A mapping of one set into another, each of which has a certain structure (defined by algebraic operations, a topology, or by an order relation). The general definition of an operator coincides with the definition of a mapping or function. Let $X$ and $Y$ be two sets. A rule or correspondence which assigns a uniquely defined element $A(x)\in Y$ to every element $x$ of a subset $D\subset X$ is called an operator $A$ from $X$ into $Y$. $$\begin{equation}A:D\to Y, \qquad \text{where } D \subset X.\end{equation}$$The term operator is mostly used in the case where $X$ and $Y$ are vector spaces. The expression $A(x)$ is often written as $Ax$.

## Contents

- 1 Definitions and Notations
- 2 Connection with Equations
- 3 Graph
- 4 Examples of operators.
- 4.1 References
- 4.2 Comments
- 4.3 References

### Definitions and Notations

- The subset $D$ is called the domain of definition of the operator $A$ and is denoted by $\operatorname{Dom}(A)$; the set $\{A(x) : x\in D\}$ is called the domain of values of the operator $A$ (or its range) and is denoted by $\operatorname{R}(A)$.
- If $A$ is an operator from $X$ into $Y$ where $X=Y$, then $A$ is called an operator on $X$.
- If $\operatorname{Dom}(A)=X$, then $A$ is called an everywhere-defined operator.
- If $A_1$, $A_2$ are operators from $X_1$ into $Y_1$ and from $X_2$ into $Y_2$ with domains of definition $\operatorname{Dom}(A_1)$ and $\operatorname{Dom}(A_2)$, respectively, such that $\operatorname{Dom}(A_1)\subset\operatorname{Dom}(A_2)$ and $A_1x=A_2x$ for all $x\in\operatorname{Dom}(A_1)$, then if $X_1=X_2$, $Y_1=Y_2$, the operator $A_1$ is called a compression or restriction of the operator $A_2$, while $A_2$ is called an extension of $A_1$; if $X_1\subset X_2$, $A_2$ is called an extension of $A_1$ exceeding $X_1$.
- If $X$ and $Y$ are vector spaces, then in the set of all operators from $X$ into $Y$ it is possible to single out the class of linear operators (cf. Linear operator); the remaining operators from $X$ into $Y$ are called non-linear operators.
- If $X$ and $Y$ are topological vector spaces, then in the set of operators from $X$ into $Y$ the class of continuous operators (cf. Continuous operator) can be naturally singled out, so are the class of bounded linear operators $A$ (operators $A$ such that the image of any bounded set in $X$ is bounded in $Y$) and the class of compact linear operators (i.e. operators such that the image of any bounded set in $X$ is pre-compact in $Y$, cf. Compact operator).
- If $X$ and $Y$ are locally convex spaces, then it is natural to examine different topologies on $X$ and $Y$; an operator is said to be semi-continuous if it defines a continuous mapping from the space $X$ (with the initial topology) into the space $Y$ with the weak topology (the concept of semi-continuity is mainly used in the theory of non-linear operators); an operator is said to be strongly continuous if it is continuous as a mapping from $X$ with the boundedly weak topology into the space $Y$; an operator is called weakly continuous if it defines a continuous mapping from $X$ into $Y$ where $X$ and $Y$ have the weak topology. Compact operators are often called completely-continuous operators. Sometimes the term "competely-continuous operator" is used instead of "strongly-continuous operator" , or to denote an operator which maps any weakly-convergent sequence to a strongly-convergent one; if $X$ and $Y$ are reflexive Banach spaces, then these conditions are equivalent to the compactness of the operator. If an operator is strongly continuous, then it is weakly continuous.

### Connection with Equations

Many equations in function spaces or abstract spaces can be expressed in theform $Ax=y$, where $y\in Y$, $x \in X$; $y$ is given, $x$ is unknownand $A$ is an operator from $X$ into $Y$. The assertion of theexistence of a solution to this equation for any right-hand side $y\inY$ is equivalent to the assertion that the range of the operator $A$ isthe whole space $Y$; the assertion that the equation $Ax=y$ has aunique solution for any $y\in\operatorname{R}(A)$ means that $A$ is aone-to-one mapping from $\operatorname{Dom}(A)$ onto$\operatorname{R}(A)$.

