In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. The most frequently met usage is a mapping between vector spaces; this kind of operator is distinguished by taking one vector and returning another. For example, consider an enlargement, say by a factor of √2; such as is required to take one size of paper to another. It can also be applied geometrically to vectors as operands.
In many important cases, operators transform functions into other functions. We also say an operator maps a function to another. The operator itself is a function, but has an attached type indicating the correct operand, and the kind of function returned. This extra data can be defined formally, using type theory; but in everyday usage saying operator flags its significance. Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the domain and codomain.
To begin with, the usage of operator in mathematics is subsumed in the usage of function: an operator can be taken to be some special kind of function. The word is generally used to call attention to some aspect of its nature as a function. Since there are several such aspects that are of interest, there is no completely consistent terminology. Common are these:
A single operator might conceivably qualify under all three of these. Other important ideas are:
These are abstract ideas from mathematics, and computer science. They may however also be encountered in quantum mechanics. There Dirac drew a clear distinction between q-number or operator quantities, and c-numbers which are conventional complex numbers. The manipulation of q-numbers from that point on became basic to theoretical physics.
Operators are described usually by the number of operands:
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, outside than functional programming, the -ary terms are more often used than the other variants. See arity for an extensive list of the -ary endings.
There are three major systematic ways of writing operators and their arguments. These are
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
This section concentrates on illustrating the expressive power of the operator concept in mathematics. Please refer to individual topics pages for further details.
Main article: Linear transformation
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:
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)
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.
Main article: Probability theory
Operators are also involved in probability theory. Such operators as expectation, variance, covariance, factorials, et al.
Calculus is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator very clearly. The key operator studied is the differential operator. It is linear, as are many of the operators constructed from it.
Main article: 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 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.
Given that integration is an operator as well (inverse of differentiation), we have some important operators we can write in terms of integration.
Main article: Convolution
The convolution of two functions is a mapping from two functions to one other, defined by an integral as follows:
If x1=f(t) and x2=g(t), define the operator Q such that;
which we write as (f * g)(τ).
Main article: 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 up an integral form:
Main article: 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:
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.
Main article: Operator (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.