For physical consistency, the mean values of any dynamical variable A must be real numbers: this implies that the mean values of the operator *A* which describes A must be real, i.e. that

⟨*ψ*, *A**ψ*⟩ = ⟨*ψ*, *A**ψ*⟩ * = ⟨*A**ψ*, *ψ*⟩, *ψ* ∈ D(*A*), where D(*A*) is the domain of the operator *A*.

It can be proved that this condition is equivalent to the following:

⟨*ψ*, *A**χ*⟩ = ⟨*A**ψ*, *χ*⟩, *ψ*, *χ* ∈ D(*A*).

Having defined the linear operator *A* † in H, which is called the *adjoint* of *A*, such that

⟨*A* † *ψ*, *χ*⟩ = ⟨*ψ*, *A**χ*⟩, *ψ*, *χ* ∈ H, it follows that *A* must be *Hermitian*, i.e. its domain must be such that D(*A*) ⊆ D(*A* † ) and *A* must coincide with *A* † in D(*A*).

*'Definition: Hermitian operators whose domain is dense in H are called symmetric.*'

In particular, if a *bounded* linear operator is symmetric, it is also a Hermitian and self-adjoint operator.

Category:Mathematical Structure Category:Handbook of Quantum Information