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] Normal operators are important because the spectral theorem holds for them.
[clarification needed] Put in another way, the kernel of a normal operator is the orthogonal complement of its range.
Every generalized eigenvalue of a normal operator is thus genuine.
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.
[3] The product of normal operators that commute is again normal; this is nontrivial, but follows directly from Fuglede's theorem, which states (in a form generalized by Putnam): The operator norm of a normal operator equals its numerical radius[clarification needed] and spectral radius.
A normal operator coincides with its Aluthge transform.
If a normal operator T on a finite-dimensional real[clarification needed] or complex Hilbert space (inner product space) H stabilizes a subspace V, then it also stabilizes its orthogonal complement V⊥.
First it is noted that Now using properties of the trace and of orthogonal projections we have: The same argument goes through for compact normal operators in infinite dimensional Hilbert spaces, where one make use of the Hilbert-Schmidt inner product, defined by tr(AB*) suitably interpreted.
[5] It follows that the Hilbert space cannot in general be spanned by eigenvectors of a normal operator.
The invariant subspaces of a shift acting on Hardy space are characterized by Beurling's theorem.
Explicitly, a closed operator N is said to be normal if Here, the existence of the adjoint N* requires that the domain of N be dense, and the equality includes the assertion that the domain of N*N equals that of NN*, which is not necessarily the case in general.
The proofs work by reduction to bounded (normal) operators.
[7][8] The success of the theory of normal operators led to several attempts for generalization by weakening the commutativity requirement.