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{{About|operators in [[mathematics]]|other uses|Operator (disambiguation)}}
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{{cleanup-rewrite|date=November 2010}}


An '''operator''' is a [[Map (mathematics)|mapping]] from one [[vector space]] or [[Module (mathematics)|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]], [[observable]]s are represented by [[hermitian operators]]. Important properties that various operators may exhibit include [[Linear operator|linearity]], [[Continuous operator|continuity]], and [[Bounded operator|boundedness]].
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<!-- I'm giving definition for vector spaces. Please generalize to modules, I'm weak in these


-->Let ''U, V'' be two [[vector space]]s. 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):
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:<math>(A + B)\mathbf{x} := A\mathbf{x} + B\mathbf{x},</math>
 
:<math>(\alpha A)\mathbf{x} := \alpha A \mathbf{x}</math>
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Additionally, operators from any vector space to itself form a [[unital algebra|unital]] [[associative algebra]]:
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:<math>(AB)\mathbf{x} := A(B\mathbf{x})</math>
with the [[identity mapping]] (usually denoted ''E'', ''I'' or id) being the unit.


=== Bounded operators and operator norm ===
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{{Main|bounded operator|operator norm|Banach algebra}}
 
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Let ''U'' and ''V'' be two vector spaces over the same [[ordered field]] (for example, <math>\mathbf{R}</math>), and they are equipped with [[norm (mathematics)|norm]]s. Then a linear operator from ''U'' to ''V'' is called '''bounded''' if there exists ''C > 0'' such that
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:<math>||A\mathbf{x}||_V \leq C||\mathbf{x}||_U</math>
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'':
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:<math>||A|| = \inf\{C: ||A\mathbf{x}||_V \leq C||\mathbf{x}||_U\}</math>.
 
 
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In case of operators from ''U'' to itself it can be shown that
 
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Any unital [[normed algebra]] with this property is called a [[Banach algebra]]. It is possible to generalize [[spectral theory]] to such algebras. [[C*-algebra]]s, which are [[Banach algebras]] with some additional structure, play an important role in [[quantum mechanics]].
  <li>[http://www.121e.cn/forum.php?mod=viewthread&tid=2375124 http://www.121e.cn/forum.php?mod=viewthread&tid=2375124]</li>
 
 
== Special cases ==
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=== Functionals ===
 
{{Main|Functional (mathematics)}}
</ul>
A functional is an operator that maps a vector space to its underlying [[Field (mathematics)|field]]. Important applications of functionals are the theories of [[generalized function]]s and [[calculus of variations]]. Both are of great importance to theoretical physics.
 
=== Linear operators ===
{{Main|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
:<math>A(\alpha \mathbf{x} + \beta \mathbf{y}) = \alpha A \mathbf{x} + \beta A \mathbf{y}</math>
for all '''x''', '''y''' in ''U'' and for all ''α, β'' in ''K''.
 
The importance of linear operators is partially because they are [[morphism]]s between vector spaces.
 
In finite-dimensional case linear operators can be represented by [[Matrix (mathematics)|matrices]] in the following way. Let <math>K</math> be a field, and <math>U</math> and <math>V</math> be finite-dimensional vector spaces over <math>K</math>. Let us select a basis <math>\mathbf{u}_1, \ldots, \mathbf{u}_n</math> in <math>U</math> and <math>\mathbf{v}_1, \ldots, \mathbf{v}_m</math> in <math>V</math>. Then let <math>\mathbf{x} = x^i \mathbf{u}_i</math> be an arbitrary vector in <math>U</math> (assuming [[Einstein convention]]), and <math>A: U \to V</math> be a linear operator. Then
: <math>A\mathbf{x} = x^i A\mathbf{u}_i = x^i (A\mathbf{u}_i)^j \mathbf{v}_j </math>.
Then <math>a_i^j := (A\mathbf{u}_i)^j \in K</math> is the matrix of the operator <math>A</math> in fixed bases. <math>a_i^j</math> does not depend on the choice of <math>x</math>, and <math>A\mathbf{x} = \mathbf{y}</math> iff <math>a_i^j x^i = y^j</math>. Thus in fixed bases n-by-m matrices are in bijective correspondence to linear operators from <math>U</math> to <math>V</math>.
 