### Graph

The set $\Gamma(A)\subset X\times Y$ defined by the relation$$\begin{equation}\Gamma(A) = \{(x,Ax) : x\in \operatorname{Dom}(A)\}\end{equation}$$is called the graph of the operator $A$.Let $X$ and $Y$ be topological vector spaces; an operator from $X$ into $Y$ is called a closed operator if its graph is closed. The concept of a closed operator is particularly useful in the case of linear operators with a dense domain of definition.

The concept of a graph allows one to generalize the concept of an operator: Any subset $A$ in $X\times Y$ is called a multi-valued operator from $X$ into $Y$; if $X$ and $Y$ are vector spaces, then a linear subspace in $X\times Y$ is called a multi-valued linear operator; the set$$\begin{equation}D(A) = \{x\in X : \text{ there exists an } y\in Y \text{ such that } (x, y)\in A \}\end{equation}$$is called the domain of definition of the multi-valued operator.

If $X$ is a vector space over a field $\mathcal K$ and $Y = \mathcal K$, then an everywhere-defined operator from $X$ into $Y$ is called a functional on $X$.

If $ X $and $ Y $are locally convex spaces, then an operator $ A $from $ X $into $ Y $with a dense domain of definition in $ X $has an adjoint operator $ A ^{*} $with a dense domain of definition in $ Y ^{*} $(with the weak topology) if, and only if, $ A $is a closed operator.

### Examples of operators.

1) The operator assigning the element $ 0 \in Y $to any element $ x \in X $(the zero operator).

2) The operator mapping each element $ x \in X $to the same element $ x \in X $(the identity operator on $ X $, written as $ \mathop{\rm id}\nolimits _{X} $or $ 1 _{X} $).

3) Let $ X $be a vector space of functions on a set $ M $, and let $ f $be a function on $ M $; the operator on $ X $with domain of definition

$$ D(A) = \{ {\phi \in X} : {f \phi \in X} \}$$

and acting according to the rule

$$ A \phi = f \phi$$

if $ \phi \in D(A) $, is called the operator of multiplication by a function; $ A $is a linear operator.

4) Let $ X $be a vector space of functions on a set $ M $, and let $ F $be a mapping from the set $ M $into itself; the operator on $ X $with domain of definition

$$ D(A) = \{ {\phi \in X} : {\phi \circ F \in X} \}$$

and acting according to the rule

$$ A \phi = \phi \circ F$$

if $ \phi \in D(A) $, is a linear operator.

5) Let $ X,\ Y $be vector spaces of real measurable functions on two measure spaces $ (M,\ \Sigma _{M} ,\ \mu ) $and $ (N,\ \Sigma _{N} ,\ \nu ) $, respectively, and let $ K $be a function on $ M \times N \times \mathbf R $, measurable with respect to the product measure $ \mu \times \nu \times \mu _{0} $, where $ \mu _{0} $is Lebesgue measure on $ \mathbf R $, and continuous in $ t \in \mathbf R $for any fixed $ m \in M $, $ n \in N $. The operator from $ X $into $ Y $with domain of definition $ D(A) = \{ {\phi \in X} : {f(x) = \int _{M} K (x,\ y,\ \phi (y)) \ dy} \} $, which exists for almost-all $ x \in N $and $ f \in Y $, and acting according to the rule $ A \phi = f $if $ \phi \in D(A) $, is called an integral operator; if

$$ K(x,\ y,\ z) = K(x,\ y)z, x \in M, y \in N, z \in \mathbf R ,$$

then $ A $is a linear operator.

6) Let $ X $be a vector space of functions on a differentiable manifold $ M $, let $ \xi $be a vector field on $ M $; the operator $ A $on $ X $with domain of definition

$$ D(A) = \{ {f \in X} : {\textrm{ the derivative } D _ \xi f \textrm{ of the function } f\textrm{ along the field } \xi \textrm{ is everywhere defined and } D _ \xi f \in X} \}$$

and acting according to the rule $ Af = D _ \xi f $if $ f \in D(A) $, is called a differentiation operator; $ A $is a linear operator.

7) Let $ X $be a vector space of functions on a set $ M $; an everywhere-defined operator assigning to a function $ \phi \in X $the value of that function at a point $ a \in M $, is a linear functional on $ X $; it is called the $ \delta $-function at the point $ a $and is written as $ \delta _{a} $.

