Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the fact that in the Dirac-von Neumann formulation of quantum mechanics, physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian
which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators.
The structure of self-adjoint operators on infinite dimensional Hilbert spaces essentially resembles the finite dimensional case, that is to say, operators are self adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinite dimensional spaces. Since an everywhere defined self adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail
Bounded symmetric operators are also called Hermitian.
The previous definition agrees with the one for matrices given in the introduction to this article, if we take as H the Hilbert space Cn with the standard dot product and interpret a square matrix as a linear operator on this Hilbert space. It is however much more general as there are important infinite-dimensional Hilbert spaces.
The spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may not have any eigenvalues.
A general version of the spectral theorem which also applies to bounded symmetric operators is stated below. If the set of eigenvalues for a symmetric operator is non empty, and the eigenvalues are nondegenerate, then it follows from the definition that eigenvectors corresponding to distinct eigenvalues are orthogonal. Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all (although the spectrum of any self adjoint operator is nonempty). The example below illustrates the special case when an (unbounded) symmetric operator does have a set of eigenvectors which constitute a Hilbert space basis. The operator A below can be seen to have a compact inverse, meaning that the corresponding differential equation A f = g is solved by some integral, therefore compact, operator G. The compact symmetric operator G then has a countable family of eigenvectors which are complete in . The same can then be said for A.
Example. Consider the complex Hilbert space L2[0,1] and the differential operator
defined on the subspace consisting of all complex-valued infinitely differentiable functions f on [0,1] with the boundary conditions:
Then integration by parts shows that A is symmetric. Its eigenfunctions are the sinusoids
with the real eigenvalues n2π2; the well-known orthogonality of the sine functions follows as a consequence of the property of being symmetric.
We consider generalizations of this operator below.
Given a densely defined linear operator A on H, its adjoint A* is defined as follows:
Notice that it is the denseness of the domain of the operator, along with the uniqueness part of Riesz representation, that ensures the adjoint operator is well defined.
A result of Hellinger-Toeplitz type says that an operator having a bounded adjoint is bounded. Therefore the adjoint of a unbounded operator is necessarily unbounded.
The condition for a linear operator on a Hilbert space to be self-adjoint is stronger than to be symmetric.
For any densely defined operator A on Hilbert space one can define its adjoint operator A*. For a symmetric operator A, the domain of the operator A* contains the domain of the operator A, and the restriction of the operator A* on the domain of A coincides with the operator A, i.e. , in other words A* is extension of A. For a self-adjoint operator A the domain of A* is the same as the domain of A, and A=A*. See also Extensions of symmetric operators.
Theorem. Let J be the symplectic mapping
Then the graph of A* is the orthogonal complement of JG(A):
A densely defined operator A is symmetric if and only if
where the subset notation is understood to mean An operator A is self-adjoint if and only if ; that is, if and only if
Example. Consider the complex Hilbert space L2(R), and the operator which multiplies a given function by x:
The domain of A is the space of all L2 functions for which the right-hand-side is square-integrable. A is a symmetric operator without any eigenvalues and eigenfunctions. In fact it turns out that the operator is self-adjoint, as follows from the theory outlined below.
As we will see later, self-adjoint operators have very important spectral properties; they are in fact multiplication operators on general measure spaces.
Partially defined operators A, B on Hilbert spaces H, K are unitarily equivalent if and only if there is a unitary operator U:H → K such that
whose domain is the space of ψ for which the right-hand side above is in L2 is called a multiplication operator.
Theorem. Any multiplication operator is a (densely defined) self-adjoint operator. Any self-adjoint operator is unitarily equivalent to a multiplication operator.
This version of the spectral theorem for self-adjoint operators can be proved by reduction to the spectral theorem for unitary operators. This reduction uses the Cayley transform for self-adjoint operators which is defined in the next section. We might note that if T is multiplication by f, then the spectrum of T is just the essential range of f.
Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h(T) is the operator of multiplication by the composition . In order for this to be well-defined, we must show that it is the unique operation on bounded real-valued Borel functions satisfying a number of conditions.
It has been customary to introduce the following notation
where denotes the function which is identically 1 on the interval . The family of projection operators ET(λ) is called resolution of the identity for T. Moreover, the following Stieltjes integral representation for T can be proved:
The definition of the operator integral above can be reduced to that that of a scalar valued Stieltjes integral using the weak operator topology. In more modern treatments however, this representation is usually avoided, since most technical problems can be dealt with by the functional calculus.
