In functional analysis, a branch of mathematics, a compact operator is a linear operator that behaves, in several important respects, like a finite-dimensional operator such as a matrix. In infinite-dimensional spaces, bounded sets are usually not compact, and bounded sequences need not have convergent subsequences. Compact operators partly restore this finite-dimensional behavior by sending bounded sets to sets whose closures are compact, or equivalently, in normed spaces, by sending bounded sequences to sequences with convergent subsequences.
Compact operators first arose in the theory of integral equations, where many integral operators have compactness properties. They play a central role in the Fredholm alternative, in the spectral theory of linear operators, and in applications to differential equations and Sobolev spaces. For example, compactness often implies that the nonzero spectrum of an operator consists of isolated eigenvalues of finite multiplicity, with possible accumulation only at zero.
Motivation: integral equations
Compact operators were first studied in connection with integral equations. A typical example is an operator of the form
where the function is called the integral kernel. Under suitable regularity assumptions on , the operator sends bounded families of functions to families that are uniformly bounded and equicontinuous. By the Arzelà–Ascoli theorem, such families have compact closure in spaces such as . Thus many integral operators are compact.
This compactness is useful because it makes some infinite-dimensional linear equations resemble finite-dimensional systems of linear equations. For example, the Fredholm equation
can often be studied by finite-dimensional approximation and by the Fredholm alternative. In this way, compact operators provide a bridge between concrete integral equations and the abstract spectral theory of operators on function spaces.
The same idea appears in other parts of analysis. Compact embeddings of function spaces, such as certain embeddings of Sobolev spaces into Lp spaces, allow differential and boundary value problems to be treated by methods similar to those used for compact integral operators.
Definitions
Normed spaces
Let and be normed vector spaces, and let be a linear operator. The operator is called compact if it maps bounded subsets of to relatively compact subsets of ; that is, if for every bounded set , the closure of in is compact.
Equivalently, is compact if the image of the closed unit ball of is relatively compact in . In metric terms, this means that every sequence bounded in has a subsequence such that converges in .
Every compact linear operator between normed spaces is bounded, and hence continuous.
Many standard results on compact operators are stated for Banach spaces. If is Banach, a subset of is relatively compact if and only if it is totally bounded. Thus, for operators into Banach spaces, compactness may also be expressed by saying that the image of every bounded subset of is totally bounded in .
TVS case Let be topological vector spaces and a linear operator.
The following statements are equivalent, and different authors may pick any one of these as the principal definition for " is a compact operator":
there exists a neighborhood of the origin in and is a relatively compact subset of ; there exists a neighborhood of the origin in and a compact subset such that ; there exists a nonempty open set in and is a relatively compact subset of .
Basic properties
Throughout this section, denote normed or Banach spaces, as specified, and denotes the space of bounded linear operators from to . The compact operators from to are denoted by .
Relation with finite-dimensional operators
Every finite-rank operator is compact. Indeed, if the range of is finite-dimensional, then the image of a bounded subset of is a bounded subset of a finite-dimensional normed space, and hence has compact closure.
Compactness is therefore automatic in finite-dimensional linear algebra. Conversely, the identity operator on an infinite-dimensional Banach space is not compact: the closed unit ball of an infinite-dimensional Banach space is not compact. Thus a Banach space is finite-dimensional if and only if its identity operator is compact.
More generally, if a Banach space admits an invertible compact operator , then is finite-dimensional.
Continuity and boundedness
Every compact linear operator between normed spaces is bounded, and hence continuous.
Algebraic properties
If and are Banach spaces, then is a closed linear subspace of in the operator norm. Equivalently, if a sequence of compact operators converges in operator norm to an operator , then is compact.
Compact operators are stable under composition with bounded operators. If is compact and , are bounded linear operators, then
is compact. In particular, is a two-sided ideal in the Banach algebra .
In a Hilbert space , the ring of bounded operators modulo the two-sided ideal is the Calkin algebra. This quotient is used to define one common form of the essential spectrum: for a bounded operator on a Hilbert space, the essential spectrum is the spectrum of the coset in the Calkin algebra, so compact perturbations do not change it.
Approximation by finite-rank operators
Since every finite-rank operator is compact, every operator-norm limit of finite-rank operators is compact. On Hilbert spaces the converse also holds: every compact operator between Hilbert spaces is the operator-norm limit of finite-rank operators.
For general Banach spaces this converse need not hold. The question of whether compact operators can always be approximated by finite-rank operators is related to the approximation property, which fails for some Banach spaces. Whether this was true in general for Banach spaces (the approximation property) was an unsolved question for many years; in 1973 Per Enflo gave a counter-example, building on work by Alexander Grothendieck and Stefan Banach.
