Self-adjoint operator

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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.

[edit] 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} x E^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} .

[edit] Examples of self-adjoint operators

As mentioned above, a simple instance of a self-adjoint operator is a Hermitian matrix.

For a more advanced example 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^{\infty}_0(\mathbb{R};\mathbb{C}) of \scriptstyle L^2(\mathbb{R};\mathbb{C}) of all infinitely differentiable complex-valued functions with compact support on \scriptstyle \mathbb{R}. Define the operators Q, P on \scriptstyle C^{\infty}_0(\mathbb{R};\mathbb{C}) as:

Q(f)(x)= xf(x) \quad \forall f \in C^{\infty}_0(\mathbb{R};\mathbb{C})

and

P(f)(x)=i \hbar \frac{d}{dx}f(x) \quad \forall f \in C^{\infty}_0(\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^{\infty}_0(\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.

[edit] 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.
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