Normal operator
(on a complex Hilbert space) continuous linear operator / From Wikipedia, the free encyclopedia
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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.[1]
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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.