Operator (mathematics)

In mathematics, an operator is generally a mapping or function that acts on elements of a 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 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 equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples)

The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from $\mathbb {R} ^{n}$ to $\mathbb {R} ^{n}$ .^[1]^[2]^[a] Such operators often preserve properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators 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)).

^ Rudin, Walter (1976). "Chapter 9: Functions of several variables". Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. p. 207. ISBN 0-07-054235-X. Linear transformations of $X$ into $X$ are often called linear operators on $X$ .
^ ^a ^b Roman, Steven (2008). "Chapter 2: Linear Transformations". Advanced Linear Algebra (3rd ed.). Springer. p. 59. ISBN 978-0-387-72828-5.

Cite error: There are <ref group=lower-alpha> tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).

[Rudin-1973-Analysis-1] Rudin, Walter (1976). "Chapter 9: Functions of several variables". Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. p. 207. ISBN 0-07-054235-X. Linear transformations of $X$ into $X$ are often called linear operators on $X$ .

[Roman-2008-LinAlg-2] Roman, Steven (2008). "Chapter 2: Linear Transformations". Advanced Linear Algebra (3rd ed.). Springer. p. 59. ISBN 978-0-387-72828-5.

[1]

[2]

[a]