# Dictionary Definition

operator

### Noun

1 (mathematics) a symbol that represents a
function from functions to functions; "the integral operator"

2 an agent that operates some apparatus or
machine; "the operator of the switchboard" [syn: manipulator]

3 someone who owns or operates a business; "who
is the operator of this franchise?"

4 a shrewd or unscrupulous person who knows how
to circumvent difficulties [syn: hustler, wheeler
dealer]

5 a speculator who trades aggressively on stock
or commodity markets

# User Contributed Dictionary

## English

### Noun

- One who operates.
- A telecommunications facilitator whose job is to establish temporary network connections.
- A function or other mapping that carries variables defined on a domain into another variable or set of variables in a defined range.
- Another name for Chinese whispers.
- A person who is adept at making deals or getting results, especially one who uses questionable methods.
- A member of a military Special Operations unit.

#### Translations

one who operates

- Croatian: operater
- Danish: operatør
- Finnish: operaattori, käyttäjä, koneenkäyttäjä, kuljettaja
- German: Betreiber
- Japanese: (, sōsasha), (operētā)

telecommunications operator

- Croatian: operater
- Danish: telefonist(inde)
- Finnish: operaattori, teleoperaattori
- German: Telefonist , Telefonistin , Vermittlung
- Japanese: (, kōkanshu)

mathematical operator

- Croatian: operator
- Danish: operator
- Finnish: operaattori
- French: opérateur
- German: Operator , Rechenzeichen
- Italian: operatore
- Japanese: (, enzanshi)

Chinese whispers

See: Chinese
whispers

## Croatian

### Noun

hr-noun m## Kurdish

### Noun

- surgeon (doctor who performs surgery)

# Extensive Definition

In mathematics, an operator is
a function
which operates on (or modifies) another function. Often, an
"operator" is a function which acts on functions to produce other
functions (the sense in which Oliver
Heaviside used the term); or it may be a generalization of such
a function, as in linear
algebra, where some of the terminology reflects the origin of
the subject in operations on the functions which are solutions of
differential
equations. An operator can perform a function on any number of
operands (inputs) though most often there is only one
operand.

An operator might also be called an operation,
but the point of view is different. For instance, one can say "the
operation of addition" (but not the "operator of addition") when
focusing on the operands and result. One says "addition operator"
when focusing on the process of addition, or from the more abstract
viewpoint, the function +: S×S → S.

## Notation

An operator name or operator symbol is a notation
which denotes a particular operator. When there is no danger of
confusion, an operator name or operator symbol may be referred to
more briefly as an "operator". Strictly speaking, however, the
operator is a mathematical object and not the syntactic entity
which denotes it. The reason for identifying it with its notation
is that there are some operators which have come to have standard
notations.

## Simple examples of operators

In linear
algebra an "operator" is a linear
operator. In analysis
an "operator" may be a differential
operator, to perform ordinary differentiation, or an integral
operator, to perform ordinary integration.

One example of a differential operator is the
derivative
itself. The corresponding operator name D, when placed before a
differentiable function f, indicates that the function is to be
differentiated
with respect to the variable.

## Operators versus functions

The word operator can in principle be applied to
any function. However, in practice it is most often applied to
functions which operate on mathematical entities of higher complexity
than real
numbers, such as vectors,
random
variables, or mathematical
expressions. The differential
and integral
operators, for example, have domains
and codomains whose
elements
are mathematical expressions of indefinite complexity. In contrast,
functions with vector-valued domains but scalar ranges are called
functionals
and forms.

In general, if either the domain or codomain (or
both) of a function contains elements significantly more complex
than real numbers, that function is referred to as an operator.
Conversely, if neither the domain nor the codomain of a function
contain elements more complicated than real numbers, that function
is likely to be referred to simply as a function. Trigonometric
functions such as cosine are examples of the latter case.

