Compact operator explained

T:X\toY

, where

X,Y

are normed vector spaces, with the property that

maps bounded subsets of

to relatively compact subsets of

(subsets with compact closure in

). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that

X,Y

be Banach, but the definition can be extended to more general spaces.

Any bounded operator

that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. Every compact operator in a Hilbert space

is a limit (in operator norm) of finite-rank operators, so the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology. 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.

The origin of the theory of compact operators is in the theory of integral equations, where integral operators supply concrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K on function spaces; 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.

Definitions

TVS case

Let

X,Y

be topological vector spaces and

T:X\toY

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":

of the origin in X

and

T(U)

is a relatively compact subset of Y

;

of the origin in X

and a compact subset

V\subseteqY

such that

T(U)\subseteqV

;

there exists a nonempty open set

in X

and

T(U)

is a relatively compact subset of Y

.

Normed case

If in addition

X,Y

are normed spaces, these statements are also equivalent to:^[1]

the image of the unit ball of

under

is relatively compact in

;

the image of any bounded subset of

under

is relatively compact in

;

for any bounded sequence

(x_n)_n\in

in X

, the sequence

(Tx_n)_n\in\N

contains a converging subsequence.

Banach case

If in addition

is Banach, these statements are also equivalent to:

the image of any bounded subset of

under

is totally bounded in
Y

.

Properties

In the following,

X,Y,Z,W

are Banach spaces,

B(X,Y)

is the space of bounded operators

X\toY

under the operator norm, and

K(X,Y)

denotes the space of compact operators

X\toY

\operatorname{Id}_X

denotes the identity operator on

B(X)=B(X,X)

, and

K(X)=K(X,X)

If a linear operator is compact, then it is continuous.

K(X,Y)

is a closed subspace of

B(X,Y)

(in the norm topology). Equivalently,

- given a sequence of compact operators

(T_n)_n

mapping

X\toY

(where

X,Y

are Banach) and given that

(T_n)_n

converges to

with respect to the operator norm,

is then compact.

In particular, the limit of a sequence of finite rank operators is a compact operator.
Conversely, if

X,Y

are Hilbert spaces, then every compact operator from

X\toY

is the limit of finite rank operators. Notably, this "approximation property" is false for general Banach spaces

and

B(Y,Z)\circK(X,Y)\circB(W,X)\subseteqK(W,Z),

where the composition of sets is taken element-wise. In particular,

K(X)

forms a two-sided ideal in

B(X)

Any compact operator is strictly singular, but not vice versa.^[2]
A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem).
- If

T:X\toY

is bounded and compact, then:

- - the closure of the range of

is separable.

- - if the range of

is closed in

, then the range of

is finite-dimensional.

is a Banach space and there exists an invertible bounded compact operator

T:X\toX

then

is necessarily finite-dimensional.

Now suppose that

is a Banach space and

T\colonX\toX

is a compact linear operator, and

T^*\colonX^*\toX^*

is the adjoint or transpose of T.

For any

T\inK(X)

{\operatorname{Id}_X}-T

is a Fredholm operator of index 0. In particular,

\operatorname{Im}({\operatorname{Id}_X}-T)

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

and

are subspaces of

where

is closed and

is finite-dimensional, then

M+N

is also closed.

S\colonX\toX

is any bounded linear operator then both

S\circT

and

T\circS

are compact operators.

λ ≠ 0

then the range of

T-λ\operatorname{Id}_X

is closed and the kernel of

T-λ\operatorname{Id}_X

is finite-dimensional.

λ ≠ 0

then the following are finite and equal:

\dim\ker\left(T-λ\operatorname{Id}_X\right)=\dim(X/\operatorname{Im}\left(T-λ\operatorname{Id}_X\right)) =\dim\ker\left(T^*-λ

\operatorname{Id}
	X^*

\right) =\dim(X^*/\operatorname{Im}\left(T^*-λ

\operatorname{Id}
	X^*

\right))

\sigma(T)

is compact, countable, and has at most one limit point, which would necessarily be the origin.

is infinite-dimensional then

0\in\sigma(T)

λ ≠ 0

and

λ\in\sigma(T)

then

is an eigenvalue of both

and

T^*

For every

r>0

the set

E_r=\left\{λ\in\sigma(T):|λ|>r\right\}

is finite, and for every non-zero

λ\in\sigma(T)

the range of

T-λ\operatorname{Id}_X

is a proper subset of

Origins in integral equation theory

A crucial property of compact operators is the Fredholm alternative in the solution of linear equations. Let

be a compact operator,

a given function, and

the unknown function to be solved for. Then the Fredholm alternative states that the equation

(\lambda K + I)u = f

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

on an infinite-dimensional Banach space has spectrum that is either a finite subset of

which includes 0, or the spectrum is a countably infinite subset of

which has

as its only limit point. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of

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 spaces, 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.^[3] 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 an infinite-dimensional separable Hilbert space form a maximal ideal, so the quotient algebra, known as the Calkin algebra, is simple. More generally, the compact operators form an operator ideal.

