Self-adjoint operator

In mathematics, a self-adjoint operator is a densely defined linear operator mapping a complex Hilbert space onto itself and which is invariant under the unary operation of taking the adjoint. That is, if A is an operator with a domain $\scriptstyle H_{0}$ which is a dense subspace of a complex Hilbert space H then it is self-adjoint if $\scriptstyle A=A^{*}$, where $\scriptstyle A^{*}$ denotes the adjoint operator of A. Note that the adjoint of any densely defined linear operator is always well-defined (in fact, the denseness of the domain of an operator is necessary for the existence of its adjoint) and two operators A and B are said to be equal if they have a common domain and their values coincide on that domain.

On an infinite dimensional Hilbert space, a self-adjoint operator can be thought of as the analogy of a real symmetric matrix (i.e., a matrix which is its own transpose) or a Hermitian matrix in (i.e., a matrix which is its own Hermitian transpose) when these matrices are viewed as (bounded) linear operators on $\scriptstyle \mathbb {R} ^{n}$ and $\scriptstyle \mathbb {C} ^{n}$, respectively.

Special properties of a self-adjoint operator

The self-adjointness of an operator entails that it has some special properties. Some of these properties include:

1. The eigenvalues of a self-adjoint operator are real. As a special well-known case, all eigenvalues of a real symmetric matrix and a complex Hermitian matrix are real.

2. By the von Neumann’s spectral theorem, any self-adjoint operator X (not necessarily bounded) can be represented as

$X=\int _{-\infty }^{\infty }xE^{X}(dx),$

where $\scriptstyle E^{X}$ is the associated spectral measure of X (in particular, a spectral measure is a Hilbert space projection operator-valued measure)

3. By Stone’s Theorem, for any self-adjoint operator X the one parameter unitary group $\scriptstyle U=\{U_{t}\}_{t\in \mathbb {R} }$ defined by $\scriptstyle U_{t}=\int _{-\infty }^{\infty }e^{-itx}\,E^{X}(dx)$, where $\scriptstyle E^{X}$ is the spectral measure of X, satisfies:

${\frac {dU_{t}}{dt}}u=-iXU_{t}u=U_{t}(-iX)u,$

for all u in the domain of X. One says that the operator -iX is the generator of the group U and writes: $\scriptstyle U_{t}=e^{-itX},\,\,t\in \mathbb {R}$.

Examples of self-adjoint operators

Consider the complex Hilbert space $\scriptstyle L^{2}(\mathbb {R} ;\mathbb {C} )$ of all complex-valued square integrable functions on $\scriptstyle \mathbb {R}$ with the complex inner product $\scriptstyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx$, and the dense subspace $\scriptstyle C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )$ of $\scriptstyle L^{2}(\mathbb {R} ;\mathbb {C} )$ of all infinitely differentiable complex-valued functions on $\scriptstyle \mathbb {R}$ vanishing at $\scriptstyle \pm \infty$. Define the operators Q, P on $\scriptstyle C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )$ as:

$Q(f)(x)=xf(x)\quad \forall f\in C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )$

and

$P(f)(x)=i\hbar {\frac {d}{dx}}f(x)\quad \forall f\in C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} ),$

where $\scriptstyle \hbar$ is the real valued Planck's constant. Then Q and P are self-adjoint operators satisfying the commutation relation $\scriptstyle [Q,P]=i\hbar I$ on $\scriptstyle C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )$, where I denotes the identity operator. In quantum mechanics, the pair Q and P is known as the Schrödinger representation, on the Hilbert space $\scriptstyle L^{2}(\mathbb {R} ;\mathbb {C} )$, of canonical conjugate position and momentum operators q and p satisfying the canonical commutation relation (CCR) $\scriptstyle [q,p]=i\hbar$.

Further reading

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980.
2. K. Parthasarathy, An Introduction to Quantum Stochastic Calculus, ser. Monographs in Mathematics, Basel, Boston, Berlin: Birkhauser Verlag, 1992.