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{{Short description|Type of continuous linear operator}}
In [[functional analysis]], a branch of [[mathematics]], a '''compact operator''' is a [[linear operator]] <math>T: X \to Y</math>, where <math>X,Y</math> are [[normed vector space]]s, with the property that <math>T</math> maps [[Bounded set|bounded subsets]] of <math>X</math> to [[relatively compact]] subsets of <math>Y</math> (subsets with compact [[closure (topology)|closure]] in <math>Y</math>). Such an operator is necessarily a [[bounded operator]], and so continuous.<ref name="Conway 1985 loc=Section 2.4">{{harvnb|Conway|1985|loc=Section 2.4}}</ref> Some authors require that <math>X,Y</math> are [[Banach space|Banach]], but the definition can be extended to more general spaces.

Any bounded operator ''<math>T</math>'' that has finite [[rank of a linear operator|rank]] is a compact operator; indeed, the class of compact operators is a natural generalization of the class of [[finite-rank operator]]s in an infinite-dimensional setting. When ''<math>Y</math>'' is a [[Hilbert space]], it is true that any compact operator is a limit of finite-rank operators,<ref name="Conway 1985 loc=Section 2.4"/> 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 [[Alexander Grothendieck]] and [[Stefan Banach]].<ref>{{harvnb|Enflo|1973}}</ref>

The origin of the theory of compact operators is in the theory of [[integral equation]]s, where integral operators supply concrete examples of such operators. A typical [[Fredholm integral equation]] gives rise to a compact operator ''K'' on [[function space]]s; 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 <math>T: X \to Y</math> between two [[topological vector space]]s is said to be '''compact''' if there exists a neighborhood ''<math>U</math>'' of the origin in ''<math>X</math>'' such that <math>T(U)</math> is a relatively compact subset of ''<math>Y</math>''.{{sfn | Schaefer|Wolff| 1999 | p=98}}

Let <math>X,Y</math> be normed spaces and <math>T: X \to Y</math> a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors<ref name=":0">{{Cite book|last=Brézis|first=H.|url=https://www.worldcat.org/oclc/695395895|title=Functional analysis, Sobolev spaces and partial differential equations|date=2011|publisher=Springer|others=H.. Brézis|isbn=978-0-387-70914-7|location=New York|oclc=695395895}}</ref>
* ''<math>T</math>'' is a compact operator;
* the image of the unit ball of ''<math>X</math>'' under ''<math>T</math>'' is [[relatively compact]] in ''<math>Y</math>'';
* the image of any bounded subset of ''<math>X</math>'' under ''<math>T</math>'' is [[relatively compact]] in ''<math>Y</math>'';
* there exists a [[neighbourhood (mathematics)|neighbourhood]] <math>U</math> of the origin in ''<math>X</math>'' and a compact subset <math>V\subseteq Y</math> such that <math>T(U)\subseteq V</math>;
* for any bounded sequence <math>(x_n)_{n\in \N}</math> in ''<math>X</math>'', the sequence <math>(Tx_n)_{n\in\N}</math> contains a converging subsequence.
If in addition ''<math>Y</math>'' is Banach, these statements are also equivalent to:
* the image of any bounded subset of ''<math>X</math>'' under ''<math>T</math>'' is [[totally bounded space|totally bounded]] in <math>Y</math>.

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

== Properties ==

In the following, <math>X, Y, Z, W</math> are Banach spaces, <math>B(X,Y)</math> is the space of bounded operators <math>X \to Y</math> under the [[operator norm]], and <math>K(X,Y)</math> denotes the space of compact operators <math>X \to Y</math>. <math>\operatorname{Id}_X</math> denotes the [[identity operator]] on <math>X</math>, <math>B(X) = B(X,X)</math>, and <math>K(X) = K(X,X)</math>.

* <math>K(X,Y)</math> is a closed subspace of <math>B(X,Y)</math> (in the norm topology). Equivalently,{{sfn | Rudin | 1991 | pp=103-115}}  
** given a sequence of compact operators  <math>(T_n)_{n \in \mathbf{N}}</math> mapping <math>X \to Y</math> (where <math>X,Y</math>are Banach) and given that <math>(T_n)_{n \in \mathbf{N}}</math> converges to <math>T</math> with respect to the [[operator norm]], ''<math>T</math>'' is then compact.
* Conversely, if <math>X,Y</math> are Hilbert spaces, then every compact operator from <math>X \to Y</math> is the limit of finite rank operators. Notably, this "[[approximation property]]" is false for general Banach spaces ''X'' and ''Y''.<ref name=":0" />
*<math>B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z),</math> where the [[function composition|composition]] of sets is taken element-wise. In particular, <math>K(X)</math> forms a two-sided [[ideal (ring theory)|ideal]] in <math>B(X)</math>.
*Any compact operator is [[strictly singular]], but not vice versa.<ref>N.L. Carothers, ''A Short Course on Banach Space Theory'', (2005) London Mathematical Society Student Texts '''64''', Cambridge University Press.</ref>
* A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (''Schauder's theorem'').{{sfn | Conway | 1990 | pp=173-177}} 
** If <math>T: X \to Y</math> is bounded and compact, then:{{sfn | Rudin | 1991 | pp=103-115}}{{sfn | Conway | 1990 | pp=173-177}}
*** the closure of the range of ''<math>T</math>'' is [[Separable space|separable]].
*** if the range of ''<math>T</math>'' is closed in ''Y'', then the range of ''<math>T</math>'' is finite-dimensional.
* If <math>X</math> is a Banach space and there exists an [[invertible]] bounded compact operator <math>T: X \to X</math> then ''<math>X</math>'' is necessarily finite-dimensional.{{sfn|Conway|1990|pp=173-177}}

