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.