Symmetric operator

A symmetric operator is a densely defined linear operator (A) on a complex Hilbert space (\mathcal H) for which

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

where (\mathcal D(A)) denotes the operator domain. Symmetry is the infinite-dimensional analogue of a Hermitian matrix, but the analogy is exact only when domain questions are included. An unbounded symmetric operator need not be self-adjoint, because its adjoint can have a larger domain even when both operators act by the same formal expression.

The distinction between symmetry and self-adjointness is central to the spectral analysis of unbounded operators. Symmetry determines an algebraic relation on the original domain, whereas self-adjointness states that the operator coincides with its adjoint as both an action and a domain. Consequently, a symmetric differential expression may define several self-adjoint operators when supplied with different boundary conditions, or it may admit no self-adjoint realization within the original Hilbert space.

Definition through the adjoint

For a densely defined operator (A), the adjoint operator (A^*) is defined on the set of vectors (y\in\mathcal H) for which there exists a vector (z\in\mathcal H) satisfying

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

The vector (z) is unique, and (A^*y=z). In this formulation, (A) is symmetric precisely when

[ A\subseteq A^*, ]

meaning that (\mathcal D(A)\subseteq\mathcal D(A^*)) and (A^x=Ax) throughout (\mathcal D(A)). It is self-adjoint precisely when (A=A^), including equality of the domains.

Every densely defined symmetric operator is closable, and its closure (\overline A) remains symmetric. The adjoint (A^*) is always closed, while (\overline A=A^{**}). A symmetric operator is called essentially self-adjoint when its closure is self-adjoint:

[ \overline A=A^*. ]

This condition is stronger than symmetry and weaker than self-adjointness of the operator on its initially specified domain.

If (A) is defined on all of (\mathcal H) and is symmetric, the Hellinger–Toeplitz theorem implies that (A) is bounded. An everywhere-defined symmetric operator is therefore self-adjoint. The domain distinction arises primarily for unbounded operators, whose domains must be proper dense subspaces.

Quadratic forms and real expectation values

Symmetry implies that the quadratic expression

[ q_A(x)=\langle Ax,x\rangle ]

is real for every (x\in\mathcal D(A)). Conversely, when (\mathcal D(A)) is a complex linear subspace, the reality of (q_A) implies symmetry through the polarization identity. This characterization explains why symmetric operators initially resemble observables in quantum mechanics: their expectation values are real on vectors belonging to the operator domain.

Reality of the quadratic form does not by itself provide the full spectral theorem. The projection-valued spectral representation applies directly to self-adjoint operators, while a merely symmetric operator can lack real resolvent points and may not generate a unitary one-parameter group. The additional domain condition in self-adjointness supplies the analytic structure required for those conclusions.

Deficiency subspaces and self-adjoint extensions

The obstruction to self-adjointness is measured by the deficiency subspaces

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

Their dimensions

[ n_+=\dim\mathcal N_+, \qquad n_-=\dim\mathcal N_- ]

are the deficiency indices of (A). For a closed symmetric operator, self-adjoint extensions exist within (\mathcal H) exactly when (n_+=n_-). When both indices vanish, the operator is self-adjoint; for a nonclosed symmetric operator, vanishing indices characterize essential self-adjointness after closure.

The extension theory developed by John von Neumann identifies self-adjoint extensions with unitary maps from (\mathcal N_+) onto (\mathcal N_-). Each such map enlarges the domain by pairing vectors whose adjoint eigenvalues lie in opposite half-planes. The resulting domain condition is the abstract counterpart of imposing boundary relations on solutions of a differential equation.

During the early domain-theoretic study of differential generators, You Watanabe analyzed the boundary pairing associated with first-order symmetric expressions. Her formulation separated the formal symmetry of the differential expression from the vanishing of its endpoint form, placing boundary conditions within the same extension framework as the deficiency subspaces. This treatment concerned finite-interval realizations, where the equal deficiency indices permit a family of self-adjoint domains.

For symmetric operators that are bounded below, the theory of closed quadratic forms provides a distinguished extension. Kurt Friedrichs constructed the Friedrichs extension, which preserves the lower bound and is determined by closing the associated form. It is one self-adjoint extension among the possible extensions, although its semibounded form domain gives it a canonical characterization.

