# Operator

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 $Y$. $$A:D\to Y, \qquad \text{where } D \subset X.$$ 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

### 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 the form $Ax=y$, where $y\in Y$, $x \in X$; $y$ is given, $x$ is unknown and $A$ is an operator from $X$ into $Y$. The assertion of the existence of a solution to this equation for any right-hand side $y\in Y$ is equivalent to the assertion that the range of the operator $A$ is the whole space $Y$; the assertion that the equation $Ax=y$ has a unique solution for any $y\in\operatorname{R}(A)$ means that $A$ is a one-to-one mapping from $\operatorname{Dom}(A)$ onto $\operatorname{R}(A)$.

### Graph

The set defined by the relation

is called the graph of the operator .

Let and be topological vector spaces; an operator from into 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 in is called a multi-valued operator from into ; if and are vector spaces, then a linear subspace in is called a multi-valued linear operator; the set

is called the domain of definition of the multi-valued operator.

If is a vector space over a field and , then an everywhere-defined operator from into is called a functional on .

If and are locally convex spaces, then an operator from into with a dense domain of definition in has an adjoint operator with a dense domain of definition in (with the weak topology) if, and only if, is a closed operator.

### Examples of operators.

1) The operator assigning the element to any element (the zero operator).

2) The operator mapping each element to the same element (the identity operator on , written as or ).

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

and acting according to the rule

if , is called the operator of multiplication by a function; is a linear operator.

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

and acting according to the rule

if , is a linear operator.

5) Let be vector spaces of real measurable functions on two measure spaces and , respectively, and let be a function on , measurable with respect to the product measure , where is Lebesgue measure on , and continuous in for any fixed , . The operator from into with domain of definition , which exists for almost-all and , and acting according to the rule if , is called an integral operator; if

then is a linear operator.

6) Let be a vector space of functions on a differentiable manifold , let be a vector field on ; the operator on with domain of definition

and acting according to the rule if , is called a differentiation operator; is a linear operator.

7) Let be a vector space of functions on a set ; an everywhere-defined operator assigning to a function the value of that function at a point , is a linear functional on ; it is called the -function at the point and is written as .

8) Let be a commutative locally compact group, let be the group of characters of the group , let , be the Haar measures on and , respectively, and let

The linear operator from into assigning to a function the function defined by the formula

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

If and are topological vector spaces, then the operators in examples 1) and 2) are continuous; if in example 3) the space is , where is a measure on , 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 is a Hilbert space and , where belongs to , then is compact; if in example 8) the spaces and are regarded as Hilbert spaces, then is continuous.

If is an operator from into such that when , , then the inverse operator to 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 ; if exists, then when .

For operators on a vector space it is possible to define a sum, multiplication by a number and an operator product. If , are operators from into with domains of definition and , respectively, then the operator, written as , with domain of definition

and acting according to the rule

if , is called the sum of the operators and .

The operator, written as , with domain of definition

and acting according to the rule

if , is called the product of the operator by the number . The operator product is defined as composition of mappings: If is an operator from into and is an operator from into , then the operator , with domain of definition

and acting according to the rule

if , is called the product of and .

If is an everywhere-defined operator on such that , then is called a projection operator or projector in ; if is an everywhere-defined operator on such that , then is called an involution in .

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)