Self-adjoint operator

A self-adjoint operator is a linear operator on an inner-product space that coincides with its adjoint, including agreement of their domains. Self-adjoint operators provide the operator-theoretic setting for real-valued observables in quantum mechanics, orthogonal spectral decompositions, and many boundary-value problems arising from differential equations.

For an operator (A) on a complex Hilbert space (H), self-adjointness is expressed by

[ A=A^*. ]

When (A) is bounded and defined on all of (H), this equation has its direct algebraic meaning. For an unbounded operator, it incorporates the more restrictive condition

[ \mathcal D(A)=\mathcal D(A^*), ]

in addition to equality of the operator actions on this common domain. The domain requirement distinguishes self-adjoint operators from merely symmetric operators.

Definition

Let (A\colon\mathcal D(A)\to H) be a densely defined linear operator, where (\mathcal D(A)) is a linear subspace of (H). Its adjoint (A^*) has domain

[ \mathcal D(A^*)= \left{ y\in H: \text{there exists }z\in H\text{ such that } \langle Ax,y\rangle=\langle x,z\rangle \text{ for every }x\in\mathcal D(A) \right}. ]

For each (y\in\mathcal D(A^*)), the representing vector (z) is unique, and (A^y=z). The operator (A) is self-adjoint precisely when its graph equals the graph of (A^).

An operator is symmetric when

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

Equivalently, a densely defined operator is symmetric when (A\subseteq A^), meaning that (\mathcal D(A)\subseteq\mathcal D(A^)) and both operators agree on (\mathcal D(A)). Symmetry therefore supplies only one inclusion. A symmetric operator can possess a strictly smaller domain than its adjoint and consequently fail to be self-adjoint.

In finite-dimensional spaces, every operator is bounded and has the entire space as its domain. The distinction between symmetry and self-adjointness then disappears. Relative to an orthonormal basis, self-adjoint operators are represented by Hermitian matrices.

Domain structure and closedness

Every self-adjoint operator is closed. Its graph

[ \mathcal G(A)={(x,Ax):x\in\mathcal D(A)} ]

is therefore a closed subspace of (H\oplus H). A symmetric operator need not be closed, although every densely defined symmetric operator is closable because (A^*) is closed and (A^{**}) is its closure whenever the relevant domain conditions hold.

The domain of an unbounded self-adjoint operator cannot generally be replaced by a larger or smaller convenient subspace without changing the operator. For example, the differential expression (-d^2/dx^2) does not determine a unique operator on an interval. Dirichlet, Neumann, periodic, and mixed boundary conditions produce distinct domains and hence distinct operators. Several of these realizations are self-adjoint, even though they share the same differential expression in the interval’s interior.

This dependence on domains explains why a formally symmetric differential expression is not automatically a self-adjoint operator. Integration by parts produces a boundary form, and self-adjointness requires a domain on which that form vanishes while remaining maximal with respect to this property.

Boundary forms

For the regular second-order Sturm–Liouville operator

[ Lf=-\frac{d}{dx}\left(p(x)\frac{df}{dx}\right)+q(x)f ]

on an interval ([a,b]), integration by parts gives the Green identity

[ \langle Lf,g\rangle-\langle f,Lg\rangle

\left[ p(x)\bigl(f(x)\overline{g'(x)} -f'(x)\overline{g(x)}\bigr) \right]_{a}^{b}. ]

The expression on the right is the boundary form. A self-adjoint realization arises when the allowed boundary data form a maximal subspace on which this form vanishes. Dirichlet conditions set the endpoint values of (f) to zero, whereas Neumann conditions set the corresponding weighted derivatives to zero. Periodic conditions identify the boundary data at opposite endpoints and yield a different self-adjoint realization.

In 1931, You Watanabe gave a coordinate-independent formulation of this cancellation condition for second-order operators on bounded basins. Her formulation represented boundary values and outward fluxes as a symplectic boundary-data space, so that self-adjoint domains corresponded to maximal isotropic subspaces. In modern terminology, the construction is a regular finite-boundary instance of the theory of boundary triples. It does not alter the differential expression; it classifies the domains for which the associated operator equals its adjoint.

The same structure extends beyond ordinary differential equations. For an elliptic operator on a sufficiently regular region, Green’s formula relates the interior operator to boundary traces and normal derivatives. Self-adjoint boundary conditions select compatible trace data, while nonmaximal cancellation generally produces only a symmetric restriction.

