Infunctional analysis, a branch ofmathematics, acompact operator is alinear operator${\displaystyle T:X\to Y}$, where${\displaystyle X,Y}$ arenormed vector spaces, with the property that${\displaystyle T}$ mapsbounded subsets of${\displaystyle X}$ torelatively compact subsets of${\displaystyle Y}$ (subsets with compactclosure in${\displaystyle Y}$). Such an operator is necessarily abounded operator, and so continuous.^[1] Some authors require that${\displaystyle X,Y}$ areBanach, but the definition can be extended to more general spaces.

Any bounded operator${\displaystyle T}$ that has finiterank is a compact operator; indeed, the class of compact operators is a natural generalization of the class offinite-rank operators in an infinite-dimensional setting. When${\displaystyle Y}$ is aHilbert space, it is true that any compact operator is a limit (inoperator norm) 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 thenorm topology. Whether this was true in general for Banach spaces (theapproximation property) was an unsolved question for many years; in 1973Per Enflo gave a counter-example, building on work byAlexander Grothendieck andStefan Banach.^[2]

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 equation gives rise to a compact operatorK onfunction 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 operator is derived from this connection.

Let${\displaystyle X,Y}$ betopological vector spaces and${\displaystyle T:X\to Y}$ a linear operator.

The following statements are equivalent, and different authors may pick any one of these as the principal definition for "${\displaystyle T}$ is a compact operator":^[3]

• there exists aneighborhood${\displaystyle U}$ of the origin in${\displaystyle X}$ and${\displaystyle T(U)}$ is a relatively compact subset of${\displaystyle Y}$;
• there exists aneighborhood${\displaystyle U}$ of the origin in${\displaystyle X}$ and a compact subset${\displaystyle V\subseteq Y}$ such that${\displaystyle T(U)\subseteq V}$;
• there exists a nonempty open set${\displaystyle U}$ in${\displaystyle X}$ and${\displaystyle T(U)}$ is a relatively compact subset of${\displaystyle Y}$.

If in addition${\displaystyle X,Y}$ are normed spaces, these statements are also equivalent to:^[4]

• the image of the unit ball of${\displaystyle X}$ under${\displaystyle T}$ isrelatively compact in${\displaystyle Y}$;
• the image of any bounded subset of${\displaystyle X}$ under${\displaystyle T}$ isrelatively compact in${\displaystyle Y}$;
• for any bounded sequence${\displaystyle (x_n{)}_{n\in \mathbb{N}}}$ in${\displaystyle X}$, the sequence${\displaystyle (T{x}_n{)}_{n\in \mathbb{N}}}$ contains a converging subsequence.

If in addition${\displaystyle Y}$ is Banach, these statements are also equivalent to:

• the image of any bounded subset of${\displaystyle X}$ under${\displaystyle T}$ istotally bounded in${\displaystyle Y}$.

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

• If a linear operator is compact, then it is continuous.
• ${\displaystyle K(X,Y)}$ is a closed subspace of${\displaystyle B(X,Y)}$ (in the norm topology). Equivalently,^[5]
  • given a sequence of compact operators${\displaystyle (T_n{)}_{n\in \mathbf{N}}}$ mapping${\displaystyle X\to Y}$ (where${\displaystyle X,Y}$are Banach) and given that${\displaystyle (T_n{)}_{n\in \mathbf{N}}}$ converges to${\displaystyle T}$ with respect to theoperator norm,${\displaystyle T}$ is then compact.
• In particular, the limit of a sequence of finite rank operators is a compact operator.
• Conversely, if${\displaystyle X,Y}$ are Hilbert spaces, then every compact operator from${\displaystyle X\to Y}$ is the limit of finite rank operators. Notably, this "approximation property" is false for general Banach spaces${\displaystyle X}$ and${\displaystyle Y}$.^[2][4]
• ${\displaystyle B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z),}$ where thecomposition of sets is taken element-wise. In particular,${\displaystyle K(X)}$ forms a two-sidedideal in${\displaystyle B(X)}$.
• Any compact operator isstrictly 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).^[7]
  • If${\displaystyle T:X\to Y}$ is bounded and compact, then:^[5][7]
    • the closure of the range of${\displaystyle T}$ isseparable.
    • if the range of${\displaystyle T}$ is closed in${\displaystyle Y}$, then the range of${\displaystyle T}$ is finite-dimensional.
• If${\displaystyle X}$ is a Banach space and there exists aninvertible bounded compact operator${\displaystyle T:X\to X}$ then${\displaystyle X}$ is necessarily finite-dimensional.^[7]

Now suppose that${\displaystyle X}$ is a Banach space and${\displaystyle T:X\to X}$ is a compact linear operator, and${\displaystyle T^{\ast }:X^{\ast }\to X^{\ast }}$ is theadjoint ortranspose ofT.

