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 $Y$. $$A:D\to Y,\text{where }D\subset X.$$ The term operator is mostly used in the case where $X$ and $Y$ are vector spaces. The expression $A(x)$ is often written as $Ax$.

Definitions and Notations[edit]

• The subset $D$ is called the domain of definition of the operator $A$ and is denoted by $\mathrm{Dom}(A)$; the set $\{A(x):x\in D\}$ is called the domain of values of the operator $A$ (or its range) and is denoted by $\mathrm{R}(A)$.
• If $A$ is an operator from $X$ into $Y$ where $X=Y$, then $A$ is called an operator on $X$.
• If $\mathrm{Dom}(A)=X$, 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 $\mathrm{Dom}(A_1)$ and $\mathrm{Dom}(A_2)$, respectively, such that $\mathrm{Dom}(A_1)\subset \mathrm{Dom}(A_2)$ and $A_{1}x=A_{2}x$ for all $x\in \mathrm{Dom}(A_1)$, 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[edit]

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 $y\in \mathrm{R}(A)$ means that $A$ is a one-to-one mapping from $\mathrm{Dom}(A)$ onto $\mathrm{R}(A)$.

Graph[edit]

The set $\mathrm{\Gamma }(A)\subset X\times Y$ defined by the relation $$\mathrm{\Gamma }(A)=\{(x,Ax):x\in \mathrm{Dom}(A)\}$$ 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 $A^{\ast }$ with a dense domain of definition in $Y^{\ast }$( with the weak topology) if, and only if, $A$ is a closed operator.

Examples of operators.[edit]

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 ${\mathrm{id}}_X$ 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

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

and acting according to the rule

$$A\varphi =f\varphi$$

if $\varphi \in D(A)$, 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

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

and acting according to the rule

$$A\varphi =\varphi \circ F$$

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

5) Let $X,Y$ be vector spaces of real measurable functions on two measure spaces $(M,{\mathrm{\Sigma }}_M,\mu )$ and $(N,{\mathrm{\Sigma }}_N,\nu )$, 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 $D(A)=\{\varphi \in X:f(x)={\int }_{M}K(x,y,\varphi (y))dy\}$, which exists for almost-all $x\in N$ and $f\in Y$, and acting according to the rule $A\varphi =f$ if $\varphi \in D(A)$, 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

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

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 $\varphi \in X$ 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 $\hat{G}$ be the group of characters of the group $G$, let $dg$, $\widehat{dg}$ be the Haar measures on $G$ and $\hat{G}$, respectively, and let

$$X=L_2(G,dg),Y=L_2(\hat{G},\widehat{dg}).$$

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

$$\hat{f}(\hat{g})=\int f(g)\hat{g}(g)dg$$

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 $L_2(M,{\mathrm{\Sigma }}_M,\mu )$, 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 $L_2(M,{\mathrm{\Sigma }}_M,\mu )$ and $K(x,y,z)=K(x,y)z$, where $K(x,y)$ belongs to $L_2(M\times M,{\mathrm{\Sigma }}_M\times {\mathrm{\Sigma }}_M,\mu \times \mu )$, 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 $Ax\ne Ay$ when $x\ne y$, $x,y\in D(A)$, 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

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

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 $I\circ I={\mathrm{id}}_X$, 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.

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