In functional analysis, a compact operator (or completely continuous 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 L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite rank operators in an infinite-dimensional setting. When X = Y and 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 Enflo gave a counter-example. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, a Banach space is said to have the approximation property (AP in short), if every compact operator is a limit of finite rank operators. ...

The origin of the theory of compact operators is in the theory of integral equations. 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. In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... In mathematical analysis, a sequence of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood (a precise definition appears below). ... In mathematics, a Fredholm operator is a bounded linear operator between two Banach spaces whose range is closed and whose kernel and cokernel are finite-dimensional. ...

The spectral theory for compact operators in the abstract was worked out by Frigyes Riesz (published 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 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 − λI has a finite-dimensional kernel for all complex λ ≠ 0). In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ... Frigyes Riesz Frigyes Riesz (January 22, 1880 – February 28, 1956) was a mathematician who was born in GyÅ‘r, Austria-Hungary (now Hungary) and died in Budapest Hungary. ... 1918 was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...

The compact operators form a two-sided ideal in the set of all operators between two Banach spaces. Indeed, the compact operators on a Hilbert space form a maximal ideal, so the quotient algebra, known as the Calkin algebra, is simple. In mathematics, the term ideal has multiple meanings. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In functional analysis, the Calkin algebra is the quotient of B(H), the set of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators. ... In mathematics, an algebra is simple if it contains no non-trivial ideals. ...

Examples of compact operators include Hilbert-Schmidt operators, or more generally, operators in the Schmidt class. In mathematics, a Hilbert-Schmidt operator is a bounded operator A on a Hilbert space H1->H2 such that there exists an orthonormal basis of H1 such that is finite. ...

Compact operator on Hilbert spaces

An equivalent definition of compact operators on a Hilbert space may be given as follows.

An operator on a Hilbert space In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...

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

where and and are (not necessarily complete) orthonormal sets. Here, is a sequence of real or complex numbers, the singular values of the operator, which tends to zero if the sequence is infinite. The bracket is the scalar product on the Hilbert space; the sum on the right hand side must converge in the norm. In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2. ...

An important subclass of comact operators are the trace-class or nuclear operators. In mathematics, a nuclear operator or a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. ...