Additionally, when functions are used so often
that they have evolved faster or easier notations than the generic
F(x,y,z,...) form, the resulting special forms are also called
operators. Examples include infix
operators such as addition "+" and division "/", and postfix
operators such as factorial "!". This usage is unrelated to the
complexity of the entities involved.

## Influences from other disciplines

Concepts from other disciplines, including in
physics and to a lesser
degree computer
science, have influenced the ways in which operators are
perceived and used.

### Physics

The mutual influence between physics and
mathematics regarding the concept of operators has been long-term,
beginning in the early 1900s, and profound in both directions.
Quantum
mechanics in particular was forced to move from classical
measurement strategies involving only simple numeric values to the
use of operators which transformed and manipulated far less
intuitive entities. These included vectors
in both real space and in generalizations of real space called
Hilbert
spaces, spinors, and
various forms of matrices.
The great physicist P.A.M.
Dirac captured the importance of the relationship between
quantum physics and mathematics by saying "Physical laws should
have mathematical beauty and simplicity."

## Examples of mathematical operators

This section concentrates on illustrating the
expressive power of the operator concept in mathematics. Please
refer to individual topics pages for further details.

### Linear operators

The most common kind of operator encountered are
linear operators. In talking about linear operators, the operator
is signified generally by the letters T or L. Linear operators are
those which satisfy the following conditions; take the general
operator T, the function acted on under the operator T, written as
f(x), and the constant a:

- T(f(x)+g(x)) = T(f(x))+T(g(x))
- T(af(x)) = aT(f(x))

Many operators are linear. For example, the
differential operator and Laplacian operator, which we will see
later.

Linear operators are also known as linear
transformations or linear mappings. Many other operators one
encounters in mathematics are linear, and linear operators are the
most easily studied (Compare with nonlinearity).

Such an example of a linear transformation
between vectors in R2 is reflection: given a vector x = (x1,
x2)

- Q(x1, x2) = (−x1, x2)

We can also make sense of linear operators
between generalisations of finite-dimensional vector spaces. For
example, there is a large body of work dealing with linear
operators on
Hilbert spaces and on
Banach spaces. See also operator
algebra.

### Operators in probability theory

Operators are also involved in probability
theory, such as expectation,
variance, covariance, factorials, etc.

### Operators in calculus

Calculus is,
essentially, the study of two particular operators: the differential
operator D = d/dt, and the indefinite
integral operator \int_0^t. These operators are linear, as are
many of the operators constructed from them. In more advanced parts
of mathematics, these operators are studied as a part of functional
analysis.

#### The differential operator

The differential
operator is an operator which is fundamentally used in calculus
to denote the action of taking a derivative. Common notations are
dy/dx, and y'(x) to denote the derivative of y(x). Here, however,
we will use the notation which is closest to the operator notation
we have been using; that is, using Df to represent the action of
taking the derivative of f.

#### Integral operators

Given that integration is an operator as well
(inverse of differentiation), we have some important operators we
can write in terms of integration.

##### Convolution

The convolution *\, is a mapping from two
functions f(t) and g(t) to another function, defined by an integral
as follows:

- (f * g)(t) = \int_0^t f(\tau) g(t - \tau) \,d\tau.

##### 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 (spatial) domain to a
function on another (frequency) domain, in a way which is
effectively invertible. Nothing
significant is lost, because there is an inverse transform
operator. In the simple case of periodic
functions, this result is based on the theorem that any
continuous periodic function can be represented as the sum of a
series of sine waves and
cosine waves:

- f(t) = + \sum_^

When dealing with general function R → C, the
transform takes on an integral form:

- f(t) = \int_^.

##### Laplacian 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:

- F(s) = (\mathcalf)(s) =\int_0^\infty e^ f(t)\,dt.

