# Hermitian operator

A linear operator $A$ on a Hilbert space $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ is a self-adjoint operator. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a Hermitian matrix.