In physics, particularly in quantum mechanics, the spectral theorem is expressed in a way which combines the spectral theorem as stated above and the Borel functional calculus using Dirac notation as follows:
If H is Hermitian (the name for self-adjoint in the physics literature) and f is a Borel function,
where the integral runs over the whole spectrum of H. The notation suggests that H is diagonalized by the eigenvalues ΨE. Such a notation is purely formal. One can see the similarity between Dirac's notation and the previous section. The resolution of the identity(sometimes called projection valued measures) formally resembles the rank-1 projections . In the Dirac notation, (projective) measurements are described via eigenvalues and eigenstates, both purely formal objects. As one would expect, this does not survive passage to the resolution of the identity. In the latter formulation, measurements are described using the spectral measure of , if the system is prepared in prior to the measurement. Alternatively, if one would like to preserve the notion of eigenstates and make it rigorous, rather than merely formal, one can replace the state space by a suitable rigged Hilbert space.
If f=1, the theorem is referred to as resolution of unity:
and write the spectral theorem as:
The following question arises in several contexts: if an operator A on the Hilbert space H is symmetric, when does it have self-adjoint extensions? One answer is provided by the Cayley transform of a self-adjoint operator and the deficiency indices. (We should note here that it is often of technical convenience to deal with closed operators. In the symmetric case, the closedness requirement poses no obstacles, since it is known that all symmetric operators are closable.)
Theorem. Suppose A is a symmetric operator. Then there is a unique partially defined linear operator
Conversely, given any partially defined operator U which is isometric on its domain (which is not necessarily closed) and such that 1 − U is dense, there is a (unique) operator S(U)
The operator S(U) is densely defined and symmetric.
The mappings W and S are inverses of each other.
The mapping W is called the Cayley transform. It associates a partially defined isometry to any symmetric densely-defined operator. Note that the mappings W and S are monotone: This means that if B is a symmetric operator that extends the densely defined symmetric operator A, then W(B) extends W(A), and similarly for S.
Theorem. A necessary and sufficient condition for A to be self-adjoint is that its Cayley transform W(A) be unitary.
This immediately gives us a necessary and sufficient condition for A to have a self-adjoint extension, as follows:
Theorem. A necessary and sufficient condition for A to have a self adjoint extension is that W(A) have a unitary extension.
A partially defined isometric operator V on a Hilbert space H has a unique isometric extension to the norm closure of dom(V). A partially defined isometric operator with closed domain is called a partial isometry.
Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:
Theorem. A partial isometry V has a unitary extension if and only if the deficiency indices are identical. Moreover, V has a unique unitary extension if and only if the both deficiency indices are zero.
We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. An operator which has a unique self-adjoint extension is said to be essentially self-adjoint. Such operators have a well-defined Borel functional calculus. Symmetric operators which are not essentially self-adjoint may still have a canonical self-adjoint extension. Such is the case for non-negative symmetric operators (or more generally, operators which are bounded below). These operators always have a canonically defined Friedrichs extension and for these operators we can define a canonical functional calculus. Many operators that occur in analysis are bounded below (such as the negative of the Laplacian operator), so the issue of essential adjointness for these operators is less critical.
Suppose A is symmetric; any symmetric extension of A is a restriction of A*; Indeed if B is symmetric
Theorem. Suppose A is a densely defined symmetric operator. Let
where the decomposition is orthogonal relative to the graph inner product of dom(A*):
These are referred to as von Neumann's formulas in the Akhiezer and Glazman reference.
We first consider the differential operator
defined on the space of complex-valued C∞ functions on [0,1] vanishing near 0 and 1. D is a symmetric operator as can be shown by integration by parts. The spaces N+, N− are given respectively by the distributional solutions to the equation
which are in L2 [0,1]. One can show that each one of these solution spaces is 1-dimensional, generated by the functions x → eix and x → e−ix respectively. This shows that D is not essentially self adjoint, but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings
which in this case happens to be the unit circle T.
This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators P on an open set M. They are determined by the unitary maps between the eigenvalue spaces
where Pdist is the distributional extension of P.
We next give the example of differential operators with constant coefficients. Let
be a polynomial on Rn with real coefficients, where α ranges over a (finite) set of multi-indices. Thus
We also use the notation