In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The term "unbounded operator" can be misleading, since
In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain.
The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible.
Remark For a distribution one define the Schwartz kernel as follow:
The theory of unbounded operators was stimulated by attempts in the late 1920s to put quantum mechanics on a rigorous mathematical foundation. The systematic development of the theory is due to John von Neumann [1] and Marshall Stone. [2] The technique of using the graph to analyze unbounded operators was introduced by von Neumann in "Über Adjungierte Funktionaloperatoren". [3] [4]
Let and be Banach spaces. An unbounded linear operator (or simply operator)
is a linear map from a linear subspace of — the domain of — to the space [5] Contrary to the usual convention, may not be defined on the whole space
An operator is said to be densely defined if is dense in [5] This also includes operators defined on the entire space since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint and the transpose (see below).
An operator is called closed if its graph of is a closed set in the direct sum . [6] This means that for every sequence in converging to such that as one has and
An operator is called closeable if the closure of is the graph of some operator In this case is unique and is called the closure of
is an extension of an operator if , i.e. and for Denote by
Two operators are equal if and or equivalent: and for
The operations of unbounded operators are more complicated than in the bounded case, since one has take care of the domains of the operators. Let and be Banach spaces over
For an operator and an scalar the operator is given by
For two operator one define the operator by
For an operators and an operator the operator is defined by
The
inverse of exists if i.e. is
injective. Then the operator is defined by
Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators.
Let be two Banach spaces. A linear operator is said to be closed if one of the following equivalent properties hold:
For a closed operator one has
Consider the derivative operator
on the Banach space of all continuous functions on an interval with the Supremum norm. If one takes its domain to be then is a closed operator. (Note that one could also set to be the set of all differentiable functions including those with non-continuous derivative. That operator is not closed!)
The Operator is not bounded. For example, for the sequence one has but for it is for
If one takes to be instead the set of all infinitely differentiable functions, will no longer be closed, but it will be closable, with the closure being its extension defined on
For two Banch spaces an operator is called closable if the following equivalent properties hold:
The operator with the graph is said to be the closure of and is denoted by It follows that is the restriction of to Note, that other, non-minimal closed extensions may exist. [8] [9]
A core of a closable operator is a subset of such that the closure of the restriction of to is
Remark Not all operators are closable as the following example shows:
Consider the Operator on defined on and . For the sequence in given by one has
but Thus, is not closable.
Let be a densely defined operator on a Banach space and Then is called to be in the
resolvent set of denoted by if the operator is bijective and is a bounded operator. It follows by the closed graph theorem that the resolvent is bounded for all if is a closed operator.
For the resolvent of is defined by
The set is called the
spectrum of denoted by
The spectrum of an unbounded operator can be divided into three parts in exactly the same way as in the bounded case:
Remark The spectrum of an unbounded operator can be any closed set, including and The domain plays an important role as the following example shows:
Consider the banach space and the operators defined by and and
If , then Thus,
For the linear differential equation exists a unique solution which defines an inverse for Therefore
Let be an densely defined operator between Banach spaces and the continuous dual space of Using the notation the transpose (or dual) of is an operator satisfying:
The operator is defined by
Remark The necessary and sufficient condition for the transpose of to exist is that is densely defined (for essentially the same reason as to adjoints, see below.)
Let be a
vector space over the field , a
linear subspace. Let be a
sublinear function and
be a
linear functional with for all (where is the real part of a complex number ).
Then, there exists a
linear functional with
Satz von Banach-Steinhaus (Uniform boundedness principle)
Let be a banach space and be a normed vector space. Suppose that is a collection of bounded linear operators from to The uniform boundedness principle states that if for all in we have , then
Let be banach spaces and surjective. Then is an
open map.
In particular:
Bounded inverse theorem If bijective und bounded, then its inverse is also bounded.
Let be banach spaces. If is linear and closed, then is bounded.
For a densely defined closed operator the following properties are equivalent:
In this section let , and be Hilbert spaces.
For an unbounded operator the definition of the adjoint is more complicated than in the bounded case, since it is necessary to take care of the domains of the operators.
