Compact operator - Citizendia

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. For functional analysis as used in psychology see the Functional analysis (psychology article Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) 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 Such an operator is necessarily a bounded operator, and so continuous. In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces

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. The column rank of a matrix A is the maximal number of Linearly independent columns of A. In Functional analysis, a finite rank operator is a Bounded linear operator between Banach spaces whose range is finite dimensional 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. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of 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. 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 Per Enflo, born in Stockholm, Sweden in 1944 is a university professor in the Department of Mathematical Sciences at Kent State University, Ohio USA

The origin of the theory of compact operators is in the theory of integral equations, where integral operators supply concrete examples of such operators. In Mathematics, an integral equation is an equation in which an unknown function appears under an Integral sign A typical Fredholm integral equation gives rise to a compact operator K on function spaces; the compactness property is shown by equicontinuity. In Mathematics, the Fredholm integral equation is an Integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. In Mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood 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, a Fredholm operator is an operator that arises in the Fredholm theory of Integral equations It is named in honour of Erik Ivar Fredholm

1 Origins in integral equation theory
2 Compact operator on Hilbert spaces
3 Equivalent formulations
4 Completely continuous operators
5 Some properties
6 Examples
7 See also
8 References

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

(λK + I)u = f

behaves much like as in finite dimensions. In Mathematics, the Fredholm alternative, name after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. The spectral theory of compact operators then follows, and it is due to Frigyes Riesz (1918). In Functional analysis, Compact operators are linear operators that map bounded sets to precompact ones Frigyes Riesz ( January 22, 1880 &ndash February 28, 1956) was a Mathematician who was born in Győr, Hungary Year 1918 ( MCMXVIII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common 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 (in that case, the operator has finite rank), or the spectrum is a countably infinite subset of C which has 0 as its only limit point. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" 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, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes 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

An important example of a compact operator is compact embedding of Sobolev spaces, 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. In Mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another In Mathematics, a Sobolev space is a Vector space of functions equipped with a norm that is a combination of ''Lp'' norms of the function In Mathematics, Gårding's inequality is a result that gives a lower bound for the Bilinear form induced by a real linear elliptic partial differential operator Weak formulations are an important tool for the analysis of mathematical equations that permit the transfer of concepts of Linear algebra to solve problems in other fields such In Mathematics, an elliptic boundary value problem is a special kind of Boundary value problem which can be thought of as the stable state of an Evolution problem ^[1] 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 in the algebra of all bounded operators on the space. iDEAL is an Internet payment method in The Netherlands, based on online banking In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with Indeed, the compact operators on a Hilbert space form a minimal ideal, so the quotient algebra, known as the Calkin algebra, is simple. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Functional analysis, the Calkin algebra is the quotient of B ( H) the ring of Bounded linear operators on a separable In Mathematics, specifically in Ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the set {

Compact operator on Hilbert spaces

Main article: Compact operator on Hilbert space

An equivalent definition of compact operators on a Hilbert space may be given as follows. In Functional analysis, Compact operators on Hilbert spaces are a direct extension of matrices in the Hilbert spaces they are precisely the closure of Finite

An operator $T$ on a Hilbert space $\mathcal{H}$

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

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

$T = \sum_{n=1}^N \lambda_n \langle f_n, \cdot \rangle g_n$

where $1 \le N \le \infty$ and $f_1,\ldots,f_N$ and $g_1,\ldots,g_N$ are (not necessarily complete) orthonormal sets. This article assumes some familiarity with Analytic geometry and the concept of a limit. Here, $\lambda_1,\ldots,\lambda_N$ is a sequence of positive numbers, called the singular values of the operator. In Linear algebra, the singular value decomposition ( SVD) is an important factorization of a rectangular real or complex matrix The singular values can accumulate only at zero. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" The bracket $\langle\cdot,\cdot\rangle$ is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.

