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

is called the integral kernel sends bounded families of functions to families that are uniformly bounded and equicontinuous. By the Arzelà–Ascoli theorem. 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

be normed vector spaces is called compact to relatively compact is compact.

is compact if the image of the closed unit ball.

Every compact linear operator between normed spaces is bounded, and hence continuous.

Many standard results on compact operators are stated for Banach spaces is relatively compact if and only if it is totally bounded.

TVS case

be topological vector spaces 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 is a relatively compact subset of ;

• there exists a neighborhood of the origin in ;

• in is a relatively compact subset of .

Basic properties

.

Relation with finite-dimensional operators

is a bounded subset of a finite-dimensional normed space, and hence has compact closure.

is compact.

is finite-dimensional.

Continuity and boundedness

Every compact linear operator between normed spaces is bounded, and hence continuous.

Algebraic properties

in the operator norm. is compact.

are bounded linear operators, then

.

is the Calkin algebra. This quotient is used to define one common form of the essential spectrum 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.

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.

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.

be a Banach space be compact. Then

is closed, and

, the operator

is Fredholm of index zero.

Fredholm alternative

, the equation

if and only if the homogeneous equation

satisfying finitely many compatibility conditions.

, these compatibility conditions can be written as

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

. 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.

be an infinite-dimensional complex Banach space belongs to the spectrum is an eigenvalue, then

is an isolated point of the spectrum.

, the set

is finite. Thus compact operators cannot have a continuous band of nonzero spectral values.

, the operator

has finite-dimensional kernel and cokernel, and these dimensions are equal:

is an eigenvalue, not because of residual or continuous spectrum away from zero.

. More precisely,

when algebraic multiplicities are taken into account.

These results are often called the Riesz–Schauder theory.

A notion related to the compactness of an operator is an operator with compact resolvent, such as a differential operator 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

is compact and self-adjoint is an eigenvalue, and the eigenspaces corresponding to distinct eigenvalues are mutually orthogonal.

. After choosing an orthonormal basis in each eigenspace and in the kernel, the operator is represented diagonally:

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

are called the singular values. They form a finite sequence or a sequence tending to zero.

, such that

has finite rank. Conversely, every compact operator between Hilbert spaces is the operator-norm limit of finite-rank operators.

This singular-value expansion is often the most useful description of a compact Hilbert-space operator that is not self-adjoint. It generalizes the singular-value decomposition of finite-dimensional matrices.

Hilbert–Schmidt and trace-class operators

Important subclasses of compact Hilbert-space operators are the Hilbert–Schmidt operators and trace-class is Hilbert–Schmidt when

and trace class when

Every trace-class operator is Hilbert–Schmidt, and every Hilbert–Schmidt operator is compact. is compact but not Hilbert–Schmidt.

Completely continuous operators

is called completely continuous if, for every weakly convergent sequence.

Compact operators between Banach spaces are always completely continuous, but the converse is a reflexive Banach space is compact.

Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though the latter is a weaker condition in modern terminology.

Examples and non-examples

is a bounded subset of a finite-dimensional normed space, and hence has compact closure. Thus finite-rank operators are compact. In finite-dimensional spaces this accounts for all bounded linear operators, but in infinite-dimensional spaces compactness is a genuine restriction.

, an operator of the form

does not tend to zero, the images of suitable coordinate vectors cannot have a convergent subsequence, so the corresponding multiplication operator is not compact.

Integral operators also provide compact operators in many important cases. If

often maps bounded families of functions to uniformly bounded and equicontinuous families. By the Arzelà–Ascoli theorem. For instance, the operator

. On Hilbert spaces, Hilbert–Schmidt integral operators, then the integral operator

.

The identity operator on an infinite-dimensional Banach space is a basic non-example. It is bounded, but not compact, because the closed unit ball of an infinite-dimensional Banach space is not compact. Equivalently, the identity operator would be compact only if every bounded sequence had a convergent subsequence, which fails in infinite-dimensional Banach spaces. The same argument shows that any nonzero scalar multiple of the identity is not compact on an infinite-dimensional Banach space.

is not relatively compact. The backward shift is non-compact for the same reason.

Every bounded linear operator from a normed space into a complete nuclear space is compact, because bounded sets in complete nuclear spaces are precompact. A complex-analytic example is given by the Cauchy integral operator, define

, and is therefore relatively compact by Montel's theorem..

Compact embeddings

Compact operators often arise as inclusion maps, the inclusion

is called an embedding when it is continuous. The embedding is called compact. Compact embeddings are often denoted

Compactness of an embedding is stronger than continuity. A continuous embedding gives an estimate of the form

.

The Arzelà–Ascoli theorem provides one example. The inclusion of a uniformly bounded and equicontinuous has compact closure. This is one reason many integral operators are compact: they send bounded sets of functions into equicontinuous families.

The Rellich–Kondrachov theorem is a bounded domain with suitable regularity, then certain Sobolev space and

then the natural inclusion

.

A non-example is the inclusion of Lebesgue spaces is a finite measure space, then by Hölder's inequality. with Lebesgue measure, the Rademacher functions. Thus compactness of embeddings such as those in the Rellich–Kondrachov theorem depends on additional regularity, not merely on the inclusion of a stronger normed space into a weaker one.

Compact embeddings are central in the study of partial differential equations and calculus of variations. They allow weak or bounded sequences of approximate solutions to be replaced, after passing to subsequences, by strongly convergent sequences in a weaker norm. This compactness is used in existence proofs, in the study of elliptic boundary value problems, and in showing that many differential operators have discrete spectral behavior analogous to that of compact operators.

Holomorphic. For example, on the unit disk, the inclusion

from the Hardy space, and hence is relatively compact in the compact-open topology by Montel's theorem.

The same principle applies to Bergman spaces,

are normal families, and the natural inclusion

.

In several complex variables, analogous compact embeddings occur for Hardy and Bergman spaces on bounded domains, and for Hardy spaces on tube domains-type bounds on translated real slices. Interior estimates again imply local boundedness on compact subsets of the tube, so bounded sets form normal families and are compact in the compact-open topology.

Notes

References

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