In functional analysis, a normal operator on a Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N*:
N N* = N* N.
The main importance of this concept is that the spectral theorem applies to normal operators.

Examples of normal operators:

  • Unitary operators (N* = N −1)
  • Hermitian operators (N* = N)
  • Normal matrices can be seen as normal operators if one takes the Hilbert space to be Cn.