Normal operator

In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ${\displaystyle H}$ is a continuous linear operator ${\displaystyle N\colon H\rightarrow H}$ that commutes with its Hermitian adjoint ${\displaystyle N^{\ast }}$, that is: ${\displaystyle N^{\ast }N=NN^{\ast }}$.

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: ${\displaystyle U^{\ast }=U^{-1}}$
• Hermitian operators (i.e., self-adjoint operators): ${\displaystyle N^{\ast }=N}$
• skew-Hermitian operators: ${\displaystyle N^{\ast }=-N}$
• positive operators: ${\displaystyle N=M^{\ast }M}$ for some ${\displaystyle M}$ (so N is self-adjoint).

A normal matrix is the matrix expression of a normal operator on the Hilbert space ${\displaystyle \mathbb {C} ^{n}}$.