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 calledsymmetric.

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