# Normal operator

Last updated

In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : HH that commutes with its hermitian adjoint N*, that is: NN* = N*N. [1]

## Contents

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.

## Properties

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. [2]

Let ${\displaystyle T}$ be a bounded operator. The following are equivalent.

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

If ${\displaystyle N}$ is a normal operator, then ${\displaystyle N}$ and ${\displaystyle N^{*}}$ have the same kernel and the same range. Consequently, the range of ${\displaystyle N}$ is dense if and only if ${\displaystyle N}$ 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 ${\displaystyle N^{k}}$ coincides with that of ${\displaystyle N}$ for any ${\displaystyle k.}$ Every generalized eigenvalue of a normal operator is thus genuine. ${\displaystyle \lambda }$ is an eigenvalue of a normal operator ${\displaystyle N}$ if and only if its complex conjugate ${\displaystyle {\overline {\lambda }}}$ is an eigenvalue of ${\displaystyle N^{*}.}$ 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 ${\displaystyle N_{1}}$ and ${\displaystyle N_{2}}$ are normal operators and if ${\displaystyle A}$ is a bounded linear operator such that ${\displaystyle N_{1}A=AN_{2},}$ then ${\displaystyle N_{1}^{*}A=AN_{2}^{*}}$.

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 PV be the orthogonal projection onto V. Then the orthogonal projection onto V is 1HPV. The fact that T stabilizes V can be expressed as (1HPV)TPV = 0, or TPV = PVTPV. The goal is to show that PVT(1HPV) = 0.

Let X = PVT(1HPV). 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 we note that

${\displaystyle XX^{*}=P_{V}T({\boldsymbol {1}}_{H}-P_{V})^{2}T^{*}P_{V}=P_{V}T({\boldsymbol {1}}_{H}-P_{V})T^{*}P_{V}=P_{V}TT^{*}P_{V}-P_{V}TP_{V}T^{*}P_{V}}$.

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

{\displaystyle {\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 ${\displaystyle \ell ^{2}}$, 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

${\displaystyle N^{*}N=NN^{*}.}$

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]

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

with

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

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

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

## Related Research Articles

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*:

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product is a linear map A that is its own adjoint: for all vectors v and w. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

In mathematics, a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces.

In functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of . Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces.

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical logic with the facts concerning the measurement of complementary variables in quantum mechanics, such as position and momentum.

In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature.

Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

The spectrum of a linear operator that operates on a Banach space consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:

In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.

The Born rule is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.

In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

In operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a normal operator.

In operator theory, a bounded operator T: XY between normed vector spaces X and Y is said to be a contraction if its operator norm ||T|| ≤ 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.

In mathematics, a symmetrizable compact operator is a compact operator on a Hilbert space that can be composed with a positive operator with trivial kernel to produce a self-adjoint operator. Such operators arose naturally in the work on integral operators of Hilbert, Korn, Lichtenstein and Marty required to solve elliptic boundary value problems on bounded domains in Euclidean space. Between the late 1940s and early 1960s the techniques, previously developed as part of classical potential theory, were abstracted within operator theory by various mathematicians, including M. G. Krein, William T. Reid, Peter Lax and Jean Dieudonné. Fredholm theory already implies that any element of the spectrum is an eigenvalue. The main results assert that the spectral theory of these operators is similar to that of compact self-adjoint operators: any spectral value is real; they form a sequence tending to zero; any generalized eigenvector is an eigenvector; and the eigenvectors span a dense subspace of the Hilbert space.

In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.

This is a glossary for the terminology in a mathematical field of functional analysis.

## References

1. Hoffman, Kenneth; Kunze, Ray (1971), Linear algebra (2nd ed.), Englewood Cliffs, N.J.: Prentice-Hall, Inc., p. 312, MR   0276251
2. Hoffman & Kunze (1971), p. 317.
3. 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.
4. Andô, Tsuyoshi (1963). "Note on invariant subspaces of a compact normal operator". Archiv der Mathematik. 14: 337–340. doi:10.1007/BF01234964. S2CID   124945750.
5. Garrett, Paul (2005). "Operators on Hilbert spaces" (PDF). Archived (PDF) from the original on 2011-09-18. Retrieved 2011-07-01.
6. Weidmann, Lineare Operatoren in Hilberträumen, Chapter 4, Section 3
7. 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
8. John B. Conway, A Course in Functional Analysis, Second Edition, Chapter X, Section §4