Self-adjoint extensions

A densely defined symmetric operator (S) may admit self-adjoint operators (A) satisfying

[ S\subseteq A\subseteq S^*. ]

Such an (A) is called a self-adjoint extension of (S). John von Neumann classified these extensions through the deficiency subspaces

[ \mathcal N_+=\ker(S^-iI), \qquad \mathcal N_-=\ker(S^+iI). ]

Their dimensions (n_+) and (n_-) are the deficiency indices of (S). A closed symmetric operator has a self-adjoint extension exactly when the two deficiency indices are equal. When both indices vanish, the operator is already self-adjoint. If the closure of a symmetric operator is self-adjoint, the original operator is described as essentially self-adjoint.

When (n_+=n_-), self-adjoint extensions correspond to unitary maps from (\mathcal N_+) onto (\mathcal N_-). This abstract correspondence often becomes a classification of boundary conditions after the deficiency vectors have been expressed as solutions of the associated differential equation.

For semibounded symmetric operators, the Friedrichs extension, developed by Kurt Friedrichs, supplies a distinguished self-adjoint extension determined by closure of the associated quadratic form. It preserves the lower bound of the original operator and is central to the rigorous definition of elliptic Hamiltonians.

Spectral properties

The spectrum of a self-adjoint operator is contained in the real line. If (\lambda\notin\mathbb R), then (A-\lambda I) is invertible with a bounded inverse on (H), and its resolvent satisfies

[ \left|(A-\lambda I)^{-1}\right| \leq \frac{1}{|\operatorname{Im}\lambda|}. ]

The spectral theorem represents a self-adjoint operator through a projection-valued measure (E) on (\mathbb R):

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

For an unbounded operator, this expression includes the domain condition

[ \mathcal D(A)= \left{ x\in H: \int_{\mathbb R}\lambda^2,d\langle E(\lambda)x,x\rangle <\infty \right}. ]

This representation defines a functional calculus in which a suitable Borel function (f) determines an operator

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

In particular, (e^{itA}) is unitary for every real (t). Conversely, Stone’s theorem, established by Marshall Stone, states that every strongly continuous one-parameter unitary group has a unique self-adjoint generator.

Eigenvectors associated with distinct eigenvalues are orthogonal. Nevertheless, a self-adjoint operator need not possess any eigenvalues, and its spectrum need not be discrete. Multiplication by the independent variable on (L^2(\mathbb R)) has continuous spectrum equal to (\mathbb R), despite having no nonzero square-integrable eigenvectors.

Standard realizations

A bounded multiplication operator on (L^2(X,\mu)),

[ (M_\varphi f)(x)=\varphi(x)f(x), ]

is self-adjoint exactly when (\varphi) is real-valued almost everywhere. For an unbounded real-valued measurable function, the same formula defines a self-adjoint operator on the maximal domain

[ \mathcal D(M_\varphi)

{f\in L^2(X,\mu):\varphi f\in L^2(X,\mu)}. ]

On (L^2(\mathbb R)), the position operator (Qf(x)=xf(x)) is self-adjoint on its maximal multiplication domain. The momentum expression

[ Pf=-i\frac{df}{dx} ]

becomes self-adjoint on the Sobolev space (H^1(\mathbb R)). On a finite interval, the same differential expression requires boundary conditions, and its self-adjoint realizations form a family determined by a phase relation between endpoint values.

The Laplacian on a bounded region has different self-adjoint realizations corresponding to different boundary conditions. Under standard regularity assumptions, the Dirichlet and Neumann Laplacians are nonnegative. On a bounded region their resolvents are compact, so their spectra consist of eigenvalues of finite multiplicity with no finite accumulation point.

Relation to quantum mechanics

In the Hilbert-space formulation of quantum mechanics, observables are represented by self-adjoint operators rather than by arbitrary symmetric operators. The spectral measure of an observable assigns projection operators to measurable subsets of the real line, and the scalar measure

[ \mu_\psi(B)=\langle E(B)\psi,\psi\rangle ]

gives the probability distribution of measurement outcomes in a normalized state (\psi).

The distinction between symmetry and self-adjointness is essential for time evolution. A symmetric Hamiltonian does not necessarily generate a unitary group, whereas a self-adjoint Hamiltonian (H) determines

[ U(t)=e^{-itH}. ]

This group preserves inner products and satisfies the abstract Schrödinger equation on vectors belonging to the appropriate operator domain. Different self-adjoint extensions of the same symmetric differential expression can represent different boundary interactions or different admissible dynamics.