra of functions holomorphic on a neighbourhood of $\sigma(A)$ onto the algebra generated by $A$ and its resolvent. In particular, for a self-adjoint $A$ the projections $E(B)$ are exactly the projections in the von Neumann algebra generated by $A$; when $A$ has simple spectrum the commutant $\{A\}'$ consists of the bounded Borel functions of $A$, and in general the commutant is described by the multiplicity decomposition, its structure being that of Operator Algebras.

The holomorphic calculus for a single bounded operator is the standard one obtained from the Cauchy integral formula, $\varphi(A)=\frac{1}{2\pi i}\oint\varphi(z)(z-A)^{-1}dz$ around $\sigma(A)$, and it agrees with the Borel calculus where both are defined. The Riesz functional calculus and its extension to several operators, and the identification of the von Neumann algebra with the commutant of the spectral projections, belong to Operator Algebras.

Polar Decomposition

Theorem (polar decomposition). Let $A$ be a closed densely defined operator. Then there is a positive self-adjoint operator $|A|=(A^*A)^{1/2}$ with $D(|A|)=D(A)$ and a partial isometry $U$ with $\ker U=\ker A$ and closed initial space $\overline{|A|(D(A))}$, such that

$$ A=U|A|, \qquad |A|=U^*A, $$

and $U$ is uniquely determined by these conditions. The decomposition separates $A$ into its "modulus" $|A|$ and its "argument" $U$, and $|A|$ has spectral measure computed from $A^*A$; the decomposition specialises to the polar decomposition of a bounded operator and to the absolute value of a self-adjoint operator.

Stone's Theorem

One-Parameter Unitary Groups

Definition. A strongly continuous one-parameter unitary group on $H$ is a family $\{U(t)\}_{t \in \mathbb{R}}$ of unitary operators with $U(0)=I$, $U(s+t)=U(s)U(t)$ and $t \mapsto U(t)x$ continuous for every $x \in H$. Its generator is the operator $A$ with

$$ D(A)=\Bigl\{x \in H:\lim_{t \to 0}\frac{U(t)x-x}{t} \text{ exists}\Bigr\}, \qquad Ax=\frac1i\lim_{t \to 0}\frac{U(t)x-x}{t} . $$

Theorem (Stone). The generator of a strongly continuous one-parameter unitary group is self-adjoint, and it is the unique self-adjoint $A$ with $U(t)=e^{itA}$ for all $t$. Conversely, for every self-adjoint $A$ the formula $U(t)=\int e^{it\lambda}\,dE(\lambda)$, with $E$ the spectral measure of $A$, defines a strongly continuous one-parameter unitary group with generator $A$.

Proof (sketch). For the direct part one first takes the limit in the weak topology, obtaining a symmetric operator $A$; unitarity gives $\|U(t)x\|=\|x\|$ and hence $\operatorname{Re}\langle Ax,x\rangle=0$ for $x \in D(A)$, so the operator is symmetric. For $y \in H$ the vectors

$$ x_+=\int_0^\infty e^{-t}U(t)y\,dt, \qquad x_-=\int_0^\infty e^{-t}U(-t)y\,dt $$

are defined by strong integrals and satisfy $(A+i)x_+=iy$ and $(A-i)x_-=-iy$, as one checks by differentiating $e^{s}\int_s^\infty e^{-u}U(u)y\,du$ and $e^{-s}\int_{-\infty}^s e^{u}U(u)y\,du$ at $s=0$. Hence $A\pm i$ have range all of $H$, and $A$ is self-adjoint by the basic criterion. The converse is the functional calculus: the multiplicativity $e^{is\lambda}e^{it\lambda}=e^{i(s+t)\lambda}$ holds under the measure, so the group law holds, and the strong continuity follows from dominated convergence.

Corollary (dynamics). Stone's theorem is the analytic form of a continuous dynamical system: a flow $t \mapsto T_t$ of measure-preserving transformations of a space corresponds to the unitary group $U(t)f=f\circ T_t$ on $L^2$, and its generator is the self-adjoint operator of differentiation along the flow; the spectral properties of the generator are the frequencies of the system. The correspondence between flows and their generators, the mean ergodic theorem and the mixing properties expressed in the spectrum are the subject,and Probability and Ergodic Theory; the case of the geodesic flow is treated.

Stone's Theorem and the Spectral Multiplicity

Example (multiplication and differentiation). On $L^2(\mathbb{R})$ the group $(U(t)f)(x)=f(x+t)$ is strongly continuous and unitary, and its generator is $D=-i\,d/dx$: for $f \in H^1$, $(U(t)f-f)/t \to f'$ in $L^2$, so $A=(1/i)f'=-if'=D$. Thus $U(t)=e^{itD}=e^{t\,d/dx}$ is the translation group, the spectral measure of $D$ is the one carried by the Fourier transform, and the group is the direct integral of the characters $e^{it\xi}$.

