Compact operator explained
, where
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
are 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. When
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 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 Grothendieck and 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.
Equivalent formulations
A linear map
between two
topological vector spaces is said to be
compact if there exists a neighborhood
of the origin in
such that
is a relatively compact subset of
.
Let
be normed spaces and
a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors
[1]
is a compact operator;- the image of the unit ball of
under
is relatively compact in
;- the image of any bounded subset of
under
is relatively compact in
;
of the origin in
and a compact subset
such that
;
in
, the sequence
contains a converging subsequence.If in addition
is Banach, these statements are also equivalent to:
- the image of any bounded subset of
under
is totally bounded in
.If a linear operator is compact, then it is continuous.
Important properties
In the following,
are Banach spaces,
is the space of bounded operators
under the
operator norm, and
denotes the space of compact operators
.
denotes the
identity operator on
,
, and
.
is a closed subspace of
(in the norm topology). Equivalently,
- given a sequence of compact operators
mapping
(where
are Banach) and given that
converges to
with respect to the
operator norm,
is then compact.
are Hilbert spaces, then every compact operator from
is the limit of finite rank operators. Notably, this "
approximation property" is false for general Banach spaces
X and
Y.
B(Y,Z)\circK(X,Y)\circB(W,X)\subseteqK(W,Z).
  In particular,
forms a two-sided
ideal in
.
- 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).
is bounded and compact, then:
- the closure of the range of
is separable.
is closed in Y
, then the range of
is finite-dimensional.
is a Banach space and there exists an
invertible bounded compact operator
then
is necessarily finite-dimensional.
Now suppose that
is a Banach space and
is a compact linear operator, and
is the
adjoint or
transpose of
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
is also closed.
is any bounded linear operator then both
and
are compact operators.
then the range of
is closed and the kernel of
is finite-dimensional.
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
\right)
=\dim(X*/\operatorname{Im}\left(T*-λ
\right))
of
is compact,
countable, and has at most one
limit point, which would necessarily be the origin.
is infinite-dimensional then
.
and
then
is an eigenvalue of both
and
.
the set
Er=\left\{λ\in\sigma(T):|λ|>r\right\}
is finite, and for every non-zero
the range of
is a
proper subset of
X.
Origins in integral equation theory
A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution of linear equations of the form
(where K is a compact operator, f is a given function, and u is the unknown function to be solved for) 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 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)
,
,
is said to be compact if it can be written in the form
T=
λn\langlefn, ⋅ \ranglegn
,
where
and
are orthonormal sets (not necessarily complete), and
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
for some
and every
, then the operator has finite rank,
i.e., a finite-dimensional range, and can be written as
T=
λn\langlefn, ⋅ \ranglegn
.
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
tends to zero for
while
.
Completely continuous operators
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.
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.
- For
and a sequence
(tn) converging to zero, the multiplication operator (
Tx)
n = tn xn is compact.
- For some fixed g ∈ C([0, 1]; R), define the linear operator T from C([0, 1]; R) to C([0, 1]; R) 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 lemma, the identity operator is a compact operator if and only if the space is finite-dimensional.
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)