The important concepts directly related to operators between finite-dimensional vector spaces are the ones of [[Matrix rank|rank]], [[determinant]], [[inverse operator]], and [[eigenspace]].
 
Linear operators also play a great role in the infinite-dimensional case. The concepts of rank and determinant cannot be extended to infinite-dimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinite-dimensional case. The study of linear operators in the infinite-dimensional case is known as [[functional analysis]] (so called because various classes of functions form interesting examples of infinite-dimensional vector spaces).
 
The space of [[sequence]]s of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinite-dimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as [[sequence space]]s. Operators on these spaces are known as [[sequence transformation]]s.
 
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 [[Spectrum (functional analysis)|spectra]] that elegantly generalizes the theory of eigenspaces.
<!-- Please expand -->
 
== Examples ==
=== Geometry ===
{{Main|general linear group|isometry}}
 
In [[geometry]], additional structures on [[vector space]]s are sometimes studied. Operators that map such vector spaces to themselves bijectively are very useful in these studies, they naturally form [[group (mathematics)|group]]s by composition.
 
For example, bijective operators preserving the structure of a vector space are precisely the [[invertible]] [[linear operator]]s. 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 ===
{{Main|Probability theory}}
 
Operators are also involved in probability theory, such as [[expected value|expectation]], [[variance]], [[covariance]], [[factorial]]s, etc.
<!-- if you know it and you know it please expand... -->
 
=== Calculus ===
{{Main|differential operator|integral operator}}
From the point of view of [[functional analysis]], [[calculus]] is the study of two linear operators: the [[differential operator]] <math>\frac{\mathrm{d}}{\mathrm{d}t}</math>, and the [[Volterra operator|indefinite integral operator]] <math>\int_0^t</math>.
 
==== Fourier series and Fourier transform ====
 
{{Main|Fourier series|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 function]]s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of [[sine wave]]s and cosine waves:
:<math>f(t) = {a_0 \over 2} + \sum_{n=1}^{\infty}{ a_n \cos ( \omega n t ) + b_n \sin ( \omega n t ) } </math>
Coefficients ''(a<sub>0</sub>, a<sub>1</sub>, b<sub>1</sub>, a<sub>2</sub>, b<sub>2</sub>, ...)'' are in fact an element of an infinite-dimensional vector space [[Sequence space|ℓ<sup>2</sup>]], and thus Fourier series is a linear operator.
 
When dealing with general function '''R''' → '''C''', the transform takes on an [[integral]] form:
 
:<math>f(t) = {1 \over \sqrt{2 \pi}} \int_{- \infty}^{+ \infty}{g( \omega )e^{ i \omega t } \,d\omega }. </math>
 
==== Laplace transform ====
 
{{Main|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:
:<math>F(s) = (\mathcal{L}f)(s) =\int_0^\infty e^{-st} f(t)\,dt.</math>
 
=== Fundamental operators on scalar and vector fields ===
{{Main|vector calculus|vector field|scalar field|gradient|divergence|curl (mathematics)|l6=curl}}
 
Three operators are key to [[vector calculus]]:
* Grad ([[gradient]]), (with operator symbol [[Del|<math>\nabla</math>]]) 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 [[Del#Divergence|<math>\nabla \cdot</math>]]) is a vector operator that measures a vector field's divergence from or convergence towards a given point.
* [[Curl (mathematics)|Curl]], (with operator symbol [[Del#Curl|<math>\nabla \times</math>]]) 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. <ref name="Vector and Tensor Operators">{{cite book |isbn= 0-393-92516-1 |url= http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161/ref=sr_1_1?s=books&ie=UTF8&qid=1388768941&sr=1-1&keywords=div+grad+curl |title=Div Grad Cural and All that |author=h.m. schey |location=New York|publisher=W W Norton|year=2005}}</ref>
 
== See also ==
* [[Operation (mathematics)|Operation]]
* [[Function (mathematics)|Function]]
* [[List of mathematical operators]]
* [[Vector space]]
* [[Dual space]]
* [[Operator algebra]]
* [[Banach algebra]]
* [[List of operators]]
* [[Operator (physics)]]
* [[Operator (programming)]]
 
== References ==
{{reflist}}
 
[[Category:Algebra]]
[[Category:Functional analysis]]
[[Category:Mathematical notation]]

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