8) Let $ G $be a commutative locally compact group, let $ \widehat{G} $be the group of characters of the group $ G $, let $ dg $, $ \widehat{dg} $be the Haar measures on $ G $and $ \widehat{G} $, respectively, and let

$$ X = L _{2} ( G ,\ dg ),Y = L _{2} ( \widehat{G} ,\ \widehat{dg} ).$$

The linear operator $ A $from $ X $into $ Y $assigning to a function $ f \in X $the function $ \widehat{f} \in Y $defined by the formula

$$ \widehat{f} ( \widehat{g} ) = \int\limits f(g) \widehat{g} (g) \ dg$$

is everywhere defined if the convergence of the integral is taken to be mean-square convergence.

If $ X $and $ Y $are topological vector spaces, then the operators in examples 1) and 2) are continuous; if in example 3) the space $ X $is $ L _{2} (M,\ \Sigma _{M} ,\ \mu ) $, where $ \mu $is a measure on $ X $, then the operator of multiplication by a bounded measurable function is closed and has a dense domain of definition; if in example 5) the space $ X=Y $is a Hilbert space $ L _{2} (M,\ \Sigma _{M} ,\ \mu ) $and $ K(x,\ y,\ z) = K(x,\ y)z $, where $ K(x,\ y) $belongs to $ L _{2} (M \times M,\ \Sigma _{M} \times \Sigma _{M} ,\ \mu \times \mu ) $, then $ A $is compact; if in example 8) the spaces $ X $and $ Y $are regarded as Hilbert spaces, then $ A $is continuous.

If $ A $is an operator from $ X $into $ Y $such that $ Ax \neq Ay $when $ x \neq y $, $ x,\ y \in D(A) $, then the inverse operator $ A ^{-1} $to $ A $can be defined; the question of the existence of an inverse operator and its properties is related to the theorem of the existence and uniqueness of a solution of the equation $ Ax = f $; if $ A ^{-1} $exists, then $ x = A ^{-1} f $when $ f \in R(A) $.

For operators on a vector space it is possible to define a sum, multiplication by a number and an operator product. If $ A $, $ B $are operators from $ X $into $ Y $with domains of definition $ D(A) $and $ D(B) $, respectively, then the operator, written as $ A+B $, with domain of definition

$$ D(A+B) = D(A) \cap D(B)$$

and acting according to the rule

$$ (A+B)x = Ax + Bx$$

if $ x \in D(A+B) $, is called the sum of the operators $ A $and $ B $.

The operator, written as $ \lambda A $, with domain of definition

$$ D( \lambda A) = D(A)$$

and acting according to the rule

$$ ( \lambda A)x = \lambda (Ax)$$

if $ x \in D( \lambda A) $, is called the product of the operator $ A $by the number $ \lambda $. The operator product is defined as composition of mappings: If $ A $is an operator from $ X $into $ Y $and $ B $is an operator from $ Y $into $ Z $, then the operator $ BA $, with domain of definition

$$ D(BA) = \{ {x \in X} : {x \in D(A) \textrm{ and } Ax \in D(B)} \}$$

and acting according to the rule

$$ (BA)x = B(Ax)$$

if $ x \in D(BA) $, is called the product of $ B $and $ A $.

If $ P $is an everywhere-defined operator on $ X $such that $ PP = P $, then $ P $is called a projection operator or projector in $ X $; if $ I $is an everywhere-defined operator on $ X $such that $ I \circ I = \mathop{\rm id}\nolimits _{X} $, then $ I $is called an involution in $ X $.

The theory of operators constitutes the most important part of linear and non-linear functional analysis, being in particular a basic instrument in the theory of dynamical systems, representations of groups and algebras and a most important mathematical instrument in mathematical physics and quantum mechanics.

#### References

[1] | L.A. [L.A. Lyusternik] Liusternik, "Elements of functional analysis" , F. Ungar (1961) (Translated from Russian) |

[2] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |

[3] | L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian) |

[4] | N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958) |

[5] | R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) |

[6] | K. Yosida, "Functional analysis" , Springer (1980) |

#### Comments

#### References

[a1] | T. Kato, "Perturbation theory for linear operators" , Springer (1976) |

[a2] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) pp. Chapt. 5 |

[a3] | F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French) |

[a4] | W. Rudin, "Functional analysis" , McGraw-Hill (1973) |

[a5] | I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981) |

**How to Cite This Entry:**

Operator. *Encyclopedia of Mathematics.* URL: http://encyclopediaofmath.org/index.php?title=Operator&oldid=44339

This article was adapted from an original article by M.A. NaimarkA.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article