Operator

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.

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Operators and levels of abstraction

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:

• To draw attention to the function domain, which may itself consist of vectors or functions, rather than just numbers. The expectation operator in probability theory, for example, has random variables as domain (and is also a functional).
• To draw attention to the fact that the domain consists of pairs or tuples of some sort, in which case operator is synonymous with the usual mathematical sense of operation.
• To draw attention to the function codomain; for example a vector-valued function might be called an operator.

A single operator might conceivably qualify under all three of these. Other important ideas are:

• Overloading, in which for example addition, say, is thought of as a single operator able to act on numbers, vectors, matrices ... .
• Operators are often in practice just partial functions, a common phenomenon in the theory of differential equations since there is no guarantee that the derivative of a function exists.
• Use of higher operations on operators, meaning that operators are themselves combined.

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.

Describing operators

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.

Notations

There are three 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 awkward and not commonly used for operators other than binary operators. Infix style is written, for example, as
x1 Q x2.

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 [itex]\mathcal{F}(f(t)) = F(\omega).[itex]

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

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:

[itex]T\ (f(x)+g(x)) = T\ (f(x))+T\ (g(x))[itex]
[itex]T\ (af(x)) = a\,T\ (f(x))[itex]

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

Main article: Probability theory

Operators are also involved in probability theory. Such operators as expectation, variance, covariance, factorials, et al.

Operators in calculus

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.

The differential operator

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.

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

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;

[itex]Q\ x_1\ x_2\ = \int f(t) g(\tau - t) dt[itex]

which we write as [itex](f*g)(\tau)[itex].

Fourier transform

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:

[itex] f(t) = {a_0 \over 2} + \sum_{n=1}^{\infty}{ a_n \cos ( \omega n t ) + b_n \sin ( \omega n t ) } [itex]

When dealing with general function R->C, the transform takes up an integral form:

[itex] f(t) = {1 \over \sqrt{2 \pi}} \int_{- \infty}^{+ \infty}{g( \omega ) \cdot \exp {( i \omega t )} \cdot d \omega } [itex]
Laplacian transform

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:

[itex]F(s) = (\mathcal{L}f)(s) =\int_0^\infty e^{-st} f(t)\,dt.[itex]

Fundamental operators on scalar and vector fields

Main articles: vector calculus, scalar field, gradient, divergence, and curl

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.

Operators in physics

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.

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