# 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 $\Gamma(A)\subset X\times Y$ defined by the relation $$$$\Gamma(A) = \{(x,Ax) : x\in \operatorname{Dom}(A)\}$$$$ 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 $$$$D(A) = \{x\in X : \text{ there exists an } y\in Y \text{ such that } (x, y)\in A \}$$$$ 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)