Normal element
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
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 T be a bounded operator. The following are equivalent.
If N is a normal operator, then N and N* have the same kernel and the same range. Consequently, the range of N is dense if and only if N 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 Nk coincides with that of N for any k. Every generalized eigenvalue of a normal operator is thus genuine. λ is an eigenvalue of a normal operator N if and only if its complex conjugate is an eigenvalue of N*. 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.
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