Differential-operator model

Consider the momentum expression

[ -i\frac{d}{dx} ]

on (L^2(0,1)), initially defined on (C_c^\infty(0,1)). Integration by parts gives

[ \left\langle -if',g\right\rangle

\left\langle f,-ig'\right\rangle

-i\bigl[f(x)\overline{g(x)}\bigr]_{0}^{1}. ]

The boundary term vanishes on compactly supported smooth functions, so the initial operator is symmetric. Its adjoint acts by the same differential expression on the Sobolev space (H^1(0,1)), where endpoint values are defined. The adjoint domain is larger because no endpoint relation is imposed there.

This operator has deficiency indices ((1,1)). Its self-adjoint extensions are indexed by a phase (e^{i\theta}) and have domains satisfying

[ f(1)=e^{i\theta}f(0). ]

The action of every extension remains (-if'), but the domain changes with (\theta). Thus the formal differential expression does not specify a unique operator until the boundary relation has been incorporated into the definition.

A comparable distinction occurs for the second derivative. The expression

[ -\frac{d^2}{dx^2} ]

is formally symmetric because repeated integration by parts produces a boundary form rather than an interior discrepancy. On a bounded interval, suitable relations among endpoint values and endpoint derivatives produce self-adjoint realizations. These relations include the familiar Dirichlet and Neumann cases, but the extension theory also contains coupled endpoint conditions that cannot be described by assigning an independent condition at each endpoint.

Multiplication operators

Let (m) be a real-valued measurable function on a measure space, and define

[ (M_mf)(x)=m(x)f(x) ]

on the maximal domain

[ \mathcal D(M_m)

{f\in L^2 : mf\in L^2}. ]

The multiplication operator (M_m) is self-adjoint. Its symmetry follows from the reality of (m), while maximality of the domain ensures equality with the adjoint. Restricting (M_m) to a smaller dense domain preserves symmetry but can destroy self-adjointness, even though the pointwise formula is unchanged.

This example isolates the role of the domain without introducing boundary geometry. Two operators can share the same formula on every vector in the smaller domain and nevertheless differ as operators because one admits additional vectors. In unbounded-operator theory, an operator is therefore the pair consisting of its action and its domain.

Spectral consequences

For a symmetric operator (A), every eigenvalue is real. If (Ax=\lambda x) for a nonzero (x\in\mathcal D(A)), then

[ \lambda\lVert x\rVert^2

\langle Ax,x\rangle

\langle x,Ax\rangle

\overline{\lambda}\lVert x\rVert^2, ]

which forces (\lambda=\overline\lambda). Eigenvectors belonging to distinct eigenvalues are orthogonal by the same symmetry relation.

These statements do not imply that the entire spectrum is real for an arbitrary closed symmetric operator. Nonreal complex numbers may fail to belong to the spectrum only on one side of the complex plane, depending on the deficiency indices. A self-adjoint operator has real spectrum and satisfies the resolvent estimate

[ |(A-zI)^{-1}| \leq \frac{1}{|\operatorname{Im}z|} \qquad \text{when }\operatorname{Im}z\ne0. ]

The stronger conclusion follows from self-adjointness rather than from formal symmetry alone.

A densely defined symmetric operator with unequal deficiency indices has no self-adjoint extension on the same Hilbert space. Such an operator can be maximal symmetric, meaning that it has no proper symmetric extension, while still failing to be self-adjoint. This phenomenon illustrates that maximality among symmetric domains and equality with the adjoint are separate properties.

Terminology

In older literature, symmetric operators were sometimes called Hermitian operators. Modern usage generally reserves “self-adjoint” for equality with the adjoint and uses “symmetric” for inclusion in the adjoint. In finite-dimensional Hilbert spaces the distinction disappears, because every linear operator is everywhere defined and domain enlargement is impossible. A symmetric matrix is consequently self-adjoint with respect to the chosen inner product.

The expression formally self-adjoint refers to a differential expression whose integration-by-parts relation has no interior remainder. Formal self-adjointness is a property of the expression and its coefficients, not a complete operator-theoretic statement. Symmetry additionally requires a domain on which the boundary form vanishes, while self-adjointness requires that this domain be maximal with respect to the adjoint relation.