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In basic mathematics, an operator is a symbol or function representing a mathematical operation.

In terms of vector spaces, an operator is a mapping from one vector space or module to another. Operators are of critical importance to both linear algebra and functional analysis, and they find application in many other fields of pure and applied mathematics. For example, in classical mechanics, the derivative is used ubiquitously, and in quantum mechanics, observables are represented by linear operators. Important properties that various operators may exhibit include linearity, continuity, and boundedness.

Definitions

Let U, V be two vector spaces. The mapping from U to V is called an operator. Let V be a vector space over the field K. We can define the structure of a vector space on the set of all operators from U to V:

$$(A + B)\mathbf{x} := A\mathbf{x} + B\mathbf{x},$$
$$(\alpha A)\mathbf{x} := \alpha A \mathbf{x}$$

for all A, B: U → V, for all x in U and for all α in K.

Additionally, operators from any vector space to itself form a unital associative algebra:

$$(AB)\mathbf{x} := A(B\mathbf{x})$$

with the identity mapping (usually denoted E, I or id) being the unit.
Bounded operators and operator norm
Main articles: bounded operator, operator norm, and Banach algebra

Let U and V be two vector spaces over the same ordered field (for example, $$\mathbf{R}),$$ and they are equipped with norms. Then a linear operator from U to V is called bounded if there exists C > 0 such that

$$||A\mathbf{x}||_V \leq C||\mathbf{x}||_U$$

for all x in U.

It is trivial to show that bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of U and V:

$$||A|| = \inf\{C: ||A\mathbf{x}||_V \leq C||\mathbf{x}||_U\}.$$

In case of operators from U to itself it can be shown that

$$||AB|| \leq ||A||\cdot||B||.$$

Any unital normed algebra with this property is called a Banach algebra. It is possible to generalize spectral theory to such algebras. C*-algebras, which are Banach algebras with some additional structure, play an important role in quantum mechanics.
Special cases
Functionals
Main article: Functional (mathematics)

A functional is an operator that maps a vector space to its underlying field. Important applications of functionals are the theories of generalized functions and calculus of variations. Both are of great importance to theoretical physics.
Linear operators
Main article: Linear operator

The most common kind of operator encountered are linear operators. Let U and V be vector spaces over a field K. Operator A: U → V is called linear if

$$A(\alpha \mathbf{x} + \beta \mathbf{y}) = \alpha A \mathbf{x} + \beta A \mathbf{y}$$

for all x, y in U and for all α, β in K.

The importance of linear operators is partially because they are morphisms between vector spaces.

In finite-dimensional case linear operators can be represented by matrices in the following way. Let K be a field, and U and V be finite-dimensional vector spaces over K. Let us select a basis $$\mathbf{u}_1, \ldots, \mathbf{u}_n$$ in U and $$\mathbf{v}_1, \ldots, \mathbf{v}_m$$ in V. Then let $$\mathbf{x} = x^i \mathbf{u}_i$$ be an arbitrary vector in U (assuming Einstein convention), and $$A: U \to V$$ be a linear operator. Then

$$A\mathbf{x} = x^i A\mathbf{u}_i = x^i (A\mathbf{u}_i)^j \mathbf{v}_j .$$

Then $$a_i^j := (A\mathbf{u}_i)^j \in K$$ is the matrix of the operator A in fixed bases. It is easy to see that a_i^j does not depend on the choice of x, and that $$A\mathbf{x} = \mathbf{y}$$ iff $$a_i^j x^i = y^j.$$ Thus in fixed bases n-by-m matrices are in bijective correspondence to linear operators from U to V.

The important concepts directly related to operators between finite-dimensional vector spaces are the ones of rank, determinant, inverse operator, and eigenspace.

In infinite-dimensional case linear operators also play a great role. The concepts of rank and determinant cannot be extended to infinite-dimensional case, and although there are infinite matrices, they cease to be a useful tool. This is why a very different techniques are employed when studying linear operators (and operators in general) in infinite-dimensional case. They form a field of functional analysis (named such because various classes of functions form interesting examples of infinite-dimensional vector spaces).

Bounded linear operators over Banach space form a Banach algebra in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra that elegantly generalizes the theory of eigenspaces.
Examples
Geometry
Main articles: general linear group and isometry

In geometry (particularly differential geometry), additional structures on vector spaces are sometimes studied. Operators that map such vector spaces to themselves bijectively are very useful in these studies, they naturally form groups by composition.

For example, bijective operators preserving the structure of linear space are precisely invertible linear operators. They form general linear group; notice that they do not form a vector space, e.g. both id and -id are invertible, but 0 is not.

Operators preserving euclidean metric on such a space form orthogonal group, and operators that also preserve orientation of vector tuples form special orthogonal group, or the group of rotations.
Probability theory
Main article: Probability theory

Operators are also involved in probability theory, such as expectation, variance, covariance, factorials, etc.
Calculus
Main articles: differential operator and integral operator

From the point of view of functional analysis, calculus is the study of two linear operators: the differential operator $$\frac{\mathrm{d}}{\mathrm{d}t}$$, and the indefinite integral operator $$\int_0^t$$.
Fourier series and Fourier transform
Main articles: Fourier series and Fourier transform

The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way 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:

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

Coefficients $$(a_0, a_1, b_1, a_2, b_2, ...)$$ are in fact an element of an infinite-dimensional vector space ℓ2, and thus Fourier series is a linear operator.

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

$$f(t) = {1 \over \sqrt{2 \pi}} \int_{- \infty}^{+ \infty}{g( \omega )e^{ i \omega t } \,d\omega }.$$

Laplace 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:

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

Fundamental operators on scalar and vector fields
Main articles: vector calculus, vector field, scalar field, gradient, divergence, and curl

Three operators are key to vector calculus:

Grad (gradient), (with operator symbol ∇) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change.
Div (divergence) is a vector operator that measures a vector field's divergence from or convergence towards a given point.
Curl is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point.