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.