The adjoint of an unbounded operator can be defined in two equivalent ways. First, it can be defined in a way analogous to how we define the adjoint of a bounded operator.
For a densely defined operator its adjoint is defined by
Since is dense in the functional extends to the whole space via the Hahn–Banach theorem. Thus, one can find a unique such that
Finally, let completing the construction of [10] and it is
Remark exists if and only if is densely defined.
The other equivalent definition of the adjoint can be obtained by noticing a general fact: define a linear operator
We then have: is the graph of some operator if and only if is densely defined. [12] A simple calculation shows that this "some" satisfies
Thus, is the adjoint of
The definition of the adjoint can be given in terms of a transpose as follow:
For any Hilbert space and its continuous
dual space there is the
anti-linear
isomorphism
given by where for and Through this isomorphism, the transpose relates to the adjoint in the following way:
where . (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.)
By definition, the domain of could be anything; it could be trivial (i.e., contains only zero) [14] It may happen that the domain of is a closed hyperplane and vanishes everywhere on the domain. [15] [16] Thus, boundedness of on its domain does not imply boundedness of . On the other hand, if is defined on the whole space then is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. [17] If the domain of is dense, then it has its adjoint [11]
For a densely defined operator
If densely defined and , then . Further if are densely defined, then and [20] In contrast to the bounded case, it is not necessary that we have: since, for example, it is even possible that doesn't exist.[ citation needed] This is, however, the case if, for example, is bounded. [21]
Some well-known properties for bounded operators generalize to closed densely defined operators.
In particular, if has trivial kernel, has dense range (by the above identity.) Moreover, is surjective if and only if there is a such that
(This is essentially a variant of the closed range theorem.)
A densely defined operator is called symmetric if for all [25]
A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself. [25]
A symmetric operator is called bounded (from) below if there exists a constant with . The operator is said to be positve if .
An operator is symmetric if it satisfies one of the following equivalent properties:
Remark The last condition does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators.
A densely defined operator is said to be self-adjoint if [25]
For a densely defined closed operator one has:
Let be a symmetric operator. Then follwing conditions are equivalent:
[33]
An operator is
self-adjoint if the following equivalent properties hold:
Remarks
A densely defined, closed operator is called normal if it satisfies the following equivalent properties : [36]
Remarks
Let a symmetric operator on a Hilbert space .
Problem When does have self-adjoint extensions?
The Cayley transform of a symmetric operator is defined by . is an isometry between and and the range is dense in
Theorem is self-adjoint if and only if is
unitary.
In particular: has self-adjoint extensions if and only if has
unitary extensions.
Friedrichs extension theorem Every symmetric operator which is bounded from below has at least one self-adjoint extension with the same lower bound.
[41]
These operators always have a canonically defined self-adjoint extension which is called
Friedrichs extension.
Remark An everywhere defined extension exists for every operator, which is a purely algebraic fact explained at General existence theorem and based on the axiom of choice. If the given operator is not bounded then the extension is a discontinuous linear map. It is of little use since it cannot preserve important properties of the given operator, and usually is highly non-unique
A symmetric operator is called essentially self-adjoint if has one and only one self-adjoint extension. [33] Or equivalent, if its closure is self-adjoint. [27]. Note, that an operator may have more than one self-adjoint extension, and even a continuum of them. [9]
Remark The importance of essentially self-adjointness is that one is often given a non-closed symmetric operator If this operator is essential self-adjoint, then there is uniquely associated to a self-adjoint operator
Let be a symmetric operator. Then follwing conditions are equivalent: [30]
Remark For a bounded operator the terms self-adjoint, symmetric and essentially self-adjoint are equivalent.
Let be complete Riemannian manifold. The Laplace operator
on with the domain the space of all smooth, compactly supported function on is essentially self-adjoint. [42]
The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous spectral theorem holds for self-adjoint operators. In combination with Stone's theorem on one-parameter unitary groups it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see Self-adjoint operator#Self adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing time evolution in classical and quantum mechanics.
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Category:Linear operators
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In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The term "unbounded operator" can be misleading, since
In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain.
The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible.
