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Self-adjoint operator

Definition: In a finite dimensional space, given any linear operator $\; A$ defined on the whole space, there exists an operator $\; A^\dagger$ such that

$\; \langle \phi, A\chi \rangle = \langle A^\dagger\phi, \chi \rangle, \quad \phi, \chi \in \mathcal{H}.$

$\; A^\dagger$ 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 $\; A$ , $\; \mathcal{D}(A)$ , is dense in $\; \mathcal{H}$ and only for those vectors $\; \phi$ for which $\; \langle \phi, A\chi \rangle$ is a continuous function of $\; \chi$ . The set of these vectors is a vectorial subset of $\; \mathcal{H}$ and by definition it will be the domain, $\; \mathcal{D}(A^\dagger)$ , of $\; A^\dagger$ .

Definition: A linear operator $\; A$ is said to be:

a Hermitian operator if $\; A \subseteq A^\dagger$ , i.e. if

$\; \langle \phi, A\phi \rangle = \langle A\phi, \phi \rangle, \quad \phi \in \mathcal{D}(A)$ ;

a symmetric operator if $\; A \subseteq A^\dagger$ and $\; \overline{\mathcal{D}(A) }=\mathcal{H}$ , where $\; \overline{\mathcal{D}(A) }$ is the complement of $\; {\mathcal{D}(A) }$ ;
a self-adjoint operator if $\; A = A^\dagger$ 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

$\; \langle \phi, A\chi \rangle = \langle A\phi, \chi \rangle, \quad \phi, \chi \in \mathcal{H}.$

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:

$\; 4\langle \phi, A\chi \rangle = \langle \phi+\chi, A(\phi+\chi) \rangle \langle \phi-\chi, A(\phi-\chi) \rangle -i\langle \phi+i\chi, A(\phi+i\chi) \rangle + i\langle \phi-i\chi, A(\phi-i\chi) \rangle.$

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:

$\; A_{m,n}^* = \langle \xi_m, A\xi_n\rangle^* = \langle A^\dagger\xi_m, \xi_n\rangle^* = \langle \xi_n, A^\dagger\xi_m\rangle = A_{n,m}^\dagger$ .