In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, a normal operator on a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
is a
continuous 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 mapping V \to W between two vector spaces that pr ...
that
commutes with its
Hermitian adjoint
In mathematics, specifically in operator theory, each linear operator A on an inner product space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where \l ...
, that is:
.
Normal operators are important because the
spectral theorem
In 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 because computations involvin ...
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.
Non-trivial examples include rotations, reflections, and the Fourier operator.
Unitary operators generalize unitar ...
s:
*
Hermitian operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
s (i.e., self-adjoint operators):
*
skew-Hermitian operators:
*
positive operators:
for some
(so ''N'' is self-adjoint).
A
normal matrix
In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
:A \text \iff A^*A = AA^* .
The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to nor ...
is the matrix expression of a normal operator on the Hilbert space
.
Properties
Normal operators are characterized by the
spectral theorem
In 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 because computations involvin ...
. A
compact normal operator (in particular, a normal operator on a
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
inner product space
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, ofte ...
) 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
[In contrast, for the important class of ]Creation and annihilation operators
Creation operators and annihilation operators are Operator (mathematics), mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilatio ...
of, e.g., quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, they don't commute
If
is a bounded 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 measure
In mathematics, particularly in functional analysis, a projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-va ...
s. 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 and are normal operators and if is a bounded linear operator such that then .
The operator norm of a normal operator equals its numerical radius and ]spectral radius
''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma ...
.
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, ofte ...
on the space of endomorphisms of ''H'', it is enough to show that tr(''XX*'') = 0. 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. 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 (or two-sided shift) acting on , 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 of an involutive algebra is said to be normal if .
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 contin ...
.
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
:
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[Weidmann, Lineare Operatoren in Hilberträumen, Chapter 4, Section 3]
:
with
:
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 operators
* Subnormal operators
See also
*
*
Notes
References
{{Functional analysis
Linear operators
Operator theory