Operator
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In mathematics, an operator is a function, usually of a special kind depending on the topic. For instance, in linear algebra an "operator" is a linear operator. In analysis an "operator" may be a differential operator, generalizing the ordinary derivative, or an integral operator, generalizing ordinary integration. Often, an "operator" is a function that 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 that are solutions of differential equations. One often thinks of an operator as dynamic, changing something into something else, whence the name; but this is only a way of thinking and not a formal definition.
An operator name or operator symbol is a notation that 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 that denotes it. The reason for identifying it with its notation is that there are some operators that have come to have standard notations.
An example of an operator, specifically 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.
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.
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[edit] Operators versus functions
The word operator can in principle be applied to any function. However, in practice it is most often applied to functions that 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 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 addition "+" and division "/", and postfix operators such as factorial "!". This usage is unrelated to the complexity of the entities involved.
[edit] 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.
[edit] 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 that 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 observing that "Physical laws should have mathematical beauty and simplicity."
[edit] Computer science
Computer science has contributed the concept of overloading to operators. An overloaded operator is one for which the same function notation is interpreted differently depending on the type signature of its arguments. Mathematically, an overloaded operator is itself an operator or "meta-operator" that maps from function labels and argument types into definite functions with well-specified domains and codomains. Since overloading is meaningful only if an operator-preserving homomorphism exists for the different types of arguments, it is most commonly used with argument types that correspond to mathematical fields—that is, types (sets) whose elements fully support addition, subtraction, multiplication, and division (except by zero). For example, the "+" operator can be overloaded to include addition of integers, real numbers, matrices, or any meaningful combination thereof. This also shows the dangers of overloading, since every possible combination of "+" and its argument types must be mapped into a definite function.
[edit] Describing operators
Operators are described usually by the number of operands:
- monadic or unary operators take one argument.
- dyadic or binary operators take two arguments.
- triadic or ternary/trinary/tertiary operators take three arguments.
The number of operands is also called the arity of the operator. If an operator has an arity given as n-ary (or n-adic), then it takes n arguments. In programming, other than functional programming, the -ary terms are more often used than the other variants. See arity for an extensive list of the -ary endings. The field of universal algebra also includes the study of operators and their arities.
[edit] Notations
There are five major systematic ways of writing operators and their arguments. These are
- prefix: where the operator name comes first and the arguments follow, for example:
-
- Q(x1, x2,...,xn).
- In prefix notation, the brackets are sometimes omitted if it is known that Q is an n-ary operator.
- postfix: where the operator name comes last and the arguments precede, for example:
-
- (x1, x2,...,xn) Q
- In postfix notation, the brackets are sometimes omitted if it is known that Q is an n-ary operator.
- infix: where the operator name comes between the arguments. This is not commonly used for operators other than binary operators. Infix style is written, for example, as
-
- x1 Q x2.
- juxtaposition, for example for multiplication of numbers, scalar multiplication, matrix multiplication, and function composition, and also for "multiplication" of a numerical value and a physical unit, and of two physical units, e.g. 3Nm.
- a superscript or subscript on the right or on the left; the main uses are selection (an index), such as a coordinate of a vector, and, in the case of a superscript on the right, for exponentiation of numbers and square matrices, and multiple function composition.
For operators on a single argument, prefix notation such as −7 is most common, but postfix such as 5! (factorial) or x* is also usual.
There are other notations commonly met. Writing exponents such as 28 is really a law unto itself, since it is postfix only as a unary operator applied to 2, but on a slant as binary operator. In some literature, a circumflex is written 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 Q and the operand a function f, we write Qf and not usually Q(f); this latter notation may however be used for clarity if there is a product — for instance, Q(fg). Later on we will use Q to denote a general operator, and xi to denote the i-th argument.
Notations for operators include 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.
Operators are often written in calligraphy to differentiate them from standard functions. For instance, the Fourier transform (an operator on functions) of f(t) (a function of t), which produces another function F(ω) (a function of ω), would be represented as 
When an operators take a function of one variable, f(t) to a function of a different variable F(ω)(as with the fourier transform) it can be useful to include the t to emphisize the functions do not have the same domain. For the case of the Fourier transform one might write 
[edit] 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.
[edit] 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.
[edit] Operators in probability theory
Operators are also involved in probability theory. Such operators as expectation, variance, covariance, factorials, etc.
[edit] Operators in calculus
Calculus is, essentially, the study of two particular operators: the differential operator D = d/dt, and the indefinite integral operator 
. 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.
[edit] 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 d/dx, and y ′(x) to denote the derivative of y(x). Here, however, 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.
[edit] 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.
[edit] Convolution
The convolution 
is a mapping from two functions f(t) and g(t) to another function, defined by an integral as follows:
[edit] 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 that 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:
When dealing with general function R → C, the transform takes on an integral form:
[edit] 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:
[edit] Fundamental operators on scalar and vector fields
Three main operators are key to vector calculus, the operator ∇, known as gradient, where at a certain point in a scalar field forms a vector which points in the direction of greatest change of that scalar field. In a vector field, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point. Curl, in a vector field, is a vector operator that shows a vector field's tendency to rotate about a point.
[edit] 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.
[edit] 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.