Adjoints and ranges
A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact; this result is known as Schauder's theorem.
If is compact, then the closure of the range of is separable. If the range of is closed, then the range is finite-dimensional.
Relation to other operator classes
Every compact operator between Banach spaces is strictly singular, but the converse is false: there are strictly singular operators that are not compact.N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cambridge University Press.
Fredholm theory
Compact operators are closely connected with Fredholm operators and the Fredholm alternative. This connection is one reason compact operators behave, in many respects, like finite-dimensional linear maps.
Let be a Banach space and let be compact. Then
is a Fredholm operator of index zero. Equivalently, the kernel of is finite-dimensional, the range of is closed, and
More generally, for every scalar , the operator
is Fredholm of index zero.
Fredholm alternative
A standard consequence is the Fredholm alternative. For a compact operator , the equation
has a unique solution for every if and only if the homogeneous equation
has only the zero solution. If the homogeneous equation has a nonzero solution, then its solution space is finite-dimensional, and the inhomogeneous equation is solvable only for those satisfying finitely many compatibility conditions.
In terms of the adjoint operator , these compatibility conditions can be written as
Thus the obstruction to solving is finite-dimensional. This is analogous to the finite-dimensional situation in which a system of linear equations is solvable precisely when the right-hand side is orthogonal to the nullspace of the transpose.
For the parameter-dependent equation
with , the same alternative applies to the operator . Failure of invertibility occurs precisely when is a nonzero eigenvalue of . Since the nonzero spectrum of a compact operator consists only of isolated eigenvalues of finite multiplicity, these exceptional parameters form a discrete set, with possible accumulation only at infinity.
Spectral theory
The spectral theory of compact operators is closer to finite-dimensional linear algebra than the spectral theory of general bounded operators.
Let be an infinite-dimensional complex Banach space, and let be a compact operator. Then belongs to the spectrum . Apart from , every spectral value of is an eigenvalue of finite multiplicity. More precisely, if and , then
is finite-dimensional, the range of is closed, and is an isolated point of the spectrum.
The nonzero spectrum is therefore either finite or countably infinite. If it is infinite, its only possible accumulation point is . Equivalently, for every , the set
is finite. Thus compact operators cannot have a continuous band of nonzero spectral values.
The finite-dimensional character of the nonzero spectrum can also be expressed in Fredholm-theoretic terms. For , the operator
has finite-dimensional kernel and cokernel, and these dimensions are equal:
In particular, fails to be invertible precisely because is an eigenvalue, not because of residual or continuous spectrum away from zero.
The adjoint has the same nonzero spectral values. If is in the spectrum of , then is also an eigenvalue of the adjoint operator . More precisely,
when algebraic multiplicities are taken into account.
These results are often called the Riesz–Schauder theory of compact operators. They generalize the elementary fact that a finite-dimensional linear map has a spectrum consisting only of eigenvalues, while allowing for the infinite-dimensional phenomenon that eigenvalues may accumulate at .
A notion related to the compactness of an operator is an operator with compact resolvent. An unbounded operator , such as a differential operator, is said to have compact resolvent if exists for some and is a compact operator. This condition implies that the spectrum of has discrete eigenvalues of finite multiplicity, with no finite accumulation point. Compact resolvents commonly arise for elliptic operators on bounded domains, where compact Sobolev embeddings provide the required compactness.
Compact operators on Hilbert spaces
Compact operators on Hilbert spaces have a particularly close relationship with finite-dimensional linear algebra. In addition to the general Banach-space spectral properties of compact operators, Hilbert-space compact operators admit orthogonal decompositions, singular-value expansions, and, in the self-adjoint case, a spectral theorem resembling the spectral theorem for finite-dimensional Hermitian matrices.
Compact self-adjoint operators
If is a Hilbert space and is compact and self-adjoint, then every nonzero element of the spectrum of is an eigenvalue of finite multiplicity. The nonzero eigenvalues are real, have no possible accumulation point except , and the eigenspaces corresponding to distinct eigenvalues are mutually orthogonal.
Equivalently, decomposes as an orthogonal direct sum of the eigenspaces of , together with the kernel of . After choosing an orthonormal basis in each eigenspace and in the kernel, the operator is represented diagonally:
where is an orthonormal family of eigenvectors, are the corresponding nonzero eigenvalues, and if there are infinitely many nonzero eigenvalues. Thus compact self-adjoint operators behave much like finite-dimensional self-adjoint matrices, except that the eigenvalues may form a sequence tending to zero.
The same description applies, with complex eigenvalues, to compact normal operators on a complex Hilbert space.
Singular-value expansion