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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, a normal operator on a complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''H'' is a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operators are important because the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
holds for them. The class of normal operators is well understood. Examples of normal operators are *
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
s: ''N*'' = ''N−1'' *
Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to i ...
s (i.e., self-adjoint operators): ''N*'' = ''N'' *
Skew-Hermitian __NOTOC__ In linear algebra, a square matrix with Complex number, complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisf ...
operators: ''N*'' = −''N'' *
positive operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \l ...
s: ''N'' = ''MM*'' for some ''M'' (so ''N'' is self-adjoint). A
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
is the matrix expression of a normal operator on the Hilbert space C''n''.


Properties

Normal operators are characterized by the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
. 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. * T is normal. * T^* is normal. * \, T x\, = \, T^* x\, for all x (use \, Tx\, ^2 = \langle T^* Tx, x \rangle = \langle T T^*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 + i T_2 with T_1 := \frac and i\,T_2 := \frac, then T_1 T_2 = T_2 T_1.In contrast, for the important class of
Creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
of, e.g.,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, they don't commute
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 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 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. 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 = A N_2, then N_1^* A = A N_2^*. The operator norm of a normal operator equals its numerical radius and
spectral radius In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
. A normal operator coincides with its Aluthge transform.


Properties in finite-dimensional case

If a normal operator ''T'' on a ''finite-dimensional'' real 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 ''PV'' be the orthogonal projection onto ''V''. Then the orthogonal projection onto ''V'' is 1''H''−''PV''. The fact that ''T'' stabilizes ''V'' can be expressed as (1''H''−''PV'')''TPV'' = 0, or ''TPV'' = ''PVTPV''. The goal is to show that ''PVT''(1''H''−''PV'') = 0. Let ''X'' = ''PVT''(1''H''−''PV''). Since (''A'', ''B'') ↦ tr(''AB*'') is an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on the space of endomorphisms of ''H'', it is enough to show that tr(''XX*'') = 0. First we note that :\begin XX^* &= P_V T(\boldsymbol_H - P_V)^2 T^* P_V \\ &= P_V T(\boldsymbol_H - P_V) T^* P_V \\ &= P_V T T^* P_V - P_V T P_V T^* P_V. \end Now using properties of the trace and of orthogonal projections we have: :\begin \operatorname(XX^*) &= \operatorname \left ( P_VTT^*P_V - P_VTP_VT^*P_V \right ) \\ &= \operatorname(P_VTT^*P_V) - \operatorname(P_VTP_VT^*P_V) \\ &= \operatorname(P_V^2TT^*) - \operatorname(P_V^2TP_VT^*) \\ &= \operatorname(P_VTT^*) - \operatorname(P_VTP_VT^*) \\ &= \operatorname(P_VTT^*) - \operatorname(TP_VT^*) && \text T \text V\\ &= \operatorname(P_VTT^*) - \operatorname(P_VT^*T) \\ &= \operatorname(P_V(TT^*-T^*T)) \\ &= 0. \end 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. However, for bounded normal operators, the orthogonal complement to a stable subspace may not be stable. It follows that the Hilbert space cannot in general be spanned by eigenvectors of a normal operator. Consider, for example, the
bilateral shift In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift o ...
(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 In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
.


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 whichWeidmann, Lineare Operatoren in Hilberträumen, Chapter 4, Section 3 :\, Nx\, =\, N^*x\, \qquad with :\mathcal(N)=\mathcal(N^*). The spectral theorem still holds for unbounded (normal) operators. The proofs work by reduction to bounded (normal) operators.Alexander Frei, Spectral Measures, Mathematics Stack Exchange
Existence

Uniqueness
John B. Conway, A Course in Functional Analysis, Second Edition, Chapter X, Section §4


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) * Hyponormal operators * Normaloids * Paranormal operators *
Quasinormal operator In operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a normal operator. Every quasinormal operator is a subnormal operator. Every quasinormal operator on a finite-dimensional Hilbert ...
s *
Subnormal operator In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators wi ...
s


See also

* *


Notes


References

{{Hilbert space Operator theory