Compact operator

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In functional analysis, a branch of mathematics, 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.

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.

The spectral theory of 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).

The compact operators from a Banach space to itself form a two-sided ideal in the algebra of all bounded operators on the space. Indeed, the compact operators on a Hilbert space form a minimal ideal, so the quotient algebra, known as the Calkin algebra, is simple.

Examples of compact operators include Hilbert-Schmidt operators, or more generally, operators in the Schmidt class.

1 Compact operator on Hilbert spaces
2 Some properties of compact operators
3 Examples
4 See also

[edit] 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.

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. Here, $\lambda_1,\ldots,\lambda_N$ is a sequence of positive numbers, called the singular values of the operator. The singular values can accumulate only at zero. The bracket $\langle\cdot,\cdot\rangle$ is the scalar product on the Hilbert space; the sum on the right hand side converges in the Hilbert space norm.

An important subclass of compact operators are the trace-class or nuclear operators.

[edit] Some properties of compact operators

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.

A bounded operator is compact if and only if any of the following is true
- $T (B X)$ 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 in Y.
- there exists a neighbourhood of 0, $U\subset X$ , and compact set $V\subset Y$ such that $T(U)\subset V$ .
- 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.
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)
$i d X$ is compact if and only if X has finite dimension
For any compact operator $T\in K(X)$ , $i d X - T$ is Fredholm operator with index 0.