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, Aψ⟩ = ⟨ψ, Aψ⟩^* = ⟨Aψ, 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, Aχ⟩ = ⟨Aψ, 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, Aχ⟩,  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