Operator
Encyclopedia
In basic mathematics, an operator is a symbol or function representing a mathematical operation
.
In terms of vector spaces, an operator is a mapping
from one vector space
or module
to another. Operators are of critical importance to both linear algebra
and functional analysis
, and they find application in many other fields of pure and applied mathematics. For example, in classical mechanics
, the derivative
is used ubiquitously, and in quantum mechanics
, observable
s are represented by linear operators. Important properties that various operators may exhibit include linearity, continuity, and boundedness
.
s. The mapping from U to V is called an operator. Let V be a vector space over the field K. We can define the structure of a vector space on the set of all operators from U to V:
for all A, B: U → V, for all x in U and for all α in K.
Additionally, operators from any vector space to itself form a unital associative algebra
:
with the identity mapping (usually denoted E, I or id) being the unit.
(for example,
), and they are equipped with norm
s. Then a linear operator from U to V is called bounded if there exists C > 0 such that
for all x in U.
It is trivial to show that bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of U and V:
.
In case of operators from U to itself it can be shown that
.
Any unital normed algebra with this property is called a Banach algebra
. It is possible to generalize spectral theory
to such algebras. C*-algebras, which are Banach algebras with some additional structure, play an important role in quantum mechanics
.
. Important applications of functionals are the theories of generalized function
s and calculus of variations
. Both are of great importance to theoretical physics.
for all x, y in U and for all α, β in K.
The importance of linear operators is partially because they are morphism
s between vector spaces.
In finite-dimensional case linear operators can be represented by matrices
in the following way. Let
be a field, and 
and 
be finite-dimensional vector spaces over 
. Let us select a basis 
in 
and 
in 
. Then let 
be an arbitrary vector in 
(assuming Einstein convention), and 
be a linear operator. Then
Then
is the matrix of the operator 
in fixed bases. It is easy to see that 
does not depend on the choice of 
, and that 
iff 
. Thus in fixed bases n-by-m matrices are in bijective correspondence to linear operators from 
to 
.
The important concepts directly related to operators between finite-dimensional vector spaces are the ones of rank, determinant
, inverse operator, and eigenspace.
In infinite-dimensional case linear operators also play a great role. The concepts of rank and determinant cannot be extended to infinite-dimensional case, and although there are infinite matrices, they cease to be a useful tool. This is why a very different techniques are employed when studying linear operators (and operators in general) in infinite-dimensional case. They form a field of functional analysis
(named such because various classes of functions form interesting examples of infinite-dimensional vector spaces).
Bounded linear operators over Banach space
form a Banach algebra
in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra
that elegantly generalizes the theory of eigenspaces.
(particularly differential geometry), additional structures on vector space
s are sometimes studied. Operators that map such vector spaces to themselves bijectively are very useful in these studies, they naturally form group
s by composition.
For example, bijective operators preserving the structure of linear space are precisely invertible linear operators. They form general linear group
; notice that they do not form a vector space, e.g. both id and -id are invertible, but 0 is not.
Operators preserving euclidean metric on such a space form orthogonal group
, and operators that also preserve orientation of vector tuples form special orthogonal group, or the group of rotations.
, variance
, covariance
, factorial
s, etc.
, calculus
is the study of two linear operators: the differential operator
, and the indefinite integral operator
.
s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine wave
s and cosine waves:
Coefficients (a0, a1, b1, a2, b2, ...) are in fact an element of an infinite-dimensional vector space ℓ2
, and thus Fourier series is a linear operator.
When dealing with general function R → C, the transform takes on an integral
form:
Given f = f(s), it is defined by:
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
.
In terms of vector spaces, an operator is a mapping
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
from one vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
or module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
to another. Operators are of critical importance to both linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, and they find application in many other fields of pure and applied mathematics. For example, in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
is used ubiquitously, and in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s are represented by linear operators. Important properties that various operators may exhibit include linearity, continuity, and boundedness
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
.
Definitions
Let U, V be two vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s. The mapping from U to V is called an operator. Let V be a vector space over the field K. We can define the structure of a vector space on the set of all operators from U to V:
for all A, B: U → V, for all x in U and for all α in K.
Additionally, operators from any vector space to itself form a unital associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
:
with the identity mapping (usually denoted E, I or id) being the unit.
