Editing Operator (mathematics) (section)

{{Short description|Function acting on function spaces}}
{{About|operators in mathematics|other uses|Operator (disambiguation)}}
{{distinguish|text=the symbol denoting a [[mathematical operation]] or [[mathematical symbol]]}}

In [[mathematics]], an '''operator''' is generally a [[Map (mathematics)|mapping]] or [[function (mathematics)|function]] that acts on elements of a [[space (mathematics)|space]] to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', but the term is often used in place of ''function'' when the [[domain of a function|domain]] is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an [[integral operator]]), and may be extended so as to act on related objects (an operator that acts on functions may act also on [[differential equation]]s whose solutions are functions that satisfy the equation). (see [[Operator (physics)]] for other examples)

The most basic operators are [[linear map]]s, which act on [[vector space]]s. Linear operators refer to linear maps whose domain and range are the same space, for example from <math>\mathbb{R}^n </math> to <math>\mathbb{R}^n</math>.<ref name=Rudin-1973-Analysis>
{{cite book
 | last=Rudin  | first=Walter
 | year=1976
 | chapter=Chapter 9: Functions of several variables
 | title=Principles of Mathematical Analysis
 | publisher=McGraw-Hill
 | edition=3rd
 | isbn=0-07-054235-X
 | page=207
 | quote=Linear transformations of {{mvar|X}} into {{mvar|X}} are often called '''linear operators''' on {{mvar|X}}&nbsp;.
}}
</ref><ref name=Roman-2008-LinAlg>
{{cite book
 | last=Roman | first=Steven
 | year=2008
 | chapter=Chapter 2: Linear Transformations
 | title=Advanced Linear Algebra
 | edition=3rd
 | publisher=Springer
 | isbn=978-0-387-72828-5
 | page=59
}}
</ref>{{efn|: (1) A linear transformation from {{mvar|V}} to {{mvar|V}} is called a ''linear operator'' on {{mvar|V}}.
The set of all linear operators on {{mvar|V}} is denoted {{math|''ℒ''(''V'')}}&nbsp;. A linear operator on a real vector space is called a ''real operator'' and a linear operator on a complex vector space is called a ''complex operator''. ... We should also mention that some authors use the term linear operator for any linear transformation from {{mvar|V}} to {{mvar|W}}. ...
:
''Definition:''
The following terms are also employed:
: (2) ''endomorphism'' for linear operator ...
: (6) ''automorphism'' for bijective linear operator.
::::— Roman (2008)<ref name=Roman-2008-LinAlg/>
}}
Such operators often preserve properties, such as [[continuous function|continuity]]. For example, [[differentiation (mathematics)|differentiation]] and [[indefinite integration]] are linear operators; operators that are built from them are called [[differential operator]]s, [[integral operator]]s or integro-differential operators.

'''Operator''' is also used for denoting the symbol of a [[mathematical operation]]. This is related with the meaning of "operator" in [[computer programming]] (see [[Operator (computer programming)]]).