Normal operator

In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its Hermitian adjoint N*, that is: NN* = N*N.

Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are

• unitary operators: N* = N⁻¹
• Hermitian operators (i.e., self-adjoint operators): N* = N
• skew-Hermitian operators: N* = −N
• positive operators: N = MM* for some M (so N is self-adjoint).

A normal matrix is the matrix expression of a normal operator on the Hilbert space Cⁿ.