Bounded operators and operator norm
Let U and V be two vector spaces over the same ordered fieldOrdered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
(for example,
), and they are equipped with normNorm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
s. Then a linear operator from U to V is called bounded if there exists C > 0 such that
for all x in U.
It is trivial to show that bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of U and V:
.In case of operators from U to itself it can be shown that
.Any unital normed algebra with this property is called a Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
. It is possible to generalize spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
to such algebras. C*-algebras, which are Banach algebras with some additional structure, play an important role in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
Functionals
A functional is an operator that maps a vector space to its underlying fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. Important applications of functionals are the theories of generalized function
Generalized function
In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
s and calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. Both are of great importance to theoretical physics.
Linear operators
The most common kind of operator encountered are linear operators. Let U and V be vector spaces over a field K. Operator A: U → V is called linear iffor all x, y in U and for all α, β in K.
The importance of linear operators is partially because they are morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s between vector spaces.
In finite-dimensional case linear operators can be represented by matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
in the following way. Let
be a field, and and be finite-dimensional vector spaces over . Let us select a basis in and in . Then let be an arbitrary vector in (assuming Einstein convention), and be a linear operator. Then
-
.
Then
is the matrix of the operator in fixed bases. It is easy to see that does not depend on the choice of , and that iff . Thus in fixed bases n-by-m matrices are in bijective correspondence to linear operators from to .The important concepts directly related to operators between finite-dimensional vector spaces are the ones of rank, determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
, inverse operator, and eigenspace.
In infinite-dimensional case linear operators also play a great role. The concepts of rank and determinant cannot be extended to infinite-dimensional case, and although there are infinite matrices, they cease to be a useful tool. This is why a very different techniques are employed when studying linear operators (and operators in general) in infinite-dimensional case. They form a field of functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
(named such because various classes of functions form interesting examples of infinite-dimensional vector spaces).
Bounded linear operators over Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
form a Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra
Spectrum (functional analysis)
In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the...
that elegantly generalizes the theory of eigenspaces.
Geometry
In geometryGeometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
(particularly differential geometry), additional structures on vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s are sometimes studied. Operators that map such vector spaces to themselves bijectively are very useful in these studies, they naturally form group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s by composition.
For example, bijective operators preserving the structure of linear space are precisely invertible linear operators. They form general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
; notice that they do not form a vector space, e.g. both id and -id are invertible, but 0 is not.
Operators preserving euclidean metric on such a space form orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
, and operators that also preserve orientation of vector tuples form special orthogonal group, or the group of rotations.
Probability theory
Operators are also involved in probability theory, such as expectationExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
, variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
, covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...
, factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
s, etc.
Calculus
From the point of view of functional analysisFunctional 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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
is the study of two linear operators: the differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
, and the indefinite integral operatorVolterra operator
In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, represents the operation of indefinite integration, viewed as a bounded linear operator on the space L2 of complex-valued square integrable functions on the interval...
.Fourier series and Fourier transform
The Fourier transform is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively invertible. Nothing significant is lost, because there is an inverse transform operator. In the simple case of periodic functionPeriodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine wave
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
s and cosine waves:
Coefficients (a0, a1, b1, a2, b2, ...) are in fact an element of an infinite-dimensional vector space ℓ2
Sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers...
, and thus Fourier series is a linear operator.
When dealing with general function R → C, the transform takes on an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
form:
Laplace transform
The Laplace transform is another integral operator and is involved in simplifying the process of solving differential equations.Given f = f(s), it is defined by:
Fundamental operators on scalar and vector fields
Three operators are key to vector calculus:- Grad (gradientGradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
), (with operator symbol ∇) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change. - Div (divergenceDivergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
) is a vector operator that measures a vector field's divergence from or convergence towards a given point. - Curl is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point.
See also
- OperationOperation (mathematics)The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
- FunctionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
- Vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
- Dual spaceDual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
- Operator algebraOperator algebraIn functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
- Banach algebraBanach algebraIn mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
- List of operators
- Operator (physics)Operator (physics)In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
- Operator (programming)Operator (programming)Programming languages typically support a set of operators: operations which differ from the language's functions in calling syntax and/or argument passing mode. Common examples that differ by syntax are mathematical arithmetic operations, e.g...