In mathematics, a self-adjoint operator on a complex vector space V with inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ is a linear map A (from V to itself) that is its own adjoint. That is, ${\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }$ for all ${\displaystyle x,y}$ ∊ V. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A^∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator ${\displaystyle {\hat {H}}}$ defined by

which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators.

The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs to be more attentive to the domain issue in the unbounded case. This is explained below in more detail.

Let ${\displaystyle H}$ be a Hilbert space and ${\displaystyle A}$ an unbounded (i.e. not necessarily bounded) linear operator with a dense domain ${\displaystyle \operatorname {Dom} A\subseteq H.}$ This condition holds automatically when ${\displaystyle H}$ is finite-dimensional since ${\displaystyle \operatorname {Dom} A=H}$ for every linear operator on a finite-dimensional space.

The graph of an (arbitrary) operator ${\displaystyle A}$ is the set ${\displaystyle G(A)=\{(x,Ax)\mid x\in \operatorname {Dom} A\}.}$ An operator ${\displaystyle B}$ is said to extend ${\displaystyle A}$ if ${\displaystyle G(A)\subseteq G(B).}$ This is written as ${\displaystyle A\subseteq B.}$

Let the inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ be conjugate linear on the second argument. The adjoint operator ${\displaystyle A^{*}}$ acts on the subspace ${\displaystyle \operatorname {Dom} A^{*}\subseteq H}$ consisting of the elements ${\displaystyle y}$ such that

The densely defined operator ${\displaystyle A}$ is called symmetric (or Hermitian) if ${\displaystyle A\subseteq A^{*}}$, i.e., if ${\displaystyle \operatorname {Dom} A\subseteq \operatorname {Dom} A^{*}}$ and ${\displaystyle Ax=A^{*}x}$ for all ${\displaystyle x\in \operatorname {Dom} A}$. Equivalently, ${\displaystyle A}$ is symmetric if and only if