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en>Jochen Burghardt: mentioned Goodman's 1946 article and Carnap's 1947 response (yet to be summarized)

2014-01-27T16:37:47Z

mentioned Goodman's 1946 article and Carnap's 1947 response (yet to be summarized)

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In [[functional analysis]], a branch of [[mathematics]], a '''compact operator''' is a [[linear operator]] ''L'' from a [[Banach space]] ''X'' to another Banach space ''Y'', such that the image under ''L'' of any bounded subset of ''X'' is a [[relatively compact]] subset of ''Y''. Such an operator is necessarily a [[bounded operator]], and so continuous.

Any bounded operator ''L'' that has finite [[rank of a linear operator|rank]] is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of [[finite-rank operator]]s in an infinite-dimensional setting. When ''Y'' 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 in the [[operator norm]] of the finite-rank operators. Whether this was true in general for Banach spaces (the [[approximation property]]) was an unsolved question for many years; in the end [[Per Enflo]] gave a counter-example.

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 [[bounded operator]] ''T'' is compact if and only if any of the following is true
* Image of the unit ball in ''X'' under ''T'' is [[relatively compact]] in ''Y''.
* Image of any bounded set under ''T'' is [[relatively compact]] in ''Y''.
* Image of any bounded set under ''T'' is [[totally bounded space| totally bounded]] in ''Y''.
* there exists a [[neighbourhood (mathematics)| neighbourhood]] of 0, <math>U\subset X</math>, and compact set <math> V\subset Y</math> such that <math>T(U)\subset V</math>.
* For any sequence <math>(x_n)_{n\in \mathbb N}</math> from the unit ball in ''X'', the sequence <math>(Tx_n)_{n\in\mathbb N}</math> contains a [[Cauchy sequence|Cauchy subsequence]].

Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.

== Important properties ==
In the following, ''X'', ''Y'', ''Z'', ''W'' are Banach spaces, B(''X'', ''Y'') is the space of bounded operators from ''X'' to ''Y'' with the [[operator norm]], K(''X'', ''Y'') is the space of compact operators from ''X'' to ''Y'', B(''X'') = B(''X'', ''X''), K(''X'') = K(''X'', ''X''), <math>id_X</math> is the [[identity operator]] on ''X''.

* K(''X'', ''Y'') is a closed subspace of B(''X'', ''Y''): Let ''T''''n'', ''n'' ∈ '''N''', be a sequence of compact operators from one Banach space to the other, and suppose that ''T''''n'' converges to ''T'' with respect to the [[operator norm]]. Then ''T'' is also compact.

* Conversely, if ''X'', ''Y'' are Hilbert spaces, then every compact operator from ''X'' to ''Y'' is the limit of finite rank operators. Notably, this is false for general Banach spaces ''X'' and ''Y''.

* <math>B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z).</math>  In particular, K(''X'') forms a two-sided operator [[ideal (ring theory)|ideal]] in B(''X'').

* <math>id_X</math> is compact if and only if ''X'' has finite dimension.

* For any ''T'' ∈ K(''X''),  <math>id_X - T</math>  is a [[Fredholm operator]] of index 0. In particular,  <math>\operatorname{im}\,(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 ''M'' and ''N'' are subspaces of a Banach space where ''M'' is closed and ''N'' is finite dimensional, then {{nowrap|''M'' + ''N''}} is also closed.

* 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>

* An operator is compact if and only if its adjoint is (Schauder's theorem).

==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 Hilbert space form a maximal ideal, so the [[quotient algebra]], known as the [[Calkin algebra]], is [[simple algebra|simple]].

==Compact operator on Hilbert spaces==
{{main|Compact operator on Hilbert space}}
An equivalent definition of compact operators on a Hilbert space may be given as follows.

An operator <math>T</math> on an infinite dimensional [[Hilbert space]] <math>\mathcal{H}</math>

:<math>T:\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 (not necessarily complete) orthonormal sets, 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. 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>

The bracket <math>\langle\cdot,\cdot\rangle</math> is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.

An important subclass of compact operators is the trace-class or [[nuclear operator]]s.

== 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.

== Examples ==
* Every finite rank operator is compact.

* For <math>\ell^p</math> 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'' by

::<math>(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'''''n'' and the integral kernel ''k'' : Ω × Ω → '''R''' is a [[Hilbert-Schmidt kernel|Hilbert—Schmidt kernel]], then the operator ''T'' on ''L''2(Ω; '''R''') defined by

::<math>(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.

==See also==

* [[Spectral theory of compact operators]]
* [[Fredholm operator]]
* [[Fredholm integral equation]]s
* [[Fredholm alternative]]
* [[Compact embedding]]
* [[Strictly singular operator]]

== Notes ==
<references/>

== References ==
*{{Cite book|first=John B.|last=Conway|title=A course in functional analysis|publisher=Springer-Verlag|year=1985|isbn=3-540-96042-2|ref=harv|postscript=}}

* {{cite book
| author = Renardy, Michael and Rogers, Robert C.
| title = An introduction to partial differential equations
| series = Texts in Applied Mathematics 13
| edition = Second
|publisher = Springer-Verlag
| location = New York
| year = 2004
| page = 356
| isbn = 0-387-00444-0
}} (Section 7.5)

* {{cite book
| author = Kutateladze, S.S.
| title = Fundamentals of Functional Analysis
| series = Texts in Mathematical Sciences 12
| edition = Second
|publisher = Springer-Verlag
| location = New York
| year = 1996
| page = 292
| isbn = 978-0-7923-3898-7
}}

{{Refimprove|date=May 2008}}

{{Functional Analysis}}

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

New riddle of induction - Revision history

en>Gilo1969: fix citation with wikilinks embedded in URL

84.175.67.112: /* Quine */ unified "resembling" --> 'similar"

en>Jochen Burghardt: mentioned Goodman's 1946 article and Carnap's 1947 response (yet to be summarized)