• For any${\displaystyle T\in K(X)}$,${\displaystyle {\mathrm{Id}}_X-T}$ is aFredholm operator of index 0. In particular,${\displaystyle \mathrm{Im}({\mathrm{Id}}_X-T)}$ 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${\displaystyle M}$ and${\displaystyle N}$ are subspaces of${\displaystyle X}$ where${\displaystyle M}$ is closed and${\displaystyle N}$ is finite-dimensional, then${\displaystyle M+N}$ is also closed.
• If${\displaystyle S:X\to X}$ is any bounded linear operator then both${\displaystyle S\circ T}$ and${\displaystyle T\circ S}$ are compact operators.^[5]
• If${\displaystyle \lambda \ne 0}$ then the range of${\displaystyle T-\lambda {\mathrm{Id}}_X}$ is closed and the kernel of${\displaystyle T-\lambda {\mathrm{Id}}_X}$ is finite-dimensional.^[5]
• If${\displaystyle \lambda \ne 0}$ then the following are finite and equal:${\displaystyle \dim \ker \left(T-\lambda {\mathrm{Id}}_X\right)=\dim (X/\mathrm{Im}\left(T-\lambda {\mathrm{Id}}_X\right))=\dim \ker \left(T^{\ast }-\lambda {\mathrm{Id}}_{X^{\ast }}\right)=\dim (X^{\ast }/\mathrm{Im}\left(T^{\ast }-\lambda {\mathrm{Id}}_{X^{\ast }}\right))}$^[5]
• Thespectrum${\displaystyle \sigma (T)}$ of${\displaystyle T}$ is compact,countable, and has at most onelimit point, which would necessarily be the origin.^[5]
• If${\displaystyle X}$ is infinite-dimensional then${\displaystyle 0\in \sigma (T)}$.^[5]
• If${\displaystyle \lambda \ne 0}$ and${\displaystyle \lambda \in \sigma (T)}$ then${\displaystyle \lambda }$ is an eigenvalue of both${\displaystyle T}$ and${\displaystyle T^{\ast }}$.^[5]
• For every${\displaystyle r>0}$ the set${\displaystyle E_r=\left\{\lambda \in \sigma (T):|\lambda |>r\right\}}$ is finite, and for every non-zero${\displaystyle \lambda \in \sigma (T)}$ the range of${\displaystyle T-\lambda {\mathrm{Id}}_X}$ is aproper subset of${\displaystyle X}$.^[5]

A crucial property of compact operators is theFredholm alternative in the solution of linear equations. Let${\displaystyle K}$ be a compact operator,${\displaystyle f}$ a given function, and${\displaystyle u}$ the unknown function to be solved for. Then the Fredholm alternative states that the equation$${\displaystyle (\lambda K+I)u=f}$$behaves much like as in finite dimensions.

Thespectral theory of compact operators then follows, and it is due toFrigyes Riesz (1918). It shows that a compact operator${\displaystyle K}$ on an infinite-dimensional Banach space has spectrum that is either a finite subset of${\displaystyle \mathbb{C}}$ which includes 0, or the spectrum is acountably infinite subset of${\displaystyle \mathbb{C}}$ which has${\displaystyle 0}$ as its onlylimit point. Moreover, in either case the non-zero elements of the spectrum areeigenvalues of${\displaystyle K}$ with finite multiplicities (so that${\displaystyle K-\lambda I}$ has a finite-dimensionalkernel for all complex${\displaystyle \lambda \ne 0}$).

An important example of a compact operator iscompact embedding ofSobolev spaces, which, along with theGårding inequality and theLax–Milgram theorem, can be used to convert anelliptic 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-sidedideal in thealgebra of 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.