Example (the Laplacian). The heat semigroup $e^{t\Delta}$, $t\ge0$, and the wave group $e^{it\sqrt{-\Delta}}$, $t \in \mathbb{R}$, are defined by the functional calculus of the self-adjoint operator $-\Delta$: the first through the bounded functions $e^{-t\lambda}$ and the second through the functions $e^{\pm it\sqrt{\lambda}}$, on the spectrum $[0,\infty)$ of $-\Delta$. The wave group is unitary and strongly continuous, its generator on an appropriate domain is $\sqrt{-\Delta}$, and the spectral measure of $-\Delta$ gives the spectral resolution of the wave propagator. The unitary group of a self-adjoint operator, and the one-parameter groups generated by functions of it, are the link between the spectral theorem and the differential equations of this Part.

Summary

A linear operator on a Hilbert space is a linear map on a dense domain; it is closed when its graph is closed, and a closed everywhere-defined operator is bounded. Its adjoint is defined on the vectors $y$ for which $x\mapsto\langle Ax,y\rangle$ is continuous, and $A$ is symmetric when $A \subseteq A^*$, self-adjoint when $A=A^*$ with equal domains, and essentially self-adjoint when its closure is self-adjoint. The same formula can be symmetric and not self-adjoint: $-i\,d/dx$ on $C_c^\infty(\mathbb{R})$ is essentially self-adjoint, on $C_c^\infty(0,\infty)$ it has unequal deficiency indices and no self-adjoint extension, and $-\frac{d^2}{dx^2}$ on $C_c^\infty(0,\infty)$ has deficiency indices $(1,1)$ and a circle of self-adjoint extensions, parametrised by the boundary condition $u'(0)=\theta u(0)$.

The deficiency indices $n_\pm=\dim\ker(A^*\mp i)$ decide the extension problem: $A$ is self-adjoint exactly when $n_+=n_-=0$, and von Neumann's theorem parametrises the self-adjoint extensions of a closed symmetric operator with $n_+=n_-$ by the unitary maps $N_+\to N_-$, the Cayley transform converting the analytic condition into the extension of an isometry to a unitary. The spectral theorem represents every self-adjoint $A$ as $A=\int\lambda\,dE(\lambda)$ with a unique projection-valued measure $E$ on $\mathbb{R}$ supported on $\sigma(A)$, and the same measure yields the Borel functional calculus $f(A)=\int f\,dE$, a $*$-homomorphism whose restriction to $C(\sigma(A))$ is isometric and whose spectral projections generate the von Neumann algebra of $A$. The operator is classified up to unitary equivalence by the multiplicity function of $E$, and the multiplication model realises it as a direct sum of multiplications by the variable. From the calculus comes the polar decomposition $A=U|A|$ of a closed operator, with $|A|=(A^*A)^{1/2}$. Finally Stone's theorem identifies the strongly continuous one-parameter unitary groups with the self-adjoint operators: the generator of $U(t)$ is self-adjoint and $U(t)=e^{itA}$, and the theorem supplies the analytic form of a continuous dynamical system.

Summary of Notation

Symbol	Meaning
$H$	complex separable Hilbert space
$\mathbb{K}$	$\mathbb{R}$ or $\mathbb{C}$
$A$, $D(A)$, $\Gamma(A)$	operator, domain, graph
$\overline A$	closure of a closable operator
$A^*$	adjoint operator
$A \subseteq A^$, $A=A^$	symmetric, self-adjoint
$N_\pm=\ker(A^*\mp i)$, $n_\pm$	deficiency subspaces and indices
$V_A=(A-i)(A+i)^{-1}$	Cayley transform
$A_U$, $U:N_+\to N_-$	von Neumann self-adjoint extension
$E$, $E(B)$, $E_{x,y}$	projection-valued measure and its scalar measures
$A=\int\lambda\,dE(\lambda)$	spectral representation
$f(A)$, $D(f)$	Borel functional calculus and its domain
$Q$, $D=-i\,d/dx$	multiplication by $x$ and differentiation
$\mu_n$, multiplicity	classification data of a self-adjoint operator
$\|A\|=(A^*A)^{1/2}$, $A=U\|A\|$	modulus and polar decomposition
$U(t)$, $e^{itA}$	strongly continuous one-parameter unitary group
$-\Delta$, $H_0^1$, $H^2$	Laplacian and its Sobolev domains
$\theta$	boundary parameter of the half-line Laplacian