Compact operator
Encyclopedia
In functional analysis
, a branch of mathematics
, a compact operator is a linear operator L from a Banach space
X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator
, and so continuous.
Any bounded operator L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting. When Y is a Hilbert space
, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure in the operator norm
of the finite-rank operators. Whether this was true in general for Banach spaces (the approximation property
) was an unsolved question for many years; in the end Enflo gave a counter-example.
The origin of the theory of compact operators is in the theory of integral equation
s, where integral operators supply concrete examples of such operators. A typical Fredholm integral equation
gives rise to a compact operator K on function space
s; the compactness property is shown by equicontinuity
. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator
is derived from this connection.
T is compact if and only if any of the following is true
Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.
, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X),
is the identity operator on X.
, which asserts that the existence of solution of linear equations of the form
behaves much like as in finite dimensions. The spectral theory of compact operators
then follows, and it is due to Frigyes Riesz
(1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a countably infinite
subset of C which has 0 as its only limit point
. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with finite multiplicities (so that K − λI has a finite dimensional kernel for all complex λ ≠ 0).
An important example of a compact operator is compact embedding of Sobolev space
s, which, along with the Gårding inequality and the Lax–Milgram theorem, can be used to convert an elliptic boundary value problem
into a Fredholm integral equation. Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist.
The compact operators from a Banach space to itself form a two-sided ideal
in the algebra
of all bounded operators on the space. Indeed, the compact operators on a Hilbert space form a maximal ideal, so the quotient algebra
, known as the Calkin algebra
, is simple.
An operator
on a Hilbert space
is said to be compact if it can be written in the form
where
and 
and 
are (not necessarily complete) orthonormal sets. Here, 
is a sequence of positive numbers, called the singular values of the operator. The singular values can accumulate
only at zero. The bracket
is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.
An important subclass of compact operators are the trace-class or nuclear operator
s.
sequence
from X, the sequence 
is norm-convergent in Y . Compact operators on a Banach space are always completely continuous. If X is a reflexive Banach space, then every completely continuous operator T : X → Y is compact.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a compact operator is a linear operator L from a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
, and so continuous.
Any bounded operator L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting. When Y is a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure in the operator norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
of the finite-rank operators. Whether this was true in general for Banach spaces (the approximation property
Approximation property
In mathematics, a Banach space is said to have the approximation property , if every compact operator is a limit of finite-rank operators. The converse is always true....
) was an unsolved question for many years; in the end Enflo gave a counter-example.
The origin of the theory of compact operators is in the theory of integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s, where integral operators supply concrete examples of such operators. A typical Fredholm integral equation
Fredholm integral equation
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm.-Equation of the first kind :...
gives rise to a compact operator K on function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
s; the compactness property is shown by equicontinuity
Equicontinuity
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein...
. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator
Fredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
is derived from this connection.
Equivalent formulations
A bounded operatorBounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
T is compact if and only if any of the following is true
- Image of the unit ball in X under T is relatively compact in Y.
- Image of any bounded set under T is relatively compact in Y.
- Image of any bounded set under T is totally boundedTotally bounded spaceIn topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of any fixed "size" . The smaller the size fixed, the more subsets may be needed, but any specific size should require only finitely many subsets...
in Y. - there exists a neighbourhoodNeighbourhood (mathematics)In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of 0,
, and compact set 
such that 
. - For any sequence
from the unit ball in X, the sequence 
contains a Cauchy subsequenceCauchy sequenceIn mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
.
Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.
Important properties
In the following, X, Y, Z, W are Banach spaces, B(X, Y) is the space of bounded operators from X to Y with the operator normOperator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X),
is the identity operator on X.- K(X, Y) is a closed subspace of B(X, Y): Let Tn, n ∈ N, be a sequence of compact operators from one Banach space to the other, and suppose that Tn converges to T with respect to the operator normOperator normIn mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
. Then T is also compact.
-
In particular, K(X) forms a two-sided operator idealIdeal (ring theory)In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in B(X).