### Fundamental operators on scalar and vector fields

Three operators are key to vector
calculus:

- ∇, known as gradient, assigns a vector at every point in a scalar field which points in the direction of greatest change of that field.
- Divergence is an operator which measures a vector field's tendency to originate from or converge upon a given point.
- Curl is a vector operator which shows a vector field's tendency to rotate about a point.

## Relation to type theory

In type theory,
an operator itself is a function, but has an attached type
indicating the correct operand, and the kind of function returned.
Functions can therefore conversely be considered operators, for
which we forget some of the type baggage, leaving just labels for
the domain and codomain.

## Operators in physics

In physics, an operator often takes
on a more specialized meaning than in mathematics. Operators as
observables are a key
part of the theory of quantum
mechanics. In that context operator often means a linear
transformation from a Hilbert
space to another, or (more abstractly) an element of a C*-algebra.

## Operators in computer programming languages

In general, the term 'operator' in computer
programming
languages has the same meaning as in mathematics. This is
particularly true in functional
programming languages, where an operator is also a
function.

### Operators as primitives

However, most programming languages distinguish
between operators and functions in that operators are a special
primitive part of the language, both syntactically and in terms of
functionality. For example, most languages provide a '+'
(addition) operator, which adds two numbers without making a
function call.

In many languages, this behaviour is totally
different from that of a function call. For example, in C
(and many derivatives such as
Java), the arithmetic operators can act on any numeric data type,
while functions are only allowed to act on a single explicit type.
However in C++
the distinction is blurred, since Operator
overloading allows operators to be defined as functions, albeit
only for data types that are not built-in.

Other languages (primarily older ones) do not
have functions which return values at all. However, they often
still have operators which do return values, widening the
distinction between operators and functions.

### Non-mathematical operators

Programming languages often feature
non-mathematical operators. These may include operators which
reference or dereference pointers,
which access array
elements, or get the size
of a data type. They may also include compound operators such
as "+=", which increments a variable by a given value.

### Operators in assembly language

In assembly
language programming, the term "operator" may refer to the
opcode of a given
instruction. This is very similar to the primitive concept of an
operator in a higher-level language.

## See also

operator in Catalan: Operador matemàtic

operator in Czech: Operátor

operator in German: Operator (Mathematik)

operator in Spanish: Operador

operator in French: Opérateur
(mathématiques)

operator in Italian: Operatore

operator in Hebrew: אופרטור

operator in Hungarian: Műveleti jel

operator in Dutch: Operator (wiskunde)

operator in Japanese: 作用素

operator in Portuguese: Operador

operator in Russian: Оператор (математика)

operator in Slovenian: operator
(matematika)

operator in Swedish: Operator

operator in Vietnamese: Toán tử

# Synonyms, Antonyms and Related Words

Machiavellian, PBX
operator, activist,
actor, administrator, adventurer, agent, architect, author, ball of fire, beaver, behind-the-scenes
operator, big operator, big wheel, big-time operator, bustler, busy bee, central, coconspirator, conductor, conniver, conspirator, conspirer, counterplotter, creator, director, doer, driver, eager beaver, eminence
grise, engineer,
enthusiast, executant, executor, executrix, exploiter, fabricator, faker, finagler, fraud, functionary, gamesman, go-getter, gray
eminence, gunslinger,
handler, human dynamo,
hustler, intrigant, intriguer, kingmaker, lame duck, live
wire, logroller, long
distance, machinator,
maker, man of action, man
of deeds, manager,
maneuverer, manipulator, margin
purchaser, medium,
militant, mover, new broom, operant, operative, operative surgeon,
opportunist,
performer, perpetrator, pilot, plotter, plunger, political activist,
pork-barrel politician, powerhouse, practitioner, prime mover,
producer, runner, sawbones, scalper, schemer, smart operator,
smoothie, speculator, stag, steersman, strategist, subject, superintendent, supervisor, surgeon, switchboard operator,
take-charge guy, telephone operator, telephonist, wheeler-dealer,
winner, wire-puller, wise
guy, worker