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

An operator${\displaystyle T}$ on an infinite-dimensionalHilbert space${\displaystyle (\mathcal{H},⟨\cdot ,\cdot ⟩)}$,

${\displaystyle T:\mathcal{H}\to \mathcal{H}}$,

is said to becompact if it can be written in the form

${\displaystyle T=\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n⟨f_n,\cdot ⟩g_n}$,

where${\displaystyle \{f_1,f_2,\dots \}}$ and${\displaystyle \{g_1,g_2,\dots \}}$ are orthonormal sets (not necessarily complete), and${\displaystyle {\lambda }_1,{\lambda }_2,\dots }$ is a sequence of positive numbers with limit zero, called thesingular values of the operator, and the series on the right hand side converges in the operator norm. The singular values canaccumulate only at zero. If the sequence becomes stationary at zero, that is${\displaystyle {\lambda }_{N+k}=0}$ for some${\displaystyle N\in \mathbb{N}}$ and every${\displaystyle k=1,2,\dots }$, then the operator has finite rank,i.e., a finite-dimensional range, and can be written as

${\displaystyle T=\sum _{n=1}^N{\lambda }_n⟨f_n,\cdot ⟩g_n}$.

An important subclass of compact operators is thetrace-class ornuclear operators, i.e., such that. While all trace-class operators are compact operators, the converse is not necessarily true. For example${\lambda }_n=\frac{1}{n}$ tends to zero for${\displaystyle n\to \mathrm{\infty }}$ while$\sum _{n=1}^{\mathrm{\infty }}|{\lambda }_n|=\mathrm{\infty }$.

Let${\displaystyle X,Y}$ be Banach spaces. A bounded linear operator${\displaystyle T:X\to Y}$ is calledcompletely continuous if, for everyweakly convergentsequence${\displaystyle (x_n)}$ from${\displaystyle X}$, the sequence${\displaystyle (T{x}_n)}$ is norm-convergent in${\displaystyle Y}$ (Conway 1985, §VI.3).

Compact operators on a Banach space are always completely continuous, but theconverse is false, because there exists a completely continuous operator that is not compact. However, the converse is true if${\displaystyle X}$ is areflexive Banach space, then every completely continuous operator${\displaystyle T:X\to Y}$ 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.

• Every finite rank operator is compact.
• The scaling operator${\displaystyle x\mapsto kx}$ for any nonzero${\displaystyle k}$ is compact if and only if the space is finite-dimensional. This can be proven directly, or as a corollary ofRiesz's lemma.^[9]
• Themultiplication operator onsequence space${\displaystyle {\ell }^p}$ with fixed${\displaystyle p\in [1,\mathrm{\infty }]}$, defined as${\displaystyle (Tx{)}_n=t_n{x}_n}$ and sequence${\displaystyle (t_n)}$ converging to zero, is compact.
• EveryHilbert–Schmidt operator is compact.
  • In particular, everyHilbert–Schmidt integral operator is compact. That is, if${\displaystyle \mathrm{\Omega }}$ is any domain in${\displaystyle {\mathbf{R}}^n}$ and the integral kernel${\displaystyle k:\mathrm{\Omega }\times \mathrm{\Omega }\to \mathbf{R}}$ satisfies, then the integral operator${\displaystyle T}$ on${\displaystyle L^2(\mathrm{\Omega };\mathbf{R})}$ defined by$${\displaystyle (Tf)(x)={\int }_{\mathrm{\Omega }}k(x,y)f(y)\mathrm{d}y}$$ is a compact operator.
• Theintegral transform on${\displaystyle C([0,1];\mathbf{R})}$ (i.e. thecontinuous function space on aclosedboundedreal interval), defined by$${\displaystyle (Tf)(x)={\int }_0^{x}f(t)g(t)\mathrm{d}t,}$$ for any fixed${\displaystyle g\in C([0,1];\mathbf{R})}$, is a compact operator by theArzelà–Ascoli theorem.
• Theinclusion map$${\displaystyle I:W^{1,p}(\mathrm{\Omega })\hookrightarrow L^q(\mathrm{\Omega }),I(x)=x,}$$compactly embedding theSobolev space${\displaystyle W^{1,p}(\mathrm{\Omega })}$ in theLebesgue space${\displaystyle L^q(\mathrm{\Omega })}$ for every and, is a compact operator by theRellich–Kondrachov theorem.

• The forward and backwardunilateral shift operators arenot compact.

• Compact embedding
• Compact operator on Hilbert space – Functional analysis concept
• Fredholm alternative – One of Fredholm's theorems in mathematics
• Fredholm integral equation
• Fredholm operator – Part of Fredholm theories in integral equations
• Strictly singular operator
• Spectral theory of compact operators

• 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.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.

This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Compact operator" – news ·newspapers ·books ·scholar ·JSTOR(May 2008) (Learn how and when to remove this message)

• Compactness (mathematics)
• Linear operators
• Operator theory

• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from May 2008
• All articles needing additional references