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
and
be two sets. A rule or correspondence which assigns a uniquely defined element
to every element
of a subset
is called an
operator
from
into
.
is called the
domain of definition of the operator
and is denoted by
;
the set
is called the
domain of values of the operator
(or its
range)
and is denoted by
.
The expression
is often written as
.
The term operator is mostly used in the case where
and
are vector spaces. If
is an operator from
into
where
,
then
is called an
operator on
.
If
,
then
is called an
everywhere-defined operator.
If
,
are operators from
into
and from
into
with domains of definition
and
,
respectively, such that
and
for all
,
then if
,
,
the operator
is called a
compression
or
restriction
of the operator
,
while
is called an
extension
of
;
if
,
is called an extension of
exceeding
.
Many equations in function spaces or abstract spaces can be expressed in the form
,
where
,
;
is given,
is unknown and
is an operator from
into
.
The assertion of the existence of a solution to this equation for any right-hand side
is equivalent to the assertion that the range of the operator
is the whole space
;
the assertion that the equation
has a unique solution for any
means that
is a one-to-one mapping from
onto
.
If
and
are vector spaces, then in the set of all operators from
into
it is possible to single out the class of linear operators (cf.
Linear operator);
the remaining operators from
into
are called
non-linear operators.
If
and
are topological vector spaces, then in the set of operators from
into
the class of continuous operators (cf.
Continuous operator)
can be naturally singled out, so are the class of
bounded linear operators
(operators
such that the image of any bounded set in
is bounded in
)
and the class of compact linear operators (i.e. operators
such that the image of any bounded set in
is pre-compact in
,
cf.
Compact operator).
If
and
are locally convex spaces, then it is natural to examine different topologies on
and
;
an operator is said to be
semi-continuous
if it defines a continuous mapping from the space
(with the initial topology) into the space
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
with the boundedly weak topology into the space
;
an operator is called
weakly continuous
if it defines a continuous mapping from
into
where
and
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
and
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.
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.