Compact operator

Infunctional analysis,a branch ofmathematics,acompact operatoris alinear operator$T:X\to Y$,where$X,Y$arenormed vector spaces,with the property that$T$mapsbounded subsetsof$X$torelatively compactsubsets of$Y$(subsets with compactclosurein$Y$). Such an operator is necessarily abounded operator,and so continuous.Some authors require that$X,Y$are Banach, but the definition can be extended to more general spaces.

Any bounded operator$T$that has finiterankis a compact operator; indeed, the class of compact operators is a natural generalization of the class offinite-rank operatorsin an infinite-dimensional setting. When$Y$is aHilbert 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 thenorm topology.Whether this was true in general for Banach spaces (theapproximation property) was an unsolved question for many years; in 1973Per Enflogave a counter-example, building on work byGrothendieckandBanach.

The origin of the theory of compact operators is in the theory ofintegral equations,where integral operators supply concrete examples of such operators. A typicalFredholm integral equationgives rise to a compact operatorKonfunction spaces;the compactness property is shown byequicontinuity.The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea ofFredholm operatoris derived from this connection.

Equivalent formulations

A linear map$T:X\to Y$between twotopological vector spacesis said to becompactif there exists a neighborhood$U$of the origin in$X$such that$T(U)$is a relatively compact subset of$Y$.

Let$X,Y$be normed spaces and$T:X\to Y$a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors

• $T$is a compact operator;
• the image of the unit ball of$X$under$T$isrelatively compactin$Y$;
• the image of any bounded subset of$X$under$T$isrelatively compactin$Y$;
• there exists aneighbourhood$U$of the origin in$X$and a compact subset$V\subseteq Y$such that$T(U)\subseteq V$;
• for any bounded sequence$(x_{n})_{n\in \mathbb {N} }$in$X$,the sequence$(Tx_{n})_{n\in \mathbb {N} }$contains a converging subsequence.

If in addition$Y$is Banach, these statements are also equivalent to:

• the image of any bounded subset of$X$under$T$istotally boundedin$Y$.

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

Important properties

In the following,$X,Y,Z,W$are Banach spaces,$B(X,Y)$is the space of bounded operators$X\to Y$under theoperator norm,and$K(X,Y)$denotes the space of compact operators$X\to Y$.$\operatorname {Id} _{X}$denotes theidentity operatoron$X$,$B(X)=B(X,X)$,and$K(X)=K(X,X)$.

• $K(X,Y)$is a closed subspace of$B(X,Y)$(in the norm topology). Equivalently,
  • given a sequence of compact operators$(T_{n})_{n\in \mathbf {N} }$mapping$X\to Y$(where$X,Y$are Banach) and given that$(T_{n})_{n\in \mathbf {N} }$converges to$T$with respect to theoperator norm,$T$is then 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 "approximation property"is false for general Banach spacesXandY.
• $B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z).$  In particular,$K(X)$forms a two-sidedidealin$B(X)$.
• Any compact operator isstrictly singular,but not vice versa.
• A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem).
  • If$T:X\to Y$is bounded and compact, then:
    • the closure of the range of$T$isseparable.
    • if the range of$T$is closed inY,then the range of$T$is finite-dimensional.
• If$X$is a Banach space and there exists aninvertiblebounded compact operator$T:X\to X$then$X$is necessarily finite-dimensional.

Now suppose that$X$is a Banach space and$T\colon X\to X$is a compact linear operator, and$T^{*}\colon X^{*}\to X^{*}$is theadjointortransposeofT.

• For any$T\in K(X)$,${\operatorname {Id} _{X}}-T$  is aFredholm operatorof 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$M$and$N$are subspaces of$X$where$M$is closed and$N$is finite-dimensional, then$M+N$is also closed.
• If$S\colon X\to X$is any bounded linear operator then both$S\circ T$and$T\circ S$are compact operators.
• If$\lambda \neq 0$then the range of$T-\lambda \operatorname {Id} _{X}$is closed and the kernel of$T-\lambda \operatorname {Id} _{X}$is finite-dimensional.
• If$\lambda \neq 0$then the following are finite and equal:$\dim \ker \left(T-\lambda \operatorname {Id} _{X}\right)=\dim {\big (}X/\operatorname {Im} \left(T-\lambda \operatorname {Id} _{X}\right){\big )}=\dim \ker \left(T^{*}-\lambda \operatorname {Id} _{X^{*}}\right)=\dim {\big (}X^{*}/\operatorname {Im} \left(T^{*}-\lambda \operatorname {Id} _{X^{*}}\right){\big )}$
• Thespectrum$\sigma (T)$of$T$is compact,countable,and has at most onelimit point,which would necessarily be the origin.
• If$X$is infinite-dimensional then$0\in \sigma (T)$.
• If$\lambda \neq 0$and$\lambda \in \sigma (T)$then$\lambda$is an eigenvalue of both$T$and$T^{*}$.
• For every$r>0$the set$E_{r}=\left\{\lambda \in \sigma (T):|\lambda |>r\right\}$is finite, and for every non-zero$\lambda \in \sigma (T)$the range of$T-\lambda \operatorname {Id} _{X}$is aproper subsetofX.