Remark For a distribution one define the Schwartz kernel as follow:
The theory of unbounded operators was stimulated by attempts in the late 1920s to put quantum mechanics on a rigorous mathematical foundation. The systematic development of the theory is due to John von Neumann [1] and Marshall Stone. [2] The technique of using the graph to analyze unbounded operators was introduced by von Neumann in "Über Adjungierte Funktionaloperatoren". [3] [4]
Let and be Banach spaces. An unbounded linear operator (or simply operator)
is a linear map from a linear subspace of — the domain of — to the space [5] Contrary to the usual convention, may not be defined on the whole space
An operator is said to be densely defined if is dense in [5] This also includes operators defined on the entire space since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint and the transpose (see below).
An operator is called closed if its graph of is a closed set in the direct sum . [6] This means that for every sequence in converging to such that as one has and
An operator is called closeable if the closure of is the graph of some operator In this case is unique and is called the closure of
is an extension of an operator if , i.e. and for Denote by
Two operators are equal if and or equivalent: and for
The operations of unbounded operators are more complicated than in the bounded case, since one has take care of the domains of the operators. Let and be Banach spaces over
For an operator and an scalar the operator is given by
For two operator one define the operator by
For an operators and an operator the operator is defined by
The
inverse of exists if i.e. is
injective. Then the operator is defined by
Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators.
Let be two Banach spaces. A linear operator is said to be closed if one of the following equivalent properties hold:
For a closed operator one has
Consider the derivative operator
on the Banach space of all continuous functions on an interval with the Supremum norm. If one takes its domain to be then is a closed operator. (Note that one could also set to be the set of all differentiable functions including those with non-continuous derivative. That operator is not closed!)
The Operator is not bounded. For example, for the sequence one has but for it is for
If one takes to be instead the set of all infinitely differentiable functions, will no longer be closed, but it will be closable, with the closure being its extension defined on
For two Banch spaces an operator is called closable if the following equivalent properties hold:
The operator with the graph is said to be the closure of and is denoted by It follows that is the restriction of to Note, that other, non-minimal closed extensions may exist. [8] [9]
A core of a closable operator is a subset of such that the closure of the restriction of to is
Remark Not all operators are closable as the following example shows:
Consider the Operator on defined on and . For the sequence in given by one has
but Thus, is not closable.
Let be a densely defined operator on a Banach space and Then is called to be in the
resolvent set of denoted by if the operator is bijective and is a bounded operator. It follows by the closed graph theorem that the resolvent is bounded for all if is a closed operator.
For the resolvent of is defined by
The set is called the
spectrum of denoted by
The spectrum of an unbounded operator can be divided into three parts in exactly the same way as in the bounded case:
Remark The spectrum of an unbounded operator can be any closed set, including and The domain plays an important role as the following example shows:
Consider the banach space and the operators defined by and and
If , then Thus,
For the linear differential equation exists a unique solution which defines an inverse for Therefore
Let be an densely defined operator between Banach spaces and the continuous dual space of Using the notation the transpose (or dual) of is an operator satisfying:
The operator is defined by
Remark The necessary and sufficient condition for the transpose of to exist is that is densely defined (for essentially the same reason as to adjoints, see below.)
Let be a
vector space over the field , a
linear subspace. Let be a
sublinear function and
be a
linear functional with for all (where is the real part of a complex number ).
Then, there exists a
linear functional with
Satz von Banach-Steinhaus (Uniform boundedness principle)
Let be a banach space and be a normed vector space. Suppose that is a collection of bounded linear operators from to The uniform boundedness principle states that if for all in we have , then
Let be banach spaces and surjective. Then is an
open map.
In particular:
Bounded inverse theorem If bijective und bounded, then its inverse is also bounded.
Let be banach spaces. If is linear and closed, then is bounded.
For a densely defined closed operator the following properties are equivalent:
In this section let , and be Hilbert spaces.
For an unbounded operator the definition of the adjoint is more complicated than in the bounded case, since it is necessary to take care of the domains of the operators.
The adjoint of an unbounded operator can be defined in two equivalent ways. First, it can be defined in a way analogous to how we define the adjoint of a bounded operator.
