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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 ${\displaystyle \scriptstyle H_{0}}$ which is a dense subspace of a complex Hilbert space H then it is self-adjoint if ${\displaystyle \scriptstyle A=A^{*}}$, where ${\displaystyle \scriptstyle A^{*}}$ 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 ${\displaystyle \scriptstyle \mathbb {R} ^{n}}$ and ${\displaystyle \scriptstyle \mathbb {C} ^{n}}$, 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

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

where ${\displaystyle \scriptstyle E^{X}}$ 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 ${\displaystyle \scriptstyle U=\{U_{t}\}_{t\in \mathbb {R} }}$ defined by ${\displaystyle \scriptstyle U_{t}=\int _{-\infty }^{\infty }e^{-itx}\,E^{X}(dx)}$, where ${\displaystyle \scriptstyle E^{X}}$ is the spectral measure of X, satisfies:

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

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

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

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

and

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

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