Type of continuous linear operator
In
functional analysis
, a branch of
mathematics
, a
compact operator
is a
linear operator
, 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.
[1]
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,
[1]
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
.
[2]
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
[
edit
]
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
[4]
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
;
- there exists a
neighbourhood
of the origin in
and a compact subset
such that
;
- for any bounded sequence
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
[
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]
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.
- Conversely, if
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
.
[4]
In particular,
forms a two-sided
ideal
in
.
- Any compact operator is
strictly singular
, but not vice versa.
[6]
- A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (
Schauder's theorem
).
- If
is bounded and compact, then:
- the closure of the range of
is
separable
.
- if the range of
is closed in
Y
, then the range of
is finite-dimensional.
- If
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
.
- For any
,
is a
Fredholm operator
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
and
are subspaces of
where
is closed and
is finite-dimensional, then
is also closed.
- If
is any bounded linear operator then both
and
are compact operators.
- If
then the range of
is closed and the kernel of
is finite-dimensional.
- If
then the following are finite and equal:
![{\displaystyle \dim \ker \left(T-\lambda \operatorname {Id} _{X}\right)=\dim {\big (}X/\operatorname {Im} \left(T-\lambda \operatorname {Id} _{X}\right){\big )}=\dim \ker \left(T^{*}-\lambda \operatorname {Id} _{X^{*}}\right)=\dim {\big (}X^{*}/\operatorname {Im} \left(T^{*}-\lambda \operatorname {Id} _{X^{*}}\right){\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbe17c58f886d8ad169886a991139ca4f075710)
- The
spectrum
of
is compact,
countable
, and has at most one
limit point
, which would necessarily be the origin.
- If
is infinite-dimensional then
.
- If
and
then
is an eigenvalue of both
and
.
- For every
the set
is finite, and for every non-zero
the range of
is a
proper subset
of
X
.
Origins in integral equation theory
[
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]
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
? λ
I
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
Garding inequality
and the
Lax?Milgram theorem
, can be used to convert an
elliptic boundary value problem
into a Fredholm integral equation.
[8]
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
[
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]
For Hilbert spaces, another equivalent definition of compact operators is given as follows.
An operator
on an infinite-dimensional
Hilbert space
,
,
is said to be
compact
if it can be written in the form
,
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
.
An important subclass of compact operators is the
trace-class
or
nuclear operators
, i.e., such that
. While all trace-class operators are compact operators, the converse is not necessarily true. For example
tends to zero for
while
.
Completely continuous operators
[
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]
Let
X
and
Y
be Banach spaces. A bounded linear operator
T
:
X
→
Y
is called
completely continuous
if, for every
weakly convergent
sequence
from
X
, the sequence
is norm-convergent in
Y
(
Conway 1985
, §VI.3). 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
[
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]
- Every finite rank operator is compact.
- For
and a sequence
(t
n
)
converging to zero, the multiplication operator (
Tx
)
n
= t
n
x
n
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
R
n
and the integral kernel
k
: Ω × Ω →
R
is a
Hilbert?Schmidt kernel
, then the operator
T
on
L
2
(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.
See also
[
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]
Notes
[
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]
References
[
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]
- Conway, John B.
(1985).
A course in functional analysis
. Springer-Verlag. Section 2.4.
ISBN
978-3-540-96042-3
.
- Conway, John B.
(1990).
A Course in Functional Analysis
.
Graduate Texts in Mathematics
. Vol. 96 (2nd ed.). New York:
Springer-Verlag
.
ISBN
978-0-387-97245-9
.
OCLC
21195908
.
- Enflo, P.
(1973).
"A counterexample to the approximation problem in Banach spaces"
.
Acta Mathematica
.
130
(1): 309?317.
doi
:
10.1007/BF02392270
.
ISSN
0001-5962
.
MR
0402468
.
- Kreyszig, Erwin (1978).
Introductory functional analysis with applications
. John Wiley & Sons.
ISBN
978-0-471-50731-4
.
- Kutateladze, S.S. (1996).
Fundamentals of Functional Analysis
. Texts in Mathematical Sciences. Vol. 12 (2nd ed.). New York: Springer-Verlag. p. 292.
ISBN
978-0-7923-3898-7
.
- Lax, Peter
(2002).
Functional Analysis
. New York: Wiley-Interscience.
ISBN
978-0-471-55604-6
.
OCLC
47767143
.
- Narici, Lawrence; Beckenstein, Edward (2011).
Topological Vector Spaces
. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
ISBN
978-1584888666
.
OCLC
144216834
.
- Renardy, M.; Rogers, R. C. (2004).
An introduction to partial differential equations
. Texts in Applied Mathematics. Vol. 13 (2nd ed.). New York:
Springer-Verlag
. p. 356.
ISBN
978-0-387-00444-0
.
(Section 7.5)
- Rudin, Walter
(1991).
Functional Analysis
. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
McGraw-Hill Science/Engineering/Math
.
ISBN
978-0-07-054236-5
.
OCLC
21163277
.
- Schaefer, Helmut H.
; Wolff, Manfred P. (1999).
Topological Vector Spaces
.
GTM
. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
ISBN
978-1-4612-7155-0
.
OCLC
840278135
.
- Treves, Francois
(2006) [1967].
Topological Vector Spaces, Distributions and Kernels
. Mineola, N.Y.: Dover Publications.
ISBN
978-0-486-45352-1
.
OCLC
853623322
.
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Basic concepts
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Main results
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Special Elements/Operators
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Spectrum
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Decomposition
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Spectral Theorem
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Special algebras
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Finite-Dimensional
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Generalizations
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Miscellaneous
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Examples
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Applications
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Basic concepts
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Main results
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Maps
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Set operations
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