Origins in integral equation theory

A crucial property of compact operators is theFredholm alternative,which asserts that the existence of solution of linear equations of the form

$(\lambda K+I)u=f$

(whereKis a compact operator,fis a given function, anduis the unknown function to be solved for) behaves much like as in finite dimensions. Thespectral theory of compact operatorsthen follows, and it is due toFrigyes Riesz(1918). It shows that a compact operatorKon an infinite-dimensional Banach space has spectrum that is either a finite subset ofCwhich includes 0, or the spectrum is acountably infinitesubset ofCwhich has 0 as its onlylimit point.Moreover, in either case the non-zero elements of the spectrum areeigenvaluesofKwith finite multiplicities (so thatK− λIhas a finite-dimensionalkernelfor all complex λ ≠ 0).

An important example of a compact operator iscompact embeddingofSobolev spaces,which, along with theGårding inequalityand theLax–Milgram theorem,can be used to convert anelliptic boundary value probleminto a Fredholm integral equation.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-sidedidealin thealgebraof all bounded operators on the space. Indeed, the compact operators on an infinite-dimensional separable Hilbert space form a maximal ideal, so thequotient algebra,known as theCalkin algebra,issimple.More generally, the compact operators form anoperator ideal.

Compact operator on Hilbert spaces

Main article:Compact operator on Hilbert space

For Hilbert spaces, another equivalent definition of compact operators is given as follows.

An operator$T$on an infinite-dimensionalHilbert space$({\mathcal {H}},\langle \cdot,\cdot \rangle )$,

$T\colon {\mathcal {H}}\to {\mathcal {H}}$,

is said to becompactif it can be written in the form

$T=\sum _{n=1}^{\infty }\lambda _{n}\langle f_{n},\cdot \rangle g_{n}$,

where$\{f_{1},f_{2},\ldots \}$and$\{g_{1},g_{2},\ldots \}$are orthonormal sets (not necessarily complete), and$\lambda _{1},\lambda _{2},\ldots$is a sequence of positive numbers with limit zero, called thesingular valuesof the operator, and the series on the right hand side converges in the operator norm. The singular values canaccumulateonly at zero. If the sequence becomes stationary at zero, that is$\lambda _{N+k}=0$for some$N\in \mathbb {N}$and every$k=1,2,\dots$,then the operator has finite rank,i.e.,a finite-dimensional range, and can be written as

$T=\sum _{n=1}^{N}\lambda _{n}\langle f_{n},\cdot \rangle g_{n}$.

An important subclass of compact operators is thetrace-classornuclear 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${\textstyle \lambda _{n}={\frac {1}{n}}}$tends to zero for$n\to \infty$while${\textstyle \sum _{n=1}^{\infty }|\lambda _{n}|=\infty }$.

Completely continuous operators

LetXandYbe Banach spaces. A bounded linear operatorT:X→Yis calledcompletely continuousif, for everyweakly convergentsequence$(x_{n})$fromX,the sequence$(Tx_{n})$is norm-convergent inY(Conway 1985,§VI.3). Compact operators on a Banach space are always completely continuous. IfXis areflexive Banach space,then every completely continuous operatorT:X→Yis 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$\ell ^{p}$and a sequence(t_n)converging to zero, the multiplication operator (Tx)_n= t_nx_nis compact.
• For some fixedg∈C([0, 1];R), define the linear operatorTfromC([0, 1];R) toC([0, 1];R) by$$(Tf)(x)=\int _{0}^{x}f(t)g(t)\,\mathrm {d} t.$$That the operatorTis indeed compact follows from theAscoli theorem.
• More generally, if Ω is any domain inRⁿand the integral kernelk:Ω × Ω →Ris aHilbert–Schmidt kernel,then the operatorTonL²(Ω;R) defined by$$(Tf)(x)=\int _{\Omega }k(x,y)f(y)\,\mathrm {d} y$$is a compact operator.
• ByRiesz's lemma,the identity operator is a compact operator if and only if the space is finite-dimensional.

See also

• Compact embedding
• Compact operator on Hilbert space
• Fredholm alternative– mathematical theoremPages displaying wikidata descriptions as a fallback
• Fredholm integral equation
• Fredholm operator– bounded operator with kernel and cokernel both having finite dimensionPages displaying wikidata descriptions as a fallback
• Strictly singular operator
• Spectral theory of compact operators

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