Now suppose that <math>X</math> is a Banach space and <math>T\colon X \to X</math> is a compact linear operator, and <math>T^* \colon X^* \to X^*</math> is the [[Hermitian adjoint|adjoint]] or [[transpose]] of ''T''.
* For any <math>T\in K(X)</math>, <math>{\operatorname{Id}_X} - T</math>&thinsp; is a [[Fredholm operator]] of index 0.  In particular, <math>\operatorname{Im}({\operatorname{Id}_X} - T)</math> 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 ''<math>M</math>'' and ''<math>N</math>'' are subspaces of ''<math>X</math>'' where <math>M</math> is closed and ''<math>N</math>'' is finite-dimensional, then <math>M+N</math> is also closed.
* If <math>S\colon X \to X</math> is any bounded linear operator then both <math>S \circ T</math> and <math>T \circ S</math> are compact operators.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>\lambda \neq 0</math> then the range of <math>T - \lambda \operatorname{Id}_X</math> is closed and the kernel of <math>T - \lambda \operatorname{Id}_X</math> is finite-dimensional.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>\lambda \neq 0</math> then the following are finite and equal: <math>\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)</math>{{sfn | Rudin | 1991 | pp=103-115}}
* The [[Spectrum (functional analysis)|spectrum]] <math>\sigma(T)</math> of ''<math>T</math>'' is compact, [[countable]], and has at most one [[limit point]], which would necessarily be the origin.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>X</math> is infinite-dimensional then <math>0 \in \sigma(T)</math>.{{sfn | Rudin | 1991 | pp=103-115}}
* If <math>\lambda \neq 0</math> and <math>\lambda \in \sigma(T)</math> then <math>\lambda</math> is an eigenvalue of both ''<math>T</math>'' and <math>T^{*}</math>.{{sfn | Rudin | 1991 | pp=103-115}}
* For every <math>r > 0</math> the set <math>E_r = \left\{ \lambda \in \sigma(T) : | \lambda | > r \right\}</math> is finite, and for every non-zero <math>\lambda \in \sigma(T)</math> the range of <math>T - \lambda \operatorname{Id}_X</math> is a [[proper subset]] of ''X''.{{sfn | Rudin | 1991 | pp=103-115}}

==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

<math>(\lambda K + I)u = f </math>

(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 [[Countable set|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 [[eigenvalue]]s of ''K'' with finite multiplicities (so that ''K'' − λ''I'' has a finite-dimensional [[kernel (algebra)#Linear operators|kernel]] for all complex λ ≠ 0).

An important example of a compact operator is [[compact embedding]] of [[Sobolev space]]s, 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.<ref name="mclean">William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000</ref> 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 (ring theory)|ideal]] in the [[algebra over a field|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 associative algebra|quotient algebra]], known as the [[Calkin algebra]], is [[simple algebra|simple]].  More generally, the compact operators form an [[operator ideal]].

==Compact operator on Hilbert spaces==
{{main|Compact operator on Hilbert space}}
For Hilbert spaces, another equivalent definition of compact operators is given as follows.

An operator <math>T</math> on an infinite-dimensional [[Hilbert space]] <math>(\mathcal{H}, \langle \cdot, \cdot \rangle)</math>,

:<math>T\colon\mathcal{H} \to \mathcal{H}</math>,

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

:<math>T = \sum_{n=1}^\infty \lambda_n \langle f_n, \cdot \rangle g_n</math>,

where <math>\{f_1,f_2,\ldots\}</math> and <math>\{g_1,g_2,\ldots\}</math> are orthonormal sets (not necessarily complete), and <math>\lambda_1,\lambda_2,\ldots</math> is a sequence of positive numbers with limit zero, called the [[singular value decomposition#Bounded operators on Hilbert spaces|singular value]]s of the operator, and the series on the right hand side converges in the operator norm. The singular values can [[limit point|accumulate]] only at zero. If the sequence becomes stationary at zero, that is <math>\lambda_{N+k}=0</math> for some <math>N \in \N</math> and every <math>k = 1,2,\dots</math>, then the operator has finite rank, ''i.e.'', a finite-dimensional range, and can be written as

:<math>T = \sum_{n=1}^N \lambda_n \langle f_n, \cdot \rangle g_n</math>.