For a densely defined operator its adjoint is defined by
Since is dense in the functional extends to the whole space via the Hahn–Banach theorem. Thus, one can find a unique such that
Finally, let completing the construction of [10] and it is
Remark exists if and only if is densely defined.
The other equivalent definition of the adjoint can be obtained by noticing a general fact: define a linear operator
We then have: is the graph of some operator if and only if is densely defined. [12] A simple calculation shows that this "some" satisfies
Thus, is the adjoint of
The definition of the adjoint can be given in terms of a transpose as follow:
For any Hilbert space and its continuous
dual space there is the
anti-linear
isomorphism
given by where for and Through this isomorphism, the transpose relates to the adjoint in the following way:
where . (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.)
By definition, the domain of could be anything; it could be trivial (i.e., contains only zero) [14] It may happen that the domain of is a closed hyperplane and vanishes everywhere on the domain. [15] [16] Thus, boundedness of on its domain does not imply boundedness of . On the other hand, if is defined on the whole space then is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. [17] If the domain of is dense, then it has its adjoint [11]
For a densely defined operator
If densely defined and , then . Further if are densely defined, then and [20] In contrast to the bounded case, it is not necessary that we have: since, for example, it is even possible that doesn't exist.[ citation needed] This is, however, the case if, for example, is bounded. [21]
Some well-known properties for bounded operators generalize to closed densely defined operators.
In particular, if has trivial kernel, has dense range (by the above identity.) Moreover, is surjective if and only if there is a such that
(This is essentially a variant of the closed range theorem.)
A densely defined operator is called symmetric if for all [25]
A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself. [25]
A symmetric operator is called bounded (from) below if there exists a constant with . The operator is said to be positve if .
An operator is symmetric if it satisfies one of the following equivalent properties:
Remark The last condition does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators.
A densely defined operator is said to be self-adjoint if [25]
For a densely defined closed operator one has:
Let be a symmetric operator. Then follwing conditions are equivalent:
[33]
An operator is
self-adjoint if the following equivalent properties hold:
Remarks
A densely defined, closed operator is called normal if it satisfies the following equivalent properties : [36]
Remarks
Let a symmetric operator on a Hilbert space .
Problem When does have self-adjoint extensions?
The Cayley transform of a symmetric operator is defined by . is an isometry between and and the range is dense in
Theorem is self-adjoint if and only if is
unitary.
In particular: has self-adjoint extensions if and only if has
unitary extensions.
Friedrichs extension theorem Every symmetric operator which is bounded from below has at least one self-adjoint extension with the same lower bound.
[41]
These operators always have a canonically defined self-adjoint extension which is called
Friedrichs extension.
Remark An everywhere defined extension exists for every operator, which is a purely algebraic fact explained at General existence theorem and based on the axiom of choice. If the given operator is not bounded then the extension is a discontinuous linear map. It is of little use since it cannot preserve important properties of the given operator, and usually is highly non-unique
A symmetric operator is called essentially self-adjoint if has one and only one self-adjoint extension. [33] Or equivalent, if its closure is self-adjoint. [27]. Note, that an operator may have more than one self-adjoint extension, and even a continuum of them. [9]
Remark The importance of essentially self-adjointness is that one is often given a non-closed symmetric operator If this operator is essential self-adjoint, then there is uniquely associated to a self-adjoint operator
Let be a symmetric operator. Then follwing conditions are equivalent: [30]
Remark For a bounded operator the terms self-adjoint, symmetric and essentially self-adjoint are equivalent.
Let be complete Riemannian manifold. The Laplace operator
on with the domain the space of all smooth, compactly supported function on is essentially self-adjoint. [42]
The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous spectral theorem holds for self-adjoint operators. In combination with Stone's theorem on one-parameter unitary groups it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see Self-adjoint operator#Self adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing time evolution in classical and quantum mechanics.
{{
citation}}
: More than one of |author1=
and |last=
specified (
help)
This article incorporates material from Closed operator on
PlanetMath, which is licensed under the
Creative Commons Attribution/Share-Alike License.
Category:Linear operators
Category:Operator theory
Category:Article Feedback 5