Operator
In
mathematics, an
operator generally is a symbolism to show a certain mapping, usually from one or more given functions to another (between function spaces), however, operators can refer to mappings between vector spaces in general as well.
Operators generally transform functions into other functions. We also say an operator
maps a function to another. In some literature, they are designated by showing a small uphat over the operator name. In certain circumstances, they are written
unlike functions, when an operator has a single
argument or
operand, for example, if the operator name is called Q and operates on a function
f, we write Q
f and not usually Q(
f), however this latter notation may be used for clarity if there is a product for instance, eg. Q(
fg). Throughout this article we will use Q to denote a general operator, and
x_{i} to denote the
ith argument.
Notations for operations on functions may also be notated as the following. If f(x) is a function of x and Q is the general operator we can write Q acting on f as:
 (Qf)(x)
also.
Functions can be considered operators, but are generally thought of differently conceptually. "Numbers" can be considered functions too, if f(x)=x^{0}, this represents the number 1. Similarly after multiplication by a constant, any number can be defined. When an operator takes some numbers as arguments, we can consistenly regard the operator as still transforming functions, since we have seen that numbers can be considered as functions.
Describing operators
Operators are described usually by the number of operands:
 monodic, or unary: one argument
 dyadic, or binary: two arguments
 triadic, or ternary: three arguments
and so on.
There are three major ways of writing operators and their arguments. These are
 prefix: where the operator name comes first and the arguments follow, for example:
Q(x_{1}, x_{2},...,x_{n}).
In prefix notation, the brackets are sometimes omitted if it is known that Q is a nary operator.
 postfix: where the operator name comes last and the arguments precede, for example:
(x_{1}, x_{2},...,x_{n}) Q
In postfix notation, the brackets are sometimes omitted if it is known that Q is a nary operator.
 infix: where the operator name comes between the arguments. This is not commonly used for operators taking greater than 2 arguments, ie binary operators. Trivially for an operator taking 1 argument, writing infix is equivalent to writing prefix. Infix style is written, for example:
x_{1} Q x_{2}
A key concept is the concept of the linear operator. Linear operators are those which satisfy the following conditions; take the general operator Q, the function acted on under the operator Q, written as f(x), and the constant a:


Such examples of linear operators are the differential operator and Laplacian operator, which we will see later.
Linear operators with respect to mappings between vector spaces are known more commonly as linear transformations or linear mappings.
Such an example of a linear transformation between vectors in R^{2} is reflection, given a vector x=(x_{1}, x_{2})
 Q(x_{1}, x_{2})=(x_{1}, x_{2})
Additive operators
An additive operator, in abstract algebra, may satisfy the
commutative and
associative laws. If there is also a predefined multiplicative operator then the operator must satisfy the
distributive law.
A multiplcative operator, in abstract algebra, may satisfy the
associative law. If there is also a predefined
multiplicative operator the operator must satisfy the
distributive law.
Arithmetic operators are binary operators that perform simple transformations that many would find familiar. It is not obvious, but addition, subtraction, etc. are in fact operators. Many of these standard arithmetic operators use symbols to denote what operations are being performed.
Addition is written using the symbol +. It transforms two numbers
x_{1} and
x_{2} into their
sum. For example:
 3 + 5 = 8
It is written most commonly as
x_{1}+
x_{2}.
In prefix notation, it may be written as +
x_{1} x_{2}, or +(
x_{1},
x_{1}), or even with + changed to a word, such as
 plus x_{1} x_{1}.
Addition follows the
field axioms.
Subtraction is written using the symbol . It transforms two numbers
x_{1} and
x_{2} into their
difference. For example:
 11  4 = 7
It is written most commonly as
x_{1}
x_{2}.
In prefix notation, it may be written as 
x_{1} x_{2}, or (
x_{1},
x_{1}), or even with  changed to a word, such as
 sub x_{1} x_{1}.
Subtraction is equivalent to addition. The identity is that:
 x_{1}  x_{2} ≡ x_{1} + (x_{2})
where  as a unary operator represents negation (see next section)
Negation is written also using the symbol , however, it is only a unary operator. Given a number α, we denote the transformation of α to its additive inverse by α. The additive inverse of a number k is an element k', such that k+k'=0.
Multiplication is written using the symbol ×. In certain circumstances, the operator symbol is
omitted usually when the arguments to × are variable quantities, eg
xy. Less commonly when representing the product of two numbers, they are placed in brackets and placed adjacently, eg. (2)(3)=6. Less commoner still, a small dot is used infix instead of ×, eg 2·3=6
Multiplication transforms two numbers x_{1} and x_{2} into their product. For example:
 6 × 2 = 12
It is written most commonly as
x_{1}x_{2}.
In prefix notation (using ×) it may be written as ×
x_{1} x_{2}, or ×(
x_{1},
x_{1}), or even with × changed to a word, such as
 mul x_{1} x_{1}.
Multiplication is equivalent to repeated addition. The identity is that:
 x_{1}x_{2} ≡ x_{1} + x_{1} + ...( x_{}2 times)...+x_{1}
Division
Division is written using the symbol /. Like multiplication, there are several ways to denote this, other than using /. If there is not much room on a page, or when typeset on a single line, the two arguments are written infix, eg 3 / 4, or
x_{1}/
x_{2}. If there is room on a page, the two arguments are usually written atop each other and a line seperating them, eg:

