Normal operator

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. 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.

Properties

Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional inner product space) is unitarily diagonalizable.^[2]

Let $T$ be a bounded operator. The following are equivalent.

$T$ is normal.
$T^{\star }$ is normal.
$\|Tx\|=\|T^{*}x\|$ for all $x$ (use $\|Tx\|^{2}=\langle T^{*}Tx,x\rangle =\langle TT^{*}x,x\rangle =\|T^{*}x\|^{2}$ ).
The self-adjoint and anti–self adjoint parts of $T$ commute. That is, if $T$ is written as $T=T_{1}+iT_{2}$ with $T_{1}:={\frac {T+T^{*}}{2}}$ and $i\,T_{2}:={\frac {T-T^{*}}{2}},$ then $T_{1}T_{2}=T_{2}T_{1}.$ ^{[note 1]}

If $N$ is a bounded 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.^{[clarification needed]} 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 $N^{k}$ coincides with that of $N$ for any $k.$ Every generalized eigenvalue of a normal operator is thus genuine. $\lambda$ is an eigenvalue of a normal operator $N$ if and only if its complex conjugate ${\overline {\lambda }}$ 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.^[3] 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.^[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):

If

N_{1}

and

N_{2}

are normal operators and if

A

is a bounded linear operator such that

N_{1}A=AN_{2},

then

N_{1}^{*}A=AN_{2}^{*}

.

The operator norm of a normal operator equals its numerical radius^{[clarification needed]} and spectral radius.

A normal operator coincides with its Aluthge transform.

Properties in finite-dimensional case

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^⊥. (This statement is trivial in the case where T is self-adjoint.)

Proof. Let P_V be the orthogonal projection onto V. Then the orthogonal projection onto V^⊥ is 1_H−P_V. The fact that T stabilizes V can be expressed as (1_H−P_V)TP_V = 0, or TP_V = P_VTP_V. The goal is to show that P_VT(1_H−P_V) = 0.

Let X = P_VT(1_H−P_V). Since (A, B) ↦ tr(AB*) is an inner product on the space of endomorphisms of H, it is enough to show that tr(XX*) = 0. First it is noted that

{\begin{aligned}XX^{*}&=P_{V}T({\boldsymbol {1}}_{H}-P_{V})^{2}T^{*}P_{V}\\&=P_{V}T({\boldsymbol {1}}_{H}-P_{V})T^{*}P_{V}\\&=P_{V}TT^{*}P_{V}-P_{V}TP_{V}T^{*}P_{V}.\end{aligned}}

Now using properties of the trace and of orthogonal projections we have:

{\begin{aligned}\operatorname {tr} (XX^{*})&=\operatorname {tr} \left(P_{V}TT^{*}P_{V}-P_{V}TP_{V}T^{*}P_{V}\right)\\&=\operatorname {tr} (P_{V}TT^{*}P_{V})-\operatorname {tr} (P_{V}TP_{V}T^{*}P_{V})\\&=\operatorname {tr} (P_{V}^{2}TT^{*})-\operatorname {tr} (P_{V}^{2}TP_{V}T^{*})\\&=\operatorname {tr} (P_{V}TT^{*})-\operatorname {tr} (P_{V}TP_{V}T^{*})\\&=\operatorname {tr} (P_{V}TT^{*})-\operatorname {tr} (TP_{V}T^{*})&&{\text{using the hypothesis that }}T{\text{ stabilizes }}V\\&=\operatorname {tr} (P_{V}TT^{*})-\operatorname {tr} (P_{V}T^{*}T)\\&=\operatorname {tr} (P_{V}(TT^{*}-T^{*}T))\\&=0.\end{aligned}}

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.^[4] However, for bounded normal operators, the orthogonal complement to a stable subspace may not be stable.^[5] It follows that the Hilbert space cannot in general be spanned by eigenvectors of a normal operator. Consider, for example, the bilateral shift (or two-sided shift) acting on $\ell ^{2}$ , which is normal, but has no eigenvalues.