Compact operator on Hilbert spaces

See main article: Compact operator on Hilbert space. For Hilbert spaces, another equivalent definition of compact operators is given as follows.

An operator

on an infinite-dimensional Hilbert space

(l{H},\langle ⋅ , ⋅ \rangle)

T\colonl{H}\tol{H}

is said to be compact if it can be written in the form

	infty
\sum
	n=1

λ_n\langlef_n, ⋅ \rangleg_n

where

\{f_1,f_2,\ldots\}

and

\{g_1,g_2,\ldots\}

are orthonormal sets (not necessarily complete), and

λ_1,λ_2,\ldots

is a sequence of positive numbers with limit zero, called the singular values of the operator, and the series on the right hand side converges in the operator norm. The singular values can accumulate only at zero. If the sequence becomes stationary at zero, that is

λ_N+k=0

for some

N\in\N

and every

k=1,2,...

, then the operator has finite rank, i.e., a finite-dimensional range, and can be written as

	N
\sum
	n=1

λ_n\langlef_n, ⋅ \rangleg_n

An important subclass of compact operators is the trace-class or nuclear operators, i.e., such that

\operatorname{Tr}(|T|)<infty

. While all trace-class operators are compact operators, the converse is not necessarily true. For example

\lambda_n = \frac

tends to zero for

n\toinfty

while

\sum_^ |\lambda_n| = \infty

Completely continuous operators

Let

X,Y

be Banach spaces. A bounded linear operator

T:X\toY

is called completely continuous if, for every weakly convergent sequence

(x_n)

from

, the sequence

(Tx_n)

is norm-convergent in

Compact operators on a Banach space are always completely continuous, but the converse is false, because there exists a completely continuous operator that is not compact. However, the converse is true if

is a reflexive Banach space, then every completely continuous operator

T:X\toY

is compact.

Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though they are not necessarily completely continuous by the definition of that phrase in modern terminology.

Examples

Every finite rank operator is compact.
The scaling operator

x\mapstokx

for any nonzero

is compact if and only if the space is finite-dimensional. This can be proven directly, or as a corollary of Riesz's lemma.

\ell^p

with fixed

p\in[1,infty]

, defined as

(Tx)_n=t_nx_n

and sequence

(t_n)

converging to zero, is compact.

Every Hilbert–Schmidt operator is compact.
- In particular, every Hilbert–Schmidt integral operator is compact. That is, if

\Omega

is any domain in

Rⁿ

and the integral kernel

k:\Omega x \Omega\toR

satisfies

\iint|k|²<infty

, then the integral operator

L^2(\Omega;R)

defined by

(T f)(x) = \int_ k(x, y) f(y) \, \mathrm y

is a compact operator.

The integral transform on

C([0,1];R)

(i.e. the continuous function space on a closed bounded real interval), defined by

(Tf)(x) = \int_0^x f(t)g(t) \, \mathrm t,

for any fixed

g\inC([0,1];R)

, is a compact operator by the Arzelà–Ascoli theorem.

W^1,(\Omega)

in the Lebesgue space

L^q(\Omega)

for every

1\leqq<\tfrac{np}{n-p}

and

1\leqp<n

, is a compact operator by the Rellich–Kondrachov theorem.

The forward and backward unilateral shift operators are not compact.

Notes

Book: Brézis, H.. Functional analysis, Sobolev spaces and partial differential equations. 2011. Springer. H.. Brézis. 978-0-387-70914-7. New York. 695395895.
N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cambridge University Press.
William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000

References

Book: Conway, John B. . John B. Conway . A course in functional analysis . Springer-Verlag . 1985 . 978-3-540-96042-3. Section 2.4.
- Enflo . P. . Per Enflo . A counterexample to the approximation problem in Banach spaces . Acta Mathematica. 130. 1 . 1973 . 309–317 . 10.1007/BF02392270. 402468 . 0001-5962 . free .
Book: Kreyszig. Erwin. Introductory functional analysis with applications. John Wiley & Sons. 1978. 978-0-471-50731-4.
Book: Kutateladze, S.S.. Fundamentals of Functional Analysis. Texts in Mathematical Sciences. 12. 2nd. Springer-Verlag. New York. 1996. 292. 978-0-7923-3898-7.
Book: Lax, Peter . Peter Lax . Functional Analysis . Wiley-Interscience . New York . 2002 . 978-0-471-55604-6 . 47767143.
- Book: Renardy. M.. Rogers. R. C.. An introduction to partial differential equations. Texts in Applied Mathematics. 13. 2nd. Springer-Verlag. New York. 2004. 356. 978-0-387-00444-0. (Section 7.5)