An important subclass of compact operators is the [[trace class|trace-class]] or [[nuclear operator]]s, i.e., such that <math>\operatorname{Tr}(|T|)<\infty</math>. While all trace-class operators are compact operators, the converse is not necessarily true. For example <math display="inline">\lambda_n = \frac{1}{n}</math> tends to zero for <math>n \to \infty</math> while <math display="inline">\sum_{n=1}^{\infty} |\lambda_n| = \infty</math>.

== Completely continuous operators ==
Let ''X'' and ''Y'' be Banach spaces.  A bounded linear operator ''T'' : ''X'' → ''Y'' is called '''completely continuous''' if, for every [[weak topology|weakly convergent]] [[sequence (mathematics)|sequence]] <math>(x_n)</math> from ''X'', the sequence <math>(Tx_n)</math> is norm-convergent in ''Y''  {{harv|Conway|1985|loc=§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 ==
* Every finite rank operator is compact.
* For <math>\ell^p</math> and a sequence ''(t<sub>n</sub>)'' converging to zero, the multiplication operator (''Tx'')''<sub>n</sub> = t<sub>n</sub> x<sub>n</sub>'' is compact.
* For some fixed ''g''&nbsp;∈&nbsp;''C''([0,&nbsp;1];&nbsp;'''R'''), define the linear operator ''T'' from ''C''([0,&nbsp;1];&nbsp;'''R''') to ''C''([0,&nbsp;1];&nbsp;'''R''') by <math display="block">(Tf)(x) = \int_0^x f(t)g(t) \, \mathrm{d} t.</math>That the operator ''T'' is indeed compact follows from the [[Ascoli theorem]].
* More generally, if Ω is any domain in '''R'''<sup>''n''</sup> and the integral kernel ''k''&nbsp;:&nbsp;Ω&nbsp;×&nbsp;Ω&nbsp;→&nbsp;'''R''' is a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt kernel]], then the operator ''T'' on ''L''<sup>2</sup>(Ω;&nbsp;'''R''') defined by <math display="block">(T f)(x) = \int_{\Omega} k(x, y) f(y) \, \mathrm{d} y</math> is a compact operator.
* By [[Riesz's lemma]], the identity operator is a compact operator if and only if the space is finite-dimensional.{{sfn|Kreyszig|1978|loc=Theorems 2.5-3, 2.5-5}}

==See also==

* {{annotated link|Compact embedding}}
* {{annotated link|Compact operator on Hilbert space}}
* {{annotated link|Fredholm alternative}}
* {{annotated link|Fredholm integral equation}}
* {{annotated link|Fredholm operator}}
* {{annotated link|Strictly singular operator}}
* {{annotated link|Spectral theory of compact operators}}

== Notes ==
<references/>

==References==

* {{cite book | last = Conway | first = John B. | author-link = John B. Conway  | title = A course in functional analysis | publisher = Springer-Verlag | year = 1985 | isbn = 978-3-540-96042-3|at=Section 2.4}}
* {{Conway A Course in Functional Analysis|edition=2}}
* {{cite journal|last1 = Enflo | first1 = P. | author-link = Per Enflo | title = A counterexample to the approximation problem in Banach spaces | journal = [[Acta Mathematica]]|volume = 130|issue = 1 | year = 1973 | pages = 309–317 | doi = 10.1007/BF02392270|mr = 402468 | issn = 0001-5962 | doi-access = free }}
* {{Cite book|last1=Kreyszig|first1=Erwin|title=Introductory functional analysis with applications|publisher=John Wiley & Sons|year=1978|isbn=978-0-471-50731-4}}
* {{cite book|last = Kutateladze|first = S.S.|title = Fundamentals of Functional Analysis|series = Texts in Mathematical Sciences|volume=12|edition = 2nd|publisher = Springer-Verlag|location = New York|year = 1996|page = 292|isbn = 978-0-7923-3898-7}}
* {{cite book | last=Lax | first=Peter | author-link1 = Peter Lax | title=Functional Analysis | publisher=Wiley-Interscience | publication-place=New York | year=2002 | isbn=978-0-471-55604-6 | oclc=47767143}} 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{cite book|last1 = Renardy|first1 = M.|last2 = Rogers|first2 =R. C.|title = An introduction to partial differential equations|series=Texts in Applied Mathematics|volume=13|edition=2nd|publisher = [[Springer-Verlag]]|location = New York|year = 2004|page = 356|isbn = 978-0-387-00444-0}} (Section 7.5)
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer|Wolff| 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->

{{Refimprove|date=May 2008}}

{{Spectral theory}}
{{Topological vector spaces}}
{{Functional analysis}}

[[Category:Compactness (mathematics)]]
[[Category:Linear operators]]
[[Category:Operator theory]]