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 hermitian operators. Important properties that various operators may exhibit include linearity, continuity, and boundedness.
Definitions
Let U, V be two vector spaces. Any 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 and B are operators):

(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
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.
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\cdotB.
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
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
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 finitedimensional case linear operators can be represented by matrices in the following way. Let K be a field, and U and V be finitedimensional 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. a_i^j does not depend on the choice of x, and A\mathbf{x} = \mathbf{y} iff a_i^j x^i = y^j. Thus in fixed bases nbym matrices are in bijective correspondence to linear operators from U to V.
The important concepts directly related to operators between finitedimensional vector spaces are the ones of rank, determinant, inverse operator, and eigenspace.
Linear operators also play a great role in the infinitedimensional case. The concepts of rank and determinant cannot be extended to infinitedimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinitedimensional case. The study of linear operators in the infinitedimensional case is known as functional analysis (so called because various classes of functions form interesting examples of infinitedimensional vector spaces).
The space of sequences of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinitedimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as sequence spaces. Operators on these spaces are known as sequence transformations.
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
In 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 a vector space are precisely the invertible linear operators. They form the general linear group under composition. They do not form a vector space under the addition of operators, e.g. both id and id are invertible (bijective), but their sum, 0, is not.
Operators preserving the Euclidean metric on such a space form the isometry group, and those that fix the origin form a subgroup known as the orthogonal group. Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group, or the group of rotations.
Probability theory
Operators are also involved in probability theory, such as expectation, variance, covariance, factorials, etc.
Calculus
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
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 infinitedimensional vector space ℓ^{2}, 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
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
Three operators are key to vector calculus:

Grad (gradient), (with operator symbol \nabla) 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), (with operator symbol \nabla \cdot) is a vector operator that measures a vector field's divergence from or convergence towards a given point.

Curl, (with operator symbol \nabla \times) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point.
As an extension of vector calculus operators to physics, engineering and tensor spaces, Grad, Div and Curl operators also are often associatied with Tensor calculus as well as vector calculus. ^{[1]}
See also
References

^ h.m. schey (2005). Div Grad Cural and All that. New York: W W Norton.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.