The invariant subspaces of a shift acting on Hardy space are characterized by Beurling's theorem.

Normal elements of algebras

The notion of normal operators generalizes to an involutive algebra:

An element x of an involutive algebra is said to be normal if xx* = x*x.

Self-adjoint and unitary elements are normal.

The most important case is when such an algebra is a C*-algebra.

Unbounded normal operators

The definition of normal operators naturally generalizes to some class of unbounded operators. Explicitly, a closed operator N is said to be normal if

N^{*}N=NN^{*}.

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.

Equivalently normal operators are precisely those for which^[6]

\|Nx\|=\|N^{*}x\|\qquad

with

{\mathcal {D}}(N)={\mathcal {D}}(N^{*}).

The spectral theorem still holds for unbounded (normal) operators. The proofs work by reduction to bounded (normal) operators.^[7]^[8]

Generalization

The success of the theory of normal operators led to several attempts for generalization by weakening the commutativity requirement. Classes of operators that include normal operators are (in order of inclusion)

Notes

^ In contrast, for the important class of Creation and annihilation operators of, e.g., quantum field theory, they don't commute

References

^ Hoffman, Kenneth; Kunze, Ray (1971), Linear algebra (2nd ed.), Englewood Cliffs, N.J.: Prentice-Hall, Inc., p. 312, MR 0276251
^ Hoffman & Kunze (1971), p. 317.
^ ^a ^b Naylor, Arch W.; Sell George R. (1982). Linear Operator Theory in Engineering and Sciences. New York: Springer. ISBN 978-0-387-95001-3. Archived from the original on 2021-06-26. Retrieved 2021-06-26.
^ Andô, Tsuyoshi (1963). "Note on invariant subspaces of a compact normal operator". Archiv der Mathematik. 14: 337–340. doi:10.1007/BF01234964. S2CID 124945750.
^ Garrett, Paul (2005). "Operators on Hilbert spaces" (PDF). Archived (PDF) from the original on 2011-09-18. Retrieved 2011-07-01.
^ Weidmann, Lineare Operatoren in Hilberträumen, Chapter 4, Section 3
^ Alexander Frei, Spectral Measures, Mathematics Stack Exchange, Existence Archived 2021-06-26 at the Wayback Machine, Uniqueness Archived 2021-06-26 at the Wayback Machine
^ John B. Conway, A Course in Functional Analysis, Second Edition, Chapter X, Section §4

[3] In contrast, for the important class of Creation and annihilation operators of, e.g., quantum field theory, they don't commute

[1] Hoffman, Kenneth; Kunze, Ray (1971), Linear algebra (2nd ed.), Englewood Cliffs, N.J.: Prentice-Hall, Inc., p. 312, MR 0276251

[FOOTNOTEHoffmanKunze1971317-2] Hoffman & Kunze (1971), p. 317.

[Naylor-4] Naylor, Arch W.; Sell George R. (1982). Linear Operator Theory in Engineering and Sciences. New York: Springer. ISBN 978-0-387-95001-3. Archived from the original on 2021-06-26. Retrieved 2021-06-26.

[5] Andô, Tsuyoshi (1963). "Note on invariant subspaces of a compact normal operator". Archiv der Mathematik. 14: 337–340. doi:10.1007/BF01234964. S2CID 124945750.

[Garrett-6] Garrett, Paul (2005). "Operators on Hilbert spaces" (PDF). Archived (PDF) from the original on 2011-09-18. Retrieved 2011-07-01.

[7] Weidmann, Lineare Operatoren in Hilberträumen, Chapter 4, Section 3

[Frei-8] Alexander Frei, Spectral Measures, Mathematics Stack Exchange, Existence Archived 2021-06-26 at the Wayback Machine, Uniqueness Archived 2021-06-26 at the Wayback Machine

[Conway-9] John B. Conway, A Course in Functional Analysis, Second Edition, Chapter X, Section §4

[1]

[2]

[note 1]

[3]

[4]

[5]

[6]

[7]

[8]

v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F