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Concept
Normal operator
Summary
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* = NN.
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.
Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable.
Let be a bounded operator. The following are equivalent.
is normal.
is normal.
for all (use ).
The self-adjoint and anti–self adjoint parts of commute. That is, if is written as with and then
If is a normal operator, then and have the same kernel and the same range. Consequently, the range of is dense if and only if is injective. Put in another way, the kernel of a normal operator is the orthogonal complement of its range. It follows that the kernel of the operator coincides with that of for any Every generalized eigenvalue of a normal operator is thus genuine. is an eigenvalue of a normal operator if and only if its complex conjugate is an eigenvalue of Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces. This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator. There is also an infinite-dimensional version of the spectral theorem expressed in terms of projection-valued measures. The residual spectrum of a normal operator is empty.
Official source
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