For any compact operator T, the compact perturbation of the identity I + T is Fredholm operator with index 0. In Mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of Integral equations It is named in honour of Erik Ivar Fredholm

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

Equivalent formulations

In the following, X,Y,Z,W are Banach spaces, B(X,Y) is space of bounded operators from X to Y, K(X,Y) is space of compact operators from X to Y, B(X)=B(X,X), K(X)=K(X,X), $B X$ is the unit ball in X, $i d X$ is the identity operator on X. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that

A bounded operator is compact if and only if any of the following is true
- $T (B X)$ is relatively compact in Y. In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces In Mathematics, a relatively compact subspace (or relatively compact subset) Y of a Topological space X is a subset whose closure
- Image of any bounded set under T is relatively compact in Y.
- Image of any bounded set under T is totally bounded in Y. In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any
- there exists a neighbourhood of 0, $U\subset X$ , and compact set $V\subset Y$ such that $T(U)\subset V$ . In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.
- For any sequence $(x_ n)_{n\in \mathbb N}$ from the unit ball $B X$ , the sequence $(Tx_n)_{n\in\mathbb N}$ contains a Cauchy subsequence. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence

Completely continuous operators

Let X and Y be Banach spaces. A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergent sequence $(x n)$ from X, the sequence $(T x n)$ is norm-convergent in Y (Conway 1985, §VI. In Mathematics, weak topology is an alternative term for Initial topology. In Mathematics, a sequence is an ordered list of objects (or events 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. In Functional analysis, a Banach space is called reflexive if it satisfies a certain abstract property involving Dual spaces Reflexive spaces turn out to

Some properties

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. In Mathematics, the operator norm is a means to measure the "size" of certain Linear operators Formally it is a norm defined on the space of Then T is also compact. Hence, K(X,Y) is closed subspace of B(X,Y).
$B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z)$ This is a generalization of the statement that K(X) forms a two-sided operator ideal in B(X). In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.
$i d X$ is compact if and only if X has finite dimension.
For any $T\in K(X)$ , $i d x - T$ is a Fredholm operator of index 0. In Mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of Integral equations It is named in honour of Erik Ivar Fredholm In particular, $\operatorname{im}\,(id_X-T)$ 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 a Banach space where M is closed and N is finite dimensional, then M + N is also closed.

Examples

For some fixed g ∈ C([0, 1]; R), define the linear operator T by

$(Tf)(x) = \int_0^x f(t)g(t) \, \mathrm{d} t.$

That the operator T is indeed compact follows from the Ascoli theorem. In Mathematics, the Arzelà–Ascoli theorem of Functional analysis gives necessary and sufficient conditions to decide whether a set of Continuous functions

More generally, if Ω is any domain in Rⁿ and the integral kernel k : Ω × Ω → R is a Hilbert-Schmidt kernel, then the operator T on L²(Ω; R) defined by

$(T f)(x) = \int_{\Omega} k(x, y) f(y) \, \mathrm{d} y$

is a compact operator. In Mathematics, a Hilbert-Schmidt integral operator is a type of Integral transform.

By Riesz's lemma, the identity operator is a compact operator if and only if the space is finite dimensional. Riesz's lemma is an lemma in Functional analysis. It specifies (often easy to check conditions which guarantee that a Subspace in a Normed linear

References

^ William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000

Conway, John B. In Functional analysis, Compact operators are linear operators that map bounded sets to precompact ones In Mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of Integral equations It is named in honour of Erik Ivar Fredholm In Mathematics, the Fredholm integral equation is an Integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels In Mathematics, the Fredholm alternative, name after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. In Mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another (1985), A course on functional analysis, Springer-Verlag, ISBN 3-540-96042-2

Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 356. ISBN 0-387-00444-0. (Section 7. 5)

Kutateladze, S. S. (1996). Fundamentals of Functional Analysis, Second edition, Texts in Mathematical Sciences 12, New York: Springer-Verlag, 292. ISBN 978-0-7923-3898-7.

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