Self-adjoint operator

In mathematics, a self-adjoint operator on a complex vector space with inner product is a linear map that is its own adjoint. That is, for all. If is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of is a Hermitian matrix, i.e., equal to its conjugate transpose. By the finite-dimensional spectral theorem, has an orthonormal basis such that the matrix of relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which 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 operator defined by
which as an observable corresponds to the total energy of a particle of mass in a real potential field. 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 to be more attentive to the domain issue in the unbounded case. This is explained below in more detail.

Definitions

Let be a Hilbert space and an unbounded linear operator with a dense domain This condition holds automatically when is finite-dimensional since for every linear operator on a finite-dimensional space.
The graph of an operator is the set An operator is said to extend if This is written as
Let the inner product be conjugate linear on the second argument. The adjoint operator acts on the subspace consisting of the elements such that
The densely defined operator is called symmetric if, i.e., if and for all. Equivalently, is symmetric if and only if
Since is dense in, symmetric operators are always closable. If is a closed extension of, the smallest closed extension of must be contained in. Hence,
for symmetric operators and
for closed symmetric operators.
The densely defined operator is called self-adjoint if, that is, if and only if is symmetric and. Equivalently, a closed symmetric operator is self-adjoint if and only if is symmetric. If is self-adjoint, then is real for all, i.e.,
A symmetric operator is said to be essentially self-adjoint if the closure of is self-adjoint. Equivalently, is essentially self-adjoint if it has a unique self-adjoint extension. In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator.
In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.

Bounded self-adjoint operators

Let be a Hilbert space and a symmetric operator. According to Hellinger–Toeplitz theorem, if then is necessarily bounded.
A bounded operator is self-adjoint if
Every bounded operator can be written in the complex form where and are bounded self-adjoint operators.
Alternatively, every positive bounded linear operator is self-adjoint if the Hilbert space is complex.

Properties

A bounded self-adjoint operator defined on has the following properties:

is invertible if the image of is dense in
The operator norm is given by
If is an eigenvalue of then ; the eigenvalues are real and the corresponding eigenvectors are orthogonal.

Bounded self-adjoint operators do not necessarily have an eigenvalue. If, however, is a compact self-adjoint operator then it always has an eigenvalue and corresponding normalized eigenvector.

Spectrum of self-adjoint operators

Let be an unbounded operator. The resolvent set of is defined as
If is bounded, the definition reduces to being bijective on. The spectrum of is defined as the complement
In finite dimensions, consists exclusively of eigenvalues. The spectrum of a self-adjoint operator is always real, though non-self-adjoint operators with real spectrum exist as well. For bounded operators, however, the spectrum is real if and only if the operator is self-adjoint. This implies, for example, that a non-self-adjoint operator with real spectrum is necessarily unbounded.
As a preliminary, define and with. Then, for every and every
where
Indeed, let By the Cauchy–Schwarz inequality,
If then and is called bounded below.

Spectral theorem

In the physics literature, the spectral theorem is often stated by saying that a self-adjoint operator has an orthonormal basis of eigenvectors. Physicists are well aware, however, of the phenomenon of "continuous spectrum"; thus, when they speak of an "orthonormal basis" they mean either an orthonormal basis in the classic sense or some continuous analog thereof. In the case of the momentum operator, for example, physicists would say that the eigenvectors are the functions, which are clearly not in the Hilbert space. Physicists would then go on to say that these "generalized eigenvectors" form an "orthonormal basis in the continuous sense" for, after replacing the usual Kronecker delta by a Dirac delta function.
Although these statements may seem disconcerting to mathematicians, they can be made rigorous by use of the Fourier transform, which allows a general function to be expressed as a "superposition" of the functions, even though these functions are not in. The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by, where is the variable of the Fourier transform.
The spectral theorem in general can be expressed similarly as the possibility of "diagonalizing" an operator by showing it is unitarily equivalent to a multiplication operator. Other versions of the spectral theorem are similarly intended to capture the idea that a self-adjoint operator can have "eigenvectors" that are not actually in the Hilbert space in question.

Multiplication operator form of the spectral theorem

Firstly, let be a σ-finite measure space and a measurable function on. Then the operator, defined by
where
is called a multiplication operator. Any multiplication operator is a self-adjoint operator.
Secondly, two operators and with dense domains and in Hilbert spaces and, respectively, are unitarily equivalent if and only if there is a unitary transformation such that:

If unitarily equivalent and are bounded, then ; if is self-adjoint, then so is.
The spectral theorem holds for both bounded and unbounded self-adjoint operators. Proof of the latter follows by reduction to the spectral theorem for unitary operators. We might note that if is multiplication by, then the spectrum of is just the essential range of.
More complete versions of the spectral theorem exist as well that involve direct integrals and carry with it the notion of "generalized eigenvectors".

Functional calculus

One application of the spectral theorem is to define a functional calculus. That is, if is a function on the real line and is a self-adjoint operator, we wish to define the operator. The spectral theorem shows that if is represented as the operator of multiplication by, then is the operator of multiplication by the composition.
One example from quantum mechanics is the case where is the Hamiltonian operator. If has a true orthonormal basis of eigenvectors with eigenvalues, then can be defined as the unique bounded operator with eigenvalues such that:
The goal of functional calculus is to extend this idea to the case where has continuous spectrum.
It has been customary to introduce the following notation
where is the indicator function of the interval. The family of projection operators E is called resolution of the identity for T. Moreover, the following Stieltjes integral representation for T can be proved:

Formulation in the physics literature

In quantum mechanics, Dirac notation is used as combined expression for both the spectral theorem and the Borel functional calculus. That is, if H is self-adjoint and f is a Borel function,
with
where the integral runs over the whole spectrum of H. The notation suggests that H is diagonalized by the eigenvectors Ψ_E. Such a notation is purely formal. The resolution of the identity formally resembles the rank-1 projections. In the Dirac notation, 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, the theorem is referred to as resolution of unity:
In the case is the sum of an Hermitian H and a skew-Hermitian operator, one defines the biorthogonal basis set
and write the spectral theorem as:
.

Formulation for symmetric operators

The spectral theorem applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless, we can at this point give a simple example of a symmetric operator that has an orthonormal basis of eigenvectors. Consider the complex Hilbert space L² and the differential operator
with consisting of all complex-valued infinitely differentiable functions f on satisfying the boundary conditions
Then integration by parts of the inner product shows that A is symmetric. The eigenfunctions of A are the sinusoids
with the real eigenvalues n²π²; the well-known orthogonality of the sine functions follows as a consequence of A being symmetric.
The operator A can be seen to have a compact inverse, meaning that the corresponding differential equation Af = g is solved by some integral 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.