Division transforms two numbers
x_{1} and
x_{2} into their
quotient. For example:
 8 / 2 = 4
It is written most commonly as
x_{1}/
x_{2}.
In prefix notation (using /) it may be written as /
x_{1} x_{2}, or /(
x_{1},
x_{1}), or even with / changed to a word, such as
 div x_{1} x_{1}.
Division is equivalent to repeated subtraction.
x_{2} is subtracted from
x_{1} until there only is a positive
remainder left. When, after application of this algorithm, there is zero remainder, we call the amount of subtractions we have performed the
quotient. If not, we can write the result of this operation as either a
fraction or as a decimal number (See those articles for further information).
Exponentiation is most generally not written using a symbol, but with the second argument written as a
superscript, for example . In certain circumstances, as in representing this operation in
programming, the symbol ^ is used.
Exponentiation transforms two numbers x_{1} and x_{2} into their repeated product. For example:
 6^{2} = 6 × 6 = 36
In prefix notation (using ^) it may be written as ^
x_{1} x_{2}, or ^(
x_{1},
x_{1}), or even with ^ changed to a word, such as
 pow x_{1} x_{1}.
Exponentiation is equivalent to repeated multiplication. The identity is that:
 x_{1}x_{2} ≡ x_{1}x_{1} ...( x_{}2 times)...x_{1}.
Generalizations
With addition as a basic operator, we can see that the extension of multiplication is an iterated addition. Exponentiation is an iterated multiplication.
We have a notation we can use to show an extension of this generality.
hyper_{4} is the operator that is defined as repeated exponentiation. If we define Q to be a binary operator, Q x_{1} x_{2} =

where
x_{1} is exponentiated
x_{2} times.
This operation has several names, viz., tetration, superpower, superdegree, or powerlog. The two most common notations for this is Knuth's uparrow notation as x_{1} ↑↑ x_{2}, and hyper^{4}. Less commonly seen, though somewhat more convenient notations are x_{1}^{(4)}x_{2}.
Only the hyper^{4}, definition is technically a different operator, since this operation can be reduced to exponentiated exponentiation (iterated exponentiation). If we again define Q x_{1} x_{2} = :
as before, then we define x_{1} ↑↑↑ x_{2} or hyper_{5}(x_{1},x_{2}) as being:

nesting
x_{2} times.
Further generalizations can be taken similarly ad infinitum.
We can generalize back addition, multiplication, and exponentiation in terms using the notations we have just described, ie.,
 hyper_{1} x_{1} x_{2} = x_{1} + x_{2}
 hyper_{2} x_{1} x_{2} = x_{1}x_{2}
 hyper_{3} x_{1} x_{2} =
Similar behaviors
Some operators aforementioned can also have other behaviours than what was previously described. In programming terms, this is known as overloading, however in mathematics the meaning of an operation is understood from the context by generally the subject matter or what the arguments are. Some examples follow.
The concept of the addition operator + has been extended to cover addition of sets,
vectorss and
matrices.
Multiplication of a vector by a particular matrix is a
unary operator or
transformation. We can regard the multiplication of the matrix to be an operator (see below).
We have seen that an operator transforms one function to another. So, we can define + to be the sum of the two functions, x_{1} and x_{2}. resulting in another function. For example, if we define Q this way;
 Q (x^{2}+3x) (5x^{2}+9) = 6x^{2}+3x+9
We can define multiplication, division, etc. in the same way.
Additionally, we have some other operators which we can define on functions. One such fundamental operator is that of function composition. Given two functions
x_{1}=f(t) and
x_{2}=g(t), define the operator Q:
 Q x_{1} x_{2} = f(g(t))
We write this operator infix using a small circle. So, with the same definitions as before,
 f(g(t))=(fog)(x)=x_{1} o x_{2}
Probability theory
Operators are also involved in probability theory.
Such operators as expectation, variance, covariance, factorials, et al.
Factorials are essential to the combination and permutation functions of probability and combinatorics, and are also the most commonly known postfix operator, being denoted by a ! placed after the number it expands. Its expansion follows the pattern,
 x! = 1 * 2 * ... * (x1) * x
Calculus and operators
Calculus is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator in great clarity. This key operator we study in Calculus is the
differential operator.
The differential operator is the symbolism used in Calculus to denote the action of taking a derivative. Common notations are such
d/dx, y'(x) to denote the derivative of y(x). However here we will use the notation that is closest to the operator notation we have been using, that is, using D
f to represent the action of taking the derivative of f.
If f is a function of n variables t_{1},t_{1},...,t_{n}, we write

to represent the action of differentiating f with respect to t_{i}.
If we differentiate f, k times, we write
How does the differential operator exemplify the idea of the operator?
Consider the function f=x^{2}. Elementary calculus tells us that D f = 2x, futhermore if f=x^{α}, D f = αx^{α1}. So we see clearly that the differential operator maps, or transforms, functions of the form x^{α} to functions αx^{α1}.
The act of integration is also equivalent somewhat to taking the derivative backwards. So, in a sense it is differentiating 1 times, so we have integration in terms of the differential operator:

It is clear that integration thus is equivalent to differentiation, so integration acts just like an operator as well  mapping functions to functions.
Given that integration is an operator as well, we have some important operators we can write in terms of integration.
The convolution of two functions is a mapping from two functions to one other, defined by an integral as follows:
If x_{1}=f(t) and x_{2}=g(t), define the operator Q such that;

which we write as .
The Fourier transform is another integral operator, and is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few.
It is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves:
f(x) = ∑ A_{1} sin ω + A_{2} sin ω/2 + A_{3} sin ω/3 + ...
See Fourier transform for more information.
The Laplacian transform is another integral operator and is involved in simplifying the process of solving differential equations.
Given f=f(s), it is defined by:

See Laplace transform for more information.
In physics, an operator often takes on a more specialized meaning that in mathematics. It often means a
linear transformation from a
Hilbert space to another or an element of a
C* algebra. See
Operator (physics).
Operators are also a key part of the theory of quantum mechanics.
Programming languages, being that computers are mathematical devices, have a set of operators that perform various functions.
The arithmetic operators are the same as the mathematical ones while the bit (binary digit) operations deal with the binary number system. The logical operators determine boolean values. The string operators manipulate strings of text and there are operators which allocate segments of memory for use.
Operators are also terms for some functionality in programming languages. Consider the C programming language syntax for pointers, using the operators * and &. sizeof is sometimes considered an operator, and in C++, new and delete are also operators.
In object oriented languages, like C++, you can define your own uses for operators.
Operators in telecommunications, who are usually women, aid
telephone users in various ways including long distance calling, directory assistance and telephone
repair. As
technology advances, human operators are becoming more often replaced by a computerized system, and the idiom is turning over to mean a
secret agent.