-
is compact if and only if X has finite dimension.
- For any T ∈ K(X),
is a Fredholm operatorFredholm operatorIn mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
of index 0. In particular,
is closed. This is essential in developing the spectral properties of compact operators. One can notice the similarity between this property and the fact that, if M and N are subspaces of a Banach space where M is closed and N is finite dimensional, then is also closed.
- Any compact operator is strictly singular, but not vice-versa.
Origins in integral equation theory
A crucial property of compact operators is the Fredholm alternativeFredholm alternative
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators...
, which asserts that the existence of solution of linear equations of the form
behaves much like as in finite dimensions. The spectral theory of compact operators
Spectral theory of compact operators
In functional analysis, compact operators are linear operators that map bounded sets to precompact sets. The set of compact operators acting on a Hilbert space H is the closure of the set of finite rank operators in the uniform operator topology. In general, operators on infinite dimensional spaces...
then follows, and it is due to Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...
(1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a countably infinite
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
subset of C which has 0 as its only limit point
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with finite multiplicities (so that K − λI has a finite dimensional kernel for all complex λ ≠ 0).
An important example of a compact operator is compact embedding of Sobolev space
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
s, which, along with the Gårding inequality and the Lax–Milgram theorem, can be used to convert an elliptic boundary value problem
Elliptic boundary value problem
In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem...
into a Fredholm integral equation. Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist.
The compact operators from a Banach space to itself form a two-sided ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in the algebra
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
of all bounded operators on the space. Indeed, the compact operators on a Hilbert space form a maximal ideal, so the quotient algebra
Quotient algebra
In mathematics, a quotient algebra, , also called a factor algebra is obtained by partitioning the elements of an algebra in equivalence classes given by a congruence, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense...
, known as the Calkin algebra
Calkin algebra
In functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of B, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K of compact operators....
, is simple.
Compact operator on Hilbert spaces
An equivalent definition of compact operators on a Hilbert space may be given as follows.An operator
on a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
is said to be compact if it can be written in the form
where
and and are (not necessarily complete) orthonormal sets. Here, is a sequence of positive numbers, called the singular values of the operator. The singular values can accumulateLimit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
only at zero. The bracket
is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.An important subclass of compact operators are the trace-class or nuclear operator
Nuclear operator
In mathematics, a nuclear operator is roughly a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis .Nuclear operators are essentially the same as trace class operators, though most authors reserve the term "trace...
s.
Completely continuous operators
Let X and Y be Banach spaces. A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergentWeak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
sequence
from X, the sequence is norm-convergent in Y . Compact operators on a Banach space are always completely continuous. If X is a reflexive Banach space, then every completely continuous operator T : X → Y is compact.Examples
- For some fixed g ∈ C([0, 1]; R), define the linear operator T by
- That the operator T is indeed compact follows from the Ascoli theorem.
- More generally, if Ω is any domain in Rn and the integral kernel k : Ω × Ω → R is a Hilbert—Schmidt kernel, then the operator T on L2(Ω; R) defined by
- is a compact operator.
- By Riesz's lemmaRiesz's lemmaRiesz's lemma is a lemma in functional analysis. It specifies conditions which guarantee that a subspace in a normed linear space is dense.- The result :...
, the identity operator is a compact operator if and only if the space is finite dimensional.
See also
- Spectral theory of compact operatorsSpectral theory of compact operatorsIn functional analysis, compact operators are linear operators that map bounded sets to precompact sets. The set of compact operators acting on a Hilbert space H is the closure of the set of finite rank operators in the uniform operator topology. In general, operators on infinite dimensional spaces...
- Fredholm operatorFredholm operatorIn mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
- Fredholm integral equationFredholm integral equationIn mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm.-Equation of the first kind :...
s - Fredholm alternativeFredholm alternativeIn mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators...
- Compact embedding
- Strictly singular operatorStrictly singular operatorIn functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator L from a Banach space X to another Banach space Y, such that it is not an isomorphism, and fails to be an isomorphism on any infinitely dimensional subspace of X...