# Self-adjoint operator

'''Definition: In a finite dimensional space, given any linear operator *A* defined on the whole space, there exists an operator *A* † such that '''

⟨*ϕ*, *A**χ*⟩ = ⟨*A* † *ϕ*, *χ*⟩, *ϕ*, *χ* ∈ H. ** A † is uniquely determined and is called the adjoint of A.**

It must be noticed that for bounded operators the adjoint operator can be defined naturally, whereas for unbounded operators this can be done only if the domain of *A*, D(*A*), is dense in H and only for those vectors *ϕ* for which ⟨*ϕ*, *A**χ*⟩ is a continuous function of *χ*. The set of these vectors is a vectorial subset of H and by definition it will be the domain, D(*A* † ), of *A* † .

**Definition:** A linear operator *A* is said to be:

- a
*Hermitian*operator if*A*⊆*A*† , i.e. if

⟨*ϕ*, *A**ϕ*⟩ = ⟨*A**ϕ*, *ϕ*⟩, *ϕ* ∈ D(*A*);

- a
*symmetric operator*if*A*⊆*A*† and $\; \overline{\mathcal{D}(A) }=\mathcal{H}$, where $\; \overline{\mathcal{D}(A) }$ is the complement of D(*A*); - a
**self-adjoint**operator if*A*=*A*† and $\; \overline{\mathcal{D}(A) }=\mathcal{H}$.

For bounded linear operators, and in particular for linear operators in finite dimensional Hilbert spaces, the three definitions coincide.

'''Proposition: A necessary and sufficient condition for the linear operator *A* to be *Hermitian* is that '''

⟨*ϕ*, *A**χ*⟩ = ⟨*A**ϕ*, *χ*⟩, *ϕ*, *χ* ∈ H.

**Proof:** This equality certainly implies the definition 1. of Hermitian operator. The converse is true because of the following identity, which can be easily verified:

4⟨*ϕ*, *A**χ*⟩ = ⟨*ϕ* + *χ*, *A*(*ϕ* + *χ*)⟩⟨*ϕ* − *χ*, *A*(*ϕ* − *χ*)⟩ − *i*⟨*ϕ* + *i**χ*, *A*(*ϕ* + *i**χ*)⟩ + *i*⟨*ϕ* − *i**χ*, *A*(*ϕ* − *i**χ*)⟩.

The matrix which represents the *adjoint* of a linear operator *A* in any orthonormal basis is the Hermitian conjugate of the matrix representing *A*. Indeed its matrix elements are given by:

*A**m*, *n* * = ⟨*ξ**m*, *A**ξ**n*⟩ * = ⟨*A* † *ξ**m*, *ξ**n*⟩ * = ⟨*ξ**n*, *A* † *ξ**m*⟩ = *A**n*, *m* † .

So, if a bounded linear operator is **self-adjoint**, it is represented by a Hermitian matrix in any orthonormal basis.